Dirac-like quasiparticles in graphene

Dirac-like quasiparticles in graphene

V.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

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 ofabout 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 switchextremely fast.

Easy control of charged carriers.

Anomalous quantum Hall effect: σxy = ±4(n + 1/2) e2

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 be important for making liquid crystal displays.

Mechanical properties. Graphene is the strongest material ever tested, siffness - 340N/m. Because ofthis 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.


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. NatureMaterials 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).


Outline

1 Dirac theory of graphene

2 Shubnikov de Haas effect

3 Quantum Hall effect in graphene

4 AC conductivity of graphene

5 Quantum Hall effect at large B (latest developments)

6 ν = 0 plateau and edge states

7 Summary


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


Allotropes of carbon

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


Field Effect Experiment

Field effect transistor


Field Effect Experiment

Field effect transistor

Vg > 0 means that weincrease the number of elec-trons with increasing Vg , whilewe control holes for Vg < 0.Close to zero Vg , 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.


Orbitals of graphene

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



Carbon has 6 electrons2 are core electrons4 are valence electrons: – one 2s and three 2p orbitalssingle 2s and two 2p orbitals hybridise forming three “σbonds” in the x-y plane



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


Lattice of graphene

H = t∑

n,δ,σ

a†n,σbn+δ,σ + h.c.,

t ≈ 3eV , a = 2.46A is the latticeconstant, σ = ±1 is the spin.

H = t∑

k,σ

(

a†k,σ b

†k,σ

)

(

0∑

δδδ e ikδδδ∑

δδδ e−ikδδδ 0

)(

ak,σ

bk,σ

)

The π bands structure of a single graphene sheet is

E (kx , ky ) = ±t

√

1 + 4 cos

√3kxa

2cos

kya

2+ 4 cos2

kya

2.


Band structure of graphene

akxaky

-2

0

2

Εt

2

0

2

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

a1

a2

∆1

∆2

∆3

A

B

b1

b2

K -K-K

HbL

(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).


Low-energy excitations in graphene

The low-energy excitations at two inequivalentK ,K ′ = (0,±4π/3a) points have a linear dispersion

Ek = ±~vF |~k | with vF = (√

3/2~)ta ≈ 106 m/s.vF plays role of the velocity of light c⇒ vF = c/300.Each K point is described by its own spinor:

ψK ,σ =

(

ψKAσ

ψKBσ

)

K ,K ′ points are also called Dirac points.

Pseudospin direction is linked to

an axis determined by electronic

momentum: for conduction and

valence band electrons ~τ~p|~p|

= ±1.


Hamiltonian for K pointThe Hamiltonian for K point

HK = ~vF

∑

σ=±1

∫

d2k

(2π)2ψ†

Kσ

(

0 kx − iky

kx + iky 0

)

ψKσ,

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


Hamiltonian for K pointThe Hamiltonian for K point

HK = ~vF

∑

σ=±1

∫

d2k

(2π)2ψ†

Kσ

(

0 kx − iky

kx + iky 0

)

ψKσ,

where the momentum k = (kx , ky ) is already given in a localcoordinate system associated with a chosen K point.Introduce ψKσ = ψ†

Kσγ0 and 2 × 2 γ matrices

γ0,1,2 = (τ3, iτ2,−iτ1):

H = ~vF

∑

σ=±1

∫

d2k

(2π)2ψKσ(t, k)(γ1kx + γ2ky )ψKσ(t, k).

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

σ = (ψKAσ, ψKBσ, ψK ′Bσ, ψK ′Aσ).Ψσ = Ψ†

σγ0 and 4 × 4 γ-matrices belonging to a reducible

representation γν = τ3 ⊗ (τ3, iτ2,−iτ1) 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

Chiral representation and chiralityIn the massless Dirac theory γ5 = iγ0γ1γ2γ3 chiral−γµ

=

(

I2 00 −I2

)

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

Indeed the spinors ΨK+ =

(

ψK+

0

)

, ΨK− =

(

0ψK−

)

that

describe the quasiparticle excitations at K±, are the eigenstates ofγ5: γ5ΨK+ = ΨK+ , γ5ΨK− = −ΨK− .


