Boundary conditions for the Néel order parameter in a chiral antiferromagnetic slab

Understanding of the interaction of antiferromagnetic solitons including domain walls and skyrmions with boundaries of chiral antiferromagnetic slabs is important for the design of prospective antiferromagnetic spintronic devices. Here, we derive the transition from spin lattice to micromagnetic nonlinear $\sigma$ model with the corresponding boundary conditions for a chiral cubic G-type antiferromagnet and analyze the impact of the slab boundaries and antisymmetric exchange (Dzyaloshinskii-Moriya interaction) on the vector order parameter. We apply this model to evaluate modifications of antiferromagnetic domain walls and skyrmions upon interaction with boundaries for different strengths of the antisymmetric exchange. Due to the presence of the antisymmetric exchange, both types of antiferromagnetic solitons become broader when approaching the boundary and transform to a mixed Bloch-Néel structure. Both textures feel the boundary at the distance of about five magnetic lengths. In this respect, our model provides design rules for antiferromagnetic racetracks, which can support bulklike properties of solitons.

Figure 1

Chiral G-type antiferromagnet. Simple cubic crystal lattice with antiferromagnetically coupled spins (a) at the edge of a slab (b). The highlighted octamer $\mathit{\rho }=\{i,j,k\}$ contains eight spins, labeled as $\mathit{A},...,\mathit{H}$ (index $\mathit{\rho }$ is omitted). Less neighbors at the edge as compared to the bulk results in a tilt of spin moments from the anisotropy direction due to the exchange and DMI. (b) AFM slab is schematically colored to show two chiral domain walls, tilted at the edges of the slab. (c) The continuum counterpart of the octamer (a) with the Néel vector $\mathit{n}$, magnetization vector $\mathit{m}$, and auxiliary vector fields ${\mathit{u}}_{x,y,z}$. (d) The spin octamer in the bulk and its continuum counterpart possess no tilt from the anisotropy direction. The DMI vector is shown for the bulk DMI type. (e) Parametrization of the Néel vector $\mathit{n}$ in the local spherical reference frame with the polar and azimuthal angles $\theta$ and $\varphi$, respectively.

Figure 2

Ground state of a chiral antiferromagnetic slab. (a) Schematic representation of the changes of the ground state in confined AFM sample (width $2w<\infty$ along ${\mathit{e}}_y$ direction). At side faces, the Néel vector is tilted with the surface twist angle ${\theta }_{\text{st}}$. The ground state in the interior of the domain is shown within the transparent part of the sample. (b) Vertical component of the Néel vector at the center of the sample along the $y$ axis: for different strength of the bulk DMI $D$. Symbols and solid lines correspond to simulations and analytics [Eq. (6)], respectively. (c) Surface twist angle ${\theta }_{\text{st}}$ for different strength of the bulk DMI. The simulations are carried out for a rectangular slab of $120\times 120\times 160$ spins ($24\ell \times 24\ell \times 32\ell$) with the magnetic length $\ell =5a_0$.

Figure 3

Chiral domain walls in antiferromagnetic slabs. (a) Schematics of the domain wall (DW) in a confined sample of a constant width $2w_y<\infty$ along ${\mathit{e}}_y$ axis. (b) At the sample's edge, the domain wall acquires a tilt indicated with the angle $\beta \ne 0$. The tilt vanishes ($\beta =0$) in the bulk. The panel (b) shows the simulation results for $D/D_c=0.8$ with geometry and other material parameters the same as in Fig. 2. Dashed line represents the center of the domain wall. Arrows schematically show the surface twist $\delta \varphi$ of the domain wall phase $\varphi$ far from the side faces. (c) Domain wall profile with $p=-1$ is shown for the component of the order parameter $n_z$ along ${\mathit{e}}_x$ axis. (d) Domain wall width $\mathrm{\Delta }$ expands at the surface $z=0$ in comparison with the bulk. Solid lines and symbols correspond to the analytics (9) and simulations, respectively. (e) Domain wall phase $\varphi =-\pi /2+\delta \varphi$ [see panel (b)] at the sample's surface far from edges. (f) Tilt angle $\beta$ for different DMI. Dashed lines are guides to the eye.

Figure 4

Skyrmion in antiferromagnetic slabs. (a) Schematics of a skyrmion in a thick AFM slab. Its radii at the surface and in the bulk are marked as $R_{\text{sk}}^{\text{surf}}$ and $R_{\text{sk}}^{\text{bulk}}$, respectively. In the simulations we consider a slab consisting of $150\times 150\times 100$ spins with $\ell =5a_0$. (b),(c) Skyrmion radius as a function of the DMI strength. “Unstable” marker shows the region where the skyrmion of small radius is unstable. Symbols and solid lines correspond to simulations and analytics (11), (13), respectively. Dashed line is guide to the eye. (d) Skyrmion radius for different axial cross sections of the sample. Solid lines correspond to the ansatz of the skyrmion of small radius. (e) Width of the skyrmion of large radius measured at $R_{\text{sk}}^{\text{bulk}}$. (f) Skyrmion isosurfaces $n_{x,y,z}\left(\mathit{r}\right)=0$ in the center and at the bottom of the slab. While this is a purely Bloch skyrmion in the bulk, the $\mathit{n}$ is tilted when approaching the surface. (g),(h) Phase $\varphi$ at the top surface for different strengths of the DMI at the distance $x=R_{\text{sk}}^{\text{surf}}$ from the origin. Symbols and solid lines correspond to simulations and analytics, respectively. Dashed line is a guide to the eye. (i),(j) Phase $\varphi$ of $\mathit{n}$ at the distance $R_{\text{sk}}^{\text{bulk}}$ from the origin for skyrmions of small and large radius, respectively. Notations are the same as in panel (d).

Figure 5

Spin-flop and spin-flip transitions. (a) Symbols correspond to spin-lattice simulations and the line is a guide to the eye. Dashed lines correspond to the analytically found values of the spin-flop and spin-flip fields $B_{\text{sf}}$ and $B_{\text{x}}$. Insets (b) and (c) show a zoom of spin-flip and spin-flop regions in (a). Solid blue and dotted orange lines correspond to the samples with dimensions $50\times 50\times 20$ and $100\times 100\times 50$ spins, respectively. Dashed red and solid green lines correspond to simulations with periodic boundary conditions (BC). The field is tilted by $1^{\circ }$ angle from ${\mathit{e}}_z$ in the $xz$ plane.