Optimising Graphene Visibility in van der Waals Heterostructures
Thanmay S. Menon Simli Mishra Vidhu Catherine Antony Kiranmayi Dixit Saloni Kakkar Tanweer Ahmed Saurav Islam Aditya Jayaraman Kimberly Hsieh Paritosh Karnatak Arindam Ghosh










^1 Undergraduate Programme, Indian Institute of Science, Bangalore 560012, India
^2 Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Kolkata, Mohanpur 741246, Nadia, West Bengal, India
^3 Department of Physics, Bangalore University, Bangalore 560 056, India
^4 Department of Physics, Indian Institute of Technology Guwahati, Guwahati 781 039, Assam, India
^5 Department of Physics, Indian Institute of Science, Bangalore 560012, India
^6 Centre for Nano Science and Engineering, Indian Institute of Science, Bangalore 560012, India


These authors contributed equally to this work. Correspondence should be addressed to K.H. (kimberly@iisc.ac.in) or P.K. (paritosh@iisc.ac.in).


^† Current address: Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA

^ Current address: Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland



Graphene constitutes one of the key elements in many functional van
der Waals heterostructures. However, it has negligible optical visibility
due to its monolayer nature. Here we study the visibility of graphene
in various van der Waals heterostructures and include the effects
of the source spectrum, oblique incidence and the spectral sensitivity
of the detector to obtain a realistic model. A visibility experiment
is performed at different wavelengths, resulting in a very good agreement
with our calculations. This allows us to reliably predict the conditions
for better visibility of graphene in van der Waals heterostructures.
The framework and the codes provided in this work can be extended
to study the visibility of any 2D material within an arbitrary van
der Waals heterostructure.

Optimising Graphene Visibility in van der Waals Heterostructures


    Thanmay S. Menon^1,*, Simli Mishra^2, Vidhu Catherine Antony^3, Kiranmayi Dixit^4,†, Saloni Kakkar^5, Tanweer Ahmed^5, Saurav Islam^5, Aditya Jayaraman^5, Kimberly Hsieh^5,*, Paritosh Karnatak^5,,* and Arindam Ghosh^5,6
    December 30, 2023
===============================================================================================================================================================================================================================





§ INTRODUCTION


The family of van der Waals materials has now expanded beyond graphene
and offers a wide range of material functionalities, such as semiconductors,
insulators, superconductors
and ferromagnets. In addition, van der
Waals heterostructures are fabricated by stacking individual 2D materials
to obtain compound materials with novel functionalities.
However, graphene is often difficult to locate through an optical
microscope during or after the heterostructure assembly due to its
near-optical transparency. While the visibility of graphene on SiO2/Si
substrate has been studied before,
its visibility in various configurations of van der Waals heterostructures
has not yet been explored. In this paper, we develop a stepwise, robust
formalism to study the visibility of graphene in various heterostructures
and include practical considerations such as the details of the source
spectrum, oblique incidence which is necessary at high magnifications,
and the spectral sensitivity of the detector. We perform the visibility
calculations for graphene-BN , graphene-BN-MoS2,
graphene-MoS2-BN and graphene-MoS2
heterostructures on SiO2 substrate and suggest conditions
for better visibility. In order to corroborate our theoretical calculations,
we experimentally determine the optical visibility of a graphene-BN-MoS2
and graphene-MoS2 heterostructures at different wavelengths.
Our methods, codes and table of information can be employed to study
the visibility of most 2D van der Waals materials in an arbitrary
heterostructure configuration.



§ METHODS


For the theoretical calculations, a thin film interference model was
assumed where the reflection coefficients were calculated using Fresnel's
equations. Reflection coefficients r_i,s and r_i,p (for s and p polarisations respectively) for the interface between i^th and i+1^th layer (see Fig.
<ref>b) can be written as:

    r_i,s=ñ_icos(θ_i)-ñ_i+1cos(θ_i+1)/ñ_icos(θ_i)+ñ_i+1cos(θ_i+1)



    r_i,p=ñ_i+1cos(θ_i)-ñ_icos(θ_i+1)/ñ_i+1cos(θ_i)+ñ_icos(θ_i+1)


where ñ_i (ñ_i+1) and θ_i(θ_i+1)
corresponds to the refractive index and angle from the normal respectively
in the i^th (i+1^th) layer. θ_i can be obtained
using Snell's law applied to the i^th layer and is complex for
an absorbent material.

Often one or more layers in the stack are optically anisotropic, i.e.,
the in-plane and out-of-plane polarisation refractive indices are
not equal. For uniaxial crystals, the calculation of r_i,s remains
trivial. However, the refractive index in the case of p-polarisation
is obtained by solving the following self-consistent equation
in θ: 

ref: Born and wolf for optical anisotropy formula



    1/ñ_p^2=cos^2θ/ñ_∥^2+sin^2θ/ñ_⊥^2


where n_∥ and n_⊥ are the in-plane and out-of-plane
polarisation refractive indices respectively.



