Abstract
Twodimensional electron gases (2DEGs) in SrTiO_{3} have become model systems for engineering emergent behaviour in complex transition metal oxides. Understanding the collective interactions that enable this, however, has thus far proved elusive. Here we demonstrate that angleresolved photoemission can directly image the quasiparticle dynamics of the delectron subband ladder of this complexoxide 2DEG. Combined with realistic tightbinding supercell calculations, we uncover how quantum confinement and inversion symmetry breaking collectively tune the delicate interplay of charge, spin, orbital and lattice degrees of freedom in this system. We reveal how they lead to pronounced orbital ordering, mediate an orbitally enhanced Rashba splitting with complex subbanddependent spin–orbital textures and markedly change the character of electron–phonon coupling, cooperatively shaping the lowenergy electronic structure of the 2DEG. Our results allow for a unified understanding of spectroscopic and transport measurements across different classes of SrTiO_{3}based 2DEGs, and yield new microscopic insights on their functional properties.
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Introduction
The ubiquitous perovskite oxide SrTiO_{3}, a widegap band insulator, hosts varied bulk properties including quantum paraelectricity, dilute dopinginduced superconductivity and high thermoelectric coefficients. These reflect a subtle competition between interactions of the underlying quantum manybody system. Intriguingly, thermodynamic and transport measurements^{1,2,3,4,5,6} indicate that the balance of these interactions can be tuned to engineer striking emergent properties when quantum confinement and doping are combined to create a twodimensional electron gas (2DEG)^{1,2,7}. A diverse and attractive array of properties have been uncovered to date in this system, including gatetuned superconductivity^{3,4,8}, its coexistence with ferromagnetism^{5,6} and enhanced Seebeck coefficients^{9}, establishing SrTiO_{3}based 2DEGs as a model platform for use in future multifunctional electronic devices^{1}.
They are most commonly realised at a polar interface to another band insulator LaAlO_{3}, creating a narrow conducting channel that resides solely within the SrTiO_{3} (ref. 10). Similar 2DEGs can also be created by interfacing SrTiO_{3} to a wide array of other band or Mott insulators including NdAlO_{3} (ref. 11), LaTiO_{3} (ref. 12) and GdTiO_{3} (ref. 13), by chemical doping of electrons into narrow SrTiO_{3} channels^{14,15}, analogous to δdoping of semiconductors such as Si, and by fieldeffect doping in a transistorstyle configuration^{8}. Moreover, the recent discovery of a 2DEG formed at the free surface of a bulk SrTiO_{3} crystal opens new avenues for its advanced spectroscopic investigation^{7,16}.
Exploiting this, here we present unified angleresolved photoemission (ARPES) measurements and tightbinding supercell calculations revealing new richness of the electronic structure of this model oxide 2DEG. We show how a pronounced orbital ordering mediates an unconventional spin splitting, giving rise to strongly anisotropic and subbanddependent canted spin–orbital textures. The orbitally enhanced Rashba effect explains the pronounced spin splittings previously inferred from magnetotransport in this system, while simultaneously revealing a breakdown of the conventional picture used to describe these. We uncover how this complex ladder of subband states are further renormalized by manybody interactions. This reconciles previous discrepancies between effective masses estimated from ARPES and quantum oscillations, unifying the properties of surface and interface SrTiO_{3} 2DEGs, and reveals a strikingly different nature of electron–phonon coupling compared with bulk SrTiO_{3}.
