Domain structures in circular ferroelectric nano-islands with charged defects: A Monte Carlo simulation

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Domain structures in circular ferroelectric nano-islands with charged defects: A Monte Carlo simulation

D. P.Chen,¹Y.Zhang,^1,a)X. M.Zhang,²L.Lin,¹Z. B.Yan,¹X. S.Gao,²and J.-M.Liu^1,2

1Laboratory of Solid State Microstructures and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

2Institute for Advanced Materials and Laboratory of Quantum Engineering and Materials, South China Normal University, Guangzhou 510006, China

(Received 27 May 2017; accepted 16 July 2017; published online 28 July 2017)

The pattern evolution of striped and vortex domain structures in circular ferroelectric nano-islands with in-plane polarization and charged defects is investigated using the Monte Carlo simulation based on the Landau-Devonshire phenomenological theory. The domain structures of islands under- going different annealing processes are compared. Given embedded charge carriers at the center of islands, the domain patterns would be markedly affected as a result of the competition and balance between the electrostatic charge energy and other free energy terms in the Landau-Devonshire phenomenological theory. The symmetry of islands with different sizes and charge quantities is also analyzed. The present work provides a simple explanation of a variety of ferroelectric nano-islands and proposes an alternative promising approach to tune the domain structures and symmetry for the applications of nano-sized ferroelectric devices.Published by AIP Publishing.

[http://dx.doi.org/10.1063/1.4996350]

I. INTRODUCTION

It is always intriguing to explore more functionalities of ferroelectric (FE) materials and improve the performances of FE materials for both fundamental investigations and diverse applications in industry and daily life.^1–3In general, two of the conventional approaches have been paid special attention to deal with this issue. One is to explore new kinds of FE materials with unusual underlying physics,⁴ for instance, improper ferroelectrics whose ferroelectricity results from various primary order parameters and their coupling rather than the FE distortions.^5,6 The other is to design functional FE components containing nanoscale/composite structures with the help of rapid-developing micro-fabrication technol- ogy. For the latter, researchers have been focusing on electromechanical transducers,⁷magnetoelectric heterostructures,^8,9 interfacial-coupled superlattices,^10,11 low-dimensional FE devices,^12,13and so on. Among them, a promising way is to downscale the materials into nanostructures such as thin films, nano-dots, and nano-wires^14–16 because these small- scale structures, including free boundaries, electromechanical interfaces, and various defects, would provide a wealth of splendid possibilities to tune FE domain patterns for specific applications.^17,18 The latter strategy has become one of the most concerned topics in recent research activities.

The most convenient way along this line is to start from well-known tetragonal perovskite ferroelectrics like BaTiO₃ and Pb(Zr_1–xTi_x)O₃(PZT) and focus on a well-aligned stripe- like 90-domain structure which is one of the most representative characteristics in bulk systems.^19,20The well-aligned stripe structure results from the competition among multifold interactions, especially the delicate balance of the long-range elastic

energy and electric (dipole-dipole interaction) energy.²¹ However, when it comes to low-dimensional systems, the abundant boundaries, strong size effects, and localized defects could not be ignored. In other words, the competition between elastic energy and electric energy terms would not be dominant in determining the evolution of domain structures for nanostructure FE materials any more. Therefore, the domain patterns of those structures may be much more complicated and on the other hand offer more opportunities for additional functionalities and effects. One may take FE nano-islands as an example, which are extensively investigated nowadays. The strong depolarization effect will not favor the formation of regular stripe-like domains near the boundaries but constrain the corresponding electric dipoles to be as parallel to the edges as possible in order to minimize the excess depolarization energy.²²As a consequence, the domain pattern of nano- islands tends to be a single-vortex domain structure for the small size of islands, while it generally evolves into a 90- striped domain in coexistence with the vortex pattern when the size gets larger.^21,23In addition, many inner defects, such as charged oxygen vacancies, should also play a more important role in the interaction with other effects due to the nanometer scale.

