Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Mapping the orbital wavefunction of the surface states in three-dimensional topological insulators

Abstract

Understanding the structure of the wavefunction is essential for depicting the surface states of a topological insulator. Owing to the inherent strong spin–orbit coupling, the conventional hand-waving picture of the Dirac surface state with a single chiral spin texture is incomplete, as this ignores the orbital components of the Dirac wavefunction and their coupling to the spin textures. Here, by combining orbital-selective angle-resolved photoemission experiments and first-principles calculations, we deconvolve the in-plane and out-of-plane p-orbital components of the Dirac wavefunction. The in-plane orbital wavefunction is asymmetric relative to the Dirac point. It is predominantly tangential (radial) to the k-space constant energy surfaces above (below) the Dirac point. This orbital texture switch occurs exactly at the Dirac point, and therefore should be intrinsic to the topological physics. Our results imply that the Dirac wavefunction has a spin–orbital texture—a superposition of orbital wavefunctions coupled with the corresponding spin textures.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Figure 1: ARPES energy–momentum intensity plots at the Γ point for s and p photon polarizations.
Figure 2: Deducing the orbital texture from constant energy surface intensity plots.
Figure 3: Sketch of the orbital texture switch deduced from the experimental and theoretical matrix elements.
Figure 4: OPRλ switches sign at the Dirac point.

Similar content being viewed by others

References

  1. Qi, X. L. & Zhang, S. C. The quantum spin Hall effect and topological insulators. Phys. Today 63, 33–38 (January, 2010).

    Article  Google Scholar 

  2. Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    ADS  Google Scholar 

  3. Qi, X-L. & Zhang, S-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    ADS  Google Scholar 

  4. Read, N. Topological phases and quasiparticle braiding. Phys. Today 65, 38–43 (July, 2012).

    Article  Google Scholar 

  5. Moore, J. E. The birth of topological insulators. Nature 464, 194–198 (2010).

    Article  ADS  Google Scholar 

  6. Basak, S. et al. Spin texture on the warped Dirac-cone surface states in topological insulators. Phys. Rev. B 84, 121401(R) (2011).

    Article  ADS  Google Scholar 

  7. Zhang, W., Yu, R., Zhang, H., Dai, X. & Fang, Z. First-principles studies of the three-dimensional strong topological insulators Bi2Te3, Bi2Se3 and Sb2Te3 . New J. Phys. 12, 065013 (2010).

    Article  ADS  Google Scholar 

  8. Yazyev, O. V., Moore, J. E. & Louie, S. G. Spin polarization and transport of surface states in the topological insulators Bi2Se3 and Bi2Te3 from first principles. Phys. Rev. Lett. 105, 266806 (2010).

    Article  ADS  Google Scholar 

  9. Hsieh, D. et al. A tunable topological insulator in the spin helical Dirac transport regime. Nature 460, 1101–1105 (2009).

    Article  ADS  Google Scholar 

  10. Pan, Z-H. et al. Electronic structure of the topological insulator Bi2Se3 using angle-resolved photoemission spectroscopy: Evidence for a nearly full surface spin polarization. Phys. Rev. Lett. 106, 257004 (2011).

    Article  ADS  Google Scholar 

  11. Xia, Y. et al. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nature Phys. 5, 398–402 (2009).

    Article  ADS  Google Scholar 

  12. Park, S. R., Kim, C. H., Yu, J., Han, J. H. & Kim, C. Orbital-angular-momentum based origin of Rashba-type surface band splitting. Phys. Rev. Lett. 107, 156803 (2011).

    Article  ADS  Google Scholar 

  13. Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).

    ADS  Google Scholar 

  14. Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).

    Article  ADS  Google Scholar 

  15. Moore, J. E. & Balents, L. Topological invariants of time-reversal-invariant band structures. Phys. Rev. B 75, 121306 (R) (2007).

