Abstract
In LaTiO_{3}/SrTiO_{3} and LaAlO_{3}/SrTiO_{3} heterostructures, the bending of the SrTiO_{3} conduction band at the interface forms a quantum well that contains a superconducting twodimensional electron gas (2DEG). Its carrier density and electronic properties, such as superconductivity and Rashba spinorbit 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 highlydoped regime. This leads to an irreversible doping regime where all the electronic properties of the 2DEG, such as its resistivity and its superconducting transition temperature, saturate. The escape mechanism can be described by the simple analytical model we propose.
Introduction
Twodimensional electron gases (2DEGs) at LaAlO_{3}/SrTiO_{3} and LaTiO_{3}/SrTiO_{3} oxide interfaces^{1} have attracted much attention since their electronic properties display a very rich physics with various electronic orders such as superconductivity^{2,3,4} and magnetism^{5,6,7,8}. In these structures, the 2DEG is confined in an interfacial quantum well that typically extends on the order of 10 nm into the SrTiO_{3} substrate at low temperature^{2,4,9,10,11}. Applying a backgate voltage enables to control electrostatically the filling of the well and thus modulate the 2DEG electronic properties^{12,13,14}. This exciting feature opens new avenues for studying the electronic orders and quantum phase transitions^{15} in theses structures, as well as for developing oxidebased electronics that could make use of them^{16,17,18,19}. Of particular interest, it was shown that adding electron to the well, increases continuously the electronic mobility and the strength of the Rashba spinorbit coupling^{14,20,21}. Also of interest, is the fact that the superconducting transition temperature of the 2DEG exhibits a domelike shape with a maximum T_{c} of 200–300 mK at optimal doping^{13,14}. Because of these remarkable properties, the highlydoped regime must be explored in further detail. However, the overdoped side of the dome seems particularly difficult to study because of unexplained saturation and hysteresis of the physical properties^{13,14,20}. As of yet, this region is not fully understood. These observations raise a fundamental question: how much one can electrostatically dope the 2DEG at oxides interfaces?
Although several theoretical descriptions of LaXO_{3}(X = Al, Ti)/SrTiO_{3} interfaces have been proposed^{22,23,24,25,26,27}, the exact interfacial band structure remains controversial. Regardless of the calculation method and for the sake of clarity, in the present report we will consider a simple generalized situation where the bending of the SrTiO_{3} conduction band defines a quantum well that accommodates discrete electronic subbands. To illustrate this point, we show in the Figure 1 the type of result that can be obtained by using a semiconductor approach in which we solve selfconsistent SchrödingerPoisson equations^{14,28}. In this example, six subbands are filled and the higher energy one extends within 5 nm in the SrTiO_{3}^{10,11}; this situation corresponds to a carrier density of approximately 7 × 10^{13} e^{−}/cm^{2}, typical of values found in the literature. The effect of a back gate voltage on the interface is twofold: (i) it adds electrons to the well as the gate voltage increases (ΔV_{G} > 0) and removes electrons from the well as the gate voltage decreases (ΔV_{G} < 0), (ii) it controls the shape of the upper part of the well by tilting the conduction band profile in the substrate. Figure 2 shows illustrations of quantum well energy profiles that describe these different situations. As long as the Fermi level remains deep in the well, electrostatic gating can reversibly empty and fill the well as in the wellknown semiconductors quantum wells. However, when the Fermi level rises to the top of the well, we no longer expect the doping to be possible. In this article we show that in LaXO_{3}(X = Al, Ti)/SrTiO_{3} heterostructures, the Fermi energy lies intrinsically near the top of the well and that, beyond a filling threshold, additional electrons irreversibly escape from the well, hence limiting the possibility to reach a highlydoped regime. This behaviour can be described by a simple analytical model based on a thermal activated mechanism.
