Download Unconventional Sequence of Fractional Quantum Hall

Unconventional Sequence of Fractional Quantum Hall
States in Suspended Graphene
Sudeep Kumar Ghosh
CCMT, Department of Physics, Indian Institute of Science.
January 31, 2013
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
1 / 36
Outline of the talk
1
2
Review of Quantum Hall Effect
QHE in 2DEG
QHE in Graphene
The single electron transistor scanning electrometer(SETSE)
QH measurement in 2DEG using SETSE
3
Unconventional FQHE seen in suspended graphene using SETSE
4
Summary of results
5
Future directions
6
References and acknowledgements
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
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31, 2013
Graphene
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Review of quantum Hall effect
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
3 / 36
Landau levels
Electrons in a magnetic field:
H=
1 ~
~ 2
(P + e A)
2m
(1)
In the Landau gauge the eigenstates are:
(y −y0 )2
e ikx x e − 2l 2 Hn (y − y0 )
ψn,kx (x, y ) =
(22n π(n!)2 )1/4
(2)
with energies
1
En = (n + )~ωC ∝ B,
2
~
eB
2
where ωC = m , l = eB and y0 = l 2 kx .
Finite sample of area A implies degeneracy of each LL =
where φ0 = h/e.
Filling factor ν = ρ/( φB0 ).
(3)
A
2πl 2
=
φ
φ0
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Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
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Edge states and broadening of Landau levels
Figure: Edge states
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Physics, IndianSequence
Institute of Fractional
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Edge states and broadening of Landau levels
Figure: Edge states
Figure: Broadening of Landau levels
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
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The quantum Hall effect of 2DES
Figure: Overview of Hall(RH ) and longitudinal(R) resistances respectively. (No
plateau is associated with f = 1/2). Source: H. L. Stormer, Rev.Mod.Phys.
71875889(1999).
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
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Understanding the plateaus
IQHE
The presence of the localized bands explains the plateaus.(Can be
seen by changing ρ at fixed B)
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
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31, 2013
Graphene
7 / 36
Understanding the plateaus
IQHE
The presence of the localized bands explains the plateaus.(Can be
seen by changing ρ at fixed B)
In the plateaus n extended edge branches carry the current.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
7 / 36
Understanding the plateaus
IQHE
The presence of the localized bands explains the plateaus.(Can be
seen by changing ρ at fixed B)
In the plateaus n extended edge branches carry the current.
FQHE
Laughlin’s brilliant guess for the ground state wave function for
ν = 1/3.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
7 / 36
Understanding the plateaus
IQHE
The presence of the localized bands explains the plateaus.(Can be
seen by changing ρ at fixed B)
In the plateaus n extended edge branches carry the current.
FQHE
Laughlin’s brilliant guess for the ground state wave function for
ν = 1/3.
Explains the Laughlin fractions ν = 1/n (n is an odd integer) and
their particle hole symmetric counter parts ν = 1 − 1/n.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
7 / 36
Understanding the plateaus
IQHE
The presence of the localized bands explains the plateaus.(Can be
seen by changing ρ at fixed B)
In the plateaus n extended edge branches carry the current.
FQHE
Laughlin’s brilliant guess for the ground state wave function for
ν = 1/3.
Explains the Laughlin fractions ν = 1/n (n is an odd integer) and
their particle hole symmetric counter parts ν = 1 − 1/n.
But can not explain ≈ 30 other fractions seen for ν < 1.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
7 / 36
Understanding the plateaus
IQHE
The presence of the localized bands explains the plateaus.(Can be
seen by changing ρ at fixed B)
In the plateaus n extended edge branches carry the current.
FQHE
Laughlin’s brilliant guess for the ground state wave function for
ν = 1/3.
Explains the Laughlin fractions ν = 1/n (n is an odd integer) and
their particle hole symmetric counter parts ν = 1 − 1/n.
But can not explain ≈ 30 other fractions seen for ν < 1.
