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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 ?

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new field dependence of Landau levels

hybrid 2D electron gas : new dispersion relation

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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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Dirac pointsenergy bandsgraphene: isotropic Dirac gas

isotropic Dirac gas

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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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anisotropic Dirac gas

energy bands moving Dirac points

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Landau levels of anisotropic Dirac gas

low field

lifts 2-fold LLdegeneracy

3 2

-1

-2-3

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Y. Hasegawa, M. Kohmoto, 2006

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anisotropic Dirac gas

moving Dirac points

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energy bands

anisotropic hybrid Schrödinger-Dirac gas

Merging of Dirac points

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“hybrid” gas

2/31 / 2( )n B

2

-1

-2

1

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hybrid Schrödinger-Dirac gas

Density of states Onsager argument

gap opening

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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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2't t

't t

2X

1

2 1n 1 n

2 1n

Quartic oscillator

Harmonic oscillator

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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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Universal features for 2D lattices with two atoms basis

model oblique lattice

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Phase diagram

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GAPGAP

GAP

Dirac spectrum graphene

GAP

“Hybrid” spectrum

/'t t

/''t t

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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

194 atoms basis, 8 hopping integrals

Dirac cones in organic conductors

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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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'i iete tE t 1 2k.a k.a

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2 3( )BEDT TTF I

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The Chocolate LEDERER Prize The NOBEL Prize

First recipient : Pascal Lederer

FRANCAIS

FRANCAIS

Lederer

Chocolate