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Doctoral Thesis

Local investigation of the classical and the quantum hall effect

Author(s): Baumgartner, Andreas

Publication Date: 2005

Permanent Link: https://doi.org/10.3929/ethz-a-004959884

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Local Investigation of the Classicaland the Quantum Hall Effect

Andreas Baumgartner

Diss. ETH No. 15923

2005

Diss. ETH No. 15923

Local Investigation of the Classical andthe Quantum Hall Effect

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH

for the degree ofDoctor of Natural Sciences

presented by

Andreas Baumgartner

Dipl. Natw. ETH

born April 2, 1975

citizen of Engi (GL)

accepted on the recommendation of:

Prof. Dr. K. Ensslin, examiner

Prof. Dr. F.M. Peeters, co-examiner

PD Dr. T. Ihn, co-examiner

February 2005

Contents

List of symbols v

Abstract vii

Zusammenfassung viii

1 Introduction 1

2 Electron transport in 2DEGs 3

2.1 Two-dimensional electron gases (2DEG) . . . . . . . . . . . . . . . . 3

2.2 Landauer-Buttiker formalism . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Conductance from transmission . . . . . . . . . . . . . . . . . 5

2.2.2 Quantum point contact (QPC) . . . . . . . . . . . . . . . . . 7

2.3 Drude model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Magnetoresistance in homogeneous samples . . . . . . . . . . 8

2.3.2 Magnetoresistance in inhomogeneous samples . . . . . . . . . 9

2.4 Quantum Hall Effect (QHE) . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.2 Electron wave functions in magnetic fields . . . . . . . . . . . 11

2.4.3 Edge states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.4 Integer quantum Hall effect (IQHE) . . . . . . . . . . . . . . . 13

2.4.5 Quantum Hall transition . . . . . . . . . . . . . . . . . . . . . 15

2.4.6 Compressible and incompressible stripes . . . . . . . . . . . . 15

3 Scanning probe microscopy 19

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Scanning tunneling microscope (STM) . . . . . . . . . . . . . . . . . 20

3.3 Atomic force microscope (AFM) . . . . . . . . . . . . . . . . . . . . . 20

3.4 Probing the local magnetic field . . . . . . . . . . . . . . . . . . . . . 22

i

3.5 Probing the local electric field . . . . . . . . . . . . . . . . . . . . . . 23

3.6 Scanning near-field microscopy (SNOM) . . . . . . . . . . . . . . . . 24

4 Experimental setup 25

4.1 General setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.2 Insert and vibration isolation . . . . . . . . . . . . . . . . . . 25

4.1.3 Cabling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Positioning system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.1 Piezo scanner and z-module . . . . . . . . . . . . . . . . . . . 27

4.2.2 xy-table and thermometry . . . . . . . . . . . . . . . . . . . . 30

4.2.3 Capacitive sensors and orientation on the sample . . . . . . . 30

4.3 Tuning fork sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.1 Harmonic oscillator model . . . . . . . . . . . . . . . . . . . . 33

4.3.2 ‘Standard’ design . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.3 Cantilever tuning-fork (CLTF) sensors . . . . . . . . . . . . . 35

4.3.4 Excitation and readout principles . . . . . . . . . . . . . . . . 36

4.3.5 Read-out scheme . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Local spectroscopy 41

5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Kelvin Probe Measurements . . . . . . . . . . . . . . . . . . . . . . . 43

5.2.1 General electrostatics . . . . . . . . . . . . . . . . . . . . . . . 43

5.2.2 Experiments at fixed position . . . . . . . . . . . . . . . . . . 44

5.2.3 Experiments at variable positions . . . . . . . . . . . . . . . . 48

5.2.4 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3 Local resistance measurements . . . . . . . . . . . . . . . . . . . . . . 50

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Scanning Gate experiments 55

6.1 The scanning gate technique . . . . . . . . . . . . . . . . . . . . . . . 55

6.2 Symmetry considerations . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.3 Measurement procedure . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.4 Sample characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7 Scanning gate experiments at zero magnetic field 63

7.1 Overview: experiments on structured 2DEGs . . . . . . . . . . . . . . 63

ii

7.2 Longitudinal resistance . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.2.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.2.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.3 Hall resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.3.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.4 Further considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8 Classical Hall effect regime 79

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.3 Models and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.3.1 Completely depleted disk . . . . . . . . . . . . . . . . . . . . . 82

8.3.2 Disk of finite electron density in the diffusive regime . . . . . . 84

8.3.3 Disk of finite electron density in the ballistic regime . . . . . . 88

8.4 Experiments in the ballistic regime . . . . . . . . . . . . . . . . . . . 91

8.5 Tip voltage dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8.6 Summary: classical transport . . . . . . . . . . . . . . . . . . . . . . 94

9 Quantum Hall effect regime 95

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

9.2 The ‘high mobility’ sample . . . . . . . . . . . . . . . . . . . . . . . . 98

9.2.1 Sample and setup . . . . . . . . . . . . . . . . . . . . . . . . . 98

9.2.2 Description of the data . . . . . . . . . . . . . . . . . . . . . . 99

9.3 QPC-model of quantum Hall transition . . . . . . . . . . . . . . . . 107

9.3.1 Transmission matrices . . . . . . . . . . . . . . . . . . . . . . 108

9.3.2 From a given potential to the scanning gate image . . . . . . . 111

9.3.3 The experiments and the QPC model . . . . . . . . . . . . . . 114

9.4 Experimental identification of scattering configurations . . . . . . . . 116

9.5 Experiments with larger tip voltage . . . . . . . . . . . . . . . . . . . 118

9.5.1 Description of the data . . . . . . . . . . . . . . . . . . . . . . 119

9.5.2 Analysis and discussion . . . . . . . . . . . . . . . . . . . . . . 121

9.5.3 Tip voltage dependence . . . . . . . . . . . . . . . . . . . . . 126

9.5.4 Comments on the measurements . . . . . . . . . . . . . . . . . 130

9.6 From classical to quantum transport . . . . . . . . . . . . . . . . . . 131

9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

iii

10 Summary and outlook 135

Appendices 137

A: Inhomogeneous samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

B: Finite Element Method (FEM) . . . . . . . . . . . . . . . . . . . . . . . 138

C: CLTF experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

D: Additional measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 144

E: Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Publications 151

Acknowledgements 162

iv

Lists of symbols

physical constant ExplanationaB Bohr radius−e electron chargeε dielectric permittivityε0 vacuum dielectric constant

Φ0 = h/e flux quantumh = 2π~ Planck’s constant

kB Boltzmann constantµB Bohr magneton

Abbreviation Explanation2DEG two dimensional electron gasAB Aharonov Bohm

AFM atomic force microscopeCB Coulomb blockadecpd contact potential difference

(L)DOS (local) density of statesFWHM full width at half maximum

LL Landau levelMBE molecular beam epitaxy

QHE, IQHE, FQHE Quantum Hall effect, integer and fractionalQPC Quantum point contactSdH Shubnikov - de Haas

TF, CLTF tuning fork, cantilever TF

v

Symbol ExplanationL, W , A system size (length, width, area)

B magnetic field~A vector potential

Φm magnetic fluxΦ electric potentialQ chargeI current in setupJ current in sampleU voltageG conductanceR resistanceEF Fermi energyvF Fermi velocitykF Fermi wavenumberλF Fermi wavelengthg∗ gyromagnetic factor`m magnetic length`el elastic mean free path`ϕ phase coherence lengthτϕ phase coherence timem∗ effective electron massµ electron mobilityn electron sheet densityν filling factorωc cyclotron frequencyRc cyclotron radiusT temperature

vi

Abstract

The scope of this work is the presentation of local investigations on the classical andthe quantum Hall effect in a two-dimensional electron gas and the description of ahome-built low-temperature scanning probe microscope (SPM).

The SPM has been designed with special care in view of thermal and vibrationisolation, easy access to the components and their commercial availability. Position-ing of the sample during a cooldown is achieved using a slip-stick piezo drive. Home-built quartz tuning fork (TF) sensors allow fully electronic excitation and read-outwith ultra-low heat dissipation. A novel type of TF-based sensor for low-temperatureexperiments and high magnetic fields has been developed in collaboration with T.Akiyama from the IMT Neuchatel.

In local probe experiments the question of the distance dependence of the contactpotential difference is addressed.

Low-temperature scanning gate experiments on Hall bars are presented focusingon the Hall resistance. At zero magnetic field a distinct pattern is found that isreproducible in various samples. It exhibits the expected symmetries of a Hallmeasurement in a diffusive sample and is interpreted as the deflection of the currentdensity by the AFM-tip-induced potential. In the ballistic transport regime thesymmetries are violated by additional structures attributed to mesoscopic effects inthe Hall cross.

In the quantum Hall regime local centers of increased Hall resistance are observed1/B-periodically in scanning gate images. At even integer filling factors essentiallyno changes in the Hall resistance can be introduced by the AFM tip. Betweenthe quantum Hall plateaus a rich structure develops, which changes its appearancecompletely at fields near the peaks of the Shubnikov - de Haas oscillations. Aninterpretation is given in the picture of percolating states, where the coupling ofcounterpropagating states takes place at saddle points in the local potential. Thesaddle points can be influenced locally by the electric potential induced by themicroscope sensor leading to a change in the resistance. Mesoscopic fluctuationsobserved in the magnetoresistance appear at magnetic fields where scanning gateimages show a series of rings, centered at one or two individual positions.

vii

Zusammenfassung

Das Ziel dieser Arbeit ist die lokale Untersuchung des klassischen und des Quanten-Hall-Effekts in einem zweidimensionalen Elektronengas und die Beschreibung einesselbstgebauten Tieftemperatur-Rastersondenmikroskops.

Beim Bau des Mikroskops wurde speziell auf die thermische und mechanische Iso-lation geachtet, sowie darauf, dass die Komponenten einfach zuganglich und kom-merziell erhaltlich sind. Die Positionierung wahrend dem Abkuhlen wird durcheinen Piezo-Ruckelmotor erreicht. Von Hand gefertigte Kraftsensoren aus Quarz-Stimmgabeln erlauben eine rein elektronische Anregung und Detektion mit extremgeringem Warmeeintrag. Ein neuartiger Sensortyp, basierend auf Quarzstimmga-beln, wurde in Zusammenarbeit mit T. Akiyama vom IMT in Neuenburg fur denEinsatz bei tiefen Temperaturen und starken Magnetfeldern entwickelt.

In ersten lokalen Experimenten wird die Frage nach der Abstandsabhangigkeitdes Kontaktpotentials in solchen Messungen angegangen. Bei den hier prasentierten‘Scanning Gate’ Experimenten auf ‘Hallbars’ bei tiefen Temperaturen wird vor allemauf den Hall Widerstand eingegangen. Ohne angelegtes Magnetfeld tritt eine aus-gepragte Struktur auf, die in mehreren Proben reproduzierbar ist. Die Struktur zeigtdie erwarteten Symmetrien einer Hallwiderstandsmessung in diffusiven Proben undwird auf eine veranderte Stromdichte zuruckgefuhrt, die durch das SpitzeninduziertePotential hervorgerufen wird. Im ballistischen Transportregime werden diese Sym-metrien durch zusatzliche Strukturen verletzt, die durch mesoskopische Effekte imHall Kreuz erklart werden konnen.

Im Regime des Quantenhalleffekts zeigen ‘Scanning Gate’ Bilder lokale Zen-tren, in denen der Hallwiderstand erhoht wird. Die Strukturen wiederholen sich1/B-periodisch. Bei geraden ganzzahligen Fullfaktoren wird der Hallwiderstandkaum durch die AFM-Spitze beeinflusst. Zwischen den Quantenhallplateaus dage-gen entwickelt sich eine vielfaltige Struktur, deren Erscheinungsbild sich komplettverandert bei Magnetfeldern nahe der Maxima der Shubnikov - de Haas Oszillatio-nen. Eine Interpretation wird im Bild von perkolierenden Kanalen gegeben, in demgegenlaufige Zustande an Sattelpunkten des lokalen Potentials gekoppelt werden.Diese Sattelpunkte werden lokal vom induzierten elektrischen Potential der Sen-sorspitze beeinflusst, was zu einem veranderten Widerstand fuhrt. MesoskopischeFluktuationen in Magnetowiderstandskurven werden an denselben Magnetfeldernbeobachtet, in denen die ‘Scanning Gate’ Bilder eine Serie von Ringen zeigen, diezentriert an einer oder zwei einzelnen Stellen auftreten.

viii

Chapter 1

Introduction

All beings in Flatland, animate or inanimate, no matterwhat their form, present to our view the same, or nearly thesame, appearance, viz. that of a straight Line. How then canone be distinguished from another...? [1]

from ‘Flatland’ by Edwin Abbott Abbott (1838-1926)

In the second half of the 20th century semiconductor physics brought a revo-lution, both in science and in every day life. Who does not listen to his favoritemusic from a CD, read by a semiconductor laser, amplified and recorded by dig-ital means and bought at a computer based cash desk - inveigled by around theworld communication and computer aided advertising? Most of this takes place in3 dimensions.

Crystal growth technology of the past 30 years developed and optimized a com-pletely new and amazing field: ‘flatlands’, inhabited by electrons, interacting witheach other and controlled by physicists. In the last decade, also ‘linelands’ and‘pointlands’ for electrons were created, which opened up novel possibilities of mate-rial designs, schemes for computation and new and fascinating physics.

The measurement of the Hall voltage is conceptually simple: one sends a currentthrough a bar of material, applies a magnetic field orthogonal to the current andmeasures the voltage drop in the third direction. The classical Hall effect wasdiscovered in 1879 [2] in three-dimensional metal plates, where this voltage drop isproportional to the applied magnetic field. The classical Hall effect is still a topicof recent research, mainly because inhomogeneities and optimized geometries canbe used to enhance the capabilities of magnetic field sensors, which themselves arecrucial for modern research and development.

Completely unexpected, the integer quantum Hall effect was found 1980 [3] in atwo-dimensional electron gas (2DEG), where electrons are confined so strongly in onedirection, that only one quantum mechanical subband is occupied, but the electrons

1

Chapter 1. Introduction

can move freely in the other directions. The linear increase of the Hall voltagedevelops into a step-wise increase at very high magnetic fields. These experimentstriggered the field of low-dimensional semiconductor structures. Soon afterwardsthe fractional quantum Hall effect [4] was discovered, with even more plateaus inthe Hall resistance curves.

Another cornerstone of modern physics is the invention of the scanning tunnelingmicroscope (STM) in 1981 [5] and the atomic force microscope (AFM) in 1986 [6],which, for example, allow to image and manipulate matter on atomic length scales.These tools give access to regimes where the macroscopic world and the microscopicquantum world meet. One speaks of mesoscopic physics if effects of individualscattering centers are not averaged entirely to the macroscopic values. One exampleof a mesoscopic system is provided by the quantum mechanical states involved inthe quantum Hall effect.

The scope of this thesis is the investigation of the classical and the quantumHall effect on the relevant length scales by the use of a scanning probe microscope.The thesis is structured as follows: chapters 2 and 3 give a short introduction tothe physics of two-dimensional electron gases, with focus on electron transport inGaAs/AlGaAs heterostructures at high magnetic fields, and to the basic principles ofscanning probe microscopy. In chapter 4 the home-built low-temperature scanningprobe microscope is described and characterized. Some details of the home-builttuning fork AFM sensors and the design of a novel type of sensor are discussed.

In chapter 5 local probe experiments are presented that address the question ofa distance-dependent contact potential difference. The scanning gate experiments,the main part of this thesis, are introduced in chapter 6, together with symmetryconsiderations for a Hall cross and the characterization of the samples. In chapter 7scanning gate experiments at zero magnetic field are presented and discussed withthe help of classical models. Experiments at small, i.e., non-quantizing magneticfields can be found in chapter 8. Also these results are described in classical termsand finite element method (FEM) calculations are presented to understand somedetails. Deviations from the established pattern are found in a quasi-ballistic sample.

The quantum Hall regime is investigated in chapter 9, where series of scans show1/B-periodic behavior and patterns that are interpreted in a percolation model ofquantum Hall edge states. An intuitive model is presented that allows the cha-racterization of the observed features in terms of transmission configurations in theLandauer-Buttiker formalism. The development of individual patterns in increasingmagnetic fields are observed and characterized. Fluctuations in the magnetic fielddependent Hall resistance, usually attributed to mesoscopic details in the sample,are found together with local features in the scanning gate images. Also data atfields between the classical and the quantum Hall regime are presented. The lastchapter contains a brief summary and an outlook.

2

Chapter 2

Electron transport in 2DEGs

2.1 Two-dimensional electron gases (2DEG)

In modern molecular beam epitaxy (MBE) it is possible to deposit layers of variousmaterials with atomic precision and thus to control the crystal growth in detail [7].In order to obtain a GaAs-heterostructure with a 2DEG one starts with growingGaAs on a GaAs substrate to get a flat (001) surface. Then, after the last layeris exactly finished1, one grows AlxGa1−xAs on top, which is also semiconductingand has almost the same lattice constant, but a different band gap. This gives anatomically precise interface between the two materials (‘heterojunction’) with littlestrain. After the doping layer an additional GaAs ‘cap’ layer is added on top. Thislayer sequence is depicted schematically in Fig. 2.1(a).

GaAsGaAs

EC

EF

EVNeutraldonors

Ionizeddonors

Al Ga Asx 1-x

Al Ga Asx 1-x

x

yz

2DEG17 nm52 nm

Figure 2.1: (a) Schematic of the layers in a heterostructure. The 2DEGforms 52 nm below the sample surface (blue) and the δ-donor layer is 17nm above the 2DEG. (b) Schematic band diagram for a heterojunction. Thehorizontal axis is the growth direction.

1This can be checked in situ by RHEED (Reflection High Energy Electron Diffraction) or byan STM [7]

3

Chapter 2. Electron transport in 2DEGs

The key ingredient is a monolayer (‘δ-doping’) with embedded Si atoms, 17 nmabove the heterojunction (‘modulation doping’), that have one loosely bound elec-tron each. The majority of these donors get ionized and a part of the electronsdiffuses into the GaAs with the lower conduction band edge, until the resulting elec-tric field stops this process. The electric field in growth direction (z-axis) keeps theelectrons at the interface. Since these electrons are almost free to move orthogo-nally to z, they constitute a so-called two dimensional electron gas (2DEG), becausetheir properties can be approximated very well by the model of free electrons in twodimensions with a renormalized mass m∗ ≈ 0.067m0. At low temperatures the scat-tering rate is usually dominated by scattering at ionized impurities. Since these arespatially separated of the electrons the mobilities can become very large [8]. Thebandstructure is distorted at the interface, as shown in Fig. 2.1(b), in order to alignthe electrochemical potential of the electrons in both materials. The valence band iscompletely occupied while for the conduction band only a very small region at theheterojunction lies below the Fermi energy and forms the 2DEG (blue). The 2DEGforms without the application of an external gate voltage, in contrast to (silicon-)MOSFET structures. Modulation doping is now used in industry for example inHEMTs (high electron mobility transistors). A review about 2DEGs is given in Ref.[9].

The confining potential can be approximated by a triangle-shaped function andthe corresponding eigenfunctions are known as Airy functions. A two-dimensionalelectron gas is given if only the ground state is occupied, i.e. at low temperaturesand low electron densities. If states with higher quantum numbers in z-direction arerelevant, one speaks of ‘higher subbands’. For an ideal 2DEG the following relationscan be deduced by counting the equidistant states in k-space and comparing withthe electron density n:

kinetic energy DOS Fermi wave number Fermi energy

E = ~2k2

2m∗ D = m∗

π~2 kF =√

2πn EF = nD

= π~2

m∗ n

Table 2.1: Characteristic quantities for a 2DEG in GaAs with spindegeneracy 2 and valley degeneracy 1.

The electrical connection to such a 2DEG is made by Ohmic contacts, for exam-ple, obtained by diffusing eutectic AuGe through the top layers. Lateral structuresin the 2DEG can be defined by photo- or electron-beam lithography, where the partof the 2DEG that is not needed is etched away. Another possibility of structuringthe 2DEG is by applying a negative voltage to top gates of the desired shape, whichpushes away the electrons beneath. AFM lithography done directly on the 2DEG isa method that allows to confine the electrons in a large variety of geometries: a largenegative voltage applied between a sharp tip of an AFM and the sample oxidizes theGaAs cap layer locally, which depletes the 2DEG below [10]. A similar techniquecan be used to structure thin metallic top gates [11].

4

2.2. Landauer-Buttiker formalism

2.2 Landauer-Buttiker formalism

2.2.1 Conductance from transmission

Electrons in a (narrow) conducting bar of 2DEG can be regarded as electrons in awave guide whose eigenfunctions can be written as 1

Leikxχ(y). In order to calculate

the current carried by a single mode connecting two electrical contacts (‘left’ and‘right’) at given electrochemical potentials µL and µR, one generally sums up theexpectation value of the current density operator of all eigenmodes p. As can alsobe seen in appendix E, one arrives for T → 0 at

Jp =e

h(µL − µR) ≡ e2

hU, (2.1)

i.e., every mode carries a current of e2

hU . Counterpropagating modes at the same

energy cancel in the net current and only the states between the two electrochemicalpotentials effectively contribute to the non-equilibrium transport.

L R

Jin

Jr

Jt

T T13

T14

T12

1

2

3R1N1

4(a) (b)

Figure 2.2: (a) A barrier with transmission T scatters a portion Jr of theincoming current Jin back into the left contact and transmits Jt. (b) Schematicof the transmission matrix elements for a four terminal device.

If one inserts a potential barrier, i.e. a scatterer described by the transmissionprobability T , into a wave guide with M independent modes, as depicted in Fig.2.2(a) for one mode, the injected current Jin is partially reflected leading to Jr, andpartially transmitted, giving Jt. Thus one finds the total current Jtot = Jin − Jr =Jt = e2

hMTU and the conductance

G =e2

hMT. (2.2)

This is the so-called Landauer formula for the two-terminal conductance. Itcan be applied to arbitrary two-terminal geometries, if T is regarded as an averagetransmission per mode. In the Landauer-Buttiker formalism [12] the previous resultis generalized to multi-terminal geometries by writing the Landauer formula for everypair of contacts using the respective transmissions Tij ≡ Ti←j from lead j to lead i.The elements Tii = Ni−Ri with Ni the number of modes and Ri the backscatteringprobability, describe the probability of an electron to enter the structure at lead i.

5


This situation is illustrated for a Hall cross in Fig. 2.2(b). For the currents in leadi one finds

Ji =∑

j

[GjiUi −GijUj] with Gij =2e2

~Ti←j (2.3)

and the transmission T including all modes. If all voltages Ui are the same, the netcurrent has to be zero, which leads to the relation

∑j Gij =

∑j Gji. Inserted in Eq.

(2.3) one finds

Ji =∑

j

[Ui − Uj] Gij or ~J = G · ~U (2.4)

Usually, the boundary conditions are given as currents, e.g. for a Hall voltagemeasurement as ~J = (J, 0,−J, 0), with the contact numbering given in Fig. 2.2(b).The goal is to find the corresponding voltage drops. Since the matrix G is singularit can not be inverted directly. Choosing contact 3 to lie on zero potential, andnoting that due to charge conservation the current through this lead is the sum ofall the other lead currents, the row and column with number 3 can be left away andthe resulting matrix G can be inverted to get the resistance matrix R.

Though in this thesis the inversion is done numerically in most cases, a formulafor a general four-terminal measurement is reproduced here [12, 13]. It is generallyknown as the ‘Buttiker formula’ for four-terminal measurements. The current forthe measurement is applied from contact n to m, while the voltage drop from l tok is measured. The corresponding resistance Rmn,kl can be expressed as function ofthe transmission coefficients in the following way:

Rmn,kl =h

e2

TkmTln − TknTlm

D(2.5)

where D is a number that is independent of the indices m, n, k and l and sym-metric in the magnetic field. The derivation of this expression does not depend on alocally defined conductivity. In fact, the Landauer-Buttiker formalism is equivalentto the Kubo-Greenwood formalism, which relates the current density ~j(~r) as re-

sponse of the system to the applied electric field ~E(~r′) by the non-local conductivitytensor σ(~r, ~r′) [14].

With Eq. (2.5) it is easy to test some symmetries: the exchange of the twocurrent leads or of the two voltage leads reverses the sign of the measured resis-tance: Rmn,kl(B) = −Rmn,lk(B) = −Rnm,kl(B). More fundamental is the followingreciprocity relation for the resistances:

Rmn,kl(B) = Rkl,mn(−B) (2.6)

which can be seen by noting that the time reversed paths have the same probability,i.e. Tij(B) = Tji(−B) and that D is symmetric in B. Equation (2.6) states that ifthe current and voltage contacts are exchanged and the magnetic field reversed, oneshould measure the same resistance.

6

2.2. Landauer-Buttiker formalism

This relation fundamentally differs from the Onsager-Casimir relations, herestated for the electrical conductivities [15]:

σαβ(B) = σβα(−B) (2.7)

where α and β refer to the coordinate system, not to leads. These relations are onlyvalid if the involved quantities are well defined as local properties. If Eq. (2.7) isestablished, Eq. (2.6) can be derived, but one can not deduce Eq. (2.7) from thefact that Eq. (2.6) holds.

In summary, in the Landauer-Buttiker formalism the calculation of the conduc-tance is transformed into the calculations of transmission coefficients. One way toobtain the Tij for classical electrons, i.e. without quantum mechanical effects, isto calculate particle trajectories numerically and make a counting statistic for theindividual contacts.

2.2.2 Quantum point contact (QPC)

If two 2DEG regions are coupled by a narrow constriction (‘point contact’) and onemeasures the conductance as a function of its width, one finds that G increases insteps of ∆G = 2e2

h, starting from ideally G = 0 for the closed constriction [16, 17].

Such a constriction is known as a quantum point contact or QPC.The stepwise increase of the conductance can be understood by considering the

interior of the QPC as a wave guide with plane wave eigenfunctions in both direc-tions. The number of transmitting modes is M ≈ W

λF /2, with W the width of the

channel. Each of these modes adds 2e2/h to the conductance (the factor 2 stemsfrom the spin degeneracy). More detailed discussions, for example of transmissionresonances, can be found in Refs. [18, 19, 20]. The modes that are not transmittedare reflected, which leads to the so-called ‘contact resistance’ even for fully ballisticsamples. Between two conductance steps the next not yet transmitting, ‘evanescent’mode penetrates the QPC more and more until transmission 1 is reached. These ef-fects are rather independent of the QPC geometry and are observed in other materialsystems as well, e.g. in break junctions [21].

7


2.3 Drude model

For a regime in which many modes are occupied and scattering introduces phasebreaking on microscopic length scales, all properties, e.g. the conductivities, become‘locally’ well defined and the electrons can be described as classical point chargeswith an effective mass m∗. This is known as Drude model, which will be brieflyintroduced now. A glimpse at the mechanisms separating classical and quantumbehavior is given in appendix E.

With the number density of electrons n, their charge −e < 0 and their meandrift velocity ~vd one has the general expression for the electrical current density ~jas a function of electric field ~E:

~j = −ne~vd = −neµ ~E = σ ~E (2.8)

The definition of the mobility is ~vd = −µ ~E and resolves in isotropic crystals to thescalar mobility µ = eτel/m

∗ with the elastic scattering rate τ−1el . The conductivity

at zero magnetic field is σ0 = enµ = ne2τel/m∗.

2.3.1 Magnetoresistance in homogeneous samples

From the classical equation of motion for an electron in a magnetic field parallel tothe z-axis and with the relaxation time approximation for the momentum distribu-tion one finds the steady state current density ~j = σ ~E with

σ =σ0

1 + ω2cτ

2el

(1 ωcτel

−ωcτel 1

)=

enµ

1 + µ2B2

(1 −µB

µB 1

)(2.9)

where the cyclotron frequency ωc = eB/m∗ was introduced.In this model only two parameters are needed for describing the transport properties:the mobility µ and the electron density n. Inverting the conductivity gives theresistivity tensor

ρ = σ−1 =1

enµ

(1 µB

−µB 1

)(2.10)

with the following diagonal and off-diagonal elements:

ρxx = ρyy =1

enµand ρxy = −ρyx =

B

en(2.11)

The longitudinal resistivity ρxx is independent of the magnetic field while thetransverse or Hall resistivity ρxy gives rise to the classical linear increase of the Hallvoltage in B. The latter is commonly used for sample characterization. Without lossof generality one can choose the x-axis parallel to the current density and calculatethe ‘Hall angle’ θ to the electric field:

tan(θ) =Ey

Ex

=ρyx

ρxx

= −µB, (2.12)

8

2.3. Drude model

which is independent of n. This tilting of the potential gradient can be observed ex-perimentally, e.g. by optical techniques [22]. The conductivity through two parallelconductors is described by the sum of the their individual conductivities σ = σ1+σ2.

2.3.2 Magnetoresistance in inhomogeneous samples

Inhomogeneous sample parameters, for example a variable electron density, are nottrivial to treat, because of two reasons: first, the statistical properties in the ho-mogeneous case are not valid in principle, especially not in regions where theirvalues change. Second, the problem has to be solved self-consistently in an iterationprocess, because the altered fields produce a backaction on the sample parametersthat have to be inserted into the calculation of the fields. Therefore, the followingideas and the solutions of the differential equations have to be considered as firstapproximations to the problem at hand.

Starting from Maxwell’s equations for the electromagnetic fields one can assumelinear response in the material and derive the continuity equation ~∇ · ~j + ρ =0 which reduces to ~∇ · ~j = 0 for stationary fields and currents. Inserting ~j =σ ~E and the equation that defines the electric potential ~E = −~∇Φ, one arrivesat the following differential equation that, together with corresponding boundaryconditions, completely determines Φ:

~∇ · (σ · ~∇Φ) = 0 (2.13)

One method to solve this differential equation is the finite element method (FEM),which is introduced in appendix B. For the calculations of the electric potential inHall bars that are presented in this thesis, the following boundary conditions arechosen: the component of the current density ~j = −σ · ~∇Φ orthogonal to the sampleedges are chosen to vanish, except for the current leads, where it is set constant andof opposite sign at source and drain.

Since ~j is a vector field the divergence used to find equation 2.13 is not enoughto completely establish it, but one also needs the rotation ~∇ ∧ ~j. The x and ycomponents are 0 and

(~∇∧~j)z = P

(σxx

σxy

)· ~∇Φ (2.14)

as can be shown by straightforward calculation. P =

(∂y ∂x

−∂x ∂y

)is an operator that

contains only spatial derivatives. For a spatially constant conductivity tensor onefinds ~∇∧~j ≡ 0 and ~j is independent of the magnetic field. If the components of σ areinhomogeneous equation 2.14 gives an additional contribution to the current densitythat is a function of the applied magnetic field. This contribution is completelydetermined by Φ from Eq. (2.13) and the sample properties given by σ(~r). Equation2.14 shows that the current distribution in an inhomogeneous sample is a functionof the applied magnetic field.

9


2.4 Quantum Hall Effect (QHE)

2.4.1 Phenomenology

Figure 2.3(a) shows schematically a Hall-bar setup for the measurement of the Hall-and longitudinal resistances, Rxy = Uy

Jand Rxx = Ux

J, respectively, with the applied

current J and the measured voltages Uy and Ux. The length and width of the bar arelabeled by L and W , respectively. At low temperatures and high magnetic fields thetransverse or Hall resistance Rxy does not increase linearly with the magnetic fieldB, as predicted by the Drude model. Instead, one observes plateaus at Rxy = h

e21ν

with ν ∈ 1, 2, 3, ... [3, 23], as shown in Fig. 2.3(b). These values are exact toa relative precision of at least 10−8 [24] and depend only on constants of nature.The effect is known as the ‘integer quantum Hall effect’ (IQHE). The accuracy ofthe quantized values is the primary reason that the QHE is used as internationalstandard for resistance since 1990 [24].

The quantization is very general: it is essentially independent of geometry, e.g.the width or the length of the sample, and it is independent of the material as longas there are free carriers confined to a two-dimensional plane. For example, theQHE has also been observed in hole-gases [25].

The quantum Hall effect is of general interest because it led to many new conceptsin solid state physics that can be applied to other systems, like, for example, to arotating gas of ultra cold atoms [26].

The longitudinal resistance Rxx is not constant under these experimental condi-tions, as predicted by the Drude model, and one observes 1/B-periodic oscillationsthat drop to zero where the QHE plateaus occur. These are known as ‘Shubnikov- de Haas’ (SdH) oscillations. An experimental curve is shown in Fig. 2.3(b) (bluecurve). In samples of very high mobility and at very high magnetic fields, otherplateaus occur at Rxy = h

e21ν

with ν = pq

with p and q integers and q odd [4]. This

Figure 2.3: (a) Setup for measuring the longitudinal and transverse resis-tance of a Hall bar. (b) Typical traces of ρxx and ρxy versus magnetic field.

10

2.4. Quantum Hall Effect (QHE)

effect is generally called ‘fractional quantum Hall effect’ (FQHE) [27]. Also at theseplateaus the longitudinal resistance drops to zero.

2.4.2 Electron wave functions in magnetic fields

Classically, an electron in a magnetic field is bent onto a circular trajectory by theLorentz force. In a semi-classical picture an integer multiple of the wavelength λF ofthe electron wave function has to fit the circumference of the circle. This conditionallows only discrete values of λF and thus leads to quantized energies E. Thequantization is absent in 3 dimensional electron gases because the motion parallelto the applied field is not quantized and the kinetic energy can take on any valueleading to a continuous spectrum.

In order to find the energy levels one has to solve Schrodinger’s equation for asingle electron in a given potential U(x, y):

(i~~∇+ e ~A

)2

2m∗+ U(x, y)

Ψ(x, y) = E ·Ψ(x, y) (2.15)

For a free electron gas U(x, y) ≡ 0 holds. For the vector potential of the magnetic

field in z-direction the Landau gauge can be chosen, ~A =

(−By

0

). The omitted

z-component is zero and the potential in this direction is incorporated in the modelof the ‘2DEG’. Separating the two variables by the ansatz Ψ(x, y) = eikx · χ(y) onefinds the equation of a harmonic oscillator for χ:[

p2y

2m∗+

1

2m∗ω2

c (y + yk)2

]χ(y) = E · χ(y) (2.16)

with the solutions χn,k(y) ≡ 〈n, k |y|n, k〉 = e−eB~ (y−yk)2 · Hn(y − yk). The Hn

are the Hermite polynomials which are centered around yk = ~keB

. The width of thegaussian envelope function gives the length scale on which the wave function spreads

in y-direction, the magnetic length `m =√

~eB

. The eigenenergies that constitute

the so-called Landau levels (LLs) are given by

En = (n +1

2) · ~ωc. (2.17)

They are independent of k, so that the group velocity v(n, k) = 1~

∂E(n,k)∂k

= 0.The degeneracy of a Landau level can be found by counting the number of statesinside the sample at this energy: the quantum numbers k of the plane wave in x-direction are separated by ∆k = 2π

Lso that the centers of the y-part are separated

11


by ∆yk = ~eB

∆k. This gives the number of states per Landau level

N =W

∆yk

=eB ·W · L

h=

WL

2π`2m

or N =φ

φ0

. (2.18)

Dividing the total number of electrons in the sample Ntot = n·WL by the numberof electrons per LL gives the ‘filling factor’

ν =Ntot

N=

h

eBn (2.19)

The energy levels are broadened due to inhomogeneities in the local potential andother scattering mechanisms. The broadening can be estimated from the quantummechanical scattering time τel to be ∆E ≈ ~

τel. In high magnetic fields the interaction

of the electron spin with the magnetic field lifts the spin degeneracy by the Zeemanterm g∗µBB.

The density of states (DOS) resulting from this model is sketched in Fig. 2.4(a).The DOS is not constant in E as at zero field and 1/B-periodic at a given energy.Between the LLs a gap opens in the electron spectrum at high enough magnetic fieldswith only a small DOS in-between, provided by the localized states. The energy gapis essential for the QHE and can be explained for the IQHE in the presented single-particle model. The only point in this picture where more than one electron getinvolved is by the Fermi-Dirac statistics.

At classical magnetic fields the Landau levels overlap strongly and form an essen-tially continuous DOS. The critical magnetic field where the LL spacing ~ωc equalstheir broadening ∼ ~/τel due to scattering is therefore

Bc ≈m∗

eτel

(2.20)

2.4.3 Edge states

In every sample there are boundaries. They can be described by a confining potentialU(x, y). In a long bar this potential is only a function of y. Treating U(y) in firstorder perturbation theory the local eigenenergies are shifted:

E(n, k) ≈ (n +1

2)~ωc+ < n, k|U(y)|n, k >≈ (n +

1

2)~ωc + U(yk). (2.21)

The last step is based on the assumption that U(y) is nearly constant on the scale

of `m =√

~eB

, the extent of the wave function around yk. Equation (2.21) bends

the originally flat Landau levels along the confining potential U(y). In the middleof the sample, where U(y) = 0, the dispersion is flat, but at the edges E(y) is bentupwards. Thus, if EF lies in a gap in the bulk of the sample, every LL below EF

provides one state at the Fermi energy at the sample edges.In general one gets a finite local density of states (LDOS) at EF where E(n, k)

pierces the Fermi energy, also at potential fluctuations inside the sample. This is

12


depicted in Fig. 2.4(b) for various EF . Integrating the LDOS over the whole samplegives the broadened DOS shown in Fig. 2.4(a). When EF is situated between twoLLs, the only delocalized states are the states at the sample edges, which are oftencalled ‘edge states’. Also note that localized states inside the sample below andabove the center of a LL occur at different positions.

Figure 2.4: (a) DOS showing the spin-split Landau levels (b) The confiningpotential (dotted line) bends the LL at the sample boundaries and at potentialfluctuations, leading to different localized states at the Fermi energy inside thesample.

The group velocity of the electrons in these states is different from zero:

v(n, k) =1

~∂E(n, k)

∂k=

1

~∂U(y)

∂y

∂yk

∂k=

1

eB

∂U(y)

∂y. (2.22)

v(n, k) has opposite signs for opposite edges due to the inversed sign in the slopeof the potential. Thus, states carrying current in opposite directions are spatiallyseparated from each other. The classical analogue is the picture of skipping orbitsdirected along equipotential lines of the local potential, known as ~E ∧ ~B-drift [19].A more elaborate treatment of the current density in the model system of ideal edgestates is given in Ref. [28].

