Improvements to Modelling. of Raman Scattering Intensity. for Molybdenum Disulfide

Transcription

1 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.

2 APPROVAL Name: Degree: Title: Yule Wang Master of Science 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

3 Abstract Single layer molybdenum disulfide (MoS 2 ) has attracted significant interest as a semiconducting two-dimensional material. In this work, ultrathin layers of MoS 2 were exfoliated on gel and wax substrates. Raman studies of the ultrathin layers of MoS 2 were carried out to characterize the thickness. During Raman measurement, an anomaly occurs: the strongest Raman intensity appears at a finite thickness of the MoS 2, a phenomenon also seen in graphite. A previous work [Y. Y. Wang et al., Appl. Phys. Lett. 92, (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 MoS 2. The simulations indicated that a constant complex index of refraction for MoS 2 of i provided the best fit to published experimental results for MoS 2 on SiO 2 /Si. This helped to estimate the thickness of ultrathin layers on gel. Seven layers MoS 2 on gel showed the strongest Raman intensity according to this simulation. Keywords: MoS 2 ; Raman spectroscopy iii

4 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

5 Contents Approval Abstract Acknowledgements Contents List of Tables List of Figures ii iii iv v vii viii 1 Introduction Overview of MoS Dissertation Organization Introduction to MoS Crystal Structures of MoS The band structure Raman spectra of the E 1 2g and A 1g modes Raman determination by intensity Background of Raman spectroscopy Basics Raman Scattering Optical Set-up Phonons Interatomic Force Constant v

6 3 Modelling of Raman Peak Intensity The Previous Model The Revised Model Revisions made to the Previous Model Application to Graphite Application to MoS Additional discussion of this model Sample Preparation and Characterization Substrate types SiO 2 /Si Gel Wax/Glass Copper Grid Transmission Electron Microscopy (TEM) Optical Results Raman Results Thickness Determination of the MoS 2 Films Raman Peak Frequency Raman Peak Intensity Raman Line Width Discussion Absorption Spectrum Conclusions 46 Bibliography 47 Appendix A Derivation of the Effective Reflection Coefficient of the Sample/SiO 2 /Si 52 vi

7 List of Tables 3.1 Experimental and simulated results of the Raman peak intensity of graphite on 300 nm SiO 2 /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 I bulk Refractive index of MoS 2 in three references Comparison of experimental and simulated Raman intensities for the E and A modes. The simulation assumed both modes have the same intensity for MoS 2 on 285 nm SiO 2 /Si and used 4 different complex refractive indices of MoS 2 reported in 3 different references Simulated Raman peak intensity for MoS 2 on gel compared to that on 285 nm SiO 2 /Si. (Strongest (layers): the thickness where the strongest Raman peak intensity appears. I strongest, I bulk and I 2L : the strongest intensity, the intensity of the bulk and two layers, respectively.) The difference in peak frequencies (cm 1 ) for the E 1 2g and A 1g modes for MoS Raman peak shift (cm 1 ) from individual measurements from ultrathin layers compared with the shifts for average measurements for MoS 2 on substrates of 285 nm SiO 2 /Si [18] and gel vii

8 List of Figures 1.1 Top view of a single layer and side view of a multilayer crystal structure of 2H and 1T MoS 2. For 2H MoS 2, the period in the z direction is two-layers, whereas, 1T MoS 2 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 Theoretical simulations show that the evolution of the band structure of 4-1 layers of MoS 2 (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) A simplified band structure of MoS 2 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 The in-plane mode E 1 2g and the out-of-plane mode A 1g for multiple-layers MoS 2. 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

9 1.5 (a) Raman peak frequencies (left vertical axis) and the frequency differencies (right vertical axis) of the E 1 2g and A 1g 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 Raman scattering in a lattice. The lattice absorbs incident photons and emits red-shifted new photons with the corresponding vibrational mode excited Optical Set-up: the Raman signal scattered from the sample will be reflected and collected by the spectrometer, for spectrum analysis 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 Transverse vibrational modes of atoms in a one-dimensional monatomic lattice with quantized k values Theoretical simulation (lines) and experimental result (dots) of the dispersion relation, the frequency of the E 1 2g and A 1g modes as a function of q for bulk MoS 2. 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 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/sio 2 /Si in a real Raman experiment, as shown in the diagram (c). The Raman scattering, however, will propagate in all directions The experimental result for G-band of graphite [6] (black dots) and the previous simulated result [6] (red line) Raman scattering upwards (orange) and downwards (red) from the same infinitesimal source in the sample interfer ix