Results
Orbital ordering
Figure 1 summarizes the generic electronic structure of SrTiO_{3} 2DEGs, as revealed by ARPES from a SrTiO_{3}(100) surface with saturated band bending^{7}. Consistent with previous reports from both surface and interface 2DEGs^{7,16,17,18,19}, we find a broad bandwidth that extends up to ≈250 meV below the Fermi level. Here we can resolve a ladder of at least three light subband states that contribute concentric circular Fermi surface sheets, coexisting with just a single heavy electron band (m*=14±3m_{e}) that has a much shallower binding energy of <50 meV and gives rise to elliptical Fermi surfaces oriented along 〈10〉. From this Fermi surface topology together with the polarization dependence of our measured intensities (Supplementary Fig. 2), we assign not only the lowest^{16} but rather the whole ladder of observed light states as having dominantly d_{xy} orbital character, while the heavy states derive from d_{xz/yz} orbitals. This immediately indicates a strong breaking of the t_{2g} orbital degeneracy that is present in the bulk electronic structure of SrTiO_{3} (ref. 20), driving a pronounced orbital ordering with a polarization which exceeds 30%, a lower limit derived from our experimentally resolved Fermi surface areas.
This is a direct consequence of the realspace anisotropy of the orbital wavefunctions combined with inversion symmetry breaking by the electrostatic potential that defines the 2DEG by creating a steep asymmetric quantum well along the z direction (Fig. 1c). As shown by our selfconsistent tightbinding supercell calculations (Fig. 1b, see Methods), the resulting quantized subbands that derive from planar d_{xy} orbitals have wavefunctions reminiscent of the envelope functions of a semiconductor quantum well, except that in SrTiO_{3} they are much more localized in real space, almost to within a single unit cell for the lowest subband state. In contrast, the potential variation acts as a much weaker perturbation on the outofplane d_{xz/yz} orbitals, which have much larger hopping amplitudes along the z direction. The resulting subbands sit close to the top of the potential well, leading to wavefunctions that penetrate much deeper into the bulk. This disparate spatial extent of the subband states is consistent with their relative spectral weight in our surfacesensitive ARPES measurements.
Unconventional Rashba spin splitting
The same breaking of inversion symmetry that drives this orbital ordering can additionally lift the spin degeneracy through a Bychkov–Rashbatype spin–orbit interaction^{21,22,23}. Focussing near the band bottom of the lowest d_{xy} band (Fig. 2a,b), we indeed find a small characteristic splitting between the calculated energy of spinup and spindown states, δE=2αk_{}, with a Rashba parameter, α=0.003 eVÅ. The nonnegligible splitting found here, despite the modest spin–orbit interaction in 3d transition metals, is indicative of the very strong electric field gradient of the confining potential. From our selfconsistent bandbending potential calculation (Fig. 1c), we estimate that this exceeds 2 × 10^{8} V m^{−1} within the first two unit cells where the lowest subband state is confined. For the more delocalized second d_{xy} subband, whose wavefunction extends into regions of shallower band bending, we find a slightly smaller Rashba parameter α=0.0014, eVÅ, confirming that the strength of this spin splitting is controlled by the confining electric field. This should therefore be directly tuneable by electrical gating, suggesting a potential route towards spintronic control in oxides.
Unlike typical Rashba systems such as the Au(111) surface, however, here the interplay between orbital ordering and spin–orbit coupling leads to a significantly richer spin structure of the 2DEG states. Close to the crossings of the light d_{xy} and heavy d_{xz/yz} subbands, the spin splitting increases by approximately an order of magnitude, concomitant with a strong mixing of their orbital character (Fig. 2a–c). This rationalizes an increased spin splitting reported from transport when the d_{xz/yz} subbands become populated in electrically gated SrTiO_{3}/LaAlO_{3} interface 2DEGs^{24}. Moreover, the crossover from klinear to strongly enhanced spin splitting that we find here readily explains the approximately k^{3} dependence of the splitting that has been reported^{25}. Our layerprojected calculations indicate that the subband wavefunctions become more delocalized in the z direction close to these band crossings, a natural consequence of the stronger overlap of neighbouring d_{xz/yz} orbitals along z (Supplementary Fig. 3). This delocalization would naively be expected to reduce the strength of the Rashba effect. In contrast, its significant enhancement here points to a dominant role of interorbital hopping in driving such surprisingly large spin splittings. Similar enhancements have recently also been observed in other calculations, mainly based on model 3band Hamiltonians^{26,27,28}, which are qualitatively entirely consistent with our results. Our calculations demonstrate how this is a direct consequence of orbital ordering in the real experimentally confirmed multisubband structure of the SrTiO_{3} 2DEG.