2.4.4 Integer quantum Hall effect (IQHE)

‘Backscattering’ of an electron means a transition from an initial state |n, +k > toa final state |n,−k >. At integer filling factors, i.e. if the Fermi energy lies betweentwo bulk Landau levels, these states are spatially separated by essentially the samplewidth, which suppresses such tunnel processes exponentially and backscattering isstrongly reduced. This causes the minima in Rxx that go to zero as T → 0. Since inthis case the edge states behave like ideal one-dimensional transmission channels, allcontacts fed by the reservoir of µR have the same electrochemical potential. Similar,the contacts connected to the other reservoir all have the potential µL. Therefore,

13


the measured Hall voltage is Uy = µL−µR

e≡ U , while the current is J = ν · e2

h· U ,

with ν the number of edge states, which corresponds to the number of fully occupiedLLs. The Hall resistance then reads

Rxy =Uy

I=

h

e2

1

ν(2.23)

This situation is illustrated in Fig. 2.5 for ν = 1.

1

2

3

4

Figure 2.5: Idealized edge statesat filling factor ν = 2. The statesare doubly spin degenerate.

A more general way to calculate the transverse resistance is to insert ν idealedge channels with transmission 1 into the Landauer-Buttiker formalism. With thecontacts numbered like in Fig. 2.5, the current applied to contacts 1 and 3 and theHall resistance measured at contacts 2 and 4, one finds

J0−J0

=e2

h

ν 0 0 −ν−ν ν 0 00 −ν ν 00 0 −ν ν

U1

U2

U3

U4

(2.24)

Setting U3 ≡ 0 and reducing the dimension of the matrix as discussed, allows toinvert the resulting matrix. Multiplication with the current vector given from theboundary conditions ~J = (J, 0, 0) (J3 = −

∑p6=3 Jp = −J) immediately shows that

U4 = U3 = 0 and U2 = U1 and

Rxy =U2 − U4

J=

h

e2· 1

ν(2.25)

This reproduces the quantized values for the Hall resistance. As long as there is nocoupling of edge states, i.e. in the gap between two LLs, these values do not change,leading to the quantum Hall plateaus. This picture explains the strongly reducedlongitudinal resistance, since all contacts on the same side of the sample are on thesame potential.

14


2.4.5 Quantum Hall transition

At integer filling of the Landau levels extended states are formed only at the sampleboundaries. The bulk contains only localized states. At the quantum Hall transitionbetween two LLs the localized states form a dense network of states that allowselectron transport by hopping or tunnel processes [29]. Around half integer fillingthe the network percolates and gives rise to the next extended state, which movesto the sample boundaries with further lowering the magnetic field. The microscopicorigin of the localized and delocalized states is the same and their quality dependson the local configuration of the potential. To prove this picture in experiments isa formidable task which will be addressed in this thesis. Another, intuitive point ofview of the underlying mechanism of the quantum Hall plateaus and the transitionsbetween them can be found in Ref. [30].

2.4.6 Compressible and incompressible stripes

The picture of edge channels is a one electron model. The electron density in thismodel increases from zero at the sample edge in steps of nL = 1

2π`2m, the density

due to one LL, cf. Eq. (2.18). This situation is illustrated in Fig. 2.6(a). It is avery unphysical configuration since a large gradient in n implies large electric fieldswhich get screened by the mobile carriers.

The screening of the confinement potential due to the ensemble of electrons hasto be taken into account self-consistently [31]. The depletion length `d for a top gatedefined sample edge of a semi-infinite 2DEG at zero magnetic field has been foundas [32]

`d =Ugε

4π2en0

(2.26)

with Ug the gate voltage, n0 the bulk electron density and ε the dielectric constant

of GaAs. For an etched structure Ug can be approximated by Ug ≈ 12

Eg

e. n then

rises to the bulk value in the form [31]

n(y) =

(y − ldy + ld

) 12

for y > ld. (2.27)

This expression is plotted in Fig. 2.6(d) as the red dashed line. It leads to completescreening of the original potential.

The the situation arising by the application of a magnetic field is depicted in Fig.2.6(c). It costs the energy ~ωc to populate the next LL, so that it is energeticallyfavourable for the system to relocate electrons from right to left to fill up the LLthere and leave the other empty. This results in a dipolar stripe at the positions yk

where n(y, B = 0) takes on integer multiples of nL [31]:

yk = `deplν2 + k2

ν2 − k2, (2.28)

15


Figure 2.6: (a) Landau levels in the single particle picture piercing theFermi energy at the sample boundary to the left, which leads to a unphysicalstep wise increase of the electron density shown in (b). (c) If one takes screen-ing self consistently into account, stripes of partially filled, compressible statesform, interrupted by stripes completely filled, incompressible states. The elec-tron density rises continuously with plateaus at the incompressible stripes.The red dashed line corresponds to B = 0 (after [31]).

where ν denotes the bulk filling factor and k ≡ bνc the number of completelyfilled LLs, which therefore corresponds to the number of dipolar stripes.

The compressibility κ of an electron gas is defined as [33]

κ−1 = n2 · ∂µelch

∂n(2.29)

Thus in the region of the dipolar stripes, where adding an electron costs the energy~ωc, the compressibility is zero. These regions are therefore called ‘incompressiblestripes’ and can be easily identified in Figs. 2.6(c) and (d), where the Landaulevels in the screened potential and the corresponding electron density are shown,respectively. The width of the stripes can be estimated by setting the potentialdrop ~ωc equal to the typical electric field in the stripe arising due to the densitygradient. The exact result is [31]:

ak =4√π

(a∗B`d)12

νk12

ν2 − k2. (2.30)

with `d the width of the depletion region from the gate that defines the sample

16


boundary. Between the incompressible stripes the DOS at the Fermi energy isnot zero and the screening capability of the 2DEG is finite. As can be seen inFig. 2.6(c) electrons can be added at infinitesimally small energies and hence thename ‘compressible stripes’. These results are confirmed in self-consistent numericalcalculations as good approximations [34]. Experiments probing the involved stripestructure can be done for example by changing the depletion length by a gate voltagein magnetocapacitance measurements [35]. Other experiments will be discussedlater.

The current distribution in the QHE regime is still rather controversial. Inthermal equilibrium the current density in the sample can be written in the form[36]

jx(y) =~

2m∗

[(2bνc+ 1) +

1

µB

∂µxcch

∂B

]· ∂n

∂y+

n

m∗ωc

∂UHxc

∂y, (2.31)

where µB denotes the Bohr magneton, µxcch portion of the chemical potential

due to exchange interactions and UHxc the self-consistent potential, which includesthe confining potential, the Hartree- and the exchange energies. The first term isproportional to the gradient of the electron density, which is positive going from theleft into the sample center and gives a current density which is diamagnetic. Thesecond term is proportional to the gradient of the self-consistent potential, whichdecreases monotonically along the y-direction and therefore produces a paramagneticcurrent density.

In the compressible regions the electron density varies, while the self-consistentpotential is pinned to a constant value and the second term in Eq. (2.31) is zero, cf.Fig. 2.6(d). In the incompressible stripes, the density is fixed and the self-consistentpotential varies. Therefore the two stripes show opposite current densities, like it isindicated in Fig. 2.6(d), where the compressible stripes are sketched as light grayregions and the incompressible as dark gray. In thermal equilibrium the currentdensities on the opposite sample edges are exactly symmetric, but with opposite signso that no net current flows between source and drain. Nevertheless, an equilibriumcurrent flows around the sample and the potential fluctuations that gives rise to thede Haas - van Alphen effect in the magnetization.

If an external bias voltage is applied, the electrochemical potential on the twoopposite sides of the sample differs by eUxy, the potential due to the Hall voltage.µelch drops mainly across the innermost incompressible stripes, since the potentialin the compressible stripes is flat. The net current is then carried dissipationless bythe unequal population (dissipation is connected to low energy excitations, whichare forbidden in these regions). Only at filling factors right below integer numbersthe incompressible stripes are very narrow and the potential drop occurs also viacompressible regions, where dissipation sets in [37].

17

Chapter 3

Scanning probe microscopy

3.1 Introduction

In scanning probe microscopy (SPM) a sensor is scanned across a sample surface,i.e., it is positioned at specific points on a predefined grid, where the sensor signalis recorded. The data are presented as 3-dimensional maps, often color-coded, asfunction of the position in the sample plane. An SPM therefore consists of threeconceptional parts: (1) A mechanism to generate the motion parallel to the samplesurface (xy-plane) and orthogonal to it (z-direction). (2) A control signal to keepthe sensor-sample distance as required. (3) A sensor and electronics for acquiringthe data.

The first part of an SPM is achieved by piezoelectric elements: when mechanicalpressure is applied to a piece of piezoelectric material, the deformation of the crystalstructure produces a voltage proportional to the pressure. Conversely, when an elec-tric field is applied to the crystal, the shape of the piezo-material is changed. Thismechanism is used for many applications, from filtering of electric signals, pressuresensors and other forms of transducers in industry to being part of qubits of pos-sible quantum computers [38]. Piezoelectric effects are also a field of fundamentalresearch, for example the giant piezoelectric effect in strontium titanate at low tem-peratures, probably due to quantum critical phenomena [39]. Since the materialsdeforme on sub-atomic length scales, this mechanism provides the ideal basis forany scanning probe applications.

The second part, a controlled and stable sensor-sample separation, is accom-plished by keeping the measurement signal constant by an electronic feedback cir-cuit or by using an additional sensor. Usually, SPM experiments rely on the STMor AFM setups discussed in the next sections. A specific implementation and someconstructional details are given in chapter 4 for the microscope built and used in thisthesis. Overviews on general SPM techniques and on STMs and AFMs in particular,can be found in Refs. [40, 41]. The remaining sections of this chapter give a roughoverview on what kind of sensors are used in current research, though space allowsonly for the most basic implementations.

19

Chapter 3. Scanning probe microscopy

3.2 Scanning tunneling microscope (STM)

Modern scanning probe microscopy started in 1981. The first sensor signal used forSPM was the tunnel current between a sharp metallic tip and a conducting samplebrought close together [5, 42]. A voltage applied between the two metals leads to atunnel current, which depends on the local density of states (LDOS) of both mate-rials and exponentially on the tip-sample distance. The latter explains the fantasticspatial resolution, since the current leaves the tip only at the very last atom of thetip. Atomic resolution was achieved already in the early experiments, for exampleresolving the 7×7 reconstruction of the Si(111) surface [43], a heavily debated topicat that time. Reviews on STM developments can be found, for example, in Refs.[44, 45].

Today STMs are important instruments in surface sciences. Low temperatureexperiments on Co-adsorbate induced 2DEGs on the (110)-surface of p-InAs showreal-space images of percolating conducting islands, turning an insulator to a metalwhen the electron density is increased [46, 47]. A technique to tunnel into localizedstates is also reported [48]. Very recent spectroscopic experiments are the spinexcitation spectra of individual manganese atoms that show variations for differentlocal environments [49].

In addition, the STM setup can be used to manipulate single atoms and moleculeson surfaces: several CO-molecules can be arranged in mechanically unstable relativepositions on a Cu(111) surface, which leads to so-called ‘molecular cascades’ thatallow the construction of classical logic gates [50]. Another example of fundamentalinterest is the construction of gold atom chains of variable length, which allows oneto observe the formation of a one-dimensional bandstructure in real space [51, 52].‘Quantum corals’ in a stadium of piled-up atoms [53] and experiments on quantumdots on semiconductor surfaces [54] demonstrate the possibility of mapping electronwave functions with this technique.

3.3 Atomic force microscope (AFM)

A severe disadvantage of the STM-technique is that only well conducting samplescan be investigated. The observation of large forces acting between the STM tip andthe sample already in the early experiments led to the invention of the atomic forcemicroscope (AFM) [6]. This technique is used in the present thesis and is thereforereviewed in more detail now:

Between a sharp tip and the sample surface various interactions with differentcharacteristic length scales are relevant [55]:

• Pauli principle (repulsive)

• Electric: van-der-Waals (attractive), capacitive (attractive)

• Magnetic (only with magnetic tip)

20

3.3. Atomic force microscope (AFM)

• Chemical bonds

• Capillary forces

• Dissipative mechanisms

Except for the dissipative mechanisms all these forces are independent of timeand vary with the tip-sample distance. A review of many experiments and onmodeling the various tip-sample interactions can be found in Ref. [55].

In order to measure the force acting on a tip, the latter is mounted on a cantileverthat can be characterized by an effective spring constant. The force acting on thetip is then detected by measuring the deflection of the cantilever, for example. Thisis usually achieved by shining a laser beam on the parts in motion and observingthe reflection of the beam by a CCD-device. While approaching the surface, the tipfirst gets attracted by the long-range van-der-Waals forces and, after a minimum inthe interaction energy, comes into the repulsive regime, when the sensor ‘touches’the surface (Pauli principle).

In the so called contact mode the topographic image is recovered by keeping thedeflection of the cantilever at a constant value. This is achieved by an electronicfeedback that controls the z-piezo voltage, commonly called ‘z-feedback’. This modeis not used very often anymore, since the rather direct contact necessary to detectthe signal influences the sample as well as the tip quality.

In the dynamic mode the cantilever is excited to an oscillation so that the am-plitude (‘amplitude detection’) or phase relative to the excitation can be observed(‘phase detection’). The resonance frequency of the cantilever depends on the forcegradient acting on the tip, as discussed in chapter 4.3.1. A curvature in the in-teraction potential shifts the resonance frequency and changes the relative phaseand the mechanical amplitude of the sensor. These ‘error signals’ can be used in afeedback-loop to adjust the tip-sample distance. If a feedback is incorporated thatholds the excitation frequency at the actual resonance frequency of the cantilever,the deviation from the original resonance frequency can be measured and used forthe z-feedback (‘frequency detection’). If this is done with the phase signal, thetechnique is called ‘phase-locked loop’ (PLL), which will be discussed in some de-tails in chapter 4.3.5. In a PLL-scheme another feedback can be incorporated tocontrol the excitation voltage for keeping the sensor amplitude constant, thus givinginformation about dissipative forces.

Concerning the tip-sample interaction, two operation regimes are commonly dis-tinguished in atomic force microscopy: in the tapping mode the oscillation amplitudeis large comapared to the scale of the interaction potential, while it is small in thenon-contact mode.

Because of the persistent photoeffect the read-out of the mechanical amplitudeand phase of the sensor oscillation by a laser beam is problematic when investigatingsemiconductor structures at low temperatures. One solution is a perfect shieldingof the laser beam. In this thesis a fully electronic read-out scheme based on piezo-

21


electric currents in quartz sensors is adopted and described in chapter 4.3. Piezo-resistive sensors have been available commercially only for a short time.

Some applications of AFMs shall be reviewed briefly now. In contrast to STMsalso insulating surfaces can be imaged by an AFM. This makes it an invaluabletool for surface science and an ideal positioning system for experiments with moreexotic sensors. AFMs are used for surface characterization in very many branchesof science, for example for studies of crystal growth [56] or, by measuring lateralforces, in the new field of nanotribology, where the microscopic origins of frictionare investigated [57, 58]. Atomic resolution, showing for example the Si 7 × 7reconstruction, was demonstrated for various operation modes [59]. AFMs alsostart to play an important role in investigations of biological systems [60], e.g. forimaging, positioning, cutting and folding single DNA strings [61].

An AFM with a metallic tip offers a wide range of possible experiments. Oneexample is to directly probe the electric potential by direct contact with the sample.This is useful in investigating molecular electronics, like in a carbon nanotube, wherethis method enables one to measure separately the resistances between the tube andthe contacts, the resistance between the tube and the tip, and also the intrinsicresistance of the tube itself [62]. Other examples are the twin-use in an STM/AFMsetup, the investigation of integrated circuits [63], AFM lithography and the localchange of sample parameters in scanning gate experiments. Related experimentsand techniques will be discussed in the main part of this thesis.

3.4 Probing the local magnetic field

In order to probe the local magnetic field three kinds of sensors have been demon-strated. The first consists of a small Hall bar on the corner of a GaAs heterostructurechip. A thin gold layer close to the Hall bar allows positioning in the STM-mode.The recorded signal in the corresponding scanning probe measurements is the Hallvoltage in the Hall bar, which is sensitive to the local magnetic field. This techniquewas used, for example, to image individual vortices in high-Tc superconductors [64],including real time imaging of them entering the material in a magnetic field cycle[65]. A resolution of 850 nm could be achieved in the latter experiments.

An other method for investigating magnetic fields is to scan a superconductingquantum interference device (SQUID) [66] across the surface. This technique wasused for example to image vortices in a superconductor [67, 68] and currents inintegrated circuits [69]. The spatial resolution is limited to about 1 µm by the tipsample separation and the diameter of the SQUID.

The third possibility to obtain informations about the local magnetic field isto use a magnetic tip in an AFM setup and measure the additional forces. Withthis technique domain walls of various ferromagnetic systems [70, 71] and the strayfields of vortices in the glas-phase of a high-Tc superconductor could be observed [72].Combined with the microwave field of electron spin resonance (ESR) experiments it

22

3.5. Probing the local electric field

is even possible to detect the magnetic moment of a single unpaired electron spin indangling bonds of silicon [73].

3.5 Probing the local electric field

The detection of electric fields without a major interaction with the sample canbe achieved in principle by a QPC [74]. Although this method is of great interestfor fixed geometries in connection with experiments on mesoscopic and quantumsystems [75, 76], it is hard to be implemented in a scanning setup due to technicaldetails. Such sensors for low temperatures and high resolution are still a matter ofcurrent research. The properties of such sensors can be tested by scanning a chargedAFM tip across the active area [77], similar to the scanning gate experiments in thisthesis.

The current in a single electron transistor (SET) is very sensitive to the ‘gatevoltage’, which makes it a very sensitive sensor for electric fields. An SET is ametallic island with two current leads and a capacitance so small that single electroncharging blocks the the electron transport (‘Coulomb blockade’), if only small biasvoltages are applied. The electron population on the island depends on the onehand on its geometrical size, giving rise to the charging energy, and at low electrondensities, on the other hand, on the electron wave function. For an SET in thelatter regime it is common to use the expression ‘quantum dot’ instead of SET.With an additional gate the total electric energy of the system can be adjustedand the discrete energy levels of the dot can be aligned to the Fermi energy ofthe leads, which allows a current to flow. Reviews on quantum dots can be found,for example, in Refs. [78, 79]. Scanning experiments with an SET are technicallyvery demanding, because of the low temperatures needed for its operation and theinvolved manufacturing processes necessary to bring an SET on an AFM sensor.The few available publications are reviewed in the introductions to chapters 7 and9.

By applying a voltage between an AFM tip and the sample, electrostatic forcemicroscopy (EFM) has been used to study many material properties like the capaci-tance, surface potential, charge or dopant distribution and dielectric properties. Forexample experiments on ∼ 5 nm wide nanocrystals are presented in Ref. [80]. InEFM, though, the sample is influenced by the tip, which has to be accounted for inthe interpretation. More examples are discussed in the introduction to chapter 9.

23


3.6 Scanning near-field microscopy (SNOM)

Many techniques were developed in recent years that allow to increase the resolutionof optical microscopy beyond the diffraction limit of λ/2, with λ the wave length ofthe used electromagnetic wave.

In the approach of the ‘scanning near-field optical microscopy’ (SNOM), thesample is illuminated by a conventional laser and a small optical probe is broughtclose to the sample surface. At distances smaller than the wavelength, also wavesthat do not propagate can be probed (‘near field’ or ‘evanescent field’). Further awayfrom the surface these waves decay exponentially and can not be observed anymore.Usually, the probe consists of an aperture through which the field couples into anoptical wave guide leading to the detector. Reviews on this and similar techniquescan be found in Refs. [81, 82]. Better resolution might be achieved, for example, byusing single molecules positioned near the surface as light sources [83].

Most applications are in the field of single molecule spectroscopy. An examplefrom solid state physics is to probe single eigenstates of low dimensional quantumsystems [84]. A very recent development is the ‘scanning near-field photolithography’allowing photolithography with a resolution down to 50 nm [85].

24

Chapter 4

Experimental setup

The low-temperature microscope setup built during this thesis consists of the fol-lowing units: (1) A standard 4He-cryostat with a variable temperature insert (VTI)and a superconducting magnet1 (2) an experimental platform with the cabling, in-sert mechanism and vibration isolation (3) a coarse and fine positioning system thatbrings the AFM sensor to well defined coordinates relative to the sample in threedimensions and (4) the AFM sensor with appropriate read-out schemes and dataacquisition. These units are described in the following.

4.1 General setup

4.1.1 Cryostat

The cooling system consists of a commercial 4He-cryostat with a large diametersample space (50 mm) to accommodate the AFM. The sample space is isolatedfrom the laboratory by an inner and an outer vacuum chamber (IVC and OVC)and by a caoutchouc mat, cf. Fig 4.1(a). The latter helps to dampen mechanicalvibrations. For cooling, liquid 4He is evaporated from the bath into into the samplespace, controlled by a needle valve inserted in the connecting capillary tube. Moredetailed accounts on the physics of cryogenics can be found for example in Ref. [86].The exhaust helium is collected in a recovery line. The magnetic field of up to ±8T is produced by the current fed through a superconducting coil with the samplespace in the center.

4.1.2 Insert and vibration isolation

The actual microscope is mounted at the end of a long stainless steel tube by anindium seal. This allows to insert the experiment into the VTI, exactly to the centerof the magnet, cf. Fig. 4.1(b). The microscope is enclosed by a vacuum beaker made

1Oxford Instruments

25

Chapter 4. Experimental setup

Figure 4.1: (a) Cryostat and gas handling system. (b) Vacuum tube withthe microscope head at the lower end, surrounded by the vacuum beaker. (c)and (d) show the vacuum feed-through for the various cables at the top endof the tube. A heavy stone plate reduces vibrations in the setup.

of CuBe and connected to the stainless steel tube for evacuation by a conical sealwith vacuum grease in-between. Pressures below 10−5 mbar and heating the sampleduring the cooldown process prevent water and air from condensing on the samplesurface. Inside the tube metal baffels reduce temperature radiation from the warmparts on top of the setup to the bottom.

In SPM experiments the relative position of the sensor and the sample has tobe controlled on length scales of 1 nm and below. Therefore damping mechanicalvibrations is a major task. Vibrations in the microscope are generated by boilinghelium in the cryostat, irregular helium flow in the recovery line (e.g. oil dropletsfrom the pump), vibrations in the laboratory, or acoustic coupling. Most of thesesources are decoupled from the experiment in the following way: the stainless steeltube with the microscope is tightly screwed via a heavy three-pod (∼ 15 kg) to amassive stone plate (∼ 50 kg, see Figs. 4.1(c) and (d)). During the experimentthis stone plate resides on the cryostat, separated by three metal cylinders for easyaccess, cf. Fig. 4.1(a). Between the cylinders and the stone plate a special rubberfoam of the ideal thickness decouples the stone and thus the microscope from theenvironment. The microscope does not touch the cryostat and the VTI is connectedto the three-pod head by a rubber seal. This setup is shown in Fig. 4.1(b).

4.1.3 Cabling

The three-pod head shown in Fig. 4.1(c) and (d) is also the central part of thevacuum feed-through. Its interior is connected to the vacuum beaker through thestainless steel tube. They get evacuated by an exterior turbo pump. The highvoltage (HV) cables, sample connections (A) and temperature sensors (B), as well as

26

4.2. Positioning system

the coax-connections are separately brought into the vacuum with commercial plugsand at room temperature. The cables are lead down to the microscope as ‘twistedpairs’ in four thin stainless steel tubes inside the larger vacuum tube. Available are3×12 teflon-insulated manganine wires (HV, A and B) and 10 coax-cables. They endin the microscope in corresponding plugs, see Fig. 4.2(e), which makes connectingand handling very simple. All plugs and cables in this setup are commerciallyavailable and can be exchanged easily.

Since only the outer vacuum tube is cooled by the passing helium gas, goodthermal anchoring is necessary to take care of the additional heat load. For thispurpose two different techniques have been chosen: (1) The manganine cables aresoldered on 10 cm tracks of printed circuit board that are glued on a copper post withx-shaped cross section, to be seen in Fig. 4.2(e). (2) The coax-cables are clampedtightly to the same copper post in a separate quadrant. The plugs are shieldedfrom each other by the quadrant ‘walls’ of the copper post, e.g. separating thehigh voltage from the measurement cables. This copper post is hard-soldered to theconical seal, where the relatively large and well cooled vacuum beaker connects tothe microscope body. This guarantees good cooling of the cables and the microscope.

In order to characterize the cabling all involved resistances and the complete47×47 capacitance matrix of all cables and the microscope body were measured. Theinductances can be neglected at frequencies lower that ∼ 10 MHz. This knowledgemakes it possible to discriminate effects of the setup from the sample response.

4.2 Positioning system

The crucial part of any scanning probe microscope is the positioning system. Thisnot only includes the scanning apparatus, but also the coarse positioning. At lowtemperature the scanning range is rather small. The sensor is shifted relative to thesample due to thermal contractions during the cooldown (factor 150 in temperature!)typically by about 50 µm in a rather reproducible way. Mechanical vibrations arecompletely random and can lead to misplacements of up to several 100 µm. Asymmetric design and the choice of materials brought the values to acceptable limits.A typical situation might be illustrated by the following scaled model: You are veryshort-sighted, so that you can investigate only an area of 2× 2m2 (scan range) per2 hours (time per scan) and somebody drops you anywhere on a soccer-field (typicalshifts of the sensor) without orientation signs. Your goal is to find the ball (4×10 µmHall bar).

4.2.1 Piezo scanner and z-module

The scanning motion is performed by a five-electrode piezo-tube as shown in theinset of Fig. 4.2(a) (length=6.35 cm, outer diameter= 0.95 cm, wall thickness= 0.5

27


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28


mm, material: lead zirconate titanate (PZT) ceramics, EBL#22). One electrode isinside the tube and four at the outside, which allows motion in all directions in space.Cables for the high-voltage are glued directly to the electrodes. The maximum rangeof the xy-motion at room temperature is 138× 138 µm2, 35× 35 µm2 at T = 78 Kand 21×21 µm2 at T = 2 K. The range in z-direction is 9.2 µm, 2.3 µm and 1.34 µm,respectively. The calibration of the displacements in all three directions was done ona calibration grid at various temperatures. The scan piezo is fixed at the top of theso-called ‘prism’ shown in Fig. 4.2(a). A platform for mounting the AFM sensor isglued to the bottom of the piezo in such a way that the scan motion is not impeded.The prism has a cross section with a three-fold symmetry and has three glidingplanes made of polycrystaline saphire (white). The prism is designed to glide insidethe z-module up and down in z-direction in order to perform a coarse approach tothe surface. The interior of the z-module is depicted in Fig. 4.2(b). The motionis produced by six commercially available piezo stacks3, like the one shown in theinset of Fig. 4.2(c). They consist of four 5×5 mm wide thin piezo plates stacked oneach other with evaporated electrodes in-between and a single crystal saphire plateon top. In this design an applied voltage leads in first order to a shear motion ofthe stack in only one direction. For the transport of the prism the so called ‘slip-stick’ motion is used: first, the voltage is increased slowly so that the prism sticksto the piezo stacks due to frictional forces and the prism gets transported alongthe z-direction. Then the voltage is sharply reduced to zero and the piezo stackssnap back into the original position. If the prism is supposed to follow this motion,the frictional forces have to be larger than the corresponding change in the prismsmomentum per time. If this does not hold, the prism ‘slips’ and stays at the sameposition. This slip-stick motion can be repeated many times which allows to movethe prism as far as possible geometrically by design.

To produce a net-motion, it is required to have

mprismdvprism

dt> Fstick ≈ αFN (4.1)

with mprism the mass of the prism, vprism its velocity, α the sticking-friction co-efficient and FN the normal force. This normal force is provided by the ‘sledge’,which consists essentially of two piezo stacks and a spring, cf. Fig. 4.2(c). Thespring presses the stacks to the gliding planes of the prism and the latter to theother stacks of the microscope body, as visible in Fig. 4.2(b). This holds the prismin position, if no voltages are applied. How these parts fit together can be seenin Fig. 4.2(f). The z-module is mainly used during the cooldown because of thethermal contraction that would press the AFM tip into the sample and destroy it.An adequate cooldown proceedure is described in chapter 6.3.

2Staveley Sensors3PI Ceramics/ Polytec PI, Inc.

29


4.2.2 xy-table and thermometry

The xy-table shown in Fig. 4.2(d) holds the sample and allows its coarse positioningin xy-direction. It is firmly screwed to the z-module by four screws that fix thewhite Shapal plate, to be seen to the right of Fig. 4.2(f). Into this platform threesmaller 5-electrode piezo tubes (length= 1.6 cm, outer diameter= 3.2 mm, wallthickness= 0.25 mm, material: EBL#4) are glued with single-crystal saphire ballsat both ends. Both ends of the tubes make the slip-stick motion as discussed for thez-module, if an appropriate voltage cycle is applied to opposite quadrant electrodes.The piezo material for this application has to be chosen with care, since in addition tothe maximum expansion coefficient like for the scan piezo, also a large AC-depolingfield is required, because of the fast switching in the slip-stick cycles.

The mobile part of the xy-table consists of two CuBe-plates at the upper andlower ends of the piezo tubes, held together by three small but rigid CuBe cylindersand a spring at the lower end of the table that provides the normal force for theslip-stick mechanism. The plates glide across the saphire balls on three 5× 5 mm2

polycrystaline saphire plates. On the upper CuBe plate a chip socket is mountedand fixed by a Cu screw for thermal contact. It is designed to accommodate astandard ceramic 32-pin chip carrier, where the sample has to be glued horizontallyand firmly for good scan quality.

The measurement contacts to the sample are made of thin varnish-isolated copperwires (red) soldered to metallic pins on the white Shapal-platform, to be seen inFig. 4.2(d). The connections to the probe cables are made by four Macor plugs, asdepicted at the right of Fig. 4.2(f). As thermometers an Allen-Bradley and a Pt100resistor are cast into the copper part with a synthetic resin4 right below the chipsocket. The changes in the Allen-Bradley resistor are strong at low temperatures buttoo weak for being used as a precise thermometer at T > 70 K. It was calibrated byfitting formula (1) from Ref. [87] to the measured 4-terminal resistances at room,liquid nitrogen and liquid helium temperatures. The effect of magnetic fields ison the order of a few percent [88] and is neglected in this thesis. The metallicPt100 resistor shows an increase of resistivity at high temperatures and is used as athermometer between T = 70 K and T = 350 K. The calibration for this resistor isgiven by the manufacturer. A long manganin wire for heating is coiled around thescrew that holds the chip socket and fixed by synthetic resin.

4.2.3 Capacitive sensors and orientation on the sample

In order to measure a change in the relative position of two metallic objects theirmutual capacitance can be used and calibrated. In the microscope described herea differential capacitance sensor is implemented for monitoring the z-motion of theprism. The principle can be described using the schematic in Fig. 4.3: two electrodes(blue) move synchronous relative to a third electrode (green). The capacitance from

4Stycast from LakeShore Cryotronics, Inc.

30


one of the blue plates to the green depends on the relative position and can beapproximated by individual plate capacitors, as it is indicated with C1 and C2. Toeach of the two moving plates a harmonic voltage is applied with amplitude Uexc andangular frequency ω, but with a 180 phase shift. The relevant signal is the inducedcurrent on the green plate which is measured via a current-to-voltage converter(I-U-converter) and a lock-in with the applied voltage as reference.

The current in this plate capacitor model can easily be calculated to

I = Uexc · ωWεε0

d· (L2 − L1) (4.2)

with W being the width of the plates. The other geometric quantities are givenin the Fig. 4.3. If the plates are arranged symmetrically, i.e. L1 = L2, the inducedcurrent is zero. Deviations ∆L from this center position lead to a linear increase ofthe current, until the green plate resides completely below one of the blue.

Figure 4.3: Schematicof a capacitive positioningsensor.

The implementation of this idea can be seen in Fig. 4.2(a) and (b): the formershows a gold film evaporated on one of the flat surfaces of the prism. Also visibleis a gap that separates the film into two electrodes. They correspond to the blueelectrodes in the sketch of Fig. 4.3 and are connected by glued wires at the side of thegold film. The counter part is made of a copper plated piece of printed circuit boardand is shown in Fig. 4.2(b). Experimentally, at low temperatures, a sensitivity of

IUexc∆Lω

≈ 0.22 nAnmV·Hz

is found, which corresponds very well to the geometry andEq. (4.2).

For orientation on the sample surface the same idea was implemented in a fourquadrant capacitive sensor, evaporated around the sample and the AFM tip ascenter electrode. Unfortunately, the standard thickness of the gold evaporation wasso large that the sensor quality was affected too strongly in fast scanning. An easierconcept for orientation is to use a grid of etched markers that shows the way to thestructure under investigation. This concept proved very useful and is adopted inmost of the experiments.

31


Data acquisition and high voltage

The core of the scan system is the control electronics from a formerly commercialavailable low-temperature STM5. A software on a standard PC generates a programcode that is loaded into the specialized hardware, where the low voltage signalsare generated, e.g. the ramp for the scan motion and the voltage sequence for theslip-stick mechanism. These signals are then amplified by the desired factors. Theresulting scan signal of ±240 V is produced by the high-voltage amplifiers. Theamplifiers for the slip-stick motion were completely rebuilt. It proved necessary tohave a maximum output of 700 V for the z-motor. The corresponding amplifier canbe tuned by an external potentiometer and by the software in parallel. In theseamplifiers fast shut-off circuits are implemented to ground the piezos on a time scaleof microseconds.

Parallel to the scan motion the inputs of eight 16-bit AD-converters are recorded.In this setup three are used for the sensor signals, as will be discussed below, andthe remaining 5 can be used for other signals, for example sample characteristicsduring a scan, as done in a scanning gate experiment.

Another new component is the compensation of the inevitable sample tilt forconstant height scanning. It is achieved by applying an individually adjustablefraction of either of the voltages that produce the scan motion to the inner electrodeof the scan piezo. The corresponding circuit can be switched on and off separately.

4.3 Tuning fork sensors

A small selection of possible SPM-sensors is discussed in chapter 3. In the home-builtsensors used in this thesis the oscillating element consist of a piezoelectric quartztuning fork (TF), as it is used as frequency standard in wrist watches and otherelectronics. The two prongs can be described as two coupled harmonic oscillators.The design of the electrodes, to which the driving voltage is applied, favors theantisymmetric eigenmode of the system at a frequency of about 32.7 kHz. Since theeigenmodes couple only very weakly if the prongs oscillate freely, as it is the casehere, the description can be reduced to a single harmonic oscillator [89].

A review of the large variety of experiments with TF-sensor is given in Ref. [59].Very recent experiments include, for example, scanning capacitance measurements[90] and SNOM-sensors based on quartz tuning forks [91].

In this section AFM-sensors based on quartz tuning forks (TF) are discussed inmore detail. In addition, a novel type of TF-based sensors is presented with somefirst applications shown in appendix C. The implemented read-out schemes and ashort analysis of the involved feedback loops can be found at the end of this chapter.

5TOPS3 system by Oxford Instruments

32

4.3. Tuning fork sensors

4.3.1 Harmonic oscillator model

Independent of how the sensing tip of an AFM-sensor is constructed (e.g. micro-machined silicon or carbon nanotubes [92]), the force sensor is usually an oscillating‘cantilever’. How the force (-gradient) can be measured by such a device is nowshown in the harmonic oscillator approximation. Neglecting damping terms, theeigenfrequency of the system can be found by solving Newton’s equation of motion

z = −k0

mz + Fts(z) (4.3)

where k0 describes the system’s spring constant and Fts the tip-sample interaction

force. For a constant force the solution is z = Amechcos(ω0t) with ω0 ≡ 2πf0 =√

k0

m

and Amech the amplitude of the oscillation. Spacially slowly varying forces can beapproximated by two terms of a Taylor expansion in z-direction, i.e. Fts ≈ Fts,0 +(

∂Fts

∂z

)z ≡ Fts,0−ktsz, where kts can be seen as an effective spring constant produced

by the interaction potential. Inserted into Eq. (4.3) the resonance frequency reads

fres ≈1

2π

√k0 + kts

mor ∆fres ≈ f0

kts

2k0

(4.4)

The last approximation is valid for kts k0 and has to be multiplied by 1/2in the case of the tuning fork sensors with two coupled oscillators. Formula (4.4)is valid as long as the force gradient is constant for all tip positions on a completeoscillation, e.g. for small mechanical amplitudes. On the other hand, fast andstable scanning is possible only for amplitudes Amech > 3 − 10 nm, due to noisein the system and a limited read-out sensitivity. This is about the length scale onwhich the interaction potential varies for PtIr tips on GaAs surfaces [93] and morecare in the analysis is required.

Equation (4.3) can be solved for an arbitrary force Fts by expanding the periodicz(t) into a Fourier series [59]. Inserting into Eq. (4.3) gives the Fourier componentsas functional of the external force. The first coefficient corresponds to a constantdeflection proportional to the force averaged over one period. It can be omitted byredefining the coordinate system. For small perturbations it is enough to keep thenext term in the expansion, i.e. z(t) ≈ Amech · cos(ωt). Solving for ω gives

fres ≈ f0

(1− 〈Fts · z〉

k0A2mech

)or ∆fres ≈ −f0

〈Fts · z〉k0A2

mech

(4.5)

where 〈...〉 denotes the average over one oscillation. The last approximation isvalid for 〈Fts · z〉 k0A

2/2, i.e. for stiff cantilevers or large mechanical amplitudes.A more general discussion can be found in Ref. [94].

33


4.3.2 ‘Standard’ design

An example for the ‘standard’ design of low-temperature AFM sensors built in ourgroup is shown in Fig. 4.4(b). The casing of the tuning fork is removed completely,except for the metal ring around the base of the TF. This ring is then used to solderthe tuning fork on a carrier plate with electrical contacts. More details can be foundin Refs. [95, 96].

The original TF-casing is ferromagnetic, including the ring for soldering. Thisleads to large forces on the sensor that may even result in mechanical shifts of thesensor platform. Therefore the measurements presented in this thesis were performedwith the sensor shown in Fig. 4.4(a), where the casing was removed completely andthe tuning fork glued onto a piece of Macor by a non-conducting epoxy glue. Themacor piece itself is fixed on the standard carrier plate with 6 separate electricalcontacts, best seen in Fig. 4.4(c). This carrier plate is then screwed tightly to theplatform on the scan piezo.

The tip is made by attaching a 15 µm diameter Pt80Ir20-wire manually to the‘upper’ prong, parallel to the TF oscillation. The other end serves as separateelectrical contact and is supported by a thick copper post. The latter is electricallyisolated from the wire and grounded during the measurements. By electrochemicaletching tip radii of 30 − 100 nm are achieved. Also experiments with the muchsmaller 74 kHz tuning forks have been performed to demonstrate the principle, butthe handling and fabrication is more tedious so that the idea was not pursued.

