Page 1

arXiv:0801.1291v2 [cond-mat.mtrl-sci] 18 Jul 2008

Chirality-induced Dynamic Kohn Anomalies in Graphene

Wang-Kong Tse1, Ben Yu-Kuang Hu1,2, and S. Das Sarma1

1Condensed Matter Theory Center, Department of Physics,

University of Maryland, College Park, Maryland 20742 and

2Department of Physics, University of Akron, Akron, Ohio 44325-4001

We develop a theory for the renormalization of the phonon energy dispersion in graphene due

to the combined effects of both Coulomb and electron-phonon (e-ph) interactions. We obtain the

renormalized phonon energy spectrum by an exact analytic derivation of the phonon self-energy,

finding three distinct Kohn anomalies (KAs) at the phonon wavevector q = ω/v,2kF ± ω/v for LO

phonons and one at q = ω/v for TO phonons. The presence of these new KAs in graphene, in

contrast to the usual KA q = 2kF in ordinary metals, originates from the dynamical screening of

e-ph interaction (with a concomitant breakdown of the Born-Oppenheimer approximation) and the

peculiar chirality of the graphene e-ph coupling.

PACS numbers: 63.22.-m,63.20.D-,63.20.kd,71.10.-w

Graphene, a two-dimensional (2D) sheet of graphite

comprising a planar hexagonal lattice of carbon atoms,

has attracted a great deal of recent interest with enor-

mous amount of experimental and theoretical activity

since its discovery. Of fundamental significance is the

study of the interaction between electrons and phonons

in this novel system. In particular, the many-body renor-

malization of the phonon spectrum due to the electron-

phonon (e-ph) interaction is an important open question.

Experimentally, Raman scattering offers a means to mea-

sure the long-wavelength phonon energy whereas X-ray

and neutron scattering provide an avenue to probe the

entire phonon energy dispersion.

induced many-body interactions, one expects the ob-

served phonon energy in a doped system to be differ-

ent from the “bare” phonon energy in an undoped sys-

tem. Physically, electrons respond to the dynamical lat-

tice vibrations by screening the lattice potential, chang-

ing the elastic constants of these vibrational modes and

thereby renormalizing the phonon energy. Recently, a

number of Raman scattering experiments [1] on extrin-

sic graphene (where the electron density can be tuned

by gating) have emerged, with the observed density de-

pendence of the Raman shift (for the long-wavelength

G-band optical phonons at the Γ point) pointing to the

inapplicability of the Born-Oppenheimer approximation

(BA) [2] in graphene. This density dependence has been

addressed for the long-wavelength phonons (i.e. q = 0)

using perturbation theory [1, 3] and density-functional

theory [4].

Due to free carrier-

An interesting question remains as to whether the

Kohn anomaly (KA) [5], which appears as a cusp in the

phonon energy dispersion at q = 2kF for ordinary 2D

metals, will be substantially modified in graphene due

to its quasi-relativistic chiral band structure. The occur-

rence of KA is entirely a many-body effect as it originates

from the screening of the e-ph interaction by electrons,

and correpsonds to the singularities of the phonon self-

energy or its derivatives as a function of q. For graphite,

the KA was recently studied in Ref. [6] as a function of

q at zero doping. Ref. [4] addresses the phonon energy

renormalization in doped graphene, but only at q = 0

relevant to Raman scattering experiments.

ally, KA is probed experimentally with X-ray or neu-

tron scattering spectroscopy by measuring the phonon

energy as a function of q, for which a systematic theory

for the phonon energy renormalization in graphene for

finite wavevector q > 0 is still lacking.

Tradition-

In this Letter, we present a theory for the renormaliza-

tion of the phonon energy dispersion in graphene due to

both Coulomb and e-ph interaction effects. From our the-

ory, we obtain two major new results: (1) We find that

direct Coulomb and phonon-mediated electron-electron

interactions decouple to all orders of perturbation theory

within the random-phase approximation (RPA), and the

electronic collective plasmon mode does not contribute

to the phonon energy renormalization; (2) we obtain the

renormalized phonon energy dispersion as a function of

q, predicting the occurrence of three distinct KAs at the

phonon wavevector q = ω/v,2kF±ω/v for the LO mode

and one at q = ω/v for the TO mode, which arise from

the chiral structure of the graphene e-ph coupling. The

novel pecularity that these KAs do not occur at q = 2kF

(as in usual metals) originates from the fact that the

phonon dynamics cannot be neglected in the screened e-

ph interaction, indicating the inapplicability of the BA.

