First-Principles Calculation of Alloy Scattering in ${\mathrm{Ge}}_x{\mathrm{Si}}_{1-x}$

First-principles electronic structure methods are used to find the rates of intravalley and intervalley $n$-type carrier scattering due to alloy disorder in ${\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x$ alloys. The required alloy scattering matrix elements are calculated from the energy splitting of nearly degenerate Bloch states which arises when one average host atom is replaced by a Ge or Si atom in supercells containing up to 128 atoms. Scattering parameters for all relevant $\Delta$ and $L$ intravalley and intervalley alloy scattering are calculated. Atomic relaxation is found to have a substantial effect on the scattering parameters. $f$-type intervalley scattering between $\Delta$ valleys is found to be comparable to other scattering channels. The $n$-type carrier mobility, calculated from the scattering rate using the Boltzmann transport equation in the relaxation time approximation, is in excellent agreement with experiments on bulk, unstrained alloys.

Figure 1

The intravalley, intervalley $f$-type, and intervalley $g$-type scattering matrix elements between the six equivalent $k$ points, $\mathbf{k}=(2\pi /a_0)(\xi 00)$, etc., as functions of crystallographic momentum $\xi$ along the $\Delta$ line in the Brillouin zone, for Ge content $x=0$.

Figure 2

Calculated $n$-type carrier mobility in ${\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x$ at 300 K (solid line), mobility without $\Delta \mathrm{\text{−}}L$ alloy scattering (clear dashed line), phonon limited mobility only (dashed line), and experimental data from Refs. 23 (triangles) and 9, 24 (squares).