Moreover, as shown in Fig. 2d,e, they reveal an exotic coupled spin–orbital texture of the resulting 2DEG Fermi surfaces. While an approximately perpendicular spinmomentum locking ensures tangential spin winding around the circular d_{xy} sections of Fermi surface, it leads to spins aligned almost perpendicular to the Fermi surface for large sections of the extremely anisotropic d_{xz} (d_{yz}) sheets. Around the crossings of d_{xy} and d_{xz/yz} states, the spins of neighbouring Fermi surfaces align (anti)parallel with a ↓〉↑〉↑〉↓〉 ordering, ensuring maximal hybridization gaps are opened. At the same time, rather than being quenched to zero as might have naively been expected, we find a finite orbital angular momentum (OAM, L) emerges. This is relatively small (≲0.05ħ) for the isolated d_{xy} and d_{xz/yz} sections of Fermi surface (Fig. 2e and Supplementary Fig. 3), but grows as large as 0.7ħ around the band crossings. Moreover, we find that the OAM is oriented either parallel or antiparallel to the corresponding spin angular momentum (SAM, S), and so this increase in OAM maximally enhances L·S, by a factor of ~14, at and in the vicinity of the hybridization gaps, comparable to the corresponding increase in spin splitting (Fig. 2b).
This therefore provides a natural basis to understand the large spin splittings inferred from magnetotransport^{29,30,31}, despite the small atomic spin–orbit interaction of SrTiO_{3}, in terms of an orbital Rashba effect^{32,33}. For isolated d_{xy} sections of Fermi surface, we find that the OAM of the inner and outer spinsplit branches have the same helicity, consistent with a weak spin–orbit coupling limit^{33}. For the d_{xz/yz} sections of Fermi surface, however, the inner and outer branches have opposite OAM, reflecting additional richness as compared with model systems such as noble metal surface states^{33}. This causes mixed helicities of the OAM around the inner branch of each Fermi surface sheet, as compared with a complete 2π winding for the outer branches (see arrows in Supplementary Fig. 4). Importantly, we find that the dominant interband interactions cause the winding direction of both the OAM and SAM of the outer d_{xy}derived Fermi surfaces to abruptly switch sign across the hybridization gaps. For several inner Fermi surfaces whose orbital character is strongly mixed, continuously evolving between d_{xy} and d_{xz/yz}like around the Fermi surface (Fig. 2c), this leads to strongly frustrated spin and orbital textures, rapidly canting from tangential to radial alignment as a function of Fermi surface angle (Fig. 2d,e). This is quite distinct from the functional form of conventional Rashba splitting^{21} and provides strong constraints for the influence of spin splitting on magnetism^{34} and superconductivity^{35}.
Manybody interactions
In Fig. 3, we further uncover a pronounced role of electron–phonon interactions on this complex hierarchy of electronic states. Unlike in bulkdoped SrTiO_{3}, where the Fermi energy is typically only a few meV and the electron–phonon interaction is thus nonretarded, the occupied widths of different subbands of the 2DEG range from almost zero up to values greater than the highest phonon frequency of ≈100 meV. This is an unusual situation, neither described by the adiabatic (ħω_{D}<<E_{F}) nor the antiadiabatic (ħω_{D}>>E_{F}) approximation, and points to a complex influence of electron–phonon coupling in this system. We extract the corresponding selfenergy, Σ_{e−ph}(ω)=Σ′(ω)+iΣ′′(ω), from our ARPES measurements of the lowest subband along the [11] direction (see methods), where we resolve an isolated band all the way up to E_{F}. The slope of Σ′ at the Fermi level yields an electron–phonon coupling strength of λ=0.7(1), while its broad maximum between ≈20−60 meV is indicative of coupling to multiple modes. Indeed, the experimentally determined selfenergy is in excellent agreement with a calculation within Eliashberg theory that assumes a coupling function α^{2}F(ω) proportional to the entire phonon density of states associated with the motion of oxygen and Ti ions^{36} and includes the realistic 2DEG electron density of states from our tightbinding calculation.
Together with a moderate correlationinduced mass enhancement of ≈1.4 that we estimate from a Kramers–Kronig transform of a Fermiliquid contribution to the imaginary part of the selfenergy, our analysis suggests an overall mass enhancement arising from manybody interactions of m*/m_{band}≈2.1, close to the values deduced for lightly doped bulk SrTiO_{3} from measurements of the electronic specific heat^{37} and optical spectroscopy^{38}. The nature of the electron–phonon coupling, however, is very different. In lightly doped bulk SrTiO_{3}, it is dominated by longrange coupling to longitudinal optical (LO) phonons as described by the Fröhlich model^{38,39}. This model predicts much weaker coupling to lowenergy modes than observed here, but a significantly stronger coupling to the highest LO phonon at 100 meV, as evident from a calculation employing coupling strengths from bulk SrTiO_{3} (refs 39, 40), which yield an electron–phonon selfenergy in clear contrast to our experimental findings (Fig. 3b).
The electric field that confines the 2DEG is known to dramatically reduce its dielectric constant^{7,41}. This, together with higher carrier densities as compared with bulk SrTiO_{3}, will lead to shorter electronic screening lengths for the 2DEG, explaining the observed suppression of longrange coupling to LO modes. The enhanced coupling to lowenergy phonons for the 2DEG instead leads to a pronounced kink in the dispersion of the d_{xy} subbands at an energy around 30 meV. We resolve these along both the [10] and [11] directions for the first two d_{xy} subbands. Crucially, the resulting enhanced quasiparticle mass, which we estimate as 1.1(2) m_{e} from our measured Fermi velocities, rectifies the discrepancy between the light masses around 0.6 m_{e} reported in earlier ARPES studies of SrTiO_{3} surface 2DEGs^{7,16} and recent quantum oscillation experiments that revealed effective masses typically around 1m_{e} (refs 42, 43).
Intriguingly, along [10] the kink energy coincides almost exactly with the crossing of the light d_{xy} and heavy d_{xz} subband states. This behaviour is well captured by our spectral function simulations calculated from our tightbinding bare dispersions and electron–phonon selfenergy. These illustrate a very different effect of electron–phonon interactions on the heavy compared with the light subbands of the 2DEG, the former coupling to phonons with frequencies ranging from below to above the bare bandwidth. Our calculations reveal that electron–phonon coupling essentially results in an overall bandwidth renormalization of these states, in agreement with our experimental data where we find the band bottom of the heavy state substantially above the value predicted by our model tightbinding calculations.
Discussion
The combination of electron–phonon coupling with orbital ordering therefore effectively pins the crossings of the d_{xy} and d_{xz/yz} subbands to the lowenergy peak in the phonon density of states. As demonstrated above, however, we additionally find orbitally enhanced spin–orbit splittings that become maximal around these band crossings. Our direct spectroscopic measurements, together with our theoretical calculations, therefore demonstrate how a cooperative effect of orbital ordering and electron–phonon coupling sets the relevant energy scale for dominant spin splitting in this system.
Together, this reveals an intricate hierarchy of interactions and orderings governing the lowenergy electronic structure of the SrTiO_{3} 2DEG (Fig. 4). Electrostatic screening in response to a surface or interface charge generates an electron accumulation layer confined to a narrow potential well. The resulting quantum size effects drive pronounced orbital ordering, creating multiple intersections of light d_{xy} and heavy d_{xz/yz}derived subband states. This leads to orbital mixing and induces a significant local orbital angular momentum, which in turn permits a pronounced spin splitting to emerge, despite modest atomic spin–orbit coupling. Manybody interactions, of strikingly different form to the bulk, enhance the quasiparticle masses of these spinsplit subbands, reduce their bandwidths and renormalize the energetic locations of their intersections, thus modulating their unconventional spin splitting.
