First atomistic studies of epitaxial growth of Na0.5Bi0.5TiO3 on SrTiO3

© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

phys. stat. sol. (b) 245, No. 12, 2649–2656 (2008) / DOI 10.1002/pssb.200844246 p s sbasic solid state physics

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First atomistic studies of epitaxial growth of Na0.5Bi0.5TiO3 on SrTiO3

Petar Petrov, Hannes Guhl, and Wolfram Miller*

Leibniz Institute for Crystal Growth (IKZ), Max-Born-Str. 2, 12489 Berlin, Germany

Received 3 June 2008, revised 20 August 2008, accepted 16 September 2008

Published online 24 October 2008

PACS 05.10.Ln, 68.35.Md, 68.47.Gh, 71.15.Mb, 81.15.Aa

* Corresponding author: e-mail [email protected], Phone: +49-30-63923074, Fax: +49-30-63923003

© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Layers of perovskites are very inter-esting for applications because of their piezoelectric and ferroelectric properties. They are grown by pulsed laser deposition, molecular beam epitaxy or metal-organic chemical vapor deposition (MOCVD). The latter is the most promising for achieving layers of high perfection. Nevertheless, for many materials of interest due to their expected excellent properties the current perfection of the layers is far away from those required for applications. Therefore, it is essential to understand the growth kinetics in order to improve the quality of the layers. One tool to study the kinetics on atomistic level is the kinetic Monte-Carlo (KMC) method in conjunction with density func-tional theory (DFT). In this paper we present first results for the system Na0.5Bi0.5TiO3 on SrTiO3 aiming to give some preliminary insight into the possible mechanisms of growth in such a system. Firstly, we will present the determination the energet-ics for some of the relevant processes by means of DFT calculations using the code Castep [1]. Then, we will de-scribe shortly the KMC method, which we have developed in order to take the perovskite structure into account. The KMC method is applied to study island growth, which can be analyzed and compared with scaling theory. All computations are done for the growth on a SrTiO3(001) surface, which is typically used in experi-ments [2, 3]. When using common preparation techniques it is known that such surfaces have a TiO2 termination [4]. Therefore, we restrict our computations on the energetics

and kinetics on a TiO2-terminated SrTiO3(001) surface without reconstruction. Though there are some hints from experiments for different kind of reconstructions [5, 6] all DFT calculations show that the reconstructions are by far less energetically favorable [7, 8]. 2 Determination of the energetics The energetics of the oxygen ad-atom has been investigated in detail on both terminations of the SrTiO3-(001)-surface. For the sake of brevity we restrict ourselves to present the results for the oxygen ad-atom which are important for the KMC simula-tion. Details and further results concerning the oxygen ad-atom will be published elsewhere. In addition we will re-port about our calculations about the single metal ad-atoms, i.e. bismuth and sodium on the TiO2-termination. For the calculation of the energetic data, we used the CASTEP-code [1], which implements the density functional theory (DFT) [9, 10]. For the results presented in this work we employed the local density approximation (LDA) for the exchange-correlation. The electronic density was repre-sented in plane waves and the atomic nuclei and the inner electronic levels were modeled by ultra-soft pseudo-potentials [11]. The TiO2-terminated surface of SrTiO3 was modeled using a slab geometry in a super-cell, to which periodic boundary conditions were applied. The slab itself consisted of five atomic layers, i.e. two SrO and three TiO2-layers, which were arranged in an alternating manner. The ad-atoms were placed on both sides of the slab, in or-der to prevail a mirror plane in the center of the slab paral-

The first stages of epitaxial growth of Na0.5Bi0.5TiO3 on

SrTiO3 substrates were studied by means of a newly de-

veloped kinetic Monte-Carlo (KMC) model employing sev-

eral types of particles, different types of neighbors and vari-

ous energy barriers. Density functional method (DFT) has

been used for computing the surface structure with adatoms

and for obtaining some of the diffusion barriers.

