Oxide superlattices and microstructures hold the promise for creating a new
class of devices with unprecedented functionalities. Density-functional studies
of the recently fabricated superlattices of lattice-matched perovskite
titanates (SrTiO3)n/(LaTiO3)m reveal a classic wedge-shaped potential
originating from the Coulomb potential of a charged sheet of La atoms. The
potential in turn confines the electrons in the vicinity of the sheet, leading
to an Airy-function localization of the electron states. Magnetism is
suppressed for structures with a single LaTiO3 monolayer, while the bulk
antiferromagnetism is recovered in the structures with a thicker LaTiO3, with a
narrow transition region separating the magnetic LaTiO3 and the non-magnetic
SrTiO3

Popovic, Z. S.

Satpathy, S.

English

arXiv

http://arxiv.org/ftp/cond-mat/papers/0409/0409281.pdfOxide superlattices and microstructures hold the promise for creating a new class of devices with unprecedented functionalities. Density-functional studies of the recently fabricated superlattices of lattice-matched perovskite titanates (SrTiO3)n/(LaTiO3)m reveal a classic wedge-shaped potential originating from the Coulomb potential of a charged sheet of La atoms. The potential in turn confines the electrons in the vicinity of the sheet, leading to an Airy-function localization of the electron states. Magnetism is suppressed for structures with a single LaTiO3 monolayer, while the bulk antiferromagnetism is recovered in the structures with a thicker LaTiO3, with a narrow transition region separating the magnetic LaTiO3 and the non-magnetic SrTiO3.Acknowledgment. This work was supported by the U. S. Department of Energy under Grant
No. DE-FG02-00ER45818

Popovic, Zoran S.