After calculating r_i (for s- and p polarisations), the reflectivity
can be obtained from the N-layer reflection formula, R̃_N
which is obtained recursively as follows:

If ϕ_i=4πñ_icosθ_i d_i/λ is the
phase shift due to the optical path difference in the i^th layer
of thickness d_i,


    R̃_N=R_Ne^iδ_N=r_0+R̃_N-1e^-iϕ_0/1+r_0R̃_N-1e^-iϕ_0


where δ_N is the total phase acquired after transmission through N layers, r_0 is the reflection coefficient of the topmost interface (see Fig. <ref>b) and is given by (<ref>) or (<ref>)
depending on the polarisation of incident light, and R̃_N-1
is the reflection from N-1 layers which is computed by applying (<ref>)
to the substrate and using r_1 and ϕ_1 instead of r_0
and ϕ_0. This method is repeated for subsequent layers till
we reach R̃_0=r_N which is the reflection from the interface
between the Nth layer and semi-infinite medium of silicon. The reflected
intensity from the entire stack is given by I=|R̃_N|^2.

Accounting for oblique incidence is especially important when viewing
the samples at large magnifications (especially 100×) because of the
high numerical aperture (α_NA) of the objective . To take this into
account, we assume that the incident beam has a gaussian profile
and integrate the reflected intensity for both polarisations over
angles from 0 to θ_M=asin(α_NA)


    I(λ)=_0^θ_mI(θ,λ) e^-2ρ^2/ρ_m^2tanθ dθ


where ρ=tanθ and ρ_m=tanθ_m (see Fig.
<ref>b)

In the visibility calculation, the normalisation constants for the
gaussian distribution cancel out eventually. As the incident beam
is unpolarised, the reflected intensity is the average of the reflected
intensities due to s and p polarisations.






The visibility (also known as Michelson's contrast)
is defined as:


    Visibility (in %)=100×I_s-I_g/I_s+I_g


I_s corresponds to the reflection from the substrate and I_g
refers to the reflection from the entire heterostucture. Positive
or negative value of visibility corresponds to graphene appearing
darker or lighter than the substrate, respectively.


Often the spectrum of the source and the spectral sensitivity of the
detector must also be included to obtain a more accurate value of
the visibility. For a typical RGB digital camera, if the red, green
and blue channel spectral sensitivities are ω_R(λ), ω_G(λ)
and ω_B(λ) and the spectrum of the source is S(λ),
then the reflected intensities picked up by the red, green and blue
channels are:


    I_j=_0^∞I(λ) ω_j(λ) S(λ) dλ

where j=R, G, B.




By substituting these intensities in the visibility formula (<ref>),
one can calculate the visibility for each colour channel. One should
also include the spectral dependence of lenses, mirrors, etc. for
more sensitive applications.

The dependence of graphene's visibility on wavelength for different
angles of incidence is shown in Figs. <ref>c and <ref>d
for graphene-SiO2 and graphene-BN-SiO2
heterostructures respectively. Although there is a shift in the peak
position for both heterostructures, the relative shifts
in the height of the peaks are different for both heterostructure
configurations. Specifically, comparing between incidence at 60^∘
and normal incidence, there is a greater difference in the height
of the peaks for graphene-BN-SiO2 than for graphene-SiO2.
This explains why accounting for oblique incidence is important for
complicated heterostructure configurations. The codes employed in this work are available on GitHub .

The heterostructures presented in this work were prepared by mechanically exfoliating the individual flakes of graphene,
BN and MoS2 on separate SiO_2 substrates followed by micro-mechanical transfer technique using a polydimethylsiloxane (PDMS) stamp spin-coated with a transparent sacrificial polymer layer. Similarly prepared graphene-based heterostructures have been previously shown to exhibit relatively high mobilities when encapsulated by BN and have also found widespread functionality as ultra-sensitive photodetectors due to their efficient interlayer charge transfer.



We have analysed and performed visibility calculations for three heterostructure
configurations: graphene-BN-SiO2 and
graphene-MoS2-SiO2, and graphene-BN-MoS2-SiO2. The images
were taken using an Olympus UC30 camera mounted on an Olympus BX51
microscope through a 100× objective. The heterostructures were illuminated
using standard light emitting diodes (LEDs) of different wavelengths at a constant power of 2 mW and the exposure time (1-2 sec) was set so as not to saturate any of the channels but also yield sufficient
signal-to-noise ratio.The images were split into three RGB channels and
were analysed using ImageJ software. The relative intensities of these
channels depend on the particular LED being used. The channel with
the maximum signal-to-noise ratio for each LED was used in the calculations. To minimise
errors due to uneven illumination, I_s was recorded at a point
on the substrate close to the point on graphene where I_g was
recorded and then visibility was calculated using (<ref>).
This was done multiple times over the whole sample for the
same LED and the average visibility is plotted. The spectrum of the LEDs and the spectral sensitivity
of the camera (see supplementary information Fig. S1) were incorporated in the calculations
using (<ref>). The experiment was performed using standard LEDs rather than the microscope's incandescent light source since it provides a better control over the choice and range of the source wavelength and also avoids complications that arise from digital image processing of a white-light image (such as white-balance), unique to each camera and its imaging software. The thickness of BN used in the stack was determined to be 11 nm using atomic force microscopy (AFM) and Raman spectroscopy was
used to verify that MoS2 and graphene were monolayers.
We used a standard SiO2-Si substrate with an SiO2
thickness of 285 nm. We have used both the in-plane and out-of-plane
refractive indices of graphene and MoS2
 and the refractive indices of BN and SiO2
(both of which have zero extinction coefficients) as well as that
of Si.