The use of a TF-based AFM sensor has several advantages in low-temperatureexperiments. The high stiffness of k0 ∼ 10 kN/m avoids problems with the so-called‘snap-in’ to the sample surface. For experiments with semiconductor samples opticalread out can not be used easily because of the persistent photoeffect. The powerdissipation (< 1 nW in the usual operation mode) is many orders of magnitude

Figure 4.4: (a) Home-built TF-sensor mounted on a piece of Macor. ThePtIr tip is glued to an insulated Cu post. The inset shows an electron mi-crograph of the tip glued to one prong of the tuning fork. (b) ‘Conventional’home-built TF-sensor. (c) Cantilever tuning fork (CLTF) sensor mounted on aplate fitting to the platform on the scan piezo. Also low-temperature I-U con-verter stage can be seen in this image. The inset shows an electron micrographof the cantilever attached to the TF.

34


smaller than in piezoresistive sensors. The observed dependence of the resonancefrequency on the applied magnetic field is small enough to be corrected before ascan. Tuning forks are very cheap and relatively big and it is possible to mountother sensor types on the scanning prong.

Drawbacks of TF-sensors are the high stiffness, which reduces the force sensitivityof the sensor and leads inevitably to a blunt tip in case of a tip crash. As will beseen below, also the electronic setup for the scan control is rather involved becauseof the small relative frequency changes (∼ 10−7). Last, but not least, all sensors arehand-made in a rather cumbersome process.

4.3.3 Cantilever tuning-fork (CLTF) sensors

A way to exploit the advantages of the TF-sensors while avoiding the drawbacks wasreported recently [97, 98]: by attaching a micro-machined cantilever to both prongsof the tuning fork, the motion of the TF is translated in an orthogonal movement, asdepicted in Fig. 4.5. The displaced prongs induce mechanical stress in the cantilever‘legs’, which is accommodated by a deformation of the cantilever orthogonal to theTF plane. An opposing force acting on the tip leads to a back-action on the TF,which changes the eigenfrequency of the system.

This novel sensor concept allows the use of any technology known from theproduction of standard cantilever sensors to fabricate the probing tip. Even batch-fabrication could be demonstrated, based on the fact that all components lie in thesame plane.

This sensor concept was developed further during this thesis in collaborationwith T. Akiyama, K. Suter and U. Staufer (University of Neuchatel) for the use inlow-temperature experiments on semiconductor structures. Several steps could be

Figure 4.5: Schematic of the working principle of the CLTF sensor. Thehorizontal motion of the prongs are transformed via the cantilever into verticalmovements of the tip. (a) and (b) show the two situations when the TFamplitude is positive and negative, respectively. The figure is kindly providedby T. Akiyama, University of Neuchatel.

35


achieved: a metallic tip with external connections was produced and successfullytested at low temperatures. A selection of these first experiments are shown inappendix C, for example scanning at all accessible temperatures (1.7 K < T <300 K), amplitude calibration, force-distance experiments, tip-sample capacitancemeasurements, investigations on the screening capabilities of a 2DEG in magneticfields up to B = 8 T, Kelvin probe measurements and scanning gate experiments atvarious magnetic fields.

The experiments can be summarized by noting that reliable scanning is possibleat all temperatures and in large magnetic fields. Handling is easy, especially forthe latest generation of sensors and, due to the soft cantilever, tip crashes are lessdestructive. The spatial resolution was limited by the electronic setup and not bythe sensor, which is satisfactory for the discussed purposes.

Nevertheless, judging from particles on the surface after scanning and from elec-tron micrographs, all tested sensors seem to suffer from a mechanical instability ofthe epitaxial layers, a problem that will be addressed in the near future. First ex-periments at low temperatures with a sensor with an electrically shielded tip wereperformed and showed problems with the excitation of the cantilever, though scan-ning was possible and the shielding worked. Further in the future lies a sensorwith a single electron transistor at the apex of the AFM tip. For this purpose firstlow-temperature tests of the structures were done during this thesis.

4.3.4 Excitation and readout principles

Self exciting and self sensing

Excitation and read out is naturally incorporated in the TF-sensors: an externalvoltage of 0.1−2 mV is applied to one of the electrodes which leads to the mechanicalexcitation of the tuning fork. The other electrode is on virtual ground of a current tovoltage converter (I −U converter), by which the piezoelectric current is measured.The signal amplitude and phase, compared to the driving voltage, can be obtainedby lock-in technique.

Figure 4.6 shows the response in the piezo current of a tuning fork sensor in afrequency sweep at a constant excitation voltage of UTF = 1 mV and under variousexternal conditions. Plotted is the amplitude ITF and, in the insets, the phase φTF

of the signal. Going from ambient conditions, shown as black curve in Fig. 4.6(a),to vacuum (red curve) the resonance is shifted by ∼ 5.7 Hz due to reduced damping.The amplitude gets larger by about a factor 3. A larger shift of ∆fres ≈ 134 Hzcan be observed when cooling down the sensor to T = 4.2 K, due to the increasingstiffnes of the materials. The amplitude increases by another factor ∼ 3.5. Thefrequency shift occuring by cooling down further to T = 2.0 K is a little less than0.5 Hz and the amplitude increases again by some fraction. This process is not ascontinuous as these images may suggest: around T = 100 K the amplitude is ratherlow and the resonance peak very broad.

36


Figure 4.6: Sensor-current amplitude ITF versus driving frequency fTF

under various conditions (a) in air (black) and vacuum (red) (b) in vacuum atT = 2 K (red) and T = 4 K (black). The insets show the corresponding phasein the same frequency interval.

The phase signal drops sharper the sharper the resonance, i.e., the Q-factor ofthe sensor becomes larger. This can be seen in the insets of the figures. Not shownis the anti-resonance due to the parallel capacitance that lets the phase go back to+90.

Frequency shifts also occur when a force gradient is applied to the AFM tip,e.g. by coming close to the sample surface. By keeping either the amplitude orthe phase constant by a feedback loop with the z-position, this fact can be used torecord the sample topography in the amplitude or phase control modes, respectively.If the excitation is kept on the actual resonance by a phase-locked loop (PLL), asdiscussed in the next section, force-distance curves can be recorded, as demonstratedin the appendix for the CLTF-sensors. Examples for the TF-sensors used here canbe found in Refs. [93, 99]. Also in appendix C, a low-temperature I − U converteris introduced that is small enough to be built onto the sensor platform, as depictedin Figs. 4.4(b) and (c).

Calibration of the mechanical amplitude at low temperatures

The current I through the tuning fork (TF) is connected to the applied voltage andthe mechanical amplitude:

I = iωC0UTF + 2iαω · Amech (4.6)

with the piezo electric coupling constant α. On resonance the parallel capacitanceC0 can be neglected and one finds Amech ≈ I

2αω. Interferometric measurements at

room temperature give a value of α = 4.2 µC/m [100].An alternative way to measure the mechanical amplitude of the sensor is by ap-

plying various driving voltages and recording the change in the z-position where apredefined frequency shift occurs, e.g. where the z-feedback starts to work. Mea-surements at low temperatures lead to α ≈ 4.8 µC/m for the sensor used here. No

37


significant change could be measured in an external magnetic field. The deviationfrom the room temperature value may be due to creep of the piezo material.

4.3.5 Read-out scheme

Phase-locked loop, amplitude and z-feedback

The actual read-out scheme used in the experiments consists of three electronicfeedback loops, shown schematically in Fig. 4.7: the piezo current from the TF-sensor is converted to a voltage with a conversion factor of 106 Ω (black arrows).This signal is fed into a lock-in amplifier6 that measures the phase and the amplitudeof the signal with reference to the excitation voltage (gray arrow). The deviation ofthe phase signal from zero is filtered by a PID-controller, which allows to adjust thesystem response function, and is used to control the excitation frequency in a voltagecontrolled oscillator (VCO). The applied frequency shift is proportional to the inputat the VCO so that the excitation frequency is kept on the resonance of the sensorand the phase gets ‘locked’ to zero (red arrows), hence the name ‘phase-locked loop’.

The so-called z-feedback uses the same signal as the VCO, i.e. the frequency shiftof the sensor, to control the z-position of the sensor. This is done via the TOPS3system, which incorporates another PID-controller. The high-voltage amplifiers thenproduce the corresponding voltage to move the scan piezo. This loop is illustratedby the blue arrows in Fig. 4.7.

Figure 4.7: Schematic of the involved feedbacks.

6Stanford SR830 DSP

38


The third feedback loop keeps the oscillation amplitude constant (green arrows):the deviations of the amplitude signal at the lock-in from an adjustable offset-valuecontrols directly the excitation amplitude.

Two details of the real setup: first, for the reason of bandwidth, the real andimaginary parts of the lock-in signal are used instead of the amplitude and thephase, because the latter two have a much slower update rate (512 Hz) than thefirst two (256 kHz) due to internal computations in the lock-in. Second, because ofnoise considerations, the excitation voltage is brought as a signal in the volt-regimeas close as possible to the sensor, before it is divided by a factor 1000.

As an example, the response of this setup to a step in the scanned surface is nowdiscussed shortly: before the step, the scan electronics applies a voltage to the scanpiezo that forwards the sensor to the sample so that the frequency error signal ∆fres

equals a fixed value chosen by the experimentalist, usually 50 mHz. If the resonancefrequency of the sensor changes abruptly at the step the phase at the lock-in changesaccordingly. The excitations frequency is adjusted according to its PID-parameters,as is the z-voltage. Unfortunately the bandwidth of the two feedbacks can not beincreased indefinitely, because the noise is amplified as well, for example in the phasesignal, which can lead to overshoots in fast scans or even to a positive feedback andthus to a tip crash.

Optimum feedback

It has become the art of the experimentalist to adjust all the involved parameters inorder to obtain an optimum feedback. However, an analysis of these nested feedbacksin the idealized case of a pure harmonic oscillator and for small mechanical ampli-tudes can be done and give some hints. The corresponding theory and experimentsfor such a setup can be found in Refs. [89, 101]. The analysis was condensed to aMatlab program that gives hints to the best parameters (PLLsettingsALL.m). Thisworks nice with the conventional TF-sensors, but very bad with the CLTF-sensors,at least at high temperatures. The difference is probably that in the latter theeigenmodes are very strongly coupled (almost in all resonance curves the symmetricmode is observed). This situation is not described by the above analysis.

The main results of the analysis can be summarized as follows:

• z-feedback and PLL need to be adjusted together to get the optimum band-width of the complete feedback. This is illustrated in Fig. 4.8, where the leftand right columns were recorded with the same overall bandwidth: the leftcolumn with a fast z-feedback and a slow PLL, the right column vice versa.The topography looks essentially the same, whereas the frequency shift imagefor the fast PLL shows only noise.

• The complicated setup does not allow to increase the overall bandwidth orscan speed. But the tracking of the resonance frequency allows to use sensorswith very high Q-factors, like TF-sensors. Their frequency shift, for example

39


during an approach or fast scanning, would bring them out of the linear part ofthe phase response. Much larger frequency ranges (set by the experimentalist,usually 2 Hz) can be covered, for example for force-distance curves.

• A PLL can provide an additional image in a scan, cf. Fig. 4.8.

• The separate phase and amplitude feedbacks and the tracking of the reso-nance frequency allow the distinction between conservative frequency shiftsand dissipative effects. The latter can also be recorded during the scan as theamplitude error signal and may be used to discriminate different materials.

Figure 4.8: Images of a 200 nm wide quantum wire at a fixed responseband width. Left column: topographic image (top) and frequency error signal(bottom) with a slow PLL. Right column: the same with a fast PLL.

40

Chapter 5

Local spectroscopy

5.1 Overview

The term ‘local spectroscopy’ is mainly known from STM experiments. Neverthelessit is used in AFM setups for experiments done at a fixed position in the xy-plane.The most common experiments are force-distance curves which contain informationsabout the tip-sample interaction and can therefore be used for surface characteriza-tions. Examples can be found in appendix C and in Refs. [93, 99]. In the chapter athand the focus will be on Kelvin probe experiments which allow to extract the localcontact potential difference (cpd), local capacitance changes and the local electrondensity in a 2DEG. Also some local resistance measurements are presented. Localtransparency and similar measurements and some preliminary measurements at highmagnetic fields are shown in appendix C.

The electrochemical potential of a particle with charge q residing on an objecti (electrode) is defined as µ

(i)elch = µ

(i)ch + qΦi with the chemical potential µ

(i)ch and

the electric potential Φi. The voltage Uij between two electrodes is proportionalto the energy difference for a particle with charge e > 0 brought from electrodej to electrode i: eUij = ∆µelch = µ

(i)elch − µ

(j)elch. The voltage corresponding to the

difference of the chemical potentials of the two materials is called ‘contact potentialdifference’ (cpd) eU

(ij)cpd = ∆µch = µ

(i)ch −µ

(j)ch . Collecting terms one finds for electrons

(q = −e) the relation

Uij − U(ij)cpd =

q

e∆Φij = −∆Φij (5.1)

Numerical and analytical models for the electrostatic tip-sample interaction forvarious geometries and models can be found in Ref. [102]. Finite element method(FEM) calculations of tips with arbitrary shape are possible as well.

The so-called ‘Kelvin method’ is widely used to determine the contact potentialdifference between two metals. In a capacitor made up from two electrodes of therespective materials the distance of the electrodes is changed periodically and thecurrent flowing from the electrodes is measured. The charge on an electrode is

41

Chapter 5. Local spectroscopy

Q = C · (U − Ucpd) with U the applied voltage and C the capacitance between thetwo electrodes. The indices ij are omitted. The induced current at a fixed voltagetherefore is

J = Q = C · (U − Ucpd) (5.2)

Adjusting the external voltage so that no current flows from the capacitor, onefinds the contact potential difference of the materials.

An alternative to the Kelvin method is to measure the force between the twoelectrodes, which is the gradient of the total (electric) energy of the system, W =12C(U − Ucpd)

2. The force acting on the tip can be nulled by applying the corre-sponding voltage to the tip. With this method, together with an electronic feedback [103], one has a non-invasive tool for measuring the cpd of different metalslocally in SPM experiments [104] and with high resolution (100 nm) [105]. Sincethis method is sensitive to changes in the electronic potential, it can also be used asa local potentiometer. In very recent experiments this technique was used to inves-tigate the charge accumulation in a polymer field-effect transistor [106], to map thedomain structure of ferroelectric crystals [107] and to observe quantum size effectsin the cpd of self-assembled InAs quantum dots on GaAs [108]. These experimentswere all done at room temperature. References for low-temperature experiments arediscussed in the introduction to chapter 9

The experiments presented here were done on a 4 µm wide Hall bar at T = 2.0 K.Since during the experiments the sample became inhomogeneous, it was illuminatedafter some time and warmed up to ∼ 70 K. Measurements before this procedureare denoted as ‘sample II, cooldown 0’, whereas the ones afterwards as ‘sampleII, cooldown 1’. The electron density found both with the Hall curve and the SdH-oscillations is n = 3.65·1015m−2 in cooldown 0 and increased during the experimentsto n = 3.85 · 1015m−2. The Hall mobility was found to be µ ≈ 70 m2

Vsbefore the first

measurements and then dropped to µ ≈ 25 m2

Vs. Considering the corresponding Hall

curves and SdH traces one can conclude that the sample became inhomogeneousduring the experiments. For cooldown 1 the following parameters were found: n =5.0 · 1015m−2 and µ ≈ 9 m2

Vs, which did not change as dramatically anymore.

42

5.2. Kelvin Probe Measurements

5.2 Kelvin Probe Measurements

Local Kelvin probe measurements on a GaAs heterostructure and at low tempera-tures have been reported recently [109]. They allowed to extract the local electrondensity and raised the question of a dependence of the cpd on the distance of theAFM sensor to the sample surface.

5.2.1 General electrostatics

Generally, the charges on a coupled system of metallic objects (electrodes) can be

written as Qi =∑

j CijΦj + Q(0)i with the matrix of mutual capacitances Cij and

charges Q(0)i induced by external charges Qext

k . The total energy of the system thenreads

W =∑

i

ΦiQi + W ext =1

2

∑i,j

ΦiCijΦj +∑

i

ΦiQ(0)i + W ext (5.3)

The following assumptions lead to a simplified expression for the total systemenergy:

1. As long as external charges, e.g. donor ions, do not change their configuration,W ext is a constant and can be left out.

2. The contact potential between two electrodes is a material constant, e.g. in-dependent of the involved geometries.

3. It is only the capacitance between tip and sample that matters. Therefore,the two indices s and t will be used. The mutual capacitance is denoted as C.

The energy at a fixed tip position takes on the form

W =1

2C · (U − Ucpd)

2 + Q(0)t · (U − Ucpd) + const. (5.4)

From Eq. (4.4), the shift of the resonance frequency of the AFM sensor isproportional to the force gradient. The latter is the negative gradient of the totalenergy of the system, Fts = −dW

dz. This leads to the following relation:

∆f = η · ∂Fts

∂z= −η

∂2W

∂z2= −η

2

∂2C

∂z2(U − Ucpd)

2 − η∂2Q

(0)t

∂z2(U − Ucpd) (5.5)

with the proportionality constant η. In the model of a harmonic oscillator forthe TF sensors one finds η = f0

4k0, and experimentally η ≈ 0.8 Hz

N/m[93, 101].

Rearranging terms gives

∆f = −η

2C”

[(U − Ucpd) +

Q(0)t ”

C”

]2

+η

2

(Q

(0)t ”

)2

C”(5.6)

43


where the ” denote the second derivative in z-direction at a given position.Varying U therefore leads to a parabolic modification of the resonance frequency

of the sensor, where the parabola is open towards negative frequencies and the apex

is shifted by Ucpd− Q(0)t ”

C”. Usually, the second term is not incorporated in an analysis

of such Kelvin parabolas and the shift is thought to originate solely from Ucpd, whichworks well for metal electrodes. In previous work [96] it was shown that in a platecapacitor model with a donor layer this second term is independent of the tip-sampledistance.

The Kelvin probe technique can be extended by applying a slowly varying ACvoltage on the tip and measuring the periodically modulated frequency shift [101,110].

5.2.2 Experiments at fixed position

Kelvin parabola

In the experiments presented here the shift in the resonance frequency ∆fres isrecorded as a function of the applied tip voltage Utip for −2 V < Utip < 1 V. Tipvoltages outside this interval have been found to cause a hysteretic behavior of thetransport properties and of the local forces. Similar effects are also reported in theliterature [111] and are attributed to rearrangements of charges in the donor or inthe surface layers.

The experiments are done at fixed xy-coordinates of the AFM tip, i.e. in thecenter of a Hall cross, as it is marked as red circle in the low temperature topographicimage in Fig. 5.1(b). The distance of the tip to the surface, z, is measured as thedifference to the position where the dissipation in the AFM sensor increases when‘touching’ the surface. In Fig. 5.1(a) ‘series B’ of cooldown 1 is presented, showingKelvin parabolas for tip-sample distances ranging from z = 1 nm to z = 26 nm,

Figure 5.1: (a) Kelvin parabola of series B for tip-sample distances between1 nm and 26 nm. The distance is measured from the onset of dissipation.(UTF = 0.1 mVrms, corresponding to Amech ≈ 2nm) (b) Tip position duringthe experiments relative to the Hall bar in a topography scan at T = 2.1 K.

44


labeled below the corresponding curves for some examples. The driving voltage ofthe tuning fork sensor is adjusted to result in a mechanical amplitude of about A ≈ 2nm. ‘Series A’ consists of a larger set of measurements under the same conditionsexcept that the mechanical amplitude is Amech ≈ 10 nm and z ranges from 1 nm to211 nm.

A general trend can already be observed in the raw data of Fig. 5.1(a): withincreasing distance the curvature and the offset of the parabolas along the frequencyaxis are reduced and a small shift of the parabola apexes towards more positivevoltages can be discriminated. Very close to the surface the fluctuations in thesignal get larger.

In order to get a more quantitative picture the following analysis, illustrated inFig. 5.2(a), is done with every Kelvin curve: the data (black curve) are fitted with aparabola (red curve) for −1 V < Utip < 1 V. From the latter the following quantitiescan be inferred: the frequency shift ∆f of the parabola apex, with the resonancefrequency at the maximum tip-sample distance as reference, the curvature c and thedisplacement Ucpd along the voltage axis. At low tip voltages, the data deviate fromthe parabolic shape. The distance from the point where this deviation has a certainvalue to the parabola apex is denoted ∆U . These quantities are plotted in Figs.5.2 - 5.4, where the blue data set stands for ‘series A’ and the red for ‘series B’ ofcooldown 0.

Figure 5.2: (a) Illustration of the fit parameters. (b) Shift of the resonancefrequency at the parabola apex compared to the value with completely with-drawn tip plotted as a function of the tip-sample distance z. The blue and reddata points represent the measurements of series A and B, respectively. Theblue points are shifted by +2 nm.

45


Frequency shift of the apex

Figure 5.2(b) shows the shift of the parabola apex in frequency, ∆f , as function ofthe tip-sample distance z. The data points of ‘series A’ (blue) are shifted by 2 nmto the left in order to match the position of the steep slope.

The ∆f vs. z curve is part of a force-distance measurement with compensatedcontact potential difference. The frequency shift is negative because the experimentare done in the attractive regime of the interaction potential. The step in ∆f atz ≈ 80 nm is due to a sudden change in the resonance frequency of the sensor,probably due to relaxation of strain in the piezoelectric material of the tuning fork,or due to a change in the mass distribution on the sensor, which is not unusual andusually compensated by adjusting the ‘zero’ of the resonance frequency.

Curvature

The curvature c of the Kelvin parabolas is plotted versus z in Fig. 5.3(a). Thecurvatures are negative as discussed in section 5.2.1 and get more extreme for thetip positioned close to the sample surface. Here, the data of ‘series A’ (blue) haveto be shifted by 17 nm to ‘match ‘series B’ (red), which corresponds roughly to theadditional mechanical amplitude in this experiment.

The inset of Fig. 5.3(a) shows the tip-sample capacitance obtained by numeri-cally integrating twice the curvature of the Kelvin parabolas, weighted by the factor2η

with η = 0.8, see Eq. (5.6). The integration was started at z = 500 nm to

where the data were linearly extrapolated (the experimental determination of thecurvature at larger distances was not possible due to the small signal differences).A comparison with direct measurements performed with a different sensor and abridge circuitry [95] shows that the result obtained here is about a factor of 10larger. The reason for this discrepancy is probabely due to the fact that the sensorin the present experiments was heavily used, which usually leads to a flattened tipand a larger capacitance.

Contact potential difference

An open question is the distance dependence of the contact potential difference Ucpd

[96, 109]. In Fig. 5.3(b) Ucpd is plotted versus z for the two measurement series Aand B. One can clearly observe a reproducible monotonic increase of the measuredcontact potential difference. After an essentially linear increase from Ucpd ≈ −10mV at z = 0 nm to Ucpd ≈ 100 mV at z ≈ 30 nm, the curve starts to flatten afterz = 50 nm and gets approximately constant at z > 100 nm. The quality of theparabolic fits at these distances is rather poor, which results in the strong scatteringof the data points.

An interpretation of the distance dependence of the contact potential differenceby taking the positively charged Si-donor layer of the heterostructure into accountin a plate capacitor model is not successful [96, 109]. One possible conclusion may

46


Figure 5.3: (a) Curvature c of the Kelvin parabolas vs. z for the twodiscussed measurement series. The data points of series A (blue) are shifted by17 nm to the right. The inset shows the change of the tip-sample capacitancefound by numerical integration of the data. (b) Contact potential differencevs. z, extracted from the shift of the parabola apex along the voltage axis.

be that the results in the measurements presented here are either due to the non-homogeneous distribution of donor ions, which occurs naturally when the Hall bar isetched, or due to inhomogeneous charging of the donor layer or the sample surface.The latter coincides with the presumably inhomogeneous transport properties.

Depletion voltage

The last parameter that can be extracted from the Kelvin parabolas is the posi-tion on the tip-voltage axis where the data deviate appreciably from the fit. Thisvoltage is interpreted as the potential where the depletion of the electron gas atthe tip position starts and the effective capacitance of the system is reduced. In aplate capacitor model [109] the difference between this voltage and the apex of theparabola, ∆U = Ucpd − Udepl, is related linearly to the local electron density.

Udepl can be found in two ways: first, one can normalize the deviations fromthe parabolas by the values of the fit. At positive tip voltages this quantity is1 and at lower voltages it falls off approximately linear with decreasing voltage.Interpolating this linear relation to larger voltages one finds the intersection withthe horizontal value 1, which indicates the onset of depletion [96, 109]. ∆U foundby this method are plotted in Fig. 5.4(a) for the two discussed measurement series.By inspection, one finds that the deviations are only well defined for z < 50 nm.Fitting a linear function to the data as suggested by the plate capacitor model [109],leads to an electron density of nloc ≈ 5.3 · 1014 m−2. For comparison, also the slopefor n ≈ 3.65 · 1015 m−2, the density extracted from both the Hall slope and theSdH-oscillations, is plotted in Fig. 5.4(a).

This method works well for data sets expanding to relatively large negative tipvoltages, but are rather questionable for fits with small voltage intervals, since twofitting procedures are involved. The second method presented here depends only

47


Figure 5.4: (a) ∆U = Ucpd − Udepl extracted by the method of Ref. [109].For comparison the slopes for the indicated electron densities from a platecapacitor model are plotted as dotted black lines. (b) Udepl − Ucpd extractedfrom the point where the fit deviated 100 mHz from the data. The fitted redcurve predicts a local electron density of n = 3.5× 1014m−2.

on one fit: the onset of depletion is defined as the voltage where the Kelvin curvedeviates by a fixed number from the fit. Here, the value 100 mHz is chosen. Theresulting ∆U is shown in Fig. 5.4(b) for ‘series B’. The electron density extractedby employing again the plate capacitor model is nloc ≈ 3.9 · 1014 m−2, similar to thevalue obtained by the previous method.

The extraction of the local electron density in these data sets is only possibleclose to the surface, because the deviations from the parabolic fit can not be resolvedwith the relatively small voltages applied, compare for example with Ref. [109].Nevertheless, one can conclude that at this specific position the electron densityis reduced. This is reasonable, since many spectroscopy experiments have beenperformed before the presented experiment with rather negative tip voltages. It isnot entirely obvious, though, that the plate capacitor model can be applied for theHall bar geometry and for an inhomogeneous sample.

5.2.3 Experiments at variable positions

Spatial fluctuations of the order of 0.3 V in the contact potential difference arereported in the literature [111] and are generally accounted for by local, non-homogeneous charging of the surface or donor layers. The charge distribution inthese layers varies between individual samples and from one cooldown to the next.In addition to these accidental charge distributions, the Hall bar structure intro-duces an inhomogeneous distribution in the donor atoms and changes the electro-static coupling to ground, compared to an unstructured 2DEG. Therefore, Kelvinprobe measurements on various position were preformed in cooldown 1 and are nowpresented.

Figure 5.5(a) and (b) show the extracted contact potential differences and thecurvature of the parabolas, respectively, as function of the tip position relative to

48


Figure 5.5: Parameters of parabolic fits to Kelvin probe measurementsdone at various positions. (a) Contact potential difference. (b) Curvature ofthe fitted parabolas. In both plots the red (1) and blue (2) curves are takenacross an etched sample edge, while the black curve (3) is taken along thecenter axis of the Hall bar. The directions of these positions are indicated inthe inset of Fig. (a).

the Hall bar. The height above the surface is constant, z = 25 nm. Three pathswere chosen that are illustrated in the inset and numbered according to the plots.The data on paths 1 and 2 cross an edge of the Hall bar, which is indicated in theplots as a gray dashed vertical line. The third curve is taken completely inside theHall bar structure. s denotes the distance on the lines.

The measured contact potential difference Ucpd increases by 2 V when crossingthe Hall bar edge on line 1. The change takes place on a length scale of ∆s ≈ 3 µm.An increase is also observed by crossing the sample edge along line 2, which is closerto other leads. Here, the slope is smaller by almost a factor of 2. Along path 3,inside the Hall bar, Ucpd shows a dip at the center of the Hall bar, where the previousexperiments have been performed. Nevertheless, even compared with this dip, theincrease at the sample boundaries is significant.

The curvature c, depicted in Fig. 5.5(b) as function of the tip position, ap-proaches zero when the sensor is positioned outside the sample, cf. curves 1 and 2.In comparison, the values recorded well inside the Hall bar stay essentially constant.

49


5.2.4 Interpretation

These experiments are consistent with the notion of the tip-sample capacitancedropping outside the Hall bar and with it its second derivative. Considering theassumptions discussed in section 5.2.1, the easiest to drop is number 3, i.e., thecoupling to the Hall bar is reduced when moving laterally away from the structure,while the coupling to the environment, e.g. to the bottom of the chip carrier, isenhanced at the same time. The latter incorporates a third chemical potentialwhich gets mixed into the total signal.

Another possibility is that the donor-induced charges on the tip diminish whenthe tip is moved away from the 2DEG, which may change the second term in theparabolic part of Eq. (5.6).

An inhomogeneously charged surface can lead to similar effects: the inducedcharge on the tip can change when the tip is moved across the surface, while thecapacitance stays constant. If the tip is withdrawn, it couples to a larger area of thesample and other charge fluctuations get involved.

Preliminary experiments at high magnetic fields are presented in appendix D,for completeness.

5.3 Local resistance measurements

Of particular interest in semiconductor nanostructures is the response of the re-sistances on local changes in the sample, for example the local electron potential.Since this is the main topic in the chapters to follow, only experiments regarding thedistance and tip-voltage dependence are presented here. All data shown here stemfrom sample II, cooldown 0. The resistances are measured by applying a currentJ = 100 nA at a frequency of 676 Hz and detecting the voltage drops by lock-inamplifiers.

Distance sweeps

Figure 5.6(a) shows the longitudinal resistance at zero magnetic field as function ofthe tip-sample distance z for a series of tip voltages. Rxx is measured at the uppertwo voltage contacts of the Hall bar. The traces from the other two probes lookessentially the same. The tip is positioned in the center of the Hall bar, the sameas shown in Fig. 5.1(b).

With Utip < 0.0 V, Rxx increases when the tip is approached to the samplesurface. The effect grows with decreasing distance. The curve shapes starts with anapproximately linear behavior at z = 300 nm. Then, about at the positions of thewhite circles, the curvature increases. Increasing the tip voltage from Utip = −2.0V, the effect gets much smaller and the resistance changes become negative forUtip > 0.0 V. The largest observed changes are of the order of 10%. The curves forthe different tip voltages come closer together at z = 300 nm, but do not coincide

50

5.3. Local resistance measurements

(a) (b)B=0.3 TB=0 -2.0

U [V]tip

-0.5

0.0

0.5

-1.0

-1.5

1.0

Figure 5.6: (a) Longitudinal resistance at B = 0 and (b) Hall resistanceat B = 0.3 T, both as function of the tip-sample distance and with the tipvoltages indicated between the two figures. The white points in (a) indicateapproximately the position where the curves deviate from a linear behavior.

at the maximum distance of the experiment.Similar experiments can be done with the Hall voltage. Figure 5.6(b) shows Rxy

at B = 0.3 T, measured at the lower right Hall cross as a function of the tip sampledistance with various tip voltages applied. The tip is positioned in the center of thesame Hall cross. Instead of only 300 nm the range the tip is withdrawn by 800 nm.

The Hall voltage is increased by positioning the tip close to the surface forUtip < 0.0 V and it is decreased for Utip > 0.0 V. As in the longitudinal resistancemeasurements the curve at Utip = 0.0 V (yellow) is the flattest. The curvatures ofall traces look essentially constant. The largest observed changes are of the order ofa few percent.

Interpretation

The experiments can be understood qualitatively by observing that the tip-samplecapacitance decreases with increasing distance. The applied voltage between tip andsample is constant and therefore the total charge on the ‘electrodes’, Q = C ·U , hasthe same distance dependence as the capacitance. If for the charging only the 2DEGis taken into account and if one assumes that the local density of states (LDOS) isspatially constant in the sample, i.e. in the classical regime, then the added chargesresult in a (local) change in the electron density. Considering the Drude results forclassical resistivities, ρxx = 1

enµand ρxy = B

en, cf. Eq. (2.11), a positive tip voltage

that induces carriers into the 2DEG decreases the two resistances, and vice versafor a negative voltage.

The tip voltage shows the least effect on the resistances at Utip ≈ 0.0. In thesimple capacitance model this implies that the contact potential difference (cpd) iszero. Since the resistances do not change at larger distances either, one can concludethat the cpd measured by this method is independent of the tip sample distance.No obvious effects of an inhomogeneous charging or donor distribution are observed.

51


Voltage sweeps

During the Kelvin probe experiments in the center of the Hall bar discussed above,the longitudinal resistance was recorded simultaneously. Figure 5.7(a), therefore,shows Rxx as a function of the applied tip voltage for various tip-sample distancesz. Far away from the surface (∼ 200 nm), the slope of the curve is constant witha value of −7 Ω/V (dark blue). While approaching the tip to the surface the slopegets generally more negative and two regimes start to develop, separated at Utip ≈ 0V: at negative tip voltages the curves are flatter than at positive, but still linear inthe respective intervals.

Figure 5.7: (a) Longitudinal resistance as a function of the tip voltage,recorded for various tip-sample distances z. (b) Inset: slope of the piece-wiselinear curves of figure (a). The main figure shows the difference between thetwo slopes.

To be more specific, in Fig. 5.7(b) the difference of the slopes between rightand left of Utip = 0 is plotted versus z. The values are extracted from linear fits tothe curves in 0.5 V intervals at the minimum and maximum of the applied voltage,respectively. The inset shows the two slopes. At z = 211 nm the value to the leftis approximately −13 Ω/V and to the right −21 Ω/V. As can be seen in the mainfigure, the difference grows essentially linearly from zero at z = 211 nm to −8 Ω/Vclose to the surface.

The voltage where the two fitted lines intersect (the ‘knee’ of the curves) shiftsfrom Utip ≈ −0.3 V at small z to Utip ≈ 0.2 V at z ≈ 30 nm, is then constant upto z ≈ 80 nm and starts to decrease again, though the data are strongly scatteredabove 100 nm (not shown).

Interpretation

The increase of the resistance at more negative tip voltages is consistent with thecapacitor model and changes in the electron density. The most striking feature,though, the kink in the curves close to the surface, has no obvious explanation.The distance dependence suggests a local effect and the reduced electron density at

52

5.4. Summary

this tip position deduced from the Kelvin probe measurements in section 5.2.2 mayindicate a connection to inhomogeneous transport characteristics.

5.4 Summary

From the presented Kelvin probe experiments the parameters of a parabolic fit areinterpreted and show a picture consistent with other experiments and theory. Thelocal Kelvin method allows, for example, to estimate the tip-sample capacitanceand the local electron density. The latter was significantly reduced at the positionof the measurements, compared to the values found in transport experiments. Aclear dependence of the measured contact potential difference on the tip-sampledistance was found and tentatively attributed to sample inhomogeneities, either inthe doping layer, due to the etched structure, or to surface charging. In order tosubstantiate this statement, position dependent measurements have been shown.Local resistance experiments are presented and interpreted as changes in the (local)electron density, induced by the ‘tip-gate’. Tip-voltage sweeps at various distancesshow a linear dependence of the resistance. Close to the surface curves with twolinear parts develop, which could not be explained satisfactorily.

53

Chapter 6

Scanning Gate experiments

In this chapter the scanning gate technique is introduced and some general aspectsof the experiments are discussed. A short overview over the experimental conditionsand the sample characteristics in the individual experiments is given in the lastsection.

6.1 The scanning gate technique

In a scanning gate experiment a structure connected to macroscopic contacts is in-vestigated the same way as in a corresponding conventional experiment, e.g. in aHall bar the longitudinal and Hall resistances are measured. In addition, the conduc-tive tip of an AFM with a DC voltage applied with respect to the sample is scannedacross, either in constant height or in z-feedback mode. The tip acts as a spatiallyvariable, local gate and couples electrostatically to the sample. This mechanismcan be used to change the local electron potential at an arbitrary position in the2DEG. The systems response to this manipulation leads to changes in the measuredquantities. These are recorded separately for various tip positions leading to theso-called scanning gate images. This so-called scanning gate technique is illustratedschematically in Fig. 6.1(a) for a scanning gate image of the Hall voltage Uy as afunction of the tip position (x, y). The Hall voltage divided by the applied currentalong the x-axis then gives the Hall resistance Rxy(x, y). The longitudinal voltageRxx(x, y) is defined accordingly. The scanning gate experiments can be understoodas either a moving local gate with only a very small lever arm to the rest of thestructure, or as a series of different samples, each with an inhomogeneity at the tipposition. Short reviews of experiments where this technique was applied to struc-tures based on 2DEGs and at low temperatures will be given in the introductionsof the following chapters. One example of a scanning gate experiment in anothermaterial system is the single-electron charging in carbon nanotubes. Defects in thematerial lead to isolated islands in the conducting tube where, depending on thetip position, the potential in these quantum dots could be altered, which led toCoulomb blockade oscillations in the conductance of the nanotube [112].

55

Chapter 6. Scanning Gate experiments

Figure 6.1: (a) Schematic of a scanning gate experiment. A conducting tipat a constant voltage Utip applied with respect to the electron gas is scannedacross the sample surface while the transverse resistance Rxy is recorded. (b)Topography image of the sample in experiment A at 56 K. The various voltagesrecorded are indicated in the schematic setup.

6.2 Symmetry considerations

Most of the data presented in this thesis were taken on Hall crosses. The geometry ofthese structures can lead to symmetries in scanning gate measurements. These con-siderations are quite fundamental and can be generalized to other structures. Theycan be seen as a third possibility to characterize and categorize patterns in scanninggate images in Hall crosses, in addition to the more general four-terminal Buttikerformulas [12] (‘global Onsager relations’), which is a non-local generalization of the(‘local’) Onsager relation, which work for locally defined transport coefficients likeelectrical conductivity [15].

