Domain Wall Motion in Perovskite Ferroelectrics Studied by the

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Domain Wall Motion in Perovskite Ferroelectrics Studied by Nudged Elastic Band Method Xiaoyu Li, Qiong Yang, Juexian Cao, Lizhong Sun, Qiangxiang Peng, Yichun Zhou, and Ruixian Zhang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b11330 • Publication Date (Web): 05 Jan 2018 Downloaded from http://pubs.acs.org on January 5, 2018

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The Journal of Physical Chemistry

Domain Wall Motion in Perovskite Ferroelectrics Studied by Nudged Elastic Band Method X. Y. Li,† Q. Yang,*,† J. X. Cao,‡ L. Z. Sun,† Q. X. Peng,*,† Y. C. Zhou,† and R. X. Zhang† †

Key Laboratory of Low Dimensional Materials and Application Technology, Ministry of

Education, School of Materials Science and Engineering, Xiangtan University, Xiangtan, Hunan 411105, China ‡

School of Physics and Optoelectronics, Xiangtan University, Xiangtan, Hunan 411105, China

ABSTRACT Due to its unique physical properties, domain wall (DW) in ferroelectrics not only plays a key role in the electrical properties of ferroelectric film, but also has shown tremendous prospect in the DW-based memory and logic devices. However, the motion mechanism of ferroelectric DW, which is of great significance for the application of ferroelectric films, has not been clearly understood. In this paper, the 180o DW motions in BaTiO3 (BTO) and PbTiO3 (PTO) were studied by using the nudged elastic band method based on the density functional theory. The evolutions of atomic structure, local polarization and energy of the system for the DW motion process between the two adjacent equilibrium positions were systematically revealed. The DW migration across the oxygen vacancy was also simulated and the corresponding potential well for the DW motion was obtained. It was also found that the DW motion barrier could be significantly influenced by the in-plane strain. The ‘activation field’ deduced from the energy barrier of DW motion in the present calculation agrees with experimental values, which may provide a fundamental understand of DW dynamics.

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1. Introduction Ferroelectric domain wall (DW) and its dynamic properties have attracted much attention in the past decades. On the one hand, the structural and kinetic properties of ferroelectric DW play an important role in the polarization switching of ferroelectric materials. Most of the ferroelectric-based electronic devices, such as the ferroelectric memory and piezoelectric sensor,1,2 work based on the polarization switching of ferroelectrics (generally films). Take the ferroelectric memory as an example, the performance parameters, including the access speed and operating voltage, all depend on the polarization switching properties. It is well known that the polarization switching in ferroelectrics is realized by the nucleation and growth of ferroelectric domain with polarization direction parallel to the applied electric field, which is essentially the motion of DW.3,4 From this principle, one can deduce that impeded DW motion could lead to failure of ferroelectric devices. On the other hand, due to the particular atomic and electronic structures, the electrical, magnetic and optical properties along the ferroelectric DW are different from those inside the domain. The dissimilar physical properties of DW not only contribute to the macroscopic performance of the ferroelectric film, but also promote the application of the DW itself.5 In the ferroelectric BiFeO3, Seidel et al. have observed the occurrence of electrical conduction at 109o and 180o DWs, and the absence of conduction at 71o DWs. Their studies show that the current can be incrementally controlled by creating or erasing the conducting DWs.6 Sharma et al. have also realized the nonvolatile ferroelectric DW memory by writing and erasing the DW in BiFeO3 thin film with the atomic force microscopy (AFM).7 "Domain wall electronics"8 has become a widely accepted concept. With a Pt electrode made by the electron-beam-induced deposition (EBID) technique, McGilly et al. found that the single and multiple branches propagation of PZT DWs can be realized by PFM voltage pulses.9 Rubio-Marcos et al. have driven the DW motion in BTO single crystal by varying the polarization angle of a coherent light source, leading to the realization of remote control of ferroelectric DWs.10 These works demonstrated that the ferroelectric DW ‘racetrack’ memory and photoelectric logic elements are possible, showing a great prospect of DW B


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devices.11,12 Tuning the mobility of DW and controlling the DW position are the key ways to extend the application of ferroelectric film. Thus, understanding the evolution of DW and its intrinsic physical mechanism are of vital importance. So far, the researches on the DW motion are mostly focused on the experimental observation by using transmission electron microscope (TEM) and atomic force microscopy (AFM). As early as 1954, Merz found that the DW speed is proportional to exp (2Ea/E), which is universally accepted as the Merz’s law. Here, the parameter Ea is the ‘activation field’ and the variable E is the applied electric field at the DW.13 In 1960, Miller and Weinreich developed a classic theoretical work, supporting Merz’s law. However the ‘activation field’ calculated from the nucleation model developed by Miller was overestimated by two orders of magnitude as compared to the experimental value.14-16 Here, the experimental value of the activation field is measured by Tybell et al..15 By using AFM, they investigated the nanoscale domain dynamics of monocrystalline Pb(Zr0.2Ti0.8)O3 thin films and verified that the lateral DW motion is a process of creep. The DW speed as a function of the inverse field was found fitting well to a creep formula: v ~ exp[-

