  In contradictin to the common concept that acoustic phonons can only travel
inside a material medium, they can in fact "tunnel" across a vacuum gap with
the help of piezoelectricity, transmitting a significantly stronger heat flux
than that of blackbody radiation. Here, we present a theoretical formulation
for the heat flux of such piezoelectrically mediated heat transfer, applicable
to any anisotropic piezoelectric crystals with an arbitrary orientation. A few
numerical results are demonstrated and compared to heat transfer driven by
other close-range mechanisms, including near-field radiative heat transfer and
other acoustic phonon tunneling mechanisms. We find that piezoelectrically
mediated heat transfer has a significant effect when the vacuum gap size is
smaller than the phonon characteristic thermal wavelength, and its heat flux
can dominate heat transfer between piezoelectric solids over all other known
heat transfer mechanisms at temperatures below 50 K.
