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Optical properties of MoGe

thin-films

Thesis

submitted in partial fulfillment of the requirements for the degree of

Bachelor of Science in

Physics

Author : Maialen Ortego Larrazabal

Student ID : s2383969

Supervisor : Michiel de Dood

2nd corrector : Jan Aarts

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Optical properties of MoGe

thin-films

Maialen Ortego Larrazabal

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

28th of June, 2019

Abstract

We investigate the optical properties of thin film (5-120nm) of amorphous MoGe with the long term goal to establish a universal relation between the resistivity and the optical properties of strongly disordered superconducting

materials. We use a direct, analytical inversion to obtain the complex dielectric constant from the measured data on thick film. A comparison of

the optical constants for thick films with a Drude model that uses the measured resistance shows that we overestimate the damping in the material. Ellipsometric measurements as a function of time show changes in the optical

properties of the films on a typical timescale of 42 days. We relate these changes to oxidation of the MoGe films and show an increase in the total

thickness of our films with time. The continued oxidation of MoGe is detrimental for superconducting nanowire single photon detectors. We show

that the oxidation can be stopped by capping the film with a thin film of SiO2.

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Chapter

1

Introduction

Superconducting single photon detectors (SSPD)[1] consist of a nanostripe of superconducting material. When the wire is cooled well-below its critical temperature and sufficient bias current is applied to such a wire, single photons can be detected. The energy of the absorbed photon is sufficient to destroy the superconductivity in a region of the nanowire. The change to the normal (resistive) state results in a measurable voltage pulse. SSPDs are of practical interest because they combine a high efficiency with low timing jitter and lower dark count rates when compared to other detection techniques for single photons in the visible and infrared range of the spectrum[2].

Amorphous superconductors such as MoSi, MoGe and WSi [3–5] are ex-tremely promising materials for use in SSPDs. The absence of structure in the material makes it easier to deposit the superconductor on a wide range of materials and is believed to lead to more homogeneous films and higher fabri-cation yield and reproducibility of devices in the production process. To date, the highest device efficiency for an SSPD of 95% is achieved for an amorphous WSi device [6]. For widely used crystalline materials, e.g. NbN and NbTiN, the micro structure of the film depends strongly on the substrate, thickness and deposition parameters of the process used. For an amorphous material the properties of the film are mostly determined by the geometry, i.e. film thickness.

For thin amorphous films of MoGe the scattering is controlled by the in-terfaces and leads to well-known relations between the film thickness and the resistivity[7] and the superconductor critical temperature [7]. If the dielectric constant of the film is well-described by the free electron contribution follow-ing a Drude model a direct relation between the dielectric constant of the film and the thickness should exist as well. If such a relation exists, a simple measurement of the sheet resistance of the superconducting material might be sufficient input to design both the electronic and optical response of an SSPD. This thesis reports ellipsometry measurements on different thicknesses of MoGe film samples with the goal to find a simple relation between the dielectric constant, resistivity and thickness. Unfortunately, we find that the MoGe films oxidize when exposed to air, which greatly complicates the task of finding this relation. In chapter 2 we introduce the method of ellipsometry as an optical technique that measures the change in the polarization state of a light beam when this is reflected from a thin-film surface. In standard ellipsometry the

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thickness of the film and the dielectric constant are determined by fitting a model to the data.

It is often undesirable to use a complex model with many fit parameters because this complicates the physical interpretation of the result. Instead we rely on alternative ways to determine the properties of our samples. Section 3.2 discusses the different options to measure the thickness of the film. Section 4.1 analyses the dielectric function for a thick film of MoGe and 4.2 compares the ellipsometry data for films of different thicknesses to a theoretical curve obtained from the dielectric function of the thick film. Section 4.3 discusses the stability of MoGe and different options to slow down the changes. Section 4.4 links the resistivity to the dielectric function through the Drude model.

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Chapter

2

Ellipsometry

Ellipsometry[8][9] is an optical technique used to study the optical properties of surfaces and thin films. This technique is based on exploiting the change of polarization that occurs when light is reflected from or transmitted through the sample. The device with which ellipsometry is done, the ellipsometer, mea-sures the initial and final states of polarization. To obtain information about the optical properties of the sample, a model-based approach is needed, where the optical properties and layer thicknesses are derived by fitting or comparing the model to the data. The measurements in this thesis have been performed using a J.A. Woollam M-2000 ellipsometer[10].

2.1

Instrumentation

Figure 1: Schematic drawing of the rotating analyzer ellipsometer used in this thesis: Light from the light source passes through a polarizer and compensator and is reflected by the sample. The reflected light passes through a second compensator and an analyzer before reaching the detector.

