Abstract
We investigate the quantum tunneling properties through a square potential barrier in the \(\alpha -T_3\) model with an effective mass term. The additional mass term can be induced by the effective magnetic field or the site energy difference on different sublattices, in which the flat band is located at the center or edge of the band gap band, respectively. The linear dispersion in the \(\alpha -T_3\) model is modified in the presence of the mass term, leading to the destroying of the Klein tunneling for the normal incidence. It is demonstrated that the additional mass term in general suppresses the transmission. When the flat band is located at the band edge, the super-Klein tunneling and the resonant tunneling are considerably suppressed with the increase in the energy gap. For the dice lattice that \(\alpha =1\), the super-Klein tunneling is preserved when the flat band is located at the center of the band gap.
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References
M.R. Setare, D. Jahani, Klein tunneling of massive Dirac fermions in single-layer graphene. Physica B 405(5), 1433–1436 (2010)
P.E. Allain, J.-N. Fuchs, Klein tunneling in graphene: optics with massless electrons. Eur. Phys. J. B 83(3), 301 (2011)
E. Illes, Properties of the \(\alpha \)-\({T}_3\)Model. Ph.D. Thesis (2017)
A. Iurov, G. Gumbs, D. Huang, Peculiar electronic states, symmetries, and berry phases in irradiated \(\alpha \)-\({T}_3\) materials. Phys. Rev. B 99(20), 205135 (2019)
K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Electric field effect in atomically thin carbon films. Science 306(5696), 666–669 (2004)
A.H.C. Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, The electronic properties of graphene. Rev. Mod. Phys. 81(1), 109 (2009)
Y. **e, Y. Tan, A.W. Ghosh, Spintronic signatures of Klein tunneling in topological insulators. Phys. Rev. B 96(20), 205151 (2017)
X. Yafang, G. **, Omnidirectional transmission and reflection of pseudospin-1 Dirac fermions in a Lieb superlattice. Phys. Lett. A 378(47), 3554–3560 (2014)
X. Hong-Ya, Y.-C. Lai, Pseudospin-1 wave scattering that defies chaos \(Q\)-spoiling and Klein tunneling. Phys. Rev. B 99(23), 235403 (2019)
L. Wang, D.-X. Yao, Coexistence of spin-1 fermion and Dirac fermion on the triangular kagome lattice. Phys. Rev. B 98(16), 161403 (2018)
B. Dey, T.K. Ghosh, Photoinduced valley and electron-hole symmetry breaking in \(\alpha \)-\({T}_3\) lattice: the role of a variable Berry phase. Phys. Rev. B 98(7), 075422 (2018)
H.-F. Lü, Y.-H. Deng, S.-S. Ke, Y. Guo, H.-W. Zhang, Quantum impurity in topological multi-Weyl semimetals. Phys. Rev. B 99(11), 115109 (2019)
V. Hung Nguyen, J.-C. Charlier, Klein tunneling and electron optics in Dirac–Weyl fermion systems with tilted energy dispersion. Phys. Rev. B 97(23), 235113 (2018)
T. Ozawa, A. Amo, J. Bloch, I. Carusotto, Klein tunneling in driven-dissipative photonic graphene. Phys. Rev. A 96(1), 013813 (2017)
J.Y. Vaishnav, C.W. Clark, Observing Zitterbewegung with ultracold atoms. Phys. Rev. Lett. 100(15), 153002 (2008)
B. Dey, T.K. Ghosh, Floquet topological phase transition in the \(\alpha \)-\({T}_3\) lattice. Phys. Rev. B 99(20), 205429 (2019)
L. Chen, J. Zuber, Z. Ma, C. Zhang, Nonlinear optical response of the \(\alpha -{T}_3\) model due to the nontrivial topology of the band dispersion. Phys. Rev. B 100(3), 035440 (2019)
M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415(6867), 39 (2002)
M.G.G. Abad, M. Valinezhad, M. Mahmoudi, Enhanced nonlinear magneto-optical rotation in cold atoms: a theoretical study. Sci. Rep. 9(1), 6312 (2019)
F. Wang, Y. Ran, Nearly flat band with Chern number \(c= 2\) on the dice lattice. Phys. Rev. B 84(24), 241103 (2011)
D. Bercioux, D.F. Urban, H. Grabert, W. Häusler, Massless Dirac–Weyl fermions in a \({\cal{T}}_3\) optical lattice. Phys. Rev. A 80(6), 063603 (2009)
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(3), 035118 (2015)
