Optical components of XUV monochromator

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Optical components of XUV monochromator

For use in laser based angle-resolved photoemission spectroscopy

ANDREAS ÖSTLIN

Master’s Thesis at MAP Supervisor: Prof. Oscar Tjernberg

Examiner: Prof. Oscar Tjernberg

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Abstract

At the division of Material Physics at KTH an angle resolved photoemission spectrometer (ARPES) is being built. This system uses a pulsed laser to create ultraviolet light through higher harmonic generation. The laser has a high output effect which puts the optical components of the system under a large heat load. This thesis investigates the use of silicon carbide (SiC) as a possible material for use in the system. A diffraction grating is modelled and then processed by use of photolithography and plasma etching. This is then characterized by different methods, finally in its working environment. It is concluded that silicon carbide is a plausible material for use in the ARPES system.

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Sammanfattning

Vid avdelningen för materialfysik vid KTH håller en laserbaserad vinkelupplöst fotoemissionsspek-trometer på att byggas upp. Denna använder ultraviolett ljus för att excitera fotoelektroner som sedan detekteras genom time of flight. Ljuskällan som ger UV-fotonerna är en pulsad laser som ger en hög ef-fekt, vilket ställer krav på att alla optiska komponenter i systemet skall klara en hög värmelast. I detta examensarbete undersöks om kiselkarbid (SiC) uppfyller dessa krav. Modellering av ett diffraktions-gitter görs, och resultatet av detta ger vägledning under processningen av gittret. Gittret tillverkas med hjälp av fotolitografi och plasmaetsning av en kiselkarbidwafer. Denna karakteriseras sedan med olika metoder och finns fungera bra för sitt ändamål.

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Introduction

At the Material Physics Division at KTH/ICT, a state of the art angle-resolved photoemission spectrometer (in short: ARPES) is being built. This system, called BALTAZAR (Bi-purpose Angle-resolving Laser-based Time-of-flight AnalyZer for Advanced Research), uses a pulsed laser in the XUV wavelength range to generate light for use in condensed matter research (high-T_c superconductors, polarized materials, topological insulators etc). To separate the right wavelength from the fundamental one a monochromator is needed. This monochromator is mainly composed of two parts, a concave mirror and a plane grating. Since the laser has an average effect of 15 W these optical components needs to handle a large heat load. In this respect, several commercial mirrors and gratings have been found wanting. An example of this can be seen in figure 1.1, where the laser beam has created a hole in the grating surface. The main goal of this thesis is to model and process optical components suitable for use in the monochromator of BALTAZAR. As a working hypothesis, the mirror will consist of a silicon substrate coated with a thin film, and the grating will be etched in silicon carbide (SiC) and fused onto a copper substrate. Since SiC has a high thermal conductivity it is a good candidate for use in a grating, as showed in [6].

At the start of this thesis work, the grating in use is a commersial aluminium grating coated with M gF₂and platinum. The grating is an echelette grating, i.e. it has a triangular pattern. The mirror is a M gF2-coated aluminium mirror.

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Figure 1.1. Destroyed grating on glass substrate. Notice the hole burned into the surface.

The chapters are as follows:

• Chapter 2 deals mainly with the theory and modelling of the grating. The theory of gratings, which rests on Maxwell’s equations, is introduced. Then the efficiency of the grating in this special case is calculated. The chapter ends with the thermal modelling of the optical components.

• The process used to create the mirror and the grating is documented in chapter 3. The grating is etched, using samples of SiC wafers to step by step find the best etching technique. The mirror is coated with thin films by sputtering.

• Characterization of the optical components is the topic of chapter four. This work is mainly done in the monochromator of BALTAZAR, where the components are to be used.

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Chapter 2

Theory and Modelling

This chapter begins with the preliminaries of grating theory, introducing several definitions and concepts. In 2.1.2 the so called grating problem is derived and stated. 2.1.3 deals with one of the methods available for solving the grating problem, the differential method. The next section uses the differential method in the specific case of a laminar dielectric grating of silicon carbide. The method is then implemented in MATLAB® to calculate the efficiency of the grating. 2.4 deals with the effect of laser heating on the grating. Finally, in 2.5 the mirror is modelled.

2.1 Preliminaries

The phenomena of diffraction gratings was first discovered during the 17th century, and the first man-made grating was constructed by David Rittenhouse in 1785 [11]. To understand how the angles of diffraction relate to the wavelength and angle of the incoming beam of light, only classical optics is needed. If one on the other hand wants to find the efficiencies of the diffracted orders, i.e. how much intensity of the incoming beam that is going into the different orders, one must use the theory of electromagnetism. One of the first to do this was Lord Rayleigh [13]. Later, after the Second World War, grating manufacture improved and the possibility to tailor-make grating profiles increased. This, together with the advent of faster computers needed to solve numerical algorithms, pushed on the development of grating theory to where it is today [12].

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2.1.1 Maxwell’s equations

Since, as is known from Maxwell’s time, light is electromagnetic waves, a good place to start is from Maxwell’s equations:

∇ · D = ρ ∇ · B = 0 ∇ × E = −∂B ∂t ∇ × H = J + ∂D ∂t

The electric and magnetic fields are related in the usual way, i.e D = E and B = µH. Assuming that we have time-harmonic fields, i.e fields ∝ e−iωt, and neither free charge or current, Maxwell’s equations becomes:

∇ · D = 0 (2.1)

∇ · B = 0 (2.2)

∇ × E = iωB (2.3)

∇ × H = −iωD (2.4)

Here, ω = 2πc_λ , where λ is the desired wavelength of light. Now, taking the curl of (2.3-2.4) and using (2.1-2.2) as well as vector identities, the four equations (2.1-4) reduces to two second order equations,

∆E + k2E = 0 (2.5)

∆H + k2H = 0 (2.6)

where k2= µω2 is the wavenumber. (2.5-6) can be recognized as Helmholtz equations. The solving of these lie at the heart of grating theory, but first some basic grating geometry needs to be stated.

2.1.2 A primer on gratings

See figure 2.1 and assume Cartesian coordinates. The profile of the grating lies in the xy-plane, and the grooves are parallel with the z-axis. The grating profile can be described by the d-periodic function y = f (x), which then divides the xy-plane in two regions. Region 1 is in the area y > f (x) and region 2 lies in y < f (x). Region 1 is filled with a medium (often vacuum) with wave number k₁, and region 2 is filled with a medium of wave number k₂. Note that the main theory of gratings is more general then this, the different media might not have homogenous dielectric constants, for example. We also define the height of the grating as h = max(f (x)).

Now, assume that the grating is illuminated by a plane wave of unit amplitude, with incidence angle θi and wave vector ki = (α, −β, 0). Here, α = k1sin θi and β = k1cos θi. Note that

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2.1. PRELIMINARIES 5 x h y = f (x) y ki θi d 2d 3d

Figure 2.1. An example of a grating

freedom still exists, namely the polarization of the incoming beam. Two fundamental cases can be recognized: P-polarization (E-field parallel with the grooves) or S-polarization (H-field parallel with the grooves). P-polarization is easier to handle mathematically, and hence this will be our choice (for the S case, see [12]).