§ RESULTS AND DISCUSSION


Figs. <ref>(a) and <ref>(c) show the experimental results
along with the theoretical calculations for two-layer graphene-BN
and graphene-MoS2 heterostructures with SiO2
as substrate. The respective thicknesses were fixed at the experimentally determined values. To the right of Figs. <ref>(a) and <ref>(c) are the microscope images of the respective heterostructures illuminated with LEDs of different wavelengths. Although we have performed the visibility experiment using seven LEDs with wavelengths spread over the visible regime, we have shown only selected images in those colour channels with maximum signal-to-noise ratio. α_NA obtained from the best fit of the visibilities was ∼0.88 which is very close to the value of 0.9 provided by Olympus. We can see very close agreement between theory and experiment,
which suggests that our model for computing visibilities is sufficiently
accurate. Minor deviations of the experimental values from the theoretical
values may be due to the incident beam not being strictly gaussian
in nature. In Figs. <ref>b and <ref>d, we have presented
the results of our visibility calculations for the graphene-BN and
graphene-MoS2 heterostructures as a function of varying substrate and underlayer thicknesses. It must be mentioned here that the calculations in Figs. <ref>(a) and <ref>(c) incorporated the spectrum of the LEDs in order to corroborate the results of the visibility experiment while the calculations in Figs. <ref>b and <ref>d have been performed assuming a typical α_NA of 0.9 and integrating over the spectrum of an incandescent light source of blackbody temperature 3100 K, the most common light source used in optical microscopes employed for the searching of suitable flakes of van der Waal materials for device fabrication. Although we have assumed the spectral sensitivity of our camera (SONY ICX252AQ CCD image sensor) in our calculations, it does not differ widely across different cameras.
Such a model gives us three values of visibility (one for each colour
channel). In these figures, we have plotted the maximum of the absolute
values of the three visibilities arising from red, green and blue
channels. From our experiments, we have arrived at an absolute visibility
threshold value of 2.5% above which the graphene would be visible.
In this way, the phase space of BN-SiO2 thicknesses
in Fig. <ref>b can be divided into `islands of visibility'
(bounded by dashed white lines in the figure), where graphene would
be visible if the BN and SiO2 thicknesses lie within
such an island. The same has been done in Fig. <ref>d for graphene-MoS2
heterostructure. In these figures, we see that using lower thicknesses
of SiO2 substrate results in higher visibility for
graphene in both graphene-BN and graphene-MoS2 heterostructures.
Fig. <ref>b also indicates that in a graphene-BN heterostructure
on SiO2 (with a typical thickness of 300 nm), the
visibility of graphene reduces with increasing BN thickness. For BN
thicknesses above ≈ 42 nm, graphene is visible only at low
SiO2 thicknesses and in the supplementary information (Fig. S2), we have
shown that a 380 nm double layer of polymethyl methacrylate (PMMA) spin-coated
on the heterostructure improves the visibility significantly for thicker
BN on 285 nm SiO2.

Fig. <ref>a shows the experimental results and theoretical
calculations for graphene-BN-MoS2 heterostructure (the right panels show the corresponding optical images under the illumination of various LEDs). We see good agreement between the theory and experiment even in a
three-layer heterostructure. The insets in Figs. <ref>a,
<ref>c and <ref>a show that a model that considers only
normal incidence is inadequate in explaining the experimental results
and therefore, one must resort to an oblique incidence model to make
reliable predictions. Figs. <ref>b and <ref>c show visibility
calculations for graphene-BN-MoS2 and graphene-MoS2-BN
heterostructures as a function of SiO2 and BN thicknesses and assuming MoS2 to be a monolayer. We observe that this figure looks similar to the graphene-BN
graph. This is because of the similarity between the refractive indices
of MoS2 and BN especially in the red and green wavelength
ranges and because we are only considering monolayer MoS2.
Similar to the graphene-BN heterostructure, lower SiO2
thickness is recommended for better visibility of graphene.

Fig. <ref> shows the maximum absolute visibility for various
heterostructure configurations on 285 nm of SiO2
along with the peak wavelengths. It also suggests various thicknesses
of van der Waals materials that can be used for viewing graphene on
285 nm SiO2. These values tend to lie in the green
wavelengths which our eyes are most sensitive to. This is not
a coincidence but a result of choosing SiO2 thickness
to be 285 nm. From our calculations, we can conclude that effects of van der Waals materials under graphene are fundamentally no different when compared to a regular dielectric. Although the effect of the substrate dominates over non-metallic van der Waals materials, using an SiO_2 substrate of an arbitrary thickness to improve the visibility is often neither desirable nor practical. Instead, our aim in this paper is to provide a general framework and codes to allow the readers to calculate the visibility of any crystal within a heterostructure. In case of poor visibility, the readers may consider choosing an underlayer (say, BN) of appropriate thickness, if the calculations predict that this improves the visibility. From the figure, we also note that graphene-Bi2Se3
has very poor visibility. This is attributed to the high extinction coefficient
of Bi2Se3, due to the existence of its metallic surface states. Hence, we expect graphene's visibility to be adversely affected by metallic single- to few-layers. A similar reduction in contrast was already observed for graphene on gold substrates <cit.>.

