**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

2*nd* corrector : Jan Aarts

**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

### 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.

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

### 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.

These parameters are defined by the relation:

*ρ*= *rp*

*rs*

=tanΨ*ei*∆ (2.1)

where*rp*and*rs*are the reflection coefficients for p-polarized and s-polarized

light, tanΨis the amplitude ratio upon reflection and ∆ is a phase difference.
Because the amplitude ratio is defined as positive, we find that 0 ≤ Ψ ≤ *π*_{2}

while 0≤ ∆≤2*π*.

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]

*rp*_{12}= *N*2cos*φ*1−*N*1cos*φ*2

*N*2cos*φ*1+*N*1cos*φ*2

(2.2)

*rs*_{12}= *N*1cos*φ*1−*N*2cos*φ*2

*N*1cos*φ*1+*N*2cos*φ*2

(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 attenuation of the electromagnetic wave when
it propagates through the material. The angles *φ*1 and *φ*2 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.

*2.2 The SiO*_{2} *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*
23*e*2*iβ*

1+*r*_{12}*p* *rp*_{23}*e*2*iβ*

*rs* = *r*
*s*

12+*r*23*s* *e*2*iβ*

1+*rs*_{12}*r*_{23}*s* *e*2*iβ*

where *β* = 2*πd _{λ}*

*N*2cos

*φ*2 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,

*N*1sin*φ*1 = *N*2sin*φ*2. For an absorbing medium, the refractive index *N*2 is

complex-valued and *φ*2 cannot be directly calculated from Snell’s law. This

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

*N*2cos*φ*2 = *u*2+*iv*2 Where *u*2 and *v*2 are real functions of *N*2, *N*1 and *φ*1

defined via:

(*u*2+*iv*2)2 =*N*22(1−*N*12sin*φ*12) (2.4)

With the help of *u*2 and *v*2 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**

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*SiO*2 on a *Si*substrate. Experimentally such

substrates are available either by thermal oxidation of*Si* wafers or by sputter
deposition of *SiO*2 on a substrate. *SiO*2 has the advantage of being a very

well-known and well-defined material that is stable over time. The *SiO*2

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 *SiO*2 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*SiO*2 film,

with the film thickness as only fit parameter. Tabulated data for the optical
properties of *Si* [8] and thermally grown *SiO*2 [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 Ψis 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.

*2.2 The SiO*_{2} *film on Si* 11

As an alternative way of fitting a model to the data,Ψand ∆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 *SiO*2 on a *Si* substrate. The

curves show a parametric plot of the expectedΨand ∆ 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*SiO*2, respectively) for the three

incident angles. Starting from a bare silicon layer, the curve goes downwards
to the right as the thickness of the*SiO*2 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*SiO*2 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.

**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◦and70◦from top to bottom curves). The errors ofΨ(left) and∆(right) for the65◦ incident angle measurements are depicted at the top of the figure.

Figure 6 shows Ψand ∆ 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 *SiO*2 film on *Si*. In

*2.3 Accuracy of Ellipsometry measurements* 13

Figure 7: Ψand∆values as function of wavelength for a 120nm thick*SiO*2film on*Si*at an

angle of incidence of65◦. 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 *SiO*2 film on *Si* can be

### 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**

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 *µ*m2
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**

*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.

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 *d _{depo}* (in nm) estimated from the constant
deposition rate.

*d _{prof}* =−1.5315+1.1396·

*d*

_{depo}### 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].

*n*2 =*n*1tan*θ*1

s

1− 4*ρ*

1+*ρ*2sin*θ*1

2 _{(4.1)}

where *ρ* = tanΨ*ei*∆, and Ψ and ∆ are the experimentally determined
ellip-someter parameters. *n*2 and *n*1 are the refractive index of the film and the

ambient medium, respectively and *θ*1 is the angle of incidence. The values of

the optical constants for the thick film are *n* = *Re*[*n*2] and *κ* =*Im*[*n*2]. 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+*i*2 = (*n*+*iκ*)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

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 dependentΨand∆values for*SiO*2.

