Infrared studies of trenches etched in silicon

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Department of Physics, Chemistry and Biology

Master’s Thesis

Infrared studies of trenches etched in silicon

Lars Karlsson

LITH-IFM-EX--07/1857--SE

Department of Physics, Chemistry and Biology Linköpings universitet

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Master’s Thesis LITH-IFM-EX--07/1857--SE

Infrared studies of trenches etched in silicon

Lars Karlsson

Supervisor: Hans Arwin

ifm, Linköpings universitet

Examiner: Kenneth Jährendahl

ifm, Linköpings universitet

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Avdelning, Institution

Division, Department Laboratory of Applied Optics

Department of Physics, Chemistry and Biology Linköpings universitet

SE-581 83 Linköping, Sweden

Datum Date 2007-11-23 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport

URL för elektronisk version

http://urn.kb.se/resolve?urn:nbn:se:liu:diva-7326

ISBN

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ISRN

LITH-IFM-EX--07/1857--SE

Serietitel och serienummer

Title of series, numbering

ISSN

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Titel

Title

Optiska studier av etsade kiselstrukturer med spektroskopisk ellipsometri i det infraröda våglängdsområdet

Infrared studies of trenches etched in silicon

Författare

Author

Lars Karlsson

Sammanfattning

Abstract

Previous studies of protein adsorption on silicon have been restricted by the choice of a simple structure or large surface for protein to adsorb on. The aim of this project was to develop an optical model for more complex nanostructures in form of trenches etched in silicon and then examine if a protein would adsorb to the surface. The method used was infrared ellipsometry.

The experimental values from measurements on the sample were used to de-velop an optical model that represent the nanostructure. A three-layered biaxial model proved to be accurate. One sample was then exposed to the protein albumin and then measured upon again. The results before and after protein adsorption were compared and a small optical signature was found were it could be expected for this specific protein. This shows that it is possible to detect adsorption in a complex nanostructure and to develop an accurate optical model for said structure.

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Nyckelord

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Abstract

Previous studies of protein adsorption on silicon have been restricted by the choice of a simple structure or large surface for protein to adsorb on. The aim of this project was to develop an optical model for more complex nanostructures in form of trenches etched in silicon and then examine if a protein would adsorb to the surface. The method used was infrared ellipsometry.

The experimental values from measurements on the sample were used to de-velop an optical model that represent the nanostructure. A three-layered biaxial model proved to be accurate. One sample was then exposed to the protein albumin and then measured upon again. The results before and after protein adsorption were compared and a small optical signature was found were it could be expected for this specific protein. This shows that it is possible to detect adsorption in a complex nanostructure and to develop an accurate optical model for said structure.

...

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Acknowledgements

This master’s thesis would not be possible without the help and support of Hans Arwin whose enthusiasm for science has made a great impact on me as a person. Of course I have to thank my friends for sticking with me trough thick and thin over the years, especially Pontus, Daniel, Markus, Roger, Jonatan and Pelle. I can not think of a better gang to ride into the sunset with now that this adventure is over.

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Introduction

1.1 Background

Recentley there has been a merging of bio- and nanotechnology. Many new applica-tions in biosensors, bioelectronics and biomaterials are found. Protein absorption onto silicon surfaces has previously been studied when trying to observe what is adsorbed and how much is absorbed. Infrared ellipsomerty is a non destructive analysis technique that can be used for identification and for obtaining structural information. There is a large number of reference spectra availably for organic molecules such as proteins.

Previously there have been studies with bioadsorption on flat surfaces and monolayers [1] or in porous silicon [2]. In porous silicon it is possible to have a large surface but exact structure of the internal surface is unknown and therefore there have not been reliable optical models for the nanostructure in those studies. For monolayers the surface structure is known but there is a very small surface for the proteins to absorb to.

In this study we will use trenches etched in silicon which gives us a known structure to build a model for and it has more surface than a flat silicon structure. Simulations show an enhanced sensitivity for studies of bioadsorption in nanos-tructures in the form of trenches etched in silicon [3]. Such samples have been prepared [4] but so far no acceptable optical model is available.

1.2 Objective

This project has the main objective to measure with IR-ellipsometry on silicon trenches and to develop and represent nanostructure to describe their reflection properties. A second objective is to expose these trenches to protein molecules which adsorb on the "walls" in the structure. Finally it will be investigated if the dielectric function of the adsorbed protein can be determined.

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Chapter 2

Theory

2.1 Maxwell’s equations, plane waves and

polar-ization

Electromagnetism was summarized by Maxwell in 1873 in his four equations [5]. Maxwell´s equations describe electromagnetic fields and their behavior when they interact with matter. The equations in the most common form are

∇ · D = ρ (2.1) ∇ · B = 0 (2.2) ∇ × H = ∂D ∂t + J (2.3) ∇ × E = −∂B ∂t (2.4)

These fields are the electric displacement field D, the electric field E, the magnetic flux density B and the magnetic field H. J is the current density and ρ in this case is the charge density. For ellipsometry the most interesting field is the E-field. One solution to Maxwell’s equations in the case of harmonic fields is the plane wave. The electric field, the magnetic field and the propagation vector q are orthogonal to each other and form a right-handed system. The amplitudes of the fields will also be proportional to each other according to

E0= cµH0

where c is the speed of light, is the electric permittivity and µ is the permeability. The amplitudes E0 and H0 are defined by

E= E0ei(q·r−ωt) (2.6) H= H0ei(q·r−ωt) (2.7) where r is the coordinate vector.

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4 Theory

2.1.1 Polarization

When studying reflection at surfaces it is necessary to describe the orientation of the fields in a propagating electromagnetic wave, i.e. the polarization. This is done by decomposing the fields into two components, usually with the plane of incidence as a reference. The plane of incidence is defined by the propagation vectors of the incident, reflected and refracted waves when studying reflection and transmission at surfaces. When no oblique reflection or incidence occurs, a plane of incidence is chosen in such a way that it simplifies the description of the polarization. The two components are the p-component, seen in fig. 2.1, for which the electric field lies in the plane of incidence, and the s-component, also seen in fig. 2.1, where the electric field is perpendicular to the plane of incidence.

N0 θ0 θ0 θ1 Etp Htp N1 Eip Hip Erp Hrp N0 θ0 Ets Hts N1 Eis His Ers Hrs θ0 θ1

Figure 2.1: p- and s-parts of the reflected, transmitted and incident fields in reflection from a single surface with complex refractive index N0 of the ambient medium and N1 of the substrate. The angle of incidence θ0 equals the angle of reflection, and θ1is the angle of refraction.

2.2 Ellipsometry

Ellipsometry has been around for quite some time and has a lot of standard ap-plications today. Paul Drude provided a theoretical basis for ellipsometry in the late 1800 [6]. He also performed experiments and measurement that determined optical properties of metals [7]. Since the mid 1970s there have been great progress in the field of ellipsometry thanks to the availability of faster computers.

