solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...

, an energy gap is an energy range in a

solid Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...

where no

electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...

states exist, i.e. an energy range where the

density of states In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...

vanishes. Especially in

condensed-matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...

, an energy gap is often known more abstractly as a

spectral gap In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to othe ...

, a term which need not be specific to electrons or solids.

Band gap

If an energy gap exists in the

band structure In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or ' ...

of a material, it is called

band gap In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (in ...

. The physical properties of

semiconductor A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...

s are to a large extent determined by their band gaps, but also for insulators and metals the band structure—and thus any possible band gaps—govern their electronic properties.

Superconductors

For superconductors the energy gap is a region of suppressed density of states around the

Fermi energy The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. In a Fermi ga ...

, with the size of the energy gap much smaller than the energy scale of the band structure. The superconducting energy gap is a key aspect in the theoretical description of superconductivity and thus features prominently in

BCS theory BCS theory or Bardeen–Cooper–Schrieffer theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory describes sup ...

. Here, the size of the energy gap indicates the energy gain for two electrons upon formation of a

Cooper pair In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Coope ...

. If a conventional superconducting material is cooled from its metallic state (at higher temperatures) into the superconducting state, then the superconducting energy gap is absent above the critical temperature

T_

, it starts to open upon entering the superconducting state at

T_

, and it grows upon further cooling. BCS theory predicts that the size

\Delta

of the superconducting energy gap for conventional superconductors at zero temperature scales with their critical temperature

T_

\Delta(T=0)=1.764 \, k_ T_

(with

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...

k_

Pseudogap

If the density of states is suppressed near the Fermi energy but does not fully vanish, then this suppression is called

pseudogap In condensed matter physics, a pseudogap describes a state where the Fermi surface of a material possesses a partial energy gap, for example, a band structure state where the Fermi surface is gapped only at certain points. The term pseudogap wa ...

. Pseudogaps are experimentally observed in a variety of material classes; a prominent example are the cuprate high-temperature superconductors.

Hard gap vs. soft gap

If the density of states vanishes over an extended energy range, then this is called a hard gap. If instead the density of states exactly vanishes only for a single energy value (while being suppressed, but not vanishing for nearby energy values), then this is called a soft gap. A prototypical example of a soft gap is the

Coulomb gap First introduced by M. Pollak, the Coulomb gap is a soft gap in the single-particle density of states (DOS) of a system of interacting localized electrons. Due to the long-range Coulomb interactions, the single-particle DOS vanishes at the chemic ...

that exists in localized electron states with Coulomb interaction.

References

{{Reflist Electronic band structures Superconductivity