Extraction of complex refractive index of absorbing films from ellipsometry measurement

Thin Solid Films 520 (2012) 5568–5574

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Extraction of complex refractive index of absorbing films from ellipsometry measurement Mickaël Gilliot Université de Reims Champagne-Ardenne, GReSPI, BP 1039, 51687 Reims Cedex 2, France

a r t i c l e

i n f o

Article history: Received 11 June 2011 Received in revised form 20 January 2012 Accepted 16 April 2012 Available online 24 April 2012 Keywords: Ellipsometry Data inversion Thin films Absorbing films Optical properties Refractive index Extinction coefficient

a b s t r a c t Numerical extraction of complex refractive index of an unknown absorbing layer inside a multilayer sample from ellipsometry measurement is discussed. The approach of point by point extraction considering all points of spectroscopic data as independent data points is investigated. This problem has typically multiple solutions and the standard method consisting in fitting calculated to experimental point is likely to converge to a wrong solution if a precise guess value is not given. An alternate method is proposed, based on the determination of contours of the ellipsometric function, to provide all solutions in an as extended as wanted range of complex refractive index values. The method is tested through different kinds of sample examples. Errors relative to any of the parameters used in the sample model are calculated and discussed. This method should be helpful in many practical cases of ellipsometry data interpretation. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Optical characterization techniques and especially spectroscopic ellipsometry are favorite techniques for surface and thin film analyses, as attested by its use for decades in semiconductor industry and the more and more extended field of applications for example to flat panel displays, photovoltaics materials, polymers, organic materials or biotechnology [1,2]. Although instrumentation has inherited much progress, analysis of data is still a delicate task because of the highly non-linear character of optical equations. Only a limited number of cases allowing unambiguous analytical inversion, such as the determination of complex refractive index of the ambient–substrate system and the determination of the layer thickness of the ambient–layer– substrate system provided that all optical constants are known. General process of spectroscopic ellipsometry data interpretation consists in building a parametric representative model of the sample where unknown materials are represented by dispersion laws and fitting generated to experimental data by varying some of the parameters. However materials cannot always be represented by theoretical laws. Fitting procedure furthermore requires the knowledge of estimates values of the parameters as a starting point. Alternatively numerical procedures referred as point by point methods can be used to extract some parameters of interest from each point of the spectrum considered as an independent data point. Numerical procedures have been developed to extract refractive index and thickness of an unknown transparent layer

E-mail address: [email protected]. 0040-6090/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2012.04.047

on known substrate [3,4] but the problem of direct data inversion is more complicated when the layer is absorbing. This paper is focused on point by point inversion processes. A widespread point by point procedure concerning simultaneous determination of refractive index and extinction coefficient (and possibly thickness) is to consider them as variables to fit ellipsometric function to experimental data for each point of the spectrum. Interesting additional works have been published to extract point by point refractive index, extinction coefficient and possible thickness at the same time by computing solution curves when one parameter is varied [5–8] and intersecting such solution curves for multiple measurements such as multiple angle of incidence data or multiple thicknesses data of a growing sample [9–11]. Other interesting works use spectroscopic data to extract at the same time thicknesses of different films inside a stack by extracting spectra of unknown complex refractive index when energy-independent parameters are varied. The solution is then selected based on smoothness criteria of the spectral dielectric function [12,13]. Direct fitting processes as well as computation of parametric solution curves however often require the knowledge of guess values. In this paper an analysis of the problem of point by point extraction of complex refractive index of a layer of known thickness in an arbitrary stack of layers from ellipsometry data is performed. This inversion case is of particular interest because a large number of materials are transparent over some spectral range, especially semiconductors for energy below the band-gap, where thickness can be accurately determined by other numerical techniques [3,4,14,15]. The difficulty is that there are typically multiple solution couples of the refractive index and extinction coefficient for a given set of sample parameters and