10 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 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 (a) Experimental results for G-band of graphite [6] (dots) and the simulated result using i (line). Inset: a magnified view of the region of layers. (b) Experimental results for G-band of graphite [6] (dots) and the simulated result using i Simulated results of Raman intensity as a function of thickness for MoS 2 with different refractive indices: i (red dashed), i (red), 6.5-3i (black) and i (blue). Inset: intensity at thinner thickness The peak layer (the number of layers where the strongest intensity apprears) as a function of the thickness of SiO 2 for MoS 2 on SiO 2 /Si An energy diagram for Raman scatterings of the E 1 2g and A 1g modes. The difference of Raman shift between the two modes is 20 cm 1. The energy difference between E 1 2g and A 1g is h ω v = 0.1kT The simulated result using a refractive index of i (blue dashed) to fit the experimental result [18] of MoS 2 on 285 nm SiO 2 /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) (a) Optical image of exfoiated MoS 2 on 100 nm thick SiO 2 /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 layers were produced by exfoliation and deposited onto gel Optical image of ultrathin MoS 2 near an edge with many layer thicknesses (a) and a single layer (b) on wax. The single layer was as large as µm x

11 4.4 A Bright field image (a) and selected area diffraction pattern (b) of the ultrathin film (red circle) on the edge. 2H MoS 2 was identified (a) and (b) are BF and SAD images of a single crystal. The periodic fringes were likely stacking faults (a) and (b) are a BF and SAD image of a polycrystalline MoS 2. The SAD image indicates a misalignment for these crystals Curve fitted results (solid lines) of original Raman spectra (dots) with a resolution of 1.5 cm 1 of the E 1 2g and A 1g modes for ultrathin layers and bulk MoS 2. The thickness was determined by Raman peak frequency difference of the two modes. (The uncertainty for the single measurement is 0.5 cm 1.) Single measurements of the peak intensity as a function of the difference of the peak frequencies in E 1 2g (black) and A 1g (red) modes for MoS 2 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 i was used i was used to predict the intensity of the bulk as it made a reasonable prediction of I peak : I bulk for MoS 2 on 285 nm SiO 2 /Si (a) Plots of simulated Raman peak intensity of the A 1g or E 1 2g modes of MoS 2 on gel as a function of thickness using the refractive indices of MoS 2 of i, i, 6.5-3i, and i reported by Ref [28] [29] [30] and a selected refractive index of i. (b) A magnified figure at thinner thickness (a) Raman peak frequency as a function of thickness in the A 1g and E 1 2g modes of MoS 2 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 A 1g and E 1 2g modes of MoS 2 on gel (black) and on wax (red). Simulation (blue solid) with the refractive index of i was used to fit the experimental results. (c) Raman peak line widths as a function of thickness for the A 1g and E 1 2g modes of MoS 2 on gel (black) and on wax (red) The integrated intensity as a function of the thickness of MoS 2 on gel and on wax xi

12 5.6 Excitonic peaks A at 2.05 and B at 1.86 ev appeared for layers at room temperature. [15] xii

13 Chapter 1 Introduction 1.1 Overview of MoS 2 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 (MoS 2 ), 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 MoS 2 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 MoS 2 was the main focus. A review of MoS 2 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

14 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 MoS 2 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) MoS 2 will be discussed in detail. Additionally, the absorption spectrum of thick MoS 2 at room temperature will be briefly presented. Chapter 6 will summarize the results. 1.3 Introduction to MoS Crystal Structures of MoS 2 Two structural phases, the hexagonal (2H) and octahedral (1T) crystal structures, are mostly found in MoS 2. Figure 1.1 presents the crystal structures of the 2H and 1T phases in top view and side view. The 1T phase of MoS 2 is metallic. But the 2H crystal structure, which has a band gap, is the main focus of my project. The semiconducting 2H phase of MoS 2 is more stable than the 1T phase at room temperature [7]. Ion intercalation will transform the 2H phase of MoS 2 to the 1T phase [8], and revert the 1T to the 2H phase by annealing at 300 C [7]. The 2H MoS 2 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] The band structure One interesting aspect of MoS 2 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 MoS 2 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

