Skip to main content
SpringerLink
Log in

Strain-Controlled Anisotropic Pseudospin Tunneling in the \(\alpha {-}T_{3}\) Model

  • Published:
Journal of Low Temperature Physics Aims and scope Submit manuscript

Abstract

We investigate the anisotropic pseudospin tunneling effect of electrons through a potential barrier in the \(\alpha {-}T_{3}\) model under an uni-axial strain. When compressing the \(\alpha {-}T_{3}\) lattice along the armchair direction, a pair of Dirac points in the energy band structure may merge into a single one along the zigzag direction. If the strain is strong enough, the energy gap will open, and the dispersion relation will be linear in the armchair direction but quadratic in the zigzag direction. The merging of Dirac points represents a topological phase transition from a metallic phase to an insulating phase. We found that in the armchair direction, the tunneling properties are similar to the undeformed \(\alpha {-}T_{3}\) lattice. For example, when the incident particle is normal to the potential barrier, we find Klein tunneling effect independent of \(\alpha\), and for the dice lattice with the incident energy is equal to half of the barrier height, there occurs super-Klein tunneling. While in the zigzag direction, we get the opposite conclusion in the deformed \(\alpha {-}T_{3}\) lattice. The Klein tunneling effect becomes the reflection effect, and the super-Klein tunneling effect is transformed into the total reflection effect, and the conductance with gap opening can be turned off using strain applied along the armchair direction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

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

Similar content being viewed by others

References

  1. Z. Dai, L. Liu, Z. Zhang, Strain engineering of 2D materials: issues and opportunities at the interface. Adv. Mater. 31, 1805417 (2019)

    Article  Google Scholar 

  2. D. Midtvedt, C.H. Lewenkopf, A. Croy, Strain–displacement relations for strain engineering in single-layer 2D materials. 2D Mater. 3, 011005 (2016)

    Article  Google Scholar 

  3. Y. Sun, K. Liu, Strain engineering in functional 2-dimensional materials. J. Appl. Phys. 125, 082402 (2019)

    Article  ADS  Google Scholar 

  4. Z.-Z. Cao, Y.-F. Cheng, G.-Q. Li, Strain-controlled electron switch in graphene. Appl. Phys. Lett. 101, 253507 (2012)

    Article  ADS  Google Scholar 

  5. C. Si, Z. Sun, F. Liu, Strain engineering of graphene: a review. Nanoscale 8, 3207–3217 (2016)

    Article  ADS  Google Scholar 

  6. M. Bharati, V. Soma, Flexible SERS substrates for hazardous materials detection: recent advances. Opto-Electron. Adv. 4, 210048 (2021)

    Article  Google Scholar 

  7. H.-F. Lü, Y. Guo, X.-T. Zu, H.-W. Zhang, Optically controlled spin polarization in a spin transistor. Appl. Phys. Lett. 94, 162109 (2009)

    Article  ADS  Google Scholar 

  8. V.M. Pereira, A.C. Neto, Strain engineering of graphene’s electronic structure. Phys. Rev. Lett. 103, 046801 (2009)

    Article  ADS  Google Scholar 

  9. M. Ohtsu, History, current developments, and future directions of near-field optical science. Opto-Electron. Adv. 3, 190046 (2020)

    Article  Google Scholar 

  10. A. McRae, G. Wei, A. Champagne, Graphene quantum strain transistors. Phys. Rev. Appl. 11, 054019 (2019)

    Article  ADS  Google Scholar 

  11. M.-Y. Li, Y. Shi, C.-C. Cheng, L.-S. Lu, Y.-C. Lin, H.-L. Tang, M.-L. Tsai, C.-W. Chu, K.-H. Wei, J.-H. He et al., Epitaxial growth of a monolayer WSe\(_2\)–MoS\(_2\) lateral pn junction with an atomically sharp interface. Science 349, 524–528 (2015)

    Article  ADS  Google Scholar 

  12. C. Zhang, M.-Y. Li, J. Tersoff, Y. Han, Y. Su, L.-J. Li, D.A. Muller, C.-K. Shih, Strain distributions and their influence on electronic structures of WSe\(_2\)–MoS\(_2\) laterally strained heterojunctions. Nat. Nanotechnol. 13, 152–158 (2018)

