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Page 2 of 10 Lu et al. Microstructures 2023;3:2023033 https://dx.doi.org/10.20517/microstructures.2023.28

INTRODUCTION
Ferroelectric and ferroelastic materials spontaneously split into domains where the order parameter
(spontaneous polarisation or spontaneous strain, respectively) is uniform. The boundaries between these
domains are called domain walls. They move in response to an applied external field (electric field or stress
[1-6]
field) . They also exhibit emergent properties that do not exist in bulk, such as a spontaneous polarisation
[7]
in ferroelastics or anomalous electrical conductivity in ferroelectrics [8-11] .
Landau theory predicts that domain walls are smooth with a hyperbolic tangent profile. However,
experimental observations demonstrate a different behaviour, showing that domain walls can exhibit
complex profiles with meanders and atomic steps called kinks [12-15] . This phenomenon is particularly evident
[19]
[16]
in various materials such as membranes , ceramics [17,18] and thin films . This internal structure has direct
consequences on the emergent properties of domain walls. In lithium niobate, scanning transmission
electron microscopy images on a lamella cut from a bulk single crystal revealed kinks and antikinks (i.e.,
atomic steps in the opposite direction compared to kinks) at domain walls. It was proposed that these kinks
and antikinks lead to localised electric charges that influence the dielectric response of the material .
[20]

Kinks are also essential elements that form during switching under an applied external field [21-23] . Clear
experimental evidence supports the notion that kink formation and local bending constitute the first stage
[24]
of domain wall motion . As such, the collective motion of domain walls in avalanche-like processes is
triggered by kinks . Kinks seem to play an even more important role in switching along non-polar
[25]
directions or in non-polar materials. In lithium niobate, switching on non-polar cuts is governed by the
generation and propagation of charged kinks . Simulations also indicate that kink movements dominate
[13]
[26]
the switching mechanism of polar domain walls in non-polar ferroelastic . In addition, the movement of
kinks has been predicted to be supersonic, opening possibilities for materials applications at GHz
[27]
frequencies .

In samples with high densities of domain walls, interactions between walls are governed by their
junctions [28,29] . In simpler but not uncommon configurations of parallel domain walls separated by distances
larger than the boundary thickness, the interactions may arise from kinks that are known to lead to
[30]
enhanced areas of strain . For nanoscopic sizes, the interaction energy between domain walls was found to
[18]
2
[30]
be dipolar, i.e., to decay as 1/d where d is the distance between the walls , which agrees well with the
energy decay of surface steps [31,32] . For large samples, the interaction energy was found to decay as 1/d .
[33]
In this work, using reasonable interatomic potentials, we show that a crossover between “dipolar” 1/d
2
interactions and “monopolar” 1/d interactions in free-standing samples occurs at lateral sizes in the order of
1,000 l.u. (l.u. = lattice unit), which corresponds to around 0.5 μm for a unit cell parameter of 0.5 nm. This
behaviour changes greatly if the sample is clamped on one side, such as in thin films, since we find that the
kink-kink interactions stay monopolar even for very thin films (~ 100 l.u.). Our findings are important for
understanding the organisation of domain walls and their response to an applied external field in
membranes, transmission electron microscopy lamellae, ceramics, and thin films.

METHODS
Ferroelastic domains and confined atomic kinks residing inside domain walls are described by a model for
[34]
ferroelastic transitions based on a Landau-type double-well potential , as schematically shown in
Supplementary Figure 1. The potential energy U(r) contains three terms: the harmonic first nearest atomic
2
interactions U(r) = 20(r - 1) (black springs), the anharmonic second-nearest interactions U(r) = -25(r - √2)
2
4
+ 20,000(r - √2) (yellow springs) along diagonals in the lattice unit and the fourth-order third-nearest