Ground State of the Magnetic Impurity Problem; Nuclear-Magnetic-Resonance Studies of Dilute Copper Alloys

We report here nuclear-magnetic-resonance (NMR) studies of dilute alloys of Cu containing $3d$ transition-element impurities at temperatures above 0.03°K. The impurity contribution $\Delta H$ to the ${\mathrm{Cu}}^{63}$ NMR linewidth was measured in applied magnetic fields up to 50 kOe for $\mathrm{Cu}\mathrm{Fe}$ and up to 11 kOe for $\mathrm{Cu}\mathrm{Cr}$, $\mathrm{Cu}\mathrm{Mn}$, and $\mathrm{Cu}\mathrm{Co}$. The $\mathrm{Cu}\mathrm{Fe}$ and $\mathrm{Cu}\mathrm{Cr}$ alloys exhibit several remarkable features. First, $\Delta H$ is greatly reduced from the value expected from high-temperature linewidth measurements. Second, $\Delta H$ is linear in concentration and in field, but independent of temperature at low temperature. Thus $\Delta H$ does not follow a free-spin Brillouin function, but is well fitted by $\frac{\Delta H}{H}$ proportional to ${(T+T_K)}^{-1\mathrm{}}$, where $T_K=14\mathrm{}°$K for $\mathrm{Cu}\mathrm{Fe}$ and $T_K=1.4\mathrm{}°$K for $\mathrm{Cu}\mathrm{Cr}$. Both values agree with Kondo temperatures obtained by other methods. Third, $\Delta H$ is found to be proportional to previously published susceptibility data for $\mathrm{Cu}\mathrm{Fe}$ alloys, indicating the persistence of Ruderman-Kittel-Kasuya-Yosida (RKKY) spin-density oscillations with magnitude proportional to $〈S_z〉$ even for $T\ll T_K$. No evidence is seen for either a $\ln T$ or a $T^{-\frac{1}{2}}$ contribution to $\Delta H$. The field dependence of $\Delta H(T\ll T_K)$ for $\mathrm{Cu}\mathrm{Fe}$ and $\mathrm{Cu}\mathrm{Cr}$ alloys changes slope at applied fields of 25 and 2.5 kOe, respectively, in disagreement with the calculation of Nam and Woo. the slope change is a factor of 2.5 for $\mathrm{Cu}\mathrm{Fe}$ and 5 for $\mathrm{Cu}\mathrm{Cr}$. The effects of short-range order upon $\Delta H$ were studied in a 200-ppm $\mathrm{Cu}\mathrm{Mn}$ alloy below the "ordering" temperature. The data show a large field-independent linewidth and are consistent with a saturation of the impurity spins along their local axis of quantization. We conclude that short-range magnetic order cannot lead to the effects seen in $\mathrm{Cu}\mathrm{Fe}$ and $\mathrm{Cu}\mathrm{Cr}$. The field dependence of $\mathrm{Cu}\mathrm{Co}$ is consistent with $T_K>1000\mathrm{}°$K. An equation-of-motion calculation of the initial susceptibility $\chi$ and the spin polarization about the impurity $\sigma \mathrm{}\left(r\right)\mathrm{}$, is presented. The theory is based upon the Kondo-Applebaum many-body singlet ground state of the magnetic impurity problem. For the static susceptibility, we find $\chi ={\chi }_{\mathrm{Pauli}}+\frac{2g^2{{\mu }_B}^2}{\left[\frac{4}{3}\ln \right(\frac{D}{k{T}_K}\left)\right]k{T}_K},$ and for the spin polarization near the impurity $〈\sigma \mathrm{}\left(r\right)\mathrm{}〉=〈S^z〉[\frac{A\left(cos2k_{F}r\right)}{{\left(k_{F}r\right)}^3}+B{\left(\frac{\left(sin{k}_{F}r\right)}{k_{F}r}\right)}^2].\mathrm{}$ The first term is the well-known RKKY spin-density oscillation. The second term arises from the polarization of the quasiparticle in an external field. Whereas the first term accounts for the field-dependent linewidths of $\mathrm{Cu}\mathrm{Fe}$ and $\mathrm{Cu}\mathrm{Cr}$, the second term is negative definite, giving rise to an excess Knight shift. The excess Knight shifts for the Cu alloys and for ${\mathrm{V}}^{51}$ in $\mathrm{Au}\mathrm{V}$ (0 to 10%) are calculated and found to be in reasonable agreement with experiment. This agreement supports the existence of the extended range of the quasiparticle polarization, and provides the first indirect measurement of the coherence length for the magnetic impurity problem.