Absorption properties of β-Sn nanocrystals in SiO2

Jonas Gunsel

Winter 2010

i

This thesis has been submitted to the faculty of science at Aarhus Uni-

versity to fulfill the requirements for obtaining a PhD degree. The work

has been carried out under supervision of professor Brian Bech Nielsen

at the department of Physics and Astronomy and the Interdisciplinary

Nanoscience Center (iNANO).

ii

List of publications

• Jonas Gunsel, Jacques Chevallier and Brian Bech Nielsen. Absorption

enhancement by a layered structure compared to randomly distributed

β-Sn nanocrystals in SiO2, in preparation for Physical Review B

iii

List of Figures

1.1 β-Sn unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Schematic representation of a RBS experiment . . . . . . . . . . . 6

2.2 Energy loss in RBS . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Depth resolution of RBS . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Principle of TEM operation . . . . . . . . . . . . . . . . . . . . . . 11

2.5 As grown random Sn sample . . . . . . . . . . . . . . . . . . . . . 14

2.6 Schematic overview of the spectrophotometer . . . . . . . . . . . . 16

3.1 Overview of the sputtering process . . . . . . . . . . . . . . . . . . 18

3.2 Quartz wafer transmittance . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Multilayered and random Sn samples . . . . . . . . . . . . . . . . . 21

3.4 Formation of nanocrystals . . . . . . . . . . . . . . . . . . . . . . . 21

3.5 Homogeneous nucleation barrier . . . . . . . . . . . . . . . . . . . 22

3.6 TEM picture of Sn nanocrystals randomly distributed in SiO2 . . . 24

3.7 RBS spectrum of Sn randomly distributed in SiO2 . . . . . . . . . 25

3.8 TEM pictures of multilayered nanocrystals . . . . . . . . . . . . . 26

3.9 Diffraction from Sn nanocrystals . . . . . . . . . . . . . . . . . . . 27

4.1 Full and reduced Mie expression . . . . . . . . . . . . . . . . . . . 34

4.2 Absorption from bulk vs nanocrystals . . . . . . . . . . . . . . . . 35

4.3 Garcia model for composite dielectric function . . . . . . . . . . . 37

4.4 Effect of the nanocrystal distribution on the refractive index . . . 38

4.5 Lorentz model for ε . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.6 Dielectric function of a free electron metal . . . . . . . . . . . . . . 42

4.7 Sn banddiagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.8 Bulk vs nanocrystal . . . . . . . . . . . . . . . . . . . . . . . . . . 47

iv

List of Figures

4.9 Sn dielectric functions . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.10 Reflection from a surface . . . . . . . . . . . . . . . . . . . . . . . . 49

4.11 Schematic overview of the matrix method . . . . . . . . . . . . . . 51

4.12 Reflection from Quartz wafer and thin film . . . . . . . . . . . . . 55

4.13 Minimization procedure . . . . . . . . . . . . . . . . . . . . . . . . 57

4.14 point by point simulation . . . . . . . . . . . . . . . . . . . . . . . 58

4.15 Refractive index of a quartz wafer . . . . . . . . . . . . . . . . . . 59

5.1 Sn peak in RBS for RSn1 . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 TEM and size distribution of RSn1 . . . . . . . . . . . . . . . . . . 65

5.3 RBS spectrum of multi layered sample . . . . . . . . . . . . . . . . 67

5.4 Size distribution of ML with different PV-TEM preparation tech-

niques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.5 TEM of two multi layered samples . . . . . . . . . . . . . . . . . . 68

5.6 Simulation structure for RSn samples . . . . . . . . . . . . . . . . 69

5.7 BaSO4 reflectance measurement . . . . . . . . . . . . . . . . . . . . 70

5.8 BaSO4 reflectance measurement . . . . . . . . . . . . . . . . . . . 71

5.9 RSn1 measurement and simulation . . . . . . . . . . . . . . . . . . 73

5.10 n and κ for RSn1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.11 σnc for RSn1 compared to MG . . . . . . . . . . . . . . . . . . . . 75

5.12 Mie vs MG for the RSn1 sample . . . . . . . . . . . . . . . . . . . 76

5.13 Comparison between absorption from RSn samples . . . . . . . . . 76

5.14 Comparison between transmittance from as grown and annealed

samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.15 Simulation structure for multi layered samples . . . . . . . . . . . 78

5.16 Refractive index of β-Sn nanocrystals . . . . . . . . . . . . . . . . 79

5.17 MLSn4 reflection and transmission . . . . . . . . . . . . . . . . . . 80

5.18 MLSn4 reflection and transmission . . . . . . . . . . . . . . . . . . 81

5.19 Comparison of σa for RSn and ML samples . . . . . . . . . . . . . 82

5.20 MLSn2 compared to MG theory . . . . . . . . . . . . . . . . . . . 83

5.21 Absorption from multi layered samples . . . . . . . . . . . . . . . . 84

5.22 Enhancement of absorption from nanocrystals in a nearby layer . . 87

5.23 Enhancement of absorption from nanocrystals within a single layer 88

5.24 Absorption enhancement relative to MG theory . . . . . . . . . . . 89

5.25 Model vs experimental absorption enhancement . . . . . . . . . . . 90

5.26 Absorption from single slab . . . . . . . . . . . . . . . . . . . . . . 92

v

List of Figures

5.27 Comparison of σa for MLSn samples from direct measurement and

the thin film modeling procedure . . . . . . . . . . . . . . . . . . . 94

5.28 Multi layered samples considered as effective media . . . . . . . . . 95

5.29 Interlayer reflections . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.30 Absorption comparison with previous work . . . . . . . . . . . . . 96

5.31 Absorption of SnO2 vs Sn nanocrystals in SiO2 . . . . . . . . . . . 97

5.32 Absorption comparison with previous work 2 . . . . . . . . . . . . 99

A.1 Field from a layer of nanocrystals a distance z0 away . . . . . . . . 102

A.2 Field from a layer of nanocrystals . . . . . . . . . . . . . . . . . . . 108

A.3 Field from randomly distributed nanocrystals . . . . . . . . . . . . 111

vi

Contents

List of Figures iv

Contents vii

Acknowledgements ix

List of abbreviations xi

1 Introduction 1

2 Characterization techniques 5

2.1 Rutherford Backscattering Spectrometry . . . . . . . . . . . . . 5

2.1.1 RBS theory . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Extracting information from experiments . . . . . . . . 9

2.1.3 Experimental details . . . . . . . . . . . . . . . . . . . . 9

2.2 Transmission Electron Microscopy (TEM) . . . . . . . . . . . . 10

2.2.1 Principle of operation . . . . . . . . . . . . . . . . . . . 11

2.2.2 Bright field imaging . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Diffraction mode . . . . . . . . . . . . . . . . . . . . . . 12

2.2.4 High Resolution TEM . . . . . . . . . . . . . . . . . . . 13

2.2.5 Sample preparation . . . . . . . . . . . . . . . . . . . . 13

2.3 Optical measurements . . . . . . . . . . . . . . . . . . . . . . . 14

3 Synthesis of Sn nanocrystals 17

3.1 RF magnetron sputtering . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 The sputtering chamber . . . . . . . . . . . . . . . . . . 18

3.1.2 Wafer wedging . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Thin films composed of Sn and SiO2 . . . . . . . . . . . . . . . 20

vii

Contents

3.2.1 Sn nanocrystals randomly distributed in SiO2 . . . . . . 20

3.3 Sn nanocrystals in a layered structure . . . . . . . . . . . . . . 25

4 Interactions between light and matter 29

4.1 The Maxwell equations and electromagnetic waves . . . . . . . 29

4.1.1 EM waves in unbounded media . . . . . . . . . . . . . . 30

4.2 Nanocrystals embedded in a host material . . . . . . . . . . . . 32

4.2.1 Origin of the dielectric function . . . . . . . . . . . . . . 38

4.2.2 The β-Sn dielectric function . . . . . . . . . . . . . . . . 47

4.3 The matrix method for determining reflection and transmission 49

4.4 Simulation based determination of absorption . . . . . . . . . . 54

5 Sn nanocrystals in SiO2 61

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2.1 Random Sn samples . . . . . . . . . . . . . . . . . . . . 63

5.2.2 Sn in a multi layered structure . . . . . . . . . . . . . . 65

5.3 Simulation based determination of absorption . . . . . . . . . . 69

5.3.1 The correction factor for reflection measurements . . . . 69

5.3.2 Refractive indices of the quartz wafer and SiO2 layers . 71

5.4 β-Sn absorption cross sections . . . . . . . . . . . . . . . . . . . 72

5.5 Comparison with previous studies . . . . . . . . . . . . . . . . . 95

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

A A simple model for the impact on nanocrystal absorption

from the surrounding nanocrystals 101

A.1 Nanocrystals in a different layer . . . . . . . . . . . . . . . . . . 102

A.1.1 Electric field from a layer of nanocrystals . . . . . . . . 103

A.2 Nanocrystals in the same layer . . . . . . . . . . . . . . . . . . 108

A.3 Randomly distributed nanocrystals . . . . . . . . . . . . . . . . 111

Bibliography 115

viii

Acknowledgements

For more than 5 years I have been working in the semiconductor group at

Aarhus University starting out doing a bachelor project before launching into

the PhD study to be described in this thesis. I have met many challenges

and obstacles during the process, which undoubtedly would have been very

hard to overcome without the help and support from a lot of people. First

and foremost I would like to thank my supervisor Brian Bech Nielsen for his

help and guidance throughout this project. His great enthusiasm towards

physics and his ability to focus on the interesting physical aspects of a given

problem has many a time send me smiling from his office eager to investigate

the matter further. Also his readiness to always take time to discuss the

recent results despite of his very busy schedule has been much appreciated.

Jacques Chevalier has prepared all the samples used in this project and has

been very helpful in discussing their structural properties. His ‘divine powers’

operating the TEM has been a big support and I am very grateful for his

help throughout my time here. Also John Lundsgaard Hansen has provided

invaluable help with the X-ray and RBS equipment which sometimes could

seem to have a mind of its own. Pia Bomholt has prepared all the TEM

samples during my study and furthermore served as a wonderful guide in

all kinds of different tasks carried out in the chemistry lab. Her help with

all this is much appreciated, just as her ability to create a pleasant working

environment where conversations do not have to be related to physics. I also

wish to thank Jesper Skov Jensen, Christian Uhrenfeldt and Amelie Tetu for

introduction to and guidance in the experimental work in the lab. Christians

detailed knowledge of obstacles and pitfalls in optical measurements has been

very helpful and Amelie has helped me on numerous occasions with TEM, PL

lab and where not. Also I appreciate her kindness to volunteer to proof read

this thesis. Duncan Sutherland is much appreciated for letting me use his

ix

Acknowledgements

spectrophotometer for optical characterization and for always being helpful

when I had questions regarding its use.

I would like to thank all my friends and fellow students for creating an

inspiring and pleasant environment and for making the past 8 years behind

the yellow walls be about more than just physics. Finally I am grateful to

my family for their support and interest and especially to Caroline Arnfeldt

for her love and support that has kept me going during the long hours of this

project.

x

List of abbreviations

Here is an alphabetically ordered list of abbreviations that will be used in the

thesis.

amu : Atomic mass unit

BF : Bright Field

CCD : Charge Coupled Device

DF : Dark Field

DFT : Density Functional Theory

EDX : Energy Dispersive X-ray

EM : Electromagnetic

HR-TEM: High Resolution Transmission Electron Microscopy

MBE : Molecular Beam Epitaxy

MG : Maxwell-Garnett

ML : Multi Layered

NC : Nanocrystal

PVD : Physical Vapor Deposition

R : Reflectance

RBS : Rutherford Backscattering Spectrometry

RF : Radio Frequency

SCCM : Standard Cubic Centimeter pr Minute

Si : Silicon

Sn : Tin

T : Transmittance

TEM : Transmission Electron Microscopy

UV : Ultra Violet

xi

Chapter 1

Introduction

One of the greatest challenges of the 21st century is to secure the worlds energy

need and preferably do it in an environmentally sustainable way. As the coal

and oil reserves on the planet will eventually be exhausted the incentive to find

renewable energy sources is right at hand, and with the current debate about

global warming and greenhouse gases directly related to the fossil fuels, the

interest in clean energy is bigger than ever. There are a number of different

areas with the potential to provide clean and renewable energy such as wind

power, waves in the sea, geothermal, and solar energy. Of these the solar

energy has by far the greatest potential, as the world combined energy need

is an insignificant fraction of the sunlight hitting the earth. For instance, it

has been calculated that covering only 4% of the global desert area with solar

panels is sufficient to account for the worlds electrical energy need [1]. On

the bottom line the important parameter is the price to pay for the energy,

and solar cells have yet to become competitive to the fossil fuels. Therefore

either the efficiency of the solar cells has to be improved, the production cost

reduced or a combination of the two strategies. This process is already ongoing

for the silicon (Si) based solar cells where the first generation was based on

thick wafers of crystalline Si. The second generation cells use thin films to

reduce the fabrication price [2]. When using sufficiently thin films however

the absorption efficiency is reduced due to the smaller distance travelled by

the sunlight in the film. Methods, such as texturing of the backside of the thin

Si layer [3, 4, 5, 6], that has been used to enhance the light path and thus the

chance of absorption, has been shown to improve the efficiency to cost ratio of

the solar cells. There is still a long way to go to reach efficiencies around 43%,

1

1. Introduction

a

a

c

Figure 1.1: β-Sn unit cell. It is a tetragonal structure with a 4 atoms basis.

which has been predicted as an upper limit for single junction solar cells [7].

One of the major issues to improve the efficiency is to use some of the energy

lost to thermalization of ‘hot’ electrons and holes. This has been adressed

recently as multi exciton generation in silicon nanocrystals has been observed

[8], but there are still some scepticism about whether this will in fact improve

the solar cell efficiency [9].

Another method which has proven to enhance the efficiency of thin film Si

solar cells exploit the surface plasmon resonance effect of metallic nanoparticles

[10, 11, 12, 13, 14]. In this approach the excitation of a surface plasmon on a

metal nanoparticle help to scatter the light into the Si layer and can thereby

increase the efficiency of the solar cell significantly. The efficiency increase

depends on the nanocrystal material, size, shape, and environment, and a lot

of effort is being put into such studies in recent years, with the vast majority

focusing on the noble metals silver or gold. Usually the nanocrystals are placed

on top of the Si film in order to avoid the creation of metallic defect states

in the band gap often seen for metals in Si [15]. As tin (Sn) belongs to the

same group in the periodic table as Si a Sn atom can substitute for a Si atom

without introducing unwanted electron states in the Si band gap which makes

Sn a possible candidate for device fabrication. Therefore knowledge of the

optical properties of Sn nanostructures is expected to be important for future

photo-voltaic devices. Another area where metal nanocrystals are gaining a

footing is in the shading area such as sunglasses, where a transparency in the

visible spectrum along with UV absorptivity are desired [16] [17].

2

Basically Sn comes in two allotropes at atmospheric conditions termed α-

Sn and β-Sn. The α form is a diamond structured semiconductor with at very

small band gap of around 0.1 eV which is only stable at temperatures below

13.2C [18]. The metallic β form has a body-centered tetragonal structure

(figure 1.1) and is the preferred form at room temperature and above. As the

transition temperature lies within the range of temperatures experienced in

the earth‘s climate effects of the α β transition has been known for many

years. The transition from metallic to semiconducting Sn has been termed

the ‘tin plague’ as it causes the Sn to become powdery and fall apart, as has

been seen for church organ pipes and for Napoleons soldiers as they marched

into the Russian winter in 1812 [19]. Even though the α → β transition is

heavily favored from a kinetic point of view [18, 20, 21] nanostructures of

alpha Sn can be kept stable at temperatures significantly above the transition

temperature by incorporating them into a diamond structured matrix of for

instance Ge [22], Si [23] or CdTe [24, 25]. These structures are very interesting

for optoelectronic devices due to their direct and tunable band gap.

As β-Sn is thermodynamically stable at room temperature nanocrystals

in that form can be studied in a wide variety of materials, though for photo-

voltaic devices those based on silicon are of greatest interest. The aim of this

project has been to investigate the optical properties of β-Sn nanocrystals in

a SiO2 matrix. The oxide has been chosen as a host material because of its

very high band gap which makes optical measurements across a wide range of

wavelengths feasible. Furthermore SiO2 is fully compatible with silicon based

photo-voltaic device fabrication. It would be ideal to study the nanocrystals

without a surrounding matrix, but if Sn gets in contact with oxygen it is

immediately oxidized so it is necessary to keep the nanocrystals inside a host

material.

The thesis has been split into 5 chapters and an appendix. First a brief

introduction to the most important experimental techniques used for struc-

tural and compositional characterization of the samples will be given. This is

followed by a chapter devoted to the synthesis of nanocrystals which describes

the technique used for thin film growth and some general features of the sam-

ples studied. After that there is a chapter dedicated to some of the basic

theory on interaction between light and matter and the geometrical effects

of the nanocrystals. Furthermore it gives a description of thin film interfer-

ence effects and how to model such effects in order not to be mislead when

3

1. Introduction

interpreting absorption spectra. In the fifth chapter the main findings of this

work will be presented and a detailed description of a model used to describe

absorption enhancement is given in the appendix.

4

Chapter 2

Characterization techniques

In order to understand and model the optical effects of the composite systems

investigated in this thesis it is of utmost importance to know the detailed

structure of the samples under study. The Rutherford Backscattering Spec-

trometry (RBS) technique provides a depth resolved chemical composition of

the sample and such quantitative information is very important in order to

compare measured data with theoretical modeling. Another important sample

parameter is the nanocrystal size which can be obtained using Transmission

Electron Microscopy (TEM), a very powerful technique for visualizing various

nanostructures. A brief discussion of these important techniques will be given

in the following paragraphs followed by a description of the setup used for

optical measurements. The RBS theory can be found in various textbooks

and the following section is mainly based on [26].

2.1 Rutherford Backscattering Spectrometry

The RBS technique is based on the famous Geiger-Marsden experiment from

the beginning of the 20th century, where the backwards scattering of He ions

from a gold foil led scientist to abandon the ‘plum-pudding’ model for the

atom. Basically positive He ions are accelerated to a kinetic energy of a few

MeV and focused onto the sample surface. The ions penetrate into the sample

and some of them will be scattered by the atomic nuclei. A multichannel

detector collect the backscattered ions in a certain angle θ and determine

their energy distribution. One very favorable aspect of RBS is the lack of

5

2. Characterization techniques

He+ ion

Detector

Sample

Ion accelerator

Θ

Figure 2.1: Sketch of the experimental setup for RBS measurements.

sample preparation needed to perform the experiment. The overall geometry

of the RBS experiment is sketched in figure 2.1.

2.1.1 RBS theory

As mentioned above, the theory behind RBS is based on elastic scattering of

alpha particles by heavier atoms, the basis of which being the Coulomb repul-

sion between the nuclei. Therefore the energy lost in the scattering event can

be derived classically by considering conservation of energy and momentum.

For an ion of mass Mi and initial energy E0 scattered into an angle θ from an

atom with mass Mt the energy of the ion after collision is

E1 = E0

Micosθ +√M2t −M2

i sin2θ

Mi +Mt

2

≡ KE0, (2.1)

where K is called the kinematic factor which, for an experiment where Mi

and θ are fixed, is seen solely to depend on the mass of the target nuclei.

6

2.1. Rutherford Backscattering Spectrometry

Ein

∆Ein

Escat

∆EoutEout

t

x

Figure 2.2: The different contributions to the energy loss of a He ion during aRBS experiment.

Placing the detector in an angle close to 180 will give the maximum energy

transfer between ion and target resulting in the optimum mass resolution of

the experiment.

For scattering by heavy nuclei, that is Mi << Mt, the scattering cross

section can be approximated by

σ (θ,Ei) =

(ZiZte

2

4Ei

)21

sin4(θ/2), (2.2)

where Zi (Zt) is the atomic number of the ion (target) and e is the elementary

charge. As implied by the Z2t behavior RBS has a much higher sensitivity for

heavy atoms than light ones.

So far only the scattering event in itself has been discussed, but there are

other equally important energy losses in the experiment to be considered, as

sketched in figure 2.2.

Both before and after scattering by a sample nuclei the He ions travel

through the sample where they lose energy from inelastic scattering by elec-

trons and small angle scattering by nuclei. Since the energy lost from the lat-

7

2. Characterization techniques

ter type is orders of magnitude much smaller than the first one, it will not be

taken into account. Although the inelastic ion-electron collisions are discrete

in nature, the energy lost is sufficiently small that the ions energy loss passing

through the sample can be considered as a continuous process as a function

of distance. The energy loss pr unit distance is termed stopping power and is

given by

−dEidx

=2πZ2

i e4

EiNZt

(Mi

me

)ln

2meviI

, (2.3)

where me is the electron mass, vi is the velocity of the He ion and N and I

are the atomic density and ionization energy of the sample respectively. Thus

on the inward trip of length t into the sample the He ion will lose

∆Ein =

∫ t

0

dEidx

dx ≈ tdEidx

∣∣∣in, (2.4)

where the last part is an approximation where the stopping power is evaluated

at an average between Ein and the energy just before backscattering. This

is a standard approximation used when studying thin films, where the low

penetration depth makes it rather good. Since the energy loss depends on the

sample electron density it is convenient to introduce the stopping cross section

ε =1

N

dE

dx. (2.5)

For a composite material the stopping cross section is a sum of the individual

atomic stopping cross sections weighted by their relative amount in the sample,

which is known as Bragg’s rule. With this, and the fact that the same story

goes for the outward path, the He ion emerges at the detector with an energy

of

Eout(t) = K (Ein − tNεin)− t

|cosθ|Nεout, (2.6)

where the first part Escat = K(Ein− tNεin) is the energy of the ion just after

scattering, and the final part is the energy loss on its way back out. The

energy difference of an ion scattered on the surface and one scattered in a

depth t (by the same type of atom) is then

∆E = Nt

(Kεin +

1

|cosθ|Nεout

)≡ [S]Nt, (2.7)

8

2.1. Rutherford Backscattering Spectrometry

where the stopping cross section factor [S] is introduced. The position of the

peaks in the spectrum identifies the element and the width of the peak is now

seen to be directly proportional to the depth of penetration t.

