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Effects of dislocations with different locations, orientations and density on domain evolution of ferroelectric thin film: A phase field study
Computational Materials Science 169 (2019) 109102
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Effects of dislocations with different locations, orientations and density on domain evolution of ferroelectric thin film: A phase field study Xuanling Liua, Xinyi Shana, Shiqi Fenga, Xiaofei Xua,b, Limei Jianga,b, a b
T
⁎
Hunan Provincial Key Laboratory of Thin Film Materials and Devices, School of Materials Science and Engineering, Xiangtan University, Hunan 411105, China Key Laboratory of Key Film Materials & Application for Equipment of Hunan Province, Xiangtan University, Hunan 411105, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Ferroelectric material Phase field method Flexoelectric effect Dislocation Polarization distribution Polarization switching
The effects of dislocations on polarization distribution and switching in ferroelectric thin films are discussed based on a modified multi-field coupling theoretical framework that combining the flexoelectric effect and the strain field caused by dislocations. First, the correctness of the model is verified through a study about the effects of different flexocoupling types on domain structures around the dislocations. Then, effects of dislocations with different locations, orientations and density are systematically studied. Dislocations in the film will induce the appearance of a-domain. And the size of the new a-domain depends on the orientations and locations of dislocations. Among [001¯], [101¯] and [100 ] oriented single dislocations, the size of the new a-domain near the [101¯] oriented dislocation is the largest. It is essential to avoid generating [100 ] and [101¯] oriented dislocations since the [101¯] oriented dislocations can cause imprint failure and [100 ] oriented dislocations bring intense pinning effects. As the density of dislocations increases, the pinning effect and imprint behavior becomes stronger. The hysteresis loop of the film with [100 ] oriented multi-dislocations shows no ferroelectricity once the density reaches a certain limit. These dislocations induced imprint behavior and pinning effect can be eliminated by a larger applied electric field when the dislocation density is low. But when the dislocation density gets higher, the situation is different.
1. Introduction With the miniaturization and high integration of ferroelectric devices, the size of the ferroelectric thin film in the thickness direction is close to the nanometer scale. Both the structure and the evolution of the polarization (domain) are more and more constrained by the microstructures including lattice defect, interface terminal type, oxygen vacancy and so on [1,2]. Therefore, to make ferroelectric devices more reliable, it is necessary to study the influence mechanism of the micro factors affecting the polarizations. As one of the most common defects [3], dislocations are inevitably produced during the fabrication of ferroelectric thin films. A dislocation is a crystallographic defect or irregularity within a crystal structure. Some types of dislocations can be visualized as being caused by the termination of an atoms plane in the middle of a crystal. Up to now, dislocations, edge dislocations in particular, with different orientations have been found inside the ferroelectric films or at the interface. Dislocations can induce polarization instability [4], domain pinning [5], or even degrading of the ferroelectric properties [6] because of the electro-mechanical coupling
between the strain / strain gradient field and polarization. Meanwhile, in order to explain the above observed experimental phenomena and reveal the intrinsic influence mechanism about how dislocations affect polarization patterns and switching properties, many phase field models have been established [7–11]. For example, Wu et al. [9] established a phase field model to discuss how dislocations lead to pinning and de-pinning in single-crystal ferroelectric films. Zheng et al. [11] studied the effect of interfacial dislocations on ferroelectric properties in a one-dimensional environment and showed that the interfacial dislocation can greatly reduce the average polarization of ferroelectric thin films at high temperature, and it can reduce the effective Curie temperature several times and have a great influence on the local dielectric properties. Even though the previous works about how dislocations affect the ferroelectric thin film’s properties have made a significant contribution, our understanding of their influence mechanism is still limited. For example, there are large strain gradients around the dislocations, which affect ferroelectric thin films’ properties through flexoelectric coupling. Flexoelectric effect implies that strain gradient can induce polarization
⁎ Corresponding author at: Hunan Provincial Key Laboratory of Thin Film Materials and Devices, School of Materials Science and Engineering, Xiangtan University, Hunan 411105, China. E-mail address: [email protected] (L. Jiang).
https://doi.org/10.1016/j.commatsci.2019.109102 Received 28 February 2019; Received in revised form 24 June 2019; Accepted 25 June 2019 0927-0256/ © 2019 Elsevier B.V. All rights reserved.
Computational Materials Science 169 (2019) 109102
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(the direct flexoelectric effect, see Eq. (1)), or polarization gradient can induce stress (the converse flexoelectric effect, see Eq. (2)) [12,14]:
Pl = μijkl ∂εij / ∂xk
(1)
εij = Ωijkl ∂Pl/ ∂xk
(2)
σij = ∂h/ ∂εij tijl = ∂h/ ∂εij, l Di = −∂h/ ∂Ei; ηi = ∂h/ ∂Pi; Λij = ∂h/ ∂Pi, j
where, σij, ηi , Di , tijl, Λij respectively represent stress, effective local electric force, electric displacement, double stress tensor that workconjugate with the strain gradient tensor and higher order local electric force. It should be noted that there is large strain around the dislocations, which can couple with the polarization gradient through the flexoelectric effect. Therefore, in the establishment of a phase field model considering dislocations, it is necessary to put the strain caused by dislocations into consideration. Hence, the total strain εijtotal should be composed of two parts, the elastic strain, εij , and the strain related to a dislocation, εijd . The dislocation strain can be deduced from dislocation stress field based on constitutive Eq. (6).
where, Pl , μijkl , εij , xk , Ωijkl denote the components of the material polarization, the flexoelectric coupling tensor components, the strain tensor components, the displacements, and the inverse flexoelectric coupling tensor components, respectively. The flexoelectric field has been proved to be as large as 1 KV/m, which is strong enough to induce polarization rotation [13], and even 180° polarization switching [15] in ferroelectric thin films without dislocations. Then when there is a dislocation, the huge strain gradient exists, and the impact of flexoelectric effect will become stronger and cannot be ignored. However, at present, few phase field studies regarding to dislocations in ferroelectric thin films have put flexoelectric effect into consideration. Furthermore, there are also few studies making systematical discussion about effects of dislocations with different orientations, locations and density on domain structures and evolution. In this paper, a phase field model that combining the flexoelectric effect and the strain field caused by dislocations is established to systematically discuss the effects of dislocations with different locations, orientations and density on polarization distribution and switching. The effect of dislocations on polarization distribution is more pronounced after considering the flexoelectricity. Among the dislocations with three orientations studied in this paper, the dislocations with [101¯] orientation have the most influence on promoting the emergence of the new adomain. What’s more, the hysteresis loops of thin films containing a single dislocation, three dislocations or seven dislocations at the interface are plotted respectively. The results show that with the dislocation density increases, the ferroelectric properties of thin films with [100 ] oriented multi-dislocations are gradually suppressed or even disappear.
εijd = σijd (1 + υ)/ E − νΘδij / E here, the stress tensor components caused by dislocations by [17,18].
are given
d σxx = −Wy (x 2 − y 2 )/(x 2 + y 2 )2 d d τxy = τ yx = Wx (x 2 + y 2 )/(x 2 + y 2 )2
(7)
For simulating the stress field around dislocations with different orientations easier, we turn to describe the stress field in the polar coordinate: d σxx = −Wsinθ (2 + cos 2θ)/ r d σyy = Wsinθcos 2θ / r d τxy = W cosθcos 2θ / r
(8)
where, W = Gb/[2π (1 − υ)], in which, G is the shear modulus, b is the mode of Burgers' vector, E is the elastic modulus, ν is the Poisson's ratio; Θ = σx + σy + σz = σkk ; δij is the kronecker delta notation; r is the distance between the dislocations and a specific point; θ is the angle between the x axis and the line connecting the dislocation and the specific point. What needs to be raised is that dislocations actually are the results of the system energy minimization. Up to now, there have been a lot of excellent works focusing on the dislocation formation and evolution [19–23]. Especially the work reported by A Skaugen et al. [24], in which a phase field crystal model was established to investigate the formation of dislocations and analyze the dislocation density quantitatively. These works have made great contributions and are instrumental to reveal the essence of the dislocation formation and evolution. However, these works put their emphasis on the dislocation itself including the dislocation motion and evolution rather than the effects of dislocations on other physical quantities. For ferroelectric materials, people are more concerned about how dislocations affect domain structures, and do not pay much attention to the source and influencing factors of dislocations. Dislocations in ferroelectric thin films usually were considered to be pre-existing and fixed in position and density [11,25–27]. The stress/strain fields of dislocations are often simplified by the following ways: a) the strain caused by dislocations is treated as the eigenstrain which will influence the elastic free energy of the system, thus affect the whole free energy of the system [25,26]; b) simulate the exist of dislocations by setting the boundary conditions of the displacement field according to the strain situation above and below the dislocations in real space [27]; c) the stress field caused by dislocation is regarded as an applied field and coupled with other physical fields to affect the free energy of the system together with the eigenstrain [11]. Our work refers to the approach of Y. Zheng et al. [11],
In this paper, the previous phase field model of flexoelectricity [16] is modified by combining the effect of dislocations.
