Graded Optical Filters in Porous Silicon
for use in MOEMS Applications
by
Sean Erik Foss
Submitted
in partial fulfilment of the requirements
for the degree of
Doctor Scientarium
Department of Physics
Faculty of Mathematics and Natural Sciences
University of Oslo
Oslo, Norway
September, 2005
Abstract
Combining optics, electronics and mechanics on one miniature platform is
an emerging reality in micro device technology. An important goal of this
is the simplification and enhancement of actions in every-day life, e.g. labon-a-chip for full characterization of blood samples with integrated loading,
transportation, manipulation and analysis on one chip. Many elements are
required for this to work, among them the control and manipulation of light.
This thesis presents a study of the use of porous silicon within this scope.
Optical filters changing the spectral characteristics of light are fabricated
in an electrochemical etching process of silicon in solutions containing hydrofluoric acid. The aim of this investigation is to fabricate high quality optical transmission and reflection filters in the near-infrared making use of the
special properties of porous silicon which are hard to achieve in other optical materials, and at the same time enhance micro-opto-electro-mechanicalsystem technology in silicon by adding the possibilities presented by porous
silicon.
Both discrete layer and graded index optical filters (rugate filters), with
and without laterally dependent filter functions, are fabricated showing the
versatility of this process. However, to obtain high quality optics with
porous silicon, a very good control of the etch process is needed. For this
reason, equipment has been developed for monitoring the most important
etch parameters in situ; depth/time dependent porosity, etch rate, and
interface roughness. The technique is based on interference effects in an
infrared laser beam partly reflected off the different interfaces in a sample
during etching of a porous layer. The information obtained is later used to
control the etching of the designed structures.
Several device designs and ideas incorporating multilayer or graded index
porous silicon are included at the end of the thesis.
i
Acknowledgements
During the last few years which have brought me to the point where I am
writing these acknowledgements, I have had the great fortune of receiving
help in one form or other from many people. My adviser, Terje G. Finstad,
has guided me patiently through the latest part of my education as a scientist. He has done this by always giving me time and sharing his abundance
of knowledge. I am greatly indebted to him for the opportunity he gave
me! A great thanks is also due my second adviser, Åsmund Sudbø. He has
given me an interesting insight into the enlightening world of optics.
Fortunately, I attended a few conferences in connection with my studies. This led me to some interesting discussions with Hans Bohn of
Forschungszentrum Jülich, Germany, who gave me some suggestions on
rugate filters in porous silicon, and Gilles Lèrondel at UTT, France, who
shared some of his indepth understanding of the porous silicon multilayer
etching process.
As ideas evolved and I wanted to test new things in the lab, having the
mechanical workshop and the electronics lab and the people at these places
has been invaluable. Thanks for all the help.
Had I been been alone in the lab or by my computer day after day I had
surely gone mad, so I owe much of my still fairly sound sanity to my colleagues with whom I have discussed everything between science and the
weather. Especially thanks to Ingelin Clausen, Chenglin Heng, Klaus Magnus Johansen, to name a few. Thanks also to Håvard Alnes for being my
bad conscience and keeping me reasonably fit. Erik Marstein deserves a
special thanks for being a good friend and also introducing me into the
secrets of porous silicon. With such an enthusiasm, how can one not think
that whatever he is doing is the most important thing in the world?
My family has always supported me and lent me a helping hand whenever
needed, which I am very grateful for. Thank you mom and mormor. Part
of the reason why I ever thought of doing a PhD is my late uncle Larry. He
has always been an inspiration in both character and career. Thank you for
giving me the opportunity to know you. Last but definity not least I am
forever indebted to my wife, Hilde, for her patience and unfailing confidence
in me. This could not have been done without you my dear.
iii
Contents
Abstract
i
Acknowledgements
iii
1 Introduction
1
2 Porous silicon formation
5
2.1
Porous silicon history . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Porous silicon basics . . . . . . . . . . . . . . . . . . . . . .
6
2.2.1
Formation . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2.1.1
Chemistry . . . . . . . . . . . . . . . . . . .
7
2.2.1.2
I-V characteristics . . . . . . . . . . . . . .
7
2.2.1.3
Morphology . . . . . . . . . . . . . . . . . .
8
2.2.1.4
Formation theories . . . . . . . . . . . . . .
9
Influence of formation parameters . . . . . . . . . . . . . . .
10
2.3.1
Sample . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.3.1.1
Doping . . . . . . . . . . . . . . . . . . . .
10
2.3.1.2
Preparation . . . . . . . . . . . . . . . . . .
10
2.3.1.3
Resistivity variations . . . . . . . . . . . . .
10
2.3.1.4
Drying . . . . . . . . . . . . . . . . . . . . .
13
Electrolyte properties . . . . . . . . . . . . . . . . . .
14
Etch setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.4.1
Optimizing the etch setup . . . . . . . . . . . . . . .
16
Etch setup for graded filter etching . . . . . . . . . . . . . .
17
2.3
2.3.2
2.4
2.5
3 Thin-film calculations
3.1
21
Effective medium theory . . . . . . . . . . . . . . . . . . . .
v
22
vi
3.2
3.3
3.4
Reflectance calculation . . . . . . . . . . . . . . . . . . . . .
25
3.2.1
Characteristic matrix . . . . . . . . . . . . . . . . . .
25
3.2.2
Admittance matrix . . . . . . . . . . . . . . . . . . .
26
Roughness calculation . . . . . . . . . . . . . . . . . . . . .
28
3.3.1
Davies-Bennett theory . . . . . . . . . . . . . . . . .
29
Optical multilayer interference filters . . . . . . . . . . . . .
31
3.4.1
Discrete, homogeneous layers . . . . . . . . . . . . .
31
3.4.2
Inhomogeneous layers . . . . . . . . . . . . . . . . . .
33
4 In situ interferometry experiment
4.1
4.2
41
Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.1.1
Usage . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.1.2
Laser . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.1.3
Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.1.4
Beam to sample coupling . . . . . . . . . . . . . . . .
46
4.1.5
Other equipment . . . . . . . . . . . . . . . . . . . .
49
Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.2.1
Chemical etching . . . . . . . . . . . . . . . . . . . .
49
4.2.2
Effect of irregular sampling . . . . . . . . . . . . . .
51
4.2.3
Frequency analysis . . . . . . . . . . . . . . . . . . .
51
4.2.4
Etch rate and porosity calculation . . . . . . . . . . .
53
4.2.4.1
Measurement of the effect of limited HF diffusion . . . . . . . . . . . . . . . . . . . . .
53
4.2.4.2
Etch calibration
. . . . . . . . . . . . . . .
55
4.2.4.3
Possibility of real-time monitoring . . . . .
57
Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Paper III
83
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Filter fabrication
97
5.1
Basic filter etching . . . . . . . . . . . . . . . . . . . . . . .
98
5.2
Deviations from the basic assumptions . . . . . . . . . . . .
99
5.2.1
Effect of HF diffusion . . . . . . . . . . . . . . . . . . 100
5.2.2
Effect of temperature . . . . . . . . . . . . . . . . . . 102
vii
5.2.3
Chemical etching . . . . . . . . . . . . . . . . . . . . 104
5.3
Etch calibration . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4
Prepared filters . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.5
5.4.1
Reflectance measurement setup . . . . . . . . . . . . 106
5.4.2
Reflectance analysis . . . . . . . . . . . . . . . . . . . 107
5.4.2.1
Discrete filters . . . . . . . . . . . . . . . . 107
5.4.2.2
Rugate filters . . . . . . . . . . . . . . . . . 111
5.4.2.3
Graded filters . . . . . . . . . . . . . . . . . 114
Improvements of the process . . . . . . . . . . . . . . . . . . 115
Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6 Porous silicon applications for MOEMS and passive optics133
6.1
6.2
Passive optical elements . . . . . . . . . . . . . . . . . . . . 133
6.1.1
Schottky barrier spectroscopic IR detector . . . . . . 133
6.1.2
2D photonic crystal . . . . . . . . . . . . . . . . . . . 134
6.1.3
GRIN optics . . . . . . . . . . . . . . . . . . . . . . . 136
6.1.4
Novel optical filter . . . . . . . . . . . . . . . . . . . 138
MOEMS devices . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2.1
Membrane based MEMS pressure sensors . . . . . . . 139
6.2.2
MOEMS optical scanner and switch . . . . . . . . . . 140
6.2.3
Multispectral MOEMS pixel array . . . . . . . . . . . 141
6.2.4
Holographic scanner . . . . . . . . . . . . . . . . . . 142
7 Conclusion
145
Bibliography
147
Chapter 1
Introduction
Roughly 50 years of development has made silicon the material of choice
in the electronics industry. As so much effort and investment has been
put into silicon technology, an increasing focus is now directed towards silicon photonics. There are other materials available, and with fairly mature
technologies, that are technically superior to silicon for specific uses in photonics. However, to be able to integrate the different elements of a photonic
circuit, which is where a synergy is possible, as well as making the technology commercially viable, only silicon technology has the potential of solving
all the challenges. Why photonic circuits? The basic assumption is that
photons move faster than electrons and can therefore move data quicker,
hence faster data processing. A second assumption is that there is less
of a heating problem with photons, leading to lower power consumption.
Before we see a wholly integrated photonic/optical circuit there are many
problems which need to be addressed. This thesis deals with a technology
which may broaden the usage of silicon for optical elements and thereby
help solve some of the details of these challenges. The last couple of years
have shown great improvement in silicon photonics, with a couple of significant breakthroughs such as an all silicon Raman laser [1, 2] and silicon
optical modulators [3].
By porosifying silicon, several material properties of silicon undergo change.
This may be used to expand the application possibilities of silicon, both by
introducing new concepts to silicon technology and by improving already
applied ideas. A note on terminology is appropriate here. Porosified or
porous silicon has had many abbreviations, in this thesis PS will be used
and where any ambiguity is possible this will be explicitly noted.
Engineered pores in silicon may be in many forms and sizes. This leaves us
with a material in one extreme where the pores do not change any material
properties but act more as a mechanical construction, in the other extreme
one reaches the quantum limit and the material properties are affected by
quantum effects, namely a widening of the band-gap due to quantum confinement in the nm-size silicon crystalline dots and rods. The properties
1
2
affected by porosification are the electrical and thermal resistivity, refractive index, luminescence, stiffness, chemical reactivity and so on. Of these
properties the control of refractive index will be the focus of this thesis.
In principle, by changing the porosity from 0 to 100% (no or all material
removed) the refractive index for a given photon wavelength will change
from that of the silicon bulk material to that of air. This, together with the
fact that the porosity value may be controlled in space and that the size
of the pores is also controllable, most importantly in this case to be much
smaller than the photon wavelength of interest, makes it possible to have
control of the photon trajectory in the material.
The thesis and work done is divided in two parts: the processing/fabrication
of porous silicon and the use of this knowledge for fabrication of devices or
elements of devices. This work has in practice been carried out in parallel with feedback between the two areas indicating where better control
of parameters is needed and which parameters are important for fabrication. Chapter 2 will go into some basics of porous silicon processing and
physics. The next chapter will discuss some of the theory needed for designing optical filters. Chapter 4 presents the work done on etch parameter
monitoring while Chapter 5 discusses some experimental details of the processing of porous silicon. The sixth chapter takes a look at the possibilities
of porous silicon in electronic/photonic devices with an emphasis on microopto-electro-mechanical-systems. A conclusion will be presented in the last
chapter.
Five papers are included in the thesis, presenting the main results:
Paper I:
S.E. Foss, P.Y.Y. Kan and T.G. Finstad
Single beam determination of porosity and etch rate in situ
during etching of porous silicon
J. Appl. Phys., 97, 114909 (2005)
Paper II:
P.Y.Y. Kan, S.E. Foss and T.G. Finstad
The effect of etching with glycerol, and the interferometric
measurements on the interface roughness of porous silicon
Phys. Stat. Sol. (a), 202, 8, 1533 (2005)
Paper III:
S.E. Foss, P.Y.Y. Kan and T.G. Finstad
In situ porous silicon interface roughness characterization by
laser interferometry
Accepted for publication in the Proceedings of the 3rd Pits
and Pores symposium, 206th Meeting, ECS, Hawaii, 2004
Paper IV:
S.E. Foss and T.G. Finstad
Multilayer interference filters with non-parallel interfaces
Proceedings of the Nordic Matlab Conference, Copenhagen,
Denmark, 2003
Paper V:
S.E. Foss and T.G. Finstad
Laterally graded rugate filters in porous silicon
3
Mat. Res. Soc. Symp. Proc., 797, W1.6.1 (2004)
Paper I deals with the setup of a fiber optic based in situ infrared reflectance experiment and the theory behind the analysis of the obtained
data. Etch rate, porosity depth profiles and PS-substrate interface roughness are obtained. Some measured data is presented showing the accuracy
of the technique.
Paper II presents experiments on the effect of glycerol and HF concentration in the electrolyte on PS-substrate interface roughness based on in
situ IR reflectance measurements during etching. The roughness showed a
dependence on both glycerol and HF concentration, while porosity showed
only a weak dependence on glycerol concentration.
Paper III discusses in detail the calculation of roughness from in situ infrared reflectance data. Results are presented showing the dependence of
PS-substrate interface roughness on HF and glycerol concentration, temperature and formation current density, and the relative importance of these
parameters.
Paper IV presents calculations of the effect of a laterally graded optical
filter on the filter characteristics. The calculations are based on ray-tracing
through the graded layer stack. It is shown that both a small gradient in
layer thicknesses and a small divergence in the incident beam widens and
reduces the stop band reflectance.
Paper V describes fabrication of laterally graded rugate reflection filters.
Filters with a shift in the reflection band center wavelength of up to 100
nm pr. mm across the filter surface are realized. The shift is close to linear
with position, but a broadening of the reflection band is observed with a
greater gradient.
Contributions to other papers were made during the work on this thesis,
however, these papers are not included as the subject matter is besides the
focus of the thesis. These papers are:
P.Y.Y. Kan, T.G. Finstad, H. Kristiansen and S.E. Foss
Porous silicon for chip cooling applications
Physica Scripta, T114, 2004
P.Y.Y. Kan, S.E. Foss and T.G. Finstad
Thick etch-through macroporous Si membrane from p- & n-Si, and fast
pore etching and tuning the pore size from n-Si
Submitted to the E-MRS 2005 Spring Meeting, Strasbourg
Chapter 2
Porous silicon formation
2.1
Porous silicon history
In 1956 Ingeborg and Arthur Uhlir at Bell Telephone Laboratories were
working on electrochemical etching of silicon using hydrofluoric acid (HF)
solutions [4]. This was done with the intent of polishing and shaping microstructures in silicon, however, the silicon was polished only above a threshold current density, whereas below this current density the surface turned
red or black. However, these films were of relatively little interest at the
time. Similar experiments were also performed by Turner [5]. The porous
nature of the film was first reported in 1971 by Watanabe and Sakai [6]
and subsequently by Theunissen in 1972 [7]. The main focus of research up
until the ’90s was on the use of oxidized PS as a dielectric isolator [6, 8, 9].
However, a wider interest in PS was sparked by a paper by Canham in 1990
on efficient photoluminescence (PL) in PS at room temperature [10]. This
led to a flurry of activity in the PS research with focus on the active optical
properties. Shortly after, electroluminescence in PS was reported [11] and
numerous groups attempted to make PS based LEDs with emissions at different wavelengths [12]. This research has led to a general interest in PS,
resulting in research into many other properties and uses.
Even before the discovery of PL, it was known that PS have very diverse
morphologies [13]. The passive optical properties of PS, i.e. refractive
index and absorption, were also subjects of some research before 1990 [14].
This formed some of the background for the PS multilayer (PSM) optical
filter structures reported by both Vincent [15] and Berger et al. [16] in
1994. The formation of multilayer structures for use as optical filters has
since become nearly a standard technique. In 1995 Mazzoleni and Pavesi
reported the use of PS Fabry-Pérot filters (two stacks of pairs of layers
fulfilling the Bragg condition (optical density = λ/4) with a spacer layer
of thickness λ/2 between) to tune and narrow the PL emission from the
PS [17]. Shortly after, Pavesi et al. reported an enhancement in the PL
emission line using the same type of filters [18] indicating a coupling of the
5
6
PS spontaneous emission with the cavity mode of the multilayer structure.
This again led to the incorporation of materials into the porous structure
to take advantage of the increased coupling between field and matter in
the micro-cavity. The doping with erbium ions in a Fabry-Pérot structure
resulted in an enhanced infrared (IR) PL at a peak wavelength of 1.536 µm
[19].
Due to the flexibility and relative ease of fabricating multilayer structures
in PS, many different structures have been reported. The basic optical multilayer structures such as Bragg mirrors and Fabry-Pérot filters have been
mentioned. These may be constructed such that the optical characteristics
change due to an external effect. As the spectral features of these structures
may be very sharp and narrow, slight changes in layer optical thickness will
significantly change the spectral features. Incorporation of liquid crystals
into the pores of a PSM structure lead to the ability of modulating the filter characteristics by controlling a voltage [20, 21, 22], or alternatively, the
filter characteristics could be controlled by a temperature modulation [23].
This may be used for optical switching. Alternatively, one may have polarization dependent filter characteristics as in the case of dichroic filters
etched in h110i Si [24]. These optical elements may also be used as sensors,
either by utilizing the change in spectral characteristics as an indication
of filling ratio, e.g. measure amount of air moisture, amount of liquid in
the pores, or the pore walls may be sensitized, or activated, with different materials which react to more specific molecules [25, 26]. There are
many other structures possible as well; Lèrondel et al. [27] have fabricated
a diffraction grating in PS by light assisted etching using two interfering
laser beams incident on the sample surface during normal etching. Volk et
al. [28] have reported lateral PSMs for use as ultraviolet diffraction gratings
using a buried n-doped region to control the current distribution close to
the surface. Further, as will be an important part of this thesis, the possibility of arbitrarily controlling the refractive index with depth has resulted
in the fabrication of rugate filters of different kinds [29, 30].
2.2
Porous silicon basics
Porous silicon is most often fabricated by an electrochemical reaction where
the Si sample is placed in an HF based electrolyte and an external bias is
applied. The porosifying reaction depends on an availability of holes at the
electrolyte-Si interface. An alternative method is ”stain-etching” where HF
is combined with a strong oxidizing agent, such as nitric acid, HNO3 , and
no external bias is used [31]. However, this method will not be discussed
here.
7
2.2.1
Formation
2.2.1.1
Chemistry
To drive the porosification, the Si-sample is positively biased (anode) and
in contact with the electrolyte in which a negatively biased Pt-electrode
(cathode) is placed. The mechanism of pore initiation is still under debate,
however, there are suggestions that defects or slight variations in surface
potential due to defects or doping atoms are the starting point of the pores.
When a bias is turned on, holes from the sample and F− ions in the electrolyte will move towards the electrolyte-substrate interface and react. The
exact reaction kinetics are not well understood, and may very well vary
quite significantly depending on formation parameters as is evident in the
many different morphologies of PS obtainable. Good reviews of the reaction
kinetics and PS formation are given in Refs. [32, 33, 34].
The main reaction during PS formation, assuming a hydrogen terminated
Si surface, is suggested by Lehmann and Gösele [35] to be:
SiH2 + 2F− + 2h+ → SiF2 + H2 (divalent dissolution)
SiF2 + 4HF → 2h+ + SiF2−
(in solution).
6 + H2
(2.1)
(2.2)
Here the SiH2 is bound to the the Si surface. In this reaction hydrogen gas
is formed which may interfere with the etching. In this general reaction
two holes are needed for each Si atom dissolved, hence the valence of the
reaction is two. It may vary, with normal values for PS formation between
2 and 2.8. This reflects both the model used to calculate the valence and
also the complexity of the reaction. The overall valence of the reaction may
be roughly calculated by using the etch rate and porosity:
ν=
j · Ar,Si
.
NA · ρSi · e · r · P
(2.3)
Here ν is the valence, j the current density (A/m2 ), NA = 6.02 · 1023 mol−1
the Avogadro number, Ar,Si = 28.09 g·mol−1 the molar mass of silicon,
ρSi = 2.33 g·cm−3 the density of silicon, e = 1.60 · 10−19 C the electron
charge, r the etch rate (m/s) and P the porosity (absolute values).
2.2.1.2
I-V characteristics
PS is formed in a limited range of current density or bias as reflected in
I-V curves measured in the electrolyte-Si system, see Fig. 2.1. These I-V
curves are taken from Ref. [34] and show the current-voltage relationship in
a system consisting of a p-type Si sample in a HF based electrolyte under
forward and reverse bias, with illumination and without. The I-V curves
for n-type Si under the same conditions will be somewhat different, but
are omitted here as p-type PS is the focus in this thesis. If the sample
8
is illuminated during etching there will be photon generated holes which
may react with the F− ions. This is often used for n-type PS etching,
while, as can be seen in Fig. 2.1, for anodically biased p-type PS formation
this has only a small, if any, effect as the photon generated holes have
an insignificant concentration. However, to avoid any uncertainty, samples
presented in this thesis have been etched in the dark. The I-V curves have
some similarities with a Schottky diode I-V curve. However, there are a
few important differences, most importantly the two peaks on the curve
under forward bias. The first peak signifies the start of electropolishing,
while the second marks the onset of current oscillation. Electropolishing is
driven by a slightly different reaction than that indicated in the reaction
represented by Eq. 2.1, namely a reaction of valence four. In this case the
dissolution of Si is not direct, but goes through an oxidation step first.
There are some general trends which are seen in I-V curves of this system,
such as Fig. 2.1. By increasing the HF concentration, the first peak shifts to
higher current values (higher electropolishing current), while increasing the
substrate doping concentration shifts the first peak towards lower voltages.
Figure 2.1: The IV curves
of the silicon-electrolyte system closely resembles a Schottky diode IV curve. Some important differences can be seen
in the two peaks in the forward
bias region. The plot is taken
from Smith and Collins [34]
2.2.1.3
Morphology
The result of the electrochemical etching of Si within the limitations discussed above is in general a porous structure. Generally, little will happen
with the pore walls of the already etched structure as the electrochemical
reactions take place at the pore-front. However, depending on the parameters of the etching, the structure will vary greatly. One property which will
be sensitive to several parameters, like current density, sample resistivity,
HF concentration and solvent composition, is the pore size. The classification used is defined by the International Union of Pure and Applied
Chemistry and describes porous materials in general: pore-sizes less than
2 nm are denoted micro, between 2 and 50 nm are meso and above 50 nm
are macro. The PS films fabricated for this thesis are mostly meso-porous
as may be seen in the scanning-electron-microscope (SEM) micrograph in
Fig. 2.2. This picture shows the surface of a typical filter structure with
9
a median pore-size of approximately 15 nm at the surface. The structural
morphology in general is very sensitive to the formation parameters. A
short summary and categorization of different pore morphologies is given
in Ref. [33]. Pores may be sponge-like, straight or branched, random or
aligned along the h100i crystalline axis of the sample. Macro-pores may be
filled by micro-pores or empty, just to mention a few possibilities.
Figure 2.2: This SEM image shows the surface of a
typical filter structure. The
size distribution of the pores
is shown in the histogram.
The diameter is calculated assuming circular pores. The
mean pore size of 14.9 nm corresponds well with what has
been reported in the literature
under similar etch conditions.
2.2.1.4
Formation theories
How the pores are formed is also a question still under some discussion.
There are three predominant models; the Beale model, the quantum confinement model and the diffusion-limited model. The Beale model [13]
proposes that the pore-walls in meso- and micro-PS are depleted of charge
carriers due to overlapping depletion layers resulting in a concentration of
the electrical field at the pore tips, hence an increased concentration of holes
with a resulting etching at the pore tips. The quantum confinement model
suggested by Lehmann and Gösele [35] is based on the quantum confinement of charge carriers in the nanometer sized Si pore-walls of micro- and
meso-PS. This quantum confinement will lead to an increase in the band
gap compared to bulk Si. This introduces a barrier for the holes going from
the bulk to the porous Si-structure whereby the hole concentration increases
close to the pore tips resulting in a dissolution of Si. The diffusion-limited
model [34] describes the formation of pores as a result of a random walk
process of the holes. In this model the holes moving towards the electrolytesubstrate interface will most likely reach a pore tip first, hence the formation
of PS is limited by the diffusion of the holes. Carstensen et al. [36] have also
introduced a model called the ”current-burst” model to explain pore growth
in Si. Etching in this case occurs in bursts, both temporal and spatial (i.e.
at discrete positions). The passivation of the pore walls is in this model
a result of hydrogen termination. These models may describe micro-, and
10
to some extent, meso-PS, however, macro-PS usually does not have overlapping depletion layers in the pore walls, so the formation mechanism is
somewhat different [37]. Formation of macro-PS will not be discussed here.
2.3
Influence of formation parameters
2.3.1
Sample
2.3.1.1
Doping
The doping concentration and type of the sample are crucial parameters
for PS formation. As there is no need for external lighting when etching ptype Si and the fairly low PS-substrate interface roughness obtained, most
PSM structures are etched in p-type samples. The obtained morphology
and porosity ranges are dependent on the resistivity of the sample. Samples of high resistivity tend to give microporous PS which are very brittle
and the controllable porosity range is rather narrow. With lower resistivity
samples, the interface roughness (microscopic) tends to decrease, although
macroscopic roughness, i.e. due to striations, tends to increase. The porosity range of highly doped samples is quite large. The samples presented in
this work are p-type, boron doped with a nominal resistivity of <0.1 Ω·cm,
measured to be around 0.018 Ω·cm. Nominal sample thickness was 520 µm.
2.3.1.2
Preparation
Before etching, an Al-back contact is evaporated on the samples and annealed to give a good ohmic contact. This is crucial for a homogeneous
current density distribution over the etched area, also for highly doped samples. Different back-contact geometries may be used as will be discussed
in Sec. 2.4.1. The samples are ultrasonically cleaned in trichloroethylene,
acetone and DI-water before etching. This process seems crucial, as badly
cleaned samples show inhomogeneities in the spectral characteristics of the
etched optical filters.
2.3.1.3
Resistivity variations
As current density is an important factor in the etching of PS, the local
sample resistivity will have an impact on the resulting porous structure.
The resistivity of the sample is controlled by the doping and as the dopant
distribution usually is slightly inhomogeneous, the resulting local etch rate
and porosity will be locally inhomogeneous. For many applications this is
acceptable, but in the case of optical elements, both an inhomogeneity in
11
the refractive index and rough layer interfaces will be detrimental to the
optical quality.
Both refractive index inhomogeneities and PS-substrate interface roughness
have been observed. In Fig. 2.3 the reflectance spectrum of a rugate optical
reflectance filter is measured. Details of the optical filters are discussed in
Chap. 3. Two measurements are made at different positions, 1 mm apart,
on the filter surface. The reflection bands are shifted relative to each other
which indicates different conditions for interference. This is most likely due
to local differences in etch conditions, e.g. resistivity differences. These
inhomogeneities are often visible on the surface of the optical filters, fabricated for this thesis, as slight deviations in color. These deviations take
the form of concentric circles often roughly coinciding with the center of
the wafer. This may be seen in Paper V where a grayscale optical microscope image clearly shows local differences in color. The difference in color
stems from porosity and layer thickness variations. By selectively removing
the PS by etching in a concentrated alkaline solution, e.g. 40 % NaOH,
the PS-substrate interface is revealed. Figure 2.4 shows a 3D surface plot
of height data obtained by white-light interferometry from a test sample.
One may clearly see ridges caused by spatially inhomogeneous etch rates.
These ridges coincide with the color deviations. This is discussed more in
Paper V. The spatial period with which these ridges occur along the radial
direction is in the 100 µm to 1 mm range.
The cause of these inhomogeneities in resistivity is most likely striations,
or fluctuations in dopant concentration caused by the Si-ingot production
process. This is a well known problem in Si-technology and is thoroughly
discussed in technical papers on Si wafer material quality, e.g. Ref. [38]
gives a summary of semiconductor crystal growth specifically discussing
striation formation. The striation induced roughness has rather long spatial
periods and may in p+ samples give quite large surface height fluctuations.
In lower doped p-type Si samples the striation induced roughness is less
pronounced [39]. This may be understood considering that the same dopant
fluctuation relative to the average dopant concentration is likely to occur in
both highly doped and low doped Si, with the result that a different absolute
change in etch rate is observed. However, there will be interface roughness
with smaller spatial periods (micro-roughness) which is more pronounced in
p− samples than in p+ samples where this type of roughness is very small.
This results in locally very good optical quality of p+ PS based optics,
and promises good quality optics on larger area when striation effects are
controlled.
The resistivity, ρ, does not only change at a local (µm) level, but also on a
wafer level. This has an impact on the reproducibility of filter fabrication.
In Fig. 2.5 the result of a four-point-probe resistivity measurement across
a typical wafer is shown. Standard geometrical correction factors from
Ref. [40] for thin, circular disks are used to calculate the resistivity. Distance
to wafer edge, orientation of the four-point-probe, wafer thickness and probe
12
0.5
Shift: 11 nm
Reflectance (abs)
0.4
0.3
FWHM = 26 nm
0.2
0.1
0.0
550
600
650
700
Wavelength (nm)
750
Figure 2.3: Reflectance spectra of a rugate filter measured at two different positions, 1 mm apart, on the filter surface. There is a small
wavelength shift of the reflection band indicating different
etch conditions due to sample
resistivity inhomogeneities.
Figure 2.4: A surface plot
of white-light interferometry
measurement data from the
PS-substrate interface after removal of the PS by an alkaline etch.
The PS film
was about 117 µm thick. The
ridges due to local etch rate
differences are clearly visible.
These are caused by striations,
or dopant inhomogeneities in
the substrate.
spacing were all taken into account using
ρ = G·
U
I
π
G =
· t · T2
ln 2
s
t
∆ d
· C0
· K2
,
· F4 (t, s)
s
d
d s
,
(2.4)
where U is the measured voltage and I is the measured current, T2 , C0 , K2 ,
and F4 are correction factors, t is the thickness, s is the probe spacing, d is
the wafer diameter,and ∆ is the probe displacement from the wafer center.
The correction factor T2 accounts for the effect of finite thickness, C0 is
a factor taking into account the distance to the edge when measured in
the center, while K2 is an additional factor adjusting C0 for displacements
towards the edge. F4 takes into account both thickness and closeness to the
edge.
The maximum resistivity difference in Fig. 2.5 is about 7 %. However, the
variation is relatively small within the area of a typically etched filter (1 cm
diameter circle). The range of local current densities within the filter area
is therefor roughly independent of the sample position in the wafer. On the
other hand the resulting morphology may be slightly different from filter
to filter across a wafer as the sample resistivity and the necessary bias is
13
different. Different resistivity and bias will likely change the depletion layer
at the pore front which may change the resulting porosity or structure of the
PS. There have been few, if any, systematic investigations of small changes
in wafer resistivity on the morphology of PS with optical applications in
mind. A similar example to that in Fig. 2.3 of the effect of local resistivity
variations on filter characteristics is given by Lérondel et al. in Ref. [41].
2.3.1.4
21
-3
cm)
20
Resistivity (10
Figure 2.5: The resistivity
measured at different positions
across a typical wafer used for
PS etching.
The measurements are obtained by a fourpoint-probe. The change in
resistivity across the wafer is
significant and will affect the
reproducibility of PS etching.
Error bars show standard error
based on three measurements
at different currents. Edge effects are taken into account.
The line is only a guide.
19
18
17
16
0
20
40
60
80
100
Distance from edge (mm)
Drying
After etching, before the samples are taken out of the etch-bath, the bath
with the sample in it is rinsed out with ethanol. The sample is then taken
out to air dry. Because of the size of the pores the capillary stress within the
pores may be quite high. Depending on the size of the structure (porosity)
and the surface tension of the liquid, cracking of the PS layer may occur.
This limits the maximum porosity obtainable and also suggests a procedure
for drying [42, 43, 44]. When drying in air, a meniscus will always form
in the pores which will result in a stress on the pore walls. This makes it
important to have a very low surface tension liquid in the pores when drying.
As suggested this may be done by rinsing out with pure ethanol which has
a lower surface tension than water (22 mJ/m2 compared to 72 mJ/m2 ), an
alternative is to use pentane (with a surface tension of 14 mJ/m2 ). Pentane,
however, is not water-soluble so the sample is usually rinsed in ethanol first.
The best results, however, have been obtained by supercritical drying [44]
in CO2 (>95 % porosity) where drying is performed above the supercritical
point of a liquid, usually CO2 .
In the electrolyte and immediately after drying, the pore walls are mostly
H-terminated [34]. The hydrogen will be replaced by oxygen to form native
oxide quite rapidly in air. This will change the properties of the PS over
time. Due to the large surface area of the PS the silicon-oxygen ratio may
be quite large resulting in a significant impact on the properties of PS. For
many applications this instability is not acceptable and several methods
14
for surface passivation have been reported in the literature. Among these
are controlled oxidation by anodic or chemical oxidation, rapid thermal
oxidation, capping of the PS layer by a dielectric or metal [45], thermal
nitridation or thermal carbonization [46].
2.3.2
Electrolyte properties
The electrolyte contents used for PS etching may vary substantially, however, the electrolyte is generally based on aqueous HF. For all experiments
reported here, a 40 % aqueous HF has been used as the base. It is quite
possible to etch PS with this base diluted in water, however, to facilitate extraction of hydrogen bubbles formed during etching, ethanol is usually used
as a surfactant. Compared to water, ethanol has better wettability and
lower surface tension which results in better infiltration in the nanometersized pores. Different additions or substitutions may be made to change the
properties of the electrolyte, e.g. to increase viscosity which is thought to influence the PS-substrate interface roughness, glycerol may be added. Other
substitutions include other organic solvents, especially dimethyl formamide
(DMF) and dimethyl sulfoxide (DMSO) which result in p-type macro-PS
for certain parameters. A short overview of the different electrolyte compositions reported in the literature is given in Ref. [32].
In papers II and III the effect of different electrolyte parameters is discussed.
Electrolytes containing different ratios of glycerol are used while measuring
the PS-substrate roughness evolution during etching. A comparison of room
temperature and low temperature etch is also made. Both temperature and
glycerol content seem to affect the interface roughness, however, the degree
depends on other parameters like HF concentration. For some parameter ranges the roughness decreases. It has been suggested [39, 47] that
the reduction in roughness with decreasing temperature, down to -40 ◦ C,
and increasing glycerol ratio is due to an increase in viscosity. Data from
Ref. [48] suggest that a mix of water and glycerol (25 %) at 20 ◦ C has the
same viscosity as an equivalent mix of water and ethanol at about 12 ◦ C.