The model implemented here consists of a Hall cross defined in a 2DEG with acurrent carrying bar of constant width. The voltage leads have both the same width,but not necessarily the same as the current leads. They are placed exactly oppositeeach other and orthogonal to the current channel. The basic symmetry operationsthat can be performed in real space are the 180 rotations about the x-, y- and z-axis.The coordinate system is depicted in the upper right of Fig. 6.2(a). A homogeneousmagnetic field B is applied orthogonal to the sample and an originally homogeneouscurrent density is assumed. In addition, a scatterer, e.g. an AFM-tip inducedpotential, is introduced at an arbitrary position (x, y), indicated by the red spot inFig. 6.2(a). The scatterer is assumed to have at least the symmetry of the Hall cross,i.e., it is not distinguishable in the measurement, if one of the symmetry operationsis applied to the scatterer alone. Also time reversal symmetry should not be broken

56

6.2. Symmetry considerations

Figure 6.2: (a) Measurement setup before and (b) after the rotation aboutthe x−axis (coordinate system is indicated). (c) Interchanged voltage contactsinvert the measured Hall voltage, which recovers the original setup. (d) At zeromagnetic field one finds this general pattern of changes in the Hall voltage.The lines of symmetry where the Hall voltage cannot be changed, are alsoindicated.

by the scatterer. The transverse voltage Uy(x, y, B, I) ≡ U0y then depends on the

parameters given as arguments. A 180 rotation of the entire arrangement about thex-axis does not change the measured voltage, Fig. 6.2(b). If the voltage contacts areinterchanged, which inverses the sign of the signal, one recovers the original setupbut with reversed magnetic field and the scatterer positioned at (x,−y), Fig. 6.2(c).The last point is only true if the symmetry of the scatterer is as mentioned above.This and similar procedures using the other symmetries of the Hall cross and thefact that the measured voltages are inversed upon inversion of the current direction(in the linear transport regime) lead to the following expressions:

Uy(x,−y,−B, I) = −Uy(x, y, B, I) (6.1)

Uy(−x, y,−B, I) = −Uy(x, y, B, I) (6.2)

Uy(−x,−y, B, I) = Uy(x, y, B, I) (6.3)

Equation 6.3 follows from the other two, but it is put down here because itconnects measurements at different points at the same magnetic field. In a Hall

57


cross with 4 leads of equal width this corresponds to a mirror symmetry along thediagonals. At B = 0 these equations reduce to

Uy(x,−y) = −Uy(x, y) (6.4)

Uy(−x, y) = −Uy(x, y) (6.5)

Uy(−x,−y) = Uy(x, y) (6.6)

which leads to the schematic scanning gate image of Fig. 6.2(d), if the Hall voltage inone quadrant U0

y is given. One finds two lines of symmetry where the Hall voltagecan not be changed. In addition an alternating pattern of resistance changes isexpected, if one goes around the Hall cross.

When does it make sense to use these equations? If there is no scatterer at allone recovers the relation for an ideal Hall cross Rxy(B) = −Rxy(−B), which canalso be deduced with the 4-terminal relation of Buttiker, Eq. (2.5). This also holdsfor a Hall cross that is truly diffusive, i.e., where there are so many scatterers thatthe sample is homogeneous on the scale of the averaging in the measurement dueto the width of the voltage contacts. In this case the result corresponds to the‘local’ Onsager relation, Eq. (2.7). Introducing the artificial scatterer in a trulyballistic or a completely diffusive Hall cross leads exactly to the situation discussedabove and Eqs. (6.1)-(6.3) are expected to hold. Only if the sample is in a ‘quasiballistic’ regime, where there are few scatterers present within the Hall cross andthe measurement can distinguish between the setups after a symmetry operation,these relations are no longer valid. Experiments in the diffusive regime at B = 0are shown in chapter 7 and scanning gate images performed in the diffusive regimeat finite magnetic fields as well as on a ballistic sample are presented in chapter 8.

These equations can also be applied to a local magnetic field at otherwise zerofield if the local field is treated as parameter B which reverses its sign the same wayas the outer magnetic field before. Other sample symmetries should be as easy toobtain, for example geometries with voltage probes of different dimensions, as theyare of some interest in sensor optimization [113].

58

6.3. Measurement procedure

6.3 Measurement procedure

The sample is mounted in the sample holder of the low-temperature scanning forcemicroscope described in chapter 4. In all the scanning gate experiments discussed inthe remaining chapters, tuning fork sensors with an attached PtIr-wire were used,like they are discussed and characterized in chapter 4.

Some critical steps should be remembered for a successful cooldown, in additionto the usual procedure with semiconductor samples:

1. Put all contacts to ‘ground’, also the ones not used.

2. Fix potential of the AFM tip.

3. Position at ambient conditions by binoculars and scanning

4. Evacuate the sample chamber to at most 10−5 mbar while heating the sampleto 60− 80C for one night

5. Important: keep heating during cooldown (∼ 30 K difference to insert)

6. Put microscope into cryostat: do not shake!

7. Important: position again!

8. Check step size of step motor and withdraw accordingly (∼ 180 µm)

9. Cool down slowly (1 day to LN2)

10. Approach and position at LN2 temperature or at about 50 K.

11. Check sample, then all contacts to ‘ground’ !

12. Check step size and withdraw accordingly (∼ 40 µm)

13. Cool down to base temperature

14. Approach, check sample then position the tip relative to the structure

It is essential to estimate the workfunction difference between the sample and theAFM tip before the cooldown and fix the applied tip voltage accordingly. Goodtopography images of the sample are necessary for orientation during the positioning.Also the relative orientation of the sample to the scanning directions should bechecked.

The time it takes for one scanning gate image is mainly given by the settings ofthe lock-in time constants and the number of points in the scan. Usually the timeconstant is set to τLI = 30 ms. For most of the experiments presented here this ledto a time per scan of about 2 hours. The time constant was adjusted to τLI = 300 msfor the very weak signals in some of the experiments at low magnetic fields, leading

59


to scanning times of about 20 hours. Even during these extreme measurement timesthe temperature was usually stable to within 30 mK, read at the carbon resistorbeneath the sample.

6.4 Sample characteristics

In the following chapters measurements under 5 different experimental conditionsare presented, abbreviated as experiments A1, A2, B, C and D. The correspondingparameters and sample characteristics are given in tables 6.1 and 6.2, respectively.Some more details can be found in appendix A.

Exp. sample (cooldown) T [K] Utip [V] d or ∆fres fm [Hz] J [nA]A1 I (1) 56.0 -1.0 50 mHz 29.6 100A2 I (1) 2.0 -1.0 50 mHz 223.4 100B II (1) 1.9 -0.7 120 nm 680.9 100C II (2) 1.9 0.0 120 nm 680.9 100D II (3) 1.9 0.3 120 nm 680.9 100

Table 6.1: Samples and experimental details for the experiments.

All scanning gate experiments presented here were performed on Hall bars pro-duced in Ga(Al)As heterostructures with a 2DEG formed 52 nm below the samplesurface. The structures were defined by standard photo lithography techniques andwet chemical etching. The longitudinal (Ux) and transverse (Uy) voltage drops inthe W = 4 µm wide and L = 10 µm long Hall bars were measured with lock-intechnique at a frequency fm and a constant current amplitude J . These parameterswere optimized in the course of the experiments to increase the signal to noise ratioand the scan speed. The experiments were performed at a given temperature T .The measurement setup is sketched in Fig. 6.1(b), together with a topography scantaken at T = 56 K in experiment A1. For every scanning gate image the contour ofthe sample was extracted from such a topography scan and overlaid on the scanninggate image for orientation.

Exp. ρxx(0) [Ω] n [1015m−2] µ [m2

Vs] lmfp [µm] B(ρxx = ρxy) [mT] quality

A 180.8 3.3 10 1.0 181 inh.B 137.8 5.0 9 1.0 114 inh.C 30.7 3.6 66 6.5 20 hom.D 236.0 3.9 7 0.7 130 inh.

Table 6.2: Sample characteristics at T = 2 K. The last column abbreviates thesample quality, as discussed in appendix A, as homogeneous or inhomogeneous.

60

6.4. Sample characteristics

In experiment A it became clear that performing many scans in z-feedback modeand Utip = −1.0 V generates inhomogeneities in the sample, which can be identifiedby their characteristic Rxx and Rxy traces in the magnetic field, cf. appendix A.Thus experiments B were done with better adjusted parameters and in constantheight scans on a new sample, on the cost of signal strength and resolution. Beforethese scanning gate experiments, many local spectroscopy experiments were per-formed and the sample ended up inhomogeneous. After illumination and warmingup to 100 K the electron density was increased as indicated. In the experimentsC, which were done with the same sample as experiments B after warming up toroom temperature, the contact potential difference was compensated better and thesample characteristics did not show signs of inhomogeneities.

In table 6.2 the zero magnetic field resistivity ρxx(0), the electron density n, theirmobility µ, the estimated mean free path lmfp = ~µ

e

√2πn and the magnetic field at

which ρxx = ρxy holds, are listed for reference.

61

Chapter 7

Scanning gate experiments at zeromagnetic field

7.1 Overview: experiments on structured 2DEGs

It is of fundamental interest to perturb a macroscopically well known system onmesoscopic scales and in a controlled manner to understand the reaction of the sys-tem. Local investigations on 2DEGs, and at zero magnetic field in particular, areimportant for device applications in the course of further miniaturization of inte-grated circuits and designs of novel composite materials. The devices may becomeso small that individual potential fluctuations, e.g. due to donor atoms or singleimpurities, are no longer statistically averaged and govern the properties of the de-vice. Optimization of magnetic field sensors, for example for scanning Hall probemicroscopy, is another very active field [114].

Only few scanning probe experiments at zero magnetic field and low temper-atures were done on 2DEGs or on other semiconductor nanostructures. A 2DEGwith metallic gates was investigated by scanning an SET across the sample, whichallowed direct imaging of single ionized donors and the confining electron potentialat the edge of the sample [115, 116].

Scanning gate experiments were performed on quantum point contacts definedby metallic top gates [117] showing that the electrons outside of the QPC flow inchannels made up by local potential fluctuations. This phenomenon is known as‘branched flow’ [118]. The individual modes in the QPC are coupled with differentstrength to the single ‘branches’, which could be shown experimentally by placingthe AFM tip at different exquisite positions at the exit of the QPC. For each tipposition the QPC-conductance was then recorded as function of the gate voltage.Each gate sweep showed a reduction in the value of one single conductance plateaus[119], which can be explained by selective backscattering of electrons into the modesof the QPC, induced by he AFM tip. The same scanning gate images exhibitspatial oscillations with a period of half the Fermi wave length, which proves thatthe electron transport is coherent on the investigated length scales. The locally

63

Chapter 7. Scanning gate experiments at zero magnetic field

varying wave length can be used to map approximately the local electron density[120]. The decaying amplitude of the fringes allows to estimate inelastic scatteringtimes in the 2DEG. Also a ‘Snell’s law’ for electrons passing regions of differentFermi energy was demonstrated [121].

In these experiments the QPC is essential in the first place to produce a smallconstriction in the 2DEG that reduces the otherwise dominating effect of indirectbackscattering. Only a small fraction of the electrons that are influenced by thepresumably large tip induced potential are scattered back through the QPC, whichexplains the excellent resolution of a few nm in the resistance images. Exceptfrom the different QPC modes coupling into different potential ‘branches’, not muchinformation about the electron transport inside the QPC can be gained. How thebranched flow affects the overall transport in mesoscopic structures is still debated.

The electron density distributions corresponding to the individual modes insidea quantum wire were observed by a slightly modified scanning gate technique, inwhich an AC voltage applied to the tip modulates the interior of the wire. Theconductance is then measured at this frequency (‘local transconductance’ ∂G

∂Utip(x,y))

[122]. In other experiments microconstrictions in a quantum wire could be observedand led to estimates of the involved barrier heights [123]. In a quantum wire definedby AFM-lithography oscillations on the scale of a Fermi wave length were attributedto quantum interference of scattered electron waves at local potential fluctuations[124]. An other topic of mesoscopic physics is telegraph noise, i.e. noise arising ifthe conductance of the system flips between two possible values. This phenomenonis observed in many systems. In 2DEGs it often originates from defects that changetheir internal two-level state at random. Scanning a charged tip in the region of sucha defect changes its local potential and thus its preferred state and the flipping rate,which leads to different average resistances, for example in a pseudo 1-dimensionalquantum wire [125].

Single electron charging as in quantum dots could be imaged in scanning gateexperiments in isolated islands formed as electron puddles in the donor layer of a2DEG [126]. On an artificially defined and controlled quantum dot in a heterostruc-ture this technique led to rings of Coulomb blockade peaks around the dot [127].Vice versa, these experiment give the opportunity to map directly the electrical po-tential of the tip, a crucial prerequisite if one wants to learn more about the internalstructure of the quantum dot.

In small structures with only few eigenstates occupied by electrons a local changein the potential can lead to strong variations in the overall behavior of the electronsystem. This is different in a Hall bar, which is wider and more modes are occupiedand get coupled to each other. Small changes in the interference pattern or couplingof single modes do not necessarily lead to large changes in the measured signal. Onthe other hand, a large homogeneous density of states at the Fermi energy allowsan analysis in classical terms. Though a Hall bar is the most basic structure in a2DEG, no scanning gate experiments on the Hall resistance have been reported inthe literature so far.

64

7.2. Longitudinal resistance

An other goal is pursued in experiments where a magnetic tip is scanned abovethe structure: this was done to find optimized Hall cross geometries for magneticfield sensors with better spatial resolution and better sensitivity [128]. To designdevices with semiconductor nanostructures with ferromagnetic materials, for exam-ple deposited and structured on top of a Hall cross [129, 130], it is necessary tounderstand the response of the different structures in the 2DEG to inhomogeneousmagnetic fields. Therefore, device application require experiments in the classicaltransport regime, i.e. at low or zero magnetic field.

Calculations of the Hall voltage for ballistic [131] and diffusive samples [113, 132,133] with inhomogeneous magnetic fields applied incorporated small disks of a localmagnetic field at different positions in the Hall cross area. The influence of the Hallcross geometry was investigated for example in connection with the quenching ofthe Hall effect for narrow Hall crosses [134] or for sensor optimization [113].

In experiments with inhomogeneous magnetic fields it is hard to discriminate theeffects of the local magnetic field and the effects of a changed electron density inthe 2DEG due to differences in the work functions and capacitive coupling, cf. Eqn.(2.9) and (2.11). Theoretical work on the effects of an inhomogeneous potentialprofile in a Hall cross was performed for the ballistic regime [135]. This topic willbe discussed in detail in chapter 8. In conclusion experiments are needed wheremagnetic materials are not relevant and only n is varied, which is exactly what isdone with the scanning gate technique.

In this chapter scanning gate experiments are presented on a 4 µm wide Hall barwith 2 µm wide voltage contacts attached. More information about the experimentalparameters are given in table 6.1, experiment A2 and A1. The corresponding samplecharacteristics can be found in table 6.2, experiment A. At the end of the chapterother experiments are presented, performed under similar conditions, denoted as‘experiment D’ in the sample overview of chapter 6.

7.2 Longitudinal resistance

7.2.1 Experiments

In Fig. 7.1(a) and (b) scanning gate images of the longitudinal resistances Rxx,1

(lower two contacts) and Rxx,2 (upper two contacts) at zero magnetic field are pre-sented (cf. Fig. 6.1(a) in chapter 6 for the setup and notations). The temperaturewas T = 2.0 K and the images were taken in z-feedback mode. The tip voltage wasUtip = −1.0 V, while the sample was at zero potential. The images are redrawnfurther below in Fig. 7.9 without the lines and arrows.

Generally, Rxx is changed only if the tip is positioned between or very near themeasurement contacts. The resistance pattern is not homogeneous but one candistinguish the corresponding voltage probes for Rxx,1 and Rxx,2 by the enhancedvalues in the regions of the contacts in use. Near the unused voltage probes acrossthe Hall bar a drop of Rxx is induced by the tip, pointed out exemplarily for Rxx,2

65


by white arrows in 7.1(b). Similar structures are also visible in Fig. 7.1(a) for Rxx,1

at the other contacts. The black arrow in Fig. 7.1(b) points to a position inside theHall bar where both Rxx are not influenced by the tip. Around this spot the largestchanges in Rxx are observed. Further away from this feature the changes in Rxx,2

get more homogeneous than for Rxx,1 in (a). Other features are the sharper drop ofRxx at the right side of the Hall bar for Fig. 7.1(a) and the different distance of thechanges to the upper and lower edge of the Hall bar. These features are also visiblein constant height scans.

Figure 7.1: (a) Rxx,1, (b) Rxx,2 in a scanning gate experiment at B = 0 andT = 2.0 K. The straight black lines in (a) indicate the cross sections presentedin Fig. 7.2, with the starting points indicated by black dots. The white andblack arrows in (b) point out features discussed in the text. The numbers in(a) identify the corners of a Hall cross.

Some more details are visible in cross sections through these data. The blacklines in Fig. 7.1(a) indicate the cross sections s1 to s4 presented in Fig. 7.2 forRxx,1, with the starting points indicated by black dots. Figure 7.2(a) shows the15 µm long cross section s1 orthogonal to the Hall bar. The peak in the middle from5 µm to 9 µm is the changed resistance when the tip scans across the Hall bar. Themaximum change compared to the constant value to the left is D1 ≈ 37.5 Ω, cf. Fig.7.2(a). On the other side of the Hall bar Rxx does not go back to the original valuebut drops additionally by D2 ≈ 18.5 Ω. This feature seems to recover very slightlyfurther away from the Hall bar. Zooming in on the peak of the Hall bar in Fig.7.2(a), one finds in Fig. 7.2(b) the cross sections s2 and s3. The latter shows anearlier drop at the right hand side due to the spot where the influence of the AFMtip is reduced. The maximum of both curves lies not in the middle of the Hall barat s = 2 µm, but about 500 nm to the left. Both curves have about the same height.The typical distance at which the signal rises from the left to the maximum valueis L1 ≈ 880 nm.

Figure 7.2(c) shows the cross section s4 parallel to the Hall bar indicated in Fig.7.1(a). The black lines at s = 2 µm and s = 10 µm indicate the edges of the voltageprobes closer to the Hall bar. On the right hand side one sees the sharp drop exactlyat the end of the Hall bar on the length scale L2 ≈ 2.36 µm. On the left the slopeis similar but goes to smaller values. At s ≈ 4 µm the slope of Rxx,1 decreases andrecovers again to the previous value at s ≈ 5 µm. Between s ≈ 6 µm and s ≈ 9 µm

66

7.2. Longitudinal resistance

s4s1

D2

D1

L1

L2

(a) (b) (c)s2

s3

Figure 7.2: Cross sections through scanning gate images of Rxx,1 shown inFig. 7.1(a). (a) Long orthogonal cross section s1. (b) Short cross sections s2

and s3. (c) Cross section parallel to the Hall bar, s4.

Rxx,1 is approximately constant.

7.2.2 Discussion

To get a first estimate for the influence of the AFM tip on Rxx in a diffusive sampleone can think of the tip completely depleting the 2DEG within a disk-shaped areain the 2DEG. This reduces the effective channel width and changes Rxx. For furthersimplification this disk is replaced by a square of dimension b = 2Rtip. If one ig-nores inhomogeneous current densities one then can think of three serial resistances,schematically drawn in Fig. 7.3(a). R1 and R3 are the resistances describing theunperturbed sections, R2 the part with the smaller cross section. This leads to thefollowing resistance change:

∆Rxx = Rtipxx −R0

xx = ρxx

(b

W − b− b

W

)(7.1)

Solving for b and inserting ∆Rxx = 37.5 Ω, ρxx = 180 Ω and the ideal, litho-graphical width of the Hall bar W = 4 µm one obtains b ≈ 1.5 µm, correspondingto a disk radius Rtip ≈ 750 nm.

Qualitatively one can understand the shape in the orthogonal cross section bymoving a disk of depleted 2DEG into the Hall bar, like it is depicted in Fig. 7.3(b).In Fig. 7.3(c) the blue curve shows the area of the disk inside the Hall bar asfunction of the distance s of the disk center to the edge of the Hall bar. Convertingthis area to a square with the same area this leads to a change in Rxx by Eq. (7.1),which then is plotted as red curve for Rtip = 750 nm in Fig. 7.3(c). The length onwhich the signal rises can be interpreted as the diameter of the tip induced disk.The turning point on the other hand can be shifted from the center as indicated bythe black arrow and is thus not a good indicator.

The induced change in Rxx parallel to the Hall bar in the Hall cross region (cf.L2 in Fig. 7.2(c)) is not only governed by the radius and the shape of the tip inducedpotential, but also by the width of the contacts. The simplest assumption is that thepotential inside the probes is the average of the potential at their entrance. Then

67


b(a) (b) (c)

J

L

W R1 R2 R3

Figure 7.3: (a) Serial resistor model for Rxx. (b) Model for the shape at theedge of the Hall bar: the tip region enters the 2DEG and reduces the effectivewidth of the Hall bar. (c) The area Ain of the disk inside the Hall bar as afunction of the tip position. The black line at s = 0 is the sample boundary.An estimate for the induced ∆Rxx is given, as discussed in the text.

it can be described as the convolution of a rectangular function of 2 µm length and12µm−1 height, otherwise zero, with the real potential drop. This smears out a sharp

drop to a linear decrease of 2 µm length. If the real potential drop is not sharp, themeasured decrease gets more broadened, like it is observed in the measurements.

The discrepancy in Rxx between the left and the right side of the Hall bar canprobably be attributed to a capacitive coupling of the rest of the AFM sensor tothe 2DEG which decreases the electron density if the back of the senor is positionedover the Hall bar. Since this is very far away this effect does not influences thescanning gate image locally, though it might lead to some smearing at the edgesof the structures. In later experiments with smaller tip voltages this effect was notobserved anymore.

The small feature pointed out by the black arrow in Fig. 7.1(b), where Rxx is notinfluenced by the tip, results from an already depleted region in the 2DEG1. Sincethis region decreases the effective width of the Hall bar at this position, the currentdensity is larger in the remaining cross section so that the effect of the tip aroundthis position is increased, cf. Fig. 7.2(c). Because one would expect the currentdensity to recover only on the length scale of the mean free path (lmfp ≈ 1 µm forthis experiment), this defect might be responsible for the differences between theright and left ends of the Hall bar.

The resistance changes in the contact regions that allow to identify the measure-ment contacts in the Rxx,1 and Rxx,2 images can be explained as follows: As will beseen later on, the current penetrates some distance into all of the contacts, leadingto an additional voltage drop (this depends on electrostatics and mainly geometry,and thus is not some non-local ballistic transport). If the tip is positioned nearone of the voltage probes, the current into it and thus the additional voltage drop

1Since the feature persists also in scans at constant height it is not due to a larger distancefrom a reaction of the z-feedback. Also a metal-like particle that could screen the tip would haveto be connected to a large reservoir, like ground, which is very unlikely.

68

7.3. Hall resistance

is inhibited and the measured voltage is increased. Also the decreased resistancespointed out by the white arrows in Fig. 7.1(b) originate from the altered currentflow around the tip position. Both will be discussed below in more detail.

In all of the above estimates the depletion length in the lithographically definedstructure (about 200 nm) was neglected. It should lead to a smaller effective width ofthe Hall bar and to a different change of the scanning gate image at the edges of thestructure. Since both geometries, the one of the tip-induced potential and the one ofthe Hall bar itself are not known exactly, these two effects can not be discriminatedwithout further measurements. Nevertheless the above estimates should give theright order of magnitude, since the involved length scales differ by almost a factorof ten.

7.3 Hall resistance

7.3.1 Experiments

Figures 7.4(c) and (d) show the tip induced changes in the transverse resistance Rxy

on the left and on the right Hall cross, respectively. The images are redrawn furtherbelow in Fig. 7.9 without indicators. For both Rxy measurements one finds thesame, very distinct pattern between the corresponding voltage contacts: Withoutthe tip the Hall voltage is zero2.

If the tip is positioned at the quadrant of corner 1 or 3 (cf. Fig. 7.4(d)) of theHall cross the measured Hall resistance becomes positive, while it turns negativein the other two quadrants. Because of the general shape of this feature we haveadopted the term ‘butterfly’ to refer to it.

Figure 7.4: (a) Rxy,1 and (b) Rxy,2 in a scanning gate experiment at B = 0and T = 2.0 K. The straight black lines in (a) indicate the cross sectionspresented in Fig. 7.5, with the starting points indicated by black dots. Thenumbers in (b) identify the corners of a Hall cross.

2The constant background seen here can have two reasons: Either a missalignment of thevoltage contacts of the order ∆l ≈ ∆Rxy

ρxx· W ≈ 100 nm, or a residual magnetic field of about

Bres = en · ∆Rxy ≈ 3 mT. Another possibility are inhomogeneities that change the effectivegeometry.

69


(a) (b)

Figure 7.5: (a) Cross sections s1 and s2 parallel to the Hall bar through thetwo maxima of the ‘butterfly’, as indicated in Fig. 7.4(a). (b) Cross sections3 along the diagonal from corner 3 to corner 1 of the Hall bar, as indicatedin Fig. 7.4(a).

The pattern shows well defined maxima and minima that can be seen more clearlyin the cross sections s1 and s2 presented in Fig. 7.5(a). The positions of the slicesare indicated in Fig. 7.4(a). They lie inside the Hall cross region on the diagonals,although the structure itself spreads also outside. The heights of the maxima areabout 21 Ω, the depths of the minima about −10 Ω. The two minima and the twomaxima show also about the same shape along these cross sections. The widths attheir base is 4.0 µm and the distance from the maximum to the minimum along theHall bar is about 2.0 µm, which corresponds to the width of the voltage probes. Theshape of one of these peaks in these measurements looks like a ‘half moon’ alignedalong the rounded corners of the Hall cross. The cross section in Fig. 7.5(b) is takenalong the diagonal from corner 3 in the upper left of the Hall cross to corner 1 inthe lower right. One recognizes two peaks of the same height H ≈ 21Ω, as indicatedin the figure. The dotted vertical line indicates approximately the position of theedge of the sample. The distance to the maximum is L ≈ 0.6 µm.

A qualitative different observation are the lines of symmetry along the center ofthe Hall bar and along the center of the measurement contacts, where the tip doesnot influence Rxy. These lines together with the sample boundaries result in thetwo ‘butterflies’ (red and blue).

7.3.2 Discussion

On the intuitive level one can think about the ‘butterfly’ pattern in the followingway: electrons ‘flow’ from the right to the left. If there is no scatterer at all, the samenumber of particles gets deflected into the upper and the lower voltage contacts andthe Hall resistance is zero. If one positions a barrier at corner 1 of the Hall cross, theprobability of an electron to enter the lower voltage contact is reduced and the onefor them entering the upper contact enhanced: more electrons at the upper contactmeans a positive Hall voltage since it is measured as the potential difference from

70


the lower to the upper contacts. The same argument can be applied at the othercorners of the Hall cross.

Since the sample is not in a truly ballistic regime in these experiments the aboveargument is in principle not applicable. A similar argument can be made for thecurrent density being deflected into the contacts, but to obtain some more insight,the following model for the diffusive regime is now presented:

General arguments

The AFM-tip couples electrostatically to the sample and changes the local potential,e.g. from Utip < Ucpd usually follows a larger potential energy and the electrons arepushed away. To keep the reasoning simple one can assume a completely depleteddisk beneath the tip with radius Rtip, though the argument presented here alsoapplies for other density distributions. The current density in the Hall bar is thenchanged, like it is shown schematically in Fig. 7.6(a) for the tip residing at the lowerleft corner of the Hall bar. The Hall voltage can be found formally by integratingthe electric field along any path from a point B on one side of the Hall bar to apoint A straight across on the other side:

Uy = ΦA − ΦB = −∫ A

B

~E · ~ds (7.2)

The points A and B need to lie well inside the contacts to ensure they measurethe appropriate potential. This expression can be further developed by introducing alocal coordinate system whos x−axis always points along the chosen path (ds ≡ dx,dy ≡ 0), as in Fig. 7.6(a). Since ~j is varying smoothly one can always choose a pathfirst running parallel to the current density, from B to C and then orthogonal fromC to A, as shown in Fig. 7.6(a). If the local resistivity tensor ρ is well defined theparallel part picks up a voltage drop originating from the longitudinal resistivity andthe orthogonal part collects a potential difference due to the off-diagonal elements.This leads to the following expressions for the measured transverse voltage betweenthe two contacts:

Uy = −∫ A

B

(ρxx · jx + ρxy · jy)dx = −∫ C

B

ρxx · jx · dx−∫ A

C

ρxy · jy · dx (7.3)

The classical Hall resistivity ρxy = Ben

is zero for zero magnetic field and thus forall electron densities, which makes the integral from C to A zero, not only in thecase of a completely depleted region beneath the tip. Only the path from B to Cgives a contribution, which is negative because ρxx > 0 and the path runs parallelto the local current density. The same line of reasoning also produces the changesin the other quadrants.

For the tip being positioned on the symmetry lines and well inside the Hall crossone can choose a path vertical to the boundary of the disk, around it and again

71


(a) (b)

Figure 7.6: (a) Schematical current distribution in a Hall cross geo-metry with a locally changed electron density. The path used for theintegration in the text is indicated as white line from point A to B (b)FEM-simulation of a Hall cross with n0 = 3 · 1015 m−2 and nt = 0.

vertically to the other side of the Hall cross. The path around it exactly cancels,e.g. running first along and then opposite the current density, which is symmetric forB = 0 or for a completely depleted disk. The latter is only true as long as the sampleboundaries do not alter the current density at the disk boundary. This enables oneto estimate the length scale on which the current density is influenced by the tip:by noting that the lines of symmetry are not very broad and by assuming a diskradius of Rtip ≈ 750 nm, one can conclude that the current density is significantlyaltered on the scale of W/2−Rtip ≈ 1.25 µm.

FEM simulations

The qualitative arguments above were confirmed by calculations using the FiniteElement Method (FEM, Femlab 3.0, see appendix B). The basic equation to besolved is given in Eq. (2.13). The model used here consists of the Hall cross geometryin a two dimensional metal with the conductivity tensor calculated with an electrondensity of n0 = 3.5 · 1015 m2 and a mobility of µ = 10m2

Vs, according to Eqs. (2.9).

The potential of the AFM tip is modeled as a disk-shaped region of zero electrondensity nt = 0 and radius Rtip. Also outside the Hall bar n = 0 holds. As boundaryconditions all the current densities orthogonal to the sample boundaries are chosento be 0, except the boundaries of the current leads that are held at a constant currentdensity of j = 25 nA

µmin x-direction.

The result of such a calculation with the tip residing in the lower left corner isshown in Fig. 7.6(b). The color code represents the electrical potential, the whitelines are the corresponding equipotential lines. The black arrows show the currentdensity. Clearly visible as different colors is the potential drop between the upperand lower voltage contacts, as it is observed in the experiments. This picture isqualitatively the same as the schematic to the left that was used for the modelabove. One particular point of this figure is that one sees how the current densitydoes not extend into both measurement contacts equally.

72


a) b)

Rtip

Figure 7.7: FEM-calculations for B = 0, n0 = 5 ·1015 m−2, electron densitybeneath the tip nt = 0 and different disk radii, as indicated. (a) Calculatedline scan along the Hall bar at a distance to the sample edge correspondingto the respective disk representing the tip (see schematic Hall cross to theupper right of the plot). The values for the larger tip was divided by 3.5 forcomparison of the curve shapes. The black vertical lines represent the interiorof the Hall cross. (b) Calculated line scans along the diagonal from corner 3to corner 1, as indicated in the schematic Hall cross to the lower left of theplot. The black vertical lines show the sample boundaries and the dashed linesindicate a distance of the corresponding tip radius to the sample edge.

These calculations were repeated for several tip positions on different straightlines, thus simulating line scans. The resulting curves are shown in Fig. 7.7 for a tipradius of Rtip = 0.3 µm and Rtip = 0.75 µm, respectively. One recovers the distinctpattern of resistance changes as measured in the experiments, see Fig. 7.5(a). Theexact shape of the peaks in the simulation depends on the geometry of the tip, thesample and their distance. A large tip results in larger resistance changes (the curvefor Rtip = 0.75 µm is divided by 3.5).

Figure 7.7(a) shows two simulated line scans along the Hall bar in a distancecorresponding to the radius of the depleted area beneath the tip. The black verticallines indicate the interior of the Hall cross and thus the width of the voltage contacts.One also sees that the peaks are broader for the larger disk. For line scans alongthe diagonal in the Hall cross area, from corner 3 to corner 1, shown in Fig. 7.7(b),more information can be gained directly from the data. Again the black vertical linesindicate the sample edges, while the dashed lines are drawn in the distance of onedisk radius to it. One sees immediately that the maximum of the structure is exactlyone disk radius away from the edge, which can be identified in the experiments as thepoint with half of the maximum value, for all disk radii. Taking the left maximumin Fig. 7.5(b) for this analysis one arrives at a disk diameter Rtip ≈ 600 nm.

In these simulations it is easy to check the current density flowing into the contactthat is closer to the tip position. This reveals that the maxima occur exactly at theposition where no more current flows between the nearer corner and the depleteddisk. This is quite obvious if one considers that in this position the geometricalbarrier is largest and the least current enters the contact. In Fig. 7.8 this situation

73


(a) (b) (c)

Figure 7.8: Current flow (red) and equipotential lines (black) for two almostidentical tip positions: (a) some current extends into the lower voltage probeand below the depleted disk, while in (b), this is not possible. (c) Line scansalong the diagonal, with Rtip = 0.75 µm with depleted disk (blue) and a diskwith an electron density nt = 8.1 · 1014 m2

Vs , adjusted to arrive at a peak valuearound 20Ω.

is shown for the tip in the lower right corner: in (a) some current can flow pastbelow the disk, but if the corner of the Hall bar inhibits this, very few currentflows into the contact (The red lines depict the current flow and the black are theequipotential lines). Current densities, on the other hand, come along with a voltagedrop, or more intuitively, less current means less charge in the contact and thus aless positive Hall voltage.

This insight allows one to describe the shape of a ‘butterfly-wing’: the tip has thebiggest effect if the depleted disk just touches the sample boundary. This explainsthe maximum running in parallel to the corner in the experiment. The distancethen should correspond to the disk radius, as discussed before. The amplitude ofthe effect decreases far away from the contact entrances, since the tip is no longerrelevant for the current entering the voltage probes.

Another property to note when comparing the two artificial scans is that in thecentral region of the Hall cross the induced changes get smaller with reduced diskradius. This in turn should be seen in broadened symmetry lines in the experimentaldata. Since the depleted disk in a real experiment is connected with the appliedtip voltage, this effect should be observable in a series corresponding scans. Somepreliminary experiments are shown in appendix D.

These calculations reproduce the right amplitudes of the peaks and minima onlywith a much lower disk radius than estimated from the shape of the ‘butterfy’ andfrom the longitudinal resistance. If one reduces in the calculations the electrondensity beneath the tip to nt = 8.1 · 1014 m2

Vs, i.e. about a factor of 3.7 smaller than

the 2DEG density, one arrives at the right order of magnitude of the peaks. This isshown in Fig. 7.8(c) for the disk radius Rtip = 0.75 µm as red curve for a simulatedline scan along the diagonal from corner 3 to corner 1, like in Fig. 7.7(b). Forcomparison, also the curve for the depleted disk with the same radius is given inblue, but divided by 3.5. The drop in the red curve takes place again at the distance

74

7.4. Further considerations

Rtip away from the sample boundary, which gives the same estimates for Rtip asabove. Only between the sample edge and the drop outside the sample a plateaudevelops, probably with some more structure. Although the latter could not beobserved in the experiments, the qualitative picture described above stays useful.

7.4 Further considerations

Kirchhoff’s law

The altered current path has to be incorporated also in the integration for calculatingthe longitudinal voltage drop. This can be done in exactly the same manner as forthe transverse resistance and explains the tails entering the measurement contactsand the reduced longitudinal voltage on the opposite side of the Hall bar in thescanning gate images of Rxx (chapter 7.2). For simplicity the scanning gate imagesof the previous sections are shown again in Figs. 7.9(a)-(d).

By considering Kirchhoff’s laws for a Hall bar one can show that the describedeffects are closely related. For every tip position ~r the following relation has to hold:

δ := Rxx,1(~r)−Rxy,1(~r)−Rxx,2(~r) + Rxy,2(~r) = 0 (7.4)

Since the non-zero pattern of Rxy,1 and Rxy,2 are well separated, plotting Rxx,1−Rxx,2 should allow to recognize the pattern for Rxy,1 and Rxy,2, the latter withreversed sign. This quantity is plotted in Fig. 7.9(e). In addition, Fig. 7.9(f) showsthe complete sum of Eq. (7.4). Except for the noise and some weak structure on

Figure 7.9: (a) Rxx,1, (b) Rxx,2 (c) Rxy,1 and (d) Rxy,2 in a scanning gateexperiment at B = 0 and T = 2.0 K. The numbers in (d) identify the cornersof a Hall cross. (e) Rxx,1 −Rxx,2 and (f) δ := Rxx,1 −Rxy,1 −Rxx,2 + Rxy,2 asfunction of the tip position.

75


the left Hall cross, the signal is zero (the residues on the left Hall cross are probablydue to the different lock-in (shield on ground) used for the measurement of Rxy,2.It is also the reason for the larger noise floor in these measurements). Thus, as longas the two respective Hall voltage probes are well separated, i.e. without non-localeffects, it is in principle enough to measure only the two longitudinal resistances toobtain all the information.

Reciprocity relations

In Fig. 7.10 additional measurements on the ‘butterfly’ at zero magnetic field arepresented. They were taken under the conditions of ‘experiment D’, e.g. with a tipvoltage of Utip = 0.3 V. In Fig. 7.10(a) the current direction and the direction givenby the voltage measurement ΦA − ΦB are the same as in the previous data. The‘butterfly’ structure is clearly visible, though a slight asymmetry in the peak heightcan be observed. The stripes in this data are due to a technical problem solved inthe next images. Figure 7.10(b) shows a scanning gate image with exactly the sameparameters, except for the inversed current direction. The sum of the data of image(a) and (b) is presented in Fig. 7.10(d): except for the stripes and the noise the twosignals are essentially the inverse of each other. In Fig. 7.10(c) the experiment wasdone with current and voltage leads interchanged and in Fig. 7.10(e) the differenceto the data in (a) is plotted. The two data set are essentially identical.

Every point in these in these images is a 4-terminal Hall resistance measurement.General symmetries, for example the ones following from the 4-terminal resistance

~

Uy

A

B

(b)(a)

Current direction

Measurementdirection: A-B

(b) (c)

(d) (e)

Figure 7.10: (a) Original setup for measuring the Hall voltage. The blackarrow identifies the current direction, the blue stands for the measurementdirection Uy = ΦA−ΦB. (b)-(d) Rotated setup: as in (a), the arrows indicatethe current and measurement directions. (e)-(g) Difference from scans in (b)-(d) to the scan in (a). There is essentially no difference in the scanning gateimages if the current and voltage probes are rotated.

76

7.4. Further considerations

formula Eq. (2.5) discussed in chapter 2, have to hold for every data point in thescans. The two trivial examples are the exchange of the voltage probes (not shown)and reversing the current direction. Both lead to a reversed sign in the signal atevery point. The latter example demonstrates that the experiments were done in thelinear transport regime, also on local scales. A little less trivial is the interchange ofthe current contacts with the voltage probes, which is described by Eq. (2.6), thereciprocity relation of the 4-terminal resistances. These measurements can be seenas a demonstration of the ‘global’ Onsager relation in 128 × 256 = 32′768 sampleswith each having a slightly different potential landscape.

