Coulomb blockade

From: zigoteau (zigoteau_at_yahoo.com)
Date: 01/23/05 

Date: Sun, 23 Jan 2005 14:57:05 +0000 (UTC)

Hi, All,

I have just contributed to a thread on sci.physics initiated by Sam
Wormley, but think that its topic is important enough to rate its own
thread in this group. It is appropriate to provide a bit more
overview, and to correct a few typos I made.

I think that current descriptions of the Coulomb blockade are to some
extent blinding with science. It is possible to visualize the effect
in much simpler terms, and it is then apparent that some of the claims
made about it are not sustainable.

The seminal paper was Averin and Likharev, J. Low Temp. Phys. 62
(1986) 345 (PDFAOR). There's nothing (much) wrong with their math
derivation, but it is possible to describe it in much simpler terms
allowing greater intuition.

An unfortunate misprint makes it unclear, but their references and
other literature confirm, that the theory completely neglects
interactions between electrons. That's perfectly justified for
electrons in metallic nanoparticles. It is however an important point.
The phenomenon is not a subtle quantum effect. It's zeroth-order,
in-your-face, involving only single-particle analysis.

The blinding with science involves an analysis in which the
independent variable is taken to be the charge Q on the nanoparticle.
That's highly unusual, and, I would estimate, loses 99.99% of readers.
However the effect still falls out of an analysis in which the
independent variable is the voltage V of the nanoparticle. Let's
analyze an SET in which the nanoparticle is surrounded by source,
drain and gate electrodes. The source and drain are separated from the
nanoparticle by tunneling gaps whose capacitance is kept as low as
possible. The gate subtends a much larger solid angle at the
nanoparticle so that the gate-nanoparticle capacitance is much larger,
but the gap is so great that the tunneling probability is negligible.

Because of the coupling to the electrodes, there are no electron
eigenstates completely localized on the nanoparticle. However since
you need to work with localized functions, the Hamiltonian will have
off-diagonal elements. The simplest basis set to work with is the
eigenstates of the isolated nanoparticle. You then need to consider
two things (1) the charge in each of these basis states and (2) the
current between the source and drain electrodes. Since it's all
one-electron stuff it doesn't involve too much effort.

Both the occupation of each basis state and its condtribution to the
low-voltage conductivity are described by Lorentzian functions. The
occupation probability is multiplied by the Fermi-Dirac function and
integrated over all energies. The low-voltage conductivity is
multiplied by the derivative of the Fermi-Dirac function and
integrated over all energies.

At low enough temperatures and small enough tunneling probabilities
the Lorentzians don't overlap with one another (A & L appear to have
forgotten about spin degeneracy in non-magnetic systems, but let's
move on). You get a peak of the conductivity when the Fermi level of
the source electrode coincides with the center of one basis state.
When this happens, the basis state is on average half-occupied. Other
basis states are either completely full, or completely empty. Hence
the expectation charge Q on the nanoparticle has the form (n+1/2)*e,
where n is an integer.

The SET is often described as "a turnstile for electrons". But from
the above analysis it is clear that there is nothing going "click,
click". It's more like a series of doors for electrons, like one of
those airport trains. You can get from the "In" platform to the "Out"
platform only when there is a train in the station and the carriage
doors are open on both sides. In the nanoparticle, current can get
from the source to the drain only when one of the nanoparticle levels
is lined up with the Fermi levels on both sides. And the number of
electrons taking advantage of the opportunity is a random thing,
described by Poisson statistics.

The only quantization in the system is related to the parameter Q. But
Q is the expectation charge, not the instantaneous charge. There is
nothing to stop the instantaneous charge from fluctuating.

There is another aspect disguized by the complexity of the A&L choice
of Q as independent variable. The A&L derivation is only manageable
when you do it in various limits of low temperature, weak coupling,
etc. However if you take V as the independent parameter you can make
things a lot more realistic. In particular the expectation charge Q
is controlled by the voltage VG on the gate electrode. The A&L
analysis assumes that Q/VG is the capacitance CG between the gate and
nanoparticle, but with the independent-V analysis you see that in fact
 Q = CG*(VG-VN), where VN is the voltage on the nanoparticle, which
increases as you fill up successive nanoparticle states. If QN is the
charge on the nanoparticle then you can define a value Cd with the
dimensions of a capacitance equal to e/DeltaVN, where DeltaVN is the
voltage gap between successive states of the nanoparticle. Typically
Cd is much larger than CG, but for the SET to work you have to try to
maximize CG, and for a 10nm nanoparticle the ratio is not many orders
of magnitude. People have proposed that an SET be used as a highly
precise way of measuring small charges and currents, but to make the
difference between the states as distinct as possible at as high a
temperature as possible you need as small a nanoparticle as possible,
and this effect will complicate precise work.

Cheers,

Zigoteau.