Hall effect in dilute magnetic alloys

We review the theory and the experimental results on the Hall effect in noble metals containing magnetic impurities of transition metals. In order to illustrate the various types of observed effects, we focus succesively on selected systems: $\text{Cu}$Mn, with only enhancement of the ordinary Hall effect due to the existance of different spin-up and spin-down currents; $\text{Au}$Fe and $\text{Au}$Cr, with skew scattering by magnetic impurities; $\text{Cu}$MnT ternary alloys (where T is a non-magnetic impurity), with skew scattering effects due to combined spin—orbit scattering by non-magnetic impurities and spin scattering by Mn impurities. The skew scattering in $\text{Au}$Fe and $\text{Au}$Cr can be ascribed to the orbital character of the impurity moments and accounted for in an orbitally degenerate virtual bound state model. However, the anomalous temperature dependence of the skew scattering in Kondo alloys at low temperature is not well understood. We also present some magnetoresistance data in order to describe the links between the Hall effect and the magnetoresistance in magnetic alloys. In particular, we relate the skew scattering and the magnetoresistance anisotropy observed in $\text{Au}$Cr alloys.     

Pseudogap-state superconductivity in a “hot-point” model: Gor’kov equations

The specific features of the superconducting state (with s and d pairing) are considered in terms of a pseudogap state caused by short-range order fluctuations of the “dielectric” type, namely, antiferromagnetic (spin density wave) or charge density wave fluctuations, in a model of the Fermi surface with “hot points.” A set of recurrent Gor’kov equations is derived with inclusion of all Feynman diagrams of a perturbation expansion in the interaction between an electron and short-range order fluctuations causing strong scattering near hot points. The influence of nonmagnetic impurities on superconductivity in such a pseudogap state is analyzed. The critical temperature for the superconducting transition is determined, and the effect of the effective pseudogap width, correlation length of short-range-order fluctuations, and impurity scattering frequency on the temperature dependence of the energy gap is investigated.     

Influence of nonmagnetic impurities on the kondo effect

The influence of scattering by nonmagnetic impurities is studied in perturbation theory. While the finite lifetime of the electrons in intermediate states due to scattering by nonmagnetic impurities does not lead to a change in the logT-behaviour of the third-order self-energy, certain vertex-corrections give rise to an additional term which varies like 1/√T at low temperatures. Similar correction terms are found to occur in the higher order self-energy contributions. Although these terms diverge more strongly atT=0 than the logarithmic contributions they are quite small at finite temperatures since they depend on the lifetime τ of the electrons through a factor of (? _F τ)^?5/2 (? _F Fermi energy). The possibility of observing these interference effects experimentally is discussed.     

Temperature and concentration dependences of the electrical resistivity for alloys of plutonium with americium under normal conditions

The temperature and concentration dependences of the electrical resistivity for alloys of americium with plutonium are analyzed in terms of the multiband conductivity model for binary disordered substitution-type alloys. For the case of high temperatures (T > Θ_D, Θ_D is the Debye temperature), a system of self-consistent equations of the coherent potential approximation has been derived for the scattering of conduction electrons by impurities and phonons without any constraints on the interaction intensity. The definitions of the shift and broadening operator for a single-electron level are used to show qualitatively and quantitatively that the pattern of the temperature dependence of the electrical resistivity for alloys is determined by the balance between the coherent and incoherent contributions to the electron-phonon scattering and that the interference conduction electron scattering mechanism can be the main cause of the negative temperature coefficient of resistivity observed in some alloys involving actinides. It is shown that the great values of the observed resistivity may be attributable to interband transitions of charge carriers and renormalization of their effective mass through strong s-d band hybridization. The concentration and temperature dependences of the resistivity for alloys of plutonium and americium calculated in terms of the derived conductivity model are compared with the available experimental data.     

Theory of impurity scattering: Interstitial impurities

A framework for studying impurity scattering in dilute, non-magnetic, metal alloys can be developed from a knowledge of the exact electronic eigenstates of a single impurity in an otherwise perfect lattice of muffin tin potentials. Such an approach has been developed for systems in which the impurity occupies a substitutional site of the lattice, as will be discussed by Coleridge, Lee, Harris, and other speakers of this conference. In this paper, motivated by recent experimental studies of Dingle temperature anisotropies induced by hydrogen impurities in copper, we will discuss the analogous treatment of scattering by interstitial impurities. In contrast to a substitutional impurity, an interstitial impurity introduces an additional scattering site into the lattice. Whereas the substitutional impurity wavefunctions can be described in terms of the same structure factors as can the Bloch wavefunctions for the pure host lattice, the interstitial impurity wavefunctions depend upon additional structure factors appropriate to the new scattering geometry. These additional structure factors appear in the transition matrix for impurity-induced scattering between Bloch states of the host lattice, and consequently in the weight factors involved in a partial wave analysis of the Dingle temperature anisotropies.     

