= Solution
{c}
Near a parabolic <valence band> maximum $E_v$, the missing-electron <effective mass> is $m_h^*>0$. The same two-dimensional state counting gives
$$
\boxed{p=\frac{m_h^*k_BT}{\pi\hbar^2}e^{-\beta(\mu-E_v)}\equiv N_v e^{-\beta(\mu-E_v)}.}
$$
Multiplication eliminates the <chemical potential>, giving the <semiconductor mass-action law>
$$
\boxed{np=N_cN_v e^{-E_g/(k_BT)}=m_e^*m_h^*\left(\frac{k_BT}{\pi\hbar^2}\right)^2e^{-E_g/(k_BT)},\qquad E_g=E_c-E_v.}
$$
For an <intrinsic semiconductor>, <charge neutrality> imposes $n=p=n_i$, so
$$
n_i=\frac{\sqrt{m_e^*m_h^*}\,k_BT}{\pi\hbar^2}e^{-E_g/(2k_BT)}.
$$
The intrinsic <chemical potential> is not exactly midgap when the two <effective masses> differ: $\mu_i=(E_c+E_v)/2+(k_BT/2)\log(m_h^*/m_e^*)$. The exponential suppression still justifies the assumed nondegenerate regime here.
Back to article page
