Optical properties of MoGe
submitted in partial fulfillment of the requirements for the degree of
Bachelor of Science
Author : Maialen Ortego Larrazabal
Student ID : s2383969
Supervisor : Michiel de Dood
2nd corrector : Jan Aarts
Optical properties of MoGe
Maialen Ortego Larrazabal
Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
28th of June, 2019
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
Superconducting single photon detectors (SSPD) 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.
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 . 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.
Ellipsometry 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.
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:
whererpandrsare 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 ≤ 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
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 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.
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 andφi are the refractive index and angles of incidence in each layer anddis
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:
rp = r p 12+r
rs = r s
where β = 2πdλ N2cosφ2 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,
N1sinφ1 = N2sinφ2. 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:
N2cosφ2 = u2+iv2 Where u2 and v2 are real functions of N2, N1 and φ1
(u2+iv2)2 =N22(1−N12sinφ12) (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.
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 ofSiO2 on a Sisubstrate. Experimentally such
substrates are available either by thermal oxidation ofSi 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 aSiO2 film,
with the film thickness as only fit parameter. Tabulated data for the optical properties of Si  and thermally grown SiO2  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 forrs and rp contain the optical thickness, i.e.
2.2 The SiO2 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 SiO2 on a Si substrate. The
curves show a parametric plot of the expectedΨand ∆ values calculated with the thicknessd 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 (bareSi, 25nm, 90nm and 155nmSiO2, respectively) for the three
incident angles. Starting from a bare silicon layer, the curve goes downwards to the right as the thickness of theSiO2 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 aSiO2 film on aSi 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.
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 SiO2 film on Si. In
2.3 Accuracy of Ellipsometry measurements 13
Figure 7: Ψand∆values as function of wavelength for a 120nm thickSiO2film onSiat 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 SiO2 film on Si can be
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 superconductingM 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.
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.
Scanning Electron Microscopy
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 thickM 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.
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 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 thicknessdprof (in nm) measured by
the profilometer and the thickness ddepo (in nm) estimated from the constant deposition rate.
Results and discussion
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 indexn+iκof bulk MoGe directly from ellipsometry measurements, an analytic inversion formula can be implemented.
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
The dielectric constant can be calculated from the refractive index:
=1+i2 = (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. 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†.
Ellipsometry on thin film MoGe
Figure 5 in chapter 2 showed the thickness dependentΨand∆values forSiO2.
Here, the same idea is applied to absorbing M oGe films on Sisubstrate.
†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 fornandκ, 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 ofSiandSiO2were taken from literature [18, 19]. The
calculations for three interfaces (Si-SiO2-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.
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.
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 onSi 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.
Capping the film with SiO2
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.
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 electronsvfollow a harmonic time dependence proportional toe−iωt. In the absence of magnetic field (B =0) the following expression can be obtained for the frequency-dependent conductivity:
σ(ω) = ne 2τ/m
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:
(ω) =1+ iσ(ω)
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
ln 2Rmeasured (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 700-1000µΩcm 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 .
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 ofτ = 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 usingrho= 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.
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+30.24iat 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
SiO2 show no measureable changes in the properties of the layer and suggest
 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).
 R. H. Hadfield Single-photon detectors for optical quantum information applications Nature Photonics 3, 696 (2009).
 D. Bosworth, S.L. Sahonta, R. H. Hadfield, and Z. H. Barber, Amorphous molybdenum silicon superconducting thin films, AIP Advances 5, 087106 (2015).
 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).
 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).
 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).
 J.M.Graybeal Competition between superconductivity and localization in two-dimensional ultrathin a-MoGe films, Physica B+C 135, 113 (1985).
 Harland G. Tompkins, A user’s guide to ellipsometry. Academic Press, Inc., San Diego, 1993.
 R.M.A. Azzam, N.M. Bashara.Ellipsometry and polarized light. North Hol-land, 1979.
 J. A. Woollam Ellipsometry Studios, https://www.jawoollam.com/
 P. Markos, C.M. SoukoulisWave Propagation. From electrons to photonic crystals and left-handed materials. Princeton University Press, 2008.
 D. van Klink, Optical properties of amorphous thin-film MoGe, Leiden University, 2018.
 J. M. Graybeal and M. R. Beasley, Localization and interaction effects in ultrathin amorphous superconducting films Phys. Rev. B 29, 4167 (1984).
 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).
 W. R. White, A. Kapitulnik, and M. R. BeasleyCollective vortex motion in a-MoGe superconducting thin films, Phys. Rev. Lett. 70, 670 (1993).
 M. P. Marder Condenser Matter Physics Wiley, 2010.
 D. E. Aspnes and A. A. StudnaDielectric 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).
 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 SiO2 and Ta2O5 films,