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Electron-Phonon Interactions and Quantum Confinement Effects on Optical Transitions in Nanoscale Silicon Films

ProQuest Dissertations and Theses, 2011
Dissertation
Author: Vimal Kumar Kamineni
Abstract:
Theoretical studies have attributed the temperature dependence of the linear optical response (dielectric function) of bulk semiconductors to electron-phonon interactions and thermal expansion of the lattice. However, the role of phonons in the optical properties of nanoscale structures is often overlooked. This thesis systematically investigates the impact of both carrier confinement and electron-phonon interactions using nanoscale films of silicon in crystalline silicon quantum wells ( c -Si QW). Spectroscopic ellipsometry (SE) is a linear optical technique used to of extract the dielectric function and thickness of very thin films. X-ray reflectivity (XRR) was used as the complementary thickness metrology method. The dielectric function of c -Si QWs with thicknesses ranging from 2 nm to 10 nm is found to have a significant dimensional dependence. The major differences in the dielectric function with thickness were observed at the critical points (direct gap transitions) in silicon. Critical points (CP) are the vertical band-to-band transitions with maximum probability due to high joint density of states. Specifically, a blue shift (higher energy shift) in the critical point energy (E CP ) and enhanced lifetime broadening (Γ) of the excitonic E 1 (interband transition in the direction) critical point as a function of thickness was observed. Direct space analysis was used to extract ECP and Γ from the dielectric function. Low-temperature (Liquid- 4 He) SE measurements were performed to study the contribution of electron-phonon interactions on the ECP and Γ. In contrast to bulk silicon, the E1 direct gap transition of sub-9 nm silicon films becomes more intense than the non-excitonic E2 direct gap transition as temperature decreases. This is evidence of the importance of electron-phonon interactions on the direct gap transitions of nanoscale semiconductor structures. The temperature dependence of the critical point energy was modeled with the Bose-Einstein statistical factor to extract the contribution of acoustic phonon frequency. An elastic continuum theory approach was used to calculate the acoustic phonon energy with a change in the thickness of the c -Si QW and with a change in the dielectric layer on top of the QW. The phonon dispersion of the quantum confined layers was further altered by changing the dielectric layer above the nanoscale silicon. Change in the stress, interface and crystal quality due to sample fabrication steps in the nanoscale silicon films can also alter its dielectric function. High-resolution x-ray diffraction was used to measure the strain and crystal quality, while photoreflectance (a surface sensitive characterization method) was used as the non-linear optical technique to probe the surface of the nanoscale films to measure the energy and lifetime broadening of the E1 critical point.

TABLE OF CONTENTS

DEDICATION

................................ ................................ ................................ ...................

iii

ACKNOWLEDGEMENTS

................................ ................................ ...............................

iv

ABSTRACT

................................ ................................ ................................ .......................

vi

LIST OF FIGURES

................................ ................................ ................................ ...........

xi

LIST OF TABLES

................................ ................................ ................................ ...........

xvi

LIST OF ACRO NYMS

................................ ................................ ................................ ..

xvii

GLOSSARY

................................ ................................ ................................ ....................

xix

CHAPTER I

................................ ................................ ................................ ........................

1

1.1 Introduction to Silicon - on - insulator Technology

................................ .............

2

1.2 Dielectric Response of Semiconductors

................................ ...........................

5

1.2.1 Linear Optical Response

................................ ................................ ....

7

1.2.2 Non - linear Optical Response

................................ .............................

9

1.3 Electronic Band Structure of Semiconductors

................................ ................

10

1.3.1 Electronic Band Structure from

Metho d for Si

.......................

13

1.3.2 Direct Interband Transitions

................................ ............................

14

1.3.3 Indirect Interband Transitions

................................ ..........................

19

1.4 Phonon Dispersion of Semiconductors

................................ ...........................

20

1.5 Electron - phonon Interactions

................................ ................................ ..........

24

1.6 Nanoscale Effects in Semiconductors

................................ .............................

32

1.6.1 Carrier Confinement

................................ ................................ .......

32

1.6.2 Modifications of the Electronic Band Structure

..............................

34

ix

1.6.3

Phonon Confinement

................................ ................................ .......

