Download Absorption properties of β-Sn nanocrystals in SiO

Absorption properties of β-Sn nanocrystals in SiO2
Jonas Günsel
Winter 2010
i
This thesis has been submitted to the faculty of science at Aarhus University 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 Günsel, 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
2.2
2.3
2.4
2.5
2.6
Schematic representation of a RBS experiment
Energy loss in RBS . . . . . . . . . . . . . . . .
Depth resolution of RBS . . . . . . . . . . . . .
Principle of TEM operation . . . . . . . . . . .
As grown random Sn sample . . . . . . . . . .
Schematic overview of the spectrophotometer .
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Overview of the sputtering process . . . . . . . . . . . .
Quartz wafer transmittance . . . . . . . . . . . . . . . .
Multilayered and random Sn samples . . . . . . . . . . .
Formation of nanocrystals . . . . . . . . . . . . . . . . .
Homogeneous nucleation barrier . . . . . . . . . . . . .
TEM picture of Sn nanocrystals randomly distributed in
RBS spectrum of Sn randomly distributed in SiO2 . . .
TEM pictures of multilayered nanocrystals . . . . . . .
Diffraction from Sn nanocrystals . . . . . . . . . . . . .
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
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SiO2
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Full and reduced Mie expression . . . . . . . . . . . . . . . .
Absorption from bulk vs nanocrystals . . . . . . . . . . . . .
Garcia model for composite dielectric function . . . . . . . .
Effect of the nanocrystal distribution on the refractive index
Lorentz model for ˜ . . . . . . . . . . . . . . . . . . . . . . . .
Dielectric function of a free electron metal . . . . . . . . . . .
Sn banddiagram . . . . . . . . . . . . . . . . . . . . . . . . .
Bulk vs nanocrystal . . . . . . . . . . . . . . . . . . . . . . .
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2
List of Figures
4.9
4.10
4.11
4.12
4.13
4.14
4.15
Sn dielectric functions . . . . . . . . . . . .
Reflection from a surface . . . . . . . . . . .
Schematic overview of the matrix method .
Reflection from Quartz wafer and thin film
Minimization procedure . . . . . . . . . . .
point by point simulation . . . . . . . . . .
Refractive index of a quartz wafer . . . . .
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5.1
5.2
5.3
5.4
Sn peak in RBS for RSn1 . . . . . . . . . . . . . . . . . . . . . . .
TEM and size distribution of RSn1 . . . . . . . . . . . . . . . . . .
RBS spectrum of multi layered sample . . . . . . . . . . . . . . . .
Size distribution of ML with different PV-TEM preparation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TEM of two multi layered samples . . . . . . . . . . . . . . . . . .
Simulation structure for RSn samples . . . . . . . . . . . . . . . .
BaSO4 reflectance measurement . . . . . . . . . . . . . . . . . . . .
BaSO4 reflectance measurement . . . . . . . . . . . . . . . . . . .
RSn1 measurement and simulation . . . . . . . . . . . . . . . . . .
n and κ for RSn1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
σnc for RSn1 compared to MG . . . . . . . . . . . . . . . . . . . .
Mie vs MG for the RSn1 sample . . . . . . . . . . . . . . . . . . .
Comparison between absorption from RSn samples . . . . . . . . .
Comparison between transmittance from as grown and annealed
samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation structure for multi layered samples . . . . . . . . . . .
Refractive index of β-Sn nanocrystals . . . . . . . . . . . . . . . .
MLSn4 reflection and transmission . . . . . . . . . . . . . . . . . .
MLSn4 reflection and transmission . . . . . . . . . . . . . . . . . .
Comparison of σa for RSn and ML samples . . . . . . . . . . . . .
MLSn2 compared to MG theory . . . . . . . . . . . . . . . . . . .
Absorption from multi layered samples . . . . . . . . . . . . . . . .
Enhancement of absorption from nanocrystals in a nearby layer . .
Enhancement of absorption from nanocrystals within a single layer
Absorption enhancement relative to MG theory . . . . . . . . . . .
Model vs experimental absorption enhancement . . . . . . . . . . .
Absorption from single slab . . . . . . . . . . . . . . . . . . . . . .
64
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5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
5.26
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v
List of Figures
5.27 Comparison of σa for MLSn samples from direct measurement and
the thin film modeling procedure . . . . . . . . . . . . . . . . . . .
5.28 Multi layered samples considered as effective media . . . . . . . . .
5.29 Interlayer reflections . . . . . . . . . . . . . . . . . . . . . . . . . .
5.30 Absorption comparison with previous work . . . . . . . . . . . . .
5.31 Absorption of SnO2 vs Sn nanocrystals in SiO2 . . . . . . . . . . .
5.32 Absorption comparison with previous work 2 . . . . . . . . . . . .
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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
2.1 Rutherford Backscattering Spectrometry . . . . .
2.1.1 RBS theory . . . . . . . . . . . . . . . . .
2.1.2 Extracting information from experiments
2.1.3 Experimental details . . . . . . . . . . . .
2.2 Transmission Electron Microscopy (TEM) . . . .
2.2.1 Principle of operation . . . . . . . . . . .
2.2.2 Bright field imaging . . . . . . . . . . . .
2.2.3 Diffraction mode . . . . . . . . . . . . . .
2.2.4 High Resolution TEM . . . . . . . . . . .
2.2.5 Sample preparation . . . . . . . . . . . .
2.3 Optical measurements . . . . . . . . . . . . . . .
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3 Synthesis of Sn nanocrystals
3.1 RF magnetron sputtering . . . . . .
3.1.1 The sputtering chamber . . .
3.1.2 Wafer wedging . . . . . . . .
3.2 Thin films composed of Sn and SiO2
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Contents
3.3
3.2.1 Sn nanocrystals randomly distributed in SiO2 . . . . . .
Sn nanocrystals in a layered structure . . . . . . . . . . . . . .
20
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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
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Experimental details . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Random Sn samples . . . . . . . . . . . . . . . . . . .
5.2.2 Sn in a multi layered structure . . . . . . . . . . . . .
5.3 Simulation based determination of absorption . . . . . . . . .
5.3.1 The correction factor for reflection measurements . . .
5.3.2 Refractive indices of the quartz wafer and SiO2 layers
5.4 β-Sn absorption cross sections . . . . . . . . . . . . . . . . . .
5.5 Comparison with previous studies . . . . . . . . . . . . . . . .
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A A simple model for the impact on nanocrystal
from the surrounding nanocrystals
A.1 Nanocrystals in a different layer . . . . . . . . . .
A.1.1 Electric field from a layer of nanocrystals
A.2 Nanocrystals in the same layer . . . . . . . . . .
A.3 Randomly distributed nanocrystals . . . . . . . .
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111
Bibliography
viii
absorption
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115
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 Amélie Têtu 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 Amélie 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 αSn 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
13.2◦ C [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 photovoltaic 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 structural 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 samples 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 interference 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 Spectrometry (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
Ion
accelerator
r
to
ec
t
De
Sample
He+ ion
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 repulsion 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 
q
2
Mt2 − Mi2 sin2 θ
 ≡ KE0 ,
Mi + M t
Mi cosθ +
(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
x
∆Ein
Ein
Escat
∆Eout
Eout
t
Figure 2.2: The different contributions to the energy loss of a He ion during a
RBS 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 ) =
Zi Zt e 2
4Ei
2
1
,
sin (θ/2)
4
(2.2)
where Zi (Zt ) is the atomic number of the ion (target) and e is the elementary
charge. As implied by the Zt2 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 electrons and small angle scattering by nuclei. Since the energy lost from the lat7
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
dEi
2πZi2 e4
−
N Zt
=
dx
Ei
Mi
me
ln
2me vi
,
I
(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
Z
∆Ein =
0
t
dEi dEi
dx ≈ t
,
dx
dx 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
1 dE
.
(2.5)
N dx
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 − tN in ) −
t
N out ,
|cosθ|
(2.6)
where the first part Escat = K(Ein − tN in ) 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
1
∆E = N t Kin +
N out ≡ [S]N t,
(2.7)
|cosθ|
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 4 He+ 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
a)
1.5
1.7
b)
1.5
1.7
1.9
Recoil energy [MeV]
Figure 2.3: Sn peak in a RBS spectrum of a multi layered structure with 5
Sn layers separated by SiO2 layers of 15nm (a) and 75nm (b) respectively. The
layered 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 determination 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 influencing 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)
LaB6
cathode
High voltage
acceleration
Condenser
lenses
Condenser
aperture
Objective
lenses
Sample
Objective
aperture
Projector
lenses
CCD camera /
fluorescent screen