It should be mentioned that the variety of domain patterns in FE nano-islands are beneficial for multifarious applications of FE materials. If the relationships among those energy terms are understood, one might be able to regulate the movement and rotation of domains caused by both external stimuli and inner conditions. Therefore, it is necessary to pay close attention to the domain structures of FE nano-islands and attempt to explain them. Recently, some experimental results showed that the domains in BiFeO₃nano-islands, given specific conditions of fabrication, exhibit various kinds of spontaneous FE topo- logical patterns (in real space), such as the center-convergent

a)Author to whom correspondence should be addressed: zhangyang919@

gmail.com.

0021-8979/2017/122(4)/044103/7/$30.00 122, 044103-1 Published by AIP Publishing.

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domains, center-divergent domains, double-center domains, and so on.^12,13,24Analyses together with some numerical simulations revealed that those different patterns may come from the inside elastic and electric interactions, surface strain, and charge accumulation. These reversible domains are promising for potential applications in high density memory devices. At the same time, many simulations also analyzed the evolution of domain patterns in single crystal BaTiO₃ with oxygen vacancies and found that those charged oxygen vacancies would have a great influence on the domain structures as well as FE and piezoelectric properties.^25,26

Inspired by these results, we are particularly interested in the role of charged defects in FE nanostructures. This motivation is straightforward and could be of general interest since the most favored ordered FE nanostructures are fabricated via the top-down approaches.² Typically, one starts from well-prepared epitaxial FE thin films and obtains well- aligned FE nano-dot/nano-island arrays by microscopic fabrication techniques such as ion-etching with mask. In this sense, the as-fabricated structures inevitably receive substan- tial external beaming which thus implants charged defects such as oxygen vacancies most typically.²⁴While the role of charged defects in FE materials has been a common source for a number of non-intrinsic phenomena,²⁷ it has not yet received sufficient attention from theoretical aspects. Our major task in this paper is to investigate the domain structures in FE nano-islands with charged defects.

As a representative case, we choose circular nano-islands of different sizes that include one charged defect at each center, in order to investigate the role of this charge in the domain structure formation. Although mobile charges are also possible, they may make domain patterns more complicated, and we only consider fixed charges for the sake of simplification.

We first perform Monte Carlo simulations on the two- dimensional (2D) square FE lattice with periodic boundary conditions as an approximation to an infinite tetragonal FE thin film. When this 2D lattice is extended to nano-islands, it is assumed that the nano-islands have a sufficient diameter over height ratio, and then, the FE polarization would be in- plane aligned. Subsequently, we crop this whole lattice into an ordered circular island array to simulate the top-down nanoscale fabrication process and introduce a point charge in each island at the center to check how these nanostructure and charged defects modulate the evolution of FE domains. It should be noted that extension to the three-dimensional (3D) lattice is direct.

II. MODEL AND SIMULATION METHODOLOGY

We start from a 2D LLsquare lattice, where on each sitei, the energy can be represented by two order parameters:

electric dipole vector P(r) ¼ (Px, Py) and elastic displacement vectoru(r)¼(ux,uy), withxandybeing the Cartesian coordinates of each site. Similar to those treatments in the earlier literature, the free energy of a FE lattice can be divided into four terms: bulk free energyFbulk, gradient free energyFgra, elastic free energyFela, and electric free energy Fele.^27,28 For such a lattice, the total free energy density is written as

ftotal¼fbulkþfgraþfelaþfele: (1) For the Landau free energy term, we consider a Landau- Devonshire polynomial extended to the sixth-order

fbulk¼A1ðP²_xþP²_yÞ þA11ðP⁴_xþP⁴_yÞ þA12P²_xP²_y þA111ðP⁶_xþP⁶_yÞ þA112ðP⁴_xP²_yþP⁴_yP²_xÞ; (2) where A1, A11, A12, A111, and A112 are the Landau coefficients which are temperature-independent except A1. Here, A1¼A10(T–T0) with critical point T0 for the FE transition and positive A1and negativeA11are necessary for the first- order transitions.

