Structure of amorphous BaTiO3 by molecular dynamics simulations using a shell model

Journal Pre-proof Structure of amorphous BaTiO3 by molecular dynamics simulations using a shell model Tamotsu Hashimoto, Hiroki Moriwake PII:

S0921-4526(19)30695-7

DOI:

https://doi.org/10.1016/j.physb.2019.411799

Reference:

PHYSB 411799

To appear in:

Physica B: Physics of Condensed Matter

Received Date: 24 June 2019 Revised Date:

26 September 2019

Accepted Date: 18 October 2019

Please cite this article as: T. Hashimoto, H. Moriwake, Structure of amorphous BaTiO3 by molecular dynamics simulations using a shell model, Physica B: Physics of Condensed Matter (2019), doi: https:// doi.org/10.1016/j.physb.2019.411799. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

APS/123-QED

Structure of Amorphous BaTiO3 by Molecular Dynamics Simulations Using a Shell Model Tamotsu Hashimoto1, ∗ and Hiroki Moriwake2, 3 1

Research Center for Computational Design of Advanced Functional Materials (CD-FMat), National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba Central 2, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan 2

Nanostructures Research Laboratory,

Japan Fine Ceramics Center (JFCC), 2-4-1 Mutsuno, Atsuta-ku, Nagoya, 456-8587, Japan 3

Center for Materials research by Information Integration (CMI2), National Institute for Materials Science (NIMS), 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan (Dated: September 25, 2019)

Abstract We studied the structure of amorphous BaTiO3 by molecular dynamics (MD) simulations. The local bonding units (LBUs) were mainly TiO6 in agreement with an X-ray absorption fine-structure (XAFS) study. An LBU was linked to its neighboring ones with their apexes, edges, and/or faces shared. The first peak of the Ti-Ti radial distribution function (g(r)) was located at ∼ 3 ˚ A with a trimodal distribution that results from the three linkage types, supporting extended X-ray absorption fine structure (EXAFS) experiments. If the number of Ba atoms around a Ti atom was larger than the average, the number of non-bridging oxygen (NBO;

[1] O)

atoms was larger around

that Ti atom. In the opposite case, the number of tricluster oxygen (TO; [3] O) atoms was increased around it, and the number of edge or face-sharing linkages with the neighboring Ti atoms was also increased. Keywords: amorphous BaTiO3 , structure, dielectric susceptibility, molecular dynamics simulation, shell model

∗

[email protected]

1

1.

INTRODUCTION

Amorphous BaTiO3 can be formed by sputtering or molecular beam epitaxy (MBE) onto a substrate[1–5], and also is observed between nanosized particles formed by aerosol deposition[6]. The amorphous BaTiO3 can be turned into quasi-amorphous (QA) phase by pulling a thick film through a steep temperature gradient[3] or by annealing a thin film deposited on the SrTiO3 substrate[5]. The QA phase was shown to be a pyroelectric with a little larger dielectric susceptibility than that for the amorphous phase[3], and even a ferroelectric[5]. The amorphous perovskites are assumed to have random networks of LBUs[2, 7–9]. By XAFS spectroscopy, the LBU in the amorphous phase was shown to be TiO6 [2]. The EXAFS experiments show the evidence for the edge or face-sharing LBU linkages for amorphous SrTiO3 [9]. However, as for amorphous BaTiO3 the LBU linkage is not well understood because of the proximity of the Ba L3 edge to the Ti K edge[2, 9], and it is only expected to be similar to that for SrTiO3 [1]. To the authors’ knowledge, the number of theoretical studies on the structure of amorphous BaTiO3 is very small. By classical MD simulations, the LBU for amorphous BaTiO3 is predicted to be TiO4 [10].

In this study, we made some models of amorphous BaTiO3 and analyzed the structures. The LBU in amorphous BaTiO3 was mainly TiO6 , whitch is in agreement with the XAFS study[2]. The nearest Ti-Ti distance in the amorphous phase was about 1 ˚ A shorter than that in the crystalline phase, and it had a trimodal distribution. It was shown that the trimodal distribution corresponds to apex, edge, and face-sharing LBUs, as expected from the experimental results for amorphous SrTiO3 [9]. Furthermore, we found that the apex:edge:face linkage ratio clearly depends on the local environment around the Ti atom. If there are more Ba atoms around a Ti atom than the average, the Ti atom is surrounded by fewer Ti and O atoms than their respective averages, and it tends to be [5] Ti with many [1] O atoms and fewer edge or face-sharing linkages, while in the opposite case, it is surrounded by more Ti and O atoms than their respective averages, and it tends to be [3]

[7]

Ti with many

O atoms and many edge or face-sharing linkages with surrounding LBUs. It is also shown

that the dielectric susceptibility for the amorphous phase was much smaller than that for the crystalline phase in agreement with experiments. 2

2.

