Domain-Wall Skyrmion and Anti-Domain-Wall Skyrmion Pairs

• Domain-wall–skyrmion–anti-domain-wall–skyrmion pairs are composite topological defects formed by skyrmions and anti-skyrmions bound to intersecting domain walls in chiral magnets.
• Their formation and stability are governed by exchange, anisotropy, and Dzyaloshinskii–Moriya interactions, analyzed via micromagnetic and sigma-model soliton frameworks.
• Insights from their controlled nucleation and dynamics offer practical avenues for designing advanced spintronic memory and logic devices.

A domain-wall–skyrmion–anti-domain-wall–skyrmion pair consists of two topological spin defects (skyrmion and anti-skyrmion) trapped at the intersection or juxtaposition of a domain wall and an anti-domain wall in a chiral magnet or related spin system. The physics of these pairs is governed by the competition between exchange, anisotropy, and Dzyaloshinskii–Moriya interactions (DMI), and their stability, formation, and dynamics are parameterized via micromagnetic or field-theoretic soliton models. These composite structures are pivotal for understanding fundamental topological transitions and for engineering spintronic memory and logic architectures that exploit nontrivial defect-bound states.

1. Micromagnetic Framework and Soliton Definitions

Chiral magnets hosting skyrmions and domain walls are described by a two-dimensional micromagnetic free-energy density,

$$E = A \int (\nabla \mathbf{m})^2 \,d^2r - D \int \mathbf{m}\cdot (\nabla \times \mathbf{m}) \,d^2r - K \int (m_z)^2 \,d^2r - \mu_0 M_s \int \mathbf{H}\cdot \mathbf{m}\,d^2r,$$

where $\mathbf{m}(x,y)$ is the unit magnetization, $A$ is exchange stiffness, $D$ is Dresselhaus-type DMI, $K$ is out-of-plane easy-axis anisotropy, $M_s$ the saturation magnetization, and $\mathbf{H}$ an externally applied field (critical for manipulating the domain-wall internal phase $\alpha$) (Gudnason et al., 2024).

A straight domain wall, with internal U(1) phase $\alpha$, binds to the ground state $\alpha=\pi/2$ (Bloch wall configuration). Skyrmions are characterized by a localized swirl in $\mathbf{m}(x,y)$0 carrying a topological charge $\mathbf{m}(x,y)$1, while merons, as half-skyrmions, account for $\mathbf{m}(x,y)$2 charge density.

The effective DMI coupling $\mathbf{m}(x,y)$3 controls the balance of wall width and stability, and is typically $\mathbf{m}(x,y)$4 for thin-film heterostructures such as Pt(Co/Ni)/Ir.

2. Mechanisms of Pair Creation

The controlled creation or annihilation of skyrmions and anti-skyrmions at domain walls depends on the manipulation of the domain-wall internal phase $\mathbf{m}(x,y)$5 and the relative separation $\mathbf{m}(x,y)$6 between the skyrmion and the wall. Applying in-plane magnetic fields ($\mathbf{m}(x,y)$7) enables the wall phase $\mathbf{m}(x,y)$8 to be tuned away from the ground state, concretely transforming the skyrmion–wall interaction from repulsive to attractive.

At the antipodal wall phase ($\mathbf{m}(x,y)$9), approaching skyrmions nucleate a meron–antimeron pair on the wall. The subsequent relaxation coalesces these half-charges into two full skyrmions, or leads to the formation of a metastable anti-domain-wall–skyrmion bound state before detaching as bulk skyrmions (Gudnason et al., 2024).

A key “nucleation” condition for defect pair formation is

$A$0

with $A$1 the meron-pair creation energy barrier, and $A$2 the domain-wall width (Gudnason et al., 2024).

In the Landau–Lifshitz–Gilbert formalism, a dynamic “Kibble–Zurek” quench at the unstable wall phase causes the wall’s phase to spontaneously break symmetry, nucleating pairs of domain-wall skyrmions and anti-domain-wall skyrmions (Gudnason et al., 26 Dec 2025).

3. Model-Specific Pair Structures

In the analytic O(3) sigma-model with DMI, domain-wall skyrmions are constructed as sine-Gordon kinks confined to the wall. The effective low-energy action on the wall is

$A$3

where $A$4 and $A$5 emerge from the moduli approximation, with kink solutions corresponding to confined skyrmion and anti-skyrmion pairs (Ross et al., 2022).

For two well-separated walls (domain and anti-domain), superposition of stereographic coordinates $A$6 encodes a pair of kinks (skyrmion and anti-skyrmion) localized on each wall. Their interaction energy decays exponentially with separation, and both signs of kink are equally stable on the wall, contrary to the bulk where chirality selects only one topological charge (Ross et al., 2022).

