Kondo Theory Research Articles

ESR Study of the s-d Exchange Scattering

The precise measurements of the ESR linewidth of Mn in dilute CuMn and AuMn alloys (0.05∼1.5 at.% of Mn) with Fe, Co, V and Ni impurities have been made over the temperature range from 1.4°K to about 200°K. The Mn spin relaxation due to magnetic impurities is well understood by the s - d exchange scattering of conduction electrons by these impurities. The temperature dependence of the spin-flip scattering probability is studied. From this study, T K of Co in CuCo alloy is estimated to be about 1000°K. The effect of fhe s - d scattering on the spin relaxation remains even below T K . It is clarified that the first Born approximation gives a good agreement with experimental results at \(T{\gtrsim}T_{\text{K}}\), at least with respect to the relaxation problem. The results at \(T{\lesssim}T_{\text{K}}\) are discussed by Suhl and Kondo's theory of the s - d exchange scattering.

S−dExchange Interaction in Amorphous Cr-Pd-Si and Mn-Pd-Si Alloys

The electrical and magnetic properties of amorphous alloys obtained by rapid quenching from the liquid state have been studied. The compositions of these alloys are ${\mathrm{Cr}}_{x}{\mathrm{Pd}}_{80\ensuremath{-}x}{\mathrm{Si}}_{20}$ and ${\mathrm{Mn}}_{x}{\mathrm{Pd}}_{80\ensuremath{-}x}{\mathrm{Si}}_{20}$, in which $x$ is the atomic concentration and ranges between 0 and 7. A well-defined minimum in the resistivity versus temperature is observed for these alloys except for the case where $x=0$. Below the resistivity-minimum temperature ${T}_{m}$, the resistivity difference $\ensuremath{\Delta}\ensuremath{\rho}$ varies as -$\mathrm{ln}T$ in agreement with Kondo's theory, and tends to level off at lower temperatures. The resistivity-minimum phenomenon persists in amorphous alloys containing as much as 7 at.% or more of transition metals like Cr or Mn. Also, the resistivity-minimum temperature for some of the amorphous alloys studied can be as high as 580\ifmmode^\circ\else\textdegree\fi{}K. The resistivity-minimum anomaly is always accompanied by a negative magneto-resistivity approximately proportional to the square of the magnetization and a susceptibility obeying the Curie-Weiss law for a wide temperature range. These observations lead to the conclusion that a Kondo-type $s\ensuremath{-}d$ exchange interaction exists in the amorphous Pd-Si alloys containing Cr or Mn. In terms of the $s\ensuremath{-}d$ exchange model, the following statements can be made: (a) The results of susceptibility and magnetoresistivity measurements suggest that the $d\ensuremath{-}d$ spin correlation between magnetic atoms is, in general, weaker in the amorphous alloys than in the corresponding crystalline alloys. This is consistent with the fact that a Kondo-type resistivity anomaly has been observed in amorphous alloys containing 7 at.% Cr or Mn. (b) The spin-compensated state has been observed in the amorphous alloys containing Cr. From the resistivity data, it is concluded that the Kondo temperature of these alloys depends linearly on the Cr concentration. For the Mn-Pd-Si alloys, the spin-compensated state is masked by the $d\ensuremath{-}d$ spin correlation. (c) The $s\ensuremath{-}d$ exchange integrals for the Cr and Mn amorphous alloys are estimated and found to be consistent with Friedel's virtual-bound-state model.

Magnetic field dependence of the thermoelectric power at low temperatures for very dilute alloys of Fe in Au

Fe–Fe interactions are shown to be responsible for an increase in thermoelectric power with applied magnetic field. A simple model, with Kondo's theory, accounts qualitatively for the observations. Observation only of effects of isolated Fe atoms requires concentrations < 1 p.p.m.

Magnetic field dependence of the thermoelectric power of Au + 0.03% Fe at low temperatures

Measurements of the thermoelectric power of gold + 0.03 atomic percent iron from 0.35°K - 10°K and 0–77 kOe are presented. The data are seen to disagree with Kondo's theory of thermopower in dilute magnetic alloys.

Electrical Resistivity of Some Dilute Gold-Rare-Earth Alloys

The electrical resistivity has been measured from 1.3° to 20°K on solid solutions of Gd, Tb, and Ho in Au. It was found that the resistivity decreased with decreasing temperature with no evidence of a resistance minimum. However, the resistivity showed a more pronounced decrease in the temperature range 6°–3.5°K. These results are discussed in terms of the Kondo theory. It was also found that the residual resistivity, ρ(4.2°K), is proportional to the concentration of rare earths for the Gd, Tb, and Ho alloys. Furthermore, the proportionality constant is the same for the three sets of alloys indicating ionic (trivalent) scattering as the main contribution to the residual resistivity.

Resistance Anomaly in Palladium-Silver Alloys Containing Rare Earth Impurities

Electrical resistivities of Pd and three Pd–Ag alloys (Pd0.5Ag0.5, Pd0.35Ag0.65, and Pd0.25Ag0.75) containing small amounts of Gd and Nd have been determined at temperatures between 1.2° and 40°K. All lanthanidedoped alloys were observed to have resistance anomalies which were similar in character for the heavy and light rare earth solutes. However, the anomalies were found to vary drastically with the host lattice. With decreasing temperature, the resistivity-temperature curves show (1) a sudden increase in slope in the matrix of Pd, (2) a minimum in Pd0.5Ag0.5, and (3) a minimum and a maximum in Pd0.35Ag0.65 and Pd0.25Ag0.75. The present alloys, which constitute a continuous series of solid solutions having a common magnetic impurity, have thus exhibited a gradual shift from one to another of the three possible types of anomalous resistance behavior predicted for dilute magnetic alloys by Kondo and other workers. The shift signifies that a dominant role is played by the host lattice in the mechanism leading to a resistance anomaly. The difficulties in reconciling the present results with predictions on the Kondo theory as related to the sign of the effective s-f exchange integral and earlier EPR measurements are discussed.

