Ljubljana, april 2013

First, the basic theory of superconductivity will be introduced, leading to an explanation of the superconducting proximity effect and microscopic Andreev reflection. Experimental scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS) techniques will be described as standard tools for studying this effect. Finally, some measurements of proximity effect in two different systems of a superconductor in contact with a normal metal will be presented.

STM Studies of Superconducting

Proximity Effect

Seminar 1b

Author: Tjaša Parkelj

Mentor: doc. dr. Miha Škarabot

Co-Mentor: dr. Erik Zupanič

Ljubljana, April 2014

Department of physics

Abstract

1

................................................................................................................................................................. 0

1. Introduction ..................................................................................................................................... 1

2. Superconductivity ............................................................................................................................ 2

London Theory .................................................................................................................................... 2

BCS Theory ......................................................................................................................................... 2

Ginzburg - Landau Theory .................................................................................................................. 3

Inhomogeneous System ................................................................................................................... 3

Surface ............................................................................................................................................. 4

NS interface ..................................................................................................................................... 4

3. Superconducting Proximity Effect and Andreev reflection ............................................................ 5

Andreev Reflection .............................................................................................................................. 5

4. Scanning Tunneling Microscopy ..................................................................................................... 6

Scanning Tunneling Spectroscopy ...................................................................................................... 7

5. STM and STS Observations of Superconducting Proximity Effect ................................................ 8

NbSe2-Au Junctions Studies ................................................................................................................ 8

STM Study of the Proximity Effect in a Disordered Two-Dimensional Metal ................................... 9

6. Conclusions ................................................................................................................................... 11

7. References ..................................................................................................................................... 11

1. Introduction

In the year 1911, shortly after the first production of liquid helium, Kamerlingh Onnes observed a new

state of matter. When studying the electrical resistance of solid mercury at cryogenic temperatures, he

noticed that at the temperature of 4.2 K the resistance dropped to zero. Superconductivity was

discovered [1].

Since then, many other metals and alloys, so-called conventional superconductors, were found to be

superconducting. The theory of conventional superconductors has been well developed [1, 2]. The

same theory however does not apply for unconventional superconductors like heavy fermion

superconductors discovered in 1978 [3]. Until now about 30 heavy fermion superconductors were

found in materials based on Ce and U, with a critical temperature up to 2.3 K. Superconductivity was

also observed in some organic materials like (TMTSF)2ClO4. In 1986 the first high temperature

superconductor was found. These cuprate and perovskite materials have critical temperatures up to

135 K which is the highest critical temperature ever recorded at standard pressure [4, 5].

Superconductors exhibit different interesting phenomena. In 1932 Holm and Meissner observed zero

resistance in superconductor - normal metal - superconductor (SNS) pressed contacts, in which two

superconducting metals are separated by a thin film of a non-superconducting normal metal [6]. When

placing a normal metal (N) in contact with a superconductor (S) the N acquires superconducting

properties and the same properties are lowered in the S. This phenomenon is known as the proximity

effect [7, 8].

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Characterization of this superconducting effect can be achieved by different conventional techniques

using tunnel junctions, proximity-effect sandwiches, or point contacts. One of the best techniques for

investigation of this effect are scanning tunneling microscopy (STM) and scanning tunneling

spectroscopy (STS) [9].

2. Superconductivity

The main properties that define superconductivity are zero electrical resistivity, expulsion of weak

magnetic fields from superconductors called the Meissner effect and the energy gap in the density of

states of electrons. There are different theories that describe superconductivity, starting with the

phenomenological theory developed by the London brothers in 1935 [1].

London Theory

The theory is based on an assumption that under a certain critical temperature some of the conduction

electrons in a superconducting material become superconducting, while the rest remain “normal”. The

“normal” electrons continue to act like they have a finite resistivity while the superconducting

electrons move without dissipation.

Using the Maxwell equations and taking into account the fact that there are no magnetic fields in a

superconductor interior far from its surface, they found that the magnetic field in a superconductor

decays exponentially from the surface with a characteristic length :

(2.1)

(2.2)

where is the density of superconductive electrons and is called the penetration depth [1]. For

most superconductors the penetration depth is in the range from a few 10 to a few 100 nm (Table 1).

