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February 1, 2011 12:7 World Scientific Review Volume - 9in x 6in Chap7
124 A. Das, B. Chakraborty and A. K. Sood
and the experimental position of the 2D peak is shown in Fig. 11(c) by
a solid line. Indeed, as experimentally determined, the position of the 2D
peak is predicted to decrease for an increasing electron concentration in the
system. This allows to use the 2D peak to discriminate between electron
and hole doping. The FWHM(2D) (Fig. 11(d)) increases at higher doping
and this is not related to disorder as we do not see any appearance of the
D mode (defect mode) in the Raman spectra even at higher doping level.
We have not yet understood this result.
Intensity ratio
Figure 16(a) shows how the G mode and 2D mode Raman intensities in
single layer graphene change with doping. For this we have normalized
the Raman spectra with respect to the polymer peak at ∼ 2890 cm
−1
to
exclude any external effects like power change or defocusing etc. The G peak
increases with doping and then decreases at higher doping level. However,
2D peak decreases sharply with doping. Figure 16(b) plots the variation
of 2D and G intensity ratio (I(2D)/I(G)) as a function of doping. It shows
a strong dependence on doping. Hence the 2D/G intensity ratio is an
important parameter to estimate the doping concentration. Figures 16(b)
and 11(a) also show that the I(2D)/I(G) ratio and G peak position should
not be used to estimate the number of graphene layers.
Now, for BLG:
58
Π(E
F
)
BLG
= α
∞
0
γ
2
kdk
s,s
j,j
φ
+
jj
×
[f(
sjk
) − f (
s
j
k
)][
sjk
−
s
j
k
]
(
sjk
−
s
j
k
)
2
− (ω
0
+ iδ)
2
(15)
where s,s
are the band indices and j,j
are the subband indices. It can
be seen from Eq. (15) that for BLG both interband and intraband tran-
sitions contribute. The φ
+
jj
is weightage of the electron-phonon coupling
(EPC) for transitions between different subbands in a bilayer graphene.
The ‘+’ sign is for the symmetric mode which is Raman active G mode.
The values of φ
+
jj
are given in Ref. [58] and can be expressed as:
φ
11
= φ
22
=0.5
(γk)
2
(γ
1
/2)
2
+(γk)
2
(16)
φ
12
= φ
21
=0.5
(γ
1
/2)
2
(γ
1
/2)
2
+(γk)
2
February 1, 2011 12:7 World Scientific Review Volume - 9in x 6in Chap7
Pr obing Single and Bilayer Graphene Field Effect Transistors 125
Fig. 16. (a) Intensity of G and 2D as a a function of doping. (b) I(2D)/I(G) as a
function of hole and electron doping.
50
Equation (15) can be decoupled for different transitions and shown picto-
rially in Figs. 17(a),(b), where the blue lines and red lines correspond to
interband and intraband transitions, respectively. The contribution of in-
terband and intraband transitions to the Eq. (15) are plotted separately
in Fig. 17(c) as a function of Fermi energy shift at T=4K and δ=0.001eV.
The dashed line, dotted line and solid line in Fig. 17(c) correspond to the
interband, intraband and total (interband plus intraband) contribution, re-
spectively. We see a change of slope at ∼ 0.4 eV, which is the signature of
the second subband filling. It is clear from Fig. 17(c) that the sharp change
of slope at ∼ 0.4 eV becomes pronounced due to intraband contribution.
Note that the slope of ∆Pos(G)
dyn
just above E
F
= γ
1
is double than that
just below γ
1
.
Figure 18 shows the theoretical results for the change in peak posi-
tions of G mode and FWHM for both SLG and BLG at different tem-
peratures (T) and broadening factors (δ) using the value of parameter
February 1, 2011 12:7 World Scientific Review Volume - 9in x 6in Chap7
126 A. Das, B. Chakraborty and A. K. Sood
(c)
T
T
5
T
T
2
T
1
T
3
T
4
G
ω
T
6
(a)
(b)
Fig. 17. (a) Phonon decay into real (solid circle) and virtual (dotted circle) electron-
hole excitations. The gap between the vertical arrows corresponds to the phonon energy.
(b) Different transitions. Blue lines and red lines for interband (T
1
,T
2
,T
3
and T
4
)
and intraband (T
5
and T
6
) transitions, respectively. (c) Frequency shift of G peak as
a function of Fermi energy. Dashed line and dotted line correspond to interband and
intraband transition, respectively. Black line is the total contribution.
SLG
(c)
(d)
BLG
(a)
(b)
Fig. 18. (a) Pos(G) for (a) SLG; (c) BLG as a function of Fermi energy. FWHM(G)
of (b) SLG; (d) BLG as a function of Fermi energy at different temperatures (T) and
broadening factors δ.
α
=4.4 × 10
−3
, where the value of α
is obtained from the density func-
tional theoretical values of EPC(Γ) = 45.6 (eV)
2
/
˚
A
2
and v
F
=0.84×10
6
m/sec
29,56
). It can seen from Fig. 18 that the logarithmic divergences at
E
F
= ω
G
/2 are only visible at low temperatures but at room temperature
with cleaner sample (δ ∼ 0) it will be of the order of ∼−1cm
−1
. Therefore,
at room temperature the frequency position remains constant upto 0.1eV
February 1, 2011 12:7 World Scientific Review Volume - 9in x 6in Chap7
Pr obing Single and Bilayer Graphene Field Effect Transistors 127
and then increases linearly with E
F
. For BLG there is a change of slope at
E
F
∼ 0.4 eV, as seen in Fig. 18(c).
The FWHM of SLG and BLG decreases with E
F
,asshownin
Figs. 18(b),(d). Only the real transitions contribute to the life time of
phonon. Therefore, with Fermi energy shift (E
F
>ω
0
/2) there will be
blockage of interband real transitions due to Pauli exclusion principle and
as a result FWHM will decrease. Note that there is a kink in FWHM at
E
F
∼ 0.4 eV for BLG with δ=0.1eV, as seen in Fig. 18(d). This is due
to the blockage of real intraband transitions. However, there are no real
intraband transitions in BLG for δ=0.
4.3.1. Comparison between the experiment and theory
(Bilayer)
To compare calculations with the experimental data, we need to know exact
value of δ in our sample. The parameter δ is obtained by fitting the ex-
perimental FWHM(G) to FWHM(G)= FWHM(G)
EPC
+FWHM(G)
0
,with
FWHM(G)
0
a constant accounting for non-EPC effects (e.g. resolution and
anharmonicity). From Fig. 14(b) and Fig. 14(d) we determine δ =0.13eV
for SLG and δ =0.03eV for BLG. We get FWHM(G)
0
=4.3cm
−1
and
5.1cm
−1
for SLG and BLG, respectively. These δ values are then used to
compute Pos(G). Note that the relation between n and E
F
implies that
same amount of charge inhomogeneity causes different E
F
broadening in
SLG and BLG (e.g. δn∼10
12
cm
−2
would give δ = 0.13eV and 0.03eV in
SLG and BLG,respectively).
The solid lines in Fig. 14 are the theoretical Pos(G) and FWHM(G)
trends at 300K. The FWHM(G) of SLG and BLG increases at higher dop-
ing, similar to the FWHM(2D) of SLG. The kink at ∼ 0.4eVinBLGis
the most striking difference between SLG and BLG. It is the signature of
the second subband filling in BLG.
4.3.2. Physical interpretation
AshiftofE
F
changes f() in Eq. (15) thereby modifying the type and
number of transitions contributing to Π. A positive contribution to Π
arises when |
s,j,k
−
s
,j
,k
| < ω
0
, i.e. a subset of those between (s =
−1; j =1)and(s =1;j = 1) (interband transitions, solid blue lines in
Fig. 19). On the other hand interband transitions with |
s,j,k
−
s
,j
,k
| > ω
0
(solid red lines in Fig. 19) and all intraband (between (s = ±1; j =1)and
(s = ±1,j = 2), dashed red lines in Fig. 19) contribute to Π as negative
February 1, 2011 12:7 World Scientific Review Volume - 9in x 6in Chap7
128 A. Das, B. Chakraborty and A. K. Sood
Fig. 19. Phonon renormalization for BLG: (I) E
F
< ω
0
,(II)ω
0
<E
F
<γ
1
, (III)
E
F
>γ
1
. Blue and red arrows correspond respectively to positive and negative contri-
butions to Π. Solid and dashed arrows correspond to interband and intraband processes
respectively.
terms. Let see what happen to Π with Fermi energy shift. We discuss three
different cases: (I), |E
F
| < ω
0
,(II)ω
0
< |E
F
| <γ
1
, and (III) |E
F
| >γ
1
.
Taking E
F
> 0 (the same applies for E
F
< 0). Case (I), |E
F
| < ω
0
:the
positive contributions from interband transitions are suppressed, whereas
new negative intraband transitions are created, resulting in strong phonon
softening. Experimentally it has been observed at low temperatures in
Ref. [69]. At T=300K, these effects are blurred by the broadening of the
Fermi function, resulting in an almost doping independent phonon energy.