2650 P. Petrov et al.: First atomistic studies of epitaxial growth of Na0.5Bi0.5TiO3 on SrTiO3

© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

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lel to the surface, which is necessary to avoid the dipole in-teractions between the slab and its periodic image. All ge-ometries were optimized until the absolute value of the largest ionic force was smaller than 2 eV/Å. During these geometry optimizations, only the positions of the ions in the three central layers were kept fixed at their ideal bulk positions. All other atoms were allowed to move based on the condition that the overall symmetry properties of the respective configuration did not change. We confirmed the reliability of our computational set-up by carrying out calculations for both SrTiO3-termina-tions. Comparing the surface rumpling (relative displace-ment of the oxygen atom with respect to the metal atom) and the relative interlayer distance of the surface layer with recently published results of computations with the hybrid functional B3PW [12] we find an agreement better than 6%. Furthermore, we calculated the excesses of Sr and O atom in the surface and the free energy of formation for SrTiO3 surfaces according to [8] (See Table VII in this Ref.). We found a very good agreement of our results with those of Heifets et al. (differences within 3%). Since we are interested in a comparison between the energetics of different species on SrTiO3 rather than a high numerical accuracy, we consider our approach to be sufficient to ob-tain at least a coarse-grained picture. For bismuth and so-dium we used a plane-wave basis cut-off energy of 350 eV and integrations over the Brillouin-zone were carried out on a grid of (5 × 5 × 1) k-points. The vacuum space in the super-cell was 11.57 Å wide, which is large enough to suppress any spurious interaction of the surface with its pe-riodic image in direction perpendicular to the surface. We computed the binding energy of the three species on the high-symmetry positions on the TiO2-termination. These positions are depicted in Fig. 1 and the binding en-ergy is calculated according to:

1

Bind Conf Slab Adatom2E E E EÈ ˘

Î ˚= - - , (1)

where Conf

E , Slab

E and Adatom

E represent the total energies of whole configuration, the slab alone, i.e. the surface without the adatom, and the single isolated adatom, respectively. It is clear that for an accurate determination of the diffu-sion barriers these energies calculated at these few points may be not sufficient. Nevertheless, it can be seen from symmetry considerations that at Pos. I, Pos. II and Pos. III

(see Fig. 1) there must be either a local energetic maximum, a local minimum or a saddle point. Consequently, we ex-pect that the binding energies at these positions provide at least a good guess for the entire energetics of the species on the surface. The actual slabs in the production calcula-tions had always (2 × 2) surface unit meshes in order to screen the interaction of the adatoms with their periodic images along the surface. Since the bulk lattice constant of SrTiO3 was determined to be 3.858 Å, the actual size of the super-cell was (7.716 × 7.716 × 21.219) Å. In Table 1 we present the actual binding energies and the diffusion barri-ers for the metal and the oxygen adatoms. It is remarkable that the binding of the all the investigated species to the surface is quite strong. Furthermore, it is worth to mention that the oxygen atom is not sitting in its position of the perovskite structure but almost top on the oxygen atom in the TiO2 layer (see Fig. 2). The binding between the two oxygen atoms is similar to those of an oxygen molecule. For the binding energy of the latter the LDA computation yields 3.6 eV compared to 3.69 eV for the binding to the surface. As well known, LDA overestimates the binding energies. We also used the exchange-correlation functional introduced by Perdew, Burke, and Ernzerhof (PBE) [13]. Performing the same calculations results in 2.82 eV, which is close to 2.56 eV found in the experiment. The corre-sponding PBE computation for an oxygen adatom on the TiO2-terminated surface gives 2.64 eV for the binding en-ergy, which is smaller than the value obtained by LDA but still very large. Therefore, a desorption of such atoms is unlikely at process temperatures typical for MOCVD or MBE. This strong binding to the surface is accompanied by high energetic barriers for diffusions, which are found to be approximately half of the values of the absolute binding en-ergy for oxygen and sodium. Only for bismuth the barrier for diffusion is much smaller (about one tenth of the abso- Table 1 Binding energies and barriers for diffusion ΔE for the

examined species on the TiO2-terminated surface.

–Bind

E ED

Bi –3.28 eV 0.30 eV

Na –2.84 eV 1.59 eV

O –3.69 eV 1.62 eV

Figure 1 (online colour at: www.pss-b.com) Schematic illustration of adsorption of the adatom on the high symmetry positions of the

TiO2 terminated surface. Depicted is only one unit mesh of the surface and only the two topmost atomic layers. Green spheres: Ti, red

spheres: Sr, blue spheres: O.

phys. stat. sol. (b) 245, No. 12 (2008) 2651

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Figure 2 (online colour at: www.pss-b.com) Side view on the

equilibrium configuration for an oxygen adatom (white sphere)

on a TiO2-terminated surface. Green spheres: Ti, red spheres: Sr,

blue spheres: O. lute values of its binding energy), because the bismuth atom is much bigger than the other species investigated. Therefore, it does not actually “fit” into one unit mesh of the