15 Figure 1.1: Top view of a single layer and side view of a multilayer crystal structure of 2H and 1T MoS 2. For 2H MoS 2, the period in the z direction is two-layers, whereas, 1T MoS 2 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 MoS 2 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 MoS 2. Figure 1.3(b) shows peaks A and B in the absorption spectrum for MoS 2 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

16 Figure 1.2: Theoretical simulations show that the evolution of the band structure of 4-1 layers of MoS 2 (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 MoS 2 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

17 1.3.3 Raman spectra of the E 1 2g and A 1g 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 E 1 2g and A 1g modes are two vibrational modes commonly used to identify the thickness of ultrathin layers of MoS 2. Figure 1.4 shows that E 1 2g 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. A 1g 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 E 1 2g and A 1g modes would rise with increasing thickness due to the interaction of more layers. Experimental results confirm this prediction for the A 1g mode [18]. But for the E 1 2g 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 E 1 2g and the out-of-plane mode A 1g for multiple-layers MoS 2. 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

18 (a) (b) Figure 1.5: (a) Raman peak frequencies (left vertical axis) and the frequency differencies (right vertical axis) of the E 1 2g and A 1g 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 Raman determination by intensity Figure 1.5 shows the Raman peak frequency of the E 1 2g and A 1g modes (left vertical axis) and their frequency difference (right vertical axis) as a function of MoS 2 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 MoS 2, 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 MoS 2 (and graphite). Figure 1.5(b) [18] shows the integrated intensities of the two modes as a function of the thickness for MoS 2 on 285 nm SiO 2 /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

19 However, there are questionable details in this model. model and present a modified theory in Chapter 3. I will address the deficiencies of this 7

20 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 MoS 2 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 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 Optical Set-up The Raman Spectroscopy set-up is shown in Figure 2.2. A laser beam with wavelength λ 0 (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

21 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 λ 1 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 ω v. It is proportional to the difference between wavenumbers of the excitation and scattering light (wavelengths λ 0 and λ 1, respectively), defined as ω n (cm 1 ), where c is the speed of light. ω v 2πc = ω n = ( 1 λ 0 1 λ 1 ), (2.1) 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 ω v will generate a new state S 1. When the lattice (molecule) absorbs an incident photon, it is excited from the ground state S 0 to a virtual state S v, a very short-lived state. If it decays back to the ground state S 0, it undergoes Rayleigh scattering. If instead the excited state relaxes from S v to S 1, red-shifted photons will be emitted and Raman scattering (Stokes scattering) occurs. The difference in energy between S 1 (E 1 ) and S 0 (E 0 ) is related to the frequency ω v of the associated phonon: hω v = (E) = E 1 E 0. (2.2) 9

22 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

23 Blue-shifted anti-stokes scattering, in contrast, happens when the S 1 state is excited to the virtual state, and then decays to the ground state S 0. Because the S 1 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 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: then the lattice waves have quantized k in the Brillouin zone. Φ(r + N i a i ) = Φ(r), (2.3) Figure 2.4 describes an example of the vibrations of atoms with frequency ω v 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

24 As compared to real particles, the creation and annihilation of phonons is described by a virtual momentum hq: hk = hk ± hq. (2.4) And the energy levels dependent on k space, are described by E(k ) = E(k) + hω v. (2.5) Interatomic Force Constant The interatomic force constant C kα,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 E tot : C kα,kβ (a, b) = 2 E tot ukα a, (2.6) ub k β where uk a β is the displacement of atom k in cell a in the α direction. Its Fourier transform C kα,k β is correlated to the angular frequency ω of the lattice wave, C(q) ω 2 = 0, (2.7) Mk M k where M k and M k are the masses of the two atoms. From the force constant matrix C(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 E 2g and A 1g modes as [21]: C kα,kβ (a, b) = Z kz k ( δ αβ ϵ d 3 3d αd β ), (2.8) d5 where Z k and Z k 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 MoS 2. 12

25 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 E 1 2g and A 1g modes as a function of q for bulk MoS 2. 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

