Thermal conduction in single-layer black phosphorus: highly anisotropic?

The single-layer black phosphorus is characteristic for its puckered structure, which has led to distinct anisotropy in its optical, electronic, and mechanical properties. We use the non-equilibrium Green's function approach and the first-principles method to investigate the thermal conductance for single-layer black phosphorus in the ballistic transport regime, in which the phonon-phonon scattering is neglected. We find that the anisotropy in the thermal conduction is very weak for the single-layer black phosphorus--the difference between two in-plane directions is less than 4%. Our phonon calculations disclose that the out-of-plane acoustic phonon branch has lower group velocities in the direction perpendicular to the pucker, as the black phosphorus is softer in this direction, leading to a weakening effect for the thermal conductance in the perpendicular direction. However, the longitudinal acoustic phonon branch behaves abnormally; i.e., the group velocity of this phonon branch is higher in the perpendicular direction, although the single-layer black phosphorus is softer in this direction. The abnormal behavior of the longitudinal acoustic phonon branch is closely related to the highly anisotropic Poisson's ratio in the single-layer black phosphorus. As a result of the counteraction between the out-of-plane phonon mode and the in-plane phonon modes, the thermal conductance in the perpendicular direction is weaker than the parallel direction, but the anisotropy is pretty small.