Ellipsometry based study of the optical properties of thin molybdenum germanium films

Ellipsometry based study of the

optical properties of thin

molybdenum germanium films

THESIS

submitted in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

PHYSICS

Author : Thom Boudewijn

Student ID : 1512714

Supervisor : Dr. M.J.A. de Dood

2ndcorrector : Prof. dr. M.P. van Exter Leiden, The Netherlands, July 20, 2020

Ellipsometry based study of the

optical properties of thin

molybdenum germanium films

Thom Boudewijn

Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands

July 20, 2020

Abstract

Ellipsometry measurements were done on thin amorphous MoGe layers with a Ge content of∼20%. Capping the MoGe layer with at least 20 nm of sputtered SiO2prevents oxidation of the layer.

We find that the refractive index of the sputtered oxide for wavelengths longer than 680 nm is given by

n(λ) =1.44606+60, 682/λ2. An estimate of the optical properties

of MoGe as a function of wavelength is made using an iterative procedure applied to the data of a ∼300 nm thick MoGe film capped with∼60 nm of sputtered oxide. A comparison with ellipsometry data for thin film MoGe samples show that the data

are inconsistent with the model despite the fact that multiple sources for systematic errors were eliminated.

Contents

1 Introduction 1 2 Experimental 3 2.1 Sample preparation 3 2.2 Ellipsometry 4 3 Ellipsometry 5 3.1 Theoretical description 5 3.2 Sample oxidation 6

4 Results & Discussion 11

4.1 Refractive index for thin SiO2films 11

4.2 Statistical error of the ellipsometer 11

4.3 Preventing MoGe oxidation 12

4.4 Ellipsometry of sputtered SiO2layers 14

4.5 Ellipsometry of Molybdenum Germanium layers 17

5 Outlook 23

6 Conclusions 25

A Appendix A 27

A.1 Scattering matrix formalism 32

Chapter

1

Introduction

Amorphous molybdenum-germanium (Mo1−xGex) is a superconducting

material with a critical temperature that depends on the atomic fraction of Ge. In this thesis we study aim for a fraction x ∼ 21% to get a relatively high critical temperature for superconductivity, Tc ∼7 K for bulk material

[1]. The amorphous nature of the material ensures that the atomic struc-ture of thin films and nanowires is independent of the thickness and shape of the nanowire. This allows to study the effect of the geometry of the nanowire on the superconducting properties of the wire. Hence, super-conducting nanowire single photon detectors (SSPDs or SNSPDs) made out of an amorphous superconductor provide a material system where the effect of device geometry on photon detection efficiency can be studied.

It is well-known that the superconducting properties and the normal state resistivity of thin-film superconductors depend on the layer thickness with many materials experiencing a superconductor-to-insulator transi-tion (SIT) [2]. In principle, a measurement of the square resistance of a MoGe film at room temperature gives a good prediction of the critical tem-perature Tc of the film and would be sufficient to estimate the electrical

performance of an SSPD. Because free electrons in a metal determine both the resistivity and the optical properties through a Drude model a ques-tion arises if the optical properties of MoGe are directly correlated with film thickness and the measured square resistance. If this can be realised measuring the square resistance of a film at room temperature would be sufficient to design an optimal SSPD for single photon detection at a par-ticular wavelength, provided that a good model for the working principle of the SSPD exists.

Due to the Corona measures the experimental plans for this project had to be changed. The consequences of this are that electrical measurements

2 Introduction

were not possible anymore and that no new samples for optical measure-ments could be made.

Previous work on the optical properties of MoGe is available [3, 4]. The original thesis by van Klink [3] reports transmission and reflection measurements of unprotected MoGe films on transparent glass substrates. The thesis of Ortego Larrazabal [4] expanded on these results by doing ellipsometry of unprotected MoGe films and identified the oxidation of the film as a limiting factor to determine the optical properties. In this work we investigate the use of ellipsometry on protected MoGe films.

This thesis is organised as follows: in Chapter 2 experimental informa-tion about the sample fabricainforma-tion and the ellipsometry technique is given. The theory of ellipsometry is explained and supplemented by the analysis of a capping oxide layer in Chapter 3. Chapter 4 contains results of the measurements on MoGe samples with varying thickness. Chapter 5 gives an outlook where the suggestions for measurements in further research are discussed. The conclusions of this project can be found in Chapter 6.

Chapter

2

Experimental

To determine the optical constants of thin-film MoGe, various samples have been made that were studied by spectroscopic ellipsometry. In this chapter we first describe the sample fabrication procedure and give an overview of the different samples. The details of the spectroscopic ellip-someter and ellipsometry measurement are given in Chapter 2.2. A theo-retical description of the ellipsometry measurement in terms of the ellip-sometry parametersΨ and ∆ can be found in Chapter 3.

2.1

Sample preparation

Thin MoGe films are sputter deposited on a substrate using a composite Mo target with pieces of Ge. Before sputtering the vacuum chamber is pumped down below 1×10−5 mbar. The MoGe layer is sputtered in an Argon plasma at 5×10−3mbar using an RF power of 120 W resulting in an approximate deposition rate of 6.5 nm/min. Samples are fixed using silver paint to ensure that the substrate temperature is close to room temperature during the deposition process.

A variety of samples is made using different substrates: bare silicon and silicon with a wet thermal oxide layer of 300 nm. After deposition of the MoGe layer the samples are either stored in a box or transferred to a second sputtering system where a layer of silicon dioxide is sputtered on top of the MoGe layer to prevent oxidation. Sputtering of SiO2 is done in

an Ar atmosphere at 5×10−3mbar and an RF power of 200 W leading to a typical deposition rate of 3.5 nm/min.