26 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 d 1. Since the beam will be infinitely internally reflected between the sample/sio 2 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/sio 2 interface is given by: t 1 e β 1y e i(2πñ 1y/λ), (3.1) where t 1 = 2n 0 /(n 0 + ñ 1 ) (n 0 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: t 1 r e β 1(2d 1 y) e i(2πñ 1(2d 1 y)/λ 1 ), (3.2) 14

27 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/sio 2 /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/sio 2 /Si interface, r = (r 2 + r 3 e 4iπn 2d 2 /λ )/(1 + r 2 r 3 e 4iπn 2d 2 /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 r 1 r e 4πñ 1 d 1 λ 1 e 2βd 1, (3.4) where r 1 = (ñ 1 ñ 0 )/( n 0 + ñ 1 ) 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 = t 1e βy e i(2πñ1y/λ) + t 1 r e β(2d1 y) e i[2πñ 1(2d 1 y)/λ]. (3.5) 1 r 1 r e 4πñ 1 d 1 λ e 2βd 1 15

28 signal: In stage (b), γ is used to describe the electric field due to the non-interfering scattering γ = (eβ 1y + r e β 1(2d 1 y) )t 1 1 r 1 r e 2β 1d 1, (3.6) where t 1 = 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 which depends on the thickness d 1 of the sample. 0 tγ 2 dy, (3.7) 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 nm) on 300 nm SiO 2 /Si [6] (black dots) and the simulated result (red line). In this simulation, the refractive index of air, graphite, SiO 2 of thickness d 2 and Si were n 0 = 1, ñ 1 = i, ñ 2 = 1.46, and ñ 3 = i, respectively. The authors incorrectly used these complex refractive indices including the imaginary component for equations ( ) 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 ( 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 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

29 G-band Raman Intensity(a.u) 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 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) t 1 e 2πk λ y e i(2πñ 1y/λ 1 ), (3.1) 17

30 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/sio 2 /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 = (eβ 1y + r e β 1(2d 1 y) )t 1 1 r 1 r e 2β 1d 1. (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 = ( eβ1y 2 + r 2 e β 1(2d 1 y) 2 ) t r 1 r e 2β, (3.8) 1d 1 2 (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= t 1 eβ1y e i[2πn 1y/λ 1 ] + t 1 r e β 2(2d 1 y) e i[2πn 1(2d 1 y)/λ 2 ] 1 r 1 r e 2β 2(2d 1 y) e 2i f. (3.9) 2 I will now show why the optical interference of Raman signal from the same infinitesimal source should be coherent. 18

31 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 10 7 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 Application to Graphite I chose mostly the same parameters as in the simulation of the previous model. The refractive indices of SiO 2 (thickness: 300 nm) and Si were unchanged: n 2 = 1.46 and n 3 = The excitation light of wavelength λ 1 = 532 nm will scatter a λ 2 = 580 nm Raman signal in the G band. The only difference was that the refractive index of graphite ñ 1G = i (calculated 19

32 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 ñ 1g = i, 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 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 ñ 1G = i) 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 ñ 1g = i 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 i 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 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 i 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 i in Figure 3.6(a) was inferred by the optical reflectance measurements [26]) and i 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

33 layers, the simulation result using i 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 SiO 2 /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 I bulk. Strongest (L) Minimum (L) I strongest : I bulk I 2L : I bulk Experimental results [6] 9-15 around Previous simulated results [6] 22 N.A No interference Eq. (3.8) Interference ñ 1G = i Eq. (3.9) ñ 1g = i Application to MoS 2 As the revised model was successful when applied to graphite, I will apply it to MoS 2 in this section. 21

34 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 MoS 2 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 respectively, and for thick layers, the variations are and , respectively. (In Ref [29], it was claimed that the sample was a thin layer but there was no precise indication of the thickness of MoS 2. 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 MoS 2, I will simulate using all these parameters. The thickness of SiO 2 was 285 nm and the wavelength of incident light was 514 nm, the same as the parameters in Ref [18]. The frequencies of the E 1 2g and A 1g modes are and 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 λ 2 = 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

35 (a) G-band Raman Intensity(a.u) bulk Number of layers (b) Figure 3.6: (a) Experimental results for G-band of graphite [6] (dots) and the simulated result using i (line). Inset: a magnified view of the region of layers. (b) Experimental results for G-band of graphite [6] (dots) and the simulated result using i. 23