Ellipsometry is defined as the measurement of the state of the change in po-larization of a polarized light wave [8]. For the measurements carried out in this project IR-ellipsometry was used. The interaction between the sample and the light is in reflectionmode which is defined as reflection ellipsometry. When the change of the polarization after it is reflected from a sample is analyzed, one can get information about layers that are thinner than the wavelength of the light. The change in polarization of the light beam depends on the surface and the thin film properties. With the help of ellipsometry one can determine optical properties of materials in terms of spectral dependence of refractive indices and determine

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2.2 Ellipsometry 5

microstructural parameters such as crystal orientation, layer thickness and poros-ity.

The technique has many standard applications in the semiconductor industry today but lately it has become interesting for researchers in such fields as biology and medicine. In these new areas ellipsometry faces the challenge to measure on unstable surfaces and to develop microscopic and imaging ellipsometry [9].

2.2.1 Reflection ellipsometry

Reflection ellipsometry is a technique based on measuring polarization changes occurring upon reflection at oblique incidence of a polarized monochromatic plane wave. Figure 2.2 shows the principle of ellipsometry.

p-plane s-plane E p-plane s-plane E Plane of incidence Sample θ

Figure 2.2: The principle of ellipsometry: In oblique reflection the plane of inci-dence defines the angle of inciinci-dence and the complex amplitudes of the p- and s-polarized complexed-valued electric field components before and after reflection, respectively.

The basic quantity measured with an ellipsometer is the ratio

ρ=χr

χi (2.8)

where χrand χiare the complex-number representations of the states of

polariza-tion of the reflected and incident beams. In a cartesian coordinate system with the p- and s-direction parallel and perpendicular to the plane of incidence the quantity

χis found to be

χ=Ep Es

(2.9) where Ep and Es are the complex-valued representations of the electric fields in

the p- and s-directions. For light reflected from optically isotropic samples no coupling will occur between the orthogonal p- and s-polarizations. The

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6 Theory

valued amplitude reflection coefficients Rpand Rsfor light polarized in the p- and

s-direction then become

Rp= Epr Epi (2.10) Rs= Esr Esi (2.11) χi= Epi Esi (2.12) χr= Epr Esr (2.13) Equation (2.8) becomes ρ= Epr Esr Esi Epi = Rp Rs

= tan Ψei(δp−δs) (2.14)

and the difference in phase is

∆ = δp− δs (2.15)

Ψ and ∆ are introduced as parameters in a polar description of ρ and are called ellipsometric angles and

tan Ψ = Rp Rs (2.16)

This gives us the definitions to the two parameters Ψ and ∆ [8].

2.2.2 Generalized ellipsometry

We can see that Eq.(2.14) is valid for optically isotropic samples and for anisotropic samples having diagonal Jones matrices. In the general case of anisotropic samples Ψ and ∆ depend on the polarization state of the incident beam. This can be described with a non-diagonal Jones matrix.

Rr= Rpp Rsp Rps Rss (2.17) The first index refers to incident polarization mode and the second to the emerging polarization mode.

An ellipsometric characterization then requires measurement of at least three values on ρ at three different χi and three pairs of Ψ and ∆ are defined. This is

re-ferred to as generalized ellipsometry. The complex-valued generalized ellipsometer parameters are then defined according to

ρpp = Rpp Rss

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2.3 Optical modeling 7 ρps= Rps Rpp = tan Ψpsei∆ps (2.19) ρsp= Rsp Rss = tan Ψspei∆sp (2.20)

This gives us data in six different parameters Ψpp, Ψsp, Ψps, ∆pp, ∆spand ∆ps.

De-pending on the sample properties and orientation, the off-diagonal normalized ele-ments ρps and ρspmay be symmetric, antisymmetric, Hermitian, anti-Hermitian,

completely different or zero [10].

2.2.3 Infrared ellipsometry

Ellipsometry in the visual light region is very common for analysis of thin films. It can be used to determine the thickness of the film with great accuracy. Spec-troscopic infrared ellipsometry can be used for characterization of vibrations in molecules. At first glance one might think that larger wavelengths of the infrared light might reduce the sensitivity of the measurement but it is actually reliable down to nanometer thick layers. Specific chemical information about the the sam-ple can be determined in a spectrum that represents molecular vibrations.

This makes IR-ellipsometry a very useful tool for characterization of organic materials such as proteins, polymers etc. When looking at proteins it is interesting to note that the IR-spectra is divided into four different regions [11].

1. The region with vibrations between hydrogen and other elements such as oxygen, nitrogen and carbon. This happens within the wavenumber range 4000 to 2500 cm−1_.

2. The region with vibrations from triple bindings, 2500 to 2000 cm−1_{, where} C ≡ C and C ≡ N occur.

3. The region with double bindings, 2000-1500 cm−1_{, were usually C=C, C=O}

and C=N occur.

4. The region under 1500 cm−1_{, also known as the fingerprint region where}

similar molecules have different absorption patterns.

2.3 Optical modeling

Ellipsometry provides us with information about Ψ and ∆ for the sample. These parameters depend also on wavelength and angle of incidence. To be able to analyze the information we need an optical model. What kind of model depends on the sample. With optical modeling we will develop a model for the sample and then determine the unknown model parameters by non-linear regression analysis.

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8 Theory

The linear optical response of materials can be described with the refractive index

N(λ) = n(λ) + ik(λ) (2.21)

where n(λ) and k(λ) are the wavelength dependent real and imaginary part of

N(λ).

As an alternative one can also use the dielectric function

(λ) = 0(λ) + i00(λ) = N(λ)2 (2.22)

Often the objective of analyzis of ellipsometric data is to determine N(λ) of one or several constituents of the sample.

In many cases a wavelength-by-wavelength approach is used. This means that the N -value is extracted from the experimental data for each wavelength and independent of all other spectral points. For this the thickness and N -spectra of all the remaining sample constituents have to be known. However, in this project dispersion models are used for the materials optical properties

Model

Fit

Results

Modify

Data

Figure 2.3: Illustration of the different steps in ellipsometric data analysis Once an optical model is defined the data collected from the ellipsometry mea-surement can be fitted to the model. This is done by generating ellipsometry values for the same condition as the measured data. In this work some parameters are known since the sample is manufactured after specifications.

The Levenburg-Marquard algorithm is used to minimize the difference between the generated values that the model gives us and the experimental values. This algorithm is based on Mean Squared Error(MSE) and is carried out by altering the unknown parameters in the optical model until a minimum in MSE is reached. MSE is defined as

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2.3 Optical modeling 9 M SE = 1 2N − M X " Ψmod₋_Ψexp σΨ,i 2 + _∆mod₋_∆exp σ∆,i 2# (2.23) where N is the number of measured Ψ and ∆ couples, M is the number of pa-rameters that have been fitted and σΨ and σ∆ are the standard deviation of Ψ and ∆, respectively. The general procedure is shown in Fig. (2.3). In practice the user and/or program varies the fitting parameters until MSE has a minimum. The fitting parameters values are then the result of the analysis.

