First-principles investigation of the alloy scattering potential in dilute Si${}_{1-x}$C${}_x$

A first-principles method is applied to find the intra and intervalley $n$-type carrier scattering rates for substitutional carbon in silicon. The method builds on a previously developed first-principles approach with the introduction of an interpolation technique to determine the intravalley scattering rates. Intravalley scattering is found to be the dominant alloy scattering process in Si${}_{1-x}$C${}_x$, followed by $g$-type intervalley scattering. Mobility calculations show that alloy scattering due to substitutional C alone cannot account for the experimentally observed degradation of the mobility. We show that the incorporation of additional charged impurity scattering due to electrically active interstitial C complexes models this residual resistivity well.

Figure 1

Calculations for the scattering parameters in unrelaxed Si${}_{1-x}$C${}_x$ using the $6\times 6$ matrix method of Eq. (33). Points at $k_z=1/2$ and $k_z=1$ were obtained from a cubic 64-atom supercell. For the points at $k_z=2/3$ and $k_z=4/3$ an fcc 54-atom supercell was used. The criterion used for the number of supercell states to include in the scattering matrix was based on the absolute difference of the perturbed supercell energies $\epsilon$ and the originally sixfold degenerate unperturbed Bloch-states energies ${\epsilon }_0$ being less than or equal to 4 eV. The white triangles mark the polynomial interpolation to the $\Delta$ valley minimum, with values of $-$0.4, $-$0.6, and $-$1.8 eV for intravalley, $g$-type, and $f$-type scatterings, respectively.

Figure 2

Calculations for the scattering parameters in relaxed Si${}_{1-x}$C${}_x$ using the $6\times 6$ matrix method of Eq. (33). The same comments given in the caption to Fig. 1 apply here. The polynomial interpolations to the $\Delta$-valley minimum yield values of $-$4.5, $-$1.2, and $-$0.7 eV for intravalley, $g$-type, and $f$-type scatterings, respectively.

Figure 3

Comparison of the calculations for $g$-type scattering in Si${}_{1-x}$C${}_x$ using (i) the $6\times 6$ matrix method and (ii) from the elements of the unreduced scattering matrix of a cuboid 128-atom supercell. The data points for method (i) are the blue squares (unrelaxed structure) and red diamonds (relaxed structure), also plotted in Figs. 1 and 2, respectively. For method (ii), the down-pointing cyan triangles are for the unrelaxed structure and the upright green triangles are for the relaxed structure. Interpolating to the $\Delta$-valley minimum for the second set of calculations gives values of $-$0.6 and $-$1.1 eV for the unrelaxed and relaxed structures, respectively.

Figure 4

Calculations for intravalley scattering in Si${}_{1-x}$C${}_x$ using the elements of the unreduced scattering matrix for a cuboid 128-atom supercell. Here, the matrix elements are for the $\mathbf{q}$-point scattering vector from the $\Delta$-valley minimum. Hence, the interpolation is to $\mathbf{q}=0$, giving values of 0.2 and $-$4.5 eV for the unrelaxed and relaxed structures, respectively. Also shown for comparison are the values interpolated to the $\Delta$-valley minimum calculated via the $6\times 6$ matrix method, $-$0.4 and $-$4.5 eV for the unrelaxed and relaxed structures, respectively (c.f Figs. 1 and 2).

Figure 5

Experimental Hall mobilities by Osten et al. (symbols) and calculated mobilities for the same C doping. The scattering processes in pure Si have been incorporated by fitting the experimental mobility to combined scattering rates with ${\epsilon }^{-3/2}$, ${\epsilon }^{1/2}$, and ${\epsilon }^{3/2}$ dependencies, including a screening factor on the ${\epsilon }^{-3/2}$ rate (representing ionized impurity scattering). These are then combined with the rates for alloy scattering due to substitutional carbon (dashed lines) and both alloy scattering and additional charged impurity scattering due to electrically active defects (solid lines). At the high-temperature side (right of graph), the lines decrease in mobility with increasing $x$. The highest dashed line is the fit to the pure Si mobility.