Mattis–Bardeen theory

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

${\displaystyle \alpha _{s}=\int {\left|M_{s}\right|^{2}N_{s}(E)N_{s}(E+\hbar \omega )\times [f(E)-f(E+\hbar \omega )]{\rm {}}}dE}$

where ${\displaystyle N_{s}}$ is the density of states. And ${\displaystyle M_{s}}$ is the matrix element of an interaction Hamiltonian ${\displaystyle H_{1}}$ where

${\displaystyle H_{1}=\sum \limits _{k\sigma ,k'\sigma '}{B_{k'\sigma ',k\sigma }c_{k'\sigma '}^{*}}c_{k'\sigma '}}$

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 ${\displaystyle M_{s}}$ into the matrix element ${\displaystyle M}$ of single electron and the coherence factors F(Δ,E,E').

${\displaystyle F(\Delta ,E,E')={\frac {1}{2}}\left(1\pm {\frac {\Delta ^{2}}{EE'}}\right)}$

Then, the transition rate is

${\displaystyle \alpha _{s}=\int {\left|M\right|^{2}F(\Delta ,E,E+\hbar \omega )N_{s}(E)N_{s}(E+\hbar \omega )\times [f(E)-f(E+\hbar \omega )]{\rm {}}}dE}$

where the transition rate can be translated to real part of the complex conductivity, ${\displaystyle \sigma _{1}}$, because the electrodynamic energy absorption is proportional to the ${\displaystyle \sigma _{1}E^{2}}$.

${\displaystyle {\frac {\alpha _{s}}{\alpha _{n}}}={\frac {\sigma _{1s}}{\sigma _{n}}}}$

In finite temperature condition, the response of electrons due to the incident electromagnetic wave can be regarded as two parts, the “superconducting” and “normal” electrons. The first one corresponds to the superconducting ground state and the next to the thermally excited electrons from the ground state. This picture is the so-called "two-fluid" model. If we consider the “normal” electrons, the ratio of the optical conductivity to the one of the normal state is

${\displaystyle {\frac {\alpha _{s}}{\alpha _{n}}}={\frac {2}{\hbar \omega }}\int _{\Delta }^{\infty }{{\frac {\left[{{\text{E(E + }}\hbar \omega {\text{)}}+\Delta ^{2}}\right][f(E)-f(E+\hbar \omega )]}{(E^{2}-\Delta ^{2})^{1/2}[(E+\hbar \omega )^{2}-\Delta ^{2}]^{1/2}}}dE}{-}\Theta (\hbar \omega -2\Delta ){\frac {1}{\hbar \omega }}\int _{\Delta -\hbar \omega }^{-\Delta }{{\frac {\left[{{\text{E(E + }}\hbar \omega {\text{)}}+\Delta ^{2}}\right][1-2f(E+\hbar \omega )]}{(E^{2}-\Delta ^{2})^{1/2}[(E+\hbar \omega )^{2}-\Delta ^{2}]^{1/2}}}dE}}$

where ${\displaystyle \Theta (x)}$ is the Heaviside theta function. The first term of the upper equation is the contribution of "normal" electrons, and the second term is due to the superconducting electrons.

The calculated optical conductivity breaks the sum rule that the spectral weight should be conserved through the transition. This result implies that the missing area of the spectral weight is concentrated in the zero frequency limit, corresponding to the dirac delta function (which covers the conduction of the superconducting condensate, i.e. the Cooper pairs). Many experimental data supports the prediction. This story on electrodynamics of superconductivity is the starting point of optical study. Because any superconducting T_c never exceeds 200K and the superconducting gap value is about the 3.5 k_BT, microwave or far-infrared spectroscopy is suitable technique applying this theory. With the Mattis–Bardeen theory, we can derive fruitful properties of the superconducting gap, like gap symmetry.

• Michael Tinkham, Introduction to Superconductivity. Second edition.
• Shu-Ang-Zhou, Electrodynamics of Solids and Microwave Superconductivity.