Jain’s idea of composite fermions ⇒ FQHE of electrons is like IQHE
of composite fermions.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
7 / 36
Idea of composite fermions
A composite fermion is the bound state of an electron and an even
number of quantised vortices.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
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31, 2013
Graphene
8 / 36
Idea of composite fermions
A composite fermion is the bound state of an electron and an even
number of quantised vortices.
Figure: Cartoon of composite fermions(taken from Jain’s book).
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
8 / 36
Idea of composite fermions
A composite fermion is the bound state of an electron and an even
number of quantised vortices.
Figure: Cartoon of composite fermions(taken from Jain’s book).
Other flavors are 4 CF , 6 CF etc.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
8 / 36
FQHE as IQHE of composite fermions
Composite fermions =
electrons + 2p flux quanta.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
9 / 36
FQHE as IQHE of composite fermions
Composite fermions =
electrons + 2p flux quanta.
Experience an effective
magnetic field:
B ∗ = B − 2pρφ0
(4)
where, p is an integer.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
9 / 36
FQHE as IQHE of composite fermions
Composite fermions =
electrons + 2p flux quanta.
Experience an effective
magnetic field:
B ∗ = B − 2pρφ0
(4)
where, p is an integer.
CFs fill Λ (Landau like) levels
in this effective magnetic levels.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
9 / 36
FQHE as IQHE of composite fermions
Composite fermions =
electrons + 2p flux quanta.
Experience an effective
magnetic field:
B ∗ = B − 2pρφ0
(4)
where, p is an integer.
CFs fill Λ (Landau like) levels
in this effective magnetic levels.
Then in the above equation
0
using B = ρφν 0 and B ∗ = ρφ
ν∗
we obtain the standard CF
sequence,
ν=
ν∗
.
2pν ∗ ± 1
(5)
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
in Suspended
31, 2013
Graphene
9 / 36
FQHE as IQHE of composite fermions
Composite fermions =
electrons + 2p flux quanta.
Experience an effective
magnetic field:
B ∗ = B − 2pρφ0
(4)
where, p is an integer.
CFs fill Λ (Landau like) levels
in this effective magnetic levels.
Then in the above equation
0
using B = ρφν 0 and B ∗ = ρφ
ν∗
we obtain the standard CF
Figure: Left column shows ellctron g.s
sequence,
of IQHE and the right column shows
∗
C.F g.s of FQHE with ν = n/(2n ± 1)
ν
ν=
.
(5)
(taken from Jain’s book).
2pν ∗ ± 1
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall StatesJanuary
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QHE in graphene
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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10 / 36
Graphene band structure and effective Hamiltonian
Figure: Band structure of graphene
Near the Dirac points :
ξ
Heff
= ξ~vF (qx σx + qy σy ),
(6)
0
where, the valley pseudospin ξ = +1(−1) denotes the K (K ) point.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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11 / 36
Landau levels of Dirac electrons
Dirac Hamiltonian in a magnetic field :
H = ξvF (Πx σ x + Πy σ y )
(7)
where, Πi = (Pi + eAi ) and ξ is the valley pseudospin.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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Landau levels of Dirac electrons
Dirac Hamiltonian in a magnetic field :
H = ξvF (Πx σ x + Πy σ y )
(7)
where, Πi = (Pi + eAi ) and ξ is the valley pseudospin.
The LL spectrum is
λ,n = λ
√
~vF √
2n ∝ B
l
where, l 2 =
(8)
~
eB .
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
12 / 36
Landau levels of Dirac electrons
Dirac Hamiltonian in a magnetic field :
H = ξvF (Πx σ x + Πy σ y )
(7)
where, Πi = (Pi + eAi ) and ξ is the valley pseudospin.
The LL spectrum is
λ,n = λ
√
~vF √
2n ∝ B
l
where, l 2 =
(8)
~
eB .
Each LL is four fold degenerate
⇐ 2-fold spin + 2-fold valley
degeneracy.
Figure: Relativistic Landau levels.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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The relativistic QHE
SU(4) symmetry of each LL
implies changing the filling by
4 b/w adjacent plateaus.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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13 / 36
The relativistic QHE
SU(4) symmetry of each LL
implies changing the filling by
4 b/w adjacent plateaus.
