1.4.3 Strong-Coupling Theory of Superconductivity

As a preface to the discussion of the superconducting pairing mechanism in A₃, a brief summary of the strong-coupling theory of superconductivity is included here.

The ``strength'' of the electron-electron attractive coupling is measured by the dimensionless coupling constant, $\lambda_e$.In the BCS theory $\lambda_e = V g(0)$, and the boundary between the ``weak'' and ``strong'' coupling regimes is[49] about $\lambda_e = 0.25$ with the weak-coupling limit being $\lambda_e \rightarrow 0$ and the infinitely strong-coupling limit being $\lambda_e \rightarrow 1$.

The theory of Bardeen, Cooper and Schrieffer treated only the simplest form of attractive interaction between electrons, i.e. the interaction V(E) was simply a constant (independent of energy, direction or temperature) for E within the Debye energy of the Fermi energy and zero outside of this range. A more realistic and general treatment of the electron-phonon interaction, which allows both for structure to the interaction and strong electron-phonon coupling was first accomplished by Eliashberg[50]. For general reviews of this topic the reader is referred to the works of Scalapino[49], McMillan and Rowell [51], and the more recent review of Carbotte [52].

The general electron-phonon interaction hamiltonian is

where $b_{{\bf q}\lambda}$ is the annihilator for a phonon at wavevector ${\bf q}$ with branch index $\lambda$,$c_{{\bf k}\sigma}$ is the annihilator for an electron at wavevector ${\bf k}$ and spin $\sigma$, M is the interaction, and ${\bf q} = {\bf k'} - {\bf k}$.The phonon density of states is

where $E_{{\bf q},\lambda}$ is the $\lambda$-branch phonon dispersion curve. In the Eliashberg theory, this interaction is included via a Fermi Surface (FS) average coupling-constant phonon density of states product:

where A_FS is the area of the Fermi Surface and $\tilde{M}$ is the ``dressed'' electron-phonon interaction M which includes the effect of Coulomb renormalization (see p.481 of [49]). The function $\alpha^2F(E)$ contains all the relevant information about the electron-phonon coupling giving rise to the effective attractive interaction between electrons which produces the superconductivity. However it turns out that in many cases the important information in this distribution is not in its details, but simply in two of its ``moments'': the mass enhancement parameter $\lambda_m$ and the logarithmic moment, $E_{\log}$ which are defined by:

The effective electron mass modified by the electron-phonon interaction is just $m^*/m = 1 + \lambda_m$.

The theory of superconductivity based on this general approach to the electron-phonon interaction is summarized in the Eliashberg equations, which are coupled non-linear self-consistent equations which take the place of the BCS gap equation and include the BCS gap equation as a special case. From these equations, McMillan (see e.g. [51] and references therein) developed an equation for T_c analogous to the BCS equation (Eq. 1.2) which was subsequently improved by others (see [52]). It reads

The as yet undefined parameter in the above $\mu^*$ represents the electron-electron Coulomb interaction and is discussed further below. The parameterization of the dimensionless coupling $\lambda_e$ in terms of V and g(0) has evidently been replaced. In fact, correspondence with the BCS theory gives

The argument of the exponential function in the McMillan equation (Eq. 1.9) is not simply $-\lambda_e^{-1}$, as it was in the BCS theory. This is because the the form $e^{-1/\lambda_e}$ is only approximate. From Eq. 1.10, we see that for an infinite electron-phonon mass renormalization ($\lambda_m \rightarrow \infty$), the dimensionless coupling parameter $\lambda_e \rightarrow 1$, i.e. this is the infinitely strong-coupling limit. A representative range for $\lambda_m$ in conventional superconductors is 0.4-3.0[52].

For the range of parameters found in real materials, the following approximate form for the important ratio of the energy gap to T_c (analogous to the BCS result 1.3) has been found (see [52])

For real superconductors this ratio is, for the weak-coupling limit, close to the BCS value of 3.52, and for stronger coupling materials it ranges up to about 5.1. Thus the broad range of reported values (Table 1.1) encompasses both the weak-coupling and strong-coupling regimes.