2.1. Thermodynamic potential The Landau-Devonshire free energy density expression is: (3)
where hlandau , hstrain , hcoup, hgrad , helec and hflexo indicate the landau free energy density, elastic energy density, polarizations and strain coupling energy density, polarization gradient energy density and flexoelectricity energy density, respectively. The expression for flexoelectricity energy density is:
hflexo = fijkl (εkl ∂Pi/ ∂x j − Pk ∂εij / xl )/2
(6)
σijd
d σxx = −Wy (3x 2 + y 2 )/(3x 2 + y 2 )2
2. Theoretical method
h = hlandau + hstrain + hcoup + hgrad + helec + hflexo
(5)
(4)
and fijkl represent the flexoelectric coefficients. 2.2. Constitutive equations of each physical field Generally, in order to describe the electro-mechanical coupling behaviors of ferroelectric thin films, three physical fields are required, which are electric field, displacement field and polarization field. And the constitutive equation of these three physical fields can be deduced from the free energy density [16]. 2
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which considered the strain field of dislocations as a linear applied field. The strain field of dislocations in our model is coupled with polarization to change the strain-polarization coupling energy h coup and ultimately change the total free energy of the system. Although the results may be biased away from the reality, it still reveals the influence mechanism of the dislocations on domain structures to some extent like the results of previous studies [11,25–27]. And the feasibility of these qualitative studies is proved in the following Section 3.
completely compensated by external charges through a short circuit boundary condition. Since the size effect is too strong to be ignored owing to the ultrathin thickness of thin film, the polarization boundary conditions of the top and bottom surface of the thin film are both assumed as −dP2/ dx2 = 0 . And the initial condition of the polarization field is P1 = 0 , P2 = 1. In the simulations, we discretize the ferroelectric thin film into 2400 (80 × 30) quadrilateral elements, and the size of each element is Δx1∗ = Δx 2∗ = 1, which corresponds to the real size Δx1 = Δx2 = 1 nm . We set every dimensionless time step as Δt ∗ = 0.05, and set the total number of steps to be 5000 in order to ensure that the evolution of domain structures reaches a equilibrium state. For the cubic point-group materials like PZT, the flexocoupling coefficients have three independent components: f1111, f1122 , f1212 ( f2121) [40], which describe three different flexocoupling types. Our previous studies found that flexocoupling types can make different effects on polarization distribution. So, do different types of flexoelectricity play different roles in the redistribution of polarizations near dislocations? Taking the ferroelectric thin films containing a single dislocation with [001¯] orientation as an example, we simulate 7 situations: (1) f11 ≠ 0 , f12 = f44 = 0 ; (2) f12 ≠ 0 , f11 = f44 = 0 ; (3) f44 ≠ 0 , f11 = f12 = 0 ; (4) f11 ≠ f12 ≠ 0 , f44 = 0 ; (5) f11 ≠ f44 ≠ 0 , f12 = 0 ; (6) f12 ≠ f44 ≠ 0 , f11 = 0 ; (7) f11 ≠ f12 ≠ f44 ≠ 0 . We compare the polarization distributions of these 7 situations to find out the role of each flexoelectricity type in the redistribution of polarizations near dislocations. Where, f11 , f12 and f44 respectively represent the flexoelectric coefficients in Voigt form f1111, f1122 and f1212 . As shown in Fig. 1, a small a-domain emerges in the single c-domain near the dislocation. Compared with the domain structures of thin films considering f12 or f44 related flexocoupling type, the new a-domain of the thin film considering f11 related flexocoupling is the broadest which is more coincide with experimental observations. And it should be noticed that the polarization switching area of films considering only one flexocoupling type is obviously smaller than those observed by experiments [41,42]. Therefore, a further discussion about the effect of dislocations on polarization distribution under the collective effect of different flexocoupling types is indispensable. As shown in Fig. 1(f), when f12 ≠ f44 ≠ 0 , f11 = 0 , the size of the emerging a-domain in the thin film is smaller than that in the film considering f11 only (see Fig. 1(a)), which again demonstrates that these two flexocoupling types have much smaller effects on polarizations near the dislocation. However, from Fig. 1 (d) and (e), we can see that when consider f11 and f12 together or f11 and f44 together, the emerging a domain become larger than that of f11 (Fig. 1 (a)), and the simulation results become closer to the experimental results [41,42]. It implies that although the f12 and f44 related flexocoupling types have much smaller effects on the polarizations near the dislocation, their effects cannot be neglected. Thus, it can be speculated that the redistribution of polarizations near dislocations are owing to the collective effect of three flexocoupling types. As shown in Fig. 1(g), due to the presence of a dislocation located at (0,0), a large a-domain occurs in the single cdomain structure. That is, the polarization near the dislocation has a very significant 90° switching so that the domain structure changes into a c/a/c multi-domain structure. And the simulation result that an evident 90° polarization switching caused by dislocations agrees with the experimental results well [41,42]. Therefore, the dislocation phase field model considering the collective effect of three flexocoupling types is the most accurate. The following works are based on this model which ensures that the simulations correctly predict the effects of dislocations on domain structures.
2.3. The equilibrium equations for each physical field The mechanical equilibrium equation including the contribution of strain gradient is given based on the Mindlin’s strain gradient theory [28,29]:
(σij − tijk, k ), j + bi = 0,
(9)
where bi are the components of a body force per unit volume. The temporal evolution of the polarization field is governed by the time dependent Ginzburg–Landau equations [30–35]:
∂Pi/ ∂t = −L (ηi − Λij, j)
(10)
The electric field is governed by the Maxwell’s (or Gauss’) equation:
Di, i − ξ = 0
(11)
where ξ is the volume charge density. For the equations listed above have to be solved simultaneously under appropriate boundary and initial conditions, the finite element methods are employed based on the PDE part of the finite element software COMSOL [30]. And the details of finite element formulations can be found in Ref. [36]. 3. Effects of different flexocoupling modes on domain structures around dislocations In this paper, the tetragonal Pb(Zr0.1Ti0.9)O3 (PZT) material is used as an example. For the sake of simplicity, 2D finite element simulations of a single c-domain are carried on, as shown in Fig. 1. The x1 and x2 axes respectively denote the length and thickness directions of thin films. Here, the directions of Burgers’ vectors are used to describe the orientations of dislocations. For example, the dislocation with Burgers’ vector [100 ] is called as the dislocation with orientation [100 ]. As known, the orientations of dislocations should obey the structure and energy conditions of crystals. The PZT can be regarded as the coexistence of face-centered cubic and body-centered cubic. According to the Burgers’ vectors of perfect dislocations and partial dislocations in common crystals, the Burgers’ vectors of perfect dislocations existed in 2D-plane of PZT are < 100> and < 110>. Therefore, this paper discusses three directions derived from the < 100> and < 110> crystal family, namely [100 ], [001¯] and [101¯]. The material coefficients including the phenomenological Landau–Devonshire coefficients, the elastic constants, the gradient coefficients, and the electrostrictive coefficients that required for expressing the system energy in Eq. (3) are consistent with those used by Qiu et al. [37], and Xu et al. [38]. In order to improve the compute speed, all the quantities in this paper are nondimensionalized based on the spontaneous polarization Ps (C / m2) at room temperature, the gradient energy coefficient g110 (m4N / C −2) and the free energy coefficient of Landauα 0 (m2N / C −2) . All the formulations of conversion between dimensioned and dimensionless quantities can be found in Ref. [39]. The boundary conditions of each field are set as follows: To imitate a real ferroelectric thin film, we assume periodic boundary conditions for displacement, polarization, and electric field along the x1 direction. The top and bottom surfaces are assumed to be traction free, and the mechanical initial values are u 0 = v0 = 0 . For the electric field, the electric potential of the top and bottom surface of the film is taken as zero, which means the surface charges induced by polarization are
4. Effects of single interfacial or internal dislocations on domain structures In the process of fabricating ferroelectric thin films, dislocations are often formed at the interface of the film, and as the temperature increases, dislocations can enter the interior of the films. In this section, 3
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(a)
(d)
(b)
(e)
(c)
(f)
(g)
Fig. 1. Domain structure of ferroelectric thin films containing a single dislocation with [001¯] orientation when (a) f11 ≠ 0 , f12 = f44 = 0 ; (b) f12 ≠ 0 , f11 = f44 = 0 ; (c) f44 ≠ 0 , f11 = f12 = 0 ; (d) f11 ≠ f12 ≠ 0 , f44 = 0 ; (e) f11 ≠ f44 ≠ 0 , f12 = 0 ; (f) f12 ≠ f44 ≠ 0 , f11 = 0 ; (g) f11 ≠ f12 ≠ f44 ≠ 0 .
