Structure of Periodic Crystals and Quasicrystals in Ultrathin Film Ba-Ti-O
,, Structure of Periodic Crystals and Quasicrystals in Ultrathin Film Ba-Ti-O
Eric Cockayne ∗ Material Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 USA
Marek Mihalkoviˇc
Institute of Physics, Slovak Academy of Sciences, 84511 Bratislava, Slovakia
Christopher L. Henley † Department of Physics, Cornell University, Ithaca, New York 14850 USA (Dated: September 1, 2018)We model the remarkable thin-film Ba-Ti-O structures formed by heat treatment of an initialperovskite BaTiO thin film on a Pt(111) surface. All structures contain a rumpled Ti-O networkwith all Ti threefold coordinated with O, and with Ba occupying the larger. mainly Ti O , pores.The quasicrystal structue is a simple decoration of three types of tiles: square, triangle and 30 ◦ rhombus, with edge lengths 6.85 ˚A, joined edge-to-edge in a quasicrystalline pattern; observedperiodic crystals in ultrathin film Ba-Ti-O are built from these and other tiles. Simulated STMimages reproduce the patterns seen experimentally, and identify the bright protrusions as Ba atoms.The models are consistent with all experimental observations. Quasicrystals have fascinated the materials world sincetheir discovery[1] due to their noncrystallographic sym-metries and quasiperiodic translational order. The firstquasicrystal reported was in an Al-Mn alloy. While nu-merous families of quasicrystals have since been found[2],until recently, all known physically realized quasicrystalswere either intermetallic alloys or soft matter systems.[3,4] This changed in 2013 with the report of a quasicrystal oxide by F¨orster et al. [5] in a thin film Ba-Ti-O struc-ture created by a multistep heat treatment of perovskiteBaTiO deposited on a Pt(111) surface, among other,periodic, structures. The quasicrystalline structure isidentified as a dodecagonal quasicrystal by its twelve-fold electron diffraction pattern, and by scanning tunnel-ing microscopy (STM) images that show bright protru-sions separated by a distance of about 6.85 ˚A, in posi-tions characteristic of a dodecagonal rectangle-triangle-30 ◦ rhombus tiling[6]. Additional experimental detailsdistinguish the phase from perovskite BaTiO . The Tiions appear to have charge 3+ via photoemission spec-troscopy. The overall stoichiometry (including both thequasicrystalline layer and BaTiO islands that remainafter the heat treatment used to create the quasicrys-tal) is Ba . TiO . . While these observations give in-triguing information about the nature of the quasicrys-talline structure, a complete structure determination islacking. In this paper, we give tiling decoration modelsfor the atomic structure of thin film quasicrystalline Ba-Ti-O and its related periodic crystalline structures, fullyconsistent with experimental observations. ∗ Electronic address: [email protected] † Deceased 29 June 2015
Our tile decoration model for the quasicrystal is shownin Fig. 1(a). The Ba, Ti, and O atoms overlay the top-most Pt layer and project into a plane as shown. Baatoms occupy the vertices of the tiles. All Ti atoms are3-fold coordinated with oxygen. Most oxygen atoms aretwofold coordinated with Ti except for those inside the30 ◦ rhombi (henceforth “rhombs”). The tiles join edge-to-edge. The atomic positions of the ideal tile structuresmatch perfectly for triangles to join rhombs and squares;the other tile combinations require merger of the tile Opositions near the shared edge.Squares, triangles, and rhombs can be combined toproduce numerous periodic and aperiodic tilings. Thesquare-triangle-rhomb tiling pattern formed by connect-ing the bright spots in the STM images[5] bears a strik-ing resemblance to the “ C a ” tiling pattern found byG¨ahler[6], using projection from a higher dimensionalspace with a concave acceptance region. This tiling (andits sufficiently large periodic approximants), containsonly square-triangle, triangle-triangle, and triangle-thinrhomb edges. The G¨ahler tiling has squares, triangles,and rhombs in the relative frequency √ √ √ − / √ / . TiO . . From X-rayphotoelectron spectroscopy measurements[5], it was de-duced that there were both Ti and Ti ions in theirsample in the ratio 5:1. Assuming that all the Ti ionthe quasicrystalline film have charge 3+, all the Ti inthe BaTiO islands charge 4+, and that the quasicrys-talline film has the composition found in our model, theoverall composition rounds to Ba . TiO . , exactly as de-termined experimentally.