Question #43719

A semiconductor rod of length $L$ is placed between two heat reservoirs maintained at temperatures $T_1$ (at $x = 0$) and $T_2$ (at $x = L$). The thermal conductivity of the rod is uniform and constant, and steady-state heat flow is established. The electron concentration in the rod is maintained as $n(x) = n_0 e^{-\alpha x}$, where $n_0$ and $\alpha$ are positive constants. The generalized one-dimensional electron charge current density is given by $J = e n \mu E + e \frac{d}{dx}(D n)$, where $e$ is the elementary charge, $\mu$ is the constant electron mobility, $E$ is the electric field, and $D$ is the diffusion coefficient. The diffusion coefficient satisfies the local Einstein relation $D(x) = \frac{k_B T(x)}{e} \mu$, where $k_B$ is the Boltzmann constant. Assuming the net electron current density is zero in the steady state, what is the built-in potential difference $\Delta V = V(L) - V(0)$ across the rod?