Spectroscopic Studies of Optical Second-Harmonic Generationfrom Si(001) Surfacesby

YONG QIANG AN

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2003

This thesis entitled:
Spectroscopic Studies of Optical Second-Harmonic Generation from Si(001) Surfaces
written by Yong Qiang An
has been approved for the Department of Physics_______________________________
Dr. Steven T. Cundiff_______________________________
Dr. Charles T. Rogers

Date _______________________

The final copy of this thesis has been examined by the signatories, and we find thatboth the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.

iii

An, Yong Qiang (Ph. D., Physics)
Spectroscopic Studies of Optical Second-Harmonic Generation from Si(001) Surfaces
Thesis directed by Professor Steven T. Cundiff

I present a spectroscopic study of optical second-harmonic generation (SHG)
from Si(001) surfaces and interfaces in the vicinity of the direct two-photon E1
transition using tunable femtosecond lasers. The samples investigated are oxidized Si,hydrogen terminated Si, and Cr-SiO2-Si structures. I first use a phenomenologicaltheory and susceptibility tensors to predict the symmetry properties of several
different SHG contributions and present methods for separating bulk and surface
SHG contributions and uniquely determining susceptibility tensor elements. By
measuring polarization selected rotational-anisotropy SHG (RA-SHG), I show thatboth bulk and surface SHG contributions display resonances and that interference
between these contributions can shift the apparent resonance energy. The strength ofbulk and surface SHG contributions varies with photon energy. Linear optics also
plays a role in SHG spectroscopy. For certain photon energies, the peak locations of
the RA-SHG signals from oxidized and hydrogen terminated Si(001) surfaces differ.
This indicates phase shift between surface SHG fields. For appropriate polarizations,peaks of the RA-SHG signals from oxidized Si surfaces can be turned into valleys byvarying the photon energy, and eightfold symmetric RA-SHG signals can be observed
at certain photon energies. Comparison of RA-SHG signals from Cr-SiO2-Si

iv
structures and oxidized Si samples also shows a difference in the peak location atcertain photon energies. Further experimental results show that an ultrathin Cr coating
film on oxidized Si introduces additional sources of SHG, which modify the spectra
and time-dependence of SHG. I also study the effect of thermal oxidation of Si(001)samples on SHG and show that SHG is sensitive to interface width. RA-SHG signals
with eightfold symmetry are found for several different polarizations and thecorresponding photon energies are sensitive to interface conditions. Thermaloxidation affects the time-dependence and spectroscopy of SHG. These results
indicate that SHG spectroscopy is a powerful tool for characterizing Si surfaces orinterfaces.

Dedication

To the memory of my grandfather.

vi
Acknowledgments
There are many people who have made this thesis and this work possible.
I wish first to thank my advisor, Steven Cundiff, for his support and guidance
during my graduate studies. His ability to instruct clear physics and to explain it
concisely has set a standard to which I will always aspire. His willingness to promoteme professionally through teaching me English and sending me to conferences has
been most generous. He has always been available to talk to me. Through his interest
and enthusiasm for my work, he has driven me to do my personal best in the lab. An
advisor can do little more for a student.
I would like to thank my committee members: Charles Rogers, Daniel Dessau,
Henry Kapteyn, and Kelvin Wagner, for many helpful comments and questions on
my thesis.
I am grateful for the help of many incredibly talented scientists, postdocs, andstudents at JILA during my graduate studies. I thank Prof. Stephen Leone for help in
my Comps III exam. I thank Prof. David Nesbitt for critical reading of mymanuscripts. JILA is a wonderful place to do research because of the wide array oftalented, knowledgeable, and helpful support staff. I am thankful for the electronics
shop, the computing, the instrument shop, and the administration staff for their help. I
thank David Alchenberger for help in preparing experiment samples, and Barbara
Tennis and Marilee DeGeode for helpful English classes.
I would like to thank the whole Cundiff group for making the lab an enjoyable
place to work and to live. These members are Justin Shacklette, Amy Spivey, Tara
Fortier, Kelda Furbush, Virginia Lorenz, Zhigang Chen, Qudsia Quraishi, Tianhao

vii
Zhang, David Jones, Camelia Borca, Rudolf Bratschitsch, Pete Roos, Martin Griebel,
Soobong Choi, and Jessi Ames. I enjoyed and would remember our group skiing trips
to Eldora and Keystone.
Finally, I would like to thank all my family members for their constant loveand support. Especially, I thank my wife, Chuanxia, for all of her kind support,
encouragement, and unconditional love, and my son, Rocky, for the love and joy hebrings.

viii
CONTENTS
Abstract, Acknowledgments, Contents, List of Tables, and List of Figures

i-xv

Chapter
1 Introduction

1

1.1 Historical Overview of Second-Harmonic Generation (SHG)

1

1.2 This Work4

1.3 Experimental Setup

8

2 Phenomenological Theory of SHG from Si Surfaces

14

2.1 Introduction

14

2.2 The Model

14

2.3 Rotational-Anisotropy SHG (RA-SHG)

18

2.3.1 Bulk and Surface Second-Harmonic (SH) Contributions

18

2.3.2 RA-SHG from Vicinal Si Faces

21

2.3.3 RA-SHG from the Si(001) Face32

2.4 Separation of Bulk and Surface SH Contributions

35

2.5 Summary

38

3 Bulk and Surface Contributions to Resonant SHG

40

3.1 Introduction

40

3.2 Bulk and Surface SHG Contributions

42

3.3 Sample Preparation and Experimental Methods

43

3.4 Experimental Results

44

3.4.1 Spectroscopy of the Bulk Anisotropic Contribution

44

3.4.2 Spectroscopy of the Isotropic Contribution

47

ix
3.5 Discussion

51

3.6 Summary

52

4 Phase Inversion in RA-SHG

54

4.1 Introduction

54

4.2 Experimental Conditions and Sample Preparation

55

4.3 Observed Phase Inversions55

4.3.1 Phase Inversion due to Photon Energy Variation

55

4.3.2 Phase Inversion due to Surface Modification

57

4.4 Phase and Amplitude of Susceptibility Tensors

59

4.5 Discussion

63

4.6 Summary66

5 SHG from Cr-SiO2-Si(001) Structures68

5.1 Introduction

68

5.2 Theoretical Background71

5.3 Experimental Conditions and Sample Preparation

73

5.4 Experimental Results

73

5.4.1 Comparison of RA-SHG from Cr-SiO2-Si and SiO2-Si Surfaces 73
5.4.2 Spectroscopic Study of RA-SHG from Cr-SiO2-Si

79

5.4.3 Comparison of TD-SHG from Cr-SiO2-Si and SiO2-Si Surfaces 81
5.5 Discussion and Further Experiments

83

5.5.1 Phase Inversion due to Additional SHG Sources

83

5.5.2 Further Experiments to Detect for SHG Sources84

5.5.3 Insight into the Charge Trapping Process88x
5.6 Summary
6 SHG from Thermally Oxidized Si(001) Surfaces

91
93

6.1 Introduction

93

6.2 Experimental Conditions and Sample Preparation96

6.3 Variation of SHG with Oxide Thickness

97

6.3.1 Isotropic and Anisotropic SHG Contributions

97

6.3.2 Phase Inversion Photon Energy

101

6.3.3 Time-dependent SHG

104

6.3.4 Effect of Thermal Oxidation on SHG Spectroscopy

105

6.4 Discussion

108

6.5 Summary

109

7 Conclusions

111

Bibliography

113

Appendix: SHG from Vicinal Si(001) Surfaces

121

xi
LIST OF TABLES
Table
2.1 Angular functions Φ m (α )25

2.2 b00BQ,( g , h ) and bmBQ, ( g , h ) as functions of Fresnel factors

26

2.3 b00d ,( g , h ) and bmd , ( g , h ) as functions of Fresnel factors

28

2.4 smSD,( g ,h ) as combinations of Fresnel factors and tensor elements

30

xii
LIST OF FIGURES
Figure1.1 Experimental setup for surface SHG

9

2.1 Geometry and unit vectors for the propagating fundamental and SH
fields15

2.2 Diagram of the beam frame and the vicinal surface with respect to the
cubic crystal axes

22

2.3 Definition of the linear polarization configuration with the incident
fundamental beam (a) and the reflected SH beam (b)33

3.1 Spectra of the anisotropic SHG contribution a4,( p , s ) , the linear optics
coefficient L( p , s ) ,and the magnitude ( | ζ | )of the anisotropic

element ζ

46

3.2 RA-SHG intensities from modified Si(001) surfaces (NO-Si, TO-Si andH-Si) at a two-photon energy of 3.26 eV for different polarization
configurations: (p, p), top panel and (p, s), bottom panel

48

3.3 Spectra of the isotropic amplitudes a0,r for NO-Si and TO-Si (top panel)and the interfering and non-interfering isotropic amplitudes a0,r and a0,ifor H-Si (bottom panel)50

4.1 RA-SHG signals from both TO-Si and NO-Si samples for the polarization(s, p) at several two-photon energies

56

xiii
4.2 RA-SHG intensities from modified Si(001) surfaces (NO-Si, TO-Si, andH-Si) at a two-photon energy of 3.26 eV for different polarizations: (s, p),
top panel, and (q, s), bottom panel

58

4.3 Spectra of the amplitudes and relative phases of the tensor elements ∂ 31
(upper panel) and ∂ 15 (lower panel) for all three samples: TO-Si, NO-Si
and H-Si

62

5.1 RA-SHG signals from Cr-NO-Si and Cr-TO-Si samples for the (p, p), (s,p), and (q, s) polarizations at a two-photon energy of 3.40 eV

75

5.2 RA-SHG signals from both NO-Si and TO-Si samples for the (p, p), (s, p),and (q, s) polarizations at a two-photon energy of 3.40 eV

76

5.3 Comparison of the (p, s) polarized RA-SHG signals from TO-Si and CrTO-Si samples at a two-photon energy of 3.40 eV

78

5.4 (p, p) polarized RA-SHG intensities from the Cr-TO-Si sample at twophoton energies. Note that the RA-SHG with eightfold symmetry appears
at the two-photon energy of 3.35 eV80

5.5 Comparison of the TD-SHG signals for the (p, p) polarization at a twophoton energy of 3.40 eV for different samples: NO-Si and TO-Si (upper
panel); Cr-NO-Si and Cr-TO-Si (lower panel)82

5.6 Isotropic SHG signals variation with two-photon energy from the surfacesof bulk Cr, thin Cr coated silica, and silica covered Cr

86

6.1 Ratio of the isotropic to anisotropic SHG components for the (p, p)
polarization (upper panel) and anisotropic SHG components for the (p, s)xivpolarization (lower panel) as a function of oxide thickness at several twophoton energies: 3.44 eV, 3.40 eV, and 3.26 eV

100

6.2 Upper panel: oxide thickness dependence of the two-photon energy at
which the RA-SHG signal shows eightfold symmetry for (p, p), (q, s), and(s, p) polarizations. Lower panel: RA-SHG signals from a thermally
oxidized Si(001) sample with 57.9 nm thick oxide for the (s, p)polarization, showing either eightfold or fourfold symmetry102

6.3 Time-dependent SHG signal for the (p, p) polarization at a two-photon
energy of 3.26 eV for several thermally oxidized Si(001) surfaces withdifferent thicknesses of the oxide layer

104

6.4 SH spectra from a thermally oxidized Si(001) sample with 57.9 nm thickoxide at a fixed azimuthal angle of 22.5o for different polarization
configurations: (p, p), (s, p), and (q, s)

107

A.1 (p, p) polarized RA-SHG signals at a two-photon energy of 3.22 eV from
natively oxidized vicinal Si(001) surfaces (NO-Si V) (upper panel) andthermally oxidized vicinal Si(001) surfaces (TO-Si V) (lower panel) withdifferent vicinal angles of 0o, 1o, 2o, 3o, 4o, and 5o

123

A.2 (p, p) polarized RA-SHG signals from a natively oxidized Si sample withvicinal angle of 5o (NO-Si V5) (upper panel) and a thermally oxidized Si
sample with vicinal angle of 5o (TO-Si V5) (lower panel) at two-photon
energies 4.34 eV, 4.41 eV, and 4.48 eV

124

A.3 RA-SHG signals from a natively oxidized Si sample with vicinal angle of
5o (NO-Si V5) at several two-photon energies of 3.10 eV 3.26 eV, 3.40 eV,

xv
and 3. 49 eV for different polarizations: (p, p), (s, p), (p, s), and(s, s)

125

A.4 RA-SHG signal from a thermally oxidized Si sample with vicinal angle of
5o (TO-Si V5) at several two-photon energies of 3.10 eV 3.26 eV, 3.40 eV,
and 3. 49 eV for different polarizations: (p, p), (s, p), (p, s), and(s, s)

126

A.5 RA-SHG signal from a hydrogen terminated Si sample with vicinal angleof 5o (H-Si V5) at several two-photon energies of 3.10 eV 3.26 eV, 3.40
eV, and 3. 49 eV for different polarizations: (p, p), (s, p), (p, s),and (s, s)127

1
Chapter 1
Introduction
1.1

Historical overview
A new branch of physics, nonlinear optics, was born in 1961 when Franken etal. [1] first demonstrated second-harmonic generation in a quartz crystal. Numerous
nonlinear optical phenomena have been discovered since then. As laser technology
progressed, nonlinear optics has become increasingly more mature and several
comprehensive text books have been written on this subject [2-5]. Within the vast
area of nonlinear optics, second-harmonic generation (SHG), or the frequency
doubling of light, plays an essential role.
In 1962, theoretical investigations dealing with the behavior of light waves bysolving the Maxwell’s equations in a nonlinear dielectric and at the boundary of
nonlinear media were perform by Armstrong et al. [6] and Bloembergen et al. [7],respectively. The classical laws of optical reflection and refraction were generalized
to treat the nonlinear optical response. The predicted laws of nonlinear reflection
were verified by Ducuing et al. [8] and both the real and imaginary parts of thecomplex nonlinear susceptibility were measured by Chang et al. [9].
SHG in a medium with a center of inversion symmetry was first observed by
Terhune et al. in calcite [10]. They introduced a nonlinear term of quadrupolar origin
in the form of a second-harmonic (SH) polarization proportional to the product of thefundamental field and its gradient. The most careful observation of the quadrupoleeffect in a phase matched propagation geometry in calcite was carried out by

2
Bjorkholm and Siegman [11]. In cubic and isotropic media with inversion symmetry,the quadrupole term does not give rise to transmitted harmonic radiation as thepolarization source of the quadrupolar origin has only a longitudinal component.
Initial experiments on cubic centrosymmetric materials, particularly Si and Ge,
were carried out by Bloembergen et al. [12]. The resulting SH signal was believed tooriginate solely from the nonlinear quadrupolar source term and independent of
surface conditions. Moreover, the SH signal was found to be independent of theorientation of the surface cut with respect to the crystallographic axes. Wang first
proposed that SHG from isotropic media originated from a surface dipole layer byinferring from studies on liquid-air interfaces [13]. The exist of a dipole layer at theinterface responsible for the observed signal implies the surface sensitivity of thistechnique. The surface sensitivity of SHG was demonstrated by Brown and Matsuoka
in 1969 [14] and by Chen et al. in 1973 [15] through observation of dramatic change
in SHG upon surface modification.
The development of nonlinear optics during the decade of the sixties wasfollowed by a decade with relatively little activity. Since 1980, the subject has
experienced a period of continuous growth. As described by Bloembergen in ahistorical overview paper [16], for this topic one may designate the decade of thesixties as the period of “classical antiquity”, the seventies as the Middle Ages, withthe renaissance starting in 1980. Downer described that today’s nonlinear optics isprobably in “a new low” and a new era is about to begin [17].
The potential of SHG as a surface-specific tool was not fully exploited until
the decade of the eighties. Shen [18, 19] and Richmond et al. [20] have reviewed the

3progress on SHG at interfaces of media with inversion symmetry made during theeighties. SHG as a surface probe has received much attention because of it simplicity,
surface specificity, and versatility. In 1983, the potential of SHG was demonstrated
for its surface-specific spectroscopy [21], and the ability to measure molecularadsorbate orientation [22]. The discovery of the dependence of SHG on crystal
orientation by Guidotti et al. [23], however, showed that SHG has a bulk contribution.
The anisotropic SH signal from both Si and Ge was observed by measuring the SH
signal reflected from these surfaces while azimuthally rotating the substrates. The SH
anisotropy was attributed to the bulk electric-dipole mechanism being permitted
through inversion-symmetry breaking by high-density photo-induced carriers. The
rotational-anisotropy SHG (RA-SHG) from Si surfaces was also demonstrated by
Tom et al. [24], but the anisotropy of SHG was claimed to originate from the bulk
quadrupole effect. The explanation of Litwin et al. [25] agreed with the bulk
quadrupole model and it was generally accepted later.
If one intends to use the SHG technique as a probe for surface-specificproperties, one must be able to distinguish surface and bulk contributions. GuyotSionnest et al. [26] discussed various conditions under which the surface contribution
is expected to be large relative to the bulk contribution. Sipe et al. [27] developed aphenomenological theory of SHG for cubic centrosymmetric crystals and discussed
the possibility of bulk and surface discrimination. It has been shown that separation of
bulk and surface SH contributions is a problem of fundamental difficulty in the use of
SHG as a strictly surface probe [28]. However, in most cases strict separation is notneeded because the surface SHG usually dominates over the bulk SHG.

4
In recent years, SHG has matured into a versatile and powerful technique forprobing the electronic and structural properties of surface or interfaces, as describedin several excellent review papers [29-31]. With the advent of the mode-locked
Ti:sapphire

laser

[32]

and

the

tunable

short

pulse

opticalparametricamplifier/oscillator systems, spectroscopic SHG studies could be carried out over awide wavelength range on important semiconductor interfaces. The demonstration ofresonant enhancement of SHG at the Si-SiO2 interface provided insight into
electronic structures at the interface [33]. Microscopic theories have been developed
to predict and explain the SH spectra [34, 35]. Among the later significant
developments in Si surfaces, the studies demonstrating the dc field enhancement of
SHG [36, 37] and the time-dependence of SHG [38] are related to this work.
1.2

This work
SHG has been recognized and used for more than two decades in basic

research on the physical and chemical properties of surfaces or interfaces. Using the
SHG technique as a surface-specific probe is based on the principle that SHG iselectric-dipole forbidden in the bulk of media with inversion symmetry, such as Si
and Ge, but allowed at the surface where the inversion symmetry is broken. Existenceof the bulk SHG in the total SH signal is against the basic principle of surface
selectivity of SHG, and it is generally believed to be a fundamental problem for using
the surface SHG technique. Most previous surface SHG studies neglected orinadequately addressed the bulk SH contribution. However, we take advantage of thenatural co-existence of bulk and surface SH contributions and use the interference

5
between them to monitor the phase of the surface SH field. We also investigate therelative size of bulk and surface SH contributions by varying with photon energy.
We use the SHG technique to study the buried SiO2-Si interface. Since the
SHG technique is an optical probe, it can access buried interfaces if the top medium istransparent. The significant advantages of the SHG probe include capabilities of noncontact, non-invasive, and in situ sampling. Spatial and/or temporal resolution of thelaser beam is also potentially attractive. The SiO2-Si interface is of enormous
technological importance to integrated circuit manufacturers. It occurs in the channel
region of metal-oxide-semiconductor field-effect-transistors, which are the basic
building blocks of modern integrated circuits.
Crystalline Si is one of the most intensively studied media, thus the medium
properties can be easily related to the measured SH signal. Physical properties(density: 2.33 g/cm3, melting point: 1415 oC, band gap: 1.12 eV, electron mobility:
1350 cm /Vs, hole mobility: 480 cm /Vs, resistivity: 2.5 x 105 Ω-cm, etc.) [39] andlattice structures and symmetry (structure: diamond structure with a=5.42 Å, spacegroup: Fd 3 m , crystal class: m3m (international notation) or Oh7 (Schonfliess

notation), symmetry formula: 3L44L36L29PC) [40] are well known for single crystal
Si. Our Si samples were provided by Virginia Semiconductor, Inc.
We study the SHG mainly from Si surfaces with (001) orientation. The SHG
response from this surface can be characterized by a relatively small number ofsusceptibility tensor elements, which is an advantage in separating the bulk andsurface SH contributions. Practically speaking, microelectronic circuits are built
exclusively on wafers with the (001) orientation. The structural quality of SiO2-

6
Si(001) interfaces is superior over SiO2-Si(111) interfaces. The SH response of theSi(001) interface is usually much weaker than that from the more intensity studied
Si(111) surface. SHG studies of the Si(001) surface have been comparatively few innumber, incomplete, or controversial in significant ways. It is timely and important toperform a comprehensive spectroscopic SHG study on the technologically important
Si(001) surface.
A good SiO2-Si interface should be nearly atomically smooth, have as few
electrically active defects as possible, and have a minimal concentration of carrier
traps that can be charged, either permanently or temporarily. Charge trapping canmodify the electronic state at the interface, which can be monitored by timedependent SHG (TD-SHG). For oxidized Si surfaces, especially those with thin oxide
layers, phenomenological susceptibility tensors are not constant values, because laserinteraction with the interfacial medium may cause quasi-static time-dependent effects.
For example, charge transfer by photo-injection across the SiO2-Si interface maybuild up an effective dc electric field, which causes time-dependence of SHG.
This thesis is mainly a spectroscopic study of SHG from principal (flatcut)
Si(001) surfaces. Some results of SHG on vicinal (miscut by a small angle fromprincipal face) Si(001) surfaces are presented in an appendix. In addition to the SH
intensity, we observe the variation of the phase of SH field with photon energy. By
comparing SH signals from different surfaces, we investigate the relationshipbetween the observed SH signal and the interfacial properties. Some of these results
are also presented elsewhere in publications [41, 42].

