Free electron-gas model

1D model and density of states

Assume an electron is confined in a infinite square wall, the electron wave function will be

where $k_n = n\pi / L$.

density of states is by definition “state per $\Delta k$”, and in 1D case it is

3D case

F-D distribution

Because $E_\text{f} \gg k_\text{B}T$, $f(E) = \mathrm{H}(E-E_\text{f})$ ($H(x)$ is the Heaviside function).

Number of electrons

1D: $N = k_\text{f} \, D(k)$

3D: $N = D_{k_\text{F}^3} \frac{1}{8} \frac{4}{3}\pi k_\text{F}^3$

Fermi energy

Fermi surface

$E(k_\text{f})$

“Energy” density of states

The energy density of states is $D(E)$, $D(E) \propto \sqrt E$

Because $k_n \propto E$, density of states gives $D(E)$ or $D(k)$,

Electronic energy and electronic specific heat

See also lattice specific heat.

Sommerfeld expansion…?

References

• Wikipedia contributors. (2021, February 12). Electronic specific heat. In Wikipedia, The Free Encyclopedia. Retrieved 20:09, February 17, 2021, from https://en.wikipedia.org/w/index.php?title=Electronic_specific_heat&oldid=1006278621