M. Gilliot / Thin Solid Films 520 (2012) 5568–5574

experimental data point. Inversion processes then require a good knowledge of guess value or reduced searching range, otherwise they could fall into local minima and provide wrong solutions. When materials is completely unknown, and knowledge of guess too approximate, then a large range of values must be considered and all solutions within this range must be considered. A method is proposed in this paper, based on the determination of contours of the ellipsometric function, to return all solutions in a chosen range (Section 3). The interest of this method is illustrated through different examples having multiple solutions for which the result returned by a standard minimization inversion process is skipped to a wrong solution, depending on the guess value (Section 4). The proposed method does not require guess values and returns the set of all solutions within the considered range but additional work is necessary to choose the good one. The choice of a solution, which can be done for example thanks to well-known techniques of using complementary measurement or using spectral considerations, is discussed (Section 5). Errors related to input parameters of this inversion process are also presented (Section 6). 2. Background Ellipsometry measures the change of polarization state between incident and reflected light on a sample. The ellipsometric angles Ψ and Δ are related to Fresnel coefficients of the samples rp and rs, respectively for p-polarized (parallel to the plane of incidence) and s-polarized lights (perpendicular to the plane of incidence) by rp iΔ ¼ tanΨ e : rs

ρ¼

ð1Þ

The transformation of polarization is the result of different phenomena such as interferences due to the thin film structure of the sample, optical properties of the medium and interfaces. It also depends on photon energy and angle of incidence. For a stack of an arbitrary number of layers on substrate, the ellipsometric response can be calculated by standard well-known matrix formulation [16,17]. Let's consider a stack of m layers, each of them referred by the subscript j and characterized by its thickness dj and complex refractive index Nj, between an ambient medium referred by the index 0 and a substrate referred by the subscript m + 1. A scattering matrix is calculated for each of p and s linear polarizations by: S

p;s

p;s p;s p;s p;s

p;s

p;s

p;s

¼ I01 L1 I12 L2 …Iðj−1Þj Lj …Lm Imðmþ1Þ ;

ð2Þ

p, s p, s are interface matrix between media j − 1 and j where I(j − 1)j and Lj and propagation matrix inside medium j for p and s polarizations. These two matrices are expressed as:

p;s Iðj−1Þj

¼

1

r p;s t p;s ðj−1Þj ðj−1Þj

 jβ Lj ¼ e 0 β¼

"

1

0

−jβ

e

# r ðp;s j−1Þj ; 1

 ;

2πdj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N2j −N20 sin2 ϕ0 ; λ

ð3Þ

ð4Þ

ð5Þ

where ϕ0 and λ are the angle of incidence and wavelength of light on p, s p, s the sample, and r(j − 1)j and t(j − 1)j are reflection and transmission Fresnel coefficients. Finally the ellipsometric response is obtained by

ρ¼

r p Sp21 Ss11 ¼ : r s Sp11 Ss21s

ð6Þ

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3. Inversion method Assuming the investigated system is represented by a stack of thin homogeneous layers on a substrate which only unknowns are refractive index n and extinction coefficient k of one of the layers (thickness of this layer is supposed to be known), the ellipsometric response can be calculated as a function of n and k. For a single point data or for each point data of a spectroscopic measurement, the problem to be solved is then a problem with two unknowns n and k for two measured parameters: e ρr ðn; kÞ ¼ ρr e ρi ðn; kÞ ¼ ρi

