30 May 2025
Fermi-Dirac Distribution and Carrier Equilibrium
Fermi occupation, equilibrium electron and hole concentrations, and the mass-action law.
The Fermi level $E_F$ is the chemical potential for electrons. Thermal equilibrium gives the occupation probability
\[\boxed{f(E)=\frac{1}{1+\exp[(E-E_F)/(k_BT)]}}.\]At $E=E_F$, $f=1/2$. At $T=0$, every state below $E_F$ is occupied and every state above it is empty. Holes require an empty valence state, so their probability is
\[1-f(E)=\frac{1}{1+\exp[(E_F-E)/(k_BT)]}.\]Electron concentration
The electron density is
\[n=\int_{E_c}^{\infty}g_c(E)f(E)\,dE.\]For a non-degenerate semiconductor, $E_c-E_F\gtrsim3k_BT$, and
\[f(E)\simeq\exp[-(E-E_F)/(k_BT)].\]Substituting the parabolic-band density of states and setting $\epsilon=E-E_c$ gives
\[n=\frac{1}{2\pi^2}\left(\frac{2m_n^*}{\hbar^2}\right)^{3/2} e^{-(E_c-E_F)/(k_BT)} \int_0^\infty\epsilon^{1/2}e^{-\epsilon/(k_BT)}d\epsilon.\]With $u=\epsilon/(k_BT)$ and $\Gamma(3/2)=\sqrt\pi/2$,
\[n=2\left(\frac{2\pi m_n^*k_BT}{h^2}\right)^{3/2} e^{-(E_c-E_F)/(k_BT)}.\]Defining $N_c=2(2\pi m_n^*k_BT/h^2)^{3/2}$,
\[\boxed{n=N_c e^{-(E_c-E_F)/(k_BT)}}.\]An identical calculation for empty valence states gives
\[\boxed{p=N_v e^{-(E_F-E_v)/(k_BT)}},\qquad N_v=2\left(\frac{2\pi m_p^*k_BT}{h^2}\right)^{3/2}.\]Multiplication eliminates $E_F$:
\[\boxed{np=N_cN_v e^{-E_g/(k_BT)}=n_i^2}.\]This mass-action law applies at thermal equilibrium. For an intrinsic semiconductor, $n=p=n_i$, hence
\[n_i=\sqrt{N_cN_v}\,e^{-E_g/(2k_BT)},\]and
\[\boxed{E_i=\frac{E_c+E_v}{2}+\frac{k_BT}{2}\ln\frac{N_v}{N_c}}.\]The symbolic derivation is checked in carrier-statistics-check.mac; its mass-action and intrinsic-neutrality residuals are both zero.
Discussion