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

Abstract

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 B(t) = B 0 + 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 B 0 both in the normal state, and by quantizing the conduction electrons or holes with Landau orbits arising from 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 resultwhere 2|∆(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.

The Zeeman, spin-orbit, and quantum spin Hall interactions in anisotropic and low-dimensional conductors

“…Microscopically, the nucleus of an atom moves slowly inside a 3D electronic shell, and as for the Dirac equation of a free electron, the 3D relativistic motion of each of its neutrons and protons leads to it having an overall spin I and a nuclear Zeeman energy that can be probed by a time t-dependent external B(t) in nuclear magnetic resonance (NMR) and in Knight shift measurements when in a metal [45,46]. The orbital electrons bound to that nucleus also move in a nearly isotropic 3D environment, and have a much larger Zeeman interaction with B(t), modified only by the V(r) = −|e|Φ(r) of nearby atoms.…”

Quantum Spin Hall Effect in Electric and Magnetic Fields without Spin-Orbit Coupling

“…2A. Microscopically, however, its nucleus moves slowly inside a 3D electronic shell, and as for the Dirac equation of a free electron, the 3D relativistic motion of each of its neutrons and protons leads to it having an overall spin I and a nuclear Zeeman energy that can be probed by a time t-dependent external magnetic field H(t) in nuclear magnetic resonance (NMR) and in Knight shift measurements when in a metal [19,20]. The orbital electrons bound to that nucleus also move in a nearly isotropic 3D environment, and have a much larger Zeeman interaction with H(t), modified only by the V (r) = −eΦ(r) of nearby atoms.…”

“…In both cases, the nuclear spin of an atom interacts with that of one of its orbital electrons via the hyperfine interaction. But when that atom is in a metal, the orbital electron can sometimes be excited into the conduction band, travelling throughout the crystal, and then returning to the same nuclear site, producing the leading order contribution to the Knight shift [19,20]. The dimensionality of the motion of the electron in the conduction band is therefore crucial in interpreting Knight shift measurements of anisotropic materials, as first noticed in the anisotropic three-dimensional superconductor, YBa 2 Cu 3 O 7−δ [24].…”

The Zeeman, Spin-Orbit, and Quantum Spin-Hall Interactions in Anisotropic and Low-Dimensional Conductors

“…1(a). Microscopically, however, its nucleus moves slowly inside a 3D electronic shell, and as for the Dirac equation of a free electron, the 3D relativistic motion of each of its neutrons and protons leads to it having an overall spin I and a nuclear Zeeman energy that can be probed by a time t-dependent external B(t) in nuclear magnetic resonance (NMR) and in Knight shift measurements when in a metal [28,29]. The orbital electrons bound to that nucleus also move in a nearly isotropic 3D environment, and have a much larger Zeeman interaction with B(t), modified only by the V (r) = −eΦ(r) of nearby atoms.…”

“…In both cases, the nuclear spin of an atom interacts with that of one of its orbital electrons via the hyperfine interaction. But when that atom is in a metal, the orbital electron can sometimes be excited into the conduction band, travelling throughout the crystal, and then returning to the same nuclear site, producing the leading order contribution to the Knight shift [28,29]. The dimensionality of the motion of the electron in the conduction band is therefore crucial in interpreting Knight shift measurements of anisotropic materials, as first noticed in the anisotropic three-dimensional superconductor, YBa 2 Cu 3 O 7−δ [9].…”