Figure 1 shows a schematic drawing of an ellipsometer. The fixed polarizer converts the unpolarized light that comes from the light source into linearly polarized light. Then, the light is reflected from the sample and passes through a rotating analyzer to measure the light intentsity at the different angles of the analyzer. From these measurements of the intensity at different angles the ellipsometric parameters, Ψ and ∆, can be obtained.

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These parameters are defined by the relation:

ρ= rp

rs

=tanΨei∆ (2.1)

where rpand rsare the reflection coefficients for p-polarized and s-polarized

light, tanΨ is the amplitude ratio upon reflection and ∆ is a phase difference. Because the amplitude ratio is defined as positive, we find that 0 ≤ Ψ ≤ π2 while 0 ≤∆ ≤ 2π.

For a single interface the ratio of the amplitude of the reflected wave to the amplitude of the incident wave is given by Fresnel coefficients[11]

rp12= N2cos φ1− N1cos φ2 N2cos φ1+N1cos φ2 (2.2) rs12= N1cos φ1− N2cos φ2 N1cos φ1+N2cos φ2 (2.3) Where the superscript refers to the polarization of the waves, being either parallel (p) or perpendicular (s) to the plane of incidence as depicted in Figure 2. The subscript 12 refers to an interface between media 1 and 2 with the incident wave coming from medium 1. Ni is the complex refractive index of

the two media, N = n+iκ. The real part of the refractive index is denoted

by n and gives the phase velocity of the electromagnetic waves. The extinc-tion coefficient κ indicates the attenuaextinc-tion of the electromagnetic wave when it propagates through the material. The angles φ1 and φ2 correspond to the

angle in each medium with respect to the surface normal.

Figure 2: Schematic drawing of reflection of an electromagnetic wave from an interface at an oblique angle. The big arrows represent the wavevector of the incoming and reflected light. The electric field vector for s and p polarization is indicated in the figure.

The discussion above concerned a single interface, while most practical situations in ellipsometry involve a thin film with two interfaces or multilayers with multiple interfaces. The simpler case of two interfaces is depicted in Figure 3. The light reflected to medium 1 consists of light directly reflected from the first interface plus multiple reflections in the film. The resultant wave is the addition of this infinite series of reflections.

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2.2 The SiO2 film on Si 9

Figure 3: Contributions to the reflection and transmission of light from a thin film on a substrate. Ni and φi are the refractive index and angles of incidence in each layer and d is

the thickness of the film.

The ratio of amplitude of the total reflected light to the amplitude of the incident light is obtained as the sum of the geometric series and it is given by the reflection coefficients[8]:

rp = r p 12+r p 23e2iβ 1+r12p rp23e2iβ rs = r s 12+r23s e2iβ 1+rs12r23s e2iβ

where β = 2πdλ N2cos φ2 is the propagation phase in the films and d is the

thickness of the film.

When the index of refraction of medium 2 is real-valued, the incident angle in medium 2 can be obtained applying Snell’s law between medium 1 and 2,

N1sin φ1 = N2sin φ2. For an absorbing medium, the refractive index N2 is

complex-valued and φ2 cannot be directly calculated from Snell’s law. This

can be solved by redefining the angle in the following way:

N2cos φ2 = u2+iv2 Where u2 and v2 are real functions of N2, N1 and φ1

defined via:

(u2+iv2)2 =N22(1 − N12sin φ12) (2.4)

With the help of u2 and v2 the Fresnel coefficients between absorbing media

can be calculated as well.

The calculation of reflection coeffiicients can be extended to multilayers us-ing a transfer matrix formalism[12, 13]. In this method the properties of a multilayer are found via matrix multiplication of 2×2 matrices. The elements of these matrices are given by the relations given in this chapter.

2.2

The SiO

2

film on Si

The standard ellipsometry method requires a model to fit the data. From this fit the film thickness and optical constants of the film and substrate are obtained. For materials with unknown optical properties this fitting proce-dure often results in complex fitting functions with many fit parameters that characterize the sample. Giving a proper physical interpretation to these fit parameters is highly non-trivial. To avoid these difficulties we demonstrate an alternative method to represent ellipsometry dataΨ and ∆ that makes it more

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intuitive to compare the data to a model.

In this section we illustrate the method by exploring a well-known model sys-tem of a thin transparent film of SiO2 on a Si substrate. Experimentally such

substrates are available either by thermal oxidation of Si wafers or by sputter deposition of SiO2 on a substrate. SiO2 has the advantage of being a very

well-known and well-defined material that is stable over time. The SiO2

ma-terial is non-absorbing (κ=0) for most wavelengths, making it a good system to explore ellipsometric measurements.