Chiral representation and chiralityIn the massless Dirac theory γ5 = iγ0γ1γ2γ3 chiral−γµ

=

(

I2 00 −I2

)

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

Indeed the spinors ΨK+ =

(

ψK+

0

)

, ΨK− =

(

0ψK−

)

that

describe the quasiparticle excitations at K±, are the eigenstates ofγ5: γ5ΨK+ = ΨK+ , γ5ΨK− = −ΨK− .

In 2D p is ingraphene plane,while the onlyreal rotationdescribed byτ3 ⊥ to plane

For massless particle in 3D the helicity(characterizes the projection of its spin on thedirection of momentum) corresponds to chirality:γ5Ψ = ±pΣΣΣ/|p|Ψ.The operator τττ = (τ1, τ2) 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.


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

3ky

kx

ky

kx

ky

kx

BZ

RBZ

Γ

HDSC =

∫

d2k

(2π)2ψ†(k) [τ3(ǫ(k) − µ) + τ1∆(k)]ψ(k),

where Pauli matrices τ act on the particle-hole indices ofthe Nambu-Gorkov spinor.

ǫ(k)=−2t (cos kxa + cos kya) , E (k)= ±√

(ǫ(k)− µ)2+∆2(

(Eγ0 − vFkxγ1 − v∆kyγ2) Ψ = 0.


Introducing a gap

HK+ =∑

σ=±1

∫

d2k

(2π)2ψ†

K+σ

(

∆ ~vF (kx − iky )~vF (kx + iky ) −∆

)

ψK+σ,

where the momentum k = (kx , ky ) is already given in a localcoordinate system associated with a chosen K+ point.The presence of ∆ 6= 0 makes the spectrumE (k) = ±

√~2v 2Fk2 + ∆2 with the gap ∆.


Introducing a gap

HK+ =∑

σ=±1

∫

d2k

(2π)2ψ†

K+σ

(

∆ ~vF (kx − iky )~vF (kx + iky ) −∆

)

ψK+σ,

where the momentum k = (kx , ky ) is already given in a localcoordinate system associated with a chosen K+ point.The presence of ∆ 6= 0 makes the spectrumE (k) = ±

√~2v 2Fk2 + ∆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 SiCsubstrate at the K point.(a) Structure of graphene in the real andmomentum space.(b) ARPES intensity map taken alongthe black line in the inset of (a).


QED3 form of Lagrangian

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

L =

∫

d2r Ψσ(t, r)[

γ0(~i∂t + µ− σgL/2µBB) –Zeeman term

+vFγ1(~i∂x −

e

cAext

x

)

+ vFγ2(~i∂y −

e

cAext

y

)

− ∆]

Ψσ(t, r)

− e2

2ǫ

∫

d2r d2r′ Ψσ(t, r)γ0Ψσ(t, r)1

|r − r′|Ψσ(t, r′)γ0Ψσ(t, r′).

µ _ sgn(Vg )√

|Vg | ∈ [−3600K, 3600K] when Vg ∈ [−100V, 100V];µ > 0 – electrons. gL ≈ 2 in graphene.


QED3 form of Lagrangian

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

L =

∫

d2r Ψσ(t, r)[

γ0(~i∂t + µ− σgL/2µBB) –Zeeman term

+vFγ1(~i∂x −

e

cAext

x

)

+ vFγ2(~i∂y −

e

cAext

y

)

− ∆]

Ψσ(t, r)

− e2

2ǫ

∫

d2r d2r′ Ψσ(t, r)γ0Ψσ(t, r)1

|r − r′|Ψσ(t, r′)γ0Ψσ(t, r′).

µ _ sgn(Vg )√

|Vg | ∈ [−3600K, 3600K] when Vg ∈ [−100V, 100V];µ > 0 – electrons. gL ≈ 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).


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~Bmec

∼ 1.35B[T].