Particle - hole symmetry leads
to half filled zero energy LL at
n = 0.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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31, 2013Graphene
13 / 36
The relativistic QHE
SU(4) symmetry of each LL
implies changing the filling by
4 b/w adjacent plateaus.
Particle - hole symmetry leads
to half filled zero energy LL at
n = 0.
In the absence of Zeeman
effect and electronic interaction
no QHE at ν = 0.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
13 / 36
The relativistic QHE
SU(4) symmetry of each LL
implies changing the filling by
4 b/w adjacent plateaus.
Particle - hole symmetry leads
to half filled zero energy LL at
n = 0.
In the absence of Zeeman
effect and electronic interaction
no QHE at ν = 0.
Completely filled n = 1 LL
occurs only for ν = ±2, hence
the sequence is,
1
ν = ±4(n + ) .
2
(9)
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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31, 2013Graphene
13 / 36
The relativistic QHE
SU(4) symmetry of each LL
implies changing the filling by
4 b/w adjacent plateaus.
Particle - hole symmetry leads
to half filled zero energy LL at
n = 0.
In the absence of Zeeman
effect and electronic interaction
no QHE at ν = 0.
Completely filled n = 1 LL
occurs only for ν = ±2, hence
the sequence is,
1
ν = ±4(n + ) .
2
Figure: RQHE observed in exfoliated
graphene.Adapted from Novoselov,
(9) Geim, Morozov et al., 2005, Nature
438, 197.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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The single electron transistor scanning electrometer(SETSE)
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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The Single Electron Transistor
Figure: SET
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The Single Electron Transistor
Figure: SET
Figure: Coulomb oscillations(period
e/Cg )
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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SET scanning electrometer over 2DEG
SETSE, uses the SET as a probe to sense the electrically induced
charge on its small (100nm) metal island held in proximity to the
sample surface. It can detect ≈ 1% of an electron charge.
Figure: SETSE over 2DEG
Figure: Magnified view of SETSE
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Typical data
Figure: Typical current oscillations
ISET (Vb ) of a SET.
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Physics, IndianSequence
Institute of Fractional
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Typical data
Figure: Typical current oscillations
ISET (Vb ) of a SET.
Figure: Complete data set of ISET (Vb )
for 2µm by 2µm raster scan. The top
of the data block maps the surface
electric field.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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Surface potential map
Figure: VS measured along edge.
(Inset)VS map in this region.
Figure: VS map showing two metal
stripes and the intermediate 2DEG
region.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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QH measurement in 2DEG using SETSE
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Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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Density profile near edge
Figure: Density profile near edge
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
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Institute of Fractional
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Incompressible strips
Figure: (A) and (G ) are surface potential maps. Others are transparency images
measure of the 2DEG compressibility.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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QH measurement in suspended graphene using SETSE
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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SETSE over suspended graphene
Suspended graphene
Typically have linear dimension 1 − 2µm.
Mobility ranges from 20000 to 150000cm2 /Vs at low
temperature.(The low temperature mobility of devices on Si/SiO2
substrates are typically 2000 to 15000cm2 /Vs).
Figure: Typical device
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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SETSE over suspended graphene
Suspended graphene
Typically have linear dimension 1 − 2µm.
Mobility ranges from 20000 to 150000cm2 /Vs at low
temperature.(The low temperature mobility of devices on Si/SiO2
substrates are typically 2000 to 15000cm2 /Vs).
Figure: Typical device
Figure: Measurement setup
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
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Measurement of local electronic compressibility
Modulate the charge carrier density using the back gate and monitor
the resulting change in the SET current. Measure the local chemical
potential and the change in carrier density.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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Measurement of local electronic compressibility
Modulate the charge carrier density using the back gate and monitor
the resulting change in the SET current. Measure the local chemical
potential and the change in carrier density.
Obtain the inverse compressibility(κ) using κ−1 = n2 (dµ/dn).
Figure: Landau fan
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
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Observations
B = 0 has incompressible peak due to vanishing d.o.s at the charge
neutrality point.
