The two-dimensional electron system at the interface between the insulating oxides LaAlO3 and SrTiO3 exhibits ferromagnetism, superconductivity and a range of unique magnetotransport properties. An open experimental challenge is to identify, out of the multitudinous energy bands predicted to exist at the interface, the key ingredients underlying its emergent transport phenomena. Here we show, using magnetotransport measurements, that a universal Lifshitz transition between d orbitals of different symmetries lies at the core of the observed phenomena. We find that LaAlO3/SrTiO3 systems generically switch from one- to two-carrier transport at a universal carrier density, which is independent of the LaAlO3 thickness and electron mobility. Interestingly, the maximum superconducting critical temperature occurs also at the Lifshitz density, indicating a possible connection between the two phenomena. A simple band model, allowing for spin-orbit coupling at the atomic level, connects the observed transition to a variety of previously reported properties. Our results demonstrate that the fascinating behaviour observed so far in these oxides follows from a small but fundamental set of bands.
A key feature of the electron gas at the LaAlO3/SrTiO3 (LAO/STO) interface1 is that its ground state can be tuned using field-effect gating2. Ferromagnetic3,4 and superconducting5,6 ground states have been demonstrated in this system, as well as their coexistence7,8,9. Superconductivity was shown to emerge at a certain gate voltage, concomitant with the appearance of strong spin–orbit (SO) interactions10. At higher gate voltages, the critical temperature of superconductivity was shown to slowly decrease6,11 together with a similar decrease in SO strength12. A growing body of theoretical works13,14,15,16,17 predict that at the interface there would be a large number of electronic sub-bands, suggesting that the above transport characteristics might be rather sensitive to the details of the disorder and of the confining potential at the interface. This complexity is further supported by transport11,18,19,20, optical21 and heat conductivity22 measurements that proposed that at high-carrier densities this system has multiple types of carriers existing at different distances from the interface, possibly on different TiO2 planes11,13,14,17,21,23. Thus, to date, a simple relation between specific energy bands and the fundamental transport phenomena observed in this system is still missing.
In this paper, we show using magnetotransport experiments that the physics at the interface follows from a universal picture with a minimal number of energy bands, implying that only a few among the manifold of states are important. Specifically, we show that a variety of behaviour of this two-dimensional (2D) system can be traced back to a special point where the system undergoes a Lifshitz transition between populations of d orbitals of different symmetries.
Observation of a special point in transport of LAO/STO system
All measured LAO/STO samples showed a critical gate voltage across which their transport characteristics changed significantly. We have measured four samples differing in their LAO layer thickness and spanning an order of magnitude in mobility, all showing similar behaviour. Figure 1a shows the measured ρXY as a function of magnetic field, B, at various gate voltages for sample 1. Starting from the lowest gate voltage VG=−40 V and up to a critical value VC=40 V, ρXY is linear in B with a slope that decreases with increasing VG. This behaviour is expected from the field-effect tuning of carrier density. However, as we cross VC the shape of the Hall traces changes considerably; they become kinked, with the kink position progressively moving to lower B with increasing VG. Zooming-in on the low-field Hall slopes (Fig. 1b) we find that they too undergo a significant change at VC, from slopes that gradually decrease with increasing VG below VC, to slopes that are roughly constant and even slightly increasing above VC (inset Fig. 1b). Thus, both the low- and high-field Hall characteristics signify that a distinct transition occurs in the system at VC.
This transition becomes even more striking when we plot the Hall coefficient RH=ρXY/B as a function of B and VG (Fig. 1c). Below VC, RH is roughly a constant as a function of B, whose value decreases with increasing VG. Above VC, however, the curves acquire a qualitatively different form: they first increase with field at low fields, then peak at an intermediate field, Bp, (see Supplementary Methods for more details about this peak) and finally drop down at high fields yielding an overall bell-like shape, whose width we define as BW (Fig. 1c caption). The characteristic field, BW, progressively decreases as VG is increased. This trend in RH is echoed by the longitudinal resistance, ρXX (Fig. 1d). Below VC, ρXX is practically independent of B, but above VC, ρXX remains flat only up to an intermediate characteristic field after which it rises continuously. Just like BW, this characteristic field becomes smaller with increasing VG.
The self-similarity of the transport traces on each side of the transition point suggests that they can be grouped into two generic forms. Below VC, the curves have almost identical shapes up to an offset (Fig. 2a,b). Above VC, the curves (Fig. 1c,d) differ substantially in their dependence on B. However, if for every gate voltage we add a y-offset, scale the curve height and plot it as a function of the scaled magnetic field, B/BW, we observe that all RH curves perfectly collapse onto a single trace (Fig. 2c). This suggests that the Hall resistance above VC is described by a generic functional form. Interestingly, if we use the field with which we scaled RH to scale the ρXX trace at the same gate voltage, we find that all the ρXX traces also collapse perfectly (Fig. 2d). The fact that ρXY and ρXX scale together demonstrates their strong connection.
