Domain walls in finite-width nanowires with interfacial Dzyaloshinskii-Moriya interaction

It is widely known that the interfacial Dzyaloshinskii-Moriya interaction (DMI) may stabilize Néel walls rather than Bloch walls in magnetic thin films. When the DMI is weak, it results in a “tilted” Bloch wall. However, for most applications, domain walls are in nanowires rather than thin films. Here we present a semianalytic two-parameter calculation for the static domain wall in a nanowire of finite width and thickness, with DMI. The DMI strength that is needed to force a Néel wall is smaller in nanowires than in films due to demagnetizing energy. Even nanowires that are hundreds of nanometers wide may have different domain wall solutions than thin films and so their finite size must be considered. The impact of this result on current experiments is briefly discussed. We extend the model to show that applying a weak magnetic field allows the domain wall type to be tuned.

Figure 1

The geometry of the magnetic nanowire (infinite in the $z$ direction) with width $w$ and thickness $d<w$. The magnetization direction in the two domains are shown. The local magnetization direction can be represented by a polar angle $\theta$ and an azimuthal angle $\varphi$. An interface with a strong spin-orbit material such as Pt gives rise to an interfacial DMI. The characteristic domain wall length is given by $L$.

Figure 2

Contour plot of (a) tilt angle $\varphi$ and (b) domain wall length $L$, as a function of nanowire width $w$ and DMI strength $D$ for a 3.4 nm Co/Ni nanowire with parameters given in the text, calculated using Eqs. (10) and (11), a two-parameter model assuming that the tilt angle is constant in space and with simplified demagnetizing energy. The values of each contour region are given by the legends to the right. Regions of different domain wall type are labeled in (a). There is no applied field.

Figure 3

The same plot as in Fig. 2, but calculated using a better approximation for the demagnetizing energy, given by Eq. (14). The contour plots show (a) tilt angle $\varphi$ and (b) domain wall length $L$, as a function of nanowire width $w$ and DMI strength $D$ for parameters given in the text. The values of each contour region are given by the legends to the right. Regions of different domain wall type are labeled in (a). There is no applied field. By comparing Figs. 2 and 3, one sees that the Bloch-Néel transition is moved to larger nanowire widths (which match experiment) and that the domain wall lengths are generally lower, compared to calculations that use simple demagnetizing factors.

Figure 4

The tilt angle $\varphi$ as a function of DMI strength for a 3.4 nm thick, 100 nm wide Co/Ni nanowire with $H=10$ kA/m (${\mu }_{0}H=12.6$ mT). The results of the semianalytic two-parameter model [Eqs. (20) and (21)] are shown by the dashed line. The results of numerically minimizing the energy density Eq. (18) and letting $\varphi$ vary with $z$ are given by the dots, in which case the tilt in the center of the domain wall is plotted.

Figure 5

Contour plot of (a) tilt angle $\varphi$ and (b) domain wall length $L$, as a function of nanowire width $w$ and DMI strength $D$ for a 3.4 nm Co/Ni nanowire with parameters given in the text, calculated using Eqs. (20) and (21), a two-parameter model assuming that the tilt angle is constant in space. Regions of different domain wall type are labeled in (a). There is a weak applied field $H=10$ kA/m (${\mu }_{0}H=12.6$ mT) along $z$.

Figure 6

The same plot as shown in Fig. 5 but calculated using better calculations for the demagnetizing energy, as mentioned in Eq. (14). The contour plots show (a) tilt angle $\varphi$ and (b) domain wall length $L$, as a function of nanowire width $w$ and DMI strength $D$ for parameters given in the text. There is a weak applied field $H=10$ kA/m (${\mu }_{0}H=12.6$ mT) along $z$. Figures 5 and 6 are qualitatively the same but the domain wall lengths $L$ (b) calculated here are in general shorter than those calculated using simple approximations for the demagnetizing factors.