In the supplementary information (Fig. S2), we have computed the visibility for other common
heterostructure configurations such as BN-graphene-BN, PMMA-graphene-BN,
PMMA-graphene-MoS2 and a recently proposed photocatalyst, graphene-ZrS_2 <cit.> using the same method and have identified optimal thicknesses of different layers for maximum visibility.

In conclusion, we have studied the conditions for the optimal visibility of graphene
in van der Waals heterostructures. We have also performed experiments
to demonstrate the accuracy of our predictions. Our methods and codes
may be directly employed to calculate the visibility of a van der
Waals material in a heterostructure.



 §.§ Acknowledgements


The authors thank Am. Ghosh for use of a spectrometer and SERC, IISc for their computational facilities. The authors also acknowledge G. Singh and S. Majumder for useful discussions.









44
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Paul et al.(2016)Paul,
  Ghatak, and Ghosh]Paul2016
  author author T. Paul, author S. Ghatak,  and  author A. Ghosh, 10.1088/0957-4484/27/12/125706 journal journal Nanotechnology volume 27, pages 125706 (year 2016)NoStop
[Duan et al.(2015)Duan,
  Wang, Pan, Yu, and  Duan]Duan2015
  author author X. Duan, author C. Wang, author A. Pan, author
  R. Yu,  and author
  X. Duan, 10.1039/c5cs00507h journal journal Chem. Soc.
  Rev. volume 44, pages 8859 (year 2015)NoStop
[Bhattacharjee et al.(2017)Bhattacharjee, Ganapathi, Chandrasekar,
  Paul, Mohan, Ghosh,
  Raghavan, and Bhat]Bhattacharjee2017
  author author S. Bhattacharjee, author K. L. Ganapathi, author H. Chandrasekar, author T. Paul,
  author S. Mohan, author A. Ghosh, author
  S. Raghavan,  and author
  N. Bhat, 10.1002/aelm.201600358 journal journal Adv.
  Electron. Mater. volume 3, pages
  1600358 (year 2017)NoStop
[Britnell et al.(2012)Britnell, Gorbachev, Jalil, Belle, Schedin, Katsnelson, Eaves, Morozov, Mayorov, Peres, Castro Neto, Leist, Geim, Ponomarenko, and Novoselov]Britnell2012
  author author L. Britnell, author R. V. Gorbachev, author R. Jalil,
  author B. D. Belle, author F. Schedin, author
  M. I. Katsnelson, author
  L. Eaves, author S. V. Morozov, author A. S.  Mayorov, author N. M. R.  Peres, author A. H. Castro Neto, author J. Leist, author A. K. Geim,
  author L. A. Ponomarenko,  and  author K. S. Novoselov, 10.1021/nl3002205 journal journal Nano
  Lett. volume 12, pages 1707 (year 2012)NoStop
[Duong et al.(2017)Duong,
  Yun, and Lee]Duong2017
  author author D. L. Duong, author S. J. Yun,   and author Y. H. Lee, 10.1021/acsnano.7b07436 journal journal
  ACS Nano volume 11, pages 11803
  (year 2017)NoStop
[Kusmartseva et al.(2009)Kusmartseva, Sipos, Berger, Forró, and Tuti šš]Kusmartseva2009
  author author A. F. Kusmartseva, author B. Sipos,
  author H. Berger, author L. Forró,  and author E. Tuti šš, 10.1103/PhysRevLett.103.236401 journal journal Phys. Rev. Lett. volume 103, pages 236401 (year
  2009)NoStop
[Xi et al.(2015)Xi,
  Zhao, Wang, Berger,
  Forró, Shan, and Mak]Xi2015
  author author X. Xi, author L. Zhao, author Z. Wang, author
  H. Berger, author L. Forró, author J. Shan,  and author K. F.  Mak, 10.1038/nnano.2015.143 journal journal Nat. Nanotechnol. volume 10, pages 765 (year 2015)NoStop
[Huang et al.(2017)Huang,
  Clark, Navarro-Moratalla, Klein, Cheng, Seyler, Zhong, Schmidgall, McGuire, Cobden, Yao, Xiao, Jarillo-Herrero, and Xu]Huang2017
  author author B. Huang, author G. Clark,
  author E. Navarro-Moratalla,
  author D. R. Klein, author R. Cheng, author
  K. L. Seyler, author
  D. Zhong, author E. Schmidgall, author M. A.  McGuire, author D. H.  Cobden, author W. Yao, author D. Xiao, author P. Jarillo-Herrero,  and author X. Xu, 10.1038/nature22391 journal journal Nature  volume 546, pages 270 (year
