Thin film characterisation with a simple Stokes

Thin film characterisation with a simple Stokes ellipsometer Khosrow Hassani and Kolsum Abbaszadeh Optics Research Lab. 3, Department of Physics, University of Tehran, 1439955961, Tehran, Iran E-mail: [email protected] Abstract. Ellipsometry is an important and non-destructive technique to measure the optical constants and thickness of layered materials. This technique is based on the simple idea of changes in the polarisation state of light upon reflection from the sample surface. However, commercial ellipsometers are rather complicated and expensive devices that are not the best choices for learning the principles of ellipsometry. They usually require precise measurements of the angular position of polarisation-sensitive components, such as polarisers and wave retarders. In this work we demonstrate a simple ellipsometer which uses the concept of Stokes parameters to fully determine the polarisation state of the reflected light by measuring the relative intensities at four positions of the analyser. We also describe the essential steps in analysing the ellipsometry data, and provide three real-world examples. The film thickness and index of refraction of layered samples are measured with reasonable accuracy. This work is the result of a graduate-level physics project and can easily be implemented in an educational optics laboratory to help graduate students better understand and use the concepts of light polarisation, Stokes parameters, and ellipsometry.

Optical testing techniques, Stokes ellipsometry, Thin film structure and morphology Submitted to: European Journal of Physics

Thin film characterisation with a simple Stokes ellipsometer

2

1. Introduction Many physical properties of thin films are controlled by their optical constants and thickness. Therefore, In applications such as optical coating, semiconductor industry, solar cells, and biological sensors it is important to be able to measure these quantities with reasonable or high accuracy. Ellipsometry is a non-destructive optical technique widely used to measure the optical constants and thicknesses of single or multilayer thin films, simultaneously [1]. This technique is especially suitable for thin films of materials with a complex index of refraction (semiconductors, metals, etc.), and its sensitivity can be as good as detecting a single layer of atoms on a substrate [2]. The working principles of ellipsometry concern with studying the changes in the polarisation state of a beam of monochromatic light with known polarisation, after being reflected from the sample surface. However, because of technical considerations, the commercial ellipsometers look like rather complicated devices that hide the simple idea of light polarisation and its measurement. Although, some relevant articles exist that focus on the educational aspects of the ellipsometry and polarimetry techniques [3, 4, 5, 6], almost all of them follow the conventional routine of angle measurements to determine the polarisation state of light. Stokes parameters [7], on the other hand, provide an alternative and elegant way to grasp a deep understanding of the concept of light polarisation. The connection between ellipsometry and Stokes parameters is mainly present in technical literatures [8], and applications such as studying anisotropic materials [9], remote sensing [10], biology [11], and many more, but less discussed in educational texts oriented toward students at graduate or advanced undergraduate levels. Several other variants of the ellipsometry techniques such as, spectroscopic, generalised, white-light, and imaging ellipsometry have also been developed, with their specific setup and applications [12, 13, 14]. Also, interesting properties of thin films such as, surface roughness [15, 16], anisotropy [17], porosity [18, 19, 20, 21], and growth rate [22] have been studied by some of these methods. However, these apparatuses are expensive and not easily affordable in the educational labs with limited resources. Students who are going to learn the ellipsometry principles, or use this technique in their future professional work, do not usually get this opportunity in an educational lab at school. In this work we aim to demonstrate that Stokes ellipsometry can be used as an effective and reliable technique for thin film characterisation with less complexity and cost. It can also help the graduate students to comprehend the basic ideas in light polarisation measurements and ellipsometry, without too much complication. We realise this by a simple Stokes ellipsometry setup and measuring the thickness and optical constants of three thin film systems; a thermal silica, a magnesium fluoride thin film on glass, and a cadmium sulfide thin film also on glass substrate. Our setup uses modest equipments usually available in an educational optics lab. An essential, and not always clear, part of every ellipsometry experiment is the analysis of the raw data. Usually, a model in which the optical constants and thicknesses of the layers and substrate are taken as free parameters, is fitted to the experimental data to obtain the best values for the


