Hall Effects and Berry Phase
The chapter gives basic information about the Berry phase and parameters that differentiate TI from conventional dielectrics. The integer quantum Hall and intrinsic anomalous Hall effects are discussed in relation to the topology of Hilbert space in cryst
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Hall Effects and Berry Phase
The classification of topological phases requires parameters that differentiate TI from conventional dielectrics. Solid state phases that differ in crystalline symmetry can be characterized by the order parameter. That’s what the Landau theory of phase transitions deals with. The onset of the nonzero order parameter signals the breaking of symmetry and the transition to a low-symmetry phase when temperature or other system parameters change. TI cannot be described this way as no symmetry breaking occurs at the topological phase transition and thus no order parameter exists. The topologically non-trivial phase can be distinguished by topological invariants, the numbers that characterize the topology of the Hilbert space, in other words, certain properties of electron wave functions as they flow during the movement of electron momentum across the first Brillouin zone (BZ). The state with a nonzero topological invariant reveals unusual properties of a crystal boundary where trivial and non-trivial regions touch each other: backscattering is forbidden for electrons pinned to the interface (edge, surface). This protects topological interfaces against electron scattering on crystal imperfections. An example topological interface is the edge of a two-dimensional Quantum Hall Effect (QHE) system where the time reversal symmetry is broken by a perpendicular magnetic field and the Chern number plays the role of a topological invariant that protects the chiral electron edge states. Another example is the TI surface, also called Quantum Spin Hall Effect (QSHE) system, where time reversal symmetry holds and spin-resolved counterpropagating surface states (see Chap. 1) are protected by the Z2 invariant [1]. Below we discuss various Hall effects and their common features related to the topological properties of the energy bands.
© Springer Nature Switzerland AG 2020 V. Litvinov, Magnetism in Topological Insulators, https://doi.org/10.1007/978-3-030-12053-5_2
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2 Hall Effects and Berry Phase
Hall Effects
The brief discussion of Hall effects in this section is preliminary in nature and concerns phenomenological relations. More details will be given throughout the text. From here onward in the text we focus on intrinsic mechanisms that are not related to the specifics of impurity scattering. The conventional Hall effect (HE) is normally studied with the help of the Boltzmann equation [2, 3]; however, the basics can be qualitatively understood from the classical picture of electron motion in an external magnetic field. In the Hall setting, an external magnetic field is perpendicular to the driving electric field E, as shown in Fig. 2.1. The Hall voltage proportional to EH stems from the Lorenz force that deflects electrons in the direction perpendicular to the drift velocity, EH ¼ v B, where B is the magnetic field. In an isotropic resistive media, the Hall conductivity σ Hxy enters Ohm’s law as follows: J x ¼ σ Ω E x þ σ Hxy E H , J y ¼ σ Hxy Ex þ σ Ω E H ,
ð2:1Þ
where e2 nτ , σΩ ¼ m 1 þ
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