Download Full text - Simon Fraser University

Improvements to Modelling
of Raman Scattering Intensity
for Molybdenum Disulfide
by
Yule Wang
B.Sc., Harbin University of Science and Technology, 2012
Thesis Submitted in Partial Fulfillment
of the Requirements for the Degree of
Master of Science
in the
Department of Physics
Faculty of Science
© Yule Wang 2015
SIMON FRASER UNIVERSITY
Summer 2015
All rights reserved.
However, in accordance with the Copyright Act of Canada, this work may be
reproduced without authorization under the conditions for ”Fair Dealing”.
Therefore, limited reproduction of this work for the purposes of private study,
research, criticism, review and news reporting is likely to be in accordance
with the law, particularly if cited appropriately.
APPROVAL
Name:
Yule Wang
Degree:
Master of Science
Title:
Improvements to Modelling of Raman Scattering Intensity for Molybdenum Disulfide
Examining Committee:
Chair: Dr. Eldon Emberly
Professor
Dr. Karen Kavanagh
Senior Supervisor
Professor
Dr. George Kircznow
Supervisor
Professor
Dr. Michael Chen
Internal Examiner
Senior Lecturer
Date Approved:
August 24th, 2015
ii
Abstract
Single layer molybdenum disulfide (MoS2 ) has attracted significant interest as a semiconducting two-dimensional material. In this work, ultrathin layers of MoS2 were exfoliated on gel
and wax substrates. Raman studies of the ultrathin layers of MoS2 were carried out to characterize the thickness. During Raman measurement, an anomaly occurs: the strongest Raman
intensity appears at a finite thickness of the MoS2 , a phenomenon also seen in graphite. A
previous work [Y. Y. Wang et al., Appl. Phys. Lett. 92, 043121 (2008)] theoretically explained
this unexpected phenomenon but with some questionable details. To improve their model, the
complex index of refraction was properly used in the optical equations, and the assumption of
optical interference of Raman signals from an infinitesimal source was added. This modified
model successfully predicted the experimental results when applied to graphite and was then applied to MoS2 . The simulations indicated that a constant complex index of refraction for MoS2
of 6.5-1.7i provided the best fit to published experimental results for MoS2 on SiO2 /Si. This
helped to estimate the thickness of ultrathin layers on gel. Seven layers MoS2 on gel showed
the strongest Raman intensity according to this simulation.
Keywords: MoS2 ; Raman spectroscopy
iii
Acknowledgements
I would first express my sincere gratitude to my supervisor, Dr. Karen Kavanagh. Her insight
and patience has guided me throughout my research. Her friendship and mentorship helped me
adapt to living in this foreign land.
I also thank all the professors who give me ideas for physics and research. Thanks to Dr.
George Kirczenow’s valuable suggestions for my project. I am very grateful for Dr. Malcolm
Kennett’s patient help for my physics study, answering my endless questions. I appreciate Dr.
Mike Hayden and Dr. Andrew DeBenedictis about the ideas of electromagnetic theory so that
I can better understand the model in this thesis.
I am also thankful for the invaluable thesis writing advice I received from my generous friends
Christoph, Philip, Clayton, Nicholas, Etienne, and Alex!
I appreciate the help of Li Yang who showed me how to master nanoimaging techniques.
I also would like to acknowledge Simon Fraser University, the Department of Physics, NSERC
(Natural Sciences and Engineering Research Council of Canada) and 4D labs.
iv
Contents
Approval
ii
Abstract
iii
Acknowledgements
iv
Contents
v
List of Tables
vii
List of Figures
viii
1 Introduction
1
1.1
Overview of MoS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.3
Introduction to MoS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3.1
Crystal Structures of MoS2 . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3.2
The band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3.3
Raman spectra of the E12g and A1g modes . . . . . . . . . . . . . . . .
5
1.3.4
Raman determination by intensity . . . . . . . . . . . . . . . . . . . .
6
2 Background of Raman spectroscopy
2.1
8
Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.1
Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.2
Optical Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.3
Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.1.4
Interatomic Force Constant . . . . . . . . . . . . . . . . . . . . . . . .
12
v
3 Modelling of Raman Peak Intensity
14
3.1
The Previous Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.2
The Revised Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.2.1
Revisions made to the Previous Model . . . . . . . . . . . . . . . . . .
16
3.2.2
Application to Graphite . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2.3
Application to MoS2
. . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.2.4
Additional discussion of this model . . . . . . . . . . . . . . . . . . . .
25
4 Sample Preparation and Characterization
4.1
4.2
Substrate types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.1.1
SiO2 /Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.1.2
Gel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.1.3
Wax/Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.1.4
Copper Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
Transmission Electron Microscopy (TEM) . . . . . . . . . . . . . . . . . . . .
32
5 Optical Results
5.1
5.2
29
35
Raman Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
5.1.1
Thickness Determination of the MoS2 Films . . . . . . . . . . . . . . .
35
5.1.2
Raman Peak Frequency . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5.1.3
Raman Peak Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
5.1.4
Raman Line Width . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
5.1.5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Absorption Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
6 Conclusions
46
Bibliography
47
Appendix A Derivation of the Effective Reflection Coefficient of the Sample/SiO2 /Si 52
vi
List of Tables
3.1
Experimental and simulated results of the Raman peak intensity of graphite on
300 nm SiO2 /Si with and without considering the interference of Raman signals.
Listed are the number of layers where the strongest and minimal Raman peak
intensity occur, as well as intensity ratios compared to the bulk Ibulk .
. . . . .
21
. . . . . . . . . . . . . . . . . .
24
3.2
Refractive index of MoS2 in three references
3.3
Comparison of experimental and simulated Raman intensities for the E and A
modes. The simulation assumed both modes have the same intensity for MoS2
on 285 nm SiO2 /Si and used 4 different complex refractive indices of MoS2
reported in 3 different references1 . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
25
Simulated Raman peak intensity for MoS2 on gel compared to that on 285 nm
SiO2 /Si. (Strongest (layers): the thickness where the strongest Raman peak
intensity appears. Istrongest , Ibulk and I2L : the strongest intensity, the intensity
of the bulk and two layers, respectively.) . . . . . . . . . . . . . . . . . . . . .
39
5.2
The difference in peak frequencies (cm−1 ) for the E12g and A1g modes for MoS2 .
40
5.3
Raman peak shift (cm−1 ) from individual measurements from ultrathin layers
compared with the shifts for average measurements for MoS2 on substrates of
285 nm SiO2 /Si [18] and gel. . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
41
List of Figures
1.1
Top view of a single layer and side view of a multilayer crystal structure of 2H
and 1T MoS2 . For 2H MoS2 , the period in the z direction is two-layers, whereas,
1T MoS2 repeats itself every single layer. Also in a single layer of 1T phase, the
sulfur atoms in the upper and bottom sublayers do not sit at the same positions.
This can be seen by rotating the lower sulfur sublayer 60◦ compared to the upper
sublayer.
1.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Theoretical simulations show that the evolution of the band structure of 4-1 layers
of MoS2 (a→ d). The direct gap at the K point hardly changes while the indirect
gap gets bigger as the thickness decreases, resulting in the transformation from
an indirect to direct gap from bilayer to single-layer. Reprinted with permission
from Ref [11]. Copyright 2010 American Chemical Society. . . . . . . . . . . .
1.3
4
(a) A simplified band structure of MoS2 near the K point. Excitons A and B
correspond to the direct band gap at the K point. The splitting at the valence
band is caused by spin-orbit splitting. (b) Exciton excitation peaks A and B in
the absorption spectrum at 135K. Panel b reprinted with permission from [Ref
[13] as follows: RF Frindt, Physical Review, 140, A536 and 1965.] Copyright
(1965) by the American Physical Society. . . . . . . . . . . . . . . . . . . . .
1.4
4
The in-plane mode E12g and the out-of-plane mode A1g for multiple-layers MoS2 .
Reprinted with permission from [Ref [19] as follows: Alejandro Molina-Sanchez
and Ludger Wirtz, Physical Review B, 84, 15 and 2011.] Copyright (2011) by
the American Physical Society. . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
5
1.5
(a) Raman peak frequencies (left vertical axis) and the frequency differencies
(right vertical axis) of the E12g and A1g modes as a function of thickness. (b)
The integrated Raman peak intensity of both modes (left vertical axis) and their
ratio (right vertical axis). Panels a and b reprinted with permission from Ref
[18]. Copyright 2010 American Chemical Society. . . . . . . . . . . . . . . . .
2.1
Raman scattering in a lattice. The lattice absorbs incident photons and emits
red-shifted new photons with the corresponding vibrational mode excited. . . .
2.2
9
Optical Set-up: the Raman signal scattered from the sample will be reflected
and collected by the spectrometer, for spectrum analysis. . . . . . . . . . . . .
2.3
6
10
Energy diagrams of Rayleigh, Stokes, and Anti-Stokes Scatterings. Elastic Rayleigh
scattering takes no frequency shift; Stokes scattering scatters red-shifted light
whose frequency shift is the frequency of the excited vibrational mode; while
anti-Stoke scattering is blue-shifted. . . . . . . . . . . . . . . . . . . . . . . .
2.4
Transverse vibrational modes of atoms in a one-dimensional monatomic lattice
with quantized k values.
2.5
10
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Theoretical simulation (lines) and experimental result (dots) of the dispersion
relation, the frequency of the E12g and A1g modes as a function of q for bulk
MoS2 . Reprinted with permission from [Ref [19] as follows: Alejandro MolinaSanchez and Ludger Wirtz, Physical Review B, 84, 15 and 2011.] Copyright
(2011) by the American Physical Society. . . . . . . . . . . . . . . . . . . . .
3.1
13
Raman measurement process: the absorption and multiple reflections of excitation light (a) and scattering signal (b). The direction of excitation light is normal
to the surface of the sample/SiO2 /Si in a real Raman experiment, as shown in
the diagram (c). The Raman scattering, however, will propagate in all directions. 15
3.2
The experimental result for G-band of graphite [6] (black dots) and the previous
simulated result [6] (red line). . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
17
Raman scattering upwards (orange) and downwards (red) from the same infinitesimal source in the sample interfer. . . . . . . . . . . . . . . . . . . . . .
ix
19
3.4
Three simulated results of Raman peak intensity as a function of thickness for
graphite: the previous simulated result (red) [6], the simulated result ignoring
interference of Raman signals scattering from the same infinitesimal point (blue
dashed), and a simulated result considering this interference (black). Inset: a
magnified plot at thinner thickness. . . . . . . . . . . . . . . . . . . . . . . . .
3.5
Experimental results for G-band of graphite [6] (dots) and the simulated result
of the revised model (line). Inset: a magnified view of the thinner layer region.
3.6
21
22
(a) Experimental results for G-band of graphite [6] (dots) and the simulated result
using 2.675-1.35i (line). Inset: a magnified view of the region of 40-150 layers.
(b) Experimental results for G-band of graphite [6] (dots) and the simulated
result using 2.52-1.94i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7
23
Simulated results of Raman intensity as a function of thickness for MoS2 with
different refractive indices: 3.1-0.65i (red dashed), 4.8-1.15i (red), 6.5-3i (black)
and 5.25-1.7i (blue). Inset: intensity at thinner thickness.
3.8
24
The peak layer (the number of layers where the strongest intensity apprears) as
a function of the thickness of SiO2 for MoS2 on SiO2 /Si.
3.9
. . . . . . . . . . .
. . . . . . . . . . .
26
An energy diagram for Raman scatterings of the E12g and A1g modes. The
difference of Raman shift between the two modes is 20 cm−1 . The energy
difference between E12g and A1g is h̄∆ωv = 0.1kT.
. . . . . . . . . . . . . . .
