Controlling Skyrmion Lattices via Strain: Elongation, Tilting, and Collapse Mechanisms

TL;DR

This work addresses how three-dimensional skyrmion strings respond to uniaxial strain by modifying the Dzyaloshinskii-Moriya interaction (DMI). An analytical framework shows strain-induced anisotropy in DMI, predicting elongation and bidirectional tilting with $k^2(\eta)$ scaling, including a critical point $\eta_c \approx 0.2929$ where string rupture and lattice collapse occur; this is corroborated by micromagnetic simulations solving the LLG equation with thermal noise, revealing multi-domain to single-domain tilt transitions and bobber-mediated rupture into a conical phase, with a temperature-dependent first-to-second-order-like collapse near $T_c$. Experimental validation via in situ LTEM on ${\mathrm{Co}}_{8}{\mathrm{Zn}}_{8.5}{\mathrm{Mn}}_{3.5}$ confirms strain-induced elongation and collapse to a conical phase with anti-cluster formation, supporting strain-modulated DMI as the dominant mechanism over magnetocrystalline anisotropy. Overall, the study establishes a low-energy, strain-guided route to manipulate 3D topological spin textures, offering a pathway toward strain-engineered spintronic devices.

Abstract

This study establishes a comprehensive framework for the three-dimensional strain control of magnetic skyrmion strings. We integrate analytical modeling, micromagnetic simulations, and \textit{in situ} Lorentz transmission electron microscopy experiments to demonstrate that externally applied strain is a powerful stimuli for manipulating three-dimensional magnetic skyrmion strings. Analytical models predict that strain induces both elongation and bidirectional tilting of skyrmion strings in bulk systems, a finding corroborated by numerical simulations. These simulations further reveal that strain drives the system from fragmented multi-domain states toward unified single-domain configurations and facilitates skyrmion string rupture via bobber formation at critical strain levels. The collapse of the skyrmion lattice exhibits a temperature-dependent character, shifting from first-order to second-order behavior near the critical temperature $T_c$. Reducing sample thickness significantly increases the critical strain required for annihilation due to the suppression of tilting. Experimental validation on a $\text{Co}_8\text{Zn}_{8.5}\text{Mn}_{3.5}$ sample confirms strain-induced elongation and subsequent collapse into a conical phase via anti-cluster formation, directly implicating strain-modulated Dzyaloshinskii-Moriya interaction (DMI) as the primary mechanism in this system, over magnetocrystalline anisotropy. These findings provide a mechanistic understanding of strain-mediated control in three-dimensional magnetic systems, demonstrating its feasibility for energy-efficient spintronic applications.

Figures (6)

• Figure 1: Analytically derived relationship between the square of slope $k^2$ and the effective strain $\eta$ (solid black line). This relationship demonstrates linearity for small values of $\eta$ (dotted red line), yet exhibits a transition to inverse behavior as $\eta$ approaches the singular point $\eta=1-\sqrt{2}/2\approx0.3$ (dashed blue line). The inset provides an enlarged view of the small-$k^2$ region, which highlights the linear regime.
• Figure 2: Three-dimensional micromagnetic simulations of skyrmion string dynamics under uniaxial stress at $T=0.02T_\text{C}$ and $H=0.55H_D$. The thickness of the sample $L_z=1.5L_D$ for (a--d), $L_z=0.83L_D$ for (e--f). Skyrmion cores (where $m_z < -0.9$) are rendered as tubes. (a--c) Strain-dependent 3D configurations in a thick sample ($L_z>L_D$): (a) $\eta=0$ (vertical alignment), (b) $\eta=0.15$ (opposing-direction tilting with domain formation), (c) $\eta=0.48$ (merged uniform tilt). A preference for opposing tilt directions drives Ising-like domain formation in (a--b), which transitions to a uniformly ordered tilt at $\eta=0.48$ in (c). (d) Individual string evolution: initial tilt $\rightarrow$ rupture $\rightarrow$ surface annihilation; arrows indicate in-plane magnetization directions in the shell region ($m_z=0$). (e--f) Strain-dependent 3D configurations in a thin sample ($L_z<L_D$): (e) $\eta=0.48$ [tilting is significantly suppressed compared to (c)], (f) $\eta=0.95$ (weak tilting remains visible).
• Figure 3: Evolution of the in-plane magnetic structure under uniaxial stress from micromagnetic simulations, conducted at temperature $T = 0.6T_c$, magnetic field $H = 0.55H_D$, and sample thickness $L_z = 1.5L_D$. Strain values are $\eta = 0$ (a), $\eta = 0.40$ (b), $\eta = 0.56$ (c), and $\eta = 0.61$ (d) and (e). Panels (a)--(d) show cross-sectional views in the $x$-$y$ plane at the midplane ($z = L_z/2$), viewed along $z$. Panel (e) shows the $y$-$z$ cross-section viewed along $-x$. (a)-(d) are colored by the $z$-component of magnetization $m_z$; in (e), color represents the $x$-component $m_x$. Arrows indicate spin orientation. Insets in (a)--(c) show FFT analyses of the corresponding skyrmion lattice structures; yellow lines mark the two crystallographic directions closest to perpendicular to the strain. Panels (f)--(i) show the $z$-dependence of the in-plane average magnetization components $\langle m_x \rangle$ and $\langle m_y \rangle$, corresponding to the structures in (a)--(d), respectively. The sinusoidal magnetization profiles in (h) and (i), consistent with the spin structure in (g), confirm the formation of a conical phase.
• Figure 4: The topological number $N_{sk}$(serves as an indicator of the skyrmion count) as a function of effective strain $\eta$ across various temperatures. A rapid decrease in $N_{sk}$ marks the onset of skyrmion annihilation. Notably, at lower temperatures (e.g., $T=0.52T_\text{C}$), this transition is sharp and abrupt, resembling a first-order transition. In contrast, at higher temperatures (e.g., $T=0.96T_\text{C}$), the transition becomes smoother, characteristic of a second-order-like phenomenon.
• Figure 5: Magnetic structures and phase diagram in a $\sim$190-nm-thick (001)$\text{Co}_8\text{Zn}_{8.5}\text{Mn}_{3.5}$ thin plate with applied strain along the [110] direction at room temperature (293 K). (a) Skyrmion fraction $N_f$ as a function of magnetic field, defined as $N_f = N(B)/N_\text{max} \times 100\%$, where $N(B)$ is the number of skyrmions at field $B$ and $N_\text{max}=N$($B$=70mT) is the maximum observed number. Both the vertical position and color represent $N_f$. (b) Underfocused ($-2$ mm) LTEM image of randomly oriented helical spin structure at zero field. (c) Skyrmion lattice structure at 70 mT. (d) Saturated ferromagnetic phase with no contrast at 145 mT.