Role of Surface-Segregation-Driven Intermixing on the Thermal Transport through Planar $\mathrm{Si}/\mathrm{Ge}$ Superlattices

It has been highly debated whether the thermal conductivity $\kappa$ of a disordered SiGe alloy can be lowered by redistributing its constituent species so as to form an ordered superlattice. By ab initio calculations backed by systematic experiments, we show that Ge segregation occurring during epitaxial growth can lead to $\kappa$ values not only lower than the alloy’s, but also lower than the perfect superlattice values. Thus we theoretically demonstrate that $\kappa$ does not monotonically decrease as the Si- and Ge-rich regions become more sharply defined. Instead, an intermediate concentration profile is able to lower $\kappa$ below both the alloy limit (total intermixing) and also the abrupt interface limit (zero intermixing). This unexpected result is attributed to the peculiar behavior of the phonon mean free path in realistic Si/Ge superlattices, which shows a crossover from abrupt-interface- to alloylike values at intermediate phonon frequencies of $\sim 3\mathrm{THz}$. Our calculated $\kappa$’s quantitatively agree with the measurements when the realistic, partially intermixed profiles produced by segregation are considered.

Figure 1

(a) Sketch of the $(\mathrm{Si}{)}_m/(\mathrm{Ge}{)}_n$ SL structures. (b) Total thermal resistance $R_{\mathrm{tot}}$ of SLs with fixed Ge amount per layer (3 ML) and variable Si-spacer thickness. The Ge period number $N$ is 21. (c) Total thermal resistance and (d) single-barrier thermal resistance as a function of Ge-layer thickness of SLs with fixed Si-spacer thickness (43 ML). Different symbols are used for the results obtained by $3\omega$ and TDTR methods. The vertical dashed and dotted lines in (b) and (d) separate regions of planar SLs, SLs with coherently strained nanodots (huts) and SLs with dislocated nanodots (superdomes). All measurements were performed at room temperature.

Figure 2

(a) Comparison of experimental and theoretical cross-plane $\kappa$ for $(\mathrm{Si}{)}_{43}/(\mathrm{Ge}{)}_n$ SLs. (b) Comparison of experimental and theoretical $\kappa$ for $(\mathrm{Si}{)}_m/(\mathrm{Ge}{)}_3$ SLs. $N$ is the period number. (c) Calculated phonon mean free path spectrum for the $(\mathrm{Si}{)}_{43}/(\mathrm{Ge}{)}_4$ SL with (solid line) and without (dotted line) Ge segregation and the corresponding alloy (dashed line). The structure models used for ab initio calculations are illustrated in the inset. (d) Expected concentration profiles of a single SL period for different $(\mathrm{Si}{)}_{43}/(\mathrm{Ge}{)}_n$ SLs, according to the model in Ref. 37.

Figure 3

$\kappa$ of $(\mathrm{Si}{)}_m/(\mathrm{Ge}{)}_n$ SLs as a function of (a) mean Ge concentration $x\sim n/(n+m)$ and (b) total film thickness. The thermal conductivities are obtained by averaging the $3\omega$ and TDTR results. Our theory results as well as bulk alloy and SL data from the indicated Refs. are also included in (a). $\kappa$’s for thin ${\mathrm{Si}}_{0.8}{\mathrm{Ge}}_{0.2}$ alloy films from Ref. 22 (“thin-film alloy limit”) and the TDTR measurement of an alloy film grown in this work are shown in (b). In (b) the mean Ge concentration is indicated near each data point.