Wave-particle duality of electrons with spin-momentum locking

Double-slit interference and single-slit diffraction effects of electrons on the surface of three-dimensional topological insulators


We investigate the effects of spin-momentum locking on the interference and diffraction patterns due to a double- or single-slit in an electronic Gedankenexperiment. We show that the inclusion of the spin-degree-of-freedom, when coupled to the motion direction of the carrier—a typical situation that occurs in systems with spin–orbit interaction—leads to a modification of the interference and diffraction patterns that depend on the geometrical parameters of the system.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Data availability statement

This manuscript has associated data in a data repository. [Authors’ comment: All data generated or analysed during this study are included in this published article.]


  1. 1.

    R.M. Eisberg, R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (Wiley, Hoboken, 2009)

    Google Scholar 

  2. 2.

    R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 3 (Addison-Wesley, Reading, 1965)

    MATH  Google Scholar 

  3. 3.

    C. Jönsson, Zeitschrift für Physik 161(4), 454 (1961). https://doi.org/10.1007/bf01342460

    Article  ADS  Google Scholar 

  4. 4.

    C. Jönsson, Am. J. Phys. 42(1), 4 (1974). https://doi.org/10.1119/1.1987592

    Article  ADS  Google Scholar 

  5. 5.

    P.G. Merli, G.F. Missiroli, G. Pozzi, Am. J. Phys. 44(3), 306 (1976). https://doi.org/10.1119/1.10184

    Article  ADS  Google Scholar 

  6. 6.

    A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, H. Ezawa, Am. J. Phys. 57(2), 117 (1989). https://doi.org/10.1119/1.16104

    Article  ADS  Google Scholar 

  7. 7.

    J. Steeds, P.G. Merli, G. Pozzi, G.F. Missiroli, A. Tonomura, Phys. World 16(5), 20 (2003). https://doi.org/10.1088/2058-7058/16/5/24

    Article  Google Scholar 

  8. 8.

    M. Malgieri, P. Onorato, A. De Ambrosis, Phys. Rev. Phys. Educ. Res. 13, 010101 (2017). https://doi.org/10.1103/PhysRevPhysEducRes.13.019901

    Article  Google Scholar 

  9. 9.

    R. Sayer, A. Maries, C. Singh, Phys. Rev. Phys. Educ. Res. 13, 010123 (2017). https://doi.org/10.1103/PhysRevPhysEducRes.13.010123

    Article  Google Scholar 

  10. 10.

    M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, A. Zeilinger, Nature 401(6754), 680 (1999). https://doi.org/10.1038/44348

    Article  ADS  Google Scholar 

  11. 11.

    O. Nairz, M. Arndt, A. Zeilinger, J. Mod. Opt. 47(14–15), 2811 (2000). https://doi.org/10.1080/09500340008232198

    Article  ADS  Google Scholar 

  12. 12.

    O. Nairz, M. Arndt, A. Zeilinger, Am. J. Phys. 71(4), 319 (2003). https://doi.org/10.1119/1.1531580

    Article  ADS  Google Scholar 

  13. 13.

    A.D. Cronin, J. Schmiedmayer, D.E. Pritchard, Rev. Mod. Phys. 81(3), 1051 (2009). https://doi.org/10.1103/revmodphys.81.1051

    Article  ADS  Google Scholar 

  14. 14.

    K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter, M. Arndt, Rev. Mod. Phys. 84(1), 157 (2012). https://doi.org/10.1103/revmodphys.84.157

    Article  ADS  Google Scholar 

  15. 15.

    S. Eibenberger, S. Gerlich, M. Arndt, M. Mayor, J. Tüxen, Phys. Chem. Chem. Phys. 15(35), 14696 (2013). https://doi.org/10.1039/c3cp51500a

    Article  Google Scholar 

  16. 16.

    E.G. Carnio, H.P. Breuer, A. Buchleitner, J. Phys. Chem. Lett. 10(9), 2121 (2019). https://doi.org/10.1021/acs.jpclett.9b00676

    Article  Google Scholar 

  17. 17.

    Y. Hasegawa, K. Saitoh, N. Tanaka, S. Tanimura, M. Uchida, J. Phys. Soc. Jpn. 82(3), 033002 (2013). https://doi.org/10.7566/jpsj.82.033002

    Article  ADS  Google Scholar 

  18. 18.

    V. Bazhenov, M. Soskin, M. Vasnetsov, J. Mod. Opt. 39(5), 985 (1992). https://doi.org/10.1080/09500349214551011

    Article  ADS  Google Scholar 

  19. 19.