Both the MoGe and sputtered oxide thickness is controlled through the sputtering time. A number of Si samples is covered with sputtered oxide

4 Experimental

only. These samples are used to determine the optical constants of the sputtered oxide in the ellipsometer.

2.2

Ellipsometry

In this project a J. A. Woollam VASE Ellipsometer is used for ellipsometry measurements. A light beam from one arm is reflected on a sample after which it is detected in the other arm. This ellipsometer uses a spectrum of wavelengths between 240 nm and 1000 nm. In this thesis, ellipsometry parameters are measured at fixed angles of incidence of 55◦, 65◦and 75◦.

Chapter

3

Ellipsometry

3.1

Theoretical description

The optical properties of thin films and multilayers can be investigated using ellipsometry. An ellipsometer measures the (complex valued) ratio of the amplitude reflection coefficients for s and p polarised light. The data should be compared to a model of the optical properties of the sample that involves the dielectric constant e and the thickness d of each layer. It is important to note that we consider complex e with a real and an imaginary part to include optical loss in our model. As a result each layer has three independent physical parameters.

In an ellipsometer a light beam is incident onto the sample and the re-flected beam is collected by a detector. The light source and detector are placed on a moving arm to vary the angle of incidence. The detector and light source are equiped with (rotating) polarisers to measure polarisation dependent reflectivity from which the ellipsometer determines the param-etersΨ and ∆ that are defined as

ρ= r

(p)

r(s) =tan(Ψ)e

i∆_{, (3.1)}

where r(p)and r(s)are the Fresnel reflection coefficients for parallel (p) and perpendicular (s) polarisation, respectively. We define the xz-plane as the plane of incidence with the x- (z-)direction being parallel (perpendicular) to the interface. For a sample with two interfaces (film on a substrate) the

6 Ellipsometry

reflection coefficients can be given in closed form by r(p) = r (p) 12 +r (p) 23 e2ik2zd 1+r(₁₂p)r(₂₃p)e2ik2zd , (3.2) r(s) = r (s) 12 +r (s) 23 e2ik2zd 1+r(₁₂s)r(₂₃s)e2ik2zd , (3.3)

where r_ij(p) and r_ij(s) are the reflection coefficients for the interface between medium i and medium j. The z-component of the wave-vector of medium 2 is given by k2z. A more extensive version of the derivation of these

coef-ficients can be found in Appendix A.

The convention used by J. A. Woollam [5] leads to a sign difference in the exponential factors in the definitions of r(p) and r(s). Following the standard definition of the transfer matrix in a layered system shows that the P2-matrix in equations A.23 and A.25 contain this sign difference that

leads to an alternative definition of r(p) and r(s). We emphasise that the definition given by J.A. Woollam leads to a problematic interpretation of gain and absorption in the system and should be avoided. Unfortunately, this is not possible because this inconvenient definition is embedded in the software of the ellipsometer.

Figure 3.1 shows the Fresnel reflection coefficients rpand rsfor the field

amplitude reflected from a medium with a refractive index n = 3.45 as a function of angle of incidence θ. The angle of incidence is defined to be θ = 0 (θ = π

2) at normal (parallel) incidence. At the Brewster angle

θB the sign of rp changes and therefore the phase ∆ changes by π. It is

important to note that our sign convention leads to a different sign for the field components at normal incidence.

3.2

Sample oxidation

In the research of Ortego Larrazabal [4] the optical properties of thin MoGe films change on a typical timescale of 42 days. A possible explanation for these changes is continued oxidation of the MoGe films. In order to prevent this oxidation we cover the MoGe films with a film of SiO2.

An extra layer on a sample makes the model of the optical properties of the sample more complicated. Therefore it is important that the dielec-tric constant e of SiO2 is well-defined and known from an independent

measurement. In general the dielectric constant e is defined by

3.2 Sample oxidation 7

Figure 3.1: Fresnel reflection coefficients rp and rs for the field amplitude as

a function of angle of incidence θ. At Brewster angle θB the reflection for

p-polarised light vanishes. Our sign convention implies that rp = −rs at normal

incidence.

where η is the refractive index and κ is the extinction coefficient.

Using the definition of the dielectric constant e and a positive thick-ness, the convention used by J. A. Woollam [5] leads to a passive system with gain, which is unphysical. To resolve this situation we redefine equa-tion 3.1 to be

ρ = r

(p)

r(s) =tan(Ψ)e

−i∆_{. (3.5)}

This is allowed due to symmetry reasons and makes our model predic-tions consistent with the data from the Woollam ellipsometer. While this resolves the impossibility of passive layers with gain the definition

re-8 Ellipsometry

mains inconsistent with Woollam’s own definition for ρ.

We assume that the extinction coefficient for a SiO2film is κ =0, which

can be justified by the fact that SiO2is a transparent material. For the

re-fractive index we use data from measurements of Herzinger et al. [6]. Fig-ure 3.2 shows experimental ellipsomety data (Ψ,∆) at a fixed wavelength for sputtered SiO2 layers of various thickness (points). The data is

com-pared to a calculation of a model of a thin transparant SiO2layer on silicon

(line).