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Physics, IndianSequence
Institute of Fractional
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Observations
B = 0 has incompressible peak due to vanishing d.o.s at the charge
neutrality point.
For B > 0 strong incompressible behavior occurs at ν = 4(n + 1/2)
⇒ monolayer nature of the sample.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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Observations
B = 0 has incompressible peak due to vanishing d.o.s at the charge
neutrality point.
For B > 0 strong incompressible behavior occurs at ν = 4(n + 1/2)
⇒ monolayer nature of the sample.
Additional peaks at intermediate integer filling factors ν = 0, 1, 3,
4, 5, 7, 8 and 9 are seen.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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Observations
B = 0 has incompressible peak due to vanishing d.o.s at the charge
neutrality point.
For B > 0 strong incompressible behavior occurs at ν = 4(n + 1/2)
⇒ monolayer nature of the sample.
Additional peaks at intermediate integer filling factors ν = 0, 1, 3,
4, 5, 7, 8 and 9 are seen.
These broken symmetry states arise from interactions among electrons
and are visible well below 1T indicating high quality of the sample.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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Observations
B = 0 has incompressible peak due to vanishing d.o.s at the charge
neutrality point.
For B > 0 strong incompressible behavior occurs at ν = 4(n + 1/2)
⇒ monolayer nature of the sample.
Additional peaks at intermediate integer filling factors ν = 0, 1, 3,
4, 5, 7, 8 and 9 are seen.
These broken symmetry states arise from interactions among electrons
and are visible well below 1T indicating high quality of the sample.
Most intriguing is the appearance of the incompressible peaks at
fractional filling factors, the strongest of which emerges around
B = 1T .
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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Compressibility as a function of filling factor
Figure: The vertical features correspond to QH states, whereas localized states
curve as the magnetic field is changed.
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Physics, IndianSequence
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Finer compressibility image for ν between 0 and 1
Incompressible peaks occur at
ν = 13 , 32 , 52 , 35 , 73 , 74 and 49
reproducing standard
composite fermion sequence
n
ν = 2n±1
observed in GaAs.
Figure: Finer measurement of κ for
filling factor ν between 0 and 1
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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Finer compressibility image for ν between 0 and 1
Incompressible peaks occur at
ν = 13 , 32 , 52 , 35 , 73 , 74 and 49
reproducing standard
composite fermion sequence
n
ν = 2n±1
observed in GaAs.
Strongest incompressible states
at ν = 13 and 23 can be resolved
down to B ≈ 1T although
ν = 23 state weakens
considerably below 4T .
Figure: Finer measurement of κ for
filling factor ν between 0 and 1
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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Finer compressibility image for ν between 0 and 1
Incompressible peaks occur at
ν = 13 , 32 , 52 , 35 , 73 , 74 and 49
reproducing standard
composite fermion sequence
n
ν = 2n±1
observed in GaAs.
Strongest incompressible states
at ν = 13 and 23 can be resolved
down to B ≈ 1T although
ν = 23 state weakens
considerably below 4T .
Figure: Finer measurement of κ for
filling factor ν between 0 and 1
With the increase in the filling
factor denominator the field at
which the state emerges also
increases, with ν = 49 being
apparent only above B ≈ 9T .
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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Finer compressibility image for ν between 1 and 2
Different pattern of
incompressible behavior is
observed.
Figure: Finer measurement of κ for
filling factor ν between 1 and 2
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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Finer compressibility image for ν between 1 and 2
Different pattern of
incompressible behavior is
observed.
No FQH states with odd
numerators occur in this
regime.
Figure: Finer measurement of κ for
filling factor ν between 1 and 2
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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Finer compressibility image for ν between 1 and 2
Different pattern of
incompressible behavior is
observed.
No FQH states with odd
numerators occur in this
regime.
Instead the FQH states occur
14
only at ν = 34 , 85 , 10
7 and 9 .
Figure: Finer measurement of κ for
filling factor ν between 1 and 2
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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31, 2013Graphene
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Finer compressibility image for ν between 1 and 2
Different pattern of
incompressible behavior is
observed.