The London theory gives us an explanation for the Meissner effect: the superconducting material near

the surface responds to the external magnetic fields with electric currents. The magnetic field of these

currents cancels out the magnetic field inside the superconductor [1, 4].

BCS Theory

The first microscopic theory of superconductivity was given in 1957 by Bardeen, Cooper and

Schrieffer and it gave a complete explanation of the properties of conventional superconductors [1].

The theory predicted the inverse relation between the critical temperature TC and the mass of the

isotope in the crystal lattice - the isotope effect, which gives an upper limit for critical temperatures for

most superconductors.

The second main prediction of the BCS theory was the existence of an energy gap at the Fermi

energy. The energy gap is a consequence of the bound state of two electrons called a Cooper pair.

Electrons near the Fermi level are paired by an attractive force due to coupling between the electrons

and the phonons of the underlying crystal lattice [1]. When paired the electrons lower their energy

bellow the Fermi level and so the energy gap is formed [2]. In the first approximation the energy gap

is correlated with the critical temperature with equation:

(2.3)

3

Ginzburg - Landau Theory

Before the microscopic BCS theory, in 1950, Ginzburg and Landau proposed another

phenomenological theory of superconductivity. It describes the superconductive phase transition from

a thermodynamic point of view. Although the description within the GL theory is less informative,

especially for the homogeneous state, it is an interesting approach for describing a non-homogeneous

state, for example at the edges of a superconductor or at the interface between a superconductor and a

normal metal.

It is based on an order parameter which is zero in the normal state and non-zero in the

superconducting state, describing a second order phase transition. The free energy density can be

expanded in powers of the order parameter for temperatures around the critical temperature . For a

system in a magnetic field we have to add two more terms to the free energy density. This way we

can write the free energy density of the superconductor as [10]:

(2.4)

Here and are the superconductive state and normal state free energy densities,

respectively. The two parameters ) and are temperature dependent

phenomenological parameters of the theory, the parameter is the effective mass, effective

charge and is the vector potential.

Inhomogeneous System

In an inhomogeneous system the order parameter depends on position . If there are no

external magnetic fields we can change the equation (2.4) by keeping the first three terms and adding a

term with the gradient of . The total free energy can then be written as:

(2.5)

at point [1].

The order parameter can be found by minimizing the total free energy. By using the variational

approach and substituting

,

the variation of free energy of the superconductive state can be written as:

.

. (2.6)

The two terms involving gradients can be integrated by parts, to obtain

(2.7)

If the variation of the free energy density is to equal zero the following equation must be true:

(2.8)

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We have found that minimizing the total free energy leads to this Schrödinger like equation for .

Surface

We can now use equation 2.8 to study the response of the superconductive order parameter at the

surface of a superconductor.

Suppose that the surface lies in the plain with the superconductor in the region. The

superconductive order parameter at the surface must be zero. Assuming that must be

continuous, we have to solve the nonlinear Schrödinger equation:

(2.9)

in the region with the boundary conditions and . The resulting function

is:

, (2.10)

where is the value of the order parameter in the bulk far from the surface and the parameter is

defined by

(2.11)

This quantity, which has dimensions of length, is called the coherence length and it is a measure of the

distance from which the order parameter has recovered back to nearly its bulk value (Figure 1). In

BCS theory the coherence length relates to the physical size of a single Cooper pair. Its size varies for

different materials (Table 1).

Figure 1: Order parameter of a superconductor near a

surface. It recovers to its bulk value over a length scale of the

coherence length. [1]

NS interface

We can use this same approach to study how the order parameter changes at a normal metal-

superconductor (NS) interface. The order parameter is a continuous function. It is zero in the bulk of a

normal metal and it exponentially approaches the value at the interface. In a superconductor the order

parameter exponentially decreases from the bulk value towards the value at the junction (Figure 2).

Substance ξ [nm] λ [nm]

Al 1550 45

Sn 180 42

Nb 39 52

Pb 87 39

Nb3Ge 3 90

YBa2Cu

3O

7 1.65 156

Table 1: Coherence length (ξ) and penetration depth

(λ) at zero temperature for some important

superconductors [1].

