We present experimental and theoretical investigations of phonon thermal transport in (LaMnO$_3$)$_m$/(SrMnO$_3$)$_n$ superlattices (LMO/SMO SLs) with the thickness of individual layers $m,n = 3 - 10;$ u.c. and the thickness ratio $m/n = 1, 2$. Optical transient thermal reflectivity measurements reveal a pronounced difference in the thermal conductivity between SLs with $m/n = 1$, and SLs with $m/n = 2$. State-of-the art electron microscopy techniques and ab-initio density functional calculations enables us to assign the origin of this difference to the absence ($m/n = 1$) or presence ($m/n = 2$) of spatially periodic, static oxygen octahedral rotation (OOR) inside the LMO layers. The experimental data analysis shows that the effective thermal conductance of the LMO/SMO interfaces strongly changes from $0.3$ GW/m$^2$K for $m/n = 2$ SLs with OOR to a surprisingly large value of $1.8$ GW/m$^2$K for $m/n = 1$ SLs without OOR. An instructive lattice dynamical model rationalizes our experimental findings as a result of coherent phonon transmission for $m/n = 1$ versus coherent phonon blocking in SLs with $m/n = 2$. We briefly discuss the possibilities to exploit these results for atomic-scale engineering of a crystalline phonon insulator. The thermal resistivity of this proposal for a thermal metamaterial surpasses the amorphous limit, although phonons still propagate coherently.
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