Question #42898

In a non-degenerate n-type semiconductor, the electron mobility $\mu_e$ depends on the absolute temperature $T$ as $\mu_e(T) = \mu_0 (T/T_0)^{-\alpha}$, where $\mu_0$ and $\alpha$ are positive constants ($\alpha$ is dimensionless), and $T_0$ is a reference temperature. The semiconductor is subjected to an electric field with an $x$-component $E$ and maintains a constant electron concentration gradient $\frac{dn}{dx} > 0$ along the $x$-axis. The electron concentration at the point of observation is $n$. Assume the Einstein relation $D_e = \frac{k_B T}{e} \mu_e$ holds valid, where $k_B$ is the Boltzmann constant and $e$ is the elementary charge. If the total electron current density $J_n$ along the $x$-axis is independent of small temperature variations around $T = T_0$ (i.e., $\left. \frac{dJ_n}{dT} \right|_{T=T_0} = 0$), what is the required value of $E$?