Abstract
Interplay of spin, charge, orbital and lattice degrees of freedom in oxide heterostructures results in a plethora of fascinating properties, which can be exploited in new generations of electronic devices with enhanced functionalities. The paradigm example is the interface between the two band insulators LaAlO_{3} and SrTiO_{3} that hosts a twodimensional electron system. Apart from the mobile charge carriers, this system exhibits a range of intriguing properties such as field effect, superconductivity and ferromagnetism, whose fundamental origins are still debated. Here we use softXray angleresolved photoelectron spectroscopy to penetrate through the LaAlO_{3} overlayer and access charge carriers at the buried interface. The experimental spectral function directly identifies the interface charge carriers as large polarons, emerging from coupling of charge and lattice degrees of freedom, and involving two phonons of different energy and thermal activity. This phenomenon fundamentally limits the carrier mobility and explains its puzzling drop at high temperatures.
Introduction
Coupling of the electron and lattice degrees of freedom in solids through electron–phonon interaction (EPI) is a key concept in electron transport and many other phenomena of condensed matter physics. An electron moving in the lattice can displace atoms from their equilibrium positions in response to the EPI. Such an electron (or hole) dragging behind a local lattice distortion—or phonon ‘cloud’—forms a composite charge carrier known as polaron^{1,2}. The increased effective mass m* of this quasiparticle (QP) fundamentally limits its mobility μ∝1/m* beyond incoherent scattering processes. The polarons are key players in many technological devices, a widespread example of which are highelectron mobility transistors (HEMTs) utilized in highfrequency devices such as mobile phones. In typical HEMTs, a donor layer of ndoped AlGaAs injects electrons into the channel layer of intrinsic GaAs where, escaping scattering on the dopant impurities, the electrons are limited in their mobility only by the polaronic coupling enhanced by spatial confinement in the GaAs quantum well (QW)^{3}.
Angleresolved photoelectron spectroscopy (ARPES) is a unique method to measure the single particle spectral function A(ω,k) in crystalline solids resolved in electron energy ω and momentum k. Containing all manybody (electron–electron, electron–phonon, and so on) interactions, A(ω,k) reveals the formation of polarons by a characteristic peakdiphump (PDH) lineshape, where the sharp peak corresponds to a QP and the broad hump, extending to higher binding energies, corresponds to the cloud of entangled phonons with frequencies ω_{0} (refs 1, 2). However, the extreme surface sensitivity of conventional ARPES with photon energies hν below ∼100 eV sets the buried interfaces out of its reach. The crucial feature of our experiment is the use of softXray ARPES (SXARPES) operating in the hν range of hundreds eV (for a recent review see ref. 4). The longer photoelectron mean free path enables SXARPES to penetrate through the top layers and access A(ω,k) at buried interfaces.
Complex oxide interfaces are presently at the forefront of fundamental research in view of their enhanced functionalities achieved by exploiting electron correlations^{5,6}. The twodimensional electron system (2DES) in LaAlO_{3}/SrTiO_{3} (LAO/STO)^{7} is confined within a narrow region of a few nanometres on the STO side^{8,9,10}, where the mobile electrons populate the t_{2g}derived d_{xy}, d_{xz} and d_{yz}states of Ti ions acquiring reduced valence compared with bulk Ti^{4+}. Confinement in the interface QW further splits these states into a ladder of subbands^{8,11,12,13,14}. This complex energy structure based on the correlated 3d orbitals, very different from conventional semiconductor heterostructures described as free particles embedded in the meanfield potential, is the source of a rich and nontrivial phenomenology. Here high 2DES mobility μ_{2DES} typical of uncorrelated ESs coexists with superconductivity^{5,6}, ferromagnetism^{15,16}, large magnetoresistance^{17} and other phenomena typical of localized correlated electrons. Other intriguing puzzles in this intricate physics are why μ_{2DES} measured in transport falls short of estimates based on meanfield theories, and what causes the dramatic drop of μ_{2DES} with increase of temperature^{18}.