Satpathy, Sashi Sekhar, 1956-

University of Missouri: MOspace

1Airy-function electron localization in the oxide superlatticesZ. S. Popovic* and S. SatpathyDepartment of Physics, University of Missouri, Columbia, MO 65211, USAOxide superlattices and microstructures hold the promise for creating anew class of devices with unprecedented functionalities. Density-functionalstudies of the recently fabricated superlattices of lattice-matched perovskitetitanates (SrTiO3)n/(LaTiO3)m reveal a classic wedge-shaped potentialoriginating from the Coulomb potential of a charged sheet of La atoms. Thepotential in turn confines the electrons in the vicinity of the sheet, leadingto an Airy-function localization of the electron states. Magnetism issuppressed for structures with a single LaTiO3 monolayer, while the bulkantiferromagnetism is recovered in the structures with a thicker LaTiO3,with a narrow transition region separating the magnetic LaTiO3 and thenon-magnetic SrTiO3.PACS: 73.20.-r, 73.21.-bWhile the growth of the atomically abrupt, lattice-matched interfaces betweensemiconductors has long been perfected and such structures are now widely used inelectronic devices, the growth of such high-quality oxide structures has not been possible fora long time. In a recent work, Ohmoto et al.1 reported the growth of atomically precise,lattice-matched superlattices made up of alternating layers of LaTiO3 and SrTiO3. What isstriking is that the quality of the interfaces easily matched the quality of similarsemiconductor structures. This has raised the hope of being able to grow quality structuresusing other oxides2 as well, opening them up for new physics and potentially novel deviceconcepts.An understanding of the interface electron states played a key role in the development ofsemiconductor electronics and revealed many novel features not found in the bulkconstituents, e. g., the formation of the two-dimensional electron gas and the subsequentdiscovery of the Quantum Hall effect. Analogously, new electronic properties may beanticipated for the oxide heterostructures, where strong correlations between electrons willundoubtedly lead to new physics, not found in the semiconductor counterparts. In this paper,we study the interface electronic structure of the recently fabricated perovskite titanatesuperlattices using calculations based on the first-principles density-functional theory. Aninteresting finding is the wedge-shaped potential well caused by the positive sheet-chargedensity at the interface and the Airy function localization of the electron states in thepotential well. The Airy function localization has been observed1 in the form of anexponential-like spread of the Ti3+ fraction near the interface. To our knowledge, this is thefirst time that a wedge-shaped potential has been found in any system, which opens up thepossibility for studying the physics of a new two-dimensional Airy gas.3The bulk electronic structure of both the constituents of (SrTiO3)n/(LaTiO3)m, the subjectof this paper, has been well studied. Both form in the perovskite structure, with a three-2dimensional network of corner-sharing TiO6 octahedra.4,5 While SrTiO3 is a band insulatorwith an empty Ti (d0) bands, LaTiO3, in contrast, is a Mott insulator with Ti (d1) occupancy.In a simple picture, the latter can be thought of within the context of the half-filled Hubbardmodel, leading to an antiferromagnetic insulator as observed. However, the insulating stateis very quickly destroyed with the introduction of a small concentration of holes via theaddition of extra oxygen6 or via Sr substitution (with as little as x ≈ 0.05 for La1-xSrxTiO3).7The quick destruction of the insulating state by the hole doping is reminiscent of theNagaoka state where a single hole destroys both the antiferromagnetism and the insulatingbehavior in the half-filled Hubbard model.8The behavior of the electrons in the superlattices such as (SrTiO3)n/(LaTiO3)m is expectedto be fundamentally different from that in the random alloy with the same composition. Asopposed to some effective average potential seen by the conduction electrons (the stripped-off La (d1) electrons) in the alloy, the potential in the heterostructure is due to a coherentsuperposition of the Coulomb potentials of the La ions organized on lattice planes. The La(d) electrons become detached from the sheet but become confined in the wedge-shapedCoulomb potential of the sheet charge. Strong screening due to the embedding SrTiO3material weakens the Coulomb potential, spreading thereby the bound electron into severallayers of SrTiO3 on either side of the interface. A Hartree-Fock model study by Okamotoand Millis9 has shown that this delocalization of the electron state leads to a metallicbehavior.To study the electronic structure of the (SrTiO3)n/(LaTiO3)m superlattices, we haveperformed density-functional calculations within the local-spin-density approximation(LSDA) as well as the “LSDA+U” method.10 Since the LSDA+U method is necessary todescribe the correlation physics for bulk LaTiO3, albeit in an approximate way,11,12 themajority of calculations reported here are performed with this method. The linear muffin-tinorbitals (LMTO) method13 is used throughout to solve the Kohn-Sham equations. Forsuperlattices with a thick LaTiO3 part, we have used the full octahedral distortions of bulkLaTiO3, while for superlattices with a single LaTiO3 layer, we have used undistortedoctahedral structure of the embedding SrTiO3.Consider the (SrTiO3)n/(LaTiO3)1 superlattice grown along the (100) direction as in theexperiment. Here a single LaTiO3 layer is embedded in a thick SrTiO3 part and with the La(d1) electrons stripped from it, the layer behaves like a two-dimensional sheet of positivecharge. The Coulomb potential of the sheet charge is determined using elementaryelectrostatics, where according to the Gauss’ law, the electric field is given by E = σ ε/ 2 0,where σ = 1 |e|/S is the sheet charge density and S is the interface surface