Advertisement

Communications in Mathematical Physics

, Volume 367, Issue 3, pp 941–989 | Cite as

Magnetic Oscillations in a Model of Graphene

  • Simon Becker
  • Maciej ZworskiEmail author
Article
  • 23 Downloads

Abstract

We consider a quantum graph as a model of graphene in constant magnetic field and describe the density of states in terms of relativistic Landau levels satisfying a Bohr–Sommerfeld quantization condition. That provides semiclassical corrections (with the magnetic flux as the semiclassical parameter) in the study of magnetic oscillations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We gratefully acknowledge support by the UK Engineering and Physical Sciences Research Council (EPSRC) Grant EP/L016516/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis (SB), by the National Science Foundation under the Grant DMS-1500852 and by the Simons Foundation (MZ).We would also like to thank Nicolas Burq for useful discussions, Semyon Dyatlov for help with MATLAB coding and insightful comments and Hari Manoharan for introducing us to molecular graphene and for allowing us to use Figs. 1 and 7(b).

References

  1. BHJ18.
    Becker, S., Han, R., Jitomirskaya, S.: Cantor spectrum in graphene. arXiv:1803.00988 (2018)
  2. BGP07.
    Brüning J., Geyler V., Pankrashkin K.: Cantor and band spectra for periodic quantum graphs with magnetic fields. Commun. Math. Phys. 269(1), 87–105 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. CU08.
    Carmier P., Ullmo D.: Berry phase in graphene: a semiclassical perspective. Phys. Rev. B 77, 245413 (2008)ADSCrossRefGoogle Scholar
  4. CM01.
    Champelde T., Mineev VP.: The de Haas–van Alphen effect in two-and quasi-two-dimensional metals and superconductors. Philos. Mag. B 81, 55–74 (2001)ADSCrossRefGoogle Scholar
  5. CdV80.
    de Verdière YC.: Spectre conjoint d’opérateurs pseudo-différentiels qui commutent. II. Le cas intégrable. Math. Z. 171, 51–73 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. DS99.
    Dimassi M., Sjöstrand J.: Spectral Asymptotics in the Semi-Classical Limit. Cambridge University Press, Cambridge (1999)CrossRefzbMATHGoogle Scholar
  7. DyZw2.
    Dyatlov, S., Zworski, M.: Mathematical theory of scattering resonances, book in preparation. http://math.mit.edu/dyatlov/res/
  8. FW12.
    Fefferman C., Weinstein M.: Honeycomb lattice potentials and Dirac points. J. Am. Math. Soc. 25, 1169–1220 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. G*12.
    Gomes K.K., Mar W., Ko W., Guinea F., Manoharan H.C.: Designer Dirac fermions and topological phases in molecular graphene. Nature 483, 306–310 (2012)ADSCrossRefGoogle Scholar
  10. GA03.
    Gat, O., Avron, J.E.: Semiclassical analysis and the magnetization of the Hofstadter model. Phys. Rev. Lett. 91(18), (2003)Google Scholar
  11. GS05.
    Gusynin V., Sharapov S.: Magnetic oscillations in planar systems with the Dirac-like spectrum of quasiparticle excitations. II. Transport properties. Phys. Rev. B 71, 125124 (2005)ADSCrossRefGoogle Scholar
  12. GS06.
    Gusynin V., Sharapov S.: Transport of Dirac quasiparticles in graphene: Hall and optical conductivities. Phys. Rev. B 73, 245411 (2006)ADSCrossRefGoogle Scholar
  13. HKL16.
    Helffer, B., Kerdelhué, P., Royo-Letelier, J.: Chambers’s formula for the graphene and the Hou model with Kagome periodicity and applications. Ann. H. Poincaré 17(4) (2016)Google Scholar
  14. Ho03.
    Hörmander L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  15. HR84.
    Helffer B., Robert D.: Puits de potentiel généralisés et asymptotique semi-classique. Ann. Inst. H. Poincaré Phys. Théor. 41, 291–331 (1984)MathSciNetzbMATHGoogle Scholar
  16. HS88.
    Helffer, B., Sjöstrand, J.: Analyse semi-classique pour l’équation de Harper (avec application à l’équation de Schrödinger avec champ magnétique). Mém. Soc. Math. France (N.S.) 34 (1989)Google Scholar
  17. HS89.
    Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper. In: Schrödinger operators (Sønderborg, 1988). Lecture Notes in Phys., vol. 345, pp. 118–197, Springer, Berlin (1989)Google Scholar