For the relativistic-like system (graphene),

En = ±√

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

the energy scale, characterizing the distance between Landau levels,

is ~ωL[K] = vF

√~2eBc

∼ 424K ·√

B[Tesla] for vF ≈ 106m/s!

This is why QHE effect is observed at room temperature!


Dirac Landau levels

B = 0 andzero mass, ∆ = 0

K K'

HaL

k

EHkL

B=0

ΜEHkL=±ÑvFk

(a) The low-energylinear-dispersion.µ = 0 as incompensatedgraphene’s plane.


Dirac Landau levels


K K'

HaL

k

EHkL

B=0

ΜEHkL=±ÑvFk


B = 0 andfinite mass, ∆ 6= 0

K K'

HbL

k

EHkL

B=0

Μ

EHkL=±"###############################

D2 + Ñ2 vF2 k2

(b) A possiblemodification of thespectrum by the finitegap ∆. µ is shiftedfrom zero by the gatevoltage.


Dirac Landau levels


K K'

HaL

k

EHkL

B=0

ΜEHkL=±ÑvFk


B = 0 andfinite mass, ∆ 6= 0

K K'

HbL

k

EHkL

B=0

Μ

EHkL=±"###############################

D2 + Ñ2 vF2 k2

(b) A possiblemodification of thespectrum by the finitegap ∆. µ is shiftedfrom zero by the gatevoltage.

B 6= 0 andfinite mass, ∆ 6= 0

K K'

HcL

k

En

B¹0

Μ

E0=-D

E0=D

En=±"##########################################

D2 + 2 n ÑeB vF2 c

(c) Landau levels En.Notice that for Kpoint E0 = −∆ andfor K ′ point E0 = ∆.Then the degeneracyof n = 0 level is halfof the degeneracy ofLL with n 6= 0.


de Haas van Alphen effect

Landau’s (1930) paper on diamagnetism: a prediction thatmagnetization at low T should oscillate as the field varies. Alreadyin 1930 de Haas and van Alphen reported oscillatory magneticbehavior in Bi.


de Haas van Alphen effect

Landau’s (1930) paper on diamagnetism: a prediction thatmagnetization 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 essentiallyquantum-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 directconsequence of the existence of theFermi energy. The magnetization andother thermodynamic quantitiesexperience oscillations, too.


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

√

2n|eB|~v 2F/c)

D0(ǫ) = −1

πImtr

1

ǫ+ i0 − H=

2eB

π~c×[

δ(ǫ) +

∞∑

n=1

(δ(ǫ− En) + δ(ǫ+ En))

]

⇒ D0(ǫ) =2|ǫ|π~2v 2

F

, for B = 0.

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

DΓ(ǫ) =1

πsgn(ǫ)

d

dǫ

[

ǫ2

+ 2eB

∞∑

k=1

1

πksin

πkǫ2

eBexp

(

−2πk|ǫ|ΓeB

)

]

.


Shubnikov de Haas effect in 2DEG

SdH – transport: Lifshitz-Kosevich (LK) formula

σxx =σ0

1 + (ωcτ)2

1 + 2

∞∑

k=1

cos

[

2πk

(

µ~ωc

+1

2

)]

RT (k) RD(k)

,

where ωc = eBm∗c

cyclotron frequency (for ξ(k) = ~2k2

2m∗− µ ),




σxx =σ0

1 + (ωcτ)2

1 + 2

∞∑

k=1

cos

[

2πk

(

µ~ωc

+1

2

)]

RT (k) RD(k)

,

where ωc = eBm∗c


2m∗− µ ),

RT (k) = 2π2kT/~ωc

sinh 2π2kT/~ωcis temperature amplitude factor,




σxx =σ0

1 + (ωcτ)2

1 + 2

∞∑

k=1

cos

[

2πk

(

µ~ωc

+1

2

)]

RT (k) RD(k)

,

where ωc = eBm∗c


2m∗− µ ),



RD(k) = exp(

−2πk Γ~ωc

)

is Dingle factor—

amplitude reduction due to impurities, Dingle temperature TD = Γπ;

