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It is easy to understand that for a free electron, we can easily define the energy state density, and by doing the integration of the State density* Fermi-Dirac distribution we will be able to figure out the chemical potential at zero kelvin, which is the Fermi-Energy. Hence, we can define the Fermi Momentum correspondingly.

However, I feel rather struggling to define the fermi-momentum and Fermi energy for a non-parabolic dispersion.

Say, For example, A 2D system, I have a dispersion

E = A*k^2 +B*k^3

So k(E) will be in general complicated due to its cubic relations.

So If we still want to evaluate the Fermi Energy, Do still solve the integration

int _0 ^E_f state density*Fermi-Dirac Distribution dE = Number of Particles

to evaluate E_f?

And I think this integration will be very complicated.

Also, Under this circumstance, how do we define the Ferim-Momentum? Do we still solve

E_f = A k_F^2 + B k_F^3 to find k_F?

Thanks!