METHOD OF CALCULATION

We used the isotropic shell model developed by Tinte et al[11, 12]. In the shell model, each atom is composed of a core and a shell. The intra-atomic core-shell interaction is expressed by V (r) = c2 r2 /2 + c4 r4 /24, where r is the core-shell distance and c2 and c4 are parameters. The inter-atomic interaction is through the Buckingham type shell-shell interaction V (r) = A exp(−r/ρ) − C/r6 , where r is the inter-atomic shell-shell distance and A, ρ, and C are parameters, and the Coulomb interaction. The cutoff length for the nonbonded interactions were 10.0 ˚ A. The computations were carried out in constant temperature and constant pressure (NPT) ensembles using the code developed by us[13–15]. The pressure was controlled by the Parrinello-Rahman method[16]. The temperature was controlled by the massive Nose-Hoover chain method[17–19]. The externally applied pressure was set to 0 Pa throughout the simulations. Dielectric susceptibilities were calculated by the fluctuation formula   1 ∂Pi χij = ε0 ∂Ej σ,T 1 ' h∆Mi ∆Mj i . ε0 < Ω > kB T

(1)

E, σ, T , and kB are the electric field, the stress, the temperature, and the Boltzmann’s constant. P is the polarization. M = ΩP is the total dipole moment of the MD cell. ∆Mi represents Mi − hMi i.

3.

RESULTS 3.1.

models

We used MD cells made up with 12×12×12 unit cells. We melted the crystal BaTiO3 and annealed it for 6 ps at 2500 K and 0 Pa. Then we cooled the systems to 300 K at varying cooling rates (Table I) with decreasing the target temperature of the thermostat every time step. Then we annealed them for 400 ps at 300 K and then analyzed the structure using data for 3 ps. Table I summarizes the densities of the obtained amorphous models together with their cooling rates. If we started from the same amorphous structure, the densities of the amorphous phase showed slight cooling rate dependence, and the slower cooling rate 3

resulted in a denser structure. The density of all models is close to the experimental one which is 82 to 84 % of the crystal density[1, 20, 21]. Since this model almost reproduces the experimental density, we assumed that it can predict to some extent the structural properties of the amorphous phase. To check the classical MD results, we also performed first-principles molecular dynamics (FPMD) simulations with 135 atoms. The results showed similar trend to the results of the classical MD, which will be shown in elsewhere. Figure 1(a) shows the snapshot of the amorphous BaTiO3 obtained in a simulation. In crystalline BaTiO3 , all the TiO6 octahedra are linked with their apexes shared (Fig. 1(b)), while in the amorphous phase, some TiOm -TiOn polyhedra are linked with their edges or faces shared (Figs. 1(c) and (d)). Also, in the crystalline phase, all the oxygen atoms are bridging oxygen (BO;

[2]

O) atoms, and each is bonded to two Ti atoms, while in the

amorphous phase, there are NBO, BO, TO, and beyond-tricluster oxygen ([n>3] O) atoms. In the snapshot, the central LBU has an apex-sharing linkage with its left LBU, an edge-sharing linkage with its upper LBU, and a face-sharing linkage with its lower-right LBU.

3.2.

Ti-O g(r)s and n(r)s

Figure 2(a) shows Ti-O g(r)s and the running coordination numbers (n(r)s) for crystalline and amorphous BaTiO3 . The bimodal first peak for the crystalline phase corresponds to instantaneous three shorter and three longer Ti-O bonds in the tetragonal phase. The first peak of the Ti-O g(r) for the amorphous phase has a unimodal distribution. The Ti-O (and other) g(r)s for Models 1-3 are similar to each other. Therefore, we focus on Model 3 for detailed analyses. Figure 2(b) shows the decomposed first peak of the Ti-O g(r)s for the amorphous phase. They are decomposed so that the sum of [m] Ti-[n] O g(r)s becomes the total Ti-O g(r). This decomposition method is used throughout this paper. There is a tendency that the bond lengths for

[m]

Ti-[1] O is shorter than that for

[m]

