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Compatible domain structures and the poling of single crystal ferroelectrics
Mechanics of Materials 42 (2010) 740–753
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Compatible domain structures and the poling of single crystal ferroelectrics N.T. Tsou *, J.E. Huber Department of Engineering Science, University of Oxford, Parks Rd., Oxford OX1 3PJ, UK
a r t i c l e
i n f o
Article history: Received 9 November 2009 Received in revised form 19 February 2010
Keywords: Ferroelectric Domain configuration Compatibility Disclination Single crystal
a b s t r a c t Ferroelectric crystals are known to adopt low energy, compatible configurations of microstructure. Many models have been proposed to describe these low-energy domain configurations, mainly based on the idea of fine phase mixtures and average compatibility. The present work develops criteria for the exact compatibility of domain structure in the form of a periodic multi-rank laminate. Exactly compatible structure is expected to be energetically favourable and does not require the concept of a fine mixture. A hierarchical binary tree representation is used to describe the relations among the domains in a multi-rank laminate, and this enables a concise statement of the conditions for exact compatibility. The disclinations associated with domain junctions are also taken into account. The resulting method is a rapid and systematic procedure for finding exactly compatible domain arrangements. This is used to explore minimum rank compatible microstructure in the tetragonal and rhombohedral crystal systems. The results reveal routes in polarization and strain space along which the microstructure can continuously evolve, including lowrank poling paths for single crystals. The method generates several well known domain configurations and many new structures. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Ferroelectric domain structures can be organized or engineered to give properties, such as piezoelectricity and remanent polarization. The relationship between induced properties and domain configuration has been studied extensively (Arlt, 1990; Arlt and Sasko, 1980; Liu and Lynch, 2006; Park and Shrout, 1997; Pertsev and Arlt, 1991). Observations show that ferroelectric crystals adopt certain low-energy domain configurations, such as herringbone and multi-rank laminate structures (Arlt, 1990; Arlt and Sasko, 1980). To understand microstructures in ferroelectric crystals and their ability to evolve under the applied loads, theoretical models of the microstructure play an essential role. Recently, studies of domain configuration in ferroelectrics have been carried out using phase-field simulations (Choudhury et al., 2005; Dayal and Bhattacharya, 2007; Müller et al., 2007). Although the equilibrium patterns and evolution of microstructure are described well * Corresponding author. Tel.: +44 1865283487. E-mail address: [email protected] (N.T. Tsou). 0167-6636/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2010.04.004
by models of this type, the computational demands have limited most studies either to 2-dimensions, or to very small regions of material. Moreover, models of this type treat the domain wall as a diffuse interface and thus require a mesh size finer than the domain wall width which is typically a few lattice parameters (Eng et al., 1997; Jona and Shirane, 1962). In this work a sharp-interface approach is used, and there is a consequent saving in computation. Several approaches to predicting the equilibrium structure of ferroelectric crystals and related systems are based on energy minimization (Ball and James, 1987; DeSimone, 1993; Shu and Bhattacharya, 2001). Central to these studies is the idea that domain pairs adopt orientations that eliminate internal charge and residual stress across domain walls and thus achieve a low energy, compatible state. By using the compatibility principles, an elegant model of domain structure was proposed by Li and Liu (2004) following Bhattacharya (1993) and DeSimone and James (2002). Their approach models the microstructure as a periodic, multi-rank laminate of ferroelectric domains in which compatibility requirements are satisfied in an average sense at each level of lamination. We shall use
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the term ‘‘compatible on average” (COA) to describe such structures, which are also known as b-type structure (Arlt and Sasko, 1980). Note that, although COA structures achieve compatibility at some internal interfaces, they still have local incompatibilities between sub-laminates. To resolve this issue, the concept of a fine phase mixture (Ball and James, 1987) and a separation of length scales are introduced, wherein the sub-laminations are taken to be sufficiently fine that the resulting laminate can be treated as a homogeneous medium. The model of Li and Liu (2004) generally results in high rank laminates, with rank n 1, when n crystal variants are present. Combining this with separation of length scales gives extremely fine microstructure, such that the domain wall energy, which is commonly neglected, may then become significant. Thus it appears likely that, among compatible laminate structures, those of lower rank are energetically favourable. Direct observation reveals that real microstructures are commonly inhomogeneous, containing multiple regions of low-rank compatible laminate structure, with differing length scales and arrangement (Arlt, 1990; Liu and Lynch, 2006). Several recent models also focus on low rank laminates. For example, Rodel (2007) discussed the compatibility of rank-2 hierarchical domain patterns; Weng and Wong (2009) and Yen et al. (2008) modeled the evolution of a rank-2 microstructure. This motivates the study of compatible structures with a view to finding the minimum rank solutions to the compatibility equations. In this work, we also study ‘‘exactly compatible” (EC), or a-type microstructure (Arlt and Sasko, 1980). In EC microstructure, every domain wall satisfies the compatibility equations. It is reasonable to expect that EC structures have low energy by comparison with COA structures of similar rank. Furthermore, Zhang et al. (2009) recently showed that exact compatibility is an important factor in ferroelectric hysteresis. Finally, the disclinations that form at the junctions of domain walls (Fousek and Mokry, 2005; Pertsev and Arlt, 1991; Savytskii and Bismayer, 2008; Shu and Bhattacharya, 2001) should be considered. Either the disclination vanishes, to give a stress-free compatible arrangement, or continuity is preserved by a state of stress at the domain junction. We will describe as ‘‘disclinationfree” (DF) any laminate structure in the class of EC structures which has no disclinations. The goal of the present work is finding minimum rank EC and DF microstructures in 3-dimensions, that may be of technological interest from the point of view of designing or engineering domain structure. A sharp-interface approach is utilized for rapid computation, recognising the potential of diffuse interface models to extend our findings. The present method gives insight into possible poling routes for single crystals, and has potential as a tool for microstructural design.
2. Theory of compatibility 2.1. Compatibility of domain pairs When the loads and boundary conditions favour the coexistence of more than one crystal variant, the total
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energy of the microstructure is minimized by domain configurations that satisfy compatibility, based on the theory of energy minimization (Shu and Bhattacharya, 2001). For a pair of domains i and j with strain states ei, ej and corresponding polarization vectors pi, pj, the unit interface normal vector n of a compatible domain wall must satisfy (Ball and James, 1987; DeSimone and James, 2002; Shu and Bhattacharya, 2001):
1 2 ðpi pj Þ n ¼ 0:
ei ej ¼ ða n þ n aÞ;
ð1Þ ð2Þ
where a is an arbitrary vector chosen to satisfy Eq. (1). A compatible domain wall is possible if there is a solution for n. In the present work, consideration is limited to sets of crystal variants which are mutually related by pure rotations. Eq. (1) can then be solved by using the eigenvalues kk and eigenvectors ek (k = 1, . . . , 3) of the matrix representation M = ei ej, noting that at least one eigenvalue is zero. Let k3 = 0; then two solutions for the interface normal n are given by:
n¼
e2 e1 pffiffiffi : 2
ð3Þ
The resulting vector n must also satisfy Eq. (2) if polarization compatibility is to be achieved, and this commonly produces a unique interface normal. In the special case when M = ei ej = 0, Eq. (1) is trivially satisfied by setting a = 0, and Eq. (2) then provides a continuous set of solutions for interface normal n. This special case arises with 180° domains, which have no habit plane and result in ‘‘watermark” domain patterns (Arlt and Sasko, 1980). If only two crystal variants are present, a simple EC structure consisting of a rank-1 laminate can always be formed. However, groups of more than two crystal variants may require a higher rank configuration, giving rise to further compatibility conditions. 2.2. Compatibility of multi-rank laminate structures In laminated domain structure, pure variants can be regarded as rank-0 laminations, while a pair of variants can be laminated to form a rank-1 lamination. Similarly, pairs of laminates can themselves be laminated to form a higher rank lamination. Here, we make use of a hierarchical binary tree diagram to represent the arrangement of domains in a laminate of rank-R (Goldsztein, 2001). For example, Fig. 1 shows the tree diagram representing a rank-2 lamination. A simple top-down numbering of the nodes is used. The root node (node 1) represents the entire microstructure while nodes at lower levels in the tree represent structures that are conjoined to form higher rank laminations. At the lowest level, the nodes represent pure crystal variants. The tree diagram of a rank-R domain configuration contains 2R+1 1 nodes and R + 1 levels; it gives a convenient representation of laminated domain structure for computational purposes. Each node represents a material characterized by two properties: an average strain, and an average polarization.
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Fig. 1. A schematic binary tree for rank-2 (R = 2) laminate structure where f1 is the volume fraction associated with the ith node and ni is the interface normal vector. Levels of the tree are numbered r = 0. . .R corresponding to the rank of nodes at that level.
Nodes at levels other than the lowest level are associated with an interface normal vector that specifies the orientation of the lamination. The ith parent node (i = 1, . . . , 2R 1) is associated with volume fraction fi, strain ei and polarization pi directly related to its child nodes at the next level down.
fi ¼ f2i þ f2iþ1 ;
ei ¼ f2i e2i þ f2iþ1 e2iþ1 ;
pi ¼ f2i p2i þ f2iþ1 p2iþ1 :
ð4Þ
If boundary conditions are specified as a set of displacements and surface charges consistent with an average polarization P and average remanent strain er, the total volume fraction f(k) of each pure variant can be found. Here, a bracketed subscript has been introduced to distinguish the total volume fraction f(k) of the kth crystal variant from the volume fraction fi associated with node i in the tree diagram. Under conditions of zero internal charge and stress-free domain structure, the macroscopic values P and er are given exactly by volume averages of the corresponding local quantities. Thus:
P¼
N X
fðkÞ pðkÞ ;
ð5Þ
k¼1
er ¼
N X
fðkÞ eðkÞ ;
ð6Þ
k¼1
1¼
N X
fðkÞ ;
ð7Þ
k¼1
where N is the number of variant types and all f(k) are subjected to the condition 0 6 f(k). Eqs. (5)–(7) correspond to a system of 10 linear equations. It is worth noting that Eqs. (5)–(7) may be invertible, singular or overdetermined, depending on the particular set of crystal variants. Once the total volume fractions of the variants are determined, there remains the problem of constructing a compatible domain configuration – that is, distributing the volume fractions among the nodes of a tree diagram in such a way as to produce a compatible structure overall. Three types of compatibility are discussed: compatibility on average (COA), exact compatibility (EC) and disclinationfree (DF) structure:
2.2.1. Compatibility on average (COA) Consider a prescribed macroscopic state of polarization and strain and let Eqs. (5)–(7) produce a set of n non-zero volume fractions f(i)(i = 1, . . . , n) in the crystal. It is always possible to construct COA structure according to the theory developed by Li and Liu (2004), such that:
P ¼ l1 pð1Þ þ l2 ð1 l1 Þpð2Þ þ þ ln1
n2 Y
ð1 lm Þpðn1Þ
m¼1
þ
n1 Y
ð1 lm ÞpðnÞ ;
ð8Þ
m¼1
er ¼ l1 eð1Þ þ l2 ð1 l1 Þeð2Þ þ þ ln1
n2 Y
ð1 lm Þeðn1Þ
m¼1
þ
n1 Y
ð1 lm ÞeðnÞ ;
ð9Þ
m¼1
where lm are coefficients related to the volume fractions f(i) as follows (Bhattacharya, 1993):
lm ¼
8 > > < 0; > > :
m1 P
fðiÞ ¼ 1;
i¼1 f
ð10Þ
PðmÞm1 ; otherwise:
1
i¼1
fðiÞ
This domain configuration generates a rank n 1 laminate structure. An example of a tree diagram corresponding to a rank-3 laminate generated by the procedure of Li and Liu is shown in Fig. 2. There are four energetically favoured variants (f(k) – 0, k = 1, . . . , 4). Following Eqs. (8)–(10), the resulting rank-3 laminate structure is ensured to satisfy the average compatibility criteria by distributing the volume fraction of variant 1 across four nodes (numbers 8, 10, 12 and 14), and variant 2 across two nodes (numbers 9 and 13) at the lowest level of the tree. To understand this, note that the values of l1, l2, and l3 generated by Eq. (10) cause the volume fraction ratio of variant 1 and 2 at node 4 to match exactly it of variant 1 and 3 at node 5. This guarantees that a COA laminate can be formed between the materials represented by nodes 4 and 5. A similar pattern of COA lamination is generated throughout the tree diagram. It is
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Fig. 2. The domain configuration of a rank-3 laminate constructed by Li and Liu’s method.