Effect of diffusive transport

The explanation of the ‘butterfly’ in the previous section requires that the resistivitytensor ρ, the potentials and the fields are locally well defined quantities. This isonly the case in diffusive samples where the mean free path is much smaller than thesample dimensions. Since in these experiments lmfp ≈ 1 µm, the Hall cross is in anintermediate regime between diffusive and ballistic. To prove experimentally thatthese patterns can arise from diffusive transport alone, scanning gate experimentsdone at T = 56 K are shown in Fig. 7.11(a) and (c). At this temperature themean free path is much smaller due to phonon scattering. One observes essentiallythe same features as at T = 2 K. Only two differences are obvious: The patternsare weaker in absolute magnitude and Rxx drops very much below the unperturbedsignal on both sides of the Hall bar, as can be seen in the cross section indicatedin Fig. 7.11(a) and plotted in (b). Both effects are probably connected with somecomplicated screening effects.

(a) (c)(b)

Figure 7.11: Scanning gate images at 56 K and B = 0 showing ∆Rxx,2 andRxy,1, respectively.

77


7.5 Summary

In summary the measured changes in the longitudinal and transverse resistance aregenerated by the altered current flow induced by the AFM-tip. The effects canbe understood as ‘adding longitudinal resistance to the Hall resistance’ and viceversa, due to the deflected current density. Such phenomena can be produced alsonaturally in inhomogeneous samples. Here, the inhomogeneity is brought aboutdeliberately and in a controlled fashion. With appropriate modeling and controlledexperimental conditions information about the sample, but also about the sensor,can be gained. Qualitative agreement with FEM-simulations could be shown, butthe model with only a completely depleted area beneath the tip is not fully consistentwith the experiment: the disk radii extracted from the various measurements aresimilar, but inserted into the simulations give too large changes in the Hall resistance.A reduced electron density in the disk area helps to solve this problem. Alreadythese experiments show the complexity arising from the combined system of tip andsample.

78

Chapter 8

Classical Hall effect regime

8.1 Introduction

Scanning gate experiments in small, i.e. non-quantizing, magnetic fields are rare.The following two can be found in the literature:

Electrons exiting a QPC are typically collimated in a cone of 15−30 around thenormal to the QPC. This can be observed in the so-called ‘magnetic steering’ exper-iment (see for example [136]), in which a second QPC (collector) is located oppositeand collects the electrons emerging from the first (source). When a magnetic fieldis applied perpendicular to the 2DEG, the trajectories are bent corresponding totheir cyclotron radius. If the collector is within the mean free path of the sourceQPC, the bending can be observed as a maximum of the collector voltage aroundzero magnetic field. In a scanning gate experiment an AFM-tip induced potentialbarrier positioned within the path of the majority of the electrons leads to a deflec-tion of the electron beam. Since a QPC has a very small entry region, the scatteredelectrons do not enter the collector anymore, which leads to a reduced voltage onthis QPC. The bent paths of electrons and of positive charged quasi-particles couldbe mapped directly with this technique [137].

In a quantum billiard defined by ‘erasable electrostatic lithography’ (EEL) withthe same setup in situ before the actual experiment, some indications of so-called‘scarred wave functions’ could be observed in real space [138]. In a semiclassicalpicture the wave function can be understood as electron interferences in a stadiumbilliard where a small change in the magnetic field can have a strong effect on theinterference pattern. The latter has a different sensitivity to changes in the localpotential which leads to the observed scanning gate images.

No experiments are reported on standard geometries with wide voltage probescontaining many modes, like a Hall bar, for small magnetic fields.

79

Chapter 8. Classical Hall effect regime

8.2 Experiments

In this chapter scanning gate experiments of the Hall resistance on a 4 µm wideHall bar with a small magnetic field applied orthogonal to the sample plane arepresented. The parameters and sample characteristics are given as ‘experiment B’in chapter 6.4, tables 6.1 and 6.2. For example the mean free path extracted fromthe low field data is about 1 µm, like in the sample discussed in chapter 7.

(b)

1 m

(a)

4

32

1

Figure 8.1: (a) Raw data of the Hall resistance measured on sample II,cooldown 1 at B = 0. Also here one recognizes the ‘butterfly’-shaped patternas in sample I, but much weaker. (b) The same data as in (a), but convolutedwith a Gaussian window function for filtering.

One essential difference to the experiments discussed in chapter 7 is that themaximum change in Rxy introduced by the AFM tip at zero magnetic field is abouta factor of 3 smaller. Possible reasons are the reduced tip voltage of Utip = −0.7 Vand the increased tip sample separation. In addition, the enhanced electron densityin the 2DEG and probably also at the surface and in the donor layer, might screenthe tip more strongly. In order to extract the prominent features more clearly, thedata are filtered by convolution with a Gaussian window function of width σ = 31nm, corresponding to 3×2 pixels in the original images. This procedure was appliedto all data in the same way and is denoted simply as ‘filtered’. This filtering distortsthe edges of the scans, so they are omitted, which leads to a slightly smaller image.Also the extracted contours are treated in this way. As an example the data forB = 0 are reproduced in Fig. 8.1, showing the ‘butterfly’ pattern discussed inchapter 7.3, with two broad minima and two broad maxima at the corners of theHall cross. The different resistance scales in the images stem from the fact that thespikes in the data are flattened by the filtering.

In Figs. 8.2(a)-(f) scanning gate measurements are presented for a series of smallpositive magnetic fields perpendicular to the sample plane. They may be comparedto the zero field feature shown in Fig. 8.1(b). Already at B = 25 mT the twopositive peaks occurring at corner 1 and 3 have merged across the center of theHall cross. The minima at the other corners are still visible. A similar situation isgiven for B = 50 mT, though the features at the corners are less prominent. AtB = 75 mT the two minima are clearly weaker and the former maxima can not bedistinguished anymore. Instead, a global maximum is found near the center of the

80

8.2. Experiments

Hall cross, a little displaced to corner 3. Any minima or maxima at the cornershave vanished at B = 100 mT, though the measured signal difference is still of thesame order of magnitude. In this image one finds a maximum elongated along thediagonal from corner 1 to 3 with a smooth drop outside the Hall cross area. Upto this field, the maximum changes introduced by the AFM-tip are always around∆Rxy ≈ ±3 Ω. At B = 250 mT, and more clearly at B = 375 mT, the area ofchanged resistance gets smaller and more localized around corner 4. The amplitudeof this feature increases approximately linearly with magnetic field. More scansillustrating the latter development are shown in chapter 9.6 and will be discussedthere in detail.

The symmetry statements of Eqs. (6.1)-(6.3) in chapter 6.2 suggest a comparisonto scanning gate measurements at reversed magnetic fields. A selection of images isshown in Figs. 8.2(g)-(i): at B = −50 mT the negative peaks of the zero field featureare connected. Then the positive peaks disappear completely between B = −50 mTand B = −250 mT and a global minimum near the center forms.

Figure 8.2: Filtered data of scanning gate experiments at small magneticfields. The development from the zero field case in Fig. 8.1(b) is explained inthe text. The corner numbering is given in Fig. (i).

81


The symmetries in the scanning gate measurement of the Hall voltage describedin chapter 6.2 can be observed qualitatively at low magnetic fields, though not in thedetailed structures. At higher fields the pattern of changed resistance gets localizedat corner 3 instead of corner 4 like at positive fields. This violates relation (6.3) andEq. (6.2), while Eq. (6.1) still holds. It will be shown in chapter 9.6 that theseobservations are connected to the onset of electron transport in quantized states athigh magnetic fields, which makes the resistances more sensitive to details in thelocal electron potential.

8.3 Models and discussion

Judging from the mean free path of about 1 µm the Hall cross is in an intermediateregime between diffusive and ballistic. Since in chapter 7 the ‘butterfly’ pattern atzero magnetic field was found also at elevated temperatures, the main focus in thischapter will be on the further development of the diffusive model of a depleted diskthat simulates the tip-induced potential. Calculations in the ballistic regime will bepresented in a subsequent section.

8.3.1 Completely depleted disk

In order to understand the scanning gate experiments at low magnetic fields onecan start with the same model for the diffusive regime as in chapter 7.3.2: the effectof the tip on the 2DEG is modeled as a disk of depleted electron gas with radiusRtip. The Hall voltage can be calculated by integrating the electric field along achosen path from one voltage probe to the other. This path can be split into partsthat run parallel and orthogonal to the current density. In this model the currentdensity for a given tip position is independent of the magnetic field, since all thecurrent flows in regions of constant sample parameters, i.e. ~∇·~j = 0 and ~∇∧~j = 0,which completely determines the vector field ~j, independent of B. This allows oneto choose the same path of integration for every magnetic field, for example thesame as at B = 0 in chapter 7.3.2 in Fig. 7.6(a), and to split the path into a partparallel to the current density, from B to C, and one othogonal to ~j, from C to A.This gives the following Hall voltage:

Uy = −∫ C

B

ρxx · jx · dx︸︷︷︸‘Butterfly’ as at B=0

−∫ A

C

ρxy(B) · jy · dx︸︷︷︸Uy of unperturbed sample

(8.1)

The first integral is exactly the same as at B = 0, since ρxx is independent ofthe magnetic field in the Drude model, cf. Eq. (2.11). The second integral leadsexactly to the Hall voltage of the unperturbed Hall cross since the path traverses thecomplete current orthogonally exactly once, independent on the tip position. Thismodel thus predicts a scanning gate image that consists of the structure measured

82

8.3. Models and discussion

at zero magnetic field, the ‘butterfly’, plus a constant background that correspondsto the Hall voltage in the unperturbed sample. Therefore, this model predicts thatthe following quantity is zero at any tip position and magnetic field:

∆Rxy(x, y, B) = Rxy(x, y, B)−Rxy(x, y, B = 0)−Rxy(z →∞, B) (8.2)

where Rxy(x, y, B) is the scanning gate image at the magnetic field B, Rxy(x, y, B =0) the pattern at zero field and Rxy(z →∞, B) the Hall voltage in the unperturbedsample, which is a constant for a given scan. The experimental ∆Rxy are plottedin Fig. 8.3. The manipulation was done with the original data and the result wasfiltered like discussed before. A small background value had to be substracted to getcomparable colorscales, which can probably be attributed to a small gating effect ofthe sensor setup on the entire Hall bar.

B=25 mT B=50 mT B=75 mT

B=100 mT B=-50 mT

(a) (b) (c)

(d) (e)

4

32

1

Figure 8.3: Scanning gate data after substracting the Hall voltage and thezero field feature, as described by Eq. (8.2). The filtering was done after themanipulations and then the minimum was substracted.

The data clearly show structures in the plotted quantity: at small fields a peakin the middle of the Hall cross appears that has about the same height for B = 25mT to 75 mT. The peak is negative with the same strength for B = −50 mT. AtB = 75 mT some additional structure becomes visible at the lower left corner (‘4’)which is the dominant feature above 100 mT. At higher fields the scanning gateimages look essentially the same as in Fig. 8.2, because the ‘butterfly’ feature isvery small compared to the rest of the signal. Below |B| = 75 mT the difference dataare dominated by the maximum in the middle which is approximately independentof the magnetic field and at higher fields the maximum is given by the structure atcorner ‘4’. The amplitude of the latter increases linearly in B with a slope of aboutd(∆R)

dB≈ 80Ω

T.

83


8.3.2 Disk of finite electron density in the diffusive regime

In the model presented above the images should show no structure at all. Sincethe above argument also holds for an arbitrary shape of the tip induced potential,the only change in the model that can be introduced in the diffusive regime is toallow for a spatially more variable conductivity tensor. The simplest adjustmentis to introduce a constant, but finite electron density in the disk of radius Rtip,which simulates the AFM tip. This idea is explored in the next sections using finiteelement method (FEM) calculations in two-dimensional models (cf. appendix B).Before specific results are presented, an intuitive picture of the underlying physicsis developed.

Intuitive picture

In a sample with a known resistivity tensor ρ(~r) = σ−1(~r) the electric field is

completely determined by the current density by ~E = ρ~j. If the current is forcedto flow around a depleted disk, the electric field is modified. For illustration, Fig.8.4(a) shows a FEM simulation of the electric field with the bias field substracted.

Intuitively, one can think of the formation of a dipole due to the accumulationof electrons on the right of the disk and the lack of electrons on the left. Hence,the additional electric field will, from now on, be referred to as ‘dipole field’, thoughthe exact dependence may be different. This dipole is not to be mixed up with theLandauer resistivity dipole, which originates from a the density modulation inducedby scattering of quantum mechanical electron waves at a potential barrier [14].

Figure 8.4: FEM-calculations of the electric field for a region with a diskof a different conductivity tensor than the surrounding. The current is shownas red lines and flows from the left to the right. In (a), no magnetic field isapplied and the electron density in the disk region is zero. The difference ofthe electric field to the external field is plotted as black arrows. (b) Contourplot of | ~E| around a depleted disk at finite magnetic field. The ‘dipole’ pointsalong the x-axis (black arrow). (c) Same as in (b) but with the finite electrondensity nt = 1

3n0 beneath the tip, which leads to the tilted ‘dipole’ (blackarrow) and a finite current density in the disk region.

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When a magnetic field is applied to the model sample which is perturbed by thedepleted disk, one does not expect a change in the current density, because ~j differsfrom zero only in regions with the original sample parameters, cf. Eq. (2.14). Thetotal electric field for this situation is plotted in Fig. 8.4(b) as gray lines of constantfield strength and the black arrow indicates the direction of the dipole field.

Figure 8.5: (a) FEM-calculations of the current density (colorscale andcontour plot) for B = 1 T, nt = 0.75n0 with sample parameters as in theexperiment and Rtip = 750 nm. The red lines represent the current flow. Apath for the integration mentioned in the text is shown in white. (b) Crosssections through the current density at the positions indicated in the inset.The parameters are B = 50 mT and nt = 0.3n0. The black rectangle standsfor the current distribution without the AFM tip.

If at finite magnetic fields a current is allowed to pass through the disk region,e.g. by an increase of the electron density under the tip, an additional electric fieldis created orthogonal to the this current. Intuitively this leads to a rotation of thedipole field around the center of the disk. This rotation is illustrated by the FEMsimulation in Fig. 8.4(c). Because the angle between the current density and theelectric field depends only on the mobility (‘Hall angle’, cf. Eq. (2.12)), the currentdoes not flow through the disk in parallel to the x-axis, rather it is deflected relativeto the current density outside the disk. Such a situation is illustrated in Fig. 8.5(a).Both, the colorscale and the contour plot represent the norm of the current densityat a large magnetic field and a large electron density in the disk. ~j is asymmetricwith respect to the symmetry axis of the Hall cross. The same effect can also beseen with the more realistic parameters of B = 100 mT and nt = 0.3n0 in crosssections orthogonal to the Hall bar, shown in Fig. 8.5(b). The current enters on theupper left and leaves the disk at the lower right.

The measured Hall resistance is enhanced by a disk of low, but finite electrondensity placed in the center of the Hall bar. This can be understood in two ways:(1) By considering an integral of the electric field on a path like it is depicted in Fig.8.5(a): the path from B to A traverses the complete current exactly once orthogonalto ~j, leading to a Hall voltage as in the unperturbed sample. In addition, a positivevoltage is collected on the paths parallel to the current density, because the latter

85


is larger on part b of the path than on part a. (2) By considering the drop of theelectric potential along the ‘dipole’: If it is not rotated the two voltage probes of theHall cross are at the same potential and only the Hall voltage of the unperturbedsample is measured. A counter clockwise rotation of the dipole leads to a reducedpotential in the upper and an increased potential in the lower contact and thus toa larger Hall voltage.

FEM simulations

The low-field evolution of the scanning gate image is determined by the peak that isfound near the center of the Hall bar in the measurements shown in Fig. 8.3. Thisfeature can be reproduced within the model of a tip induced disk of finite electrondensity. Several FEM simulations are now presented where effects of variations of themodel parameters on the measured Hall voltage are investigated. The characteristicsof the unperturbed sample are chosen to be the same as in the experiments.

Figure 8.6(a) shows a simulated line scan through the Hall cross along the x-axis, as illustrated in the inset. The parameters are B = 25 mT, Rtip = 0.75 µmand nt = 0.3n0. In the Hall resistance a maximum develops in the center of theHall bar. The FWHM corresponds approximately to the sum of the width of thevoltage probes and the disk diameter (Uxy = 31.75 Ω with the tip far away). Thesefindings can be understood intuitively by considering the overlap of the spatiallyshifted ‘dipole’ with the voltage contacts.

In Fig. 8.6(a) the absolute value of the maximum resistance change is not exactlythe experimentally observed ∼ 3 Ω. Therefore the dependence of the signal on thedisk radius is plotted in Fig. 8.6(b) to show that a larger tip radius can accountfor this finding. The required change in Rxy is found at Rtip ≈ 1 µm. Increasingthe disk radius further leads to large effects around Rtip =

√5 µm (dashed vertical

Figure 8.6: (a) FEM simulation for a tip scanned along the x-axis. Theparameters are Rtip = 0.75 µm, B = 25 mT, nt = 0.3n0 and the sampleparameters of the experiments. (b) Hall resistance versus disk radius for thetip positioned in the center of the Hall cross (inset) with otherwise equalparameters as in (a).

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Figure 8.7: (a) FEM-calculations of ∆Rxy versus the electron density inthe disk, nt, for the tip in the center of the Hall bar at the three indicatedmagnetic fields and Rtip = 0.75 µm. The dashed line shows the correspondingvalues obtained by a parallel resistor model at B = 25 mT, where only currentsparallel to the x-axis are allowed. The full line shows the values for the ballisticmodel discussed in the next section at the same field. (b) FEM simulationof the central maximum extended to densities nt > n0 for otherwise equalparameters and B = 25 mT.

line), where the disk ‘touches’ the edges of the Hall cross. At even larger tip inducedstructures the Hall response flattens out and the entire structure has the new electrondensity, corresponding to Rxy ≈ 106 Ω.

The other parameter in the model is the electron density in the disk, nt. In Fig.8.7(a) calculations of the change in the Hall resistance versus nt are shown for thethree indicated magnetic fields and for 0 ≤ nt ≤ n0. At nt = 0 and nt = n0 theinduced changes are zero. In-between a maximum develops, which gets shifted tohigher densities with increasing field. At a fixed electron density beneath the tip ofnt ≈ 0.3n0, Rxy shows similar values for a large range of magnetic fields, pinpointedby the intersection of the curves at B = 75 mT and B = 0.5 T, respectively. Thismechanism can account qualitatively for the constant peak height in the experimentsat low magnetic fields.

For B = 25 mT also the curves for a parallel resistor model (dashed red line)and for the ballistic model discussed below (full red line) are given for comparison.The parallel-resistor model neglects current densities not parallel to the x-axis andtherefore does not reproduce the numerical model very well.

The maximum in the ∆Rxy vs. nt curves can be understood intuitively byconsidering the electric dipole formed by the current flow: by increasing the densityin the disk from 0, the dipole is rotated and the Hall resistance increases. At thesame time the ‘dipole strength’ is reduced, since the charge pile-up is weaker. Atthe maximum of the curve the latter process starts to dominate and the dipoledisappears completely at nt = n0. At high magnetic fields the Hall voltage is largerand therefore also the rotation of the dipole, so that the maximum is shifted tohigher fields. Is the tip induced density enhanced to nt > n0 the current gets

87


focused into the disk and the dipole grows, but with reversed sign and the measuredHall voltage is reduced. At very high densities almost the complete current flowsthrough the disk and the process saturates. FEM calculations showing Rxy also forlarge nt-values are presented in Fig. 8.7(b).

A finite density in the disk region has also an effect on the evolution of the‘butterfly’ pattern. Figure 8.8 shows a simulated scan along the Hall bar, as drawnschematically in the inset, for B = 0.5 T and nt as indicated. The dips in Rxy atthe lower left corner disappears for nt close to n0. This prediction is not observedin the data, probably due to the lack of resolution.

Figure 8.8: Calculations ofthe effect of a finite electron den-sity in the disk on the ‘butterfly’structure for line scans as indi-cated in the inset.

8.3.3 Disk of finite electron density in the ballistic regime

Because the sample in the experiments discussed above is not clearly in the diffusiveregime, also calculations and an intuitive picture for the ballistic regime are nowdiscussed. Experiments on a sample with a larger mean-free path are presented inthe subsequent section.

In order to calculate resistances in the ballistic regime, electrons can be consid-ered as classical charged point particles that move in an external potential. Fromthe trajectories calculated by solving Newtons equations of motion, one can inferthe transmission coefficients of a given structure. For illustration few trajectoriesare plotted in Fig. 8.9(a) for a Hall bar with a tip modeled as a circle. From thetransmission coefficients the involved resistances can be calculated by the Buttikerformula Eq. (2.5).

In similar calculations the following statement for a Hall cross geometry with4 identical leads was shown [135]: as long as (1) the disk of constant potentialrepresenting the tip resides well inside the Hall cross, (2) the diameter of the diskd is smaller than half the lead width 2W and (3) the potential change in the disk,V0 is smaller than half the Fermi energy EF , the Hall resistance depends only on

88


the average potential and therefore on the average electron density in the Hall cross(slightly reformulated) [135]:

Rxy ≈B

e 〈n〉with 〈n〉 = n0

[1− πd2

4W 2

V0

EF

](8.3)

This formula can be rewritten in the terms used in this thesis, with Atip and AHC

the areas of the disk and the (effective) Hall cross, respectively.

Rxy ≈B

en0

1

1− Atip

AHC

n0−nt

n0

(8.4)

This extrapolation to the geometry with smaller voltage probes is not trivialand should be considered in future calculations. The change in Rxy in this model isplotted in Fig. 8.6(a) as a full red line for B = 25 mT and the same disk radius, forcomparison with the other models.

Figure 8.9: (a) Many electron trajectories are calculated to get the trans-mission coefficients. (b) Electron trajectory for an arbitrary electron thatwould have missed the upper voltage contact of the Hall cross if the tip wasnot present.

Intuitively the effect of a disk with an electron density different to the rest of the2DEG can be understood with an ‘electronic version’ of Snell’s Law and the fact thatthe cyclotron radius is smaller for slower electrons, i.e. at reduced densities. ‘Snell’sLaw’ can be deduced by considering that the parallel component of the electronmomentum is conserved, whereas the orthogonal component changes due to thechange of the Fermi energy at the interface. This leads to the following formula:

sin θ2

sin θ1

=

√n1√n2

(8.5)

with the two involved electron densities n1 and n2 and the angles of incidenceand transmission θ1,2.

Like it is shown schematically in Fig. 8.9(b), electrons that enter the disk arebent more strongly and are focused into the upper voltage lead. Due to ‘Snell’s Law’electrons with a flat incidence angle are reflected at the surface of the disk. Because

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the trajectories are bent by the Lorentz force, a flat incidence is more probable at theupper part of the disk. Therefore the disk leads to effectively enhanced transmissioncoefficients Ti,i+1, whereas all other Tij, j 6= i + 1, are reduced. Inserting into theButtiker formula, Eq. (2.5), with the enumeration of the leads given in Fig. 8.9gives:

Rxy ≡ R13,24 =

larger︷︸︸︷T12T34−

smaller︷︸︸︷T14T32

D(8.6)

Snell’s Law makes also clear why at a stronger depletion of the disk a qualitativechange occurs: the angle of incidence when the electron enters the disk area mustnot be too large, otherwise the electron is scattered back from the surface, i.e.

sin(θ1) <√

nt

n0. At nt < 0.5n0 this happens for the majority of electrons.

Very recently calculations in the completely ballistic regime of a Hall cross withfour identical leads have been performed in the group of Prof. F.M. Peeters (Univer-sity of Antwerp) with an electron density of n = 3.65 ·1015 m2, which corresponds tothe sample in ‘experiment C’ discussed in the next section. The simulations in Fig.8.10(a) and (b) show results for B = 50 mT and B = 100 mT, respectively. Thepotential induced by the tip is assumed to have a Gaussian shape with a width of14

of the lead dimensions and a maximum potential modulation of 12EF . At B = 50

mT the ‘butterfly’ structure is reproduced, while it has disappeared completely atB = 100 mT. Calculations with a stronger potential modulation and larger widthof the tip induced potential lead to larger resistance changes and the two maximamerge at lower magnetic fields. Also the observation that it is the minima in thescanning gate images that merge at negative fields is reproduced (not shown).

Figure 8.10: Simulated scanning gate images for a Hall cross (black lines)in the ballistic regime at (a) B = 50 mT and (b) B = 100 mT (with courtesyof F.M. Peeters, University of Antwerp, Belgium).

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8.4. Experiments in the ballistic regime

8.4 Experiments in the ballistic regime

Figure 8.11 shows a series of scanning gate images at small magnetic fields taken inthe cooldown ‘experiment C’ described in tables 6.1 and 6.2. The main differenceto the previous measurements is that the mean free path of the electrons is about6.6 µm, which exceeds the Hall cross dimensions. At zero magnetic field the Hallresistance is enhanced by the AFM tip at the corners 1 and 3 (the enumeration isgiven in Fig. 8.11(e)) and reduced at the other corners. This pattern is similar tothe ‘butterfly’ features in the previous measurements, though here the features arebetter separated and become even better pronounced at B = ±50 mT.

Another difference to the experiments on the sample with smaller mean freepaths is very prominent: the symmetry lines along the axis of the Hall cross do notexist, due to some additional structure orthogonal to the diagonal from corner 2 to4. In a positive magnetic field this central structure gets more pronounced, while atnegative fields it disappears. The maximum changes in resistance are much smallerthan in the previous measurements, in all the images about 1 Ω.

B=0

B=50 mT B=100 mT

B=-50 mT B=-100 mT

(a)

(b) (c)

(d) (e)

1

2

4

3

Figure 8.11: Scanning gate experiments at small magnetic fields on a bal-listic Hall cross (sample II, cooldown 2). Still the pattern like in the diffusivestructure is visible, but some new structure appears, inducated by the arrows.

The features at the corners are reproduced by the simulations for the ballisticregime, cf. Fig. 8.10. It can be explained in an intuitive picture by the notion thatat zero magnetic field a disk with a different electric potential scatters electrons fromone side of the Hall bar to the other symmetrically to the Hall bar axis. Is the diskpositioned on a corner of the Hall bar, no electrons enter the disk from outside thestructure, so that the focusing effect on the impinging electrons is not compensated.

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The advent of a central structure at finite magnetic field is discussed in chapter8.3.3. The additional features observed in the measurements are attributed to tworeasons: first, the sample might be in a quasi-ballistic regime, i.e., few individualscatterer can be present inside or near the Hall cross. The effect of such scatter-ers is not averaged by many collisions and the effective sample geometry loses thesymmetry of the Hall cross. Second, in this experiment the applied tip voltage ismuch smaller than in the other experiments. Therefore, a certain potential changeis achieved in a smaller region of the 2DEG, which explains the better resolutionand the smaller effect on Rxy.

It would be interesting to conduct calculations of ballistic electrons in a Hallcross with fixed scattering centers placed at well chosen locations to compare withthe experiments.

8.5 Tip voltage dependence

The models presented in the previous sections suggest that the effect of the scannedAFM tip on the Hall resistance can be inverted by enhancing the electron densitybeneath the tip. This is generally achieved by changing the applied tip voltage.

Figure 8.12 shows a series of scanning gate images taken at B = 300 mT and adistance of the tip to the surface of d = 300 nm, taken with different tip voltages Utip

as indicated. These measurements were done on sample II at the start of cooldown

Figure 8.12: Scanning gate images at a magnetic field of B = 0.3 T fordifferent tip-sample voltages (sample II, cooldown 1, d = 300 nm). At Utip ≈ 0the induced change in Hall resistance turns from enhanced Rxy at negative todecreased Rxy at positive voltages. The black line in Fig. (a) indicates thecross section shown in Fig. 8.13(b).

92

8.5. Tip voltage dependence

2, where ns = 3.86 · 1015 m−2. One finds an ellipsoidal shape where Rxy is changedinside of the Hall cross. At Utip = −2.0 V the transverse voltage is clearly increasedwhen the tip is positioned inside the Hall cross, while at Utip = +1.0 V it is reducedbelow the value without the tip. Between these two voltages the pattern gets weakerand the inversion of the sign of the resistance changes takes place between Utip = 0.0V and Utip = 0.5 V. The width of the feature seems to be given essentially by theHall cross geometry along the large semi axis of the elliptic shape, while the smallsemi axis is smaller for small applied effective tip voltages. The latter also is smallerat Utip = −2.0 V, which is not expected. A cross section along the smaller semi axisat this tip voltage is shown in Fig. 8.13(b).

In Fig. 8.13 the mean value of the Hall resistance in the scanning gate images ofFig. 8.12 are plotted as function of the tip voltage (red curve). The curve increaseswith decreasing Utip due to the peak in th center of the Hall cross. In contrast, thevalues outside the structure, for example at the lower left corner of the scanninggate images, are essentially constant (blue curve). The light gray band marks theminima and maxima in the individual data sets and the dark gray band indicates therespective standard deviations from the mean values. Both have a minimum aroundUtip ≈ 0.5 V and broaden when the tip voltage deviates from this value. If one takesthe blue line as orientation one clearly sees that an enhanced Hall resistance can beachieved with Utip < 0.5 V and a reduced resistance with Utip > 0.5 V, e.g. the graybands lie above and below the blue curve, respectively.

In order to arrive at estimates for the tip induced electron densities within thediscussed models, one has to assume a tip radius. In the diffusive model the observedresistance change of about 30 Ω can not be explained by small disk areas (cf. Fig.

Figure 8.13: (a) Statistical data of the scans with different tip voltages inFig. 8.12: the red curve shows the mean Hall resistance while the value in thelower left corner of the scanning gate image is plotted in blue. The light grayband indicates the maximum and minimum values in the scans, while the darkgray band is the standard deviation from the mean value. (b) Cross sectionthrough the filtered image 8.12(a) (Utip = −2.0 V) along the smaller axis ofthe ellipsoid, as indicated in Fig. 8.12(a).

93


8.7(a)), which is no contradiction, since the tip sample distance is larger than in theprevious measurements and one can expect the disk radius to be larger as well. Theballistic model with Eq. (8.4), on the other hand, can reproduce the required value.For example with Rtip = 1 µm one arrives at nt = 0.85n0 and with Rtip = 2 µmat nt = 0.96n0. Both values seem rather large, but this has to be assigned to theunnatural model of a homogeneous potential in the disk.

8.6 Summary: classical transport

Scanning gate measurements at zero and small magnetic fields in the classical regimeare presented. They reveal many features that can be understood on a qualitativelevel with models based on a local conductivity tensor for the diffusive regime andsymmetry considerations. On the quantitative level FEM simulations can help toget more detailed insights. The model presented here can be extended to morecomplex potential variations, like a Lorentzian.

At zero magnetic field a ‘butterfly’-shaped pattern is found in the measurementsand is explained in the ballistic and the diffusive regime. The development of thisfeature in small magnetic field is described and attributed to the appearance of amaximum in the center of the Hall cross. This feature could only be explained byintroducing a region with a changed, but finite conductivity tensor beneath the tipin both transport regimes.

A short summary of the models shows why the principal features occur bothin the ballistic and in the diffusive regime. The ‘butterfly’ pattern is produced bytwo ingredients: (1) a potential perturbation in the 2DEG that alters the paths ofthe electrons, or more general of the current density. (2) An asymmetric electronflux impinging on its surface due to other geometric constraints, e.g. the Hall crosscorners. In a magnetic field the electrons that traverse the disk are not distributedsymmetrically anymore, which leads to the central structure in both regimes. Intruly ballistic or diffusive structures the presented symmetry relations are expectedto hold. Therefore, the observed violation in one of the samples has to be attributedto single scatterers in the Hall bar and the transport regime is thought to be ‘quasi-ballistic’. The decomposition of the observed patterns as a superposition of the zerofield pattern and the central peak at finite magnetic fields is not exact, because theyboth depend on the current density, which in turn is a function of the magnetic fieldfor inhomogeneous electron density distributions.

More precise and controlled measurements are required to disentangle unambigu-ously the discussed effects. Experiments on larger Hall crosses are easier to achieveand would prove the importance of the corner geometry. Experiments on smallerHall bars or quantum wires should lead to the observation of ‘universal’ conductancefluctuations in real space as a function of magnetic field, which would prove theirinterference character. For this goal, it is necessary to know the non-coherent effectslike they are discussed in this chapter.

94

Chapter 9

Quantum Hall effect regime

9.1 Introduction

The transport properties of a 2DEG at high magnetic field are well established andcan be understood in the picture of edge states and compressible and incompressiblestripes. These states have very long coherence times and can be used as buildingblocks for more complicated structures like interference devices. For example it wassuggested to generate entangled edge states in order to test Bell’s inequalities withmassive particles [139].

In the literature many experiments with fixed geometries are reported for theQHE regime that are interpreted as coupling of edge channels at (macroscopic) po-tential barriers and constrictions [140, 141, 142, 143]. Also some insight in the stripestructure can be gained by such experiments [144] and by using a single-electrontransistor (SET) as local potential sensors in order to investigate the electrostaticsin the 2DEG [145]. In the formation of quantum Hall plateaus and in the transitionbetween them, individual potential fluctuations play a major role. Conventionaltransport experiments can provide informations only about averaged properties andhave to be interpreted with statistical models [146]. Direct experimental access tothe microscopic mechanisms of the quantum Hall transition can be gained by theuse of scanning probe microscopy.

The local density of states (LDOS) of a 2DEG was investigated locally by twomethods. In the sub-surface charge accumulation (SCA) method [147] an AC volt-age applied to the 2DEG leads to the accumulation of charges in the 2DEG whichinduces a current on the metallic STM tip that is scanned across the sample. Mea-surements in the QHE regime show regions of low signal strength surrounded bymeanders of alternating high and low signals. The low-signal areas are interpretedas incompressible stripes that can not be charged because of the gap in the density ofstates [148, 149, 150]. It was also shown that the resistivity across an incompressiblestripe is very large and that such stripes can enclose and electrically insulate areasfrom the rest of the sample [151].

The second method demonstrated recently [116] is to scan an SET across the

95

Chapter 9. Quantum Hall effect regime

2DEG surface. Mapping the electrostatic potential in the QHE regime showed diffe-rent electrochemical potentials at two opposite sample boundaries [152]. In addition,the local screening capabilities of the 2DEG can be observed in the same scan byapplying an AC voltage to a back gate and measuring the potential modulationat the sample surface. With this technique extended and localized states could beidentified in real space and with high resolution in the IQHE regime [152, 153, 154]as well as in the FQHE regime [155]. These states are mapped as stripes of highsignal strength corresponding to the reduced screening capabilities of incompressiblestripes.

An open question is the current density distribution in the QHE regime. Theelectro-chemical potential is carried by the compressible stripes which allows one touse the Landauer-Buttiker formalism and the picture of ideally transmitting edgestates. The latter is due to the fact that the Hall resistance only depends on thedrop of the electro-chemical potential between the two sample edges and the totalcurrent. The current distribution does not enter this picture. In equilibrium thecompressible stripes carry a diamagnetic, the incompressible stripes a paramagneticcurrent (see chapter 2.4.6), and since the current density is symmetric in referenceto the sample center the total current is zero. Nevertheless, this circulating currentdensity gives rise to a 1/B-periodic magnetization [156]. In the non-equilibriumsituation of an applied bias voltage a net current is carried by these states. Sincethe highest occupied compressible stripes at the edges are pinned to the Fermi energythese stripes do not contribute to the net current, but they are necessary for thecoupling to the metallic leads.

A direct measure for the current distribution is the electric potential distributionin the 2DEG plane. In GaAs-heterostructures this potential has been obtained intwo ways: (1) optically, by exploiting the linear electro-optic effect which changesthe polarization of low energy photons in the presence of an electric field [22]. Non-equilibrium electron populations along the sample edge could be mapped recently bylocally collecting the photons from electron-hole recombination [157]. (2) A betterspatial resolution (∼ 100 nm) can be achieved by using a low temperature AFM:the additional force gradient due to the electric potential leads to a shift in theresonance frequency of the sensor (see also chapter 5.2). Line scan experimentsacross Hall bars show that the potential drop is essentially linear at filling factorsbelow an integer number and smooth, but rather arbitrary around integer filling.Above integer filling nearly no change in the potential inside the sample could beobserved, but relatively sharp potential steps occur near the sample edges. Thispattern occurs 1/B-periodically [158, 159, 160]. The voltage drop can be understoodif one considers the difference of the electro-chemical potential on the two sides ofthe sample and the insulating behavior of incompressible stripes, which increaseswith the width of a stripe.

From Eq. (2.22) one can deduce the additional current density from the voltagedrop. As it is discussed in chapter 2.4.6 in detail, the current density depends onthe local density of states at the local Fermi energy. From the measurements it

96

9.1. Introduction

is concluded that at integer filling factors (neglecting spin degeneracy) the currentis carried without dissipation in the interior of the sample in the incompressiblestripes. Above an integer filling an also dissipationless current is deduced to flowin the innermost incompressible stripe at the sample edge. Just below an integerfilling factor the current is evenly distributed and flows also via compressible stripes,which leads to backscattering and dissipation [37].

In samples with a fixed geometry experiments with selectively populated edgestates have shown the importance of individual impurities in edge state equilibrationand backscattering [161, 162]. An ensemble averaged model of impurity scattering istherefore inadequate in mesoscopic samples. To identify individual scattering sitesand microscopic mechanisms, scanning probe micriscopes provide a versatile tool.In the so-called ‘scanning gate’ technique a DC-voltage is applied to the metallictip of an AFM. This tip acts like a gate but on a relatively local scale. When itis scanned across the sample the quantity under consideration, usually a resistance,is recorded for every tip position. The observed structure in these scanning gateimages reveal the dependence of the signal on a change of the local potential atindividual positions (see chapter 6).

Three scanning gate experiments in the QHE-regime of a 2DEG have been re-ported in the literature. The first direct measurement of edge state coupling atindividual scattering sites was performed on a Hall bar with two metallic gatesacross. When the tip was scanned across the structure the measured equilibrationlength was strongly enhanced at single tip positions at the edges of the sample [111].The tip enhanced longitudinal resistance in some regions of the sample was inter-preted as ‘weak links’ in the local potential where the overlap of the edge state wavefunctions is larger and leads to a change in the resistance [111]. In these experimentsRxx could be lowered or raised by the tip. It is still a controversy if the number ofobserved scattering centers is high enough to explain all observed couplings.