2TiO -termination. This is strongly supported by test

calculations using slabs having only one unit mesh and not (2 × 2), where a very strong interaction of the bismuth ad-atom with its periodic images along the surface is found. In contrast to oxygen, both investigated metal atoms are found to be most stable on Pos. I (see Fig. 1), which corresponds to the bulk position of the strontium atom in the SrTiO3 bulk structure and is consequently the right po-sition both species would assume in the idealized structure of Na0.5Bi0.5TiO3. For sodium and bismuth Pos. II. is very likely the transition state, which determines the diffusion barrier. On Pos. III, both species are being rather repelled into the vacuum than bound to the surface. While the oxygen adatom binds only to the other oxy-gen atoms on the surface, in a covalent manner, the va-lence states of bismuth and sodium overlap with the 3d-states of the titanium atoms on the surface. This leads to a non-vanishing density of state (DOS) at the Fermi-level, which means that the sodium and bismuth produce an me-

tallic over-layer on the SrTiO3 surface. Having in mind that we are here still looking at one adatom per 2 × 2 surface unit meshes, it becomes clear that bismuth and sodium in-fluence the electronic properties of the SrTiO3 surface on a remarkable range.

3 KMC model for perovskites In order to study theoretically the first stages of epitaxial growth of Na0.5Bi0.5TiO3 on a substrate of SrTiO3, we have developed a KMC technique which is based on the standard solid-on-solid KMC model designed for MBE growth but adjusted to the typical structure of perovskites and enriched with new features (several types of particles, different type of neighbors – lateral or diagonal, various energy barriers, variable adsorption rate etc.). The structure of the model is quite complex. There are two types of layers, and for every particle the neighbors for diffusion and neighbors for en-ergy calculations could be different (see Fig. 3). Thus, after deposition, atoms diffuse in a random walk, but the direc-tion and the length of moving are specified for every parti-cle type differently. In general, with some obvious modifi-cations our method can be applied to epitaxial growth of different perovskite compounds on a substrate with similar structure. More details about this model will be published elsewhere. As a typical KMC model, perovskite KMC model maintains a list of processes allowed for the different at-oms on the surface. Only processes regarded as important are included in the simulation: deposition of atoms onto the surface, diffusion of surface adatoms as a hopping process to the in-plane or out-of-plane nearest or next-nearest neighbors at the rate

0 B( ) exp ( )R E T k E k TD , = -D / ( ED

being the diffusion barrier, T – substrate temperature and

Bk – Boltzmann’s constant), and desorption. At any dis-crete moment of the physical time, a random generator [14] supplies a number

1(0 1)r Œ , and an event is chosen ac-

cording to its probability and then executed. The probabili-ties of all events are proportional to their rates. After every execution of an event the list of processes and their rates is updated since the coordination of the corresponding atom

Figure 3 (online colour at: www.pss-b.com) Structure of the two different layer types. Small blue circles represent sites with oxygen.

The larger circles represent Me1 (Ti, left) and Me2 (Na or Bi, right). The stroke denotes sites, which cannot be occupied. Counting of

the sites starts in the left bottom corner with (0, 0). In the text a site (p, q) will be referred to as even-even (e-e), odd-even (o-e), even-

odd (e-o) or odd-odd (o-o) according to whether the number p (resp. q) is even or odd.



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and its neighbors might have been changed. The time is in-creased by a step tD that is computed via [15, 16]

2 totallnt r RD = - /

using a second random number 2

(0 1)r Œ , . Here total

R is the sum of all rates in the system. The diffusion barrier ED which defines the rate of a process has four contributions: a term

SE representing the

barrier for a free adatom on a terrace, a contribution N

E from a neighbor (energy neighbor), and additional barriers for hopping up across a step edge

upE (step edge barrier)

and down

E for hopping down across a step edge (known also as Ehrlich–Schwoebel barrier). Thus, ED is given by

S N step desE E nE E ED = + + + ,

where n is the number of neighbors contributing to ED and

stepE is equal to one of the values 0, down

E or up

E . des

E is an additional small barrier representing energy barrier for de-sorption. As usual, periodic boundary conditions have been applied with respect to energy neighbors and neighbors for moving.

