##Dirac-Like Quasi Particles in Graphene_Gusynin_talk_Dubna 2009

8/3/2019 ##Dirac-Like Quasi Particles in Graphene_Gusynin_talk_Dubna 2009

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Dirac-like quasiparticles in grapheneV.P. Gusynin

Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine

July 9, 2009

V.P. Gusynin (BITP, Kiev) Dirac-like quasiparticles in graphene Dubna 1 / 61

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http://goback/



Why graphene is so interesting?Atomic structure. Graphene is an one-atom-thick layer of graphite packed in the honeycomb lattice –a two dimensional crystal. Free suspended graphene shows “rippling” of the flat sheet with amplitude of

about one nanometer.The ability to sustain huge (> 108A/cm2) electric currents make it a promising candidate forapplications in devices such as nanoscale field effect transistors.

Graphene exhibits the highest electronic quality among all known materials. It has a high electronmobility at room temperature, µ ∼ 15000cm2/V · s . In free suspended graphene µ ∼ 150000cm2/V · s

that implies that ballistic transport takes place even at room temperature: the quasiparticles in graphenetake less than 0.1 ps to cover the typical distance between source and drain electrods in a transistor.The very high mobility of graphene makes it promising for applications in which transistors must switch

extremely fast.Easy control of charged carriers.

Anomalous quantum Hall effect: σxy = ±4(n + 1/2) e 2

h.

Thermal properties. The thermal conductivity κ ∼ 5 · 103W /m · K is larger that for carbon nanotubesor diamond.

Optical properties. Graphene has high optical transparency: it absorbs only πα ≈ 2.3% of white light(α = 1/137 - fine structure constant). This can b e important for making liquid crystal displays.

Mechanical properties. Graphene is the strongest material ever tested, siffness - 340N /m. Because of this graphene remains stable and conductive at extremely small scales of order several nanometers.

Prototype devices have already been made from graphene. The first graphene transistor wasdemonstrated by researchers from AAchen University in 2007. In 2009 the MIT researchers built anexperimental graphene chip known as frequency multiplier.

Graphene is today one of the materials being considered as a potential replacement of silicon for futurecomputing.


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Reviews

A.C. Neto, F. Guinea and N.M. Peres. Drawing conclusionsfrom graphene. Physics World, 19, 33 (2006).

M. Katsnelson, Graphene: carbon in two dimensions. Materialstoday, 10, 20 (2007).

A.Geim and K. Novoselov. The rise of graphene. Nature

Materials 6, 183 (2007).V.P. Gusynin, S.G. Sharapov, J.P. Carbotte. AC conductivityof graphene: from tight-binding model to 2+1-dimensionalquantum electrodynamics. Int.J.Mod.Phys.B21, 4611-4658

(2007).A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.

Novoselov, A. K. Geim. The electronic properties of graphene.

Rev. Mod. Phys. 81, 109 (2009).


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One atomic layer of graphene

The structures were made by simply rubbing the freshly cleavedsurface of a layered crystal onto another surface, like drawing chalkon a blackboard. This micromechanical "peeling" created flakes,some of which were – unexpectedly – just one layer thick. Thecrystals are stable and could be used to make transistors and sensors.

Novoselov et al.,Science 306, 666 (2004)

Photographin normal white lightof multilayer graphene flake


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Allotropes of carbon

Well-known forms of carbon all are derived from graphene.


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Field Effect Experiment

Field effect transistor


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Field Effect Experiment

Field effect transistor

V g > 0 means that weincrease the number of elec-trons with increasing V g , while

we control holes for V g < 0.Close to zero V g , this film is acompensated semimetal.

Longitudinal conductivityas a function of gate voltage.

High mobility µ = σ/ne ∼ 15000cm2V −1s −1.

Novoselov et al., Nature 308, 197 (2005) and other groups.


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Orbitals of graphene

Carbon has 6 electrons2 are core electrons4 are valence electrons: – one 2s and three 2p orbitals


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Carbon has 6 electrons

2 are core electrons4 are valence electrons: – one 2s and three 2p orbitalssingle 2s and two 2p orbitals hybridise forming three “σ

bonds” in the x-y plane


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Carbon has 6 electrons

2 are core electrons4 are valence electrons: – one 2s and three 2p orbitalsremaining 2p z orbital (“π” orbital) is perpendicular to thex −

y plane

only π orbital relevant for energies of interest for transportmeasurements, so keep only this one orbital per site in thetight binding model


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Lattice of graphene

H = t n,δ,σ

a†n,σb n+δ,σ + h.c .,

t ≈ 3eV , a = 2.46A is the lattice

constant, σ = ±1 is the spin.

H = t k,σ

a†k,σ b

†k,σ

0

δδδ e i kδδδ

δδδ e −i kδδδ 0

ak,σb k,σ

The π bands structure of a single graphene sheet is

E (k x , k y ) = ±t

1 + 4cos

√3k x a

2cos

k y a

2+ 4cos2

k y a

2.


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Band structure of graphene

akxaky

2

0

2

Εt

Two bands touch each other andcross the Fermi level in six Kpoints located at the corners of the hexagonal 2D Brillouin zone.

a1

a2

∆1

∆2

∆3

A

B

b1

b2

K KK

b

(a) Graphene hexagonal latticecan be described in terms of twotriangular sublattices, A and B.(b) Hexagonal and rhombicextended Brillouin zone (BZ).

Two non-equivalent K points inthe extended BZ, K− = −K+.P.R. Wallace, PR 71, 622 (1947).


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Low-energy excitations in graphene

The low-energy excitations at two inequivalent

K , K ′ = (0,±4π/3a) points have a linear dispersionE k = ±

v F | k | with v F = (√

3/2

)ta ≈ 106 m/s.v F plays role of the velocity of light c ⇒ v F = c /300.Each K point is described by its own spinor:

ψK ,σ =

ψKAσ

ψKB σ

K , K ′ points are also called Dirac points.