    Article  ADS  Google Scholar 

  16. Zhang, H. et al. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nature Phys. 5, 438–442 (2009).

    Article  ADS  Google Scholar 

  17. Liu, C-X. et al. Model Hamiltonian for topological insulators. Phys. Rev. B 82, 045122 (2010).

    Article  ADS  Google Scholar 

  18. Damascelli, A., Hussain, Z. & Shen, Z-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003).

    Article  ADS  Google Scholar 

  19. Wang, Y. H. et al. Observation of a warped helical spin texture in Bi2Se3 from circular dichroism angle-resolved photoemission spectroscopy. Phys. Rev. Lett. 107, 207602 (2011).

    Article  ADS  Google Scholar 

  20. Ishida, Y. et al. Common origin of the circular-dichroism pattern in angle-resolved photoemission spectroscopy of SrTiO3 and CuxBi2Se3 . Phys. Rev. Lett. 107, 077601 (2011).

    Article  ADS  Google Scholar 

  21. Park, S. R. et al. Chiral orbital-angular momentum in the surface states of Bi2Se3 . Phys. Rev. Lett. 108, 046805 (2012).

    Article  ADS  Google Scholar 

  22. Bian, G. et al. Illuminating the surface spin texture of the Giant-Rashba quantum-well system Bi/Ag(111) by circularly polarized photoemission. Phys. Rev. Lett. 108, 186403 (2012).

    Article  ADS  Google Scholar 

  23. Liang, F. Hexagonal warping effects in the surface states of the topological insulator Bi2Te3 . Phys. Rev. Lett. 103, 266801 (2009).

    Article  Google Scholar 

  24. Zhang, H-J., Liu, C-X. & Zhang, S-C. Spin-orbital texture in topological insulators. Preprint at http://arxiv.org/abs/1211.0762 (2012).

  25. Cao, Y. et al. Coupled spin-orbital texture in a prototypical topological insulator. Preprint at http://arxiv.org/abs/1211.5998 (2012).

  26. Bansal, N. et al. Epitaxial growth of topological insulator Bi2Se3 film on Si(111) with atomically sharp interface. Thin Solid Films 520, 224–229 (2011).

    Article  ADS  Google Scholar 

  27. Luo, J-W. & Zunger, A. Design principles and coupling mechanisms in the 2D quantum well topological insulator HgTe/CdTe. Phys. Rev. Lett. 105, 176805 (2010).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We acknowledge helpful discussions with S-C. Zhang, S-R. Park, M. Hermele, A. Essin and G. Chen. The ARPES work was carried out at the Advanced Light Source, LBL, and was supported by the DOE Office of Basic Science by grant DE-FG02-03ER46066 and by the NSF under DMR-1007014. A.Z., X-W.Z. and J-W.L. were supported as part of the Center for Inverse Design, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under award number DEAC 36-08GO28308. X-W.Z. also acknowledges the administrative support of REMRSEC under NSF grant number DMR-0820518, Colorado School of Mines, Golden, Colorado. The Rutgers work was supported by IAMDN of Rutgers University, National Science Foundation (NSF DMR-0845464) and Office of Naval Research (ONR N000140910749), and the Brookhaven work was supported by the DOE under contract number DE-AC03-76SF00098. Both LBL and BNL are supported by the DOE, Office of Basic Energy Sciences.

Author information

Authors and Affiliations

Authors

Contributions

Y.C. led the experimental data taking and analysis. J.A.W., Q.W. and T.J.R. helped with the data taking, and S.K.M. with the instrument and decapping procedure. X-W.Z., J-W.L. and A.Z. carried out the density functional calculations. Y.C., X-W.Z., J-W.L. and A.Z. analyzed the results from the first-principles calculations. Z.X., A.Y., J.S. and G.D.G. prepared the single crystal samples, and M.B., N.B. and S.O. prepared the thin film samples. Y.C. and D.S.D. did the majority of the paper writing (with contributions from all coauthors) and D.S.D. directed the overall project.

Corresponding authors

Correspondence to Yue Cao or D. S. Dessau.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 1997 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cao, Y., Waugh, J., Zhang, XW. et al. Mapping the orbital wavefunction of the surface states in three-dimensional topological insulators. Nature Phys 9, 499–504 (2013). https://doi.org/10.1038/nphys2685

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys2685

This article is cited by

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing