Barbaroux, Jean-Marie; Cornean, Horia D.; Zalczer, Sylvain

Localization for Gapped Dirac Hamiltonians with Random Perturbations: Application to Graphene Antidot Lattices

Doc. Math. 24, 65-93 (2019)
DOI: 10.25537/dm.2019v24.65-93
Communicated by Heinz Siedentop

Summary

In this paper we study random perturbations of first order elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities. The unperturbed operator \(H_0 := D_S + V_0\) is the sum of a Dirac-like operator \(D_S\) plus a periodic matrix-valued potential $V_0$, and is assumed to have an open gap. The random potential \(V_\omega\) is of Anderson-type with independent, identically distributed coupling constants and moving centers, with absolutely continuous probability distributions. We prove band edge localization, namely that there exists an interval of energies in the unperturbed gap where the almost sure spectrum of the family \(H_\omega := H_0 +V_\omega\) is dense pure point, with exponentially decaying eigenfunctions, that give rise to dynamical localization.

Mathematics Subject Classification

81Q10, 46N50, 34L15, 47A10

Keywords/Phrases

Dirac operators, random potentials, localization

References

  • 1. P. Anderson. Absence of diffusion in certain random lattices. Phys. Rev., 109:1492, 1958. http://dx.doi.org/10.1103/PhysRev.109.1492.
  • 2. J.-M. Barbaroux, J.-M. Combes, and P.D. Hislop. Localization near band edges for random Schrödinger operators. Helvetica Physica Acta, 70:16-43, 1997. http://dx.doi.org/10.5169/seals-117008; zbl 0866.35077; MR1441595.
  • 3. J.-M. Barbaroux, H.D. Cornean, and E. Stockmeyer. Spectral gaps in graphene antidot lattices. Integral Equations and Operator Theory, 89:631-646, 2017. http://dx.doi.org/10.1007/s00020-017-2411-9; zbl 1381.81044; MR3735513; arxiv 1705.09187.
  • 4. H. Boumaza and H. Najar. Lifshitz tails for matrix-valued anderson models. J. Stat. Phys, 160:371-396, 2015. http://dx.doi.org/10.1007/s10955-015-1255-4; zbl 1360.82039; MR3360464; arxiv 1111.3121.
  • 5. P. Briet and H.D. Cornean. Locating the spectrum for magnetic Schrödinger and Dirac operators. Comm. Partial Differential Equations, 27(5-6):1079-1101, 2002. http://dx.doi.org/10.1081/PDE-120004894; zbl 1009.35058; MR1916557.
  • 6. S.J. Brun, M.R. Thomsen, and T.G. Pedersen. Electronic and optical properties of graphene antidot lattices: comparison of Dirac and tight-binding models. J. Phys. Condens. Matter, 26:265301, 2014. http://dx.doi.org/10.1088/0953-8984/26/26/265301; arxiv 1404.6899.
  • 7. A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim. The electronic properties of graphene. Rev. Mod. Phys., 81:109-162, 2009. http://dx.doi.org/10.1103/RevModPhys.81.109; arxiv 0709.1163.
  • 8. J.-M. Combes and P.D. Hislop. Localization for some continuous, random hamiltonians in d-dimensions. Journal of Functional Analysis, 124:149-180, 1994. http://dx.doi.org/10.1006/jfan.1994.1103; zbl 0801.60054; MR1284608.
  • 9. M. Disertori, W. Kirsch, A. Klein, F. Klopp, and V. Rivasseau. In Random Schrödinger Operators, volume 25 of Panoramas et Synthèses. Société Mathématique de France, 2008. MR2516524.
  • 10. M. Dvorak, W. Oswald, and Z. Wu. Bandgap opening by patterning graphene. Sci. Rep., 3:2289, 2013. http://dx.doi.org/10.1038/srep02289.
  • 11. A. Figotin and P. Kuchment. Band-gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals. SIAM J. Appl. Math., 56(6):1561-1620, 1996. http://dx.doi.org/10.1137/S0036139995285236; zbl 0868.35009; MR1417473.
  • 12. J.A. Fürst, J.G. Pedersen, C. Flindt, N.A. Mortensen, M. Brandbyge, T.G. Pedersen, and A.-P. Jauho. Electronic properties of graphene antidot lattices. New J. Phys., 11:095020, 2009. http://dx.doi.org/10.1088/1367-2630/11/9/095020; arxiv 0907.0122.
