Klein tunneling in phononic crystals: A step case

Abstract

In a recent paper [Jiang, et al., Science 370, 1447 (2020)], it was reported that zero reflection or Klein tunneling can be observed for normally incident quasiparticles upon a potential barrier constructed by two phononic crystals (PCs) with Dirac cone band structures. Here, we develop a first-principles approach for accurate computation of the reflection of quasiparticles by a potential step with two PCs at normal incidence. Strikingly, it is found that minimal reflection of quasiparticles (R_m = 0.8%) occurs at an energy below the potential step, while moderate reflection (R_d = 17%) remains at the center energy of the potential step, even with the PCs adopted by Jiang et al. A physical model is presented to understand such phenomena, where the zigzag interface in the step serves as a graded antireflection layer and reflection increases dramatically near the band gap of PCs. Two solutions are also shown for realizing lower R_d or even ideal Klein tunneling in PCs, with reducing the difference between the two PCs or enhancing the antireflection effect of the interface. Our work reveals the effects of the zigzag interface and band gap on Klein tunneling in PCs, which will be inspiring for exploring more fascinating phenomena of quasiparticles in classical wave systems.

Graphical abstract

Keywords

Klein tunneling / phononic crystals

References

[1] O. Klein, The reflection of electrons at a potential jump to the relative dynamics of Dirac, Z. Phys. 53(3−4), 157 (1929)

[2] A. Calogeracos and N. Dombey, History and physics of the Klein paradox, Contemp. Phys. 40(5), 313 (1999)

[3] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films, Science 306(5696), 666 (2004)

[4] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Chiral tunnelling and the Klein paradox in graphene, Nat. Phys. 2(9), 620 (2006)

[5] C. H. Park, L. Yang, Y. W. Son, M. L. Cohen, and S. G. Louie, Anisotropic behaviours of massless Dirac fermions in graphene under periodic potentials, Nat. Phys. 4(3), 213 (2008)

[6] C. H. Park and S. G. Louie, Energy gaps and Stark effect in boron nitride nanoribbons, Nano Lett. 8(8), 2200 (2008)

[7] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81(1), 109 (2009)

[8] B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang, and D. Goldhaber-Gordon, Transport measurements across a tunable potential barrier in graphene, Phys. Rev. Lett. 98(23), 236803 (2007)

[9] R. V. Gorbachev, A. S. Mayorov, A. K. Savchenko, D. W. Horsell, and F. Guinea, Conductance of p-n-p structures with “Air-Bridge” top gates, Nano Lett. 8(7), 1995 (2008)

[10] A. F. Young and P. Kim, Quantum interference and Klein tunnelling in graphene heterojunctions, Nat. Phys. 5(3), 222 (2009)

[11] P. E. Allain and J. N. Fuchs, Klein tunneling in graphene: Optics with massless electrons, Eur. Phys. J. B 83(3), 301 (2011)

[12] X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials, Nat. Mater. 10(8), 582 (2011)

[13] J. Mei, Y. Wu, C. T. Chan, and Z. Q. Zhang, First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals, Phys. Rev. B 86(3), 035141 (2012)

[14] Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, Topological acoustics, Phys. Rev. Lett. 114(11), 114301 (2015)

[15] A. B. Khanikaev, R. Fleury, S. H. Mousavi, and A. Alú, Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice, Nat. Commun. 6(1), 8260 (2015)

[16] M. Xiao, W. J. Chen, W. Y. He, and C. T. Chan, Synthetic gauge flux and Weyl points in acoustic systems, Nat. Phys. 11(11), 920 (2015)

[17] C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, Acoustic topological insulator and robust one-way sound transport, Nat. Phys. 12(12), 1124 (2016)

[18] J. Lu, C. Qiu, M. Ke, and Z. Liu, Valley vortex states in sonic crystals, Phys. Rev. Lett. 116(9), 093901 (2016)

[19] Y. Liu, C. S. Lian, Y. Li, Y. Xu, and W. Duan, Pseudospins and topological effects of phonons in a Kekule lattice, Phys. Rev. Lett. 119(25), 255901 (2017)

[20] Z. Zhang, Q. Wei, Y. Cheng, T. Zhang, D. Wu, and X. Liu, Topological creation of acoustic pseudospin multipoles in a flow-free symmetry-broken metamaterial lattice, Phys. Rev. Lett. 118(8), 084303 (2017)

[21] M. Dubois, C. Shi, X. Zhu, Y. Wang, and X. Zhang, Observation of acoustic Dirac-like cone and double zero refractive index, Nat. Commun. 8(1), 14871 (2017)

[22] F. Li, X. Huang, J. Lu, J. Ma, and Z. Liu, Weyl points and Fermi arcs in a chiral phononic crystal, Nat. Phys. 14(1), 30 (2018)

[23] X. J. Zhang, M. Xiao, Y. Cheng, M. H. Lu, and J. Christensen, Topological sound, Commun. Phys. 1(1), 97 (2018)

[24] C. Xu, J. Mei, G. Ma, and Y. Wu, Type-II Dirac phonons in a two-dimensional phononic crystal, APL Mater. 12(4), 041128 (2024)

[25] X. Jiang, C. Shi, Z. Li, S. Wang, Y. Wang, S. Yang, S. G. Louie, and X. Zhang, Direct observation of Klein tunneling in phononic crystals, Science 370(6523), 1447 (2020)

[26] J. Li and C. T. Chan, Double-negative acoustic metamaterial, Phys. Rev. E 70(5), 055602(R) (2004)

[27] L. Liu, H. Chang, C. Zhang, and X. Hu, Single-channel labyrinthine metasurfaces as perfect sound absorbers with tunable bandwidth, Appl. Phys. Lett. 111(8), 083503 (2017)

[28] Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures, Nat. Nanotechnol. 2(12), 770 (2007)

[29] See supplemental material for more results on the topic.

Declarations

The authors declare that they have no competing interests and there are no conflicts.

Electronic supplementary materials

The online version contains supplementary material available at https://doi.org/10.15302/frontphys.2025.022204.

Acknowledgements

This work was supported by the National Key Research and Development Program of China (Nos. 2023YFA1406901 and 2018YFA0306201).

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2024 Higher Education Press