Temperature Dependence of the Low-Field Galvanomagnetic Coefficients of Bismuth

Abstract

The linear Hall and quadratic magnetoresistance coefficients of bismuth have been measured as functions of temperature in the range 4-16°K. The sensitivity (^10~1 2 V) and accuracy (1 part in 10 4 ) necessary for the experiment required the construction of an automatically balancing superconducting-chopper picovolt potentiometer, together with a cryogenic system which was stable to 1 part in 10 6 at any value of temperature in the range 4-16°K. The zero-field resistivities pn° and p 3 3°, normal and parallel to the trigonal direction, respectively, have been measured to 26°K. Both p 33° and pn° are closely proportional to T 2 between 8 and 20°K. All eight magnetoresistance coefficients have an approximate T~2 dependence, while the large Hall term P23, I decreases approximately 7% as the temperature increases from 6 to 16°K. A least-squares fit of the data to a model based on the accepted band structure of bismuth was made at each temperature. From these, experimental values for the carrier density and the components of the mobility tensors for electrons and holes were obtained as a function of temperature. The carrier density, constant with temperature, is 2.7X10 17 electrons per cm 3 , and an equal hole density. All the mobility components varied as T~2 in the temperature range 8-16°K. At 4.2 the electron mobilities are (in 10 7 cm 2 /V sec) jui= 11, p,2 = 0.3, /* 3 = 6.7, p;4= -0.71, v\ = 2.2, and *> 3 = 0.35. The mobility tilt angle is a constant, 0 M =6.2°, in the temperature range 4.2-16°K. The components of the conductivity relaxation-time tensor were calculated for the electrons and holes at each temperature. At 4.2°K the maximum anisotropy of the electron relaxation-time tensor was found to be 5:1, decreasing rapidly as the temperature increased, while the anisotropy of the hole tensor was 2:1 over the entire temperature range. At 4.2°K the diagonal components of the electron and hole relaxation-time tensors are (in units of 10~1 0 sec): ri e =4.4, T2 e =22, r 3e =4.4, TIA = 8.5, and T 3 A=15. Because the conductivity varies as T~2, we argue that the dominant scattering is not deformation-potential scattering, but rather is between carriers in separate valleys. The carriers in different valleys interact via the Coulomb interaction, each remaining in its respective valley, conserving energy and momentum in the center-of-mass system, though not individually. For carriers of differing charge or of sufficient anisotropy, this mechanism contributes to the resistivity. In support of this mechanism, the electron and hole mobilities at 4.2°K were estimated, from the known ionized-impurity scattering, to be ju e =9X 10 7 cm 2 /V sec and //^=0.6X 10 7 cm 2 /V sec, in very good agreement with the measured mobilities. 5

Low Temperature Transport Properties of the Group V Semimetals

“…For bismuth, Zitter (1962) was the first to study rigorously this low-field condition in his investigation of the galvanomagnetic tensor of bismuth at 4·2 K. Hartman (1969) extended this work by carefully exploring the range 4·2-15·7 K. His results were consistent with the known Fermi surface. Abeles and Meiboom (1956) and Okada (1957) explored a higher temperature range.…”

“…At lower temperatures, there is the r 2 variation first reported by White and Woods (1958) for polycrystalline material and further confirmed by many investigators on single crystal samples (Bhagat and Manchon 1967;Friedman 1967;Fenton et al 1969;Hartman 1969;Chopra et al 1971;Kopylov and Mezhov-Deglin 1974;Kukkonen and Sohn 1977;Uher and Pratt 1977). But this variation still remains a matter for conjecture (Anagnostopoulos and Aubrey 1976;Kukkonen and Maldague 1976).…”

“…The theory of such effects has been reported with full details by many authors (Juretschke 1955;Abeles and Meiboom 1956;Drabble and Wolfe 1956;Okada 1957;Hartman 1969), and we shall only point out here the essential features. If Ei is the electric field and J i the current density, the resistivity Pi/B) is given by In the limit of weak fields (fiB ~ 1, where f1 is a carrier mobility), only the terms in Band B2 need be retained in the expansion of this tensor as a power series.…”

Spin-orbit interaction and phase coherence in lithographically defined bismuth wires

“…The electrons, however, are confined to three anisotropic Fermi surfaces, and µ then represents an effective transport mobility through many randomly oriented grains. In pure crystalline Bi, the electron and hole densities should be equal due to charge neutrality 49,50 . Our films are slightly n-doped, likely a result of lattice strain and vacancies due to the substrate mismatch and polycrystalinity.…”

Magnetic-field-induced far-infrared transmission in bismuth

“…The frequency 1.4 dependence of w was assumed because it had given the best fit 312 at lower frequencies, slightly better than the w fits shown earlier, and because it gave the best fit at the higher frequencies as well. 41 …”