Figure 3.2: Ellipsometry of a thin transparent layer on a silicon substrate. The dots represent experimental data (Ψ,∆) at λ = 900.63 nm for various SiO2layers

sputtered on silicon. The solid line represents a calculation ofΨ and ∆ as a func-tion of thickness for a layer with ηSiO2 =1.4568 on a substrate with ηSi =3.6177.

Dashed lines represent calculations for ηSiO2 =1.4568±0.05.

The calculated ellipse is for layers without absorption. For converging layers with κ > 0 the reflection coefficients change from that of a thin

3.2 Sample oxidation 9

film on a substrate to the reflection of a surface when the film thickness becomes much larger than the absorption length. Hence the ellipsometry data converges to one (Ψ,∆)-point instead. The sensitivity of Ψ and ∆ to the refractive index of a thin film on the SiO2-on-silicon-substrate is highly dependent on the starting position on the ellipse.

Chapter

4

Results & Discussion

4.1

Refractive index for thin SiO

2

films

In the analysis of a thin SiO2 film on silicon we assume that the optical constant of the thin film is independent of the thickness of the film. Re-ports in literature [7] show that for very thin films (less than 20 nm) of SiO2on a Si substrate this assumption is not true due to extreme stress in

these thin films. Furthermore, the assumption of a sharp transition in the optical properties of the thin film and the substrate is not valid for thin SiO2films on a Si substrate.

4.2

Statistical error of the ellipsometer

In order to be able to evaluate the deviation of the data from the model, an estimation of the measurement error has to be made. We replace and re-align the same silicon substrate multiple times and measure the ellipsom-etry parameters. Figure 4.1 shows the difference inΨ (top) and ∆ between the first measurement and subsequent measurements.

The data in figure 4.1 for the Ψ-data the average deviation is less than 1 mrad, while for the∆-data the deviation is∼5 mrad. The∆-data shows a systematic deviation between measurements indicating that the align-ment and placealign-ment of the sample causes a significant systematic error. Typically, the fluctuations around these values are of∼1 mrad, compara-ble to the error in Ψ. It should be noted that these error bars will depend significantly on the point (Ψ, ∆) where the measurements are done and are thus only a rough estimate of the experimental error.

12 Results & Discussion

Figure 4.1:Measurement error of the ellipsometer expressed as the difference be-tween subsequent ellipsometry measurements on the same sample (silicon sub-strate).

4.3

Preventing MoGe oxidation

To prevent oxidation of thin film MoGe we use a sputtered silicon oxide as a capping layer. To quantify the difference between MoGe layers with and without a sputtered layer of SiO2on top, we measure them during twelve

weeks.

Figure 4.2 shows ellipsometry data (Ψ, ∆) as a function of wavelength for the unprotected sample (left column) and protected sample (right col-umn). The different line colours correspond to different exposure times. The measurements of the sample without capping SiO2 layer (4.2a and

4.3 Preventing MoGe oxidation 13

exposure time. The changes in the ellipsometry data for the capped sam-ple (4.2b and 4.2d) are minimal compared to the uncapped samsam-ple. Please note that the measurements of the capped and uncapped sample cannot be compared directly because the capping layer changes theΨ and ∆ values.

(a) (b)

(c) (d)

Figure 4.2: Ellipsometry data (Ψ,∆) evolution of a thin MoGe layer with a thin SiO2layer on top for 4.2b and 4.2d, on a silicon substrate. Each line represents a

measurement ofΨ- or ∆-values for multiple wavelengths in a different week. The change of (Ψ,∆) in time is significantly higher for the film without a SiO2layer on

top of the sample.

To gain better insight in the time evolution of the data, theΨ,∆-points at λ = 900.63 nm are plotted against time in figure 4.3 for the uncapped sample (solid symbols) and the capped samples (open symbols). The lines

14 Results & Discussion

in the graph serve to guide the eye. As can be seen in the figure a small shift of the data for the capped sample, especially during the first weeks, is visible. Most notably, the capped sample is clearly more stable than the uncapped sample and confirms the hypothesis that the change in optical properties of the uncapped sample is due to oxidation. We attribute the small initial changes in the capped sample to small changes to the MoGe layer or a slight oxidation of the sputtered SiO2 capping layer.

(a) (b)

Figure 4.3:Time evolution of ellipsometry data (Ψ,∆) of a thin MoGe layer

with-out (open symbols) and with (closed symbols) a thin SiO2layer on top, on a silicon

substrate. The lines through the data serve to guide the eye. Measured Ψ- and ∆-values as a function of time are taken at λ = 900.63 nm. Covering the sample with a thin SiO2layer causes the change ofΨ and ∆ to be decreased compared to

the uncovered sample.

We will use capped samples in the remaining experiments of this thesis and aim for a thickness of the SiO2capping layer of 60 nm to avoid issues

with the optical constants of very thin SiO2 films as discussed in section 4.1. For the same reason we use substrates with 300 nm thermal SiO2 on

top instead of Si wafers with a natixe oxide.

4.4

Ellipsometry of sputtered SiO

₂

layers

Figure 4.4 shows the measured ellipsometry parameters (Ψ, ∆) as a para-metric plot where each point corresponds to a different wavelength. Fig-ure 4.4a compares the measFig-ured data to a model using tabulated values

4.4 Ellipsometry of sputtered SiO2layers 15

for the refractive index of silicon and SiO2 [6]. This model is fitted to the

data using the layer thickness as the only fitting parameter resulting in a thickness d = 224.5±0.2 nm. As can be seen, a small deviation between the model and data is found.

