Instruction to the Ellipsometry

Materialkundliches Praktikum MP13 Instruction to the Ellipsometry

Chair of Materials Science (CMS) Otto Schott Institute for Materials Research Faculty of Physics and Astronomy Friedrich Schiller University Jena Contents

1 Introduction 1

2 Tasks 1

3 Basic Principles of Ellipsometry 2 3.1 Jones Vector and Jones Matrix ...... 2 3.2 The Ellipsometry Setup ...... 5 3.3 The Ellipsometry Equation ...... 6

4 Reflection and Refraction at the Interface 8 4.1 Reflection Coefficient for p-Polarized Light ...... 9 4.2 Reflection Coefficient for s-Polarized Light ...... 9 4.3 Complex Reflection Coefficient ...... 10

5 Basic Principle of Spin-coating 10

6 Experimental Details 12 6.1 Fabrication of Thin Films via Spin-Coating Method ...... 12 6.2 Measurement and Data Fitting ...... 12

7 Questions for Preparation 13

Reference 14

This document is for private use by students of the Friedrich Schiller University Jena (termed FSU in the following) only. You should not alter, modify, copy, or otherwise distribute this document to people outside the FSU. No liability whatsoever for any damages incurred by you resulting from errors in or omissions from the information included herein is assumed by the authors. They do not assume any liability for infringement of patents, copyrights, or other intellectual property rights of third parties by or arising from the use of the document or the technical information described in this document. 2 Tasks 1

1 Introduction

The Ellipsometry provides a noninvasive technique for measuring the film thickness, characterizing the micro morphology of a film and its optical con- stants. Due to the high accuracy and the contact free measurement the ellipsometer is particularly suitable for detecting the ultra thin film of polymer, metal and semiconductor. A standard ellipsometry consist of five parts: a laser source, a polarizer, a compensator(quarter wave plate), an analyzer and a detector. The linear polarized light going through the polarizer is incident on the sample and its polarized state will be changed by the light-matter interaction. The change of polarized state, which contains the information of thickness d and the properties like refractive index N , can be determine by calculating the ellipsometry parameters Ψ and Δ through the angle between the incident plain and the axis of polarizer, compensator and analyzer. In order to determine d and N, normally an optical model of the film will be built, and a series of incident angle are also required to give experimental sampling of Ψexp and Δexp. After the numerical calculation from some algo- rithm the data can be fit and d and N can be obtained. In this experiment we prepare the sample on the silicon wafer via spin-coating and dip-coating method, and then observe and measure the optical constant of the sample via ellipsometry and data fitting. This lab course aims to study and understand the principles of ellipsometry and the interaction of polymer films and polarized light.

2 Tasks

1. Prepare 3 samples of polystyrene ﬁlm with diﬀerent thicknesses via spin-coating.

2. Look-up the literature value of the refractive index n for polystyrene.

3. Obtain the experimental sampling of Ψexp and Δexp via ellipsometry, use the existing model, to optimize the thickness d and the refractive index n for all samples.

4. Estimate a value for α in the spin-coating equation (Eq. 18).

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 3 Basic Principles of Ellipsometry 2

3 Basic Principles of Ellipsometry

3.1 Jones Vector and Jones Matrix The Jones Vector is a The ellipsometry is based on the change of the polarization of the laser beam when it is reﬂected by the interface of the medium. In this experiment the laser beam can be considered as a monochromatic plane wave, therefore the electric ﬁeld of the light can be described as [1]:

i k·r−ωt+δ E(r, t)=E0e ( ), (1) where k is the wave vector, E0 is the amplitude, ω is the circle frequency. When we have totally polarized light, we can also use the Jones vector to describe the electric ﬁeld propagating in z direction, which is decomposed into two orthogonal components, i.e. p and s polarized components: E , eiδp E z,t 0 p ei(kz−ωt), ( )= iδs (2) E0,s e where the vector in Eq. 2 is the Jones vector. The direction of p and s components are shown in the coordinate system of the laser beam. See Fig. 1.

Figure 1: The coordinate system of the laser beam (p, s, z) and the coordinate system of the sample (X, Y, Z), taken from [2].

Depending on the phase diﬀerence and the ratio of the amplitude of the p and s components the polarization states can be grouped into diﬀerent categories, see Tab. 1.

