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:

Direct determination of spin–orbit interaction coefficients and realization of the persistent spin helix symmetry

Subjects

Abstract

The spin–orbit interaction plays a crucial role in diverse fields of condensed matter, including the investigation of Majorana fermions, topological insulators, quantum information and spintronics. In III–V zinc-blende semiconductor heterostructures, two types of spin–orbit interaction—Rashba and Dresselhaus—act on the electron spin as effective magnetic fields with different directions. They are characterized by coefficients α and β, respectively. When α is equal to β, the so-called persistent spin helix symmetry is realized. In this condition, invariance with respect to spin rotations is achieved even in the presence of the spin–orbit interaction, implying strongly enhanced spin lifetimes for spatially periodic spin modes. Existing methods to evaluate α/β require fitting analyses that often include ambiguity in the parameters used. Here, we experimentally demonstrate a simple and fitting parameter-free technique to determine α/β and to deduce the absolute values of α and β. The method is based on the detection of the effective magnetic field direction and the strength induced by the two spin–orbit interactions. Moreover, we observe the persistent spin helix symmetry by gate tuning.

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

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

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

Figure 1: Weak localization anisotropy to the relative angle between Beff and Bin.
Figure 2: Numerical results for conductance in quasi-one-dimensional quantum wires.
Figure 3: Schematic illustration of measurement configuration.
Figure 4: Weak localization anisotropy in the [100] direction wire at Vg = −5 V.
Figure 5: Polar plots of weak localization amplitudes as a function of θin.
Figure 6: Numerically calculated and experimental weak localization amplitude as a function of |Bin|/|Beff| ratio.

Similar content being viewed by others

References

  1. Mourik, V. et al. Signatures of majorana fermions in hybrid superconductor–semiconductor nanowire devices. Science 336, 1003–1007 (2012).

    Article  CAS  Google Scholar 

  2. Chen, Y. L. et al. Experimental realization of a three-dimensional topological insulator, Bi2Te3 . Science 325, 178–181 (2009).

    Article  CAS  Google Scholar 

  3. Datta, S. & Das, B. Electronic analog of the electro-optic modulator. Appl. Phys. Lett. 56, 655–657 (1990).

    Article  Google Scholar 

  4. Nitta, J., Akazaki, T., Takayanagi, H. & Enoki, T. Gate control of spin–orbit interaction in an inverted In0.53Ga0.47As/In0.52Al0.48As heterostructure. Phys. Rev. Lett. 78, 1335–1338 (1997).

    Article  CAS  Google Scholar 

  5. Engels, G., Lange, J., Schäpers, T. & Lüth, H. Experimental and theoretical approach to spin splitting in modulation-doped InxGa1−xAs/InP quantum wells for B → 0. Phys. Rev. B 55, R1958–R1961 (1997).

    Article  CAS  Google Scholar 

  6. Kohda, M. et al. Spin–orbit induced electronic spin separation in semiconductor nanostructures. Nature Commun. 3, 1038 (2012).

    Article  Google Scholar 

  7. König, M. et al. Direct observation of the Aharonov–Casher phase. Phys. Rev. Lett. 96, 076804 (2006).

    Article  Google Scholar 

  8. Koga, T., Sekine, Y. & Nitta, J. Experimental realization of a ballistic spin interferometer based on the Rashba effect using a nanolithographically induced square loop array. Phys. Rev. B 74, 041302(R) (2006).

    Article  Google Scholar 

  9. Bergsten, T., Kobayashi, T., Sekine, Y. & Nitta, J. Experimental demonstration of the time reversal Aharonov–Casher effect. Phys. Rev. Lett. 97, 196803 (2006).

    Article  Google Scholar 

  10. Nowack, K. C., Koppens, F. H. L., Nazarov, Y. V. & Vandersypern, L. M. K. Coherent control of a single electron spin with electric fields. Science 318, 1430–1433 (2007).

    Article  CAS  Google Scholar 

  11. Frolov, S. M. et al. Ballistic spin resonance. Nature 458, 868–871 (2009).

    Article  CAS  Google Scholar 

  12. Sanada, H. et al. Manipulation of mobile spin coherence using magnetic-field-free electron spin resonance. Nature Phys. 9, 280–283 (2013).

    Article  CAS  Google Scholar 

  13. Brüne, C. et al. Evidence for the ballistic intrinsic spin Hall effect in HgTe nanostructures Nature Phys. 6, 448–454 (2010).

    Article  Google Scholar 

  14. D'yakonov, M. I. & Perel', V. I. Spin relaxation of conduction electrons in noncentrosymmetric semiconductors. Sov. Phys. Solid State 13, 3023–3026 (1971).

    Google Scholar 

  15. Rashba, E. I. Properties of semiconductors with an extremum loop. Sov. Phys. Solid State 2, 1224–1238 (1960).

    Google Scholar 

  16. Dresselhaus, G. Spin–orbit coupling effects in zinc blende structures. Phys. Rev. 100, 580–586 (1955).

    Article  CAS  Google Scholar 

  17. Schliemann, J. & Loss, D. Anisotropic transport in a two-dimensional electron gas in the presence of spin–orbit coupling. Phys. Rev. B. 68, 165311 (2003).

    Article  Google Scholar 

  18. Bernevig, B. A., Orenstein, J. & Zhang, S-C. Exact SU(2) symmetry and persistent spin helix in a spin–orbit coupled system. Phys. Rev. Lett. 97, 236601 (2006).

    Article  Google Scholar 

  19. Koralek, J. D. et al. Emergence of the persistent spin helix in semiconductor quantum wells. Nature 458, 610–613 (2009).

