Thinning down of thermal conductivity in ultrashort period superlattices

Abstract

We present numerical results for the reduction of the lattice thermal conductivity tensor components ${{\ensuremath{\kappa}}_{\ensuremath{\alpha},\ensuremath{\beta}}}$ in ultrashort period (Si)${}_{n}$(Ge)${}_{n}$[001] superlattices, with $1\ensuremath{\le}n\ensuremath{\le}8$, where $n$ represents the number of atomic bilayers. The calculations are made within the single-mode relaxation-time approximation, accounting for interatomic bond length relaxation and employing phonon dispersion relations obtained from density functional perturbation theory, a model anharmonic Hamiltonian to deal with three-phonon interactions involving acoustic as well as optical phonons in a two-material superlattice structure, and an improved scheme for phonon scattering due to mass smudging at interfaces. The cross-plane component of the conductivity is around 4.1 times smaller than the in-plane component for the $n=8$ case at room temperature and an interface mass mixing (IMS) scattering strength of 0.05. Both the in-plane and cross-plane components of the conductivity decrease sharply with the superlattice period when the strength of the IMS scattering is kept constant. Incorporating physical considerations into the behavior of the IMS scattering, we predict a minimum of the thermal conductivity for $n\ensuremath{\approx}4$. We estimate that a small amount of interface mass smudging results in a reduction of around 3%--14% in the $zz$ conductivity component for temperatures of 100 to 700 K when boundary scattering is relatively weak. We estimate relevant phonon scattering rates to explain available experimental conductivity measurements on a system comparable in size to the (8,8) superlattice.