area per La atom.The field is reduced due to screening and leads to a wedge-shaped potential with theminimum located at the plane of the sheet charge. This potential in turn binds the La (d1)electron in its Coulomb field.The variation of the potential seen by the electron in the solid may be calculated from thevariation of some reference energy in the density-functional calculation. A convenientreference energy that we have successfully used for the semiconductor interfaces is the cell-3averaged point-charge Coulomb potential V.14 It may be calculated by first averaging thepotential parallel to the plane near the interface (planar-averaged potential) and then byaveraging over a period normal to the plane. We can, alternatively, calculate V by firstaveraging over the volume of the Wigner-Seitz atomic spheres:             V qsqr riiiji jj= +−∑32 | |/, (1)and then by averaging over all spheres with a weight factor proportional to their volumes: 15V Vi ii ii= ∑ ∑Ω Ω/ , where Ωi is= 4 33π /  is the sphere volume, si  is the sphere radius, ri  isthe sphere position, and qi  is the total charge, nuclear plus electronic, and i is the sphereindex. In Eq. (1), the first term is the sphere average of the potential of the point chargelocated at the center of the muffin-tin sphere and the second term is the Madelung potentialdue to all other spheres in the solid.       (a) (b) (c)Figure 1 (color online) The (SrTiO3)n/(LaTiO3)1 superlattice structure (a) with amonolayer of LaTiO3 embedded in bulk SrTiO3. The positively charged Lasheet produces a wedge-shaped potential (b) following Gauss’ law inelementary electrostatics. In the bottom part of (b), dots denote the cell-averaged electrostatic potential V(z) calculated from Eq. (1) with distance zfrom the interface in units of the SrTiO3 monolayer thickness, while the solidline is a fit corresponding to a uniformly charged sheet. The potentialconfines the stripped-off La (d1) electrons in the interface region (c). Thecontours show the electron charge-density, integrated between theconduction bottom and the Fermi energy, indicating the electron leakage intoabout three Ti layers on either side of the interface. Contour values are:ρ ρ δn n= 0 10 , where ρ0 3 33 4 10= −. /x e Å , δ = 0.31, and n labels the contours.The top part of (b) shows the electron density n(z), which is integrated over aperovskite layer at the position z, with the solid line being a fit using thelowest-energy Airy state. All results were calculated with the local spin-density approximation in the density-functional theory.4The cell-averaged potential V(z) as a function of the distance z from the interface isshown in Fig. 1. The potential has a remarkable, text-book like wedge shape, V = -eE | z |, asexpected for the constant electric field near a sheet of charge, leading to a novel wedge-shaped quantum well. If we compare the computed electric field with that in the vacuumelectric field, E = σ/2ε0, we infer a large dielectric constant of ε ≈ 23. It is well known thatSrTiO3 is close to being ferroelectric with an unusually large long-wavelength dielectricresponse of ε ≈ 30 x 103 at low temperatures.16,17 However, the screening of the sheet chargeinvolves the short-range dielectric response, with a smaller expected ε, consistent with thecalculated electric field. The truly wedge-shaped potential occurs for a theoretical sheetcharge of zero thickness. The finite thickness, of one atomic layer in the present case, causesthe bottom of the potential to flatten out at the center of the sheet, a feature that is present inour calculated potential. Indeed, the inversion symmetry dictates that the electric field isstrictly zero at the center of the sheet.The stripped-off La(d1) electrons which produce the positive La sheet charge in the firstplace, become bound in turn in the wedge-shaped potential well. The eigenstates of theelectrons are obtained from the Schrödinger equation,−+ =h2 222mddzV z EΨ Ψ Ψ( ) ,                                (2)where V(z) = F |z| is the potential of the sheet charge and F = -eE is a positive constant.Defining the scaled length l mF= ( / ) /h2 1 32 , energy λ = ( / ) /2 2 2 1 3m F Eh , and coordinateξ λ= −z l/ , the Schrödinger equation takes the form d d2 2 0Ψ Ψ/ ξ ξ− = , whose solutionsare the well known Airy functions Ai(ξ). The complete solutions are obtained by joining theAiry functions at the interface (z = 0) and then by matching the function or its derivative:Ai(-λ)=0 or dAi d( ) / |ξ ξ ξ λ=− = 0, for the odd and even parity states, respectively. The lowestthree eigenstates are shown in Fig. 2.Figure 2 Lowest eigenstates of the electron in the wedge-shaped potential well of auniform sheet-charge density, with length and energies in scaled units (seetext).5The lowest of the Airy states is occupied in the solid, resulting in a rapid decay of thecharge of the electron away from the LaTiO3 layer, as may be seen from the densities-of-states (Fig. 3). The electronic charge is predominantly Ti (d) like, located mostly on the Ti1atoms and decaying rapidly as one moves away from the sheet (see also Figs. 1(b) and 1(c)).The calculated core level energies systematically track the wedge-shaped potential as well.Figure 3 Energy- and atom-resolved densities-of-states, indicating the rapiddecay of the Ti conduction charge (mostly Ti (d)) as one moves away from the Lasheet. In fact, very little conduction charge (conduction DOS integrated up to Ef) isvisible from the figure for Ti atoms beyond the first layer.The number of electrons occupying different layers may be obtained by integrating thedensity-of-states (DOS) from the conduction bottom to the Fermi energy for each perovskitelayer as a function of its distance from the La sheet. The layer charges, so calculated, areshown by circles in the top part of Fig. 1, and they fit quite well to the shape of the lowest-energy Airy function (solid line). The length scale of the Airy function obtained from this fitis l=3.0 Å in excellent agreement