27
3.10 The simulated result using a refractive index of 6.5-1.7i (blue dashed) to fit the
experimental result [18] of MoS2 on 285 nm SiO2 /Si near the peak in integrated
intensity. Since it cannot explain the lower intensity for a monolayer, a transition downwards in y axis was made in the simulated result to better fit the
experimental result (black solid).
4.1
. . . . . . . . . . . . . . . . . . . . . . . .
28
(a) Optical image of exfoiated MoS2 on 100 nm thick SiO2 /Si substrate. (b)
The Raman Spectrum of this thin region (red circle region in the optical image).
The difference of the two modes may indicate that this might be three or four
layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.2
1-4 layers were produced by exfoliation and deposited onto gel.
31
4.3
Optical image of ultrathin MoS2 near an edge with many layer thicknesses (a)
. . . . . . . .
and a single layer (b) on wax. The single layer was as large as 10× 40 µm2 .
x
.
32
4.4
A Bright field image (a) and selected area diffraction pattern (b) of the ultrathin
film (red circle) on the edge. 2H MoS2 was identified.
4.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
(a) and (b) are a BF and SAD image of a polycrystalline MoS2 . The SAD image
indicates a misalignment for these crystals.
5.1
33
(a) and (b) are BF and SAD images of a single crystal. The periodic fringes
were likely stacking faults.
4.6
. . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
34
Curve fitted results (solid lines) of original Raman spectra (dots) with a resolution
of 1.5 cm−1 of the E12g and A1g modes for ultrathin layers and bulk MoS2 . The
thickness was determined by Raman peak frequency difference of the two modes.
(The uncertainty for the single measurement is 0.5 cm−1 .) . . . . . . . . . . .
5.2
36
Single measurements of the peak intensity as a function of the difference of the
peak frequencies in E12g (black) and A1g (red) modes for MoS2 on gel (a) and
on wax (b). The thickness of 1-4 layers and bulk were confirmed. As for thicker
layers, the thickness of layers were roughly determined by the curve fit. The curve
fit (purple) was the simulated result in the revised model discussed in Chapter 2,
in which the refractive index of 6.5-1.7i was used. 4.8-1.15i was used to predict
the intensity of the bulk as it made a reasonable prediction of I peak : Ibulk for
MoS2 on 285 nm SiO2 /Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
37
(a) Plots of simulated Raman peak intensity of the A1g or E12g modes of MoS2 on
gel as a function of thickness using the refractive indices of MoS2 of 3.1-0.65i,
4.8-1.15i, 6.5-3i, and 5.25-1.71i reported by Ref [28] [29] [30] and a selected
refractive index of 6.5-1.7i. (b) A magnified figure at thinner thickness. . . . .
5.4
38
(a) Raman peak frequency as a function of thickness in the A1g and E12g modes
of MoS2 on gel (black) and on wax (red) (left vertical axis) and their difference
(gel grey and wax red right vertical axis). (b) Raman peak intensity as a function
of thickness for the A1g and E12g modes of MoS2 on gel (black) and on wax (red).
Simulation (blue solid) with the refractive index of 6.5-1.7i was used to fit the
experimental results. (c) Raman peak line widths as a function of thickness for
the A1g and E12g modes of MoS2 on gel (black) and on wax (red). . . . . . . .
5.5
42
The integrated intensity as a function of the thickness of MoS2 on gel and on
wax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
44
5.6
Excitonic peaks A at 2.05 and B at 1.86 eV appeared for 83-95 layers at room
temperature. [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
45
Chapter 1
Introduction
1.1
Overview of MoS2
Two-dimensional materials, single-layer crystals of atomic thickness, including graphene and
boron nitride, have attracted much interest in the last few decades. When electrons are confined
to move in two dimensions, several extraordinary applications arise due to this physical effect.
For example, the quantum Hall effect happens in a two-dimensional systems usually at low
temperatures. Recently, however, an integer quantum Hall effect was realized in graphene at
room temperature [1] and theoretically a prediction of the existence of anomalous quantum Hall
effect in honeycomb materials at higher temperatures was made [2]. Graphene, one of today’s
most interesting two-dimensional materials, has a zero overlap semi-metal band structure, which
makes it a metallic conductor. To achieve semiconducting graphene, band gap engineering for
graphene is required [3], but the decrease of mobility to one tenth is an issue [4]. Thus, a
single-layer molybdenum disulfide (MoS2 ), another honeycomb lattice material, which has a
large intrinsic direct band gap of 1.8 eV, is of interest as it gives us the possibility of a twodimensional semiconductor. Single-layer MoS2 transistors were made with high mobility similar
to that of graphene [5].
1.2
Dissertation Organization
In this work, a Raman spectroscopy study of ultrathin layers of MoS2 was the main focus. A
review of MoS2 and its properties will be presented in Chapter 1. Raman scattering theory will
be introduced in Chapter 2. In Chapter 3, the model proposed in Ref [6] for predicting Raman
1
intensity will be introduced. However, this model has made a few physical and mathematical
mistakes. I developed a revised model to fix these problems and an assumption was proposed
to include interference of the Raman scattering signal, which made possible predictions more
consistent with the experimental results. Chapter 4 describes transmission electron microscope
defect analysis of raw MoS2 specimens and the preparation of samples on different substrates for
Raman analysis (and absorption spectrum analysis). In Chapter 5, my experimental result for
ultrathin layer (and bulk) MoS2 will be discussed in detail. Additionally, the absorption spectrum
of thick MoS2 at room temperature will be briefly presented. Chapter 6 will summarize the
results.
1.3
1.3.1
Introduction to MoS2
Crystal Structures of MoS2
Two structural phases, the hexagonal (2H) and octahedral (1T) crystal structures, are mostly
found in MoS2 . Figure 1.1 presents the crystal structures of the 2H and 1T phases in top view
and side view. The 1T phase of MoS2 is metallic. But the 2H crystal structure, which has
a band gap, is the main focus of my project. The semiconducting 2H phase of MoS2 is more
stable than the 1T phase at room temperature [7]. Ion intercalation will transform the 2H phase
of MoS2 to the 1T phase [8], and revert the 1T to the 2H phase by annealing at 300 ◦ C [7].
The 2H MoS2 layers are bonded by weak Van der Waals forces, with a thickness of 0.65 nm for
each layer [9]. Therefore, mechanical exfoliation is a very common method to produce few and
single layers [10].
1.3.2
The band structure
One interesting aspect of MoS2 is the transition from the indirect band gap as small as 1.2
eV in multiple layers to the direct band gap of 1.9 eV in the single layer. Theoretical simulations
[11](2010) as shown in Figure 1.2 give the evolution of the band structure of MoS2 from multiple
layers to a single layer. Clearly, a direct band gap exists at the K point in the Brillouin zone in
multiple layers, and this gap remains at about 1.9 eV, hardly affected by the number of layers.
(It only changes by about 0.05 to 0.1 eV.) As the number of layers decreases, the top edge of
the valence band at the Γ point declines, and the bottom edge of the conduction band between
2
Figure 1.1: Top view of a single layer and side view of a multilayer crystal structure of 2H and
1T MoS2 . For 2H MoS2 , the period in the z direction is two-layers, whereas, 1T MoS2 repeats
itself every single layer. Also in a single layer of 1T phase, the sulfur atoms in the upper and
bottom sublayers do not sit at the same positions. This can be seen by rotating the lower sulfur
sublayer 60◦ compared to the upper sublayer.
the Γ and K points rises, enlarging the indirect band gap. For the single layer, the direct gap at
the K point is smaller; therefore the transition from indirect to direct gap happens.
Experimental evidence of the band gap can be found in absorption and photoluminescence
spectra. At room temperature, exciton peaks in the absorption spectrum of MoS2 were first
observed by Frindt and Yoffe in 1963 [12][13]. There are two exciton peaks A and B at 1.88
and 2.06 eV at room temperature associated with the direct-gap transitions at the K point in
the Brillouin zone. They correspond to the spin-orbit splitting of the valence band at the K
point [14][15] as shown in Figure 1.3(a). This experiment was repeated and will be discussed
in Chapter 4. Photoluminescence studies, on the other hand, confirm the indirect band gap for
the multiple layers [11][16] .
Frindt also made an effort [13] to understand the temperature evolution of excitons A and
B in thick MoS2 . Figure 1.3(b) shows peaks A and B in the absorption spectrum for MoS2 at
135 K. As for a thin film, photoluminescence with changing temperature from as low as 4K to
room temperature has been observed [17], which will help understand the band structure at a
low temperature.
3
Figure 1.2: Theoretical simulations show that the evolution of the band structure of 4-1 layers
of MoS2 (a→ d). The direct gap at the K point hardly changes while the indirect gap gets
bigger as the thickness decreases, resulting in the transformation from an indirect to direct gap
from bilayer to single-layer. Reprinted with permission from Ref [11]. Copyright 2010 American
Chemical Society.
(a)
(b)
Figure 1.3: (a) A simplified band structure of MoS2 near the K point. Excitons A and B
correspond to the direct band gap at the K point. The splitting at the valence band is caused by
spin-orbit splitting. (b) Exciton excitation peaks A and B in the absorption spectrum at 135K.
Panel b reprinted with permission from [Ref [13] as follows: RF Frindt, Physical Review, 140,
A536 and 1965.] Copyright (1965) by the American Physical Society.
4
1.3.3
Raman spectra of the E12g and A1g modes
Raman scattering describes an inelastic scattering process in which photons are absorbed
by the lattice and new photons are then emitted with a red-shifted frequency, caused by the
harmonic vibrations of the atoms. The E12g and A1g modes are two vibrational modes commonly
used to identify the thickness of ultrathin layers of MoS2 . Figure 1.4 shows that E12g is an
in-plane mode where sulfur atoms in the upper and lower sub-sheets in one single layer are
vibrating in phase, while the Mo atoms between them have opposite vibrations. Meanwhile,
corresponding atoms in adjacent layers have inverted behaviours. A1g is an out-of-plane mode,
in which Mo atoms do not vibrate, while S atoms in two sublayers are moving in the opposite
phases. Also, S atoms in adjacent layers are moving out of phase.
We expected that the frequency of both E12g and A1g modes would rise with increasing
thickness due to the interaction of more layers. Experimental results confirm this prediction for
the A1g mode [18]. But for the E12g mode, additional layers reduce its frequency [18]. Work in
Ref [19] demonstrated that a significant rise of the dielectric tensor (dielectric screening) was
the reason for the reduction of the frequency with increasing thickness.
Figure 1.4: The in-plane mode E12g and the out-of-plane mode A1g for multiple-layers MoS2 .
Reprinted with permission from [Ref [19] as follows: Alejandro Molina-Sanchez and Ludger
Wirtz, Physical Review B, 84, 15 and 2011.] Copyright (2011) by the American Physical
Society.
5
(a)
(b)
Figure 1.5: (a) Raman peak frequencies (left vertical axis) and the frequency differencies (right
vertical axis) of the E12g and A1g modes as a function of thickness. (b) The integrated Raman
peak intensity of both modes (left vertical axis) and their ratio (right vertical axis). Panels a
and b reprinted with permission from Ref [18]. Copyright 2010 American Chemical Society.
1.3.4
Raman determination by intensity
Figure 1.5 shows the Raman peak frequency of the E12g and A1g modes (left vertical axis) and
their frequency difference (right vertical axis) as a function of MoS2 thickness. A Raman shift of
a specific vibrational mode equals the frequency of the corresponding vibrational mode, which
is an intrinsic property caused by interactions in the periodic lattice. For MoS2 , this provides an
effective way to identify 1-4 layers and the bulk. However, for a film of more than four layers,
the shift of both modes becomes tiny. Figure 1.5 indicates that the Raman frequency differences
between the two modes are 18.9 ± 0.6, 21.6 ± 0.3, 23.3 ± 0.2, 24.2 ± 0.3, and 25.2 ± 0.2 cm−1
for 1-4 layers. But for a thin film of more than four layers, the increment of an additional layer
is less than 0.3 cm−1 . Therefore, other Raman information, such as the Raman intensity, should
be investigated to help with determining the thickness.