    Article  ADS  Google Scholar 

  13. Y. Hu, D. Liang, R. Beausoleil, Flexible SERS substrates for hazardous materials detection: recent advances. Opto-Electron. Adv. 4, 200094 (2021)

    Article  Google Scholar 

  14. Z.H. Ni, T. Yu, Y.H. Lu, Y.Y. Wang, Y.P. Feng, Z.X. Shen, Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening. ACS Nano 2, 2301–2305 (2008)

    Article  Google Scholar 

  15. O.B. Aslan, I.M. Datye, M.J. Mleczko, K. Sze Cheung, S. Krylyuk, A. Bruma, I. Kalish, A.V. Davydov, E. Pop, T.F. Heinz, Probing the optical properties and strain-tuning of ultrathin Mo\(_{1-x}\)W\(_{x}\)Te\(_{2}\). Nano Lett. 18, 2485–2491 (2018)

    Article  ADS  Google Scholar 

  16. A. McCreary, R. Ghosh, M. Amani, J. Wang, K.-A.N. Duerloo, A. Sharma, K. Jarvis, E.J. Reed, A.M. Dongare, S.K. Banerjee et al., Effects of uniaxial and biaxial strain on few-layered terrace structures of MoS\(_{2}\) grown by vapor transport. ACS Nano 10, 3186–3197 (2016)

    Article  Google Scholar 

  17. P. Gant, P. Huang, D.P. de Lara, D. Guo, R. Frisenda, A. Castellanos-Gomez, A strain tunable single-layer MoS\(_{2}\) photodetector. Mater. Today 27, 8–13 (2019)

    Article  Google Scholar 

  18. S. Qiu, V. Krishnan, S. Padula, R. Noebe, D. Brown, B. Clausen, R. Vaidyanathan, Measurement of the lattice plane strain and phase fraction evolution during heating and cooling in shape memory NiTi. Appl. Phys. Lett. 95, 141906 (2009)

    Article  ADS  Google Scholar 

  19. E. Khestanova, F. Guinea, L. Fumagalli, A. Geim, I. Grigorieva, Universal shape and pressure inside bubbles appearing in van der Waals heterostructures. Nat. Commun. 7, 1–10 (2016)

    Article  Google Scholar 

  20. J. Zabel, R.R. Nair, A. Ott, T. Georgiou, A.K. Geim, K.S. Novoselov, C. Casiraghi, Raman spectroscopy of graphene and bilayer under biaxial strain: bubbles and balloons. Nano Lett. 12, 617–621 (2012)

    Article  ADS  Google Scholar 

  21. S. Cronin, A. Swan, M. Ünlü, B. Goldberg, M. Dresselhaus, M. Tinkham, Measuring the uniaxial strain of individual single-wall carbon nanotubes: resonance Raman spectra of atomic-force-microscope modified single-wall nanotubes. Phys. Rev. Lett. 93, 167401 (2004)

    Article  ADS  Google Scholar 

  22. A. Sharbirin, S. Akhtar, J. Kim, Light-emitting MXene quantum dots. Opto-Electron. Adv. 4, 200077 (2021)

    Article  Google Scholar 

  23. C. Lee, X. Wei, J.W. Kysar, J. Hone, Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321, 385–388 (2008)

    Article  ADS  Google Scholar 

  24. C. Martella, C. Mennucci, E. Cinquanta, A. Lamperti, E. Cappelluti, F. Buatier de Mongeot, A. Molle, Anisotropic MoS\(_{2}\) nanosheets grown on self-organized nanopatterned substrates. Adv. Mater. 29, 1605785 (2017)

    Article  Google Scholar 

  25. Z. Li et al., Light-triggered interfacial charge transfer and enhanced photodetection in CdSe/ZnS quantum dots/MoS\(_{2}\) mixed-dimensional phototransistors. Opto-Electron. Adv. 4, 210017 (2021)

    Article  Google Scholar 

  26. S. Wise, J. Lowengrub, J. Kim, K. Thornton, P. Voorhees, W. Johnson, Quantum dot formation on a strain-patterned epitaxial thin film. Appl. Phys. Lett. 87, 133102 (2005)

