Importance of Fermi energy in semiconductors

The Fermi level $E_F$ is the energy at which the probability of occupation by an electron is exactly one half, from the Fermi-Dirac function

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

In a semiconductor it usually lies in the band gap, where there are no states, but it still fixes how many electrons and holes are present.

Importance

1. Sets carrier concentrations. For a non-degenerate semiconductor:

$$n = N_c\exp\left[-\frac{E_c - E_F}{kT}\right], \qquad p = N_v\exp\left[-\frac{E_F - E_v}{kT}\right]$$

The closer $E_F$ is to $E_c$, the more electrons; the closer to $E_v$, the more holes.

2. Shows the type of semiconductor.

 Intrinsic       n-type          p-type
 --- Ec          --- Ec          --- Ec
                 ... EF
 ... EF = EFi    (near Ec)
 (mid-gap)                       ... EF
 --- Ev          --- Ev          --- Ev  (near Ev)

• Intrinsic: $E_{Fi}$ near mid-gap, $E_{Fi} = \frac{E_c+E_v}{2} + \frac{3}{4}kT\ln\frac{m_h^*}{m_e^*}$.
• n-type: $E_F - E_{Fi} = kT\ln(N_d/n_i)$, above mid-gap.
• p-type: $E_{Fi} - E_F = kT\ln(N_a/n_i)$, below mid-gap.

3. Measures doping. The shift of $E_F$ from $E_{Fi}$ tells the doping level; heavy doping pushes $E_F$ into a band (degenerate semiconductor).

4. Constant in equilibrium. In any system in thermal equilibrium, $E_F$ is the same everywhere. This rule gives band bending, the built-in potential of a pn junction ($eV_0 = E_{Fn} - E_{Fp}$ before contact) and metal-semiconductor contact potentials.

5. Applied voltage. An applied voltage $V$ separates the Fermi levels on the two sides by $eV$. $dE_F/dx = 0$ means no net current; a gradient in $E_F$ drives current.

6. Temperature behaviour. As $T$ rises, $E_F$ of an extrinsic semiconductor moves towards mid-gap, showing the change to intrinsic behaviour.

7. Work function. $E_F$ fixes the work function $\Phi = E_{vac} - E_F$, which decides contact behaviour (ohmic or Schottky).