The exact viscosity values of the electrolytes will differ from these, but the
closeness of the tabulated data indicates that viscosity may be a critical
parameter. However, it is not obvious that an increase in viscosity itself is
the only reason why lower roughness is obtained. Especially in the case of
low temperature etching, the reaction kinetics will most likely be affected.
The tentative explanation why a change in electrolyte viscosity affects the
interface roughness of PS, and also the refractive index inhomogeneity, is
that a situation closer to that of electropolishing is reached. During electropolishing the holes diffuse faster to the interface than do active electrolyte
species (e.g. F− ), which results in a ”guaranteed” availability of holes at
the surface. This has the consequence that ”peaks” are etched first, hence
the resulting surface is locally flat, where the extent of the locality depends
15
on a characteristic length (e.g. diffusion length of holes). By reducing the
diffusion of ions in the electrolyte during PS etching, the local differences of
hole availability caused by an inhomogeneous resistivity will be reduced. In
the extreme case of ion diffusion controlled etching, the etch rate and porosity should be independent of resistivity. Some results of etching in glycerol
containing electrolytes and low temperature etching will be presented in
Chapters 4 and 5.
2.4
Etch setup
There are usually two ways of setting up a sample for etching. These are
normally referred to as single etch cell and double etch cell. In the single
etch cell the sample is usually horizontal with a solid back contact, often
a Cu-plate, and the front side is in contact with a reservoir containing the
electrolyte. In the electrolyte there will be a Pt-cathode. In the double cell
the sample is vertically placed between two separate reservoirs containing
the electrolyte, both with Pt-electrodes, one working as cathode and one
as anode. Here the electrolyte reservoir works as a back side contact. All
samples fabricated for this thesis were made with a single cell setup. A
vertical single cell setup was used for some tests, but this resulted in etch
rate and porosity inhomogeneities caused by H2 bubble formation and trapping. Vertical ridges were observed at the PS-substrate interface after PS
stripping by concentrated NaOH which were most likely caused by bubble
induced change in etch rate.
A sketch of a basic etch cell used in this thesis is shown in Fig. 2.6. The
back contact upon which the Si sample is placed is made of copper, on top
of the sample, between the sample and the top part of the cell, is a sealing
ring or a sheet with an opening. The cell is made of Teflon or another
material inert to HF. Several different variations of this basic cell have
been used. The etch current is supplied by a computer-controlled Keithley
2400 sourcemeter. Etching is normally performed under galvanostatic, or
constant current, conditions, where current is the control parameter.
The biasing voltage was monitored during single PS layer fabrication to
detect anomalies, such as significant changes in voltage due to leakage.
In Fig. 2.7 a typical voltage monitored during an etching is shown. It is
included here due to the curious curve shape. The plot contains surprisingly
many features considering the sample was etched with a constant current.
There is a transient region in the beginning which may be ascribed to a build
up of charge before etching begins, e.g. due to an activation energy. The
irregular sawtooth pattern may be due to local oxide build-up and etching,
as in the current-burst model [36] or due to hydrogen bubbles interfering
with electrolyte flow.
16
Figure 2.6: A sketch of the
basic etch cell used. The Si
sample is placed on a Cu-plate
which is positively biased, an
o-ring or silicon rubber sheet
(HF-resistant) is placed on top
of the sample before the top
part of the cell is fastened.
The cell material is either
Teflon or PVC. The electrolyte
is filled into the reservoir and
a Pt-electrode is placed in the
electrolyte. The Pt-electrode
is negatively biased. A computer controlled current source
controls the current/bias.
Voltage (V)
1.50
1.48
1.46
1.44
0
20
40
60
80
100
120
Time (s)
2.4.1
Figure 2.7: During etching with constant current the
varying voltage is measured.
There is a transient period at
the start which after a few seconds goes over into an irregular saw tooth pattern. This oscillation may be linked to oxide build up and etch in the
pores, however, this is not well
understood.
Optimizing the etch setup
The geometry of the etch cell will influence the current density distribution
in the sample. In an effort to optimize the etch cell for homogeneous etching, a simplified 2D finite element method (FEM) calculation was done in
Femlab [49] to understand the current density distribution. Three factors
were tested, the geometry of the top part of the cell, i.e. the electrolyte
reservoir, the size of the back-contact and the influence of an opening in the
back-contact. The sketch in Fig. 2.8b) shows the cell geometries tested. The
wide, funnel-shaped cell has a top opening of 2 cm diameter. The sample
opening for both designs is 1 cm diameter, while the height is 2 cm for the
funnel design and 4 cm for the straight reservoir design. For the FEM calculations, an electrostatic model described in Cartesian coordinates was used.
The geometries consist of the sample, with the measured resistivity value,
electrolyte, with a measured resistivity value and an isolating cell material.
Only the potential and current density distributions based on different geometries and voltages were considered in the calculation, disregarding all
17
chemical and electrochemical effects. The results from these calculations
are rough approximations as the potential drop across the Helmholz-layer
close to the Si-surface will normally be quite large. This probably results in
an overestimate of the significance of the placements of the electrodes and
of the cell geometry, however, the trends shown should still be valid.
Figure 2.8a) shows the current density distribution at a depth of 20 or 50 µm
in the sample from the center out to the side of the etched area. The top
plot shows a comparison between the two cell geometries. There is a clear
inhomogeneity of the current density when the wide cell is used. In this case
the modeled samples had back-contacts covering the whole back-side and
the current density was calculated for a depth of 20 µm. The middle plot
shows a comparison between two samples having back contacts with the
same size as the front opening. One of the samples has a 0.5 mm opening in
the center of the back contact to make possible interference measurements as
will be described in Sec. 4. Current densities were calculated for a depth of
50 µm. It is clear that the influence of the back contact opening is significant
at this depth. The selected opening size is the minimum obtainable due to
mechanical constraints in the fabrication of the back plate.
The bottom plot shows a comparison of two samples with wide and narrow
back contacts, i.e. the back contact has the same size as the front opening.
The homogeneity of the current density is significantly better in the case of
the narrow back contact. The current density distribution close to the edge
of the electrolyte in the two cases may also be seen in the cross-sections
shown in Fig. 2.9. In the case of the narrow back contact, the current
density distribution will be more homogeneous with depth also. As can be
seen in Fig. 2.9, the current spreads out more towards the back contact in
the wide contact case. There will be a certain under-etching due to the
spreading of current in the sample. The effect of this is a slight decrease in
current density with depth which may be compensated for by an increase
in the current with time.
2.5
Etch setup for graded filter etching
To make optical filters where the filter characteristics are dependent on the
position on the filter, a voltage was set up between two contacts, 1.2 cm
apart, on the back side of the sample. This resulted in a position dependent
current density during etching with in turn gave a change in refractive index
modulation with depth at different positions. Some results on etched filter
structures using this effect are presented in Paper V. The etch cell used
for this is schematically shown in Fig. 2.10 and is basically the same as in
Fig. 2.6 with a difference in the back contact. Two thin Cu-sheets with a
spacing of about 12 mm were used instead of the single Cu-plate. A FEM
simulation of the current density in the sample at a depth of 2 µm for a
typical set of parameters is shown in Fig. 2.11. The model used is similar
18
A
24
B
Current density (a.u.)
22
20
C
24
D
22
20
C
24
E
22
20
0
1
2
3
4
5
Position (mm)
a)
b)
Figure 2.8: a) A comparison of current density profiles is shown for different etch cell and back contact geometries from FEM calculations of a 2D
electrostatic model in Femlab. The top figure shows profiles calculated at a
substrate depth of 50 µm while the two figures below are calculated at 20 µm.
The profiles are plotted from the center of the etch area to the edge of the
reservoir. The top figure shows a comparison between a cell with a funnel
shaped reservoir and a back contact wider than the electrolyte contact area
(curve A, the geometry to the right) with a tall and narrow reservoir geometry with a back contact the same size as the front contact area (curve B,
the geometry to the left). The middle figure shows a comparison of two profiles obtained with the narrow geometry with the same narrow back contact
(curve C) save for a 0.5 mm opening for in situ IR reflectance measurements
(curve D). The bottom figure shows a comparison between a narrow (curve
C) vs. wide (curve E) back contact with the tall reservoir geometry. Figure
b) shows the cell geometries.
to that in Sec. 2.4.1 The potential difference across the sample in this case
is 0.2 V while the potential difference between the back contact with the
highest potential and the Pt-cathode is 2 V. The conductivities used for the
materials in the simulation are realistic, although not necessarily measured.
19
2
Reservoir wall
2
Electrolyte reservoir
0
Position (10−1 mm)
Position (10−1 mm)
Electrolyte reservoir
Reservoir wall
1
1
−1
−2
Substrate
−3
−4
−2
Substrate
−3
−4
−5
−5.4
0
−1
−5
Back−contact
−5.2
−5
−4.8 −4.6 −5.4 −5.2
Position (10−1 mm)
−5
−4.8
−4.6
−5.4
a)
Back−contact
−5.2
−5
−4.8 −4.6 −5.4 −5.2
−1
Position (10 mm)
−5
−4.8
−4.6
b)
Figure 2.9: Screen shots from a Femlab calculation showing the current
density distribution and direction for the narrow (a) and wide (b) back contact with a narrow electrolyte reservoir. The effect of the back contact geometry is larger towards the edges of the electrolyte contact area and towards
the back contact.
Figure 2.10: The etch cell for
the graded samples is basically
the same as in Fig. 2.6. However, the back contact is split
in two with a gap between.
A constant bias is applied between these two contacts. One
of the contacts is connected to
the computer controlled current source.
30
Current density (a.u.)
Figure 2.11:
The current density profile calculated
20 µm into the sample from
the electrolyte-sample interface. The two back contacts
are biased to 2 and 1.8 V while
the Pt electrode is grounded.
The calculation is made with
a 2D model in Femlab. In
this calculation the electrolyte
is assumed to be a conducting
material and the electrochemistry is disregarded.
25
20
15
10
-5
-4
-3
-2
-1
0
1
Position (mm)
2
3
4
5
Chapter 3
Thin-film calculations
For many years multilayer dielectric thin film optical interference filters
have been used [50]. Obtaining interference effects by using layers of optical
thickness in the order of the wavelength of the light of interest have been
exploited in many applications. In this thesis multilayer and inhomogeneous
structures in PS are used to some extent to control light. Interference
effects with electromagnetic (EM) fields are obtained only with very specific
structures. Layer optical and physical thicknesses or layer refractive index
modulation periods must be in the order of the wavelength at which one
wants interference. In the visible and near infrared wavelength regions this
translates to structures with elements down to below 100 nm with strict
demands on size accuracy and definition. It is very important to understand
the optical properties of the material, which in this case is PS, to be able
to design such interference systems. In the first section the correspondence
between the porous structure of PS and refractive index will be discussed.
As the correspondence between PS fabrication parameters and obtained
structures is not trivial, a detailed knowledge of the intended structures and
their optical response is needed to be able to understand the actual response
from fabricated structures. The background for calculation of interference
filter responses will be presented in Sec. 3.2. In addition to the most basic multilayer optical system a few physical effects are taken into account.
These include dispersion, absorption and interface roughness. The incorporation of interface roughness into optical response calculations is described
in Sec. 3.3. This will be used both in optical filter calculations as well as
in the analysis of in situ laser reflectance measurements described in detail
in Chap. 4. In the last section, different multilayer and inhomogeneous
refractive index optical filter structures will be presented together with
calculations of optical responses based on the methods described. These
structures are experimentally realized and presented in Chap. 5.
21
22
3.1
Effective medium theory
A porous medium will exhibit different optical properties than the same
material in bulk. If the typical feature sizes (e.g. pore size) are much smaller
than the wavelengths of the incident electromagnetic field, the field in the
porous medium encounters an effective dielectric function. This effective
dielectric function is dependent on the dielectric functions of both the bulk
material and the filling material (e.g. air) in a ratio controlled by, amongst
other parameters, the porosity.
The theory describing the dielectric function of the mixed media is referred to as effective medium theory. There are several prominent effective
medium formulas, e.g. Bergman [51], Maxwell-Garnett [52], Looyenga [53]
and Bruggeman [54]. The main difference between these formulas lies in
how the microtopology of the pores are taken into account. The optical
response of a porous medium will change with the degree of ”connectedness” (percolation strength) of the network and the sizes of the segments of
material left in the medium. The dependence on microtopology makes the
problem of finding a correspondence between porosity and effective dielectric function non-trivial. Maxwell-Garnett, Looyenga and Bruggeman all
assume certain microtopologies resulting in a more or less limited validity
when considering the effective dielectric function of PS as the microtopology is greatly dependent on formation parameters. An example of this is
reported by Setzu et al. [47] where the same porosity obtained by different
formation parameters gives significantly different refractive indexes.
The Bergman formula is general and takes into account the microtopology
by a spectral density function, however, this function is usually not known
and must be expressed specifically for all the different microtopologies of
different PS films, hence this approach is quite involved. The use of effective medium theory to describe the optical properties of PS is extensively
discussed by Theiß in Ref. [55, 56]. Figure 3.1 is taken from [56] and shows
a comparison of several different effective medium formulas.
Figure 3.1: Comparison of
different effective medium formulas giving refractive index
as a function of porosity. The
plot is taken from Ref. [55].
Notably the most cited effective medium formula in association with PS
23
is the Bruggeman formula, often referred to as the Effective Medium Approximation (EMA). The popularity of the EMA is based on a paper by
Aspnes [57] where spectroscopic ellipsometry measurements were performed
on rough amorphous Si. The model describing the roughness which gave
the best fit was the EMA. The EMA is also used in this thesis for all refractive index-porosity calculations. The samples presented in this thesis
should have a narrow range of topologies caused by a narrow range of substrate resistivities and a relatively narrow range of HF concentrations in the
electrolyte. Any error introduced by the choice of the EMA as an effective
medium formula will then be systematic and may be easily adjusted for.
The Bruggeman formula is given by:
p
− eff
M − eff
+ (1 − p)
=0
M + 2eff
+ 2eff
(3.1)
where p is the porosity, , M and eff are the dielectric functions of the
embedded material (Si), the host material (air/vacuum) and the effective
medium (PS) respectively.
By using the EMA for all porosities and PS structures, it is clear from the
preceding discussion that for most situations there will be some error in the
calculation due to different microtopologies. From the literature it seems
that this error is small and, in most cases, tolerable for the applications and
measurements in mind. Some papers [58, 59] suggest that the Looyenga formula better describes the dielectric function of highly porous meso-porous
PS. We may calculate the effect of using different effective medium formulas
on the optical response of a simple Bragg reflector. We see from Fig. 3.1
that there is a maximum difference in refractive index of about 0.08 for
80 % porosity, or equivalently a difference in porosity of about 4.5 % for a
refractive index of 1.3. Given a Bragg reflector designed for a reflection
band around a wavelength of 600 nm, the change introduced by going from
the Bruggeman to the Looyenga effective medium formula will result in a
shift of the reflection band (given unchanging layer thicknesses) of about
40 nm. This error may be critical for some types of applications, hence the
choice of model, and possible corrections, must be kept in mind.
If we use a complex, frequency dependent dielectric function (dispersive
media), ˆ(ω), the absorption in the material will be taken into account.
The effective medium models may be used with these complex values. The
complex dielectric function is defined in the following as well as its relation
to the complex refractive index, n̂(ω). We assume for all calculations that
the materials are non-magnetic, i.e. the magnetic permeability, µ, equals 1
(from [60]):
ˆ(ω) = r (ω) + ii (ω)
n̂(ω) = n(ω) + ik(ω)
p
ˆ(ω)µ
n̂(ω) =
(3.2)
24
Solving the equation for n̂ analytically is not necessary. Note that the frequency dependence will be considered implicit in the following discussion.
The real refractive index, n, and the extinction coefficient, k, are usually
referred to as the optical constants of a material. By solving the Maxwell
equation for a plane wave using the complex dielectric function and considering the field intensity, I, the attenuation of the field as it is transported
in a medium is evident:
4πk
→
λ0
I = I0 e−αz
absorption coefficient
α =
(3.3)
where z is the distance traversed by the field in the medium. By using a
non-optimal effective medium formula with the complex dielectric function,
both the resulting effective refractive index and extinction coefficient, hence
absorption, will be inaccurate, see [55] for a thorough discussion of this. As
the absorption in silicon decreases with increasing porosity, the use of PS
for optical filters in the visible is made possible.
The different dielectric functions used in the calculation of a PS effective dielectric function or refractive index are taken from the literature. For air, a
constant value of 1 is used for both dielectric function and refractive index.
The dielectric function of crystalline Si as a function of EM field wavelength
or energy is tabulated in several reviews. One standard reference for Si optical constants is a collection made by Palik [61]. Values from this reference
will be used for calculations in this thesis. The value from Ref. [61] are for
intrinsic Si. The relative change in extinction coefficient with increasing
doping is very small for energies above the band gap, however, for energies lower than, but close to, the band gap, there is a small but significant
effect of doping due to increased free carrier absorption. Data extracted
from Ref. [62] are used to adjust the data from Palik to better reflect the
situation in the material used. The tabulated data for the refractive index
used in the following calculations are plotted in Fig. 3.2.
7
0.4
6
0.3
5
0.2
4
0.1
0.0
3
0.4
0.6
0.8
4
6
Wavelength ( m)
8
10
Extinction coefficient, k (imaginary)
Refractive index, n (real)
0.5
Figure 3.2:
This shows
the tabulated data from Palik [61] of the refractive index (blue line - left axis) and
the extinction coefficient (red
line - right axis) used in all
reflectance and transmittance
calculations.
25
3.2
Reflectance calculation
Two equivalent ways of mathematically defining an optical multilayer system will be presented here. Both of these methodologies will be described
and used as they have different advantages. For these calculations we will
assume that the material is homogeneous for each layer, i.e. the refractive
index is constant and identical in all directions: n̂(x, y, z) = n̂, and that the
planes are parallel.
3.2.1
Characteristic matrix
A good derivation of the characteristic matrix approach is given
in [63](Chapter 1). A short summary will be given in the following.
Each layer in a multilayer system may be represented by a ”characteristic
matrix”, M, describing its optical properties. To obtain the optical characteristics of a stack of layers, a matrix multiplication is performed with the
characteristic matrixes of all layers. The normal to the stacks is along the
z−axis. To relate the EM field vectors on both sides of a stack of i number
of layers we get:
Q0 = M1 M2 · · · Mi Q = MQ ,
(3.4)
where Q and Q0 contain the x- and y-components of the EM field at position
z and z0 = 0, respectively. The characteristic matrix describing one layer
in a multilayer system for a transverse electric (TE) field (s-polarized) is
given by


i
− sin (k0 ni di cos θi )
 cos (k0 ni di cos θi )
pi
 .

(3.5)
Mi (di ) = 

−ipi sin (k0 ni di cos θi )
cos (k0 ni di cos θi )
p
Here pi = (ˆi /µi ) cos θi and k0 = 2π/λ0 , where λ0 is the incident wavelength in vacuum, and di is the layer thickness. To obtain the same characteristic
p matrix in the case of a transverse magnetic (TM) field (p-polarized),
qi = (µi /ˆi ) cos θ is used instead of pi . With the components of the total
characteristic matrix of the stack denoted by


m11 m12
 ,
M=
(3.6)
m21 m22
the reflection and transmission coefficients of the system are then given by
r =
(m11 + m12 pl ) p1 − (m21 + m22 pl )
(m11 + m12 pl ) p1 + (m21 + m22 pl )
,
(3.7)
t =
2p1
(m11 + m12 pl ) p1 + (m21 + m22 pl )
.
(3.8)
26
The reflectivity and transmissivity are given by
R = |r|2 ,
pl 2
T =
|t|
,
p1
(3.9)
where p1 and pl are for the first and last layers, respectively. A sketch of the
described layered system with notation for both the characteristic matrix
approach and the admittance matrix approach described below is shown in
Fig. 3.3
Figure 3.3: The system described in the text, with the
used notation. Usually light is
considered to travel from left
to right.
To calculate the reflection of transmission spectrum, the equations in Eq. 3.9
must be calculated for the wavelength range of interest. This description of
the transfer matrix method is the easiest to implement, but also the least
flexible. The alternative method presented below gives identical results with
the same assumptions, but has the added advantage of easy implementation
of interface roughness.
3.2.2
Admittance matrix
A different approach to describe an optical system of dielectric layers is to
start with the optical admittance. The description presented below is taken
from Knittl [64] and Mitsas and Siapkas [65]. It easily facilitates the direct
introduction of the Fresnel coefficients of the interfaces, hence the addition
of factors describing interface roughness as will be shown. The notation
is such that the ith interface corresponds to the ith layer just to the right
of the interface and the interfaces are numbered left to right. From the
boundary conditions of an interface between two media, the characteristic
optical admittance of a medium may be defined as
Y =
H tR
H tL
=−
,
E tR
E tL
(3.10)
where H tR and E tR are the tangential vector components of the incident
magnetic and electrical field respectively, both going right. The same goes
27
for the field going left. For each polarization this becomes
Ys = −n cos θ
n
Yp =
.
cos θ
(3.11)
The boundary conditions of the interface between the layers i and i − 1
using Eq. 3.10 becomes
E i,interface = E Ri + E Li = E Ri + E Li
H i,interface = Yi−1 E Ri − Yi−1 E Li = Yi E 0Ri − Yi E 0Li ,
(3.12)
where the prime denotes that the quantities are on the right side of the
boundary. This may be written in matrix form, relating the field amplitudes
to the right and the left of the interface;
0 E Ri
E
Vi−1
= Vi Ri
,
(3.13)
E Li
E 0Li
with
1
1
Vi =
,
Yi −Yi
This can be rewritten
0 E Ri
E
= Wi−1/i Ri
,
E Li
E 0Li
Vi−1
1 Yi−1
=
.
1 −Yi−1
(3.14)
−1
Wi−1/i = Vi−1
Vi .
(3.15)
The matrix Wij is known as the refractive matrix. In this case we work
from the rightmost layer toward the left, although the incident light will
usually come from the left. However, the expressions are general and light
may come from the left and/or the right. By finding the Fresnel coefficients
from the admittance we get
ci−1/i 1
rLi
,
(3.16)
Wi−1/i =
tRi ri tRi tLi − rRi rLi
where rLi and tLi are the reflection and transmission Fresnel coefficients,
respectively, of the ith interface with the incident field coming in from the
right. ci−1/i is chosen for the correct polarization and is given by
cos θi−1 / cos θi
for p-polarization
ci−1/i =
.
(3.17)
1
for s-polarization
The discussed expressions only relate to what happens at the interfaces,
however the fields at each ”end” of a layer are not independent. This can
be expressed as
0 iφ
E Ri
e i
0
E R,i+1
E R,i+1
=
= Ui
(3.18)
E 0Li
0 e−iφi
E L,i+1
E L,i+1
28
Ui is called the phase or propagation matrix. The phase-shift, φi , is given
by
2π
ni di cos θi
(3.19)
φi =
λ0
To obtain the total field transformation of a layered system, with layers
from 1 to i, the matrices are multiplied to give, in the general case:
0
E R,i+1
E R1
= W01 U1 W12 U2 W23 . . . Wi−1,i Ui Wi,i+1
E L1
E 0L,i+1
0
E R,i+1
(3.20)
= S
,
E 0L,i+1
Here S is the system transfer matrix, and
s11 s12
S=
.
s21 s22
(3.21)
The system transfer matrix transforms the tangential components of the
incident fields, from both sides of the layer stack, at one end of the stack
to the exiting field at the other end. To find the Fresnel coefficients of the
system we use the definitions which gives
rRi
E Li
=
,
E Ri
rLi
E 0Ri
,
=
E 0Li
tRi
E 0Ri
=
,
E Ri
tLi
E Li
=
.
E 0Li
(3.22)
Together with Eq. 3.20 we get for the system:
rR =
s21
,
s11
tR
=
1
,
s11
(3.23)
rL
s12
= − ,
s11
tL
det S
=
.
s11
By using the complex refractive index, absorption will be taken account of
with this method also.
As will become clear in the next section, it is quite simple to add roughness
coefficients to this description. These coefficients are added as pre-factors
to the Fresnel coefficients and takes into account the roughness at each
interface, given certain assumptions.
3.3
Roughness calculation
Roughness in the context of this thesis is based on the height function of the
interface of interest, h(x, y). We assume that the xy plane is parallel to the
29
sample surface. This function describes the deviation of the interface from
a perfectly flat surface. It is more practical to work with a single number
for the roughness than the height function, so we define an interface height
average. It is assumed that the interface height function is isotropic, hence
it is enough to find the average in one direction, which may then be given
by
Z
1 L
h(x)dx.
(3.24)
a=
L 0
The value used for roughness characterization is usually the root mean
square (rms) value of the height function, σ, given by:
s
σ=
1
L
Z
L
(h(x) − a)2 dx.
(3.25)
0
L is a characteristic distance, e.g. scan length. This length is quite important as average values (large radius bending of surface) and rms values
(different spatial period of roughness/fluctuations) may change considerably with the scale of L. Roughness may be measured by several methods,
i.e. stylus profilometry, white light interferometry or optical scattering (diffuse and specular). Pertaining to this thesis, the consequences of roughness
on the optical field is of most importance, both for determining roughness
and also for evaluating the effect of roughness. A method to measure the
roughness evolution in situ during PS etching, based on the theory described
here, has been developed and will be presented in detail in Chapter 4 and
in Paper I.
3.3.1
Davies-Bennett theory
A much used theory describing the effects of a randomly rough surface on
the propagation of an EM field is the theory described by Bennett and
Porteus [66]. This theory is based on work by Davies [67] who modeled
the scattering of perfectly conducting random rough surfaces. Bennett and
Porteus modified this theory to include materials of finite conductivity,
hence the theory is often referred to as Davies-Bennett theory. These papers
only discuss normal incidence reflection from rough surfaces, but the theory
is easily expanded to include transmission [68] as well as EM fields incident
at oblique angles.
The principle is that of the rough surface introducing a fluctuation in the
reflected or transmitted phase of the field such that when the field intensity
is measured there will be destructive interference between different parts
of the field front. It should be noted that the effect of roughness on the
intensity depends on the area which is illuminated. Interface height fluctuations typically have a correlation length describing over what length scales
the fluctuations occur. If the EM field intensity measured is from a smaller
30
area than covered by the correlation length of an interface, the phase fluctuations will be minimal. On the other hand, if the area measured is larger
than that covered by the interface correlation length, the phase fluctuations
will be stronger. A schematic representation of the principle is shown in
Fig. 3.4
Figure 3.4: The roughness of
an interface between two media of different refractive index
will introduce perturbations in
the wavefront. The characteristic measure of the roughness
is the root-mean-square height
from a flat base plane.
The phase difference between two different parts of the wavefront reflected
from a rough surface where there is a height difference, ∆h, between the
positions of the surface the wave strikes, in a medium with refractive index
n, is given by
2π
2n∆h cos θ.
(3.26)
δr =
λ0
For the ”average” phase difference over the surface area of interest, ∆h is
replaced by the rms height difference, σ, as defined in Eq. 3.25. In Eq. 3.26
it is assumed that ∆h < λ, from this follows that σ λ. For σ & λ the EM
field will be fully incoherent and the measured intensity follows a different
relation to σ than for a partially coherent field. In addition, a Gaussian
interface height distribution is assumed and a plane wave EM field.
The phase difference results in a modification of the Fresnel reflection coefficient of
1 2
r s = r 0 e − 2 δr ,
(3.27)
following [67, 66]. Here r0 is the reflection coefficient of a smooth surface.
In the case of transmission from medium 1 to 2 the phase difference and
modified Fresnel transmission coefficient are
δt =
2π
∆h (n2 cos θ2 − n1 cos θ1 ) ,
λ0
(3.28)
31
1 2
ts = t0 e− 2 δt .
(3.29)
This results in the following adjusted Fresnel coefficients which will be used
in simulations with the proposed methods in 3.2:
0
0
rRi = rRi
exp −2 (2πσi ni−1 cos θi−1 /λ0 )2 = αrRi
0
0
rLi = rLi
exp −2 (2πσi ni cos θi /λ0 )2 = βrLi
tRi = t0Ri exp −1/2 (2πσi /λ0 )2 (ni cos θi − ni−1 cos θi−1 )2 = γt0Ri
tLi = t0Li exp −1/2 (2πσi /λ0 )2 (ni−1 cos θi−1 − ni cos θi )2 = γt0Li
(3.30)
The optical power lost in the specular direction (both reflected and transmitted) because of roughness is regained in the diffuse scattering, i.e. scattering in other directions than specular. For the calculations done in this
thesis, only specular reflection and transmission is considered.
As the roughness, as discussed above, will be incorporated into a calculation of reflectance/transmittance of multilayer structures, the correlation
of interface profiles from one layer to another must be considered. It is
reasonable to assume, due to the nature of PS fabrication, that there will
be a certain degree of correlation of roughness between layers. This would
somewhat decrease the effect of interface roughness compared to fully uncorrelated interface profiles. However, the refractive index contrast between
layers is smaller than between the layer stack and the substrate which results
in a greater effect of roughness at the PS-substrate interface than elsewhere.
This is especially true for rugate filters where the refractive index contrast
between layers is very small. Due to the simplicity of incorporation and the
relative small difference the two options should make on the outcome, only
the effect of fully uncorrelated interface roughness scattering/incoherence
is incorporated into the calculations.
3.4
3.4.1
Optical multilayer interference filters
Discrete, homogeneous layers
The simplest type of a multilayer thin film interference filter is a Bragg
reflector or a Bragg stack. It is based on a stack of paired layers, where
each layer satisfies the Bragg condition,
nd = λ0 /4 ,
(3.31)
with one layer having a low refractive index, nL , and the other layer having
a high refractive index, nH . A compact way of denoting the design is
HLHLHL . . . HL = (HL)i , with i being the number of pairs. The thickness
of the layer is d while λ0 is the wavelength of the incident EM wave in
32
vacuum for which the maximum reflection will occur (design wavelength).
Plane waves are assumed. This condition results in the reflected EM wave
at the design wavelength constructively interfering in each layer and one
may reflect up to 100 % of the incident energy. The reflectance of a Bragg
reflector designed for a maximum reflectance at a wavelength of 1550 nm
is shown in Fig. 3.5. In this case the stack consists of 10 layer pairs. The
material used is 50 % and 80 % porosity PS. All the calculations in the
following are based on the discussion in Sec. 3.2.
1.0
1.0
0.8
0.8
Reflectance (abs)
Reflectance (abs)
By increasing the contrast, i.e. the difference between the high and low refractive index, one may widen the reflected wavelength band. By increasing
the number of pairs in the stack, the reflection band edges will be sharper,
as shown in Fig. 3.6. The stack is similar to that used in Fig. 3.5 except
a stack of 20 layer pairs was used. In both cases interface roughness, absorption and dispersion were disregarded. In most applications the ideal
reflector would be one where there is no reflection outside the band and
100 % reflection within the band, i.e. a square reflection band.
0.6
0.4
0.2
0.6
0.4
0.2
0.0
0.0
1000
2000
3000
4000
Wavelength (nm)
Figure 3.5: Calculated reflection
spectrum of a Bragg reflector consisting of 10 layer pairs of 50 % and
80 % porosity on a Si-substrate, the
designed band is centered at a wavelength of 1550 nm. Absorption, dispersion and interface roughness are
not taken into account.
1000
2000
3000
4000
Wavelength (nm)
Figure 3.6: The design is comparable to Fig. 3.5 except it consists of 20 layer pairs. Note the
sharper band edges due to the increased number of layers. Absorption, dispersion and interface roughness are not taken into account.
The layers may be combined in any way, and many types of filters are
possible for many different uses. Anti-reflection coatings, polarizers, beamsplitters and band-pass filters may all be made by use of multilayer thin
films. A Fabry-Pérot band-pass filter is obtained if a spacer layer is introduced between two mirrored Bragg stacks. An example of this is the
structure (HL)i (LH)j where the spacer consists of two L layers. This will
give a sharp resonance peak in the transmission spectra at the design wavelength.
33
3.4.2
Inhomogeneous layers
The types of optical multilayer filters mentioned above are based on discrete layers with homogenous refractive indexes and uniform thicknesses.
It is possible, however, to generalize this to layers of controlled inhomogeneous refractive index (both in depth and laterally). The process of continuously varying the refractive index of a layer with depth has until recently
been quite difficult. There are a few systems in which this is possible,
such as plasma enhanced chemical vapor deposition of silicon oxynitride
(SiOx Ny ) [69] where the refractive index changes with the stoichiometry
which may be controlled with the ratio of formation gases. Other methods include magnetron sputtering deposition with variable control of the
formation gases [70] or deposition of quasi-inhomogeneous layers consisting of many, very thin, homogeneous layers [71], glancing angle deposition
to fabricate porous structures of depth dependent porosity [72, 73], and
codeposition of two oxides by evaporation [74]. However these systems are
usually relatively expensive and complex and have a limited range of refractive indexes available.
With PS, a practically obtainable refractive index contrast in the NIR of
about ∆n = 3.0 − 1.2 = 1.8 is possible and the method is comparably
straightforward and inexpensive. This flexibility of PS enables both novel
optical elements in intimate connection to Si technology as well as a testing
bed for novel stand-alone optical filters. There are several arguments for
fabricating optical elements with inhomogeneous refractive index. By avoiding internal interfaces, the structure becomes mechanically more stable, e.g.
against scratches, and the probability of delamination is reduced. Fewer interfaces also result in a decrease in the scattering of the EM field within the
structure. One typical use of inhomogeneous refractive index layers is in
anti-reflection coatings. Another possibility is varying the refractive index
sinusoidally with layer depth. This results in a rugate reflection filter. The
refractive index profile is given by
2πz
np
sin
+φ ,
(3.32)
n (z) = na +
2
na d
with the design wavelength equal to 2na d. Here na is the average of the
maximum (nH ) and minimum (nL ) refractive index used in the layer, np is
the peak-to-peak difference between minimum and maximum values and d
is the physical thickness period of the refractive index sine profile, z is the
depth in the layer and φ is the phase. The resulting spectral characteristics
of the filter are quite similar to a Bragg reflector. However, in the case of
small np , there are no higher order harmonics (at normal incidence) and by
further exploiting a continuous variation in refractive index one may reduce
sidebands as well. To avoid higher harmonics in the reflectance spectrum for
larger modulations in the refractive index profile, Eq. 3.32 must be modified
so that the exponential of the sinusoidally modulated refractive indexes are
34
obtained [29, 75]:
ln nH + ln nL ln nH − ln nL
2πz
n (z) = exp
+
sin
+φ .