R E0 µ ( ) ] , where µ = 1 and the effective ‘activation k BT E

energy’ [R/kBT]1/µE0 are equal to 1.32, 1.30, and 0.50 MV/cm for 290, 370, and 810 Å thick films, respectively. Since the activation energies in this experiment disagree with those extracted from the nucleation model based on the DW energy of PTO,17 the creep behavior was attributed to the glassy characteristics of randomly pinned DWs in a disordered system. In 2007, Shin et al. performed molecular dynamics and Monte Carlo simulations of the ferroelectric DW motion.18 Results show that the speed also follows Merz’s law with an activation field Ea equal to 0.8–3.2 MV/cm over the 200–300 K temperature range. In the work of Meyer and Vanderbilt17 and Beckman et al.,19 they also tried to reveal the energy barriers for ferroelectric DW motion by calculating the energy difference between the two configurations with the DW fixed on the planes through the A (Pb) atom and B (Ti) atom, respectively. In recent years, many nice experimental works on the DW dynamics have also been done by TEM C



observation, but the DW speed and the DW motion barriers have not been systematically studied.20,21 There are also no proper techniques to measure the barriers for the DW motion in ferroelectrics, which is of fundamental significance in the comprehension of the underlying DW motion mechanism and materials properties.19 It can be concluded that, despite of the many works that have been done to study the DW dynamics, the intrinsic mechanism of DW motion and the associated activation energy have not been well understood. Actually, the ferroelectric DWs not only move in the periodic potential of crystal lattice, their motion is also affected by various lattice defects. Many researches have demonstrated that, the ferroelectric failure, such as fatigue and imprint, originates from the DW pinning induced by the lattice defects, especially the oxygen vacancies.22-25 Many works have been done to reveal the detailed effects of the defects on the DW dynamics. Statistical theory has treated the DW pinning as the propagation of elastic objects in disordered media. As a result of the competition between elastic energy and pinning potential, the DW undergoes a continuous pinning-depinning transition, leading to a sequence of discrete and erratic DW jumps.26 The in-situ TEM observation on the real-time DW motion in recent years have proved that, the presence of lattice defects would impede the nucleation of ferroelectric domain and pin the DW motion, leading to the polarization fatigue in ferroelectric film.27-29 Through first-principle calculation, He and Vanderbilt have found that oxygen vacancies exhibit a minimum energy at the DW location, indicating that the energy coupling of the oxygen vacancy and DW could be the origin of the DW pinning.24 However, the intrinsic mechanism and microscopic process of the DW pinning and depinning are still not clear. The influence of defects on the activation energy is also unknown. Overall, besides pure scientific interest, understanding the fundamental mechanism of DW motion in ferroelectric materials is of great significance for engineering applications. There is a lack, however, on the study of DW motion mechanism both in the perfect and defective ferroelectrics. The DW is a special atomic-scale interface and its dynamic is a rapid evolution process, which make the D


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in-depth studies of such problem by experimental and macroscopic theoretical approaches difficult. The quantum-mechanical calculation has been proved to be an effective method in the study of the ferroelectric oxides, especially for system with atomic-scale interface30 or vacancies.31 Though the computational scale of quantum-mechanical calculation is still limited, the “nudged elastic band” approach (NEB),32 aiming to find the minimum energy path (MEP), has been maturely used in reconstructing the chemical reaction path between the reactants and products and simulating the ion migration process from the initial state to the final state in recent years.31,33 This method also enables us to find the reasonable transition states (TS) during the evolution process. Therefore, density functional theory (DFT) study with the NEB approach is an ideal method for the ferroelectric DW research and was adopted in this work in order to shed light on the intrinsic mechanism of the DW dynamics. The 180o DW was chosen as a typical DW to study the DW motion mechanism in this paper. The structural and energetic properties of the 180o DW in typical tetragonal perovskites BTO and PTO were firstly studied by the total energy method. After that, the quasi-static evolution process of the 180o DW motion was reproduced by using the NEB method and the activation energy for the DW motion was also analyzed. In order to reveal the DW pinning effect of the defects, the 180o DW motion across an oxygen vacancy was also simulated and the influence of oxygen vacancy on the DW migration energy was investigated. In addition, due to the strain engineering, which has shown remarkable effects on the polarization switching behaviors,34,35 was considered to be an effective approach to improve the electrical properties of ferroelectric films, a part of researches have been arranged to explore the possible influences of the mechanical strain on the DW migration and the consequent polarization switching.

