Piezocapacitance measurements of phosphorous- and antimony-doped silicon: Uniaxial strain-dependent donor polarizabilities

Piezocapacitance measurements on high-purity Si, P-doped $\mathrm{Si} (N_D\sim 6.8\times \frac{{10}^{16}}{{\mathrm{cm}}^3} \mathrm{to} 1.9\times \frac{{10}^{18}}{{\mathrm{cm}}^3})$, and Sbdoped $\mathrm{Si} (N_D\sim 6.3\times \frac{{10}^{16}}{{\mathrm{cm}}^3})$ samples, with an uniaxial tensile stress applied along [110] and [100] axes and electric field along the [001] axis, were made from $T=4.2\mathrm{}$ to 1.1 K with a low-frequency three-terminal capacitance bridge. A value of ${\epsilon }_h=11.40\mathrm{}\pm 0.06$ is obtained for the static dielectric of pure Si as $T\to 0.{\epsilon }_{h,zz}$ varies linearly with a [110] axis stress ${\sigma }_s$ for ${\sigma }_s$ up to 610 kg/${\mathrm{cm}}^2$ and yields $\frac{\left(\frac{1}{{\epsilon }_{h,zz}}\right)\Delta {\epsilon }_{h,zz}}{\Delta {\sigma }_s}=-(3.37\mathrm{}\pm 0.07)\times {10}^{-7\mathrm{}}$ ${\mathrm{kg}}^{-1\mathrm{}}$ ${\mathrm{cm}}^2$. The temperature variation obtained was $\frac{\left(\frac{1}{{\epsilon }_h}\right)d{\epsilon }_h}{\mathrm{dt}}=(1.12\mathrm{}\pm 0.05)\times {10}^{-4\mathrm{}}$ ${\mathrm{K}}^{-1\mathrm{}}$. The stress-dependent ${\epsilon }_{\mathrm{zz}}(N_D,x_{100})$ values always showed a minimum for a reduced valley strain $x_{100}^{min}$ in the range 0.4 to 0.6 for Si:P and $x_{100}^{min}\approx 0.9$ for Si:Sb. For a [110] stress there was no evidence of a minimum for $x_{110}$ up to 0.5. Values of ${\alpha }_D(N_D,x)$ were obtained from the Clausius-Mossotti relationship. The stress-dependent behavior of $\epsilon (N_D,x)$ and ${\alpha }_D\mathrm{}\left(x\right)\mathrm{}$ is very similar. The initial slopes are donor dependent and such that $\frac{\left[\frac{1}{{\alpha }_D\mathrm{}\left(0\right)\mathrm{}}\right]d{\alpha }_D}{d{x}_{110}}\approx \frac{2\left[\frac{1}{{\alpha }_D\mathrm{}\left(0\right)\mathrm{}}\right]d{\alpha }_D}{d{x}_{100}}$ in agreement with theory. For Si:P $\frac{\left[\frac{1}{{\alpha }_D\mathrm{}\left(0\right)\mathrm{}}\right]d{\alpha }_D}{d{x}_{100}}=-0.13\mathrm{}\pm 0.01$ while for Si:Sb $\frac{\left[\frac{1}{{\alpha }_D\mathrm{}\left(0\right)\mathrm{}}\right]d{\alpha }_D}{d{x}_{100}}=-0.07\mathrm{}$. For Si:P the position of the minimum $x_{100}^{min}$ seems to decrease slightly with increasing donor density $N_D$. The valley repopulation model, including valley-dependent changes in the Bohr radius with strain, cannot explain the ${\alpha }_D\mathrm{}\left(x\right)\mathrm{}$ results quantitatively. However, the data can be explained quantitatively if a strain-dependent variation of the valley-valley coupling matrix is also included. Reliable values of ${\alpha }_D(N_D,\omega ,T=0,x=0)$ were obtained by extrapolating the finite-$T$ data to $T=0\mathrm{}$. The dilute-limit values of ${\alpha }_D\mathrm{}(T=0\mathrm{})\mathrm{}$ approximating the isolated donor polarizabilities are (1.2±0.2)×${10}^5$ ${\mathrm{\AA }}^3$ and (1.9±0.6)×${10}^5$ Å for P and Sb, respectively. ${\alpha }_D(N_D,T=0)$ shows an enhancement with increasing $N_D$, but the enhancement is less than inferred from previous work. The related stress-dependent ac conductivity data have not been analyzed quantitatively, but agree qualitatively with the dielectric constant polarizability data.