2
2
na d
(3.33)
This ensures that the optical thickness of the positive sine half is not larger
than the negative sine half, hence there will a be perfect match between
the structure and the incident EM wave. By using Eq. 3.32 with relatively
large np there will be higher order harmonics in the reflectance/transmission
spectrum. Similar to Bragg reflectors the reflection band of a rugate filter
will widen with a greater refractive index contrast (nL to nH ) and the
number of periods gives the ”quality” of the reflection band.
By adding an index matching region between the filter and the main interfaces of air and substrate, the ”base” reflectance is reduced. By apodizing,
i.e. adding a windowing function to the refractive index profile, the sidebands are reduced. The index matching consists of a smooth transition in
the refractive index between two media, i.e. from air to the filter ”medium”,
and from the filter to the substrate, where the filter refractive index is considered to be na . There are many possible gradient functions, but one
which has been shown to give good results is the 5th order polynomial
(quintic) [75, 76]:
n = nL + (nH − nL )(10t3 − 15t4 + 6t5 ),
t ∈ [0, 1],
(3.34)
where t is a normalized parameter proportional to depth. There are many
windowing functions that may be used for apodization, e.g. triangular,
sine, Gaussian, polynomial and other windowing functions used in signal
analysis.
Figure 3.7 shows the calculated reflectance of a rugate filter designed for
maximum reflection at 1550 nm. In this case, to show the ideal reflectance,
the outer and substrate media have the same refractive index as the filter
average index and no dispersion, absorption or roughness is taken into account. The filter consists of 20 periods with no apodization. Maximum
and minimum refractive index correspond to a porosity of 50 % and 80 %
respectively. Compared to Figs. 3.5 and 3.6 the higher order harmonics are
clearly gone. By adding apodization, the sidebands are completely removed
which is shown in Fig. 3.8. The apodization function used was a quintic
polynomial.
To get more realistic calculations, an outer medium of air and a Si-substrate
were added in Fig. 3.9 as well as dispersion in the substrate and filter refractive indexes. The design parameters are otherwise the same as in Fig. 3.7.
The sideband reflection in this case is comparable to the Bragg reflector,
however, the higher order harmonic is still gone. By adding quintic apodization and index matching to the refractive index profile the sidebands greatly
reduce, but there is a slight increase in the ”base” reflectance compared to
Fig. 3.8 due to the difference in average refractive indexes between air, filter
1.0
1.0
0.8
0.8
Reflectance (abs)
Reflectance (abs)
35
0.6
0.4
0.2
0.6
0.4
0.2
0.0
0.0
1000
2000
3000
4000
Wavelength (nm)
Figure 3.7: Calculated reflectance
spectrum of an ideal rugate reflection filter designed for maximum reflectance at 1550 nm. 20 refractive
index periods were used with no index matching as the outer media
and substrates both had the same
refractive index as the average filter refractive index. No apodization was used. The refractive index
range used corresponded to 50 % to
80 % porosity. Absorption, dispersion and interface roughness are not
taken into account.
1000
2000
3000
4000
Wavelength (nm)
Figure 3.8: The same filter as in
Fig. 3.7, but with a quintic apodization function used on the refractive index profile. The sideband oscillations are practically gone when
apodization is used. Absorption,
dispersion and interface roughness
are not taken into account.
and substrate. This is shown in Fig. 3.10. Narrow filters are obtained by
decreasing the refractive index range. This is shown in Fig. 3.11 which is
based on the same parameters as in Fig. 3.10 but with a refractive index
range corresponding to 50 % to 60 % porosity.
The refractive index profile of rugate filters lends itself to combinations of
different kinds. It is possible to put any number of profiles with different design wavelength in series such that the reflection spectrum will show
corresponding reflection bands [74, 75]. An example of a multiband rugate filter is shown in Fig. 3.12. To obtain this reflectance spectrum three
partly overlapping refractive index profiles are used. Each profile consists
of 40 periods and varies between refractive indexes corresponding to 75 %
and 78 % porosity. The profiles are designed for maximum reflectance at
1000 nm, 2000 nm, and 3000 nm. The calculated spectrum is for the ideal
case with matching outer medium, filter, and substrate and with no absorption, roughness, or dispersion.
The wavelengths in a multiband rugate filter may be so close to each
other that the bands fully or partly overlap resulting in one wide reflection band [77] or a narrow gap transmission band [78], respectively. The
1.0
1.0
0.8
0.8
Reflectance (abs)
Reflectance (abs)
36
0.6
0.4
0.2
0.6
0.4
0.2
0.0
0.0
1000
2000
3000
4000
Wavelength (nm)
Figure 3.9: The calculated reflectance spectrum of a more realistic situation with a Si-substrate and
air as outer medium. The parameters are otherwise similar to those
used in Fig. 3.7. The sideband reflectance is much higher in this case
due to the sharp transitions in refractive index from air to filter and
filter to substrate. Dispersion is
taken into account.
1000
2000
3000
4000
Wavelength (nm)
Figure 3.10: The situation here is
similar to that in Fig. 3.9 but both
quintic index matching and quintic
apodization is used. The sharp refractive index transitions still cause
some sideband reflectance. Dispersion is taken into account.
profiles may also be in parallel or partly shifted relative to each other, i.e.
the sine modulation part of the profiles are added to each other keeping
na as a common base. This results in a thinner filter with the same functionality. However, the total profile is limited by the range of refractive
indexes available. This may be amended by using a partial overlap/shift
of apodized profiles such that the sum of refractive indexes at any position
never goes outside the available range. An example of this is shown in
Fig. 3.13 where two profiles partly overlap. Another possibility of making
thin filters with multiple bands is to add the profiles so that the refractive index range available is exceeded and the resulting profile clipped to
fit within the constraints. How much clipping can be accepted depends on
the tolerance limit of the filter application as this approach will introduce
sidebands and ”noise” in the reflection/transmission spectrum.
With the above in mind, a possible design for a narrow transmission band
rugate filter was designed for use as a graded filter in conjunction with a
strip Schottky detector on the back side of the substrate. This device will be
discussed more in Chaps. 5 and 6. In the calculation of the reflectance spectrum of this filter, shown in Fig. 3.14, both absorption and dispersion were
included. Air and Si-substrate media were used. The filter was designed
with four reflection bands: at 950 nm, 1130 nm, 1450 nm, and 1800 nm,
each refractive index profile consisting of 30 periods with the profiles partly
1.0
1.0
0.8
0.8
Reflectance (abs)
Reflectance (abs)
37
0.6
0.4
0.2
0.6
0.4
0.2
0.0
0.0
1000
2000
3000
4000
Wavelength (nm)
Figure 3.11:
Calculated reflectance spectrum for a narrow rugate reflector similar to the case in
Fig. 3.10 but a variation in refractive index corresponding to porosities between 50 % and 60 %. Due
to absorption and apodization it is
difficult to get unit reflectance with
narrow band rugate filters as in this
case. Dispersion is taken into account.
1000
2000
3000
4000
Wavelength (nm)
Figure 3.12:
Calculated reflectance spectrum of a three band,
narrow band rugate reflector. The
reflector is designed for maximum
reflectance at 1000 nm, 2000 nm and
3000 nm. Each band correspond to
a section in the refractive index profile consisting of 40 periods with
porosities between 75 % and 78 %
nominally.
The total calculated
porosity profile will have a larger
range as the different sections in
the profile will partly or fully overlap. The situation is the same as in
Fig. 3.7 with outer media and substrate being identical, no apodization used, and absorption and dispersion not taken into account.
overlapping. Each profile was apodized with a quintic function and the refractive indexes corresponded to a porosity range of 50 % to 80 %. A quintic
index matching was used with the maximum and minimum refractive index
corresponding to what is obtainable for the etch setup presented earlier in
Sec. 2.4.1. The order of the profiles with respect to the surface is significant
for the reflectance due to the absorption. The physically thinnest profile
should be at the top resulting in more equal significance of each profile compared to the opposite case. In Fig. 3.14 absorption values from Palik [61]
are used, according to Sec. 3.1, for curve C (red) and a 10 % fraction of these
to see the incremental effect are used for curve B (green). The transmitted
spectrum in the latter case is shown in the inset. A narrow band centered
at 1290 nm is clearly visible.
For all the discussed filter types, the reflection/transmission spectrum is
dependent on the incident angle. The spectral features for Bragg reflectors,
Fabry-Pérot filters and rugate filters will shift towards shorter wavelengths
38
1.9
Refractive index
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
0
5
10
15
20
25
Thickness ( m)
Figure 3.13: Refractive index profile of a double band rugate reflectance
filter. The shown profile consists of two profiles partly overlapping, one for
each band.
with larger incident angle, i.e. blue-shift. This places restrictions on the
usable wavelength range and angular range of a filter. It is possible, however, to device reflection filters having a constant band of reflection for all
angles, as in the PS based omni-directional mirror of Bruyant et al. [79].
39
1.0
A
0.6
0.8
C
0.6
Transmittance (abs)
Reflectance (abs)
B
0.4
0.2
B'
0.4
0.0
1000
2000
3000
4000
Wavelength (nm)
0.2
0.0
1000
2000
3000
4000
Wavelength (nm)
Figure 3.14: A comparison of different parameters used in a calculation of
a four band rugate filter designed for narrow pass band use. The reflection
bands are close to each other so they will overlap except at a narrow band
around 1290 nm. The reflection bands are centered at 950 nm, 1130 nm,
1450 nm, and 1800 nm and result from four refractive index profiles partly
overlapping, each consisting of 30 periods. Both quintic index matching and
apodization are used. Porosities vary between 50 % and 80 %. Curve A (blue)
shows the case when the outer medium, the substrate and the average filter
refractive index matches. Curve C (red) shows the case where dispersion and
tabulated absorption are taken into account, air is used as outer medium
and a Si substrate is used. Curve B (green) shows the situation with an
absorption 10 % of the tabulated values. The inset shows the transmittance
(B’) corresponding to the case with 10 % absorption. The narrow pass band
is clearly visible.
Chapter 4
In situ interferometry
experiment
For the efficient and successful fabrication of optical filters it is essential to
have good control of properties of the deposited material. The most important properties are refractive index (also density/porosity, homogeneity
and absorption) and interface roughness when discrete layers are deposited.
When making optical elements in PS, especially filters, this translates to
having a good (instantaneous) control of etch rate, porosity, roughness and,
to some extent, microtopology as discussed in Sec. 3.1.
Etching of PS is a dynamical exercise. The parameters of the system change
during etching which necessitates an in situ monitoring of the most important parameters so the changes may be counteracted, or enable a real-time
feedback to the etching process. The feedback method is often used to obtain optimum results in other conventional optical material systems [69].
Often feedback in this situation is based on interference effects with monoor poly-chromatic coherent light. Interference at single wavelengths and ellipsometry give very good optical density/physical thickness resolution and,
in the case of interference, is easily implemented in a deposition system.
In the PS fabrication setup already described, the sample is in contact
with a liquid electrolyte on one side. In this case it is easier to set up a
system where the optical probing is done from the opposite side than to
have the optics on the electrolyte side which may be closed off or corrosive.
This presupposes a transparent sample which is obtained when using an
IR laser with a wavelength longer than that corresponding to the silicon
bandgap energy (λlaser & 1130 nm). With this setup the IR laser beam
will encounter several interfaces, i.e. back side, PS-substrate, and frontside, with the resulting partially reflected beams phase shifted thus giving
rise to interference in the measured reflected beam. Similar systems are
described by Steinsland et al. [80] for tetramethyl ammonium hydroxide
etching of silicon and by Thönissen et al. [81] and Gaburro et al. [82] for
PS fabrication monitoring.
41
42
The change in interference conditions during PS etching results in a signal
containing information on relative interface movement and the different
refractive indexes. By performing frequency analysis on this signal it is
possible to obtain several key parameters of a PS film during formation.
The parameters depend on a few assumptions about the formation process
and substrate material. Given the correctness of these assumptions, we
obtain the refractive index and porosity profile of the PS film, and the
time dependent etch rate. From this we can calculate the refractive index
profile of the film in air, average porosity and refractive index, PS-substrate
interface roughness, film thickness and instantaneous valence values during
formation.
Most of the work done with the setups presented here is described in Papers I-III, which are included at the end of this chapter. These include
background theory, analysis, experimental setup, and some results. Some
details not discussed in the papers are included in the following sections.
4.1
Setup
4.1.1
Usage
The goal of this work was to design an in situ measurement setup which
would fit in with a standard etch cell as detailed in Sec. 2.4. As the analysis
of the data obtained from the setup gives all the critical parameters without
further processing of the sample, a quick, non-destructive measurement of
the standard calibration parameters for constant current conditions is made
possible. These data, porosity and etch rate versus time and depth at
constant current, may be used for controlling the etch current to obtain
layers of constant porosity. The setups used are schematically shown in
Fig. 4.1. The main descriptions of the setups are given in Paper I and some
of the concepts in the following discussion are defined in this paper.
In addition to the mentioned material properties, the evolution of interface
roughness is also monitored by measuring the introduced partial incoherence
in the reflected laser beam. The incoherence results in a decreasing signal
intensity with increasing roughness. This is needed to better understand
the optical properties of the interfaces encountered in the multilayer PS
optical filters. By changing formation parameters, an optimal parameter
range may be found for filter fabrication with regards to interface roughness.
This understanding is crucial for optimizing the quality of the PS optical
elements.
It is possible to test the calibration data by calculating the needed current profile for a constant porosity layer and measure the actual obtained
porosity with the setup in situ. An example of this will be given later.
One challenge when using calibration data for a multilayer or inhomoge-
43
neous refractive index structure, is that the conditions at the pore front
depend on the ”history” of the etch. The conditions during multilayer
etching is different from those present at the single layer calibration etch.
This effect may be seen by etching a single layer and changing the current
after a certain time. Comparing the porosity obtained after the current
change with porosities obtained by etching a layer at the same current as
this from the start, a shift in porosity is observed.
The measurement of the back side reflected signal will also give information on any change in refractive index or layer thickness of the porous
layer after the etch current is turned off. As the layer thickness is unlikely
to change, one must assume all change comes from the refractive index.
This only applies if all else in the setup is constant, i.e. no thermal expansion/contraction or movement of the sample relative to the optics. A change
in refractive index after the electrochemical etch has stopped may be caused
by a purely chemical etching of the porous structure. This chemical etching
of silicon by HF is very slow for the bulk material, but due to the large
internal surface area of PS the relative time dependent change in Si volume
may be significant, thus will give a measurable change in refractive index
with the in situ reflectance setup. However, there are several possible explanations to the change in refractive index after current is turned off, such
as relaxation of the structure due to decreasing temperature, electrolyte refractive index change due to diffusion, and oxidation of the structure in the
electrolyte. Similar measurements have been performed by Navarro-Urrios
et al. [83].
It is possible to obtain a complete calibration curve, i.e. porosity versus
current density, by performing a current sweep during etching while measuring the interference signal. However, this will only be an approximation
as it does not take into account the time dependent change in etching conditions. For many applications this approach may prove efficient and accurate
enough. This technique is briefly discussed in Ref. [82].
One possible improvement of the current setup is to extend the data analysis
capabilities to real-time operation, hence facilitate the possibility of feedback to the PS formation. With the knowledge of an approximate range
for the starting porosity value, this technique could prove quite accurate.
4.1.2
Laser
Two laser systems were used. For the free space setup a 1310 nm wavelength laser diode (LD), LD1087, with drive electronics, LDP201, from
Power Technolgy was used. The diode was in a housing containing the
collimating optics. The output optical power was 8 mW and the beam diameter was about 2 mm with a divergence of about 10◦ . By monitoring
the beam the quality of the measurement would increase by increasing the
signal-to-noise ratio. However, this was not necessary for obtaining satis-
44
Figure 4.1: Schematic of the different etch setups. The setup denoted A
is based on free space beam transmission. The laser diode is denoted LD.
The schematics of both setup A and B shows a cross-section of the etch
bath. Setup B shows the wide beam optical fiber variant. MM denotes the
multimode optical fiber and SM denotes single mode optical fiber. A coupler
is shown; this splits the fiber and guides the reflected signal from the sample
to the detector. A graded index lens, as indicated, is used to collimate and
collect the light close to the sample. The third setup, C, shows a fiber variant
where no collimating lens is used, but where the fiber is close enough to the
sample back side (< 500 µm) to collect a significant amount of the light
reflected from the sample. This results in a probed area about the size of
the fiber core cross section.
factory results.
For the optical fiber based setup a pigtailed laser system assembled by
Thorlabs was used. The diode used was of type ML976H11F from Mitsubishi, an InGaAsP, multiple quantum well, distributed feedback diode
laser. The output wavelength was 1550 nm with an optical power of 2 mW
when coupled to the single mode fiber.
As the measurement of the interference effect in the reflected signal is dependent on a constant phase of the input beam, and also, for the calculation
of roughness, a constant amplitude, a well regulated system is needed. The
diode lasers are sensitive to temperature fluctuations and temporal effects.
To keep the output power and phase constant the diode needs to be stable
in a specific laser mode. A small temperature change may introduce a mode
jump which may give a change in phase and also a more noisy output as
there will be transient signals, both in the diode itself, and also in the regulation of the diode. Therefor careful regulation of temperature and output
power was needed. This was done using an ILX Lightwave Technologies
LDC 3722 laser diode driver controlling the current by feedback from a
45
monitoring diode in the laser package. The same driver regulated the diode
temperature with a thermo-electric element connected to the laser diode
housing with thermal paste. The diode lasing mode was very sensitive to
changes in temperature which resulted relatively often in unusable periods
in the measured data. This problem was improved by allowing the laser
warm up for a few hours before use.
The lasing in the diode is also very sensitive to back-reflected light from
the pigtailed fiber. To minimize back-reflectance an optical isolator was
connected to the fiber from the laser diode. By adding a second isolator
the stability of the LD seemingly improved further. Because of a small but
significant time delay between an amplitude change in the LD and an amplitude change in the signal reflected off the opposite end of the connected
fiber to the LD, instabilities in the regulation of the output-power may be
introduced. By introducing a light-”valve” so that the light is not reflected
back to the diode the problem of regulating the power may be reduced.
Even though back-reflectance is very small with two isolators, there may
still be enough to introduce a slight oscillation in the output power which
may explain one persisting artefact in the measurements. A long wavelength oscillation of equal or higher amplitude compared to the measured
interference oscillation was present in all measurements. One possible cause
may be instability in the laser, however, this was most likely ruled out by
monitoring the output signal by using a 2x2 coupler, with a monitoring detector and sample setup opposite the LD, and a reflectance signal detector
on the same side as the LD. This enabled the monitoring of the LD output concurrently with etching to see if the long wavelength oscillation was
present in the output. The conclusion was that the oscillation is introduced
by the etch setup ”arm” of the 2x2 coupler.
The standard design of a polarization insensitive optical isolator uses two
birefringent plates with a magnetic garnet crystal used as a Faraday rotator
sandwiched between. The beam from the fiber is collimated at input and
output by lenses. The birefringent plates are wedge shaped such that the
ordinary and extraordinary rays at the input side are parallel but spatially
separated and the Faraday rotator rotates the polarization plane of each
beam by 45◦ , while the output birefringent plate maintain the spatial relationship between the two beams which are collected and coupled to the
output fiber by a collimating lens. The returning beam will see a system
where the two birefringent plates do not keep the two beams parallel thus
dissipating the beams in the cladding of the isolator. A typical isolation
effect of such a device will be over 40 dB for the optimized wavelength. The
design wavelength depends on the length of the Faraday rotator.
46
4.1.3
Fiber
Due to availability, all connectors for the optical fibers were FC/PC. This
ensured a satisfactory coupling between fibers and relatively little interface
back reflection. Possibly, by using FC/APC couplers the back reflectance
seen at the LD might be reduced somewhat, resulting in a more stable LD
output.
Both multimode (MM) and single mode (SM) fiber were used in the setup.
As the fiber pigtailed to the LD was SM, it would be optimal to only
use SM fibers due to back reflectance at connectors. However, it proved
difficult to couple the light reflected from the sample back into the SM
fibers due to the small fiber core diameter and the small acceptance angle
or numerical aperture. Because of the larger numerical aperture and core
diameter a MM fiber was used on the sample side of the setup. The SM fiber
coupled to the LD was a standard SMF-28 type fiber with a core diameter
of 8.3 µm, while the multimode fiber used had a 62.5 µm diameter core.
Coupling between the two fiber types is not optimal. The core material
may be different leading to Fresnel reflection losses, while the MM fiber
will have many available, unfilled modes which makes the fiber sensitive to
movement/bending etc. as the beam from the SM fiber may change mode
as a response to external influences. To avoid these problems efforts were
made to keep all fibers as rigid and stable as possible by taping fibers and
fastening all measurement equipment to the same metal frame. However, a
short length of the fiber from the 2x1 coupler to the etch setup was free as
the etch setup was inside a flow box with the rest of the equipment outside.
One possible further remedy would be to fill all the modes of the MM fiber
by scrambling the single model from the SM fiber, spreading the beam over
several modes in the MM fiber.
The 2x1 coupler used was connected as shown in Fig. 4.1B. The split ratio
between the two split fibers was 50/50.
4.1.4
Beam to sample coupling
How the ”probe” beam interacts with the sample is critical to the quality of
the measurement. The three different setups used, free-space, wide-beam,
and narrow-beam, all have different advantages which mainly depend on
how the beam couples to the sample.
In the case of the free-space setup, as shown in Fig. 4.1A, the ”probe”
beam goes through air from the LD to the sample to the detector. The
beam propagation direction is controlled by mirrors, and to simplify the
setup the beam is at an angle of about 12◦ to the sample normal. With this
setup there will be no back reflectance to the laser diode, hence, the laser
should be quite stable. However, by traveling through air, the beam may
be distorted by dust particles, changes in air density, moisture content and
47
temperature, thus introducing noise in the measured signal. At the same
time the mechanical positioning of the LD, the detector and the sample
may drift slightly with time introducing amplitude shifts due to misalignment. The beam is easily moved and changed in size. This may be used to
probe different areas of the sample, even scan the whole sample area during
etching by swiping the beam with a high speed beam movement system,
e.g. bar-code scanner setup. By changing the beam size, the same setup
may probe roughness at different spatial scales, although the minimum obtainable beam diameter will be significantly larger than the effective beam
diameter obtained with the narrow beam fiber setup.
It is important for PS layer homogeneity, both in porosity and etch rate,
that the electrical potential in the Si-sample is evenly distributed. This
necessitates a good, and optimally shaped, back contact, even for highly
doped Si-samples. See Sec. 2.4.1 for a discussion on this. However, as the
contact used is non-transparent aluminum a hole must be etched to let
the beam through. Depending on the size of this hole, the homogeneity of
the PS film will be more or less affected. With the beam sizes obtainable
with the free-space setup this will always be a problem. An advantage of
the free-space setup compared with the fiber based setups is that all the
interference comes from interactions in the sample, as opposed to interference between fiber end and sample back side. This assumes that the LD
amplitude and phase is stable or oscillates very slowly compared to the
interference frequencies of interest.
In the wide-beam fiber setup the beam is transported to and from the sample by optical fiber, thus avoiding many of the potential noise sources of the
free-space setup. However, as already noted, movement, bending or temperature gradients may also introduce noise in the fiber, but nonetheless the
beam in the fiber setup is relatively stable. As in the free-space setup, there
is a compromise between beam size and optimal back contact. However in
the fiber case the beam is more rigid. As can be seen in Fig. 4.2, to obtain
a different beam a different graded index (GRIN) lens must be chosen and
fitted.
Once a lens has been fitted, the Cu-plate must be leveled so that the beam
reflected off the sample will be coupled back into the lens. This need only
be done once in a while as there is some reshaping of the materials used
in the sample holder when the sample is clamped down. This setup is
very compact and can be used as a standard addition to the etching of PS.
There are a couple of challenges however. There is always the danger of
leakage of electrolyte, so all parts of the etch setup should be HF resistant.
For the setup used, this means a possible degradation of the the long term
stability of the system as a non-HF-resistant GRIN lens was used as well
as an aluminum lens holder. There is also a possibility of interference
effects introduced in the signal from movement of the sample relative to the
lens, due to either temperature changes or bending of the sample during
etching [84]. This may be one explanation for the long period oscillation
48
observed in the measured reflectance.
By avoiding the GRIN lens setup and only using the cut fiber end, the
problem of PS homogeneity is reduced as only a very narrow hole in the Al
back contact is needed. In this setup a MM fiber is cut straight and placed
as close to the sample as possible. In the used setup a small hole of about
0.5 mm diameter was drilled in a Cu-contact plate in which the bare fiber
end was positioned, see Fig. 4.3.
Figure 4.2: A zoom-in on the coupling between fiber and sample for
the GRIN lens in situ setup. The
Cu-plate with a center hole used as
back contact is shown on top. The
GRIN lens is placed in an aluminum
holder which is shown. There are
screws for adjusting the plane of the
Cu-plate in the top flange of the Al
lens holder so back coupling from
the sample may be maximized. The
fiber is in contact with the end of
the GRIN lens for minimum backreflectance.
Figure 4.3: The bare fiber end
setup uses wax to hold the fiber in
place. As can be seen, the outer
cladding of the fiber is stripped away
close to the fiber end so the diameter of the hole in the Cu-contact may
be as small as possible. This results
in as small as possible influence of
the hole on the current distribution
in the sample.
The fiber was pushed through as far as practical without leaving the end
free on the front side so it could break when the sample was positioned.
This gave a distance between fiber end and sample of < 0.5 mm. On the
other side of the Cu-plate the fiber was fastened with wax. This was stable
enough and also made it easy to fix and recut the fiber in case of damage or
wear to the end, e.g. by HF etching. This setup is able to measure the most
interesting parameters without interfering with layer homogeneity, and as
the case with the wide beam fiber setup, can be an integral part of the
etch setup. As the probed area of the sample in this setup has a diameter
49
roughly the same as the core diameter, assuming relatively flat interfaces,
the roughness measured may have spatial wavelengths smaller than the
probe diameter and thus will not be able to give a broad characterization
of the interface roughness of the sample. However, to get a good idea of
the roughness, several different beam sizes must be used. The alignment
of this setup is very robust as the fiber has a wide acceptance angle, which
means the fiber end may still capture much of the interference even though
it is somewhat tilted. To improve on back reflectance from the fiber end,
the end may be cut at a small angle. However, the results were satisfactory
with a straight cut.
4.1.5
Other equipment
For both the free-space and fiber based setups, the sensor used was a New
Focus 2011 detector with built in amplification. It was based on a InGaAs
PIN diode. This was connected to a multimeter, either a Keithley 199 DMM
or a HP 34970A. This again was connected to a computer. The electrolyte
temperature was measured with a Pt-based thermocouple for all calibration
measurements. Both temperature and signal was then logged by a LabView
program.
4.2
4.2.1
Data analysis
Chemical etching
In addition to etching at the pore tips there will also be some time dependent
etching of the PS layer as discussed briefly in Sec. 4.1.1. This results in a
gradient in the porosity with depth opposite to the gradient observed by
the in situ measurements. The chemical etching results in an increasing
porosity towards the surface which also will increase with time. The rate of
this etching will change with time as the structure of the pores change due
to etching. A possible result of this is show in Fig. 4.4 where the back side
reflected signal is measured both during etching as well as after etch current
is turned off. The point where the etch current is turned off can be seen
where the short period oscillation stops. A relatively slow oscillation is still
present indicating a continuing change in refractive index of the PS layer.
In the upper right plot the oscillation present in the signal after etch current
is turned off is isolated and the slow variations are filtered out. As can be
seen in the upper left plot of this figure, there is a very significant long
period oscillation present throughout the measurement. This oscillation is
most likely caused by thermal gradients, slow movement of the fiber close
to the sample or bending of the sample as discussed in Sec. 4.1.4.
It is possible that the end-of-etching oscillation is caused by these effects,
50
0
50
100
120
125
130
2.0
73.2
73.0
72.8
1.8
72.6
87.6s
410.6s
181.2s
Porosity, %
Time, min
1.04
1.02
1.6
1.00
0.98
1.4
2.0
2.0
1.8
1.8
1.6
1.6
100
110
120
130
Reflectance, a.u.
Reflectance, a.u.
39.4s
140
Time, min
Figure 4.4: Different aspects of a signal recorded during an in situ IR laser
reflection measurement. The sample was etched at about 4.3 ◦ C in a 26 % HF
electrolyte at 30 mA/cm2 for 117 min. The etch setup used was the bare fiber
end setup. The upper left figure shows the whole signal measured during the
electrochemical etching. The bottom figure shows the transition of the signal
from short period oscillation due to electrochemical etching to a longer period
oscillation possibly due to chemical etching. The middle right figure shows
the long period oscillation high pass filtered so peaks are easier to quantify.
The period between peaks is shown above the signal. The upper right figure
shows the calculated change in porosity based on the signal period obtained
from the figure below.
however, the period and amplitude of the the most noticeable oscillation
during etching is quite different from the end-of-etch oscillation. There
are other plausible explanations to the latter besides the external effects
and chemical etching. During etching the current through the electrolyte
and the sample will drive up the temperature, possibly changing refractive
indexes of both the electrolyte and PS layer. When the etch current is
turned off this change will relax back to the values at ambient temperature.
The refractive index of electrolyte may also change due to diffusion. Sirich chemical species will diffuse out of the PS layer while HF will diffuse
51
towards the pore front. Assuming that the end-of-etch oscillation is only
due to chemical etching, the average porosity change in the layer is easily
calculated. The data shown in Fig. 4.4 is measured during formation of a
PS layer with an electrolyte containing 26 % HF at 30 mA/cm2 and at an
average temperature of 4.3 ◦ C. The final etched thickness is 207.6 µm, the
sample was etched for 117 min, while the average porosity of the layer is
72.6 %. The optical thickness difference of the porous layer between two
adjacent oscillation peaks in the upper right plot of Fig. 4.4 corresponds to
half the incident wavelength:
nP S,t1 d − nP S,t2 d = λ0 /2
(4.1)
with nP S,t being the refractive index of the porous layer in the electrolyte
at a time t and d is the layer thickness. This gives a change in the average
refractive index of the layer of about 0.0037 for each period of the oscillation.
The resulting change in porosity is shown in the upper right plot in Fig. 4.4.
4.2.2
Effect of irregular sampling
A necessary condition for obtaining an optimal spectrogram, which will
be explained below, is that the sampling rate is constant throughout the
measurement. The FFT function presumes regular sampling, such that the
sampling time of the data points of irregularly sampled data will be shifted
and thereby introduce errors and noise in the FFT spectrum. Some irregularity is acceptable, and will at any rate most likely be introduced by the
numerical handling of the data. When the sampling period deviation from
the constant/average sampling period is random, the noise introduced in
the FFT spectrum will be white noise, however, when there is a periodicity
in the deviation this may introduce significant artifacts in the FFT spectrum. Due to a discrepancy between the software set timestamp and the
actual measurement time for a set of measurements, spurious signals were
introduced. The periodic timestamp discrepancy probably had a sine-like
time dependence resulting in the spectrogram of Fig. 4.5b. The spectrogram of Fig. 4.5a is obtained from the measurement of a sample fabricated
under nearly the same conditions as for Fig. 4.5b, however, the timestamp
for each data point was set by hardware and the sampling rate was nearly
constant. The spurious partials in Fig. 4.5b are clearly seen, especially
around the partial marked 1.
4.2.3
Frequency analysis
Short time Fourier transform (STFT) is used for analysis of the obtained
signal to give both temporal and frequency resolution. The details of the
analysis is discussed in Paper I. When selecting the parameters for the
STFT analysis, it is important to find an optimal balance giving the best
52
Figure 4.5: A comparison between spectrograms obtained by analysis of
measurements with data timestamped by a) the measurement hardware and
by b) the measurement software. The small periodic error in the timestamps
introduced by the software results in spurious signals in the spectrogram as
clearly seen in b).
result. This is critical for the choice of window, both which window function
and what length it should have. As the partials’ frequencies are likely to
change with time, increasing a window length will work towards broadening
the observed peak, while at the same time the more periods of the partials
represented within the window, the better frequency resolution one should
get. These effects oppose each other, hence for a partial with only a slight
change in frequency with time, represented in Fig. 4.6 as curve B, the effect
of an increase in the number of oscillation periods overcomes the effect of
the frequency change within the window. On the other hand, for curve A
in Fig. 4.6 the frequency change within the window is too large, resulting in
the broadening of the peak width with an increasing window length. The
data in Fig. 4.6 shows the measured peak widths of the two main partials
of the in situ measured interference signal shown in the spectrogram of Fig.
5 in Paper I varying the window width. Curve A corresponds to the partial
marked 1 while curve B corresponds to the partial marked 2. As can be
seen in Fig. 4.6, for a decreasing window width under a certain threshold
value the peak width increases sharply. This is due to the lack of periods
present within the window, hence the determination of a frequency becomes
more uncertain.
The choice of window function depends on which frequency peak features
are most critical. Some window functions will give very sharp main peaks
Figure 4.6: A comparison
of the effect of window function length on the full width
at half maximum (FWHM) of
the peaks corresponding to the
two main partials of the spectrogram in Fig. 5 in Paper
I. The frequency of the partial corresponding to curve B
changes little while the frequency of the partial corresponding to curve A changes
more resulting in change in
FWHM as seen. The FWHM
is taken at 40 min.
Full width at half maximum (a.u.)
53
200
150
A
100
50
B
0
0
20
40
60
80
100
120
Window function width (min)
but will also introduce a high level of noise, or sidebands, not containing any
information of interest. The shape of three different window functions are
shown in Fig. 4.7a, these are the square window marked A, i.e. the result of
not applying any particular window function, the Blackman window marked
B and the Blackman-Harris window marked C. In the case of STFT analysis,
the selected window is multiplied with the signal of a selected range before
the FFT of the product is computed. By computing a FFT of the window
function itself one can see which effect a particular window function will
have on the result of the signal analysis. The FFT of the window functions
in Fig. 4.7a is shown in Fig. 4.7b. In general a window function giving a
very narrow main peak will have high sidebands, while a window function
with a wide main peak will have more subdued sidebands. For the present
analysis, a balance between the two seems the best. We would like to be
able to determine the frequency-trace of a partial as certain as possibly
requiring sharp peaks, but, due to the tracing procedure, the height of the
peak ridge relative to the noise floor should be as large as possible. As
a compromise the Blackman window function was selected for the STFT
analysis.