of the internal dislocation. Fig. 2(d–f) shows that the polarization switching zones near the [001¯] and [101¯] oriented dislocations are obviously larger than that near the [100 ] oriented dislocation. Polarization vectors near the dislocation with [100 ] orientation only have small angle of rotation, while polarizations near the [001¯] and [101¯] oriented dislocations experience 90° switching. What’s more, it can be seen from the Fig. 2(d) that the polarizations above the dislocation turn to be negative, while the polarizations below the dislocation are just strengthened rather than inversion. Same as the situation of the thin film containing a single interfacial dislocation, the polarization switching zone near the [101¯] oriented dislocation is the largest. This may be due to the strongest piezoelectric effect and flexoelectric coupling effect between the [101¯] oriented dislocation strain and polarizations.
effects of single interfacial and internal dislocations with different orientations on domain structures are discussed. With the thin film containing a single [100 ] oriented dislocation at the position (0,0) as the initial state, and α as the clockwise rotation angle of the dislocation, we deduce the stress field near dislocations with different orientations. The α of 0°, 45° and 90° respectively correspond to the dislocation orientations of [100 ], [101¯] and [001¯]. d σxx = −2W (ycosα + xsinα )·(x 2 + y 2 )/(x 2 + y 2 )2 − W (ycosα + xsinα )[(x 2 + y 2 ) cos 2α − xysin2α ]/(x 2 + y 2 )2 d σyy = W (ycosα + xsinα )·[(x 2 − y 2 ) cos 2α − xysin2α ]/(x 2 + y 2 )2 d σxy = W (xcosα + ysinα )·[(x 2 − y 2 ) cos 2α − xysin2α ]/(x 2 + y 2 )2
(12)
As shown in Fig. 2(a)–(c), the interfacial dislocations in a single cdomain tend to lead to the emergence of the a-domain, thus the domain structures of thin films change from a single c domain to be an a/c/a domain. The region of the appeared a domain is the largest in the thin film with the interfacial [101¯] oriented dislocation. For the reason that interfacial dislocations can enter the interior of the films with the temperature increases during the fabrication of ferroelectric thin films, the effects of internal dislocations on domain structures of ferroelectric thin films are also needed to be discussed. The stress field around the internal dislocation is deduced from Eq. (12) by means of replacing x and y with x + x1 and y + y1, respectively. Here, x1 and y1 respectively represent the horizontal and vertical displacements
5. Effects of periodic dislocations on domain structures In fact, there are always many dislocations in the ferroelectric thin films. Therefore, in order to be closer to the actual situation, the phase field model is further modified to contain multi-dislocations. To simplify the multi-dislocations model, it is assumed that the dislocations are periodic and the distances between adjacent dislocations are equal. The stress field equations around the periodic dislocations are deduced according to the stress field equations around the single dislocation [43]: 4
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(a)
(d)
(b) (e)
(c) (f) Fig. 2. Stable domain structures of ferroelectric thin films with various dislocations. Interfacial dislocations with different orientations: (a) [100 ], (b) [101¯], (c) [001¯]. Internal dislocations with different orientations: (d) [100 ], (e) [101¯], (f) [001¯].
6. Polarization response of ferroelectric thin films with dislocations
d = −W ∑ y (3x m2 + y 2 )/(x m2 + y 2 )2 σxx m d = W ∑ y (x m2 − y 2 )/(x m2 + y 2 )2 σyy m d = W ∑ x m (x m2 − y 2 )/(x m2 + y 2 )2 σxy m
Polarization switching is also a very important property for ferroelectric materials. Thus, the effects of dislocations on the polarization responses are studied in this section. The polarization–electric field hysteresis loops (P-E curves) of ferroelectric thin films containing interfacial single dislocations with orientations [001¯], [101¯] and [100 ] are plotted under the applied sinusoidal electric field E0∗ = 2sin(2πt ∗/200) along the ferroelectric film’s thickness direction. While the time step is set as Δt ∗ = 0.05, and the total time is set to be 250 in the simulations. It can be seen from Fig. 4 that the effects of the [001¯] and [101¯] oriented interfacial single dislocations on the P-E curves are different. The positive and negative remnant polarizations and the coercive field shown in the [001¯] oriented dislocation related hysteresis loop are symmetrical. However, under the effect of the interfacial single dislocation with [101¯] orientation, though the positive and the negative remnant polarizations of the P-E curve are symmetrical, the positive and negative coercive fields show some deviations. The positive coercive field is slightly larger than the negative coercive field, in other words, the hysteresis loop shifts to the right. This phenomena implies that interfacial single dislocations with [101¯] orientation would induce slight imprint failure of ferroelectric thin film. And the imprint behavior of the ferroelectric thin film becomes more serious under the effect of [100 ] oriented dislocations. Both the remnant polarization and the coercive field shown in the hysteresis loop are not symmetrical. That is, the value of the positive remnant polarization is greater than the negative one and the value of the positive coercive field is smaller than the negative one, indicating that the interfacial dislocation with [100 ] orientation can suppress the switching of positive polarizations, and a larger coercive field is needed for polarization switching. We then plot the P-E curves of ferroelectric thin film with seven [001¯], [101¯] or [100 ] oriented interfacial dislocations, as shown in Fig. 5. The applied electric field and time step are the same as above. Among
(13)
where x m = mx 0 , and x 0 is the distance between two adjacent dislocations, m = ⋯−2, −1, 0, 1, 2⋯. The stable domain structures of thin films containing interfacial multi-dislocations with different orientations are shown in Fig. 3(a-c). The positions of seven interfacial dislocations are: (0,0), (−10,0), (10,0), (−20,0), (20,0), (−30,0), (30,0). Through comparing the polarization switching zones shown in Fig. 3 with that of thin films containing single dislocations, we find that the polarization switching zone in the film with periodic interfacial dislocations is far broader. The stable domain structures of thin films containing internal multidislocations with different orientations are displayed in Fig. 3(d–f). The position of seven dislocations are: (0,15), (−10,15), (10,15), (−20,15), (20,15), (−30,15), (30,15). From the figure, it can be readily seen that the polarization 90° switching occurs above the internal dislocations with orientation [100 ]. But polarizations below the [100 ] oriented internal dislocations maintain their initial state. When there exist periodic internal dislocations with orientation [101¯], the 90° domain walls appear both above and below the internal dislocations, so the initial c-domain changes to be an irregular and complicated a-c domains. It is very interesting that polarization vortices are found near the dislocations with orientation [001¯]. And polarizations far away the [001¯] oriented internal dislocations experience 90° switching. In summary, the internal dislocations with different orientations all can cause polarization switching and lead to the appearance of 90° domain walls. However, the polarization switching zone near each dislocation of the multi-dislocations array is smaller than that of single dislocations. This is because the more dislocations in the ferroelectric thin film, the stronger pinning effect occurs which will suppress the polarization switching and the evolution of domain structures. 5
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(a)
(d)
(b)
(e)
(c)
(f)
Fig. 3. Stable domain structures of ferroelectric thin film containing various periodic dislocations. Periodic interfacial dislocations with different orientations: (a) [100 ] (b) [101¯] (c) [001¯]. Periodic internal dislocations with different orientations: (d) [100 ] (e) [101¯] (f) [001¯].
Fig. 4. The polarization–electric field hysteresis loops of ferroelectric thin films with [001¯], [101¯] or [100 ] oriented single interfacial dislocations.