ð7Þ

where ρre and ρie represent real and imaginary parts of the ellipsometric ratio of the experimental data point and ρr(n, k) and ρi(n, k) represent the calculated real and imaginary parts of the ellipsometric ratio. The typical approach for solving problems of this type is to do a least-squares. Such minimization approach has been used and discussed in different works [6,18–20] and has become a standard procedure to extract point by point values of n and k for a long time. The minimization operation can be performed using different algorithms, such as Newton–Raphson, downhill simplex or Levenberg–Marquardt [21]. One starts with an estimate for n and k, then iterates this estimate to obtain a better estimate from the previous estimate to minimize the error that results between the calculated and measured values. The difficulty here is that there are typically multiple solutions (n,k) for a given set of sample parameters and given ellipsometric experimental point and hence a possibility that the algorithm converges to a wrong solution if the guess value is not given close enough to the good solution. Because guess values cannot always be given with enough accuracy, a scan of the different possible solutions in an acceptable range of (n, k) values is useful. A method named contours method is proposed in this paper to provide exhaustively all possible solutions in the chosen (n, k) range. The proposed method simply performs calculation of contours in the (n, k) plane of constant real and imaginary parts of the ellipsometric function ρr(n, k) and ρi(n, k) corresponding to constant experimental point values ρre and ρie, and returns intersections of the two contours as all possible solutions of the problem a in the chosen (n, k) range. In our implementation the contours are calculated for both real and imaginary parts of the ellipsometric ratio function (ρr and ρi), but could also be calculated on the ellipsometric angles (Ψ and Δ) with additional attention to the 180°/−180° additional discontinuity of angles that can introduce misleading points. For each component of the function, the (n, k) space is mapped with a chosen resolution, typically 100 × 100 points, and contour points are calculated using a Brent method [21] between two consecutive points of the grid when they are found to bracket the experimental value of the considered function. As the ellipsometric ratio can have singular points when the denominator rs tends to zero because of interference effects for s-polarized light, the contour lines may have sharp features and must be calculated with thinner resolution over a restricted area around these singular points. Finally the intersections between the two resulting sets of contours are calculated to yield the set of solutions to the problem with iterated resolution to improve solutions' accuracy until the difference between computed and experimental values of Ψ and Δ is less than a certain tolerance fixed to be 5 × 10− 4° in the results presented throughout this paper. The contours method shows differences in solving the ellipsometric problem in comparison to the typical least-squares approach. First this method treats independently the two different information associated to the two measured ellipsometric parameters, the solutions are searched for both components, then only the common solutions are accepted, whereas minimization process treat both components in one error function. Second, least-squares minimization can yield a

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M. Gilliot / Thin Solid Films 520 (2012) 5568–5574

point anywhere away from the initial guess, whereas the points of the contours are detected by scanning the points of a grid (with resolution fitted to local details of the function) and calculated using iterative process bounded to this grid points. In brief, the contours method performs a scan of the input (n, k) range and returns all possible solutions in this range. For illustration of the contours method principle, the example case of a silicon nitride (SiN) layer on silicon (Si) substrate for the incident wavelength of 213.8 nm (5.8 eV) is considered. Values of experimental ellipsometric angles for this sample are simulated data at the angle of incidence of 70∘ and are calculated to be Ψe = 29.3606∘ and Δe = 58.7186∘ when the properties of the SiN are n = 2.485, k = 0.092 and d = 100 nm, and the properties of silicon are n = 1.136, k = 3.057. The contours are calculated when the complex refractive index of the SiN layer is considered as unknown. The searching range can be with no restriction as wide as wanted when the materials are completely unknown. Calculations are here performed in the range (0–5) for n and (0–5) for k using a 100 × 100 points initial grid and a 50 times enhanced resolution over the singular points restricted areas. The contours of the ellipsometric function are represented and their intersection locations specified in Fig. 1. As expected the contours have sharp features and especially tightened loci close to the k = 0 area where the singularity peaks are located. Among all intersections, two points are identified as singular points where the real and imaginary part of the ellipsometric function change from minus infinity to plus infinity, acting as asymptotic points where closed contour lines tend to start in one direction and terminate in the opposite direction. Such points are excluded from the contours they should not be mistaken for solutions. The algorithm finds actually six different solutions including the real one, which shows the appropriateness of the contours method, and which also shows the number of possible solutions and the usefulness of such a scanning process when good estimates cannot be given. The point of estimates and necessity of scanning are discussed in the following section. 4. Numerical examples In this section the problem of guess values and the usefulness of a general (n, k) inversion process such as the contours method are illustrated. For that different problems are considered with one materials layer considered as unknown, and the contours method as well as the typical least-squares inversion are used to solve them. In the least-squares approach the problem is solved by minimizing the error function:   2 e 2 e 2 χ ¼ ρr ðn; kÞ−ρr þ ρi ðn; kÞ−ρi ;

ð8Þ

5

ρr contour ρi contour

4

k

3

1.121, 1.103

2

0.94, 0.06

1

singular 1.530, point 0.137

3.484, 0.056

2.485, 0.093 singular point

4.50, 0.04

0 0

1

2

3

n

4

5

Fig. 1. Contours in the (n, k) plane of constant real and imaginary parts of the ellipsometric function for an unknown 100 nm thick layer on silicon substrate at 5.8 eV. Intersection points are indicated on the figure.

when guess values for n and k are provided using the python implementation of the Levenberg–Marquardt of the Open Source Library of Scientific Tools SciPy. In the contours approach the problem is treated by solving the set of equations: e

