spin-orbit effects [@BenShalom:2010kv; @Caviglia:2010jv], a metal-insulator transition [@thiel06; @Liao:2011bk], gate-controlled superconductivity [@Caviglia:2008uh], and a possible nematic transition at (111) interfaces [@Miao:2016hr; @Davis:2017; @Boudjada:2018; @Boudjada:2019]. Furthermore, STO’s proximity to the ferroelectric state has led to suggestions that quantum fluctuations shape its band structure [@atkinson17] and support superconductivity [@Edge:2015fj; @Dunnett:2018]. More generally, there has been a growing appreciation that lattice degrees of freedom play a key role in shaping the electronic structure near LAO/STO interfaces [@Behtash:2016dt; @Lee:2016dj; @Gazquez:2017bu; @raslan18]. With this in mind, the recent discovery that ferroelectric-like properties persist in some metallic perovskites [@Rischau:2017vj] naturally leads one to explore the effects of Ca or Ba doping on LAO/STO interfaces and, as described above, the TIM provides a natural framework in which to do this.

We found, however, that the TIM as it is usually formulated in equation  cannot reproduce the interfacial 2DEG and therefore fails to describe even the simple LAO/STO interface. In this work, we explain the reason for this failure and propose a modification to the TIM. In , we introduce the modified model and by comparison with the standard Landau-Ginzburg-Devonshire (LGD) expansion, illustrate why the failure arises and how we fix it. As a simple example, we apply 