Abstract
The conductance and electronic transmission of Dirac electrons and holes across multibarrier Cantor-like graphene are investigated using on the transfer matrix method and Landauer–Buttiker formalism. Electric and magnetic fields are applied to the top of a monolayer graphene to generate multiple electromagnetic barriers separated by quantum wells. The impact of the magnetic and electric fields as well as the quantum size on the behavior of the transmission coefficient and conductance is discussed. The results indicate that the transmission coefficients exhibit oscillations indicating the existence of resonant states in miniband energies separated by minigap energies. This phenomenon known as the bifurcation process is more pronounced for a higher number of barriers. The behavior observed in the conductance variation reflects of the transmission coefficient especially for lower energies. Furthermore, the contour plot of the transmission coefficient shows the predominant impact of the incidence angle on the symmetry of the minigaps and minibands. These results are expected to be beneficial for experiments that improve the performance of new generations of devices based on multibarrier Cantor-like graphene systems.
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References
Barbier, M., Vasilopoulos, P., Peeters, F.M.: Kronig-Penney model on bilayer graphene: spectrum and transmission periodic in the strength of the barriers. Phys. Rev. B 82, 235408 (2010). https://doi.org/10.1103/PhysRevB.82.235408
Biswas, R., Biswas, A., Hui, N., Sinha, C.: Ballistic transport through electric field modulated graphene periodic magnetic barriers. J. Appl. Phys. 108, 043708 (2010). https://doi.org/10.1063/1.3467778
Biswas, R., Maiti, S., Mukhopadhyay, S., Sinha, C.: Electron transmission through a periodically driven graphene magnetic barrier. Phys. Lett. A 381(18), 1582–1591 (2017). https://doi.org/10.1016/j.physleta.2017.02.045
Bliokh, Y.P., Freilikher, V., Nori, F.: Tunable electronic transport and unidirectional quantum wires in graphene subjected to electric and magnetic fields. Phys. Rev. B 81(7), 075410 (2010). https://doi.org/10.1103/PhysRevB.81.075410
Calogeracos, A., Dombey, N.: History and physics of the Klein Paradox. Contemp. Phys. 40, 313–321 (1999). https://doi.org/10.1080/001075199181387
Dell’Anna, L., De Martino, A.: Magnetic superlattice and finite-energy Dirac points in graphene. Phys. Rev. B 83(15), 155449 (2011). https://doi.org/10.1103/PhysRevB.83.155449
Dell’Anna, L., De Martino, A.: Multiple magnetic barriers in graphene. Phys. Rev. B 79(4), 045420 (2009). https://doi.org/10.1103/PhysRevB.79.045420
De Martino, A., Dell’Anna, L., Egger, R.: Magnetic confinement of massless dirac fermions in graphene. Phys. Rev. Lett. 98(6), 066802 (2007a). https://doi.org/10.1103/PhysRevLett.98.066802
De Martino, A., Dell’Anna, L., Egger, R.: Magnetic barriers and confinement of Dirac-Weyl quasiparticles in graphene. Solid State Commun. 144(12), 547–550 (2007b). https://doi.org/10.1016/j.ssc.2007.03.062
De Martino, A., Dell’Anna, L., Egger, R.: Magnetic barriers and confinement of Dirac-Weyl quasiparticles in graphene. Solid State Commun. 144, 547–550 (2007c). https://doi.org/10.1016/j.ssc.2007.03.062
El-Shafai, N.M., Ramadan, M.S., Alkhamis, K.M., Aljohani, M.M., El-Metwaly, N.M., El-Mehasseb, I.M.: A unique engineering building of nanoelectrodes based on titanium and metal oxides nanoparticles captured on graphene oxide surface for supercapacitors and energy storage. J. Alloys Compd. 939, 168685 (2023). https://doi.org/10.1016/j.jallcom.2022.168685
Ghosh, S., Sharma, M.: Electron optics with magnetic vector potential barriers in graphene. J. Phys. Condens. Matter 21, 292204 (2009). https://doi.org/10.1088/0953-8984/21/29/292204
Ghosh, T.K., De Martino, A., Häusler, W., DellAnna, L., Egger, R.: Conductance quantization and snake states in graphene magnetic waveguides. Phys. Rev. B 77(8), 081404 (2008). https://doi.org/10.1103/PhysRevB.77.081404