36 Table 3.2: Refractive index of MoS 2 in three references Substrate Thickness n 1 k 1 [28](2014) <5 L L [29](2010) Thin [30](1992) Thick Figure 3.7: Simulated results of Raman intensity as a function of thickness for MoS 2 with different refractive indices: i (red dashed), i (red), 6.5-3i (black) and i (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 : I bulk, 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 i was 5 times as much as that of i. Thus, the refractive index is likely to change considerably from ultra-thin layers to the bulk. 24

37 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 MoS 2 on 285 nm SiO 2 /Si and used 4 different complex refractive indices of MoS 2 reported in 3 different references 1. Experimental result [18] Simulated results Strongest (L) I strongest : I bulk I strongest I bulk E 1 2g A 1g 4 8 ñ i i i i The Effect of Changing the Thickness of SiO 2 on Si The thickness of SiO 2 will periodically affect the intensity as a function of the thickness of MoS 2 (as the effective reflection coefficient of MoS 2 /SiO 2 /Si is a periodic function of the thickness of SiO 2 ). For an incident light of 514 nm and a Raman signal of 525 nm, the periodicity of the thickness of SiO 2 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 SiO 2 using ñ 1 = 6.5 3i (as this refractive index yielded a better prediction of the peak position for MoS 2 on 285 nm SiO 2 /Si). The highest intensity can occur for MoS 2 thicknesses up to 16.5 layers when the thickness of SiO 2 is ( n) nm, n is a non-negative integer 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 MoS 2 so that the experimental data in Chapter 5 can be properly fitted. 1ñ 1 = i and ñ 1 = i respectively are average values of the results in Ref [28] for a thin layer and a thicker layer i is the average value of the result in Ref [29] i is the average value of the result in Ref [30]. 25

38 18 The highest peak number is 16.5 L The peak layer (L) The periodicity is 179 nm The thickness of SiO 2 (nm) Figure 3.8: The peak layer (the number of layers where the strongest intensity apprears) as a function of the thickness of SiO 2 for MoS 2 on SiO 2 /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

39 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 E 1 2g and A 1g modes of MoS 2, as the experimental observations [18] and my experimental results (in Chapter 5) indicated. This difference means that the E 1 2g mode with a Raman shift of around 382 cm 1 excites more easily than the A 1g 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 E 1 2g and A 1g of h ω v = 0.1kT (where k is the Boltzmann constant and T is the room temperature). Thus, the ratio between the number of states exciting the A 1g and E 1 2g 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 E 1 2g and A 1g modes. The difference of Raman shift between the two modes is 20 cm 1. The energy difference between E 1 2g and A 1g is h ω v = 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 i. It was best suited to fit the experimental integrated intensities of the E 1 2g (black dots) and A 1g (red dots) modes for MoS 2 on 285 nm SiO 2 /Si as shown in Figure This simulation gave the strongest intensity at 4 layers. However, simulated results indicated (I 1 I 0 )/(I 2 I 1 ) > 1, which cannot explain 27

40 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 A 1g (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 MoS 2 on gel (or wax), the thickness of some thin layers was determined by comparing the measured intensity to the simulated intensity. Integrated intensity (arb. u.) 22 E 1 2g Number of layers Figure 3.10: The simulated result using a refractive index of i (blue dashed) to fit the experimental result [18] of MoS 2 on 285 nm SiO 2 /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). A 1g 28

41 Chapter 4 Sample Preparation and Characterization In this chapter, I will introduce the preparation methods for MoS 2 samples on different substrates for use in Raman and absorption spectrum analysis. Then, the transmission electron microscope (TEM) analysis results of the MoS 2 sample will be briefly discussed. 4.1 Substrate types Mechanical exfoliation is widely used as an approach to exfoliate ultrathin layers from bulk mineral MoS 2 [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 MoS 2. Ultrathin films were successfully deposited on wax, and gel. Samples were then analyzed using Raman spectroscopy. Thick MoS 2 on wax/glass was investigated in order to understand excitonic dynamics of the absorption spectrum. For SiO 2 /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 SiO 2 /Si A piece of adhesive scotch tape was put on the bulk mineral MoS 2 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 SiO 2 /Si substrate.to identify potential ultrathin layers, an optical microscope was used to find transparent regions. 29