4 Island size distribution 4.1 Introduction It is of great importance to gain in-sight into the first stages of the epitaxial growth of Na0.5Bi0.5TiO3 on SrTiO3. This means we like to understand island formation and growth in submonolayer regime for different set of parameters. This kind of problem has been studied in detail for other systems, especially in molecular-beam epitaxy (MBE) concerning structures with one type of particle and constant adsorption and diffusion rates. Studying the mean island density and the shape of island size distribution has a broad technological importance since the submonolayer structures usually influence the morphology of corresponding multilayer film. It is well known that island formation undergoes three different regimes of growth – nucleation, aggregation, and coalescence. In the early stages of the submonolayer growth, in the regime of nucleation, the number of adatoms is much

greater than the number of islands and the most probable event is the formation of new islands. Their stability is re-lated to the so called critical island size [17, 18]. The basic definition of critical island size i is the size that is one less than the number of atoms needed to form the smallest stable island. For instance, if 0i = the monomers are immobile, if

1i = monomers diffuse but the dimers are stable, etc. In recent years much theoretical effort has been done in studying scaling properties of island size distribution in the precoalescence regime [17–30]. Let ( )

nρ θ be the island

size density function at coverage θ , which gives the den-sity (per site) of islands of size n, where n is the number of atoms in the island. Then the average island size N can be written as follows

2

2

( )

( )

n

n

n

n

nρ θ

ρ θ

≥

≥

= .

Â

ÂN

It is known from the scaling theory that in the case of MBE growth there exists only one characteristic size in the problem which is the mean island size ( )θN and one may write

2( ) ( ) ( 2)n

f n nρ θ θ -

= / ≥ .N N

The scaling function f depends on many factors, in par-ticular on the relation between diffusivity and adsorption, island morphology, and the temperature. The scaling the-ory predicts [19, 23] that the shape of island scaling func-tion is related also to the critical island size [24]. To this end dynamic scaling of island size distribution for the two different layers (see Fig. 3) is studied.

4.2 Results 4.2.1 Island counting and statistics Due to the special perovskite structure of our model in the counting of islands we take into account neighboring in diagonal direc-tion as well as in lateral direction. For every type of inves-tigation we have made 20 trials with different seeds of the random generator. Then the values old{ }

kf of scaling func-

Figure 4 (online colour at: www.pss-b.com)

Scaled island-size distribution for Me2O layer

with the simplified set of energy parameters

S0 75 eVE = . , EN = 0.4 eV.

phys. stat. sol. (b) 245, No. 12 (2008) 2653


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tion corresponding to sizes { }k from every data file were smoothed to new{ }

kf using the central arithmetic mean for-

mulae:

new old new old old old1

2 2 3 2 3 43

6

new old1

13

6

( ) . . .

8

k

k

k

f f f f f f

f f k

�

�

�

= +

= -

= , = + + , ,

= , ≥ .Â

Finally, the mean value was taken with respect to the 20 data files containing values new{ }

kf .

4.2.2 Runs with simplified parameters In the first kind of simulations we used simplified energy barriers (one and the same energy barrier to the substrate

S0 75E = . eV

for all particles) in order to investigate the presumable scaling behavior of island size distribution. Our trials confirmed the diffusion mediated scaling be-havior of the island size distribution. We observed that for both layers at temperature 800 K distributions collapse onto one curve. Figure 4 presents the island size scaling distribution of layer type 0 for the coverages 15–25%. The X-axis indicates the normalized island size and the Y-axis shows the normalized density of islands of such sizes. In agreement with the classical theory for the one-particle KMC models we observed for layer type 0 of our perovski-te model a bell-shaped scaling function with strictly posi-tive derivative at the origin corresponding to critical island size 1i = similar to the results in [31]. The only difference is that the maximum is attained at the point 1/2 conforming to island size 2n = /N rather than 1 (n = N ). Sample view of the results from simulation of layer type 0 shows a ma-jority of large compact islands (see Fig. 6 (top)). In Fig. 5 the scaling functions for the layer type 1 at different temperatures are given. The monotonicity of the curves indicates critical island size 0i = [31], that is, after arriving on the surface single atoms become almost immo-bile. This explains also the weak dependence of the curve with respect to the temperature in the range 800–1000 K.

4.2.3 Island-size distribution with parameters from ab initio computations In the second type of runs we used the structural and energetic information from the DFT calculations described in section 2. In particular, we know the equilibrium position and the diffusion barriers

SE

for a Na, Bi, and O atom on the TiO2-terminated surface (see Tables 2 and 1). Because the oxygen atom is sitting at a position different than that in the bulk of the material, we had to modify the rules in the KMC. In layer type 0 we al-low that oxygen atoms can sit on odd-even and even-odd sites. In order to make this favorable compared to the even-even site, we have chosen a small

SE for an oxygen atom

Figure 6 (online colour at: www.pss-b.com) Views of the sur-

face at 25% coverage with simplified model (top) and parameters

suggested from ab initio calculations (bottom).

