Antiferromagnetic skyrmion crystals: Generation, topological Hall, and topological spin Hall effect

Skyrmions are topologically nontrivial, magnetic quasiparticles that are characterized by a topological charge. A regular array of skyrmions, a skyrmion crystal (SkX), features the topological Hall effect (THE) of electrons, which, in turn, gives rise to the Hall effect of the skyrmions themselves. It is commonly believed that antiferromagnetic skyrmion crystals (AFM-SkXs) lack both effects. In this Rapid Communication, we present a generally applicable method to create stable AFM-SkXs by growing a two-sublattice SkX onto a collinear antiferromagnet. As an example we show that both types of skyrmion crystals, conventional and antiferromagnetic, exist in honeycomb lattices. While AFM-SkXs with equivalent lattice sites do not show a THE, they exhibit a topological spin Hall effect. On top of this, AFM-SkXs on inequivalent sublattices exhibit a nonzero THE, which may be utilized in spintronics devices. Our theoretical findings call for experimental realization.

Figure 1

(a) Skyrmion and (b) antiferromagnetic skyrmion crystal on a honeycomb lattice. The spins at each site are represented by arrows. The lower hexagon represents (a) a ferromagnet and (b) a collinear antiferromagnet, on which the (antiferromagnetic) skyrmion layer has been deposited. Gray lines, forming the honeycomb lattice, represent exchange interactions with constant $J_1^{\mathrm{AB}}$; see text. White thin lines visualize the exchange coupling within a sublattice (among second-nearest sites) $J_1$.

Figure 2

Antiferromagnetic skyrmion crystals on a honeycomb lattice of size $\lambda =4.5a$ [see (b); $a$ lattice constant] characterized by vorticity $v$ and helicity $\gamma$ (azimuth of a spin $\mathrm{\Phi }=v\varphi +\gamma$, where $\varphi$ is the azimuth of the position vector with respect to the skyrmions center). Blue (red) circles denote a positive (negative) $z$ component of the spins, whereas arrows represent their in-plane components (modulus and direction). (a) Crystal of antiskyrmions ($v=-1;\gamma \approx -\pi /3$). (b) Crystal of antiferromagnetic Néel-type skyrmions (sublattice skyrmions $v=+1;\gamma \approx \frac{\pi }{2}\pm \frac{\pi }{2}$). (c) Crystal of antiferromagnetic Bloch-type skyrmions (sublattice skyrmions $v=+1;\gamma \approx \pi \pm \frac{\pi }{2}$). Parameters: $J_1=1.63042J;J_3=-J$, and $k_{\mathrm{B}}T=2.5J$; for (a) $J_1^{\mathrm{AB}}=+0.05J$ and $B_{\mathrm{A}}=+B_{\mathrm{B}}=0.9J$, while for (b) and (c) $J_1^{\mathrm{AB}}=-0.05J$ and $B_{\mathrm{A}}=-B_{\mathrm{B}}=0.9J$. This staggered magnetic field corresponds to a coupling to a collinear antiferromagnet with strength $0.9J$. Cluster size: $36\times 36$ sites per sublattice. (d) Realistic antiferromagnetic antiskyrmion crystal. Parameters: $J_1=0.42J;J_3=-0.665J;J_1^{\mathrm{AB}}=-1.751J$ on a $32\times 32$ cluster; $T$ and $B$ are as in (b) and (c). The size of the antiferromagnetic skyrmions is reduced ($\lambda =4a$).

Figure 3

Topological Hall conductivities in skyrmion crystals [black: charge conductivity (SkX); orange: spin conductivity (${\mathrm{SkX}}_S$)] and in antiferromagnetic skyrmion crystals [red: charge conductivity (AFM-SkX); blue: spin conductivity (AFM-${\mathrm{SkX}}_S$)] with 72 sites in the unit cell. The tight-binding parameters read (a) $t_1=t;t_2=0$ [case (i)] and (b) $t_1=0;t_2=t$ [case (ii)]; the coupling to the skyrmion texture equals $m=5t$. The hopping strengths are sketched in the insets. Conductivities are quantized in units of ${\sigma }_0=e^2/h$ (charge) and ${\sigma }_0=e/\left(4\pi \right)$ (spin) (see Refs. [26, 27]).

Figure 4

Topological spin Hall effect in an antiferromagnetic skyrmion crystal of generic case (ii) with 32 sites in the unit cell. (a) Electronic structure at the edge of the semi-infinite sample computed by Green's function renormalization [47, 48]. Black: bulk states, orange: edge states. (b) Deflection of electrons with opposite spins (blue and red arrows) in an AFM-SkX (schematic). (c) Deflection of electrons with equal spins in a SkX.

Figure 5

Topological Hall conductivity (SkX: black; asymmetric AFM-SkX: red; in units of ${\sigma }_0=e^2/h$). (a) Conductivity ${\sigma }_{xy}$ versus energy for differing on-site energies and second-nearest-neighbor strengths: $t_1=0.75t;t_2^{\mathrm{A}}=0.2t\ne t_2^{\mathrm{B}}=t;\delta \epsilon =2t$. (b) Same as (a) but with $\delta \epsilon =0$ and $t_1=0$. The coupling to the skyrmion texture is $m=5t$ in all cases. (c) DOS of the zero-field band structure for the parameters of (b).