Ti-[n>1] O. The shorter Ti-[1] O

bond for the amorphous phase is in agreement with the preceding MD simulation study[10] and our FPMD study. Figure 3 shows the probability distribution for TiOn . Here, we assumed that an O atom within 2.7 ˚ A from a Ti atom was bonded to the Ti atom, because the Ti-O g(r) is close to zero at that distance (Fig. 2). About 80 % of the polyhedra are TiO6 . However, several percent of TiO5 and TiO7 also exist. The predominance of TiO6 is in agreement with XAFS 4

study[2]. The probability for finding TiO8 and TiO4 were very small. Therefore, due to insufficient statistics, the latter was ignored in the figures for the decomposed radial and (dihedral) angle distribution functions. Figure 3 also shows the probability distribution for the number of Ti atoms bonded to an O atom (nTiO ) for each nOTi . nTiO = 1, 2, 3, ... correspond to [1] O(NBO), [2] O(BO), [3] O(TO), ..., respectively. The numbers of NBOs and BOs account for more than 1/2 for the TiO6 , and decrease with increasing nOTi . The numbers of

[3]

Os and

[4]

Os accounted for less than

1/2 for the TiO6 , and increase with increasing nOTi . This result can be understood as the charge compensation mechanism discussed for the Na2 O-Al2 O3 -SiO2 system[22], assuming that the atoms have nominal ionic charges. Table II shows the averaged charge on TiOn calculated using the data for the probability distribution for nTiO in Fig. 3. If all O atoms are BOs and the charge on each O atom is apportioned onto the surrounding LUBs, the charge of the TiO6 LBU is –2. If we use data in Fig. 3, it is calculated to be –1.8. However, if all O atoms are BOs, the charges of the TiO5 , TiO7 and TiO8 LBUs are –1, –3, and –4, respectively. To compensate for the deficiency or excess of the negative charge with respect to –2, the numbers of

[n≤2]

Os and

[n>2]

Os increased and decreased compared to those for

TiO6 , respectively, so that the charge of the TiO5 LBU becomes –2.7, and they decreased and increased compared to those for TiO6 , respectively, so that the charges of the TiO7 and TiO8 LBUs become –1.1 and –0.8, respectively, for Model 3. The same trend is shown for the FPMD simulations.

3.3.

Ti-Ti g(r)s

Figure 4(a) shows Ti-Ti g(r)s and n(r)s for crystalline and amorphous BaTiO3 . The first peak of the Ti-Ti g(r) for the crystalline phase is located at 4.0 ˚ A, while for the amorphous phase it is located at ∼ 3 ˚ A with a trimodal distribution. The short Ti-Ti distance for the amorphous phase is also reported by the EXAFS experiments for SrTiO3 [9]. The authors suggested that the peak of the amorphous phase is due to the edge-sharing LBU linkages. Figure 4(b) shows the Ti-Ti g(r)s and n(r)s for amorphous BaTiO3 decomposed into apex, edge, and face-sharing LBU linkages. It is shown that the three Ti-Ti g(r) peaks from right to left correspond to apex, edge, and face-sharing LBU linkages, respectively. A Ti atom has six Ti neighbors in the crystalline phase, while it has less than six Ti neighbors 5

in the amorphous phase. It is shown that the percentage of the apex-sharing LBU linkages are dramatically reduced compared to the crystalline phase, while that of edge- and facesharing ones account for a considerable proportion. These apex, edge, and face-sharing LBU linkages are also reported for BaTi2 O5 by MD simulations, with fractions of 78 %, 20 %, and 3 %, respectively[7]. Figure 4(b) also shows further decomposed Ti-Ti g(r)s into TiOm -TiOn components. The (m, n) = (6, 6) pair mainly contribute to each peak.

3.4.

Ti-Ti and Ba-Ti n(r)s [n]

Figure 5 shows the number of [m]

Ti around

[m]

Ti (n[n] Ti[m] Ti ) that corresponds to the

Ti-[n] Ti n(r) in Fig. 4(b) at ∼4.5 ˚ A that is not shown to avoid the figure becoming too

complicated. Each bar for the (m, n) pair is decomposed into apex, edge, and face-sharing linkage components. Because [6] Ti is dominant, the numbers of [6] Ti around [m] Ti (n[6] Ti[m] Ti ) are the larger components. There is a trend that if m is large, the number of Ti atoms P around [m] Ti, n n[n] Ti[m] Ti , is large. This implies that if a Ti atom is bonded by many O atoms, there are many Ti atoms around that Ti atom. There is also a trend that if m increases from 5 to 7, the number of the edge or face-sharing linkages around an