ρr ðn; kÞ ¼ ρr ; ρi ðn; kÞ ¼ ρei ;

ð9Þ

using the previously described contours method which does not require guess value but returns a set of possible solutions over the desired range of exploration. To illustrate on the influence of the initial guess of the least-squares approach, this initial guess is varied and the result is reported as a function of n and k guess over the same range of exploration, and a map of the solution reached by the least-squares method is drawn. The set of solutions found by the contours method is also reported. It is observed that, depending on the position of the guess on the (n, k) map the least-squares method converges to either one of the possible solutions returned by the contours or sometimes does not converge in the chosen range of exploration. The considered problems consist of single film on substrate or multiple layers on substrate. As input experimental data, we use simulated data using standard optical constants of materials so that obtained results can be compared to the exact initial values of optical constants. In each of these problems the complex refractive index of one layer is considered as unknown. The map of the solutions reached by the least-squares method as a function of guess values is then drawn when the (n, k) exploration range is mapped over a 1000 × 1000 points grid. The contours are calculated over the same exploration range using a 100 × 100 points initial grid and a 50 times enhanced resolution over the irregular restricted areas. In these examples, the exploration range for n and k solutions is specified for each example and is intentionally chosen with a large extent as one would do if the materials were completely unknown. The first problem considered is the previously mentioned case of a film of silicon nitride (SiN) layer on silicon (Si) substrate for the incident wavelength of 213.8 nm (5.8 eV). The properties of the SiN sample are n = 2.485, k = 0.092 and d = 100 nm. The properties of silicon are n = 1.136 and k = 3.057. Values of ellipsometric angles for this sample at the angle of incidence of 70 ∘ are calculated to be Ψe = 29.3606∘ and Δe = 58.7186∘. Using these data, the contours method and the leastsquares map are done when complex refractive index of the SiN layer is considered as unknown. The values of n are considered to fall within the range of (1–4), and k within the range of (0–5). The solutions found by the contours method are given in the legend of Fig. 2. Four different solutions are found, including the good one. The map obtained using the least-squares method is shown in Fig. 2. For all the points corresponding to the n and k guess coordinates on both axes, the map indicates the solution yielded among the four possibilities given in the legend of the figure. It can be observed that the resulting map is quite complicated. There is a dominating area for the solution presenting the largest value of k. The black area representing convergence out of the range of investigation is second in importance. On the whole for initial guesses around each of the solution points, the least-squares converges to this solution but only if the guess value is not too far from the solution, typically in a range of 0.4 width for n and k. Especially the solution of interest labeled as no. 2 can be yielded only on a very narrow range of initial values for n in the range between 1.15 and 1.75 and for k in the range between 0 and 0.4. Surprisingly, there are other possible areas away from the solutions allowing convergence. The good solution can for example be reached from an area around the value n = 1.7, k = 3.5. This is the consequence of the complicated shape of the least-squares function that can present local extrema and singular points. Another example of a simple film–substrate system is considered. It is made of a 300 nm thick TiO2 layer on glass for the incident wavelength of 370 nm (3.35 eV). The properties of the materials are

M. Gilliot / Thin Solid Films 520 (2012) 5568–5574

5.0 4.5 4.0 3.5 3.0

k 2.5 2.0 1.5 1.0 0.5 0.0 1.0

1.3

1.6

1.9

2.2

2.5

2.8

3.1

3.4

3.7

4.0

n solution 3 n=2.485, k=0.092

solution 2 n=1.530, k=0.137

solution 1 n=1.121, k=1.103

solution 4 n=3.484, k=0.056

no solution

Fig. 2. Map of the solutions reached by the least-squares method as a function of the initial values of n and k for an unknown 100 nm SiN on silicon substrate at 5.8 eV.

n = 2.625, k = 0.015 for TiO2 and n = 1.523, k = 0 for glass. Values of ellipsometric angles for this sample at the angle of incidence of 70 ∘ are calculated to be Ψ e = 16.3647 ∘ and Δ e = 11.3289 ∘. Using these data, the contours method and the least-squares map are done when complex refractive index of the TiO2 layer is considered as unknown. The values of n are considered to fall within the range of (1–4), and k within the range of (0–5). The solutions found by the contours method are given in the legend of Fig. 3. Five different solutions are found, including the good one. The map obtained using the leastsquares method is shown in Fig. 3. Similar remarks can be formulated as the ones formulated on the SiN/Si sample. The map is even more complicated because of the more important number of possible solutions. There is a dominating area for the solution presenting the largest value of k. The different solutions can be reached if the guess value is not too far from the solution. Especially the solution of interest labeled as no. 3 can be yielded only on a very narrow range of initial