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 collected 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 wavelength 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 corresponds to a wavelength of
h
λ= r
2me eV 1 +
eV
2me c2
= 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 Å 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 electron 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 nanostructures 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 nanocrystal 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 nanocrystals, 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 electron 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 substrate 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 conditions 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
1
The 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 randomly
distributed 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 spec14
2.3. Optical measurements
trophotometer, which is sketched in figure 2.6. The measurements were performed 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
Slit
Gratings
Gratings
Halogen
lamp
Slit
Slit with
filter
Entrance
window
= Mirror
D2
lamp
Sample compartment
Reference beam
Integrating
sphere
Exit windows
Chopper mirror
Sample beam
Figure 2.6: Schematic overview of the Shimadzu UV-3600 spectrophotometer
used to perform transmission and reflection measurements.
16
Chapter 3
Synthesis of Sn nanocrystals
In this work Radio Frequency (RF) magnetron sputtering has been the preferred 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 epitaxy [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
Substrate
Target atom
Ar+
Target material
Target
Magnet
Figure 3.1: Schematic representation of the sputtering process used to grow
thin 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 15◦ C 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 2·10−3 mbar. 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 interference 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 mechanically 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
Transmittance [%]
110
100
90
80
70
60
50
200
400
600
800
1000
1200
1400
Wavelength [nm]
Figure 3.2: Transmittance spectrum of a 10 µm thick quartz wafer calculated
by the matrix method described in the following chapter (blue line). This is
compared to a calculation on a wedge-shaped wafer of the same thickness (red
dashed 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 crystallization process, which is sketched in figure 3.4. There are three different
20
3.2. Thin films composed of Sn and SiO2
SiO2
Sn nanocrystal
(a)
(b)
Figure 3.3: Cross sectional schematic slice through a) Sn layers sandwiched between SiO2 in a multilayered structure and b) randomly distributed Sn nanocrystals in SiO2
Sn
atom
Sn
nanocrystal
G0
G0 + ∆G
Figure 3.4: Homogeneous nucleation of Sn nanocrystals in SiO2 . The nanocrystal will form if there is an overall decrease in energy, meaning ∆G must be
negative.
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
∆G*
0
R*
Volume and
strain term
R
Figure 3.5: Barrier in homogeneous nucleation and the different thermodynamic factors causing it. The cluster has to pass the barrier in order to get the
critical 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 3
R (∆Gv − ∆Gs ) + 4πR2 γ.
3
(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∗ =
∆G∗ =
2γ
(∆Gv − ∆Gs )
(3.4)
16πγ 3
.
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]
− ∆G∗
k T
C ∗ = CSn e
b
(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 nucleation 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 400◦ C and 1100◦ C [38, 39,
40] have been used to form Sn nanocrystals in SiO2 depending on the deposition 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 400◦ C 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 400◦ C resulted
in a more uniform distribution of nanocrystals than at higher annealing temperatures. 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 400◦ C. The remaining Ar was found to be less than 0.2 at% for
samples annealed for 1 hour at 400◦ C. This low concentration is expected to
24
3.3. Sn nanocrystals in a layered structure
Measured
Simulation
Normalized Yield
O
Si from
wafer
Blow up of
the Sn peak
5
Si from film
Ar
Sn
0
1.5
1.7
1.9
Recoil Energy [MeV]
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 lay25
3. Synthesis of Sn nanocrystals
16
Number of observations
14
50 nm
12
10
8
6
4
2
0
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
Diameter [nm]
a)
b)
c)
Figure 3.8: Cross sectional (a) and plane view (b) TEM pictures of a sample
with 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 400◦ C 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
700◦ C 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 800-1000◦ C. Therefore the annealing temperature was kept at 400◦ C which proved sufficient to
form nanocrystals. TEM was used to find the nanocrystal size, crystal structure 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 nanocrystals
and 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 they
are 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 layered 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
2A − 4πR2 n
=
Atotal
2A
(3.7)
If all the Sn from the layer is converted into nanocrystals then 4πR3 n/3 = tA
(assuming the same Sn density in layer and nanocrystals) which will turn
equation 3.7 into
∆A
3t
=1−
.
(3.8)
Atotal
2R
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 400◦ C 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 nonmagnetic and current free material the Maxwell equations are given by [35]
29
4. Interactions between light and matter
~ · (˜
~ = ρ
∇
E)
0
(4.1)
~ ·B
~ =0
∇
(4.2)
~
~ ×E
~ = − ∂B
∇
∂t
(4.3)
~ ×B
~ = µ0 0
∇
~
∂(˜
E)
∂t
!
(4.4)
~ is the electric field, B
~ is the magnetic induction, ρ is the free charge
where E
density and 0 is the vacuum permittivity. These equations must be satisfied
at any point and time in the material. The dielectric function ˜, 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~ = 0 (˜
− 1)E
(4.5)
and the dielectric function can be written as ˜ = r + ii . 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 ˜ 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
1
For 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 equations
~ r, t) = E
~ 0 ei(~k·~r−ωt)
E(~
(4.6)
~ r, t) = B
~ 0 ei(~k·~r−ωt)
B(~
(4.7)
~ 0 and B
~ 0 are mutual perpendicular amplitude vectors, each also perwhere E
pendicular to the wave vector ~k. In order to satisfy the Maxwell equations ~k
and the frequency ω must be related by
2
~ k = µ0 0 ˜ω 2 .
Introducing the speed of light in vacuum c =
(4.8)
√1
0 µ0
this reduces to
ω√
~ ˜.
(4.9)
k =
c
For bulk materials it is often convenient to introduce the refractive index
√
Ñ = n + iκ = ˜. The electric field of a homogeneous plane wave traveling
in the z direction is then given by
2π
~ t) = E
~ 0 e− λ0 κz ei
E(z,
2π
nz−ωt
λ0
(4.10)
where the free space wavelength λ0 = 2πc
ω 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) = I0 e−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
2
Such that all possible fields satisfying the Maxwell equations can be expanded in plane
waves.
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
κ.
λ0
(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=
32
2πRnh
,
λ0
(4.14)
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
m2 − 1
x2 m2 − 1 m4 + 27m2 + 38
= πR 4xIm
1+
m2 + 2
15 m2 + 2
2m2 + 3
(
)
2
m2 − 1
28 4
+ πR x Re
3
m2 + 2
2
where the nanocrystal dielectric function relative to the host dielectric function
for clarity. If |m| x << 1 this can be further
has been replaced by m2 = ˜˜nc
h
reduced and the scattering and absorption cross sections can be identified as
128π 5 R6 n4h ˜nc − ˜h 2
σscat =
˜nc + 2˜
h 3λ40
˜nc − ˜h
8π 2 R3 nh
Im
.
σabs =
λ0
˜nc + 2˜
h
(4.15)
(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 geometry and dielectric surroundings. Here the atomic absorption cross section
3
As the host material used in this work is SiO2 the refractive index is purely real in the
wavelength range studied.
33
4. Interactions between light and matter
σnc [arb. unit]
Full Mie
Reduced Mie
200
200
400
600
800
Full Mie
Reduced Mie
400
600
Wavelength [nm]
800
Figure 4.1: Nanocrystal absorption cross section for Sn nanocrystals in
SiO2 with a diameter of 5 nm (top) and 50 nm (bottom).
The red
curve labeled ‘Reduced Mie’ is calculated from equation 4.16 wheres the
black ‘Full Mie’ curve is calculated using the MiePlot software available at
http://www.philiplaven.com/mieplot.htm and based on the computer code presented 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 section 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 absorp4
The density of Sn in nanocrystals is assumed to be the bulk density, which is reasonable
since they share the bulk crystal structure. On another note, the comparison in figure 4.2
is only possible for arbitrarily sized nanocrystals because the Mie absorption cross section is
directly proportional to the nanocrystal volume, which cancels out from the equation when
converting to atomic absorption cross section.
34
4.2. Nanocrystals embedded in a host material
-16
Bulk
Sn in vacuum
Sn in SiO
σ [cm2]
a
1
x 10
0
2
500
1000
Wavelength [nm]
1500
Figure 4.2: Atomic absorption cross sections from bulk Sn compared to Sn
nanocrystals surrounded by either vacuum or SiO2 . The origin of the dielectric
functions 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 (˜
nc + 2˜
host ) 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 (˜
nc + 2˜
host ) ≈ 0 is satisfied at a
different wavelength.
The Mie theory accurately describe a single particle embedded in an infinite 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

