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