Pseudospin direction is linked toan axis determined by electronic

momentum: for conduction and

valence band electrons τ p

| p |= ±1.


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Hamiltonian for K pointThe Hamiltonian for K point

H K =

v F

σ=±1

d

2

k (2π)2

ψ†K σ

0 k x − ik y

k x + ik y 0

ψK σ,

where the momentum k = (k x , k y ) is already given in a localcoordinate system associated with a chosen K point.


f

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Hamiltonian for K pointThe Hamiltonian for K point

H K =

v F

σ=±1

d

2

k (2π)2

ψ†K σ

0 k x − ik y

k x + ik y 0

ψK σ,

where the momentum k = (k x , k y ) is already given in a localcoordinate system associated with a chosen K point.

Introduce ψK σ = ψ†K σγ 0 and 2× 2 γ matricesγ 0,1,2 = (τ 3, i τ 2,−i τ 1):

H =

v F σ=±1 d 2k

(2π)2ψK σ(t , k)(γ 1k x + γ 2k y )ψK σ(t , k).

Spinors can be also combined in one four-component Dirac spinorΨT

σ = (ψKAσ, ψKB σ, ψK ′B σ, ψK ′Aσ).Ψσ = Ψ†

σγ 0 and 4× 4 γ -matrices belonging to a reduciblerepresentation γ ν = τ 3

⊗(τ 3, i τ 2,

−i τ 1) Dirac algebra.

Semenoff, PRL 53, 2449 (1984); DiVincenzo and Mele, PRB 29, 1685 (1984).V.P. Gusynin (BITP, Kiev) Dirac-like quasiparticles in graphene Dubna 12 / 61

Ch l d h l

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Chiral representation and chiralityIn the massless Dirac theory γ 5 = i γ 0γ 1γ 2γ 3

chiral −γ µ

=

I 2 00 −I 2

commutes with the Hamiltonian, it introduces the conservingchirality quantum number which corresponds to the valley index.

Indeed the spinors ΨK+ =

ψK+

0

, ΨK− =

0

ψK−

that

describe the quasiparticle excitations at K±, are the eigenstates of γ 5: γ 5ΨK+ = ΨK+ , γ 5ΨK− = −ΨK− .


Ch l d h l

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Chiral representation and chiralityIn the massless Dirac theory γ 5 = i γ 0γ 1γ 2γ 3

chiral −γ µ

=

I 2 00 −I 2

commutes with the Hamiltonian, it introduces the conservingchirality quantum number which corresponds to the valley index.

Indeed the spinors ΨK+ =

ψK+

0

, ΨK− =

0

ψK−

that

describe the quasiparticle excitations at K±, are the eigenstates of γ 5: γ 5ΨK+ = ΨK+ , γ 5ΨK− = −ΨK− .

In 2D p is ingraphene plane,while the onlyreal rotationdescribed by

τ 3 ⊥ to plane

For massless particle in 3D the helicity(characterizes the projection of its spin on thedirection of momentum) corresponds to chirality:

γ 5Ψ = ±pΣΣΣ/|p|Ψ.The operator τ τ τ = (τ 1, τ 2) does not have thephysical meaning of the usual spin operator. Stillmay use pseudohelicity which is not related to theLorentz group and the real space rotations.


O h d d (HTSC)

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Other condensed matter systems (HTSC)

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3k y

k x

k y

k x

k yk x

BZ

RBZ

Γ

H DSC =

d 2k

(2π)2ψ†(k ) [τ 3(ǫ(k )− µ) + τ 1∆(k )] ψ(k ),

where Pauli matrices τ act on the particle-hole indices of

the Nambu-Gorkov spinor.

ǫ(k )=−2t (cos k x a + cos k y a) , E (k )= ±

(ǫ(k )− µ)2+∆2

(E γ 0−

v F k x γ 1−

v ∆k y γ 2) Ψ = 0.


I d i

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Introducing a gap

H K +

= σ=±1

d 2k

(2π)2ψ†

K +σ ∆

v F (k x

−ik y )

v F (k x + ik y ) −∆ψK

+σ,

where the momentum k = (k x , k y ) is already given in a localcoordinate system associated with a chosen K + point.The presence of ∆

= 0 makes the spectrum

E (k) = ±

2v 2F k2 + ∆2 with the gap ∆.


I d i

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Introducing a gap

H K +

= σ=±1

d 2k

(2π)2

ψ†

K +σ ∆

v F (k x

−ik y )

v F (k x + ik y ) −∆ψK

+σ,

where the momentum k = (k x , k y ) is already given in a localcoordinate system associated with a chosen K + point.The presence of ∆

= 0 makes the spectrum

E (k) = ±

2v 2F k2 + ∆2 with the gap ∆.

For B = 0 mass can be induced by interaction with substrate.

S.Y. Zhou et al., Nature Mat. 6, 770 (07).

Observation of the gap opening insingle-layer epitaxial graphene on a SiC

substrate at the K point.(a) Structure of graphene in the real andmomentum space.(b) ARPES intensity map taken along

the black line in the inset of (a).V.P. Gusynin (BITP, Kiev) Dirac-like quasiparticles in graphene Dubna 15 / 61

QED f f L i

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QED3 form of Lagrangian

An external field B ⊥ to the plane Aext = (−By /2, Bx /2)

L =

d 2r Ψσ(t , r)

γ 0(

i ∂ t + µ− σg L/2µB B ) –Zeeman term

+v F γ 1 i ∂ x − e

c Aext

x + v F γ 2 i ∂ y − e

c Aext

y − ∆ Ψσ(t , r)

− e 2

2ǫ

d 2r d 2r′ Ψσ(t , r)γ 0Ψσ(t , r) 1

|r− r′|Ψσ(t , r′)γ 0Ψσ(t , r′).

µ sgn(V g ) |V g | ∈ [−3600K, 3600K] when V g ∈ [−100V, 100V];

µ > 0 – electrons. g L ≈ 2 in graphene.


