Determination of the Chemical Potential for Transport Coefficients

The accurate determination of the chemical potential is one of the key ingredients in the computation of electronic transport coefficients such as conductivity, mobility, and thermopower. Transport coefficients depend sensitively on the occupation of electronic states around the Fermi energy. Because the occupation is governed by the Fermi–Dirac distribution, an accurate evaluation of the chemical potential as a function of temperature and doping is essential.

Why the chemical potential matters

At zero temperature in an undoped system, the chemical potential coincides with the Fermi energy. However, in real materials and realistic conditions:

The system may be doped, i.e. it contains additional electrons or holes.
The system may be at finite temperature, which modifies the balance between electron and hole occupations.
The transport coefficients are dominated by states within a narrow energy window around the chemical potential, so even small inaccuracies can lead to large errors.

Thus, in any calculation of electron–phonon interaction and related transport properties, it is crucial to have a consistent way of specifying and determining the chemical potential.

Different ways to specify carriers

There are several equivalent, but practically distinct, ways of describing the carrier concentration in a solid. Each corresponds to a different way of constraining or shifting the chemical potential in the calculation:

Shift of the chemical potential: One can directly specify a shift of the chemical potential with respect to the undoped zero-temperature Fermi energy. This approach is straightforward when the doping is weak and the Fermi level remains inside the same band.

Carrier density: Alternatively, one can specify the additional carrier density (per volume). This is the natural way to connect with experimental conditions, where doping levels are often given as carrier densities (e.g. 10^18 cm^-3). The calculation must then solve self-consistently for the chemical potential that yields the specified carrier density.

Extra carriers per unit cell: Finally, one may work in terms of the number of additional carriers per unit cell. This is a natural choice in periodic first-principles calculations, where the fundamental unit is the primitive cell. Again, a self-consistent procedure determines the chemical potential consistent with the requested number of carriers.

Each of these perspectives can be translated into the others, but depending on the context, one choice may be more convenient. For example, experiments usually quote carrier density, while theoretical band-structure calculations naturally yield the relation between chemical potential and extra carriers per cell.