worth noting that variant 1 is distributed such that it is present in all four nodes in the rank-1 level of the tree. Similarly, variant 2 is present in all nodes at the rank-2 level. This elegant configuration ensures that every laminating process is equivalent to conjoining of a pair of pure variants, since the sub-laminates always contain common crystal variants whose polarization and strain states cancel in Eqs. (1) and (2). Note that the Li–Liu domain structures commonly have high rank and local incompatibility. It is often possible to find a lower rank lamination than that generated by the procedure of Li and Liu. Furthermore, EC arrangements can be found where the Li–Liu method would produce COA structures. The contrast between COA and EC structures is illustrated in Fig. 3. Whereas, the COA structure of Fig. 3a has some regions with ‘‘head to head” or ‘‘tail to tail” polarization which violate Eq. (2), the EC structure of Fig. 3b, has low energy, one to one perfect domain alignment with ‘‘head to tail” polarization directions across interfaces. Thus low energy microstructures can be found by considering the conditions for exact compatibility. 2.2.2. Exact compatibility (EC) By considering the tree diagram of a rank-R laminate, it can be seen that there are three criteria for achieving EC
structure. First, the volume fraction ratios of all domain pairs in each level r = 0, . . . , R 2 must be identical:
f Rr f2Rr ¼ 2 þ2n2 ; f2Rr þ1 f2Rr þ2n1
n ¼ 2; . . . ; 2Rr1 :
ð11Þ
Here, 2Rr is the first node number in level r of the tree, and the maximum value of n is the number of laminate pairs in level r. This condition ensures that domains that meet across a high level interface have the same spacing at that interface. A similar constraint was described by Rodel (2007). Interestingly, the volume fraction arrangement proposed by Li and Liu always satisfies Eq. (11), as illustrated in the tree diagram of Fig. 2. A second requirement for EC structure is that, where domain walls meet a higher level interface, their projection into that interface matches in orientation with the projection of the corresponding domain wall on the opposite side of the interface. This is illustrated in Fig. 4 where an EC structure of rank-3 is shown in Fig. 4a, and the same structure is split open along a high level interface in Fig. 4b. Where the structure is split, the domain patterns exposed on the interface must match. In terms of the tree diagram, for every child node, the projection of its interface normal vector onto all interfaces of its higher level parent nodes must be parallel to that of a corresponding child node on
Fig. 3. Rank-2 microstructures constructed with (a) compatibility on average and (b) exact compatibility.
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Fig. 4. (a) An EC structure of rank-3 and (b) the structure split open along a high level interface showing a matching set of domain wall orientations.
the opposite side of the tree. The relationship between the interface normal vectors of parent node i = 1 . . . , 2R1 1 and every child node p and q must satisfy:
ðnp ðnp ni Þni Þ ðnq ðnq ni Þni Þ ¼ 0;
ð12Þ
which is equivalent to
np nq ni ¼ 0;
ð13Þ
where the node numbers p and q are given by
p ¼ 2jþ1 i þ k;
q ¼ p þ 2j ;
ðj ¼ 0; . . . ; r 2; k ¼ 0; . . . ; 2j 1Þ:
ð14Þ
In Eq. (14), the tree level r of node i is given by r ¼ R blog2 ic, where bxc indicates the greatest integer less than or equal to x. For EC structure, Eq. (12) must be satisfied over all the permutations of p and q indicated by Eq. (14). This ensures that all layers and their sub-layers have the same orientation on their common interface. Thus, a microstructure can have ‘‘one to one” matched domain configuration if its corresponding tree diagram satisfies Eqs. (11)–(14). Even though geometrically matched structures are achieved by having equal domain spacing and orientation, local polarization and strain compatibility requirements must also be satisfied to give EC structure. This avoids incompatibilities such as ‘‘head to head” polarization. Thus, a third condition for EC structure specifies that all pairs of adjacent domains must satisfy compatibility across the higher rank interface that separates them. Again, this condition is readily expressed in terms of the nodes of the tree diagram. The interface normal vector ni (i = 1, . . . , 2R 1) of each parent node i (layer r) and the polarization and strain states pp, pq, ep, eq, of the pure crystal variants with node numbers p and q in the lowest level (rank-0) of the tree must satisfy the basic compatibility Eqs. (1) and (2). Here node numbers p, q are limited to:
p ¼ 2r i þ k;
q ¼ p þ 2r1 ;
ðk ¼ 0; . . . ; 2r1 1Þ:
ð15Þ
Structures satisfying Eqs. (11)–(14) and Eqs. (1) and (2) for all nodes specified by Eq. (15) are exactly compatible at all
interfaces: they minimize the total energy without the need for separation of length scales. 2.2.3. Disclination-free compatibility (DF) Next consider the compatibility issues arising due to the disclination associated with certain domain walls. Let the true rotation of the crystal lattice across a given domain wall be hr. The angles hr in several ferroelectric crystal systems are given in the work of Shu and Bhattacharya (2001). The net rotation of lattice planes, h, where the pair of crystal variants i and j meet, is given by
^i p ^ j Þ; h ¼ hr cos1 ðp
ð16Þ
^ i and p ^ j are the unit vectors along the pair of the where p idealized polarization directions in the reference frame of one of the crystal variants. When several domains are assembled into a composite, the lattice rotations may result in disclinations. To illustrate this, consider tetragonal barium titanate, in which there is a shear strain of magnitude about 1% (Arlt and Sasko, 1980) between 90° domains resulting in a lattice rotation of 0.62° (Shu and Bhattacharya, 2001) across the domain wall. A disclination can form at the junction of four 90° domain walls. Fig. 5a shows a group of four domains in a DF configuration, with the lattice rotation exaggerated. This configuration is readily extended to form a rank-2 DF laminate (see Fig. 5b). By contrast, the group of four domains shown in Fig. 5c produces a disclination. They are shown in a relaxed state, producing an overlap at one domain wall. The group can be fitted together in a closed continuous form only by deforming the crystal lattice, introducing stress or electric field. The resulting structure (Fig. 5d) is EC but not DF. This vortex structure have been observed in related ferroic system, such as ferromagnets (Schilling et al., 2007) and are also generated by simulations such as phase field and ab initio models (Naumov et al., 2004; Schrade et al., 2007; Slutsker et al., 2008). The present work is concerned with finding DF multirank laminate structures and thus the global compatibility of an arrangement of domains must be considered. By using the tree diagram, the junctions of domains in a multi-rank laminate can be identified, and thus the full set of
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Fig. 5. Disclinations in rank-2 structures: (a) a disclination-free structure, and (b) a corresponding laminate configuration; (c) a disclination at the junction of four domains, showing a relaxed state with domains overlapping and (d) a corresponding laminate configuration, in stressed state.
domain junctions in the structure can be taken into account. In an EC structure, a pair of domains p, q which are children of node i in level r = 1, can be rotated from the ideal, or reference lattice configuration, into their exact configuration by a rotation Ri about the axis pp pq satisfying:
cos1 ðRi pp RTi pq Þ ¼ hrp;q ;
ð17Þ
where p and q are node numbers generated from Eq. (15); hrp;q is the true rotation of lattice planes at the domain wall between domains p and q. The rotation matrix Ri is given explicitly by:
2
u21 þð1 u21 Þc
6 Ri ¼ 4 u1 u2 ð1 cÞ þu3 s
u1 u2 ð1 cÞ u3 s u1 u3 ð1 cÞ þu2 s u22 þð1 u22 Þc
u1 u3 ð1 cÞ u2 s u2 u3 ð1 cÞ þu1 s
3
7 u2 u3 ð1 cÞ u1 s 5; u23 þð1 u23 Þc
ð18Þ where u = (pp pq)/|pp pq|, c = cos(h/2) and s = sin(h/2). Rotation Ri is used to turn the polarization at node p, while RTi turns the polarization at node q in the opposite sense. These rotations can be carried out at all pairs of nodes in the lowest level of the tree diagram, producing a new, rotated set of strain and polarization states for each domain. At higher levels of the tree, further conditions arise
because of domains meeting across high level interfaces. Then for each node i in levels r = 2 . . . , R, a rotation matrix Ri must be found that satisfies Eq. (17), and additionally the rotated interface normals must align, giving a condition similar to Eq. (13):
ðRi ns RTi nt Þ ni ¼ 0:
ð19Þ
For all node numbers s and t in the set
s ¼ 2jþ1 i þ k;
t ¼ s þ 2j ;
ðj ¼ 0; . . . ; r 2; k ¼ 0; . . . ; 2j 1Þ:
ð20Þ
A matrix Ri that satisfies conditions (17)–(20) can counterrotate the laminates meeting across the interface defined at node i into a compatible DF configuration. However, this rotation could disturb the strain compatibility across the domain wall if the axis of rotation is not perpendicular to the interface normal ni. Thus, in addition to Eq. (19), the basic compatibility between adjacent domains must be checked after the rotation. In general, Eqs. (17)–(20) are strongly overdetermined and solutions only exist in special cases. Some examples of rank-2 DF structures are discussed in Section 3. Note that all of the conditions of microstructural compatibility described in Section 2 are independent of length
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scale. In sufficiently fine-scale microstructure, the domain wall energy becomes non-negligible and can control spacing. In coarser microstructure the interaction of disclination fields may play a key role in spacing. However, these interactions are neglected in the present theory; we thus design compatible structural states without specifying precise scale limitations. Observations of domain structure suggest that domain wall spacing arise naturally over a wide range of length scales from 106 m or less up to 102 m or more. 3. Minimum rank compatible laminates High rank structures often cannot satisfy the exact compatibility conditions, leading to COA structures with separation of length scales. However, this results in very fine domains, such that the domain wall energy, which commonly ignored, becomes significant. While the model of Li and Liu (2004) gives rank n 1 structures when there are n crystal variants present, the minimum possible rank is dlog2 ne, where dxe indicates the least integer greater than or equal to x. Thus there is the potential to find structures of lower rank than that of the Li–Liu construction whenever n P 4. Fig. 6a shows an example of a rank-5 laminate which is a COA structure for a tetragonal crystal. This structure was determined using the model of Li and Liu, by setting the average polarization and strain state of the entire crystal to zero; this models the unpoled state of the material, as cooled from above the Curie temperature. Fig. 6b and c show two alternative configurations for this polarization and strain state: Fig. 6b is a rank-3 COA structure, while Fig. 6c is a rank-3 EC structure. Of these three possible structures, it is expected that Fig. 6c represents the lowest energy state. In order to discover other low energy structures, we have used a systematic procedure to search for minimum rank domain configurations which satisfy the criteria of exact compatibility or disclinationfree compatibility. 3.1. Application to tetragonal ferroelectrics A tetragonal crystal has six crystal variants in three pairs of opposite polarization. The strain states e(±i), (i = 1, . . . , 3) and corresponding polarization states p(±i) are given by:
2
b
0
0
3
2
pt
3
7 6 7 eð1Þ ¼ 6 4 0 a 0 5; pð1Þ ¼ 4 0 5; 0 0 a 0 2 3 2 3 0 a 0 0 7 6 7 eð2Þ ¼ 6 4 0 b 0 5; pð2Þ ¼ 4 pt 5; 0 0 0 a 2 3 2 3 0 a 0 0 6 7 7 eð3Þ ¼ 6 4 0 a 0 5; pð3Þ ¼ 4 0 5: 0
0
b
ð21Þ
pt
Here a, b, and pt are material properties, but there is choice over the reference state from which strains are measured. Strains of the unit cell can be defined relative to a cubic state of equal volume, such that tr(e) = 0 and a = b/2. The true polarization rotation angle hr of the crystal lattice at a 90° domain wall is then (Shu and Bhattacharya, 2001):
hr ¼ 2 tan1
1þb 1 b=2
ð22Þ
and the net rotation of lattice planes at the domain wall can be calculated by using Eq. (16). The problem to be solved may be stated as follows: Let a tetragonal single crystal have average remanent strain er and average polarization P. We seek an exactly compatible multi-rank laminate comprising the crystal variants defined in Eq. (21) with the given average strain and polarization state, and with minimum rank. In the tetragonal system, Eqs. (5)–(7) give a unique solution for the total volume fractions of each of the six crystal variants. Let n be the number of variants with non-zero volume fractions. Now consider distributing the volume fractions of these n variants into 2dlog2 ne parts to fill the lowest level of a minimum rank binary tree diagram. Whenever 2dlog2 ne > n there is freedom over how the volume fractions are divided among the nodes at the lowest level of the tree because at least one variant must have its volume fraction split between more than one node. There are then multiple permutations of the volume fraction arrangement. However, the requirements of exact compatibility, Eq. (11), constrain the distribution of volume fractions. For example, in a rank-3 construction, the volume fractions must have the form: l1l2l3, (1 l1)l2l3, l1(1 l2)l3, (1 l1)(1 l2)l3, l1l2(1 l3), (1 l1)
Fig. 6. A tetragonal crystal with all 6 variants present can be constructed by (a) rank-5 compatibility on average (b) rank-3 compatibility on average and (c) rank-3 exact compatibility.
