論文 / 著書情報
Article / Book Information
題目(和文)

マイクロ・ナノラマン分光によるナノマテリアルの局所物性評価に関する研究

Title(English)

Characterization of local physicochemical properties of nano-sizedmaterials by micro and nano-Raman spectroscopy

著者(和文)

BaloisMariaVanes

Author(English)

Maria Vanessa Balois

出典(和文)

学位:博士（理学）,学位授与機関:東京工業大学,
報告番号:甲第9959号,
授与年月日:2015年9月25日,
学位の種別:課程博士,
審査員:林 智広，早澤 紀彦，原

正彦，北村

房男，尾笹

Citation(English)

Degree:,Conferring organization: Tokyo Institute of Technology,
Report number:甲第9959号,
Conferred date:2015/9/25,
Degree Type:Course doctor,
Examiner:

学位種別(和文)

博士論文

Type(English)

Doctoral Thesis

Powered by T2R2 (Science Tokyo Research Repository)

一成

Characterization of local physicochemical
properties of nano-sized materials by
micro and nano-Raman spectroscopy

byMaria Vanessa Cases Balois

Submitted to the Department of Electronic Chemistry
for the degree of

DOCTOR OF SCIENCE
at theTOKYO INSTITUTE OF TECHNOLOGYTable of Contents
Chapter 1: Introduction ……………………………………………………… 1
1.1 Transistor channel materials ……………………………………....... 1
1.1.1 Strained Silicon (ε-Si) technology ……………………………… 1
1.1.2 Carbon Nanotubes: a future possibility ……………………...... 3
1.2 Raman spectroscopy as a characterization technique fornano-sized materials ………………………………………………….. 4
1.2.1 The Raman effect and Raman scattering ……………………... 4
1.2.2 Micro-Raman Spectroscopy …………………………………….. 7
1.2.3 Nano-Raman Spectroscopy …………………………………….. 8
1.3 The Scope of this Thesis …………………………………………….. 9
1.3.1 The Remaining Chapter of this Thesis ………………………… 9
1.4 References ………………………………………………………….. 10
Chapter 2: Properties of a Focused Three-Dimensional Field acrossan Interface for Micro-Raman Spectroscopy ………………………….. 13
2.1 Focusing an electric field using an aplanatic lens ………………. 13
2.2 Diffraction of a focused electric field across an interface ………. 17
2.2.1 Linear Polarization ………………………………………………. 19
2.2.2 Radial Polarization ………………………………………………. 27
2.2.3 Azimuthal Polarization ………………………………………….. 30
2.3 Polarization dependence of a focused electric field across anInterface ……………………………………………………………… 32
2.3.1 Linear Polarization ………………………………………………. 32
2.3.2 Radial Polarization ………………………………………………. 34
2.3.3 Azimuthal Polarization …………………………………………... 35
2.4 References …………………………………………………………… 35
Chapter 3: Strained Silicon: a Theoretical Study …………………... 37
3.1 Raman scattering in crystals ……………………………………….. 37
3.2 Excitation and detection of phonon modes ………………………..38
3.2.1 Silicon Raman tensors in crystal and sample coordinates …. 39

	
  

i	
  

3.2.2 Using a high numerical aperture objective lens to excite
transverse optical phonon modes …………………………….. 41
3.2.3 Importance of transverse optical phonon mode detection for
stress determination in Silicon ………………………………… 42
3.3 Three-dimensional electric field calculations: polarization
Dependence …………………………………………………………. 43
3.4 Calculation of Fresnel coefficients ………………………………… 49
3.5 Phonon mode visualization in strained Silicon via threedimensional Raman intensity calculation ………………………… 51
3.5.1 Nanomembrane ………………………………………………… 52
3.5.2 Nanowire ………………………………………………………… 53
3.6 References ………………………………………………………….. 57
Chapter 4: Strained Silicon: a micro-Raman Spectroscopy
Characterization Study of Stress after Nanopatterning via
Precise Polarization Control …………………………………………… 59
4.1 Sample fabrication ……………………………………………………. 61
4.2 Experimental setup …………………………………………………. 63
4.2.1 Stability of experimental system ……………………………… 64
4.3 Analysis of stress I: Longitudinal optical phonon approach ……. 67
4.3.1 LO: Theoretical Approach ……………………………………... 67
4.3.2 LO: Experimental Results and Discussion …………………... 71
4.4 Analysis of stress II: Transverse optical phonon approach …….. 83
4.4.1 TO: Theoretical Approach ……………………………………... 83
4.4.2 TO: Experimental Results and Discussion …………………... 86
4.5 Comparison of LO and TO analysis ………………………………. 94
4.6 References …………………………………………………………... 97
Chapter 5: Carbon Nanotubes: Theory, Properties andTip-Enhanced Raman Spectroscopy ………………………………..... 99
5.1 Tip-enhanced Raman scattering (TERS) …………………………. 99
5.1.1 Electric field enhancement at the tip ………………………... 100
5.1.2 System geometry ………………………………………………103

ii	
   	
  

5.1.3 Fabrication of metallic probes used for TERS ……………... 103
5.1.4 Taking care of TERS tips …………………………………….. 104
5.2 Carbon nanotubes …………………………………………………. 105
5.2.1 The Raman spectrum of CNTs ……………………………… 106
5.2.2 CNT and TERS ………………………………………………...107
5.3 References …………………………………………………………. 109
Chapter 6: Carbon Nanotubes: Tip-Enhanced TeraHertz Raman
Scattering, a nano-Raman Spectroscopy Study ………………….. 113
6.1 Temperature determination based on the Boltzmann
distribution ………………………………………………………………… 114
6.2 Sample Preparation ……………………………………………….. 116
6.2.1 Au/Ge substrate ………………………………………………. 116
6.2.2 Carbon nanotube on substrate preparation ………………... 118
6.3 Experimental setup for tip-enhanced THz Raman scattering … 118
6.4 Local temperature detection of carbon nanotube samples by
TE-THzRS ………………………………………………………….. 120
6.5 Comparison and discussion with analytical results ……………. 122
6.6 TE-THzRS and carbon nanotubes: a possible next step ……... 127
6.7 References …………………………………………………………. 128
Chapter 7: Summary and Conclusions …………………………….. 131
List of Publications …………………………………………………….. 135
List of Papers …………………………………………………………….. 135
List of Presentations (International conference) ……………………… 135
List of Presentations (Domestic conference) …………………………. 136
Awards and Distinctions ………………………………………………….137
Acknowledgements …………………………………………………… 139
Appendix A : Copyright Permissions ……………………………… 143

	
  

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Chapter 1: Introduction
Modern day computing is constantly getting smarter and faster because of theunrelenting rule of Moore’s law – which states that the number of transistors that canbe placed on an integrated circuit doubles approximately every two years. In order tokeep pace, ingenious engineers have devised new methods to manipulate silicon(Si), such as strain engineering, and are constantly on the look out for uniquematerials that can be the next transistor channel, should Si reach its limit.
1.1.

TRANSISTOR CHANNEL MATERIALS

1.1.1. Strained Silicon (ε-Si) Technology
Strain in semiconductors such as Si is important because it enhances device
performance and reduces power consumption of electronic device [1.1]. Present-day
transistors found in a computer’s central processing unit (CPU) (Fig. 1.1) employ ε-Si
as channel material using high amounts of strain and stress to increase carrier
mobility [1.1, 1.2]. The increase in carrier mobility is mainly due to strain-induced
band warping. By applying strain to Si, its crystal symmetry is broken causing theband edge energy to shift and altering the band structure away from the energy
minimum. When more stress is applied along a low-symmetry axis, the crystal
symmetry is further destroyed resulting to greater band warping compared to whenstress is applied along a high symmetry axis [1.3]. Strain in silicon can be induced via
heteroepitaxy, a process of growing a film whose equilibrium lattice constant isdifferent from the substrate [1.4]. For tensile strained Si, a bulk-Si film is grown on topof a bulk-Si1-xGex film (Fig. 1.2a). The lattice mismatch between Si and Si1-xGex
causes the tensile strain in the bulk-Si film (Fig. 1.2b). Compressive strain can also
be induced using epitaxy by using a substrate that has a smaller lattice constantcompared to Si. The manipulation of strain done through precise strain and stressnanoengineering has been used as a powerful strategy in fabrication and processingof variety of electronic, optoelectronic and photovoltaic devices [1.1, 1.5 – 1.11].

	
  

1	
  

Figure 1.1. (a) Typical CPU with pins. (b) CPU without pins (removed by polishing). (c)Typical transistor in CPU located in the area indicated by the white square and point C in (b).

Figure 1.2. Illustration of applying strain to Silicon: (a) Bulk lattice constant of a bulk-Si film tobe grown on top of a bulk-Si1-xGex film and (b) the two films are placed together, wherein the
top Si film is tensile strained.

Engineers are constantly shrinking the size of transistors leading to complex
structures in order to accommodate Moore’s law. As a consequence, the stresspresent in these structures is no longer isotropic / uniaxial but more anisotropic. In
order to probe the stress and strain in the nanoscale, various characterization
techniques have been developed. Each of these techniques suffers, however, anumber of limitations such as being invasive or insensitive to all strain components,or having a limited spatial resolution.

Transmission electron microscopy-based

techniques, including among others holographic interferometry [1.12] and nanobeam
electron beam diffraction, [1.13] provide the ultimate spatial resolution in mapping
stress, but they are invasive, as they require a special sample preparation. This
involves cutting the materials and thinning it down to ~100-500 nm, which can induce
partial stress relaxation in addition to possible lattice local distortion during theanalysis. X-ray-based techniques have immensely benefited from the availability ofhigh-brilliance X-ray sources in the 3rd generation synchrotron sources or free

2	
   	
  

electron lasers leading to development of lensless coherent imaging andnanodiffraction techniques. These techniques have recently succeeded in mapping
the stress in sub-micrometer structures and devices [1.13 – 1.19]. One such example
of an effective X-ray-based technique was the use of X-ray nanodiffraction combinedwith finite element modeling in probing the local stress in single quantum dot-based
devices [1.19]. However, they are still not yet ready for routine analyses because of
the need for synchrotron facilities besides requiring special sample preparation toeliminate the background signal from the underlying substrate. Another techniquethat shows a lot of promise is micro-Raman spectroscopy, which is highly sensitive,
nondestructive and simple to implement. The major drawback of this technique is its
inability to measure the individual components of stress thus making the average
stress value the only information accessible. If the individual components ofstress (in the x, y, and z-directions) can be measured, the actual stress rather
than the average stress value can be determined. Thus, providing a better
understanding of the interplay between stress and basic properties and performanceof strained semiconductors based nanoscale materials and devices.
1.1.2. Carbon Nanotubes: a Future Possibility
Moore’s law is now slowly taking its toll on e-Si based transistors, presenting new
problems as a consequence to transistor miniaturization such as current leakage andpower wastage in the form of heat [1.20]. Preparing for the inevitable, scientists andengineers have been looking for alternative materials that are scalable and still follow
Moore’s law. A promising material is carbon nanotubes due to their superior electricalproperties and ultrathin body [1.20 – 1.24]. CNTs have exhibited ballistic carriertransport and, due to their utlrathin body, are able to maintain gate control of current
better than other competing structures, such as nanowires and fins - preventing
short-channel effects [1.22]. There have already been studies on carbon nanotube
transistors with sub-10 nm channel length [1.23]. And recently, the first CNT
computer has been demonstrated by engineers from Stanford University [1.24]. The
dimensions of single CNTs are typically 1 - 2 nm in diameter. To study these nanosized structures, nano-Raman spectroscopy must be utilized. Function optimization
can be realized through characterization of CNTs’ local electronic properties via
nano-Raman techniques such as TERS and tip-enhanced THz-Raman spectroscopy(TE-THzRS). Although nano-Raman spectroscopy is said to be non-invasive
technique, it is not entirely true. In TERS, a strong localized optical field is generated
at the apex of a metallic tip when illuminated [1.25]. This localized field produces

	
  

3	
  

heat, which in turn can heat up the sample. Sample heating may affect its physical orchemical properties, which thus affects the analysis of the sample itself. For heatsensitive samples, heating may even cause sample damage. From this point of view,
TERS can be classified as an invasive technique. If the actual temperature
generated by the localized optical field can be controlled, sensed anddetermined during TERS experiments, experimental parameters such as laser
power and exposure time can be optimized in order not to damage or change
the physical or chemical state of the sample. Such kind of work can also be
extended to studying temperature related behavior of the sample.
1.2.

RAMAN SPECTROSCOPY AS A CHARACTERIZATION TECHNIQUE FOR

NANO-SIZED MATERIALS
Raman spectroscopy is a vibrational spectroscopy that obtains direct information on
the structure and orientation of molecules (in particular their vibrational structure) andcrystals via molecular or lattice vibrations, respectively [1.26]. It is based on the
inelastic scattering of a monochromatic excitation source, typically a laser. Wheninelastic scattering occurs, the energy of the incident photon is changed. The change
in energy is manifested as a change in frequency of the scattered light relative to theincident light [1.27]. This phenomenon is called Raman scattering, which is based on
the Raman effect.
1.2.1 The Raman Effect and Raman Scattering
The Raman effect is a phenomenon that results from the interaction of light andmatter [1.28]. It was discovered by Chandrasekhara Venkata Raman [1.29] in 1928 inIndia. Raman’s Nobel Prize winning discovery was motivated by his fascination withthe beautiful blue color of the Mediterranean Sea while he was on a boat returning to
India from England. He did not accept Lord Rayleigh’s explanation that the blue colorof the seawater was caused by reflection of light from the blue sky. This was because
when he observed the blue seawater through a Nicol prism set at the Brewster angle(to eliminate reflected light from the seawater surface), the seawater was still blue.
He then surmised that light scattering of the seawater caused the seawater to beblue [1.27, 1.30]. When he returned to India, he conducted an experiment using a
focused beam of sunlight onto a liquid. He filtered the incident sunlight before theliquid with a green filter to produce a monochromatic light source. By placing a yellow

4	
   	
  

filter right after the sample, he then observed yellow light being scattered by the
sample. This was evidence of the existence of inelastic scattering.
As mentioned earlier, inelastic scattering results in a change in frequency and energy
between the incident and scattered photon. The change in frequency of the scatteredphoton can be either higher or lower than the incident photon, as shown in Fig. 1.3.
Let ω 0 be the frequency of the incident photon with energy E0 that encounters amolecule. From the photon-molecule interaction, the scattered photon is produced
having a frequency ω s and energy Es. If the scattered photon loses energy to themolecule (ES < EL), the molecule is excited to a higher energy level, moving from thevibrational ground state, v = 0, to the first excited state, v = 1, of a vibrational mode.
The vibrational energy would be !ω M = EL − ES , where ω M is associated to thefrequency of the vibrational level. As a result, the frequency of the scattered photon
will be lower than the incident photon ( ω s = ω 0 − ω M ) and this is called Stokes
Raman scattering. On the other hand, if the scattered photon gains energy from themolecule (ES > EL), the molecule relaxes from an excited vibrational state (v = 1) tothe ground state (v = 0) with vibrational energy !ω M = ES − EL . In this case, thefrequency of the scattered photon will be higher than the incident photon (ω s = ω 0 + ω M ) and this is called anti-Stokes Raman scattering. For anti-Stokes
Raman scattering to occur, the molecule must initially be in the excited vibrational
state. This usually only occurs through thermal excitation, implying that at equilibrium
the anti-Stokes signal depends on temperature (T) by the Boltzmann factor,()

exp ⎡⎣ −!ω M / kBT ⎤⎦ . As a consequence, the anti-Stokes lines are much weaker
than the Stokes lines and become exponentially weaker at higher vibrationalenergies [1.26, 1.30].
Another type of scattering occurs along with Raman scattering and this is Rayleighscattering. Unlike Raman scattering, Rayleigh scattering is a type of elastic
scattering. The incident photon does not lose nor gain energy from the molecule (ES= EL). Therefore, if the molecule was initially at the ground state, it will return to theground state. The frequencies of the incident and the scattered photons are the same
( ω s = ω 0 ). In Raman spectroscopy, all three scattered photon frequencies –Rayleigh, Stokes and anti-Stokes – can be detected and recorded onto a spectrum

	
  

5	
  

as shown in Fig. 1.4. The most intense peak is from Rayleigh scattering. Surrounding
the Rayleigh peak are those of the Stokes and anti-Stokes peaks.

Figure 1.3. Energy diagram for Rayleigh, Stokes and anti-Stokes scattering.

Figure 1.4. Raman spectra with Rayleigh, Stokes and anti-Stokes peaks.

There are two kinds of Raman spectroscopy techniques that can be used forcharacterization.

These

are

micro-Raman

spectroscopy

andnano-Raman

spectroscopy. Although both techniques rely on Raman scattering, their maindifference is the attainable spatial resolution and is illustrated in Fig. 1.5.

6	
   	
  

Figure 1.5. Configuration of (a) micro-Raman and (b) nano-Raman spectroscopic system.

1.2.2 Micro-Raman spectroscopy
In conventional Raman spectroscopy, laser light is focused onto the sample by a lensand detected by a spectrometer. Micro-Raman spectroscopy integrates a microscopethat is coupled to a Raman spectrometer to acquire Raman spectra from microscopic
samples. It is a sensitive and non-destructive technique that does not require any
complex sample preparations [1.31]. Typically, a confocal microscope is used toattain higher resolution and contrast by introducing a pinhole that can spatially filter
out unwanted stray light from out-of-focus regions of the sample [1.28]. However, themaximum spatial resolution that can be attained by a micro-Raman spectroscopy
system cannot exceed the diffraction limit, given by

1.22λ
,
NA

(1.1)

NA = nsinθ ,

(1.2)

d=

where d is the resolution length, λ is the wavelength of light, NA is the numericalaperture of the objective lens defined in Eq. (1.2), n is the index of refraction of thematerial being imaged and θ is the half-angle subtended by the objective lens [1.32].
Therefore, fine structures in the sample that are separated by less than a wavelength
cannot be resolved (Fig. 1.5a). If there are nanoscale structures (red and green) in

	
  

7	
  

the sample, the detected Raman signal would be the average of these two
structures. The resolution of a micro-Raman system ranges from a few micrometers
to hundreds of nanometers depending on the wavelength of light and the NA of the
microscope objective. Micro-Raman spectroscopy is classified as a far-fieldtechnique because it is diffraction limited and the lens that illuminates and collects
light from the sample is many wavelengths away from the sample.
1.2.3. Nano-Raman spectroscopy
To observe samples with nanoscale structures (typically only tens of nanometers insize), nano-Raman spectroscopy is necessary. In nano-Raman spectroscopy, amicro-Raman spectroscopy system is combined with probe-based techniques thatcan break the resolution barrier wherein the spatial resolution is dictated by the probe
diameter (less than 100 nm). This technique is called near-field [1.33]. Near-fieldtechniques use an optical probe that enhances the incident light and acts as a nanolight source. Using this nano-light source, it is possible to probe the individual
structures in the sample and obtain information about that particular area of thesample, rather than an average of a large area, as is the case for micro-Raman
spectroscopy. Higher spatial resolution is achieved and is dependent on the size ofthe optical probe (Fig. 1.5b). Prime examples of nano-spectroscopic techniques are:tip-enhanced Raman scattering (TERS) [1.34-1.36] to measure physical andchemical composition, and tip-scattering techniques using infrared light to measureelectromagnetic field distributions [1.37].

8	
   	
  

1.3.

THE SCOPE OF THIS THESISIn this thesis, characterization and analysis of the local physicochemical properties ofε-Si and carbon nanotubes are conducted via micro and nano-Raman spectroscopy,
respectively. Each technique had certain limitations that were overcome by thedevelopment of novel methods, which upon further improvement, can be utilized forcharacterization of actual transistors.
For ε-Si, the behavior of anisotropic stress relaxation upon nanopatterning fornanowire structures of various dimensions was investigated. Initially, micro-Raman
spectroscopy using a backscattering geometry was insensitive to anisotropy. But
through the use of a high numerical aperture objective lens and polarized microRaman spectroscopy, it is now possible to detect anisotropy.
For CNTs, the temperature change induced by nano-Raman techniques and theactive control of temperature was studied. In order to determine the temperature andchange in temperature at the sample, both the low frequency Stokes and anti-Stokes
Raman (THz-Raman) intensities were detected. These intensities are related to theRaman shift and temperature via the Boltzmann distribution. THz-Raman detection
was possible through the development of a novel nano-Raman technique called
tip-enhanced THz-Raman spectroscopy.
1.3.1. The Remaining Chapters of this Thesis
In Chapter 2, the theoretical foundation on diffraction of focused electric fields acrossan interface for three different polarizations (linear, radial and azimuthal) ispresented. This is essential in understanding the behavior of light as it moves fromone medium to another, which is the case for any microscope-based system. The
effect of polarization on the diffracted beam is also discussed. Through proper
polarization control, it is possible to achieve better spatial resolution even though thesystem is still diffraction limited.
Chapters 3 and 4 focus on concepts and micro-Raman spectroscopy experiments
related to ε-Si. Chapter 3 discusses the behavior of light as it is scattered by Si. It isshown theoretically and by numerical simulations that through the use a highnumerical aperture lens and appropriate polarization it is possible to excite and

	
  

9	
  

detect transverse optical phonon modes that are thought to be forbidden based on
the Raman selection rules. Chapter 4 shows the first reported experimental
observation of transverse optical modes from a ε-Si nanowire and nanomembrane
samples. Stress analysis using longitudinal optical phonons and transverse opticalphonons are made.
Chapter 5 and 6 focus on concepts and nano-Raman spectroscopy experiments
related to carbon nanotubes. Chapter 5 discusses the theory and properties ofcarbon nanotubes and TERS. Chapter 6 discusses the theory of a novel temperature
determination method by a nano-Raman spectroscopy technique, TE-THzRS.
Experimental results are presented showing the potential of this technique to sense
local temperature at the nanoscale and its possible applications.
Finally, Chapter 7 gives the summary and conclusions.
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[1.9] Süess, M. J.; Minamisawa, R. A.; Schiefler G.; Frigerio, J.; Chrastina, D.; Isella,G.; Spolenak, R.; Faist, J.; Sigg, H. Nature Photon. 2013, 7, 466-472.

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[1.10] Boucaud, P.; El Kurdi, M.; Sauvage, S.; de Kersauson, M.; Ghrib, A.;
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[1.20] J. Appenzeller, Carbon nanotubes for high-performance electronics – progress
and prospects, Proceedings of the IEEE, 96, 201-211, 2008.
[1.21] P. Collins, M.S. Arnolds, and P. Avouris, Engineering Carbon nanotubes andnanotube circuits using electrical breakdown, Science, 292, 706, 2001.
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Lundstrom, and W. Haensch, Sub-10 nm Carbon nanotube transistor, Nano
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[1.23] S.J. Tans, A.R.M. Verschueren, and C. Dekker, Room-temperature transistor
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Carbon nanotube computer, Nature 501, 526, 2013.
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[1.28] G. Turrell, J. Corset, Raman Microscopy: developments and applications,Elsevier, London, 1996.
[1.29] C.V. Raman and K.S. Krishnan, Nature, 121, 501, 1928.
[1.30] D.A. Long, The Raman Effect: a unified treatment of the theory of Raman
scattering by molecules, Wiley: New York, 2002.
[1.31] M. Becker, H. Scheel, S. Christiansen, and H.P. Strunk, Grain orientation,
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[1.32] M. Born and E. Wolf, Principles of Optics: Seventh (Expanded) Edition,
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[1.36] M.S. Anderson, Appl Phys Lett 76, 3130, 2000.
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Chapter 2: Properties of a Focused Three-Dimensional
Electric Field across an Interface for Micro-Raman
Spectroscopy
Knowledge of the diffracted-light distribution of focused electromagnetic fields plays
an important role in various applications – from lens design to microscopy. These
applications mainly deal with coherent beams strongly focused in the vicinity ofplanar interfaces. Such systems are common in biology and chemistry, wherein theyuse confocal microscopes with high numerical aperture (NA) objective lenses tofocus light on a sample mounted on a glass slide. The sample usually has a different
index of refraction than glass resulting to an interface. The interface itself can change
various properties of the incident beam, such as the intensity of the individualcomponents that make the beam [2.1]. These changes vary depending on thepolarization of the incident beam. Exploring the behavior of light for different
polarizations has been a topic of interest because through proper selection ofincident polarization, it is possible to attain higher spatial resolution because of atighter focus [2.2].
In this chapter, I discuss the properties of a focused electric field traversing aninterface for different polarizations, namely, linear, radial and azimuthal. I begin withthe theory of focusing light through a lens. A rigorous derivation on the mathematicalrepresentation of a diffracted focused field across an interface is shown. The
derivations follow the formalism by Richards and Wolf [2.3]. After the derivations, I
discuss and compare the electric field distributions in a simple glass/air system foreach polarization that were computed via numerical simulations based on theequations derived. An air-glass system is used as proof of principle of themathematical concepts explained. This is in preparation for Chapter 3 wherein athree-layer system consisting of oil/ε-Si/air is studied in detail.
2.1 FOCUSING AN ELECTRIC FIELD USING AN APLANATIC LENS
In order to thoroughly discuss focused electric fields across an interface, a theoreticalunderstanding of how light is focused through an aplanatic lens is established. The
discussion presented follows that of Novotny in Ref. 2.1.

	
  

13	
  

Using geometrical optics, the fields near the optical lens can be represented
mathematically. There are two rules that need to be followed when dealing with anaplanatic lens. These are 1) the sine condition and 2) the intensity law as shown in
Fig. 2.1a and 2.1b, respectively. The sine condition states that the incident ray
travelling parallel to the optical axis (also called the “conjugate ray”) intersects withthe refracted ray that emerges from or converges to the focus F of the aplanatic
optical system on a sphere of radius f (called the Gaussian reference sphere), where
f is the focal length of the lens. The height between the optical axis and the incident
ray is:h = f sinθ

(2.1)

where θ is the divergence angle of the incident ray. The behavior of the refracted
optical rays is dictated by the sine condition. The intensity law states that the energy
flux along each ray must remain constant. Thus, the energy entering the lens must
be equal to the energy exciting the lens. The fields before and after refraction are
related by

E2 = E1

n1
n2

1/2
µ2
cosθ
µ1

(

)

(2.2)

where the magnetic permeability is equal to one ( µ = 1 ) at optical frequencies. This is

(

)

2

−1
E dA . Z µε is the wave
based on the power associated by a ray, dP = 1/ 2 Z µε

impedance and dA is an infinitesimal cross-section perpendicular to the ray
propagation.

Figure 2.1. (a) The sine condition and (b) the intensity law of geometrical optics.

14	
   	
  

A geometrical representation of the aplanatic optical system is shown in Fig. 2.2. The

(

) ( )
following definitions are made. ( x ,y ,z ) is an arbitrary point on the surface of thereference sphere, while ( x,y,z ) , an arbitrary field point near the focus. n , n , andfocus is located at the origin x,y,z = 0,0,0 . To facilitate the discussion, the
∞

∞

∞

ρ

φ

nθ are unit vectors of a cylindrical coordinate system. nθ and nφ are unit vectors of aspherical coordinate system. The reference sphere of radius f refracts the incident
rays. In doing so, it transforms a cylindrical coordinate system (incoming beam) into aspherical coordinate system (focused beam). In reality, these rays represent fields
and if the incident field, Einc , passes through the reference sphere, it is convenient todecompose the field into two components, E(s)and E(p). These represent the sincinc
polarized and p-polarized vectors, respectively. From the unit vectors defined earlier,the two fields can be defined asE(p)= ⎡⎣Einc • nρ ⎤⎦ nρ .
inc

E(s)
= ⎡⎣Einc • nφ ⎤⎦ nφ ,
inc

(2.3)

The physical interpretation of Eq. (2.3) is E(s)and E(p)refract differently. nφ remainsinc
inc
the same while nρ , is mapped into nθ . As the lens refracts the two fields, reflection
and transmission occurs, for now, the only field of interest is the field transmitted
through the lens. Hence, the total transmitted electric field, EFF , is{EFF = ts ⎡⎣Einc • nφ ⎤⎦ nφ + t p ⎡⎣Einc • nρ ⎤⎦ nθ} nn (cosθ ) .
1/2

1

(2.4)

2

where ts and tp are the Fresnel transmission coefficients defined as [2.4]:() µ k2µ+kµ k ,t p k x ,k y =() µµ kk −+ µµ kk ,rp k x ,k y =ts k x ,k y =

2 z1

2 z1

rs k x ,k y =

1 z2

2 z1

1 z2

2 z1

1 z2

(

) ε k2ε+kε k() εε kk −+ εε kk .

2 z1

2 z1

1 z2

2 z1

1 z2

2 z1

1 z2

µ 2ε 1
,
µ1ε 2

(2.5)

(2.6)

rs and rp are the Fresnel reflection coefficients which will be of much use in the latterpart of this discussion. It has been defined here along with the transmission
coefficients for ease of reference to the reader. Both transmission and reflection
coefficients are functions of k z1 and k z2 . This implies that the coefficients can be

	
  

15	
  

expressed in terms of the angle of incidence θ . The intensity law is obeyed by thepresence of the(n1 / n2 cosθ)

1/2

factor.

Figure 2.2. Aplanatic system and definition of the coordinates.

The unit vectors nρ , nφ , and nθ can now be expressed as Cartesian unit vectors nx ,ny , and nz using spherical coordinates θ and φ :nρ = cos φnx + sin φny ,(2.7)

nφ = − sin φnx + cos φny ,(2.8)

nθ = cosθ cos φnx + cosθ sin φny − sinθ nz .

(2.9)

Substituting Eqs. (2.7) – (2.8) into Eq. (2.9),⎡⎛ − sin φ ⎞ ⎤ ⎛ − sin φ ⎞
⎢
1/2
⎜
⎟ ⎥⎜
⎟ n1
EFF = ts θ ⎢Einc θ ,φ • ⎜ cos φ ⎟ ⎥ ⎜ cos φ ⎟
cosθ
⎢
⎜
⎟ ⎥⎜
⎟ n2
0
0
⎝
⎠ ⎥⎦ ⎝⎠⎢⎣⎡⎛ cos φ ⎞ ⎤ ⎛ cos φ cosθ ⎞
⎢
1/2
⎜
⎟ ⎥⎜
⎟ n1
+t p θ ⎢Einc θ ,φ • ⎜ sin φ ⎟ ⎥ ⎜ sin φ cosθ ⎟cosθ .
⎢⎜ 0 ⎟ ⎥⎜
⎟ n2
− sinθ
⎝⎠ ⎥⎦ ⎝
⎠
⎢⎣

()

()

( )

( )

(

)

(

(2.10))

This field, EFF, is the field to the right of the reference sphere of the aplanatic lens in
Cartesian vector components form and is at the core of the calculations of the focal
fields. As noted by Novotny, this field corresponds to the field on the surface of thereference sphere of the focusing lens.