4.2.4
Etch rate and porosity calculation
4.2.4.1
Measurement of the effect of limited HF diffusion
During constant current etch measurements with the in situ interferometry
setup the porosity and etch rate is observed to change with etched depth.
This is thought to be caused by diffusion restrictions on electrolyte constituents to and from the pore tips changing the conditions of etching. This
effect will also be present when etching multilayers, and it is also likely that,
since this effect is caused by diffusion, the conditions change depending on
the pore structure of the layers already etched. Hence, after a given time
54
6
1.0
10
A
A
3
10
B
0.5
-3
10
B
Power
Amplitude
0
10
-6
10
C
-9
C
10
0.0
-12
0.0
0.5
1.0
0.0
Length
a)
0.5
10
1.0
Frequency, a.u.
b)
Figure 4.7: Three normalized window functions are shown in a); a square
function (curve A), a Blackman function (curve B) and a Blackman-Harris
function (curve C). The resulting Fourier transform of these functions are
shown in b) in the corresponding colors.
the porosity at a given current density will be different in a layer etched at
constant current density compared to a layer etched at varying current density. If one is to calibrate the etching of PS layers based on porosity profile
and etch rate versus time data this change in etching conditions must be
taken into account. If the variation in porosity is small through the layer
this effect may not need to be taken into account, however, if the variation
is significant the difference between designed and obtained porosity profile
may be significant.
An indication of the change in conditions at the pore front depending on
the ’etch history’ is shown in the following experiment. The reflection signal
was measured during the etching of a sample in a 26 % HF solution at 6 ◦ C.
Etching was done first with 40 mA for 15 min then abruptly changed to
20 mA for another 15 min. This change is clearly seen in the spectrogram in
Fig. 4.8 at about 13 min. Note that the time axes in the spectrograms are
slightly offset as the time here denotes the starting point of the windowed
signal for each window position along the signal. Hence, the first power
spectrum plotted in the spectrogram, denoted t = 0, encompasses data from
t = 0 to t = window length. Figure 4.9 shows the calculated porosity of
this sample. Note the large change in porosity within the first 15 min; from
55
56.5 % to 64.5 %. The porosity changes abruptly as expected at t = 15 min.
The second curve plotted after t = 15 min is the porosity obtained after
15 min for a sample etched under the same conditions but with a constant
etch current of 20 mA from t = 0.
The porosity of the dual current etched sample has shifted toward lower
values compared to the constant current etched sample. This may be understood as a consequence of a change in HF concentration assuming a
simplistic view of the electrolyte chemistry at the pore front. This concentration depends both on diffusion of HF to the pore front as well as on
the usage of HF depending on e.g. current density. The constant current
density etched sample has a certain porosity profile, resulting in a certain
diffusion of HF to the pore front, where the diffusion constant will change
with position through the layer. For the dual current density etched sample
the first half of the etching results in a different porosity profile than that
in the sample of constant current density etched after the first 15 min. As
the current is higher in the first half, the porosity will be higher, hence a
higher diffusion of HF will result. From this one may assume that the HF
concentration at the pore front is higher for the dual etched sample after 15
min of etching than for the constant etched sample after the same time. In
p-type silicon, it is well known [45] that a higher HF concentration, given
the same current density, will result in a lower porosity. This gives a good
understanding of the results in Fig. 4.9.
4.2.4.2
Etch calibration
It is possible to use porosity and etch rate time dependence data to calibrate
etching so that a known porosity with a known etch rate is obtained at any
given time of continuous etching by changing the current density correctly.
A constant time step has been used in the calculation of the needed current
density. The current density needed to obtain the desired porosity is found
by interpolating the porosity data at a given time for all the measured
current densities and extracting the current density for the desired porosity.
This is done for each time step until a given film thickness i reached. The
etched thickness for each time step based on the found current density is
also interpolated from the measured etch rate calibration data.
This procedure is based on the assumption that the etch condition at the
pore front is a function of time and instantaneous etch current density only,
independent of the earlier etch conditions. This is only an approximation,
as exemplified by Fig. 4.9.
The data needed for this procedure are obtained by fabricating several PS
layers at constant current condition for different current densities. This
has been done for two separate electrolyte conditions. A low temperature
etch in a 26 % HF solution was used for the results in Figs. 4.10 and 4.12.
The etching was done in a refrigerator with an average temperature of 5 ◦ C,
56
Porosity (%)
65
Current change
40 mA
60
Constant current from
t = 0 min
55
20 mA
50
Current change from
15 min
25
30
40 mA at t =
0
5
10
15
20
Time (min)
Figure 4.8: The spectrogram of the reflectance
measurements of a sample etched with an abrupt
change in etch current,
the shift in the partials
frequencies is clearly visible at around 14 min.
Figure 4.9: The calculated porosity time
profile of the sample used in Fig. 4.8 compared to the porosity calculated for a sample
etched from the start at the changed etch current. There is a notable shift in porosity between the two profiles possibly indicating that
the HF concentration changes with time during etching.
however, the temperature regulation was slow and the temperature in the
electrolyte varied ±3 ◦ C. The results shown in Figs. 4.11 and 4.13 were obtained during etching at room temperature with an electrolyte consisting of
15 % HF and 10 % of the ethanol replaced by glycerol. These results show
the generally accepted trends for PS formation in p-type Si; higher current
density leads to higher porosity and etch rate, and for a given current density the porosity increases with a decreasing HF concentration. Also, clearly
the conditions for etching changes with time. This also follows what has
been reported earlier [85, 81, 82] in p-type PS. The trend is toward higher
porosities and lower etch rates with depth/time. Both observations fit with
a picture of decreasing HF concentration with depth/time due to diffusion
limitations. Note that some exceptionally high porosity values are shown
in Figs. 4.10 and 4.12. Porosity values above ≈90 % are hard to obtain
after drying. Possible reasons for the shown porosity values are the use of
the EMA effective medium formula which may give slightly shifted porosity
values and that highly porous PS may still be mechanically intact during
etching. Highly porous PS is unstable mostly due to the capillary forces
which occur during drying, resulting in breakage of the internal structure.
To test the calibration data, a PS layer was fabricated where the current had
been calculated beforehand to give a layer of uniform porosity throughout
the depth. The back side reflection was measured during this fabrication
57
100
100
40 mA/cm
Porosity (%)
80
15 mA/cm
60
5 mA/cm
40
1 mA/cm
2
2
10 mA/cm
2
2
60
30 mA/cm2
15 mA/cm2
10 mA/cm2
5 mA/cm2
60
40
2
20
40
50 mA/cm2
80
2
20 mA/cm
20
2
2
30 mA/cm
0
70mA/cm
2
Porosity (%)
55 mA/cm
80
100
120
140
20
0
20
Time (min)
Figure 4.10: The obtained porosity time profiles for different current
densities. The electrolyte used is
26 % HF without glycerol and etching is done at approximately 5 ◦ C.
Porosities increase with current density. Plotted are data for 1, 5, 10,
15, 20, 30, 40, and 55 mA/cm2
40
60
80
100
120
140
Time (min)
Figure 4.11: The obtained porosity time profiles for different current
densities with an electrolyte consisting of 15 % HF and 10 % ethanol
substituted with glycerol. Etching is done at room temperature.
Porosities increase with current density. Plotted are data for 5, 10, 15,
30, 50, and 70 mA/cm2 .
so etch rate and porosity could be calculated. The porosity profile is shown
in Fig. 4.14. Due to the short window used in the STFT analysis there
are some irregularities which would be smoothed out with a longer window. The etch duration was designed to give a layer of 10 µm thickness
and the current profile was designed to give a uniform 50 % porosity using
26 % HF at 5 ◦ C. The uniformity of the porosity in Fig. 4.14 shows this
is a feasible way of obtaining uniform layer porosity, however the porosity
measured is roughly 3 % (abs.) below the designed value and the thickness
obtained according to the reflectance measurement was 10.69 µm. The average porosity was measured by gravimetry which gave a porosity of 52.0 %
and a thickness of 9.56 µm. These discrepancies likely show the uncertainty
of the gravimetric method and the error introduced by the choice of an
effective medium approximation as discussed in Sec. 3.1.
4.2.4.3
Possibility of real-time monitoring
As seen in Fig. 4.4 and in the Papers I-III, the measured reflection signal
has a good signal-to-noise ratio. Assuming that the problem of the slow
signal oscillation is likely due to mechanical or thermal instabilities of the
setup/fiber, as discussed, may be resolved, the setup may well be used for
real-time monitoring of parameters or feedback control of etching parameters. In the case of real time monitoring, the use of STFT is appropriate,
however, prior knowledge of porosity and etch rate ranges will increase the
58
4
4
2
Etch rate ( m min )
-1
-1
Etch rate ( m min )
55 mA/cm
3
40 mA/cm
2
2
30 mA/cm
2
20 mA/cm
1
2
15 mA/cm
2
10 mA/cm
5 mA/cm
0
0
20
40
60
80
100
70mA/cm
2
3
50 mA/cm
2
30 mA/cm
2
2
15 mA/cm
1
2
10 mA/cm
2
2
5 mA/cm
2
2
120
140
0
Time (min)
0
20
40
60
80
100
120
140
Time (min)
Figure 4.12: Etch rates corresponding to Fig. 4.10. Etch rates
increase with current density.
Figure 4.13: Etch rates corresponding to Fig. 4.11. Etch rates
increase with current density.
60
Porosity (%)
55
50
45
40
35
30
1.0
1.5
2.0
2.5
Time (min)
3.0
3.5
Figure 4.14: The porosity
time profile of a sample etched
with a preset current profile calculated to give constant
porosity with depth based on
the calibration data in Figs.
4.10-4.13. The calculated uncertainty based on the FWHM
of the peak in the spectrogram
is plotted with dotted lines.
usability of such a system. This knowledge will ensure that the parameter space is limited such that the two main peaks in the STFT are found
quickly at onset of etching, see paper I for a description of the data analysis.
In addition, optimization of the peak tracing algorithm is needed. In the
case of feedback operation, the use of STFT for frequency determination is
only appropriate if the current density used is constant with time or varies
slowly, i.e. significant variations over much longer time than window function length. In the case of multilayer etching, in which case the feedback
operation would be most useful, the etching time for each layer will often
be too short for STFT based analysis. As seen in Fig. 4.15, which is from
the etching of an infrared Bragg filter with layer thicknesses in the order
of a few 100 nm, each layer etching results in around one period, or less,
of the main interference oscillation (main partial). With prior knowledge
of the system and an appropriate fitting algorithm, the information in this
signal may be used for feedback. Note that in Fig. 4.15 there are three
periods; the high current layer with a high etch rate, the low current layer
59
with a lower etch rate and a break period with no current. This is used to
regenerate the electrolyte at the pore tips instead of adjusting the current
density to give the set porosity with changing electrolyte conditions.
High current layer
Figure 4.15: The measured
in situ reflectance signal during etching of a multilayer
structure. One etch period
consists of a high current layer,
one low current layer and
one break period. These are
marked in the figure.
Signal amplitude (a.u.)
0.8
Low current layer
0.6
Break
0.4
0.2
0.0
24.5
25.0
25.5
26.0
Time (min)
26.5
27.0
I
Paper I
S.E. Foss, P.Y.Y. Kan and T.G. Finstad
Single beam determination of porosity and etch rate
in situ during etching of porous silicon
J. Appl. Phys., 97, 114909 (2005)
JOURNAL OF APPLIED PHYSICS 97, 114909 共2005兲
Single beam determination of porosity and etch rate in situ during etching
of porous silicon
S. E. Foss, P. Y. Y. Kan, and T. G. Finstada兲
Department of Physics, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norway
共Received 14 January 2005; accepted 1 April 2005; published online 31 May 2005兲
A laser reflection method has been developed and tested for analyzing the etching of porous silicon
共PS兲 films. It allows in situ measurement and analysis of the time dependency of the etch rate, the
thickness, the average porosity, the porosity profile, and the interface roughness. The interaction of
an infrared laser beam with a layered system consisting of a PS layer and a substrate during etching
results in interferences in the reflected beam which is analyzed by the short-time Fourier transform.
This method is used for analysis of samples prepared with etching solutions containing different
concentrations of HF and glycerol and at different current densities and temperatures. Variations in
the etch rate and porosity during etching are observed, which are important effects to account for
when optical elements in PS are made. The method enables feedback control of the etching so that
PS films with a well-controlled porosity are obtainable. By using different beam diameters it is
possible to probe interface roughness at different length scales. Obtained porosity, thickness, and
roughness values are in agreement with values measured with standard methods. © 2005 American
Institute of Physics. 关DOI: 10.1063/1.1925762兴
I. INTRODUCTION
Porous silicon 共PS兲 has been studied intensively for well
over a decade. The optical properties of PS have been of
particular interest. The application of PS for passive optical
devices came with the development of multilayer optical
Bragg reflectors by Vincent1 and Berger et al.2 in 1994.
Multilayer films in PS have shown great potential for a wide
range of applications. A variety of applications have lately
been fabricated, such as Si-based integrated optical circuits,3
chemical microsensors,4 and broadband laser mirrors.5
Most applications depend critically on the material properties. Many of the material properties of PS, such as the
optical, mechanical, and electrical ones, depend on the porosity. The porosity depends on the process parameters.
Hence to achieve well-controlled properties a tight control of
process parameters is necessary. Ex situ characterization of
these properties is normally employed. Some characterization techniques that are common are gravimetry for porosity
measurement, cross-section scanning electron microscopy
for thickness measurement, and profilometry for interface
roughness determination. These ex situ characterization techniques are all destructive and give no direct information of
the etch history or porosity gradients.
One critical aspect of using PS for many optical devices
is the inherent roughness at the different interfaces developed
during the etching process, especially the PS-substrate interface. This roughness results in a nonoptimal optical quality
of the device. One mechanism of degradation, compared to
the ideal case, is scattering. The degradation in optical quality will be strong for short wavelengths as the scattering
power at a given roughness is inversely proportional to the
wavelength. The roughness is often described by a surface
a兲
Electronic mail: terje.finstad@fys.uio.no
0021-8979/2005/97共11兲/114909/11/$22.50
height function of which a root-mean-square 共rms兲 value is
obtained. Silicon has a very large absorption for wavelengths
below about 1.1 ␮m, but freestanding transmission filters
and reflection filters on substrates are still possible for this
range. However, the low absorption and relatively smaller
scattering in the near-infrared spectral region above 1.1 ␮m
make this range the best suited for optical filters based on
PS. Still, a tight control of the interface roughness is necessary to obtain the optical quality needed for a given application. Roughness in PS has been studied extensively by
Lérondel et al.6 as well as by Setzu et al.7 and Servidori
et al.8
A method for monitoring several parameters important
for optical element fabrication during etching of PS films
will be presented in this paper. The method is based on interferometry where the oscillation frequency and amplitude
of a backside reflected monochromatic infrared 共IR兲 laser
beam are measured in situ during PS formation. From this
single signal, and the following analysis, the PS film thickness, the etch rate, the refractive index, the porosity, profile,
the average porosity, and the interface roughness may be
obtained. The calculations used for the analysis are based on
Airy summation and Davies–Bennett theory.9–11 The analysis
of the measured signal is quite extensive and uses the fast
Fourier transform algorithm which easily facilitates an
implementation of an automated feedback system. The
method is quite robust and intuitive and may be adapted to
many different PS etching cells. The use of interferometry
techniques for monitoring parameters in situ during processing of semiconductors has been reported before. Steinsland
et al.12 used IR laser backside reflection interferometry to
monitor the etch rate for tetramethyl ammonium hydroxide
共TMAH兲 etching of silicon, and the present work is an extension of that work. Thönissen et al.13 and Gaburro et al.14
applied a front side technique with a visible laser to monitor
97, 114909-1
© 2005 American Institute of Physics
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114909-2
J. Appl. Phys. 97, 114909 共2005兲
Foss, Kan, and Finstad
FIG. 1. 共Color online兲 Schematic of the different etch setups. The setup
denoted 共A兲 is based on free-space beam transmission. The laser diode is
denoted LD. The schematics of both setups 共A and B兲 show a cross section
of the etch bath. Setup 共B兲 shows the wide beam optical-fiber variant. MM
denotes the multimode optical fiber and SM denotes single mode optical
fiber. A coupler is shown; this splits the fiber and guides the reflected signal
from the sample to the detector. A graded index lens, as indicated, is used to
collimate and collect the light close to the sample. The third setup 共C兲 shows
a fiber variant where no collimating lens is used, but where the fiber is close
enough to the sample backside 共⬍500 ␮m兲 to collect a significant amount
of the light reflected from the sample. This results in a probed area about the
size of the fiber core cross section.
etch rate and porosity. The addition of the interface roughness measurement to the refractive index and etch rate measurements in the presented method results in an extended
characterization ability.
We will describe the experimental setup first in Sec. II.
There we also include the details of the samples where the
method has been used, supplementary characterization techniques, and the determination of supplementary parameters
needed by the method. Then in Sec. III we outline the
method used and the theory behind it. In Sec. IV we present
some examples where the method has been used. These measurements are briefly discussed in Sec. V.
II. EXPERIMENTAL DETAILS
A. Interferometric measurement setup
The preparation of PS films was done in a standard upright etch cell with a solid Cu back contact. Through a hole
cut out in the Cu back contact an IR diode laser was directed
at the sample and the reflected beam was detected with an
InGaAs detector. Three different measurement geometries
were used, with the main difference being the beam diameter. Figure 1 shows a schematic of the different setups. By
using different beam diameters it is possible to probe interface roughness with different spatial wavelengths and also
obtain spatial averaging over probe areas of different sizes.
The latter implies a compromise between a large electrical
contact area on the sample backside, which is needed for
homogenous etching over the etch area, and large beam diameter, needed for obtaining spatially averaged etch rate and
porosity data. Two of the setups were based on fiber optics
while the third was based on free-space optics. By using
optical fibers a compact setup was obtained and beam alignment and positioning were simple.
For both fiber setups, which are shown in Fig. 1 共setups
B and C兲, the monochromatic, coherent light source used
was a diode laser pigtailed to a single mode fiber with an
optical isolator. The wavelength was 1550 nm and the output
power was 2 mW. A 2 ⫻ 1 fused coupler was used to couple
the incident beam to the sample backside and the reflected
beam to the detector. The coupler used was based on a multimode fiber. The beam collection in a multimode fiber are
easier than in a single mode fiber as both the core and acceptance angle are larger in the multimode fiber. For the
narrow beam fiber setup, Fig. 1 共setup C兲, the bare multimode fiber end was positioned close 共⬍500 ␮m兲 to the
sample backside through a hole in the Cu backplate. The
fiber core with a diameter of 62.5 ␮m then collected light
from a probing area on the sample interfaces equal to the
fiber core area. This setup was used to measure roughness
with a short spatial wavelength. The other fiber setup, shown
in Fig. 1 共setup B兲, used a collimating graded index lens to
give a collimated beam of 2-mm diameter. The beam was
oriented normal to the sample and the lens both collimated
the incident beam and collected the reflected beam and
coupled it back into the fiber. The free-space setup, shown in
Fig. 1 共setup A兲, used a collimated beam from a laser diode
with a wavelength of 1310 nm and an output power of 8 mW.
The beam diameter was between 1 and 2 mm, however, it
could easily have been magnified to cover a larger area. The
beam was at an angle of 12° to the sample normal. The
sampling frequencies of the reflected light for all setups were
between 0.78 and 10 Hz.
B. Sample preparation
The wafers used for PS preparation were boron-doped
共p-type兲 Czochralski-grown 具100典-oriented, double side polished with a thickness of about 520 ␮m and a resistivity of
0.01–0.02 ⍀ cm. The electrochemical etching of the samples
was performed in an electrolyte made from 40% aqueous HF
diluted with ethanol and glycerol. The HF concentrations
used were 15%, 20%, and 26% while the glycerol-to-ethanol
ratio varied from 0% to 70%. Constant current densities applied were from 5 to 30 mA/ cm2. Samples were etched up to
120 min at both room temperature 共RT兲 and at a low temperature 共LT兲 of 5 °C.
C. Other experimental methods
Porosities were determined both by analysis of the reflected signal and by gravimetry. The layer thicknesses after
etching were determined by the reflected signal, by crosssection observation in an optical microscope or by stylus
surface profilometry after stripping away the PS film in concentrated NaOH. The interferometrically measured roughness values were compared to values obtained by white-light
interferometry measurements 共WYKO NT-2000兲 after stripping away the PS.
Refractive indices of the different electrolytes used were
measured by the amount of parallel shift of a laser beam
transmitted through a holder made of Plexi-glass containing
the electrolyte while it was rotated and using Snell’s law.
Different concentrations of both HF and glycerol were used.
The inset in Fig. 2 shows a schematic of the experimental
setup. The laser beam used had a wavelength of 1310 nm.
The refractive index was found to change little between 1310
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114909-3
J. Appl. Phys. 97, 114909 共2005兲
Foss, Kan, and Finstad
FIG. 4. 共Color online兲 Measured signal during etching of a sample with
20% HF and 10% glycerol at 20 mA. Oscillation caused by changing conditions for interference due to a moving PS-substrate interface is evident.
The expanded view clearly shows that there are superposed partials.
FIG. 2. 共Color online兲 Measured refractive index of the electrolyte with
15% HF as a function of glycerol content. Glycerol content given in % of
total ethanol/glycerol content. The inset shows a schematic of the setup used
for the measurement. The laser beam will be refracted twice 共disregarding
the holder walls, however, this is accounted for in the refractive index calculation兲, into and out of the electrolyte, resulting in a measurable parallel
shift when the holder is rotated. This shift gives the refracted angle, which is
dependent on the difference in refractive index between electrolyte and air.
and 1550 nm. A black-and-white charge-coupled device
共CCD兲 video camera 共SONY兲 was used for measuring the
beam position.
III. THEORY AND METHOD
A. Determination of etch rate and porosity by
interferometry
We consider the situation sketched in Fig. 3 consisting of
three interfaces which all reflect and transmit a part of the
incident laser beam. The interaction of the beam with the
layers can be represented by individual rays, each having an
associated phase and amplitude. The signal to be detected
consists of the part of laser light being reflected into the
detector at the same side of the sample as the laser. This total
signal will have contributions from many rays that interfere.
As the porous layer thickness increases as etching
progresses, the optical thickness of each layer changes and
the phase of each ray will vary. Thus the reflection signal
will vary in intensity with etching time. An example of an
experimental reflection intensity signal is shown in Fig. 4. If
we ignore absorption in the layers, the amplitude of each
partially reflected and transmitted ray is given by the product
of Fresnel coefficients from the encountered interfaces. As
will be shown later, the time varying signal can be decomposed into partials of specific frequencies and the frequencies
of these partials are those arising from the interference of ray
pairs. The rate of change of the phase difference between two
detected rays will give a partial with a specific instantaneous
frequency of oscillation. There are many rays contributing to
the detected signal which results in many possible partials of
different frequencies. However, most rays, and thus most
partials, will have very small amplitudes due to multiple reflections and transmissions. By extracting the appropriate
frequencies and their time dependence from the experimental
signal, it is possible to calculate the optical thickness and the
etch rate of each layer at any given time.
The schematic of the system in Fig. 3 shows the principal rays and partials used in the analysis. The phase difference between the ray transmitted at the air-substrate interface, reflected once at the substrate-PS interface, and
transmitted back out 共ray II in Fig. 3兲 and the ray reflected at
the substrate 共ray I兲 is given by
␦1共t兲 =
2 2
冑n − sin2␪ dsub共t兲 + ␸0 − ␸dsub ,
␭0 sub
while the interference frequency is
␯1共t兲 =
FIG. 3. 共Color online兲 Ray trace through the sample during etching. Interference between rays I and II is denoted as situation 1 and between I and III
as situation 2. Scattering of light due to a rough PS-substrate interface is
indicated.
冏
冏
⳵ ␦1共t兲 2 2
d dsub共t兲
= 冑nsub − sin2␪
.
⳵t
␭0
dt
共1兲
共2兲
Here nsub is the refractive index of the substrate, ␭0 is the
vacuum wavelength of the incident beam, ␪ is the incident
angle, while ␸0 and ␸dsub are the phase changes on reflection
as the rays are reflected at the backside and at the PSsubstrate interface, respectively, these are assumed constant.
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114909-4
J. Appl. Phys. 97, 114909 共2005兲
Foss, Kan, and Finstad
The substrate thickness, dsub共t兲, and the PS layer thickness,
dPS共t兲, are time dependent. The etch rate is found by the time
derivative of either thicknesses. By defining the etch rate to
be positive when moving into the substrate and thereby
avoiding the absolute value in Eq. 共2兲, it may be written as
1
d dsub共t兲 d dPS共t兲 ␭0
=
=
␯1共t兲.
2
2 冑nsub
dt
dt
− sin2␪
r共t兲 = −
共3兲
If the interface does not move with a constant speed or the
porosity of the PS layer changes, the frequency will also
change. This information is present in the reflected signal.
Equation 共2兲 contains no information of the porous layer.
The partial with this information is found when looking at
the interference between rays I and III in Fig. 3. The phase
difference here is given by
␦2共t兲 =
1
关2ODsub共t兲 + 2ODPS共t兲兴 + ␸0 − ␸dsurf ,
␭0
共4兲
where the constant phase change contribution is equivalent
to those occurring in Eq. 共1兲. The substrate contribution to
the optical path difference between the two rays, ODsub共t兲, as
in Eq. 共1兲, is given by
2
− sin2␪ dsub共t兲,
ODsub = 冑nsub
共5兲
while the PS layer contribution, ODPS共t兲, is slightly more
complex if the PS refractive index varies with depth. The PS
refractive index is modeled to have only depth dependence,
nPS共l兲, where l is the depth or thickness of the layer, and not
an explicit time dependence as will be motivated in Sec. V.
The optical path difference caused by the PS layer is then
given by
ODPS =
冕
dPS共t兲
2
冑nPS
共l兲 − sin2␪ dl.
共6兲
0
Here the porous layer, dPS共t兲 thick, is divided into an infinite
number of sublayers of infinitesimal thickness 共dl兲. Each sublayer has a certain refractive index nPS共l兲. The integral over
the optical path contribution of each sublayer then gives the
total optical path. Taking the time derivative of the phase
difference in Eq. 共4兲 necessitates the use of the Leibniz’s rule
for differentiation of integrals15 to derivate Eq. 共6兲, solving
for nPS then results in
nPS关dPS共t兲兴 =
冑冋 冑
2
nsub
− sin2␪ −
␭0␯2共t兲
2r共t兲
册
2
+ sin2␪ . 共7兲
Using the measured refractive indeces for the different electrolytes it is now possible to calculate the porosity profile
and average layer porosity as a function of PS layer depth.
Equations 共3兲 and 共7兲 together give the most important parameters of the PS etching process.
B. Interface roughness obtained by interferometry
To obtain information on the PS-substrate interface
roughness, the interference between the ray reflected from
the substrate backside and the ray reflected from the
substrate-PS interface is of most interest. This corresponds to
situation 1 in Fig. 3. The intensity of the combined reflection
of these two rays, Iref, is given by
2
2
Iref = Asub
+ APS
+ 2AsubAPScos共␸sub − ␸PS兲,
共8兲
where Asub and APS are the amplitudes and ␸sub and ␸PS are
the phases of the rays reflected from the substrate backside
and the PS-substrate interface, respectively. The cosine component of Iref gives the amplitude and frequency of the interference oscillation. The oscillation amplitude is given by the
cosine prefactor. Asub is constant while APS contains information on the PS-substrate interface scattering, substrate absorption, and PS refractive index 共in the electrolyte兲,
APS共t兲 = tair-sub␣sub共t兲sR,sub-PS共t兲rsub-PS共t兲␣sub共t兲tsub-air . 共9兲
Here tair-sub and tsub-air are the transmission amplitude coefficients, rsub-PS共t兲 the time-dependent reflection amplitude coefficient at the substrate-PS interface, sR,sub-PS共t兲 the timedependent scattering factor as the roughness changes with
time, and ␣sub共t兲 the absorption factor which changes due to
a decrease in substrate thickness. The time dependence of the
reflection amplitude coefficient is due to the change in the PS
porosity with time.
Transmission and reflection amplitude coefficients, as
well as the absorption factor, are calculated using published
data for the complex refractive index of bulk Si. Normal
Fresnel relations are used for the transmission and reflection
coefficients, while the absorption factor is given by ␣sub共t兲
= exp关−2␲kz共t兲 / ␭兴, where k is the imaginary part of the complex refractive index and z共t兲 is the time-dependent substrate
thickness. The scattering factor value will be extracted from
the interference data and is in the following assumed known.
This extraction is explained in Sec. III C below.
The Davies–Bennett theory9,10 attempts to describe the
local phase change in the reflected plane wave front introduced by the height irregularities of the interface. These
phase changes result in a reduced intensity in the specular
direction as conditions for destructive interference will develop between different parts of the wave front resulting in a
loss of coherence. The theory assumes a rms irregularity
height 共roughness兲 value, ␴, much smaller than the wavelength of the incident light in the medium, ␭, and that the
height function describing the roughness has a Gaussian distribution. In this case the Fresnel reflection coefficient for the
rough PS-substrate interface may be written as
冋冉
2
= R0 exp −
Rsub = R0sR,sub-PS
4␲␴sub-PSnsub
␭0 cos ␪sub
冊册
2
,
共10兲
given the reflection coefficient for the perfectly flat surface,
R0. Here ␪sub is the incident angle in the medium, nsub is the
refractive index of the incident medium, ␭0 the wavelength
in vacuum, and ␴sub-PS the PS-substrate interface roughness.
Note that the spatial wavelength of the roughness does not
enter into this equation, so both long period 共for example,
striations兲 and short period roughness as will have an equal
effect, depending only on ␴sub-PS.
To calculate the rms roughness of the front surface, a
scattering factor for transmission, sT,sub-PS, must be calcu-
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114909-5
J. Appl. Phys. 97, 114909 共2005兲
Foss, Kan, and Finstad
lated based on the obtained ␴sub-PS. The transmission coefficient at the PS-substrate interface has been shown by
Filiński11 to be
2
Tsub = T0sT,sub-PS
再冋
= T0exp −
2␲␴sub-PC共nsubcos ␪sub − nPScos ␪PS兲
␭0
册冎
2
,
共11兲
for the same conditions as for the reflection coefficient in Eq.
共10兲. Here T0 is the transmission coefficient for a perfect
interface, while nPS is the refractive index of the PS and ␪PS
is the angle in the PS layer. Then the reflected amplitude
from the front becomes
2
2
2
共t兲sR,PS-front
Afront共t兲 = tair-sub␣sub
共t兲sT,sub-PS
共t兲tsub-PS共t兲␣PS
⫻共t兲rPS-front共t兲tPS-sub共t兲tsub-air ,
共12兲
where most of the parameters are the same as in Eq. 共9兲 with
the addition of a time-dependent transmission amplitude coefficient for the PS-substrate interface, tsub-PS共t兲, a timedependent reflection amplitude coefficient for the PS-front
interface, rPS-front共t兲, and an absorption factor in the PS layer,
␣PS共t兲, defined as for ␣sub共t兲 with kPS calculated by an
effective-medium theory using the complex refractive index
of bulk Si. The reflected amplitude is given by the same
calculations on the extracted amplitude of the partial in situation 2 of Fig. 3 as for APS. The calculation of the scattering
factor, sR,PS-front, is the same as for sR,sub-PS and from this the
rms roughness value of the front surface is calculated.
C. Analysis
For the analysis of the reflected signal, the short-time
Fourier transform 共STFT兲 was utilized to extract the different
frequency components. This gives both the frequency versus
time and the signal amplitude development. An example of a
reflection signal transformed with STFT is shown in the
spectrogram in Fig. 5共a兲. The signal was measured during
etching of a sample in 26% HF at RT and 25 mA/ cm2 with
the wide beam fiber setup. In the spectrogram several curves
are traced. These curves represent the observable and readily
understood partials of the signal and correspond to the situations schematically shown in Fig. 5共b兲. In this figure, different ray trajectories and combinations are indicated which
will give rise to the partials of the signal. The two main
partials in Fig. 5共a兲 are drawn as solid lines. These correspond to, as labeled, situations 1 and 2 of Fig. 3 having
frequencies f and F, respectively. The dashed lines drawn in
Fig. 5共a兲 are calculated by the relation: n1 f ± n2F, where n1
and n2 are positive integers.
The analysis was done using the spectrogram and the
fast Fourier transform function in the software package
MATLAB.16 The STFT method uses a movable time window, where a Fourier transform is performed on the windowed signal for each window position along the signal with
the assumption that signals have constant frequencies within
the window. Both the time resolution and the frequency resolution depend on the chosen window size, however, the dependence follows a Heisenberg-type uncertainty relation; a
FIG. 5. 共Color online兲 共a兲 An example of a spectrogram. The peaks drawn
by the solid lines are the two dominant partials, corresponding to situations
1 and 2 in Fig. 3, as denoted on the spectrogram. The change of frequency
with time is evident. The dashed lines are the readily understandable higherorder partials, calculated by n1 f ± n2F, where n1 and n2 are positive integers
and f and F are the frequencies of curves 1 and 2, respectively. Most higherorder partials are not detectable. The sample was etched at RT for 120 min
with 25 mA/ cm2 and 26% HF. The length of the Blackman windowing
function used for the STFT analysis was 6 min 共3600 samples兲 and the
overlap between consecutive windows was 95%. 共b兲 The corresponding ray
traces of the different higher-order reflections shown in the spectrogram.
finer frequency resolution results in a coarser time resolution
and vice versa. With a narrow window, frequency peaks will
be relatively broad compared with larger windows. Different
window functions have different uses in signal processing. A
Blackman windowing function was used for this analysis as
it gave the best compromise between frequency resolution
and sidelobe suppression. To obtain the required detail of the
traced curves, consecutive windows overlapped by 95%.
Two window sizes were used when analyzing the data; 6 and
12 min of data. This resulted in comparable spectrograms
and peak widths between different samples and it was a good
compromise between time resolution and peak width.