2.1.2 Extracting information from experiments

The output of an RBS measurement is the yield of backscattered ions at

an angle θ as a function of energy and in order to extract the composition

and depth profile some computer modeling is necessary. One option is the

RUMP [27] software package where a structure containing different layers with

given compositions and thicknesses is entered and simulated to obtain a RBS

spectrum. By comparing the simulated spectrum to the measurement one can

adjust the composition and thickness of the layers in the modeled structure

until it matches the measurement. RUMP has a database of atomic stopping

cross sections and atomic densities it uses to calculate stopping powers of the

user proposed layers exploiting Bragg’s rule for composite layers.

The accuracy of the final output is influenced both by measurement and

simulation. In the measurement a certain amount of total charge is collected

by the detector. The more charge the better signal to noise ratio is obtained

but it comes at the expense of prolonged measurement time. On the other

hand the simulations is based on tabulated atomic densities which may not

be exactly the same as for the sputtered films investigated. With the values

used in this study the accuracy of the chemical composition is expected to be

around 10% [28].

2.1.3 Experimental details

All RBS measurements have been performed with a 5 MeV van de Graff

accelerator supplying 2 MeV 4He+ ions incident on the sample at normal angle

(the sample was tilted up to 2 during measurement to avoid channeling). A

silicon solid state detector, with an energy resolution of about 40 keV placed at

an angle of θ = 161 with respect to the incoming beam, as shown in figure 2.1,

collected the backscattered alpha particles. The penetration depth of the alpha

particles was significantly higher than the film thickness, so compositional

information throughout the sample was available. Due to the high penetration

depth the silicon substrate signal can be used to normalize the measurement

to the simulation, as the substrate has a well described density. A 400 V

9

2. Characterization techniques

Recoil energy [MeV]

a) b)

1.5 1.7 1.5 1.7 1.9

Figure 2.3: Sn peak in a RBS spectrum of a multi layered structure with 5Sn layers separated by SiO2 layers of 15nm (a) and 75nm (b) respectively. Thelayered structure is not resolved for the smallest distance between the layers.

electron suppressor ensured reliable counting of the 20 µC charge directed at

the sample. With the experimental settings used the depth resolution in RBS

is about 40 nm. In order to resolve smaller features or to obtain a higher

precision in the thickness estimate other techniques such as TEM are used.

An example of the depth resolution in RBS measurements is shown in figure

2.3. Here the part of the RBS spectrum showing the Sn peak of two samples

with alternating layers of Sn and SiO2 is shown where the difference between

the samples is the separation of the Sn layers. As the Sn layers come closer

together they become unresolvable by RBS.

2.2 Transmission Electron Microscopy (TEM)

As noted in the introduction TEM is an indispensable tool for size determi-

nation when working with nanocrystals both embedded in a solid host or in

a solution. The huge advantage of the technique is the ability to actually see

the structures in the sample without having to extract the information from

elaborate simulation procedures. On the other hand the sample preparation

necessary is time consuming and destructive in addition to potentially influ-

encing the sample structure. A description of TEM can be found in a number

of textbooks since the technique has been used for decades, but the following

10

2.2. Transmission Electron Microscopy (TEM)

Condenser aperture

Condenser lenses

Sample

Objective aperture

Objective lenses

Projector lenses

CCD camera / fluorescent screen

LaB6

cathode

High voltage acceleration

Figure 2.4: Schematic overview of a Transmission Electron Microscope.

part is mainly based on [29].

2.2.1 Principle of operation

In a transmission electron microscope electrons are emitted at a cathode and

accelerated through a voltage difference of a few hundred keV. The electrons

are focused through a set of magnetic lenses and apertures both before and

after hitting a sample as sketched in figure 2.4. Finally the electrons are col-

lected on a fluorescent screen or by a CCD camera. In general the technique

does not differ much in concept from conventional optical microscopes, but

the superiority of the electron microscope is the low electron de Broglie wave-

length compared to visible light, which results in a significant enhancement in

resolution.

11

2. Characterization techniques

In this study a V = 200keV acceleration voltage was applied which corre-

sponds to a wavelength of

λ =h√

2meeV(

1 + eV2mec2

) = 0.0024 nm. (2.8)

Here h is Plank’s constant, me is the electron mass, c is the speed of light and

e is the electron charge. The wavelength is seen to be orders of magnitude

less than visible light, but the limiting factor in resolution turns out to be

aberration in the lenses [30]. The point resolution of the Philips CM20 system

used in this study is 2.7 A which is sufficient for the systems studied. A

pressure of ∼ 10−7 mbar is sustained in the column to avoid electron scattering

and sample contamination. The microscope can be operated in a number of

different modes which offer different kinds of information and some of those

will be reviewed in the following paragraphs.

2.2.2 Bright field imaging

The bright field (BF) imaging mode is the one in closest resemblance with

an optical microscope. The image is made from the central spot of the elec-

tron beam, as all electrons scattered on their way through the sample have

been removed by the objective aperture. In that way the picture will consist

of bright and dark regions corresponding to areas of the sample where few

or many electrons are scattered respectively. As the dominant mechanism

for scattering is interactions with core electrons which increases with atomic

number, higher atomic numbers will look increasingly dark in BF mode. In

order to get the optimal contrast the apertures are set to remove as much of

the scattered light as possible, both for amorphous and crystalline structures

present in the sample.

2.2.3 Diffraction mode

In diffraction mode the screen shows the diffraction pattern formed in the

back focal plane of the objective lens and this mode is used to determine the

crystallinity and crystal structure of the sample. When crystalline nanostruc-

tures are present in an amorphous environment the electron scattering will

be most intense whenever the Bragg conditions in the nanocrystals are met.

12

2.2. Transmission Electron Microscopy (TEM)

This happens when a lattice plane in the nanocrystal happens to align with

the electron beam in such a way that elastically scattered electrons have a

change in wave vector equal to the reciprocal lattice vector. For a single nano-

crystal a pattern of bright spots will appear on the fluorescent screen and their

distance from the center spot determine the plane spacing and thus the crystal

structure. When looking at a large number of randomly oriented nanocrys-

tals, as is present in the samples studied in this thesis, the diffraction pattern

will instead become concentric circles but the plane distance is measured in

the same way. In order to get rid of diffraction from the silicon substrate a

selected-area aperture can be inserted and the beam is then focused only on

a small spot. The diffraction rings are rarely extremely well defined when

looking at nanocrystals due to the small size that limits the number of elec-

tron scattered and thereby the brightness of the diffraction spots. This would

result in an inaccurate determination of the inter planar distance but since the

atomic nature of the nanocrystals is often already known, the precision only

needs to be good enough to discriminate between different crystal structures.

2.2.4 High Resolution TEM

In High Resolution TEM (HR-TEM) the aperture used to block out the

diffracted beam in BF mode is widened enough to include some of the diffracted

electrons as well. This is done on a spot where no diffraction from the sub-

strate is included. The interference between the direct and diffracted beam

will produce a picture of the periodic charge distribution seen by the electrons

(the lattice planes) superimposed on the BF image. The experimental condi-

tions for doing HR-TEM is very demanding. The focus has to be perfect and

the lenses have to be corrected for astigmatism for the interference effects to

become visible, and in general these conditions are fairly hard to meet.

2.2.5 Sample preparation

The samples used for TEM measurements has to be very thin in order for the

majority of the electrons to pass through the specimen, that is ∼100 nm. The

first step is a rough polishing of a small piece of sample until it is about 10

µm thick followed by ion milling with 5 keV Ar+ ions, which sputters away

material1. Two different types of samples have been prepared; cross sectional

1The sputtering process is explained in chapter 3

13

2. Characterization techniques

SiO2

SiO2

Figure 2.5: Cross sectional TEM picture of an as grown sample of randomlydistributed Sn in SiO2.

and planar view samples. When studying thin films a planar view sample is

thinned in a direction perpendicular to the film as opposed to cross sectional

where the thinning occurs parallel to the film. Thus cross sectional view

provides depth resolution of the thin film whereas the planar view provides

information about a thin layer of the film. The samples were coated with

carbon after ion milling in order to prevent charging of the SiO2 layers.

The ion milling step potentially raises the temperature in the TEM sample

enough for the Sn to form nano-clusters, as seen in figure 2.5. The formed

clusters however, did not show any diffraction pattern, so they are considered

to be amorphous. Whether the formation is in fact a result of the sample

preparation or originate from the sputtering process is impossible to determine

from the TEM pictures, but it is a thing to keep in mind when interpreting

the pictures.

2.3 Optical measurements

Transmission and reflection measurements on the samples prepared on quartz

substrates were performed using a Shimadzu UV-3600 double beam spec-

14

2.3. Optical measurements

trophotometer, which is sketched in figure 2.6. The measurements were per-

formed in the wavelength range from 200-1500 nm (6.2-0.83 eV). A deuterium

lamp is used for wavelengths from 200-325 nm whereas a halogen lamp covers

the rest of the spectrum. From the lamp compartment the light is led into

the main body of the spectrophotometer through the entrance window. After

hitting the first grating a slit limits the beam divergence and after another

grating the following slit is also equipped with a filter to remove higher order

diffracted light. The (ideally) monochromatic beam is led through a chopper

mirror which is alternating between reflecting the beam and letting it pass,

which gives rise to two beams termed the sample beam and the reference beam.

These are let through the exit windows into the sample compartment where

an integrating sphere is installed. The walls of the sphere are coated with

BaSO4 white paint which is essentially 100% reflecting across a wide range

of wavelengths. The detectors used are a photomultiplier tube and a PbS

solid state detector placed in the top and bottom of the integrating sphere

respectively. The reference beam enters the integrating sphere at an 8 angle

such that the specular reflection from a sample placed on the opposite side

of the sphere can be measured. In order to calculate the 100% reflection or

transmission line compressed BaSO4 white powder was used as a reference.

Reflection data for such powders has been measured previously [31, 32], but as

they are somewhat dependent on the exact nature and thickness of the paint,

a measurement of its reflectance has been performed.

The sensitivity of the spectrophotometer is ±0.003 absorbance and it can

measure up to 6 absorbances. At high absorbances the measurement is very

sensitive towards microscopic holes in the sample as the absorbance would

level off at some value depending on how big the hole is compared to the

beam profile. In this work the measured samples never have an absorbance

much above 1 so microscopic holes will not have a big influence. In any case,

each sample was measured at different spots and the spectra were seen to be

in accordance for all samples. Besides the lamp change at 325 nm a grating

and detector change occur at 900 nm, which often give rise to fluctuations in

the spectra at these wavelengths.

15

2. Characterization techniques

D2 lamp

Halogen lamp

Entrance window

Slit

Gratings

Slit

Gratings

Slit with filter

Chopper mirrorExit windows

Sample compartment

Reference beam

Sample beam

Integra-ting sphere

= Mirror

Figure 2.6: Schematic overview of the Shimadzu UV-3600 spectrophotometerused to perform transmission and reflection measurements.

16

Chapter 3

Synthesis of Sn nanocrystals

In this work Radio Frequency (RF) magnetron sputtering has been the pre-

ferred technique for nanocrystal synthesis. This physical vapor deposition

(PVD) technique was chosen due to its ability to grow thin amorphous layers

with good uniformity and thickness controllability in a relatively short time

[33]. The technique is furthermore better suited for large scale production

compared to other thin film growth techniques such as molecular beam epi-

taxy [34]. The magnetron sputtering technique will be presented here and the

description is mainly based on [29]. This will be followed by a description of

the different types of samples produced in this work.

3.1 RF magnetron sputtering

The basis of sputtering is the ability for high energy ions to knock off atoms or

molecules from a target upon impact. The released atoms with the appropriate

direction will condense on the substrate and form a film. An overview of the

process is given in figure 3.1. Basically a vacuum chamber is flooded with

an inert gas such as argon and the target is placed at a negative potential

compared to the substrate. This will accelerate the positive Ar ions towards

the target, ionizing additional Ar atoms on their way. Upon collision with the

target the Ar atoms will knock out target atoms in different directions as well

as secondary electrons. The electrons are accelerated towards the substrate

and ionize more Ar atoms. This will result in a self sustaining ion plasma

and due to the magnetic field generated by the magnet below the target, the

17

3. Synthesis of Sn nanocrystals

Target material

Magnet

Substrate

Target atom

Ar+

Target

Figure 3.1: Schematic representation of the sputtering process used to growthin films in this work.

plasma will be located right above the target. The magnetic field causes the

electrons to move in a helical trajectory which increases their path length

towards the substrate and thus their probability of ionizing Ar atoms. In

that way a lower Ar pressure is sufficient to sustain a plasma which facilitates

higher sputtering rates, as both target atoms on their way to the substrate

and Ar+ ions moving in the opposite direction undergo less collisions with Ar

atoms. Sputtering of insulating materials, such as SiO2 used in this work,

requires an alternating potential between substrate and target, otherwise the

target surface will accumulate positive charge which eventually terminates the

sputtering process. The alternating potential ensures that the target is hit by

alternating periods of Ar+ ions and electrons, and as the electrons are much

easier to set in motion due to their lower mass, more electrons than Ar+ ions

will hit the target during a full cycle keeping the target at a negative potential.

3.1.1 The sputtering chamber

The sputtering equipment used in this work was a homebuilt system with four

separate targets, so each sample can consist of up to four different materials.

18

3.1. RF magnetron sputtering

The targets were placed 70 mm from the substrate which was water cooled to

a temperature of about 15C during sputtering. To ensure the quality of the

sputtered films the chamber was initially pumped down to a base pressure of

10−7 mbar. A 30 sccm flow of 99.999% pure Ar gas was let into the chamber

and the Ar pressure during sputtering was kept fixed at 2·10−3mbar. By tuning

the sputtering power and the deposition time the thickness of a given layer

can be controlled. 20x20 mm pieces of both silicon and quartz were used as

substrates. Those on silicon were used for RBS measurements and to prepare

TEM samples whereas those on quartz were used for optical measurements.

In order to deposit mixed layers of Sn and SiO2 small pieces of Sn were put on

top of a SiO2 target covering a carefully calculated area in order to produce

a desired atomic ratio in the film. The substrate is rotated at a speed of ∼2

rounds per minute during sputtering to ensure a homogeneous film growth.

3.1.2 Wafer wedging

As will be discussed in a later chapter multiple reflections give rise to interfer-

ence fringes in the transmission and reflection spectra in thin films, but such

effects may also occur in thick non-absorbing samples, such as a quartz wafer.

In figure 3.2 the transmittance of a 10 µm quartz wafer has been calculated

(blue line) by the matrix method revealing a high frequency oscillation on top

of the smooth transmission (red dashed line). Such oscillations would obscure

the measurement, but can be removed by either measuring at a sufficiently

low resolution or by polishing the wafers in a wedge shaped profile, as pointed

out in [35]. In order for the oscillations to be removed the difference in height

δh across the quartz wafer must satisfy

δh >>λ

4nq(3.1)

where nq is the quartz refractive index. After the quartz wafers were mechani-

cally ground into a wedged shape they were thoroughly polished in order to get

a smooth and clean surface. All samples for optical measurements were grown

on wedge-shaped quartz wafers in order to prevent substrate oscillations from

contaminating the optical measurements.

19

3. Synthesis of Sn nanocrystals

Wavelength [nm]

Tra

nsm

itta

nc

e [

%]

200 400 600 800 1000 1200 140050

60

70

80

90

100

110

Figure 3.2: Transmittance spectrum of a 10 µm thick quartz wafer calculatedby the matrix method described in the following chapter (blue line). This iscompared to a calculation on a wedge-shaped wafer of the same thickness (reddashed line). The quartz dielectric function is taken from [36].

3.2 Thin films composed of Sn and SiO2

Two different types of samples have been investigated in this work: one with Sn

nanocrystals randomly distributed in SiO2 and one with the Sn nanocrystals

arranged in layers. The two types are sketched in figure 3.3, and will be

explained in more details in the remainder of this chapter.

3.2.1 Sn nanocrystals randomly distributed in SiO2

First of all the basic theory covering nanocrystal formation will be presented.

Reference [37] has a very thorough description of the thermodynamics involved

in cluster formation and serves as the basis for the following section. After

that the heat treatment will be described and in the end some of the structural

parameters of the samples will be discussed.

Thermodynamically driven nucleation

When Sn atoms are initially randomly distributed in SiO2 a driving force is

needed in order for them to aggregate and form nanocrystals. This driving

force originates from the decrease in Gibbs free energy involved in crystal-

lization process, which is sketched in figure 3.4. There are three different

20

3.2. Thin films composed of Sn and SiO2

SiO2

Sn nano-crystal

(a) (b)

Figure 3.3: Cross sectional schematic slice through a) Sn layers sandwiched be-tween SiO2 in a multilayered structure and b) randomly distributed Sn nanocrys-tals in SiO2

Sn atom

Sn nano-crystal

G0 G0 + ∆G

Figure 3.4: Homogeneous nucleation of Sn nanocrystals in SiO2. The nano-crystal will form if there is an overall decrease in energy, meaning ∆G must benegative.

contributions to the Gibbs free energy change in the formation of a cluster of

volume V and surface area A as seen in equation 3.2.

∆G = −V∆Gv +Aγ + V∆Gs (3.2)

The first term describes the decrease in free energy from atoms joining to form

a cluster, the second term is the interface energy between the cluster and the

matrix and the last term is the induced strain energy if the newly formed

21

3. Synthesis of Sn nanocrystals

Surface term

∆G*

R

∆G

0

Volume and strain term

R*

Figure 3.5: Barrier in homogeneous nucleation and the different thermody-namic factors causing it. The cluster has to pass the barrier in order to get thecritical size necessary to start growing to become a nanocrystal.

cluster does not fit perfectly into the host matrix. It should be noted that

with the form of the surface free energy term γ introduced in the second term

in equation 3.2 it is assumed to be isotropic, which is a valid assumption if the

nanocrystals are spherical. In addition the concentration of Sn atoms in the

neighborhood of the clusters is assumed to be unchanged by their formation.

For spherical clusters this can be rewritten in terms of the nanocrystal radius

R to

∆G = −4π

3R3(∆Gv −∆Gs) + 4πR2γ. (3.3)

If the misfit strain energy i small (∆Gs < ∆Gv) which is often the case in

an amorphous matrix such as SiO2, the behavior of ∆G will look as sketched in

figure 3.5. From this it is evident that a critical radius R∗ exist, which defines

the number of Sn atoms needed to cluster together in order to overcome the

barrier where growth is thermodynamically favorable. By differentiation of

eq. 3.3 the critical radius R∗ and the barrier height ∆G∗ becomes

22

3.2. Thin films composed of Sn and SiO2

R∗ =2γ

(∆Gv −∆Gs)(3.4)

∆G∗ =16πγ3

3 (∆Gv −∆Gs)2 . (3.5)

The energy needed to surmount the barrier can be supplied in form of

a post growth heat treatment, thus nanocrystal formation is often said to be

thermally activated. If the concentration of Sn atoms is CSn the concentration

of clusters reaching the critical size C∗ at a given temperature T can be shown

to be [37]

C∗ = CSne−∆G∗kbT (3.6)

where kb is the Boltzmann constant. Thus depending on the hight of the

barrier heat treatment may be a necessity or just a means of speeding up the

nanocrystal formation. This section has described how homogeneous nucle-

ation occurs, but if there are impurities or other irregularities present in the

SiO2 the cluster formation may begin at specific sites. This would result in a

lowering of the energy barrier ∆G∗ which would encourage cluster formation,

without influencing the growth kinetics.

Annealing procedure

In the literature annealing temperatures between 400C and 1100C [38, 39,

40] have been used to form Sn nanocrystals in SiO2 depending on the deposi-

tion method and annealing atmosphere. It was found in [40] that increasing

the annealing temperature led to formation of larger nanocrystals, which was

also seen in this work. This could seem in contradiction to the conclusions of

the previous section, where higher temperature lowers the barrier for cluster

formation which would indicate the formation of many small nanocrystals.

The phenomenon which can be accredited for this is the Ostwald Ripening

effect [41] which describes how larger nanocrystals grow at the expense of

smaller ones. In this work an annealing temperature of 400C in vacuum for

1 hour was chosen, as it turned out to be sufficient to produce nanocrystals.

Furthermore Huang et al. [40] discovered that annealing at 400C resulted

in a more uniform distribution of nanocrystals than at higher annealing tem-

peratures. Annealing was performed in a vacuum furnace at a pressure of P

23

3. Synthesis of Sn nanocrystals

20 nm

Figure 3.6: BF-TEM picture of randomly distributed Sn nanocrystals in SiO2.The crystallinity was determined by looking at the diffraction pattern.

< 10−4 mbar in order to avoid oxidation of the nanocrystals, which can occur

when annealing in a N2 atmosphere [42, 43, 44, 45].

In figure 3.6 a TEM picture of Sn nanocrystals randomly distributed in

SiO2 is shown. TEM diffraction analysis showed that the nanocrystals were

in the β-Sn phase as would be expected [40]. The Sn content in the films was

measured by RBS before annealing and such a spectrum together with the

RUMP simulation is shown in figure 3.7.

The RBS spectrum has been normalized to the RUMP simulation in the

part originating from the silicon wafer in order to be able to extract the areal

density Ω for the thin film constituents. In that way the amount of Sn in

the sample can be determined by integration of the area under the Sn peak.

