Abstract
Twodimensional electron gases (2DEGs) at transitionmetal oxide (TMO) interfaces and boundary states in topological insulators, are being intensively investigated. The former system harbors superconductivity, large magnetoresistance and ferromagnetism. In the latter, honeycomblattice geometry plus bulk spinorbit interactions lead to topologically protected spinpolarized bands. 2DEGs in TMOs with a honeycomblike structure could yield new states of matter, but they had not been experimentally realized, yet. We successfully created a 2DEG at the (111) surface of KTaO_{3}, a strong insulator with large spinorbit coupling. Its confined states form a network of weaklydispersing electronic gutters with 6fold symmetry, a topology novel to all known oxidebased 2DEGs. If those pertain to just one Ta(111) bilayer, model calculations predict that it can be a topological metal. Our findings demonstrate that completely new electronic states, with symmetries not realized in the bulk, can be tailored in oxide surfaces, promising for TMObased devices.
Introduction
The realization of 2DEGs at surfaces or interfaces of transitionmetal oxides is a field of bursting activity^{1,2,3,4,5,6,7,8,9,10,11,12,13}. These materials are usually correlatedelectron systems presenting a wealth of unique properties, such as hightemperature superconductivity, colossal magnetoresistance, metaltoinsulator transitions, or multiferroic behaviour. In the search for new functionalities and future electronic device applications, a crucial challenge is to find original ways to design the oxidebased 2DEGs, so as to endow them with the exotic physics of their parent compounds and to tailor novel states of matter.
Research in nontrivial topological edge states is also a very active field^{14,15,16,17,18,19}. The possibility of realizing these in oxidebased 2DEGs, raised in several recent theoretical works^{20,21,22,23,24}, is attracting much interest. A practical proposal was based on a key insight: that the cubic ABO_{3} perovskite structure, common to many correlated oxides, realizes a honeycomb lattice when a bilayer along the [111] direction is considered^{20,23,24}. Such artificial structure could, in principle, be digitally engineered as a sandwich between two oxide insulators^{20}. However, this approach currently remains a technological challenge.
Here we take an alternative route towards the same goal. Beyond the realization of 2DEGS at LaAlO_{3}/SrTiO_{3} interfaces oriented along [001]^{1} and lately [110] and [111] cristallographic directions^{25,26}, it was recently discovered that 2DEGs can be simply obtained at the vacuumcleaved (001) surface of insulating ABO_{3} perovskites^{10,11,12,13} opening an exciting perspective to create novel 2DEGs at the surface of correlated oxides. Following this methodology, here we demonstrate that the Fermi sea of the 2DEG can be crafted by properly choosing the cleaving plane of the perovskite crystal. Specifically, we obtain a 2DEG at the (111) surface of the strong spinorbit coupled insulator KTaO_{3} (KTO) and determine directly its electronic structure using angleresolved photoemission spectroscopy (ARPES). We observe an open Fermi surface of 6fold symmetry with weakly dispersing branches, which is unique among all other previously known oxidebased 2DEGs. Using model calculations we show that it originates from electron hopping between consecutive layers of Ta atoms along the [111] direction. Furthermore, by analyzing the symmetries of this electronic structure, we demonstrate that confining this novel 2DEG to a bilayer of Ta atoms, hence a honeycomb lattice, will lead to a topological metal.
Results
Basic considerations: structure of KTaO_{3} along the [111] direction
Figure 1(a) presents the crystal structure of KTaO_{3}. Along the [111] direction, the system has 3fold symmetry, as any cubic system and consists of alternating layers of Ta and KO_{3}. There are three different Ta(111) layers per unit cell, labelled TaI, TaII and TaIII. Figure 1(b) shows the Ta atoms in the 3D structure seen from the [111] direction. Arrows represent the lattice vectors of a 2D unit cell in the (111) plane, which corresponds to a centered hexagonal network with 6fold symmetry. When only two layers are considered (TaI and TaII, for example), the lattice formed by nearest neighbors is a honeycomb lattice (thick black lines), similar to the bilayer structure proposed in Ref. 20. As we shall see later, our ARPES data are consistent with the formation of a 2DEG with the periodicity and the symmetry of an unresconstructed (111) plane.
