The drift diffusion equations

3. Free charge carrier statistics

In OghmaNano you can choose how free carriers are described statistically: either by the classical Maxwell–Boltzmann (MB) approximation or by full Fermi–Dirac (FD) statistics. The appropriate choice depends on the material, carrier densities, and whether degeneracy or a non-parabolic / disordered density of states (DOS) must be captured.

Under the MB approximation (valid when quasi-Fermi levels lie several \(kT\) away from the band edges), the carrier densities are

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

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

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 \(E_c\), \(E_v\) are the local band edges.

When degeneracy or DOS shape matters, use full FD statistics. The electron and hole densities are then computed from

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

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

with the Fermi–Dirac distribution

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

and (for a 3D parabolic band) the DOS

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

where \(m^*\) is the effective mass and \(\hbar\) is the reduced Planck constant. In disordered or non-parabolic systems you may instead supply a custom DOS \(\rho(E)\).

The average carrier energy (useful for scattering/recombination models) is

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

In the MB limit with a parabolic DOS this reduces to the familiar \(\bar{W}=\tfrac{3}{2}kT\). For irregular or disordered DOS (e.g. organics, a-Si, hybrid perovskites), the integral must be evaluated and the average energy generally deviates from \(\tfrac{3}{2}kT\).