16	
   	
  

2.2. DIFFRACTION OF A FOCUSED ELECTRIC FIELD ACROSS AN INTERFACE
Wolf presented an integral representation to obtain the electromagnetic field in theimage space of an optical system that has only a single medium. His work was
extended by Torok, et. al. [2.5] to an image space that had two media with different
indices of refraction. The work of Torok is briefly summarized here and focuses only
on the electric field, as it is of primary interest. For a full derivation including themagnetic field, the reader is encouraged to refer to these references [2.3, 2.5, 2,6,
2.7].
The optical system is illustrated in Figure 2.3 [adapted from Ref. 2.5 with permission
from the Optical Society of America]. It is a system of revolution in a (x, y, z)
rectangular coordinate system whose origin O is at the Gaussian focus and theoptical axis is along z. All rays travel from the negative to positive z direction, unlessindicated otherwise. The system consists of two media with refraction indices of n1
and n2 separated by a planar interface located z = z0. A monochromatic point source
is fixed in the object space at z = −∞ . It radiates a coherent beam that is focused by
an aplanatic lens with aperture, Σ , producing a perfect converging spherical wave inimage space. It is assumed that the lens is aberration free, since current real-life
microscope objectives corrected for spherical aberration are well designed. Consider
a point V in image space, at a certain time, t, where

( )

( )

(

)

! V,t = Re ⎡E V exp −iω t ⎤E⎣⎦(2.11)

is the time-dependent electric field. The field of interest is the electric field vector E. It
is generally complex and the Re denotes the real part. The location of V is defined by(its position vector, mv = (x, y, z), originating from O to V. Further, s! = s x ,sy ,sz)

represents a unit vector along a geometrical ray in image space.
The vector E satisfies the time-independent wave equation in homogeneous image
space. It is assumed to be a superposition of plane waves and has an integral
representation eloquently formalized by Wolf [2.6]

( )

EV =−

	
  

(

)

{

}

a s x ,syik
exp ik ⎡ Φ s x ,sy + s! ⋅mv ⎤ × ds xdsy ,
∫∫
⎣
⎦
2π Ω
sz

(

)

(2.12)17	
  where a is the vector field amplitude [2.8]; Φ is the wave aberration function, which iszero in this case; Ω is the solid angle formed by all s! geometrical rays in image
space exiting through aperture Σ . The wavenumber is defined as

k=

2π
2π
=n
= nk0
λ
λ0

(2.13)

where n is the refractive index of the medium in image space, and λ and λ0 are thewavelengths in image and in vacuum, respectively. For z < z0 , the field is in material
1, so k = k1 , while for z > z0 , the field is in material 2, so k = k 2 . Also, since thewaves are homogeneous in image space:sz = + 1− s x 2 − sy 2 when s x2 + sy2 ≤ 1.

(2.14)

Figure 2.3. Schematic of an electric field focused across an interface between two different
media.

A focused field Ei irradiates the interface from the left ( z < z0 ) at an angle ofincidence, θ . The interface causes reflection and transmission to occur. Hence, thetotal field can be expressed as [2.1]⎪⎧E + Er , z < z0 ,E=⎨ iz > z0 ,⎩⎪E t ,

18	
   	
  

(2.15)

where Er and Et are the reflected and transmitted fields, respectively. The refractionof a field at a planar interface is described by the Fresnel reflection (rs, rp) andtransmission (ts, tp) coefficients. As discussed in Section 2.1, a field encountering alens or interface can be decomposed into two components, E(s) and E(p) , that resultsto

E = E(s) + E(p)(2.16)

Here, E(s) is parallel to the interface and E(p) is perpendicular to the wavevector k andE(s) . This decomposition has also been thoroughly explained in Section 2.1. EFF, Eq.
(2.10), is the sum of the s- and p-polarized fields in terms of θ and φ . Although Eq.
(2.10) describes a transmitted field, it can also describe a reflected field by simply
changing the transmission coefficients into reflection coefficients.
2.2.1. Linear Polarization

Calculating for the incident electric field, Ei . The derivation begins with solving for

Ei in material 1. Eq. (2.12) can be rewritten to represent Ei as()

Ei x,y,z = −{}ik1 EFF,i
exp ik1 ⎡⎣s1x x + s1y y + s1z z ⎤⎦ × ds1xds1y ,
2π ∫∫
s1z
Ω

(2.17)

1(

where EFF,i is the transmitted field right after the aplanatic lens and the term s!1 ⋅mv)

has been expanded. To get an expression for EFF,i , a few assumptions are maderegarding the incoming wave, Einc , before the lens and the lens itself. Let Einc belinearly polarized along the x-axis,

Einc = Eincnx .

(2.18)

Next it is assumed that the lens has a good antireflection coating, so that the Fresnel
transmission coefficients are()()

ts θ = t p θ = 1.

(2.19)

Also, both sides of the lens are in the same medium. EFF,i is represented as

	
  

19	
  

( )

(

)
⎡ cosθ + sin φ (1− cosθ ) ⎤⎢⎥= E ⎢ − (1− cosθ ) sin φ cos φ ⎥ ( cosθ ) ,⎢⎥EFF,i θ ,φ = Einc ⎡⎣cos φnθ − sin φnφ ⎤⎦ cosθ

1/2

2

1/2

(2.20)

inc

⎢
⎣

⎥
⎦

− sinθ cos φwhere the final expression is expressed in Cartesian vector components. EFF,i is interms of θ and φ because of the necessary coordinate transformations to project

EFF,i along unit vectors nθ and nφ . Therefore, Eq. (2.16) must first be written in terms()

of spherical polar coordinates, r, θ , φ r > 0, 0 ≤ θ ≤ π , 0 ≤ φ ≤ 2π as defined in Fig.
2.3:

s1x = sinθ cos φ, s1y = sinθ sin φ, s1z = cosθ ,(2.21)

and because of symmetry, x, y, and z can be expressed as

x = ρ cosϕ , y = ρ sinϕ , z = z.

(2.22)

where ρ is the magnitude of vector mv and ϕ is the angle between the z-axis andvector mv.
Therefore,()

s1x x + s1y y + s1z z = ρ sinθ cos φ − ϕ + z cosθ .

(2.23)The integral variables, ds1xds1y / s1z , must also be expressed in terms of θ and φgiven byds1xds1y
s1z= dΩ = sinθ dθ dφ,(2.24)

where dΩ is an element of the solid angle Ω.
Letting E inc = 1 and, substituting Eqs. (2.20) to (2.24) into Eq. (2.17), the expression
for Ei in Cartesian components is:

20	
   	
  

⎫
ik1 θmax 2πcosθ sinθ ⎡⎣cosθ + 1− cosθ sin2 φ ⎤⎦⎪∫∫2π 0 0⎪× exp ⎡⎣ ik1ρ sinθ cos φ − ϕ ⎤⎦ exp ⎡⎣ ik1z cosθ ⎤⎦ dθ dφ,⎪⎪ik1 θmax 2π⎪
E iy =cosθ sinθ 1− cosθ sin φ cos φ⎪∫∫
2π 0 0⎬ (2.25)
× exp ⎡⎣ ik1ρ sinθ cos φ − ϕ ⎤⎦ exp ⎡⎣ ik1z cosθ ⎤⎦ dθ dφ,⎪⎪⎪ik1 θmax 2π
E iz =cosθ sin2 θ cos φ⎪∫∫
2π 0 0⎪× exp ⎡⎣ ik1ρ sinθ cos φ − ϕ ⎤⎦ exp ⎡⎣ ik1z cosθ ⎤⎦ dθ dφ,⎪⎭(

E ix = −

(

(

)
)

(

)

(

)

)

where θ max is the maximum incident angle defined as Arc Sin(NA / n1) .
Equation (2.25) can be further simplified by expressing the φ integrals in terms ofBessel functions by the following formulae∫ cosnφ exp ⎡⎣ iU cos (φ − γ )⎤⎦ dφ = 2π i J (U ) cosnγ ,⎪ 	
  	
  	
  
⎬
⎡
⎤
⎪
sinn
φ
exp
iU
cos
φ
−
γ
d
φ
=
2
π
i
J
U
sinn
γ
.
(
)
(
)
∫
⎣
⎦
⎭
2π

⎫

n

n

0

2π

(2.26)nn0where n is an integer and Jn(U) is the Bessel function of the first kind and order n. A
derivation of Eq. 2.26 is found in Ref. [2.3]. By using the double-angle identities:

sin φ cos φ =11
sin2φ and sin2 φ = 1− cos 2φ ,
2
2

(

)

(2.27)

and applying Eq. (2.23), the simplified expressions of each component of Ei at point
V are:⎫
ik1⎡Ii 0 + Ii 2 cos 2ϕ ⎤ ,⎪
⎣
⎦
2
⎪
ik1
⎪
E iy = − ⎡⎣Ii 2 sin 2ϕ ⎤⎦ ,⎬2⎪⎪
E iz = −k1 ⎡⎣Ii1 cosϕ ⎤⎦ ,
⎪
⎭

( )

E ix = −( )(2.28)

where the integrals Iin (n = 0, 1, 2) are grouped by Bessel function order:

	
  

21	
  

Ii 0 = ∫θmax

Ii1 = ∫

θmax

Ii 2 = ∫

θmax

(

) (
) (
) (

)

(
)
(

)

cosθ sinθ 1+ cosθ J0 k1ρ sinθ exp ik1z cosθ dθ ,⎫⎪⎪2cosθ sin θ J1 k1ρ sinθ exp ik1z cosθ dθ ,⎬⎪cosθ sinθ 1− cosθ J2 k1ρ sinθ exp ik1z cosθ dθ .⎪
⎭

0

(

0

(

0

)

(2.29))

Calculating for the reflected electric field, Er . Now that an expression for Ei has()

been derived, an expression for Er can be formulated. At the interface z = z0 inmaterial 1, the incident electric field Ei is()

Ei x,y,z0 = −{}ik1 EFF,i
exp ik1 ⎡⎣s1x x + s1y y + s1z z0 ⎤⎦ × ds1xds1y .
2π ∫∫
s1z
Ω

(2.30)

1

This field, obeying Fresnel’s law of refraction, is reflected near the interface( z = z + δ ) and is expressed as0()

Er x,y,z0 = −

{

}

E
ik1
Tr(e) FF,i exp ik1 ⎡⎣s1x x + s1y y + s1z z0 ⎤⎦ × ds1xds1y (2.31)
∫∫
2π Ω
s1z
1

for the case of δ → 0, wherein no sign change has occurred yet for s1z since the field
can still be considered approximately at the interface. T(e) is an operator thatmodulates EFF,i and is a function of the angle of incidence θ and n1. The reflected
field in material 1 can be represented as a superposition of plane waves and is asolution of the time-independent wave equation, given by:()

Er x,y,z = −{}ik1 EFF,r
exp ik1 ⎡⎣s1x x + s1y y − s1z z ⎤⎦ × ds1xds1y .
2π ∫∫
s
Ω
1z

(2.32)

1

Note that in Eq. (2.32), a change in sign of s1z was made to ensure that the reflected
field travels in the backward direction. Using Eq. (2.32) as a boundary condition atz = z0 for Eq. (2.33), EFF,r can be expressed as

EFF,r = Tr(e)EFF,i exp ⎡⎣ 2ik1z0 ⎤⎦ .

22	
   	
  

(2.33)

The factor Tr(e)EFF,i is solved in the same manner as EFF,i . The field, however, isreflected by the interface. Therefore, the reflection of the s- and p-polarizedcomponents via the Fresnel reflection coefficients must be taken into consideration. It
must be noted that all the Fresnel coefficients are functions of the incident angle, θ ,()()

such as rs θ . But the θ has been dropped for the sake of brevity. The factor

Tr(e)EFF,i is now defined as(

Tr(e)EFF,i = E inc ⎡⎣ −rp cos φnθ − rs sin φnφ ⎤⎦ cosθ

(

)

)

1/2

⎡ r cosθ − r + r cosθ sin2 φ ⎤
s
p
⎢ p
⎥
1/2
⎢
⎥
= −E inc ⎢rs + rp cosθ sin φ cos φ⎥ cosθ .
⎢⎥−rp sinθ cos φ
⎢
⎥
⎣
⎦

(

)

(

(2.34))

Substituting Eq. (2.34) into Eq. (2.33), EFF,r is now()⎡ r cosθ − sin2 φ r + r cosθ
s
p
⎢ p
⎢
EFF,r = −E inc exp 2ik1z0 ⎢rs + rp cosθ sin φ cos φ⎢−rp sinθ cos φ
⎢
⎣

(

(

)

) ⎤⎥

1/2
⎥
⎥ cosθ . (2.35)
⎥
⎥
⎦

(

)

Eqs. (2.35) and Eqs. (2.21) – (2.24) are then substituted into Eq. (2.32) andassuming E inc = 1 ,⎫ik1 θmax 2πcosθ sinθ ⎡rp cosθ − rs + rp cosθ sin2 φ ⎤
⎪
∫
∫
⎣
⎦
2π 0 0⎪× exp ⎡⎣ ik1ρ sinθ cos φ − ϕ ⎤⎦ exp −ik1z cosθ exp 2ik1z0 dθ dφ,⎪⎪ik1 θmax 2π⎪
Ery =cosθ sinθ rs + rp cosθ sin φ cos φ
⎪
∫
∫
0
0
2π
⎬ (2.36)⎡⎤
× exp ⎣ ik1ρ sinθ cos φ − ϕ ⎦ exp −ik1z cosθ exp 2ik1z0 dθ dφ,⎪⎪⎪ik1 θmax 2π2Erz = −cosθ sin θ (rp )cos φ⎪∫∫
2π 0 0⎪× exp ⎡⎣ ik1ρ sinθ cos φ − ϕ ⎤⎦ exp −ik1z cosθ exp 2ik1z0 dθ dφ.⎪
⎭
Erx =

(

(

(
(

	
  

(

)

)

(

)
)

)

)

(

)

(

)

(

)

(

)

(

)

23	
  

Applying Eqs. (2.26) and (2.27) to Eq. (2.36), gives a representation of the individualcomponents of Er.

⎫
ik1⎡Ir 0 + Ir 2 cos 2ϕ ⎤ ⎪
⎦
2 ⎣
⎪
ik1
⎪
Ery = − ⎡⎣Ir 2 sin 2ϕ ⎤⎦⎬2⎪⎪Erz = k1 ⎡⎣Ir 1 cosϕ ⎤⎦
⎪
⎭

( )

Erx = −

( )(2.37)

where

Ir 0 = ∫θmax

Ir 1 = ∫θmax

Ir 2 = ∫

θmax

(

) (

)

⎫
⎪
× exp −ik1z cosθ exp 2ik1z0 dθ ,⎪⎪⎪2cosθ sin θ (rp )J1 k1ρ sinθ⎪⎬× exp −ik1z cosθ exp 2ik1z0 dθ ,⎪⎪⎪cosθ sinθ rs + rp cosθ J2 k1ρ sinθ⎪× exp −ik1z cosθ exp 2ik1z0 dθ .⎪⎭cosθ sinθ rs − rp cosθ J0 k1ρ sinθ

0

0

(
(

(

0

(

)

) (

(

)

(

)

(

)

(

)

)

(2.38)))

Calculating for the transmitted electric field, E t . Next, the transmitted field, E t , is()

calculated. At the interface z = z0 in material 1, the incident electric field Ei is given()

by Eq. (2.30). When the field is transmitted near the interface z = z0 + δ , it obeys
Fresnel’s law and is expressed as()

Et x,y,z0 = −

{

}

E
ik1
T (e) FF,i exp ik1 ⎡⎣s1x x + s1y y + s1z z0 ⎤⎦ × ds1xds1y (2.39)
∫∫
2π Ω
s1z
1

for the case of δ → 0. T(e) is an operator that modulates the amplitude a and is afunction of the angle of incidence θ and n1 and n2. The transmitted field in material 2
can be represented as a superposition of plane waves and is a solution of the timeindependent wave equation, given by:()

Et x,y,z = −{}ik 2 EFF,t
exp ik 2 ⎡⎣s2 x x + s2y y + s2z z ⎤⎦ × ds2 xds2y .
2π ∫∫
s2z
Ω

(2.40)

2

Note that in Eq. (2.40), the subscript 2 indicates that the transmitted field is inmaterial 2. This now becomes a boundary condition problem wherein Eq. (2.40) must

24	
   	
  

satisfy the boundary condition imposed by Eq. (2.39). From the vectorial law ofrefraction, the relationship between s!1 and s!2 is [2.5]k 2s2 x = k1s1x , k 2s2y = k1s1y .

(2.41)()

A coordinate transformation is made for s!2 = f s!1 . The Jacobian for thistransformation is defined as()=⎛k ⎞ ,J =⎜ ⎟∂( s ,s ) ⎝ k ⎠2∂ s2 x ,s2y

1

1x

2

(2.42)

01ywhich was obtained from Eq. (2.41). Equation (2.40) in terms of s!1 becomes()

Et x,y,z = −{ik 2
EFF,t exp ik1 ⎡⎣s1x x + s1x y ⎤⎦
2π ∫∫
Ω

}

1

2

⎛k ⎞
× exp ik 2s2z z ⎜ 1 ⎟ ds1xds1y .
⎝ k2 ⎠()(2.43)At z = z0 , Eq. (2.43) will satisfy the boundary condition, Eq. (2.39) whenE⎛k ⎞EFF,t = ⎜ 2 ⎟ T (e) FF,i exp ⎡⎣ iz0 k1s1z − k 2s2z ⎤⎦ .
s1z
⎝ k1 ⎠()(2.44)

Substituting Eq. (2.44) into Eq. (2.43), the transmitted electric field E t in material 2 is()

Et x,y,z = −E
ik 2 2T (e) FF,i exp ⎡⎣ iz0 k1s1z − k 2s2z ⎤⎦∫∫2π k1 Ω
s1z

(

1

(

)

)

(

)

(2.45)

× exp ⎡ ik1 s1x x + s1y y ⎤ exp ik 2s2z z ds1xds1y .
⎣⎦The factor T (e)EFF,i uses the same assumptions as when solving for EFF,i . The case of
the transmitted field, has been derived in Section 2.1 and is given by Eq. (2.10).
Solving for this equation gives

	
  

25	
  

1/2
n1
cosθ
n2

(

T (e)EFF,i = E inc ⎡⎣cos φt pnθ − sin φtSnφ ⎤⎦)()⎡ t cosθ + sin2 φ t − t cosθ
s
p
⎢ p
⎢
= E inc ⎢ − ts − t p cosθ sin φ cos φ⎢−t p sinθ cos φ
⎢
⎣

(

) ⎤⎥

1/2
⎥ n1
⎥ n cosθ .
⎥ 2
⎥
⎦

(

)

(2.46)

Substituting Eqs. (2.46) and Eqs. (2.21) – (2.24) into Eq. (2.45) and assuming thatE inc = 1 , E t can be expressed as

Etx = −⎫⎪⎪⎪
× exp ⎡⎣ ik1ρ sinθ cos φ − ϕ ⎤⎦ exp ik 2z cosθ ⎪⎪× exp ⎡⎣ iz0 k1 cosθ − k 2 cosθ ⎤⎦ dθ dφ, ⎪⎪⎪cosθ sinθ ts − t p cosθ sin φ cos φ⎪⎪⎪× exp ⎡⎣ ik1ρ sinθ cos φ − ϕ ⎤⎦ exp ik 2z cosθ ⎬ (2.47)⎪
× exp ⎡⎣ iz0 k1 cosθ − k 2 cosθ ⎤⎦ dθ dφ, ⎪⎪⎪2cosθ sin θ (t p )cos φ⎪⎪× exp ⎡⎣ ik1ρ sinθ cos φ − ϕ ⎤⎦ exp ik 2z cosθ ⎪⎪⎡⎤× exp ⎣ iz0 k1 cosθ − k 2 cosθ ⎦ dθ dφ. ⎪
⎪
⎪⎭

(

(

Ety =

ik 22 n1 θmax 2π
2π k1 n2 ∫0 ∫0(

ik 22 n1 θmax 2π
2π k1 n2 ∫0 ∫0

(

)

(

(

)

(

(

)

(

)

)

)

(

Etz =

)

ik 22 n1 θmax 2πcosθ sinθ ⎡t p cosθ + ts − t p cosθ sin2 φ ⎤⎣⎦2π k1 n2 ∫0 ∫0

(

)

)

)

)

Applying Eqs. (2.26) and (2.27) to Eq. (2.47), gives a representation of the individualcomponents of E t⎫
ik 22 n1
⎡It 0 + It 2 cos 2ϕ ⎤ ,⎪⎣⎦2k1 n2
⎪
⎪
2
ik
n1
⎪
⎡It 2 sin 2ϕ ⎤ ,Ety = − 2⎬⎣⎦
2k1 n2⎪⎪2
k 2 n1⎪⎡I cosϕ ⎤⎦ ,Etz = −⎪
k1 n2 ⎣ i1⎭

( )Etx = −

( )

26	
   	
  

(2.47)

where

It 0 = ∫θmax

It1 = ∫

θmax

It 2 = ∫

θmax

0

(

) (

⎫
)
⎪
⎪
× exp ( ik z cosθ ) exp ⎡⎣ iz ( k cosθ − k cosθ ) ⎤⎦ dθ ,⎪⎪cosθ sin θ (t )J ( k ρ sinθ )⎪⎬ (2.48)
× exp ( ik z cosθ ) exp ⎡⎣ iz ( k cosθ − k cosθ ) ⎤⎦ dθ ,⎪⎪⎪cosθ sinθ ( t − t cosθ ) J ( k ρ sinθ )⎪⎡⎤× exp ( ik z cosθ ) exp ⎣ iz ( k cosθ − k cosθ ) ⎦ dθ . ⎪⎭cosθ sinθ ts + t p cosθ J0 k1ρ sinθ
2

0

1

2

0

1

2

0

1

2

2

p

1

1

2

0

0

s

p

2

1

2

These equations along with Eqs. (2.28), (2.29), (2.37), and (2.38) are the necessary
equations to visualize an electric field diffracted at an interface for linearly polarized
illumination.
2.2.2. Radial Polarization
In studying radially and azimuthally polarized light, the same system described in
Sections 2.1 and 2.2 is used along with the same geometry. But, the incident beam isexpressed differently compared to linearly polarized light. The electric field prior to
the lens is resolved into radial and azimuthal components [2.8 – 2.10],

(

)

(

)

Einc = E inc,rad cos φnx + sin φny + E inc,azi − sin φnx + cos φny .

(2.48)For radially polarized light, the azimuthal component, E inc,azi = 0 in Eq. (2.48).
Therefore, the incident light is completely p-polarized. Using the same methodology
in Section 2.2.1, the incident, reflected, and transmitted electric fields on the interface
can be computed.

Incident electric field, Erad . The field right after the aplanatic lens, EFF,i,rad , using Eq.
(2.48) as the field before the lens, is⎡ cosθ cos φ ⎤⎢⎥1/2
EFF,i,rad = E inc,i,rad ⎢ cosθ sin φ ⎥ cosθ .
⎢⎥− sin φ
⎢⎣
⎥⎦

(

	
  

)

(2.49)27	  From this point onwards, it is assumed that E inc,i,rad = 1, the resulting incident electricfield, Ei, in individual Cartesian components is

E ix,rad = −ik1 ⎡⎣ iIi1,rad cosϕ ⎤⎦ ,⎫⎪⎪E iy ,rad = −ik1 ⎡⎣ iIi1,rad sinϕ ⎤⎦ , ⎬⎪E iz,rad = ik1 ⎡⎣ iIi 0,rad sinθ ⎤⎦ , ⎪⎭(2.50)

where
θmax

0

θmax

) sinθJ (k ρ sinθ ) exp ( ik z cosθ ) dθ ,⎪⎪()

Ii1,rad = ∫ cosθ0⎫(

Ii 0,rad = ∫ cosθ

1/2

0

3/2

1

1

(2.51)⎬⎪
sinθJ1 k1ρ sinθ exp ik1z cosθ dθ . ⎪
⎭

(

)

(

)

Reflected electric field, Er ,rad . The vector field amplitude EFF,r,rad , is expressed as⎡ cosθ cos φ ⎤⎢⎥1/2
EFF,r,rad = −E inc,i,rad rp ⎢ cosθ sin φ ⎥ cosθ ,⎢⎥− sin φ
⎢⎣
⎥⎦

(

)

(2.52)

where the negative sign is a result of the reflection of the field and rp is the Fresnel
reflection coefficient. The field reflected by the interface, Er ,rad , isErx,rad = ik1 ⎡⎣ iIr 1,rad cosϕ ⎤⎦ , ⎫⎪⎪Ery ,rad = ik1 ⎡⎣ iIr 1,rad sinϕ ⎤⎦ , ⎬⎪Erz,rad = −ik1 ⎡⎣ iIr 0,rad sinθ ⎤⎦ ,⎪⎭(2.53)

where
θmax

0

θmax

) sinθ (r )J (k ρ sinθ ) exp ( −ik z cosθ ) exp ( 2ik z ) dθ ,⎪⎪()

Ir 1,rad = ∫ cosθ
0

28	
   	
  

⎫

(

Ir 0,rad = ∫ cosθ

1/2

p

3/2

0

1

1

1 0

⎬
⎪
sinθ rp J1 k1ρ sinθ exp −ik1z cosθ exp 2ik1z0 dθ . ⎪
⎭

( ) (

)

(

)

(

)

(2.54)

Transmitted electric field, Et,rad . The vector field amplitude, EFF,t,rad , is expressed as⎡ cosθ cos φ ⎤
⎢
⎥ n
1/2
EFF,t,rad = E inc,i,rad t p ⎢ cosθ sin φ ⎥ 1 cosθ .
⎢⎥ n2− sin φ
⎣⎢
⎦⎥

(

)

(2.55)The transmitted electric field diffracted across the interface, Et,rad , can now be
expressed as⎫n1⎡ iIt1,rad cosϕ ⎤ ,⎪
⎦
n2 ⎣
⎪
⎪
ik 22 n1⎪
⎡ iIt1,rad sinϕ ⎤ , ⎬Ety ,rad = −
⎣
⎦
k1 n2
⎪
⎪
2
ik
n1
⎡ iIt 0,rad sinθ ⎤ , ⎪Etz,rad = 2⎦ ⎪k1 n2 ⎣⎭
ik 22
Etx,rad = −k1(2.56)

where

⎫
⎪
p
0
1
⎪
0
⎪
⎡
⎤
× exp ik 2z cosθ exp ⎣ iz0 k1 cosθ − k 2 cosθ ⎦ dθ ,⎪
⎬
θmax
3/2
⎪
It1,rad = ∫ cosθsinθ t p J0 k1ρ sinθ⎪0⎪× exp ik 2z cosθ exp ⎡⎣ iz0 k1 cosθ − k 2 cosθ ⎤⎦ dθ .⎪
⎭
θmax

(

It 0,rad = ∫ cosθ) sinθ (t ) J (k ρ sinθ )
1/2

(

(

)

( ) (

)

(

(

)

(

)

)

)

(2.57)

To construct the radial, azimuthal, and longitudinal components of the incident,
reflected and transmitted fields, the following transformations are used [2.8]:

E ρ ,* = E jx,* cos φ + E jy ,* sin φ, ⎫⎪⎪Eφ ,* = −E jx,* sin φ + E jy ,* cos φ,⎬⎪
E z,* = E jz,* ,⎪⎭(2.58)

where ρ , φ, and z represent the radial, azimuthal and longitudinal components,
respectively, and the * indicates the type of polarization – radial or azimuthal. Forradial polarization, the Eφ ,rad = 0.

	
  

29	
  

2.2.3. Azimuthal Polarization
For azimuthally polarized light, the radial component, E inc,rad = 0 in Eq. (2.48).
Therefore,theincident

light

is

completelys-polarized.

The

mathematicalrepresentations of the incident, reflected, and transmitted electric fields on theinterface are summarized below.

Incident electric field, Ei,azi . The field right after the aplanatic lens, EFF,i,azi , using Eq.
(2.48) as the field before the lens, is⎡ − sin φ ⎤⎢⎥1/2
EFF,i,azi = ⎢ cos φ ⎥ cosθ .
⎢
⎥
0
⎢⎣
⎥⎦

(

)

(2.59)The incident electric field, Ei,azi , in Cartesian coordinates is

E ix,azi = ik1 ⎡⎣ iIi1,azi sinϕ ⎤⎦ , ⎫⎪⎬E iy ,azi = −ik1 ⎡⎣ iIi1,azi cosϕ ⎤⎦ ,⎪⎭(2.60)

where
θmax

(

Ii1,azi = ∫ cosθ0) sinθ J (k ρ sinθ ) exp ( ik z cosθ ) dθ .
1/2

1

1

1

(2.61)

Reflected electric field, Er ,azi . The vector field amplitude, EFF,r ,azi , is given by⎡ − sin φ ⎤⎢⎥1/2
EFF,r ,azi = rs ⎢ cos φ ⎥ cosθ .
⎢
⎥
0
⎣⎢
⎦⎥

(

)

(2.62)The resulting reflected electric field is

Erx,azi = ik1 ⎡⎣ iIr 1,azi sinϕ ⎤⎦ , ⎫⎪⎬Ery ,azi = −ik1 ⎡⎣ iIr 1,azi cosϕ ⎤⎦ ,⎪
⎭

30	
   	
  

(2.63)

whereθmax(

Ir 1,azi = ∫ cosθ0) sinθ (r ) J (k ρ sinθ ) exp ( −ik z cosθ ) exp ( 2ik z ) dθ . (2.64)
1/2

s

1

1

1

1 0

Transmitted electric field, Et,azi . The vector field amplitude, EFF,t,azi , is given by⎡ − sin φ ⎤
⎢
⎥ n
1/2
EFF,r ,azi = ts ⎢ cos φ ⎥ 1 cosθ .
⎢
⎥ n2
0
⎢⎣
⎥⎦

(

)

(2.65)The transmitted electric field in Cartesian coordinates can be represented as

Etx,azi =

ik 22
k1

⎫
⎪
⎪
⎬
n1
⎪
⎡ iIr 1,azi cosϕ ⎤ ,⎪
⎣
⎦
n2
⎭

n1
⎡ iI
sinϕ ⎤⎦ ,n2 ⎣ r 1,azi

ik 2
Ety ,azi = − 2
k1(2.66)

where
θmax

(

It1,azi = ∫ cosθ0) sinθ J (k ρ sinθ )
1/2

1

(

1

)

(

)

(2.67)

× exp ik 2z cosθ exp ⎡⎣ iz0 k1 cosθ − k 2 cosθ ⎤⎦ dθ .
As shown above, azimuthally polarized light has no longitudinal component andpropagates as a purely transverse polarization field [2.8-2.9].
The results presented in this entire section can be extended to multi-layer interfaces
by using generalized Fresnel reflection/transmission coefficients that take into
consideration the entire structure of the media. The generalized Fresnel reflection
coefficient between region i and region i + 1 for an N-layered medium is defined as[2.11]

R! i,i+1 =

(
(

)
)

Ri,i+1 + R! i+1,i+2 exp ⎡⎣ 2ik i+1,z d i+1 − d i ⎤⎦,1+ Ri,i+1R! i+1,i+2 exp ⎡⎣ 2ik i+1,z d i+1 − d i ⎤⎦(2.68)

where di and di+1 are the thicknesses of regions i and i + 1, respectively. The
generalized transmission coefficient is defined as [2.11]

	
  

31	
  

N−1

(

)

T!1N = ∏ exp ⎡ ik jz d j − d j −1 ⎤S j ,j +1,
⎣
⎦
j =1

(2.69)

where d 0 = d1 and S j ,j +1 has the form of

Si−1,i =Ti−1,i()

1− Ri,i−1R! i,i+1 exp ⎡⎣ 2ik iz d i − d i−1 ⎤⎦.