3

unknown sample parameters. In practice, this part is usually accomplished by a software accompanying the commercial ellipsometer. In our work, we also explain the algorithm and computer code we have developed ourselves to analyse the data and get the useful ellipsometry results. Data acquisition is not automatic and somewhat time consuming, but the results are in good agreement with other reported data within about 0.3 % of relative uncertainties. In the following sections we first briefly review the theoretical principles of the Stokes method and its connection to the ellipsometry data. Then we describe our experimental works, data analysis, and results. At the end, we summarise our work and present a list of useful references. 2. Fundamental principles of ellipsometry In general, the transverse (perpendicular to the propagation direction) components of the electric field of a harmonic electromagnetic wave with angular frequency ω and wave number k that propagates in the z direction can be described by the following Eqs.: Ep (z) = E0p ei(kz−ωt+φp )

(1)

Es (z) = E0s ei(kz−ωt+φs ) ,

(2)

where, t is the time. Here, according to the ellipsometry conventions [12], E0s and φs represent the amplitude and initial phase of the component of the electric field of the incident light, perpendicular to the plane of incidence. Similarly, E0p and φp are those of the field component parallel to this plane. It is more convenient to combine the amplitude and initial phase of the wave into a single complex amplitude E˜0 = E0 eiφ . It can be shown [7] that when the relative phase δ = φs − φp is stationary (a polarised wave), then at any fixed point z along the propagation direction the tip of the electric ˆ + Esˆs traces a polarisation ellipse in the plane of oscillation which can be field E = Ep p characterised by δ and the amplitude ratio tan(α) = E0p /E0s . The polarisation state of the light can also be specified with a single complex quantity, X = E˜0s /E˜0p = E0s /E0p eiδ . The same ellipse can, alternatively, be specified by an azimuth angle ψ and ellipticity angle χ = tan−1 (b/a), where b and a are the minor and major semi-axes of the ellipse (see Fig. 1). The two sets of angles are related by the ellipsometry equations. [7]: tan(2ψ) = tan(2α) cos(δ)

(3)

sin(2χ) = sin(2α) sin(δ).

(4)

When light with a known polarisation state reflects off a surface, in general, the polarisation of the reflected light changes. The principle quantity of ellipsometry is defined as the ratio of X for the incident and reflected lights: ρ≡

tan(αi ) eiδi Xi . = Xr tan(αr ) eiδr

The right-hand side can be re-arranged in terms of more familiar quantities: E˜0p,r E˜0s,i rp Xi · = . = ρ≡ Xr rs E˜0p,i E˜0s,r

(5)

(6)

4


Y

S E os

E

b

−E op

α

χ

X a

ψ

P E op

−Eos Figure 1. Polarisation ellipse can be specified either by the azimuth ψ and ellipticity χ angles, or by the amplitude ratio tan(α) and relative phase δ (not shown).

Where, rp = E˜0p,r /E˜0p,i and rs = E˜0s,r /E˜0s,i are the Fresnel reflection coefficients for the p and s components of the electric field. Connection between the ellipsometry quantities and the sample properties can readily be made by observing that the right hand side of Eq. 6 depends on the properties of the object from which light is reflected. In practice, the ratio ρ = rp /rs is calculated by considering a model for the sample (a structure of different layers on a substrate). The left hand side of Eq. 5, on the other hand, can experimentally be determined by measuring the polarisation state of the reflected light, Xr , for a known polarisation state of the incident light, Xi . The model is then, fitted against the experimental data taking the sample properties, such as layers thicknesses and optical constants, as the fit parameters. The principle quantity of ellipsometry is usually expressed in terms of its amplitude and phase as: rp (7) ρ(˜ nj , dj ) = tan(Ψ) ei∆ = . rs This Eq. explicitly states that ρ is a function of the complex refractive index n ˜ j = nj +iκj and thickness dj of the layers in the sample. 3. Stokes parameters In 1852 Stokes showed that the polarisation state of a monochromatic electromagnetic wave can fully be determined by measuring only four quantities with irradiance (intensity) dimensions. These quantities, known as Stokes parameters, are defined by the following Eqs. [7]: 2 2 S0 = Ep Ep∗ + Es Es∗ = E0p + E0s

(8)

2 2 S1 = Ep Ep∗ − Es Es∗ = E0p − E0s

(9)

S2 =

Ep Es∗

+

S3 =

i(Ep Es∗

Es Ep∗

−

= 2E0p E0s cos(δ)

Es Ep∗ )

= 2E0p E0s sin(δ).

