Effects of spatially engineered Dzyaloshinskii-Moriya interaction in ferromagnetic films

As the simplest case of a heterochiral film, we first consider a 1D model with DMI strength $D_{1}$ on the left $(x<0)$ and DMI strength $D_{2}<D_{1}$ on the right $(x>0)$, as depicted in Fig. 1. After relaxing the uniform magnetized state, significant canting of the magnetization is observed at the interface between regions with different DMI (in the present case, at $x=0$). We denote the canting angle of magnetization at the interface as $\theta_{0}$.

After minimizing the energy functional (5) with the Euler-Lagrange method, we obtain the expression for the magnetization

which bears similarities with the expression for a domain wall. Note that one still has to determine the angle $\theta_{0}$. In order to do this, we calculate the energy difference with respect to the energy of the uniform state:

The first term is the increase in exchange and anisotropy energy due to the canting with angle $\theta_{0}$ at the interface. The second term is the DMI energy which depends on the difference between DMI strengths $D_{1}$ and $D_{2}$. Minimizing the energy difference, by varying the angle $\theta_{0}$, yields

The canting of the magnetization at the interface depends on the difference between DMI strengths $D_{1}$ and $D_{2}$. Even for DMI strengths below $D_{c}$, canting will occur. Increasing the exchange or the anisotropy will lower the canting of the magnetization $\theta_{0}$, i.e., would tend to make the state more uniform. Since the DMI can be positive or negative, the largest canting angle is obtained when $D_{1}=-D_{c}$ and $D_{2}=D_{c}$, and is equal to $\arcsin(2/\pi)\approx 0.69$ rad.

The magnetic ground state of a chiral ferromagnetic film is cycloidal in the case of a strong DMI $|D|>D_{c}$ and uniform otherwise. Therefore, in a heterochiral film, it is possible to confine a cycloid in a high-DMI region. To illustrate this, we consider a high-DMI strip in the film, surrounded by an extended region without DMI. The energy of the confined cycloid is the sum of the energy in the outer regions, which we calculated analytically in the previous section [Eq. (8)], and the energy inside the high-DMI strip. We can rewrite the 1D energy functional for this case as

for a cycloid confined in a high-DMI strip with DMI strength $D_{\mathrm{in}}$ and width $w=2a$, centered at $x=0$. We minimize the energy $E^{\mathrm{1D}}$ using a conjugate gradient method. This yields a stable magnetization $\theta(x)$, including the magnetization at the interfaces $\theta_{\pm a}$. The stabilized number of domain walls in the confined cycloid depends on the initial guess of the magnetization. The energy and magnetization profiles of the confined cycloids, corresponding with the lowest energy states, are shown in Fig. 2.

The spin canting at the interface, discussed in the previous section, is clearly visible for the quasiuniform state. This spin canting also occurs when the number of domain walls is small in a wide high-DMI strip [see single-wall case in Fig. 2(b)]. The energy of the cycloidal state with $n$ domain walls converges to the sum of wall energies $E_{\mathrm{wall}}$ and the energy of canted spins at the interface  $E_{\mathrm{cant}}$, i.e., $nE_{\mathrm{wall}}+E_{\mathrm{cant}}$ for large confinement widths $w$. In other words, adding walls lowers the energy for large $w$. On the other hand, a large number of domain walls is not preferred in narrow confinement regions due to the resulting high exchange energy. These competing effects are notable in the phase diagram of the ground state, shown in Fig. 2(d). The ground state is quasiuniform for DMI strength $D_{\mathrm{in}}$ below $D_{c}$, or for small $w$. The number of domain walls in the ground state increases for increasing DMI strength or an increasing confinement width $w$ of the high-DMI strip.

This phase diagram is similar to the phase diagram of cycloids confined in finite ferromagnetic films with homogeneous DMI Mulkers et al. (2016). However, there is a fundamental difference between the two design methods regarding the stability of domains. The energy barrier related to the annihilation of domain wall at the edge of a monochiral ferromagnet is smaller than the energy needed to collapse an entire domain in a high-DMI strip within a low-DMI film. A Bloch point is unavoidable when collapsing an entire domain in an extended film. This leads to a very high energy barrier (infinite exchange energy in the continuum approximation). Pushing a domain out of a film can be done in a continuous manner without the formation of a Bloch point, corresponding with a finite energy barrier. In an atomistic model, the energy barrier to annihilate a skyrmion becomes finite, but it is reasonable to assume that this barrier is still much higher than the energy barrier for the escape of a skyrmion through sample boundary.

