Cluster expansion and optimization of thermal conductivity in SiGe nanowires

We investigate the parametrization and optimization of thermal conductivity in silicon-germanium alloy nanowires by the cluster-expansion technique. ${\text{Si}}_{1-x}{\text{Ge}}_x$ nanowires are of interest for thermoelectric applications and the reduction in lattice thermal conductivity $\left({\kappa }_L\right)$ is desired for enhancing the thermoelectric figure of merit. We seek the minimization of ${\kappa }_L$ with respect to arrangements of Si and Ge atoms in 1.5 nm diameter [111] ${\text{Si}}_{1-x}{\text{Ge}}_x$ nanowires, by obtaining ${\kappa }_L$ from equilibrium classical molecular-dynamics (MD) simulations via the Green-Kubo formalism, and parametrizing the results with a coarse-grained cluster expansion. Using genetic algorithm optimization with the coarse-grained cluster expansion, we are able to predict configurations that significantly decrease ${\kappa }_L$ as verified by subsequent MD simulations. Our results indicate that superlatticelike configurations with planes of Ge show drastically lowered ${\kappa }_L$.

Figure 1

(Color online) An example SiGe nanowire simulation cell (side view). Red, larger spheres denote Ge while blue, smaller spheres denote Si.

Figure 2

Histogram of lattice thermal conductivity calculated using MD for the training set of 104 nanowires with different Si/Ge configurations.

Figure 3

Lattice thermal conductivity of nanowires with different Si/Ge configurations vs Ge composition. For SiGe nanowires in the training set, there is stronger dependence of ${\kappa }_L$ on configurations within the same Ge composition range than on Ge composition.

Figure 4

Examples of clusters considered equivalent in the coarse-grained cluster expansion: (left, side view) two pair clusters that are both surface and along, and (right, end view) two triplet clusters that are both intermediate and across.

Figure 5

After performing the fitting procedures, the values of ${\kappa }_L$ predicted from coarse-grained cluster expansion [Eq. (5)] vs those calculated from MD.

Figure 6

Convergence of ${\kappa }_L$ with generation number in genetic algorithm optimization.

Figure 7

(Color online) An example of a configuration found by cluster expansion and genetic algorithm optimization and verified by MD to have low ${\kappa }_L$ (side view).

Figure 8

Histogram of ${\kappa }_L$ calculated with MD for (a) the original training data (dark gray), (b) predicted low-${\kappa }_L$ configurations from cluster expansion and genetic algorithm optimization (light gray), and (c) configurations with perfect planes of Ge (white).

Figure 9

(Color online) Superlatticelike configuration with the lowest value of ${\kappa }_L$ as computed by MD (side view).

Figure 10

(Color online) A 10-nm-long simulation cell with surface roughness at 2.5 nm intervals (side view). The SiGe alloy configuration is superlatticelike as in Fig. 9.