We calculate the collective charge density excitation dispersion and spectral
weight in bilayer semiconductor structures {\it including effects of interlayer
tunneling}. The out-of-phase plasmon mode (the ``acoustic'' plasmon) develops a
long wavelength gap in the presence of tunneling with the gap being
proportional to the square root (linear power) of the tunneling amplitude in
the weak (strong) tunneling limit. The in-phase plasmon mode is qualitatively
unaffected by tunneling. The predicted plasmon gap should be a useful tool for
studying many-body effects.Comment: 10 pages, 6 figures. to appear in Phys. Rev. Let