(2.70)The dummy indices are substituted from Eq. (2.68) into Eq. (2.69).
2.3. POLARIZATION DEPENDENCE OF A FOCUSED ELECTRIC FIELD ACROSSAN INTERFACE
From the theory discussed in the previous sections, the nature of the focused beam
is different depending on the polarization of the incident beam. In this section, thebehavior of the focused beam for the three types of polarizations by visualizing thediffracted electric field through numerical simulations is discussed. The equations
used for the simulations are those derived in Section 2.2. A glass/air interface waschosen as proof-of-principle of the derived equations.
The numerical simulations model a system wherein a laser beam with a λ = 532 nm
is focused by an aplanatic microscope objective lens with a NA of 1.49. A high NA(NA > 1) objective lens is chosen to obtain higher spatial resolution as compared to a
low NA (NA < 1) objective lens. The focused beam illuminates glass/air interface (n1= 1.525, n2 = 1) located at z0 = 0 . The polarization of the laser beam could bechosen to be linear, radial or azimuthal.
2.3.1. Linear Polarization
Figures 2.4a – 2.4b show a linearly polarized beam being focused on a glass/air
interface. The distribution of the beam is different below the interface (glass) andabove the interface (air). A standing wave pattern is observed in the glass material,
which is a denser material compared to air. This pattern is due to the superposition of
the incident and reflected plane wave components in the glass. It can also be
observed that the reflected waves away from the focused spot are not transmitted
across the interface. This is because the reflection at the interface is caused by totalinternal reflection, when the angle of incidence θ is larger than the critical angle θ c .
It is also observed that the profile of the focused spot is different in Fig. 2.4a

32	
   	
  

compared to Fig. 2.4b. This is because the focused spot is elliptical in shape, as
2

2shown in Fig. 2.4c. The focused spot is the sum of the E x , E y , and Ez

2fields(Figs. 2.4d – 2.4f). The strongest of the three components is E x . The weakest of thethree is the depolarized E y component. However, the longitudinal field, E z , has arelatively strong contribution to total field, being only half of E x . There are two
reasons why this is so. The first reason is the use of a high NA lens, which can aidein generating a stronger E z field. The second reason is due to the presence of theinterface. [2.1] The boundary conditions at the interface state that the transverse fieldcomponents E x and E y must be continuous across the interface. The longitudinal
field, however, scales as [2.1]E z1ε1 = E z2ε 2 .

(2.71)The longitudinal field is composed of two maxima. These two maxima are along thepolarization direction, which in this case is the x-axis. Therefore, the resulting
focused spot is elongated.

Figure 2.4 Electric field distribution of a linearly polarized beam focused by an aplanatic lens(NA = 1.49) on a glass/air interface (n1 = 1.525, n2 = 1). Each image has dimensions of 1 µm
x 1 µm. Images (a) and (b) show a cross-sectional view of the field as it moves from glass(bottom) to air (top). The white horizontal indicates the interface located at z0 = 0. The
22

intensity of the (c) total electric field is composed of the (d) |Ex| , (e) |Ey| , and (f) |Ez|components. Bright areas indicate high intensity and dark areas indicate low intensity.

	
  

2

33	
  

2.3.2. Radial Polarization
Figures 2.5a and 2.5b show a focused radially polarized beam impinging on theglass/air interface. The transverse field or radial component has an on axis nullthrough the focus. The longitudinal component, however, has a bright central spot at
the focus. The reason is due to the distribution of each component. The radial
component (Fig. 2.5c) has a donut-like shape as it propagates, wherein the centerhas zero intensity. This is a consequence of the beam being out of phase at opposite
sides [2.9]. The longitudinal component, on the other hand, has a center lobe withhigh intensity that is surrounded by a halo of weaker intensity (Fig. 2.5d). Theirphases are opposite causing a null area between them. Comparing the intensities of
the radial and the longitudinal component, the longitudinal component is stronger
than the radial component. This is because of the high NA of the objective lens.
For applications that utilize the longitudinal component to excite the sample, radialpolarization is a good choice because of its intense single lobe feature. Linear
polarization can also be used, but probing the sample would be more difficult due tothe double lobes.

Figure 2.5 Electric field distribution of a radially polarized beam focused by an aplanatic lens(NA = 1.49) on a glass/air interface (n1 = 1.525, n2 = 1). Each image has dimensions of 1 µm
x 1 µm. Cross-sectional views of the (a) transverse and (b) longitudinal fields as they movefrom glass (bottom) to air (top) are shown. The white horizontal indicates the interface locatedat z0 = 0. The intensity distribution of the (c) transverse and (d) longitudinal fields in the x-yplane.

34	
   	
  

2.3.3. Azimuthal Polarization
The transverse field of azimuthally polarized light exhibits the same on axis nullthrough the focus as shown in Fig. 2.6a. This is because transverse field exhibits thesame donut-like shape as that of radially polarized light (Fig. 2.6b). Azimuthally
polarized light, though, has no longitudinal component as shown in theory.

Figure 2.6 Electric field distribution of an azimuthally polarized beam focused by an aplanaticlens (NA = 1.49) on a glass/air interface (n1 = 1.525, n2 = 1). Each image has dimensions of 1
µm x 1 µm. Cross-sectional view of the (a) transverse field as it moves from glass (bottom) toair (top) are shown. The white horizontal indicates the interface located at z0 = 0. The
intensity distribution of the (c) transverse field in the x-y plane.

2.4. REFERENCES[2.1] L. Novotny and B. Hecht, Principles of Nano-Optics: Second Edition,
Cambridge University Press: New York, 2012.
[2.2] S. Quabis, R. Dorn, M. Eberler, O. Glöckl and G. Leuchs, Focusing light to atighter spot, Optics Communications 179, 1-7, 2000.
[2.3] B. Richards and E. Wolf, Electromagnetic diffraction in optical systems II.
Structure of the image field in an aplanatic system, Proceedings of the Royal
Society A: Mathematical, Physical and Engineering Sciences, 253, 358-379,
1959.
[2.4] M. Born and E. Wolf, Principles of Optics: Seventh (Expanded) Edition,
Cambridge University Press: New York, 2009.
[2.5] P. Török, P. Varga, Z. Laczik and G.R. Booker, Electromagnetic diffraction oflight focused through a planar interface between amterials of mismatchedrefractive indices: an integral representation, J. Opt. Soc. Am. A, 12, 325-332,
1995.

	
  

35	
  

[2.6] E. Wolf, Electromagnetic diffraction in optical system I. An integral
representation of the image field, Proceedings of the Royal Society A:
Mathematical, Physical and Engineering Sciences, 253, 349-357, 1959.
[2.7] J.J. Stamnes, Waves in Focal Regions: propagation, diffraction and focusing oflight, sound and water waves, IOP Publishing Limited: Bristol, 1986.
[2.8] K.S. Youngworth and T.G. Brown, Focusing of high numerical aperturecylindrical-vector beams, Optics Express, 7, 77-87, 2000.
[2.9] D.P. Biss and T.G. Brown, Cylindrical vector beam focusing through a dielectricinterface, Optics Express, 9, 490-497, 2001.
[2.10] L.E. Helseth, Roles of polarization, phase and amplitude in solid immersion
lens systems, Optics Communications, 191, 161-172, 2001.
[2.11] W.C. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press: New
York, 1995.

	
  

36	
   	
  

Chapter 3: Strained Silicon: a Theoretical StudyIn the semiconductor industry, strained Silicon (ε-Si) is utilized in transistors by
inducing strain to increase electron flow. This chapter discusses the relationship ofstrain and stress to the crystal structure and phonon modes of Si from a theoreticalpoint of view. Phonon mode detection based on the Raman tensors is also explored.
An experimental system using a backscattering geometry was modeled. The
distribution of the electric field in the ε-Si sample is also discussed. Numericalsimulations are conducted based on the equations derived in Chapter 2.
3.1. RAMAN SCATTERING IN CRYSTALS
Measuring the Raman spectrum of a crystal structure, such as silicon (Si), via microRaman spectroscopy, is one of the main methods for obtaining information about its
lattice vibration frequencies [3.1]. These quantized lattice vibrations are calledphonons [3.2-3.4] and mathematically expressed as [3.2]()

Q j = Aj exp ⎡ ±i q j ⋅ r − ω j t ⎤
⎣
⎦

(3.1)

where Qj is the normal coordinate of the vibration, Aj is a constant, qj is the
wavevector, r is the position vector, ωj is the frequency at a time t. Raman scattering
can occur when there is a change in the electrical susceptibility of the crystal, whichis caused by lattice vibrations. The classical treatment of Raman scattering in
crystals is explained in brief [3.2, 3.5]. An incident monochromatic light withfrequency, ωi, irradiates a crystal in a direction ki, generating an electric field, E, at apoint r, expressed as()

E = E0 exp ⎡⎣ i ki ⋅ r − ω i t ⎤⎦ .

(3.2)

This electric field induces an electric moment P given by()

P = ε 0 χ ⋅ E0 exp ⎡⎣ i ki ⋅ r − ω i t ⎤⎦(3.3)

where χ is the susceptibility tensor describing the response of the crystal to theelectric field. The susceptibility can be expressed as a function of vibrations, having
normal mode of vibration j,⎛ ∂χ ⎞⎛ ∂χ ⎞
χ = χ0 + ⎜⎟ Qj + ⎜⎟ Q jQk + ...
⎝ ∂Q j ⎠ 0⎝ ∂Q j ∂Qk ⎠ 0(3.4)

Combining Eqs. (3.3) and (3.4), we have

	
  

37	
  

(

)

P = ε 0 χ 0 ⋅ E0 exp ⎡⎣ i ki ⋅ r − ω i t ⎤⎦(3.5)
⎛ ∂χ ⎞⎡⎤⎡⎤+ ε 0 χ 0 ⋅ E0 ⎜
A
exp
−i
ω
±
ω
t
exp
i(k
±
q
)
⋅
r
.
⎟ j
i
j
j
⎣ i
⎦
⎣
⎦
⎝ ∂Q j ⎠ 0()

This shows that, in a crystal, the induced moment will scatter light with three distinct
frequencies: ωi, Rayleigh scattering, ωi + ωj, anti-Stokes and ωi – ωj, Stokes Raman
scattering. The anti-Stokes and Stokes propagate in the direction determined byki ± q j . The Raman effect will occur when ωj is an optical frequency of the crystal.
The Raman scattering process observes the conservation of energy. More
importantly, it also observes the conservation of momentum, as shown from ouranalysis above. As a consequence, lattice waves have directional properties. Hence,in crystals, the Raman spectrum is dependent on the orientation of thecrystallographic axes with respect to the direction and polarization of both theexcitation and the scattered light [3.5].
3.2. EXCITATION AND DETECTION OF PHONON MODES
In a cubic crystal of Si, there are three propagation directions of great interest. These
are the [100], [110], and [111] directions corresponding to the cube edge, facediagonal, and body diagonal, respectively. When a wave travels along any of these
directions, the phonons can be described to have longitudinal polarization (Fig. 3.1a)or transverse polarization (Fig. 3.1b) depending on whether their displacements areparallel or perpendicular to the direction of the wave vector [3.3, 3.6]. For Si, there
are three optical phonon modes: one longitudinal optical (LO) phonon mode and two
transverse optical (TO1 and TO2) modes, whose names describe their polarization.
Physical observation of these phonon modes, in the form of scattered Raman
intensity, is governed by the polarization selection rules dictated by the Raman
tensors.

38	
   	
  

Figure 3.1. Displacement of atoms when encountering (a) longitudinal and (b) transverse
waves. Dashed lines indicate planes of atoms when in equilibrium. Solid lines indicate
displacement. The coordinate r is the measure of displacement of the planes. [Adapted from
Ref. 3.3]

3.2.1. Silicon Raman Tensors in Crystal and Sample Coordinates
The intensity of scattered Raman light, I, is related to the Raman tensor Rj by [3.2]2I ∝ ∑ esT R j e i ,

(3.6)

j

where Rj is the Raman tensor for the jth active phonon mode, ei and es are thepolarization unit vectors of the incident and scattered light, respectively. The
superscript T indicates a transpose of the scattered electric field vector. The Raman
tensors, Rj, for stress-free silicon were derived by Loudon [3.1] in the crystal
coordinate system x = [100], y = [010], and z = [001]:⎛ 0 0 d ⎞⎛ 0 d 0 ⎞⎛ 0 0 0 ⎞
⎜
⎟
⎜
⎟
⎜
⎟
R x = ⎜ 0 0 d ⎟ , R y = ⎜ 0 0 0 ⎟ , Rz = ⎜ d 0 0 ⎟ ,⎜⎝ 0 d 0 ⎟⎠⎜⎝ d 0 0 ⎟⎠⎜⎝ 0 0 0 ⎟⎠(3.7)

where d is a constant dependent on the Raman polarizability of the sample. For the
micro-Raman experiments, a backscattering geometry is employed, wherein theincident laser beam is traveling along the +z-axis and the scattered light is collectedin the –z direction by the same microscope objective lens. In the case ofbackscattering from a (001) surface, Rx and Ry are the Raman tensors corresponding
to TO1 and TO2 phonons, respectively, while Rz corresponds to LO phonon. These
assignments are relative and depend on the surface from which scattering isobserved (e.g. in backscattering from a (100) surface, Rx corresponds to the LO
phonon [3.2]).

	
  

39	
  

In this study, the edges of the nanowire are aligned along the [110] direction. Thus,
the Si Raman tensors expressed in the crystal axes (x, y, z) need to be transformed
to the sample coordinate (x’, y’, z’) axes using the tensor rotation, wherein thesample is rotated 45˚ from the crystal axes (Figure 3.2). For the sample coordinate
axes, the Raman tensors are [3.7]:⎛ 0 0 d ⎞
⎛ 0 0
1 ⎜
1 ⎜
⎟
Rx ' =
0 0 d , Ry ' =0 0
2 ⎜⎜ d d 0 ⎟⎟2 ⎜⎜ d −d
⎝
⎠
⎝

⎛ d 0
d ⎞
⎟
⎜
−d ⎟ , Rz ' = ⎜ 0 −d⎜⎝ 0 00 ⎟⎠0 ⎞⎟0 ⎟ . (3.8)
0 ⎟⎠From the polarization selection rule, Eq. (3.6), and the Raman tensors, only the LO
phonons (Rz and Rz’) can be observed for backscattering geometry along the z andz’-axes in both crystal and sample coordinate system. On the other hand, TO phononmodes, theoretically, cannot be detected when using the same backscattering
configuration.

Figure 3.2. Crystal (x, y, z) and sample (x’, y’, z’) coordinate systems. The sample coordinate
system is rotate 45˚ around the z-axis with respect to the crystal coordinate system [Adaptedfrom Ref. 3.8 with permission from ACS Publications].

To facilitate the discussion of the polarization and scattering configuration from the Si
crystal, we shall use the Porto notation for the rest of this text, e.g. [3.9]

40	
   	
  

The bar on top of the propagation direction indicates backscattering geometry.

3.2.2. Using a High Numerical Aperture Objective Lens to Excite Transverse
Optical Phonon Modes
From the discussion in the previous section, TO phonon modes cannot be detected,
in theory, because the backscattering geometry limits the excitation polarization only
to x’- or y’-polarization. If it were possible to generate z’-polarization, then TO phononmodes can be detected. A solution to this problem would be the utilization of a highNA objective lens.
For a low NA lens, Raman intensity calculation is typically based on the simplified
Raman signal collection efficiency (ei and es) from x’- or y’- polarization. The z’polarization is ignored, whereas the x’- and y’- polarizations are set either to unity(detectable) or zero (undetectable) depending on the analyzer setting. It was found,however, that when the NA of the objective lens is sufficiently high (i.e., NA > 1.0),
the z’- component is no longer negligible, making it possible to detect the TO phonon[3.10, 3.11]. Moreover, the depolarized (y’-component) light at the focus also
increases and becomes comparable to the z’-component for linearly x’-polarized
incident light as shown in Fig. 3.3. Therefore, contributions from the depolarized lightcannot be ignored in the analysis of the scattered Raman signal. The configuration()

z ' x ',y ' z ' is an example for this case in polarized micro-Raman spectroscopy.

	
  

41	
  

Figure 3.3. Change in polarization due to high NA objective lens.

3.2.3. Importance of Transverse Optical Phonon Mode Detection for Stress
Determination in Silicon
The phonon mode frequencies of crystalline Si are selectively modified when Si issubjected to mechanical stress or strain [3.2]. Figure 3.4a – c illustrates the phononmodes at different stress states. When no strain is present (Fig. 3.4a), only a single
triply degenerated phonon mode can be detected [3.8], which, in the case of (001)oriented Si surface, has a peak frequency of about ω0 ≈ 520 cm-1 at room
temperature [3.2]. In the presence of a bi-isotropic in-plane strain ( ε 'xx = ε 'yy ),however, the degeneracy is lifted and the original single phonon will split into two(Fig. 3.4b): the singlet LO and the doublet TO. When anisotropic strain ( ε 'xx ≠ ε 'yy ) ispresent, the doublet TO further splits into two singlet phonon modes (Fig. 3.4c)
resulting to three phonon modes in total. To get a complete picture of stressexperienced by a Si structure, such as ε-Si, all three phonon modes must be
measured, either individually or collectively. It is theoretically possible to detect all
three phonon modes either when the incident or scattered electric fields are notparallel to the sample surface or when the sample surface is not (001)-oriented [3.4,3.12]
The three optical phonon modes in the presence of strain can be obtained by solving
the secular equation [3.2, 3.13]:()

pε11 + q ε 22 + ε 33 − λ

42	
   	
  

2r ε12()

2r ε132r ε12pε 22 + q ε 33 + ε11 − λ

2r ε 232r ε132r ε 23pε 33 + q ε11 + ε 22 − λ(= 0, (3.9))

where p, q, and r are the phonon deformation potentials (PDP) related to the materialand εij are the components of the strain tensor. Following the formalism of de Wolf
[3.2], the eigenvalues λj are defined as

λ j = ω 2j − ω 2j 0
Δω j = ω j − ω j 0 ≈(3.10)λj2ω j 0.

(3.11)

Here, ωj (j = 1, 2, 3) and ωj0 represent the Raman frequencies in the presence andabsence of stress, respectively. Hooke’s Law relates the strain tensors, ε’ijkl, to thestress tensors, σ’ijkl, represented as

σ 'ij = ∑ c 'ijkl ε 'kl ,(3.12)

ijkl

where c’ijkl is the stiffness tensor. This serves as the starting point for calculating thestress in ε-Si nanostructures, which will be discussed further in Chapter 4.

Figure 3.4. Phonon mode splitting for various types of induced strain. (a) In the absence ofstrain, the detected phonon frequency is triply degenerate, consisting of the LO and TO
phonon modes. (b) In the case of bi-isotropic strain (nanomembrane), the triply degeneratemode is split into two singlet LO and doublet TO. (c) When anisotropic strain is present, thedoublet TO is further split into two singlet TO phonon modes resulting to three phonon modes
in total: one singlet LO and two singlet TO. [Taken from Ref. 3.8 with permission from ACS
Publications.]

3.3. THREE-DIMENSIONAL ELECTRIC FIELD CALCULATIONS: POLARIZATION
DEPENDENCE
To accurately describe the scattering process involving illumination and collection bya high NA lens (NA > 1), the ei cannot be assumed to be constant across the focal
volume. Considering all the non-negligible effects discussed in Section 3.2.2., theintensity of the Raman mode can be rewritten as [3.10]:

	
  

43	
  

(
(
(

⎡
⎛ E
θ ,φ,η
2π θmax ⎢⎜ scaX
Ii = I0 ∑ ∫ ∫ ⎢R α , β ,γ ⎜ EscaY θ ,φ,η
⎢
⎜
i 0 0
⎢
⎜ EscaZ θ ,φ,η
⎝
⎢⎣

(

)

2

T

)
)
)

⎞⎤ ⎡
⎛ E' ⎞⎤
x
⎟⎥ ⎢
⎜
⎟⎥
⎥
⎢
⎟ × Ri • R α , β ,γ ⎜ E 'y ⎟ ⎥ sinθ dθ d φ
⎟⎥ ⎢
⎜
⎟⎥
⎜⎝ E 'z ⎟⎠ ⎥⎟⎥ ⎢⎠ ⎦⎥ ⎣⎦()(3.13)
where I0 is the incident intensity, η is the angle of the analyzer with respect to the x’ –axis, and θ and φ are the polar and azimuthal angles, respectively. E’x, E’y and E’z are
the incident electric fields in the crystal coordinate system (x,y,z). Ri is the Raman
tensor also in the crystal coordinate system, which is transformed into the sample()

coordinate system (x’,y’,z’) using the Euler rotation matrix [3.14], R α , β ,γ , definedas:⎛ cos α cos γ − sin α cos β sin γ⎜
R α , β ,γ = ⎜ sin α cos γ + cos α cos β sin γ⎜
sin β sin γ⎝()− cos α sin γ − sin α cos β cos γ− sin α sin γ + cos α cos β cos γ
sin β cos γ⎞⎟− cos α sin β ⎟
⎟
cos β
⎠
sin α sin β(3.14)As discussed in Chapter 2, the incident electric field, Ei, can be generated
numerically for different kinds of polarization. When linearly polarized light,()

E'Lin = x ',y ',z ',d , is focused onto ε-Si, the electric field distribution within the ε-Si isrepresented as [3.15 – 3.21]:⎡⎢⎢ −i F0 x ',y ',z ',d + F2 x ',y ',z ',d⎢2x ' y 'E'Lin x ',y ',z ',d = ⎢−iF2 x ',y ',z ',d 2⎢
x ' + y '2⎢x'⎢−2F1 x ',y ',z ',d⎢x '2 + y '2
⎣

{ (

(

(

)

)

(

)

)

(

(

)

(

)

(

)

)}

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

where F0 x ',y ',z ',d , F1 x ',y ',z ',d , and F2 x ',y ',z ',d are written as

44	
   	
  

(3.15)

F0 (x ',y ',z ',d) = ∫

(

θmax

)

cosθ sinθJ0 k x '1(θ ) x '2 + y '2 ×

θmin

2
⎧⎛
⎫
⎞
⎛ ε
⎞
⎪⎜
⎪
⎟
0
sinθ ⎟ + ts01(θ ,d) exp ⎡⎣ ik z '1(θ )z ' ⎤⎦ +⎪⎜ t p01(θ ,d) 1− ⎜⎜
⎪
⎟
⎟⎠
⎝ ε1
(3.16)
⎪⎜⎝
⎪
⎟⎠
⎪
⎪
d
θ
⎨
⎬
2
⎞
⎪⎛
⎪
⎛ ε
⎞
⎪⎜ −r (θ ,d) 1− ⎜ 0 sinθ ⎟ + r (θ ,d)⎟ exp ⎡ −ik (θ )z + 2idk (θ ) ⎤ ⎪s01
z '1⎣ z '1⎦⎪⎟⎪⎜ p01
⎜⎝ ε
⎟⎠
1
⎟⎠
⎪⎩⎜⎝
⎪⎭

F1(x ',y ',z ',d) = ∫

θmax

θmin

cosθ sin2 θ

ε0
ε1

(

)

J1 k x '1(θ ) x '2 + y '2 ×{t (θ ,d)exp ⎡⎣ik (θ )z '⎤⎦ − r (θ ,d)exp ⎡⎣−ik (θ )z '+ 2idk (θ )⎤⎦} dθ
p01

z '1

F2 (x ',y ',z ',d) = ∫

θmax

θmin

p01

z '1

(

)

(3.17)

z '1cosθ sinθJ2 k x '1(θ ) x '2 + y '2 ×

⎧⎛
⎛
⎪⎜
t
(
θ
,d)
−
t
(
θ
,d)
1−
⎜
⎪⎜ s01
p01
⎜⎝
⎪⎜⎝
⎪
⎨
⎪⎛
⎛
⎪⎜ r (θ ,d) + r (θ ,d) 1− ⎜p01⎪⎜ s01
⎜⎝
⎪⎩⎜⎝

⎫
⎪
⎪
⎪
⎪
⎬ dθ
2⎞
⎪
⎞
ε0
⎟
sinθ ⎟ exp ⎡⎣ −ik z '1(θ )z '+ 2idk z '1(θ ) ⎤⎦ ⎪
⎪
⎟⎠ ⎟
ε1
⎟⎠
⎪⎭
2⎞
⎞
sinθ ⎟ ⎟ exp ⎡⎣ ik z '1(θ )z ' ⎤⎦ +
⎟⎠ ⎟
ε1
⎟⎠

ε0

(3.18)

where θ is the incident angle of light, ε0 and ε1 are the dielectric constants of oil andε-Si, respectively. θmin and θmax are the minimum and maximum angles corresponding
to sin−1(NA / ε 0 ) .

()()

The k x '1 θ and k z '1 θ are the x’- and z’-components of the

( )

( )

wavenumbers in ε-Si, respectively. The t p01 θ ,d and rp01 θ ,d are the transmission
and reflection Fresnel coefficients for p-polarized light at the ε-Si/SiO2 interface andε-Si/oil interface, respectively. Fresnel coefficients of the three layers were calculated
by taking into account multiple reflections at different interfaces as outlined in thenext section (Section 3.4) and in Chapter 2. In order to consider the attenuation due
to the absorption of light crossing the ε-Si layer, Fresnel coefficients are also
expressed as a function of thickness d. For high NA lens illumination, the z’component is small but not negligible. This means that the scattered light comprises
both LO and TO modes. When the scattered Raman passes through the analyzer
that is oriented parallel to the incident polarization, LO phonons could be suppressed.

	
  

45	
  

However, the analyzer cannot block the scattered light of the depolarized,()

F2 x ',y ',z ',d , signal, which excites LO as well. Thus, even under linearly polarized
Raman measurement, both LO and TO are excited and detected.
In addition to the use of linear polarization, radially polarized light can also be utilized
to further increase the z’-component of the electric field – a prerequisite for TO
excitation – while also achieving tighter focus [3.22]. This indicates that by using
radially polarized light [3.10], both TO and LO phonons can be excited.

In order toenhance the intensity of TO phonon relative to LO phonon mode, it is necessary toincrease the longitudinal electric field relative to the transverse field.

Radialpolarization is the promising approach for this purpose combined with a high NA lens.
When radially polarized light is focused onto the planar surface (x’y’ plane) through ahigh NA oil immersion lens, the electric field distribution inside the ε-Si is given by[3.15 – 3.21]:⎡x'⎢ iFrad (x ',y ',z ',d)2x ' + y '2⎢⎢y'Erad (x ',y ',z ',d) = ⎢ iFrad (x ',y ',z ',d)2⎢x ' + y '2⎢⎢
−Grad (x ',y ',z ',d)
⎢
⎣

(

⎤
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦

(3.19))

where Erad x ',y ',z ',d is the electric field vector having x’, y’ and z’- component of
the light at each depth d ( 0 ≤ d ≤ 15 nm) of the ε-Si sample. The explicit forms of theterms Frad (x ',y ',z ',d) and Grad (x ',y ',z ',d) are given by:

2

Frad (x ',y ',z ',d) = ∫

θmax

θmin

⎛ ε
⎞
cosθ sinθ 1− ⎜ 0 sinθ ⎟ J1 k x '1(θ ) x '2 + y '2 ×⎜⎝ ε⎟⎠
(3.20)1(){t (θ ,d)exp ⎡⎣ik (θ )z '⎤⎦ + r (θ ,d)exp ⎡⎣−ik (θ )z '+ 2idk (θ )⎤⎦} dθ
p01

46	
   	
  

z '1

p01

z '1

z '1

Grad (x ',y ',z ',d) = ∫θmaxcosθ sinθ

θmin

ε0
ε1

(

)

sinθJ0 k x '1(θ ) x '2 + y '2 ×{t (θ ,d)exp ⎡⎣ik (θ )z '⎤⎦ − r (θ ,d)exp ⎡⎣−ik (θ )z '+ 2idk (θ )⎤⎦} dθ
p01

z '1

p01

z '1

(3.21)z '1

In the case of azimuthally polarized light, the z-component of the electric field is zero.
From the Raman polarization selection rules [3.23, 3.24], this indicates that only LO
phonons can be detected simply because there is no z-polarized component to excite
the TO phonons. However, the focus spot profile shows a donut shape [3.25], which
degrades the spatial resolution. This is because when the donut shaped focus
illuminates the sample, it essentially irradiates two different areas of the sample, andit has a larger focus spot as compared to radial polarization, which has a tighter focus
spot. The components of the electric field distributions created by azimuthally
polarized, Eazi (x,y,z,d) , light are given by [3.15 – 3.21]:⎡⎢ Fazi x ',y ',z ',d⎢⎢Eazi x ',y ',z ',d = ⎢ Fazi x ',y ',z ',d
⎢
⎢
⎢
0
⎢
⎣

(

)

(

)

(

)

y'
x '2 + y '2x'x ' + y '2
2

⎤
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦

(3.22)

where the corresponding form of Fazi (x ',y ',z ',d) is described below: 	
  Fazi (x ',y ',z ',d) = ∫

θmax

θmin

{

(

)

cosθ sinθJ1 k x '1(θ ) x '2 + y '2 ×}ts01(θ ,d)exp ⎡⎣ ik z '1(θ )z ' ⎤⎦ + rs01(θ ,d)exp ⎡⎣ −ik z '1(θ )z '+ 2idk z '1(θ ) ⎤⎦ dθ

(3.23)

where ts01(θ ,d) and rs01(θ ,d) are the Fresnel coefficients for s-polarized light.
In the backscattering configuration, the same lens is used for the collection of the

(
(
(

⎛ E
θ ,φ,η
⎜ scaX
scattered electric field, ⎜ EscaY θ ,φ,η⎜⎜ EscaZ θ ,φ,η⎝) ⎞⎟) ⎟⎟ , which consists of all three phonon modes.
) ⎟⎠The dependence of these collected phonon modes (LO and TO) on the NA of thelens need to be integrated to accurately assess the intensity of each phonon modes.

	
  

47	
  

For this reason, to describe the scattering collection process, it is necessary to define( )an objective transfer matrix, Tobj θ ,φ , for the scattered Raman polarization vector,

es , and integrate over the entire polar angle θ subtended by the objective. This
calculation is similar to that of the incident light. The scattered Raman polarization
vector is defined as:⎛ cos η ⎞⎜⎟es η = ⎜ sin η ⎟
⎜ 0 ⎟
⎝
⎠

( )

(3.24)

where h is the angle setting of the analyzer with respect to the x’ – axis (h = 0 forparallel to x’ – axis and h = 90 for parallel to y’ – axis). The expression for 3×3( )transfer matrix Tobj θ ,φ for the scattered Raman polarization vector es is [3.26, 3.27]:

T⎛ cosθ cos φ ⎞ ⎛ cos φ ⎞⎛ sin φ ⎞ ⎛ sin φ ⎞
⎜
⎟⎜
⎟
⎜
⎟⎜
⎟
Tobj (θ ,φ ) = t p10 (ξ ) ⎜ cosθ sin φ ⎟ ⎜ sin φ ⎟ + ts10 (ξ ) ⎜ − cos φ ⎟ ⎜ − cos φ ⎟⎜⎟⎜ 0 ⎟⎜⎟⎜⎟− sinθ
0
0
⎝
⎠⎝
⎠
⎝
⎠⎝
⎠

T

(3.25)
where t p10 (ξ ) and ts10 (ξ ) are the Fresnel transmission coefficients for p- and spolarized (parallel to light traveling from ε-Si to oil), respectively. Since the scattering
process takes place inside the ε-Si, the Fresnel coefficients are expressed in terms ofξ = sin−1( ε ε sin(θ )) , which is a function of the incident angle of light that varies
01

within 0 ≤ θ ≤ θ max ( θ max = 78˚ for NA = 1.49).

