Large-Area Layer Counting of Two-Dimensional Materials Evaluating

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Large-Area Layer Counting of Two-Dimensional Materials Evaluating the Wavelength Shift in Visible-Reflectance Spectroscopy Andreas Hutzler, Christian David Matthus, Christian Dolle, Mathias Rommel, Michael P.M. Jank, Erdmann Spiecker, and Lothar Frey J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b00957 • Publication Date (Web): 08 Mar 2019 Downloaded from http://pubs.acs.org on March 10, 2019

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The Journal of Physical Chemistry

Large-Area Layer Counting of Two-Dimensional Materials Evaluating the Wavelength Shift in Visible-Reflectance Spectroscopy Andreas Hutzler1,*, Christian D. Matthus2, Christian Dolle3, Mathias Rommel2, Michael P. M. Jank2, Erdmann Spiecker3, Lothar Frey1,2 1

Electron Devices (LEB), Department of Electrical, Electronic

and Communication Engineering, Friedrich- Alexander University of Erlangen-Nürnberg, Cauerstraße 6, 91058 Erlangen, Germany 2

Fraunhofer Institute for Integrated Systems and Device

Technology IISB, Schottkystraße 10, 91058 Erlangen, Germany 3

Institute of Micro- and Nanostructure Research (IMN) & Center

for Nanoanalysis and Electron Microscopy (CENEM), Department of Materials Science and Engineering, Friedrich-Alexander University of Erlangen- Nürnberg, Cauerstraße 3, 91058 Erlangen, Germany

ACS Paragon Plus Environment

1

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Page 2 of 45

ABSTRACT An advanced and highly scalable approach for determining the number of layers of 2D materials via optical spectroscopy is introduced. Based

on

appropriate

subjacent

layer

stacks,

the

reflectance

spectra of the 2D material assemblies exhibit wavelength shifts in distinct minima which are linearly related to the number of layers. A linear correlation enables straightforward data interpretation which is essential for implementing simple and comparatively fast measurement routines for process control on wafer scale. The structure of the optical layer stacks as well as the complex refractive indices of 2D materials were found to strongly influence the spectral position of the reflectance minima as well as the magnitude

and

the

linearity

of

the

wavelength

shifts.

We

experimentally proof this method being applicable for large-area layer counting of subsequently stacked CVD graphene films on a layer stack consisting of silicon nitride and silicon oxide. The measurement results are confirming the calculated wavelength shift of the reflection minimum around 540 nm equaling approx. 3 nm per layer. Numerical analysis shows that comparable behavior is also achievable by the tailored design of subjacent layer stacks for graphene oxide, hexagonal boron nitride, and more complex 2D materials

like

transition

metal

dichalcogenides.

For

the

achievement of linear relations between wavelength shifts of the


2

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respective

minimum

and

analytical

design

rules

the are

layer

count

derived

of

the

considering

2D

material,

the

optical

properties of the underlying layer stack as well as oscillator frequencies within the complex refractive index of the 2D material. The largest signal response of 12 nm per layer was calculated for MoSe2 on an optimized layer stack.


3


1.

Page 4 of 45

Introduction

Two-dimensional materials like graphene (Gr), graphene oxide (GO), hexagonal boron nitride (hBN), or layered transition metal dichalcogenides (TMDs) e.g. MeX2, with Me = Mo, W or Sn and X = S, Se or Te substantiated a new field of research during the last decade due to their unique electrical, optical and mechanical properties high

1–6.

In the case of graphene these properties include

conductivity

and

optical

transmissivity

making

it

e.g.

suitable for use as contact material in organic photovoltaics

1.

Moreover, the linear magnetoresistance of graphene enables an application

in

magnetic

sensor

devices

2

and

its

mechanical

strength and impermeability to liquids and gases allows for a utilization in acoustic resonators microscopy

1,4,5.

3

and liquid cell electron

The mentioned properties are strongly influenced

by the number of layers of the respective 2D material. Even more critical, the actual number of layers is fundamental for TMDs, because these materials turn from an indirect semiconductor in bulk to a direct semiconductor in the monolayer configuration

7.

Accordingly, the control and monitoring of the actual number of 2D sheets is vital for their application and facile and precise methodologies for its correct determination have to be supplied. Today, the determination of the number of layers of 2D materials may be done for example by transmission electron microscopy (TEM) in

selected

area

electron

diffraction

mode

(SAED)

or

high


4

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resolution (HRTEM) mode 10,

8,9,

low energy electron microscopy (LEEM)

or scanning electron microscopy (SEM)

11.

These approaches

benefit from a high resolution but are time consuming and not large-area

capable.

Alternative

photoelectron spectroscopy (XPS) (SPM)

14.

12,13

approaches

include

X-ray

and scanning probe microscopy

However, the data evaluation is not always conclusive and

straightforward. In the case of SPM one major problem is the inherent surface contamination which interferes the measurement results. Another possibility is Raman spectroscopy where complex evaluation algorithms have to be performed for more than 2 layers or 2D heterostructures, especially if individual layers of the 2D material

are

turbostratically

electronically decoupled

15–18.

rotated

against

each

other

or

In addition, information about the

stacking order and thickness of 2D materials can be gained by Raman spectroscopy

in

a

low-frequency

range

19.

Furthermore,

spectroscopic ellipsometry has been proven to be an extremely accurate approach for determining the thickness of various 2D materials like graphene

13,20,

hBN

20,21

and MoS2

20,22.

However, this

particular technique is time consuming because spectra have to be acquired in incidence.

large

spectral

Moreover,

ranges

spectroscopic

and

in multiple angles of

ellipsometry

requires

deep

knowledge of the electronic structure of a material for physical modeling

and

data

interpretation.