Figure 4 shows the ellipsometer measurements of Ψ (left fig.) and ∆ (right fig.) as function of wavelength for a 155nm thick SiO2 film on a Si substrate

at incident angles of 65◦ and 60◦. The data (red and green curves) are fitted with a model (orange and blue curves) of a silicon substrate and a SiO2 film,

with the film thickness as only fit parameter. Tabulated data for the optical properties of Si [8] and thermally grown SiO2 [8] are used. The reliability of

the fit can be quantified with the value of the normalized Mean Squared Error (MSE). The lower this MSE value is, the better the model agrees with the data. An ideal model fit has a MSE value of approximately 1.

Figure 4: Ellipsometry data of Ψ (left) and ∆ (right) as function of wavelength for 155nm SiO2 film on top of a Si substrate. The green lines correspond to the experimental data at an incident angle of 65◦, and the red at an angle of 60◦. The blue and orange lines are the fit models used for this sample, at incident angles of 60◦ and 65◦, respectively.

As can be seen in Figure 4, the value of Ψ is maximum for a wavelength of approximately 700nm, while the curve of ∆ passes through 180◦, indicating a change in sign for the ratio ρ=rp/rs.

Fitting the data with a model requires some knowledge about the properties of the sample. The expressions for rs and rp contain the optical thickness, i.e.

the product of thickness and refractive index, making it impossible to indepen-dently determine both the index and the thickness from a single measurement. To determine both the thickness and the refractive index ellipsometry mea-surements at multiple angles are required. The process can be simplified if the thickness or the refractive index of the layer is known from independent measurements.

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2.2 The SiO2 film on Si 11 As an alternative way of fitting a model to the data,Ψ and ∆ can be plotted at a specific wavelength and angle if incidence as a function of film thicknesses. The thickness of a particular film can then be estimated by comparingΨ and ∆ to different samples of the same material.

Figure 5 shows Ψ and ∆ for a thin film of SiO2 on a Si substrate. The

curves show a parametric plot of the expectedΨ and ∆ values calculated with the thickness d as a parameter. The innermost green curve corresponds to an angle of incidence of 60◦, the middle red curve to 65◦ and exterior purple curve to 70◦. The black dots indicate the measured values of Ψ and ∆ for different thicknesses (bare Si, 25nm, 90nm and 155nm SiO2, respectively) for the three

incident angles. Starting from a bare silicon layer, the curve goes downwards to the right as the thickness of the SiO2 film grows and ends closing the circle

when the optical path inside the film equals the wavelength. For a single mea-surement of a film one could retrieve the thickness modulo a constant. Since the optical path inside the film depends on the angle of incidence the exact film thickness can be obtained from data at multiple angles of incidence or multiple wavelengths.

Figure 5: Parametric plot ofΨ and ∆ for a SiO2 film on a Si substrate with the thickness

t as a parameter for incident angles of 60◦ (inside green curve), 65◦ (middle red curve) and 70◦(exterior purple curve). Experimental data shown by the black dots.

For a thin film, the reflected intensity is, to first order, independent of film thickness. Figure 5 clearly shows the rapid change in∆ for very thin films. All curves start at values of ∆ ≈ 180◦ and values of Ψ between5 and 30 degrees. This makes thin-film samples distinguishable one from each other, which is a characteristic advantage of ellipsometry. In addition, Figure 5 shows that there are no points where∆ and Ψ are independent of thickness, showing that the sensitivity of ellipsometry does not strongly depend on film thickness.

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2.3

Accuracy of Ellipsometry measurements

The purpose of our research is to investigate the optical constants of thin film MoGe samples. The accuracy of our results will depend on how accurate one can measure the ellipsometer parameters Ψ and ∆. To test the accuracy and reproducibility of our measurements we repeated the same measurement on the same MoGe sample (30 nm thick MoGe on Si). In between measurements the sample was repositioned and the ellipsometer was realigned.

Figure 6: Ψ (left) and ∆ (right) as function of wavelength measured on a 30-nm-thick MoGe film on a Si substrate. Several sets of measurements are shown for three different angles of incidence (60◦, 65◦and 70◦from top to bottom curves). The errors ofΨ (left) and ∆ (right) for the 65◦ incident angle measurements are depicted at the top of the figure.

Figure 6 shows Ψ and ∆ values for three different angles of incidence(60◦, 65◦ and 70◦). All measurements were done the same day to limit possible effects due to aging of the sample. The top figures show the estimated mea-surement error for Ψ (left) and ∆ (right) obtained as the difference between the different repeated measurements. From these data we estimate an error below 0.2◦ for both Ψ and ∆.

A similar procedure was followed with the 120-nm-thick SiO2 film on Si. In

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2.3 Accuracy of Ellipsometry measurements 13

Figure 7: Ψ and ∆ values as function of wavelength for a 120nm thick SiO2film on Si at an

angle of incidence of 65◦. Ellipsometry measurements (graphs on the right side) performed on different days, within a month. The low MSE values (left top and bottom plots) indicate the accuracy of these measurements.

