Reconsidering the origins of Forsbergh birefringence patterns

A. Schilling, A. Kumar, R. G. P. McQuaid, A. M. Glazer, P. A. Thomas, and J. M. Gregg

Phys. Rev. B 94, 024109 – Published 12 July 2016

Abstract

In 1949, Forsbergh, Jr. reported spontaneous spatial ordering in the birefringence patterns seen in flux-grown $BaTi O_{3}$ crystals under the transmission polarized light microscope [Phys. Rev. 76, 1187 (1949)]. Stunningly regular square-net arrays were often only found within a finite temperature window and could be induced on both heating and cooling, suggesting genuine thermodynamic stability. At the time, Forsbergh rationalized the patterns to have resulted from the impingement of ferroelastic domains, creating a complex tessellation of variously shaped domain packets. However, no direct evidence for the intricate microstructural arrangement proposed by Forsbergh has subsequently been found. Moreover, there are no robust thermodynamic arguments to explain the finite region of thermal stability, its occurrence just below the Curie temperature, and the apparent increase in entropy associated with the loss of the Forsbergh pattern on cooling. Despite decades of research on ferroelectrics, this ordering phenomenon and its thermodynamic origin have hence remained a mystery. In this paper, we reexamine the microstructure of flux-grown $BaTi O_{3}$ crystals, which show Forsbergh birefringence patterns. Given an absence of any obvious arrays of domain polyhedra or even regular shapes of domain packets, we suggest an alternative origin for the Forsbergh pattern in which sheets of orthogonally oriented ferroelastic stripe domains simply overlay one another. We show explicitly that the Forsbergh birefringence pattern occurs if the periodicity of the stripe domains is above a critical value. Moreover, by considering well-established semiempirical models, we show that the significant domain coarsening needed to generate the Forsbergh birefringence is fully expected in a finite window below the Curie temperature. We hence present a much more straightforward rationalization of the Forsbergh pattern than that originally proposed in which exotic thermodynamic arguments are unnecessary.

Received 19 February 2016
Revised 27 April 2016

DOI:https://doi.org/10.1103/PhysRevB.94.024109

Physics Subject Headings (PhySH)

Birefringence Ferroelasticity Magnetic domains Microstructure

Ferroelectrics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

A. Schilling¹, A. Kumar¹, R. G. P. McQuaid¹, A. M. Glazer^2,3, P. A. Thomas³, and J. M. Gregg^1,*

¹Centre for Nanostructured Media, School of Mathematics and Physics, Queen's University Belfast, Belfast, BT7 1NN, United Kingdom
²Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
³Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom

^*Corresponding author: m.gregg@qub.ac.uk

X-ray white beam topography of self-organized domains in flux-grown $BaTi O_{3}$ single crystals

D. Walker, A. M. Glazer, S. Gorfman, J. Baruchel, P. Pernot, R. T. Kluender, F. Masiello, C. DeVreugd, and P. A. Thomas

Phys. Rev. B 94, 024110 (2016)

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Vol. 94, Iss. 2 — 1 July 2016