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l2(1 l3), l1(1 l2)(1 l3), and (1 l1)(1 l2)(1 l3) in nodes 8–15, respectively. Here li has the same meaning as defined in Section 2.2.1, Eqs. (8)–(10). For each permutation of the split volume fractions, a set of n simultaneous nonlinear equations in li can be found to ensure that the nodal volume fractions consistently sum to the variant volume fractions. These equations of volume fraction continuity may be written as: fðiÞ ¼
2Rþ1 X1
C ij fj ;
ð23Þ
j¼2R
where Cij is a permutation matrix such that Cij = 1 if the crystal variant at node j is variant number i and zero otherwise. The nodal volume fractions fj are given by
fj ¼
R Y
ð1 2bÞlk þ b;
ð24Þ
k¼1
where
b¼
$ % j 2R 2k1
2
$ % j 2R : 2k
ð25Þ
The binary digit b takes on a value of zero or unity depending on whether the jth node is reached by taking the left hand or right hand fork at the level r = k 1 of
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the binary tree. The non-linearity of Eqs. (23)–(25) causes significant complication; however, the appearance of common factors of li in these equations simplifies the solution process. We make a systematic search of the set of permutations, in each case checking for a consistent solution to Eqs. (23)–(25). If a solution exists, the resulting arrangement is taken forward as a candidate structure and the remaining criteria for EC structure, Eqs. (1), (2), (13), (14), (15), are checked. In addition, it is possible that Eqs. (1) and (2) allow a domain wall orientation to vary continuously, as is the case for 180° domain walls. This provides extra degrees of freedom that can allow the EC conditions to be satisfied. In practice, where an extra degree of freedom arises, it normally allows Eq. (12) to be solved, thus increasing the chance of satisfying all the conditions of exact compatibility. The domain structure shown in Fig. 6c is a typical example of minimum rank EC configuration; in this case er = 0 and P = 0 which gives six variants, each with equal volume fraction 1/6. It is worth noting that the laminates shown in Fig. 6b and c have exactly the same tree diagram, except that the interface normal vectors of the structure in Fig. 6b do not satisfy the EC conditions. Now consider applying the method described above to study the practical problem of poling a ferroelectric single
Fig. 7. (a) The space of possible remanent strain states in a tetragonal single crystal with P1 = P2 = 0. (b) and (c) Illustrate curved spaces of 6 variant rank-3 structures.
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region ABCD require at least five variants. These points generally require a lamination of rank-4 or greater and only allow laminate structures that are compatible on average. However, points in the triangle ABC and on the curved surfaces AGCD and GBDC (shown in Fig. 7b, 7c) admit rank-3 exactly compatible structures (see Table 2 for details). The curved surfaces AGCD and GBDC contain both the unpoled initial state, O, and the fully-poled state D. Thus a tetragonal crystal can be poled, while remaining in an exactly compatible state, with laminate structure of rank three or less. This is a significant result as it implies that tetragonal crystals could be poled by continuous motion of domain walls (without nucleation of new domains) and in exactly compatible states that do not necessitate fine domain structure. Some poling paths for tetragonal single crystals can be suggested. For example, the route O ? C ? D consists of axisymmetric straining along OC (er11 ¼ er22 ¼ er33 =2) as a rank-3 laminate, followed by strain-free 180° switching along CD. It is interesting to note that the domain configuration jumps two rank levels from rank-3 to rank-1 at point C, where variants ±1 and ±2 vanish. We speculate that this poling path could be achieved by a two stage loading process consisting of axisymmetric stress loading, followed by the application of an electric field parallel to the x3 direction. The domain evolution during this poling process is shown in Fig. 8. Interestingly, this poling procedure for tetragonal single crystals is similar to a poling method given by Granzow et al. (2006) in the context of maximizing polarization in polycrystalline ferroelectrics. An alternative poling path O ? H ? D consists of a plane strain deformation, holding er33 ¼ 0, in rank-3, followed by a second plane strain deformation, holding er11 ¼ b=2, in rank-2. The polarization evolves continually along this path and so combined stress and electric field boundary conditions could be needed. Note that the domain configurations corresponding to a given state of polarization and strain are generally non-unique. For example, the unpoled state O in Fig. 7a can be realized in many different ways. Three distinct ways can be seen by considering the expressions for the arrangement of nodal volume fractions corresponding to point O in surfaces ABC, AGCD, and GBDC in Table 2. We should also emphasize that the compatible solutions found by the present method are restricted to periodic multi-rank laminate configurations. Other compatible configurations can exist, such as the crossing domain patterns identified by Shu and Bhattacharya (2001). Furthermore,
Table 1 The crystal variants and volume fraction arrangement on line ED (FD) in Fig. 7, for the tetragonal crystal system. Line ED (FD) 0 x 14
pffiffiffiffiffiffiffiffiffi 14x
a ¼ ð12xÞþ 2x
Node no. Crystal variant
4 1 (2)
5 3
6 3
7 1 (2)
Volume fraction
x
a
x
ax
x
crystal. A natural question is whether there exists a continuous path that the microstructure may follow, from an initial, unpoled state (P1 = P2 = P3 = 0; er11 ¼ er22 ¼ er33 ¼ 0), to a fully poled, single variant state (P1 = P2 = 0, P3 = pt; er11 ¼ er22 ¼ b=2; er33 ¼ b). If such a path exists, what is the minimum rank route to poling? In order to resolve this question in the tetragonal crystal system, it is instructive to explore the polarization-strain space (P3, er11 ; er33 ) with P1 = P2 = 0, P3 P 0 as shown in Fig. 7a. The tetrahedral region ABCD in Fig. 7a is the possible space of all P3, er11 ; er33 values, with the volume fractions constrained by Eq. (7) and by the condition 0 6 f(i). A general point in region ABCD requires all six variants to be present, and so the crystal could form a laminate of rank-3 (minimum possible rank) to rank-5 (Li and Liu’s model). However, certain special points in ABCD can have fewer than six variants. Point D represents an ideally poled state consisting of a single variant (variant number +3 in Eq. (21)) and thus has a rank-0 structure. Points A, B, C and line CD everywhere except at D require only two variants, giving rank-1 structure. The line CD represents laminates of variant 3 and variant 3; thus the average polarization, P3, varies along line CD without change in the average remanent strain. Points on lines AD and BD need three variants and form rank-2 exactly compatible structures. Note that the minimum rank laminates for these points have identical domain configurations to those provided by the model of Li and Liu (2004). Typical points in the triangular surfaces ACD and BCD need four variants and rank-3 structures. However, the edges AB, AC, BC and the curves ED and FD in Fig. 7a comprise a set of strain and polarization states that correspond to rank-2 exactly compatible laminates. The detail of the crystal variants and volume fraction in the lowest level of the tree diagram for points on the curves ED and FD is shown in Table 1. By varying the parameter x in the range 0 6 x 6 1/4 all of the polarization and strain states along these curves can be obtained. All remaining points in
Table 2 The crystal variants and volume fraction arrangement of surfaces ABC, AGCD, and GBDC in Fig. 7, for the tetragonal crystal system. Surface ABC 0 6 x 6 12 ; 0 6 y 6 ð12 xÞ
Node no. Crystal variant
8 1
9 1
10 2
11 2
12 1
1 ; y ¼ 16 At O, x ¼ 12
Volume fraction
x
x
y
y
x 2ðxþyÞ
AGCD (GBDC) 1 0 x 1; 0 y 2xþ2
Node no. Crystal variant
8 1 (2)
9 1(2)
10 3
11 1(2)
12 2(1)
13 2(1)
14 3
1 At O, x ¼ 1; y ¼ 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4y 2 a ¼ 3y þ y þ 1þx 2
Volume fraction
a+y
y
a þ ay
a
x(a + y)
xy
xða þ ay Þ
2
13 1 x
x 2ðxþyÞ
14 3 x
y 2ðxþyÞ
15 3 y
y 2ðxþyÞ
y
15 2(1) 2
xa
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Fig. 8. The domain evolution in the poling route O ? C ? D of Fig. 7. (a) Unpoled state. (b) A typical point on OC. (c) Variants ±1 and ±2 vanishing as point C is reached. (d) Rank-1 structure at point C. (e) A typical point on CD. (f) Fully-poled state.