2.3.1 Reflectance

In optical measurements the parameter usually obtained is the irradiance, which will be denoted I. Now we can calculate the reflectance

<=Ir Ii = Er Ei 2 = |R|2 _(2.24)

where Ir and Ii are incident and reflected irradiances and R the reflection

coeffi-cient.

Two-phase systems

The two-phase system, see Fig. 2.4, is the most simple case, i.e. a substrate with refractive index N1 in an ambient with index N0, and the Fresnel equations are used in their original form . The Fresnel reflection coefficients will be denoted r

rp= Erp Eip = N1cos θ0− N0cos θ1 N1cos θ0+ N0cos θ1 = tan (θ0− θ1) tan (θ1+ θ0) (2.25) rs= Ers Eis = N0cos θ0− N1cos θ1 N0cos θ0+ N1cos θ1 = sin (θ1− θ0) sin (θ1+ θ0) (2.26)

These equations relate the p- and s-components of the reflected fields, Erpoch Ers

to components of the incident fields, Eip och Eis

From this we can with the help of Snell’s law derive an expression for the complex refractive index

N1= N0sin θ0 s 1 + _{1 − ρ} 1 + ρ 2 tan2_θ 0 (2.27) and since = N2 (2.28)

it follows that for the medium nr 1

1= 0sin2θ0 1 + 1 − ρ 1 + ρ 2 tan2_θ 0 ! (2.29)

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main: 2007-12-5 15:34 — 10(22) 10 Theory d1 N₁ N₂ N0 N₀ N₁ N0 N₁ Nn-2 Nn-1 dn-2 d1 1 θ0 θ θ0 1 θ 2 θ θ_n-1 n-2 θ θ1 θ0

Figure 2.4: Reflection and transmission in the 2-phase-, 3-phase- and n-phasemodel.

Three-phase systems

First we restrict ourselves to discuss the case when the principal axes coincide with an xyz coordinate system defined with the xz-plane equal to the plane of incidence, with the x-axis along the surface and the z-axis pointing into the substrate. In the three-phase-model a thin layer with the thickness d and refractive N1= n1+ ik1 between two semi-infinite media are considered. The top layer is called the ambient and has the real-valued refractive index N0= n0. The bottom layer is the substrate and the refractive index for that layer is N2= n2+ ik2.

As usual we will assume a plane wave incident in the xz-plane at an angle of incidence θ0. When exploring the three-phase system we have to account for two reflecting interfaces due to the interaction between the electromagnetic field of the incidence wave and the thin film covered substrate.

What we are trying to do here is to find relations between the incident wave and the reflected and transmitted waves. When trying to find these transmission and reflection coefficients it is important to remember that the boundary conditions apply differently for the s- and p-polarizations so we need to treat these separately. Reflection coefficients will be denoted Rpand Rsand the transmissions coefficients

will be denoted Tp and Ts.

The general form of the plane-wave solutions to the wave equation in the xz-plane becomes

E(r) = E(z)ei(qxx−ωt) (2.30)

where q is the propagation vector q = (qx,0, qz)

If we assume an s-polarization we only have an y-component of E which gives us

E(r) = (0, E0y,0) ei(qxx−ωt) (2.31)

In the ambient, the layer and in the substrate we will get different equations for how E varies in the ambient

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2.3 Optical modeling 11

In the layer

E1y(z) = Ceiq1zz+ De−iq1zz 0 < z < d (2.33) In the substrate

E2y(z) = F eiq2z(z−d) z < d (2.34)

where qz and −qz are the z-components of the wave vectors of the incident and

reflected waves and A, B, C, D and F are their complex amplitudes. Using this we can define a reflection coefficient as the ratio B/A between the reflected and incident fields. A transmission coefficient can be defined as F/A between the transmitted and incident fields.

The boundary conditions on the tangential components of the E and H-field require that Ex, Ey, Hx and Hy must be continuous at the interfaces z = 0 and z = d. For the s-polarization there is no Hy -component so the only thing that

remains is to find the Hx-component. Using equation (2.4) and

B= µµ0H (2.35) implies that Hx= i (ωµ0) ∂E ∂z (2.36)

and then using what we learned in Eqs. (2.32 - 2.34) we find that

Hx(z) =      −q0z ωµ0(Ae iq0zz_{− Be}−iq0zz), z < 0 −q1z ωµ0(Ce iq1zz_{− De}−iq1zz), 0 < z < d −q2z ωµ0F e iq2z(z−d)_, _{z < d} (2.37) Considering that the boundary conditions saying that Ey and Hxmust be

contin-uous at the interfaces z = 0 and z = d gives us

A+ B = C + D (2.38) q0z(A − B) = q1z(C − D) (2.39) Ceiq1zd+ De−iq1zd= F (2.40) q1z Ce iq1zd+ De−iq1zd= q 2zF (2.41)

Now we eliminate C and D and if we solve the equations for the ratios B/A and F/A we obtain B A = (q0z− q1z) (q1z+ q2z) + (q0z+ q1z) (q1z− q2z) ei2q1zd (q0z+ q1z) (q1z+ q2z) + (q0z− q1z) (q1z− q2z) ei2q1zd (2.42) F A = 4q0Zq1Zeiq1Zd (q0z+ q1z) (q1z+ q2z) + (q0z− q1z) (qz− q2z) ei2q1zd (2.43) For real wave vectors we have that

qiz =

2π

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12 Theory

(i =0,1,2) and gives us the Fresnel coefficients for the s-polarizations as

r01= q0z− q1z q0z+ q1z (2.45) r12= q1z− q2z q1z+ q2z (2.46) t01= 2q 1z q0z+ q1z (2.47) t01= 2q1z q1z+ q2z (2.48) and now the reflection and transmission coefficients can be rewritten as

R ≡ B A = r01+ r12ei2β 1 + r01r12ei2β (2.49) T ≡ B A = t01t12ei2β 1 + r01r12ei2β (2.50) where rlm is the Fresnel’s coefficient rp between the phases l and m and β is the

film phase thickness, which is given by

β= 2πd

λ N1cos θ1 (2.51)

Here N1is the complex index of refraction of the middle medium, d is the thickness of the layer and θ1is the angle of refraction. If the same derivation is performed for the p-polarization this leads to the same expressions except that the coefficients

r01, r12, t01and t12will be associated with the p-versions of the Fresnel equations. This is the final piece of the puzzle and we can now write

Rp= Erp Eip = r01p+ r12pei2β 1 + r01pr12pei2β (2.52) Rs= Ers Eis = r01s+ r12sei2β 1 + r01sr12sei2β (2.53) n-phase systems

When adding more layers to the model the complexity of the analytical expression for the reflectivity will increase. For solving these systems matrix-based methods is used with the help of computers. The optical system involves m layers indexed 1, 2, 3..m of different materials between the semi-infinente ambient indexed 0 and the semi-infinite substrate indexed m + 1. We assume that all phases are homogeneous and isotropic. All boundaries are parallel and abrupt. At any point in the system we can mathematically resolve the electric field of the light into two subfields, one corresponding to a wave traveling in the +z-direction, E+ _{and one}

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2.3 Optical modeling 13

travelling in the -z-direction, E−_{. The total fields in any given z-plane is given by} E(z) = E+(z) + E−(z).