(a) (b)

Figure 4.4: Ellipsometry data for a sputtered SiO2 layer on silicon (symbols) for

different wavelengths between 680 nm and 1000 nm. The blue lines correspond to a best fit of the model to the data for a thermal oxide (left) and a Cauchy model (right). The orange and green curves are calculations where the index is changed by±0.02.

A better fit of the model is obtained in figure 4.4b where we use tabu-lated data for the index of the silicon substrate and a simple Cauchy model for the sputtered SiO2layer. According to the Cauchy model the refractive

index n is given by

n=n0+ A 2

λ2, (4.1)

where n0 and A are fit parameters. From the best fit to the model we find

d =225.7±0.1 nm, n0 =1.446±0.02 and A =60.68±3 nm. These latter

values can be compared to the value of n0and A for thermal oxide. From a

fit of the Cauchy model to the tabulated data we obtain n0 =1.453±0.02

and A=60.64±3 nm for thermal oxide.

To obtain a best fit we calculated a least squares τ2for each data point defined as

τ2 = (Ψmodel−Ψdata)2+ (∆model−∆data)2. (4.2)

The average τ2 for all data points is normalised in such a way that the minimum of the parabola is at τavg2 = 1. The error bar in the fit

parame-16 Results & Discussion

ter can be found by using the definition of the least squares distribution. The error in the fit parameter is found from the expected variance of the least squares distribution, i.e. the points where τavg2 = 1+

q

2

n with n the

number of degrees of freedom.

In figure 4.5 the results of the τ2 calculations are for the fit in figure 4.4a (red line) and figure 4.4b (blue line). The values of τ2 are normalised to the value for the fit with Cauchy parameters. As can be deduced from the lower value of τ2the fit to the Cauchy model provides a better fit, con-sistent with the observations in figure 4.4. As can be seen in section 4.5 the optimal thickness has shifted due to the change of refractive index used in the model. Similar calculations of τ2 as a function of n0 and A result

in an estimate for the standard deviation of the fit parameters n0 (±0.02)

and A (±3). We remark that the deviation between the tabulated and cal-culated refractive index for the capping layer is smaller than the standard deviation estimated based on the variance in the parameters n0 and A.

Nevertheless, the fit to the tabulated parameters has a significantly larger minimum value of τ2 and the results in figure 4.4a show as systematic deviation between model and experiment.

In the analysis of the data we have limited the data to wavelengths above 680 nm because the deviations between model and data for lower wavelengths are too large to correct for in the fitting procedure, i.e. they cannot be described using the simple Cauchy model with two fit param-eters. A discussion using more fit parameters in the model is beyond the scope of this thesis: it will complicate the data analysis but does not alter the important conclusions. In addition at shorter wavelengths substantial differences in the tabulated optical properties of silicon exists depending on the source of the data which further complicates the analysis of the data for short wavelengths.

We attribute the difference between the tabulated values for thermal oxide and the Cauchy model for sputtered oxide to a slight difference in optical properties between the materials. However, without a reference measurement on a thermal oxide with a similar layer thickness we cannot rule out that the differences are caused by a difference in the ellipsometer measurement itself. To avoid that this difference propagates to the analysis of the capped MoGe samples we assume that the capping layer is well described by the Cauchy parameters found for the sputtered oxide on a silicon wafer.

4.5 Ellipsometry of Molybdenum Germanium layers 17

Figure 4.5: The calculated least squares τ2 for a model using the tabulated re-fractive index of SiO2(red) and the calculated refractive index based on a Cauchy

model (blue). The values are normalised to the minimum value for the fit with the Cauchy model. The black lines indicate the τ2-values at one standard deviation away from the minimum.

4.5

Ellipsometry of Molybdenum Germanium

lay-ers

Samples of MoGe with varying thickness were made on substrates with 300 nm thermal SiO2on silicon. These samples are covered with a layer of

sputtered SiO2of approximately 60 nm thickness. The optical properties of

these samples were studied using ellipsometry with the goal to determine the dielectric constant of the MoGe.

parame-18 Results & Discussion

ters should best be minimised. To this end one sample without a MoGe layer is put in the sputtering system and serves to determine the thick-ness of the oxide layer via an independent measurement. This reference sample is produced at the same time as the capping layer of the MoGe samples to ensure that all samples in a measurement series have the same oxide thickness. The remaining unknown parameters for each sample are the complex refractive index and the thickness of the MoGe layer. Unfor-tunately, the thickness of the MoGe is controlled by the deposition time and only gives a rough estimate of the thickness because the deposition rate varies between different depositions even if the depositions are done in consecutive order on the same day.

The dielectric constant e can be calculated in an iterative procedure that goes over multiple values for η and κ (defined in equation 3.4). At each wavelength the procedure minimises the difference between the data and the model. For samples with a MoGe thickness beyond ∼ 120 nm, the MoGe layer can be approximated as infinitely thick.

The refractive index found via this iterative procedure is plotted in fig-ure 4.6.

Figure 4.6:Optical properties η and κ as a function of wavelength for three differ-ent angles of incidence obtained from an iterative procedure using ellipsometry data on a thick (∼300 nm) MoGe layer capped with∼60 nm sputtered oxide.

With the now known estimate of the dielectric properties of MoGe we can verify our assumption of an infinite layer. The result is shown in fig-ure 4.7 that shows the calculated Ψ and ∆ values as a function of layer thickness for a wavelength of 407.43 nm and 900.63 nm. As can be seen in the figure,Ψ and ∆ become independent of layer thickness for a thickness above∼100 nm and thus validates our initial assumption.

