We build a theoretical model for the electronic properties of the two-dimensional (2D) electron gas that forms at the interface between insulating SrTiO3 and a number of polar cap layers, including LaTiO3, LaAlO3, and GdTiO3. The model treats conduction electrons within a tight-binding approximation, and the dielectric polarization via a Landau-Devonshire free energy that incorporates strontium titanate's strongly nonlinear, nonlocal, and temperature-dependent dielectric response. The self-consistent band structure comprises a mix of quantum 2D states that are tightly bound to the interface, and quasi-three-dimensional (3D) states that extend hundreds of unit cells into the SrTiO3 substrate. We find that there is a substantial shift of electrons away from the interface into the 3D tails as temperature is lowered from 300 K to 10 K. This shift is least important at high electron densities (∼ 10 14 cm −2 ), but becomes substantial at low densities; for example, the total electron density within 4 nm of the interface changes by a factor of two for 2D electron densities ∼ 10 13 cm −2 . We speculate that the quasi-3D tails form the low-density high-mobility component of the interfacial electron gas that is widely inferred from magnetoresistance measurements.

We study a model SrTiO3 interface in which conduction t2g electrons couple to the ferroelectric (FE) phonon mode. We treat the FE mode within a self-consistent phonon theory that captures its quantum critical behavior, and show that proximity to the quantum critical point leads to universal tails in the electron density of the form n(z) ∼ (λ + z) −2 , where λ ∼ T 2−d/z , with d = 3 the dimensionality and z = 1 the dynamical critical exponent. Implications for the metal-insulator transition at low electron density are discussed.

Multiple experiments have observed a sharp transition in the band structure of LaAlO3/SrTiO3 (001) interfaces as a function of applied gate voltage. This Lifshitz transition, between a single occupied band at low electron density and multiple occupied bands at high density, is remarkable for its abruptness. In this work, we propose a mechanism by which such a transition might happen. We show via numerical modeling that the simultaneous coupling of the dielectric polarization to the interfacial strain ("electrostrictive coupling") and strain gradient ("flexoelectric coupling") generates a thin polarized layer whose direction reverses at a critical density. The Lifshitz transition occurs concomitantly with the polarization reversal and is first-order at T = 0. A secondary Lifshitz transition, in which electrons spread out into semiclassical tails, occurs at a higher density.LaAlO 3 (LAO) and SrTiO 3 (STO) are band insulators; however, when four or more monolayers of LAO are grown on top of a STO substrate, a mobile twodimensional electron gas (2DEG) forms on the STO side of the interface [1, 2]. One compelling feature of these interfaces is that the character of the 2DEG changes dramatically with the application of a gate voltage. Indeed, for (001) interfaces there is a narrow doping range over which the superconducting transition temperature [3][4][5][6][7], the spin-orbit coupling [7-10], and the metamagnetic response [11] change by an order of magnitude. Furthermore, the superconducting gap and resistive transition appear at different temperatures at low electron densities, n 2D , but track each other closely at high n 2D [12]. This qualitative distinction between low and high doping has also been seen in quantum dot transport experiments, which reveal a crossover from attractive to repulsive pairing interactions with increasing n 2D [13]. While there is general agreement that the sensitivity to doping is connected to an observed Lifshitz transition [5, 14-17] between a single occupied band at low density and multiple occupied bands at high density [6,[17][18][19][20][21][22][23][24], the mechanism by which this transition happens is not established.Density functional theory (DFT), while instrumental in establishing fundamental interface properties [25][26][27][28], finds electron densities that are an order of magnitude larger than the Lifshitz transition density n L ∼ 0.02-0.05 electrons per 2D unit cell, and cannot easily be tuned through the transition. Schrödinger-Poisson calculations, for which n 2D can be continuously tuned, persistently find multiple occupied bands even for n 2D n L [5,15,[29][30][31]. Indeed, we showed previously that, because of STO is close to a ferroelectric (FE) quantum critical point [32], electrons become deconfined from an ideal interface as n 2D → 0, and form a dilute quasi-threedimensional (quasi-3D) gas extending far into the STO substrate [33]. This result is incompatible with experiments and raises the question, why is only a single band occupied at low n 2D ? Furthermore, t...