In the case that the constant frequency assumption does
not hold and the frequency of a signal changes within a window, the frequency peak will increasingly broaden with window length, hence there will be an optimum window length
giving the narrowest peaks. As changing signal frequencies
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114909-6
Foss, Kan, and Finstad
are assumed in this analysis, the signal peak at a given time
will be an average over signal frequencies within the window
and the true position of the peak at that time is not certain
unless the frequency shift is linear with time. However, the
rate of change of the frequency of a signal may change during a measurement which results in a change in the optimum
window length. In the present analysis two window lengths
were used for all samples as a compromise between peak
definition and time resolution. For simplicity, it is assumed
that the true peak position is unknown but within the full
width at half maximum of the calculated peak. Calculations
of porosity, etch rate, and thickness include this uncertainty.
This assumption is likely to overestimate the actual uncertainty of the peak position as the Fourier transforms of consecutive time windows will correlate. Uncertainty introduced
by the experiment is much smaller than the uncertainty introduced by the analysis and the model used. Experimental
uncertainty includes uncertainty in the incident angle, refractive index of the substrate, flatness of the interfaces, measurement sampling rate, and the laser wavelength. There is
an additional uncertainty in the calculation of the porosity as
this value depends on the effective-medium model used.
To find the etch rate, porosity profile, and interface
roughness, it is only necessary to track the two strongest
partials in the spectrogram and obtain their history. The ray
combinations shown in Fig. 3, giving rise to situations 1 and
2, will have the fewest possible interface interactions of the
possible oscillation producing combinations in the total reflected signal. The amplitude of each ray will decrease with
the number of interface interactions due to both scattering/
coherence loss and partial reflection/transmission. Based on
the presented system model, the two curves with the largest
amplitudes in the obtained spectrogram will correspond to
the two situations denoted in Fig. 3. Of the two main curves,
the curve with the lowest frequency will always correspond
to situation 2 as the total optical thickness for ray III changes
slower than that for ray II of Fig. 3. Equation 共2兲 gives the
frequency of the interference between rays I and II. The
curve corresponding to this situation 共curve 1兲 will not always have the largest amplitude compared to curve 2 when
the ray is reflected off a rough substrate-PS interface even
though it has gone through the least number of interface
interactions. The reason for this is that the amplitude of the
reflected ray is affected more by interface roughness than the
transmitted amplitude.11 The traces of these two curves are
performed assuming small variations in the peak frequencies
from one window to another. This assumption is well
founded based on the measurements performed. Since each
time slice in the spectrogram is a frequency versus power
spectrum, the starting points of the two strongest partials are
found from the spectrum for zero time 共t = 0兲. This procedure
is well suited for real-time implementation with an expectation of the first frequency value as input in the tracking routine. With this a feedback control could be realized.
The procedure for calculating the scattering factor of
Eqs. 共9兲–共12兲 starts by smoothing the extracted partials amplitude. This is done by fitting the amplitude to a double
exponential. The function was chosen because it showed a
good fit to most of the obtained amplitudes. Smoothing is
J. Appl. Phys. 97, 114909 共2005兲
done to avoid the amplitude fluctuations present in a partial
around the position where the frequency of another partial
momentarily crosses. Crossing partials are present in the
spectrogram in Fig. 5共a兲 as can be seen for both main curves
at about 88 min. The fitted amplitude is then scaled so that
the extrapolated value at t = 0 corresponds to the theoretical
amplitude with zero scattering. The scaling is necessary because the measured data are not normalized to unit reflectance. Because of the windowing of the measured signal the
amplitude values are averaged over the time span the window covers, hence the STFT amplitude values calculated for
the lowest time value are correct for the partial at a time
around the middle of the first window. The exact position
depends on the shape of the amplitude change within the
window, but is for all calculations presented here assumed to
be at the middle of the window. This necessitates an extrapolation of amplitude to t = 0. The fitted and scaled amplitude
functions correspond to the cosine prefactor of Eq. 共8兲 from
which APS共t兲 and Afront共t兲 can be calculated.
D. Electrolyte refractive index
To get the right porosity value of the PS layer, knowledge of the refractive index of the etchant is necessary as the
value obtained from frequency analysis is the refractive index of the PS layer when it is immersed in the etchant. An
effective-medium model must be used to approximate the
porosity, which both accounts for the refractive index of the
silicon substrate and the etchant. The Bruggeman approximation theory was chosen as this is most often used for PS in
the literature.17 The refractive index of glycerol in the visible
is 1.466 while for ethanol it is 1.365 and for water it is 1.350,
hence glycerol will influence the etchant refractive index significantly. Refractive index values for solutions with concentrations different from those measured are extrapolated from
the measurements assuming a linear dependency between refractive index and both HF and glycerol concentration. Some
data with error bars are shown in Fig. 2. For the analysis
presented in Secs. III A–III C the electrolyte refractive index
is assumed constant and independent on time and thickness
of the PS layer. This assumption could be violated if the
electrolyte composition changes close to the pore tips due to,
e.g., diffusion limitations on the HF concentration as well as
buildup of Si-rich chemical species. However, the use of the
assumption seems well justified based on a comparison
between porosities obtained by interferometry and by
gravimetry.
IV. EXPERIMENTAL RESULTS
Using the method presented above some results from
analyzing measured reflection signals will be presented in
the following. The parameters obtained were the etch rate,
PS layer thickness, average porosity versus time and etched
thickness, porosity versus depth 共porosity profile兲, and interface roughness. Firstly, the effect of the beam size on the
measured amplitude will be shown. Following this is an example of the change in porosity and etch rate with time.
Further, a comparison between average porosities obtained
by interferometry and by gravimetry will be given. The ef-
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114909-7
Foss, Kan, and Finstad
FIG. 6. The oscillation amplitudes of the signals corresponding to situation
1 in Fig. 3 and curve 1 in Fig. 5共a兲 for measurements prepared with different
setups. The solid curve represents a measurement made with the narrow
beam fiber-optic setup. The amplitude decreases slightly in the beginning as
a result of a slight increase of PS-substrate interface roughness, while later
the amplitude increases because the effect of a thinning substrate, i.e., less
absorption, overtakes the amplitude loss due to scattering. For the measurement made with the wide beam fiber-optic setup 共----兲 the effect from the
thinning substrate is not large enough to overcome the decrease in amplitude
due to more scattering at the interface as the beam area is large enough to
encompass striations. Etching is done at room temperature with 26% HF and
15 mA/ cm2.
fect on average porosity of different HF and glycerol concentrations will be shown next. After this some PS-substrate
interface roughness measurement results will be given as
well as the time dependence of the valence of a few samples.
Some of the measurements used to test this method have
been reported earlier.18 Those measurements were all done
with the free-space optics setup.
The difference in using the wide beam and narrow beam
setups can be seen in Fig. 6. Here two experimental runs are
compared. They use the same parameters; 26% HF,
15 mA/ cm2 at RT, but were performed in different setups.
The traced amplitude of the partial corresponding to curve 1
in Fig. 5共a兲 is plotted for both experimental runs. The amplitudes have been scaled so that their starting values are identical. For both cases the general trends in the amplitudes are
the same; there is a decrease in the beginning due to rapidly
increasing roughness. This is overtaken by an increase in
amplitude due to less absorption in the substrate as the PS
film grows. There is a clear difference in the relative importance of these effects, loss due to scattering/coherence loss
and loss due to absorption, between the two experimental
runs using different beam sizes. In the case using the narrow
beam setup the initial decrease is much less, indicating a
significantly smaller measured roughness within the beam
spot than for the wide beam case, hence the relative effect of
the decreased absorption is greater. This is a strong indication that there are different spatial wavelengths of roughness
of importance at different spot sizes. In most of the following
measurements the wide beam free-space or fiber-optic setup
has been used as the signal-to-noise ratio is better.
All the p+ samples etched at constant current conditions
in this study have shown an increase in porosity with longer
etching time. This increase is the result of the increasing
porosity with depth which is exemplified in Fig. 7 where the
porosity profile of one of the samples is calculated and plotted. To be able to compare the porosity measured by gravim-
J. Appl. Phys. 97, 114909 共2005兲
FIG. 7. An example of a porosity profile 共solid line兲 in depth 共upper horizontal axis兲/time 共lower horizontal axis兲 and the accompanying etch rate
change. The time development of the average porosity is also indicated
共dashed line兲. Data are from the same sample as in Fig. 5共a兲. The dotted
lines are an indication of uncertainty based on the peak width of the signal
line in the spectrogram.
etry with the interferometrically obtained data, the average
porosity as a function of time is also calculated. The average
porosity values are plotted with the porosity profile in Fig. 7.
The etch rate also changes with time. This is plotted in Fig. 7
as well. Note that in these examples the largest change for
both porosity and etch rate is at the beginning of the etching.
For all three parameters upper and lower limits are shown.
These curves correspond to the full width at half maximum
values of the traced peaks discussed earlier in determining
the true peak position, hence gives an upper limit on the
uncertainty in the calculation. The sample data shown in Fig.
7 are the result of an analysis of the spectrogram in Fig. 5共a兲.
The sampling frequency for this sample was 10 Hz and the
window length used in the analysis was 6 min 共3600
samples兲. The sample was etched at RT with 26% HF and
25 mA/ cm2 and the wide beam fiber-optic setup was used.
The average porosity data value for five different
samples was measured by the interferometry method described. The samples were etched in an electrolyte containing 26% HF and ethanol at different current densities. The
average porosity value at three different times for all five
FIG. 8. 共Color online兲 The average porosity as a function of current density
given at three different times. Samples are etched with 15% HF. j denotes
15-min etching, . denotes 25-min etching, and s denotes 45-min etching.
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114909-8
J. Appl. Phys. 97, 114909 共2005兲
Foss, Kan, and Finstad
TABLE I. Comparison between average gravimetrically measured and interferometrically measured porosities
共%兲. The fit is quite good, values obtained by interferometry even give a more consistent change in porosity
with change in current density. The electrolyte used contains only HF and ethanol. Etching was done at RT.
Average porosity 共%兲
after etching in 26% HF
Average porosity 共%兲
after etching in 15% HF
Current density
共mA/ cm2兲
Gravimetry
Interferometry
Gravimetry
Interferometry
5
10
15
20
30
36
43
38
48
53
32
40
¯
46
53
69
65
66
80
83
65
66
71
79
¯
samples is plotted against current density in Fig. 8. It is seen
that the porosity increases smoothly with current density, as
expected. In addition the average porosity increases with
time for all the measurement series, as is also exemplified in
Fig. 7. The increase of porosity with time has been reported
earlier for p+ Si,13 so this behavior is expected. The final
average porosities obtained by interferometry for a different
set of measurements are shown in Table I where the values
can be compared with those measured gravimetrically. It can
be seen that the porosity values agree well within the errors
given. It should also be noted that the interferometrically
obtained porosity values increase more consistently with current density than the gravimetrical measurements.
The absolute change in average porosity with time seems
to be identical for different concentrations of HF, about 10%
over 50 min, this implies a substantially larger difference in
porosity between PS layer surface and bottom, as can be seen
in the porosity profile data in Fig. 7. A comparison of the
average porosities of PS etched with different HF concentrations is shown in Fig. 9. The uncertainty for all the porosity
calculations is about ±3%. The error bars are not shown for
clarity. Figure 10 shows the average porosity obtained after
etching as a function of glycerol concentration for samples
etched in 15% and 26% HF solutions at the same current
density. This plot clearly indicates a varying effect of glycerol on porosity dependent on the HF concentration.
Roughness estimates were also made based on the
method discussed. The plot shown in Fig. 11 is representative of the samples prepared in this experiment. Here the rms
roughnesses of two samples are plotted against layer thickness during etching. The samples are etched with identical
parameters except for a difference in glycerol content. The
sample with 10% glycerol shows a power-law dependence of
roughness on thickness, however, with values substantially
smaller than for the sample etched without glycerol. The
latter sample shows a saturation occurring at around 10 ␮m
after a similar power-law dependence. A saturation of roughness has been reported before in the case of p and p− PS.6 In
Table II the interferometrically obtained maximum roughness values of several samples are compared with white-light
interferometry 共WLI兲 roughness values obtained from the
PS-substrate interface after stripping of the PS layer with
NaOH. The data shown are from samples measured with the
wide beam fiber setup. A WLI spot size similar to the spot
size obtained with the interferometry setup was used. As
there was some curvature over the whole etched interface as
well as fluctuations in thickness with spatial wavelengths
longer than the spot diameter the rms roughnesses when
measured over the whole etched area for these samples were
significantly larger.
From the local porosity data and the etch rate data as
determined by interferometry and the measured anodic current, it is possible to calculate the valence of the reaction,
FIG. 9. 共Color online兲 Comparison of the evolution of average porosities
with time for different HF concentrations. All three measurements are made
on samples etched with 20 mA/ cm2 and no glycerol at RT. As expected,
porosities decrease with increasing HF concentration. The time evolution is
quite similar for all samples.
FIG. 10. The effect of the glycerol content in the electrolyte on average
porosity for different concentrations of HF. The points from samples etched
in 15% HF are marked with s and 26% are marked with j. The glycerol
ratio is compared to the total ethanol/glycerol content. There is a clear
difference between the 15% and 26% HF cases.
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114909-9
J. Appl. Phys. 97, 114909 共2005兲
Foss, Kan, and Finstad
FIG. 12. Valence calculated based on etch rate and porosity as a function of
time for samples etched with several variations in etching parameters. It is
evident that the different parameters have a large effect.
FIG. 11. Roughness 共nm rms兲 obtained from calculations using the reflection signal amplitude data for two samples etched with 15% HF at RT with
different glycerol content. 10% glycerol gives the lowest roughness in the
case of 15% HF of all concentrations tested. The roughness saturation of the
0% glycerol data has been reported earlier 共Ref. 6兲 for lower-doped p-type
PS.
i.e., the charge required to remove one silicon atom from the
substrate. It is assumed that the etching only occurs at the
pore tips,14,19
␯=
j
,
Nerp
共13兲
where ␯ is the valence, j the current density, N the numbers
of silicon atoms per unit volume, e the elementary charge, r
the etch rate, and p the porosity at the interface. Figure 12
shows the valence determined this way as a function of time
for several samples etched at different conditions. Temperature, current density, HF concentration, and glycerol content
have all been varied to see the effect on the valence.
V. DISCUSSION
The presented interferometric method has been tested.
From the etch rate data, PS layer thicknesses may be calculated which corresponds well with stylus surface-profile
measurements. The obtained refractive index at the PSsubstrate interface gives a film porosity profile from which
the average layer porosity may be calculated. This correTABLE II. Comparison of measured roughness rms values by interferometry with the wide beam fiber setup and values obtained with white-light
interferometry 共WLI兲 on a 2-mm-diameter circular area at the center of each
sample after stripping of PS. LT is low temperature 共5 °C兲 and RT is room
temperature. Note that thicknesses will not be the same for different
samples.
Etch condition
Small area WLI
共rms nm兲
Reflectance
measurement
共rms nm兲
26% HF, 15 mA/ cm2, RT, 100 min
15% HF, 15 mA/ cm2, RT, 120 min
26% HF, 15 mA/ cm2, RT, 120 min
26% HF, 15 mA/ cm2, LT, 120 min
26% HF, 30 mA/ cm2, LT, 120 min
157
173
168
155
187
157
170
169
153
186
sponds well with gravimetrically obtained porosities. The obtained rms roughness also corresponds well with other measurements. Thus, we consider the obtained results for good.
The present setup is slightly different than setups presented
before12–14 and the analysis is more extensive.
In the present setup as well as in Ref. 12, the beam is
transmitted through the sample using an IR laser beam for
which Si is transparent, hence there will be no free-carrier
generation in the sample. When preparing PS it is of great
importance for reproducibility to control the access of charge
carriers, specifically holes, to the etched surface. By using an
IR laser, the beam intensity may be quite high without influencing the etching. This gives a good signal-to-noise ratio.
The use of an IR beam also facilitates the measurement of
thick layers in comparison with a beam in the visible range
which will be absorbed within the first few microns. On the
other hand a shorter wavelength will improve the resolution
of thickness and etch rate. The transmission setup may easily
be adapted to a liquid back contact etch cell. By directing the
beam from the backside, disturbances from hydrogen
bubbles in the beam path through the electrolyte are avoided.
To obtain both etch rate and porosity either one or two beams
may be applied. In the two-beam case, the beams must have
different wavelengths or different incident angles. Only the
top layer needs to be probed in this case, hence a simple
reflection signal is obtained, however, for this to happen
there must be enough absorption or a highly scattering interface to avoid reflection from the back surface.14 In the onebeam case the beam must be able to probe more layers and
this results in a more complex reflection signal also making
it more complicated to analyze. However, having only one
beam simplifies the setup. Further, by using fiber optics, the
footprint of the setup is minimized making it more mobile
and space efficient and making alignment easier.
When preparing multilayer structures for optical filter
applications in PS, thicknesses less than 20 ␮m are normally
used to minimize absorption, for filters used in the visible
range even thinner filters are made. It is clear that the variation in porosity within the first few microns of etching will
be detrimental to the properties of the optical filters if not
taken into account in the design.20 The results shown in Fig.
7 suggest that the greatest change in porosity with depth
occurs within the first 20–30 ␮m. By changing current den-
Downloaded 01 Jun 2005 to 129.240.153.219. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
114909-10
Foss, Kan, and Finstad
sity during etching according to this knowledge it is possible
to obtain PS layers of uniform porosity through the layer.13
An alternative is to use etch stops to regenerate the electrolyte to its original condition and thus avoid a change in etching condition at the pore tip.20
As shown with the presented method, a porosity gradient
is a common characteristic of the fabrication technique used
for PS preparation today, i.e., constant current conditions and
a closed, constant electrolyte volume. However, both positive and negative gradients have been reported in the
literature.8,14,21 The gradient seems to depend on the sample
doping and formation conditions. The etch rate will typically
also change with time, as shown in Fig. 7. Both of these
effects are routinely overlooked, with a few notable exceptions where very good filters have been fabricated.22,23 When
not accounting for these effects there will be noticeable discrepancies between the designed filter characteristics and the
experimentally obtained, especially for thick filters, e.g., IR
filters. The observation of a porosity gradient has usually
been performed by extensive modeling on data obtained by
variable angle spectroscopic ellipsometry24–26 or synchrotron
x-ray reflectivity8 as a more direct measurement of the gradient has proven difficult, e.g., by electron microscopy techniques. The models used for ellipsometry or x-ray reflectivity
data are not always transparent and the necessary fitting
could give only local minima and not necessarily the best
values. The method presented in this paper is based on a
fairly simple model, in addition, the measurements are done
in situ resulting in the formation history being part of the
model input. Thus, one need only to find the porosity of the
last infinitesimal homogenous layer at any given time to have
the complete porosity profile. However, to obtain the best
possible porosity profile, a combination of the in situ interferometric method with an ex situ ellipsometry measurement
would be necessary.
The choice of p+ Si for etching PS for optical applications is based on the large obtainable porosity range and the
reported comparably low interface roughness. There are,
however, a few challenges with p+ PS. Because of the high
dopant concentration, absorption will be slightly higher compared to p and p− PS. The high concentration also gives rise
to larger spatial fluctuations in the dopant concentration, often referred to as striations, hence spatially varying etch rates
and porosities are obtained. These fluctuations are often observed as concentric circles in the PS, as the refractive index
is affected during etching, or as ridges on the interface surface after PS stripping.6 The radial distance between ridges is
most often in the order of 250 ␮m. By using a probing area
smaller than this the ridges will spread little of the light,
hence this roughness will be filtered out and roughness
caused by other effects will be measured, hence the need for
beams of different diameters.
In the literature13,21 two different causes for a depth dependence of the refractive index of the porous layer are
given: 共i兲 etching of the porous structure not considered to be
caused by electrochemical etching, i.e., chemical etching,
which leads to a time-dependent porosity increase in already
etched parts of the PS film, and 共ii兲 an increase or decrease in
the porosity at the etch front with time with otherwise con-
J. Appl. Phys. 97, 114909 共2005兲
stant conditions caused by changing local conditions for
etching with time. This change in local conditions at the etch
front is thought to be caused by diffusion limitations on the
local HF concentration. In p+ PS, chemical etching has been
reported to be very small for the electrolytes used here,
hence this effect has been neglected in the present analysis.
Using the same interferometric setup as in Ref. 14,
Navarro-Urrios et al.27 have shown the effect of chemical
etching on the detected interference signal; an oscillation is
still present even after the etch current has been turned off,
possibly indicating a change in the refractive index, and
hence the porosity, of the PS layer. A slow oscillation after
turning off the etch current has also been observed in the
measurements discussed in the present paper, although this
may have other causes than chemical etching, such as thermal effects. A rough estimate of the possible chemical etching effect on the refractive index of a typical sample 共etch
conditions: RT, 26% HF, 15 mA/ cm2, 100-min etching,
150 ␮m thick兲 using one period of the oscillation in the measured signal after the current is turned off gives an average
change in the refractive index for the whole thickness of
about 0.07/ h immediately after etching. This translates to a
porosity change of roughly 2.5% / h. This estimate shows
that chemical etching in this case is indeed a small effect and
gives a maximum uncertainty in the porosity profile values
of the order of the uncertainties already discussed.
In the model considered in Sec. III, the etching is considered to occur only at the pore tip. This is an approximation to the actual situation where several factors will determine the reaction distribution over the surface of the pore.19
Note that this assumption will influence the porosity profile
determination, but not that of the average porosity.
There are several methods available to perform joint
time-frequency analysis of a signal besides STFT; these are
mainly the wavelet transform and the Wigner–Ville distribution. The Wigner–Ville distribution was not used because of
problems with cross terms. The measured signal obtained
through the discussed setup shows relatively low-frequency
oscillations, hence the requirement of stationarity within
each window of the STFT is close to satisfied. It is of importance to have the best compromise between time and frequency resolution in the frequency range of interest. As the
wavelet transform gives an increasing frequency resolution
and decreasing time resolution for a decreasing signal frequency the choice of resolution is limited. With the STFT the
resolution is the same for all frequencies and it may be set to
optimize the resolution and uncertainty of the calculated parameters. Further the STFT is easily implemented and more
intuitive than the wavelet transform.
VI. CONCLUSION AND SUMMARY
The method presented here shows the possibility of
monitoring multiple process parameters simultaneously during the formation of a porous silicon layer with a fairly
simple setup. The extracted values using the method are in
good agreement with those obtained using other ex situ or
destructive methods. The validity of the method is further
justified by giving the same trends as reported in the litera-
Downloaded 01 Jun 2005 to 129.240.153.219. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
114909-11
ture for porosity versus depth. Analysis of the mean porosity
and the etch rate evolution during etching caused by a gradient in the porosity with depth was discussed as well as the
effect of different HF concentrations has on these parameters. The effect of glycerol in the electrolyte was also
shown, looking at both porosity change and roughness evolution. The spectrogram calculation can be done in real time
and this has potential for feedback control of the etching
process using the measured parameters in the feedback loop.
ACKNOWLEDGMENTS
This work is supported by the Research Council of Norway. The authors are grateful for help by Chetna Schukla and
Erik S. Marstein during the initial experiments.
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S. Billat, M. Thönissen, R. Arens-Fischer, M. Berger, M. Krüger, and H.
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Downloaded 01 Jun 2005 to 129.240.153.219. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
Paper II
P.Y.Y. Kan, S.E. Foss and T.G. Finstad
The effect of etching with glycerol, and the interferometric measurements on the interface roughness
of porous silicon
Phys. Stat. Sol. (a), 202, 8, 1533 (2005)
II
Original
Paper
phys. stat. sol. (a) 202, No. 8, 1533 – 1538 (2005) / DOI 10.1002/pssa.200461173
The effect of etching with glycerol, and the interferometric
measurements on the interface roughness of porous silicon
P. Y. Y. Kan*, S. E. Foss, and T. G. Finstad
Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway
Received 23 July 2004, revised 21 September 2004, accepted 27 January 2005
Published online 8 June 2005
PACS 68.35.Ct, 68.55.Jk, 81.05.Rm, 81.40.Tv, 82.45.Vp
We have carried out interferometric measurements of interface roughness in-situ during electrochemical
etching of p-type porous silicon (PS) at room temperature. We found that at a certain porosity (~70%) and
with an electrolyte where a low fraction (10%) of the ethanol was replaced with glycerol, there was a significant decrease of the interface roughness. However, a higher content of glycerol (>10%) increased the
surface roughness. We have varied the current density in the electrolytic cell and the HF concentration of
the electrolyte. We also found that the porosity of the PS varied only slightly when glycerol at various
concentrations was used. This investigation shows that an interferometric technique could be a useful tool
for measuring the etch rate and the interface roughness of the PS.
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1
Introduction
Porous silicon (PS) can be easily used to fabricate into optical structures such as waveguides [1], optical
filters [2, 3] or luminescent microcavities [4] because its refractive index can be easily modulated.
Chemical and biological sensors based on modulation of the optical properties of PS have also been
made [5]. The interface roughness between PS and silicon (Si) plays a very important role in PS optical
devices. Light transmission can undergo severe losses through scattering if the interfaces are not smooth.
There are many parameters that can affect the roughness of the PS. For example, the type and resistivity
of the Si substrate, the composition of the electrolyte, i.e. the ratio between HF, H2O and ethanol (EtOH),
the current density (J) and the temperature of the electrolyte. The latter two parameters have been studied recently [6, 7] where it was shown that lowering the temperature decreases the interface roughness of
PS which was attributed to the increase of viscosity. Glycerol will also increase the viscosity and was
reported to reduce the roughness of PS-substrate interface [7].
2
Experimental methods and results
In this study we used glycerol as a partial replacement for ethanol in an electrolyte with ratios from
10 –70%. The electrolyte consisted of HF:(EtOH:glycerol). The effect of different current densities and
HF concentrations was also measured. The etching solution was prepared from 40% HF diluted with
EtOH in a ratio depending on the desired percentage of HF. The PS-substrate interface movement and
roughness were measured in-situ during etching by a simple interferometric technique with an infrared
diode laser beam incident from the dry backside of the sample. Experiments were conducted at room
*
Corresponding author: e-mail: y.y.kan@fys.uio.no
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1534
P. Y. Y. Kan et al.: The effect of etching with glycerol, and the interferometric measurements
-
Pt
Fig. 1 (online colour at: www.pss-a.com) Schematic diagram of
interferometric measurement on the interface roughness of PS. The
wavelength of the infrared laser is 1.31 µm.
HF:EtOH
Si wafer
Copper
+
Teflon
detector
IR Laser
mirror
temperature with p+-type Si wafers. The average porosity for each experiment was determined by the
gravimetric method. The obtained average thickness was typically around 50 µm. The starting material
was p+-type Cz (100) silicon, double-side polished with a thickness of 520 µm. The resistivity was 0.01 –
0.02 Ω cm.
The experimental setup for in-situ interferometric measurement during etching is shown schematically
in Fig. 1. A two-electrode Teflon cell was placed on a stand equipped with two inclined mirrors that
guided the infrared (IR) laser beam (λ = 1.31 µm, maximum power is 8 mW, spot size is 1 – 2 mm in
diameter). The copper electrode had the centre opened so that the laser light could reach the Si wafer and
interference could be produced from the bulk Si and the PS layer while etching. A constant current density of 5 to 30 mA/cm2, supplied from a Keithley 2400 current source, was applied for 1 hour while the
interference signal was sampled in each experimental run. An example of the measured interference
signal is shown in Fig. 2. The signal amplitude clearly falls off with time which is an indication of increasing roughness.
For easy parameterization we have defined ‘roughness’ here as a percentage which is calculated, as
illustrated in Fig. 3a, from the difference between the modulation start amplitude, Amax, and that at time t,
A(t):
Roughness = (Amax – A(t))/Amax · 100 .
(a)
30% glycerol
Intensity (a.u)
15%HF 10mA
(1)
0
10
20
30
40
50
60
Time (min)
(b)
Intensity (a.u)
Intensity (a.u)
(c)
0
2
4
6
Fig. 2 Enlarged views for 60 minute
measurement of IR laser reflection from the
backside of the sample during etching: (a)
original interference pattern; (b) zoomed
fragment of first 6 minutes; (c) last 6 minutes.
t
54
Time (min)
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
56
58
Time (min)
60
Original
Paper
phys. stat. sol. (a) 202, No. 8 (2005) / www.pss-a.com
1535
2
PS
Amax
Fig. 3 (a) Diagram showing a method for
determining the PS roughness. Amax is the
maximum amplitude, A(t) is the amplitude at
time t. (b) Reflection from various interfaces
in a PS sample.
3
Si
A(t)
1
laser
Time
a)
b)
Figure 3b illustrates the reflection from various interfaces in a PS sample. Interference between the partially reflected IR beams (1, 2, 3) can easily be measured. The amplitude of the signal will be dependent
on the roughness of the interfaces. Beams (2) and (3) may get weaker as etching proceeds and as the
reflected light undergoes diffuse scattering which gives weaker interference signals. Thus the modulation
amplitude decreases (narrower width) with time (see Fig. 2a). The waveform shape and periodicity can
be analyzed by Fourier decomposition to monitor etching rate and porosity as a function of time, which
will be presented in future works. The roughness parameter as here defined is influenced by uneven etch
rate over the whole area of the laser beam, fluctuations in the effective dielectric constant of the PS layer
due to bubbles and by scattering of the beam. Both interfaces will contribute. A good review of common
ways of quantizing roughness can be found in Ref. [8].
A plot of the roughness against the PS thickness is shown in Fig. 4. There was not much difference
in the roughness for the 26% HF sample (Fig. 4a) as the glycerol percentage was varied. However,
for the sample in Fig. 4b (15% HF, porosity of 70%), there was a large difference in the roughness between samples with 0% and 10% glycerol. The sample with a 10% glycerol replacement of ethanol indicates a relatively smooth interface, whereas the one without glycerol shows a very high roughness percentage.
The 30 and 70% glycerol samples also show a smoother surface than the 0% glycerol sample
(Fig. 4b), but these are comparatively rougher than the 10% case. This implies that glycerol is an effective agent for smoothing the interface roughness. However, the results also indicate that a high glycerol
content (>10%) would not be as effective as the 10% sample. This could be due to the fact that the content of ethanol also plays an important role in the etching solution, which can be speculated is related to
Time (min)
10
100
30
Time (min)
40
50
20
60
40
60
80
100
Roughness (%)
0% glycerol
10%
"
30%
"
50%
"
70%
"
80
Roughness (%)
20
60
40
80
0% glycerol
10%
"
30%
"
70%
"
60
40
porosity ~45%
20
20
26%HF
2
10mA/cm
(a)
0
0
10
20
30
Thickness (um)
40
50
60
0
0
15%HF
2
10mA/cm
porosity ~70%
(b)
10
20
30
40
50
60
Thickness (um)
Fig. 4 Graphs showing the calculated roughness change in samples etched with different electrolytes. (a) An electrolyte with 26% HF is used with changing glycerol content, and (b) an electrolyte with 15% HF is used. In the 15%
case it is clear that 10% glycerol gives the lowest roughness. In the 26% HF case the effect is not so large.
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
P. Y. Y. Kan et al.: The effect of etching with glycerol, and the interferometric measurements
(a)
15%HF
(b)
0% glycerol
10%glycerol
5mA
5mA
Intensity (a.u)
15%HF
Intensity (a.u)
1536
10mA
12mA
10mA
12mA
15mA
15mA
20mA
20mA
0
10
20
30
40
50
0
60
10
20
30
40
50
60
Time (min)
Time (min)
Fig. 5 Interferometric results from the15% HF samples at different current densities, 5 – 20 mA/cm2 for 60 minutes.
Plot (a) with 0% and (b) with 10% glycerol.
the dependency of bubble formation on viscosity and surface energy during the etching, where ethanol
has a favourable effect.
Figure 5 shows the interferometric results from the 15% HF (porosity 70%) samples, one with 10%
glycerol and one without, and the current density varied from 5 to 20 mA/cm2. By observing the modulation amplitude of the interference graphs for the 0% glycerol samples (Fig. 5a), it is evident that
the roughness has an irregular trend. This means that the roughness as measured here varies for nominally identical conditions. However, the trend seems to be steady for the 10% glycerol samples. For all
samples investigated we always produce low roughness with that percentage of glycerol. Detailed analysis indicates that the 10% samples did indeed smooth this irregular trend, as is shown in Fig. 6 (15%
HF).
Other samples with different HF concentration (13– 26% HF, porosity 85– 50%) are also presented in
Fig. 6 which all indicate a steady trend from the 10% glycerol samples. Largely varying roughness for
nominally identical experimental conditions can only be observed from the samples with the higher porosity and without glycerol replacement (13% and 15% HF samples). This indicates that the roughness of
the high porosity samples, at least in respect to the current measurement method, is critically dependent
13%HF
15%HF
26%HF
20%HF
0% glycerol
10% "
roughness
80
40
0
0
10
20
10
20
30
10
20
30
10
20
30
2
J (mA/cm )
Fig. 6 (online colour at: www.pss-a.com) A plot of roughness against current density at 13% to 26% HF which
correspond to the porosity of 85 to 50%, respectively. The 15% HF sample with 0% glycerol has two attempts of
measurements.
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Original
Paper
phys. stat. sol. (a) 202, No. 8 (2005) / www.pss-a.com
1537
100
100
(a)
13% HF
15% "
26% "
80
Porosity (%)
Porosity (%)
80
60
40
13% HF
15% "
26% "
20
60
40
0.01 ohm·cm
2
10mA/cm
20
0.01 ohm-cm
(b)
solid line - 0%glycerol
dot line - 10%glycerol
0
0
0
5
10
15
20
2
J (mA/cm )
25
30
0
10
20
30
40
50
60
70
% glycerol
Fig. 7 A plot of porosity against current density (a), and against glycerol percentage (b), at 13% to 26% HF. Differences between 0% and 10% glycerol are also shown in (a).
on the etching parameters. The addition of glycerol seems to yield conditions which are in a part of the
parameter space which is much less sensitive to either the etching conditions or parameters of the roughness measurements. It has been suggested [7] that the smoothness of interfaces of PS and Si can be understood in terms of diffusion limited asperity smoothing. The nature of the roughness on several length
scales should be investigated further. Lérondel et al. [8] has made measurements of the evolution of
interfaces which suggest a saturation. The present experiments were not designed to test this.
From an optical application point of view a key issue is whether the same porosity and the same optical properties can be achieved with and without glycerol. To ensure that the 0% and 10% glycerol replacement is comparable, their porosities should be in a narrow range. We thus carried out a porosity
analysis on each sample and the results are shown in Fig. 7a. Interestingly, they only show a little difference, the porosities between samples with 0% and 10% glycerol varied only slightly, indicating that the
results were comparable.