Fig. 5. The polarization–electric field hysteresis loops of ferroelectric thin films containing 7 interfacial dislocations with [001¯], [101¯] or [100 ] orientation.
these three oriented interfacial dislocations, the P-E curve is more symmetrical for [001¯] orientation case. The curve turns to be asymmetrical for the case of dislocations with [101¯] orientation, in other words, an imprint behavior occurs here. The P-E curve of the thin film containing interfacial dislocations with [100 ] orientation is a linear line and shows no features of hysteresis loops. This indicates that, in this situation, the ferroelectric thin film is at a modulation phase and shows no ferroelectricity. Previous studies show that dislocations in ferroelectric thin films would pin the ferroelectric domain and hinder the domain switching [25]. The dislocations with [100 ] orientation here make polarizations above the dislocations tend to point to the horizontal direction and can’t switch under the applied electric field. Therefore, the ferroelectric film eventually loses its ferroelectricity.
Whether the pinning effect emerged here can be eliminated by a larger applied electric field? In order to answer this question, the applied electric field is then set as E0∗ = 5sin(2πt ∗/200) along the thickness direction of the ferroelectric thin film (x2 direction). The simulation result is shown in Fig. 6. For the thin film containing 7 interfacial dislocations with [101¯] orientation, as the amplitude of the electric field increases, some of the polarizations are de-pinned and take part in the switching process, resulting in an increase in the positive and negative remnant polarization. For the thin film containing 7 interfacial dislocations with [100 ] orientation, the hysteresis loop begins to appear when the electric field is strong enough (> 2 or < -3), and it’s obvious that the “hysteresis phenomenon” is more apparent when the electric field is positive. Although a double hysteresis loop appears in this situation, the
6
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Fig. 6. The polarization–electric field hysteresis loops of ferroelectric thin films containing 7 interfacial dislocations with [001¯], [101¯] or [100 ] orientation. The amplitude of the applied electric field is 5.
Fig. 8. The polarization–electric field hysteresis loops of ferroelectric thin films containing 3 interfacial [001¯], [101¯] or [100 ] oriented dislocations. The amplitude of the applied electric field is 2.
polarizations still not switch completely. So, whether the pinning effect emerged in the film containing seven [100 ] oriented interfacial dislocations can be eliminated by an even larger electric field? The P-E curve under an electric field E0∗ = 10sin(2πt ∗/200) is shown in Fig. 7. More polarization vectors take part in the switching process when the amplitude of the electric field increases to 10, but the double hysteresis loop still exist. From the above analysis, it is found that the presence of 7 dislocations will pin the polarization switching strongly. The pinning effect of the [100 ] oriented interfacial dislocations is the strongest and cannot be eliminated by a very large electric field. So, if the number of dislocations decreases from 7 to 3, will the pinning effect disappear? The simulation results are shown in Fig. 8. The remnant polarization and coercive field of hysteresis loop corresponding to three interfacial dislocations with [101¯] orientation are also asymmetry, meaning that the imprint phenomenon emerges at a lower dislocation density. And the P-E curve corresponding to 3 interfacial dislocations with [100 ] orientation shows a double polarization–electric field hysteresis loop, implying that the pinning effect emerges at a lower dislocation density. Comparing Fig. 7 with Fig. 8, it can be concluded that the pinning effect of [100 ] oriented dislocation becomes weaker as the density of
Fig. 9. The polarization–electric field hysteresis loops of ferroelectric thin films containing 3 interfacial [001¯], [101¯] or [100 ] oriented dislocations. The amplitude of the applied electric field is 5.
dislocations decreases. These dislocations induced imprint behavior and pinning effect can almost be eliminated by a larger applied electric field when the dislocation density is lower. As displayed in Fig. 9, when the amplitude of the electric field increases from 2 to 5, the double hysteresis loop corresponding to three interfacial dislocations with [100 ] orientation disappear, and it become a single hysteresis loop that polarizations have switched completely. It again proves that the pinning effect of seven interfacial dislocations with [100 ] orientation is stronger than that of 3 interfacial dislocations. 7. Conclusion In this paper, the previous phase field model of flexoelectric effect is modified to investigate the effects of dislocations with different locations, orientations and density on domain structures and evolution. Dislocations in the film lead to the appearance of a-domain. And the size of the new a-domain depends on the orientations and locations of dislocations. Among [001¯], [101¯] and [100 ] oriented single dislocations, the new a-domain near the [101¯] oriented dislocation is the broadest. The polarization switching properties with different dislocations in thin
Fig. 7. The polarization–electric field hysteresis loop of ferroelectric thin film containing 7 interfacial [100 ] oriented dislocations. The amplitude of the applied electric field is 10. 7
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film are also studied. It is found that the [001¯] oriented dislocation has weak effect on polarization switching, while the [101¯] oriented dislocations can result in imprint failure, and the [100 ] oriented dislocations bring intense pinning effect on polarization switching. As the density of dislocations increases, the pining effects become stronger, and the ferroelectricity of ferroelectric thin film is gradually suppressed, especially that the existence of seven dislocations with [100 ] orientation can even suppress the ferroelectricity of thin films completely. These dislocations induced imprint behavior and pinning effect cannot be eliminated by a larger applied electric field when the dislocation density is high. Therefore, in order to acquire ferroelectric thin film with higher quality and longer lifetime, it is necessary to avoid the existence of high density [101¯] oriented dislocations which can lead to imprint failure and high density [100 ] oriented dislocations that show the strongest pinning effect.
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Yang, Strain tunability of the downward effective polarization of mechanically written domains in ferroelectric nanofilms, RSC Adv. 6 (2016) 80946–80954. [37] Q.Y. Qiu, S.P. Alpay, V. Nagarajan, Phase diagrams, dielectric response, and piezoelectric properties of epitaxial ultrathin (001) lead zirconate titanate films under anisotropic misfit strains, J. Appl. Phys. 107 (2010) 114105. [38] X.F. Xu, L.M. Jiang, Y.C. Zhou, Reduction of leakage currents in ferroelectric thin films by flexoelectricity: a phase field study, Smart Mater. Struct. 26 (2017) 115024. [39] L.M. Jiang, J.Y. Tang, Y.C. Zhou, Q. Yang, Y.Z. Hang, Simulations of local -mechanical -stress-induced ferroelectric polarization switching by a multi-field coupling model of flexoelectric effect, Comput. Mater. Sci. 108 (2015) 309–315. [40] R. Ahluwalia, A. Tagantsev, P. Yudin, N. Setter, N. Ng, D. Srolovitz, Influence of flexoelectric coupling on domain patterns in ferroelectrics, J. Phys. Rev. B 89 (2014) 174105. [41] M.W. 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CRediT authorship contribution statement Xuanling Liu: Data curation, Formal analysis, Methodology, Software, Visualization, Writing - original draft. Xinyi Shan: Investigation, Resources. Shiqi Feng: Validation. Xiaofei Xu: Project administration, Supervision. Limei Jiang: Conceptualization, Funding acquisition, Writing - review & editing. Acknowledgements This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 11502224, 11032010, 11372266, 11172257 and 11202054), and Key research and development plan of Hunan Province (Grant No. 2017GK2040). Data available The data used to support the findings of this study are available from the corresponding author upon request. References [1] Q. Yang, J.X. Cao, Y. Ma, Y.C. Zhou, L.M. Jiang, X.L. Zhong, Strain effects on formation and migration energies of oxygen vacancy in perovskite ferroelectrics: a first-principles study, J. J. Appl. Phys. 113 (2013) 184110. [2] Q. Yang, J.X. Cao, Y. Ma, Y.C. Zhou, First principles study of polarization-strain coupling in SrBi2Ta2O9, J. AIP Adv. 3 (2013) 052134. [3] S. Ramanathan, Thin Film Metal-Oxides, Springer, US, 2010. [4] K.J. Choi, et al., Enhancement of ferroelectricity in strained BaTiO3 thin film, J. Sci. 306 (2004) 1005–1009. [5] T. Kiguchi, K. Aoyagi, Y. Ehara, H. Funakubo, T. Yamada, N. Usami, T.J. Konno, Configuration and local elastic interaction of ferroelectric domains and misfit dislocation in PbTiO3/SrTiO3 epitaxial thin films, J. Sci. Technol. Adv. Mater. 12 (2011) 034413. [6] Y. Liu, Y.L. Tang, Y.L. Zhu, W.Y. Wang, X.L. Ma, Spatial coupling of ferroelectric domain walls and crystallographic defects in the PbTiO3 films, J. Adv. Mater. Interfaces 3 (2016) 1600342. [7] H.H. Wu, J. Wang, S.G. Cao, L.Q. Chen, T.Y. Zhang, The unusual temperature dependence of the switching behavior in a ferroelectric single crystal with dislocations, J. Smart Mater. Struct. 23 (2014) 025004. [8] Y.L. Li, et al., Influence of interfacial dislocations on hysteresis loops of ferroelectric films, J. J. Appl. Phys. 104 (2008) 104110. [9] H.H. Wu, J. Wang, S.G. Cao, L.Q. Chen, T.Y. Zhang, Micro-/macro-responses of a ferroelectric single crystal with domain pinning and depinning by dislocations, J. J. Appl. Phys. 114 (2013) 164108. [10] H.H. Wu, S.G. Cao, J.M. Zhu, T.Y. Zhang, The frequency-dependent behavior of a ferroelectric single crystal with dislocation arrays, J. Acta Mech. 228 (2017) 2811–2817. [11] Y. Zheng, B. Wang, C.H. Woo, Simulation of interface dislocations effect on polarization distribution of ferroelectric thin films, J. Appl. Phys. Lett. 88 (2006)
8
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Effects of dislocations with different locations, orientations and density on domain evolution of ferroelectric thin film: A phase field study Xuanling Liua, Xinyi Shana, Shiqi Fenga, Xiaofei Xua,b, Limei Jianga,b, a b
T
⁎
Hunan Provincial Key Laboratory of Thin Film Materials and Devices, School of Materials Science and Engineering, Xiangtan University, Hunan 411105, China Key Laboratory of Key Film Materials & Application for Equipment of Hunan Province, Xiangtan University, Hunan 411105, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Ferroelectric material Phase field method Flexoelectric effect Dislocation Polarization distribution Polarization switching
The effects of dislocations on polarization distribution and switching in ferroelectric thin films are discussed based on a modified multi-field coupling theoretical framework that combining the flexoelectric effect and the strain field caused by dislocations. First, the correctness of the model is verified through a study about the effects of different flexocoupling types on domain structures around the dislocations. Then, effects of dislocations with different locations, orientations and density are systematically studied. Dislocations in the film will induce the appearance of a-domain. And the size of the new a-domain depends on the orientations and locations of dislocations. Among [001¯], [101¯] and [100 ] oriented single dislocations, the size of the new a-domain near the [101¯] oriented dislocation is the largest. It is essential to avoid generating [100 ] and [101¯] oriented dislocations since the [101¯] oriented dislocations can cause imprint failure and [100 ] oriented dislocations bring intense pinning effects. As the density of dislocations increases, the pinning effect and imprint behavior becomes stronger. The hysteresis loop of the film with [100 ] oriented multi-dislocations shows no ferroelectricity once the density reaches a certain limit. These dislocations induced imprint behavior and pinning effect can be eliminated by a larger applied electric field when the dislocation density is low. But when the dislocation density gets higher, the situation is different.