2
3
2
3
3.2. Application to rhombohedral ferroelectrics
pt a b b 7 6 7 eð1Þ a b 5; pð1Þ ¼ 4 pt 5; pt b a 3 2 3 pt a b b 7 6 7 eð2Þ ¼ 6 b 5; pð2Þ ¼ 4 pt 5; 4 b a a pt b b 2 3 2 3 pt a b b 6 7 eð3Þ ¼ 6 a b 7 4 b 5; pð3Þ ¼ 4 pt 5; p b b a 2 3 2 t 3 pt a b b 7 6 7 eð4Þ ¼ 6 4 b a b 5; pð4Þ ¼ 4 pt 5: pt b b a
In the rhombohedral crystal system, there are eight crystal variants with the strain e(±i), and polarization p(±i), (i = 1, . . . , 4) given by
Again, a, b, and pt are material properties and we define the strain states relative to a reference configuration of equal volume, such that a = 0.
non-periodic, or macroscopically inhomogeneous configurations are not considered. Finally, consider disclination-free domain configurations in the tetragonal crystal system. The additional constraint placed on the domain arrangement in order to produce disclination-free structure eliminates all points in region ABCD of Fig. 7a, except the edges AB, AC, BC and CD. Here, disclination-free structures of rank two or less, having pairs of domains with anti-parallel polarization, can be found. Since the unpoled state, O, does not lie within the set of points where disclination-free structure is possible, there is no disclination-free poling path.
6 ¼ 4b b 2
ð26Þ
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Fig. 9. The space ABCD of possible remanent strain states in a rhombohedral single crystal with P = 0. (a) Edges and vertices of the tetrahedron, corresponding to structures of rank-2 or less. (b) A curved surface showing the strain states of exactly compatible rank-3 structures with all 8 variants present.
Considering disclinations, there are two types of domain walls that contribute to disclinations in the rhombohedral crystal system. The polarization rotations in the rhombohedral system across these two types of domain wall are approximately 71° and 109°. However, the true rotations hr(I), hr(II) are given by (Shu and Bhattacharya, 2001):
hrðIÞ ¼ cos1
1 8b 2b2 3 þ 6b2
hrðIIÞ ¼ cos1
! ð27Þ
;
1 þ 10b þ 7b2
!
3ð1 þ 2b þ 3b2 Þ
ð28Þ
:
The rotation of lattice planes can be calculated using Eq. (16) and thus the criteria for disclination-free structure can be tested. For a rhombohedral single crystal with average remanent strain er and average polarization P, Eqs. (5)–(7) expand into seven linear equations for the eight unknown volume fractions. The general solution of Eqs. (5)–(7) gives volume fractions in the form:
fð1Þ
fð1Þ
fð2Þ
fð2Þ
fð3Þ
fð3Þ
fð4Þ
fð4Þ
T
¼ x0 þ c½ 1 1 1 1 1 1 1 1 T :
ð29Þ
Here x0 is a particular solution and 0 6 c 6 cmax; the value of cmax is set by the constraint that 0 6 f(i). In the present work, for simplicity, only the cases c = 0, c = cmax and c = cmax/2 are considered. When c = 0 or cmax, at least one variant has zero volume fraction, and the number of variants present in the laminate is minimized. When c = cmax/2, the number of variants present is maximized, generally meaning all eight variants are present. This latter choice of c also maximizes the least volume fraction of any variant and leads to at least two volume fractions being equal. Now consider the ferroelastic straining of a rhombohedral crystal, maintaining P1 = P2 = P3 = 0. The only non-zero
strain components are er12 ; er13 and er23 , and the space of possible strain states is limited to the tetrahedral region ABCD shown in Fig. 9a. The state O in Fig. 9a has er = 0 and is representative of a rhombohedral crystal in an unpoled state, as cooled from above the Curie temperature. This point allows many exactly compatible configurations of either rank-2 or rank-3 with four or eight variants present, respectively. Interestingly, one of the rank-3 laminate solutions at point O is identical to a result provided by a recent non-conventional phase field model (Shu et al., 2008) as shown in Fig. 10a. Phase field simulations such as those of Shu et al. (2008) produce minimum energy configurations and this commonly results in exactly compatible configurations. It is likely that the exactly compatible configurations have lower energy than configurations with average compatibility. An advantage of our approach is that the full set of compatible solutions is generated. The detail of the domain arrangement in the lowest level of the tree can be obtained from Table 3 by setting the parameters x = 1, y = 1/8. An alternative laminate at point O, of rank-2, is shown in Fig. 10b. This arrangement was also found using a phase field model by Shu (2010). For its detail, see Table 4. Returning to Fig. 9, Points A, B, C, and D require two variants to be present, and allow rank-2 exactly compatible structures. Each of these four points in strain space corresponds to a structure consisting of a pair of domains with anti-parallel polarization. The six edges AB, AC, AD, BC, BD, and CD need four variant structures and produce minimum rank-2, disclination-free domain configurations. Points on the surfaces ABC, ABD, ACD and BDC require six variants and give rank-3 exactly compatible structures. The points inside region ABCD generally require at least seven variants and lead to complex configurations of laminate. However, we can find several special cases of minimum rank exactly compatible laminates within ABCD. For example, a surface of the form er12 ¼ er23 er13 (and cyclic permutations) is shown in Fig. 9b. Points on this surface
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Fig. 10. (a) One of the possible rank-3 exactly compatible laminates in the unpoled state which is identical to a result provided by Shu et al. (2008). (b) An alternative rank-2 exactly compatible laminate in the unpoled state.
Table 3 The crystal variants and volume fraction arrangement of surfaces ABDC (er23 ¼ er12 er13 ) and ABCD (er12 ¼ er23 er13 ) in Fig. 9, for the rhombohedral crystal system. Surface ABDC, er23 ¼ er12 er13 (ABCD, er12 ¼ er23 er13 Þ
Node no.
8
1 0 6 x; 0 6 y 6 ð2xþ2Þ
Crystal variant
2(1)
4
At Rank-3 O, x ¼ 1; y ¼ 18
Volume fraction
1 2xþ2
x 2xþ2
9
y
10
11
2(1) xy
admit rank-3 exactly compatible structures with all eight variants present (see the detail in Table 3). The resulting rank-3 laminates do not satisfy the conditions for disclination-free structure; however, it is interesting to note that the angular errors from disclination-free configuration are very small (of order 104 b) so that the corresponding structures are low energy configurations. This surface connects the unpoled state, O, with each of the vertices A–D. As with the tetragonal system, several poling paths for rhombohedral crystals can be suggested which allow the domain configuration to remain in an exactly compatible state with rank-3 or less. For example, one poling route consists of ferroelastic straining O ? E ? D in Fig. 9a followed by a strain-free polarizing stage. Stage O ? E is polarization-free shearing, which increases er23 . The details of domain evolution on O ? E are given in Table 3 (surface ABCD with x = 1 and 1/8 P y P 0). This is followed by stage E ? D, comprising further ferroelastic straining with constant er23 , and er12 ¼ er13 (see Table 4 for details). Once point D is reached, the structure has simplified to a rank-1 laminate of variants +1 and 1. This can be poled without fur-
1 2xþ2
12
4 y
x 2xþ2
xy
13
14
15
3(2)
1(3)
3(2)
1(3)
y
xy
y
xy
ther straining by moving the domain wall that separates the two variants. The domain evolution along line O ? E ? D and from point D to the fully-poled state is shown in Fig. 11. 4. Conclusions Three dimensional exact and disclination-free compatibility conditions for periodic, multi-rank laminate domain configurations of ferroelectric single crystals have been developed, and methods have been given for applying the compatibility conditions to the full set of domain walls that exist in a multi-rank laminate. A hierarchical tree diagram was used to represent the laminates, and to define the relations between domains. An algorithm for finding minimum rank compatible laminate structures was used to study the tetragonal and rhombohedral crystal systems. While, many exactly compatible structures were found, the conditions of disclination-free structure prove highly restrictive and few such structures were found. A key result is that
Table 4 The crystal variants and volume fraction arrangement of the rank-2 unpoled state, O, and line ED in Fig. 9, for the rhombohedral crystal system. Point Rank-2 O
Line Node no. Crystal variant Volume fraction
4 1 (1)
5 3 (3)
6 2 (2)
7 4 (4)
1 4
1 4
1 4
1 4
ED 0 6 x 6 14
4 1 1 2x
5 4 x
6 1 1 2
x
7 4 x
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Fig. 11. The domain evolution in the poling route O ? E ? D ? fully-poled state in Fig. 9. (a) Unpoled state. (b) A typical point on OE. (c) Rank-2 structure at point E. (d) Rank-1 structure at point D. (e) Rank-1 laminate evolves between point D and fully-poled state. (f) Fully-poled state.
domain laminates in both the tetragonal and rhombohedral crystal systems can be designed, which allow the crystal to be poled from a state of zero strain and polarization to a single domain state, while the rank of the laminate never exceeds three. Several poling paths are suggested and details of the domain topology along these paths have been found. The theory can be readily applied to any ferroelectric crystal system, and provides a rapid means of computing low energy structural configurations. The theory also has potential applications to other ferroic systems such as ferromagnetic shape memory alloys. Acknowledgement This authors wish to acknowledge the support of EPSRC, project number EP/E026095/1. References Arlt, G., 1990. Twinning in ferroelectric and ferroelastic ceramics – stress relief. J. Mater. Sci. 25, 2655–2666.
Arlt, G., Sasko, P., 1980. Domain configuration and equilibrium size of domains in BaTiO3 ceramics. J. Appl. Phys. 51, 4956–4960. Ball, J.M., James, R.D., 1987. Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100, 13–52. Bhattacharya, K., 1993. Comparison of the geometrically nonlinear and linear theories of martensitic-transformation. Continuum Mech. Therm. 5, 205–242. Choudhury, S., Li, Y.L., Krill, C.E., Chen, L.Q., 2005. Phase-field simulation of polarization switching and domain evolution in ferroelectric polycrystals. Acta Mater. 53, 5313–5321. Dayal, K., Bhattacharya, K., 2007. A real-space non-local phase-field model of ferroelectric domain patterns in complex geometries. Acta Mater. 55, 1907–1917. DeSimone, A., 1993. Energy minimizers for large ferromagnetic bodies. Arch. Ration. Mech. Anal. 125, 99–143. DeSimone, A., James, R.D., 2002. A constrained theory of magnetoelasticity. J. Mech. Phys. Solids 50, 283–320. Eng, L.M., Fousek, J., Gunter, P., 1997. Ferroelectric domains and domain boundaries observed by scanning force microscopy. Ferroelectrics 191, 419–426. Fousek, J., Mokry, P., 2005. Stress-free domain quadruplets in ferroics. Ferroelectrics 323, 3–9. Goldsztein, G.H., 2001. The effective energy and laminated microstructures in martensitic phase transformations. J. Mech. Phys. Solids 49, 899–925. Granzow, T., Kounga, A.B., Aulbach, E., Rodel, J., 2006. Electromechanical poling of piezoelectrics. Appl. Phys. Lett. 88, 252907.
