Mobility of Electrons and Holes in the Polar Crystal, PbS

Abstract

From measurements of the Hall effect and resistivity on single crystals of PbS, the mobility of electrons and holes has been found over the temperature range 77-600\ifmmode^\circ\else\textdegree\fi{}K. The crystals had donor or acceptor concentrations in the range ${10}^{16}$-${10}^{19}$/${\mathrm{cm}}^{3}$. These data are used to critically examine theoretical work on the mobility in polar crystals and to obtain estimates of the effective masses of electrons and holes. The perturbation theory of Fr\"ohlich and Mott (F-M), and Howarth and Sondheimer (H-S) for the scattering of electrons by the polar (optical) modes of the lattice vibration is first considered. The expansion parameter of the perturbation theory is evaluated as 0.28 ${(\frac{{{m}_{e}}^{*}}{{m}_{e}})}^{\frac{1}{2}}$ from published data on PbS; ${m}_{e}$ is the free and ${{m}_{e}}^{*}$ is the effective electron mass. The polaron theory of mobility of Low and Pines, which has been developed to replace the perturbation theory when the expansion parameter approaches or exceeds unity, is then discussed. $\frac{{{m}_{e}}^{*}}{{m}_{e}}$ is the only unknown parameter in the two theories. The mobility data is first compared with the F-M, H-S theory from which $\frac{{{m}_{e}}^{*}}{{m}_{e}}=0.33$, $\frac{{{m}_{h}}^{*}}{{m}_{e}}=0.36$. These effective masses, when substituted in the Hall equation, provide for reasonable agreement with the Hall data. However, at high and low temperatures, discrepancy exists between the theoretical and experimental mobility curves. Comparison of the mobility data with the polaron theory shows that the polaron theory is nearly identical with the F-M, H-S theory at low temperatures. The polaron theory has not been completed for the high temperature region.Since the data does not indicate the presence of impurity scattering, an analysis is made combining polar (F-M, H-S theory) and acoustical scattering, which yields $\frac{{{m}_{e}}^{*}}{{m}_{e}}=0.22$ and $\frac{{{m}_{h}}^{*}}{{m}_{e}}=0.10$. These effective masses give theoretical Hall curves in reasonably good agreement with the data. While the theoretical and experimental mobility curves now agree at low and intermediate temperatures, discrepancy still exists at high temperatures. Possible reasons for the discrepancy are discussed. It is concluded that the most likely source of error lies in perturbation theory expressions for the scattering cross sections for electron energies greater than that corresponding to the frequency of the polar vibration. Comparison of the high-temperature data with the polaron theory awaits further development of the theory and should provide a sensitive test of the general polaron theory, as well as being of interest in the theory of mobility in polar crystals.