In

return,

it

enables

the

determination of the complex refractive index of a material which


5


Page 6 of 45

is a prerequisite for many other optical methods. The most widely used technique for the characterization of the thickness of 2D materials

is

reflectance

light

microscopy

spectroscopy

which

in

combination

profits

from

with

being

visible

fast,

non-

destructive, and large-area capable. It is widely utilized that the

optical

substrate

contrast

can

be

between

strongly

a

2D

enhanced

material by

and

proper

a

subjacent

design

subjacent layers, which was shown for distinct dielectric semiconducting

26,33,

and conductive layer stacks

34.

of

the

6,23–33,

However, the

contrast-based method is a relative measurement technique, which requires a reference spectrum for calculating a contrast function describing the variation of the light reflected from a substrate with and without the 2D material. Contrast functions often depend non-linearly on the number of layers

35

which complicates data

evaluation. However, the contrast for reflectance values near zero strongly depends on the sensitivity of the detector due to the use of the absolute reflectance for calculation of the number of layers 32.

Within this study a novel characterization method is shown which aims not only on the characterization of ideal but also defective or partially contaminated 2D materials. Defects can arise when the 2D material is grown by distinct deposition techniques like CVD. Moreover, some 2D materials, especially CVD grown 2D materials, are often transferred onto the substrate of choice utilizing


6

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polymer-assisted approaches, introducing surface contaminations. It will be demonstrated that the direct evaluation of the spectral position of distinct extrema of reflectance is appropriate for the determination of the layer number and is even suited for the characterization

of

non-ideal

samples

(i.e.

defective

or

contaminated), fully coated with a 2D material of homogeneous nature

if

the

underlying

layer

stack

is

sufficiently

pre-

characterized. This method is independent of distinct steps in the number of 2D sheets that, by definition, cannot be achieved by the contrast-based method. The general principle to determine the number of 2D layers is a wavelength shift Δλ of a distinct reflectance minimum of the layer stack that is related to the spectral position of the reflectance minimum of the layer stack without the 2D material, namely λ0. The magnitude and the linearity of this wavelength shift can be tailored by adapting the underlying optical layer stack. Both, the spectral position of the reflectance minimum for n layers of the 2D material (λn), and their difference Δλ are predicted by an analytical calculation using the generalized transfer matrix method (TMM)

32,36,37.

If appropriate underlying

stacks are used, the relation between the wavelength shift and the number of individual layers of the 2D material is almost perfectly linear for several 2D materials considered making the approach very

versatile.

It

is

exemplarily

shown

that

layer

stacks

consisting of 300 nm SiO2, which are commonly used in literature


7


6,25,

or of 53.5 nm Si3N4 on 11 nm SiO2

32,

Page 8 of 45

both on silicon substrate,

are applicable for determining the layer number of graphene and other 2D materials like GO or hBN. These particular 2D materials do

not

exhibit

oscillation

frequencies

caused

e.g.

by

band

transitions or quasiparticle resonances, while the respective extrema and inflection points are present in the complex refractive indices of TMDs within the visible wavelength range

38.

Hence,

different layer stacks are required for the proposed approach for layer counting individually for the respective 2D materials which have to be designed following distinct rules that are discussed in the following sections. 2.

Methods

2.1. Algorithm for evaluating the number of layers via wavelength shifts of local reflectance minima Generally, the method introduced in this work is based on the following

steps:

The

optical

properties,

i.e.

the

real

and

imaginary part of the spectral complex refractive index of the investigated 2D material are considered with special attention to oscillator frequencies. Here, as a first approximation and because according reliable data sets do not exist, it is assumed that the complex refractive index does not change significantly for a number of layers below 10. This assumption might introduce an error when band structures of 2D materials change with increasing number of layers or different stacking orders. The subjacent layer stack as


8

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a prerequisite has to be designed in order to find a suitable reflectance minimum with its spectral position λmin,0 preferably afar from oscillator frequencies within the selected wavelength range, i.e. the visible range in this particular case. Considering that, the reflectance spectra of the particular layer stack for zero up to i layers of the 2D material are individually calculated using the generalized TMM

32.

For each calculated spectrum the

wavelengths of local extrema, in this case reflectance minima, are determined. The spectral positions of the reflectance minima shift in relation to the layer stack without the 2D material for one up to

i

layers

of

the

2D

material

(λmin,0…λmin,i).

In

order

to

characterize this wavelength shift, the wavelengths of the minima are evaluated against the number of layers n. If the coefficient of determination R2 of a linear regression for the obtained data is large enough, e.g. if R2 exceeds 90%, the wavelength of the minimum for n layers λmin,n can be described in a good approximation by the following equation: λmin,n = λmin,0 + Δλ∙n. (1) The wavelength shift Δλ depends on the complex refractive index and the effective optical thicknesses of the 2D material and the subjacent refractive subjacent

layer index layer

stack, and stack

whereas

the

λmin,0 depends

effective

only.

The

optical

effective

on

the

thickness optical

complex of

the

thickness


9


Page 10 of 45

additionally depends on the physical thicknesses and the angle of incidence, polarization and wavelength of the incident light. Finally, the actual number of layers is determined by measuring the spectral position of the reflectance minimum λmin,n and solving Eqn.1 for n. However, residuals adsorbed at the 2D material under investigation can interfere the data evaluation as they lead to additional wavelength shifts. For a correct data interpretation, these residuals have to be taken into account by considering them as additional layers in the analytical model. Otherwise, this might lead to an inaccurate calculation of the number of layers to be determined.

Furthermore,

for

a

lens

optics

based

measurement

system as used in this study, the numerical aperture (NA) of the objective lens has to be considered as well. The analytical model as well as the data evaluation is implemented in MATLAB.