Experimental electronic structure
Figure 2(a) shows the Fermi surface (color map) of the 2DEG measured at the (111) surface of transparent and bulk insulating KTO, pictured in Fig. 2(b). Experimental details of the extraction of this Fermi surface are given in the Supplementary Material. The black hexagons in figure 2(a) represent the Brillouin zone of the unreconstructed (111) surface. Note that the observed Fermi surface has a well defined 6fold symmetry and a periodicity corresponding to that of the unreconstructed KTO(111) layers. By Bloch theorem, we infer that the 2DEG is formed by itinerant conduction electrons experiencing an inplane potential compatible with unreconstructed (111) layers. In turn, this strongly suggests that, while one might expect that the highlypolar KTO(111) surface reconstructs, the observed 2DEG is formed by electrons confined in unreconstructed subsurface layers.
More specifically, the Fermi surface of Fig. 2(a) consists of 6 branches extending out from to the six points. From the area enclosed by this Fermi surface, we obtain a carrier density n_{2D} ~ 10^{14} cm^{−2}. The observation of such a large Fermi surface fully supports the quasi2D (i.e., confined) character of the electron gas. Indeed, if the Fermi surface was a crosssection of an hypothetical 3D Fermi surface, the ensuing electron density would be, by simple scaling, . For such density, only one order of magnitude smaller than that of pure gold, the sample would be highly conducting and its aspect would be mirrorlike, in stark contrast with the transparent character of the bulk crystal shown in figure 2(b), indicative of an insulating state.
Different energymomentum intensity maps across the Fermi sea provide valuable complementary information about the 2DEG at the (111) surface of KTO. Figures 2(c, d) show the dispersion along the direction, dashed green line in figure 2(a), in the form of, respectively, intensity map and stack of momentum distribution curves (MDCs). This dispersion indicates that the Fermi surface branches are formed by electronlike bands coming close to each other near . The effective mass of these electronlike bands, deduced from a parabolic fit (red dashed lines), is (m_{e} is the free electron mass). Furthermore, as shown by the stack of energy distribution curves (EDCs) in figure 2 (e), the bottom of each of these electron bands is essentially nondispersive along the direction. Accordingly, the Fermi surface forms a star with six open branches, each branch being composed of two quasiparallel sheets. In other words, the electronic structure of the 2DEG at the (111) surface of KTO consists of a network of weakly dispersing “electron gutters”.
Since we observe only the first subband (E_{1}) of the 2DEG, we can only estimate an upper limit for the extension L of the 2DEG. We make the reasonable assumptions that E_{1} ≈ −V_{0}, where V_{0} is the depth of the confining potential well and that the next quantum level (E_{2}) of the same nature as the measured subband lies at an energy E_{2} − E_{1} ≈ V_{0}, at or above E_{F}. The result, detailed in the Supplementary Material, is L < 16 Å, or equivalently, L < 7 Talayers along (111).
Note, from figures 2(c–e), that there is an intense background below E = −260 meV trailing behind the bands forming the 2DEG. The most likely origin of this background is inelastic scattering of the electrons off impurities or disorder. This can be either inplane scattering, which will broaden the spectral lines, or scattering of the electron on its way out of the surface during the photoemission process. For instance, we expect that the fractured native surface will be rugged and will contain a large density of vacancies. Thus, a large number of photoemitted electrons will loose some kinetic energy, due to inelastic scattering, on their way out of the solid. All these electrons will contribute to an intense inelastic background trailing behind the broadened parabolic quasiparticle peak. As this background is extrinsic to the electronic structure, it has no effect on the quasiparticle spectral function, in agreement with the observation, noted earlier, that the experimental effective mass of the 2DEG's band is essentially the same as the one given by noncorrelated tightbinding calculations (Supplementary Material and Ref. 13).
Discussion
The observed electronic structure can be understood in terms of electron hopping between neighbouring Ta layers. To see how, we modeled the 2DEG using a tight binding (TB) hamiltonian on a Ta(111) bilayer with the hopping parameters determined from our previous study of KTO(001)^{13} –see details in the Supplementary Material. Indeed, as schematically indicated by the double arrows in figure 1(b) and in Ref. 20, the largest hopping occurs between the Ta atoms of consecutive (111) layers, as they correspond to nearest neighbors along [001] in the cubic 3D lattice. In figure 3(a) we compare the experimental ARPES Fermi surface to the calculated Fermi surface. We observe that the model correctly gives the bandstructure of the weakly dispersing electron gutters. The comparison between the data and calculations close to the point is more difficult due to the experimental resolution. We checked that including extra Ta layers or considering the hopping of electrons between Ta atoms on the same (111) layer, which are nextnearest neighbors, will mainly change the computed bandstructure around the point, but not the structure of the gutters, so that the comparison above remains valid.