Experiments have shown that the strongest Raman peak intensity peaks at a finite number
of layers for MoS2 (and graphite). Figure 1.5(b) [18] shows the integrated intensities of the
two modes as a function of the thickness for MoS2 on 285 nm SiO2 /Si: the strongest Raman
intensity appears at four layers. This anomalous behavior of the Raman intensity contradicts our
intuition that the strength of the Raman signal should always increase with increasing thickness
of the material. Ref [19] successfully resolved this paradox when applied to graphite. The
model (Y. Y. Yang et al. (2008)) succeeded by considering multiple reflections in the material.
6
However, there are questionable details in this model. I will address the deficiencies of this
model and present a modified theory in Chapter 3.
7
Chapter 2
Background of Raman spectroscopy
Raman Spectroscopy is a technique based on an inelastic scattering effect, called Raman
scattering. The effect was discovered by C.V Raman in 1928, who was later awarded the Nobel
Prize for his contributions. It is widely used to analyze the structure of a material, as Raman
scattering is affected by the vibrational modes of the atoms. The number of layers of graphite
and MoS2 in a film can be effectively determined [18][20] by their Raman shift, which is the
frequency shift between the incident and scattering photons. In this chapter, I will discuss the
mechanism of Raman scattering.
2.1
2.1.1
Basics
Raman Scattering
Raman scattering describes an inelastic scattering process in which photons are absorbed
by the lattice (or molecules) and new photons are then emitted with a red-shifted frequency
(Figure 2.1). This red-shift is caused by harmonic vibrations of the atoms in the lattice. In lattice
dynamics, we call the collective excitations of such vibrations phonons. As the vibrational modes
depend on the interactions in the lattice, the technique of Raman Spectroscopy is very well-suited
to the identification of different materials or to different structures within the same materials.
2.1.2
Optical Set-up
The Raman Spectroscopy set-up is shown in Figure 2.2. A laser beam with wavelength λ0
(a 50 mW laser of 1 mm beam diameter at laser aperture (1% power) at 514 nm was used as
the excitation light in these experiments in my project.) is normally incident on the samples
8
Figure 2.1: Raman scattering in a lattice. The lattice absorbs incident photons and emits
red-shifted new photons with the corresponding vibrational mode excited.
(observed by an optical microscope of an objective with an magnification of 50X), and Raman
scattered photons with wavelength λ1 are emitted. The Raman signal is then collected and
analyzed by a spectrometer. The observed frequency shift is equal to the frequency of the
Raman active vibrational mode ωv . It is proportional to the difference between wavenumbers
of the excitation and scattering light (wavelengths λ0 and λ1 , respectively), defined as ∆ωn
(cm−1 ),
ωv
1
1
= ∆ωn = ( − ),
2πc
λ0
λ1
(2.1)
where c is the speed of light.
Raman signals arising from inelastic, Stokes and anti-Stokes scattering, are weak relative to
elastic Rayleigh scattering.
Rayleigh, Stokes and Anti-Stokes scattering
There are three types of scattering: Rayleigh, Stokes and Anti-Stokes as shown in Figure
2.3. An excitation of a vibrational mode of frequency ωv will generate a new state S1 . When
the lattice (molecule) absorbs an incident photon, it is excited from the ground state S0 to a
virtual state Sv , a very short-lived state. If it decays back to the ground state S0 , it undergoes
Rayleigh scattering. If instead the excited state relaxes from Sv to S1 , red-shifted photons will
be emitted and Raman scattering (Stokes scattering) occurs. The difference in energy between
S1 (E1 ) and S0 (E0 ) is related to the frequency ωv of the associated phonon:
h̄ωv = ∆( E) = E1 − E0 .
9
(2.2)
Figure 2.2: Optical Set-up: the Raman signal scattered from the sample will be reflected and
collected by the spectrometer, for spectrum analysis.
Figure 2.3: Energy diagrams of Rayleigh, Stokes, and Anti-Stokes Scatterings. Elastic Rayleigh
scattering takes no frequency shift; Stokes scattering scatters red-shifted light whose frequency
shift is the frequency of the excited vibrational mode; while anti-Stoke scattering is blue-shifted.
10
Blue-shifted anti-Stokes scattering, in contrast, happens when the S1 state is excited to the
virtual state, and then decays to the ground state S0 . Because the S1 state is less populated
than the ground state, this process happens less often compared to Stokes scattering. Coherent
anti-Stokes Raman spectroscopy is a technique based on this scattering mechanism.
2.1.3
Phonons
A phonon is a collective excitation of harmonic vibrations of atoms in a periodic lattice. If
the wavefunctions Φ(r) of atoms satisfies the Born-von Karman boundary condition:
Φ(r + Ni ai ) = Φ(r),
(2.3)
then the lattice waves have quantized k in the Brillouin zone.
Figure 2.4 describes an example of the vibrations of atoms with frequency ωv in a periodic
one-dimensional monoatomic lattice, in which k can only take
2π
Na
(where a is the spacing between
two atoms, N is the number of the atoms in the lattice, and n is an integer). Only when N is
very large, does the ∆k go to zero, where ∆k is the spacing between the two quantized k values.
Figure 2.4: Transverse vibrational modes of atoms in a one-dimensional monatomic lattice with
quantized k values.
11
As compared to real particles, the creation and annihilation of phonons is described by a
virtual momentum h̄q:
h̄k′ = h̄k ± h̄q.
(2.4)
And the energy levels dependent on k space, are described by
E(k′ ) = E(k) + h̄ωv .
2.1.4
(2.5)
Interatomic Force Constant
The interatomic force constant Ckα,kβ ( a, b) describes the correlations between atom k in unit
cell a and atom k′ in unit cell b in a lattice. It is strongly affected by the energy of the lattice
Etot :
Ckα,kβ ( a, b) =
∂2 Etot
,
a ∂ub
∂ukα
k′ β
(2.6)
where uka′ β is the displacement of atom k in cell a in the α direction. Its Fourier transform C̃kα,k′ β
is correlated to the angular frequency ω of the lattice wave,
C̃ (q)
2
√
−
ω
M M′
= 0,
k k
(2.7)
where Mk and Mk′ are the masses of the two atoms.
From the force constant matrix C̃ (q) corresponding to a specific mode, the dispersion
relation and the frequency of that mode as a function of q, can be calculated. (For further
derivations please refer to Ref [21].) The vibrational modes at the Γ point where q → 0
contributes to the first-order Raman scattering. Higher-order Raman scattering (non-zero q)
also happens, but the intensity is much weaker than the first order Raman scattering.
Interatomic interactions are mainly dominated by dipole-dipole interactions (and quadrupolequadrupole interactions), thus the interactomic constant is described in the E2g and A1g modes
as [21]:
Ckα,kβ ( a, b) =
dα d β
Zk Zk′ δαβ
( 3 − 3 5 ),
ϵ
d
d
(2.8)
where Zk and Zk′ are effective charges between two atoms, d is the distance between two atoms
and ϵ is the dielectric constant.
Figure 2.5 is a plot of the ab-initio calculated dispersion relation, which is the vibrational
frequencies versus q for different vibrational modes in the Brillouin zone (lines) for bulk MoS2 .
12
The higher-order experimental Raman scattering results are presented as dots [19]. The strong
first-order Raman scattering happens at Γ (0,0,0).
Figure 2.5: Theoretical simulation (lines) and experimental result (dots) of the dispersion relation, the frequency of the E12g and A1g modes as a function of q for bulk MoS2 . Reprinted
with permission from [Ref [19] as follows: Alejandro Molina-Sanchez and Ludger Wirtz, Physical
Review B, 84, 15 and 2011.] Copyright (2011) by the American Physical Society.
13
Chapter 3
Modelling of Raman Peak Intensity
3.1
The Previous Model
To explain why the strongest intensity appears at finite thickness for graphite, a model (Y.
Y Yang et al. (2008)) [6] succeeded by considering multiple reflections of incident light and
scattering Raman signal. Two stages were considered (Figure 3.1): (a) The incident photons
are absorbed by the sample so that atoms in the lattice are excited to higher energy state(s).
(b) The Raman signal caused by the excitation of a specific vibrational mode is emitted.
In stage (a): A laser beam of wavelength λ perpendicularly irradiates the surface of the
sample with thickness d1 . Since the beam will be infinitely internally reflected between the
sample/SiO2 and the sample/air interfaces, every infinitesimal point at depth y is assumed to
have an electric field caused by infinite reflections of the excitation light.
The electric field caused by the transmitted light propagating downwards before the first
reflection at the sample/SiO2 interface is given by:
t1 e β1 y e−i(2π ñ1 y/λ) ,
(3.1)
where t1 = 2n0 /(n0 + ñ1 ) (n0 and n˜1 = n + ik are the complex refractive indices of the
air and the sample, respectively) is the electric field transmission coefficient at the air/sample
interface, and β depends on the extinction coefficient k (k = 1.3): β = −2πk/λ, describing
the absorption of the electric field in a material.
After that, the light reflects upwards, and the electric field becomes:
t1 r ′ e β1 (2d1 −y) e−i(2π ñ1 (2d1 −y)/λ1 ) ,
14
(3.2)
Figure 3.1: Raman measurement process: the absorption and multiple reflections of excitation
light (a) and scattering signal (b). The direction of excitation light is normal to the surface
of the sample/SiO2 /Si in a real Raman experiment, as shown in the diagram (c). The Raman
scattering, however, will propagate in all directions.
where r ′ is the effective field reflection coefficient at the sample/SiO2 /Si interface,
r ′ = (r2 + r3 e−4iπ n˜2 d2 /λ )/(1 + r2 r3 e−4iπ n˜2 d2 /λ ).
(3.3)
(Please see the derivation of the effective field coefficient k in Appendix A. The derivation
assumes an infinite thickness of silicon [22].)
To describe the electric field due to the light travelling one more round between the interfaces,
Eq. (3.1) and (3.2) should be multiplied by
r1 r ′ e
−4π ñ1 d1
λ1
e2βd1 ,
(3.4)
where r1 = (ñ1 − ñ0 )/(n˜0 + ñ1 ) is the electric field reflection coefficient at the sample/air
interface. Thus, the electric field t at depth y is described by a geometric progression:
t=
t1 e βy e−i(2π ñ1 y/λ) + t1 r ′ e β(2d1 −y) e−i[2π ñ1 (2d1 −y)/λ]
1 − r1 r ′ e
−4π ñ1 d1
λ
15
e2βd1
.
(3.5)
In stage (b), γ is used to describe the electric field due to the non-interfering scattering
signal:
γ=
(e β1 y + r ′ e β1 (2d1 −y) )t1′
,
1 − r1 r ′ e2β1 d1
(3.6)
where t1′ = 2n˜1 /(n˜0 + n˜1 ) is the electric field transmission coefficient at the sample/air interface. (The authors [6] assumed light interference did not exist for the Raman scattering signal.
However, the expression in Eq. (3.6) does not correctly reflect this assumption. In addition, the
assumption is not physical, as will be discussed in section 3.2.)
The probability of emission of a Raman signal at depth y is proportional to the intensity of
the incident light at that point, so the total Raman signal intensity becomes
I=
∫ d1
0
|tγ|2 dy,
(3.7)
which depends on the thickness d1 of the sample.