7
We first discuss, in the next section, the general experimental setup usedduring the SHG experiments. In Chapter 2, we use a phenomenological theory of
SHG to predict the symmetry property of SHG and extend to calculation of bulk andsurface electric-field-induced SHG (EFISH) effects from either principal or vicinal
Si(001) surfaces. In addition, we discuss the possibility of separating bulk and surface
SH contributions and present methods for uniquely determining susceptibility tensor
elements. In Chapter 3, we study bulk and surface contributions to resonant SHG
from Si(001) surfaces and show that interference between bulk and surface
contributions can modify the apparent spectrum obtained for a fixed azimuthal angle.
In Chapter 4, we present the observation of the phase inversion in rotationalanisotropy SHG at Si(001) interfaces, which can be induced by either varying thephoton energy or by surface modification. In Chapter 5, we compare RA-SHG
spectroscopy and time-dependent SHG results from Cr-SiO2-Si(001) structures with
an ultrathin Cr coating film and SiO2-Si(001) surfaces with coating. The difference in
the SHG signal between Cr coated and uncoated samples was attributed to additional
SH sources caused by the ultrathin Cr film. In Chapter 6, we study the effect ofthermal oxidation of Si(001) samples on SHG. We show that RA-SHG, TD-SHG, andSHG spectroscopy depend strongly on the interface width caused by thermaloxidation. For the same sample, the resonant behaviors of SHG are polarization
dependent. In Chapter 7, we summarize the results of SHG on Si(001) surfaces andtheir implications, and present our view of the weakness and strongpoint of thesurface SHG technique. In the appendix, we present the results of SHG on vicinal
Si(001) surfaces.

8
1.3

Experimental setup

SHG from Si surfaces is a very weak effect because the main SHG
contribution comes from only a few monolayers of atoms with broken inversion
symmetry. This requires the use of short pulsed lasers to generate high peak powers
and photon-counting to detect the weak signals. SHG signals are collected as afunction of the azimuthal angle of the sample and as a function of time with the same
basic apparatus.
The apparatus for measuring SHG is shown in Fig. 1.1. This apparatus canmeasure both rotational-anisotropy SHG (RA-SHG) and time-dependent SHG (TDSHG). Both of these are discussed in more detail in the following chapters. RA-SHG
provides information about the symmetry of the interface and the interference ofdifferent SHG contributions. TD-SHG is sensitive to the carrier dynamics at theinterface, which affects the SHG signal.

9Computer

Focus Lens

Filter
Purge
Gas

QuartzPMT

Sampleon
Rotation
Stage

PMT

fs LASER

Wave Plate
Polarizer

ReferenceArm

Shutter Filter Beam Splitter

Fig. 1.1. Experimental setup for surface SHG. Components are described in text.

10
The laser pulses are generated by a Kerr-lens-modelocked Ti:sapphire laser(Coherent Mira 900). The Ti:sapphire laser is pumped by either an argon-ion laser(Coherent Innova 300) or a diode-pumped solid-state laser (Coherent Verdi V10).
The pulse width is typically about 150 fs, but increases at edges of the spectral range
to about 200 fs (more strongly at the blue edge). The pulse repetition rate is about 76
MHz. Output of the Ti:sapphire laser can be tuned from 700 to 900 nm (photon
energies of 1.38-1.77 eV). An optical parametric oscillator (OPO) (Coherent APE
OPO Basic) synchronously pumped by the Ti:sapphire laser is used to realize
additional spectroscopic tuning. It is based on a collinear, noncritical phasematchedprocess in KTP crystals and made for a frequency transformation from theTi:sapphire range to 1050~1600 nm. To further extend the spectroscopic tuning range,a BBO crystal is used to double the frequency of the output pulses from the OPO toobtain pulses around 550 nm. A prism pair is used to compensate for the groupvelocity-dispersion and simultaneously to eliminate unwanted spectral bands. If sucha prism pair is not used, an important consideration in experimental design is tominimize the amount of material that the pulses traverse. Material group-velocitydispersion stretches the pulses, thereby reducing the peak intensity and hence thenonlinear signal.
The laser pulses pass through a filter to remove any residual scattered
background pump light. They are then split between signal and reference arms by adielectric beam splitter that is designed to be 50% reflective at 750 nm. The beam in
the signal arm passes through a half-wave plate and a polarizer before being focusedon the sample. The focusing lens is a gradient index lens with a 10 mm focal length.

11This lens is chosen to give the tightest possible focus while passing through thesmallest possible amount of glass. It can produce close to a diffraction-limited spot of
4 µm. The beam is incident on the sample at an angle of 45o from the surface normal.
The reflected beam, which now includes SH light, is collimated by a UV-fused-silica
lens. The reflected beam passes through a polarizer and then a filter that transmits theSH light while absorbing the fundamental light. The SH signal is detected using aphotomultiplier tube (PMT) that has a photocathode with a large work function sothat it is insensitive to the fundamental light (nevertheless multi-photon processes atthe photocathode can result in a signal from the fundamental if it is too strong). The
optical path of the reference arm, which is used for normalization, is essentially thesame as the signal arm, with the exception that the half-wave plate is omitted. Neutral
density filters are included in the signal arm before the sample to control the incident
power onto the silicon samples. A neutral density filter is included in the reference
arm after the sample to avoid overloading the detection electronics; there is no risk ofdamaging the quartz sample in the reference arm with the available power levels.
SHG from the optical elements is unmeasurable, even if they had an SHG efficiency
as high as silicon, the fact that they are only exposed to an unfocused beam meansthat their response would be several orders of magnitude less than that from thesample.
The average power on the sample is between 40 and 80 mW (varies withwavelength). At this power level, heating of the Si samples is negligible. Because of
the tight focus and short pulses, these modest average power levels correspond to apeak intensity of 3 GW/cm2.

12
The signals from both PMTs are recorded by a computer interfaced withphoton-counting electronics. The computer controls a rotation stage on which thesample in the signal arm is mounted. The rotation axis is in the surface normal
direction. This allows data to be collected as a function of the sample azimuthal angle
for the RA-SHG measurements. The computer also controls a shutter in the incident
laser beam. This is used for the TD-SHG measurements. It is also used to allow thesample to discharge in the dark before and between RA-SHG scans.
The polarization optical elements are used to select orientation of theexcitation beams and to analyze the SHG beams. The polarizers in the incident beams
are either borosilicate glass with aligned silver particles or Glan Thompson prisms.
The half-wave plate is zero order to obtain the greatest possible bandwidth and ismade from a birefringent polymer stack on BK7 glass to obtain as thin an element as
possible. The polarizers in the reflected beams are Glan Taylor prisms.
The test sample is enclosed in a chamber that can be purged. This is required
because different gases in the ambient may have different influences on the oxidecharging phenomena and on the properties at the medium-ambient interface, which inturn affect the SHG signal. The laser beams enter and exit the purge chamber via 1.5
mm thick UV-fused silica windows. The stream of the purging gas is directed towards
the sample in order to improve the purity.
A reference arm is introduced for spectroscopic calibration, as shown in Fig.
1.1. Variation of the pulse width and intensity with tuning of wavelength isnormalized out by measuring the ratio of the SHG from the sample and the SHG froma z-cut quartz plate in the reference arm. Quartz is chosen here to generate the SHG

13for calibration because it is a wide bandgap (8.9 eV) medium [43, 44], thus dispersionof the linear and SHG susceptibility is small in the tuning range. In the reference arm,
the polarization configuration is fixed to be p-in/p-out and the orientation of thequartz crystal is fixed at where the SHG signal is maximized as the crystal is rotatedabout its surface normal. The quartz plate is 1.5 mm thick. For one incident beam,
there are two spatially separated linear or SH beams reflected from the plate: one
from the front surface and the other from the back surface. The SHG signal reflectedfrom the back surface is used for the spectroscopic calibration, because our measured
SHG signal from the back surface is about 500 times stronger than that from the front
surface. However, SHG spectroscopic behaviors are about the same for both beams.

14
Chapter 2
Phenomenological theory of SHG from Si surfaces
2.1

Introduction

The incident beam at frequency ω creates a polarization at the harmonicfrequency 2ω , which radiates second-harmonic (SH) light. Nonlinear susceptibility
tensors connect the fundamental field and the generated SH field. The tensor elementsthat are allowed by both symmetry of the medium and geometry of the beams aretreated as sources of SH polarization. Using a phenomenological theory, we predict
the symmetry properties of rotational-anisotropy SHG (RA-SHG) in reflection fromeither principal or vicinal Si(001) surfaces. We consider bulk and surface SH
contributions and SHG from the electric-field-induced SHG (EFISH) contribution if
present. We show that under certain circumstances bulk and surface SH contributions
can be separated; moreover, individual tensor elements can be uniquely determinedby combining polarization selection and RA-SHG.
2.2

The Model

Consider a laser beam, idealized as single incident plane wave at frequency ωrand wave vector vi , incident from the ambient with dielectric constant ε = 1 on themedium with ε (ω ) ≠ 1 of interest at an angle θ 0 , as shown in Fig. 2.1.

15

Pi
ε=1
ε(ω)

S

z

Pr

θ0

κ
s
θ

wt

νt
Pt

Fig. 2.1. Geometry and unit vectors for the propagating fundamental and SH fields.
Note that the diagram assumes that the refractive index of the medium is real.

16v

We study the SH field E ( 2 ω) in reflection geometry. The incident field

r r
v
v v
E i (r ,t) , the reflected field E r (r,t) , and the transmitted field E t (r ,t) are written in thegeneral formvv v v
E n ( r , t ) = E n e iv n ⋅r − iωt + c.c. ,(2.1)

where the subscript n can be replaced by i, r, or t to represent the incident, thereflected, or the transmitted field, respectively. Each of the amplitudes of the electricfields can be expressed as a superposition of its p- and s-polarized components asv
E n = E np pˆ n + E ns sˆ .

(2.2)

We take the normal to the surface to be the ẑ direction and the wave-vector
component perpendicular to ẑ as κv = κˆ vi sin θ 0 . In terms of these, the wave vectors

for these three fields arevv
vn = κ − wn zˆ .

(2.3)

For all three fields, the components of the wave vector normal ( wn ) and parallel ( κ )to the medium surface are related bywi = (ω~ 2 − κ 2 )1 / 2 = − wr ,(2.4a)

wt = (ω~ 2ε (ω ) − κ 2 )1/ 2 ,(2.4b)

where ω~ = ω / c and ε (ω ) is the dielectric constant of the medium. In Eq. (2.4), wechoose the root with Im(ω ) ≥ 0 , and if Im(ω ) = 0 , we take Re(ω ) ≥ 0 .
The reflected and transmitted fields are obtained from the incident field asvv
E r = ( sˆrs sˆ + pˆ r rp pˆ i ) ⋅ Ei ,(2.5a)v
v
Et = ( sˆt s sˆ + pˆ t t p pˆ i ) ⋅ Ei ,(2.5b)

17
where rs and rp ( t s and t p ) are the usual Fresnel coefficients of reflection(transmission) from the ambient into the medium for s- and p-polarized light,
respectively,rs =wi − wt,
wi + wt

2 wi
,
ts =
wi + wtrp =

wiε (ω ) − wt;wiε (ω ) + wt

2nwi
,
tp =
wiε (ω ) + wt

(2.6)

while p̂t is the direction of polarization of p-polarized light in the medium given bypˆ t =

κzˆ + wtκˆ= f s zˆ + f cκˆ .
nω~

(2.7)

where n = ε (ω ) is the complex refractive index of the medium, and f s and f c are
the Fresnel factors. Note that if n is real, f s and f c are simply the sine and cosine of
the angle of beam propagation in the medium, respectively.
We introduce the analogous equations to Eqs. (2.4)-(2.7) for the SH fields,
viz.,
~ 2ω
, K = 2κ , N = ε (2ω ) ,Ω=c~~
Wt = [Ω 2ε (2ω ) − K 2 ]1 / 2 , Wi = [Ω 2 − K 2 ]1 / 2 ,Kzˆ + WtκˆPˆt =~ = Fs zˆ + Fcκˆ ,NΩPˆ = Fs zˆ − Fcκˆ ,
Ts =

(2.8)

2Wt
2 NWt
, Tp =.
Wi + Wt
Wiε (2ω ) + Wt

Again, if N is real, Fs and Fc are the sine and cosine of the angle of SH beam
propagation in the medium. P̂t

and P̂ are the directions of polarization of p-polarized SH light in the medium and vacuum, respectively. Ts and T p are the

18Fresnel transmission coefficients from the medium into the ambient for s- and ppolarized SH light, respectively.
The amplitude of the generated SH field in vacuum can be expressed as asuperposition of p- and s-polarized components asv
E ( 2 ω ) = E p( 2 ω ) Pˆ + E s( 2 ω ) sˆ .

(2.9)

If the dielectric constant of the medium and the wave-vector of the incident beam are
known, all of the Fresnel factors and coefficients should be readily calculated using
these equations.
2.3

Rotational-anisotropy SHG

2.3.1

Bulk and surface SH contributions

For a centrosymmetric medium, SHG due to the electric dipole response isforbidden in the bulk because of inversion symmetry, but is allowed at the surface
because of broken symmetry. At the surface, in addition to the dipole response, adiscontinuity in the normal component of the electric field can produce SHG due tothe non-local response as well. Both effects have been discussed in details before [26]and combined through the definition of an effective dipole response characterized bya surface susceptibility tensor χ ijk( 2 ) . In the bulk, although SHG due to the dipole
response is zero, higher-order terms of polarization, such as the electric quadrupole
response [12, 45] give a contribution to SHG, which could be quite significant incomparison with the surface SH contribution. In the presence of an external dc
electric field in the medium and at the surface, SHG can have a contribution arising19
from dc electric-field-induced SH (EFISH) effect. The EFISH effect should be split
into a bulk and a surface contribution, because the symmetry at the surface isdifferent from that in the bulk.
v
For a centrosymmetric medium, second-order nonlinear polarization P ( 2ω ) at

the SH frequency 2ω is given by
v
v
v
v
v
P ( 2ω ) = P SD , ( 2ω ) + P BQ , ( 2ω ) + P BE , ( 2ω ) + P SE , ( 2ω ) .

(2.10)

These four terms correspond to the surface dipole, bulk quadrupole, bulk EFISH, andsurface EFISH polarizations, respectively.
The bulk quadrupole response can be written in terms of an effective
polarization [24, 27], as

v
v
v
(2)
Pi BQ ,( 2ω ) ( r ) = Γijkl
E j ( r )∇ k E l ( r ) ,(2.11)

where the gradient is determined with respect to the field coordinates and thevsummation convention is used. E i ( r ) is the fundamental field inside the medium as av
( 2)
is a 4th rank susceptibility tensor that connects thefunction of position r . Γijkl

fundamental field and the SH field.
For a crystal with bulk cubic symmetry, such as Si, when the crystal axes are( 2)
are giventaken to be the standard cubic axes, all distinct elements for the tensor Γijkl

by [46]( 2)
Γijkl= a0δ ijkl + a1 (δ ijδ kl + δ il δ jk ) + a2δ ik δ jl ,(2.12a)where ai are non-zero phenomenological constants. In summation of indices, δ ij δ kl
implies i, j ≠ k , l . If such a restriction is eliminated, the tensor is written as

20
(2)
Γijkl
= ( a0 − 2a1 − a2 )δ ijkl + a1δ ijδ kl + a1δ il δ jk + a2δ ik δ jl . (2.12b)

Here, the relation a0 ≠ 2a1 + a2 holds for cubic symmetric media.
( 2)
under the symmetry
We consider the transformation characteristics of Γijkl

operation of rotating the crystal about its surface normal. The last three terms are all
isotropic, because each term keeps it own form under an arbitrary rotation with thetransformation matrix R, i.e.,δ ijδ kl = Rio R jp Rkq Rlr δ opδ qr .

(2.13)

The first term of Eq. (2.12b) is anisotropic because such an arbitrary rotation
operation does not exist in general.
By using Eq. (2.12b), Eq. (2.11) can then be written in the formvv
v v
Pi BQ ,( 2 ω ) = ζ E i ∇ i E i + a 1 E i [ ∇ ⋅ E ] + a 1 [ E ⋅ ∇ ] E i + γ ∇ i ( E ⋅ E ) , (2.14)v
where Ei = Ei ( r ) for simplicity andζ = a0 − 2a1 − a2 ,and γ =

1a2 .
2

(2.15)

Notice that Eq. (2.14) is similar to the previous usual form [27], but thesecond-order nonlinear polarization of magnetic dipole origin is not included here.
However, adding of the magnetic dipole effect is equivalent to adjusting of thephenomenological constants.
For excitation of a homogeneous medium by a single transverse plane wave,
the middle two terms of Eq. (2.14) are zero, thus, we recover the polarization to theprevious results [27]. The bulk susceptibility tensor is split into an isotropic piece,( 2 ),i( 2 ),aΓijkl= 2γδ ik δ jl , and an anisotropic piece, Γijkl= ζδ ijkl .

21
The isotropic nonlinear polarization is given by

v
v v
v v
v
P ( 2ω ),i = γ∇( E ⋅ E ) = 2iγ (κ − wzˆ)(E s2 + E p2 )e 2i(κ ⋅R − wz) ,(2.16)v
v v )where R = ( x, y ) , r = R + zz , and w = wt .
The SH fields arising from the bulk isotropic source are identical for all
crystal faces and independent of the surface orientation. Following Sipe [27], thegenerated isotropic fields at 2ω outside of the medium for general linear polarized
light and for either s- or p-polarized harmonic light are
E sBQ ,( 2ω ),i = 0 ,(2.17)E pBQ ,( 2ω ),i = A0T p Fs γ ( E s2 + E p2 ) .

(2.18)~
Here A0 = 2πiΩ / NFc is a constant.