The gradient free energy density or the domain wall energy density can be expressed as

fgra¼1

2 G11P²_x;xþP²_y;y

þG12Px;xPy;y

h

þG44ðPx;yPy;xÞ²þG⁰44ðPx;yþPy;xÞ²i

; (3)

where Gij>0 are the gradient energy coefficients and Pi,j¼@Pi/@rj.

The elastic free energy, which is the third term on the right hand of Eq.(1), is given by

fela¼1

2cijklheije⁰_iji

ekle⁰_kl

; (4)

where cijkl is the elastic stiffness tensor, total strain eij¼eij

þ1/2(ui,jþui,j) with homogeneous strain eij, ui,j¼@ui/@rj, and eigenstraine⁰_ij¼QijklPkPlwith electrostrictive coefficients Qijkl. With regard to the given conditions in this lattice (2D and no homogeneous strain), the elastic free energy term can be divided into three parts: the first related only to the gra- dients of elastic displacement vector ui,j, the second related only to electric dipole vectorPiwhich can be merged into the bulk free energy term, and the last related to bothui,jandPi.²⁹ Therefore, the elastic free energy can be rewritten as

fela¼1

2C11u²_x;xþu²_y;y

þC12ux;xuy;yþ2C44u²_x;y ðq11ux;xþq12uy;yÞP²_xðq11uy;yþq12ux;xÞP²_y q44ðux;yþuy;xÞPxPy; (5) whereq11¼C11Q11þ2C12Q12,q12¼C11Q12þC12(Q11þQ12), q44¼2C44Q44, andCijand Qij are related tocijklandQijklby Vogit’s notation.

Finally, the electric free energy density is simply calculated by

fele¼ ðExPxþEyPyÞ; (6) where electric field E¼(Ex, Ey)¼–r/ with potential / at siter¼(rx,ry). In general, the total electric potential comes from multiple sources, including other dipoles (i.e., dipole- dipole interactions), charge carriers (e.g., oxygen vacancies), and external applied electric field. Thus, densityfeleat sitei can be rewritten as

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fele¼ 1 4pe

X

j6¼i

P_iP_j

jrirjj³3PiðrirjÞPiðrirjÞ jrirjj⁵

" #

1 4pe

X

j

QjP_iðr_iR_jÞ

jriR_jj³ E^extPi; (7) where the pre-factor (1/4pe)¼1 is taken for normalizing purposes,R_jand Qjare the coordinates and charge quantities of charge carriers, respectively, andE^ext is the external electric field. For the long-range dipole-dipole interaction, precise methods like the Ewald summation scheme are usually tedious.³⁰Instead, we adopt a more tractable calculation by choosing a preset truncating distanceR0from central site i(R0¼8 in our simulation). Fortunately, computed energy values via this kind of scheme are sufficiently reliable for such a 2D lattice compared with the results by the Ewald summation method, while it is known that this truncation does not work well for 3D lattices.

For the practical simulations, we submit an intact lattice for annealing from a highTT0to a lowTT0with sufficient relaxation. Mathematically, a lattice iteration concern- ing mechanical and electric equilibrium conditions can be solved by the time-dependent Ginzburg-Landau (TDGL) equation^27,29 and the spectral iterative perturbation (SIP) method.³¹However, following earlier work, we still employ the Monte Carlo simulation to track the domain evolution for the sake of convenience and comparison, and thus, a detailed simulation process is not given here.^32,33

After introducing an independent FE nano-island array, we assumeP(r)0 and ui,j(r)0 (i,j¼x,y) for the sites outside islands, which means that the intervals between islands are empty (full of air). In the present simulation, we choose the BaTiO₃thin film as the object lattice. The parameters of relevant materials are taken from the literature.^34,35 Lattice constanta0, Landau coefficienta₀, and polarizationP0

atT¼T1far belowT0are selected as reference values to nor- malize other parameters.³⁶The dimensionless parameters for simulations are listed in TableI. Here, it should be mentioned that the electric dipole at each site does not correspond to a realistic lattice unit polarization, and it is a coarse-grained dipole for computations.