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




σxx =σ0

1 + (ωcτ)2

1 + 2

∞∑

k=1

cos

[

2πk

(

µ~ωc

+1

2

)]

RT (k) RD(k)

,

where ωc = eBm∗c


2m∗− µ ),



RD(k) = exp(

−2πk Γ~ωc

)

is Dingle factor—

amplitude reduction due to impurities, Dingle temperature TD = Γπ;

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

PHASE FACTOR: cos[

2πk(

µ~ωc+ 1

2

)]

with the phase π


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

σij = limΩ→0

ImΠRij (Ω + i0)/Ω,

σxx osc =σ0

1 + (ωcτ)2

∞∑

k=1

cos

[

πkµ2~v 2F |eB|/c

]

RT (k, µ)RD(k, µ),

where Dingle and temperature amplitude factors (∆ = 0):

RD(k, µ) = exp(

−2πkc|µ|Γ

v2Fe~B) , RT (k, µ) =

2π2kTc|µ|/(v2Fe~B)

sinh2π2kTc|µ|

v2F~eB

.

RT , RD depend on µ (ρ = µ2

πv2F~2 sgnµ)- there is a dependence on

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

RNR

D (k) _ exp(

−2πk Γ~ωNRc

)

, RNR

T (k) = 2π2kT/~ωNRc

sinh 2π2kT/~ωNRc,

ωNR

c = eBmec

. For the relativistic system:~ωc =e~Bv2

F

c|µ|= e~B

cmc, mc = |µ|

v2F

_√|Vg |!V.P. Gusynin (BITP, Kiev) Dirac-like quasiparticles in graphene Dubna 22 / 61

Manifestation of Berry′s PhaseOscillating conductivity

σxx _ ∞∑

k=1

cos

[

2πk

(

Bf

B+

1

2+ β

)]

RT (k)RD(k),

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

En = e~Bmec

(

n + 12

)

;β = 0.

⇐ ⇒

En = ±vF

√~2|n|eBc

;

β = −1/2 – Dirac.


Manifestation of Berry′s PhaseOscillating conductivity

σxx _ ∞∑

k=1

cos

[

2πk

(

Bf

B+

1

2+ β

)]

RT (k)RD(k),

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

En = e~Bmec

(

n + 12

)

;β = 0.

⇐ ⇒

En = ±vF

√~2|n|eBc

;

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 2

F

Semiclassical quantization condition: S(ε) = 2π~eBc

(n + 12

+ β)

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


Oscillating conductivity - experiment

K.S. Novoselov, et al., Nature

438, 197 (2005)

a) Dependence of BF on carrier concentration n




438, 197 (2005)

a) Dependence of BF on carrier concentration nb) Landau fan diagrams used to find BF . N isthe number associated with different minima ofoscillations. The curves extrapolate to differentorigins: to N = 1/2 (in one layer) and 0 (twoand more layers).




438, 197 (2005)

a) Dependence of BF on carrier concentration nb) Landau fan diagrams used to find BF . N isthe number associated with different minima ofoscillations. The curves extrapolate to differentorigins: to N = 1/2 (in one layer) and 0 (twoand more layers).c) The behavior of SdHO amplitude as afunction of T for two different carrierconcentrations.d) Cyclotron mass mc(= Fermi energy |µ| forDirac quasiparticles! ) of electrons and holes asa function of their concentration.mc = |µ|/v 2

F =√

π~2|n|/v 2F The effective

carrier mass = 0, because E (k) = ±vF~|k|.V.P. Gusynin (BITP, Kiev) Dirac-like quasiparticles in graphene Dubna 24 / 61

Oscillating conductivity towards QHE

Filling factor ν =density of particles

Landau level densitycos [πkcµ2/~v 2

F |eB|] → cos (πkν/2).

νclassic ≡ |ρ||eB|/hc

, ρ = µ2

π~2v2F

sgnµ

ρ _ Vg is the carrier imbalance; ρ ≡ n − n0 = n+ − n−, where n+

and n− are the densities of electrons and holes.