4.5 Ellipsometry of Molybdenum Germanium layers 19

We explore the thickness dependence of the optical properties of MoGe layers in figure 4.8 at three representative wavelengths. The values for

η and κ used in figure 4.8 are given in table 4.1. In this procedure the

model is fit through the thickest sample. Assuming the refractive index is independent on the layer thickness, the calculated refractive index should result in a model which corresponds to the data for all thickness.

Figure 4.7:Calculated ellispometry parametersΨ and ∆ as a function of the thick-ness of the MoGe layer capped with∼60 nm of sputtered oxide. For thicker lay-ers the expected Ψ and ∆ become independent of layer thickness. Calculations are shown for a wavelength of 407.43 nm (orange curves) and 900.63 nm (blue curves).

The yellow stars correspond to samples without MoGe. It makes sense that these points, together with the data points for the thickest sample, are in good correspondence with the model. As can be seen the data points for

20 Results & Discussion

Table 4.1: Optical constants η and κ of MoGe at different representative wave-lengths λ=820.82, λ=900.63 and λ=980.12.

Wavelength λ (nm) η κ

820.82 5.108±0.2 4.035±0.1

900.63 5.349±0.2 4.191±0.1

980.12 5.543±0.2 4.308±0.1

the other MoGe thickness (nominally 5, 10 and 30 nm) are not on the theo-retical curve. The dashed lines in figure 4.8 take into account the estimated error for the optical constants. As can be seen in figure 4.8 the measured data points are not consistent with the model also when we take into ac-count the experimental error. This is most notable for the sample of 30 nm nominal thickness.

In our analysis and sample preparation we have been very careful to prepare samples with identical capping layers. We have also prepared MoGe samples on the same day and stored samples in vacuum before the deposition of the capping layer. The thickness of the bottom thermal oxide layer and the capping layer thickness have been carefully selected to op-timise both the sensitivity of the ellipsometry measurement and to avoid problems with an unknown refractive index of thin SiO2 layers. Despite

these precautions our measurements are not consistent with the model and we conclude that some, yet unknown, parameter plays an important role.

4.5 Ellipsometry of Molybdenum Germanium layers 21

Figure 4.8:Comparison of the model calculation (solid lines) to ellipsometry data (symbols) for MoGe layers. In each graph the solid line represents the theoretical evolution ofΨ and ∆ for varying thickness of the MoGe layer. The dashed lines represent the evolution ofΨ and ∆ at one standard deviation of η and κ. The stars represent data points of samples of different MoGe layer thickness (nominally 0, 5, 10, 30 and 300 nm). The columns correspond to an angle of incidence of 55◦, 65◦ and 75◦. The rows correspond to wavelengths λ = 820.82, λ = 900.63 and

Chapter

5

Outlook

The results of this research leave multiple questions unanswered, which is partly due to measures taken to deal with the corona crisis. Measurements of the resistivity could not be done during this project, and it will definitely be interesting to investigate this in further research. Such measurements could be used to verify the thickness of samples using a known relation between film resistance and film thickness. This may reveal differences between samples that we are currently not aware of.

The measurements of the MoGe samples in this project are not done with multiple samples. The size of the sputter system limits the number of samples that can be capped at the same time to ∼5 samples. It would be interesting to repeat the experiments with several thicker samples. If this would be done the spread in the resulting refractive indices for sim-ilar samples can be analysed. This gives information about the accuracy of the technique used to determine the refractive index. The accuracy of the current experimental data is mostly based on a statistical analysis of a single data set.

In the calculation of the refractive index of the sputtered oxide layer using a Cauchy model, the resulting refractive index is always a real value. The imaginary part could be tested by sputtering a much thicker layer or by sputtering on a glass substrate and verify the transparency of the sample.

Also repeating measurements for the refractive index of sputtered ox-ide should decrease the error in the calculation. Different thicknesses of oxide should result in similar results for the refractive index, as long as the layer is thick enough (above 20 nm thickness). In a layered model the error of each layer adds up which makes it important to aim for small errors in the reference measurements.

24 Outlook

At the end, the absorption is important in building a SSPD device. Do-ing specific measurements on the absorption and comparDo-ing the results with the results for the refractive index, would be more valuable for fu-ture research.

Chapter

6

Conclusions

As a first step in this project the oxidation of the MoGe samples was inves-tigated. The optical properties of MoGe samples capped with sputtered oxide are stable over time, while uncapped samples change over time. We link this to a continued oxidation and estimate that a 20 nm thick sputtered oxide layer is sufficient to provide protection.

The next task was to determine the refractive index of the capping SiO2

layer. It is important to provide an accurate description of the optical prop-erties of this layer since the error made in the optical propprop-erties for the reference sample propagates to the final estimation for the refractive in-dex of the MoGe layer. We find that a fit to a Cauchy layer with two fit parameters minimises the error in the refractive index of the oxide layer and find n(λ) =1.44606+60.682/λ2where λ is the wavelength measured

in nm. This relations holds for wavelengths above 680 nm which leads to restrictions in further calculations.

After identifying the properties of the capping layer, ellipsometry mea-surements were done on the MoGe layers. The refractive index of MoGe is calculated from the measurement of a thick MoGe layer. A model curve for MoGe layers of varying layer thickness is plotted and compared to mea-sured data points. We find significant deviations of this curve with respect to the data points that cannot be explained by the statistical error in our analysis. Further research is needed to achieve more conclusive results.

