Abstract
We study the electronic properties of a position-dependent effective mass electron on a bilayer graphene catenoid bridge. We propose a position-dependent mass (PDM) as a function of both Gaussian and mean curvature. The Hamiltonian exhibits parity and time-reversal steaming from the bridge symmetry. The effective potential contains the da Costa, centrifugal, and PDM terms which are concentrated around the catenoid bridge. For zero angular momentum states, the PDM term provides a transition between a reflectionless to a double-well potential. As a result, the bound states undergo a transition from a single state around the bridge throat into two states each one located at rings around the bridge. Above some critical value of the PDM coupling constant, the degeneracy is restored due to double-well tunneling resonance.
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This manuscript has no associated data or the data will not be deposited. [Authors’ comment: All the data is in the paper.]
References
A.K. Geim, K.S. Novoselov, Nat. Mater. 6, 183 (2007)
A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81, 109 (2009)
M. Katsnelson, Graphene: Carbon in Two Dimensions (Cambridge University Press, Cambridge, 2012)
S. Berber, Y.K. Kwon, D. Tomanek, Phys. Rev. Lett. 84, 4613 (2000)
A. Carvalho, M. Wang, X. Zhu, A.S. Rodin, H. Su, A.H. Castro Neto, Nat. Rev. Mater. 1, 11 (2016)
C. Furtado, F. Moraes, A.M. de M. Carvalho, Phys. Lett. A 372, 5368 (2008)
R. Dandoloff, T.T. Truong, Phys. Lett. A 325, 233 (2004)
V. Atanasov, R. Dandoloff, A. Saxena, Phys. Rev. B 79, 033404 (2009)
V. Atanasov, A. Saxena, Phys. Rev. B 92, 035440 (2015)
F. Guinea, A.K. Geim, M.I. Katsnelson, K.S. Novoselov, Phys. Rev. B 81, 035408 (2010)
F. de Juan, A. Cortijo, M.A.H. Vozmediano, Phys. Rev. B 76, 165409 (2007)
V. Atanasov, A. Saxena, Phys. Rev. B 81, 205409 (2010)
H. Jensen, H. Koppe, Ann. Phys. 63, 586 (1971)
R.C.T. da Costa, Phys. Rev. A 23, 1982 (1981)
R.C.T. da Costa, Phys. Rev. A 25, 2893 (1982)
S. Matsutani, J. Phys. Soc. Jpn. 61, 3825 (1992)
Luiz C.B. da Silva, Cristiano C. Bastos, Fábio G. Ribeiro, Ann. Phys. 379, 13 (2017)
G. Ferrari, G. Cuoghi, Phys. Rev. Lett. 100, 230403 (2008)
Y.L. Wang, L. Du, C.T. Xu, X.J. Liu, H.S. Zong, Phys. Rev. A 90, 042117 (2014)
Mark Burgess, B. Jensen, Phys. Rev. A. 48(3), 1861 (1993)
J. González, J. Herrero, Nucl. Phys. B 825, 426 (2010)
R. Pincak, J. Smotlacha, Quantum Matter 5, 114 (2016)
R. Dandoloff, Phys. Lett. A 373, 2667–2669 (2009)
R. Dandoloff, A. Saxena, B. Jensen, Phys. Rev. A 81, 014102 (2010)
M. Spivak, A Comprehensive Introduction to Differential Geometry (Publish or Perish, Houston, 1999)
H. Terrones, A.L. Mackay, Chem. Phys. Lett. 207, 45 (1993)
J.E.G. Silva et al., Phys. Lett. A 384, 126458 (2020)
Y.N. Joglekar, A. Saxena, Phys. Rev. B 80, 153405 (2009)
A. Sinner, K. Ziegler, Ann. Phys. 400, 262–278 (2019)
M.G. Burt, J. Phys. Condens. Matter 4(32), 6651 (1992)
S.Y. Ren, Y.-C. Chang, Ann. Phys. 325(5), 937–947 (2010)
J.M. Lévi-Le-Blond, Phys. Rev. A 52, 1845 (1995)
Gang Chen, Zi-dong Chen, Phys. Lett. A 331, 312 (2004)
F. Serafim et al., Phys. E 108, 139 (2019)
Pedro H. Souza et al., Ann. Phys. 530, 1800112 (2018)
T.J. Willmore, J. Lond. Math. Soc. 3, 307 (1971)
W. Helfrich, Z. Nat. C 28, 693 (1973)
M.A. Khamehchi et al., Phys. Rev. Lett. 118, 155301 (2017)
D. Yiqun et al., Phys. Rev. Lett. 99, 093904 (2007)
D. Strasser et al., Phys. Rev. Lett. 89, 283204 (2002)
C.M. Bender, Rept. Prog. Phys. 70, 947 (2007)
C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)
H.F. Jones, J. Mateo, Phys. Rev. D 73, 085002 (2006)
A.A. Andrianov, Phys. Rev. D 76, 025003 (2007)
A.A. Andrianov, Ann. Phys. 140, 82 (1982)
A.C.A. Ramos, G.A. Farias, N.S. Almeida, Phys. E 43, 1878 (2011)
M. Novaes, N. Studart, Mecânica Quântica Básica - São Paulo: Editora Livraria da Física (Série MNPEF), (2016)
Acknowledgements
J. E. G. Silva thanks the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Grants n\(^{\underline{\hbox {o}}}\) 312356/2017-0 for financial support.
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Silva, J.E.G., Furtado, J. & Ramos, A.C.A. Position-dependent mass effects on a bilayer graphene catenoid bridge. Eur. Phys. J. B 94, 127 (2021). https://doi.org/10.1140/epjb/s10051-021-00138-3
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DOI: https://doi.org/10.1140/epjb/s10051-021-00138-3