(10) (11)


5

The physical interpretation of the Stokes parameters is evident from Eqs. 8–11; S0 is simply proportional to the total irradiance of the light. S1 represents the difference between the irradiance of the p and s components of the light field. S2 measures the difference between the field components along a polariser with its transmission axis at 45 and 135 degrees with respect to the p axis. The last parameter, S3 , is the difference between intensities of the right and left circular polarisation components of the light field. It can be shown [7] that for a partially polarised light, S02 ≥ S12 + S22 + S32 , which turns to equality for a completely polarised state of light. In practice, these parameters can easily be measured using their physical interpretations as follows [7]: S0 = I(0, 0) + I(90, 0) = I0

(12)

S1 = I(0, 0) − I(90, 0) = I0 cos(2α)

(13)

S2 = I(45, 0) − I(135, 0)

(14)

= I0 sin(2α) cos(δ)

(15)

S3 = I(45, 90) − I(135, 90) = I0 sin(2α) sin(δ).

(16) (17)

In these Eqs. I(β, φ) is the intensity measured after light passes through a linear analyser at angle β (with respect to the p direction), and a wave retarder that introduces a relative phase shift φ between the p and s components. The right sides of these equations can be obtained by substituting E0p = E0 cos(α), E0s = E0 sin(α), and I0 = E02 in Eqs. 8–11. Therefore, the ellipsometry variables α and δ can be obtained from the measured Stokes parameters using the following simple equations: S1 π 1 (18) α = cos−1 ( ) , 0 ≤ α ≤ 2 S0 2 S3 δ = tan−1 ( ) , 0 ≤ δ ≤ 2π. (19) S2 4. Experimental work To demonstrate the capability of the Stokes ellipsometry in thin film characterisation, we prepared three different thin film systems: a thermal silica, a magnesium fluoride film step on glass substrate (prepared by evaporation for another experiment), and a cadmium sulfide thin film evaporated also on glass substrate. We carried out our measurements using the experimental setup shown in Fig. 2. The light source is a 5 mW single-mode He-Ne laser with wavelength λ = 632.8 nm polarised normal to the plane of incidence (s direction). In the incident arm, like other common ellipsometry methods [12], we used a linear polariser P with its transmission axis aligned at 45 degrees with respect to the plane of incidence. The linearly polarised light is then incident on the sample at angle θ, and the specularly reflected beam is analysed by a quarter-wave plate, C followed by a linear analyser, A to determine its polarisation state. The reflected light intensities required to measure S0 , S1 , and S2 can all be determined simply by setting the analyser’s axis at 0, 45, 90, and 135 degrees with respect to the plane of incidence.


6

Detector A

P C

θ

S

(a)

(b) Figure 2. (a) Multi angle of incidence Stokes ellipsometry setup: a He-Ne laser beam transmitted through a linear polariser P with its transmission axis at 45 degree with respect to the plane of incidence, is incident on the sample S at angle θ. The polarisation state of the specularly reflected beam is determined by measuring the Stokes parameters using the combination of a quarter-wave plate C, a linear analyser A, and a photodiode detector. (b) a real photograph of the experimental setup.