From the figure it is also clear that some Ar is present in the sample. This

is an unavoidable side effect from the sputtering process that some of the Ar

atoms get incorporated into the thin film. Most of the Ar is released from

the samples during annealing, but due to the low annealing temperature a

very small amount is still present. Annealing at higher temperatures would

probably remove all of the Ar [46, 47] but in order to avoid complications with

the film quality (to be described in the following section) the temperature

was kept at 400C. The remaining Ar was found to be less than 0.2 at% for

samples annealed for 1 hour at 400C. This low concentration is expected to

24

3.3. Sn nanocrystals in a layered structure

SnAr

Si from film

Si from wafer

O MeasuredSimulation

Recoil Energy [MeV]

Nor

mal

ized

Yie

ld

1.5 1.7 1.90

5

Blow up of the Sn peak

Figure 3.7: RBS spectrum of Sn randomly distributed in SiO2 before annealing.The inset to the right is a blow up of the Sn peak in the spectrum.

have minimal influence on the film parameters.

By comparing the Sn peak in the spectrum with the RUMP simulation

(the blow up in the right part of figure 3.7) one can see that the experimental

curve is slightly higher than the simulation at the left part of the peak. This

indicates that the Sn distribution is not completely homogeneous across the

layer, as it is assumed in the simulation. It does not affect the fact that the

nanocrystals are randomly distributed, but it will cause the filling fraction to

vary in depth and it should be kept in mind when modeling such structures.

3.3 Sn nanocrystals in a layered structure

Samples with multiple layers of Sn sandwiched between SiO2 (see figure 3.3(a))

were produced and annealed under the same conditions as the random samples

described above. This was done in order to investigate how the distribution

of the nanocrystal affected their absorption. Ge nanocrystals have shown an

enhanced absorption for an equivalent amount of material when placed in lay-

25

3. Synthesis of Sn nanocrystals

50 nm

a) b)

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.002468

10121416

Num

ber o

f obs

erva

tions

Diameter [nm]

c)

Figure 3.8: Cross sectional (a) and plane view (b) TEM pictures of a samplewith layers of nanocrystals. The size corresponding size distribution is shown in(c).

ers compared to random distributions [48], and the effect of the layer distance

was also an element of interest. The layers were produced by sputtering of

individual targets of Sn and SiO2 and the number of Sn layers were kept at 5

in order to limit the total film thickness to approximately 500 nm.

Just as the randomly distributed Sn samples the multilayered structures

were annealed in vacuum at 400C for 1 hour in order to form nanocrystals. At

this temperature the nanocrystals stay in the layered structure, and their mean

diameter is 4-5 times the thickness of the sputtered Sn layer. Annealing at

700C showed loss of the layer structure as nanocrystals migrated into the SiO2

layers leaving voids in the structure and earlier heat treatments performed in

an N2 atmosphere showed cracked films when annealed at 800-1000C. There-

fore the annealing temperature was kept at 400C which proved sufficient to

form nanocrystals. TEM was used to find the nanocrystal size, crystal struc-

ture and the distance between neighboring layers whereas the total amount

of Sn was measured by RBS. BF-TEM pictures of a multilayered structure is

presented in figure 3.8 along with the corresponding size distribution and the

diffraction from such a structure is seen in figure 3.9.

The distance from the center spot to the diffraction rings is connected to

26

3.3. Sn nanocrystals in a layered structure

(200) (101) (211)

Figure 3.9: Diffraction pattern from a sample with layers of Sn nanocrystals.The ring like structure is a result of the random orientation of the nanocrystalsand by looking carefully the innermost ring is actually two rings close together.The 3 most intense rings have been identified and labeled on the figure, as theyare the ones most easily identified.

the plane distance in the nanocrystals, so from pictures like this the crystal

structure can be verified. All samples showed the nanocrystals to be in the β

form.

The physical process driving nanocrystal formation in the case of the lay-

ered structure can be considered to be the reduction of surface area between

Sn an SiO2 by converting a Sn layer into nanocrystals. If a layer of area A

and thickness t is converted into n nanocrystals with a radius R the relative

change in surface area is given by

∆A

Atotal=

2A− 4πR2n

2A(3.7)

If all the Sn from the layer is converted into nanocrystals then 4πR3n/3 = tA

(assuming the same Sn density in layer and nanocrystals) which will turn

equation 3.7 into

∆A

Atotal= 1− 3t

2R. (3.8)

Thus if R > 1.5t one gets a positive number signifying a surface area reduction

and ss mentioned above the diameter observed in TEM was 4-5 times the

layer thickness. In this calculation strain energies have not been considered

at all even though the increase in nanocrystal size compared to the Sn layer

27

3. Synthesis of Sn nanocrystals

thickness could very well introduce a strain in the SiO2 layers. Films with

thicker Sn layers compared to the separating SiO2 layers resulted in cracked

and partially peeled off films when annealed at 400C and higher, so strain

effects may certainly play a role in such structures. The films presented in this

work, however, all looked smooth and showed no sign of such strain related

effects.

28

Chapter 4

Interactions between light

and matter

Before venturing into the area of optical interactions in composite systems

some of the basic theory of the interaction between light and matter will be

discussed. At first the Maxwell equations and the resulting fields in bulk

media will be introduced followed by a discussion of the geometrical effects of

nanocrystals embedded in a host material. After an overview of the dielectric

function and its connection to the electronic states in the material, the matrix

model for determining reflection and transmission through a multi layered

structure will be presented. Finally there will be given a description of how

the matrix method can be applied to extract information about the individual

layers in such multi layered structures.

4.1 The Maxwell equations and electromagnetic

waves

The physical laws obeyed by an electromagnetic (EM) wave traveling in any

medium are summarized in the Maxwell equations. These equations describe,

in macroscopic terms, which electric and magnetic fields that are allowed to

propagate given the electronic structure of the material, and are found in

almost any textbook related to light-matter interactions. Assuming a non-

magnetic and current free material the Maxwell equations are given by [35]

29

4. Interactions between light and matter

~∇ · (ε ~E) =ρ

ε0(4.1)

~∇ · ~B = 0 (4.2)

~∇× ~E = −∂~B

∂t(4.3)

~∇× ~B = µ0ε0

(∂(ε ~E)

∂t

)(4.4)

where ~E is the electric field, ~B is the magnetic induction, ρ is the free charge

density and ε0 is the vacuum permittivity. These equations must be satisfied

at any point and time in the material. The dielectric function ε, which is

connected to the electronic structure of the material, describes the materials

response to an electric field. For linear isotropic materials the polarization ~P

caused by an electric field1 is given by

~P = ε0(ε− 1) ~E (4.5)

and the dielectric function can be written as ε = εr + iεi. The form of the

dielectric function depends on the crystal structure of the material in question.

A cubic crystal lattice would result in an isotropic dielectric function and it

can be represented by a scalar whereas for anisotropic materials ε has to be

represented by a tensor. For materials with a tetragonal structure such as

β-Sn there are two crystallographic axes along which the dielectric function

will differ. However when studying the optical responses of nanocrystals with

a random crystallographic orientation the dielectric function can be described

as an average of the components along each crystallographic axis.

4.1.1 EM waves in unbounded media

Among the solutions to the Maxwell equations in a homogeneous unbounded

medium the plane waves are probably those most commonly encountered due

1For very large electric fields the polarization is no longer linearly related to the field,but these fields are mostly encountered when working with very intense beams such as lasers.For normal optical measurements equation 4.5 is obeyed.

30

4.1. The Maxwell equations and electromagnetic waves

to their simple form and the fact that they form a complete set of functions2.

The E and B fields for a plane wave can be described by the following equa-

tions

~E(~r, t) = ~E0ei(~k·~r−ωt) (4.6)

~B(~r, t) = ~B0ei(~k·~r−ωt) (4.7)

where ~E0 and ~B0 are mutual perpendicular amplitude vectors, each also per-

pendicular to the wave vector ~k. In order to satisfy the Maxwell equations ~k

and the frequency ω must be related by∣∣∣~k∣∣∣2 = µ0ε0εω2. (4.8)

Introducing the speed of light in vacuum c = 1√ε0µ0

this reduces to∣∣∣~k∣∣∣ =ω

c

√ε. (4.9)

For bulk materials it is often convenient to introduce the refractive index

N = n + iκ =√ε. The electric field of a homogeneous plane wave traveling

in the z direction is then given by

~E(z, t) = ~E0e− 2πλ0κzei(

2πλ0nz−ωt

)(4.10)

where the free space wavelength λ0 = 2πcω has been introduced. As shown

in equation 4.10 the complex part of the refractive index can be associated

with the field attenuation during propagation through a material. Since light

intensity is what typically is measured experimentally the attenuation can be

described in terms of the initial intensity I0 and the intensity after traversing

a length z through a given material I(z) through

I(z) = I0e−αz (4.11)

where the attenuation is described by α. In bulk materials the dominant

mechanism for attenuation is absorption in the material and for that reason α

is known as the absorption coefficient. From equation 4.10 and 4.11, using the

2Such that all possible fields satisfying the Maxwell equations can be expanded in planewaves.

31

4. Interactions between light and matter

fact that intensity is proportional to the electric field squared, the absorption

coefficient can be described by

α =4π

λ0κ. (4.12)

It is evident from equation 4.12 that the absorption coefficient is a pure

material property related to its electronic structure since it depends only on κ.

To get a quantity related to the individual atoms instead of a whole ensemble

the atomic absorption cross section σa is often used

σa =α

ρ. (4.13)

Here ρ is the material density, and the introduction of σa provides a way to

compare measurements on samples where the amount of absorbing material

is important.

4.2 Nanocrystals embedded in a host material

So far bulk materials have been considered and the endless repetition of unit

cells have imposed little boundary conditions on the Maxwell equations. When

dealing with composite materials there are interfaces between the constituents

where the material symmetry is broken and boundary conditions must be

applied in order to determine the electromagnetic fields. This applies to a

nanocrystal embedded in a host medium and for spherical nanocrystals the

problem is described in a theory developed in the early 1900’s and accredited to

Gustav Mie [49]. The Mie theory, which is treated thoroughly in [35], describes

how a plane wave is scattered and absorbed by a single spherical particle in a

host material by expanding the wave in spherical Bessel functions and impose

continuity of the electric and magnetic fields across the material interfaces.

Although the complete analytical solution can be found, an approximation

valid for small particles such as nanocrystals is often used. This approximation

is imposed by expanding the Bessel functions in the parameter x, which is

the nanocrystal circumference divided by the wavelength in the surrounding

medium

x =2πRnhλ0

, (4.14)

32

4.2. Nanocrystals embedded in a host material

where nh is the real part of the host refractive index3 and R is the nanocrystal

radius. Keeping only terms up to x4 in the expansion the extinction cross

section becomes [35]

σext = πR24xIm

m2 − 1

m2 + 2

[1 +

x2

15

(m2 − 1

m2 + 2

)m4 + 27m2 + 38

2m2 + 3

]+ πR2 8

3x4Re

(m2 − 1

m2 + 2

)2

where the nanocrystal dielectric function relative to the host dielectric function

has been replaced by m2 = εncεh

for clarity. If |m|x << 1 this can be further

reduced and the scattering and absorption cross sections can be identified as

σscat =128π5R6n4

h

3λ40

∣∣∣∣ εnc − εhεnc + 2εh

∣∣∣∣2 (4.15)

σabs =8π2R3nh

λ0Im

εnc − εhεnc + 2εh

. (4.16)

Reformulated in words the assumption that |m|x << 1 can be phrased

like the dimension of the nanocrystals is much smaller than the wavelength

inside it. Thus the field across the nanocrystal is homogeneous at a given

time which is called the electrostatic approximation. Equations 4.15 and 4.16

can be shown to be identical to the scattering and absorption cross sections

obtained from an oscillating dipole [35] so the nanocrystals can be viewed

as small dipoles in a host material. In figure 4.1 the Mie absorption cross

section from the full Mie expansion is compared to equation 4.16 for two

different sizes of Sn nanocrystals in SiO2, and it is clear that for sufficiently

large nanocrystals equation 4.16 is no longer valid. It has been verified that

|m|x < 0.2 for the sizes and wavelengths use in this work so the formulas

for σscat and σabs given above can be used with sufficient accuracy. It can

be noted how the scattering cross section in equation 4.15 is proportional to

x4 whereas the absorption cross section is proportional to x. Thus for small

nanocrystals the absorption will dominate the extinction.

From figure 4.2 it is clear that the Sn absorption depends heavily on the ge-

ometry and dielectric surroundings. Here the atomic absorption cross section

3As the host material used in this work is SiO2 the refractive index is purely real in thewavelength range studied.

33

4. Interactions between light and matter

200 400 600 800

Full MieReduced Mie

200 400 600 800

Full MieReduced Mie

Wavelength [nm]

σ nc [

arb

. un

it]

Figure 4.1: Nanocrystal absorption cross section for Sn nanocrystals inSiO2 with a diameter of 5 nm (top) and 50 nm (bottom). The redcurve labeled ‘Reduced Mie’ is calculated from equation 4.16 wheres theblack ‘Full Mie’ curve is calculated using the MiePlot software available athttp://www.philiplaven.com/mieplot.htm and based on the computer code pre-sented in [35]. Dielectric functions for Sn and SiO2 respectively are taken from[50] and [36].

of bulk Sn found from equation 4.13 is compared to Sn nanocrystals embedded

in a SiO2 host or in a vacuum where the dielectric function from Sn and SiO2

respectively has been taken from [50] and [36]. As the absorption cross sec-

tion from the Mie expression in equation 4.16 is for a nanocrystal it has been

converted to an atomic cross section by dividing with the number of atoms pr

nanocrystal4 ρVnc, where Vnc is the nanocrystal volume. The shape of the

bulk absorption differ substantially from the spherical nanocrystals both in

SiO2 and in vacuum which in turn are mostly distinct regarding the absorp-

4The density of Sn in nanocrystals is assumed to be the bulk density, which is reasonablesince they share the bulk crystal structure. On another note, the comparison in figure 4.2is only possible for arbitrarily sized nanocrystals because the Mie absorption cross section isdirectly proportional to the nanocrystal volume, which cancels out from the equation whenconverting to atomic absorption cross section.

34

4.2. Nanocrystals embedded in a host material

500 1000 15000

1x 10-16

Wavelength [nm]

σ a [

cm

2]

BulkSn in vacuumSn in SiO

2

Figure 4.2: Atomic absorption cross sections from bulk Sn compared to Snnanocrystals surrounded by either vacuum or SiO2. The origin of the dielectricfunctions used is explained in the text.

tion onset. The most pronounced effect going from bulk to nanocrystal is the

appearance of a peak in the spectrum. This is best seen for the Sn in SiO2

around 230 nm and is a result of the (εnc + 2εhost) denominator in equation

4.16. When this approaches 0 the absorption increases rapidly, a phenomenon

known as a Mie plasmon [51]. The position of the absorption peak can be

varied by choosing a host material where (εnc + 2εhost) ≈ 0 is satisfied at a

different wavelength.

The Mie theory accurately describe a single particle embedded in an infi-

nite host matrix, but in practice measurements will often be conducted on a

large number of nanocrystals, so interactions between those has to be taken

into account. This has been done in the Maxwell-Garnet (MG) theory [52],

which describe a random distribution of spherical particles in a host medium

using an average dielectric function for the composite medium given by

εave = εh

1 +3f(εnc−εhεnc+2εh

)1− f εnc−εh

εnc+2εh

. (4.17)

Here εh is the host dielectric function and f = Vncρnc is the filling factor

or volume fraction occupied by the nanocrystals. From equation 4.13 and

35

4. Interactions between light and matter

4.12, which can be reformulated in terms of εi in place of κ to α = 2πλ0

εin , the

nanocrystal absorption cross section in MG theory is given by

σnc =α

ρnc

=2πVncλ0navef

Imεave (4.18)

where nave = Re√εave is the real part of the refractive index of the com-

posite layer. Assuming that the host dielectric function has no complex part

in the wavelength range of interest (which is true for SiO2 used in this work)

this can be reduced to

σnc =6πVncλ0nave

Im

εh

εnc−εhεnc+2εh

1− f εnc−εhεnc+2εh

. (4.19)

In the MG theory the spherical inclusions are, as in Mie theory, assumed to

be dipoles and the interaction between them arises from mutual polarization

fields. Thus the MG theory will only be accurate up to a certain filling factor

above which initially neglected multipole effects will become important. For

a filling factor below f = 0.3-0.5 the simple expression in equation 4.17 is

sufficiently accurate [53, 54, 55]. The MG formula assumes the nanocrystal size

to be much smaller than the wavelength in the same way as described above

for the Mie theory, and by expanding equation 4.17 in the volume fraction f

keeping only the leading term, the Mie result from equation 4.16 emerges. For

a low filling factor the nanocrystals will be far apart and interactions between

them will be negligible which support that the two theories should be related.

For nanocrystals arranged in a layered structure the ordering of the nanocrys-

tals need to be taken into consideration. This has been done for a single layer

by Toudert et al. [56] who used ellipsometric measurements in conjunction

with thin film modeling to extract the dielectric function from a layer contain-

ing nanocrystals and relate it to their morphology. For more layers, a method

is described in [57] to account for the spatial arrangement of nanocrystals.

This method, originally proposed by Garcia et al. [58, 59], is essentially an

extension of the MG theory to account for the non-random distribution of

the nanocrystals. Here the local electric field experienced by a nanocrystal in

36

4.2. Nanocrystals embedded in a host material

rij

RL

i j

Figure 4.3: Imaginary sphere in a plane of nanocrystals used for calculation ofthe electric field exerted on particle i in the center of the sphere.

the point i, Eiloc, is the sum of the external field and the field from all other

nanocrystals (treated as dipoles)

~Eiloc = ~Eext + ~Einc. (4.20)

The field from the nanocrystals can be calculated by setting up an imag-

inary sphere, as seen in figure 4.3, (it is usually termed the Lorentz sphere)

and adding the contributions from the nanocrystals inside (Eiin) and outside

(Eiout) the sphere

~Einc = ~Eiin + ~Eiout. (4.21)

The size of the sphere has to be large enough to be representative of the

nanocrystal distribution in a layer. By considering an external electric field

directed along the x-axis the author in [57] derive an expression for the effective

dielectric function describing the composite medium given by

εave = εm

(1 +

f(εnc − εm)

εm + S(εnc − εm))

). (4.22)

Here f is the filling fraction previously introduced and S = L − f3 −

fK4π

is a factor dependent on the structural and geometrical arrangement of the

nanocrystals. It has been assumed that the nanocrystals are similar in respect

to dielectric function, volume, shape and surroundings. L is the depolarization

factor which equals 1/3 for spherical clusters and K is given by

37

4. Interactions between light and matter

200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

Wavelength [nm]

κ ave

MGPositive KNegative K

Figure 4.4: The average refractive index κave of spherical Sn nanocrystals inSiO2 calculated by the MG formula (equation 4.17) compared to the modified MGexpression (equation 4.22) with a positive and negative value for K respectively.As the absorption cross section is directly proportional to κ this demonstrateshow a distribution giving rise to a positive value of K will shift the absorptiontowards higher wavelengths compared to the MG case.

K =∑j

[3x2

ij

r5ij

− 1

r3ij

]pxjP. (4.23)

and describes the dependence on the spatial arrangement of the nanocrystals.

The summation is over the nanocrystals with xij being the x component of

the position vector ~rij between them. pxj is the x component of the dipole

moment and P is the polarization density. If K=0 equation 4.22 reduces to

the MG expression in equation 4.17, so K is related to how much the distri-

bution deviates from random. Figure 4.4 shows how the imaginary part of the

refractive index for Sn nanocrystals in SiO2 changes with the sign of K. Al-

though the derivation above was made for randomly distributed nanocrystals

it was shown in [57] that this approach could adequately describe the response

from samples with silver nanocrystals placed in parallel layers.

4.2.1 Origin of the dielectric function

In the beginning of this chapter when the Maxwell equations were introduced,

it was noted that the influence of the medium was contained solely in the

dielectric function (assuming non-magnetic media). It seems now only appro-

38

4.2. Nanocrystals embedded in a host material

priate to introduce the dielectric function in a more thorough way. As already

mentioned the refractive index and the dielectric function are related quanti-

ties linked to the same physical properties and since these relations are widely

used throughout this thesis they will be summarized here.

ε = N2 = (n+ iκ)2

εr = n2 − κ2

εi = 2nκ

(4.24)

For historical reasons n and κ are often called ‘optical constants’, which

can look rather puzzling since they are connected to the ‘dielectric function’.

As this thesis is probably not going to change the terminology on this matter

it will just be noted that both the dielectric function and refractive index are

(often strongly) dependent on the wavelength.

In the following sections the classical and quantum mechanical origins of

the dielectric function are described. As the classical formulation contains

some intuitive physics this approach is still useful as an introduction to the

quantum mechanical description.

Classical formulation of the dielectric function

The contributions to the classical description of the dielectric function were

largely made by Lorentz and Drude who addressed different aspects of the

electronic structure of a solid. Lorentz considered the force attaching the

electrons in an atom to the nucleus to be like small springs, which would then

describe the bound electrons in a metal. The free electrons on the other hand

can be described by the Drude model, which is basically a special case of the

Lorentz spring model.

