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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)
Single electron tunneling 4
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
Single electron tunneling 5
Resulting I-V curve
Experiment: Al-Al203-Al, 10 nm x10 nm (superconductivity was destroyed by magnetic field)
Coulomb blockage
Single electron tunneling 6
SET: Basic circuit and devices
Single electron tunneling 7
Basic tunneling circuits
Isolated island:
At the energy cost vanishes!
Background charge
Single electron tunneling 8
Residual capacitance Tunneling barriers
Island between the barriers
Double barrier structure: Circuitry
Single electron tunneling 9
Circuitry
Single electron tunneling 10
Electrostatics: 4 different charge transfer events are relevant
V=0
q0=0
S D
induced charge
Single electron tunneling 11
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
Single electron tunneling 12
What happens when an electron reaches the island?
During each cycle a single electron is transferred!
D
Single electron tunneling 13
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.
Single electron tunneling 14
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.
Single electron tunneling 15
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
Single electron tunneling 16
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
Single electron tunneling 17
Finally,
Temperature
Heaviside
With induced charges, SET
Single electron tunneling 18
Calculations for different background charges
Thermal smearing
Coulomb staircase
Experiment: STM of surface clusters
Single electron tunneling 19
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
Single electron tunneling 20
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
Single electron tunneling 21
Experimental test: Al-Al203 SET, temperature 30 mK
V=10 V
Coulomb blockade oscillations
Single electron tunneling 22
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.
Single electron tunneling 23
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.
Single electron tunneling 24
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
Single electron tunneling 25
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!
Single electron tunneling 26
The current is then, I=-ef
This is an excellent current standard!
Accuracy check: 10-6
Opposite phase shifts
Single electron tunneling 27
Single electron tunneling 28
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
Single electron tunneling 29
Single electron tunneling 30
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
Single electron tunneling 31
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.
Single electron tunneling 32
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