Heat transport by acoustic phonons in 2D materials is fundamentally different from that in 3D crystals [1][2][3][4] because the out-of-plane phonons propagate in a unique way that strongly depends on tension and bending rigidity. Since in-plane and out-of-plane phonon baths are decoupled, initial studies suggested they provide independent pathways for heat transport and storage in 2D materials 5,6 . Here, we induce tension in freestanding graphene membranes by electrostatic force, and use optomechanical techniques to demonstrate that it can change the rate of heat transport by as much as 33%. Using a ballistic Debye model, we account for these observations and extract the average bending rigidity of the flexural acoustic phonons, which increases approximately linearly with the membrane's areal mass density, in contrast to the cubic dependence seen in bulk structures. Thus, we not only elucidate phononic heat transport mechanisms in suspended 2D materials, but also provide a promising route for controlling nanoscale heat transport by tension.Although in most bulk materials the propagation speed of different types of acoustic phonons is of similar magnitude, the situation is vastly different in 2D materials 7,8 . In these atomically thin materials, in-plane phonons have a propagation speed that is determined by the atomic bond stiffnesses, whereas out-of-plane flexural phonons, exhibit a speed that is dominated by tension 1

Heat transport by acoustic phonons in 2D materials is fundamentally different from that in 3D crystals because the out-of-plane phonons propagate in a unique way that strongly depends on tension and bending rigidity. Since in-plane and out-of-plane phonon baths are decoupled, initial studies suggested they provide independent pathways for heat transport and storage in 2D materials. Here, we induce tension in freestanding graphene membranes by electrostatic force, and use optomechanical techniques to demonstrate that it can change the rate of heat transport by as much as 33%. Using a ballistic Debye model, we account for these observations and extract the average bending rigidity of the flexural acoustic phonons, which increases approximately linearly with the membrane's areal mass density, in contrast to the cubic dependence seen in bulk structures. Thus, we not only elucidate phononic heat transport mechanisms in suspended 2D materials, but also provide a promising route for controlling nanoscale heat transport by tension.