daniel e. sheehy - lsu · coulomb interaction in graphene daniel e. sheehy work in collaboration...
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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)