Influence of Hydrostatic Pressure and Temperature on the Deep Donor Levels of Sulfur in Silicon

The Hall effect and resistivity of silicon doped with sulfur have been measured as a function of temperature in the range between 300 K and temperatures as low as 50 K (the latter depending on the sample in question) and as a function of hydrostatic pressure up to 8 kbar at various temperatures in this range. From the Hall-effect data, three of the four known levels of sulfur in silicon were found to have ionization energies in meV as follows: $A: (105\mathrm{}\pm 15)-(0.22\pm 0.03)P+(0.25\pm 0.05)T$ $B: (190\mathrm{}\pm 20)-(0.55\pm 0.05)P+(0.057\pm 0.012)T$ $C: (360\mathrm{}\pm 30)-(1.1+0.1)P-(0.61\mathrm{}\pm 0.12)T$ where $P$ is in kbar and $T$ is in K. The temperature dependence of the ionization energies may be due to electron-phonon interaction, but neither the sign nor magnitude of the effect has been calculated theoretically. With changing pressure, each level shifts with respect to both conduction- and valence-band edges at a rate that decreases with its distance from that band edge. This "rubber-band effect" is discussed qualitatively. A group-theoretical analysis is made of the Bloch waves that can appear in expansions of localized-impurity wave functions of various symmetries. This provides a basis for drawing some inferences concerning the symmetry of impurity centers from the presence or absence of the rubber-band effect. The observed effects are consistent with the idea that the $D$ center is a substitutional ${\mathrm{S}}^{+}$ at a $T_d$ site; that the $C$ center is an ${S_2}^{+}$-center with $D_{3d}$ symmetry; that the $B$ center is the neutral version of the $C$ center; and that the $A$ center is not simply the neutral version of the $D$ center, but may represent as well the effects of interstitial sulfur at $D_{3d}$ sites. With no applied pressure, the Hall mobility was found to have a magnitude and temperature dependence in samples prepared from undoped silicon different from that in samples prepared from boron-doped silicon. A quantitative explanation of the mobility is lacking, but it appears that scattering by agglomerations of sulfur might be important, in addition to scattering by lattice vibrations, ionized impurities, and neutral impurities. The pressure dependence of the Hall mobility is very small and can be attributed, at least in part, to the decrease of effective mass with increasing pressure.