Weak coupling theory predicts the critical temperature of a phonon superconductor to be Tc=1.13e-3/2ωDe-1/λ, where ωD is the Debye frequency, λ is the dimensionless electron-phonon coupling constant, and the factor e-3/2 comes from fermionic self-energy and frequency dependence of the interaction. Other corrections are small either in ωD/EF, by Migdal's theorem, or in λ. However, this formula assumes that ωDEF, where EF is the Fermi energy. We obtain Tc in the dilute regime, when the Fermi energy is smaller than ωD. We argue that in this situation Migdal's theorem is no longer valid, and Kohn-Luttinger-type corrections to the pairing interaction must be included to obtain the correct prefactor for Tc.
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We are thankful to M. Christensen, R. Fernandes, M. Gastiasoro, A. Klein, A. Millis, N. Prokofiev, and B. Svistunov for useful discussions. This work was funded by the Department of Energy through the University of Minnesota Center for Quantum Materials, under DE-SC-0016371.
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