In Figure 7, Ψ and ∆ values for a 120nm thick SiO2 film on Si can be

seen. The measurements were done on the sample sample, but were carried out on different days.As can be seen in the figure the error in Ψ and ∆ is approximately 1◦. It should be noted that the error in Ψ becomes small close to the extremes inΨ, at wavelengths of ∼250 and 710 nm. As can be seen, the value of ∆ changes rapidly around these wavelength and gives rise to a much larger error in ∆ of approximately 6◦.

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Chapter

3

Sample preparation

The samples in this thesis were made by sputter-deposition of M oGe on a Si substrate using a composite target consisting of pieces of Ge glued to a M o sputter target. The composition of the target is carefully tuned to deposit films with 21% Ge content to create thin films of superconducting M oGe with the highest possible critical temperature Tc[14–16] The sputter process uses

Ar ions to form a thin layer of M oGe on the Si substrate that is kept at a potential of +1000V. The target and the substrate are at an approximate dis-tance of 3cm leading to a deposition rate of 4.5 nm per minute.

The thickness of the MoGe was varied by changing the the deposition time. During deposition some samples were partially covered with vacuum-compatible adhesive tape, in order to create samples with a clear step in thickness after the adhesive tape is removed. Following this procedure a set of films with a MoGe thickness of 5, 10, 15, 20, 25, 30, 45, 120 and 200nm was created. The thickness of the film is indicative and is based on the deposition time.

3.1

Thickness measurement

A prominent issue with ellipsometry is that it is difficult to distinguish be-tween the effects of an increase in film thickness and an increase in refractive index. This undesirable correlation can be removed by an independent mea-surement of the thickness of the film. This reduces the experimental challenge of determining the value of the optical constants and thickness of the material from ellipsometry. The thickness of the film is a required parameter in order to determine the optical values of the material from ellipsometry. To this end we explore several methods to measure the thickness of the M oGe films.

3.1.1

Scanning Electron Microscopy

Scanning Electron Microscopy (SEM) is capable of producing high resolution images by scanning the surface of a material using an electron beam and col-lecting the emitted secondary electrons as a function of the position of the

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beam.

In order to measure the thickness of a film, the Si substrate needs to be cleaved accurately in a direction that is exactly perpendicular to the sample surface. Figure 8 shows a SEM image of a 120nm thick M oGe sample.

Figure 8: SEM image of a 120-nm-thick MoGe film on a Si substrate.

Although the resolution of the SEM of approximately 5 nm is sufficient to measure the thickness of the MoGe layer, several factors limit the usefulness of this technique. The process of cleaving the Si substrate with a diamond scribe does not necessarily create straight, 90 degrees, angles that allow to unambiguously look at the cross-section of the film. In addition, we find that the MoGe film detaches from the substrate during the cleaving procedure. As a result, the angle between the electron beam and the MoGe layer is ill-defined. Nevertheless, a rough estimate of the layer thickness results in values of ∼100 nm, comparable to the estimate based on the deposition time.

3.1.2

Atomic force microscopy

Atomic Force Microscopy is a high resolution (less than 1 nm) scanning probe microscopy method that uses a tip touching or approaching the sample(cite) to create an image of the surface of the sample.

The AFM used to do the measurement is capable of scanning a 30×30 µm2 area of the surface. The AFM measurements on the 120-nm-thick film do not show a clear step and the observed height differences are limited to ∼40 nm (data not shown). Most likely, the method of making the samples with the adhesive tape creates a gradual transition instead of a sharp step and greatly complicates an AFM measurement of the film thickness.

3.1.3

Profilometer

A profilometer scans a tip in contact with the sample along a line to create a one-dimensional scan of the height. The advantadge of the profilometer is that it spans a larger distance compared to the AFM, which makes possible to

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3.1 Thickness measurement 17

measure the step height. We find that the profilometer is the most appropriate method for our purpose. Unfortunately, the resolution of the profilometer is about 5 nm and it proved to be impossible to reliably measure the thickness for the thinnest films (below 15 nm).

Figure 9: Typical profilometer scan for a sample partially covered by a 25-nm-thick MoGe layer. Note that the image is not to scale: the height on the vertical axis in in nanometers, while the distance on the horizontal axis is in millimeters.

Figure 9 compares the thickness measured by the profilometer (vertical axis) as a functions of the thickness estimated from the deposition time as-suming a constant deposition rate. As can be seen, the measured values do not exactly match the predicted values.

Figure 10: Measured thickness of the MoGe film as a function of expected thickness esti-mated from the total deposition time. The measurements were done with 15, 20, 30, 45, 120 and 200 nm thick MoGe films on a Si substrate. The lines through the data are linear fits to the data (see text). The picture on the right is a zoom of the picture of the left side. All measurements were performed the same day as the sample was made.