  • 13. F. Germinet and S. de Bièvre. Dynamical localization for discrete and continuous random Schrödinger operators. Commun. Math. Phys, 194:323-341, 1997. http://dx.doi.org/10.1007/s002200050360; zbl 0911.60099; MR1627657.
  • 14. F. Ghribi, P.D. Hislop, F. Klopp. Localization for Schrödinger operators with random vector potentials. Contemp. Math., 447: 123-138, Amer. Math. Soc., Providence, RI, 2007. http://dx.doi.org/10.1090/conm/447/08687; zbl 1137.81012; MR2423576.
  • 15. F. Germinet and A. Klein. Bootstrap multiscale analysis and localization in random media. Commun. Math. Phys., 222:415-448, 2001. http://dx.doi.org/10.1007/s002200100518; zbl 0982.82030; MR1859605.
  • 16. I. Goldsheid, S. Molchanov, and L.A. Pastur. Pure point spectrum of stochastic one dimensional Schrödinger operators. Funct. Anal. App., 11:1-8, 1977. MR0470515.
  • 17. T. Kato. Perturbation Theory for Linear Operators. Classics in Mathematics. Springer Berlin Heidelberg, 1995. zbl 0836.47009; MR1335452.
  • 18. W. Kirsch. Random Schr\``odinger operators. A course. In Schr\''odinger operators (Sønderborg, 1988), volume 345 of Lecture Notes in Phys., pages 264-370. Springer, Berlin, 1989. http://dx.doi.org/10.1007/3-540-51783-9_23; zbl 0712.35070; arxiv 0712.35070.
  • 19. A. Klein and A. Koines. A general framework for localization of classical waves. II. Random media. Math. Phys. Anal. Geom., 7(2):151-185, 2004. http://dx.doi.org/10.1023/B:MPAG.0000024653.29758.20; zbl 1054.35102; MR2057123.
  • 20. A. Klein, A. Koines, and M. Seifert. Generalized eigenfunctions for waves in inhomogeneous media. Journal of Functional Analysis, 190(1):255-291, 2002. http://dx.doi.org/10.1006/jfan.2001.3887; zbl 1043.35097; MR1895534.
  • 21. L.A. Pastur. Spectra of random self-adjoint operators. Russian Math. Surv., 28, 1973. http://dx.doi.org/10.1070/RM1973v028n01ABEH001396; zbl 0277.60049; MR0406251.
  • 22. J.G. Pedersen and T.G. Pedersen. Band gaps in graphene via periodic electrostatic gating. Phys. Rev. B, 85:235432, 2012. http://dx.doi.org/10.1103/PhysRevB.85.235432; arxiv 1203.3629.
  • 23. R.A. Prado and C.R. de Oliveira. Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator. J. Math. Phys., 46:1-17, 2005. http://dx.doi.org/10.1063/1.1948328; zbl 1110.81079; MR2153538; arxiv math-ph/0505046.
  • 24. R.A. Prado, C.R. de Oliveira, and S.L. Carvalho. Dynamical localization for discrete Anderson Dirac operators. J. Stat. Phys., 167:260-296, 2017. http://dx.doi.org/10.1007/s10955-017-1746-6; zbl 1372.35044; MR3626630.
  • 25. M. Reed and B. Simon. Methods of Modern Mathematical Physics, volume I: Functional Analysis. Elsevier Science, 1981. zbl 0459.46001.
  • 26. B. Simon. Trace Ideals and Their Applications. Mathematical Surveys and Monographs. American Mathematical Society, 2nd ed., 2005. zbl 1074.47001; MR2154153.
  • 27. B. Thaller. The Dirac Equation. Springer-Verlag, Berlin, 1992. zbl 0765.47023; MR1219537.
  • 28. M.R. Thomsen, S.J. Brun, and T.G. Pedersen. Dirac model of electronic transport in graphene antidot barriers. J. Phys. Condens. Matter, 26(33):335301, 2014. http://dx.doi.org/10.1088/0953-8984/26/33/335301.
  • 29. S. Yuan, R. Roldán, A.-P. Jauho, and M. Katsnelson. Electronic Properties of Disordered Graphene Antidot Lattices. Physical Review B, 87:085430, 2013. http://dx.doi.org/10.1103/PhysRevB.87.085430; arxiv 1211.5432.

Affiliation

Barbaroux, Jean-Marie
Centre de Physique Theorique, UMR 7332 CNRS, Université de Toulon, France
Cornean, Horia D.
Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg Ø, Denmark
Zalczer, Sylvain
Aix Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France

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