7

The quarter-wave plate is only needed to introduce a π/2 phase shift required to measure S3 . However, to take into account the plate’s absorption, as suggested in ref. [23], in all of our measurements we keep this plate in place (after the analyser). The back side of the glass substrates were made diffusive and colored black to minimise the stray light reflection. The reflected light intensity is measured by a home-made detector consisting of a silicon photodiode equipped with a current-to-voltage converter (built as part of the project by students) and a digital voltmeter. Since, only relative intensities are needed to calculate the Stokes parameters, the detector only needs to output a voltage proportional to the incident light intensity. This was confirmed by a simple test using the Malus’ law [24]. In order to reduce the error due to the input light fluctuations, the laser was turned on at least two hours before the measurements started. Samples were mounted on an educational rotating goniometer with 0.1 degree resolution, and data were collected at several angles of incidence. To estimate the statistical uncertainties, measurements were repeated five times for each sample. The averaged Stokes parameters obtained from these data were used in the fitting process, and their maximum variations were taken as uncertainties. 5. Data analysis As mentioned in sec. 1, data analysis is an important part of every ellipsometry analysis. The procedure usually includes a computer model where sample characteristics (optical constants, film thickness, etc.) are regarded as free parameters, and optimum values for these parameters are obtained by fitting the experimental data to the model. The main ideas can be understood by considering a simple model consisting of a thin layer on top of an infinitely thick substrate with sharp interface between them. It can be shown that when a linearly polarised and monochromatic light wave with wavelength λ is incident on a single-layer system composed of a flat layer of thickness d and index of refraction n on top of an infinitely thick substrate, the Fresnel amplitude reflection coefficient of the system can be calculated using the following equation: rj =

r01,j + r12,j e−2iφ , 1 + r01,j r12,j e−2iφ

(20)

where j = s, p stands for the polarisation direction of the incident light, φ = 2πndcos(θ)/λ is the phase shift between the light reflected from the air-layer and layersubstrate interfaces introduced by traversing the layer [θ is the angle of incidence], and r01 and r12 are the Fresnel amplitude reflection coefficients at these interfaces, respectively. We don’t take into account effects such as surface roughness, porosity, and interfacial regions in our model, as they make the analysis more complicated and introduce extra fit parameters. Although, this simplified model may introduce potential systematic errors into the final results, the main purpose of this study, i.e. learning the ellipsometry data analysis is still fulfilled. An essential element of any commercial ellipsometer is a software that is capable of producing appropriate theoretical models with the desired number of layers and as many parameters as required for the physical


8

system under study. The fitting algorithm also plays a crucial role in obtaining meaningful results. In our study we developed the ellipsometry code ourselves. The code has three main constituents: (i) a ’model’ program to model the sample, (ii) a ’stokes’ program to calculate Stokes parameters from the data, (iii) a ’fit’ program to to fit the data to the model. The ’model’ program takes the film thickness d and its (complex) index of refraction n, the index of refraction of the substrate ns , the polarisation state, and the angle of incidence θ, as its input variables. The function then calculates the reflection coefficients r01 and r12 for the specified polarisation state using the Fresnel equations [24], and then returns the combined amplitude reflection rj using eq. 20. The theoretical value of the fundamental ellipsometry quantity, ρc , can easily be calculated from eq. 6. The ’stokes’ program takes the four experimental relative intensities as its input data, and returns the Stokes parameters using eqs. 12. The angle αr and phase δr for the reflected light are then calculated from eqs. 18. Since the incident light is linearly polarised at 45◦ with respect to the plane of incidence, we have Xi = 1. The experimental value for the fundamental ellipsometry quantity, ρm , is then obtained from eq. 5. In the ’fit’ program we use the least squares method based on the Marquet–Levenberge algorithm to minimise the following quantity: C 2 = (Ψm − Ψc )2 + (∆m − ∆c )2 ,

(21)

where, Ψm and ∆m are the amplitude and phase of ρm , and Ψc and ∆c are those of ρc [see eq. (7)]. All required computer programs were written in the Yorick interpreting language [25], but any high-level programming language can be used to develop similar codes. It is also possible (and recommended) to generalise the code by using the matrix method [24] to calculate the Fresnel reflection coefficients, rp and rs . This way, it will be possible to analyse a multilayer system with complex refractive indices and different thicknesses. Therefore, in principle, it is possible to add an interfacial layer or surface roughness using the effective-medium approximation [12]. This extension would, probably, fit into a graduate-level physics project. 6. Results Fit results for the three samples are shown Fig. 3; thermal silica (top), magnesium fluoride on glass (middle), and cadmium sulfide on glass substrate (bottom). The optical constants and film thicknesses obtained from the fits for these samples are summarised in table 1. The magnesium fluoride layer had a step-shape profile covering only half of the underlying glass substrate. So, we were able to confirm its thickness measurement with a stylus profilometer (DekTac Surface) available in our university. Since this sample was prepared by evaporation, it is likely that the layer’s thickness is not uniform all over the step. The profilometer measures the step height at a single point, whereas in