Here, we directly access the nature of the LAO/STO interface carriers through their A(ω,k) measured by SXARPES at ultrahighenergy resolution. We discover that the LAO/STO interface forms a polaronic metal state involving at least two active phonons. While polaronic coupling to hard LO3 phonons fundamentally limits μ_{2DES} at low temperatures, coupling to soft TO1 phonons with increasing temperature provides the microscopic mechanism of μ_{2DES} drop observed in transport.
Results
Spectroscopic signatures of the polaronic metal state
Our LAO/STO(001) samples with an LAO overlayer thickness of ∼18 Å corresponding to 5 unit cells (u.c.) were grown using Pulsed Layer Deposition (PLD), and subsequently annealed in oxygen atmosphere to minimize the concentration of oxygen vacancies (V_{O}s) and the related extrinsic charge carriers^{7,19}. SXARPES with its crucial advantage of enhanced probing depth is ideally suited to study this buried system where the 2DES only develops with the LAO layer thickness >3 u.c. (ref. 20) The extremely small 2DES signal, however, has to be boosted using resonant photoemission^{21} locked to the interface Ti ions^{14,22,23}. For details of the sample growth and SXARPES experiment, see Methods.
Our lowtemperature experimental data set in Fig. 1 was measured at 12 K using spolarized Xrays (the parallel ppolarization data are given in the Supplementary Note 1). The resonance map of (angleintegrated) photoemission intensity, Fig. 1a, was recorded under variation of hv across the Ti 2p absorption edge ∼460 eV. We identify there the 2DES signal at the Fermi level E_{F} blowing up near the two Ti^{3+} L_{3} and L_{2}resonances and vanishing everywhere else (we notice that strong suppression of the ingap states around −1.2 eV, related to the V_{O}s (refs 14, 22, 23), confirms the prevalence of the intrinsic interface charge carriers). Tuning hv onto the stronger L_{3}resonance produces the Fermi surface (FS) map in Fig. 1b where, by comparison with the superimposed theoretical FS contours, we recognize the manifold of merged circular d_{xy}derived FS sheets and the elliptical d_{yz}sheet extending in the k_{x}direction^{14,23}.
The ARPES images measured along the ΓX (k_{y}=0) line of the square 2D Brillouin zone (BZ) at the L_{3} and L_{2}resonances are shown in Fig. 2a,b, respectively. With a highenergy resolution of 40 meV, these images resolve individual interface bands. The use of spolarization selects the d_{xy} and d_{yz}derived states, which are antisymmetric relative to the ΓX line^{14} (although the selection rules are slightly relaxed by remnant structural distortions). By comparison with the overlaid E(k) dispersions calculated for LAO/STO interface with pseudoselfinteraction correction DFT (see Methods) one can recognize the lower d_{yz}band with its flat dispersion. Already at this point, we note signs of a QP interaction, which reduces the band dispersion compared with the overlaid DFT prediction as characterized by an effective mass ratio of m*/m_{0}∼2.5. The d_{xy}bands are not visible due to vanishing matrix elements, but the lowest d_{xy}band appears as two bright spots where it hybridizes with the d_{yz}band.
The most striking visual aspect of the experimental E(k) is, however, the two vertical waterfalls extending down from these highintensity d_{xy} spots. In Fig. 2c,d we show the energy distribution curves (EDCs)—that is, ARPES intensity as a function of binding energy for a given k—extracted from the images in Fig. 2a,b, respectively. These EDCs reveal, remarkably, a pronounced PDH structure of A(ω,k), where the peak reflects the QP and the hump at ∼118 meV below the peak its coupling to bosonic modes (such as magnon, plasmon, phonon) whose nature will be identified later on.
We note in passing that the EDC representing the whole d_{xy}band, Fig. 2c, exhibits a smaller but broader QP peak in comparison to its d_{yz}counterpart in Fig. 2d. Their nearly equal integral QP weight indicates that, nontrivially, the bosonic coupling is quite insensitive to different spatial distribution of the d_{xy} and d_{yz}states^{8}. The larger broadening of d_{xy}EDC can reflect larger defect scattering, because the lowest d_{xy}state is localized closest to the interface where the concentration of defects generated by the nonequilibrium PLD growth is maximal, whereas the d_{yz}state extends deeper into the defectfree STO bulk. This observation is consistent with the recent Shubnikov–de Haas experiments^{15}, which have found smaller d_{xy}mobility compared with d_{xz/yz}one.
