MIGDAL-ELIASHBERG THEORY

In this section we describe the key equations entering the anisotropic Migdal-Eliashberg theory for electron-phonon superconductors [1] as implemented in EPW. For a detailed derivation and a more in-depth discussion see [2].

The key equations to be solved in the Migdal-Eliashberg theory are:

\begin{eqnarray} \qquad Z({\bf k},i\omega_n) = 1 + \frac{\pi T}{N_{\rm F}\omega_n} \sum_{{\bf k}' n'} \frac{ \omega_n' }{ \sqrt{\omega_n'^2+\Delta^2({\bf k}',i\omega_n')} } \lambda({\bf k},{\bf k}',n\!-\!n') \delta(\epsilon_{{\bf k}'}), \end{eqnarray} \begin{eqnarray} \!\!Z({\bf k},i\omega_n) \Delta({\bf k},i\omega_n) &=& \frac{\pi T}{N_{\rm F}} \sum_{{\bf k}' n'} \frac{ \Delta({\bf k}',i\omega_n') }{ \sqrt{\omega_n'^2+\Delta^2({\bf k}',i\omega_n')} } \\ &&\times\left[ \lambda({\bf k},{\bf k}',\!n-\!n')-N_{\rm F} V({\bf k}-{\bf k}')\right] \delta(\epsilon_{{\bf k}'}), \end{eqnarray}

In these equations \(T\) is the absolute temperature, and the functions \(Z\) and \(\Delta\) represent the renormalization function and the superconducting gap, respectively. \(N_{\rm F}\) is the density of electronic states at the Fermi level, and \(\delta(\epsilon_{\bf k})\) is the Dirac delta function (the zero of energy is set to the Fermi level). \({\bf k}\) denotes the composite band and wavevector index, and \(i\omega_n=i(2n+1)\pi T\) (with \(n\) integer) are the fermion Matsubara frequencies. The quantity \(\lambda({\bf k},{\bf k}',\!n-\!n')\) represents the anisotropic electron-phonon coupling matrix and is given by:

\begin{equation} \lambda({\bf k},{\bf k}',n - n') = \int_{0}^{\infty} d\omega \frac{2\omega}{(\omega_n - \omega_n')^2+\omega^2}\alpha^2F({\bf k},{\bf k}',\omega), \end{equation}

with \(\alpha^2F({\bf k},{\bf k}',\omega)\) the Eliashberg electron-phonon spectral function:

\begin{equation} \alpha^2F({\bf k},{\bf k}',\omega) = N_{\rm F} \sum_{\nu} | g_{mn,\nu}^{SE}({\bf k,q})|^2 \delta(\omega-\omega_{{\bf k}-{\bf k}',\nu}). \end{equation}

The notation \(g_{{\bf k}{\bf k}'\nu}\) is a short for the electron-phonon matrix element. In the present case the band index is incorporated inside the wavevector for ease of notation. The frequency \(\omega_{{\bf k}-{\bf k}',\nu}\) corresponds to a phonon of branch index \(\nu\) and wavevector \({\bf q}={\bf k}-{\bf k}'\). In EPW-3.0.0 the static screened Coulomb interaction \(V({\bf k}-{\bf k}')\) is replaced by the standard Coulomb pseudotential \(\mu_{\rm c}^*\) (a fully ab initio calculation of the Coulomb term has not been implemented yet).

The equations for the renormalization function and the superconducting gap form a coupled nonlinear system and are solved by EPW self-consistently. The renormalization function and the superconducting gap are evaluated for each Matsubara frequency along the imaginary energy axis. After calculating \(Z({\bf k},i\omega_n)\) and \(\Delta({\bf k},i\omega_n)\), EPW performs an analytic continuation to the real axis. This continuation can be performed exactly, using the procedure described in [3], or approximately, using Pade’ functions.

Once determined the mass renormalization function \(Z({\bf k},\omega)\) and the superconducting gap \(\Delta({\bf k},\omega)\) along the real frequency axis, one can obtain the quasiparticle energies in the superconducting state by determining the poles of the normal Green’s function (i.e. the \(11\) component of the Nambu-Gor’kov matrix Green’s function):

\begin{eqnarray} G({\bf k},\omega) = \frac{ \omega Z({\bf k},\omega) + \epsilon_{\bf k} } { [\omega Z({\bf k},\omega)]^2 - \epsilon_{\bf k}^2 - [ Z({\bf k},\omega) \Delta({\bf k},\omega)]^2}. \end{eqnarray}

The poles \(E_{\bf k}\) of this Green’s function are given by:

\begin{eqnarray} E_{\bf k}^2 = \left[ \frac{\epsilon_{\bf k}}{Z({\bf k},E_{\bf k})} \right]^2 + \Delta^2({\bf k},E_{\bf k}). \end{eqnarray}

At the Fermi level \(\epsilon_{\bf k}=0\) by construction and the quasiparticle shift is \(E_{\bf k}=\textrm{Re}\Delta({\bf k},E_{\bf k})\). This identity defines the leading edge \(\Delta_{\bf k}\) of the superconducting gap.

Footnotes

[1]

16. 2. Allen and B. Mitrovic, Theory of superconducting Tc, Solid State Phys. 37, 1 (1982).

[2]

5. 18. Margine and F. Giustino, Anisotropic Migdal-Eliashberg theory using Wannier functions, Phys. Rev. B 87, 024505 (2013).

[3]

6. Marsiglio, M. Schossmann, and J. P. Carbotte, Iterative analytic continuation of the electron self-energy to the real axis, Phys. Rev. B 37, 4965 (1988).