Results
First positive polarization
In these experiments we used LaTiO_{3} and LaAlO_{3} epitaxial layers grown on TiO_{2}terminated SrTiO_{3} single crystals by Pulsed Laser Deposition as described in the Methods section. The samples had typical dimensions of 1 × 2 mm^{2} and each had a metallic backgate deposited at the rear of the 0.5 mm thick SrTiO_{3} substrate. Before cooling, samples are kept in the dark for more than twelve hours to suppress any photoconductive effects. We studied the transport properties of LaXO_{3}(X = Al, Ti)/SrTiO_{3} heterostructures at 4.2 K while applying a positive gate voltage for the first time, referred to as the first positive polarization. We first focus on the response of the LaTiO_{3}/SrTiO_{3} sample shown in Figure 3. In this sample, increasing the gate voltage causes the resistance of the 2DEG, after decreasing slightly, to saturate quickly and become independent of gate voltage. This behaviour is unexpected because, according to electrostatic laws, more electrons are added to the 2DEG and therefore the resistance should decrease. This first positive polarization is irreversible: when the gate voltage is decreased from its maximum at , the reverse resistance curve deviates from the first forward curve. The carrier density n was extracted from a twocarrier analysis of the Hall effect at high magnetic field (45 T) as described in reference [14] and Supplementary material. It was found to be constant during the first positive polarization, a behaviour consistent with the saturation of the resistance. Similar to the resistance curve, when the gate voltage is decreased after being increased to , the reverse carrier density curve does not follow the first forward curve. These two behaviours are in agreement with the irreversible situation described in Figure 2d. After the heterostructure is initially cooled, its Fermi level lies very close to the top of the well. Increasing the gate voltage adds electrons at the interface that quickly fills the highest energy subbands at the top of the well. After a certain time, these electrons eventually escape into the conduction band of the SrTiO_{3} substrate, causing the saturation of the carrier density and the resistance of the 2DEG. Note that this saturation is not associated with an increase of the gate current, suggesting that charges are trapped in the system.
Beyond our initial experiments, we studied the irreversibility of the first positive polarization in further detail. After cooling a LaTiO_{3}/SrTiO_{3} heterostructure to 4.2 K, we measured its sheet resistance as a function of gate voltage by using different polarization procedures (Fig. 4a). Different from our previous measurements, in this experiment we applied a negative first polarization down to . Doing so increased the resistivity – an expected behaviour, as electrons are removed from the 2DEG – and we observed no saturation. As the voltage returns to V_{G} = 0 V, the forward resistance curve appears to match the first reverse curve. However, when the gate voltage is further increased to the value and then decreased back to V_{G} = −200 V, the reverse resistance curve deviates from the first forward curve. This new curve is fully reversible as long as the gate voltage is not increased above . We replicated this pattern with increasing maximum gate voltages of V_{G} ( and ).
These results show that the well can be emptied (ΔV_{G} < 0) and filled (ΔV_{G} > 0) reversibly as long as the gate voltage is not increased beyond a critical value corresponding to the maximum value previously applied to the metallic gate, a situation in which the Fermi level reaches the top of the well. Beyond this maximum value, we expect electrons to escape irreversibly from the 2DEG into the SrTiO_{3} substrate. We performed the same measurements on an LaAlO_{3}/SrTiO_{3} sample and we observed similar results (Fig. 5a). For this sample, the irreversible regime is reached at a gate voltage of 50 V indicating that the Fermi level after the initial cooling was slightly below the top of the well. As a consequence, at lower temperatures, the superconducting transition temperature saturates beyond this gate value (Fig. 5b and inset). To suppress the undesirable hysteresis effects, a first positive polarisation can be used as a forming process of the quantum well prior to other measurements^{13,14,20}. However, in any case, the heavy doping of the 2DEG will always be limited because the Fermi level cannot exceed the top of the quantum well.
