Abstract
This article provides a pedagogical review on Klein tunneling in graphene, i.e. the peculiar tunneling properties of two-dimensional massless Dirac electrons. We consider two simple situations in detail: a massless Dirac electron incident either on a potential step or on a potential barrier and use elementary quantum wave mechanics to obtain the transmission probability. We emphasize the connection to related phenomena in optics, such as the Snell-Descartes law of refraction, total internal reflection, Fabry-Pérot resonances, negative refraction index materials (the so called meta-materials), etc. We also stress that Klein tunneling is not a genuine quantum tunneling effect as it does not necessarily involve passing through a classically forbidden region via evanescent waves. A crucial role in Klein tunneling is played by the conservation of (sublattice) pseudo-spin, which is discussed in detail. A major consequence is the absence of backscattering at normal incidence, of which we give a new shorten proof. The current experimental status is also thoroughly reviewed. The Appendix contains the discussion of a one-dimensional toy model that clearly illustrates the difference in Klein tunneling between mono- and bi-layer graphene.
Similar content being viewed by others
References
G. Gamow, Z. Phys. 51, 204 (1928)
R.W. Gurney, E.U. Condon, Nature 122, 439 (1928)
R.W. Gurney, E.U. Condon, Phys. Rev. 33, 127 (1929)
A. Messiah, Quantum mechanics (Dover, 1999)
O. Klein, Z. Phys. 53, 157 (1929)
A. Calogeracos, N. Dombey, Contemp. Phys. 40, 313 (1999)
P.A.M. Dirac, Proc. R. Soc. Lond. 117, 610 (1928)
P.A.M. Dirac, The Principles of quantum mechanics (Oxford University Press, 1930)
M.I. Katsnelson, K.S. Novoselov, A.K. Geim, Nature Phys. 2, 620 (2006)
V.V. Cheianov, V. Fal’ko, Phys. Rev. B 74, 041403 (2006)
J.M. Pereira, V. Mlinar, F.M. Peeters, P. Vasilopoulos, Phys. Rev. B 74, 045424 (2006)
B. Huard et al., Phys. Rev. Lett. 98, 236803 (2007)
R.V. Gorbachev, A.S. Mayorov, A.K. Savchenko, D.W. Horsell, F. Guinea, Nano Lett. 8, 1995 (2008)
N. Stander et al., Phys. Rev. Lett. 102, 026807 (2009)
A.F. Young, P. Kim, Nature Phys. 5, 222 (2009)
A. Calogeracos, N. Dombey, Int. J. Mod. Phys. A 14, 631 (1999)
C.W.J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008)
J.M. Pereira Jr, F.M. Peeters, A. Chaves, G.A. Farias, Semicond. Sci. Technol. 25, 033002 (2010)
P.R. Wallace, Phys. Rev. 71, 622 (1947)
C. Bena, G. Montambaux, New J. Phys. 11, 095003 (2009)
A. Castro-Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A. Geim, Rev. Mod. Phys. 81, 109 (2009)
A. Geim, P. Kim, Sci. Amer. 298, 90 (2008)
J.N. Fuchs, M.O. Goerbig, Pour la Science 367, 36 (2008)
N.H. Shon, T. Ando, J. Phys. Soc. Jpn 67, 2421 (1998)
T. Ando, T. Nakanishi, J. Phys. Soc. Jpn 67, 1704 (1998)
T. Ando, T. Nakanishi, R. Saito, J. Phys. Soc. Jpn 67, 2857 (1998)
E. McCann, K. Kechedzhi, V.I. Falko, H. Suzuura, T. Ando, B.L. Altshuler, Phys. Rev. Lett. 97, 146805 (2006)
J.H. Bardarson, J. Tworzydlo, P. Brouwer, C.W.J. Beenakker, Phys. Rev. Lett. 99, 106801 (2007)
K. Nomura, M. Koshino, S. Ryu, Phys. Rev. Lett. 99, 146806 (2007)
V. Jakubský, L.M. Nieto, M.S. Plyushchay, Phys. Rev. D 83, 047702 (2011)
V.V. Cheianov, V. Fal’ko, B.L. Altshuler, Science 315, 1252 (2007)
J.B. Pendry, D.R. Smith, Sci. Amer. 295, 60 (2006)
A.G. Aronov, G.E. Pikus, Sov. Phys. JETP 24, 188 (1967)
J. Cayssol, B. Huard, D. Goldhaber-Gordon, Phys. Rev. B 79, 075428 (2009)
X. Chen, J.W. Tao, Appl. Phys. Lett. 94, 262102 (2009)
A.V. Shytov, M.S. Rudner, L.S. Levitov, Phys. Rev. Lett. 101, 156804 (2008)
M.I. Katsnelson, Eur. Phys. J. B 51, 157 (2006)
J. Tworzydlo et al., Phys. Rev. Lett. 96, 246802 (2006)
E.B. Sonin, Phys. Rev. B 79, 195438 (2009)
M.C. Lemme, T.J. Echtermeyer, M. Baus, H. Kurz, IEEE Electron Device Lett. 28, 282 (2007)
J.R. Williams, L. DiCarlo, C.M. Marcus, Science 317, 638 (2007)
B. Özyilmaz, P. Jarillo-Herrero, D. Efetov, D.A. Abanin, L.S. Levitov, P. Kim, Phys. Rev. Lett. 99, 166804 (2007)
G. Liu, J. Velasco, W. Bao, C.N. Lau, Appl. Phys. Lett. 92, 203103 (2008)
K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306, 666 (2004)
M.M. Fogler, D.S. Novikov, L.I. Glazman, B.I. Shklovskii, Phys. Rev. B 77, 075420 (2008)
L.M. Zhang, M.M. Fogler, Phys. Rev. Lett. 100, 116804 (2008)
E. Rossi, J.H. Badarson, P.W. Brouwer, S. Das Sarma, Phys. Rev. B 81, R121408 (2010)
M.R. Setare, D. Jahani, Physica B Condens. Matter. 405, 1433 (2010)
S. Park, H.-S. Sim, arXiv:1103.3331 (2011)
O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901 (2010)
M. Ramezani Masir, P. Vasilopoulos, F.M. Peeters, Phys. Rev. B 82, 115417 (2010)
C.H. Park, L. Yang, Y.W. Son, M.L. Cohen, S.G. Louie, Nature Phys. 4, 213 (2008)
M. Barbier, P. Vasilopoulos, F.M. Peeters, Phys. Rev. B 80, 205415 (2009)
S. Ghosh, M. Sharma, J. Phys.: Condens. Matter 21, 292204 (2009)
M. Bocquet, Nucl. Phys. B 546, 621 (1999)
N. Mott, W.D. Twose, Adv. Phys. 10, 107 (1961)
F. Sauter, Z. Phys. 73, 547 (1931)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Allain, P.E., Fuchs, J.N. Klein tunneling in graphene: optics with massless electrons. Eur. Phys. J. B 83, 301 (2011). https://doi.org/10.1140/epjb/e2011-20351-3
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2011-20351-3