N.T. Tsou, J.E. Huber / Mechanics of Materials 42 (2010) 740–753 Jona, F., Shirane, G., 1962. Ferroelectric Crystals. Pergamon Press, Oxford. Li, J.Y., Liu, D., 2004. On ferroelectric crystals with engineered domain configurations. J. Mech. Phys. Solids 52, 1719–1742. Liu, T., Lynch, C.S., 2006. Domain engineered relaxor ferroelectric single crystals. Continuum Mech. Therm. 18, 119–135. Müller, R., Gross, D., Schrade, D., Xu, B., 2007. Phase field simulation of domain structures in ferroelectric materials within the context of inhomogeneity evolution. Int. J. Fract. 147, 173–180. Naumov, I.I., Bellaiche, L., Fu, H.X., 2004. Unusual phase transitions in ferroelectric nanodisks and nanorods. Nature 432, 737–740. Park, S.E., Shrout, T.R., 1997. Ultrahigh strain and piezoelectric behavior in relaxor based ferroelectric single crystals. J. Appl. Phys. 82, 1804–1811. Pertsev, N.A., Arlt, G., 1991. Internal-stresses and elastic energy in ferroelectric and ferroelastic ceramics – calculations by the dislocation method. Ferroelectrics 123, 27–44. Rodel, J., 2007. Effective intrinsic linear properties of laminar piezoelectric composites and simple ferroelectric domain structures. Mech. Mater. 39, 302–325. Savytskii, D., Bismayer, U., 2008. Strain at junctions in multidomain configurations. Phase Trans. 81, 431–447. Schilling, A., Adams, T.B., Saad, M.M., Baxter, P., Morrison, F.D., Catalan, G., Scott, J.F., Bowman, R.M., Gregg, J.M., 2007. Nanoscale ferroelectrics machined from single crystals. Integr. Ferroelectr. 92, 53–64. Schrade, D., Mueller, R., Xu, B., Gross, D., 2007. Domain evolution in ferroelectric materials: a continuum phase field model and finite
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element implementation. Comput. Method Appl. Mech. Eng. 196, 4365–4374. Shu, Y.C., 2010. Application of energy-minimizing laminates to the modeling of ferroelectric domain and switching. In: Khan, A.S., Farrokh, B. (Eds.), Finite Plasticity and Visco-plasticity of Conventional and Emerging Materials. NEAT Press, Maryland, pp. 142–144. Shu, Y.C., Bhattacharya, K., 2001. Domain patterns and macroscopic behaviour of ferroelectric materials. Philos. Mag. B 81, 2021–2054. Shu, Y.C., Yen, J.H., Chen, H.Z., Li, J.Y., Li, L.J., 2008. Constrained modeling of domain patterns in rhombohedral ferroelectrics. Appl. Phys. Lett. 92, 052909. Slutsker, J., Artemev, A., Roytburd, A., 2008. Phase-field modeling of domain structure of confined nanoferroelectrics. Phys. Rev. Lett. 100, 087602. Weng, G.J., Wong, D.T., 2009. Thermodynamic driving force in ferroelectric crystals with a rank-2 laminated domain pattern, and a study of enhanced electrostriction. J. Mech. Phys. Solids 57, 571–597. Yen, J.H., Shu, Y.C., Shieh, J., Yeh, J.H., 2008. A study of electromechanical switching in ferroelectric single crystals. J. Mech. Phys. Solids 56, 2117–2135. Zhang, Z., James, R.D., Müller, S., 2009. Energy barriers and hysteresis in martensitic phase transformations. Acta Mater. 57, 4332–4352.
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Compatible domain structures and the poling of single crystal ferroelectrics N.T. Tsou *, J.E. Huber Department of Engineering Science, University of Oxford, Parks Rd., Oxford OX1 3PJ, UK
a r t i c l e
i n f o
Article history: Received 9 November 2009 Received in revised form 19 February 2010
Keywords: Ferroelectric Domain configuration Compatibility Disclination Single crystal
a b s t r a c t Ferroelectric crystals are known to adopt low energy, compatible configurations of microstructure. Many models have been proposed to describe these low-energy domain configurations, mainly based on the idea of fine phase mixtures and average compatibility. The present work develops criteria for the exact compatibility of domain structure in the form of a periodic multi-rank laminate. Exactly compatible structure is expected to be energetically favourable and does not require the concept of a fine mixture. A hierarchical binary tree representation is used to describe the relations among the domains in a multi-rank laminate, and this enables a concise statement of the conditions for exact compatibility. The disclinations associated with domain junctions are also taken into account. The resulting method is a rapid and systematic procedure for finding exactly compatible domain arrangements. This is used to explore minimum rank compatible microstructure in the tetragonal and rhombohedral crystal systems. The results reveal routes in polarization and strain space along which the microstructure can continuously evolve, including lowrank poling paths for single crystals. The method generates several well known domain configurations and many new structures. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Ferroelectric domain structures can be organized or engineered to give properties, such as piezoelectricity and remanent polarization. The relationship between induced properties and domain configuration has been studied extensively (Arlt, 1990; Arlt and Sasko, 1980; Liu and Lynch, 2006; Park and Shrout, 1997; Pertsev and Arlt, 1991). Observations show that ferroelectric crystals adopt certain low-energy domain configurations, such as herringbone and multi-rank laminate structures (Arlt, 1990; Arlt and Sasko, 1980). To understand microstructures in ferroelectric crystals and their ability to evolve under the applied loads, theoretical models of the microstructure play an essential role. Recently, studies of domain configuration in ferroelectrics have been carried out using phase-field simulations (Choudhury et al., 2005; Dayal and Bhattacharya, 2007; Müller et al., 2007). Although the equilibrium patterns and evolution of microstructure are described well * Corresponding author. Tel.: +44 1865283487. E-mail address: [email protected] (N.T. Tsou). 0167-6636/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2010.04.004
by models of this type, the computational demands have limited most studies either to 2-dimensions, or to very small regions of material. Moreover, models of this type treat the domain wall as a diffuse interface and thus require a mesh size finer than the domain wall width which is typically a few lattice parameters (Eng et al., 1997; Jona and Shirane, 1962). In this work a sharp-interface approach is used, and there is a consequent saving in computation. Several approaches to predicting the equilibrium structure of ferroelectric crystals and related systems are based on energy minimization (Ball and James, 1987; DeSimone, 1993; Shu and Bhattacharya, 2001). Central to these studies is the idea that domain pairs adopt orientations that eliminate internal charge and residual stress across domain walls and thus achieve a low energy, compatible state. By using the compatibility principles, an elegant model of domain structure was proposed by Li and Liu (2004) following Bhattacharya (1993) and DeSimone and James (2002). Their approach models the microstructure as a periodic, multi-rank laminate of ferroelectric domains in which compatibility requirements are satisfied in an average sense at each level of lamination. We shall use
N.T. Tsou, J.E. Huber / Mechanics of Materials 42 (2010) 740–753
the term ‘‘compatible on average” (COA) to describe such structures, which are also known as b-type structure (Arlt and Sasko, 1980). Note that, although COA structures achieve compatibility at some internal interfaces, they still have local incompatibilities between sub-laminates. To resolve this issue, the concept of a fine phase mixture (Ball and James, 1987) and a separation of length scales are introduced, wherein the sub-laminations are taken to be sufficiently fine that the resulting laminate can be treated as a homogeneous medium. The model of Li and Liu (2004) generally results in high rank laminates, with rank n 1, when n crystal variants are present. Combining this with separation of length scales gives extremely fine microstructure, such that the domain wall energy, which is commonly neglected, may then become significant. Thus it appears likely that, among compatible laminate structures, those of lower rank are energetically favourable. Direct observation reveals that real microstructures are commonly inhomogeneous, containing multiple regions of low-rank compatible laminate structure, with differing length scales and arrangement (Arlt, 1990; Liu and Lynch, 2006). Several recent models also focus on low rank laminates. For example, Rodel (2007) discussed the compatibility of rank-2 hierarchical domain patterns; Weng and Wong (2009) and Yen et al. (2008) modeled the evolution of a rank-2 microstructure. This motivates the study of compatible structures with a view to finding the minimum rank solutions to the compatibility equations. In this work, we also study ‘‘exactly compatible” (EC), or a-type microstructure (Arlt and Sasko, 1980). In EC microstructure, every domain wall satisfies the compatibility equations. It is reasonable to expect that EC structures have low energy by comparison with COA structures of similar rank. Furthermore, Zhang et al. (2009) recently showed that exact compatibility is an important factor in ferroelectric hysteresis. Finally, the disclinations that form at the junctions of domain walls (Fousek and Mokry, 2005; Pertsev and Arlt, 1991; Savytskii and Bismayer, 2008; Shu and Bhattacharya, 2001) should be considered. Either the disclination vanishes, to give a stress-free compatible arrangement, or continuity is preserved by a state of stress at the domain junction. We will describe as ‘‘disclinationfree” (DF) any laminate structure in the class of EC structures which has no disclinations. The goal of the present work is finding minimum rank EC and DF microstructures in 3-dimensions, that may be of technological interest from the point of view of designing or engineering domain structure. A sharp-interface approach is utilized for rapid computation, recognising the potential of diffuse interface models to extend our findings. The present method gives insight into possible poling routes for single crystals, and has potential as a tool for microstructural design.
2. Theory of compatibility 2.1. Compatibility of domain pairs When the loads and boundary conditions favour the coexistence of more than one crystal variant, the total
741
energy of the microstructure is minimized by domain configurations that satisfy compatibility, based on the theory of energy minimization (Shu and Bhattacharya, 2001). For a pair of domains i and j with strain states ei, ej and corresponding polarization vectors pi, pj, the unit interface normal vector n of a compatible domain wall must satisfy (Ball and James, 1987; DeSimone and James, 2002; Shu and Bhattacharya, 2001):
1 2 ðpi pj Þ n ¼ 0:
ei ej ¼ ða n þ n aÞ;
ð1Þ ð2Þ
where a is an arbitrary vector chosen to satisfy Eq. (1). A compatible domain wall is possible if there is a solution for n. In the present work, consideration is limited to sets of crystal variants which are mutually related by pure rotations. Eq. (1) can then be solved by using the eigenvalues kk and eigenvectors ek (k = 1, . . . , 3) of the matrix representation M = ei ej, noting that at least one eigenvalue is zero. Let k3 = 0; then two solutions for the interface normal n are given by:
n¼
e2 e1 pffiffiffi : 2
ð3Þ
The resulting vector n must also satisfy Eq. (2) if polarization compatibility is to be achieved, and this commonly produces a unique interface normal. In the special case when M = ei ej = 0, Eq. (1) is trivially satisfied by setting a = 0, and Eq. (2) then provides a continuous set of solutions for interface normal n. This special case arises with 180° domains, which have no habit plane and result in ‘‘watermark” domain patterns (Arlt and Sasko, 1980). If only two crystal variants are present, a simple EC structure consisting of a rank-1 laminate can always be formed. However, groups of more than two crystal variants may require a higher rank configuration, giving rise to further compatibility conditions. 2.2. Compatibility of multi-rank laminate structures In laminated domain structure, pure variants can be regarded as rank-0 laminations, while a pair of variants can be laminated to form a rank-1 lamination. Similarly, pairs of laminates can themselves be laminated to form a higher rank lamination. Here, we make use of a hierarchical binary tree diagram to represent the arrangement of domains in a laminate of rank-R (Goldsztein, 2001). For example, Fig. 1 shows the tree diagram representing a rank-2 lamination. A simple top-down numbering of the nodes is used. The root node (node 1) represents the entire microstructure while nodes at lower levels in the tree represent structures that are conjoined to form higher rank laminations. At the lowest level, the nodes represent pure crystal variants. The tree diagram of a rank-R domain configuration contains 2R+1 1 nodes and R + 1 levels; it gives a convenient representation of laminated domain structure for computational purposes. Each node represents a material characterized by two properties: an average strain, and an average polarization.
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Fig. 1. A schematic binary tree for rank-2 (R = 2) laminate structure where f1 is the volume fraction associated with the ith node and ni is the interface normal vector. Levels of the tree are numbered r = 0. . .R corresponding to the rank of nodes at that level.