A linear fit to the data in Figure 10 (blue line) shows that the data is well-described by a straight line that does not go through the origin. For com-parison, the red line is a line through the origin and the point for the largest thickness. This deviation between the two lines is most clearly visible in the

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right panel of Figure 10. We attribute this deviation to the fact that the de-position does not start at time t = 0s, so that the thickness of the MoGe layer is not exactly proportional to the deposition time. From the linear fit we determine a relation between the real thickness dprof (in nm) measured by

the profilometer and the thickness ddepo (in nm) estimated from the constant deposition rate.

dprof =−1.5315+1.1396 · ddepo

This relation is especially important when doing measurements on thin films because the thickness of the film is estimated from the calibration curve presented in Figure 10. The thickness values stated in this thesis refer to the estimated thickness from the deposition time, unless stated otherwise.

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Chapter

4

Results and discussion

4.1

The optical constant of a thick MoGe film

In this section the optical values of thick M oGe layers that show bulk be-haviour will be discussed. The thickness of a metal film can be considered as infinitely thick if the layer thickness is much larger than the skin depth of the metal. This skin depth is expected to be 10-30nm for most materials for wavelengths in the visible and near-infrared. We consider all films thicker than 100 nm to display bulk behavior.

To determine the complex refractive index n+iκ of bulk MoGe directly from

ellipsometry measurements, an analytic inversion formula can be implemented[9][11].

n2 =n1tan θ1

s

1 −

1+ρ2sin θ1

2 (4.1)

where ρ = tanΨei∆, and Ψ and ∆ are the experimentally determined ellip-someter parameters. n2 and n1 are the refractive index of the film and the

ambient medium, respectively and θ1 is the angle of incidence. The values of

the optical constants for the thick film are n= Re[n2] and κ=Im[n2]. The

above formula is only valid for a thick enough film so that multiple reflections and interference coming from the interface with the substrate (see Figure 3, Chapter 2) do not occur. We tested the inversion formula by determining the index of refraction for bare SiO2 and found excellent agreement with literature

values[19].

The dielectric constant can be calculated from the refractive index:

 =1+i2 = (n+)2

where the subscripts 1 and 2 refer to the real and complex part of the dielectric constant, respectively. Figure 11 shows the real (1) and complex (2) parts

of the dielectric function obtained by applying the inversion formula to data for a 120nm M oGe film on Si. As can be seen, 2 >> 1, for M oGe and the

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Figure 11: Real (orange) and complex (blue) values of the dielectric constant of bulk MoGe as a function of wavelength. These values are calculated from the measuredΨ and ∆ for a 120-nm-thick MoGe film using the inversion formula (Eq. 4.1).

If the optical constants of a thin film are known, an analytical formula can be used to find the thickness of the film[9]. In principle, this formula can be used to obtain the refractive index via a numerical point-by-point∗ approach. Such a routine is available as part of the ellipsometer software. Unfortunately, this numerical method is not very consistent and appears to be rather unphysical†.

4.2

Ellipsometry on thin film MoGe

Figure 5 in chapter 2 showed the thickness dependentΨ and ∆ values for SiO2.

Here, the same idea is applied to absorbing M oGe films on Si substrate.

point-by-point:

The thicknesses as well as the imaginary part of the dielectric constant of the films were

sometimes negative. The imaginary part should be always positive corresponding to the energy gain.

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4.2 Ellipsometry on thin film MoGe 21

Figure 12: MeasuredΨ and ∆ for MoGe thin films of increasing thickness (red points). Data are shown for a MoGe thickness, from left to right of 5, 10, 15, 20, 30, 45, 120 and 200 nm. The data are compared to a calculation using the optical constants of bulk MoGe (120 nm) on a bare Si substrate (solid line) and a calculation taking into account the native oxide of the Si substrate (dashed line). The purple and grey points correspond to bare Si, measured from a sample and calculated from database values for n and κ, respectively.

Figure 12 shows measured Ψ and ∆ values (red dots) for various MoGe films with different thickness (nominal thickness of 5, 10, 15, 20, 30, 45, 120 and 200nm). Data are shown for one incident angle (60◦) and one wavelength (930, 88nm). The measured Ψ and ∆ values for a Si substrate with native oxide (0 nm MoGe) is represented by the purple dot close to ∆ = 180◦ and Ψ=22◦. The measured data points are all on a continuous curve connecting the point for bare Si to that for a bulk MoGe sample.

The curves in Fig. 12 are calculations of Ψ and ∆ using a simple model for the optical properties of MoGe layers as a function of film thickness. In these calculations the values for n and κ were calculated using equation 4.1 for the data of a 120nm M oGe film. The blue symbols correspond to a calculation that assumes a pure Si substrate, without native oxide, and a MoGe film of varying thickness. In the calculation the thickness is varied from 0 to 200 nm in 1 nm steps and it is assumed that the optical properties of the MoGe layer are independent of thickness. As can be seen in the figure there is a large difference between the measured and calculated points. Moreover, the calculated curve does not start at the measured point for a Si wafer with native oxide.