If we assume that an electron is attached to the nucleus with a spring like

force and is acted upon by a periodically varying field such as a plane wave

described in equation 4.10 the equation of motion for a small displacement ~r

is [60]

md2~r

dt2+mΓ

d~r

dt+K~r = e ~E (4.25)

where m and e are the mass and charge of the electron, Γ is the damping

coefficient necessary for dissipating energy from the system and K = mω20

is the restoring force. A number of assumptions have been done here. First

39

4. Interactions between light and matter

of all it is assumed that the electron interacts negligibly with the magnetic

field of the incident light wave which is justifiable as the interaction is given

by e~v × ~B/c and the speed of the electron is much smaller than the speed of

light c. Secondly the mass of the nucleus is assumed to be infinite compared

to the electron. This could be sorted out by using the reduced mass instead,

but as the purpose of this section is mainly to give some qualitative insight

the use of the electron mass will suffice. For the same reason the electric

field responsible for the displacement is taken to be the incident plane wave

even though it should be the local field experienced by the electron. Finally

interactions between the electron under consideration and others are neglected

too.

As the time variation of the electric field is e−iωt (see equation 4.6) the

solution to equation (4.25) becomes

~r =e ~E

m(ω20 − ω2 − iΓω)

(4.26)

where the natural frequency of the oscillator ω0 has been substituted for the

restoring force K. The movement of the electron with respect to the nucleus

will induce a dipole moment ~p proportional to the displacement ~r given by

~p = e~r =e2 ~E

m(ω20 − ω2 − iΓω)

= α(ω) ~E (4.27)

where α(ω) is the polarizability of the one electron atom. The total contribu-

tion for N of such oscillators pr unit volume becomes

~P = Nα(ω) ~E. (4.28)

Using equation (4.5) the resulting dielectric function becomes

ε = 1 +Nα(ω)

ε0

= 1 +Ne2

mε0(ω20 − ω2 − iΓω)

(4.29)

which can be separated into a real and a complex part. Before doing so

the theory is usually extended to bulk structure by allowing for more than

one spring constant since electrons are bound with different strength. If the

40

4.2. Nanocrystals embedded in a host material

2 3 4 5 6 7 8-4

-2

0

2

4

6

8

10

εrε i

0

ω

ω0

Figure 4.5: Real and imaginary part of the dielectric function from the Lorentzoscillator model. The imaginary part which is connected to absorption in thematerial is seen to peak at the natural frequency of the oscillators ω0

density of oscillators with the natural frequency ωj is Nj the dielectric function

becomes [60]

εr = 1 +e2

mε0

∑j

Nj(ω2j − ω2)

(ω2j − ω2)2 + Γ2ω2

(4.30)

εi =e2

mε0

∑j

NjΓω

(ω2j − ω2)2 + Γ2ω2

(4.31)

Equations 4.30 and 4.31 are then the classical contribution to the dielectric

function from electrons bound the the nucleus. The behavior of the εr and εiare sketched for a single oscillator with natural frequenzy ω0 in figure 4.5.

In the beginning of this section the damping coefficient Γ was introduced

as a mean for energy to dissipate from the system. For bulk materials this

happens primarily by absorption and from equation (4.31) this is seen to be

associated with the imaginary part of the dielectric function. The absorption

is strongest at the natural frequency of the oscillator, ω0, as seen in figure 4.5.

The contribution to ε from free electrons in a metal follow directly from

the Lorentz model. As the electrons are not attached to any nucleus ω0 = 0

and equations 4.30 and 4.31 turn into

εr = 1− Ne2

mε0

1

ω2 + Γ2(4.32)

41

4. Interactions between light and matter

1 2 3 4 5 6 7-10

-5

0

5

εrε i

ε r , ε

i ωp

0

ωFigure 4.6: Real and imaginary contribution to the dielectric function from freeelectrons in the classical model.

εi =Ne2

mε0

Γ

ω(ω2 + Γ2). (4.33)

It should be noted that Γ here is still a damping coefficient, but it is related

to the free electron scattering responsible for classical resistivity, and is as such

related to another mechanism than for the bound electrons. Therefore in its

place it makes sense to introduce the lifetime τ describing the mean time

between an electron undergoes a scattering event. As τ = 1/Γ [60] the above

equations turn into

εr = 1−ω2pτ

2

1 + ω2τ2(4.34)

εi =ω2pτ

ω(1 + ω2τ2)(4.35)

where the plasma frequency ωp = Ne2

mε0has been introduced. These equations

are known as the Drude model for free electron metals and the behavior of

the real and complex part of the dielectric function is sketched in figure 4.6.

The total dielectric function is the sum of the free and bound electron con-

tributions, but it is not always possible to distinguish the two. By comparing

figure 4.5 and 4.6 it can be noted that for low frequencies εr for the Lorentz

42

4.2. Nanocrystals embedded in a host material

ZK

0

5

10

-5

-10

eV

Figure 4.7: β-Sn band structure taken from [50]. Interband and intrabandtransitions corresponding to bound and free electrons respectively are sketchedwith a red and blue arrow. The dashed line is the Fermi energy which is placedat 0 eV in the figure.

model approaches a constant, whereas the free electron Drude contribution

approaches −∞ so the behavior in that range can be expected to be governed

by the free electrons. This can be understood by looking at the band dia-

gram5 for Sn in figure 4.7. The free electron contribution to the dielectric

function describe intraband transitions, meaning that the electron is excited

into a vacant state within the same energy band, as shown with a blue arrow

in the figure, whereas the interband transitions shown in red are contained in

the bound electron oscillator model. For low energies bound electrons cannot

jump to another band so the behavior is expected to be free electron like.

Quantum theory of the dielectric function

The quantum mechanical approach rely on the interaction between the applied

field and the multi electron wave function describing an atom. To perform a

5The concept of energy bands is borrowed from quantum mechanics.

43

4. Interactions between light and matter

complete quantum mechanical treatment the electromagnetic waves should

be quantized into photons, but as this approach become unnecessarily com-

plicated for the purpose here (see for instance [61]), a simpler semi-classical

approach is given. Here the field is treated classically while the electrons are

described by their quantum mechanical wave functions. By treating the in-

teraction between the field and an electron as a small perturbation on the

electron states, the quantum mechanical analog to the Lorentz model describ-

ing interband transitions can be derived [62].

For convenience the electromagnetic field is described by a vector potential~A given by

~A =1

2A0r

(ei(~k·~r−ωt) + e−i(

~k·~r−ωt))

(4.36)

where r is a unit polarization vector and the constant A0 can be chosen such

that |A0| = |E(ω)|ω . This way the vector potential is related to the electric field

by

~E = −∂~A

∂t. (4.37)

The two exponentials in equation 4.36 describe absorption and stimulated

emission respectively, and since the absorption is the interesting part here6

only the first part is considered in the following. The perturbation to the

electronic Hamiltonian describing an electron in an external field in the weak

field regime becomes [62]

Hpert =e

m~A · ~p (4.38)

where ~p is the momentum operator for the electron. Applying first order per-

turbation theory results in the Fermi golden rule for the transition probability

Wif for an electron between an initial state ψi(~r) and a final state ψf (~r) via

interaction with the field [60]

Wif =2π

~|〈ψf |Hpert|ψi〉|2 δ(Ef − Ei − ~ω). (4.39)

Here the delta function is the condition for energy conservation, and the matrix

element

6It could be assumed that the atom is initially in its ground state, making stimulatedemission impossible.

44

4.2. Nanocrystals embedded in a host material

|〈ψf |Hpert|ψi〉|2 =e2

m2

∣∣∣⟨ψf ∣∣∣ ~A · ~p∣∣∣ψi⟩∣∣∣2 (4.40)

describes the transition amplitude between states. If the electron states are

described by Bloch functions with uk being a lattice periodic function

ψk(~r) = uk(~r)ei~k·~r (4.41)

and the dipole approximation is invoked, the transition probability becomes

[62]

Wif =2π

~

( e

2mω

)2|E(ω)|2

∑~k

|Pif |2 δ(Ef (~k)− Ei(~k)− ~ω) (4.42)

where |Pif |2 = |〈uf |r · ~p|ui〉|2 has been inserted. This is the probability for

a vertical band to band transition in a crystal as the summation is over all

the filled electron states in the valence band. By considering the continuity

equation for energy lost from absorption in a unit volume of the crystal the

imaginary part of the dielectric function is readily obtained7[62]

εi(ω) =πe2

m2ω2ε0

∑~k

|Pif |2 δ(Ef (~k)− Ei(~k)− ~ω). (4.43)

The real and imaginary parts of ε are related via the Kramers-Kronig

relations given by [60]

εr − 1 =2

πPV

∫ ∞0

ω′εi(ω′)

(ω′)2 − ω2dω′ (4.44)

εi =2ω

πPV

∫ ∞0

εr(ω′)− 1

(ω′)2 − ω2dω′ (4.45)

where PV denotes the principal value of the following integral. Using equation

4.44 the real part of the dielectric function turn out to be

εr(ω) = 1 +e2

mε0

∑~k

2 |Pif |2

m~ωif1

ω2if − ω2

(4.46)

7The summation is now over allowed k vectors per unit volume of the crystal.

45

4. Interactions between light and matter

where ωif =Ef (~k)−Ei(~k)

~ has been inserted. The current form of equation

4.46 has been chosen to emphasize the resemblance with the classical Lorentz

oscillator model. If Γ is assumed to be zero, equation 4.30 reduces to

εr = 1 +e2

mε0

∑j

Nj

ω2j − ω2

. (4.47)

The quantity fif =2|Pif |2m~ωif , which is referred to as the oscillator strength of

an optical transition, can be interpreted as the number of oscillators with the

frequency ωif and is seen to be the quantum mechanical analog to the density

of oscillators in the Lorentz model.

The contribution from the free electrons to the dielectric function via in-

traband transitions can also be calculated quantum mechanically. This is done

for instance in [60] where the results for a free electron gas is generalized to a

solid by the use of Bloch wave functions and in the dipole approximation the

real part of the dielectric function is given by

εintrar = 1− e2

~2ω2ε0

∑k,l

F (Ek,l)∂2Ek,l∂k2

(4.48)

where l is the band index and F is the Fermi function. This is seen to be

identical to the Drude result in equation 4.32 when a monovalent metal is

considered8. Introducing the reduced mass of the electron 1m∗ = 1

~2∂2E∂k2 and

the average value of the Fermi function across a Brillouin zone (=12) and

carrying out the summation over k points in a Brillouin zone results in

εintrar = 1− Ne2

m∗ε0ω2(4.49)

which, besides from the damping factor, is identical to equation 4.32. Equa-

tions for the imaginary part of ε can of course be derived as well, but the

comparison to the classical counterparts is not easily done. Often a single

expression encompassing both the real and imaginary parts are used, such as

for instance in [50] where the total dielectric function is written as

ε(ω) =e2~2

ε0m2

∑i,f

[f(Ei)− f(Ef )] |Pif |2

Eif (E2if − ~2ω2)

(4.50)

8In this discussion lifetime broadening has been ignored so the damping coefficient Γ isabsent.

46

4.2. Nanocrystals embedded in a host material

Unit cell

Bulk Nanocrystal

Figure 4.8: Difference between bulk matter and nanocrystals in terms of thenumber of unit cells. The small number of unit cells present in a nanocrystalchange the electronic structure compared to bulk.

for each crystallographic axis. By introducing the Bloch functions the dielec-

tric function becomes a bulk parameter, as the Bloch functions describe the

symmetry across a large number of unit cells. Therefore using the bulk dielec-

tric function can be misleading when studying sufficiently small nanocrystals,

as the translational symmetry described by the Bloch functions is not present.

The difference between bulk and a nanocrystal in terms of the number of unit

cells is sketched in figure 4.8. Using bulk dielectric functions, either measured

or calculated, to describe optical properties for structures on the nanometer

scale could thus very well be misleading and the results should at the very

least be given proper consideration.

4.2.2 The β-Sn dielectric function

As the following section will explain how modeling based on layers with a

known dielectric function can be used to extract the absorption coefficient

from a thin film sample, a brief description of the β-Sn dielectric function

will be given here. There have been several papers aimed at finding the Sn

dielectric function [63] [64] [65] [66] including more recent experimental [67]

and theoretical studies [50].

The data for the Sn dielectric function from Takeuchi [67] and Pedersen

[50] are compared in figure 4.9. As Takeuchi only measures the component

along the c axis (see figuer 1.1) of the β-Sn unit cell, this is the one shown

in figure 4.99. The two are seen to be in fair agreement only for wavelengths

9In the theoretical calculations by Pedersen the components along both axis of the unit

47

4. Interactions between light and matter

200 400 600 800 1000 1200 14000

20

40

60

80

PedersenTakeuchi

200 400 600 800 1000 1200 1400-80

-60

-40

-20

0

20

40

PedersenTakeuchi

Wavelength [nm]

εr εi

Figure 4.9: Comparison between real (left) and imaginary (right) part of theβ-Sn dielectric function from two recent studies by Pedersen [50] and Takeuchi[67]. The intraband contribution in [50] has been added by using the values forplasma frequencies and damping coefficient given in the article. It should benoted that it is only the part of the dielectric function parallel to the c axis ofthe β-Sn unit cell that is shown here.

below 400 nm, which may reflect their different origin. Takeuchi uses spec-

troscopic ellipsometry on a 25 nm β-Sn film at room temperature together

with some thin film modeling to extract a bulk dielectric function. Pedersen

on the other hand uses DFT band structure calculations to get the real and

imaginary parts of the dielectric function for both crystallographic directions.

The calculated data are then found to agree well with previous measurements

at -200C performed in [65] and the components does not show significant dif-

ferences between the two crystallographic directions. Therefore the dielectric

function found by Takeuchi [67] is considered to be representative for the total

Sn dielectric function, which for randomly oriented nanocrystals will be given

by an average of the crystallographic directions. As Takeuchi extract his di-

electric function from room-temperature measurements they can be expected

to be somewhat different from the low temperature calculations of Pedersen,

such as seen in [65]. As optical measurements in this work have been con-

ducted at room temperature the dielectric function given by Takeuchi can be

expected to best fit the conditions.

cell are given.

48

4.3. The matrix method for determining reflection and transmission

Θ1

Θ2

Θ1 Θ2

a) b)

n1 n2 n1 n2

d

Figure 4.10: Reflection contributions from a) a thick absorbing slab and b)thin weakly absorbing film. Multiple reflections contribute to the total reflectionin the latter.

4.3 The matrix method for determining reflection

and transmission

Whenever light is incident on an interface a part of it will be reflected and

a part of it will be transmitted and continue to travel through the material.

This situation is sketched in figure 4.10a.

For a thick absorbing slab all the transmitted part will be absorbed and

the reflection from such a slab would simply be given by the Fresnel equation

for the front interface, which for a plane wave incident on a plane interface

looks like

R =

(±n1 cos(θ1)∓ n2 cos(θ2)

n1 cos(θ1) + n2 cos(θ2)

)2

(4.51)

where the ± depends on the polarization. For light incident normal to the sur-

face θ1 = θ2 the cosines and the distinction between polarizations disappear,

and equation 4.51 reduces to

R =

(n1 − n2

n1 + n2

)2

. (4.52)

The reflection from such a surface is easily deduced from the refractive

indexes of the involved materials and similar equations exist for the trans-

mission. If the slab is not completely absorbing a part the transmitted light

49

4. Interactions between light and matter

will be reflected from the back side of the slab and contribute to the total

reflection. If the slab is sufficiently thin or weakly absorbing the light may

be reflected multiple times, as shown in figure 4.10b. When looking at thin

films this will very often be the case and the different reflection contributions

may interact and create interference effects in the reflection and transmission

measurements. For thin films containing nanocrystals it is thus important to

be able to separate the effects of such interferences from those related to the

nanocrystal properties as pointed out in [68].

One way to address the effect of multiple reflections is simply by adding

all the contributions for each layer. The simplest example is a single slab

of material as in figure 4.10b, where the successive contributions to the final

reflection10 have traversed the slab an increasing number of times before at-

tenuated enough to be negligible. Assuming normal incidence and that the

slab has an absorption coefficient α the sum of these contributions can be

written as a geometric series [69]

Rslab = R12 +T12R21T21e

−2αd

1−R12R21e−2αd(4.53)

where T is the transmittance across an interface. As R12 = R21 from equation

4.52 and the same applies to the transmission equation 4.53 is rather simple.

For more than one layer the summation approach becomes increasingly incon-

venient as reflections across layers need to be taken into account and it gets

difficult to keep track of all the different contributions.

Another way to treat the multiple reflections is by the matrix formalism

described in [48] and [70]11. This method keeps track of the different trans-

mission and reflection contributions across all boundaries by itself and is thus

much easier to work with for a large number of layers. The matrix method

can be shown to be equivalent to the summation method described above [71]

and is chosen for the simulations performed in this work.

Figure 4.11 shows a schematic example of a sample consisting of N layers

where the i´th layer is described by a refractive index Ni = ni + iκi. In each

layer the electric field can be divided into parts traveling left and right and a

prime will be used to distinguish between the field in each end of the layers.

10Transmission could be considered equivalently.11The two citations create the basis for the following part.

50

4.3. The matrix method for determining reflection and transmission

i j N......1

di dj

Eri E'ri

E'li

Eli

Elj E'lj

E'rj

Erj

HijL

iL

j

Figure 4.11: Overview of the notation used in the matrix method. The differentoperators are explained in the text.

The notation for the field at the left and right side of layer i, ( ~Ei) and ( ~E′i)

are

~Ei =

(EliEri

)

~E′i =

(E′li

E′ri

)If we limit the discussion to incident light normal to the interfaces the

Fresnel coefficients for transmission and reflection from the ij´th interface is

given by

τij =2Ni

Ni + Nj

(4.54)

rij =Ni − Nj

Ni + Nj

. (4.55)

Each of the fields in figure 4.11 can be considered as a sum of two contri-

butions. For instance E′li is the transmitted part of Elj across the ij interface

plus the reflected part of E′ri from the same interface. In that way all the fields

are connected and the exercise is to describe the link between the initial field

incident on the first interface and the field exiting in the N´th layer. This can

be accomplished by applying the symmetry relations of the Fresnel coefficients

51

4. Interactions between light and matter

which follow directly from equations 4.54 and 4.55 to the connected fields just

described.

rij = −rji

τij = 1 + rij

τijτji + (rij)2 = 1

In that way the correlation between fields across a layer can be found.

Introducing the interface transition matrix

Hij =1

τij

[1 rijrij 1

]the relation between fields across a boundary can be described by

~E′i = Hij

~Ej . (4.56)

In the same way a propagation matrix relating the fields across a single

layer can be defined as

Li =1

τij

[eiβi 0

0 e−iβi

](4.57)

resulting in the field relation

~Ei = Li~E′i . (4.58)

βi is given by

βi =2πNidiλ0

where di is the layer thickness and λ0 is the vacuum wavelength. From equa-

tions 4.56 and 4.58 it is evident that the field in one layer can be related to

the field in another layer. The relation between layer 1 and the N´th layer is

given by

~E′1 = H12L2H23L3...HN−2,N−1LN−1HN−1,N

~EN ,

52

4.3. The matrix method for determining reflection and transmission

which is more conveniently written as

~E′1 =

[S11 S12

S21 S22

]~EN = S ~EN . (4.59)

S is the stack matrix and contains all the contributions to the total reflectance

and transmittance for all the layers (often called a stack). The boundary

conditions for layer 1 and layer N

~E′1 =

(ErEi

)

~EN =

(0

Et

)can be inserted in equation 4.59 to isolate the reflection and transmission

coefficient for the entire stack. These state that in layer N there is only a field

component moving away from the stack, which is the transmitted portion Et,

in contrary to layer 1 where there are both the incident Ei and reflected field

Er. This results in the following equations for the transmission and reflection

coefficient respectively

rstack =ErEi

=S12

S22

τstack =EtEi

=1

S22.

By definition the reflectance (R) and transmittance (T ) then become

R = |rstack|2 (4.60)

T =nNn1|τstack|2 . (4.61)

In most cases the sample is surrounded by air, which means nN = n1 = 1

and the refractive indexes in equation 4.61 can be disregarded. With the

procedure outlined here it is possible to calculate the reflectance and trans-

mittance for an arbitrary number of layers where multiple reflections in and

between the layers are fully accounted for. It can be done analytically, but

53

4. Interactions between light and matter

for more than a few layers the equations become terribly cumbersome making

computer assistance crucial.

In order to calculate the reflectance (R) and transmittance (T ) at a given

wavelength the complex refractive index N = n+ iκ along with the thickness

of each layer comprising the stack need to be known. On the other hand, if R

and T can be measured the matrix method can be used to find the refractive

index of a given layer. Since the thickness of any given layer can be determined

from BF-TEM pictures the only two unknown parameters are the real and

imaginary part of its refractive index. With independent measurements of

R and T it should be possible to deduce n and κ for a layer by comparing

measurement with simulation, for instance, by varying the refractive index

input to the simulation until measurement and simulation are in agreement.

This procedure works better on paper than in practice, since not all possible

combinations of n and κ can be examined. In order to find values that are

likely to be close to the actual ones, an iterative process with a carefully chosen

start point has been applied. The process is described in more detail in the

following section.

4.4 Simulation based determination of absorption

Extracting the absorption coefficient for a given sample from a measurement of

the transmitted part of a beam of light has been performed for many years. If

the transmission can be measured without any reflection taking place across a

certain range of wavelengths experimentalists have often used the absorbance

A defined by

A = − log(T ) = − log

(ItIi

)(4.62)

where Ii and It are the incident and transmitted intensity. If the thickness of

the absorbing material is d and using equation 4.11 equation 4.62 becomes

A = αd log(e) (4.63)

and the absorption coefficient is readily acquired by measuring the transmis-

sion. Since a reflection free measurement of the transmission is not always

possible this method has its limitations. If the reflection is sufficiently con-

stant across the measured wavelength range the correct spectral dependence

54

4.4. Simulation based determination of absorption

300 400 500 600 700 800 900 1000 1100 12000

5

10

15

Wavelength [nm]

Re

fle

cta

nc

e [

%]

Quartz waferQuartz wafer + thin film

Figure 4.12: Measured reflectance spectra of a 0.4 mm thick wedge shapedquartz wafer and a similar wafer with a 500 nm thick SiO2 thin film on top.Whereas the pure quartz wafer has an almost constant reflection the thin filmshows substantial interference effects. The feature at 900 nm is a result of agrating change in the instrument.

for the absorption coefficient is found. The reason these simple equations have

found wide applications throughout the scientific world is the simplicity of the

experiment required. Simply take a beam of light and measure how much the

intensity is reduced upon introducing the sample in the light path.

