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Klein tunnelling and electron trapping in nanometre-scale graphene quantum dots

Nature Physics volume 12, pages 1069–1075 (2016)Cite this article

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Abstract

Relativistic fermions that are incident on a high potential barrier can pass through unimpeded, a striking phenomenon termed the ‘Klein paradox’ in quantum electrodynamics. Electrostatic potential barriers in graphene provide a solid-state analogue to realize this phenomenon. Here, we use scanning tunnelling microscopy to directly probe the transmission of electrons through sharp circular potential wells in graphene created by substrate engineering. We find that electrons in this geometry display quasi-bound states where the electron is trapped for a finite time before escaping via Klein tunnelling. We show that the continuum Dirac equation can be successfully used to model the energies and wavefunctions of these quasi-bound states down to atomic dimensions. We demonstrate that by tuning the geometry of the barrier it is possible to trap particular energies and angular momentum states with increased efficiency, showing that atomic-scale electrostatic potentials can be used to engineer quantum transport through graphene.

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Figure 1: Klein tunnelling in a continuous graphene sheet.
Figure 2: Quasi-bound states in a graphene quantum dot.
Figure 3: Spectrum of quasi-bound states of a GQD.
Figure 4: Trapping times in quasi-bound states.
Figure 5: Atomic-scale GQDs and sublattice symmetry breaking.

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  • 26 August 2016

    In the version of this Article originally published, refs 30, 32 and 33 were mistakenly cited at the end of a sentence that now reads: 'Such a particle would mimic a massless, spinless particle, and thus could be described by the Klein-Gordon equation9, which has effectively described finite-sized GQD islands31.' This has now been corrected in the online versions of the Article.

References

  1. Klein, O. Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac. Z. Phys. 53, 157–165 (1929).

    Article  ADS  Google Scholar 

  2. Dombey, N. & Calogeracos, A. Seventy years of the Klein paradox. Phys. Rep. 315, 41–58 (1999).

    Article  ADS  Google Scholar 

  3. Katsnelson, M. I., Novoselov, K. S. & Geim, A. K. Chiral tunnelling and the Klein paradox in graphene. Nature Phys. 2, 620–625 (2006).

    Article  ADS  Google Scholar 

  4. Cheianov, V. V. & Fal’ko, V. I. Selective transmission of Dirac electrons and ballistic magnetoresistance of n–p junctions in graphene. Phys. Rev. B 74, 041403 (2006).

    Article  ADS  Google Scholar 

  5. Pereira, J. M., Mlinar, V., Peeters, F. M. & Vasilopoulos, P. Confined states and direction-dependent transmission in graphene quantum wells. Phys. Rev. B 74, 045424 (2006).

    Article  ADS  Google Scholar 

  6. Cheianov, V. V., Fal’ko, V. & Altshuler, B. L. The focusing of electron flow and a Veselago lens in graphene pn junctions. Science 315, 1252–1255 (2007).

    Article  ADS  Google Scholar 

  7. Shytov, A. V., Katsnelson, M. I. & Levitov, L. S. Atomic collapse and quasi-Rydberg states in graphene. Phys. Rev. Lett. 99, 246802 (2007).

    Article  ADS  Google Scholar 

  8. Shytov, A. V., Rudner, M. S. & Levitov, L. S. Klein backscattering and Fabry–Pérot interference in graphene heterojunctions. Phys. Rev. Lett. 101, 156804 (2008).

    Article  ADS  Google Scholar 

  9. Barbier, M., Peeters, F. M., Vasilopoulos, P. & Pereira, J. M. Dirac and Klein–Gordon particles in one-dimensional periodic potentials. Phys. Rev. B 77, 115446 (2008).

    Article  ADS  Google Scholar 

  10. Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).

    Article  ADS  Google Scholar 

  11. Vakil, A. & Engheta, N. Transformation optics using graphene. Science 332, 1291–1294 (2011).

    Article  ADS  Google Scholar 

  12. Young, A. F. & Kim, P. Quantum interference and Klein tunnelling in graphene heterojunctions. Nature Phys. 5, 222–226 (2009).

    Article  ADS  Google Scholar 

  13. Stander, N., Huard, B. & Goldhaber-Gordon, D. Evidence for Klein tunneling in graphene p–n junctions. Phys. Rev. Lett. 102, 026807 (2009).

    Article  ADS  Google Scholar 

  14. Lee, G.-H., Park, G.-H. & Lee, H.-J. Observation of negative refraction of Dirac fermions in graphene. Nature Phys. 11, 925–929 (2015).

    Article  ADS  Google Scholar 

  15. Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in few-electron quantum dots. Rev. Mod. Phys. 79, 1217–1265 (2007).

    Article  ADS  Google Scholar 

  16. Matulis, A. & Peeters, F. M. Quasibound states of quantum dots in single and bilayer graphene. Phys. Rev. B 77, 115423 (2008).

    Article  ADS  Google Scholar 

  17. Hewageegana, P. & Apalkov, V. Electron localization in graphene quantum dots. Phys. Rev. B 77, 245426 (2008).

    Article  ADS  Google Scholar 

  18. Bardarson, J. H., Titov, M. & Brouwer, P. W. Electrostatic confinement of electrons in an integrable graphene quantum dot. Phys. Rev. Lett. 102, 226803 (2009).

    Article  ADS  Google Scholar 

  19. Heinisch, R. L., Bronold, F. X. & Fehske, H. Mie scattering analog in graphene: lensing, particle confinement, and depletion of Klein tunneling. Phys. Rev. B 87, 155409 (2013).