42 Previous work demonstrated that for certain thicknesses of SiO 2 ( nm), monolayers of MoS 2 were successfully deposited on SiO 2 /Si substrate [5][32][9][33][34][35][36][37]. In the present work, no monolayers were found for samples prepared on SiO 2 /Si substrate, which may be due to the weak optical contrast between the bare substrate surface and a single layer MoS 2 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 A 1g and E 1 2g modes. As the Raman peak frequency differences of three, four layers and bulk MoS 2 were 23.6, 24.5 and 25.0 cm 1, respectively, as indicated in a previous experimental result [18] on 285 nm SiO 2 /Si, this region might be 3-4 layers thick. (a) 5x cm -1 4x10 4 Intensity (counts) 3x10 4 2x10 4 1x Raman shift (cm -1 ) (b) Figure 4.1: (a) Optical image of exfoiated MoS 2 on 100 nm thick SiO 2 /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

43 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 MoS 2 to exfoliate MoS 2 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 µm 2 were observed. Figure 4.2: 1-4 layers were produced by exfoliation and deposited onto gel Wax/Glass In this method, flat bulk MoS 2 was adhered to melted wax (with a refractive index of 1.46 [39] ) supported on a glass substrate. The MoS 2 /wax/glass samples were heated to 373 K for 20 minutes and subsequently cooled in air for 30 minutes. After ensuring that the MoS 2 was in good contact with the wax, the scotch tape was used to peel off thick layers, leaving ultrathin MoS 2 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 µm 2 was attained as shown in Figure 4.3(b). 31

44 (a) (b) Figure 4.3: Optical image of ultrathin MoS 2 near an edge with many layer thicknesses (a) and a single layer (b) on wax. The single layer was as large as µm Copper Grid To prepare MoS 2 samples for TEM analysis, thin pieces of MoS 2 exfoliated from the mineral bulk 2H MoS 2 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 MoS 2 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 MoS 2 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 MoS 2. 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

45 (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

46 (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

47 Chapter 5 Optical Results In this chapter, Raman spectroscopy results for MoS 2 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 E 1 2g and A 1g modes. Figure 5.1 shows examples of Raman spectra with Gaussian fits (solid lines) to raw data points (dots) for 1-4 layers of MoS 2 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 Thickness Determination of the MoS 2 Films Figure 5.2 shows the absolute Raman peak intensity plotted versus peak frequency difference of each measurement of the E 1 2g (black) and A 1g (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

48 Figure 5.1: Curve fitted results (solid lines) of original Raman spectra (dots) with a resolution of 1.5 cm 1 of the E 1 2g and A 1g modes for ultrathin layers and bulk MoS 2. 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 i was used in the simulation as it had the best fit for MoS 2 on 285 nm SiO 2 /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 MoS 2 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 MoS 2 for the E 1 2g and A 1g 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

49 Intensity (counts) L E 1 2g mode A 1g mode Simulated fit result 2 L 3 L 4 L qqsf qqsf 7 L 12 L 18 L bulk The frequency difference of two modes (cm -1 ) (a) Intensity (counts) L E 1 2g mode A 1g mode Simulated fit result 2 L 3 L 4 L qqsf qqqqqsf The frequency difference of two modes (cm -1 ) (b) qqqqqsf 5 L 18 L bulk Figure 5.2: Single measurements of the peak intensity as a function of the difference of the peak frequencies in E 1 2g (black) and A 1g (red) modes for MoS 2 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 i was used i was used to predict the intensity of the bulk as it made a reasonable prediction of I peak : I bulk for MoS 2 on 285 nm SiO 2 /Si. 37

50 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 MoS 2, five refractive indices were used in the simulation: i, i, 6.5-3i, i from Ref [28] [29] [30] and the index of i obtained in section which was better suited to fit the experimental result of MoS 2 on 285 nm SiO 2 /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 MoS 2 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 A 1g or E 1 2g modes of MoS 2 on gel as a function of thickness using the refractive indices of MoS 2 of i, i, 6.5-3i, and i reported by Ref [28] [29] [30] and a selected refractive index of i. (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 SiO 2 /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 SiO 2 /Si. As the refractive index of i was better at predicting the peak number for MoS 2 on 285 nm SiO 2 /Si as shown in Figure 3.10, I assumed that it would also apply to MoS 2 on gel (or wax). However, there were two problems when applied to MoS 2 on 285 nm SiO 2 /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 (I 1 I 0 )/(I 2 I 1 ) > 1. b. When using the refractive index of i, I peak : I bulk = 18.8 was rather high compared to 8 in the experimental result, 38

