Anomalous behavior of the electronic structure of (${\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{In}}_{x}{)}_{2}{\mathrm{Se}}_{3}$ across the quantum phase transition from topological to trivial insulator

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Anomalous behavior of the electronic structure of ( ${Bi}_{1 - x} {In}_{x})_{2} {Se}_{3}$ across the quantum phase transition from topological to trivial insulator

J. Sánchez-Barriga, I. Aguilera, L. V. Yashina, D. Y. Tsukanova, F. Freyse, A. N. Chaika, C. Callaert, A. M. Abakumov, J. Hadermann, A. Varykhalov, E. D. L. Rienks, G. Bihlmayer, S. Blügel, and O. Rader

Phys. Rev. B 98, 235110 – Published 5 December 2018

Abstract

Using spin- and angle-resolved photoemission spectroscopy and relativistic many-body calculations, we investigate the evolution of the electronic structure of ${({Bi}_{1 - x} {In}_{x})}_{2} {Se}_{3}$ bulk single crystals around the critical point of the trivial to topological insulator quantum-phase transition. By increasing $x$ , we observe how a surface gap opens at the Dirac point of the initially gapless topological surface state of ${Bi}_{2} {Se}_{3}$ , leading to the existence of massive fermions. The surface gap monotonically increases for a wide range of $x$ values across the topological and trivial sides of the quantum-phase transition. By means of photon-energy-dependent measurements, we demonstrate that the gapped surface state survives the inversion of the bulk bands which occurs at a critical point near $x = 0.055$ . The surface state exhibits a nonzero in-plane spin polarization which decays exponentially with increasing $x$ , and which persists in both the topological and trivial insulator phases. Our calculations reveal qualitative agreement with the experimental results all across the quantum-phase transition upon the systematic variation of the spin-orbit coupling strength. A non-time-reversal symmetry-breaking mechanism of bulk-mediated scattering processes that increase with decreasing spin-orbit coupling strength is proposed as explanation.

Received 3 July 2018
Revised 20 September 2018

DOI:https://doi.org/10.1103/PhysRevB.98.235110

Physics Subject Headings (PhySH)

Electronic structure First-principles calculations Quantum phase transitions Spin-orbit coupling Surface states Topological materials

Topological insulators

Angle-resolved photoemission spectroscopy Density functional calculations GW method Spin-resolved photoemission spectroscopy Tight-binding model Wannier function methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

J. Sánchez-Barriga^1,*, I. Aguilera², L. V. Yashina³, D. Y. Tsukanova³, F. Freyse¹, A. N. Chaika⁴, C. Callaert⁵, A. M. Abakumov⁶, J. Hadermann⁵, A. Varykhalov¹, E. D. L. Rienks¹, G. Bihlmayer², S. Blügel², and O. Rader¹

¹Helmholtz-Zentrum Berlin, Albert-Einstein-Str. 15, D-12489 Berlin, Germany
²Peter Grünberg Institute and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany
³Department of Chemistry, Moscow State University, Leninskie Gory 1/3, 119991 Moscow, Russia
⁴Institute of Solid State Physics RAS, Academician Ossipyan str. 2, Chernogolovka, 142432 Moscow District, Russia
⁵EMAT, Department of Physics, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium
⁶Skolkovo Institute of Science and Technology, 3 Nobel Street, 143026 Moscow, Russia

^*Corresponding author: jaime.sanchez-barriga@helmholtz-berlin.de

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Vol. 98, Iss. 23 — 15 December 2018

