Coulomb interaction in graphene

Daniel E. Sheehy

Work in collaboration with J. Schmalian (Iowa State U. )

Phys. Rev. Lett. 99, 226803 (2007)Phys. Rev. B 80, 193411 (2009)

• Low-energy theory: Coulomb-interacting Dirac fermions

• Theoretical research: Often neglects effect of Coulomb interactionsSee however:Gonzalez et al Nucl. Phys. B 94, ibid. PRB 99, Gorbar et al PRB 06, Khveshchenko PRB 06, Son PRB 07, Das Sarma et al PRB 2007, Vafek PRL 07, Barlas et al PRL 07, Biswas et al PRB 07, Hwang et al PRL 2007,…

• Compute interaction corrections for several quantities

Renormalization Group Hertz PRB 76, Millis PRB 1993

Graphene: at a quantum critical point

• Why are interaction effects negligible in optical transparency? Nair et al Science 2008

Outline

• Graphene: One-atom thick sheet of graphite

Graphene• Single-atom thick layer of graphite

–Theory: Wallace 47, Semenoff 84

– Exp’t: Novoselov et al 2004 Zhang et al 2005

• Model:– Coulomb-interacting fermions on honeycomb lattice– Half-filled: One fermion/site

• Kinetic energy

!

H0 = "t ci# c j#i, j ,#$ †

Nearest-neighbor hopping Connects two sublattices!

a

fermions on two sublattices

!

ak" ,bk"

Next: Low energy

Low-energy theory of graphene• Kinetic energy

• Near nodes:

!

i =1,...4Spin, node

!

k << " # a$1 Next: Dirac fermions• Applies at low momenta:

!

E

!

kx!

ky• Eigenvalues ! Energy bands

!

a =1.42 Angstrom

• Two inequivalent nodes: Linear dispersion

!

E(k) " ±v |k |

!

v = 3ta /2VelocityFilled

Empty

Free fermions on the honeycomb lattice

!

H = "i(p)[vp# $]"i(p)p,i%

†• “Relativistic” low-energy Hamiltonian

Two-componentspinor

Pauli matrices.

• Condensed-matter phenomena:

Dirac fermions in B-fieldNovel Quantum Hall effect

Photoemission, Infrared spectroscopy

Zhang et al Nature 2005

Bostwick Nat. Phys 2007; Li Nat. Phys. 2008

!

v " c /300Observe Novoselov Nature 2005

• What about Coulomb interaction? Unscreened

Next: Full Hamiltonian(No Fermi surface)

• “Relativistic” quantum field theory phenomena

Zitterbewegung Katsnelson 2006

Schwinger pair production Allor et al PRD 2008

“jittery motion” of Dirac fermions - uncertainty principle

particle-hole production in an electric field

Fine structure constant

!

"QED =e2

ch=1137

• Is the Coulomb interaction important?

!

" =e2

vhDimensionless

interaction strength

(Not small!)

!

" = 300"QED # 2.2• Effective fine structure constant

Full Hamiltonian: Coulomb

!

+12

d2rd2r'" n(r)n(r ') e2

# r - r'

!

H = "i(p)[vp# $]"i(p)p,i% †

Kinetic energy

!

n(r) = "i(r)"i(r)i=1

N

# †Coulomb interaction

• Broken relativistic invariance:

Interactions: photons at velocity c

Fermions: velocity v Next: Dimensional analysis

Dimensional analysis

!

+12

d2rd2r'" n(r)n(r ') e2

# r - r'

!

H = "i(p)[vp# $]"i(p)p,i% †

Kinetic energy

!

n(r) = "i(r)"i(r)i=1

N

# †Coulomb interaction

!

EKE "#Fn " n2

!

" p =p2

2m• Usual 2-D electron gas: Kinetic energy

Kinetic energy per particle:

!

ECoulomb " n n " n3 / 2Potential energy per particle:

Interparticle spacing

Relative importance ofKinetic & Coulomb depends on density!

• Low density:

!

ECoulomb >> EKinetic Wigner crystal

!

ECoulomb << EKinetic• High density: Free electronsNext: Dirac fermions

• Dirac fermion case

!

ECoulomb " n3 / 2

– Gas of density :

!

n

!

EKinetic " n3 / 2

Relative importance ofKinetic & Coulomb independent of n!

!

+12

d2rd2r'" n(r)n(r ') e2

# r - r'

!

H = "i(p)[vp# $]"i(p)p,i% †

Kinetic energy

!

n(r) = "i(r)"i(r)i=1

N

# †Coulomb interaction

Dimensional analysis 2

– Applied chemical potential:

!

"F #1EF

– Applied Magnetic field:

!

l B "1B

• Interactions are marginal

– No typical length scale

Perturbations break this symmetry

Next: RG analysis near QCP• Quantum critical point

!

T = 0

2) Rescale momenta & fermion fields

!

"<(p) = Z#"(bp)

!

"(b) ="

1+14" lnb Iterate:

!

T(b) =Tb

1+14" lnb

• Original theory:

!

b =1

3) New theory: same as original, with new coupling & temperature

!

T

!

"

• Increase b: Trace over high-momenta

• , grows

!

"(b)#0

!

T(b)

Renormalized theory has small coupling and high temperature Next: density

Wilsonian renormalization groupSee also Gonzalez Nucl. Phys. B 94, Khveshchenko PRB 06,Son PRB 07, Herbut et al PRL 08, …

1) Trace over states in a thin high-p shell: †

!

Z = Trexp "#H[$(p),$(p)][ ]

Prescription:

!

" =1/T

!

" /b < p < "

!

"(p) =

!

">(p)

!

"<(p)

!

p < " /b Partial trace…

Effective theory for

!

"<(p)!

kx!

ky

!

"

• Chemical potential scaling:

!

µ(b) =µb

1+14" lnb

Grows under RG

Geim & Novoselov Nat. Mat. 2007

Electron density• Fermion density: adjustable via doping or external gate Vg – Impose chemical potential

!

µ" Vg

Intrinsic:

!

µ = 0,n = 0 (No Fermi surface)

Hole doped: (h Fermi surface)

!

µ < 0,n < 0Electron doped: (e Fermi surface)

!

µ > 0,n > 0

!

n(µ,T,") = b#2n(µ(b),T(b),"(b))• Density scaling relation:

What we want In renormalized system

!

"(b)

!

T(b) large

small

• Need: Choice for b renormalization condition Next: RG condition

Renormalization condition

Magnetic field

• Graphene: critical point at

!

T = µ = B = n = 0

– Terminate RG flow when perturbation reaches UV scale (bandwidth)

• Relevant perturbations: Grow under RG, leave vicinity of QCP

!

"#1 $%v 2

4T ln21+14& lnT0

T'

( )

*

+ , 2

Result:

Free Dirac Fermions Log-T enhancementNext: Density

KT 40 108!" Characteristic temp.

!

"(b*) # 0

Scaling:

!

"#1(µ,T,$) = b2 TT(b)

"#1(µ(b),T(b),$(b))

RG condition:

!

T(b*) = v"

Electronic compressibility

!

"#1 = $µ$n Assume

!

µ << T

Compressibility at low T

• Finite density, low T: RG condition

!

n(b*) = n0 " a#2

Renormalized density lattice scale

2150 104 !"# cmn

!

"#1 $ v %4 n

1+&4ln n0

n

'

( ) )

*

+ , ,

Log correction depends on density

See Also: Hwang et al PRL 07

Free Dirac Fermions

weakly parameter dependent!

Next: Recent exp’ts

• Interactions: Alter velocity in n or T dependent way

–Different experiments should measure distinct velocities

–Log prefactors: NOT small

• Note: Result is always

!

v" v 1+#4ln(...)

