  We investigate the effects of two electronic bands at the negative electronic
compressibility (NEC) in a two-dimensional electron gas (2DEG). We use a simple
homogeneous model with Coulombic interactions and first-order multi-band
coupling to examine the role of effective mass and relative permittivity in
relation to the critical carrier density, where compressibility turns negative.
We demonstrate that the population of a second band, along with the presence of
inter-band coupling, can dramatically change the cross-over carrier density.
Given the difficulty in determining and confirming multi-band electronic
systems, this model provides a potential method for identifying multi-band
electronic systems using precise bulk electronic properties measurements. To
help illustrate this method, we apply our results to the observed NEC in the 2D
electron gas at the interface of LaAlO$_3$/SrTiO$_3$ (LAO/STO) and determine
that, for the known parameters of LAO/STO, the system is likely a realization
of a two-band 2D electron gas. Furthermore, we provide general limits on the
inter-band coupling with respect to the electronic band population.
