Spin-Electron (s-d) Interaction in TI Thin Films
In this chapter, we propose the effective surface model, in which the interaction matrix depends on the position of localized spins relative to the slab surfaces. The s-d interaction constants are specific to a particular TI depending on whether it is a t
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Spin-Electron (s-d) Interaction in TI Thin Films
Conducting surface states in topological insulators (TIs) present a playground for studying the response of massless Dirac fermions to various perturbations such as applied fields, ternary alloy variation, magnetic doping, as well as superconducting and ferromagnetic proximity effects. The ability to control the magnetic properties of surface electrons is key to the technology that might bring TI into the current discourse of spintronic device applications. The way surface ferromagnetism acts on TI is that it opens an energy gap in the Dirac spectrum on a single surface. Ferromagnetism itself is affected by the surface electrons as the indirect exchange interaction, mediated by Dirac electrons and holes, may or may not cause ferromagnetic ordering, depending on the details of the energy spectrum. One of the approaches to studying the indirect exchange is based on a model that describes massless Dirac electrons interacting with localized spins by contact s-d interaction [1–6]. This model is simplistic in the sense that it is comprised of two terms of different origins: the Dirac model which is an effective theory obtained by projection of the bulk Hamiltonian onto surface states (see Chap. 1) and the s-d interaction on the surface which is normally assumed to be equivalent to that in the bulk. As a result, the s-d interaction constants are not specific to surface states in that they do not depend on the position of localized spin while one would expect the interaction to fade with the distance from the surface. This factor might be important as in real materials the magnetic ions are not pinned to a surface, but are rather distributed in the whole sample, making the simplified model not adequate for real situation. Besides, most models account for a single Dirac cone, and are thus not applicable to a thin film where the two Dirac cones on opposite surfaces couple with each other. A consistent approach to surface magnetism should operate with the effective s-d coupling calculated as a projection of the bulk Hamiltonian that comprises electrons, localized spins, and the contact s-d interaction between them. In this chapter we discuss an effective surface model which serves as a background for the analytical study of surface magnetism [7]. Within the effective surface model, the interaction matrix depends on the position of localized spins relative to the slab surfaces. © Springer Nature Switzerland AG 2020 V. Litvinov, Magnetism in Topological Insulators, https://doi.org/10.1007/978-3-030-12053-5_6
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6 Spin-Electron (s-d ) Interaction in TI Thin Films
It results in variable impurity spin textures if we move away from the surfaces. Also, s-d interaction constants depend on the parameters of the electron spectrum that makes the constants specific to a particular TI whether it is a thick sample with a single massless Dirac cone or a thin slab with gapped massive fermions. The interaction matrix is a function of the gate bias and is affected by the Rashba spin splitt
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