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}

*4.2 Ellipsometry on thin film MoGe* 21

Figure 12: MeasuredΨand∆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 *M oGe*

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 Ψand∆ 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*SiO*2were taken from literature [18, 19]. The

calculations for three interfaces (*Si*-*SiO*2-*M oGe*-Air) were performed using the

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.

*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).

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

*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*
2_{τ}_{/}_{m}

1−*iωτ* =*σ*0

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. *σ*0 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+ *iσ*(*ω*)

*ω*0

(4.3)

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

*σ*0 and *τ*. The value of *σ*0 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:

*R _{sheet}* =

*ρ*

*d* =

2*π*

ln 2*Rmeasured* (4.4)

where *d* is the known thickness of the measured film.

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*µ*Ωcm for films below 50 nm,
and a value of 500 *µ*Ωcm 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].

*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*=575*µ*Ω*cm*). 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 18: Comparison of the dielectric constant according to the Drude model using*rho*=
115*µ*Ω*cm* and *tau* = 0.34*f 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.

### 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+30.24*i*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.

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

*SiO*2 show no measureable changes in the properties of the layer and suggest

### Bibliography

[1] G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K.
Smirnov, B. Voronov, A. Dzardanov, C. Williams and R. Sobolewski,
*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*
*fabricated from an amorphous Mo0.75Ge0.25 thin film*. Appl. Phys. Lett.
105, 022602 (2014).

[5] X. Zhang, A. Engel, Q. Wang, A. Schilling, A. Semenov, M. Sidorova,
H.-W. H ˜AŒbers, I. Charaev, K. Ilin, M. Siegel, *Characteristics of *
*supercon-ducting tungsten silicide WxSi1-x for single photon detection*, Phys. Rev.
B 94, 174509 (2016).

[6] F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits,
I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin and S. W. Nam,*Detecting*
*single infrared photons with 93% system efficiency*, Nature Photonics 7, 210
(2013).

[7] J.M.Graybeal *Competition between superconductivity and localization in*
*two-dimensional ultrathin a-MoGe films*, Physica B+C 135, 113 (1985).

[8] Harland G. Tompkins, *A user’s guide to ellipsometry*. Academic Press,
Inc., San Diego, 1993.

[9] R.M.A. Azzam, N.M. Bashara.*Ellipsometry and polarized light*. North
Hol-land, 1979.

[10] J. A. Woollam Ellipsometry Studios, https://www.jawoollam.com/

[12] P. Markos, C.M. Soukoulis*Wave Propagation. From electrons to photonic*
*crystals and left-handed materials*. Princeton University Press, 2008.

[13] D. van Klink, *Optical properties of amorphous thin-film MoGe*, Leiden
University, 2018.

[14] J. M. Graybeal and M. R. Beasley, *Localization and interaction effects in*
*ultrathin amorphous superconducting films* Phys. Rev. B 29, 4167 (1984).

[15] J. M. Graybeal and M. R. Beasley *Observation of a new universal *
*resis-tive behavior of two-dimensional superconductors in a magnetic field*, Phys.
Rev. Lett. 56, 173 (1986).

[16] W. R. White, A. Kapitulnik, and M. R. Beasley*Collective vortex motion*
*in a-MoGe superconducting thin films*, Phys. Rev. Lett. 70, 670 (1993).

[17] M. P. Marder *Condenser Matter Physics* Wiley, 2010.

[18] D. E. Aspnes and A. A. Studna*Dielectric functions and optical parameters*
*of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV*

Physical review. B, Condensed matter 27, 985 (1983).

[19] Luis V. Rodr´ıguez-de Marcos, Juan I. Larruquert, Jos´e A. M´endez, and
Jos´e A. Azn´arez *Self-consistent optical constants of SiO*2 *and Ta*2*O*5 *films*,