2. CALCULATION DETAILS 2.1. Computational methods and boundary conditions. In this paper, tetragonal BTO and PTO were taken as the two typical ferroelectrics to study the DW dynamics. Our first-principle studies were carried out using the Vienna Ab initio simulation E



package (VASP) based on the density functional theory (DFT) with a projector augmented wave (PAW) method.36 The local density approximation (LDA) has been used for the exchange correlation function, as it was successfully applied in the previous studies of the physical properties of ferroelectric DW.17,37 In the calculation, electrons in the following orbitals: Ba-5s 5p 6s, Ti-3s 3p 3d 4s, O-2s 2p,38 Pb-6s 6p, were treated as valence electrons for BTO and PTO, respectively. The cutoff energy for the plane-wave expansion used in all the calculation in this study were 31 Ry and 33 Ry for BTO and PTO, respectively, both leading to well converged results. The structural relaxation was performed with the conjugate gradient algorithm, and the Hellmann-Feynman forces on the ions were converged to be less than 0.005 eV/Å. The Brillouin zone (BZ) for single elementary cell calculations was sampled using a 12×12×12 Monkhorst–Pack k-mesh. The in-plane lattice parameters of tetragonal phase BTO and PTO were calculated to be 3.946 and 3.854 Å, and the c/a ratios (tetragonality) are 1.013 and 1.05, respectively. The spontaneous polarizations for the optimized BTO and PTO single crystals calculated using the Berry phase method39 were 26.1 and 82.1 µC/cm2. In order to show the evolution of local polarization of the following DW configurations during the DW motion, the Born effective charge method similar to that in our previous studies40 was used to calculate the polarization magnitude of each single cell. Here, by using the Born effective charge method, the spontaneous polarizations for the aforementioned optimized BTO and PTO unit cells were calculated to be 27.6 and 90.9 µC/cm2, respectively. The computed structural parameters and ferroelectric polarizations values for the BTO and PTO are typical for LDA calculation and are in reasonable agreement with the experiments.41 In order to study the 180o DW, tetragonal BTO and PTO supercells which consist of N×1×1 perovskite unit cells stacked in the x (100) direction with N/2 unit cells polarizing “upward” (along the +z direction) and N/2 unit cells “downward” were constructed (see Figure 1), including 2 DWs. Only the DWs along the (100) crystal plane were taken into consideration because the energy of (110)-oriented 180o DW is much higher than that of (100)-oriented one, according to the study of Lawless.42 Here, N/2 unit cells indicate the width of each ferroelectric 180o stripe domain. N was F


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set to be 10, which is large enough for the precision of DW energy according to the study of Meyer and Vanderbilt,17 no matter the 180o DW is A(Ba, Pb)- or B(Ti)-centered. The lattice constants for these supercells are the integer multiples of those for the optimized single cells. The periodic boundary conditions were adopted throughout the whole study, which means the DW supercells were constrained by the invariable lattice constants and ideally short circuited. The DW position and DW energy were investigated by the same method described in refs 15 and 17. The atomic configuration of the 180o DW was relaxed with an inversion symmetry centered on a specific atom located in a given DW, which prevents the DW from shifting away from the pre-set position. Except for the fixed atoms in the given DW, the other atoms were all fully relaxed in all three degrees of freedom. The DW energies for these 180o stripe domain

structures

formula38: J dw =

were

Etot , DW − Etot , SD

2A

calculated

by

using

the

following

, where Etot,DW is the total energy of the supercell with

the presence of 2 DWs, Etot,SD is the total energy of the single domain supercell and 2A is the area of that 2 DWs. The results will be given in the Section 3.

2.2. Calculation details for domain wall motion. Though DW motion was recognized as the basic form for the polarization switching of ferroelectric materials, its microscopic physical mechanism has not been well understood. It will be shown that the A (Ba or Pb) atom-centered (100) plane is the equilibrium position of 180o DW for the lower energy of DW located in this plane, as compared with the one located on B (Ti) atom. Therefore, the 180o DW motion is actually a sideward jump of DW from an equilibrium position to another, i.e. the DW jump from the two adjacent A atom-centered (100) planes is the basic step of DW migration. Naturally, clarifying the exact process of this basic step is of essential importance to understand the DW motion. In order to reproduce the evolution process of DW moving from one equilibrium position to another, climbing image Nudged Elastic Band (NEB) method was used to find the minimum energy path (MEP) for the DW motion and transition states (TS) on this path. The 10×1×1 supercells were applied in the DW motion study, which contains a fixed DW and a movable DW. The initial state consisted of 5 upward G



