Diagrammatic Analysis of the Dynamics of Localized Moments in Metals

Using the conventional exchange Hamiltonian ${\mathcal{H}}_{\mathrm{es}}$, together with an additional model Hamiltonian which provides a source of lattice relaxation for the conduction-electron magnetization, the coupled local-moment-conduction-electron transverse dynamic susceptibility is calculated. Feynman temperature-ordered Green's functions are used and are "dressed" with self-energies which are correct to second order in interaction parameters. A pair of coupled vertex equations are constructed which includes all diagrams correct to second order in the interaction parameters. In order to obtain the desired two-particle characteristics of the response function, some new manipulative and mathematical methods are introduced. These enable the vertex equations to be reduced to a coupled pair of linear equations which determine the coupled susceptibility. At sufficiently high temperatures the Kondo "$g$ shifts" and the weak frequency dependence of the self-energies can be ignored. These linear equations are then equivalent to the linearized version of the following Bloch equations: $\frac{d}{\mathrm{dt}}{\overrightarrow{\mathrm{M}}}_s=g_s[{\overrightarrow{\mathrm{M}}}_s\times ({\overrightarrow{\mathrm{H}}}_{\mathrm{ext}}+\lambda {\overrightarrow{\mathrm{M}}}_e\left)\right]-\left(\frac{1}{T_{\mathrm{se}}}\right)[{\overrightarrow{\mathrm{M}}}_s-{\chi }_s^0({\overrightarrow{\mathrm{H}}}_{\mathrm{ext}}+\lambda {\overrightarrow{\mathrm{M}}}_e\left)\right]+\left(\frac{g_s}{g_e{T}_{\mathrm{es}}}\right)[{\overrightarrow{\mathrm{M}}}_e-{\chi }_e^0({\overrightarrow{\mathrm{H}}}_{\mathrm{ext}}+\lambda {\overrightarrow{\mathrm{M}}}_s\left)\right]$, $\frac{d}{\mathrm{dt}}{\overrightarrow{\mathrm{M}}}_e=g_e[{\overrightarrow{\mathrm{M}}}_e\times ({\overrightarrow{\mathrm{H}}}_{\mathrm{ext}}+\lambda {\overrightarrow{\mathrm{M}}}_s\left)\right]-(\frac{1}{T_{\mathrm{es}}}+\frac{1}{T_{\mathrm{e}1\mathrm{}}})[{\overrightarrow{\mathrm{M}}}_e-{\chi }_e^0({\overrightarrow{\mathrm{H}}}_{\mathrm{ext}}+\lambda {\overrightarrow{\mathrm{M}}}_s\left)\right]+\left(\frac{g_e}{g_s{T}_{\mathrm{se}}}\right)[{\overrightarrow{\mathrm{M}}}_s-{\chi }_s^0({\overrightarrow{\mathrm{H}}}_{\mathrm{ext}}+\lambda {\overrightarrow{\mathrm{M}}}_e\left)\right]+D{\nabla }^2[{\overrightarrow{\mathrm{M}}}_e-{\chi }_e^0({\overrightarrow{\mathrm{H}}}_{\mathrm{ext}}+\lambda {\overrightarrow{\mathrm{M}}}_s\left)\right]$, where $T_{\mathrm{se}}$, $T_{\mathrm{es}}$, and $T_{\mathrm{e}1}$ are the relaxation times for the local-moment-conduction-electron-spin, the conduction-electron-spin-local-moment, and the conduction-electron-spin-lattice systems. The diffusion constant is $D=\frac{1}{3}{v}_F^2{T}_i$, where $T_i$ is the nonmagnetic-impurity scattering time, and ${\overrightarrow{\mathrm{M}}}_e$ and ${\overrightarrow{\mathrm{M}}}_s$ are, respectively, the conduction-electron and local-moment magnetizations. The spin-orbit scattering in the presence of large-potential-impurity scattering results in relaxation effects identical to those obtained above with the model electron-lattice Hamiltonian. ${\overrightarrow{\mathrm{H}}}_{\mathrm{ext}}$ is the external static and rf field, ${\chi }_e^0$ and ${\chi }_s^0$ are the conduction-electron and local-moment-unenhanced (by the $s-d$ exchange) susceptibilities, and $g_e$ and $g_s$ are the respective $g$ factors ($\hslash$ and ${\mu }_B$ have been set equal to unity). These Bloch equations show that the relaxation destination vectors are those appropriate to the total internal field. The presence of the $g$-factor ratios are consistent with the fact that ${\mathcal{H}}_{\mathrm{es}}$ transfers spin, rather than magnetization, from the conduction electrons to local moments and vice versa. For temperatures $\mathrm{kT}<\mathrm{}{\omega }_s$, where ${\omega }_s$ is the local-moment Zeeman energy, these equations fail, the self-energies become frequency dependent, and there are modifications to ${\chi }_s^0$. Most important, the electronic self-energy becomes dependent on the magnitude of the electronic momentum, so that a single equation for the total electron magnetization cannot be written. A microscopic derivation of the detailed-balance condition is given.