  2017)NoStop
[Gong et al.(2017)Gong,
  Li, Li, Ji, Stern, Xia, Cao, Bao,
  Wang, Wang, Qiu,
  Cava, Louie, Xia, and  Zhang]Gong2017
  author author C. Gong, author L. Li, author Z. Li, author
  H. Ji, author A. Stern, author Y. Xia, author T. Cao, author W. Bao, author C. Wang, author
  Y. Wang, author Z. Q. Qiu, author R. J.  Cava, author S. G. Louie, author J. Xia,  and  author X. Zhang, 10.1038/nature22060 journal journal
  Nature volume 546, pages 265
  (year 2017)NoStop
[Liu et al.(2016)Liu,
  Weiss, Duan, Cheng,
  Huang, and Duan]Liu2016
  author author Y. Liu, author N. O. Weiss,
  author X. Duan, author
  H.-C. Cheng, author
  Y. Huang,  and author
  X. Duan, 10.1038/natrevmats.2016.42 journal journal Nat.
  Rev. Mater. volume 1, pages 16042
  (year 2016)NoStop
[Georgiou et al.(2013)Georgiou, Jalil, Belle, Britnell, Gorbachev, Morozov, Kim, Gholinia, Haigh, Makarovsky, Eaves, Ponomarenko,
  Geim, Novoselov, and Mishchenko]Georgiou2013
  author author T. Georgiou, author R. Jalil,
  author B. D. Belle, author L. Britnell, author
  R. V. Gorbachev, author
  S. V. Morozov, author
  Y.-J. Kim, author A. Gholinia, author S. J.  Haigh, author O. Makarovsky, author L. Eaves,
  author L. A. Ponomarenko,
  author A. K. Geim, author K. S. Novoselov,  and author A. Mishchenko, 10.1038/nnano.2012.224 journal journal Nat.
  Nanotechnol. volume 8, pages 100
  (year 2013)NoStop
[Roy et al.(2013a)Roy, Padmanabhan, Goswami, Sai, Kaushal, and Ghosh]Roy2013_1
  author author K. Roy, author M. Padmanabhan,
  author S. Goswami, author T. P. Sai, author
  S. Kaushal,  and author
  A. Ghosh, 10.1016/J.SSC.2013.09.021 journal journal Solid
  State Commun. volume 175-176, pages
  35 (year 2013a)NoStop
[Roddaro et al.(2007)Roddaro, Pingue, Piazza, Pellegrini, and Beltram]Roddaro2007
  author author S. Roddaro, author P. Pingue,
  author V. Piazza, author V. Pellegrini,  and author F. Beltram, 10.1021/nl071158l journal journal Nano Lett.  volume 7, pages 2707 (year
  2007)NoStop
[Blake et al.(2007)Blake,
  Hill, Castro Neto, Novoselov, Jiang, Yang, Booth, and Geim]Blake2007
  author author P. Blake, author E. W. Hill,
  author A. H. Castro Neto,
  author K. S. Novoselov, author D. Jiang, author
  R. Yang, author T. J. Booth,  and author A. K. Geim, 10.1063/1.2768624
  journal journal Appl. Phys. Lett. volume 91, pages 063124 (year
  2007)NoStop
[Casiraghi et al.(2007)Casiraghi, Hartschuh, Lidorikis,
  Qian, Harutyunyan, Gokus,
  Novoselov, and Ferrari]Casiraghi2007
  author author C. Casiraghi, author A. Hartschuh, author E. Lidorikis, author H. Qian,
  author H. Harutyunyan, author T. Gokus, author
  K. S. Novoselov,  and author
  A. C. Ferrari, 10.1021/nl071168m journal journal Nano Lett.  volume 7, pages 2711 (year
  2007)NoStop
[Ni et al.(2007)Ni,
  Wang, Kasim, Fan,
  Yu, Wu, Feng, and  Shen]Ni2007
  author author Z. H. Ni, author H. M. Wang,
  author J. Kasim, author H. M. Fan, author
  T. Yu, author Y. H. Wu, author Y. P.  Feng,  and author Z. X.  Shen, 10.1021/nl071254m journal journal Nano Lett. volume
  7, pages 2758 (year 2007)NoStop
[Li et al.(2013)Li,
  Wu, Huang, Lu, Yang, Lu, Xiong, and Zhang]Li2013rapid
  author author H. Li, author J. Wu, author X. Huang, author
  G. Lu, author J. Yang, author X. Lu, author Q. Xiong,  and  author H. Zhang, @noop 
  journal journal ACS Nano volume 7, pages 10344 (year
  2013)NoStop
[Bing et al.(2018)Bing,
  Wang, Bai, Du, Wu, and Liu]Bing2018optical
  author author D. Bing, author Y. Wang, author J. Bai, author
  R. Du, author G. Wu,  and author L. Liu, @noop  journal journal Opt. Commun. volume 406, pages 128 (year 2018)NoStop
[Huang(2019)]Huang2019optical
