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Current equation

Dear professor Datta, I’m sorry to bother you to ask a question puzzling me for a long time. I have read some papers on electron transport (such as PRB, 56, 3296) in metals and one of the central formula to calculate the electron current is, “j = sigma nabla mu”———equ. (1)

i.e. the electron current is equal to the product of the conductance sigma and the gradient of chemical potential mu (or electrochemical potential, it is hard to distinguish these two concepts). This formula is considered as a more universal one than that of the drift and diffusion form,

“j = sigma nabla V + D nabla n”———equ. (2), where V is the external voltage supply, D is the diffusion coefficient and n is the electron density.

equ. (2) is easy to understand and I try to deduce equ. (1) from equ. (2), but I failed. Would you please help me to interpret equ. (1) in a pellucid way and could you tell me the quantitative relationship between the external supply V(x), chemical potential mu(x) and electron density n(x) ?

Thank you very much!

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    Supriyo Datta

    I believe you can write (D(E) : Density of states) n(x) = Integral dE D(E) f(E-mu(x)-qV(x)) (3) where f(y) = 1/(1+exp(y/kT))

    I believe Eq.(1) is correct and other versions like Eq.(2) are derived by combining it with Eq.(3) .. possibly with additional approximations like the Boltzmann limit: f(y) ~ exp(-y/kT) ..

    Hope this helps

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