For many purposes the reflection is not sufficiently constant (see figure

4.12) for equation 4.62 to be useful and other means of extracting the ab-

sorption coefficient must be used. Especially for thin films the interference

effects become important and if they are not properly taken into account it

can result in misinterpretation of absorption properties, as pointed out in [72].

Therefore a multitude of ways to remove interference and retain the correct

absorption coefficient have been developed. In [73], a method for determining

n and α as a function of wavelength for an amorphous silicon film using the in-

terference fringes in the transmission spectrum, is developed. This procedure

however, seem useful only for a homogeneous film with a very uniform thick-

ness. Others [68] [74] use both T and R to compute 1−RT , which for certain

sufficiently absorbing films removes the interference and allows determination

of the absorption coefficient. Often computer simulation based methods aid

in extracting the parameters of interest. One method is to remove the inter-

ference fringes from the transmission measurement as in [48] or alternatively

55

4. Interactions between light and matter

do as in [68] where the authors simulate R and compare it to the experimental

curve to check their values for α. As the methods described failed to produce

an interference free absorption coefficient for the samples used in this work an

iterative process based on the matrix method was used instead.

In the previous section it was pointed out that the only unknown param-

eters for determining the reflection and transmission of a multi layered film

is the real and imaginary part of the refractive index (or equivalently the

dielectric function) for the layers12. By constructing a stack of layers with

composition and distances determined from RBS and TEM measurements

the resulting reflectance and transmittance can be calculated by the matrix

method for a suitable input of the refractive indexes. The calculated val-

ues are then compared to the measured ones and the refractive index values

are optimized to ensure a good agreement. This is done separately for the

real and imaginary part by use of the fminsearch function in MATLAB. The

fminsearch function is based on the Nelder-Mead algorithm [75] [76] which

is a direct search method for minimizing a given function. The function to

minimize is given by

F = (Rm −Rs)2 + (Tm − Ts)2 (4.64)

where m and s refer to measured and simulated respectively. Thus by min-

imizing the function F the simulated R and T will approach the measured

ones. The minimization is performed with respect to n and κ for the layers

containing Sn and for all layer thicknesses involved, di. The function requires

an initial parameter guess for the optimization algorithm to start from and

some care must be taken concerning this choice. As the Nelder-Mead algo-

rithm performs a direct search for the minimum it can potentially end up in

local minima if such exist closer to the start point than the global minimum.

This situation is sketched on figure 4.13. Therefore the start point for the op-

timization should be chosen close to the expected value to increase the chance

for the optimization algorithm to end up in the global minimum.

The optimization of n, κ and di were performed individually and as one

parameter had been optimized it was held fixed during optimization of the

other two. In this way the optimization iteratively approaches a set of values

that are consistent with the experimentally measured R and T .

12Their thicknesses can be determined by TEM, but as will be explained later this pointwill also be considered in the simulations.

56

4.4. Simulation based determination of absorption

- 1 - 0 . 5 0 0 . 5 1 1 . 5 2

0

0 . 5

1

1 . 5

2

2 . 5

3

x

F

x0 xminx1x2

Figure 4.13: Schematic illustration of the importance of selecting a good start-ing point for the optimization process with a arbitrary parameter x. The func-tion value F is described in equation 4.64, but the function showed here is justa random function. Starting out in the point x0 will result in the Nelder-Meadalgorithm finding the correct minimum xmin, but on the other hand, starting inx1 will lead to the local minimum in x2 instead.

It turned out that κ could be precisely determined on a wavelength to

wavelength basis for all the samples studied resulting in nice smooth curves.

For some of the samples this was the case for n as well, but for others the

simulations jumped abruptly at some wavelength and did not return to the

expected value immediately after. An example of this behavior is seen in

figure 4.14(a). At wavelengths around 220 nm and 310 nm the otherwise

nicely continuous values from the optimization procedure change abruptly.

The reason behind this can be seen in the parts (b) and (c) of the same figure,

showing simulations of the transmittance and reflectance from a stack of thin

film layers where n has been varied with ±10%. At the wavelengths mentioned

R and T are almost identical despite the large variation in n, which is why

minimizing equation 4.64 with respect to n fails to produce a valid output at

these wavelengths.

This unsmooth behavior was deemed unphysical as the refractive index is

expected to be smooth in the wavelength interval studied. To overcome this

problem n was fit as a polynomial across the full wavelength range for those

samples in question. In figure 4.15 the optimization of n for a quartz wafer

performed both on a wavelength to wavelength basis and with a polynomial

57

4. Interactions between light and matter

5

10

15

20

nn + 10%n - 10%

200 250 300 350 400 450 500 550 60020

40

60

80

nn + 10%n -10%

1.2

1.4

1.6

1.8

2

OptimizedInput

Wavelength [nm]

T [%

]R

[%]

n

a

b

c

Figure 4.14: a) Real part of the refractive index resulting from the minimizationof equation 4.64 for a stack of thin film layers (black line) compared to the inputguess (red dashed line). The resulting reflectance (b) and transmittance (c)computed for the stack with n varied within 10%.

58

4.4. Simulation based determination of absorption

200 300 400 500 600 700 800 900 1000 1100 1200

1.5

1.55

1.6

1.65

1.7

PointwisePolynomial

Wavelength [nm]

n

Figure 4.15: Refractive index from a quartz wafer obtained from minimizingequation 4.64 from measured R and T spectra by the procedure outlined in thissection. The comparison is made between the results of a minimization where n isfound on a wavelength by wavelength basis and using the polynomial descriptionfrom equation 4.65.

fit. The two are seen to be in excellent agreement.

n(λ) = A+B(λ− λ0) + C(λ− λ0)2 +D(λ− λ0)3...O((λ− λ0)N ) (4.65)

The polynomial is given by equation 4.65 with λ0 chosen in the center of the

wavelength interval and the minimization in equation 4.64 is then performed

with respect to the coefficients A,B,C etc. The order of the polynomial has to

be chosen high enough to retrieve the characteristics of n, but not too high as

the computational time would be too long and the result would be dominated

be the high order terms at high and low wavelength. Typically a polynomial

of order 8-10 has been chosen as they fit the purpose well.

59

Chapter 5

Sn nanocrystals in SiO2

This chapter presents the main results of the investigations of β-Sn nanocrys-

tals in SiO2 in order to gain insight into their optical properties. The size of

the nanocrystals was chosen small enough that scattering could be disregarded

in the optical extinction leading to the absorption cross section as the main

physical parameter of interest. The absorption cross section was determined

from the measured reflectance (R) and transmittance (T ) spectra together

with structural information from TEM and RBS characterization by the sim-

ulation based method described in chapter 4. The absorption cross sections

are then compared to Mie and MG theory with the use of bulk dielectric func-

tions and to previous experimental work. It is also discussed how care should

be taken when applying effective medium theories to layered structures.

5.1 Introduction

During the past 15 years the interest in structures on the nanometer scale

has exploded [77] and today there are few research areas which are not in

some way exploiting the opportunities of nanotechnology. With the current

technology providing the opportunity to tailor materials to acquire various

desirable properties there are a vast number of technological applications in

sight. The computer industry has been interested in semiconductor nanocrys-

tals for a long time due to their potential use in optoelectronic devices [78] [79].

The quantum confinement effect [80] relaxes the requirement of momentum

conservation in optical transitions and nanocrystals of semiconductor materi-

61

5. Sn nanocrystals in SiO2

als with an indirect band gap becomes optically active. Nanocrystals of the

semiconducting form of Sn, α-Sn, has been extensively studied in recent years

due to their direct tunable band gap [22, 23, 81, 82, 83] but their usefulness

is somewhat limited by the precautions needed to stabilize the α-Sn. Metal-

lic nano-structures have been attracting interest due to their potential use in

single electron memory devices [84, 85] and the majority of studies into β-Sn

nanocrystals are related to that aspect [86, 87]. In recent years the potential

for metallic nano particles to improve solar cell efficiency has led to a more

pronounced focus on optical properties [10, 88].

A wide variety of techniques have been used to fabricate Sn nanocrystals in

a host material including ion implantation [38, 39, 42, 86, 89], molecular beam

epitaxy (MBE) [22, 23, 90], evaporation condensation [91, 92] and sputtering

[40, 93, 94]. As explained earlier the sputtering technique has been chosen in

this work since the grown films need not be crystalline and it is very suitable

to produce amorphous layers with well defined thicknesses. Previous work

investigating the optical absorption properties of β-Sn nanocrystals in SiO2

have been done by Huang et al. [40] and Kjeldsen et al. [94]. Huang et al.

produce β-Sn nanocrystals randomly distributed in SiO2 by co-sputtering of

individual Sn and SiO2 targets and subsequent vacuum annealing. In the op-

tical measurements they look for band to band absorption in order to identify

quantum confinement effects for the Sn nanocrystals. As they see absorption

in the UV range they ascribe that to oxidized Sn clusters that they identify

in their samples as their β-Sn nanocrystals are considered to big to show con-

finement effects and thus absorption in the UV range. They do not consider

the absorption to be related to the composite medium, as described in sec-

tion 4.2, but entirely as related to the individual tin oxide nanocrystals or Sn

related defects. They report their absorption as the percentage of absorbed

light instead of absorption cross section and they do not make a quantitative

comparison of the absorption to Mie or MG theory.

Kjeldsen et al. [94] produce a single layer of β-Sn nanocrystals in SiO2

by the sputtering technique and subsequent annealing in a N2 atmosphere at

450C for 30 minutes. The size of the nanocrystals is varied by increasing

the thickness of the sputtered Sn layer and they report a slight redshift in

the position of the extinction peak as the nanocrystal size is increased. Again

there is no comparison to Mie or MG theory for the extinction properties.

In this work the first experimental atomic absorption cross section for

62

5.2. Experimental details

Sample Mixedlayerthick-ness(nm)

SurroundingSiO2 layerthickness(nm)

Sncon-tent(at%)

Sn Arealden-sity Ω(1/cm2)

Meandiam-eter(nm)

Fillingfrac-tionf

RSn1 72±5 220±10 4.1 2.15·1016 5.0 0.08

RSn2 295±15 105±6 0.78 2.10·1016 - 0.02

Table 5.1: TEM and RBS parameters for the two RSn samples. Mixed layerrefers to the layer containing both Sn and SiO2. No nanocrystals were identifiedin BF-TEM for the RSn2 sample and no diffraction pattern was seen either, thusno nanocrystal size is given for this sample.

β-Sn nanocrystals in SiO2 is reported. This is done both for nanocrystals

randomly distributed throughout SiO2 and for nanocrystals arranged in a

layered structure and the absorption cross section is compared to Mie and

MG theory. It is shown that the absorption increases when the nanocrystals

are arranged in layers and a simple model is developed in order to understand

this increase.

5.2 Experimental details

As described in chapter 3 two types of samples have been investigated. The

Sn nanocrystals have been formed either randomly distributed throughout an

amorphous SiO2 (a-SiO2) layer or in layers separated by different thickness of

a-SiO2. The samples with randomly distributed nanocrystals will be labeled

RSn whereas MLSn will be used to denote those with layers of nanocrystals.

5.2.1 Random Sn samples

Two different samples with Sn randomly distributed in a-SiO2 were fabricated

as sketched in figure 3.3b. Both had a total film thickness of ∼500 nm but

the thickness of the Sn containing layer was varied and consequently also the

thicknesses of the surrounding oxide layers.

Various structural details for the random Sn samples are summarized in

table 5.1. TEM measurements did not show any nanocrystals in the RSn2

63

5. Sn nanocrystals in SiO2

Recoil energy [MeV]

Yiel

d

Sn peak in RBS spectrum

MeasuredSimulated

Figure 5.1: Part of the RBS spectrum of the RSn1 sample showing the Snpeak. From the very good agreement between the measured curve and the onesimulated by the RUMP software [27] the Sn atoms are expected to be evenlydistributed throughout the layer.

sample in BF mode and in diffraction mode no diffraction pattern was ob-

served. A doubling of the annealing time compared to the RSn1 sample was

attempted, but still no nanocrystals were identified. For the RSn1 sample the

amount of Sn was determined from the RBS spectrum, a part of which is shown

in figure 5.1. From the agreement between the measurement and simulation

of the Sn peak it is concluded that the Sn is evenly distributed throughout

the layer. The Sn content in the RSn2 sample was also determined by RBS

and the areal density was very close to that of the RSn1 sample. Therefore

it is concluded that the Sn content is present either in nanometer scale struc-

tures too small to be seen in TEM or possibly as randomly scattered Sn atoms

throughout the sample.

For the other RSn1 sample the individual nanocrystals were visible in

TEM and a BF picture of this sample together with its size distribution is

given in figure 5.2. The average diameter is 5.0±0.1 nm. The Sn nanocrystals

was confirmed to by in the β phase by TEM diffraction. There was no signs

64

5.2. Experimental details

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.002468

10121416

Num

ber o

f obs

erva

tions

Diameter [nm]a) b)

10 nm

Figure 5.2: a) BF-TEM image of randomly distributed Sn nanocrystals in SiO2

for the RSn1 sample. b) The corresponding size distribution for RSn1 where theaverage diameter is 5.0±0.1 nm.

of crystalline tin oxide diffraction seen in either TEM or X-ray diffraction

measurements. Based on Energy Dispersive X-ray (EDX) studies in [86] it

was concluded that the oxygen stay on the SiO2 after heat treatment in a

Sn implanted SiO2 layer. As the Si=O double bond is stronger than the

Sn=O bond thermodynamics also expect the oxygen to stay attached to the

Si atoms [95]. It could be speculated that oxygen could diffuse from the air

during storage and oxidize the Sn atoms but this has been shown in [94] to be

unlikely. SiO2 has also previously proved to be an efficient diffusion barrier

[96, 97]. The relatively long annealing time of 1 hour should give all Sn atoms

enough time to diffuse to a growing nanocrystal and therefore it is expected

that the majority of the Sn is in the form of β-Sn nanocrystals.

5.2.2 Sn in a multi layered structure

A series of multi layered samples with approximately the same Sn content

but with a different distance between the layers was also produced. The

structural parameters of the multi layered samples based on TEM and RBS

measurements are summarized in table 5.2.

The RBS spectra were compared to RUMP [27] simulations, as explained

in chapter 3, such that the Sn areal density could be extracted. For the

samples with a Sn layer separation less than 45 nm the layer structure was

65

5. Sn nanocrystals in SiO2

Sample SeparatingSiO2 layerthickness(nm)

Surroun-ding SiO2

layerthickness(nm)

SnArealden-sity Ω(1/cm2)

MeanNC di-ameter(nm)

Standarddevi-ation(nm)

MLSn1 3±0.5 230±15 2.27·1016 6.0 0.81

MLSn2 11±2 224±8 2.06·1016 6.1 0.65

MLSn3 24±4 181±10 2.14·1016 6.2 0.71

MLSn4 42±3 158±8 2.46·1016 6.1 0.62

MLSn5 54±4 135±7 2.43·1016 6.3 0.53

MLSn6 68±5 90±6 2.34·1016 6.2 0.58

Table 5.2: Structural data for the multi layered samples. The Sn areal densityand atomic content are found from RBS whereas the nanocrystal diameter andlayer thicknesses are measured by TEM.

not resolved by RBS and they could be fitted with a mixed Sn and SiO2 layer

as for the RSn samples. For the samples with a larger Sn layer separation

the layers were resolved, as shown for sample MLSn6 in figure 5.3. In this

the RUMP simulation is based on a structure of alternating layers of Sn and

SiO2 and whereas the layer structure in the Sn peak is clearly resolvable, the

simulation does not agree fully with the measurement. This agreement can be

improved considerably by expanding the thickness of the Sn layers a bit and

include some SiO2 in their composition. This is can be justified by comparison

with TEM pictures, where the Sn is seen to form clusters already in the as

grown samples. If this cluster formation occurs during sputtering the following

SiO2 layer will ‘cover the grooves’, and the layer can best be described as a

mixed layer of Sn and SiO2. In this way the TEM and RBS measurements

are in accordance. The Sn clusters observed in as grown TEM pictures, as

mentioned in section 2.2.5, thus appear to be a result of the sputtering process

and not of the ion milling step in the TEM sample preparation. To ensure

that the ion milling step is not responsible for nanocrystal formation plane

view TEM samples for the MLSn2 sample were prepared both by the ion

milling procedure previously described and by etching in a hydrofluoric acid

solution. Comparison between the size distributions from the two approaches

are seen in figure 5.4 and did not suggest that the ion milling step influence

66

5.2. Experimental details

Recoil energy [MeV]

MeasuredSimulated

Figure 5.3: RBS spectrum of MLSn6 as grown. The correspondence betweenmeasurement and simulation can be increased by expanding the thickness of theSn layers and letting them be composed of both Sn and SiO2.

the nanocrystal size1.

TEM characterization provides a more precise estimate of different layer

thicknesses in the ML samples compared to RBS and additionally the size

distribution of the nanocrystals can be determined. The layer thicknesses are

directly extracted from cross sectional TEM samples, whereas the size distri-

bution are most easily derived from plane view pictures. There will always

be a slight ambiguity in this approach since nanocrystals from more than one

layer can be taken into account where ideally only a single layer should be

considered. This effect should be more pronounced when the layers are close,

but as the average size of the nanocrystals turn out to be very consistent

throughout all the samples, as seen from table 5.2, the size distributions are

considered to satisfactorily determined.

In figure 5.5 cross sectional TEM pictures of the MLSn2 and MLSn4 sam-

ples are presented. The layer thicknesses are measured at different places and

between the different layers in order to get the most precise values and an

1A standard test for difference in mean diameter was performed based on the size distri-butions in figure 5.4 and the result was within the 95% confidence interval for equal diameter.Although the test assumes the diameters so be normally distributed, which can be debatedgrounded on the figure, it is concluded that the sample preparation does not influence thenanocrystal size.

67

5. Sn nanocrystals in SiO2

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.00

2

4

6

8

10

12

14

16

Nu

mb

er

of

ob

serv

ati

on

s

Diameter [nm]

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.00

2

4

6

8

10

12

14

16

18

20N

um

be

r o

f o

bse

rva

tio

ns

Diameter [nm]

Figure 5.4: Size distributions from the MLSn2 sample from PV-TEM samplesprepared by chemical etching in a hydrofluoric acid solution (left) and by ionmilling (right). The average sizes are 5.9±0.11 and 6.1±0.09 respectively.

50 nm 50 nm

Figure 5.5: Cross sectional TEM picture of MLSn2 (left) and MLSn4 (right)showing Sn nanocrystals surrounded by SiO2 layers of different thickness.

68

5.3. Simulation based determination of absorption

d2 d3 d4

1 2 3 4 5 6

d5

Air SiO2 Sn nanocrystal Quartz

Layer index

Figure 5.6: Overview of the different layers introduced to describe the RSnsample in the modeling procedure. The different layers are each characterizedby a thickness and a complex refractive index.

estimate of the uncertainty in the numbers presented in table 5.2. All samples

showed very consistent layer thicknesses across different spots at the sample.

5.3 Simulation based determination of absorption

In order to extract the correct absorption coefficient from the samples a thin

film modeling procedure, as described in section 4.3, was employed. This way,

the oscillations in the reflection and transmission spectra originating from in-

terference in the beam could be modeled. For the RSn sample the simulations

have been performed on a 6 layers structure as sketched in figure 5.6. As the

refractive indices need to be known for the SiO2 layers and the quartz wafer

in order to extract information about the layer containing nanocrystals these

have been determined from measurements described in the following sections.

5.3.1 The correction factor for reflection measurements

In order to model the layered structure correctly within the matrix model,

measurements were performed to acquire the refractive index of the layers not

containing Sn in figure 5.6. First of all the correction to the reflection from

the spectrophotometer needed to be found. The spectrophotometer measures

the reflectance relative to a BaSO4 powder standard. Since the correction is

mainly due to the deviation from 100% reflectance of the powder, the reflective

69

5. Sn nanocrystals in SiO2

Wavelength [nm] Wavelength [nm]

200 300 400 500 600 700 80030

40

50

60

70

80

R [

%]

MeasuredSimulated

200 300 400 500 600 700 8000.8

0.85

0.9

0.95

1

1.05

Cor

rect

ion

fact

or

BaSO4 table value

Measurement

Figure 5.7: The correction factor compared to data for the BaSO4 reflectancetaken from Grum [31] (left) determined by comparing the measured reflectancefrom a Si wafer to a simulation (right). Data for the Si refractive index are takenfrom [36].

properties of the BaSO4 powder used in the integrating sphere was put under

study. The tabulated data for BaSO4 reflectance in [31] and [32] do not cover

the entire wavelength range used in this work. The reflectance within the

entire range of interest was determined by measuring the reflection from a

polished silicon wafer and by comparing it to a simulation using the matrix

method and bulk values for the Si refractive index. As the wafer is thick

enough that it absorbs practically all light below the Si band gap of 1120 nm

the only reflection contribution in the wavelength range of interest originate

from the front surface. Although the Si wafer is p-type doped to about 1019

atoms/cm3 this is not expected to influence the reflection. Also, the surface

layer of natural oxide developing on top of the Si wafer when exposed to air

has also been taken into account in the simulation. A layer thickness of 2 nm

[98] [99] was confirmed by ellipsometry.