Together, this interplay strongly entangles the charge, spin, orbital and lattice degrees of freedom of the SrTiO_{3} 2DEG on an energy scale 30 meV. This will dominate both its transport and thermodynamic properties. Already we have shown how this explains enhanced quasiparticle masses observed from quantum oscillations as well as signatures of spin splitting in magnetotransport, unifying electrical and spectroscopic measurements from surface and interfacebased SrTiO_{3} 2DEGs. More generally, it establishes how quantum size effects can dramatically manipulate the underlying electronic landscape of interacting electron liquids, setting the stage for engineering new emergent properties by dimensional confinement in transition metal oxides.
Methods
Angleresolved photoemission
ARPES measurements were performed at the CASIOPEE beamline of SOLEIL synchrotron, the SIS beamline of the Swiss Light Source, and beamline 10.0.1 of the Advanced light source using Scienta R4000 hemispherical electron analysers, and with base pressures <5 × 10^{−11} mbar. Single crystal SrTiO_{3} commercial wafers were cleaved in situ at the measurement temperature of T=20–30 K along notches defining a (100) plane. Measurements were performed on stoichiometric transparent insulating samples as well as very lightly Ladoped samples (Sr_{1−x}La_{x}TiO_{3} with x=0.001) to help avoid charging. 2DEGs were induced at the bare surface by exposure to intense UV synchrotron light^{7}. The samples were exposed to irradiation doses ≳1,000 J cm^{−2} to saturate the formation of the 2DEG, and we experimentally confirmed that saturation was reached before starting any of the measurements presented here. The data shown here was measured using spolarized photons of 51 or 55 eV, except for the Fermi surface maps shown in Supplementary Fig. 2 that used 43eV s and ppolarized light. All measurements were performed in the second Brillouin zone.
Selfenergy determination
To determine the electron–phonon selfenergy experimentally, we fit momentum distribution curves (MDCs) and energy distribution curves (EDCs) of the lowest d_{xy} subband measured along the [11] direction. We chose this band as its dispersion does not intersect that of the heavy d_{xz/yz} subbands up to the chemical potential along this direction (Fig. 1b), allowing us to perform a quantitative analysis over an extended energy range, free from complications associated with the hybridization of different subbands. The real part of the selfenergy, Σ′(ω), is given by the difference between our extracted dispersion and that of a ‘bare’ band. In order to derive a Kramers–Kronig consistent selfenergy, we take the cosine bare band shown in Fig. 3a, which includes a moderate bandwidth renormalization due to electron correlations. We extract the imaginary part of the selfenergy, Σ′′(ω)=Δk(ω)/2·∂ɛ/∂k, where Δk is the full width at half maximum of Lorentzian fits to MDCs, and ∂ɛ/∂k is the bare band dispersion. This results in an imaginary part that includes a contribution from electron–electron interactions, which we approximate by the expression for a twodimensional Fermi liquid Σ″_{e−e}(ω)=βω^{2}(1+0.53ln(ω/E_{F})). Subtracting this contribution with realistic parameter values of β=1.5 eV^{−1} and E_{F}=0.25 eV from our total extracted Σ′′ (see Fig. 3b) yields the imaginary part of the electron–phonon selfenergy.