III. RESULTS AND DISCUSSION

We start from a FE lattice after annealing with lattice sizeL¼64, whose domain pattern is illustrated in Fig.1(a).

The well-aligned stripe-like 90 domain structure with walls along the [11] direction is observed, and the results are in

accord with those based on the phase-field simulations using the TDGL equation.^37,38 Subsequently, a nano-island array with an island diameter ofD¼6 30 is introduced into the lattice. After a sufficiently long-time relaxation, the domain structure of each island in this array would reach the energy- minimum state eventually. It should be pointed out that the interactions between these islands are rather small since both the bulk free energyfbulkand the gradient free energyfgraare short-range interactions and the dipole-dipole free energy reduces as the distance increases by the third power. As for elastic free energyfela, the values ofui,j at the sites outside islands are set to equal zero, and so,felais fully released.

TABLE I. Parameters chosen for the simulation (Boltzmann constantkB¼1).

Parameter (unit) Value Parameter (unit) Value Parameter (unit) Value

A10(a10) 0.10 A11(a11P²₀/ja0j) –0.24 A12(a12P²₀/ja0j) 4.50

A111(a₁₁₁P⁴₀/ja0j) 0.49 A112(a₁₁₂P⁴₀/ja0j) 1.20 G11(g11/a²₀ja0j) 1.60 G12(g12/a²₀ja0j) 0.00 G44(g44/a²₀ja0j) 0.80 G⁰44(g⁰44/a²₀ja0j) 0.80 C11(C*11/ja0jP²₀) 2.75 C12(C^*12/ja0jP²₀) 1.79 C44(C^*44/ja0jP²₀) 0.54

Q11(Q^*11/ja0j) 0.142 Q12(Q^*12/ja0j) –0.0074 Q44(Q^*44/ja0j) 0.0157

L(a0) 64 T0(Tc/(Tc–T1)) 5.00

FIG. 1. Simulated in-plane domain patterns for the intact lattice (a), lattice with a circular nano-island array of sizesD¼8 (b),D¼10 (c),D¼12 (d), andD¼14 (e). The red, blue, green, and yellow domains indicate directions [10], [01], [10], and [01], respectively. (f) The sketches of four types of vortices.

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As an example, we present a 44 circular island array in Figs.1(b)–1(e) withD¼8, 10, 12, and 14, respectively. It is obvious that these islands either transform into vertices or maintain stripe-like 90 domains. The larger the islands are, the more striped domains remain. A simple explanation is that the dipole vectors near the boundaries of islands tend to be parallel to edges due to the strong depolarization effect, and so, the vortex pattern will lower the energy of islands significantly.

Actually, this kind of vortex pattern is the ground state.

However, since the initial states of islands are cropped from a whole lattice made of striped domains (temperature TT0

during the evolution process of islands), different nano-islands have different initial domain configurations. Some islands may be caught in a local energy-minimum state, i.e., stripe-like 90 domains with only local deformation near boundaries, instead of the energy-lowest vortex state.

For reference, the domain patterns of individual islands with different sizes that have undergone the whole annealing process are shown in Fig. 2. These domain structures all reach the lowest energy states by choosing and comparing different annealing paths. WhenDis small, the ratio between the area near boundaries of one island and that of the whole island is quite high, and so, the domain structures will be affected by boundaries easily. In contrast, asD gets larger, such a size effect will not dominate the evolution of domain patterns but work together with other energy terms. This kind of delicate competition and balance finally lead to more complicated domain structures. As illustrated in Fig. 2(d), the stripe-like 90 domains in coexistence with the vortex structure are observed, rather than a single vortex structure.

