Question

Consider a two-dimensional electron gas in a square box of side L, with periodic boundary conditions.

a) Derive the density of states per unit energy, D(ε).

b) Find the Fermi wavevector, k_F, and the Fermi energy, ε_F, in terms of the number of electrons N and the area of the box, A=L² .

c) Calculate the density of states at the Fermi energy, D(ε_F), expressed in terms of N and A. Calculate the heat capacity, C_V.

Answer 1

A)

B)

C)

The fermi wave factor k_F is defined in terms of the Fermi energy E_F

E_F h^{cut 2} k_f² /2m

At zero temperature all states are occupied upto the fermi level

by periodic boundary condition

The seperation between states in the reciprocal lattice is 2/L_{i in each direction of length}

the electron concentration is

n₂g 1/(2)² k_F²k_F²/2

The wave factor k_F (2n₂)^1/2

Energy factor N 1/A