Electronic structure and selfenergy calculations
To calculate the subband structure, we start from a relativistic density functional theory calculation of bulk SrTiO_{3} using the modified Becke–Johnson exchange potential and Perdew–Burke–Ernzerhof correlation functional as implemented in the WIEN2K programme^{44}. The muffintin radius of each atom R_{MT} was chosen such that its product with the maximum modulus of reciprocal vectors K_{max} become R_{MT}K_{max}=7.0. The Brillouin zone was sampled by a 15 × 15 × 15 kmesh. We downfold this using maximally localized Wannier functions to generate a set of bulk tightbinding transfer integrals, and then incorporate these into a 30unit cell supercell with additional onsite potential terms to account for band bending via an electrostatic potential variation. We solve this selfconsistently with Poisson’s equation, incorporating an electric fielddependent dielectric constant^{41}, to yield the bare band dispersions including Rashbatype spin splitting^{45} of the 2DEG. We stress that the total magnitude of the band bending is the only adjustable parameter, and yields a realistic electronic structure in good agreement with our spectroscopic measurements apart from our observed signatures of electron–phonon interactions that are not included at the level of density functional theory. To incorporate these, we calculate the selfenergy Σ_{e−ph}(ω) in an Eliashberg model,
where N(ɛ) is the bare electronic density of states determined from our tightbinding calculations and f(ɛ,T) and are the Fermi and Bose occupation factors. For the coupling function we use two different models. The blue line in Fig. 3b assumes proportional to the entire O and Tiderived phonon density of states from ref. 36, while the black line is a calculation for the coupling strengths given in refs 39, 40 for the three LO phonons that were found to dominate the electron–phonon interaction in lightly doped bulk samples. We then calculated the spectral function
where the bare band dispersion ɛ(k) is taken from our tightbinding calculation. To better compare with our experimental data in Fig. 3c, we project this onto different atomic orbitals, and include contributions from d_{xy} and d_{xz} but not d_{yz} orbitals to account for transition matrix elements in our experimental geometry. We additionally project the calculation onto different layers of our supercell, and incorporate an exponential attenuation of signal with depth below the surface in photoemission, assuming an inelastic mean free path of 5 Å, into our simulation. Finally, we convolve the simulated spectral function with a 2D Gaussian to account for an experimental energy and momentum resolution of 0.01 eV and 0.015 Å^{−1}, respectively.
Additional information
How to cite this article: King, P. D. C. et al. Quasiparticle dynamics and spin–orbital texture of the SrTiO_{3} twodimensional electron gas. Nat. Commun. 5:3414 doi: 10.1038/ncomms4414 (2014).
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Acknowledgements
This work was supported by the UK EPSRC (EP/I031014/1), the ERC (207901), the SNSF (200021146995), the Scottish Funding Council, The Thailand Research Fund (RSA5680052), Office of the Higher Education Commission, Suranaree Univerisity of Technology and the Japan Society for the promotion of Science (JSPS), through the ‘Funding Program for WorldLeading Innovative R&D on Science and Technology (FIRST Program)’, initiated by the council for Science and Technology policy (CSTP). P.D.C.K. acknowledges support from the Royal Society through a University Research Fellowship (UF120096). We acknowledge SOLEIL (beamline CASSIOPEE), the ALS (beamline 10.0.1) and SLS (SIS beamline) for provision of synchrotron radiation facilities, and in particular N.C. Plumb, M. Radovic′ and M. Shi (SLS) and P. Le Fèvre, F. Bertran and A. TalebIbrahimi (SOLEIL) for technical assistance. The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under Contract No. DEAC0205CH11231. We gratefully acknowledge C. Bell. C. Berthod, V. Cooper, A. Fête, H.Y. Hwang, M. Kim, J. Mannhart, D. van der Marel and J.M. Triscone for useful discussions.
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The experimental data was measured by P.D.C.K., S.M.W., A.T., A.d.l.T., T.E., P.B., W.M. and F.B., and analysed by P.D.C.K., S.M.W., A.T. and F.B. P.D.C.K. and M.S.B. performed the electronic structure calculations and A.T. performed the electron–phonon selfenergy calculations. S.K.M. maintained the ARPES endstation at the Advanced Light Source and provided experimental support. P.D.C.K. and F.B. were responsible for overall project planning and direction, and wrote the manuscript with input and discussion from all coauthors.
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King, P., McKeown Walker, S., Tamai, A. et al. Quasiparticle dynamics and spin–orbital texture of the SrTiO_{3} twodimensional electron gas. Nat Commun 5, 3414 (2014). https://doi.org/10.1038/ncomms4414
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Nature Communications (2019)
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