(
(
(

⎛ E
θ ,φ,η
⎜ scaXThe components of the collected scattered field, ⎜ EscaY θ ,φ,η⎜⎜ EscaZ θ ,φ,η⎝) ⎞⎟) ⎟⎟ , of a high NA lens
) ⎟⎠at different analyzer settings can be obtained by performing a matrix multiplication of
Eqs. (3.24) and (3.25):

(
(
(

⎛ E
θ ,φ,η
⎜ scaX⎜ EscaY θ ,φ,η⎜⎜ EscaZ θ ,φ,η
⎝

48	
   	
  

⎛ T (θ ,φ ) cos η + T (θ ,φ ) sin η ⎞) ⎞⎟⎜⎟) ⎟⎟ = T (θ ,φ ) ⋅ e (η ) = ⎜⎜ T (θ ,φ ) cosη + T (θ ,φ ) sinη ⎟⎟ (3.26)⎜ T (θ ,φ ) cos η + T (θ ,φ ) sin η ⎟
) ⎟⎠
⎝
⎠
s

11

12

21

22

31

32

( )

where T11 , T12 , T21 , T22 , T31 , and T33 are the non-zero elements of the Tobj θ ,φobjective transfer matrix expressed as:( ) ()()T (θ ,φ ) = T (θ ,φ ) = cos φ sin φ ⎡⎣t (ξ ) cosθ − t (ξ ) ⎤⎦T (θ ,φ ) = t (ξ ) cosθ sin φ + t (ξ ) cos φT (θ ,φ ) = −t (ξ ) sinθ cos φT (θ ,φ ) = −t (ξ ) sinθ sin φ
T11 θ ,φ = t p10 ξ cosθ cos2 + ts10 ξ sin2 φ
12

21

p10

2

22

p10

31

p10

32

p10

s10

2

s10

(3.27)

3.4. CALCULATION OF FRESNEL COEFFICIENTSWhen a light is focused by a high NA lens through a three-layer system consisting ofoil (0th ε 0 = 2.356 ), ε-Si (1st ε1 = 17.2207 + 0.3569i ) and SiO2 (2nd ε 2 = 2.136 ), theamplitude of the p and s-polarized light changes. The change in the amplitude can bederivedfromSnell’s

Law

and

the

Fresnelequations[3.28]givenasθ = sin−1(NA / ε 0 ) , where θ is the incident angle of light. The wave number,k = 2π / λ , expressed in terms of wavelength, λ =532nm, can be obtained from eachlayer as: k0 = ε 0 k (oil), k1 = ε1k (e-Si), and k 2 = ε 2 k (SiO2).
Since the components of this wavenumber are dependent on θ , we can represent
them as:k x0 (θ ) = k x1(θ ) = k x2 (θ ) = k0 sinθ ⎫⎪k z0 (θ ) = k0 cosθ
⎪
⎪
⎛ ε0
⎞
⎪
2
k z1(θ ) = k1 ⎜ 1− sin θ ⎟
⎬
ε
⎝
⎠
1
⎪
⎪
⎛ ε0
⎞
2
⎪
k z2 (θ ) = k 2 ⎜ 1− sin θ ⎟
⎪
⎝ ε2
⎠
⎭

(3.28)The transmission, t p01(θ ,d) , and the reflection, rp01(θ ,d) , Fresnel coefficients for ppolarized light are given by:

t p01(θ ,d) =rp01(θ ,d) =t p01(θ )1+ rp01(θ )rp12 (θ )exp ⎡⎣ 2ik z1(θ )d ⎤⎦t p01(θ )rp12 (θ )1+ rp01(θ )rp12 (θ )exp ⎡⎣ 2ik z1(θ )d ⎤⎦(3.29)(3.30)

where, d (=15nm) is the thickness of ε-Si. Note that the equations are express incomplex Fresnel coefficients to take into account the effect of extinction coefficient,

	
  

49	
  

which is not negligible for shorter laser wavelengths. This is important because some
information on the change of the stress with depth can be obtained by using different
excitation wavelengths.
The reflection and transmission coefficients at different interfaces of the three-layer
system are as follows:

ε 0k z1(θ ) − ε1k z0 (θ ), Reflection coefficient from oil (0th) to ε-Si (1st)ε1k z0 (θ ) + ε 0k z1(θ )rp01(θ ) =t p01(θ ) =rp12 (θ ) =2 ε 0 ε1k z0 (θ )ε1k z0 (θ ) + ε 0k z1(θ )(3.31), Transmission coefficient from oil (0th) to ε-Si (1st) (3.32)

ε1k z2 (θ ) − ε 2k z1(θ ), Reflection coefficient from ε-Si (1st) to SiO2 (2nd). (3.33)
ε 2k z1(θ ) + ε1k z2 (θ )The Fresnel coefficients for s-polarized light are given by:ts01(θ ,d) =rs01(θ ,d) =ts01(θ )(3.34)

1+ rs01(θ )rs12 (θ )exp ⎡⎣ 2ik z1(θ )d ⎤⎦ts01(θ )rs12 (θ )1+ rs01(θ )rs12 (θ )exp ⎡⎣ 2ik z1(θ )d ⎤⎦(3.35)

where, d (=15nm) is the thickness of ε-Si andrs01(θ ) =k z0 (θ ) − k z1(θ ), Reflection coefficient from oil (0th) to e-Si (1st)k z0 (θ ) + ε 0k z1(θ )(3.36)

ts01(θ ) =2k z0 (θ ), Transmission coefficient from oil (0th) to ε-Si (1st)k z0 (θ ) + k z1(θ )(3.37)

rs12 (θ ) =k z1(θ ) − k z2 (θ ), Reflection coefficient from ε-Si (1st) to SiO2 (2nd).
k z1(θ ) + k z2 (θ )(3.38)For scattered photons traveling from ε-Si to oil, the Fresnel coefficients are:t p10 (ξ ) =2 ε1 ε 0 k z10 (ξ ) k z11(ξ ) k z10 (ξ )ε 0k z10 (ξ ) + ε1k z11(ξ )the transmission coefficient from ε-Si (1st) to oil (0th)

50	
   	
  

,

(3.39)

ts10 (ξ ) =2k z10 (ξ ) k z11(ξ ) k z10 (ξ )k z10 (ξ ) + k z11(ξ ),(3.40)

the transmission coefficient from ε-Si (1st) to oil (0th)where ξ = sin−1( ε ε sin(θ ))01

k z10 (ξ ) = ε1k cos ξk z11(ξ ) = ε 0 k 1−

ε1 2
sin ξ
ε0

(3.41)(3.42)

3.5. PHONON MODE VISUALIZATION IN STRAINED SILICON VIA THREEDIMENSIONAL RAMAN INTENSITY CALCULATION
Figure 3.5 shows the results of numerically simulated 1 x 1 µm2 electric field
distributions inside the ε-Si surface at different polarizations. Three kinds ofpolarization are studied, namely, linear, radial and azimuthal. These distributions
were obtained for a 532 nm light source focused using the full aperture of a highnumerical aperture (NA) (0 ≤ NA ≤ 1.49) oil immersion lens. To simplify thecomparison, the electric field distributions were normalized at each polarization.
The incident electric field (Fig. 3.5) is the field that has exited the microscope
objective and illuminates the ε-Si sample. For each polarization, the incident field has
been decomposed into the x’, y’, and z’-components. In all polarizations, the threecomponents of the incident electric field are non-zero – except for the z’-componentof azimuthal polarization. This indicates that the three Raman modes of Si (TO1, TO2
and LO) are excited for high NA illumination based on Eq. (3.13). Clear differences
can be seen among the different polarization types when the incident field illuminates
either a nanomembrane or nanowire. The intensity distribution of the incident field
determines the intensity profile of the excited phonon modes. The intensity profile of
the LO mode is dictated by either x’ or y’ depending on the analyzer setting, while theintensity profiles of the TO modes are that of z’. In the case of azimuthal polarization,
the TO modes’ intensity profiles are the sum of the x’ and y’.

( )

(

)

The Porto notations z ' *,x ' z ' and z ' *,y ' z ' are used to indicate when the analyzer
is set parallel to the x’-axis (η = 0˚) or y’-axis (η = 90˚), respectively. The “ * ” implies
linear, radial or azimuthal incident polarization. The TO/LO ratio in the numerical

	
  

51	
  

simulations is computed by adding the intensities of the TO1 and TO2 modes anddividing this sum by the intensity of the LO signal. The individual intensities of thephonon modes are taken by integrating the intensity in three dimensions over the 15
nm thick ε-Si nanomembrane.
3.5.1. Nanomembrane
In the case of the ε-Si nanomembrane (Fig. 3.5), the Raman intensity is normalizedwith respect to the maximum Raman intensity of the LO signal at each analyzer
setting.
For linearly polarized light, the incident field has a strong x’-polarized component ascompared to the depolarized y’- and the normal z’-components. Thus, LO phonon are()

mostly excited. In the z ' x ',x ' z ' setting, the LO intensity is strong compared to the
TO1 and TO2 intensities even after scaling the TO value 40 times. This is reflected by()

the TO/LO ratio which is ~ 0.1. In the z ' x ',y ' z ' setting, the TO1 and the TO2
phonon modes can be detected because the LO mode is mostly filtered out by the
analyzer. This results to a higher TO/LO ratio of ~ 0.94 as compared to the( )( )z ' ( x ',y ') z ' to be TO-active. For radial polarization, the x’ and y’-components havez ' x ',x ' z ' setting. Hence, we designate the z ' x ',x ' z ' to be LO-active andsimilar intensities. But, the z’-component of the incident electric field is tightly focused
and, comparing this feature to the two lobes in linear polarization, indicates better
spatial resolution. Also, the intensity of the radially polarized z’-component isrelatively higher – by around twice – than the linearly polarized incident field (Fig.
3.6a). This makes radially polarized light TO-active and suitable for TO
characterization. For azimuthal polarization, there is no z’-component. Therefore, theTO1 and TO2 Raman signals are very weak, making this polarization LO-active.
The integrated intensities of TO1 and TO2 are assumed to be equal because ofsymmetry in the sample (for both nanomembrane and nanowire), but differences inthe distribution can be seen between the two modes. This is due to the polarization
selection rules [3.1, 3.2, 3.10]. The behavior of the transverse electric fields are
determined by the Raman tensors, Rx and Ry for TO1 and TO2, respectively. For TO1,
the transverse fields add up resulting to a circular distribution. For TO2, thetransverse fields cancel each other resulting to a more oval-shaped distribution.

52	
   	
  

3.5.2. Nanowire
From the nanomembrane simulation data, the Raman intensity distribution on the εSi sample is not uniform and the location of the most intense areas vary dependingon the polarization of the incident electric field. This information is important whencharacterizing strain in nanostructures such as nanowires because, depending on
the location of the focused light on the nanowire, the detected Raman signal can beeither LO or TO phonon modes.
Figure 3.5 shows the Raman intensity distribution on ε-Si nanowires having 30 nm
width, 1 mm length and 15 nm thickness within a diffraction-limited focused spot
given by 1.22λ/NA (~ 436 nm). In the simulation of the ε-Si nanowires, the Raman
intensity is normalized with respect to the maximum Raman intensity of either the LO
or TO1 signal at each analyzer setting.

(

)

(

)

In the TO-active z ' x ',y ' z ' and z ' rad,y ' z ' settings, there is a considerable
increase in the TO/LO ratio for both polarizations compared to their nanomembranecounterparts. This is because, as it can be seen from the LO profile of thenanomembrane, when using a nanowire that is located parallel to the x’-axis, the LOcomponents induced by the depolarized field ( Ey ) can be drastically reduced()

resulting in a high contrast of TO/LO ratio. For z ' x ',y ' z ' , the TO/LO ratio is 84.57()

while for z ' rad,y ' z ' , the TO/LO ratio is 12.62. This change in TO/LO ratio is also
dependendent on the width of the nanowire as shown in Fig. 3.7a. The data shown is

(

)

(

)

from the z ' x ',y ' z ' setting. The same behavior is also observed for the z ' rad,y ' z 'setting. The TO/LO ratio peaks at a nanowire width of 20 nm and then decreases
approaching a value of 0.94 as the width reaches infinity (nanomembrane case). The
increase and then decrease for widths larger than 20 nm is due to the change of theoverlap between the electric field distribution of the tight focus and the nanowire
structure. Precisely, this rapid increase of the LO intensity as the width increases is()

caused by the depolarized electric field contribution in the z ' x ',y ' z ' configuration(Fig. 3.7b). Based on the simulation, it should be pointed out that the TO/LO ratio issufficiently high allowing for selective detection of TO Raman signals particularly
when the width is smaller than 100 nm because of less contribution by the
depolarized electric field induced LO Raman signal. Continuing the comparison of

(

)

(

)

z ' x ',y ' z ' and z ' rad,y ' z ' , the probed area is different for the two polarizations. In

	
  

53	
  

terms of TO detection sensitivity, linear polarization is more sensitive than radialpolarization judging from the TO/LO ratio. But when similar laser power is used, theTO intensity is stronger for radial polarization than for linear polarization, as shown in
Fig. 3.6b. Therefore, radial polarization is useful for low laser power TO
characterization in ε-Si nanowires. With the improved sensitivity of TO phonon()

modes for nanowires, it is reasonable to fit the spectra of z ' x ',y ' z ' configuration
with double Lorentzian functions for the split TO1 and TO2.

54	
   	
  

Figure 3.5. Incident electric field distribution illuminating the surface of the ε-Si nanomembrane and nanowires with the calculated Raman intensities using2
Eq. 3.13 at various polarizer and analyzer settings. Raman intensities were normalized at each setting. Image size is 1 x 1 µm . The maximum Raman
intensity is observed to be at the SiO2 and ε-Si interface.

	
  

55	
  

Figure 3.6. Comparison of line profiles across the center of (a) Ez’ component of the incident
electric field for linear and radial polarization in a nanomembrane. (b) TO1 intensity

(

)

(

)

comparison for TO-active z ' x ',y ' z ' and z ' rad,y ' z ' configuration in a nanowire. The
smaller area reflects the diffraction-limited focus spot illuminating the sample.	
  Figure 3.7 (a) Calculated Raman intensity ratio of TO and LO phonon signal as a function of
nanowire width. The arrows indicate the widths used in experiments. Inset shows the TO/LO
ratio for nanowires greater than 250 nm up to infinity. The TO/LO ratio reaches a value of
0.94. (b) Calculated Raman intensity of each phonon mode as a function of nanowire width.
LO intensity drastically increases with larger width because of the depolarized electric field()

contribution (e.g. Ey’ field in z ' x ',y ' z ' configuration).

56	
   	
  

3.6. REFERENCES[3.1] R. Loudon, The Raman effect in crystals, Adv. Phys., 13, 423-482, 1964.
[3.2] I. De Wolf, Micro-Raman spectroscopy to study local mechanical stress insilicon integrated circuits, Semicond. Sci. Technol., 11, 139-154, 1996.
[3.3] C. Kittel, Introduction to Solid State Physics: Eighth Edition, John Wiley &Sons, Inc: USA, 2005.
[3.4] R. Ossikovski, Q. Nguyen,G. Picardi and J. Schreiber, Determining the stresstensor in strained semiconductor structures by using polarized micro-Raman
spectroscopy in oblique backscattering configuration, J. Appl. Phys., 103,
093525, 2008.
[3.5] G. Turrell, J. Corset, Raman Microscopy: developments and applications,Elsevier, London, 1996.
[3.6] P. Yu, M. Cardona, Fundamentals of Semiconductors: Physics and Materials
Properties 3rd Edition, Springer: Berlin, 2001.
[3.7] I. De Wolf, H.E. Maes, and S.K. Jones, Stress measurements in silicon devices
through Raman spectroscopy: bridging the gap between theory andexperiment. J. Appl. Phys. 79, 7148-7156, 1996.
[3.8] M.V. Balois, N. Hayazawa, A. Tarun, S. Kawata, M. Reiche and O.
Moutanabbir. Direct optical mapping of anisotropic stresses in nanowires usingtransverse optical phonon splitting. Nano Lett. 14, 3793-3798, 2014.
[3.9] P.V. Huong, A.L Verma, J.P. Chaminade, L. Nganga and J.C. Frison,
Characterization of materials by micro-Raman spectroscopy, Materials Scienceand Engineering, B5, 255-260, 1990.
[3.10] A. Tarun, N. Hayazawa, H. Ishitobi, S. Kawata, M. Reiche and O. Moutanabbir,
Mapping the “forbidden” transverse-optical phonon in single strained silicon(100) nanowire, Nano Lett. 11, 4780-4788, 2011.
[3.11] D. Kosemura and A. Ogura, Transverse-optical phonons excited in Si using ahigh-numerical aperture lens, Appl. Phys. Lett. 96, 212106, 2010.
[3.12] K.

Mizoguchi,

Orientations

and
in

S.

Nakashima.

SiliconFilms

by"Determinationof

Crystallographic

Raman-Microprobe

Polarization

Measurements", J. Appl. Phys., 65, pp.2583-2590 (1989).
[3.13] S. Ganesan, A.A. Maradudin, and J. Oitmaa, A lattice theory of morphic effects
in crystals of the diamond structure, Ann. Phys. 56, 556-594, 1970.
[3.14] J.B. Hopkins and L.A. Farrow, Raman microprobe determination of local crystal
orientation, J. Apply. Phys. 59, 1103-1110, 1986.

	
  

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[3.15] B. Richards and E. Wolf, Electromagnetic diffraction in optical systems II.
Structure of the image field in an aplanatic system, Proceedings of the Royal
Society A: Mathematical, Physical and Engineering Sciences, 253, 358-379,
1959.
[3.16] P. Török, P. Varga, Z. Laczik and G.R. Booker, Electromagnetic diffraction oflight focused through a planar interface between amterials of mismatchedrefractive indices: an integral representation, J. Opt. Soc. Am. A, 12, 325-332,
1995.
[3.17] J.J. Stamnes, Waves in Focal Regions: propagation, diffraction and focusing oflight, sound and water waves, IOP Publishing Limited: Bristol, 1986.
[3.18] K.S. Youngworth and T.G. Brown, Focusing of high numerical aperturecylindrical-vector beams, Optics Express, 7, 77-87, 2000.
[3.19] D.P. Biss and T.G. Brown, Cylindrical vector beam focusing through a dielectricinterface, Optics Express, 9, 490-497, 2001.
[3.20] L.E. Helseth, Roles of polarization, phase and amplitude in solid immersion
lens systems, Optics Communications, 191, 161-172, 2001.
[3.21] H. Ishitobi, I. Nakamura, N. Hayazawa, Z. Sekkat, and S. Kawata, Orientational
imaging of single molecules by using azimuthal and radial polarizations, J.
Phys. Chem. B, 114, 2565-2571, 2010.
[3.22] S. Quabis, R. Dorn, M. Eberler, O. Glöckl and G. Leuchs, Focusing light to atighter spot, Optics Communications 179, 1-7, 2000.
[3.23] A. Tarun, N. Hayazawa, M. Motohashi, and S. Kawata, Rev. Sci. Instrum., 79,
013706, 2008.
[3.24] R. A. Minamisawa, M. J. Süess, R. Spolenak, J. Faist, C. David, J. Gobrecht,
K. K. Bourdelle, and H. Sigg, Nat. Comm., 3, 1096, 2012.
[3.25] N. Hayazawa, K. Furusawa, A. Taguchi, and S. Kawata, J. Apply. Phys., 106,
113103, 2009.
[3.26] Deb, S. K.; Bansal, M. L. Roy, A. P. Appl. Spectrosc. 1984, 38, 500.
[3.27] F. Cerdeira, C. J. Buchenauer, F. H. Pollak, and M. Cardona. Phys. Rev. B.
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[3.28] M. Born and E. Wolf, Principles of Optics: Seventh (Expanded) Edition,
Cambridge University Press: New York, 2009.

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Chapter 4: Strained Silicon: a micro-Raman spectroscopy
characterization study of stress after nanopatterning via
precise polarization control [4.1 – 4.2]
Silicon (Si) electronics currently have a nanowire-based architecture to achieve lower
power consumption, higher performance and scalable devices whose constant trendof miniaturization is dictated by Moore’s law [4.2]. Si nanowires are quasi-onedimensional nanostructures that can overcome the limitations faced in twodimensional planar devices [4.3 – 4.8]. Therefore, transistors employ nanowire-likechannels to create high-performance microprocessors and memory devices for
wireless systems such as mobile phones and radars [4.2]. Such new technologies
utilize nanowires that have three free facets with the fourth facet forming an interface
with the underlying substrate [4.7, 4.8]. But due to the quantum confinement andscattering at the sidewalls, these nanowires suffer degradation in carrier velocity that
decreases device performance [4.9, 4.10]. Stress nano-engineering has emerged asa powerful technique to alleviate these limitations and extend the capabilities ofnanowire-based electronics [4.11].
Strain engineering has been used in the microelectronics industry since the 90-nm
technology node [4.1]. This approach employs deposition of stressor layers on top oftransistors. The dimensions of nanowire-based transistors, however, are becoming
smaller than the required thickness of the stressor overlayer, making it difficult to
employ this method for smaller nodes or other nanowire-based architectures.
Recently, ultrathin, globally strained silicon layers (called nanomembranes) havebeen the base material to generate strained Si nanowires (ε-SiNWs) via top-down
nanofabrication processes [4.11 – 4.13]. A crucial step in the fabrication of ε-SiNWs
involves nanoscale patterning (nanopatterning) of biaxially strained nanomembranes,
but this step results in the formation of free surfaces. The free surfaces cause a local
relaxation of strain due to the rearrangement of lattice atoms near the newly formed
edges [4.12, 4.14 – 4.18]. Generally speaking, the extent of this phenomenon
depends on dimension and geometry [4.14 – 4.18]. It is widely accepted, however,that due to nanowire geometry (i.e., the high aspect ratio) the post-patterning stressbecomes uniaxial, which means that the stress is fully relaxed along the width of the
nanowire [4.11, 4.14]. But this is not the case for nanowires directly fabricated onoxide as discussed in this chapter. The nanowire-substrate interface makes the strain
redistribution rather complex due to the underlying oxide layer that stabilizes one of

	
  

59	
  

the four facets [4.1]. After nanopatterning, it is found that the ε-SiNWs experience
anisotropic stress relaxation. The amount of relaxation is very much dependent on
the dimensions of the ε-SiNWs [4.2]. Another important aspect that is investigated isthe effect of the nanomembrane edge on the stress distribution of the ε-SiNWs [4.19].
The proximity of the ε-SiNW to the nanomembrane edge can affect the stresspresent in the nanowire. Exploring and understanding these subtle but importantphenomena is crucial for accurate strain nano-engineering and a precise predictionof the performance of strained Si nanowire-based devices.
To probe the evolution of strain and stress in ε-SiNWs, Tarun et. al. recently
developed a high-resolution polarized micro-Raman spectroscopic technique thatallows the detection and analysis of the individual contributions of longitudinal optical(LO) phonons and transverse optical (TO) phonons [4.12]. In this work, to explore theeffects of the interface, I employ two excitation wavelengths with linear polarizations
corresponding to different penetration depths in silicon thereby achieving LO phonon
detection as a local stress probe at different depths within a single nanowire. To
probe the effects of the nanomembrane edge on the stress redistribution in ε-SiNWs,
I employ a single excitation wavelength (532 nm) using linear polarization. This isbecause of the stronger signal-to-noise ratio that can be achieved when using 532nm illumination. Direct detection of TO phonon modes can characterize anisotropic
stress relaxations at each position by using a high NA lens and proper polarization. I
experimentally demonstrated for the first time the splitting of TO phonon modes andwas able to analyze the dimension dependence of anisotropic stress relaxation. I
also discuss and take into consideration the effects of the scattering at the nanowire
sidewalls on the obtained Raman data and present two methods for accurate
determination of stress in nanoscale structures. The first method is LO phonon
analysis, while the second method is TO analysis. A comparison of the two is madeat the end of this chapter.
The work presented in this chapter is adapted from Balois and Tarun, et. al. (2013)[4.1] and Balois, et. al. (in preparation) [4.19] for the analysis of stress redistributionin ε-SiNWs and nanomembranes using longitudinal optical (LO) phonon analysis andBalois, et. al. (2014) [4.2] for the analysis of anisotropic stress relaxation in ε-SiNWs
using transverse optical (TO) phonon analysis.

60	
   	
  

4.1 SAMPLE FABRICATION
There are two kinds of samples used in all the experiments discussed in this chapter.
The first kind of sample is a ε-Si nanomembrane (Figure 4.1) that has a 1 µm length,
15 nm thickness and 10 µm width. The second kind of sample is ε-Si nanowires witha fixed 1 µm length, 15 nm thickness and varying widths of 80 nm, 50 nm and 30 nm(Fig. 4.1a – only 80 nm width is shown). Figure 4.1b shows the transmission electron
microscopy (TEM) cross-sectional image of the ε-Si sample.

Figure 4.1 (a) Optical reflection image of the nanowires aligned near the unpatterned
nanomembrane and the scanning electron micrograph image of the aligned nanowires (inset).
The scale bar indicates 2 mm. (b) TEM of the cross-sectional profile of the ε-Si sample. The
scale bar indicates 75 nm. (c) Comparison of Raman spectra with and without Ge layer. The
Ge layer perfectly suppresses the background Raman signal of the silicon substrate peaked-1
at 520.5 cm and allows selective detection of the ε-Si layer.

From a 15 nm thick biaxially tensile ε-Si nanomembrane, the ε-SiNWs were formedby nanopatterning and electrochemical etching. The fabrication process is shown in
Figs. 4.2a – f. To generate the ε-Si nanomembrane, an ultrathin layer of Si was
epitaxially grown on a ~500 nm thick Si0.84Ge0.16 relaxed buffer layer. This buffer layer
is grown on a Si (001) substrate via reduced pressure chemical vapor deposition. A
ε-Si/Si0.84Ge0.16/Si heterostructure was made and the ε-Si layer was coated with a
SiO2 layer via plasma-enhanced chemical vapor deposition. The Si (001) host

	
  

61	
  

substrate was prepared separately which consists of a ~120 nm thick Ge layer
deposited directly on top of the Si host substrate via solid molecular beam epitaxy.
The Ge layer was then coated with a ~200 nm thick SiO2 layer. The ε-Si layer wastransferred to the host substrate via direct wafer bonding. Ion-induced slicing wasused to remove the Si0.84Ge0.16/Si wafer and selective chemical etching was
conducted to remove the excess Si0.84Ge0.16 relaxed buffer layer. The resulting wafer
was a ε-Si/SiO2/Ge/Si heterostructure with a 15 nm thick ε-Si nanomembrane layer.
The role of the Ge is to act as a shield that prevents the laser from reaching the Si
substrate, in the case of visible excitation. The existence of the Ge layer allows us toselectively detect the ε-Si layer without any background signal from the underlying Si
host substrate. Ordered arrays of rectangular nanostructures, spaced 500 nm apart,were patterned onto a negative resist using electron-beam lithography. Reactive ion
etching (RIE) was used to transfer the pattern to the strained layer, forming ordered
arrays of rectangular ε-Si nanowires directly on SiO2. The patterned islands arealigned along the [110] direction. The RIE process was done at a pressure of 6 mTorr
using a mixture of SF6 (100 sccm) and O2 (5 sccm) with a rate of ~2 nm s-1 [4.20].

Figure 4.2. Illustration of the process flow for the fabrication of background-free ultrathin εSiNWs using thin layer transfer. (a) Growth of Si0.84Ge0.16 relaxed buffer layer on Si-(001)substrate; (b) growth of biaxially tensile strained Si on Si0.84Ge0.16 and subsequent hydrogenion implantation into the grown heterostructure; (c) growth of Ge layer and SiO2 deposition on
Si host substrate and bonding of hydrogen-implanted heterostructure onto host substrate; (d)exfoliation via thermal annealing around the hydrogen implantation depth; (e) polishing to
remove excess Si0.84Ge0.16 relaxed buffer layer; (f) Nanopatterning of ε-Si nanostructuresusing electron beam lithography and RIE. [Adapted from Ref. 4.12 with permission from ACS
Publications]

62	
   	
  

4.2 EXPERIMENTAL SETUP
Figure 4.3 shows the experimental setup used for both LO and TO experiments.
Laser light (λ = 355 nm, 442 nm, for LO multiwavelength experiment and 532 nm for
TO analysis and comparative LO analysis) passes through a linear polarizer and ahalf wave plate. The linear polarizer sets the incoming polarization from the laser.
The half-wave plate is used to align the incident polarized beam either parallel (x’) or
perpendicular (y’) to the nanowire long axis. The beam is then expanded and focused
onto the nanowires using an oil-immersion objective lens (numerical aperture, NA =1.4, x125 for 355 nm and 442 nm experiments, and NA = 1.49, x100, for 532 nm).
The surface of the ε-Si is directly immersed in oil. This configuration improves thelaser focusing and minimizes index mismatch-induced spherical aberrations.
According to the Rayleigh criterion, the diameter of the laser spot is λ/2. The
backscattered Raman signal is collected by the same objective lens and passes
through a pinhole (φ = 50 mm), which leads to an edge filter to block the strong
Rayleigh signal in order for the backscattered Raman signal to be detected. The
backscattered light goes through an analyzer before reaching a spectrometer (grating= 1800 g mm-1, slit width = 50 µm) equipped with a thermo-electronically cooledcharge coupled device (CCD) camera. The samples with nanowire arrays were
mounted on an x’-y’ translation stage and were scanned with a 25 nm step while
exposed to the focused laser beam. At each step, a Raman spectrum was recorded.
To prevent laser heating, Raman spectra were acquired at 5 mW power. For the LO
data, a 6 µm x 6 µm area was scanned with an exposure time of 0.5 s/pixel (total
scan time ~ 12 hours / image). For the TO data, a 3.5 µm x 2.75 µm area wasscanned with an exposure time of 5 s/pixel (total scan time ~ 22 hours / image). A
longer exposure time was needed for the TO data because of its relatively weakersignal compared to the LO data. Acquiring both LO and TO data required scan time
ranging from half a day to almost a day. For long duration experiments like these,stability of the experimental system is important, especially thermal stability, which
can induce both defocusing and sample drift. To address these issues, anautofocusing system [4.21] has been installed to the microscope objective assembly
and a thermal control booth enclosed the entire confocal microscope system.

	
  

63	
  

Figure 4.3 Experimental setup for LO and TO detection.

