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Single Electron Transistor (SET)Cg dotVge-e- gate source drain
channelA single electron transistor is similar to a normal
transistor (below), except the channel is replaced by a small
dot.the dot is separated from source and drain by thin
insulators.An electron tunnels in two steps: source dot drainThe
gate voltage Vg is used to control the charge on the gate-dot
capacitor Cg .How can the charge be controlled with the precision
of a single electron?
Kouwenhoven et al., Few Electron Quantum Dots, Rep. Prog. Phys.
64, 701 (2001).
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Nanoparticle attracted electrostatically to the gap between
source and drain electrodes.The gate is underneath.Designs for
Single Electron Transistors
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Charging a Dot, One Electron at a TimeSweeping the gate voltage
Vg changes the charge Qg on the gate-dot capacitor Cg . To add one
electron requires the vol- tage Vg e/Cg since Cg=Qg/Vg.The
source-drain conductance G is zero for most gate voltages, because
putting even one extra electron onto the dot would cost too much
Coulomb energy. This is called Coulomb blockade .Electrons can hop
onto the dot only at a gate voltage where the number of electrons
on the dot flip-flops between N and N+1. Their time-averaged number
is N+ in that case.The spacing between these half-integer
conductance peaks is an integer.
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The SET as Extremely Sensitive Charge DetectorAt low
temperature, the conductance peaks in a SET become very
sharp.Consequently, a very small change in the gate voltage
half-way up a peak produces a large current change, i.e. a large
amplification. That makes the SET extremely sensitive to tiny
charges.The flip side of this sensitivity is that a SET detects
every nearby electron. When it hops from one trap to another, the
SET produces a noise peak.
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Gate Voltage versus Source-Drain VoltageThe situation gets a bit
confusing, because there are two voltages that can be varied, the
gate voltage Vg and the source-drain voltage Vs-d .Both affect the
conductance. Therefore, one often plots the conductance G against
both voltages (see the next slide for data). Schematically, one
obtains Coulomb diamonds, which are regions with a stable electron
number N on the dot (and consequently zero conductance).
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Including the Energy Levels of a Quantum DotContrary to the
Coulomb blockade model, the data show Coulomb diamonds with uneven
size. Some electron numbers have particularly large diamonds,
indicating that the corresponding electron number is particularly
stable. This is reminiscent of the closed electron shells in atoms.
Small dots behave like artificial atoms when their size shrinks
down to the electron wavelength. Continuous energy bands become
quantized (see Lecture 8). Adding one electron requires the Coulomb
energy U plus the difference E between two quantum levels (next
slide) . If a second electron is added to the same quantum level
(the same shell in an atom), E vanishes and only the Coulomb energy
U is needed.
The quantum energy levels can be extracted from the spacing
between the conductance peaks by subtracting the Coulomb energy U =
e2/C .
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Quantum Dot in 2D (Disk)Filling the Electron Shells in 2D
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Shell Structure of Energy Levels for Various PotentialsMagic
Numbers (in 3D)Potentials:
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Two Step Tunnelingsource dot drain(For a detailed explanation
see the annotation in the .ppt version.)
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Coulomb energy U=e2/C of a spherical dot embedded in a medium
with dielectric constant , with the counter electrode at infinity :
2e2/ dCoulomb Energy U
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Conditions for a Coulomb Blockade1) The Coulomb energy e2/C
needs to exceed the thermal energy kBT.Otherwise an extra electron
can get onto the dot with thermal energy instead of being blocked
by the Coulomb energy. A dot needs to be either small ( h/e2 26
k
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A superconducting SET sample with a 2 mm long island and 70 nm
wide leads. The gate at the bottom allows control of the number of
electrons on the island.Superconducting SET
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Current vs. charge curves for a superconducting dot with normal
metals as source and drain. At low temperatures (bottom) the period
changes from e to 2e, indicating the involvement of Cooper
pairs.Superconducting SET
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Single Electron Turnstile
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Precision Standards from Single ElectronicsCount individual
electrons, pairs, flux quanta (f = frequency)I = e fV = h/2e fV/I =
R = h/e2
This picture is for a small negative source voltage (and an
equally small positive drain voltage) which draws electrons from
source to drain. Both voltages are assumed to be negligible
compared to the gate voltage.Note the different voltage scales (V
versus mV). The same This is a more sophisticated version of the
energy level diagram of a single electron transistor which combines
the Coulomb energy with the quantum energy. The Coulomb energy is
the large energy difference between the blocks of narrowly-spaced
quantum energy levels. When changing the number of electrons from N
(left) to N+1(right), all quantum levels are shifted up by the
Coulomb energy U.
Left: An electron gets ready to tunnel (hop) across the energy
barrier of the insulator into the lowest empty quantum level of the
dot (the LUMO in chemistry language).Right: When the electron sits
on the dot, we have to use the level diagram for N+1 electrons,
where all the levels are shifted up by the charging energy (right).
The extra electron now occupies the highest occupied quantum level
(the HOMO in chemistry language). Eventually it tunnels out to the
drain.
The gate voltage shifts these energy levels up and down and
thereby makes it possible to access levels with different electron
numbers N. Some nitty-gritty details:
The actual charging energy of a capacitor is given by QV (Q =
charge, V = Voltage). The factor comes from the fact that not all
of the charge is added at full voltage when charging a capacitor.
Use the definition of the capacitance C=Q/V and set Q=e.
The energy difference U between the levels with N and N+1
electrons on a dot is twice the charging energy. This can be seen
from the Coulomb diamonds in Slide 5. To change the most stable
charge state at the center of each diamond by 1 requires twice the
gate voltage Vg than to get conductance across the dot (see the
half-integer conductance peaks).
In Lecture 2, Slide 2, the dielectric constant of silicon was
used to arrive at the length scale of 9 nm.
Another source of confusion is the choice of the capacitance. A
SET has (at least) two capacitances to choose from, the gate-dot
capacitance Cg and the capacitance C between the gate and all the
other electrodes (gate, source, drain). Roughly speaking, one needs
to choose Cg when varying the gate voltage and C when varying the
source-drain voltage. That gives rise to the different voltage
scales in the Coulomb diamond plot (Slide 6, note the different
scales, V versus mV). Adding the same charge to the dot requires a
much larger gate voltage Vg than source-drain voltage Vs-d . This
is due to the relation V=e/C (definition of the capacitance) and
the fact that Cg is always smaller than C (parallel capacitances
add up).