37

1.6.4 Changes in the Phonon Dispersion Curves

................................ ......

38

1.6.5 Electron - phonon interactions

................................ ...........................

39

1.7 Motivation

................................ ................................ ................................ .......

40

1.8

Content of Thesis

................................ ................................ ............................

41

CHAPTE R II

................................ ................................ ................................ .....................

43

2.1 Sample Preparation of c - Si QWs

................................ ................................ ....

43

2.2 Material Characterization

................................ ................................ ...............

45

2.2.1 Spectroscopic Ellipsometry

................................ .............................

45

2.2.1.1 Reflection of p - and s -

Polarized Light Waves ...................

46

2.2.1.2 Reflection Coefficients of Thin Films

................................

48

2.2.1.3 Optical Properties and Thickness

................................ .....

49

2.2.1.4

Instrumentation

................................ ................................ .

50

2.2. 1.5 Data Collection and Analysis

................................ ...........

51

2.2.2

Photoluminescence Spectroscopy

................................ ....................

53

2.2.3

Photoreflectance

................................ ................................ ...............

56

2.2.4

X - ray Reflectivity

................................ ................................ ............

59

2.2.5 High Resolution X - ray Diffraction

................................ ..................

63

CHAPTER III

................................ ................................ ................................ ...................

66

3.1 Experimental Data for c - Si QWs

................................ ................................ ....

66

3.2

Thickness Dependent Dielectric Function

................................ ......................

68

3.3 Extraction of Parameters through Direct Space Analysis

...............................

69

3.4 Energy Shift and Lifetime Broadening in CP for c - Si QWs

...........................

71

x

3.5 Possible Effects for the Observations in the Nanoscale Si Films

...................

72

3.5.1 Effects of Crystal Quality

................................ ................................

73

3.5.2 Impact of Stress on the Optical Response ................................ ........

76

3.5.3 Influence of Interface Quality on E CP

and Γ

................................ ....

78

3.6 Quantum Confinement Effects and Electron - phonon Interactions

.................

81

3.7 Temperature Dependent Dielectric Function

................................ ..................

83

3.7.1 Critical Point Energy and Lifetime Bro adening

..............................

83

3.7.2 Electron - Phonon Interactions and Average Phonon Energy

...........

87

3.8 Acoustic Phonon Mode Calculations for SOI

................................ .................

88

3.9

Impact of Surrounding Dielectric Layers

................................ .......................

90

CHAPTER VI

................................ ................................ ................................ ...................

94

REFERENCES…………………………………………………………………………..98

xi

LIST OF FIGURES

CHAPTER I

Figure 1.1:

Growth in the consolidated sales of SOI wafers in millions of euros. 2

...........

2

Figure 1.2:

Smart - Cut ®

SOI wafer fabrication steps.

................................ ........................

3

Figure 1.3:

NMOSFET and PMOSFET built on a SOI wafer.

................................ ..........

4

Figure 1.4:

Illustration of an electron bound to a nucleus in the ball - and - spring model.

..

5

Figure 1.5:

Real ( ε 1 ) and imaginary ( ε 2 ) parts of the dielectric function.

..........................

8

Figure 1.6:

(a) First Brillouin zone of a Si FCC lattice and (b)

electronic band structure of Si calculated using

method. 19

................................ ................................ ...............

14

Figure 1.7:

Direct band gap transition in a semiconductor.

................................ .............

15

Figure 1.8: CPs shown (a) in the electronic band structure 19 of Si and (b) imaginary part of the dielectric function of bulk c - Si at room temperature between 1 eV to 6 eV.

.........

17

Figure 1.9:

Dielectric function calculated by Rohlfing and Louie of Si with (sol id lines) and without (dashed lines) excitonic interactions. 21

................................ .........................

18

Figure 1.10: Optical energy contour in the (110) plane for the b and structure of silicon. 28

The energy values (eV) are indicated as numbers and stars indicate optical CPs.

...........

19

Figure 1.11:

Phonon assisted indirect band gap transition in a semiconductor.

..............

19

Figure 1.12:

Models to calculate phonon dispersion curves in

semiconductors

(a) force constant model, (b) shell model, (c) bond model, and (d) adiabatic bond charge method (adapted figure). 19

................................ ................................ ................................ .............