In order to improve the estimation of the refractive index of MoGe, po-tential sources of error in the procedure and model should be eliminated. Possible ways to do this are a better defined sputtered oxide layer and doing more measurements to average out the error. A possible outcome is that it is impossible to further improve the estimation for the refractive index.

26 Conclusions

We found an important inconsistency in the models used by J.A. Wool-lam that leads to unphysical results. We adjusted the model to make sure the model is physically consistent. The differences between different mod-els should be kept in mind for further research. Sign differences could lead to models describing the wrong physics, i.e. gain in a passive medium.

Appendix

A

Appendix A

In this appendix a complete derivation of the relevant formulas for ellip-sometry is given. Our notation is based on the wavevector in the x- and z-direction being parallel and perpendicular to the interface, respectively. In a medium with dielectric constant eithe wavevectors are given by

k2_i =k2_ix+k2_iz (A.1) kix = 2π√ei λ sin(θi) (A.2) kiz = 2π√ei λ cos(θi) (A.3)

where θi is the angle of the beam with the normal to the interface. The

incident medium 1 has a real-valued dielectric constant e = e1 so that

both k1x and k1z are real values. The Fresnel reflection coefficients for the

interface between medium 1 and 2 are given by r(₁₂p) = e2k1z−e1k2z

e2k1z+e1k2z (A.4)

r₁₂(s) = k1z−k2z k1z+k2z

. (A.5)

To obtain k2zwe use Snell’s law k1x =k2x, so that

k2z =

q

k2₂−k2_1x. (A.6)

Ellipsometry measures the complex ratio ρ defined by

ρ = r

(p)

r(s) =tan(Ψ)e

28 Appendix A

whereΨ and ∆ are refered to as the ellipsometry parameters. The fields E1

(H1) and E2(H2) are related via a transfer matrix formalism

E+ 2 E₂−  =M(s)E + 1 E₁−  = 1 2 1+k1z k2z 1− k1z k2z 1−k1z k2z 1+ k1z k2z ! E+ 1 E−₁  (A.8) and  H+ 2 H₂−  = M(p) H + 1 H₁−  = 1 2 1+e2k1z e1k2z 1− e2k1z e1k2z 1−e2k1z e1k2z 1+ e2k1z e1k2z !  H+ 1 H₁−  (A.9) for TE (TM) polarisation. The subscripts+and−denote fields travelling to the right and left, respectively. This is valid for one interface between two media. To extend the method to two interfaces, three transfer matrices are needed: M12 through interface 1→2, P2 through medium 2 and M23

through interface 2→3. In general, when going from medium 1 to medium n, the resulting expression will look like

E+ n E_n−  = T1nE + 1 E₁−  . (A.10)

The total transfer matrix T1nis the result of the multiplication of all transfer

matrices involved. It is important that the order of multiplication is the same as the order of light travelling through the media. We already know that E+ 2 E−₂  = M12E + 1 E−₁  (A.11) and applying the same problem to the next interface results in

E+ 3 E−₃  = M23E + 2 E−₂  . (A.12)

It is important to note that E+₂ and E₂−are not the same in the two formulas above. One is the E-field just after the 1→2 interface and the other is just before the 2→3 interface. Therefore we will rename the fields as E2,topand

E2,bottom. Then we are left with

E+_2,top E−_2,top ! =M12E + 1 E₁−  (A.13) and E+ 3 E−₃  = M23 E+_2,bottom E−_2,bottom ! . (A.14)

29

The fields E2,topand E2,bottomare related via the transfer matrix of medium

2 E_2,bottom+ E_2,bottom− ! =P2 E+_2,top E−_2,top ! (A.15) where P2is given by P2 =e ik2zd ₀ 0 e−ik2zd  . (A.16)

The total transfer matrix T13 relates the fields in medium 1 and 3 and is

given by E+ 3 E₃−  = M23 E_2,bottom+ E_2,bottom− ! =M23P2 E+_2,top E−_2,top ! = M23P2M12E + 1 E−₁  =T13E + 1 E₁−  . (A.17)

Using induction the general expression for a system of n layers becomes E+ n E_n−  =T1nE + 1 E−₁  = M(n−1)nP(n−1)...M23P2M12E + 1 E₁−  . (A.18) The Fresnel reflection coefficients for the multi-layer are given by

r(s) = E − 1 E+₁ (A.19) and r(p) = H − 1 H₁+. (A.20)

For s-polarisation, writing out the matrix multiplication gives E+ n En−  =T1nE + 1 E−₁  =  λ1 λ2 λ3 λ4  E+ 1 E₁−  =  λ1E1++λ2E − 1 λ3E₁++λ4E₁−  . (A.21) For reflection we set E−n =0 and obtain

r(s) = E − 1 E₁+ = − λ3 λ4 . (A.22)

