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Optimal skyrmion stability in antisymmetric ultrathin ferromagnetic bilayers

Anne Bernand-Mantel, Valeriy V. Slastikov, Cyrill B. Muratov

Abstract

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.

Optimal skyrmion stability in antisymmetric ultrathin ferromagnetic bilayers

Abstract

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 systems. Our results provide a new pathway for experimental observations of 10 nm radius zero-field skyrmions with lifetimes compatible with information technology applications.

Paper Structure

This paper contains 16 sections, 1 theorem, 73 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Theory of antisymmetric bilayers
  4. Thin film energy
  5. Antisymmetric profiles
  6. Bilayer skyrmions
  7. Skyrmion bursting
  8. Thermal stability
  9. Beyond bursting
  10. Numerical skyrmion solutions and energy barriers
  11. Comparison with synthetic antiferromagnets and ferromagnetic monolayers
  12. Synthetic antiferromagnet
  13. Ferromagnetic monolayers
  14. Comparison of skyrmion solutions for a typical ferromagnetic material
  15. Conclusions
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\bar{\delta} \geq 0$ and $\bar{\kappa} \geq 0$ be such that Then there exists a minimizer of $\bar{E}^\pm$ among all $\mathbf m \in H^1_\mathrm{loc}(\mathbb R^2; \mathbb S^2)$ such that $q(\mathbf m) = 1$, $\int_{\mathbb R^2} |\nabla \mathbf m|^2 d^2 r < 16 \pi$, and $\mathbf m +\hat{\mathbf{z}} \in L^2(\mathbb R^2; \mathbb R^3)$. $\blacktriangleleft$$\black

Figures (8)

  • Figure 1: Schematics of the ferromagnetic multilayer structures with a skyrmion present: (a) antisymmetric ferromagnetic bilayer consisting of two identical ferromagnetic layers separated by a single non-magnetic layer (NM2) and capped by a different non-magnetic material (NM1); (b) a SAF consisting of two ferromagnetic layers (FM) capped by two different non-magnetic layers (NM1 and NM2) and separated by an exchange coupler (RKKY); (c) single ferromagnetic monolayer (FM) capped by two different non-magnetic layers (NM1 and NM2).
  • Figure 2: The dependence of $\bar{E}^\pm(\mathbf{m}_{\bar{\rho}})$ from \ref{['eq:EpmR']} on $\bar{\rho}$ for $\bar{\delta} = 0.5$ and $\bar{\kappa} = 0.4$, $\bar{\kappa} = 0.4293$, and $\bar{\kappa} = 0.5$, respectively.
  • Figure 3: Summary of the numerical results obtained from Mumax3 simulations (see Sec.\ref{['sec:numer-simul-antisymm']} for details) of an antisymmetric ferromagnetic bilayer with $A = 20$ J/m, $M_s = 10^6$ A/m, $d = 1$ nm, and $K_u$ and $D$ varied in accordance with \ref{['eq:DK']} on a $614.4 \times 614.4$ nm computational domain with periodic boundary conditions in the plane: (a) the dimensionless radius $\bar{\rho}_2^\mathrm{sky}$; (b) the dimensionless skyrmion energy $\bar{E}_2^\mathrm{sky}$; (c) the dimensionless collapse energy barrier $\Delta \bar{E}_2^c$; (d) the dimensionless bursting energy barrier $\Delta \bar{E}_2^b$; (e) the dimensionless effective energy barrier $\Delta \bar{E}_2$. The solid line shows $\bar{\kappa}_2^\mathrm{opt}(\bar{\delta})$ and the dashed line shows $\bar{\kappa}_2^b(\bar{\delta})$ governed, respectively, by \ref{['eq:kappa2opt']} and \ref{['eq:kappab2']}.
  • Figure 4: Comparison of the skyrmion energy obtained numerically (open circles) with the prediction of \ref{['eq:dEbc0']} (solid lines): (a) the graph of $\bar{E}^{\pm,0}(\mathbf{m}^\mathrm{sky})$ vs. $\bar{\kappa}$; (b) the graph of $8 \pi - \bar{E}^{\pm,0}(\mathbf{m}^\mathrm{sky})$ vs. $\bar{\kappa}$.
  • Figure 5: Summary of the numerical results obtained from Mumax3 simulations (see Sec. \ref{['sec:monolayers']} for details) of a ferromagnetic monolayer with the same parameters as in Fig. \ref{['fig:rhoE2']}: (a) the dimensionless radius $\bar{\rho}_1^\mathrm{sky}$; (b) the dimensionless skyrmion energy $\bar{E}_1^\mathrm{sky}$; (c) the dimensionless collapse energy barrier $\Delta \bar{E}_1^c$; (d) the dimensionless bursting energy barrier $\Delta \bar{E}_1^b$; (e) the dimensionless effective energy barrier $\Delta \bar{E}_1$. The solid line shows $\bar{\kappa}_1^\mathrm{opt}(\bar{\delta})$, while the dashed line shows $\bar{\kappa}_1^b(\bar{\delta})$, governed, respectively, by \ref{['eq:kappa1opt']} and \ref{['eq:kb1']}.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1