The left part of figure 5.7 shows the correction factor resulting from com-

paring the measured to the simulated reflectance of the Si wafer which are

shown on the right. The spectrophotometer uses the compressed BaSO4 pow-

der as a baseline such that the measured reflectance is normalized to it. The

BaSO4 reflectance is then extracted by dividing the measured spectrum with

the simulated one. The correction factor becomes larger than 1 for wave-

lengths between 300 nm and 450 nm thus it is not only related to the BaSO4

reflectance as a reflectance above 100% would be unphysical. The correction

factor found in this way did however improve the correspondance between

70

5.3. Simulation based determination of absorption

n kWavelength [nm]

200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1x 10-5

Table valueOptimized

200 300 400 500 600 700 8001.52

1.56

1.6

1.64

1.68

Table valueOptimized

Figure 5.8: Real (left) and imaginary (right) part of the refractive index for awedged shaped quartz wafer. Tabulated values from [36] are compared to thoseextracted from measured R and T .

measurements and simulations for other samples specifically in the 300-400

nm range. The effects that ultimately contribute to the correction factor were

not investigated further but the correction factor has been used to normalize

all further reflectance measurements correctly.

5.3.2 Refractive indices of the quartz wafer and SiO2 layers

After settling the correction factor the refractive index of the quartz wafer

could be determined. Tabulated values are available [36] [100], but as the

quartz wafers are mechanically ground into a wedged shape and subsequently

polished their optical properties may (though hopefully do not) change. Trans-

mission and reflection from a wedged shaped quartz wafer were measured and

compared to simulations, and the refractive index of the wafer was optimized

to give the best overall correspondence on a wavelength to wavelength basis.

The result is seen in figure 5.8, and the most notable difference between the

tabulated values and those extracted by the optimization procedure is that κ,

and thereby the absorption, sets in at a higher wavelength for the quartz used

in this study compared to the tabulated values in [36]. The optimized values

for n and κ for the wafer were used in subsequent simulations.

The SiO2 refractive index was determined in the same way as for the

71

5. Sn nanocrystals in SiO2

quartz wafer using a quartz wafer upon which a layer of SiO2 was sputtered

for the measurements. The results obtained in this way agreed very nicely

with tabulated values for amorphous SiO2 (within 1%, which indicates a good

oxide quality) and were also used in the following simulations.

All measurements were conducted on wedge shaped quartz wafers in order

to remove interference oscillations from the substrate. Those oscillations need

to be removed in the simulations as well to get a better comparison with the

measurements. As the layers introduced in the matrix method are considered

having a fixed thickness di (see figure 4.11) interference oscillations from all the

layers including the substrate will show up in the calculated R and T spectra.

Removing these oscillations can be accomplished in different ways. One way is

to simply remove the e−iβ part from the propagation matrix in the quartz layer

(equation 4.57) so no electric field component travel to the left in the layer.

Doing this, however, neglects the reflection contribution from the backside of

the substrate, which is considerable for weakly absorbing films. Another way

would be to consider an infinitely thick or thin wafer, but then the thickness

has to be correlated with the imaginary part of the quartz refractive index

to account for its absorption. The most favorable way to remove the wafer

oscillations turned out to be closely related to the actual wedged shape used

in measurements. A series of calculations were made where the thickness of

the quartz wafer was successively increased by a small amount followed by an

averaging of the resulting calculated R and T . It was found that averaging

over 500 thicknesses was sufficient to remove the interference from the wafer

completely. This procedure is closely linked to the measurement where the

interfering rays have traveled through a slightly different thickness of quartz

due to its wedged shape.

5.4 β-Sn absorption cross sections

Returning to the RSn samples sketched in figure 5.6 the refractive index for

the Sn containing layer (layer 3) can now be determined from the simulation

procedure. This has been done for both RSn samples, where the starting

point for the simulation was taken as the MG values for n and κ found from

equations 4.17 and 4.24 by using bulk values for the Sn dielectric function from

[50] or [67]. By letting the simulation procedure independently determine n, κ,

and the thickness of the layers di, a good agreement with experimental R and

72

5.4. β-Sn absorption cross sections

5

10

15

20R

[%

]

MeasurementSimulation

200 300 400 500 600 700 80020

40

60

80

100

Wavelength [nm]

T [

%]

MeasurementSimulation

Figure 5.9: Measured and simulated R and T spectra for the RSn1 sample aftern, κ and di have been optimized.

T was found. It should be noted that whereas the experimental R is measured

at an 8 degree angle, normal incidence light is considered in the simulations.

The effect of this is mainly to introduce a small uncertainty in the simulation

of the layer distances of ∼ 1% which is considered to be insignificant. The

results of the optimization procedure was also found to be independent of

which Sn dielectric function was used for the input guess. The measured and

simulated R and T for the RSn1 sample is shown in figure 5.9 and the resulting

values for n and κ are shown in figure 5.10. The layer distances were included

as a parameter for optimization as it is related to the interference oscillations

through β in equation 4.57 and, as for all other parameters, it had no imposed

boundaries. Therefore it was important to check that the values determined

from the simulation are consistent with TEM measurements in order to assure

that the simulations represent the samples correctly.

When κ for the layer with Sn nanocrystals randomly distributed in SiO2

is determined, the absorption cross section for the Sn atoms can be calculated

through equation 4.12 and 4.13 where ρ is the density of Sn atoms in the layer

given by

ρ =Ω

d(5.1)

73

5. Sn nanocrystals in SiO2

1

1.5

2

n

OptimizedMG input

200 300 400 500 600 700 8000

0.5

1

Wavelength [nm]

k

OptimizedMG input

Figure 5.10: n and κ values for RSn1 corresponding to the simulated R andT in figure 5.9 (red dashed line) compared to the MG values used as an initialguess (full black line). The Sn dielectric function used was from Pedersen [50]and the one for SiO2 was found in the section above.

where Ω is determined from RBS and d from TEM or simulations. The atomic

absorption cross section in the MG theory can be calculated from

σa =σncVncρB

(5.2)

where σnc is given by equation 4.18, Vnc is the nanocrystal volume and ρB is

the bulk Sn density. In figure 5.11 the atomic absorption cross section for the

RSn1 sample calculated from equation 5.2 is compared to that obtained from

the MG theory using bulk values for the β-Sn dielectric function. The curve

extracted from experiments is seen to resemble the one using the dielectric

function from Takeuchi [67] much better than the one from Pedersen [50]. As

both this work and the work in [67] was performed at room temperature as

opposed to the low temperature calculations in [50] the better correspondence

may in part be temperature related. Another thing is that Takeuchi extracted

his dielectric function from a thin film of β-Sn and as discussed in section 4.2.1

the dielectric function of nano scale structures might be different from bulk.

The quantitative agreement between the measured absorption cross section

and the Takeuchi MG curve support the conclusion that the Sn is in the

form of β-Sn nanocrystals. The disagreement at very low wavelengths (below

220nm) may be related to the precise reading of the dielectric function from the

74

5.4. β-Sn absorption cross sections

200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1

1.2x 10

-16

Wavelength [nm]

σ a S

n

RSn1MG PedersenMG Takeuchi

[cm

2]

Figure 5.11: Nanocrystal absorption cross section for the RSn1 sample calcu-lated from the κ deduced in figure 5.10 compared to MG theory with the bulkSn dielectric constant taken from Pedersen [50] and Takeuchi [67].

graphs given in [67]. The structural dissimilarities between the experimental

curve and the Takeuchi MG curve can be ascribed to the use of bulk dielectric

functions to describe the absorption properties of nanocrystals. Previously

experimental absorption curves for semiconductor nanocrystals have shown a

smooth behavior lacking the kinks seen by the use of bulk dielectric functions

in the MG theory [48].

In the RSn1 sample the interaction between the nanocrystals is not ex-

pected to be significant due to their small size and the relatively low filling

fraction seen in table 5.1. This can be verified by plotting the absorption cross

section from Mie and MG theory, as has been done in figure 5.12. The differ-

ences between them are minute. In the low wavelength range the MG curve

is however seen to be slightly higher than the Mie curve. This suggests that

when the nanocrystals start to interact the effect is to increase the absorption

from each nanocrystal (at least in the wavelength interval studied here) and

this effect becomes more pronounced if the filling fraction is increased.

The same procedure as described above has been used to extract the ab-

sorption cross section from the RSn2 sample which can be compared to the one

from RSn1 already discussed. This has been shown in figure 5.13(a) together

with the absorbed percentage of the incident light (A), which is available di-

rectly from the measurements through A = 1−R− T . From figure 5.13(a) it

75

5. Sn nanocrystals in SiO2

200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1

1.2x 10

-16

Wavelength [nm]

σ a S

n

MieMG

[cm2 ]

Figure 5.12: Atomic absorption cross section for Sn atoms in the RSn1 samplecalculated by Mie and MG theory using the bulk Sn dielectric function given in[67].

200 300 400 500 600 700 8000

1

2

3

4

5

6x 10-17

Wavelength [nm]

σ a [

cm

2]

RSn1RSn2

(a) Atomic absorption cross section for the RSnsamples

200 300 400 500 600 700 8000

10

20

30

40

50

60

70

Wavelength [nm]

Ab

sorb

ed

pe

rce

nta

ge

[%

]

RSn1RSn2

(b) Absorbed percentage for the RSn samples

Figure 5.13: Comparison of the absorption from the two random Sn samples.The absorbed percentage is found directly from measured data whereas the ab-sorption cross section for the Sn atoms is extracted from the simulation basedprocedure.

76

5.4. β-Sn absorption cross sections

200 300 400 500 600 700 80020

30

40

50

60

70

80

90

100

Wavelength [nm]

Tra

nsm

itta

nce

[%]

as grown

400° C annealed

(a) Transmittance of the RSn1 sample

200 300 400 500 600 700 80020

30

40

50

60

70

80

90

100

Wavelength [nm]

Tra

nsm

itta

nc

e [

%]

as grown

400° C annealed

(b) Transmittance of the RSn2 sample

Figure 5.14: Measured transmittance spectra for the RSn samples before andafter annealing. Whereas the differences between the as grown and the 400Cannealed sample are small for the RSn2 sample the RSn1 sample clearly shows adecrease in transmittance particularly in the low wavelength range after anneal-ing.

is clear that the absorption from each Sn atom is larger for the RSn1 sample

than for the RSn2 sample. It was previously concluded that the Sn in the

RSn1 sample was in the form of β-Sn nanocrystals whereas the form of Sn

in the RSn2 sample has not been determined. This suggests that the forma-

tion of nanocrystals is a way to enhance the absorption from Sn containing

materials. A further indication of an increased nanocrystal absorption can

be seen by comparing the transmittance for the as grown and annealed RSn

samples. This has been done for the RSn1 sample in figure 5.14(a) and the

RSn2 sample in figure 5.14(b). For the RSn1 sample there is a pronounced

decrease in transmittance upon annealing whereas the transmittance for the

RSn2 sample almost shows the opposite effect2. It should be noted that the re-

flectance spectra (not shown) are essentially unchanged after annealing so the

differences in the transmittance spectra represent the absorption differences

well.

Turning to the multi layered samples the Sn containing layer could be fit

either with a single layer such as the RSn1 sample, as explained in section 4.2,

or by taking the individual layers into account. In figure 5.15 the multi layer

stack is shown and layers which are given similar properties in the simulation

2The multi layered samples also showed a decrease in transmittance upon annealing,though the effect was much smaller than for the RSn1 sample.

77

5. Sn nanocrystals in SiO2

Air SiO2 Sn nanocrystal Quartz

1 2 3 4 3 4 3 4 3 4 3 5 6 1

d2 d3 d4 d5 d6

Layer index

Figure 5.15: Overview of the different layers introduced to describe the MLSnsamples in the modeling procedure. As the layer index in the top of the figureindicates the five nanocrystal layers are characterized by the same thickness andrefractive index, just as the four SiO2 layers separating them.

are assigned the same layer index. Thus, simulating the multi layer stack,

the layers with nanocrystals are all assumed to have the same thickness and

are described with the same refractive index. The same goes for the SiO2

layers separating them and from TEM pictures these restrictions seem to be

appropriate.

To obtain an initial guess of the refractive index of the layers containing Sn

nanocrystals they are described as MG layers with Sn and SiO2 as mentioned

in figure 5.3. By letting the layer thickness be equal to the diameter of the

nanocrystals the filling fraction f can be estimated from the Sn areal density

as

f =Ω

dtotρB(5.3)

where dtot is the combined thickness of the Sn nanocrystal layers. With this it

is possible to make a good initial guess for the refractive index of the nanocrys-

tals layers. The filling fraction for the layers is close to 20% so within the range

where the MG theory remains applicable as stated in section 4.2. Although

it can be debated whether the nanocrystals can be thought of as randomly

distributed this will serve as a good first guess of the parameters for the sim-

ulation procedure. The thickness of the surruonding SiO2 layers were almost

78

5.4. β-Sn absorption cross sections

200 250 300 350 400 450 500 550 6000

1

2

3

4

5

Wavelength [nm]

n

NC from RSn1Bulk Sn

(a) n for the nanocrystals from RSn1

200 250 300 350 400 450 500 550 6000

2

4

6

8

Wavelength [nm]

k

NC from RSn1Bulk Sn

(b) κ for the nanocrystals from RSn1

Figure 5.16: n and κ for the β-Sn nanocrystals deduced from the RSn1 sampleas explained in the text compared to data for bulk tin from [67]. The wavy be-havior of the nanocrystal parameters may very well be related to the polynomialfit for n in the RSn1 sample.

identical, as seen in TEM, but they were nevertheless treated seperately in

order for the simulation to have another free parameter to model the oscilla-

tions. According to figure 5.11 the Sn dielectric function from [67] was seen

to give the best correspondense between MG theory and the randomly dis-

tributed nanocrystals in the RSn1 sample, and would thus be appropriate to

use to find the initial guess for n and κ. Because of the disagreement for small

wavelengths, however, a refractive index for the Sn nanocrystals was extracted

from the RSn1 sample. This can be done from the n and κ values determined

for the layer using the MG formula in equation 4.17 ‘in reverse’. The resulting

values for n and κ for the Sn nanocrystals are seen in figure 5.16(a) and (b)

where they are compared to the values from [67]. They are only shown up

to a wavelength of 600 nm as it can be doubted whether useful information

can be extracted at higher wavelengths given the very low absorption in that

region, as seen in figure 5.11. They were used together with the SiO2 dielectric

function to produce the initial guess for n and κ for the nanocrystal layers

from equation 4.17.

In figure 5.17 the measured R and T from the MLSn4 sample are compared

to a simulation with the values of n, κ and di extracted from the minimization

process. For the transmission the two are in excellent agreement, whereas

the reflection show minor disagreements. This was the general trend seen

throughout the line of samples and the overall agreement represented by the

79

5. Sn nanocrystals in SiO2

15

50

100

T [

%]

SimulationMeasurement

200 300 400 500 600 700 800

5

10

15

Wavelength [nm]

R [

%]

SimulationMeasuerment

Figure 5.17: Transmission and reflection spectra of the MLSn4 sample togetherwith the simulated spectra from the optimized n, κ and d parameters. The cor-respondence between simulation and measurement is very good for the transmis-sion whereas there are minor discrepancies in the reflection. Notice however thedifference in scale bar between the two.

value of the minimized function F (equation 4.64) was more or less the same

for all samples. The corresponding n and κ for the Sn layers are shown in

figure 5.18. Whereas n seem to be in very nice agreement with the initial

guess where the Sn dielectric function was extracted from the RSn1 sample,

the values for κ show pronounced differences.

Table 5.3 provides an overview of the thicknesses resulting from the sim-

ulation procedure and all the values can be seen to be consistent with TEM

measurements which were summarized in table 5.1 and 5.2. From table 5.3

the thicknesses of the surrounding SiO2 layers for the RSn1 sample are seen to

be in good agreement with the MLSn2 sample. The total thickness of the Sn

containing part of the samples is also approximately equal so the main differ-

ence between the two samples is that the nanocrystals are arranged randomly

or in layers respectively. The atomic absorption cross section from these two

samples are compared in figure 5.19(a). It is noted that the absorption sets in

at approximately the same wavelength for the two samples, but that the peak

absorption from the multi layered sample is much larger than for the random

80

5.4. β-Sn absorption cross sections

1

1.5

2

2.5

n

OptimizedMG input

200 300 400 500 600 700 8000

0.5

1

Wavelength [nm]

k

OptimizedMG input

Figure 5.18: n and κ corresponding to the simulated R and T in figure 5.17for the MLSn4 sample compared to the MG values used as an initial guess.Whereas n seem to be well described by the MG input the values for κ areheavily underestimated in the low wavelength range.

Sample Thicknessof Sn con-taininglayers (nm)

Thicknessof top SiO2

layer (nm)

Thicknessof bottomSiO2 layer(nm)

Thicknessof interme-diate SiO2

layers (nm)

MLSn1 6.1 242 238 3.2

MLSn2 6.3 221 223 11.2

MLSn3 6.0 192 188 27.5

MLSn4 6.1 155 151 41.6

MLSn5 6.2 129 137 55.3

MLSn6 6.0 100 95.6 70.5

RSn1 70.4 228 222 -

RSn2 307 107 98 -

Table 5.3: A summary of the layer thicknesses resulting from the optimizationprocedure for both the random and multi layered samples. The different layersreferred to can be identified in figure 5.6(RSn) and 5.15(MLSn). All values canbe seen to be consistent with the TEM measurements summarized in table 5.1and 5.2.

81

5. Sn nanocrystals in SiO2

200 300 400 500 600 700 8000

1

2

3

4

5

6

7

8x 10

-17

Wavelength [nm]

σ a

RSn1MLSn2

(a) σa for RSn1 compared to MLSn2

200 300 400 500 600 700 8000.6

0.8

1

1.2

1.4

1.6

1.8

2

Wavelength [nm]

En

ha

nc

em

en

t

(b) Enhancement of absorption from multi layers

Figure 5.19: a)Atomic absorption cross section for the MLSn2 show a higherabsorption than for the RSn1 sample. b) The MLSn2 absorption relative to thatof RSn1 is termed the enhancement and it is found by division of the two curvesin a).

nanocrystals. The absorption enhancement from arranging the nanocrystals

in a layered structure, as compared to a random distribution, is shown in figure

5.19(b). The peak value is close to 2 and the enhancement occur exclusively

in the ultraviolet region from 200-400 nm.

In figure 5.20 the absorption cross section for the MLSn2 atoms is com-

pared to MG theory using the Sn dielectric function derived from the RSn1

sample described above for both a multi layered structure (figure 5.15) and a

structure with a single effective layer (figure 5.6). It is clear that neither can

explain the increased absorption which indicates that the basic MG theory is

not suitable to describe a system with nanocrystals in a layered structure, at

least with the filling fractions used here.

The question now is whether the increase in absorption is a consequence

of the arrangement of the nanocrystals in individual layers or it is related to

the interaction between different layers. If the latter is the most important

effect then the absorption should be sensitive to the interlayer distance which

is exactly the parameter varied between the multi layered samples. In figure

5.21 the absorption cross section for all multi layered samples are compared.

The interlayer distances can be found in table 5.2, but the overall trend is that

the Sn layer separation increases from MLSn1 to MLSn6. From the figure it is

clear that there are differences between the absorption cross sections between

the multi layered samples, but there is no apparent trend in going from a low

82

5.4. β-Sn absorption cross sections

200 300 400 500 600 700 8000

1

2

3

4

5

6

7

8x 10-17

Wavelength [nm]

σ a [

cm

2]

MLSn2MG with a layered structureMG with a single effective layer

Figure 5.20: Atomic absorption cross section of the MLSn2 sample comparedto MG theory for both a multi layered structure such as seen in figure 5.15 anda structure with a single effective layer as seen in figure 5.6. The Sn dielectricfunction used for the MG theory was the one extracted from the RSn1 sample(figure 5.16(a) and 5.16(b)).

layer distance to a higher. Based on this finding it is concluded that it is more

likely that the absorption enhancement is related to interactions in the layers

themselves.

To account for the absorption enhancement seen between the RSn1 and

MLSn2 samples a calculation based on the dipolar character of the nanocrys-

tals has been performed. A test nanocrystal is placed in the center of a

coordinate system and the absorption enhancement due to the surrounding

nanocrystals is calculated for different nanocrystal configurations. A more

detailed description of this is given in appendix A. In chapter 4 the interband

absorption was described by the electron-radiation interaction Hamiltonian in

equation 4.38 in terms of the vector potential ~A. In the dipole approximation

it can be described equivalently in terms of the electric field by [62]

Hpert = −e~r · ~E = ~E · ~d (5.4)

where ~d is the dipole moment involved in the transition. As evident from the

Fermi Golden Rule in equation 4.39 it is the square of ~E · ~d that becomes

83

5. Sn nanocrystals in SiO2

200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1x 10-16

Wavelength [nm]

σ a [

cm

2]

MLSn1MLSn2MLSn3MLSn4MLSn5MLSn6

Figure 5.21: Comparison of the atomic absorption cross section for the multilayered samples.

important for the absorption, and thus calculating

⟨(~E · ~d

)2⟩

(the brackets

denote the average value) would give insight into the absorption efficiency.

