Generic skyrmion phase diagram in ferrimagnetic films

The gist The authors investigate how variations in inter-sublattice exchange coupling govern relaxed stable and metastable ferrimagnetic skyrmion configurations and introduce a dimensionless parameter ζe f f to characterize the crossover between strong and weak inter-sublattice locking in ferrimagnetic systems.

Model and Methodology

The study considers an ultra-thin chiral FiM film of thickness d in the xy-plane, consisting of two antiferromagnetically coupled sublattices with unit vectors m⃗ 1 and m⃗ 2, where the total energy is given by Eq. (1) The effective parameters emerge as thickness-weighted averages Ae f f = (A1t1 + A2t2)/d and De f f = (D1t1 + D2t2)/d, and the dimensionless ratios al = (Al tl)/(Ae f f d) and dl = (Dl tl)/(De f f d) quantify the relative exchange stiffness and DMI strength of each sublattice with respect to the effective FiM values The stationary magnetization configurations satisfy Eq. (3).

Separation of Strong and Weak Inter-sublattice Coupling Regimes

The crossover between strong and weak coupling regimes is characterized by the normalized order parameter ∆rms = p m⃗ 1 + m⃗ 22 / √2. In the strong-coupling limit, ∆rms ≈ 0, reflecting coherent locking of the sublattices. As intersublattice exchange coupling is reduced, this locking progressively weakens, leading to a breakdown of collective behavior and allowing the sublattices to relax independently according to their own material parameters. This crossover is marked by a rapid increase of ∆rms beyond a critical value of ζe f f, signaling the breakdown of inter-sublattice locking.

Strong Coupling Regime and Ferrimagnetic Skyrmions

In the strong inter-sublattice coupling regime, the two antiferromagnetically coupled sublattices remain locked, forming FiM skyrmions that behave as a single effective magnetic structure. Under strong coupling, the total energy reduces to the sum of the two sublattice energies evaluated for a common spin texture, and the effective perpendicular anisotropy is given by Ke f f = K1e f f + K2e f f. The stability in this regime is governed solely by κe f f, with the critical condition remaining κe e f = 1 separating isolated and condensed skyrmion phases.

Weak Coupling Regime and Breakdown of Locking

In the weak inter-sublattice coupling limit, the AFM exchange is insufficient to rigidly lock the two sublattices, and the magnetizations m⃗ 1 and m⃗ 2 relax largely independently. This regime is characterized by a small parameter ε ≡ 1/ζe f f ≪ 1, which controls the strength of intersublattice locking. In this limit, skyrmion (meta)stability and phase boundaries are governed solely by the intrinsic control parameter κl of each sublattice, and the critical condition κl = 1 separating the isolated and condensed phases.

Skyrmion Stabilization in a DMI-free Sublattice by Inter-sublattice Locking

The paper demonstrates that skyrmions can be stabilized even when one sublattice is DMI-free (D2 = 0) while the other sublattice retains finite DMI (D1, 0), provided the inter-sublattice exchange coupling is sufficiently strong. In the strong-coupling regime, this effective description provides a stabilization mechanism enabled by inter-sublattice locking. However, as the coupling weakens and the system enters the weak-coupling regime, this effective description breaks down because the DMI-free sublattice lacks intrinsic chiral interactions and therefore can no longer sustain a skyrmion, leading to its collapse.

Conclusion

The analysis demonstrates that the crossover between strong and weak coupling regimes is governed primarily by ζe f f, and κle f f primarily governs the skyrmion phase stability while determining where the crossover occurs together with al and dl. The findings provide a unified framework linking inter-sublattice exchange to skyrmion phase stability and offer practical design principles for coupling-engineered spintronic devices based on FiM systems.

How it works

1. In the strong inter-sublattice coupling regime, the system behaves collectively, and the effective perpendicular anisotropy is given by Ke f f = K1e f f + K2e f f.

2. The critical condition separating isolated and condensed skyrmion phases in this regime is κe e f = 1, which allows the construction of a phase diagram in the κ1e e f κ2e e f-plane.

3. In the weak-coupling regime, each sublattice independently satisfies its own metastability condition governed by its intrinsic control parameter κl, with the critical condition being κl = 1 separating isolated and condensed phases.

4. Skyrmions can be stabilized in a DMI-free sublattice if the inter-sublattice exchange coupling is strong enough to provide an effective DMI strength De f f = (D1t1 + D2t2)/d.

Summary of Key Findings

The generic skyrmion phase diagram in ferrimagnetic films

This paper establishes a unified framework linking inter-sublattice exchange and skyrmion phase stability in ferrimagnetic systems. It shows that the crossover between strong and weak coupling regimes is governed primarily by ζe f f, and κle f f primarily governs the skyrmion phase stability while determining where the crossover occurs together with al and dl. The effective description in the strong-coupling regime simplifies to a chiral ferromagnet governed by renormalized coefficients leading to a single effective equation identical in structure to that of a chiral ferromagnet, but with parameters emerging from thickness-weighted averaging, κe f f that separate isolated (κe f f 1) skyrmion phases. In the weak-coupling regime, the two sublattices are governed independently by their intrinsic control parameters κl = 1 separating isolated (κl 1) skyrmion phases. The authors further demonstrate that skyrmions can be stabilized in a DMI-free sublattice even when the other sublattice has finite DMI under strong inter-sublattice exchange locking.

How it works

The transition between isolated and condensed skyrmion phases is governed by the single parameter κe f f = 1 in the strong-coupling regime. In the weak-coupling regime, the lines κ1 = 1 and κ2 = 1 separate condensed (κl > 1) and isolated (κl < 1) skyrmions in sublattices 1 and 2, respectively. The effective length scale Le f f = (4Ae f f)/(πDe f f) determines the size of the (meta)stable skyrmions in both regimes.

This paper provides a complete and general description for FiM films at zero temperature by presenting phase diagrams in the κ1e f f κ2e f f-plane for the strong-coupling regime and in the κ1κ2-plane for the weak-coupling regime. The analysis demonstrates that thickness does not act as an independent control parameter for the phase transition itself, but rather modifies the effective material parameters entering κe f f. The findings offer a clear and unified framework to understand and organize both experimental observations and numerical results.

1. Improved AI systems can accurately predict skyrmion phase stability in ferrimagnetic films by determining if a system is in a strong-coupling regime or weak-coupling regime based on the effective inter-sublattice coupling strength ζe f f, as this parameter controls the crossover.

2. The AI can calculate whether a DMI-free sublattice will sustain a skyrmion by assessing the condition derived from Section IV: In the strong-coupling regime, even when one sublattice is DMI-free (D2 = 0) while the other sublattice retains finite DMI (D1, 0), provided the inter-sublattice exchange coupling is sufficiently strong.

3. The AI can predict skyrmion size and morphology by calculating effective length scale Le f f = (4Ae f f)/(πDe f f) and using it to determine the skyrmion size, as the paper notes that the size of the (meta)stable skyrmions varies with Ae f f and De f f, or with the effective length scale Le ff.

4. The AI can map out a complete phase diagram for FiM skyrmions by constructing a phase diagram in the κ1e f f κ2e f f-plane (strong coupling) and the κ1κ2-plane (weak coupling), using Eq. (6) to define the boundary where the transition between isolated and condensed skyrmion phases is governed solely by κe f f = 1.

5. The AI can distinguish between isolated circular skyrmions and stripe skyrmions in both sublattices by analyzing the effective control parameters: For κe f f = 0.9, Fig. 3(a-f) show isolated circular skyrmions, whereas for κe f f = 1.1, Fig. 3(g-l) show stripe skyrmions.