The results are similar to those in the former work.²¹In fact, it can also account for the decrease of vortex structures as the island size grows as shown in Figs.1(b)–1(e), that is, if island sizeDbecomes pretty large, the depolarization effect around boundaries of islands will not have a great influence on the dipoles near the center, and so, more islands may be trapped in the striped state and vice versa.

In the practical experiments, above standard FE nano- islands rarely if ever exist. Instead, these islands may be easily affected by external stimuli such as electric field and stress or internal conditions such as oxygen vacancies and disloca- tions, which make the domain structures much more perplex- ing. Here, we focus only on the roles that charged defects play in the evolution of domain patterns. For simplification, we place one hypothetical point charge with quantityQin the center of each nano-island as an approximation to a charge inside the island. It should be noted that the charge quantity Q has an arbitrary unit since all the parameters used in our simulation have been normalized in advance, and the obtained values are useful only at the semi-quantitative level.

First, the domain patterns of nano-islands withD¼8 that contain differentQvalues in the center are shown in Fig.3.

The arrows represent the dipole vectors (both the amplitude

FIG. 2. Simulated in-plane domain patterns with the lowest energy of island sizesD¼8 (a),D¼12 (b),D¼16 (c), and D¼30 (d), respectively. The red, blue, green, and yellow domains indicate directions [10], [01], [10], and [01], respectively.

FIG. 3. Simulated in-plane domain patterns with an island size ofD¼8 at charge quantities Q¼0 (a), Q¼–10 (b), Q¼10 (c), Q¼–30 (d), and Q¼30 (e). The arrows indicate the dipole vectors at corresponding sites, and the colors indicate the directions of dipoles.

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and the direction) at corresponding sites, while the colors again indicate vectors’ directions. At first glance, the exis- tence of charges has a marked impact on the domain structures, which is predictable. WhenQ¼0 as shown in Fig.3(a), the single-vortex domain structure is one of the energy-lowest states shown in Fig.1(b). ForQ<0, the charge-induced electric field points to the center of islands, and hence, dipoles prefer to aim at the core, especially the ones near the center.

If the absolute value ofQis large enough, the energy as gen- erated,fQ, will predominate in the formation of the vortex, as shown in Fig.3(d). For Q>0, one can observe the opposite results. However, one phenomenon in common is that the ori- entations of the dipole rotate in the clockwise direction when we look into the dipoles around the center by the clockwise direction. According to definition,^39,40such a kind of vortex is called the “vortex,” rather than the “anti-vortex.” Therefore, four sketches in Fig.1(f)and five patterns in Fig.3all belong to the “vortex.”

For further discussion on the influences of different charge values in bigger nano-islands, we then present the domain patterns of nano-islands atD¼14 withQ¼0, 8, 20, 50, 200, and 600, respectively, in Fig. 4. The initial state (i.e.,Q¼0) is one energy-minimum structure composed of striped domains. AsQgets larger and larger, the total free energy except charge energyfQ, let us sayf⁰total¼ftotal–fQ, becomes less and less important. Finally, the domain pattern is nearly center symmetric when Q is extremely large, as shown in Fig.4(f). Meanwhile, the spatial contours off⁰total

andfQare also calculated as shown in Fig.5for islands with D¼14 and Q¼50 at the beginning (just after islands are introduced from the intact lattice) and the end of evolution, where the grey arrows indicate the dipole vectors. Figure 5(a)shows the contour of termf⁰totalfor initial patterns. One can observe that the energy insides domains is the lowest and the energy near the boundaries of the island is higher than that at other sites, while the energy along the domain walls remains in the middle. However, when the chargeQis added, the value offQat the left side of the center would get so large that f⁰total no longer determines the domain structures there, as shown in Fig.5(b). As a consequence, dipoles near the center point to the opposite directions of the core and dipoles far away from the center are affected by both f⁰totalandfQ. Figures 5(c)and 5(d)present the energy contours of termsf⁰totalandfQfor the final pattern in Fig.4(d). It is clear that the energy f⁰total gets a bit higher especially around the center, yetfQ is significantly lower than that of the initial structure. In other words, the dipole vectors in the center have almost been “fixed” by the charge carrier orfQ, while this kind of control decays rapidly with the increasing distance from the center. Nevertheless, those “unfixed”