  author author F. Huang, @noop  journal journal J.
  Phys. Chem. C  (year 2019)NoStop
[Jessen et al.(2018)Jessen,
  Whelan, Mackenzie, Luo,
  Thomsen, Gammelgaard, Booth,  and Bøggild]Jessen2018
  author author B. S. Jessen, author P. R. Whelan,
  author D. M. Mackenzie, author B. Luo, author
  J. D. Thomsen, author
  L. Gammelgaard, author
  T. J. Booth,  and author
  P. Bøggild, 10.1038/s41598-018-23922-1 journal journal Sci.
  Rep. volume 8, pages 2 (year 2018)NoStop
[Bruna and Borini(2009)]Bruna2009
  author author M. Bruna and author S. Borini,  10.1088/0022-3727/42/17/175307 journal
  journal J. Phys. D Appl. Phys. volume
  42, pages 175307 (year 2009)NoStop
[Katzen et al.(2018)Katzen,
  Velickỳ, Huang, Drakeley,
  Hendren, Bowman, Cai,
  Chen, Li, and Huang]Katzen2018
  author author J. M. Katzen, author M. Velickỳ,
  author Y. Huang, author S. Drakeley, author
  W. Hendren, author R. M. Bowman, author Q. Cai, author Y. Chen, author L. H. Li,  and  author F. Huang, @noop 
  journal journal ACS Applied Materials &
  Interfaces volume 10, pages 22520
  (year 2018)NoStop
[Velickỳ et al.(2018)Velickỳ, Hendren, Donnelly,
  Katzen, Bowman, and Huang]Velicky2018
  author author M. Velickỳ, author W. R. Hendren, author G. E. Donnelly, author J. M. Katzen, author R. M. Bowman,
   and author F. Huang,  @noop  journal journal Nanotechnology  volume 29, pages 275205 (year 2018)NoStop
[Hecht(2016)]hecht2016optics
  author author E. Hecht, https://books.google.co.in/books?id=Bv1RrgEACAAJ title Optics (publisher Pearson, address New York, year 2016)NoStop
[Born et al.(1999)Born,
  Wolf, Bhatia, Clemmow,
  Gabor, Stokes, Taylor,
  Wayman, and Wilcock]Born1999
  author author M. Born, author E. Wolf, author A. B. Bhatia, author
  P. C. Clemmow, author
  D. Gabor, author A. R. Stokes, author A. M.  Taylor, author P. A.  Wayman,  and author W. L.  Wilcock, 10.1017/CBO9781139644181 title Principles of Optics (publisher
  Cambridge University Press, address Cambridge, year 1999)NoStop
[Anders(1965)]Anders65
  author author H. Anders, @noop  title Thin Films in
  Optics (publisher The Focal Press, address
  London, year 1965)NoStop
[Knittl(1976)]knittl1976optics
  author author Z. Knittl, https://books.google.co.in/books?id=vCu9AQAACAAJ title Optics of Thin Films: An Optical Multilayer Theory  (publisher Wiley, address London, year 1976)NoStop
[Jung et al.(2007)Jung,
  Pelton, Piner, Dikin,
  Stankovich, Watcharotone, Hausner, and Ruoff]InhwaJung2007
  author author I. Jung, author M. Pelton,
  author R. Piner, author D. A. Dikin, author
  S. Stankovich, author
  S. Watcharotone, author
  M. Hausner,  and author
  R. S. Ruoff, 10.1021/nl0714177 journal journal Nano Lett.  volume 7, pages 3569 (year
  2007)NoStop
[Michelson(1995)]Michelson1927
  author author A. Michelson, https://books.google.co.in/books?id=m2tUZ4_8WGMC
  title Studies in Optics (publisher Dover
  Publications, address New York, year
  1995)NoStop
[Stigell et al.(2007)Stigell, Miyata, and Hauta-Kasari]Stigell2007
  author author P. Stigell, author K. Miyata,   and author M. Hauta-Kasari,  10.1134/S1054661807020101 journal journal Pattern Recogn. Image Anal. volume 17,  pages 233 (year 2007)NoStop
[Menon(2019)]GitHub
  author author T. Menon, @noop  title Calculating visibility of
  van der Waals heterostructures (year 2019), note
  retrieved from
  <https://github.com/Thanmay-Menon/visibility>NoStop
[Zomer et al.(2011)Zomer,
  Dash, Tombros, and van
  Wees]Zomer2011
  author author P. J. Zomer, author S. P. Dash,
  author N. Tombros,  and  author B. J. van Wees, 10.1063/1.3665405 journal journal
  Appl. Phys. Lett. volume 99, pages
  232104 (year 2011)NoStop
[Roy et al.(2013b)Roy, Padmanabhan, Goswami, Sai, Ramalingam, Raghavan, and  Ghosh]Roy2013