Nodes at levels other than the lowest level are associated with an interface normal vector that specifies the orientation of the lamination. The ith parent node (i = 1, . . . , 2R 1) is associated with volume fraction fi, strain ei and polarization pi directly related to its child nodes at the next level down.
fi ¼ f2i þ f2iþ1 ;
ei ¼ f2i e2i þ f2iþ1 e2iþ1 ;
pi ¼ f2i p2i þ f2iþ1 p2iþ1 :
ð4Þ
If boundary conditions are specified as a set of displacements and surface charges consistent with an average polarization P and average remanent strain er, the total volume fraction f(k) of each pure variant can be found. Here, a bracketed subscript has been introduced to distinguish the total volume fraction f(k) of the kth crystal variant from the volume fraction fi associated with node i in the tree diagram. Under conditions of zero internal charge and stress-free domain structure, the macroscopic values P and er are given exactly by volume averages of the corresponding local quantities. Thus:
P¼
N X
fðkÞ pðkÞ ;
ð5Þ
k¼1
er ¼
N X
fðkÞ eðkÞ ;
ð6Þ
k¼1
1¼
N X
fðkÞ ;
ð7Þ
k¼1
where N is the number of variant types and all f(k) are subjected to the condition 0 6 f(k). Eqs. (5)–(7) correspond to a system of 10 linear equations. It is worth noting that Eqs. (5)–(7) may be invertible, singular or overdetermined, depending on the particular set of crystal variants. Once the total volume fractions of the variants are determined, there remains the problem of constructing a compatible domain configuration – that is, distributing the volume fractions among the nodes of a tree diagram in such a way as to produce a compatible structure overall. Three types of compatibility are discussed: compatibility on average (COA), exact compatibility (EC) and disclinationfree (DF) structure:
2.2.1. Compatibility on average (COA) Consider a prescribed macroscopic state of polarization and strain and let Eqs. (5)–(7) produce a set of n non-zero volume fractions f(i)(i = 1, . . . , n) in the crystal. It is always possible to construct COA structure according to the theory developed by Li and Liu (2004), such that:
P ¼ l1 pð1Þ þ l2 ð1 l1 Þpð2Þ þ þ ln1
n2 Y
ð1 lm Þpðn1Þ
m¼1
þ
n1 Y
ð1 lm ÞpðnÞ ;
ð8Þ
m¼1
er ¼ l1 eð1Þ þ l2 ð1 l1 Þeð2Þ þ þ ln1
n2 Y
ð1 lm Þeðn1Þ
m¼1
þ
n1 Y
ð1 lm ÞeðnÞ ;
ð9Þ
m¼1
where lm are coefficients related to the volume fractions f(i) as follows (Bhattacharya, 1993):
lm ¼
8 > > < 0; > > :
m1 P
fðiÞ ¼ 1;
i¼1 f
ð10Þ
PðmÞm1 ; otherwise:
1
i¼1
fðiÞ
This domain configuration generates a rank n 1 laminate structure. An example of a tree diagram corresponding to a rank-3 laminate generated by the procedure of Li and Liu is shown in Fig. 2. There are four energetically favoured variants (f(k) – 0, k = 1, . . . , 4). Following Eqs. (8)–(10), the resulting rank-3 laminate structure is ensured to satisfy the average compatibility criteria by distributing the volume fraction of variant 1 across four nodes (numbers 8, 10, 12 and 14), and variant 2 across two nodes (numbers 9 and 13) at the lowest level of the tree. To understand this, note that the values of l1, l2, and l3 generated by Eq. (10) cause the volume fraction ratio of variant 1 and 2 at node 4 to match exactly it of variant 1 and 3 at node 5. This guarantees that a COA laminate can be formed between the materials represented by nodes 4 and 5. A similar pattern of COA lamination is generated throughout the tree diagram. It is
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Fig. 2. The domain configuration of a rank-3 laminate constructed by Li and Liu’s method.
worth noting that variant 1 is distributed such that it is present in all four nodes in the rank-1 level of the tree. Similarly, variant 2 is present in all nodes at the rank-2 level. This elegant configuration ensures that every laminating process is equivalent to conjoining of a pair of pure variants, since the sub-laminates always contain common crystal variants whose polarization and strain states cancel in Eqs. (1) and (2). Note that the Li–Liu domain structures commonly have high rank and local incompatibility. It is often possible to find a lower rank lamination than that generated by the procedure of Li and Liu. Furthermore, EC arrangements can be found where the Li–Liu method would produce COA structures. The contrast between COA and EC structures is illustrated in Fig. 3. Whereas, the COA structure of Fig. 3a has some regions with ‘‘head to head” or ‘‘tail to tail” polarization which violate Eq. (2), the EC structure of Fig. 3b, has low energy, one to one perfect domain alignment with ‘‘head to tail” polarization directions across interfaces. Thus low energy microstructures can be found by considering the conditions for exact compatibility. 2.2.2. Exact compatibility (EC) By considering the tree diagram of a rank-R laminate, it can be seen that there are three criteria for achieving EC
structure. First, the volume fraction ratios of all domain pairs in each level r = 0, . . . , R 2 must be identical:
f Rr f2Rr ¼ 2 þ2n2 ; f2Rr þ1 f2Rr þ2n1
n ¼ 2; . . . ; 2Rr1 :
ð11Þ
Here, 2Rr is the first node number in level r of the tree, and the maximum value of n is the number of laminate pairs in level r. This condition ensures that domains that meet across a high level interface have the same spacing at that interface. A similar constraint was described by Rodel (2007). Interestingly, the volume fraction arrangement proposed by Li and Liu always satisfies Eq. (11), as illustrated in the tree diagram of Fig. 2. A second requirement for EC structure is that, where domain walls meet a higher level interface, their projection into that interface matches in orientation with the projection of the corresponding domain wall on the opposite side of the interface. This is illustrated in Fig. 4 where an EC structure of rank-3 is shown in Fig. 4a, and the same structure is split open along a high level interface in Fig. 4b. Where the structure is split, the domain patterns exposed on the interface must match. In terms of the tree diagram, for every child node, the projection of its interface normal vector onto all interfaces of its higher level parent nodes must be parallel to that of a corresponding child node on
Fig. 3. Rank-2 microstructures constructed with (a) compatibility on average and (b) exact compatibility.
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Fig. 4. (a) An EC structure of rank-3 and (b) the structure split open along a high level interface showing a matching set of domain wall orientations.
the opposite side of the tree. The relationship between the interface normal vectors of parent node i = 1 . . . , 2R1 1 and every child node p and q must satisfy:
ðnp ðnp ni Þni Þ ðnq ðnq ni Þni Þ ¼ 0;
ð12Þ
which is equivalent to
np nq ni ¼ 0;
ð13Þ
where the node numbers p and q are given by
p ¼ 2jþ1 i þ k;
q ¼ p þ 2j ;
ðj ¼ 0; . . . ; r 2; k ¼ 0; . . . ; 2j 1Þ:
ð14Þ
In Eq. (14), the tree level r of node i is given by r ¼ R blog2 ic, where bxc indicates the greatest integer less than or equal to x. For EC structure, Eq. (12) must be satisfied over all the permutations of p and q indicated by Eq. (14). This ensures that all layers and their sub-layers have the same orientation on their common interface. Thus, a microstructure can have ‘‘one to one” matched domain configuration if its corresponding tree diagram satisfies Eqs. (11)–(14). Even though geometrically matched structures are achieved by having equal domain spacing and orientation, local polarization and strain compatibility requirements must also be satisfied to give EC structure. This avoids incompatibilities such as ‘‘head to head” polarization. Thus, a third condition for EC structure specifies that all pairs of adjacent domains must satisfy compatibility across the higher rank interface that separates them. Again, this condition is readily expressed in terms of the nodes of the tree diagram. The interface normal vector ni (i = 1, . . . , 2R 1) of each parent node i (layer r) and the polarization and strain states pp, pq, ep, eq, of the pure crystal variants with node numbers p and q in the lowest level (rank-0) of the tree must satisfy the basic compatibility Eqs. (1) and (2). Here node numbers p, q are limited to:
p ¼ 2r i þ k;
q ¼ p þ 2r1 ;
ðk ¼ 0; . . . ; 2r1 1Þ:
ð15Þ
Structures satisfying Eqs. (11)–(14) and Eqs. (1) and (2) for all nodes specified by Eq. (15) are exactly compatible at all
interfaces: they minimize the total energy without the need for separation of length scales. 2.2.3. Disclination-free compatibility (DF) Next consider the compatibility issues arising due to the disclination associated with certain domain walls. Let the true rotation of the crystal lattice across a given domain wall be hr. The angles hr in several ferroelectric crystal systems are given in the work of Shu and Bhattacharya (2001). The net rotation of lattice planes, h, where the pair of crystal variants i and j meet, is given by
^i p ^ j Þ; h ¼ hr cos1 ðp
ð16Þ
^ i and p ^ j are the unit vectors along the pair of the where p idealized polarization directions in the reference frame of one of the crystal variants. When several domains are assembled into a composite, the lattice rotations may result in disclinations. To illustrate this, consider tetragonal barium titanate, in which there is a shear strain of magnitude about 1% (Arlt and Sasko, 1980) between 90° domains resulting in a lattice rotation of 0.62° (Shu and Bhattacharya, 2001) across the domain wall. A disclination can form at the junction of four 90° domain walls. Fig. 5a shows a group of four domains in a DF configuration, with the lattice rotation exaggerated. This configuration is readily extended to form a rank-2 DF laminate (see Fig. 5b). By contrast, the group of four domains shown in Fig. 5c produces a disclination. They are shown in a relaxed state, producing an overlap at one domain wall. The group can be fitted together in a closed continuous form only by deforming the crystal lattice, introducing stress or electric field. The resulting structure (Fig. 5d) is EC but not DF. This vortex structure have been observed in related ferroic system, such as ferromagnets (Schilling et al., 2007) and are also generated by simulations such as phase field and ab initio models (Naumov et al., 2004; Schrade et al., 2007; Slutsker et al., 2008). The present work is concerned with finding DF multirank laminate structures and thus the global compatibility of an arrangement of domains must be considered. By using the tree diagram, the junctions of domains in a multi-rank laminate can be identified, and thus the full set of
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745
Fig. 5. Disclinations in rank-2 structures: (a) a disclination-free structure, and (b) a corresponding laminate configuration; (c) a disclination at the junction of four domains, showing a relaxed state with domains overlapping and (d) a corresponding laminate configuration, in stressed state.