30 Appendix A

For a system with two interfaces (e.g. a thin film on a substrate) we calcu-late T13 T₁₃(s) = M₂₃(s)P2M (s) 12 = 1 4 1+k2z k3z 1− k2z k3z 1−k2z k3z 1+ k2z k3z ! eik2zd ₀ 0 e−ik2zd  1+k1z k2z 1− k1z k2z 1−k1z k2z 1+ k1z k2z ! = 1 4 (1+k2z k3z)e ik2zd ₍₁−k2z k3z)e −ik2zd (1−k2z k3z)e ik2zd ₍₁₊k2z k3z)e −ik2zd ! 1+ k1z k2z 1− k1z k2z 1− k1z k2z 1+ k1z k2z ! = 1 4  A B C D  (A.23) where A =  1+k2z k3z   1+k1z k2z  eik2zd₊  1−k2z k3z   1−k1z k2z  e−ik2zd B =  1+k2z k3z   1−k1z k2z  eik2zd₊  1−k2z k3z   1+k1z k2z  e−ik2zd C =  1−k2z k3z   1+k1z k2z  eik2zd₊  1+k2z k3z   1−k1z k2z  e−ik2zd D =  1−k2z k3z   1−k1z k2z  eik2zd₊  1+k2z k3z   1+k1z k2z  e−ik2zd

and find the Fresnel coefficient r(s) = −λ3 λ4 = −(1− k2z k3z)(1+ k1z k2z)eik2zd+ (1+ k2z k3z)(1− k1z k2z)e −ik2zd (1−k2z k3z)(1− k1z k2z)e ik2zd+ (1+ k2z k3z)(1+ k1z k2z)e −ik2zd = −(k3z−k2z)(k2z+k1z)e ik2zd_{+ (k} 3z+k2z)(k2z−k1z)e−ik2zd (k3z−k2z)(k2z−k1z)eik2zd+ (k3z+k2z)(k2z+k1z)e−ik2zd = k2z−k3z k2z+k3ze 2ik2zd₊k1z−k2z k1z+k2z k1z−k2z k1z+k2z k2z−k3z k2z+k3ze 2ik2zd+1 = r (s) 12 +r (s) 23e2ik2zd 1+r(₁₂s)r₂₃(s)e2ik2zd. (A.24)

31

For p-polarisation we obtain T₁₃(p) = M(₂₃p)P2M (p) 12 = 1 4 1+e3k2z e2k3z 1− e3k2z e2k3z 1−e3k2z e2k3z 1+ e3k2z e2k3z ! eik2zd ₀ 0 e−ik2zd  ₁₊ e2k1z e1k2z 1− e2k1z e1k2z 1− e2k1z e1k2z 1+ e2k1z e1k2z ! = 1 4 (1+ e3k2z e2k3z)e ik2zd ₍₁−e3k2z e2k3z)e −ik2zd (1− e3k2z e2k3z)e ik2zd ₍₁₊e3k2z e2k3z)e −ik2zd ! 1+e2k1z e1k2z 1− e2k1z e1k2z 1−e2k1z e1k2z 1+ e2k1z e1k2z ! = 1 4  A B C D  (A.25) where A =  1+e3k2z e2k3z   1+e2k1z e1k2z  eik2zd+  1−e3k2z e2k3z   1− e2k1z e1k2z  e−ik2zd B =  1+e3k2z e2k3z   1−e2k1z e1k2z  eik2zd+  1−e3k2z e2k3z   1+ e2k1z e1k2z  e−ik2zd C =  1−e3k2z e2k3z   1+e2k1z e1k2z  eik2zd+  1+e3k2z e2k3z   1− e2k1z e1k2z  e−ik2zd D =  1−e3k2z e2k3z   1−e2k1z e1k2z  eik2zd+  1+e3k2z e2k3z   1+ e2k1z e1k2z  e−ik2zd and r(p) = −λ3 λ4 = −(1− e3k2z e2k3z)(1+ e2k1z e1k2z)e ik2zd_{+ (1+}e3k2z e2k3z)(1− e2k1z e1k2z)e −ik2zd (1−e3k2z e2k3z)(1− e2k1z e1k2z)e ik2zd+ (1+e3k2z e2k3z)(1+ e2k1z e1k2z)e −ik2zd = −(e2k3z−e3k2z)(e1k2z+e2k1z)e ik2zd_{+ (e} 2k3z+e3k2z)(e1k2z−e2k1z)e−ik2zd (e2k3z−e3k2z)(e1k2z−e2k1z)eik2zd+ (e2k3z+e3k2z)(e1k2z+e2k1z)e−ik2zd = e3k2z−e2k3z e3k2z+e2k3ze 2ik2zd₊e2k1z−e1k2z e2k1z+e1k2z e2k1z−e1k2z e2k1z+e1k2z e3k2z−e2k3z e3k2z+e2k3ze 2ik2zd₊₁ = r (p) 12 +r (p) 23 e2ik2zd 1+r₁₂(p)r(₂₃p)e2ik2zd . (A.26)

32 Appendix A

Using these Fresnel coefficients we find the expression of ρ

ρ= r (p) r(s) = r₁₂(p)+r(p)₂₃ e2ik2zd 1+r(p)₁₂ r₂₃(p)e2ik2zd r(s)₁₂+r(s)₂₃e2ik2zd 1+r(s)₁₂r₂₃(s)e2ik2zd . (A.27)

The result of a measurement is a value forΨ and ∆ defined by

ρ=tan(Ψ)ei∆. (A.28)

The expressions forΨ and ∆ as a function of ρ are thus given by

Ψ=arctan(|ρ|) (A.29) ∆ = −i ln  ρ |ρ|  (A.30) where∆ is a phase and we impose 0≤∆ ≤2π.

