Free charge carrier statistics in drift–diffusion models

3. Free charge carrier statistics

Free charge carrier statistics define the relationship between quasi-Fermi levels and carrier densities. For a given electrostatic potential and quasi-Fermi level, they determine the local electron and hole concentrations \(n\) and \(p\). In OghmaNano, this relation may be described using either the classical Maxwell–Boltzmann (MB) approximation or the full Fermi–Dirac (FD) formalism, depending on the degree of degeneracy and the form of the density of states (DOS).

The MB approximation is valid when the quasi-Fermi levels lie several \(kT\) away from the band edges, such that occupation probabilities are small and Pauli exclusion effects can be neglected. Under this assumption, the carrier densities are exponentially related to the quasi-Fermi levels:

\[n_{l}=N_c \exp\!\left(\frac{F_n-E_{c}}{kT}\right)\]

\[p_{l}=N_v \exp\!\left(\frac{E_{v}-F_p}{kT}\right)\]

where \(N_c\) and \(N_v\) are the effective densities of states in the conduction and valence bands, \(F_n\) and \(F_p\) are the electron and hole quasi-Fermi levels, and \(E_c\), \(E_v\) define the local band edges. This formulation implicitly assumes a parabolic band structure and a thermalized carrier population.

When carrier densities become sufficiently high that the quasi-Fermi level approaches or enters the band, or when the DoS deviates from a simple parabolic form, the MB approximation breaks down and the full FD formalism must be used. In this case, carrier densities are obtained by integrating the DOS weighted by the Fermi–Dirac occupation function:

\[n_{\mathrm{free}}(E_{f},T)=\int_{E_{\min}}^{\infty} \rho(E)\, f(E,E_{f},T)\, dE\]

\[p_{\mathrm{free}}(E_{f},T)=\int_{E_{\min}}^{\infty} \rho(E)\, \bigl[1-f(E,E_{f},T)\bigr]\, dE\]

where the Fermi–Dirac distribution is given by

\[f(E,E_f,T)=\frac{1}{1+\exp\!\bigl((E-E_f)/kT\bigr)}\]

and \(\rho(E)\) is the density of states. For a three-dimensional parabolic band, the DOS takes the form

\[\rho(E)_{3D}=\frac{\sqrt{E}}{4\pi^2}\left(\frac{2m^{*}}{\hbar^2}\right)^{3/2}\]

where \(m^*\) is the carrier effective mass and \(\hbar\) is the reduced Planck constant. In practical device simulations, \(\rho(E)\) may instead be specified explicitly to capture exponential tails, Gaussian disorder, or other non-parabolic features characteristic of amorphous or organic systems.

The statistical description also determines the average carrier energy, which enters scattering and recombination models. This quantity is obtained as

\[\label{eq:energy} \bar{W}(E_{f},T)= \frac{\int_{E_{\min}}^{\infty} E\,\rho(E)\,f(E,E_{f},T)\, dE} {\int_{E_{\min}}^{\infty} \rho(E)\,f(E,E_{f},T)\, dE} \]

In the non-degenerate MB limit with a parabolic DoS, this expression reduces to \(\bar{W}=\tfrac{3}{2}kT\), reflecting equipartition of energy. However, in systems with energetic disorder or strong degeneracy, the carrier distribution is distorted, and the mean energy deviates from this classical value, directly impacting recombination kinetics and transport coefficients.