$

% &

'

( ) At low energies,

velocity grows

Leading-order…

Recent compressibility dataJ. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing, A. Yacoby, Nat. Phys. (2007)

• Scanning Single electron transistor: Locally measure

!

"#1 = $µ$n

Substrate: Altersdielectric const.

!

" =e2

#vh

!

"#1 $ v %4 n

1+&4ln n0

n

'

( ) )

*

+ , ,

• Interactions necessary to understand data

Graphene in vacuum

!

" = 0 Doesn’t fit….

–Best fit

!

" = 5.5• Difficult to observe ln(n) dependence

–Uncertainty in velocity Next: Heat capacity

Free energy & specific heat

!

F(T,",µ) = b#2 TT(b)

F(T(b),"(b),µ(b))• Scaling:

Vafek PRL 07RG: True low-T behavior

!

C" T 2

ln(v# /T)[ ]2 Next: Phase Diagram

electron or hole F.S.

!

C " #T

!

" #n

v 1+$4ln n0

n

%

& '

(

) *

!

C = "T #2F#T 2• Two regimes

Metallic

!

µ >> T– Fermi liquid ( )

!

C "9N#(3)2$v 2

T 2

1+14% ln(v& /T)

'

( ) *

+ ,

2– Dirac liquid ( )

!

µ << T

Perturbation theory:

!

C "9N#(3)2$v 2

T 2 1% 12& ln(v' /T)

(

) * +

, - Unphysical at low T

• RG: Sums leading logarithms

Phase diagram: Crossover behavior

!

C" T 2

log2 T

TC !"• Dashed line: Crossover between two regimes Next: Transparency

Aperture partiallycovered by graphene

Transmittance:

!

t = 97.7%

• Theory*:

!

t(") =1

(1+ 2#$(") /c)2Measures conductivity in

optical regime!

*Stauber et al PRB 2008 Next: Are interactions important?

Optical transparency of graphene

• Recent Experiments: Graphene nearly transparent

Nair et al, Science 2008: “Fine structure constant defines visual transparency of graphene”

See also Kuzmenko et alPRL 2008 -- graphite

Optical transparency of graphene

• Nair et al results: Consistent with noninteracting Dirac Fermions

!

t

!

"(nm)

• Question: Why can we neglect Coulomb interaction?

– No log prefactors in

– Small perturbative correction Next: Are interactions important?

!

" #( )!

t(") =1

(1+ #$QED /2)2• Transmission Fine structure constant!

!

" 0 =e2

4h

“Universal”Ludwig et al PRB 94!

0.98

!

0.97

!

0.96

!

400

!

500

!

600

!

700

Conductivity scaling

!

"(#,T) =1#

jx (q,#) jx ($q,$#)q%0

• Kubo formula:

current-current correlator

• Scaling of : Constrained by a Ward identity

!

jµ (q,") Gross, Les Houches, 1975

!

"(b) ="T(b)T

!

r j (q,") = b

r j R (bq,"(b))

exact

• Quantum field theory: No anomalous dimension of conserved currents

–Valid to all orders in perturbation theory

• Conductivity scaling:

!

"(#,T,$) =" (#(b),T(b),$(b))

No factor…

Next: Higher-order correction

tiny correction for ; NOT small in optical range!

!

" # 0!

"(b*) # "

1+"4ln v$ /%( )

Herbut, Juricic, Vafek, PRL 08Mishchenko Europhys. Lett. 08

!

" =e2

4h1+

c1#

1+14# ln v$ /%( )

&

'

( ( (

)

*

+ + +

• Low-T regime:

!

"(b*) = v#

• Perturbation theory in renormalized:

!

"(#,T,$) %" 0(#,T) +$(b)"1(#,T) +$(b)2" 2(#,T) + ...keep…

Conductivity of clean graphene

!

"(#,T,$) =" (#(b),T(b),$(b))• Scaling:

Previously: in renormalized theory

!

"(b) # 0

• What is the coefficient ?

!

c1Calculate some diagrams…

Next: Two different results!

!

t

!

