The exchange interactions in insulators depend on the orbital state of magnetic ions, obeying certain phenomenological principles, known as Goodenough-Kanamori-Anderson rules. Particularly, the ferro order of alike orbitals tends to stabilize antiferromagnetic interactions, while the antiferro order of unlike orbitals favors ferromagnetic interactions. The Kugel-Khomskii theory provides a universal view on such coupling between spin and orbital degrees of freedom, based on the superexchange processes: namely, for a given magnetic order, the occupied orbitals tend to arrange in a way to further minimize the exchange energy. Then, if two magnetic sites are connected by the spatial inversion, the antiferro orbital order should lead to the ferromagnetic coupling and break the inversion symmetry. This constitutes the basic idea of our work, which provides a pathway for designing ferromagnetic ferroelectrics: the rare but fundamentally and practically important multiferroic materials. After illustrating the basic idea on toy-model examples, we propose that such behavior can be indeed realized in the van der Waals ferromagnet VI3, employing for this analysis the realistic model derived from first-principles calculations for magnetic 3⁢𝑑 bands. We argue that the intra-atomic interactions responsible for Hund's second rule, acting against the crystal field, tend to restore the orbital degeneracy of the ionic 𝑑2 state in VI3 and, thus, provide a necessary flexibility for activating the Kugel-Khomskii mechanism of the orbital ordering. In the honeycomb lattice, this orbital ordering breaks the inversion symmetry, stabilizing the ferromagnetic-ferroelectric ground state. The symmetry breaking leads to the canting of magnetization, which can be further controlled by the magnetic field, producing a huge change of electric polarization.

中文翻译：

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由 Kugel-Khomskii 轨道排序机制引起的铁磁铁电性，由原子 Hund 第二规则效应辅助


绝缘体中的交换相互作用取决于磁离子的轨道状态，遵循某些现象学原理，称为 Goodenough-Kanamori-Anderson 规则。特别是，相似轨道的铁序倾向于稳定反铁磁相互作用，而不同轨道的反铁序有利于铁磁相互作用。Kugel-Khomskii 理论基于超交换过程，为自旋和轨道自由度之间的这种耦合提供了普遍的观点：即，对于给定的磁序，被占用的轨道倾向于以进一步最小化交换能量的方式排列。然后，如果两个磁位点通过空间反转连接，则反铁轨道级应导致铁磁耦合并打破反转对称性。这构成了我们工作的基本思想，它为设计铁磁铁电体提供了一条途径：铁电材料是稀有但具有根本性和实践意义的多铁材料。在说明了玩具模型示例的基本思想之后，我们提出这种行为确实可以在范德华铁磁体 VI3 中实现，并在此分析中使用从磁性 3d 带的第一性原理计算中得出的真实模型。我们认为，负责 Hund 第二规则的原子内相互作用，与晶场相反，倾向于恢复 VI3 中离子 d2 态的轨道简并性，从而为激活轨道排序的 Kugel-Khomskii 机制提供了必要的灵活性。 在蜂窝晶格中，这种轨道排序打破了反转对称性，稳定了铁磁-铁电基态。对称性的打破导致磁化强度的倾斜，这可以通过磁场进一步控制，从而产生巨大的电极化变化。