Optimal skyrmion stability in antisymmetric ultrathin ferromagnetic bilayers

The gist Optimal skyrmion stability in antisymmetric ultrathin ferromagnetic bilayers demonstrates that stray-field-mediated stabilization, through synergistic action between DMI and dipolar interaction, allows for compact and stable skyrmions in exchange-decoupled systems.

Model and Energy Functional

The study begins by considering the micromagnetic energy of a stack of identical ferromagnetic layers, which is then non-dimensionalized to obtain an effective thin film energy functional E(m) (Page 3). This model accounts for intralayer exchange, bulk perpendicular magnetocrystalline anisotropy, interfacial DMI, and magnetostatic interaction (Page 3). The resulting dimensionless energy E(m) is simplified in the ultrathin film limit to a reduced thin-film energy EN(m1,..., mN) (Page 4). This reduced energy obeys a scaling symmetry where E2(m1, m2) = E¯2(m¯ 1, m¯ 2), which is determined by only two dimensionless parameters: the reduced DMI strength and the reduced film thickness (Page 5).

Antisymmetric Bilayer Analysis

For antisymmetric bilayers, the system is characterized by N=2, Q1 = Q2 = Q > 1, and κ1 = -κ2 = κ ≥ 0 (Page 5). The thin-film energy E2(m1, m2) is invariant under a scaling transformation leading to the simplified dimensionless energy E¯2(m¯ 1, m¯ 2) (Page 5). An ansatz where the in-plane components of magnetization are anti-parallel and out-of-plane components are parallel leads to an effective energy E¯±(m), which is equivalent to a single ferromagnetic layer of thickness 2¯δ with an effective DMI constant κ + 1/2¯δ (Page 5).

Skyrmion Existence and Stability

Theorem 1 establishes the existence of a minimizer of E¯± among all m in H1 loc(R squared; S 2) such that q(m) = 1, under the condition that 0 < 2¯κ + 3¯δ ≤ √2 (Page 5). The asymptotic behavior for small DMI and small film thickness is given by a formula for the dimensionless skyrmion radius ρsky2 of the skyrmion in an antisymmetric bilayer: ρ¯sky2 ≃ 32¯κ + (16 + π/2)¯δ − 64W−1(-32¯κ+(16+π/2)δ¯)/128 e(1+γ) (Page 5).

Skyrmion Bursting and Thermal Stability

The analysis investigates the range of existence beyond Theorem 1, leading to the identification of a bursting line defined by κb2(¯δ) (Page 6). The optimal thermal stability is found at κ≃ κbar opt2(¯δ) for every fixed ¯δ (Page 6). The skyrmion lifetime in the Arrhenius regime is given by τ = τ0 exp(Ad/kB T∆E¯2), where ∆E¯2 = min(∆Ebar c,02, ∆Ebar b,02) (Page 7).

Comparison with Other Systems

The paper compares the antisymmetric bilayer with synthetic antiferromagnets (SAF) and ferromagnetic monolayers using Pt/Co/AlOx stacks as a building block (Page 10). The results show that the antisymmetric bilayer provides a broadening of the range of existence of skyrmion solutions, which could potentially result in a significant decrease of the degree of fine-tuning necessary to obtain very compact and stable skyrmions (Page 15).

Conclusion

The study concludes that stray field effects can exert a pronounced stabilizing effect on magnetic skyrmions, forming a chiral pair of skyrmions in a flux-closure pattern in exchange-decoupled antisymmetric bilayers (Page 20). The importance of the obtained phase diagrams lies in providing, for the first time, the region of skyrmion stability via a quantitative prediction of skyrmion bursting and an effective energy barrier that determines skyrmion stability against thermal noise (Page 20). This confirms that the antisymmetric bilayer is a promising platform for 10 nm radius zero-field skyrmions with lifetimes compatible with information technology applications (Page 20).

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Optimal skyrmion stability in antisymmetric ultrathin ferromagnetic bilayers Anne Bernand-Mantel, Valeriy V. Slastikov and Cyrill B. Muratov demonstrate the stray-field-mediated skyrmion stabilizing capabilities of ultrathin exchangedecoupled antisymmetric ferromagnetic bilayers based on conventional transition metal materials (Page 1). Using an asymptotically exact micromagnetic model valid in the ultrathin film limit, we show that the antisymmetric tailoring of the bilayer allows the Dzyaloshinskii-Moriya interaction and the dipolar interaction to act synergistically to stabilize skyrmions, in contrast to the monolayer case, in which these energies compete (Page 1). To obtain optimal stability of these skyrmions against collapse and bursting – the two fundamental processes determining skyrmion lifetime, we carry out an asymptotic analysis of the saddle point solution that separates the skyrmion from the demagnetized state (Page 1). The result is an optimal stability line for compact skyrmions in the non-dimensional parameter space of the effective Dzyaloshinskii-Moriya interaction strength and the effective film thickness (Page 1). Our predictions are confirmed by extensive micromagnetic simulations of antisymmetric bilayers, using magnetic parameters of the conventional Pt/Co/AlOx systems (Page 1). Our results provide a new pathway for experimental observations of 10 nm radius zero-field skyrmions with lifetimes compatible with information technology applications (Page 1).

