## Extrinsic semiconductors (Boltzmann approximation)

The chemical potential μ of an extrinsic semiconductor is plotted as a function of temperature. At each temperature the chemical potential was calculated by requiring that charge neutrality be satisfied.

 μ [eV] T [K]
 Nc(300 K) = 1/cm³ Semiconductor Nv(300 K) = 1/cm³ Eg = eV Nd = 1/cm³ Donor Ec - Ed = eV Na = 1/cm³ Acceptor Ea - Ev = eV T1 = K T2 = K

Once the Fermi energy is known, the carrier densities $n$ and $p$ can be calculated from the formulas, $n=N_c\left(\frac{T}{300}\right)^{3/2}\exp\left(\frac{\mu-E_c}{k_BT}\right)$ and $p=N_v\left(\frac{T}{300}\right)^{3/2}\exp\left(\frac{E_v-\mu}{k_BT}\right)$.

The intrinsic carrier density is $n_i=\sqrt{N_c\left(\frac{T}{300}\right)^{3/2}N_v\left(\frac{T}{300}\right)^{3/2}}\exp\left(\frac{-E_g}{2k_BT}\right)$.

 $\log_{10}$ $n,p,n_i$ [cm-3] T [K]
 $\log_{10}$ $n,p,n_i$ [cm-3] 1/T [K-1]

See www.ioffe.rssi.ru/SVA/NSM/Semicond/index.html for the bandgaps and donor and acceptor states of various semiconductors.