Electric field control of three-dimensional vortex states in core-shell ferroelectric nanoparticles

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The fundamental question whether the structure of curled topological states, such as ferroelectric vortices, can be controlled by the application of an irrotational electric field is open. In this work, we studied the influence of irrotational external electric fields on the formation, evolution, and relaxation of ferroelectric vortices in spherical nanoparticles. In the framework of the Landau-Ginzburg-Devonshire approach coupled with electrostatic equations, we performed finite element modeling of the polarization behavior in a ferroelectric barium titanate core covered with a “tunable” paraelectric strontium titanate shell placed in a polymer or liquid medium.A stable two-dimensional vortex is formed in the core after a zero-field relaxation of an initial random or poly-domain distribution of the polarization, where the vortex axis is directed along one of the core crystallographic axes. Subsequently, sinusoidal pulses of a homogeneous electric field with variable period, strength, and direction are applied. The field-induced changes of the vortex structure consist in the appearance of an axial kernel in the form of a prolate nanodomain, the growth of the kernel, an increasing orientation of the polarization along the field, and the onset of a single-domain state. We introduced the term “kernel” to name the prolate nanodomain developed near the vortex axis and polarized perpendicular to the vortex plane. In ferromagnetism, this region is generally known as the vortex core.Unexpectedly, the in-field evolution of the polarization includes the formation of Bloch point structures, located at two diametrically opposite positions near the core surface. After removal of the electric field, the vortex recovers spontaneously; but its structure, axis orientation, and vorticity can be different from the initial state. As a rule, the final state is a stable three-dimensional polarization vortex with an axial dipolar kernel, which has a lower energy compared to the initial purely azimuthal vortex. The nature of this counterintuitive result is a significant gain of the negative Landau energy in the axial region of the vortex due to the formation of a kernel, which is only partially compensated by an increase in positive energy of the depolarization field, polarization gradient, and elastic stress for a vortex with a prolate single-domain kernel.The analysis of the torque and electrostatic forces acting on the core-shell nanoparticle in an irrotational electric field showed that the torque acting on the vortex with a kernel tends to rotate the nanoparticle in such way that the vortex axis becomes parallel to the field direction. The vortex (with or without a kernel) is electrostatically neutral, and therefore the force acting on the nanoparticle is absent for a homogeneous electric field, and nonzero for the field with a strong spatial gradient.The vortex states with a kernel possess a manifold degeneracy, appearing from three equiprobable directions of vortex axis, clockwise and counterclockwise directions of polarization rotation along the vortex axis, and two polarization directions in the kernel. This multitude of the vortex states in a single core is promising for applications of core-shell nanoparticles and their ensembles as multi-bit memory and related logic units. The rotation of a vortex kernel over a sphere, possible for the core-shell nanoparticles in a soft matter medium with controllable viscosity, may be used to imitate qubit features.