22

Figure 1.13:

Theoretical phonon dispersion curves 34

(solid lines) and experimental data 35

(circles) of bulk Si.

................................ ................................ ................................ .

23

xii

Figure 1.14:

Feynman diagrams of the temperature dependent electron - phonon interactions (a) Fan term, (b) lowest order DW term, and (c) higher order DW term.

.....

27

Figure 1.15:

Spectral function

g 2 F (Ω) for the (a) highest VB and (b) lowest CB states. 41 ................................ ................................ ................................ ................................ ...........

29

Figure 1.16:

Spectral function

g 2 B (Ω) for the (a) highest VB and (b) lowest CB states. 40 ................................ ................................ ................................ ................................ ...........

31

Figure 1.17:

Band diagram of SOI structures.

................................ ................................ .

33

Figure 1.18: Electronic band structure of 1 nm

(8 atoms)

thick (100) Si nanofilm. 48

.....

35

Figure 1.19:

VBM and CBM energy shifts in nanoscale silicon films with respect to the bulk energies taken as zero. 53 ................................ ................................ ............................

36

Figure 1.20: Phonon dispersion curve of a silicon nanofilm of thickness 0.543 nm. 64

...

39

Figure 1.21:

Moore’s Law and the use of ETSOI. ................................ ...........................

40

Figure 1.22:

Flowchart o f the experimental characterization performed on c - Si QWs.

..

42

CHAPTER II

Figure 2.1:

Sequence of steps to prepare c - Si QWs.

................................ .......................

44

Figure 2.2:

Electric and magnetic field vectors for s -

and p -

polarized light.

.................

47

Figure 2.3:

Reflection and transmission by an air - film - substrate system. .......................

48

Figure 2.4:

Optical configuration of the (a) Woollam VUV - VASE and (b) Woollam RC2 (P –

Polarizer, AR –

Auto retarder, A –

Analyzer and C –

Compensator).

......................

51

Figure 2.5:

Imaginary part of the dielectric function of bulk single crystal silicon at 125 K extracted after using different ice op tical constants.

................................ .....................

52

Figure 2.6:

Customized window block layout of the cryostat configured for PL measurements.

................................ ................................ ................................ ...................

54

xiii

Figure 2.7: Photoluminescence setup for measuring indirect interband transitions in c - Si QWs.

................................ ................................ ................................ ................................ .

55

Figure 2.8:

Schematic diagram of the photoreflectance effect in an n - type semiconductor.

................................ ................................ ................................ ..................

56

Figure 2.9:

Experimental setup for photoreflectance measurements.

..............................

59

Figure 2.10:

A typical x - ray reflectivity profile demonstrating the effects of thickness, density, and roughness.

................................ ................................ ................................ .....

60

Figure 2.11:

Simulated HRXRD rocking curve of a 300 Å epitaxial Si 0.8 Ge 0.2

layer on Si. ................................ ................................ ................................ ................................ ...........

64

Figure 2.12: HRXRD in (a) out - of - plane and (b) in - plane geometry.

.............................

65

CHAPTER II I

Figure 3.1:

Experimental data of the (a) real (tanΨ) and (b) imaginary parts ( e i Δ )

of the ratio ( ρ ) of the Fresnel reflection coefficients at 70° for ETSOI.

................................ .....

67

Figure 3.2: Experimental data and simulated data at 70° for a 2 nm ETSOI.

.................

68

Figure 3.3: Imaginary part of the dielectric function of c - Si QWs at room temperature. The data is for c - Si QWs with a native oxide layer at the surface.

................................ ...

69

Figure 3.4:

(a) Dielectric function and (b) second derivative of the dielectric function of bulk Si at a local region of the E 1

CP. ................................ ................................ ...............

70

Figure 3.5: Second derivative of the imaginary part of the dielectric function of c - Si QWs at room temperature near the E 1

CP.

................................ ................................ .......

71

Figure 3.6:

(a) Blue shift in the energy and (b) enhanced lifetime broadening of the E 1

CP with decreasing thickness. ................................ ................................ ...........................

72

xiv

Figure 3.7: Silicon - on - insulator wafers with the top Si layer showing (a) tilt and (b) twist compared to the bulk silicon.