2.3.2

RA-SHG from vicinal Si faces

We first study a vicinal surface, which is disoriented by a small angle α fromthe Si(001) face but still has a mirror plane of symmetry normal to the surface plane,
as shown in Fig. 2.2. The offset direction is from the [001] axis toward [110]. For
cubic centrosymmetric crystals, Lüpke presented two ways of miscutting to form asurface with symmetry of one mirror plane in the vicinal face [47]: the misorientationfrom [001] toward [011] and the misorientation from [001] toward [110]. It is
important to note that for crystals with diamond structure, which preserve cubic
centrosymmetry, the first way of miscut does not produce a vicinal surface with one
mirror plane of symmetry.

22

z
[001]
α

y

α

ψ

x

[110]

Fig. 2.2. Diagram of the beam frame and the vicinal surface with respect to the cubic
crystal axes.

23
The angles α =0, arccos(1 / 3 ) , and π / 2 yield the low-index (001), (111),
and (110) crystal faces, respectively; therefore the results of RA-SHG from such ageneral face is applicable to low-index crystal faces by adjusting the vicinal angle.
To study the transformation properties of anisotropic susceptibility tensors, weestablish three coordinate systems. The coordinates ( xˆ0 , yˆ 0 , zˆ0 ) are chosen as thenatural crystallographic coordinate system corresponding to the [100], [010], and[001] directions, respectively. The coordinates ( xˆ1 , yˆ1 , zˆ1 ) are defined to be thecrystallographic directions corresponding to the [110], [ 1 10] , and [001] directions,
respectively. The coordinates ( xˆ , yˆ , zˆ ) are defined to be a fixed beam frame with theẑ axis to be the normal of the macroscopic vicinal surface, as shown in Fig. 2.2. By
virtue of these definitions, the azimuthal angle ψ is defined to between x̂ and the
downward miscut direction.
The susceptibility tensor is transformed from the natural crystallographic
coordinates to the final beam coordinates following three steps. First, is the rotationof coordinate counterclockwise about the ẑ 0 axis by an angle ψ 0 = 45o from thecoordinates ( xˆ0 , yˆ 0 , zˆ0 ) to ( xˆ1 , yˆ1 , zˆ1 ) with the transformation matrix

 cos(ψ 0 ) sin(ψ 0 ) 0 
R z (ψ 0 ) =  − sin(ψ 0 ) cos(ψ 0 ) 0  .

0
0
1 

(2.19)

Second, is the rotation counterclockwise about the ŷ1 axis by an angle α
from ( xˆ1 , yˆ1 , zˆ1 ) to ( xˆ , yˆ , zˆ ) when ψ =0 following the transform matrix

24

cos(α ) 0 − sin(α )
R y (α ) =  01
0  .
 sin(α ) 0 cos(α ) 

(2.20)

Third, is the rotation of coordinate counterclockwise about the ẑ axis for anangle ψ from the coordinates ( xˆ , yˆ , zˆ ) when ψ =0 to the coordinates ( xˆ , yˆ , zˆ ) whenψ >0. The transform matrix is Rz (ψ ) , which has the same matrix form as Rz (ψ 0 ) but
with a different rotation angle. Thus, the final transformation matrix is the ordered
product of the three matrixes, written as
R = Rz (ψ ) R y (α ) Rz (ψ 0 ) .

(2.21)( 2 ),a( 2 ), a
The tensor Γopqr
in the crystallographic coordinate is transformed to Γ'ijkl
inthe beam coordinate according to the tensor transformation rule( 2 ),a( 2 ),aΓ'ijkl= Rio R jp Rkq Rlr [Γopqr].

(2.22)

With respect to the final beam coordinates ( xˆ , yˆ , zˆ ) , we obtain the bulk
anisotropic nonlinear polarization
v v
v v
v
( 2)
Pi ( 2ω ), a ( r ) = Γ'ijkl
E ' j ( r )∇'k E 'l ( r ) .

(2.23)

Following Sipe [27, 48], the SH fields outside of the medium generated with ahalf-space filled with air (or vacuum) are calculated. For simplicity, the polarizationsof both the fundamental and harmonic fields are limited to s and p. The notation (g, h)
is established to represent g polarized fundamental and h polarized harmonic
radiation, where g and h=s or p.
BQ , ( 2 ω ), a

For the (g, p) cases, the SH fields E g , p
quadrupole source arearising from bulk anisotropic

25
4

E gBQ, p , ( 2 ω ), a = A0 T p ζ Γ [ b00BQ, ( g , p ) + ∑ b mBQ, ( g , p ) Φ m (α ) cos( m ψ ) ][ E g( ω ) ] 2 ,m=0(2.24a)and for the (g, s) cases, the SH fields are
4

E gBQ, s , ( 2 ω ), a = A 0 T s ζ Γ [ ∑ b mBQ, ( g , s ) Φ m (α ) sin( m ψ ) ][ E g( ω ) ] 2 . (2.24b)m =1~Here, Γ = nΩ / 8(2 wt + Wt ) . Φ m (α ) are the angular functions for the specificmisorientation as shown in Fig. 2.2, which are listed in Table 2.1 in the same form asthat from Lüpke [47]. The coefficients b00BQ,( g , h ) and bmBQ, ( g , h ) are the combinations of
Fresnel factors specific to each Fourier coefficient, as listed in Table 2.2. These
results are from a complete and systematic calculation rather than a repeat of previous
report [47].

Table 2.1. Angular functions Φ m (α )
Φ 0 (α )1[25 + 4 cos(2α ) + 3 cos(4α )]32

Φ1 (α )

1
[2 sin(2α ) + 3 sin( 4α )]16Φ 2 (α )−1[1 − 4 cos(2α ) + 3 cos(4α )]16Φ 3 (α )−1[14 sin(2α ) − 3 sin( 4α )]16Φ 4 (α )−1[7 + 28 cos(2α ) − 3 cos(4α )]32

26
Table 2.2. b00BQ,( g , h ) and bmBQ, ( g , h ) as functions of Fresnel factors(s, s)(p, s)(s, p)(p, p)

b00BQ

0

0

4 f c Fs

4[ Fc f s (−2 f c2 + f s2 ) + Fs f c ( f c2 − 3 f s2 )]

b0BQ

0

0

Fc f s − 4 f c Fs

Fc f s (11 f c2 − 4 f s2 ) − 4 Fs f c ( f c2 − 4 f s2 )]

b1BQ

3 fc

f c ( f c2 − 6 f s2 )− f c Fc − f s Fs

Fc f c (10 f s2 − 3 f c2 ) + Fs f s (4 f s2 − 11 f c2 )]

b2BQ

fs

f s (5 f c2 − 2 f s2 )− 2 f c Fs

2 Fc f s ( f s2 − 3 f c2 ) + 2 Fs f c ( f c2 − 2 f s2 )]

b3BQ

− fc

f c ( f c2 − 2 f s2 )f c Fc + f s Fs

f c (− f c2 Fc + 2 Fc f s2 − f c f s Fs )]

b4BQ

fs

− f c2 f s− Fc f s

f c2 Fc f s

We apply the formalism of the bulk anisotropic SHG to calculate theanalogous equations for bulk EFISH in reflection with emphasis on dependence of the
EFISH field on crystal symmetry. To our knowledge, a theoretical expression for the
EFISH effect in the bulk of a medium has not been developed. The bulk dc field
induced dipole polarization can be written phenomenologically as,( 3)
Pi BE ,( 2ω ) = χ ijkl
E j E kd El ,(2.25)v
where E d is the dc electric field, which is a function of normal position measuredv
from the surface and directed along the surface normal, written as E d ( z ) = zˆE d ( z ) .
For a cubic centrosymmetric medium, when the coordinate axes are taken to
( 3)
( 2)
be the standard cubic axes, χ ijkl
preserves the same symmetry properties as Γijkl, and

it is written as,( 3)
χ ijkl= (b0 − 2b1 − b2 )δ ijkl + b1 (δ ij δ kl + δ il δ jk ) + b2δ ik δ jl .

(2.26)

27
Equation (2.25) can be written in the formv v
v v
Pi BE ,( 2ω ) = ζ d E i E id E i + η d E i ( E d ⋅ E ) + γ d E id ( E ⋅ E ) , (2.27)
where
1
2

ζ d = b0 − 2b1 − b2 ,η d = 2b1 , and γ d = b2 .

(2.28)

Notice that there is a difference between the bulk quadrupole and the bulk
EFISH polarizations. The middle term of the latter is non-zero in general, even for
excitation of a homogeneous medium by a single transverse plane wave. The SH
polarization arising from the middle term is isotropic, written as

v
v v
v
v
P BE ,( 2 ω ),η ,i = η d E ( E d ⋅ E ) = E d (z)f s η d E p E .

(2.29)The generated fields at 2ω from this source outside of the medium for eithers- or p-polarized harmonic light are
E sBE ,( 2ω ),η ,i = A0Ts Γ d f sη d E p E s ,(2.30a)E pBE ,( 2ω ),i = A0T p Γ d f s Fsη d E p2 .

(2.30b)0Here, Γ d = ∫ E d ( z ) exp[−i (2 wt + Wt ) z ]dz and the integration is over the half infinite−∞

medium.
Similarly, the polarization arising from the last term is

v
v v v
v
v v
d
P BE,( 2 ω), γ ,i = γ d E d ( E ⋅ E ) = γ d E d (E s2 + E p2 )e 2 i( κ ⋅R − wz) , (2.31)
and the generated fields at 2ω from this source outside of the medium for either s- or
p-polarized harmonic light ared(2.32a)d(2.32b)

EsBE ,( 2ω ),γ ,i = 0 ,E pBE ,( 2ω ),γ ,i = A0T p Γ d Fs γ d ( E s2 + E p2 ) .

28
For the anisotropic term and with the polarization of the fundamental andharmonic fields limited to s and p states, the calculation proceeds the same way as
BE , ( 2 ω ), abulk quadrupole SH source. For the (g, p) cases, the SH fields E g , parisingfrom anisotropic EFISH source are
3

E gBE, p, ( 2 ω ), a = A0T p ζ d Γ d [b00d , ( g , p ) + ∑ bmd , ( g , p ) Φ m (α ) cos( m ψ ) ][ E g( ω ) ] 2 ,m =0(2.33a)and for the (g, s) cases, the SH fields are
3

E gBE, s , ( 2 ω ), a = A 0 T s ζ d Γ d [ ∑ b md , ( g , s ) Φ m (α ) sin( m ψ ) ][ E g( ω ) ] 2

.

m =1

(2.33b)Here, b00d ,( g , h ) and bmd , ( g , h ) are the combinations of Fresnel factors specific to each
Fourier coefficient, as listed in Table 2.3.

Table 2.3. b00d ,( g , h ) and bmd , ( g , h ) as functions of Fresnel factors(s, s)(p, s)(s, p)(p, p)

b00d

0

0

4 Fs

4(−2 Fc f c f s + f c2 Fs − f s2 Fs )

b0d

0

0

− 4 Fs− 4(−2 Fc f c f s + f c2 Fs − 2 f s2 Fs )b1d3

f c2 − 4 f s2

− Fc

− 3 f c2 Fc + 4 Fc f s2 − 8 f c f s Fs

b2d

0

4 fc fs− 2 Fs

2 f c (−2 Fc f s + f c Fs )

b3d

fc

f c2

Fc

− f c2 Fc

229
Symmetry properties of the bulk SH contribution are predicted from the above
calculation. Now we consider the surface SH contribution. The SH polarization
arising from the surface dipole response is given by

Pi SD ,( 2 ω ) = χ ijkS ,( 2 ) E j E k(2.34)For a surface with one mirror plane of symmetry, with the ŷ axis
perpendicular to the plane of symmetry, Eq. (2.34) can be rewritten in the usual
piezoelectric contracted notation as [49, 50]

 Px( 2ω )   ∂11
 ( 2ω )   Py  =  0
 Pz( 2ω )  ∂ 31



∂12

∂13

0

∂15

0

0

∂ 24

0

∂ 32

∂ 33

0

∂ 35

 Ex 2 

2 
Ey 

0 
 E2 
∂ 26  ×  z  , (2.35)
2E E 0   y z 
2E E x z
2 E x E y 

where ∂ mn are independent tensor elements of χ ijkS ,( 2) . Note that the x̂ axis should also
be specified to relate the surface to the bulk susceptibilities. For the vicinal Si surface,
the x̂ axis is toward the downward miscut direction when ψ =0, as shown in Fig. 2.2.
For the ideal low-index (001) surface under perfect flatcut condition, the ∂ mn
simplify considerably to only the nonzero elements, ∂ 31 , ∂ 32 , ∂15 , ∂ 24 , and ∂ 33 . For apractical Si(001) surface, to a good approximation it is macroscopic fourfold
symmetric; therefore, these nonzero elements can be simplified again to∂ 31 = ∂ 32 , ∂15 = ∂ 24 , and ∂ 33 [27].
With the fundamental and SH beams limited to s- or p-polarized states, the SH
SD , ( 2 ω )fields E g , h

for the (g, p) cases are30
3

E gSD, p,( 2 ω ) = A0 T p [ ∑ s mSD,( g , p ) cos( m ψ ) ][ E g ] 2 ,(2.36)

m =0

and for the (g, s) cases are
3

E gSD, s ,( 2 ω ) = A0 T s [ ∑ s mSD,( g , s ) sin( m ψ ) ][ E g ] 2 .

(2.37)

m =1

Here, smSD,( g ,h ) are the combinations of Fresnel factors and independent tensor elements,
as listed in Table 2.4. In the table, the following expressions have been introduced fornotational convenience [47]:1(3∂11 + ∂12 + 2∂ 26 ),
4
1
(1)
∂12
= (∂11 + 3∂12 − 2∂ 26 ),
4
1
( 3)
∂11
= (∂11 − ∂12 − 2∂ 26 ).
4

(1)
∂11
=

(2.38)

Table 2.4. smSD,( g ,h ) as combinations of Fresnel factors and tensor elements(s, s)(p, s)(s, p)

s0SD

0

0

1
ε (2ω ) Fs (∂ 31 + ∂ 32 )
2

s1SD

(1)
∂ 11

(1)
f c2 ∂12+ f s2 ∂13(1)− Fc ∂12(p, p)1 2
f c (∂ 312+ ∂ 32 )] − Fc f c f s (∂15 + ∂ 24 )ε (2ω ) Fs [ f s2 ∂ 33 +2ε (2ω ) Fs f c f s ∂ 35
(1)
− Fc [ f s2 ∂13 + f c2 ∂11
)

s2SD

0

f c f s (∂15 − ∂ 24 )1
− ε (2ω ) Fs (∂ 31 − ∂ 32 )21
ε (2ω ) Fs f c2 (∂ 31 − ∂ 32 )2
− Fc f c f s (∂15 − ∂ 24 )

s3SD

( 3)
− ∂11

( 3)
f c2 ∂11( 3)
Fc ∂11( 3)− f c2 Fc ∂11

31
If there is a dc electric field at the surface (or interface), SHG has av
contribution from the surface EFISH polarization P SE ,( 2ω ) , given by
S ,( 3 )Pi SE , ( 2 ω ) = χ ijkl
E j E kd E l .

(2.39a)At the surface or interface, the symmetry is different from in the bulk. The dc
field here is limited to the surface normal direction. Considering the small size of theinterfacial region, we treat the EFISH polarization as a surface effect by integrating
the z-dependent variables across the interfacial layer, written as

Pi d , s ,( 2 ω ) = ∫ χ ijs ,3(k3 ) ( z ) E 3d ( z ) dz E j E k = M ijkS ,( 2 ) E j E k .

(2.39b)I

If χ ijkl

S ,( 3 )S ,( 2)is a mathematical tensor, the new variable M ijk

is not a tensor,

strictly speaking; however, it possesses the same symmetry properties as χ ijk

S ,(2)

when the symmetry operation is limited to rotation about the surface normal.
Therefore, the surface EFISH effect can be combined into the surface SH without thedc field by defining a new set of field-dependent tensor elements ∂ dmn and adding
them one by one to the field-independent tensor ∂ mn , written as ∂ mn + ∂ dmn . The
symmetry properties of the rotational-anisotropy SHG should be the same with orwithout the EFISH effect.
(2ω )The total SH field E g , harising from both the bulk and surface for the (g, p)cases are4

E g( 2, ωp ) = [ ∑ C m ,( g , p ) cos( m ψ ) ][ E g ] 2 ,m =0

and for the (g, s) cases are(2.40a)32
4

E g( 2, sω ) = [ ∑ C m ,( g , s ) sin( m ψ ) ][ E g ] 2 .

(2.40b)m =1

The fourfold Fourier coefficient C 4 is unique in that it originates from only
anisotropic bulk quadrupole contribution. The SH intensity I ( 2 ω ) is proportional tothe magnitude squared of the SH field, written as I ( 2 ω ) ∝ | E ( 2 ω ) | 2 .

2.3.3

RA-SHG from the Si(001) face

In calculating SHG from vicinal Si surfaces, the polarizations of both thefundamental and the SH fields are limited to either s or p in order to simplify theexpressions of the SH fields for a complicated surface susceptibility tensor. In
contrast, for the Si(001) surface, the surface susceptibility tensor is much simpler thanthat of vicinal surfaces; therefore, we consider a general linear polarization
configuration for both the fundamental and the SH fields. The notation (∆α , ∆β ) is
introduced to represent a specific linear polarization configuration, as shown in Fig.
v
2.3, where ∆α represents the polarization direction of the incident field E0 (∆α ) thatis an angle ∆α counterclockwise from its s-polarization, and ∆β represents thevpolarization of the SH field E ( 2ω ) (∆β ) that is an angle ∆β counterclockwise from its

s-polarization. Here, counterclockwise is with respect to the forward beam
propagation direction.

33

P

E0 (∆α)

P

(2ω)

E

(∆β)

∆α

∆β

S

S

(a)

(b)

Fig. 2.3. Definition of the linear polarization configuration with the incident
fundamental beam (a) and the reflected SH beam (b).

34
Under the excitation of a ∆α polarized fundamental beam, the generated spolarized SH field due to the bulk anisotropic source isE∆( 2αω, s), ani = A0Ts Γζf s {− f ctst p sin(2∆α ) +[ts cos2 (∆α ) − f c t p sin 2 (∆α )] sin(4φ ) + f ctst p sin(2∆α ) cos(4φ )}[ E0 ]222

2, (2.41a)

and the generated p-polarized SH field isE∆( 2αω, p), ani = A0Tp Γζf s {[ Fcts cos 2 (∆α ) + (3Fc f c + 4 Fs f s f c )t p sin 2 (∆α )] +2

2

2

f c Fctst p sin( 2∆α ) sin( 4φ ) − Fc [ts cos 2 (∆α ) − f c t p sin 2 (∆α )] cos(4φ )}[ E0 ]2
2

2

2

.

(2.41b)Here, φ is the azimuthal angle between the [100] crystal axis and the incident plane.
For the (∆α , ∆β ) polarization, the total SH field arising from bulk anisotropic
source isE∆( 2αω,∆),βani = E∆( 2αω,s),ani cos(∆β ) + E∆( 2αω, p),ani sin( ∆β ) .

(2.42)The isotropic SH field arising from both bulk and surface sources is written
together asE∆( 2αω,∆),βiso = ( A0Ts ∂15 f s t s t p sin(2∆α ) cos(∆β ) + A0T p {[ε (2ω )∂ 31 + γ ]Fs [t s cos 2 (∆α )2+ t p sin 2 (∆α )] + [ε (2ω )(∂ 33 − ∂ 31 ) Fs f s − 2∂15 f s Fc f c ]t p sin 2 (∆α )}sin(∆β ))[E0 ]222

2. (2.43)
Thus, the total SH field under the (∆α , ∆β ) polarization isE∆( 2αω,∆) β = E∆( 2αω,∆),βani + E∆( 2αω,∆),βiso .