A review of: “Classical electricity and magnetism”

The article reviews the considerable progress that has been made recently in the experimental determination of the electronic structure of metals and in particular the determination of Fermi surfaces and neighbouring surfaces of constant energy. In Part I the concept of electronic structure is briefly explained and this is followed by a simple analysis of the dynamics of electron motion in a magnetic field. The geometry of the orbit carried out by an electron in real space is related to the geometry of surfaces of constant energies in k-space and it is shown how the dimensions of orbits may be inferred from size-effect and ultrasonic experiments if perfect enough samples are available. Finally the frequency of rotation of the electrons in a magnetic field is related to a differential property of the constant energy surfaces and it is shown how this frequency can be measured in experiments on cyclotron resonance.

Part II starts by considering the effect of purification of the orbital motion and it is shown that this leads to an oscillatory field dependence of the magnetic properties known as the de Haas-van Alphen effect. The conditions for practical observation of this effect are discussed and it is shown that it can provide valuable information about the electronic structure. The frequency of the oscillations gives extreme areas of the Fermi surface, the temperature dependence of amplitude gives information similar to that from cyclotron resonance, while the field dependence of amplitude gives information about scattering time. As an illustration of the experimental methods a fairly detailed account is given of the determination of the Fermi surface of copper and of the variation of electron velocity over this Fermi surface. The article concludes with a brief mention of the more complicated Fermi surfaces of a few polyvalent metals.     

Evidence from the de Haas-van Alphen effect

The amplitude of the de Haas-van Alphen effect is reduced by collision broadening of the Landau levels approximately as if the temperature were raised by an amountx (the Dingle temperature) which is inversely related to an appropriate relaxation timeτ. Following a discussion of what is involved in obtaining significant estimates ofτ from experimental measurements ofx, the rather meagre available results are reviewed. For fairly pure samples, values ofx are often found which are as much as 50 times what would be expected from the resistive relaxation time; the evidence suggests that such high values ofx are due to small-angle scattering by imperfections, though they may also be due partly to phase smearing effects not connected with collision broadening. The most significant results come from studies of the increases ofx due to controlled additions of small amount of impurity. These increases are generally of much the same order of magnitude as would be expected from the increase or residual resistance, though detailed correlation with resistivity is hardly possible where thex measurements refer to a small part of a complicated Fermi surface. Recent results on impurities in the noble metals are beginning to give some indication of the anisotropy of scattering and because of the relative simplicity of the Fermi surface, a more detailed correlation with resistivity is possible. Particularly interesting anomalous behaviour is obtained for transition metal impurities which give rise to the Kondo effect. The possibility of studying phonon scattering through its effect onx is briefly discussed.     

Model calculations of anisotropic electron lifetimes due to dilute substitutional impurities in molybdenum

The de Haas-van Alphen technique forms a sensitive probe of the electronic structure of dilute alloys because of its dependence on the seattering rate in the forward direction. We report the results of a first principle model calculation of the scattering rate of dilute impurities in molybdenum which focuses on the anisotropy of the host wave function on the Fermi surface. This simplified model treats the impurity potential as an atomic potential screened by a Thomas-Fermi function. Substitutional impurities in bcc molybdenum have been studied using the Greens function (KKR) wave functions. The band structure calculated via the KKR method was fitted to a Fourier series representation in order to accurately determine a sufficiently large number of states on the Fermi surface. The KKR wavefunctions were used to calculate scattering rates for the substitutional impurity since the impurity potential can be best described in an angular momentum representation which is inherent to the Greens function method. Our detailed results, to be presented, suggest additional experiment to be done.     

Theory of impurity scattering: Interstitial impurities

A framework for studying impurity scattering in dilute, non-magnetic, metal alloys can be developed from a knowledge of the exact electronic eigenstates of a single impurity in an otherwise perfect lattice of muffin tin potentials. Such an approach has been developed for systems in which the impurity occupies a substitutional site of the lattice, as will be discussed by Coleridge, Lee, Harris, and other speakers of this conference. In this paper, motivated by recent experimental studies of Dingle temperature anisotropies induced by hydrogen impurities in copper, we will discuss the analogous treatment of scattering by interstitial impurities. In contrast to a substitutional impurity, an interstitial impurity introduces an additional scattering site into the lattice. Whereas the substitutional impurity wavefunctions can be described in terms of the same structure factors as can the Bloch wavefunctions for the pure host lattice, the interstitial impurity wavefunctions depend upon additional structure factors appropriate to the new scattering geometry. These additional structure factors appear in the transition matrix for impurity-induced scattering between Bloch states of the host lattice, and consequently in the weight factors involved in a partial wave analysis of the Dingle temperature anisotropies.This work was supported by the National Science Foundation through the Materials Research Laboratory at the University of Chicago.     