    Article  CAS  Google Scholar 

  20. Walser, M. P., Reichl, C., Wegscheider, W. & Salis, G. Direct mapping of the formation of a persistent spin helix. Nature Phys. 8, 757–762 (2012).

    Article  CAS  Google Scholar 

  21. Kohda, M. et al. Gate-controlled persistent spin helix state in (In,Ga)As quantum wells. Phys. Rev. B 86, 081306(R) (2012).

    Article  Google Scholar 

  22. Cartoixà, X., Ting, D. Z-Y. & Chang, Y-C. A resonant spin lifetime transistor. Appl. Phys. Lett. 83, 1462–1464 (2003).

    Article  Google Scholar 

  23. Schliemann, J., Egues, J. C. & Loss, D. Nonballistic spin-field-effect transistor. Phys. Rev. Lett. 90, 146801 (2003).

    Article  Google Scholar 

  24. Kunihashi, Y. et al. Proposal of spin complementary field effect transistor. Appl. Phys. Lett. 100, 113502 (2012).

    Article  Google Scholar 

  25. Miller, J. B. et al. Gate-controlled spin–orbit quantum interference effects in lateral transport. Phys. Rev. Lett. 90, 076807 (2003).

    Article  CAS  Google Scholar 

  26. Ganichev, S. D. et al. Experimental separation of Rashba and Dresselhaus spin splittings in semiconductor quantum wells. Phys. Rev. Lett. 92, 256601 (2004).

    Article  CAS  Google Scholar 

  27. Meier, L. et al. Measurement of Rashba and Dresselhaus spin–orbit magnetic fields. Nature Phys. 3, 650–654 (2007).

    Article  CAS  Google Scholar 

  28. Studer, M., Salis, G., Ensslin, K., Driscoll, D. C. & Gossard, A. C. Gate-controlled spin–orbit interaction in a parabolic GaAs/AlGaAs quantum well. Phys. Rev. Lett. 103, 027201 (2009).

    Article  CAS  Google Scholar 

  29. Ishihara, J., Ohno, Y. & Ohno, H. Direct imaging of gate-controlled persistent spin helix state in a modulation-doped GaAs/AlGaAs quantum well. Appl. Phys. Exp. 7, 013001 (2014).

    Article  Google Scholar 

  30. Meijer, F. E., Morpurgo, A. F., Klapwijk, T. M. & Nitta, J. Universal spin-induced time reversal symmetry breaking in two-dimensional electron gases with Rashba spin–orbit interaction. Phys. Rev. Lett. 94, 186805 (2005).

    Article  CAS  Google Scholar 

  31. Scheid, M., Kohda, M., Kunihashi, Y., Richter, K. & Nitta, J. All-electrical detection of the relative strength of Rashba and Dresselhaus spin–orbit interaction. Phys. Rev. Lett. 101, 266401 (2008).

    Article  Google Scholar 

  32. Kettemann, S. Dimensional control of antilocalization and spin relaxation in quantum wires. Phys. Rev. Lett. 98, 176808 (2007).

    Article  Google Scholar 

  33. Mal'shukov, A. G. & Chao, K. A. Waveguide diffusion modes and slowdown of D'yakonov–Perel' spin relaxation in narrow two-dimensional semiconductor channels. Phys. Rev. B 61, R2413–R2416 (2000).

    Article  CAS  Google Scholar 

  34. Schäpers, Th. et al. Suppression of weak antilocalization in GaxIn1–xAs/InP narrow quantum wires. Phys. Rev. B 74, 081301(R) (2006).

    Article  Google Scholar 

  35. Kunihashi, Y., Kohda, M. & Nitta, J. Enhancement of spin lifetime in gate-fitted InGaAs narrow wires. Phys. Rev. Lett. 102, 226601 (2009).

    Article  Google Scholar 

  36. Wimmer, M. & Richter, K. Optimal block-tridiagonalization of matrices for coherent charge transport. J. Comput. Phys. 228, 8548–8565 (2009).

    Article  CAS  Google Scholar 

  37. Scheid, M., Adagideli, İ., Nitta, J. & Richter, K. Anisotropic universal conductance fluctuation in disordered quantum wires with Rashba and Dresselhaus spin–orbit interaction and applied in-plane magnetic field. Semicond. Sci. Technol. 24, 064005 (2009).

    Article  Google Scholar 

Download references

Acknowledgements

The authors acknowledge support from the Strategic Japanese–German Joint Research Program. K.R. thanks the DFG for support within Research Unit FOR 1483. T.D. acknowledges support by the DFG within research project SFB 689. This work was financially supported by Grants-in-Aid from the Japan Society for the Promotion of Science (JSPS; no. 22226001).

Author information

Authors and Affiliations

Authors

Contributions

A.S., S.N. and Y.K. performed device fabrication and measurements. T.B., T.D. and K.R. performed numerical calculations. A.S. and M.K. wrote the main part of the manuscript. T.D. and K.R. wrote the theoretical part. All authors discussed the results and worked on the manuscript at all stages. M.K., K.R. and J.N. planned the project. J.N. directed the research.

Corresponding author

Correspondence to J. Nitta.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary information

Supplementary Information (PDF 996 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sasaki, A., Nonaka, S., Kunihashi, Y. et al. Direct determination of spin–orbit interaction coefficients and realization of the persistent spin helix symmetry. Nature Nanotech 9, 703–709 (2014). https://doi.org/10.1038/nnano.2014.128

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nnano.2014.128

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