................................ ................................ ............................

73

Figure 3.8: Out - of - plane diffraction of 004 symmetric reflection (a) ω scan and (b) ω/2θ scan for a 10 nm c - Si QW.

................................ ................................ ................................

74

Figure 3.9:

In - plane diffraction of (a) 400 and (b) 220 reflection for a 10 nm c - Si QW.

75

Figure 3.10: TEM image of a 6.5 nm silicon layer.

................................ .........................

76

Figure 3.11:

Reciprocal space maps near the Si 004 reflection for a 10 nm c - Si QW.

...

77

Figure 3.12: Experimental and simulated data of the photoreflectance spectra of bulk silicon at room temperature (T=290 K).

................................ ................................ ...........

78

Figure 3.13:

Photoreflectance spectrum of bulk Si and c - Si QWs at room temperature (290 K) near the E 1

CP. ................................ ................................ ................................ .....

80

Figure 3.14: The experimental measured E 1

CP shift and the calculated E 1

CP shift due to confinement of carriers with film thickness.

................................ ................................

82

Figure 3.15:

Temperature dependent imaginary part of the dielectric function (  2 ) for (a) 9 nm, (b) 7 nm and (c) 2 nm of c - Si QW. 112

................................ ................................ .....

83

Figure 3.16: Blue shift in the energy with respect to temperature near the E 1

CP.

.........

84

Figure 3.17:

Second derivative of the temperature dependent real part of the dielectric function for (a) bulk Si and (b) 2.5 nm c - Si film.

................................ .............................

85

Figure 3.18:

The thickness and temperature dependence of the (a) energy and (b) lifetime broadening of E 1

CP (95 % confidence limits for the energy and broadening are less than 10 meV). 112

................................ ................................ ................................ .......................

86

Figure 3.19:

Acoustic phonon energy of she ar modes near the zone center for (a) free standing c - Si and (b) 4 nm SiO 2 / c - Si/4 nm SiO 2

at various c - Si thicknesses.

..................

90

x v

Figure 3.20:

Imaginary part of the refractive index of c - Si QWs (~ 5 nm) with native oxide, 20 nm SiO 2 , and 10 nm HfO 2

with a SiO 2

interfacial layer.

................................ ..

91

Figure 3.21:

Specular x - ray reflectivity of hafnium oxides (~ 10 nm) on ETSOI with different c - Si QW thicknesses.

................................ ................................ .........................

92

Figure 3.22: Second derivative of the imaginary part of the dielectric function of c - Si QW with a HfO 2

surface layer. Inset: (a) Energy and (b) lifetime broadening (Γ) of the E 1

CP extracted using direct space analysis. 113

................................ ................................ .

93

CHAPTER IV

Figure 4.1:

Room temperature (RT) photoluminescence spectra of (a) p - type InP (b) n - type GaAs (c) GaN + QW and (d)

GaN defect states taken from the system described in section 2.2.2.

................................ ................................ ................................ .....................

96

Figure 4.2: Forming gas annealing effects on the RT PL of Si with 25 nm thermal oxide. ................................ ................................ ................................ ................................ ...........

97

Figure 4.3:

Photoluminescence spectra at 10 K of a ~ 7.6 nm c - Si QW with a 12.7 nm thick thermal o xide after a forming gas anneal.

................................ ................................

98

xvi

LIST OF TABLES

Table 1.1:

The Hamiltonians under Born - Oppenheimer approximation ……………… ..25

Table 3.1:

The E 1

critical point energy extracted from the photoreflectance data ……...81

Table 3.2 :

Average phonon frequency

(90 % confidence limits) at various

c - Si film thickness es ……………………………………………………………………………….87

xvii

LIST OF ACRONYMS

AOI:

Angle of Incidence

BE:

Bound Exciton

BOX:

Buried Oxide

BZ:

Brillouin Zone

CB:

Conduction Band

CCD:

Charge - coupled D evice

CMOS:

Complementary Metal Oxide Semiconductor

CP:

Critical Point

CW:

Continuous Wave

DC:

Direct current

DFT:

Density Functional T heory

EHD:

Elec tron - hol e

Drops

EHL:

Electron - hole Liquid

EHP:

