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VOLUME 85, NUMBER 24 PHYSICAL REVIEW LETTERS11 DECEMBER 2000

Double Resonant Raman Scattering in Graphite

C. Thomsen and S. Reich

Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany

(Received 9 August 2000)

We find that the electronic dispersion in graphite gives rise to double resonant Raman scattering for

excitation energies up to 5 eV. As we show, the curious excitation-energy dependence of the graphite

D mode is due to this double resonant process resolving a long-standing problem in the literature and

invalidating recent attempts to explain this phenomenon. Our calculation for the D-mode frequency shift

(60 cm21?eV) agrees well with the experimental value.

PACS numbers: 78.30.–j, 81.05.Tp

Single incoming or outgoing resonances are widely

known in Raman spectroscopy and frequently used to

study the electronic and vibrational properties of crystals

or molecules. They occur if the energy of the incoming or

the scattered photon matches the transition energy of an al-

lowed electronic transition leading to a large enhancement

of the Raman cross section [1]. Closely related is the idea

of double resonant Raman scattering, where, in addition to

the incoming or outgoing resonances, the elementary exci-

tation makes a real transition. Double resonances are much

stronger than single resonances. They were, however, only

observed under very specific experimental conditions:

The energetic difference between two electronic bands

was adjusted to the phonon energy by applying electric

or magnetic fields, uniaxial stress, or by a proper choice

of the parameters of semiconductor quantum wells [2–5].

Double resonant conditions were thereby realized for

distinct excitation energies.

In this Letter we study the double resonant Raman pro-

cess for linearly dispersive bands as in semimetals. We

show that double resonances are responsible for the ob-

servation of the defect induced D mode in graphite and

its peculiar dependence on excitation energy. The double

resonance considered has a much stronger enhancement

than simple incoming or outgoing resonances explaining

why the defect mode (and its second order peak) is so

strong compared to the graphite G point vibration.

The first order Raman spectra of graphite show besides

the G point modes an additional defect induced peak, the

so-called D mode [6–9]. The D mode is related to the

finite crystallite size and disappears for perfect crystals

[6,10]. Its frequency was found to shift with excitation

energy at a rate of 40 50 cm21?eV over a wide excita-

tion energy range [7,11–13], a phenomenon which has

not been understood for almost 20 years. Tan et al. found

that there is a curious discrepancy between the Stokes and

anti-Stokes frequencies of the D mode, which they were

unable to explain [14]. Various groups have recently at-

tempted to explain the unusual excitation-energy depen-

dence which was found also for the second order spectra,

where the mode shifts at approximately twice the rate and

is not defect induced. Sood et al. proposed a disorder in-

duced double resonance above a gap D ? 1 eV in the band

structure leading to a dependence of the phonon wave vec-

tor q and hence the phonon frequency on the energy of

the incoming light E1 as q ? ?E12 D?1?2[12]. There

is, however, no such gap in the electronic structure of

graphite; it is a semimetal with valence and conduction

band crossing the Fermi level at the K point of the Bril-

louin zone. Pócsik et al. introduced a new Raman mecha-

nism for which the wave vector of the electron which is

excited by the incoming resonant photon supposedly de-

fines the wave vector of the scattered phonon [13]. This

ad hoc k ? q quasiselection rule was applied to particu-

lar branches of the phonon band structure by Matthews

et al. and Ferrari and Robertson [15,16]. However, these

explanations required a mysterious coupling of the opti-

cal branches to a transverse acoustic branch in the phonon

band structure or did not yield the correct shift of the D

mode. Single resonances have not been identified in the

graphite Raman spectra because the linear dispersion of

the electronic bands allows these resonances to occur at

all energies E1and for the entire D band independently of

q. There is no reason why the quasimomentum where the

electronic transitions occur is transferred selectively to the

phonon seen in the Raman spectra, and it is impossible to

understand the difference between Stokes and anti-Stokes

frequencies. In other words, the “quasiselection rule” in-

voked by Refs. [13,15,16] has no physical basis and cannot

explain the experimental observations.

To study the resonant Raman effect in a semimetal like

graphite we first consider a one-dimensional example as

depicted in Fig. 1, where we have shown two linear bands

with different Fermi velocities which cross at the Fermi

level. The peculiarity of this electronic dispersion is that,

in addition to single resonances, a double resonant transi-

tion is possible for a wide variety of excitation energies.

The first step of the double resonance for a particular inci-

dent laser energy E1is to create an electron-hole pair at the

k point matching the energy difference between the con-

duction and the valence band (i ! a). It is obvious that

for a monotonically increasing phonon dispersion vph?q?

there exists a (vph,q) combination which can scatter the

electron to a state on the second band (a ! b).This

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VOLUME 85, NUMBER 24 PHYSICAL REVIEW LETTERS 11 DECEMBER 2000

is the phonon for which double resonance occurs. Then

the electron is elastically scattered back by the lattice de-

fects (b ! c) and recombines conserving k in the process

(c ! i). For a larger incoming photon energy a larger

phonon quasimomentum is required for this transition and

hence a different phonon energy, if the phonon band is dis-

persive. The double resonance occurs also for two emitted

phonons at twice the energy where quasimomentum is con-

served by equal and opposite q of the two emitted phonons,

i.e., it is not defect induced. The anti-Stokes process for a

particular incoming energy is seen to be doubly resonant

at a larger phonon momentum (and hence energy) than the

Stokes process.

