Fermi level

The Fermi level is the chemical potential of electrons in a solid, defined as the energy required to add one electron to the system at constant volume and entropy. At absolute zero, it coincides with the Fermi energy, the highest occupied energy level. At finite temperatures, the Fermi level shifts slightly with temperature and doping, reflecting the change in electron density. In a metal, the Fermi level lies within a conduction band, allowing electrons to move freely and conduct electricity. In semiconductors, it sits in the band gap, and its position determines the type of conductivity: closer to the conduction band for n-type, closer to the valence band for p-type.

In semiconductors, the Fermi level governs the population of charge carriers. Doping with donor atoms raises the Fermi level toward the conduction band, while acceptor doping lowers it toward the valence band. The built-in potential at a p–n junction arises from the difference in Fermi levels across the junction, which aligns at equilibrium. This alignment is crucial for diode and transistor operation. The Fermi level also determines the work function, the minimum energy needed to remove an electron from the surface, which is key to field emission and thermionic emission. In heterojunctions, band offsets are set by aligning the Fermi levels of the two materials.

The Fermi level is not directly measurable but is inferred from techniques such as ultraviolet photoelectron spectroscopy (UPS) and Kelvin probe force microscopy (KPFM), which probe the work function. In metals, the Fermi surface can be mapped via angle-resolved photoemission spectroscopy (ARPES) or de Haas–van Alphen oscillations. In semiconductors, the Fermi level position is extracted from capacitance–voltage (C–V) measurements or from the temperature dependence of carrier concentration. These methods are essential for characterizing electronic materials and devices.

Beyond standard applications, the Fermi level plays a role in exotic phenomena. In topological insulators, the Fermi level lies in a bulk gap but crosses surface states that are protected by time-reversal symmetry. In graphene, the Fermi level can be tuned by an electric field, enabling the observation of the quantum Hall effect at room temperature. In thermoelectric materials, the Fermi level is optimized to maximize the Seebeck coefficient. The concept also extends to nuclear physics, where the Fermi level of nucleons in a nucleus influences nuclear stability. Historically, the term was coined by Arnold Sommerfeld in the 1920s, building on Enrico Fermi's statistical work.

Chemical potential

The energy required to add one particle to a system at constant entropy and volume.

Fermi energy

The highest occupied energy level at absolute zero.

Band gap

The energy range in a solid where no electron states exist.

Work function

The minimum energy needed to remove an electron from a solid surface.