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The electrical resistivity anisotropy of Cd, Zn and Mg
Abstract
The authors have measured eddy-current decay times in orientated single crystals of Cd, Zn and Mg between 1.3K and 300K to obtain the residual and temperature-dependent resistivity and anisotropy. Deviations from Matthiessen's rule have been measured for Cd and Mg in order to estimate the temperature range in which one can reliably obtain values of the 'ideal' phonon resistivity and its anisotropy, to be compared with the theory of phonon-limited resistivity. The characteristic residual scattering anisotropy, which is very difficult to obtain from a conventional resistance measurement, has been determined for a dilute homovalent Cd(Zn) alloy, and the results indicate that the electron relaxation time is not isotropic for this system.
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It is possible to obtain an approximate solution of the Boltzmann transport equation which corresponds to assigning a definite scattering time for each electron on the Fermi Surface (F.S.). We have made an estimate of the scattering times when the mechanism for electron scattering is the electron-phonon interaction. Information on the lattice dynamics is taken from the phonon dispersion curves which have been measured by the technique of inelastic neutron scattering. For the electron-ion pseudopotential we employ the model form factors developed by Stark and Falicov. From this information the scattering times are calculated as a function of temperature and position on the F.S. They show considerable variation over the F.S. The electrical resistivity (ϱ) with electric field parallel (ϱ) and perpendicular (ϱ⊥) to the c-axis is discussed. The temperature variation of the ratio ϱ/ϱ⊥ is found to be in qualitative agreement with experiment.RésuméIl est possible d'obtenir une solution approximative de l'équation de transport de Boltzmann qui correspond à definir un temps de parcour pour chaque éléctron à la surface de Fermi. Nous avons estimés ces temps de parcour dans le cas ou le méchanism qui déflect les élétrons est l'interaction éléctron-phonon. Les informations sur la dynamic des ions sont tirées des mesures de diffusion inelastique de neutron.Pour l'interaction éléctron-phonon nous employons la forme du pseudopotential proposee par Stark et Falicov. A partir de ces données nous avons calculés la variation sur la Surface de Fermi des temps de parcour. Cette variation est très importante. La resistivity (ϱ) du zinc avec champ éléctric parallèle (ϱ// et perpendiculaire (ϱ⊥) à l'axe c est calculée et nous trouvons que la variation en temperadu raport ϱ///ϱ⊥ est en accord qualitative avec les resultats experimentaux.
Eddy current decay in a single-crystal sphere of an anisotropic metal is investigated as a function of the angle between the crystal C axis and the magnetic field. By comparison with resistivity measurements obtained using the conventional four-probe method it is shown that an eddy current technique may be used to obtain the resistivity anisotropy alpha , at least for materials with anisotropies less than 1.5. The residual resistivity anisotropy is measured for samples of zinc and cadmium and results are presented for the ideal anisotropy between 4K and 300K.
Geometric resonances in the ultrasonic attenuation have been studied in magnesium for sound propagated in the three principle crystallographic directions. A total of twenty different geometric resonance branches have been obtained. The assignment of these branches to Fermi-surface calipers shows that the magnesium Fermi surface is much more free-electron-like than currently thought. Three of the first four local pseudopotential Fourier expansion coefficients are estimated from data to be 5 × 10-4 1.0 × 10-2, and 1.7 × 10-2 Ry for the absolute value of the [101¯0], [0002], and [101¯1] coefficients, respectively. A two orthogonalized-plane-wave model is given for the lens-shaped piece of the Fermi surface in the third Brillouin zone, and various physical properties of this model are calculated and compared with experimental data.
Ultrasonic attenuation as a function of magnetic field intensity has been measured in very pure Cd at liquid-helium temperatures. The results are related to the dimensions of the Fermi surface using the usual phenomenological model for the magnetoacoustic effect. New resonances are observed, and an explanation in terms of cylindrical sections of the Fermi surface is offered. Finally, evidence for acoustic cyclotron resonance is presented, along with comments on the observation and interpretation of such phenomena.
The normal-mode frequencies of magnesium at 290 K have been determined by the inelastic scattering of slow neutrons. The frequencies of about 3500 phonons with wavevectors in the (0001) and (01{1}0) planes have been measured with a time-of-flight apparatus, and an interpolation technique has been used to deduce the dispersion curves along closely spaced lines in the two planes. Measurements with a triple-axis crystal spectrometer have been used to supplement the time-of-flight data along several symmetry directions. In order to generate the frequencies throughout the entire Brillouin zone a simple force-constant model extending to eight-nearest neighbours has been used. The values of the force constants were obtained from the phonons along five symmetry directions. Although the model has little physical significance, it gives frequencies in good agreement with the experimental values over the two planes for which measurements have been made. The model has been used to calculate the phonon density of states, and hence the Debye temperatures for the specific heat. The latter are in satisfactory agreement with the values deduced from calorimetric measurements.
Using pseudopotential theory for the phonons, electrons, and electron-phonon interaction of zine, we have calculated several electronic properties of this metal. Among these are the electron lifetime at three positions on the Fermi surface, the anisotropic electrical resistivity, the anisotropic thermal conductivity, and the ultrasonic attenuation, all due to the electron-phonon interaction, and all as a function of temperature. For the electrical resistivity, two solutions of the Boltzmann equation were obtained, one consisting of a single spherical harmonic, and the other made up of six spherical harmonics. The single spherical harmonic is not adequate at low temperatures.
The electrical resistivity observed along and perpendicular to the hexagonal axis has been measured in monocrystals of Cd, Zn, and Mg over the range ≈(3-300)°K [≈(3-590)°K for Cd]. The temperature dependence of the anisoptropy parameter (α=ρ∥/ρ⊥) has been examined in the light of the Case-Gueths simplified model of conductivity in an anisotropic metal. In the case of Cd there is good quantitative agreement with the predictions of the model, while in Zn there is at least good qualitative agreement with them. But for Mg (which alone has α<1) quite the wrong behavior of α(T) is observed. It is suggested that the discrepancy reflects the oversimplified view of the Fermi surface which is used, particularly since it neglects the conductivity in the basal plane arising from the third-band "needles"—which of course are absent in Cd and genuinely negligibly small in Zn.
The authors have measured the electrical resistivities from 2K to 300K for eight Cd single crystals and eight Cd-Mg single crystals of two different concentrations. The measured densities of their specimens were typically 21/2% below the accepted X-ray densities. This is attributed to small bubbles in the specimens and the authors have made an appropriate correction to obtain the resistivity values. The ice point resistivities for Cd were found to be rho i//=7.81(+or-0.03) mu Omega cm and rho i perpendicular to =6.30(+or-0.03) mu Omega cm. Deviations from Matthiessen's rule Delta which were calculated from these measurements both confirm and clarify earlier experimental results for polycrystalline specimens and agree qualitatively with a theoretical calculation for single crystals.
A method for measuring the resistivity of metallic specimens is described. The measurement is made by noting the rate of decay of flux from a bar situated in an external magnetic field that has been rapidly reduce to zero. The method is suitable for specimens greater than 5×10-3 cm in diameter. For a specimen 1 cm in diameter, resistivities from 10-11 to 10-3 ohm‐cm may be measured with an error of less than three percent. The method requires no contact to the specimen, and local values of resistivity may be measured. Several applications are described.