We consider a harmonically trapped rotating spin-1 Bose–Einstein condensate with SU(3) spin–orbit coupling subject to a gradient magnetic field. The effects of SU(3) spin–orbit coupling, rotation, and gradient magnetic field on the ground-state structure of the system are investigated in detail. Our results show that the interplay among SU(3) spin–orbit coupling, rotation, and gradient magnetic field can result in a variety of ground states, such as a vortex ring and clover-type structure. The numerical results agree well with our variational analysis results.
We consider a binary dipolar Bose-Einstein condensate confined in a rotating harmonic plus quartic potential trap. The ground-state vortex structures are numerically obtained as a function of the contact interactions and the dipole-dipole interaction in both slow and rapid rotation cases. The results show that the vortex configurations depend strongly on the strength of the contact interactions, the relative strength between dipolar and contact interactions, as well as on the orientation of the dipoles. A variety of exotic ground-state vortex structures, such as pentagonal and hexagon vortex lattice, square vortex lattice with a central vortex, annular vortex lines, and straight vortex lines, are observed by turning such controllable parameters. Our results deepen the understanding of effects of dipole-dipole interaction on the topological defects.
Trivalent rare earth erbium ion (Er<sup>3+</sup>) doped titanium oxide (TiO<sub>2</sub>) can produce a very wide range of applications due to its excellent optoelectronic properties, standing out among many rare-earth-doped luminescent crystals. However, these issues regarding local structure and electronic properties have not been finalized. To address these problems, the the Crystal structure AnaLYsis by Particle Swarm Optimization (CALYPSO) method combined with the first-principles calculations are employed, and many converged structures of Er<sup>3+</sup>-doped TiO<sub>2</sub> was successfully obtained. Further structural optimization was performed by using the Vienna Ab Initio Simulation Package (VASP) software package, and we report for the first time that the lowest energy structure of Er<sup>3+</sup>-doped TiO<sub>2</sub> has the <em>P</em><span style="text-decoration:overline">4</span><em>m2</em> symmetry. It can be observed that the doped Er<sup>3+</sup> ions enter the host crystal and occupy the position of Ti<sup>4+</sup> ions, resulting in structural distortion, which eventually leads to the reduction of the local Er<sup>3+</sup> coordination site symmetry from D<sub>2d</sub> to C<sub>2v</sub>. We speculate that there are two reasons: 1. the difference in charge between Er<sup>3+</sup> ions and Ti<sup>4+</sup> ions leads to charge compensation; 2. the difference in electron radius is obvious, Er<sup>3+</sup> ions and Ti<sup>4+</sup> ions are 0.0881 and 0.0605 nm, respectively. In addition, during the structural search process, we also find many metastable structures that may exist at a special temperature or pressure, which play an important role in the study of structural evolution. When calculating the electronic band structure of the Er<sup>3+</sup>-doped TiO<sub>2</sub> system, we adopted the method of local density approximation (LDA) combined with the on-site Coulomb repulsion parameter <em>U</em> to accurately describe the strongly correlated system. For the specific value of U, we adopted 3.5 eV and 7.6 eV to describe the strong correlation of 3<em>d</em> electrons of Ti<sup>4+</sup> ions and 4<em>f</em> electrons of Er<sup>3+</sup> ions, respectively. According to the calculation of electronic properties, the band gap value of Er<sup>3+</sup> doped TiO<sub>2</sub> is about 2.27 eV, which is lower than that of the host crystal (<em>Eg</em> = 2.40 eV). The results show that the reduction in the band gap is mainly caused by the f state of Er<sup>3+</sup> ions. The doping of Er ion does reduce the band gap value, but it does not change the conductivity of the system, which have great application prospect in diode-pumped laser. These findings not only provide data for further exploration of the properties and applications of Er<sup>3+</sup>:TiO<sub>2</sub> crystals, but also provide a systematic approach for the study of other rare-earth-doped crystalline materials.
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