We will now address the nature of the involved bosonic modes. Considering the extremely small ferromagnetic response of the LAO/STO interface^{15}, the magnons can safely be ruled out. Plasmons can also be excluded since the energy of the hump does not depend on the interfacial carrier concentration n_{s} varied via manipulation of V_{O}s (see Supplementary Note 2) while the plasma frequency ω_{p} is proportional to . These bosonic modes can therefore be assigned to phonons coupling to electron excitations and forming polarons^{24,25,26}. The hump apex, located at ∼118 meV below the QP peak, identifies the main coupling phonon frequency ω_{0}′. The 2DES at the LAO/STO interface realizes therefore a polaronic metal state.
To identify the phonon modes forming the observed polaron, we used DFT to calculate the phonon dispersions (see Methods) for cubic bulk STO at different electron doping concentrations n_{v}; the results are shown in Fig. 3a. Although these initial calculations are for cubic STO, we will address shifts of the phonon frequencies at the cubic to tetragonal phase transition when discussing the temperature dependence. The response of the calculated dispersions quantifies the EPI strength as a function of phonon mode and q. With the actual electron density distribution^{8,9,10}, our n_{s} measured by the Luttinger area of the experimental FS roughly corresponds to n_{v}=0.12 electrons per u.c. Since the carriers reside on the STO side of the LAO/STO interface, we expect these results to be relevant for the interface as well. The polaron can be associated with the hard longitudinal optical phonon LO3, which is the only mode available in this highenergy range and, moreover, has the largest coupling constant λ among all LO modes^{27,28} as observed by Raman^{29} and neutron spectroscopy^{30}. At finite q vectors, the LO3 mode represents a breathing distortion of the octahedral cage around a Ti site, Fig. 3b, which is typical of polaron formation driven by the (Holsteintype) shortrange EPI, barely sensitive to electron screening in a metallic system. This character of EPI is confirmed by the increase of with increase of q away from the Γpoint. Recent optical studies on bulk STO^{31} have not only confirmed the involvement of the LO3 phonon in EPI but also found the corresponding effective mass renormalization m*/m_{0} ∼3.0 close to our value. This mode has also been observed by ARPES on (doped) bare STO(001)^{32,33,34} with the strength of the polaronic structure depending on n_{s}. We note that, our theoretical LO3 energy of ∼100 meV perfectly matches that found experimentally for the bare STO bulk and surface, but it differs from our ω_{0}′∼118 meV measurement at the LAO/STO interface. To gain some insight on this discrepancy, we investigated the possible role of strain by performing additional calculations with the STO inplane lattice constants constrained to that of LAO. However, only a minor frequency shift to 102 meV was found. This implies that other interfacial factors such as the electric field, phonon coupling across the interface, propagation of the LAO distortion into STO^{35}, structural changes due to electrostatic doping^{36} or weaker coupling to additional phonon modes could also play a role. We also note that the EPI at LAO/STO is enhanced by the tight 2D electron confinement in the interface QW^{2,3}, which was recently evidenced by huge oscillations of thermopower as a consequence of large phonon drag^{37}.
Next, we perform a theoretical analysis of the PDH structure to estimate the strength of the EPI governing the polaron formation. With the total A(ω,k) spectral weight normalized to unity, we define Z_{0} as the integral weight of the QP part. The observed A(ω,k) of a polaron with momentum k can be expressed as a sum of two terms^{1,2}
The first term represents the sharp QP peak with the dispersion E(k) and effective mass m* resulting from the renormalization of the noninteracting singleparticle band with dispersion ɛ(k) and mass m^{0}. The second term arises due to phonons coupling to the excited photohole. The EPI has a twofold effect on A(ω,k): it reduces Z_{0} below unity, and builds up , with its dispersion following ɛ(k), as a Frank–Condon series of phonon peak replicas at energy separations ω_{n}=nω_{0} from the QP peak, where ω_{0} is the phonon frequency and n indexes the replicas.