5571

values for n in the range between 2.5 and 2.8 and for k in the range between 0 and 0.3. Surprisingly, there are other possible areas away from the solutions allowing convergence. The good solution can for example be reached from an area around the value n = 1.4, k = 2.0 or n = 2.2, k = 4.0. In the next test, the problem of samples having an arbitrary number of layers is considered. The method is the same but the films overlying the unknown layer must not be too absorbing to let a fraction of light reach the layer of interest. A sample made of a polysilicon layer sandwiched between two silica layers on silicon substrate is considered at wavelength of 632.8 nm (1.96 eV). The top silica layer is 30 nm thick, the intermediate polysilicon layer is 120 nm thick and the bottom silica layer is 50 nm thick. The properties of the materials are n = 3.870, k = 0.037 for polysilicon, n = 1.457, k = 0 for silica and n = 3.882, k = 0.019 for silicon. The ellipsometric angles are computed as Ψe = 21.0483∘ and Δe = 175.0763 ∘ at the angle of incidence of 70 ∘. Using these data, the contours method and the least-squares map are done when complex refractive index of the polysilicon layer is considered as unknown. The values of n are considered to fall within the range of (1–5), and k within the range of (0–5). The solutions found by the contours method are given in the legend of Fig. 4. Only one solution, which is the good one, is found. The map obtained using the least-squares method is shown in Fig. 4. The map is this time more simple because of the limited possibilities of solutions. The possible range of guess values to reach the solution is more important but the probability to converge to a solution out of the range of investigation is still dominant. The solution can be yielded only for initial values of n in the range between 3 and 5 and k in the range between 0 and 2.5 or n in the range between 1 and 3 and k in the range between1 and 1.5. A final test is considered regarding with metallic materials. The sample consists of a 20 nm thick layer of gold on a 50 nm thick layer of alumina on aluminum substrate for wavelength of 500 nm (2.48 eV). The properties of the materials are n = 0.665, k = 1.910 for gold, n = 1.775, k = 0 for alumina and n = 0.768, k = 6.078 for aluminum. The ellipsometric angles are computed as Ψ e = 40.1014 ∘ and Δ e = 70.1856 ∘ at the angle of incidence of 70 ∘. Using these data, the contours method and the least-squares map are done when complex refractive index of the gold layer is considered as unknown. The values of n are considered to fall within the range of (0–5), and k within the range of (0–5). The solutions found by the contours method are given in the legend of Fig. 5. Two different solutions are found, including the good one. The map obtained using the least-squares method is shown in Fig. 5. The map is more simple because of the limited possibilities of solutions. The good solution has the largest of k

5.0 4.5

5.0

4.0

4.5

3.5

4.0

3.0

3.5

k 2.5

3.0

2.0

k 2.5

1.5

2.0

1.0

1.5

0.5

1.0

0.0 1.0

0.5 1.3

1.6

1.9

2.2

2.5

2.8

3.1

3.4

3.7

4.0

n solution 1 n=1.678, k=0.167

solution 2 n=2.041, k=0.030

solution 3 n=2.625, k=0.015

solution 4 n=3.214, k=0.010

solution 5 n=3.810, k=0.0080

no solution

Fig. 3. Map of the solutions reached by the least-squares method as a function of the initial values of n and k for an unknown 300 nm TiO2 layer on glass.

0.0 1.0

1.4

1.8

2.2

2.6

3.0

3.4

3.8

4.2

4.6

5.0

n solution 1 n=3.870, k=0.037

no solution

Fig. 4. Map of the solutions reached by the least-squares method as a function of the initial values of n and k for an unknown 120 nm polysilicon layer between silica layers on silicon.