[m]

Ti atom

increases. Around [7] Ti atoms, the number of [n>2] O atoms are larger (Fig. 3), so it is natural that they form more edge or face sharing linkages to the neighboring LBUs. Figure 5 also shows the number of Ba atoms around [m] Ti (nBa[m] Ti ). Here, we counted the number of Ba atoms within 5.2 ˚ A from a Ti atom, because the Ba-Ti g(r) has a minimum at that distance. The Ba density clearly depends on m=nOTi . Around

[5]

Ti, the number of

Ti atoms is smaller than the average and the number of Ba atoms is larger than the average. On the other hand, around [7] Ti and [8] Ti, the number of Ti atoms is larger than the average and the number of Ba atoms is smaller than the average. If a Ti atom is surrounded by fewer Ti atoms and more Ba atoms than their respective averages, because the charges of Ti and Ba ions are 4+ and 2+, respectively, fewer O atoms than the average should be attracted to the Ti atom so that it becomes

[5]

Ti. In such a case, these O atoms tend to be NBOs

(Fig. 3). On the other hand, if a Ti atom is surrounded by more Ti atoms and fewer Ba atoms than their respective averages, more O atoms than the average should be attracted to the Ti atom so that it becomes

[7]

Ti or

[8]

Ti.

6

3.5.

Ba-O and other g(r)s and n(r)s

Figure 6(a) shows the Ba-O g(r)s. The first peak of it for the amorphous phase is much broader than that for the crystalline phase. The coordination number of Ba around O is 4 in the crystalline phase, while it is 2.6 at 3.7 ˚ A in the amorphous phase. Figure 6(b) shows the decomposed Ba-O g(r)s into Ba-[1] O and Ba-[n>1] O components. There is a tendency that the Ba-[1] O distance is shorter than the Ba-[n>1] O distance. Thus, compared to

[n>1]

Os, the

NBOs are closer to both Ti (Fig. 2(b)) and Ba atoms within this model. Similar results are also observed in our FPMD calculations as well as in other glass materials such as calcium aluminosilicate [CaO-Al2 O3 -SiO2 ] (CAS) glass by classical MD simulations[23, 24], CAS melt by first-principles MD simulations[25], and Na2 O-SiO2 glass by classical MD simulations[26].

Figure 7 shows the probability distribution for the number of Ba and Ti atoms bonded to an O atom (nBaO and nTiO , respectively). Here, we counted the number of Ba (Ti) atoms within 3.7 (2.7) ˚ A from an O atom. 2.5 %, 3.0 %, 4.5 % of O atoms were free oxygen atoms ([0] O) for Models 1, 2, and 3, respectively, and they were surrounded only by on average 4.5 Ba atoms. The free oxygen atoms are also observed in the classical MD simulations of CAS glasses[23, 27]. The probability distribution of nTiO and nBaO has a maximum at nTiO =nBaO =2, and if nTiO increases, nBaO decreases almost linearly. The same trend is also observed in our FPMD simulations and in the MD simulation of BaTi2 O5 glass[7].

Figure 8 shows g(r)s for Ba-Ba, Ba-Ti and O-O. The half widths of the first peaks of the g(r)s in the amorphous phase for Ba-Ba, Ba-Ti and Ba-O (Fig. 6) are 4.3, 4.1, and 2.6 times wider than those in the crystalline phase, respectively. Those for O-O and Ti-O (Fig. 2(a)) show less than 1.4 times wider distributions than the corresponding crystalline phase. g(r)s for Ba-Ba and Ba-Ti are also decomposed as done for Ti-Ti in the lower panels of Figs. 8 (a) and (b), respectively. Ba-Ba is four O sharing in the crystalline phase, but the component is decreased and other components dominates in the amorphous phase, which results in the very wide first peak. Ba-Ti is face sharing in the crystalline phase, but the component is decreased and other components appear in the amorphous phase, which also results in the very wide first peak. 7

3.6.

