What Is the Fermi Level?

The Fermi level, also called the Fermi energy in the zero-temperature case, is the energy at which the probability of finding an electron state occupied is exactly one half. It is the electrochemical potential of the electron population in a solid, and it is the reference against which every other electronic energy in a device is measured. The concept follows from Fermi-Dirac statistics, formulated by Enrico Fermi and Paul Dirac in 1926, which govern particles obeying the Pauli exclusion principle. Because no two electrons may share a quantum state, filling a solid's available states from the bottom upward produces a sharp occupancy boundary rather than the smooth Boltzmann tail that classical particles would show.

The occupation probability is given by the Fermi function, in which the difference between a state's energy and the Fermi level is scaled by the thermal energy kT. At absolute zero the function is a step: every state below the Fermi level is full, every state above it is empty. At finite temperature the step softens over a window a few kT wide, kT itself being about 26 meV at room temperature, and only electrons within that window can respond to an applied field or absorb a small quantum of energy. The teaching module on the Fermi-Dirac distribution from Cambridge's DoITPoMS project works through this behavior for both metals and semiconductors.

Metals, Semiconductors, and Doping

In a metal the Fermi level lies inside a partially filled band, so states are available immediately above it and conduction requires no activation. Fermi energies in common metals fall between roughly 2 and 12 eV, corresponding to Fermi temperatures far above any temperature at which the metal remains solid, which is why the electron gas in a metal is degenerate and its heat capacity is small. In a semiconductor the Fermi level sits within the forbidden gap, where no states exist, and its position determines carrier concentrations. For an intrinsic material it lies near midgap, shifted slightly by any asymmetry in the effective densities of states of the conduction and valence bands. Donor doping raises it toward the conduction band edge and acceptor doping lowers it toward the valence band edge, and in heavily doped material it can move into a band entirely, producing degenerate semiconductor behavior.

Equilibrium and Band Alignment

The practical importance of the Fermi level comes from a single rule: when two materials are placed in contact and allowed to reach thermal equilibrium, charge flows until their Fermi levels coincide. Everything else about the junction follows. The mismatch in initial Fermi positions becomes the built-in potential, band edges bend near the interface over a screening length set by doping, and a depletion region forms. This is the basis of the p-n junction, the Schottky contact, the heterojunction, and the field-effect transistor channel, where a gate voltage separates the electron and hole quasi-Fermi levels and modulates conductivity. Under illumination or forward bias the population is no longer in equilibrium, and separate quasi-Fermi levels for electrons and holes describe each carrier type, with their separation setting the open-circuit voltage of a solar cell.

Fermi Level Pinning

At many metal-semiconductor interfaces the barrier height refuses to track the metal work function as simple band alignment predicts. Interface states, whether metal-induced gap states or defect and dangling-bond states, hold the semiconductor Fermi level near a fixed charge-neutrality position, an effect called Fermi level pinning. Pinning limits contact engineering because choosing a different metal barely changes the barrier. Two-dimensional materials have provided a route around it, and work on van der Waals metal-semiconductor junctions with weak pinning showed that a transferred contact restores tunability of the Schottky barrier. Related studies of interface passivation that reduces pinning and of barrier height control at oxide Schottky interfaces continue to refine the picture.

Applications

The Fermi level is central to work in a range of areas, including:

• Semiconductor device design, including diodes, transistors, and contacts
• Photovoltaics and photodetector voltage and quantum efficiency analysis
• Thermoelectric materials, where transport depends on states near the Fermi surface
• Electrochemistry and catalysis, where electrode potential is an electron energy
• Superconductivity and correlated electron physics