Supplementary Information for “Interference effect on
Raman spectrum of graphene on SiO2/Si”
Duhee Yoon,1 Hyerim Moon,1 Young-Woo Son,2* Jin Sik Choi,3 Bae Ho Park,3 Young Hun Cha,4
Young Dong Kim,4 and Hyeonsik Cheong1§
1
2
3
Department of Physics, Sogang University, Seoul 121-742, Korea,
School of Computational Sciences, Korea Institute for Advanced Study, Seoul 130-722, Korea,
Division of Quantum Phases & Devices, School of Physics, Konkuk University, Seoul 143-701,
Korea,
4
1.
Department of Physics, Kyung Hee University, Seoul 130-701, Korea,
Calculation for the Raman enhancement factor as a function of the numerical aperture
(N.A.) of the objective lens
Figure S-1. Schematic diagrams of multiple reflection interference in the (a) absorption and (b)
scattering processes. (c) Optical beam path of the incident laser beam to the objective lens.
As shown in Fig. S-1(a) and (b), the incident laser beam and the Raman scattered light go through
multiple reflections inside the graphene layer as well as in the SiO2 layer. To include the effect of the
large N.A. in the calculation of the Raman enhancement factor, one should consider two points.
First, the laser beam focused by the objective lens has many optical beam paths with incident angle
(  ) between the normal and maximum incident angle ( max ) as in Fig. S-1(c). For oblique incidence
(   0 ) one should consider both the s- and p-polarizations of the incident laser beam and the
scattered light. Second, the intensity of the incident laser beam has a Gaussian distribution with a
weighting factor ( g ( ) ), which is a function of  .
The reflection and transmission Fresnel coefficients are different for the s- and p-polarizations.1
The Fresnel coefficients for the s-polarization are
r1 
no cos  0  ( p1  iq1 )
( p  iq2 )  ( p3  iq3 )
( p  iq2 )  ( p2  iq2 )
, r2  1
, r3  2
,
no cos  0  ( p1  iq1 )
( p1  iq1 )  ( p2  iq2 )
( p2  iq2 )  ( p3  iq3 )
2no cos  0
2( p1  iq1 )
t1 
, t1 
,
no cos  0  ( p1  iq1 )
no cos 0  ( p1  iq1 )
.
and those for the p-polarization are
(n12  k12 ) cos  0  n0 p1  i (2n1k1 cos 0  n0 q1 )
r1 
,
(n12  k12 ) cos  0  n0 p1  i (2n1k1 cos 0  n0 q1 )
r2 
( p1  iq1 )(n2  ik2 ) 2  ( p2  iq2 )(n1  ik1 ) 2
,
( p1  iq1 )(n2  ik2 ) 2  ( p2  iq2 )(n1  ik1 ) 2
r3 
( p2  iq2 )(n3  ik3 ) 2  ( p3  iq3 )(n2  ik 2 ) 2
,
( p2  iq2 )(n3  ik3 ) 2  ( p3  iq3 )(n2  ik2 ) 2
t1 
2n0 ( n1  ik1 ) cos  0
2( p1  iq1 )(n1  ik1 )
, t1 
,
2
(n1  ik1 ) cos  0  n0 ( p1  iq1 )
(n1  ik1 ) 2 cos  0  n0 ( p1  iq1 )
where
p1 
1
1
(11  13  (11  13 ) 2  12 2 ), q1 
(13  11  (11  13 ) 2  12 2 ),
2
2
p2 
1
1
( 21   23  ( 21   23 ) 2   22 2 ), q2 
( 23   21  ( 21   23 ) 2   22 2 ),
2
2
p3 
1
1
( 31   33  ( 31   33 ) 2   32 2 ), q3 
( 33   31  ( 31   33 ) 2   32 2 ),
2
2
and
11  n12  k12 , 12  2n1k1 , 13  (n0 sin 0 )2 ,
 21  n2 2  k2 2 ,  22  2n2 k2 ,  23  (n0 sin 0 )2 ,
 31  n32  k32 ,  32  2n3k3 ,  33  (n0 sin 0 )2 .
The refractive indices of the air, graphene, SiO2, and Si are n0  1 , n1  n1  ik1 , n2  n2  ik2 ,
and n3  n3  ik3 , respectively. The phase differences are given by
 x  2 x( p1  iq1 ) /  , 1  2 d1 ( p1  iq1 ) / , 2  2 d2 ( p2  iq2 ) / .
The 2-dimensional weighting factor g ( ) reflects the Gaussian distribution of the incident laser
beam into the objective lens.2,3 If one assumes that 99.4 % (3) of the incident laser beam enters into
the aperture of the objective lens, the weighting factor is expressed as3
g ( ) 
1
2
  3sin  /sin  max 
9
2
e
,
2 sin 2  max
where N.A.  n0 sin max .
The total enhancement factor is then expressed as
F  N
d1
0

 max
0
1
4

j s, p k s, p
2
Fabj ( x,  ) Fsck ( x,  ) g ( ) 2 sin  cos  d dx.
We calculated the Raman enhancement factors of G (FG) and 2D (F2D) bands as a function of the
excitation laser wavelength and the thickness of SiO2 layer for several N.A. values of the objective
lens. Fig. S-2 (a) and (b) plots the maxima of the enhancement factors for different N.A. values. The
maxima of the Raman enhancement factors shift slightly as the N.A. increases.
Figure S-2. Maxima points of the Raman enhancement factors for (a) G and (b) 2D bands as a
function of the excitation laser wavelength and the thickness of SiO2 layer for the several N.A.
values for the objective lens.
Reference
(1) I. Jung, M. Pelton, R. Piner, D. A. Dikin, S. Stankovich, S. Watcharotone, M. Hausner, and R. S.
Ruoff, Nano Lett. 7, 3569 (2007).
(2) L. Gao, W. Ren, F. Li, and H. Cheng, ACS Nano 2, 1625 (2008).
(3) G. Turrell, M. Delhaye, and P. Dhamelincourt, in Raman Microscopy edited by G. Turrell, and J.
Corset, Academic Press, London, 1996.
To whom correspondence should be addressed. E-mail: hand@kias.re.kr (Y.-W.S.); hcheong@sogang.ac.kr (H.C.).