Consider the relation between the total fields in two different planes z1and z2. Light propagation is described by linear equations and thus we geneally have

E+(z1) = S11E+(z2) + S12E−(z2) (2.54) E−(z1) = S21E+(z2) + S22E−(z2) (2.55) or in matrix form E(z1) = S(z1, z2)E(z2) (2.56) where E(z1) = E+_(z 1) E−(z1) E(z2) = E+_(z 2) E−(z2) (2.57) are called generalized field vectors. The matrix S(z1, z2) is defined by

S(z1, z2) = S11 S12 S21 S22 (2.58) The properties of the part of the system that lies between the planes z1 and z2 enter the matrix in the form of coefficients Sij In the case of two optical systems

the relation between the fields of the first and second optical system is related by

E1= S1E2 (2.59)

E2= S2E3 (2.60)

where S1 _{and S}2 _{are the matrices containing the properties of the two optical} systems. If we eliminate the intermediate field vector E2 we can write

E1= S1S2E3= SE3 (2.61)

We call S = S1_S2 _{the scattering matrix of various optical systems. To derive the} reflection and transmission coefficients from the S-matrix we write equation (2.56)

as E₀+ E₀− = S11 S12 S21 S22 E_m+1+ E_m+1− (2.62) where E0 is the field vector in the ambient immediately adjacent to the interface, Em+1is the corresponding field vector in the substrate and S the total scattering

matrix for the whole system containing m layers on a substrate. E+

0 is the incident wave, E−

0 is the reflected wave, E +

m+1the transmitted wave and E −

1+m= 0 because there is no backscattered wave in the substrate. From this follows

E₀+ E₀− = S11 S12 S21 S22 E_m+1+ 0 (2.63) The overall complex transmission coefficient is given by

T = E + m+1 E₀+ = 1 S11 (2.64)

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14 Theory

and the overall complex reflection coefficient by

R=E − 0 E₀+ = S21 S11 (2.65)

Scattering matrices are generally defined independently for the p- and s-polarizations,

Spand Ss, respectively. The reflection and transmission coefficients for the p- and

s-directions become Rp= S21p S11p Tp= 1 S11p (2.66) Rs= S_S21s 11s Ts= 1 S11s (2.67)

2.3.2 Reflection at an anisotropic surface

Biaxially anisotropic substrate

We restrict ourselfs to discuss the case when the principal axes coincide with an xyz coordinate system defined with the xz-plane equal to the plane of incidence with the x-axis along the surface and the z-axis pointing into the substrate. There are three different indices of refraction in an biaxial medium and they will be denoted N1x, N1y and N1z. The reflection matrix is diagonal in this symmetry and the diagonal elements are

rpp= N1xN1zcos φ0− N0 N1z2 − N 2 0sin 2_φ 0 1₂ N1xN1zcos φ0+ N0 N1z2 − N02sin 2_φ 0 1 2 (2.68) rss= N0cos φ0− N1y2 − N02sin 2_φ 0 1 2 N0cos φ0+ N1y2 − N02sin 2_φ 0 12 (2.69)

2.4 Effective medium approximation

A heterogeneous material can be modeled with a so called effective medium ap-proximation generally known as EMA. The goal is to find an effective dielectric function EM Aexpressed in terms of microstructure and complex-valued dielectric

functions of the components of a composite material. This EM Acan account for

the essential features of the heterogeneous structure. The various optical quanti-ties like reflection and transmission can be determined in a similar manner as for homogeneous materials.

Initially we assume that heterogeneous materials are macroscopically uniform and we also assume that components of the heterogeneous material are isotropic so they can be characterized by a scalar dielectric function.

Consider a simple hypothetical two-component composite of materials A and B, see fig. 2.5. The macroscopic average dielectric function k in the case the

boundaries are parallel to the applied field is given by

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2.4 Effective medium approximation 15

where fAand fBare the materials volume fractions. When the field and the sample

boundaries are perpendicular, the macroscopic average is given by

⊥= _f 1 A A + fB B (2.71) Due to the limitations on the validity of the effective medium approximations there are many different theories which are applicable to different microstructures.

A

B

ε

B

A

B

ε

B A A

f

A A B B

E

_E

Figure 2.5: Microstructures for a two-component composite. To the left all the boundaries are perpendicular to E and to the right the boundaries are parallel to the applied field E

2.4.1 The Bruggeman effective medium

One theory that is often used in optical analysis is the Bruggeman effective medium [12]. This theory describes an aggregate microstructure where materials A and B are randomly mixed. The approximation assumes that the microstructural dimensions are much smaller than the wavelength λ.

In this theory we approximate that the unit cell is a sphere whose dielectric function is A with the probability fA and B with the probability fB = 1 − fA.

Bruggeman gives us this equation for the composite dielectric function 

fA A−  A+ 2 + (1 − fA) B−  B+ 2 = 0 (2.72)

One of the great advantages of the Bruggeman model is that it is symmetric and can be used for all values of fA.

So far all particles discussed have been assumed to be spherical. However, the shape of the particle will affect the local field. To coupe with this problem there are some correction terms called depolarization factors, table 2.1. When a dielectric object is placed in an electric field E0 the surface bound charge creates a depolarization field E1 which reduces the total field in the object. The electric field can be described as

E1= E0− qP

0

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16 Theory

where q is the depolarization factor and P is the polarization inside the object. The factor q is dependent on the geometric shape of the object and 0is the electric constant.

Depolarization Factors

Shape Axis q

Sphere any 1/3

Thin slab normal 1

Thin slab in plane 0

Long circular cylinder longitudinal 0 Long circular cylinder transverse 1/2

Table 2.1: Depolarization factors

2.5 Euler angles

In this project measurements are done at many different sample angles (rotational azimuths) and it will be necessary to figure out what movement respond to which Euler angle. Any rotation can be described using three angles fig. 2.6. The angles that give the three rotation matrices are called Euler angles.

The intersection of the xy and the x´y´ coordinate planes is called the line of nodes. Φ is the angle between the x-axis and the line of nodes. θ is the angle between the z-axis and the z´-axis. Ψ is the angle between the line of nodes and the x´-axis. ψ θ Φ z y x x’ z’ y’ Line of nodes

Figure 2.6: Euler angles

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2.6 Capillary action 17 M =   cos φ sin φ 0 −sin φ cos φ 0 0 0 1     (1 0 0 0 cos θ sin θ 0 − sin θ cos θ     cos ψ sin ψ 0 −sin ψ cos ψ 0 0 0 1   (2.74) The problem is that it has never been agreed upon the order in which the rotations are applied and even the axes about which they are applied. This means that I must somehow define which rotation affects the results in my measurements.

2.6 Capillary action

Capillary action is the force that drives liquids through capillaries. It is caused by the relationship between adhesion which is the liquids attractive force against the capillary wall and the cohesive force between the liquids molecules. This is the same phenomenon that helps trees to suck up water from the ground.

