Fixed-chemical-potential Sommerfeld density increase
= Fixed-chemical-potential Sommerfeld density increase
For a three-dimensional parabolic band and fixed positive $\mu-E_c$, the <Sommerfeld expansion> gives $n(T)=n(0)[1+\pi^2(k_BT/(\mu-E_c))^2/8+O(T^4)]$. This increase differs from the usual fixed-density calculation, in which the <chemical potential> decreases with <temperature>. The exact density is proportional to $T^{3/2}F_{1/2}[(\mu-E_c)/(k_BT)]$.