Angle distributions

Figure 9(a) shows the O-[n] Ti-O angle distributions. The angle distributions within TiO5 and TiO6 LBUs have maxima near 90◦ and 170◦ , suggesting that their average shapes are close to square pyramid and octahedron, respectively. The peak near 90◦ is composed of more than one component. It is shown that the to the smaller angle component, while the

[m>1]

to the larger angle component. In the case of

[m>1]

O-Ti-[m>1] O pairs mainly contribute

O-Ti-[1] O and [m>1]

[1]

O-Ti-[1] O pairs contribute

O-Ti-[m>1] O, some of the

[m>1]

Os are

involved in the edge or face-sharing linkage with neighboring LBUs, and in such cases, the [m>1]

O-Ti-[m>1] O angle becomes smaller than 90◦ due to the Ti-Ti repulsion. The angle

distribution within TiO7 LBU is shifted in the smaller angular direction because there are many O atoms around the Ti atom. Figure 9(b) shows the Ti-O-Ti angle distributions. The Ti-O-Ti angles for the apexsharing LBUs are widely distributed between 110◦ -180◦ . In agreement with this, the first√ peak of the Ti-Ti g(r) for the apex-sharing LBUs is also widely distributed roughly from 3l to 2l, where l is the Ti-O bond length (Fig. 4(b)). The peak position near 120◦ corresponds √ to the Ti-[3] O-Ti angle, and the Ti-Ti distance in this structure ( 3l) corresponds to the first-peak position of the Ti-Ti g(r) for the apex-sharing LBUs (Fig. 4(b)). The Ti-O-Ti angles for the edge-sharing LBUs are narrowly distributed around 95◦ . This angle is close to the ideal angle of 90◦ but it is slightly wider than that because of the Ti-Ti repulsion (Fig. 1(c)). The Ti-O-Ti angles for the face sharing LBUs are narrowly distributed around 85◦ . This is also wider than the ideal angle of 70.5◦ because of the Ti-Ti repulsion (Fig. 1(d)). In edge and face-sharing LBU pairs, there is a tendency that the larger the average of m and n, the larger the

[m]

Ti-O-[n] Ti angle because the O-[m] Ti-O or O-[n] Ti-O angles are smaller

for larger m or n (Fig. 9(a)). The face sharing linkage is composed of the larger (m, n) pairs (Fig. 5). Figure 9(c) shows the Ti-O· · · O-Ti dihedral angle distributions for edge and face-sharing LBUs, where O· · · O is the edge shared by the two LBUs. The dihedral angles for the edgesharing LBUs are distributed around 180◦ , and the four atoms involved in the angle tend to be on the same plane. The dihedral angles for the face-sharing LBUs are distributed around 120◦ , which is slightly larger than the ideal angle of 109.5◦ when two octahedra are linked without deformation because of the Ti-Ti repulsion (Fig. 1(d)). The dihedral angles 8

are almost independent of coordination numbers m and n. Figure 10 shows the calculated dielectric susceptibility χij of amorphous models of BaTiO3 . Because amorphous BaTiO3 is isotropic, the calculated dielectric susceptibility tensors are nearly diagonal with almost equal diagonal elements. The experimental dielectric susceptibilities for the amorphous phases are 13 [4] and 9 [3], which are much smaller than the experimental values of the crystalline phase, which are 4500 and 200 for χ11 and χ33 , respectively[28]. The calculated dielectric susceptibilities for the amorphous phases were 14 for all the models in good agreement with experiments.

4.

CONCLUSIONS

We analyzed the structures of amorphous BaTiO3 within classical shell model. The densities of the amorphous phases were close to the experimental ones. The LBUs in amorphous BaTiO3 were mainly TiO6 in agreement with XAFS study. The TiOn polyhedra were randomly linked with their apexes, edges, and/or faces shared. If there are more Ba atoms around a Ti atom than the average, the Ti atom is surrounded by fewer Ti and O atoms than their respective averages, and it tends to be

[5]

Ti with many NBO atoms, while in

the opposite case, it is surrounded by more Ti and O atoms than their respective averages, and it tends to be

[7]

Ti with many TO atoms and many edge or face-sharing linkages with

other LBUs. The dielectric susceptibilities for the amorphous phases were much smaller than those for the crystalline phase, which is in agreement with experiments.

ACKNOWLEDGMENT

This work was supported in part by “Materials research by Information Integration” Initiative (MI2I) project of the Support Program for Starting Up Innovation Hub from Japan Science and Technology Agency (JST) and JSPS KAKENHI Grant-in-Aid for Scientific Research (B) No. 18H01710. The computations were carried out at ISSP Supercomputer Center, University of Tokyo, and at AIST.