No FQH states with odd
numerators occur in this
regime.
Instead the FQH states occur
14
only at ν = 34 , 85 , 10
7 and 9 .
The peaks at ν = 43 and 85 are
the most robust, persisting
down to ≈ 1T to ≈ 1.5T .
Figure: Finer measurement of κ for
filling factor ν between 1 and 2
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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Finer compressibility image for ν between 1 and 2
Different pattern of
incompressible behavior is
observed.
No FQH states with odd
numerators occur in this
regime.
Instead the FQH states occur
14
only at ν = 34 , 85 , 10
7 and 9 .
The peaks at ν = 43 and 85 are
the most robust, persisting
down to ≈ 1T to ≈ 1.5T .
Figure: Finer measurement of κ for
filling factor ν between 1 and 2
The even numerator fractions
can be explained by considering
the standard CF sequence and
ν ∗ = 2 − ν.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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Average compressibility
Averaging over magnetic field reduces the influence of localized states
and shows clear incompressible peaks centered at ν = 13 , 25 , 37 , 49 , 47 , 35 ,
2 4 8 10
14
3 , 3 , 5 , 7 and 9 .
Figure: dµ/dn between ν = 0 and 1, averaged over 9 to 11.9 T (blue), and
between ν = 1 and 2, averaged over 4.9 to 6.4 T (red).
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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Possible explanation of the unconventional hierarchy
The absence of odd-numerator fractions indicates that a robust
underlying symmetry enables low-lying excitations, preventing the
formation of incompressible states.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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31, 2013Graphene
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Possible explanation of the unconventional hierarchy
The absence of odd-numerator fractions indicates that a robust
underlying symmetry enables low-lying excitations, preventing the
formation of incompressible states.
The differing behavior above and below ν = 1 suggests an intriguing
interplay between the inherent symmetries of graphene and electronic
correlations in the lowest LL.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
30 / 36
Possible explanation of the unconventional hierarchy
The absence of odd-numerator fractions indicates that a robust
underlying symmetry enables low-lying excitations, preventing the
formation of incompressible states.
The differing behavior above and below ν = 1 suggests an intriguing
interplay between the inherent symmetries of graphene and electronic
correlations in the lowest LL.
May be the Zeeman effect lifts spin degeneracy, but valley symmetry
remains intact, allowing large valley skyrmions to form with a minimal
energy penalty at odd-numerator filling factors.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
30 / 36
Possible explanation of the unconventional hierarchy
The absence of odd-numerator fractions indicates that a robust
underlying symmetry enables low-lying excitations, preventing the
formation of incompressible states.
The differing behavior above and below ν = 1 suggests an intriguing
interplay between the inherent symmetries of graphene and electronic
correlations in the lowest LL.
May be the Zeeman effect lifts spin degeneracy, but valley symmetry
remains intact, allowing large valley skyrmions to form with a minimal
energy penalty at odd-numerator filling factors.
The total electron density, and not just the filling fraction, may play
an important role in electronic interactions in the lowest LL.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
30 / 36
Steps in chemical potential
Integrate the inverse compressibility w.r.t carrier density to extract
the step in chemical potential ∆µν associated with each FQH state
and thereby determine the corresponding energy gap ∆ν .
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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31, 2013Graphene
31 / 36
Steps in chemical potential
Integrate the inverse compressibility w.r.t carrier density to extract
the step in chemical potential ∆µν associated with each FQH state
and thereby determine the corresponding energy gap ∆ν .
√
∆µν are expected to scale as B but the behavior seen is unclear.
Also the extracted energy gaps do not match with that of theory.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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31, 2013Graphene
31 / 36
Spatial dependence of compressibility
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
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31, 2013Graphene
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Summary of results
For filling factor ν < 1 the standard FQH sequence is observed.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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Summary of results
For filling factor ν < 1 the standard FQH sequence is observed.
Between ν = 0 and 1, the compressibility is approximately symmetric
about ν = 1/2, suggesting that the fourfold spin and valley
degeneracy is fully lifted.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
33 / 36
Summary of results
For filling factor ν < 1 the standard FQH sequence is observed.
