  We report on strong enhancement of mid-infrared second harmonic generation
(SHG) from SiC nanopillars due to the resonant excitation of localized surface
phonon-polaritons within the Reststrahlen band. The magnitude of the SHG peak
at the monopole mode experiences a strong dependence on the resonant frequency
beyond that described by the field localization degree and the dispersion of
linear and nonlinear-optical SiC properties. Comparing the results for the
identical nanostructures made of 4H and 6H SiC polytypes, we demonstrate the
interplay of localized surface phonon polaritons with zone-folded weak phonon
modes of the anisotropic crystal. Tuning the monopole mode in and out of the
region where the zone-folded phonon is excited in 6H-SiC, we observe a
prominent increase of the already monopole-enhanced SHG output when the two
modes are coupled. Envisioning this interplay as one of the showcase features
of mid-infrared nonlinear nanophononics, we discuss its prospects for the
effective engineering of nonlinear-optical materials with desired properties in
the infrared spectral range.
