Phase-Factor-Controlled Surface Spirals in the Magnetic Conical Phase: The Role of In-Plane Directionality

TL;DR

This work addresses how in-plane directionality of conical-phase spirals can be harnessed via a surface phase factor $φ_0$. By combining micromagnetic simulations of 1D flat and 2D FCIs with LTEM observations in Co$_8$Zn$_{10}$Mn$_2$, the authors show that surface spirals (SSs) form near ferromagnetic-cone interfaces and that their existence, shape, and topological charge are controlled by $φ_0$, including discontinuities and multi-SS states. They also demonstrate SS formation at skyrmion-cluster edges in the conical phase and map two experimental formation pathways: thermally activated co-growth and field-driven transformation from residual helices. The results establish $φ_0$ as a fundamental control parameter enabling phase-tunable, multi-state spin textures with potential applications in high-density memory (HD-PMS) and neuromorphic computing.

Abstract

In chiral magnets, the magnetic textures surrounding domain walls exhibit a rich variety of structures, offering insights into fundamental physics and potential applications in spintronic devices. Conical spirals and related structures possess intrinsic in-plane directionalities governed by phase factors $φ_0$, which are often obscured in long spirals due to cylindrical symmetry but become prominent in short spirals or thin films. Using micromagnetic simulations, we systematically studied magnetic textures at ferromagnetic-conical interfaces (FCI), including 1D and 2D FCIs with various shapes. Surface spirals (SS) emerge adjacent to these FCIs, closely linked to the cone's in-plane reorientation. In 1D FCIs, reorientation controls the presence, shape, and topological charge of the SS, with a discontinuity point observed where spirals with opposite charges form on opposite sides. In 2D FCIs, eyebrow-like SS are evident. The reorientation angle between top and bottom SS is controlled by the film thickness, similar to stacked spirals reported previously. We further demonstrate that SSs form at the facets of skyrmion clusters within the conical phase, as confirmed by both simulations and Lorentz transmission electron microscopy observations in Co$_8$Zn$_{10}$Mn$_2$ thin films. The experiments specifically reveal two distinct formation pathways: thermally activated co-growth and field-driven transformation from residual helices. These findings establish $φ_0$ as a fundamental control parameter for magnetic states, enabling promising spintronic functionalities such as multi-state memory through SS polymorphism and energy-efficient neuromorphic computing via controlled topological transitions.

Figures (8)

• Figure 1: (a) An FCI comprising of a FM domain (left section) and a cone domain (right section). (b) The in-plane directionality of the cone characterized by the angle $\phi(z)$ between the normal vector of the FCI (aligned with the $\hat{x}$ axis), and the in-plane projection of the magnetization vector of the cone, denoted as ${\bf m}_{{\it \parallel}}=(m_x,m_y,0)=(\cos\phi(z),\sin\phi(z),0)$.
• Figure 2: Magnetic structure of SSs spontaneously formed near the FCI, illustrated for the top-right quadrant at a field $H/H_D=0.41$. The structures are shown for varying surface phase factor: (a) $\phi_0=0$, (b) $\phi_0=0.53\pi$, (c) $\phi_0=0.66\pi$, (d) $\phi_0=\pi$, (e) $\phi_0=1.03\pi$, and (f) $\phi_0=1.59\pi$. Each panel displays only the superficial layers of the top-right quadrant to clearly resolve the structural details. The color scale represents the $z$-component of magnetization $m_z$, and the arrows indicate the in-plane spin orientation.
• Figure 3: (a) Energy line density $E_{xz}$ and (b) topological charge $N_{\text{tc}}$ versus the surface phase factor $\phi_0$ at $H/H_D = 0.41$. Data are presented for the top-right (red circles), top-left (blue triangles), and bottom-left (black squares) quadrants. The curves for the top-left and bottom-left quadrants follow the symmetry relations given by Eqs. (7)--(10), as described in the text. Selected points corresponding to the magnetic configurations shown in panels (a) to (f) of Fig. \ref{['fig:SS']} are marked with labels a to f.
• Figure 4: The order parameter $S$, characterizing the size of SS formed at top-right (a) or bottom-right (b) quadrant of the film, as a function of external field $H$ and phase factor $\phi_0$.
• Figure 5: Topological charge distribution $n_\text{tc}(x,z)$ before (a) and after (b) the discontinuity at $\phi_0\approx\pi$, corresponding to Fig. \ref{['fig:SS']}(d) and Fig. \ref{['fig:SS']}(e) respectively. The color plots represent $n_\text{tc}$; arrows show spin orientation projected in x-z plane; black circles mark the positions of SS. (c) Magnetic structure of inner-twists next to 1D FCI in an infinite sample at $H/H_D=0.41$. (d) Topological charge per unit length $s_\text{tc}$ as a function of reorientation angle $\phi(z)$ for the left-side FCI A to C and the right-side FCI D to F. The applied field $H/H_D=0.41$ for A&D, $0.61$ for B&E, and $0.81$ for C&F.