    Y.A. Bychkov, E.I. Rashba, J. Phys. C Solid State Phys. 17(33), 6039 (1984). https://doi.org/10.1088/0022-3719/17/33/015

    Article  ADS  Google Scholar 

  20. 20.

    K. Shimizu, M. Mochizuki, Phys. Rev. B 101, 045301 (2020). https://doi.org/10.1103/PhysRevB.101.045301

    Article  ADS  Google Scholar 

  21. 21.

    D. Bercioux, A. De Martino, Phys. Rev. B 81, 165410 (2010). https://doi.org/10.1103/PhysRevB.81.165410

    Article  ADS  Google Scholar 

  22. 22.

    D. Bercioux, D.F. Urban, F. Romeo, R. Citro, Appl. Phys. Lett. 101(12), 122405 (2012). https://doi.org/10.1063/1.4753975

    Article  ADS  Google Scholar 

  23. 23.

    J.H. Bardarson, J.E. Moore, Rep. Prog. Phys. 76(5), 056501 (2013). https://doi.org/10.1088/0034-4885/76/5/056501

    Article  ADS  Google Scholar 

  24. 24.

    D. Bercioux, P. Lucignano, Rep. Prog. Phys. 78(10), 106001 (2015). https://doi.org/10.1088/0034-4885/78/10/106001

    Article  ADS  Google Scholar 

  25. 25.

    M. Born, E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 2019)

    Book  Google Scholar 

  26. 26.

    E. Bocquillon, V. Freulon, F.D. Parmentier, J.M. Berroir, B. Plaçais, C. Wahl, J. Rech, T. Jonckheere, T. Martin, C. Grenier, D. Ferraro, P. Degiovanni, G. Fève, Ann. Phys. 526(1–2), 1 (2013). https://doi.org/10.1002/andp.201300181

    Article  Google Scholar 

  27. 27.

    J.M. Edge, J. Li, P. Delplace, M. Büttiker, Phys. Rev. Lett. 110, 24 (2013). https://doi.org/10.1103/physrevlett.110.246601

    Article  Google Scholar 

  28. 28.

    A. Inhofer, D. Bercioux, Phys. Rev. B 88, 23 (2013). https://doi.org/10.1103/physrevb.88.235412

    Article  Google Scholar 

  29. 29.

    P.P. Hofer, M. Büttiker, Phys. Rev. B 88(24), 241308 (2013). https://doi.org/10.1103/PhysRevB.88.241308

    Article  ADS  Google Scholar 

  30. 30.

    D. Ferraro, C. Wahl, J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 89, 075407 (2014). https://doi.org/10.1103/PhysRevB.89.075407

    Article  ADS  Google Scholar 

  31. 31.

    D. Ferraro, T. Jonckheere, J. Rech, T. Martin, Physica Status Solidi (b) 254(3), 1600531 (2016). https://doi.org/10.1002/pssb.201600531

    Article  ADS  Google Scholar 

  32. 32.

    N. Avraham, J. Reiner, A. Kumar-Nayak, N. Morali, R. Batabyal, B. Yan, H. Beidenkopf, Adv. Mater. 30(41), 1707628 (2018). https://doi.org/10.1002/adma.201707628

    Article  Google Scholar 

  33. 33.

    P. Roushan, J. Seo, C.V. Parker, Y.S. Hor, D. Hsieh, D. Qian, A. Richardella, M.Z. Hasan, R.J. Cava, A. Yazdani, Nature 460(7259), 1106 (2009). https://doi.org/10.1038/nature08308

    Article  ADS  Google Scholar 

  34. 34.

    P. Sessi, F. Reis, T. Bathon, K.A. Kokh, O.E. Tereshchenko, M. Bode, Nat. Commun. 5, 1 (2014). https://doi.org/10.1038/ncomms6349

    Article  Google Scholar 

  35. 35.

    Y.L. Chen, J.G. Analytis, J.H. Chu, Z.K. Liu, S.K. Mo, X.L. Qi, H.J. Zhang, D.H. Lu, X. Dai, Z. Fang, S.C. Zhang, I.R. Fisher, Z. Hussain, Z.X. Shen, Science 325(5937), 178 (2009). https://doi.org/10.1126/science.1173034

    Article  ADS  Google Scholar 

  36. 36.

    L. Fu, Phys. Rev. Lett. 103(26), 266801 (2009). https://doi.org/10.1103/physrevlett.103.266801

    Article  ADS  Google Scholar 

  37. 37.

    K.K. Gomes, W. Mar, W. Ko, F. Guinea, H.C. Manoharan, Nature 483(7389), 306 (2012). https://doi.org/10.1038/nature10941

    Article  ADS  Google Scholar 

  38. 38.

