Electron-Hole Recombination in Bismuth

We have measured the electron-hole recombination time ${\tau }_R$ in bismuth at temperatures from 2 to 50°K for two single-crystal samples with residual resistivity ratios, $\frac{{{\rho }_{300}}^{\circ }\mathrm{K}}{{\rho }_{4.2}}$ °K, of 260 and 560. Above ∼6°K, the value of ${\tau }_R$ is the same for both samples and decreases rapidly as the temperature increases from ∼${10}^{-8\mathrm{}}$ sec at 6°K. We postulate a model in which the absorption or emission of a single phonon provides for momentum conservation in the recombination of electrons and holes. The data above ∼6°K can be fitted with two phonons, one of energy (43±4) °K, the other (130±15) °K. We have determined, by group-theoretical methods, the selection rules for the phonons involved, and have shown our data to be consistent with them. At lower temperatures, ${\tau }_R$ becomes a function of sample purity. Below ∼3°K, the value of ${\tau }_R$ was found to be temperature-independent for both samples and equal to 1.3×${10}^{-8\mathrm{}}$ and 2.5×${10}^{-8\mathrm{}}$ sec, respectively, the ratio of which equals the ratio of the residual resistivities. The results were obtained from measurements of the acoustomagnetoelectric effect (AME) at frequencies ranging from 6 to 35 MHz, in which high-frequency ultrasound sent longitudinally along a sample in a transverse magnetic field generates a dc electric field normal to both the magnetic field and the sound-propagation direction. The dependence of the AME on frequency and on the magnitude and direction of the magnetic field was measured and compared with the theory of Yamada. The temperature dependence of the ultrasonic attenuation coefficient $\alpha$ was also obtained. For $T\le 20$ °K, the attenuation is mainly due to the interaction of the sound wave with carries via the deformation potential, which interaction also produces the AME. For large magnetic fields, quantum oscillations similar to the de Haas-van Alphen effect are observed in both $\alpha$ and the AME voltage. Electron periods in the trigonal plane are identified. Finally, a lower bound for the deformation potential that describes the change of overlap of the electron and hole bands due to a trigonal compression is obtained: $|E_n-E_p|\ge 1.5$ eV.