(2.44)

In case that there is an EFISH effect, we treat it as an additional SH
contribution. The surface EFISH field is easily treated by replacing ∂15 , ∂ 31 , and ∂ 33ddd
, ∂ 31 + ∂ 31
, and ∂ 33 + ∂ 33
, respectively.
with ∂15 + ∂15

35
For the (∆α , ∆β ) polarization, the bulk EFISH field is calculated following
Green’s function formalism [48, 51], written as
E ∆BEα ,,∆( 2βω ) = ( A0Ts Γ dη d f s t s t p sin( 2∆α ) cos( ∆β ) + A0T p Γ d Fs {γ d [t s cos 2 (∆α )2+ t p sin 2 (∆α )] + (2η d f s + f s ζ d )t p sin 2 (∆α )}sin( ∆β ))[ E0 ]2
2

2

2

.

(2.45)
2.4

Separation of bulk and surface SH contributions

Using SHG as a surface-specific probe for centrosymmetric materials is basedon the fact that SHG is forbidden under the dipole approximation in the bulk but
allowed at the surface. The involvement of a bulk SH contribution complicates theinterpretation of SHG data. Separation of bulk and surface contributions is a problem
of fundamental importance if one intends to use SHG as a strict surface probe. Greateffort has been put on the bulk-surface discrimination [26, 52, 53], but it has beenshown that it is fundamentally difficult under certain circumstances, even for static
medium properties [28]. If there exists any time-dependence of SHG or EFISH effect,
discrimination of different SH contributions would be considerably complicated. The
theoretical predictions also show that there is no advantage in using vicinal Si
surfaces to distinguish different SH contributions because the surface susceptibility
tensor for vicinal surfaces is much complicated.
For simplicity, we consider SHG from a flatcut Si(001) face without any timedependent effect of SHG and EFISH effect. If all of the five independent tensor
elements ζ , γ , ∂15 , ∂ 31 , and ∂ 33 are known, SH response from the Si(001) surface
would be fully determined, and then different SH contributions would be neatly

36
separated. We consider here the possibilities of separation of these elements bycombining polarization selection and RA-SHG. From our theoretical predictions, theh-polarized SH fields E g( 2, hω ) are related to the g-polarized incident fundamental fields
Eg by the following equations,iδEg( 2, sω ) = a4,( g , s ) sin(4φ )e g ,s Eg2 ,(2.46)iδEg( 2, ωp ) = [a0,( g , p ) + a4, ( g , p ) cos(4φ )]e g , p Eg2 ,(2.47)

where the polarization (g or h) of both fields is limited to be s or p. On the other hand,for a special polarization (q, s), this relation takes the form,iδEq( 2, sω ) = [a0,( q , s ) + a4s,( q , s ) sin(4φ ) + a4, ( q , s ) cos(4φ )]e q ,s Eq2 .

(2.48)

Here, the letter q (-q) represents the linear polarization that is 45o counterclockwise(clockwise) from s when facing in the propagation direction.
The Fourier coefficients in Eqs. (2.46)-(2.48) are functions of susceptibility
tensor elements and linear optical preparation factors, as shown in Eqs. (2.41)-(2.43).
We choose the anisotropic coefficient a4 be positive and real (with its phase includedin δ g , h ), then the isotropic term a0 is complex in general and can be split into real andimaginary parts, written as a0 = a0,r + ia0,i . Because all of the a 4 ’s for different
polarizations are a function of the bulk anisotropic tensor element ζ , they are related
by linear optical coefficients. The SH intensity is proportional to the magnitude
square of the SH field. For cubic centrosymmetric media, ζ is usually nonzero, so is
a 4 . From Eq. (2.46), the (p, s) or (s, s) RA-SHG intensity should show eightfold

symmetry. From Eqs. (2.47) and (2.48), the (p, p), or (s, p), or (q, s) RA-SHG

37
intensity may show either eightfold or fourfold symmetry, depending on whether a0,ris zero or not. For the (p, p) polarization, if the RA-SHG intensity shows fourfold
symmetry, the peak locations of RA-SHG discerns the sign of a0,r , i.e., if a peak
appears at φ =0o, a0,r > 0 , and if a peak appears at φ =45o, a0,r < 0 .
From one scan of either (s, s) or (p, s) RA-SHG signal, the absolute value ofζ can be determined by fitting the RA-SHG to Eq. (2.46). We note that it also
possible to avoid the ζ SH contribution by combining polarization and RA-SHG, forexample, by measuring the (s, p) polarized SH signal at a fixed azimuthal angle φ =0o.
The relative phase between the SH field and fundamental field is unknown from only
intensity measurement. The surface tensor element ∂15 is determined from the (q, s)
RA-SHG signal with the known ζ . For the (q, s) polarized RA-SHG, bulk andsurface SH contributions can be strictly separated in theory. As shown in Eq. (2.43),∂ 31 and γ are always together, thus thorough separation of ∂ 31 , γ , and ∂ 33 is
impossible from this theory. However, if the γ contribution is negligible, ∂ 31 isdetermined from (s, p) RA-SHG, and then ∂ 33 is determined from (p, p) RA-SHG.
The anisotropic coefficient a4,( p , s ) can be obtained from fitting the measured(p, s) RA-SHG signal to Eq. (2.46), the remaining a4,( g , h ) of other polarizations anda4s,( q , s ) are derived using optics coefficients that take into account linear propagation

and beam geometry. Both the real part ( a0,r ) and the imaginary part ( a0,i ) of a0 canbe extracted from the measured RA-SHG signals using Eqs. (2.47) and (2.48);
however, the value of a0 can not be uniquely determined by only one scan of RA-

38
SHG. For example, both a0,r ± ia0,i give the same RA-SHG intensity as shown from
Eq. (2.47), i.e.,| E g( 2, ωp ) | 2 =| [a0,r ,( g , p ) ± ia 0,i ,( g , p ) + a 4,( g , p ) cos(4φ )]E g2 | 2(2.49)

We present a theoretical method to solve the ambiguity in fit coefficients bycombining two scans of differently polarized RA-SHG signals. By subtracting the (-q,s) RA-SHG signal from the (q, s) RA-SHG signal, as shown from Eqs (2.41)-(2.44),
we obtain the difference between these two scans of RA-SHG signal, which is
proportional to| E q( 2,sω ) |2 − | E −( 2qω,s) |2 = {4[a0,r ,( q ,s ) + a 4,( q ,s ) cos( 4φ )]a 4s,r ,( q ,s ) sin( 4φ )+ 4a0,i ,( q ,s ) a 4s,i ,( q ,s ) cos( 4φ )} | E q4 |.

(2.50)

This is a linear function of the fit coefficients a0,r ,( q ,s ) and a0,i ,( q ,s ) ; therefore, ∂15 canbe uniquely determined. To determine ∂ 31 uniquely, which is impossible from only
one scan of the (s, p) RA-SHG, we need combine a differently polarized scan of RASHG. A convenient example is subtracting the (s, -q) polarized RA-SHG signal fromthe (s, q) polarized RA-SHG signal, then we obtain a similar linear fit equation as Eq.
(2.50). Similarly, by combining appropriate RA-SHG scans ∂ 33 can be determineduniquely.
2.5

Summary

We have used a phenomenological theory of SHG to predict the symmetry
properties of RA-SHG from Si surfaces. Four SH sources (surface dipole, bulk
quadrupole, surface EFISH, and bulk EFISH) have been systematically calculated.
For vicinal Si(001) surfaces, the polarizations of both the fundamental and harmonic

39
fields were limited to s or p in the calculation. For the Si(001) surface, we considered
configurations of mixed linear polarization.
Both the surface dipole and surface EFISH contributions can be combined
into an effective surface dipole response, which is isotropic for the Si(001) surface
but anisotropic for vicinal Si(001) surfaces. The bulk EFISH effect, which ischaracterized by ζ d , η d , and γ d , can not be combined into the bulk quadrupole
response, which is characterized by ζ and γ . The ζ contribution is anisotropic forboth major and vicinal Si(001) surfaces, but the ζ d contribution is anisotropic forvicinal Si(001) surfaces but isotropic for the Si(001) surface. The remaining tensor
elements η d , γ d , and γ are all isotropic.
For the Si(001) surface, we showed that bulk and surface SH contributions
can be separated exactly both in theory and in experiment by combining polarization
selection and RA-SHG. In addition, we presented methods of uniquely determining
both the amplitude and the phase of individual susceptibility tensor elements from
RA-SHG scans.
Compared with other crystalline Si surfaces, the Si(001) surface ischaracterized by the simplest surface susceptibility tensor. Furthermore, for theSi(001) surface, it is possible to separate out or eliminate the bulk anisotropic SH
contribution, which is independent of surface conditions and can be used as a reliablereference signal for surface SHG. Both properties are significant advantages of using
the Si(001) surface for SHG studies.

40
Chapter 3
Bulk and surface contributions to resonant SHG
3.1

Introduction

Surface SHG has proven to be a powerful diagnostic technique for studying
the properties of buried Si-SiO2 interfaces [29, 54], which are important tosemiconductor technology. It has been shown that SHG is very sensitive to medium
surface conditions, such as crystal orientation [24], miscut [55, 56], and roughness[56-58]. It is also well known that SHG is sensitive to laser parameters, such aspolarization and wavelength [33, 59]. Thus, care must be taken when interpreting
results or inter-comparing published results.
The SHG signal from Si(001) consists of a strong electric-dipole allowed
contribution from the surface (or interface) and a weaker one due to the electricquadrupole and magnetic-dipole terms in the bulk [60] (also see Chapter 2). The
expectation that the bulk contribution is relatively independent of surface conditions
suggests that it can be used as a reference to aid in comparison between samples, as
has been done in studying the effect of interface roughness [56-58]. However, by
combining spectroscopic and rotational-anisotropy SHG (RA-SHG) measurements,
we have discovered that the bulk contribution displays resonant behavior. This was
not reported in earlier spectroscopic studies of SHG from oxidized [33, 59] orhydrogen terminated [61] silicon. The resonant behavior must be accounted for if thebulk contribution is used as a reference to compare samples. Furthermore, the bulk

41
contribution can interfere with the surface contribution, thereby shifting the apparent
resonance position for data taken at a single azimuthal angle [33, 59].
The SHG response is described by nonlinear optical susceptibility tensors thatconnect the generated signal to the electric field of the incident beam, as shown in
Chapter 2. Ideally, it would be possible to completely measure all of the tensor
elements of the susceptibility tensors that describe the bulk and surface contributions.
For spectroscopic measurements, the variation of the tensor elements as function of
the photon energy of the emitted (or incident) light would be measured. However, it isfundamentally difficult to separate the isotropic bulk and surface contributions [28].
Nevertheless, by careful use of RA-SHG in combination with polarization selective
SHG, we are able to provide some separation of the tensor elements, which improvescomparison among samples. While full phase information can be obtained bydispersion detection with a reference [62] or by homodyne detection with a reference[63], we can obtain relative phase information from our simpler experiment. We
observe that the isotropic response, which includes contribution from both surface
and bulk, displays a resonance that is sensitive to surface preparation. The two photon
resonant energy is 3.39 eV for native oxide silicon (NO-Si), while it shifts to 3.35 eV
for a thermal oxide (TO-Si) sample. The anisotropic component, which in a simple
picture is only due to the bulk, has a resonance at 3.42 eV, for NO-Si, TO-Si, andhydrogen terminated Si (H-Si). Prior work observed resonances around 3.3 eV andascribed them to the E1 critical point (CP) due to the band structure of bulk siliconwith a red shift due to sub-interface strain of the Si-Si bonds by the oxide layer [33].
Based on studying the effect of surface modification, this earlier work concluded that42
the resonance was purely in the surface contribution and not due to the bulk
contribution. This is contrary to our conclusions based on RA-SHG; however, we can
explain this difference as arising from interfering surface and bulk contributions thatcan distort the spectroscopic results taken for a single azimuthal angle.
3.2

Bulk and surface SHG contributions

In bulk crystalline Si, SHG arises only from the nonlocal response
characterized by two phenomenological constants ζ and γ , which are anisotropic
and isotropic respectively [27]. SHG from a (001) surface or interface is isotropic.
These symmetry properties are systematically presented in Chapter 2. With thepolarizations limited to s and p states, we write the total SH fields E g( 2, hω ) from aSi(001) crystal face as
E g( 2, ωp ) (φ ) = [ a 0 ,( g , p ) eiθ g , p+ a 4 ,( g , p ) cos( 4 φ )] e

E g( 2, sω ) (φ ) = a 4 , ( g , s ) sin( 4 φ ) eiδ g , s

E g2 .

iδ g , p

E g2 ,

(3.1)
(3.2)

Here, Eg is the incident fundamental field, and φ is the azimuthal angle between theincident plane and [100] direction in the sample surface plane. The notation (g, h)represents g polarized fundamental and h polarized harmonic radiation, where g and h= s or p. The SHG intensity I g( 2, hω ) (φ ) is proportional to the magnitude square of theSH field, i.e., I g( 2, hω ) (φ ) ∝| Eg( 2, hω ) (φ ) |2 . Both θ and δ provide phase information, but δis irrelevant when only the intensity is measured. The constants a0 and a4 are themagnitudes of the isotropic and anisotropic contributions to the RA-SHG signal,
respectively. They are determined by the relevant susceptibility elements, Fresnel

43factors, and dielectric functions. The dependence of a4 on the laser frequency, ω,involves only ζ and two dielectric functions ε (ω ) and ε (2ω ) , as shown in Chapter
2. It can be written as

a 4 ,( g , h ) (ω ) = f ( g , h ) (ζ (ω ), ε (ω ), ε ( 2ω )) = L( g , h ) (ω ) | ζ (ω ) | , (3.3)
where L( g ,h ) (ω ) is a function of frequency arising from linear optics. The polarization
dependence in a 4 and L results from the geometry of experiment.
3.3

Sample preparation and experimental method

The samples were prepared from the same undoped (ρ >20 Ω ⋅ cm), on-axis
Si(001) wafer. The NO-Si sample, covered with 2 nm thick native oxide, wasinvestigated as received from the manufacturer. The TO-Si sample was prepared byoxidation of chemically cleaned NO-Si at 1000 oC in a dry oxygen atmosphere toyield an oxide of 14 nm thickness. Before the thermal oxidation process, natively
oxidized Si wafers were cleaned followed the standard the RCA cleaning process [64].
Two H-Si samples were obtained by dipping NO-Si and TO-Si samples in an
NH4F:HF buffered oxide etch (BOE). The durations of the BOE were controlled, sothat oxides were just removed but the samples were not over-etched [65]. The
difference in the SH signals from these two H-terminated samples are negligible, so
we do not distinguish between them.
The experimental setup is described in detail in Chapter 1. We used amodelocked laser with ~150 fs pulse width, 76 MHz repetition rate, and wavelength
tunable from 700 to 810 nm. The ~60 mW beam was focused to <30 µm on thesamples at 45o incident angle. During the experiment, the samples were held in a

44chamber purged by dry N2 to reduce the charging effect [66] and oxidation of the HSi samples. The SHG signals in reflection were detected by photomultiplier tubes
with 1 sec gate time. The SHG spectra were calibrated against the SHG signal inreflection from a quartz sample. The uncertainty at each point is estimated by taking
multiple scans and calculating the standard deviation at each point. These uncertainty
estimates are then used in the fitting procedure to estimate the uncertainty of the fit
coefficients. The uncertainty in the photon energy is due to the laser bandwidth,
which was measured at each wavelength and varies between 20 and 30 meV (twophoton energy). Error bars are omitted in the figures if they are smaller than the plot
symbol.
3.4

Experimental results

3.4.1

Spectroscopic results of the bulk anisotropic contribution

To separate out the bulk anisotropic contribution, we use a combination ofpolarization and RA-SHG. For Si(001) surfaces, the anisotropic coefficient, a4 , isonly due to the bulk while the isotropic coefficient, a0 , is mainly due to the surface,but includes a weak bulk contribution. To provide a clean measurement of the bulk
anisotropic contribution, we measured RA-SHG in the (p, s) configuration as afunction of photon energy. The spectra obtained this way showed negligible
dependence on surface modification. The spectrum of a4, ( p , s ) for NO-Si is shown in
Fig. 3.1, which shows a peak at 3.42 eV two-photon energy. Using Eq. (3.3), we canobtain a spectrum for ζ (ω ) . The function L(p,s)(ω) is calculated using thephenomenological theory in Chapter 2 and tabulated values for the dielectric

45functions of crystalline silicon [67]. L(p,s)(ω) displays a weak minimum around 3.35
eV, which shifts the resonance of ζ (ω ) compared to that of a4, ( p , s ) , yielding a peak at
3.38 eV for ζ (ω ) as shown in Fig. 3.1. This resonance energy is between the E0' CP
energy of 3.33 eV and E1 CP energy of 3.42 eV [68], which suggests contributions
from both critical points. We note that the data in Fig. 3.1 are normalized so thata4,( s , s ) (3.26 eV) = 1 by using Eq. (3.3), i.e., the appropriate ratios of L(g,h). All peak
positions have an uncertainty of ±0.01 eV due to the laser bandwidth in the vicinity ofthe peaks.

46

-4

3

1x10
a4, (p, s)|ζ|a4,(p, s)2-53.42±0.01 eV 5x10|ζ|13.38±0.01 eV

00L(p, s)3.35 eV44x10

L(p, s)

4

3.1

3.2

3.3

3.4

3.5

3x10

Two-Photon Energy (eV)Fig. 3.1. Spectra of the anisotropic SHG contribution a4,( p , s ) (normalized to a 4,( s , s ) at
3.26 eV), the linear optics coefficient L( p , s ) and the magnitude ( | ζ | ) of theanisotropic element ζ . The normalization of a4,( p , s ) determines the magnitude of | ζ | .
The vertical error bars come from data fitting and the horizontal error bars come fromthe laser bandwidth.

47
3.4.2

Spectroscopic results of the isotropic contribution

Based on this measurement of the spectrum of the anisotropic bulk component
from a4, ( p , s ) , we obtain the spectrum of a 4,( p , p ) by using the calculated spectrum ofL(p,s) and L(p,p). Combined with measured (p, p) polarized RA-SHG spectrum, thisallows us to accurately determine both the amplitude and relative phase of theisotropic component, which includes an inseparable combination of bulk and surface
contributions. Previous spectral measurements have not used a combination of (p, s)and (p, p) measurements but rather only the latter [69, 70]. However, we find the fits
to the RA-SHG data are not robust when only (p, p) data are used, resulting in a large
uncertainty in the fit coefficients. Typical RA-SHG data at a two-photon energy of
3.26 eV (fundamental wavelength of 760 nm) for all three samples are shown in Fig.
3.2. Clearly the azimuthal dependence of the RA-SHG signal for H-Si displays aphase shift with respect to the other two signals. This is indicative of a phase shift of
the isotropic response with respect to the anisotropic bulk response, which is
confirmed by our analysis. Based on complex phasor notation, we designate thequadrature components of the isotropic response with respect to the anisotropic bulkiθ

response as a0,r and a0,i, i.e., a 0,r + ia 0,i = a 0,( g , p )e g , p in Eq. (3.1). This conveniently
breaks the isotropic response into interfering (a0,r) and non-interfering (a0,i)
components. Note that there is an ambiguity in sign of a0,i obtained from our analysis.

48(p, p)SH Intensity (a. u.)

800

600

NO-Si

TO-Si

H-Si

400

200
(p, s)

2
0
0

90
180
270
o
Azimuthal Angle φ ( )360

Fig. 3.2. RA-SHG intensities from modified Si(001) surfaces (NO-Si, TO-Si and HSi) at a two-photon energy of 3.26 eV for different polarization configurations: (p, p),
top panel and (p, s), bottom panel. The error bars are the standard deviation obtained
from averaging successive scans.