dipoles would interact with others by long-range elastic and electric free energy terms, and the whole island finally evolves into the energy-lowest state. In addition, whenQis fairly small, f⁰total and fQ are on the same magnitude and only a part of domains in the island is modified by the charge Qand interactions between dipoles in the center and them- selves. If Q is sufficiently large, fQ dominates the domain structures, which have the radial symmetry.

In order to analyze the symmetry of each nano-island quantitatively, we here introduce a simple scaling parameter xwhich is written as:

x¼ he⁰_iPii_island¼1 N

X^N

i¼1

r⁰_iPi

jr⁰_ij ; (8) whereh irepresents the spatial averaging over the whole island,r⁰_iis the Cartesian coordinate of dipole vectorP_iwith the originO⁰at the center of the island,e⁰_iis the unit vector corresponding tor⁰_i, andNis the total number of dipole vectors in the island, as illustrated in Fig. 6(a). It needs to be noted that the range of amplitudes ofP_iis (0, 1], and so, the value domain ofxis from –1 to 1 according to the definition.

The scaling parameterxequaling zero indicates that orienta- tions of dipole vectors on average express no tendency along the originO⁰. In our case, the island should have the fourfold rotational symmetry, and the directions of dipoles are parallel

FIG. 4. Simulated in-plane domain patterns with an island size ofD¼14 at charge quantitiesQ¼0 (a),Q¼8 (b),Q¼20 (c),Q¼50 (d),Q¼200 (e), and Q¼600 (f). The arrows indicate the dipole vectors at corresponding sites, and the colors indicate the directions of dipoles.

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to the axes, which is illustrated as vortex type-(i) and type-(ii) in Fig.1(f). Usually, it occurs whenQ¼0. Forx>0, P_i is apt to point outwards fromO⁰, whose typical domain patterns are shown in Figs. 3(c) and 3(e). Usually, it occurs when

Q>0. Forx<0, the situation is just the opposite, as shown in Figs. 3(b) and 3(d). Particularly, x¼61 means that the dipole vector P_iand its position vectorr⁰_iare all parallel or antiparallel, and the amplitude of Pi equals 1 for each site (Pi//r⁰iandjPij ¼1 for alli) as vortex type-(iii) or type-(iv) in Fig. 1(f). We then plot the curves of x as a function of Q withD¼8, 18, and 28, respectively, in Fig. 6(b). Naturally, scaling parameter x and charge quantity Q exhibit strong positive correlation for each island size D. When D is smaller, the whole island on average would be affected byfQ

more easily, and so,xis larger.

Finally, it should be mentioned that the present simulations are too simplified. We only deal with the domain structures of 2D circular FE nano-islands. One may consider various shapes of islands depending on the preparation tech- nology. Also, the 3D cases where in-plane domains coexist with out-of-plane domains are worth further investigation since they would be more realistic. Moreover, the charged defects would not exactly remain in the center of islands as point charges. Instead, they could distribute in a certain range of islands. Nevertheless, the underlying physics should be similar for these extended cases. The main task here is to explain the formation of various kinds of domain patterns with different symmetries for FE nano-islands containing charged defects so that one can try to regulate the movement and even the flip of domains. In fact, some experiments revealed a variety of FE nano-islands.²⁴ Those experiment findings as well as our results here allow us to believe that analyzing and modifying the domain structures of FE nano-islands are quite promising for the applications of FE materials.

