The phonon dispersion and anharmonic phonon-phonon scattering rates can be derived on the basis of the harmonic and anharmonic interatomic force constants (IFCs). We first fit the IFCs for pure GaAs and AlAs, respectively, based on the first principles data regarding forces acting on different atoms and their displacements in a large supercell (2 × 2 × 2 conventional unit cells, 64 atoms), with imposed translational and rotational invariances. Harmonic IFCs up to the fifth nearest neighbor and third-order anharmonic IFCs up to the first nearest neighbor were considered. The force-displacement data were computed using the density functional theory as implemented in the QUANTUM ESPRESSO package (45), for which we used the norm-conserving pseudopotential with the Perdew and Zunger (46) local density approximation for the exchange-correlation functional, a cutoff energy of 60 rydberg, and a 16 × 16 × 16 k-mesh. We then took the average of the IFCs (both harmonic and anharmonic) corresponding to pure GaAs and AlAs as the IFCs for the SL, which is a good approximation due to the small lattice mismatch between GaAs and AlAs and has previously been used for calculating the thermal conductivity of SLs (13). For further calculating the phonon dynamics, the unit cell lattice constant for the GaAs/AlAs SL was taken to be the average (5.5722 Å) of the calculated values in pure GaAs and AlAs.

The phonon dispersion can be computed from the dynamical matrix, whose matrix elements, in real space, take the form Embedded Image, where φij represents the harmonic IFC that couples the ith and jth atoms of mass mi and mj, respectively. Although this model did not describe the local strain effects from either the interfaces or the ErAs nanodots, the dispersion of the phonon frequency for an SL with a period of 5.57 nm (fig. S9) agreed well with the previously calculated dispersion for a 24-nm period GaAs/AlAs SL using semiempirical force constants in the work of Luckyanova et al. (13)

The phonon MFP due to anharmonic phonon-phonon scattering was calculated by Λqλ = vqλτqλ, where vqλ is the phonon group velocity and τqλ is the phonon relaxation time. The group velocity was obtained from the phonon dispersion, while the relaxation times were calculated using the lowest-order three-phonon scattering process via (47, 48)Embedded Image(1)where Embedded Image are the three-phonon coupling matrix elements and depend on the third-order IFCs (47). For the MFP calculation, due to the computational complexity, we chose the SL structure to have three conventional cells for each layer (GaAs or AlAs) perpendicular to the SL interface (dimensions: 5.5722 Å by 5.5722 Å by 33.43 Å). A 16 × 16 × 3 q-mesh was used for calculating the phonon relaxation times (Eq. 1), and the convergence with respect to the mesh was checked. We also checked the phonon relaxation times of the SL with smaller period (dimensions: 5.5722 Å by 5.5722 Å by 22.29 Å) and found that the phonon MFPs have small differences. Therefore, our calculated results should be close to the experimental configuration, which has a slightly larger period.

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