In a similar experiment the coupling of two edge states could be influenceddirectly by the scanned tip in a more controlled way due to a more complex samplegeometry [163]. More detailed substructures could be observed indicating that alsophase coherence might be of importance in the details of the edge state coupling.

In neither of the two described experiments the 1/B periodicity expected fromthe filling of Landau levels was shown. Also an analysis of the involved scatteringmechanisms is not reported. In the most recent scanning gate experiments some ofthese questions have been addressed with measurements of the longitudinal resis-tance on a Hall bar [164]. For example the 1/B-periodic dependence of scatteringrates could be shown, though not for individual scattering centers. Edge statecoupling could be quantified and was treated in the general frame of the Landauer-Buttiker formalism for the backscattering inside the Hall bar.

The effect of edge state coupling on the Hall resistance at individual potentialfluctuations, either natural or induced by the AFM tip, has not been investigatedexperimentally up to now.

In this chapter scanning gate experiments in the QHE regime of a conventional

97


Hall bar are presented, focusing mainly on the Hall resistance. Corresponding mea-surement series at reversed magnetic fields and of the longitudinal resistance can befound in appendix D. If the two longitudinal resistances are included in the ana-lysis, as well as measurements at reversed magnetic fields, scattering configurationsat individual scattering centers can be distinguished, differing in their transmis-sion matrices. Scanning gate images are presented, showing a 1/B-periodicity. Theinfluence of the width of the edge states on the experiments will be discussed. Meso-scopic fluctuations in the magnetic field trace of the Hall resistance will be tracedback to individual scattering centers. Such experiments are of fundamental inter-est since they provide the (experimentally) missing connection between the edgechannel picture and the general view of a percolating network of localized states.

9.2 The ‘high mobility’ sample

9.2.1 Sample and setup

The parameters of the measurements presented here are summarized in tables 6.1and 6.2 as ‘experiment C’. The Hall plateaus and the Shubnikov - de Haas oscillationsare both well defined, the latter showing pronounced minima at ν = 2, 4, for example.The Shubnikov - de Haas oscillations show rather asymmetric peaks. The quantumHall plateau values are exact within the measurement precision. For these conditionsthe sample characteristics are n0 = 3.6 × 1015 m−2 and µ = 66 m2

V s(lmfp ≈ 6.6 µm),

while the scan parameters are Utip = 0.0 V and d = 120 nm. Figure 9.1(a) showsthe magnetic field traces of Rxx in red and Rxy in blue.

During the scanning gate experiments in constant height not only the voltagedrops in the sample are recorded, but also the shift of the resonance frequency ofthe AFM sensor. These images of the force gradient at the tip position can help tocontrol and observe changes in the surface topography during the constant height

(a) (b)

Figure 9.1: (a) Rxx and Rxy vs. magnetic field. (b) Zoom in the inter-vall between ν = 8 and ν = 4. The colored points show the measurementspresented in Fig. 9.3.

98

9.2. The ‘high mobility’ sample

scans. A detailed discussion of this technique is given in chapter 5. These images canbe understood as the most simple version of a scanning Kelvin probe experiment.

Such an image is shown in Fig. 9.2(b) for one example of the measurement seriespresented here. For comparison the topography is given in Fig. 9.2(a), from whichthe contours were extracted and inserted in Fig. 9.2(b). In the frequency shift imageone recognizes the shape of the Hall cross. The fact that this does not coincide withthe extracted shape from the topography scan can be due to a deformed tip shape:it has to be elongated along the y-axis, as shown schematically in Fig. 9.2(c). Thesample tilt is estimated to be smaller than 0.05 in all directions. The contour linessuperimposed in all the following images correspond to the topography images.

The most prominent feature in Fig. 9.2(b) is a circular region in the center of theHall cross where the resonance frequency is reduced by about 10 mHz, correspondingto an attractive force. This can be explained by positively charged areas on thesurface or in the donor layer or by image charges in some metallic islands. Thecomparison with the topography image shows that the reduced frequency shift isnot correlated with the particles on the surface. The small particle at the upperedge of this feature rather reduces the effect.

(a) (b)

(c)

Tip

Sample

Figure 9.2: (a) Topographic AFM image at T = 1.9 K showing the regionof the experiment. (b) Frequency shift in a constant height scan of d = 120 nmabove the Hall bar surface. The black contour lines stem from the topographyimage in Fig. (a). (c) Schematic of an asymmetric AFM tip scanning an objectwith sharp edges. The red dashed line is the resulting topographic image.

9.2.2 Description of the data

Even integer filling factors

Measurements were done between filling factor 8 and 4, at the magnetic fields indi-cated in Fig. 9.1(b) as points on the magnetoresistance curves. The measurementsare distributed equidistantly in Rxx from the minimum to the maximum of the Shub-nikov - de Haas (SdH) peak between ν = 6 and ν = 4. The idea is that the amountof backscattering is measured by Rxx which thus indicates the interesting physicstaking place. The other measurement points are chosen to be the 1/B-periodicextrapolation of the first series.

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The corresponding scanning gate images of Rxy,1, the Hall resistance measuredat the left Hall cross, are shown in Fig. 9.3. For identification, the filling factor ν isgiven below the images, which are arranged with increasing magnetic field over twoSdH oscillations. At exact filling factor ν = 4 no structure can be resolved. Thesame is true for the longitudinal resistances, though not shown here. The averagevalue of the Hall resistance does not correspond exactly to the ideal value, probablydue to the higher measurement frequency. In this image also the numbering of thecorners used in the text is given. The same observations are made for filling factorν = 2 shown and discussed below, see Fig. 9.8.

At the other even integer filling factors 6 and 8 the situation is quite similar: theobserved pattern is almost featureless. At ν = 6 it only consists of an ellipsoidal dipin Rxy of about 10 Ω at the upper right corner. It has two semiaxes of about 300nm and 750 nm, respectively, and is oriented at an angle of about 45 to the y-axis.This feature also shows up in the longitudinal resistances.

At filling factor ν = 8 a similar dip of approximately the same depth occursnear corner 2 as at ν = 6. In addition two broad but similarly weak positive peaksappear at the corners 1 and 3. They both also have ellipsoidal shape, but both withapproximately ±90 rotated semiaxes with respect to the dip in corner 2.

Quantum Hall transition: from ν = 8 to ν = 6

The development of the scanning gate images in the magnetic field interval betweenthe quantum Hall plateaus at ν = 8 and ν = 6 is now studied, starting out fromν = 8 (left oscillation in Fig. 9.3).

At ν = 7.87 a minimum develops also at the lower left corner, 4, similar to theone at corner 2. This pattern is bent around the corner given by the frequency-shiftimage, rather than around the topographic corner. The ellipsoid in the upper leftcorner is more pronounced with a maximum change of about 20 Ω. It also showssome internal structure along the longer semiaxes. The feature at corner 1 seems notto change much compared to ν = 8, though also here more structure is visible. Thesame tendencies apply to the filling factors ν = 7.81 and ν = 7.76. The maximumchanges in the Hall resistance due to the AFM tip are very similar in these scans.Substracting one from the other shows a slight growth of the minima and maximaon the way to the odd filling factor.

At ν = 7.69 the large structure in corner 3 splits into two regions which them-selves also have internal structures. Also the maximum at corner 1 is split into atleast two distinct parts. The maxima are much better differentiated and the minimaare barely visible. New structures start to evolve to the left of corner 4 and at corner2. These spots are very weak compared to the other structures in the image.

Approximately in the middle of the two plateaus, at ν = 7.52 on the top ofthe SdH oscillation, one can identify at least 6 well separated regions where Rxy isincreased by the tip. The feature near corner 4 is quite localized and shows now thelargest resistance change in the scanning gate image. Along the diagonal at corner

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Figure 9.3: Scanning gate images of the Hall resistance for two well definedquantum Hall plateaus including their transitions. The numbers at ν = 4indicate the corners, the numbers at ν = 7.87 the positions used for Fig.9.4(b) Please note the different scales for individual experiments.

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2 a new local maximum is present and at corner 3 a very sharp maximum can bediscriminated between the other structures. The feature near corner 1 is washedout more and is elongated into the lower voltage contact. This scanning gate imageshows the largest variety of features in this series.

Decreasing the filling factor further, from the middle of the quantum Hall tran-sition at ν = 7.52 to ν = 7.28, changes the scanning gate image dramatically: theimage is now dominated by the feature at corner 4, though some weak pattern re-main at the other positions and disappear continuously when lowering the fillingfactor further. Also at corner 2 some structure in Rxy is visible. At ν = 6.97 avery weak minimum at corner 3 can be discriminated. The most obvious changein the images at the high-field end of the transition is that the feature at corner 4gets more localized with decreasing filling factor until it disappears at ν = 6. Themaximum change in Rxy of this feature is essentially constant on this slope of theShubnikov - de Haas oscillation, i.e. about 100 Ω.

Transition from ν = 6 to ν = 4

Scanning gate images of the transition from ν = 6 to ν = 4 are also presented inFig. 9.3. In the interval from ν = 6 to ν = 5.75 very much the same pattern as onthe start of the previous transition is observed, demonstrating the 1/B-periodicity.Also here the maximum induced changes in Rxy are about 20 Ω, as in the transitionbefore. The main difference is that the features are better separated and sharper.The most individually discriminated features are found at ν = 5.69. This imageshows essentially the same patterns as at ν = 7.52, though it is not on the maximumof the SdH oscillation. Like in the previous transition, the structures in the lower leftand upper right corners become dominant if the filling factor is reduced further fromthis field and the patterns from the left of the transition essentially disappear. Alsohere, the change in the Hall resistance takes place on virtually the same absolutescale as at ν = 7.52, namely about 60 Ω. But the dominating feature at corner 4 isvery broad in comparison.

At filling factor ν = 5.28 the scanning gate image is very similar to the exper-iment at ν = 7.28, including weak structures in the Hall cross. The prominentstructure at corner 4 shows more details and the shape is defined more sharply. Inaddition a small peak appears to the right, at the sample corner. In the previoustransition this structure became smaller and more localized. Here, this does nothappen: at ν = 5.13 it is washed out strongly and few details can be recognized,though some very sharp dips can be discriminated. Lowering the filling factor toν = 5.0, an even stronger discrepancy to the previous transition occurs: the featureat corner 4 is very broad and barely visible, but at corner 3 a strong dip of about40 Ω appears. This dip gets even deeper and more localized at ν = 4.8. It clearlydominates the scanning gate image in this magnetic field interval. Finally, at ν = 4,all structure has disappeared again.

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Deviations from mean values between ν = 8 and ν = 4

A different view on the data can be gained by considering the average influence ofthe AFM tip on the Hall resistance in a scan. In Fig. 9.4(a) the mean values ofindividual scanning gate images are plotted versus magnetic field (blue curve). Theerrorbars (very tiny, in blue) represent the standard deviation of the data in everyscan. For comparison, also the magnetic field trace of the Hall resistance is shown inred, taken before the scan series. An overall gating effect due to the sensor platformleads to an increase of Rxy in the classical regime, as it is discussed in chapter 5.3.In the quantum regime a density change should lead to a shift of the plateaus in B.Therefore the deviations of the mean values from the original Hall curve have to beattributed to some charge rearrangements during the first scans1. The deviationsfrom the mean value is very small, of the order of a few percents. This shows thatthe electron system is not perturbed strong enough to change the overall behaviorof the sample, i.e., the introduced changes are only local or very small.

(b)(a)

Figure 9.4: (a) Hall resistance versus magnetic field gained from a sweepbefore the scanning gate measurements (red curve). The blue curve shows themean values of the scanning gate images and the (tiny) error bars show thestandard deviation for every scan. (b) Deviations from the mean value of theHall resistance in the scans. The light blue symbols stand for the resistancechanges of feature 1, indicated in Fig. 9.3, ν = 7.87, the dark blue symbols forfeature 5, the gray for the global minimum and black for the global maximum.For orientation the Hall resistance trace is given in red.

The deviations from the mean value are produced by individual features in thescanning gate images. In Fig. 9.4(b) resistance changes at several fixed tip positionsare plotted versus magnetic field. For orientation the Hall trace is given as a redcurve. The light blue curve labeled ‘feature 1’ shows the deviations for the tippositioned at the upper left corner, as indicated in Fig. 9.3, ν = 7.87. This patternis relatively weak compared to other structures. Nevertheless, it shows maxima atthe high-field ends of the QHE plateaus, i.e. at filling factors slightly below eveninteger numbers. The negative values of the light blue curve can be attributed to

1No curve is available after the scans due to a power failure

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an enhanced mean value due to the other features, e.g. the maximum at the lowerleft corner. The latter is characteristic for the scanning gate images at the low-field end of the next plateau. This is documented with the dark blue curve in Fig.9.4(b), labeled ‘feature 5’. For this curve the tip is positioned to the left of corner4, indicated in Fig. 9.3, ν = 7.87. The minimum in this curve at B ≈ 2.7 T canprobably be explained the same way as that in the light blue curve. The black curve,which represents the global maximum value in the scans, shows how the involvedfeatures get dominant at different filling factors. For example around B = 2.6 T,feature 1 becomes the dominating effect, whereas in the other parts of the transitionfeature 5 is important. The global minimum shown as gray curve is not completelygiven by the change of the average by the positive features, but has also some smallstructures on its own due to the minima discussed above.

A similar analysis is presented for the longitudinal resistance Rxx,2 in Fig. 9.5.The corresponding series of scanning gate images is presented in appendix D. Rxx,1

exhibits essentially the same behavior. In Fig. 9.5(a) the magnetic field traceof Rxx,2 is shown as red curve for the discussed interval. The blue curve showsthe mean values from the scanning gate images. The tiny error bars signify thestandard deviations in the corresponding scanning gate image, which show againthat the applied tip voltage does not change the sample characteristics strongly.The deviations from the magnetic field trace can be attributed to the fact thatthe red curve was taken before any scan was performed and charges might havebeen rearranged during the first measurements. In Fig. 9.5(b) the deviations fromthe mean value are plotted as black dashed curve for the tip positioned in point 2indicated in Fig. 9.3, ν = 7.87. This position is almost on the Hall bar itself and the

(b)(a)

Figure 9.5: (a) Longitudinal resistance versus magnetic field from a fieldsweep made before the scanning gate measurements (red curve). The bluecurve shows the mean values and the (tiny) error bars are the standard devi-ation for every scan. (b) Deviations from the mean value of the longitudinalresistance in the scans. The black dashed curve shows the deviation of Rxx,2

from the mean value at position 2, indicated in Fig. 9.3, ν = 7.87. The darkblue symbols are the global maximum and the light blue the global minimumdeviations observed in the scans. For orientation the Rxx,2 is given in red.

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curve has maxima where the magnetic field trace of Rxx,2 is largest (red). The globalmaxima in the Rxx-scans (dark blue) show two peaks, one at the high-field end of aplateau and one on the low-field end of the next plateau. The position where thesemaxima occur in the scans are at the corner of the Hall cross. The global minima,shown in light blue, have dips at the same field values as the SdH-maxima occur.

1/B periodicity at larger filling factors

In order to show the 1/B-periodicity of the patterns observed in the transitionregion between ν = 8 and ν = 6 also at higher filling factors a selection of additionalscanning gate images are presented in Fig. 9.6. At ν = 10.09 ≈ 10 the observedpattern is very weak and has a very similar structure as the corresponding imagesat ν = 8 and ν = 6 in Fig. 9.3. In contrast to the other even integer filling factors,the symmetry expected from the symmetry considerations in chapter 6.2 can beidentified. At ν = 11.31 and ν = 9.33, on the right slope of the correspondingSdH-oscillation, one recognizes the resistance changes at corner 2 and 4 and somereminiscent structures inside the Hall cross. A comparison with the images at ν =7.28 and ν = 5.28 of Fig. 9.3 clearly shows the 1/B-periodic relation. However,the strong feature at the lower left corner is shifted to the right by almost 1 µm,compared to the lower filling factors. At ν = 9.89, at the start of the quantum Halltransition, the pattern corresponds to the experiments at ν = 7.87 and ν = 5.86,though the structures are washed out more strongly.

=9.33=9.89=10.09=11.31

(a) (b) (c) (d)

Figure 9.6: Scanning gate images at lower magnetic fields for comparisonwith Fig. 9.3. (a) and (d) were taken at filling factors at the high-field end ofa quantum Hall transition and (c) at the start. (b) shows measurements onthe plateau ν ≈ 10.

Transition from ν = 4 to ν = 2

The magnetic field trace of the longitudinal resistance between filling factor ν = 4and ν = 2 shows a local minimum around ν = 3 that is usually attributed to spinsplit Landau levels, see Fig. 9.7. It is expected that the Hall resistance shows aplateau at this magnetic field. Here, instead of a plateau, an oscillation is formedthat is slightly above the theoretical plateau value, as can be seen in Fig. 9.7(a)

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(a) (b)

Figure 9.7: (a) Longitudinal- and Hall resistance versus magnetic field. (b)shows a zoom on the region between filling factor ν = 4 and ν = 2 with themeasurement points of the scanning gate experiments of Fig. 9.8.

and in the enlarged Fig. 9.7(b). In the latter the measurement points of the nowpresented scanning gate images are indicated.

In Fig. 9.8(a) scanning gate data at ν ≈ 3.78 are displayed. The image consistsof 6 to 7 very well defined maxima and looks very similar to the images at ν = 7.52and ν = 5.69 in Fig. 9.3, though this filling factor is essentially positioned halfway from the SdH minimum at ν = 4 to the SdH maximum. At the Rxx-maximumitself (ν = 3.56) only the contribution at corner 4 is visible, which is essentially the

=2.00=2.63=2.85=2.93

=3.01=3.25=3.56=3.78

(a) (b) (c) (d)

(e) (h)(f) (g)

Figure 9.8: Scanning gate measurements in the interval of spin-split Landaulevels, like they are shown as points in Fig. 9.7(a). (a) is taken at the startof the quantum Hall transition from ν = 4 to ν = 2, in the left slope of theSdH-oscillation, (b) between the ν = 4 and ν = 3 plateau, (c) between (b) andthe spin resolved SdH-minimum of (d). (e) is on the left slope of the secondspin resolved SdH-peak, for which an image is shown in (f). (g) is taken onthe right slope to the ν = 2 quantum Hall plateau.

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9.3. QPC-model of quantum Hall transition

structure observed at the high-field side of both quantum Hall transitions in Fig.9.3. This feature gets broader with increasing field, as can be seen in the scan ofν = 3.25. This looks similar to the images at around ν = 5.3 in Fig. 9.3.

At the spin-split minimum of the SdH-oscillation, at ν ≈ 3.0, two broad stripes ofresistance changes are visible in the scanning gate experiments, cf. Fig. 9.8(d). Thispattern stays essentially the same climbing the next SdH-peak at ν = 2.85 plateau.This structure can not be identified as one occurring at the previous quantum Halltransition. At best it may remind of the elongated structures entering the voltageprobes at ν = 7.52 and ν = 5.49.

At ν = 2.85, at the second SdH-peak in the transition, a dip in Rxy appears atcorner 3, cf. Fig. 9.8(f). At ν = 2.63 this is the only feature observed. This remindsone of the feature described in the previous subsection at filling factors 5.0 and 4.8.Figure 9.8(h) then shows again the featureless image of an even integer filling, likeat ν = 4.

Summary

In scanning gate experiments on a Hall cross the AFM tip influences the quantumHall effect at quite localized positions. The field-position correlation is 1/B-periodicand a dramatic change in the pattern occurs together with the SdH-maxima. Nochanges in the Hall resistance can be introduced on well defined quantum Hallplateaus.

9.3 QPC-model of quantum Hall transition

In order to obtain insight into the rich structures of the experiments one can startwith the idealized quasi one-dimensional edge states in the Landauer-Buttiker for-malism. The impact of the broader self-consistent stripe-structure will be discussedfurther below. In chapter 2 the quantization of the Hall plateaus is understoodin terms of ideal edge channels with the transmission matrix given in Eq. (2.24).The transition from one quantum Hall plateau to the next is usually interpretedas a localization-delocalization transition in a percolating network of ‘internal’ edgestates localized at potential fluctuations. In these models, the analysis of the di-vergence of the localization length at the transition show that quantum tunnelingbetween edge states at saddle points of the local potential has to be taken intoaccount in order to explain the (universal) critical exponent [146, 165]. The QHEtransition in a large system is then found as percolation of states through a networkof such saddle points everywhere in the sample. The network can then be describedby the statistics of random QPC-parameters.

In this chapter a model is developed that makes a connection between the coup-ling of edge states at a quantum point contact (QPC) and the quantum Hall tran-sition without taking ensemble averages.

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9.3.1 Transmission matrices

The Hall bar under investigation is a six-terminal device, which can be analyzedeasily in the Landauer-Buttiker formalism. Backscattering in the Hall bar is alreadydiscussed in the literature [164] and is achieved by coupling the edge states likeit is shown in Fig. 9.9. The ‘coupling constant’ δ is the change of the involvedtransmission matrices and is the only free parameter in the problem. Calculatingthe longitudinal resistance changes gives

∆Rxx,1 = ∆Rxx,2 =h

e2ν

δ

ν − δ≡ R0

xy

δ

ν − δ(9.1)

where R0xy denotes the unperturbed value of the corresponding quantum Hall plateau.

This mechanism has no effect on the Hall resistance: ∆Rxy,1 = ∆Rxy,2 = 0.

Figure 9.9: Scattering inside the Hall bar: the transmissions from contact5 to 6, T65, and from 2 to 3, T32, are reduced by δ, while the coupling of theedge states in the Hall bar is increased, i.e. T62 = T35 = δ.

For the present experiments it is enough to consider only one Hall cross, sinceno effect of the AFM tip scanned on one Hall cross could be observed in the Hallvoltage of the other Hall cross. This is not an entirely obvious result since edgestates introduce very long equilibration (up to 1 mm!) and phase coherence lengthsin very pure 2DEGs, which can lead to non-local resistances [142].

The transmission matrix of a Hall cross has 16 elements and 7 linearly inde-pendent sum rules from the boundary conditions. Therefore one needs 9 linearlyindependent basis matrices T B

k and 9 parameters β(m)k to completely describe the

scattering problem. The general transmission matrix T m for the states of Landaulevel m can then be written as

T (m) =∑

k

β(m)k T B

k (9.2)

Since the choice of the basis matrices is arbitrary they can be constructed in-tuitively by considering Fig. 9.10. Figure (a) shows schematically the edge statesin a Hall cross with backscattering in lead 1. Rotation of the Hall cross gives 3additional scattering configurations which are categorized as ‘backscattering in one

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11 HB HB

22

66

(a)

(c)

(b)

HB1

2

6

Figure 9.10: (a) Coupling ofedge states with backscattering inonly one contact does not changethe Hall resistance. (b) Coupling ofedge states with backscattering intwo leads and (c) four leads altersRxy.

lead’. Only the one with backscattering in the Hall bar (HB) has any effect on themeasured resistances: it describes the same situation as in Fig. 9.9.

Figure 9.10(b) shows a configuration that leads to Rxy,1 = he2

1ν−δ

. Thus increasingδ from 0 to 1 describes formally a transition from one Hall plateau value to the next.The change in Rxy,1 then reads

∆Rxy,1 =h

e2ν

δ

ν − δ≡ R0

xy

δ

ν − δ(9.3)

If Rxy,1 is changed, Kirchhoff’s rules require that at least one of the longitudinalresistances has to be altered as well. From the 6-terminal analysis one finds

∆Rxx,1 = R0xy

δ

ν − δand ∆Rxx,2 = 0 (9.4)

The schematic of this scattering configuration can be rotated to the other leadswhich gives additional 3 basis matrices. This category is labeled for obvious reasons‘backscattering in two leads’. There are no configurations with reflected edge statesin three contacts. The last category is ‘backscattering in four leads’ and consists ofonly one matrix. Its schematics is shown in Fig. 9.10(c). The resistance changescorresponding to these 9 scattering configurations are summarized in table 9.1 for

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positive and inversed magnetic fields. They can be described by one parameter δonly. One finds that the 9 configurations can in principle be distinguished in mea-surements by recording Rxx,1, Rxx,2 and Rxy,1 at a certain magnetic field and at itsinverse. One has to keep in mind, though, that these are only scattering config-urations concerning the transmission matrices. There are infinitely many possiblemicroscopic scattering processes that lead to the same configuration.

Nb. Refl. in leads ∆RB>0xx,1 ∆RB>0

xx,2 ∆RB>0xy,1 ∆RB<0

xx,1 ∆RB<0xx,2 ∆RB<0

xy,1

1 1 0 0 0 0 0 02 2 0 0 0 0 0 03 6 0 0 0 0 0 04 HB w w 0 w w 0

5 HB, 6 w 0 w w w 06 6, 1 0 0 0 0 w −w7 1, 2 w 0 w 0 0 08 2, HB w w 0 0 w −w

9 1, 2, 3, HB w 0 w 0 w −w

Table 9.1: Resistance changes for the 9 scattering configurations describedin the text and depicted in Fig. 9.10. The following abbreviation is used:w = R0

xyδ

ν−δ . ‘HB’ means the contact made up by the Hall bar.

Any (non-trivial) scattering configuration involves the branching or coupling ofseveral edge states. This generally happens at a saddle point like structure, orQPC, in the local potential. In edge channels the distance from the edge dependson the position of the Fermi energy relative to the QPC-potential. Figure 9.11(a)shows schematically three possibilities: two essentially uncoupled (blue), coupled(red) and deflected (green) edge states at a QPC. The first four basis matricescorresponding to the scattering configuration in Fig. 9.10(a) are achieved by asingle QPC. With adequate linear combinations among the remaining basis matricesit might be possible to find a basis that consists only of matrices corresponding toa single QPC.

In the self-consistent picture of compressible and incompressible stripes this pic-ture stays the same, even though the coupling then also depends on the structureof the states. Scattering of electrons happens in the compressible stripes at theFermi energy, if one assumes energy conservation. This adds a dependence of thescattering rates on the width of the incompressible stripe.

In random network models, networks of QPCs are connected and used to describedisorder-induced quantum phase transitions, not only in the QHE regime [165].

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9.3.2 From a given potential to the scanning gate image

Quantum Hall transition in the QPC model

In this section the quantum Hall transition and scanning gate experiments are illus-trated in a computer model. For this purpose a number of points in the Hall crossregions are chosen at random, around which the Coulomb potential of a point chargeat a distance of 17 nm is added. This donor density is chosen to ND ≈ 3× 1013m−2,much smaller than in reality, but resulting in less crowded illustrations. No screeningor other correlations are taken into account, either. A cross section through such apotential is shown in Fig. 9.11(b) and the potential itself is plotted in the inset.

A complete quantum Hall transition from the spin degenerate filling factor 2 tofilling factor 4 for this model is shown in Fig. 9.12. The edge states are plotted asequipotential lines of the potential at various levels. The direction of the edge statesis found by using the gradient of the potential.

At ν = 2 and in a sample with potential fluctuations smaller than half the cy-clotron energy, one finds a finite density of states at EF only at the sample bound-aries, as depicted in Fig. 9.12(a). While increasing the filling factor, localized statesform at isolated local potential minima, Fig. 9.12(b). They can be understood asinternal edge states that lead to a non-zero density of states and broaden the initialLandau level to Landau bands. They percolate to more extended states and finallyconnect voltage and current contacts. This leads to the new value of Rxy, which iscalculated for every figure in the Landauer-Buttiker formalism.

A close inspection shows that it is the constellation depicted in Fig. 9.12(c) wherethe transition observed in Rxy takes place. The red arrow marks the correspondingsaddle point in the local potential where the edge channels of the lower voltagecontact gets coupled to the rest of the sample. The coupling of these two states is

Figure 9.11: (a) Schematic of edge states at different energies near a QPC.(b) Cross section through the potential landscape shown in the inset. Theyellow line indicates the cross section. Also depicted are the standard deviationσ (dashed gray) of the potential fluctuations, the LL spacing ~ωc and thesample dimensions.

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Figure 9.12: A continuous increase of the filling factor from ν = 2 in (a)to ν = 4 in (f) leads to a percolating network of edge channels. The fillingfactors are indicated by the black bar at the side of the plots, the resistancesare given in units of h/e2 and the red arrows indicate the QPCs discussed inthe text.

exactly the situation described by Eq. (9.3). In contrast, the QPC formed in Fig.9.12(d) pointed out by the red arrow has no influence on the Hall resistance.

Further increasing ν leads to localized states where potential maxima pierce theFermi energy, Fig. 9.12(e), all states below are occupied. This shows intuitivelyhow the density of states is reduced again inside the sample when the filling factorapproaches the next integer number. At ν = 4 the next Landau level is completelyfilled and the only states available at the Fermi energy are the extended edge statesat the sample boundaries, 9.12(f). This results in the suppression of backscatteringand zero longitudinal resistance, but Rxy is reduced to the next even integer plateauvalue because of the additional transmission channel.

Simulating scanning gate experiments in the QPC model

The AFM-tip in a scanning gate experiment couples electrostatically to the sample.This coupling might be different in details to the classical regime, because screeninggenerally depends on the DOS at the Fermi energy which varies spatially in theQHE regime. But generally the tip modifies the local potential and with it theequipotential lines so that the paths of the edge states are changed locally.

The edge states in the same simulated potential landscape used in the previoussection are shown in Fig. 9.13 for different tip positions. The tip is modeled as a

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Figure 9.13: If the AFM tip is positioned on top of the 2DEG the localpotential landscape gets distorted and with it the equipotential lines where theedge states run along. (a) and (b) show situations where the tip introducedno change in Rxy, while in (c) the edge channel of the lower voltage contactgets disconnected from the rest of the sample. The red arrow points out thecritical QPC. In all images the approximate value of Rxy is given in the upperright corner in fractions of e2/h.

disk of large potential for simplicity. The filling factor is chosen so that the edgestates couple at the critical QPC of the quantum Hall transition in Fig. 9.12(c). Inmost positions the tip has no influence on the Hall resistance, e.g. Fig. 9.13(a), sincethe changed paths of the edge states does not lead to more coupling of the relevantstates. In Fig. 9.13(b) the distance of the important edge states is reduced, whichgives an enhanced Hall resistance. In Fig. 9.13(c) the state in the lower voltagecontact even gets disconnected from the rest of the sample, which increases Rxy tothe value of ν = 2. The result of this artificial scanning gate experiment is shownin Fig. 9.14(a).

One can take these simulations as a hint for the involved mechanisms that lead tothe experimentally observed shapes of the features: if the saddle-point like structureis narrow compared to the tip-induced potential, it behaves like a QPC with atransmission function depending only on the local Fermi energy. The tip can thenbe seen as gate so that the scanning gate image reflects the geometry and electricpotential of the AFM tip. This can be seen in the already discussed simulatedscanning gate image of Fig. 9.14(a). On the other hand, if the local potential formsa saddle point that is elongated, then the scanning gate image is dominated by thepotential landscape. An example is the artificial scanning gate image presented inFig. 9.14(b), where a potential landscape was created that leads to two critical QPCat this filling factor. They are indicated by the red arrows in Fig. 9.14(c).

The QPC model makes plausible why localized features in scanning gate exper-iments occur. The categories due to the basis matrices is applicable independentlyof the physical scattering mechanisms and it might even serve as guiding idea forfuture experiments. For example a saddle point is always created by the scannedtip and the corner of the Hall cross. Another effect is that the electron density is

113


RH

[h/e

2]

0.25

0.50(b)(a) (c)

Figure 9.14: (a) Artificial scanning gate image at the Landau level filling ofFig. 9.13(c). In this case the critical QPC is narrow so that the tip diameterdefines the width of the feature. (b) Artificial scanning gate image of a differentrandom distribution of scattering centers at its critical filling factor. Here, thecritical QPC is elongated and thus dominates the structure in the scanninggate image. The red arrows in (c) point out the involved QPCs.

smaller at the sample edges, which leads to weaker screening of the tip potential.Therefore experiments on a larger Hall bar might show more genuine effects of theQHE transition inside the sample. Smaller Halls bars, on the other hand, may begoverned by only one QPC.

The QPC model has many limitations, in practice. The edge state picture strictlyonly holds in the adiabatic limit, i.e. if mode mixing can be neglected. This criterioncan be violated for example if the Landau bands overlap, which is well possible forthe strong disorder in these samples. Nevertheless, coupling of the electron statestakes place at QPCs, but in a less intuitive manner. The tip-induced potentialshould be taken into account self-consistently at every magnetic field, starting alsowith a more realistic potential landscape. It is not obvious if this would only changedetails or if new features could be generated. For a quantitative model also thedetailed transmission functions of the individual QPCs have to be known, includingphase coherence effects and quantum tunneling.

9.3.3 The experiments and the QPC model

This section provides a discussion of the most striking observations in the scanninggate measurements shown in Fig. 9.3 on the basis of the QPC model.

• The inability to change the Hall resistance at (even) integer filling factors:all electron states are occupied and compressible stripes for scattering canbe found only at the sample edges. Scattering across the whole Hall crossis exponentially reduced. Or in one provocative sentence: there are no QPCsavailable for edge state coupling! This matches the general notion of a quantumHall state as being very robust against details in the sample geometry or thelocal potential fluctuations.

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• The largest variety of features occurs where the SdH oscillation has its maxi-mum: at this filling factor the localized bulk states start to form the percolatingnetwork and scattering is maximized. At the percolation threshold the largestnumber of QPCs is formed at the Fermi energy.

• At about the field where a SdH maximum occurs, the characteristics of thescanning gate images change abruptly: the edge states at the Fermi energycircle the minima in the first and the maxima in the second case. Since maximaand minima can not occur at the same position the scattering configurationcompletely flips.

• In QHE-transitions at high magnetic fields the image where the percolationtakes place is shifted to higher filling fractions. This effect is strongest whenspin splitting is observed in the transport measurements. The spin degeneracygets lifted and additional, non-degenerate edge states get spatially separated.This argument then naturally accounts for the obviously different couplingthat leads to the dip at ν = 2.63. Nevertheless spin splitting is not largeenough to produce independent LLs, i.e., the equilibration length is expectedto be smaller than in the degenerate case and the features get broader, as itis observed at much higher filling factors.

• Equations (6.1)-(6.3) describing symmetries in a Hall measurement are sup-posed to hold also at high magnetic fields in the regimes where individualpotential fluctuations can be neglected. This is also true if edge channels areformed. Clearly, most of the observed features do not fulfill any of these equa-tions (the measurements at inversed fields can be found either in Fig. 9.15or in appendix D, Fig. 14). It is only the pattern that occurs at, or rightbelow, even integer filling factors that approximately shows all three symme-tries. This finding can be explained by noting that at these filling factors thepotential fluctuations do not lead to an appreciable density of states at theFermi energy and no QPCs are formed. The observed coupling might even bedue to the AFM tip alone.

• The small deviations brought about by the AFM tip may indicate that therelevant scattering configuration is still the same as without the tip and onlythe transmission coefficient are changed, as it is assumed in the QPC model.

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9.4 Experimental identification of scattering con-

figurations

Table 9.1 is now taken as motivation to compare ∆Rxx,1, ∆Rxx,2 and ∆Rxy,1 atpositive and negative magnetic fields for individual features in the scanning gateexperiments. This is done exemplarily for features at filling factors ν = 8.0, ν = 7.86on the low-field end of the quantum Hall transition, at ν = 7.52 in the middle andat ν = 6.97 at the high-field end of the transition. In Fig. 9.15 scanning gateimages for these filling factors and the three relevant quantities are presented forpositive magnetic fields in the first 3 rows and for the inversed fields in the last 3rows. The negative fields were chosen to give formally the same filling factor as atthe positive fields, though this led to somewhat different positions relative to theSdH-oscillations. The discussed features are labeled by numbers from 1 to 6 in theimages. All six features show a different behavior in the analysis.

Feature 1 is the weak positive pattern observed on the quantum Hall plateauat ν = 8 at positive magnetic field. This feature is also visible in both longitudinalresistances, though it is a factor of about 2 smaller in Rxx,2 and the change there isnegative. All the features at this filling factor are even in magnetic field. Even if oneneglects the smaller resistance changes, this pattern does not fit to one of the purebasis matrices of table 9.1. Thus it has to be constructed by a linear superposition.The same behavior can be observed when this feature has grown at ν = 7.81.

In feature 2 changes take place only in the two longitudinal resistances and notin the Hall resistance. This also holds at the inversed magnetic field. This behavioris described by configuration number 4 of table 9.1, where backscattering in theHall bar takes place. The feature gets broadened in the middle of the quantum Halltransition as it is expected in the picture of edge states, since they move into thesample when one reaches the center of the bulk Landau level. This pattern is evenin magnetic field, like it is expected for the longitudinal resistance.

Feature 3 is the dip observed at ν = 8 at the opposite corner as the positivepeaks. This pattern is only observed in Rxx,2, both at positive and negative mag-netic fields. Also this feature is even in magnetic field for all involved resistances.Nevertheless, like in the positive counterpart on the other corner of the Hall cross,it is not possible to assign a single transmission matrix to this process.

Another qualitative different behavior is found in feature 4. First, it is odd inmagnetic field: at positive fields the change in Rxy is positive, whereas it is negativeat inversed fields. This might be expected for the Hall resistance, but not for Rxx,1.Second, no change can be found in Rxx,2 at any sign of the field. The behavior atpositive field fits well to the case of backscattering in the contacts 1 and 2 and thecase of backscattering in lead 6 and the Hall bar. But both cases do not show theobserved behavior at inversed field. Thus, it is a qualitatively new feature, thoughit can not be described with only one parameter in the chosen matrix basis.

The prominent feature 5 at ν = 6.97 in the lower left corner of the Hall resis-tance measurement at positive fields also appears in Rxx,1 and, by about a factor 4

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9.4. Experimental identification of scattering configurations

Figure 9.15: Scanning gate images of the Hall resistance (top row) and thetwo longitudinal resistances Rxx,1 and Rxx,2 (second and third rows), takenat the indicated filling factors. The columns are also labeled by their relativeposition to the center of the QHE-transition in a magnetic field sweep. Thecomplete measurement series of Rxx,1 at positive fields and Rxy at negativefields can be found in appendix D.

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weaker, in Rxx,2. It is much weaker at the inversed magnetic field. If one neglectsthe latter the characteristics fits very well to the transmission basis matrix 7, wherebackscattering takes place in the two contacts 1 and 2.

Feature 6 in the upper left corner is only visible at reversed magnetic fields andthe longitudinal resistance changes have opposite signs. Also here one can only statethat the involved scattering has a different origin than the previously discussed anda novel scattering configuration relevant.

In summary the identification of single scattering processes is only possible inspecial cases. Other observed changes in the resistances by the scanned gate haveto be accounted for by superpositions of the basis matrices or by a different choiceof the latter. Nevertheless the categories allow the classification of single scatteringsites. The identification of the configurations works in principle. The fact that thetransmission matrices can be produced by infinitely many microscopic configurationsmakes statements about mesoscopic details in the structure very complicated.

