  We present a theoretical study of the modification of the near-field
radiative heat transfer due to phonon-plasmon coupling in bilayer systems made
of a doped semiconductor and a polar dielectric. By tuning the surface-plasmon
mode of the former material in resonance with the phonon modes of the latter
one, the near-field spectral radiative heat transfer can be suppressed in a
frequency band gap. We distinguish between the interlayer mode coupling within
each bilayer and the intragap one between the two bilayers. We elucidate the
role of each of them in the formation of the band gap and we determine the
distance range at which one mechanism dominates over the other. Furthermore, we
show that the surface plasmon polariton of the top-most layer allows optimizing
the total heat transfer as a function of the separation of the two bilayers.