The purple dashed curve corresponds to a similar calculation using a Si substrate with a native oxide of approximately 2 nm as a starting point. Values for the optical constants of Si and SiO2were taken from literature [18, 19]. The

calculations for three interfaces (Si-SiO2-M oGe-Air) were performed using the

transfer matrix formalism[12]. As can be seen this calculation starts and ends at the right point and gives a reasonable description only for the very thin MoGe film of 5 nm thickness. The measurements for the thicker MoGe films clearly deviate from the measured curve.

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By starting at the right point for a bare Si wafer with native oxide the green solid curve is close to the data points for the thinnest samples. By construction the curve ends at the thickest films used to derive the optical constants of bulk MoGe. As can be seen the calculated curve deviates significantly from the measured data for thicknesses between 20 and 50 nm. This can be due to several reasons that have not been taken into account so far. Samples were measured within a few hours after being made. There is a possibility that in that time the samples have reacted and changed while they are exposed to air. This idea is further developed in section 4.3. Because MoGe is metallic it would be convenient to see if there are other paths to obtain the dielectric constant and compare them. For instance, this can be achieved making use of the Drude theory of metals, that uses the resistivity of the material as input. This is discussed in section 4.4. Better agreement with the data can be obtained if the dielectric constant is assumed to be dependent on the film thickness. However, the films are much thicker than the electron mean free path and we lack a good physical picture why the dielectric constant depends on film thickness.

4.3

Degradation of MoGe films

One of the possible reasons for the deviation of the experimental data shown in Figure 12 is degradation of the material with time. In order to have a better understanding of this process in an amorphous metallic alloy, the evolution of the physical properties of MoGe over time was tested.

Figure 13: Time evolution ofΨ (left) and ∆ (right) values for an initially 10-nm-thick MoGe film on a Si substrate. The green line through the data serves to guide the eye.

Figure 13 shows the measured ellipsometry parameters for a 10nm-thick MoGe film on Si substrate as a fucntion of time over a total time of 42 days. The blue line connects the data points from the measurements and shows that the optical properties of the sample change. The dashed green line serves to guide the eye and is given by the function of √Ct where C is a constant. This

functional dependence in the change of Ψ and ∆ over time is inspired by the diffusion equation. For diffusion limited oxidation the square root behavior

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4.3 Degradation of MoGe films 23

give the oxide thickness over time. If the ellipsometer parameters Ψ and ∆ are not at a maximum or a minimum they can be approximated as depending linearly on the thickness of the oxide.

Based on the ellipsometry data of Fig. 13, the oxidation of MoGe is at best suggestive and needs to be supported by additional data. We measured the thickness of an approximately 1-year-old sample that was exposed to air during that entire period. The sample was created using a sputtering time of 2 minutes, and the original film was expected to be around 10 nm in thickness. The measured values from the profilometer were close to 50 nm, which implies a fivefold increase in thickness in a year. We have also attempted profilometry on the sample in Figure 13. Unfortunately, these data are not conclusive due to the limited accuracy of the profilometer.

4.3.1

Sample in vacuum

If MoGe degrades due to oxidation, the process should stop when the sample is stored in vacuum. To test this idea two identical samples of 25nm M oGe film on Si were made and measured. One sample was stored in vacuum while the other was kept in air. The samples were measured immediately after fab-rication and one week after making the samples. The ellipsometry parameters for both samples were found to change. However, the optical properties for the sample stored in vacuum had changed much less than the properties of the sample stored in air. The corresponding ellipsometry measurements are summarized in Figure 14.

Figure 14: Ψ (left) and ∆ (right) measurements as function of wavelength for a 25nm MoGe film on Si. The green and blue lines correspond to the initial measurements done immediately after the samples were made. The green curve sample was stored in vacuum while the blue curve sample was stored in air. After a week measurements were repeated for both samples. The yellow (red) curve represents the data for the sample stored in vacuum (air).

Figure 14 shows four different measurements of Ψ and ∆ as function of wavelength. The green and blue lines are the consecutive measurements done for both sample immediately after they were made. After the measurement, the sample of the green curve was stored in vacuum, while the other was stored in air. One week later measurements were repeated and the corresponding data is represented by the yellow and red curves. The yellow and red curve corresponds

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to measured data for the sample stored in vacuum and air, respectively. As can be seen, the sample stored in air has changed appreciably while the changes in ellipsometry parameters for the sample stored in vacuum are much smaller.

4.3.2

Capping the film with SiO

2

Another procedure to stop the oxidation behavior of MoGe films is to cover the MoGe sample with a thin film (∼10 nm) of SiO2. This film is grown by sputter

deposition. This process is of technological interest: when successful MoGe based devices can be covered with transparent oxide to prevent oxidation.