2.2. Experimental procedures In order to verify the approach, different samples were prepared. These comprise one blank substrate containing the layer stack only and four graphene-coated samples (G1-G4) with different numbers of graphene layers n. The layer stack consisting of 53.5 ± 0.5 nm of Si3N4 on 10.9 ± 0.2 nm of SiO2 (measured by ellipsometry) on silicon substrate was fabricated as described elsewhere

32.

After dicing,

graphene was deposited by PMMA-assisted transfer of commercially available CVD graphene (ACS Material) onto the layer stack. CVD


10

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has

shown

quality

2D

compared

to

mechanical exfoliation techniques because of its scalability

39.

materials

to

be

for

advantageous

next-generation

in

depositing

device

high

fabrication

However, most 2D materials cannot be directly deposited onto arbitrary substrates. Thus, transfer processes like the PMMAassisted approach are used for this purpose. One major problem of this technique is that PMMA cannot be entirely removed leaving residuals on the 2D material

40.

In our case the PMMA protection

layer was removed by immersing the samples in acetone. In order to prepare samples containing graphene flakes with different numbers of layers, single-layer and bi-layer graphene were used. Samples G1

and

G2

consist

of

one

single-layer

and

one

bi-layer

(predominantly AB stacked, see Fig. S4c and d) graphene film on the substrate (G0), respectively. Three layer graphene (G3) was realized by superimposing one single-layer graphene film and one bi-layer graphene film. The four-layer sample (G4) was prepared by superimposing

two

bi-layer

graphene

films,

respectively.

Furthermore, two samples with unknown layer distribution were investigated, namely G3/5 and G6/8. These are specified consisting of

3-5 layers and 6-8

layers

of natively

grown turbostratic

graphene. The preparation was conducted by transferring few-layer graphene sheets onto the layer stack as in the case of G1 and G2. The optical characterization of these samples was conducted in a spectral range from 430 nm to 750 nm by reflectometry using the


11


Page 12 of 45

integrated grating spectrometer of a Zeta 300 optical profiler system from Zeta Instruments. The light source is an ultra-bright white LED which emits non-polarized light (assumed as 50% TE (transverse electric) and 50% TM (transverse magnetic) for all calculations).

The

spatial

resolution

of

the

measurement

is

limited by the utilized objective lens and can be adjusted to be 30 µm (20x objective, NA = 0.4), 12 µm (50x objective, NA = 0.8) or 6.25 µm (100x objective, NA = 0.9). However, the numerical aperture (NA) of the different objectives has to be taken into account for the calculation of the effective reflectance spectra as this adds distinct angular collection ranges. This can be achieved by a weighted numerical integration of the angular range of the NA. In this work, we used the according approach presented 41.

by Saigal et al.

Furthermore, the spectral resolution of the

spectrometer is 1 nm. In this work, raster scans were performed using the 20x (NA 0.4) objective lens exclusively by recording the spectral

reflectance

averaged

over

ten

successively

recorded

spectra with a raster of the same size as the spot size of 30 µm per

position

wavelength

across

shift.

all

Smaller

samples features

to can

verify only

the be

calculated measured

by

averaging the reflectance spectrum across this spot size. The position of each graphene sheet with about 1 cm2 in size could be determined easily due to the high optical contrast measurements,

an

internal

integration

time

of

29.

500 ms

For all and

a


12

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bandwidth of 320 nm were utilized in the Zeta 300 instrument, so that hundred spectra at ten positions per minute could be recorded. The large area scans discussed in this work took from about 90 minutes (9,000 spectra for a total scanned area of approximately 1 mm²) until up to 16.5 hours (100,000 spectra for a total scanned area of approximately 9 mm²).

3. Results and Discussion 3.1. Verification of method - graphene layer counting on 53.5 nm Si3N4 / 10.9 nm SiO2 / Si The calculated reflectance of normally incident non-polarized light is shown in Figure 1(a) for different numbers of graphene layers on the referenced nitride/oxide stack. The reflectance minima for zero to up to 10 graphene layers are depicted by inverted triangles. The according reflectance values vary between 10-2% for zero or one graphene layer and up to 1.2% for ten graphene layers. All considered minima are located in the wavelength range between 539 nm without graphene (λmin,0) and 569 nm for ten graphene layers (λmin,10), whereas the related spectral position of the minimum (λmin,n) changes linearly with increasing number of atomic layers n as shown in Figure 1(b). In this particular case, the calculated wavelength shift Δλ has a value of 2.9 nm per graphene layer

towards

larger

wavelengths

with

a

coefficient

of

determination R2 of 99.93%. Additionally, Figure 1(b) shows the


13


NA-corrected minima of the spectra calculated according to the approach of Saigal et al.

41

discussed in section 2b). The optical

properties of the four materials, i.e. graphene, Si3N4, SiO2 and Si (Figure S7), were experimentally verified in a previous study reflectance minimum

1.4 1.2 1.0 0.8 0.6

n

0.4 0.2 0.0

normal incidence 500

520

540

560

580

wavelength  in nm

600

graphene layers w/o n=1 2 3 4 5 6 7 8 9 10 minima

10

number of graphene layers

1.6

reflectance R in %

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 45

(a)

32.

normal incidence NA corrected

8 6 4 2 0 530

540 550 560 570 spectral position of min in nm

(b)

Figure 1. (a) Calculated spectral reflectance (normal incidence) of graphene on a layer stack consisting of 53.5 nm of Si3N4 on 10.9 nm of SiO2 on a silicon substrate with different number of graphene layers and (b) calculated minima for the case of normal incidence and including NA correction (NA = 0.4).