49
The spectra of a 0,r (for NO-Si and TO-Si), a0, r and a0,i (for H-Si) are shownin Fig. 3.3. The measurements show that for NO-Si and TO-Si θ ~ π, thus a0,i isnegligible and a 0,r < 0 (we actually plotted - a 0,r ). The a 0,r for NO-Si peaks at twophoton energy of 3.39 eV, while for TO-Si, it peaks at 3.35 eV. Both of these show ashift to lower energy as compared to the bulk resonance, with a larger shift for the
thicker oxide. The shift direction is consistent with previous results [33, 59]. As
expected from the phase shift in Fig. 3.2, a0, r for H-Si is opposite sign (> 0) as
compared to the oxide samples. In addition, the non-interfering quadrature, a0,i , iscomparable in magnitude to a0, r for H-Si although their relative strength varies withphoton energy.

-a0,r (a. u.)

40

30

20

SH Intensity

50

2(a0+a4)2(a0-a4)

3.2 3.3 3.4
Two-Photon
Energy3.39±0.01 eV

3.35±0.01 eV

a0, x (a. u.)

10

10

NO-Si

TO-Si

H-Si
a0,r
a0,i

0
3.1

3.2
3.3
3.4
Two-Photon Energy (eV)

3.5

Fig. 3.3. Spectra of the isotropic amplitudes a0,r for NO-Si and TO-Si (top panel) andthe interfering and non-interfering isotropic amplitudes a0,r and a0,i for H-Si (bottom
panel). The inset in the top panel shows the resonance energy shift due to interference
between a0 and a4 for TO-Si for fixed azimuthal angles of 0o (a0-a4) and 45o (a0+a4).
The uncertainties in energy are the same as Fig. 3.1.

51
3.5

Discussion

The stronger signal for the NO-Si, as compared to the TO-Si, is in part due toan induced electric field across the Si-SiO2 interface [66]. This occurs because photoexcited electrons become trapped in the oxide or at the interface, yielding a timedependent enhancement of the signal. In the NO-Si sample, this enhancement
happens within a few seconds, and thus can cause a significant enhancement within
the 1 second gate time of the photon-counting electronics. In the TO-Si sample, theenhancement takes tens of seconds to occur and thus is weak within the 1 sec gate
time (the rotation brings a new portion of the sample into the focal spot for eachmeasurement). It is important to note that the enhancement itself also displays
resonant behavior, which further complicates interpretation of the spectrum obtained
for a sample with a thin oxide. The original work on this time-dependent increase
observed that it is enhanced when oxygen was present in the ambient [66]. Based on
this, we used a nitrogen purge to try to reduce the effect, but nevertheless observe astrong time-dependent signal on resonance. This suggests that additional processes
may contribute to the time-dependent enhancement for on-resonance excitation.
The remarkable phase difference of the SHG signal from the H-Si sample,
compared to the oxidized samples, is indicative of the different properties of thehydrogen terminated surface compared to an oxidized surface. An ideal truncated Si
surface would be electrically neutral; however the electronic structure is modified by
the presence of hydrogen or oxygen, which have differing electronegativity. The peakof the signal for H-Si sample is very close to that of the bulk anisotropic response,
consistent with a lack of the strain present in the oxidized samples.

52
The interference between the bulk anisotropic term and the isotropic terms cancause a shift in the apparent resonance position of the SHG signal if spectral data are
only taken at a single azimuthal angle, as has been done in some experiments [33, 59].
This arises because of the shift between the bulk resonance and the surface resonance.
As illustrated in the inset to Fig. 3.3, the peak position for the net resonance depends
on whether the interference is constructive or destructive. Although a small shift, it
can be comparable to the shifts due to sample preparation. Furthermore, the π phaseshift of a0,r between oxidized silicon and H-Si may lead to misleading conclusions
about the relative strength of the SHG signals for data taken at a fixed angle.
3.6

Summary

We have used a careful combination of rotational anisotropy SHG andpolarization dependent SHG from Si(001) samples to separate anisotropic bulk terms
from isotropic terms, which are dominated by the surface. This shows that the bulk
response does display a resonance. Using this information, we are then able toidentify the amplitude and phase of the isotropic terms. A bulk anisotropic resonanceis observed at a two-photon energy of 3.42 eV, which results from the interaction of
the bulk susceptibility tensor element ζ (peak at 3.38 eV) and linear opticalpropagation factors. The isotropic surface contributions for native oxide Si andthermal oxide Si show peaks at 3.39 eV and 3.35 eV, respectively. Interference
between these contributions and the bulk signal can shift the apparent resonanceposition if they are not separated. Oxidation clearly induces a red shift of theresonance relative to the bulk, which is ascribed to interface strain induced by the53oxide layer. For the (p, p) polarization, oxidized surfaces only produce a signal thatinterferes with the anisotropic bulk signal, whereas H-Si produces both interferingand non-interfering (in-quadrature) signals. Furthermore, there is a phase shift
between the interfering terms for oxidized Si as compared to H-Si. These effects canmodify the apparent spectrum obtained for a fixed azimuthal angle.

54
Chapter 4
Phase inversion in rotational-anisotropy SHG
4.1

Introduction

We present the observation of phase inversion in RA-SHG signals.
Specifically, the azimuthal angle at which a peak occurs in the RA-SHG signal
switches to being one at which a valley occurs. The phase inversion can be inducedby either tuning the energy of the incident photons or by surface modification,
specifically the phase of H-terminated Si is inverted with respect to that of oxidized
silicon at certain photon energies. The phase inversion is observed for the specificpolarization configuration of s polarized incident light and detection of the ppolarized SH. Such phase inversion in the presence of an applied electric field acrossthe interface in metal-oxide-silicon structures has been reported for (p, p) polarization[69, 71]. RA-SHG phase inversion, without an applied electric field but rather
induced by a combination of tuning of the photon energy and proper selection of thepolarization configuration, has not been presented previously. We ascribe the phase
inversion as function of photon energy to the presence of a resonance in the surface
contribution. Because the phase is actually a relative phase between surface and bulk
contributions, we conclude that this resonance must be purely a surface state that does
not arise from the bulk band structure. Earlier spectroscopic studies have foundresonances with this nature [59]. Furthermore, by comparing (s, p) data to (q, s) data,
it is possible to distinguish the resonant behavior of ∂ 15 from that of ∂ 31 . The phase
inversion due to H-termination is attributed to the change in interfacial electronic

55
structure as compared to the Si-SiO2 interface. This is most pronounced for (s, p)polarization, although we have also observed it for (p, p) polarization, as shown in
Chapter 3.
4.2

Experimental conditions and sample preparation

The experimental setup was the same as described in Chapter 1. RA-SHG
curves were obtained by rotating the sample about the surface normal. The azimuthal
angle is defined as being between plane of incidence and the [100] crystal direction.
The same three samples as that in Chapter 3 were studied. They are a native oxide Si(NO-Si), a thermal oxide Si (TO-Si), and a hydrogen terminated Si (H-Si). All wereprepared from the same undoped Si(001) wafer.
4.3

Observed phase inversions

4.3.1 Phase inversion due to photon energy variation

The (s, p) RA-SHG curves for the TO-Si and NO-Si samples are shown in Fig.
4.1 for a range of two-photon energies between 3.40 and 3.52 eV. The phase
inversion is obvious between the curves for the lowest and highest photon energies,
specifically the location of a peak in one corresponds to a valley in the other. Bytaking finer steps in photon energy, we can observe the phase inversion taking place.
Most strikingly, we observe eightfold symmetry at one specific photon energy foreach oxidized sample. Aside from the intensity, we find that the appearance of the
RA-SHG signal depends sensitively on the photon energy.

56(s, p)
160

3.44
(eV)

SH Intensity (arb. units)

120

3.47
NO-Si

80

3.50
3.52

40
160

3.40

120

TO-Si

80

3.44
3.46
3.47
3.49

40
0

90
180
270
Azimuthal Angle (deg.)360

Fig. 4.1. RA-SHG signals from both NO-Si and TO-Si samples for the polarization (s,p) at several two-photon energies. The vertical displacement reflects the changing
signal strength with two-photon energy. Note that the RA-SHG signals show
eightfold symmetry at 3.50 and 3.46 eV for the NO-Si and TO-Si samples,
respectively. The solid lines show the fits.

57
4.3.2

Phase inversion due to surface modification

In Fig. 4.2, we show the RA-SHG curves for all three samples at a fixed twophoton energy of 3.26 eV for both (s, p) and (q, s) polarization configurations. Here
we observe that for both polarizations the phase of the H-Si sample is inverted
compared to that of either oxidized sample. We have also observed that this happensfor (p, p) polarization in Chapter 3 as well. To verify that doping does not play a role,
we have also taken data on NO-Si samples with n and p doped substrates. They havethe same phase as the oxidized samples shown in Fig. 4.2.

58

100

SH Intensity (arb. units)75

50(s, p)
200
150
3.26 eV
100

NO-Si

50

TO-Si

H-Si

(q, s)
0

90
180
270
Azimuthal Angle (deg.)360

Fig. 4.2. RA-SHG intensities from modified Si(001) surfaces (NO-Si, TO-Si, and HSi) at a two-photon energy of 3.26 eV for different polarizations: (s, p), top panel, and(q, s), bottom panel. Solid lines show the fits.

59
4.4

Phase and amplitude of susceptibility tensors

To analyze the RA-SHG data, we fit the curves to theoretical predictions from
Chapter 2, which shows that the SH field, E g(2,hω ) , is related to the incident field, E g ,by

E s(,2pω ) = [ a 0 ( s , p ) + a 4 ( s , p ) cos (4φ )]e ( s , p ) E s2 ,
iθ

(4.1)

E q( 2, sω ) = [ a 0 (q , s ) + a 4′ (q , s ) sin (4φ ) + a 4 (q , s ) cos (4φ )]e ( q , s ) E q2 ,
iθ

(4.2)

where φ is the azimuthal angle, the a n ( g ,h ) ’s are the fit coefficients describing theanisotropic bulk component, a 4( g ,h ) , and the isotropic component that includes bothbulk and surface contributions, a 0( g ,h ) . The intensity of the measured SH signal is
proportional to the magnitude squared of the SH fields given in Eq. (4.1) or (4.2). The
a 4( g ,h ) coefficients are related to each other by expressions that only involve the linear
optical properties, as shown in Chapter 2. We find that the most robust procedure is toobtain a 4( p , s ) from (p, s) data (not shown), which suppresses isotropic contributions,and then use the values of a 4(s , p ) , a 4 (q , s ) and a 4′ (q , s ) derived from a 4( p , s ) in fitting theother data. The bulk anisotropic contribution was utilized as a phase reference andtherefore we choose a 4( g ,h ) to be real and positive by including its phase, δ 4( g ,h ) , in

θ ( g ,h ) . Since θ ( g ,h ) represents an overall phase, our intensity measurements areinsensitive to it. The relative phase between the bulk and surface contribution is
included by allowing a 0( g ,h ) to be complex. For a given polarization configuration, weiδwrite a 0( g ,h ) = a0 ( g ,h ) e 0 ( g , h ) , i.e., in terms of the magnitude and phase relative to the60anisotropic bulk component. Note that we do not determine the sign of the imaginary
part of a 0( g ,h ) , which makes the sign of δ 0( g ,h ) ambiguous. The eightfold symmetry
occurs when a 0( g ,h ) is exactly 90o out of phase with a 4( g ,h ) , in which case there is nocross term when the intensity is calculated from Eqs. (4.1) and (4.2). The cross term
has the fourfold symmetry observed in most of the traces, whereas the eightfold
symmetry arises from taking the square of the cos(4φ ) terms. Similarly, an eightfold
symmetry is observed in (p, s) data where the isotropic term is suppressed entirely,
also eliminating the cross term.
Based on the theory, the fit coefficients can be related to the susceptibility
tensor elements. Although the actual equations, as shown in Eqs. (2.41)-(2.43) in
Chapter 2, are complicated, it is sufficient to note that a 0(q , s ) is a function of thesurface element ∂ 15 and the bulk element ζ , while a 0( s , p ) is a function of the linear
combination of surface and bulk elements ε (2ω )∂ 31 + γ and the bulk element ζ ,where ε (2ω ) is the complex dielectric constant at the frequency of the SH. The
advantage of these two polarization configurations is that they isolate the two surface
tensor elements ∂ 15 and ∂ 31 , whereas (p, p) yields a combination of elements. Also,
since we observe that the data for the (s, p) configuration strongly depends on surface
modification, we can make some separation of ∂ 31 and γ , because the latter isinsensitive to the surface. Specifically, the phase inversion of the (s, p) signal withsurface modification leads us to conclude that it is dominated by ∂ 31 ; thus we ignore

γ in our analysis. It has been shown fundamentally that such a separation cannot bedone otherwise [28].

61
The phase and amplitude of ∂ 15 and ∂ 31 as a function of two-photon energy
obtained from the fitting are shown in Fig. 4.3 for all three samples. The phase anglesof ζ , ∂ 15 , and ∂ 31 are designated by δ ζ , δ 15 , and δ 31 , respectively. Since there isambiguity in the sign of the imaginary part of a 0( g ,h ) , we plot both possible values ofthe phases at each two-photon energy. Note that, in principle, the sign ambiguity inthe fit coefficients can be lifted so that the phases of ∂ 31 and ∂15 can be determineduniquely, as shown in Eq. (2.50) in Chapter 2. However this requires virtually perfectpolarization alignment and a better signal to noise ratio than we achieved in most of
our data. By choosing to make a 4( g ,h ) real and positive, we are measuring the phasesof ∂ 15 and ∂ 31 with respect to the phase of the bulk component ζ . To obtain thesetensor elements from the fit coefficients, we have calculated the linear opticalcoefficients from published values for the linear optical susceptibility of silicon [67].
These coefficients are complex, i.e., they result in phase shifts, thus the point of phase
inversion in the signal does not correspond to 90o phase difference between the tensor
elements. Furthermore, it means that two possible phase angles are not just different
by a minus sign.

6x10

-7

3x10

-7

NO-Si
TO-Si
H-Si

0
180
0
-180

3x10

-5

2x10

-5

1x10

-5

δ15- δζ (deg.)|∂15|

δ31- δζ (deg.)

|∂31|

62

0
180

0

-180

3.1

3.2

3.3

3.4

Two-Photon Energy (eV)

3.5

Fig. 4.3. Spectra of the amplitudes and relative phases of the tensor elements ∂ 31
(upper panel) and ∂ 15 (lower panel) for all three samples: TO-Si, NO-Si and H-Si.
The amplitudes are normalized so that a4(s,s) = 1 at 3.26 eV. Both possible phases,
arising from the ambiguity in the sign of the imaginary part of the fit coefficient, areplotted as open and filled symbols. Lines are to guide the eye.

63
4.5

Discussion

Comparing the present experimental results of SHG spectroscopy with theprevious microscopic theory of SHG at Si(001) surfaces [35], we find someagreement in the resonant energy but considerable discrepancy in the magnitudes ofthe surface susceptibility tensor elements. Previous microscopic theories [34, 35]
usually neglected the bulk SH contribution, but our results show it may significantly
alter the SH spectroscopy through interference effect. From this point of view, someprevious consistences between theory and experiment should be revised. For the SiSiO2 interface, we show here that SHG spectroscopy is sensitive to the sample
preparation conditions and the sample azimuthal angle, and we will show in Chapter
6 that it is also sensitive to the polarization configuration. Therefore, a feasible
microscopic theory to calculate the SH spectrum from a realistic Si-SiO2 interface
would be hard to develop and to apply to various circumstances. For hydrogenterminated surfaces, however, the SH spectra of ∂15 , ∂ 31 , and ∂ 33 have been
calculated microscopically [35]; therefore, we can conveniently compare them to ourexperimental results. The theory and our experimental results are in agreement withthat the (p, p) polarized SH contributions from all three surface tensor elements arecomparable. However, our measured relative magnitude of ∂15 and ∂ 31 is very
different from the theoretical prediction. Our results show that | ∂ 31 | is about twoorders of magnitudes smaller than | ∂15 | , but the microscopic theory predicted that
∂15 , ∂ 31 , and ∂ 33 are all comparable in magnitude. As evident in Fig. 4.2, the (s, p)and (q, s) SH intensities, which are due to ∂ 31 and ∂15 , respectively, are comparable

64
because the SH intensity is proportional to the susceptibility multiplied by its
radiation factors, and not the bare susceptibility alone. Our observed peak energies ofthe magnitudes of ∂ 31 and ∂15 are very close to the theoretical prediction.
The data show the clear trend that δ 31 varies significantly with respect to δ ζ ,for increasing photon energy for the oxidized samples. The variation of δ 31 is weaker
for the hydrogen terminated sample. The variation of δ 15 is much weaker than that ofδ 31 for all the samples, although it appears to begin varying at the highest photon
energies in the plot. Note that the magnitudes for both ∂ 15 and ∂ 31 show some degreeof resonant behavior at approximately 3.38 eV for all three samples.
Change in phase with respect to driving frequency (here, the photon energy) is
generally due to the presence of resonant behavior. For a simple driven oscillator, theoscillator is in phase with the driving term below resonance, lags it by 90o on
resonance, and lags by 180o if the driving frequency is significantly higher than theresonance frequency. Since we are measuring the relative phase between surface andbulk terms, which also have resonances (see Chapter 3), care must be taken ininterpreting the meaning of the phase; in particular, a 90o phase shift does notnecessarily occur exactly on resonance. The observed phase inversion by photon
energy has been predicted by the calculation of SH spectra using polarizable-bondmodel [34]. However, this phase inversion was not experimental reported before,
probably due to an ambiguity in the origin of the azimuthal angle. We suggest that in
the future experiments the initial direction against which one measure φ should
remain fixed throughout the whole set of frequencies.

65
The fact that the relative phase between ∂ 31 and ζ is shifting with photon
energy is evidence that one has a resonance that is not present in the other tensor
element. Since ∂ 15 and the H-terminated sample show smaller or no phase shifts,
although the bulk response should be identical, we conclude that the resonance ispresent in ∂ 31 . Based on the magnitude, all the tensor elements show a resonant peakaround 3.38 eV. This peak has been attributed to the E1 critical point in the bulk band
structure, which also appears in the surface terms due to strain imposed by the oxide
layer. Since the E1 resonance is also present in the bulk response (see Chapter 3), wewould expect ζ to display a similar resonance and thus no photon energy dependent
phase shifts, as occurs in the H-terminated sample. Thus ∂ 31 must have a resonancethat is not present in the bulk.
Spectroscopic measurements that only measured the amplitude of the SHG
signal, and not its phase, showed the existence of resonances at higher photon
energies [59]. Based only on the energetics of these resonances, it was concluded thatthey must be due to transitions that were unique to the interface as there are noequivalent bulk transitions. Since (p, p) polarization was used, the resonant behaviorof the SH intensity was a combined effect of the interface (including ∂ 31 , ∂ 33 , and∂ 15 ) and bulk contributions. Our separation of individual tensor elements reveals theeffect of excitation orientation. As shown in Fig. 4.3, the resonant energy of ∂ 31
discriminates different strains at the interface because the resonant energies of ∂ 31
increases from TO-Si to NO-Si then to H-Si. The amplitude of ∂ 15 is most sensitive
to the difference between oxidized and hydrogen terminated surfaces. This indicates

66
that in-plane excitation probes the interfacial strain while out-of-plane excitation
probes the interfacial bond. The presence of interface only transitions explains theshift in the relative phase between the surface and bulk contributions that we observe.
Furthermore, our phase measurements prove that there are transitions that are only
due to the interface and do not occur in the bulk, otherwise there would not be a phaseshift.
For low photon energy, i.e., below the E1 resonance, there is a consistent
phase shift between the oxidized samples and the H-terminated sample. We attribute
this to the difference in the electronic structure of the H-Si interface compared to thatof the SiO2-Si interface. Specifically, O is substantially more electronegative than H.
Although the actual interface bonds are not resonant at these photon energies, it has
been shown theoretically that simply changing the electronegativity of ad-atoms candramatically change the spectrum of SHG [72]. Although this study did not address
the phase of the SHG, it would be expected that it would also be affected. This ispresumably due to modification of the band structure of the first few monolayers of
the crystal, which is consistent with the fact that phase inversion can be induced by adc electric field across the SiO2-Si interface [69, 71].
4.6

Summary

We have shown that the RA-SHG signals provide information about therelative phase of the surface or interface tensor elements compared to the bulk
elements. The phase of the azimuthal dependence of second harmonic generationfrom Si(001) interfaces is observed to undergo an inversion for appropriate

67
polarization configurations. The azimuthal dependence is due to interference between
second harmonic generation from bulk anisotropic tensor elements and isotropic
elements, which include both bulk and surface terms. The inversion can be inducedby either varying the photon energy or by surface modification. In the former case,the inversion is ascribed to the presence of a resonance in the surface contribution and
it enhances the ability to identify resonance positions. The photon energy dependenceof the phase reveals resonances that are only present in the Si-SiO2 interface and haveno equivalent in the band structure of the bulk. The careful analysis presented here
shows that significantly more information can be extracted from RA-SHG scans,particularly with regard to separating the bulk and surface contributions, than has
been done heretofore.

