  Exciton optomechanics, bridging cavity exciton polaritons and optomechanics,
opens new opportunities for the study of light-matter strong interactions and
nonlinearities, due to the rich nonlinear couplings among excitons, phonons,
and photons. Here, we propose to entangle two exciton modes in an
exciton-optomechanical system, which consists of a semiconductor optomechanical
microcavity integrated with two quantum wells. The quantum wells support two
exciton modes, which simultaneously couple to an optical cavity mode via a
linear coupling, and the cavity mode also couples to a mechanical vibration
mode via a dispersive optomechanical interaction, accounting for both the
radiation pressure and the photoelastic effect. We show that by strongly
driving the microcavity with a red-detuned laser field and when the two exciton
modes are respectively resonant with the Stokes and anti-Stokes sidebands
scattered by the mechanical motion, stationary entanglement between the two
exciton modes can be established under currently available parameters. The
entanglement is robust against various dissipations of the system and can be
achieved at room temperature for a mechanical quality factor higher than
$\sim10^4$.