[5] a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n a b c d FIG. 1: (a) Decoration of square, triangle, and 30 ◦ rhom-bus (rhomb) tiles. Ba atoms are large blue; Ti small green,and oxygen medium red. (b-d) DFT-relaxed structures of Ba-Ti-O quasicrystalline approximants on a Pt-111 surface. Ptatoms gray. (a) 25.6 ˚A approximant; top view. (b) 25.6 ˚A ap-proximant; front view. (c) 49.4 ˚A approximant; top view. To test the stability of our model, refine the atomicpositions, and determine their optimal heights about thePt(111) surface, we turn to density functional theory(DFT) calculations. DFT calculations were performedusing the code[7] VASP.[8] The PBEsol GGA exchange-correlation functional[9] was used, and the plane-wavecutoff was 500 eV. Calculations were performed using onek-point, at the origin. A previous hybrid density func-tional theory study of Ti O in the corundum structureshows a spin-paired electronic ground state.[10] To re-produce these results as closely as possible at lower com-putational cost, we ran spin-unpolarized GGA+U calcu-lations, after tuning the U parameters for Ti and O toreproduce the volume and bandgap of corundum Ti O as accurately as possible (Ti U = 2.35 eV; O U = 0.30eV).Quasicrystalline structures are not compatible with pe-riodic boundary conditions. To investigate the stabilityof our model, we turn instead to periodic approximantsto the quasicrystalline structure. These periodic approx-imants have the same local structures as the quasicrys-tals. One such approximate has a unit cell 25.6 ˚A on anedge with γ = 120 ◦ . We created an approximate to theG¨ahler C a tiling and decorated it as shown as Figure 1.We placed this structure above a model Pt-111 trilayer,strained to have the same periodicity as the approximant.The lowest Pt layer had all atomic coordinates fixed, thenext layer z was allowed to vary, and the topmost layerall atomic coordinates were allowed to move. A vacuumlayer was created between the structure and periodic im-ages by using a periodicity of 20 ˚A in the z-direction.The structure was relaxed until all forces were relaxed toless than 0.1 eV/ ˚A. An additional, square, approximantof side length 49.4 ˚A was created and relaxed on a fixedPt monolayer until all forces were less than 0.1 eV/ ˚A.The relaxed structures are shown in Fig. 1(b-d). Thestructures retain their ideal tile geometries quite well.The most significant distortion is the clockwise or coun-terclockwise rotation of 3 O around a Ti that frequentlyoccurs. This rotation is often driven by the formation ofmore favorable O-Ti-O bond angles for the Ti in the thinrhombs; ab initio molecular dynamics reveal frequent li-bration of TiO units at room temperature. The averageheights above the topmost Pt layer in the relaxed struc-ture are 3.1 ˚A, 2.2 ˚A, and 3.1 ˚A for Ba, Ti, and O, re-spectively. Ti-O distances range from 1.80 ˚A to 1.95 ˚A.The rumpling of the Ti-O network (Fig. 1(c)) is drivenby transfer of electrons from Ba to Pt; the negative sur-face charge on the Pt layer then attracts the positive Tiions and repels the negative O ions.[11]STM topographs were simulated using the Tersoff-Hanann approximation.[12] In this approximation, mea-suring a contour of constant current is equivalent to mea-suring a contour of constant electron density, projectedin the energy range between E F and E F + eV , with E F the Fermi level and V the bias voltage. The projected FIG. 2: Simulated scanning tunneling microscopy (STM) to-pograph of the quasicrystal approximant shown in Fig. 1(b).Simulated bias voltage − .
