  We show that phononic frequency conversion can be enhanced by orders of
magnitude in piezoelectric systems by heterogeneous integration of
high-mobility semiconductor films. A lithium niobate and indium gallium
arsenide heterostructure is utilized to demonstrate efficient three-wave mixing
processes at microwave frequencies, including 16% phononic power conversion
efficiency for sum-frequency generation and 1% phononic power conversion
efficiency for difference-frequency generation, as well as the most efficient
degenerate four-wave phononic mixing to date. We present a theoretical model
that accurately predicts the sum-frequency and difference-frequency generation
processes and we show that the conversion efficiency can be further enhanced by
the application of semiconductor bias fields. Laser Doppler vibrometry is then
applied to examine many three-wave and four-wave mixing processes
simultaneously in the same device. Through the use of our developed model, we
show that these nonlinearities can be enhanced far beyond what is demonstrated
here by confining phonons to smaller dimensions in waveguides and optimizing
semiconductor material properties or using 2D semiconductors.