˜ave = ˜h 1 +
3f
1−

˜nc −˜
h
˜nc +2˜
h
˜nc −˜
h
f ˜nc +2˜
h
.
(4.17)
Here ˜h is the host dielectric function and f = Vnc ρnc 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 i in place of κ to α =
nanocrystal absorption cross section in MG theory is given by
σnc =
=
α
ρnc
2πVnc
Im{˜
ave }
λ0 nave f
2π i
λ0 n ,
the
(4.18)
√
where nave = Re{ ˜ave } is the real part of the refractive index of the composite 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
)
(
˜nc −˜
h
6πVnc
˜nc +2˜
h
=
Im ˜h
.
h
nc −˜
λ0 nave
1 − f ˜˜nc
+2˜
h
(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 nanocrystals 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 containing 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
i
rij
j
RL
Figure 4.3: Imaginary sphere in a plane of nanocrystals used for calculation of
the electric field exerted on particle i in the center of the sphere.
i , is the sum of the external field and the field from all other
the point i, Eloc
nanocrystals (treated as dipoles)
~
~i
~i = E
E
ext + Enc .
loc
(4.20)
The field from the nanocrystals can be calculated by setting up an imaginary sphere, as seen in figure 4.3, (it is usually termed the Lorentz sphere)
i ) and outside
and adding the contributions from the nanocrystals inside (Ein
i
) the sphere
(Eout
~i = E
~i + E
~i .
E
nc
in
out
(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
f (˜
nc − ˜m )
˜ave = ˜m 1 +
.
(4.22)
˜m + S(˜
nc − ˜m ))
K
Here f is the filling fraction previously introduced and S = L − f3 − f4π
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
0.5
MG
Positive K
Negative K
0.4
κ ave
0.3
0.2
0.1
0
200
300
400
500
Wavelength [nm]
600
700
800
Figure 4.4: The average refractive index κave of spherical Sn nanocrystals in
SiO2 calculated by the MG formula (equation 4.17) compared to the modified MG
expression (equation 4.22) with a positive and negative value for K respectively.
As the absorption cross section is directly proportional to κ this demonstrates
how a distribution giving rise to a positive value of K will shift the absorption
towards higher wavelengths compared to the MG case.
K=
"
X 3x2ij
j
5
rij
1
− 3
rij
#
pxj
.
P
(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 distribution 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. Although 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 appro38
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 quantities linked to the same physical properties and since these relations are widely
used throughout this thesis they will be summarized here.
˜ = Ñ 2 = (n + iκ)2
r = n2 − κ2
i = 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]
d2~r
d~r
~
(4.25)
+ mΓ + K~r = eE
dt2
dt
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 = mω02
is the restoring force. A number of assumptions have been done here. First
m
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
~ and the speed of the electron is much smaller than the speed of
by e~v × B/c
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 =
~
eE
m(ω02 − ω 2 − 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
~
= α(ω)E
2
m(ω0 − ω 2 − iΓω)
(4.27)
where α(ω) is the polarizability of the one electron atom. The total contribution for N of such oscillators pr unit volume becomes
~
P~ = N α(ω)E.
(4.28)
Using equation (4.5) the resulting dielectric function becomes
˜ = 1 +
= 1+
N α(ω)
0
N e2
m0 (ω02 − ω 2 − 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
10
8
ω0
εr
εi
6
4
2
00
-2
-4
2
3
4
5
ω
6
7
8
Figure 4.5: Real and imaginary part of the dielectric function from the Lorentz
oscillator model. The imaginary part which is connected to absorption in the
material 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]
r = 1 +
i =
Nj (ωj2 − ω 2 )
e2 X
m0
(ωj2 − ω 2 )2 + Γ2 ω 2
j
Nj Γω
e2 X
2
m0
(ωj − ω 2 )2 + Γ2 ω 2
j
(4.30)
(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 r and i
are 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 ˜ 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
r = 1 −
N e2
1
2
m0 ω + Γ2
(4.32)
41
4. Interactions between light and matter
5
ωp
εr , εi
00
-5
εr
-10
1
εi
2
3
4
ω
5
6
7
Figure 4.6: Real and imaginary contribution to the dielectric function from free
electrons in the classical model.
i =
N e2
Γ
.
2
m0 ω(ω + 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
ωp2 τ 2
r = 1 −
1 + ω2τ 2
(4.34)
ωp2 τ
ω(1 + ω 2 τ 2 )
(4.35)
i =
2
e
where the plasma frequency ωp = N
m0 has 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 contributions, 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 r for the Lorentz
42
4.2. Nanocrystals embedded in a host material
10
eV
5
0
-5
-10
K
Z
Figure 4.7: β-Sn band structure taken from [50]. Interband and intraband
transitions corresponding to bound and free electrons respectively are sketched
with a red and blue arrow. The dashed line is the Fermi energy which is placed
at 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 diagram5 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
5
The 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 complicated 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 interaction between the field and an electron as a small perturbation on the
electron states, the quantum mechanical analog to the Lorentz model describing interband transitions can be derived [62].
For convenience the electromagnetic field is described by a vector potential
~
A given by
~ = 1 A0 r̂ ei(~k·~r−ωt) + e−i(~k·~r−ωt)
A
(4.36)
2
where r̂ 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
~
~ = − ∂A .
E
(4.37)
∂t
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]
e ~
A · p~
(4.38)
m
where p~ is the momentum operator for the electron. Applying first order perturbation 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]
Hpert =
2π
|hψf |Hpert | ψi i|2 δ(Ef − Ei − ~ω).
(4.39)
~
Here the delta function is the condition for energy conservation, and the matrix
element
Wif =
6
It could be assumed that the atom is initially in its ground state, making stimulated
emission impossible.
44
4.2. Nanocrystals embedded in a host material
e2 D ~ E2
(4.40)
ψf A · p~ ψi m2
describes the transition amplitude between states. If the electron states are
described by Bloch functions with uk being a lattice periodic function
|hψf |Hpert | ψi i|2 =
~
ψk (~r) = uk (~r)eik·~r
(4.41)
and the dipole approximation is invoked, the transition probability becomes
[62]
Wif =
X
2π e 2
|E(ω)|2
|Pif |2 δ(Ef (~k) − Ei (~k) − ~ω)
~ 2mω
(4.42)
~k
where |Pif |2 = |huf |r̂ · p~| ui i|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]
i (ω) =
πe2 X
|Pif |2 δ(Ef (~k) − Ei (~k) − ~ω).
m2 ω 2 0
(4.43)
~k
The real and imaginary parts of ˜ are related via the Kramers-Kronig
relations given by [60]
2ω
i =
PV
π
∞
Z
2
r − 1 = P V
π
0
Z
∞
0
ω 0 i (ω 0 )
dω 0
(ω 0 )2 − ω 2
r (ω 0 ) − 1 0
dω
(ω 0 )2 − ω 2
(4.44)
(4.45)
where P V denotes the principal value of the following integral. Using equation
4.44 the real part of the dielectric function turn out to be
r (ω) = 1 +
e2 X 2 |Pif |2
1
2
m0
m~ωif ωif − ω 2
(4.46)
~k
7
The summation is now over allowed k vectors per unit volume of the crystal.
45
4. Interactions between light and matter
E (~k)−E (~k)
where ωif = f ~ i
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
r = 1 +
e 2 X Nj
.
m0
ωj2 − ω 2
j
(4.47)
2
2|Pif |
, which is referred to as the oscillator strength of
The quantity fif = m~ω
if
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 intraband 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
intra
=1−
r
∂ 2 Ek,l
e2 X
F
(E
)
k,l
~2 ω 2 0
∂k 2
(4.48)
k,l
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
2
considered8 . Introducing the reduced mass of the electron m1∗ = ~12 ∂∂kE2 and
the average value of the Fermi function across a Brillouin zone (= 21 ) and
carrying out the summation over k points in a Brillouin zone results in
N e2
(4.49)
m∗ 0 ω 2
which, besides from the damping factor, is identical to equation 4.32. Equations for the imaginary part of ˜ 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
=1−
intra
r
(ω) =
e2 ~2 X [f (Ei ) − f (Ef )] |Pif |2
2 − ~2 ω 2 )
0 m2
Eif (Eif
(4.50)
i,f
8
In this discussion lifetime broadening has been ignored so the damping coefficient Γ is
absent.
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 the
number of unit cells. The small number of unit cells present in a nanocrystal
change the electronic structure compared to bulk.
for each crystallographic axis. By introducing the Bloch functions the dielectric function becomes a bulk parameter, as the Bloch functions describe the
symmetry across a large number of unit cells. Therefore using the bulk dielectric 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
9
In the theoretical calculations by Pedersen the components along both axis of the unit
47
4. Interactions between light and matter
40
20
80
εr
60
εi
0
40
-20
-40
-60
-80
200
20
Pedersen
Takeuchi
400
600
800
1000
1200
1400
0
200
Pedersen
Takeuchi
400
600
800
1000
1200
1400
Wavelength [nm]
Figure 4.9: Comparison between real (left) and imaginary (right) part of the
β-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 for
plasma frequencies and damping coefficient given in the article. It should be
noted that it is only the part of the dielectric function parallel to the c axis of
the β-Sn unit cell that is shown here.
below 400 nm, which may reflect their different origin. Takeuchi uses spectroscopic 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 -200◦ C performed in [65] and the components does not show significant differences 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 dielectric 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 conducted 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
n1
n2
n1
n2
Θ2
Θ1
Θ1
Θ2
a)
d
b)
Figure 4.10: Reflection contributions from a) a thick absorbing slab and b)
thin weakly absorbing film. Multiple reflections contribute to the total reflection
in 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 surface 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 transmission. 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 attenuated 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 +