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M. Gilliot / Thin Solid Films 520 (2012) 5568–5574

5.0 4.5 4.0 3.5 3.0

k 2.5 2.0 1.5 1.0 0.5 0.0 0.0

0.5

1.0

1.5

20

2.5

3.0

3.5

4.0

4.5

5.0

n solution 1 n=0.665, k=1.910

solution 2 n=0.448, k=0.233

no solution

Fig. 5. Map of the solutions reached by the least-squares method as a function of the initial values of n and k for an unknown 20 nm gold layer on alumina/aluminum.

and can this time be reached with the largest probability. Three areas of non convergence are present on the edges of the maps with complicated shapes. Especially values of n in the range between 0 and 2.5 and k in the range between 0 and 0.5 should not be used to reach the good solution. These different examples show the multiple solution problems that can occur in ellipsometry data inversion. The number of solutions can be relatively high, depending on the sample, on materials properties and on thicknesses. Especially for large values of thickness, interference effects are more important, and ellipsometry function is more complicated, presents more peaks and singularities and has many more solutions. The contours method returns all solutions, including the good one with accuracy better than 10 − 4 simultaneously on n and k in these examples. The convergence of the least-squares really depends on guess value. The map of solution reached as a function of guess value is complicated and does present irregular geometrical shape. In most cases the correct solution can be yielded only if a guess value inside a narrow surrounding range is provided. In brief the least-squares type method should be used only in the case when guess values are relatively well known. For an unknown material or with too approximately known values of n and k, it appears necessary to look for all possible solutions in a desired range. The contours method appears as a good toll to provide all solutions in a desired range. A drawback of this method is that when different solutions are provided, additional work is necessary to choose the good one. This point is discussed in the next section.

previous works [5,6,8,18,25]. Using the contours methods, all solutions are exhaustively known in the desired range, the additional work is then only to select the good solution among all the proposed possibilities. Using for example spectroscopic data or an additional measurement allows to make a good selection of the solution. Considering spectroscopic ellipsometry measurement, the contours method is applied to the case of the first example namely SiN layer on Si substrate over the whole 4–6 eV spectral range. Values of Ψ and Δ are computed using standard optical constants values [26]. Results of the contours method are given in Fig. 6 over this spectral range. As expected several solutions can be found for each wavelength. The spectral results are made of 6 different branches (one of the branches on the k graph with high k values is not plotted for the sake of clarity of the figure). One of the branches (branch labeled 3) is perfectly superimposed to the real value, which shows the efficiency of the contours method. The other branches are also solutions but branches 1 and 2 have too high n values to be physical solutions and they are reduced to one part of the spectrum, branch 4 has a non physical oscillating behavior on k and too low n values, branch 5 has too low n values, and branch 6 has a non physical decreasing behavior for n. Another way to choose the correct solution among the different possibilities consists in using an additional measurement, for example an ellipsometry measurement at an other angle of incidence or a photometric measurement either in transmission or in reflection. The correct solution can be selected by comparing this additional experimental result to the ones generated using the different solutions. The correct solution is then the one being closest to experimental value. Ellipsometry response at the angle of incidence of 50∘ and reflectometry intensity measurement at the angle of incidence of 70∘ for p and s-polarized incident lights are considered for the previously studied SiN on Si sample at wave energy of 5.8 eV. Note that such reflectometry measurement can also be provided by some of the commercially available ellipsometers. Generated data for the four possibilities of n and k are presented in Table 1. Concerning ellipsometry at 50 ∘ the expected values are Ψ = 28.0506 ∘ and Δ = 102.8939 ∘. This couple of value can easily be distinguished from the three other couples of values; either one or both ellipsometric angles differ by a few degrees. Concerning reflectometry measurement, the expected values are Rp = 0.1659 and Rs = 0.5241. The closest value among the three other ones is 0.1727 for Rp, which makes differences of 0.07, and 0.5456 for Rs, which makes a difference of 0.02. These two differences are well above the detection threshold of any available reflectometry measurement setup. The use of either complementary ellipsometry measurement at 50∘ or of only one of these two reflectometry components or of reflectometry measurement for unpolarized incident light (which is the average of Rp and Rs) would undoubtedly lead the right solution among the four by selecting the calculated value matching this additional measurement.

