Galvanomagnetic Studies of Sn-Doped Bi. II. Negative Fermi Energies

Abstract

Studies of the Shubnikov-de Haas effect and the Hall effect in single crystals of bismuth doped with tin are extended to excess hole concentration beyond 2×${10}^{18}$ ${\mathrm{cm}}^{-3\mathrm{}}$, the upper limit achieved in earlier reported work. For dopings greater than about 3×${10}^{18}$ ${\mathrm{cm}}^{-3\mathrm{}}$, the Fermi level lies below the bottom of the $L$-point conduction band, so that the Fermi energy measured from this band edge is negative. The passage of the Fermi level into the forbidden gap at the $L$ point is accompanied by the disappearance of electron quantum oscillations and by a decrease by two orders of magnitude in the magnetoresistance at 4.2°K. At about 5×${10}^{18}$ excess holes/${\mathrm{cm}}^3$, a large magnetoresistance effect reappears, and low-field quantum oscillations, which are attributed to light holes at the $L$ point, are observed. The dependence of the light-hole periods on magnetic field orientation suggests that the longitudinal mass is smaller for $L$-point holes [$L_a\mathrm{}\left(3\right)\mathrm{}$] than for electrons [$L_s\mathrm{}\left(3\right)\mathrm{}$], as is predicted by Golin's band-structure calculation. $T$-point hole oscillations are observed for excess hole concentrations up to ${10}^{19}$ ${\mathrm{cm}}^{-3\mathrm{}}$ and measurements have been extended to fields of 90 kOe to study their anisotropy. The anisotropy is consistent with a $T$-point hole surface which is a prolate ellipsoid of revolution for excess hole concentrations up to 3×${10}^{18}$ ${\mathrm{cm}}^{-3\mathrm{}}$, but the surface becomes less prolate as tin is added. Data on this surface are compared with the predictions of a six-band $\mathrm{k}·\pi$ calculation derived from Golin's pseudopotential theory, using the matrix elements at $T$ which he calculated; good agreement is obtained. It is pointed out that the analysis of band nonparabolicity in an earlier paper which makes use of the Abrikosov-Falkovski dispersion relation underestimates the magnitude of the ${T_{45}}^{-}\mathrm{}\left(1\right)-{T_6}^{+}\mathrm{}\left(3\right)\mathrm{}$ direct gap at $T$. On the basis of the six-band model, the observed nonparabolicity is found to be consistent with the gap estimated by Golin.