4.2.1 Stability of the Experimental System
As mentioned in the previous section, image acquisition during the LO and TO
experiments require a considerable amount of time. In imaging experiments, it isimportant that the illuminating laser remain focused onto the sample during the entire
experiment to avoid image degradation and erroneous data analysis. Stability
becomes a key issue since fluctuations in the experimental system and its
environment are not negligible as they compound over time. Fluctuations can beelectrical, vibrational or thermal in nature. Electrical and vibrational fluctuations canbe dealt with by state-of-the-art circuits and ingenious vibrational isolation designs
that are commercially available. Thermal fluctuations are more complex because
these fluctuations can come from a variety of sources, even from the person
operating the experimental system. Two unavoidable consequences arise fromthermal fluctuations: 1) thermal drift of the sample in the x’ and y’-directions and 2)
defocusing caused by drifting of the microscope objective in the z’-direction. To
minimize sample drift (x’-y’), a thermally controlled booth was installed to enclose theentire confocal microscope system. Figure 4.4a shows the temperature booth
controller (left-side) and the temperature boot (right side). This booth regulates thetemperature inside the booth to be constantly at 22˚C with a minimal fluctuation of ±0.1˚C, as shown in Fig. 4.4b. When the temperature booth is turned OFF, thetemperature is not stable and either increases or decreases. As for the defocusing,an autofocusing system [4.21] has been installed and attached to the microscope

64	
   	
  

objective (Fig. 4.5a). Figure 4.5b shows the configuration of the autofocusing system.
It is composed of a capacitive sensor (Mess-tek TRA616-200-V2) that can be
attached to any microscope objective mounted on a Z-PZT stage (Nanocontrol,
PFHC252R-020U-N). A microscope objective with a high NA is preferable to attain atightly focused spot. The capacitive sensor has a maximum dynamic range of 100µm. A polyimide tape (thickness ~ 50 µm) is used as a spacer to form a gap between
the capacitive sensor and the grounded sample stage, which serves as a counterelectrode. The capacitive sensor recognizes the gap and the signal is converted to
DC voltage (low-pass-filter: 1 kHz). An analog-digital converter (16-bit resolution)
constantly monitors the DC voltage from the capacitive sensor. After setting the
focused position of the objective lens, the DC voltage of the sensor is locked. If there
are changes in the voltage, which can be due to the movement of the objective lensin the z’-direction or tilting of the sample, a feedback voltage is applied to the Z-PZT(dynamic range: 2 µm; bandwidth: DC ~ 1 kHz) to compensate for the displacement.
This autofocusing system constrains the gap fluctuation within ± 1 nm [4.21].

Figure 4.4 (a) Temperature booth controller and temperature booth (black structure at theright side of picture). (b) Temperature fluctuation when temperature booth is ON and OFF.

	
  

65	
  

Figure 4.5 (a) Autofocusing system installed onto the microscope objective lens where (A) isthe capacitive sensor and (B) is the Z-PZT. (b) Schematic of the autofocusing system[Adapted from Ref. 4.21 with permission from IOP Publishing].

Figure 4.6 Nanomembrane and nanowire area (6 µm x 6 µm) scanned with different
environmental parameters: (a) room air conditioning OFF, autofocusing system OFF andthermal control booth OFF; (b) room air conditioning ON, autofocusing system ON andthermal control booth OFF; (c) all three systems are turned ON.

66	
   	
  

To demonstrate the influence of the environment and the stability of the system, a 6µm x 6 µm area was scanned which consisted of part of the nanomembrane and 80
nm width nanowires (Figs. 4.6a – 4.6c). A laser wavelength of 532 nm was used.
During each scan, the environmental conditions were different. For Fig. 4.6a, the air
conditioner of the experimental room, the thermally controlled and the autofocusing
system were turned off. This resulted to a gradually degraded image. The initially
focused image of the nanowire region became defocused as the scanning
progressed. Not even half-way through the scan, the contrast between the nanowires
decreased to the point that the nanowires cannot be individually resolved. This wasdue to the thermal drift present in all three directions. The direction of the drift was
influenced by the increase in room temperature, since the air conditioning of theroom was also turned off. For Fig. 4.6b, the air conditioning was turned on along withthe autofocusing system, but the thermally controlled booth was still turned off.
Through the use of the autofocusing system, the nanomembrane and the nanowire
areas were clearly identified and the amount of drift lessened in the x’ and y’directions. But periodic artifacts can be seen in the nanomembrane and in the
nanowire areas. This was due to the periodic on and off cycle of the air-conditioner tomaintain a constant room temperature. For Fig. 4.6c, the thermally controlled booth
was turned on, along with the room air conditioning and the autofocusing system.
The thermally controlled booth helped maintain a constant temperature within thevicinity of the microscope and eliminated the unwanted artifacts caused by the cycleof the room air conditioning system.
4.3 ANALYSIS OF STRESS I: LONGITUDINAL OPTICAL PHONON APPROACH[4.1]
4.3.1. LO: Theoretical Approach
The local stress in Si lattice is obtained from Raman shifts by solving the well-known
secular equation. In the case of the unpatterned nanomembrane, the equi-biaxial
stress ( σ xx = σ yy = σ 0 ) model of stress-induced Raman shift for doublet ( Δω xy : TO
phonons) and the singlet ( Δω z : LO phonons) are [4.22]:

(

) (

)

⎡ p S11 + S12 + q S11 + 3S12 ⎤ σ 0⎦ = −3.42 × σ ,Δω xy = ⎣
0
2ω 0

	
  

(4.1)

67	
  

(

)(

)

⎡ pS12 + q S11 + S12 ⎤ σ xx + σ yy⎦Δω z = ⎣= −4.60 × σ 0 ,
2ω 0

(4.2)

where Δω xy = ω ε −SiTO − ω Si and Δω z = ω ε −SiLO − ω Si are expressed in the Raman
frequency in each mode in the presence ( ω ε −Si ) and absence ( ω Si ) of stress. S11 =7.68 and S12 = –2.14 (in GPa) are the elastic compliance tensor elements, while p =–1.85 and q = –2.31 are the phonon deformation potentials for bulk Si at room
temperature [4.23]
The stress relaxation ratio upon patterning at any position in the nanowire is

( )

( )

calculated using R x,y = 1− σ nanowire x,y / σ 0 , where σ0 and σnanowire correspond to
the initial stress before nanopatterning and the measured stress in nanowires,
respectively [4.24]. The initial stress, σ0, in the nanomembrane is calculated from themeasured Raman shift using Eq. (4.2). However, it is important to note that Eq. (4.2)
cannot be used for the extraction of stress in the nanowires, σnanowire, mainly because
the assumption that shear stress is zero used to obtain the stress is no longer valid
as the stress is no longer bi-isotropic in the nanowire due to geometrical constraints
and non-uniform rearrangements of atoms near the newly formed free surfaces. In
fact, in a nanowire the shearing strain terms at the crystal coordinate are no longer
zero due to anisotropic strain relaxation along the nanowire axes. In order to obtainthe nanowire stress, I first calculated the strain and stress components in the sample
coordinates where shearing terms are all zero due to the symmetry of the structure at
the sample coordinate system and then transform them into crystal coordinate [4.24].
At the center of the nanowire, the strain component along the nanowire length (()ε 'xx x ',d ) has almost the value of the original membrane, ε 0 , but the stress along()

the width ( σ 'yy x ',d ) will be dramatically relaxed due to the reduced dimension inthis direction. d represents the width of the nanowires. The shear strain component
becomes zero in the middle of the nanowire, and the stress component along z’ isequal to zero due to free surface boundary conditions; the strain and stress tensors
are then expressed as [4.24]

68	
   	
  

{ε '( x ',d )}

⎡
⎢
⎢ 1
0
⎢
2
sα x ',d C11 + C12− C11 C12 − H / 2⎢= ε 'xx x ',d ⎢ 0
2
C
C
+
H
/ 2 − C12
11
11
⎢
⎢
⎢ 0
0
⎢
⎣

(

)

(

)

(

)

)

(

(

0

)

0

(
(

)

(

)

−sα x ',d C12 − 2C12C44)

2
C11 C11 + H / 2 − C12)(⎡ C ' ε ' x ',d + C ' ε ' x ',d + C ' ε ' x ',d
11
xx
12
yy
13
zz
⎢
⎢
=
0
⎢
⎢
0
⎣

{σ '( x ',d )}
(

(

⎤
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦

)

0

(

)

α x ',d σ 0
0

0 ⎤
⎥
0 ⎥,
⎥
0 ⎥
⎦

(4.3)

(4.4)

)

where α x ',d = σ 'yy x ',d / σ 0 , s = C11 +C12 – (2C212/C11) and H = 2C44 + C12 – C11.
The elastic constants C11 = C22 = 166 GPa, C12 = C21 = 64 GPa, C44 = 79.6 GPa, C’12
= C12 – 0.5H = 35.4 GPa, C’11 = C11 + 0.5H = 194.6 GPa and C’13 = C12 = 64 GPa are()

obtained from Refs. [4.24] and [4.25]. The constant α x ',d varies between 0 and 1()

and depends upon the diameter of the nanowire. At the limits, α x ',d equals 0 for
uniaxial stress and 1 for bi-isotropic stress. After substitution, the strain tensor in thesample coordinate reduces to:{ε '( x ',d )}

⎡
⎤
0
0
⎢ 1
⎥
⎥
= ε 'xx x ',d ⎢ 0 1.06α x ',d − 0.06
0
⎢
⎥
⎢ 0
0
−0.41α x ',d − 0.36 ⎥
⎣
⎦

(

)

(

)

(

(4.5)

)

and transformation to the crystal coordinate system results to:{ε '( x ',d )}

( )
( )

( )
( )

⎡ 0.53α x ',d + 0.47 −0.53α x ',d + 0.53⎤
0⎢⎥⎥ . (4.6)= ε 'xx x ',d ⎢ −0.53α x ',d + 0.53 0.53α x ',d + 0.47
0
⎢
⎥
⎢
0
0
−0.41α x ',d − 0.36 ⎥
⎣
⎦

(

)

(

)

Using the strain components above, the solution to the well-known secular equation
for the LO phonon mode is expressed asΔω z = ω ε −SiLO − ω Si =

	
  

ω Si
⎡ pε x ',d + 2qε yy x ',d ⎤ ,⎦2 ⎣ zz

(

)

(

)

(4.7)

69	
  

Δω z = ω ε −SiLO − ω Si =(−ω Si ε 'xx x ',d2) ⎡1.70α ( x ',d ) + 1.50 ⎤ .
⎣

⎦

(

(4.8)

)

In Eq. (4.6), there are two unknown parameters, strain ε 'xx x ',d along the nanowire()

long x’-axis and the α x ',d , stress relaxation factor along the nanowire width, which
I have to extract from the Raman measurement. However, assuming I detect only LO
phonon mode by conventional backscattering geometry, it is impossible to determine
the two unknown parameters from the single phonon mode.

To circumvent thislimitation and obtain the stress in nanowires, it is widely accepted to assume that thestress along the width of the nanowire can be assumed fully relaxed, that is()

α x ',d = 0 , (see for e.g., Refs. 4.15, 4.24 and 4.25 and references therein). Herein,
I demonstrate that this assumption is too simplistic leading to an inaccurate analysis()

of stress. Using Eq. (4.5) and at α x ',d = 0 , the stress along the x’-axis is given by:

(

)

(

)

σ 'xx x ',d = 168.25 × ε 'xx x ',d . By substituting the strain components in Eq. (4.6),

(

)

(

)

ε zz = −0.36 × ε 'xx x ',d and ε xx = 0.47 × ε 'xx x ',d , into Eq. (4.7), the stress (in GPa)in the lon axis can be expressed in terms of the measured Raman shift as:()

σ 'xx x ',d = −225.67 ×

ω ε −Si − ω Si
.
ω Si

(4.9)

In an alternative approach, I propose the introduction of an edge structure (edgeparallel to y’-axis) consisting of a patterned nanomembrane with a practically semiinfinite dimension along y’ (˃ 1 cm), which can be assumed as a hypotheticalnanowire with an infinite width ( d = ∞ ).

Eq. (4.8) indicates that there are two

unknown parameters, ε 'xx (x ',d) and α (x ',d) , while the experiment only provides thevalues of Δω z (Fig. 4.7b). For α (x ',d) ≠ 0 , ε 'xx (x ',d) can be determined using thereference structure (d = ∞). Here, the nanowire strain profile along x’-axis, ε 'xx (x ',d) ,can be taken as comparable to the profile along the same direction in the reference
structure, ε 'xx (x ',d = ∞) = ε 'xx (x ',d) , because at an identical length (1 µm) the effectof the newly formed free surfaces should be similar in both systems. In y’ axis, thereference sample is infinite and thus the initial strain is preserved (i.e., no relaxation

70	
   	
  

does take place along this axis) suggesting that ε 'yy (x ',d = ∞) is equivalent to theinitial strain in the unpatterned nanomembrane, ε 0 .
Analogously to Eq. (4.8), I introduce an additional anisotropic stress relaxation factor,

β (x ',d = ∞) = σ 'xx (x ',d = ∞) / σ 0 , which represents the relative stress relaxation alongx’-axis in the reference structure. Thus, the equation of Raman shift as a function ofβ (x ',d = ∞) can be written as:Δω z = ω ε −SiLO − ω Si =(where ε 'yy x ',d = ∞(−ω Si ε 'yy x ',d = ∞2) ⎡1.70β x ',d = ∞ + 1.50 ⎤()⎣⎦(4.10)) is obtained from Raman signal of the unpatterned

nanomembrane, whereas Δω z is obtained from Raman scans along the x’-axis from()

the edge. By using the obtained values of ε 'yy x ',d = ∞ and Δω z , I can determine

β. Thus, the strain along the x’-axis of the edge sample can be extracted similarly to
Eq. (4.5) using β:

(

)

(

)

(

)

ε 'xx x ',d = ∞ = ε 'yy x ',d = ∞ ⎡⎣1.06β x ',d = ∞ − 0.06 ⎤⎦ .

(4.11)

This strain is only a function of x’ and varies from x’ = 0 (edge) to an x’ = 0.5 mm()

around the nanowire center. The strain component, ε 'yy x ',d = ∞ (parallel to theedge) remains unchanged. This approach is analogous to the semi-infinitely wideline structures used in the theoretical treatment developed earlier by Jain et al. [4.24].
The measured Raman shifts from the edge structures were attributed mostly to the()

strain along the x’, ε 'xx x ',d = ∞ and the constant strain component in the y’direction, ε 0 which is the same as the strain obtained from the nanomembrane. In
the following, I test the validity of these two approaches.
4.3.2 LO: Experimental Results and Discussion
Multiwavelength experiments utilizing 442 nm and 355 nm lasers for interface
effects investigation [4.1]. Fig. 4.7a shows the Raman spectra obtained with the()

442 nm excitation at the center of the nanowires using z ' x ',x ' z ' configuration. I

	
  

71	
  

reported recently that TO phonons can be effectively observed using high NA
illumination [4.12]. The observed TO-active Raman shift are upshifted relative to LO
phonons as predicted by the association of Eq (4.1) and (4.2) [4.12]. This upward
shift in the Raman spectra arising from TO phonon can be easily mixed with LO
phonon signal and consequently overestimate the level of stress relaxation. This isbecause the spectral split between LO and TO phonons is small to be resolved bypeak function fits. Proper selection of sample azimuth, incident polarization andanalyzer orientation is therefore necessary to accurately uncover the stress behavior()

in strained Si nanowires. I observed that the z ' x ',x ' z ' is constantly lower than the

(

)

(

)

z ' y ',y ' z ' configuration for all regions in the nanowires. Under z ' x ',x ' z ' , that is
the incident polarization is parallel to the nanowire axis, the observed Raman is
predominantly due to LO-phonon.

()

For the case of z ' y ',y ' z ' , where incident

polarization is perpendicular to the nanowire axis, the observed Raman consists of()

both LO- and TO-phonons [4.22]. This is because in z ' y ',y ' z ' configuration thetransverse (Ey) electric field is efficiently converted to Ez-field when introduced to the(110) and (-110) sidewalls, which results in an effective excitation TO phonons [4.12].
The TO scattered Raman is efficiently collected by backscattering geometry [4.22].
From previous work [4.12], I experimentally verified that TO phonon in nanowires has
higher Raman peak shift relative to LO-phonons. This explains why the Raman peak

(

)

(

)

shift for z ' y ',y ' z ' is higher and broader (data not shown) than z ' x ',x ' z ' . Hence,in order to obtain an accurate in-plane tensile stress from LO-phonons, it is importantto choose the incident polarization parallel to the nanowire.
Here, the Raman signal is collected from the whole thickness of the nanowire (i.e., 15nm). The measured spectra are “background-free” and contain only the intrinsic SiSi Raman shift peak of the probed nanowire because of the introduction of Geinterlayer described above. It should be noted that the dominant contribution to themeasured Raman shift originated from the LO phonon mode from both the top (001)surface and the long axis sidewalls, [110], based on the Raman tensor calculation
under x’-polarized light.

For the sake of comparison, Raman spectrum of initialstrained nanomembrane (solid black line) as well as the Si-phonon peak position inbulk Si (broken vertical lines) is also shown. Table 4.1 summarizes the Raman peakand spectral width for all the observed structures. As the width decreases, theRaman peaks shift upward with respect to the position of the Si-phonon peak of theinitial strained Si layer.

72	
   	
  

Table 4.1. Summary of Raman peak and spectral width at center of single nanowires.

Initial strained layer
Nanowire width, d (nm)15nm-thick ( ω ε −SiLO = 515.42 cm−1 Δυ = 4.86 cm−1 )
80

50

30

Peak, ω ε −SiLO (cm-1)

516.43516.99517.42

Width, Δυ (cm-1)

5.09

5.20

5.16

Figure 4.7. (a) Raman spectra recorded at the center of nanowires with lateral dimensions of
30 nm (triangles), 50 nm (squares) and 80 nm (circles). The Raman spectrum of theunpatterned strained Si is shown in solid black lines. The vertical dashed line denotes the Si–
phonon peak position in bulk (unstrained) Si. (b) Profile of the Raman peak for nanowire
obtained under the z(x’x’)z configuration. The solid and dashed line represents the Raman-1peak position of the initial strained Si nanomembrane using 442 nm (515.75 cm ) and 355nm-1(515.42 cm ) excitation, respectively [Taken from Ref. 4.1 with permission from IOP
Publishing].

	
  

73	
  

No significant broadening in the nanowire Si-Si Raman mode is, however, observed.
It is noticeable that Si-Si peak position of the investigated nanowires is upshifted withrespect to its position in the initial ( ω ε −SiLO = 515.42 cm−1 ) strained nanomembrane.
This shift is indicative of partial relaxation of stress during the process of nanowire
pattering using reactive ion etching.
To obtain more insights into the complex redistribution of stress as nanowire
dimension shrinks, the profiles of the Raman shift for nanowires of varying width areplotted in Fig. 4.7b. The nanowires were scanned along the x’-axis with a step size()

of 25 nm using z ' x ',x ' z ' configuration. The Raman shift for d = 30 nm using 355
nm (UV) excitation is also displayed in Fig. 4.7b. At this excitation, only the top 10
nm of the nanowire thickness is probed.

For comparison, the Raman shifts

measured from the original membrane are indicated with solid (442 nm) and dashed(355 nm) lines.

()

Fig. 4.8 shows the stress profiles using the (a) first, α x ',d = 0 , and (b) second,

ε 'xx (x ',d = ∞) = ε 'xx (x ',d) , approaches.

Regardless of the approaches used, theprofiles measured for all the nanowires display a qualitatively similar behavior
characterized by a more pronounced relaxation of stress near the nanowire edge.
Moreover, as general trend, the residual stress profile shows plateau-like profiles in
the region around the center. The breadth of this region is sensitive to the nanowire
width and varies from ~400 nm at d = 50 or 80 nm to ~700 nm at d = 30 nm. The
enhanced relaxation near the edges results from the additional free surface ascompared to the rest of the nanowire (four vs. three facets). As described below, thisconstant stress values away from the edges can be attributed to the nearly constantand highly retained stress at the buried strained-Si/SiO2 interface.
The obtained initial stress from the nanomembrane using Eq. (4.2) was plotted (solid
black) to evaluate the validity of the first approach based on the assumption of a full()

relaxation along the nanowire width ( α x ',d = 0 ).

Strikingly, the stress profiles

along the long-axis obtained using this approach is higher than the original stress inthe nanomembrane.

This discrepancy indicates that the assumption of a fullrelaxation along the width is inaccurate and that stress is not uniaxial.
following, I focus my analysis on the second approach.

74	
   	
  

In theFigure 4.8.
Stress Profiles in single nanowires obtained under different excitation
wavelengths using the first (a) and second (b) approaches described in the text. (c) Stressrelaxation dependence estimated at the center of the nanowire as a function of the nanowire
diameter [Taken from Ref. 4.1 with permission from IOP Publishing].

I first calculated the strain from Eqs. (4.10) and (4.11) using the measured Raman
shift scanned from x’ = 0 (edge) to x’ = 0.5 mm (towards the center) sample withsemi-infinite width.

The strain values are then used in Eqs. (4.8) and (4.9) to

calculate the stress at each position.

(

σ 'xx x ',dFig. 4.8b shows the obtained stress profiles) and σ ' ( x ',d ) . The calculated stress values along the x’-direction areyy

below the original stress of the membrane. It is also interesting to note that the profileof the stress along x’-direction of the nanowires is similar and with stress values very
close to stress of the edge sample, suggesting that stress is preserved at the center.
Moreover, the stress along x’-direction at edges (x’ = 0 and x’ = 1.0 µm) is the same (σ 'xx (x 'd) ≈ 0.85 GPa ) for all nanowires regardless of excitation wavelengths. This is
not the case when a full relaxation is assumed along the width, which indicates that
this assumption is fraught with large uncertainties.

	  On the other hand, I can clearly

75	
  

(

)

see from the profile along the width, σ 'yy x ',d , that the nanowire is still under abiaxial stress. The stress profile along the width is relatively constant through thewhole length of the nanowire unlike along the long-axis where pronounced relaxation
is observed at the edges. Furthermore, a full relaxation of stress is only observed at
the edge of the nanowire with d = 30 nm. For this set of nanowires, the stress at thecenter reaches a value of ~200 MPa. This behavior is observed for both excitation
wavelengths. At d = 80 nm and d = 50 nm, the stress is well above the fully relaxed
region (gray dashed zero line). This result indicates that the overall average stress inthe nanowire is biaxial but not bi-isotropic. Note that at an excitation wavelength of
442 nm the buried strained-Si/SiO2 interface is also probed.

Here, the stressbetween 200 ≤ x’ ≤ 800 nm is comparable to the initial stress in the nanomembranes,
whereas at 355 nm the initial stress in only preserved in the region 450 ≤ x’ ≤ 550
nm. Fig. 8c exhibits the relaxation ratio at the center of the nanowire as a function of
nanowire diameter. It is noticeable that the extent of this relaxation along the width
exhibitsalinearbehaviorΔσ 'yy / σ 0 = 1.01− 5.77 × 10 −3 × d .

asafunctionofthenanowire

width:The relaxation along the long-axis is small andremains unchanged with the nanowire width.
Interestingly, the combination of two excitation wavelengths provides new insights
into the complex behavior of stress in nanowires.

Fig. 4.9 shows the depth

dependence of stress profiles along the two in-plane axes for nanowire with a width()

of d = 30 nm under z ' x ' x ' z ' condition using different excitation wavelength(penetration depth) namely: 442 nm (~168 nm) and 355 nm (~10 nm).

It isnoteworthy that Raman shifts measured using the 355 nm laser are centered aroundhigher wavenumbers as compared to those obtained using the 442 nm excitation asshown in Fig. 4.7b. The Fig. 4.9a also displays the original stress (lines) in theunpatterned nanomembrane obtained using the two-excitation wavelengths as well.
The small difference in the measured stress using the two excitations suggests thatthe region near the surface of the nanomembranes is slightly relaxed.
After patterning of the nanowires, the stresses are observed to be more relaxed inboth 442 nm and 355 nm excitations. However, the interesting observation is thatthe stress relaxation behavior as obtained by the 355 nm excitation decreases
monotonously towards the center of the nanowire, which remains highly strained. At
the 442 nm excitation, the stress relaxation plateaus in the region 200 ≤ x’ ≤ 850 nm.

76	
   	
  

Figure 4.9. Stress profiles along the (a) x’- and (b) y’-axis for nanowire with a width of d =30
nm at different depths obtained using the two excitation wavelengths (penetration depths):
442 nm (~168 nm) and 355 nm (~10 nm). The horizontal solid (442 nm) and dashed (355nm) lines marked the initial strain in Si nanomembrane. (c) Stress profiles at different depths
in the reference sample with a semi-infinite edge.

	
  

77	
  

This dissimilarity in stress behavior between the two-excitation wavelengths isindicative of the non-uniform distribution of the in-plane stress along the nanowire
thickness. More precisely, the fact that the stress measured using a laser with theshallower penetration depth is systematically smaller than the value averaged over
the whole thickness provides direct evidence that the region near the nanowire-oxideinterface maintains a high level of stress. By combining the stress profiles measured
at 442 nm ( σ '15nm) and 355 nm ( σ '10nm) excitations, I extract the stress profile in the
xx
xx

(

σ '5nm= 3 × σ '15nm− 2 × σ '10nm
bottom 5 nm of the nanowire along the x’-direction σ '5nm
xx
xx
xx
xx

)

(Fig. 9a). Similarly, the stress profile along y’-direction is depicted in Fig. 4.9b. For
the two in-plane axes, the obtained stress profiles are qualitatively identical but
remarkably different from the profiles measured for the top 10 nm or over the wholenanowire thickness. Particularly, the part of the nanowire near the interface becomesunder a higher stress especially in the region about 100 to 200 nm away from theedge in a qualitative agreement with finite element simulations (not shown). In thisregion, close stress values are recorded for the two in-plane directions indicating thatthe interface with the oxide preserves almost the initial isotropy (i.e., a biaxial strain).
Interestingly, the stress in x’-direction reaches values that are higher than the initialstress indicating that the contraction of the lattice near the newly formed freesurfaces is accompanied by a strong distortion close to the interface with the oxide.
This means that when the lattice near the edges moves inwards it drags theunderlying substrate leading to a complex redistribution of stress. Above ~200 nm
away from the edge, the stress decreases slightly along the nanowire length tostabilize around 1.2 GPa. The decrease is more significant along the width. For thisdirection, the in-plane stress remains the same independently of the depth in the
nanowire. A similar behavior is also observed for the edge sample as shown in Fig.
9c. The only noticeable difference in the bottom 5 nm stress values along the longaxis between the d = 30nm nanowire and reference sample is located at the edges.
The bottom 5 nm stress at the edge is ~1.06 GPa, which is just above the stressmeasured with 355 nm excitation whereas for d = 30 nm nanowire, the bottom 5 nm
stress is 0.83 GPa.

Towards the center, the bottom 5 nm stress values remainunchanged.
Nanomembrane edge induced stress field [4.19]. I scanned a 12 µm x 6 µm area
and Figure 4.10 shows the intensity and the Raman shift images of the 30 nm, 50 nm
and 80 nm width ε-SiNWs. From the intensity images, various localized defects areobserved, particularly in the 50 nm (Fig. 10b) and 30 nm (Fig. 10c) ε-SiNWs. These

78	
   	
  

defects become more pronounced in the Raman shift image. It is interesting to pointout that there are defects only evident in the Raman shift image when compared its
corresponding intensity image. Examples of such occurrences are in the Raman shift
image of the 80 nm width ε-SiNWs and some part of the 50 nm width ε-SiNWs. This
localized defect has significant effect on the Raman shift distribution in the ε-SiNWs
as it lowers the supposed Raman shift, which represents higher stress. For proper
analysis of the data, the defect areas were avoided. I also observed that the Raman
shifts of the nanowires are blue shifted from the nanomembrane. This means that the
Raman shifts of the ε-SiNWs are higher than the nanomembrane and that stressupon patterning occurs due to the formation of free surfaces.

Figure 4.10. Intensity images of NWs with length of 1 µm and width of (a) 80 nm, (b) 50 nm
and (c) 30 nm while (d), (e) and (f) are the corresponding Raman shift images. Image size is
12 µm x 6 µm with step size of 25 nm/pixel.

From the images in Fig. 4.10, the behavior of the Raman shift and the intensity signal
as the NW approaches the edge was studied. Horizontal line profiles were taken at
specific areas of the images, avoiding the defects. The black lines in Figure 4.11
denote the intensity line profile while the red lines denote the Raman shift line profilefor (a) 80 nm, (b) 50 nm and (c) 30 nm width NWs. As seen from Fig. 4.11, a local
maximum in the intensity line profile corresponds to a local minimum in the Raman
shift line profile, which is indicative of the center location of each ε-SiNW. In the
Raman shift, a redshift occurs for ε-SiNWs closer to the nanomembrane edge ascompared to the ε-SiNWs farther from the edge. The affected nanowires are within 4µm from the nanomembrane edge and exhibits the tuning of the stress that is
induced by the nanomembrane edge. It should be noted that the half portion of εSiNWs at the left most row of the 50 nm width ε-SiNWs

are embedded in thenanomembrane edge while the ones in 30 nm and 80 nm width are separated fromthe edge due to the fabrication process. This is also confirmed by the separate
measurement by optical reflection imaging.

	
  

79	
  

Three regions of interests have been identified and for ease of discussion shall becalled: near edge, middle region and far from edge. Intensity images of the ε-SiNWs
with 80 nm and 30 nm widths (Fig. 4.11a and 4.11c) show distinct edges of the εSiNW row. I denote the column of the NWs with 80 nm and 30 nm width at theboundary of the nanomembrane to be the near edge region while the second column
from the edge of the 50 nm width ε-SiNWs to be the near edge region. The middle
region of each ε-SiNW was taken at the column right after the near edge column
while the far from edge region was taken at the 5th column from membrane for the
80 nm and 30 nm width NW and 6th column from the membrane for the 50 nm width
ε-SiNWs.

Figure 4.11. Intensity (black) and Raman shift (red) line profiles of ε-SiNWs with width of (a)80 nm, (b) 50 nm and (c) 30 nm. The defects observed in Figure 4.10 were avoided when
acquiring this data.

80	
   	
  

Using the Raman shift at the three identified regions, I determined the stress profile(σ’xx) using the method outlined in Section 4.3.1. This method is usually applicable tothe ε-SiNWs located far from the edge of the nanomembrane, since thenanomembrane affects the stress distribution of the NWs near it. I assumed that thestress distribution along the x’-axis of the ε-SiNWs will be modified as the ε-SiNWs
approach the edge due to the higher stress induced through the substrate. As wasshown in the multiwavelength experiment, the interface has a large effect on the
stress in the ε-SiNWs. The stress distribution along the y’-axis, however, will remainunchanged –

(

)

(

)

(

)

σ 'yy x 'far from edge ,d = σ 'yy x 'middle region ,d = σ 'yy x 'near edge ,d – since there

is no edge present perpendicular to ε-SiNWs y’-axis.
Figure 4.12a shows the Raman shift of 30 nm, 50 nm and 80 nm width ε-SiNWs
used for stress along x’-axis calculation, with the latter shown in Fig. 4.12b. For agiven ε-SiNW width, the Raman shift decreases as one moves from the far edge
region to the near edge region. All profiles show an increase in the Raman shift
towards the center of the nanowire, indicating that a high amount of stress ismaintained at the center of the nanowire, with the exception of the Raman shiftprofile of 30 nm width ε-SiNW, which is characterized by a steady increase in the
Raman shift from the near nanomembrane side to the other edge of the ε-SiNW.
The stress behavior along the length of the nanowire (Fig. 4.12b) displays position
dependence relative to the nanomembrane, particularly in the nanowire closest to thenanomembrane except the 30 nm width ε-SiNW near the edge. The side of the εSiNW near the nanomembrane edge exhibits higher stress than the center, indicatingthat the edge has bigger effect on it than its intrinsic stress. The effect of thenanomembrane edge on the ε-SiNWs near the edge also depends on the ε-SiNW
width, as the stress along the x’-axis relaxes as the width increases. Also, as the εSiNW is farther away from the nanomembrane edge, the behavior of the stress along
the length of the wire becomes symmetric, which is consistent with themultiwavelength experiments explained in the earlier part of this section. Independentof the ε-SiNWs’ width, I found that the stress relaxes as the nanowire moves farther
away from the nanomembrane. This is is because the stress from the highly stressnanomembrane is transferred to the smaller and less stressed ε-SiNW s via thesubstrate. This effect lessens as the ε-SiNW moves away from the nanomembrane.