Graphene, behaving as a 2D zero-gap semiconductor,

has a G-band optical phonon energy ω0 = 200meV at

the Γ point which is comparable to the Fermi energy

εF ∼ 110 − 370meV at the usual extrinsic carrier den-

sity n = 1012− 1013cm−2. This approximate equality of

the phonon and electron energy scales ω0∼ εF implies

a breakdown of the static approximation for the phonon

degree of freedom [7], naturally explaining the violation

of the BA in the recent Raman scattering experiments

[1] since the BA implicitly assumes the phonon dynam-

ics to be much slower than the electron dynamics. In

addition, plasmon-phonon coupling, which has been ex-

Page 2

2

tensively studied in doped semiconductor systems (e.g.,

GaAs, SiC), is expected to occur whenever the phonon

dynamics is at a comparable time scale as the electron

motion. It follows that one cannot take the screening

of the e-ph interaction to be simply static while keep-

ing the screening of the Coulomb interaction to be dy-

namic; instead, one has to take into account the dynami-

cal screening of both Coulomb and e-ph interactions, i.e.,

direct electron-electron (e-e) and e-ph interactions must

be treated on an equal footing [8].

Near the Brillouin zone corner K (i.e. the Dirac point),

graphene is described by the effective chiral Hamiltonian

H = vσ·k, where v ≈ 106ms−1is the quasiparticle veloc-

ity and σ is the set of Pauli matrices describing the two

A and B sublattice degrees of freedom. The quasiparti-

cle energy dispersion as obtained from this Hamiltonian

is ǫkλ= λǫk, where ǫk= vk and λ is the chirality label

representing the conduction band (λ = 1) and valence

band (λ = −1); the corresponding eigenstate is denoted

as |kλ?. The e-ph interaction vertex for optical phonons

is given by [3, 9, 10] gM, where g is the coupling constant

characterizing the magnitude of the e-ph interaction, and

M is the off-diagonal matrix

M(q) =

?

0MABe−iφq

0

MBAeiφq

?

, (1)

with MAB= −1 or i and MBA= 1 or i for LO or TO

phonons, respectively, and φq= tan−1(qy/qx) the azith-

muthal angle of the momentum q. The chiral structure

of the e-ph coupling is peculiar to graphene, describing

the bond stretching and bending between neighbouring

carbon atoms of the A-sublattice and B-sublattice.

In the presence of direct e-e interaction and e-

ph interaction, the phonon Green function is renor-

malized by both, as represented in the diagram-

matic language in Fig. 1.

grams leads to the renormalized phonon Green func-

tion D(q,ω) = D0(q,ω)/[1 − D0(q,ω)Πpp(q,ω)], where

D0(q,ω) = 2ω0/(ω2−ω2

function with ω0the optical phonon energy, Πpp(q,ω) is

the RPA-screened phonon self-energy given by the fol-

lowing equation [Fig. 1(b)]:

Summation of these dia-

0+i0+) is the bare phonon Green

Πpp(q,ω) = Πpp

0(q,ω)+Πpc

0(q,ω)Vc

ee(q,ω)Πcp

0(q,ω), (2)

where Vc

screened Coulomb interaction [11] (here Πcc

tronic polarizability), Πpp

energy, and Πpc

bubbles with one Coulomb interaction vertex and one

e-ph interaction vertex.In regular metals or doped

polar semiconductors, the phonon self-energy and the

hydrid bubbles are the same (up to a factor given by

the e-ph coupling constant) as the electronic polarizabil-

ity; the Dyson equation for the renormalized phonon

Green function (Fig. 1) simply reduces to the usual

ee(q,ω) = Vq/[1−VqΠcc

0(q,ω)] is the usual RPA-

0is the elec-

0(q,ω) the bare phonon self-

0(q,ω), Πcp

0(q,ω) are the bare “hybrid”

(a)

(b)

FIG. 1: (Color online) (a) Dyson equation for the renormal-

ization of the phonon Green function. The zigzag lines denote

the phonon Green function and the crosses denote the e-ph in-

teraction vertex. The shaded bubble with two cross vertices

stands for the renormalized phonon self-energy. (b) Equa-

tion for the renormalized phonon self-energy. The unshaded

bubble with two cross vertices denotes the bare phonon self-

energy, the two bubbles with one cross vertex and one dot

vertex are “hybrid bubbles” with one e-ph interaction vertex

and one Coulomb interaction vertex. The wavy line stands

for the usual RPA-screened Coulomb interaction.