Oscillating conductivity towards QHE

Filling factor ν =density of particles

Landau level densitycos [πkcµ2/~v 2

F |eB|] → cos (πkν/2).

νclassic ≡ |ρ||eB|/hc

, ρ = µ2

π~2v2F

sgnµ

ρ _ Vg 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

ρxx =σxx

σ2xx + σ2

xy

, ρxy = − σxy

σ2xx + σ2

xy .

1.σxx = 0,

2.σxy = −nec

B= −e2

hν, ν =

[

nch

eB

]

= 0, 1, . . . .


Quantum Hall effect in graphene

Clean limit Γ → 0:

σxy = − e2

hsgn(eB)

[

tanhµ+ ∆

2T+ tanh

µ− ∆

2T

+ 2∞∑

n=1

(

tanhµ+

√∆2 + 2eBn

2T+ tanh

µ−√

∆2 + 2eBn

2T

)]

.

When ∆ = 0 and T → 0 we obtain

σxy = − 2e2

hsgn(eB)sgnµ

[

1 + 2

∞∑

n=1

θ(

|µ| −√

2eBn)

]

= − 2e2

hsgn(eB)sgnµ

(

1 + 2

[

µ2c

2~|eB|v 2F

])

.


Quantum Hall effect in graphene - theory

-11 -9 -7 -5 -3 -1 0 1 3 5 7 9 11Ν sgn@ΜD

-22

-18

-14

-10

-6

-2

2

6

10

14

18

22

Σh

e2

Σxy classic

Σxy

Σxx

The Hall conductivity σxy and the diagonal conductivity σxx

measured in e2/h units as a function of the filling νB = ν/2. Thestraight line – classical dependence σxy = −ec|ρ|/B .

σxy = − e2

hν, ν = 4(n + 1

2) = ±2,±6, . . . , n = 0,±1, . . .

The n = 0 Landau level is special: En = ±√

2v 2F~|neB|/c ⇒ E0 = 0

and its degeneracy is half of the degeneracy of LL with n 6= 0.V.P. Gusynin, S.G. Sharapov, PRL 95, 146801 (2005).


Hall effect in graphene - experiment

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

Hall conductivity σxy = (4e2/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

s2s3

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

H = − 1

2m

(

0 (π†)2

π2 0

)

, π = kx + iky

En = ±~ωc

√

n(n − 1), ωc = eB/mc .

Two states E0, E1 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 × (4e2/h)) across the region µ = 0.E. McCann, V.I. Fal’ko, PRL 96, 086805 (2006).


Three types of integer QHE

Three types of the integerQHE:(a) conventional QHE in 2Dsemiconductors(b) QHE in 2L graphene(c) QHE for massless Diracfermions in 1L graphene.


AC conductivity of graphene:zero magnetic field case, B = 0

In addition to intraband(Drude) transitions, thereareinterband transitions( between positive andnegative Dirac cones).

0 100 200 300 400W @GHzD

0

1

2

3

4

5

6

Σxx@WD

he2

T=1K D=5K

T=5K D=0K

T=1K D=0KΜ=0 K

G=0.1 K

The microwave conductivity σxx(Ω,T ) inunits e2/h vs frequency Ω in GHz. Inaddition to Drude peak (intraband) thereis a constant background (interband).Black curve is for ∆ 6= 0: Drude peak issuppressed and increase for Ω = 2∆.V.P. Gusynin, S.G. Sh., J.P. Carbotte, PRL 96, 256802 (2006).


Interband termThe strict Γ = 0 limit and T = 0

σxx(Ω) =2πe2

h|µ|δ(Ω) +

πe2

2hθ

( |Ω|2

− |µ|)

.

Universal AC conductivity: Reσxx(Ω) ≃ πe2

2h, Ω ≫ µ,T .


Interband termThe strict Γ = 0 limit and T = 0

σxx(Ω) =2πe2

h|µ|δ(Ω) +

πe2

2hθ

( |Ω|2

− |µ|)

.

Universal AC conductivity: Reσxx(Ω) ≃ πe2

2h, Ω ≫ µ,T .

