Density of states and occupancy statistics

Density of states

The density of states $g(E)$ is the number of available electron states per unit energy per unit volume at energy $E$. So $g(E)\,dE$ is the number of states (per m³) between $E$ and $E + dE$. For free electrons in a metal

$$g(E) = \frac{8\sqrt{2}\,\pi\,m^{3/2}}{h^3}E^{1/2} \propto \sqrt{E}$$

Fermi–Dirac statistics

$g(E)$ only says how many states exist. To know how many are occupied, quantum mechanics uses Fermi–Dirac (FD) statistics, which applies to electrons because they are identical, indistinguishable particles that obey the Pauli exclusion principle (at most one electron per state).

The probability that a state of energy $E$ is occupied at temperature $T$ is

$$f(E) = \frac{1}{1 + \exp\left(\dfrac{E - E_F}{kT}\right)}$$

where $E_F$ is the Fermi energy and $k$ is Boltzmann's constant.

Properties:

• At $T = 0$ K: $f(E) = 1$ for $E < E_F$ and $f(E) = 0$ for $E > E_F$ (a sharp step).
• At $T > 0$: $f(E_F) = \frac{1}{2}$ at every temperature; the step becomes smooth over a range of a few $kT$ around $E_F$.
• For $E - E_F \gg kT$, $f(E) \approx e^{-(E - E_F)/kT}$ (Boltzmann approximation, used for semiconductors).

 f(E)
  1 +----------.          T = 0 K: step
    |           \
 0.5|- - - - - - x         T > 0: smooth
    |             \___
  0 +----------|-------------> E
              E_F

Number of occupied states

The number of electrons per unit volume per unit energy is the product:

$$n_E(E) = g(E)\,f(E)$$

and the total electron concentration is $n = \int_0^\infty g(E)f(E)\,dE$. At 0 K this gives $E_F = \frac{h^2}{8m}\left(\frac{3n}{\pi}\right)^{2/3}$.

Example: probability of occupation of a state $3kT$ above $E_F$ is $1/(1 + e^3) = 0.047$.