Correlation between phase compatibility and efficient energy conversion in Zr-doped Barium Titanate

Recent demonstrations of both heat-to-electricity energy conversion devices and electrocaloric devices based on first-order ferroelectric phase transformations identify the lowering of hysteresis and cyclic reversibility of the transformation as enabling criteria for the advancement of this technology. These demonstrations, and recent studies of the hysteresis of phase transformations in oxides, show that satisfying conditions of supercompatibility can be useful for lowering hysteresis, but with limitations for systems with only a few variants of the lower symmetry phase. In particular, it is widely accepted that in a classic cubic-to-tetragonal phase transformation, with only three tetragonal variants having only six twin systems, tuning for improved crystallographic compatibility will be of limited value. This work shows that, on the contrary, the tuning of lattice parameters in Ba(Ti1-xZrx)O3 for improved crystallographic compatibility, even at low doping levels of Zr (x ≤ 0.027), give significant improvement of transformation and ferroelectric energy conversion properties. Specifically, the transformation hysteresis is lowered by 25%, and the maximum value of the polarization/temperature ratio dP/dT at the phase transformation is increased by 10%.


Figure S 2.
Temperature dependent lattice parameters of tetragonal and cubic phase for the system Ba(Ti1-xZrx)O3 with (a) x=0.006, (b) x=0.009, (c) x=0.013 and (d) x=0.027 with error bars equal to a standard deviation of 0.036 %.

Derivation of the cofactor conditions (CC1), (CC2), (CC3) and the function q(f)
A brief derivation of the cofactor conditions ((CC1), (CC2) and (CC3) in the text) is given, which explains the occurrence of the function q(f). Further details and implications of these conditions can be found in the references. 3,22,25 The cofactor conditions (CC1), (CC2), (CC3) represent a degeneracy of the Crystallographic Theory of Martensite in which there exist infinitely many low energy interfaces between austenite and twinned martensite. The generic case is 4 such interfaces per twin system. (This gives altogether 24 interfaces in the cubic-to-tetragonal case.) Furthermore, when the cofactor conditions are satisfied, many of the new interfaces have zero elastic energy, rather than just low elastic energy. Supercompatibility refers to the presence of these zero elastic energy interfaces.
The usual low energy interfaces have an elastic transition layer whose energy competes with, and balances, the total interfacial energy on the twin boundaries.
The condition (CC1) alone (without (CC2) and (CC3)) is the condition λ2 = 1. This is a necessary and sufficient condition that there exists a perfect, unstressed interface between undistorted austenite and a single undistorted variant of martensite. In the cubic-to-tetragonal case this is simply the condition λ2 = a/a0 = 1. This condition arises in the following way. In the cubic-totetragonal case we have three variants of martensite described by the three stretch tensors here written in the orthonormal cubic basis. A perfect interface between the cubic phase and, say, variant 1 of the tetragonal phase is described by the compatibility equation to be solved for the rotation tensor Q and vectors a and n. Here, n is a normal to the interface, a describes the shear of the transformation, and Q describes the rigid rotation of variant 1 needed to secure perfect compatibility with the cubic phase. This equation is solved in reference 24 (Prop. 4). A necessary and sufficient condition that there is a solution Q, a, n is λ2 = η1 = a/a0 = 1; this is the main result used in the text. In the usual case η2 ≠ 1, there are in fact two solutions.
An equation of the same form as (2) describes the twin boundaries, i.e., the compatible interfaces between variants. For example, variants 1 and 2 have compatible interfaces described by where is a rotation tensor. There are also two solutions , c, d of (3) and these exist under the mild condition η1 ≠ η2. They describe the compound twins of variants 1 and 2. Similar statements can be made about variants 1 and 3 or 2 and 3.
The Crystallographic Theory of Martensite describes low energy interfaces between a twinned laminate (blue and green in Figure 3(b)) and the austenite (red in Figure 3(b)). Supposing that the blue and green are variants 1 and 2, and that equation (3) has been solved so that , c, d are known. The deformation of this twinned laminate is described on average by a linear transformation given by By saying "on average" it is meant that the deformation of the twinned laminate (measured from the cubic phase) is described by the linear transformation Hf except for little zig-zags due to the twins; these zig-zags get smaller and smaller as the twin spacing gets finer and finer, so the deformation of the laminate gets closer and closer to the linear transformation described by (4).
In this process of making the twins finer and finer, the volume fraction f of blue vs. green is held constant. With this physical picture in mind it is clear that, if Hf is compatible with I (i.e., the undistorted cubic phase) the elastic energy in the transition layer between austenite and martensite can be made arbitrarily small by making the twins finer and finer. The mathematical condition for this to occur is again a "middle eigenvalue condition", that is, the middle eigenvalue of $ = % (5) is 1. Here T denotes the transpose. So the problem of the Crystallographic Theory is to find a volume fraction f such that the middle eigenvalue of the positive-definite, symmetric tensor Gf is 1.
To find f such that the middle eigenvalue of Gf is equal to 1, the zeros of det(Gf −I) have to be found. Gf is clearly quadratic in f and the determinant is a cubic polynomial, so it looks like det*$ − + = 0 is a 6 th order polynomial equation. But underlying symmetries in the problem 25 imply that it is in fact only quadratic, and, moreover, it is symmetric about f = 1/2. The q(f) of the paper is this quadratic function, i.e., The condition (CC2) is the condition that