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##### Density functional theory for superconductors: exchange and correlation potentials for inhomogeneous systems

**Authors**: M.A.L. Marques

**Ref.**: Ph. D. thesis, University of Würzburg (2000)

**Abstract**: Almost a century ago, Kamerlingh Onnes discovered that the resistivity of some metals dropped to zero very quickly below a certain temperature [1]. The phenomenon, that we now call superconductivity, resisted any microscopic explanation for more than forty years. In 1957, Bardeen, Cooper and Schrieffer proposed a revolutionary microscopic theory that clarified the nature of superconductivity [2], and predicted quite accurately several material independent quantities. Their theory was based on the concept of Cooper pairs [3], electron pairs formed due to the instability of the electron gas in the presence of an attractive interaction. The origin of this interaction was phononic, and had already been studied by Froehlich [4] and by Bardeen and Pines [5] some years before. Another major step in the theoretical description of superconductors was done in 1960 by
Eliashberg [6]. His theory, a generalisation of Migdal treatment of the electron-phonon interaction in solids [7], allowed accurate calculations of material dependent quantities, like the gap at
zero temperature, and an appropriate description of the so called strong-coupling superconductors (like Nb or Pb.) A serious drawback of this approach was the description of the Coulomb interaction between the electrons - it was usually replaced by a number, mu^{*}, which was fitted to the experimental transition temperature, making the theory semi-phenomenological. Nevertheless, it was commonly accepted that superconductivity in the simple metals was a perfectly understood phenomenon.

The situation changed dramatically in 1986, when Georg Bednorz and Alex Mueller reported finding superconductivity in an oxide material at a temperature 12K higher than previously known. Their discovery of the high-Tc ceramics, prompted active research on these materials, both experimental and theoretical. During the past 15 years, we have witnessed the birth (and death) of several theories aimed to describe theoretically the superconducting ceramics. Several aspects of these systems are now understood, but important issues, like what is the mechanism that drives the ceramics superconducting at such high temperatures, are still unknown. The discovery of other superconducting systems, like the heavy-fermions or the organic superconductors, also raised questions that could not be answered by the old theory.

In 1988, triggered by the remarkable discovery of the high-Tc materials, Oliveira, Gross and Kohn proposed a Density Functional Theory (DFT) for the superconducting state [9] (SCDFT.)
Their theory, a generalisation of the traditional DFT for the normal (non-superconducting) state [10, 11], aimed at a unified treatment of correlation and inhomogeneity effects in superconductors.
In the original formulation, Coulomb interactions were treated at an exact level, while electron-phonon interactions were only included in an approximate way. It is quite tempting to try such a DFT approach, if one looks at the advances and successes of traditional DFT methods both in chemistry and in physics. In fact, although DFT methods cannot compete with traditional Quantum Chemistry for few-electron systems, its numerical simplicity makes it the method of choice
for larger systems, and it is the de facto standard in solid-state band structure calculations. In DFT, most of the complexities of the many-body problem are hidden behind an innocent looking quantity, the exchange-correlation (xc) potential. In the traditional DFT, several approximations for this quantity have appeared over the past 30 years. The first, and simplest, of these functionals, is the Local Density Approximation (LDA), which relates the xc energy of an inhomogeneous
system to the one of the homogeneous electron gas. By its construction, one would expect the LDA to be a good approximation only in systems with slowly varying densities, but surprisingly,
it performs quite well even in highly inhomogeneous environments, like atoms or molecules. The following generation of functionals were the so-called Generalised Gradient Expansions (GGA).
In the GGAs, the xc energy of the inhomogeneous system does not depend only locally on the density, but also on its gradient. Recently, a new class of functionals has become quite popular: In the Optimised Effective Potential (OEP) method [12], the xc energy is written as an explicit functional of the Kohn-Sham orbitals (although it is still an implicit functional of the density.)

Although a wealth of xc functionals exist for normal DFT, the case is quite different in the case of DFT for superconductors. In fact, no functional existed until quite recently [13, 14, 15]. It is the
purpose of this thesis to address this problem.

In chapter 2, we provide some background information on the Eliashberg equations. Two different derivations are given: the traditional one, using Nambu-Gorkov propagators, and another, using the concept of Off-Diagonal Long Range Order (ODLRO.) The linearised version of these
equations will then be solved for simple metals, in the case of phonon-only superconductivity (i.e., neglecting the Coulomb interaction.) These results will provide us with a benchmark for our DFT phonon-only calculations, and will enable us to assess the quality of the phonon part of the xc potentials.

The next chapter is devoted to the development of the SCDFT formalism. First the original Oliveira, Gross and Kohn [9] treatment is discussed, and then we proceed to show a possible way to include strong-coupling electron-phonon effects in the theory [15], by developing a multi-component DFT for the electron-nuclear system. Both the Hohenberg-Kohn theorem and the Kohn-Sham construction are then explained. The Kohn-Sham equations for the system consist of an (interacting) equation for the nuclei, and a Bogoliubov-de Gennes [16] like equation for the electrons. All xc effects are included in these equations through the xc potentials.

In chapter 4 we show two systematic ways to construct these xc potentials. Both make use of Kohn-Sham perturbation theory but, while the first uses the definition of the xc potentials as functional derivatives of the xc free energy, the second generalises Sham and Schlueter ideas [17, 18, 19] to the case of SCDFT. In first-order perturbation theory, the xc potentials are the same in both approaches. We then analyse the structure of these potentials, and solve the DFT linearised gap equation to determine Tc for simple metals. At the end of the chapter, we discuss several ways
to improve the simple first-order xc potentials.

A different approach to the construction of the xc potentials for superconductors is described in chapter 5 where we develop an LDA functional. This functional depends on the xc energy of the homogeneous electron gas, forced into the superconducting state by the presence of an external pairing field. This energy is then evaluated using Kohn-Sham perturbation theory at the level of an RPA.

Finally, in chapter 6, we take a brief digression into the realm of relativistic effects in superconductors [20]. We provide a derivation of the Dirac-Bogoliubov-de Gennes (DBdG) equation for the most general pair potential [21], an analysis of the symmetry properties of these pair potentials, and a weakly relativistic expansion of the DBdG equation. The most important results of this chapter are the relativistic correction terms for triplet superconductors.