Energy Bands in Topological Insulators
The chapter describes a real-life example of the topological insulator by calculating the surface energy spectrum in bismuth chalcogenides. The spin-momentum locking of the massless and massive Dirac fermions and the inverse spin-galvanic effect are discu
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Energy Bands in Topological Insulators
Topological insulators (TI) are identified by their gapless surface electronic states with the energy lying inside a bandgap of the bulk band structure. This makes the crystal conducting on the surface and insulating in the bulk. To study the magnetic and electrical properties of TI materials and to understand what features make them different from conventional semiconductors, we start by studying the energy spectrum of an example material system (Bi, Sb)2(Te, Se)3. Bismuth chalcogenides is not the only class of materials that shows a topological response to external electric and magnetic fields: we could also mention other TI materials such as II–VI compounds (Cd, Hg)Te and also IV–VI semiconductor alloys (Pb, Sn)(Te, Se) which are called crystalline topological insulators. Semiconductors from II–VI and IV–VI families are TI examples that are less convenient for experimental studies due to the small energy gaps and also because of the strongly non-stoichiometric growth that makes it difficult to pin the Fermi energy inside the energy gap of bulk crystal, where the topological effects become observable. The electrical properties of Bismuth chalcogenides and their device applications in thermoelectricity have been intensively studied and thus all the details of their energy spectrum, electron, and phonon scattering mechanisms have long been known. What makes them special as TI solids? We start from the microscopic approach, analyzing the band structure of bulk Bi2Se3 and the specific features of the surface electron states which differentiate them from conventional solids. A phenomenological approach that relates TI general properties to band topology and symmetry will be discussed in the next chapters.
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Spin–Orbit Interaction in Bismuth Chalcogenides
The bismuth chalcogenides family crystallizes in a rhombohedral structure with The structure consists of the set of quintuple point group D3d and space group R3m. layers illustrated in Fig. 1.1. Quintuple layers are coupled by the weak Van der © Springer Nature Switzerland AG 2020 V. Litvinov, Magnetism in Topological Insulators, https://doi.org/10.1007/978-3-030-12053-5_1
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1 Energy Bands in Topological Insulators
Fig. 1.1 Bismuth chalcogenide crystal structure. QL stands for a quintuple layer
Fig. 1.2 Spin–orbit coupling inverted energy levels corresponding to hybridized Bi (| P1i) and Se (Te) (jP2i) p-orbitals
Waals interaction that results in easy cleavage along planes {0001} perpendicular to axis z. Relevant electron energy bands have energy close to the Fermi level, they stem from Bi- and Se- p-orbitals and have been calculated in [1, 2]. The origin of conduction and valence energy levels are shown in the left panel of Fig. 1.2,
1.2 Electron Spectrum and Band Inversion
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where wave functions jP1 " #i and jP2 " #i stand for double spin degenerate linear combinations of the atomic Bi- and Te- p-orbitals, respectively. Wave functions have parity () with respect to inversion symmetry and also account for the e
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