9

Thin film characterisation with a simple Stokes ellipsometer (a)

(b) ∆ (degree)

Ψ (degree)

−20 40

35

−40

−60 30 30

40

50

60

30

θ (degree)

50

60

(a)

(b) ∆ (degree)

Ψ (degree)

40

30

20

10

−50

−100

−150

20

40

60

20

θ (degree) 40

40

60

θ (degree) (a)

(b) ∆ (degree)

Ψ (degree)

40

θ (degree)

30

20

10

−50

−100

−150 40

60

40

60

θ (degree)

θ (degree)

Figure 3. Ellipsometry fit results for: thermal silica (top), magnesium fluoride on glass (middle), and cadmium sulfide on glass substrate (bottom). Squares are the experimental data, and solid curves are the fit results.

Table 1. Fit results for the film thicknesses and optical constants of the three samples. The second lines for each sample show measurements by other methods or reported values.

Sample

d(nm)

n

k

SiO2 /Si

83.1 ± 0.2 –

1.348 ± 0.004 1.45–1.47 [26]

0.0 0.0

MgF2 /glass

90.7 ± 0.03 156.0a

1.378 ± 0.003 1.36–1.38 [27]

0.0005 0.0

CdS/glass

109.0 ± 0.5 –

1.902 ± 0.003 2.416 [27]

< 1 × 10−4 0.17

a

stylus profilometer (DekTac Surface)


10

ellipsometry the average hight along the step is measured. This might be the reason for the discrepancy between the two results. The two other samples did not have this feature, and were not suitable for this measurement. All refractive indices obtained were compared with the existing values in the literature. Where applicable, independent values are also listed in table 1 for comparison. As shown in the table, the uncertainties in the fit parameters are at most about 0.3%. However, as mentioned before, since this analysis is model-based, potential systematic errors may occur. 7. Discussion We have demonstrated a simple Stokes ellipsometer with the experimental setup and analysis software all developed as a graduate physics project. The thickness and refractive index of three different thin film systems were measured with reasonable agreement with the existing values in other studies. There is no real restriction on applying this technique to study other samples with a complex refractive index (metallic or semiconductor, for example). The main advantage of the Stokes method, compared to other conventional ellipsometry techniques, is that in this method only the relative light intensities at four configurations of the analyser are necessary to fully determine the polarisation state of the reflected light. Since our setup uses modest optical and mechanical equipments, it is possible for students in an educational optics lab at graduate or advanced undergraduate level to use this setup for a hands-on experiment and learn the basic principles of ellipsometry experiments and data analysis. Our study suggests that Stokes ellipsometry can effectively be used as an accurate thin film characterisation method in educational and research labs with limited resources and equipments. 8. Reference list [1] R. M. A. Azzam and N. M. Bashara. Ellipsometry and Polarized Light. North-Holland, Amsterdam, 1987. [2] Ulrich Wurstbauer, Christian R¨oling, Ursula Wurstbauer, Werner Wegscheider, Matthias Vaupel, Peter H Thiesen, and Weiss Dieter. Imaging ellipsometry of graphene. Applied Physics Letter, 97:231901, 2010. [3] W. E. J. Neal and S. J. Petraitis. An ellipsometer for student experiments. European Journal of Physics, 2(2):69, 1981. [4] W. E. J. Neal and R. W. Fane. Ellipsometry and its applications to surface examination. Journal of Physics E, 6(5):409, 1973. [5] L. L´evesque. Refractive index determination of materials on thin transparent substrates using ellipsometry. Physics Education, 35(5):359, 2000. [6] E. Salik. Quantitative investigation of fresnel reflection coefficients by polarimetry. Am. J. Phys., 80(3):216, 2012. [7] Max Born and Emil Wolf. Principles of Optics. Pergamon Press, Oxford, 7 edition, 2007. [8] L. Bakshi, S. Eliezer, G. Appelbaum, N. Nissim, L. Perelmutter, and M. Mond. A full stokes vector ellipsometry measurement system for in situ diagnostics in dynamic experiments. Rev. Sci. Inst., 83:053904, 2012.


