Spin-orbit terms in multi-subband electron systems: a bridge between bulk and two-dimensional Hamiltonians
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OW-DIMENSIONAL SYSTEMS
Spin–Orbit Terms in Multi-Subband Electron Systems: a Bridge between Bulk and Two-Dimensional Hamiltonians1 K. V. Kavokina and M. E. Portnoib aIoffe
Physicotechnical Institute, St. Petersburg, 194021 Russia e-mail: [email protected] bSchool of Physics, University of Exeter, Exeter EX4 4QL, United Kingdom e-mail: [email protected] Received February 6, 2008; accepted for publication February 11, 2008
Abstract—We analyze the spin–orbit terms in multi-subband quasi-two-dimensional electron systems, and how they descend from the bulk Hamiltonian of the conduction band. Measurements of spin–orbit terms in one subband alone are shown to give incomplete information on the spin–orbit Hamiltonian of the system. They should be complemented by measurements of inter-subband spin–orbit matrix elements. Tuning electron energy levels with a quantizing magnetic field is proposed as an experimental approach to this problem. PACS numbers: 73.21.Fg, 71.70.Ej, 73.90.+f DOI: 10.1134/S1063782608080198
Spin-dependent phenomena in semiconductors was one of the favorite research topics of Vladimir Idelevich Perel’ since the early 1970s [1]. From the mid-1980s his interests shifted towards spin-related effects in lowdimensional systems, starting with optical orientation and polarization properties of hot photoluminescence in quantum-well structures, which was closely connected to experiments carried out by the group of Mirlin [2]. Some of this work was done with one of us (MEP) [3]. In the series of more recent papers, the transition from the two-dimensional to quasi-three dimensional case was considered [4]. Several late publications of Vladimir Idelevich were focused on spindependent tunneling and the role played in it in spin– orbit interaction [5]. This has defined the subject choice for our contribution to the special issue devoted to his memory. The spin–orbit interaction in semiconductors has been widely discussed recently in relation to some proposals of spin-electronic and quantum-computing devices. It is of considerable physical interest in itself, because due to strong gradients of atomic potentials within the crystal unit cell, the spin–orbit terms in the effective-mass Hamiltonian are often greatly enhanced with respect to those of a free electron [6]. In addition, the reduced crystal symmetry brings about new spin– orbit terms unknown for free particles, the so-called Dresselhaus terms [7]. In two-dimensional systems, spin–orbit effects are known to be even stronger than in bulk semiconductors. 1 The
text was submitted by the authors in English.
In particular, in the effective-mass two-dimensional (2D) Hamiltonian there appear spin–orbit terms which are linear in the 2D wave vector k. They may exist in structures where the spatial inversion symmetry is broken. There are so-called bulk inversion asymmetry (BIA) terms, which appear in averaging the bulk Dresselhaus terms over the envelope function of the corresponding size-quantization level, and structure inversion asymmetry (SIA), or Ra
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