Abstract:
The Hamiltonian of an electron in a two-dimensional lattice of
point potentials in a transverse magnetic field is considered. An
expression is found for the Green's function of this Hamiltonian,
and from it an equation that determines the dispersion laws
Es(k) is obtained in the case of rational magnetic flux.
A detailed investigation of the spectrum is made for the case of
integral flux. A criterion is obtained for the conditions under
which the Landau levels enter the spectrum of the lattice
Hamiltonian.
Citation:
V. A. Geiler, V. A. Margulis, “Spectrum of the bloch electron in a magnetic field in a two-dimensional lattice”, TMF, 58:3 (1984), 461–472; Theoret. and Math. Phys., 58:3 (1984), 302–310
\Bibitem{GeiMar84}
\by V.~A.~Geiler, V.~A.~Margulis
\paper Spectrum of the bloch electron in a magnetic field in a two-dimensional lattice
\jour TMF
\yr 1984
\vol 58
\issue 3
\pages 461--472
\mathnet{http://mi.mathnet.ru/tmf4678}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=752077}
\transl
\jour Theoret. and Math. Phys.
\yr 1984
\vol 58
\issue 3
\pages 302--310
\crossref{https://doi.org/10.1007/BF01018053}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1984TJ86800013}
Linking options:
https://www.mathnet.ru/eng/tmf4678
https://www.mathnet.ru/eng/tmf/v58/i3/p461
This publication is cited in the following 18 articles:
L. I. Danilov, “O spektre dvumernogo operatora Shredingera s odnorodnym magnitnym polem i periodicheskim elektricheskim potentsialom”, Izv. IMI UdGU, 51 (2018), 3–41
S.H. Lee, C.W. Chiu, Y.H. Ho, M.F. Lin, “Uniaxial-stress effects on electronic structures of monolayer and bilayer graphenes”, Synthetic Metals, 160:23-24 (2010), 2435
Tatsuya Nakajima, Hideo Aoki, “Landau quantization of graphene including diamagnetic shift and shrinkage of wave function”, Physica E: Low-dimensional Systems and Nanostructures, 40:5 (2008), 1354
N. G. Galkin, V. A. Geyler, V. A. Margulis, “Quasiballistic electron transport in a three-dimensional microconstriction”, J. Exp. Theor. Phys., 91:1 (2000), 197
Sergey Gredeskul, Masha Zusman, Yshai Avishai, Mark Ya. Azbel, The IMA Volumes in Mathematics and its Applications, 96, Wave Propagation in Complex Media, 1998, 95
V. A. Geiler, V. A. Margulis, L. I. Filina, “Conductance of a quantum wire in a longitudinal magnetic field”, J. Exp. Theor. Phys., 86:4 (1998), 751
V.A. Geyler, I.Yu. popov, “Eigenvalues imbedded in the band spectrum for a periodic array of quantum dots”, Reports on Mathematical Physics, 39:2 (1997), 275
S.A. Gredeskul, M. Zusman, Y. Avishai, M.Ya. Azbel', “Spectral properties and localization of an electron in a two-dimensional system with point scatterers in a magnetic field”, Physics Reports, 288:1-6 (1997), 223
V. A. Geiler, V. A. Margulis, “Point perturbation-invariant solutions of the Schrödinger equation with a magnetic field”, Math. Notes, 60:5 (1996), 575–580
V. A. Geiler, I. Yu. Popov, “Ballistic transport in nanostructures: explicitly solvable models”, Theoret. and Math. Phys., 107:1 (1996), 427–434
V. A. Geyler, B. S. Pavlov, I. Yu. Popov, “Spectral properties of a charged particle in antidot array: A limiting case of quantum billiard”, Journal of Mathematical Physics, 37:10 (1996), 5171
A. Gramada, M. E. Raikh, “Short-range impurity in the vicinity of a saddle point and the levitation of the two-dimensional delocalized states in a magnetic field”, Phys. Rev. B, 54:3 (1996), 1928
V. A. Geiler, V. V. Demidov, “Spectrum of three-dimensional landau operator perturbed by a periodic point potential”, Theoret. and Math. Phys., 103:2 (1995), 561–569
M. Ya. Azbel', B. I. Halperin, “Landau levels in the presence of dilute short-range scatterers”, Phys. Rev. B, 52:19 (1995), 14098
S. A. Gredeskul, M. Ya. Azbel', “Two-dimensional short-range scatterer in a magnetic field”, Phys. Rev. B, 49:4 (1994), 2323
Y. Avishai, M. Ya. Azbel', S. A. Gredeskul, “Electron in a magnetic field interacting with point impurities”, Phys. Rev. B, 48:23 (1993), 17280
V. A. Geiler, V. A. Margulis, “Anderson localization in the nondiscrete maryland model”, Theoret. and Math. Phys., 70:2 (1987), 133–140
V. A. Geiler, V. A. Margulis, “Structure of the spectrum of a bloch electron in a magnetic field in a two-dimensional lattice”, Theoret. and Math. Phys., 61:1 (1984), 1049–1056
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