The variables W and
Wh are defined such that
E = Ec + W and
Eh = Ev −Wh. The density of states for electrons and holes then becomes (from
Equation 3-21 and
Equation 3-22):
The jth order
Fermi-Dirac integral (
Fj(
η)) is defined in the following manner:
where Γ is the gamma function and
Li the polylogarithm (see
Ref. 15 for a brief review on the properties of the Fermi-Dirac integral). Note that
Γ(3/2)
= π(1/2)/2. The electron and hole densities can be written in the compact forms:
One of the properties of the Fermi-Dirac integral is that Fj(
η)
→eη as
η→ −∞ (this result applies for all
j). In semiconductors this limit is known as the
nondegenerate limit and is often applicable in the active region of semiconductor devices. In order to emphasize the nondegenerate limit,
Equation 3-47 is rewritten in the form:
In the nondegenerate limit, the Fermi-Dirac distribution reverts to the Maxwell-Boltzmann distribution and
γn and
γp are 1. By default, COMSOL Multiphysics
uses Maxwell-Boltzmann statistics for the carriers, with
γn and
γp set to 1 in
Equation 3-49, irrespective of the Fermi level. When Fermi-Dirac statistics are selected
Equation 3-50 is used to define
γn and
γp.
In the Semiconductor Equilibrium study step, the above equations are used to express the carrier concentrations as functions of the electric potential, and the electric potential is solved via the resulting partial differential equations.
For the density-gradient formulation (Ref. 45), the formulas above are modified with additional contributions from the quantum potentials
VnDG and
VpDG (SI unit: V):
where the quantum potentials VnDG and
VpDG are defined in terms of the density gradients:
The density-gradient coefficients bn and
bp (SI unit: V m^2) are given by the inverse of the density-gradient effective mass tensors
mn and
mp (kg):