  18. HS90a.
    Helffer, B., Sjöstrand, J.: Analyse semi-classique pour l’équation de Harper. II. Comportement semi-classique près d’un rationnel. Mém. Soc. Math. France (N.S.) 40 (1990)Google Scholar
  19. HS90b.
    Helffer, B., Sjöstrand, J.: On diamagnetism and de Haas–van Alphen effect. Ann. Inst. H. Poincaré Phys. Théor. 52(6), 303–375 (1990)Google Scholar
  20. KF17.
    Küppersbusch C., Küppersbusch C., Küppersbusch C.: Modifications of the Lifshitz–Kosevich formula in two-dimensional Dirac systems. Phys. Rev. B 96, 205410 (2017)ADSCrossRefGoogle Scholar
  21. KH14.
    Kishigi K., Hasegawa Y.: Quantum oscillations of magnetization in tight-binding electrons on a honeycomb lattice. Phys. Rev. B 90, 085427 (2014)ADSCrossRefGoogle Scholar
  22. KL14.
    Kerdlhué, P., Royo-Letelier, J.: On the low lying spectrum of the magnetic Schrödinger operator with kagome periodicity. Rev. Math. Phys. 26(10) (2014)Google Scholar
  23. KP07.
    Kuchment P., Post O.: On the spectra of carbon nano-structures. Commun. Math. Phys. 275(3), 805–82 (2007)Google Scholar
  24. KS03.
    Kostrykin V., Schrader R.: Quantum wires with magnetic fluxes. Commun. Math. Phys. 237, 161–179 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. L11.
    Luk’yanchuka A.: De Haas–van Alphen effect in 2D systems: application to mono- and bilayer graphene. Low Temp. Phys. 37, 45 (2011)ADSCrossRefGoogle Scholar
  26. O52.
    Onsager L.: Interpretation of the de Haas–van Alphen effect. Philos. Mag. 7, 43 (1952)Google Scholar
  27. Pa06.
    Pankrashkin K.: Spectra of Schrödinger operators on equilateral quantum graphs. Lett. Math. Phys. 77(2), 139–154 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. P*13.
    Polini M., Guinea F., Lewenstein M., Manoharan H.C., Pellegrini V.: Artificial honeycomb lattices for electrons, atoms and photons. Nat. Nanotechnol. 8, 625–633 (2013)ADSCrossRefGoogle Scholar
  29. RS78.
    Reed, M., Simon, B.: Analysis of Operators. Methods of Modern Mathematical Physics, vol. IV. Elsevier, Amsterdam (1978)Google Scholar
  30. S84.
    Shoenberg D.: Magnetic Oscillations in Metals. Cambridge University Press, Cambridge (1984)CrossRefGoogle Scholar
  31. Sch12.
    Schmüdgen, K.: Unbounded Self-Adjoint Operators on Hilbert Space. Graduate Texts in Mathematics, Springer, Berlin (2012)Google Scholar
  32. SGB04.
    Sharapov S.G., Gusynin V.P, Beck H.: Magnetic oscillations in planar systems with the Dirac-like spectrum of quasiparticle excitations. Phys. Rev. B. 69, 075104 (2004)ADSCrossRefGoogle Scholar
  33. Sj89.
    Sjöstrand, J.: Microlocal analysis for periodic magnetic Schrödinger equation and related questions. In: Bony J.-M., Grubb G., Hörmander L., Komatsu H., Sjöstrand J. (eds.) Microlocal Analysis and Applications. Lecture Notes in Mathematics, vol. 1495, Springer, berlin (1989)Google Scholar
  34. SZ07.
    Sjöstrand J., Zworski M.: Elementary linear algebra for advanced spectral problems. Ann. Inst. Fourier 57, 2095–2141 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  35. S17.
    Stauber T., Parida P., Trushin M., Ulybyshev M.V., Boyda D.L., Schliemann J.: Interacting electrons in graphene: fermi velocity renormalization and optical response. Phys. Rev. Lett. 118, 266801 (2017)ADSCrossRefGoogle Scholar
  36. Tan11.
    Tan Z., Tan C., Ma L., Liu G., Lu L., Yang C.: Shubnikov–de Haas oscillations of a single layer graphene under dc current bias. Phys. Rev. B 84, 115429 (2011)ADSCrossRefGoogle Scholar
  37. W11.
    Waldmann D. et al.: Bottom-gated epitaxial graphene. Nat. Mater. 10, 357–360 (2011)ADSCrossRefGoogle Scholar
  38. We77.
    Weinstein A.: Asymptotics of the eigenvalues clusters for the laplacian plus a potential. Duke Math. J. 44, 883–892 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  39. Zw12.
    Zworski, M.: Semiclassical Analysis. Graduate Studies in Mathematics, vol. 138. AMS, Providence (2012)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.DAMTPUniversity of CambridgeCambridgeUK
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations