Shubnikov-de Haas Measurements in ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Te}$

Oscillatory-magnetoresistance measurements have been carried out on various samples of ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Te}$. Compositions ranged from $x=0.16\mathrm{}$ to $x=0.32\mathrm{}$, carrier concentrations from 3 × ${10}^{16}$ to 1.8 × ${10}^{18}$, and most of the samples were $p$ type. The Fermi surface was found to consist of prolate ellipsoids oriented along the [111] axis at the $L$ points. The ellipsoid-anisotropy ratio $K$ has values between 10.5 and 11.2, and within an accuracy of about 10% is independent of carrier concentration and of $x$ within the above range. Effective masses and $g$ factors were also determined. For example, a sample with $x=0.216\mathrm{}$ and with a hole density of 8.5 × ${10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$ was found to have a transverse mass $m_{111}^{*}=0.0225m$, where $m$ is the free electron mass. The mass was observed to decrease with decreasing carrier density. Clear spin splitting was observed in the [110] and [100] directions in this sample with $g_{110}=52\mathrm{}$ and $g_{100}=37\mathrm{}$. The ratio of spin splitting to Landau-level spacing was found to be approximately 0.68 for both directions. The carrier density as determined by Hall measurements agrees with the number of states calculated from the measured Fermi-surface dimensions. Extra frequency branches such as those observed by Burke et al. in SnTe and in ${\mathrm{Pb}}_{0.87}$${\mathrm{Sn}}_{0.13}$Te were not found.