In the case of the silicon trenches the capillary force will keep the water out of the trenches since the height of the liquid column is given by

h= 2γ cos θ

ρgr (2.75)

where γ is the surface tension (J/m2_{), θ is the contact angle, ρ is the liquid density,} g is the acceleration due to gravity and r is the with of the trench.

2.7 Proteins

The word protein comes from the greek word "prota" which means "of primary importance" and was first described and named by the Swedish chemist Jakob Berzelius in 1838. Proteins are large organic compounds made of amino acids that are arranged in a linear chain and joined together by peptide bonds between the carboxyl and amino acid residues.

Proteins are linear polymers built from 20 different L-α-amino acids. All amino acids share common structural features including an α-carbon to which an amino group, a carboxyl group and a variable side chain are bonded.

The amino acids in a polypeptide chain are linked by peptide bonds formed in a dehydration reaction. Once linked in the protein chain, an individual amino acid is called a residue and the linked series of carbon, nitrogen, and oxygen atoms are known as the main chain or protein backbone. The peptide bond has two resonance forms that contribute some double bond character and inhibit rotation around its axis, so that the α carbons are roughly coplanar.

The isoelectric point is the pH at which a protein carries no net electrical charge. It is important to know the isoelectric point to determine what kind of buffer should be used for protein adsorption. This is to minimize the electrostatic forces that counteract the adsorption [13].

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main: 2007-12-5 15:34 — 18(30)

18 Theory

2.7.1 Albumin

Albumin is the most abundant protein in the human blood plasma. It is produced in the liver. Albumin comprises about half of the blood serum protein. The protein has many functions such as maintaining osmotic pressure, transporting thyroid hormones that control how fast the body burns energy, transporting fat soluble proteins, transporting many drugs, competitively bind calcium ions and buffering pH. Figure 2.7 shows the structure of human albumin.

Albumin is synthesized in the liver as preproalbumin which has an N-terminal peptide that is removed before the protein is released from the rough endoplasmic reticulum. The product, proalbumin, is in turn cleaved in the Golgi vesicles to produce the secreted albumin.

The reference range for albumin concentrations in blood is 30 to 50 g/L. Low blood albumin levels (hypoalbuminemia) can be caused by the liver dis-ease cirrhosis of the liver (most commonly), decrdis-eased production (as in star-vation), excess excretion by the kidneys (as in nephrotic syndrome), excess loss in bowel (protein losing enteropathy) malnutrition, malabsorption, neoplasia and pregnancy.

Molecular dimensions of human albumin is 3 × 3 × 8 nm. This means that it should not have a problem fittning in the treanches. It has a serum half-life of approximately 20 days and a molecular mass of 67 kDa. The isoelectric point is located at pH 5.8.

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2.7 Proteins 19

2.7.2 Protein adsorption

Adsorption is a process that occurs when a gas or liquid solute accumulates on the surface of a solid or, more rarely, a liquid (adsorbent), forming a molecular or atomic film (the adsorbate). It is different from absorption, in which a substance diffuses into a liquid or solid to form a solution. The term sorption encompasses both processes, while desorption is the reverse process.

Adsorption is operative in most natural physical, biological, and chemical sys-tems, and is widely used in industrial applications such as activated charcoal, synthetic resins and water purification. Adsorption, ion exchange and chromatog-raphy are sorption processes in which certain adsorptive are selectively transferred from the fluid phase to the surface of insoluble, rigid particles suspended in a vessel or packed in a column.

Protein adsorption is controlled by the characteristics of the surface, the protein and by the buffer. Properties such as electrical charge, molecular orientation and pH of the buffer all influence the adsorption process [14].

Two main factors that control protein adsorption are • Hydrophobic interaction in the proteins

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main: 2007-12-5 15:34 — 20(32)

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Chapter 3

Experimental and

instrumentation

3.1 Ellipsometer

The ellipsometer used for the measurements is an IR-VASE from J.A. Woollam Co. Inc., USA. IR-VASE is a spectroscopic ellipsometer that covers a spectral range from 300 to 5000 cm−1 _{The IR-ellipsometer is based on the concept of a}

rotating compensator [9].

The input unit of the IR-VASE contains mirrors that focus the polarized radia-tion on the sample that is being measured. Polarizaradia-tion is achieved by passing the radiation through a wire grid polarizer. This polarizer is mounted to a goniometer which allows the polarization of the radiation to be adjusted relative to the plane of incidence.

The input unit also contains an alignment laser, a tool that helps to set the sample surface perpendicular to the plane of incidence. For the analysis of the experimental data the WVASE program from J.A. Woollam Co. Inc., USA was used.

3.2 Silicon samples

The samples used for the measurements were made at KTH by Xavier Badel. The samples were made using the lithography and electrochemical etching method [15]. This is a technique that makes it easy to make 3-D etchings in silicon. The idea is to have trenches etched in the sample. The walls are about 300 nm wide, 9 µm high and spaced with a distance center-to-center of 1500 nm. The sample is shown in Fig. 3.1. The etched area of the sample is circular with a diameter of about 1.6 cm. As it turns out the periphere of the sample can not be used for measurements.

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main: 2007-12-5 15:34 — 22(34)

22 Experimental and instrumentation

Figure 3.1: The trenches in the silicon sample

3.2.1 Euler angles

In the measurements the Euler angles were defined in according to fig. 3.2. This is how the angles are represented in the WVASE-program that was used to process the experimental data and also used to build the model. The Φ-direction is the in the a wall of the sample. θ is 90◦ _{and is the direction along the normal of the}

wall. Ψ is the sample rotation. There is a build in idiosyncrasy in the WVASE system. The legends on WVASE experimental data graphsis the sample rotation angle is shown as θ = xx◦ _{, but is more closely related to Ψ in the Euler angles.}

Therefor θ will be the sample rotation from here on.

3.3 Protein adsorption

Before applying the protein the trenches must be filled with water. Because of the capillary forces we first need to put the sample in alcohol, which have different characteristics than water and therefore has no problem making its way down into the trenches. The sample was put in 30 ml of methanol that was diluted with 15 ml of water every 24:th hour until the alcohol percentage was down under 0.1%.

10 mg of human serum albumin was dissolved in 10 ml PBS ( Phosphate-Buffered Saline). The PBS has a pH of 7.4 which is close to the pH in the human body. The silicon sample was put in 26 ml of PBS since that is the amount it took to cover it completly. Then 3 ml of the protein solution was added. The sample was left in the solution för 60 minutes whereafter it was dried with nitrogen gas.

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3.4 Measurements 23

θ ψ

Φ

z

Wall in the sample

Substrate x

y

Figure 3.2: Definition of euler angles during measurements in the WVASE-program.