1. Introduction With the miniaturization and high integration of ferroelectric devices, the size of the ferroelectric thin film in the thickness direction is close to the nanometer scale. Both the structure and the evolution of the polarization (domain) are more and more constrained by the microstructures including lattice defect, interface terminal type, oxygen vacancy and so on [1,2]. Therefore, to make ferroelectric devices more reliable, it is necessary to study the influence mechanism of the micro factors affecting the polarizations. As one of the most common defects [3], dislocations are inevitably produced during the fabrication of ferroelectric thin films. A dislocation is a crystallographic defect or irregularity within a crystal structure. Some types of dislocations can be visualized as being caused by the termination of an atoms plane in the middle of a crystal. Up to now, dislocations, edge dislocations in particular, with different orientations have been found inside the ferroelectric films or at the interface. Dislocations can induce polarization instability [4], domain pinning [5], or even degrading of the ferroelectric properties [6] because of the electro-mechanical coupling
between the strain / strain gradient field and polarization. Meanwhile, in order to explain the above observed experimental phenomena and reveal the intrinsic influence mechanism about how dislocations affect polarization patterns and switching properties, many phase field models have been established [7–11]. For example, Wu et al. [9] established a phase field model to discuss how dislocations lead to pinning and de-pinning in single-crystal ferroelectric films. Zheng et al. [11] studied the effect of interfacial dislocations on ferroelectric properties in a one-dimensional environment and showed that the interfacial dislocation can greatly reduce the average polarization of ferroelectric thin films at high temperature, and it can reduce the effective Curie temperature several times and have a great influence on the local dielectric properties. Even though the previous works about how dislocations affect the ferroelectric thin film’s properties have made a significant contribution, our understanding of their influence mechanism is still limited. For example, there are large strain gradients around the dislocations, which affect ferroelectric thin films’ properties through flexoelectric coupling. Flexoelectric effect implies that strain gradient can induce polarization
⁎ Corresponding author at: Hunan Provincial Key Laboratory of Thin Film Materials and Devices, School of Materials Science and Engineering, Xiangtan University, Hunan 411105, China. E-mail address: [email protected] (L. Jiang).
https://doi.org/10.1016/j.commatsci.2019.109102 Received 28 February 2019; Received in revised form 24 June 2019; Accepted 25 June 2019 0927-0256/ © 2019 Elsevier B.V. All rights reserved.
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(the direct flexoelectric effect, see Eq. (1)), or polarization gradient can induce stress (the converse flexoelectric effect, see Eq. (2)) [12,14]:
Pl = μijkl ∂εij / ∂xk
(1)
εij = Ωijkl ∂Pl/ ∂xk
(2)
σij = ∂h/ ∂εij tijl = ∂h/ ∂εij, l Di = −∂h/ ∂Ei; ηi = ∂h/ ∂Pi; Λij = ∂h/ ∂Pi, j
where, σij, ηi , Di , tijl, Λij respectively represent stress, effective local electric force, electric displacement, double stress tensor that workconjugate with the strain gradient tensor and higher order local electric force. It should be noted that there is large strain around the dislocations, which can couple with the polarization gradient through the flexoelectric effect. Therefore, in the establishment of a phase field model considering dislocations, it is necessary to put the strain caused by dislocations into consideration. Hence, the total strain εijtotal should be composed of two parts, the elastic strain, εij , and the strain related to a dislocation, εijd . The dislocation strain can be deduced from dislocation stress field based on constitutive Eq. (6).
where, Pl , μijkl , εij , xk , Ωijkl denote the components of the material polarization, the flexoelectric coupling tensor components, the strain tensor components, the displacements, and the inverse flexoelectric coupling tensor components, respectively. The flexoelectric field has been proved to be as large as 1 KV/m, which is strong enough to induce polarization rotation [13], and even 180° polarization switching [15] in ferroelectric thin films without dislocations. Then when there is a dislocation, the huge strain gradient exists, and the impact of flexoelectric effect will become stronger and cannot be ignored. However, at present, few phase field studies regarding to dislocations in ferroelectric thin films have put flexoelectric effect into consideration. Furthermore, there are also few studies making systematical discussion about effects of dislocations with different orientations, locations and density on domain structures and evolution. In this paper, a phase field model that combining the flexoelectric effect and the strain field caused by dislocations is established to systematically discuss the effects of dislocations with different locations, orientations and density on polarization distribution and switching. The effect of dislocations on polarization distribution is more pronounced after considering the flexoelectricity. Among the dislocations with three orientations studied in this paper, the dislocations with [101¯] orientation have the most influence on promoting the emergence of the new adomain. What’s more, the hysteresis loops of thin films containing a single dislocation, three dislocations or seven dislocations at the interface are plotted respectively. The results show that with the dislocation density increases, the ferroelectric properties of thin films with [100 ] oriented multi-dislocations are gradually suppressed or even disappear.
εijd = σijd (1 + υ)/ E − νΘδij / E here, the stress tensor components caused by dislocations by [17,18].
are given
d σxx = −Wy (x 2 − y 2 )/(x 2 + y 2 )2 d d τxy = τ yx = Wx (x 2 + y 2 )/(x 2 + y 2 )2
(7)
For simulating the stress field around dislocations with different orientations easier, we turn to describe the stress field in the polar coordinate: d σxx = −Wsinθ (2 + cos 2θ)/ r d σyy = Wsinθcos 2θ / r d τxy = W cosθcos 2θ / r
(8)
where, W = Gb/[2π (1 − υ)], in which, G is the shear modulus, b is the mode of Burgers' vector, E is the elastic modulus, ν is the Poisson's ratio; Θ = σx + σy + σz = σkk ; δij is the kronecker delta notation; r is the distance between the dislocations and a specific point; θ is the angle between the x axis and the line connecting the dislocation and the specific point. What needs to be raised is that dislocations actually are the results of the system energy minimization. Up to now, there have been a lot of excellent works focusing on the dislocation formation and evolution [19–23]. Especially the work reported by A Skaugen et al. [24], in which a phase field crystal model was established to investigate the formation of dislocations and analyze the dislocation density quantitatively. These works have made great contributions and are instrumental to reveal the essence of the dislocation formation and evolution. However, these works put their emphasis on the dislocation itself including the dislocation motion and evolution rather than the effects of dislocations on other physical quantities. For ferroelectric materials, people are more concerned about how dislocations affect domain structures, and do not pay much attention to the source and influencing factors of dislocations. Dislocations in ferroelectric thin films usually were considered to be pre-existing and fixed in position and density [11,25–27]. The stress/strain fields of dislocations are often simplified by the following ways: a) the strain caused by dislocations is treated as the eigenstrain which will influence the elastic free energy of the system, thus affect the whole free energy of the system [25,26]; b) simulate the exist of dislocations by setting the boundary conditions of the displacement field according to the strain situation above and below the dislocations in real space [27]; c) the stress field caused by dislocation is regarded as an applied field and coupled with other physical fields to affect the free energy of the system together with the eigenstrain [11]. Our work refers to the approach of Y. Zheng et al. [11],
In this paper, the previous phase field model of flexoelectricity [16] is modified by combining the effect of dislocations.