polarized unit cells and 5 downward polarized ones. Accordingly, the final state involved 6 upward and 4 downward unit cells. A chain of 4 or 8 original images between the initial and finals states were constructed by linear interpolation method and relaxed to the required accuracy (Hellmann-Feynman forces on each image are below 0.02 eV/Å) to obtain the MEP of DW motion. In order to investigate the influence of oxygen vacancy on the 180o DW migration, the DW motion in the 180o domain system with one oxygen vacancy was also studied by using the NEB method. In this calculation, a 7 × 2 × 3 perovskite supercell with one part of polarization pointed along the −z direction and the other part of polarization along the +z direction was taken into consideration. The oxygen vacancy is located in the Ti-O chain along the polarization direction (c-axis) of perovskite cell and the 180o DW is moving along the (100) direction. So that, the movable 180o DW will certainly move across the preset oxygen vacancy. As the DW has been proved to move across the unit cells one by one, the whole process of DW migration across the oxygen vacancy was separated into several individual DW moving sub-processes. Here, each individual sub-process was a DW motion process across only one unit cell and was arranged head-to-tail in the NEB calculation. In order to avoid the interaction of two adjacent DWs, the sub-process, in which the two DWs are too close, is actually modified by translating the movable DW and oxygen vacancy towards the center of the super cell simultaneously relative to the fixed DW. The Brillouin zone (BZ) in this case was sampled using a 1 × 3 × 2 Monkhorst-Pack k-mesh. Each equilibrium state, with the DW located on a given A atom-centered (100) plane, was relaxed to the required accuracy. Then, each sub-process of DW motion between two adjacent equilibrium states was simulated by using the NEB method. A chain of 8 images were constructed between the two adjacent equilibrium states when performing the NEB calculation and the convergence criteria is the same as the situation without the oxygen vacancy. The strain effect on the 180o DW migration was also studied in this paper. As the general misfit strain condition in epitaxial film, only the biaxial strain in the ab-plane was taken into consideration. The lattice constants in the ab-plane of the stress free H


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PTO and BTO single unit cells were altered to induce biaxial strain, while the third lattice constant was relaxed, as was done in our previous studies.43 The magnitude of the applied biaxial strain was set to range from -2% to 1%, and the lattice constants of the 180o domain supercells were arranged based on each strained unit cells. The 10×1×1 supercells with 180o domain structures were also adopted in this part and the NEB calculation procedure of the DW motion in the strained 180o domain structure was as same as the aforementioned unstrained situation.

3. RESULTS AND DISCUSSION 3.1. The structure and energy of 180o DW. 180o DWs in tetragonal BTO and PTO were studied in a 10×1×1 supercell with the total energy method. The A(Ba, Pb)- or B(Ti)-centered 180o DWs were calculated with the fixed DW to determine the exact positions and structures of the DWs, as described in ref 15. The atomic structures of the 180o domain were relaxed and plotted in Figure 1a,b. The energy of the A(Ba, Pb)- and B(Ti)-centered 180o DWs were investigated first. For BTO, the energy of Ba-centered DW is found to be 5.6 mJ/m2, and the energy of Ti-centered DW is found to be 11.5 mJ/m2. The energy difference between these two types of DWs is 5.9 mJ/m2, which was recognized as the energy barrier of 180o DW motion by Beckman et al..19 The 180o DW energies for the Pb-centered and Ti-centered cases are 153.3 mJ/m2 and 182.3 mJ/m2, respectively, which lead to a 29.0 mJ/m2 energy difference. From the consideration of energy difference between these two types of DWs, the A(Ba, Pb)-centered DW is the more likely case for 180o DW. These results are in good agreement with the work of Meyer and Vanderbilt.17 In order to further confirm the position of the 180o DW, the two types of fixed DW were then relaxed fully without any symmetry constraint. Results show that the B(Ti)-centered DW is not stable and will automatically transform into the A(Ba, Pb)-centered DW, while the A(Ba, Pb)-centered DW remains unchanged after releasing the symmetry constraint. This indicates that the A(Ba, Pb)-centered plane is indeed the equilibrium position of 180o DW for BTO and PTO. Based on the relaxed structures of 180o domains, the local polarization distribution for BTO and PTO calculated using the Born effective I



charge method were plotted in the Figure 1c,d, respectively. From the evolution of local polarization, it can be found that the DW motion is essentially the polarization switching of the unit cell at the right of original DW. The local atomic displacements of A(Ba, Pb) ions of each unit cell relative to the O ion in the 180o domains, which represent the ferroelectricity of each cell, were analyzed as shown in Figure 1e,f. Results show the typical 180o stripe-domains with ideal DWs, for which the thickness is only an atomic layer.