With these assumptions made, (2.5) is reduced to a scalar Helmholtz equation

∆Ez(x, y) + k2(x, y)Ez(x, y) = 0 (x, y) ∈ R2 (2.7)

with k2(x, y) defined as k2(x, y) := (

k₁2 for y > f (x) k₂2 for y < f (x)

The boundary condition will, in the case of P-polarization, be that the electric field and its derivative should be continuous at y = f (x). The problem of solving (2.7) is usually called the grating problem. There exists several ways of dealing with this problem, the one chosen for this thesis is called the differential method, dealt with in the next section.

Since f (x) is a periodic function, it seems plausible to Fourier expand the solution. Begin by defining u(x, y) = e−iαxEz(x, y), which will be our ansatz. u is d-periodic and will also

satisfy equation (2.7), hence we can expand it in Fourier series: Ez(x, y) = eiαx ∞ X n=−∞ un(y)einΩx= ∞ X n=−∞ un(y)eiαnx (2.8)

where Ω = 2π_d is the spatial frequency and αn = α + nΩ. This is often called a Rayleigh

expansion of the field, since Lord Rayleigh was the first to use this method [13]. If αn is

divided by k₁, an old acquaintance is found: αn/k1 = sin θ + nΩ/k1 := sin θn

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⇒ ν₁sin θ_n= ν₁sin θ + nλ/d

(assume vacuum) ⇒ d(sin θn− sin θ) = nλ

This is the classical grating formula. The θn’s will correspond to the deviation angle in

the n’th diffraction order.

2.1.3 The differential method

Insert (2.8) into (2.7) and Fourier expand k2(x, y) in the einΩx-basis to obtain

∞ X n=−∞ _d2_u n dy2 − α 2 nun+ ∞ X m=−∞ cn−m(y)um · einΩx = 0

and we get a differential system of n equations in the bracketed term. This system can be written in matrix form as

u00+ A(y)u = 0 (2.9)

where u is the column vector consisting of the functions un(y) and A is the square matrix

consisting of elements Anm= cn−m(y) − α2nδnm.

In the region y > h and y < 0 the solving of the system becomes easy, since only one medium exist in each region. The solutions will simply be exponential functions as usual from Helmholtz equation:

un(y) = Tne−iβ

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n y _for _{y < 0} _(2.10)

un(y) = δn0e−iβy+ Rneiβ

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n y _for _{y > h} _(2.11) where βn(1) = (k12− α2n)1/2 and β

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n = (k22− α2n)1/2. Note that (2.10) has no incoming wave

in the solution. Already now it is easy to see that Tn and Rn are the transmission and

reflection coefficients, respectively, of the transmitted and diffracted orders. If (2.10)-(2.11) is differentiated one can through some algebra show the relations

u0_n(0) + iβ_n(2)un(0) = 0

u0_n(h) − iβ_n(1)un(h) = −2iβδn0e−iβh

or in matrix form

u0(0) + L0u(0) = 0 (2.12)

u0(h) + Lhu(h) = s (2.13)

where (L0)nm= iβn(2)δnm, (Lh)nm = −iβ(1)n δnm and sn= −2iβδn0e−iβh.

As can be seen from (2.10) and (2.11), a knowledge of u(0) and u(h) will give the coefficients Tn and Rn, one of the goals on the way of finding the efficiencies of the reflected (and

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2.2. THE LAMINAR GRATING 7

by solving (2.9). Hence, the grating problem is reduced to finding the solution of (2.9) in the interval (0, h), which is the point of the differential method. The differential system (2.9) is in fact infinite, and hence needs to be truncated to some order N , giving P = 2N + 1 coupled differential equations. This is no problem though, since N can be chosen so that only the evanescent orders (which carry no energy) are left out.

One way to solve this problem is to use the “shooting method”, a standard method in nu-merical analysis to turn a boundary value problem into an initial value problem. Start with a value of u(0) = c. (2.12) then gives u0(0). Now u(h) and u0(h) can be found by a suitable, in general numerical, method from (2.9). Further, v := u0_{(h) + L}hu(h) can be calculated. Due

to the linearity of the differential system there must exist a matrix M such that Mc = v. But this matrix m:th column is simply the result for v when c has elements c_n = δ_nm. If this procedure is done P times, looping through P different values for c, then M can be constructed column by column. Now, the “real” field u(0) is found by Mu(0) = s. From this field u(h) can be found by the same method as before and the coefficients R_n and T_ncan be obtained.

The sought after efficiencies e_n, defined as [flux of n’th diffracted wave/incoming flux], can be calculated from the Poynting vector:

en= RnRn βn(1) β (reflection case) (2.14) en= TnTn βn(2) β (transmission case) (2.15)

It should be noted that the differential method is only one way to solve the grating problem, there exists several suitable under different conditions. The differential method can also be generalized to deal with more complex grating structures.

2.2 The laminar grating

In the preceding section the differential method was introduced to solve the grating problem in the case of a grating consisting of a single medium. From now on, this medium will be assumed to be dielectric silicon carbide, and the medium above the grating profile f (x) will be vacuum. The grating profile will be a so called laminar (or corrugated) profile (see figure 2.2). The grating period d and the height h are defined as before, but now the aspect ratio a is introduced. It is a real number 0 ≤ a ≤ 1, and defines the groove width (and hence also the laminar width) of the grating.

The material constants enter into the problem through the wave numbers k₁and k₂. Assuming that µ ≈ µ0, only the dielectric constant (and hence the optical index) will differ between the

media. For SiC, call the dielectric constant , giving k₂2 = 0µ0ω2. Since the region above

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x y

ad d

h y = f (x)

Figure 2.2. A laminar grating

x k2(x, ∀y) k2 2 k2 1 ad d

Figure 2.3. The k2 _function

2.2.1 Solving the grating problem

As was shown in section 2.1.3, the essence of the grating problem lies in the solving of equation (2.9) in the region 0 < y < h. The crux is to find the matrix A(y), which in turn means finding the Fourier coefficients cn−m of the function k2(x, y) (see figure 2.3). In general, the Fourier

coefficients of k2(x, y) needs to be calculated ∀y ∈ (0, h), or at least for a discrete number of y in the case of numerical integration. In the case of the laminar grating, k2_{(x, y) = k}2_{(x) is}

independent of y and only one calculation is needed. The coefficients are found in the usual way:

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2.2. THE LAMINAR GRATING 9 • n 6= 0 cn= 1 d Z d 0 k2(x)e−inΩxdx = 1 d " Z ad 0 k2₁e−inΩxdx + Z d ad k2₂e−inΩxdx # = =1 d " − k 2 1 inΩ e −inΩad_{− 1} − k 2 2 inΩ e −inΩd_{− e}−inΩad # = = − k 2 1 2πin e −2πina_{− 1} − k 2 2 2πin e −2πin_{− e}−2πina • n = 0 c0= 1 d Z d 0 k2(x)dx = 1 d " Z ad 0 k₁2dx + Z d ad k₂2dx # = 1 d k2₁(ad) + k2₂(d − ad)= =a(k₁2− k2₂) + k₂2 This gives cn−m= ( −k21(exp(−2πia(n−m))−1)+k 2 2(exp(−2πi(n−m))−exp(−2πia(n−m))) 2πi(n−m) if n − m 6= 0 a(k2 1− k22) + k22 if n − m = 0