    Article  ADS  Google Scholar 

  20. Schulz, C., Heinisch, R. L. & Fehske, H. Electron flow in circular graphene quantum dots. Quantum. Matter 4, 346–351 (2015).

    Article  Google Scholar 

  21. Zhao, Y. et al. Creating and probing electron whispering-gallery modes in graphene. Science 348, 672–675 (2015).

    Article  ADS  Google Scholar 

  22. Brown, L. et al. Polycrystalline graphene with single crystalline electronic structure. Nano Lett. 14, 5706–5711 (2014).

    Article  ADS  Google Scholar 

  23. Gutiérrez, C. et al. Imaging chiral symmetry breaking from Kekulé bond order in graphene. Nature Phys. http://dx.doi.org/10.1038/nphys3776 (2016).

  24. Sandl, P. & Bertel, E. Surface states, local bonding, and surface reconstruction: Na on Cu (110). Surf. Sci. 302, L325–L330 (1994).

    Article  ADS  Google Scholar 

  25. Memmel, N. Monitoring and modifying properties of metal surfaces by electronic surface states. Surf. Sci. Rep. 32, 91–163 (1998).

    Article  ADS  Google Scholar 

  26. Lin, C. L. et al. Manifestation of work function difference in high order Gundlach oscillation. Phys. Rev. Lett. 99, 216103 (2007).

    Article  ADS  Google Scholar 

  27. Borca, B. et al. Potential energy landscape for hot electrons in periodically nanostructured graphene. Phys. Rev. Lett. 105, 036804 (2010).

    Article  ADS  Google Scholar 

  28. Pauly, C., Grob, M., Pezzotta, M., Pratzer, M. & Morgenstern, M. Gundlach oscillations and Coulomb blockade of Co nanoislands on MgO/Mo(100) investigated by scanning tunneling spectroscopy at 300 K. Phys. Rev. B 81, 125446 (2010).

    Article  ADS  Google Scholar 

  29. Chang, H. W., Wu, B. F., Yao, Y. D., Su, W. B. & Chang, C. S. Co nanoislands on Au (111) and Cu (111) surfaces studied by scanning tunneling microscopy and spectroscopy. J. Nanosci. Nanotech. 10, 4663–4666 (2010).

    Article  Google Scholar 

  30. Phark, S.-h. et al. Direct observation of electron confinement in epitaxial graphene nanoislands. ACS Nano 5, 8162–8166 (2011).

    Article  Google Scholar 

  31. Hämäläinen, S. K. et al. Quantum-confined electronic states in atomically well-defined graphene nanostructures. Phys. Rev. Lett. 107, 236803 (2011).

    Article  ADS  Google Scholar 

  32. Subramaniam, D. et al. Wave-function mapping of graphene quantum dots with soft confinement. Phys. Rev. Lett. 108, 046801 (2012).

    Article  ADS  Google Scholar 

  33. Jolie, W. et al. Confinement of Dirac electrons in graphene quantum dots. Phys. Rev. B 89, 155435 (2014).

    Article  ADS  Google Scholar 

  34. Zhang, Y. et al. Giant phonon-induced conductance in scanning tunnelling spectroscopy of gate-tunable graphene. Nature Phys. 4, 627–630 (2008).

    Article  ADS  Google Scholar 

  35. Natterer, F. D. et al. Strong asymmetric charge carrier dependence in inelastic electron tunneling spectroscopy of graphene phonons. Phys. Rev. Lett. 114, 245502 (2015).

    Article  ADS  Google Scholar 

  36. Lee, J. et al. Imaging electrostatically confined Dirac fermions in graphene quantum dots. Nature Phys. http://dx.doi.org/10.1038/nphys3805 (2016).

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Acknowledgements

We thank I. Aleiner for helpful discussions. Work at Columbia is supported by the Office of Naval Research (award number N00014-14-1-0501, C.G.) and by the Nanoelectronics Research Corporation (NERC), a wholly owned subsidiary of the Semiconductor Research Corporation (SRC), through the Institute for Nanoelectronics Discovery and Exploration (INDEX) (A.N.P.). Support for microscopy measurements is provided by the Air Force Office of Scientific Research (AFOSR) (award number FA9550-11-1-0010, A.N.P.) and by the NSF through the Columbia Center for Precision Assembly of Superstratic and Superatomic Solids (DMR-1420634, A.N.P.). Work at Cornell University is supported by the NSF through the Cornell Center for Materials Research (NSF DMR-1120296). Additional funding was provided by AFOSR (FA9550-16-1-0106, FA2386-13-1-4118) and the Nano Material Technology Development Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and Future Planning (2012M3A7B4049887).

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Authors and Affiliations

  1. Department of Physics, Columbia University, New York, New York 10027, USA

    Christopher Gutiérrez & Abhay N. Pasupathy

  2. Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA

    Lola Brown, Cheol-Joo Kim & Jiwoong Park

  3. Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA

    Jiwoong Park

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  1. Christopher Gutiérrez
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Contributions

C.G. performed and analysed all STM measurements and theory calculations. L.B. and C.-J.K. performed CVD growth of graphene samples. J.P. supervised the CVD sample growth. A.N.P. supervised STM measurements. All authors participated in writing the manuscript.

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Correspondence to Abhay N. Pasupathy.

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Gutiérrez, C., Brown, L., Kim, CJ. et al. Klein tunnelling and electron trapping in nanometre-scale graphene quantum dots. Nature Phys 12, 1069–1075 (2016). https://doi.org/10.1038/nphys3806

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