RPA series. In graphene, due to the presence of a chi-

ral structure of the e-ph interaction Eq. (1), the bub-

bles Πpp

0 are not equal to one an-

other. In particular, Πpc

0

and Πcp

pling effects of the Coulomb and e-ph interactions within

the RPA, and describe the renormalization effect of the

phonon energy by the Coulomb interaction.

ingly, we find that with the graphene e-ph interaction

Eq. (1), these hybrid bubbles vanish identically: Πpc

Πcp

0

=kBTg?

qλ′|M(q)|kλ??kλ|k + qλ′? = 0, where G0

1/(ikn− ξkλ) is the quasiparticle Green function with

ξkλ = ǫkλ− εF the quasiparticle energy rendered from

the Fermi level. Eq. (2) then implies that Πppis simply

given by the bare phonon self-energy Πpp

interaction does not contribute to the screening of the e-

ph interaction within the RPA. The phonon energy dis-

persion, which is given by the pole of the real part of the

renormalized phonon Green function,

0, Πpc

0, Πcp

0, and Πcc

0

incorporate the cou-

Interest-

0 =

ikn,kG0

kλ(ikn)G0

k+qλ′(ikn + iqn)?k +

kλ(ikn) =

0, and Coulomb

ω2= ω2

0+ 2ω0ReΠpp

0(q,ω), (3)

is therefore only renormalized by the e-ph interaction

but not by Coulomb interaction.

pled plasmon-phonon modes do not arise in graphene,

with the phonon and plasmon modes having separate

branches of energy dispersion in the ω − q phase space

despite comparable energy scales for the phonon and

electron dynamics.In addition, direct Coulomb and

phonon-mediated e-e interactions are simply additive,

with the RPA-screened total e-e interaction given by

It follows that cou-

Page 3

3

Vtot

ee

is the unscreened phonon-mediated e-e interaction [10].

= Vq/[1 − VqΠcc

0] + Vph

ee/[1 − D0Πpp

0], where Vph

ee

0

0

1

1

2

2

3

3

4

4

q/kF

−0.8

−0.4

0

−0.75

−0.25

0.25

0.75

1.25

Π

pp

+(q,ω0)

LO

TO

FIG. 2: Dynamic phonon self-energy˜Πpp

q/kF at u = ω0/εF for the LO (solid line) and TO (dashed)

modes at a density n = 1013cm−2. Inset: Static phonon self-

energy˜Πpp

+(x,u) versus x = q/kF at u = 0.

+(x,u) versus x =

The phonon self-energy Πpp= Πpp

from the e-ph interaction vertex Eq. 1 as:

0can then be derived

Πpp(q,iqn) = 41

2g2?

[1 ∓ λλ′cos(φk+q+ φk)],

λλ′

?

k

nF(ξkλ) − nF(ξk+qλ′)

iqn+ ξkλ− ξk+qλ′

(4)

where the sign −(+) corresponds to LO(TO) phonons,

the factor of 4 counts the spin and valley de-

generacies, and φk

= tan−1(ky/kx) is the azith-

muthal angle of the momentum k measured from

q.The real part of the phonon self-energy com-

prises two contributions ReΠpp≡ Πpp

pingthe notation ‘Re’

after), where Πpp

µ

= 2g2?

fµλ(k,q) = [1 ∓ λcos(φk+q+ φk)]/[ω + µξk+− ξk+qλ]−

[1 ∓ λcos(φk+ φk−q)]/[ω − µξk++ ξk−qλ]. The contri-

bution Πpp

+is due to the extrinsic conduction band elec-

trons whereas Πpp

−is solely due to the intrinsic valence

band electrons. In the renormalization of the phonon

energy in extrinsic graphene, Πpp

from the total Πppto avoid overcounting of the intrinsic

contribution, since the “bare” phonon energy (i.e. when

graphene is undoped), by definition, already includes the

effect of Πpp

into account in the phonon energy renormalization by

free carriers in extrinsic graphene. We also note that,

for Πpp

+, the largest energy scale is the Fermi energy εF

whereas for Πpp

−it is much higher, of the order of the

cutoff energy Λ = 2π?v/a for the graphene linear band

dispersion (a = 2.46˚ A is the graphene lattice spacing).

Therefore, although the BA is inapplicable for extrinsic

graphene with ω0/εF ∼ O(1), for intrinsic graphene the

BA is strictly valid since ω0/Λ ≪ 1.

++ Πpp

real

−

(drop-

there- forthe

knF(ξkµ)fµλ(k,q) with

part

λ

?

