Theory of Raman Scattering from Leggett's Collective Mode in a Multiple Band Superconductor: Application to MgB$_{2}$

Using an extension of BCS theory to a two-band superconductor, Leggett showed that if the relevant parameters obeyed certain conditions a collective mode would exist corresponding to the counter flow of the two condensates.$^{1}$ I have extended earlier work on electronic Raman in superconductors$^{2}$ to the multiple band case in order to incorporate Leggett's theory. The following effects have been included: (a) Vertex correction in the particle/hole channel where the Raman vertex acts. (b) Realistic parameters that apply to MgB$_{2}$ yielding a counter flow mode that decays into the pair-breaking continuum associated with the lower gap $\pi $ band. (c) Large finite wave-vector effects due to the relatively large Fermi velocity of the $\pi $ band. (d) Integration over the wave-vector in part (c) necessitated by the exponential decay of the photon fields traveling into and out of the metallic sample. A comparison to the results of Blumberg$^{3}$ will be given. $^{1}$A.J. Leggett, Progr. Theor. Phys. \textbf{36}, 901 (1966). $^{2}$M.V. Klein and S.B. Dierker, Phys. Rev. B\textbf{29}, 4976 (1984). $^{3}$G. Blumberg et al., Phys. Rev. Lett. \textbf{99}, (2007); arXiv:0710.2803.