	
  

81	
  

Figure 4.12. Raman shift along the x’-axis of the ε-SiNWs show that the Raman shift (a)increases from the near edge to the middle region up to the far from edge. (b) The stressalong the x’-axis exhibits relaxation from the center to the going to the edge of the NW, with
NWs near the edge having higher stress compared to the middle part and far from edge
region. Gray dashed line indicates the stress on an unpatterned nanomembrane.

82	
   	
  

4.4 ANALYSIS OF STRESS II: TRANSVERSE OPTICAL PHONON SPLITTINGAPPROACH [4.2]
4.4.1 TO: Theoretical Approach
In this method, both TO phonon modes are simultaneously excited using the crosspolarization configuration by rotating the polarizer 90º. It is possible to detect thesephonon modes because of the use of the high NA objective lens. Since both phononmodes are present, it is necessary to separate the peaks of the two TO phononmodes during data analysis. This is done through double Lorentzian fitting. In thecase of nanowires, the stress is anisotropically relaxed as discussed in Chapter 3. In
this case, the degenerate doublet mode Δω xy (TO phonon modes) in Eq. (4.1) is split
into two modes, Δω TO1 and Δω TO2 , which are given by [4.26]:

( )
( )

( )
( )

( )
( )

Δω TO1 x ' = −2.88σ 'xx x ' − 0.54σ 'yy x ' , ⎫⎪⎬Δω TO2 x ' = −0.54σ 'xx x ' − 2.88σ 'yy x ' ,⎪⎭(4.12)

where the numerical values were derived using the PDPs reported in Ref. 4.27 andthe elastic compliance tensor reported in Ref. 4.28. The values for Δω TO1 and Δω TO2
can be obtained experimentally and through simple algebraic manipulation, bothstresses σ 'xx and σ 'yy can be directly solved. Eq. (4.12) was derived by Kosemura,et. al. in Ref. [4.29] and their derivation is briefly summarized in this section.
I assume a linear relationship between the force constant ΔK and strain in the
crystal coordinate system, wherein x : [100], y : [010] and z : [001]. I define ΔK as

ΔK = Aε ,

⎛ p
⎜
⎜ q
⎜
A=⎜ q
⎜ 0
⎜ 0
⎜
⎝ 0

	
  

(4.13)

q q

0

0

p q

0

0

q

0

0

0 0 2r
0 0 0
0 0 0

0
2r
0

p

0 ⎞
⎟
0 ⎟
0 ⎟,
⎟
0 ⎟
0 ⎟
⎟
2r ⎠

(4.14)83	
  where ε is a strain tensor and A is a fourth-rank tensor with PDPs p, q, and r. The
coordinate transformation for ε-SiNWs fabricated on a (001) Si in the direction of[110] (just like my sample and those found in actual transistors), use second-rank
and fourth-rank tensors to move from the crystal coordinates to sample coordinates.
In the sample coordinate system of x’ : [110], y’ : [-110], and z’ : [001]. These tensor
equations are:Tij = aik a jlTkl ,(4.15)

Tijkl = aima jnako alpTmnop ,(4.16)

where T and a are second- or fourth- rank tensors and a transformation matrix,
respectively. Therefore, Eq (4.13) becomes

ΔK ' = A' ε ',(4.17)

where the prime refer to the sample coordinate system in (x’,y’,z’). The secularequation of ΔK ' is then defined as

ΔK 'xx − λΔK 'xyΔK 'xz

ΔK 'xy

ΔK 'yy − λ

ΔK 'yz

ΔK 'xz

ΔK 'yz

ΔK 'zz − λ= 0,(4.18)

where λ is the eigenvalues. For anisotropic biaxial stress, the second-rank stresstensor is given by

⎛ σ'
0
xx
⎜
σ ' = ⎜ 0 σ 'yy
⎜
⎜⎝ 0
0

⎞
⎟
0 ⎟,
⎟
σ 'zz ⎟⎠
0

(4.19)

where σ 'xx and σ 'yy are the stress components in the [110] and [-110] directions,
respectively. As explained in Chapter 3, the presence of stress can change theoptical-phonon frequencies and also the Raman tensors. But in the special casewherein the stress tensor has only diagonal components, the Raman tensor does notchange. Therefore, the polarization selection rules for stress-free Si and Si
experience biaxial stresses σ xx and σ yy are the same. The sample coordinate
Raman tensors are given by Eq. (3.8) in Chapter 3. The stress tensors are related tothe strain tensors by Hooke’s law:

ε ' = S 'σ ',

84	
   	
  

(4.20)⎛ S⎞
S12 S12
0
0
0
11
⎜
⎟
0
0
0
⎜ S12 S11 S12⎟⎜⎟S
S12 S11000⎟,
S = ⎜ 12⎜ 0⎟
0
0 S44 / 4
0
0
⎜
⎟
0
0
0
S44 / 4
0
⎜ 0
⎟
⎜
⎟
⎜⎝ 0
0
0
0
0
S44 / 4 ⎟⎠(4.21)

where S is the elastic compliance tensor and its individual components, S11, S12, andS44 were defined earlier along with Eqs. (4.1) and (4.2). The elastic compliancetensor S is transformed to sample coordinates using Eq. (4.16) to solve for the strain
tensor ε ' . Then solving the secular equation, Eq. (4.18), the eigenvalues λ j can becalculated. The eigenvalues take the same form as Eqs. (3.10) and (3.11) in Chapter
3. I can then get expressions that relate the shift of the Raman shift to the amount ofstress present in Si. These are

λ1
2ω 0

Δω TO1 =

(

) (

)

⎡ p S11 + S12 + q S11 + 3S12 + rS44 ⎤=⎢⎥ × σ 'xx
4ω 0
⎢⎣
⎥⎦
⎡ p S11 + S12 + q S11 + 3S12 − rS44 ⎤+⎢⎥ × σ 'yy ,
4ω 0
⎢⎣
⎥⎦

(

Δω TO2 =

λ2
2ω 0

(

) (

(4.22))) ()⎡ p S11 + S12 + q S11 + 3S12 − rS44 ⎤=⎢⎥ × σ 'xx
4ω 0
⎢⎣
⎥⎦
⎡ p S11 + S12 + q S11 + 3S12 + rS44 ⎤+⎢⎥ × σ 'yy ,
4ω 0
⎢⎣
⎥⎦

(

Δω LO =

λ3
2ω 0

(

) (

) (

(4.23)))⎡ p S11 + S12 + q S11 + 3S12 + rS44 ⎤=⎢⎥ × σ 'xx + σ 'yy .
2ω 0
⎢⎣
⎥⎦

(

)

(4.24)

Solving Eqs (4.22) – (4.24) using the PDPs reported by Anastasskis, et. al. in 1990,
results to Eq. (4.12).

	
  

85	
  

4.4.2 TO: Experimental Results and Discussion()

Linear polarization in the z ' x ' y ' z ' is used. It was shown in Chapter 3 that in thispolarization configuration, TO phonon modes are predominantly detected, particularlyin the case of nanowires. To extract the TO phonon peaks, double Lorentzian fitting
was used, based on the results of the simulation suggesting that the detected signal
consists only of TO1 and TO2 (negligible LO intensity) with equal intensity due tosymmetry of the nanowire with respect to its orientation in the sample coordinate
system. This observation was supported by numerical simulations, wherein the()

TO/LO ratio was computed for varying nanowire widths using the same z ' x ' y ' z 'polarization configuration as that in experiment. From the simulations, it was foundthat the TO/LO ratios for the 80 nm, 50 nm and 30 nm nanowires are 23.56, 54.89
and 84.57, respectively. These values are very large as compared to the TO/LO ratio
for the nanomembrane, which is 0.94. This means that for nanowires, the LOcontribution is very small to the point that it is negligible. In contrast, previous work()

based on z ' x ' x ' z ' predominantly detects only LO phonon mode [4.12].

Verifying the validity of the numerical simulations. To verify the validity of thenumerical simulations, experiments were made on ε-Si nanomembranes under both

(

)

(

)

z ' x ' x ' z ' and z ' x ' y ' z ' configurations. For a nanomembrane, it is reasonable toshow that it is experiencing bi-isotropic strain. From the simulations discussed in
Chapter 3, two peaks – LO and TO – are present. The LO peak was determined()

using the LO-active configuration z ' x ' x ' z ' , ω ε −SiLO = 514.97 cm-1 as shown in Fig.
4.13a. Thus the TO frequency can be extracted to be 	
   ω ε −SiTO = 516.26 cm-1 by Eqs.
(4.1) and (4.2). In order to obtain the experimental TO/LO ratio, these LO and TO
Raman frequencies are used for double Lorentzian fitting of TO-active configurationin Fig. 4.13b. The experimental TO/LO ratio was computed to be 0.944, which is ingood agreement with the calculated value of 0.94 extracted in Chapter 3. Thus
validating the soundness of my simulations.

86	
   	
  

Figure 4.13 (a) Raman spectrum taken on the unpatterned ε-Si nanomembrane using()

z ' x ',x ' z ' configuration. Only LO Raman signal is detected which peaked at 514.97 cm-1.

()(b) Raman spectrum taken on the unpatterned ε-Si nanomembrane using z ' x ',y ' z '-1configuration. Singlet LO – peaked at 514.97 cm – and doublet TO – peaked at 516.26 cm– are detected in this configuration.

-1

Characterization of anisotropic stress relaxation through the detection of TO
phonon splitting. Figures 4.14a – 4.14c show the experimentally obtained Raman
spectra at the center of the nanowire for 80 nm, 50 nm and 30 nm widths,
respectively. These spectra were obtained under TO-active configuration wherein
the contributions from the LO peak are negligible as supported by numericalsimulations. The red vertical lines in Fig. 4.14a indicate the TO1 (515.96 cm-1) andthe TO2 (517.71 cm-1) Raman shift peak for the 80 nm width nanowire.

The

difference between the two peaks is ∆ωTO_80 =1.75 cm-1. This difference in the TO
peaks is related to the strain anisotropy in the nanowire as illustrated in Fig. 3.3c in
Chapter 3. Similarly, for the 50 nm nanowire, the blue vertical lines in Fig. 4.14b
indicate the TO1 Raman shift peak at 515.84 cm-1 and the TO2 Raman shift peak at
517.75 cm-1. The difference between the two peaks is ∆ωTO_50 = 1.91 cm-1 and isfound to be slightly higher than that of 1.75 cm-1 for the 80 nm-wide nanowire. Figure
4.14c shows the Raman spectra from the 30 nm-wide nanowire.

The signal isrelatively weaker than the Raman spectra obtained from the 80 nm and 50 nm
nanowires due to the smaller volume probed as compared to the wider nanowires.
The green vertical lines denote the TO1 (515.6 cm-1) and TO2 (518.32 cm-1) Raman
shift peak. The splitting is more apparent at this width and the difference between
the two peaks is ∆ωTO_30 = 2.72 cm-1. This indicates that the difference in the TO
peaks tend to increase with smaller widths of the nanowire. Note that this is the first

	
  

87	
  

experimental demonstration of TO1 and TO2 splitting in (001)-oriented silicon. The
observed behavior is summarized in Fig. 4.14d displaying the TO1 and TO2 Raman
shift peak as a function of the nanowire width. As a reference, the Raman peak of
the nanomembrane is also shown in this figure. Based on the experiment on theunpatterned strained silicon nanomembrane, which can be considered as an infinitely
wide nanowire, the extracted TO phonon frequency ωTO_∞ = 516.26 cm-1. I see aslight increase in the TO1 peak position (within a ±0.5 cm-1 range) as the nanowire
width increases.

In contrast, the TO2 Raman peak decreases with increasing

nanowire width. Besides, TO2 was also found to have a higher slope compared to
TO1, which suggest that TO2 is more sensitive to the changes in the nanowire
dimension.

For both TO1 and TO2, the evolution of the peak positions as the

nanowire width increases exhibits a linear but opposite trend.

It is also worth-1mentioning that the TO1 and TO2 peaks converge at 516.26 cm , which corresponds
to the Raman peak of the nanomembrane (i.e., a nanowire with an infinite width).
These results agree very well with theoretical predictions shown in Fig. 3.3b (Chapter3), where the detected TO is doubly degenerate for the nanomembrane.

Asillustrated in Fig. 3.3, this degeneracy is lifted in the case of the nanowire. From thelinear trends, one can extrapolate that at a width of 155 nm the structure issufficiently broad to possibly preserve the bi-isotropic character at the center, similar
to the original nanomembrane.

Looking at the difference between TO1 and TO2

Raman shift peaks for each nanowire width, we see that ∆ωTO_30 > ∆ωTO_50 > ∆ωTO_80.
The increase in the splitting between the TO1 and TO2 is due to the narrowing of thewidth of the nanowires and is a manifestation of the increase in the anisotropy as the
nanowire width decreases.

To the best of my knowledge, this is the first

demonstration of TO mode splitting.
Figures 4.15a – 4.15c display the TO1 and TO2 Raman shift images of the 80 nm, 50
nm, and 30 nm nanowires, respectively.

These images were obtained throughfiltering based on the integrated intensity in order to identify the nanowires (see Fig.
4.16 for the intensity profile). This visualization of the fitted Raman shifts validates
further the numerical simulations and demonstrates the successful detection of both
TO1 and TO2 phonons. To examine the behavior of these modes along a nanowire,
Figure 4.15d exhibits the profiles of the LO, TO1 and TO2 phonon frequencies along
the x’ – axis of the nanowire. Each profile is averaged over 10 different
measurements of single nanowires.

88	
   	
  

Figure 4.14. Raman spectra from the center of the nanowire with widths of a, 80 nm, b, 50
nm and c, 30 nm. Each Raman spectrum represents averaged data from 10 nanowires. The
black hollow circles represent the actual experimental data, while the solid curves represents
the fitted data. For fitting, the background signal was subtracted and it was assumed that theintensities of TO1 and TO2 are equal due to symmetry. Splitting becomes more evident as the
nanowire becomes narrower. To further support the idea that splitting is indeed occurring, the-1spectral width (∆u) of the experimental data was determined. They are ∆u80 = 5.38 cm , ∆u50
-1
-1
= 5.46 cm , and ∆u30 = 5.93 cm , for 80 nm, 50 nm and 30 nm nanowire widths, respectively.
Spectral width broadening occurs as the nanowire width decreases, which coincides with theincrease of the splitting between the TO phonon peaks. d, The Raman shift of each nanowire-1is plotted with the degenerated phonon frequency (=516.26 cm ) of the unpatterned
nanomembrane corresponding to infinity width (see Fig. 4.13b).

	
  

89	
  

Figure 4.15 The TO1 and TO2 Raman shift derived from the double Lorentzian fitting wereused to visualize the Raman distribution for a, 80 nm, b, 50 nm, and c, 30 nm width
nanowires. The areas without any nanowires present were set to be black to improve thecontrast of the image (see also the corresponding Raman intensity image in Fig. 4.16). The
scale bar is equivalent to 1 µm. The edges of the nanowires are rounded as compared to the
SEM image shown in Fig. 4.1a. This is due to the convolution of the edge of the nanowire
and the focus spot, which is probing the nanowire. d, The LO, TO1 and TO2 Raman shifts
along the x’ – axis of a nanowire is plotted and compared to those of the nanomembrane (LO:-1
-1514.97 cm , TO: 516.26 cm ) (see Fig. 4.13b). These data were averaged from 10
nanowires for each set.

90	
   	
  

The nanomembrane is used as a reference indicating the initial Raman shifts of LO
and TO phonon modes prior to patterning, which are 514.97 cm-1 and 516.26 cm-1,
respectively (Fig. 4.13b). The LO peaks at each position of the nanowires were()

obtained via single Lorentzian fitting in the LO – active z ' x ',x ' z 'polarization

setting. It can be seen that for the Raman shifts, TO2 > LO > TO1. This behavior
reflects the case of anisotropic strain illustrated in Fig. 3.3c. For TO1, the Raman
shift does not vary much as the nanowire width decreases. TO2, on the other hand,exhibits an increase in the Raman shift as the width decreases. For both cases, theRaman shift around the center of the nanowire ( x’ – position: 200 nm to 800 nm) isrelatively constant and increases towards the edges. Based on the obtained TO1 andTO2 Raman shifts – ∆ωTO1 and ∆ωTO2, respectively – the stress along the length (σ’xx)
and along the width (σ’yy) can now be directly evaluated using Eq. (4.12).
Substituting the two TO modes’ frequencies from Fig. 4.15 to Eq. (4.12), σ’xx and σ’yy
along the nanowires are determined as shown in Fig. 4.16a.

Interestingly, thisdemonstrates that stress in-plane components can be resolved using micro-Raman
scattering spectroscopy in backscattering configuration. The general trend is that thestress is higher at the center of the nanowire and decreases as it approaches theedges of the nanowire. Interestingly, and against the prevalent belief that patterning
induces strain relaxation in all directions [4.15, 4.24], we observe that the stressalong the nanowire axis, σ’xx, is higher than that of the original stress in thenanomembrane before patterning, which is around 1.2 GPa. However, the stressdecreases significantly along y’-direction, σ’yy.

The strong contraction along thesmallest dimension is attributed to a higher free surface to volume ratio along y’direction.

Fig. 4a also provides strong evidence that this contraction in the y’-direction is accompanied by a stretch along the x’-direction as expected from the
Poisson effect. This lattice expansion translates into an increase in the stress along
this direction. This subtle but important observation cannot be achieved based on
the sole detection of LO phonons. Therefore, it is of paramount importance to detectboth TO1 and TO2 to precisely characterize the stress distribution in nanowires andother nanoscale systems with anisotropic stresses. In order to probe the behavior of
the stress anisotropy, Fig. 4.16b exhibits the anisotropy ratio σ’xx / σ’yy as a functionof the nanowire width.

Expectedly, the narrower the width, the higher is theanisotropy ratio. Interestingly, the anisotropy appears to be more significant at theedge of the nanowire where the contraction along the shortest dimension is more
pronounced.

	
  This heterogeneous distribution of stress in nanowires and its

91	
  sensitivity to the nanowire width are very important for the design and fabrication ofstrained silicon-based devices as the electrical, optical and mechanical properties of
the nanowire depend on them.
Using a high NA lens and polarized Raman spectroscopy, we have realized thedetection of TO phonons, which is essential for a more comprehensive
characterization of stress in nanostructures such as nanowires. Based on numericalsimulations, I was able to infer that the LO contribution in the Raman signal isnegligible and the detected signal is mainly composed of TO1 and TO2. Based on themeasured Raman signals from strained silicon nanowires, TO mode splitting isobserved for the first time. The splitting between the modes becomes larger as the
nanowire width becomes smaller therefore showing higher anisotropic stressrelaxation for narrower nanowires. From Raman shift imaging of the TO1 and TO2
modes, the stress profiles in single nanowires are directly obtained and it was seenthat the stress along the x’ – axis is higher than the stress of the nanomembranes asa result of the strong lattice contraction along the shortest dimension. Through thedetection of the two TO modes, a more detailed stress analysis can be made which
will provide a better understanding of the interplay between stress and basic
properties and performance of strained semiconductors based nanoscale materials
and devices.

92	
   	
  

Figure 4.16 a The stress in the x’ – axis, s’xx, and y’ – axis, s’yy, along the x’ – direction of a
nanowire for varying width, as compared to the stress in a nanomembrane (1.2 GPa). The
circular, square and triangular markers are the computed stresses for the 80 nm, 50 nm and30 nm nanowire widths, respectively. The solid lines are the corresponding fitting from thecomputed stresses. b, The behavior of the anisotropy ratio as a function of the width both at
the center and at the edge of nanowires.

	
  

93	
  

4.5. Comparison of LO and TO Analysis
Using the same experimental setup described in the main text, a 6 µm x 6 µm area()

was scanned using the z ' x ',x ' z ' polarization configuration with a step size of 25
nm. At each step, a Raman spectrum was recorded using an exposure time of 500
ms and incident laser power of 5 mW prior to the microscope. Single Lorentzian
fitting was used to extract the LO Raman Shift peaks. Figures 4.17a to 4.17c show
the LO intensity images while Figs. 4.17d to 4.17f shows their corresponding LO
Raman shift images for the 80 nm, 50 nm and 30 nm width nanowires, respectively.
Filtering was used based on the LO intensity images in order to obtain the exact
position of the nanowires. It can be seen that as the nanowire width decreases, theRaman shift peak increases.

Figure 4.17. Raman intensity and Raman shift images of LO phonon mode detected by()

z ' x ',x ' z ' configuration. The Raman shift clearly increases with decreasing nanowire width.
Figure 4.18 shows the extracted stress based on the experimentally observed LO
phonon peaks. It is observed that all the stress values do not exceed the originalstress in the unpatterned nanomembrane, which indicates that stress relaxation
occurs in both directions. Comparing the stress relaxation in each direction, the y’ –axis, σ’yy, is more relaxed than the x’ – axis, σ’xx. It is also observed that, similar to
the TO data (Fig. 4.16a), the variation of σ’xx is not much as compared to that of σ’yy,

94	
   	
  

wherein the stress relaxation increases as the nanowire width decreases. This is
primarily due to the change in dimension in the y’ – axis wherein the width is varied
as compared to the constant length dimension of 1 µm in the x’ – axis. What iscommon though in both σ’xx and σ’yy is the behavior of the stress. The stress ismaintained at the center of the nanowires and quickly relaxes as it moves to theedges of the nanowires due to the formation of free surfaces. The stress relaxation
for a nanomembrane edge, which is assumed as a nanowire with semi-infinite width,
was also computed based on experimental measurement from the edge (x’ = 0) to x’= 0.5 mm. The computed σ’xx is plotted at 0 < x’ < 0.5 mm and its mirror image isplotted at 0.5 < x’ < 1.0 mm, resulting to a symmetric plot. Its σ’xx is almost similar tothat of the nanomembrane, except that there is some relaxation at the edges. This is
expected since one can imagine an infinitely wide nanowire with constant length of 1
µm to be a minute slice of the nanomembrane. Due to its infinite width, the stress at
the center of nanowire is nearly preserved, but since it has a finite length, relaxation
at the edges occurs. But compared to the relaxation of the 80 nm, 50 nm and 30 nm
nanowires, this relaxation is relatively small.
Comparing the stress calculated using LO analysis (Fig. 4.18) and TO analysis (Fig.
4.16a), the main difference is, as mentioned earlier, that the stress does not exceedthat of the nanomembrane when LO analysis is used. This is due to the initialassumption made that the nanomembrane is similar to a semi-infinite bulk material.
Although this assumption is not wrong, it limits itself to the analysis of stress in the x’direction and to structures experiencing bi-isotropic stress. TO analysis on the other
hand does not utilize any assumptions that can limit its applications. Also, the TO
analysis method is well suited for nanowire-like structures as shown in this work.
The strain was also computed using the LO method (Fig. 4.19) and it was seen thatthe derived strain values of strain are comparable to the values used in thesemiconductor industry [4.30]. This shows that micro-Raman spectroscopy is indeed
a useful technique for strain engineering.

	
  

95	
  

Figure 4.18. Experimentally extracted stress profile along nanowires with different diameters()

based on LO phonon modes using z x ',x ' z ' configuration. All the stress values are smaller
than the initial unpatterned nanomembrane stress (1.2 GPa) meaning stress relaxation occurs
in both directions.

Figure 4.19. Calculated strain experienced by the ε-SiNWs from the LO data obtained.

96	
   	
  

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[4.17] V. Poborchii, T. Tada, K. Usuda, and T. Kanayama, Appl. Phys. Lett. 99,
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[4.18] O. Moutanabbir, M. Reiche, A. Hähnel, M. Oehme, and E. Kasper, Appl. Phys.
Lett. 97, 053105 (2010).
[4.19] M.V. Balois, L.L. Dasallas, N. Hayazawa, A. Tarun, S. Kawata and O.
Moutanabbir, in preparation.
[4.20] O. Moutanabbir, M. Reiche, N. Zakharov, F. Naumann, and M. Petzold,Observation of free surface-induced bending upon nanopatterning of ultrathinstrained silicon layer, Nanotechnology 22, 045701, 2011.
[4.21] N. Hayazawa, K. Furusawa, and S. Kawata, Nanometric locking of the tightfocus for optical microscopy and tip-enhanced microscopy, Nanotechnology 23,465203, 2012.
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[4.23] T. Miyatake and G. Pezzotti, J. Appl. Phys. 110, 093511, 2011.
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52, 6247, 1995.
[4.25] M. Chandrasekhar, J.B. Renucci, and M. Cardona Phys. Rev. B 17, 1623,
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[4.26] D. Kosemura, M. Tomita, K. Usuda, and A. Ogura, Jpn. J. Appl. Phys. 51,
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[4.28] W.A. Brantley, J. Appl. Phys. 44, 534-535, 1973.
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effect devices, Taylor and Francis: New York, 2007.

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Chapter 5. Carbon Nanotubes: Theory, Properties and TipEnhanced Raman Spectroscopy
As silicon is reaching its limit to maintain the continuity of Moore’s law, chip
manufacturers have invested time and money to develop new computing
architectures and processor designs, wherein some are based on new materials[5.1]. There are a number of materials and technologies being innovated by different
chip manufacturing companies. Some of these candidates are graphene [5.2], carbon
nanotubes [5.3], memristors [5.4], and cognitive computers [5.5, 5.6]. Among thesepromising candidates, carbon nanotubes (CNTs) are of particular interest because of
their intrinsic electronic properties and ultrathin body [5.7] and also because of therecent innovation of a CNT computer developed by engineers in Stanford University[5.8]. To learn more about the characteristics of this nano-sized material whose
diameter ranges from around 0.7 – 10 nm in diameter [5.9], it is necessary to use
nano-Raman spectroscopy techniques to spatially resolve individual CNTs or
bundled CNTs. In this work, a nano-Raman spectroscopy technique called tipenhanced Raman scattering is utilized to study CNTs.
In this chapter, the theory and basic concepts of TERS are first discussed. Numericalsimulations by finite-difference time-domain calculations are presented to show theevolution of the electric field found in the gap between the metal-coated tip and asubstrate, which is either glass or a metal-coated glass substrate. Afterwards, thetheory and properties of CNTs are discussed. The Raman spectrum of CNTs are
explained and results of TERS experiments on CNTs are presented.
5.1. TIP-ENHANCED RAMAN SCATTERING (TERS)
The basic concept of TERS was proposed in 1985 by Wessel – utilizing a single
metal nanoparticle that can produce a constant field enhancement to investigate asurface [5.10]. The role of the sharp metal tip was to act as an active site to probe thesurface point by point. By integrating this probe with scanning probe microscopy(SPM) techniques, it would be possible to map and analyze the entire surface. This
concept became a reality through the development of the TERS system andexperimentally demonstrated independently by Hayazawa [5.11], Stöckle [5.12], andAnderson [5.13]. After the initial development of TERS, the spatial resolution was
improved being limited only by the diameter of the tip. Such high resolution wasdemonstrated by Hartschuh, et. al. [5.14] by imaging of single-walled CNTs using

	
  

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TERS. Aside from achieving high spatial resolution, TERS has proven itself to be a
powerful analytical tool because of its high chemical sensitivity. Owing to thissensitivity, TERS has then been used to study various samples and phenomena,
reaching even the fields of biology and chemistry. There are a number of applications
worth mentioning. The nucleobases of DNA [5.15-5.16] and RNA [5.17] have been
identified through their unique vibrational modes. Staphylococcus epidermidis
bacterium [5.18] has been chemically characterized and the different components of
its cell wall [5.19] have been identified that may help further understand the biologicalprocesses occurring on the surface of cells. Viruses like the tobacco mosaic virus
have also found an application for TERS to provide vibrational spectroscopic
information that can eventually lead to fast detection and identification of single virus
particles [5.20]. Proteins, such as cytochrome c [5.21], and peptides [5.22] have been
analyzed via TERS and TERS gave more chemical information compared to surfaceenhanced Raman scattering (SERS) and scanning tunneling microscopy (STM),
respectively. Changes in the molecular structure of single-molecule junctions fordifferent conductance states can be studied using a derivative of TERS called“fishing-mode” TERS [5.23]. Photocatalytic reactions can be monitored using timeresolved TERS [5.24], and, quite recently, mapping of catalytic activity using TERS
has been achieved [5.25]. This demonstrates the potential and versatility of TERS as
an analytical tool.
5.1.1 Electric Field Enhancement at the Tip
An enhanced electric field is generated at the tip of a metallic probe having ananometric diameter by first irradiating the tip with an electric field (e.g. laser light).
This incident electric field causes conductive free electrons to collectively oscillate at
the surface of the metal (Fig. 5.1) called surface plasmon polaritons (SPP) [5.26,
5.27]. Both electrons and positive charges are concentrated at the tip and generate astrong external electric field. The photon energy is confined at the local vicinity of thetip, making the tip work as a photon reservoir [5.27]. The local electric field serves asa nanolight source that, when positioned near a sample surface excites andenhances the Raman signal from molecules near the tip apex giving access tospectroscopic information of the sample [5.28].

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Figure 5.1. Formation of enhanced electric by local SPP at the metallic tip apex.

The strength of the enhanced electric field is dependent on the polarization of theincident beam. For tip enhancement to occur, the incident electric field must be ppolarized [5.14, 5.29-5.31]. In particular, if radially polarized light illuminates the tip, astrong electric field shall be generated. This is due to the presence of the strong
longitudinal field that couples with the tip [5.31]. Figure 5.2 shows finite-differencetime-domain (FDTD) calculations of the electric field generated at the tip apex
situated on top of a substrate for different polarizations. The tips simulated are silver(Ag, diameter = 35 nm) and gold (Au, diameter = 60 nm) coated Si tips. These tips
were chosen to simulate the tips used in actual TERS experiments. The substrates
simulated are glass and Au/Ge coated substrates. The Au has a thickness of 10 nm
while the Ge has a thickness of 1 nm. The Au/Ge substrate was used to simulate theactual substrate used in Chapter 6 of this thesis. This substrate is an ultrasmooth
substrate that is highly transparent and has no SERS background. For thesimulations, two wavelengths were used – 532 nm for Ag simulations and 633 nm forAu simulations. This is to excite the localized surface plasmon resonance on themetal surface. From the simulations, it can be seen that when s-polarization is used
(Figs. 5.2a – 5.2b and Figs 5.2g – 5.2h) the generated electric field is weak and thecentral part has minimum enhancement. This can be correlated to the profile of thelongitudinal field of linearly polarized light, simulated in Chapter 2 and Chapter 3.
When p-polarization is used (Figs. 5.2c – 5.2d and Figs. 5.2i – 5.2j), the generated
electric field is stronger compared to that of s-polarization. But when radialpolarization is used (Figs. 5.2e – 5.2f and Figs. 5.2k and 5.2l), the generated electricfield is more than 10 times higher compared to p-polarization. Therefore, in terms ofgenerating a strong electric field, radial polarization is the best choice. Comparing theuse of a glass substrate and a metal-coated substrate, it can be seen that the metal
coated substrate helps in generating a stronger electric field. The presence of themetallic thin film enables a tighter confinement around the apex.

	
  

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Figure 5.2 Electric field distribution at the tip apex illuminated by either 532 nm for Ag or 633 nm for Au coated tips. Metallic tips on glass
substrates using (a) – (b) s-polarized, (c) – (d) p-polarized and (e) – (f) radially polarized illumination. Metallic tips on Au/Ge substrate using (g)– (h) s-polarized, (i) – (j) p-polarized and (k) – (l) radially polarized illumination.

