Rashba Effect in Topological Quantum Wells
The chapter deals with the zero magnetic field spin-splitting in topological quantum wells. It is shown that the splitting acquires specific features if the bulk is in the non-trivial topological phase.
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Rashba Effect in Topological Quantum Wells
Zero magnetic spin-splitting in quantum wells is an important tool in semiconductor spintronics. The Rashba coefficient which quantitively describes the splitting has been calculated based on a microscopic theory in two instances, both regarding topologically trivial materials: cubic [1–3] and wurtzite [4, 5] III-V semiconductors. The near-surface region in the topological insulator Bi2Se3 is characterized by a large Rashba coefficient of (0.79 1.3)1010eVm [6, 7]. Topological invariants establish the connection of surface electrons to the topology of the TI bulk band structure, predicting stable and non-trivial surface states. A question arises whether and how the non-trivial bulk topology affects the bulk electronic properties of a solid. In this chapter, we consider the zero magnetic field spin-splitting of bulk origin that acquires specific features when the bulk is in the non-trivial topological phase.
5.1
Rashba Interaction in Normal Semiconductors
Spin–orbit interaction creates zero magnetic field spin-splitting in crystals and heterostructures if their symmetry does not include spatial inversion (for review, see [8, 9] and [10] for III-V cubic and wurtzite semiconductors, respectively). The theory of linear-in-k spin-splitting of electron states was developed first for wurtzite bulk materials [11–13] and then for non-centrosymmetric heterostructures [14–17]. Spin-splitting follows from symmetry considerations and can be described by the phenomenological constant called the Rashba coefficient. The model Hamiltonian was first proposed to describe spin–orbit interaction in wurtzite semiconductors, where linear-k spin-splitting is caused by bulk inversion asymmetry:
© Springer Nature Switzerland AG 2020 V. Litvinov, Magnetism in Topological Insulators, https://doi.org/10.1007/978-3-030-12053-5_5
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5 Rashba Effect in Topological Quantum Wells
H R ¼ Ec þ
h2 k 2 h2 k 2 þ αR k y σ x k x σ y ¼ E c þ þ αR ½k σ z : 2m 2m
ð5:1Þ
Since (5.1) accounts for the conduction band only and neglects coupling to other bands, one can consider αR to be a phenomenological constant. Its microscopic origin is discussed in [10]. The effective mass m is introduced to account for k2-order contributions from the remote energy bands. It should be noted that spin-orbit linear-k terms also exist in zinc-blende materials for wave vectors in the [111] and [110] directions. There, the terms appear in the expanded set of initial basis functions that accounts for d-states [18]. The Hamiltonian (5.1) is writtenin the spinorbasis that is the pair of eigen1 0 vectors of Pauli matrix σ z : j"i ¼ , j #i ¼ . These spinors are not 0 1 eigenstates of the Hamiltonian, so the Hamiltonian is non-diagonal in spin indexes. Eigenvalues of HR represent the spin-dependent electron energy spectrum referenced to the edge of the conduction band: E ¼ E0 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 k 2 mα2 αR k, k ¼ k2x þ k2y , E0 ¼ 2 : 2m h
ð5:2Þ
Two branches of the spectrum are illustrated in Fig. 5.1. Time reversa
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