QED f f L i

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QED3 form of Lagrangian

An external field B ⊥ to the plane Aext = (−By /2, Bx /2)

L =

d 2r Ψσ(t , r)

γ 0(

i ∂ t + µ− σg L/2µB B ) –Zeeman term

+v F γ 1 i ∂ x − e

c Aext

x + v F γ 2 i ∂ y − e

c Aext

y − ∆ Ψσ(t , r)

− e 2

2ǫ

d 2r d 2r′ Ψσ(t , r)γ 0Ψσ(t , r) 1

|r− r′|Ψσ(t , r′)γ 0Ψσ(t , r′).

µ sgn(V g ) |V g | ∈ [−3600K, 3600K] when V g ∈ [−100V, 100V];

µ > 0 – electrons. g L ≈ 2 in graphene.∆ is a possible excitonic gap (Dirac mass) generated due toCoulomb interaction. Magnetic field favors gap opening: Magnetic catalysis

phenomenon: V.P. G., V.A. Miransky, I.A. Shovkovy PRL 73, 3499 (1994).


L d l l

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Landau levels

Free electrons in a magnetic field: Landau levels (LL).

L.D. Landau, Z.fur Physik, 64, 629 (1930);M.P. Bronstein, Ya.I. Frenkel, ЖРФХО, 62, 485 (1930); I. Rabi, Z.fur Physik, 49, 507 (1928).

In nonrelativistic case the distance between LL coincides with the

cyclotron energy,

ωc [K] =e B me c ∼ 1.35B [T].

For the relativistic-like system (graphene),

E n = ± 2n v 2F |eB |/c , n = 0, 1, . . . .

the energy scale, characterizing the distance between Landau levels,

is

ωL[K] = v F

2eB

c ∼ 424K ·

B [Tesla] for v F ≈ 106m/s!

This is why QHE effect is observed at room temperature!


Dirac Landau levels

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Dirac Landau levels

B = 0 and

zero mass, ∆ = 0

K K'

a

k

EkB0

ΜEkvFk

(a) The low-energylinear-dispersion.µ = 0 as in

compensatedgraphene’s plane.


Dirac Landau levels

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Dirac Landau levels

B = 0 and

zero mass, ∆ = 0

K K'

a

k

EkB0

ΜEkvFk



B = 0 and

finite mass, ∆ = 0

K K'

b

k

Ek

B0

Μ

Ek2 2vF

2k2

(b) A possiblemodification of thespectrum by the finite

gap ∆. µ is shiftedfrom zero by the gatevoltage.


Dirac Landau levels

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Dirac Landau levels

B = 0 and

zero mass, ∆ = 0

K K'

a

k

EkB0

ΜEkvFk



B = 0 and


K K'

b

k

Ek

B0

Μ

Ek2 2vF

2k2

(b) A possiblemodification of thespectrum by the finite

gap ∆. µ is shiftedfrom zero by the gatevoltage.

B = 0 and


K K'

c

k

En

B0

Μ

E0

E0

En2 2n eBv F

2 c

(c) Landau levels E n.Notice that for K point E 0 = −∆ and

for K ′

point E 0 = ∆.Then the degeneracyof n = 0 level is half of the degeneracy of LL with n

= 0.


de Haas van Alphen effect

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Landau’s (1930) paper on diamagnetism: a prediction that

magnetization at low T should oscillate as the field varies. Alreadyin 1930 de Haas and van Alphen reported oscillatory magneticbehavior in Bi.



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Landau’s (1930) paper on diamagnetism: a prediction that

magnetization at low T should oscillate as the field varies. Alreadyin 1930 de Haas and van Alphen reported oscillatory magneticbehavior in Bi.

DOS oscillations

The dHvA effect is essentially

quantum-mechanical in nature, beingdue to Landau quantization of electronenergy in an applied magnetic field. Thefree energy of the system oscillates withmagnetic field and this is a direct

consequence of the existence of theFermi energy. The magnetization andother thermodynamic quantitiesexperience oscillations, too.


Oscillations of DOS for Dirac quasiparticles

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Oscillations of DOS for Dirac quasiparticlesDOS is the sum over LL (E n =

2n|eB | v 2F /c )

D 0(ǫ) = − 1πImtr 1

ǫ + i 0− H = 2eB

π

c

×

δ(ǫ) +

∞

n=1

(δ(ǫ− E n) + δ(ǫ + E n))

⇒ D 0(ǫ) =

2|ǫ|π

2v 2F

, for B = 0.

Using the Poisson summation formula and in presence of impuritiesthe DOS is

D Γ(ǫ) =

1

π sgn(ǫ)

d

d ǫ

ǫ

2

+ 2eB

∞k =1

1

πk sin

πk ǫ2

eB exp

−2πk |ǫ|Γ

eB

.


Shubnikov de Haas effect in 2DEG

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SdH – transport: Lifshitz-Kosevich (LK) formula

σxx =σ0

1 + (ωc τ )2

1 + 2

∞k =1

cos

2πk

µ

ωc

+1

2

R T (k ) R D (k )

,

where ωc =eB

m∗c cyclotron frequency (for ξ(k) =

2k2

2m∗ − µ ),



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σxx =σ0

1 + (ωc τ )2

1 + 2

∞k =1

cos

2πk

µ

ωc

+1

2

R T (k ) R D (k )

,

where ωc =eB


2k2

2m∗ − µ ),R T (k ) = 2π2kT / ωc

sinh 2π2kT /

ωc is temperature amplitude factor ,



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σxx =σ0

1 + (ωc τ )2

1 + 2

∞k =1

cos

2πk

µ

ωc

+1

2

R T (k ) R D (k )

,

where ωc =eB


2k2

2m∗ − µ ),R T (k ) = 2π2kT / ωc

sinh 2π2kT /


R D (k ) = exp−2πk Γ

ωc

is Dingle factor

amplitude reduction due to impurities, Dingle temperature T D = Γ

π

;Impurity scattering rate, Γ = 1/(2τ ), τ being a mean free time of quasiparticles.



