Theory of Raman scattering from Leggett's collective mode in a multiband superconductor: Application to MgB$_2$

TL;DR

This paper extends the theory of Raman scattering to multiband superconductors, specifically applying it to MgB$_2$, to analyze Leggett's collective mode and compare with experimental observations.

Contribution

It develops a multiband Raman scattering theory that includes Leggett's collective mode and applies it to MgB$_2$, matching experimental data.

Findings

Leggett's mode peak broadens with increased interband potential.

The Raman peak energy agrees with experimental results.

Finite momentum transfer effects are incorporated into the model.

Abstract

In 1966 Leggett used a two-band superconductor to show that a new collective mode could exist at low temperatures, corresponding to a counter-flow of the superconducting condensates in each band. Here, the theory of electronic Raman scattering in a superconductor by Klein and Dierker (1984) is extended to a multiband superconductor. Raman scattering creates particle/hole pairs. In the relevant $A_{1g}$\ symmetry, the attraction that produces pairing necessarily couples excitations of superconducting pairs to these p/h excitations. In the Appendix it is shown that for zero wave vector transfer $% q$ this coupling modifies the Raman response and makes the long-range Coulomb correction null. The 2-band result is applied to MgB$_{2}$ where this coupling activates Leggett's collective mode. His simple limiting case is obtained when the interband attractive potential is decreased to a value well below that given by LDA theory. The peak from Leggett's mode is studied as the potential is increased through the theoretical value: With realistic MgB$_{2}$\ parameters, the peak broadens through decay into the continuum above the smaller ($\pi $ band) superconducting gap. Finite $q$ effects are also taken into account, yielding a Raman peak that agrees well in energy with the experimental result by Blumberg \textit{et el.} (2007). This approach is also applied to the $q=0$, 2-band model of the Fe-pnictides considered by Chubukov \textit{et al.}(2009).