The approach was evaluated experimentally by analysis of the reflectance minima of samples G0-G4 with a focus on the exact determination of their spectral position. For doing so, each local reflectance minimum was fitted by a Gaussian function. Systematic tool variability and local inhomogeneities of the graphene stack were accounted for by averaging over 10 spectra per position as


14

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well as raster scanning over larger sample areas. In the latter case, the spectral position of the reflectance minimum was again extracted by Gauss fitting over the spatial distribution (Figure 2(a)). The

distributions of the

mean

values

of the

measured

reflectance minima for samples G0-G4 are clearly shifting towards larger wavelengths for increasing number of graphene layers. For one up to four graphene layers, a linear regression of the obtained data reveals a wavelength shift Δλ of 3.5±0.1 nm per layer with a coefficient Figure 2(b)).

of

determination

This

is

R2

slightly

of

99.8%

larger

than

(orange the

line

in

analytically

extracted value. Based on according NA-corrected calculations, a shift

of

approximately

2.9 nm

per

layer

and

wavelengths

of

reflectance minima of 536.7 nm up to 545.3 nm for one up to four layers of graphene are expected (see black points in Figure 2(b)). Thus, the wavelength shifts per layer show a good correlation between calculation and measurements. With respect to the minima of the graphene coated samples, the bare reference stack shows a strong deviation, i.e. the wavelength of its minimum (λmin,0 of 535.6 nm) lies about 8 nm below the extrapolated value. However, it is obvious that this measurement perfectly fits the calculated NA corrected value of 533 nm for the bare reference stack (Figure 1(b) and black diamonds in Figure 2(b)). This suggests the assumption that rather the graphene coated samples than the bare layer stack show anomalous behavior. The


15


measurements show an additional offset for the wavelength of the minimum of approximately 8 nm (543.9 nm up to 554.4 nm) with respect to the calculations (see last paragraph). In order to investigate the origin of this mismatch, a deeper study of the samples (SEM),

was

performed

atomic

force

including

scanning

microscopy

(AFM),

electron

microscopy

transmission

electron

microscopy (TEM) and Raman spectroscopy. The results of these measurements are shown in the supplementary material (Figures S2 to S4 and S9). 575 1

G4: 4 Gr layer

0 1

G3: 3 Gr layer

0 1

G2: 2 Gr layer

570

NA corrected TMM results: 53.5 nm Si3N4

565

min in nm

relative frequency

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 45

0 1

G1: 1 Gr layer

560 555

53.5 nm Si3N4 + 3 nm PMMA measurement: G0 - G4 G3/5 G6/8

550 545 linear fit calculation:  = 2.9 nm/layer calculation:  = 2.8 nm/layer measurement:  = 3.5 nm/layer

540

0 1

G0: w/o graphene Gauss fits

0 530

535 530

540

550 min in nm

560

0

2 4 6 8 number of graphene layers n

(a)

Figure 2.

(a)

10

(b)

Measured

spectral

reflectance

of

graphene

with

different numbers of layers and (b) calculated wavelengths of reflectance minima without (black) and with 3 nm of PMMA (green) in

comparison

with

measured data

at 900 positions (red) and

measurement of 3-5 and 6-8 layer graphene at 104 positions (cyan and blue). Horizontal error bars (number of layers) are deduced


16

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from

the

measured

spectral

error

(standard

deviation

of

the

measured spectral distribution of the minima, cf. Figure 3(c) and Figure S1(c)).

The surface roughness was determined by AFM for the bare layer stack, i.e. G0, and the graphene covered samples (G1, G2). Without graphene the surface roughness (root-mean square value) is as low as 0.2 nm while it increases strongly to approximately 1.61 nm for all graphene coated samples. Concerning the sample properties, the following three aspects have to be considered for data evaluation: Firstly, the graphene used in this study was grown by chemical vapor

deposition

(CVD)

on

copper

substrates.

As

known

literature, graphene grows in individual isolated grains

from

42–44.

For

the preparation of single-layer graphene this process is manually stopped after a fixed time, assumed to complete the process of covering the whole copper surface. At this point, several regions (typically less than 1 µm2) are already present where islands of a second layer started to grow below the first layer

44.

For bi-layer

graphene, this process is continued until the mean number of layers is two

43.

That means that the bi-layer islands are larger (tens

of µm2), but do not cover the whole area and three-layer graphene is already grown below the second layer in distinct regions

42–44.

Moreover, the measurement spot size of 30 µm used in this work is


17


Page 18 of 45

much larger than the typical island size which prevents a direct measurement.

The

island

structure

was

confirmed

by

SEM

measurements (supplemental information Figure S2). Although this growth behavior slightly increases the surface roughness, its magnitude is expected to be in the range of the graphene layer thickness, i.e. below 1 nm. Secondly, cracks and wrinkles cause a locally higher or lower number of graphene layers and thus a higher surface

roughness.

Furthermore,

the

whole

process

was

not

conducted under ideal conditions such as a cleanroom environment. Hence, particles may also increase the surface roughness but this would be also valid for sample G0 where a very low surface roughness was measured. As all these effects are minimized with respect to standard lab conditions by a proper sample preparation, they are assumed to be not mainly responsible for the difference between

measurements

and

calculation.

As

a

third

point

PMMA

residuals can be assumed as a reason for the increase of the surface roughness and the spectral shift of the measurement data with respect to the calculation. PMMA is utilized as protection layer on graphene during the transfer process (see section 2a). After the graphene transfer the samples are exposed to acetone in order

to remove

this PMMA layer. However, it

is practically

impossible to remove this protection layer without leaving any residuals

40.