domain junctions in the structure can be taken into account. In an EC structure, a pair of domains p, q which are children of node i in level r = 1, can be rotated from the ideal, or reference lattice configuration, into their exact configuration by a rotation Ri about the axis pp pq satisfying:
cos1 ðRi pp RTi pq Þ ¼ hrp;q ;
ð17Þ
where p and q are node numbers generated from Eq. (15); hrp;q is the true rotation of lattice planes at the domain wall between domains p and q. The rotation matrix Ri is given explicitly by:
2
u21 þð1 u21 Þc
6 Ri ¼ 4 u1 u2 ð1 cÞ þu3 s
u1 u2 ð1 cÞ u3 s u1 u3 ð1 cÞ þu2 s u22 þð1 u22 Þc
u1 u3 ð1 cÞ u2 s u2 u3 ð1 cÞ þu1 s
3
7 u2 u3 ð1 cÞ u1 s 5; u23 þð1 u23 Þc
ð18Þ where u = (pp pq)/|pp pq|, c = cos(h/2) and s = sin(h/2). Rotation Ri is used to turn the polarization at node p, while RTi turns the polarization at node q in the opposite sense. These rotations can be carried out at all pairs of nodes in the lowest level of the tree diagram, producing a new, rotated set of strain and polarization states for each domain. At higher levels of the tree, further conditions arise
because of domains meeting across high level interfaces. Then for each node i in levels r = 2 . . . , R, a rotation matrix Ri must be found that satisfies Eq. (17), and additionally the rotated interface normals must align, giving a condition similar to Eq. (13):
ðRi ns RTi nt Þ ni ¼ 0:
ð19Þ
For all node numbers s and t in the set
s ¼ 2jþ1 i þ k;
t ¼ s þ 2j ;
ðj ¼ 0; . . . ; r 2; k ¼ 0; . . . ; 2j 1Þ:
ð20Þ
A matrix Ri that satisfies conditions (17)–(20) can counterrotate the laminates meeting across the interface defined at node i into a compatible DF configuration. However, this rotation could disturb the strain compatibility across the domain wall if the axis of rotation is not perpendicular to the interface normal ni. Thus, in addition to Eq. (19), the basic compatibility between adjacent domains must be checked after the rotation. In general, Eqs. (17)–(20) are strongly overdetermined and solutions only exist in special cases. Some examples of rank-2 DF structures are discussed in Section 3. Note that all of the conditions of microstructural compatibility described in Section 2 are independent of length
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scale. In sufficiently fine-scale microstructure, the domain wall energy becomes non-negligible and can control spacing. In coarser microstructure the interaction of disclination fields may play a key role in spacing. However, these interactions are neglected in the present theory; we thus design compatible structural states without specifying precise scale limitations. Observations of domain structure suggest that domain wall spacing arise naturally over a wide range of length scales from 106 m or less up to 102 m or more. 3. Minimum rank compatible laminates High rank structures often cannot satisfy the exact compatibility conditions, leading to COA structures with separation of length scales. However, this results in very fine domains, such that the domain wall energy, which commonly ignored, becomes significant. While the model of Li and Liu (2004) gives rank n 1 structures when there are n crystal variants present, the minimum possible rank is dlog2 ne, where dxe indicates the least integer greater than or equal to x. Thus there is the potential to find structures of lower rank than that of the Li–Liu construction whenever n P 4. Fig. 6a shows an example of a rank-5 laminate which is a COA structure for a tetragonal crystal. This structure was determined using the model of Li and Liu, by setting the average polarization and strain state of the entire crystal to zero; this models the unpoled state of the material, as cooled from above the Curie temperature. Fig. 6b and c show two alternative configurations for this polarization and strain state: Fig. 6b is a rank-3 COA structure, while Fig. 6c is a rank-3 EC structure. Of these three possible structures, it is expected that Fig. 6c represents the lowest energy state. In order to discover other low energy structures, we have used a systematic procedure to search for minimum rank domain configurations which satisfy the criteria of exact compatibility or disclinationfree compatibility. 3.1. Application to tetragonal ferroelectrics A tetragonal crystal has six crystal variants in three pairs of opposite polarization. The strain states e(±i), (i = 1, . . . , 3) and corresponding polarization states p(±i) are given by:
2
b
0
0
3
2
pt
3
7 6 7 eð1Þ ¼ 6 4 0 a 0 5; pð1Þ ¼ 4 0 5; 0 0 a 0 2 3 2 3 0 a 0 0 7 6 7 eð2Þ ¼ 6 4 0 b 0 5; pð2Þ ¼ 4 pt 5; 0 0 0 a 2 3 2 3 0 a 0 0 6 7 7 eð3Þ ¼ 6 4 0 a 0 5; pð3Þ ¼ 4 0 5: 0
0
b
ð21Þ
pt
Here a, b, and pt are material properties, but there is choice over the reference state from which strains are measured. Strains of the unit cell can be defined relative to a cubic state of equal volume, such that tr(e) = 0 and a = b/2. The true polarization rotation angle hr of the crystal lattice at a 90° domain wall is then (Shu and Bhattacharya, 2001):
hr ¼ 2 tan1
1þb 1 b=2
ð22Þ
and the net rotation of lattice planes at the domain wall can be calculated by using Eq. (16). The problem to be solved may be stated as follows: Let a tetragonal single crystal have average remanent strain er and average polarization P. We seek an exactly compatible multi-rank laminate comprising the crystal variants defined in Eq. (21) with the given average strain and polarization state, and with minimum rank. In the tetragonal system, Eqs. (5)–(7) give a unique solution for the total volume fractions of each of the six crystal variants. Let n be the number of variants with non-zero volume fractions. Now consider distributing the volume fractions of these n variants into 2dlog2 ne parts to fill the lowest level of a minimum rank binary tree diagram. Whenever 2dlog2 ne > n there is freedom over how the volume fractions are divided among the nodes at the lowest level of the tree because at least one variant must have its volume fraction split between more than one node. There are then multiple permutations of the volume fraction arrangement. However, the requirements of exact compatibility, Eq. (11), constrain the distribution of volume fractions. For example, in a rank-3 construction, the volume fractions must have the form: l1l2l3, (1 l1)l2l3, l1(1 l2)l3, (1 l1)(1 l2)l3, l1l2(1 l3), (1 l1)
Fig. 6. A tetragonal crystal with all 6 variants present can be constructed by (a) rank-5 compatibility on average (b) rank-3 compatibility on average and (c) rank-3 exact compatibility.
N.T. Tsou, J.E. Huber / Mechanics of Materials 42 (2010) 740–753
l2(1 l3), l1(1 l2)(1 l3), and (1 l1)(1 l2)(1 l3) in nodes 8–15, respectively. Here li has the same meaning as defined in Section 2.2.1, Eqs. (8)–(10). For each permutation of the split volume fractions, a set of n simultaneous nonlinear equations in li can be found to ensure that the nodal volume fractions consistently sum to the variant volume fractions. These equations of volume fraction continuity may be written as: fðiÞ ¼
2Rþ1 X1
C ij fj ;
ð23Þ
j¼2R
where Cij is a permutation matrix such that Cij = 1 if the crystal variant at node j is variant number i and zero otherwise. The nodal volume fractions fj are given by
fj ¼
R Y
ð1 2bÞlk þ b;
ð24Þ
k¼1
where
b¼
$ % j 2R 2k1
2
$ % j 2R : 2k
ð25Þ
The binary digit b takes on a value of zero or unity depending on whether the jth node is reached by taking the left hand or right hand fork at the level r = k 1 of
747
the binary tree. The non-linearity of Eqs. (23)–(25) causes significant complication; however, the appearance of common factors of li in these equations simplifies the solution process. We make a systematic search of the set of permutations, in each case checking for a consistent solution to Eqs. (23)–(25). If a solution exists, the resulting arrangement is taken forward as a candidate structure and the remaining criteria for EC structure, Eqs. (1), (2), (13), (14), (15), are checked. In addition, it is possible that Eqs. (1) and (2) allow a domain wall orientation to vary continuously, as is the case for 180° domain walls. This provides extra degrees of freedom that can allow the EC conditions to be satisfied. In practice, where an extra degree of freedom arises, it normally allows Eq. (12) to be solved, thus increasing the chance of satisfying all the conditions of exact compatibility. The domain structure shown in Fig. 6c is a typical example of minimum rank EC configuration; in this case er = 0 and P = 0 which gives six variants, each with equal volume fraction 1/6. It is worth noting that the laminates shown in Fig. 6b and c have exactly the same tree diagram, except that the interface normal vectors of the structure in Fig. 6b do not satisfy the EC conditions. Now consider applying the method described above to study the practical problem of poling a ferroelectric single
Fig. 7. (a) The space of possible remanent strain states in a tetragonal single crystal with P1 = P2 = 0. (b) and (c) Illustrate curved spaces of 6 variant rank-3 structures.
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region ABCD require at least five variants. These points generally require a lamination of rank-4 or greater and only allow laminate structures that are compatible on average. However, points in the triangle ABC and on the curved surfaces AGCD and GBDC (shown in Fig. 7b, 7c) admit rank-3 exactly compatible structures (see Table 2 for details). The curved surfaces AGCD and GBDC contain both the unpoled initial state, O, and the fully-poled state D. Thus a tetragonal crystal can be poled, while remaining in an exactly compatible state, with laminate structure of rank three or less. This is a significant result as it implies that tetragonal crystals could be poled by continuous motion of domain walls (without nucleation of new domains) and in exactly compatible states that do not necessitate fine domain structure. Some poling paths for tetragonal single crystals can be suggested. For example, the route O ? C ? D consists of axisymmetric straining along OC (er11 ¼ er22 ¼ er33 =2) as a rank-3 laminate, followed by strain-free 180° switching along CD. It is interesting to note that the domain configuration jumps two rank levels from rank-3 to rank-1 at point C, where variants ±1 and ±2 vanish. We speculate that this poling path could be achieved by a two stage loading process consisting of axisymmetric stress loading, followed by the application of an electric field parallel to the x3 direction. The domain evolution during this poling process is shown in Fig. 8. Interestingly, this poling procedure for tetragonal single crystals is similar to a poling method given by Granzow et al. (2006) in the context of maximizing polarization in polycrystalline ferroelectrics. An alternative poling path O ? H ? D consists of a plane strain deformation, holding er33 ¼ 0, in rank-3, followed by a second plane strain deformation, holding er11 ¼ b=2, in rank-2. The polarization evolves continually along this path and so combined stress and electric field boundary conditions could be needed. Note that the domain configurations corresponding to a given state of polarization and strain are generally non-unique. For example, the unpoled state O in Fig. 7a can be realized in many different ways. Three distinct ways can be seen by considering the expressions for the arrangement of nodal volume fractions corresponding to point O in surfaces ABC, AGCD, and GBDC in Table 2. We should also emphasize that the compatible solutions found by the present method are restricted to periodic multi-rank laminate configurations. Other compatible configurations can exist, such as the crossing domain patterns identified by Shu and Bhattacharya (2001). Furthermore,
Table 1 The crystal variants and volume fraction arrangement on line ED (FD) in Fig. 7, for the tetragonal crystal system. Line ED (FD) 0 x 14
pffiffiffiffiffiffiffiffiffi 14x
a ¼ ð12xÞþ 2x
Node no. Crystal variant
4 1 (2)
5 3
6 3
7 1 (2)
Volume fraction
x
a
x
ax
x
crystal. A natural question is whether there exists a continuous path that the microstructure may follow, from an initial, unpoled state (P1 = P2 = P3 = 0; er11 ¼ er22 ¼ er33 ¼ 0), to a fully poled, single variant state (P1 = P2 = 0, P3 = pt; er11 ¼ er22 ¼ b=2; er33 ¼ b). If such a path exists, what is the minimum rank route to poling? In order to resolve this question in the tetragonal crystal system, it is instructive to explore the polarization-strain space (P3, er11 ; er33 ) with P1 = P2 = 0, P3 P 0 as shown in Fig. 7a. The tetrahedral region ABCD in Fig. 7a is the possible space of all P3, er11 ; er33 values, with the volume fractions constrained by Eq. (7) and by the condition 0 6 f(i). A general point in region ABCD requires all six variants to be present, and so the crystal could form a laminate of rank-3 (minimum possible rank) to rank-5 (Li and Liu’s model). However, certain special points in ABCD can have fewer than six variants. Point D represents an ideally poled state consisting of a single variant (variant number +3 in Eq. (21)) and thus has a rank-0 structure. Points A, B, C and line CD everywhere except at D require only two variants, giving rank-1 structure. The line CD represents laminates of variant 3 and variant 3; thus the average polarization, P3, varies along line CD without change in the average remanent strain. Points on lines AD and BD need three variants and form rank-2 exactly compatible structures. Note that the minimum rank laminates for these points have identical domain configurations to those provided by the model of Li and Liu (2004). Typical points in the triangular surfaces ACD and BCD need four variants and rank-3 structures. However, the edges AB, AC, BC and the curves ED and FD in Fig. 7a comprise a set of strain and polarization states that correspond to rank-2 exactly compatible laminates. The detail of the crystal variants and volume fraction in the lowest level of the tree diagram for points on the curves ED and FD is shown in Table 1. By varying the parameter x in the range 0 6 x 6 1/4 all of the polarization and strain states along these curves can be obtained. All remaining points in
Table 2 The crystal variants and volume fraction arrangement of surfaces ABC, AGCD, and GBDC in Fig. 7, for the tetragonal crystal system. Surface ABC 0 6 x 6 12 ; 0 6 y 6 ð12 xÞ
Node no. Crystal variant
8 1
9 1
10 2
11 2
12 1
1 ; y ¼ 16 At O, x ¼ 12
Volume fraction
x
x
y
y
x 2ðxþyÞ
AGCD (GBDC) 1 0 x 1; 0 y 2xþ2
Node no. Crystal variant
8 1 (2)
9 1(2)
10 3
11 1(2)
12 2(1)
13 2(1)
14 3
1 At O, x ¼ 1; y ¼ 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4y 2 a ¼ 3y þ y þ 1þx 2
Volume fraction
a+y
y
a þ ay
a
x(a + y)
xy
xða þ ay Þ
2
13 1 x
x 2ðxþyÞ
14 3 x
y 2ðxþyÞ
15 3 y
y 2ðxþyÞ
y
15 2(1) 2
xa
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Fig. 8. The domain evolution in the poling route O ? C ? D of Fig. 7. (a) Unpoled state. (b) A typical point on OC. (c) Variants ±1 and ±2 vanishing as point C is reached. (d) Rank-1 structure at point C. (e) A typical point on CD. (f) Fully-poled state.