As can be seen the c’s are independent of y. Now the A-matrix can be constructed with elements Anm= cn−m− αn2δnm. Note that the matrix is independent of y and also Hermitian

if the dielectric constant is real valued. This will come of use later on. The system (2.9) simplifies to

u00+ Au = 0 (2.16)

a system of coupled ordinary second-order differential equations with constant coefficients. It is tempting to uncouple the equations by a diagonalization of A. This can be achieved by eigenvalue decomposition through A = ξ−1Dξ, where D = diag[λ−n· · · λn] is the diagonal

matrix of eigenvalues λ_k and ξ is the matrix consisting of columns of eigenvectors. Using the substitution u = ξx, (2.16) becomes

x00+ Dx = 0 (2.17)

and we now have a system of independent ordinary differential equations of the form x00(y) + λkx(y) = 0. These can be solved by regular methods (characteristic equations etc). Since A

is Hermitian, all its eigenvalues λk must be real, and this will simplify the matter of solving

the uncoupled equations. Only three different solutions exists (assume r = √λ): xk= C1ery+ C2e−ry for λk< 0

xk= C1y + C2 for λk= 0

xk= C1cos(ry) + C2sin(ry) for λk> 0

C1 and C2 are constants that can be determined from the (transformed) boundary conditions

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x h y Mb ψA ψB ψC Ma

Figure 2.4. Definitions of matrices in the Numerov method

also x_k(h). The real field u is now found from the substitution u = ξx, and the efficiencies are obtained as in section 2.1.3.

A good check on the result is to see if all efficiencies add up to one,

X RnRn βn(1) β + X TnTn β(2)n β = 1 (2.18)

Here the sums should run through all n where βn(1) and βn(2) respectively are non-imaginary,

since the evanescent waves carry no energy.

The drawbacks of this method is that diagonalization of A and computing the inverse of ξ is needed, and since these are in practice large matrices cumbersome computations are needed. On the other hand these are relatively simple and fast to do on a computer, compared with using numerical methods.

For this thesis, the above method is implemented in MATLAB®-code, which can be found in appendix A.

2.2.2 Numerical solution

The differential method can also be implemented in a more general way using numerical algorithms, and actually has to be solved numerically if the A-matrix from (2.9) is non-constant. Since (2.9) lacks terms of first order the Numerov method can be used. This method has seen use in nuclear physics due to its ability to handle the Schrödinger equation for arbitrary potentials [8], but is also suitable for grating efficiency calculations. First some modifications on the definitions in section 2.1.3 needs to be made. Redefine the solutions

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2.2. THE LAMINAR GRATING 11 (2.10) and (2.11) as: un(y) = Cne−iβ (2) n y _for _{y < 0} _(2.19) un(y) = Ane−iβ (1) n y_{+ B} neiβ (1) n y _for _{y > h} _(2.20) and let ψA, ψB and ψC be vectors consisting of, respectively, the coefficients An, Bn and

Cn. These vectors are related linearly as ψA = MaψC, ψB = MbψC and ψB = RψA; giving

R = MbM−1a (see figure 2.4). As can be seen R and T := M−1a will be the reflection and

transmission matrix respectively. The numerical integration will have to start at y = 0 where the coefficients Cn are not yet known, but as in section 2.1.3 this can be solved by doing

P = 2N + 1 numerical integrations; using P linearly independent ψC’s to build up the Ma

and Mb matrices.

The Numerov method uses a constant integration step ` and works according to the following scheme, beginning from the equation

u00= Au (2.21)

Define a vector ζ and use the above equation to obtain: ζ := u − ` 2 12u 00₌ _{I −} `2 12A ! u (2.22)

The Numerov algorithm then uses the formula ζ(y) = 2I + `2_{A(y − `) +} `

4

12A

2_{(y − `)}

!

ζ(y − `) − ζ(y − 2`) + O(`6) (2.23)

to step by step reach u(h). As can be noted the formula needs the value of ζ at the two preceding points, and hence we need both u(0) and u(`) to start the algorithm. u(0) and its derivative is found from the boundary condition at y = 0:

u(0) = Cn (2.24)

u0(0) = −iβ_n(2)Cn (2.25)

(2.22) will then give ζ(0). u(`) is found from a Runge-Kutta algorithm [4]

u(`) = I +` 2 6A(0) + `2 3A(`) + `4 24A(`)A(0) ! u(0)+ (2.26) +` I +` 2 6A(`) ! u0(0) + O(`5) (2.27)

Again, (2.22) gives ζ(`) and the algorithm (2.23) can start. At the end of the numerical integration ζ needs to be transformed back into u. This is done by using the inversion

u(h) = I + ` 2 12A(h) + `4 144A 2_(h) ! ζ(h) + O(`6) (2.28)

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The derivative is somewhat more tricky to find, one needs the seven preceding points of ζ(h):

u0(h) = [10ζ(h − 7`) + 28ζ(h − 6`) − 485ζ(h − 5`) + 1778ζ(h − 4`)− 3325ζ(h − 3`) + 3740ζ(h − 2`) − 3150ζ(h − `) − 360ζ(h)]/720`+ 147u(h)/60` + O(`6)

It can be noted that since we are still dealing with a laminar grating in this thesis, A is constant and this will simplify the above calculations. Using the continuity of the derivative at y = h and (2.20) we get the relation

u0(h) = iβ_n(1)(Bneiβ (1) n h_{− A} ne−iβ (1) n h₎ _(2.29)

This together with (2.20) gives An= 1 2 u(h) + iu0(h)/β_n(1)eiβ(1)n h _(2.30) Bn= 1 2 u(h) − iu0(h)/β_n(1)e−iβ(1)n h _(2.31) Running the Numerov algorithm P times the Ma and Mb matrices can be built column by

column from An and Bn, respectively. From these matrices R = MaM−1a is found and its

N + 1 column (representing the zeroth order) will be equivalent to the reflection coefficients Rnfound by analytical means in section 2.2.1. Note that this method also can handle complex

dielectric constants.

For practical purposes the differential system needs to be cut at a certain order to save computation time. This can be done giving a good approximation only if the efficiencies neglected are negligible, a good check is to see if the efficiencies add up to one. Using the Numerov method we now have an independent (but computationally cumbersome) solution of the grating problem to compare with the analytical one.

2.3 Design of the grating

With the program code found in Appendix A, simulations of various laminar gratings can be made. The aim is to find a grating profile that best suits the needs of BALTAZAR. First, this means that as much efficiency as possible should be concentrated into a specific diffraction order, namely the first order. Second, the grating should be a good compromise between two wavelengths, 71 nm and 118 nm.

2.3.1 Results

There are four degrees of freedom for a given wavelength that can affect the grating effi-ciency, the grating period d, the aspect ratio a, the groove depth h and the incidence angle θi. The incidence angle θi has a negligible effect on the efficiencies for small angles (which

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2.3. DESIGN OF THE GRATING 13

Figure 2.5. Efficiency plots n = 1

The grating period d, which is often given in grooves/mm, is as can be seen from the classical grating equation important for determining the diffraction angles θn. Keeping everything else

fixed and trying different d-values, it can be seen that the grating period has no significance on the efficiency. The grating period is therefore decided to be d = 2 µm, or equivalently 500 grooves/mm.

