zbMATH — the first resource for mathematics

Infinite mass boundary conditions for Dirac operators. (English) Zbl 1419.81014
Denote by \(\Omega\subset {\mathbb R}^2\) a bounded and simply-connected domain with boundary \(\partial\Omega\in C^3\). Let \(T\) be the differential expression associated with the massless Dirac operator, i.e., \[ T=\frac{1}{i}(\partial_1\sigma_1+\partial_2\sigma_2), \sigma_1=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_2=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \] and \(H_\infty\) is self-adjoint realization of \(T\) under the infinite mass boundary conditions (proposed by M. V. Berry & R. J. Mondragon). The authors consider also the Dirac operator in \(L^2({\mathbb R}^2,{\mathbb C}^2)\) defined on \({\mathbb R}^2\) with a mass term supported outside \(\Omega\): \[ H_M=T+i\sigma_2\sigma_1 M(1-\mathbf{1}_\Omega), M>0, \] where \(\mathbf{1}_\Omega\) is the characteristic function on \(\Omega\).
The main result of the paper under review is the convergence (as \(M\to \infty\)), in the sense of spectral projections, of \(H_M\) towards \(H_\infty\). In particular, the eigenvalues of \(H_M\) converge towards the eigenvalues of \(H_\infty\) and any eigenvalue of \(H_\infty\) is the limit of eigenvalues of \(H_M\).
The assumption \(\partial\Omega\in C^3\) allowed the authors to prove also regularity of eigenfunctions \(\{\varphi_j\}\) of \(H_\infty\): \(\{\varphi_j\}\subset H^2(\Omega,{\mathbb C}^2)\).

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
46N50 Applications of functional analysis in quantum physics
81Q37 Quantum dots, waveguides, ratchets, etc.
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
Full Text: DOI arXiv
[1] R. A. Adams and J. F. Fournier, Sobolev spaces. Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. MR 2424078 Zbl 1098.46001
[2] A. R. Akhmerov and C. W. J. Beenakker, Boundary conditions for Dirac fermions on a terminated honeycomb lattice. Phys. Rev. B 77 (2008), no. 8, 085423.
[3] W. Arendt and C. J. K. Batty, Absorption semigroups and Dirichlet boundary conditions. Math. Ann. 295 (1993), no. 3, 427–448.MR 1204830 Zbl 0788.47031 · Zbl 0788.47031
[4] R.. Benguria, S. Fournais, E. Stockmeyer, and H. Van Den Bosch, Self-adjointness of two-dimensional Dirac operators on domains. Ann. Henri Poincaré 18 (2017), no. 4, 1371–1383.MR 3626307 Zbl 1364.81117 · Zbl 1364.81117
[5] R. Benguria, S. Fournais, E. Stockmeyer, and H. Van Den Bosch, Spectral gaps of Dirac operators describing graphene quantum dots. Math. Phys. Anal. Geom. 20 (2017), no. 2, Art. 11, 12 pp.MR 3625007 Zbl 06921600 · Zbl 1424.81011
[6] M. V. Berry and R. J. Mondragon, Neutrino billiards: time-reversal symmetrybreaking without magnetic fields. Proc. Roy. Soc. London Ser. A 412 (1987), no. 1842, 53–74.MR 0901725
[7] B. Booß-Bavnbek, M. Lesch, and C. Zhu, The Calderón projection: new definition and applications. J. Geom. Phys. 59 (2009), no. 7, 784–826.MR 2536846 Zbl 1221.58016
[8] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene. Rev. Modern Phys. 81 (2009), 109–162.
[9] H. De Raedt and M. I. Katsnelson, Electron energy level statistics in graphene quantum dots. JETP letters 88 (2009), no. 9, 607–610.
[10] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, Boron nitride substrates for highquality graphene electronics. Nature Nanotechnology 5 (2010), no. 10, 722–726.
[11] L. C. Evans, Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, R.I., 2010.MR 2597943 Zbl 1194.35001
[12] C. L. Fefferman and M. I. Weinstein, Honeycomb lattice potentials and Dirac points. J. Amer. Math. Soc. 25(2012), no. 4, 1169–1220.MR 2947949 Zbl 1316.35214 · Zbl 1316.35214
[13] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften, 224. SpringerVerlag, Berlin, 1983.MR 0737190 Zbl 0562.35001 Infinite mass boundary conditions for Dirac operators599 · Zbl 0562.35001
[14] G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly, and J. Van Den Brink, Substrate-induced band gap in graphene on hexagonal boron nitride: Ab initio density functional calculations. Phys. Rev. B 76 (2007), no. 7, 073103.
[15] R. Hempel and I. Herbst, Strong magnetic fields, Dirichlet boundaries, and spectral gaps. Comm. Math. Phys. 169 (1995), no. 2, 237–259.MR 1329195 Zbl 0827.35031 · Zbl 0827.35031
[16] I. W. Herbst and Z. X. Zhao, Sobolev spaces, Kac-regularity, and the Feynman– Kac formula. In E. Çinlar, K. L. Chung, R. K. Getoor and J. Glover (eds.), Seminar on Stochastic Processes, 1987.Papers from the Seventh Seminar held at Princeton University, Princeton, N.J., March 26–28, 1987. Progress in Probability and Statistics, 15. Birkhäuser Boston, Boston, MA, 1988, 171–191.MR 1046416 Zbl 0656.60089
[17] T. Kato, Perturbation theory for linear operators. Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.MR 1335452 Zbl 0836.47009
[18] J. R. F. Lima, Electronic structure of a graphene superlattice with massive Dirac fermions. J. of App. Phys. 117 (2015), no. 8, 084303.
[19] G. Marko and T. Milan, Electronic states and optical transitions in a graphene quantum dot in a normal magnetic field. Serbian Journal of Electrical Engineering 8 (2011), 53–62.
[20] 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(2004), no. 5696, 666–669.
[21] T. Paananen and R. Egger, Finite-size version of the excitonic instability in graphene quantum dots. Phys. Rev. B 84 (2011), 155456.
[22] M. Prokhorova, The spectral flow for Dirac operators on compact planar domains with local boundary conditions. Comm. Math. Phys. 322 (2013), no. 2, 385–414. MR 3077920 Zbl 1281.58015 · Zbl 1281.58015
[23] A. Qaiumzadeh, F. K. Joibari, and R. Asgari, Effect of gap opening on the quasiparticle properties of doped graphene sheets. The European Physical Journal B 74, no. 4 (2010), 479–485.
[24] S. Schnez, K. Ensslin, M. Sigrist, and T. Ihn, Analytic model of the energy spectrum of a graphene quantum dot in a perpendicular magnetic field. Phys. Rev.B 78 (2008), 195427.
[25] B. Thaller, The Dirac equation. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992.MR 1219537
[26] P.R. Wallace, The band theory of graphite. Phys. Rev. (2) 71 (1947), no. 9, 622–634. Zbl 0033.14304 · Zbl 0033.14304
[27] M. Zarenia, A. Chaves, G. A. Farias, and F. M. Peeters, Energy levels of triangular and hexagonal graphene quantum dots: A comparative study between the tight-binding and dirac equation approach. Phys. Rev. B 84 (2011), no. 24, 245403. 600E. Stockmeyer and S. Vugalter Received February 14
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.