The Crucial Role of Coulomb Repulsion in Superconducting Gap Anisotropy — Epoche C1
The claim: repulsion changes the shape of the gap, not only its size The superconducting energy gap $\Delta(\mathbf{k})$ — the minimum energy needed to break a Cooper pair of electrons with momenta $\mathbf{k}$ and $-\mathbf{k}$ — is in almost every real superconductor a function of direction on the Fermi surface, not a constant. This essay concerns where that directional structure comes from, and argues that the screened Coulomb repulsion between electrons is one of its sources rather than a uniform penalty subtracted from an otherwise attractive interaction. The reason is structural: in the equation that determines the gap, the phonon-mediated attraction and the Coulomb repulsion both enter as kernels — functions of two momenta — and the gap is an eigenfunction of their difference. Change one of two operators being subtracted and you change the eigenfunction, not only the eigenvalue. Treating the repulsion as a scalar suppresses this fact by construction. Two corrections to the usual short statement of this point are made below, because the expansion exposes them. The Coulomb interaction does not 'screen' the phonon attraction; it is screened itself, by the conduction electrons, and what remains adds to the pairing kernel with the opposite sign. And in a multi-band metal, repulsion does not in general drive a gap to exactly zero: it drives it small, or reverses its sign, and exact vanishing requires a decoupling that only symmetry can supply. The gap equation, and where the repulsion enters Start from the result an undergraduate course supplies. In the theory of Bardeen, Cooper and Schrieffer (1957), electrons near the Fermi surface — the surface in momentum space separating occupied from empty states at zero temperature — bind into pairs under any net attraction, however weak, and the pairing amplitude satisfies a self-consistent equation. Linearised at the transition temperature $T_{\mathrm c}$, and restricted to states on the Fermi surface, it reads $$\Delta(\hat{\mathbf{k}}) \;=\; -\!\int\! \frac{d\hat{\mathbf{k}}'}{4\pi}\, N(0)\,V(\hat{\mathbf{k}},\hat{\mathbf{k}}')\, \Delta(\hat{\mathbf{k}}')\, \ln\!\left(\frac{1.13\,\hbar\omega_{\mathrm c}}{k_{\mathrm B}T_{\mathrm c}}\right),$$ where $\hat{\mathbf{k}}$ is a direction on the Fermi surface, $N(0)$ the electronic density of states per spin at the Fermi energy, $V(\hat{\mathbf{k}},\hat{\mathbf{k}}')$ the effective interaction scattering a pair from $\hat{\mathbf{k}}'$ to $\hat{\mathbf{k}}$, $\omega_{\mathrm c}$ the energy cut-off over which that interaction acts, and $k_{\mathrm B}$ Boltzmann's constant. The logarithm is the signature of the Cooper instability: it diverges as $T_{\mathrm c}\to 0$, which is why any net attraction eventually wins. Written this way the equation is an eigenvalue problem. Define the dimensionless kernel $\Lambda(\hat{\mathbf{k}},\hat{\mathbf{k}}') = -N(0)V(\hat{\mathbf{k}},\hat{\mathbf{k}}')$, positive where the interaction is attractive. Then $T_{\mathrm c}$ is fixed by the largest eigenvalue $\Lambda_{\max}$ of $\Lambda$, through the familiar weak-coupling form $k_{\mathrm B}T_{\mathrm c} = 1.13\,\hbar\omega_{\mathrm c}\exp(-1/\Lambda_{\max})$, and the momentum dependence of the gap is the corresponding eigenfunction. The effective interaction is a difference of two pieces, $\Lambda = \lambda - \mu$: the phonon-mediated attraction $\lambda(\hat{\mathbf{k}},\hat{\mathbf{k}}')$, and the screened Coulomb repulsion $\mu(\hat{\mathbf{k}},\hat{\mathbf{k}}')$, defined as $N(0)$ times the Coulomb matrix element averaged over the two directions. The isotropic BCS solution — $\Delta$ the same in every direction — is what one obtains if both kernels are constants. That is a special case, not the general one. Why the repulsion is weaker than it looks: retardation and the pseudopotential Before the anisotropy can be discussed, the size of the repulsion must be settled, because the bare numbers look fatal to superconductivity. The dimensionless bare Coulomb parameter $\mu$ in a simple metal is of order $0.5$, while the electron-phonon coupling $\lambda$ is typically $0.3$ to $1.5$; if the two simply subtracted, superconductivity in weakly coupled metals would not exist. The resolution, given by Morel and Anderson (1962), is retardation : the two interactions act over completely different energy ranges. The Coulomb interaction is effectively instantaneous, so it connects electronic states across the whole occupied bandwidth, an energy scale of several electronvolts. The phonon-mediated attraction arises because a passing electron leaves behind a lattice distortion that persists for a time of order $1/\omega_{\mathrm{ph}}$, so it acts only between states within about $\hbar\omega_{\mathrm{ph}}$ of the Fermi energy — typically tens of millielectronvolts. The consequence is that the gap function $\Delta(\xi)$, regarded as a function of the electronic energy $\xi$ measured from the Fermi level, need not have one sign. Inside the phonon window it is driven by the attraction; outside it, in the region between $\hbar\omega_{\mathrm{ph}}$ and $E_{\mathrm F}$, the only source term is repulsive, so $\Delta$ there takes the opposite sign. Feeding that sign-reversed high-energy tail back into the equation for the low-energy part partially cancels the direct repulsion. Carrying the elimination through with a constant density of states gives the Coulomb pseudopotential $$\mu^{*} \;=\; \frac{\mu}{1 + \mu\,\ln\!\left(E_{\mathrm F}/\hbar\omega_{\mathrm{ph}}\right)}.$$ The logarithm is not decorative and its origin is worth naming: the elimination integral runs over the high-energy shell, and with $N(\xi)$ approximately constant near the Fermi level the integrand goes as $d\xi/\xi$, whose integral from $\hbar\omega_{\mathrm{ph}}$ to $E_{\mathrm F}$ is exactly $\ln(E_{\mathrm F}/\hbar\omega_{\mathrm{ph}})$. Put numbers in: with $\mu = 0.5$, $E_{\mathrm F} = 5\,\mathrm{eV}$ and $\hbar\omega_{\mathrm{ph}} = 30\,\mathrm{meV}$, the logarit