The Hall coefficient and other transport properties of dilute alloys of copper and silver

The low field Hall coefficient of a number of polycrystalline foils of dilute (2%) alloys of copper and silver has been measured in the temperature range 1.5–50°K, and at room temperature. The alloys chosen wereCu-Au andAg-Au (uncharged impurity),Cu Ge andAg-Sn (charged impurity), andCu-Ni andAg-Pd (transition metal impurity). At 20°K and below, the Hall coefficients of the different copper alloys differ widely from each other,Cu-Ge giving the highest (negative) values (up to twice the room temperature value for pure copper), andCu-Au the lowest (down to 0.7 of this value). There are also significant concentration dependences. The silver alloys show corresponding but smaller changes. A relationship, due to Tsuji, gives the Hall coefficient as a function of the Fermi velocityν and the mean curvature 1/ϱ of the Fermi surface, for the case of an isotropic relaxation time. The integrals over the Fermi surface have been numerically estimated using the known Fermi surface and electron velocities. For both Cu and Ag the results agree with the experimental room temperature values, which we take as evidence thatτ(k) for phonon scattering is here close to isotropic. On the other hand, to account for the Hall coefficients of the alloys, it is necessary to assume that the relaxation timeτ varies over the Fermi surface. It is seen that in Cu and Ag the neck regions contribute relatively little toR since both 1/ϱ andν are small there. The main change inR in different alloys arises from the variation in the relative weighting given to the belly regions by different kinds of impurity scattering. A closer analysis shows that the bulges in the Fermi surface of copper in the 〈100〉 directions contribute relatively heavily because of their high positive curvature. The anisotropy ofτ deduced from the Hall coefficient is compared with that deduced from other measurements.     

Nuclear spin-lattice relaxation in germanium single crystals

The temperature dependence of the spin-lattice relaxation time corresponding to the inelastic scattering of phonons by the⁷³Ge quadrupole moment in Ge single crystals is calculated in the framework of the adiabatic bond charge model. The results obtained agree with the experimental data.     

To the question on electron-electron scattering in metals

Temperature dependence of the scattering rate in copper and aluminium has been investigated by the radio frequency size effect with the aim of extracting the contribution of electron-electron collisions. The scattering rate is varied by the cubic law on the wide temperature range (1.2–10°K), which is due to the electron-phonon interaction. On the basis of modern representations on the electron-electron interactions, the absence of the quadratic term in the probability of scattering of charge carriers in these metals, close to the transition ones, is discussed.     

Interpretation of isomer shifts of substitutional119Sn and129I in group IV semiconductors

The measurements of the isomer shift in Mössbauer effect of substitutionally implanted¹¹⁹Sn and¹²⁹I atoms in group IV semiconductors show that the contact density of 5s electrons at the impurity ńucleus decreases when going from Ge to Si and diamond. A satisfactory interpretation of this dependence can be given based on a simple perturbation approach: Interaction of the host crystal with the bonding valence electrons of the impurity atom causes a charge redistribution of these electrons and results in decrease of occupancy of the 5s level at the impurity atom. Some general aspects of this problem with substitutional isoelectronic and donor impurities are discussed.     

Renormalization group and Fermi liquid theory

We give a Hamiltonian-based interpretation of microscopic Fermi liquid theory within a renormalization group framework. The Fermi liquid fixed-point Hamiltonian with its leading-order corrections is identified and we show that the mean field calculations for this model correspond to the Landau phenomenological approach. This is illustrated first of all for the Kondo and Anderson models of magnetic impurities which display Fermi liquid behaviour at low temperatures. We then show how these results can be deduced by a reorganization of perturbation theory, in close parallel to that for the renormalized φ⁴ field theory. The Fermi liquid results follow from the two lowest order diagrams of the renormalized perturbation expansion. The calculations for the impurity models are simpler than for the general case because the self-energy depends on frequency only. We show, however, that a similar renormalized expansion can be derived also for the case of a translationally invariant system. The parameters specifying the Fermi liquid fixed-point Hamiltonian are related to the renormalized vertices appearing in the perturbation theory. The collective zero sound modes appear in the quasiparticle-quasihole ladder sum of the renormalized perturbation expansion. The renormalized perturbation expansion can in principle be used beyond the Fermi liquid regime to higher temperatures. This approach should be particularly useful for heavy fermions and other strongly correlated electron systems, where the renormalization of the single-particle excitations are particularly large.

