  In this Letter, we report on quantitative measurements of the absolute
near-field mediated heat flux between a gold coated near-field scanning thermal
microscope (NSThM) tip and a planar gold sample at nanometer distances of 0.2
nm- 7 nm. We find an extraordinary large heat flux which is more than five
orders of magnitude larger than black-body radiation and four orders of
magnitude larger than the values predicted by conventional theory of
fluctuational electrodynamics. Additionally, we compare our data with different
theories of phonon tunneling which might explain a drastically increased heat
flux, but are found not to be able to reproduce the distance dependence
observed in our experiment. The findings demand modified or even new models of
heat transfer across vacuum gaps at nanometer distances.
