Theoretical study on stabilization and destabilization of magnetic skyrmions by uniaxial-strain-induced anisotropic Dzyaloshinskii-Moriya interactions

Magnetic skyrmions in chiral-lattice ferromagnets are currently attracting enormous research interest because of their potential applications in spintronic devices. However, they emerge in bulk specimens only in a narrow window of temperature and magnetic field. This limited stability regime is recognized as an obstacle to technical applications. Recent experiments demonstrated that the thermodynamic stability of magnetic skyrmions is enhanced or suppressed by the application of a uniaxial strain depending on its axial direction in bulk chiral-lattice ferromagnets MnSi [Y. Nii et al., Nat. Commun. 6, 8539 (2015); A. Chacon et al., Phys. Rev. Lett. 115, 267202 (2015)] and ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ [S. Seki et al., Phys. Rev. B 96, 220404(R) (2017)]. Motivated by these experimental discoveries, we theoretically investigated the effects of anisotropic Dzyaloshinskii–Moriya interactions on the stability of magnetic skyrmions caused by this uniaxial strain. We find that magnetic skyrmions are significantly stabilized (destabilized) in the presence of anisotropic DM interactions when an external magnetic field lies perpendicular (parallel) to the anisotropy axis, along which the DM coupling is strengthened. Our results account completely for the experimentally observed strain-induced stabilization and destabilization of magnetic skyrmions and provide a firm ground for possible strain engineering of skyrmion-based electronic devices.

Figure 1

[(a)–(c)] Three different cases were examined in the present study. The external magnetic field $\mathit{H}$ was applied parallel to the $z$ axis for all cases. (a) Case A with isotropic DM interactions, $D_x=D_y=D_z$, which corresponds to a system without strain. (b) Case B with anisotropic DM interactions, $D_x>D_y=D_z$. This condition corresponds to a system to which a uniaxial strain $\mathit{\sigma }$ perpendicular to the $\mathit{H}$ field ($\mathit{\sigma }\perp \mathit{H}$) is applied in the $x$ direction. (c) Case C with anisotropic DM interactions, $D_z>D_x=D_y$. This condition corresponds to a system to which a uniaxial strain $\mathit{\sigma }$ parallel to the $\mathit{H}$ field ($\mathit{\sigma }\parallel \mathit{H}$) is applied in the $z$ direction. [(d)–(f)] Calculated $H$ profiles of relative energies of various magnetic states at $T$=0 for (d) case A with $D_x=D_y=D_z=0.727$, (e) case B with $D_x=0.8$ and $D_y=D_z=0.727$, and (f) case C with $D_z=0.8$ and $D_x=D_y=0.727$. [(g)–(i)] Calculated $H$ profiles of net magnetization for (g) case A, (h) case B, and (i) case C. From these $H$ profiles of relative energies and net magnetizations, we drew the phase diagrams as function of $H$ at $T=0$ for the three cases. [(j)–(m)] Magnetization configurations of examined magnetic states: (j) helical A, (k) helical B, (l) skyrmion crystal, and (m) conical states. The layered magnetic structures are stacked uniformly along the $z$ axis.

Figure 2

Calculated $H$ profiles of energies $E_{\mathrm{DM}}^{\alpha }$ associated with the DM interactions on the bonds along the $\alpha$ axis ($\alpha =x,y,\text{and}z$) at $T=0$ for case B with $D_x=0.8$ and $D_y=D_z=0.727$.

Figure 3

(a) Theoretical phase diagram in the plane $D_x$ and $H$ for case B with $D_x>D_y=D_z$ where $D_y$ and $D_z$ are fixed at 0.727. The condition corresponds to a system to which a uniaxial strain $\mathit{\sigma }(\parallel \mathit{x}$) is applied perpendicular to $\mathit{H}(\perp \mathit{z}$). (b) Theoretical phase diagram in the plane $D_z$ and $H$ for case C with $D_z>D_x=D_y$ where $D_x$ and $D_y$ are fixed at 0.727. The condition corresponds to a system to which a uniaxial strain $\mathit{\sigma }(\parallel \mathit{z}$) is applied parallel to $\mathit{H}(\parallel \mathit{z}$).

Figure 4

Temperature profiles of (a) specific heats and (b) magnetic susceptibilities calculated using the replica-exchange Monte Carlo techniques for selected values of $H$ for case B with $D_x=0.8$ and $D_y=D_z=0.727$, which corresponds to the condition $\sigma \perp \mathit{H}$. Inverted triangles in (a) indicate transition points identified by the anomalies of the magnetic susceptibilities in (b) where no remarkable anomalies appear in the specific heats. The right panel of (a) magnifies the area indicated by the gray rectangle in the left panel of (a).

Figure 5

(a) Theoretical $T\text{−}H$ phase diagram for case A with $D_x=D_y=D_z=0.727$, which corresponds to an unstrained system with isotropic DM coupling (reproduced from Ref. [52]). (b) Theoretical $T\text{−}H$ phase diagram for case B with $D_x=0.8$ and $D_y=D_z=0.727$, which corresponds to a system to which a uniaxial strain $\mathit{\sigma }(\perp \mathit{H}$) is applied. (c) Theoretical $T\text{−}H$ phase diagram for case C with $D_z=0.8$ and $D_x=D_y=0.727$, which corresponds to a system to which a uniaxial strain $\mathit{\sigma }(\parallel \mathit{H}$) is applied. Circles and squares indicate transition points identified by anomalies in specific heats and magnetic susceptibilities, respectively. (d) Experimental $T\text{−}H$ phase diagram for ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ without strain. (e) Experimental $T\text{−}H$ phase diagram for ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ under $\mathit{H}\parallel \left[100\right]$ to which a uniaxial strain $\mathit{\sigma }\parallel \left[001\right]$ ($\perp \mathit{H}$) is applied. (f) Experimental $T\text{−}H$ phase diagram for ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ under $\mathit{H}\parallel \left[100\right]$ to which a uniaxial strain $\mathit{\sigma }\parallel \left[100\right]$ ($\parallel \mathit{H}$) is applied. The experimental phase diagrams in (d)–(f) are reproduced from Ref. [47].