Figure 1. The atomic structures and polarization patterns of the 180o DW structures. (a) The atomic structures of Ba-centered DW and Ti-centered DW of BTO. The red, blue and grey circles represent O, Ba and Ti atoms, respectively. The arrows show the polarization direction of each ferroelectric domain. (b) The atomic structures of Pb-centered DW and Ti-centered DW of PTO. The red, black and grey circles represent O, Pb and Ti atoms, respectively. (c) and (d) show the local polarization patterns of 180o DW structures of BTO and PTO. (e) and (f) give the A-O and B-O relative displacements in 180o DW structures of BTO and PTO. B-O relative displacement is calculated from the difference between the z coordinates of B atom and the neighboring O atom in the same xy-plane. A-O relative displacement is calculated from the difference between the z coordinate of A atom and the average of z coordinates of the two neighboring O atom in the same xy-plane.

3.2. NEB calculation of DW motion. According to the total energy method calculation, the A (Ba or Pb) atom-centered (100) plane is the actual position of the 180o DW in BTO and PTO, while the B (Ti) atom-centered DW does not exist. J


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However, how does the DW move from an equilibrium position to another is still not clear. Meyer and Vanderbilt17 and Beckman et al.19 have studied the DW of perovskite ferroelectric PTO and regarded the energy difference between the fixed Ti-centered DW and the Pb-centered DW as the ideal barrier (denoted by EB(∆) in this paper) of DW motion. The energy of the Ti-centered DW was regarded as the saddle point (SP) of energy profile along the estimated path for the DW motion. Since the Ti-centered DW is theoretically proposed to explain the DW motion, the actual process involved in DW motion could be different and EB(∆) may be inaccurate to represent the energy barrier for the DW motion. As such, we studied the DW motion between the two equilibrium positions by performing the NEB calculation. As described in Figure 2a,b, the 180o DW is moving from the initial equilibrium position (denoted by the blue solid line) to the final equilibrium position (denoted by the blue dash line). With this DW motion process, the downward polarized domain on the left side is expanding from 4 unit cells to 5 unit cells, while the upward polarized domain on the right side is shrinking from 6 unit cells to 5 unit cells. After the NEB calculation, the MEPs for the DW motion for BTO and PTO were obtained. Through the energy profile along the MEP as shown in Figure 2c (denoted by the blue line), the energy barrier (EB(NEB)) for the DW motion of BTO is determined by the energy difference of the saddle point and the initial (or final) state, which is 5.6 meV. Since many researchers took the energy barrier of the uniform polarization switching, i.e. the switching of the single unit cell under periodic boundary condition, to represent the switching behavior of ferroelectrics, the switching of the single unit cell of BTO was also studied by the NEB method for comparison. The switching barrier of the uniform polarization switching (EB(uni)) was calculated to be 12 meV per unit cell as shown by the red curve in Figure 2c. In order to compare the NEB result with the aforementioned fixed DW situation, which was adopted in the ref 17, EB(∆) for BTO was also plotted in Figure 2c with the black line. It can be seen that the EB(∆) is 5.8 meV, very close to the barrier of EB(NEB) which is half of the EB(uni). For PTO, EB(uni), EB(∆) and EB(NEB) are 143 meV, 28.2 meV and 30.4 meV, respectively, as shown in Figure 2d. The polarization switching barriers of PTO are greater than those of BTO, however, K



the relative magnitudes of the three types of energy barriers for BTO and PTO are in good agreement. These results indicate that sideward DW motion will be the mode of polarization switching for the perovskite ferroelectrics due to the much smaller energy needed for the polarization switching per unit volume. Our results also proved that, the DW motion barrier studied by NEB method (EB(NEB)) is generally the same as the barrier calculated from the energy difference between A-centered DW and B-centered DW (EB(∆)).

Figure 2. The energy barriers of the DW motion. (a) and (b) are the sketches of DW motion for BTO and PTO, respectively. (c) and (d) show the energy barriers of different polarization switching modes for BTO and PTO. Blue symbols and lines show the energy evolution of the MEP relaxed by NEB method. Black symbols and lines show the energy evolution of the path between the A-centered DW and B-centered DW obtained by the linear interpolation. Red symbols and lines show the energy evolution of the polarization switching of a single cell under periodic boundary condition relaxed by NEB method.

