The phononic band structure and thermal conductivity of a family of a two-atom unit cell Lennard-Jones crystals are predicted using molecular dynamics simulations. The structure consists of alternating layers of atoms with different masses, leading to anisotropic thermal properties. An increase in the mass ratio results in an increase in the width of the band gap and a decrease in the value of its central frequency. The thermal conductivity decreases with an increase in the mass ratio, and in all cases is lower than that for a monatomic unit cell. The thermal conductivity increases with an increase in the ratio of the central gap frequency to its width. The rate of this increase decreases and becomes less temperature-dependent at lower mass ratios. The results could be utilized towards the development of guidelines for the design of materials with extreme thermal transport properties.

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