T12 R21 T21 e−2αd
1 − R12 R21 e−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 inconvenient 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 transmission 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 Ñi = 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.
10
11
50
Transmission could be considered equivalently.
The two citations create the basis for the following part.
4.3. The matrix method for determining reflection and transmission
1
i
...
Eli
j
E'li
Elj
di
Eri
...
N
E'lj
dj
E'ri
Erj
Hij
Li
E'rj
Lj
Figure 4.11: Overview of the notation used in the matrix method. The different
operators are explained in the text.
~0)
~ i ) and (E
The notation for the field at the left and right side of layer i, (E
i
are
~i =
E
Eli
Eri
~0 =
E
i
Eli
0
Eri
0
!
!
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 =
2Ñi
Ñi + Ñj
(4.54)
rij =
Ñi − Ñj
.
Ñi + Ñj
(4.55)
Each of the fields in figure 4.11 can be considered as a sum of two contri0
butions. For instance Eli is the transmitted part of Elj across the ij interface
0
plus the reflected part of Eri 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
τij τji + (rij )2 = 1
In that way the correlation between fields across a layer can be found.
Introducing the interface transition matrix
"
#
1
1 rij
Hij =
τij rij 1
the relation between fields across a boundary can be described by
~ 0 = H E~ .
E
ij j
i
(4.56)
In the same way a propagation matrix relating the fields across a single
layer can be defined as
#
"
1
eiβi
0
(4.57)
Li =
τij
0 e−iβi
resulting in the field relation
~0.
~ i = Li E
E
i
(4.58)
βi is given by
βi =
2π Ñi di
λ0
where di is the layer thickness and λ0 is the vacuum wavelength. From equations 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
0
~N ,
E~1 = H12 L2 H23 L3 ...HN −2,N −1 LN −1 HN −1,N E
52
4.3. The matrix method for determining reflection and transmission
which is more conveniently written as
0
E~1 =
"
S11 S12
S21 S22
#
~ N = SE
~N .
E
(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
!
0
E~1 =
Er
Ei
~N =
E
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 =
S12
Er
=
Ei
S22
τstack =
Et
1
=
.
Ei
S22
By definition the reflectance (R) and transmittance (T ) then become
R = |rstack |2
T =
nN
|τstack |2 .
n1
(4.60)
(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 transmittance 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 Ñ = 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
It
A = − log(T ) = − log
Ii
(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 transmission. Since a reflection free measurement of the transmission is not always
possible this method has its limitations. If the reflection is sufficiently constant across the measured wavelength range the correct spectral dependence
54
4.4. Simulation based determination of absorption
Reflectance [%]
15
Quartz wafer
Quartz wafer + thin film
10
5
0
300
400
500
600
700
800
Wavelength [nm]
900
1000
1100
1200
Figure 4.12: Measured reflectance spectra of a 0.4 mm thick wedge shaped
quartz 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 film
shows substantial interference effects. The feature at 900 nm is a result of a
grating 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 absorption 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 interference fringes in the transmission spectrum, is developed. This procedure
however, seem useful only for a homogeneous film with a very uniform thickness. Others [68] [74] use both T and R to compute 1−R
T , 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 interference 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 parameters 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 values 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 minimizing 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 algorithm 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 optimization 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 .
12
Their thicknesses can be determined by TEM, but as will be explained later this point
will also be considered in the simulations.
56
4.4. Simulation based determination of absorption
F
3
2 .5
2
1 .5
1
0 .5
0
-1
-0 .5
0
x2
0 .5
x1
1
x0
1 .5
xmin
2
x
Figure 4.13: Schematic illustration of the importance of selecting a good starting point for the optimization process with a arbitrary parameter x. The function value F is described in equation 4.64, but the function showed here is just
a random function. Starting out in the point x0 will result in the Nelder-Mead
algorithm finding the correct minimum xmin , but on the other hand, starting in
x1 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
2
a
Optimized
Input
n
1.8
1.6
1.4
1.2
R [%]
20
15
b
n
n + 10%
n - 10%
10
5
T [%]
80
60
n
n + 10%
n -10%
40
20
200
c
250
300
350
400
450
500
550
Wavelength [nm]
Figure 4.14: a) Real part of the refractive index resulting from the minimization
of equation 4.64 for a stack of thin film layers (black line) compared to the input
guess (red dashed line). The resulting reflectance (b) and transmittance (c)
computed for the stack with n varied within 10%.
58
600
4.4. Simulation based determination of absorption
1.7
Pointwise
Polynomial
n
1.65
1.6
1.55
1.5
200
300
400
500
600
700
800
900
1000
1100
1200
Wavelength [nm]
Figure 4.15: Refractive index from a quartz wafer obtained from minimizing
equation 4.64 from measured R and T spectra by the procedure outlined in this
section. The comparison is made between the results of a minimization where n is
found on a wavelength by wavelength basis and using the polynomial description
from 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 nanocrystals 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 simulation based method described in chapter 4. The absorption cross sections
are then compared to Mie and MG theory with the use of bulk dielectric functions 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 nanocrystals 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 materi61
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. Metallic 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 optical 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 confinement effects and thus absorption in the UV range. They do not consider
the absorption to be related to the composite medium, as described in section 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
450◦ C 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
RSn1
RSn2
Mixed
layer
thickness
(nm)
72±5
295±15
Surrounding
SiO2 layer
thickness
(nm)
Sn
content
(at%)
Sn Areal
density
Ω
(1/cm2 )
Mean
diameter
(nm)
Filling
fraction
f
220±10
105±6
4.1
0.78
2.15·1016
2.10·1016
5.0
-
0.08
0.02
Table 5.1: TEM and RBS parameters for the two RSn samples. Mixed layer
refers to the layer containing both Sn and SiO2 . No nanocrystals were identified
in BF-TEM for the RSn2 sample and no diffraction pattern was seen either, thus
no 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
Yield
Sn peak in
RBS
spectrum
Measured
Simulated
Recoil energy [MeV]
Figure 5.1: Part of the RBS spectrum of the RSn1 sample showing the Sn
peak. From the very good agreement between the measured curve and the one
simulated by the RUMP software [27] the Sn atoms are expected to be evenly
distributed throughout the layer.
sample in BF mode and in diffraction mode no diffraction pattern was observed. 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 structures 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
Number of observations
5.2. Experimental details
10 nm
16
14
12
10
8
6
4
2
0
3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
Diameter [nm]
a)
b)
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 the
average 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
Separating
SiO2 layer
thickness
(nm)
MLSn1
MLSn2
MLSn3
MLSn4
MLSn5
MLSn6
3±0.5
11±2
24±4
42±3
54±4
68±5
Surrounding SiO2
layer
thickness
(nm)
230±15
224±8
181±10
158±8
135±7
90±6
Sn
Areal
density Ω
(1/cm2 )
2.27·1016
2.06·1016
2.14·1016
2.46·1016
2.43·1016
2.34·1016
Mean
NC diameter
(nm)
Standard
deviation
(nm)
6.0
6.1
6.2
6.1
6.3
6.2
0.81
0.65
0.71
0.62
0.53
0.58
Table 5.2: Structural data for the multi layered samples. The Sn areal density
and atomic content are found from RBS whereas the nanocrystal diameter and
layer 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
Measured
Simulated
Recoil energy [MeV]
Figure 5.3: RBS spectrum of MLSn6 as grown. The correspondence between
measurement and simulation can be increased by expanding the thickness of the
Sn 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 distribution 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 samples 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
1
A standard test for difference in mean diameter was performed based on the size distributions 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 debated
grounded on the figure, it is concluded that the sample preparation does not influence the
nanocrystal size.
67
20
16
18
14
16
Number of observations
Number of observations
5. Sn nanocrystals in SiO2
14
12
10
8
6
4
2
0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
12
10
8
6
4
2
0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
Diameter [nm]
Diameter [nm]
Figure 5.4: Size distributions from the MLSn2 sample from PV-TEM samples
prepared by chemical etching in a hydrofluoric acid solution (left) and by ion
milling (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
1
Layer index
3
4
2
d2
Air
d3
SiO2
d4
Sn nanocrystal
5
6
d5
Quartz
Figure 5.6: Overview of the different layers introduced to describe the RSn
sample in the modeling procedure. The different layers are each characterized
by 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 interference 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
1.05
80
1
70
0.95
60
R [%]
Correction factor
5. Sn nanocrystals in SiO2
0.9
0.85
0.8
200
300
400
500
50
BaSO table value
40
Measurement
600
700
30
200
4
Wavelength [nm]
800
Measured
Simulated
300
400
500
600
700
800
Wavelength [nm]
