Spin-orbit interaction of electrons on a curved surface

Éntin, M. V.; Magarill, L. I.

doi:10.1103/physrevb.64.085330

Cited by 111 publications

(82 citation statements)

References 16 publications

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“…39 that the interface contribution to the Rashba spin-orbit term in the 2D Hamiltonian 41 is related to discontinuity of the band parameters at the semiconductor heterointerface and that this contribution is an additional one to those connected with the space charge and/or the external electric field. The importance of the effects described by the Rashba term for the 2D electrons confined near the curved surface 42,43 and cylindrical semiconductor quantum dots 44 has been emphasized recently. The Rashba spin-orbit term in our spherical dots is a direct consequence of the GBCs for the envelope function.…”

Section: Discussionmentioning

confidence: 99%

Effect of the surface on the electron quantum size levels and electrongfactor in spherical semiconductor nanocrystals

2003

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The structure of the electron quantum size levels in spherical nanocrystals is studied in the framework of an eight-band effective mass model at zero and weak magnetic fields. The effect of the nanocrystal surface is modeled through the boundary condition imposed on the envelope wave function at the surface. We show that the spin-orbit splitting of the valence band leads to the surface-induced spin-orbit splitting of the excited conduction band states and to the additional surface-induced magnetic moment for electrons in bare nanocrystals. This additional magnetic moment manifests itself in a nonzero surface contribution to the linear Zeeman splitting of all quantum size energy levels including the ground 1S electron state. The fitting of the size dependence of the ground state electron g factor in CdSe nanocrystals has allowed us to determine the appropriate surface parameter of the boundary conditions. The structure of the excited electron states is considered in the limits of weak and strong magnetic fields.

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Section: Discussionmentioning

confidence: 99%

Effect of the surface on the electron quantum size levels and electrongfactor in spherical semiconductor nanocrystals

2003

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“…The curved geometry of nanotubes induces SO coupling on the order of 10 −1 meV among the single-particle levels of the electrons. 18 While the curvature-induced SO coupling was envisioned previously for semiconductors, 25,26 for nanotubes it is the dominant mechanism. It was Ando 27 and others 28,29 who developed the first theories of SO coupling in nanotubes.…”

Section: Introductionmentioning

confidence: 99%

Spin-orbit effects in carbon-nanotube double quantum dots

Weiss¹,

Rashba²,

Kuemmeth³

et al. 2010

Phys. Rev. B

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We study the energy spectrum of symmetric double quantum dots in narrow-gap carbon nanotubes with one and two electrostatically confined electrons in the presence of spin-orbit and Coulomb interactions. Compared to GaAs quantum dots, the spectrum exhibits a much richer structure because of the spin-orbit interaction that couples the electron's isospin to its real spin through two independent coupling constants. In a single dot, both constants combine to split the spectrum into two Kramers doublets while the antisymmetric constant solely controls the difference in the tunneling rates of the Kramers doublets between the dots. For the two-electron regime, the detailed structure of the spin-orbit split energy spectrum is investigated as a function of detuning between the quantum dots in a 22-dimensional Hilbert space within the framework of a single-longitudinalmode model. We find a competing effect of the tunneling and Coulomb interaction. The former favors a left-right symmetric two-particle ground state while in the regime where the Coulomb interaction dominates over tunneling, a left-right antisymmetric ground state is found. As a result, ground states on both sides of the ͑11͒-͑02͒ degeneracy point may possess opposite left-right symmetry, and the electron dynamics when tuning the system from one side of the ͑11͒-͑02͒ degeneracy point to the other is controlled by three selection rules ͑in spin, isospin, and left-right symmetry͒. We discuss implications for the spin-dephasing and Pauli blockade experiments.

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“…Band splitting was found considering surface curvature effects [9], as well as in the electron spin resonance spectra of achiral CNTs derived by low-energy theory [10]. In an earlier work, we showed that the inclusion of the full lattice symmetry is essential for deriving spin-orbit (SO) effects in CNTs [11].…”

mentioning

confidence: 99%

“…As an example for chiral tubes, the band structures calculated including the SOI for three particular tubes, (6,4), (9,3), and (8,4) belonging to each of the three families, are shown in Fig. 4 around the Fermi level.…”

mentioning

confidence: 99%

Curvature-induced anisotropic spin-orbit splitting in carbon nanotubes

2009

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We have theoretically explored the spin-orbit interaction in carbon nanotubes. We show that, besides the dependence on chirality and diameter, the effects of spin-orbit coupling are anisotropic: spin splitting is larger for the higher valence or the lower electron band depending on the specific tube. Different tube behaviors can be grouped in three families, according to the so-called chiral index. Curvature-induced changes in the orbital hybridization have a crucial role, and they are shown to be family-dependent. Our results explain recent experimental results which have evidenced the importance of spin-orbit effects in carbon nanotubes. PACS numbers: 71.20.Tx, 71.70.Ej Improvements in the quality of carbon nanotubes (CNTs) have enabled the fabrication of quantum dots aiming at the realization of spintronics devices [1,2,3,4]. CNTs present a high Fermi velocity and a twofold orbital degeneracy originating from the topology of the honeycomb lattice. The unique fourfold degeneracy of CNTs energy states (spin plus orbital moment) has been observed in CNT quantum dots (QDs) by magnetic field spectroscopy measurements [5] and makes them particularly interesting since, besides the spin degree of freedom, they present the orbital moment to allow for quantum manipulation. In a recent experiment [6], spin-orbit coupling has been directly observed in CNT as a splitting of the fourfold degeneracy of a single-electron energy level in ultra-clean QDs. This important finding seems to be in contradiction with the interpretation of earlier experiments in defect-free CNTs, from which independent spin and orbital symmetries and electron-hole symmetry have been deduced [7]. Besides showing the importance of spin-orbit effects in carbon nanotubes, Kuemmeth et al.[6] point out an unexplained anisotropic splitting of electron and holes in carbon nanotube quantum dots, which deserves further exploration.On theoretical grounds, spin-orbit interaction (SOI) has been investigated on CNTs by deriving an effective mass Hamiltonian including a weak SOI in carbon orbitals to the lowest order in perturbation theory [8]. Band splitting was found considering surface curvature effects [9], as well as in the electron spin resonance spectra of achiral CNTs derived by low-energy theory [10]. In an earlier work, we showed that the inclusion of the full lattice symmetry is essential for deriving spin-orbit (SO) effects in CNTs [11]. Employing an empirical tightbinding model, we demonstrated an intrinsic symmetry dependence of SOI effects. As confirmed by recent experimental results [6], we showed that, in the absence of a magnetic field, CNTs present spin-orbit split bands at the Fermi level. In addition, SOI induces zero-field spin splitting in chiral CNTs, while Kramers theorem on time-reversal symmetry alongside the inversion symmetry preserve the spin-degeneracy in achiral-i.e., (n, n) armchair and (n, 0) zigzag-nanotubes [12]. More recent works [13] have indicated the importance of curvature in the SOI effects investigated with a continuum ...

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Spin-orbit interaction of electrons on a curved surface

Cited by 111 publications

References 16 publications

Effect of the surface on the electron quantum size levels and electrongfactor in spherical semiconductor nanocrystals

Effect of the surface on the electron quantum size levels and electrongfactor in spherical semiconductor nanocrystals

Spin-orbit effects in carbon-nanotube double quantum dots

Curvature-induced anisotropic spin-orbit splitting in carbon nanotubes

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