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9

[2] A. I. Frenkel, Y. Feldman, V. Lyahovitskaya, E. Wachtel, and I. Lubomirsky, Phys. Rev. B 71, 024116 (2005). [3] V. Lyahovitskaya, I. Zon, Y. Feldman, S. Cohen, A. Tagantsev, and I. Lubomirsky, Adv. Mater. 15, 1826 (2003). [4] B.-S. Chiou and M.-c. Lin, Thin Solid Films 248, 247 (1994). [5] J. L. Wang, A. Pancotti, P. Jgou, G. Niu, B. Gautier, Y. Y. Mi, L. Tortech, S. Yin, B. Vilquin, and N. Barrett, Phys. Rev. B 84, 205426 (2011). [6] Y. Imanaka, H. Amada, F. Kumasaka, N. Awaji, and A. Kumamoto, J Nanopart Res 18, 102 (2016). [7] H. Inoue, A. Masuno, S. Kohara, and Y. Watanabe, J. Phys. Chem. B 117, 6823 (2013). [8] J. Yu, S. Kohara, K. Itoh, S. Nozawa, S. Miyoshi, Y. Arai, A. Masuno, H. Taniguchi, M. Itoh, M. Takata, T. Fukunaga, S.-y. Koshihara, Y. Kuroiwa, and S. Yoda, Chem. Mater. 21, 259 (2009). [9] A. I. Frenkel, D. Ehre, V. Lyahovitskaya, L. Kanner, E. Wachtel, and I. Lubomirsky, Phys. Rev. Lett. 99, 215502 (2007). [10] P. P. Phule, P. A. Deymier, and S. H. Risbud, J. Mater. Res. 5, 1104 (1990). [11] M. Sepliarsky, A. Asthagiri, S. R. Phillpot, M. G. Stachiotti, and R. L. Migoni, Curr. Opin. Solid. St. M. 9, 107 (2005). [12] S. Tinte, M. Stachiotti, S. Phillpot, M. Sepliarsky, D. Wolf, and R. Migoni, J. Phys.: Condens. Matter 16, 3495 (2004). [13] T. Hashimoto and H. Moriwake, Mol. Simulat. 41, 1074 (2015). [14] T. Hashimoto and H. Moriwake, Physica B: Condensed Matter 485, 110 (2016). [15] T. Hashimoto and H. Moriwake, J. Phys. Soc. Jpn. 85, 034702 (2016). [16] M. Parrinello and A. Rahman, Phys. Rev. Lett. 45, 1196 (1980). [17] G. Martyna, M. Tuckerman, D. Tobias, and M. Klein, Mol. Phys. 87, 1117 (1996). [18] T.-Q. Yu, J. Alejandre, R. Lpez-Rendn, G. J. Martyna, and M. E. Tuckerman, Chem. Phys. 370, 294 (2010). [19] D. J. Tobias, G. J. Martyna, and M. L. Klein, J. Phys. Chem. 97, 12959 (1993). [20] V. Lyahovitskaya, Y. Feldman, I. Zon, E. Wachtel, K. Gartsman, A. K. Tagantsev, and I. Lubomirsky, Phys. Rev. B 71, 094205 (2005). [21] D. Ehre, H. Cohen, V. Lyahovitskaya, and I. Lubomirsky, Phys. Rev. B 77, 184106 (2008).

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[22] C. Le Losq, D. R. Neuville, P. Florian, G. S. Henderson, and D. Massiot, Geochim. Cosmochim. Ac. 126, 495 (2014). [23] L. Cormier, D. Ghaleb, D. R. Neuville, J.-M. Delaye, and G. Calas, J. Non-Cryst. Solids 332, 255 (2003). [24] P. Ganster, M. Benoit, W. Kob, and J.-M. Delaye, J. Chem. Phys. 120, 10172 (2004). [25] M. Benoit, S. Ispas, and M. E. Tuckerman, Phys. Rev. B 64, 224205 (2001). [26] N. M. Vedishcheva, B. A. Shakhmatkin, M. M. Shultz, B. Vessal, A. C. Wright, B. Bachra, A. G. Clare, A. C. Hannon, and R. N. Sinclair, J. Non-Cryst. Solids 192-193, 292 (1995). [27] M. Bauchy, J. Chem. Phys. 141, 024507 (2014). [28] W. J. Merz, Phys. Rev. 75, 687 (1949).

11

FIGURES

12

(a)

(b)

(c)

(d)

FIG. 1. (Color online) (a) A snapshot of the amorphous BaTiO3 . The large green and blue circles represent Ba and Ti atoms, respectively. The small red, yellow, green, and gold circles represent NBO, BO, TO, and beyond-tricluster oxygen ([n>3] O) atoms, respectively. The TiOm LBUs are shown by polyhedra. (b), (c), and (d) represent apex, edge, and face-sharing ideal TiO6 LBUs, respectively.