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Figure 1
(a)–(f) High-resolution ARPES dispersions ( $h ν = 50 eV$ ) for $x$ ranging from (a) 0 to (f) 0.087. The vertical white spectra on each panel represent EDCs obtained in normal emission ( $k_{∥}$ $= 0$ ). (g) Size of the band gaps relative to the transition point. Red (green) solid symbols are the experimental bulk (surface) band gap, and solid lines the corresponding calculations referenced to the SOC strength. The red dashed line is the bulk band gap as extracted from photoluminescence experiments. The region of the transition point is emphasized in red. (h) STM image ( $x = 0.049$ ) taken at $U = - 0.8$ V and $I = 60$ pA revealing three different types of In-related defects, protrusions, and depressions (triangular or circular). Inset: atomically resolved STM image taken at $U = - 20$ mV and $I = 57$ pA. (i) High-resolution averaged EDX map ( $x = 0.049$ ): HAADF-STEM image (top left), partial composition maps (top right), and individual Bi, Se, and In maps (bottom).
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Figure 2
(a), (b) Full ARPES maps obtained with 50-eV photons around the $\bar{Γ}$ point of the SBZ for (a) $x = 0.049$ and (b) $x = 0.065$ . In each case, the surface gap $Δ$ is also indicated. (c)–(h) Photon-energy dependence of the ARPES dispersions, indicating that the gapped surface states are two dimensional. This is clearly seen at photon energies above 37 eV where the intensity of the surface states is enhanced with respect to the one of bulk states. The surface gap is highlighted by horizontal dashed lines [(c)–(e) $x = 0.049$ and (f)–(h) $x = 0.065]$ . (i) Dependence of the bulk band gap on In content. The data were obtained after sampling the dispersion of the bulk bands as a function of $k_{⊥}$ for photon energies below 36 eV, where the intensity of the bulk bands is considerably enhanced. The resulting dispersions were then integrated over $k_{⊥}$ around $k_{∥} = 0$ . The evolution of the (direct) bulk band gap is indicated by vertical solid lines, which are guides to the eye.
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Figure 3
Band structure of a 15-QL slab of ${Bi}_{2} {Se}_{3}$ obtained with a tight-binding model based on $G W$ with different SOC strengths: (a) and (f) 100% SOC (nontrivial); (b) and (g) 90% (nontrivial); (c) and (h) 77% (nontrivial, very close to the transition); (d) and (i) 70% (trivial); (e) and (j) 60% (trivial). The color scheme in panels (a)–(e) represents the localization of the states on the topmost QL. The bulk states are displayed as a gray background. The color in panels (f)–(j) shows the in-plane spin polarization, and the black dots represent the wave vector at which the spin polarization is discussed in Figs. 4 and 5 ( $k_{∥} = 0.07 Å^{- 1}$ ).
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Figure 4
(a)–(d) Selected spin-resolved ARPES measurements taken for In contents $x$ of (a), (c) 0 and (b), (d) 0.049 at 50 eV, revealing a decrease of the in-plane spin polarization. (a), (b) Spin-resolved EDCs obtained at $k_{∥} = - 0.07$ $Å^{- 1}$ . The blue (red) curves show spin-up (-down) EDCs, and the orientation of the in-plane spin polarization is perpendicular to $k_{∥}$ . (c), (d) Corresponding in-plane spin polarizations (green curves), which reverse at opposite momentum (gray curves). (e) Absolute magnitude of the in-plane spin polarization across the topological to trivial quantum-phase transition at $k_{∥} = 0.07$ $Å^{- 1}$ . The green solid symbols are experimental data, and the red solid line the corresponding calculation. The TSS exhibits an unconventional transformation to a gapped surface state which is spin polarized and persists on the trivial phase. Top-right inset in (e): out-of-plane spin polarization for $x = 0.049$ at $k_{∥} = 0$ .
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Figure 5
(a) Wave functions (wfns) decay in real space: contribution from atoms of each QL (horizontal axis) to the wave function of the surface state at $\sim 0.07 Å^{- 1}$ , corresponding to the black circles in Figs. 3–3. The symbol size indicates the contribution from atoms of each QL to the in-plane spin polarization. Results are represented for different SOC strengths as indicated by the color scale: green, nontrivial; gray, transition; purple, trivial. Inset: zoom-in on the lower right part of the figure. (b), (c) Sketch of the proposed gapping mechanism on the topological side of the phase transition, depicting how trivial inhomogeneities are responsible for the opening of a surface gap (see text).
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Physical Review B

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Anomalous behavior of the electronic structure of ( ${Bi}_{1 - x} {In}_{x})_{2} {Se}_{3}$ across the quantum phase transition from topological to trivial insulator

J. Sánchez-Barriga, I. Aguilera, L. V. Yashina, D. Y. Tsukanova, F. Freyse, A. N. Chaika, C. Callaert, A. M. Abakumov, J. Hadermann, A. Varykhalov, E. D. L. Rienks, G. Bihlmayer, S. Blügel, and O. Rader

Phys. Rev. B 98, 235110 – Published 5 December 2018

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Anomalous behavior of the electronic structure of (Bi1−xInx)2Se3 across the quantum phase transition from topological to trivial insulator

J. Sánchez-Barriga, I. Aguilera, L. V. Yashina, D. Y. Tsukanova, F. Freyse, A. N. Chaika, C. Callaert, A. M. Abakumov, J. Hadermann, A. Varykhalov, E. D. L. Rienks, G. Bihlmayer, S. Blügel, and O. Rader

Phys. Rev. B 98, 235110 – Published 5 December 2018

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Anomalous behavior of the electronic structure of ( ${Bi}_{1 - x} {In}_{x})_{2} {Se}_{3}$ across the quantum phase transition from topological to trivial insulator