Understanding the onset of negative electronic compressibility in single-band and two-band two-dimensional electron gases: Application to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>LaAlO</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>SrTiO</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:math>

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