Numerical simulations confirm that a chain of alternating domain-wall–skyrmion and anti-domain-wall–anti-skyrmion structures is locally stable in the ferromagnetic phase, with stability lost near the transition to the chiral soliton lattice (CSL), where the wall cusp height diverges and pair unbinding occurs (Amari et al., 2023).

4. Numerical Results and Stability Regimes

Micromagnetic and field-theoretic simulations employ grids with $A$7 nm mesh and physically realistic parameters: $A$8 J/m, $A$9 J/m$D$0, $D$1 J/m$D$2 (Gudnason et al., 2024, Gudnason et al., 26 Dec 2025). The domain-wall width $D$3 nm and bulk skyrmion radius $D$4.

Characteristic outcomes plotted in $D$5 phase diagrams distinguish regimes of

• repulsion (bulk skyrmion escapes),
• capture (DW–skyrmion bound state),
• annihilation (skyrmion destroyed by wall),
• meron-pair creation leading to two bulk skyrmions,
• transient or metastable skyrmion–anti-domain-wall–skyrmion pairs (Gudnason et al., 2024).

Metastability is ensured when defect separation greatly exceeds the soliton width; in open boundary conditions, pairs can be long-lived under LLG dynamics with realistic damping ($D$6) (Gudnason et al., 26 Dec 2025).

5. Energy Barriers, Forces, and Interaction Laws

The far-field skyrmion–wall interaction energy is

$D$7

with $D$8 for $D$9. The sign of $K$0 determines repulsion or attraction, with critical attractive force needed for nucleation in the $K$1–$K$2nm range (Gudnason et al., 2024).

At the CSL phase boundary ($K$3), domain-wall cusps diverge and isolated skyrmions unbind; stable pairs require $K$4 (Amari et al., 2023). In sigma-model language, both $K$5 solitons are stable on a wall, but stability flips in the bulk (Ross et al., 2022).

6. Variants, Multilayer and Network Realizations

Composite skyrmion–anti-skyrmion pairs are generalizable to multilayer synthetic antiferromagnets (SAF). Dual domain walls in Co/Ru/Co, with alternating Néel and transverse types, host bi-meronic defects (bi-vortex/bi-antivortex pairs) at their junctions. These can coalesce into composite skyrmions or skyrmion–anti-skyrmion “molecules,” modifiable by applied magnetic fields. Each composite-defect event releases quantized energy into the wall, causing discrete jumps and enhanced wall mobility (Kolesnikov et al., 2017).

Networked domain-wall architectures, engineered with multiple segments and controlled phase configurations, facilitate robust and tunable creation of skyrmion–anti-skyrmion pairs, as observed in Y-junction and multi-wall collision simulations (Winyard, 2015).

7. Potential Applications in Spintronics

The ability to nucleate, annihilate, and manipulate domain-wall–skyrmion–anti-domain-wall–skyrmion pairs underpins advanced spintronic strategies:

• "Racetrack" memory devices benefit from encoding composite skyrmion pair bits, extending information density beyond single-skyrmion architectures (Gudnason et al., 2024).
• Spin-logic gates can exploit controlled meron-pair injection and domain-wall driven annihilation for "write" and "erase" cycle engineering.
• Locally tunable wall phase or current-driven approaches enable reconfigurable nanomagnetic networks with selective defect manipulation capabilities (Gudnason et al., 2024, Gudnason et al., 26 Dec 2025).

The demonstrated stability and dynamic control of these composite objects via external fields and domain-wall engineering suggest broad utility in nonvolatile memory, logic, and programmable magnonic circuits.

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Tables: Representative parameters and stability criteria

Material System	DMI Strength ($K$6)	DW Width ($K$7)	Skyrmion Radius ($K$8)	Key Phase Boundaries
Pt(Co/Ni)/Ir (thin film)	0.4	6.3 nm	5.8 nm	$K$9, $M_s$0
Co/Ru/Co (SAF)	negligible	500–600 nm	—	$M_s$1, $M_s$2
Chiral FM (sigma model)	—	—	—	$M_s$3

Parameters are material-specific and tunable via film thickness, alloy composition, and external field orientation.

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Domain-wall–skyrmion–anti-domain-wall–skyrmion pairs constitute a rich, analytically tractable, and numerically validated class of topological defects in chiral magnets and related systems, with stability and dynamics dictated by a controllable set of micromagnetic and sigma-model parameters. Their integration into spintronic architectures is supported by recent experimental and theoretical results (Gudnason et al., 2024, Gudnason et al., 26 Dec 2025, Amari et al., 2023, Winyard, 2015, Kolesnikov et al., 2017, Ross et al., 2022).