In Fig. 2e we plot the dependence of BW on the carrier density, extracted from the high-field Hall slope. This field increases with decreasing density and becomes immeasurably high when we approach the transition point. If we plot its inverse, 1/BW, as a function of the density (Fig. 2f), we see that it extrapolates to zero at nC, the critical density that marks the onset of the observed transition.
What is the nature of the observed transition point? Below VC ρXY is linear with B, its slope is inversely proportional to VG, and ρXX hardly changes with B, which is the classical behaviour expected from single-carrier transport. Above VC, on the other hand the S-shaped Hall traces and the strong positive magnetoresistance in ρXX are suggestive of multi-carrier transport. Most studies so far focused on this high-carrier density regime, and interpreted the S-shaped Hall traces11,12,19,22 to result from transport by two or more kinds of carriers. The transition that we observe in our measurement is therefore from one- to multiple-carrier transport, the latter being so far the more prevalent regime in studies of this system. The observation of a density regime in which transport is given by only a single-carrier type is in fact very surprising in light of the multitudinous d-bands predicted by theory13,14, and intimates that any additional bands that may exist in the system must have localized carriers that are irrelevant to transport.
The simple generic form observed for the transport above VC suggests that even above VC the transport might be simpler than originally imagined. In fact it can be shown that the S-shaped transport traces above the transition point, as well as the observed correlation between ρXY and ρXX, could be nicely explained by a simple two-carrier transport model. A careful analysis of our data shows however that the transition field BW between one- and two-carrier transport is not accounted for by the simple two-carrier model, often used in the literature24, which assumes that the mobilities and densities of the two carriers are independent of magnetic field (described in Supplementary Methods). We further show there that this assumption makes the simple model inadequate for extracting the two carrier mobilities, but that it can still reliably determine their densities. Specifically, if the two carriers have significantly different mobilities, the asymptotic value of RH at zero field gives the density of the high mobility carriers, nhi=1/eR0, whereas its value at large fields yields the total density of both carriers, ntotal=1/eR∞, (e is the electron charge). In Fig. 3a we extract these two quantities, nhi and ntotal, from the Hall coefficients measured at B=0 T and 14 T, and plot them against VG. For voltages below the transition point we see that all the carriers are of the high mobility type, implied by the fact that ntotal≈nhi, and that their density is continuously tuned by the gate up to a critical value, nC. After crossing the transition point, nhi saturates at nC, whereas ntotal continues to rise, implying that at high densities only lower mobility carriers are added by the gate, in contrast to the notion that its main effect is to add carriers populating higher mobility TiO2 planes at increasing distances from the interface.
An underlying universal Lifshitz transition
How universal is the observed critical density? One expects that this density depends on the disorder and inhomogeneities in the system4,9,25, as well as on the different confinement potential profiles that can form at the interface. Indeed, for samples with low mobilities we find that the critical density can change substantially from sample to sample. The surprising observation is, however, that in all high mobility samples that we have measured we observe a striking universality in the critical density. In Fig. 3b–e, we show measurements on additional samples whose LAO layer thicknesses varied from 6 to 10 unit cells and whose mobilities varied by an order of magnitude, analysed like in Fig. 3a. In all cases, we observed similar Hall and longitudinal resistance traces and comparable transition points, where the transport switched from one to two kinds of carriers (see also Supplementary Fig. S1). Whereas the critical gate voltage and the mobility at the transition, μC, vary substantially between samples, the critical density in these high mobility samples is almost the same. Plotting the critical density versus the critical mobility measured in several independent Hall bars on four different samples (Fig. 3f), we see that these densities have a universal value nC=1.68±0.18×1013 cm−2 independent of disorder, pointing to an intrinsic origin. The observed critical density is therefore fundamentally different from the recently determined metal–insulator transition density26, both in its experimental signature (below the metal–insulator transition the I–V characteristics are non-ohmic, whereas we see ohmic behaviour throughout except at the lowest densities, which are much smaller than nC), as well as in its underlying source: the metal–insulator transition density is non-universal26, being set by the extrinsic disorder, whereas the critical density that we observe is universal, pointing to an intrinsic source.