102	
  	
  

5.1.2. System Geometry
A typical TERS setup consists of laser light that enters an inverted microscope and is
focused by an objective lens (NA = 1.49) as shown in Fig. 5.3a. The metal-coated tip isthen moved to the focus point to create a localized and enhanced electric field. The useof a high numerical aperture (NA) objective lens helps generate a strong longitudinal fieldalong the tip axis that is p-polarized [5.32]. The spatial resolution is determined by thediameter of the tip, which dictates the size of the enhanced field. This kind of
configuration is called “transmission mode” [5.11, 5.12] and is widely used fortransparent samples. Another configuration is called “reflection mode” [5.13] and is used
for opaque samples (Fig. 5.3b). Although reflection mode is advantageous for opaquesamples, it has a relatively higher background signal compared to transmission mode
TERS. This is because of the low NA objective lens with long working distance that isused to illuminate the tip.

Figure 5.3 (a) Transmission mode and (b) reflection mode TERS.

5.1.3. Fabrication of Metallic Probes used for TERSThe tips used for the TERS experiments are contact mode atomic force microscopy(AFM) silicon cantilever tips (Fig. 5.4a), CSG10 (NT-MDT, force constant = 0.11 N/m).
They are baked for 10 hours to form a SiO2 layer (thickness > 200 nm) (Fig.5.4b). The
diameter of the tips are around 30 nm after baking. The tips are baked in order to: (i)blue shift the resonant wavelength, (ii) the resulting SiO2 is transparent and notabsorptive compared to silicon and (iii) moderately dulls the tip end to aid in theformation of the appropriate metal grain size during evaporation [5.33]. Usually the noble

	
  

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metals, Au or Ag, are deposited onto the oxidized tips via thermal evaproation. The
thickness to attain high enhancement varies for each metal. For Ag, typically 60 nm is
evaporated while for Au, 90 nm is evaporated (Fig. 5.4c).

Figure 5.4. SEM images of the cantilever tips in each step of the metallization process: (a)unoxidized Si tip, (b) oxidized tip, and (c) Au-coated (metallized) tip.

5.1.4 Taking Care of TERS tip
The life of a TERS experiment relies heavily on how the tip is prepared and how it ishandled during experiment. When the tip is prepared, it must be in a clean environment.
A contaminated tip cannot be used for TERS experiments because the Raman
generated by the contamination (called contamination peaks) would overpower the
Raman from the sample. During TERS experiments, the power of the illuminating beam
must be strong enough to detect Raman signal from the sample, but weak enough so as
not to damage the tip. When too strong a power is used on the tip, the tip and the metal
will melt as shown in Fig. 5.5a. Care must also be made when approaching the tip onto
the sample surface. If the tip crashes onto the sample surface, the tip end may be
broken (Fig. 5.5b). Any appreciable damage to the tip end caused by melting or
breakage will render the tip useless for the rest of the experiment. In principle, tips must
be used from within 24 hours to a few days after fabrication to ensure that the tip is still
TERS active. When storing tips, it is advisable to put them in vacuum-sealed containers
and store these containers in a vacuum cabinet. This will minimize the amount ofcontamination that can contaminate the tip.

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Figure 5.5 SEM images of damaged Au-coated SiO2 tips: (a) melted tip due to high illumination
power (~ 3 mW) and (b) broken tip.

5.2. CARBON NANOTUBES (CNTs)
Single walled CNTs are defined as graphene sheets rolled up into a seamless cylinder,
which is usually one atom thick [5.9, 5.34]. A nanotube is identified by its chiral vector Ch[5.34],

Ch = na1 + ma 2 ≡ (n,m),

(

(5.1)

)

where the paired indices n,m indicate the number of unit vectors na1 and ma 2 in the
hexagonal honeycomb lattice contained in the vector Ch. Ch makes a chiral angle θ withthe zigzag or a1 direction (Fig. 5.6).

Figure 5.6 Unrolled honeycomb lattice of a CNT. When lattice sites O and A, and lattice sites B
and B’ are connected, a nanotube is constructed. OA represents the chiral vector, Ch. OB
represents the translational vector T. The rectangle OAB 'B is the unti cell of the CNT. In this

(

) ( )

illustration, the CNT is n,m = 4,2 . [Taken from Ref. 5.9]

The diameter of the CNT, dt, is defined as [5.9]d t = L / π , L = Ch = Ch ⋅Ch = a n 2 + m2 + nm,

	
  

(5.2)

105	
  

where L is the circumferential length of the CNT.
The chiral angle θ is computed by taking the inner product of Ch and a1,cosθ ==

Ch ⋅ a1Ch a1
2n + m2 n 2 + m2 + nm

(5.3)

.

From this equation, the zigzag nanotube has a chiral angle of θ = 0° , while the armchair
nanotube has θ = 30° . There are two kinds of CNTs depending on their electronic
properties. These are metallic or semiconducting CNTs based on their on dt and θ .
5.2.1 The Raman Spectrum of CNTs
The Raman spectrum of CNTs can give valuable information about the electronic andstructural state of the CNT. Figure 5.7 shows a typical micro-Raman spectrum of asemiconducting CNT. There are 4 distinct Raman peaks and these are the G band, 2D
band, D band and radial breathing mode (RBM).

Figure 5.7 Typical micro-Raman spectrum from a semiconducting CNT, the G-band, D-band, 2Dband and RBM (RBM’ is for anti-Stokes) are identified.

The G band is related to the longitudinal optical (LO) phonon mode whose peak occurs
around 1580 cm-1 [5.35]. This is related to the tangential vibration mode of the C atoms
and is useful for 1) diameter characterization, 2) distinguishing between metallic andsemiconducting CNTs through their differences in Raman lineshapes, 3) probing charge

106	
  	
  

transfer from doping of CNT and 4) studying Raman selection rule for various scattering
processes and geometries [5.36]. The 2D band arises from a two-photon, intervalley,second-order Raman scattering process. Its peak position is dependent on the excitation
laser used. In the case of Fig. 5.7, a 532 nm laser was used. The intensity of the 2Dband determines the quality of the CNT. If the CNT is of high-quality, the 2D band’s
intensity is comparable to that of the G band. The D band is induced by disorders in the
carbon. It indicates atomic defects within the layer of carbon. The relative intensity of D
to G bands can determine the CNT purity [5.35]. While changes in the D and 2D bands
can be used to probe structural modifications of the nanotube sidewalls due to defects or
attachment of chemical species onto the walls [5.36]. The RBM is useful to determine the()

dt via its RBM frequency ω RBM . It is also possible to extract the temperature of the CNT
with knowledge of the Stokes and anti-Stokes intensities [5.28]. Further discussion on
this aspect of the RBM can be found in Chapter 6 of this thesis. From the dt, the chiralityof the CNT being studied can also be determined. The intensity of the RBM can also giveinformation regarding the electronic structure of the CNT. Along with the Kataura plot
[5.37], the RBM can identify whether the CNT is metallic or semiconducting. Any
structural changes in the CNT can have implications on the electronic properties of the
CNT. Through the use of Raman spectroscopy, it is possible detect such changes.
5.2.2 CNT and TERSFrom Section 5.2.1, using micro-Raman spectroscopy, it is possible to detect the Raman
spectra from CNTs. But since micro-Raman spectroscopy is diffraction limited, it is notpossible to detect whether the signal is coming from a single CNT or a group of CNTs
spaced close together. Since CNTs are nanometer in size, they cannot be seen via a
conventional microscope. In order to resolve this nano-sized material, TERS isnecessary. Figure 5.8a shows the far-field image obtained via micro-Raman
spectroscopy. The bright area indicates the presence of CNTs (semiconducting in thiscase), but no clear structure of the CNTs can be seen. Figure 5.8b and 5.8c show the
TERS and topography image of the same area shown in Fig. 5.8a, respectively. Both
TERs and topography images were taken simultaneously during experiment. The whitehorizontal lines on the right side of the topography image (Fig. 5.8c) are artifacts from themeasurement. From the TERS image, the semiconducting CNT can be clearly seen.
Even the bending of the tube is resolved and shows good correspondence with the
topography image. Figure 5.9 shows both the far-field (gray line) and the TERS (red line)
spectra taken at the same point A with the same power (150 µW) and exposure time (10

	
  

107	
  

s). It can be clearly seen that TERS can enhance the signal from the sample. The
contrast to determine the enhancement is defined as [5.38]⎛I⎞
I
Contrast = near = ⎜ total − 1⎟
Ifar
⎝ Ifar⎠(5.4)

where Inear is the intensity of the near-field signal, Ifar is the intensity of the far-field signal
and Itotal is the measured intensity during experiment, which is the sum of the near-field
and far-field Raman intensities. The enhancement factor (EF) is defined as [5.27]⎛ d2 ⎞
EF = Contrast × ⎜ focus,
2 ⎟⎝ d tip ⎠(5.5)

where d focus and d tip are the diameters of the focus and tip, respectively. For the G-band,the contrast is around 12. The diameter of the focus using a 532 nm laser and a NA =1.49 objective lens is 435.6 nm. The diameter of the tip is 35 nm. Therefore, the EF for
the G-band is 1760. With this high enhancement and high spatial resolution, TERS canprovide information on the local changes in structure of CNTs which can be useful inanalyzing the CNTs’ electronic properties.

Figure 5.8 (a) Far-field, (b) near-field (TERS) and (c) topography image of bundled
semiconducting CNTs. Scale bar indicates 200 nm.

108	
  	
  

Figure 5.9 Far field (gray line) and TERS (red line) spectra taken at point A of Fig. 5.8b. Far-field
is when the tip is retracted from the sample. TERS is when the tip has approached the sample.

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Rodriguez V, Bonhommeau (2014) Tip-enhanced Raman spectroscopy of combed
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strands: towards a novel direct-sequencing method. Angew Chem Int Ed 47:16581661[5.18] Neugebauer U, Schmid U, Baumann K, Ziebuhr W, Kozitskaya S, Deckert V,
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catalytic activity using tip-enhanced Raman spectroscopy. Nanoscale 7:7133-7137[5.26] N. Hayazawa, A. Tarun, A. Taguchi, and S. Kawata, Development of tip-enhancednear-field optical spectroscopy and microscopy, Jpn. J. Appl. Phys. 48, 08JA02,
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[5.27] Kawata S, Shalaev VM (2007) Tip enhancement. Elsevier, The Netherlands[5.28] M.V. Balois, N. Hayazawa, F.C. Catalan, S. Kawata, T. Yano, and T. Hayashi, Tipenhanced THz Raman spectroscopy for local temperature determination at thenanoscale, Analytical and Bioanalytical Chemistry, accepted, Vol. xx, pp. xx-xx,
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112	
  	
  

Chapter 6. Carbon Nanotubes: Tip-Enhanced TeraHertz Raman
Scattering, a nano-Raman spectroscopy study [6.1]One consequence of the enhancement brought about by TERS is the increase intemperature, particularly in the vicinity of the tip apex [6.2-6.3]. The joule heating
generated at the tip apex is caused by the absorption of light – represented by a value
proportional to the imaginary part of the dielectric function of the tip. Temperature plays a
key role when studying physicochemical properties of materials, especially thermallysensitive ones. Variations in the temperature of the material and its environment mayalter its molecular structure, chemical composition or induce thermal damage. Therefore,
it is important to know the temperature on the sample for a certain amount of laser power
used during TERS experiments. There are only several works reported so far on thetemperature issue in TERS. Downes et al. analytically predicted the increase of the local
temperature at the tip apex based on the finite element method [6.2], in which a large tipenhancement effect induces a temperature rise over 3 orders of magnitude higher than
without the tip. Zenobi’s group reported an experimental verification of local heating at
the tip apex by observing the morphological change of metallic nano-grains even at low
temperatures (less than 100 °C) and the subsequent loss of the tip-enhancement [6.3].
This observation, however, was an indirect method and the exact temperature was notknown. Sokolov’s group reported on the direct determination of the local temperature
based on the Boltzmann distribution of the anti-Stokes/Stokes Raman intensity ratio[6.4]. The extracted temperature had a minimum value of 100 °C, but they used a
relatively thick film as a sample (thickness ~ 50 nm) making the extracted temperature
an average over the sample volume. The detection sensitivity was also quite low due tothe high energy of their observed Raman shift (1000 cm-1 ~ 1600 cm-1) resulting in
extremely weak anti-Stokes signal. In addition, the difference of detection efficiency forboth anti-Stokes and Stokes frequencies was not considered in their calculation. In 2010,
Kawata’s group reported another approach to determine the temperature at a metalcoated silicon cantilever tip [6.5]. The temperature was determined by the shift of theRaman peak originating from the silicon cantilever [6.6]. Although this method is useful, it
is limited to silicon cantilever based tips and is not applicable to recently reported
oxidized tips for higher tip-enhancement [6.7]. Moreover, the detectable temperature wasrather high (greater than 100 ºC) because the spectral shift at low temperatures (lessthan 100 ºC) was too weak to be detected. In this thesis chapter, all the issues of theprevious works are circumvented. The proposed approach is based on the direct
determination of the temperature extracted by the Boltzmann distribution of

	
  

113	
  

simultaneously observed anti-Stokes and Stokes Raman scattering in the terahertz(THz) Raman frequency range (less than 200 cm-1). Let us focus on the low frequency
feature of the radial breathing mode (RBM) of single walled carbon nanotubes (CNTs) bywhich the difference of detection efficiency, tip-enhancement, and resonant Raman
efficiency at anti-Stokes and Stokes frequencies could be negligible due to the small
wavelength gap between the two frequencies. Furthermore, the intensity of anti-Stokes
Raman scattering is sufficiently high to be even comparable to Stokes Raman scattering
because of the low energy of THz Raman modes. Here the local temperature of thesample located underneath the tip apex at a low temperature range (less than 100 ºC) is
determined and showed that the temperature was controlled by the incident laser powerwith high sensitivity.
6.1

TEMPERATURE

DETERMINATION

BASED

ON

THE

BOTLZMANN

DISTRIBUTION
The temperature at the sample is related to the ratio of the intensities of the anti-Stokes(IAS) and Stokes (IS) Raman signal according to the Boltzmann distribution defined as[6.8]
4⎡ −hcω R ⎤IAS ⎛ ω 0 + ω R ⎞
=⎜
exp
⎢
⎥
IS ⎝ ω 0 − ω R ⎟⎠⎣ kBT ⎦

(6.1)

where ω0 is the laser excitation frequency, ωR is the Raman scattering frequency, h is
Planck’s constant, c is the speed of light, kB is Boltzmann’s constant and T is thetemperature at the sample. The factor raised to the fourth power corrects the frequency
dependent scattering cross section. However, Eq. 6.1 is only valid if IAS and IS aremeasured directly. For experiments involving photon counting using a charge couple
device (CCD) as a detector, a conversion factor is needed. This conversion factor is the
Planck relationship that relates the photon count rates to the intensities and is defined as[6.9]

E = hω

(6.2)

(

)

IAS ∝ PAS = n! AS h ω 0 + ω R ,⎫⎪⎬IS ∝ PS = n!S h ω 0 + ω R . ⎪⎭

(

)

(6.3)

In Eq. 6.3, PAS and PS are the power propagated by the scattered light (both anti-Stokes
and Stokes), while n! AS and n!S are the photon count rates. Substituting Eq. 6.3 into Eq.
6.1 results into the more applicable form

114	
  	
  

3

⎡ −hcω R ⎤n! AS ⎛ ω 0 + ω R ⎞
=⎜
exp ⎢
⎥.
⎟
n!S ⎝ ω 0 − ω R ⎠⎣ kBT ⎦

(6.4)

Henceforth, the intensity being described is the intensity based on the photon countrates, n! AS and n!S .

Figure 6.1 shows the behavior of the n! AS and n!S ratio ( n! AS / n!S ) as a function oftemperature up to around 100 ˚C (373.15 K) based on Eq. 6.4 for different ωR. As the ωR
increases, the change of n! AS / n!S as a function of temperature also increases; however,n! AS becomes extremely weak, compared to n!S , which makes it nearly impossible toexperimentally detect the signal at the given low temperature range. In this sense, the Dand the G-bands of CNTs are not good choices for temperature calculation. Similarly, as
the Raman shift gets lower than 100 cm-1, n! AS becomes higher and even comparable ton!S ; however, the change of n! AS / n!S becomes very small and is experimentally difficult to
detect. In contrast, the Raman shift between 150 ~ 200 cm-1, which is the typicalfrequency range of RBM, shows comparable intensity while exhibiting more than 30% of
the change in n! AS / n!S within the plotted temperature. This signal level and the change in

n! AS / n!S make RBM detection sensitive enough to experimentally obtain the local
temperature in the probed volume. Previous works have shown the temperaturedependence of the RBM frequency [6.10-6.11], but these measurements have only been
done in the far-field. It is advantageous to use the RBM to determine in situ thetemperature of the CNTs during TERS measurements because both the anti-Stokes andthe Stokes can be measured simultaneously with high spectral resolution. Strictly
speaking, the experimentally detected n! AS and n!S are affected by the detection
efficiency of the detector such as diffraction efficiency of the grating and camera
sensitivity. However, owing to the THz feature of RBM, the wavelength gap between
anti-Stokes and Stokes is 10 nm – for the case of 532 nm excitation with 180 cm-1
Raman shift – which makes the difference of the detection efficiency negligible.
Moreover, the possible frequency dependent tip-enhancement effect and resonant
Raman condition could be negligible.

	
  

115	
  

Figure 6.1. Calculated between anti-Stokes/Stokes intensity ratio as a function of temperature at-1
various Raman shifts. ω0 is set to 15802.8 cm corresponding to 632.8 nm.

6.2 SAMPLE PREPARATION
6.2.1. Au/Ge substrate
In TERS, a metallic gap induces much higher electric field enhancement [28-30], which
would result to higher temperature rise. Among the metallic substrates, on whichmolecules are adsorbed, gold is one of the preferable choices for its high chemical
stability. In order to apply transmission configuration for the illumination described below,
an ultrathin and smooth Au (gold) / Ge (germanium) coated cover glass is used as a
substrate [6.12], which is similar to the concept of a single crystal Au nanoplate [6.13].
First, the Ge layer with a thickness of 1 nm was deposited onto a cover glass in athermal evaporator (K-Science Inc. KS-807RK) at an evaporation rate of 0.1 Å/s.
Afterwards, the Au layer with a thickness of 10 nm was evaporated at a rate of 0.2 Å/s.
The Ge layer acted as a wetting layer that improved the adhesion of Au onto the glass
surface, resulting to a smooth Au film.

116	
  	
  

(a)

(b)

Transmission [%](c) 80

Au
Au/Ge

70
60
50
40

400

600
800
Wavelength [nm]1000

Figure 6.2. Scanning electron microscope images (SEM) of (a) Au and (b) Au/Ge substrates. The
scale bars indicate 200 nm length. (c) Transmission spectra of Au (red line) and Au/Ge (blue line)
substrates.

Figures 6.2(a) and 6.2(b) show the scanning electron microscope images of the surfacesof an Au thin film (thickness = 10 nm) and an Au/Ge thin film (Au thickness = 10 nm / Ge
thickness = 1 nm), respectively. Without the Ge layer (Fig. 6.2(a)), gaps across the Au
film were observed. These gaps are potential sources of hot spots that can generate
SERS signals and affect the TERS measurements. On the other hand, when the Gelayer was introduced, a smooth and almost gap-free Au film was fabricated (Fig. 6.2(b)).
This Au/Ge substrate had no SERS effect, which was also confirmed by dark field imageof the substrate [6.12]. The change in surface morphology also reflected in the films’
optical properties. Fig. 6.2(c) shows the transmission spectra of the Au (red line) and the
Au/Ge (blue line) thin films. It should be noted that the transmission of Ge film wasseparately measured and was subtracted from the Au/Ge spectrum. While the Au/Ge
thin film reflected the dielectric function of gold and exhibited a broad transmission fromvisible to near infrared wavelengths, the Au thin film showed a dip in the transmission
from 550 ~ 800 nm caused by absorption due to local surface plasmon polariton
excitation. Thus, the high transmission of the SERS free Au/Ge film makes it a goodsubstrate for TERS in the transmission configuration and can be used for biological[6.13] and self-assembled monolayers (SAMs) samples.

	
  

117	
  

6.2.2. Carbon nanotube on substrate preparation
The samples used in this study are CNTs made via arc-discharge method (Meijo Nano
Carbon Co., Ltd), in which the diameter distribution is centered at 1.4 nm. Based on the
Kataura plot [6.14], semiconducting CNTs (s-CNTs) are in resonance with 532 nm
excitation and metallic CNTs (m-CNTs) are in resonance with 632.8 nm excitation.
Accordingly, two separate samples were made: one containing only s-CNTs and theother containing only m-CNTs. Both samples were prepared in a similar manner. An
amount of 0.1 mg of CNTs was dissolved in 10 mL of dichloroethane through sonication.
From this solution, 50 µL was taken and spin coated on an Au/Ge thin film on coverglass following a sequence of 500 rpm for 10 s and 3000 rpm for 30 s. The sample wasplaced in a vacuum (0.13 KPa) drying oven and baked at 150 ˚C for 1.5 hours tominimize carbon-based contamination. Baking also ensured the evaporation of residuals,
which is essential to affix the CNTs onto the substrate surface and prevent dragging
during contact-mode AFM scanning.
6.3.

EXPERIMENTAL

SETUP

FOR

TIP-ENHANCEDTERAHERTZ

RAMAN

SCATTERING
I have recently developed a TERS system to detect THz-Raman modes that have beendifficult to observe by conventional edge filters due to the filters’ cutoff frequency. In thiswork, tip-enhanced THz-Raman spectroscopy (TE-THzRS) is explored as a means todetermine the local temperature of a sample directly underneath the tip apex. The
temperature of CNTs is determined through the detection of the RBM using low laser
power on the sample. The experimental setup of the TE-THzRS system (Figure 6.3(a))
consists of an inverted optical microscope (NIKON, TE2000) equipped with an XY-PZT
stage for stage scanning the sample and an atomic force microscope (AFM) head forcontrolling the a metal coated cantilever tip. The beam from either a continuous-wave
He-Ne (λ = 632.8 nm) or solid-state (λ = 532 nm) laser is introduced into the polarization
convertor (ARCoptix) to generate radially polarized laser light for efficient tipenhancement effect [6.14] and focused onto the sample via a high numerical aperture(NA) oil-immersion objective lens (NA = 1.49, ×100). The scattered light from the tip andthe sample is collected using the same objective lens. To detect both the Stokes and theanti-Stokes Raman signals from the sample, the scattered light passes through two
notch filters (OptiGrate, BragGrateTM Notch Filter) that block the strong Rayleighcomponent. The beam is then guided to a spectrometer and detected by either a cooledcharge coupled device (CCD) or by a single-photon counting avalanche photodiode

118	
  	
  

(APD).

Spectroscopic

measurements

aredetected

via

CCD.

Opticalimagingmeasurements are detected via APD while simultaneously obtaining topographic
images. Thermally oxidized silicon cantilever tips (NTMDT, CSG10) [6.8] are coated witheither Au (thickness = 90 nm) or Ag (silver) (thickness = 60 nm) by evaporation. The
consequent diameters of the tips after evaporation are typically 60 nm for Au (Fig. 6.3(b))
and 35 nm for Ag (Fig. 6.3(c)). These Au- and Ag-coated tips are experimentallyconfirmed to show sufficiently high reproducibility and tip-enhancement for 632.8 nm and532 nm excitation, respectively.

(a)

Ag or Au coated tip
SampleObj.Lens(N.A. 1.49, x100)

Distance control
Stage scan (XY-PZT)
notch filter

Excitation（Radial Pol.）532nm or 633nm pinhole

(b)

AFM
Controller

APD

Imaging

Spec.

Spectroscopy

PC

CCD

(c)

Figure 6.3. (a) Schematic of TE-THzRS experimental setup. Scanning electron microscope
images of (b) Au- and (c) Ag-coated SiO2 cantilever tips. Scale bars indicate 100 nm length.

	
  

119	
  

6.4. LOCAL TEMPERATURE DETECTION OF CARBON NANOTUBE SAMPLES BY
TE-THzRS
Figures 6.4(a) and 6.4(b) show the experimental results using 632.8 nm illumination on
an m-CNT sample. The laser power illuminating the sample is 50 µW. Fig. 6.4(a) is the
tip-enhanced Raman scattering (TERS) image resulting from the detection of the G-band(ωG-632.8 = 1580.05 cm-1). Fig. 6.4(b) is the corresponding topography image obtained
simultaneously with the TERS image. Based on the topographical height, the observed
m-CNT sample is a bundle consisting of several individual m-CNTs. In both images, theintersection of two m-CNTs is clearly visualized. It should be pointed out that the width ofthe m-CNT in topography image is wider than the one in TERS image. This is attributed
to the large grain diameter of the Au-tip apex (see Fig. 6.3(b)). Fig. 6.4(c) and 6.4(d)show, respectively, the TERS image of G-band (ωG-532 = 1591.21 cm-1) and thecorresponding topography image of s-CNTs excited by 532 nm with the power of 36.7
µW. With this power, the photon density (and the resultant electric field) at the tight focus
is comparable with the 632.8 nm measurement. The area with two parallel s-CNTs isclearly resolved beyond the diffraction limit in both images. Due to the relatively smaller
diameter of the Ag-tip (see Fig. 6.3(c)) and the high tip-enhancement factor, the width of
the TERS image is analogous to the topography image. It should be noted thatregardless of the different widths of the topography images in Fig. 6.4(b) and 6.4(d), theheight is still comparable, meaning that both the observed m-CNTs and s-CNTs arebundles comprising several CNTs. For TE-THzRS spectroscopy of RBM, an area withstrong TERS intensity in Fig. 6.4(a) was chosen. Fig. 6.4(e) shows the TE-THzRS
spectrum (red line) and the far-field spectrum (blue line) from Point A in Fig. 6.4(a). The
corresponding topography height is around 6 nm. To obtain the anti-Stokes and Stokes
intensities, single Lorentzian fitting is made for the TE-THzRS spectrum from which the
far-field spectrum is subtracted. The calculated intensities are taken to be thebackground free integrated intensities of the fitted Lorentzian curves. The RBM
frequency, ωRBM-632.8, is 187.70 cm-1. The calculated n! AS / n!S is 0.44. Using Eq. (1),wherein the Raman scattering frequency of interest is the RBM frequency, the sample
temperature is calculated to be 33.7 ˚C. The rise in temperature with respect to room
temperature (17.0 ˚C) is ∆T632.8 = 16.7 ˚C. Fig. 6.4(f) shows the TE-THzRS spectrum (redline) and the far-field spectrum (blue line) from Point B in Fig. 6.4(c). The corresponding
topography height is also around 6 nm. The RBM frequency, ωRBM-532 is 173.83 cm-1. The

n! AS / n!S ratio is 0.48 so that the derived sample temperature is 51.6 ˚C. The rise intemperature from the room temperature is ∆T532 = 34.6 ˚C.

120	
  	
  

Figure6.4. (a) TERS image of G-band from m-CNT using 632.8 nm illumination. (b)Corresponding topography image of m-CNT simultaneously acquired with the TERS image. (c)TERS image of s-CNT using 532 nm illumination. (d) Corresponding topography image of s-CNT
also simultaneously acquired with the TERS image. The scale bars indicate 200 nm length. (e)TE-THzRS Raman spectrum of RBM and (f) TERS G and D band Raman spectra from Point A ofm-CNT (Fig. 6.4(a)). (g) TE-THzRS Raman spectrum of RBM and (h) TERS G and D band
Raman spectra from point B of s-CNT (Fig. 6.4(c)). Near-field data (TE-THzRS and TERS) areboth indicated by red lines. Raman spectrum without the tip is also plotted in each figure (bluelines).

	
  

121	
  

Although the expected number of CNTs in both cases is similar based on the
topography, the observed TERS and TE-THzRS intensity in 632.8 nm are quite weaker
than the one in 532 nm. There could be two reasons for this. First, the m-CNT has aweaker Rayleigh scattering efficiency (632.8 nm illumination) as compared to the s-CNT(532 nm illumination). Second, from the Kataura plot [6.14], the m-CNTs of 1.4 nm indiameter are not in perfect resonance with the 632.8 nm laser whereas the s-CNTs of
1.4 nm in diameter exhibit a good overlap with the 532 nm laser, leading to strong
resonance Raman efficiency. This can be confirmed by the observed RBM frequencies.
The diameter of the CNT was determined using the relation [6.16]:

ωRBM = A / dt + B(2)

where dt is the diameter of the CNT, A = 234 cm-1 and B = 10 cm-1. In the case of ωRBM532 = 173.83 cm-1

dt is 1.43 nm while ωRBM-632.8 = 187.7 cm-1 gives dt = 1.32 nm, which isquite smaller than the expected diameter centered at 1.4 nm. Based on the Kataura plot,
at around 1.4 nm diameter using 632.8 nm excitation, m-CNTs with smaller diameter (~1.3 nm) are in resonance condition. Hence, only a few m-CNTs having such small
diameters contribute to the observed TERS and TE-THzRS intensity of 632.8 nm
resulting in a weaker signal compared to the 532 nm on s-CNTs.
Regardless of the different resonant Raman condition for 632.8 nm and 532 nm, theexperimentally obtained temperature rise of the m-CNT sample (∆T632.8) was
considerably lower than the temperature rise of the s-CNT sample (∆T532) under thecomparable photon density at the tight focus. In order to further investigate the differencein the temperature rise in both excitation wavelengths, analytical calculations have been
conducted and are shown in the next section.
6.5 Comparison and Discussion with Analytical Results
I carried out electromagnetic and thermal simulations of TERS using a finite element
software package (COMSOL Multiphysics) to predict the temperature rise at different
particle sizes and gap distances. In order to mimic and simplify the real experimental
system, I assumed the metallic tip as a metallic particle (Au for 632.8 nm and Ag for 532nm) with a diameter comparable to the size observed in the SEM images (Figs. 6.2(b) –6.2(c)) and modeled the Au substrate as an Au particle for both excitation wavelengths in
the analytical calculation. The initial phase of the analysis consisted of high frequency
electromagnetic calculations to obtain the Joule heat generation caused by theabsorption of light, which is proportional to the imaginary part of the dielectric function

122	
  	
  

and the enhanced electromagnetic field between the particles. The Joule heat generated
was then used in the steady state thermal simulation to predict the temperature at the
gap using conduction and radiation heat transfer. The initial temperature in thecalculation was set to 20 ºC (293.15 K). The complex permittivity of Au and Ag wereobtained from literature [6.17]. Also the thermal constants such as conductivity and heat
capacity, and density of the materials used in the simulation were taken from the
COMSOL library.

400

D=60nmAu

g=5 nm250
200

E2/E02

k

300

Au

100

350

E2/E02

E0

50

0

14
12
ΔT [ºC]

Au
Au

10
8

6
4

50nm

2

140
120
100
80
60
40
20
0

40
30
20
10
0

1

2

3 4 5 6gap(g) [nm]

7

(f)

1

2

3 4 5 6gap(g) [nm]

7

1

2

3 4 5 6gap(g) [nm]

7

250
200

ΔT [ºC]

(d)

k

60

20

(b)
E0

80

100

x103
50

D=70 nm
D=60 nm
D=50 nm
D=40 nm
D=30 nm40150E0=42300 V/m
λinc=632.8 nm 50nm

(e)

x103

E2/E02(c)

ΔT [ºC]

(a)

150
100
50

1

2

3 4 5 6gap(g) [nm]70Figure 6.5. Calculated distributions of (a) electric field intensity enhancement and (b) temperaturerise at Au-Au particles’ gap. The diameter of particle and the gap are set to 60 nm and 5 nm,
respectively. (c) and (d) are intensity enhancement and temperature rise at the Au-Au gap as afunction of gap distance with various particle diameter for 632.8 nm excitation. (e) and (f) are thecases of Ag-Au gap for 532 nm excitation.

