Theory for superconductivity in a magnetic field: A local approximation approach

We present a microscopic theory for superconductivity in a magnetic field based on a local approximation approach. We derive an expression for free energy density $F$ as a function of temperature $T$ and vector potential {\bf a}, and two basic equations of the theory: the first is an implicit solution for energy gap parameter amplitude $|\Delta_{\bf k}|$ as a function of wave vector {\bf k}, temperature $T$ and vector potential {\bf a}; and the second is a London-like relation between electrical current density {\bf j} and vector potential {\bf a}, with an ``effective superconducting electron density'' $n_s$ that is both $T$- and {\bf a}-dependent. The two equations allow determination of spatial variations of {\bf a} and $|\Delta_{\bf k}|$ in a superconductor for given temperature $T$, applied magnetic field ${\bf H}_a$ and sample geometry. The theory shows the existence of a ``partly-paired state,'' in which paired electrons (having $|\Delta_{\bf k}|>0$) and de-paired electrons (having $|\Delta_{\bf k}|=0$) co-exist. Such a ``partly-paired state'' exists even at T=0 when $H_a$ is above a threshold for a given sample, giving rise to a non-vanishing Knight shift at T=0 for $H_a$ above the threshold. We expect the theory to be valid for highly-local superconductors for all temperatures and magnetic fields below the superconducting transition. In the low-field limit, the theory reduces to the local-limit result of BCS. As examples, we apply the theory to the case of a semi-infinite superconductor in an applied magnetic field ${\bf H}_a$ parallel to the surface of the superconductor and the case of an isolated vortex in an infinite superconductor, and determine, in each case, spatial variations of quantities such as {\bf a} and $|\Delta_{\bf k}|$. We also calculate...