51 which implies that ultra-thin layers and bulk MoS 2 had considerably different refractive indices. Assuming the same problems also exist in MoS 2 on gel (or wax), a translation of the simulated results along the y axis was applied to address a. Also, another refractive index of i, which better predicted I peak : I bulk for 285 nm SiO 2 /Si, was applied to simulate the intensity for bulk on gel to address b. Table 5.1: Simulated Raman peak intensity for MoS 2 on gel compared to that on 285 nm SiO 2 /Si. (Strongest (layers): the thickness where the strongest Raman peak intensity appears. I strongest, I bulk and I 2L : the strongest intensity, the intensity of the bulk and two layers, respectively.) Substrate ñ Strongest (L) I strongest : I bulk I 1L : I bulk i nm SiO 2 /Si [18] i (Experiment shows 6.5-3i peak layer is 4 L.) i i i gel (n = 1.46) i (Probably the peak layer 6.5-3i should be more than four layers) i i Based on the above assumptions, the thickness of thin layers and bulk MoS 2 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 MoS 2 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 Raman Peak Frequency Figure 5.4(a) plots the Raman peak frequencies of the E 1 2g and A 1g modes for MoS 2 of different thickness on gel (black) and 1-3 layers and bulk for MoS 2 on wax (red), as compared to a reported experimental result [18] (Figure 1.5). Both on gel and wax, the frequency of the 39

52 A 1g mode was generally blue-shifted as the thickness increased agreeing with Ref [18], except for the frequency of 4-7 layers. For the E 1 2g 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 E 1 2g mode was claimed to be continuously red-shifted as the thickness increased. The peak frequencies for MoS 2 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 E 1 2g and A 1g modes for MoS 2 both on gel and on wax in comparison with the experimental result on SiO 2 /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 E 1 2g and A 1g modes for MoS 2. Substrate 1 L 2 L 3 L 4 L More than 4 L Bulk cm nm SiO 2 /Si [18] ±0.6 ±0.3 ±0.2 ±0.3 ±0.2 gel ±0.1 ±0.1 ±0.1 ±0.2 ±0.1 wax ±0.1 ±0.1 ±0.1 ±0.2 ±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 MoS 2 on substrates of 285 nm SiO 2 /Si [18] and gel. The errors for single measurements are ±0.5 cm 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 MoS 2 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 A 1g and E 1 2g modes of MoS 2 on gel (black) and on wax (red), with a magnified plot at lower thicknesses (inset). The intensities had a roughly linear 40

53 Table 5.3: Raman peak shift (cm 1 ) from individual measurements from ultrathin layers compared with the shifts for average measurements for MoS 2 on substrates of 285 nm SiO 2 /Si [18] and gel. Single measurement on gel Average measurements on gel Average measurements on 285 nm SiO 2 /Si [18] 1 L 2 L 3 L 4 L More than 4 L bulk cm ± ± ± ± ± ±0.1 ±0.1 ±0.1 ±0.2 ± ±0.6 ±0.3 ±0.2 ±0.3 ±0.2 relationship at 1-4 layers. For MoS 2 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 Raman Line Width Figure 5.4(c) shows line widths of both modes as a function of MoS 2 thickness on gel (black) and on wax (red) for 1-3 layers and bulk. Line widths of the A 1g mode were wider than those of the E 1 2g mode. Line widths of bulk in both modes for both substrates had the smallest line widths consistent with the published result for 285 nm SiO 2 /Si substrate [18]. The line widths of the A 1g mode were approximately twice as those of the E 1 2g mode. However, unlike the reported result [18], on gel, the A 1g mode had a continuous decline for the first four layers while the trilayer presented the highest width in the E 1 2g mode. This inconsistency was due to differences of the substrates, as the medium (material) will expand the optical Gaussian width. 41