In order to clarify the detailed process of DW motion simulated by the NEB method, the evolution of atomic structure and local polarization along the MEP of 180o DW motion were then studied. As shown in Figure 3a, the partial atomic structures of the PTO 180o domain, including the movable DW along the MEP, are presented. In the initial state (image 1, I1), the DW is located at the PbO plane between cell 4 and cell 5. In the final state (image 6, I6), the DW is located at the PbO plane between cell 5 and cell 6. From the series of relaxed transition images, the main change of the atomic structure along the MEP is the atomic displacement of cell 5, L


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which represents the polarization switching of cell 5. The cation displacements relative to the coplanar oxygen ion gradually vary from positive to negative, with image 3 (I3) being the saddle point. To show the evolution of local polarization along the MEP, the polarization value of every unit cell in each image was calculated by the Born effective charge method and the polarization patterns of each image are shown in Figure 3b,c. It can be seen that the polarization direction of cell 5 changes from upward to downward during the DW motion process. From the evolution of atomic structure and local polarization, it can be concluded that the DW motion across a unit cell is actually the polarization switching of the unit cell at the right side of the initial DW.

Figure 3. The evolutions of atomic structure and local polarization during the DW motion process studied by NEB method. (a) The atomic structure of each images (I1 to I6) along the MEP (only PTO is presented). (b) and (c) show the evolution of local polarization of each unit cell along the MEP for BTO and PTO, respectively. SP in the pictures denotes image of saddle point.

As the polarization evolutions in the DW motion process and the uniform polarization switching process are both characterized by the polarization switching of a unit cell, it is meaningful to clarify the difference between these two cases, which may induce the difference between these two types of energy barriers. As in our previous work,44 the relative displacements of cation to the coplanar oxygen ions M



along the polarization direction, i.e. the ion rumpling of each unit cell, are plotted in Figure 4. As in Figure 3a, since the atoms near the fixed DW are static, only the part that contains the movable DW (cell 3 to cell 7) is presented. Figure 4a,c shows the Ba-O and Ti-O relative displacements for BTO. For the initial state, the DW located at the BaO-plane in the right of cell 4, so the Ba-O relative displacement at this plane is zero. The Ti-O relative displacement in the middle of cell 5 and Ba-O relative displacement in the right of cell 5 are both approximately equal to the bulk value. With the evolution of DW motion process, the Ba-O relative displacement in the right of cell 4 varies from zero to negative bulk value, while the Ba-O relative displacement in the right of cell 5 reduces from positive bulk value to zero. The Ti-O relative displacement in the middle of cell 5 varies from positive to negative bulk value. This indicates that DW has moved from the right of cell 4 to the right of cell 5. As compared with the uniform polarization reverse, in which the Ti-O and Ba-O relative displacements both vary from positive bulk value to negative bulk value, the Ba-O relative displacements in two sides of cell 5 move only half of the distance along the c-axis. This means that the initial and final states of the cell 5 are not stable as the unit cell in the single domain area in energy. This character of atomic displacements leads to the smaller barrier of DW motion than the uniform polarization switching, which indicates that the DW motion is a more practical mode of polarization switching than the uniform domain switching. For the saddle point, which determines the energy barrier of the DW motion, the Ti-O relative displacement in the middle of cell 5 is zero and Ba-O relative displacements at its both sides show opposite values. By examining the atomic structure and total energy, it was proved that the saddle point on the MEP is substantially the Ti-centered 180o DW. The results of PTO are similar to those of BTO. This means that the 180o DW motion is indeed a DW translation across the B (Ba, Pb)-centered and A (Ti)-centered planes. Our results also confirmed that the energy difference between the B (Ba, Pb)-centered and A (Ti)-centered DWs can be used to represent energy barrier of 180o DW motion.

N


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Figure 4. The relative atomic displacements (rumpling) of each image along the MEP. (a) and (b) are the Ba-O and Ti-O relative atomic displacements, respectively, from cell 3 to cell 7 in each images along the DW motion process of BTO. (c) and (d) are the Pb-O and Ti-O relative atomic dispalcements , respectively, for the DW motion of PTO.

The DW was assumed to cross one unit cell at one time in the DW motion process simulated above. However, it is possible that the DW crosses two unit cells at one time, i.e. the two unit cells on the right of DW may switch simultaneously. In order to verify this assumption, the DW motion across two unit cells were designed and simulated by using the NEB method. As shown in Figure 5a, the initial state contains 4 downward polarized unit cells and 6 upward polarized unit cells, while the final state contains 6 downward polarized unit cells and 4 upward polarized unit cells. The energy profiles of the MEPs for BTO and PTO are shown in Figure 5b. From the energy profile, there is an intermediate state between the initial state and the final state. By analyzing the evolution of the local polarization and atomic structure, it is found that the intermediate state actually consists of 5 downward polarized unit cells and 5 upward polarized unit cells. Therefore, there are two saddle points along the energy O



