Energy of band of d-dimensional semiconductor when voltage V is applied across

Apr 2019
1
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Let's say we have a one-dimensional semiconductor and I apply voltage V across it, I want to calculate the energy of a parabolic band, when a source and drain voltage is applied across it. I expect it to be $U = \Sigma_k \frac{\hbar^2 k^2}{2 m^*}f(k)$ where f(k) is the fermi function.

When no voltage is applied $U_0 = \Sigma_k \frac{\hbar^2 k^2}{2 m^*}f_0(k)$

(I am just confused what the fermi-level should be in the case when voltage is applied and I want to do a full integral including temperature dependence)

My question basically is, what is the expression for $f(k)$ and $f_0(k)$. Would it be different in the ballistic and diffusive regime? And how do I extend this for d-dimensional semiconductor?