Interplay of electric and magnetic fields in skyrmion phases of the classical Heisenberg model on a square lattice

TL;DR

This work addresses how combined magnetic and electric fields influence skyrmion phases in a square-lattice classical Heisenberg magnetoelectric model with DM interaction. Using Monte Carlo simulations, the authors map $(B_z,E)$ phase diagrams for two orientations of $oldsymbol{E}$, identifying FM, FE, spiral, SkX, SkG, and bimeron phases and the transitions between them. They demonstrate that electric fields can destroy or reshape skyrmion lattices, induce SkG and bimeron-rich states, and that magnetization and polarization co-evolve in a magnetoelectric–topological landscape, revealing a Skyrmion–quadrupole interplay. The results provide a microscopic, dual spin–dipolar view of ME coupling in multiferroics and suggest routes for electric-field control of topological textures with potential spintronic applications.

Abstract

Magnetic skyrmions, topologically stable spin textures, have attracted significant interest due to their potential applications in information storage and processing. They are typically stabilized by the Dzyaloshinskii-Moriya interaction in the presence of a magnetic field and can be manipulated by electric fields in magnetoelectric systems. Here we investigate, using Monte Carlo simulations, the behavior of skyrmions in a classical Heisenberg magnetoelectric model on the square lattice under combined magnetic and electric fields. We analyze spin and dipolar textures, structure factors, magnetization, chirality, and polarization for different field directions and magnitudes, identifying ferromagnetic, ferroelectric, spiral, skyrmion crystal, skyrmion gas, and bimeron phases, as well as the field-induced transitions between them. We find that the competition between electric and magnetic fields can destroy or transform skyrmion lattices into skyrmion-gas or bimeron phases. While magnetic fields induce chiral phases even in the presence of an electric field, electric fields strongly reshape the chiral region and deform skyrmion textures. This reciprocal influence between magnetic and electric orders reflects the intrinsic magnetoelectric coupling characteristic of multiferroic materials. Specifically, we observe the simultaneous sudden growth in magnetization with a switch-off in the polarization, typically observed in experiments. In this context, localized magnetoelectric entities, such as skyrmions carrying electric quadrupoles, exemplify the intertwined nature of spin and charge degrees of freedom, providing a microscopic basis for the control of topological states in ME systems and their potential use in spintronic applications.

Figures (12)

• Figure 1: Square lattice scheme. The blue arrows represent the primitive translation vectors $\Vec{e}_1 = (1,0)$ and $\Vec{e}_2 = (0,1)$, the green arrows are the DM vectors $\Vec{D} = D\delta\hat{\mathrm{r}}$, and the positions $r_i, r_j, r_k$ indicate the sites involved in the calculation of the local chirality in the triangle $\Delta$.
• Figure 2: Spin textures and perpendicular (to z) structure factors obtained at $E=0$ for representative values of $B_z$, showing, from left to right, the spiral , bimeron , skyrmion crystal , skyrmion gas , and ferromagnetic phases.
• Figure 3: (a) Magnetization ($M_z$), chirality ($\chi$), and polarization parallel to the electric field ($P_{xy}$) are shown as functions of the fields $B_z$ and $E_{xy}$ in the square lattice, for $J=1$ and $D=1$ and $\mathrm{T} = 0.0009$. Black lines in the chirality are the boundaries between phases determined from $\chi$ and $\Vec{S_\perp}(\Vec{q})$. The red rectangles mark the position of the textures presented in the second part of this figure. (b) Representative spin and dipolar moment textures and structure factors for $B_z = 0.2$ and $B_{z} = 0.4$.
• Figure 4: Spin and dipole moment textures of isolated skyrmions for fields $B_z = 0.5$ and $E_{xy} = 0, 1, 2$. A and B squares are used to highlight the behavior of applying an electric field.
• Figure 5: Specific heat and chirality per site for $B_z=0.4$ and $E_{xy}=0.6$ for lattice sizes L = 48, 60, 72.