Electron - hole Plasma

EM:

Electromagnetic

ETSOI:

Extremely Thin Silicon - on - insulator

FCC:

Face

C entered Cubic

FE:

Free Exciton

GPIB:

General Purpose Interface Bus

GWA:

Green Function (G) Columbic Interaction (W) A pprox imation (A )

HRTEM:

High Resolution Transmission Electron Microscope

xviii

HRXRD:

High Resolution X - ray Diffraction

IC:

Integrated Circuit

IR:

Infrared

ITRS: International Technology Roadmap for Semiconductors

LDA :

Local Density A pproximations

MOS:

Metal Oxide Semiconductor

MOSFET:

Metal Oxide Semiconductor Field Effect T ransistor

OPSL:

Optically Pumped Semiconductor Laser

PL:

Photoluminescence

PMT:

Photomultiplier Tube

PR:

Photoreflectance

QM:

Quantum Mechanics

QW:

Quantum Well

RC2:

Dual Rotating Compensator

SE:

Spectroscopic Ellipsometry

SIMOX:

Separation by Implantation of Oxygen

SLD:

Scattering Length Density

SOI:

Silicon - on - insulator

TEM:

Transmission Electron Microscope

VB:

Valence Band

VUV:

Vacuum Ultraviolet

XRR:

X - ray Reflectivity

xix

GLOSSARY

m : mass

e : electronic charge

Γ :

electronic collision rate

or lifetime broadening

ω : frequency

ω 0 : resonant frequency

̅ : dipole moment

̅ ( ω): atomic polarizability

̅ : polarization

̅ : electric field

̅ : magnetic field

χ : susceptibility

̅ : displacement field

N : complex refractive index ( N

=

n

+

ik )

n : refractive index

k : extinction coefficient

or wave vector

α : absorption coefficient

ε 1 : dielectric constant

ε : dielectric function ( ε

=

ε 1

+

i ε 2 )

ε 1 : real part of the dielectric function

ε 2 : imaginary part of the dielectric function

d : thickness

xx

H : Hamiltonian

p : momentum

V : potential energy

E : e nergy

E C : conduction band energy

E V : valence band energy

: reduced Planck constant

R p : reflection coefficient of p - polarized light

R s:

reflection coefficient of s - polarized light

θ i : incident angle

θ t : refracted angle

θ r : reflected angle

β : phase angle

Ψ: amplitude ratio of the reflection coefficients

Δ : change in phase difference

R : reflect ivity

T : transmitted light

: reduced interband mass

A : amplitude

θ : phase

E CP : critical point energy

c : speed of light

λ : wavelength of light

xxi

r e : Bohr electron radius

α c : critical angle

1

CHAPTER I

Introduction

Since

the advent of electronic technology, novel

consumer electronics have been

manufactured

with the goal of attracting

customers. These electronic devices have improved performance with respect to their

predecessor devices with higher m emory storage capacity,

faster

computing performa nce,

lower cost per tra nsistors ,

and higher transistor density . These developments in the

field of semiconductor manufacturing

along with the overall minimization of the size of transistors require cutting - edge metrology tools for characterization.

To impr ove the capabilities of current metrology methodologies,

it is imperative to understand the semiconductor physics at nanoscale dimensions. Currently, f undamental properties of dimensionally confined

crystalline semiconductor materials

(nanofilms, nanowires and nanodots)

are being explored by theoretical and experimental research groups . The

device rese arch efforts in the semiconductor

industry

are moving towa rds the use of nanoscale single crystal

silicon quantum wells ( c - Si QW) on ins ulator structures , which are

referred to as

extremely thin silicon - on - insulator (ETSOI). 1

A brief background including the advantages of

SOI technology will be provided in this chapter

along with

introducing

the necessary concepts

of semiconductor physics and the optical response of semiconductors ,

which will enable us to draw conclusive results from the experiments

on c - Si QWs . The nanoscale effects on electronic band structure and phonon dispersion curves are

also

presented . Furthermore, the dimensional c onfinement of carrier s

and phonon s

along with their inter actions are discussed. In addition, this chapter will present the motivation behind this

2

research work and the outline of the thesis.