68
Chapter 5
Surface SHG from Cr-SiO2-Si(001) structures
5.1

Introduction

Silicon is the dominant semiconductor material for electronic circuits anddevices. Ongoing miniaturization of electronic systems drives critical regions ofintegrated circuits ever closer to silicon interfaces. SiO2-Si interfaces are of particularimportance due to their almost ubiquitous presence in metal-oxide-semiconductor(MOS) structures and MOS field-effect transistors. The thickness of the gate insulator(commonly SiO2) for MOS transistors is being scaled down. However, there exists a
scaling limit because tunneling current through the gate oxide increases drasticallywith reduced thickness. In addition, dopants may penetrate from the heavily doped
polysilicon gate into the substrate, which causes instability in the threshold voltage.
Performance and reliability of MOS structures depends more and more on the
microscopic quality of dielectrics and their interfaces. Traditionally, the properties of
SiO2-Si interfaces were investigated by means of electrical characterization, such as
capacitance voltage (C-V), current voltage (I-V) and constant voltage stress (I-t)
measurements [73, 74]. The nonlinear optical method of second-harmonic generation(SHG) has been developed into a sensitive surface or interface probe [19, 20, 54].
Among the many advantages of this technique, the features of non-contact sampling
and accessibility to buried interfaces are most attractive to the microelectronics
industry. It has been shown that surface SHG is sensitive to a number of properties of

69
the SiO2-Si interface, such as crystalline orientation [24], interfacial structure [33, 75],
preparation and roughness [57, 58], and charge transfer [66, 76].
The Cr-SiO2-Si(001) structures have been used to study the SHG dependence
on the bias dc field across the SiO2-Si interface [37, 69-71], but the sources of SHG in
reflection from such complicated MOS structures were not systematicallyinvestigated. It was found that the SHG signals produced in reflection from silicon orsilver surface showed a significant variation with a dc bias electric field applied
normal to the surface [77]. The dc field was applied by immersing the sample in anelectrolyte solution with a bias voltage between it and an electrode. SHG at the SiO2Si interface was also found to be very sensitive to a dc field across the interface. The
enhancement of SHG is caused by the electric-field-induced SHG (EFISH) effect. In
the original experiment on the EFISH effect at the SiO2-Si interface [36], the dc field
was applied between the Si substrate and a ring shaped metal electrode on top of the
SiO2 layer, and SHG was measured in transmission geometry through the metal ring;
therefore, the propagation of light was not affected by the electrode. Several further
EFISH experiments were performed on Cr-SiO2-Si structures using a reflection
geometry [37, 69-71]. In these studies, the dc biases were applied between the Si
substrate and an ultrathin semitransparent Cr film, which fully covered the SiO2 layer,
so the incident light was transmitted through the Cr coating film to reach the SiO2-Si
interface. The measured SHG signals and the bias dependent SHG spectroscopy weretreated as if they came from the SiO2-Si interface alone with no contribution from the
Cr coating. With the experimental evidence for high sensitivity of SHG to surface
conditions, it is reasonable to question the pre-condition that the SHG signal in

70reflection from such a Cr-SiO2-Si(001) structure has negligible contribution from the
Cr coating film. Several regions of the MOS structure, such as the outer surface of the
Cr film, the Cr-SiO2 interface, or even the SiO2 or Cr bulk, could be possible SHG
sources. The experimental evidence of strong enhancement of SHG at metal surfaces
by a dc bias [77] also suggests that the SHG bias dependence could come from the
Cr-SiO2 interface as well.
In this chapter, we present a comprehensive study of the effect of ultrathin Cr
coating film on the SHG response in reflection from the Cr-SiO2-Si(001) MOS
structures. In addition, we study the influence of the thickness of the SiO2 layer on the
SHG response. The aims of this work are to identify the sources of SHG from MOS
structures and to explore the potential for using the surface SHG to characterize theelectronic dynamics in MOS structures. We compare SHG signals from Cr coated anduncoated SiO2-Si surfaces and compare SHG signals from thin oxide and thick oxide
covered Si surfaces in several respects, including rotational-anisotropy SHG (RASHG), SHG spectroscopy, and time-dependent SHG (TD-SHG). We find that the RASHG peak locations (azimuthal angles) of the RA-SHG signals from Cr coated anduncoated SiO2-Si surfaces differ at higher photon energies but are the same at lower
photon energies for several polarizations. The modulation of peaks and valleys in
RA-SHG signals arises from interference between isotropic and anisotropic SHG
contributions (see Chapters 3 and 4), whereas the anisotropic contribution is
generated only from the bulk and is independent of surface preparation. Therefore,
RA-SHG signals carry phase information about the surface SHG relative to the bulk
anisotropic SHG. The altered peak location of the RA-SHG signals indicates that the

71
Cr layer on oxidized Si(001) surfaces produces an additional SHG contribution
compared to uncoated surfaces. The strength of SHG contribution originated from the
Cr coating varies with photon energy. The observed SHG signal from the Cr coatednatively oxidized Si surface is weaker than the Cr coated thermally oxidized Si
surface; but this relation is reversed for samples before the Cr coating. The observed
TD-SHG signals decrease with time for Cr coated SiO2-Si surfaces while increase foruncoated surfaces. To further identify the additional sources of SHG, we investigate
SHG from the surface of a very thick Cr film and from the SiO2-Cr interface with andwithout a dc bias. We show that SHG contributions from all these surfaces (orinterfaces) are non-negligible at certain photon energies. We attribute the differencein the SHG spectrum from Cr coated and uncoated samples to interference between
additional and original SHG contributions.
5.2

Theoretical background

From the phenomenological theory in Chapter 2, both bulk and surface
contribute to the SHG. We take advantage of this natural co-existence of bulk andsurface SHG and show that we can actually obtain reliable phase information forsurface SHG, which can be easily utilized to distinguish different surface conditions.
This is realized by combining polarization selection and RA-SHG. The fact that thebulk anisotropic SHG can be independently detected from the Si(001) face makes it agood choice for studying surface SHG.
From Chapter 2, the h-polarized SHG fields E g( 2, hω ) are related to the gpolarized incident fundamental fields Eg by,

72iδ

Eg( 2, sω ) = a4,( g , s ) sin(4φ )e g ,s Eg2 ,

(5.1)
iδ

Eg( 2, ωp ) = [a0,( g , p ) + a4, ( g , p ) cos(4φ )]e g , p Eg2 ,(5.2)

where φ is the azimuthal angle measured between the plane of incidence and the [100]axis in the surface. The coefficients an ’s are functions of related susceptibility
tensors elements and linear optical factors (see Chapter 2). Here, the polarization (gor h) of both fields is limited to be s or p.
On the other hand, for a special polarization (q, s), for which the bulk andsurface SHG can be exactly separated in theory, this relation takes the form,iδEq( 2, sω ) = [a0,( q , s ) + a4s,( q , s ) sin(4φ ) + a4, ( q , s ) cos(4φ )]e q ,s Eq2 .

(5.3)

The SHG intensity is proportional to the magnitude squared of the SHG field.
If the anisotropic coefficient, a4 , is chosen to be positive and real (with its phaseincluded in δ g , h ), then the isotropic term, a0 , is complex in general. As shown in
Chapter 2, if the γ SHG contribution is neglected, all other tensor elements can be
determined from polarization selected RA-SHG signals, specifically, ζ from (s, s) or(p, s), ∂15 from (q, s), ∂ 31 from (s, p), and ∂ 33 from (p, p). Thus, the SHG response
from the Si(001) surface is fully characterized by this set of measurements.
If there is a dc electric field along the surface normal, SHG has an additionalsource from the EFISH effect, as shown in Chapter 2. The surface EFISH effect canbe effectively combined into the field-independent surface SHG effect by defining addd, ∂ 33, and ∂15
, and then adding them to the original fieldnew set of elements, ∂ 31ddd
independent elements one by one, written as ∂ 31 + ∂ 31
, ∂ 33 + ∂ 33
, and ∂15 + ∂15.

73
Therefore, the surface EFISH effect is isotropic for the (Si(001) surface. The bulk
EFISH polarization is treated by introducing additional tensor elements, ζ d , η d , andγ d . SHG from the bulk EFISH effect is still isotropic for the Si(001) surface.
However, it is not appropriate to combine it into either bulk or surface fieldindependent effect, because ζ d and η d have no equivalents in field-independent
elements. Moreover, the presence of the EFISH effect greatly complicates theseparation of different SHG contributions.
5.3

Experimental conditions and sample preparation

The experimental setup is described in Chapter 1. Four samples were preparedfrom the same undoped silicon wafer ( ρ > 20 Ω cm ), <100> oriented. The natively
oxidized Si (NO-Si) and the thermally oxidized Si (TO-Si) samples are the same as in
Chapter 3. Two MOS structures were prepared by depositing Cr on oxidized Si
substrates to form Cr coated NO-Si (Cr-NO-Si) and Cr coated TO-Si (Cr-TO-Si),with both Cr coatings 1.5 nm thick. The Cr coating was performed in an Edwards
Auto 306 vacuum chamber (~1×10-6 Torr) using thermal evaporation. The substratetemperature was kept at 20~45 oC during deposition.
5.4

Experimental results

5.4.1

Comparison of RA-SHG from Cr-SiO2-Si and SiO2-Si surfaces

At a two-photon energy of 3.40 eV (fundamental wavelength 730 nm), the
RA-SHG signals from the Cr-NO-Si and Cr-TO-Si samples are shown in Fig. 5.1, andthose from the NO-Si and TO-Si samples are shown in Fig. 5.2, for (p, p), (s, p), and74(q, s) polarization configurations. All these patterns show a fourfold symmetry, but
the RA-SHG signals from Cr coated and uncoated samples consistently differ in peaklocations for all these polarizations. The (p, p) RA-SHG signal shows a valley at

φ =45o for Cr coated samples while a peak for uncoated samples. The SHG signal
from the NO-Si is stronger than that from the TO-Si, while this relation reverses after
the Cr coating. Clearly, the Cr coating produces a large change in the RA-SHG
response compared to that before coating. The question arises here whether the Cr
coating process affects the structure at the SiO2-Si interface. Based on the deposition
condition of thermal evaporation at room temperature, such a possibility is not likely.
An additional way to check this is to vary the thickness of the SiO2 layer on Si(001)surfaces and measure the RA-SHG signal from each sample. A series of oxidized
Si(001) surfaces with oxide thickness up to 40 nm were prepared and it was foundthat the peak locations of the (p, p) RA-SHG signals are all consistent with theobserved trends, which further demonstrate that the Cr coating does not affect thestructure at the SiO2-Si interface.

75

300

SHG Intensity (arb. units)

200
100
3.40 eV0(p, p)

300
200
100
0

Cr-NO-Si

Cr-TO-Si(s, p)

100
0(q, s)
0

90
180
270
Azimuthal Angle φ (deg.)360

Fig. 5.1. RA-SHG signals from Cr-NO-Si and Cr-TO-Si samples for the (p, p), (s, p),and (q, s) polarizations at a two-photon energy of 3.40 eV.

76

2000

SHG Intensity (arb. units)

1000(p, p)

3.40 eV

0
200
100

NO-Si

0

TO-Si

(s, p)

300
0(q, s)
0

90
180
270
Azimuthal Angle φ (deg.)360

Fig. 5.2. RA-SHG signals from both NO-Si and TO-Si samples for the (p, p), (s, p),and (q, s) polarizations at a two-photon energy of 3.40 eV. Note that the peaklocations are different from that of Cr coated samples consistently for all thesepolarizations.

77
Besides the difference in the peak locations of RA-SHG, the SHG intensities
are also very different, especially in the case of (p, p) polarization. However, weshould keep in mind that the thin Cr layer changes the linear optical properties at theoutside surface of the oxide, which is verified by the comparison of the (p, s) RASHG signals from the TO-Si and Cr-TO-Si samples, as shown in Fig. 5.3. The (p, s)
RA-SHG signals from the NO-Si is also weaker than that from the Cr-NO-Si (datanot shown). Based on the phenomenological theory in Chapter 2, bulk SHG is the
only contribution to observed SHG signals. The bulk Si is untouched by surface
coating. The weaker SHG signal from Cr coated samples is easily understood to becaused by decreased transmission of both the fundamental and the SHG light throughthe coating layer, which also suggests that comparison of SHG intensities before andafter the Cr coating should take into account of linear optics effects.

78

SHG Intensity (arb. units)

10

TO-Si

Cr-TO-Si(p, s)

5

0

0

90
180
270
Azimuthal Angle φ (deg.)360

Fig. 5.3. Comparison of the (p, s) polarized RA-SHG signals from TO-Si and Cr-TOSi samples at a two-photon energy of 3.40 eV. The solid lines show the fits.

79
5.4.2 Spectroscopic study of RA-SHG from Cr-SiO2-Si

The relationship of the peak positions in the RA-SHG signals for Cr coatedand uncoated samples does not stay the same with variation of the photon energy. The(p, p) RA-SHG signals from the Cr-TO-Si samples at several two-photon energies areshown in Fig. 5.4. At the azimuthal angle φ =45o, RA-SHG signals show a valley athigher photon energies but a peak at lower photon energies. The RA-SHG signal has
eightfold symmetry at the two-photon energy 3.28 eV (fundamental wavelength 755nm), which is significant because absolute SHG intensity is not needed to see theeight peaks. A similar trend is observed for the Cr-NO-Si sample, but the photon
energy where the eightfold symmetry appears slightly shifts to blue (data not shown).
We showed in Chapter 3 that the peak locations of the (p, p) RA-SHG signals fromboth the NO-Si and TO-Si samples does not change in the two-photon energy rangeof 3.06-3.54. This inconsistency in peak locations clearly indicates that the Cr layer
affects the spectroscopic behavior of SHG compared to before coating.

80(p, p)SHG Intensity (arb. units)

300

200

100

0

3.44
3.26

3.35
3.18

3.28
3.02

(eV)

Cr-TO-Si
0

90

180

270

360

Azimuthal Angle φ (deg.)

Fig. 5.4. (p, p) polarized RA-SHG intensities from the Cr-TO-Si sample at several
two-photon energies. Note that the RA-SHG with eightfold symmetry appears at thetwo-photon energy of 3.35 eV. Note that these curves are different from the RA-SHG
signal from the TO-Si sample for corresponding photon energy.

81
5.4.3

Comparison of the TD-SHG responses

The Cr coating significantly changes the time-dependent SHG (TD-SHG).
Figure 5.5 shows the comparison of the (p, p) TD-SHG signals from all four samples
at a two-photon energy of 3.40 eV. To observe the TD-SHG, we moved the sample sothat a fresh spot, i.e., one that the laser beam has been kept away for a long period oftime, was used for each measurement. Using the shutter, we started to illuminate thesample at time zero and then began recording the signal over a series of 0.5 sec
counting intervals. After 80 sec, when the signal was close to its asymptotic value, theshutter was closed. To observe the variation of the SHG signal with time when thelaser beam is blocked, we repeated the following procedure. After 30 sec, the shutter
was opened for a 0.5 sec gate and the SHG signal was recorded and then shutter was
closed again. At the initial charging stage, the TD-SHG signals increase with time forsamples without coating while they decrease with time for Cr coated samples. During
the measurement, the sample azimuthal angle was fixed at a peak RA-SHG signal foreach sample. We note that these trends of increase and decrease in the TD-SHG
signals are the same at all other azimuthal angles. The measured time-dependent
effect for the (p, s) TD-SHG is negligible for all of the samples.

82

φ=0 deg

3000

SHG Intensity (arb. units)

2000
1000
NO-Si

TO-Si

0
φ=45 deg

300
200
100
0

Cr-NO-Si
050

Cr-TO-Si

100
150Time (sec.)

200

Fig. 5.5. Comparison of the TD-SHG signals for the (p, p) polarization at a twophoton energy of 3.40 eV for different samples: NO-Si and TO-Si (upper panel); CrNO-Si and Cr-TO-Si (lower panel). Note that the sample azimuthal angle is chosen to
be where the RA-SHG signal shows a peak, i.e., φ =0 deg for uncoated samples and

φ =45 deg for Cr coated samples.

83
5.5

Discussion and further experimental results

5.5.1

Phase inversion due to additional SHG sources

The observed azimuthal dependence of the SHG intensity follows directlyfrom the phenomenological prediction based on symmetry consideration. As shownin Eqs. (5.1), (5.2), and (5.3), the peaks and valleys in the RA-SHG signal comesfrom interference between isotropic and anisotropic SHG contributions. The latter
originates from only the bulk and is independent of surface preparation; therefore, it
serves as a reference for both the phase and the amplitude of surface SHG, as we
showed in Chapters 3 and 4. We split the isotropic coefficient, a0 , in Eqs. (5.2) and(5.3) into a real part and an imaginary part, written as a0 = a0, r + ia0,i , and designatethem as interfering and non-interfering isotropic SHG contributions, respectively.
They are so named because only the real part interferes with the bulk anisotropic
SHG contribution, while the imaginary part functions as an azimuthal independent
increase in the SHG signal. For the (p, p) polarization, the isotropic SHG is usuallymuch stronger than the anisotropic SHG, thus the RA-SHG scan usually shows
fourfold symmetry. If a0, r and a4 are the same sign, a valley appears at φ =45o in
RA-SHG, and if a0, r and a4 are opposite in sign, a peak appears at φ =45o. If a0, r =0,
the RA-SHG is eightfold symmetric, as shown in one the RA-SHG scans in Fig. 5.4.
Compared with the eightfold symmetry in the (p, s) RA-SHG, the dc level of the RASHG signal on which the modulation rides is much higher for the (p, p) polarization
than that for the (p, s) polarization, which is about zero for the latter. It should be84
noted that the measured peak locations of the (p, p) RA-SHG signal from Cr coatedsamples at the two-photon energy 3.40 eV is consistent with previous SHG results ona similar MOS structure at zero dc bias [69] but inconsistent with our results on
uncoated samples in Chapter 3.
5.5.2

Further experiments to determine the sources of SHG

In order to investigate the reason for such an apparent difference of the SHG
responses from the surfaces of Cr coated and uncoated samples, we prepared a seriesof new samples to detect possible SHG sources from silica, Cr, and Cr-Silica
interfaces. Figure 5.6 shows the results of the SHG signals from different samples atseveral two photon energies. Based on previous results of SHG on metal surfaces [77,78], it is reasonable to predict that the Cr layer might introduce an additional SHG
contribution. The first sample, designated as Silica, was a 1mm thick fused silica slab
without Cr coating. The measured SHG signal in reflection from it is very weak (notshown in Fig. 5.6). The second sample, designated as Cr-Silica, was prepared bycoating a 1.5 nm thick Cr film on top of the fused silica sample, using the same
coating condition as that for the Cr coated Si samples. The measured SHG signal
from the Cr-Silica surface is about two orders of magnitude stronger than that fromthe Silica surface. This is a clear evidence that the Cr coating has an effect on the totalreflected SHG signal; however, we should expect that the additional SHG signal
originated from the Cr coating on the Cr-TO-Si is even stronger than the measured
SHG signal from the Cr-Silica sample, because Si substrate reflects much more SHG
light than silica does, due to different indices of refraction between Si and silica. The
third sample, designated as Cr, was prepared by depositing a 125 nm thick Cr film on

85
a 0.16 mm thick fused silica substrate. The measured SHG signal in reflection fromthe upper surface of the Cr film is very strong, as shown in Fig. 5.6. Because the thick
Cr layer is opaque, light can not penetrate to the Cr-Silica interface, thus no SHG
signal is generated from it. This signal is strong, but we should consider that the SHG
response from a 1.5 nm thick Cr film is different from that from a very thick Cr film.
The trend of increasing SHG signal with increasing coating thickness is expected tobe similar to previously measurements on silver films [78]. To study the SHG
response from the Cr-Silica interface, we just turned over the Cr sample (125 nm
thick Cr coated on a 0.16 mm thick silica substrate) and measured the SHG signal inreflection from the upper surface of silica, which we designate as Silica-Cr. As shownin Fig. 5.6, this signal is about one order of magnitude weaker than that from the Cr
sample; however, we expect to see similar SHG signals from Silica-Cr and Cr
surfaces. Differences in interfacial morphology and electron mobility between these
two interfaces are possibly responsible for the discrepancy. Combining these facts,
we conclude that the Cr coating adds an SHG contribution to original SHG fromsamples without the Cr coating and the outermost surface of Cr film is likely the main
source of additional SHG. The change of peak locations in RA-SHG signals at thetwo-photon energy 3.40 eV is caused by interference effect from additional SHG
sources.