13

(a)

20 Ti-O crystal amorphous Model 1 amorphous Model 2 amorphous Model 3

g(r), n(r)

15

10

5

0

g(r)

(b)

0

1

2

3

4

5

6

7

r/Å

14 Ti-O (Total)

[6]Ti-[n>1]O

12

[4]Ti-[1]O

[6]Ti-O

[4]Ti-[n>1]O

[7]Ti-[1]O

10

[4]Ti-O

[7]Ti-[n>1]O

[5]Ti-[1]O

[7]Ti-O

8

[5]Ti-[n>1]O

[8]Ti-[1]O

[5]Ti-O

[8]Ti-[n>1]O

6

[6]Ti-[1]O

[8]Ti-O

4 2 0

0

0.5

1

1.5

2

2.5

3

r/Å

FIG. 2. (Color online) Ti-O g(r)s for crystalline and amorphous BaTiO3 . (a) Total g(r)s (solid lines) and n(r)s (dashed and dotted lines for O around Ti and Ti around O, respectively). (b) Total and decomposed Ti-O g(r)s for the amorphous phase.

14

1 nTiO

Probability

0.8

1 2 3 4 5 6

0.6 0.4 0.2 0

0

1

2

3

4

5

6

7

8

9

10

nOTi

FIG. 3. (Color online) The probability distribution for TiOn (a thick line) and the probability of the number of Ti atoms around an O atom (nTiO ) for each TiOn (bars).

15

(a)

20 Ti-Ti crystal amorphous Model 1 amorphous Model 2 amorphous Model 3

g(r), n(r)

15

10

5

0

(b)

8 7

g(r), n(r)

6 5 4 3 2

0

1

2

3

4

5

6

7

r/Å Ti-Ti amorphous (total) apex sharing edge sharing face sharing (m,n)=(5,5) (5,6) (5,7) (5,8) (6,6) (6,7) (6,8) (7,7) (7,8) (8,8)

1 0

1

2

3

4

5

r/Å

FIG. 4. (Color online) (a) Ti-Ti g(r)s (solid lines) and n(r)s (dot-dashed lines) for crystalline and amorphous BaTiO3 . (b) Ti-Ti g(r)s (solid lines) and n(r)s (dashed lines) for amorphous BaTiO3 decomposed into apex, edge, and face-sharing LBU linkages. The g(r)s are further decomposed into

[m] Ti-[n] Ti

components which are shown in thin solid, dashed, and dot-dashed lines for apex,

edge, and face-sharing LBU linkages, respectively.

16

m 5

6

7

8

n[n]Ti[m]Ti

apex 4.5 edge 4 face 3.5

11 10 9

3

8

2.5

7

2

6

1.5

5

1

4

0.5 0

nBa[m]Ti

4

5

44 45 46 47 48 54 55 56 57 58 64 65 66 67 68 74 75 76 77 78 84 85 86 87 88

3

mn

FIG. 5. (Color online) The number of [n] Ti around [m] Ti (n[n] Ti[m] Ti ; bars). Each bar is decomposed into apex, edge, and face-sharing linkage components. The number of Ba atoms around

[m] Ti

(nBa[m] Ti ) is also shown in black circles. m and n represent nOTi , and m in the horizontal axis corresponds to nOTi in the horizontal axis of Fig. 3.

17

(a)

20 Ba-O crystal amorphous Model 1 amorphous Model 2 amorphous Model 3

g(r), n(r)

15

10

5

0

(b)

0

1

2

3

4

5

6

7

r/Å

3.5

Ba-O (Total) Ba-[1]O Ba-[n>1]O

3

g(r)

2.5 2 1.5 1 0.5 0

0

1

2

3

4

5

6

7

r/Å

FIG. 6. (Color online) Ba-O g(r)s for crystalline and amorphous BaTiO3 . (a) Total g(r)s (solid lines) and n(r)s (dashed and dotted lines for O around Ba and Ba around O, respectively). (b) Total and decomposed Ba-O g(r)s for the amorphous phase.

18

FIG. 7. (Color online) The probability distribution for the number of Ba and Ti atoms bonded to an O atom (nBaO and nTiO , respectively).