This has been done in order to investigate how the absorption of a nanocrystal

would be affected by a layer of nanocrystals placed a distance z0 away. The

details of the calculation is given in appendix A. If the nanocrystals are treated

as point dipoles in a host material with dielectric function εh, the end result

becomes

⟨(~E · ~d

)2⟩

=1

3E2

0d2 +

nsp2d2

2304πε2h

60a4 + 36z40 + 91a2z2

0(a2 + z2

0

)4 (5.5)

where ns is the areal density of nanocrystals in a layer, p is the size of the

dipole moment of a nanocrystal, a is an integration constant and E0 is the

external electric field inside the host. This equation shows two contributions

to the matrix element, the first term is only related to the applied external

field and would thus represent the situation described in Mie theory where the

interactions between nanocrystals are neglected. The second term describes

the effect from all nanocrystals in a layer, placed a distance z0 away. In figure

5.12 it was shown that the Mie and MG theory were in very good agreement

84

5.4. β-Sn absorption cross sections

due to the low filling fraction in the RSn1 sample so the enhancement of the

nanocrystal absorption due to a neighboring layer of nanocrystals compared

to a random distribution is assumed to be well described by dividing through

with 13E

20d

2 in equation 5.5

ABSlayerABSMie

= 1 +nsp

2

768πε2hE20

60a4 + 36z40 + 91a2z2

0(a2 + z2

0

)4 . (5.6)

Until now the nanocrystals have been treated as point dipoles. To take

their actual shape into account the dipole moment p can be introduced as that

of a dielectric sphere (nc) surrounded by a host material (h) given by [101]

p = 4πεhR3

∣∣∣∣ εnc − εhεnc + 2εh

∣∣∣∣E0 (5.7)

where R is the nanocrystal radius. With that the absorption enhancement

becomes

ABSlayerABSMie

= 1 +nsπR

6∣∣∣ εnc−εhεnc+2εh

∣∣∣248

60a4 + 36z40 + 91a2z2

0(a2 + z2

0

)4 . (5.8)

The areal density of nanocrystals, ns, can be estimated from plane view TEM

pictures or calculated from the atomic areal density found by RBS. Using

TEM pictures to do the estimate has the disadvantage that nanocrystals from

more than one layer may be observed leading to a larger value of ns than the

actual one so this estimate can at best be used as an upper limit. Therefore

the nanocrystal density has been determined from RBS data by

ns =Ω

5ρBVnc(5.9)

where the factor of 5 represents the number of Sn layers in the sample. The

values of ns are close to 0.01 nm−2. To get the enhancement of the absorp-

tion from a nanocrystal due to a neighboring layer of randomly distributed

nanocrystals the integration constant a is set equal to zero and the radius

of the nanocrystals is R = 3 nm. The dielectric functions for Sn and SiO2

were found earlier in this chapter and the resulting enhancement is shown in

figure 5.22 for different values of z0. It is clear from equation 5.8 that the

85

5. Sn nanocrystals in SiO2

z−40 dependence reduces the influence of a nearby layer of nanocrystals sig-

nificantly the further away it is. As seen in figure 5.22 the enhancement is

already negligible when z0 is 6 nm and the lowest center to center distance of

the nanocrystal layers is for the MLSn1 sample for which it is close to 9 nm.

This supports the conclusion from figure 5.21 that it is not the interaction

between different layers that is responsible for the increased absorption seen

in the multi layered samples compared to the randomly distributed nanocrys-

tals. It could be argued that the nanocrystals are not very well described

by point dipoles as they have a diameter of ∼6 nm and that the effective

value z0, at least for the MLSn1 sample, should be significantly smaller. The

areal coverage of nanocrystals in a layer3 is about 25% so there may not be a

nanocrystal directly above the test nanocrystal and the integration constant

a can be different from zero. Also it should be kept in mind that the direc-

tion of the dipole moment is taken perpendicular to the layer separation. In

reality the situation is more complicated than in the simple picture of ideal

dipoles, but the result in equation 5.8 is nevertheless a strong indication that

the main contribution to the absorption enhancement seen in figure 5.19(b) is

not interactions between different nanocrystal layers.

For that reason the increase in absorption in the multi layered samples com-

pared to the RSn samples must be due to the interaction between nanocrystals

within the same layer. As explained in appendix A the absorption enhance-

ment can not simply be determined by letting z0 → 0 in equation 5.8 but it

can be calculated using the formalism described in the appendix. The calcu-

lations have been performed in appendix A and he result is given in equation

A.40 and is rewritten here

ABSlayerABSMie

=

1 +nsπR

3∣∣∣ εnc−εhεnc+2εh

∣∣∣a

2

+5nsπR

6∣∣∣ εnc−εhεnc+2εh

∣∣∣24a4

. (5.10)

where equation 5.7 has again been used for the dipole moment of the nanocrys-

tals. a is again an integration constant, but this time it is always larger than

zero as there is a limit to how close the nanocrystals can be4. It can be noticed

3This can be estimated by multiplying ns with the cross sectional area of a nanocrystalπR2.

4As the nanocrystals are not point dipoles in reality and there can not be anothernanocrystal arbitrarily close to the test nanocrystal.

86

5.4. β-Sn absorption cross sections

200 300 400 500 600 700 8000.98

1

1.04

1.08

Wavelength [nm]

En

ha

nc

em

en

t

z0 = 6 nm

z0 = 8 nm

z0 = 10 nm

Figure 5.22: Enhancement of the nanocrystal absorption from a layer ofnanocrystals a distance z0 away. The Sn dielectric function used is the oneextracted from the RSn1 sample.

that the term proportional to a−4 is identical to the second term in equation

5.8 if z0 = 0, but an additional enhancement arises from the nanocrystals

within the layer.

In figure 5.23 the absorption enhancement given by equation 5.10 is shown

for a few different values of the integration constant a. This represents the

closest distance between neighboring nanocrystals and the exact value can be

hard to establish. Knowing the nanocrystal areal density ns and their radius

it is possible to calculate the average nanocrystal separation by, for instance,

assuming that they are arranged in a square lattice. This results in an average

distance of ∼4 nm between the nanocrystals5. As the calculations leading to

equation 5.10 assume the nanocrystals to be point dipoles but the actual dipole

moment is somewhat smeared out across the size of the nanocrystal a value of

a somewhat larger than 4 nm can be expected. By setting a equal to 6 nm the

comparison between the absorption enhancement due to nanocrystals within

a layer and nanocrystals located in another layer can be done by comparing

figure 5.22 with figure 5.23. From this it is clear that the absorption is largely

5This is the distance from the rim of one nanocrystal to the rim of the neighboringnanocrystal, not the center to center distance.

87

5. Sn nanocrystals in SiO2

200 300 400 500 600 700 8001

1.4

1.8

2.2

2.6

3

Wavelength [nm]

En

hanc

eme

nt

a = 4 nma = 6 nma = 8 nm

Figure 5.23: Enhancement of the nanocrystal absorption from nanocrystalswithin the same layer for a few different values of the integration constant a.

affected by nanocrystals within the same layer.

Until now the absorption enhancement has been calculated with reference

to the absorption of a single nanocrystal that only feel the external field (as is

the case in Mie theory). This was done because the interaction between the

nanocrystals in the RSn sample was expected to be insignificant due to the low

filling fraction and therefore the Mie and MG results are almost similar (figure

5.12). This should of course also be evident from the dipole calculations so

the absorption enhancement due to a random distribution of dipoles (as in the

RSn samples) has been calculated in the last part of appendix A. The result

is

ABSrandomABSMie

= 1 +2nvπR

6∣∣∣ εnc−εhεnc+2εh

∣∣∣2a3

(5.11)

where nv is the nanocrystal volume density and the dipole moment from equa-

tion 5.7 has been used. nv can be calculated from

nv =Ω

dρBVnc(5.12)

88

5.4. β-Sn absorption cross sections

200 300 400 500 600 700 8001

1.04

1.08

1.12

1.16

1.2

Wavelength [nm]

Enh

ance

men

t

(a)

200 300 400 500 600 700 800

1

1.5

2

2.5

3

Wavelength [nm]

Enha

ncem

ent

Different layer z0 = 6 nm

Same layer a = 6 nm

(b)

Figure 5.24: a) Absorption enhancement from randomly distributed nanocrys-tals compared to a single nanocrystal. The integration constant a has been setto 6 nm. b) Absorption enhancement relative to a).

where d is the thickness of the Sn containing layer in the RSn samples (see

table 5.1). With this the increase in absorption from randomly distributed

Sn nanocrystals relative to the effect of the external field (Mie theory) can be

calculated. This is shown in figure 5.24(a) where the parameter a, which in

this case represents the average distance between nanocrystals in three dimen-

sions, is again set to 6 nm. This is consistent with the density of nanocrystals

in the RSn1 sample. The absorption enhancement due to the other nanocrys-

tals is seen to be very small when they are randomly distributed, which is

consistent with the small difference between Mie and MG theory for the RSn1

sample shown in figure 5.12. The theoretical approach used to describe the

nanocrystal layers is thus consistent with the MG theory.

Figure 5.24(b) shows the absorption enhancement relative to the randomly

distributed nanocrystals in figure 5.24(a). The enhancement is not very dif-

ferent from figure 5.22 and 5.23 due to the small difference between Mie and

MG theory for the RSn1 sample, but it may be noticed how the z0 = 6 nm

line drops below 1. This signifies that the absorption enhancement due to a

layer sufficiently far away is actually less than for a random distribution of

nanocrystals which is a result of the strong dependence on distance.

In figure 5.25 the comparison between the model and the experimental

data for the absorption enhancement of a layered structure is shown. The two

89

5. Sn nanocrystals in SiO2

200 300 400 500 600 700 800

0.8

1.2

1.6

2

Wavelength [nm]

Enha

ncem

ent

ExperimentModel

Figure 5.25: Comparison of the absorption enhancement from MLSn2 relativeto RSn1 from experimental data (shown previously in figure 5.19(b)) and thedipole model described in appendix A. The model adequately describe the max-imum absorption increase but not the general trend from the experiment. As itwas shown above that the nanocrystals within the layer dominate the enhance-ment the effects from other layers has been ignored.

are seen only to agree around the maximum for the experimental curve. This

directly reveals some of the limitations of this very simple model. Treating

the nanocrystals as point dipoles is only a good approximation when they are

far apart so this poses a problem as the interaction between them is seen to

be strongly dependent on distance so the ones close by contribute the most.

The point dipole approach is to some extent remedied by the introduction of

the dipole moment in the form of equation 5.7, which takes the size of the

nanocrystals into account, but it is still by no means a thorough treatment.

Another element is that in this approach the dipole moment of the nanocrys-

tals is assumed to be caused solely by the external field (equation 5.7). Thus

the contribution to the dipole moment from the field from all other nanocrys-

tals has been neglected even though it is by no means insignificant. The

exact value for the integration constant a can also be debated. In a more

thorough model the integration would be carried out from a certain distance

large enough that the nanocrystals beyond it can be described as randomly

distributed. The nanocrystals within this perimeter should then be randomly

90

5.4. β-Sn absorption cross sections

placed and the resulting terms in the

⟨(~E · ~d

)2⟩

equation ( equation A.28

given in appendix A) should be summed. Averaging this over a large number

of distributions would probably improve the model. This summation approach

is in line with what was proposed by Garcia et al. [58] as described in section

4.2. As the main purpose was to reveal whether the increased absorption in

the multi layered samples was a result of the layer structure itself or caused

by interaction between the layers the current model suffices.

An interesting comparison can be made between the values for σa ex-

tracted from the simulation procedure and an approximate expression derived

from simple energy conservation considerations. When light is incident on a

sample it will either be reflected, transmitted, absorbed or scattered. As the

nanocrystals are small the scattering is expected to be much smaller than the

absorption and can be ignored6. The percentage of absorbed light A can then

be written as

A = 1−R− T. (5.13)

From the definition of the absorption cross section σabs it can be shown

that

σncabs =

number of absorbed photons/number of absorption centers

number of incident photons/area

σncabs =

A/dNncA

1/A

σaabs =

A

Ω(5.14)

where in the last line it has been converted to an atomic absorption cross

section by introducing the atomic areal density Ω instead of the nanocrystal

density Nnc, A has been identified as the number of absorbed photons divided

by the number of incident photons and A is a unit area. This is a crude

estimate based on the assumption that there is no overlap of the absorption

cross sections of the individual nanocrystals, which is very questionable at least

6See the size dependence in equations 4.15 and 4.16. It can also be argued that the scat-tering has been taken into account in the measurement of transmission (forward scattering)and reflection (backward scattering) due to the geometry of the measurement. Either wayit is disregarded here.

91

5. Sn nanocrystals in SiO2

sample

I0

I0(1-Ras)(1-e -αd)

I0(1-Ras)(1-e -αd)R2

ase-2αd

I0(1-Ras)(1-e

-αd)Rase-αd

d

Figure 5.26: The absorption from a weakly absorbing sample can be dividedinto contributions from successive reflections of the continuously attenuatedbeam. The absorption from each crossing is written above the line represent-ing the ray path, which is assumed to be perpendicular to the sample surfaces.The sample is assumed to be characterized by a uniform absorption coefficient αand a reflection coefficient Ras between air and sample.

for the multi layered geometry used. Another approach which is a little more

elaborate treat the sample as an effective medium and add the contributions to

the absorption for multiply reflected rays, as sketched in figure 5.26. Each time

a ray traverses the layer some of it will be absorbed and for each encounter

with the sample/air interface some of it will be reflected and transmitted much

like the description in section 4.3. By adding the absorbed portion for each

passage (the first few are shown in figure 5.26) the result sums up to

A =(1−Ras)(1− e−αd)

1−Rase−αd(5.15)

where Ras is the reflectance from the air-sample interface. Since α is related

to the absorption cross section by αd = σaΩ this can be rewritten as

σa =1

Ωln

(1− A

1−RasRas

1− A1−Ras

)(5.16)

By using the measured reflectance for Ras, σa can be estimated directly from

the measured R, T and Ω through equations 5.13 and 5.16. This has been

done in figure 5.27(a)-(f) where it is compared to the absorption cross section

92

5.4. β-Sn absorption cross sections

deduced from the simulations based on the matrix formalism. The two ap-

proached are seen to be in relatively good agreement as long as the interlayer

distance is below 45 nm, but for the MLSn5 and MLSn6 samples the qualita-

tive behavior is not reproduced. This suggest that interference effects arising

from the reflection between different Sn layers become important, and if they

are not properly accounted for it could lead to wrong conclusions for the ab-

sorption properties. In order to take the interference effects into account the

electric fields must be considered which is done in the matrix formalism but

not in equation 5.16.

The same conclusions can be drawn based on the approach proposed by

Garcia [58] which was presented in the last part of section 4.2. Another author

[57] claims that a layered structure can be described as an effective medium

with a dielectric function given by equation 4.22 by choosing a suitable value

for K (equation 4.23). The simulation based procedure was thus performed

on the multi layered samples where they were described with an effective layer

in the same way as the RSn samples (figure 5.6). The initial guess for n and

κ was found from equation 4.22 with different values for K and the resulting

values were fairly consistent. The same dielectric functions for SiO2 and Sn

as for the multi layered approach were used. By this method it was seen (not

shown here) that the experimental R and T could be reproduced relatively

well for the MLSn1-MLSn4 samples whereas there were large deviations for

the samples with the largest distance between layers, MLSn5 and MLSn6.

Figure 5.28(a) and (b) show the comparison between the measured re-

flectance and the result of the optimization process using an effective layer as

in figure 5.6. The MLSn2 sample is well described whereas the strong peak

in the reflection for the MLSn6 sample around 240 nm is completely unre-

producible with the effective layer description. This suggest that the peak is

related to an interlayer reflection and by considering the lowest order Bragg

condition for constructive interference between two layers

λ = 2dnSiO2 (5.17)

this can be verified. Inserting for the distance d between the front of two

adjacent layers the sum of the layer distance and the thickness of a nanocrystal

layer from table 5.3 (70.5 nm + 6.0 nm) and the refractive index of SiO2 at

this wavelength (1.55) positive interference should occur at 237 nm. This is

further supported by the MLSn5 sample where a similar calculation places

93

5. Sn nanocrystals in SiO2

200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1 x 10-16 MLSn1

Wavelength [nm]

σ a

SimulationMeasurement

(a)

200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1 x 10-16 MLSn2

Wavelength [nm]

σ a

SimulationMeasurement

(b)

200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1 x 10-16 MLSn3

Wavelength [nm]

σ a

SimulationMeasurement

(c)

200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1 x 10-16 MLSn4

Wavelength [nm]

σ a

SimulationMeasurement

(d)

200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1 x 10-16 MLSn5

Wavelength [nm]

σ a

SimulationMeasurement

(e)

200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1 x 10-16 MLSn6

Wavelength [nm]

σ a

SimulationMeasurement

(f)

Figure 5.27: Comparison between the atomic absorption cross section derivedfrom energy conservation (equation 5.16) and from the iterative optimizationprocedure based on the matrix method.

94

5.5. Comparison with previous studies

200 300 400 500 600 700 8000

5

10

15

20

25

Wavelength [nm]

R [

%]

MeasurementSimulation

(a) MLSn2 considered as an effective medium

200 300 400 500 600 700 8000

5

10

15

20

25

Wavelength [nm]

R [

%]

MeasurementSimulation

(b) MLSn6 considered as an effective medium

Figure 5.28: Multi layered samples analyzed as effective media just as theRSn samples. A good correspondence is seen for the MLSn2 sample where thelayers are relatively close together, whereas the measured reflectance can not bereproduced for the MLSn6 sample.

200 300 400 500 600 700 8000

10

20

30

Wavelegth [nm]

R [

%]

MLSn6MLSn5

Figure 5.29: Measured reflectance spectra for the MLSn5 and MLSn6 samples.The strong peak around 240 nm for the MLSn6 sample has moved down below200 nm for the MLSn5 sample consistent with the decrease in Sn layer separation.

the interlayer reflection peak just below 200 nm and in figure 5.29 the peak

is clearly seen. The effective layer approach is thus not very useful when the

distance between layers becomes large.

5.5 Comparison with previous studies

It was noted in the beginning of this chapter that Huang et al. [40] produced

samples similar to the RSn samples in this work. As they measured the

95

5. Sn nanocrystals in SiO2

300 400 500 600 700 8000

20

40

60

80

Wavelength [nm]

Ab

sorb

ed

pe

rce

nta

ge

[%

]

RSn1Huang et al.

(a) Absorption comparison with Huang et al.[40]

300 400 500 600 700 8000

0.5

1

1.5

2

2.5x 10-17

Wavelength [nm]

σ a [

cm

2]

RSn1Huang et al.

(b) Absorption cross sections from (a)

Figure 5.30: a) Comparison of the absorbed percentage given in [40] with theone obtained in this work. Both samples have been annealed in vacuum at 400C.b) Atomic absorption cross sections calculated from equation 5.14 for the curvesin (a).

absorbed percentage A this is compared to the RSn1 sample, where A is given

by equation 5.13, in figure 5.30(a). It is noted that their absorbed percentage

is higher than in this work, but that could very well be due to the amount

of absorbing material in the sample which has not been taken into account.

By converting the absorbed percentage to an absorption cross section using

equation 5.14 7 the comparison on a per atom scale can be done. The filling

fraction and film thickness given in [40] along with the amount of Sn being in

the β-Sn phase from their XPS measurements are used to determine the atomic

absorption cross section due to the β-Sn nanocrystals by use of equations 5.3

and 5.14. In figure 5.30(b) the sample annealed at 400C from Huang et al.

is compared to the absorption cross section for the RSn1 sample annealed at

the same temperature in this work. Both were calculated using equation 5.14.

The resemblance of the two curves is very good so quite possibly Huang et al.

are actually measuring the absorption from β-Sn nanocrystals.

In their samples a significant part of the Sn is oxidized which is why

they ascribe their absorption to SnO2 nanocrystals. The absorption of SnO2

nanocrystals in SiO2 calculated by MG theory is shown in figure 5.31 where it

7The more elaborate equation 5.16 would require knowledge of the reflection which isnot given in [40].

96

5.5. Comparison with previous studies

300 400 500 600 700 800

0

1

2

3x 10

-17

Wavelength [nm]

σ a [

cm

2]

SnSnO

2

Figure 5.31: Comparison of MG theory for nanocrystals of β-Sn or SnO2 inSiO2 with the same nanocrystal density and filling fraction. The dielectric func-tion of β-Sn is taken from [67] and the one for SnO2 is from [102].

is compared to that of β-Sn nanocrystals. It is seen that for the same amount

of material the absorption due to SnO2 is much smaller than for β-Sn which is

consistent with the good agreement in figure 5.30(b). If the absorption were

to originate from band to band transitions in the SnO2 clusters, as claimed in

the paper, the absorbed percentage of light from these transitions should be

larger than that from the Sn nanocrystals. The absorption coefficient α for

SnO2 is below 105 cm−1 for wavelengths above 250 nm [102, 103, 104] and the

equivalent thickness of SnO2 in the samples described is close to 10 nm. The

corresponding absorbance of such a layer (equation 4.63) is less than 0.04 so

less than 10% of the light would be absorbed by SnO2 at 250 nm and even less

at higher wavelengths. Assuming this analysis is valid Huang et al. [40] could

be measuring the absorption of β-Sn nanocrystals. If that is the case their

measurements seem to be in very good agreement with what was measured in

this work.

Uhrenfeldt [48] studied the absorption from different semiconductor nanocrys-

tals embedded in SiO2 and saw that the absorption cross section of both Ge,

Si and InSb nanocrystals randomly distributed in SiO2 were well described by

MG theory with bulk refractive indices. This agrees well with the observations

in this work and another indication that Huang et al. are actually measuring

the absorption of β-Sn nanocrystals.

97

5. Sn nanocrystals in SiO2

Kjeldsen et al. [94] measured the extinction from a single layer of β-Sn

nanocrystals in SiO2 for different nanocrystal sizes. In figure 5.32 the sample

with the smallest nanocrystals (15 nm diameter) are compared to one of the

multi layered samples in this work. The sample from [94] was chosen as the

size is expected to be small enough for absorption to be the dominant form of

extinction. As the extinction in [94] is calculated directly from transmission

spectra through equation 4.62 this in combination with αd = σaΩ and equation

4.63 makes it possible to calculate an atomic absorption cross section through

σa =− log(T )

Ω log(e)(5.18)

where T is the measured transmittance. A large discrepancy between the two

studies is seen in the low wavelength range where the absorption is largest.

