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I'm trying to learn the concept of quantum dots and coulomb diamonds, and I'm trying to read this but I have encountered a problem which I couldn't find answer online. On the 4th page it mentioned:

By tuning the gates it is possible to tune $\tilde{\mu}_{N+1}$ to lie between the electro chemical potentials in source and drain, allowing electrons to tunnel on and off the dot one at a time

I think I can understand the meaning of chemical potential of the island/dot, as it is the energy required to put another electron onto the island/dot. But here the chemical potential of the source and drain doesn't seem to be clearly defined and I couldn't find any related information about it. Can anyone explain with more details? Also, are there any recommended textbooks that cover these details? Thanks!

Frank Wang
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The source and the drain are just electron reservoirs, they have a chemical potential in the same sense as a piece of metal. The difference between these two chemical potentials is what drives a current through the island between them. The assumption here is that the tunneling time is much longer than the relaxation time in the reservoirs, which allows us to consider them as equilibrium, even though the whole system is not - the tunneling through the central island can be viewed as a relaxation process.

Note also that defining the chemical potential for the central island is in fact more problematic than for the leads, because adding electrons cost different energy - depending on how many electrons are already in the island. This is why it is better to view the chemical potential as the parameter entering the Fermi function, rather than use literally the definition from the statistical mechanics.

There are exist many reviews about the Coulomb blockade, but the subject is quite old, and most of them are out of date with the state of the technology and therefore rather confusing. I could suggest looking up the Kouwenhoven's review on the double quantum dots and following the references.

Roger V.
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It's a fancy way of saying that they hooked up a battery (or voltage source) across the device, and thus the source and drain are at different voltages. Note that they're refering to the electrochemical potential, which is not quite the same thing as the chemical potential (altho the terms are sometimes used interchangeably). In particular, the electrochemical potential is not constant in space. Per Ashcroft and Mermin Eq. 29.7, the electrochemical potential

$$\mu_e\left(\vec{x}\right) = \mu + e \phi\left(\vec{x}\right),$$

where $\mu$ is the chemical potential and $\phi\left(\vec{x}\right)$ is the electric potential (i.e. the voltage due to the battery).

A better reference for your purposes might be the first chapter of "Quantum Transport" by Supriyo Datta, altho any book on transport or semiconductor devices will cover it. The topic is often briefly covered in solid state physics books --- usually in the context of a simple semiconductor device like a diode (as in Ashcroft and Mermin).

PS. The reference you cite is a very good one. In particular, it's precise with terminology. A lot of the time, the electrochemical potential will just be called the Fermi level, which I think is a confusing term. Also, as Vadim notes, the electrochemical potential in the island is actually a much thornier issue than in the source and drain --- large reservoirs, which can be treated as near equlibrium and thus have a well-defined chemical potential.

lnmaurer
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