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σxx =σ0

1 + (ωc τ )2

1 + 2

∞k =1

cos

2πk

µ

ωc

+1

2

R T (k ) R D (k )

,

where ωc =eB


2k2

2m∗ − µ ),R T (k ) = 2π2kT / ωc

sinh 2π2kT /


R D (k ) = exp−2πk Γ

ωc

is Dingle factor

amplitude reduction due to impurities, Dingle temperature T D = Γ

π

;Impurity scattering rate, Γ = 1/(2τ ), τ being a mean free time of quasiparticles.

PHASE FACTOR: cos

2πk

µ ωc

+ 12

with the phase π


Shubnikov de Haas effect: Dirac case

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Shubnikov de Haas effect: Dirac caseWe use the Kubo formalism:

σij = limΩ→0 ImΠ

R

ij (Ω + i 0)/Ω,

σxx osc =σ0

1 + (ωc τ )2

∞k =1

cos

πk µ2

v 2F |eB |/c

R T (k , µ)R D (k , µ),

where Dingle and temperature amplitude factors (∆ = 0):R D (k , µ) = exp

−2πk

c |µ|Γv 2

F e B

, R T (k , µ) =

2π2kTc |µ|/(v 2F

e

B )

sinh2π2 kTc |µ|

v 2F

eB

.

R T , R D depend on µ (ρ = µ2

πv 2F

2 sgnµ)- there is a dependence on

carrier concentration ρ. In nonrelativistic (NR) case it was:R NR

D (k ) exp−2πk Γ

ωNRc

, R NR

T (k ) = 2π2kT / ωNRc

sinh 2π2kT / ωNRc ,

ωNR

c = eB me c

. For the relativistic system:

ωc =e

Bv 2F

c |µ|= e B

cmc , mc = |µ|

v 2

F

|V g

|!


Manifestation of Berry′s Phase

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Manifestation of Berry′s PhaseOscillating conductivity

σxx

∞k =1

cos

2πk

B f

B + 1

2+ β

R T (k )R D (k ),

where β is Berry′s phase. If β = 0 ⇒ (−1)k - “nonrelativistic” LK

formula.E n = e B

me c

n + 1

2

;

β = 0.

⇐ ⇒

E n = ±v F

2|n|eB

c ;

β = −1/2 – Dirac.


Manifestation of Berry′s Phase

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Manifestation of Berry s PhaseOscillating conductivity

σxx

∞k =1

cos

2πk

B f

B + 1

2+ β

R T (k )R D (k ),

where β is Berry′s phase. If β = 0 ⇒ (−1)k - “nonrelativistic” LK

formula.E n = e B

me c

n + 1

2

;

β = 0.

⇐ ⇒

E n = ±v F

2|n|eB

c ;

β = −1/2 – Dirac.

Dirac quasiparticles are phase shifted by π!Cross-sectional area of the orbit in k -space:nonrelativistic S (ε) = 2πmε and relativistic S (ε) = πε2/v 2F

Semiclassical quantization condition: S (ε) = 2π eB c

(n + 12

+ β )

G.P. Mikitik and Yu. V. Sharlai, PRL 82, 2147 (1999).


Oscillating conductivity - experiment

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O g y p

K.S. Novoselov, et al., Nature

438, 197 (2005)

a) Dependence of B F on carrier concentration n



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g y p


438, 197 (2005)


b) Landau fan diagrams used to find B F . N isthe number associated with different minima of oscillations. The curves extrapolate to differentorigins: to N = 1/2 (in one layer) and 0 (two

and more layers).



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g y p


438, 197 (2005)


b) Landau fan diagrams used to find B F . N isthe number associated with different minima of oscillations. The curves extrapolate to differentorigins: to N = 1/2 (in one layer) and 0 (two

and more layers).c) The behavior of SdHO amplitude as afunction of T for two different carrierconcentrations.d) Cyclotron mass mc (= Fermi energy

|µ

|for

Dirac quasiparticles! ) of electrons and holes asa function of their concentration.mc = |µ|/v 2F =

π

2|n|/v 2F The effectivecarrier mass = 0, because E (k) = ±v F |k|.


Oscillating conductivity towards QHE

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g y Q

Filling factor ν =density of particles

Landau level densitycos [πkc µ2/ v 2F |eB |] → cos (πk ν/2).ν classic ≡ |ρ|

|eB |/hc , ρ = µ2

π

2v 2F

sgnµ

ρ

V g is the carrier imbalance; ρ ≡ n − n0 = n+ − n−, where n+and n− are the densities of electrons and holes.


Oscillating conductivity towards QHE

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g y

Filling factor ν =density of particles

Landau level densitycos [πkc µ2/ v 2F |eB |] → cos (πk ν/2).ν classic ≡ |ρ|

|eB |/hc , ρ = µ2

π

2v 2F

sgnµ

ρ

V g is the carrier imbalance; ρ ≡ n − n0 = n+ − n−, where n+and n− are the densities of electrons and holes.Unusually the minima of σosc at the double odd fillings (k=1)

ν = 2(2n + 1).

This inspired experimentalists K.S. Novoselov, et al., Nature 438, 197 (2005); Y. Zhang, et

al., Nature 438, 201 (2005) to look for an unconventional Quantum Hall Effectpredicted in:

V.P. Gusynin, S.G.Sharapov, PRL 95, 146801 (2005); N.M.R. Peres, F. Guinea, A.H. Castro Neto, PRB 73,

125411 (2006).


Quantum Hall Effect in 2DEG

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ρxx =σxx

σ2xx + σ2

xy

, ρxy = − σxy

σ2xx + σ2

xy .

1.σxx = 0,

2.σxy = −nec

B = −e 2

hν, ν =

nch

eB

= 0, 1, . . . .


