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Page 4 of 10 Lu et al. Microstructures 2023;3:2023033 https://dx.doi.org/10.20517/microstructures.2023.28

Figure 1. Domain wall configurations for kink interactions. (A) Domain wall configurations containing one or two kinks. The structural
details of the atomic kinks, kink pairs and strain fields near the kinks are shown in (B and C). The strain maps in (A-C) are colour-coded
by the atomic-level strain: ɛ . Green dashed lines in (A) indicate the position of domain walls. Red solid lines in (A) indicate the atomic
xx
steps of the kink structures.

We then calculate the self-energy of kinks in domain walls with different sample thicknesses. The self-
energy of kinks is calculated as the energy difference between samples with and without kinks inside the
wall. The self-energy is dependent on the sample thickness, as shown in Figure 2D. In analogy with
dislocations, the self-energy of kinks consists of two components, namely the core energy of the kink and
the elastic energy around the core. The core energy is found to be 0.17 eV, similar to values found for kinks
[41]
in dislocations in silicon (~0.12 eV) . The evolution of the total energy with the sample thickness Δ follows
a logarithmic size dependence for thicknesses > 1,000 l.u [Figure 2D]. At lower distances, the energy is lower
and follows a power law E = E - A/Δ with E = 0.17 eV and A = 6 eV l.u. The kink energy per unit cell
core
kink
core
is defined as E divided by the total number of unit cells in the sample and decays with a power law
kink
dependence from 10 eV/unit cell in the smallest sample with Δ = 150 l.u. to 2 × 10 eV/unit cell for a
-7
-8
sample with a thickness of Δ = 1,601 l.u.
The size dependence of kink-kink and kink-antikink pair interactions is then investigated. Two parallel
domain walls with one kink or one antikink in each wall are created [Figure 1A and C]. The two walls were
initialised at symmetric positions with respect to centre of the sample, with a kink (or an antikink) at the
centre of each wall. The samples range from small (L = 201 l.u., L = 200 l.u.) to large sizes (L = 1,601 l.u., L
x
y
x
y
= 1,600 l.u.). To calculate the interaction energy of the kink-kink configuration, the total potential energy is
reduced by the potential energy of two noninteracting kinks E kink-kink = E - 2E . Figure 3 shows the
kink
total
sample size dependence of kink-kink interactions. For small sample sizes, the kink-kink interaction energies
(black symbols and fitted lines in Supplementary Figure 2) are similar to Lu et al., and the wall-wall
interaction shows an atypical “dipolar” character with a scaling exponent of -2 [Figure 3A] . As the sample
[30]