  Fabrication of devices made by isolated Graphene or Graphene-like single
layers (such as h-BN) has opened up possibility of examining highly correlated
states of electron systems in parts of their phase diagram that is impossible
to access in their counterpart devices such as semiconductor heterostructures.
An example of such states are Graphene (or Graphene like) double layer
electron-hole systems under strong magnetic fields where the separation between
layers can be of the order of one magnetic length with interlayer tunneling
still suppressed. In those separations correlations between electrons and holes
are of crucial importance and must be included in determination of observable
quantities. Here we report a thorough mean-field study of the coherent and
crystalline ground states of the interacting balanced electron-hole Graphene
systems in small and intermediate separations with each layer occupying up to
four lowest lying Landau levels. We calculate the capacitance of such states as
a function of layer separation and filling factor. Our calculations show
significant enhancement of the capacitance compared to geometrical value due to
quantum mechanical corrections.