It may be partially related to the fact that not all the Sn is in the form of

β-Sn nanocrystals in [94] as suggested in their BF-TEM pictures. As discussed

above this would result in a lower absorption than expected, but this alone

probably can not account for the differences. Although they do not observe

any oxidation of the Sn in their TEM pictures it has previously been reported

for post growth annealing in a N2 atmosphere as discussed in section 3.2.1.

Also they use the same commercially available sputtering system as was used

by Huang et al. [40] where they saw a significant portion of the Sn oxidate

during sputtering.

A different consideration is that the nanocrystals in [94] are so far apart

that the absorption increase for the layer of nanocrystals as compared to a

random distribution in equation 5.10 should be very small. Using the struc-

tural data given in [94] the radius R of the nanocrystals increase by a factor

of ∼2.5 compared to the MLSn samples in this work but at the same time

the nanocrystal layer density ns decrease by a factor of 4 and the increase in

nanocrystal separation would increase the integration constant a by approx-

imately the same amount. Therefore the atomic absorption cross section for

the β-Sn nanocrystals in [94] would be expected to be similar to the MLSn

samples investigated in this work. The reason for the difference seen in figure

5.32 is thus expected to be mainly due to differences in the relative amount

of Sn being in the form of β-Sn nanocrystals. In the calculation of the atomic

absorption cross section in figure 5.32 it is assumed that all of the Sn is present

as β-Sn nanocrystals.

98

5.6. Conclusion

200 300 400 500 600 700 8000

2

4

6

x 10-17

Wavelength [nm]

σ a[c

m2]

MLSn4Kjeldsen et al.

Figure 5.32: Absorption cross section from the MLSn4 sample compared tothat from Kjeldsen et al. Both have been calculated by equation 5.18.

5.6 Conclusion

The atomic absorption cross sections for β-Sn nanocrystals embedded in SiO2

has been determined in the visible to soft UV spectral range both for nanocrys-

tals randomly distributed and for nanocrystals arranged in layered structures.

The general form of the absorption from these nanostructures is significantly

different from bulk Sn. For the randomly distributed nanocrystals the absorp-

tion cross sections were well described by both Mie and MG theory if the bulk

Sn dielectric function from Takeuchi [67] was used. As the volume fraction

of Sn was small the Mie and MG theories were essentially equal. Also it was

found that the absorption cross section for the RSn1 sample where the Sn was

seen to be in the form of nanocrystals was higher than the RSn2 sample where

no nanocrystals was seen. This suggests that forming nanocrystals increases

the absorption for the Sn atoms. From the position of the absorption peak

β-Sn nanocrystals in SiO2 could seem as a good candidate for use as UW

protection in products such as sunglasses etc.

The absorption of the multi layered nanocrystals exceeded both the ran-

dom distributions and the MG theory for the individual layers. It was seen

that the increased absorption was independent of the layer separation so a

simple model to explain this observation was developed (appendix A). This

confirmed that the increased absorption is a result of the individual layers and

99

5. Sn nanocrystals in SiO2

not from interaction between the layers. The difference in absorption cross

section between the multi layered samples seen in figure 5.21 though is not

accounted for. One could expect it to be related to differences in the areal

density between the samples, as a higher areal density would mean a smaller

separation between nanocrystals8 but this was not consistent with the RBS

data. Another issue would be if a different portion of the Sn content in the

samples were in the form of nanocrystals. As all samples have been produced

and subsequently handled in exactly the same way there would be no apparent

reason for such differences, but further investigations are needed in order to

account for this.

It was suggested in [57] that a layered structure of nanocrystals could be

well described by effective medium theory, such as an extended MG method,

by taking the geometrical distribution of the nanocrystals into account. This

was seen to work well for the samples with a small interlayer distance, but when

the layers get far enough apart for interlayer interference to become impor-

tant the effective medium description becomes inadequate. A more thorough

description of the samples, such as one using the matrix formalism, is thus

necessary in order to be sure not to misinterpret absorption spectra.

It could be interesting for future experiments to address the dipolar cou-

pling of the nanocrystal in a more systematic way. For instance, if the

nanocrystals could be arranged in lines instead of planes measuring the absorp-

tion properties for light polarized parallel and perpendicular to these nanocrys-

tal lines could verify if the dipole approach used in appendix A is applicable.

By making lines with nanocrystals of the same size with different separation

the measured absorption differences could be compared to the model predic-

tions.

8As their sizes are almost equal.

100

Appendix A

A simple model for the

impact on nanocrystal

absorption from the

surrounding nanocrystals

This appendix describes how a nanocrystal is influenced by the electric field

produced by other nanocrystals in the sample. The procedure is closely related

to what was discussed in section 4.2. At first the influence of a layer of

nanocrystals located a distance z0 away when the system is acted upon by an

external field will be calculated. Next the impact of nanocrystals from within

the same layer will be discussed and finally nanocrystals randomly distributed

such as usually described by the MG theory. The general procedure is to place

a test nanocrystal in the center of the coordinate system O and then calculate

the contribution to the absorption of this nanocrystal by the external field

and by the other nanocrystals in the sample. The nanocrystals are described

as point dipoles as they are much smaller than the wavelength of the incident

light. This is assumed to be a good approximation for nanocrystals far from

the test nanocrystal, but for those close by the approximation may be more

questionable. As this is simply meant to illustrate the origin of the enhanced

absorption seen for the MLSn samples compared to the RSn1 sample the

simple point dipole description remain adequate.

101

A. A simple model for the impact on nanocrystal absorption from thesurrounding nanocrystals

y

x

y

z

p

φz0

D

r

O

Figure A.1: A layer of dipoles parallel to the (x,y) plane will produce an electricfield, which will contribute to the field felt by a nanocrystal located at the originof the coordinate system O.

A.1 Nanocrystals in a different layer

Here the absorption enhancement due to nanocrystals in a different layer than

the test nanocrystal is described. The general geometry of the problem is

sketched in figure A.1 where the test nanocrystal is placed in the origin of the

coordinate system and a layer of nanocrystals is placed a distance z0 above.

The dipoles are assumed to be directed along the y-axis with a dipole

moment p given by

~p =

0

p

0

(A.1)

and the position vector from the highlighted dipole in figure A.1 to the origin

of the coordinate system is given by

102

A.1. Nanocrystals in a different layer

~r = − ~D =

−r cosφ

−r sinφ

−z0

. (A.2)

The electric field at the origin of the coordinate system from a dipole (~p)

placed at ~D as shown in figure A.1 (black highlighted) surrounded by a host

material with dielectric function εh is given by [105]

~Edip(~r) =1

4πεh

1

r3(3(~p · r)r − ~p) (A.3)

By inserting ~p and ~r into equation A.3 the field at O from a dipole at D

becomes

ExEyEz

=1

4πεh

− 3rp sinφ

(√r2 + z2

0)5

−r cosφ

−r sinφ

−z0

− 1

(√r2 + z2

0)3

0

p

0

.

(A.4)

A.1.1 Electric field from a layer of nanocrystals

This section is intended to describe how a nanocrystal will be influenced by

the other nanocrystals located in a layer above it, as sketched in figure A.1.

The field at a given point is the sum of the external field and the dipole fields

from all other nanocrystals

~E = ~E0 +∑i

ni ~Ei (A.5)

where ~E0 is the external field experienced by a nanocrystal and ni is the num-

ber of dipoles giving rise to the dipole field ~Ei. What is ultimately interesting

is the nanocrystal absorption, and the interband absorption is given by the

Fermi golden rule in equation 4.39. The perturbation Hamiltonian is often

written as in terms of the electronic dipole operator ~d as [62]

Hpert = −e~r · ~E = ~E · ~d (A.6)

103

A. A simple model for the impact on nanocrystal absorption from thesurrounding nanocrystals

which is identical to equation 4.38 in the dipole approximation. Inserting this

in the Fermi golden rule will show that the absorption essentially depends on

the square of this perturbation Hamiltonian. As the number of nanocrystals

in a sample is very large the quantity of interest is⟨(~E · ~d

)2⟩

(A.7)

where the brackets imply the average value. Inserting ~E from equation A.5

and equating the terms produces

⟨(~E · ~d

)2⟩

=

⟨(~E0 · ~d

)2+ 2

(~E0 · ~d

)∑i

ni ~Ei · ~d

⟩

+

⟨(∑i

ni ~Ei · ~d

)∑j

nj ~Ej · ~d

⟩ . (A.8)

The second term can be seen to contain the factor∑

i 〈ni〉 ~Ei · ~d. The sum-

mation is the total average electric field from all other nanocrystals which for

a layer of randomly distributed nanocrystals becomes zero and equation A.8

reduces to

⟨(~E · ~d

)2⟩

=(~E0 · ~d

)2+

⟨∑i

ni

(~Ei · ~d

)∑j

nj

(~Ej · ~d

)⟩

=(~E0 · ~d

)2+∑i

∑j

〈ninj〉(~Ei · ~d

)(~Ej · ~d

)(A.9)

where the quantity 〈ninj〉 has to be determined. This can be done by a bit

of statistical analysis. Assuming that the probability si of finding a dipole in

the small area element ∆Ai of the layer with total area A will be given by

si =∆AiA

. (A.10)

If the total number of dipoles in the layer is N the probability for finding nidipoles in ∆Ai and nj dipoles in ∆Aj is given by the multinomial distribution

p(ni, nj , N−ni−nj) =N !

ni!nj !(N − ni − nj)!snii s

njj (1−si−sj)N−ni−nj (A.11)

104

A.1. Nanocrystals in a different layer

for which the covariance (COV) of two variables is given by

COV (ni, nj) = −Nsisj , i 6= j

COV (ni, ni) = V AR(ni) = Nsi(1− si) , i = j.

(A.12)

where V AR is the variance. This can be combined with the general definition

of the covariance [106]

COV (ni, nj) = 〈(ni − 〈ni〉)(nj − 〈nj〉)〉

= 〈ninj〉 − 〈ni〉〈nj〉 (A.13)

for the two occasions where i = j and i 6= j equation

〈ninj〉 = (N2 −N)sisj , i 6= j

〈ninj〉 = Nsi(1− si) +N2s2i , i = j.

(A.14)

This can now be inserted into equation A.9 and the transition matrix

element can be calculated

⟨(~E · ~d

)2⟩

=(~E0 · ~d

)2+∑i

∑j 6=i

(N2 −N)sisj

(~Ei · ~d

)(~Ej · ~d

)+∑i

(Nsi(1− si) +N2s2

i

) (~Ei · ~d

)2. (A.15)

This equation can be further simplified by looking at the second and third

term individually. Using that Nsi = 〈ni〉 the second term can be written as

105

A. A simple model for the impact on nanocrystal absorption from thesurrounding nanocrystals

∑i

∑j 6=i

(N2 −N)sisj

(~Ei · ~d

)(~Ej · ~d

)=

(1− 1

N

)∑i

∑i 6=j〈ni〉〈nj〉

(~Ei · ~d

)(~Ej · ~d

)=

(1− 1

N

)∑i

〈ni〉(~Ei · ~d

)∑i 6=j〈nj〉

(~Ej · ~d

)=

(1− 1

N

)∑i

〈ni〉(~Ei · ~d

)∑j

〈nj〉 ~Ej

· ~d− 〈ni〉( ~Ei · ~d) (A.16)

Again the term∑

j〈nj〉 ~Ej must be zero when the summation is over a full

layer so the second term in equation A.15 is equal to

−(

1− 1

N

)∑i

(〈ni〉

(~Ei · ~d

))2. (A.17)

In the same way the third term in equation A.15 can be rewritten to

∑i

(Nsi(1− si) +N2s2

i

) (~Ei · ~d

)2=

∑i

(〈ni〉(1− si)

(~Ei · ~d

)2+(〈ni〉

(~Ei · ~d

))2)

=

∑i

(〈ni〉

(1− 〈ni〉

N

)(~Ei · ~d

)2+(〈ni〉

(~Ei · ~d

))2)

=(1− 1

N

)∑i

(〈ni〉

(~Ei · ~d

))2+∑i

〈ni〉(~Ei · ~d

)2(A.18)

where it has been used that si = 〈ni〉N . Adding up the terms in equation A.15

now simplifies to⟨(~E · ~d

)2⟩

=(~E0 · ~d

)2+∑i

Nsi

(~Ei · ~d

)2. (A.19)

The first term is the effect on the nanocrystal directly from the external

field and the second term is the effect from the other nanocrystals. It is this

106

A.1. Nanocrystals in a different layer

part that is most interesting. It can be calculated by turning the summation

into an integration. As si = ∆AiA the second term can be written as∑

i

Nsi

(~Ei · ~d

)2=∑i

N

A

(~Ei · ~d

)2∆Ai (A.20)

and as ∆Ai is a small area element in the plane the summation can be turned

into an integration

∑i

N

A

(~Ei · ~d

)2∆Ai ≈

∫ ∞a

∫ 2π

0

N

A(Exdx + Eydy + Ezdz)

2 rdφdr. (A.21)

Equating the ~E · ~d term yields 6 terms

(Exdx + Eydy + Ezdz)2 = E2

xd2x + E2

yd2y + E2

zd2z + 2ExdxEydy

+ 2ExdxEzdz + 2EydyEzdz. (A.22)

As the components of the dipole field along the cartesian axis is given by

equation A.4 the off diagonal elements 2ExdxEydy + 2ExdxEzdz + 2EydyEzdzcan all be shown to be zero due to the φ integration, so all that remains is

ns

∫ ∞a

∫ 2π

0

(E2xd

2x + E2

yd2y + E2

zd2z

)rdφdr (A.23)

where ns = NA has been inserted. The integration over r is carried out from

a distance a to infinity. a can always be set to zero if necessary. The integral

can be calculated by inserting the electric dipole field from equation A.4 and

carrying out the integration. The dipole moment of the nanocrystals is a

constant that can be taken out from the integration and if the orientation of ~d

is random, d2x = d2

y = d2z = d2/3 , the three contributions can be put together.

The result of some lengthy calculations is

⟨(~E · ~d

)2⟩

=1

3E2

0d2 +

nsp2d2

2304πε2h

60a4 + 36z40 + 91a2z2

0(a2 + z2

0

)4 (A.24)

where the factor of 1/3 in the first term is a consequence of the assumed y

polarization of the incident light. Before too much confusion arises between d

107

A. A simple model for the impact on nanocrystal absorption from thesurrounding nanocrystals

x

ypφ

aD

Figure A.2: The electric field from a layer of nanocrystals experienced by ananocrystal placed at the origin of the coordinate system is determined by takingthe field from a single nanocrystal (highlighted in the figure) and integrating overthe entire layer.

and p both used as a dipole moment it should be noted that they are related to

different quantities. Whereas p is the dipole moment of a nanocrystal giving

rise to a dipole field, d is the dipole moment involved in the optical transition

in the nanocrystal.

A.2 Nanocrystals in the same layer

Turning to the absorption enhancement due to nanocrystals within the same

layer the geometry of this problem is sketched in figure A.2 which is generally

the same as in figure A.1 if the distance z0 approaches zero. Unfortunately

the general result can not be found simply by letting z0 approach zero in

equation A.23. This is because the term∑

i

⟨ni ~Ei

⟩· ~d is not zero as assumed

in equation A.16 when the nanocrystals are located in the same layer as the

test nanocrystal.

Assuming again that the dipole moment of the nanocrystal is oriented

along the y-axis

~p =

(0

p

)(A.25)

108

A.2. Nanocrystals in the same layer

and the position vector of a nanocrystal is given by

~r = − ~D =

(−r cosφ

−r sinφ

). (A.26)

the electric field felt by a nanocrystal in the center of the coordinate system

from the highlighted nanocrystal in figure A.2 becomes

(ExEy

)=

1

4πεh

(−3p sinφ

r4

(−r cosφ

−r sinφ

)− 1

r3

(0

p

)). (A.27)

By using the same approach as in section A.1.1 it is possible to determine

the absorption enhancement from the layer. The calculations are identical to

equations A.5 to A.8 but as the average field from the layer does not cancel out

the calculations become a bit more cumbersome. Essentially all three parts of

equation A.8 has to be calculated and the result is

⟨(~E · ~d

)2⟩

=(~E0 · ~d

)2+ 2

(~E0 · ~d

)∑i

〈ni〉(~Ei · ~d

)(A.28)

+∑i

〈ni〉(~Ei · ~d

)2+

(1− 1

N

)(∑i

〈ni〉(~Ei · ~d

))2

.

These different contributions can be calculated individually and summed, but

equation A.28 can be simplified by introducing the term δ ~E =∑

i〈ni〉 ~Ei which

is the field from the nanocrystals

⟨(~E · ~d

)2⟩

=((

~E0 + δ ~E)· ~d)2

+∑i

〈ni〉(~Ei · ~d

)2− 1

N

(δ ~E · ~d

)2. (A.29)

The last term in equation A.29 will be disregarded due to the large number

of nanocrystals in a layer N . The second term has already been calculated in

the previous section and the first term can be rewritten using

δ ~E = χ~E0 (A.30)

109

A. A simple model for the impact on nanocrystal absorption from thesurrounding nanocrystals

where χ is a dimensionless constant. Here it has been used that as δ ~E is

directed along the y-axis1 it will be parallel to the external field E0. With this

in mind equation A.29 can be reduced to⟨(~E · ~d

)2⟩

=(~E0 · ~d

)2(1 + χ)2 +

∑i

〈ni〉(~Ei · ~d

)2. (A.31)

By inserting the second term in equation A.24 with z0=0 and dividing

through with(~E0 · ~d

)2= 1

3E20d

2 the enhancementABSlayerABSMie

from nanocrystals

within the same layer becomes

ABSlayerABSMie

= (1 + χ)2 +

5nsd2p2

192πε2ha4

13E

20d

2. (A.32)

This can be further reduced by inserting the dipole moment p. Although the

nanocrystals have been described as point dipoles their spherical character

can be taken into account by using the dipole moment for a dielectric sphere

in a host material given by equation 5.7.

p = 4πεhR3

∣∣∣∣ εnc − εhεnc + 2εh

∣∣∣∣E0. (A.33)

χ can be calculated from equation A.30 by turning the summation into an

integration in the same way as it was done in equation A.21. Only the com-

ponent of the electric field in equation A.27 along the y-axis will contribute.

χ =N

AE0

∑i

Ei∆Ai

≈ nsE0

∫ ∞a

∫ 2π

0Eyi rdφdr

=nsE0

∫ ∞a

∫ 2π

0

1

4πεhr3

(3psin2φ− p

)rdφdr

=nsp

4πεhE0

∫ ∞a

1

r2dr

∫ 2π

0

(3sin2φ− 1

)dφ

=nsp

4εhE0a. (A.34)

1This can be verified by calculation or by symmetry arguments.

110

A.3. Randomly distributed nanocrystals

y

x

y

z

p

φ

D

Oθ

Figure A.3: The electric field from a random distribution of nanocrystals ex-perienced by a nanocrystal placed in the origin of the coordinate system is de-termined by taking the field from a single nanocrystal (highlighted in the figure)and integrating over the volume.

Inserting this into equation A.32 along with p from equation A.33 the

enhancement becomes

ABSlayerABSMie

=

1 +nsπR

3∣∣∣ εnc−εhεnc+2εh

∣∣∣a

2

+5nsπR

6∣∣∣ εnc−εhεnc+2εh

∣∣∣24a4

. (A.35)

A.3 Randomly distributed nanocrystals

Here the absorption enhancement from nanocrystals randomly distributed

throughout a SiO2 layer will be calculated. This corresponds to the RSn sam-

ples and are thus expected to be equivalent to the MG theory. The geometry

of the problem is shown in figure A.3.

The dipole moments are again considered to be along the y-axis (equation

A.25) and the position vector ~D of a dipole is given by

111

A. A simple model for the impact on nanocrystal absorption from thesurrounding nanocrystals

~D =

rcosφsinθ

rsinφsinθ

rcosθ

. (A.36)

From equation A.3 the electric field at O from the dipole highlighted in

figure A.3 is given by

ExEyEz

=1

4πεh

−3psinφsinθ

r3

−cosφsinθ

−sinφsinθ

−cosθ

− 1

r3

0

p

0

. (A.37)

As equation A.31 applies to this situation as well the enhancement is found

by calculating the two terms individually. Again this is done by converting the

sum to an integration the only difference being that this is a 3 dimensional

problem so the summation is no longer over areal elements ∆Ai but small

volume elements ∆Vi. The integration is over all space. As the average field

from randomly distributed dipoles is zero (χ = 0) only the second term in

equation A.31 needs to be calculated

∑i

N

V

(~Ei · ~d

)2∆Vi ≈

∫ ∞a

∫ 2π

0

∫ π

0

N

V(Exdx + Eydy + Ezdz)

2 r2sinθdθdφdr.

Again the terms ExdxEydy, ExdxEzdz and EydyEzdz will integrate to zero

so only the three terms E2xd

2x, E

2yd

2y, E

2zd

2z must be calculated. By assuming

that d2x = d2

y = d2z = d2/3 the final result becomes

∑i

N

V

(~Ei · ~d

)2∆Vi ≈

nvd2p2

24πε2ha3

(A.38)

where the nanocrystal density nv = NV has been inserted. With this the

absorption from randomly distributed nanocrystals becomes⟨(~E · ~d

)2⟩

=(~E0 · ~d

)2+

nvd2p2

24πε2ha3. (A.39)

Inserting p from equation A.33 and dividing through with 13E

20d

2 to get the

absorption enhancement due to the nanocrystals the result becomes

112

A.3. Randomly distributed nanocrystals

ABSrandomABSMie

= 1 +2nvπR

6∣∣∣ εnc−εhεnc+2εh

∣∣∣2a3

. (A.40)

113

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