For the linear bands in Fig. 1 we can calculate the Ra-

man matrix element K2f,10 explicitly by evaluating the

usual expression [1]:

K2f,10? MfMbaMcbMo

Here Mo,f are the (constant) transition matrix elements

for the incoming and outgoing photons, Mba,Mcbrep-

resents the phonon or impurity which scatters the elec-

tron from state a to b and from state b to c, and g,

the broading parameter, has been taken to be the same

for all transitions. For a semimetal the electronic en-

ergies are Ee

Ee

Fermi velocities. In one dimension the sum may be con-

verted to an integral over k:

X

a,b,c

1

bi2 i ¯ hg??E12 ¯ hvph2 Ee

?E12 Ee

ai2 i ¯ hg??E12 ¯ hvph2 Ee

ci2 i ¯ hg?. (1)

ai? jkj?y22 y1? and Ee

ci? jkj?y22 y1? with y1, 0 and y2. 0 being the

bi? 7qy1?2, and

K2f,10?

MfMbaMcbMo?2k22 q?

?y22 y1?3?k22 q

Z `

0

y2

y22y1??k21 q

y1

y22y1?

3

dk

?k12 k??k22 k?,(2)

with

¯ hvph2 i ¯ hg???y22 y1?.

wardly evaluated to

k1? ?E12 i ¯ hg???y22 y1?

and

k2? ?E12

This integral is straightfor-

K2f,10?

aMfMbaMcbMo

?k22 q

y2

y22y1??k21 q

y1

y22y1?, (3)

c

b

a

i

V1

V2

∆ωph

k

E

FIG. 1.

for linearly dispersive bands with Fermi velocities y1 and y2.

For a given incident energy E1and a monotonically increasing

phonon dispersion relation ¯ hvph?q? there is at most one q and

¯ hvphleading to double resonance. In this example the electron

is scattered back to band 1 by an impurity.

Schematic Raman double resonance in one dimension

where a ? ln?k2?k1??2k22 q????y22 y1?2¯ hvph? is a

slowly varying function of q. Physically Eq. (3) says that

there is double resonance when the phonon quasimomen-

tum is equal to

q ?E12 ¯ hvph

y2

Thedouble resonanceis particularlystrongsincetwoterms

in the denominator of Eq. (1) go to zero simultaneously,

and much stronger than the incoming resonance consid-

ered by Refs. [13,15,16], where only one term in the de-

nominator of Eq. (1) vanishes.

In Fig. 2 we plot jK2f,10j of Eq. (3) as a function of

phonon quasimomentum for two incoming photon ener-

gies. For each photon energy there are two maxima, their

separation depending on the Fermi velocities [Eq. (4)]. For

different incoming E1the resonances occur at different q

again as given by Eq. (4); for E1between 1 and 4 eV the

center q varies between 0.13 and 0.6 Å21(we chose as

Fermi velocities y1? 27 and y2? 6 eVÅ adapted from

graphite) as compared to ?0.003 Å21for a nondefect in-

duced process excited at 3 eV. For a dispersive phonon,

if it is single valued and increasing with q, there are two

or

E12 ¯ hvph

2y1

. (4)

0.0 0.20.4 0.60.8

E1= 3 eV

E1= 2 eV

|K2f,10| (arb. units)

qph (Å

-1)

FIG. 2.

in Eq. (3) for two incident photon energies and a model dis-

persion. See text for details. By far the strongest enhancement

occurs for the double resonance as given by Eq. (4).

The absolute magnitude of the Raman matrix element

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VOLUME 85, NUMBER 24 PHYSICAL REVIEW LETTERS11 DECEMBER 2000

pairs ?q,vph? which fulfill condition (4). To give a rough

estimate for a hypothetical linear dispersion of an opti-

cal phonon we adapted from the K and G point values

of graphite ¯ hvK

the double resonance for an incoming photon energy at

E1? 3 eV according to Eq. (4) occurs for the center q ?

0.44 Å21, and hence ¯ hvph? 1350 cm21is the phonon

energy actually observed. For a different E1the phonon

energy shifts accordingly. Note the large phonon quasi-

momentum which can normally be observed only with

neutron scattering. A difference in Stokes and anti-Stokes

frequencies follows naturally from Eq. (4) when replacing

vphby 2vph.

ph? 1270 cm21and ¯ hvG

ph? 1580 cm21;

After illustrating the principle of a double resonance

with linearly dispersive electronic bands near the Fermi

surface we now turn to a two-dimensional realistic de-

scription of the bands in graphite and demonstrate that our

interpretation explains quantitatively the experimental de-

pendence of the D mode in graphite on excitation energy.