Figure 5.7: The correction factor compared to data for the BaSO4 reflectance
taken from Grum [31] (left) determined by comparing the measured reflectance
from a Si wafer to a simulation (right). Data for the Si refractive index are taken
from [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 comparing the measured to the simulated reflectance of the Si wafer which are
shown on the right. The spectrophotometer uses the compressed BaSO4 powder 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 wavelengths 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
1.68
1
Table value
Optimized
x 10
-5
Table value
Optimized
0.8
1.64
k
n
0.6
1.6
0.4
1.56
1.52
200
0.2
300
400
500
600
700
800
0
200
300
400
500
600
700
800
Wavelength [nm]
Figure 5.8: Real (left) and imaginary (right) part of the refractive index for a
wedged shaped quartz wafer. Tabulated values from [36] are compared to those
extracted 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. Transmission 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
20
Measurement
Simulation
R [%]
15
10
5
100
T [%]
80
60
40
20
200
Measurement
Simulation
300
400
500
Wavelength [nm]
600
700
800
Figure 5.9: Measured and simulated R and T spectra for the RSn1 sample after
n, 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
Optimized
MG input
n
2
1.5
1
k
Optimized
MG input
0.5
0
200
300
400
500
Wavelength [nm]
600
700
800
Figure 5.10: n and κ values for RSn1 corresponding to the simulated R and
T in figure 5.9 (red dashed line) compared to the MG values used as an initial
guess (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 =
σnc
Vnc ρB
(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
-16
1.2
x 10
RSn1
MG Pedersen
MG Takeuchi
σ a Sn
[cm2]
1
0.8
0.6
0.4
0.2
0
200
300
400
500
Wavelength [nm]
600
700
800
Figure 5.11: Nanocrystal absorption cross section for the RSn1 sample calculated from the κ deduced in figure 5.10 compared to MG theory with the bulk
Sn 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 expected 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 differences 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 absorption 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 (Â), which is available directly from the measurements through  = 1 − R − T . From figure 5.13(a) it
75
5. Sn nanocrystals in SiO2
-16
σ a Sn [cm2]
1.2
x 10
Mie
MG
1
0.8
0.6
0.4
0.2
0
200
300
400
500
Wavelength [nm]
600
700
800
Figure 5.12: Atomic absorption cross section for Sn atoms in the RSn1 sample
calculated by Mie and MG theory using the bulk Sn dielectric function given in
[67].
-17
70
x 10
RSn1
RSn2
5
3
a
σ [cm2]
4
2
1
0
200
300
400
500
600
Wavelength [nm]
700
800
Absorbed percentage [%]
6
RSn1
RSn2
60
50
40
30
20
10
0
200
300
400
500
600
Wavelength [nm]
700
800
(a) Atomic absorption cross section for the RSn (b) Absorbed percentage for the RSn samples
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 absorption cross section for the Sn atoms is extracted from the simulation based
procedure.
76
100
100
90
90
Transmittance [%]
Transmittance [%]
5.4. β-Sn absorption cross sections
80
70
60
50
40
as grown
30
400 C annealed
20
200
°
300
400
500
600
Wavelength [nm]
700
(a) Transmittance of the RSn1 sample
800
80
70
60
50
40
as grown
30
400° C annealed
20
200
300
400
500
600
Wavelength [nm]
700
800
(b) Transmittance of the RSn2 sample
Figure 5.14: Measured transmittance spectra for the RSn samples before and
after annealing. Whereas the differences between the as grown and the 400◦ C
annealed sample are small for the RSn2 sample the RSn1 sample clearly shows a
decrease in transmittance particularly in the low wavelength range after annealing.
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 formation 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 reflectance 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
2
The 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
1
2
3 4
d2
d3 d4
Air
Layer index
3 4 3 4 3
SiO2
4
3
Sn nanocrystal
5
6
d5
d6
1
Quartz
Figure 5.15: Overview of the different layers introduced to describe the MLSn
samples in the modeling procedure. As the layer index in the top of the figure
indicates the five nanocrystal layers are characterized by the same thickness and
refractive 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
dtot ρB
(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 nanocrystals 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 simulation procedure. The thickness of the surruonding SiO2 layers were almost
78
5.4. β-Sn absorption cross sections
5
8
NC from RSn1
Bulk Sn
NC from RSn1
Bulk Sn
4
6
n
k
3
4
2
2
1
0
200
250
300
350
400
450
Wavelength [nm]
500
550
(a) n for the nanocrystals from RSn1
600
0
200
250
300
350
400
450
Wavelength [nm]
500
550
600
(b) κ for the nanocrystals from RSn1
Figure 5.16: n and κ for the β-Sn nanocrystals deduced from the RSn1 sample
as explained in the text compared to data for bulk tin from [67]. The wavy behavior of the nanocrystal parameters may very well be related to the polynomial
fit 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 oscillations. According to figure 5.11 the Sn dielectric function from [67] was seen
to give the best correspondense between MG theory and the randomly distributed 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
T [%]
100
50
Simulation
Measurement
R [%]
15
Simulation
Measuerment
10
5
200
300
400
500
Wavelength [nm]
600
700
800
Figure 5.17: Transmission and reflection spectra of the MLSn4 sample together
with the simulated spectra from the optimized n, κ and d parameters. The correspondence between simulation and measurement is very good for the transmission whereas there are minor discrepancies in the reflection. Notice however the
difference 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 simulation 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 difference 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
2.5
Optimized
MG input
n
2
1.5
k
1
Optimized
MG input
0.5
0
200
300
400
500
Wavelength [nm]
600
700
800
Figure 5.18: n and κ corresponding to the simulated R and T in figure 5.17
for 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 are
heavily underestimated in the low wavelength range.
Sample
MLSn1
MLSn2
MLSn3
MLSn4
MLSn5
MLSn6
RSn1
RSn2
Thickness
of Sn containing
layers (nm)
6.1
6.3
6.0
6.1
6.2
6.0
70.4
307
Thickness
of top SiO2
layer (nm)
242
221
192
155
129
100
228
107
Thickness
of bottom
SiO2 layer
(nm)
238
223
188
151
137
95.6
222
98
Thickness
of intermediate SiO2
layers (nm)
3.2
11.2
27.5
41.6
55.3
70.5
-
Table 5.3: A summary of the layer thicknesses resulting from the optimization
procedure for both the random and multi layered samples. The different layers
referred to can be identified in figure 5.6(RSn) and 5.15(MLSn). All values can
be seen to be consistent with the TEM measurements summarized in table 5.1
and 5.2.
81
5. Sn nanocrystals in SiO2
-17
8
x 10
2
RSn1
MLSn2
7
1.8
6
Enhancement
1.6
σa
5
4
3
1.2
1
2
0.8
1
0
200
1.4
300
400
500
Wavelength [nm]
600
700
(a) σa for RSn1 compared to MLSn2
800
0.6
200
300
400
500
Wavelength [nm]
600
700
(b) Enhancement of absorption from multi layers
Figure 5.19: a)Atomic absorption cross section for the MLSn2 show a higher
absorption than for the RSn1 sample. b) The MLSn2 absorption relative to that
of RSn1 is termed the enhancement and it is found by division of the two curves
in 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 compared 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
800
5.4. β-Sn absorption cross sections
8
x 10
-17
MLSn2
MG with a layered structure
MG with a single effective layer
7
a
2
σ [cm ]
6
5
4
3
2
1
0
200
300
400
500
600
Wavelength [nm]
700
800
Figure 5.20: Atomic absorption cross section of the MLSn2 sample compared
to MG theory for both a multi layered structure such as seen in figure 5.15 and
a structure with a single effective layer as seen in figure 5.6. The Sn dielectric
function 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 nanocrystals 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
~ In the dipole approximation
equation 4.38 in terms of the vector potential A.
it can be described equivalently in terms of the electric field by [62]
~ =E
~ · d~
Hpert = −e~r · E
(5.4)
where d~ is the dipole moment involved in the transition. As evident from the
~ · d~ that becomes
Fermi Golden Rule in equation 4.39 it is the square of E
83
5. Sn nanocrystals in SiO2
1
x 10
-16
MLSn1
MLSn2
MLSn3
MLSn4
MLSn5
MLSn6
0.6
a
σ [cm2]
0.8
0.4
0.2
0
200
300
400
500
600
700
Wavelength [nm]
800
Figure 5.21: Comparison of the atomic absorption cross section for the multi
layered samples.
important for the absorption, and thus calculating
2 ~ · d~
E
(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 ˜h , the end result
becomes
2 1
ns p2 d2 60a4 + 36z04 + 91a2 z02
~ · d~
E
= E02 d2 +
4
3
2304π˜
2h
a2 + z 2
(5.5)
0
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 31 E02 d2 in equation 5.5
ABSlayer
ns p2 60a4 + 36z04 + 91a2 z02
=1+
.
4
ABSM ie
768π˜
2h E02
a2 + z 2
(5.6)
0
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]
˜nc − ˜h 3 E0
p = 4π˜
h R (5.7)
˜nc + 2˜
h where R is the nanocrystal radius. With that the absorption enhancement
becomes
ABSlayer
ABSM ie
nc −˜h 2
ns πR6 ˜˜nc
+2˜
h 60a4 + 36z04 + 91a2 z02
=1+
.
4
48
a2 + z02
(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
5ρB Vnc
(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 absorption 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
z0−4 dependence reduces the influence of a nearby layer of nanocrystals significantly 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 nanocrystals. 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 direction 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 compared to the RSn samples must be due to the interaction between nanocrystals
within the same layer. As explained in appendix A the absorption enhancement 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 calculations have been performed in appendix A and he result is given in equation
A.40 and is rewritten here