In the next consideration, we assume that the magnetization $\theta(r)$ has cylindrical symmetry and the direction of the magnetization is radial, as done previously in Ref. Rohart and Thiaville, 2013. In that case, the energy functional in polar coordinates becomes

with allowed radially dependent DMI strength $D(r)$. Here, we consider a strong DMI in a central circular region of radius $R$, surrounded by an extended region without DMI [$D(r)=D_{\mathrm{in}}\Theta(r-R)$, with $\Theta(r)$ the Heaviside step function]. We then relax the quasiuniform state (no walls to begin with), a confined skyrmion (single closed wall), and a ring domain (two concentric closed walls) by minimizing the energy numerically. Comparing the energies of the three configurations yields the phase diagram of the ground state shown in Fig. 3(a). The quasiuniform state is the ground state for DMI strengths below $D_{c}$ or for strong confinement (small $R$). For DMI strengths above $D_{c}$, there is a range of confinement size $R$ for which the magnetization with a skyrmion in the high-DMI region is the ground state. For a strong DMI and a loose confinement (large $R$), the circular domain is the lowest energy state of the three configurations considered. For larger $R$, one finds higher order ring domains or other cycloidal-like domains (e.g. S-shaped ones, beyond our cylindrical approximation), within the parametric area labeled ‘other’ in Fig. 3(a).

The radius of the confined skyrmion, after minimization of the energy functional (11), is shown in Fig. 3(b). The figure shows clearly the effect of the confinement: The stronger the confinement, the smaller the skyrmion radius. In the continuum approximation, there is no limit on how small one can confine a skyrmion. In real samples, however, substantial shrinking of a skyrmion makes it increasingly unstable, and eventually the skyrmion will collapse.Rohart et al. (2016); Lobanov et al. (2016)

In this section, we examine the confinement effects and resulting deformation of a skyrmion within a high-DMI track. A single skyrmion was placed at the center of the track and subsequently relaxed, using the minimizer in Mumax3, for different widths $w$ and DMI strengths $D_{\mathrm{in}}$ of the track (outside the track, DMI was held at zero). The obtained change of geometry and size of the relaxed skyrmion is shown in Fig. 4.

For low DMI strength ($|D_{\mathrm{in}}|<D_{c}$), the effect of the confinement is only visible for narrow tracks. The size of the skyrmion converges to its expected size in an infinite film with given DMI, when increasing the width of the high-DMI track. For a stronger DMI in the track, a skyrmion will be elongated in the direction of the track. If the width of the track is above a threshold value, the skyrmion will expand along the track and convert into a stripe domain. This transition is represented in Fig. 4(d) by the ‘divergent’ size of the skyrmion in the $x$ direction.

In an extended ferromagnetic film with a strong DMI ($|D|>D_{c}$), the ground state is cycloidal. However, it is possible that the ground state becomes a triangular skyrmion lattice when applying an external field.Muhlbauer et al. (2009) For a high-DMI strip, we see a similar phenomenon. For example, the ground state in a high-DMI track of width $10\xi$ and $D_{\mathrm{in}}=1.3D_{c}$ is a single stripe domain. In the presence of an external magnetic field $B=0.3K_{\mathrm{eff}}/M_{\mathrm{sat}}$, the ground state becomes a skyrmion chain confined in the center of the track (see Fig. 5). For larger width of the track, the zigzag instability of the skyrmionic chain is expected, in analogy to similar studies on quasi-1D colloidalPiacente et al. (2010) and superconducting vortex systemsKarapetrov et al. (2009).

Considering a ferromagnetic strip with a spatially inhomogeneous DMI strength, a rotation in its magnetization is more favorable in regions with a strong DMI. We already showed that cycloidal states, including a single Néel domain wall, can be very effectively confined in straight high-DMI strips. We find that this is also the case for curved high-DMI strips, which means that a spatially-engineered DMI can be used to fix the location of the contained domain wall(s) and thereby design domains of arbitrary shape and size. Figure 6 demonstrates the proof of principle of such domain design approach by showing the lowest energy states of ferromagnetic films with (curved) high-DMI strips of different widths $w$ and shape.

