The Mattis–Bardeen theory is a theory that describes the electrodynamic properties of superconductivity. It is commonly applied in the research field of optical spectroscopy on superconductors. It was derived to explain the anomalous skin effect of superconductors. Originally, the anomalous skin effect indicates the non-classical response of metals to high frequency electromagnetic field in low temperature, which was solved by Robert G. Chambers. At sufficiently low temperatures and high frequencies, the classically predicted skin depth (normal skin effect) fails because of the enhancement of the mean free path of the electrons in a good metal. Not only the normal metals, but superconductors also show the anomalous skin effect which has to be considered with the theory of Bardeen, Cooper and Schrieffer (BCS).
Response to an electromagnetic wave The most clear fact the BCS theory gives is the presence of the pairing of two electrons (Cooper pair). After the transition to the superconducting state, the superconducting gap 2Δ in the single-particle density of states arises, and the dispersion relation can be described like the one of a semiconductor with band gap 2Δ around the Fermi energy. From the Fermi golden rule, the transition probabilities can be written as
α s = ∫ | M s | 2 N s ( E ) N s ( E + ℏ ω ) × [ f ( E ) − f ( E + ℏ ω ) ]
d E {\displaystyle \alpha _{s}=\int {\left|M_{s}\right|^{2}N_{s}(E)N_{s}(E+\hbar \omega )\times [f(E)-f(E+\hbar \omega )]{\rm {}}}dE}
where N s {\displaystyle N_{s}} is the density of states. And M s {\displaystyle M_{s}} is the matrix element of an interaction Hamiltonian H 1 {\displaystyle H_{1}} where
H 1 = ∑ k σ , k ′ σ ′ B k ′ σ ′ , k σ c k ′ σ ′ ∗ c k ′ σ ′ {\displaystyle H_{1}=\sum \limits _{k\sigma ,k'\sigma '}{B_{k'\sigma ',k\sigma }c_{k'\sigma '}^{*}}c_{k'\sigma '}}
In the superconducting state, each term of the Hamiltonian is dependent, because of the superconducting state consists of a phase-coherent superposition of occupied one-electron states, whereas it is independent in the normal state. Therefore, there appear interference terms in the absolute square of the matrix element. The result of the coherence changes the matrix element M s {\displaystyle M_{s}} into the matrix element M {\displaystyle M} of single electron and the coherence factors F(Δ,E,E').
F ( Δ , E , E ′ ) = 1 2 ( 1 ± Δ 2 E E ′ ) {\displaystyle F(\Delta ,E,E')={\frac {1}{2}}\left(1\pm {\frac {\Delta ^{2}}{EE'}}\right)}
Then, the transition rate is
α s = ∫ | M | 2 F ( Δ , E , E + ℏ ω ) N s ( E ) N s ( E + ℏ ω ) × [ f ( E ) − f ( E + ℏ ω ) ]
d E {\displaystyle \alpha _{s}=\int {\left|M\right|^{2}F(\Delta ,E,E+\hbar \omega )N_{s}(E)N_{s}(E+\hbar \omega )\times [f(E)-f(E+\hbar \omega )]{\rm {}}}dE}
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