We argue that the observed transition is a Lifshitz transition that occurs when the density exceeds a critical value at which new bands of a different symmetry are populated. It is well established that interfacial electrons reside in the t2g conduction bands of STO27,28,29,30. In the bulk, these three bands are degenerate, having ellipsoidal Fermi surfaces centered at the Γ point and oriented along the main lattice directions. At the interface this degeneracy is lifted due to tetragonal distortion and quantum confinement. Specifically, the dXY band, which has a larger effective mass in the z-direction than the dXZ and dYZ bands, splits toward lower energy27 (Fig. 3g). Therefore, below a critical carrier density all electrons are expected to reside in the dXY band and only one type of carrier exists in the system. At higher densities the Fermi energy enters the dXZ and dYZ bands, and as they both get populated we expect to observe contributions from carriers of a new type.
The Lifshitz transition into the new bands is accompanied by a large change in the density of states (DOS), determined by the geometric mean of the effective masses in the 2D plane, . As the dXZ and dYZ bands each has one light and one heavy mass in the plane whereas the dXY band has two light masses, the latter has the lowest DOS (Fig. 3h). Using effective masses determined from angle-resolved photoemission spectroscopy28, the combined DOS of the dXZ and the dYZ bands is approximately ten times higher than that of dXY band. Therefore, after raising the Fermi energy above the critical density, we expect the two newly occupied heavy bands to take up most of the added carriers while the density of the light dXY carriers remains almost constant. This scenario perfectly fits our observations plotted in Fig. 3a–e and excludes the possibility of an upper band being a more mobile, sub-band replica with the same symmetry as the lower band. The effect of SO coupling on the band structure (Fig. 3i–k) is discussed below.
We can now address the observed universality of the critical density. Within the above band model, the critical density is determined by the number of carriers that the dXY band can accommodate before the higher bands become available, . Here, ΔE is the energy splitting between the dXY and the higher two bands, and ml is the light mass. Using the X-ray measured splitting27, ΔE=50 meV, and angle-resolved photoemission spectroscopy measured light mass28, ml=0.7 me (me is the electronic mass), we estimate that nC=1.46×1013 cm−2, in good agreement with our experimentally determined value of nC=1.68±0.18×1013 cm−2. In general, ΔE can have non-universal contributions from gate-induced electric fields, the nonlinear dielectric constant of STO and charge traps, and a universal contribution from a charge or a dipole layer at the LAO/STO interface, as implied by recent measurements31. Our observation of a universal transition density suggests that an intrinsic mechanism, such as the proposed dipole layer, is the dominant component of the confinement potential at the interface (Supplementary Discussion).
The body of evidence presented above lends strong support to the interpretation that the observed Lifshitz transition occurs when d-bands of a different symmetry begin to be populated. It is interesting to see how this physics comes to bear on superconductivity in the system. Figure 4 shows the longitudinal resistivity ρXX measured down to T=50 mK, divided by its normal-state value at T=0.38 K, as a function of temperature and gate voltage. As reported earlier6,7,10,11, we find that superconductivity is dome-shaped as a function of VG. The superconducting transition temperature, TC, first increases rapidly with VG up to a peak value, TCmax=320 mK, after which it slowly decreases with increasing VG. Comparing these measurements to the critical density (dashed line in Fig. 4), we find that the latter corresponds to the peak of the dome. A similar correlation could also be deduced for published data11 if we apply to it the analysis described above. Although the full details of 2D superconductivity in this system are still not fully understood the observed correlation hints that the Lifshitz transition is probably having an important role in this phenomenon.
We now turn to discuss how the band picture that emerges from our experiments can readily explain the strong gate tunability of SO interactions observed in previous experiments. Earlier transport measurements10 have used weak anti-localization analysis to show that the strength of SO interactions rises sharply at a certain gate voltage, concomitant with the rise of TC. Measurements of the superconducting critical field12 have further indicated that at higher gate voltages, SO interaction strength reduced gradually. So far, the existence of SO in this system has been explained to result from Rashba interactions that are induced by the electric field of the confining potential10,12,17. However, in a simple Rashba picture it is unclear why SO will suddenly increase at a certain density, nor why similar SO effects are observed in δ-doped STO32, whose confining potential is symmetric.
We argue that the observed dependence of SO on the carrier density can be naturally understood within the band picture discussed above. The key is to realize that the strongest SO effect in this system is the atomic SO (ASO) coupling33,34. ASO is especially prominent when the d-bands are degenerate, so that the coupling strongly mixes the orbitals and entangles the spin and orbital degrees of freedom35. When we add ASO to the band calculation (Fig. 3i,j), we indeed see that its most marked effect is near the bottom of the dXZ and dYZ bands. Interestingly, we find that the calculated ASO strength as a function of energy (Fig. 3k, see caption for details) peaks around the transition point in a striking resemblance to the experimental observation10, suggesting that intrinsic ASO at the band transition is responsible for the strong SO observed in this system. We note that other band-structure effects could also mix the dXZ and dYZ bands, where the leading contribution, not excluded by symmetry, is due to second-nearest neighbour hopping between dXZ and dYZ orbitals along a diagonal. However, unlike ASO, which mixes the bands even at the Γ point, the diagonal hopping is proportional to the crystal momentum squared and thus is very small around the Lifshitz transition at the Γ point. In the Supplementary Methods (see also Supplementary Fig. S2), we estimate the strength of the diagonal hopping by fitting a tight binding model to density functional theory calculations without ASO13, and show that its inclusion in our Hamiltonian has negligible effects on the band-structure in the relevant energy window around the Lifshitz point (Fig. 3i–k). We further show how the intimate connection between the Lifshitz transition and ASO may also explain other observations in this system, most notably the low-carrier densities measured by Shubnikov–de Haas18,19,36,37,38.