"(nm)

!

0.98

!

0.97

!

0.96

!

400

!

500

!

600

!

700

Conductivity of clean graphene

!

"(#) =e2

4h1+

c1$

1+14$ ln v% /#( )

&

'

( ( (

)

*

+ + +

!

t(") =1

(1+ 2#$(") /c)2

Transparency:

Herbut, Juricic, Vafek:

!

c1 =25 " 6#12

$ 0.512

Mishchenko:

!

c1 =19 " 6#12

$ 0.012

Almost indistinguishable from noninteracting result

• What is the source of this discrepancy? Two ways to compute!

!

"(#) =1#Limq$0 jx (q,#) jx (%q,%#)Kubo formula:

!

"(#) = Limq$0#q2

%(q,#)%(&q,&#)Via polarization:

Next: Polarization… charge density

Diagrams inperturbation theory….

Density vertexSelf energy Vertex correction

– Diagrams separately finite for Result:

!

v"#$

!

c1 =19 " 6#12

$ 0.012(Almost indistinguishable from noninteracting result)

– Sum: Mishchenko’s result

Polarization approach

!

"(#) = Limq$0#q2

%(q,#)%(&q,&#)

Kubo formula approach

!

"(#) =1#Limq$0 jx (q,#) jx (%q,%#)

Similar Diagrams…

Current vertexSelf energy Vertex correction

– Diagrams separately divergent for Problem:

!

v"#$

– Sum is finite… depends on regulation

Ward Identity and conservation laws

Continuity:

!

" # j+ $%$t

= 0

!

Qµ jµ (Q) j" (#Q) = 0

Diagrams must satisfy!

at each order in

!

"

!

G(p,")interaction

!

V (p)propagator

How to regulate divergent diagrams?

!

Q = (",qx,qy )

!

" = j0DEFINE

and

!

Qµ jµ (Q) = 0

Restrict momenta of Green functions:

!

G(p,")#G(p,")$(% & p)

step function

!

Qµ jµ (Q) j" (#Q) $ 0 Ward identity violated

(simple to show!)

!

c1 =25 " 6#12

$ 0.512

Conductivity has

Restrict momenta of Coulomb interaction:

!

Qµ jµ (Q) j" (#Q) = 0

!

c1 =19 " 6#12

$ 0.012Ward identity preserved

!

V (p)"V (p)#($ % p)

Agrees with polarization approach….

Optical transparency: Best hope for observing Coulomb…

!

" =e2

4h1+

c1#

1+14# ln v$ /%( )

&

'

( ( (

)

*

+ + +

• High frequency: Coupling close to bare value

!

"(#)

!

" # 2.2

• Unfortunately:

!

c1 " 0.0125 Probably not observable!

Mishchenko : Europhys. Lett. 08

• Kubo formula: Sum of divergent diagrams depends on regularization

• Restrict momenta in Green functions Violates charge conservation

• Restrict transferred momenta Coulomb Obeys conservation

• Modify Coulomb

!

e2

r"

e2

r1#$

!

" # 0 Obeys conservation with

Phys. Rev. B 80, 193411 (2009)

Next: Concluding Remarks

Concluding remarks

• Graphene: Dirac fermions with Coulomb interaction (Marginal)

– Interaction effects: Log corrections to free case (Dirac fermions)

– Regularization method must preserve conservation laws

• We resolved discrepancy with earlier Kubo formula calculation

• Optical transparency probes conductivitylarge!

!

"

!

" =e2

4h1+

c1#

1+14# ln v$ /%( )

&

'

( ( (

)

*

+ + +

!

c1 =19 " 6#12

$ 0.012ButSmall correction!

Mishchenko Europhys. Lett. 2008

– Interactions only renormalize velocity:

!

v" v 1+#4ln v$ /T[ ]

%

& '

(

) *

• Renormalization group: Scaling equations for various quantities

– Specific heat, compressibility, diamagnetic susceptibility, dielectric function, …

(Leading order)