  author author K. Roy, author M. Padmanabhan,
  author S. Goswami, author T. P. Sai, author
  G. Ramalingam, author
  S. Raghavan,  and author
  A. Ghosh, 10.1038/nnano.2013.206 journal journal Nat.
  Nanotechnol. volume 8, pages 826
  (year 2013b)NoStop
[Karnatak et al.(2016)Karnatak, Sai, Goswami, Ghatak, Kaushal, and Ghosh]Karnatak2016
  author author P. Karnatak, author T. P. Sai,
  author S. Goswami, author S. Ghatak, author
  S. Kaushal,  and author
  A. Ghosh, @noop  journal journal Nat. Commun. volume
  7, pages 13703 (year 2016)NoStop
[Aamir et al.(2018)Aamir,
  Karnatak, Jayaraman, Sai,
  Ramakrishnan, Sensarma, and Ghosh]Aamir2018
  author author M. A. Aamir, author P. Karnatak,
  author A. Jayaraman, author T. P. Sai, author
  T. Ramakrishnan, author
  R. Sensarma,  and author
  A. Ghosh, @noop  journal journal Phys. Rev. Lett. volume 121, pages 136806 (year
  2018)NoStop
[Roy et al.(2018)Roy,
  Ahmed, Dubey, Sai,
  Kashid, Maliakal, Hsieh,
  Shamim, and Ghosh]Roy2018
  author author K. Roy, author T. Ahmed, author H. Dubey, author
  T. P. Sai, author R. Kashid, author S. Maliakal, author K. Hsieh, author S. Shamim,  and author A. Ghosh, @noop  journal journal Adv. Mater. volume 30, pages
  1704412 (year 2018)NoStop
[Djurišić and Li(1999)]Djurisic1999
  author author A. B. Djurišić and author E. H.  Li, 10.1063/1.369370 journal
  journal J. Appl. Phys. volume 85,  pages 7404 (year 1999)NoStop
[Hieu et al.(2018)Hieu,
  Ilyasov, Vu, Poklonski,
  Phuc, Phuong, Hoi, and  Nguyen]Hieu2018
  author author N. N. Hieu, author V. V. Ilyasov,
  author T. V. Vu, author N. A. Poklonski, author
  H. V. Phuc, author
  L. T. Phuong, author
  B. D. Hoi,  and author
  C. V. Nguyen, @noop  journal journal Superlatt. Microstruct. volume 115, pages 10 (year 2018)NoStop
[Gorbachev et al.(2011)Gorbachev, Riaz, Nair, Jalil, Britnell, Belle, Hill, Novoselov, Watanabe, Taniguchi, Geim, and Blake]Gorbachev2011
  author author R. V. Gorbachev, author I. Riaz,
  author R. R. Nair, author R. Jalil, author
  L. Britnell, author
  B. D. Belle, author
  E. W. Hill, author
  K. S. Novoselov, author
  K. Watanabe, author
  T. Taniguchi, author
  A. K. Geim,  and author
  P. Blake, @noop  journal journal Small volume 7,  pages 465 (year 2011), http://arxiv.org/abs/1008.2868 1008.2868 NoStop
[Rodríguez-de Marcos et al.(2016)Rodríguez-de Marcos, Larruquert, Méndez, and Aznárez]Rodriguez-deMarcos2016
  author author L. V. Rodríguez-de Marcos, author J. I.  Larruquert, author J. A.  Méndez,  and author J. A. Aznárez, 10.1364/OME.6.003622 journal journal Opt. Mater.
  Express volume 6, pages 3622
  (year 2016)NoStop
[Aspnes and Studna(1983)]PhysRevB.27.985
  author author D. E. Aspnes and author A. A. Studna, 10.1103/PhysRevB.27.985 journal
  journal Phys. Rev. B volume 27,  pages 985 (year 1983)NoStop
[Jiang et al.(2013)Jiang,
  Liu, Gu, and Susstrunk]Jiang2013
  author author J. Jiang, author D. Liu, author J. Gu,  and author
  S. Susstrunk, in 10.1109/WACV.2013.6475015 booktitle Proceedings of IEEE
  Workshop on Applications of Computer Vision (year 2013)  pp. pages 168–179NoStop
[Eddrief et al.(2016)Eddrief, Vidal, and Gallas]Eddrief_2016
  author author M. Eddrief, author F. Vidal,   and author B. Gallas, 10.1088/0022-3727/49/50/505304 journal journal J. Phys. D Appl. Phys. volume 49,  pages 505304 (year 2016)NoStop
[Zhang et al.(2016)Zhang,
  Meng, Rao, Wang,
  Shi, Liu, Wu, Deng, Liu, and Lu]Zhang2016
  author author X. Zhang, author Z. Meng,
  author D. Rao, author
  Y. Wang, author Q. Shi, author Y. Liu, author H. Wu, author K. Deng, author
  H. Liu,  and author
  R. Lu, @noop  journal
  journal Energy Environ. Sci. volume
  9, pages 841 (year 2016)NoStop