A.1

Scattering matrix formalism

Following the definition of the conversion from transfer to scatter matrix given [8] S= − m21 m22 1 m22 m11−m12_mm₂₂21 m_m12₂₂ ! (A.31) where mij are the elements of the transfer matrix and S is the scattering

matrix, for s-polarisation the input and output fields are related as E−_i E+_j  = − m21 m22 1 m22 m11−m12m21_m22 m12_m22 ! E+_i E−_j  . (A.32)

Using the definition of M(s)we obtain

S(s) =   kiz−kjz kiz+kjz 2kjz kiz+kjz 2kiz kiz+kjz kjz−kiz kiz+kjz  =   r_ij(s) t(_jis) t(_ijs) r(_jis)  . (A.33)

In a similar way, for the p-polarisation holds  H_i− H+_j  = − m21 m22 1 m22 m11−m12_m22m21 m_m12₂₂ !  H_i+ H_j−  (A.34)

A.1 Scattering matrix formalism 33 and S(p) =   r(_ijp) t(_jip) t(_ijp) r(_jip)  . (A.35)

For Pi we can follow the same procedure to get:

SPi =  0 eikizd eikizd ₀  (A.36) This results in a self-consistent calculation of the S-matrix of an arbitrary multi-layer by computing the transfer matrix of the system. This transfer matrix can be decomposed in a product of interface matrices Mijand

prop-agation matrices Pi for each layer. The matrix M for a general system can

be obtained via equation A.18. It is important to note that the P-matrices contain a choice of the sign and we follow the convention defined by equa-tion A.16 that gives a positive phase when the field propagates through a layer. Unfortunately, the convention used by J. A. Woollam [5] is opposite to our convention. The explanation given on their website constructs the S-matrix of the multi-layer instead of the transfer matrix and their formu-las for the ellipsometer parameters differ by a sign. This difference can be expressed in terms of the propagation matrix where J. A. Woollam uses a matrix P_i0instead of the matrix Piused here.

P_i0 =e

−ikizd ₀

0 eikizd



(A.37) To compare our model calculation to the dataΨ and ∆ produced by the el-lipsometer we correct for the difference in sign by using P_i0as propagation matrices in our calculations of all other chapters in this thesis.

A.1.1

Scattering matrix approach for a two-layer system

To illustrate the use of the scattering matrix as well as the difference in the choice of sign, we give an explicit example of the scattering matrix cal-culation for a two-interface system of a thin film on a substrate following the method introduced by J. A. Woollam. This example calculates what the outcome will be when incoming light is transmitted through inter-face 1→2 (point A), then reflected at interface 2→3 (point B) and finally transmitted through interface 2→1 (point C). The starting signal will be, looking at s-polarisation: E_1,A+ E_2,A−  =1 0  . (A.38)

34 Appendix A

The calculation will be done step-by-step. A: E−_1,A E+_2,A  = r (s) 12 t (s) 21 t₁₂(s) r(₂₁s) ! E_1,A+ E_2,A−  = r (s) 12 t (s) 21 t(₁₂s) r(₂₁s) ! 1 0  = r (s) 12 t(₁₂s) ! (A.39) A→B: E−_2,A E+_2,B  =  0 eik2zd eik2zd 0  E_2,A+ E_2,B−  =  0 eik2zd eik2zd ₀  t(₁₂s) 0 ! = 0 t(₁₂s)eik2zd ! (A.40) B: E_2,B− E_3,B+  = r (s) 23 t (s) 32 t(₂₃s) r₃₂(s) ! E_2,B+ E_3,B−  = r (s) 23 t (s) 32 t(₂₃s) r₃₂(s) ! t(₁₂s)eik2zd 0 ! = r (s) 23t (s) 12eik2zd t(₁₂s)t(₂₃s)eik2zd ! (A.41) B→C: E_2,C− E+_2,B  =  0 eik2zd eik2zd 0  E+_2,C E_2,B−  =  0 eik2zd eik2zd ₀  0 r(₂₃s)t(₁₂s)eik2zd ! = r (s) 23t (s) 12e2ik2zd 0 ! (A.42) C: E−_1,C E+_2,C  = r (s) 12 t (s) 21 t(₁₂s) r(₂₁s) ! E_1,C+ E_2,C−  = r (s) 12 t (s) 21 t(₁₂s) r(₂₁s) ! 0 r₂₃(s)t(₁₂s)e2ik2zd ! = r (s) 23t (s) 12t (s) 21e2ik2zd r₂₁(s)r₂₃(s)t(₁₂s)e2ik2zd ! (A.43)

On the Woollam-website they obtain

E−_1,C =r(₂₃s)t₁₂(s)t(₂₁s)e−2ik2zd_{. (A.44)}

As can be seen the difference in the sign is consistent with an alternative choice of the matrices Pi.

References

[1] J. M. Graybeal and M. R. Beasley, Localization and interaction effects in ultrathin amorphous superconducting films, Phys. Rev. B. 29, 4167 (1984). [2] C. Chen, P. Das, E. Aytan, and W. Zhou, Superconductor to insulator

transition in wafer-scale NbSe2, (2019).

[3] D. van Klink, Optical properties of amorphous thin-film MoGe, MSc Thesis, Leiden University (2018).

[4] M. Ortego Larrazabal, Optical properties of MoGe thin-films, BSc Thesis, Leiden University (2019).

[5] J. A. Woollam, Interaction of Light and Materials.

[6] C. M. Herzinger, B. Johs, W. A. McGahan, and J. A. Woollam, Ellip-sometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investiga-tion, Journal of applied physics (1998).

[7] H. G. Tompkins and E. A. Irene, Handbook of ellipsometry, William Andrew Publishing & Springer, 2005.

[8] P. Markos and C. M. Soukoulis, Wave propagation, Princeton University Press, 2008.