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

(a) The real part of the 2D opticalconductivity σ1(ω) at VCN 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 vF :µ = sgn(ρ)~vF

√

π|ρ| ∼ sgn(Vg )√

|Vg |.V.P. Gusynin (BITP, Kiev) Dirac-like quasiparticles in graphene Dubna 33 / 61

Fine-structure constant in graphene

A curious consequence is that the optical transmittance of a freestanding monolayer graphene is also frequency independent and isexpressed solely via the the fine structure constant, α = e2/~c:

Topt =∣

∣1 + 2πcσ(Ω)

∣

∣

−2=(

1 + πα2

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


Fine-structure constant in graphene

A curious consequence is that the optical transmittance of a freestanding monolayer graphene is also frequency independent and isexpressed solely via the the fine structure constant, α = e2/~c:

Topt =∣

∣1 + 2πcσ(Ω)

∣

∣

−2=(

1 + πα2

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

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

Transmittance spectrum of single-layergraphene (open circles). The red line isthe transmittance Topt expected fortwo-dimensional Dirac fermions, whereasthe green curve takes into account anonlinearity and triangular warping ofgraphene’s electronic spectrum.


Optical probe of the Dirac quasiparticles:finite magnetic field case, B 6= 0

HbL

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 — E0 < µ < E1;middle — E1 < µ < E2;right — E2 < µ < E3.


Optical probe of the Dirac quasiparticles:finite magnetic field case, B 6= 0

HbL

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 — E0 < µ < E1;middle — E1 < µ < E2;right — E2 < µ < E3.

0 200 400 600 800 1000 1200W@cm-1D

0

2

4

6

ReΣ

xx@WDhe

2

Μ=660K B=1T

Μ=510K B=1T

Μ=50K B=1T

Μ=50K B=0.0001T

HaLT=10K

G=15K

D=0K

Re σxx(Ω) in units of e2/h vs Ω.red – µ = 50K and B = 10−4T,black – µ = 50K, blue – µ = 510K,green – µ = 660K all three for B = 1T.


Quantum Hall effect at large B

-80 -60 -40 -20 0 20 40 60 80-10

-8

-6

-4

-2

0

2

4

6

8

10

-40 0 40-20

0

20

40

-20 0 20

-6

-4

-2

0

2

4

6

xy (e2/h)

xy (e

2 /h)

Vg (V)

Vg (V)

R (k

)

Vg (V)

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 = ±4which originates from spin splittingleaving two fold degeneracy.


Landau levels’ splitting

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

are determined from the minimum of resistivity

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

been observed at ν = 0!

∆E (ν = 4) ∼ Btot - is inagreement with Zeemansplitting EZ = gLµBB . Also,the gap ∆E (ν = 4) dependson total magnetic field Btot .∆E (ν = 1) ∼

√B⊥. The gap

magnitude ∆E (ν = 1) >> EZ

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

B = 45 T: EZ ≈ 60K,

∆E (ν = 1) ≥ 4∆E (ν = 4).


Theoretical predictions

-!!!!

2 L- ΜBB-!!!!

2 L+ ΜBB

- ΜBBΜBB

!!!!2 L- ΜBB

!!!!2 L+ ΜBB

HcL

Σ

E

-6-4-2

246

-!!!!!!!!!!!!!!!!!!!!!

2 L2 + D2 - ΜBB-!!!!!!!!!!!!!!!!!!!!!

2 L2 + D2 + ΜBB

-D- ΜBB-D+ ΜBBD- ΜBBD+ ΜBB

!!!!!!!!!!!!!!!!!!!!!2 L2 + D2 - ΜBB

!!!!!!!!!!!!!!!!!!!!!2 L2 + D2 + ΜBB

HdL

Σ

E

-6-4-2

246

-!!!!

2 L

0

!!!!2 L

HaL

Σ

E

-6-4-2

246

-!!!!!!!!!!!!!!!!!!!!!