IV. CONCLUSION

In summary, we have investigated the domain structures and symmetry in circular ferroelectric nano-islands, especially the effects of charged defects. It has been found that the domain patterns are possible to be caught in a striped energy- minimum state after being cut down from an intact lattice consisting of stripe-like 90domains, while the energy-lowest state is the vortex pattern for the islands with appropriate sizes. At the same time, charge carriers at the center of islands could notably vary the evolution of domain patterns. By analyzing the introduced scaling parameterx, we found that the domain symmetry of FE nano-islands would be determined by the quantities and signs of charged defects together with the sizes of islands. The present work provides a possible approach for practically tuning domain structures in FE nano- islands for specific electric applications.

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51431006, 11374147, and 11234005) and the National Key Research Program of China (Grant Nos. 2015CB654602 and 2016YFA0300101).

1N. Setter, D. Damjanovic1, L. Eng, G. Fox, S. Gevorgian, S. Hong, A.

Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K.

Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer,J. Appl. Phys.100, 051606 (2006).

2J. F. Scott,Science315, 954 (2007).

FIG. 5. Simulated energy contours for the total free energy except the charge energyf⁰total(a) and (c) and the charge energyfQ(b) and (d) at the initial and final moment. The grey arrows indicate the dipole vectors at corresponding sites.

FIG. 6. (a) An illustration for calculating scaling parameterx. (b) Simulated scaling parameterxas a function of charge quantityQwith island sizes D¼8, 18, and 28, respectively.

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3D. W. Bondurant and F. P. Gnadiger,IEEE Spectrum26, 30 (1989).

4K. Xu, X. Z. Lu, and H. Xiang,npj Quantum Mater.2, 1 (2017).

5Y. S. Oh, X. Luo, F. T. Huang, Y. Wang, and S. W. Cheong,Nat. Mater.

14, 407 (2015).

6H. Pang, F. Zhang, M. Zeng, X. Gao, M. Qin, X. Lu, J. Gao, J. Dai, and Q.

Li,Quantum Mater.1, 16015 (2016).

7J. Narvaez, F. Vasquez-Sancho, and G. Catalan,Nature538, 219 (2016).

8J. M. Hu, C. G. Duan, C. W. Nan, and L. Q. Chen,npj Comput. Mater.3, 18 (2017).

9P. Jain, A. Stroppa, D. Nabok, A. Marino, A. Rubano, D. Paparo, M.

Matsubara, H. Nakotte, M. Fiebig, S. Picozzi, E. S. Choi, A. K. Cheetham, C. Draxl, N. S. Dalal, and V. S. Zapf, npj Quantum Mater. 1, 16012 (2016).

10P. Wu, X. Ma, Y. Li, C. B. Eom, D. G. Schlom, V. Gopalan, and L. Q.

Chen,Appl. Phys. Lett.107, 122906 (2015).

11A. K. Yadav, C. T. Nelson, S. L. Hsu, Z. Hong, J. D. Clarkson, C. M.

Schlep€utz, A. R. Damodaran, P. Shafer, E. Arenholz, L. R. Dedon, D.

Chen, A. Vishwanath, A. M. Minor, L. Q. Chen, J. F. Scott, L. W. Martin, and R. Ramesh,Nature530, 198 (2016).

12G. Tian, F. Zhang, J. Yao, H. Fan, P. Li, Z. Li, X. Song, X. Zhang, M.

Qin, M. Zeng, Z. Zhang, J. Yao, X. Gao, and J. Liu,ACS Nano10, 1025 (2016).

13L. Zhao, Z. Lu, F. Zhang, G. Tian, X. Song, Z. Li, K. Huang, Z. Zhang, M.

Qin, S. Wu, X. Lu, M. Zeng, X. Gao, J. Dai, and J. M. Liu,Sci. Rep.5, 9680 (2015).

14D. Chen, Z. Chen, Q. He, J. D. Clarkson, C. R. Serrao, A. K. Yadav, M. E.