Supplementary Information







Optimising Graphene Visibility in van der Waals Heterostructures


    Thanmay S. Menon^1,*, Simli Mishra^2, Vidhu Catherine Antony^3, Kiranmayi Dixit^4,†, Saloni Kakkar^5, Tanweer Ahmed^5, Saurav Islam^5, Aditya Jayaraman^5, Kimberly Hsieh^5,*, Paritosh Karnatak^5,,* and Arindam Ghosh^5,6
    December 30, 2023
===============================================================================================================================================================================================================================







§ TABLE OF REFERENCES FOR REFRACTIVE INDICES OF VARIOUS 2D MATERIALS




S.I No.     Material     Reference1     Graphene     Djuri¨ic et al., J. Appl. Phys. 85, 7404 (1999)2     BN     Gorbachev et al., Small 7, 465 (2011)3     MoS2     Hieu et al., Superlatt. Microstruct. 115, 10 (2018)4     Bi2Se3     Eddrief et al., J Phys. D Appl. Phys. 49, 505304 (2016)5     SiO2     Rodríguez-de Marcos et al., Opt. Mater. Express 6, 3622
(2016)6     Si     Aspnes et al., Phys. Rev. B 27, 985 (1983)






§ EXPERIMENTAL DETAILS






The visibility of graphene-BN and graphene-BN-MoS2
heterostructure stacks were measured on different parts of the same sample. Figs. S1(a)
and (b) show the images of graphene-BN-MoS2 and
graphene-MoS2 sample under different light sources.
The light sources used were LEDs of different colours and their intensities
were calibrated to ensure uniform intensity of the LEDs. We use a gaussian beam approximation for the LEDs since the incident light beam is narrower than the input aperture of the objective lens (some high-end microscope light sources provide uniform illumination). The spectra of the LEDs, recorded using an Ocean optics spectrometer,
is given in Fig. S1(e). The images were taken using an Olympus UC30
camera, which was equipped with a SONY ICX 252 AQ CCD image sensor.
This sensor's spectral sensitivity (obtained from the sensor's datasheet)
is given in Fig. S1(f). These images were then split into RGB channels
and the channel with maximum signal-to-noise ratio was analysed using
ImageJ software. Figs. S1(c) and (d) show the same sample under an ordinary
white light source (blackbody of temperature 3100 K).





§ RESULTS FOR OTHER HETEROSTRUCTURES




Figs. S2(a) and S2(b) show the effect of polymethyl methacrylate
(PMMA) polymer on the maximum absolute visibility of graphene-BN and graphene-MoS2 heterostructures.
A standard double layer of PMMA-495K (A4) followed by  PMMA-950K (A4), spin-coated at ∼4500 rpm, gives a thickness of ∼380 nm <cit.>. We have assumed the same refractive indices for both
coatings of PMMA <cit.>. We see that in case of graphene-BN heterostructure, PMMA increases the range of BN thicknesses which are optically visible under a microscope,
especially at ∼300 nm and ∼90 nm thicknesses of SiO2. However,
in the case of graphene-MoS2 heterostructure, at
285 nm SiO2 thickness, graphene is just barely visible
on monolayer MoS2. 


Fig. S2(c) shows maximum absolute visibility of BN-graphene-BN heterostructure
where the thickness of the top layer of BN is fixed at 10 nm and the
thickness of the bottom BN layer below graphene is allowed to vary. This
figure shows features very similar to that of graphene-BN heterostructure
except that the maximum BN thickness for graphene to be visible is lesser
here. Fig S2 (d) shows the maximum absolute visibility map of graphene-Bi2Se3
heterostructure. Here the maximum absolute visibility is calculated
from 10 nm-thick Bi2Se3 onwards
as there is evidence that the refractive index of Bi2Se3
abruptly changes below 10 nm <cit.>. We can see that
in the range of thicknesses we have considered for Bi2Se3, the visibility of graphene is very poor. This is attributed to the high extinction coefficient of Bi2Se3, due to its metallic surface states.


From Fig. S3a, we can see that the maximum visibility of graphene on monolayer ZrS_2 (SiO_2 thickness being 285 nm) is close to 6% at wavelength λ≈ 527 nm. This is on par with graphene's visibility on BN which would be very helpful during device fabrication. The refractive index and monolayer thickness of ZrS_2 were taken from Ref. <cit.> and Ref. <cit.> respectively.









5
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[MicroChem(2001)]2001Microchem
  author author MicroChem,  @noop   (year 2001), note retrieved from
  <http://microchem.com/pdf/PMMA_Data_Sheet.pdf>NoStop
[Sultanova et al.(2009)Sultanova, Kasarova, and Nikolov]2009AcPPA.116..585S
  author author N. Sultanova, author S. Kasarova,  and author I. Nikolov, @noop  journal journal
  Acta Phys. Pol. A volume 116, pages
  585 (year 2009)NoStop
[Vajner et al.(2016)Vajner,
  Yan, Guo, Mathews,
  Kuhlman, Benefield, Ulrich,
  Zolghadr, Kung, Li,
  Araujo, and Wang]Vajner_2016
  author author C. Vajner, author H. Yan,
  author L. Guo, author
  M. Mathews, author M. Kuhlman, author S. Benefield, author S. Ulrich, author E. Zolghadr, author P. Kung, author L. Li, author P. T. Araujo,   and author H.-T. Wang,  @noop  journal journal 2D Mater.  volume 3, pages 021010 (year
  2016)NoStop
[Vu et al.(2019)Vu,
  Lavrentyev, Thuan, Nguyen,
  Khyzhun, Gabrelian, Tran,
  Luong, Tung, Pham et al.]Vu2019
  author author T. V. Vu, author A. Lavrentyev,
  author D. V. Thuan, author C. V. Nguyen, author
  O. Khyzhun, author B. Gabrelian, author K. C.  Tran, author H. L. Luong, author P. D. Tung,
  author K. D. Pham,  et al., @noop  journal journal
  Superlattices Microstruct. volume 125, pages 205 (year 2019)NoStop
[Wang et al.(2016)Wang,
  Huang, Jiang, Li,
  Wei, and Li]Wang2016
  author author X. Wang, author L. Huang,
  author X.-W. Jiang, author Y. Li, author
  Z. Wei,  and author
  J. Li, @noop  journal
  journal J. Mater. Chem. C volume 4,  pages 3143 (year 2016)NoStop
















