Fermi level

The Fermi level $E_F$ is the energy at which the Fermi-Dirac distribution function equals 1/2, the energy at which an electronic state has 50% probability of being occupied at thermal equilibrium. It is the central concept of semiconductor statistics and determines carrier concentrations, band alignment at junctions, and the thermal equilibrium behavior of all electronic devices.

Defining property. From the Fermi-Dirac distribution:

$$\begin{array}{c}{\displaystyle f(E)=\frac{1}{1+\exp [(E-E_F)\mathrm{/}{k}_{B}T]},}\\ {\displaystyle f(E_F)=1\mathrm{/}2.}\end{array}$$

States below $E_F$ are mostly occupied; states above are mostly empty. The transition width is $\sim k_{B}T=25.9$ meV at 300 K.

Equilibrium carrier concentrations. Combining the Fermi-Dirac distribution with the density of states:

$$\begin{array}{rl}{\displaystyle n}& {\displaystyle ={\int }_{E_c}^{\mathrm{\infty }}{g}_c(E)f(E)dE,}\\ {\displaystyle p}& {\displaystyle ={\int }_{-\mathrm{\infty }}^{E_v}{g}_v(E)[1-f(E)]dE\mathrm{.}}\end{array}$$

In the non-degenerate limit (Fermi level well within bandgap, several $k_{B}T$ from band edges):

$$\begin{array}{rl}{\displaystyle n}& {\displaystyle =N_c\exp [-(E_c-E_F)\mathrm{/}{k}_{B}T],}\\ {\displaystyle p}& {\displaystyle =N_v\exp [-(E_F-E_v)\mathrm{/}{k}_{B}T]\mathrm{.}}\end{array}$$

These are the standard formulas; together with $np=n_i^2$ (mass-action law), they determine equilibrium carrier statistics.

Intrinsic Fermi level. For an intrinsic (undoped) semiconductor, the Fermi level lies near mid-gap:

$$E_F^{\text{intrinsic}}=\frac{E_c+E_v}{2}+\frac{k_{B}T}{2}\ln (N_v\mathrm{/}{N}_c)\mathrm{.}$$

The slight offset from exact mid-gap depends on the asymmetry of conduction- and valence-band DOS. For GaAs ($N_v\mathrm{/}{N}_c=15$): $E_F^{\text{intrinsic}}$ is shifted $\sim 35$ meV above mid-gap toward the conduction band.

Doping-controlled Fermi level.

For degenerate doping ($N_D>N_c$ or $N_A>N_v$), the Fermi level enters the conduction band (n-type) or valence band (p-type) and the Boltzmann approximations no longer apply.

Fermi level and chemical potential. The Fermi level is the electrochemical potential of electrons. In a system without external fields, electrons flow from high $E_F$ to low $E_F$ until $E_F$ is uniform. This is the condition of thermal equilibrium. At equilibrium, $E_F$ is constant throughout the device:

• In a p-n junction at equilibrium: $E_F$ is flat; the band edges $E_c$ and $E_v$ bend to maintain the uniform $E_F$ as doping changes
• In a heterojunction at equilibrium: $E_F$ is flat; the band offsets $\mathrm{\Delta }{E}_c$ and $\mathrm{\Delta }{E}_v$ appear as abrupt steps in $E_c$ and $E_v$ at the interface, on top of the band bending in the depletion regions

Quasi-Fermi levels under bias. When a semiconductor is driven away from thermal equilibrium (forward bias, optical excitation), the single $E_F$ splits into separate "quasi-Fermi levels" for electrons ($E_{Fn}$) and holes ($E_{Fp}$). Their splitting characterizes the degree of non-equilibrium and determines optical gain:

$$\mathrm{\Delta }\mu =E_{Fn}-E_{Fp}\ge h{\nu }_{\text{photon}}\mathrm{.}$$

This Bernard-Duraffourg condition is the formal requirement for net optical gain.

Fermi level in contacts.

Fermi-level pinning at metal-semiconductor interfaces is a major design challenge for low-resistance contacts. Methods to mitigate:

• Heavy doping at the contact (improve tunneling)
• Selecting metals with appropriate work functions
• Inserting transitional barrier layers
• Compositional grading

Work function. The work function of a semiconductor is the energy difference between the Fermi level and the vacuum level:

$$\mathrm{\Phi }=\chi +(E_c-E_F),$$

where $\chi$ is the electron affinity (vacuum to $E_c$). Work function is critical for:

• Predicting Schottky barrier heights (${\mathrm{\Phi }}_{Bn}={\mathrm{\Phi }}_M-\chi$ Schottky-Mott formula)
• Designing heterojunction band alignment
• Selecting contact metals
• Photoemission spectroscopy interpretation

Hot-carrier Fermi level. Under strong injection or optical excitation, the carrier distribution can be non-thermal, characterized by an effective temperature $T_c>T_{\text{lattice}}$. The quasi-Fermi-level concept still applies but with the elevated carrier temperature.

Fermi level in 2D materials. For 2D materials (graphene, MoS₂, etc.) and 2DEGs (modulation-doped heterostructures), the Fermi level is referenced to the 2D band structure:

• Graphene: $E_F$ can be tuned through the Dirac point; carrier density varies dramatically with $E_F$ position
• 2DEG in GaAs/AlGaAs heterostructure: $E_F$ in the 2DEG plane is set by the donor depletion in the supply layer

Why Fermi level matters for photonics.

• Laser threshold: requires a quasi-Fermi-level separation greater than the photon energy $h\nu$ at the lasing wavelength, itself at least $E_g$
• Photodetector response: junction Fermi-level offset sets dark current and photocurrent
• Modulator efficiency: index/absorption changes track quasi-Fermi-level separation
• Solar cell efficiency: $V_{\text{oc}}=\mathrm{\Delta }{E}_F\mathrm{/}q$ where $\mathrm{\Delta }{E}_F$ is the quasi-Fermi-level splitting under illumination
• Carrier injection efficiency: alignment of metal Fermi level with semiconductor band edge determines contact resistance

Temperature dependence. The intrinsic Fermi level changes weakly with T (through the $T\ln (N_v\mathrm{/}{N}_c)$ term). For doped semiconductors at moderate doping, the Fermi level changes more strongly:

• Ionization of dopants: $E_F$ moves away from the dopant level toward mid-gap as the dopants ionize with rising $T$
• Carrier statistics: $E_F$ shifts toward mid-gap as $T$ increases ("intrinsic limit")
• Freezeout: at low T, dopants un-ionize and $E_F$ moves toward dopant level

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 16; Sze & Ng, Physics of Semiconductor Devices (3rd ed., 2007), Ch. 1 (canonical Fermi-level treatment); Pierret, Semiconductor Device Fundamentals (1996) for the comprehensive carrier-statistics derivation.