ABSlayer
= 1 +
ABSM ie
2
nc −˜h ns πR3 ˜˜nc
+2˜
h a
 +
˜nc −˜h 2
6
5ns πR ˜nc +2˜h .
4a4
(5.10)
where equation 5.7 has again been used for the dipole moment of the nanocrystals. 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
3
This can be estimated by multiplying ns with the cross sectional area of a nanocrystal
2
πR .
4
As the nanocrystals are not point dipoles in reality and there can not be another
nanocrystal arbitrarily close to the test nanocrystal.
86
5.4. β-Sn absorption cross sections
z0 = 6 nm
z0 = 8 nm
z0 = 10 nm
Enhancement
1.08
1.04
1
0.98
200
300
400
500
Wavelength [nm]
600
700
800
Figure 5.22: Enhancement of the nanocrystal absorption from a layer of
nanocrystals a distance z0 away. The Sn dielectric function used is the one
extracted 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
5
This is the distance from the rim of one nanocrystal to the rim of the neighboring
nanocrystal, not the center to center distance.
87
5. Sn nanocrystals in SiO2
3
a = 4 nm
a = 6 nm
a = 8 nm
Enhancement
2.6
2.2
1.8
1.4
1
200
300
400
500
Wavelength [nm]
600
700
800
Figure 5.23: Enhancement of the nanocrystal absorption from nanocrystals
within 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
ABSrandom
=1+
ABSM ie
nc −˜h 2
2nv πR6 ˜˜nc
+2˜
h a3
(5.11)
where nv is the nanocrystal volume density and the dipole moment from equation 5.7 has been used. nv can be calculated from
nv =
88
Ω
dρB Vnc
(5.12)
1.2
3
1.16
2.5
Enhancement
Enhancement
5.4. β-Sn absorption cross sections
1.12
1.08
Different layer z0 = 6 nm
Same layer a = 6 nm
2
1.5
1.04
1
1
200
200
300
400
500
Wavelength [nm]
(a)
600
700
800
300
400
500
Wavelength [nm]
600
700
800
(b)
Figure 5.24: a) Absorption enhancement from randomly distributed nanocrystals compared to a single nanocrystal. The integration constant a has been set
to 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 dimensions, 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 nanocrystals 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 different 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
Experiment
Model
Enhancement
2
1.6
1.2
0.8
200
300
400
500
Wavelength [nm]
600
700
800
Figure 5.25: Comparison of the absorption enhancement from MLSn2 relative
to RSn1 from experimental data (shown previously in figure 5.19(b)) and the
dipole model described in appendix A. The model adequately describe the maximum absorption increase but not the general trend from the experiment. As it
was shown above that the nanocrystals within the layer dominate the enhancement 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 nanocrystals 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 nanocrystals 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
2 ~
~
E·d
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 extracted 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  can then
be written as
 = 1 − R − T.
(5.13)
From the definition of the absorption cross section σabs it can be shown
that
number of absorbed photons/number of absorption centers
number of incident photons/area
Â/dN
A
nc
nc =
σabs
1/A
Â
a =
σabs
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 , Â 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
nc =
σabs
6
See the size dependence in equations 4.15 and 4.16. It can also be argued that the scattering has been taken into account in the measurement of transmission (forward scattering)
and reflection (backward scattering) due to the geometry of the measurement. Either way
it is disregarded here.
91
5. Sn nanocrystals in SiO2
sample
I0
I0 (1-R
as
)(1-e -αd
)
e-α
-αd
1-e )R as
)(
R
I 0(1 as
I0 (1-R
as
d
)(1-e -αd 2
)R
as
e -2αd
d
Figure 5.26: The absorption from a weakly absorbing sample can be divided
into contributions from successive reflections of the continuously attenuated
beam. The absorption from each crossing is written above the line representing 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 α
and 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
(1 − Ras )(1 − e−αd )
(5.15)
1 − Ras e−αd
where Ras is the reflectance from the air-sample interface. Since α is related
to the absorption cross section by αd = σa Ω this can be rewritten as
 =
1
σa = ln
Ω
1−
1
Â
1−Ras Ras
Â
− 1−R
as
!
(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 approached 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 qualitative 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 absorption 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 reflectance 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 unreproducible 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
1
x 10
MLSn1
-16
1
Simulation
Measurement
0.8
0.6
0.6
MLSn2
-16
Simulation
Measurement
σ
σ
a
a
0.8
x 10
0.4
0.4
0.2
0.2
0
200
300
400
500
600
Wavelength [nm]
700
0
200
800
300
400
(a)
1
x 10
600
700
1
Simulation
Measurement
0.6
0.6
x 10
MLSn4
-16
Simulation
Measurement
σ
σ
a
0.8
a
0.8
0.4
0.4
0.2
0.2
0
200
300
400
500
600
Wavelength [nm]
700
0
200
800
300
400
1
x 10
600
700
800
(d)
MLSn5
-16
500
Wavelength [nm]
(c)
1
Simulation
Measurement
0.8
0.6
0.6
x 10
MLSn6
-16
Simulation
Measurement
σ
σ
a
a
0.8
0.4
0.4
0.2
0.2
0
200
300
400
500
600
Wavelength [nm]
(e)
700
800
0
200
300
400
500
600
Wavelength [nm]
700
(f)
Figure 5.27: Comparison between the atomic absorption cross section derived
from energy conservation (equation 5.16) and from the iterative optimization
procedure based on the matrix method.
94
800
(b)
MLSn3
-16
500
Wavelength [nm]
800
5.5. Comparison with previous studies
25
Measurement
Simulation
20
20
15
15
R [%]
R [%]
25
10
10
5
0
200
Measurement
Simulation
5
300
400
500
Wavelength [nm]
600
700
0
200
800
(a) MLSn2 considered as an effective medium
300
400
500
Wavelength [nm]
600
700
800
(b) MLSn6 considered as an effective medium
Figure 5.28: Multi layered samples analyzed as effective media just as the
RSn samples. A good correspondence is seen for the MLSn2 sample where the
layers are relatively close together, whereas the measured reflectance can not be
reproduced for the MLSn6 sample.
MLSn6
MLSn5
R [%]
30
20
10
0
200
300
400
500
Wavelegth [nm]
600
700
800
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 below
200 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
-17
2.5
x 10
RSn1
Huang et al.
RSn1
Huang et al.
2
2
[cm ]
60
a
40
σ
Absorbed percentage [%]
80
20
0
1.5
1
0.5
300
400
500
600
Wavelength [nm]
700
800
(a) Absorption comparison with Huang et al.
[40]
0
300
400
500
600
Wavelength [nm]
700
800
(b) Absorption cross sections from (a)
Figure 5.30: a) Comparison of the absorbed percentage given in [40] with the
one obtained in this work. Both samples have been annealed in vacuum at 400◦ C.
b) Atomic absorption cross sections calculated from equation 5.14 for the curves
in (a).
absorbed percentage  this is compared to the RSn1 sample, where  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 400◦ C 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
7
The more elaborate equation 5.16 would require knowledge of the reflection which is
not given in [40].
96
5.5. Comparison with previous studies
3
x 10
-17
Sn
SnO2
σ
a
2
[cm ]
2
1
0
300
400
500
600
Wavelength [nm]
700
800
Figure 5.31: Comparison of MG theory for nanocrystals of β-Sn or SnO2 in
SiO2 with the same nanocrystal density and filling fraction. The dielectric function 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 nanocrystals 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 = σa Ω 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 structural 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 approximately 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
-17
x 10
MLSn4
Kjeldsen et al.
σ a[cm2]
6
4
2
0
200
300
400
500
600
Wavelength [nm]
700
800
Figure 5.32: Absorption cross section from the MLSn4 sample compared to
that 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 nanocrystals 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 absorption 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 random 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 important 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 coupling of the nanocrystal in a more systematic way. For instance, if the
nanocrystals could be arranged in lines instead of planes measuring the absorption properties for light polarized parallel and perpendicular to these nanocrystal 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 predictions.
8
100
As their sizes are almost equal.
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 the
surrounding nanocrystals
z
p
D
O
φ
r
z0
y
x
Figure A.1: A layer of dipoles parallel to the (x,y) plane will produce an electric
field, which will contribute to the field felt by a nanocrystal located at the origin
of 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