2
3
2
3
3.2. Application to rhombohedral ferroelectrics
pt a b b 7 6 7 eð1Þ a b 5; pð1Þ ¼ 4 pt 5; pt b a 3 2 3 pt a b b 7 6 7 eð2Þ ¼ 6 b 5; pð2Þ ¼ 4 pt 5; 4 b a a pt b b 2 3 2 3 pt a b b 6 7 eð3Þ ¼ 6 a b 7 4 b 5; pð3Þ ¼ 4 pt 5; p b b a 2 3 2 t 3 pt a b b 7 6 7 eð4Þ ¼ 6 4 b a b 5; pð4Þ ¼ 4 pt 5: pt b b a
In the rhombohedral crystal system, there are eight crystal variants with the strain e(±i), and polarization p(±i), (i = 1, . . . , 4) given by
Again, a, b, and pt are material properties and we define the strain states relative to a reference configuration of equal volume, such that a = 0.
non-periodic, or macroscopically inhomogeneous configurations are not considered. Finally, consider disclination-free domain configurations in the tetragonal crystal system. The additional constraint placed on the domain arrangement in order to produce disclination-free structure eliminates all points in region ABCD of Fig. 7a, except the edges AB, AC, BC and CD. Here, disclination-free structures of rank two or less, having pairs of domains with anti-parallel polarization, can be found. Since the unpoled state, O, does not lie within the set of points where disclination-free structure is possible, there is no disclination-free poling path.
6 ¼ 4b b 2
ð26Þ
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Fig. 9. The space ABCD of possible remanent strain states in a rhombohedral single crystal with P = 0. (a) Edges and vertices of the tetrahedron, corresponding to structures of rank-2 or less. (b) A curved surface showing the strain states of exactly compatible rank-3 structures with all 8 variants present.
Considering disclinations, there are two types of domain walls that contribute to disclinations in the rhombohedral crystal system. The polarization rotations in the rhombohedral system across these two types of domain wall are approximately 71° and 109°. However, the true rotations hr(I), hr(II) are given by (Shu and Bhattacharya, 2001):
hrðIÞ ¼ cos1
1 8b 2b2 3 þ 6b2
hrðIIÞ ¼ cos1
! ð27Þ
;
1 þ 10b þ 7b2
!
3ð1 þ 2b þ 3b2 Þ
ð28Þ
:
The rotation of lattice planes can be calculated using Eq. (16) and thus the criteria for disclination-free structure can be tested. For a rhombohedral single crystal with average remanent strain er and average polarization P, Eqs. (5)–(7) expand into seven linear equations for the eight unknown volume fractions. The general solution of Eqs. (5)–(7) gives volume fractions in the form:
fð1Þ
fð1Þ
fð2Þ
fð2Þ
fð3Þ
fð3Þ
fð4Þ
fð4Þ
T
¼ x0 þ c½ 1 1 1 1 1 1 1 1 T :
ð29Þ
Here x0 is a particular solution and 0 6 c 6 cmax; the value of cmax is set by the constraint that 0 6 f(i). In the present work, for simplicity, only the cases c = 0, c = cmax and c = cmax/2 are considered. When c = 0 or cmax, at least one variant has zero volume fraction, and the number of variants present in the laminate is minimized. When c = cmax/2, the number of variants present is maximized, generally meaning all eight variants are present. This latter choice of c also maximizes the least volume fraction of any variant and leads to at least two volume fractions being equal. Now consider the ferroelastic straining of a rhombohedral crystal, maintaining P1 = P2 = P3 = 0. The only non-zero
strain components are er12 ; er13 and er23 , and the space of possible strain states is limited to the tetrahedral region ABCD shown in Fig. 9a. The state O in Fig. 9a has er = 0 and is representative of a rhombohedral crystal in an unpoled state, as cooled from above the Curie temperature. This point allows many exactly compatible configurations of either rank-2 or rank-3 with four or eight variants present, respectively. Interestingly, one of the rank-3 laminate solutions at point O is identical to a result provided by a recent non-conventional phase field model (Shu et al., 2008) as shown in Fig. 10a. Phase field simulations such as those of Shu et al. (2008) produce minimum energy configurations and this commonly results in exactly compatible configurations. It is likely that the exactly compatible configurations have lower energy than configurations with average compatibility. An advantage of our approach is that the full set of compatible solutions is generated. The detail of the domain arrangement in the lowest level of the tree can be obtained from Table 3 by setting the parameters x = 1, y = 1/8. An alternative laminate at point O, of rank-2, is shown in Fig. 10b. This arrangement was also found using a phase field model by Shu (2010). For its detail, see Table 4. Returning to Fig. 9, Points A, B, C, and D require two variants to be present, and allow rank-2 exactly compatible structures. Each of these four points in strain space corresponds to a structure consisting of a pair of domains with anti-parallel polarization. The six edges AB, AC, AD, BC, BD, and CD need four variant structures and produce minimum rank-2, disclination-free domain configurations. Points on the surfaces ABC, ABD, ACD and BDC require six variants and give rank-3 exactly compatible structures. The points inside region ABCD generally require at least seven variants and lead to complex configurations of laminate. However, we can find several special cases of minimum rank exactly compatible laminates within ABCD. For example, a surface of the form er12 ¼ er23 er13 (and cyclic permutations) is shown in Fig. 9b. Points on this surface
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Fig. 10. (a) One of the possible rank-3 exactly compatible laminates in the unpoled state which is identical to a result provided by Shu et al. (2008). (b) An alternative rank-2 exactly compatible laminate in the unpoled state.
Table 3 The crystal variants and volume fraction arrangement of surfaces ABDC (er23 ¼ er12 er13 ) and ABCD (er12 ¼ er23 er13 ) in Fig. 9, for the rhombohedral crystal system. Surface ABDC, er23 ¼ er12 er13 (ABCD, er12 ¼ er23 er13 Þ
Node no.
8
1 0 6 x; 0 6 y 6 ð2xþ2Þ
Crystal variant
2(1)
4
At Rank-3 O, x ¼ 1; y ¼ 18
Volume fraction
1 2xþ2
x 2xþ2
9
y
10
11
2(1) xy
admit rank-3 exactly compatible structures with all eight variants present (see the detail in Table 3). The resulting rank-3 laminates do not satisfy the conditions for disclination-free structure; however, it is interesting to note that the angular errors from disclination-free configuration are very small (of order 104 b) so that the corresponding structures are low energy configurations. This surface connects the unpoled state, O, with each of the vertices A–D. As with the tetragonal system, several poling paths for rhombohedral crystals can be suggested which allow the domain configuration to remain in an exactly compatible state with rank-3 or less. For example, one poling route consists of ferroelastic straining O ? E ? D in Fig. 9a followed by a strain-free polarizing stage. Stage O ? E is polarization-free shearing, which increases er23 . The details of domain evolution on O ? E are given in Table 3 (surface ABCD with x = 1 and 1/8 P y P 0). This is followed by stage E ? D, comprising further ferroelastic straining with constant er23 , and er12 ¼ er13 (see Table 4 for details). Once point D is reached, the structure has simplified to a rank-1 laminate of variants +1 and 1. This can be poled without fur-
1 2xþ2
12
4 y
x 2xþ2
xy
13
14
15
3(2)
1(3)
3(2)
1(3)
y
xy
y
xy
ther straining by moving the domain wall that separates the two variants. The domain evolution along line O ? E ? D and from point D to the fully-poled state is shown in Fig. 11. 4. Conclusions Three dimensional exact and disclination-free compatibility conditions for periodic, multi-rank laminate domain configurations of ferroelectric single crystals have been developed, and methods have been given for applying the compatibility conditions to the full set of domain walls that exist in a multi-rank laminate. A hierarchical tree diagram was used to represent the laminates, and to define the relations between domains. An algorithm for finding minimum rank compatible laminate structures was used to study the tetragonal and rhombohedral crystal systems. While, many exactly compatible structures were found, the conditions of disclination-free structure prove highly restrictive and few such structures were found. A key result is that
Table 4 The crystal variants and volume fraction arrangement of the rank-2 unpoled state, O, and line ED in Fig. 9, for the rhombohedral crystal system. Point Rank-2 O
Line Node no. Crystal variant Volume fraction
4 1 (1)
5 3 (3)
6 2 (2)
7 4 (4)
1 4
1 4
1 4
1 4
ED 0 6 x 6 14
4 1 1 2x
5 4 x
6 1 1 2
x
7 4 x
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Fig. 11. The domain evolution in the poling route O ? E ? D ? fully-poled state in Fig. 9. (a) Unpoled state. (b) A typical point on OE. (c) Rank-2 structure at point E. (d) Rank-1 structure at point D. (e) Rank-1 laminate evolves between point D and fully-poled state. (f) Fully-poled state.
domain laminates in both the tetragonal and rhombohedral crystal systems can be designed, which allow the crystal to be poled from a state of zero strain and polarization to a single domain state, while the rank of the laminate never exceeds three. Several poling paths are suggested and details of the domain topology along these paths have been found. The theory can be readily applied to any ferroelectric crystal system, and provides a rapid means of computing low energy structural configurations. The theory also has potential applications to other ferroic systems such as ferromagnetic shape memory alloys. Acknowledgement This authors wish to acknowledge the support of EPSRC, project number EP/E026095/1. References Arlt, G., 1990. Twinning in ferroelectric and ferroelastic ceramics – stress relief. J. Mater. Sci. 25, 2655–2666.
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