  Superfluidity has recently been reported in double electron-hole bilayer
graphene. The multiband nature of the bilayers is important because of the very
small band gaps between conduction and valence bands. The long range nature of
the superfluid pairing interaction means that screening must be fully taken
into account. We have carried out a systematic mean-field investigation that
includes (i) contributions to screening from both intraband and interband
excitations, (ii) the low-energy band structure of bilayer graphene with its
small band gap and flattened Mexican hat-like low-energy bands, (iii) the large
density of states at the bottom of the bands, (iv) electron-hole pairing in the
multibands, and (v) electron-hole pair transfers between the conduction and
valence band condensates. We find that the superfluidity strongly modifies the
intraband contributions to the screening, but that the interband contributions
are unaffected. Unexpectedly, the net effect of the screening is to suppress
Josephson-like pair transfers and to confine the superfluid pairing entirely to
the conduction band condensate even for very small band gaps, making the system
behave similarly to a one-band superfluid.