A p-polarized 632.8 nm and 532 nm plane wave with electric field magnitude of 4.23 x
105 V/m and 4.32 x 105 V/m (equivalent to optical powers of 50 µW and 36.7 µW used in
Figs. 6.4(e) and 6.4(f)), respectively, and a propagation vector k parallel to the x-axis
were used. Figures 6.5(a) and 6.5(b) show the electric field intensity and the temperature
distribution around the gold particle gap using 632.8 nm light irradiation. The enhanced
electric field localized at the gap in Fig. 6.5(a) induced the temperature rise of theparticles and exhibited 15 ˚C temperature increase at the gap particle surface. The
surrounding media (including the gap) was modeled as air in the calculation. When a

	
  

123	
  

carbon nanotube with high thermal conductivity [6.18] filled the gap, however, theequilibrium temperature at the gap slightly decreased, although, the general trend of thedependence on gap and particle diameter was unchanged (data not shown). The
experimental value is comparable to that of simulation with d = 60 nm but is less than thesimulated value for d = 70 nm, which is reasonable since the diameter of the Au particle
on the tip is within this range. The enhancement factor and the temperature rise aresummarized in Fig. 6.5(c)-6.5(d) for 632.8 nm illumination and Fig. 6.5(e)-6.5(f) for 532
nm illumination. It was found that both the enhancement factor and the temperature rise
were sensitive to the particle diameter and to the gap distance equivalent to the diameterof the observed carbon nanotubes. As a general trend, the higher the enhancement
factor is, the higher the temperature becomes, which can be seen either when the gap
becomes smaller or when the particle size gets larger. Moreover, Ag particles in 532 nm
excitation showed relatively higher temperature as compared with Au particles in 632.8
nm owing to the higher enhancement of the Ag particle and the higher absorption of theAu particle at 532 nm. This trend is in good agreement with the experimentally obtained
temperature rise in both excitation wavelengths. When the gap becomes larger, theenhancement factor is drastically decreased whilst the temperature rise shows moderatedecrease. For both enhancement factor and temperature rise, the values converged tothe case of a single isolated particle (gap = infinity), which is also similar to the case of spolarized excitation for the gap (data not shown). Although it is practically difficult tocontrol the gap and the size of the tip diameter in the experiment, the experimentallyobtained temperature in this study is reasonably within the analytically expected
temperature under the same excitation power. The relatively lower temperature in theexperiment could originate from two factors: 1) the high thermal conductivity of CNTs
located at the gap and 2) the higher thermal conductivity of the flat metallic substratecompared to metallic nanoparticles [6.18], both of which improve the heat dissipation of
the experiment compared to the simulation.
When the incident laser power becomes very high, the tip enhancement factor can be
reduced due to the temperature dependent Drude function [6.5], resulting in a nonlinear
dependence of the temperature rise as a function of incident power density [6.20]. Whenthe incident power is low, as used in our experiment, the temperature rise could benearly proportional to the incident power. Power dependence experiments wereconducted to see the effect of incident power on the temperature rise. Figure 6.6
compares the TE-THzRS spectra (far-field contribution are subtracted) of the s-CNT
RBM taken at the same point (Point C in Fig. 6.4(c)) on the s-CNT sample, but using

124	
  	
  

three different incident powers, 50 µW (red line), 36.7 µW (blue line) and 25 µW (greenline). Single Lorentzian fitting was used for both cases.

Figure 6.6. Comparison of TE-THzRS Raman spectra taken at point C of s-CNT for different
incident powers. Solid lines indicate experimentally obtained data and dashed lines indicate fitting
results.

Figure 6.7. Plot of temperature and change in temperature versus power. Left-axis shows thecalculated temperature while the right-axis shows the change in temperature.

For 50 µW, ωRBM-50 was 172.87 cm-1. The n! AS / n!S is 0.52 and the computed sample
temperature is 90.8 ˚C. The temperature rise was ∆T50 = 73.8 ˚C. For 36.7 µW, ωRBM-36.7
was 177.8 cm-1. The n! AS / n!S is 0.49 and temperature is 59.0 ˚C. The temperature rise

	
  

125	
  

was ∆T36.7 = 42.0 ˚C. For 25 µW, ωRBM-25 was 166.98 cm-1. The n! AS / n!S ratio was 0.46,
yielding a lower sample temperature of 33.8 ˚C. The temperature rise was ∆T25 = 16.8
˚C. Figure 6.7 summarizes these results displaying temperature and the change intemperature as a function of laser power. The change in temperature exhibits a linear
relationship with the incident laser power, which is within the expectation when low laser
power is used. It is observed that the temperature rise from 25 µW to 50 µW is quite
large. One reason for the large temperature rise is a decrease in the gap distance andan increase in the enhancement of the electric field at the gap. A decrease in the gap
distance can be brought about by drift. Although our TE-THzRS system is stable, a small
amount of drift due to slight changes in the experimental environment (e.g. room
temperature and mechanical vibrations) is unavoidable. Hence, small drift changed theobserved positions with different gaps during the TE-THzRS measurements. An
indication of this was that the obtained RBM frequencies for the 50 µW and 25 µW
experiments were different. Temperature dependent RBM shift [6.21] is typically too
small to exhibit such a large spectral shift observed between ωRBM-50 and ωRBM-25. The
dependence of the enhancement factor on the gap distance is illustrated in the FEM
calculations (Fig. 6.5(e)). Even just a 1 nm decrease in the gap distance can result to a
two-fold increase in the enhancement factor. Accordingly, the higher enhanced signal
due to the smaller gap increases the temperature. A further validation of the decrease in
the gap distance for the case of 50 µW is the higher contrast between TE-THzRS andfar-field spectra (a contrast of 6-fold and 3-fold for 50 µW and 25 µW, respectively. Data
not shown), indicating that the enhancement was higher for the 50 µW data. The same
experiments were conducted using the 632.8 nm illumination and m-CNT sample, andshowed similar trends.
By developing TE-THzRS to see low frequency Raman modes in the nanoscale it ispossible to determine the local sample temperature of s-CNT and m-CNT through thesimultaneous detection of the anti-Stokes and Stokes Raman peaks of the RBM. FEM
calculations showed that the temperature rise at the gap was dependent on the gap
distance. This was because the gap distance dictated how high the enhancement factor
would be. A higher enhancement factor meant a stronger electric field in the gap, thus
resulting to a higher rise in the temperature. Experimental results were within theanalytically predicted results from the FEM calculations. Through power dependence
experiments, it was shown that the temperature on the sample for nanoscale sizedvolumes can be controlled well below 100 ºC by changing the incident power while
keeping sufficiently high enhancement factor. The findings of this work are applicable to

126	
  	
  

other samples of interest, particularly biological samples that are prone to thermal
damage. This work assures that TERS and TE-THzRS are one of the promisingbioanalytical techniques for studying heat sensitive materials [6.22] and the various
intermolecular and intramolecular interactions that occur in biological samples [6.236.24].
6.6. TE-THzRS AND CARBON NANOTUBES: A POSSIBLE NEXT STEP
A possible next step for utilizing TE-THzRS to study carbon nanotubes is the observationof ultra-low frequency modes whose frequencies are less than 100 cm-1, as seen in Figs.
6.4(e) and 6.4(g). These peaks can be indicative of inter-tube interaction. It has beentheoretically predicted that the E2g mode of CNTs has low frequency peaks (around 20
cm-1) [6.25, 6.26]. But this low frequency E2g mode is greatly dependent on the chiralityof the CNT [6.26]. Through TE-THzRS, thorough studies of these ultralow frequency
peaks can be made and more knowledge about the properties of CNTs can be gathered.

	
  

127	
  

6.7. REFERENCES[6.1] M.V. Balois, N. Hayazawa, F.C. Catalan, S. Kawata, T. Yano, and T. Hayashi, Tipenhanced THz Raman spectroscopy for local temperature determination at thenanoscale, Analytical and Bioanalytical Chemistry, in press, Vol. xx, pp. xx-xx,
2015.
[6.2] Downes A, Salter D, Elfick A (2006) Heating effects in tip-enhanced optical
microscopy. Optics Express 14:5216-5222[6.3] Zhang W, Schmid T, Yeo BS, Zenobi R (2008) Near-field heating, annealing, andsignal loss tip-enhanced Raman spectroscopy. J Phys Chem C 112:2104-2108[6.4] Malkovskiy, AV, Malkovsky VI, Kisliuk AM, Barrios CA, Foster MD, Sokolov AP(2009) Tip-induced heating in apertureless near-field optics. J Raman Spectrosc
40:1349-1354[6.5] Tarun T, Hayazawa N, Yano T, Kawata S (2011) Tip-heating-assisted Raman
spectroscopy at elevated temperatures. J Raman Spectrosc 42:992-997[6.6] Hart TR, Aggarwal L, Lax B (1970) Temperature dependence of Raman scatteringin silicon Phys Rev B 1:638[6.7] Hayazawa N, Yano T, Kawata S (2012) Highly reproducible tip-enhanced Raman
scattering using an oxidized and metallized silicon cantilever tip as a tool foreveryone. J Raman Spectrosc 43:1177-1182[6.8] Long, DA (1976) Raman Spectroscopy. McGraw-Hill, New York
[6.9] Rassat SD and Davis EJ (1994) Temperature measurements of single levitatedmicroparticles using Stokes/anti-Stokes Raman intensity ratios, Appl Spectrosc
48:1498-1505[6.10] Raravikar NR, Keblinski P, Rao AM, Dresselhaus MS, Schadler LS, Ajayan PM(2002) Temperature dependence of radial breathing mode Raman frequency of
single-walled carbon nanotubes. Phys Rev B 66:235424[6.11] Chiashi S, Murakami Y, Miyauchi Y, Murayama S (2008) Temperaturedependence of Raman scattering from single-walled carbon nanotubes: undefinedradial breathing mode peaks at high temperatures. Jpn J Appl Phys 47:2010-2015[6.12] Oguchi M, Mochizuki M, Yano T, Hara M, Hayashi T (2014) Light-transmittableultrasmooth gold film for gap-mode tip-enhanced Raman scattering spectroscopy.
Chem Lett 43:808-810[6.13] Deckert-Gaudig T, Deckert V (2009) Ultraflat transparent gold nanoplates – ideal
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[6.14] Kataura H, Kumazawa Y, Maniwa Y, Umezu I, Suzuki S, Ohtsuka Y, Achiba Y(1999) Optical properties of single-wall carbon nanotubes. Synthetic Metals
103:2555-2558[6.15] Hayazawa N, Saito Y, Kawata S (2004) Detection and characterization of
longitudinal field for tip-enhanced Raman spectroscopy. Appl Phy Lett 85:62396241[6.16] Dresselhaus MS, Dresselhaus G, Saito R, Jorio A (2005) Raman spectroscopy ofcarbon nanotubes. Physics Reports 409:47-99[6.17] Palik ED (1998) Handbook of Optical Constants of Solids. Academic Press,
London[6.18] Khan WA, Khan ZH, Rahi M (2014) Fluid flow and heat transfer of carbon
nanotubes along a flat plate with Navier slip boundary. Appl Nanosci 4:633-641[6.19] Feng B, Zhixin L, Zhang X (2009) Prediction of size effect on thermal conductivityof nanoscale metallic films. Thin Solid Films 517:2803-2807[6.20] Albella A, de la Osa RA, Moreno F, Maier SA (2014) Electric and magnetic fieldenhancement with ultralow heat radiation dielectric nanoantennas: considerations
for surface-enhanced spectroscopies. ACS Photonics 1:524-529[6.21] Ci L, Zhou Z, Song L, Yan X, Liu D, Yuan H, Gao Y, Wang J, Liu L, Zhou W, WangG, Xie S (2003) Temperature dependence of resonant Raman scattering in doublewall carbon nanotubes. Appl Phys Lett 82:3098[6.22] Hermann P, Hermelink A, Lausch V, Holland G, Möller L, Bannert N, Naumann D(2011) Evaluation of tip-enhanced Raman spectroscopy for characterizing different
virus strains. Analyst 136:1148[6.23] Lazarevic JJ, Uskokovic-Markovic S, Jelikic-Stankov M, Radonjic M, Tanaskovic D,
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Chapter 7. Summary and Conclusions
This thesis deals with analyses of local physicochemical properties of strained-silicon
nanowires (e-SiNWs) and carbon nanotubes (CNTs) via micro and nano-Raman
spectroscopy, respectively, to improve transistor technology. This work’s significance isaddressing the effects of size reduction and complexity increase of transistor geometry –a trend dictated by Moore’s law – on transistor performance. Present-day transistors
employ e-Si as channel material using high stress to increase carrier mobility. The
amount of stress is altered, however, during nanopatterning inducing dimensiondependent anisotropic stress relaxation, suggesting reduced carrier mobility and size
limitations. To overcome these problems, CNTs as alternative materials are considered
because of their intrinsic electronic transport properties and ultrathin body. Function
optimization can be realized through characterization of CNT’s local electronic properties
utilizing nano-Raman.
Chapter 1 briefly discussed the background, purpose and significance of this work.
Present-day transistors utilize strained silicon technology as channel material to improve
performance by increasing electron mobility. It is constantly assumed that isotropic(uniaxial) stress is used since the channels are very narrow. In order to keep pace with
Moore’s law, however, transistors are becoming smaller and more complicated. As aresult, the stress is no longer isotropic but more anisotropic. Scaling of silicon causes
other problems such as current leakage and the size limitation to maintain strain. It is thelatter of these problems that is pushing scientists and engineers to find alternate channelmaterials in order to maintain Moore’s law. One possible channel material is carbon
nanotubes. Carbon nanotubes have superior electronic transport properties and an
ultrathin body, which can address the problems of leakage and size limitation. There
have already been works resulting to carbon nanotube transistors and even a functional
carbon nanotube computer, but all these are still in the development stage. A deeper
understanding of the structural and electronic properties of carbon nanotubes is needed
to help further the development of carbon nanotube research for transistor technology.
Both strained silicon and carbon nanotubes are nano-sized materials and in order tostudy them, special characterization techniques with high sensitivity and spatial
resolution are needed. These techniques are micro and nano-Raman spectroscopy. For
strained silicon, micro-Raman spectroscopy is utilized because of its high sensitivity tostrain. Carbon nanotubes, on the other, utilize nano-Raman spectroscopy because of its

	
  

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spatial resolution. But these characterization techniques are not without problems. MicroRaman spectroscopy is insensitive to anisotropy making it a challenge to study
anisotropic stress in strained silicon. Nano-Raman spectroscopy is susceptible to heating
making it an invasive technique that can alter and damage the sample. Solutions tothese problems were presented in this thesis, making it possible to do precise
characterization and analysis of the aforementioned nano-sized materials.
Chapter 2 gave a theoretical background on how focused light is diffracted across aninterface. This chapter begins with how the electric field component of light is focusedusing an aplanatic lens. The discussion then moves to the integral representation of anelectric field and how this representation changes as the field travels between two media
of different refractive index. The resulting field that exits the final medium is asuperposition of the incident, reflected and transmitted fields. The effect of polarizationon the profile of the individual components of the electric field were also discussed. As
proof of principle, a two-layer system consisting of glass/air was studied for thenumerical simulations, and was extended to a more complex system (three-layers) in
Chapter 3. It was shown that different polarizations (linear, radial and azimuthal)
generated different electric field distributions in the transverse (Ex + Ey) and longitudinal(Ez) components. Only linear and radial polarization generated the Ez component. For
the longitudinal component, radial polarization generated a strong and tightly focused
spot. The strength of this field is dependent on the numerical aperture (NA) of theobjective lens. Linear polarization also generated an Ez field but the distribution
composed of two lobes, which would affect spatial resolution.
Chapter 3 discussed the theory of Raman scattering in Silicon and the polarization
selection rules. Stress in Silicon (particularly strained Silicon) can be quantified throughthe detection of longitudinal optical (LO) and transverse optical (TO) phonons. The
detection of these phonon modes is dictated by their respective Raman tensor. If all
three phonons are detected, a clear picture of the stress distribution in strained Silicon
can be made. From the polarization selection rules, it was found that TO modes could
not be detected in a backscattering geometry, a limitation when such geometry is used.
In order to find a way to detect the undetectable TO modes and determine thecontributions of both LO and TO modes, three-dimensional electric field calculations
were conducted based on the electric field distribution inside silicon using the theory
developed in Chapter 2. A three-layer system consisting of oil/strained-silicon/air wasused. After calculating the electric field inside silicon the Raman tensor of both LO andTO modes was applied. It was shown via numerical simulations that through the use of a

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high NA lens and appropriate polarization, it is possible to directly detect theundetectable TO phonons.

Therefore making the initially insensitive micro-Raman

spectroscopy sensitive anisotropy. This makes it possible to accurately determine thestress in strained Silicon structures such as nanowires. Nanowire structures werechosen because strained Silicon based transistors have nanowire-like channels.
Chapter 4 focuses on the redistribution of stress in strained Silicon nanowires after
nanopatterning. These strained Silicon nanowires mimic the strained Silicon channels
found in present-day transistors. Micro-Raman spectroscopy was utilized for its
sensitivity to stress and fast data acquisition. By implementing the experimental
conditions simulated in Chapter 3, TO phonon splitting was experimentally observed forthe first time. Through the detection of the two TO phonons, it was found that the stressin nanowires along the x-axis (long axis) was high (even higher than the bulk material
from where it came from). It was also found that the stress along the y-axis (short axis) is
non-zero. These findings are important because the prevalent beliefs are 1) when thedimensions of strained Silicon nanowires become smaller, the stress decreases and 2)
the stress along nanowire structures in uniaxial (stress along y axis = 0). The work in
Chapter 4 presents a new method of stress characterization that is sensitive toanisotropic stress and will be very useful for strain nanoengineering of transistors.
Chapter 5 shifts the focus of this work to carbon nanotubes giving a brief background on
their properties. The Raman spectra of CNTs was also discussed and what information
can be extracted from these spectra was explained. To study CNTs whose diameters
range from 0.7 to 10 nm, nano-Raman spectroscopy techniques such as TERS must be
employed to spatially resolve them. This chapter demonstrates the power of TERS interms of spatial resolution and signal enhancement.
Chapter 6 discusses the problem of heating in TERS. Since TERS generates a strong
enhanced electric field, it is unavoidable that this electric field can generate heat. This
heating can affect the sample, tip and substrate (if it is metallic), thus affecting sample
analysis. In order to establish TERS as a truly non-intrusive analytical tool, the local
temperature generated at the tip apex must be determined. This is most especially
important when studying heating effects in CNTs, which is an important aspect fortransistors. This chapter presents a novel temperature determination technique that can
detect the temperature and temperature changes at the nanoscale. This technique iscalled tip-enhanced THz Raman spectroscopy. Through the detection of the RBM of
CNTs and the Boltzmann distribution, the temperature of the CNTs was determined. It

	
  

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was also found that at low temperatures (less than 100˚C) the temperature can becontrolled by controlling the power of the illuminating laser. TE-THzRS is not only useful
for CNTs but can be extended to other fields and applications as well, such as biologicalsamples.

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List of Papers:
1.

M.V. Balois, N. Hayazawa, F.C. Catalan, S. Kawata, T. Yano, and T. Hayashi, Tipenhanced THz Raman spectroscopy for local temperature determination at thenanoscale, Analytical and Bioanalytical Chemistry, accepted, Vol. xx, pp. xx-xx,
2015.

2.

M.V. Balois, N. Hayazawa, A. Tarun, S. Kawata, M. Reiche and O. Moutanabbir,“Direct optical mapping of anistropic stresses in nanowires using transverse opticalphonon splitting”, Nano Lett. 14, 3789-3798, 2014.

3.

A. Tarun, N. Hayazawa, M.V. Balois, S. Kawata, M. Reiche and O. Moutanabbir,“Stress redistribution in individual ultrathin strained silicon nanowires: a highresolution polarized Raman study”, New Journal of Physics 15, 053042, 2013.

List of Presentations (International conference):
1.

M.V. Balois, N, Hayazawa, A. Tarun, O. Moutanabbir and S. Kawata, “Direct
Determination of Stress Anisotropy in Strained Silicon Nanowires via Phonon Mode
Splitting”, 24th International Conference on Raman Spectroscopy, Jena, Germany,
August, 2014. (POSTER)

2.

A. Tarun, N. Hayazawa, M.V. Balois, S. Kawata, M. Reiche, and O. Moutanabbir,"Unravelling the complexity of anisotropic stress in ultrathin strained silicon
nanowires" XXIV International Conference on Raman Spectroscopy (ICORS), p.57
(August 8-13, Jena), 2014.

3.

M.V. Balois, N, Hayazawa, A. Tarun, O. Moutanabbir and S. Kawata, “Anisotropic
Stress Relaxation in Strained Silicon Nanowires Characterized by High Precision
Polarized Raman Microscopy”, International Conference on Vibrational
Spectroscopy 7, Kobe, Japan, August, 2013. (ORAL)

4.

N. Hayazawa, A. Tarun, M.V. Balois, S. Kawata, and O. Moutanabbir, “Local Stressin Nanowires Unraveled by Raman Spectroscopy”, 7th International Conference on
Advanced Vibrational Spectroscopy (ICAVS), Kobe, Japan, August 25-30, 2013.

5.

A. Tarun, N. Hayazawa, M. V. Balois, S. Kawata, and O. Moutanabbir, “Stress
Profiles in Single Ultrathin Strained Silicon Nanowires", The 2013 MRS Spring
Meeting (April 1-5, Moscone, USA) 2013.

6.

M.V. Balois, N, Hayazawa, A. Tarun, O. Moutanabbir and S. Kawata, “Dimension
dependence of strain-relaxation of strained silicon nanostructures”, 29th Conferenceof the Physics Society of the Philippines, October, 2011. (ORAL)

7.

M.V. Balois and P. Almoro, “Wavefront sensing using correlations of two speckle
intensity measurements and a Shack Hartmann-based algorithm”, 28th Conferenceof the Physics Society of the Philippines, October, 2010. (ORAL)

8.

M.V. Balois and P. Almoro, “Technique for controlling chromatic dispersion of fullcolor holograms”, 22nd Conference of the Physics Society of the Philippines,
October, 2004. (ORAL)

	
  

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List of Presentations (Domestic conference):
1.

M.V. Balois, N, Hayazawa, T. Hayashi and S. Kawata, “Tip-enhanced THz Raman
spectroscopy for local temperature determination at the nanoscale”, Noyori School
2014, Hayama, Japan, December, 2014. (POSTER)

2.

M.V. Balois, N, Hayazawa, T. Hayashi and S. Kawata, “Tip-enhanced THz Raman
spectroscopy for local temperature determination at the nanoscale”, RIKEN Center
for Advanced Photonics Symposium, Sendai, Japan, November, 2014. (POSTER)

3.

M.V. Balois, L. Dasallas, N. Hayazawa, O. Moutanabbir and S. Kawata, “Tuning
Stress of Nanowires via Nanomembrane Edge Induced Stress Field Study by
Polarized Raman Microscopy”, Annual Meeting of the Spectroscopical Society ofJapan, Wako, Japan, May, 2014. (POSTER)

4.

M.V. Balois, N, Hayazawa, A. Tarun, O. Moutanabbir and S. Kawata,“Characterization of Anisotropic Stress Relaxation in Strained Silicon Nanowires via
Polarized Raman Microscopy using a High NA Lens”, Annual Meeting of theSpectroscopical Society of Japan, Osaka, Japan, November, 2013. (ORAL)

5.

M. V. Balois, N. Hayazawa, A. Tarun, O. Moutanabbir, and S. Kawata, "Precise andstable polarization control in a tight focusing system for accurate characterization ofstrained silicon nanostructures" JSAP-OSA Joint Symposia 2013 Abstracts, (OpticalSociety of America, 2013) paper 19a_D5_9 (Sep. 16-20, Kyoto, Japan), 2013.
(ORAL)

6.

M.V. Balois, N, Hayazawa, A. Tarun, S. Kawata and O. Moutanabbir, “Anisotropic
stress relaxation in strained silicon nanowires via polarized Raman microscopyusing a high NA lens”, RIKEN Center for Advanced Photonics Symposium, Wako,
Japan, October, 2013. (POSTER)

7.

M.V. Balois, N, Hayazawa, A. Tarun, O. Moutanabbir and S. Kawata, “Raman
Analysis of the Complex Nanopatterning Induced Redistribution of Stress in Single
Ultrathin Strained Silicon Nanowires”, Noyori Summer School 2013, Kobe, Japan,
September, 2013. (POSTER)

8.

M.V. Balois, N, Hayazawa, A. Tarun, O. Moutanabbir and S. Kawata, “Complex
nanopatterning induced redistribution of stress in single ultrathin strained silicon
nanowires”, Annual Meeting of the Spectroscopical Society of Japan, Tokyo, Japan,
November, 2012. (POSTER)

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D. Awards and Distinctions
1.

Best Poster - Analytical and Bioanalytical Chemistry: 24th International Conferenceon Raman Spectroscopy, “Direct Determination of Stress Anisotropy in Strained
Silicon Nanowires via Phonon Mode Splitting”, Jena, Germany, August, 15, 2014.

2.

2013 Young Scientist Presentation Award of the Spectroscopical Society ofJapan: “Characterization of Anisotropic Stress Relaxation in Strained Silicon
Nanowires via Polarized Raman Microscopy using a High NA Lens” Annual Meetingof the Spectroscopical Society of Japan, Osaka, Japan, November 21, 2013.

3.

Noyori Prize (Best Poster Award): RIKEN – Noyori Summer School, “Raman
Analysis of the Complex Nanopatterning Induced Redistribution of Stress in Single
Ultrathin Strained Silicon Nanowires”, September 7, 2013.

	
  

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Acknowledgements
I would first like to express my gratitude to my Academic Supervisor, Associate
Professor Tomohiro Hayashi, for giving me the opportunity to pursue my doctoral studies
here in Tokyo Institute of Technology. Without his belief in my abilities and desire to doresearch here in Japan, all this would not be possible. I am also thankful for his guidance
not only in research, but also in the other important aspects of life.
I am also very grateful to my RIKEN Supervisor, Dr. Norihiko Hayazawa, for being mymentor, who introduced me to the field of near-field scanning probe microscopy and
Raman spectroscopy. Although going through this research path was and still is filledwith challenges, his constant advice, constructive criticism and giving me the freedom ofwhat I want to do has helped mold me to become a better researcher. “If you want tolearn, you will learn,” are his words of wisdom that have been and will continue to be mymantra, as I start off on my own in this journey of science.
I would like to thank the my panel and reviewers of my doctoral thesis, Professor
Masahiko Hara, Associate Professor Fusao Kitamura and Dr. Kazunari Ozasa, for theirquestions during my defense, which have helped me gain an even deeper understandingof my own work and seeing my work through the eyes of other researchers from otherscientific fields. Their insightful comments and suggestion have helped further improve
the structure and content my doctoral thesis.
I would also like to thank my colleagues and collaborators, Dr. Alvarado Tarun and Dr.
Oussama Moutanabbir, for introducing me to the concept and material called strained
Silicon. I have learned a lot from our Skype discussions and realized the true value of
having colleagues and friends in the field of Science. Interdisciplinary work andcollaboration is an important factor in improving not only one’s research but also in
gaining a broader view of the world.
I appreciate and enjoyed the discussions I had with Assistant Professor Taka-aki Yano
during my progress reports, JDI and those random times when I have scientific questions
or I just feel like chatting about our common interests such as my home country, thePhilippines. I would also like to thank Dr. Atsushi Taguchi for teaching me how to use the
FDTD program called Lumerical. I know this knowledge will go a long way, especially if I
want to work on theoretical modeling. I would like to thank Dr. Francesca Celine Catalan
for her help and advice during my TERS experiments.

	
  

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My research work would not have gone smoothly without the help our previous secretaryin Tokyo Tech, Ms. Mari Miura. She has been my “mother” in Tokyo Tech for the twoand-a-half years of my PhD student life. I am very thankful for her help especially with
important announcements and documents, since I am always in RIKEN. I enjoyedchatting with her about various things, both in English and Japanese. I would also like tothank our current secretary, Ms. Nobue Ueda, who has also been a great help in
handling all the administrative related requirements I need in the last half year of my PhD
course.
I would like to thank Dr. Satoshi Kawata for letting me be a member of his
Nanophotonics Laboratory (NaPs) and his Near-Field Nanophotonics Research Team in
RIKEN. His way of thinking and his aggressive nature in science has been a strong
influence on how I do my research and what kind of mindset I must have as I tackle thechallenges of science in the international community.
I would like to thank the generosity of Dr. Yousoo Kim, Chief Scientist of the Surface andInterface Science Laboratory, for letting me stay in his lab in the last six months of my
PhD course. I would also like to thank Dr. Emiko Kazuma for letting me use her
Lumerical program and for teaching me how to allocate the resources of the computerfor faster computation. Her patience and kindness have encouraged me a lot.
My life in Tokyo Tech has been filled with fun and exciting experiences, not just inscience, because of my lab mates in the Hara-Hayashi Laboratory. My thanks go to mylab mates from the Hayashi Lab: Mr. Masahiro Oguchi, Mr. Masahito Mochizuki, Mr.
Taito Sekine, Ms. Kaori Takano, Mr. Ryo Hamamoto, Ms. Syifa Asatyas, Mr. Tetsu
Ogawa, Mr. Kosuke Miyaue, Mr. Takashi Nyu, Ms. Ganchimeg Lkhamsuren, Ms. Rina
Ohashi, Mr. Ryongsok Chang, Mr. Makoto Hirohara, Mr. Tatsuhiro Maekawa, Mr. Yoshiki
Mizushita, and Ms. Azuho Tsunoi; and to my lab mate from the Hara Lab: Ms. Tika
Kusbandiah, Mr. Kosuke Hara, Ms. Narangerel Ganbaatar, Mr. Yuya Nakazawa, Mr.
Keisuke Yoshida, Mr. Yuta Tsuchimoto, Ms. Nina Matsuzaki, Ms. Kanae Imai, Ms. Arie
Senda, and Mr. Yasuhiro Morita. I would like to thank them all – those who are current
students and those who have graduated – for the friendships that have been made,which I hope lasts a lifetime. The questions you keep asking me have helped improve
my analysis skills and have pushed me to study even more. I feel that I have lived mystudent life here in Japan to the fullest because of these wonderful people.

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I would like to thank RIKEN’s Program for Junior Scientists and for the financial support
provided to me as an International Program Associate. This program made it possible forme to do research in RIKEN and be able to interact with world-class researchers, whom I
aspire to become someday.
I would like to thank Mr. Masahiro Oguchi, who has walked the PhD path with me, for his
constant intellectual and emotional support especially at the times when I felt like I have
hit a wall or reached a new low. His irreplaceable company has kept my loneliness atbay as I am far away from my family and his encouragement has made me aspire forgreater heights. I wish him all the best as he finishes his PhD.
I dedicate this thesis to my family, who has been the solid foundation of my life eversince the very first day of my existence. I am very much thankful for their patience as I
told them stories of my life here in Japan on an almost nightly basis. And I am mostthankful for their love and understanding that have kept me going no matter what.

Maria Vanessa Cases Balois
August 2015

	
  

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APPENDIX A: COPYRIGHT PERMISSIONS

	
  

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