The aspect ratio a governs the width of the grooves, and is a number between 0 and 1 (see figure 2.2). If a = 0 or 1, the “grating” in fact becomes a homogenous plane surface which reflects and transmits only the zeroth order. It is a good check to see if the code returns the same value as the Fresnel equations in this special case:

eR= _{cos θ} i− ν cos θT cos θ_i+ ν cos θ_T 2 eT = 1 − eR

Here θT is the angle of the only transmitted wave. Using normal incidence and wavelength

λ = 355 nm the result is eR = 0.2699 and eT = 0.7301, which is in perfect agreement with

the Fresnel equations. Keeping everything else fixed and plotting the efficiency for different values of a it is seen that the optimal value of a is a = 1/2. Hence both the grooves and the “hills” should have the same width, 1 µm in the case of d = 2 µm. This is a well known fact

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from grating theory [9].

Now only the height h is left, and as will be seen this is the most important factor when considering the grating efficiency. A plot is made where efficiency curves for different heights are plotted versus wavelength, see figure 2.5, using the (non-complex) high-frequency dielec-tric constant [7] as an approximation. From this plot it seems that 23 nm should be a good compromise between the wavelengths, and this value was used in the Numerov method. The data for the dielectric constant was taken from [15]. The results for the desired wavelengths 71 nm and 118 nm was roughly about 12% efficiency (at the first order of diffraction) for 71 nm and 17% for 118 nm. Hence it was decided that 23 nm would be a good groove depth to aim for during the grating processing.

Since we have two different methods to obtain the efficiencies, the analytic one from sec-tion 2.2.1 and the numeric one from secsec-tion 2.2.2, we can make a comparison between these two. To this end the efficiencies are calculated for a grating with profile a = 0.5, h = 23 nm and d = 2 µm. The incoming beam has wavelength λ = 355 nm and incidence angle θ_i = 0. The numeric solution is done with a cut at N = 30. The result is as follows: This shows that

n = −3 n = −2 n = −1 n = 0 n = 1 n = 2 n = 3 Analytic 0.0018 0.0000 0.0171 0.2277 0.0171 0.0000 0.0018 Numeric 0.0018 0.0000 0.0171 0.2277 0.0171 0.0000 0.0018

the agreement between the different methods is good.

2.4 Thermal effects

The laser of BALTAZAR works at a mean effect of 15 W, and hence it is essential that all components in its monochromator can deal with the heat load associated with this effect. In this respect, several commercial gratings and mirrors have been found insufficient. The gratings have mainly been epoxy filled ruled gratings, modelled after a master grating. To better cope with the heat, it was decided to manufacture the new grating in silicon carbide and mount it on a copper substrate. The first subsection investigates the propagation of heat in the grating by numerical computations, and the second section deals with the thermal expansion of the grating period.

2.4.1 Heat transfer

Classically, the propagation of heat in a body is modelled by the heat equation: ∂T

∂t − κ∆T = Q (2.32)

Here T (r, t) is the temperature in a domain Ω with boundary ∂Ω, κ is the thermal diffusivity and Q is a possible heat source in Ω. For simple geometries of Ω, (2.32) can be solved

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2.4. THERMAL EFFECTS 15

analytically, but for more complicated geometries numerical analysis is needed. For this thesis the software COMSOL® is used to solve the heat equation. COMSOL® utilizes finite element methods to solve partial differential equations, like (2.32).

The geometry of the grating is created as follows: A cylinder with radius 2.5 cm and height 1 cm is drawn and placed with its axis base point at the origin. This will be the copper substrate. On top of this cylinder another cylinder with equal radius but with a height of 10 mm is placed. This will represent the SiC grating, and is an overestimation of its real thickness (250 µm) due to difficulties in meshing smaller dimensions. This will be no problem though, as both copper and SiC have similar thermal coefficients, and it was checked that the upper cylinder could be changed into copper with no significant change in the analysis. To model the copper plaid that will be attached to the copper mounting a 15 cm tall cylinder of radius 1 cm is placed behind the larger cylinders.

After the geometry is drawn, the material specific constants can be specified for the different subdomains. To mimic the heat sink effect of the copper plaid, the end of the plaid is given a Dirichlet boundary condition T = 0. The top of the grating is given a Neumann condition of inward heat flux, representing the heat of the Gaussian laser beam. Since the beam has a mean effect P = 15 W and a half-width of 5 mm, it can be modelled as P = R

IdA, where I = I0e−2r

2_/w2

is the intensity [W/m2]. w is the Gaussian radius of the beam, in this case equal to w = 0.005/0.59 m, since the Gaussian half width of the laser (0.005 m) is related to the beam radius as = 0.59w. To find the amplitude I₀, integrate in polar coordinates:

Z IdA = Z 2π 0 dφ Z ∞ 0 I0e−2r 2_/w2 rdr = 2πI0 −w 2_e−2r2_/w2 4 ∞ 0 =I0πw 2 2

Hence, I0 = 2P/πw2. The remaining boundaries are set to be thermally insulated. Once

this is done COMSOL can mesh the geometry automatically and solve 2.32, giving both the transient and the stationary solution. The stationary solution can be seen in 2.6, with the maximum value of 74 K in the top center of the model. Doing a transient analysis of the model, the transient time to reach the stationary solution will be 30 min, i.e. it will take about half an hour for the grating to reach its stationary solution.

To get a feel for how the stationary solution evolves with an increasing number of meshes to following converging analysis was made at the point in the middle of the interface between copper plaid and copper mounting:

No. of meshes Temperature [K] Solution time [s]

10700 71.504 2.0

41688 71.511 7.453

132238 71.512 24.155

422845 71.513 174.29

A try to mesh further lead to shortage of memory. It would have been interesting to continue with further meshing, but it seems that the solution will converge.

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Figure 2.6. Heat transfer in grating. The color scale is between 72 K (red) and 0 K (blue).

2.4.2 Thermal expansion

Most materials expand when heated, silicon carbide is no exception. This could in theory cause difficulties due to changes in the grating period d. A change ∆d could lead to new diffraction angles θn. This effect has been used to build an optical thermometer [2]. It is

hence of interest to make a quantitative study of the effects of thermal expansion on the grating. According to the laws of solid mechanics, a change in length ∆d is related to a change in temperature ∆T by

∆d = αd∆T (2.33)

where α is the thermal expansion coefficient [1/K]. Hence the expanded grating period simply becomes d + αd∆T . By manipulating the classical grating equation, the new diffracted angles can be found: θn= arcsin _nλ d(1 + α∆T ) − sin θi (2.34) If ∆T = 0, i.e. no change in temperature, the angles remain the same. It is also of interest to find the deviation ∆` of the light spot at the sample holder in the analysis chamber (see figure 2.7 for a sketch of the monochromator, a more thorough description can be found in

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2.5. MIRROR 17

grating L

θ ∆`

Figure 2.7. Top view of monochromator

section 4.2). This is found by trigonometry to be

∆` = L(tan θT0− tan θTnew) (2.35) assuming the first order of diffraction. For λ = 118 nm and an estimated L = 1 m we get that ∆` ≈ 20 µm, which is smaller then 50 µm as was desired.