Gusynin, V.P., Sharapov, S.G.: Unconventional Integer Quantum Hall Effect in Graphene. Phys. Rev. Lett. 95, 146801 (2005). https://doi.org/10.1103/PhysRevLett.95.146801
Häusler, W., De Martino, A., Ghosh, T.K., Egger, R.: Tomonaga-Luttinger liquid parameters of magnetic waveguides in graphene". Phys. Rev. B 78(16), 165402 (2008). https://doi.org/10.1103/PhysRevB.78.165402
Itzykson, C., Zuber, J.-B.: Quantum Field Theory. ISBN-10: 9780486445687, Dover, New York (2006)
Kormanyos, A., Rakyta, P., Oroszlany, L., Cserti, J.: Bound states in inhomogeneous magnetic field in graphene: semiclassical approach. Phys. Rev. B 78(4), 045430 (2008). https://doi.org/10.1103/PhysRevB.78.045430
Lima, J.R.F.: Controlling the energy gap of graphene by Fermi velocity engineering. Phys. Lett. A 379(3), 179–182 (2015). https://doi.org/10.1016/j.physleta.2014.11.005
Liu, H., Zhang, H., Liu, D., Kong, X.: Spin transport and magnetoresistance in Thue-Morse graphene superlattice with two ferromagnetic graphene electrodes. J. Appl. Phys. 114, 163715 (2013). https://doi.org/10.1063/1.4827380
Markos, P., Soukoulis, C.M.: Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials. Princeton University Press, ISBN-10: 0691130035 (2008)
Masir, M.R., Vasilopoulos, P., Matulis, A., Peeters, F.M.: Direction-dependent tunneling through nanostructured magnetic barriers in graphene. Phys. Rev. B 77(23), 235443 (2008b). https://doi.org/10.1103/PhysRevB.77.235443
Masir, M.R., Vasilopoulos, P., Peeters, F.M.: Wavevector filtering through single-layer and bilayer graphene with magnetic barrier structures. Appl. Phys. Lett. 93, 242103 (2008a). https://doi.org/10.1063/1.3049600
Masir, M.R., Vasilopoulos, P., Peeters, F.M.: Magnetic Kronig–Penney model for Dirac electrons in single-layer graphene. New J. Phys. 11, 095009 (2009). https://doi.org/10.1088/1367-2630/11/9/095009
Masir, M.R., Vasilopoulos, P., Peeters, F.M.: Graphene in inhomogeneous magnetic fields: bound, quasi-bound and scattering states. J. Phys. Condens. Matter 23, 315301 (2011). https://doi.org/10.1088/0953-8984/23/31/315301
McCann, E., Falko, V.I.: Landau-level degeneracy and quantum hall effect in a graphite bilayer. Phys. Rev. Lett. 96(8), 086805 (2006). https://doi.org/10.1103/PhysRevLett.96.086805
Myoung, N., Ihm, G.: Tunneling of Dirac fermions through magnetic barriers in graphene. Physica E: Low-Dimens Syst Nanostruct. 42(1), 70–72 (2009). https://doi.org/10.1016/j.physe.2009.09.001
Myoung, N., Ihm, G., Lee, S.J.: Magnetically induced waveguide in graphene. Phys. Rev. B 83(11), 113407 (2011). https://doi.org/10.1103/PhysRevB.83.113407
Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Katsnelson, M.I., Grigorieva, I.V., Dubonos, S.V., Firsov, A.A.: Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005). https://doi.org/10.1038/nature04233
Oroszlany, L., Rakyta, P., Kormanyos, A., Lambert, C.J., Cserti, J.: Theory of snake states in graphene. Phys. Rev. B 77(8), 081403 (2008). https://doi.org/10.1103/PhysRevB.77.081403
Park, S., Sim, H.S.: Magnetic edge states in graphene in nonuniform magnetic fields. Phys. Rev. B 77(7), 075433 (2008). https://doi.org/10.1103/PhysRevB.77.075433
Park, C.-H., Yang, L., Son, Y.-W., Cohen, M.L., Louie, S.G.: New generation of massless Dirac fermions in graphene under external periodic potentials. Phys. Rev. Lett. 101(12), 126804 (2008b). https://doi.org/10.1103/PhysRevLett.101.126804
Park, C.-H., Yang, L., Son, Y.-W., Cohen, M.L., Louie, S.G.: Anisotropic behaviours of massless Dirac fermions in graphene under periodic potentials. Nat. Phys. 4, 213–217 (2008a). https://doi.org/10.1038/nphys890
Pereira, V.M., Castro Neto, A.H.: Strain engineering of graphene’s electronic structure. Phys. Rev. Lett. 103(4), 046801 (2009). https://doi.org/10.1103/PhysRevLett.103.046801