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Images

Figure 1
Examination of a flux-grown $BaTi O_{3}$ crystal, using the METRIPOL microscope, shows the spontaneous formation of the square-net pattern reported by Forsbergh [23], Matthias and von Hippel [24], and Lambert et al. [28]. On heating, the periodic array appears spontaneously at around $111^{\circ} C$ and disappears at the Curie temperature of approximately $120^{\circ} C$ (as determined from the collapse in birefringence). The images show the orientation $ϕ$ (top panels) and $| sin δ |$ (bottom panels) distributions at three temperatures. The crystal thickness is on the order of 300 μm.
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Figure 2
Room-temperature lateral PFM imaging of the surface of a region of a flux-grown $BaTi O_{3}$ crystal in which the Forsbergh birefringence pattern is found [amplitude (a) and phase (b)] shows evidence for packets of domains. However, they are not regularly distributed and do not obviously correspond to the specific polyhedral shapes envisaged by Forsbergh [23]. For completeness, the inset shows the orientation of the cantilever axis, and the arrows indicate the directions of sensitivity to in-plane polarization components. The crystal surface is ${001}_{pseudocubic (pc)}$ .
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Figure 3
As noted by Forsbergh [23], square-net birefringence patterns only occur where perpendicular sets of $a - c$ ferroelastic stripe domains meet as can be seen in (a). The $a - c$ domain contrast is indicated by line features parallel to the ${〈 100 〉}_{pc}$ directions (highlighted by fine blue arrows) consistent with lines of intersection of ${(101)}_{pc}$ and ${(011)}_{pc} a - c$ ferroelastic domain walls with the ${(001)}_{pc} BaTi O_{3}$ surface. The ${110}_{pc} a_{1} - a_{2}$ domain walls would show stripe contrast features parallel to ${〈 110 〉}_{pc}$ directions on the same ${(001)}_{pc}$ surface and so can be discounted as responsible for the contrast. Where the perpendicular sets of lines meet, a square feature in birefringence can be seen. This transmission optical image was taken approximately $2^{\circ} C$ below $T_{C}$ (crystal temperature estimated as $118^{\circ} C$ ) under crossed polars using a white-light source. Rather than consider hypothetical structures resulting from the impingement of $a - c$ domain packets as responsible for the Forsbergh birefringence, an alternative microstructure was considered theoretically, in which two slabs (labeled slab 1 and slab 2) of perpendicularly oriented $a - c$ domains are simply superposed (b). In this schematic, the orientation of polarization within each domain in the slabs is indicated by the coarse squat blue arrows.
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Figure 4
The optical retardation $(Δ)$ through each slab of $a - c$ stripe domains will start to show spatial modulation when the ratio of domain periodicity $(ω)$ to slab thickness ( $d$ ) is greater than or equal to $\sqrt{2} / 2$ . The amplitude of the optical retardation will locally reach a maximum when $ω / d \geq \sqrt{2}$ . The case where $ω / d = \sqrt{2}$ is considered schematically in (a) with a cross section of the $a - c$ stripe domains within a single slab (top panel) and the corresponding relative optical retardation $(Δ / Δ_{\max})$ variation as a function of the ratio of distance $x$ divided by domain period $ω$ [bottom panel in (a)]. When two such slabs of $a - c$ domains are placed one above the other and at $90^{\circ}$ to each other, the spatial variation in retardation is such that the Forsbergh pattern is reproduced when viewed from above (b). Here, dark blue corresponds to the minimum relative retardation (0), and dark red corresponds to the maximum relative retardation (1) as indicated on the color scale. Even if the slabs are not of equal thickness, square-net birefringence still occurs: As an illustration, in (c) the thickness of one slab is kept the same as that considered in (a), but the thickness of the second slab is increased by a factor of 1.5.
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Figure 5
STEM image of a single crystal $BaTi O_{3}$ pillar (beam parallel to the ${〈 100 〉}_{pc}$ direction), similar to those published by Schilling et al. in Ref. [33] and fabricated by focused ion-beam (FIB) patterning in the same manner as described in Ref. [33] (a); the image illustrates the way in which ferroelastic domain periodicity depends on the local dimensions of the domain packet containing the stripe domains: Note how the domain periodicity decreases as the packet dimensions become progressively more constrained towards the points of the triangles in individual domain packets. For this image, the local domain period $(ω)$ has been examined as a function of the local domain packet width ( $d$ ), and the Landau-Kittel scaling relationship was found to hold reasonably well (b). By considering the way in which $ω / d$ varies with temperature, using Eq. (6) and literature values for Young's modulus, spontaneous strain, and domain wall energy (c), it can be seen that the $ω / d$ ratio in slabs of $a - c$ stripe domains only exceeds the two critical values of $\sqrt{2} / 2$ and $\sqrt{2}$ in a narrow region just below $T_{C}$ . The width of this region grows as the slab thickness decreases. However it is much narrower than that observed experimentally in Fig. 1, for example. Recalculating the expected $ω / d$ values using measurements by Walker et al. in Ref. [41] for the specific crystals studied herein (d), shows that the sudden collapse in spontaneous strain evident from x-ray diffraction causes a dramatic increase in the window of stability of the Forsbergh patterns (visible when the $ω / d$ ratio exceeds $\sqrt{2} / 2$ ).
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