SUPPLEMENTARY MATERIAL
Generalized approach to design multi-layer stacks for enhanced
optical detectability of ultrathin layers
A. Hutzler1,a,b), C. D. Matthus1,b), M. Rommel2, L. Frey1,2
1 Chair
of Electron Devices (LEB), Department of Electrical, Electronic and Communication Engineering,
Friedrich-Alexander University Erlangen-Nuremberg (FAU), Cauerstraße 6, 91058 Erlangen, Germany
2 Fraunhofer
Institute for Integrated Systems and Device Technology (IISB), Schottkystraße 10, 91058 Erlangen,
Germany
Calculation method:
The transfer matrix method is a more general formulation of the dielectric properties of multilayer systems that
rises from the boundary conditions of the material interfaces from the Maxwell-theory11. For contrast
enhancement of specific layers or antireflective properties of multilayer systems, which contain dissipative
materials, the optical model can be enhanced to complex refractive functions11,12. Therefore, a system with n
homogeneous isotropic layers on a semi-infinite substrate is assumed (cf. Figure S1a) on which an incident polychromatic light beam is partially reflected and partially transmitted at every layer-interface (cf. Figure S1b).
FIG. S1. a) stack of optical layers and b) reflected and transmitted light beam
The wavelength dependent complex refractive function of a single layer determines an effective optical path
length and thus the interference of the single light-components. Every optical layer i can be described via a
characteristic matrix containing its complex refractive index ni, its film thickness ti and the optical properties of
the light beam like the wavelength as well as the incident angle θ by11-13:
cos(𝑘0 𝑛𝑖 𝑡𝑖 cos 𝜃𝑖 )
𝑴𝒊 = [
−𝑗𝑝𝑖 sin(𝑘0 𝑛𝑖 𝑡𝑖 cos 𝜃𝑖 )
𝑗
−
𝑝𝑖
sin(𝑘0 𝑛𝑖 𝑡𝑖 cos 𝜃𝑖 )
cos(𝑘0 𝑛𝑖 𝑡𝑖 cos 𝜃𝑖 )
]
(S1)
The term 𝑘0 𝑛𝑖 𝑡𝑖 cos 𝜃 describes the effective optical thickness of a material where the vacuum wave vector k0 is
defined by 2𝜋 divided by the vacuum wavelength 𝜆0 :
𝑘0 =
2π
(S2)
𝜆0
The polarization p depends on the incident angle of the light beam and the relative permittivity and permeability
of the i-th material and is given by:
𝜀𝑟,𝑖 cos 𝜃𝑖
√𝜇
𝑝𝑖 =
𝑟,𝑖
for TE waves
𝑍0
(S3)
𝜇
{
𝑟,𝑖
√ 𝜀 𝑍0 cos 𝜃𝑖
𝑟,𝑖
for TM waves
for transverse electric (TE) and transverse magnetic (TM) polarized light, respectively. For normal incidence,
cos 𝜃 is equal to unity. The characteristic wave impedance Z0 is a constant calculated from the square root of the
fraction of the vacuum permeability and the vacuum permittivity:
𝜇0
𝑍0 = √
(S4)
𝜀0
Generally, the refractive index is described by a complex, wavelength dependent function
𝑛 = 𝑛(𝜆) + 𝑗𝑘(𝜆)
(S5)
where the extinction coefficient 𝑘 is equal to zero for lossless dielectric media. The correlation between the