Several of these concepts in optical processes were introduced to

th e author through course work and will be reviewed in this chapter.

1.1 Introd uction to Silicon - on - insulator T echnology

SOI technology has been prevalent in day - to - day life in the last decade. SOI wafers are currently being used across various industries utilizing microelectronics. Chips

built on SOI wafers are widely used in computing, gaming, networking, ultra low power electronics, automotive industry,

phot onics ,

etc. 2

The consolidated sales of SOI wafers have increased immensely in the last few years as shown in Fig. 1. 1.

These sa les will continue to increase along with

the need for improved performance of microelectronic devices . We note that the requireme nts for SOI are described in the International Technology Roadmap for Semiconductors (ITRS).

Figure 1 . 1:

Growth in the consolidated sales of SOI wafers in millions of euros . 2

3

Currently, SOI wafer s

are

fabricated using the methods developed by the research efforts of numerous material scientists over the past 30 years. 3

Some of the key SOI wafer fabricati on

techniques are

wafer bonding, separation by implantation of oxygen

(SIMOX) , recryst allization and epitaxial growth . The front

runner of these techniques is the method

developed by SOITEC called the Smart - Cut ®

process. 4 , 5 , 6

Figure 1. 2 :

Smart - Cut ®

SOI wafer fabricati on

steps .

The process steps for fabricating the SOI wafer using the Smart - Cut process are shown in Fig. 1. 2. Intially ,

a device wafer ( on which the final device is grown) is oxidized to create a buried oxide (BOX) layer and im planted (>10 16 cm - 2

dose) with hydrogen ions to create a hydrogen - rich layer. The device and handle wafers (bulk silicon in SOI) are then cleaned and bonded through hydrophilic bonding of the hydroxyl group (OH - ) present on the surface of the wafers. 7

Foll owing the bonding step, the pair of wafers are thermally treated through a two - step process: ( 1) a low - temperature annealing (400 - 600 o C) step to

promote the fracture of the bonded pair in the plane of the hydrogen - rich layer due to the formation of brittl e silicon hydride ,

and ( 2) a high - temperature annealing (~1000 o C) step to strengthen the hydrophilic bonding. The final device thickness, non - un iform ity and smooth ness of the

Smart - Cut process was further

4

improved by S. V. Srikrishnan by creating an etch stop layer via chemical vapor deposition or implantation. 7

The performance of bulk CMOS (conventional bulk silicon) can be improved by replac eme nt

with SOI technology. The CMOS devices in SOI technology are built on a silicon layer over the BOX layer. The insulating layer isolates the chips from the bulk of the substrate

and reduces the parasitic capacitance by eliminating the space charge layer e xtending into the substrate .

The decrease in parasitic capacitance lowers the power consumption

(35 -

70

%)

and improv es

the device performance

(20 -

35 %) . 8 , 9

Figure 1. 3 :

NMOSFET and PMOSFET

built on a SOI wafer.

NMOSFETs and PMOSFETs built

on SOI wafers are isolated

both laterally

( via shallow trench isolation )

and vertically

( via BOX layer ) as shown in Fig. 1. 3. Furthermore , the electron - hole pairs created in the substrate

of microelectronics

due to cosmic radiat ion

in

space applications

are isolated from the transistors;

thereby reducing the soft errors in SOI based devices. Additionally , SOI wafers completely eliminate

inter - device leakage and

latch - up due to th e excellent isolation of CMOS devices.

Precise cont rol of the thickness of the SOI wafers has also made it feasible to envision low cost manufacturing of f ully - depleted

SOI (FD - SOI) transistors and

FinFET structures .

5

1.2 Dielectric Response of Semiconductors

Dielectric response of semiconductors

can be der ived from classical mechanics, statistical mechan ics

( a density matrix approach ) ,

and quantum mechanics. 10 , 11 , 12 , 13

A comprehensive description of the dielectric response of semiconductors will require ( 1 ) a self - consistent quantum mechanical approa ch that accounts for all the time dependent (Liouville equation) microscopic details of all the possible interactions of the incident light with the material ,

( 2 ) followed by a solid state physics

approach to relate the microscopic properties of individual

atoms to the macroscopic bulk properties. 12

The purpose of this section is

not to give full quantum mechanical

derivations of the fundamental equ ations governing the dielectric response of semiconductors ,

but to derive

the dielectric function from the

simple

equations of motion (classical mechanics) model ,

which

are

useful in understand ing

the

necessary concepts of the optical response of semiconductors. A ball - and - spring model ,

shown in Fig. 1. 4 ,

is used to calculate the equations of motion for a negatively charged electron (mass m ) bound to a positively charged atomic nucleus (mass M ) with a spring. Since M

>> m , the p osition of the atomic nucleus is fixed. 14

Figure 1. 4 :

Illustration

of an electron

bound to a nucleus in the ball - and - spring model .

