  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.