0
 
p~ =  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 cos φ

~ =
~r = −D
 −r sin φ  .
−z0
(A.2)
The electric field at the origin of the coordinate system from a dipole (~
p)
~ as shown in figure A.1 (black highlighted) surrounded by a host
placed at D
material with dielectric function ˜h is given by [105]
~ dip (~r) =
E
1 1
(3(~
p · r̂)r̂ − p~)
4π˜
h r3
(A.3)
By inserting p~ and ~r into equation A.3 the field at O from a dipole at D
becomes




 

Ex
−r cos φ
0
1 
3rp sin φ 
1


 

 Ey  =
− p 2
 −r sin φ  − p 2
 p  .
4π˜
h
( r + z02 )5
( r + z02 )3
Ez
−z0
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
~0 +
E
X
~i
ni E
(A.5)
i
~ 0 is the external field experienced by a nanocrystal and ni is the numwhere E
~ i . What is ultimately interesting
ber of dipoles giving rise to the dipole field E
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]
~ =E
~ · d~
Hpert = −e~r · E
(A.6)
103
A. A simple model for the impact on nanocrystal absorption from the
surrounding 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
2 ~
~
E·d
(A.7)
~ from equation A.5
where the brackets imply the average value. Inserting E
and equating the terms produces
2 ~
~
E·d
=
*
+
X
2
~ 0 · d~
~ i · d~
~ 0 · d~ + 2 E
ni E
E
i
*
+
+
!
X
X
~ i · d~ 
~ j · d~ .
ni E
nj E
i
(A.8)
j
P
~ The sum~ i · d.
The second term can be seen to contain the factor i hni i E
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
2 2
~ · d~
~ 0 · d~ +
E
=
E
*
X
ni
i
=
X ~ i · d~
~ j · d~
E
nj E
j
2 X X
~ 0 · d~ +
~ i · d~ E
~ j · d~
E
hni nj i E
i
+
(A.9)
j
where the quantity hni nj i 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
∆Ai
.
(A.10)
A
If the total number of dipoles in the layer is N the probability for finding ni
dipoles in ∆Ai and nj dipoles in ∆Aj is given by the multinomial distribution
si =
p(ni , nj , N −ni −nj ) =
104
N!
n
sni s j (1−si −sj )N −ni −nj (A.11)
ni !nj !(N − ni − nj )! i j
A.1. Nanocrystals in a different layer
for which the covariance (COV) of two variables is given by
COV (ni , nj ) = −N si sj
, i 6= j
(A.12)
COV (ni , ni ) = V AR(ni ) = N si (1 − si )
, i = j.
where V AR is the variance. This can be combined with the general definition
of the covariance [106]
COV (ni , nj ) = h(ni − hni i)(nj − hnj i)i
= hni nj i − hni ihnj i
(A.13)
for the two occasions where i = j and i 6= j equation
hni nj i = (N 2 − N )si sj
, i 6= j
(A.14)
hni nj i = N si (1 − si ) +
N 2 s2i
, i = j.
This can now be inserted into equation A.9 and the transition matrix
element can be calculated
2 2 X X
~
~
~ 0 · d~ +
~ i · d~ E
~ j · d~
=
(N 2 − N )si sj E
E·d
E
i
+
X
j6=i
N si (1 − si ) + N 2 s2i
2
~ i · d~ .
E
(A.15)
i
This equation can be further simplified by looking at the second and third
term individually. Using that N si = hni i the second term can be written as
105
A. A simple model for the impact on nanocrystal absorption from the
surrounding nanocrystals
XX
1−
~ i · d~ E
~ j · d~ =
(N 2 − N )si sj E
i
j6=i
1
N
XX
i
~ i · d~ E
~ j · d~ =
hni ihnj i E
i6=j
X
X
1
~ i · d~
~ j · d~ =
hni i E
hnj i E
N
i
i6=j



