Twin boundary profiles with linear-quadratic coupling between order parameters

A new type of twin boundary was found when two order parameters interact by linear-quadratic coupling QP 2. In this solution, we find that a domain wall consists of two layers in which in one layer both order parameters Q and P are active while in the second layer only Q is active. The adjacent domains are equally asymmetric (Q, P ) and (Q, 0) so that one phase could be polar and/or magnetic and contain a ferroelastic strain while the second layer is ferroelastic only without polar or magnetic properties. The two layers represent a stepwise transition between the two domains. We analyze the full phase diagram depending on the coupling constant and anisotropy of the gradient term, and show that in a certain regime the order parameter Q becomes activated only in the interfacial region. A common solution contains kinks and breathers whereby the width of the interface can be very wide in agreement with the first order character of the transition.


Introdcution
A major progress in solid state science is the understanding that interfaces are not (always) just the 'zip' between adjacent domain structures which have no properties and cannot be found in the bulk of the domains. Until 1955, the idea of twins as 'zips' between domains dominated, then more advanced theories, such as Landau Ginzburg theory became popular and the trajectories were calculated including the minimization of the local strain energy and surface tension [1][2][3][4][5][6].
Today, the emphasis lies on the question: can we modify interfaces in such a way that they display novel properties which have nothing to do with the properties we find in the bulk of the domains. The idea that such properties exist is new [4,6]. 'Exotic' interfaces were previously either rejected Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. as wrong observations or taken as exceptions to the 'normal' behaviour, which could be safely ignored. The fact that we can generate interfaces such as twin boundaries in an engineering fashion, which are good conductors in an insulating matrix [7][8][9][10], or ferroelectric interfaces when the matrix is nonpolar [11][12][13][14][15], or when magnetic interfaces are expected in non-magnetic materials [16][17][18], adds emphasis to the field of Domain Boundary Engineering [6], which has as its goal, to produce interfaces which have properties that do not exist in the bulk and which are useful as memory devices and other device applications. There are only a few confirmed materials where the domain boundary engineering has succeeded: WO 3 (superconducting walls [8]), BiFeO 3 and LiNbO 3 (conducting walls [9,19]), SrTiO 3 , and CaTiO 3 (polar walls [12][13][14][15]) are certainly amongst them.
The most successful approach to develop a proper theory of domain boundary engineering uses the following concept: we know that phase diagrams depend sensitively on the competition of various phases whereby a stable phase is determined by its absolute minimum of some energy functional such as the Gibbs free energy. For infinite solids the construction of such energies is well understood within the framework of Landau theory. Modifications are needed for boundary conditions and are also generally understood for most relevant geometrical scenarios. The problem becomes more complicated when interfaces are considered because the energy minimization needs to be done under the appropriate boundary conditions and surface terms need to be added explicitly to describe the fact that we want to know what phases exist in the quasi-two dimensional lattice of the interface. Polar twin boundaries in CaTiO 3 are a typical example. No polar phases exist in CaTiO 3 but polarity is a physical property closely related to the structure type of CaTiO 3 . The approach is then to assume that an unstable (or metastable) phase with a well-defined polarity interacts with the structural state of the stable state. If we define each state by their thermodynamic order parameter (e.g. Q for the stable deformation and P for the polarity) then we can formulate the Gibbs free energy as an energy including a coupling term between deformation and polarity of the kind Q n P m where the order of the coupling is given by the exponents n and m. The parameters n and m are determined by symmetry, in our example n = m = 2 for CaTiO 3 .
The case of bi-linear coupling n = m = 1 is rare but well documented for the bulk [20,21]. The bi-quadratic coupling n = m = 2 is the most common and was first identified by Houchmanzadeh et al [5], and explored in detail by Conti et al [22]. The case of linear-quadratic coupling (n = 1, m = 2) is new and has been first analyzed as a new type of interaction by Salje and Carpenter [23]. Here we explore the consequences of such coupling for interfacial properties where Q forms a twin boundary and P reacts to these boundary conditions. The paper is organized as follows: we formulate the Gibbs free energy in part 2, we then discuss the solutions in part 3 and compare the results with possible applications in part 4.