Figure 15: Ellipsometry dataΨ (right) and ∆(left) as a function of wavelength for a MoGe film covered with 10nm of SiO2. The green curves show measurements performed two weeks

after the initial measurements (blue curves) were done.

The measurements shown in Figure 15 test this idea for a 30nm MoGe film covered with a 10nm layer of SiO2. Two different measurements can be seen

(green and blue curves), performed two weeks after the other. The results from the ellipsometry measurements are almost indistinguishable within ex-perimental error. From this we conclude that the coating of the film stops or significantly slows down the change in properties of MoGe over time.

4.4

Comparison of the dielectric constant with

the Drude model

The previous section shows that the timescale for a measurable change in MoGe films is of the order of days. All measurements shown in the Figure 12 were performed within hours after the samples were made. Therefore, we conclude that the deviation between data and theory in Fig. 14 cannot be explained by the gradual change in optical constant that occurs on a timescale of 1-10 days. The starting point of our theoretical model is given by the inversion formula for a thick MoGe film. If this inversion formula is not correct, a slightly different curve may result. Similarly, the model assumes a multilayer Si-SiO2

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4.4 Comparison of the dielectric constant with the Drude model 25

To further the discussion on obtaining the dielectric function of MoGe we compare the dielectric constant from the inversion formula to the Drude theory of metals using the value of the resistivity of MoGe.

The Drude model describes the response of the free electrons in a metal by taking the number of electrons per unit volume that are accelerated by external electric and magnetic fields and slowed down by friction of the electrons with the lattice and with each other. The external electric field E and the velocity of the electrons v follow a harmonic time dependence proportional to e−iωt. In the absence of magnetic field (B=0) the following expression can be obtained for the frequency-dependent conductivity[17]:

σ(ω) = ne

2τ/m

1 − iωτ =σ0 1

1 − iωτ (4.2)

where n is the electron density, e the charge of the electron, τ the average scattering time of an electron, m the effective mass of the electron and ω the frequency of the applied field. σ0 is the DC conductivity.

Following the linear response of Maxwell’s equations, the conductivity is re-lated to the dielectric function via[17]:

(ω) =1+ (ω)

ω0

(4.3)

The dielectric function given by the Drude model contains two unknowns:

σ0 and τ . The value of σ0 can be obtained measuring the sheet resistance of

a thin film using a four-point probe and calculating the inverse of the resis-tivity from there. For the linear probe used in our studies the measured sheet resistance and resistivity are linked via:

Rsheet = ρ

d =

ln 2Rmeasured (4.4)

where d is the known thickness of the measured film.

The resistance of MoGe samples was measured using a SRM four point probe head. This device has four tips on a straight line at a fixed distance.The two outermost tips are connected to the current source of a Keithley 2400 multimeter in 4-wire resistance mode. The remaining two innermost pins are connected to the voltage terminals to measure the resistance of the probe configuration.

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Figure 16: Sheet resistance (left) and resistivity (right) calculated from resistance measure-ments of the 5, 10, 15, 20, 30, 45, 120 and 200nm thick MoGe films as function of the thickness.

Figure 16 shows sheet resistance (left) and resistivity (right) values ob-tained values from resistance measurements as a function of the thickness of the measured films (5, 10, 15, 20, 30, 45, 120 and 200nm). As expected the sheet resistance depends strongly on film thickness. The resistivity derived from these data shows typical values of 700-1000 µΩcm for films below 50 nm, and a value of 500 µΩcm for the thickest film, suggesting a decrease in resis-tivity with film thickness. In general the measured resisresis-tivity is significantly higher than the values reported in literature for similar films [7].

To calculate the dielectric constant within the Drude model requires a value of the scattering time τ . The value of τ can be obtained by taking the dielectric function from the inversion formula at a certain wavelength and calculating the constant value of the scattering time.

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4.4 Comparison of the dielectric constant with the Drude model 27

Figure 17: Dielectric constant of MoGe as a function of wavelength calculated from the Drude model (Eq. 4.3) using the experimental value of the resitivity (ρ=575µΩcm). The best value of τ = 0.04 fs was estimated for a wavelength of 409.04 nm. The dashed (solid) line corresponds to the real (imaginary) part of the dielectric function.

Figure 17 shows the dielectric constant as a function of wavelength for bulk MoGe using the Drude model. In this calcualtion we used the experimental value of the resisitivity (see Fig. 16) of the 120 nm thick film and estimate the value of τ = xx fs from the experimentally determined dielectric constant at 409.04 nm (Fig 11). As can be seen these calculated values are inconsistent with the result of the inversion shown in Figure 11. We note that the mea-sured resistivity values shown in Fig.18 are significantly higher than values of the resistivity of similar MoGe films reported in literature[7]. We repeat the calculation of the Drude model using a value of ρ =115µΩcm corresponding to the value reported in literature and use τ = 0.34 fs to match the experi-mental dielectric constant at 409.04 nm. The calculated dielectric constant is shown in Figure 18 that compares the calculation (lines) to the result from the inversion formula (symbols).

