The quantum correction of the Einstein relation for high frequencies

For nondegenerate semiconductors the low-frequency Einstein relation qD₀ = kTμ₀ must be replaced by qD′(ω) = kTμ′(ω)p(ω) at high frequencies, where D′ and μ′ are the real parts of the complex diffusion coefficient D and the mobility μ, respectively, and p(ω) is the quantum correction factor for thermal noise. This is shown, first by treating thermal noise as a case of velocity-fluctuation noise, and secondly by using a simple statistical model. The results are extended to degenerate semiconductors and metals.