X
X
1
~ i · d~  hnj iE
~ j  · d~ − hni i E
~ i · d~ 
1−
hni i E
(A.16)
N
1−
i
j
P
~ j must be zero when the summation is over a full
Again the term j hnj iE
layer so the second term in equation A.15 is equal to
2
1 X
~ i · d~
hni i E
.
− 1−
N
(A.17)
i
In the same way the third term in equation A.15 can be rewritten to
X
N si (1 − si ) + N 2 s2i
2
~ i · d~
E
=
i
X
2 2 ~ i · d~ + hni i E
~ i · d~
hni i(1 − si ) E
=
i
2 hni i ~ ~2 ~
~
Ei · d + hni i Ei · d
=
hni i 1 −
N
i
2 X
2
1 X
~ i · d~
~ i · d~
1−
hni i E
+
hni i E
N
X
i
where it has been used that si =
now simplifies to
(A.18)
i
hni i
N .
Adding up the terms in equation A.15
2 X
2
2 ~
~ i · d~ .
~
~ 0 · d~ +
= E
N si E
E·d
(A.19)
i
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
i
into an integration. As si = ∆A
A the second term can be written as
2 X N 2
~ i · d~ ∆Ai
~ i · d~ =
N si E
E
A
X
(A.20)
i
i
and as ∆Ai is a small area element in the plane the summation can be turned
into an integration
XN i
A
Z
2
~
~
Ei · d ∆Ai ≈
∞ Z 2π
a
0
N
(Ex dx + Ey dy + Ez dz )2 rdφdr.
A
(A.21)
~ · d~ term yields 6 terms
Equating the E
(Ex dx + Ey dy + Ez dz )2 = Ex2 d2x + Ey2 d2y + Ez2 d2z + 2Ex dx Ey dy
+ 2Ex dx Ez dz + 2Ey dy Ez dz .
(A.22)
As the components of the dipole field along the cartesian axis is given by
equation A.4 the off diagonal elements 2Ex dx Ey dy + 2Ex dx Ez dz + 2Ey dy Ez dz
can all be shown to be zero due to the φ integration, so all that remains is
Z
∞ Z 2π
ns
a
Ex2 d2x + Ey2 d2y + Ez2 d2z rdφdr
(A.23)
0
where ns = N
A 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 = d2y = d2z = d2 /3 , the three contributions can be put together.
The result of some lengthy calculations is
2 1
ns p2 d2 60a4 + 36z04 + 91a2 z02
~ · d~
E
= E02 d2 +
4
3
2304π˜
2h
a2 + z 2
(A.24)
0
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 the
surrounding nanocrystals
a
D
φ
y
p
x
Figure A.2: The electric field from a layer of nanocrystals experienced by a
nanocrystal placed at the origin of the coordinate system is determined by taking
the field from a single nanocrystal (highlighted in the figure) and integrating over
the 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 simplyD by letting
z0 approach zero in
E
P
~ i · d~ is not zero as assumed
equation A.23. This is because the term i ni E
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~ =
108
0
p
!
(A.25)
A.2. Nanocrystals in the same layer
and the position vector of a nanocrystal is given by
−r cos φ
−r sin φ
~ =
~r = −D
!
.
(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
Ex
Ey
!
1
=
4π˜
h
3p sin φ
−
r4
−r cos φ
−r sin φ
!
1
− 3
r
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
2 2
X
~
~
~ 0 · d~ + 2 E
~ 0 · d~
~ i · d~
E·d
=
E
hni i E
(A.28)
i
+
X
i
X
2 1
~
~ i · d~
~
hni i E
hni i Ei · d + 1 −
N
!2
.
i
These different contributions can be calculated individually and summed, but
~ i which
~ = P hni iE
equation A.28 can be simplified by introducing the term δ E
i
is the field from the nanocrystals
2 X
2
2
2 ~
~ i · d~ − 1 δ E
~
~ 0 + δE
~ · d~ +
~ · d~ . (A.29)
= E
hni i E
E·d
N
i
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
~ = χE
~0
δE
(A.30)
109
A. A simple model for the impact on nanocrystal absorption from the
surrounding nanocrystals
~ is
where χ is a dimensionless constant. Here it has been used that as δ E
1
directed along the y-axis it will be parallel to the external field E0 . With this
in mind equation A.29 can be reduced to
2
2
2 X
~
~ i · d~ .
~ 0 · d~ (1 + χ)2 +
~
hni i E
= E
E·d
(A.31)
i
By inserting the second term in equation A.24 with z0 =0 and dividing
2
~ 0 · d~ = 1 E 2 d2 the enhancement ABSlayer from nanocrystals
through with E
3
0
ABSM ie
within the same layer becomes
ABSlayer
= (1 + χ)2 +
ABSM ie
5ns d2 p2
192π˜
2h a4
1 2 2
3 E0 d
.
(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.
˜nc − ˜h E0 .
p = 4π˜
h R ˜nc + 2˜
h 3
(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 component of the electric field in equation A.27 along the y-axis will contribute.
χ =
≈
=
=
=
1
110
N X
Ei ∆Ai
AE0
i
Z Z
ns ∞ 2π y
Ei rdφdr
E0 a
0
Z Z
ns ∞ 2π
1
3psin2 φ − p rdφdr
3
E0 a
4π˜
h r
0
Z ∞
Z 2π
ns p
1
2
dr
3sin
φ
−
1
dφ
4π˜
h E0 a r2
0
ns p
.
4˜
h E0 a
This can be verified by calculation or by symmetry arguments.
(A.34)
A.3. Randomly distributed nanocrystals
z
p
θ
O
φ
D
y
x
Figure A.3: The electric field from a random distribution of nanocrystals experienced by a nanocrystal placed in the origin of the coordinate system is determined 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
2
nc −˜h 2
h 6 ˜nc −˜
ns πR3 ˜˜nc
5n
πR
˜nc +2˜h s
+2˜
h  +
= 1 +
.
a
4a4

ABSlayer
ABSM ie
A.3
(A.35)
Randomly distributed nanocrystals
Here the absorption enhancement from nanocrystals randomly distributed
throughout a SiO2 layer will be calculated. This corresponds to the RSn samples 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
~ of a dipole is given by
A.25) and the position vector D
111
A. A simple model for the impact on nanocrystal absorption from the
surrounding nanocrystals


rcosφsinθ

~ =
D
 rsinφsinθ  .
rcosθ
(A.36)
From equation A.3 the electric field at O from the dipole highlighted in
figure A.3 is given by


 


−cosφsinθ
0
Ex
1  3psinφsinθ 
1  



−
 −sinφsinθ  − 3  p  .
 Ey  =
4π˜
h
r3
r
−cosθ
0
Ez

(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
XN i
V
Z
2
~
~
Ei · d ∆Vi ≈
∞ Z 2π
a
0
Z
0
π
N
(Ex dx + Ey dy + Ez dz )2 r2 sinθdθdφdr.
V
Again the terms Ex dx Ey dy , Ex dx Ez dz and Ey dy Ez dz will integrate to zero
so only the three terms Ex2 d2x , Ey2 d2y , Ez2 d2z must be calculated. By assuming
that d2x = d2y = d2z = d2 /3 the final result becomes
XN i
V
2
2 2
~ i · d~ ∆Vi ≈ nv d p
E
24π˜
2h a3
(A.38)
where the nanocrystal density nv = N
V has been inserted. With this the
absorption from randomly distributed nanocrystals becomes
2 2
2 2
~
~
~ 0 · d~ + nv d p .
E·d
= E
24π˜
2h a3
(A.39)
Inserting p from equation A.33 and dividing through with 13 E02 d2 to get the
absorption enhancement due to the nanocrystals the result becomes
112
A.3. Randomly distributed nanocrystals
ABSrandom
=1+
ABSM ie
nc −˜h 2
2nv πR6 ˜˜nc
+2˜
h a3
.
(A.40)
113
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