Figure 3.2 shows the experimental result of Raman spectroscopy [6] with a laser beam of
532 nm. It shows the G-band of graphite (with the thickness of a single layer being 0.335 nm)
on 300 nm SiO2 /Si [6] (black dots) and the simulated result (red line). In this simulation, the
refractive index of air, graphite, SiO2 of thickness d2 and Si were n0 = 1, ñ1 = 2.6 − 1.3i,
ñ2 = 1.46, and ñ3 = 4.15 − 0.044i, respectively. The authors incorrectly used these complex
refractive indices including the imaginary component for equations (3.1-3.6) and the equations
for reflection and transmission coefficients at the interfaces.
This model successfully predicted the intensity at thinner layers, but it could not explain the
intensity of thicker layers (80-150 layers). The experimental results indicated that there was
a minimum at around 150 layers, which was weaker than the intensity of bulk. However, the
model failed to predict this minimum. The reason for this failure is due to some questionable
details in this model, which will be discussed in the next section.
3.2
3.2.1
The Revised Model
Revisions made to the Previous Model
Some mistakes were made in the previous model described above. In this section, I will
discuss these problems in detail and correct them. Mistake 1 was a minor problem, likely due
to the simplification of the model. However, mistakes 2 and 3 were non-trivial, which must be
corrected as they were mathematically and physically wrong.
16
140000
G-band Raman Intensity(a.u)
120000
100000
80000
60000
40000
20000
0
0
100
200
300
400
1000
Number of layers
Figure 3.2: The experimental result for G-band of graphite [6] (black dots) and the previous
simulated result [6] (red line).
1. A minor problem was that the authors assumed that the optical parameters do not depend
on the change of frequency between the incident light and the Raman signal. For example,
the field absorption coefficient depends on the wavelength, as can be seen in β = −2πk/λ.
For the G-band the Raman shift is 1580 cm−1 , so incident light with a wavelength of 532
nm will scatter a Raman signal at 580 nm. Considering this change of frequency the
simulation shows that the intensity will peak at 23.5 layers compared to the peak at 22
layers in the original simulation. For a larger Raman shift, e.g. 2700 cm−1 for the 2D-band
of graphite, the effect on β was underestimated by 30%. However, for a specific mode
with small vibrational frequency, the change of wavelength can be ignored.
2. The imaginary component of the complex refractive index is the extinction coefficient k,
which represents the optical absorption in the material. In Eq. (3.1)
t1 e
−2πk
λ y
e−i(2π ñ1 y/λ1 ) ,
17
(3.1)
e
−2πk
λ y
already represents the loss of the electric field. Using the imaginary part of the
refractive index will doubly consider the absorption. Furthermore, the reflection and transmission coefficients at an interface do not depend on the optical absorption. (k may participate in the effective coefficient of 3-medium interfaces if the intermediate medium has
a non-zero extinction coefficient. But this does not hold for the effective coefficient for
graphite/SiO2 /Si. See its derivation in Appendix A.) Therefore, all the complex refractive
indices ñ in all equations must be converted to the refractive indices containing only the
real component n.
3. The authors intended to make the assumption that the Raman signal from the same
infinitesimal source is not interfering with itself. But there were two problems with their
approach: (a) an incorrect mathematical expression for this assumption and (b) the fact
that this assumption is not physical.
(a) The authors mistakenly constructed the field equation by making a direct summation
of electric field amplitudes as in Eq. (3.6):
γ=
(e β1 y + r ′ e β1 (2d1 −y) )t1′
.
1 − r1 r ′ e2β1 d1
(3.6)
Eq. (3.6) is not an expression of non-interfering light, but rather an expression of the
interference of in-phase light. The right expression for the Raman intensity of a noninterfering signal at depth y is the summation of each individual light intensity from
different optical paths, as
| γ |2 = |
(|e β1 y |2 + |r ′ |2 |e β1 (2d1 −y) |2 )|t1′ |2
,
1 − |r1 r ′ e2β1 d1 |2
(3.8)
(b) Even after γ was modified to Eq. (3.8), a good agreement with the experimental result
could not be achieved (Figure 3.4 blue), indicating that it was not physical to neglect the
interference of the scattering Raman signal. Thus, in order to include the interference,
the field should be constructed considering the phase, which gives
γ=
t1′ e β1 y e−i[2πn1 y/λ1 ] + t1′ r ′ e β2 (2d1 −y) e−i[2πn1 (2d1 −y)/λ2 ]
.
1 − r1 r ′ e2β2 (2d1 −y) e−2i f2
(3.9)
I will now show why the optical interference of Raman signal from the same infinitesimal
source should be coherent.
18
Is the Raman Signal Coherent?
Classically, the phase of an optical wave will be coherent if light of the same frequency
is emitted from an infinitesimal source. Even from an atomic physics standpoint, where
the typical time scale of emission from a specific state in an atom is 10−8 s (and hence
the coherence length is about 3 m), the optical path length can be considered infinite
compared to the sample dimensions and therefore infinite reflections interfere coherently.
Figure 3.3 shows multiple reflections of scattering upwards and downwards (normal to
the surface of the material) from an infinitesimal source at depth y. A coherence length of
3 m means that 107 reflections will interfere with each other in a 100 nm thick material.
Thus, γ can be attained by the summation of all reflections.
Figure 3.3: Raman scattering upwards (orange) and downwards (red) from the same
infinitesimal source in the sample interfer.
Overall, the right mathematical expressions for the intensity, where interference is not
considered, and the electric field, where interference is considered, are Eq. (3.8) and Eq.
(3.9). In next section, I will compare the simulated results when applied to graphite of
these two models to justify that the assumption of the interfering signal is physical.
3.2.2
Application to Graphite
I chose mostly the same parameters as in the simulation of the previous model. The refractive
indices of SiO2 (thickness: 300 nm) and Si were unchanged: n2 = 1.46 and n3 = 4.15. The
excitation light of wavelength λ1 = 532 nm will scatter a λ2 = 580 nm Raman signal in the G
band. The only difference was that the refractive index of graphite ñ1G = 2.88 − 1.5i (calculated
19
by density field theory[23]) was chosen, as the experimental result [6] indicated that the intensity
peaked at a thickness of more than 10 layers. The previous simulation used ñ1g = 2.6 − 1.3i,
but this is just the refractive index of graphene [24] (although the authors did not reference
its source). Though as a comparison I will still show the simulated result using both refractive
indices to show that the success of the revised model is not just due to this change of parameters.
Figure 3.4 shows the simulated results from the previous model (red), the results considering
(black) and ignoring (blue dashed) the interference of Raman signals from an infinitesimal
source in comparison with the experimental results [6]. The inset shows a magnified plot of
the lower thickness region. As Figure 3.2 indicated, the previous simulated result could not
explain the minimum intensity at a finite thickness in the experimental result. This was due to
the mistakes made in the previous model. In contrast, the corrected simulated results with or
without consideration of interference successfully predict a minimum at around 130-150 layers.
However, the simulation ignoring the interference underestimated the ratio between the peak
intensity and the intensity of the bulk.
Table 3.1 lists the details of these simulated results in comparison with the experimental
results, which persuasively indicates that a model considering interference (with ñ1G = 2.88 −
1.5i) was in good agreement with the experimental results. This indicates that the assumption
of the existence of interference is physical. The success of the revised model is not due to the
change of refractive index of graphite, since even using ñ1g = 2.6 − 1.3i still gives a much closer
result compared to the simulation not considering interference.
Figure 3.5 shows the simulated result (line) using the refractive index of 2.8-1.5i and the
experimental results (dots) [6]. The intensity of the simulation was adjusted to fit the experimental data. Clearly, a close agreement is made for 1-9 layers and for the bulk. As for the region
of 15-150 layers, the deviation between the simulated result and the experimental observations
is possibly due to the way the authors determined the thickness [6]. (They determined them
by optical contrast [25].) As compared to the simulation using the refractive index of 2.8-1.5i
predicted by theoretical calculation [23], Figure 3.6(a) and 3.6(b) show the simulated results
(lines) using the refractive indices determined by experimental methods versus the experimental
results (dots) [6]. The refractive index of 2.675-1.35i in Figure 3.6(a) was inferred by the optical reflectance measurements [26]) and 2.52-1.94i in Figure 3.6(b) was inferred by ellipsometric
measurements [27]). Inset figure in Figure 3.6(b) even shows that for the region of 15-150
20
layers, the simulation result using 2.675-1.35i has a closer fit for the experimental result. This
again shows that the assumption of the existence of interference is physical.
Figure 3.4: Three simulated results of Raman peak intensity as a function of thickness for
graphite: the previous simulated result (red) [6], the simulated result ignoring interference of
Raman signals scattering from the same infinitesimal point (blue dashed), and a simulated result
considering this interference (black). Inset: a magnified plot at thinner thickness.
Table 3.1: Experimental and simulated results of the Raman peak intensity of graphite on
300 nm SiO2 /Si with and without considering the interference of Raman signals. Listed are
the number of layers where the strongest and minimal Raman peak intensity occur, as well as
intensity ratios compared to the bulk Ibulk .
Strongest (L)
Minimum (L)
Istrongest : Ibulk
I2L : Ibulk
Experimental results [6]
9-15
around 150
4.7
1.2
Previous simulated results [6]
22
N.A.
5.0
1.7
No interference Eq. (3.8)
27.5
135
1.7
0.34
Interference
ñ1G = 2.88 − 1.5i
18.3
154
4.6
1.3
Eq. (3.9)
ñ1g = 2.6 − 1.3i
22.5
170
4.8
0.9
3.2.3
Application to MoS2
As the revised model was successful when applied to graphite, I will apply it to MoS2 in this
section.
21
Figure 3.5: Experimental results for G-band of graphite [6] (dots) and the simulated result of
the revised model (line). Inset: a magnified view of the thinner layer region.
Table 3.2 lists the range of complex refractive indices reported for MoS2 near 514 nm optical
wavelength as a function of thickness in different references [28][29][30]. For thin layers, n and
k vary from 3.0 to 4.9, and 0.5-1.3 respectively, and for thick layers, the variations are 0.5-1.3
and 1.0-3.8, respectively. (In Ref [29], it was claimed that the sample was a thin layer but there
was no precise indication of the thickness of MoS2 . In Ref [30], no thickness was indicated, but
I assumed it was a thick layer, even bulk.)
As there is no consensus on the refractive index for thin or thick MoS2 , I will simulate using
all these parameters. The thickness of SiO2 was 285 nm and the wavelength of incident light
was 514 nm, the same as the parameters in Ref [18]. The frequencies of the E12g and A1g modes
are 380-384 and 403-408 cm−1 , respectively. To simplify the model, I assumed their frequencies
were both 400 cm−1 , so the wavelengths of the Raman signal of both modes in the simulation
were λ2 = 525.3 nm.
Figure 3.7 shows simulations for the Raman intensity as a function of thickness using four
different refractive indices from three references and a magnified view of the lower thickness region (inset). Since the refractive indices vary significantly, the curves also show a large variation.
Nevertheless, for few-layer thicknesses all curves have a roughly linear relationship.
22
(a)
G-band Raman Intensity(a.u)
140000
120000
100000
80000
60000
40000
20000
0
0
100
200
300
400
1000
bulk
Number of layers
(b)
Figure 3.6: (a) Experimental results for G-band of graphite [6] (dots) and the simulated result
using 2.675-1.35i (line). Inset: a magnified view of the region of 40-150 layers. (b) Experimental
results for G-band of graphite [6] (dots) and the simulated result using 2.52-1.94i.