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 3 Basic Principles of Ellipsometry 3

Table 1: The diﬀerent polarization states.

polarization phase diﬀerence E0,p = E0,s E0,p = E0,s P

S δp − δs =0

Linear

π 0 <δp − δs < S 2

Elliptical

P P

π δp − δs = S S 2

Elliptical Circular

π <δp − δs <π S 2

Elliptical

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 3 Basic Principles of Ellipsometry 4

Table 1: The diﬀerent polarization states.

polarization phase diﬀerence E0,p = E0,s E0,p = E0,s P

S δp − δs = π

Linear

3π π<δp − δs < S 2

Elliptical

P P

3π δp − δs = S S 2

Elliptical Circular

3π <δp − δs < 2π S 2

Elliptical

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 3 Basic Principles of Ellipsometry 5

The optical components of the ellipsometry can be also described with Jones matrices (see Eq. 3 - 5), whereby a Jones matrix describes the change in polarization by various optical components: ⎛ ⎞ ⎝10⎠ polarizer: Tˆp = (3) 00 ⎛ ⎞ ⎝10⎠ quarterwaveplate: Tˆc = (4) 0 −i ⎛ ⎞ ⎝rp 0 ⎠ sample: TˆS = (5) 0 rs The polarization state after the polarization active optical component can be described by a multiplication of the Jones matrix for the respective component with the original Jones vector J of the beam before it passed the optical component. Jout = TˆJin (6)

3.2 The Ellipsometry Setup The setup of an null-ellipsometry is shown in Fig.2. The laser beam is incident on the surface of the sample with an angle of ϕ0. When linear polarized light with an axis pointing somewhere but not along s or p direction is incident on the sample, the reflected light will in general exhibit an elliptical state of polarization. The other way around, the same elliptical state of polarization (but with reversed sense of rotation) incident on the surface will generate a linear polarized reflection. More general, using the polarizer-compensator combination we can always find an ellipse that produces an exactly linear polarized reflection (if the sample is not depolarizing). This has a nice consequence: we can easily detect this particular state by using a second polarizer as an analyzer in the reflected beam. For a linear polarized beam it is possible to extinguish the beam by setting the analyzer toa90◦ position with respect to the axis of the linear polarization. Doing this is called ”finding the Null” or ”Nulling”. In practice this is equivalent to finding a minimum in the signal of a photo detector. Now the question is, how can we describe the change of the polarized state caused by the sample? The answer is using the elliptical angle Ψ and

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 3 Basic Principles of Ellipsometry 6

Figure 2: The setup of a null-ellipsometry. A laser beam is incident on the surface of the sample with an angle of ϕ0. When the beam has a suitable elliptical state of polarization (but with reversed sense of rotation) incident on the surface, a linear polarized reﬂection will be generated. If the linear polarized direction is perpendicular to the axis of the compensator, then the intensity of the light incident on the detector is 0. Taken from [2].

Δ, which can be obtained by solving the ellipsometry equation. (see section 3.3, Eq. 9)

3.3 The Ellipsometry Equation The ellipsometry equation is calculated by using the Jones vector and matrix of all the respective components, but they have to be transformed into the same coordinate system before we can do the matrix multiplication steps. Here we choose the lab coordinate system (see Fig. 1). The rotation matrix Rˆ can be used for the transformation: ⎛ ⎞ cos θ − sin θ Rˆ(θ):=⎝ ⎠ (7) sin θ cos θ

Now it is possible to rewrite the optical components introduced in section 3.1 in the lab coordinate system. The expressions for the polarizer, compensator, analyzer and sample (which is already in the lab coordinate system) read now [3]:

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 3 Basic Principles of Ellipsometry 7

⎧ ⎪ ⎪ polarizer: Rˆ(αP )TˆP Rˆ(−αP ) ⎪ ⎨⎪ compensator: Rˆ(αC )TˆC Rˆ(−αC ) (8) ⎪ ⎪ analyzer: Rˆ(αA)TˆARˆ(−αA) ⎪ ⎩⎪ sample: Rˆ(0)TˆSRˆ(0)

where αP , αC , αA are the angles between the incident plane and the axis of polarizer, compensator and analyzer, respectively, and TˆA, TˆC , TˆP , TˆS refer to the original Jones matrix of polarizer, compensator and analyzer, respectively. Now the ellipsometry equation can be composed by describing the change in the polarization state of a laser beam (Ein) going through the polarizer, compensator, interacting and reﬂected by the sample, and ultimately passing through the analyzer as an outgoing beam (Eout) [3]:

Eout = Rˆ(αA)TˆARˆ(−αA) TˆS Rˆ(αC )TˆC Rˆ(−αC ) ⎛ ⎞ ⎝EP ⎠ Rˆ(αP )TˆP Rˆ(−αP ) Ein, with Ein = (9) ES

After some algebra calculation the expression of Eout can be derived as follows: ⎛ ⎞ ⎝1⎠ Eout = (Ω1 +Ω2)(EP cos αP + ES sin αP ) (10) 0

with Ω1 = rP cos αA[cos αC cos(αC − αP ) − i sin αC sin(αC − αP )] (11) Ω2 = rS cos αA[sin αC cos(αC − αP ) − i cos αC sin(αC − αP )]