D.D. Edwall, J.S. Chen, J. Bajaj, E.R. Gertner, MOCVD Hg1-xCdxTe/GaAs for IR detectors. Semicond. Sci. Technol. 5(3S), S221 (1990)
Y. Betancur-Ocampo, V. Gupta, Perfect transmission of 3D massive Kane fermions in HgCdTe Veselago lenses. J. Phys. Condens. Matter. 30(3), 035501 (2017)
E. Illes, E.J. Nicol, Klein tunneling in the \(\alpha -{T}_3\) model. Phys. Rev. B 95(23), 235432 (2017)
K. Kim, Super-Klein tunneling of Klein–Gordon particles. Results Phys. 12, 1391–1394 (2019)
A. Jellal, E.B. Choubabi, H. Bahlouli, A. Aljaafari, Transport properties through double barrier structure in graphene. J. Low Temp. Phys. 168(1), 40–56 (2012)
F. Guinea, Models of electron transport in single layer graphene. J. Low Temp. Phys. 153(5), 359–373 (2008)
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(7), 075301 (2018)
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(3), 035414 (2017)
Y. Sun, A.-M. Wang, Magneto-optical conductivity of double Weyl semimetals. Phys. Rev. B 96(8), 085147 (2017)
T. Biswas, T.K. Ghosh, Magnetotransport properties of the \(\alpha -{T}_3\) model. J. Phys. Condens. Matter. 28(49), 495302 (2016)
E. Illes, E.J. Nicol, Magnetic properties of the \(\alpha \)-\({T}_3\) model: Magneto-optical conductivity and the Hofstadter butterfly. Phys. Rev. B 94(12), 125435 (2016)
X. Yong, L.-M. Duan, Unconventional quantum Hall effects in two-dimensional massive spin-1 fermion systems. Phys. Rev. B 96(15), 155301 (2017)
A. Raoux, M. Morigi, J.-N. Fuchs, F. Piéchon, G. Montambaux, From dia-to paramagnetic orbital susceptibility of massless fermions. Phys. Rev. Lett. 112(2), 026402 (2014)
J.J. Wang, S. Liu, J. Wang, J.-F. Liu, Valley-coupled transport in graphene with Y-shaped Kekulé structure. Phys. Rev. B 98(19), 195436 (2018)
J. Romhányi, K. Penc, R. Ganesh, Hall effect of triplons in a dimerized quantum magnet. Nat. Commun. 6, 6805 (2015)
R. Shen, L.B. Shao, B. Wang, D.Y. **ng, Single Dirac cone with a flat band touching on line-centered-square optical lattices. Phys. Rev. B 81(4), 041410 (2010)
X. Changqing, G. Wang, Z.H. Hang, J. Luo, C.T. Chan, Y. Lai, Design of full-k-space flat bands in photonic crystals beyond the tight-binding picture. Sci. Rep. 5, 18181 (2015)
C. Shang, Y. Zheng, B.A. Malomed, Weyl solitons in three-dimensional optical lattices. Phys. Rev. A 97(4), 043602 (2018)
D. Walkup, J.A. Stroscio, Helical level structure of Dirac potential wells. Phys. Rev. B 96(20), 201409 (2017)
D.A. Khokhlov, A.L. Rakhmanov, A.V. Rozhkov, Scattering on a rectangular potential barrier in nodal-line Weyl semimetals. Phys. Rev. B 97(23), 235418 (2018)
B. Dóra, J. Kailasvuori, R. Moessner, Lattice generalization of the Dirac equation to general spin and the role of the flat band. Phys. Rev. B 84, 195422 (2011)
D.F. Urban, D. Bercioux, M. Wimmer, W. Häusler, Barrier transmission of Dirac-like pseudospin-one particles. Phys. Rev. B 84, 115136 (2011)
R. Zhu, P.M. Hui, Shot noise and fano factor in tunneling in three-band pseudospin-1 Dirac–Weyl systems. Phys. Rev. A 381(23), 1971–1975 (2017)
R. Shen, L.B. Shao, B. Wang, D.Y. **ng, Single Dirac cone with a flat band touching on line-centered-square optical lattices. Phys. Rev. B 81, 041410 (2010)
A. Fang, Z.Q. Zhang, S.G. Louie, C.T. Chan, Nonuniversal critical behavior in disordered pseudospin-1 systems. Phys. Rev. B 99(1), 014209 (2019)
L.L. Chang, L. Esaki, R. Tsu, Resonant tunneling in semiconductor double barriers. Appl. Phys. Lett. 24(12), 593–595 (1974)
Acknowledgements
This work was supported by the Natural Science Foundation of China under Grant Nos. 61474018, 11574173 and 61505023, 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. KF201709).
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Ye, X., Ke, SS., Du, XW. et al. Quantum Tunneling in the \(\alpha -T_3\) Model with an Effective Mass Term. J Low Temp Phys 199, 1332–1343 (2020). https://doi.org/10.1007/s10909-020-02440-3
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DOI: https://doi.org/10.1007/s10909-020-02440-3