  Engineering thermal transport in two dimensional materials, alloys and
heterostructures is critical for the design of next-generation flexible
optoelectronic and energy harvesting devices. Direct experimental
characterization of lattice thermal conductivity in these ultra-thin systems is
challenging and the impact of dopant atoms and hetero-phase interfaces,
introduced unintentionally during synthesis or as part of deliberate material
design, on thermal transport properties is not understood. Here, we use
non-equilibrium molecular dynamics simulations to calculate lattice thermal
conductivity of (Mo|W)Se$_2$ monolayer crystals including Mo$_{1-x}$W$_x$Se$_2$
alloys with substitutional point defects, periodic MoSe$_2$|WSe$_2$
heterostructures with characteristic length scales and scale-free fractal
MoSe$_2$|WSe$_2$ heterostructures. Each of these features has a distinct effect
on phonon propagation in the crystal, which can be used to design fractal and
periodic alloy structures with highly tunable thermal conductivities. This
control over lattice thermal conductivity will enable applications ranging from
thermal barriers to thermoelectrics.