15 eV, charge density 0.1 e nm − ,and topographic range 1 ˚A. electron density was calculated via DFT, starting withthe DFT-relaxed structure in Fig. 1(b). The result isshown in Fig. 2. By comparison with the atomic struc-ture, we see that only the Ba atoms are visible and thatthe Ti-O network is invisible. The simulated STM imagestrongly resembles the experimental ones.[13] The sizes ofthe simulated bright protrusions match even better withexperiment if the computed projected charge density isconvoluted by an in-plane factor exp( − ρ / (2 σ )), σ =1.0 ˚A, to mimic experimental resolution.The presence of three-fold coordinated Ti atoms in thequasicrystal Ba-Ti-O structure model along with Ti n O n polygons with n = 4, 5, and 7 is reminiscent of three-fold coordinated network structure observed in other two-dimensional systems such as graphene[14] and bilayerSiO .[15, 16] The Ti-O-Ti links in the Ba-Ti-O systemare analogous to the C-C bonds in graphene. The Ba-Ti-O counterpart to the ideal graphene structure is shownin Fig. 3(a). This structure has actually been observedby Wu et al. [11], from thin film Ba-Ti-O on a Au-111surface, although the Ba site is not fully occupied.By assuming that the bright protrusions seen in theSTM images of the periodic Ba-Ti-O thin film struc-tures observed by F¨orster et al. [13] were Ba, and thatthe Ti and O formed a Ti-O network with threefold co-ordinated Ti, we were able to devise a structure modelfor each case (Fig. 3). A bright spot that is sometimespresent and sometimes absent in the STM image of the“wagon wheel” structure (Fig. 3(f)) is easily interpretedas partial Ba occupancy of a Ti O pore. The same a b c d e f FIG. 3: Structure models for periodic crystalline thin film Ba-Ti-O. (a) Structure model proposed by Wu et al. [11] for thinfilm Ba-Ti-O on Au-111. Light blue color indicates partialBa occupancy. (b-e): Structure models proposed in presentwork for various periodic crystalline Ba-Ti-O structures ob-served by F¨orster et al. [13]: (b) Y-rows; (c) Six-network; (d)Double-Y; (e) Six-tiling; (f) Wagon wheel. There is a partiallyoccupied Ba site shown in light blue. kinds of tiles that occur in the quasicrystal model, withthe same decorations, occur in other structures, but sev-eral additional tiles are seen, each with a characteristicdecoration: a thin hexagon, a pentagon, and an elon-gated curved decagon. The “Y-rows” and, arguably, the“Wagon wheel” structures are periodic approximants tothe quasicrystal, but the structures in Fig. 3(c-e) are un-related to the quasicrystal. The Ba:Ti ratios of the peri-odic crystalline Ba-Ti-O structures in Fig. 3 range from3:13 ≈ ≈ n O n ringswith n = 4, 5, 6, and 7, but not n = 8 or greater; (2)The n = 7 rings are very common; and the n = 6 ringsrelatively rare; (3) The 7-rings always share a Ti-O-Tiedge with two or more other 7-rings to create a larger-scale network that can be represented as a tiling; (4) The7-rings are always occupied by a Ba ion, the 6-rings par-tially occupied, and smaller rings empty.The local structure associated with the rhomb in thequasicrystal, its approximants, and the wagon wheelstructure can be viewed as the fusion of two Ti O -typepores to make a dumbbell-shaped pore. The two oxygenatoms that are coordinated with only one Ti serve as abuffer against Coulomb repulsion of the relatively closeBa-Ba pair. An analogous three-dimensional dumbbell-shaped pore was recently found in a zeolite structure,[18]and the best structural model in that case similarly hadoxygen atoms inside the pore that were bound to only oneSi each, instead of the normal two. The analogy with ze-olites also suggests one possible application of the struc-tures proposed here: if freestanding monolayers with thestructures of the Ti-O networks shown here could be cre-ated, they could act as ultrathin filters with a uniformpore size for gas separation, etc.It remains an open question how the quasicrystal struc-ture of Ba-Ti-O is stabilized relative to approximant andnon-approximant arrangements that can also occur. Thefact that these structures form at high temperature sug-gests that entropy (vibrational and perhaps tiling[19])may play a key role. In any case, the existence of de-tailed structure models for all these structures is a firststep toward solving this problem.The authors thank Terrell A. Vanderah (NIST) forhelpful discussions. M. M. was supported by Slovakgrants VEGA 2/0189/14 and APVV-0076-11. C. L. H.received support from DOE grant DE-FG02-89ER45405. [1] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn,Phys. Rev. Lett. , 1951 (1984).[2] A.-P. Tsai, Chem. Soc. Rev. , 5352 (2013). [3] X. B. Zeng, G. Ungar, Y. S. Liu, V. Percec, S. E. Dulcey,and J. K. Hobbs, Nature , 157 (2004).[4] R. Lifshitz and H. Diamant, Philos. Mag. , 3021(2007).[5] S. F¨orster, K. Meinel, R. Hammer, M. Trautmann, andW. Widdra, Nature (London) , 215 (2013).[6] F. G¨ahler, in Quasicrystalline Materials , edited byC. Janot and J. M. Dubois (World Scientific, Singapore,1988), pp. 272–284.[7] Certain commercial software is identified in this paper toadequately describe the methodology used. 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