Since there is no graphene and also no PMMA on sample


18

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G0, this wavelength shift does not occur for the bare sample which supports this assumption. With these information, another calculation was performed using a modified layer stack containing an additional layer of PMMA on top. A reasonable guess for the thickness of this PMMA layer would be

the

double

of

the

route-mean-square

roughness

of

1.61 nm

measured by AFM (Figure S3). Thus we choose a value of 3 nm for a refinement of the stack models including a PMMA top layer with an assumed coverage of 50% of the surface. HRTEM of a cross section of

the

layer

stack

(Figure S5)

additionally

confirms

this

assumption. The data and fitting curve for the refined models are shown in Figure 2(b). As can be seen, the calculated wavelengths of the reflectance minima including the PMMA residuals of 545.6 nm up to 553.9 nm for one up to four graphene layers are in good agreement to the measurement data. The mismatch of the magnitude of the wavelength shift between the measured data (3.5 nm/layer) and the calculated values (2.8 nm/layer with additional PMMA) can again be attributed to additional PMMA residues inside the layer stack of samples G3 and G4 compared to samples G1 and G2 as the former samples are prepared by two transfer steps. Nevertheless, considering these effects, the introduced method is shown to be suitable for the straightforward large-area determination of the number of graphene layers even using non-ideal graphene flakes.


19


Page 20 of 45

In order to qualify the method for measuring real specimens, two further samples, one with a graphene film consisting of nominally 6-8 layers (G6/8) and another with nominally 3-5 layers (G3/5) purchased from ACS Materials were prepared in the same way as G1 and

G2,

i.e.

with

only

one

transfer

process.

The

as-grown

multilayer CVD graphene films are not homogenous in thickness and show a high amount of turbostratical disorder (Figure S5). Raster scans were performed on sample G6/8 (and also G3/5, see SI Figure S1). A large field (3 mm x 3 mm) micrograph of sample G6/8 and the corresponding map of the spectral position of the reflectance minima are depicted in Figure 3(a)-(b). Individual features in the spatial

distribution

of

reflectance

minima

can

be

directly

correlated to cracks and wrinkles which are obvious from the micrograph. The spectral distribution of the reflectance minima is represented by a narrow peak at a wavelength of 557.9 nm ± 1.6 nm. In order to obtain the corresponding number of graphene layers the spectral position of the reflectance minimum for a single layer of graphene λmin,1 is taken as fixed reference value and the shift of the wavelength per additional layer Δλ derived from the refined model including PMMA residues gives the instruction for extracting the number of excess graphene layers. The total difference of 14.0 nm ± 1.6 nm between G6/8 and G1 corresponds to a mean number of graphene layers of 6 ± 0.6 (Figure 3(c)). The determined mean number is within the specified value of 6-8 layers. Nevertheless,


20

Page 21 of 45

the number of layers is subject to strong local variation. This is related to the island-like structure which was already discussed for single-layer and bi-layer graphene (Figure S2,S4) and was experimentally proven by HRTEM of a focused ion beam (FIB) cross section (Figure S5) which was lifted out from sample G6/8. The results (Figure S5(d)) show obvious deviations of the number of layers in both directions compared to the nominal value of 6-8. These

measurements

additionally

revealed

an

inhomogeneous

amorphous carbon layer on top of the graphene flakes with a thickness of several nanometers which is assumed to correspond to PMMA residuals.

3.0

575

2.5

565

2.0

min in nm

y coordinate in mm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.5 1.0

545 535

0.5 0.0

555

0.0

0.5

1.0 1.5 2.0 2.5 x coordinate in mm

(a)

3.0

525

(b)


21


1.0

relative frequency

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 45

measurement Gauss fit

0.8 0.6

557.9 nm ± 1.6 nm

0.4 0.2 0.0 540 560 580 600 wavelength of minimum (nm)

(c) Figure 3. (a) Optical micrograph of 6-8 layer graphene, (b) map of the reflectance minima for (a). Three characteristic cracks are highlighted in (a) and (b). (c) spectral distribution of the reflectance minima (10,000 spectra recorded).

3.2. Influence of the subjacent layer stack on wavelength shift Δλ and linearity (R2) In order to investigate the influence of the subjacent layer stack on the wavelength shift Δλ and its linearity, reflectance spectra were calculated for different layer stacks using the analytical model shown in our previous work

32.

For this purpose,

three layer stacks were chosen. The first and the second stack are the previously mentioned stacks consisting of SiO2 on Si and Si3N4 on SiO2 on Si. Another frequently used high-k dielectric which is also used in anti-reflective coatings is Al2O3

45.

This material

can be deposited with a well-defined thickness via atomic layer deposition (ALD) even at low temperatures

46.

Hence, the third


22

Page 23 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


considered layer stack consists of Al2O3 on Si3N4 on SiO2 on Si. Figures

4(a)-(c) I)

show

calculated

reflectance

plots

of

the

mentioned layer stacks consisting of silicon dioxide, silicon nitride on silicon dioxide and aluminum oxide on silicon nitride on silicon dioxide with a variable thickness of the topmost layer in each case. In order to determine the wavelength shift and its linearity, distinct thickness ranges are chosen which result in reflectance minima (for 0 up to 10 graphene layers on top) in the visible spectral

range

(i.e.

400 nm

up

to

800 nm).

In

detail,

the

considered layer thicknesses were 0 nm up to 300 nm in case of SiO2 for the first layer stack (SiO2 / Si substrate), 0 nm up to 200 nm in case of Si3N4 for the second layer stack (Si3N4 / 11 nm SiO2 / Si substrate), and 0 nm up to 100 nm in case of Al2O3 and the third layer stack (Al2O3 / 53.5 nm Si3N4 / 11 nm SiO2 / Si substrate), respectively.

Depending

on the layer thicknesses,

different reflectance minima occur and are clearly visible as violet regions in the reflectance plots. For all layer stacks only the first minimum within the visible wavelength range (indicated by

dashed

line

in

Figures 4(a)-(c) I))

is

examined

in

this

particular case which is defined as the minimum appearing in the thinnest

layer

thickness

range.