2 L2 + D2

-D

D

"####################2 L2+ D2

HbL

Σ

E

-6-4-2

246

Illustration of the spectrum andthe Hall conductivity in the n = 0and n = 1 Landau levels:(a) ∆ = 0 and EZ = 0 (noZeeman term),ν = ±2,±6,±10, . . . .(b) ∆ 6= 0 and EZ = 0,ν = 0,±2,±6,±10, . . . .(c) ∆ = 0 and EZ 6= 0,ν = 0,±2,±4,±6, . . . .(d) ∆ 6= 0 and EZ 6= 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, ∆ 6= 0

and Zeeman term, EZ 6= 0 are necessary to explain plateaux

ν = 0,±1, σxy = ν e2

h.


Symmetry

The Hamiltonian H = H0 + HC possesses “flavor”U(4) symmetry

16 generators read (spin ⊗ pseudospin)

σα

2⊗ I4,

σα

2i⊗ γ3,

σα

2⊗ γ5, and

σα

2⊗ γ3γ5

The Zeeman term breaks U(4) down toU(2)+ ⊗ U(2)−Dirac mass breaks U(2)s down to U(1)s


Order parameters

Goal: searching for solutions of gap equation withspontaneously 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 liftedby 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 wavefunction 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

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

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

〈Ψ†γ3γ5PsΨ〉 = ψ†KAsψKAs + ψ†

KBsψKBs

− ψ†K ′AsψK ′As − ψ†

K ′BsψK ′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 γ3γ5Ps

(P± = (1 ± σ3)/2).


Suspended clean graphene: gap equationThe gap equation in the random phase approximation has the form:

∆(p) = λ

∫ Λ

0

dqq∆(q)K(p, q)√

q2 + (∆(q)/vF )2, λ =

αg

1 + παg/2, Λ = t/vF ,

where αg = e2/(4π~vF ) ≈ 2.19 is the “fine structure constant” forgraphene. The kernel

K(p, q) =2

π

1

p + qK

(

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) ≃ ΛvF 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

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

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

∆ ∼√

|eB | : En =√

2n|eB | ⇒ En =√

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

2ν0G

)

, ν0 =|eB|4π2

−

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

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

〈ΨPsΨ〉 = ψ†KAsψKAs + ψ†

K ′AsψK ′As

− ψ†KBsψKBs − ψ†

K ′BsψK ′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

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


Two types of order parameters

Singlets under SU(2)s : µs and ∆s .If µ+ 6= µ− and (or) ∆+ 6= ∆−, they break U(4)symmetry like the Zeeman term. These orderparameters 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 toU(1)s . They describe ν = ±1 plateaus on LLL.


Schwinger Dyson equationSD 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 6= 0.

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

√

2~|eB⊥|v 2F/c ≃ 424

√

B⊥[T ] K

Zeeman energy: Z ≃ µBB = 0.67B[T ]K

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

A =

√πλǫ2B4Λ

, M =A

1 − λ, λ =

GintΛ

4π3/2~2v 2F

In our calculations, M = 4.84 · 10−2ǫB and A = 3.90 · 10−2ǫB .V.P. Gusynin (BITP, Kiev) Dirac-like quasiparticles in graphene Dubna 49 / 61

Solutions of Gap EquationProcess of filling LLs is described by varying “bare” chemicalpotential µ0 (or gate voltage).Results at T = 0 (µ0 ≥ 0):Dispersion relation for LLL:

Eσs = −µs + σ[µs + ∆s ] + ∆s ,

σ = ±1 - eigenvalues of pseudospin matrix γ3γ5.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 halffilled. The spin gap ∆E0 = E+ − E− corresponds to ν = 0 plateau,it is

∆E0 = 2(A + Z ) + 2M ∼√

B⊥, because bothA,M ∼√

B⊥.


Solutions of Gap Equation

ii) Hybrid (H, i.e., singlet + triplet) solution, with mass ∆ for spinup and mass ∆ for spin down, is the most favorable for2A + 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

∆E1 = E(+1)+ − ∆E1 = E

(−1)+ = 2(M + A) ∼

√B⊥ corresponds to

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

is exact.V.P. Gusynin (BITP, Kiev) Dirac-like quasiparticles in graphene Dubna 51 / 61

Solutions of Gap Equationiii)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 ∆E2 =√

2ǫB between LLL and n = 1 LL. Both SU(2)+ and

SU(2)− are now exact.