We briefly look at the breakdown of Fermi liquid theory from a renormalized perturbation theory point of view. We show how a modified version of the approach can be used in some situations, such as the spinless Luttinger model, where Fermi liquid theory is not applicable. Other examples of systems where the Fermi liquid theory breaks down are also briefly discussed.     

Superconductivity in the pseudogap state in the hot spot model: The influence of impurities and the phase diagram

We analyze the peculiarities of the superconducting state (s- and d-wave paring) in the model of the pseudogap state induced by Heisenberg antiferromagnetic short-range order spin fluctuations. The model is based on the pattern of strong scattering near hot spots at the Fermi surface. The analysis is based on the microscopic derivation of the Ginzburg-Landau expansion with the inclusion of all Feynman diagrams of perturbation theory for the interaction of an electron with short-range order fluctuations and in the ladder approximation for the scattering by normal (nonmagnetic) impurities. We determine the dependence of the critical superconducting transition temperature and other superconductor characteristics on the pseudogap parameters and the degree of impurity scattering. We show that the characteristic shape of the phase diagram for high-temperature superconductors can be explained in terms of the model under consideration.     

Pseudopotential charge distributions for characteristic k-states on the Fermi surface of aluminium

Stimulated by the existence of very anisotropic scattering effects in aluminium we have calculated wave functions and charge distributions for some selected k-state on the Fermi surface. For this purpose the pseudopotential method together with Ashcroft's potential has been used. The results show marked and characteristic differences in the various charge distributions and allow a qualitative interpretation of the scattering anisotropies observed in Hall effect experiments.     

Electric field gradients in cubic alloys

The theoretical investigations of electric field gradient (EFG) and asymmetry parameter (ν) for cubic dilute alloys are reviewed. The valence and size EFG's and the Sternheimer antishielding factor are discussed in detail. The calculations of EFG and ν for simple and transition metal (TM) dilute alloys of aluminium and copper, TM dilute alloys of vanadium and trinary dilute alloys of noble metals are summarized. It is emphasized that the size EFG is as important as the valence EFG to calculate the total EFG and ν, which are compared with the experimental values. An epilogue for further investigations is added.     

Semiclassical treatment of quadrupole shape effects in the elastic scattering of heavy ions below the Coulomb barrier

A perturbative semiclassical approximation for the elastics-matrix is used to derive simple and accurate formulae describing the effects of a nuclear quadrupole deformation on the elastic scattering of aligned nuclei. Expressions are derived for the second rank tensor components of the analysing power, the ratios of which turn out to be simple trigonometric functions of the scattering angle in agreement with experimental observations. The approximations are discussed in some detail and higher order corrections are derived. In an appendix we derive a semiclassical first-order perturbation formula describing the effect of a non-central complex perturbation in the presence of a non-perturbatively treated central term. Our formula is in disagreement with some earlier published formulae which fail to treat the real part of the perturbation correctly.     

Crystalline field effects in superconducting La1−xTbxAl2 under pressure

The pressure dependence of the superconducting transition temperature of LaAl₂ and alloys with Tb and Gd impurities has been measured. The results are compared with a theoretical calculation of the pressure dependence ofT _c , which is based on the variation by pressure of the crystalline field experienced by the Tb ions. Measurements onLaGdAl₂ where crystal field effects are absent have been used to study the influence of pressure on the scattering rate.     

Mössbauer effect in iron and dilute iron based alloys

Fifteen years of Mössbauer Effect (ME) studies have significantly widened the insight into the physical properties of iron and iron based alloys. In this review article the various contributions to the hyperfine interactions as measured with the ME technique, namely the isomer shift, the magnetic hyperfine interaction and the quadrupole interaction are summarized. Further the impurity effects as the Friedel type oscillations in the charge and the RKKY type oscillations in the spin density distribution are discussed. Special attention is paid to the peculiar magnetic properties of metallic iron and its alloys. From a comparison of the magnetic hyperfine fields and bulk magnetizations as a function of the impurity concentration and from the temperature dependence of the magnetic hyperfine fields at the various sites in iron alloys, it is concluded that the 3d magnetic moments in iron are largely localized. Further the exchange interaction is provided by the remaining small percentage of itinerant 3d electrons. Finally, from a comparison of magnetic hyperfine field and isomer shift data in alloys an overall picture of the electronic changes involved in alloying has been developed.