In the Nulling case, the intensity of Eout is 0, that means: ⎛ ⎞ ⎝1⎠ Eout = (Ω1 +Ω2)(EP cos αP + ES sin αP ) = 0 (12) 0

rP tan αC − i tan(αP − αC ) i.e. = − tan αA (13) rS 1 − i tan αC tan(αP − αC )

◦ ◦ Normally, we set αc =45 or −45 , therefore according to Eq.13 we arrive at the following conclusion:

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 4 Reﬂection and Refraction at the Interface 8

π r i(−2αP + ) α ◦ P − α e 2 for C =45 , r = tan A S π i(2α + ) ◦ rP P αC =−45 , = + tan αAe 2 rS

That means if the value of αP ,αC ,αA are already known, we can calculate the experimental elliptical angle Ψexp and Δexp which are deﬁned, as follows: r P iΔexp = ∓ tan Ψexpe (14) rS ⎧ π ⎪ ◦ ⎨ −2αP + αC =45 with Ψexp = |αA|, Δexp = π2 ⎪ ◦ ⎩ +2αP + αC = −45 2 4 Reﬂection and Refraction at the Interface

An ideal sample for the measurement can be a film(layer) with a homogeneous thick and smooth plane surface. We assume that the thickness and the refractive index of the film is d1 and n1, respectively, and the film is on a half unlimited thick substrate, which is shown in Fig. 3. If a laser beam is incident on the surface of the sample with an angle of ϕ0, the intensity of the reflected light can be calculated with the help of the Fresnel equation [1, 3].

Figure 3: Single layer model. Taken from [2].

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 4 Reﬂection and Refraction at the Interface 9

4.1 Reflection Coefficient for p-Polarized Light r According to the Fresnel equation the reflection coefficient 01P on the air-film interface is [1]: n ϕ − n ϕ r 1 cos 0 0 cos 1 01P = n1 cos ϕ0 + n0 cos ϕ1 r The reflection coefficient 12P on the film-substrate interface is : n ϕ − n ϕ r 2 cos 1 1 cos 2 12P = n2 cos ϕ1 + n1 cos ϕ2

As shown in Fig. 3, a series of reflection exists between the interface of air- film and film-substrate, all with different phase shifts. The total intensity of the reflected p-polarized light is the superposition of them. When applying the summation, the Snell’s law and trigonometric identities the following result can be obtained: r r e−2iβ r 01P + 12P P = r r e−2iβ (15) 1+ 01P 12P d β π 1 n2 − n 2 ϕ with =2 λ 1 0 sin 0

4.2 Reflection Coefficient for s-Polarized Light r Analogous to section 4.1 the reflection coefficient 01S on the air-film interface is : n ϕ − n ϕ r 0 cos 0 1 cos 1 01S = n0 cos ϕ0 + n1 cos ϕ1

r The reflection coefficient 12S on the film-substrate interface is : n ϕ − n ϕ r 1 cos 1 2 cos 2 12S = n1 cos ϕ1 + n2 cos ϕ2

The total intensity of the reﬂected s-polarized light is: r r e−2iβ r 01S + 12S S = r r e−2iβ (16) 1+ 01S 12S

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 5 Basic Principle of Spin-coating 10

4.3 Complex Reﬂection Coeﬃcient

Using Eqs. 15 and 16 we can calculate the quotient rp/rS in analogy to Eq. 14 in section 3.3: r r e−2iβ 01P + 12P −2iβ rP 1+r01 r12 e P P eiΔsim r = r r e−2iβ := tan Ψsim (17) S 01S + 12S r r e−2iβ 1+ 01S 12S

But this time the variables are ϕ0 , n0, n1 and d1. ϕ0, n0 are always known in the measurement, only n1 and d1 are unknown, sometimes even only d is unknown. Each quotient rp/rs determined by n1 and d1 corresponds to a pair of simulated curves Ψsim(ϕ0) and Δsim(ϕ0). The sampling point Ψexp and Δexp can be obtained by measuring αP ,αC ,αA, so we just need to vary n1 and d1 until the simulated curves Ψsim(ϕ0) and Δsim(ϕ0) match the sampling point Ψexp and Δexp, and then the best value of n1 and d1 are obtained. Therefore, it is extremely helpful in ellipsometry, when you can already estimate the values you want to determine precisely beforehand.

