Semiconductor Carrier Concentration

In an intrinsic (undoped) semiconductor, thermal excitation promotes electrons from the valence band to the conduction band, creating equal numbers of electrons (n) and holes (p).

Intrinsic Carrier Concentration

$n_i = \sqrt{N_c N_v} \exp\left(-\frac{E_g}{2k_BT}\right)$

where the effective density of states for the conduction band is:

$N_c = 2\left(\frac{2\pi m^* k_B T}{h^2}\right)^{3/2}$

For simplicity we assume N_c = N_v (equal effective masses for electrons and holes).

Temperature Dependence

The exponential factor dominates: doubling T dramatically increases n_i. This is why semiconductors become better conductors at high temperature (opposite to metals).

Note: Using free electron mass gives higher n_i than experimental values because real effective masses are smaller.

Conductivity

$\sigma = n e (\mu_e + \mu_h)$

where μ_e and μ_h are the electron and hole mobilities (m²/V·s).

Your Task

import math

def effective_dos_m3(T_K, m_eff_kg):
    # N_c = 2 * (2*pi*m*k_B*T / h^2)^(3/2)
    # k_B=1.381e-23, h=6.626e-34
    pass

def intrinsic_carrier_m3(E_g_eV, T_K, m_eff_kg):
    # n_i = sqrt(N_c * N_v) * exp(-E_g_J / (2*k_B*T))
    # N_c = N_v = effective_dos_m3(T_K, m_eff_kg)
    # e=1.602e-19 (to convert eV to J)
    pass

def conductivity_S_m(n_m3, mu_e_m2_V_s, mu_h_m2_V_s):
    # sigma = n * e * (mu_e + mu_h), e=1.602e-19
    pass