with the expected value l mF= ( / ) /h2 1 32 = 2.4 Å. Since theground-state Airy function spreads to about ~3 l on either side of the potential well, theelectron wave function spreads to two to three layers of SrTiO3 (layer thickness ≈ 3.91 Å).This is clearly seen from the charge-density contour plot (Fig. 1). The spread is in excellentagreement with the same obtained from the experimental EELS (electron energy lossspectroscopy) profile of Ti3+ after deconvoluting it using the EELS profile for La. The latterserves as an effective resolution function,18 as suggested by a comparison between theEELS data and the annular-dark-field image for La. 1The Fermi surface is shown in Fig. 4, which consists of five interpenetrating bands, thewave function characters of which are indicated in the lower right corner of the figure forthe Γ  point in the Brillouin zone. The orbital components of these levels are easilyunderstood using insights from band calculations for the bulk materials.12,19,20 Theoctahedral crystal field splits t2g below eg for Ti, while for La, eg is below t2g. Furthermore,since the d orbitals of La have somewhat higher energy than Ti, it is clear that the Ti1(t2g)6electrons should have the lowest energy on account of their placement near the bottom ofthe wedge-shaped potential well. Of these, the two dxy states, corresponding to the two Ti1layers on either side of the La sheet, have the lowest energy. Next come the doubly-degenerate Ti1 dyz and dzx orbitals, four in total, which are split into bonding and anti-bonding states, the latter occurring above the Fermi energy Εf. The next one up, the highestof the five states below Εf, is the La(z2-1) state, again, helped by its location at the bottom ofthe potential well. Above that is the La(x2-y2) state and so on.Figure 4 The Fermi surface of (SrTiO3)5/(LaTiO3)1 shown in the surface Brillouinzone of the superlattice and its wave function character indicated in the lower rightpart of the figure.The magnetic behavior of the superlattices show some interesting features as well. Forthe single LaTiO3 layer structures (m=1), we find the paramagnetic state to be stable (both inthe LSDA and the LSDA+U calculations), rather than the AF state of the bulk LaTiO3. Theparamagnetism of these superlattices is understandable in light of the fact that bulk LaTiO3becomes paramagnetic with a small amount of hole doping6,7 and the leakage of the electroninto the SrTiO3 part dilutes the electron density and is tantamount to hole doping in theLaTiO3 part. Along the same lines, we may reason that for a sufficiently thick LaTiO3 layer,the inner part would be antiferromagnetic (bulk-like behaviour), while the outer Ti layerswould have either reduced moments or may even become paramagnetic.To test these ideas, we have performed an LSDA+U calculation for the(SrTiO3)5/(LaTiO3)6 superlattice structure, which has a six-monolayer thick LaTiO3 part,taking the Coulomb and exchange parameters to be U = 5 eV and J = 0.64 eV, respectively,following earlier authors.12 The results show the LaTiO3 part to be antiferromagnetic, withthe Ti magnetic moments equal to the theoretical LDA bulk value (~0.9 µB). In addition, thetwo adjacent Ti layers at the interface with the SrTiO3 region acquire a sizeable magneticmoment (about half of the bulk value) due to their proximity to the magnetic region. Themagnetic moments of the rest of the Ti layers in the SrTiO3 region are zero.7In conclusion, we have shown the formation of a quantum well with a text-book-likewedge-shaped potential originating from a uniformly charged LaTiO3 sheet in the(SrTiO3)n/(LaTiO3)1 superlattices. The electronic structure is explained in terms of the Airyfunction localization of the electrons leaking out of the LaTiO3 layer, but which becomebound in the potential well. With a LaTiO3 region of several layers thick, the inner layersshow bulk-like antiferromagnetic behaviour, while the magnetism disappears in theembedding SrTiO3 layers with a narrow transition region between them.Acknowledgment. This work was supported by the U. S. Department of Energy under GrantNo. DE-FG02-00ER45818.References* Permanent address: Institute for Nuclear Sciences-“Vinca”, PO Box: 522, 11001 Belgrade,Serbia & Montenegro.1 A. Ohmoto, D. A. Muller, J. L. Grazul, and H. Y. Hwang, Nature 419, 378 (2002)2 H. Yamada, Y. Ogawa, Y. Ishii, H. Sato, M. Kawasaki, H. Akoh, and Y. Tokura, Science305, 646 (2004)3 W. Kohn and A. E. Mattsson, Phys. Rev. Lett. 81, 3487 (1998)4 T. Mitsui and S. Nouma, Editors. Numerical Data and Functional Relations in Scienceand Technology – Crystal and Solid State Physics, Landolt-Börnstein New Series, GroupIII, Vol. 16, Part (a), Springer, Berlin (1982)5 M. Cwik, et al., Phys. Rev. B 68, 060401 (2003)6 Y. Taguchi, et al., Phys. Rev. B 59, 7917 (1999)7 Y. Tokura, Y. Taguchi, Y. Okada, Y. Fujishima, T. Arima, K. Kumagai, and Y. Iye, Phys.Rev. Lett. 70, 2126 (1993)8 Y. Nagaoka, Phys. Rev. 147, 392 (1966)9 S. Okamoto and A. J. Millis, Nature 630, 630 (2004)10 For a review, see: V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein, J. Phys.:Condens. Matter 9, 767 (1997)11 I. Solovyev, N. Hamada, and K. Terakura, Phys. Rev. B 53, 7158 (1996)12 E. Pavarini, S. Biermann, A. Poteryaev, A. I. Lichtenstein, A. Georges, and O. K.Andersen, Cond-mat/0309102 (19 Jan 2004)13 O. K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571 (1984)14 S. Satpathy, Z. S. Popovic, and W. C. Mitchel, J. Appl. Phys. 95, 5597 (2004)15 W. R. L. Lambrecht, B. Segall, and O. K. Andersen, Phys. Rev. B 41, 2813 (1990)16 T. Sakudo and H. Unoki, Phys. Rev. Lett. 26, 851 (1971)17 K. A. Müller and H. Burkard, Phys. Rev. B 19, 3593 (1979)18 H. Y. Hwang, private communication.19 S. Kimura, J. Yamauchi, M. Tsukada, and S. Watanabe, Phys. Rev. B 51, 11049 (1995)820 I. V. Solovyev, Phys. Rev. B 69, 134403 (2004)

Airy-function electron localization in the oxide superlattices

Abstract

Similar works

Full text

Available Versions

University of Missouri: MOspace