5 Basic Principle of Spin-coating

Spin-coating is widely employed for the highly reproducible fabrication of thin film coatings over large areas with high structural uniformity. In the spin-coating process a viscous fluid is at first deposited on a horizontal ro- tating disc and produces a uniform liquid film. Then the disc is rapidly accelerated to a high angular velocity (spin speed). The adhesive forces at the liquid/substrate interface and the centrifugal forces acting on the rotat- ing liquid result in strong sheering of the liquid which causes a radial flow in which most of the polymer solution is rapidly ejected from the disc (see Fig. 4 )

This process combined with subsequent evaporation of the liquid causes the thickness of the remaining liquid film to decrease. For a solution, e.g. a polymer solution, the evaporation process causes the polymer concentration to increase (and thus the viscosity) at the liquid/vapor interface, i.e. a concentration gradient is formed through the liquid film, which, after evaporation of most of the remaining solvent, consequently results in the formation of a uniform practically solid polymer film. The spin-coating process is complex in nature due to the many mechanisms involved, but we can use some ana-

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 5 Basic Principle of Spin-coating 11

Figure 4: The diﬀerent ”stages” of spin coating. a) Dispensation (not modeled). b) Acceleration (not modeled). c) Flow dominated. d) Evaporation dominated. Taken from [4]. lytic model to show, that the thickness h of the ﬁlm depends on the viscosity η of the solution and the angular velocity ω of the disc [5]:

1 −α h ∝ η 3 ω (18)

The exponent α varies between 1/2 and 2/3 , this depends on which mathe- matics model we use to describe the evaporation of the solvent. Besides, the absolute thickness of the ﬁlm is also determined by the concentration of the solution, the molar mass of the polymer, the temperature, and the interfacial energy of substrate. Especially the acceleration of the spin coat should be large enough, so that the maximum angular velocity is reached before the solvent evaporates too much. In this experiment we use polystyrene dissolved in p-Xylene.

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 6 Experimental Details 12

6 Experimental Details

6.1 Fabrication of Thin Films via Spin-Coating Method (a) Prepare 1 wt.% polystyrene solution in p-Xylene (stir for 1 hour at temperature of 60◦).

(b) Load the polished silicon wafers (about 1 cm×1 cm) on the spin-coater, drop the solution carefully onto the center of the wafer, afterward spin for 2 min. See (c) for the speeds to use.

6.2 Measurement and Data Fitting

(a) Measure Δexp and Ψexp of the samples via ellipsometry and the data-ﬁt program. Please follow the manual of ellipsometry.

(b) Use the standard optical model, which has already been implemented to determine the thickness and the refractive index of the samples, by simulating the corresponding Δsim and Ψsim.

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 7 Questions for Practice 13

7 Questions for Practice

1. Explain the Fresnel equation, Brewster angle and (un)polarized light.

2. How to observe the stress distribution in the polymer by using the principle of photo elasticity? Why can we see diﬀerent colors when we use white polarized light?

3. In which condition can the dark/bright Maltese cross be seen on the Spherulite? What is the position and the direction of the dark/bright Maltese cross according to the devices and what is the principle for the concentric circles on the Spherulite? Why?

4. What is the polarized state of a out-coupling laser beam from a normal optical resonator without any polarization ﬁlter or Brewster window?

5. Additional question Please give an explanation for question 4. You can begin from the eigenstate of a photon and ultimately consider the statistical polarized state of the laser beam in the optical resonator.

6. What is the principle of ellipsometry? Draw a sketch for an Ellipsome- try and explain the function of each component. Explain the principle of data ﬁtting.

7. Additional question Draw a sketch for the ellipsometry and use the Jones vector or matrix to describe the light through the polarizer, Compensator, Sample and Analyzer, and set up an equation about the PCSA System ( the System consists of Polarizer, Compensator, Sample, Analyzer).What is the physical meaning for each parameter? In which condition can we use the Jones vector and Jones matrix ? Can we still measure the thickness of a sample if the sample is anisotropic? If the answer is yes, how to operate?

8. What is the advantage for setting an angle of 45◦ between the plane of incidence and the fast axis of the compensator? What is the physical meaning for that?

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015 14

References

[1] M. Born, E. Wolf, A.B. Bhatia, P.C. Clemmow, D. Gabor, A.R. Stokes, A.M. Taylor, P.A. Wayman, and W.L. Wilcock. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diﬀraction of Light. Cambridge University Press, 1999.

[2] Physikalisches Praktikum f¨ur Fortgeschrittene : Ellipsometrie. 2002. http://www.uni-potsdam.de/u/physik/fprakti/ANLEIF9.pdf.

[3] H.G. Tompkins and E.A. Irene. Handbook of Ellipsometry. William An- drew Pub., 2005.

[4] S.L. Hellstrom. Basic Models of Spin Coating. pages 1–6, 2007. http://large.stanford.edu/courses/2007/ph210/hellstrom1/.

[5] David B. Hall, Patrick Underhill, and John M. Torkelson. Spin coating of thin and ultrathin polymer ﬁlms. Polym. Eng. Sci., 38(12):2039–2045, December 1998.

Bojia He, BSc.; Dipl.-Phys. Matthias Arras April 2015