    Article  ADS  Google Scholar 

  27. E. Illes, E. Nicol, Klein tunneling in the \(\alpha -t_3\) model. Phys. Rev. B 95, 235432 (2017)

    Article  ADS  Google Scholar 

  28. E. Illes, Properties of the \(\alpha -T_3\) Model, Ph.D. thesis (2017)

  29. Y.-R. Chen, Y. Xu, J. Wang, J.-F. Liu, Z. Ma, Enhanced magneto-optical response due to the flat band in nanoribbons made from the \(\alpha -t_3\) lattice. Phys. Rev. B 99, 045420 (2019)

    Article  ADS  Google Scholar 

  30. A. Iurov, G. Gumbs, D. Huang, Peculiar electronic states, symmetries, and berry phases in irradiated \(\alpha -t_3\) materials. Phys. Rev. B 99, 205135 (2019)

    Article  ADS  Google Scholar 

  31. A. Raoux, M. Morigi, J.-N. Fuchs, F. Piéchon, G. Montambaux, From dia-to paramagnetic orbital susceptibility of massless fermions. Phys. Rev. Lett. 112, 026402 (2014)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  33. P.E. Allain, J.-N. Fuchs, Klein tunneling in graphene: optics with massless electrons. Eur. Phys. J. B 83, 301–317 (2011)

    Article  ADS  Google Scholar 

  34. D.F. Urban, D. Bercioux, M. Wimmer, W. Häusler, Barrier transmission of Dirac-like pseudospin-one particles. Phys. Rev. B 84, 115136 (2011)

    Article  ADS  Google Scholar 

  35. J. Malcolm, E. Nicol, Frequency-dependent polarizability, plasmons, and screening in the two-dimensional pseudospin-1 dice lattice. Phys. Rev. B 93, 165433 (2016)

    Article  ADS  Google Scholar 

  36. W.-X. Qiu, S. Li, J.-H. Gao, Y. Zhou, F.-C. Zhang, Designing an artificial Lieb lattice on a metal surface. Phys. Rev. B 94, 241409 (2016)

    Article  ADS  Google Scholar 

  37. J.D. Malcolm, E.J. Nicol, Magneto-optics of massless Kane fermions: role of the flat band and unusual berry phase. Phys. Rev. B 92, 035118 (2015)

    Article  ADS  Google Scholar 

  38. F. Wang, Y. Ran, Nearly flat band with Chern number c = 2 on the dice lattice. Phys. Rev. B 84, 241103 (2011)

    Article  ADS  Google Scholar 

  39. N.A. Franchina Vergel, L.C. Post, D. Sciacca, M. Berthe, F. Vaurette, Y. Lambert, D. Yarekha, D. Troadec, C. Coinon, G. Fleury et al., Engineering a robust flat band in iii–v semiconductor heterostructures. Nano Lett. 21, 680–685 (2020)

    Article  ADS  Google Scholar 

  40. Á.D. Kovács, G. Dávid, B. Dóra, J. Cserti, Frequency-dependent magneto-optical conductivity in the generalized \(\alpha {-}t_3\) model. Phys. Rev. B 95, 035414 (2017)

    Article  ADS  Google Scholar 

  41. X. Zhao, W. Deng, Printing photovoltaics by electrospray. Opto-Electron. Adv. 3, 190038 (2020)

    Article  Google Scholar 

  42. Y. Sun, A.-M. Wang, Magneto-optical conductivity of double Weyl semimetals. Phys. Rev. B 96, 085147 (2017)

    Article  ADS  Google Scholar 

  43. T. Biswas, T.K. Ghosh, Dynamics of a quasiparticle in the \(\alpha {-}t_3\) model: role of pseudospin polarization and transverse magnetic field on zitterbewegung. J. Phys. Condens. Matter 30, 075301 (2018)

    Article  ADS  Google Scholar 

  44. T. Biswas, T.K. Ghosh, Magnetotransport properties of the \(\alpha {-}t_3\) model. J. Phys. Condens. Matter 28, 495302 (2016)

    Article  Google Scholar 

  45. E. Illes, E. Nicol, Magnetic properties of the \(\alpha {-}t_3\) model: magneto-optical conductivity and the Hofstadter butterfly. Phys. Rev. B 94, 125435 (2016)