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Figure 2: The order parameter as a function of x at the NS interface. The order parameter spreads across the interface

from the superconductor (S) into the normal metal (N). [10]

3. Superconducting Proximity Effect and Andreev reflection

When a superconductor and a normal metal are in a good electrical contact we can observe the

superconducting proximity effect, where superconductivity is weaken in the superconductor and

induced in the normal metal.

The order parameter defining the superconducting state is a continuous function in space. Therefore, if

an unordered medium is placed in a strong contact with an ordered one, the order parameter does not

abruptly change from its bulk value to zero at the contact, but leaks into the unordered material,

changing its properties. An alternative viewpoint is given by a microscopic mechanism called Andreev

reflection [8].

Andreev Reflection

When two metals are brought into electrical contact, the Fermi levels align themselves in such a way

that they are in equilibrium. For both metals the density of states at the Fermi energy is non-zero,

therefore electrons can be transmitted from one conduction band into another. For a NS interface the

situation becomes slightly more complicated (Figure 3a).

In the ground state of a superconductor there is an energy gap in the excitation spectrum. This is a

direct result of the attractive interaction which bounds electrons near the Fermi surface into Cooper

pairs. The transmission of charge carriers across the NS interface occurs in two different ways

depending on the Fermi energy of the charge carriers [10].

- : If the energy of the incoming electron (or hole or quasi-particle) in the normal

metal is higher or lower than , the density of states is non-zero and the electron can be

transmitted into the superconductor.

- : If the energy of the incoming electron in the normal state lies within the energy

gap of the superconductor, the transmission is more complex. Electrons cannot be transmitted

into the superconductor as the density of states at is zero.

The only possibility for an electron to be transmitted across the NS boundary into the superconductor

is to undergo an Andreev reflection. An incoming electron forms a Cooper pair with a second electron

of the normal conductor. The two electrons are taken from opposite corners of the Brillouin zone, in

order to allow the Cooper pair to carry zero total momentum. This corresponds to s-wave pairing,

common in conventional superconductors. To maintain charge conservation, a hole has to be reflected

into the normal metal (Figure 3b).

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Figure 3: a) Band structure of a NS interface at zero bias across the junction and at zero temperature. Blue indicates the

valence band and pink indicates the conduction band in the single quasi-particle picture. Ds in this image indicates the

density of states[10]. b) An electron (red) meeting the interface between a normal conductor (N) and a superconductor (S)

produces a Cooper pair in the superconductor and a retroreflected hole (green) in the normal conductor. Vertical arrows

indicate the spin band occupied by each particle [6].

Looking from the other side, the paired electrons may “leak” from the superconductor to the normal

metal before they completely lose their coherence due to scattering events. The leaking of Cooper

pairs into the normal metal leads to a modification of the band structure near the NS interface. The

energy gap in the superconductor decreases continuously while approaching the NS interface. On the

other hand a small energy gap builds up in the normal metal approaching the NS interface, resulting in

the proximity effect.

The modification of the energy gap in the electron density of states near the NS interface can be

measured using different techniques. An experimental tool that provides high spatial and high energy

resolution is a scanning tunneling microscope.

4. Scanning Tunneling Microscopy

Scanning tunneling microscope is a precise instrument for imaging surfaces of metals, semiconductors

and superconductors at the atomic scale. An atomically sharp tip is brought into close proximity of the

sample surface. When a bias voltage is applied between the sample and the tip a small electrical

current starts to flow through the vacuum gap that separates the two conductors due to quantum

tunneling effect [9].

When the tip and the sample electron wave functions overlap, there is a finite probability that the

electrons will tunnel across the energy barrier presented by the vacuum between them. In a one–

dimensional case we can describe the sample, as well as the tip, as an ideal metal with electron states

filled up to the Fermi energy . The probability for an electron to be found in the barrier region or in

the sample at the distance from the tip is given by [9, 12]:

, (4.1)

, (4.2)

where is the wave function of the electron, and are work functions1 of the tip and the

sample, respectively. is the applied bias voltage that shifts the energy levels of the sample electrons

with respect to the tip electrons by the energy . The exponential dependence in equation 4.1 is the

origin of large vertical resolution of STM.