Figure 7b shows the porosity of the samples made with different percentage of glycerol replaced with
ethanol, measured at three different concentrations of HF. The lower percentage of HF (13% HF) has a
higher porosity (85%), whereas the higher percentage of HF (26% HF) has a lower porosity (45%). The
porosity change with varying glycerol percentage at a particular HF concentration does not vary too
much, but seems to be in an acceptable range. For the 15% HF sample, the porosity was only varied from
60 to 70% as the glycerol content changed from 0 to 50%.
3
Conclusion
The simplicity and the usefulness of the IR laser in-situ interferometric measurement of etch rate and
roughness have been demonstrated, and the effect of glycerol on PS interface roughness have been
measured. These measurements show that glycerol is effective for smoothing the interface roughness of
the PS for a small range of glycerol concentrations and for certain HF concentrations, while glycerol
replacement has little or no effect when higher concentrations of glycerol, up to 70% replacement of
ethanol, are used. The 10% glycerol replacement in 15% HF solution seems to be the most effective in
smoothing the interface compared to no glycerol.
Acknowledgements We thank Chetna Shukla and Erik Marstein for help during the initial stages. This work is
supported by the Norwegian Research Council.
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1538
P. Y. Y. Kan et al.: The effect of etching with glycerol, and the interferometric measurements
References
[1]
[2]
[3]
[4]
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[8]
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G. Lammel, S. Schweizer, and Ph. Renaud, Sens. Actuators A 92, 52 (2001).
L. Pavesi, C. Mazzoleni, A. Tredicucci, and V. Pellegrini, Appl. Phys. Lett. 67, 3280 (1995).
M. Cazzanelli, C. Vinegoni, and L. Pavesi J. Appl. Phys. 85, 1760 (1999).
V. S.-Y. Lin, K. Motesharei, K.-P. S. Dancil, M. J. Sailor, and M. R. Ghadiri, Science 278, 840 (1997).
S. Setzu, G. Lérondel, and R. Romestain, J. Appl. Phys. 84, 3129 (1998).
M. Servidori, C. Ferrero, S. Lequien, S. Milita, A. Parisini et al., Solid State Commun. 118, 85 (2001).
G. Lérondel, R. Romestain, and S. Barret, J. Appl. Phys. 81, 6171 (1997).
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Paper III
S.E. Foss, P.Y.Y. Kan and T.G. Finstad
In situ porous silicon interface roughness characterization by laser interferometry
Accepted for publication in the Proceedings of the
3rd Pits and Pores symposium, 206th Meeting,
ECS, Hawaii, 2004
III
IN SITU POROUS SILICON INTERFACE ROUGHNESS
CHARACTERIZATION BY LASER INTERFEROMETRY
S.E. Foss, P.Y.Y. Kan and T.G. Finstad
Department of Physics, University of Oslo
P.O.Box 1048 Blindern, N-0316 Oslo, Norway
ABSTRACT
In situ laser reflection measurements during etching of porous silicon
(PS) films are used for analyzing the time dependency of interface
roughness, etch rate and porosity. The interaction of an IR laser beam
with a time changing layered system of PS and substrate results in an
interference effect in the reflected beam which is analyzed by Short-Time
Fourier Transform (STFT). Using this method, the effect on roughness of
different temperatures, different etchant solutions and different formation
current densities is measured. Calculated roughness values are in
agreement with other methods.
INTRODUCTION
After nearly 15 years of intense research into the optical properties of porous silicon
(PS) there is still a great activity today. This activity has increasingly been focused on the
application of PS in devices such as Si-based integrated optical circuits (IOC) (1) and
chemical microsensors (2). The first application of PS for passive optical devices came
with the development of multilayer optical Bragg reflectors by Vincent (3) and Berger et
al. (4).
One critical and limiting aspect of using PS for many optical devices is the inherent
roughness at interfaces developed during the etching process, especially the PS/substrate
interface. This roughness will scatter light and degrade the optical quality of the device.
The roughness is usually described by a root-mean-square (rms) surface height function.
Silicon has a very large absorption for wavelengths below about 1.1 µm, but freestanding transmission filters and reflection filters on substrates are still possible for this
range. However, the low absorption and less scattering in near-infrared (NIR) above 1.1
µm makes PS best suited for optical filters in this range. Still, a tight control of the
interface roughness is necessary to obtain the optical quality needed for a given
application. This problem has been studied extensively by Lérondel et al. (5,6) as well as
by Setzu (7) and Servidori (8).
It is apparent that the interface roughness is influenced by many factors during
etching, such as substrate doping (concentration and distribution), temperature,
electrolyte composition (such HF concentration and glycerol content), formation current
density and time. With so many parameters, and the formation mechanism of PS still
somewhat unresolved (9), a full understanding of PS interface roughness has not been
presented. This paper deals with roughness evolution, in PS etched in p+-type Si varying
electrolyte composition, temperature and formation current density. The aim of this study
will be to find an optimal combination of parameters to make thick NIR transmission
filters in PS. The method used is based on measuring the roughness dependent decrease
in specular reflection from the PS/substrate interface using a monochromatic beam from
an IR diode laser and analyzing these results by applying the Davies-Bennett theory
described in (5,10-12). This measurement is done in situ during PS formation and also
gives porosity and etch rate data. The principle of in situ reflection measurement was
used by Steinsland (13) for Si-etching in TMAH and by Gaburro (14) and Thönissen (15)
for etch rate and porosity monitoring in PS. Some of the measurements analyzed here
have been used earlier in (16), but there rms values were not obtained.
The choice of p+ Si for etching PS for optical applications is done based on the large
obtainable porosity range and the reported comparably low interface roughness. There
are, however, a few challenges with p+ PS. Because of the high dopant concentration,
absorption will be slightly higher compared to p and p- PS. The high concentration also
gives rise to larger spatial fluctuations in the dopant concentration, often referred to as
striations, hence spatially varying etch rates and porosities are obtained. These
fluctuations are often observed as concentric rings in the PS, as the refractive index is
affected during etching (6), or as ridges on the interface surface after PS stripping.
EXPERIMENTAL DETAILS
Boron doped p+ Cz-Si with a resistivity of about 0.018 Ohm·cm, double side polished
and a thickness of 525 µm was used for PS formation. HF concentration, current density,
glycerol contents and temperature were all varied to determine the effects of these
parameters on roughness. HF concentrations used were 15 and 26 volume % made from
40 % aqueous HF, while the rest of the electrolyte consisted of ethanol or an
ethanol/glycerol mix. Glycerol contents used were up to 70% of the total ethanol/glycerol
volume. Current density was varied from 5 to 30 mA/cm2, and the measurements were
done at two temperatures, room temperature (23 °C) and 4 ±2 °C. Samples were etched
up to 120 minutes. PS was etched on a 1 cm diameter circular area on the samples. The
back sides were metallized with Al to form an Ohmic contact except for an area in the
center where the contact was removed to give access to the laser beam. Depending on the
chosen beam diameter the opening in the contact was either 0.5 or 3 mm in diameter.
A standard, upright, etch cell with a Cu-plate back-contact was used for the PS
etching. To measure roughness with different spatial wavelengths, two laser beam
diameters were used. For the narrow beam, a multimode graded index fiber with a
diameter of 62.5 µm was cut and placed as close to the substrate back side as possible
through a small hole in the contacting Cu-plate. Most of the reflected signal would come
from an area on the interface with the same shape as the cut off fiber core as the
reflecting plane is essentially flat. This setup was used to measure roughness with a
higher spatial frequency than that caused by striations, as the beam in this case would
most likely fall between two striation ridges. In one paper the distance between striations
(spatial wavelength) was measured to be about 250 µm (5).
The other setup used a collimating GRIN lens in connection with the fiber to give a
collimated beam of 2 mm diameter passing through a larger hole in a contacting Cu-plate.
The beam was oriented normal to the sample and the lens both collimated the incident
beam and collected the reflected beam. A sketch of the setup is shown in Fig. 1. A series
of experiments used a setup where the beam directed at the sample was not coupled
through fiber. This free space laser interferometer has been described before (16).
Figure 1 Sketch of the measurement setup. A standard etch cell is modified so that an IR diode laser
(LD) beam may be directed from a multimode (MM) optical fiber towards the back side of the sample
and the reflected beam collected back into the fiber, either via a collimating GRIN lens (left image) or
by direct coupling (right). The return signal is returned by a 2x1coupler to a GaAs PIN detector. The
LD is first connected to a single-mode fiber with an optical isolator, then it is connected with one input
of the MM 2x1 coupler. While etching, the PS/substrate interface moves giving rise to interference in
the reflected beam.
An IR laser diode controlled to have constant temperature and power at a wavelength
of 1550 nm pigtailed to a single-mode optical fiber was used as a monochromatic,
coherent light source. At this wavelength Si is partially transparent. To minimize back
reflectance to the diode and keep the phase stable, optical isolators were added. At the
sample end the single mode fiber was connected to a multimode fused 2x1 coupler to
make back reflectance from the sample back side into the fiber end easier (large core) and
to facilitate reading of the reflected signal, see Fig. 1. A fiber coupled PIN GaAs
photodiode was used to detect the reflected signal.
For comparison, the roughness and surface profile of a few samples were measured by
white-light interferometry (WLI) (WYKO NT-2000). To cover as large area as possible
of the sample, a low resolution of 6.22 µm per pixel was chosen. This would filter out
any high frequency component, but this kind of roughness was not expected to be
significant. These measurements were done after stripping the PS layer away by etching
in NaOH. To get a reasonable etching time with the thick layers, up to 200 µm, a
concentrated solution was used, even though this attacked the area around the PS as well.
THEORY AND METHOD
During etching, the reflected laser beam from the sample contains a combination of
beams partly reflected from several interfaces; sample backside, PS/substrate interface
and PS/electrolyte interface. As the PS/substrate interface is moving an interference
pattern will appear in the signal. The analysis of this signal is based on the Short-Time
Fourier Transform method (STFT), which gives a time-resolved frequency
decomposition of the signal. From this it is possible to extract interferences between
different partially reflected beams. The principle is shown in Fig. 2 where the most
important beams and combinations are shown. The extracted frequencies and amplitudes
of the different interference signals give information on porosity, etchrate and interface
scattering (14,16). In Fig. 3 a spectrogram from one such STFT calculation is shown,
with traces of the two lowest order reflections shown as thick black lines and higher order
reflections shown as thin black lines. These two lower order traces correspond to the two
cases in Fig. 2. To obtain information on the PS/substrate interface roughness, the
interference between the beam reflected from the substrate backside and the beam
reflected specularly from the substrate/PS interface was chosen, corresponding to case 2
in Fig. 2
The intensity of the combined reflection of these two beams, Iref, is given by
2
2
I ref = Asub
+ APS
+ 2 Asub APS cos(ϕ sub − ϕ PS ) ,
[1]
where Asub and APS are the amplitudes and φsub and φPS are the phases of the beams
reflected from the substrate back side and the PS/substrate interface respectively. As the
optical path length of the beam within the substrate changes during etching, φPS changes.
This will lead to an oscillation in Iref with a frequency given by d (ϕ sub − ϕ PS ) dt . This
frequency may not be constant as the etch rate generally will change with time or depth.
The cosine component of Iref gives the amplitude and frequency of this oscillation and is
traced in the spectrogram. The oscillation amplitude is given by the cosine prefactor. Asub
is constant while APS contains information on the PS/substrate interface scattering,
substrate absorption and PS refractive index (in the electrolyte):
APS (t ) = t air − subα sub (t ) s R , sub − PS (t ) rsub − PS (t )α sub (t )t sub − air .
[2]
Here tair-sub and tsub-air are the transmission amplitude coefficients, rsub-PS(t) the time
dependent reflection amplitude coefficient as the PS refractive index changes during
ethcing, sR,sub-PS(t) the time dependent scattering factor as the roughness changes with
time and αsub(t) the absorption factor which changes due to a decrease in substrate
thickness.
The scattering factor is calculated by fitting the extracted amplitude to a double
exponential as in Fig. 4, where a reflection amplitude from a wide beam measurement has
been extracted from the spectrogram and fitted. Fitting is done to avoid fluctuations
caused by crossing, interfering signals in the STFT as can be seen in Fig. 3. The fitted
amplitude is then scaled so that the extrapolated value at t=0 corresponds to the
theoretical amplitude with zero scattering. The scaling is necessary because the measured
data are not normalized to unit reflectance. The STFT uses a movable windowing
function which results in average amplitude values over the time range the window
covers, hence the STFT amplitude values for t=0 are correct for the signal at a time in the
middle of the first window. This necessitates an extrapolation of amplitude to t=0. The
fitted and scaled amplitude function corresponds to APS(t). Transmission and reflection
amplitude coefficients, as well as the
absorption factor, are calculated using
published data for the complex refractive
index of bulk Si. Normal Fresnel relations
are used for transmission and reflection
coefficients, while the absorption factor is
given by α sub (t ) = exp(− 2π kz (t ) λ ) where k
is the imaginary part of the complex
refractive index and z(t) is the time
dependent substrate thickness.
Figure 3 Spectrogram from STFT of an IR
reflectance signal from the back of a sample
during PS etching. Thick black line indicates
traces of the two lowest order beam
interferences used for roughness, porosity and
etch rate calculations, while thin black lines
indicate traces of higher order beam
interferences. The numbering of the traces
correspond to the two cases in Fig. 2.
0.14
0.12
Amplitude, a.u.
Figure 2 A diagram showing the reflected
beam composition. The reflected signal will
contain interference oscillations from the
combination of beams I and II (case 2) and
beams I and III (case 1) as well as other, higher
order combinations. For PS/substrate
roughness analysis, the signal from case 2 is
extracted and used in calculations. This
corresponds to the upper main trace (thick line)
in the spectrogram in Fig. 3.
0.10
0.08
Davies-Bennett theory (10,11) attempts
0.06
to describe the local phase change in the
0.04
reflected plane wave front introduced by the
height irregularities of the interface. These
0.02
phase changes results in a reduced intensity
in the specular direction as conditions for
0.00
0
10 20 30 40 50 60
destructive interference will develop
between different parts of the wave front.
Time, min
The theory assumes an rms irregularity
Figure 4 Reflection amplitude from the upper trace
height (roughness) value, σ, much smaller
(marked 2) of a spectrogram equivalent to Fig. 3.
than the wavelength of the incident light in
The dotted line is the fitted double exponential done
the medium, λ, and that the height function
to avoid amplitude fluctuations caused by
describing the roughness has a Gaussian
interfering signals.
distribution. In this case the Fresnel
reflection coefficient for a perfectly flat surface, R0, may be altered to incorporate the
effect of interface scattering at an incident angle in the medium, θsub:
Rsub = R s
2
0 R , sub − PS
⎡ ⎛ 4πσ
sub − PS n sub
= R0 exp ⎢− ⎜⎜
⎢⎣ ⎝ λ0 cos θ sub
⎞
⎟⎟
⎠
2
⎤
⎥,
⎥⎦
[3]
where nsub is the refractive index of the incident medium, λ0 the wavelength in vacuum
and σsub-PS the PS/substrate interface roughness. Note that the spatial frequency of the
roughness does not enter into this equation, so both long period (striations) and short
period (small scale) roughness will have an equal effect, depending only on σsub-PS.
To calculate the rms roughness of the front surface, a scattering factor for
transmission, sT,sub-PS, must be calculated based on the obtained σsub-PS. The transmission
coefficient at the PS/substrate interface has been shown by Filiński (12) to be
Tsub = T0 sT2 , sub − PS
2
⎡ ⎛
⎞ ⎤
n sub
n PS
⎞
⎛
−
⎢ ⎜ 2πσ ⎜
cos θ sub
cos θ PS ⎟⎠ ⎟ ⎥
⎝
⎟ ⎥,
= T0 exp ⎢− ⎜
⎟ ⎥
λ0
⎢ ⎜
⎟ ⎥
⎢ ⎜⎝
⎠ ⎦
⎣
[4]
for the same conditions as for the reflection coefficient in Eq. 3. Here T0 is the
transmission coefficient for a perfect interface, while nPS is the refractive index of the PS
and θPS is the angle in the PS layer. Then the reflected amplitude from the front becomes
2
2
A front (t ) = t air − subα sub
(t )sT2,sub− PS (t )t sub− PS (t )α PS
(t )s R, PS − front (t )rPS − front (t )t PS −sub (t )t sub−air
[5]
where most of the parameters are the same as in Eq. 2 with the addition of a time
dependent transmission amplitude coefficient for the PS/substrate interface, tsub-PS(t), a
time dependent reflection amplitude coefficient for the PS/front interface, rPS-front(t), and
an absorption factor in the PS layer, αPS(t), defined as for αsub(t) with kPS calculated by
effective medium theory using the complex refractive index of bulk Si. The reflected
amplitude is given by the same calculations on the extracted amplitude of the signal in
case 1 in Figs. 2 and 3 as for APS. The calculation of the scattering factor, sR,PS-front, is the
same as for sR,sub-PS and gives the rms roughness value at the front interface.
RESULTS
Temperature effect
Assuming that the reflectance measurements give a good estimate of the roughness at
the beam spot, and that the roughness in this limited area is an indication of both
roughness time evolution and the roughness of the whole sample, the data plotted in Fig.
5 shows that roughness of samples etched with 15 % HF are influenced by etchant
temperature while those samples etched with 26 % HF are not significantly influenced.
The porosity and etchrate data for the 15 % samples are calculated and shown in Fig. 6,
as well as data for low temperature etching with 15 % HF and 1:9 glycerol:ethanol. There
is an increase in porosity and decrease of etch rate with decreased temperature. The
porosity data describe the instantaneous porosity closest to the interface, and therefore
also gives a porosity profile of the PS layer with depth. When calculating these data the
dissolution of PS closer to the surface, i.e. chemical dissolution, has been disregarded as
this is very small for p+ PS. Note the sharper increase in the RT porosity profile
compared to the low temperature profile. A similar effect has been reported by Servidori
in (8) for PS in p-type Si.
15%,RT
26%,RT
26%,LT
1.4
0.80
15%,LT
Porosity, abs.
Roughness, rms nm
120
100
15 % HF, LT, 0 % Gly
15 % HF, LT, 10 % Gly
0.85
80
60
40
1.2
0.75
15 % HF, RT, 0 % Gly
0.70
1.0
15 % HF, RT, 0 % Gly
0.65
15 % HF, LT, 10 % Gly
20
0.8
0.60
0
Etch rate, µm/min
140
15 % HF, LT, 0 % Gly
0.55
0
20
40
60
80
100
Thickness, µm
Figure 5 Roughness dependence on temperature
for different HF concentrations. In the 15 % HF
case there is a decrease in roughness with
temperature while the 26 % HF case show no
such dependence.
0
20
40
60
80
100
120
0.6
Time, min
Figure 6 Comparison of porosity and etch rate
between low temperature (4 °C) and RT etched
samples, with electrolyte consisting of 15 % HF and
0 % or 10 % glycerol. Current density used was 15
mA/cm2 for all three.
Glycerol effect
The effect of replacing 10 % of the ethanol volume with glycerol on roughness at low
temperature is shown in Fig. 7 where a 15 % HF solution is used. In this case roughness
increases with the added glycerol compared to roughness measured without. A
comparison with results obtained at RT, shown in Fig. 9, may indicate an explanation.
These samples are etched with varying glycerol percentage from 0 to 70 % and HF
concentrations of 15 and 26 %. It is evident that a significant decrease in roughness is
obtained when replacing 10 % ethanol with glycerol at RT in the 15 % HF case. At low
temperature the viscosity of the electrolyte increases compared to at RT. By adding 10 %
glycerol to the electrolyte at low temperature, the viscosity increases further and the
roughness evolution is comparable to the evolution with a glycerol content higher than 10
% at RT. In Fig. 9 it is evident that roughness increases with glycerol content above 10
%. As seen for the low temperature porosity and etch rate data in Fig. 6, there is very
little difference between the samples with and without glycerol. This differs from the
same situation at RT in Figs. 8 and 9, where roughness decreases, porosity increases and
etch rate decreases by adding 10 % glycerol.
For other glycerol concentrations than 10 % in the 15 % HF case, the effect of adding
glycerol is minimal. Below 25 µm the roughness is greatest with no glycerol and
1.4
0.80
0.75
15%, 10 % Gly.
120
100
15%, 0 % Gly.
80
60
Porosity, abs.
Roughness, rms nm
140
1.3
10 % Gly
0.70
1.2
0 % Gly
0.65
1.1
10 % Gly
0.60
1.0
40
0 % Gly
0.55
20
Etch rate, µm/min
160
0.9
0
0.50
0
20
40
60
80
100
Thickness, µm
Figure 7 Rms roughness plotted vs. PS layer
thickness. There is a change in the roughness
evolution when adding glycerol to a 15 % HF
electrolyte at low temperature. Samples are
etched at 15 mA/cm2.
0
10
20
30
40
50
Time, min
Figure 8 Porosity and etch rate obtained by reflection
measurements showing the dependence on
temperature and time. Samples were etched at RT
with a 15 % HF solution.
decreases with decreasing glycerol concentration from 70 % and down, however, for
higher concentrations of glycerol or no glycerol at all the roughness seems to stabilize,
while for lower concentrations of glycerol the roughness steadily increases with PS
thickness. A similar comparison is done in the 26 % HF case. Here the final roughness
increases with increasing glycerol concentration, with the sample etched with no glycerol
having the lowest roughness, although for thicknesses up to about 20 µm the electrolyte
with 20 % glycerol gives the lowest roughness.
HF concentration, current density effect and roughness saturation
Comparing the data for 15 and 26 % HF in Fig. 9 a general tendency is apparent, most
of the data for 26 % HF shows a lower rms roughness than for 15 % HF, with the
exception of the 15 % HF + 10 % glycerol plot. However, HF concentration is reported to
have no effect on roughness in p and p- PS (5). When etching with 15 % HF and 10 %
glycerol and changing current density from 5 to 30 mA/cm2 a smaller spread in roughness
values is obtained compared to the same experiment done with no glycerol (16). In fact,
at a 20 µm PS layer thickness the rms roughness varies from 30 to 70 nm with 10 %
glycerol and from 40 to 110 nm without. However, there is no apparent systematic effect
of current density on roughness in these data. This differs with the case of etching whit a
26 % HF solution. Here the roughness seems to increase with current density. At 60 µm
thickness rms values are 66 nm, 75 nm and 92 nm for 10, 20 and 30 mA/cm2
respectively. According to (5), roughness decreases with an increase in current density in
p and p- PS.
Roughness, rms nm
15 % HF
26 % HF
120
100
80
60
0 % Gly
10 % Gly
20 % Gly
30% Gly
70 % Gly
0 % Gly
20 % Gly
30 % Gly
40 % Gly
40
20
0
0
20
40
60
80
0
20
40
60
Figure 9 Comparison of
rms roughness data for
different combinations of
HF concentrations and
glycerol amounts. All
samples are etched at RT
and with 10 mA/cm2.
80
Thickness, µm
In (5) two distinct regimes are shown to exist in the roughness evolution for p and ptype PS. With roughness plotted against layer thickness in a log-log plot, it is clear that
roughness enters a saturation regime after a linearly increasing regime. In the data
discussed here, roughness clearly increase faster in the beginning, however, in a log-log
plot some show a clear correspondence with the two-regime picture, while some are
closer to linear. In Fig. 10 a sample etched at RT in 15 % HF with 10 mA/cm2 shows a
saturation regime while a sample etched with 20 mA/cm2 does not show any such
saturation. The saturation regime is explained as a result of a transition from hole limited
to a diffusion limited etching. A diffusion limited etching will be dominant when
diffusion of reaction species is decreased due to increased thickness of the porous layer,
hence fluctuations in resistivity at the PS/substrate interface have no influence on hole
availability and an asperity smoothening effect occurs. Another indication of the change
in diffusion is the porosity depth profile and etch rate change with time obtained for all
the samples investigated in this study, as in Figs. 6 and 8.
2
The effect of spot size
Roughness, rms nm
By using different beam widths it is
possible to probe different roughness
spatial wavelengths which are not
accounted for in Eq. 3. The period of the
striations as reported in (5) and WLI
measurements reported here, is in the
order of a few hundred µm. By using a
spot size large enough to cover several of
these oscillations, good statistics are
obtained on the scattering effect.
However, a spot size smaller in diameter
than the striation wavelength, as obtained
with a cut fiber end giving effectively a
62.5 µm spot, will not be scattered much
by this roughness. Only roughness of
shorter wavelengths will scatter the
incident light. By using this spot size it
15 %HF, 0 % Glycerol, RT, 10 mA/cm
100
90
80
70
60
50
40
30
20
2
15 %HF, 0 % Glycerol, RT, 20 mA/cm
10
1
10
100
Thickness, µm
Figure 10 Plot of roughness showing clearly the
increase and the saturation regime in the low current
case, and the lack of saturation in the high current
case. Both samples etched in 15 % HF at RT without
glycerol. The line is only a guide for the eye.
was not possible to obtain a value for the rms roughness, indicating that short period
roughness was below the detection limit of about 10 nm (5) roughness determination by
specular reflection, hence practically all of the surface irregularity is caused by striations.
Roughness of the PS/electrolyte interface
Analysis of the PS/electrolyte interface roughness was attempted. As explained in the
theory and method part, both the signal from case 1 and case 2 in Figs. 2 and 3 are used
to obtain this roughness. Due to the small roughness (<10 nm) at this interface as
reported by Servidori et al. (8), no reliable data were obtained. The absorption in PS
during etching will likely be large due to H2 bubble formation during etching of PS, this
is not accounted for in the calculation of the absorption factor.
Comparison between reflection and WLI
measurement
To verify that the method of roughness
calculation by specular reflection gives a good
indication of roughness, WLI measurements
were performed on a few samples that were
stripped of PS. A 2 mm diameter circular area
in the center of the samples where the laser
beam was thought to have been, was extracted
Figure 11 Surface plot of PS/substrate
from the WLI data. The rms roughness values
interface height data obtained by WLI
from these areas are compared to the
measurement. Parallel ridges with at least two
maximum roughness values obtained by
distinct spatial repeatability wavelengths are
reflection in Table 1 and show an excellent fit. clearly visible. These are caused by slightly
The true position of the beam is not known, so different etch rates due to inhomogeneities in
resistivity, i.e. striations. Rms irregularity
these values are only indicative of what is
height is in this case about 180 nm.
likely. These samples were all etched for long
times, up to 130 minutes, and were therefore quite deep, 2 00 µm. Figure 11 shows a
surface plot of one such measurement. The parallel ridges caused by striations are clearly
visible. The rms roughness values obtained by this method show a very strong positional
dependence with a total rms value over the whole etched area up to a factor 2 larger than
that measured by reflection.
Figure 11 indicates that the roughness has several spatial wavelengths. A comparably
small period roughness, with a wavelength of about 200 µm in the direction normal to the
ridges, is evident in the smaller ridges in the surface plot, as well as a much larger period
roughness with a wavelength in the order of mm. This large period roughness is very
irregular compared with the smaller and gives rise to the positional dependence of the
discussed WLI measurements, but both are most likely caused by an irregular dopant
distribution. The rms roughness values for the whole etched area are also tabulated in
Table 1. Note that this value is obtained after correcting for interface curvature, which is
caused by an inhomogeneous current distribution due to the opening in the Al back
contact and the design of the etch cell. This was done by an algorithm in the WYKO
analysis program. The large period roughness was not observed in WLI measurements
done on similar samples etched much shorter, to depths of about 10 µm. At this depth,
only striations similar to the short period ones in Fig. 11 were present, and the rms
roughness was a few tenths of nm. This may indicate a slower development of the long
wavelength roughness.
Table 1 Comparison between measured roughness rms values by reflectance, data from a 2 mm diameter
circular area at the center of each sample with white light interferometry (WLI) and data from the whole
sample by WLI. LT is low temperature (4 °C) and RT is room temperature. Note that thicknesses will not
be the same for different samples.
Sample
number
46
47
48
52
53
Etch condition
Small area
WLI (nm)
26%HF,15mA/cm2,RT,100min 157
15%HF,15mA/cm2,RT,120min 173
26%HF,15mA/cm2,RT,120min 168
26%HF,15mA/cm2,LT,120min 155
26%HF,30mA/cm2,LT,120min 187
Reflectance
measurement(nm)
157
170
169
153
186
Full area
WLI (nm)
272
336
273
429
339
Reflection measurements will give a good indication of the roughness caused by
striations of short period, but not of the longer period roughness. This is probably
satisfactory when monitoring the interface quality of optical elements with small feature
sizes, of the order of the beam diameter, or when etching PS layers some tens of µm
thick, however, to monitor the roughness evolution of interfaces larger than a few mm in
diameter a wider incident beam is needed.
Roughness, rms nm
Plots of calculated rms roughness values vs. depth from the reflectance measurements
for the samples that were analyzed with
WLI are shown in Fig. 12. The data
obtained for 26 % HF show a remarkable
consistency. It seems the roughness
15%RT 15 mA
160
evolution does not depend on current
density or temperature. The full area WLI
26%RT/5 deg/15 and 30 mA
data do not follow the same trends,
80
however, indication that the larger period
roughness is affected by the parameters in
a different way. Perhaps there is a
maximum correlation length over which
0
saturation of roughness is achieved, while
for distances larger than this there is little
0
80
160
correlation between diffusion constants in
PS layer thickness, µm
the electrolyte or local hole concentration
at the interface.
DISCUSSION AND CONCLUSION
Figure 12 Rms roughness calculated from reflection
measurements comparing the influence of
temperature, HF concentration and current density.
There seems, as shown, as if there is only a very narrow region of parameter space
where an increase in electrolyte viscosity by addition of glycerol is beneficial for
PS/substrate interface roughness in p+ PS, and even this is different for room temperature
and low temperature. All the parameters studied may affect the viscosity of the
electrolyte, or rather reaction species diffusion, as has been suggested (7,8). However, it
seems a too high “diffusion constant” increases roughness and the same for a too low
“constant”, at least in the case of p+ PS. The reduction of roughness seems to be limited
to a factor of maximum two for fairly thick layers. Substrate quality is of importance, as
the roughness discussed here is more or less completely dependent on dopant
distribution. It seems more difficult to control roughness of p+ PS compared to what has
been published on p and p- PS.
The laser reflection interference roughness measurement method presented in this
paper gives a good estimate of the roughness rms value within the laser beam spot,
whether this value is representative for the whole sample depends on the spatial
wavelength of the roughness. In the case of some of the p+ Si wafers used in this study,
dopant variations with several wavelengths are present in WLI measurements, at least
with wavelengths of about 200 µm and 1 mm. This may explain some discrepancies or
unexpected results within the presented data. PS/substrate roughness on a smaller scale
(micro-roughness) and low rms value striation roughness are undetectable with the
presented method. This is in agreement with other reports stating that microscale
roughness is very small at a p+ PS/substrate interface and that specular reflection
methods are reliably for rms roughness larger than 10 nm. There are both similarities and
differences between roughness progression in p+ PS and other p-type PS. There are
indications of a saturation regime for some of the samples, where the rms roughness
value increases little more with PS layer depth, while for others a saturation regime was
not reached within the measurement time/depth.
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3. G. Vincent, Appl. Phys. Lett., 64, 2367 (1994).
4. M. G. Berger, C. Dieker, M. Thönissen, L. Vescan, H. Lüth, H. Münder, W. Theiß, M. Wernke
and P. Grosse, J. Phys. D, 27, 1333 (1994).
5. G. Lérondel, R. Romestain and S. Barret, J. Appl. Phys., 81, 6171 (1997).
6. G. Lérondel, P. Reece, A. Bryant and M. Gal, Mat. Res. Soc. Symp. Proc., 797, 15 (2004).
7. S. Setzu, G. Lérondel and R. Romestain, J. Appl. Phys., 84, 3129 (1998).
8. M. Servidori, C. Ferrero, S. Lequien, S. Milita, A. Parisini, R. Romestain, S. Sama, S. Setzu
and D. Thiaudière, Solid State Comm., 118, 85 (2001).
9. X.G. Zhang, J. Electrochem. Soc., 151, C69 (2004).
10. H. Davies, Proc. Inst. Electr. Eng., 101, 209 (1954).
11. H.E. Bennett and J.O. Porteus, J. Opt. Soc. Amer., 51, 123 (1961).
12. I. Filiński, Phys. Stat. Sol. (b), 49, 577 (1972).
13. E. Steinsland, T. Finstad and A. Hanneborg, J. Electrochem. Soc., 146, 3890 (1999).
14. Z. Gaburro, C.J. Oton, P. Bettotti, L. Dal Negro, G. Vijaya Prakesh, M. Cazzanelli and L.
Pavesi, J. Electrochem. Soc., 150, C281 (2003).
15. M. Thönissen, M.G. Berger, S. Billat, R. Arens-Fischer, M. Krüger, H. Lüth, W. Theiß, S.
Hillbrich, P. Grosse, G. Lerondel and U. Frotscher, Thin Solid Films, 297, 92 (1997).
16. P.Y.Y. Kan, S.E. Foss, T.G. Finstad, Phys. Stat. Sol. a/c (in press).
Chapter 5
Filter fabrication
The goal of producing multilayer structures, or refractive index modulated
structures, by porosification of silicon, is to add the possibility of controlling
light, its spectral composition, phase, and movement, through a device.
In this thesis the focus of the ”device” fabrication is on the control of
the spectral component of incident light, while many of the findings and
techniques may be used for other elements within silicon photonics.
The principles of optical filter design has been described earlier in Chapter 3.
From these principles we may draw the conclusion that a good refractive
index control, layer thickness control, and maximum interface smoothness
is imperative to obtain filters of good quality. In PSM fabrication this
translates to control of porosity with depth and time, control of etch rate
with depth and time and minimization of interface roughness. To a certain
degree the control, or at least a knowledge of, the microtopology of the
porous structure and the relation to refractive index is also necessary as
discussed in Sec. 3.1.
A specific goal of this thesis has been to obtain good quality IR optical
filters of different kinds in PS. A parallel goal has been to use the flexibility
of PSM fabrication to extend the usability of the standard filter. In the
case of graded filter structures the obtained quality of the filters depends
strongly on the level of understanding of flat filter fabrication in PS as some
quality will be lost to the extra design freedom obtained. Ideally, a filters
spectral features will become sharper with an increasing number of layers.
This assumes no or very weak absorption. With PS, this is possible in the
IR, especially for wavelengths around 1500 nm where the absorption is at
a minimum.
