Asymptotically exact formulas for the stripe domains period in ultrathin ferromagnetic films with out-of-plane anisotropy

TL;DR

This work tackles the problem of predicting equilibrium stripe periods in ultrathin ferromagnetic films with out-of-plane anisotropy. By deriving a reduced two-dimensional micromagnetic model for $\delta \ll 1$ and performing a careful asymptotic analysis, the authors obtain asymptotically exact formulas for Bloch and Néel stripe periods, with a prefactor proportional to the Bloch wall width and a dipolar-corrected exponential in film thickness. The key contributions are the explicit expressions $L_{opt}^{\mathrm{Bloch}} = \pi^2 e^{-\gamma} L_B \exp(\pi \sigma_B /(\mu_0 M_s^2 d))$ and $L_{opt}^{\mathrm{N\acute{e}el}} = 2 \pi^2 e^{-\gamma} L_B \exp(\pi \sigma_N /(\mu_0 M_s^2 d))$, together with accurate energy densities for each case and a rigorous comparison to micromagnetic simulations. The results enable precise extraction of magnetic parameters from measured stripe periods and robust predictions across ultrathin regimes, improving upon prior formulas that used inappropriate thickness-dependent prefactors or neglected wall-width effects; simulations confirm the theory up to thicknesses near the Bloch width.

Abstract

We derive asymptotically exact formulas for the equilibrium magnetic stripe period in ultrathin films with out-of-plane anisotropy that include the full domain wall magnetic dipolar energy. Starting with the reduced two-dimensional micromagnetic model valid for thin films, we obtain the leading order approximation for the energy per unit volume in the vanishing film thickness limit in the case of Bloch and Néel wall rotations. Its minimization in the stripe period leads to an analytical expression for the equilibrium period with a prefactor proportional to the Bloch wall width. The constant in the prefactor, related to the long-range dipolar interactions, is carefully evaluated. This results in a remarkable agreement of the stripe domain energy density and stripe period predicted by our analytical formulas with micromagnetic simulations. Our formula can be used to accurately deduce magnetic parameters from the experimental measurements of the stripe period and to systematically predict the equilibrium stripe periods in ultrathin films.

Figures (5)

• Figure 1: Schematic representation of an ultrathin film of thickness $d$ with periodic stripe magnetic domains of period ${\mathscr L}$. The black and white colors represent magnetization pointing up and down in the $z$-direction, while blue and red represent the in-plane magnetization pointing left and right in the $x$-direction (Néel domain walls).
• Figure 2: Equilibrium stripe period ${\mathscr L}_\mathrm{opt}^\mathrm{Bloch}$ as a function of the film thickness $d$ in the regime of Bloch walls (zero DMI). The material parameters are $A_\mathrm{ex}=10$ pJ$/$m, $M_s=1$ MA$/$m and $K_{u1}=0.75$ MJ$/$m$^3$ (regime 1). The asymptotic equilibrium stripe period in Eq. \ref{['LoptB']} is represented by a solid line. The equilibrium stripe period obtained by micromagnetic simulations using MuMax3vansteenkiste14 (see Sec. \ref{['sec:micromag']} for details) is represented by black dots. The equilibrium stripe periods from previous workskashuba93sukstanskii97skomski98meier17kaplan93 are also shown (see inset for details).
• Figure 3: Equilibrium stripe period ${\mathscr L}_\mathrm{opt}^\mathrm{N\acute{e}el}$ as a function of the film thickness $d$ in the regime of Néel walls. The thin film parameters are $A_\mathrm{ex}=10$ pJ$/$m, $M_s=1$ MA$/$m, $K_{u2}=1$ MJ$/$m$^3$ and $D_2=2$ mJ$/$m$^2$ (regime 2). The asymptotic equilibrium stripe period in Eq. \ref{['LoptN']} is represented by a solid line. The equilibrium stripe period obtained by micromagnetic simulations using MuMax3 (see Sec. \ref{['sec:micromag']} for details) and in previous worksmeier17kaplan93 are also presented (see inset for details).
• Figure 4: Equilibrium stripe period ${\mathscr L}_\mathrm{opt}^\mathrm{N\acute{e}el}$ as a function of the film thickness $d$ in the regime of of Néel walls for an ultrathin film with thickness-dependent volume magnetocrystalline anisotropy $K_{u3}=K_{s3} / d$ and thickness-dependent volume DMI $D_{3}=D_{s3} / d$. The thin film parameters are $A_\mathrm{ex}=10$ pJ$/$m, $M_s=1$ MA$/$m, $K_{s3}=0.6$ mJ$/$m$^2$, and $D_{s3}=1.2$ pJ/m (regime 3). The asymptotic equilibrium stripe period in Eq. \ref{['LoptN']} is represented by a solid line. The equilibrium stripe period obtained by micromagnetic simulations using MuMax3vansteenkiste14 (see Sec. \ref{['sec:micromag']} for details) and in previous worksmeier17kaplan93 are also presented (see inset for details).
• Figure 5: Comparison between the numerical simulations in dots ( MuMax3vansteenkiste14) and dimensional version of the asymptotic energy ${\mathcal{F}}({\mathscr L}) = f(\mathscr L / \ell_\mathrm{ex}) K_d$, where $f(L)$ can be found in Eq. \ref{['energy_neel']}, represented as a solid line. The parameters are $A_\mathrm{ex}=10$ pJ/m, $M_s=1$ MA/m, $K_{u2}=1$ MJ$/$m$^3$ and $D_2=2$ mJ$/$m$^2$. The film thicknesses are (starting from the bottom curve) $d=4$ nm, $d=3$ nm, $d=2$ nm, $d=1.4$ nm and $d=1$ nm.