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 . 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 . 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. 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 . 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
5214 0031-9007?00?85(24)?5214(4)$15.00© 2000 The American Physical Society
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
For the linear bands in Fig. 1 we can calculate the Ra-
man matrix element K2f,10 explicitly by evaluating the
usual expression :
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
Fermi velocities. In one dimension the sum may be con-
verted to an integral over k:
bi2 i ¯ hg??E12 ¯ hvph2 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
?y22 y1?3?k22 q
?k12 k??k22 k?,(2)
¯ hvph2 i ¯ hg???y22 y1?.
wardly evaluated to
k1? ?E12 i ¯ hg???y22 y1?
This integral is straightfor-
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
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
E12 ¯ hvph
0.0 0.20.4 0.60.8
E1= 3 eV
E1= 2 eV
|K2f,10| (arb. units)
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
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
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  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 :
w?k? ? ?3 2 cosk ? R12 cosk ? R22 cosk ? ?R12 R2?
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 . In the isotropic limit (for
small k and neglecting s) the incoming transition occurs
photon energies in the visible and higher the incoming
resonance does not occur close to the K point, and the
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
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  or
ab initio  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 . 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? 3.2 eV.
?a,b,c?ithe full electronic dispersion curves of Eq. (5)
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
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. .
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)
for different incoming photon energies E1.
Calculated Raman spectrum of the D mode in graphite
VOLUME 85, NUMBER 24 PHYSICAL REVIEW LETTERS 11 DECEMBER 2000
1.5 2.02.53.0 3.5
this work; 60 cm
Pócsik et al.; 44
Wang et al.; 47
Matthews et al.; 51
Raman shift D-mode (cm
Excitation energy (eV)
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
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