To conclude this thermal analysis, silicon carbide seems to be a good and robust material for use as a grating in BALTAZAR.

2.5 Mirror

In this section the modelling of the mirror is discussed. Substrates of silicon where available at the beginning of this thesis and the idea was to coat these with thin films by use of sputtering (see the section on Processing and Manufacturing). One substrate was coated with 30 nm of platinum as a first test, and later a second substrate were to be coated by silicon carbide. To estimate the optimum film thickness the following relation from [10] for the reflectivity was used:

r = rvf+ rf sexp(2iβ) 1 + rvfrf sexp(2iβ)

(2.36) where r_vf is the reflectivity between vacuum and film, r_{f s} is the reflectivity between film and substrate (both derived from the Fresnel equations) and β = 2πzνf

λ is the phase factor (z is

the film thickness and ν_f is the refractive index of the film). This assumes normal incidence. The total reflection will the be R = rr.

Using data for SiC and Si from [15], the reflection vs. film thickness can be calculated and plotted, see figure 2.8.

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Chapter 3

Processing and Manufacturing

With the dimensions of the grating decided, the processing of the grating can be started. Since the grating should be done in silicon carbide, a semiconductor material, the process will be done in close collaboration with the department of Integrated Devices and Circuits (EKT) at KTH.

The first section will give an introduction to atomic force microscopy, AFM, since this will be a characterization tool during the whole process. 3.2 will outline the different steps in the processing of the SiC samples, which in turn will be a guideline when the final grating is made. 3.3 then deals with the results of the measurements. In section 3.4 the process of fusing the grating together with its copper mounting is outlined. To conclude, 3.5 describes the technique of sputtering used to coat the mirrors.

3.1 Atomic force microscopy

The atomic force microscopy (AFM) is a surface scanning method that has been in use since the 1980’s. The main component is a cantilever with an attached tip on its one side. This tip is traced over the xy-plane of a surface to be scanned, during which it’s deflected due to repulsive atomic forces. This deflection causes the cantilever to move up and down in the z-direction, which in turn can be measured by a laser beam. In this way a 3-dimensional picture of the surface can be constructed.

The AFM can be used in different tracking modes, the most simple being direct contact between the tip and the sample surface. In this thesis the so called tapping mode is used. Here, a piezoelectric element causes the tip to oscillate over the surface (hence “tapping” the surface) and the amplitude of these oscillations can then be used to build the image of the surface. A feedback system is used to control the frequency of oscillation.

The AFM measurements are done on a Dimension 3100 machine used in Electrum at Kista. The main use of these measurements will be to control the groove depth of the grating during the different fabrication steps. The tips are of TESPA type from the company Veeco, silicon tips with reflective aluminium coating on the back of the cantilever.

A more thorough explanation of atomic force microscopy, as well as its history can be found in [5].

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+

RF

Figure 3.1. Schematic of a plasma etch chamber

3.2 Outline of fabrication steps

The final goal is create a laminar grating in a 2” wafer of silicon carbide with 500 grooves/mm (d = 2 µm) and a depth of 23 ± 2 nm. The grooves should be 1 µm wide. To this end two different methods was to be tested, plasma etching and selective oxidation. Both methods are commonly used in semiconductor processing and should be able to create the desired laminar patterning in silicon carbide. The two methods are based on different physical phenomena, but both needs masks of oxide deposited on the silicon carbide. These are created by lithography on photoresist masks and by oxide wet etching. A good reference for semiconductor processing can be found in [3].

The process steps are performed in the clean room at the Electrum building in Kista.

3.2.1 Plasma etching

The idea behind plasma etching is to use ion bombardment to physically remove matter from a sample. A rough sketch of a plasma etch chamber can be seen in figure 3.1. The main parts of the chamber are two electrodes placed opposite to each other, the upper electrode is grounded and the lower electrode is connected to a RF (radio frequency) power source, as well as being fitted with a capacitor. The sample to be etched is placed on the lower electrode. The chamber is also connected to two tubes, one tube leads low pressure gas (often argon or fluoride) into the system and the other tube pumps residue material out of the system. When the power is switched on an electric field is created between the electrode plates, stripping electrons from the gas and creating ions. The electrons (which have a lighter mass then the ions) will then resonate between the plates, and some will be absorbed on the lower electrode. The capacitor will let the electrons accumulate and hence a negative voltage is superimposed

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3.2. OUTLINE OF FABRICATION STEPS 21

on the RF potential. The more massive ions on the other hand will now be attracted to the lower plate and impinge on the sample surface. In doing this, they will transfer momentum that will eject atoms from the sample, creating an etched surface. The mask will protect any surface part not to be etched.

Since the alignment of the plates creates a vertical electric field the ions will descend antipar-allel towards the surface, etching out straight vertical walls. This means that plasma etching creates an anisotropic etch, i.e. an etch with a defined directionality. Since our grating in this thesis is laminar, the anisotropy of plasma etching should be advantageous. A possible downside of plasma etching is that it lacks selectivity, i.e it will etch all materials without concern of their chemical composition.

3.2.2 Selective oxidation

Selective oxidation is one of the cornerstones of semiconductor processing, integral in the creation of MOSFET’s, for example. When silicon is involved, this technique is often called LOCOS (LOCal Oxidation of Silicon). The idea is to create a local oxide layer on the substrate, using a mask to protect the areas that are not to be oxidized. The local oxide will consume the substrate, and when the layer is later stripped of (together with the mask), the substrate surface will be patterned.

In the case of silicon carbide the chemical reaction behind the creation of the oxide is unclear, but there are two proposed candidates:

SiC + 3

2O2 ←→ SiO2+ CO and

SiC + O2←→ SiO2+ C

The thickness of consumed silicon carbide, t_SiC, and the thickness of the formed SiO₂ layer, tSiO, are related as

tSiC

tSiO

≈ 0.46 (3.1)

and this can be used to calculate the desired oxide layer thickness.

The mask used to protect the areas not to be patterned is made of silicon nitride Si3N4.

Since oxygen moves slowly through the nitride, the mask will protect the underlying silicon carbide from oxidizing. There is a drawback though, since the silicon nitride induces thermal stress on the silicon carbide substrate during heating. To avoid this, a thin layer of silicon oxide is grown between the nitride and the substrate. This layer is often called pad oxide and uses the amorphous structure of SiO2 to relieve the stress caused by the overlying nitride.

There is a drawback to this solution too though, and that is the creation of so called bird beaks. These are lateral extensions of the local oxide into the pad oxide, see figure 3.2, which can be pictured as a birds beak with some imagination.

If these bird beaks can be avoided, selective oxidation might be a good candidate for the fabrication of the grating.