2.1. Thermodynamic potential The Landau-Devonshire free energy density expression is: (3)
where hlandau , hstrain , hcoup, hgrad , helec and hflexo indicate the landau free energy density, elastic energy density, polarizations and strain coupling energy density, polarization gradient energy density and flexoelectricity energy density, respectively. The expression for flexoelectricity energy density is:
hflexo = fijkl (εkl ∂Pi/ ∂x j − Pk ∂εij / xl )/2
(6)
σijd
d σxx = −Wy (3x 2 + y 2 )/(3x 2 + y 2 )2
2. Theoretical method
h = hlandau + hstrain + hcoup + hgrad + helec + hflexo
(5)
(4)
and fijkl represent the flexoelectric coefficients. 2.2. Constitutive equations of each physical field Generally, in order to describe the electro-mechanical coupling behaviors of ferroelectric thin films, three physical fields are required, which are electric field, displacement field and polarization field. And the constitutive equation of these three physical fields can be deduced from the free energy density [16]. 2
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which considered the strain field of dislocations as a linear applied field. The strain field of dislocations in our model is coupled with polarization to change the strain-polarization coupling energy h coup and ultimately change the total free energy of the system. Although the results may be biased away from the reality, it still reveals the influence mechanism of the dislocations on domain structures to some extent like the results of previous studies [11,25–27]. And the feasibility of these qualitative studies is proved in the following Section 3.
completely compensated by external charges through a short circuit boundary condition. Since the size effect is too strong to be ignored owing to the ultrathin thickness of thin film, the polarization boundary conditions of the top and bottom surface of the thin film are both assumed as −dP2/ dx2 = 0 . And the initial condition of the polarization field is P1 = 0 , P2 = 1. In the simulations, we discretize the ferroelectric thin film into 2400 (80 × 30) quadrilateral elements, and the size of each element is Δx1∗ = Δx 2∗ = 1, which corresponds to the real size Δx1 = Δx2 = 1 nm . We set every dimensionless time step as Δt ∗ = 0.05, and set the total number of steps to be 5000 in order to ensure that the evolution of domain structures reaches a equilibrium state. For the cubic point-group materials like PZT, the flexocoupling coefficients have three independent components: f1111, f1122 , f1212 ( f2121) [40], which describe three different flexocoupling types. Our previous studies found that flexocoupling types can make different effects on polarization distribution. So, do different types of flexoelectricity play different roles in the redistribution of polarizations near dislocations? Taking the ferroelectric thin films containing a single dislocation with [001¯] orientation as an example, we simulate 7 situations: (1) f11 ≠ 0 , f12 = f44 = 0 ; (2) f12 ≠ 0 , f11 = f44 = 0 ; (3) f44 ≠ 0 , f11 = f12 = 0 ; (4) f11 ≠ f12 ≠ 0 , f44 = 0 ; (5) f11 ≠ f44 ≠ 0 , f12 = 0 ; (6) f12 ≠ f44 ≠ 0 , f11 = 0 ; (7) f11 ≠ f12 ≠ f44 ≠ 0 . We compare the polarization distributions of these 7 situations to find out the role of each flexoelectricity type in the redistribution of polarizations near dislocations. Where, f11 , f12 and f44 respectively represent the flexoelectric coefficients in Voigt form f1111, f1122 and f1212 . As shown in Fig. 1, a small a-domain emerges in the single c-domain near the dislocation. Compared with the domain structures of thin films considering f12 or f44 related flexocoupling type, the new a-domain of the thin film considering f11 related flexocoupling is the broadest which is more coincide with experimental observations. And it should be noticed that the polarization switching area of films considering only one flexocoupling type is obviously smaller than those observed by experiments [41,42]. Therefore, a further discussion about the effect of dislocations on polarization distribution under the collective effect of different flexocoupling types is indispensable. As shown in Fig. 1(f), when f12 ≠ f44 ≠ 0 , f11 = 0 , the size of the emerging a-domain in the thin film is smaller than that in the film considering f11 only (see Fig. 1(a)), which again demonstrates that these two flexocoupling types have much smaller effects on polarizations near the dislocation. However, from Fig. 1 (d) and (e), we can see that when consider f11 and f12 together or f11 and f44 together, the emerging a domain become larger than that of f11 (Fig. 1 (a)), and the simulation results become closer to the experimental results [41,42]. It implies that although the f12 and f44 related flexocoupling types have much smaller effects on the polarizations near the dislocation, their effects cannot be neglected. Thus, it can be speculated that the redistribution of polarizations near dislocations are owing to the collective effect of three flexocoupling types. As shown in Fig. 1(g), due to the presence of a dislocation located at (0,0), a large a-domain occurs in the single cdomain structure. That is, the polarization near the dislocation has a very significant 90° switching so that the domain structure changes into a c/a/c multi-domain structure. And the simulation result that an evident 90° polarization switching caused by dislocations agrees with the experimental results well [41,42]. Therefore, the dislocation phase field model considering the collective effect of three flexocoupling types is the most accurate. The following works are based on this model which ensures that the simulations correctly predict the effects of dislocations on domain structures.
2.3. The equilibrium equations for each physical field The mechanical equilibrium equation including the contribution of strain gradient is given based on the Mindlin’s strain gradient theory [28,29]:
(σij − tijk, k ), j + bi = 0,
(9)
where bi are the components of a body force per unit volume. The temporal evolution of the polarization field is governed by the time dependent Ginzburg–Landau equations [30–35]:
∂Pi/ ∂t = −L (ηi − Λij, j)
(10)
The electric field is governed by the Maxwell’s (or Gauss’) equation:
Di, i − ξ = 0
(11)
where ξ is the volume charge density. For the equations listed above have to be solved simultaneously under appropriate boundary and initial conditions, the finite element methods are employed based on the PDE part of the finite element software COMSOL [30]. And the details of finite element formulations can be found in Ref. [36]. 3. Effects of different flexocoupling modes on domain structures around dislocations In this paper, the tetragonal Pb(Zr0.1Ti0.9)O3 (PZT) material is used as an example. For the sake of simplicity, 2D finite element simulations of a single c-domain are carried on, as shown in Fig. 1. The x1 and x2 axes respectively denote the length and thickness directions of thin films. Here, the directions of Burgers’ vectors are used to describe the orientations of dislocations. For example, the dislocation with Burgers’ vector [100 ] is called as the dislocation with orientation [100 ]. As known, the orientations of dislocations should obey the structure and energy conditions of crystals. The PZT can be regarded as the coexistence of face-centered cubic and body-centered cubic. According to the Burgers’ vectors of perfect dislocations and partial dislocations in common crystals, the Burgers’ vectors of perfect dislocations existed in 2D-plane of PZT are < 100> and < 110>. Therefore, this paper discusses three directions derived from the < 100> and < 110> crystal family, namely [100 ], [001¯] and [101¯]. The material coefficients including the phenomenological Landau–Devonshire coefficients, the elastic constants, the gradient coefficients, and the electrostrictive coefficients that required for expressing the system energy in Eq. (3) are consistent with those used by Qiu et al. [37], and Xu et al. [38]. In order to improve the compute speed, all the quantities in this paper are nondimensionalized based on the spontaneous polarization Ps (C / m2) at room temperature, the gradient energy coefficient g110 (m4N / C −2) and the free energy coefficient of Landauα 0 (m2N / C −2) . All the formulations of conversion between dimensioned and dimensionless quantities can be found in Ref. [39]. The boundary conditions of each field are set as follows: To imitate a real ferroelectric thin film, we assume periodic boundary conditions for displacement, polarization, and electric field along the x1 direction. The top and bottom surfaces are assumed to be traction free, and the mechanical initial values are u 0 = v0 = 0 . For the electric field, the electric potential of the top and bottom surface of the film is taken as zero, which means the surface charges induced by polarization are
4. Effects of single interfacial or internal dislocations on domain structures In the process of fabricating ferroelectric thin films, dislocations are often formed at the interface of the film, and as the temperature increases, dislocations can enter the interior of the films. In this section, 3
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(a)
(d)
(b)
(e)
(c)
(f)
(g)
Fig. 1. Domain structure of ferroelectric thin films containing a single dislocation with [001¯] orientation when (a) f11 ≠ 0 , f12 = f44 = 0 ; (b) f12 ≠ 0 , f11 = f44 = 0 ; (c) f44 ≠ 0 , f11 = f12 = 0 ; (d) f11 ≠ f12 ≠ 0 , f44 = 0 ; (e) f11 ≠ f44 ≠ 0 , f12 = 0 ; (f) f12 ≠ f44 ≠ 0 , f11 = 0 ; (g) f11 ≠ f12 ≠ f44 ≠ 0 .