Timedependent measurement
We performed timedependent resistivity measurements to assess how the 2DEG responds to gate voltage steps of ΔV_{G} = ±10 V. The expected corresponding modification of the carrier density is Δn = C_{a}ΔV_{G}/e where C_{a} is the capacitance per unit of area of the SrTiO_{3} substrate. Representative results of these measurements are shown in Figure 4b and 4c for different filling situations, labelled “A”, “B”, “C”, “D” and “E” in Figure 4a. In the reversible regime, the resistance shows clears ΔR jumps before reaching a stable value (Fig. 4b). As expected, ΔR is positive when electrons are removed (ΔV_{G} = −10 V, labels “A” and “B”) and negative when electrons are added (ΔV_{G} = +10 V, label “C”). In contrast, after applying a voltage step of ΔV_{G} = +10 V in the irreversible regime, the initial negative jump is followed by a slow increase of the resistance (Fig. 4c, labels “D” and “E”). We interpret this behaviour as the sign that electrons added to the 2DEG by the gatevoltage step, eventually escaped from the well. By a first approximation, the resistance relaxation follows a logarithmic time dependence with the form α + β log(t). This relaxation must not be confused with the charging time of the capacitor R_{G}C_{a}A (A is the area of the sample) which is always present at a much shorter time scale (see Supplementary Information).
Model and Discussion
To analyze the relaxation in the irreversible regime we propose a model that describes the dynamics of electrons escaping from the well. We consider a 2D quantum well at the interface with an infinite barrier on the LaXO_{3}(X = Al, Ti) side and a barrier of finite height E_{B} on the SrTiO_{3} side (Inset of Fig. 6). A number n_{L} of 2D parabolic subbands with energy E_{i} (i = 1,…,n_{L}) and density of states are filled. We assume that at a temperature T, electrons at the Fermi level E_{F} can jump over the barrier with first order kinetics:
where n is the carrier density of the 2DEG and k is the kinetic factor. This latter follows an Arrhenius law: where the activation energy is Δ = E_{B} − E_{F} and ν is a characteristic frequency factor. In two dimensions, the electron density is given by
where N_{F} = n_{L}N is the total density of states at the Fermi energy and . This situation is formally equivalent to the one with a single band of energy E_{L} and a density of state N_{F}. For a small variation of n, the temporal evolution of the Fermi energy is
At low temperature (k_{B}T ≪ E_{F} − E_{L}), a good approximate solution to equation (3) is
where is the Fermi level at t = 0^{+} (immediately after the voltage step, neglecting the short charging time R_{G}C_{a}A) and t_{E} is the characteristic escape time given by
where is the carrier density at t = 0^{+}. Therefore, the Fermi level is constant for t < t_{E}, after which it decreases logarithmically. From (2) and (4) we can obtain the temporal dynamics of the 2DEG Drude resistivity as
where is the resistivity at t = 0^{+}. In Figure 6 we show the resistance relaxation after a ΔV_{G} = +10 V step; this relaxation agrees very well with equation (6) over more than six decades of time (10 ms to 14 hours). A direct consequence of the logarithmic relaxation is the absence of an asymptotic value. However, on a linear scale the resistance changes very slowly after a few minutes, which can give a false impression of saturation. We emphasise here that the peculiar form of relaxation given by Eq. 6 is specific to the case of a well that empties itself and cannot describe other thermally activated mechanisms.
To validate this model, we systematically studied how the relaxation depends on the polarization parameters. In particular, we measured the resistance relaxation after a ΔV_{G} = +10 V step at different V_{G} values (Fig. 7a) and for different steps of ΔV_{G} = 5, 10, 20 and 40 V (Fig. 7c). The experimental data from both experiments agree with the theoretical equation (6), confirming that the model describes the phenomena we observed very well. We also observed the same agreement between experimental data and theoretical expectations for the LaAlO_{3}/SrTiO_{3} sample (Supplementary Material). To understand how the escape time depends on V_{G} and ΔV_{G}, we can express equation (5) as
where γ and κ are constants whose expressions can be found in the Supplementary Material. Because the dielectric constant of SrTiO_{3} is electricfielddependent, the capacitance C_{a} changes with gate voltage^{29}. Therefore, the number of charges added by a constant voltage step ΔV_{G} depends on the absolute value of the gate voltage. Figure 7b shows the linear variation of ln t_{E} as a function of C_{a} as expected from equation (7). We also found ln t_{E} to vary linearly with ΔV_{G} for small gate voltage step ΔV_{G} (figure 7d). For larger steps, the electrons are injected very high in the well complicating our determination of the short t_{E} values.