54 Peak frequency (cm -1 ) E 1 2g Bulk Thickness (L) (a) A 1g Frequency difference (cm -1 ) Intensity (counts) E 1 2g A 1g for gel E 1 2g A 1g for wax The curve fit Thickness (L) Bulk (b) 8 7 Line width (cm -1 ) A 1g E 1 2g A 1g 3 E 1 2g Bulk Thickness (L) (c) Figure 5.4: (a) Raman peak frequency as a function of thickness in the A 1g and E 1 2g modes of MoS 2 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 A 1g and E 1 2g modes of MoS 2 on gel (black) and on wax (red). Simulation (blue solid) with the refractive index of i was used to fit the experimental results. (c) Raman peak line widths as a function of thickness for the A 1g and E 1 2g modes of MoS 2 on gel (black) and on wax (red). 42

55 5.1.5 Discussion Thin layers and bulk MoS 2 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 A 1g mode and E 1 2g modes, and the simulated intensity predictions. 1. The Raman peak frequency: a. The differences of the peak frequencies of the A 1g mode and E 1 2g mode for films less than four layers and bulk MoS 2 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 E 1 2g mode should be continuously red shifted as the thickness increases. But after the decrease of the peak frequency of the E 1 2g mode from one to three layers, it began to rise as the film of MoS 2 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 MoS 2 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 A 1g mode were significantly larger than that of the E 1 2g mode. 4. Figure 5.5 shows the integrated intensity as a function of the thickness of MoS 2 on the gel and wax substrates for the E 1 2g and A 1g 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 However, the integrated intensity of the E 1 2g mode was stronger than that of the A 1g mode compared to the reported result [18]. The integrated intensity of the A 1g mode was stronger, which cannot be explained by the fact that E 1 2g is a lower state. 43

56 Integrated Intensity (arb. u.) E 1 2g A 1g for gel E 1 2g A 1g for wax Thickness (L) Figure 5.5: The integrated intensity as a function of the thickness of MoS 2 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 MoS 2. 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 I 0 of wax/glass, by the transmitted light I of sample MoS 2 /wax/glass. This eliminates the background leaving only the absorption spectrum of MoS 2 (Figure 5.6). The thickness y of the sample can be calculated by: I(y) = I 0 exp( 4πky ) (5.1) λ where k = 1.5 is the extinction coefficient at λ = 600 nm for thicker MoS 2 [28]. 44

57 nm 668 nm ln (I 0 /I) Wavelength (nm) Figure 5.6: Excitonic peaks A at 2.05 and B at 1.86 ev appeared for layers at room temperature. [15] T is, The reflection on the surface of air/mos 2 cannot be ignored. The transmission coefficient T = 4n 0n 1 (n 0 + n 1 ) 2, (5.2) where n 0 = 1, and n 1 = 1.6 are the refractive indices of air and MoS 2 at λ = 660 nm [29]. Moreover, considering that 94.7% of light will get transmitted through the interface of air/mos 2, a correction of was made on the ln( I 0 I ). The number of layers of MoS 2 is then calculated to be 83-95, which is a thin bulk MoS 2. 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 MoS 2, a need for the right refractive indices for certain thicknesses was discussed. Thus, more optical analysis should be done in the future. 45

58 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 MoS 2. To test the new model, ultra-thin layers of MoS 2 were successfully made on gel and wax substrates by mechanical exfoliation from bulk MoS 2. The thickness of MoS 2 was determined by the difference of the Raman peak frequencies of the A 1g and E 1 2g 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 MoS 2. But the simulation indicated that the most consistent index of refraction for MoS 2 was i. The ratio of the number of states between A 1g and E 1 2g 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 A 1g mode at a higher energy level excites more easily compared to the E 1 2g 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 E 1 2g mode was not continuously red-shifted as other publications[18] [19] have indicated. Absorption spectra were attained for MoS 2 of thickness of 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

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63 [39] Tables of refractive indices. oils, fats and waxes. chemistry/tables-of-refractive-indices/, [Refractive index of wax.]

64 Appendix A Derivation of the Effective Reflection Coefficient of the Sample/SiO 2 /Si Figure A.1 describes the process whereby the light from the sample transmits into SiO 2 of zero extinction value (no absorption) and undergoes infinite reflections in SiO 2, 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/sio 2 /Si structure, assuming the silicon layer is thick enough. (For silicon, with a small extinction of 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/sio 2, SiO 2 /sample and SiO 2 /Si are as shown in Eq. (A.1), Eq. (A.2) and Eq. (A.3), respectively: r 1 = n 1 n 2 n 1 + n 2, (A.1) 52