For our analysis, we used the angleintegrated EDC at the L_{3}resonance (Fig. 2c), which is dominated by the d_{xy}intensity and thus insensitive to the dispersion effects in the d_{yz}band (certain admixture of the latter is not important because its dispersion range is much smaller than our experimental resolution of 40 meV and, moreover, the d_{xy} and d_{yz}bands have the same peaktohump relative weight). Gaussian fitting of its QP peak yields Z_{0}∼0.4. For the shortrange EPI, forming the LO3 polaron, exact diagrammatic quantum Monte Carlo (QMC) calculations^{38} have shown that the relation m*/m_{0}∼1/Z_{0} holds with high accuracy. With the determined Z_{0}, this yields m*/m_{0}=2.5, which coincides with the value extracted above from the band dispersions, leaving not much room for electron correlations to contribute to the band renormalization. The latter is confirmed by theoretical analysis of tunnelling experiments^{39} where the local electron correlations were described with the Hubbard parameter U. According to (ref. 40), already at U=4t (t is the hopping integral) the density of states significantly renormalizes and forms a pseudogap at E_{F}. The absence of the latter in the tunnelling spectra^{39} indicates U<4t. Then the renormalization expected from the correlation effects (U_{c}=8t is the critical Mott value) is <1.3, which is much smaller than our experimental m*/m_{0}. Hence, the mass renormalization at the LAO/STO interface is dominated by EPI.
Furthermore, the coupling strength λ for our shortrange EPI can be estimated as λ∼1−Z_{0}, which yields λ≈0.6. This value is wellbelow the limit λ∼0.9 separating the weak and strongcoupling regimes, which implies that we observe large polarons, where the lattice distortion extends over several unit cells^{1,2,41}. This is perfectly consistent with the clear dispersion of the hump, which tracks the noninteracting ɛ(k) dispersion of the d_{yz}band in Fig. 2d. Importantly, the small λ also excludes that selftrapping of small polarons^{42,43} can be responsible for the discrepancy between the observed mobile charge and the 0.5 electrons per u.c. required for full compensation of the polar field in the LAO overlayer^{5,6,43}. However, the EPI can assist charge trapping on shallow defects^{44} created by nonequilibrium PLD growth or V_{O}s, in addition to the deep level trapping.
The polaronic reduction of μ_{2DES} fundamentally limits the application potential of the STObased heterostructures. As illustrated in Supplementary Note 2, this limit can possibly be circumvented through manipulation of V_{O}s. The V_{O}s inject into the 2DES extrinsic charge carriers^{43}, which increase the electron screening and thus reduce the EPI. This trend is consistent with the recent study at bare STO(100) surfaces^{33} and, furthermore, explains the recent results on the γAl_{2}O_{3}/STO^{45}, where the V_{O}s have not only increased n_{s} but also μ_{2DES} by almost 2 orders of magnitude. On the other hand, the V_{O}s can assist the EPI due to charge trapping on shallow defects^{44} and also increase the defect scattering rate, both effects counteracting the above positive trend. Further experiments on oxygendeficient LAO/STO will allow a better understanding of the role of V_{O}s and ways to optimize μ_{2DES}.