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Si3N4 SiO2 SiC

bird beak

Figure 3.2. A caricature of the bird beak effect

3.2.3 Process schedule

To test which method that complies the best with the demands of the grating, a SiC 2” wafer is used. The manufacturer of the wafer is the company SiCrystal AG and the wafer crystal structure is 6H, i.e with the hexagonally packed plane facing upwards. The wafer is cut into 6 pieces, of which 4 main pieces are used, A,B,C and D. A and B will be processed by plasma etching and C and D will be subjected to selective oxidation. The pieces will be handled in order, each piece giving information about the processing of the next piece.

The A and B pieces, Plasma etch

First, the protective mask on the A piece needs to be grown. To this end thermal oxidation is done to create a SiO2 layer. Once the oxide is grown, photoresist can be spun on and the

pattern can be lithographed using a 1:1 mask.

After this, the A piece is dry etched and then stripped of its photoresist and oxide. The surface is then measured with AFM to get the etched depth and hence an estimate of the etch rate. This estimate is then used to etch the B piece. The B piece will be etched without using an oxide layer, i.e. it will only be coated by photoresist.

The C and D pieces, Selective oxide

In the case of C and D, the mask will be created by first growing a thin SiO₂ layer, and then a thicker Si3N4 layer on top. These pieces are then lithographed in the same way as A and

B.

After that the C piece is stripped of its oxide layers by wet etching, and the surface is measured.

The result is then used when determining the time needed for oxidation of D. After D is selectively oxidized all oxide is removed by wet etch and the surface is measured.

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3.3. MEASUREMENTS AND RESULTS 23

3.3 Measurements and results

3.3.1 Cleaning and oxidation

Before any processing was done, all pieces was cleaned in a solution of H2O2 and H2SO4in

re-lation 1 : 2.5, called “7-up” in the Electrum clean room. This is to remove metal contaminants that might be left from manufacturing and cleaving the silicon carbide wafers. This was done for 5 min, and the temperature of the bath became about 120◦C. To wash away any leftover chemicals the samples were dipped in a bath of water mixed with nitrogen. The downside of using 7-up is that it oxidizes the surface, so to get rid of any oxide layer the pieces were dipped shortly into a mixture of hydrofluoric acid (HF), propanol and water (called IMEC in the Electrum clean room). The bath consisted of 7 l of H2O, 70 ml propanol and 70 ml HF

(50%)

The A piece was oxidized in a Bruce diffusion furnace (C2-Dry in the clean room) at a temperature of 1150◦C and recipe T-248. The most important goal was to get a uniform layer. The layer thickness was measured by use of interferometry using a Leitz MPV-SP, giving a thickness of 43 nm.

There is a risk that the natural surface oxide will form hydrogen bonds with the water vapor in the air. This will make any photoresist spun onto the surface to adhere to the water vapor rather then the surface. To avoid this the substrate was exposed to vapor priming of hex-amethyldisilazane (in short: HMDS). This was done in the APL-HMDS, a YES-5E Vacuum Bake/Vapour Prime Processing System.

3.3.2 Lithography and etching

The piece was then covered by photoresist by spinning. This was done in the Sabina machine (an OPTIspin SST20 from SSE). The lithography was done on the Emma machine (a mask aligner MA6/BA6 Karl Suss) using hard contact mode (i.e the mask is in direct contact with the sample). The mask used in this initial stage is a test mask from Acreo, containing small size patterns of different line widths. This will make it possible to compare different grating dimensions using the same test sample. The Emma machine uses a 1:1 mask patterning, meaning that the mask pattern dimension should equal the lithographed pattern on the sub-strate. Further, by trial and error the exposure time was decided to be 3.2 seconds. The mask alignment gap was set to 120 µm. The photoresist used was the standard photoresist in the Electrum clean room S1813 (Shipley 1813 positive photoresist).

The sample was developed using a MF-CD-26 developer.

After the lithography was done the A piece was etched in the P5000 plasma etch machine (an Applied Materials Precision 5000 Mark II). The choice of this machine was made due to it already having several recipes for SiC etching. The recipe used was the Standard oxide etch

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which is flourine-based, having the following composition: Pressure [mTorr] 150 RF power [W] 350 CHF3 [sccm] 15 CF4 [sccm] 5 Ar [sccm] 50

After etching the piece was stripped of its photoresist and oxide. The photoresist was removed by first submerging the sample in an acetone bath, and then by using the Tepla machine (a TePla Model 300 Plasma System) for 10 min. The Tepla uses dry etching (but with no applied voltage as in the case of the P5000) to remove any leftover photoresist without affecting the silicon carbide surface. An AFM image of the resulting surface can be seen in figure 3.3. The

Figure 3.3. AFM image of A sample, 1.0 µm + 1.0 µm pattern

depth is roughly 18 nm, giving an etch rate of about <10 nm/min, but as can be seen the dimension of the grating (wanted: 1 µm + 1 µm) is not correct and the lines are somewhat undulated, contrary to what would be expected for 1:1 lithography. Redoing the lithography, it was seen through optical microscopy that this effect was due to a mismatch of the mask pattern and the actual pattern lithographed onto the sample. A possible explanation of this is that the Emma machine is working on its resolution limit (≈ 1 µm). To solve this problem it was decided to alter the dimensions of the grating by extending the grating period. This will of course alter the diffraction angles, but will not have an effect on the grating efficiency as shown in section 2.3.1. Another possibility would have been to change to another lithography machine with a better resolution limit. In the case of the Electrum clean room, this would mean using one of the steppers. A stepper is a lithography machine that lithographs a small pattern and repeats it throughout a larger area. In theory this could have been used in our case, but there would have been a risk of misalignment between lines and hence it was decided not to try this method.

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Figure 3.4. SEM picture of lithographed B sample, 1.0 µm + 1.75 µm pattern

Also visible in 3.3 (and in all further AFM images) is random scratches. These are arte-facts from the cutting of the crystal done by the vendor, and even though polishing is done these are hard to remove completely. According to specification from the vendor the surface roughness should be 3 nm, and the AFM confirms this. How these scratches will affect the grating efficiency is hard to tell, but it is probable that they will not be beneficiary. Optical characterization needs to be done to investigate this, and if need be wafers of better polish can be bought.

With these changes in mind, the lithographed B piece was examined in a scanning elec-tron microscope (SEM) in Electrum (the Gemini Ultra 55 Zeiss). After inspection off the different patterns it was decided that a mask pattern of 1 µm width and 1.75 µm spacing would be a good try, giving a 1.4 µm + 1.4 µm pattern (and hence a grating period of 2.8 µm). See picture 3.4. With this decision in mind the B piece was etched, aiming for a depth of 20 nm (etch time: 2 min). The resulting surface can be seen in the AFM image 3.5.

To show the some of the problems encountered when using the lithography in the limit of resolution of Emma, see 3.6. As can be seen small “islands” are present in the grooves, these are artifacts from photoresist which has not been developed as expected from the mask. In the picture 3.7 it is seen that the lithography has failed to work as expected, the grooves have an undesired wave-like distortion. Hence uttermost care needs to be made if one wants to construct a grating with a period close to the resolution limit, thorough testing should be made in each unique case.