profile, with each energy barrier equal to the single barrier as shown in Figure 2c,d. The polarization patterns of the initial state, intermediate state, final state and two saddle points are plotted in Figure 5c. The polarization patterns and atomic structures show that the two saddle points are the DW structures with the DW at Ti-centered planes in cell 5 and cell 6, respectively. From the results above, it can be concluded that the DW motion process across two unit cells actually undergoes two successive sub-processes, which are the DW motion processes across only one unit cell as shown in Figure 2. It indicates that the DW motion process across one unit cell is the basic step of DW migration in BTO and PTO. The origin of this fact can be understood from the comparison of energy barriers in Figure 2c,d. As the energy barrier for the uniform polarization reverse is much higher than that of DW motion, the simultaneous polarization switching of two unit cells will need more activation energy than the polarization switching of one unit cell by one unit cell.

Figure 5. The evolutions of local polarization and energy for the DW motion across two unit cells studied by NEB method. (a) Sketch of DW motion across two unit cells at one time. (b) The energy evolution of the DW structures along the MEPs for BTO and PTO, respectively. (c) The evolution of local polarization along the MEP for PTO.

3.3. The influences of oxygen vacancy and in-plane strain on the DW motion. As the DW migration was considered to be affected by the lattice defects and internal stress in the ferroelectrics, which may have influence on the electrical properties, the DW motion including the effects of oxygen vacancy and lattice strain were then studied. In this paper, only the oxygen vacancy located on the Ti-O chain along the polarization direction, which is the most important type of oxygen vacancies in perovskite ferroelectrics,43 was taken into consideration. As sketched in Figure 6a, the P


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oxygen vacancy (Vo) was set on the Ti-O chain in cell 4 in the middle of the 7×2×3 supercell (marked by the red dashed circle). And the DW was arranged to cross the oxygen vacancy from its left side to the right side. By fully relaxing each sub-process and combining all the sub-processes together, the whole DW motion process and its energy profile as plotted in Figure 6b,c can be obtained. From the energy profile, there is also an energy barrier which should be overcome when the DW crosses every cell plane. The energy barriers of cell 1 and cell 7 are the values of DW motion in perfect lattices as calculated above. When the DW moves across cell 2, though there is a bit asymmetry on the energy curve, the energy barrier is basically similar to that in cell 1. As the DW moves in cell 3, which is just outside the cell with oxygen vacancy, the energy of the system decreases rapidly. When the DW is at the BaO- (or PbO-) plane near the oxygen vacancy, the energy of the system reaches the minimum. It can be seen from Figure 6b,c, the energy differences between the equilibrium states, which are far from and nearby the oxygen vacancy, for BTO and PTO are 23 meV and 299 meV, respectively. This means that DW would fall into a potential well of 23 meV (299 meV) for BTO (PTO) and be trapped in it when approaching to the oxygen vacancy. When the DW keeps moving forward, the energy of the system increases again. This indicates that, once the DW was trapped by the oxygen vacancy, the same energy should be provided for the DW to be depinned. This may be the reason for the polarization fatigue, which would suppress the switchable polarization, lower the switching speed and increase the energy cost.

Q



Figure 6. The energy evolution for 180o DW motion across an oxygen vacancy. (a) The sketch of DW motion across the oxygen vacancy. The red dashed circle denotes the position of oxygen vacancy. (b) and (c) show the energies of evergy image along the whole DW motion processes for BTO and PTO, respectively.

The internal stress (strain), which can be introduced in the film during the growth and annealing process, has shown a lot of effects on the ferroelectric properties and was also recognized as one of the most important methods to regulate the physical properties of film materials. In this paper, by performing the biaxial strain on the R


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lattices of BTO and PTO, the effect of strain on the DW motion properties were studied using the NEB method. Figure 7a,b show the energy barriers of DW motion in strained BTO and PTO, respectively. It can be seen that, the energy barriers are significantly increased by the biaxial compressive strain. The values of the energy barriers (EB(NEB)) for BTO varying with the strain magnitude are given in Figure 7c. For comparison, the energies of Ti-centered DW (EDW(Ti)) and Ba-centered DW (EDW(Ti)), and their difference (EB(∆)) are also provided. From the data, EB(NEB) basically coincides with EB(∆) in all the strain range. The same conclusion can be found in PTO as shown in Figure 7d. We also compared the calculated results of PTO with EB(∆) in ref 17 (denoted by open symbols). It can be seen that, although there is a little difference between the DW energies in the present work and in ref 17, all the EB(∆) and EB(NEB) are in good agreements. From the above calculation, the energy needed for the 180o DW motion is increased in the compressively strained perovskite ferroelectrics and decreased in the tensile strained ones. The variation trends of the energy barriers for the DW motion with the in-plane strain magnitude are in accordance with the switching barriers for the single unit cell as shown in our previous work. The lager ratio of bond strengths for Ti-O long and short bonds along the polarization direction and the smaller in-plane lattice constants, which may make the cation-anion relative displacement more difficult, may be the reason for the increase of energy barriers under the compressive strain.

