Limit of the electrostatic doping in two-dimensional electron gases of LaXO3(X = Al, Ti)/SrTiO3

In LaTiO3/SrTiO3 and LaAlO3/SrTiO3 heterostructures, the bending of the SrTiO3 conduction band at the interface forms a quantum well that contains a superconducting two-dimensional electron gas (2-DEG). Its carrier density and electronic properties, such as superconductivity and Rashba spin-orbit coupling can be controlled by electrostatic gating. In this article we show that the Fermi energy lies intrinsically near the top of the quantum well. Beyond a filling threshold, electrons added by electrostatic gating escape from the well, hence limiting the possibility to reach a highly-doped regime. This leads to an irreversible doping regime where all the electronic properties of the 2-DEG, such as its resistivity and its superconducting transition temperature, saturate. The escape mechanism can be described by the simple analytical model we propose.

Measurements performed at high magnetic field (45 T) during the first positive polarisation reveal that the Hall resistance is not linear with magnetic field ( Figure S1). Two distinct slopes are evidenced at low and high field respectively, suggesting a multi-band transport scenario. Following the analysis of reference [1] the Hall resistance has been fitted with a two-band model :  Figure S1: Hall resistance as a function of magnetic field for the first positive polarization measured at 4.2 K on the LaTiO3/SrTiO3 sample. Open symbols correspond to experimental data, and full lines to fits with expression (1). An offset, indicated by a black horizontal segment and an arrow, has been added to separate the curves.
where n 1 and n 2 are the 2D electron densities and, µ 1 and µ 2 the corresponding mobilities, with the constraint 1/R s = en 1 µ 1 + en 2 µ 2 . Figure S2 shows the evolution of the different fitting parameters with gate voltage. A majority of low and constant mobility µ 1 carriers (density n 1 ) is present for all gate biases, whereas a minority of mobile electrons (density n 2 ) with a mobility µ 2 is present for positive gate voltages only. In reference [1] we proposed that the carriers of density n 1 populating the lowest energy sub-bands are confined close to the interface and have a low-mobility.
When the Fermi energy is closed to the top of the well and when a positive gate voltage is applied, we expect carriers of density n 2 to fill the highest energy sub-bands and delocalize deeper in the SrTiO 3 substrate where they recover a higher mobility (less scattering and better screening due to a stronger dielectric constant) [1]. During the first positive polarisation, the total carrier density does not change significantly since the Fermi level is already close to the top of the well. However, the conduction band of SrTiO 3 bends slightly and some carriers are pulled away from the interface.
As a result we inject electrostatically a very small number of highly-mobile carriers (density n 2 ) which produce a small decrease of the resistance before reaching saturation. The slight decreases of the density n 1 is compensated by a slight increase of the n 2 one. The inset of figure S2 shows the evolution of the mobilities µ 1 and µ 2 during the first positive polarisation, which is consistent with the behaviour of the carrier densities. Indeed, the mobility µ 1 of the carriers confined deep in the well is found to be rather constant. On the other hand, the mobility µ 2 increases rapidly for positive gate voltage before saturating, indicating that the electron added fills the last subbands accessible before escaping from the well.

II. Characteristic charging time of the sample
The 2-DEG carrier density at the interface can be electrostatically tuned by a metallic gate deposited at the back of the SrTiO 3 substrate. The heterostructure forms a capacitor, whose plates consist of the metallic gate on one side and the 2-DEG on the other side, and whose dielectric material is the SrTiO 3 substrate. As shown by the scheme in Figure S3, the heterostructure of capacitance C is placed in serial with an external resistor of resistance R G defining a simple serial RC circuit. In a simple Drude model, the resistance of the 2-DEG after a voltage step reach it's final value with a time dependence exp(−t/τ ) where τ = R G C. Figure S3 shows the resistance of the 2-DEG as a function of time, after applying a gate voltage step ∆V G = 10 V, for two different values of the external resistance R G . The blue curve corresponds to the short τ = R G C time (R G =100 kΩ) used for all the experiments reported in the main text. It shows a sharp decrease of resistivity, followed by the logarithmic relaxation corresponding to electrons escaping from the well. On the other hand, the red curve corresponds to a longer τ = R G C time (R G =40 MΩ) and shows that the initial decrease of the resistance is much slower. However, the logarithmic relaxation remains basically unchanged. We thus see that in our experiment the τ = R G C time at which the electrons are added to the 2DEG do not interfere with the thermal escape of electrons into the SrTiO 3 substrate.

III. Thermal escape of electrons from the quantum well
We have built a model to account for the relaxation of resistivity in the irreversible regime based on the thermal escape of the electrons from the well. The band bending and sub-bands energies of the LaTiO 3 /SrTiO 3 interface in the irreversible regime are shown in Figure S4 as an illustration [1]. However, this model is very general and does not depend on the details of the well such as its shape.
We consider a 2D potential well with a finite barrier height E B and a finite number of parabolic sub-bands of energy E i (i=1,...,n L ). We assume that electrons at the Fermi level E F can thermally jump over the barrier with a first order kinetics : where n is the density of electrons in the well and k is the kinetic factor. For thermally activated hoping the latter follows an Arrhénius law : and ν is a characteristic frequency factor. Hence, considering that the density of state of a 2D subband is constant and that the Fermi level always stay above the highest sub-band level, the equation on E F is : which is equivalent to a problem with only one level E L = n L i=1 E i /n L but with a density of state multiplied by n L . Considering low temperatures (k B T << E F − E L ), the variations of E F can be neglected in (E F − E L ) at first order, which gives the solution : where E 0 + F is the Fermi level after the voltage step at t=0 + and t E is the characteristic escape time : where N F = n L m/π 2 is the total density of state at the Fermi level. It follows that the Fermi level is constant for t < t E and then decreases logarithmically at longer time. Hence, the conductivity calculated from a simple Drude model with a constant mobility µ is given by At first order we obtain the time evolution of the resistivity where R 0 + = (eµn 0 + ) −1 is the resistivity just after the step. As shown in Figure 7 of the main text and in Supplementary Figure S5, this time dependence of the resistance fits the experimental data with a very good accuracy for both samples. To analyse the experimental behaviour of the escape time t E with gate voltage, we now consider a positive gate voltage step ∆V G at t = 0. Starting from a density of electrons n 0 − , the density after the step is : where C a (V G ) is the capacitance per unit area of the sample. For small ∆V G , the variations of C a with V G can be neglected, and the Fermi level is given by When replaced in equation (5), the variation of t E with C a and ∆V G is given by where : In the irreversible regime, the variations of ln(t E ) follow a linear dependence with both C a and ∆V G , as shown in Figure 7 of the main text.