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\ifmmode^\circ\else\textdegree\fi{}K. The sensitivity (\ensuremath{\sim}${10}^{\ensuremath{-}12}$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\ifmmode^\circ\else\textdegree\fi{}K. The zero-field resistivities ${{\ensuremath{\rho}}_{11}}^{0}$ and ${{\ensuremath{\rho}}_{33}}^{0}$, normal and parallel to the trigonal direction, respectively, have been measured to 26\ifmmode^\circ\else\textdegree\fi{}K. Both ${{\ensuremath{\rho}}_{33}}^{0}$ and ${{\ensuremath{\rho}}_{11}}^{0}$ are closely proportional to ${T}^{2}$ between 8 and 20\ifmmode^\circ\else\textdegree\fi{}K. All eight magnetoresistance coefficients have an approximate ${T}^{\ensuremath{-}2}$ dependence, while the large Hall term ${\ensuremath{\rho}}_{23,1}$ decreases approximately 7% as the temperature increases from 6 to 16\ifmmode^\circ\else\textdegree\fi{}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\ifmmode\times\else\texttimes\fi{}${10}^{17}$ electrons per ${\mathrm{cm}}^{3}$, and an equal hole density. All the mobility components varied as ${T}^{\ensuremath{-}2}$ in the temperature range 8-16\ifmmode^\circ\else\textdegree\fi{}K. At 4.2 the electron mobilities are (in ${10}^{7}$ ${\mathrm{cm}}^{2}$/V sec) ${\ensuremath{\mu}}_{1}=11$, ${\ensuremath{\mu}}_{2}=0.3$, ${\ensuremath{\mu}}_{3}=6.7$, ${\ensuremath{\mu}}_{4}=\ensuremath{-}0.71$, ${\ensuremath{\nu}}_{1}=2.2$, and ${\ensuremath{\nu}}_{3}=0.35$. The mobility tilt angle is a constant, ${\ensuremath{\theta}}_{\ensuremath{\mu}}=6.2\ifmmode^\circ\else\textdegree\fi{}$, in the temperature range 4.2-16\ifmmode^\circ\else\textdegree\fi{}K. The components of the conductivity relaxation-time tensor were calculated for the electrons and holes at each temperature. At 4.2\ifmmode^\circ\else\textdegree\fi{}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\ifmmode^\circ\else\textdegree\fi{}K the diagonal components of the electron and hole relaxation-time tensors are (in units of ${10}^{\ensuremath{-}10}$ sec): ${\ensuremath{\tau}}_{1e}=4.4$, ${\ensuremath{\tau}}_{2e}=22$, ${\ensuremath{\tau}}_{3e}=4.4$, ${\ensuremath{\tau}}_{1h}=8.5$, and ${\ensuremath{\tau}}_{3h}=15$. Because the conductivity varies as ${T}^{\ensuremath{-}2}$, 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\ifmmode^\circ\else\textdegree\fi{}K were estimated, from the known ionized-impurity scattering, to be ${\ensuremath{\mu}}_{e}=9\ifmmode\times\else\texttimes\fi{}{10}^{7}$ ${\mathrm{cm}}^{2}$/V sec and ${\ensuremath{\mu}}_{h}=0.6\ifmmode\times\else\texttimes\fi{}{10}^{7}$ ${\mathrm{cm}}^{2}$/V sec, in very good agreement with the measured mobilities.