magnetoelectronic properties of graphites

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1. Introduction2. Magnetoelectronic properties of the AB-

stacked graphite3. Magnetoelectronic properties of the AA-

and ABC-stacked graphites4. Conclusion

Magnetoelectronic properties of graphitesMagnetoelectronic properties of graphites

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Recent studies on the layered Recent studies on the layered graphitesgraphitesTheoretical :

J.-C. Charlier, et al.• Phys. Rev. B 43, 4579 (1991) • Phys. Rev. B 44, 13237 (1991) • Phys. Rev. B 46, 4531 (1992) • Phys. Rev. B 46, 4540 (1992) • Carbon 32, 289 (1994) 1. the ab initio method.2. the tight-binding model.

Effects of Stacking sequences

1. Introduction1. Introduction

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Experiments: K. S. Novoselov, et al.

• Science 306, 666 (2004)effects of electric field

• Nature 438, 197 (2005)

a tiny overlap between a tiny overlap between conductance and valence conductance and valence bands.bands.ambipolarambipolar field effectfield effect

4Quantum Hall effect for massless Dirac fermions.

B=14 T, T=4 K


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graphitesgraphitesspsp22(s(s--ppxx--ppyy) and ) and ππ((ppzz) ) bondingsbondings

ABAB--stacked graphite (graphite or hexagonal graphite)stacked graphite (graphite or hexagonal graphite)

ABCABC--stacked graphite (stacked graphite (rhombohedralrhombohedral graphite)graphite)

AAAA--stacked graphite (Simple hexagonal graphite)stacked graphite (Simple hexagonal graphite)

π band electronic properties of graphite

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( ) ( ), ,, i ji j A j A Bk A

jr a R rψ τΨ = − −∑v

vv v1av

2av

b

A

B

( ) ( ),

0*

0

2

*0 ,,

00

2 cos 2y

y

k A

ik bik b x

A B A Bk B

fH

f

k af e e

R r H R r dr

γγ

γ ψ ψ τ

−

⎛ ⎞= ⎜ ⎟⎝ ⎠

⎛ ⎞= + ⎜ ⎟⎝ ⎠

= − − −∫ vv

K K : linear bands intersecting at EF=0.

ГГ : Maximum and minimum at ±3γ0.

M M : saddle points near ± γ0

←←EEFF=0=0

7kx

ky

π0

1

2

3

.

.

m

.

.

q

( ) ( )

( )

, ,,

,

j

i ji

i k Rj A j A Bk A

j

j

r a e R r

e A R r drc

θ ψ τ

θ

⎡ ⎤− ⋅ +⎣ ⎦Ψ = − −

=

∑

∫

vvv v

v v v

h

BB

π/4

( )00, ,0,0 , hcA Byq e

φφ φ= − =v


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0φ =(II) linear band along kx and ky intersects at EF=0.

(IA) Maximum or minimum of parabolic band along kx and ky.

14φ=

(IB) saddle points of parabolic band along kx and ky.(IA) Maximum or minimum of parabolic band along kx and ky.

(II) Landau level along kx and parabolic band along ky

(parabolic band effectivlly; 2D→1D).

(IB) saddle points of parabolic band along kx and ky.


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Energy bands near EF: linear bands :

Landau level : small

parabolic bands : large

0φφφ

=

DOS

(III) Landau level at EF=0 along kx and ky, (zero-dimensional discrete states)


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(IA) maximum or minimum of parabolic band, finite DOS.

(II) linear bands along Kx and Ky , zero DOS.

(II) 1D subband, Square-root-divergent DOS.

(IB) saddle points of parabolic band,

logarithmically divergent DOS

(IB) saddle points of parabolic band,logarithmically divergent DOS

(IA) maximum or minimum of parabolic band, finite DOS.

( )( )

( )( )22x ydk dk

D A E kω δ ωπ

= −∫v

Density of States1. Introduction1. Introduction

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(III) 0D subband at EF=0, Delta-Function-like DOS.

DOS at EF=0: zero :

divergent : small

finite : large

0φφφ

=


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AB

A B A B

2. Magnetoelectronic properties of the AB-stacked graphite


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( ) ( ), ,, i ji j A j A Bk A

jr a R rψ τΨ = − −∑v

vv v

( ) ( )( ) ( )( ) ( )( ) ( )( )

,

1 1, 1

2 2, 2

1 1, 1

1, 1

*0 ,,

*1 ,,

*2 ,,

*3 ,,

*4 ,,

i i i ik A ii

i i i ik A ii

i i i ik B ii

i i i ik B ii

i i i ik A ii

A B A Bk B

A A A Ak A

B B B Bk B

B B B Bk B

A B A Bk B

R r R r dr

R r R r dr

R r R r dr

R r R r dr

R r R r

γ ψ ψ τ

γ ψ ψ τ

γ ψ ψ τ

γ ψ ψ τ

γ ψ ψ τ

+ ++

+ ++

+ ++

+ ++

= − − −

= − − −

= − − −

= − − −

= − − −

∫∫∫∫

v

v

v

v

v

v

v

v

v

v ( )( ) ( )

1

2 2, 2

*5 ,,i i i ik A ii

A A A Ak A

dr

R r R r drγ ψ ψ τ+ ++

= − − −

∫∫ v

v

AB



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Effects of magnetic fieldthe Landau levels (0D or 1D)oscillating Landau levels (3D)

Layered graphitestrong anisotropic.strong energy dispersions

in kx, ky plane and weak in kzdirections.

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Effects of magnetic field

• the Landau levels (0D or 1D)

• oscillating landau levels



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Butterfly-like energy spectrum



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3D energy bands3D energy bandsDelta function like divergence Delta function like divergence ↔↔ 0D0D Landau levelsLandau levelssquaresquare--root divergenceroot divergence ↔↔ 1D1D Landau levelsLandau levelscusps cusps ↔↔ maximum, minimum; saddle points maximum, minimum; saddle points

(osci(oscillating Landau) llating Landau)



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As As øø decreased, more divergent peaks.decreased, more divergent peaks.



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A

C

B

B

3. Magneto-electronic properties of the AA- and ABC-stacked graphites


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• the Landau levels

• oscillating Landau levels



linear bands

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• the Landau levels

• oscillating Landau levels



linear bands



22Butterfly-like energy spectrum



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3D energy bands3D energy bandssquaresquare--root divergenceroot divergence ↔↔ 1D1D Landau levelsLandau levelscusps cusps ↔↔ maximum, minimum; saddle points maximum, minimum; saddle points




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3D energy bands3D energy bandsDelta function like divergence Delta function like divergence ↔↔ 0D0D Landau levelsLandau levelssquaresquare--root divergenceroot divergence ↔↔ 1D1D Landau levelsLandau levelscusps cusps ↔↔ maximum, minimum; saddle points maximum, minimum; saddle points




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4. Conclusion4. Conclusion

Magnetic field significantly modifies electronic properties of the layered graphites.

▪ 3D energy band into 1D or 0D Landau

subbands.

▪ Three kinds of special structures in DOS.

▪ Drastic change in low energy DOS.

magnetoelectronic properties of graphites

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