−should be subtracted

−[3, 9]. Therefore, only Πpp

+should be taken

For clarity, we express our results in the follow-

ing dimensionless quantities:

energy u = ω/εF, and phonon self-energy ˜Πpp

Πpp/(2g2

phonon-mediated e-e coupling constant (A is the sample

area). We have obtained the following asymptotic re-

sults for the phonon self-energy in the limit q/kF ≪ 1:

˜Πpp

LO,TO+(x,u) = (1/2)[1 + (u/4)ln|(2 − u)/(2 + u)|] +

∆˜Πpp

LO,TO+(x,u), where the next order correction to the

long wavelength result is given by,

wavevector x = q/kF,

=

eeεF/π) with g2

ee= g2A/?2v2the dimensionless

∆˜Πpp

LO+(x,u) =

1

2

1

2

?8 − 4u2− u4

?24 − 12u2+ u4

2u2(u2− 4)2

2u2(u2− 4)2−

1

8uln

????

2 − u

2 + u

????

????

?

x2, (5)

∆˜Πpp

TO+(x,u) =

+

1

8uln

2 − u

2 + u

????

?

x2.

(6)

At x = 0, the long wavelength result [1, 4, 12] is the

same for LO and TO phonons, the two phonon modes

being degenerate at the Γ point. At finite wavevector

x, this degeneracy is lifted with the leading-order cor-

rection going as x2given by Eqs. (5)-(6). We have also

obtained the following analytic results for the phonon

self-energy in the static case u = 0:˜Πpp

˜Πpp

xtan−1(2/√x2− 4)]θ(x − 2).

The above static phonon self-energy results are de-

picted in the inset of Fig. 2, which clearly shows the

presence of a non-analyticity at q = 2kF corresponding

to the KA for LO phonons. We also note that this non-

analyticity is entirely absent in the static electronic polar-

izability [13] of graphene. This is in contrast to the situa-

tion for regular materials with a parabolic energy disper-

sion where the phonon self-energy and the polarizability

are equal up to a factor given by the e-ph coupling. This

distinctive difference between the phonon self-energy and

the polarizability in graphene is a direct result of the pres-

ence of a chiral structure in the graphene e-ph coupling

Eq. (1), which leads to a different Berry phase depen-

dence in the expression of the phonon self-energy Eq. (4)

compared with the polarizability [13]. Therefore, KAs in

graphene originate entirely from the special chiral struc-

ture of the e-ph coupling Eq. (1).

We have evaluated the full expression for Πpp

eral q and ω dependence analytically [14], which is how-

ever too cumbersome to be shown here. The main plot of

Fig. 2 shows the evaluated Πpp

phonon energy ω = ω0, from which three cusps occurring

at vq = ω0, vq = 2εF± ω0for the LO mode are clearly

discernible. For the TO mode, there is a divergence of

Πpp

+at vq = ω0. These non-analyticities correspond to

the values of q where the denominator of the integrand of

Πpp

+vanishes at the Fermi surface k = kF, i.e. the zeros

of the equation ω ± vkF± v|kF± q| = 0.

With the calculated phonon self-energy, the renormal-

ized phonon energy spectrum can be obtained by self-

TO+(x,0) = 0, and

LO+(x,0) = −(π/8)xθ(2 − x) + (1/4)[(2/x)√x2− 4 −

+with gen-

+as a function of x at the

Page 4

4

consistently solving Eq. (3) for ω, which is shown in

Fig. 3 for the LO mode and Fig. 4 for the TO mode.

For LO phonons, three KAs which correspond to the

non-analyticities of Πpp

+ are evident, occurring at the

wavevector vq = ω0, vq = 2εF± ω0. For TO phonons,

the divergence of Πpp

+at vq = ω0is removed due to the

self-consistency condition for ω in Eq. (3), but the KA

remains as a sharp but finite peak at vq = ω0. In addi-

tion, we note the KA at vq = ω0 for both the LO and

TO modes, unlike the other two KAs for the LO mode,

is independent of electron density, an interesting conse-

quence of the quasi-relativistic linear dispersion peculiar

to graphene. For both LO and TO phonons, our re-

sults suggest that the phonon energy first increases (i.e.,

phonon hardening) with density up to a certain phonon

wavevector, and then decreases (i.e., phonon softening)

with density. The critical wavevector for this transition

from phonon hardening to softening is different for LO

and TO phonons, and we find numerically q ≃ 5×108m−1

for LO phonons and q = ω0/v = 3 × 108m−1(i.e., the

KA) for TO phonons.

0 0.51.0

9 m

1.5 2.0

q (10

−1)

0.194

0.195

0.196

0.197

0.198

0.199

0.2

ωph (eV)

4 x 10

8 x 10

1.2 x 10

12 cm

12 cm

13 cm

−2

−2

−2

FIG. 3: Renormalized LO phonon energy spectrum ωphversus

q at different electron densities n. The bare phonon energy

is shown as the horizontal solid line. The range of phonon

wavevector q shown corresponds to [0,0.08(2π/a)] away from

the Γ point.

