Single electron tunneling 1

Building blocks for nanodevices

Two-dimensional electron gas (2DEG)

Quantum wires and quantum point contacts

Electron phase coherence

Single-Electron tunneling devices

- Coulomb blockage

Quantum dots (introduction)

Single Electron Tunneling Devices

Introduction 3

Gate

Dot Electron

Attraction to the gate

Repulsion at the dot

Cost

At

the energy cost vanishes !

Coulomb blockade

Single-electron transistor (SET)

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Coulomb blockade in a tunnel barrier

At |q|< e/2 the electron tunneling will increase the energy stored in the barrier one has to pay for the tunneling by the bias voltage

Energy stored is q2/2C

Why R matters?

time delay

duration

Because of environment capacitances it is

difficult to observe CB in single junctions

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Resulting I-V curve

Experiment: Al-Al203-Al, 10 nm x10 nm (superconductivity was destroyed by magnetic field)

Coulomb blockage

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SET: Basic circuit and devices

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Basic tunneling circuits

Isolated island:

At the energy cost vanishes!

Background charge

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Residual capacitance Tunneling barriers

Island between the barriers

Double barrier structure: Circuitry

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Circuitry

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Electrostatics: 4 different charge transfer events are relevant

V=0

q0=0

S D

induced charge

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Symmetric system:

For n=0 and q0=0, all transfers are suppressed until

V=0

Coulomb blockade of transport

Electrons can tunnel both from drain onto island and from island to source

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What happens when an electron reaches the island?

During each cycle a single electron is transferred!

D

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First observation:

Giver and Zeller, 1968 granular Sn film, superconductivity was suppressed by magnetic filed

Differential resistance

Coulomb gap manifests itself as increased low-bias differential resistance

The background charges, q0, influence Coulomb blockade and can even lift it.

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I-V curves: Coulomb staircase e g c

+ -

Master equation

How one can calculate I-V curve? For simplicity, we will do it only for a stationary case.

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The partial probabilities, , can be calculated from the Fermi Golden Rule: the probability is

To get the rate we have to multiply the probability by

and then sum over initial and final states.

Since only the vicinity of the Fermi level matters we can take the densities of states and matrix elements at the Fermi level and express the results through tunneling conductance, G, of the junctions.

Fermi function

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As a result, for, e. g. , for the transition between the emitter and the grain

We use the tunneling Hamiltonian

that gives the following expression for the tunneling conductance

DOS Volume

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Finally,

Temperature

Heaviside

With induced charges, SET

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Calculations for different background charges

Thermal smearing

Coulomb staircase

Experiment: STM of surface clusters

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The SET transistor

An extra electrode (gate) defined in a way to have very large resistance between it and the island.

That allows to tune induced charges by the gate voltage

Fulton & Dolan, 1987

The so-called orthodox theory discussed before is valid; we have just to remember that the energy cost is

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In this way we arrive at the so-called stability diagram of Single Electron Transistor (SET)

Coulomb diamonds: all transfer energies inside are positive

Conductance oscillates as a function of gate voltage Coulomb blockade oscillations

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Experimental test: Al-Al203 SET, temperature 30 mK

V=10 V

Coulomb blockade oscillations

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The single electron pump

Two islands each one can be tuned by a nearby gate electrode.

The structure is symmetric.

Six electron transfers are important.

Equalities between direct and reverse processes define lines at the stability diagram.

It define the regions of stable configurations characterized specific charges of the island.

The inter-island capacitance C12 is responsible for the interplay of the contributions from VA and VB.

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At C12=0, the diagram is a set of squares, unaffected by the transitions of 3d type the corresponding lines just touch corners

In the general situation there are triple points, where and electron can transfer the whole system for free.

We will show that it allows one to pump electrons one by one.

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Experiment:

SETs 3 and 4 work as electrometers to measure charges at the islands 1 and 2

Conductance of the double island

Conductance of el. 3

Conductance of el. 4

Difference signal

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How one can pump the electrons?

Bias voltage moves lines at the stability diagram.

It creates triangles where the energy is relatively large, and CB is impossible.

Let us adjust the gate voltages to start within a triangle, and then apply to the gates AC voltages shifted in phase. Then the path in the phase space is

Exactly one electron has passed through the device!

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The current is then, I=-ef

This is an excellent current standard!

Accuracy check: 10-6

Opposite phase shifts

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Only integer number of electrons can be trapped in the potential wells drag current is quantized in units of ef, depending on the gate voltage.

Talyanskii et al., 1996

Another current standard: Quantized electron drag in quantum wires

+ + + - - - Piezoelectric

Sample

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How accurate are the Coulomb-blockade devices? Are there principle limitations?

We discussed only sequential tunneling through a grain, which is exponentially suppressed by the Coulomb blockade.

The grain states are involved only in virtual transitions!

In addition, there is a coherent transfer. Consider the initial and final states in different leads. Then the transition rate in the second order of the perturbation theory is

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This process is called the quantum co-tunneling. Its rate is

We pay by additional small tunneling transparency (one more factor containing conductance).

However, the energy costs enter as powers rather than exponents.

Due to its importance, the quantum co-tunneling has been thoroughly studied. It can lead to the contributions to the current proportional to V3 and V.

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Open questions

In the previous lecture we discussed electrons in terms of waves. However, in this lecture we spoke about particles, their charge, etc. Are we running two horses at the same time? How single-electron effects interplay with quantum interference? These problems are solved to some extent and we will discuss them later.

Slide Number 1Single Electron Tunneling DevicesSlide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18The SET transistorSlide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24How one can pump the electrons?Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32