Temperature dependence of polaronic effects
An intriguing peculiarity of the charge carriers at the LAO/STO interface is the drop of μ_{2DES} by more than 1 order of magnitude as the temperature increases above 200 K (ref. 18). To unveil the underlying microscopic mechanism, we measured the temperature dependence of the L_{3} angleintegrated EDC from Fig. 2c (the angle integration makes our analysis robust against extrinsic thermal scattering of highenergy photoelectrons, which averages the ARPES signal in kspace^{46}). These results are shown in Fig. 4a in comparison with experimental temperature dependence of μ_{2DES} in Fig. 4b, characteristic of the LAO/STO samples^{18}, which was derived from Hall effect measurements. The experimental A(ω,k) in Fig. 4a shows that with increase of temperature, first, the polaronic hump broadens and propagates to higher binding energies. This trend is accompanied by increase of the QP peak width, Fig. 4d. These effects can be attributed to incoherent scattering on thermally populated phonon modes. A striking effect beyond this weightconserving mechanism is, however, that the QP peak loses its integral weight Z_{0} (Fig. 4c) and completely dissolves in the phonon hump of A(ω,k) towards 190 K (Fig. 4a). The decrease of Z_{0}, following the trend of the temperaturedependent μ_{2DES}, unambiguously signals that the QP spectral weight leaks to soft phonon modes, different from the hard LO3, with concomitant further increase of m*. This phenomenon provides the microscopic mechanism of the puzzling μ_{2DES} drop with increasing temperature^{18,41}. Interestingly, ARPES on the TiO_{2}(001) surface shows similar temperature effects^{24} despite different crystallographic structure of TiO_{2}. In line with our results, an optical study of bulk STO^{31} has also revealed reduction of the Drude weight with temperature. Surprisingly, the bare STO(100) surface at large n_{v} (ref. 32) does not show any systematic temperature effects in the QP weight; clearly, the LAO overlayer and 2DES significantly alter the EPI in the LAO/STO system.
Numerical analysis of the temperaturedependent ARPES spectral shape allows us to identify the frequency ω_{0}″ of the soft phonon mode dominating the EPI. From the normalized EDCs, Fig. 4a, we evaluate the temperature dependence of the spectral weight Z_{0}(T) by Gaussian fitting of the QP peak. The resulting Z_{0}(T) in Fig. 4c can then be fitted with the analytic formula for the independent boson model^{1}
where is the Bose filling factor, which describes the population number as a function of the mode frequency ω_{0} at T, and I_{0} is the modified Bessel function. This equation, neglecting momentum dependence and thus valid for our momentumintegrated EDCs, describes transfer of the spectral weight from the QP peak to the hump with temperature. The constant gfactor, measuring the EPI strength, was set to 0.95 to reproduce the experimental value Z_{0}(12 K)∼0.4. Fitting Z_{0}(T) in the temperature range below ∼120 K yields ω_{0}″≈18 meV (solid line in Fig. 4c), while the highT range yields ω_{0}″∼14 meV (dashed line in Fig. 4c). This crossover of ω_{0}″ can be linked to the secondorder tetragonal to cubic phase transition in STO at 105K (refs 32, 47). One may therefore argue that the μ_{2DES} drop in LAO/STO has structural origin. Interestingly, the fastest change of the QP peak width in Fig. 4d falls into the same temperature region. All other possible temperature effects in the QP linewidth, including the Bloch–Grüneisen contribution due to acoustic phonons, are weaker for our system dominated by the optical phonon, and obscured by the phase transition and limited energy resolution. We note that, strictly speaking, our Z_{0}(T) model (2) had to include EPI with both low and highenergy phonons, contributing to the total EPI. However, the result would not change significantly because the Bose filling factor N, determining the Z_{0}(T) dependence, is most sensitive to small ω_{0}. In other words, the Z_{0}(T) dependence is most sensitive to the lowenergy sector of the phonon spectrum.