The Gibbs free energy and the interfacial energy
As in [23], we write the Gibbs free energy as G a,c,λ (Q, P ) = 1 2 where a, c, λ ∈ R, λ = 0 and a normalization constant g a,c,λ 0 is added such that the minimum of G a,c,λ is zero. Taking the leading-order gradient term for each order parameter into account, the total energy of an interface is where κ 1 , κ 2 > 0 and u, v : R → R represent Q and P , respectively. This energy has to be minimized over all pairs (u, v) that attain stable points of the Gibbs free energy as x → ±∞. The rescalings with µ ∈ R and ν > 0 together with the identities allow to fix the coupling constant λ = −1 and to consider only the relative strength of the gradient terms κ = κ 2 /κ 1 > 0. It remains to consider the energy For conciseness, we will hereafter write G = G a,c,λ and E = E a,c,κ when the notation is unambiguous.

The stable points of the Gibbs free energy
The interfaces link stable points of G. Thus, the first step in characterizing all possible profiles is a precise characterization of the stable points and their dependence on a and c. Inspection of G = G a,c,−1 yields that if (Q, P ) is one of its stable points, then Possible minimizers of G are its critical points, i.e. real solutions of These are (0, 0), ± √ −a, 0 if a 0, and real solutions of If real solutions (Q, P ) of (2) exist, the largest such Q shall be denoted as Q 0 = Q 0 (a, c) and the corresponding non-negative P as P 0 = P 0 (a, c).
Then: Furthermore, in the interior of each set the stable points enumerated above are the only stable points. Within each set the stable points depend continuously on a and c. However:

the stable points of (ii) and (iii)
coincide if and only if a −1/4.

Results
We now discuss different interfaces that arise depending on whether the parameters (a, c) lie in the sets A 1 , A 2 , A 3 or their pairwise intersections. The findings are illustrated by results of finite-element simulations.
First consider the limiting case κ = 0. Then A straightforward computation yields arg min Since (a, c) ∈ A 2 implies √ −a c, the minimization problem left to solve is which has the well-known solutions By (3) The interfaces are plotted in figure 2.

(a, c) ∈ A 3 such that only
In this case, all interfaces up to symmetry lie in In contrast to the previous case, we are not aware of the existence of closed expressions for the minimizers. However, they can be computed numerically and two typical profiles are plotted in figure 3. For all κ > 0 the length scale of the transition is of order √ κ. If κ << 1 'jumps' of v are only weakly penalized and v changes quickly while u remains roughly constant. As κ → ∞ the maximal distance of u from its boundary value Q 0 increases and its minimal value approaches √ max{−a, 0} continuously. In the limiting case κ = ∞, which can be realized by setting κ 1 = 0 in the original formulation (1) of E, the interface passes through the point ( √ max{−a, 0}, 0). This can be confirmed by an explicit calculation along the lines of the previous subsection. A 3 such that (0, 0) and (Q 0 , ±P 0 ) are stable As in subsection 3.3., we consider interfaces that link the states (Q 0 , ±P 0 ), i.e. profiles in the set A given in (5). For small κ, the picture is similar and the interfaces are well separated from the now stable point (0, 0).

