Temperature Dependence of the Low-Field Galvanomagnetic Coefficients of Bismuth

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^-12V) and accuracy (1 part in 10⁴) 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 10⁶ at any value of temperature in the range 4-16^∘{}K. The zero-field resistivities ρ₁₁⁰ and ρ₃₃⁰, normal and parallel to the trigonal direction, respectively, have been measured to 26^∘{}K. Both ρ₃₃⁰ and ρ₁₁⁰ are closely proportional to T² between 8 and 20^∘{}K. All eight magnetoresistance coefficients have an approximate T^-2 dependence, while the large Hall term ρ23,1 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 2.7×{}10¹⁷ electrons per cm³, 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 10⁷ cm²/V sec) μ₁=11, μ₂=0.3, μ₃=6.7, μ₄=-0.71, ν₁=2.2, and ν₃=0.35. The mobility tilt angle is a constant, θ_μ=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^-10 sec): τ₁ₑ=4.4, τ₂ₑ=22, τ₃ₑ=4.4, τ₁ₕ=8.5, and τ₃ₕ=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 μₑ=9×10⁷ cm²/V sec and μₕ=0.6×10⁷ cm²/V sec, in very good agreement with the measured mobilities.