Although

the

fabrication

of

thinner layers is preferable from a technological point of view, higher order minima, appearing for larger film thickness ranges,


23


Page 24 of 45

i.e. above 200 nm for SiO2 (Figure 4(a) I)), may be used with the same method, introducing a smaller wavelength shift Δλ. For each individual layer stack the reflectance minimum is extracted for the case of zero up to ten layers. The spectral positions of all minima are linearly fitted revealing the wavelength shift Δλ as well as the linearity R2 of the shift which is calculated from the deviations of the calculated minima from the linear fit. Figures 4(a)-(c) II)

show

the

wavelength

of

the

reflectance

minima without (λmin,0) and with an overlying graphene flake with 10 layers (λmin,10), the corresponding wavelength shift Δλ and its linearity (R2) for the three different layer stacks. The wavelength shift Δλ increases with increasing layer thickness t. This is caused by an increase of the effective optical path length. For 10 layers of graphene λmin,10 is shifted into the infrared range (i.e. wavelengths above 800 nm) for SiO2 thicknesses above 124 nm in case of the first layer stack, more than 82 nm Si3N4 in case of the second layer stack, and more than 45 nm Al2O3 in case of the third layer stack, respectively. The maximum wavelength shift Δλ in the visible range of 5.3 nm per layer can be achieved using an Al2O3 layer thickness of 45 nm on 53.5 nm Si3N4 and 11 nm SiO2. Moreover, the trend of the linearity is larger for multilayer systems than for single layer systems for the considered layer stacks. For the three-layer system it even exceeds values of 99.9% for an Al2O3 thickness range from 0 nm up to 40 nm. Hence, both magnitude and


24

Page 25 of 45

linearity of the wavelength shift are controllable by adapting the optical layer stack. a)

SiO2 / Si substrate

0 in nm

1000

b)

Si3N4 / 11 nm SiO2 / c) Al2O3 / 53.5 nm Si3N4 / Si substrate 11 nm SiO2 / Si substrate 1000 1000 I) 1.0 I)

I)

800

800

800

600

600

600

400

400

400

200

200 0

100 200 tSiO in nm

300

50 100 150 200 tSi N in nm

R2 in %

 in nm/layer

min in nm

2

800

3

800

II)

600

0

2

800

II)

min,0 min,10

400 6

400 6

4

4

4

2

2

2

100

100

100

95

95

95

90 80 90 100 110 120 130 tSiO in nm 2

3

II)

600

400 6

90

25 50 75 100 tAl O in nm

4

600

min,0 min,10

0.0

200 0

min,0 min,10

90 40 50 60 70 80 90 tSi N in nm 3

4

1.0

reflectance

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 10 20 30 40 50 tAl O in nm 2

3

Figure 4. I) Reflectance plots of (a) SiO2 on Si, (b) Si3N4 / 11 nm SiO2 on Si, and c) Al2O3 / 53.5 nm Si3N4 / 11 nm SiO2 on Si for different thicknesses of the topmost layer. II) top: wavelength of reflectance minimum of the substrate without and with 10 layers of graphene (i.e., min,0 and min,10, respectively), center: wavelength shift per graphene layer, and bottom: coefficient of determination for the layer stacks shown in a)-c). Calculations were done for an


25

0.0


Page 26 of 45

ideal measurement system, i.e. for normal light incidence without NA correction.


26

Page 27 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.3. Application to the 2D materials graphene oxide and hexagonal boron nitride The introduced method is not limited to graphene, but can be used for other 2D materials like GO and hBN as well. In case of graphene oxide, the optical properties measured by spectral ellipsometry and published recently by Schöche et al. was reported to be 0.7 nm per layer

48.

47

were used. The thickness

For hexagonal boron nitride

the optical data reported by Zunger et al.

49

0.33 nm per layer introduced by Golla et al.

and the thickness of 50

were used for the

calculations. Spectra of complex refractive indices of Gr, GO and hBN are shown in Figure S8 in the supporting information. The corresponding results for the wavelength shifts min,0 and min,10,

 and R² are shown in Figure 5 as a function of the Al2O3 thickness for the third layer stack (Al2O3 / 53.5 nm Si3N4 / 11 nm SiO2 / Si substrate) introduced above in comparison to the corresponding results for graphene. The maximum thickness of the Al2O3 layer was limited by the requirement that all minima up to λmin,10 should be located within the visible wavelength range. This condition is fulfilled for an Al2O3 thickness of up to 45 nm in case of graphene. For

GO

and

hBN

the

respective

maximum

Al2O3

thicknesses

are

determined to 48 nm and 51 nm. In case of GO the wavelength shift Δλ increases with higher Al2O3 thickness and a maximum value of 4.4 nm per layer for 48 nm Al2O3 can be achieved. The wavelength shift

does

not

increase

continuously

with

increasing

Al2O3


27


thickness in case of hBN. In this case, two minima can be observed for Al2O3 thicknesses of 46 nm and 52 nm, respectively. The maximum wavelength shift in the visible wavelength range of 2.3 nm per layer can be achieved for an Al2O3 thickness of 49 nm. Thus, it is about 50% smaller than the corresponding values for Gr and GO. However, the calculated linearity represented by R2 is extremely high for both 2D materials, GO and hBN. For 0 nm up to 40 nm Al2O3 it exceeds 99.99% as can be seen in Figure 5. Thus, the introduced method is applicable for Gr, GO and hBN and it is supposed that other 2D materials like germanene, silicene or black phosphorous can be characterized, too. In any case, adequate knowledge

of

the

optical

properties

of

those

materials

is

mandatory. Issues with more complex 2D materials like transition metal dichalcogenides (TMD) will be addressed in the next section. a)