The weakly dispersing gutters along the lines are the most salient feature of the observed Fermi surface. They can be intuitively understood from the directional overlaps between t_{2g} orbitals in neighboring Ta sites, which give rise to large nearestneighbor hopping amplitudes along the [001] (and equivalent) directions. When seen from the [111] direction, these appear as directional zigzag chains at 120° from each other, as depicted in Fig. 3(b) for two consecutive Ta (111) layers. Each set of parallel hopping chains on the 2D lattice will lead to open weakly dispersing Fermi sheets in reciprocal space, i.e., to the observed gutters.
Our results demonstrate that a 2DEG with hexagonal symmetry can be easily created at the (111) surface of KTO. As an outlook for nearfuture investigations, an exciting feature arises when such a 2DEG is confined to a Ta(111) bilayer, which forms a honeycomb lattice –as seen from the lattice formed by layers TaI and TaII in figure 1(b). In this case, following Ref. 27, one can demonstrate (details in the Supplementary Material) that the electron wavefunction of the ground state has an odd parity at one of the points (and its timereversal partner), but an even parity at all other points, implying a non trivial topological index Z_{2} = 1. In this case, the 2DEG at the KTO (111) surface would correspond to a topological 2D metal, similar to some of the nontrivial states predicted in Ref. 20, with the 2DEG itself corresponding to the “bulk” states and the nontrivial states appearing at the 1D edges of the sample.
From a broader perspective, our results show that different surface orientations can be used to tailor Fermi seas and to radically change the microscopic state, hence the physical properties, of oxide perovskitebased 2DEGs. Of particular interest is the prospect for engineering 2DEGs of strongly correlated insulators on honeycomb latticestructures, which could lead to correlated topological and edgestates with new physical properties not shown by semiconducting topological insulators. Thus, in perspective, our results provide some initial bridges between the fields of topological matter, correlated electrons and oxide electronics.
Methods
The angle resolved photoemission spectroscopy (ARPES) experiments were done at the Synchrotron Radiation Center (SRC, University of Wisconsin, Madison) and at the Synchrotron SOLEIL (France), using linearly polarized photons in the energy range 20–100 eV and Scienta R4000 detectors with vertical slits (henceforth the k_{y} direction). The momentum and energy resolutions were 0.25° and 15 meV, respectively. The mean diameter of the incident photon beam was smaller than 50 μm (SOLEIL) and about 150 μm (SRC).
The samples studied were undoped transparent highgrade laser crystals of KTaO_{3} (SurfaceNet, GmbH), of purity 99,9995%, grown by a modified top seeded solution growth (TSSG) using KO_{2} as solvent. The estimated amount of oxygen vacancies is less than 10 ppm, being mostly concentrated in the seeding area, which was not used for the final cut crystals. Similarly, the estimated amount of defects is less than 100 cm^{−3}.
The crystals were mounted with and fractured insitu along the (111) surface at 25 K (SRC) and 10 K (SOLEIL), in pressure lower than 6 × 10^{−11} Torr. The fractured surfaces had homogeneous electron photoemittance over a large area.
The results have been reproduced in 6 different cleaves.
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Acknowledgements
We thank R. Weht, J. Guevara, G. Montambaux, F. Piéchon, J.N. Fuchs and M.O. Goerbig for discussions. A.F.S.S. and M.G. acknowledge support from the Institut Universitaire de France. This work was supported by the ANR (project LACUNES) and the LabEX PALM (project ELECTROX).
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Project conception: A.F.S.S.; ARPES measurements: C.B., T.R., F.F., F.B. and A.F.S.S.; infrastructure for ARPES experiments at Soleil: F.B., A.T.I. and P.L.F.; sample characterizations: M.B., A.B., T.M. and P.L.; data analysis, interpretation, calculations: C.B., A.F.S.S., M.G., O.H.C. and M.J.R.; writing of the manuscript: A.F.S.S., with input from C.B., M.G. and M.J.R. All authors discussed extensively the results and the manuscript.
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Bareille, C., Fortuna, F., Rödel, T. et al. Twodimensional electron gas with sixfold symmetry at the (111) surface of KTaO_{3}. Sci Rep 4, 3586 (2014). https://doi.org/10.1038/srep03586
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DOI: https://doi.org/10.1038/srep03586
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