3.4 Measurements

Three different silicon samples were measured with the IR-VASE. The angles of incidence were 25◦_{, 45}◦_{and 65}◦_{in the range of 300 - 5000 cm}−1_{and the resolution}

was 8 cm−1_{. The sample was rotated 360}◦ _{and the measurements for all three}

angles of incidence was carried out every 45◦_{. One silicon sample was prepared}

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main: 2007-12-5 15:34 — 24(36)

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Chapter 4

Results and discussion

4.1 Experimental results

As we can see in Fig. 4.1 and 4.2 there are three different Ψ- and ∆-curves for each angle af incidence and θ = 0◦_{. In Fig. 4.3 and Fig. 4.4 we can see the results for}

the three different incidence angles. In the appendix Fig. A.1 to Fig. A.28 show the curves for different θ. AnE is the Rpp/Rss-ratio, Aps is the Rps/Rpp-ratio and

Asp is the Rsp/Rss-ratio. Information is located between 6 µm and 18 µm. Above

and below these limits the noise was to large for any usable information to be seen.

Experimental Data

Wavelength (µm) 6 8 10 12 14 16 18

Ψ

in d eg re es 0 20 40 60 80 100 Exp AnE 25° θ=0° Exp Aps 25° θ=0° Exp Asp 25° θ=0°

Figure 4.1: Experimental values of Ψ for the incident angle 25◦ _{and θ = 0}◦_.

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main: 2007-12-5 15:34 — 26(38)

26 Results and discussion

Experimental Data

Wavelength (µm) 6 8 10 12 14 16 18

∆

in d eg re es -100 0 100 200 300 Exp AnE 25° θ=0° Exp Aps 25° θ=0° Exp Asp 25° θ=0°

Figure 4.2: Experimental values of ∆ for the incident angle 25◦_{and θ = 0}◦_.

Experimental Data

Wavelength (µm) 6 8 10 12 14 16 18

Ψ

in d eg re es 0 20 40 60 80 100 Exp AnE 25° θ=0° Exp AnE 45° θ=0° Exp AnE 65° θ=0° Exp Aps 25° θ=0° Exp Aps 45° θ=0° Exp Aps 65° θ=0° Exp Asp 25° θ=0° Exp Asp 45° θ=0° Exp Asp 65° θ=0°

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4.2 Optical model devolepment 27

Experimental Data

Wavelength (µm) 6 8 10 12 14 16 18

∆

in d eg re es -100 0 100 200 300 Exp AnE 25° θ=0° Exp AnE 45° θ=0° Exp AnE 65° θ=0° Exp Aps 25° θ=0° Exp Aps 45° θ=0° Exp Aps 65° θ=0° Exp Asp 25° θ=0° Exp Asp 45° θ=0° Exp Asp 65° θ=0°

Figure 4.4: Experimental values of ∆ for all three incident angles and θ = 0◦_.

4.2 Optical model devolepment

4.2.1 Silicon trenches

For the optical model of the silicon trenches one could easily make a variety of different models that may vary in complexity. While using the Bruggeman model we assumed that trench walls are like thin slabs which gives us the depolarization factors 1 and 0 for the optical properties normal to and in plane eith the walls respectively, see table 2.1.

The general idea is to build the model from silicon substrate that has the thickness 1 mm and then try to simulate the voids in the sample with the help of EMA-materials. The model is shown in Fig. 4.5 with the silicon substrate as layer 0. Since the sample is etched in a way that could be described like walls the model must have different characteristics in different directions and therefore a biaxial model was to be used. Early on it was realized that just one biaxial was not enough so the model was expanded into a three layer biaxial model. This is because the walls do not have the same distance between them at the bottom compared to in the center and the top of the sample. These three biaxial layers are illustrated in the model in Fig. 4.5 and are numbered 13, 14 and 15.

Each of the three biaxial-layers is connected to two different EMA-materials that have the same percentage of void. These two EMA-materials are coupled to each other so that the void always is the same for both of them when they are fitted to the experimental data. Since there are three biaxial-layers the model gives us three pairs of EMA-materials. Each of the six EMA-materials is connected to

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main: 2007-12-5 15:34 — 28(40)

different p-doped silicon materials. The p-doped silicon is connected to silicon with free carriers-materials. For these materials different resistances have to be fitted on account of the direction. The layers marked 1 to 6 and -1 to -8 are help layers. Some of the offsets in these layers are to high to have a physicall explanation. To get a better model it is needed to develop specific layers adapted for silicon trenches. In Figs. 4.6 to 4.7 the experimental values and the model are shown at an angle of incidence of 25◦ _{when θ is 0. In Figs. B.1 to B.2 the results}

are shown for 45◦_{and 65}◦_{when θ is 0}◦_{. In figs. 4.10 and 4.11 we can observe that}

the model is still accurate when θ is changed to 45◦_{.In appendix B Figs. 4.10 and}

B.24 the model and the experimental values for all three angles of incidence are shown when θ vary from 45◦ _{to 135}◦ _{in intervals of 45}◦_{. We only need to follow}

the rotation of the sample to 135◦ _{because of the symmetry of the structure the}

data will just be a repetition of the first 180◦_.

The thickness of the three biaxial layers seen in Fig 4.5 is what we can expect from the specifications of the sample. Since we know the specifications for the sample we know that the height should be 9 µm. Knowing that the walls are 300 nm thick and the distance center to center of the walls is 1500 nm this gives us a void that is 80 %. From the model Fig. 4.5 we see that adding the thickness of the three biaxial layers gives us the total height of 8.67 ± 0.08 µm. When calculating the void percentage average we get that in the model it is 77.9 %. When the model that was constructed for the experimental data from the measurements on the silicon sample was fitted the Mean square error was 3.15 which is considered a good result.

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-8 silicon with free carriers8

0.000000 µm

-7 silicon with free carriers7

0.000000 µm

-6 silicon with free carriers6

0.000000 µm

-5 silicon with free carriers5

0.000000 µm

-4 silicon with free carriers2

0.000000 µm

-3 silicon with free carriers1

0.000000 µm

-2 silicon with free carriers

0.000000 µm

-1 si_p

0.000000 µm

0 si_p_doped

1 mm

1 si_p_doped1

0.000000 µm

2 si_p_doped2

0.000000 µm

3 si_p_doped3

0.000000 µm

4 si_p_doped4

0.000000 µm

5 si_p_doped5

0.000000 µm

6 si_p_doped6

0.000000 µm

7 ema (si_p_doped1)/77.5% void

0.000000 µm

8 ema2 (si_p_doped2)/77.5% void 0.000000 µm

9 ema3 (si_p_doped3)/83% void

0.000000 µm

10 ema4 (si_p_doped4)/83% void 0.000000 µm

11 ema5 (si_p_doped5)/59.6% void 0.000000 µm

12 ema6 (si_p_doped6)/59.6% void 0.000000 µm

13 biaxial

0.826717 µm

14 biaxial

7.356988 µm

15 biaxial

0.484542 µm

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main: 2007-12-5 15:34 — 30(42)

Generated and Experimental

Wavelength (µm) 6 8 10 12 14 16 18

Ψ

in d eg re es 0 20 40 60 80 100 Model Fit Exp AnE 25° θ=0° Exp Aps 25° θ=0° Exp Asp 25° θ=0°

Figure 4.6: Experimental and model values for Ψ for the incident angle 25◦ _{and θ}

is 0◦_.