1the work needed to remove an electron from a solid to a point in the vacuum

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Because of the Pauli exclusion principle the electrons near the Fermi level will tunnel out of the filled

states only if there are empty levels on the other side of the barrier. Thus the total tunneling current is

a function of the voltage difference between the sample and the tip , the tip distance from the

surface and the local density of states (LDOS) of the tip and LDOS of the sample :

, (4.3)

where is the Fermi-Dirac temperature dependent distribution for electrons and M is an evaluation

of the tunneling matrix element, which describes the interaction of the electron wave functions in the

tip with those in the sample:

(4.4)

Determining the matrix element in a more realistic approximation is almost impossible since the actual

geometry and the chemical structure of the tip are generally not known. An ideal STM tip would

consist of a point source of the tunneling current, which would result in a best possible resolution and

in no tip-surface interaction [12].

It can be seen from equations 4.3 and 4.4 that the image of the surface, recorded with STM is always a

combination of the sample topography and the local electronic structure of the surface.

There are two basic modes of imaging sample surfaces with an STM: constant height (CH) and

constant current (CC) mode. In both cases the tunneling voltage is set constant while the tip scans the

surface line by line. In CH mode, we observe the tunneling current as a function of lateral x and y

position of the tip, while keeping the height constant. In CC mode the tunneling current is held

constant by a feedback mechanism, which controls the height of the tip (Figure 4). The image is

formed by measuring the vertical tip displacement as a function of x and y position. The CH mode is

faster, as adjusting the tip's height in CC mode takes some time, but it requires a very flat surface so

the tip does not crash into the sample surface.

Figure 4: The principle of the STM operation in constant current (on the left) and constant height (on the right) imaging

modes. [12].

Scanning Tunneling Spectroscopy

The STS measurements give us information on the electron DOS of a sample directly under the STM

tip. The derivative of equation 4.3 with respect to the tunneling voltage, assuming a constant LDOS of

the tip and in the low-temperature limit, can be written as [12]:

. (4.5)

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The differential tunneling current2 can be obtained numerically by calculating the derivative

or directly by using a lock-in technique, which gives us better signal-to-noise ratio. The technique

adds a small sinusoidal modulation to the constant bias voltage, which causes a sinusoidal

response in the tunneling current (Figure 5).

Figure 5: Left: A small modulation of the tunneling voltage (blue) results in a modulated tunneling current (red). The

amplitude depends on the slope of the I-V curve. Right: A schematic presentation of the experimental setup for STS

measurements with the lock-in amplifier [12].

The resulting output signal from the lock-in is proportional to and therefore the LDOS of the

sample.

The measured tunneling current is typically in the order of a few pA to a few nA. In order to achieve

stable tunneling conditions the mechanical and electrical noise need to be reduced as much as possible.

The STM microscopes are usually combined with an ultra-high vacuum (UHV) system that prevents

the samples from getting contaminated during the experiment. Measurements can be taken at different

temperatures but working at cryogenic temperatures gives better spatial and energy resolution.

Cooling can be done using liquid nitrogen (to its boiling temperature 77 K). Lower temperatures can

be reached using liquid helium (to its boiling temperature 4.2 K) and even lower temperatures using

Joule-Thomson (down to 1 K) or dilution refrigerators (down to a few 10 mK).

The tips used in STM are usually made from W or Pt/Ir alloy that are non-superconductive. We can

also use tips made of superconductors like Nb. By bringing a superconducting tip in proximity of a

superconductor (in the tunneling regime) we make a Josephson junction (superconductor-insulator (I)-

superconductor junction). Cooper pairs can tunnel from one superconductor to the other while the

vacuum between the tip and the sample serves as an insulator [1]. By changing the distance between

the sample and the tip we can measure the change in the tunneling current across a SIS junction [13].

There are different cleaning procedures usually used for in situ preparation of clean and well ordered

surfaces, suitable for STM measurements. By using precise evaporation sources it is also possible to

grow high quality samples in UHV. Preparation of fresh samples in the STM system is one of the big

advantages of using UHV LT-STM for investigation of proximity effect.