3.3.3 The full mask

From the above results it was decided that a full size mask should be ordered, using the dimensions from the test mask. The mask was manufactured by Compugraphics. Using this

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Figure 3.5. AFM image, B sample, 1.0 µm + 1.75 µm pattern

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Figure 3.7. AFM of B sample, 1.0 µm + 2.0 µm pattern

Figure 3.8. AFM picture

mask, a sample could be created that was completely patterned by the sought after topology. During the lithography of this sample a new complication was observed; due to the sample piece being non-circular the spun on photoresist will have a different thickness on the edges, resulting in a different grating pattern then that in the middle of the sample. This should be no problem once the circular 2 inch wafer is processed.

Hence, for a non-circular grating, there will be boundary effects due to non-uniform photore-sist layers. Since the light spot will mainly hit the middle of the grating, this effect should have no large impact during usage of the grating.

At this stage it was decided that the selective oxide process described in section 3.2.2 would not be further pursued due to the success of the plasma etch method and because of

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Figure 3.9. Section cut through grating

time shortage.

A sample AFM image of the test piece that was lithographed with the new mask can be seen in 3.8. It is this piece that will be characterized optically in chapter 4. In figure 3.9 a cross section of the same test piece is shown.

To summarize, the final process recipe for the grating is the following:

1. Clean wafer with a solution of H₂O2 and H2SO4 in relation 1 : 2.5 (7-up).

2. Wash with IMEC.

3. HMDS bake for 25 min at 149◦C.

4. Spin photoresist (S1813) at 4000 rpm for 30 s. 5. Bake at 90◦C for 1 min.

6. Lithography with exposure time 3.2 s. Lamp power 350 W. 7. Developer MF-CD-26. Development time 30 s.

8. Plasma etching P5000, Standard Oxide recipe. 9. Remove rest of photoresist by acetone and plasma.

3.4 Indium fusing

As discussed in section 2.4 the contact between the grating and its copper mounting needs to be as good as possible to get a good transfer of heat. It is also important that the fusing medium is vacuum compatible, i.e. that it will not contaminate the low pressure environment

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3.5. MIRROR 29

in the monochromator. A good candidate material that reach these requirements is the element indium (In), which is malleable and has a melting point of 156◦C. It has been used as a fusing material for mirrors and gratings (made of silicon) at the ESRF (European Synchrotron Radiation Facility) in Grenoble.

As a first test a nitrogen atmosphere (1 bar) oven was used. The copper mounting was placed horizontally and a thin indium film and then a piece of SiC were placed on top. The oven was then heated in a time span of 30 min to a temperature of about 180◦C to ensure that the indium had melted. After reaching 180◦C the oven was switched off and let to cool. When room temperature was reached it was noted that the indium had melted, but not fused with the SiC sample on top. A likely explanation for this failure could be that the sample was not put under any pressure, besides its own weight, and hence could not get a good contact with the melted indium.

A second try to get the indium to fuse was made by using a heating plate to make the copper mounting reach a temperature of about 180◦C. This was done in a normal atmosphere. The benefit of using this method is that it will be easier to observe and control the process, compared to what could be done using an oven. Once the mounting had reached the desired temperature the indium film was applied and quickly melted. Using razor blades and tweezers the indium melt could then be smeared out and the grating was applied and put under pressure. After this procedure everything was left to cool down, and afterwards it was seen that the grating had fused together with its mounting.

3.5 Mirror

As mentioned in chapter 2, the mirror will be manufactured by sputtering a thin film onto a silicon substrate. The substrates are from the company Optarius, having a diameter of 25.4 mm and a thickness of 3 mm. The mirrors are concave with a 1000 mm radius of curvature, making it plausible to assume it plane during modelling.

Sputtering is closely related to the plasma etch technique, but instead of etching it deposits material onto a surface. Consider picture 3.1. Let, instead of having a substrate placed at the lower electrode, the film material takes its place. This is called the sputtering target. The mirror substrate will be placed on the other electrode. Now the ion bombardment will detach material from the target, which in turn will be deposited on the substrate at the other electrode. This can be done either by high-energy ballistic impact or by low-energy thermal diffusion, depending on the chamber pressure. In this way a thin film from the sputtering target will be grown upon the substrate.

The sputtering was done in house at Electrum on an ATC Orion 8 from AJA International. First the platinum was coated as a test, this mirror was then used in the characterization in section 4. The SiC mirror was made at the end of the writing of this thesis, but was not tested due to time shortage.

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Chapter 4

Characterization

Once the lithography is found to work, a full size mask can be created and a macroscopic grating can be etched. It is now possible to do an optical characterization of the grating, which was impossible while using the test mask. Before a full wafer is etched, a test piece can be made and examined in the visible light range. This is done in section 4.1. Once this is done, a full wafer is etched and fused onto a copper mounting. The grating is then installed into the monochromator of BALTAZAR. This is done in section 4.2.

4.1 Visible range

With the macroscopic grating now available, the grating period can be determined optically by measuring the diffraction angles of light in the visible range. The light source used in this section will be a 635 nm (red) laser.

It is of interest to measure if the grating period is constant throughout the whole grating to see if the lines are uniform. This can be achieved by fixing the light source at a constant distance and constant angle while moving the light spot to different locations on the grating. The grating period d can then be estimated by measuring one of the diffracted orders (in this case n = 1) and inserting the value in the grating equation:

d(θ1) =

λ sin θi+ sin θ1

(4.1)

Assuming normal incidence (θ_i = 0), we will get an estimated standard deviation: σd(θ1) =

λ cos θ1

sin2θ1

σθ1 (4.2)

which is used during the data analysis. The angle θ1 is obtained by measuring the distance

between the n = 1 and n = −1 orders, and then dividing by 2 (the orders will be symmetric about the normal in this case) to get one of the legs in a right triangle. The other leg is found by measuring the distance from the grating to the plane of the measured diffracted orders, and then using trigonometry to find θ₁.

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The result after measuring θ₁ for different points of incidence on the grating is: ¯

d = 2.70µm σd= 30nm

This is an underestimation compared to what was obtained from the AFM measurements. A possible explanation is that the incidence was not exactly normal, neither in the xy-plane or in the z-direction, giving systematic errors. On the other hand, the small standard deviation indicates that the grating pattern is uniform throughout the sample.

The laser used in this section has a to small effect to let any intensity measurements be made, to do this the main laser of BALTAZAR needs to be used instead.

4.2 XUV range

4.2.1 The monochromator of BALTAZAR

The high intensity light needed for the photoemission spectroscopy used by BALTAZAR is provided by a pulsed laser generated by higher order harmonics. This is achieved by using an N d : Y V O4 (neodymium-doped yttrium orthovanadate) laser with 1064 nm wavelength as a

basis. In the next stage the light passes by a third harmonic generator to give a wavelength of 355 nm and an average power of 15 W. After passing through a polarization controller the light enters the monochromator which is filled with xenon gas. It is here the higher order generation is achieved, giving wavelengths of 118 nm and 71 nm, roughly.

The light will first strike the mirror, which can be readjusted during linearization by use of remote controlled motors. The light can thus be made to reflect on the grating at the opposite end of the monochromator tank. The grating is also mounted on a holder which can be operated from outside the monochromator. This holder will make it possible to rotate the grating around the vertical axis, making it easy to control the incidence angle and hence the diffracted angles. The diffracted beam will now be reflected into the analysis chamber through a LiF output window. In the analysis chamber there is also a gold foil that can be inserted into the path of the beam. This foil is connected to an ampere meter.