electric and magnetic fields E and H at the surface of the multilayer system (between air and the first layer) and
the final interfacial layer (between the last layer and the substrate) can therefore be expressed as:
[
𝐸(0,1)
𝐸(𝑛,𝑠𝑢𝑏)
𝐸(𝑛,𝑠𝑢𝑏)
] = 𝑴1 𝑴2 … 𝑴𝒏 [
] = 𝑴𝑡𝑜𝑡 [
]
𝐻(0,1)
𝐻(𝑛,𝑠𝑢𝑏)
𝐻(𝑛,𝑠𝑢𝑏)
(S6)
The reflection coefficient r and the transmission coefficient t can be calculated from the elements of the transfer
matrix Mtot to:
(𝑀
𝑟 = (𝑀11
+𝑀12 𝑝𝑠𝑢𝑏 )𝑝0 −(𝑀21 +𝑀22 𝑝𝑠𝑢𝑏 )
11 +𝑀12 𝑝𝑠𝑢𝑏 )𝑝0 +(𝑀21 +𝑀22 𝑝𝑠𝑢𝑏 )
𝑡 = (𝑀
2𝑝0
11 +𝑀12 𝑝𝑠𝑢𝑏 )𝑝0 +(𝑀21 +𝑀22 𝑝𝑠𝑢𝑏 )
(S7)
(S8)
They describe the amplitude ratio between the reflected and the transmitted, respectively, and the incident
electromagnetic wave. The reflectivity R and transmissivity T is incidental from:
𝑅 = |r|2 , 𝑇 =
𝑝𝑠𝑢𝑏
𝑝0
|t|2
(S9)
For non-polarized light as used in this study the overall reflection and transmission was assumed to consist of
50 % TE polarized and 50 % TM polarized light. The analytical calculation is implemented in a MATLAB
script. The graphene layer-thickness was calculated following equation:
𝑡𝐺𝑟𝑎𝑝ℎ𝑒𝑛𝑒 = 𝑛𝐺𝑟𝑎𝑝ℎ𝑒𝑛𝑒𝐿𝑎𝑦𝑒𝑟𝑠 ∙ 0.335 nm
(S10)
with 𝑛𝐺𝑟𝑎𝑝ℎ𝑒𝑛𝑒𝐿𝑎𝑦𝑒𝑟𝑠 being the number of graphene layers, i.e. one in the case of single-layer graphene as
investigated in this letterS1.
Figures 2 and 3 in linear scale:
w/o graphene
1
0.5
0
200
0.5
Reflectance
Reflectance
0.5
0
1
w/o graphene
1
with
Wavelength in
nmgraphene
Zeta 300
Lambda 9
Calculation
400
600
Wavelength in nm
800
0
1
with graphene
0.5
Zeta 300
Lambda 9
Calculation
0
200
(a)
400
600
Wavelength in nm
800
(b)
FIG. S2. Calculation and measurement results of the spectral reflectance of samples (a) O1 and (b) O2 without and with
graphene, local minima offering the highest contrast indicated by arrows.
1
1
w/o graphene
0.5
0
1
Reflectance
0.5
Reflectance
w/o graphene
with graphene
Zeta 300
Lambda 9
Calculation
0.5
0
200
400
600
Wavelength in nm
0
1
with graphene
Zeta 300
Lambda 9
Calculation
0.5
0
200
800
400
600
Wavelength in nm
(a)
800
(b)
FIG. S3. Calculation and measurement results of the spectral reflectance of samples (a) N1 and (b) N2 without and with
graphene, global minima offering the highest contrast indicated by arrows.
Values of complex refractive indices used in this work:
-3
8
Silicon
1.58
8
Silicon Dioxide
1.56
6
x 10
5
4
6
1.54
1.52
k
n
4
k
n
3
4
2
1.5
2
1
1.48
0
200
300
400
500
600
700
0
800
1.46
200
300
400
3.5
0.14
2.6
0.12
2.5
0.1
2.4
0.08
2.3
0.06
2.2
0.04
2.1
0.02
3
3
2.5
2.5
2
2
1.5
1.5
1
2
200
300
400
500
600
700
0
800
1
0.5
200
0.5
300
400
500
600
700
FIG. S4. Values of complex refractive indices of silicon, SiO2, Si3N4, and SL-graphene used in this work.
References Supplementary Material:
S1
Z.H. Ni et al., Nano Lett 7, 2758 (2007).
0
800
700
Single-Layer Graphene
0.16
2.7
600
n
Silicon Nitride
k
n
2.8
500
0
800
k
2