23
Table 3.2: Refractive index of MoS2 in three references
Substrate
Thickness
n1
k1
<5 L
3-3.2
0.5-0.8
8-31 L
4.7-4.9
1-1.3
[29](2010)
Thin
5.6-7.4
2.2-3.8
[30](1992)
Thick
5.2-5.3
1.6-1.8
[28](2014)
Figure 3.7: Simulated results of Raman intensity as a function of thickness for MoS2 with
different refractive indices: 3.1-0.65i (red dashed), 4.8-1.15i (red), 6.5-3i (black) and 5.25-1.7i
(blue). Inset: intensity at thinner thickness.
Table 3.3 analyzes the curves in Figure 3.7, and compares them with the experimental results
[18]. The simulation predicted the strongest intensity at 3.5 layers when using ñ = 6.5 − 3i
compared to the reported result of 4 layers. But it fails to predict I peak : Ibulk , most likely
due to an underestimation of the intensity of the bulk. Using different refractive indices in the
simulation had a small effect on the strongest intensity (within 15%), whereas the intensity of
the bulk changed significantly: The intensity of the bulk using 3.1-0.65i was 5 times as much
as that of 5.25-1.7i. Thus, the refractive index is likely to change considerably from ultra-thin
layers to the bulk.
24
Table 3.3: Comparison of experimental and simulated Raman intensities for the E and A modes.
The simulation assumed both modes have the same intensity for MoS2 on 285 nm SiO2 /Si and
used 4 different complex refractive indices of MoS2 reported in 3 different references1 .
Experimental result [18]
Strongest (L)
Istrongest : Ibulk
E12g
4
8.4
A1g
4
8
3.1-0.65i
17.9
4.8-1.15i
Istrongest
Ibulk
3.0
13.1
4.3
7.4
9.2
12.8
1.39
5.25-1.7i
5.8
12.9
11.7
0.91
6.5-3i
3.5
24.9
10.2
0.41
ñ1
Simulated results
The Effect of Changing the Thickness of SiO2 on Si
The thickness of SiO2 will periodically affect the intensity as a function of the thickness
of MoS2 (as the effective reflection coefficient of MoS2 /SiO2 /Si is a periodic function of the
thickness of SiO2 ). For an incident light of 514 nm and a Raman signal of 525 nm, the periodicity
of the thickness of SiO2 is 179 nm.
Figure 3.8 shows a rough prediction of the thickness where the strongest intensity occurs
as a function of the thickness of the underlying SiO2 using ñ1 = 6.5 − 3i (as this refractive
index yielded a better prediction of the peak position for MoS2 on 285 nm SiO2 /Si). The
highest intensity can occur for MoS2 thicknesses up to 16.5 layers when the thickness of SiO2
is (5 + 179n) nm, n is a non-negative integer.
3.2.4
Additional discussion of this model
Several discussions are presented in this section. Points 1 and 2 will go into more detail
about the assumptions used in this model. Understanding the assumption in point 2 will bring
up a problem related to the integrated intensities of two modes in the observed results, which
has to be investigated further. Point 3 deals with finding a refractive index of MoS2 so that the
experimental data in Chapter 5 can be properly fitted.
1 = 3.1 − 0.65i and ñ1 = 4.8 − 1.15i respectively are average values of the results in Ref [28] for a thin layer
and a thicker layer. 6.5-3i is the average value of the result in Ref [29]. 5.25-1.7i is the average value of the
result in Ref [30].
1 ñ
25
18
The highest peak number is 16.5 L
16
The peak layer (L)
14
12
10
8
6
4
2
The periodicity is 179 nm
0
50
100
150
200
250
300
The thickness of SiO (nm)
2
Figure 3.8: The peak layer (the number of layers where the strongest intensity
apprears) as a function of the thickness of SiO2 for MoS2 on SiO2 /Si.
1. When this model was constructed, the interference of Raman signals from the same
infinitesimal source was considered. But the coherence of the scattered field from different
sources was ignored. This assumption is valid for common Raman spectroscopy. However,
for coherent anti-Stokes Raman spectroscopy, the signals radiated from different sources
in a material are also coherent.
2. The intensity calculated in this model was a relative intensity, the ratio of an intensity of
one thickness to an intensity of another thickness, since the following two assumptions
were made:
a. A two-dimensional model with equal probability of scattering a signal upwards and
downwards, disregarding the loss of signal in other directions.
b. The emission of a Raman signal of a specific mode after the lattice absorbed an excitation photon occurs with 100% probability, which means it was assumed to be constant and
independent of different vibrational modes. But if we are trying to understand the Raman
scattering process from an energy diagram point of view, the different energy levels will
26
cause different probabilities of decay from the virtual state to corresponding vibrational
states. Thus, the intensities of different modes will vary.
This may be a reason for the large difference between the integrated intensities of the
E12g and A1g modes of MoS2 , as the experimental observations [18] and my experimental
results (in Chapter 5) indicated. This difference means that the E12g mode with a Raman
shift of around 382 cm−1 excites more easily than the A1g mode with a Raman shift
of around 405 cm−1 . The difference between the Raman shifts is about 20 cm−1 (See
energy diagram in Figure 3.9), resulting in an energy difference between E12g and A1g of
h̄∆ωv = 0.1kT (where k is the Boltzmann constant and T is the room temperature).
Thus, the ratio between the number of states exciting the A1g and E12g modes is 0.9. This
indicates that the probability of excitation is different for the two modes.
Figure 3.9: An energy diagram for Raman scatterings of the E12g and A1g modes. The
difference of Raman shift between the two modes is 20 cm−1 . The energy difference
between E12g and A1g is h̄∆ωv = 0.1kT.
3. In order to find a refractive index which can be used to fit the experimental Raman peak
intensity in Chapter 5, I have chosen 6.5-1.7i. It was best suited to fit the experimental
integrated intensities of the E12g (black dots) and A1g (red dots) modes for MoS2 on 285
nm SiO2 /Si as shown in Figure 3.10. This simulation gave the strongest intensity at 4
layers. However, simulated results indicated ( I1 − I0 )/( I2 − I1 ) > 1, which cannot explain
27
the low intensity for a monolayer in the experimental observations. This may be due to
noise in the data collection process. Hence, the curve fit for A1g (black solid) is presented
with a translation downwards in the y axis to account for the signal reduction due to
noise. Assuming the simulation using the same parameters also applies to MoS2 on gel
(or wax), the thickness of some thin layers was determined by comparing the measured
Integrated intensity (arb. u.)
intensity to the simulated intensity.
22
E
20
A
1
2g
1g
18
16
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
7
Number of layers
Figure 3.10: The simulated result using a refractive index of 6.5-1.7i (blue dashed)
to fit the experimental result [18] of MoS2 on 285 nm SiO2 /Si near the peak in
integrated intensity. Since it cannot explain the lower intensity for a monolayer, a
transition downwards in y axis was made in the simulated result to better fit the
experimental result (black solid).
28
Chapter 4
Sample Preparation and Characterization
In this chapter, I will introduce the preparation methods for MoS2 samples on different
substrates for use in Raman and absorption spectrum analysis. Then, the transmission electron
microscope (TEM) analysis results of the MoS2 sample will be briefly discussed.
4.1
Substrate types
Mechanical exfoliation is widely used as an approach to exfoliate ultrathin layers from
bulk mineral MoS2 [31]. The method utilizes the weak van der Waals forces between the
2-dimensional layers, and usually a piece of scotch tape is used to break the molecular bond.
In this research project, mechanical exfoliation was involved in preparing ultrathin layers of
MoS2 . Ultrathin films were successfully deposited on wax, and gel. Samples were then analyzed
using Raman spectroscopy. Thick MoS2 on wax/glass was investigated in order to understand
excitonic dynamics of the absorption spectrum. For SiO2 /Si substrate, which were widely used
in previous work [18], high quality, large area ultrathin layers failed to deposit, or they were
made but could not be identified.
4.1.1
SiO2 /Si
A piece of adhesive scotch tape was put on the bulk mineral MoS2 to peel off a thick film.
Subsequently, the tape was repeatedly folded and unfolded to produce thin flakes on the tape’s
surface. The tape was then pressed onto a 100 nm SiO2 /Si substrate.To identify potential
ultrathin layers, an optical microscope was used to find transparent regions.
29
Previous work demonstrated that for certain thicknesses of SiO2 (270-300 nm), monolayers
of MoS2 were successfully deposited on SiO2 /Si substrate [5][32][9][33][34][35][36][37]. In the
present work, no monolayers were found for samples prepared on SiO2 /Si substrate, which may
be due to the weak optical contrast between the bare substrate surface and a single layer MoS2
film on Si substrates with thinner oxide over layers (100 nm) [38].
Some multilayer regions were observed. The optical image in Figure 4.1(a) shows an interesting region which might be an ultrathin film. Raman analysis of this region (Figure 4.1(b))
reveals a peak frequency difference of 24.0 ±0.5 cm−1 for the A1g and E12g modes. As the
Raman peak frequency differences of three, four layers and bulk MoS2 were 23.6, 24.5 and 25.0
cm−1 , respectively, as indicated in a previous experimental result [18] on 285 nm SiO2 /Si, this
region might be 3-4 layers thick.
(a)
5x104
-1
24 cm
Intensity (counts)
4x104
3x104
2x104
1x104
0
350
360 370
380 390
400 410
Raman shift (cm
420 430 440
450
-1
)
(b)
Figure 4.1: (a) Optical image of exfoiated MoS2 on 100 nm thick SiO2 /Si substrate. (b) The
Raman Spectrum of this thin region (red circle region in the optical image). The difference of
the two modes may indicate that this might be three or four layers.
30
4.1.2
Gel
Gel wafer films were obtained from a company named Gel-Pak (31398 Huntwood Avenue
Hayward, CA 94544). The value of the refractive index of 1.41 at an optical wavelength of
around 514 nm was provided by the company. Similar to the ”scotch tape” method, adhesive
gel was pressed onto the bulk MoS2 to exfoliate MoS2 films of varied thicknesses. Raman
spectroscopy observations indicates that ultrathin layers of 1-4 layers (determined by Raman
shift) were successfully deposited on gel as shown in the optical image depicted in Figure 4.2.
However, only small regions of monolayer film of less than 2 × 10 µm2 were observed.
Figure 4.2: 1-4 layers were produced by exfoliation and deposited onto gel.
4.1.3
Wax/Glass
In this method, flat bulk MoS2 was adhered to melted wax (with a refractive index of 1.46
[39] ) supported on a glass substrate. The MoS2 /wax/glass samples were heated to 373 K for
20 minutes and subsequently cooled in air for 30 minutes. After ensuring that the MoS2 was in
good contact with the wax, the scotch tape was used to peel off thick layers, leaving ultrathin
MoS2 regions attached to the wax. Figure 4.3(a) shows an optical image of large ultrathin strips
of different thickness. A single layer area of more than 10× 40 µm2 was attained as shown in
Figure 4.3(b).
31
(a)
(b)
Figure 4.3: Optical image of ultrathin MoS2 near an edge with many layer thicknesses (a) and
a single layer (b) on wax. The single layer was as large as 10× 40 µm2 .
4.1.4
Copper Grid
To prepare MoS2 samples for TEM analysis, thin pieces of MoS2 exfoliated from the mineral
bulk 2H MoS2 were transferred onto copper grids. Electron transparent regions were found on
the edges of those samples.
4.2
Transmission Electron Microscopy (TEM)
TEM was used to analyze the nanostructure of thin MoS2 specimens. The microscope
(Hitachi 8000) had a thermal source operating at an acceleration voltage of 200 kV. The electrons
were focused onto the sample and passed through it by elastic and inelastic scattering. The
CCD camera below the sample collected images and diffraction patterns.