5. STM and STS Observations of Superconducting Proximity Effect

NbSe2-Au Junctions Studies

In 1999 Truscott et al. studied a superconducting proximity effect of thin non-superconductive metal

layers on a thick superconductor using UHV low-temperature STM and STS techniques [14]. They

prepared the samples in situ by careful deposition of Au on a NbSe2 single crystal substrate.

2 Which can also be seen as conductance

9

Figure 6: A schematic representation of a sample with Au islands (N) on a NbSe2 crystal (S)

NbSe2 is an anisotropic superconductor with short coherence lengths of 7.7 nm parallel and 2.3 nm

perpendicular to the layers. It has a superconducting transition temperature at 7.2 K and zero

temperature superconducting energy gap of 1.1 meV. An atomically flat surface of this superconductor

was prepared under vacuum by cleaving the crystal and imaged with the STM to determine surface

quality and structure.

Both STM and STS measurements were made at a temperature of 2.5 K. The STM images revealed

Au islands with average thickness ranging from 0.1 to 60 nm and diameters around 10 nm, separated

by large areas of bare NbSe2 as shown in Figure 7a.

STS measurement of the pristine surface far from any gold deposits revealed a spectrum identical to

that obtained on the bulk sample, having the normal BCS shape and an energy gap of 1.0 meV.

Additional spectroscopy measurements were taken on several Au islands of different thicknesses. The

resulting STS spectra for each average thickness can be seen in Figure 7b.

Figure 7: a) 100 nm STM image of 0.1 nm thick gold islands on NbSe2. b) STS measured at T=2.5 K of Au islands of different

thicknesses (as Indicated). Solid lines are proximity model best fits [14].

It can be seen that the STS spectrum of a 0.1 nm thick Au layer has an energy gap of 1.0 meV, which

is of the same magnitude as observed on the bare superconductor, indicating “total proximity effect”.

On the other hand for a 10 nm thick island, the energy gap reduces to only 0.6 meV. Thus we can

observe a significant decrease in the magnitude of the energy gap with increasing thickness of the

metal layer, as expected by the proximity effect.

STM Study of the Proximity Effect in a Disordered Two-Dimensional Metal

In 2013 L. Serrier-Garcia, et al. published a low-temperature STM and STS study of the proximity

effect between a disordered two-dimensional (2D) metal and a superconductor [15]. The measured

system consisted of superconducting single crystalline Pb islands, interconnected by a non-

superconducting atomically thin disordered Pb wetting layer (WL). Bulk Pb becomes superconductive

at a transition temperature of 7.2 K. The experiments were conducted at a temperature of 320 mK

using a dilution refrigeration system.

10

They prepared the samples in situ by deposition of Pb onto an atomically clean undoped Si(111)

substrate. When evaporated Pb grows first as a 2D wetting layer and after reaching certain thickness it

starts to form 3D islands (Stranski-Krastanov growth mode) [16].

Figure 8: A scheme of Pb islands (S) and a wetting layer (N) grown on a Si(111) single crystal serving as a base.

The STM images revealed a Pb network consisting of atomically flat 7-11 ML thick Pb islands,

interconnected by a 1-2ML thick wetting layer of Pb as seen in Figure 9a. The STS measurements

were taken at different points along a line going from the island edge to the neighboring part of the

wetting layer. In Figure 8b the experimental geometry is sketched. The resulting STS spectra as a

function of the distance Y from the island edge are shown in Figure 9b.

Figure 9: a) Topographic STM image of 7-13 monolayers (ML) high Pb islands interconnected by a 1-2ML thick Pb wetting

layer covering the underlying Si substrate. b) Schematic representation of the STM and STS experiment. The fully gaped

spectra on the Pb island appear in homogeneous red, the normal wetting layer in blue, and the proximity region in yellow.

The dashed line indicates the position of the flat top edge of the Pb island. c) Spatial variations of the STS spectra as a

function of the distance from the island edge. The dashed line corresponds to the spectrum taken right at the island edge

[15].

The superconducting state in the islands is characterized by a fully opened energy gap. The spectrum

of the wetting layer far from the islands exhibits a spectrum with a V-shaped dip at zero bias which

will be referred to as a zero-bias anomaly (ZBA). The absence of a clear gap is evidence of non-

superconducting character of the WL [14].