S



Figure 7. The influence of biaxial strain on the energy barriers of DW motion. (a) and (b) shows the energies of each image along the MEP under different strains for BTO and PTO, respectively. (c) and (d) give the values of DW energies and DW motion energy barriers under different strains for BTO and PTO, respectively. Solid symbols are the datas of the present work and the open symbols are the values extracted from ref 17. The DW motion barriers calculated above is a microscope properties of the ferroelectrics, which is not convenient to be measured directly in experiment. Many researchers have developed empirical theories of DW dynamics, in order to understand the evolution laws of the DW. Merz’s law has become the most widely accepted rule to characterize the DW motion, in which the DW speed is determined by the ‘activation field’ Ea and the applied electric field E. However, the physical meaning of the Ea is not clearly understood, and the values of Ea from the previously proposed nucleation model and experiment disagree with each other 14,15. Here we try to understand the activation field from the DW motion barriers. As the polarization switching is actually the ionic displacement under the applied electric field, we can treat the DW as a charged object moving under the electric field. Take PTO as an example, the DW needs 29 meV energy to move half of the lattice constant (2 Å) from the equilibrium position to the peak of the energy barrier. If we consider the DW is equivalently 2 e to 4 e charged as compared to the ionic charge in PTO, the applied T


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electric field should be 0.75 MV/cm to 0.38 MV/cm, which can drive the DW moving sideward. This electric field basically agrees with the activation field fitted by the AFM results in ref 13 (0.50 MV/cm for 810 Å thick PZT film) in order of magnitude. Therefore, our calculated results may provide a fundamental understanding of the activation field (energy) of the DW motion. As can be seen from our calculation, the energy barriers of DW motion would be significantly affected by the presence of lattice defects and internal strain. Therefore, the local activation field of the ferroelectric can also be influenced. For example, the local activation field will be increased by the oxygen vacancy and the in-plane compressive strain, and this would increase the driving electric field and suppress the DW speed of the ferroelectric films.

4. CONCLUSION In summary, we have explored the 180o DW motion mechanism by using the nudged elastic band method, and the effects of oxygen vacancy and in-plane strain on the DW motion were also investigated. It was found that the DW motion process between the two adjacent equilibrium positions, i.e. the adjacent A(Ba, Pb)-centered (100) planes, is the basic step for the sideward migration of DW. The DW would move from A(Ba, Pb)-centered plane to B(Ti)-centered plane and reach the maximum of energy barriers, which are 5.6 and 30.4 meV per unit cell for BTO and PTO, respectively. These energy barriers for DW motion are much smaller than the energy barriers for the polarization reverse of a single unit cell, which may be the reason for the polarization switching mode of DW motion rather than uniform domain switching. For comparison, we discussed the relation between the DW motion barriers and the ‘activation field’ of the phenomenological Merz's law. It is found that the ‘activation field’ deduced from the calculated energy barrier basically agrees with the values extracted from experimental measurement. The DW motion with the presence of oxygen vacancy and internal strain was also studied. Results show that the DW would fall into a 23 meV (299 meV) potential well when coming across an oxygen vacancy for BTO (PTO). Therefore, a corresponding energy is needed for the DW to get away U



from the oxygen vacancy, which is one of the important origins for the polarization fatigue in ferroelectric film. It was also found that the energy barriers for the DW motion are increased under the in-plane compressive strain, while decreased under the in-plane tensile strain. As the energy barriers of DW motion were affected by the defects and strain, the ‘activation field’ of the DW and consequently the DW speed will be influenced. Though only the 180o DW was considered in this paper, the study method can also be applied to other types of DWs. Our results may provide a fundamental understanding on the DW migration, which is of great importance for the application of DW-based devices.

AUTHOR INFORMATION

Corresponding Author *E-mail: [email protected]. *E-mail: [email protected].

ORCID Q.Yang: 0000-0002-3235-1986 Q. X. Peng: 0000-0003-2959-6648

Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENT This work was financially supported by National Natural Science Foundation of

China (Grant Nos. 11402221, and 11502224), Natural Science Foundation of Hunan Province, China (Grant No. 2017JJ3289) and Hunan Provincial Key Research and Development Plan, China (Grant No. 2016WK2014). Calculations were partly performed at the Research Center of Supercomputing Application, National University of Defense Technology.

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Domain Wall Motion in Perovskite Ferroelectrics Studied by the

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