• グラフェン量子ホール系の発光• 量子ホール系の光学ホール伝導度

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青木研究室 M2森本高裕

KK

KK’

K’K’

Graphene quantum Hall effect

Landau level:

Cyclotron energy:

10 μm (courtesy of Geim)

rxx

sxy

(Novoselov et al, Nature 2005; Zhang et al, Nature 2005)

sxy =2(n+1/2)(-e2/h)

In the effective-mass picture the quasiparticle is described by massless Dirac eqn.

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Landau-level spectroscopy in graphene

(Sadowski et al, PRL 2006)

Uneven Landau level spacings

01

-12-23

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Peculiar selection rule |n||n|+1 (usually, nn+1)

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Basic idea

Population inversion cyclotron emissionPossibility of graphene “Landau level laser”

Uneven Landau levels √n∝+|n||n|+1 Population inversion

Ladder of excitations

Tunable wavelength

-n n+1 excitation

(Aoki, APL 1986)

Ordinary QHE systems Graphene Landau levels

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(Ando, Zheng & Ando, PRB 2002)

Optical conductivity s(w): method

Green’ s f SCBA

Level broadening by impurity is considered through Born approximation with self-consistent Green’s function.

Solve self-consistently by numerical method

s(w)Optical conductivity is calculated from Kubo formula :

current matrix elements

Singular DOS makes the calculation difficult .

short range Impurity potential

Cf. Gusynin et al. (PRB 2006) no self-consistent treatment of impurity scattering 5

Optical conductivity : result

-12

01

12

higher T

(Sadowski et al, 2006)

higher T

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Uneven Landau levels ∝

n=0 Landau level stands alone, while others form continuum spectra

Population inversion is expected between n=0 and continuum.

excitationCyclotron radiation

rapid decay

Population inversion

Density of states suitable for radiation

Impurity broadening

photoemission vs other relaxation processes (phonon)

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Orders of magnitude

more efficient photoemission

in graphene

Relaxation process : photon emission

Spontaneous photon emission rate is calculated from Fermi’s golden rule.

Singular B dependence of Dirac quasiparticle in graphene

Magnetic field:1T

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Competing process : phonon emission

Ordinary QHE system Chaubet et al., PRB 1995,1998 discussed phonon emission is the main relaxation channel.

Graphene Also obtained from golden rule and factor with and , phonon emission is exponentially small in graphene as well.

2DEG

Wavefunction with a finite thickness

Phonon ^ 2DEG

Effect of phonon ^ 2DEG same order as photoemission in conventional QHE (Chaubet et al. PRB 1998)

Graphene is only one atom thick phonon does not compete with photoemission. However, atomic phonon modes ^ graphene will have to be examined

q

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2D electron gas

2DEG

ρxyρxxB

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(Paalanen et al, 1982)

THz spectroscopy of 2DEG

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Faraday rotation

(Sumikura et al, JJAP, 2007)

Ellipticity

Resonance structure at cyclotron energy

Motivation

●conventional results - Hall conductivity quantization at w=0 - Faraday rotation measurement in finite w

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● How peculiar can optical Hall conductivity s xy (eF, w) be?● Is ac QHE possible?

Only Drude form treatment

Calculating s xy (eF, w) from …

● Kubo formula ● Self-consistent Born approximation

(O'Connell et al, PRB 1982)

sxy (w) in GaAs

●3D plot of s xy (eF, w) against Fermi energy and frequency

Hall step still remains in ac regime

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0

2

4

6

energy

0

1

2

3

4

frequency

20

0

20

xy0

2

4

6

energy

2 4 6 8F

2

4

6

8

xy

1 2 3 4

10

5

5

10

15xy

w=0.4wC

sxy (w)

eF

wResonance at cyclotron frequency

eF

w

sxy (w)

sxy (w)

sxy (w) in graphene

● sxy (eF, w) of graphene

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2 1 1 2F

10

5

5

10

xy

0.5 1 1.5 2 2.5

4

2

2

4

xy

w=0

2 1

01

2

energy

0

0.5

1

1.5

2

2.5

frequency

2010

01020

xy 2

10

1

2

energy

Reflecting massless Dirac DOS structureHall step remains

Resonance at cyclotron frequency

sxy (w)

eF

w

eF

w

sxy (w)

電子正孔対称

Consideration with Kubo formula

●Why does Hall step remain in ac region?●How robust is it?

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THz Hall 効果Hall step structure in clean system (not disturbed so much by impurity)

Clean ordinary QHE system

(Peng et al, PRB 1991)では ac の取り扱いが不十分

□ Future problem• Effect of long-range impurity• Localization and delocalization in ac field• Relation to topological arguement