In the following, we provide a schematic picture for

understanding the occurrence of the KAs. The inset of

Fig. 4 shows the ω−q phase space in which single-particle

excitation can occur through virtual phonon exchange.

The behavior of the phonon self-energy is characterized

by six different regions of the ω−q phase space where Πpp

is analytically continuous, separated by the boundaries

(indicated by the solid lines) corresponding to the set

of values of (ω,q) where Πpp

+is non-analytic. Necessary

(but by no means sufficient) conditions for the KAs to

occur are given by the intersection points between the

phonon dispersion line ω = ω0and the boundaries for the

different regions. Within the BA, phonons are treated as

static with ω = 0 in the expression of the phonon self-

energy, the only intersection points therefore occur at q =

+

0 0.51.0

9

m

1.52.0

q (10

0123

−1

4

q/k

)

F

0

2

4

ω / εF

0.194

0.196

0.198

0.2

0.202

0.204

0.206

ωph (eV)

FIG. 4: Renormalized TO phonon energy spectrum ωph ver-

sus q. The legends are the same as in Fig. 3. Inset: Different

regions for the analytical behavior of Πpp

dicate the boundaries for these regions, and the dashed line

shows the phonon energy ω = ω0 at a density n = 1013cm−2.

+. The solid lines in-

0 and q = 2kF, as in the case of, e.g., a usual metal. In

graphene where the BA is invalid, phonon dynamics must

be included with ω = ω0in the phonon self-energy, the

intersection points now occur at vq = ω0, vq = 2εF−ω0,

and vq = 2εF + ω0. For the LO mode, we find that

KAs occur at all three intersection points. This however

does not apply for the TO mode, and we find two equal

but opposite contributions in the expression for Πpp

which cancel the effects of two KAs, yielding only one

KA at vq = ω0in this case. In ordinary metals, the KA

q = 2kF corresponds to backscattering with maximum

phonon wavevectorq [5]; this KA becomes q = 2kF∓ω0/v

in graphene because of the phonon non-adiabaticity, with

q = 2kF− ω0/v,2kF+ ω0/v corresponding, respectively,

to emission and absorption of a phonon through electron

backscattering.The KA q = ω0/v does not have an

analogue in ordinary metals, and corresponds to forward

scattering of the electron through absorption of a phonon.

TO+

In conclusion, we have developed a theory for the

interaction-induced phonon renormalization in graphene,

and discovered new and multiple KAs in the renormalized

phonon dispersion. The peculiarity and the distinction of

these KAs from the usual KAs in metals are signatures

of the renormalized dynamically screened e-ph interac-

tion and the special chiral structure of the e-ph coupling

in graphene.The graphene phonon energy dispersion

can be measured with double resonance Raman scatter-

ing or electron energy loss spectroscopy on a monolayer

graphene, and the experimental verification of our pre-

dictions would establish that graphene has a very unique

e-ph many-body coupling.

This work is supported by US-ONR, NSF-NRI, and

SWAN SRC.

Page 5

5

[1] J. Yan et al., Phys. Rev. Lett. 98, 166802 (2007); S.

Pisana et al., Nat. Mater. 6, 198 (2007); A. Das et al.,

arXiv:0709.1174v1; J. Yan et al., arXiv:0712.3879v1.

[2] The BA (also going by the name of static or adiabatic

approximation) requires the phonon energy to be much

smaller than the characteristic energy scale for the elec-

trons, usually the Fermi energy.

[3] T. Ando, J. Phys. Soc. Jpn. 75, 124701 (2006).

[4] M. Lazzeri and F. Mauri, Phys. Rev. Lett. 97, 266407

(2006).

[5] W. Kohn, Phys. Rev. Lett. 2, 393 (1959).

[6] S. Piscanec et al., Phys. Rev. Lett. 93, 185503 (2004).

[7] In this connection, we note that the BA also breaks down

in usual doped semiconductors at low enough densities

(i.e., εF ∼ ω0) when plasmon-phonon coupling becomes

important.

[8] R. Jalabert and S. Das Sarma, Phys. Rev. B 40, 9723

(1989).

[9] H. Suzuura and T. Ando, Phys. Rev. B 65, 235412

(2002).

[10] W.-K. Tse and S. Das Sarma, Phys. Rev. Lett. 99,

236802 (2007).

[11] S. Das Sarma, E.H. Hwang, and W.-K. Tse, Phys. Rev.

B 75, 121406(R) (2007).

[12] A.H. Castro Neto and F. Guinea, Phys. Rev. B 75,

045404 (2007).

[13] E.H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418

(2007).

[14] W.-K. Tse, unpublished.