Electrical Resistivity of Some Dilute Gold-Rare Earth Alloys

The electrical resistivity has been measured from 1.3° to 20°K on solid solutions of Ce, Nd, Gd, Tb, and Ho in Au. It was found that the resistivity decreased monotonically with decreasing temperature with no evidence of a resistance minimum. However, for the Gd, Tb, and Ho in Au alloys, the resistivity showed a pronounced decrease in the temperature range 3.5° to 6°K. These results are discussed in terms of the Kondo theory. It was also found that the residual resistivity (4.2°K) is proportional to the concentration of rare earths for the Gd, Tb, and Ho alloys. Furthermore, the proportionality constant is the same for the three sets of alloys, indicating ionic (trivalent) scattering as the main contribution to the residual resistivity.

Electron-Tunneling Measurements on Lanthanum and Lanthanum-Lutetium Alloy Films

Results of electron-tunneling measurements on evaporated films of fcc lanthanum and dhcp lanthanum-lutetium alloys are presented. The ratio of the zero-temperature energy gap to the temperature at which the energy gap vanishes for both the pure lanthanum and lanthanum-lutetium alloy samples varied from 3.41 to 3.58. If one fits the lowest-temperature data for the energy gap with a curve of the BCS temperature dependence, the values at intermediate temperatures fall below the weak-coupling BCS prediction. The conductance maxima for the (pure La)-${\mathrm{Al}}_{2}$${\mathrm{O}}_{3}$-Al diodes are larger than predicted by the weak-coupling BCS theory. The conductance maxima for the lutetium alloy samples are more nearly equal to the weak-coupling BCS values than are those of the pure samples. They were not significantly altered by the presence of small zero-voltage anomalies. Hence zero-voltage anomalies are not enhanced at temperatures below the superconducting transition temperature. No change in conductance as large as 0.1% was observed which could be associated with the second energy gap predicted by the multiband-superconductor theory of Kuper, Jensen, and Hamilton. Kondo's multiband-superconductor theory is consistent with the experimental results. It is shown that if the $f$ band in Kondo's theory is approximately 20 meV or more higher than the Fermi level, then Kondo's theory reduces to a single-parameter theory having a gap equation identical in form to the BCS gap equation.

Localized Magnetic Impurity States in Ti, Zr, and Hf

An experimental study of localized moments on "magnetic" impurities (\ensuremath{\approx}0.2 at.% of Cr, Mn, Fe, Co, or Gd) in the group IV transition elements Ti, Zr, and Hf has been carried out. From magnetic-susceptibility measurements we find a localized moment on Mn of \ensuremath{\approx}3.5 ${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ in each of the solvents Ti, Zr, and Hf. Chromium forms a moment in Zr (0.2 ${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$) and in Hf (1.7 ${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$) but not in Ti. Iron forms a moment in Hf (0.7 ${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$) but not in Ti or Zr. Thus the tendency towards localized-moment formation on these $3d$ ions increases in the sequence Ti-Zr-Hf, possibly because of the decreasing solvent electronic-state density. Negative magneto-resistance is produced by each $d$-shell impurity for which magnetic-susceptibility measurements indicate a localized moment. The magnetoresistivity and susceptibility data are shown to be correlated in the weak-field limit by the elementary molecular-field description and the assumption that the magnetoresistance is proportional to the square of the magnetization. Resistivity minima are observed in "pure" (arc-cast, iodide process) Ti and Ti-Cr, -Mn, and -Fe; Zr-Mn; and Hf-Cr, -Mn, and -Fe. It thus appears that there is no strict correlation of resistivity minima and localized moments for the $d$-shell impurities. The resistivity-versus-temperature curves are not explained by the recent theory of Kondo. It is suggested that this apparent disagreement can be accounted for in part if one assumes a substantial temperature dependence of the nonmagnetic impurity-potential scattering, which was not included in Kondo's theory. For Gd in the group IVB solvents, a nearly free-atom moment is observed in Zr (7.0 ${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$) and Hf (6.1 ${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$), (Ti-Gd was not examined). No resistivity minima and only a very small negative magnetoresistance are observed in the alloys with Gd, suggesting a much weaker conduction-electron interaction with the localized magnetic states of the $f$-shell ions than with the localized moments on the $d$-shell impurities.

Self-Consistent Treatment of Kondo's Effect in Dilute Alloys

We investigate how conduction electrons in dilute alloys are affected by the exchange interaction with localized spins of impurities. It is shown that, if the interaction is antiferromagnetic, the perturbational treatment breaks down below a critical temperature, and that near the Fermi surface there appears a quasibound state between the conduction-electron spin and the localized spin. Because of the appearance of this quasibound state, the resistivity increases with decreasing temperature, but has a finite value at $T=0$. There is no logarithmic term in the resistivity at low temperatures, in contrast to Kondo's theory of the resistance minimum. There also appears an anomaly in the specific heat at low temperatures.

Anomalous Electron Scattering in Dilute Magnetic Alloys

The Kondo theory of the resistance minimum has been extended to show the following: (1) The resistance anomaly is not affected by nonmagnetic scattering. (2) The magnetoresistance shows a logarithmic field dependence at strong fields and low temperatures. (3) The anomalous scattering affects the superconducting properties of the alloy in nearly the same way as the normal scattering. So the theory based on normal scattering is still qualitatively correct.