 in nm/layer min in nm

Graphene 800 700 600 500

c) b) GO Al2O3 / 53.5 nm Si3N4 / 11 nm SiO2 / Si substrate

min,0 min,10

800 700 600 500

min,0 min,10

5

4

4

2

3

100.0

100.0

100.0

99.8

99.8

99.8 99.6 10 20 30 40 50 tAl O in nm 2

3

min,0 min,10

2.5 2.0

99.6 0

hBN

800 700 600 500

6

R2 in %

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 45

99.6 0

10 20 30 40 50 tAl O in nm 2

3

0

10 20 30 40 50 tAl O in nm 2

3


28

Page 29 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 5. (a) Graphene, (b) graphene oxide, and c) hexagonal boron nitride, each on Al2O3 / 53.5 nm Si3N4 / 11 nm SiO2 on Si for different thicknesses of the topmost layer. Top: wavelength of reflectance minimum of the optical layer stack without and with 10 layers of 2D material (i.e., min,0 and min,10, respectively), center: wavelength

shift

per

layer,

and

bottom:

coefficient

of

determination. Calculations were done for an ideal measurement system, i.e. for normal light incidence without NA correction.


29


Page 30 of 45

3.4. Application to transition metal dichalcogenides: MoTe2, MoSe2, and MoS2 TMDs are more critical because they exhibit multiple oscillator frequencies in their complex refractive indices within the visible spectral range

38.

The optical properties of MoS2, MoSe2, and MoTe2

have been investigated by Beal and Hughes

51.

More recently, Zhang

et al. reported on the difference in the refractive indices of bulk and monolayer MoS2

52.

Additional studies concerning MoS2,

MoSe2, WS2, and WSe2 were published by Li et al.

38.

In this study, the previously mentioned molybdenum based TMDs were

exemplarily

chosen.

However,

the

introduced

method

is

adaptable for any other TMD based on adequate knowledge of its optical properties and thickness per layer. The thicknesses of the TMDs studied in this work were supposed to be 0.63 nm, 0.65 nm, and 0.697 nm for each of MoS2, MoSe2, and MoTe2 layers, respectively 52–54.

Due to the high refractive indices (Figure S8), remarkably

larger wavelength shifts of distinct minima Δλ occur compared to the 2D materials discussed above (Gr, GO, hBN). However, the oscillator

frequencies

result

in

a

superposition

of

several

reflectance minima. This superposition is more pronounced for small absolute reflectance values near zero. Thus, for TMDs the previously introduced layer stacks utilizing the first minimum of reflectance within the visible wavelength range are rarely suited for counting the number of layers by a


30

Page 31 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


linear wavelength shift. Therefore, in a further optimization step another, i.e. the second or third reflectance minimum, occurring for a significantly higher optical path length (controllable via film thickness adjustment) was utilized. In this case, the absolute value of reflectance is higher and a linear wavelength shift can be achieved. For this purpose, the Si3N4 layer thickness was increased for the third layer stack system leading to a layer stack improved for TMDs consisting of a variable thickness of Al2O3 on 150 nm

Si3N4

on

11 nm

SiO2

on

a

Si

substrate.

The

resulting

reflectance spectra for MoX2 (X = S, Se, and Te) are depicted in Figure 6. The corresponding wavelength shifts Δλ, reflectance minima

for

zero

and

ten

layers

(λ0

and λ10)

as well as the

linearities R2 of the wavelength shifts for these three TMDs are shown. For this layer stack, several Al2O3 layer thicknesses can be found in order to achieve a high linearity (R2) and a high wavelength shift Δλ within the visible wavelength range as shown in Figure 6. In detail, suitable Al2O3 thicknesses are 5 nm, 35 nm, and 45 nm for MoS2, MoSe2, and MoTe2, respectively. The wavelength shifts Δλ for those layer stacks are as high as 8.4 nm per layer for MoS2, 12 nm per layer for MoSe2, and 7.8 nm per layer for MoTe2. The corresponding values for R2 are 98.59%, 98.64%, and 98.48%, respectively. Thus, the linearity represented by R2 is slightly lower for the TMDs compared to the other 2D materials considered previously. Due to the pronounced wavelength shift the actual


31


number of layers can be determined very precisely for TMDs, using the introduced simple optical method and an optimized multi-layer stack system. The same is valid for the other introduced layer stack systems. Figure S6(a)-(c) shows exemplarily the results for SiO2 with a thickness between 0 nm and 400 nm on silicon substrate utilizing the second and third reflectance minimum. The effect of interfering oscillator frequencies is clearly visible where the data of λmin show discontinuities in the case of TMDs having band gaps within the visible spectral range. a)

10 in nm

800

b) c) MoS2 MoSe2 MoTe2 layer stack: Al2O3 / 150 nm Si3N4 / 11 nm SiO2 / Si substrate 800 800 I) I) 700 700

I)

700 600

600

600

500

500

500

400

400

400

0

25 50 75 100 tAl O in nm 2

R2 in %

 in nm/layer

min in nm

800 700 600

0

3

min,0 min,10

2

800 II)

25 50 75 100 tAl O in nm

700

min,0 min,10

700

600

600

500

500

500

400 8

400 12

400 8

6

10

4

8

2 100

6 100

100

95

95

95

0.000

25 50 75 100 tAl O in nm 2

800 II)

1.000

0.0 0

3

1.0

reflectance

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 45

3

min,0 min,10

II)

100.0

6

99.5 99.0

4

40 60 8 100.0 99.8 99.6

90

90 0 10 20 30 40 50 60 tAl O in nm 2

3

0

90 0 10 20 30 40 50 60 tAl O in nm 2

3

0 10 20 30 40 50 60 tAl O in nm 2

3


32

Page 33 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 6. I) Reflectance plots of 10 layers (a) MoS2, (b) MoSe2, and (c) MoTe2 on Al2O3 / 150 nm Si3N4 / 11 nm SiO2 on Si substrate for different thicknesses of the topmost layer. II) Top: wavelength of reflectance minimum of the substrate without and with 10 layers of 2D material (i.e., min,0 and min,10, respectively), center: wavelength

shift

determination.

per

layer,

Calculations

were

and

bottom:

done

for

coefficient ideal

of

measurement

system, i.e. for normal light incidence without NA correction.

