Magnetoelectric effect and orbital magnetization in skyrmion crystals: Detection and characterization of skyrmions

Skyrmions are small magnetic quasiparticles, which are uniquely characterized by their topological charge and their helicity. In this Rapid Communication, we show via calculations how both properties can be determined without relying on real-space imaging. The orbital magnetization and topological Hall conductivity measure the arising magnetization due to the circulation of electrons in the bulk and the occurrence of topologically protected edge channels due to the emergent field of a skyrmion crystal. Both observables quantify the topological Hall effect and distinguish skyrmions from antiskyrmions by sign. Additionally, we predict a magnetoelectric effect in skyrmion crystals, which is the generation of a magnetization (polarization) by application of an electric (magnetic) field. This effect is quantified by spin toroidization and magnetoelectric polarizability. The dependence of the transverse magnetoelectric effect on the skyrmion helicity fits that of the classical toroidal moment of the spin texture and allows one to differentiate skyrmion helicities: It is largest for Bloch skyrmions and zero for Néel skyrmions. We predict distinct features of the four observables that can be used to detect and characterize skyrmions in experiments.

Figure 1

Core message of this Rapid Communication. (a) Skyrmions and antiskyrmions are distinguished by the topological Hall effect. (b) The helicity of skyrmions (e.g., Bloch and Néel skyrmions) is differentiated by the magnetoelectric effect. With these quantities, (c) Bloch skyrmions, (d) Néel skyrmions, and (e) antiskyrmions can be distinguished. The color scale in (c)–(e) indicates the in-plane orientation of the spins (arrows).

Figure 2

Properties of a skyrmion crystal. Parameters read $n_{\mathrm{b}}=36$ (sites in the skyrmion unit cell), coupling $m=5t$. A Bloch $(N_{\mathrm{Sk}}=+1,\gamma =\pi /2$; topological charge and helicity), a Néel $\left(N_{\mathrm{Sk}}=+1,\gamma =0\right)$, and an antiskyrmion $\left(N_{\mathrm{Sk}}=-1,\gamma =0\right)$ are compared. (a) Band structure, (b) TH conductivity ${\sigma }_{xy}$, (c) orbital magnetization $M_z$, (d) magnetoelectric polarizability ${\alpha }_{xy}$, and (e) spin toroidization $T_z$ are separated into blocks in which the electron spins are aligned parallel [red in (a)] or antiparallel (blue) with the skyrmion magnetic texture $({\sigma }_0\equiv e^2/h,M_0\equiv te/\hslash ,{\alpha }_0\equiv g{\mu }_{\mathrm{B}}e/at$, and $T_0\equiv g{\mu }_{\mathrm{B}}/a$; $a$ is the lattice constant). The band structure is identical for all skyrmion types. In (d) results for an intermediate skyrmion with $\gamma =\pi /6$ are shown in addition (green). Colored dots refer to Fig. 3. (f) ${\alpha }_{xy}$ in the strong-coupling limit $m=900t$ and (g) for larger skyrmions $n_b=48$ on a different lattice (triangular).

Figure 3

Dependence of the classical toroidal moment $t_z$ (blue, red) and the magnetoelectric polarizability ${\alpha }_{xy}$ on the helicity of a skyrmion for selected Fermi energies $E_{\mathrm{F}}$ [distinguished by color, as indicated; also marked in Fig. 2]. $t_z$ is proportional to ${\alpha }_{xy}$, with the proportionality factor depending on $E_{\mathrm{F}}$. $t_0\equiv g{\mu }_{B}a,{\alpha }_0\equiv g{\mu }_{\mathrm{B}}e/at$.