11

[9] Yu-Lung Lo, Thi-Thu-Hien Pham, and Po-Chun Chen. Characterization on five effective parameters of anisotropic optical material using stokes parameters—demonstration by a fibertype polarimeter. Opt. Express, 18(9):9133–9150, 2010. [10] John D. Perreault. Triple wollaston-prism complete-stokes imaging polarimeter. Opt. Lett., 38(19):3874–3877, Oct 2013. [11] M. P. Gorsky, L. Y. Kushneryk, L. Y. Tryphonyuk, and M. Sidor. Fourier stokes polarimetry of laser radiation scattered fields for diagnostics of dystrophic changes of biological tissues histological sections. Appl. Opt., 51(10):C170–C175, 2012. [12] R. M. A. Azzam. Ellipsometry. In Michael Bass, editor, Handbook of Optics, volume 2, chapter 16, pages 16.1–16.25. McGraw Hill, 2007. [13] R. M. A. Azzam. Polarization optics of interfaces and thin films. Physica Status Solidi (a), 205(4):709–714, 2008. [14] R. J. King and S. P. Talim. A comparison of thin film measurement by guided waves ellipsometry and reflectometry. Optica Acta, 28(8):1107–1123, 1981. [15] T. V. Vorburger and K. C. Ludema. Ellipsometry of rough surfaces. Applied Optics, 19(4):561– 573, 1980. [16] Shankar Krishnan. Mueller-matrix ellipsometry on electroformed rough surfaces. Journal of Modern Optics, 42(8):1695–1706, 1995. [17] Hiroyuki Fujiwara. Spectroscopic Ellipsometry Principles and Applications. John Wiley and Sons, Ltd, 2007. [18] Kate Kaminska, Aram Amassian, Ludvik Martinu, and Kevin Robbie. Growth of vacuum evaporated ultraporous silicon studied with spectroscopic ellipsometry and scanning electron microscopy. Journal of Applied Physics, 97:013511–8, 2005. [19] Chunxiao Yue, Zuyao Sun, Lanfang Yao, and Kaiming Jiang. Spectroscopic ellipsometry characterization of optical properties for ti-doped sio2 mesoporous films. Integrated Ferroelectrics, 127:15–20, 2011. [20] Leif A. A Pettersson, Lars Hultman, and H Arwin. Porosity depth profiling of thin porous silicon layers by use of variable-angle spectroscopic ellipsometry: a porosity graded-layer model. Applied Optics, 37(19):4130–4136, 1998. [21] A. Marino, G. Abbate, V. Tkachenko, I. Rea, L. De Stefano, and M. Giocondo. Ellipsometric study of liquid crystal infiltrated porous silicon. Molecular Crystals and Liquid Crystals, 465(1):359– 370, 2007. [22] T. W. H. Oates, L. Ryves, and M. M. M. Bilek. Dielectric functions of a growing silver film determined using dynamic in situ spectroscopic ellipsometry. Optics Express, 16:2302–2314, 2008. [23] D. Goldstein. Polarized Light: Fundamentals and applications. Marcel Decker (New York), 2nd edition, 2003. [24] L. Pedrotti. Introduction to Optics. Pergamon Press, Oxford, 3 edition, 1993. [25] http://yorick.sourceforge.net/. [26] R. J. archer. Determination of the properties of films on silicon by the method of ellipsometry. J. Opt. Soc. Am., 52(9):970–977, 1962. [27] H. J. Pulker. Coating on glass. Elsevier, 2 edition, 1999.