For the electronic bands it is sufficient to consider only the

bonding and antibonding bands which, up to about 6 eV,

are the only bands involved in real optical transitions. The

asymmetries between bonding (lower sign) and antibond-

ing (upper sign) states we took into account [17] according

to Ec,y? 6g0w?k???1 7 sw?k??, where w?k? is the tight

binding band centered around the K point of the Brillouin

zone of graphite [18]:

w?k? ? ?3 2 cosk ? R12 cosk ? R22 cosk ? ?R12 R2?

1

where R1,2are the unit vectors of the graphene cell, g0?

3.03 eV, and s ? 0.129 for a close approximation to the

graphite band structure [17]. In the isotropic limit (for

small k and neglecting s) the incoming transition occurs

at Ee

photon energies in the visible and higher the incoming

resonance does not occur close to the K point, and the

anisotropyofthebandstructuremustbetakenintoaccount.

For example, for k ? 0.23 Å21in the KM direction Ee

2.7 eV, while in the KG direction Ee

To obtain a realistic as possible two-dimensional nu-

merical evaluation of the sum in Eq. (1) we included for

Ee

and integrated jK2f,10j2over k space.

dispersion in the energy range (1270, 1620 cm21) we

modeled by simple functions such that they represent

closely the ones calculated from force constant [19] or

ab initio [20] calculations.In particular, we fixed the

G point frequency to 1580 cm21, the M and K point

ones to 1480 and 1270 cm21, respectively, and included

the overbending to 1620 cm21typical for the vicinity of

the G point in graphite [10]. Our results are not depen-

dent on the details of these curves. We then proceeded

to evaluate Eq. (1) by searching the Brillouin zone for

incoming resonances and, where we found them, by in-

tegrating ?j? ? ?E1?ko? 2 ?Ec?kf? 2 Ey?ko?? 2 ¯ hvph?kf2

ko? 2 i ¯ hg? ? ??¯ hvph?kf2 ko? 2 i ¯ hg?j2?21over kfin the

entire Brillouin zone.

We plotted the Raman intensities jK2f,10j2as a function

of phonon frequency vphfor three different incident laser

energies in Fig. 3. The peak is seen to shift to higher en-

ergies with increasing E1. In Fig. 4 we show the maxima

of the intensities as a function of incident energy together

with the experimental results on the D-mode’s excitation

dependence of various groups. The agreement is found to

be excellent, in particular, in view of the fact that no un-

known parameter was introduced in our derivation and no

fitting was performed. The slope for the D-mode’s exci-

tation energy dependence calculated by our model if taken

p3sink ? R12

p3sink ? R22

p3sink ? ?R12 R2??1?2, (5)

ai? Ec2 Ey?p3a0g0k. Note, though, that for

ai?

ai? 3.2 eV.

?a,b,c?ithe full electronic dispersion curves of Eq. (5)

The phonon

to be linear is 60 cm21?eV, the experimental slopes being

slightly lower, ranging from 46 to 51 cm21?eV [11,13,15].

The absolute values of the D-mode frequencies agree also

excellently.

The relative strength of single and double resonances

may be determined by the relative area A of the two-

phonon peaks of the G point mode and the D mode, which

are both Raman allowed. Experimentally we find A2D?

A2G? 40 which independently confirms our interpreta-

tion of the shifting D mode as due to a double resonance.

We obtained the difference in Stokes and anti-Stokes fre-

quencies as ?15 cm21(for E1? 2 eV) slightly depend-

ing on E1compared to ?7 cm21as reported in Ref. [14].

The remaining discrepancy suggests a somewhat too small

slope of our model phonon dispersions or a slightly too

large electron dispersion.

In conclusion, we investigated Raman double reso-

nances, a mechanism which leads to a strong Raman

signal at variable q vectors well within the Brillouin zone

of solids. Our interpretation resolves the long-time not

1300 1400 1500

E1 = 4 eV

E1 = 3 eV

E1 = 2 eV

|K2f,10|2 (arb. units)

Raman Shift (cm-1)

FIG. 3.

for different incoming photon energies E1.

Calculated Raman spectrum of the D mode in graphite

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VOLUME 85, NUMBER 24 PHYSICAL REVIEW LETTERS 11 DECEMBER 2000

1.5 2.02.53.0 3.5

1300

1350

1400

this work; 60 cm

Pócsik et al.; 44

Wang et al.; 47

Matthews et al.; 51

-1

Raman shift D-mode (cm

-1)

Excitation energy (eV)

FIG. 4.

a function of the excitation energy. The open symbols corre-

spond to experimental data, and the closed squares to the calcu-

lated phonon energies in double resonance. The line is a linear

fit to the theoretical values with a slope of 60 cm21?eV, the

numbers give the corresponding slopes for the data.

Measured and calculated frequencies of the D band as

understood excitation-energy dependence of the D mode

in graphite. Using a realistic electron and phonon band

structure we calculated the absolute value as well as the

rate at which the D mode shifts without adjusting any

parameter to 60 cm21?eV compared to ?50 cm21?eV as

observed experimentally. The recently introduced quasi-

selection rule “k ? q” for the explanation of this phe-

nomenon may be dismissed.

We thank M. Cardona for pointing out an error in the

initial version of this manuscript.

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