To list some of the points we must take care of to be able to fabricate good
quality PSMs:
• Porosity must be known.
• Etch rate must be known.
97
98
• PS morphology should be known.
• Change in parameters/factors during etching (as a function of time,
depth, etch area geometry, structural ”history”) should be known.
• A model for calculating the effective refractive index as a function of
porosity (and morphology) must be known.
• The transition of effective refractive index across an interface between
two layers should be understood and controlled.
• The roughness of an interface should be understood and controlled.
• The effect of chemical etching of the porous structure should be taken
account of.
• A good understanding of the etching process is helpful; where and how
(only pore tip or partly up the walls of the pore, oxidation, passivation by hydrogenation of walls etc., the effect of incident light during
etching and bubble formation).
Most of these points have been or will be addressed in this thesis. Those
points not addressed are assumed to play relatively small roles in the overall
results, however, they are still likely to be measurable.
5.1
Basic filter etching
As a first approximation to filter etching, we may assume that the obtained
porosity during etching is only dependent on the applied current density
and the given etch conditions. Likewise, the etch rate may be assumed
constant with time and layer depth and only dependent on current density.
A standard way of obtaining these data is to etch several samples at different
current densities for a given depth or for a given time. Porosity may be
measured by gravimetry and etch rate may be derived from the etched
thickness which may be obtained by SEM or profilometry. From these data
a porosity versus current density curve and etch rate versus current density
curve may be plotted. The porosity curve fits well with a log function of
the type:
j
+1 ,
(5.1)
P (j) = P0 + γ ln
β
where P (j) and P0 are the current density dependent and constant contribution porosities respectively, j is the current density and γ and β are
fitting parameters. Porosity vs. current density data with a fit using Eq. 5.1
is shown in Fig. 5.1. The data was obtained from etching p+ samples in
13.3 % HF for 60 seconds at room temperature. The fit parameters in this
case are P0 =15, γ=6.5 and β=1.6. A similar approximation to the etch
99
rate versus current density may be obtained by using the simple model of
the PS formation described by Eq. 2.3 with porosity as a parameter. By
assuming the valence to be constant with current density and using it as a
fitting parameter, a relatively good fit of the etch rate may be obtained.
Figure 5.1: Porosity vs. current density data fitted to log
function, using Eq. 5.1. The
samples were etched at room
temperature with a 13.3 % HF
electrolyte.
Porosity (%)
40
30
Equation: P1+P2*ln(x/P3+1)
Chi^2/DoF
R^2
20
P1
10
0
20
40
= 0.29219
= 0.99545
15
±0
P2
6.49619
±0.38145
P3
1.60483
±0.33501
60
80
100
120
2
Current density (mA/cm )
A current-time profile used for the filter etching may be constructed on
the basis of the refractive index corresponding to the porosity curve as well
as the needed etch time for each layer based on the etch rate curve. The
resulting filters show acceptable characteristics compared to the designed
characteristics as long as the total etched thickness is relatively small. An
example is shown in Fig. 5.2, where the reflectance from a 10 layer pair
Bragg reflector designed for peak reflectance at 1500 nm has been measured
at 45◦ . The layer stack consists of layers of 53 % and 83 % porosity and
183 nm and 301 nm thickness, respectively. The calculated spectrum shown
in Fig. 5.2 is shifted towards the blue compared to the measured spectrum.
The most likely explanation for this, in this case, is that the fitting of the
porosity and etch rate curve was not optimal together with a small error in
the refractive index calculation caused by the use of the EMA as discussed
in Sec. 3.1.
The filter characteristics shown in Fig. 5.2 may be improved by increasing
the number of layers, and thereby the total thickness. The time or depth
dependence of the porosity and etch rate have already been presented in
Figs. 4.10–4.13. When etching thick filters using the method described
above, the introduced shift between upper and lower layer pair, or period,
will give a significant distortion compared to the designed optical characteristics. There are also other effects than the drift of porosity and etch
rate that will factor in as will be discussed below.
5.2
Deviations from the basic assumptions
A good quality IR filter designed with sharp spectral features may reach
a thickness of 50 µm or more. With an average etch rate, from Fig. 4.13,
100
1.0
Reflectance (abs.)
0.8
0.6
0.4
0.2
0.0
1000
1200
1400
1600
W avelength (nm)
Figure 5.2: Reflectance of a
Bragg reflector designed for a
peak reflectance at 1500 nm.
Consists of 10 pairs of layers of 53 % and 83 % porosity
and 183 nm and 301 nm thickness respectively. No adjustment of current density with
time was used. A shift of the
measured spectrum compared
to the calculated spectrum is
present. Filled squares denote
measured spectrum and open
circles denote the calculated
design spectrum.
of about 1.5 - 2 µm/min, this thickness is reached after about 25 - 35 min.
The change in porosity and etch rate may be substantial within this time.
This should be taken into account in the design of the current profile.
Besides the drift in porosity and etch rate with time which is suggested in
Sec. 4.2.4 to be caused by limitations in diffusion of electrolyte species to and
from the etch front, a change in temperature during etching may also introduce a change in porosity and etch rate from the expected values. Another
effect is the chemical etching discussed briefly in Sec. 4.2.1. These factors
may influence both the etching of each filter as well as the reproducibility of filter etching. The etch cell geometry may also have a small effect
on reproducibility as the Pt-electrode may be positioned slightly different
for each etch, thereby changing the potential distribution and possibly the
structure, however, this variation has not been quantized.
5.2.1
Effect of HF diffusion
A closer look at the HF diffusion is in order. Because of a finite system
to start with and a finite, limiting diffusion of reaction species to and from
the etch front caused by the porous structure, the electrolyte composition
will change in the active region, i.e. etch front, during etching. This will
introduce a change in etch rate, structure and porosity with time. How this
influences the fabrication of filters is quite complex.
As F− ions are bound to Si following the reaction in Eq. 2.1, more HF
molecules are dissociated and new HF molecules diffuse toward the etch
front. In effect, the HF concentration changes at the etch front, always
decreasing during etching with a constant current. There are two main
factors controlling the HF concentration at the etch front; the usage of F− ions, mainly give by the current density, but also by the valence, and the
diffusion of HF molecules to the etch front, given by a diffusion constant
101
and the concentration gradient. The valence is sensitive to current density,
temperature and HF concentration. The diffusion coefficient function is
likely to be dependent on the electrolyte properties, e.g. viscosity, as well
as on the porous structure.
In this discussion a couple of factors are not taken into account, namely the
effect of a finite electrolyte reservoir and the behavior of other electrolyte
species, especially those resulting from the reactions at the etch front. A
simplification in a discussion like this would be to assume that the HF
concentration at the surface of the PS is constant, however, with a finite
reservoir without stirring this is not likely to be the case. A constant surface
concentration of HF could probably be used as a first approximation in a
model, however. The behavior of other electrolyte species will most likely
be linked to the HF concentration and not be rate limiting in themselves. In
an empirical model it should be possible to only use the HF concentration
and diffusion coefficient as the electrolyte parameters.
We may assume that the changes in etch rate and porosity with time already
shown in the constant current plots are only due to changes in the electrolyte
composition at the etch front. Thus, according to the discussion above, it
should be possible to correlate each porosity value at any given time to a
starting porosity for a certain HF concentration. However, to project the
time evolution of etch rate and porosity, the microtopology must also be
known, as this will influence the HF diffusion.
In Fig. 4.9 the complicating effect of different microtopologies on the porosity evolution for otherwise equivalent parameters is shown. This has a significant effect on the etching of multilayer structures. The effect of changing
the current density from high to low will be twofold: the concentration at
the etch front will most likely be lower than if the layer had been etched
at the same low current density for the same amount of time, this is due
to the higher usage of HF. However, the concentration also depends on the
diffusion which will be higher because of a steeper gradient and because the
pores are larger. The diffusion to the etch front will be higher compared to
a constant low current density etch which means that the evolution of the
following etch will be different.
To get an idea of the effect this drift in porosity and etch rate will have on
PS optical filters we may use the thick Bragg mirror mentioned above as
an example. Say that we make one filter where the current densities used
for both high and low refractive index are constant throughout the etching
assuming that there is no drift in etch parameters. For a layer etched at
38 mA/cm2 to get a porosity of 80 % (n=1.32) for 6 s to get a thickness of
200 nm, at the top of the filter, a layer etched with the same current density
and etch time after 30 min of etching will give a layer of about 90 % porosity
(n=1.14) and a thickness of 190 nm, shifting the wavelength fulfilling the
Bragg condition λBragg = 4nd (Bragg wavelength) by an amount given by
102
(assuming small changes):
∆λcenter = 4 (n ∆d + ∆n d)
.
(5.2)
Inserting the above values gives a shift of about 200 nm. When the Bragg
wavelength shifts from front to back in a Bragg mirror the resulting reflectance band broadens and ”chirps”.
5.2.2
Effect of temperature
For all chemical reactions, temperature is an important factor. Etching
of PS is no exception. The kinetics of the reaction at the etch front will
change with temperature as well as the diffusion of the different electrolyte
species. With an increasing temperature the diffusion is affected by both a
decrease in viscosity and a higher thermal activity of the diffusing species.
An example of this is shown in the comparison of etch rates and porosities
obtained at two different temperatures in Figs. 5.3 and 5.4. The samples
measured here were fabricated by etching for 100 to 120 min at 15 mA/cm2
in an electrolyte consisting of 26 % HF and ethanol. The temperatures
used were 4±1 ◦ C and 23±1 ◦ C. Two samples were etched for each set of
parameters several days apart, both measurements at each temperature are
shown in the plot. The reproducibility seems quite good, the greatest adverse effect on the reproducibility probably being temperature variations
during etching. After 100 min etching the data for the same temperatures
show a 1.6-1.7 % relative difference. The relative difference between the
4 and 23 ◦ C plots in etch rate after 100 min is 17.8 % with the etch rate
increasing with temperature, while for the porosity the relative difference
is 9.2 % with the porosity decreasing with temperature. If we assume a
linear temperature dependence, the relative difference for the etch rate is
0.93 %◦ C−1 and for the porosity it is 0.48 %◦ C−1 . These values are large
enough to possibly explain the difference between the two measurements
at the same temperature as being due to small temperature differences. In
absolute values this translates to 16.8 (nm/min)◦ C−1 for the etch rate and
-0.32 %(abs)◦ C−1 for the porosity. For simplicity we may approximate the
difference in refractive index between two porosities with a linear relation.
In the IR a 1 %(abs) difference in porosity would yield a (3.8-1)/100=0.028
difference in refractive index. Hence, the change in refractive index with increasing temperature is 0.0090 ◦ C−1 . Notice that both the etch rate change
and the refractive index change result in an increase in optical thickness
with increasing temperature for a constant current density.
For a Bragg reflector in the IR with a low index layer of 60 % porosity
(n=1.84) etched for 10 s (nominally 200 nm thick), the shift in reflectance
band center wavelength due to a change in the optical thickness of this layer,
using the values found above and Eq. 5.2 would be about 28 nm/◦ C. This
is obviously a significant shift which shows that temperature control during
103
etching of PSMs is important. These calculations only hold for the case of
15 mA/cm2 current density with the described electrolyte. The temperature dependence of PS etching will most likely depend on the electrolyte
composition and the current density.
70
Porosity (%)
-1
Etch rate ( m min )
2.0
1.5
23 C
60
4 C
23 C
50
4 C
40
1.0
0
20
40
60
80
100
120
Time (min)
Figure 5.3: Measured etch rate
as a function of time for four samples etched under the same conditions, 15 mA/cm2 with 26 % HF,
but at two different temperature as
labeled. Note the reproducibility of
the results for similar temperatures.
The etch rate increases with temperature.
0
20
40
60
80
100
120
Time (min)
Figure 5.4: Measured porosity as
a function of time for four samples
etched as in Fig. 5.3. The porosity
decreases with temperature which
means that the refractive index increases with temperature.
As current passes through the electrolyte and the sample, resistive heating
will occur. This will in the same way as above introduce changes in etch
rate and porosity, but in this case the changes occur during etching. The
temperature was measured together with the data shown in Figs. 4.10–4.13
showing the temperature change due to resistive heating and ambient temperature variations. In Fig. 5.5 the temperature profiles measured during
constant current etching with 70, 30, and 5 mA/cm2 in 15 % HF with 10 %
glycerol, at room temperature are shown. One would expect that the temperature change would be dependent on the current density with resistive
heating. This is observed in Fig. 5.5 with the 70 mA/cm2 etching increasing
in temperature from ≈18 ◦ C to ≈23 ◦ C, while the 30 mA/cm2 etching has a
fairly stable temperature. The decrease in temperature for the 5 mA/cm2
etching is most likely due to a decrease in ambient temperature. This etch
current dependent temperature change is likely to have a significant impact
on optical filter characteristics. The obtained temperature evolution will
depend on the current profile needed for each specific PSM. Following the
discussion relating to temperature differences between etches, a change in
temperature from start to finish of 5 ◦ C during etching of a Bragg mirror
could result in a substantial ”chirping” or broadening of the filter optical
104
response due to a shift of ∼140 nm in the Bragg wavelength between front
and back.
24
Temperature ( C)
A
22
20
B
18
C
0
20
40
60
80
100
Time (min)
5.2.3
120
140
Figure 5.5: Temperature vs.
time for selected samples also
used in Figs. 4.11 and 4.13.
Curve A was measured during etching with 70 mA/cm2 ,
curve B with 30 mA/cm2 and
curve C with 5 mA/cm2 . The
resistive heating is evident in
the 70 mA/cm2 case while the
two other cases are closer to a
thermal balance with the environment. The reason for
the temperature decrease in
the 5 mA/cm2 case is likely
to be a change in ambient
temperature.
Chemical etching
Chemical etching has been briefly discussed in Sec. 4.2.1. The change in
refractive index in p+ Si due to this effect should be small, but for very
good quality PS filters in p+ Si it may still be significant. To take chemical
etching into account, the porosity profiles obtained by the in situ reflectance
method must be calculated with this in mind. When calculating a current
profile for filter etching, an iterative process may be adopted where the
a profile is calculated disregarding chemical etching, the total time each
layer is in the etchant is calculated and subsequently the change in porosity
due to chemical etching. Next, a new profile is calculated attempting to
counteract the porosity change due to chemical etching.
Assuming that the oscillation in the signal shown in Fig. 4.4 is a result of
chemical etching, we have in this case (4.3 ◦ C, 30 mA/cm2 and 26 % HF
) that the average porosity of the film changes from 72.6 to 73.2 % over
12.5 min. For simplicity we assume that the corresponding refractive index
change (in air) is independent of porosity and constant, hence we have a
constant refractive index change due to chemical etching of 2 · 10−5 s−1 .
The topmost layer in a Bragg mirror, similar to the one discussed above,
will then have a change in refractive index due to long exposure (∼30 min)
to the electrolyte. This change would be about 0.036 in this case. This
would shift the reflection band center wavelength about 7 nm. Chemical
etching has been reported to depend strongly on the substrate used as the
obtained microtopology will change with doping. The dependence on HF
concentration is also significant [86] with higher concentrations resulting in
less etching. There will most likely be a temperature dependence as well.
105
5.3
Etch calibration
To improve the control of porosity and layer thickness during filter etching
compared to the basic filter etching discussed above in Sec. 5.1, the constant
current porosity and etch rate curves will be used to take into account the
drift of these parameters with time.
A first approximation to a complete calibration of a current profile may be to
assume that the observed drift in the constant current plots is independent
of etch history or microtopology and dependent only on time and current
density. Thus the needed current density at a given time for a given porosity
is obtained by taking a slice from the porosity constant current plot at the
given time and interpolating the porosities to get a porosity vs. current
density plot from which the current density is obtained. To obtain a current
density profile for filter etching, this is done for any number of discrete time
steps depending on the wanted time resolution. The necessary duration at
each porosity depends on the corresponding etch rate which is found in a
similar way. When the current density is found, this is used to find the
etch rate in an interpolated etch rate versus current density plot. For each
time step the total etched thickness and layer thickness is calculated. When
the designed layer thickness is reached, the porosity is changed according
to the design. By using small enough time steps a good approximation to
the designed refractive index profile is reached. However, there will to some
degree always be discretization errors with this method.
A porosity versus current density versus time plot for the 15 % HF, 10 %
glycerol electrolyte at room temperature is shown in Fig. 5.6. The data are
the same as in Fig. 4.11 but interpolated and smoothed. Comparing with
Fig. 5.1, it is clear that the porosity versus current plots for given times are
not as smooth. This is most likely due to the observed variation in ambient
temperature as well as resistive heating during etching.
Figure 5.6: The smoothed
porosity vs. time and current density used for etch current profile calculation. The
data used are the same as in
Fig. 4.11.
One way to take into account the etch history when calculating the current
profile for a filter, is to shift the time axis in the constant current data
106
in Figs. 4.10–4.13 with an amount depending on the current density. We
may assume that the high current density etch will be the controlling factor
for HF concentration at the etch front. If the etching starts with a high
current density etched layer, the etch front HF concentration at the end of
this etch will correspond to a concentration at a much later time for a lower
current density etch due to greater HF usage, hence the time axis is shifted
compared to the constant current plot. We may assume that the change
in etch front concentration is negligible for the low current etch, hence the
etching at the next high current density layer picks up at the same time
the last high current density etch stopped. A change in concentration due
to the low current etch may also be taken into account by shifting the
time axis of the high current density etch correspondingly. The necessary
added shift may be found by fitting the reflectance data for a PSM structure
etched assuming no drift in etch parameters with a model describing these
effects, where the time axis shift is a fitting parameter. In this conceptual
model it is assumed that etch front HF concentration is always decreasing,
however this may not be correct in all situations. Changing from high to
low current density may result in an increase in HF concentration due to
higher diffusion than usage. This calculation has not been implemented for
current profile calculations, but, as will be shown below, the idea has been
used to understand some measured reflectance spectra.
In all the examples and discussions above the emphasis has been on discrete
filters. This is only due to the ease of illustration, the same arguments apply
to the case of an inhomogeneous, e.g. rugate, filter, especially when this is
approximated by a number of discrete layers.
5.4
5.4.1
Prepared filters
Reflectance measurement setup
The reflectance experiments were performed with a standard setup utilizing
a monochromator to resolve the spectral components of the light reflected
from the sample. The setup is shown in Fig. 5.7. A broad band quartz
tungsten halogen lamp was used as a light source. The beam from the
lamp was collimated and polarized, if necessary, and sent through a chopper and an iris, for beam narrowing, before being reflected off the sample or
an Al-mirror as a reference. The reflected light was collected by a lens and
passed through a monochromator (SpectraPro 275 by Acton Research Corporation) using either a 600 line/mm or a 1200 line/mm diffraction grating
depending on the wavelength range measured. At the output a standard Si
or Ge detector was placed, depending on the wavelength range measured.
The reflectance spectrum measured from a sample was normalized to the
spectrum measured from the Al-mirror.
107
To obtain an optimal wavelength resolution, a narrow slit was used at the
output of the monochromator. Also, to reduce the contribution of light
at different angles due to a diverging reflected beam and to reduce the
measured spot size as the filters at times were somewhat inhomogeneous, a
narrow slit at the monochromator input was used. The narrow slits together
with the beam narrowing by the iris resulted in a fairly low light intensity
reaching the detector. For some measurements this is observed as a nonlinear intensity response of the detector with higher normalized reflectance
than expected for wavelengths outside the optimal wavelength range of the
detector.
Figure 5.7: Setup used for
reflectance measurements. A
broad band quartz tungsten
halogen lamp was directed at
the sample and the reflected
light was spectrally resolved
by a monochromator before
being detected by a Si or Ge
detector. The chopper and
lock-in amplifier was used to
minimize ambient noise.
5.4.2
Reflectance analysis
5.4.2.1
Discrete filters
In Fig. 5.8 the calculated current profile of a Bragg reflector using the
discussed calibration is shown. The reflector is designed for a wide reflection
band at a center wavelength of 1300 nm at normal incidence with 10 layer
pairs where a pair consists of one 60 %, 175 nm layer and one 83.75 %,
260 nm layer, nominally. The resulting reflected spectral characteristics of
the etched reflector measured at 45◦ is shown in Fig. 5.9. In the same figure
the calculated spectrum based on the design is also plotted. This calculation
is based on the assumption of equal amounts of s- and p-polarized light.
Interface roughness is not taken into account.
The fit between the calculated spectrum and the measured spectrum in
Fig. 5.9 is quite good. As indicated in the plot, the measured reflectance
band is about 80 nm shifted towards the red compared with the calculated
(designed) peak. The most likely causes of this are the effects discussed
above, including the potential error in the effective medium function used
(see Sec. 3.1), a possible difference in ambient temperature at the time of
108
Current (mA)
60
40
20
0
0
50
100
150
Time (s)
Reflectance (abs.)
1.0
0.8
0.6
0.4
0.2
0.0
1000
1200
1400
W avelength (nm)
1600
Figure 5.8: The calculated
current profile used to obtain
the filter measured in Fig. 5.9.
Note the change in high and
low current with time.
Figure 5.9: Reflectance of a
Bragg reflector (filled squares)
with a designed reflectance
band center at 1300 nm. The
reflector consists of 10 pairs
of layers, each pair consisting
of a 83.75 % layer and a 60 %
layer, nominally. The measurement is made at 45◦ using
unpolarized light. The calculated reflectance is shown with
open circles. The measured
reflectance is shifted relative
to the calculated spectrum,
which is also shown shifted for
illustration (dashed line).
etching compared to the calibration data and the use of a simplified model
for calibration. A red shift indicates optically thicker layers; physically
thicker and/or lower porosity. The observed shift of about 80 nm could be
accounted for by a difference in porosity of about 5 % (absolute difference)
compared to the designed value. This number compares well with the discussion in Sec. 3.1 and above. There are other features in the spectrum
indicating that the current profile is not perfectly adjusted for the effect
of electrolyte species diffusion. A distortion in the spectrum may also be
caused by the striation induced lateral inhomogeneity of the filter if the
beam spot is large enough, which is the case with the spot size of about
1 mm used for most of the reflection measurements.
Figures 5.2, 5.10 and 5.11 compare reflection spectra from Bragg reflectors
designed to reflect at 1500 nm. The spectrum in Fig. 5.2 is from a structure
consisting of 10 pairs of layers with nominally 53 % and 83 % porosities and
thicknesses of 183 and 301 nm, respectively, etched without time calibration.
The peak is shifted about 80 nm towards the red from the designed peak
109
which is similar to Fig. 5.9. The fit between the calculated and the measured
spectrum is quite good. This is expected as the filter is rather thin, about
5 µm, so the upper layer pair is not much different from the lowest layer pair.
In Figs. 5.10 and 5.11 the reflectors measured contain 40 layer pairs and one
may clearly see tendencies of drift in layer optical thickness within the layer
stack. In Fig. 5.10 the spectrum from a filter etched without calibration is
shown. In Fig. 5.11 a filter etched with current density adjusted for porosity
and etch rate drift is shown. Note that the calculated spectra in these two
figures take into account a 70 nm PS-substrate interface rms roughness and
an interlayer interface rms roughness of 15 nm. These numbers are used
based on measurements presented in Paper III. The sample etched with
adjustments for drift show a pronounced broadening indicating a possible
over-adjustment. These three spectra are obtained with unpolarized light.
1.0
Reflectance (abs.)
Figure 5.10: The measured
reflectance (filled squares) of
a filter with design parameters nominally the same as
in Fig. 5.2 except that 40
layer pairs are used. A fitted spectrum based on a simplified model of the porosity
and etch rate drift is shown
(open circles).
The calculated spectrum based on the
fit takes interface roughness
into account, 70 nm rms between substrate and multilayer
and 15 nm rms between each
layer. Adjacent averaging is
performed to indicate the effect of wavelength resolution
limitations of the monochromator and inhomogeneities on
a length scale similar to the incident beam diameter.
0.8
0.6
0.4
0.2
0.0
1000
1200
1400
1600
W avelength (nm)
To evaluate the structure of the etched filters and try and quantize the
amount of drift and over-compensation, a fitting procedure was used. The
measured reflectance spectra were fitted to the calculated spectra of a simple model of the structure incorporating a linear drift effect. As most of the
layer information is in the optical thickness, the variation in layer optical
thickness was parameterized by a variation in physical layer thickness. By
doing this, some information is lost because individual layer reflectances
are not changed due to constant refractive indexes. The resulting fit, using
the Levenberg-Marquardt algorithm as implemented in the IMD software
for thin film calculations by David L. Windt, for the structure measured in
Fig. 5.10 is shown in the same figure with open circles. The resulting optical
110
Reflectance (abs.)
1.0
0.8
0.6
0.4
0.2
0.0
1000
1200
1400
1600
Bragg wavelength (nm)
W avelength (nm)
53 % layers
1600
1500
1400
83.75 % layers
1300
0
20
40
Layer index
60
80
Figure 5.11: The measured
reflectance (filled squares) of
a filter with design parameters nominally the same as in
Fig. 5.10 except that the current profile is adjusted according to the calibration procedure discussed. A calculated
spectrum based on the design
is shown (open triangles). A
fitted spectrum base on a simplified model of the porosity
and etch rate drift is shown
(open circles). The calculation
based on the fit takes interface roughness into account,
30 nm rms between substrate
and multilayer and 5 nm rms
between each layer.
Figure 5.12: The resulting bragg wavelength (optical thickness×4) of the layers
of the filter corresponding to
Fig. 5.10 after fitting a simplified model to the measured
spectrum. The drift in porosity and etch rate with time
is assumed to result in a linear change in optical thickness
with depth in this model.
thickness for each layer, transformed to Bragg wavelengths (4×optical thickness), is shown in Fig. 5.12. The resulting spectrum fits the measurement
quite well. Note that adjacent averaging is used with the fitted spectrum
to simulate the effect of wavelength resolution limitations in the monochromator. The layer optical thicknesses shown in Fig. 5.12 show that there is
a marked drift in optical thickness which may be caused by one or both of
etch rate and porosity drift. The optical thickness of all the layers should
have been the same according to the design, in this case corresponding to
a Bragg wavelength of 1500 nm, but the resulting optical thicknesses differ
quite markedly between high and low porosity layers. This may be that
the starting point of the etch was incorrect, i.e. that either the calibration
curves were slightly off true value because of e.g. temperature differences, or
that the effective medium function introduced an error as discussed earlier.
The same procedure was performed for the filter measured in Fig. 5.11. The
111
Bragg wavelength (nm)
1800
Figure 5.13:
Same as
Fig. 5.12 for the case where the
current profile was calibrated
(Fig. 5.11).
83.75 % layers
1600
1400
60 % layers
1200
1000
0
20
40
60
80
Layer index
resulting fitted spectrum is shown in the same figure with open circles. Not
all features seem explained, but the widening and the general band edge
gradients seem explained quite well. Figure 5.13 shows the layer Bragg
wavelengths of the fitted spectrum. It is clear that the calibration procedure
has had a significant effect on the structure. The low porosity layers seem
most affected by the calibration, while the high porosity/high current layers
seems well adjusted and nearly constant throughout the structure. This
may indicate that a time shift of the calibration curves when designing a
filters is appropriate. The etch rate decrease or porosity increase with time
for the low porosity layers seem much stronger than that taken into account.
5.4.2.2
Rugate filters
Rugate filters were fabricated in the same manner with a time calibration
done on the current profile. In Figs. 5.14 and 5.15 the measured and calculated reflection spectra are shown for two fabricated rugate reflectors. The
sample used for Fig. 5.14 was designed for a narrow reflection band at 600
nm. Index matching was used while no apodization was used. The porosity
varied from 49 % to 53 % nominally between the index matching regions.
The filter contained 65 periods in the refractive index profile resulting in a
thickness of about 10 µm. A low temperature of about 6 ◦ C was used along
with an aqueous electrolyte consisting of 26 % HF and ethanol. The low
temperature was used in an attempt to minimize roughness as discussed in
Paper III. The values of the measured data are only indicative as a baseline was subtracted from the data as very a low light intensity resulted in
a non-linear detector response. The same trend as discussed for the other
measured spectra is present in Fig. 5.14 with the measured spectrum redshifted compared to the calculated spectrum, however the band shapes are
quite similar, with a measured full width at half maximum of 10 nm, indicating little drift of porosity and etch rate. The effect of interface roughness is
clearly seen comparing the two calculated spectra; one taking into account
interface roughness, one without roughness. In Fig. 5.15 the reflectance of a
thick IR rugate filter is measured. This filter was designed with a medium
112
width reflection band at 1550 nm with 75 periods in the refractive index,
both index matching and quintic apodization were used. The total thickness was about 30 µm. A time calibration of the current profile was done in
this case also. This filter was etched at low temperature (4 ◦ C) with a 26 %
HF electrolyte. A calculated spectrum based on the design is shown. It is
evident that the refractive index profile is not optimal as the peak is much
too wide and non-uniform. To test the suggestion that this widening is due
to a drift in the layer optical thickness as discussed, a calculation of the
reflection spectrum with a linear increase in the period was done. This is
shown as a dashed line in Fig. 5.15. The correspondence is good indicating
an over-adjustment due to a non-optimal calibration. In both Figs. 5.14
and 5.15 the calculated spectra shown in whole line are without roughness
taken into account, while the calculations shown in dashed lines are with a
rms roughness at the interface of 70 nm and an interlayer rms roughness of
15 nm.
Reflectance (abs.)
0.4
0.3
0.2
0.1
0.0
500
600
700
800
W avelength (nm)
900
1000
Figure 5.14: The measured
reflectance spectrum (filled
squares) of a rugate reflectance
filter designed for a narrow
peak at wavelength of 600 nm.
The calculated spectrum with
(solid curve, open circles) and
without (dashed curve) an rms
interface roughness of 70 nm
is shown also.
Both measurement and calculations are
for an incident angle of 45◦ .
The same red shift is present
here as in most other filters
measured.
In Fig. 5.16 the measured reflectance spectrum of a three band rugate spectrum is shown. The filter was designed as a narrow band pass filter with a
wide blocking band on both sides of the pass band. This was done similar
to the design calculated in Fig. 3.14 with three reflectance bands very close
to each other. The calculated current profile, adjusted for drift in porosity
and etch rate, is shown in Fig. 5.17. The regions in the current profile corresponding to the different bands overlap in such a way as to minimize the
thickness without allowing the total refractive index to go over a certain
threshold value. The lower current values reach a limiting threshold several
places whereby the current is clipped. This procedure introduces some features in the spectral characteristics, however, with little clipping the result
is acceptable. As can be seen in Fig. 5.16 the main spectral characteristics
in the designed filter are recognizable in the measured spectrum. Again,
due to roughness and overcompensation for drift, the result is not optimal.
Note that the calculated spectrum in this case is only for s-polarized light.
113
Figure 5.16:
Reflectance
measurement of a three peak
rugate filter (filled squares).
The filter was designed to have
a narrow transmittance band
as can be seen in the calculated spectrum (open circles). The calculated spectrum is for s-polarized light
while the measurement is unpolarized. The pass band is
present in the measured spectrum, but much wider than designed due to drift in porosity
and etch rate with depth and
interface roughness.
1.0
Reflectance (abs.)
0.8
0.6
0.4
0.2
0.0
1000
1200
1400
1600
W avelength (nm)
1.0
0.8
Reflectance (abs.)
Figure 5.15: A thick rugate reflection filter is compared with calculated spectra.
The design was for a peak
at 1550 nm, using 75 refractive index periods with porosities varying between 30 % and
53 %. The etch current profile was adjusted for drift resulting seemingly in an overcompensation. A calculated
spectrum incorporating a shift
in design peak wavelength is
shown as the dashed red curve.
This compares well with the
measured spectrum.
0.6
0.4
0.2
0.0
1000
1200
1400
1600
W avelength (nm)
Etch brakes have been used quite successfully by Billat et al. [87] and Reece
et al. [59]. The etch breaks allows the electrolyte at the etch front to
regenerate, avoiding problems caused by diffusion of the electrolyte species.
However, the etch time of a filter increases substantially compared with a
continuous etch as typically break times are in the order of 10 to 20 times
the time it takes to etch the thickest of the layers in the layer pair [88]. This
results in an etch time of about one hour for a 20 layer pair Bragg stack
with peak wavelength at 1300 nm. By increasing the etch time this much
the effect of chemical etching must be considered, even for p+ samples.
114
50
Current (mA)
40
30
20
10
0
0
500
1000 1500 2000 2500 3000
Time (s)
5.4.2.3
Figure 5.17: The current
profile used to etch the filter measured in Fig. 5.16.
The three current profile regions corresponding to the
three peaks is partly overlapping, minimizing the total filter thickness and etch time.
Graded filters
The setup for etching graded filters is explained in Sec. 2.5. A detailed
analysis of a series of graded rugate reflection filters is presented in Paper
V, while some possibilities of this technique will be discussed in Chap. 6.
The measured spectra from a near-infrared graded rugate reflection filter
is shown in Fig. 5.18. These data are from measurements at an incident
angle of 22◦ . The current profile used was based on a filter designed for a
reflection peak at 1000 nm, porosities varying from 75 % to 85 %, and a
total thickness of about 10 µm. A calculated reflection spectrum based on
this design is shown in the figure. The lateral voltage used during etching
was 1 V, resulting in a shift of the peak wavelength of about 35 nm per mm
across the filter.
1.0
Pos.3
Pos.4
Pos.5
2mm
5mm
7mm
Reflectance (abs)
Pos.2
0.6
Designed
1mm
0.8
Edge
0.4
0.2
0.0
500
600
700
800
900
Wavelength (nm)
1000
1100
Figure 5.18: Reflection spectra measured at different positions on a graded rugate
reflection filter. The calculated spectrum of a filter using the designed current profile is marked ”Designed”. The
parameters for this filter is a
peak reflectance wavelength of
1000 nm with porosities varying between 75 % and 85 %
and a total thickness of 10 µm.
The measurements are made
at an incident angle of 22◦ .
Paper IV discuss some of the effects the grading of the filter have on the
optical response using a simple single ray model. Particularly if the graded
layers introduce an angular dispersion in the reflected beam and how the
spectral response change with graded layers. The conclusion seems to be
115
the errors introduced by an angular dispersion in the incident beam overshadows any effect of the graded layers.
An indication of the regularity of the periods of the fabricated filters may
be obtained by high resolution field emission SEM (FESEM) images. Figure 5.19 shows a cross-section FESEM image of the reference (non-graded)
rugate reflection filter presented in Paper V. The average image intensity
profile taken across the image is overlayed in yellow. A sine like profile is
clearly recognizable. The obtained sine profile is not accurate enough to
show a difference in period from top to bottom which seems present when
considering the ”chirping” of the spectral responses of the filters in the
paper.
