  We investigate the evolution of the cross-plane thermal conductivity $\kappa$
of superlattices (SLs) as interfaces change from perfectly abrupt to totally
intermixed, by using non-equilibrium molecular dynamics simulations in
combination with the spectral heat current calculations. We highlight the role
of surface-interdiffusion-driven intermixing by calculating the $\kappa$ of SLs
with changing interface roughness, whose tuning allows for the $\kappa$ values
much lower than the "alloy limit" and the abrupt interface limit in same cases.
The interplay between alloy and interface scattering in different frequency
ranges provides a physical basis to predict a minimum of thermal conductivity.
More specifically, we also explore how the interface roughness affects the
thermal conductivities for SLs materials with a broad span of atomic mass and
bond strength. In particular, we find that (i) only when the "spacer" thickness
of SLs increases up to a critical value the $\kappa$ of rough SLs can break the
corresponding "alloy limit". (ii) Whether the $\kappa$ changes monotonically as
interface roughness strongly depends on the period length and intrinsic
behavior of phonon transport for SLs materials. Especially, for the SL with
large period length, there exists an optimal interface roughness which can
minimize the thermal conductivity. (iii) Surface-interdiffusion-driven
intermixing is more effective in achieving the low $\kappa$ below the alloy
limit for SL materials with large mass mismatch than with small one. (iv) It's
possible for SLs materials with large lattice mismatch (i.e., bond strength) to
design an ideally abrupt interface structure with $\kappa$ much below the
"alloy limit". These results have a clear implications for optimization of
thermal transport for heat management and for the development of thermoelectric
materials.
