  Topological phononics offers numerous opportunities in manipulating elastic
waves that can propagate in solids without being backscattered. Due to the lack
of nanoscale imaging tools that aid the system design, however, acoustic
topological metamaterials have been mostly demonstrated in macroscale systems
operating at low (kilohertz to megahertz) frequencies. Here, we report the
realization of gigahertz topological valley Hall effect in
nanoelectromechanical AlN membranes. Propagation of elastic wave through
phononic crystals is directly visualized by microwave microscopy with
unprecedented sensitivity and spatial resolution. The valley Hall edge states,
protected by band topology, are vividly seen in both real- and momentum-space.
The robust valley-polarized transport is evident from the wave transmission
across local disorder and around sharp corners, as well as the power
distribution into multiple edge channels. Our work paves the way to exploit
topological physics in integrated acousto-electronic systems for classical and
quantum information processing in the microwave regime.
