# What is a superconducting energy gap

## Energy gap

**Energy gap**, *Band gap*, *engl. Gap*, Energy range in which the presence of electrons is energetically forbidden. Since the energy bands in the semiconductor are functions of the electron pulse (wave number vector), the distance between the valence and conduction bands is not the same everywhere in k-space. The energy gap is the difference between the highest energy in the valence band and the lowest energy in the conduction band. If the wavenumber vector is the same for both extremes, then one speaks of *direct band gap*, otherwise from *indirect band gap* (Fig. 1). The gaps are on the order of 0.1 to 10 eV. The energy gap depends on many substances *Δ* with the transition temperature *T*_{c} according to *Δ* ≈ 3,5 *k*_{B.}*T*_{c} together.*Energy gap of a superconductor*: Within the BCS theory, the excitation spectrum of a superconductor is through the relationship

given (* k*: Wavenumber vector,

*Δ*

*(*

_{k}*T*):

*- and*

**k***T*-dependent energy gap;

*ε*

*:*

_{k}*-dependent band energy of the conduction electrons). Even low concentrations of impurities lubricate the*

**k***-Dependence from:*

**k***Δ*

*(*

_{k}*T*) ≡

*Δ*(

*T*). That resulting from the BCS theory

*Δ*(

*T*) is shown in Fig. 2, above the transition temperature

*T*

_{c}disappears

*Δ*(

*T*). In certain cases the loophole can also be used for

*T*<>

*T*

_{c}disappear completely (complete superconductivity) or at least some of them*-Points have nodes (e.g. at***k***d-wave superconductivity*, Pair wave function of a superconductor). Within the BCS theory, 2 applies*Δ*(0)/*k*_{B.}*T*_{c}= 3.52, but for strongly coupling superconductors (Eliashberg theory) this ratio can reach values of up to about five. In this case, the energy gap has an additional frequency dependency, which allows direct conclusions to be drawn about the coupling mechanism (electron-phonon mechanism of superconductivity). The energy gap determines the thermodynamic properties of the superconductor at low temperatures (e.g. exponential decay of the electronic specific heat) and manifests itself as a threshold behavior e.g. in ultrasonic absorption or*Infrared absorption*, whereby it can also be determined experimentally (at*T*= 0 and for quantum energies*H**ν*<>*Δ (0) no more absorption occurs). The most precise method for determining the energy gap is the one-electron tunnel effect.***Energy gap 1: **In the a) direct band gap, the conduction band edge and the valence band edge have the same wavenumber vector ** k**. In the b) indirect band gap, the two wavenumber vectors are different. The threshold energy is in the indirect transition around

increased since both a photon and a phonon are required.

*E.*

_{G}Energy gap,

*Ω*Frequency of a phonon with the wavenumber vector

*k*_{c}.

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