Ab initio theory of superconductivity. I. Density functional formalism and approximate functionals

An approach to the description of superconductors in thermal equilibrium is developed within a formally exact density functional framework. The theory is formulated in terms of three “densities:” the ordinary electron density, the superconducting order parameter, and the diagonal of the nuclear $N$-body density matrix. The electron density and the order parameter are determined by Kohn-Sham equations that resemble the Bogoliubov–de Gennes equations. The nuclear density matrix follows from a Schrödinger equation with an effective $N$-body interaction. These equations are coupled to each other via exchange-correlation potentials which are universal functionals of the three densities. Approximations of these exchange-correlation functionals are derived using the diagrammatic techniques of many-body perturbation theory. The bare Coulomb repulsion between the electrons and the electron-phonon interaction enter this perturbative treatment on the same footing. In this way, a truly ab initio description is achieved which does not contain any empirical parameters.

Figure 1

Schematic flow chart for the iterative Kohn-Sham scheme within the decoupling approximation.

Figure 2

Lowest-order electronic (a) and phononic (b), (c) contributions to $F_{\mathrm{xc}}$.

Figure 3

The function $-N\left(0\right){\mathcal{K}}^{\mathrm{ph}}(\xi ,\xi )$ for Al, Nb, and Pb, calculated at $T=0\mathrm{K}$.

Figure 4

The dependence of ${\mathcal{Z}}_i^{\mathrm{ph},\mathrm{sym}}$ on temperature for niobium.