Theory of spin-orbit and many-body effects on the Knight shift

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We derive an expression for the Knight shift ($K$) in solids, including spin-orbit and many-body effects. We construct in $\stackrel{\ensuremath{\rightarrow}}{k}$ space, using the Bloch representation, the equation of motion of the Green's function in the presence of a periodic potential, spin-orbit interaction, external magnetic field, and electron-nuclear hyperfine interaction. We use a finite-temperature Green's-function method where the thermodynamic potential is expressed in terms of the exact one-particle propagator $G$, and we derive a general expression for $K$. Our result for the Knight shift is expressed as $K={K}_{o}+{K}_{s}+{K}_{\mathrm{so}}$, where ${K}_{o}$ and ${K}_{s}$ are the usual orbital and spin contributions to $K$ modified by the spin-orbit and many-body contributions and where ${K}_{\mathrm{so}}$, which is nonzero only when spin-orbit interaction is taken into account, is a new contribution to $K$ which had been overlooked in the earlier theories. If we make simple approximations for the self-energy, our expression for ${K}_{o}$ reduces to the earlier results. If we make drastic assumptions while solving the matrix integral equations for the field-dependent part of the self-energy, our expression for ${K}_{s}$ is equivalent to the earlier results for the exchange-enhanced ${K}_{s}$ but with the free-electron $g$ factor replaced by the effective $g$ factor. A novel feature of our analysis is that while some of the terms in ${K}_{\mathrm{so}}$ have exchange enhancement effects similar to those of ${K}_{s}$, except that the exchange enhancement parameters are different, the other terms in ${K}_{\mathrm{so}}$ become modified similar to ${K}_{o}$. Thus because of the mixed character of these terms, the exchange and correlation effects on ${K}_{\mathrm{so}}$ cannot be interpreted in an intuitive way. In order to calculate the importance of the new contribution ${K}_{\mathrm{so}}$, we apply our theory to calculate the Knight shift of $^{207}\mathrm{Pb}$ in $p$-type PbTe with small hole concentrations. Our results, which agree with experimental results, indicate that ${K}_{\mathrm{so}}$ is of the same order of magnitude and has the same sign as ${K}_{s}$ and is about 3 orders of magnitude larger than ${K}_{o}$. Thus ${K}_{\mathrm{so}}$, the new contribution to the Knight shift that we have calculated, is important for solids with large effective $g$ factors and should contribute a significant fraction of their total Knight shift.