Between ν = 0 and 1, the compressibility is approximately symmetric
about ν = 1/2, suggesting that the fourfold spin and valley
degeneracy is fully lifted.
In contrast, between ν = 1 and 2 only even numerator fractions are
observed. This indicate that one symmetry persists between ν = 1
and 2.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
33 / 36
Summary of results
For filling factor ν < 1 the standard FQH sequence is observed.
Between ν = 0 and 1, the compressibility is approximately symmetric
about ν = 1/2, suggesting that the fourfold spin and valley
degeneracy is fully lifted.
In contrast, between ν = 1 and 2 only even numerator fractions are
observed. This indicate that one symmetry persists between ν = 1
and 2.
√
The chemical potential steps do not show expected B dependence.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
33 / 36
Summary of results
For filling factor ν < 1 the standard FQH sequence is observed.
Between ν = 0 and 1, the compressibility is approximately symmetric
about ν = 1/2, suggesting that the fourfold spin and valley
degeneracy is fully lifted.
In contrast, between ν = 1 and 2 only even numerator fractions are
observed. This indicate that one symmetry persists between ν = 1
and 2.
√
The chemical potential steps do not show expected B dependence.
The theoretically predicted energy gaps do not match with the ones
obtained.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
33 / 36
Summary of results
For filling factor ν < 1 the standard FQH sequence is observed.
Between ν = 0 and 1, the compressibility is approximately symmetric
about ν = 1/2, suggesting that the fourfold spin and valley
degeneracy is fully lifted.
In contrast, between ν = 1 and 2 only even numerator fractions are
observed. This indicate that one symmetry persists between ν = 1
and 2.
√
The chemical potential steps do not show expected B dependence.
The theoretically predicted energy gaps do not match with the ones
obtained.
Graphene provides a rich platform to investigate correlated electronic
states.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
33 / 36
Future directions
Further experiments are necessary to elucidate the exact spin and
valley ordering of each state; for example, tilted field measurements
decouple Zeeman splitting from orbital effects and could provide
insight into spin polarization.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
34 / 36
Future directions
Further experiments are necessary to elucidate the exact spin and
valley ordering of each state; for example, tilted field measurements
decouple Zeeman splitting from orbital effects and could provide
insight into spin polarization.
Rigorous theoretical explanation is necessary for the unconventional
sequence of FQH states observed.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
34 / 36
Future directions
Further experiments are necessary to elucidate the exact spin and
valley ordering of each state; for example, tilted field measurements
decouple Zeeman splitting from orbital effects and could provide
insight into spin polarization.
Rigorous theoretical explanation is necessary for the unconventional
sequence of FQH states observed.
Theoretical understanding is necessary for the unusual behavior of the
chemical potential steps observed.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
in Suspended
31, 2013Graphene
34 / 36
References and acknowledgements
References
1 Feldman et.al., Science 337, 1196-1199 (2012).
2
Yacoby et.al., Solid state communications 111, 1-13 (1999).
3
Csaba et.al., PRB 74, 235417(2006).
4
M. O. Goerbig, RMP 83, 1193(2011).
5
Castro Neto et. al., RMP 81, 109(2009).
6
Composite Fermions – Jainendra K. Jain.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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31, 2013Graphene
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References and acknowledgements
References
1 Feldman et.al., Science 337, 1196-1199 (2012).
2
Yacoby et.al., Solid state communications 111, 1-13 (1999).
3
Csaba et.al., PRB 74, 235417(2006).
4
M. O. Goerbig, RMP 83, 1193(2011).
5
Castro Neto et. al., RMP 81, 109(2009).
6
Composite Fermions – Jainendra K. Jain.
Acknowledgements
I acknowledge Prof. Vijay B. Shenoy and Subhomay Ghatak for many fruitful
discussions.
Sudeep Kumar Ghosh (CCMT, Department ofUnconventional
Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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Thank You!
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Physics, IndianSequence
Institute of Fractional
Science.) Quantum Hall States
January
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31, 2013Graphene
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