Stress-Induced Band Gap and Related Phenomena in Gray Tin

Abstract

The conductivity and low-field Hall coefficient of high-purity (${N}_{D}\ensuremath{-}{N}_{A}\ensuremath{\sim}5\ifmmode\times\else\texttimes\fi{}{10}^{14}$ ${\mathrm{cm}}^{\ensuremath{-}3}$) and lightly doped ($2\ifmmode\times\else\texttimes\fi{}{10}^{15}\ensuremath{\le}{N}_{D}\ensuremath{\le}2\ifmmode\times\else\texttimes\fi{}{10}^{17}$ ${\mathrm{cm}}^{\ensuremath{-}3}$) $n$-type gray tin subjected to oriented uniaxial compressions have been measured between 1.4 and 100 \ifmmode^\circ\else\textdegree\fi{}K. Stress ($\ensuremath{\chi}$) exceeding 3 \ifmmode\times\else\texttimes\fi{} ${10}^{9}$ ${\mathrm{d}\mathrm{y}\mathrm{n}/\mathrm{c}\mathrm{m}}^{2}$ was achieved in both [001] and [111] orientations. Density-of-states expressions are developed to account for the severe band anisotropies imposed by the strain in the normally degenerate ${\ensuremath{\Gamma}}_{8}^{+}$ conduction and valence bands, and these are employed to determine the band splittings at $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}=0$ from the Hall coefficient of the high-purity samples above 15 \ifmmode^\circ\else\textdegree\fi{}K. Shear deformation potentials of $b=\ensuremath{-}2.3\ifmmode\pm\else\textpm\fi{}0.5$ eV and $d\ensuremath{\simeq}\ensuremath{-}4.1$ eV are obtained by this procedure. The Hall coefficient of three high-purity samples below 10 \ifmmode^\circ\else\textdegree\fi{}K is analyzed to find the stress-dependent impurity-ionization energy ${E}_{D}(\ensuremath{\chi})$, and from the measured ${E}_{D}(\ensuremath{\chi})$ for the highest-purity sample an independent determination of $b=\ensuremath{-}2.4$ eV is obtained if ${E}_{D}(\ensuremath{\chi})$ is interpreted as reflecting donor-to---conduction-band activation. However, the measured ${E}_{D}(\ensuremath{\chi})$ for this sample is also found to be consistent with activation from the donor ground state into a ${D}^{\ensuremath{-}}$ band. The stress dependence of the impurity mobility in two of these samples is explained in terms of Sladek's model for exchange jumping between filled and unfilled impurity sites. The piezoresistance of lightly doped samples is attributed to the increased effectiveness of ionized impurity scattering caused by a stress enhancement of the ${\ensuremath{\Gamma}}_{8}^{+}$ density-of-states mass.