Spin-Orbit Coupling and the Knight Shift in Nontransition-Metal Superconductors

The Knight shift $K_s$ of nontransition-metal superconductors is discussed in terms of three contributions, namely: (a) The Van Vleck part of the contact shift attributed to the spin-orbit coupling force of the periodic potential which, in the presence of an external magnetic field, causes virtual high-energy rearrangements of conduction electrons near and inside the Fermi surface; (b) that part of the contact shift which is attributed to low-energy rearrangements of conduction electrons and affected by spin-reversing scattering; and (c) the diamagnetic orbital shift. For the calculation of (a), Wannier's theory of a Bloch electron in a magnetic field is generalized to the original Pauli Hamiltonian ${\mathcal{H}}_0$, describing the relativistic dynamical behavior of a conduction electron in the effective periodic potential of the lattice. This leads to an effective Hamiltonian which couples only Bloch-type spinors of the same band index, but of the same and of different spin indices. With the help of the eigenfunctions of ${\mathcal{H}}_0$, the hyperfine contact interaction is treated by perturbation theory. To arrive at simple expressions for the corresponding Knight shift and nuclear spin-relaxation time, valid for arbitrary strength of spin-orbit coupling, the energy-band function in the absence of the field is approximated by a parabola. Formulas for (b) and (c) are taken from the literature. The relative importance of the three contributions to $K_s$ is discussed for Al, Sn, and Hg, where the Knight shift has been observed in the normal and in the superconducting states. If one assumes that spin-reversing scattering is caused merely by spin-orbit interactions at atomic imperfections such as displaced surface atoms of small particles, and not by paramagnetic impurities, one finds that in Al neither of the two spin-orbit coupling effects is sufficiently strong to account for more than ∼2% of the residual shift $K_s\mathrm{}\left(0\right)\mathrm{}$. For the experimental Sn sample, (a) and (b) have the ratio 1:3 and, together with the orbital shift, can account for $\frac{2}{3}$ of the observed $K_s\mathrm{}\left(0\right)\mathrm{}$. For Hg, (a) and (b) are of comparable magnitude at $T=0\mathrm{}$ and together account for more than ½ of the observed shift $K_s\mathrm{}\left(0\right)\mathrm{}$.