of the internal dislocation. Fig. 2(d–f) shows that the polarization switching zones near the [001¯] and [101¯] oriented dislocations are obviously larger than that near the [100 ] oriented dislocation. Polarization vectors near the dislocation with [100 ] orientation only have small angle of rotation, while polarizations near the [001¯] and [101¯] oriented dislocations experience 90° switching. What’s more, it can be seen from the Fig. 2(d) that the polarizations above the dislocation turn to be negative, while the polarizations below the dislocation are just strengthened rather than inversion. Same as the situation of the thin film containing a single interfacial dislocation, the polarization switching zone near the [101¯] oriented dislocation is the largest. This may be due to the strongest piezoelectric effect and flexoelectric coupling effect between the [101¯] oriented dislocation strain and polarizations.
effects of single interfacial and internal dislocations with different orientations on domain structures are discussed. With the thin film containing a single [100 ] oriented dislocation at the position (0,0) as the initial state, and α as the clockwise rotation angle of the dislocation, we deduce the stress field near dislocations with different orientations. The α of 0°, 45° and 90° respectively correspond to the dislocation orientations of [100 ], [101¯] and [001¯]. d σxx = −2W (ycosα + xsinα )·(x 2 + y 2 )/(x 2 + y 2 )2 − W (ycosα + xsinα )[(x 2 + y 2 ) cos 2α − xysin2α ]/(x 2 + y 2 )2 d σyy = W (ycosα + xsinα )·[(x 2 − y 2 ) cos 2α − xysin2α ]/(x 2 + y 2 )2 d σxy = W (xcosα + ysinα )·[(x 2 − y 2 ) cos 2α − xysin2α ]/(x 2 + y 2 )2
(12)
As shown in Fig. 2(a)–(c), the interfacial dislocations in a single cdomain tend to lead to the emergence of the a-domain, thus the domain structures of thin films change from a single c domain to be an a/c/a domain. The region of the appeared a domain is the largest in the thin film with the interfacial [101¯] oriented dislocation. For the reason that interfacial dislocations can enter the interior of the films with the temperature increases during the fabrication of ferroelectric thin films, the effects of internal dislocations on domain structures of ferroelectric thin films are also needed to be discussed. The stress field around the internal dislocation is deduced from Eq. (12) by means of replacing x and y with x + x1 and y + y1, respectively. Here, x1 and y1 respectively represent the horizontal and vertical displacements
5. Effects of periodic dislocations on domain structures In fact, there are always many dislocations in the ferroelectric thin films. Therefore, in order to be closer to the actual situation, the phase field model is further modified to contain multi-dislocations. To simplify the multi-dislocations model, it is assumed that the dislocations are periodic and the distances between adjacent dislocations are equal. The stress field equations around the periodic dislocations are deduced according to the stress field equations around the single dislocation [43]: 4
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(a)
(d)
(b) (e)
(c) (f) Fig. 2. Stable domain structures of ferroelectric thin films with various dislocations. Interfacial dislocations with different orientations: (a) [100 ], (b) [101¯], (c) [001¯]. Internal dislocations with different orientations: (d) [100 ], (e) [101¯], (f) [001¯].
6. Polarization response of ferroelectric thin films with dislocations
d = −W ∑ y (3x m2 + y 2 )/(x m2 + y 2 )2 σxx m d = W ∑ y (x m2 − y 2 )/(x m2 + y 2 )2 σyy m d = W ∑ x m (x m2 − y 2 )/(x m2 + y 2 )2 σxy m
Polarization switching is also a very important property for ferroelectric materials. Thus, the effects of dislocations on the polarization responses are studied in this section. The polarization–electric field hysteresis loops (P-E curves) of ferroelectric thin films containing interfacial single dislocations with orientations [001¯], [101¯] and [100 ] are plotted under the applied sinusoidal electric field E0∗ = 2sin(2πt ∗/200) along the ferroelectric film’s thickness direction. While the time step is set as Δt ∗ = 0.05, and the total time is set to be 250 in the simulations. It can be seen from Fig. 4 that the effects of the [001¯] and [101¯] oriented interfacial single dislocations on the P-E curves are different. The positive and negative remnant polarizations and the coercive field shown in the [001¯] oriented dislocation related hysteresis loop are symmetrical. However, under the effect of the interfacial single dislocation with [101¯] orientation, though the positive and the negative remnant polarizations of the P-E curve are symmetrical, the positive and negative coercive fields show some deviations. The positive coercive field is slightly larger than the negative coercive field, in other words, the hysteresis loop shifts to the right. This phenomena implies that interfacial single dislocations with [101¯] orientation would induce slight imprint failure of ferroelectric thin film. And the imprint behavior of the ferroelectric thin film becomes more serious under the effect of [100 ] oriented dislocations. Both the remnant polarization and the coercive field shown in the hysteresis loop are not symmetrical. That is, the value of the positive remnant polarization is greater than the negative one and the value of the positive coercive field is smaller than the negative one, indicating that the interfacial dislocation with [100 ] orientation can suppress the switching of positive polarizations, and a larger coercive field is needed for polarization switching. We then plot the P-E curves of ferroelectric thin film with seven [001¯], [101¯] or [100 ] oriented interfacial dislocations, as shown in Fig. 5. The applied electric field and time step are the same as above. Among
(13)
where x m = mx 0 , and x 0 is the distance between two adjacent dislocations, m = ⋯−2, −1, 0, 1, 2⋯. The stable domain structures of thin films containing interfacial multi-dislocations with different orientations are shown in Fig. 3(a-c). The positions of seven interfacial dislocations are: (0,0), (−10,0), (10,0), (−20,0), (20,0), (−30,0), (30,0). Through comparing the polarization switching zones shown in Fig. 3 with that of thin films containing single dislocations, we find that the polarization switching zone in the film with periodic interfacial dislocations is far broader. The stable domain structures of thin films containing internal multidislocations with different orientations are displayed in Fig. 3(d–f). The position of seven dislocations are: (0,15), (−10,15), (10,15), (−20,15), (20,15), (−30,15), (30,15). From the figure, it can be readily seen that the polarization 90° switching occurs above the internal dislocations with orientation [100 ]. But polarizations below the [100 ] oriented internal dislocations maintain their initial state. When there exist periodic internal dislocations with orientation [101¯], the 90° domain walls appear both above and below the internal dislocations, so the initial c-domain changes to be an irregular and complicated a-c domains. It is very interesting that polarization vortices are found near the dislocations with orientation [001¯]. And polarizations far away the [001¯] oriented internal dislocations experience 90° switching. In summary, the internal dislocations with different orientations all can cause polarization switching and lead to the appearance of 90° domain walls. However, the polarization switching zone near each dislocation of the multi-dislocations array is smaller than that of single dislocations. This is because the more dislocations in the ferroelectric thin film, the stronger pinning effect occurs which will suppress the polarization switching and the evolution of domain structures. 5
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(a)
(d)
(b)
(e)
(c)
(f)
Fig. 3. Stable domain structures of ferroelectric thin film containing various periodic dislocations. Periodic interfacial dislocations with different orientations: (a) [100 ] (b) [101¯] (c) [001¯]. Periodic internal dislocations with different orientations: (d) [100 ] (e) [101¯] (f) [001¯].
Fig. 4. The polarization–electric field hysteresis loops of ferroelectric thin films with [001¯], [101¯] or [100 ] oriented single interfacial dislocations.
