Stagraczyński S., Chotorlishvili L., Dugaev V.K., Jia C.-L., Ernst A., Komnik A., and Berakdar J.

A key feature of topological insulators is the robustness of the electron energy spectrum. At a surface of a topological insulator, the Dirac point is protected by the characteristic symmetry of the system. The breaking of the symmetry opens a gap in the energy spectrum. Therefore, topological insulators are very sensitive to magnetic fields, which can open a gap in the electronic spectrum. Concerning ?internal? magnetic effects, for example, the situation with doped magnetic impurities, is not trivial. A single magnetic impurity is not enough to open the band gap, while in the case of a ferromagnetic chain of deposited magnetic impurities the Dirac point is lifted. However, a much more interesting case is when localized magnetic impurities form a chiral spin order. Our first principle density functional theory calculations have shown that this is the case for Fe deposited on the surface of a Bi₂Se₃ topological insulator. But not only magnetic impurities can form a chiral helicoidal spin texture. An alternative way is to use chiral multiferroics (prototype material is LiCu₂O₂ ) that induce a proximity effect. The theoretical approach we present here is valid for both cases. We observed that opposite to a ferromagnetically ordered case, a chiral spin order does not destroy the Dirac point. We also observed that the energy gap appears at the edges of the new Brillouin zone. Another interesting result concerns the spin dynamics. We derived an equation for the spin density dynamics with a spin current and relaxation terms. We have shown that the motion of the conductance electron generates a magnetic torque and exerts a certain force on the helicoidal texture.

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