    M.R. Slot, T.S. Gardenier, P.H. Jacobse, G.C.P. van Miert, S.N. Kempkes, S.J.M. Zevenhuizen, C.M. Smith, D. Vanmaekelbergh, I. Swart, Nat. Phys. 13(7), 672 (2017). https://doi.org/10.1038/nphys4105

    Article  Google Scholar 

  39. 39.

    S.N. Kempkes, M.R. Slot, S.E. Freeney, S.J.M. Zevenhuizen, D. Vanmaekelbergh, I. Swart, C.M. Smith, Nat. Phys. 15(2), 127 (2018). https://doi.org/10.1038/s41567-018-0328-0

    Article  Google Scholar 

  40. 40.

    S.N. Kempkes, M.R. Slot, J.J. van den Broeke, P. Capiod, W.A. Benalcazar, D. Vanmaekelbergh, D. Bercioux, I. Swart, C.M. Smith, Nat. Mater. 18(12), 1292 (2019). https://doi.org/10.1038/s41563-019-0483-4

    Article  ADS  Google Scholar 

  41. 41.

    L.C. Davis, M.P. Everson, R.C. Jaklevic, W. Shen, Phys. Rev. B 43, 3821 (1991). https://doi.org/10.1103/PhysRevB.43.3821

    Article  ADS  Google Scholar 

  42. 42.

    M.F. Crommie, C.P. Lutz, D.M. Eigler, Nature 363(6429), 524 (1993). https://doi.org/10.1038/363524a0

    Article  ADS  Google Scholar 

  43. 43.

    Y. Hasegawa, P. Avouris, Phys. Rev. Lett. 71, 1071 (1993). https://doi.org/10.1103/PhysRevLett.71.1071

    Article  ADS  Google Scholar 

  44. 44.

    P. Sessi, R.R. Biswas, T. Bathon, O. Storz, S. Wilfert, A. Barla, K.A. Kokh, O.E. Tereshchenko, K. Fauth, M. Bode, A.V. Balatsky, Nat. Commun. 7, 1 (2016). https://doi.org/10.1038/ncomms12027

    Article  Google Scholar 

Download references


We acknowledge useful discussions with Alessandro De Martino, Erwann Bocquillon, M. Reyes Calvo, Geza Giedke, Andreas Inhofer, and Ingmar Swart. The work of TB and DB is supported by the Spanish Ministerio de Ciencia, Innovation y Universidades (MICINN) through the Project FIS2017-82804-P, and by the Transnational Common Laboratory Quantum-ChemPhys. DB thanks the University of Aix-Marseille for hosting him during the genesis of this work.

Author information



Corresponding author

Correspondence to Dario Bercioux.

Appendix A: On the diffraction formula

Appendix A: On the diffraction formula

In this appending, we present the derivation for the diffraction formula in the leading order in \(L^{-1}\) presented in Eq. (8) or (20). The main step is to expand all the \(\cos [(n-m)\varphi ]=\cos \ell \varphi \) into exponential functions so that

$$\begin{aligned} \mathcal {D}_\text {SE}&\longrightarrow \text {e}^{-\text {i} (N-1)\varphi } \left( \sum _{m=0}^N \text {e}^{\text {i} m \varphi }\right) ^2, \end{aligned}$$
$$\begin{aligned}&=\text {e}^{-\text {i} (N-1)\varphi } \left( \frac{1-\text {e}^{\text {i} N \varphi }}{1-\text {e}^{\text {i}\varphi }}\right) ^2, \end{aligned}$$
$$\begin{aligned}&= \text {e}^{\text {i} (N-1)\varphi }\left( \text {e}^{\text {i}\frac{(N-1)\varphi }{2}}\frac{\sin (N\varphi /2)}{\sin (\varphi /2)}\right) ^2, \end{aligned}$$
$$\begin{aligned}&= \left[ \frac{\sin \left( \frac{N\varphi }{2}\right) }{\sin \left( \frac{\varphi }{2}\right) }\right] ^2, \end{aligned}$$

where in (A.1a) we have used the properties of geometric series: \(\sum _{n=0}^N r^n=(1-r^{N+1})/(1-r)\). A similar expression is obtained starting by Eq. (19) with the introduction of the spin parameter \(\chi \).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bercioux, D., van den Berg, T.L., Ferraro, D. et al. Wave-particle duality of electrons with spin-momentum locking. Eur. Phys. J. Plus 135, 811 (2020). https://doi.org/10.1140/epjp/s13360-020-00837-3

Download citation