  Phonon transport across a vacuum gap separating intrinsic silicon crystals is
predicted via the atomistic Green's function method combined with
first-principles calculations of all interatomic force constants. The overlap
of electron wave functions in the vacuum gap generates weak covalent
interaction between the silicon surfaces, thus creating a pathway for phonons.
Phonon transport, dominated by acoustic modes, exceeds near-field radiation for
vacuum gaps smaller than ~ 1 nm. The first-principles-based approach proposed
in this work is critical to accurately quantify the contribution of phonon
transport to heat transfer in the extreme near field.