Peres, N.M.R., Guinea, F., Castro Neto, A.H.: Electronic properties of disordered two-dimensional carbon. Phys. Rev. B 73, 125411 (2006). https://doi.org/10.1103/PhysRevB.73.125411
Redouani, I., Jellal, A.: Periodic barrier structure in AA-stacked bilayer graphene. Mater. Res. Express 3, 065005 (2016). https://doi.org/10.1088/2053-1591/3/6/065005
Reyes-Villagrana, R.A., Carrera-Escobedo, V.H., Suarez-Lopez, J.R., Madrigal-Melchor, J., Rodríguez-Vargas, I.: Energy minibands degeneration induced by magnetic field effects in graphene superlattices. Superlattices Microstruct. 112, 561–573 (2017). https://doi.org/10.1016/j.spmi.2017.10.014
Rodríguez-González, R., Rodríguez-Vargas, I.: The role of fractal aperiodic order in the transmittance, conductance and electronic structure of graphene-based systems. Physica E 69, 177–185 (2015). https://doi.org/10.1016/j.physe.2015.01.037
Rodríguez-González, R., Rodríguez-Vargas, I.: Transmission and transport properties in Cantor graphene structures: the case of magnetoelectric modulation. Physica B: Condens. Matter 510, 109–116 (2017). https://doi.org/10.1016/j.physb.2017.01.022
Sun, L., Fang, C., Song, Y., Guo, Y.: Transport properties through graphene-based fractal and periodic magnetic barriers. J. Phys. Condens. Matter 22, 445303 (2010b). https://doi.org/10.1088/0953-8984/22/44/445303
Sun, L., Fang, C., Guo, Y.: Controlling the energy gap of graphene by Fermi velocity engineering. J. Appl. Phys. 108, 063715 (2010a). https://doi.org/10.1063/1.3488647
Tan, L.Z., Park, C.-H., Louie, S.G.: Graphene Dirac fermions in one-dimensional inhomogeneous field profiles: transforming magnetic to electric field. Phys. Rev. B 81(19), 195426 (2010). https://doi.org/10.1103/PhysRevB.81.195426
Wang, D., Jin, G.: Magnetically confined states of Dirac electrons in a graphene-based quantum annulus. Europhys. Lett. 88, 17011 (2009). https://doi.org/10.1209/0295-5075/88/17011
Wei-Tao, Lu., Wang, S.-J., Wang, Y.-L., Jiang, H., Li, W.: Transport properties of graphene under periodic and quasiperiodic magnetic superlattices. Phys. Lett. A 377, 1368–1372 (2013). https://doi.org/10.1016/j.physleta.2013.03.035
Xu, H., Heinzel, T., Evaldsson, M., Ihnatsenka, S., Zozoulenko, I.V.: Resonant reflection at magnetic barriers in quantum wires. Phys. Rev. B 75(20), 205301 (2007). https://doi.org/10.1103/PhysRevB.75.205301
Xu, H.Z., Feng, S., Zhang, Y.: Resonant peak splitting in finite periodic superlattices with an unit cell of two barriers and two wells on monolayer graphene". Opt. Quant. Electron. 51, 158 (2019). https://doi.org/10.1007/s11082-019-1873-1
Yeh, P.: Optical Waves in Layered Media. Wiley-Interscience, ISBN-10: 0471731927 (2005)
Zhang, Y., Tan, Y.-W., Stormer, H.L., Kim, P.: Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005). https://doi.org/10.1038/nature04235
Acknowledgements
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: (22UQU4331235DSR01)
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This work was funded by the Deanship of Scientific Research at Umm Al-Qura University by Grant Code: (22UQU4331235DSR01).
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Conceptualization, WB, HD, OHS and FU; Methodology, WB, HD, OHS and FU; Software, WB, HD and OHS; Validation WB and HD; Formal analysis, WB, HD, OHS and FU; Investigation, WB, HD, OHS and FU; Resources, WB, HD, OHS and FU; Data curation, HD; Writing – original draft, HD and FU; Writing – review & editing, WB and OHS; Visualization, WB, HD, OHS and FU; Supervision, WB. All authors read and approved the final manuscript.
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Belhadj, W., Dakhlaoui, H., Alsalmi, O.H. et al. Impacts of electric and magnetic fields on the optical and electronic characteristics of graphene- based multibarrier structure. Opt Quant Electron 55, 1171 (2023). https://doi.org/10.1007/s11082-023-05430-3
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DOI: https://doi.org/10.1007/s11082-023-05430-3