Figure 5.19: Cross-section
SEM of a rugate reflection filter. The measured reflectance
spectrum of this filter is presented in Paper V. A plot of
the average pixel intensity values in the gray scale SEM image taken parallel to the surface is shown to the right,
overlayed on the image. The
sinusoidal intensity variation
coincides with the designed refractive index variation.
5.5
Improvements of the process
Several potential and known causes of errors in the PSM structures have
been discussed. Some possible remedies for these problems are suggested
in the following:
• stirring to keep sample surface electrolyte composition constant as
well as reduce temperature gradients/changes, pump induced flow for
the same reasons or bubling of e.g. nitrogen gas.
• temperature control of electrolyte (main volume) and of sample.
• introduce breaks in etching to regain original electrolyte composition
in active region as briefly discussed.
• the effect of bubble formation in pores during etching may be reduced
by using a different wetting agent (may be improved further compared
with ethanol).
116
• (micro-) roughness may be reduced by a limited viscosity increase
(lower temp or addition of glycerol) as indicated in Papers II and III.
• use a cathode integrated in the etch cell to keep the position constant
relative to the sample.
Paper IV
S.E. Foss and T.G. Finstad
Multilayer interference filters with non-parallel interfaces
Proceedings of the Nordic Matlab Conference,
Copenhagen, Denmark, 2003
IV
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S.E. Foss and T.G. Finstad
Laterally graded rugate filters in porous silicon
Mat. Res. Soc. Symp. Proc., 797, W1.6.1 2004
V
Mat. Res. Soc. Symp. Proc. Vol. 797 © 2004 Materials Research Society
W1.6.1
Laterally Graded Rugate Filters in Porous Silicon
Sean E. Foss and Terje G. Finstad
Department of Physics, University of Oslo,
POBox 1048 Blindern, 0316 Oslo, Norway
ABSTRACT
Rugate optical reflectance filters with position dependent reflectance peaks in the visible to
near infrared spectrum were realized in porous silicon (PS). Filters with strong reflection peaks,
near 100%, no detectable higher order harmonics and suppressed sidebands compared to discrete
layer filters were obtained by varying the current density continuously and periodically during
etching. An in-plane voltage up to 1.5 V was used to obtain refractive index and periodicity
change along the filter surface resulting in reflectance peak shifts of up to 100 nm/mm in the
direction of the voltage drop. The effect of the lateral change in optical parameters on the filter
characteristics is studied by varying the gradient and comparing measurements at different
positions with measurements on a non-graded filter. We have observed extra features in the
reflectance spectrum of these graded filters compared with reflectance from a non-graded filter
which is likely caused by the gradient.
INTRODUCTION
The process of discretely varying the current density during etching of porous silicon (PS) to
obtain thin layers with different refractive indexes was first reported by Vincent [1] and Berger
et al. [2] in 1994. With this method Bragg reflectors and Fabry-Perot filters are now routinely
made. By applying an in-plane voltage across the sample during etching, Hunkel et al. have
shown that it is possible to produce PS discrete filters with laterally dependent filter
characteristics [3,4]. In addition to the discrete layer filters (i.e. Bragg and Fabry-Perot) one may
also realize structures in PS that are uniquely simple to this system. By varying the current
density continuously and periodically, the refractive index into the PS layer will vary accordingly
[5]. The refractive index may be calculated from the porosity by the effective medium
approximation. With this approach one obtains reflecting rugate filters which may have narrow
reflection peaks, no higher order harmonics and suppressed sidebands. For an overview of rugate
filter theory, see Bovard [6]. Here we report on rugate filters with laterally varying
characteristics which have been made for the visible to near infrared optical spectrum.
EXPERIMENTAL DETAILS
The Si substrate used was 0.018 Ω·cm B-doped, p-type Czochralski-grown single crystal
with <100> orientation polished on both sides. The HF-based solution used for etching consisted
of 1:2 HF(40 %):ethanol. The etching current was controlled by a computer, in this case with a
time step of 1 second which sufficiently reproduced the refractive index sinusoid. The used
current profile is shown in Fig. 1. The etching occurs mostly at the pore tips which is why
porosity may be modulated by the current. There will also be a small chemical etching which is
W1.6.2
dependent on time. This is corrected for by etching the deeper part of the layer with a slightly
higher current.
The starting point for the design was a 1 cm diameter, circular, non-graded rugate filter with
a reflectance peak-wavelength at 738 nm measured with light incident at 24 deg. The filter was
made relatively thin with 23 periods of the porosity sinusoid over 5 µm. Etching was done with
current density varying between 9.4 mA·cm-2 and 19.7 mA·cm-2, which corresponds to refractive
indexes at the peak wavelength of 1.78 and 1.48 respectively. Index matching for the air-PS
interface and the PS-substrate interface was employed to reduce sidebands. This may be
observed in the beginning and end of the current profile of Fig. 1 where there are large slopes.
A schematic of the etching setup is shown in Fig. 2. A lateral gradient in porosities and etch
rates was obtained by applying a constant in-plane voltage up to 1.5 V between two contacts on
the sample back side while etching with the same current profile as for the starting point design.
By doing this the local current density varies. Ohmic contacts were made from evaporating Al on
the samples followed by a short heat treatment. The resistance between the contacts varied
between 0.6 and 0.7 Ω for the different samples. Contact resistance is a substantial fraction of
this, so good, reproducible contacts were important to achieve control of the potential drop
within the sample.
Reflectance measurements were conducted with a 0.275 m focal-length monochromator with
a Si-detector. The focused probe beam had a diameter of less than 1 mm and was directed at the
filter at an angle of about 24 deg. Both the size of the beam and the angle contribute to a
widening of the reflectance peak to some extent compared to the peak from the non-graded filter.
The effect depends on the gradient of the filter. Reflectance spectra are only plotted from 600 nm
to 1100 nm because of the combination of the detector and diffraction grating used in the
monochromator.
For the optical images a microscope with a mounted digital camera was used. Several images
were stitched together to get a better overview. To show the striation effects on the filter
Figure 1. Applied current during etching
of all filters. The slopes at the beginning
and end are for index matching.
Figure 2. Sketch of the etch setup. The Si
sample with Al back contacts is pressed on to
two Cu-plates on the back side so an in-plane
constant voltage can be set up. The currentsource is connected to a computer.
W1.6.3
reflectance clearly in these images, the RGB colors were split into separate images and only the
green was used for analysis as this had the largest contrast between striation minimum and
maximum.
A white light interferometer, WYKO NT-2000 by Veeco, was used to measure surface
topography profiles of the samples after the PS was stripped away with a concentrated NaOH
etch. This instrument has a large dynamical range and can measure µm and nm height
differences in the same measurement. Even with the gradient in the samples present,
measurements showing the ridges caused by striations, in the order of 100 nm high, were
possible.
RESULTS AND DISCUSSION
The reflectance spectrum of the non-graded starting point filter is shown in Fig. 3. Often
when designing rugate filters one employs apodization to further reduce sideband reflection [6].
This was not done on the presently reported filters. Observed sidebands show a periodicity
dependent on the average optical thickness, n·d, of the PS film at the measured position which
may be used to calculate the average refractive index. Because of the refractive index dispersion
in Si the sidebands in the reflectance plot will decrease in frequency with increasing wavelength.
Reflectance spectra at four different positions along a line through the center of a filter made
with an in-plane voltage of 1.0 V are shown in Fig. 4. Comparing these peaks with the one from
the non-graded filter in Fig. 3 it is clear that the grading affects the shape, width and height. The
position of the maximum reflected wavelength shifts along the filter as expected. The amplitude
of the maximum reflectance decreases towards shorter wavelengths. This is most likely due to
increased absorption [3]. There is also a broadening of the peaks away from the mid position
towards shorter wavelengths caused by the fact that the refractive index does not have a linear
Figure 3. Reflectance spectrum of a nongraded rugate filter used as a starting point
for graded filters. Measurements are taken
at 24 deg. incident angle. FWHM is 100
nm.
Figure 4. Plot of reflectance spectra from a
filter made with an in-plane voltage of 1.0 V.
Marked positions are along a centerline across
the circular filter in the direction of the
gradient. Note the spreading of the peak as it
moves towards lower wavelengths.
W1.6.4
dependence on current density [3]. Therefore the amplitude and period in the refractive index
sinusoid will only be optimal at one point along the filter surface. All the graded filters show
similar behavior. A decrease of amplitude between samples with increasing grading is also
observed. This is most likely caused by the finite spot size of the probe beam. When this covers a
too large area compared to the gradient at that point, the resulting measurement will be an
average of spectra and hence the filter window will be smeared out.
Figure 5 shows the peak wavelengths at different positions for a series of filters with
changing in-plane voltage. As expected, the shift increases with the voltage caused by the
increase in the porosity and etch rate gradient. The horizontal line across the plot shows the peak
wavelength of the non-graded filter. The reflectance peak is clearly not at 738 nm for the mid
position for any of the graded filters. Thus the current density does not vary linearly with
position. This is as expected considering the geometry of the etch setup.
Although the shift in peak wavelength seems close to linear, depth profiles of the PS layer
made by white light interferometry after stripping, shown in Fig. 6, show tendencies towards
non-linearity. This is also supported by [7] where Bohn and Marso describe an equivalent circuit
model for the etch situation. A non-linear current density distribution will result in a varying
position of the crossing point between the peak shift curves and the non-graded peak wavelength.
One would, however, expect that at some point the filter would have the characteristics of the
non-graded filter as the total current is the integral of the local current density at all positions on
the filter area. This is true for the filters of Fig. 5 except for the 0.1 V case. A slight difference in
process parameters is most likely the cause. In Fig. 7 the change in peak shift with applied inplane voltage is shown. The line corresponds to a linear fit which seems appropriate for this
range of voltages. A non-linearity might be expected for higher voltages as the necessary current
0
1.5 V
Depth, µm
-2
0.5 V
-4
0.1 V
-6
-8
-10
0
2
4
6
8
10
12
Distance from edge of filter, mm
Figure 5. The wavelength of reflection peaks
measured at different positions on samples with
different gradients, (○) for an in-plane voltage of
0.1 V, () for 0.5 V, (■) for 1.0 V and () for
1.5 V. The shift of the peaks increases with
increasing in-plane voltage.
Figure 6. Depth profiles of the PS-substrate
interface measured with white light
interferometry after removal of the PS layer.
One clearly sees the increasing gradient with
in-plane voltage. Curves are composed of
several single measurements stitched together
which cause drift in measured height, therefore
only tendencies are discussed.
W1.6.5
reaches amperes, hence heating will probably affect the process. The geometry of the filter has
an effect on the local current density, as may be observed visually. Perpendicular to the gradient
direction reflection colors shift somewhat away from the center line, indicating a non-constant
current density towards the edges.
Visible stripes of slightly different colors are present in all the filters made. These seem
circular and centered around the center of the 4” wafers used for samples. After stripping away
the PS with an NaOH etch these stripes are still present as ridges. Figure 8 shows a compilation
of a light microscope image representing the intensity of the green component from a color
image (to increase contrast) and the subsequent intensity plot of the 2-dimensional height profile
from white light interferometry measurements at the same position after stripping of PS. Peak to
peak height of the ridges are about 100 nm. These ridges indicate locally different etch rates.
Similar results have been reported earlier by Lérondel et al. [8] as likely being caused by
striations, i.e. radially symmetric resistivity inhomogeneities in the substrate due to an
inhomogeneous dopant distribution. One way of smoothing the PS-substrate interface is
suggested by Setzu et al. [9] where etching is done at low temperatures down to -35 ○C.
Figure 7. Plot showing the rate of change of
the shift of the reflection maximum along the
grading of the filters as a function of applied
voltage.
Figure 8. Image of the 0.1V filter with an
overlay (between the two white crosses) of
an intensity plot of the corresponding
surface topography profile of the PSsubstrate interface measured by white light
interferometry. The white square in the
intensity plot is missing data.
CONCLUSION
We have shown it is possible to make a laterally graded rugate filter with good reflectance.
The filter may be improved by optimizing the parameters used, especially by using a smaller
difference between minimum and maximum refractive index and more periods, hence a thicker
PS film. Apodization may also be used to optimize the reflectance characteristics. Striations
causing locally differing etch rates have been shown.
W1.6.6
ACKNOWLEDGMENTS
This work has been carried out under the MOEMS and MEMS research program of the
Research Council of Norway. The authors would like to thank Maaike Taklo Wisser at SINTEF
for help with the WYKO measurements, and H. G. Bohn at Forschungszentrum Jülich, Germany,
for help with the program for designing the rugate filter.
REFERENCES
1. G. Vincent, Appl. Phys. Lett. 64, 2367 (1994).
2. M. G. Berger, C. Dieker, M. Thönissen, L. Vescan, H. Lüth, H. Münder, W. Theiss, M.
Wernke and P. Grosse, J. Phys. D: Appl. Phys. 27, 1333 (1994).
3. D. Hunkel, R. Butz, R. Arens-Fischer, M. Marso and H. Lüth, J. Lumin. 80, 133 (1999).
4. D. Hunkel, M. Marso, R. Butz, R. Arens-Fischer and H. Lüth, Mater. Sci. Eng. B 69-70, 100
(2000).
5. M. G. Berger, R. Arens-Fischer, M. Thönissen, M. Krüger, S. Billat, H. Lüth, S. Hilbrich, W.
Theiss and P. Grosse, Thin Solid Films 297, 237 (1997).
6. B. G. Bovard, Applied Optics 32, 5427 (1993).
7. H. G. Bohn and M. Marso, (unpublished report).
8. G. Lérondel, R. Romestain and S. Barret, J. Appl. Phys. 81, 6171 (1997).
9. S. Setzu, G. Lérondel and R. Romestain, J. Appl. Phys. 84, 3129 (1998).
Chapter 6
Porous silicon applications for
MOEMS and passive optics
The techniques and results presented in this thesis may be considered as
part of a toolbox to build novel devices in silicon microtechnology. Some
possibly new and untested ideas will be presented in the following showing some of the many possibilities of porous silicon as an optical material.
The two main areas these ideas describe are passive optical elements and
micro-opto-electro-mechanical-systems (MOEMS). Several different passive
optical elements and applications have been presented in the literature,
some of which have been mentioned in Sec. 2.1. In the area of MOEMS
there are very few reported devices employing PS as a critical ”material”.
The most striking device is the spectrometer by Lammel [89].
6.1
Passive optical elements
Passive optical elements are here thought of as elements made with PS
where no activation is needed for them to work , neither by absorbed light
nor by an applied current/voltage. Such elements may be, e.g. lenses and
filters. One example of an active device would then by a PS-light emitting
diode (LED).
6.1.1
Schottky barrier spectroscopic IR detector
One aim of the presented work has been to prepare graded optical band-pass
filters in PS, which for example could be used in a monolithically integrated
sensor-array system. A fairly simple sensor design could be based on several
separate Schottky barrier sensors on the back side with the graded band
pass filter, either Fabry-Pérot or rugate, on the front side. This detector
would be for use in the near- to mid-IR as photons with wavelengths below
the absorption edge of the silicon substrate would be absorbed and would
133
134
not reach the Schottky barrier sensors. The sensors would be in the form of
parallel strips with the length of each strip perpendicular to the filter gradient direction. In this way each strip would respond to a specific transmitted
band of photon wavelengths, with the position of each strip relative to the
filter deciding which band will be detected. The total of the strips would
than give a spectroscopic detector. A schematic drawing of the design is
shown in Fig. 6.1.
To obtain sharp features in the filter spectral characteristics, many layers or
periods are needed resulting in fairly thick filters. This necessitates a very
good control of etch rate, porosity and interface roughness. A standard
Bragg reflector of 50 layer pairs with a peak reflecting wavelength of 1.5 µm
would have a total thickness of roughly 20 µm. Reasonable filter thicknesses
for more complex rugate filters may reach 50 µm. The good control of the
parameters is also necessary to minimize some of the problems introduced
by grading the filter.
The Schottky barrier sensor strips may be fabricated by different metals
depending on the wavelength range of interest. Deposition of Ti, Ir and
Pt with a subsequent annealing produces silicides with low enough work
functions so that photons not absorbed by the substrate may induce a
current. The possible wavelength range of the detector is determined on
the high energy side by the absorbtion edge of the Si substrate, about
1.1 eV, and on the low energy side by the barrier height, which for Al on
p-type Si is about 0.55 eV and for PtSi on p-Si is normally 0.3 eV, but has
been reported for special cases to be as low as 0.13 eV [90]. This results
in a detectable wavelength range from 1.127 µm to 2.254 µm for Al and
from 1.127 µm to 4.133 µm or 9.5 µm for PtSi. The performance of such a
spectroscopic detector would depend on the filter gradient and number and
width of the sensor strips for wavelength resolution and on the filter size
and Schottky barrier material for wavelength range.
By miniaturizing each element it may be possible to fabricate arrays of
such detectors, with each identical detector detecting a range of discrete
wavelength bands. One would then have an imaging IR spectroscopic device. Instead of basing the sensing on the Schottky effect, one may use
pyroelectric materials, such as BaTi, for heat sensitive detector arrays. It
is possible to use PS on the contact side also. Raissi and Far reported in
Ref. [91] that electroplating of Pt within the pores of PS with a subsequent
anneal produces Schottky barrier diodes with low barrier height and high
efficiency.
6.1.2
2D photonic crystal
A subject of much research recently has been photonic crystals. A multilayer thin film optical filter can be said to be a 1D photonic crystal with a
photonic band gap, or forbidden band, i.e. reflection band. 2D and even
135
Figure 6.1: A conceptual sketch of a Schottky barrier spectroscopic IR
detector. The graded band-pass rugate filter on the front of the substrate
is fabricated as described earlier and optimized for transmission of near to
medium infrared photons. On the back side, Al or another fitting metal is
deposited. The metal-silicon junction forms a Schottky barrier which will
emit charge carriers when hit by photons within a certain energy range. The
position of the different contacts will define each contacts wavelength range
of highest sensitivity.
3D photonic research has been proposed and tested for applications such
as waveguides in photonic circuits and resonators for enhancing LED emission [92]. Macro PS has been used for 2D photonic crystals [93] due to
the well ordered, high aspect ratio pores. Most 2D photonic crystal structures are based on two materials, air and a dielectric. This is mostly due
to fabrication limitations, but also the high contrast in refractive index obtainable. However, it could be interesting to have a controllable refractive
index contrast, as shown by Weiss in Ref. [22] in the case of 1D photonic
crystals. By fabricating macro-PS with pores filled with micro-PS instead
of air, and filling this again with liquid crystals, a controllable band gap
may be realized for 2D photonic crystals as well. The fabrication of this
structure may be done by micro-PS etching on a masked Si-substrate or by
etching macro-PS under certain conditions [94] obtaining filled macropores.
The obtained photonic crystals could be used as reconfigurable waveguides
where, in principle, each column of liquid crystal filled micropores could be
addressed individually. This would enable switching, modulation, and beam
shaping of the light in the crystal. Potential applications could be within
lab-on-a-chip technology (beam steering) or light sources (beam shaping
and directing). A conceptual drawing of the discussed design is shown in
Fig. 6.2.
136
Figure 6.2: A suggested design for a reconfigurable 2D photonic crystal
based on micropore filled macropores filled with liquid crystal. In principle,
each column can be individually addressed. From this, differen photonic
devices, such as modulators and switches, may be realized.
6.1.3
GRIN optics
By controlling the potential distribution through the sample both temporally and spatially during etching, it is possible to form any conceivable
refractive index geometry within the limits of the etch parameters. One
possibility is to form a graded index (GRIN) lens, a plane parallel centrosymmetric structure with a given radial function describing the refractive
index. By choosing e.g. a quadratic function with the largest refractive
index at the central axis, and etching such that the refractive indexes are
constant through the structure, possibly through the sample, a collecting
lens is fabricated.
As the practical refractive index range is limited to roughly 1.15 to 2.7 (90
- 30 % porosity at 1500 nm wavelength), and the thickness of standard Siwafers is around 500 µm, the obtainable focal length will be fairly large.
The brand of SELFOC GRIN lenses uses a refractive index profile of
n2 (y) = n20 (1 − αy 2 ),
(6.1)
where y is the radial distance and α a constant. With α2 y 2 1 for all y of
interest the focal length may be given as
f=
1
,
n0 α sin (αd)
(6.2)
where d is the thickness of the sample [95]. With α = 0.385 and n0 = 2.7 the
focal length is 5 µm giving a reasonable focal length for compact detector
137
applications. The circular lens would then be etched through the sample
and have a refractive index along the central axis of 2.7 decreasing out to
the rim to 1.15 at a radius of about 2.5 mm.
The GRIN design may also be used for waveguiding, as is the case for a
certain type of optical fiber. One challenging element of combining microtechnology and optics, usually referred to as micro photonics, is the coupling of light from the transportation level (fiber) to the manipulation level
(chip/device). Aligning fibers to waveguides on an optical integrated circuit (OIC) is difficult as the areas to be aligned have a typical size in the
µm range. One way of doing this which has proven quite efficient has been
to etch grooves adjacent to the waveguide-entrance during processing of
the OIC. The fiber will then be centered upon placement and fastening.
However, there will be a substantial loss of signal as there will most likely
be an air gap between the fiber-end and the waveguide-entrance. This
method is also permanent. One potential application where the coupling
of light and OICs is non-permanent is in the processing of lab-on-a-chips.
In this case a specially designed chip may manipulate a physical sample,
e.g. liquid containing DNA, through heat treatment and transport through
micro-conduits. The measurement of parameters of interest may be done by
luminescense measurements at some position on the lab-on-a-chip with an
external excitation source and detector. By integrating some of the optics,
e.g. waveguides, it may be possible to measure more parameters and make
the measurement more selective or sensitive. To make a non-permanent,
robust coupling between a fiber and an optical circuit the GRIN properties
of PS may be utilized. It is also possible to integrate the light sources with
the lab-on-a-chip also, possibly PS LEDs or nanodot LEDs/lasers. In this
case the system will be even more compact and coupling to and from the
chip is avoided.
Waveguides fabricated with PS have been reported by Loni [96] and others [97, 98, 99]. By combining a standard waveguide with a specially designed coupling area, a good coupling with little loss may be possible. A
schematic of the idea is shown in Fig. 6.3. Guiding of light is most often
based on total internal reflection which is possible when light in a medium
is reflected off an interface to another medium with a lower refractive index
at a minimum angle. This is shown in Fig. 6.3 as a darkly colored guiding core (high index) surrounded by a lighter colored cladding layer (low
index). However, to minimize back-reflectance from the air-guide interface
the guide refractive index should match as well as possible the air refractive index. These conflicting requirements may be resolved by gradually
increasing the core index from the coupling area at the surface to the guide.
The curvature of the core in the coupling area should be small as this will
reduce the loss due to non-total internal reflection conditions close to the
surface - the core has a lower index than the cladding here. This design
seems quite robust in that there is a range of acceptable angles and there
are no movable parts.
138
Figure 6.3: A fiber-to-chip optical coupler using graded index PS areas. A
graded refractive index area is used to minimize back reflectance and steer
the beam into the waveguide. The waveguide may also consist of PS, both
high and low porosity.
6.1.4
Novel optical filter
An interesting use of PS graded index filters is in anti-reflection coatings
(ARC) for crystalline Si solar cells [100]. Graded index ARCs have the potential of being very broad band and work at a wide range of angles [101].
However, these qualities typically come at the expense of greater filter thickness. The thickness of the ARC is crucial when used in conjunction with
solar cells, as the the efficiency of the cell depends on photon absorption in
the correct location in the cell. With a thicker ARC, more absorption will
take place in the filter itself which does not generate any photo-current. A
balance may be found between the angular range for which the ARC gives
good results and the thickness in such a way that the cell proves overall
more efficient for a wide range of positions relative to the sun. With an
immovable cell, the daily energy conversion increases due to more efficient
conversion when the sun is at the horizon.
For both the ARC and the waveguide-optical filter coupler it would be beneficial if the lowest refractive index of the PS would be lowered further,
while at the same time increasing the gradient such that the highest refractive index is kept constant. This may be done by gradually oxidizing the
PS. The relative change in refractive index would be higher where more of
the total volume of Si in the PS is oxidized, as would happen in the high
porosity regions. By choosing the right oxidation conditions, the outermost
part of the PS, in a graded index layer, would be fully oxidized resulting in
porous SiO2 while the innermost would be, relative to volume, much less
139
oxidized.
6.2
MOEMS devices
Enabling optical elements, like filters, to move on a chip level opens up vast
possibilities within practical applications.
6.2.1
Membrane based MEMS pressure sensors
It has become nearly a standard to measure movement of elements in
a micro-electro-mechanical-system (MEMS) by resistivity changes in diffused/deposited piezoresistors at critical points. An alternative to this could
be to use PS optical filters.
Membrane based pressure sensors in MEMS technology often uses piezoresistors to measure the strain in the membrane as the pressure increases.
By etching a PS filter with a narrow reflection band on the membrane, the
stretching due to the strain could possibly expand the pores thus decreasing
the refractive index and shifting the reflectance band center wavelength. A
suitable structure for this could be a rugate filter. A sketch of a possible
design is shown in Fig. 6.4a. This effect has not yet been measured and it
is not known if a change in refractive index would be significant in such a
system.
Considering that with a proper design, a change in porosity of 0.04 % (abs.)
may result in a shift in the reflection band of 1 nm, it should be possible
to measure small, strain induced changes in porosity. This is based on a
thick Bragg reflector for reflectance at 1550 nm. A rough model may be
used to get a feel for the strain induced porosity change; consider a 100 µm
diameter membrane used as a pressure sensor. This membrane may have
a maximum vertical deflection in the order of 100 nm. This results in a
stretching of the membrane surface of a few percent (< 10 %). Describing
the porous silicon as consisting of disconnected pillars of silicon with air
between, a 1 % increase in the distance between columns in both lateral
directions would lead to a porosity change in the order of 0.1 %, hence, the
strain induced porosity change could be measurable.
One could make use of this effect in a pressure sensor by using a hybrid
structure where a broad band light source and a spectrometric sensor is
integrated to measure the spectral shift of the reflection band from the
deflected and stretched membrane.
Another alternative pressure sensor could be one based on interference
where the membrane functions as one of two mirrors in a Fabry-Pérot filter
and an inflexible substrate is bonded on top of the membrane forming a
narrow cavity between. To increase sensitivity either or both of the sides
140
of the cavity may be etched to form a PS reflectance filter. With a higher
reflection from the mirrors on both sides of the cavity, the cavity mode
will be sharper, hence smaller shifts in the cavity mode due to membrane
movement may be detected. A sketch showing the principle of the device
is shown in Fig. 6.4b. This type of pressure sensor is being used today as
microphones because of the quick response and high sensitivity.
a)
b)
Figure 6.4: a) A suggestion for a pressure sensor based on change in porosity
due to strain in the membrane. A reflectance filter structure is etched in PS
on the membrane. This will stretch and shift the reflectance band of the
filter when pressure exerts a force on the membrane. b) The principle of
a Fabry-Pérot microphone with PS reflectors on both sides of the cavity.
This could increase the sensitivity of the device. Acoustic waves deflect the
membrane, changing the resonance frequency and decreases the intensity of
a monochromatic beam. The transmitted beam intensity is detected with a
pn-junction in the substrate below the cavity.
6.2.2
MOEMS optical scanner and switch
By using a comb drive as shown schematically in Fig. 6.5, a filter may be
moved back and forth in the plane with a quick response rate. This device
may be used in several different ways. With a broad band light source
directed at a graded narrow band reflectance filter the device may be used
as a scanning ”monochromatic” light-source with the output light being of
a selectable wavelength. A similar device has been reported by Lammel et
al. [102], however, this was an upright filter where the filter angle could be
varied.
The known light source may be exchanged by an unknown, external light
source to be analyzed. A detector may then be placed in the path of the light
at the output. By then scanning the graded narrow band filter, the detector
output will be proportional to the intensity of the wavelength reflected from
the filter giving the spectral content of the source. An extension to this
would be to place a line array of detectors at the output, with the positioning
141
of the array such that the length is perpendicular to scan direction and the
length of the array similar to the width of the filter, hence quite a wide filter
would be necessary. This would increase the inertial mass and reaction time
suggesting that perhaps several filters in parallel could be used. This setup
would enable a multispectral line array detector which could be used for,
e.g., spatially resolved gas detection or environmental monitoring.
Instead of a graded filter it is possible to make binary filters such as reported
by Arens-Fischer et al. [103]. By fabricating an ARC on one end of the filter
and a reflection filter on the other end a switch may be realized. The input
beam to be switched on or off may be provided by fiber, and the output
beam may be coupled to a fiber. This setup should give a very high signal
on/off ratio. The switching rate of such a device should be in the single
digit to double digit kHz following the resonance frequency of reported
comb drive devices [104]. This is not high enough for data package routing
in telecom networks, but should be enough for network reconfiguration and
perhaps other applications.
Figure 6.5: A MOEMS device making use of a graded PS filter. This basic
element may be used in several different applications. The filter element is
coupled to a system of comb-drives which is able to horizontally translate
the filter. One application as suggested in the figure is an imaging spectral
scanner.
6.2.3
Multispectral MOEMS pixel array
A somewhat more complex detector based on the same technique as above,
is an array of diode detectors in plane with movable graded narrow band
transmission filters on top of each. This would give the sensor array multispectral detection capability. The filters will have an optimized gradient
and bandwidth depending on the demands for sensitivity or detection range.
The processing of this device would necessarily be quite complex and the fill
142
factor quite low as it would be necessary to have quite long filters compared
to the size of the detector for a practical measurable spectral range. The
benefits would be a very fast, compact and monolithic imaging multispectral detector.
6.2.4
Holographic scanner
The PS multilayer etching technique introduced by Volk et al. [28] may be
used to produce holographic diffraction gratings, i.e. gratings with sinusoidal groove profile. This has a potential for many different applications.
The grating fabrication is based on the formation of n-doped regions beneath p-doped regions in p-type Si and deposition of an insulating and HF
resistant Si3 N4 layer on the surface with openings above the n-doped regions
so that current is forced to run parallel with the surface above the n-doped
regions. Grooves in the Si are etched at the openings of the nitride layer
down to the n-region so that the etching occurs from a vertical surface with
a lateral homogeneous current density. By then etching as described earlier
in this thesis, one may obtain horizontal multilayers. By then removing
the nitride layer one may use this as an amplitude grating as the different
”grooves” (layer cross-sections) have different reflection coefficients due to
different refractive indexes. By etching lightly in a porosity selective alkaline etch, e.g. KOH, one would form proper grooves fabricating a phase
grating. It would be possible to form a holographic grating by etching
a rugate porosity profile with the subsequent alkaline etch improving the
diffraction properties of the grating.
This grating could be formed and released such that it would be connected
to a comb drive as explained above, similar to Fig. 6.5. By changing the
rugate period (or discrete repetition rate) during etching in a continuous
fashion the diffraction properties would change with position. The zeroth
diffraction order will always be present in the output from the grating, and
it will have the same angle as the input beam independent of grating parameters, however, the angle of the other orders depend on several parameters,
such as grating period. By focusing on the -1st order, a system may be
designed that has as an input a monochromatic laser beam at a set angle
and as output the zeroth order beam which may be attenuated, and a -1st
order beam which changes angle with the lateral position of the grating
due to different grating periods. The standard diffraction equation gives
an estimate of the relationship between incident angle, output angles for
different diffraction orders, wavelength and grating period:
sin α + sin βm = −mλ/d.
(6.3)
A rough estimate of a possible design is as follows: the diffracted beam
of order m=-1 will change the output angle, β−1 , from about -5◦ to 5◦
with a change in period, d, from 570 to 680 nm at a wavelength λ=600 nm
143
with the incident angle, α, being 75◦ . Assuming the beam to be scanned
is a monochromatic laser beam of diameter 50 µm and that the change of
grating period within one diameter is 20 nm (roughly 2◦ output difference
between maximum and minimum diffraction period within the beam) the
needed scan length would be about 250 µm. This may result in a fairly
quick and compact laser scanner. A schematic drawing showing the graded
grating is shown in Fig. 6.6.
Figure 6.6: The shown grating consists of a laterally etched PS multilayer
structure with a grading in layer period. In this case a lying down Bragg
reflector is shown. A holographic grating could be obtained with a lying
down rugate reflector. The surface structure is obtained by lightly etching
in, e.g., KOH which will etch highly porous silicon faster than silicon of lower
porosity. The diffracted beams of order6=0 will change angle depending on
the diffraction period where the incident beam hits.
Chapter 7
Conclusion
In this thesis a method for the in situ monitoring of parameters critical
for optical applications during etching of porous silicon has been described.
Data obtained by this method has been used to fabricate different types of
interference based optical filters in porous silicon.
The development of the method for in situ monitoring includes the development of a optical fiber based measurement system composed of an infrared
laser coupled to the dry side of the Si-sample in the etch cell during etching and a detector to measure the intensity of the reflected beam. As the
system is fiber based, it is compact and the system hardware may be at
a distance from any harmful chemicals. The reflected beam contains an
oscillating signal due to interference between the beams partially reflected
off the different interfaces in the porous silicon sample; front side, porous
silicon–substrate interface and back side. By analyzing the reflected interference signal with a short-time Fourier transform, the instantaneous or
depth dependent porosity is obtained along with the instantaneous or depth
dependent etch rate and porous silicon–substrate interface roughness.
Information on porosity and etch rate from both gravimetrical measurements as well as the in situ reflectance method has been used to etch both
discrete and inhomogeneous (rugate) optical interference filters in the visible and near-infrared spectral range. By applying a voltage laterally across
the Si-samples during porous silicon filter fabrication, the resulting filters
had a gradient in the filter response in the direction of the voltage drop.
This could be developed into a near-infrared spectrometer.
The porosities and etch rates obtained by the in situ reflectance method
show a very strong dependence on etch time. This affects the filter etching
such that the resulting filter response is non-optimal. Attempts at counteracting this time variation is shown. A discussion of the causes of the
non-optimality of the filter responses is given as well as possible ways of
avoiding the detrimental effects of this time variation which is important
for fabricating infrared optical filters of very good quality.
145
146
Possible uses of porous silicon in other novel applications, both passive and
in conjunction with micro-opto-electro-mechanical-systems are discussed in
the last chapter.
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