Next we return to our DFT calculations to identify the observed soft ω_{0}″ phonon mode. The phonon dispersions of the cubic (highT) phase, Fig. 3a, show a multitude of lowenergy modes. The lowest energy LO mode (LO1), which was previously linked with kinks in ARPES dispersions^{28}, has an energy of ∼22 meV. This energy is considerably higher than our highT fitted ω_{0}″∼14 meV and, moreover, our calculations do not show a significant frequency difference between the cubic and tetragonal phases of STO. These facts, together with the reported small coupling constant λ (ref. 27), make the involvement of this LO1 mode unlikely. In Fig. 3a we see, however, the TO modes strongly affected by the electron doping n_{v} due to enhanced screening. The TO1 mode, sketched in Fig. 3c, is a polar mode whose instability in undoped STO leads to its quantumparaelectric behaviour^{48,49}. Its frequency rapidly increases and becomes real with increase of n_{v}, stabilizing above our actual n_{v}∼0.12 electrons per u.c. at ω_{TO1}∼13.7 meV at the Γ point. This ω_{TO1} matches well with our ω_{0}″∼14 meV fitted in the highT range. Turning to the lowT range, previous calculations on undoped STO have shown that the tetragonal phase transition increases ω_{TO1} (ref. 50). Our computations with n_{v}=0.12 electrons per u.c. for the tetragonal phase indeed show ω_{TO1} to shift to 15.3 meV for the doubly degenerate mode and 18.1 meV for the nondegenerate mode along the octahedral rotation axis, which is in good agreement with our lowT fitted value ω_{0}″∼18 meV. Octahedral rotations different from this bulk tetragonal distortion were observed at the LAO/STO interface due to electrostatic doping^{35,36}, but since they are combinations of distortions similar to the one studied here, they are expected to also lead to an upwards shift of the TO1 frequency. Based on the t_{1u} symmetry of the TO1 mode, its good agreement with the experimental ω_{0}″ and its sensitivity to the cubic to tetragonal phase transition, we associate the experimental soft phonon with TO1. Moreover, significant polaronic coupling to this polar mode is consistent with gigantic dielectric constant ɛ_{0} of STO caused by large polar ionic displacements under electric field in this material on the verge of a ferroelectric instability^{30}. The TO1 mode is associated with longrange EPI, as evidenced by the increase of towards the Γpoint. The hard LO3 and soft TO1 modes involved in the polaron formation are therefore associated with opposite types of EPI. The EPI strength in the latter case can be estimated from the Fröhlich model which, although strictly valid only for the polar LO modes, represents the only viable model to find the longrange coupling constant α. Based on exact diagrammatic QMC calculations^{51} and Z_{0} at the top of our temperature range, we estimate α≈2 for the TO1 mode. Similar to the above LO3 phonon, this value stays below the weak to strongcoupling crossover at α≈6, which implies that the TO1 phonon is also consistent with the large polaron scenario. We note that if LAO/STO superconductivity is driven by a phonon mechanism^{52}, it can be related to the discovered polaronic activity. While the hard LO3 phonon energy much exceeds the energy scale of the superconducting transition at 0.3 K, involved in the electron pairing may be the soft TO1 phonon.
The 2DES at the LAO/STO interface realizes thus a polaronic metal state involving at least two phonons with different energies and thermal activity. The hard LO3 phonon at ω_{0}′∼118 meV, associated with shortrange EPI, is directly resolved as the characteristic hump in the experimental A(ω,k). It sets the fundamental limit of μ_{2DES} at low temperature, and exhaustively accounts for the m* renormalization without notable effects of electron correlations. Another soft phonon, likely the TO1 one associated with longrange EPI, changes its frequency from ω_{0}″∼18 to 14 meV across the phase transition in STO. This phonon causes a dramatic fading of the QP weight with temperature as the microscopic mechanism behind the μ_{2DES} drop observed in transport. The two phonons form a large polaron characterized by a lattice deformation extending over several unit cells. Physically, the involvement of both LO3 and TO1 phonons in the polaronic state means that the corresponding lattice distortion combines both breathing and polar atomic displacements. The polaronic activity is typical of oxide perovskites, reflecting their highly ionic character and easy structural transformations^{30,42}. Our discovery may have implications for other related oxide systems, including LAO/STO interfaces with different crystallographic orientations. In a methodological perspective, we have demonstrated the power of the newly emerging experimental technique of ultrahighresolution SXARPES to retrieve information about polaronic effects at buried interfaces in the most direct way as embedded in oneelectron A(ω,k).
Methods
Sample preparation
Our LAO/STO samples were grown using PLD (for details of the growth procedure see refs 19, 22) and subsequently annealed in oxygen atmosphere with a pressure of 200 mBar at 550 °C for 1 h to minimize the concentration of V_{O}s. These vacancies manifest themselves in ARPES spectra as characteristic dispersionless ingap states at a binding energy of ∼1.2 eV (refs 14, 22) which, similar to bare STO, grow with exposure to Xrays. Our resonant spectra in Fig. 1a show only traces of such spectral structures enhanced at the Ti^{3+} L_{3} and L_{2}resonances, indicating negligible concentration of V_{O}s. Measurements on samples with various n_{s} varied through the V_{O}s (Supplementary Note 2) demonstrated constant energy of the 118meV spectral peak, excluding its plasmonic origin. The temperaturedependent mobility data were derived from standard Hall effect and conductivity measurements performed in the van der Paw geometry on square samples.