Bright field (BF) images can help analyse crystallinity. A BF image is obtained when an
aperture is used to block scattered electrons. The amplitude loss which is dependent on the
thickness of the sample creates contrast in the image. Figure 4.4(a) shows a BF image of a
MoS2 flake with multiple layers as seen by the varying contrast near the edges. Figure 4.4(b)
shows the associated electron diffraction pattern. The hexagonal structure of the diffraction
pattern confirmed that the sample was a single crystal 2H MoS2 .
Defects were detected thanks to my supervisor Dr. Karen Kavanagh. Periodic fringes, as
can be seen in the BF image of Figure 4.5(a), indicate the existence of stacking faults in my
sample. However, as mechanical exfoliation was involved in the preparation of the TEM sample,
32
(a)
(b)
Figure 4.4: A Bright field image (a) and selected area diffraction pattern (b) of the ultrathin
film (red circle) on the edge. 2H MoS2 was identified.
it was impossible to tell whether the defect was an intrinsic property or just a result of the
exfoliation process. Broad bend contours due to the bending of the specimen are shown in the
BF image of Figure 4.5(a) with the corresponding hexagonal SAD pattern presented in Figure
4.5(b). Figure 4.6 (a) and (b) are BF and SAD images of a polycrystalline foil of MoS2 . In the
SAD image, the rings clearly indicate the misalignment of the crystals.
(a)
(b)
Figure 4.5: (a) and (b) are BF and SAD images of a single crystal. The periodic fringes were
likely stacking faults.
33
(a)
(b)
Figure 4.6: (a) and (b) are a BF and SAD image of a polycrystalline MoS2 . The SAD image
indicates a misalignment for these crystals.
34
Chapter 5
Optical Results
In this chapter, Raman spectroscopy results for MoS2 on gel and wax substrates will be
presented in detail. Two absorption peaks, A and B, corresponding to the direct band gaps at
room temperature will also be reported.
5.1
Raman Results
The instrumental spectral resolution was 1.5 cm−1 . Thus Gaussian curve fits were necessary
to help achieve more accurate Raman peak frequency measurement for the E12g and A1g modes.
Figure 5.1 shows examples of Raman spectra with Gaussian fits (solid lines) to raw data points
(dots) for 1-4 layers of MoS2 and bulk on gel. Based on these curve fits, Raman peak frequencies,
intensities, and line widths were measured. Uncertainties for Raman peak frequency and intensity
from the curve fit were less than 0.2 cm−1 and 5 % respectively.
5.1.1
Thickness Determination of the MoS2 Films
Figure 5.2 shows the absolute Raman peak intensity plotted versus peak frequency difference
of each measurement of the E12g (black) and A1g (red) modes for different thicknesses on gel (a)
and on wax (b), and the simulated predictions for comparison (purple). The simulation results
will be further discussed in the next few paragraphs. Uncertainties of the experimental results
for intensity and the frequency difference in this figure were from the errors of the Gaussian
curve fits. The thicknesses of 1-4 layers and bulk were positively confirmed by the frequency
differences of the two modes according to Ref [18]. For thicker layers, however, as the differences
of peak frequencies for different thickness were very close, the thickness was distinguished only
35
Figure 5.1: Curve fitted results (solid lines) of original Raman spectra (dots) with a resolution
of 1.5 cm−1 of the E12g and A1g modes for ultrathin layers and bulk MoS2 . The thickness was
determined by Raman peak frequency difference of the two modes. (The uncertainty for the
single measurement is 0.5 cm−1 .)
from a Raman simulated intensity. The refractive index of 6.5-1.7i was used in the simulation
as it had the best fit for MoS2 on 285 nm SiO2 /Si. As the simulated intensity was calculated
as a function of thickness, in order to plot the predicted intensity as a function of frequency
difference, the frequency difference with uncertainties as a function of thickness in Ref [18] was
used as a reference. The theoretical prediction of intensity versus frequency difference was given
as the purple squares. This way, the thickness of thicker layers was approximately identified.
Two methods were utilized to help determine the thickness of ultra-thin MoS2 layers: Raman
peak frequencies for ultra-thin layers reported in the reference paper [18], and the simulated
Raman peak intensities from my modified model.
Figure 1.5 reprinted from Ref [18] showed Raman frequencies as a function of the number
of layers for ultra-thin layers and bulk MoS2 for the E12g and A1g modes. The differences of
the frequencies of the two modes were 18.9 ± 0.6 (1 L), 21.6 ± 0.3 (2 L), 23.3 ± 0.2 (3 L),
24.2 ± 0.3 (4 L), 24.5 ± 0.3 (5 L), 24.8 ± 0.1 (6 L) and 25.2 ± 0.2 cm−1 (bulk). Clearly, for
a film thicker than four layers, it is difficult to determine the thickness using the Raman peak
frequencies of the two modes.
36
140000
E
120000
A
2g
1g
mode
mode
7 L
Simulated fit
100000
Intensity (counts)
1
qqsf
result
80000
3 L
60000
qqsf
12 L
4 L
2 L
18 L
40000
1 L
20000
bulk
0
18
19
20
21
22
23
24
25
26
-1
The frequency difference of two modes (cm )
(a)
120000
E
A
100000
1
2g
1g
mode
qqqqqsf
mode
5 L
Simulated fit
Intensity (counts)
result
80000
4 L
qqsf
qqqqqsf
3 L
60000
18 L
2 L
40000
bulk
20000
1 L
0
18
19
20
21
22
23
24
25
26
-1
The frequency difference of two modes (cm )
(b)
Figure 5.2: Single measurements of the peak intensity as a function of the difference of the
peak frequencies in E12g (black) and A1g (red) modes for MoS2 on gel (a) and on wax (b). The
thickness of 1-4 layers and bulk were confirmed. As for thicker layers, the thickness of layers
were roughly determined by the curve fit. The curve fit (purple) was the simulated result in
the revised model discussed in Chapter 2, in which the refractive index of 6.5-1.7i was used.
4.8-1.15i was used to predict the intensity of the bulk as it made a reasonable prediction of
I peak : Ibulk for MoS2 on 285 nm SiO2 /Si .
37
Figure 5.3(a) and Table 5.1 show the results of the simulated Raman peak intensity. The
simulations were done assuming that the gel was thick enough ([22]). As it is hard to find an
appropriate refractive index for MoS2 , five refractive indices were used in the simulation: 3.10.65i, 4.8-1.15i, 6.5-3i, 5.25-1.7i from Ref [28] [29] [30] and the index of 6.5-1.7i obtained in
section 3.2.4 which was better suited to fit the experimental result of MoS2 on 285 nm SiO2 /Si
(See Figure 3.10 in Chapter 3).
Figure 5.3(a) shows plots of simulated Raman peak intensity as a function of thickness
for MoS2 on gel using these indices. Clearly, they vary significantly, but all the results give
approximately linear relationships at thinner thickness as the magnified plot (5.3(b)) indicates.
(a)
(b)
Figure 5.3: (a) Plots of simulated Raman peak intensity of the A1g or E12g modes of MoS2 on
gel as a function of thickness using the refractive indices of MoS2 of 3.1-0.65i, 4.8-1.15i, 6.5-3i,
and 5.25-1.71i reported by Ref [28] [29] [30] and a selected refractive index of 6.5-1.7i. (b) A
magnified figure at thinner thickness.
Table 5.1 shows the details of these simulation results as compared to the simulation on
285 nm SiO2 /Si done in Chapter 3. Despite the fact that the thickness where the strongest
intensity appears in the simulation had a huge variation when using different refractive indices,
the results on gel had the strongest intensity at a thicker thickness compared to those on 285
nm SiO2 /Si. As the refractive index of 6.5-1.7i was better at predicting the peak number for
MoS2 on 285 nm SiO2 /Si as shown in Figure 3.10, I assumed that it would also apply to MoS2
on gel (or wax). However, there were two problems when applied to MoS2 on 285 nm SiO2 /Si,
as was discussed in Section 3.2.4: a. The intensity of a monolayer was relatively low, which did
not follow the simulated prediction that ( I1 − I0 )/( I2 − I1 ) > 1. b. When using the refractive
index of 6.5-1.7i, I peak : Ibulk = 18.8 was rather high compared to 8 in the experimental result,
38
which implies that ultra-thin layers and bulk MoS2 had considerably different refractive indices.
Assuming the same problems also exist in MoS2 on gel (or wax), a translation of the simulated
results along the y axis was applied to address a. Also, another refractive index of 4.8-1.15i,
which better predicted I peak : Ibulk for 285 nm SiO2 /Si, was applied to simulate the intensity for
bulk on gel to address b.
Table 5.1: Simulated Raman peak intensity for MoS2 on gel compared to that on 285 nm
SiO2 /Si. (Strongest (layers): the thickness where the strongest Raman peak intensity appears.
Istrongest , Ibulk and I2L : the strongest intensity, the intensity of the bulk and two layers, respectively.)
Substrate
285 nm SiO2 /Si [18]
(Experiment shows
peak layer is 4 L .)
gel (n = 1.46)
(Probably the peak layer
should be more than four layers)
ñ
Strongest (L)
Istrongest : Ibulk
I1L : Ibulk
3.1-0.65i
17.9
3.0
0.23
4.8-1.15i
7.4
9.2
2.9
6.5-3i
3.5
24.9
19.0
5.25-1.7i
5.8
12.9
5.3
6.5-1.7i
4
18.8
9.4
3.1-0.65i
33.5
1.3
0.084
4.8-1.15i
13.1
3.8
0.64
6.5-3i
5.9
9.4
3.44
5.25-1.7i
10.2
5.2
1.14
6.5-1.7i
7
7.6
Based on the above assumptions, the thickness of thin layers and bulk MoS2 on gel and wax
were identified. Thus, the Raman peak frequency, Raman peak intensity, and Raman line widths
as functions of thickness were obtained as shown in the following sections. The thicknesses of
1-4 layers on gel and 1-3 layers on wax and bulk was determined assuming that the layers were
individual MoS2 layers. As there were no other calibrations other than the theoretical intensity
to determine 7, 12 ,18 layers on gel and 4 layers on wax, the uncertainties were large for these
thickness.
5.1.2
Raman Peak Frequency
Figure 5.4(a) plots the Raman peak frequencies of the E12g and A1g modes for MoS2 of
different thickness on gel (black) and 1-3 layers and bulk for MoS2 on wax (red), as compared
to a reported experimental result [18] (Figure 1.5). Both on gel and wax, the frequency of the
39
A1g mode was generally blue-shifted as the thickness increased agreeing with Ref [18], except
for the frequency of 4-7 layers. For the E12g mode, however, after the frequency was red-shifted
from 1 L to around 7 L, it started to blue shift. This did not agree with the reported result
[18], as the frequency of E12g mode was claimed to be continuously red-shifted as the thickness
increased. The peak frequencies for MoS2 on wax were in good agreement with the results on
gel for 1L, 2L and bulk (while a slight inconsistency in trilayer results existed). The difference
of the frequencies of the two modes kept increasing as the film became thicker, which was
consistent with the reported result.
Table 5.2 lists the differences in the peak frequencies of the E12g and A1g modes for MoS2
both on gel and on wax in comparison with the experimental result on SiO2 /Si [18]. Generally,
the results on the three substrates agreed with each other.
Table 5.2: The difference in peak frequencies (cm−1 ) for the E12g and A1g modes for MoS2 .