0.00 0.05 0.10 0.15 0.20 0.25 0.30-0.05

-0.04

-0.03

-0.02

-0.01

0.00

Μ0ΕB

Wl2Ε B

T=0 K HanalyticalL

S1

H1

S2

S1S2

H1T

H2S3

0.00 0.05 0.10 0.15 0.20 0.25 0.30-0.05

-0.04

-0.03

-0.02

-0.01

0.00

Μ0ΕB

Wl2Ε B

T=1 K HnumericalL


Phase Diagraqm

0.0 0.5 1.0 1.5 2.0 2.5 3.0-1.0

-0.5

0.0

0.5

1.0

T M

D±M

,Μ

±M

Z=0, Μ0=0 D+ HZ=0LD- HZ=0LΜ+ HZ=0LΜ- HZ=0L

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 < Tc .

Schematic phase diagram of graphene in the plane of

temperature and electron chemical potential.


ν = 0 Plateau

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 xx (e2

/h) 1µm

b)

ρ x

x (kΩ)

Vg (V)

σxy (e

2/h)

ρxy (kΩ

)

B, T

ρxx (kΩ)

T, K

ρxx (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 nearν = 0. Note the metallic temperaturedependence of ρxx .


ν = 0 Plateau

5 10 150.0

0.1

0.2

0.0

0.5

1.0

1.5

2.01 10 100

K7 (a)

T (K)

14131211

10

89

7

0 xx (e

2 /h)

6 T

0 5 10 150

40

80

120

160

200

K7(b)

40

20

5

2

0.3 K

R0 (k

)

0H (T)

K7

K18

(c)

R0 (

M)

0H (T)

K22

T = 0.3 K

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


Theory vs Experiment


ν = 0 Plateau and Edge States

Controversy in interpreting ν = 0 state at Dirac neutralpoint 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 Ay=0

y=Wy

x

A

A

A

A

A

A

A

B

B

B

B

B

B

B

x=0 x=WThe lattice structure of a finite width graphene ribbon with zigzag (left panel) and armchair (right panel) edges.



Energy spectrum in graphene ribbon with zigzag edges:

W=10l

-2 0 2 4 6 8 10 12-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

kl

EΕ 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 12-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

kl

EΕ 0

The mass gap dominates over

the ferromagnetic gap,

insuring the absence of gapless

edge states.



Energy spectrum in graphene ribbon with armchair edges:

W=10l

-12 -10 -8 -6 -4 -2 0 2-1.0

-0.5

0.0

0.5

1.0

1.5

kl

EΕ 0

W=10l

-12 -10 -8 -6 -4 -2 0 2-1.0

-0.5

0.0

0.5

1.0

1.5

kl

EΕ 0

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

armchair edges in the case of nonzero spin splitting and nonzero

singlet masses ∆. Gapless edge states (marked by dots) are present

in both cases.



Energy spectrum in graphene ribbon with armchair edges:

W=10l

-12 -10 -8 -6 -4 -2 0 2-1.0

-0.5

0.0

0.5

1.0

1.5

kl

EΕ 0

W=10l

-12 -10 -8 -6 -4 -2 0 2-1.0

-0.5

0.0

0.5

1.0

1.5

kl

EΕ 0

Numerical results for the low-energy spectra for a ribbon witharmchair 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 differentfrom those ones in 2DEG (density dependent cyclotron massmc = |µ|/v 2

F )

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

Uneven (or half-) Integer Quantum Hall effect:

σxy = −4e2

h(n + 1

2), n = 0,±1,±2, . . .

Universal high frequency ac conductivity σxx(Ω) = πe2/2h

Both MC and QHF are responsible for dynamical symmetrybreaking and lifting the degeneracy of Landau levels ingraphene

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


Dirac-like quasiparticles in graphene

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