Transmission via triangular double barrier and magnetic fields in graphene

Document Type : Regular article


1 Laboratory of Theoretical Physics Faculty of Sciences Ibn Maachou Street

2 Physics Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia


We study the transmission probability of Dirac fermions in graphene scattered by a triangular

double barrier potential in the presence of an external magnetic field. Our system made of two

triangular potential barrier regions separated by a well region characterized by an energy gap

Gp . Solving our Dirac-like equation and matching the solutions at the boundaries we express our

transmission and reflection coefficients in terms of transfer matrix. We show in particular that the

transmission exhibits oscillation resonances that are manifestation of the Klein tunneling effect.


Main Subjects

[1] 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, 666 (2004).
[2] N. Stander, B. Huard and D. Goldhaber-Gordon, ”Evidence for Klein Tunneling in Graphene p-n Junctions”, Phys. Rev. Lett. 102, 026807 (2009).
[3] M. I. Katsnelson, K. S. Novoselov and A. K. Geim, ”Chiral tunnelling and the Klein paradox in graphene”, Nature Phys. 2, 620 (2006).
[4] H. Sevincli, M. Topsakal and S. Ciraci, ”Superlattice structures of graphene-based armchair nanoribbons” Phys. Rev. B 78, 245402 (2008).
[5] M. R. Masir, P. Vasilopoulos and F. M. Peeters, ”Magnetic KronigPenney model for Dirac electrons in single-layer graphene”, New J. Phys. 11, 095009 (2009).
[6] L. Dell’Anna and A. De Martino, ”Multiple magnetic barriers in graphene”, Phys. Rev. B 79, 045420 (2009).
[7] S. Mukhopadhyay, R. Biswas and C. Sinha, ”Resonant tunnelling in a Fibonacci bilayer graphene superlattice”, Phys. Status Solidi B 247, 342 (2010).
[8] E. B. Choubabi, M. El Bouziani and A. Jellal, ”Tunneling for Dirac Fermions in Constant Magnetic Field”, Int. J. Geom. Meth. Mod. Phys. 7, 909 (2010).
[9] H. Bahlouli, E. B. Choubabi, A. Jellal and M. Mekkaoui, ”Measurements of Torsional Oscillations and Thermal Conductivity in Solid 4He”, J. Low Temp. Phys. 169, 51 (2012).
[10] M. Mekkaoui, A. Jellal and H. Bahlouli, ”Fano resonances in gapped graphene subject to an oscillating potential barrier and magnetic field”, Physica E 127, 114502 (2021).
[11] Y. Fattasse, M. Mekkaoui, A. Jellal and A. Bahaoui, ”Gap-tunable of tunneling time in graphene magnetic barrier”, Physica E 134, 114924 (2021).
[12] A. Jellal and A. El Mouhafid, ”Dirac fermions in an inhomogeneous magnetic field”, J. Phys. A: Math. Theo. 44, 015302 (2011).
[13] A. D. Alhaidari, H. Bahlouli and A. Jellal, ”Relativistic Double Barrier Problem with Three Sub-Barrier Transmission Resonance Regions”, Advances in Mathematical Physics 2012, 762908 (2012).
[14] M. Mekkaoui, R. El Kinani and A. Jellal, ”Goos-H¨anchen shifts in graphene-based linear barrier”, Mater. Res. Express 6, 085013 (2019).
[15] H. Bahlouli, E. B. Choubabi, A. El Mouhafid and A. Jellal, ”Transmission through biased graphene strip”, Solid State Communications 151, 1309 (2011).
[16] G. W. Semenoff, ”Chiral symmetry breaking in graphene”, Phys. Scripta T 146, 014016 (2012).
[17] M. Cvetic and G. W. Gibbons, ”Graphene and the Zermelo Optical Metric of the BTZ Black Hole” Ann. Phys. 327, 2617 (2012).
[18] A. Iorio and G. Lambiase, ”Quantum field theory in curved graphene spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking effect, and all that”, Phys. Rev. D 90, 025006 (2014).
[19] B. Pourhassan, M. Faizal, and S. A. Ketabi, ”Logarithmic correction of the BTZ black hole and adaptive model of graphene”, Int. J. Mod. Phys. D 27, 1850118 (2018).
[20] A. Matulis, F. M. Peeters, P. Vasilopoulos, ”Wave-vector-dependent tunneling through magnetic barriers”, Phys. Rev. Lett. 72, 1518 (1994).
[21] M. Ramezani Masir, P. Vasilopoulos, F. M. Peeters, ”Fabry-P´erot resonances in graphene microstructures: Influence of a magnetic field”, Phys. Rev. B 82, 115417 (2010).
[22] J. Tworzydlo, B. Trauzettel, M. Titov, A. Rycerz and C. W. J. Beenakker, ”Sub-Poissonian Shot Noise in Graphene”, Phys. Rev. Lett. 96, 246802 (2006).
[23] M. V. Berry and R. J. Modragon, ”Neutrino billiards: time-reversal symmetry-breaking without magnetic fields”, Proc. R. Soc. London Ser. A 412, 53 (1987).
[24] M. Abramowitz and I. Stegum, Handbook of Integrabls, Series and Products, (Dover, New York, 1956).
[25] L. Gonzalez-Diaz and V. M. Villalba, ”Resonances in the one-dimensional Dirac equation in the presence of a point interaction and a constant electric field”, Phys. Lett. A 352, 202 (2006).
Volume 2, Issue 1
We would like to dedicate this issue to the memory of Prof. John D. Barrow.
January 2022
Pages 49-70
  • Receive Date: 13 October 2021
  • Revise Date: 12 January 2022
  • Accept Date: 20 January 2022