Substrate
1L
2L
3L
4L
More than 4 L
Bulk
cm−1
285 nm SiO2 /Si [18]
gel
wax
18.9
21.6
23.3
24.2
±0.6
±0.3
±0.2
±0.3
19.5
22.2
23.7
24.3
±0.1
±0.1
±0.1
±0.2
19.3
22.2
23.7
24.3
25.2
±0.1
±0.1
±0.1
±0.2
±0.1
24.2-24.9
24.7-25.2
25.2
±0.2
25.2
±0.1
To explain how I determined the thickness of these ultrathin layers in the optical image in
Figure 5.1, Table 5.3 lists the Raman shifts of the single measurements of these regions on gel
as compared to the Raman shifts of the repeated measurement results for MoS2 on substrates
of 285 nm SiO2 /Si [18] and gel. The errors for single measurements are ±0.5 cm−1 .
5.1.3
Raman Peak Intensity
As the refractive index of wax is 1.46, very close to 1.41 for gel, the simulation predications
in Figure 5.3(a) and Table 5.1 also applied to MoS2 on wax.
Figure 5.4(b) shows a plot of the absolute Raman peak intensity (in a Gaussian distribution)
as a function of thickness for the A1g and E12g modes of MoS2 on gel (black) and on wax
(red), with a magnified plot at lower thicknesses (inset). The intensities had a roughly linear
40
Table 5.3: Raman peak shift (cm−1 ) from individual measurements from ultrathin layers compared with the shifts for average measurements for MoS2 on substrates of 285 nm SiO2 /Si [18]
and gel.
1L
2L
3L
4L
More than 4 L
bulk
cm−1
Single measurement
19.9
22.6
24.6
25.2
25.2
on gel
±0.5
±0.5
±0.5
±0.5
±0.5
Average measurements
19.5
22.2
23.7
24.3
on gel
±0.1
±0.1
±0.1
±0.2
Average measurements
18.9
21.6
23.3
24.2
on 285 nm SiO2 /Si [18]
±0.6
±0.3
±0.2
±0.3
24.7-25.2
24.2-24.9
25.2
±0.1
25.2
±0.2
relationship at 1-4 layers. For MoS2 on gel, it peaked approximately at seven layers based on the
theoretical curve fit. The intensities of gel were weaker than that of wax for thin film probably
because the areas of thin layers of gel were smaller than the laser spot size (the diameter of the
spot size was 1µm), while the ultrathin films on wax were large enough.
As compared to the relationship of the absolute Raman peak intensity and the thickness of
a material, a relationship between the Raman integrated intensity (over a Gaussian distribution)
and the thickness is more physical. A problem with the results for the Raman integrated
intensity is that there was a significant difference between the intensities of the two modes on
both substrates, which will be discussed in Section 5.1.5.
5.1.4
Raman Line Width
Figure 5.4(c) shows line widths of both modes as a function of MoS2 thickness on gel (black)
and on wax (red) for 1-3 layers and bulk. Line widths of the A1g mode were wider than those
of the E12g mode. Line widths of bulk in both modes for both substrates had the smallest line
widths consistent with the published result for 285 nm SiO2 /Si substrate [18]. The line widths of
the A1g mode were approximately twice as those of the E12g mode. However, unlike the reported
result [18], on gel, the A1g mode had a continuous decline for the first four layers while the
trilayer presented the highest width in the E12g mode. This inconsistency was due to differences
of the substrates, as the medium (material) will expand the optical Gaussian width.
41
406
26
404
24
402
400
388
386
384
382
380
22
20
E12g
0
2
4
6
8
10
12
14
16
18
18
20
22
24
26
1000
Bulk
-1
Frequency difference (cm
)
-1
Peak frequency (cm
)
28
A1g
408
16
Thickness (L)
(a)
120000
1
E
2g
100000
A
1g
for gel
Intensity (counts)
1
E
2g
80000
A
1g
for wax
The curve fit
60000
40000
20000
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
1000
Bulk
24
26
1000
Bulk
Thickness (L)
(b)
8
-1
Line width (cm
)
7
A1g
6
5
E12g
A1g
4
E12g
3
0
2
4
6
8
10
12
14
16
18
20
22
Thickness (L)
(c)
Figure 5.4: (a) Raman peak frequency as a function of thickness in the A1g and E12g modes of
MoS2 on gel (black) and on wax (red) (left vertical axis) and their difference (gel grey and wax
red right vertical axis). (b) Raman peak intensity as a function of thickness for the A1g and
E12g modes of MoS2 on gel (black) and on wax (red). Simulation (blue solid) with the refractive
index of 6.5-1.7i was used to fit the experimental results. (c) Raman peak line widths as a
function of thickness for the A1g and E12g modes of MoS2 on gel (black) and on wax (red).
42
5.1.5
Discussion
Thin layers and bulk MoS2 were successfully made on gel and wax substrates and the
thickness was determined by Raman spectroscopy. This was done by analyzing the differences of
Raman peak frequency of the A1g mode and E12g modes, and the simulated intensity predictions.
1. The Raman peak frequency: a. The differences of the peak frequencies of the A1g mode
and E12g mode for films less than four layers and bulk MoS2 both on wax and gel, were in
good agreement with previous results [18]. Thus, the differences of the peak frequencies
of the two modes is a faithful indicator for distinguishing the ultrathin layers from bulk.
b. Both the theoretical paper [19] and experimental results [18] state that the frequency
of the E12g mode should be continuously red shifted as the thickness increases. But after
the decrease of the peak frequency of the E12g mode from one to three layers, it began to
rise as the film of MoS2 got thicker.
2. The absolute Raman peak intensity: a. The intensity of the first few layers on both
substrates were in rough agreement with the simulated prediction. But the simulation
deviated from the experimental results for thicker samples on gel. The intensity of ultrathin
layers of MoS2 on gel may have been underestimated due to their small areas.
3. The Raman line width: a. There were inconsistencies between my experimental results
on gel and wax and the published result [18], which may be due to the fact that the line
width of a specific mode is likely to depend on the substrate. b. The thinnest line widths
were observed in bulk in both modes on gel and wax, consistent with Ref [18]. Also, the
line widths of the A1g mode were significantly larger than that of the E12g mode.
4. Figure 5.5 shows the integrated intensity as a function of the thickness of MoS2 on the
gel and wax substrates for the E12g and A1g modes. Large differences in the integrated
intensity can be easily seen. Intensity difference of the two modes was also observed in the
integrated intensity result in Ref [18] and was discussed in Section 3.2.4. However, the
integrated intensity of the E12g mode was stronger than that of the A1g mode compared
to the reported result [18]. The integrated intensity of the A1g mode was stronger, which
cannot be explained by the fact that E12g is a lower state.
43
80
1
E
2g
70
A
Integrated Intensity (arb. u.)
1g
for gel
1
E
60
2g
A
1g
50
for wax
40
30
20
10
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26
1000
Thickness (L)
Figure 5.5: The integrated intensity as a function of the thickness of MoS2 on gel and on wax.
5.2
Absorption Spectrum
Exciton peaks in an absorption spectrum allow us to probe the band structure of a sample.
Frint and Yoffe [12][13] studied the absorption spectrum and observed excitons A and B with
2.05 eV and 1.86 eV at room temperature of thick MoS2 . My experimental result confirmed
this.
The sample was prepared using the wax/glass method introduced in Chapter 3. The absorption spectrum was obtained by dividing the incident light by the transmitted light. If we
consider the absorption of wax/glass to be a background, then the absorption spectrum can be
corrected by dividing the transmitted light I0 of wax/glass, by the transmitted light I of sample
MoS2 /wax/glass. This eliminates the background leaving only the absorption spectrum of MoS2
(Figure 5.6).
The thickness y of the sample can be calculated by:
I (y) = I0 exp(
−4πky
)
λ
where k = 1.5 is the extinction coefficient at λ = 600 nm for thicker MoS2 [28].
44
(5.1)
2.5
605 nm
668 nm
0
ln (I /I)
2.0
1.5
1.0
0.5
550
600
650
700
W avelength (nm)
Figure 5.6: Excitonic peaks A at 2.05 and B at 1.86 eV appeared for 83-95 layers at room
temperature. [15]
The reflection on the surface of air/MoS2 cannot be ignored. The transmission coefficient
T is,
T=
4n0 n1
,
( n0 + n1 )2
(5.2)
where n0 = 1, and n1 = 1.6 are the refractive indices of air and MoS2 at λ = 660 nm [29].
Moreover, considering that 94.7% of light will get transmitted through the interface of air/MoS2 ,
a correction of -0.0544 was made on the ln( II0 ). The number of layers of MoS2 is then calculated
to be 83-95, which is a thin bulk MoS2 .
The expected two excitonic peaks at 2.05 eV and 1.86 eV at room temperature have been
confirmed. Both peaks correlate with the direct gap at the K point which was discussed in
Chapter 1.
In the section in Chapter 3 on Raman spectrum simulation for MoS2 , a need for the right
refractive indices for certain thicknesses was discussed. Thus, more optical analysis should be
done in the future.
45
Chapter 6
Conclusions
A previous model [6] theoretically explained an anomaly that the strongest Raman intensity
appears at a finite thickness for graphite. An improvement to this model was made in this work.
By making the assumption of interference of the Raman signal scattering from the same source,
a close prediction of the intensity as a function of thickness was achieved for graphite. This
model was then applied to predict the intensity behaviours for MoS2 .
To test the new model, ultra-thin layers of MoS2 were successfully made on gel and wax
substrates by mechanical exfoliation from bulk MoS2 . The thickness of MoS2 was determined
by the difference of the Raman peak frequencies of the A1g and E12g modes, and the simulated
prediction. The maximum Raman intensities of ultrathin layers were roughly consistent with the
simulated prediction. The success of this simulation was mostly limited by the fact that it was
hard to find an appropriate refractive index for MoS2 . But the simulation indicated that the
most consistent index of refraction for MoS2 was 6.5-1.7i. The ratio of the number of states
between A1g and E12g is 0.9, which may indicate that the probability of excitation of a specific
vibrational mode should be included into this model. However, this could not explain the fact
that the A1g mode at a higher energy level excites more easily compared to the E12g mode in my
result. Concerning the Raman peak frequencies of the two modes, their differences agreed with
the reported result [18]. But the frequency of the E12g mode was not continuously red-shifted
as other publications[18] [19] have indicated.
Absorption spectra were attained for MoS2 of thickness of 85-100 layers. Two exitonic peaks
A and B were observed and exciton A indicated the direct gap was 1.9 eV at room temperature
as expected from previous reports..
46
Bibliography
[1] Konstantin S Novoselov, Z Jiang, Y Zhang, SV Morozov, HL Stormer, U Zeitler, JC Maan,
GS Boebinger, P Kim, and AK Geim. Room-temperature quantum hall effect in graphene.
Science, 315(5817):1379–1379, 2007. 1
[2] Shu-Chun Wu, Guangcun Shan, and Binghai Yan. Prediction of near-room-temperature
quantum anomalous hall effect on honeycomb materials.
Physical Review Letters,
113(25):256401, 2014. 1
[3] Melinda Y Han, Barbaros Özyilmaz, Yuanbo Zhang, and Philip Kim. Energy band-gap
engineering of graphene nanoribbons. Physical Review Letters, 98(20):206805, 2007. 1
[4] Liying Jiao, Li Zhang, Xinran Wang, Georgi Diankov, and Hongjie Dai. Narrow graphene
nanoribbons from carbon nanotubes. Nature, 458(7240):877–880, 2009. 1
[5] Branimir Radisavljevic, Aleksandra Radenovic, Jacopo Brivio, V Giacometti, and A Kis.