Quantum Hall effect in graphene

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Clean limit Γ → 0:

σxy =− e 2

hsgn(eB )

tanh µ + ∆

2T + tanh µ−∆

2T

+ 2∞

n=1

tanhµ +

√∆2 + 2eBn

2T + tanh

µ−√∆2 + 2eBn

2T .

When ∆ = 0 and T → 0 we obtain

σxy =−

2e 2

h

sgn(eB )sgnµ1 + 2∞

n=1

θ |µ| −√

2eBn=− 2e 2

hsgn(eB )sgnµ

1 + 2

µ2c

2 |eB |v 2F

.


Quantum Hall effect in graphene - theory

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11 9 7 5 3 1 0 1 3 5 7 9 11

Ν sgn Μ

22

18

14

10

6

2

2

6

10

14

18

22

Σ h e

2

Σxy classic

Σxy

Σxx

The Hall conductivity σxy and the diagonal conductivity σxx

measured in e 2/h units as a function of the filling ν B = ν/2. The

straight line – classical dependence σxy = −ec |ρ|/B .σxy = − e 2

hν, ν = 4(n + 1

2) = ±2,±6, . . . , n = 0,±1, . . .

The n = 0 Landau level is special: E n = ±

2v 2F |neB |/c ⇒ E 0 = 0and its degeneracy is half of the degeneracy of LL with n = 0.V.P. Gusynin, S.G. Sharapov, PRL 95, 146801 (2005).


Hall effect in graphene - experiment

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K.S. Novoselov, A. Geim et al., Nature 438, 197 (2005)

Hall conductivity σxy = (4e 2/h)ν and longitudinal

resistivity ρxx of graphene as a function of their

concentration at B =14T. Inset: σxy in “two-layer

graphene” where the quantization sequence occurs at

integer ν. The latter shows that the half-integer QHE is

exclusive to “ideal” graphene.

s1

s2

s3

a1

a2

Observation of IQHE is the

ultimate proof of the existence of Dirac quasiparticles in graphene.

It is a consequence of the

honeycomb lattice


Bilayer graphene

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H = − 12m

0 (π†)

2

π2 0

, π = k x + ik y

E n = ± ωc n(n − 1), ωc = eB /mc .

Two states E 0, E 1 lie exactly at zero energy.Charge carriers in bilayer graphene have a parabolic energyspectrum but the Landau quantization results in Hallconductivity where the last (zero-level) plateau ismissing.Instead, the Hall conductivity undergoes adouble-sized step (2× (4e 2/h)) across the region µ = 0.E. McCann, V.I. Fal’ko, PRL 96, 086805 (2006).


Three types of integer QHE

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Three types of the integer

QHE:(a) conventional QHE in 2Dsemiconductors(b) QHE in 2L graphene

(c) QHE for massless Diracfermions in 1L graphene.


AC conductivity of graphene:

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zero magnetic field case, B = 0

In addition to intraband(Drude) transitions, thereare

interband transitions( between positive andnegative Dirac cones).

0 100 200 300 400

GHz

0

1

2

3

4

5

6

Σ x x

h e

2T1K 5K

T5K 0K

T1K 0K Μ0 K

0.1 K

The microwave conductivity σxx (Ω, T ) inunits e 2/h vs frequency Ω in GHz. In

addition to Drude peak (intraband) thereis a constant background (interband).Black curve is for ∆ = 0: Drude peak issuppressed and increase for Ω = 2∆.V.P. Gusynin, S.G. Sh., J.P. Carbotte, PRL 96, 256802 (2006).


Interband term

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The strict Γ = 0 limit and T = 0

σxx (Ω) =

2πe 2

h |µ|δ(Ω) +

πe 2

2h θ |Ω|2 − |µ| .

Universal AC conductivity: Reσxx (Ω) ≃ πe 2

2h, Ω ≫ µ, T .


Interband term

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The strict Γ = 0 limit and T = 0

σxx (Ω) =

2πe 2

h |µ|δ(Ω) +

πe 2

2h θ |Ω|2 − |µ| .

Universal AC conductivity: Reσxx (Ω) ≃ πe 2

2h, Ω ≫ µ, T .

Z.Q. Li et al. Nature Phys. 4, 532 (08).

(a) The real part of the 2D optical

conductivity σ1(ω) at V CN and 71 V.The solid red line shows the region wheredata support the universal result.(b) The Burstein-Moss shift

(concentrational dependencies of thethreshold) is used to determine Fermienergy µ using optical methods.Optical verification of the value of v F :µ = sgn(ρ)

v F π|ρ

| ∼sgn(V g ) |

V g

|.


Fine-structure constant in graphene

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A curious consequence is that the optical transmittance of a free

standing monolayer graphene is also frequency independent and isexpressed solely via the the fine structure constant, α = e 2/

c :

T opt =1 + 2π

c σ(Ω)

−2

=

1 + πα2

−2. A. B. Kuzmenko et al. PRL 100, 117401 (08).



http://find/





c :

T opt =1 + 2π

c σ(Ω)

−2

=

1 + πα2


R.R. Nair et al. Science 320, 1308 (08).

Transmittance spectrum of single-layergraphene (open circles). The red line isthe transmittance T opt expected fortwo-dimensional Dirac fermions, whereasthe green curve takes into account a

nonlinearity and triangular warping of graphene’s electronic spectrum.



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c :

T opt =1 + 2π

c σ(Ω)

−2

=

1 + πα2


R.R. Nair et al. Science 320, 1308 (08).

Transmittance spectrum of single-layergraphene (open circles). The red line isthe transmittance T opt expected fortwo-dimensional Dirac fermions, whereasthe green curve takes into account a

nonlinearity and triangular warping of graphene’s electronic spectrum.


Optical probe of the Dirac quasiparticles:fi it ti fi ld B 0

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finite magnetic field case, B = 0

b

K K'

Μ

0

1

1

2

2

3

3

4

4

The allowed transitionsbetween LLs n = 0, . . . 4.

The pair of cones at K andK′ are combined.Left E 0 < µ < E 1;middle E 1 < µ < E 2;right E 2 < µ < E 3.