(a, c) ∈ A 1 ∩
However, for fixed a and c there exists a critical value κ * such that for κ > κ * , the point (0, 0) gets activated and the interface is divided into two separate transitions between   (Q 0 , −P 0 ) and (0, 0) and between (0, 0) and (Q 0 , P 0 ). With the class of partial interfaces We are unaware of a general explicit characterization. Figure 4 visualizes the dichotomy for a = 0.24 and c = 0.08 where 0.1 < κ * < 1.
The phenomenon can roughly be explained as follows. The Gibbs free energy G consists of a 2-4-potential for each of the order parameters Q and P and the coupling term λQP 2 . The closer the order parameters (Q, P ) pass the point (0, 0) the weaker the coupling term gets. If they are close enough, the uncoupled potentials dominate and Q and P seek their stable points which is 0 for both, since a, c 0.    Numerical evidence and the rigorous results of subsection 3.2. suggest that optimal interfaces in this class split for all κ > 0 into a transition joining the pure phases (∓ √ −a, 0) and another joining ( √ −a, 0) with (Q 0 , P 0 ). Interfaces for several values of κ are plotted in figures 5 and 6 as functions of x and in figure 7 in the Q-P -plane.
In the first transition, the order parameter v is suppressed and remains zero while u takes the form (4) as in subsection 3.2.. The first transition region has constant length and is independent of κ. In terms of the original coefficients κ 1 and κ 2 from (1), it is of order √ κ 1 and in particular, is independent of κ 2 .
The second transition is well separated from the first and the partial interface joins different values for both u and v, respectively. Hence, its length scale is determined by the stronger gradient term, i.e. of order max{ √ κ 1 , √ κ 2 } or max{1, √ κ} in terms of κ. The reason why the interfaces always split is roughly as follows. During the first transition, the coupling term is dominated by the uncoupled potentials whose stable points are ± √ −a for u and 0 for v since here a < 0 and c > 0. It is only during the second transition that the values of u are large enough to activate the coupling term.

Discussion
The various wall profiles contain simple twin walls with only one active order parameter whereby the second order parameter is uniformly depressed in the wall and in the bulk (figure 2). The second classic solution are kinks and breathers where P changes sign and Q changes only in the domain wall (figures 3 and 4). This solution is not chiral because Q reduces from an equilibrium value Q 0 , the equivalent enhancement from Q 0 is not possible and hence there is no chiral order in this configuration, in contrast to the results from bi-quadratic coupling [22]. The novel solution in figures 5 and 6 show double kinks which are exciting when u goes from − to + while v remains non-negative. This constitutes double walls: in part of the wall the changes are mainly in v, then u changes. This constitutes two layers which are stuck together. Their stability stems from the two adjacent domains where in one domain Q and P are active while the second domain has only Q but not P as an active component. Hence, the transition between these two domains is stepwise with Q and P reacting differently in different parts of the wall.
The new type of walls have been found mathematically but not experimentally. Where can we expect them to exist? Several options can be imagined: 1. If the volume strain is large (such as quartz SiO 2 ), one could imagine a Dauphine wall to have a second 'volume' wall next to it. That would have consequences for the description of the incommensurate phase [24][25][26]. The usual rigid unit calculations do not consider this effect and hence may be wrong [27]. It also means that walls have an asymmetry with the v = 0 state on one side but not the other. Walls would then have a 'direction' which again stems from the broken symmetry.
In incommensurate phases, such directions have never been considered. In a milder way, BaTiO 3 has a volume anomaly which goes with a dipole moment as driving order parameter. Does this mean that 180 • walls (in P ) have a volume component next to it? So far the evidence is against this, i.e. the walls are very thin. The usual theory does not consider volume. This is different for 90 • walls where strain effects are important and the shape of the walls may be asymmetric [28,29]. 2. Magnetic systems are Fe-O (wuestite) and MnO near the Vervey transition [16]. Here, the experimental resolution is poor. The damping of wall movements is very well measured, however, and we may be able to extract data from there. 3. In a wider context, it is possible that walls with asymmetric charge distributions fall into the same set of problems.
If charges appear on one side of the wall but not on the other, then they follow the same profiles as discussed in this paper. Such scenarios were discussed by Eliseev et al [30]. Similarly, the interfaces between SrTiO and LaAlO 3 may contain topological defects which also generate asymmetric wall profiles [31]. Effects of linearquadratic order parameter coupling are indeed expected in these cases.
Other asymmetric walls may arise from the flexoelectric effect at the walls that lead to Neel-like polarization components at the otherwise Ising-like walls [32] where asymmetries may occur so that one needs to take our current results into account if the order parameter coupling is formulated.