The Rashba SO coupling existing in this system is intimately tied to its strong ASO coupling. Microscopically, Rashba coupling emerges as a second-order perturbation in ASO when the inversion symmetry of the interface is broken (more details in Supplementary Methods). This coupling is therefore directly proportional to the strength of ASO and thus its strength will also peak at the Lifshitz point. In the presence of such coupling, the anisotropy of transport with in-plane magnetic field39 can serve as a probe of the orbital symmetry of the conduction bands40. Our recent measurements of this anisotropy not only indicate the buildup of SO coupling when the Lifshitz transition is approached from above but also show that below the critical density the system has an isotropic ground state, consistent with a circular dXY band, whereas above it the ground state acquires a strong crystalline anisotropy, consistent with the elliptical dXZ and dYZ bands that are oriented along the crystalline directions41.
In summary, we observe that the 2D system formed at the LAO/STO interface undergoes a Lifshitz transition, which reshapes its core transport properties above a critical carrier density. The value of the critical density is observed to be universal, independent of sample mobility and LAO layer thickness. This transition is understood to be between dXY and dXZ/dYZ orbital population, showing that out of the multitude of d-bands predicted theoretically, only a small subset is relevant to fully capture the physics of the conducting 2D system. We show that near the transition ASO interactions should become prominent, accounting for the peak of SO strength observed experimentally. We further observe a similar correlation between this transition and the peak of superconducting TC, which may suggest that the population of the dXZ and dYZ bands at this critical density has an important role in the emergence of superconductivity as well as possibly other correlated states in this system. These results therefore open the way to a fundamental description of the fascinating behaviours of this system in terms of an underlying, universal transition between states of different orbital symmetries.
As detailed in an earlier work6, LaAlO3 films were grown on TiO2-terminated (001) SrTiO3 single crystals by pulsed laser deposition in ~10−4 mbar of O2. The repetition rate of the laser was 1 Hz, with the fluence of each pulse being 0.6 J cm−2. The film growth was monitored in situ using reflection high-energy electron diffraction. Three samples were grown at T=800 °C and the last at T=650 °C to yield higher mobility18. After growth, the samples were annealed in 200 mbar of O2 at about 600 °C for 1 h and cooled to room temperature in the same oxygen pressure. For the high mobility sample, annealing took place at 530 °C. More information including the influence of oxygen pressure during growth can be found in Cancellieri et al.42 Hall bars were photolithographically patterned and the sample was ultrasonically bonded using Al wire.
Transport was measured using standard back-gated Hall bars with widths varying from 100 to 500 μm and AC currents of 5–50 nA. In the entire range of gate voltages studied the measured resistance was ohmic. Most measurements were done at T=4.2 K, apart from those of superconductivity that were done down to 50 mK. The longitudinal (Hall) resistivity was symmetrized (anti-symmetrized) with respect to magnetic field to eliminate small spurious effects from lithographic imperfections.
How to cite this article: Joshua A. et al. A universal critical density underlying the physics of electrons at the LaAlO3/SrTiO3 interface. Nat. Commun. 3:1129 doi: 10.1038/ncomms2116 (2012).
We acknowledge A.D. Caviglia, S. Gariglio, A. Fete and J.–M. Triscone for the samples, fruitful discussions and useful comments on the manuscript. We benefited greatly also from discussions with E. Berg, Y. Gefen, D. Goldhaber-Gordon, Y. Meir, Y. Oreg, D. Shahar, A. Stern, V. Venkataraman and A. Yacoby. S.I. acknowledges the financial support by the ISF Legacy Heritage foundation, the Minerva foundation, the EU Marie Curie People grant (IRG) and the Alon fellowship. S.I. is incumbent of the William Z. and Eda Bess Novick career development chair. E.A. acknowledges the financial support by the ISF foundation. E.A. is incumbent of the Louis and Ida Rich Career Development Chair.
Supplementary Figures S1-S3, Supplementary Methods, Supplementary Discussion and Supplementary References