Optical probe of the Dirac quasiparticles:finite magnetic field case B 0

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finite magnetic field case, B = 0

b

K K'

Μ

0

1

1

2

2

3

3

4

4

The allowed transitionsbetween LLs n = 0, . . . 4.

The pair of cones at K andK′ are combined.Left E 0 < µ < E 1;middle E 1 < µ < E 2;right E 2 < µ < E 3.

0 200 400 600 800 1000 1200

cm

1

0

2

4

6

R e Σ x x

h e

2

Μ660K B1T

Μ510K B1T

Μ50K B1T

Μ50K B0.0001T

aT10K

15K

0K

Re σxx (Ω) in units of e 2/h vs Ω.red – µ = 50K and B = 10−4T,black – µ = 50K, blue – µ = 510K,green – µ = 660K all three f or B = 1T.


Quantum Hall effect at large B

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http://find/

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- 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0

- 4 0 0 4 0

- 2 0 0 2 0

x

y

(

e

2

/

h

)

x

y

(

e

2

/

h

)

R

(

k

)

Y. Zhang, H. Stormer et al., PRL 96, 136806

(2006).

9T, 11.5T, 17.5T, 25T, 30T, 37T,

37T, 42T, 45T.The observed filling factor sequence:ν = 0 for B > 11Tesla,ν = ±1 for B > 17Tesla.

Thus the four fold (sublattice-spin)degeneracy of n = 0 Landau level istotally resolved for B > 17Tesla.The four fold degeneracy of n = 1level is partially resolved into ν =

±4

which originates from spin splittingleaving two fold degeneracy.


Landau levels’ splitting

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Activation gaps ∆E (ν = 4),∆E (ν = 2),∆E (ν = 1)

are determined from the minimum of resistivity

R minxx ∼ exp(−∆/2kT ). No activation behavior has

been observed at ν = 0!

∆E (ν = 4) ∼ B tot - is inagreement with Zeemansplitting E Z = g LµB B . Also,the gap ∆E (ν = 4) dependson total magnetic field B tot .

∆E (ν = 1) ∼ √B ⊥. The gapmagnitude ∆E (ν = 1) >> E Z

and depends on a perpendicu-lar magnetic field B ⊥ only.

B = 45 T: E Z ≈ 60K,∆E (ν = 1) ≥ 4∆E (ν = 4).


Theoretical predictions b Illustration of the spectrum and

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2 LΜ B B

2 LΜ B B

Μ B B

Μ B B

2 LΜ B B

2 LΜ B B

c

Σ

E

6 42

246

2 L2 2 Μ B B

2 L2 2 Μ B B

Μ B BΜ B BΜ B BΜ B B

2 L2 2 Μ B B

2 L2 2 Μ B B

d

Σ

E

6 42

246

2 L

0

2 L

a

Σ

E

6 42

246

2 L2 2

2 L

2 2

b

Σ

E

6 42

246

Illustration of the spectrum andthe Hall conductivity in the n = 0and n = 1 Landau levels:(a) ∆ = 0 and E Z = 0 (noZeeman term),ν = ±2,±6,±10, . . . .(b) ∆ = 0 and E Z = 0,

ν = 0,±2,±6,±10, . . . .(c) ∆ = 0 and E Z = 0,ν = 0,±2,±4,±6, . . . .(d) ∆ = 0 and E Z = 0,ν = 0,

±1,±

2,±

4, . . . .L =

v 2F |eB |/c .

V.P.G., V.A.Miransky, V.G., S.G. Sharapov, I. Shovkovy, PRB 74, 195429 (2006). Both gap, ∆ = 0

and Zeeman term, E Z = 0 are necessary to explain plateaux

ν = 0,

±1, σxy = ν e 2

h.


http://find/




Symmetry

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The Hamiltonian H = H 0 + H C possesses “flavor”U (4) symmetry

16 generators read (spin ⊗ pseudospin)

σα

2⊗ I 4, σα

2i ⊗ γ 3, σα

2⊗ γ 5, andσα

2⊗ γ 3γ 5

The Zeeman term breaks U (4) down to

U (2)+ ⊗ U (2)−Dirac mass breaks U (2)s down to U (1)s


Order parameters

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Goal: searching for solutions of gap equation with

spontaneously broken and unbroken SU (2)s ⊂ U (2)s .Two scenarios for QH effecta) Quantum Hall Ferromagnetism (QHF):

Spin/valley degeneracy of the half-filled Landau level is lifted

by the exchange (repulsive Coulomb) interaction

This is similar to the Hund’s Rule in atomic physics

In the lowest energy state, the coordinate part of the wave

function is antisymmetric(with the electrons being as far apartas possible) i.e., it is symmetric in the spin/valley indices

This is nothing else but ferromagnetism


QHF in graphene

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K.Nomura, A. MacDonald, PRL 96, 256602 (2006); J. Alicea, M.

Fisher, PRB 74, 075422 (2006). Order parameters:pseudospindensities

Ψ†γ 3

γ 5

P s Ψ = ψ†KAs ψKAs + ψ†KBs ψKBs

− ψ†K ′As ψK ′As − ψ†K ′Bs ψK ′Bs

describes the charge-density imbalance between the twovalley points in the Brillouin zone. This order parameterbreaks SU (2)s down to U (1)s with the generator γ 3γ 5P s (P ± = (1± σ3)/2).


Suspended clean graphene: gap equationTh ti i th d h i ti h th f

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The gap equation in the random phase approximation has the form:

∆(p ) = λ Λ

0

dqq ∆(q )K

(p , q ) q 2 + (∆(q )/v F )2 , λ =

αg

1 + παg /2 , Λ = t /v F ,

where αg = e 2/(4π v F ) ≈ 2.19 is the “fine structure constant” forgraphene. The kernel

K(p , q ) = 2π

1p + q

K

2√pq p + q

,

and K(x) is full elliptic integral of first kind. The nontrivial solutionexists if the coupling λ > 1/4 (αg > 1.62):

∆(p = 0) ≃ Λv F exp

− π

λ− 1/4

.