Figures 6(g)-6(i) also demonstrate that small gaps in the high-DMI strips do not affect the end result. This is important to prove that the suggested method to fix the shape and the size of domains is robust against sample imperfections, which may be crucial for experimental realization. Note that the number of domain walls in the lowest-energy state depends on the width $w$ of the high-DMI region. This dependence is already discussed in detail for straight high-DMI strips in Sec. III.2. Recall also that in Sec. III.4 we showed that, in a high-DMI strip with $|D_{\mathrm{in}}|>D_{c}$, a skyrmion expands and eventually forms a cycloidal state with two walls parallel to the borders of the strip. This expansion can be used to create worm domains of desired shape in curved, pre-engineered high-DMI strips [see e.g. Fig. 6(e)] by first nucleating a skyrmion anywhere within the strip.

Domain walls are known to act as spin waveguides.Xing and Zhou (2016); Garcia-Sanchez et al. (2014, 2015); Borys et al. (2016) As shown in the preceding subsection, it is possible to fix the position, shape, and number of domain walls using a smoothly curved high-DMI strip and thereby also engineer the path for guidance of the spin waves. An example of a curved spin waveguide, based on a single Néel domain wall confined in a curved high-DMI strip, is shown in Fig. 7.

Using a confined domain wall in a high-DMI strip of an extended, heterochiral ferromagnet instead of a domain wall confined by the boundaries of a finite, monochiral ferromagnet, is beneficial for the robustness of the waveguide against deformations of the domain wall guiding the wave. For example, consider a domain wall mostly confined in a high-DMI strip but with a meandering deformation that places the domain wall partially inside the low-DMI region around the waveguide. This deformed domain wall will then relax towards its equilibrium state, back in the center of the confining high-DMI strip. This self-recovery mechanism is illustrated in Figs. 8(a)-8(c). Note that the geometrically designed ferromagnetic strips for domain-wall based waveguides do not have this self-healing property: Once the domain wall is deformed in such a way that it crosses the boundary of the waveguide, it will not relax back to the state with a single domain wall at the center of the strip [Figs. 8(d)-8(f)].

A spin-polarized current can move skyrmions along a racetrack. Due to drifting and, more importantly, the skyrmion Hall effect, it is possible for the skyrmion to leave the track. This process, however, has to overcome an energy barrier related to the repulsive force between the skyrmion and the edge of the racetrack. For strong spin currents, the skyrmion Hall effect can become sufficiently large to expel the skyrmion out of the racetrack.

Racetracks are usually designed by shape engineering of the ferromagnetic film. We propose an alternative method, in which the racetrack is created by a high-DMI strip in an extended ferromagnetic film, with weak or no DMI outside the strip. We compare the repulsive force between skyrmions and the track’s edges for the two design methods, by calculating the energy of a skyrmion as a function of its distance $d_{\bot}$ to the edge of the racetrack. This is done by relaxing the magnetization while keeping the magnetic moment at the center of the skyrmion fixed at a certain distance $d_{\bot}$ from the edge. The obtained energies and repulsive forces are shown in Figs. 9(a) and 9(b). When we subsequently relaxed the magnetization without fixing any spins, we see that the skyrmions move back to the center of the strip and they do not collapse, as long as there is a repulsive force with the edge. This proves that our fixed-spin method, used to determine the repulsive force, is justified.

A skyrmion placed close to the boundary of a finite chiral ferromagnet ($d_{\bot}<2.4\xi$ for $D=0.8D_{c}$) will exit through the boundary, as shown by the shorter energy/force curves in Figs. 9(a) and 9(b) for the shape-engineered racetrack. In the case of our racetrack, designed by heterogeneous DMI, the skyrmion can also escape from the DMI track by moving into the low DMI regions. This, however, requires overcoming a much larger energy barrier, especially if $D_{\mathrm{out}}$ is much smaller than $D_{\mathrm{in}}$. Consequently, the high-DMI track can sustain larger spin currents, without losing the skyrmion, than the ordinary shape-engineered DMI track. Figures 9(c) and 9(d) shows the path of a skyrmion in the two types of racetracks for the same spin current and inner DMI strength. For this current the skyrmion escapes from the shape-engineered track, whereas it keeps racing on the high-DMI track. This significantly improves the performance of the racetrack memory device and dramatically reduces its volatility to skyrmion collapse.