Low-Temperature Skyrmions and Spiral Reorientation Processes in Chiral Magnets with Cubic Anisotropy: Guidelines for Bridging Theory and Experiment

TL;DR

The paper extends the phenomenological Dzyaloshinskii framework by incorporating cubic magnetocrystalline anisotropy to map how easy and hard axes reshape spiral states and stabilize low-temperature skyrmion lattices (LT-SkL) in cubic helimagnets. Through a rotated coordinate treatment and MuMax3-based energy minimization, it identifies critical fields $h_{c1}$ and $h_{c2}$, details spiral reorientation mechanisms, and constructs phase diagrams showing LT-SkL stability thresholds at $k_c \approx 0.039$. It then correlates theory with experiments on MnSi and Fe$_{1-x}$Co$_x$Si, extracting $k_c$ from angular dependencies $\Delta_1$ and $\Delta_2$, and demonstrates tunability of LT-SkL by composition and temperature. The result is a quantitative framework to predict and interpret LT-SkL stability in cubic helimagnets and to guide material design for anisotropy-engineered skyrmion phases.

Abstract

We revisit the phenomenological Dzyaloshinskii framework, a central theoretical approach for describing magnetization processes in bulk chiral magnets, and demonstrate how magnetocrystalline cubic anisotropy reshapes the phase diagrams of states and provides the key mechanism stabilizing low-temperature skyrmion phases. We show that, for magnetic field directions along the easy anisotropy axes, the phase diagrams feature stable skyrmion pockets for both signs of the anisotropy constant. We further analyze the nature of the transitions at the critical field $H_{c1}$, associated with the reorientation of stable and metastable spirals along the field. We also examine the transition at $H_{c2}$, where the conical state closes into the homogeneous state accompanied by a deviation of the wave vector from the field direction. By mapping characteristic anisotropy-dependent parameters in the theoretical phase diagrams, we provide guidelines for connecting theory with experiment and for estimating the cubic anisotropy constant in Fe$_{1-x}$Co$_x$Si and MnSi. Our results indicate that samples Fe$_{1-x}$Co$_x$Si with small $x \sim 0.1$ possess sufficiently strong cubic anisotropy to stabilize a low-temperature skyrmion phase. Overall, these theoretical findings establish a quantitative framework for predicting and interpreting skyrmion stability in other cubic helimagnets as well.

Figures (8)

• Figure 1: (a) Construction of the rotated coordinate system $x^\prime y^\prime z^\prime$: the axes are obtained by a rotation through angle $\phi$ about the $z$ axis, followed by a rotation through angle $\alpha$ about the $y^\prime$ axis. The spiral wave vector is aligned with the $z^\prime$ axis. (b) Definition of the magnetic field orientation, described by the angle $\beta$, with the field confined to the $(010)$ plane for $\psi=0$ and to the $(\overline{1}10)$ plane for $\psi=\pi/4$. (c) Critical field $h_{c2}$ as a function of magnetic-field orientation within two selected crystallographic planes. First- and second-order transitions into the homogeneous state are indicated by I and II, respectively. Results are shown for $k_c=0.05$ (red curve) and $k_c=-0.05$ (blue curve). Insets (i) and (ii) illustrate the complex character of the transition from the conical state (dark-blue curves) to an oblique homogeneous state (green curves) for intermediate field directions $\beta=0.5$ and $\beta=1.3$. (d) Corresponding critical fields $h_{c2}$ in polar coordinates.
• Figure 2: Anisotropy-dependent magnetization behavior for different field orientations. (a) Magnetization curves for the field applied along $\mathbf{h}\parallel [001]$ for different values of $k_c$. For $k_c>0$, the curves exhibit jumps from the conical to the homogeneous state (red and blue curves), whereas for $k_c<0$, the magnetization increases smoothly up to saturation. (b) sketches of the magnetization rotation in the conical phase in the presence of cubic anisotropy with hard (red arrows) and easy (green arrows) axes for both signs of $k_c$. (c) Maximal magnetization values immediately before the transition to the ferromagnetic state for $\mathbf{h} \parallel [111]$ and $\mathbf{h} \parallel [001]$. (d) Critical fields $h_{c2}$ along the three principal field directions.
• Figure 3: Reorientation of spiral states toward easy anisotropy axes at the critical field $h_{c1}$. (a) Energy landscape for $k_c > 0$ and $\mathbf{h} \parallel [001]$ in spherical coordinates, showing multidomain spiral states with mutually perpendicular wave vectors. (b) Energy density in the plane $\phi = \psi = \pi/4$ as a function of $\alpha$, used to identify the critical field $h_{c1}$ at which local minima corresponding to transverse spiral domains disappear. (c) Critical angles $\alpha$ corresponding to energy maxima (red curve) and minima (blue curves), with $h_{c1}$ indicated explicitly for $k_c = 0.05$. (d) Energy landscape for $k_c < 0$ and $\mathbf{h} \parallel [111]$, showing multidomain spiral states with energetically inequivalent $\langle 111 \rangle$ spirals. (e) Evolution of the energy landscape, illustrating the “jump” of metastable spirals into the global minimum. (f) Slight tilting of the spiral wave vectors toward the field direction for $k_c=-0.05$.
• Figure 4: Field-driven reorientation processes of spiral states along hard anisotropy axes. (a) Energy landscape of spiral states for $k_c > 0$ and $\mathbf{h} \parallel [111]$ shown in spherical coordinates, with the wave vectors $\mathbf{q} \parallel \langle 001 \rangle$ gradually canting towards the field. (b) 2D energy profiles indicating the emergence of a new minimum along the field direction at $h = 0.111$, its degeneracy with oblique spiral states at $h = 0.121$ (critical field $h_{c1}$), and the subsequent disappearance of the oblique minima at $h = 0.124$. (c) Field dependence of energy extrema illustrating the hysteretic region (shaded rectangle) where spiral and conical states coexist. (d–f) Analogous results for $k_c < 0$ and $\mathbf{h} \parallel [001]$, where a similar reorientation occurs but within a much narrower hysteretic interval: the conical minimum appears at $h = 0.116$, becomes degenerate with spiral minima at $h = 0.119$, and fully replaces them by $h = 0.120$.
• Figure 5: Phase diagrams showing the stability regions of the low-temperature skyrmion lattice in cubic helimagnets. For $k_c>0$ and $\mathbf{h} \parallel [001]$, the vast area of SkL stability is highlighted. Critical points $A_1$, $B_1$, and $C_1$ indicate, respectively, the onset of the SkL, the threshold for the second-order transition to the homogeneous state, and the anisotropy value above which metastable spirals can act as nucleation centers for skyrmions. For $k_c<0$ and $\mathbf{h} \parallel [111]$, the SkL stability exhibits qualitatively similar features, with critical points $A_2$ and $C_2$ corresponding to the onset of the SkL and the nucleation threshold via metastable spirals.