The strong-coupling limit of graphene is an insulator.V.P. Gusynin (BITP, Kiev) Dirac-like quasiparticles in graphene Dubna 43 / 61

Suspended clean graphene

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αcrit = 1.62 is the quantum critical point of phasetransition metal-insulator: E.Gorbar et al., Phys. Rev.B66, 045108 (2002).

Monte-Carlo simulations give:αcrit = 1.11 - J. Drut and D. Son, Phys. Rev. B77,075115 (2008); J. Drut and T. Lahde, arxiv:0901.0584.See, also S. Hands and C. Strouthos, Phys. Rev. B78,

165423 (2008).


Magnetic catalysis (MC) scenario

U d t l ti fi ld th i t d t

http://find/



Under an external magnetic field the gap is generated atany coupling constant - magnetic catalysis:

∆ ∼ |eB | : E n =

2n|eB | ⇒ E n =

2n|eB |+ ∆2.

I. Krive, S, Naftulin, PRD 46, 2737 (1992); V. Gusynin, V. Miransky,I. Shovkovy, PRL 73, 3499 (1994).

In 4-dimensional Nambu-Jona-Lasinio model

∆ ≃ |eB | exp

− 1

2ν 0G

, ν 0 =

|eB |4π2

−

density of states at the lowest Landau level (compare with a

superconducting gap). First proposed for graphene inD.Khveshchenko, PRL 87, 206401 (2001); ibid 87, 246802 (2001)

E. Gorbar, V. Gusynin, V. Miransky, I. Shovkovy, PRB 66, 045108

(2002)V.P. Gusynin (BITP, Kiev) Dirac-like quasiparticles in graphene Dubna 45 / 61

b) MC scenario

http://find/



V.G., V. Miransky, S. Sharapov, I. Shovkovy, PRB 74, 195429

(2006); I. Herbut, PRL 97, 146401 (2006); J. Fuchs, P. Lederer,PRL 98, 016803 (2007).

Order parameters: Dirac mass terms

ΨP s Ψ = ψ†KAs ψKAs + ψ†K ′As ψK ′As

− ψ†KBs ψKBs − ψ†K ′Bs ψK ′Bs

describes the charge-density imbalance between the twosublattices (charge density wave). This order parameterbreaks SU (2)s down to U (1)s .


Latest developmentsQHF and MC order parameters necessarily coexist which implies

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QHF and MC order parameters necessarily coexist, which implies

that they have the same dynamical origin. E. Gorbar et al., PRB 78, 085437 (2008).


Two types of order parameters

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Singlets under SU (2)s : µs and ∆s .If µ+ = µ− and (or) ∆+ = ∆−, they break U (4)symmetry like the Zeeman term. These order

parameters describe ν = 0 plateau on the lowestLandau level (LLL).

Triplets under SU (2)s : µs and ∆s .Lead to spontaneous breakdown of SU (2)s down to

U (1)s . They describe ν = ±1 plateaus on LLL.


Schwinger Dyson equationSD eqs lead to the system of 8

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SD eqs. lead to the system of 8

integral equations which were solved

analytically in clean limit Γ = 0 and

T = 0 and numerically at T = 0.

Energy scales in the problem:Landau energy scale:ǫB ≡

2

|eB ⊥|v 2F /c ≃ 424

B ⊥[T ] K

Zeeman energy: Z ≃ µB B = 0.67B [T ]K

Dynamical mass scales: (Z ≪ A ≤ M ≪ ǫB )

A =

√πλǫ2

B

4Λ, M =

A

1− λ, λ =

G int Λ

4π3/2

2v 2F

In our calculations, M = 4.84

·10−2ǫB and A = 3.90

·10−2ǫB .


Solutions of Gap EquationProcess of filling LLs is described by varying “bare” chemical

http://find/



Process of filling LLs is described by varying bare chemicalpotential µ0 (or gate voltage).

Results at T = 0 (µ0 ≥ 0):Dispersion relation for LLL:

E σs = −µs + σ[µs + ∆s ] + ∆s ,

σ = ±1 - eigenvalues of pseudospin matrix γ

3

γ

5

.i) Singlet (S1) solution is the ground state for 0 ≤ µ0 < 2A + Z :

∆± = µ± = 0, µ± = µ0 ∓ (A + Z ), ∆± = ±M .

From dispersion relation, E + > 0 and E − < 0, i.e., the LLL is half filled. The spin gap ∆E 0 = E + − E − corresponds to ν = 0 plateau,it is

∆E 0 = 2(A + Z ) + 2M ∼

B ⊥, because both A, M ∼

B ⊥.


Solutions of Gap Equationii) Hybrid (H i e singlet + triplet) solution with mass ∆ for spin

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ii) Hybrid (H, i.e., singlet + triplet) solution, with mass ∆ for spinup and mass ∆ for spin down, is the most favorable for

2A + Z < µ0 < 6A + Z :

∆+ = M , µ+ = A, µ+ = µ0 − Z − 4A, ∆+ = 0,

∆− = 0, µ− = 0, µ− = µ0 + Z − 3A, ∆− = −M .

From dispersion relation,

E (+1)+ > 0, E

(−1)+ < 0, E

(+1)− = E

(−1)− < 0,

i.e., LLL is three-quarter filled and gap∆E 1 = E

(+1)+ − ∆E 1 = E

(−1)+ = 2(M + A) ∼ √B ⊥ corresponds to

ν = 1 plateau. While SU (2)+ is now spontaneously broken, SU (2)−

is exact.


Solutions of Gap Equationiii)Singlet (S2) solution, with mass ∆+ = ∆− is the ground state for

http://find/



iii)Singlet (S2) solution, with mass ∆+ ∆− is the ground state forµ0 > 6A + Z :

∆± = µ± = 0, µ± = µ0 ∓ Z − 7A, ∆± = −M .

