Question #43790

Consider an n-type semiconductor at a low temperature $T$ (in the freeze-out regime) doped with a donor concentration $N_D$. The donor electrons can be modeled as quasi-hydrogen atoms. The semiconductor has a relative permittivity $\epsilon_r$ and an effective electron mass $m^*$. The ionization energy of a free hydrogen atom is $E_0$, and the free electron mass is $m_e$. In this regime, the conduction electron concentration is given by $n = \sqrt{N_D N_c} \exp\left(-\frac{\Delta E}{2k_B T}\right)$, where $\Delta E$ is the donor ionization energy and $N_c$ is the effective density of states in the conduction band. Assume $N_c \propto T^{3/2}$ and the electron mobility due to ionized impurity scattering scales as $\mu \propto T^{3/2}$. Which of the following expressions correctly represents the temperature coefficient of resistivity, $\alpha = \frac{1}{\rho}\frac{d\rho}{dT}$, of this semiconductor?