    Article  ADS  Google Scholar 

  46. Z. Wan, X. Chen, M. Gu, Laser scribed graphene for supercapacitors. Opto-Electron. Adv. 4, 200079 (2021)

    Article  Google Scholar 

  47. Y. Xu, L.-M. Duan, Unconventional quantum Hall effects in two-dimensional massive spin-1 fermion systems. Phys. Rev. B 96, 155301 (2017)

    Article  ADS  Google Scholar 

  48. S.F. Islam, P. Dutta, Valley-polarized magnetoconductivity and particle-hole symmetry breaking in a periodically modulated \(\alpha {-}t_3\) lattice. Phys. Rev. B 96, 045418 (2017)

    Article  ADS  Google Scholar 

  49. E. Illes, J. Carbotte, E. Nicol, Hall quantization and optical conductivity evolution with variable berry phase in the \(\alpha {-}t_3\) model. Phys. Rev. B 92, 245410 (2015)

    Article  ADS  Google Scholar 

  50. H. Pu, S. Rhim, C. Hirschmugl, M. Gajdardziska-Josifovska, M. Weinert, J. Chen, Strain-induced band-gap engineering of graphene monoxide and its effect on graphene. Phys. Rev. B 87, 085417 (2013)

    Article  ADS  Google Scholar 

  51. O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, T. Pereg-Barnea, Klein tunneling in deformed honeycomb lattices. Phys. Rev. Lett. 104, 063901 (2010)

    Article  ADS  Google Scholar 

  52. L. Mandhour , F. Bouhadida, Klein tunneling in deformed \(\alpha {-}t_3\) lattice (2020). arXiv preprint arXiv:2004.10144

  53. X. Cheng, B. Zhou, B. Zhou, G. Zhou, Strain effect on electronic structure and transport properties of zigzag \(\alpha {-}t_3\) nanoribbons: a mean-field theoretical study. J. Phys. Condens. Matter 33, 215301 (2021)

    Article  ADS  Google Scholar 

  54. L. Beliaev, O. Takayama, P. Melentiev, A. Lavrinenko, Photoluminescence control by hyperbolic metamaterials and metasurfaces: a review. Opto-Electron. Adv. 4, 210031 (2021)

    Article  Google Scholar 

  55. G.G. Naumis, S. Barraza-Lopez, M. Oliva-Leyva, H. Terrones, Electronic and optical properties of strained graphene and other strained 2d materials: a review. Rep. Prog. Phys. 80, 096501 (2017)

    Article  ADS  Google Scholar 

  56. X. Ye, S. Ke, X. Du, Y. Guo, H. Lü, Quantum tunneling in the \(\alpha {-}t_3\) model with an effective mass term. J. Low Temp. Phys. 199, 1332 (2020)

    Article  ADS  Google Scholar 

  57. D. Khokhlov, A. Rakhmanov, A. Rozhkov, Scattering on a rectangular potential barrier in nodal-line Weyl semimetals. Phys. Rev. B 97, 235418 (2018)

    Article  ADS  Google Scholar 

  58. Y.M. Blanter, M. Büttiker, Shot noise in mesoscopic conductors. Phys. Rep. 336, 1–166 (2000)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by the Natural Science Foundation of China under Grant No. 12074209, the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2019J100), and the Open Project of the State Key Laboratory of Low-Dimensional Quantum Physics (Grant No. KF202008).

Author information

Authors and Affiliations

  1. School of Physics, University of Electronic Science and Technology of China, Chengdu, 610054, China

    Xin Ye, Sha-Sha Ke, Wei Fu, Jun-Tong Ren & Hai-Feng Lü

  2. School of Physics, Harbin Institute of Technology, Harbin, 150001, China

    Xin Ye

Authors
  1. Xin Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sha-Sha Ke
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jun-Tong Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Hai-Feng Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hai-Feng Lü.

Ethics declarations

Funding

The authors have no relevant financial or non-financial interests to disclose. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. No financial or proprietary interests: The authors have no financial or proprietary interests in any material discussed in this article. The authors are responsible for correctness of the statements provided in the manuscript.

Conflict of interest

The authors have no competing interests to declare that are relevant to the content of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ye, X., Ke, SS., Fu, W. et al. Strain-Controlled Anisotropic Pseudospin Tunneling in the \(\alpha {-}T_{3}\) Model. J Low Temp Phys 209, 108–123 (2022). https://doi.org/10.1007/s10909-022-02813-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10909-022-02813-w

Keywords

Search

Navigation