9.5 Experiments with larger tip voltage

Experiments with Utip = −0.7 V were performed on sample II in cooldown 1, seetables 6.1 and 6.2 for more details. In these experiments the sample tilt was notcompensated exactly. This can be seen in Fig. 9.16 where the shift in the resonancefrequency is plotted for a ‘constant height’ scan. From topography scans an angleof about 0.4 can be estimated for the gradient approximately along the voltagecontacts, which corresponds to about 50 nm height difference from the bottom of theimage to the top. Later, this tilt was corrected and some experiments repeated: thetilt does not lead to qualitatively different scanning gate images. As it is described inappendix A the sample in this cooldown had an inhomogeneous electron density. Itcould not be inferred from the magnetoresistance measurements nor from scanningKelvin probe images where in the sample the fluctuations are located.

In this section not the complete measured series or all the measured quantitiesare described. Instead, the data are used to focus on some details that have notbeen addressed or described in the previous section.

∆fre

s [mH

z]

−150

−100

−50

0

Figure 9.16: Resonance fre-quency shift in a constant heightscan at d = 120 nm. The sampleedges and contours of some par-ticles on the surface are shown asblack lines.

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9.5. Experiments with larger tip voltage

9.5.1 Description of the data

In Fig. 9.17 a selection of scanning gate images in the QHE regime of this sampleare presented. The filling factors are determined from the electron density gainedfrom the low field Hall slope. To show explicitly the 1/B-periodicity the scans areordered so that in every column the same fraction of the highest occupied Landaulevel is filled. In this measurement series the only changes in Rxy that could beintroduced by the scanned tip are found at the lower left corner (number 4) or closeto corner 2 at the upper right of the Hall cross. The measurements of column 2 and4 are indicated in the Hall resistance trace at the lower right of the figure. Column4 at filling factors ν = n + 1.1 with n = 2, 4, 6 show data taken at the low-field edgeof a quantum Hall plateau, i.e. at the end of a quantum Hall transition, whereasthe experiments at the high-field edge of a plateau can be found in column 2. Theother data lie between these filling factors.

The first column shows data at ν ≈ 7.7, ν ≈ 5.7 and ν ≈ 3.7. Resistancechanges occur in both corners, 2 and 4, though they are much stronger in corner 4,about 600 Ω for all three filling factors. At the right of corner 4, inside the lowervoltage contact, a small spot of enhanced resistance can be seen. At the B = 5.4 Tconcentric rings of constant resistance are well resolved. They are centered aroundboth corners of the Hall cross where the main changes in Rxy are observed. Therings can be observed in scanning gate images in the two magnetic field intervals4.9 − 5.7 T, 3.4 − 3.7 T and 2.5 − 2.8 T. The strength of this effect decreases inthis order very rapidly. These intervals coincide with intervals in the Hall resistancetrace where fluctuations perturb the monotonic increase of the Hall resistance.

In column 2, at filling factors 7.1, 5.1 and 3.1, the Hall resistance is alteredessentially only at the lower left corner. In the first two images the changes are ofthe same order of magnitude as in the column to its left, in the last image they areabout 900 Ω. This might be due to the fact that this measurement is not exactly atthe indicated filling factor. Since the scales are similar, the feature at the upper rightcorner really disappears and is not just much smaller than some novel structures.The feature at the lower left corner also has an increased diameter.

Column 3 shows measurements at ν ≈ 6.5, ν ≈ 4.5 and ν ≈ 2.5. Like in the firstcolumn, resistance changes are induced at both corners. They are weaker than incolumn 2, about 300 Ω, similar as in the first column. The pattern in corner 4 hasagain a larger extension and seems to have grown monotonically. The small pointto its right is now incorporated into the large structure. The feature at corner 2shows two peaks at ν ≈ 6.5, but not at the other, lower filling factors, where thisfeature is rather weak and positioned on one of the two points. The third image isagain not exactly at the indicated filling factor.

In the last column scanning gate images at ν ≈ 6.3, ν ≈ 4.3 and ν ≈ 2.3are presented. Here, resistance changes are induced only at the upper right corner(number 2) - or rather inside the upper voltage contact. The magnitude of thepattern in both scans is about 300 Ω. The features also show some internal structure.

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Figure 9.17: Scanning gate images for a series of magnetic fields. Thewhite line at B = 4.0 T indicates the cross section through the data shown inFig. 9.19(a). The white dot marks the start of the slice. The numbering ofthe corners is given in the image of B = 8 T.

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(a) (b)

Figure 9.18: (a) Mean values in the individual scanning gate experiments(blue) and the respective standard deviations (errorbars) plotted versus mag-netic field. The red curve is the field trace of the Hall resistance with the AFMtip far away. (b) Deviations from the mean values. The red curve is for theAFM tip positioned at the upper right and the blue curve for the lower leftcorner of the Hall cross. The pink curve shows the standard deviations andthe black the Hall resistance trace.

The mean values of the scanning gate images are plotted versus magnetic fieldin Fig. 9.18(a) in blue. The (small) error bars symbolize the standard deviationof all the points in a scan to the mean value. The red curve is the magnetic fieldtrace of the Hall resistance with the tip fully withdrawn. The QHE plateaus arereproduced and the observed deviations are in the range of a few percent up to20%. Figure 9.18(b) shows the deviations from the mean value in the scans plottedversus the magnetic field. The blue curve stands for the deviations if the AFM tipis positioned at the upper right corner and the red if it resides at the lower left. Oneclearly sees the correlation of the two features to the Hall curve given in black. Inthese measurements essentially only positive resistance changes are observed. Thepink curve shows the standard deviations from the scans. Because it is much smallerthan the global maxima, the latter must be quite localized.

9.5.2 Analysis and discussion

Short overview

Figure 9.17 shows the 1/B-periodic behavior of the images. The induced changes inRxy are generally larger by a factor of 2 to 4 compared to the measurement series ofthe previous section. This can be attributed to the stronger potential modulationdue to the more negative tip voltage and will be discussed further below. Anotherstriking difference is that essentially only two single features can be observed, whichdisagrees with the QPC model since one would expect more QPCs to form at theFermi energy in the sample with the lower mobility. Here, two explanations are sug-gested: first, the fluctuations vary fast on the scale of the width of an incompressiblestripe which would lead to a very sharp QHE-transition. This is not observed in

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the magnetoresistance curves. The other possibility is that the tip potential has avery strong effect on the sample, though not on the Hall resistance. This idea issupported by the fact that the changes in the longitudinal resistances are very large,typically of the order of 30-50% of the total signal. The edge states would then bebent away from the center of the Hall cross for most of the points in the scan andonly at the edges of the sample a qualitative change occurs.

The categorization of the features according to the basis matrices summarizedin table 9.1 also works for these measurements. Corresponding data at negativefields are not available for all positive fields. Nevertheless, if one assumes that thescattering configurations are the same for 1/B-periodic filling factors, the feature inthe lower left corner can be related to the situation with backscattering in contacts 1and 6. This feature even constitutes an exceptionally nice example for this process,since ∆Rxx,2 is zero within the measurement precision for most of the scans, whichwas not exactly the case in the previous series. The feature in the upper right cornerdoes not correspond to a pure basis matrix. A peculiarity not observed before is areduction in Rxx,2 at this tip position.

Overlap of edge state wave functions

Close to the nominal even integer filling factors, i.e. the data in the first and lastcolumn of Fig. 9.17, the observed features are quite localized. In between they growin lateral size. Similar observations can be made in the previous section, e.g. in Fig.9.3, where the features grow slowly when approaching the field of the SdH-maximum.

A more quantitative view is gained by plotting the feature width L versus mag-netic field. L is determined for convenience as the distance from the lower left cornerof the Hall bar to the white stripe in a scanning gate image, which corresponds tothe position of half the maximum. The measurement is done along the white line

(a) (b)

L

Sample corner

Half maximum

Figure 9.19: (a) Slice through the scanning gate data at B = 4.0 T in-dicated in Fig. 9.17. L is the distance from Hall bar corner to the positionof half the maximum. (b) L plotted versus magnetic field. As error bars thewidth of the white region was taken, about 150 nm. Below B = 3.6 T andabove 4.7 T the discussed feature is dominated by an other or has disappeared,respectively.

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in the scan at B = 4.0 T in Fig. 9.17. The procedure is illustrated in the crosssection in Fig. 9.19(a). L is plotted versus B in Fig. 9.19(b) for all scans in thisseries between B = 3.6 T and B = 4.675 T. The error bars in Fig. 9.19(b) are theapproximate width of the white stripes.

The feature size increases monotonically and from around B = 4.0 T the slopeis essentially constant, as indicated by the dashed line. The two points at low fieldsdeviate from this behavior and L might become constant. Below and above thisfield interval the feature essentially disappears. At the high field side this happensvery sharply, on the scale of about 25 mT, as can be seen in the two scanning gateimages in Fig. 9.20(a) and (b). An analysis of the characteristic length scales in alarger field interval and with another method is presented in appendix E.

(c)

(b)

B=

4.6

75

TB

=4

.70

0T

(a)

Figure 9.20: (a) and (b) Scanning gate images at B = 4.675 T and B =4.700 T, respectively. In the latter the feature at the lower left corner hasalmost disappeared. (c) Schematic of the last incompressible stripe (gray)in the interior of the sample, calculated with Eqs. (2.28) and (2.30) withthe parameters of the experiments. At the magnetic fields between the twovertical red lines the growing feature is observed.

A possible explanation is that the position of the incompressible stripes movefurther away from the sample boundaries and potential fluctuations. This is illus-trated in Fig. 9.20(c), where the outermost incompressible stripe is plotted versusmagnetic field as a gray band. The values stem from Eq. (2.28) and Eq. (2.30)with the experimental parameters inserted. The blue region is electrically isolatedfrom the white bulk, depending on the width of the innermost incompressible stripe.Scattering between neighboring compressible stripes is reduced accordingly, but alsodue to the distance between two separate states. The feature grows because somecoupled states move closer together inside the corresponding QPC and the tip hasnot to move as close to induce the same effect. The feature on the lower left seemsto be generated if the isolation of the edge and the bulk is not complete, whereasthe feature on the upper right corner is correlated to the edges being decoupledfrom the bulk. How this happens in detail can not be inferred from these measure-ments. Since the abrupt decrease of the first feature coincides with the increase ofthe second, one can presume that these scattering constellations are related.

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Mesoscopic fluctuations

Mesoscopic fluctuations at zero or low magnetic fields have been intensively studiedfor more than a decade [166]. In the QHE regime recent interest in this phenomenoncomes from the question of the microscopic origin of resonant backscattering, e.g. inthe FQHE regime [167]. Two mechanisms are debated in the literature: interferenceof electron trajectories, analogous to universal conductance fluctuations (UCFs) andbackscattering caused by individual potential fluctuations.

In this sample fluctuations in the Hall traces are observed in 1/B-periodic in-tervals of the magnetic field. They are strongest between B = 4.9 T and B = 5.7T. A magnification of the Hall curve of this interval is shown in Fig. 9.21(a). Therespective distances in magnetic field of these local maxima are ∆B ≈ 365, 260 mTand 265 mT.

At the same magnetic fields concentric rings are observed in the scanning gateexperiments. A cross section through a scan at B = 5.1 T, as shown in the inset, ispresented in Fig. 9.21(b). One can discriminate 5 maxima with increasing mutualdistance. Three examples for the development in magnetic fields are given as grayscale plots of the absolute value of the gradient of Rxy in Fig. 9.22 at the indicatedmagnetic fields. The contour of the sample are the same as in Fig. 9.17. At B = 5.0T three rings are visible at the upper right corner of the Hall cross. The structureat the diagonal corner is rather weak. At B = 5.1 T still 3 rings at corner 2 canbe discriminated. They are stronger than in the previous image and show a largerradius. At corner 4 some more structure starts to develop. In the third image, atB = 5.3 T, the radius of the rings at the upper right corner is again increased aswell as the distance between them. Now ring-like oscillations are also present atthe lower left corner with a smaller distance between the maxima than at the othercorner.

The observed rings are centered at the positions where the AFM tip inducesthe largest changes in Rxy. No oscillations can be observed in the scanning gate

(b)(a)

Figure 9.21: (a) Section of the Hall resistance vs. magnetic field curveshowing fluctuations as discussed in the text. (b) Cross section through thescanning gate image of Rxy, is which shown in the inset, for B = 5.1 T.

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Figure 9.22: Gray scale plots of the absolute value of the gradient in thescanning gate images of Rxy at the indicated magnetic fields.

experiments of Rxx that can be accredited to backscattering in the Hall cross (con-figuration 4 in table 9.1). These two points make it plausible that it is essentiallythe same scattering configuration as without the oscillations that is modified andproduces the rings. Since the rings in the scanning gate images occur only in one ortwo positions (depending on the magnetic field) and can become quite small in dia-meter, this modifications have to take place on a rather small length scale at onlyone position. Two effects are suggested here: charging of a quantum dot, whichcouples electrostatically to the sensitive scattering region, or Aharanov-Bohm (AB)oscillations among the involved edge states.

Concentric rings can originate from single electron charging in the donor layeror on the sample surface [111] or even in isolated islands of the 2DEG. Scanninggate experiments on artificial quantum dots were reported very recently [127]. Forsuch a quantum dot the AFM tip acts as a gate and changes the potential of thedot depending on the tip position. The electrostatic back-action on the edge statecoupling then alters the measured Hall resistance. In this solution the difficulty isto explain the fluctuation in the Hall trace if no tip is present.

The other suggested explanation is field dependent electron interference like inthe AB effect. For two electron wave functions that can be described locally as planewaves in one direction and are localized in the other, like in edge states, the relativephase depends on the magnetic flux Φm enclosed by their ‘paths’ γ1 and γ2:

∆φ =

∫γ1−γ2

~kd~r =

∫γ1−γ2

(~p + e ~A)d~r = 2πΦm

Φ0

+ ∆φ(B=0) (9.5)

This flux in turn depends on the enclosed area and the applied magnetic field.Interference of edge states has been demonstrated for example in quantum dots[168, 169].

The proposed mechanism for the fluctuations is that an internal edge state formsa ring, which is coupled by (tunnel-) junctions to the other edge states. This happensin a configuration that changes the Hall resistance. The area of this ring can bechanged by altering the path of the edge states, which explains the structure inthe scanning gate experiments. If the tip is far away from the sample, the area isapproximately constant (the position of the edge states is also field dependent), so

125


that the flux through the ring depends only on the applied magnetic field. Thisleads to the fluctuations in the Hall trace. With the observed ‘periodicity’ of thelocal maxima one can estimate the radius of the area as R ≈ 60− 70 nm.

In the scattering configurations discussed above no interference can occur becausephase coherence is lost in the contacts, i.e., no internal ring of edge states are formed.In order to account for the fluctuations the basis matrices describing the underlyingfeature without oscillations have to be added in a linear superposition, but with fluxdependent coefficients. An example of such a modified scattering configuration isshown in Fig. 9.23, which is a modification of the configuration discussed above inFig. 9.10(b). The red and green paths interfere at the entrance of the Hall bar andform the required internal edge state ring. The interference can be constructive ordestructive depending on the flux Φm the two paths enclose. This affects all othertransmission rates because of current conservation and makes them a function ofthe flux: δ = δ(Φm).

The experiments presented here do not allow to decide which microscopic mecha-nism leads to the observed fluctuations. Nevertheless it could be shown that scanninggate experiments are a viable tool to address questions, where mesoscopic details inthe sample are important.

Figure 9.23: Interference oftwo partial edge states shown inred and green can lead to field de-pendent transmissions δ(Φ) (theindex m is omitted for brevity).

9.5.3 Tip voltage dependence

Experiments

The nature of the features in the scanning gate images of the Hall resistance wasprobed further by examining the tip voltage dependence. An additional series ofmeasurements of the longitudinal resistance can be found in appendix D. The ex-periments were done at a fixed magnetic field of B = 4.25 T, where the main featurecan be characterized as a well defined scattering configuration.

A series of scans with Utip between −0.7 V and +1.0 V is shown in Fig. 9.24.At Utip = −0.7 V the maximum induced change in the Hall resistance at corner

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Figure 9.24: Scanning gate images taken with different tip-sample voltagesat a magnetic field of B = 4.25 T, corresponding to ν = 3.55. The influenceof the tip gets smaller when the tip voltage compensates the effective workfunction difference at about Utip = 0.7 V.

4 is around 500 Ω, the width of the feature is approximately 1 µm and the shapeis essentially circular. The reduction of the tip voltage to Utip = +0.3 V reducesboth the maximum change of Rxy and the feature size by essentially a factor of 2.The shape has changed considerably and the position of the maximum has movedslightly. At Utip = 0.6 V the size is again a factor of 2 smaller and the shape isnow circular. At Utip = 0.7 V the area where the maximum change takes place isreduced to a few pixels, but is still clearly resolved. At Utip = 0.8 V and Utip = 1.0V the feature at the lower left corner has disappeared and only some small (∼ 40 Ω)fluctuations on a scale of about 1 µm can be observed, which are also present in theprevious scans but obscured by the main feature.

The tip-sample interaction is governed by the electric potential difference. There-fore, between Utip = 0.7 V and Utip = 0.8 V the sample and the tip are at the samepotential since the influence of the tip on the Hall resistance vanished. This voltagecorresponds to the work function difference of the two materials and agrees withother experiments reported for similar structures [127].

Comparing for example the scans with Utip = 0.8 ± 0.2 V one finds that theobserved feature at the lower left is not (ansi-) symmetric in the tip voltage withrespect to Utip = Ucpd as in the classical regime.

Figure 9.25 shows the maximum induced change in Rxy plotted versus the appliedtip voltage. As reference signal the background at the lower right corner of the scansis chosen. For Utip > 0.75 V only very small changes are recorded as discussed before.Decreasing the tip voltage leads to a monotonic increase in the feature amplitude

127


Figure 9.25: Maximum signaldifference in the images of Fig.9.24 (and some more) versus theapplied tip voltage.

at the lower left corner. The curve is approximately linear between Utip = 0 andUtip = 0.75 with a slope of −435 Ω

V. Also between Utip = −0.1 V and Utip = −0.7

V the curve is linear, but with a slope of only −165 ΩV

. No further flattening of thecurve is observed at the most negative tip voltages applied in these experiments.

The width of the feature decreases with the increase of the applied tip voltage.Figure 9.26(a) shows cross sections along the Hall bar diagonal, as indicated in in ascanning gate image in the inset, for three distinct tip voltages. As a measure forthe feature radius D the distance along the Hall cross diagonal, from the Hall crosscorner to the point where the Hall resistance increases from the unperturbed valueof about 6000 Ω to 6200 Ω. D is plotted as function of Utip in Fig. 9.26(b). Theessentially zero widths at Utip > 0.6 are not shown. The two dashed lines are guidesto the eye to indicate a piece-wise linear behavior. The inset shows the characteristicfeature radius in the scans extracted as described in appendix E. The curve lookssimilar as discussed above.

Consistency with QPC model

If one assumes that the observed feature is generated by the same scattering mech-anism for all scanning gate images at different tip voltages, the experiments can beinterpreted in the QPC model with a QPC smaller than the measurement resolution.

In this model the tip acts as a gate which raises or lowers the local potentialin the QPC with respect to the Fermi energy. At small negative tip voltages thepotential is changed only weakly, the coupling of the involved edge states is alteredaccordingly and the effect on the Hall resistance is small. The effect grows with morenegative tip voltages, which increases the maximum signal strength. Similarly thewidth of the feature can be understood by noting that the electric potential dropswith the distance from the tip position. Therefore the effect on the QPC, which canbe thought of as a sensor for the electric potential, is reduced if the tip is positionedaway from the QPC. At small effective tip voltages the change in the Hall resistancecan not be resolved anymore and the feature in the scanning gate images shrinks,

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(a) (b)

D[

m]

D

U [V]tip

U =+0.3 Vtip

U =-0.1 Vtip

U =-0.7 Vtip

Figure 9.26: (a) Cross sections for scans with the indicated tip voltages.Indicated is also the feature radius D, extracted as discussed in the text. Theinset shows the line for the cross sections in a scan with Utip = 0.5 V. (b) Das function of the applied tip voltage Utip. The two dashed lines are guidesto the eye that show the piece wise linear behavior. The inset shows thecharacteristic feature radius extracted as described in appendix E.

though in principle, neglecting screening effects, the shape, but not the amplitude,would still be the same.

It is not known a priori whether a closed or an open QPC leads to an enhancedHall resistance for an observed scattering process, since its microscopic details arenot accessible. On the other hand, a more negative tip voltage always closes a QPC.In the experiments presented here this leads to an increase in the Hall voltage. Fromthe onset of induced resistance changes one can therefore deduce that the involvedQPC is completely open at Utip > 0.75 V. Lowering the tip voltage closes the QPCwhen the saddle point energy drops below the Fermi energy. This accounts for theincreasing resistance at lower tip voltages.

An even more negative voltage should close the QPC completely, leading to asaturation in the peak values of the Hall voltage. For spin degenerate Landau levelsand a filling factor like in the experiments, the maximum change would be

∆Rxy,max =h

e2

(1

bνc− 1

2 + bνc

)= 2.15 kΩ, (9.6)

corresponding to the difference between the two quantum Hall plateaus. bνc denotesthe rounded off integer of ν. This value is not observed in the experiment. In theQPC model two reasons can explain this finding: either the applied voltage is notnegative enough to close the QPC completely, or more QPCs are involved in thecreation of this scattering configuration. The latter possibility would neverthelessbe expected to flatten out at more negative tip voltages. This question has to beinvestigated in future experiments.

The kinks observed in the width of the feature and in its height occur aroundUtip ≈ 0 V. A possible explanation may be found in the concrete form of the QPCpotential or in the screening behavior of the edge states.

129


9.5.4 Comments on the measurements

In this paragraph some comments on the measurements of sample II, cooldown 1are presented. Sample changes due to the scanning experiments were very large atthe start of the series. The sample then seemed to stabilize after some ten scans andtest scans after about 70 scans (∼ 1 month) show the same patterns as in imagestaken before.

The electron transport takes place in the linear response regime. To prove thisstatement the same experiments were repeated with a measurement frequency off = 17 Hz, instead of f = 681 Hz as used above, with adjusted integration andmeasurement time. They show the same features, though the resolution is a littlebetter in the slow scans. Another parameter is the applied current. In experimentswith I = 10 nA and I = 100 nA with otherwise equal experimental conditionsthe same changes in the resistance are observed. Only the signal to noise ratio isincreased for the larger applied current.

Figure 9.27: (a) Scanning gate experiment at B = 4.25 T with the ‘usual’measurement setup. (b) Scanning gate experiment with the same parametersbut current and voltage leads interchanged. The resistance scales in the twoimages is the same within 20 Ω.

That the observed patterns originate from mesoscopic details in the sample isshown in a very direct experiment in Fig. 9.27. Figure (a) shows data taken with thesetup connected as in the previous sections. Inversion of the current direction andthe measurement contacts leads to exactly the same scanning gate image, as it isexpected from the Buttiker formula. An example is shown in appendix D. But if thecurrent is fed through the former voltage contacts and the Hall voltage is measuredat the former current leads, then the scanning gate image looks completely different,as can be seen in Fig. 9.27(b). This experiment does not violate the global Onsagerrelations, because the magnetic field is not inverted.

130

9.6. From classical to quantum transport

In the picture of edge states without mesoscopic fluctuations in the local poten-tial similar scanning gate images would be expected. Within the QPC model theexperiment of Fig. 9.27 is easy to understand, since in the rotated setup the sameedge channel configuration leads to the same transmission coefficients, but with per-muted indices. This in turn corresponds to different scattering configurations andleads to a scanning gate image seemingly uncorrelated to the original.

9.6 From classical to quantum transport

Description of the data

In chapter 8 it was shown that between B = 250 mT and B = 375 mT the featuresof the scanning gate images start to get localized at the lower left corner (corner 4)when the magnetic field is increased further. This process proceeds also at highermagnetic fields as can be seen in the images at intermediate fields in Fig. 9.28(a)-(f).At B = 0.5 T the feature still spreads well into the Hall cross, whereas at B = 0.75 Tit only appears in the lower part of the sample, though small resistance changes arestill visible in the complete Hall cross. This is clearly different at B = 1.0 T wherethe two sides of the Hall bar axis get visibly disconnected and the Hall resistance cannot be changed anymore in the center of the Hall cross. At magnetic fields aboveB = 1.5 T the separation gets even stronger and at B = 2.3 T the Hall resistancecan be changed only at the two corners 2 and 4 and in sharply defined and localizedregions.

Figures 9.28(g)-(j) show some corresponding scanning gate images at negativemagnetic fields, where essentially the same description holds, except that the featuregets localized at corner 3, instead of corner 4. Also the shape of the features maylook different in detail.

Edge states at low magnetic fields

From the discussion of the high field data above one can infer that the scatteringprocess that takes place at corner 4 is dominant when the filling factor is betweentwo even integer numbers. In a idealized sample without fluctuations in the localpotential this is the regime where the scattering between the sample edges is largest.

Decreasing the magnetic field in such an ideal sample towards the classicalregime, the Landau bands start to overlap strongly (~ωc gets smaller) which in-creases scattering between individual edge states. At the same time the width ofthe innermost incompressible stripe decreases and the stripe moves to the samplecenter. This is depicted in Fig. 9.28(k) for the experimental sample parameters.The red curve shows the minimum distance of the innermost incompressible stripeto the sample boundary. In the classical regime the overlap of the Landau bands isso strong that the density of states becomes homogeneous and essentially the sameas at B = 0. In this situation the classical, local Drude model can be applied for

131


Figure 9.28: (a)-(j) Scanning gate images for Rxy at the indicated moderatemagnetic fields. Positive fields are found in the first 2 rows, some correspondingmeasurements at negative fields are shown below. (k) Plot of the position andwidth of the innermost incompressible stripe versus magnetic field, after Eqs.(2.28) and (2.30). The full red curve shows the minimum distance and thedashed red vertical line indicates the magnetic field where the latter curvereaches the middle of the sample.

132

9.7. Summary

description. The field where in the ideal sample the DOS becomes finite in thecomplete sample is not at B ≈ 100 mT as expected from the estimates for an ideal2DEG, but already at 300 mT due to the finite size of the sample, cf. Fig. 9.28(k).

In a real sample potential fluctuations even enhance the equilibration of the edgechannels and provide a finite DOS in the sample center. Therefore the argumenta-tion with the ideal sample can still be applied qualitatively. At intermediate fieldsthe scattering between edge states may be similar as between two even integer fill-ing factors at high fields and the scattering mechanism in the lower left corner isdominant.

In summary the experiments show a decoupling of the sample edges in an in-creasing magnetic field in the sense that in scanning gate experiments the bulk ofthe sample is not influenced anymore. The observed pattern get sharper and morelocalized. These findings can be explained by the formation of edge states that getlocalized at the sample edges and at local potential fluctuations. Therefore scatter-ing becomes more sensitive to the mesoscopic details of the real sample.

9.7 Summary

In summary experiments in the quantum Hall regime are presented in this chapter ina variety of experimental conditions. Locally enhanced Hall resistances are observedsystematically and described. At integer filling factors no resistance changes can beintroduced into the system, which is attributed to the formation of ideal quantumHall states with the sample edges isolated from each other. Changing the fillingfactor leads to scanning gate images that show different features at the high and lowfield sides of the QHE transition. This is observed in a 1/B-periodic manner. Thisbehavior is explained in the model of edge states and their coupling at saddle point-like structures in the local potential (QPC-model). This model is able to explainqualitatively many of the observed features and gives the possibility to categorizethe observed scattering centers. In a different cooldown more experiments have beenperformed that may show the growth of the compressible and incompressible stripesin magnetic field. The observation of fluctuations in the magneto resistance curvescould be related to features in the scanning gate images at singular positions inthe Hall cross. Further insight into the structure of the observed patterns could begained by experiments with variable tip voltage. The results are consistent with thepicture of the QPC model. Some aspects, for example a full QHE transition seen inthe Hall resistance and induced by the AFM tip at only one location, have to be leftto future experiments. The transition from the classical regime to the QHE regimewas investigated in the last section, showing the decoupling of the sample edges inreal space when the magnetic field is increased.

133

Chapter 10

Summary and outlook

In this thesis the classical and the quantum Hall effect are investigated in experi-ments performed with the home-built low-temperature scanning probe microscope(SPM) that also works at high magnetic fields. Specific details of the design andthe sensors are discussed.

In local Kelvin probe experiments and resistance measurements the distancedependence of the contact potential difference is accounted for by spatially inhomo-geneous sample characteristics, like donor concentration or surface charging. Theexperiments demonstrate that such ‘details’ can influence the measurements dra-matically.

Scanning gate experiments on Hall bars are presented focusing on the Hall resis-tance. At zero magnetic field a distinct pattern is found and reproduced in varioussamples. It exhibits the expected symmetries of a Hall measurement in a diffusivesample and is interpreted as deflection of the current density by the AFM-tip-inducedpotential. In the ballistic transport regime the symmetries are violated by additionalstructures attributed to mesoscopic effects in the Hall cross.

At finite, but non-quantizing magnetic fields, an additional structure is observedin the center of the Hall bar, which is ascribed to finite gradients in the electron den-sity below the AFM tip. This feature becomes dominant when the field is increasedand eventually shifts to a corner of the Hall cross, which violates the expected sym-metries.

In the quantum Hall regime local centers of increased Hall resistance are observed1/B-periodically in the scanning gate images. The presented experiments show forthe first time a continuous development of scanning gate features in magnetic fields.At even integer filling factors essentially no changes in the Hall resistance can beintroduced by the AFM tip. Between the quantum Hall plateaus, a rich structuredevelops, which changes its appearance completely at fields near the peaks of theShubnikov - de Haas oscillations. An interpretation is given in the picture of per-colating edge states, where the coupling of counterpropagating states takes place atsaddle points in the local electron potential. The saddle points can be influencedlocally by the potential induced by the AFM tip, which leads to a change in the

135

Chapter 10. Summary and outlook

resistance. The development of such features in an increasing magnetic field canbe interpreted on the same basis. Fluctuations observed in the magnetoresistanceappear at magnetic fields where scanning gate images show a series of rings, centeredat one or two individual positions. Two interpretations are suggested, based on theCoulomb blockade effect and the Aharonov-Bohm effect, respectively. Both rely onmesoscopic details of the sample.

Future measurements should concentrate on pinning down the quantum Halltransition at a single saddle point in the local potential, e.g. in a high-mobility sam-ple. The invasiveness of the sensor can still be reduced, for example by shielded AFMtips. Edge state coupling at an artificial QPC should be studied for comparison.

Conductance fluctuations in any magnetic field intervals can be tracked locallyby the scanning gate technique. Larger effects with less scatterers can be expected,for example, in quantum wires. On the other hand, effects of the sample boundariesare expected to be less important in samples much larger than the width of thetip-induced potential. For the latter case the resistance changes at non-quantizingmagnetic fields should be smaller, whereas in the QHE regime similar values shouldbe measured as observed in this thesis. The break-down of the quantum Hall effectat high current densities should lead to a qualitative change in the observed featuresat fixed magnetic fields and could give further insight into the involved mechanisms.Also voltage noise measurements could be combined with scanning gate experiments.As an example of scanning gate experiments on more complicated structures, onecould think of local fine-tuning of the coupling in a double-dot system.

The new sensor design and the experimental setup allow to mount other, moreelaborate sensors, like a single-electron transistor. The latter can be used for less-invasive measurements of the local electric potential. An interesting goal is theobservation of transmission resonances in a QPC or a quantum wire with non-adiabatic openings. Experiments at lower temperatures would allow to investigatephase-coherence effects in various semiconductor nanostructures.

136

Appendices

Appendix A:

Inhomogeneous samples

During the local spectroscopy and scanning experiments on sample I and sample II,cooldown 1, the longitudinal and Hall resistance traces in magnetic field changedfrom the usual pattern, as shows for example for sample II, cooldown 2 in the text,to the ones shown in Figs. 1(a) and (b), respectively. Fits with a two-subband modelcould not reproduce the characteristic curvatures of Rxx at zero field together withthe high-field flattening. Neither could the characteristic change in the Hall slopebe observed. A model with two parallel resistors with different electron densityand mobility provide additional parameters from the geometry for fitting. Thus,a possible explanation of the observed behavior is that the sample characteristicsbecome inhomogeneous when large local electric fields are used in the experiments.In cooldown 2 of sample II, the tip voltage was better adjusted (Utip ≈ 0.0 V) andthese effects were not observed. The corresponding curves are shown in the text.

Figure 1: Hall (blue) and longitudinal resistance traces (red) in magneticfields for (a) sample I and (b) sample II, cooldown 1.

137

Appendix B:

Finite Element Method (FEM)

The aim of a FEM calculation is to find the solution Φ of a differential equation(DEQ), e.g. of Eq. (2.13):

~∇ · (σ · ~∇Φ) = 0, (1)

which has no source terms, for simplicity. Together with the boundary conditionsthis is known as the ‘strong form’ of a DEQ. Multiplication with a weighting functionw, integration over the volume V and inserting the boundary conditions leads tothe ‘weak form’ of the DEQ, equivalent to the strong form:∫

V

σ(~∇w)(~∇Φ)dV = 0 (2)

In the Galerkin method w ≡ Φ is chosen:∫V

σ|~∇Φ|2dV = 0. (3)

An approximate solution is found by developing the solution Φ in a sum of ‘shapefunctions’ fi with coefficients φi:

Φ(~r) =∑

i

φifi(~r), (4)

which is then inserted into the DEQ. In a one-dimensional problem, this reads

0 =∑i,j

[∫V

σdfi(~r)

dx

dfj(~r)

dxdV

]· φiφj ≡ ~φT C~φ (5)

with Cij = [...], the integral in the square brackets. The goal is to find the coefficients~φ. In this case, the solution of C~φ = 0 is also a solution to the original problem.

The dimensions of Cij become very large for complex problems. In FEM, basisfunctions fi are used that are defined only on a ‘mesh’ that consists of line ‘elements’Ei, connected at ‘nodes’ Ni. This is shown in Fig. 2(a) for a one-dimensional intervaland in Fig. 2(b) for the two-dimensional geometry of a Hall cross, as used in thisthesis. Nodes and elements are distinguished by a ‘global’ numbering. The simplestbasis functions are linear on one element as shown on Fig. 2(a). If one uses a ‘local’numbering for the nodes the local matrix CEi

αβ for element Ei has the form

CEiαβ =

σi

`i

(1 −1−1 1

)(6)

where `i is the length of element i and σ is assumed to be a scalar for simplicity.Functions centered on other nodes are zero on this element.

138

Figure 2: (a) One dimensional mesh(black points and lines) with linear ba-sis functions. (b) Two-dimensional mesh(gray) for a Hall bar geometry. The bluelines signify the boundary where the currententers and leaves the structure, no currentflows orthogonal to the red boundaries.

Crucial for the power of FEM is (1) in order to construct the global matrix C oneonly has to go through the elements and add the local CEi

αβ with the global numberingof the nodes and some geometry factors of the mesh. This process allows nodes atany desired positions. Second, since only neighboring elements are connected by anon-zero basis function, the resulting global matrix contains almost only zeros andsparse matrix techniques can be applied.

The solution of the problem is the same as with any basis functions: first, onebrings all terms known from the boundary conditions to the right side of the equa-

tion, C ~Φ = ~b, where ~Φ contains all approximate values on the ‘inner’ nodes. Second,these values can be gained by inversion of C.

Setting up the geometry, the boundary conditions and the DEQ, meshing, solvingand some post-processing is implemented, for example, in the commercial softwareFEMLab, which was used in this thesis.

139

Appendix C:

CLTF experiments

The AFM sensors based on tuning forks (TFs) with a cantilever mounted betweenthe two prongs are introduced in chapter 4.3.3. The experiments presented herewere done with Al-coted tip and on a 6 µm wide Hall bar defined on a Ga[Al]Asheterostructure with a Er+ back-gate (BG) 1.3µm from the 2DEG [170]. Scanningthe topography with separate records of the error signals of the PLL and the ampli-tude feedback are possible both in tapping mode and in non-contact mode, at lowtemperatures and high magnetic fields (not shown). A selection of measurementsis now presented in order to demonstrate that the sensors are suitable for electrontransport experiments in 2DEGs under these conditions.

Kelvin probe and Force-distance experiments

Figure 3(a) shows Kelvin parabolas as discussed in chapter 5.2 at a series of tip-sample distances z. These experiments are only possible for a working metallic tip.Back- and forth sweeps give essentially identical curves. In the inset the error signalof the amplitude feedback is shown for such experiments going to lower tip voltageand for a series of mechanical amplitudes. One finds that dissipation sets in at morepositive applied voltages for lower excitations. The onset of dissipation itself maybe due to a ‘snap in’ of the cantilever to the surface.

Figure 3: (a) Kelvin parabola observed with the CLTF sensor at a seriesof tip-sample distances. The inset shows the amplitude feed-back error signaland shows the onset of dissipation at more negative tip voltages for smallermechanical amplitudes. (b) ∆fres vs. distance curves for the indicated drivingvoltages on the resonance frequency.

A series of frequency-shift versus tip-sample distance curves is presented in Fig.3(b) for various driving voltages. The shift of the curves in z-direction allows forthe calibration of the mechanical amplitude of the sensor. Here, dAmech/dUTF ≈ 10nm/mV is found.

140

Local Capacitance Measurements

Two capacitances are of interest in these specific experiments: the capacitance be-tween 2DEG and tip, C2DEG−tip, and between the back gate (BG) and the tip,CBG−tip, repectively. The measurements are done by applying an AC voltage with

f2DEG = 7.33 kHz and U2DEG = 50 mV to the 2DEG, and fBG = 11.31 kHz andUBG = 50 mV to the back gate. The setup is shown schematically in Fig. 4(c). Theinduced current on the tip is measured by an IU-converter and two lock-ins. Theout-of phase signal is proportional to the respective capacitances.

As IU-converter serves a GaAs FET with the gate attached to the tip. Thesource-drain current is then measured in essentially a conventional feedback loopwith a cold feed-back resistor and a cold capacitor for filtering. These three electroniccomponents are mounted directly on the sensor plate, as shown in Fig. 4.4(b) and(c) in chapter 4.3.

620

600

T [K]

1.84 K

5.84 K

580

560

C2DEG-tip

CBG-tip

~

2DEG

AFM tip

BG

I-U

~

(a)

(c)

(b)

Figure 4: (a) Tip-2DEG and tip-BG capacitance as function of themagnetic field. The gray curveshows the longitudinal resistance. (b)CBG−tip in field sweeps at varioustemperatures. Inset: peak maximavs. T . (c) Schematic of the setup.