Figure 5 (online colour at: www.pss-b.com)

Island-size distribution of Me1O layer with

simplified parameters at coverage of 25%.



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Table 2 Energy parameters

O(e-e) –O(o-e, e-o) Bi Na

SE 0.1 –1.5 0.3 1.6

NE 0.9 –0.7 0.2 0.2

sitting at an even-even site (see Table 2). Furthermore, we had to guarantee that odd-even and even-odd sites be-come the less favorable the more neigbour atoms there are, i.e. the energy contribution

NE should lower the total en-

ergy barrier ED . Therefore, we chose N

0E < (see Table 2). On the other hand, an oxygen at the even-even site is stabi-lized by neighboring atoms and so we set

NE to a large

value. We considered two separate cases: a) only Na and O as adsorbing particles, b) only Bi and O as adsorbing particles. This are the two extreme situations, which will clearly lead to a non-stoichometric surface but they are illustrative in order to understand the basic dynamics. In the case of adsorption of Na the final ratio Na:O was approximately 1:4 and the O diffusivity was 2 orders of magnitude larger than that of Na. In contrast with the

trials with simplified energy set surface morphology was determined by scattered islands of small sizes (see Fig. 6 (bottom)). Because of the extremely high mobility of Bi compared to oxygen (determined by the relatively low ES), in the case of adsorption of Bi the surface was preliminar-ily “oxidized” and then a quantity of Bi about 1/4 of the oxygen quantity was deposited. Similar to the case of Na single particles were relatively immobile and again the critical island size was i = 0 (see Fig. 7–8).

4.3 Changing the structure with respect to metal: oxygen ratio Since ab initio computations showed that o-e and e-o positions are preferable for single atom O on Me1O terminated surface we performed trials with different proportions of the number of metal atoms with respect to the number of oxygen atoms on the surface. The quantity of oxygen ranging from 13% to 20% had been preliminarily deposited and then the adsorption of Bi was switched on up to the total coverage of 25%. A linear increase of the part of O atoms at even-even positions (with respect to the total O coverage) had been observed as the portion of metal atoms was increasing. Thus, the expected

Figure 8 (online colour at: www.pss-b.com) Island-size distribution of BiO islands at total coverage of 25%.

Figure 7 (online colour at: www.pss-b.com) Scaled island-size distribution of Me2O layer with parameters corresponding to Na. Coverage of 25%.

phys. stat. sol. (b) 245, No. 12 (2008) 2655


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perovskite bulk structure of BiO layer has been confirmed on the basis of our theoretical experiments, which proves the reliability of the KMC model developed. 5 Conclusions In this study we present some first re-sults for the initial stage of the growth kinetics of Na0.5Bi0.5TiO3 on a TiO2-terminated SrTiO3(001) surface. For this purpose we used a newly developed KMC method adapted to the structure of perovskites. DFT calculations for adatoms on the TiO2-terminated SrTiO3(001) surface have been performed. These calculations showed that an oxygen atom is bound to surface in a manner similiar to that in an oxygen molecule. It is sitting not the same place as in the complete layer. This fact has been considered in the dynamics of the KMC model. The dynamic scaling of island size distribution was studied for two sets of parameters. Firstly, we used simpli-fied parameters in order to study the dynamic behavior of the KMC developed. In second set of runs we used the known parameters from our DFT calculations. In both cases our trials confirmed the diffusion mediated scaling behavior of the island size distribution. The curves ob-tained show critical island sizes i = 1 (layer type 0 with simplified parameters) and i = 0 (otherwise). In the first case the surface morphology was dominated by relatively large compact islands (i = 1). Using the parameters of the DFT calculations the surface morphology is dominated by scattered small islands due to the high diffusion barriers of single adatoms. The KMC simulations also show that a re-arrangement of the oxygen atom from the position of a single adatom as computed by DFT to the position in the bulk perovskite is possible in principle. Further studies will be performed in order to check the possibility of other diffusion channels – maybe enhanced by other adatoms – where the barrier is lower than for those studied in this pa-per. Corresponding experiments of deposition by means of MOCVD are on the way.

Acknowledgements We would like to thank PD Dr. Kars-ten Reuter and the members of his research group at the Fritz-

Haber-Institute in Berlin for fruitful discussions and technical support. Some of the DFT calculations were performed at the HLRN Berlin under project number bep00022.

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First atomistic studies of epitaxial growth of Na0.5Bi0.5TiO3 on SrTiO3

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Transcript of First atomistic studies of epitaxial growth of Na0.5Bi0.5TiO3 on SrTiO3