  Two-dimensional electron and hole gases separated by a few nm from each other
are produced in p-i-n diodes based upon MBE-grown GaAs/AlGaAs heterostructures.
At such interlayer distances, the exciton formation and possibly Bose-Einstein
condensation (BEC) is expected. We measure the capacitance between the layers
and find it to oscillate as a function of the bias voltage. The peak values
exceed the geometric capacitance by up to a factor of two. The surprisingly
regular periods of the oscillations are of the order of 10 to 30 mV and scale
linearly with the inverse of the thickness, between 60 and 150 nm, of the GaAs
layer placed between the barrier and the p-doped AlGaAs. The oscillations are
related to the resonant electron tunneling into high energy levels of this GaAs
layer acting as a quantum well. We suggest that long lifetimes of the electrons
in these levels are the origin of the large peak values of the capacitance. The
possible relation of the capacitance oscillations with BEC is discussed.