How it works

The model starts with the micromagnetic energy of a stack of identical ferromagnetic layers in the presence of intralayer exchange, perpendicular magnetocrystalline anisotropy, interfacial DMI, and magnetostatic interaction (Page 3). The energy is non-dimensionalized by measuring lengths in units of the exchange length lex = pA/Kd, introducing m(r) = M(lexr)/Ms and defining dimensionless energy E(m) = E(M)/(Ad) (Page 3). In the ultrathin limit, the micromagnetic energy obeys EN (m1,..., mN), which is expanded in powers of δ (Page 4).

Antisymmetric Profiles

The analysis focuses on antisymmetric bilayers where N=2, Q1 = Q2 = Q > 1 and κ1 = -κ2 = κ ≥ 0 (Page 5). The energy E¯±(m) is derived under the ansatz where in-plane components are anti-parallel and out-of-plane components are parallel (Page 5). This ansatz is motivated by the energetic advantages it provides: in a skyrmion profile the DMI and the surfacevolume charge interactions of both layers cooperate, while the energy penalty due to the volume-volume charge interactions cancels out (Page 5).

Skyrmion Bursting

The analysis investigates solutions beyond Theorem 1, leading to a bursting line defined by κb2(¯δ) (Page 6). The optimal thermal stability is found at κ≃ κbar opt2(¯δ) for every fixed ¯δ (Page 6). The skyrmion lifetime in the Arrhenius regime is given by τ = τ0 exp(Ad/kB T∆Ebar 2), where ∆Ebar 2 = min(∆Ebar c,02, ∆Ebar b,02) (Page 7).

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FM FM FM FM NM1 NM2 antisymmetric bilayer NM1 NM2 monolayer synthetic antiferromagnet NM1 NM2 RKKY (a) (b) (c)

How it works

The property of SAF having a vanishing stray field due to the antiparallel alignment of the magnetization vector in the two layers makes the experimental observation of skyrmions even more challenging than in the case of single ferromagnets and ferromagnetically coupled multilayers (Page 2). In theory, stacking several layers of compensated SAF could lead to an increase of the collapse barrier of individual compact skyrmions [22, 29] (Page 2). However, the experimentally observed skyrmions remain in a few hundred nanometer to a few micron diameter range and suffer from stripe-out instabilities (Page 2).

1. Optimization of material design for information technology applications: The improved system can predict optimal skyrmion stability lines in non-dimensional parameter space based on effective Dzyaloshinskii-Moriya interaction strength and film thickness, providing a new pathway for experimental observation of 10 nm radius zero-field skyrmions with lifetimes compatible with information technology applications.

2. Enhanced simulation of magnetic dynamics: The system can perform micromagnetic simulations using Mumax3 software to obtain detailed phase diagrams of skyrmion solutions, including their radii and energy barriers as functions of reduced DMI strength and reduced film thickness.

3. Prediction of thermal stability: The system can predict the optimal value for the dimensionless DMI strength kappa opt 2 based on fixed film thickness delta, which maximizes skyrmion lifetime by setting the two barriers to coincide, allowing for precise tuning of material parameters for room-temperature stability.

4. Skyrmion characterization across material classes: The improved AI can compare skyrmion solutions and effective energy barriers for antisymmetric bilayers, synthetic antiferromagnets (SAF), and ferromagnetic monolayers using typical Pt/Co/AlOx systems to predict which material configuration offers the broadest range of skyrmion existence.

5. Real-time stability assessment: The system can calculate the effective energy barrier as a function of film thickness and DMI strength for different material stacks, enabling real-time prediction of skyrmion stability against thermal noise in experimental setups.

We demonstrate the stray-field-mediated skyrmion stabilizing capabilities of ultrathin exchange-decoupled antisymmetric ferromagnetic bilayers based on conventional transition metal materials. Using an asymptotically exact micromagnetic model valid in the ultrathin film limit, we show that the antisymmetric tailoring of the bilayer allows the Dzyaloshinskii-Moriya interaction and the dipolar interaction to act synergistically to stabilize skyrmions, in contrast to the monolayer case, in which these energies compete. To obtain optimal stability of these skyrmions against collapse and bursting -- the two fundamental processes determining skyrmion lifetime, we carry out an asymptotic analysis of the saddle point solution that separates the skyrmion from the demagnetized state. The result is an optimal stability line for compact skyrmions in the non-dimensional parameter space of the effective Dzyaloshinskii-Moriya interaction strength and the effective film thickness. Our predictions are confirmed by extensive micromagnetic simulations of antisymmetric bilayers, using magnetic parameters of the conventional Pt/Co/AlO x systems. Our results provide a new pathway for experimental observations of 10 nm radius zero-field skyrmions with lifetimes compatible with information technology applications.