Nowakowski, Z. Fan, L. You, X. Gao, D. Zeng, L. Chen, A. Y.

Borisevich, S. Salahuddin, J. M. Liu, and J. Bokor, Nano Lett.17, 486 (2017).

15J. M. Hu, T. N. Yang, L. Q. Chen, and C. W. Nan,J. Appl. Phys.113, 194301 (2013).

16A. Schilling, R. M. Bowman, G. Catalan, J. F. Scott, and J. M. Gregg, Nano Lett.7, 3787 (2007).

17J. Mannhart and D. G. Schlom,Science327, 1607 (2010).

18S. Dong, J. M. Liu, S. W. Cheong, and Z. Ren,Adv. Phys.64, 519 (2015).

19G. Arlt and P. Sasko,J. Appl. Phys.51, 4956 (1980).

20E. K. W. Goo, R. K. Mishra, and G. Thomas, J. Appl. Phys.52, 2940 (1981).

21F. Xue, X. S. Gao, and J. M. Liu,J. Appl. Phys.106, 114103 (2009).

22I. Ponomareva, I. I. Naumov, and L. Bellaiche,Phys. Rev. B72, 214118 (2005).

23J. Wang, M. Kamlah, and T. Y. Zhang,J. Appl. Phys.105, 014104 (2009).

24Z. Li, Y. Wang, G. Tian, P. Li, L. Zhao, F. Zhang, J. Yao, H. Fan, X.

Song, D. Chen, Z. Fan, M. Qin, M. Zeng, Z. Zhang, X. Lu, S. Hu, C. Lei, Q. Zhu, J. Li, X. Gao, and J. M. Liu,arXiv:1703.10335(2017).

25L. Hong, A. K. Soh, Q. G. Du, and J. Y. Li,Phys. Rev. B77, 094104 (2008).

26Y. Zhang, J. Li, and D. Fang,Phys. Rev. B82, 064103 (2010).

27R. K. Vasudevan, Y. C. Chen, H. H. Tai, N. Balke, P. Wu, S.

Bhattacharya, L. Q. Chen, Y. H. Chu, I. N. Lin, S. V. Kalinin, and V.

Nagarajan,ACS Nano5, 879 (2011).

28J. Wang, M. Kamlah, T. Y. Zhang, Y. Li, and L. Q. Chen,Appl. Phys.

Lett.92, 162905 (2008).

29Y. L. Li, S. Y. Hu, Z. K. Liu, and L. Q. Chen, Acta Mater. 50, 395 (2002).

30K. De’ Bell, A. B. MacIsaac, and J. P. Whitehead,Rev. Mod. Phys.72, 225 (2000).

31J. J. Wang, X. Q. Ma, Q. Li, J. Britson, and L. Q. Chen,Acta Mater.61, 7591 (2013).

32Y. Zhang, P. Chu, Y. L. Xie, D. P. Chen, Z. B. Yan, and J. M. Liu,EPL 108, 27009 (2014).

33Y. Zhang, Y. L. Wang, P. Chu, Y. L. Xie, Z. B. Yan, and J. M. Liu, Comput. Mater. Sci.108, 301 (2015).

34H. L. Hu and L. Q. Chen,J. Am. Ceram. Soc.81, 492 (1998).

35A. J. Bell,J. Appl. Phys.89, 3907 (2001).

36D. P. Chen and J. M. Liu,Appl. Phys. Lett.100, 062904 (2012).

37H. L. Hu and L. Q. Chen,Mater. Sci. Eng. A238, 182 (1997).

38L. F. Wang and J. M. Liu,Appl. Phys. Lett.91, 092908 (2007).

39F. T. Huang, B. Gao, J. W. Kim, X. Luo, Y. Wang, M. W. Chu, C. K.

Chang, H. S. Sheu, and S. W. Cheong, npj Quantum Mater.1, 16017 (2016).

40N. D. Mermin,Rev. Mod. Phys.51, 591 (1979).