First-principles calculation of carrier-phonon scattering in $n$-type ${\text{Si}}_{1-x}{\text{Ge}}_x$ alloys

First-principles electronic structure methods are used to find the rates of inelastic intravalley and intervalley $n$-type carrier scattering in ${\text{Si}}_{1-x}{\text{Ge}}_x$ alloys. Scattering parameters for all relevant $\Delta$ and $L$ intra- and intervalley scattering are calculated. The short-wavelength acoustic and the optical phonon modes in the alloy are computed using the random mass approximation, with interatomic forces calculated in the virtual crystal approximation using density functional perturbation theory. Optical phonon and intervalley scattering matrix elements are calculated from these modes of the disordered alloy. It is found that alloy disorder has only a small effect on the overall inelastic intervalley scattering rate at room temperature. Intravalley acoustic scattering rates are calculated within the deformation potential approximation. The acoustic deformation potentials are found directly and the range of validity of the deformation potential approximation verified in long-wavelength frozen phonon calculations. Details of the calculation of elastic alloy scattering rates presented in an earlier paper are also given. Elastic alloy disorder scattering is found to dominate over inelastic scattering, except for almost pure silicon $\left(x\approx 0\right)$ or almost pure germanium $\left(x\approx 1\right)$, where acoustic phonon scattering is predominant. The $n$-type carrier mobility, calculated from the total (elastic plus inelastic) scattering rate, using the Boltzmann transport equation in the relaxation time approximation, is in excellent agreement with experiments on bulk, unstrained alloys.

Figure 1

(Color online) The surfaces of constant energy in the $\Delta$ valleys (upper panel) and $L$ valleys (lower panel) for SiGe alloys. The intervalley $f$-type and intervalley $g$-type scattering matrix elements are indicated by lines between the $\Delta$ valleys, and the intervalley $LL$ scattering matrix elements by a line between two of the $L$ valleys.

Figure 2

(Color online) Averages of the potential difference [Eq. (19)] for distances greater than $r_a$ from a Si atom (squares) or a Ge atom (diamonds) in a VCA $\left(x=0.5\right)$ 64-atom supercell.

Figure 3

(Color online) Calculated elastic alloy scattering parameters with relaxed and unrelaxed atomic positions around the perturbing atom (Ge), for the $X$ and $L$ points for pure silicon.

Figure 4

Phonon dispersion for zincblende structure SiGe, calculated with full DFT (solid line), using VCA interatomic forces and correct Si and Ge atomic masses (dashed line), and using VCA interatomic forces and the average atomic mass of Si and Ge for both atoms (gray line).

Figure 5

Calculated phonon density of states for a 16-atom random alloy with composition $x=0.5$ using VCA forces and the random mass approximation (thick line), and using a full DFT calculation of the relaxed geometry and dynamical matrix (thin line).

Figure 6

(Color online) Calculated phonon density of states of ${\text{Si}}_{1-x}{\text{Ge}}_x$ at Ge composition $x=\frac{1}{2}$ for a disordered alloy, using the random mass approximation (solid line) and the average mass approximation (dashed line) in a 1000-atom supercell with periodic boundary conditions.

Figure 7

Calculated density of states and $F_{\text{nm}}$ functions for the different phonons: $\Delta g$, $\Delta f$, $L-L$, $\Delta -L$, and optical $L$ using random mass approximation at Ge composition $x=0.5$. The values calculated using the average mass approximation are shown in dashed lines.

Figure 8

Calculated phonon intravalley matrix element “$d$:” $C\cdot M_{\mathbf{k},\mathbf{k}+\mathbf{q}}={\Xi }_d\left|\mathbf{q}\right|$, and “$u$:” $3C\cdot M_{\mathbf{k},\mathbf{k}+\mathbf{q}}={\Xi }_u\left|\mathbf{q}\right|$ for the $L$ valley in Ge as a function of phonon momentum $\left|\mathbf{q}\right|$, with $q$ in the directions specified by Table , interpolated with a polynomial fit. The linear term yields the deformation potentials [see Eq. (42)].

Figure 9

(Color online) Calculated $n$-type inverse carrier mobility in ${\text{Si}}_{1-x}{\text{Ge}}_x$ at 300 K (solid line), inverse mobility due to acoustic phonon scattering only (dot), intervalley and optical phonon scattering (dashed-dotted line), alloy scattering (dashed), and experimental data from Refs. 32 (triangles), 17 (squares), and 33 (stars).

Figure 10

(Color online) Calculated $n$-type carrier mobility in ${\text{Si}}_{1-x}{\text{Ge}}_x$ at 300 K (solid line), mobility without $\Delta -L$ alloy scattering (dash in inset), and experimental data from Refs. 32 (triangles), 17 (squares), and 33 (stars).