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Figure 18: Comparison of the dielectric constant according to the Drude model using rho=

115µΩcm and tau = 0.34f s (lines) and the dielectric constant obtained via analytical inversion (symbols).

The curves in Figure 18 show that the imaginary part of the dielectric func-tion obtained by the Drude model (dashed orange line) is close to the values calculated from the inversion formula (blue symbols). The correspondence for the real part of the dielectric function is less good. The calculated value from the Drude model (green dashed line) is slightly negative and independent of wavelength, while the result from the inversion formula shows clear wavelength dependence.

Based on the measured dielectric constant we conclude that the resistivity of our films is close to the value reported in literature and that our attempts to measure the sheet resistance with the four point probe were so far not suc-cessful. We note that a better match for the real part of the dielectric constant in Fig 18 can be obtained by adding an extra resonance to the model. This resonance could be due to a phonon contribution. Adding this contribution to the model adds at least three parameters to the model that describe the resonance frequency, width and amplitude. It is clear from the data in Fig 18 that these parameters cannot be determined independently nor accurately. To resolve this issue optical measurements over a much larger wavelength range could be done to capture all phonon related transitions. We estimate that this requires additional measurements in the 1-100 µm wavelength range, which is outside the scope of this project.

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Chapter

5

Conclusions and outlook

We have experimentally determined the optical constant of MoGe films on Si. Direct, analytical, inversion of the ellipsometry measurements for a thick film MoGe gives an estimate of the dielectric constant. The dielectric constant has a large imaginary part and a relatively small real part. The imaginary part depends on frequency. A typical value of the dielectric constant of MoGe is

=−0.29+30.24i at a wavelength of 655 nm. Ellipsometry measurements for thin MoGe films show that the optical constants of bulk MoGe do not correctly predict the properties of thin films, suggesting that the dielectric constant of MoGe depends on film thickness for films of 5-100 nm thickness. A compari-son with a Drude model shows that optical constant is not consistent with the measured resistivity of the films using a four-point probe. When typical resis-tivity values from literature are used the Drude model correctly predicts the measured dielectric function. This suggests that the most important contribu-tion to the dielectric constant is due to the free electron contribucontribu-tion. The real part of the dielectric function obtained from the Drude model is constant for all frequencies and does not describe the frequency dependence in the values obtained from the inversion formula. We suggest that other material reso-nances, such as phonons, are responsible for the difference. Unfortunately, the resonance frequency for these phenomena is outside our measurement range and we have not included these into our description.

The results in this report lead to questions for further research. It is, in principle, possible to find the dielectric constant for each thickness of MoGe and obtain a curve that goes through the Ψ and ∆ data. Without a deeper understanding of the origin of the effect such a model simply rephrases the orig-inal problem of a thickness dependent dielectric constant. Such an approach has been attempted before by D. van Klink leading to a scattering time τ that depends on thickness. There is no physical interpretation for this result because the effect occurs for film thicknesses much larger than the scattering length.

To make progress in this direction, adding layers that correspond to oxi-dized MoGe would be a good idea. Another possible direction is to focus on reliable resistivity measurements and insert the measured resistivity into the Drude model. This would allow to subtract the free electron response and may shed some light on the remaining contributions to the dielectric constant.

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change in the optical properties of the material. We attribute this change to oxidization of the MoGe and find easily measurable changes on the timescale from 1-40 days. Storing the sample in vacuum reduces the change in the material properties. First results on MoGe covered with a 10 nm of layer of

SiO2 show no measureable changes in the properties of the layer and suggest

that this procedure can be used to protect the MoGe layer. This result is an essential step for reliable devices based on MoGe layers and nanowires and opens the way to future investigation of the optical properties of thin film MoGe.

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Pi-cosecond superconducting single-photon optical detector, Appl. Phys. Lett.

79, 705 (2001).

[2] R. H. Hadfield Single-photon detectors for optical quantum information

applications Nature Photonics 3, 696 (2009).

[3] D. Bosworth, S.L. Sahonta, R. H. Hadfield, and Z. H. Barber, Amorphous

molybdenum silicon superconducting thin films, AIP Advances 5, 087106

(2015).

[4] V. B. Verma, A. E. Lita, M. R. Vissers, F. Marsili, D. P. Pappas, R. P. Mirin, and S. W. Nam, Superconducting nanowire single photon detectors

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crystals and left-handed materials. Princeton University Press, 2008.

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