At T = 0 the Fermi energy is equal to the chemical potential μ. This zero temperature limit is intrinsically quantum mechanical, because all states below the Fermi level have occupation number equal to 1, and above the Fermi level have occupation number equal to 0.
So what will be the chemical potential of the system (metal or noninteracting fermions) at temperature T.?
Hi Niaz,
At a nonzero temperature, the chemical potential (mu) for a system of noninteracting fermions may be written as:
mu = Ef * [1-(1/3)* ( pi*kB*T/(2*Ef)) ^2] + ...
where Ef is the Fermi energy and kB is the Boltzmann constant.
There are also higher order contributions (order O(T^4)) but those are typically small and negligible. This expansion comes from the Sommerfield theory of metals.
I hope this is helpful.
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