Cubic magnetic anisotropy in $B$20 magnets: Interplay of anisotropy and magnetic order in Fe$_{1-x}$Co$_{x}$Si

TL;DR

The paper investigates cubic magnetocrystalline anisotropy in cubic B20 magnets MnSi and Fe$_{1-x}$Co$_x$Si using angle-resolved SQUID magnetization to extract the fourth-order anisotropy constant $K_1$ via a cubic energy expansion $E_{cub}(\ound) = K_0 + K_1(m_1^2m_2^2 + m_2^2m_3^2 + m_1^2m_3^2)$. It demonstrates a strong concentration dependence of $K_1$ in Fe$_{1-x}$Co$_x$Si, with MnSi having $K_1<0$ ($\langle111\rangle$ easy axes) and Fe-rich compositions showing $K_1>0$ ($\langle100\rangle$ easy axes), including a vanishing anisotropy near $x\approx0.5$ and weaker anisotropy at high $x$; crucially, $x\approx0.08$–$0.15$ yields a unitless anisotropy $k_c$ above a theoretical threshold ($k_c=0.039$), indicating potential stabilization of a low-temperature skyrmion phase (LTS). The study cross-validates $K_1$ through a secondary analysis based on $H_{c2}$ and Dzyaloshinskii-model simulations using $\Delta=(H_{c2[111]}-H_{c2[100]})/H_{c2[100]}$, finding broad agreement and identifying Fe$_{1-x}$Co$_x$Si concentrations most favorable for LTS engineering. Altogether, the work provides a quantitative anisotropy map for Fe$_{1-x}$Co$_x$Si and MnSi, clarifying how weak cubic anisotropy interacts with magnetic order and supports anisotropy-driven skyrmion stabilization in itinerant chiral magnets.

Abstract

The metallic systems MnSi and Fe$_{1-x}$Co$_{x}$Si are known to feature a generic magnetic phase diagram primarily determined by the isotropic exchange and Dzyaloshinskii-Moriya interactions. However, additional weaker anisotropies, lowest in the hierarchy of energy scales, play a crucial role: they determine the relative order of phases in the phase diagram and may even enable skyrmion stability far below the ordering temperature. Among cubic B20 helimagnets, the insulator Cu$_2$OSeO$_3$ is currently the only known example exhibiting a low-temperature, anisotropy-induced skyrmion pocket. In this manuscript, we present a systematic study of cubic magnetocrystalline anisotropy by means of angle-resolved SQUID magnetization measurements in MnSi and Fe$_{1-x}$Co$_{x}$Si ($0.08 \leq x \leq 0.70$) single crystals and provide quantitative values of the anisotropy constants. For Fe$_{1-x}$Co$_{x}$Si, the cubic anisotropy is found to be strongly dependent on the Co concentration $x$. In particular, for low Co concentrations ($x \sim 0.10$), the anisotropy is sufficiently strong to stabilize a low-temperature skyrmion lattice, in agreement with theoretical predictions. This finding suggests that Fe$_{1-x}$Co$_{x}$Si may represent the first chiral metallic system to exhibit a low-temperature skyrmion phase controllably stabilized by cubic anisotropy for specific directions of the magnetic field.

Figures (8)

• Figure 1: Schematic depiction of the phase diagram of Fe$_{1-x}$Co$_{x}$Si as function of the Co concentration $x$. At low Co concentrations, Fe$_{1-x}$Co$_{x}$Si undergoes an insulator-to-metal transition, closely followed by a nonmagnetic-to-magnetic transition Grefe2024. For Co concentrations up to $x=0.50$, a high-temperature skyrmion phase (HTS) persists near $T_\text{HM}$grigoriev2007grigoriev2007_principal_interactionsmuenzer_2010bauer2016_history with a strong history dependence bauer2016_history as indicated by arrows. At high Co concentrations near $x\approx 0.65$ Fe$_{1-x}$Co$_{x}$Si shows a tendency to show full field polarization (FP) Grigoriev2022_FM. From studies on Cu$_2$OSeO$_3$2018Chacon2018Halder, the question arises whether the tunable system Fe$_{1-x}$Co$_{x}$Si also features a low-temperature skyrmion phase (LTS).
• Figure 2: (a) Rotatable sample holder used to conduct angle-resolved magnetization measurements and (b) exemplary measurements of the magnetic moment upon sample rotation for Fe$_{0.92}$Co$_{0.08}$Si at $T=5~\text{K}$. The magnetization direction relative to the crystal orientation is parameterized by the angles $\theta$ and $\phi$.
• Figure 3: Magnetization energies $E$ as a function of the crystal orientation at $T=5~\text{K}$ for (a) MnSi with $\phi=45^\circ$, (b) Fe$_{0.92}$Co$_{0.08}$Si with $\phi=45^\circ$, (c) Fe$_{0.50}$Co$_{0.50}$Si with $\phi=0^\circ$ and (d) Fe$_{0.35}$Co$_{0.65}$Si with $\phi=45^\circ$. Red lines represent the cubic anisotropy fit according to Eq. (\ref{['eq:cubic_anisotropy']}). (a) represents the case of $K_1<0$, (b) and (d) the case of $K_1>0$, and (c) $K_1=0$ within the resolution of our experiment.
• Figure 4: Anisotropy constants $-K_1$ for MnSi and $K_1$ for Fe$_{1-x}$Co$_{x}$Si ($0.08 \leq x \leq 0.70$) at $T=5~\text{K}$ and $T=10~\text{K}$. Dashed lines are guides to the eye.
• Figure 5: Magnetization $M$ for Fe$_{0.85}$Co$_{0.15}$Si at $T=5~\text{K}$ along the $\langle 100 \rangle$ and $\langle 111\rangle$ directions. The red-shaded region marks the area contributing with an opposite sign to the magnetic anisotropy. The critical fields are obtained from the first derivative and through linear interpolations near saturation.