To get the higher harmonic generation to 71 nm to work, the LiF output window needs to be removed so that the monochromator and analysis chamber is completely connected. This will not be possible to arrange for this thesis.

4.2.2 Third harmonic generation - 355 nm

First, an intensity measurement is done outside the monochromator. This is done by diverging the beam (having a wavelength of 355 nm) by a mirror and using a power meter from Coherent Inc., a LabMax-TOP, to measure the output power (in mW ). The grating is then placed in the path of the beam and the power is measured in the zeroth, first and second order diffractions. The ratios between the powers of the different orders and the unscattered beam can then be used as an estimate of the intensity. The result can then be compared with the result given from grating.m in appendix A, using the static dielectric constant = 10.0 [16]. The data for the different orders were sampled for 10 s.

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4.2. XUV RANGE 33

Figure 4.1. The monochromator of BALTAZAR with top lid removed. Pt grating in use.

Main n = 0 n = 1 n = 2 ¯

x (mW ) 310.9 63.0 5.3 1.8 σ (mW ) 3.9 3.8 4.5 3.7

In the above table ¯x and σ is the mean and standard deviation, respectively. From this data the intensity is found to be:

n = 0 n = 1 n = 2 Experimental 0.2026 0.0170 0.0058 Analytical 0.2278 0.0171 0.0000

4.2.3 Higher harmonic generation - 118 nm

As explained above, the higher harmonic generation needs to be done in the Xe atmosphere of the monochromator. This means that the intensity measurement done for the 355 wavelength can’t be done for the 118 nm beam. Instead, the gold foil will be used. The foil can be inserted into the path of the beam which in turn will give rise to a photocurrent in the foil. It is this current that can be measured by use of an ampere meter, and the data can then be fitted to a third order polynomial. This will give a method to compare different arrangements of gratings and mirrors. As can be seen the current rises monotonically as the harmonic generation is

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Figure 4.2. Photocurrent from Al/M gF2 mirror, using Pt grating and SiC grating

Figure 4.3. Photocurrent from Pt grating and Si/Pt mirror

created until it peaks at ≈ 7.5 mbar and then drops off due to loss of phase matching. In figure 4.2 the photocurrent from the working constellations (Pt grating/ Al/M gF2 mirror)

and (SiC grating/ Al/M gF2 mirror) are shown. As can be seen the SiC grating gives almost

a factor of fivetimes better reflectivity. This is more than what could have been expected, since platinum has roughly one third of the reflectivity of SiC around 118 nm, see [1], page 24. This assumes that the echelette grating and the laminar grating has the same intensity profile, which is an approximate equivalence rule that can be used in this case, following [12]. A possible explanation could be that the platinum grating has degraded during use, and/or that the sputtered film is not of good enough quality concerning roughness.

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4.2. XUV RANGE 35

substrate). The result can be seen in figure 4.3 and is considerably worse then when using the Al/M gF2 mirror, the peak is now smaller then one tenth of the original setup.

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Chapter 5

Summary

The following chapter will give a summary of the work that was done while working on this thesis. Suggestions on further work will also be given.

5.1

Chapter 2

In the second chapter the grating was modelled mathematically to be able to get guidelines during the processing. The grating problem was stated and an analytical solution was found. A numerical solution was also implemented to supplement the analytical one, and they were found to give the same results, but the numerical solution was more cumbersome computa-tionally. Using these calculations it was found that the groove depth of the grating would have the largest impact on the intensity of the reflected beam. It was decided that the groove depth aimed for should be 23 nm, a good compromise between the two different working light wavelengths 118 nm and 71 nm used in BALTAZAR.

Figure 5.1. A full 2 inch wafer, etched with the grating pattern. The wafer is fused with indium

to a copper mounting.

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A thermal analysis of the grating and its copper mounting was also done using the COMSOL package. Assuming that the indium fusing could be achieved with good enough precision, the analysis was positive. A maximum temperature increase of 74 K was calculated, which together with an analysis of the thermal expansion of the grating gave the conclusion that silicon carbide should be a good material for use in BALTAZAR not only optically, but also with consideration to durability during heating.

5.2

Chapter 3

In this chapter the processing of the grating was documented. Two different methods were se-lected as candidates for the grating manufacturing, but only one was actually tested, namely the plasma etch technique. The method that was discarded was that of selective oxida-tion. The main characterization technique during these stages where atomic force microscopy (AFM), which was found to be a good tool to map the topology of the test samples.

The main difficulty in the processing was the lithography, since the mask pattern did not correctly transfer to the photoresist mask. This was due to working on the resolution limit of the available lithography machine, about 1 µm. Since the original grating period of 2 µm was hard to create with good enough quality, it was decided to increase the grating period. By using a test mask several different grating patterns could be tested simultaneously and from these tests it was decided that the new grating period would be 2.75 µm.

During AFM measurements it was noted that the wafers had scratches that were left from the cutting done by the vendor. These fell into the RMS roughness of 3 nm that was specified by the vendor. The effect of these discrepancies on the efficiency of the grating is not known, but it seems likely that they would reduce the efficiency due to scattering in different directions. One method that could be used to smooth the wafer surface is hydrogen etching, which is described in [14], but since the necessary equipment was not available at KTH this method was not considered for this thesis. A simpler method that could be used instead is to buy wafers with better RMS roughness.

5.3

Chapter 4

In this section characterizations (besides AFM that was used in chapter 3) of the grating were documented. Different methods had to be used for different light wavelengths due to what was available in the lab of BALTAZAR. First the diffraction angles were tested with a 635 nm laser, changing the light incidence point onto different places on the grating. Due to systematic errors the estimated grating period was smaller then what was expected from the AFM measurements, but on the other hand the standard deviation was small, indicating that the grating period is uniform throughout the sample.

Next, the grating was tested for the wavelength 355 nm. In this case the laser effect could be adjusted to orders of hundreds of mW , making intensity measurements possible. The data gave a good fit with the calculated intensity, at least for the zeroth and first order diffractions.

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5.3. CHAPTER 4 39

Figure 5.2. SiC grating installed in the monochromator.

To test the 118 wavelength, the grating had to be installed into the monochromator of BAL-TAZAR since a higher order generation was needed. Here it was possible to test the photon flux by measuring the photocurrent created in a gold foil. This gave a possibility to compare different permutations of gratings and mirrors in the optical setup. Together with an alu-minium mirror, it was seen that the new silicon carbide grating gave roughly a factor 5 larger flux than that of the old platinum grating. Using the platinum grating and the platinum coated mirror, a decrease in flux of more then a tenth from the original setup was seen. To conclude, silicon carbide seems to be a good candidate for use in optical components working in the XUV region. An interesting investigation that could be done in the future could be to try to optimize a grating for a single wavelength. This would certainly be ben-eficial for shorter wavelengths where the possible theoretical output will be small. Another possible study would be to increase the etch time; aiming for a deeper groove depth and see if the method described in this thesis still applies.

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Optical components of XUV monochromator