Piezoresistance in Silicon at Uniaxial Compressive Stresses up to 3 GPa

The room-temperature longitudinal piezoresistance of $n$-type and $p$-type crystalline silicon along selected crystal axes is investigated under uniaxial compressive stresses up to 3 GPa. While the conductance ($G$) of $n$-type silicon eventually saturates at $\approx 45\%$ of its zero-stress value ($G_0$) in accordance with the charge transfer model, in $p$-type material $G/G_0$ increases above a predicted limit of $\approx 4.5$ without any significant saturation, even at 3 GPa. Calculation of $G/G_0$ using ab initio density functional theory reveals that neither $G$ nor the mobility, when properly averaged over the hole distribution, saturate at stresses lower than 3 GPa. The lack of saturation has important consequences for strained-silicon technologies.

Figure 1

(a) Cross-sectional view of the press, chuck assembly, and sample, along with the electrical connections for the conductance measurement. A compressive stress, $X$, is applied parallel to the direction of current flow. (b) A series of photographs of a $p$-type $⟨111⟩$ sample before each application of stress, showing the cleaving of the sides.

Figure 2

(a) Raw data obtained for $X$ parallel to the $⟨100⟩$ crystal direction in $n$-type silicon, showing discontinuous decreases in conductance due to in situ cleaving of the sample edges. (b) Data corrected for changes in sample surface area collapse onto a single curve that exhibits saturation above 1 GPa. The fit of the charge transfer model to these curves (solid line) yields ${\Xi }_u=12.1\mathrm{eV}$ and ${\mu }_{\perp }/{\mu }_{\parallel }=2.9$. The inset shows the measured ${\pi }_l$ coefficient for this sample (black diamonds) and all other samples (open circles).

Figure 3

Corrected data obtained along the (a) $⟨110⟩$ and (b) $⟨111⟩$ directions in $p$-type silicon (symbols with no connecting lines). No saturation is observed in either case. The insets show the measured ${\pi }_l$ coefficients for all samples. In both cases the result of the ab initio calculation is shown when concentration weighted band edge $m_{\mathrm{Gl}}^{*}$ for the HH and LH bands are used (lines with open circles), and when Eq. (1) is used to average over the LH and HH hole distributions (lines with closed circles).

Figure 4

(a) The LH (open circles) and HH (closed circles) band structure calculated along $⟨110⟩$ parallel to $X$ for three stress levels indicated by the circle size: 0 (small), 1.5 (medium), and 3 GPa (large). LH/HH anticrossing plays an important role in determining ensemble averaged $G$ for $X<1\mathrm{GPa}$ where the integrand of Eq. (1) is non-negligible (b). At high stresses, holes progressively occupy only the HH resulting in a gradual roll-off in $G$ without hard saturation.