Towards a Microscopic Theory of the Knight Shift in an Anisotropic, Multiband Type-II Superconductor

A method is proposed to extend the zero-temperature Hall-Klemm microscopic theory of the Knight shift $K$ in an anisotropic and correlated, multi-band metal to calculate $K(T)$ at finite temperatures $T$ both above and into its superconducting state. The transverse part of the magnetic induction ${\bf B}(t)={\bf B}_0+{\bf B}_1(t)$ causes adiabatic changes suitable for treatment with the Keldysh contour formalism and analytic continuation onto the real axis. We propose that the Keldysh-modified version of the Gor'kov method can be used to evaluate $K(T)$ at high ${\bf B}_0$ both in the normal state, and by quantizing the conduction electrons or holes with Landau orbits arising from ${\bf B}_0$, also in the entire superconducting regime for an anisotropic, multiband Type-II BCS superconductor. Although the details have not yet been calculated in detail, it appears that this approach could lead to the simple result $K_S(T)\approx a({\bf B}_0)-b({\bf B}_0)|\Delta({\bf B}_0,T)|^2$, where $2|\Delta({\bf B}_0,T)|$ is the effective superconducting gap. More generally, this approach can lead to analytic expressions for $K_S(T)$ for anisotropic, multiband Type-II superconductors of various orbital symmetries that could aid in the interpretation of experimental data on unconventional superconductors.