The results are plotted in Fig.4(a) as function of the applied magnetic field. Forreference also the longitudinal resistance is plotted in gray. C2DEG−tip exhibits adeep dip at filling factor ν = 4 and much smaller at ν = 6 and ν = 8. At the samefields CBG−tip exhibits a large maximum. The smaller oscillations, on the otherhand, show maxima at non-integer filling factors.

The findings can be understood in principle by considering the finite capacitanceof the 2DEG arising due to the low DOS. The capacitace C between a gate and the

141

2DEG can be written as [171]

1

C=

1

C0

− 1

e2D0

+1

e2D(EF )(7)

with the respective capacitance C0 and the density of states D0 at zero magneticfield and D at the given magnetic field at the Fermi energy. Conversely, the reducedDOS at even integer filling leads to reduced screening of the BG and the currentinduced by UBG on the tip is enhanced. The small oscillations in CBG−tip may bedue to a capacitive coupling of the BG to the 2DEG, which again couples to thesensor.

The CLTF-sensor also works at various temperatures, as can be seen in a mag-netic field series of measurements on CBG−tip. The inset shows how the featureheight is reduced with increasing temperature (up and down sweeps show a slightoffset in B which leads to the shifted centers of the curves).

Driving the TF with the back gate and the 2DEG

The motion of the tuning fork (TF) is transfered to the cantilever to excited theoscillation. The back-action on the TF can be seen in direct experiment: by applyingan AC voltage to the BG one can excite the cantilever positioned close (d = 25 nm)to the surface of the heterostructure. The back action also excites the TF and apiezo current ITF can be measured. Frequency sweeps with various BG-voltages areshown in Fig. 5(a). The inset is a schematic of the setup.

Figure 5: (a) Piezoelectric current ITF from the tuning fork (TF) whenexcited in frequency sweep with back gate (BG) voltages ranging from UBG = 5mV to UBG = 45 mV in equidistant steps and locked on resonance. The graycurve is the resonance curve in a direct measurement. The inset shows theschematic setup. (b) Shift in the resonance frequency versus magnetic fieldwhen the TF is excited by the back gate. The inset shows the original data,the main figure the data with a parabolic fit substracted. For orientation, alsoRxx is plotted.

142

Figure 6: (a) Schematic setup for the experiments. (b) Piezo current ampli-tude (red) from the TF when excited by the 2DEG (U2DEG = 50mV , lockedon resonance). In blue the frequency shift is given, corrected by a parabolicfit. Rxx is plotted in gray for orientation.

The signal is strong enough to run a PLL and record the frequency shift, forexample in a sweep of the magnetic field, as shown in Fig. 5(b). For referencealso Rxx of the sample is plotted. The inset shows the original data, from which aparabolic fit was substracted. Like in the capacitance measurement a peak arises ateven integer filling factors due to the reduced screening capabilities of the 2DEG.

A similar experiment, but with a voltage applied to the 2DEG to drive theCLTF is presented in Fig. 6. Figure (a) shows the schematic setup and figure (b)the frequency shift ∆fres (blue, corrected by a parabolic fit) and the amplitude ofthe TF piezo-current ITF (red) as function of the magnetic field. As reference alsoRxx is given without a scale.

Scanning gate experiments at high magnetic fields

With the TF sensors scanning in tapping mode and in non-contact mode is possible.Figures 7(a) and (b) show scanning gate images in the less invasive non-contact modewith a CLTF sensor and a PLL. Presented are Rxx at filling factors ν = 6 and ν = 4and Rxy at ν = 6 for scans on a cross of a 4µm wide Hall bar.

Figure 7: Scanning gate images of Rxx at filling factors ν = 6 and ν = 4and of Rxy at ν = 6.

143

Appendix D:

Additional measurements

Kelvin probe experiments in magnetic fields

On sample II, cooldown 1 (cf. tables 6.1 and 6.2), Kelvin parabolas were recordedas a function of the magnetic field in a fixed tip-sample distance d = 25 nm. Thecorresponding fit parameters for a parabolic in the intervall −1 V ≤ Utip ≤ 1 V areplotted in Fig. 8.

Figure 8: Parameters ofthe fitted Kelvin parabola ver-sus magnetic field. The tip-sample distance is d ≈ 25 nmand the amplitude of the sen-sor current is kept constant.

Tip voltage dependence of the ‘butterfly’

In experiments ‘D’ (cf. tables 6.1 and 6.2) the dependence of the scanning gateimages of Rxy on the applied tip voltage was investigated at zero magnetic field.Few images are shown in Fig. 9. At Utip = −0.5 V the familiar ‘butterfly’ can berecognized, whereas at higher voltages the features get weaker by almost a factorof 2. Also the shape of the maxima and minima change and get elongated andthe saddlepoint in the middle of the structure gets broader and the minima getconnected. The mechanism of this behavior is not entirely clear.

(a)U =-0.5 Vtip

(b)U =0.1 Vtip

(c)

U =0.7 Vtip

Figure 9: ‘Butterfly’ pattern in scanning gate images with different tip-sample voltages, as indicated.

144

Tip-voltage dependence of the longitudinal resistance in theQHE regime

In this section scanning gate images recording the longitudinal resistance in a Hallbar as function of the applied tip voltage are presented. Figure 10(a)-(e) shows aseries of scans for Rxx,2 at the indicated tip voltages and at B = 4.25 T for sampleII, cooldown 1, cf. tables 6.1 and 6.2.

Most individual features are observed at Utip = 0.1 V and Utip = 0.4 V. In Fig.10(f) a contour plot of the data in figure (c) with exponentially spaced contour linesis shown. About 23 local maxima can be discriminated, more concentrated andsteeper at the lower edge of the Hall bar. The distance between single features arein the range of a few 10 nm.

Figure 10: Scanning gate measurement of Rxx,2 for various tip voltages andB = 4.25 T. The unperturbed values, for example in the upper left corner ofthe images, are at a value of about 5600 Ω in all scans. Figure f) shows acontour plot of the data in figure c) with exponentially spaced contour lines.

145

Inversed current direction

Figure 11: (a) The same experiment like in Fig. 9.27 in chapter 9.5.4 and(b) with inversed current direction.

Bend resistance

Figure 12 shows scanning gate images of the bend resistance at the indicated mag-netic fields, measured as indicated in the inset of the last plot. In the latter amagnetic field trace is shown for the measured quantity. The data stem from sam-ple II, cooldown 1, with characteristics compiled in tables 6.1 and 6.2.

3.4 T

4.2 T

B=0

4.8 T

3.8 T

J

Ub

Figure 12: Scanning gate experiments of the bend resistance at the indi-cated magnetic fields. For orientation the last figure shows a magnetic fieldtrace and a schematic of the setup.

146

QHE regime: longitudinal resistance

Figure 13: The same experiment shown in Fig. 9.3, chapter 9.2, but forRxx,2.

147

QHE regime: Hall resistance, inversed magnetic fields

Figure 14: Similar experiments as in chapter 9.2, Fig. 9.3, but at reversedmagnetic fields. Please note the negative signs of the resistance scales.

148

Appendix E:

Details

Extracting length scales

An automated routine was written to extract characteristic length scales for examplein scanning gate images. For this purpose the data (z-axis in the images) are scaledto the interval between 0 and 1. The number of connected features, N , with valueslarger than a given height h ∈ [0, 1], and the respective number of pixels constitutingthe features can be extracted. These values are compared to the total number ofpixels, which leads to an average area of the features in a scan. Assuming circularshape this leads to the characteristic diameter d of the features. The inset of Fig.15(a) shows two such slices at h = 25% and h = 75%, repectively for an examplescan with a small and a larger feature.

(a) (b)

Figure 15: (a) Characteristic feature diameter d in the scans of chapter 9.5as function of magnetic field B and the height h. On top a slice at h = 75%is shown. The insets show two examples at B = 4.25 T and h = 25% andh = 75%, respectively. (b) Plot of the number of features in the same scans asin (a) with a slice at h = 75% on top. The white vertical lines indicate eveninteger fillingfactors.

d and N are plotted in Fig. 15(a) and (b), respectively, as function of magneticfield and of h for the data described in chapter 9.5. For small h very large featuresdominate and only at h > 50% the behavior of single objects gets visible. Slices ath = 75% (yellow horizontal line) are plotted on top of the graphs.

149

Current in an ideal two-dimensional wave guide

To calculate the current carried by a single mode connecting two electrical contactsL and R at fixed electrochemical potentials µL and µR one finds for every mode thefollowing relation:

Jtot = JL − JR =

∫ ∞

0

env(k) ·D(k)[fL(E(k))− fR(E(k))

]dk (8)

=

∫ ∞

0

e1

L

1

~∂E

∂k· L

2π

[fL(E(k))− fR(E(k))

]dk (9)

=e

h

∫ ∞

0

[fL(E(k))− fR(E(k))

]dE (10)

T→0→ e

h

∫ µL

µR

dE =e

h(µL − µR) =

e2

hU (11)

fL(E) and fR(E) are the electron distributions for the particles coming fromthe left and right, respectively:

Classical and quantum transport

It makes only sense to speak of a constant, averaged DC-conductivity tensor σ ifthe fluctuations in ~E(~r) and n(~r) do not introduce new physical effects, i.e. if thesample can be considered homogeneous on the length scales of interest. This ispossible if many modes of the conductor are occupied and are mixed strongly. Thediagonal elements of σ, i.e. the longitudinal conductivity, can be written, at leastfor kF `el >> 1, in the intuitive form [14]

σxx ∝∑k,k′

~k~k′|A(~k → ~k′)|2 (12)

with A(~k → ~k′) = i~〈~k|GRE|~k′〉 and GR

E the retarded Greensfunction at energy

E. A(~k → ~k′) is the quantum mechanical amplitude for an electron in state |~k〉 to

be scattered into state |~k′〉. The longitudinal conductivity is therefore essentially

the average of the quantity ~k · ~k′, weighted by A. Averaging A before squaring it,which corresponds to neglecting phase coherence effects, leads to the classical Drudemodel [14] with its characteristic importance of small angle scattering, since ~k · ~k′vanishes for larger angles. |~k · ~k′| is also significant for exactly time reversed states,~k′ = −~k, for which A is not averaged to zero if phase coherence is taken into account.This term gives rise, for example, to the negative magneto resistance in the weaklocalization regime.

150

Publications

• A. Baumgartner, T. Ihn, and K. Ensslin, ‘Lateral imaging of the filling factordependent scattering of quantum Hall edge states’, in preparation

• A. Baumgartner, T. Ihn, and K. Ensslin, ‘Hall resistance in scanned gateexperiments at classical magnetic fields’, in preparation

• T. Akiyama, A. Baumgartner, K. Suter, T. Ihn, K. Ensslin, and U. Staufer,‘Cantilever tuning fork sensors for low-temperature and high magnetic fieldscanning probe experiments’, in preparation

• A. Baumgartner, T. Ihn, K. Ensslin, K. Maranowski and A.C. Gossard, ‘LocalInvestigation of the Classical and Quantum Hall effect’, proceedings of ICPS27, accepted

• K. Suter, T. Akiyama, N.F. de Rooij, A. Baumgartner, T. Ihn, K. Ensslin, andU. Staufer, Tuning Fork AFM with Conductive Cantilever , AIP Conf. Proc.696(1), 227 (2003)

• T. Ihn, T. Vancura, A. Baumgartner, P. Studerus, and K. Ensslin, ‘Operatinga phase-locked loop controlling a high-Q tuning fork sensor for scanning forcemicroscopy’, cond-mat/0112415

151

Bibliography

[1] E. A. Abbott, Flatland: a romance of many dimensions (Princeton ScienceLibrary, 1991), reprint 6. ed.

[2] E. Hall, American Journal of Mathematics 2, 287 (1879).

[3] K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).

[4] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559(1982).

[5] G. Binnig, H. Rohrer, C. Gerber, and E. Weibel, Appl. Phys. Lett. 40, 178(1982).

[6] G. Binnig, C. F. Quate, and C. Gerber, Phys. Rev. Lett. 56, 930 (1986).

[7] K. Barnham and D. Vvedensky, eds., Low dimensional seminconductor struc-tures. Fundamentals and device applications. (Cambridge University Press,2001).

[8] P. Y. Yu and M. Cardona, Fundamentals od Semiconductors (Springer, 1996).

[9] T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982).

[10] A. Fuhrer, A. Dorn, S. Luscher, T. Heinzel, K. Ensslin, W. Wegscheider, andM. Bichler, Superlattice Microstruct. 31, 19 (2002).

[11] M. Sigrist, A. Fuhrer, T. Ihn, K. Ensslin, D. C. Driscoll, and A. C. Gossard,Appl. Phys. Lett. 85, 3558 (2004).

[12] M. Buttiker, IBM J. Res. Develop. 32, 317 (1988).

[13] M. Buttiker, Phys. Rev. Lett. 57, 1761 (1986).

[14] T. Dittrich, P. Hanggi, G.-L. Ingold, B. Kramer, G. Schon, and W. Zwerger,Quantum transport and dissipation (Wiley-VCH, 1998).

[15] L. Onsager, Phys. Rev. 38, 2265 (1931).

152

[16] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P.Kouwenhoven, D. van der Marel, and C. Foxon, Phys. Rev. Lett. 60, 848(1988).

[17] D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F.Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie, and G. A. C. Jones, J. Phys.C 21, L209 (1988).

[18] H. van Houten, C. W. J. Beenakker, P. H. M. van Loosdrecht, T. J. Thornton,H. Ahmed, M. Pepper, C. T. Foxon, and J. J. Harris, Phys. Rev. B 37, 8534(1988).

[19] C. W. J. Beenakker and H. van Houten, Quantum Transport in SemiconductorNanostructures (Academic Press, Inc., 1991), vol. 44 of Solid State Physics,pp. 1–228.

[20] R. Landauer, Z. Phys. B 68, 217 (1987).

[21] F.-Q. Xie, L. Nittler, C. Obermair, and T. Schimmel, Phys. Rev. Lett. 93,128303 (2004).

[22] R. Knott, W. Dietsche, K. von Klitzing, K. Eberl, and K. Ploog, Semicond.Sci. Technol. 10, 117 (1995).

[23] K. von Klitzing, Rev. Mod. Phys. 58, 519 (1986).

[24] B. Jeckelmann and B. Jeanneret, Rep. Prog. Phys. 64, 1603 (2001).

[25] B. Grbic, C. Ellenberger, T. Ihn, K. Ensslin, D. Reuter, and A. D. Wieck,Appl. Phys. Lett. 85, 2277 (2004).

[26] B. Paredesa, P. Zollerb, and J. I. Cirac, Solid State Commun. 127, 155 (2003).

[27] H. L. Stormer, Rev. Mod. Phys. 71, 875 (1999).

[28] B. I. Halperin, Phys. Rev. B 25, 2185 (1982).

[29] M. Buttiker (Academic Press, 1992), vol. 35 of Semiconductors and Semimet-als, p. 191.

[30] N.-Y. Cui, Solid State Commun. 127, 505 (2003).

[31] D. B. Chklovskii, B. I. Shklovskii, and L. I. Glazman, Phys. Rev. B 46, 4026(1992).

[32] L. I. Glazman and I. A. Larkin, Semicond. Sci. Technol. 6, 32 (1991).

[33] J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 50, 1760 (1994).

153

[34] K. Lier and R. R. Gerhardts, Phys. Rev. B 50, 7757 (1994).

[35] K. Oto, S. Takaoka, and K. Murase, Physica B 298, 18 (2001).

[36] M. R. Geller and G. Vignale, Physica B 212, 283 (1995).

[37] E. Ahlswede, Ph.D. thesis, Max-Planck-Institut fur Festkorperforschung,Stuttgart (2002).

[38] A. N. Cleland and M. R. Geller, Phys. Rev. Lett. 93, 070501 (2004).

[39] D. E. Grupp and A. M. Goldman, Science 276, 392 (1997).

[40] C. J. Chen, Introduction to scanning tunneling microscopy, Oxford series inoptical and imaging sciences; 4 (Oxford University Press, 1993).

[41] D. Sarid, Scanning force microscopy : with applications to electric, magnetic,and atomic forces, Oxford series in optical and imaging sciences ; 5 (OxfordUniversity Press, 1994), revised ed. ed.

[42] G. Binnig, H. Rohrer, C. Gerber, and E. Weibel, Phys. Rev. Lett. 49, 57(1982).

[43] G. Binnig, H. Rohrer, C. Gerber, and E. Weibel, Phys. Rev. Lett. 50, 120(1983).

[44] G. Binnig and H. Rohrer, Rev. Mod. Phys. 59, 615 (1987).

[45] G. Binnig and H. Rohrer, Rev. Mod. Phys. 71, S324 (1999).

[46] J. Wiebe, C. Meyer, J. Klijn, M. Morgenstern, and R. Wiesendanger, Phys.Rev. B 68, 041402 (2003).

[47] M. Morgenstern, J. Klijn, C. Meyer, M. Getzlaff, R. Adelung, R. A. Romer,K. Rossnagel, L. Kipp, M. Skibowski, and R.Wiesendanger, Phys. Rev. Lett.89, 136806 (2002).

[48] R. Ashoori and H. Chan, Solid State Commun. 127, 79 (2003).

[49] A. J. Heinrich, J. A. Gupta, C. P. Lutz, and D. M. Eigler, Science 306, 466(2004).

[50] A. J. Heinrich, C. P. Lutz, J. A. Gupta, and D. M. Eigler, Science 298, 1381(2002).

[51] N. Nilius, T. M. Wallis, and W. Ho, Science 297, 1853 (2002).

[52] G. Nazin, X. H. Qiu, and W. Ho, Phys. Rev. Lett. 90, 216110 (2003).

154

[53] G. A. Fiete and E. J. Heller, Rev. Mod. Phys. 75, 933 (2003).

[54] T. Maltezopoulos, A. Bolz, C. Meyer, C. Heyn, W. Hansen, M. Morgenstern,and R. Wiesendanger, Phys. Rev. Lett. 91, 196804 (2003).

[55] W. A. Hofer, A. S. Foster, and A. L. Shluger, Rev. Mod. Phys. 75, 1287 (2003).

[56] M. Hirooka, H. Tanaka, R. Li, and T. Kawai, Appl. Phys. Lett. 85, 1811(2004).

[57] K. Karrai and I. Tiemann, Phys. Rev. B 62, 13174 (2000).

[58] A. Socoliuc, R. Bennewitz, E. Gnecco, and E. Meyer, Phys. Rev. Lett. 92,134301 (2004).

[59] F. Giessibl, Rev. Mod. Phys 75, 949 (2003).

[60] M. A. Poggi, E. D. Gadsby, L. A. Bottomley, W. P. King, E. Oroudjev, andH. Hansma, Anal. Chem. 76, 3429 (2004).

[61] J. Hu, Y. Zhang, H. Gao, M. Li, and U. Hartmann, Nano Lett. 2, 55 (2002).

[62] Y. Yaish, J.-Y. Park, S. Rosenblatt, V. Sazonova, M. Brink, and P. L. McEuen,Phys. Rev. Lett. 92, 046401 (2004).

[63] K.-J. Chao, J. R. Kingsley, R. J. Plano, X. Lu, and I. Ward, J. Vac. Sci.Technol. 19, 1154 (2001).

[64] A. M. Chang, H. D. Hallen, L. Harriott, H. F. Hess, H. L. Kao, J. Kwo, R. E.Miller, R. Wolfe, J. van der Zielb, and T. Y. Chang, Appl. Phys. Lett. 61,1974 (1992).

[65] A. Oral, S. J. Bending, and M. Henini, Appl. Phys. Lett. 69, 1324 (1996).

[66] M. Tinkham, Introduction to Superconductivity (McGraw-Hill, 1980).

[67] L. N. Vu, M. S. Wistrom, and D. J. V. Harlingen, Appl. Phys. Lett. 63, 1693(1993).

[68] C. Veauvy, K. Hasselbach, and D. Mailly, Phys. Rev. B 70, 214513 (2004).

[69] S. Chatraphorn, E. F. Fleet, F. C. Wellstood, L. A. Knauss, and T. M. Eiles,Appl. Phys. Lett. 76, 2304 (2000).

[70] H. J. Hug, B. Stiefel, P. van Schendel, A. Moser, R. Hofer, S. Martin, H.-J.Guntherodt, S. Porthun, L. Abelmann, J. C. Lodder, et al., J. Appl. Phys.83, 5609 (1998).

155

[71] M. Bode, M. Dreyer, M. Getzlaff, M. Kleiber, A. Wadas, and R. Wiesendanger,J. Phys.: Cond. Matt. 11, 9387 (1999).

[72] A. Moser, H. Hug, I. Parashikov, B. Stefel, O. Fritz, H. Thomas, A. Baratoff,and H.-J. Guntherodt, Phys. Rev. Lett. 74, 1847 (1995).

[73] D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, Nature 430, 329(2004).

[74] A. T. Sellwood, C. G. Smith, E. H. Linfield, M. Y. Simmons, and D. A. Ritchie,Rev. Sci. Instrum. 72, 2100 (2001).

[75] M. Field, C. G. Smith, M. Pepper, D. A. Ritchie, J. E. F. Frost, G. A. C.Jones, and D. G. Hasko, Phys. Rev. Lett. 70, 1311 (1993).

[76] R. Schleser, E. Ruh, T. Ihn, K. Ensslin, D. C. Driscoll, and A. C. Gossard,Appl. Phys. Lett. 85, 2005 (2004).

[77] L. H. Chen, M. A. Topinka, B. J. LeRoy, R. M. Westervelt, K. D. Maranowski,and A. C. Gossard, Appl. Phys. Lett. 79, 1202 (2001).

[78] L. Kouwenhoven, C. Marcus, P. McEuen, S. Tarucha, R. Westervelt, andN. Wingreen, Electron transport in quantum dots, no. E345 in Proceedingsof the NATO Advanced Study Institute on Mesoscopic Electron Transport(Kluwer Series, 1997).

[79] S. M. Reimann and M. Manninen, Rev. Mod. Phys. 74, 1283 (2002).

[80] T. D. Krauss and L. E. Brus, Phys. Rev. Lett. 83, 4840 (1999).

[81] B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, andD. W. Pohl, J. Chem. Phys. 112, 7761 (2000).

[82] H. F. Hamann, Z. Phys. Chem. 215, 1025 (2001).

[83] J. Michaelis, C. Hettich, J. Mlynek, and V. Sandoghdar, Nature 405, 326(2000).

[84] J. R. Guest, T. H. Stievater, G. Chen, E. A. Tabak, B. G. Orr, D. G. Steel,D. Gammon, and D. S. Katzer, Science 293, 2224 (2001).

[85] S. Sun and G. J. Leggett, Nano Lett. 4, 1381 (2004).

[86] F. Pobell, Matter and Methods at Low Temperatures (Springer, 1996).

[87] J. Clement and E. Quinnell, Rev. Sci. Instrum. 23, 213 (1952).

[88] H. Sample, L. Neuringer, and L. Rubin, Rev. Sci. Instrum. 45, 64 (1974).

156

[89] T. Ihn and P. S. K. E. T. Vancura, A. Baumgartner, cond-mat p. 0112415(2001).

[90] Y. Naitou and N. Ookubo, Appl. Phys. Lett. 85, 2131 (2004).

[91] Y. D. Wilde, F. Formanek, and L. Aigouy, Rev. Sci. Instrum. 74, 3889 (2003).

[92] L. Chen, C. L. Cheung, P. D. Ashby, and C. M. Lieber, Nano Lett. 4, 1725(2004).

[93] J. Rychen, T. Ihn, P. Studerus, A. Herrmann, K. Ensslin, H. J. Hug, P. J. A.van Schendel, and H. J. Guntherodt, Appl. Surf. Sci. 157, 290 (2000).

[94] H. Holscher, U. Schwarz, and R. Wiesendanger, Appl. Surf. Sci. 140, 344(1999).

[95] J. Rychen, Ph.D. thesis, ETH Zurich (2001).

[96] T. A. Vancura, Ph.D. thesis, ETH Zurich (2002).

[97] T. Akiyama, U. Staufer, and N. de Rooij, Appl. Surf. Sci. 210, 18 (2003).

[98] T. Akiyama, U. Staufer, N. F. de Rooij, P. Frederix, and A. Engel, Rev. Sci.Instrum. 74, 112 (2003).

[99] G. M. King, J. S. Lamb, and G. Nunes, Appl. Phys. Lett. 79, 1712 (2001).

[100] J. Rychen, T. Ihn, P. Studerus, A. Herrmann, K. Ensslin, H. J. Hug, P. J. A.van Schendel, and H. J. Guntherodt, Rev. Sci. Instrum. 71, 1695 (2000).

[101] T. Ihn, Electronic Quantum transport in Mesoscopic Semiconductor Struc-tures, vol. 192 of Tracts in Modern Physics (Springer, 2004).

[102] S. Belaidi, P. Girard, and G. Leveque, J. Appl. Phys. 81, 1023 (1997).

[103] H. O. Jacobs, H. F. Knapp, and A. Stemmer, Rev. Sci. Instrum. 70, 1756(1999).

[104] M. Nonnenmacher, M. P. OBoyle, and H. K. Wickramasinghe, Appl. Phys.Lett. 58, 2921 (1991).

[105] W. Nabhan, B. Equer, A. Broniatowski, and G. D. Rosny, Rev. Sci. Instrum.68, 3108 (1997).

[106] L. Burgi, R. H. Friend, and H. Sirringhaus, Appl. Phys. Lett. 82, 1482 (2003).

[107] M. M. Shvebelman, A. G. Agronin, R. P. Urenski, Y. Rosenwaks, and G. I.Rosenman, Nano Lett. 2, 455 (2002).

157

[108] T. Yamauchi, M. Tabuchi, and A. Nakamura, Appl. Phys. Lett. 84, 3834(2004).

[109] T. Vancura, S. Kicin, T. Ihn, K. Ensslin, M. Bichler, and W. Wegscheider,Appl. Phys. Lett. 83, 2602 (2003).

[110] S. V. Kalinin and D. A. Bonnell, J. Appl. Phys. 91, 832 (2002).

[111] M. Woodside, C. Vale, and P. McEuen, Phys. Rev. B 64, 041310 (2001).

[112] M. T. Woodside and P. L. McEuen, Science 296, 1098 (2002).

[113] Y. G. Cornelissens and F. Peeters, J. Appl. Phys. 92, 2006 (2002).

[114] G. Boero, M. Demierre, P.-A. Besse, and R. S. Popovic, Sens. Actuators A(2003).

[115] M. J. Yoo, T. A. Fulton, H. F. Hess, R. L. Willett, L. N. Dunkleberger, R. J.Chichester, L. N. Pfeiffer, and K. W. West, Science 276, 579 (1997).

[116] H. Hess, T. Fulton, M. Yoot, and A. Yacoby, Solid State Commun. 107, 657(1998).

[117] M. A. Eriksson, R. G. Beck, M. Topinka, J. A. Katine, R. M. Westervelt, K. L.Campman, and A. C. Gossard, Appl. Phys. Lett. 69, 671 (1996).

[118] M. A. Topinka, B. J. LeRoy, R. M. Westervelt, S. E. J. Shaw, R. Fleischmann,E. J. Heller, K. D. Maranowski, and A. C. Gossard, Nature 410, 183 (2001).

[119] M. A. Topinka, B. J. LeRoy, S. E. J. Shaw, E. J. Heller, R. M. Westervelt,K. D. Maranowski, and A. C. Gossard, Science 289, 2323 (2000).

[120] B. J. LeRoy, M. A. Topinka, R. M. Westervelt, K. D. Maranowski, and A. C.Gossard, Appl. Phys. Lett. 80, 4431 (2002).

[121] B. LeRoy, J. Phys.: Condens. Matter 15, R1835 (2003), topical review.

[122] R. Crook, C. G. Smith, M. Y. Simmons, and D. A. Ritchie, J. Phys.: Cond.Matt. 12, L735 (2000).

[123] R. Crook, C. Smith, M. Simmons, and D. Ritchie, Physica E 12, 695 (2002).

[124] T. Ihn, J. Rychen, T. Cilento, R. Held, K. Ensslin, W. Wegscheider, andM. Bichler, Physica E 12, 691 (2002).

[125] R. Crook, C. G. Smith, M. Y. Simmons, and D. A. Ritchie, J. Phys.: Cond.Matt. 13, L249 (2001).

158

[126] R. Crook, C. G. Smith, W. R. Tribe, S. J. OShea, M. Y. Simmons, and D. A.Ritchie, Phys. Rev. B 66, 121301 (2002).

[127] A. Pioda, S. Kicin, T. Ihn, M. Sigrist, A. Fuhrer, K. Ensslin, A.Weichselbaum,S. E. Ulloa, M. Reinwald, and W.Wegscheider, Phys. Rev. Lett. 93, 216801(2004).

[128] H. Guilloua, A. D. Kent, G. W. Stupian, and M. S. Leung, J. Appl. Phys. 93,2746 (2003).

[129] F. G. Monzon, M. Johnson, and M. L. Roukes, Appl. Phys. Lett. 71, 3087(1997).

[130] K. S. Novoselov, A. K. Geim, S. V. Dubonos, Y. G. Cornelissens, F. M. Peeters,and J. C. Maan, Phys. Rev. B 65, 1 (2002).

[131] F. Peeters and X. Li, Appl. Phys. Lett. 72, 572 (1998).

[132] S. J. Bending and A. Oral, J. Appl. Phys. 81, 3721 (1997).

[133] I. S. Ibrahim, V. A. Schweigert, and F. M. Peeters, Phys. Rev. B 57, 15416(1998).

[134] C. J. B. Ford, S. Washburn, M. Buttiker, C. M. Knoedler, and J. M. Hong,Phys. Rev. Lett. 62, 2724 (1989).

[135] B. Baelus and F. Peeters, Appl. Phys. Lett. 74, 1600 (1999).

[136] L. W. Molenkamp, A. A. M. Staring, C. W. J. Beenakker, R. Eppenga, C. E.Timmering, J. G. Williamson, C. J. P. M. Harmans, and C. T. Foxon, Phys.Rev. B 41, 1274 (1990).

[137] R. Crook, C. G. Smith, M. Y. Simmons, and D. A. Ritchie, Phys. Rev. B 62,5174 (2000).

[138] R. Crook, C. G. Smith, A. C. Graham, I. Farrer, H. E. Beere, and D. A.Ritchie, Phys. Rev. Lett. 91, 246803 (2003).

[139] X. Maitre, W. D. Oliver, and Y. Yamamoto, Physica E 6, 301 (2000).

[140] S. Washburn, A. Fowler, H. Schmid, and D. Kern, Phys. Rev. Lett. 61, 2801(1988).

[141] B. Snell, P. Beton, P. Main, A. Neves, J. Owers-Bradley, L. Eaves, M. Henini,H. Hughes, S. Beaumont, and C. Wilkinson, J. Phys.: Cond. Matt. 1, 7499(1989).

[142] P. L. McEuen, A. Szafer, C. A. Richter, B. W. Alphenaar, J. K. Jain, A. D.Stone, R. G. Wheeler, and R. N. Sacks, Phys. Rev. Lett. 64 (1990).

159

[143] R. J. Haug, Semicond. Sci. Technol. 8, 131 (1993).

[144] S. W. Hwang, D. C. Tsui, and M. Shayegan, Phys. Rev. B 48, 8161 (1993).

[145] Y. Y. Wei, J. Weis, K. von Klitzing, and K. Eberl, Phys. Rev. Lett. 81, 1674(1998).

[146] J. Chalker and P. Coddington, J. Phys. C 21, 2665 (1988).

[147] S. H. Tessmer, G. Finkelstein, P. I. Glicofridis, and R. C. Ashoori, Phys. Rev.B 66, 125308 (2002).

[148] S. H. Tessmer, P. I. Glicofridis, R. C. Ashoori, L. S. Levitov, and M. R.Melloch, Nature 392, 51 (1998).

[149] G. Finkelstein, P. I. Glicofridis, R. C. Ashoori, and M. Shayegan, Science 289,90 (2000).

[150] G. Finkelstein, P. I. Glicofridis, S. H. Tessmer, R. C. Ashoori, and M. R.Melloch, Phys. Rev. B 61, R16 323 (2000).

[151] P. I. Glicofridis, G. Finkelstein, R. C. Ashoori, and M. Shayegan, Phys. Rev.B 65, 121312 (2002).

[152] A. Yacoby, H. Hess, T. Fulton, L. Pfeiffer, and K. West, Solid State Commun.111, 1 (1999).

[153] N. B. Zhitenev, T. A. Fulton, A. Yacoby, H. F. Hess, L. N. Pfeiffer, and K. W.West, Nature 404, 473 (2000).

[154] S. Ilani, J. Martin, E. Teitelbaum, J. H. Smet, D. Mahalu, V. Umansky, andA. Yacoby, Nature 427, 328 (2004).

[155] J. Martin, S. Ilani, B. Verdene, J. Smet, V. Umansky, D. Mahalu, D. Schuh,G. Abstreiter, and A. Yacoby, Science 305, 980 (2004).

[156] S. Wiegers, M. Specht, L. Levy, M. Y. Simmons, D. A. Ritchie, A. Cavanna,B. Etienne, G. Martinez, and P. Wyder, Phys. Rev. Lett. 79, 3238 (1997).

[157] K. Ikushima, H. Sakuma, S. Komiyama, and K. Hirakawa, Phys. Rev. Lett.93, 146804 (2004).

[158] K. L. McCormick, M. T. Woodside, M. Huang, M. Wu, P. L. McEuen, C. Du-ruoz, and J. J. S. Harris, Phys. Rev. B 59, 4654 (1999).

[159] P. Weitz, E. Ahlswede, J. Weis, K. V. Klitzing, and K. Eberl, Physica E 6,247 (2000).

160

[160] E. Ahlswede, P. Weitz, J. Weis, K. von Klitzing, and K. Eberl, Physica B 298,562 (2001).

[161] B. Alphenaar, P. McEuen, R. Wheeler, and R. Sacks, Physica B 175, 235(1991).

[162] Y. Acremann, T. Heinzel, K. Ensslin, E. Gini, H. Melchior, and M. Holland,Phys. Rev. B 59, 2116 (1999).

[163] T. Ihn, J. Rychen, T. Vancura, K. Ensslin, W. Wegscheider, and M. Bichler,Physica E 13, 671 (2002).

[164] S. Kicin, A. Pioda, T. Ihn, K. Ensslin, D. Driscoll, and A. Gossard, Phys.Rev. B 70, 205302 (2004).

[165] B. Kramer, T. Ohtsuki, and S. Kettemann (2004), cond-mat/0409625, sub-mitted to Physics Reports.

[166] Y. Imry, Introduction to mesoscopic physics (Oxford University Press, 2002),2nd ed.

[167] J. A. Simmons, H. P. Wei, L. W. Engel, D. C. Tsui, and M. Shayegan, Phys.Rev. Lett. 63, 1731 (1989).

[168] B. J. van Wees, L. P. Kouwenhoven, C. J. P. M. Harmans, J. G. Williamson,C. E. Timmering, M. E. I. Broekaart, C. T. Foxon, and J. J. Harris, Phys.Rev. Lett. 62, 2523 (1989).

[169] A. S. Sachrajda, R. P. Taylor, C. Dharma-Wardana, P. Zawadzki, J. A. Adams,and P. T. Coleridge, Phys. Rev. B 47, 6811 (1993).

[170] A. Dorn, M. Peter, S. Kicin, T. Ihn, K. Ensslin, D. Driscoll, and A. Gossard,Physica E 21, 426 (2004).

[171] V. Mosser, D. Weiss, K. v. Klitzing, K. Ploog, and G. Weimann, Solid StateCommun. 58, 5 (1986).

161

Acknowledgements

How can one thank adequately for the good times in a research group? Here arudimentary trial: First I would like to thank Prof Klaus Ensslin for giving me theopportunity for such a long-term project. His enthusiasm and personnel involve-ment is always encouraging. As the head of the group he makes possible the goodatmosphere and an intellectual environment. Thomas Ihn, my direct supervisor, isa real example in going about a problem in a calm way, both, in the lab and intheoretical questions. He is always ready to dig deep into physics to gain a clearerunderstanding. The few-minutes discussions that lasted whole days have been veryenlightening. Many thanks also to Prof Francois Peeters for the simulations in theballistic regime and for becoming co-examiner for this thesis.Cecil Barengo and Paul Studerus, the technicians of the group, can hardly bepraised too much: without their work and patient consulting, projects like thislow-temperature AFM would hardly be possible. But I also want to mention thephilosophical discussions that make life more interesting, not the least in a lab with-out windows. Many thanks also to Brigitte Abt for taking care of much of theadministrative stuff.In this lab, the legendary ‘B19’, Tobias Vancura and Slavo Kicin shared the longhours of measurements and microscope construction, with discussions about all theworld and his wife and with many different music styles in the air. AlessandroPioda and, recently, Arnd Gildemeister joined the low-temperature scanning gangand produce an increasing amount of ‘B19-news’.Christoph Ellenberger, with whom I had the luck to share the office, is not only thegreat Linux-Guru and E-beam magician, but also a great buddy in any concerns.Later, Lorenz Meier and the post-docs Kensuke Kobayashi and Renaud Leturcqfound their way to our room, which lead to many interesting conversations and dis-cussions. I would like to thank also Martin Sigrist for providing some of the samples,Roland Schleser, Davy Graf, Boris Grbic and Barbara Simovic. Also some of theformer members of the group I would like to acknowledge, namely Rainer Jaggi,Andreas Fuhrer and August Dorn. Many thanks also to Kaspar Suter, TerunobuAkiyama and Urs Staufer from IMT in Neuchatel for their commitment to the de-velopment of new AFM sensors.I would like to thank very much my family for the support and understanding of mebeing absent too often. And, last in the list, but first in my heart, Barbara Mathys,for the patience, support and her good humor.

162

Curriculum Vitae

Name: Andreas BaumgartnerBorn: April 2, 1975, citizen of Engi (GL)

1982-1986 Primary school Durrenroth

1986-1990 Secondary school Huttwil

1990-1995 Gymnasium Langenthal

1995 Matura type C (Math and Science)

1995-2000 Studies in Interdisciplinary Science at theSwiss Federal Institute of Technology (ETH) in Zurich

1995-2005 Military service and civilian service(dry stone walling and archeological excavations)

2000 Diploma thesis in the group of Prof. A.C. Mota,Laboratory of Solid State Physics (ETH)

2001-2005 PhD research work and teaching assistant positionin the group of Prof. K. Ensslin,Laboratory of Solid State Physics (ETH)

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in copyright - non-commercial use permitted rights ...27891/... · andreas baumgartner diss. eth...

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