19

(a) 20

Ba-Ba crystal amorphous Model 1 amorphous Model 2 amorphous Model 3

g(r), n(r)

15

10

5

0

0

1

2

3

2.5

5

6

7

4

5

6

7

4

5

6

7

4

5

6

7

4

5

6

7

Ba-Ba amorphous (total) apex sharing edge sharing face sharing 4 O sharing 5 O sharing 6 O sharing

2

g(r), n(r)

4 r/Å

1.5

1

0.5

0

(b)

1

2

3 r/Å

20

Ba-Ti crystal amorphous Model 1 amorphous Model 2 amorphous Model 3

g(r), n(r)

15

10

5

0

0

1

2

3 r/Å

2.5 Ba-Ti amorphous (total) apex sharing edge sharing face sharing 2 face sharing 5 O sharing 6 O sharing

g(r), n(r)

2

1.5

1

0.5

0

(c)

1

2

3 r/Å

20

O-O crystal amorphous Model 1 amorphous Model 2 amorphous Model 3

g(r), n(r)

15

10

5

0

0

1

2

3 r/Å

FIG. 8. (Color online) g(r)s (solid lines) and n(r)s (dot-dashed lines) for Ba-Ba (a), Ba-Ti (b), and O-O (c). Ba-Ba and Ba-Ti g(r)s are decomposed to apex, edge, face, and n O sharing LBU linkage components in the lower panels in (a) and (b), respectively.

20

0.030

(a)

O-[n]Ti-O angle

Probability Density

0.025

Total n=4 5 6 7 8

0.020 0.015 0.010 0.005 0.000

Probability Density

(b)

0

20

40

60

80

100

120

140

160

180

120

140

160

180

120

140

160

180

angle / degree

0.04 [m]Ti-O-[n]Ti

0.035

angle (6,6) (6,7) (6,8) (7,7) (7,8) (8,8)

apex sharing edge sharing face sharing (m,n)=(5,5) (5,6) (5,7) (5,8)

0.03 0.025 0.02 0.015 0.01 0.005 0

(c)

0

40

60

80

100

angle / degree

0.025 [m]Ti-O···O-[n]Ti

0.02 Probability Density

20

dihedral angle (6,6) (6,7) (6,8) (7,7) (7,8) (8,8)

edge sharing face sharing 2 face sharing (m,n)=(5,5) (5,6) (5,7) (5,8)

0.015

0.01

0.005

0

0

20

40

60

80

100

angle / degree

FIG. 9. (Color online) (a) The O-[n] Ti-O angle distributions. The dashed, dot-dashed, and dotted lines correspond to (b) The

[m>1] O-[n] Ti-[m>1] O, [m>1] O-[n] Ti-[1] O,

[m] Ti-O-[n] Ti

and

[1] O-[n] Ti-[1] O

angles, respectively.

angle distributions. The thin solid, dashed, and dot-dashed lines correspond

to apex, edge, and face-sharing LBU linkages, respectively, for each (m, n) pair. (c) The

[m] Ti-

O· · · O-[n] T dihedral angle distributions, where O· · · O is the edge shared by the two LBUs. The thin dashed and dot-dashed lines correspond to edge and face-sharing LBU linkages, respectively, for each (m, n) pair.

21

(a)

20

Dielectric Susceptibility

15 10 χ11 χ12 χ13 χ22

5 0

χ23 χ33 <χii>

-5 -10

Dielectric Susceptibility

(b)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time (ns)

25 20 15

χ11 χ12 χ13 χ22

10 5

χ23 χ33 <χii>

0 -5 -10

Dielectric Susceptibility

(c)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time (ns)

50

χ11 χ12 χ13 χ22

40 30

χ23 χ33 <χii>

20 10 0 -10 -20

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time (ns)

FIG. 10. (Color online) Dielectric susceptibilities of amorphous models at 300 K. (a), (b), and (c) correspond to Models 1, 2, and 3, respectively.

22

TABLES

23

TABLE I. Amorphous models. ρMD represents the density by the perfect crystal NPT MD simulation. density

cooling rate (K / fs)

Model 1

0.802 ρMD

5.5×10−1

Model 2

0.808 ρMD

5.5×10−2

Model 3

0.813 ρMD

5.5×10−3

24

TABLE II. The averaged charge on TiOn . n

Model 1

Model 2

Model 3

Ti[2] On

4

-4.0

-3.0

-4.0

0.0

5

-2.1

-2.1

-2.7

-1.0

6

-1.8

-1.8

-1.8

-2.0

7

-1.6

-1.4

-1.1

-3.0

8

-1.1

-1.3

-0.8

-4.0

-1.9

-0.8

-5.0

9

25

Declaration of interests ☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐ The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Declarations of interest: none