Case (II), ω
0
< |E
F
| <γ
1
:theshiftofE
F
decreases negative interband
contributions and creates new negative intraband transitions.
Taking into account their number and relative weight (given by
Φ
jj
/(
s,j,k
−
s
,j
,k
)), it has been shown that interband transitions out-
weight intraband ones, leading to in phonon hardening. Case (III), |E
F
| >
γ
1
: this is similar to (II), with a significant difference that the second
subband filling suppresses negative intraband transitions at k∼K,thereby
leading to the phonon hardening. Therefore, the kink in Fig. 14 is a direct
measurement of the interlayer coupling strength.
Bilayer graphene exhibits an inversion symmetry. As shown in Fig. 1,
the in plane lattice vibrations (G phonon) are classified into symmetric
(with displacement of two layers oscillating in-phase) and antisymmetric
(with displacement of two layers oscillating out-of-phase). The symmetric
mode is Raman active and the antisymmetric mode is IR active. Under
inversion operation through the inversion center of the bilayer structure, the
in- phase motion is invariant while the out-of-phase motion changes sign. In
an unperturbed bilayer graphene the in-phase phonon and the out-of-phase
February 1, 2011 12:7 World Scientific Review Volume - 9in x 6in Chap7
Pr obing Single and Bilayer Graphene Field Effect Transistors 129
Fig. 20. G mode spectra of top gated bilayer graphene for different top gate voltages.
Lines are Lorentzian fit to the experimental data denoted by squares. The dotted line
shows the evolution of the lower mode. Note, that that the Dirac point voltage is
subtracted from applied voltages.
75
phonon are the eigenstates of the bilayer.
71,72
However, as the bilayer is
doped either with electrons or with holes, owing to the existence of an inter-
layer electric field, the inversion symmetry is broken, mixing the symmetric
and the antisymmetric modes. The breaking of this inversion symmetry
is manifested in the splitting of Raman G mode, with energy and spectral
weight evolving as the doping increases.
73–75
Figure 20 shows the bilayer
G band splitting as a function of top gate voltage in our experiments.
75
The resulting phonon eigenstates can be viewed as a superposition of both
the in-phase and out-of-phase displacements and both the modes become
Raman active. Near the charge neutral point, Raman G peak is a single
line peak which is attributed to the in-phase motion of the two layers.
As the doping increases, the separation between the two bands widens,
February 1, 2011 12:7 World Scientific Review Volume - 9in x 6in Chap7
130 A. Das, B. Chakraborty and A. K. Sood
one decreasing in energy (lower mode) while the other mode increasing in
energy (higher mode) and the intensity is determined by the content of
symmetric component in each mode. This results are being understood in
the light of recent calculations of bilayer phonons.
71,72
5. Conclusions
Raman spectroscopy is a powerful probe to look into vibrational signatures
of undoped and doped graphene layers. Owing to the high electron-phonon
coupling strength in graphene, Kohn anomalies occur in graphene’s phonon
dispersion at the Γ and K point phonons. The doping reduces the screen-
ing of the and the G phonon’s energy increases for both electron and hole
doping. The FWHM of G mode decreases on doping because of the block-
age of the decay channels of phonons into electronhole pairs due to the
Pauli exclusion principle. The frequency of Raman G mode can be used
to estimate the carrier concentration whereas the 2D peak position de-
termine the sign of induced carriers. Additionally, Raman spectroscopy
can be used to probe breaking of inversion symmetry in bilayer graphene
where G peak splits for moderate doping concentration due to mixing of
symmetric and antisymmetric modes. The agreement of time dependent
perturbation calculations for the phonon frequencies with the experimental
results for single and bilayer graphene as a function of doping is rather
modest at high concentrations. This brings out a need to look into the
theory of phonon renormalization further, incorporating electron-electron
correlations and other possible effects.
In near future, it will be interesting to study doping dependence
of Raman phonons in graphene nanoribbons, functionalized graphenes,
graphanes
22
and graphenes prepared by heating of SiC, chemical vapour
deposition on metal surfaces and chemical methods. The study of hot
phonons
76
at large drain-source bias voltage will also be important from
the point of view of graphene devices.
Acknowledgments
This review is based on the thesis of Anindya Das. We thank all our
collaborators from Cambridge University, Manchester University, Indian
Institute of Science and Jawaharlal Nehru Centre for Advanced Scientific
Research for fruitful interactions.
February 1, 2011 12:7 World Scientific Review Volume - 9in x 6in Chap7
Pr obing Single and Bilayer Graphene Field Effect Transistors 131
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