1 a new field dependence of landau levels in a graphene-like structure petra dietl, ecole...
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A new field dependence of Landau levels in a graphene-like structure
Petra Dietl , Ecole Polytechnique student
Frédéric Piéchon, LPS
G. M.
PRL 100, 146802 (2008)
www.lps.u-psud.fr/users/gilles
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field dependence of Landau levels
Schrödinger electron gas Graphene: Weyl (massless Dirac) electron gas
other field dependencies for 2D gas ?
4
tight-binding problem on honeycomb lattice
hexagonal Bravais lattice+2 atoms basis
Bloch theorem:B A
' i iE t te te 1 2k.a k.a
2't t Gap
' 2t t t ?
Y. Hasegawa et al., 2006
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Landau levels of graphene
Hofstadter-Rammal butterfly
honeycomb lattice in a magnetic field low field
1
5 3
-1-3
-5
1
53
-1
-3
-5Valley degeneracy
2-fold degeneracy of LL
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Landau levels of anisotropic Dirac gas
low field
lifts 2-fold LLdegeneracy
3 2
-1
-2-3
1
Y. Hasegawa, M. Kohmoto, 2006
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2't t
't t
A simple problem of quantum mechanics
y yp p eBx
Instead of
for a linear spectrum
2 2H
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and Berry’s phase ( )n n B
1 1 1| . .
2 2 2 4C C
iu u d d
k k k k kk = k
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2 iue
k kRoth, 1966Wilkinson
't t 1.5't t 2't t. 2
C
d k k k0 0 1/ 2
. 0C
d k k k
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Summary
motion and merging of Dirac points universal features of 2D Bravais lattices with 2 atoms basis
tight binding Hamiltonian on honeycomb lattice
experimental realization?
next nearest neighbor hopping tilted cones
Landau levels with renormalized velocity and Quantum Hall effect
(J.N. Fuchs, M.O. Goerbig, F. Piéchon and G.M. arXiv:0803.0912
Realization of a graphene structure in atomic gases
', '',t t t
A. Kobayashi, S. Katayama, Y. Suzumura, H. Fukuyama (2006)Massless Dirac fermions in organic conductor a-(BEDT-TTF)2I3
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I= conicII= conic, but metallicIII= gap
tilted Dirac cones
hybrid modeltransition line2atoms basis
4 hopping integrals
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It is enough to consider…
2 atoms per cell, 4 hopping integrals4 atoms per cell, 8 hopping integrals
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2 2
( )
11y x yx
ik i k kikc p cpE e et t ett
I = conicIV= gap
1 0ct
1 0ct 2 12c ct t
1 2 1p pt t
2 2ct
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