Theory of the Dzyaloshinskii domain-wall tilt in ferromagnetic nanostrips

We present an analytical theory of domain-wall tilt due to a transverse in-plane magnetic field in a ferromagnetic nanostrip with out-of-plane anisotropy and Dzyaloshinskii-Moriya interaction (DMI). The theory treats the domain walls as one-dimensional objects with orientation-dependent energy, which interact with the sample edges. We show that under an applied field the domain wall remains straight, but tilts at an angle to the direction of the magnetic field that is proportional to the field strength for moderate fields and sufficiently strong DMI. Furthermore, we obtain a nonlinear dependence of the tilt angle on the applied field at weaker DMI. Our analytical results are corroborated by micromagnetic simulations.

Figure 1

An example of a tilted domain wall in a transverse in-plane magnetic field computed using micromagnetic simulations. The material parameters are $A={10}^{-11}\mathrm{J}/\mathrm{m}$, $K=1.25\times {10}^6{\mathrm{J}/\mathrm{m}}^3$, $M_s=1.09\times {10}^6$ A/m, $D=2.4\mathrm{mJ}/{\mathrm{m}}^2$, and $B_y={\mu }_{0}H=80$ mT (see Sec. 2 for precise definitions).

Figure 2

Comparison of the exact edge domain-wall energies in Eq. (16) with the approximate ones given by Eq. (17) for $Q=1.674$ and $\kappa =0.366$ (see Sec. 6 for the corresponding material parameters. For those parameters, the value of $h=0.1$ corresponds to ${\mu }_{0}H=132$ mT). (a) Blue and red curves correspond to ${\sigma }_{\mathrm{edge}}^{+}$ and ${\sigma }_{\mathrm{edge}}^{-}$, respectively. Solid lines correspond to the exact values from Eq. (16) and dashed lines represent the approximation of Eq. (17). (b) The relative error in approximating $\mathrm{\Delta }{\sigma }_{\mathrm{edge}}={\sigma }_{\mathrm{edge}}^{+}-{\sigma }_{\mathrm{edge}}^{-}$ using Eq. (17) is shown.

Figure 3

Schematics of a general domain-wall geometry in an infinite strip. The up/down symbols indicate the direction of the magnetization far from the wall and the strip edges.

Figure 4

A one-dimensional $y=0$ cut through the computed two-dimensional profile $\mathbf{m}$ (blue dots) vs. a one-dimensional cut through the optimal profile ${\mathfrak{m}}_{{\gamma }_{\alpha }}$ (red line). See text for details.

Figure 5

The dependence ${\sigma }_{\mathrm{wall}}\left(\alpha \right)$ obtained from the numerical minimization of $E_{\alpha }$ (blue solid curve), the analytical expressions given by Eq. (32) (red dashed curve) and Eq. (41) (green dotted curve), corresponding to the dimensionless parameters $Q=1.674$, $\kappa =0.366$, and $h=0.073$.

Figure 6

Equilibrium tilt angle $\beta$ vs applied field $B_y={\mu }_{0}H$ for several values of the DMI strength $D$, alongside with the predictions of Eq. (56) (solid lines) and Eq. (59) (dashed lines).

Figure 7

Equilibrium tilt angle $\beta$ vs DMI strength $D$ for several values of the applied field $B_y={\mu }_{0}H$, alongside with the predictions of Eq. (56) (solid lines) and Eq. (59) (dashed lines).

Figure 8

Equilibrium tilt angle $\beta$ vs magnetic field $B_y$ from simulations with and without the dipolar interactions.