As the tip moves from the island to the WL the STS spectra evolve from a superconducting gap shape

in the islands to a non-superconducting spectrum far away from the islands giving evidence of the

proximity effect on a scale of several nanometers.

This study also presented a spatial 2D STS map. In Figure 10a we can observe the topographic

constant current STM image of a region in the vicinity of a Pb-island edge along with STS map at zero

sample bias of the same region, shown in Figure 10b. Mapping is performed by slowly imaging the

surface and simultaneously measuring the conductance ( at a fixed voltage. The result of

spatially resolved STS mapping is a plot of LDOS of the scanned area at a fixed energy [12]. Mapping

the DOS at a specific energy is a good visual way to see the inhomogeneities in the density of states.

This way we can see the interplay between the proximity effect and ZBA taking place in real space.

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Figure 10: a) Topographic ( I = 0.1 nA, V = 20 mV) STM image 55nm x 15nm acquired near the edge of a Pb-island. b) STS

tunneling conductance maps dI(V,r)/dV at zero sample bias. Colorcode (rainbow pallet) corresponds to the tunneling

conductance varying in the range 0 – 36 nS. The map corresponds to the same region showed in (a)[15].

6. Conclusions

Superconducting proximity effect is a complex phenomena observed at a normal metal-superconductor

interface. Close to such interface superconducting properties are induced in the normal metal and the

same properties are weakened in the superconductor. Two precise techniques often used for the study

of proximity effect are STM and STS. They allow high spatial resolution imaging of NS systems and

high energy resolution measurements of electronic structure around NS interfaces. STS measurements

can be used to observe the evolution of the superconducting energy gap close to a NS interface. 2D

STS mapping can reveal spatial propagation of the superconducting order parameter, giving us a

visual interpretation of the superconducting proximity effect.

7. References

[1] J. F. Annett, Superconductivity, Superfluids and Condensates (Oxford University Press, New York,

2004).

[2] C. Kittel, Introduction to Solid State Physics (John Willy and Sons, USA, 2005).

[3] http://en.wikipedia.org/wiki/Unconventional_superconductor, (4. April 2014).

[4] T. Verbovšek, Scanning Tunneling Microscopy Observations of Superconductivity

(Seminar, FMF, 2013).

[5] http://en.wikipedia.org/wiki/High-temperature_superconductivity, (4. April 2014).

[6] http://en.wikipedia.org/wiki/Proximity_effect_(superconductivity), (4. April 2014).

[7] B. Pannetier and H. Courtois, J. Low Temp. Phys. 118 , 599 (2000).

[8] T. Heikkilä, Superconducting Proximity Effect in Mesoscopic Metals (Ph.D. Thesis, Helsinki

University of Technology, 2002).

[9] R. Wiesendanger, Scanning Probe Microscopy and Spectroscopy: Methods and Applications

(Cambridge University Press, 1994).

[10] P. Prelovšek, Teorija trdne snovi, (FMF, Ljubljana 1999, http://www.f1.ijs.si/~prelovse

/teotrd.pdf, 20. April 2014).

[11] C. Handschin, Andreev reflections and superconducting Proximity effect in lateral hBN/

graphene/ NbSe2 quantum Hall devices (Master thesis, Columbia University, 2013).

[12] E. Zupanič, Low-Temperature STM Study and Manipulation of Single Atoms and Nanostructures

(Ph.D. Thesis, Jožef Stefan International Postgraduate School, 2010).

[13] J. G. Rodrigo, H. Suderow, S. Vieira, Eur.Prys. J. B 40, 483 (2004)

[14] A.D. Truscott, R.C. Dynes and L.F. Schneemeyer, Phys. Rev. Lett. 83, 1014 (1999).

[15] L. Serrier-Garcia, J. C. Cuevas, T. Cren, C. Brun,V. Cherkez, F. Debontridder,

D. Fokin, F. S. Bergeret and D. Roditchev, Phys. Rev. Lett. 110, 157003 (2013).

[16] F. Grey, R. Feidenhansl, M. Nielsen, R.L. Johnson, J. Phys. Colloques 50, 154 (1989).

a)

b)