COMSOL 6.2 - Equilibrium Carrier Concentrations

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):

So the electron and hole densities are written in the form:

The following equations are obtained with the substitutions ε = W/(kBT):

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:

For the density-gradient formulation (Ref. 47), Equation 3-47 is replaced by one similar to Equation 3-53 below.

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 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.

Away from equilibrium the above equations can still be applied but instead of using the Fermi level, the quasi-Fermi level is employed in Equation 3-41 and Equation 3-42 (Ref. 11). For a detailed description of the origin of the quasi-Fermi levels for electrons and holes see The Semiconductor Equations. The final results are:

The formulas above are used to compute the carrier concentrations as functions of the quasi-Fermi levels, when the quasi-Fermi level formulation is used. In other formulations where the carrier concentrations are solved for, the formulas above are used to compute the quasi-Fermi levels as functions of the carrier concentrations.

For the density-gradient formulation (Ref. 47), 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):