Generated and Experimental

Wavelength (µm) 6 8 10 12 14 16 18

∆

in d eg re es -100 0 100 200 300 400 Model Fit Exp AnE 25° θ=0° Exp Aps 25° θ=0° Exp Asp 25° θ=0°

Figure 4.7: Experimental and model values for ∆ for the incident angle 25◦ _and θis 0◦.

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Generated and Experimental

Wavelength (µm) 6 8 10 12 14 16 18

Ψ

in d eg re es 0 20 40 60 80 100 _{Model Fit} Exp AnE 25° θ=0° Exp AnE 45° θ=0° Exp AnE 65° θ=0° Exp Aps 25° θ=0° Exp Aps 45° θ=0° Exp Aps 65° θ=0° Exp Asp 25° θ=0° Exp Asp 45° θ=0° Exp Asp 65° θ=0°

Figure 4.8: Experimental and model values for Ψ for all the incident angles at θ = 0◦_.

Generated and Experimental

Wavelength (µm) 6 8 10 12 14 16 18

∆

in d eg re es -400 -200 0 200 400 _{Model Fit} Exp AnE 25° θ=0° Exp AnE 45° θ=0° Exp AnE 65° θ=0° Exp Aps 25° θ=0° Exp Aps 45° θ=0° Exp Aps 65° θ=0° Exp Asp 25° θ=0° Exp Asp 45° θ=0° Exp Asp 65° θ=0°

Figure 4.9: Experimental and model values for ∆ for all the incident angles at θ = 0◦_.

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main: 2007-12-5 15:34 — 32(44)

Generated and Experimental

Wavelength (µm) 6 8 10 12 14 16 18

Ψ

in d eg re es 0 20 40 60 80 Model Fit Exp AnE 25° θ=45° Exp Aps 25° θ=45° Exp Asp 25° θ=45°

Figure 4.10: Experimental and model values for Ψ for the incident angle 25◦ _and θ= 45◦.

Generated and Experimental

Wavelength (µm) 6 8 10 12 14 16 18

∆

in d eg re es -200 -100 0 100 200 300 Model Fit Exp AnE 25° θ=45° Exp Aps 25° θ=45° Exp Asp 25° θ=45°

Figure 4.11: Experimental and model values ∆ for the incident angle 25◦ _{and θ =}

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4.3 Sample with protein 33

4.3 Sample with protein

After exposing the sample to proteins the same measurements ware carried out again. According to previous studies [1] we know that the proteins should show optical signatures visible at the frequency 1537 ± 3 cm−1_{. In Fig. 4.12 we show}

that there is a small peak at 1540 cm−1 _{when θ = 0}◦_.

Experimental Data

Wave Number (cm

-1

₎

1400

1500

1600

1700

1800

Ψ

in

d

eg

re

es

25

30

35

40

45

50

55

Exp AnE 45° θ=0° Exp AnE 45° θ=0°

Figure 4.12: Ψ for clean silicon sample (dashed curve) and for same sample after expsure to protein (solid curve). A small peak is visible at 1540 cm−1_.

4.4 Conclusions and suggestion of further work

The main object for this project was to develop an optical model for the silicon trenches with the help of experimental data from infrared ellipsometry and try to detect proteins in the silicon structure. For the optical model a three layered biaxial model was developed. This gave us a model which was accurate but not perfect. The MSE was calculated to 3.54 and could not be improved by adding more layers. The thicknesses of the three biaxial layers are what we expected from the specifications of the sample and therefore give us an indication that the model is correct. Perhaps a completely different type of model could be developed. It has been suggested that a model based on photonic crystals could be used instead of the biaxial model used in this project [3]. FEM-simulations could be used to develop a theoretical model for the structures .

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main: 2007-12-5 15:34 — 34(46)

good indication that it is possible to study absorption in more complex structures. The peak was not very strong and it can probably be made more visible if exposing the samples to the protein for a longer time period. An optical model for the proteins could be developed with the help of Gauss oscillators.

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Bibliography

[1] H. Arwin, A. Askendahl, P. Tengvall, D. Thompson, and J. A. Woollam, “Infrared ellipsometry studies of the thermal stability of protein monolayers and multilayers.” submitted, 2007.

[2] H. Arwin, L. M. Karlsson, A. Kozarcanin, D. Thompson, T. Tiwald, and J. Woollam, “Carbonic anhydrase adsorption in porous silicon studied with infrared ellipsometry,” phys stat soli, no. 8, 2005.

[3] D. W. Thompson, 2007. Private communication.

[4] X. Badel, Electrochemically ethed pore arrays in silicon for X-ray imaging. PhD thesis, KTH Microelectronics and Information Technology, 2005. [5] J. C. Maxwell, A treatise on electricity and magnetism. Oxford: Claredon

Press, 1873.

[6] P. Drude Ann. Phys. Chemie, vol. 32, p. 584, 1887. [7] P. Drude Ann. Phys. Chemie, vol. 38, p. 481, 1890.

[8] R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light. North-Holland, 1986.

[9] R. W. Collins, D. E. Aspnes, and E. A. I. (Eds.), “Proceedings of the 2nd inter-national conference on spectroscopic ellipsometry,” Thin Solid Films, vol. 313-314, 1998.

[10] H. Arwin, “Ellipsometry based sensor systems,” in Encyclopededia of sensors (C. A. Grimes, E. C. Dickey, and M. V. Pishko, eds.), pp. 329–357, American Scientific Publishers, 1999.

[11] B. Stuart, Biological Applications of Infrared Spectroscopy. John Wiley and sons, Ltd, University of Greenwich, 1997.

[12] D. A. G. Bruggeman, “Berechnung verschiedener physikalischer konstanten von heterogenen substanzen,” Ann. Phys. (Leipzig), vol. 24, pp. 636–679, 1935.

[13] J. Tooze, Introduction to Protein Structure 2nd ed. Garland Publishing: New York, NY, 1999.

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36 Bibliography

[14] J. Gustavsson, “Protein adsorption on porous silicon,” Master’s thesis, De-partment of Physics and Measurement Technology, Linköping, 1999.

