Question #42887

Consider a macroscopic intrinsic semiconductor in thermal equilibrium. The average electron and hole concentrations are given by the Law of Mass Action as $\langle n \rangle = \langle p \rangle = n_i$. To analyze the noise floor in the drift-diffusion balance, we model the semiconductor as a series of adjacent microscopic cubic regions, each of volume $V = L^3$. Due to the discrete nature of charge, the actual number of electrons ($N_e$) and holes ($N_h$) in each region fluctuate independently according to Poisson statistics.

The instantaneous electron and hole diffusion currents from region 1 to an adjacent region 2 are modeled as: $I_e = \frac{e D_e}{L^2} (N_{e2} - N_{e1})$ $I_h = -\frac{e D_h}{L^2} (N_{h2} - N_{h1})$ where $N_{e1}, N_{e2}$ and $N_{h1}, N_{h2}$ are the instantaneous number of electrons and holes in regions 1 and 2, respectively. $D_e$ and $D_h$ are the diffusion coefficients, and $e$ is the elementary charge. What is the root-mean-square (RMS) value of the total instantaneous diffusion current, $I_{RMS} = \sqrt{\langle (I_e + I_h)^2 \rangle}$, between the two regions?