Fig. 5. The polarization–electric field hysteresis loops of ferroelectric thin films containing 7 interfacial dislocations with [001¯], [101¯] or [100 ] orientation.
these three oriented interfacial dislocations, the P-E curve is more symmetrical for [001¯] orientation case. The curve turns to be asymmetrical for the case of dislocations with [101¯] orientation, in other words, an imprint behavior occurs here. The P-E curve of the thin film containing interfacial dislocations with [100 ] orientation is a linear line and shows no features of hysteresis loops. This indicates that, in this situation, the ferroelectric thin film is at a modulation phase and shows no ferroelectricity. Previous studies show that dislocations in ferroelectric thin films would pin the ferroelectric domain and hinder the domain switching [25]. The dislocations with [100 ] orientation here make polarizations above the dislocations tend to point to the horizontal direction and can’t switch under the applied electric field. Therefore, the ferroelectric film eventually loses its ferroelectricity.
Whether the pinning effect emerged here can be eliminated by a larger applied electric field? In order to answer this question, the applied electric field is then set as E0∗ = 5sin(2πt ∗/200) along the thickness direction of the ferroelectric thin film (x2 direction). The simulation result is shown in Fig. 6. For the thin film containing 7 interfacial dislocations with [101¯] orientation, as the amplitude of the electric field increases, some of the polarizations are de-pinned and take part in the switching process, resulting in an increase in the positive and negative remnant polarization. For the thin film containing 7 interfacial dislocations with [100 ] orientation, the hysteresis loop begins to appear when the electric field is strong enough (> 2 or < -3), and it’s obvious that the “hysteresis phenomenon” is more apparent when the electric field is positive. Although a double hysteresis loop appears in this situation, the
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Fig. 6. The polarization–electric field hysteresis loops of ferroelectric thin films containing 7 interfacial dislocations with [001¯], [101¯] or [100 ] orientation. The amplitude of the applied electric field is 5.
Fig. 8. The polarization–electric field hysteresis loops of ferroelectric thin films containing 3 interfacial [001¯], [101¯] or [100 ] oriented dislocations. The amplitude of the applied electric field is 2.
polarizations still not switch completely. So, whether the pinning effect emerged in the film containing seven [100 ] oriented interfacial dislocations can be eliminated by an even larger electric field? The P-E curve under an electric field E0∗ = 10sin(2πt ∗/200) is shown in Fig. 7. More polarization vectors take part in the switching process when the amplitude of the electric field increases to 10, but the double hysteresis loop still exist. From the above analysis, it is found that the presence of 7 dislocations will pin the polarization switching strongly. The pinning effect of the [100 ] oriented interfacial dislocations is the strongest and cannot be eliminated by a very large electric field. So, if the number of dislocations decreases from 7 to 3, will the pinning effect disappear? The simulation results are shown in Fig. 8. The remnant polarization and coercive field of hysteresis loop corresponding to three interfacial dislocations with [101¯] orientation are also asymmetry, meaning that the imprint phenomenon emerges at a lower dislocation density. And the P-E curve corresponding to 3 interfacial dislocations with [100 ] orientation shows a double polarization–electric field hysteresis loop, implying that the pinning effect emerges at a lower dislocation density. Comparing Fig. 7 with Fig. 8, it can be concluded that the pinning effect of [100 ] oriented dislocation becomes weaker as the density of
Fig. 9. The polarization–electric field hysteresis loops of ferroelectric thin films containing 3 interfacial [001¯], [101¯] or [100 ] oriented dislocations. The amplitude of the applied electric field is 5.
dislocations decreases. These dislocations induced imprint behavior and pinning effect can almost be eliminated by a larger applied electric field when the dislocation density is lower. As displayed in Fig. 9, when the amplitude of the electric field increases from 2 to 5, the double hysteresis loop corresponding to three interfacial dislocations with [100 ] orientation disappear, and it become a single hysteresis loop that polarizations have switched completely. It again proves that the pinning effect of seven interfacial dislocations with [100 ] orientation is stronger than that of 3 interfacial dislocations. 7. Conclusion In this paper, the previous phase field model of flexoelectric effect is modified to investigate the effects of dislocations with different locations, orientations and density on domain structures and evolution. Dislocations in the film lead to the appearance of a-domain. And the size of the new a-domain depends on the orientations and locations of dislocations. Among [001¯], [101¯] and [100 ] oriented single dislocations, the new a-domain near the [101¯] oriented dislocation is the broadest. The polarization switching properties with different dislocations in thin
Fig. 7. The polarization–electric field hysteresis loop of ferroelectric thin film containing 7 interfacial [100 ] oriented dislocations. The amplitude of the applied electric field is 10. 7
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film are also studied. It is found that the [001¯] oriented dislocation has weak effect on polarization switching, while the [101¯] oriented dislocations can result in imprint failure, and the [100 ] oriented dislocations bring intense pinning effect on polarization switching. As the density of dislocations increases, the pining effects become stronger, and the ferroelectricity of ferroelectric thin film is gradually suppressed, especially that the existence of seven dislocations with [100 ] orientation can even suppress the ferroelectricity of thin films completely. These dislocations induced imprint behavior and pinning effect cannot be eliminated by a larger applied electric field when the dislocation density is high. Therefore, in order to acquire ferroelectric thin film with higher quality and longer lifetime, it is necessary to avoid the existence of high density [101¯] oriented dislocations which can lead to imprint failure and high density [100 ] oriented dislocations that show the strongest pinning effect.
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CRediT authorship contribution statement Xuanling Liu: Data curation, Formal analysis, Methodology, Software, Visualization, Writing - original draft. Xinyi Shan: Investigation, Resources. Shiqi Feng: Validation. Xiaofei Xu: Project administration, Supervision. Limei Jiang: Conceptualization, Funding acquisition, Writing - review & editing. Acknowledgements This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 11502224, 11032010, 11372266, 11172257 and 11202054), and Key research and development plan of Hunan Province (Grant No. 2017GK2040). Data available The data used to support the findings of this study are available from the corresponding author upon request. References [1] Q. Yang, J.X. Cao, Y. Ma, Y.C. Zhou, L.M. Jiang, X.L. Zhong, Strain effects on formation and migration energies of oxygen vacancy in perovskite ferroelectrics: a first-principles study, J. J. Appl. Phys. 113 (2013) 184110. [2] Q. Yang, J.X. Cao, Y. Ma, Y.C. Zhou, First principles study of polarization-strain coupling in SrBi2Ta2O9, J. AIP Adv. 3 (2013) 052134. [3] S. Ramanathan, Thin Film Metal-Oxides, Springer, US, 2010. [4] K.J. Choi, et al., Enhancement of ferroelectricity in strained BaTiO3 thin film, J. Sci. 306 (2004) 1005–1009. [5] T. Kiguchi, K. Aoyagi, Y. Ehara, H. Funakubo, T. Yamada, N. Usami, T.J. Konno, Configuration and local elastic interaction of ferroelectric domains and misfit dislocation in PbTiO3/SrTiO3 epitaxial thin films, J. Sci. Technol. Adv. Mater. 12 (2011) 034413. [6] Y. Liu, Y.L. Tang, Y.L. Zhu, W.Y. Wang, X.L. Ma, Spatial coupling of ferroelectric domain walls and crystallographic defects in the PbTiO3 films, J. Adv. Mater. Interfaces 3 (2016) 1600342. [7] H.H. Wu, J. Wang, S.G. Cao, L.Q. Chen, T.Y. Zhang, The unusual temperature dependence of the switching behavior in a ferroelectric single crystal with dislocations, J. Smart Mater. Struct. 23 (2014) 025004. [8] Y.L. Li, et al., Influence of interfacial dislocations on hysteresis loops of ferroelectric films, J. J. Appl. Phys. 104 (2008) 104110. [9] H.H. Wu, J. Wang, S.G. Cao, L.Q. Chen, T.Y. Zhang, Micro-/macro-responses of a ferroelectric single crystal with domain pinning and depinning by dislocations, J. J. Appl. Phys. 114 (2013) 164108. [10] H.H. Wu, S.G. Cao, J.M. Zhu, T.Y. Zhang, The frequency-dependent behavior of a ferroelectric single crystal with dislocation arrays, J. Acta Mech. 228 (2017) 2811–2817. [11] Y. Zheng, B. Wang, C.H. Woo, Simulation of interface dislocations effect on polarization distribution of ferroelectric thin films, J. Appl. Phys. Lett. 88 (2006)
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