From dispersion relation, E + < 0, E − < 0, i.e., LLL is completely

filled. This solution corresponds to ν = 2 plateau with the energy

gap ∆E 2 = √2ǫB between LLL and n = 1 LL. Both SU (2)+ andSU (2)− are now exact.

0.00 0.05 0.10 0.15 0.20 0.25 0.300.05

0.04

0.03

0.02

0.01

0.00

Μ0Ε B

l 2 Ε

B

T0 K analytical

S1

H1

S2

S1S2

H1T

H2S3

0.00 0.05 0.10 0.15 0.20 0.25 0.300.05

0.04

0.03

0.02

0.01

0.00

Μ0Ε B

l 2

Ε B

T1 K numerical


Phase Diagraqm

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0.0 0.5 1.0 1.5 2.0 2.5 3.01.0

0.5

0.0

0.5

1.0

T M

M ,

Μ

M

Z 0, Μ00 Z 0 Z 0

Μ Z 0 Μ Z 0

Temperature dependence of the nontrivial order

parameters in the ν = 0 QH state described by the S1

solution (the results with a vanishing Zeeman energy

Z = 0). SU (4) symmetry is spontaneously broken to

SU (2)+ ⊗ SU (2)−

at T < T c .

Schematic phase diagram of graphene in the plane of

temperature and electron chemical potential.


ν = 0 Plateau6 Longitudinal and Hall conductivities

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0

2

4

6

-80 -40 0 40 800

10

20

30

-6

-2

0

2

6

-10

-5

0

5

10

0 10 20 300

10

20

30

0 100 2000

20

40

60

a)

σ x x

( e

2 / h ) 1µm

b)

ρ

x x

( k Ω )

V g (V)

σx y ( e

2 / h )

ρx y ( k Ω )

B, T

ρ xx (kΩ)

T , K

ρ xx (kΩ)

D. Abanin,K. Novoselov et. al. et al., PRL 98,

196806 (2007).

Longitudinal and Hall conductivitiesσxx and σxy (a) calculated from ρxx

and ρxy measured at 4 K and B = 30T (b) Resistivities ρxx and ρxy . Thelower insets show temperature andmagnetic field dependence of ρxx near

ν = 0. Note the metallic temperaturedependence of ρxx .


ν = 0 Plateau

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5 1 0 1 5

1 1 0 1 0 0

T ( K )

0

x

x

(

e

2

/

h

)

0 5 1 0 1 5

0 . 3 K

R

0

(

k

)

H ( T )

R

0

(

M

)

H ( T )

T = 0 . 3 K

J.Checkelsky, L.Li, N.Ong, PRL 100,206801 (2008)


Theory vs Experiment

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ν = 0 Plateau and Edge States

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Controversy in interpreting ν = 0 state at Dirac neutral

point µ0 = 0 as either QH metal or insulator.QH metal - gapless edge states [D. Abanin,K. Novoselovet. al. et al., PRL 98, 196806 (2007).]

y

x

A

A

A

A

B

B

B

B

Missing atoms of type B

Missing atoms of type Ay0

yWy

x

A

A

A

A

A

A

A

B

B

B

B

B

B

B

x0 xW

The lattice structure of a finite width graphene ribbon with zigzag (left panel) and armchair (right panel) edges.


ν = 0 Plateau and Edge StatesEnergy spectrum in graphene ribbon with zigzag edges:

http://find/



gy p g p g g g

W 10l

2 0 2 4 6 8 10 121.5

1.0

0.5

0.0

0.5

1.0

1.5

kl

E Ε 0

The ferromagnetic gap (the

full Zeeman splitting Z + A)

dominates over the mass gap∆ (∆), insuring the presence

of gapless edge states (marked

by dots).

W 10l

2 0 2 4 6 8 10 121.5

1.0

0.5

0.0

0.5

1.0

1.5

kl

E Ε 0

The mass gap dominates over

the ferromagnetic gap,

insuring the absence of gaplessedge states.



http://find/



Energy spectrum in graphene ribbon with armchair edges:

W 10l

12 10 8 6 4 2 0 21.0

0.5

0.0

0.5

1.0

1.5

kl

E Ε 0

W 10l

12 10 8 6 4 2 0 21.0

0.5

0.0

0.5

1.0

1.5

kl

E Ε 0

Numerical results for the low-energy spectra for a ribbon with

armchair edges in the case of nonzero spin splitting and nonzerosinglet masses ∆. Gapless edge states (marked by dots) are present

in both cases.



E i h ibb i h h i d

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Energy spectrum in graphene ribbon with armchair edges:

W 10l

12 10 8 6 4 2 0 21.0

0.5

0.0

0.5

1.0

1.5

kl

E Ε 0

W 10l

12 10 8 6 4 2 0 21.0

0.5

0.0

0.5

1.0

1.5

kl

E Ε 0

Numerical results for the low-energy spectra for a ribbon with

armchair edges in the case of nonzero spin splitting and nonzerotriplet masses ∆. The existence of gapless modes (marked by dots)depends on the relative magnitude of spin splitting and tripletmasses.


Summary

Thermal and Dingle factors in MO of conductivity are different

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Thermal and Dingle factors in MO of conductivity are differentfrom those ones in 2DEG (density dependent cyclotron massmc = |µ|/v 2F )

Phase shift of π of oscillations of σxx (1/B )

Uneven (or half-) Integer Quantum Hall effect:

σxy = −4e

2

h (n + 12 ), n = 0,±1,±2, . . .

Universal high frequency ac conductivity σxx (Ω) = πe 2/2h

Both MC and QHF are responsible for dynamical symmetry

breaking and lifting the degeneracy of Landau levels ingraphene

Qualitative agreement with experiment is evident, but detailsremain to be worked out

V P Gusynin (BITP Kiev) Dirac-like quasiparticles in graphene Dubna 61 / 61

http://find/

Documents

##Dirac-Like Quasi Particles in Graphene_Gusynin_talk_Dubna 2009