Fermi energy importance; degenerate semiconductors

Importance of Fermi energy

The Fermi energy $E_F$ is the energy level at which the probability of occupation by an electron is exactly 1/2 (Fermi–Dirac function $f(E) = 1/[1+e^{(E-E_F)/kT}]$). Its importance:

1. Gives carrier concentrations directly: $n = N_ce^{-(E_c-E_F)/kT}$ and $p = N_ve^{-(E_F-E_v)/kT}$. Its position tells how many electrons and holes are present.
2. Identifies the type of material: $E_F$ near mid-gap → intrinsic; near $E_c$ → n-type; near $E_v$ → p-type.
3. Equilibrium condition: in thermal equilibrium $E_F$ is constant throughout a system. This rule fixes the band bending and built-in potential of pn junctions and metal–semiconductor contacts ($eV_0 = E_{Fn} - E_{Fp}$, the difference between the Fermi levels of the separate n and p materials).
4. Bias and devices: an applied voltage splits the Fermi level into quasi-Fermi levels separated by $eV$; these are used to describe diodes, transistors and LEDs.
5. Work function and contact potential: $\Phi = E_{vac} - E_F$; differences in $E_F$ give contact potentials and thermocouple emf.
6. Temperature behaviour: its movement towards mid-gap with rising T shows when an extrinsic material becomes intrinsic.

Degenerate vs non-degenerate semiconductor

 Non-degenerate n     Degenerate n+
 Ec ---------         EF ......... (inside CB)
 EF ......  (>3kT)    Ec ---------
 Ev ---------         Ev ---------