Spin-orbit scattering visualized in quasiparticle interference

Y. Kohsaka, T. Machida, K. Iwaya, M. Kanou, T. Hanaguri, and T. Sasagawa

Phys. Rev. B 95, 115307 – Published 15 March 2017

Abstract

In the presence of spin-orbit coupling, electron scattering off impurities depends on both spin and orbital angular momentum of electrons—spin-orbit scattering. Although some transport properties are subject to spin-orbit scattering, experimental techniques directly accessible to this effect are limited. Here we show that a signature of spin-orbit scattering manifests itself in quasiparticle interference (QPI) imaged by spectroscopic-imaging scanning tunneling microscopy. The experimental data of a polar semiconductor BiTeI are well reproduced by numerical simulations with the $T$ -matrix formalism that include not only scalar scattering normally adopted but also spin-orbit scattering stronger than scalar scattering. To accelerate the simulations, we extend the standard efficient method of QPI calculation for momentum-independent scattering to be applicable even for spin-orbit scattering. We further identify a selection rule that makes spin-orbit scattering visible in the QPI pattern. These results demonstrate that spin-orbit scattering can exert predominant influence on QPI patterns and thus suggest that QPI measurement is available to detect spin-orbit scattering.

Received 28 July 2016
Revised 4 February 2017

DOI:https://doi.org/10.1103/PhysRevB.95.115307

Physics Subject Headings (PhySH)

Metals Semiconductors Spin-orbit coupling

Doped semiconductors Single crystal materials Two-dimensional electron gas

Liquid helium cooling Scanning tunneling spectroscopy k dot p method

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Y. Kohsaka^1,*, T. Machida¹, K. Iwaya¹, M. Kanou², T. Hanaguri¹, and T. Sasagawa²

¹RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan
²Laboratory for Materials and Structures, Tokyo Institute of Technology, Yokohama, Kanagawa 226-8503, Japan

^*kohsaka@riken.jp

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Issue

Vol. 95, Iss. 11 — 15 March 2017

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Images

Figure 1
(a) A $45 \times 45 - {nm}^{2} d I / d V$ image taken at the Te-terminated surface of BiTeI. The image was taken at $- 10$ mV with a lock-in modulation voltage of 5 ${mV}_{rms}$ and a setup tunneling current of 0.2 nA at a setup bias voltage of 0.2 V. (b) Fourier transform of (a). (c) Schematic figures of the band structure of quasi-two-dimensional states at the Te-terminated surface observed by ARPES [16, 17, 18]. From left to right, a three-dimensional illustration of the band structure, the band dispersion in the $\bar{Γ} - \bar{M}$ direction, and a constant energy contour. The double-headed arrows denote dominant scattering channels producing the QPI. Spin directions are depicted by the arrows and markers colored in orange and blue: in-plane components are denoted by the arrows and out-of-plane components are denoted by the markers.
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Figure 2
Fitting results to determine the parameters of Eq. (1): (a) the Fermi surface [20], (b) band dispersion in the $\bar{Γ} - \bar{M}$ direction [20], and (c) QPI dispersion in the $\bar{Γ} - \bar{M}$ direction [24]. Open circles and solid curves are experimental data and fitting results, respectively. Since there is an energy offset $Δ E_{0}$ between ARPES and QPI dispersions, $E_{0}$ is given by the sum of two fitting parameters, $E_{0} = E_{0}^{ARPES} + Δ E_{0}$ , where $E_{0}^{ARPES} = - 0.179$ eV and $Δ E_{0} = - 0.173$ eV. The eigenvalues of the model Hamiltonian are shown in (d) constant energy contours and (e) dispersion along $\bar{M} - \bar{Γ} - \bar{K}$ .
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Figure 3
Fourier-transformed images of QPI patterns at $- 10$ meV, $| ρ (q, ω = - 10 meV) |$ , calculated with different scattering centers. (a) Scalar scattering. (b) Magnetic scattering. (c) Scalar and spin-orbit scattering with $c = 60 Å^{2}$ . The blue dashed lines in (a) depict scattering vectors expected from constant energy contours of the band dispersion. The inner and outer lines denote interband scattering between the spin-split bands and intraband scattering of the outer branch, respectively.
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Figure 4
Full simulation including spin-orbit scattering with $c = 80 Å^{2}$ and the setpoint effect with $V_{set} = 0.2$ V. (a) A side-by-side comparison between the experiment (left) and the full simulation (right) at $- 10$ meV. (b, c, d) Energy dependence (dispersion) in the $\bar{Γ} - \bar{M}$ direction. (e) Line profiles of (a) in the $\bar{Γ} - \bar{M}$ direction. An exponential background is subtracted from the experimental data.
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Figure 5
Simulations with the same parameters as Fig. 3 but at lower resolutions. Since the calculation of Fig. 3 took 4 s, calculations with direct $k$ summation of (a)–(e) are estimated to take 4 s, 1 h, 1 day, 1 month, and 7 years, respectively. (e) The same as Fig. 3 shown for comparison.
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Figure 6
Fourier-transformed images of QPI patterns at $- 10$ meV, calculated by (a) JDOS and (b) spin-dependent JDOS.
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