[15] P. Kleimann, X. Badel, and J. Linnros, “Towards the formation of 3d nano-structures by electochemical etching of silicon,” Electrochemical etched pore

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Appendix A

Experimental results

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main: 2007-12-5 15:34 — 38(50) 38 Experimental results Experimental Data Wavelength (µm) 6 8 10 12 14 16 Ψ in d eg re es 0 20 40 60 80 100 Exp AnE 45° θ=0° Exp Aps 45° θ=0° Exp Asp 45° θ=0°

Figure A.1: Experimental values Ψ for the incident angle 45◦ _{and θ =}

0◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 16 ∆ in d eg re es -100 0 100 200 300 Exp AnE 45° θ=0° Exp Aps 45° θ=0° Exp Asp 45° θ=0°

Figure A.2: Experimental values ∆ for the incident angle 45◦ _{and θ =}

0◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 16 Ψ in d eg re es 0 20 40 60 80 Exp AnE 65° θ=0° Exp Aps 65° θ=0° Exp Asp 65° θ=0°

0◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 16 18 Ψ in d eg re es 0 20 40 60 80 Exp AnE 25° θ=45° Exp Aps 25° θ=45° Exp Asp 25° θ=45°

45◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 16 18 ∆ in d eg re es -100 0 100 200 300 Exp AnE 25° θ=45° Exp Aps 25° θ=45° Exp Asp 25° θ=45°

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39 Experimental Data Wavelength (µm) 6 8 10 12 14 16 Ψ in d eg re es 0 10 20 30 40 50 60 70 Exp AnE 45° θ=45° Exp Aps 45° θ=45° Exp Asp 45° θ=45°

45◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 Ψ in d eg re es 0 10 20 30 40 50 60 Exp AnE 65° θ=45° Exp Aps 65° θ=45° Exp Asp 65° θ=45°

45◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 ∆ in d eg re es -100 0 100 200 300 Exp AnE 65° θ=45° Exp Aps 65° θ=45° Exp Asp 65° θ=45°

45◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 16 18 Ψ in d eg re es 0 20 40 60 80

Figure A.11: Experimental values of Ψ for all three incident angles and θ = 45◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 16 18 ∆ in d eg re es -100 0 100 200 300 Exp AnE 25° θ=45° Exp AnE 45° θ=45° Exp AnE 65° θ=45° Exp Aps 25° θ=45° Exp Aps 45° θ=45° Exp Aps 65° θ=45° Exp Asp 25° θ=45° Exp Asp 45° θ=45° Exp Asp 65° θ=45°

Figure A.12: Experimental values of ∆ for all three incident angles and θ = 45◦_.

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main: 2007-12-5 15:34 — 40(52) 40 Experimental results Experimental Data Wavelength (µm) 6 8 10 12 14 16 18 Ψ in d eg re es 0 20 40 60 80 Exp AnE 25° θ=90° Exp Aps 25° θ=90° Exp Asp 25° θ=90°

90◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 16 18 Ψ in d eg re es 10 20 30 40 50 60 70 Exp AnE 45° θ=90° Exp Aps 45° θ=90° Exp Asp 45° θ=90°

90◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 16 Ψ in d eg re es 0 10 20 30 40 50 60 Exp AnE 65° θ=90° Exp Aps 65° θ=90° Exp Asp 65° θ=90°

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41 Experimental Data Wavelength (µm) 6 8 10 12 14 16 18 Ψ in d eg re es 0 20 40 60

80 Exp AnE 25° Exp AnE 45° θθ=90°=90° Exp AnE 65° θ=90° Exp Aps 25° θ=90° Exp Aps 45° θ=90° Exp Aps 65° θ=90° Exp Asp 25° θ=90° Exp Asp 45° θ=90° Exp Asp 65° θ=90°

Figure A.20: Experimental values of ∆ for all three incident angles and θ = 90◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 16 18 Ψ in d eg re es 0 10 20 30 40 50 60 Exp AnE 25° θ=135° Exp Aps 25° θ=135° Exp Asp 25° θ=135°

135◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 16 18 Ψ in d eg re es 0 20 40 60 80 100 Exp AnE 45° θ=135° Exp Aps 45° θ=135° Exp Asp 45° θ=135°

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main: 2007-12-5 15:34 — 42(54) 42 Experimental results Experimental Data Wavelength (µm) 6 8 10 12 14 16 18 Ψ in d eg re es 0 20 40 60 80 100 Exp AnE 65° θ=135° Exp Aps 65° θ=135° Exp Asp 65° θ=135°

135◦_. Experimental Data Wavelength (µm) 6 8 10 12 14 16 18 Ψ in d eg re es 0 20 40 60 80 100 Exp AnE 25° θ=135° Exp AnE 45° θ=135° Exp AnE 65° θ=135° Exp Aps 25° θ=135° Exp Aps 45° θ=135° Exp Aps 65° θ=135° Exp Asp 25° θ=135° Exp Asp 45° θ=135° Exp Asp 65° θ=135°

Figure A.28: Experimental values of ∆ for all three incident angles and θ = 135◦_.

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Appendix B

Experimental and model

results

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44 Experimental and model results

Generated and Experimental

Wavelength (µm) 6 8 10 12 14 16 Ψ in d eg re es 0 20 40 60 80 100 Model Fit Exp AnE 45° θ=0° Exp Aps 45° θ=0° Exp Asp 45° θ=0°

Figure B.1: Experimental and model values for Ψ for the incident angle 45◦

and θ = 0◦_.

Wavelength (µm) 6 8 10 12 14 16 ∆ in d eg re es -400 -200 0 200 400 Model Fit Exp AnE 45° θ=0° Exp Aps 45° θ=0° Exp Asp 45° θ=0°

Figure B.2: Experimental and model values for ∆ for the incident angle 45◦ _{and θ = 0}◦_.

Wavelength (µm) 6 8 10 12 14 16 Ψ in d eg re es 0 20 40 60 80 Model Fit Exp AnE 65° θ=0° Exp Aps 65° θ=0° Exp Asp 65° θ=0°

Figure B.3: Experimental and model values for Ψ for the incident angle 65◦

and θ = 0◦_.

Wavelength (µm) 6 8 10 12 14 16 ∆ in d eg re es -200 -100 0 100 200 300 Model Fit Exp AnE 65° θ=0° Exp Aps 65° θ=0° Exp Asp 65° θ=0°

Figure B.4: Experimental and model values for ∆ for the incident angle 65◦ _{and θ = 0}◦_.

Infrared studies of trenches etched in silicon

Department of Physics, Chemistry and Biology

Master’s Thesis

Infrared studies of trenches etched in silicon

Lars Karlsson

Infrared studies of trenches etched in silicon

Lars Karlsson

Abstract

Acknowledgements

Contents

Chapter 1

Introduction

1.1

Background

1.2

Objective

Chapter 2

Theory

2.1

Maxwell’s equations, plane waves and

polar-ization

2.1.1

Polarization

2.2

Ellipsometry

2.2.1

Reflection ellipsometry

2.2.2

Generalized ellipsometry

2.2.3

Infrared ellipsometry

2.3

Optical modeling

Model

Fit

Results

Modify

Data

2.3.1

Reflectance

2.3.2

Reflection at an anisotropic surface

2.4

Effective medium approximation

A

B

ε

ε

A

B

ε

ε

f

f

f

f

E

E

2.4.1

The Bruggeman effective medium

2.5

Euler angles

2.6

Capillary action

2.7

Proteins

2.7.1

Albumin

2.7.2

Protein adsorption

Chapter 3

Experimental and

instrumentation

3.1

Ellipsometer

3.2

Silicon samples

3.2.1

Euler angles

3.3

_E