6

The motion of the electron when light (EM field) is incident on the sample can

be described using a Lorentz oscillator as shown in Eq. (1.1):

(1.1)

where e

is the electronic charge ,

̅

is the time variant radial displacement between the positive and negat ive charge , Γ is the electron collision rate, ω 0

is the resonant frequency, ω

is the driving frequency and

̅

is the time variant local electric field.

The solution for Eq. (1.1) is shown in Eq. (1.2) and The

induced dipole moment ( ) due to charge separation by the incident electric field is given by Eq. (1.3):

(1.2)

(1.3)

A linear relationship between the induced dipole moment and local electric field exists when the displacement is smal l

( r

<< λ) . 10

The constant of proportionality of this linear relationship (

̅

̅

is the frequency - dependent atomic polarizability (

̅ ( ω )) given by Eq. (1.4 ). This provides a simple model that allows

further discussion on the di electric function and in Sect.

1.3.2 the quantum mechanical description of the dielectric function of a semiconductor is shown.

(1.4)

7

1.2 .1 Linear Optical Response

In the linear regime ,

the macroscopic polarization (

̅ ) is linearly dependent on the electric field

while

the optical properties are independent of the intensity of the incident electric field.

Full document contains 136 pages
Abstract: Theoretical studies have attributed the temperature dependence of the linear optical response (dielectric function) of bulk semiconductors to electron-phonon interactions and thermal expansion of the lattice. However, the role of phonons in the optical properties of nanoscale structures is often overlooked. This thesis systematically investigates the impact of both carrier confinement and electron-phonon interactions using nanoscale films of silicon in crystalline silicon quantum wells ( c -Si QW). Spectroscopic ellipsometry (SE) is a linear optical technique used to of extract the dielectric function and thickness of very thin films. X-ray reflectivity (XRR) was used as the complementary thickness metrology method. The dielectric function of c -Si QWs with thicknesses ranging from 2 nm to 10 nm is found to have a significant dimensional dependence. The major differences in the dielectric function with thickness were observed at the critical points (direct gap transitions) in silicon. Critical points (CP) are the vertical band-to-band transitions with maximum probability due to high joint density of states. Specifically, a blue shift (higher energy shift) in the critical point energy (E CP ) and enhanced lifetime broadening (Γ) of the excitonic E 1 (interband transition in the direction) critical point as a function of thickness was observed. Direct space analysis was used to extract ECP and Γ from the dielectric function. Low-temperature (Liquid- 4 He) SE measurements were performed to study the contribution of electron-phonon interactions on the ECP and Γ. In contrast to bulk silicon, the E1 direct gap transition of sub-9 nm silicon films becomes more intense than the non-excitonic E2 direct gap transition as temperature decreases. This is evidence of the importance of electron-phonon interactions on the direct gap transitions of nanoscale semiconductor structures. The temperature dependence of the critical point energy was modeled with the Bose-Einstein statistical factor to extract the contribution of acoustic phonon frequency. An elastic continuum theory approach was used to calculate the acoustic phonon energy with a change in the thickness of the c -Si QW and with a change in the dielectric layer on top of the QW. The phonon dispersion of the quantum confined layers was further altered by changing the dielectric layer above the nanoscale silicon. Change in the stress, interface and crystal quality due to sample fabrication steps in the nanoscale silicon films can also alter its dielectric function. High-resolution x-ray diffraction was used to measure the strain and crystal quality, while photoreflectance (a surface sensitive characterization method) was used as the non-linear optical technique to probe the surface of the nanoscale films to measure the energy and lifetime broadening of the E1 critical point.