In the shown calculations no decrease in linearity is observed at larger layer numbers (tested for up to 25 layers). However, the complex refractive index changes for larger number of layers which has to be taken into account. Furthermore, the presented approach is presumably not suitable for a proper determination of the number of layers of 2D heterostructures without further refinement. This is because stacked 2D materials with similar optical refractive indices

can

hardly

be

discriminated

in

an

optical

manner.

Nevertheless, for 2D materials which show apparent features (e.g. band transitions etc.) in the measurement range a determination could be supported when considering the spectral positions as well as the absolute reflectance of more than one extreme value.

4. Conclusions


33


Page 34 of 45

In this study an advanced optical method is introduced enabling the determination of the actual number of atomic layers of 2D materials. Unlike state-of-the art methods which relate on the optical contrast, the novel technique evaluates the wavelength shift of a distinct reflectance minimum. For the presented approach no

reference

spectra

are

necessary

allowing

for

the

characterization of the layer count of 2D materials on defined substrates without the need for calibration or reference sites. Furthermore, the measurement accuracy does only weakly depend on the

sensitivity

of

the

photodetector

because

the

spectral

positions of extrema rather than absolute reflectance values are measured. In addition, the accuracy of the detection of a minimum is not necessarily defined by a single measurement but can be increased by adequate modelling in conjunction with data fitting in the vicinity of the minimum. The concept was experimentally proven by the investigation of graphene flakes on a contrast optimized layer stack

32

consisting of Si3N4 on SiO2 on Si. The

measurements were performed on one sample without graphene and samples coated with nominally 1 to 4 as well as two unknown distributions of several graphene layers purchased with nominally 6-8 layers and 3-5 layers in thickness. A good correlation with theoretically calculated values was observed. The experimentally determined wavelength shift Δλ of the reflectance minimum was 3.5 nm per graphene layer with a high linearity (R2 = 99.8%) whereas


34

Page 35 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the calculations yield Δλ of 2.8 nm per layer and R2 of 99.93%. The

method

was

shown

to

be

applicable

for

large

area

scans

(demonstrated: 3 mm x 3 mm) and to deliver appropriate acquisition times

of

a

few

seconds

properties,

i.e.

materials,

remain

correctly.

Residuals

characterized

as

interpretation. linearity

of

the

per

complex

as

adsorbed in

wavelength

they

to

order

Furthermore,

However,

refractive

challenging

well

the

spectrum.

to

the

indices

have

the

to

avoid

an

was

the

to

2D

determined

have

to

incorrect

as

shown

sample

of

be

specimen

magnitude

shift

the

well be

as

be data the

strongly

influenced by the optical properties of the subjacent layer stack. Theoretical considerations show that the method is applicable to various

other

2D

materials

including

transition

metal

dichalcogenides considering their particular optical properties, i.e. the complex refractive indices. For this purpose, a sequence is presented allowing for the utilization and optimization of the determination of the number of layers of various 2D materials on arbitrary layer stacks. Calculations of reflectance spectra using the

respective

complex

refractive

indices

were

performed

exhibiting the magnitude and the linearity of Δλ. This approach is used for a theoretical calculation of the wavelength shifts of the reflectance minima of GO, hBN and the TMDs MoX2 (X = S, Se, Te) within the visible spectral range. The results show that subjacent multilayer systems exhibit linear wavelength shifts over broader


35


Page 36 of 45

spectral ranges compared to commonly utilized SiO2 layers. Both, the wavelength shift Δλ and its linearity R2 can be tailored for random 2D layer systems by appropriate design of the optical layer stack. This makes the introduced method applicable for layer counting of various 2D materials.

ASSOCIATED CONTENT Supporting Information. PDF showing additional measurement data: 

Additional reflectance spectroscopy mapping of 3-5 layer graphene



Complementary measurements (SEM, AFM, SAED, HRTEM of a FIB cross section and Raman spectroscopy) for thickness determination of the presented graphene samples



Additional data for MoX2 (X = S, Se, Te) on a layer stack of SiO2 on silicon substrate



Complex refractive indices of the investigated dielectric, semiconducting and 2D materials

This information is available free of charge via the Internet at http://pubs.acs.org

AUTHOR INFORMATION Corresponding Author *Email: [email protected] ORCID: https://orcid.org/0000-0001-5484-707X


36

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Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Acknowledgement Financial support by the DFG via the Research Training Group GRK1896 "In situ microscopy with electrons, X-rays and scanning probes" as well as the Cluster of Excellence EAM “Engineering of Advanced Materials” and the SFB 953 “Synthetic Carbon Allotropes” is gratefully acknowledged. Notes The authors declare no competing financial interest. ABBREVIATIONS TMM transfer matrix method, TMD transition metal dichalcogenide, Gr graphene, GO graphene oxide References (1) Park, H.; Brown, P. R.; Bulović, V.; Kong, J. Graphene as transparent conducting electrodes in organic photovoltaics: Studies in graphene morphology, hole transporting layers, and counter electrodes. Nano Lett. 2012, 12, 133–140, DOI: 10.1021/nl2029859. (2) Kisslinger, F.; Ott, C.; Heide, C.; Kampert, E.; Butz, B.; Spiecker, E.; Shallcross, S.; Weber, H. B. Linear magnetoresistance in mosaic-like bilayer graphene. Nat. Phys. 2015, 11, 650–653, DOI: 10.1038/nphys3368.


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Large-Area Layer Counting of Two-Dimensional Materials Evaluating

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