86

SHG Intensity (arb. units)

300
Cr
Cr-Silica
Silica-Cr(p, p)

200

ω
Cr
SiO2
Si

100

0

2ω

3.1

3.2
3.3
Two-Photon Energy (eV)

3.4

Fig. 5.6. Isotropic SHG signals from the surfaces of a thick Cr film, a thin Cr coatedsilica sample, and a silica covered Cr sample at different two-photon energies. The
inset shows that the measured SHG signal is an interfering combination of different
sources.

87With the additional sources of SHG from Cr coated samples compared touncoated samples, an additional concern for the SHG response from MOS structures
is whether the bias dependence of SHG originates from the additional SHG sources ororiginal ones. If an external bias voltage is applied between the Cr film and anelectrode underneath the Si substrate of the MOS structure, an electric field is appliedboth across the Cr-SiO2 interface and the SiO2-Si interface. To study the bias
dependence of the SHG at the Cr-SiO2 interface, we attached a copper foil on top of asilica slab that has a 125 nm thick Cr film on the backside. A capacitor was formed
between the copper foil and the Cr film. A small hole was left in the middle of thecopper foil so that both the incident and SHG light were able to travel through it
without being affected by the copper foil. The SHG signal was measured in reflection
from the silica side through the hole. Up to 40 volts of dc voltage between the Cr film
and the copper foil was applied, but a change in the SHG signal with and without thedc bias was not observable, even when the thickness of the SiO2 was decreased from1 mm to 0.16 mm. Further experiments is needed to investigate the SHG response
under a stronger dc field across the Cr-SiO2 interface, but the field used here iscomparable to previous used values, at which the bias dependence of the SHG signal
was observable [71]. Therefore, these previous results [37, 69-71] of SHG on MOS
structures characterized the bias dependence of the SHG signal at the SiO2-Si
interface, but SHG signals originated from other unwanted sources were involved in
the measured RA-SHG.
By spectroscopic comparison of RA-SHG signals from Cr coated anduncoated samples, we obtain the relative strength of the SHG contribution arising

88
from the Cr coating. As shown in Fig. 5.4, the SHG signal at φ =22.5o is only due tothe isotropic SHG signal, which peaks at the two-photon energy 3.26 eV nm for the
Cr-TO-Si sample. This energy is a significant shift from our earlier result of 3.35 eV
for the TO-Si sample in Chapter 3. The SHG signal from the Cr sample also varies
with photon energy, as shown in Fig. 5.6. Exact separation of different SHG sources
is difficult due to interference among different SHG fields, and the involvement ofadditional sources of SHG complicates the interpretation of spectroscopic results. At
higher photon energy, the effect of the Cr coating on the RA-SHG signal is verystrong; while at lower photon energy, it appears to be negligible. Figure 5.4 also
shows that the SHG spectrum from Si(001) surfaces strongly depends on theazimuthal angle at which the SHG signal is taken.
5.5.3

Insight into the Charge Trapping Process

The observed trends in the TD-SHG signals can be explained in terms of theself-induced EFISH effect. The dc electric field across the SiO2-Si interface is not anexternal field but an effective field, which arises from charge transfer by photon
injection and subsequent trapping in the charge traps at the SiO2-Si interface andinside of the bulk oxide [66, 76]. Comparing the RA-SHG signals in Figs. 5.1 and 5.2,
we find that the peak locations for Cr coated samples fall into the valley locations for
uncoated samples consistently for all of the (p, p), (s, p), and (q, s) polarizations,which suggests that each field-independent element in ∂ 31 , ∂ 33 , and ∂15 is roughlyout of phase for Cr coated and uncoated samples. We should note that the RA-SHG
signals were taken with a counting gate time of 1 sec and after the samples were fully

89discharged. The direction of the effective dc field is the same for both the Cr coatedand uncoated samples, because electrons transfer out of the Si substrate in one
direction. Therefore, the self-induced EFISH effect enhances the TD-SHG signal for
uncoated samples while it diminishes the TD-SHG for Cr coated samples. This
explains the increasing SHG signals with time from the NO-Si and TO-Si surfaces
while decreasing SHG signals from the Cr-NO-Si and Cr-TO-Si surfaces, as shown in
Fig. 5.5. The saturated TD-SHG signal is much stronger for NO-Si than for TO-Si,
but it is stronger for Cr-TO-Si than for Cr-NO-Si. These results can be understood interms of the fact that the number of charge traps at the interface between silicon andnative oxide is much denser than that after the high temperature thermal oxidation,
consistent with other results [79]. The much shorter rise time of the TD-SHG signal
for NO-Si compared with TO-Si is due to higher mobility of the hot electrons for NOSi. No noticeable variation of the (p, s) TD-SHG signal with time indicates that the
charge sources come from a very shallow region underneath the interface, comparedwith the estimated ~30 nm escape depth of the SHG light in silicon. In thecircumstance of the self-induced EFISH effect, the bulk EFISH source predicted in
Chapter 2 is negligible. However, if there is an external dc electric field across a
MOS structure, the EFISH effect possibly includes both surface and bulk sources. If
the dc field flips its sign, the EFISH field inverts its phase; therefore, peaks of the
RA-SHG signal can turn into valleys by applying a dc bias. This explains previous
results of bias-dependent RA-SHG signals from Si(001) MOS structures [69, 71].
Comparison of the time-dependent SHG responses between thin and thickoxidized and between Cr coated and uncoated samples allows insight into the carrier

90
dynamics at interfaces. Previous results suggested that electrons can be injected fromthe Si substrate to the ambient-SiO2 interface by a multi-photon process [80-82] andtrapping of electrons can be enhanced by using oxygen ambient [66, 83]. An
assumption used in these studies is that the electron band structure of the SiO2
overlayer remains unchanged with decreasing oxide thickness. When the thickness of
the SiO2 insulator is reduced below 3-4 nm, high tunneling currents could besignificant in MOS structures. This indicates that using only the band structure of
thick SiO2 to explain the charging dynamics is not applicable to ultrathin SiO2 films.
Another assumption is that the SHG contribution at the ambient-SiO2 interface isnegligible, which is based on the fact that the SHG signal from a thick silica substrateheld in air is negligible compared with that from the Si surface, as shown by earlier
results [84] and by our results in this chapter. However, if the SiO2 overlayer is verythin and if there are trapped electrons inside the oxide and at the ambient-SiO2
interface, considerable SHG signals could be generated from both sources. Previous
observed dramatic decrease of the SHG intensity from a metal surface with increasingpressure of ambient oxygen is a clear evidence of ambient influence on SHG [85].
We used N2 ambient but observed strong TD-SHG signals. This indicates that the
charge traps are not likely located at the ambient-SiO2 interface because N2 is very
inert and its potential for charge trapping is small. If the ambient is changed from N2
to O2, we found little change in the TD-SHG signal for the TO-Si but dramaticchange for the NO-Si. If electrons can tunnel across the thin SiO2 overlayer by one
photon injection, or if the charge traps are located at the SiO2-Si interface, then amultiple photon process is not likely to happen. For ultrathin oxide covered Si

91samples, further experiments are needed to identify whether the enhancement of the
SHG signal by oxygen ambient is caused by more charge traps compared to that of
the N2 ambient or caused by an additional SHG source at the O2-SiO2 interface. We
also found that the TD-SHG signals from both the Cr-NO-Si and Cr-TO-Si samples
show similar noticeable change if the ambient is changed from N2 to O2, which can beunderstood to be due to the same ambient-Cr interface. These results probably
suggest that if there are free electrons at the ambient-medium interface, the effect ofoxygen ambient on SHG is strong, otherwise it is negligible.
5.6

Summary

The combination of polarization selection and RA-SHG has been applied to
study the effect of the ultrathin (1.5 nm) Cr coating film on SHG from Cr-SiO2Si(001) structures with either native oxide or thermal oxide gate insulators. By
comparing RA-SHG signals from Cr coated and uncoated SiO2-Si(001) surfaces, wefind that the peak locations of the signal as a function of sample azimuthal angle aredifferent at a certain photon energy consistently for several polarizations. This
enabled a detailed spectroscopic comparison of the RA-SHG signals from the Cr
coated and uncoated samples, which reveals that the peak locations are different forhigher photon energies while the same for lower photon energies. In addition, we
have shown that the TD-SHG signals decrease for Cr-coated samples and while
increase for uncoated samples. These results indicate that the Cr coating has asignificant effect on the SHG from Cr-SiO2-Si(001) structures. Through further
investigation, we have identified that the Cr coating introduces additional sources of
SHG, of which the most important contribution is from the outermost surface of the

92
Cr film. The SHG intensity strongly depends on the thickness of the overlayer oxide.
For the same thinner overlay oxides, we have observed stronger SHG signal for
uncoated samples but weaker SHG signal for Cr coated samples. All these results
have been well explained by using a phenomenological theory that takes into accountof crystal symmetry and self-induced EFISH effect. In several previous studies [37,69-71], the Cr-SiO2-Si(001) MOS structures were used to study the properties of the
SiO2-Si interface and the SHG arising from the Cr coating was neglected. However,
we showed here that the Cr coating significantly changes both the spectroscopic andthe time-dependent behaviors of SHG by interfering with additional SHG sources
introduced by the ultrathin Cr coating.

93
Chapter 6
SHG from thermally oxidized Si(001) surfaces
6.1

Introduction

Silicon plays an essential role in today’s semiconductor industry. It is the
inexpensive silicon CMOS technology that initiated and maintains the information
revolution. However, the particular role played by silicon is not solely due to its
superior properties as a semiconductor. The superior properties of silicon dioxide(SiO2) and the quality of SiO2-Si interface greatly contribute to this role as well. The
SiO2 layer is a natural electrical insulator and a natural chemical protector, which is
not only easy to produce but also stable and chemically very inert. The quality of the
SiO2-Si interface depends strongly on processing conditions. The characteristics of
the interface are critical in determining integrated circuit performance. However, theinterface is buried below the oxide and thus not accessible to direct microscopic
measurement. Based on the symmetry breaking at the interface, the nonlinear opticaltechnique of surface SHG has been developed as a sensitive and versatile probe forburied interfaces. A number of experiments on SHG at the SiO2-Si interface havebeen performed showing the influence of interfacial stain [86, 87], charge and electricfield [66, 76], preparation and interface roughness [57, 58], steps on vicinal surfaces[52, 88, 89], the possible presence of a crystalline oxide interface layer [55], and theoxide thickness [90-92]. These various observation and interpretations suggest that
SHG at the SiO2-Si interface is a complex process of nonlinear polarization that is
influence by multiple factors.

94
The observed SHG dependence on a particular interfacial property may not be
attributed to that property alone but to incidental or alternative effects. For example,
the sensitivity of SHG to interface roughness [57, 58] is possibly explained in termsof charge traps and electric-field-induced SHG. This happens because the SHG
measurement is an indirect method for interface roughness. In order to use the SHG
technique, one has to relate the measured SHG signal to the roughness being detected.
It is convenient that interface roughness at the SiO2-Si interface can be measured by
x-ray scattering [93]. By combining SHG and x-ray scattering measurements, it has
been shown that there is a monotonic correlation between SHG signal and interface
roughness [57]. The observed trend is that the ratio of the isotropic to anisotropic SH
coefficients increases as interface roughness decreases, which is consistent withearlier SHG measurements, where roughness was characterized by atomic force
microscopy (AFM) [58]. However, this trend is opposite to our recent results of SHG
on a series of thermally oxidized Si(001) samples. The samples are the same samples
as before [57], but the laser used here is at shorter wavelengths. As has been shown,
the SiO2-Si interface becomes smoother with increasing oxidation time due to kinetic
smoothing of the buried interface [93]. At first glance, interface roughness is wellrelated to the SHG signal by performing SHG studies on the set of thermally oxidized
samples. However, it is still not persuasive to say that interface roughness is the main
factor of influence to SHG, because side effects, such as changes in charge traps andthermal oxides could cause variation of SHG as well.
In order to use SHG as a probe of the buried SiO2-Si interface, it is importantto understand how the oxide layer influences the total SH signal. Since the oxide

95
layer has a refractive index between air and Si, and is transparent throughout a largepart of the spectrum, multiple reflection for both the fundamental and the SH beams
should play a role. As has been shown, the observed periodic variation with oxide
thickness of the (p, s) polarized SH signal from Si(111) surfaces [91] is explained bymultiple reflection in the oxide film. It is important to note that in order to study only
the multiple reflection effect, the structure of the SiO2-Si(111) interface should beunchanged by varying oxide thickness because the (p, s) SHG from a Si(111) surface
has both bulk and surface contributions [27]. Such a condition could be realized byetching the very thick oxide covering the Si sample down to the thickness needed.
In this chapter, we present a systematic study of the effect of thermaloxidation of the Si(001) surface on the SH response from the SiO2-Si interface. The
thermal oxidation time is controlled, so that the transition width from Si to SiO2
(interface width) and the thickness of the thermal oxide are systematically controlled
at the same time. Therefore, both the multiple reflection and interface modification
effects may contribute to the dependence of SHG on thermal oxidation. To separateout only the multiple reflection effect, we use a combination of polarization selection
and rotational-anisotropy SHG (RA-SHG). The (p, s) polarized RA-SHG from theSi(001) surface is generated only from the bulk, and interface modification is anirrelevant factor, thus multiple reflection is readily known from oxide thickness
dependence of SHG. We use the (p, p) RA-SHG signal to study the influence ofinterface conditions on SHG and find that larger interface width corresponds tostronger SHG signal. By comparing time-dependent SHG (TD-SHG) signals fromsamples with different thicknesses of thermal oxide, we show that samples with

96
thinner oxide have stronger charging effect [38, 66], which is explained in terms ofinterface width and related charge traps. By performing spectroscopic studies, we findthat a peak of the RA-SHG signal can be turned into a valley by varying photon
energy. The RA-SHG signal shows eightfold symmetry at a certain critical photon
energy. This critical photon energy varies monotonically with thickness of thermaloxide thus providing a means of interface identification. To study the effect ofthermal oxidation on the spectra of SHG, we measure the SH spectra on a very thickthermal oxide Si(001) surface under different polarization configurations. By
comparison to thinner oxide Si samples, we show that both thermal oxidation andpolarization configuration affect the SH spectra.
6.2

Sample preparation and experimental conditions

To study the effect of thermal oxidation on SHG from the SiO2-Si interface,
we prepared a series of Si samples with various thermally grown oxides. The Si(001)wafers, boron doped to 1-2 Ω ⋅ cm , were first cleaned in an H2SO4/H2O2 solution,
followed by a 100:1 H2O/HF dip. The oxides were grown by rapid thermal oxidation
in a pure O2 environment at a temperature of 1000 oC. The thickness of the oxide wascontrolled by oxidation time; longer oxidation time corresponds to thicker oxide. The
thickness of the oxide was determined by ellipsometry. Interface width was measured
using x-ray scattering [94]. The crystal orientation of the Si wafers was known fromthe sample supplier prior to the experiment.
The experimental setup was the same as in Chapter 1. RA-SHG curves wereobtained by recording the SH signal as the sample was rotated about its surface
normal. TD-SHG signals were obtained by sampling the SH signal as a function of

97
the time of laser illumination on the sample. The laser illumination started from time
zero. The test sample was held in a chamber purged by N2 to reduce O2 induced
charging effect.
6.3

Effect of thermal oxidation on SHG

Surface SHG experiments have shown that the thermal history plays adeterminant role in the structures of the interfacial transition region between the
crystalline Si substrate and the amorphous SiO2 layer [95-100]. These SHG studies
were performed on the Si(111) surface. For this interface, bulk and surface SH
contributions can not been separated by selecting polarization; therefore,
interpretation of the results was difficult. Such a difficulty is partially circumvented
by using the Si(001) surface.
6.3.1

Isotropic and anisotropic SH contributions

SHG from bulk crystalline Si includes both anisotropic and isotropic
components, but SHG from the Si(001) surface or interface is only isotropic. With thepolarizations (g or h) limited to s and p states, the h-polarized SH fields E g( 2, hω ) from aSi(001) crystal face for the g-polarized incident fundamental fields Eg can be writtenas (see Chapter 2)iδE g( 2, ωp ) = [ a 0 , ( g , p ) + a 4 , ( g , p ) cos( 4φ )]e g , p E g2 ,E g( 2, sω ) = a 4 , ( g , s ) sin( 4φ ) eiδ g ,sE g2 .

(6.1)
(6.2)

Here φ is the azimuthal angle defined as being between plane of incidence and the98[100] crystal direction. The SH intensity is proportional to the magnitude square of
the SH field. The coefficients a0 and a4 are the isotropic and anisotropic
contributions to the RA-SHG signal, respectively. They are functions of the relevant
susceptibility elements, Fresnel factors, and dielectric functions, as shown in Chapter
2. Here, a4 is chosen to be real and positive (its phase is in δ ), then a0 is complex ingeneral and can be split into real and imaginary parts, written as a0 = a0, r + ia0,i .
We fit the measured (p, p) and (p, s) RA-SHG signals from the series ofdifferently thermally oxidized Si(001) samples to Eqs. (6.1) and (6.2), respectively,
and obtain the isotropic and anisotropic coefficients. Figure 6.1 showsa0, r , ( p , p ) / a4, ( p , p ) and a0,i , ( p , p ) / a4, ( p , p ) (upper panel), and a4, ( p , s ) (lower panel) as afunction of oxide thickness at different two-photon energies: 3.44 eV, 3.40 eV, and3.26 eV. For the (p, p) polarization, the multiple reflection effect is cancelled out by
dividing the isotropic coefficient by the anisotropic coefficient. This is the reason forusing the ratios a0, r , ( p , p ) / a4, ( p , p ) and a0,i , ( p , p ) / a4, ( p , p ) to characterize the SiO2-Si
interface. As shown, the coefficient a4, ( p , s ) increases with increasing oxide thickness,
which is consistent with the prediction that the oxide layer functions as anantireflection coating. At a two photon energy of 3.26 eV, the magnitude ofa0, r , ( p , p ) / a4, ( p , p ) decreases with decreasing interface width. Note that thinner interface

width corresponds to larger oxide thickness [57, 93]. Such a trend does not hold forall photon energies. As shown, at a two-photon energy of 3.44 eV, a0, r , ( p , p ) / a4, ( p , p )changes from negative to positive as the oxide thickness increases, which means thatits magnitude is not a monotonic function of oxide thickness any more. For all these

99
photon energies, the magnitude of a0,i , ( p , p ) / a4, ( p , p ) is relatively flat with increasing
interface width, except that it drops to lower values when the oxide is very thin. The
strength of the anisotropic and isotropic contributions to the total SH signal also
varies with photon energy, as shown Fig. 6.1.