SXARPES experiment
SXARPES experiments were performed at the ADRESS beamline of the Swiss Light Source, Paul Scherrer Institute, Switzerland^{53}. The experimental geometry allows symmetry analysis of the valence states using variable linear polarizations of incident Xrays. The experiments are normally performed at low sample temperatures ∼12 K to quench thermal scattering of highenergy photoelectron destructive for the coherent kresolved spectral component^{46}. The combined (beamline and analyser) energy resolution was set to 80 meV for measurements of the FS, and to 40 meV for highresolution measurements of the band dispersions. Such resolution achieved for an interface buried behind a ∼18Åthick overlayer presents the forefront of nowadays SXARPES instrumentation. The temperature dependence was measured with increasing temperature to avoid possible hysteresis effects^{54}.
Firstprinciples calculations
Band structure calculations for the LAO/STO interface were performed using the pseudoselfinteraction correction method^{55}, with a plane wave basis set and ultrasoft pseudopotentials. This ab initio approach corrects the main deficiencies of basic density functional theory for a vast range of oxides^{8}. The theoretical band structure and the FS were calculated for an interfacial carrier density of 0.115 electrons per u.c. This value is consistent with those determined using Hall effect measurements for the LAO/STO interface. The corresponding Fermi momentum k_{F} of the d_{yz}states is in good agreement with the experimental value of ∼0.29 Å^{−1} determined as the highest band dispersion point.
DFTbased phonon calculations under different electron doping were performed using the VASP code^{57}. The electron count was adjusted while adding a compensating background charge. We used the PBEsol functional, which gives reliable lattice parameters and phonon frequencies^{56}, and PAW potentials^{57} with Sr(4s,4p,5s), Ti(3p,3d,4s) and O(2s,2p) valence shells. Phonons were then computed using the frozen phonon approach^{58}. For further details, see ref. 50.
Additional information
How to cite this article: Cancellieri, C. et al. Polaronic metal state at the LaAlO_{3}/SrTiO_{3} interface. Nat. Commun. 7:10386 doi: 10.1038/ncomms10386 (2016).
Change history
26 February 2016
In the original PDF version of this Article, which was published on 27 January 2016, the publication date was incorrectly given as 27 January 2015. This has now been corrected in the PDF; the HTML version of the paper was correct from the time of publication.
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Acknowledgements
We thank P.R. Willmott, J.H. Dil, R. Claessen, J.M. Triscone, F. Baumberger, F. Bisti, Z. Wang, M. Radović, J. Mesot and N.A. Spaldin for fruitful discussions, and X. Wang for software support of the experiment. Parts of this research were supported by the ImPACT Program of the Council for Science, Technology and Innovation (Cabinet Office, Government of Japan), the Croatian QuantiXLie Center of Excellence, and the ETH Zürich and ERC Advanced Grant program, No. 291151.
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C.C. and V.N.S. have performed the experiment supported by V.A.R. and T.S. and processed the data. C.C. has grown the samples. V.N.S. and A.S.M. have developed the scientific concept. A.S.M. has performed the polaronic analysis supported by O.S.B. and N.N. A.F. and U.A. have calculated the electron and phonon band structures, respectively. All authors have discussed the results and interpretations, as well as the manuscript written by V.N.S. and A.S.M. with contributions of C.F., U.A. and C.C.
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Supplementary Figures 12, Supplementary Notes 12 and Supplementary References. (PDF 233 kb)
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Cancellieri, C., Mishchenko, A., Aschauer, U. et al. Polaronic metal state at the LaAlO_{3}/SrTiO_{3} interface. Nat Commun 7, 10386 (2016). https://doi.org/10.1038/ncomms10386
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DOI: https://doi.org/10.1038/ncomms10386
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