Mobility and chemical potential

Question

I have been studying the Boltztrap2 software to calculate the thermoelectric properties of materials. However when following the tutorial given, the Seebeck coefficient is plotted against mu as seen here under section 5 towards the bottom of the page.

However as far as I have seen (being a newbie), the symbol mu is used for both mobility as well as chemical potential. Is there a way to distinguish between the two?

Also, why is the graph plotted with mu-fermi energy? Does this give some advantage when visualizing the data? If I am working with an non-doped semiconductor, should I consider the Seebeck coefficient value at mu-fermi energy = 0? Should I be considering this mu-fermi energy axis as an indication of the doping level in the first place?

Answer 1

If you go to the definition of the Seebeck coefficient ($S$) you will find:

$$S = \frac{{e{k_B}}}{\sigma }\int {d\varepsilon \left( { - \frac{{\partial {f_0}}}{{\partial \varepsilon }}} \right)} \Xi \left( \varepsilon \right)\frac{{\varepsilon - \mu }}{{{k_B}T}}\tag{1}$$

Here: $e$ is the electronic elementary charge, $k_B$, $f_0$ and $\mu$ are the Boltzmann constant, the Fermi-Dirac function and the chemical potential, and $\Xi$ is the transport distribution (TD) function.

So, $\mu$ is the chemical potential with units of energy.

Answering your questions:

• Is there a way to distinguish between the two? Just by the symbol, no. But you can go through the used equations and check the physics behind or just check the units, for example.
• Also, why is the graph plotted with mu-fermi energy? I think that it will depend on what are you studying. If you are interested in knowing the behavior of the Seebeck coefficient with the electron concentrations, plotting against $\mu$ is a good idea, as it depend directly on the electron concentrations.
• Does this give some advantage when visualizing the data? Against, it is a choice. You can also plot it against the temperature, for example.
• Should I consider the Seebeck coefficient value at mu-fermi energy = 0? I don't thing so as $S$ depends on it directly. There are properties, like electronic bands, that you can "shift" without any problem because they don't depend on $\mu$ directly.
• Should I be considering this mu-fermi energy axis as an indication of the doping level in the first place? As $\mu$ directly depends on the electron concentration and the electron concentration can be modified by the doping level, yes, in principle, you can use $\mu$ as a parameter very related to doping level.

Answer 2

In this case, $\mu$ refers to the chemical potential. The reason it is given as $\mu-E_f$ is simply to highlight that the "zero" has been set to the Fermi-level of the reference calculation (usually, undoped bulk). In a pseudopotential calculation, there is an arbitrary, pseudopotential-dependent offset to the energies, so their absolute values are meaningless, and only differences are important.

You are correct that $\mu-E_f$ is essentially a measure of doping. BoltzTraP(2) uses the rigid-band approximation, i.e. it assumes that the electronic states (usually approximated by the Kohn-Sham states) do not change appreciably under doping, and so all that changes is the electronic chemical potential (and, hence, the band occupancies).

If you are interested in thermoelectric properties, you will never be working with an undoped semiconductor - apart from anything else, the electrical conductivity would be zero for most sensible temperature ranges, meaning that ZT would also be zero. In real materials, there are often intrinsic defects (e.g. vacancies or impurities) which act as dopants, so if you're comparing to experiment you'll need to estimate these effects and calibrate your calculations accordingly.

Having said all that, remember that the "zero" of doping is the actual simulation you did; usually, this would be the undoped semiconductor, but there's nothing stopping you putting a dopant or two in there. In that case, what BoltzTraP considers "undoped" would actually be your doped semiconductor!