100

Fit Coefficients (a. u.)0

3.44 3.40 3.26 (eV)
a0,r, (p, p)/a4, (p, p)|a0,i, (p, p)|/a4, (p, p)

-10

4

3

3.44
3.40 (eV) a4, (p, s)
3.26

2

1

0

20
40
Oxide Thickness (nm)60

Fig. 6.1. Ratio of the isotropic to anisotropic SHG components for the (p, p)
polarization (upper panel) and anisotropic SHG components for the (p, s) polarization(lower panel) as a function of oxide thickness at several two-photon energies: 3.44 eV,
3.40 eV, and 3.26 eV.

101
6.3.2

Phase inversion photon energy

The critical two-photon energy at which RA-SHG shows eightfold symmetry
differs for different interfaces. For the (p, p) polarized RA-SHG, we observe thata0, r , ( p , p ) can be either positive or negative depending on the sample and on the photon

energy. For a given sample, it is possible to find a critical photon energy, at which
a0, r , ( p , p ) is zero, in other words, the RA-SHG signal shows eightfold symmetry. We

name the critical photon energy as the phase inversion photon energy. As we showed
in Chapter 4, phase inversion by varying photon energy happens for the (s, p)polarization. We show here that it happens for the (p, p) and (q, s) polarizations as
well. In Fig. 6.2, the upper panel shows the variation of the phase inversion photon
energy with oxide thickness for (p, p), (s, p), and (q, s) polarizations, and the lower
panel shows the (s, p) RA-SHG curves from a 57.9 nm thick oxide covered Si(001)sample at different photon energies. The appearance of eightfold symmetry in RASHG curves is significant. A small deviation of photon energy from the phase
inversion photon energy leads to a large change in the relative height between two
adjacent peaks, which can be easily detected by eye. Therefore, the phase inversion
photon energy can be accurately determined. Combining with the fact that it issensitive to interface conditions (rather than oxide thickness), we can use it as areliable probe for interface identification.

102

Two-Photon Energy (eV)

3.45(p, p)
3.40(q, s)
3.35(s, p)
3.30

0

20
40
Oxide Thickness (nm)SHG Intensity (a. u.)(s, p)

60
3.40 eV200

150
Two-photon energy 3.32 eV

100

50

57.9 nm thick oxide
0

90
180
270
Azimuthal Angle (deg.)

3.26 eV360

Fig. 6.2. Upper panel: oxide thickness dependence of the two-photon energy at which
the RA-SHG signal shows eightfold symmetry for (p, p), (q, s), and (s, p)polarizations. Lower panel: RA-SHG signals from a thermally oxidized Si(001)sample with 57.9 nm thick oxide for the (s, p) polarization, showing either eightfold
or fourfold symmetry.

103
6.3.3

Time-dependent SHG

Thermal oxidation greatly affects the time-dependent SHG (TD-SHG) signal,as shown in Fig. 6.3 for the (p, p) polarized TD-SHG signals from several Si(001)samples with different thicknesses of thermally grown oxide (and different interfaces)
at a two photon energy of 3.26 eV. During the measurement, the azimuthal angle was
fixed at a peak the RA-SHG signal. Before the TD-SHG measurement, the samples
were fully discharged by leaving them in a dark environment for a long time. The
fundamental beam started to continuously illuminate the sample at time zero. The SH
signal was taken with a counting gate time of 0.5 sec. As shown, the TD-SHG signal
increases rapidly with time and gradually saturates for samples with thin oxide layers,
but it stays about the same value for samples with thick oxide layers. The saturated
SH signal decreases with increasing thickness of the oxide layer, consistent withprevious results [66]. The TD-SHG signal was measured in a N2 (research grade
purity) ambient, but no noticeable difference in the TD-SHG response was observed
when the ambient was changed from N2 to O2. This suggests that the ambient-oxideinterface is not likely responsible for the charge trapping, which makes the SH signal
time-dependent. The stronger time-dependence of SHG from thinner oxide covered Si
samples can be attributed to more charge traps at the SiO2-Si interface. The numberof charge traps is reduced by conducting high temperature oxidation for longer time.
The observation of the time-dependence in SHG response indicates that the SH
susceptibility at the SiO2-Si interface is a dynamic rather than a static value for thinoxide covered samples. In addition, the measured SH signal depends on the charging
history and the gate time of the counting system.

104

SHG Intensity (arb. units)

3.26 eVoxide
thickness
(nm)
4.5

1000

6.9
500
11.0
15.0
19.5
33.5
57.9
0

0

20

40

60

80

100

Time (Sec.)

Fig. 6.3. Time-dependent SHG signal for the (p, p) polarization at a two-photon
energy of 3.26 eV for several thermally oxidized Si(001) surfaces with different
thicknesses of the oxide layer.

105
6.3.4

Effect of thermal oxidation on SHG spectroscopy

To systematically study the effect of thermal oxidation on SH spectra, we
compare SHG from different samples. The SH spectrum is expected to change withinterface conditions if the interface SH contribution dominates. Figure 6.4 shows the
SH spectra from a Si(001) sample with 57.9 nm thick thermally grown oxide at a
fixed azimuthal angle of 22.5o for different polarization configurations: (p, p), (s, p),and (q, s). By choosing this azimuthal angle, the anisotropic SH signal is eliminatedtotally for the (p, p) and (s, p) polarizations and mostly for the (q, s) polarization.
However, the measured SH signal still includes both bulk and surface contributions.
Combining the SH spectra results in Chapters 3 and 4, we find that the resonant
energy of 3.27 eV for the (p, p) polarization for the thick oxide sample is significantly
red shifted from that for thin oxide samples, thus thicker oxide leads to a larger redshift. This resonance is known as E1 resonance because its energy is close to the E1
critical point of the band-gap of bulk Si [68]. The directions of the observed strong
energy and intensity variations of this resonance as a result of the different oxidation
procedures are consistent with previous results [59]. The redshift in resonant energy
was previously explained by oxide-induced lattice strain in the Si layers near theinterface, leading to weakening of Si-Si bonds in these layers [33]. This explanationin terms of strain sounds reasonable, but it is probably hard to convince that it is themain effect because methods of direct measurement of the interfacial strain are very
limited. The measured SH signal consists of bulk and surface contributions. In thecase that there is charging effect, the electric-field-induced SHG should also be
included and can be treated as an effective surface SHG. The bulk SH contribution

106
increases with increasing oxide thickness, but the measured total SH intensity
drastically decrease with increasing oxide thickness. Therefore, the relative strength(including phase) between bulk and surface SH contributions varies strongly withoxide thickness. Interference between these two SH contributions can shift themeasured resonant energy, as shown in Chapter 3.
In addition to the dependence of SH spectra on thermal oxidation, we find that
resonant behavior of SHG depends strongly polarization configuration, as shown in
Fig. 6.4. For the same sample, resonant energies and SH intensities clearly differ
between polarizations. For the (p, p) polarization, the resonance energy is 3.27 eV,
but for the (q, s) polarization, it is 3.33 eV. It is interesting to notice that thisresonance almost disappears for the (s, p) polarization. At higher photon energies, theSH intensities for the (q, s) and (s, p) polarizations are much stronger than that for the(p, p) polarization. If the SH response is described by susceptibility tensors,
polarization selection helps in discriminating among tensor elements. These results
indicate that different tensor elements may have different spectra, and the relative
importance of tensor elements depends strongly on photon energy. SHG responses fordifferent polarization configurations represent different types of excitation and bondorientation. In addition, different orientation of the excitation field may have different
ability on charge injection across the SiO2-Si interface, which causes different band
bending at the interface. For example, electrons can be excited more easily across the
angstrom scale interface by an out-of-plane excitation field than an in-plane
excitation field. Moreover, the relative strength between bulk and surface SH
contributions may shift the apparent resonance.

107

SHG Intensity (arb. units)

200

150

100

(p, p)(s, p)(q, s)
57.9 nmthick oxide
φ=22.5 deg.

50

0
3.0

3.33 eV

3.27 eV

3.1

3.2
3.3
3.4
Two-Photon Energy (eV)

3.5

Fig. 6.4. SH spectra from a thermally oxidized Si(001) sample with 57.9 nm thickoxide at a fixed azimuthal angle of 22.5o for different polarization configurations: (p,p), (s, p), and (q, s).

108
6.4

Discussion

To explain the apparent discrepancy in the dependence of SHG on interface
width between the new results and previous results [57, 58], we need to investigate
the differences in experimental conditions and characterization variables used in thesestudies. The tunable photon energy used in this research is larger than the previous
used the value of two-photon energy 2.92 eV [57]. However, photon energy in thevicinity probably does not affect the direction of variation of SHG with interface
width because some photon energies used in this work are detuned from theresonance and no considerable variation with photon energy in the band from thepresent and previous photon energies is expected for linear and nonlinear opticalproperties. The observed trends of the SHG signal varying with interface width aresimilar for several photon energies, as shown in Fig. 6.1. The pulse spectral widths
here are much narrower than the previously used pulse. The fit equations of the RASHG data are slightly different for the two studies. Using real isotropic fit coefficient
for a0 in the previous studies [57, 58] is apparently a crude approximation. When the(p, p) polarized RA-SHG signal shows eightfold symmetry at a certain photon energy,
the isotropic and the anisotropic coefficients are 90o out of phase. This is a clear
evidence of the phase issue in SHG. Taking into account the complex property of a0
would be closer to the physical reality. We also notice that the anisotropic coefficients
a4 should increase with increasing thin oxide thickness because of the multiple

reflection effect, but it was assumed previously to be independent of surface

109
preparation. These considerations may account for the discrepancy on whether theratio | a0, ( p , p ) | / a4, ( p , p ) increases or decreases with interface width.
The time dependence of SHG can be related to the interface width.
Macroscopically, the volume of the medium with broken symmetry affects the SH
intensity. Thermal oxidation performed with shorter time leaves more defects in thetransition region, which can cause stronger charging effect. However, even the TDSHG signal is taken with very short gate time so that the charging effect is negligible
at the initial stage, the initial SH signal from samples with thinner oxides is still
stronger than that from samples with thicker oxide. This is qualitatively understood interms of the fact that the mobility of interfacial electrons is greater for thicker
interface width than for thinner interface.
In the future work, we hope to study how the TD-SHG signals change withoxide thickness while the SiO2-Si(001) interface stays the same. This can be done byusing a series of samples prepared by etching a very thick thermally grown oxide on
the Si(001) surface to various thicknesses. If the SH signal decreases with decreasingoxide thickness and the TD-SHG signal shows similar behavior of negligible timedependence for different samples, our results in this chapter will be further confirmed.
6.5

Summary

We have studied the effect of thermal oxidation on SHG by using a series ofSi(001) samples with different oxidation conditions. By comparing the (p, s)
polarized RA-SHG, we found that multiple reflection in the oxide film enhances themeasured SH signal. By comparing the (p, p) polarized RA-SHG, we found the110interface condition can be characterized by the ratio of the isotropic coefficient to theanisotropic coefficient (multiplication effect is cancelled out). Through spectroscopic
studies, we found that the RA-SHG signal can be either fourfold symmetric oreightfold symmetric depending on photon energy. This leads to the important issuesof the relative phase between the isotropic and anisotropic SH contributions.
Eightfold symmetric RA-SHG signals were found for all the (p, p), (s, p) and (q, s)
polarizations, and the corresponding photon energy varies monotonically withinterface width, thus it provides a reliable way for interface identification. We showedthat the time-dependence of the SH signal is stronger for a larger interface width.
Thermal oxidation affects SH spectra and the resonant behaviors for the same sample
are significantly different for different polarization configuration. These results
suggest that surface SHG is a sensitive probe for oxidized Si interfaces, but a probewith multiple effects.

111
Chapter 7
Conclusions

The experiments described here have helped improve the understanding thephysics of SHG at Si surfaces or interfaces. The experimental results show that the
SH spectroscopy from Si surfaces is closely related to the band structure of the bulk
Si, but surface modification can shift the apparent resonance through interference
among or between different SH contributions. The major advantage of SHG is theextreme surface sensitivity. As shown for oxidized Si, hydrogen terminated Si, andultrathin Cr coated SiO2-Si surfaces, the measured RA-SHG signals for appropriate
polarizations at a certain photon energy are apparently different between samples.
This sensitivity can be greatly enhanced by combination with the phase informationof the surface SH field. The observed peak-to-valley transition of RA-SHG byvarying photon energy or by surface modification clearly demonstrates the phasevariation of the surface SH field. With this information, SHG from Cr-SiO2-Si
structures and thermally oxidized Si surfaces have been comprehensively studied andnew results of SHG have been obtained with respect to the sources of SHG.
A major weakness of the SHG technique is the lack of specificity for a certain
given interfacial property. The observed variation of SHG with surface condition ispossibly explained by alternative effects. Many conclusions drawn from SHG studies
are not straightforward. It is also clear that SHG from SiO2-Si interfaces is
considerably time-dependent, especially for ultrathin oxide surfaces. Measured SH
signals depend strongly on experimental conditions, thus quantitative characterization

112
is difficult. Another problem with the SHG probe is the possibility of damaging thesurface with the high field intensity required to generate an observable response. For
Si surfaces, our SHG studies so far have given adequate signals well below thethreshold for optical damage. Further understanding of the interfacial electronic
structure and a feasible microscopic theory are needed to explain microscopically thephase variation of SHG and the SH spectroscopy.
However, the high sensitivity of the SHG probe to interface properties can be
put to the best use by comparing SH signals from different surfaces, as I
demonstrated in this thesis. The use of SHG to monitor the time-evolution of thesurface or interface and the carrier dynamics at the surface or interface is also very
exciting. I have performed a new spectroscopic study of SHG from technologicallyimportant Si(001) surfaces. The SH spectroscopy with the phase information of theSH field is far more sensitive to interface properties than that with the SH intensity
only.

113
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121
Appendix
SHG from vicinal Si(001) surfaces

Here we present the results of rotational-anisotropy SHG (RA-SHG) fromvicinal Si(001) surfaces. The surface normal direction of the vicinal Si(001) sample isaway from the principal [001] crystal axis for a small offset angle, α , which isnamed as vicinal angle, as shown in Fig. 2.2 in Chapter 2. Moreover, the offset
direction is chosen to be in a mirror plane of symmetry of the Si crystal. The vicinal
angle dependence of the (p, p) polarized RA-SHG signal is shown in Fig. A.1. The
photon energy dependence of the RA-SHG signal from vicinal Si(001) surfaces fordifferent polarization configurations are shown in Figs. A.2, A.3, A4, and A.5. These
results will not only help improve the understanding of surface SHG, but will also
help with the understanding of the step structures and energies on vicinal Si surfaces
or interfaces.
The experimental conditions were the same as in Chapter 3, except that the
spectroscopic studies here were also performed at higher photon energies, which wereobtained by frequency doubling of the output of an OPO, as described in Chapter 1.
All measurements were carried out at room temperature in a N2 ambient.
Three sets of vicinal Si(001) samples were investigated. The first set were
natively oxidized vicinal Si(001) samples with different vicinal angles of 0o, 1o, 2o, 3o,
4o, and 5o, designated as NO-Si V. The [001] crystal direction of the vicinal (off axis)
Si(001) wafer was off from the surface normal toward the [110] direction for a small
vicinal angle. Two more sets of samples, the thermally oxidized vicinal Si(001)samples, designated as TO-Si V, and the hydrogen terminated vicinal Si(001) samples,122
designated as H-Si V, were prepared from the NO-Si V samples with the same
preparation conditions as in Chapter 3.
The theoretical predictions of the symmetry properties of the RA-SHG signals
from vicinal Si(001) surfaces are shown as Eqs. (2.40a) and (2.40b) in Chapter 2. The
SH intensity in all of the figures is normalized in the same way as that in Chapter 3. It
is important to note that the azimuthal angle is defined to be between the plane ofincidence and the projection of the [110] axis in the surface plane. Moreover, when
the azimuthal angle ψ =0, the incident beam is toward the downward miscut directionof the vicinal surface (or downward steps on the vicinal surface), as shown in Fig. 2.2
in Chapter 2.

123

1000
5

NO-Si V

800

4SHG Intensity (arb. units)

600

2

3

1

400

0
Vicinal Angle (deg.)

200(p, p)
3.22 eV0500
5TO-Si V

400

3

4

300
200

0
Vicinal Angle (deg.)

100
0

2

1

0

90

180

270

360

Azimuthal Angle (deg.)

Fig. A.1. (p, p) polarized RA-SHG signals at a two-photon energy of 3.22 eV from
natively oxidized vicinal Si(001) surfaces (NO-Si V) (upper panel) and thermally
oxidized vicinal Si(001) surfaces (TO-Si V) (lower panel) with different vicinal
angles of 0o, 1o, 2o, 3o, 4o, and 5o.

SHG Intensity (arb. units)

124

1x10

5

5x10

4

NO-Si V5(p, p)

0
1x10

5

5x10

4

TO-Si V5

4.48
0

0

4.41

4.34 (eV)

90
180
270
Azimuthal Angle (deg.)360

Fig. A.2. (p, p) polarized RA-SHG signals from a natively oxidized Si sample withvicinal angle of 5o (NO-Si V5) (upper panel) and a thermally oxidized Si sample withvicinal angle of 5o (TO-Si V5) (lower panel) at two-photon energies 4.34 eV, 4.41 eV,and 4.48 eV.

125(s, p)(p, p)

5000

NO-Si V5

4000
400

3000

SHG Intensity (arb. units)

3.10 eV
3.26 eV
3.40 eV
3.49 eV

600

2000
200
1000
0

0
(p, s)(s, s)

100

100

50

50

0

0
0

180
Azimuthal Angle (deg.)

360

0

180

360

Azimuthal Angle (deg.)

Fig. A.3. RA-SHG signals from a natively oxidized Si sample with vicinal angle of 5o(NO-Si V5) at several two-photon energies of 3.10 eV 3.26 eV, 3.40 eV, and 3. 49 eV
for different polarizations: (p, p), (s, p), (p, s), and (s, s).

126(s, p)(p, p)

3.10 eV
3.26 eV
3.40 eV
3.49 eV600
TO-Si V51000

SHG Intensity (arb. units)

400
500
200

0

0
(p, s)(s, s)

100

100

50

50

0

0
0

180
Azimuthal Angle (deg.)

360

0

180

360

Azimuthal Angle (deg.)

Fig. A.4. RA-SHG signal from a thermally oxidized Si sample with vicinal angle of
5o (TO-Si V5) at several two-photon energies of 3.10 eV 3.26 eV, 3.40 eV, and 3. 49
eV for different polarizations: (p, p), (s, p), (p, s), and (s, s).

127(s, p)SHG Intensity (arb. units)(p, p)
1500

600

1000

400

500

200

0

0

3.10 eV
3.26 eV
3.40 eV
3.49 eV

H-Si V5(p, s)(s, s)

100

100

50

50

0

0
0

180
Azimuthal Angle (deg.)

360

0

180

360

Azimuthal Angle (deg.)

Fig. A.5. RA-SHG signal from a hydrogen terminated Si sample with vicinal angle of
5o (H-Si V5) at several two-photon energies of 3.10 eV 3.26 eV, 3.40 eV, and 3. 49
eV for different polarizations: (p, p), (s, p), (p, s), and (s, s).

128