In the limit , equation (6) reduces to R(t) = α + β log t where . Figure 8a and 8b show that the β parameter increases linearly with temperature, as expected for a thermally activated mechanism. As already mentioned, electrons escaping into the SrTiO_{3} substrate get trapped by the defects of the crystal and no longer contribute to electronic transport. They can be released into the 2DEG if the temperature of the sample increases above two characteristic values T_{1} ≈ 70 K and T_{2} ≈ 170 K for our LaTiO_{3}/SrTiO_{3} sample (Fig. 8b). The trapping energy inferred from the temperature T_{1} is approximately 6 meV. As the electrons are trapped at a typical distance t from the interface comparable to the quantum well extension (~10 nm) which is much smaller than the thickness d = 500 μm of the SrTiO_{3} substrate, it is not possible to detrapp the electrons with a negative gate voltage of reasonable value. Indeed, the potential energy transferred the electrons eV_{G} × t/d always remains negligible compare to the trapping one. For this reason we do not observe hysteresis for negative gate voltages.
The same detrapping behaviour has also been reported in LaAlO_{3}/SrTiO_{3}^{30} at similar temperatures. The authors associated this behaviour to a thermally activated mechanism supported by exponential relaxations of conductivity near 70 K and 160 K. However, this behaviour should not be confused with the lowtemperature logarithmic relaxations observed after a gate voltage step that we described in the present article. The relaxations at 70 K and 160 K are caused by thermal escape of electrons from traps with welldefined energy barriers, giving single exponential relaxations. In contrast, the lowtemperature relaxations during the first positive polarisation are caused by electrons escaping from the quantum well with a timedependent energy barrier, leading to logarithmic time dependence after a characteristic escape time. Similarly, relaxations associated to photoconductivity^{32,33} or relaxations sometimes observed at high temperature in SrTiO_{3} based structures and attributed to anions or vacancy diffusion^{12,31} also differ from the behaviour discussed in the present article.
In this article, we have taken into account only light electron bands to illustrate the quantum well. Note that the escape mechanism described here is inherent to the presence of a quantum well at the LaXO_{3}(X = Al, Ti) SrTiO_{3} interface, disregarding the details of the band structure. It will remain valid even in the presence of a heavy band that has been theoretically predicted^{22,25,27} and seen by ARPES measurements to a certain extent^{38}, since all the subbands cross the Fermi level (see Supplementary Figure S4). The escape mechanism does not depend on the exact shape of the potential well, which can be slightly modified by the presence of nonmobile charges at the interfaces (trapped electrons, impurities…) and can vary from sample to sample. It also does not depend strongly on the absolute carrier density of the asgrown sample since the location of the Fermi energy close to the top of quantum well at zero gate voltage is a simple consequence of the Poisson's equation.
In summary, we have shown that in LaAlO_{3}/SrTiO_{3} and LaTiO_{3}/SrTiO_{3} heterostructures, the Fermi level is instrinsically close to the top of the quantum well after the cooldown. When the carrier density is increased by an electrostatic backgate voltage beyond a critical value, electrons escape into the SrTiO_{3} substrate at a rate well explained by a thermally activated leakage from the well. This phenomenon which appears both in LaAlO_{3}/SrTiO_{3} and LaTiO_{3}/SrTiO_{3} heterostructures, is directly responsible for the saturation of the 2DEG properties with gate voltage, including the mobility and the carrier density, as well as the superconducting transition temperature observed at lower temperature. The exact capacity of the well– and, thus, the maximum carrier density– is mainly determined by growth conditions and can vary from sample to sample. While it is possible to deplete reversibly the quantum well by electrostatic backgating, the filling is limited by the intrinsic location of the Fermi energy. To overcome this problem we suggest using double gated structures: A back gate could be used to engineer the shape of the quantum well which determines the carrier mobility through the bending of the SrTiO_{3} conduction band. In conjunction, a top gate can be used to add electrons to the well^{34,35}.