Single-layer MoS2 transistors. Nature Nanotechnology, 6(3):147–150, 2011. 1, 30
[6] YY Wang, ZH Ni, ZX Shen, HM Wang, and YH Wu. Interference enhancement of raman
signal of graphene. Applied Physics Letters, 92(4):043121–043121, 2008. ix, x, 1, 14, 16,
17, 20, 21, 22, 23, 46
[7] C Ataca, H Sahin, and S Ciraci. Stable, single-layer mx2 transition-metal oxides and
dichalcogenides in a honeycomb-like structure. The Journal of Physical Chemistry C,
116(16):8983–8999, 2012. 2
[8] D Yang, S Jiménez Sandoval, WMR Divigalpitiya, JC Irwin, and RF Frindt. Structure of
single-molecular-layer MoS2 . Physical Review B, 43(14):12053, 1991. 2
47
[9] Zongyou Yin, Hai Li, Hong Li, Lin Jiang, Yumeng Shi, Yinghui Sun, Gang Lu, Qing
Zhang, Xiaodong Chen, and Hua Zhang. Single-layer MoS2 phototransistors. ACS Nano,
6(1):74–80, 2011. 2, 30
[10] RF Frindt. Single crystals of MoS2 several molecular layers thick. Journal of Applied
Physics, 37(4):1928–1929, 1966. 2
[11] Andrea Splendiani, Liang Sun, Yuanbo Zhang, Tianshu Li, Jonghwan Kim, Chi-Yung Chim,
Giulia Galli, and Feng Wang. Emerging photoluminescence in monolayer MoS2 . Nano
Letters, 10(4):1271–1275, 2010. viii, 2, 3, 4
[12] RF Frindt and AD Yoffe. Physical properties of layer structures: optical properties and
photoconductivity of thin crystals of molybdenum disulphide. Proceedings of the Royal
Society of London. Series A. Mathematical and Physical Sciences, 273(1352):69–83, 1963.
3, 44
[13] RF Frindt.
Optical absorption of a few unit-cell layers of MoS2 .
Physical Review,
140(2A):A536, 1965. viii, 3, 4, 44
[14] R Coehoorn, C Haas, and RA De Groot. Electronic structure of MoSe2 , MoS2 , and WSe2 .
ii. the nature of the optical band gaps. Physical Review B, 35(12):6203, 1987. 3
[15] Sangwan Sim, Jusang Park, Jeong-Gyu Song, Chihun In, Yun-Shik Lee, Hyungjun Kim,
and Hyunyong Choi. Exciton dynamics in atomically thin MoS2 : Interexcitonic interaction
and broadening kinetics. Physical Review B, 88(7):075434, 2013. xii, 3, 45
[16] Kin Fai Mak, Changgu Lee, James Hone, Jie Shan, and Tony F Heinz. Atomically thin
MoS2 : a new direct-gap semiconductor. Physical Review Letters, 105(13):136805, 2010. 3
[17] Tobias Korn, Stefanie Heydrich, Michael Hirmer, Johannes Schmutzler, and Christian
Schüller. Low-temperature photocarrier dynamics in monolayer MoS2 . Applied Physics
Letters, 99(10):102109, 2011. 3
[18] Changgu Lee, Hugen Yan, Louis E Brus, Tony F Heinz, James Hone, and Sunmin Ryu.
Anomalous lattice vibrations of single-and few-layer MoS2 . ACS Nano, 4(5):2695–2700,
2010. vii, ix, x, 5, 6, 8, 22, 24, 25, 27, 28, 29, 30, 35, 36, 39, 40, 41, 43, 46
48
[19] Alejandro Molina-Sanchez and Ludger Wirtz. Phonons in single-layer and few-layer MoS2
and WS2 . Physical Review B, 84(15):155413, 2011. viii, ix, 5, 6, 13, 43, 46
[20] Andre K Geim and Konstantin S Novoselov. The rise of graphene. Nature Materials,
6(3):183–191, 2007. 8
[21] Xavier Gonze and Changyol Lee. Dynamical matrices, born effective charges, dielectric
permittivity tensors, and interatomic force constants from density-functional perturbation
theory. Physical Review B, 55(16):10355, 1997. 12
[22] [Such as, for silicon, with a small extinction of 0.044 and thickness of 0.5 mm, the intensity
will diminish to 1/2400 of the incident light that first strikes the upper surface of silicon.].
15, 38
[23] Mattias Klintenberg, C Ortiz, B Sanyal, J Fransson, Olle Eriksson, et al. Evolving properties
of two-dimensional materials: from graphene to graphite. Journal of Physics: Condensed
Matter, 21(33):335502, 2009. 20
[24] Sosan Cheon, Kenneth David Kihm, Hong goo Kim, Gyumin Lim, Jae Sung Park, and
Joon Sik Lee. How to reliably determine the complex refractive index (ri) of graphene by
using two independent measurement constraints. Scientific Reports, 4, 2014. 20
[25] ZH Ni, HM Wang, J Kasim, HM Fan, T Yu, YH Wu, YP Feng, and ZX Shen.
Graphene thickness determination using reflection and contrast spectroscopy. Nano Letters,
7(9):2758–2763, 2007. 20
[26] Edward D Palik. Handbook of optical constants of solids, volume 3. Academic press, 1998.
20
[27] Gerald Earle Jellison Jr, John D Hunn, and Ho Nyung Lee. Measurement of optical functions
of highly oriented pyrolytic graphite in the visible. Physical Review B, 76(8):085125, 2007.
20
[28] Chanyoung Yim, Maria O’Brien, Niall McEvoy, Sinéad Winters, Inam Mirza, James G
Lunney, and Georg S Duesberg. Investigation of the optical properties of MoS2 thin films
using spectroscopic ellipsometry. Applied Physics Letters, 104(10):103114, 2014. xi, 22,
24, 25, 38, 44
49
[29] Andres Castellanos-Gomez, Nicolas Agraït, and Gabino Rubio-Bollinger. Optical identification of atomically thin dichalcogenide crystals. Applied Physics Letters, 96(21):213116,
2010. xi, 22, 24, 25, 38, 45
[30] Jie-Fang Li, D Viehland, AS Bhalla, and LE Cross. Pyro-optic studies for infrared imaging.
Journal of Applied Physics, 71(5):2106–2112, 1992. xi, 22, 24, 25, 38
[31] Rudren Ganatra and Qing Zhang. Few-layer MoS2 : A promising layered semiconductor.
ACS Nano, 8(5):4074–4099, 2014. 29
[32] Sanfeng Wu, Jason S Ross, Gui-Bin Liu, Grant Aivazian, Aaron Jones, Zaiyao Fei, Wenguang Zhu, Di Xiao, Wang Yao, David Cobden, et al. Electrical tuning of valley magnetic
moment through symmetry control in bilayer MoS2 . Nature Physics, 9(3):149–153, 2013.
30
[33] Branimir Radisavljevic and Andras Kis. Mobility engineering and a metal–insulator transition in monolayer MoS2 . Nature Materials, 12(9):815–820, 2013. 30
[34] Hee SungáLee, Hyoung JoonáChoi, Min KyuáPark, et al. Nanosheet thickness-modulated
MoS2 dielectric property evidenced by field-effect transistor performance.
Nanoscale,
5(2):548–551, 2013. 30
[35] Meeghage Madusanka Perera, Ming-Wei Lin, Hsun-Jen Chuang, Bhim Prasad Chamlagain,
Chongyu Wang, Xuebin Tan, Mark Ming-Cheng Cheng, David Tománek, and Zhixian Zhou.
Improved carrier mobility in few-layer MoS2 field-effect transistors with ionic-liquid gating.
ACS Nano, 7(5):4449–4458, 2013. 30
[36] Han Liu, Adam T Neal, and Peide D Ye. Channel length scaling of MoS2 mosfets. ACS
Nano, 6(10):8563–8569, 2012. 30
[37] Hee Sung Lee, Sung-Wook Min, Min Kyu Park, Young Tack Lee, Pyo Jin Jeon, Jae Hoon
Kim, Sunmin Ryu, and Seongil Im. MoS2 nanosheets for top-gate nonvolatile memory
transistor channel. Small, 8(20):3111–3115, 2012. 30
[38] MM Benameur, B Radisavljevic, JS Heron, S Sahoo, H Berger, and A Kis. Visibility of
dichalcogenide nanolayers. Nanotechnology, 22(12):125706, 2011. 30
50
[39] Tables of refractive indices. oils, fats and waxes. http://chestofbooks.com/science/
chemistry/Tables-of-Refractive-Indices/, 2008. [Refractive index of wax.]. 31
51
Appendix A
Derivation of the Effective Reflection
Coefficient of the Sample/SiO2/Si
Figure A.1 describes the process whereby the light from the sample transmits into SiO2 of
zero extinction value (no absorption) and undergoes infinite reflections in SiO2 , but loses some
amplitude due to the transmission into the silicon of infinite thickness.
Figure A.1: The effective reflection of the interface of a sample/SiO2 /Si structure, assuming
the silicon layer is thick enough. (For silicon, with a small extinction of 0.044 and thickness
of 0.5 mm, the intensity will diminish to 1/2400 of the incident light at the upper surface of
silicon.)
The field reflection coefficient at the interface of sample/SiO2 , SiO2 /sample and SiO2 /Si
are as shown in Eq. (A.1), Eq. (A.2) and Eq. (A.3), respectively:
r1 =
n1 − n2
,
n1 + n2
52
(A.1)
−r1 = −
r3 =
n1 − n2
,
n1 + n2
(A.2)
n2 − n3
.
n2 + n3
(A.3)
The field transmission coefficient at the interface of sample/SiO2 and SiO2 /sample are
t2 =
2n1
n1 + n2
(A.4)
t3 =
2n2
n1 + n2
(A.5)
Assuming the electric field when it first strikes the interface of the sample/SiO2 Ẽ = 1, thus
the electric field at the surface of sample/SiO2 is
t2 r3 e −i
2π (2n2 d2 )
λ
t2 r32 (−r1 )e−i
2π (4n2 d2 )
λ
t2 r33 r12 e−i
2π (6n2 d2 )
λ
(A.6)
....
Therefore, the total electric field is
t3 t2 r3 e −i
4π (n2 d2 )
λ
1 + r1 r3 e −i
4π (n2 d2 )
λ
.
(A.7)
And thus, the effective reflection index is the sum of Eq. (A.1) and Eq. (A.7)
4π (n2 d2 )
4n1 n2 n2 −n3 −i
λ
e
n1 − n2
( n1 + n2 )2 n2 + n3
′
r =
+
n1 + n2 1 + n1 −n2 n2 −n3 e−i 4π(nλ2 d2 )
n1 + n2 n2 + n3
=
n1 − n2
n1 + n2
n2 2
+ [( nn11 −
+ n2 ) +
1+
=
n1 − n2
n1 + n2
1+
=
+
4π (n2 d2 )
4n1 n2
] n2 − n3 e − i λ
( n1 + n2 )2 n2 + n3
n1 −n2 n2 −n3 −i 4π (nλ2 d2 )
n1 + n2 n2 + n3 e
n2 −n3 −i 4π (nλ2 d2 )
n2 + n3 e
n1 −n2 n2 −n3 −i 4π (nλ2 d2 )
n1 + n2 n2 + n3 e
r2 + r3 e −i
4π (n2 d2 )
λ
1 + r2 r3 e −i
4π (n2 d2 )
λ
.
53
(A.8)
When d2 → 0,
r′ →
n1 − n2
n2 − n3
n1 + n2 + n2 + n3
n2 n2 − n3
1 + nn11 −
+ n2 n2 + n3
(n1 − n2 )(n2 − n3 ) + (n1 + n2 )(n2 − n3 )
(n1 + n2 )(n2 + n3 ) + (n1 − n2 )(n2 − n3 )
n − n3
,
→ 1
n1 + n3
→
which is exactly the reflection index of the sample/Si.
54
(A.9)