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Lu et al. Microstructures 2023;3:2023033 https://dx.doi.org/10.20517/microstructures.2023.28 Page 5 of 10

Figure 2. Dependence of the kink-induced wall tilt and kink self-energy on the sample thickness Δ. (A) Strain distribution induced by the
kink for a sample with a thickness Δ of 101 l.u. (B) Bending of the local lattice with tilt angles θ and θ . The green and blue lines indicate
1
2
the bottom and top surfaces, respectively. (C) The variation of macroscopic tilt angles as a function of sample thickness Δ. The data
c
points in (C) are fitted by θ = a + b × Δ with a ~ 0 for θ and θ , b = 2.246 for θ and b = -2.246 for θ , and c = -1 for θ and θ . The fitted
1
1
2
2
1
2
line in (C) shows the scaling exponent of -1 between the sample thickness and the tilt. (D) Relationship between kink self-energy and
B
sample thickness Δ. The data points in (D) are fitted by E = E - A × Δ with E = 0.17 eV, A = 6 eV l.u. and B = -1. The fitted line in
kink core core
(D) shows a scaling of ~1/Δ while logarithmic scaling equally fits the data for thicknesses > 1,000 l.u.

Figure 3. Interaction energies of kink-kink configurations residing inside two parallel walls as a function of the wall-wall distance d. (A)
Interaction energy on logarithmic scales with the fitted scaling exponents. (B) Scaling exponents as a function of sample sizes. The
-2 -1
thickness scaling changes from d for thin samples to d for thick samples.
size increases, the scaling exponents decrease and “monopolar” interactions with an exponent of -1 are
reached when the sample size is larger than 1,001 l.u. This “monopolar” wall-wall interaction is consistent
with previous theoretical predictions . Our energy scaling for the sample thickness reveals a wide
[42]
crossover regimen, which had not been found before, that occurs near thicknesses of 1,000 l.u [Figure 3B].
To explore further the physical processes leading to this crossover, lattice profiles [Figure 4A-D] and strain