スライド 1

青木研究室M2
森本高裕
• 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
1
Graphene quantum Hall effect
In the effective-mass picture the quasiparticle is
described by massless Dirac eqn.
10 μm
(courtesy of Geim)
Landau level:
Cyclotron energy:
K’
K
K’
sxy
K
K’
K
sxy =
2(n+1/2)(-e2/h)
(Novoselov et al, Nature 2005;
Zhang et al, Nature 2005)
rxx
2
Landau-level spectroscopy in graphene
-12
-23
Uneven Landau level
spacings
Peculiar selection rule
|n||n|+1
(usually, nn+1)
12
01
(Sadowski et al, PRL 2006)
3
Basic idea
Ordinary QHE systems
Graphene Landau levels
Ladder of
excitations
Uneven Landau
levels ∝ √n
+
|n||n|+1
-n  n+1
excitation
Population inversion
 cyclotron emission
Possibility of graphene “Landau level laser”
 Population
inversion
Tunable
wavelength
(Aoki, APL 1986)
4
Optical conductivity s(w): method
Green’ s f  SCBA
s(w)
Singular DOS makes the calculation
difficult .
Optical conductivity is calculated from
Kubo formula :
 Level broadening by impurity is
considered through Born
approximation with self-consistent
Green’s function.
current matrix elements
short range
Impurity potential
Solve self-consistently by numerical
method
(Ando, Zheng & Ando, PRB 2002)
Cf. Gusynin et al. (PRB 2006)
 no self-consistent treatment of
impurity scattering
5
Optical conductivity : result
01
12
-12
higher T
higher T
(Sadowski et al, 2006)
6
Density of states suitable for radiation
Uneven Landau levels ∝
rapid
decay
excitation
Population inversion
Cyclotron
radiation
Impurity
n=0 Landaubroadening
level stands
alone, while others form
continuum spectra
Population inversion
is expected between
n=0 and continuum.

photoemission vs
other relaxation
processes (phonon)
7
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
Orders of
magnitude
more efficient
photoemission
in graphene
8
Competing process : phonon emission
q
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.
Effect of phonon ^ 2DEG
 same order as photoemission in
conventional QHE
(Chaubet et al. PRB 1998)
2DEG
Phonon ^ 2DEG
Graphene is only one atom thick
 phonon does not compete with photoemission.
However, atomic phonon modes ^ graphene
will have to be examined
Wavefunction with a
finite thickness
9
2D electron gas
ρxy
B
ρxx
(Paalanen et al, 1982)
10
THz spectroscopy of 2DEG
Ellipticity
Faraday
rotation
Resonance structure at
cyclotron energy
(Sumikura et al,
JJAP, 2007)
11
Motivation
●conventional results
- Hall conductivity quantization at w=0
- Faraday rotation measurement in finite w
Only Drude form
treatment
● How peculiar can optical Hall
conductivity sxy (eF, w) be?
● Is ac QHE possible?
(O'Connell et al, PRB 1982)
Calculating sxy (eF, w) from …
● Kubo formula
● Self-consistent Born
approximation
12
sxy (w) in GaAs
●3D plot of sxy (eF, w) against Fermi energy and frequency
xy (w)
sxy
Hall step still remains in ac regime
w=0.4wC
8
6
4
2
sxy (w)
xy (w)
sxy
4
6
8
1
2
3
4
eF
F
15
20
xy
2
4
10
0
5
3
20
0
2
2
1
energy
eF
4
6
0
frequency
w
5
w
10
Resonance at cyclotron
frequency
13
sxy (w) in graphene
● sxy (eF, w) of graphene
xy
電子正孔対称
10
w=0
5
2
sxy (w)
1
1
2
eF
F
5
20
10
xy
10
2.5
0
2
10
20
1.5
Resonance at
cyclotron
frequency
sxy (w)
xy
4
2
1
1
0
energy
eF
0.5
frequency
w
2
1
0.5
2
0
Reflecting massless Dirac DOS structure
Hall step remains
1
1.5
2
2.5
w
2
4
14
Consideration with Kubo formula
●Why does Hall step remain in ac region?
●How robust is it?
Clean ordinary
QHE system
Hall step structure in clean system
(not disturbed so much by impurity)
□ Future problem
• Effect of long-range impurity
• Localization and delocalization in ac field
• Relation to topological arguement
THz Hall 効果
(Peng et al, PRB 1991)
ではacの取り扱いが不十分
15

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