Methods
Growth of the heterostructures
LaTiO_{3}/SrTiO_{3} heterostructures were grown at ITT Kanpur (India) using excimer laser based PLD system on commercially available (Crystal GmbH Germany) single crystal substrates of SrTiO_{3} (100) oriented. The substrates were treated with buffered HF to expose TiO_{2} terminated surface. Before deposition, the substrates were heated to 850–950°C for one hour in an oxygen pressure of 200 mTorr to realize surface reconstruction. The source of LaTiO_{3} was a stoichiometric sintered target of 22 mm in diameter which was ablated in an oxygen partial pressure of 1 × 10^{−4} Torr with energy fluence of 1 J/cm^{2} per pulse at a repetition rate of 3 Hz to achieve a growth rate of 0.12 Å/s. Under these conditions, the LaTiO_{3} phase is grown on SrTiO_{3} substrates, as shown by XRays diffraction patterns^{4}. In this study, we used 15 u.c. thick LaTiO_{3} layers on 0.5 mm thick SrTiO_{3} substrates.
LaAlO_{3}/SrTiO_{3} heterostructures were fabricated at UMR CNRS/Thales (Paris, France). A thin LaAlO_{3} film was deposited by PLD (Surface PLD system) on a TiO_{2}terminated (001)oriented SrTiO_{3} substrate (Crystec and SurfaceNet). A buffered HF treatment followed by annealing, as described in Ref. [36], was used to obtain the TiO_{2} termination required to obtain the conducting electronic system at the interface. The KrF excimer (248 nm) laser ablates the singlecrystalline LaAlO_{3} target at 1 Hz, with a fluence between 0.6 and 1.2 J/cm^{2} in an O_{2} pressure of 2 × 10^{−4} mbar. The substrate was typically kept at 730C° during the growth, monitored in realtime by RHEED. As the growth occurs layerbylayer, it allows us to control the thickness at the unit cell level. After the growth of the film, the sample is cooled down to 500°C in 10^{−1} mbar of O_{2}, where the oxygen pressure is increased up to 400 mbar. To reduce the presence of oxygen vacancies (in both the substrate and the film), the sample stays in these conditions for 30 minutes before being cooled down to room temperature^{37}. The substratetarget distance was about 57 mm, leading to a growth rate of about 0.2 Å/s in the above conditions. In this study, we used 5 u.c. thick LaAlO_{3} layers on 0.5 mm thick SrTiO_{3} substrates.
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Acknowledgements
The authors gratefully thank M. Grilli and S. Caprara for stimulating discussions. Highmagnetic field measurements were performed at the LNCMI Toulouse with the help of D. LeBoeuf and C. Proust. This work was supported by the french ANR and the Région IledeFrance through CNano IdF and Sesame programs, as well as partially supported by Euromagnet II. The work at IIT Kanpur was funded by Council of Scientific and Industrial Research. R.C.Budhani acknowledges the J.C. Bose Fellowship from the Department of Science and Technology, Government of India.
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A.R. and R.C.B. prepared the LaTiO_{3}/SrTiO_{3} samples. N.R. and E.L. prepared the LaAlO_{3}/SrTiO_{3} samples. J.B. and S.H. performed the measurements, assisted by C.F.P., J.B., J.L. and N.B. carried out the analysis of the results and wrote the article.
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Biscaras, J., Hurand, S., FeuilletPalma, C. et al. Limit of the electrostatic doping in twodimensional electron gases of LaXO_{3}(X = Al, Ti)/SrTiO_{3}. Sci Rep 4, 6788 (2014). https://doi.org/10.1038/srep06788
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DOI: https://doi.org/10.1038/srep06788
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