5. Discussion 5

1

real n

n

4

6

2

3

3

2

4

1 4,0 0,20

5 4,5

5,0

5,5

6,0

real k

0,15

k

As illustrated by the different examples, the contours method has good convergence performances for different kinds of problems. It works for single film on substrate as well as multilayer samples. Although examples have been presented for single wavelengths, the technique can also be applied to the inversion point by point of spectroscopic data. The method provides all possible solutions to the problem in a given (n, k) range, and is an alternate to least-squares methods that can be trapped by local minima providing wrong or possible non-physical solution [22]. It is a global solution finding method like for example genetic algorithms [23] or particle swarm optimization algorithms [24], with the difference that it can accept wider range of exploration of initial (n, k) values. As a drawback, the wider the range of exploration is, the larger is the probability of finding multiple solutions. In a given range for n and k, multiple solutions that exactly solve the ellipsometric equation may exist. Additional information is then necessary to select the correct solution. The use of multiple data has been suggested in

5

1

0,10

4 2

0,05

3

0,00 4,0

4,5

5,0

5,5

6,0

wave energy (eV) Fig. 6. n and k values resulting from the contours method (scatter) and real values (black line).

M. Gilliot / Thin Solid Films 520 (2012) 5568–5574 Table 1 Ellipsometric angles at the angle of incidence of 50° and reflectometric components at the angle of incidence of 70° generated with the four n and k solutions obtained by the contours method for the SiN/Si sample. n

Ψ(∘)

k

Δ(∘)

Rp

Incidence 50° 1.1206 1.5300 2.4850 3.4841

1.1031 0.1369 0.0920 0.0558

28.6096 35.3553 28.0506 25.7260

Rs

Incidence 70° 126.3296 107.1383 102.8939 103.4721

0.1954 0.1383 0.1659 0.1727

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Table 3 n and k solutions found for the TiO2/glass sample using exact values (top lines) and erroneous values (bottom lines) for Ψ, Δ, and d input parameters. Input parameters Ψ(∘)

Output parameters Δ(∘)

d (nm)

n

k

16.3647

11.3289

300.0

16.4605

11.4199

301.3

1.6777 2.0412 2.6250 3.2137 3.8102 1.6733 2.0335 2.6149 3.2009 3.7947

0.1675 0.0303 0.0150 0.0103 0.0080 0.1680 0.0293 0.0146 0.0100 0.0077

0.6175 0.4371 0.5241 0.5456

6. Error analysis As observed in the different examples, exact input data make the algorithm converge to exact solution. Errors on input data would make the algorithm converge to a solution, corresponding to the exact solution with error propagated from the input data errors. Errors on the n and k solutions mainly occur because the algorithm uses model and parameters that are experimentally determined and known with uncertainties. In any experimental setup parameters are known with uncertainties. For instance angle of incidence has uncertainties because of mechanical limitations of the setup. Ellipsometric angles also are measured with uncertainties, resulting from misalignments and mechanical positioning errors of polarizers or possible compensators as well as imperfections of these components or also detection errors. Obviously thicknesses of the different layers of the sample as well as optical constants of known specimen media may be known with errors. The error on the final solution is the distance between value that would be found using exact parameters and value found using erroneous parameters. The performances of a good ellipsometric inversion algorithm not only rely on its ability to converge to accurate solutions but also on its sensitivity to errors. To further determine the performance of the algorithm, the same problems as used in the previous section are considered where noise having a standard deviation of 0.1 ∘ on both ellipsometric angles is added. A random error on the value of d is also added with standard deviation of 2 nm. Practically available ellipsometers are able to measure angles with accuracy as good as 0.01°, and an input error of 1 nm on thickness can be easily obtained by different techniques, especially by ellipsometry in the range of transparency of materials. The complex refractive index of the film is again computed with the contours method, using these erroneous data for the fours test cases. Results are presented in Tables 2, 3, 4, and 5 respectively for the SiN/Si, TiO2/glass, silica/polysilicon/silica/silicon and gold/alumina/ aluminum samples. The tables present solution values for n and k previously obtained with exact input data and values obtained with erroneous data. Exact and erroneous input data are also given in the

Table 2 n and k solutions found for the SiN/Si sample using exact values (top lines) and erroneous values (bottom lines) for Ψ, Δ, and d input parameters. Input parameters Ψ(∘)

table. For each case, the same number of solutions is found. All solutions are only slightly shifted with respect to the exact ones. For the SiN/Si sample, the solution of interest is found to be n = 2.4970 and k = 0.0924, which makes a deviation of 0.012 and 0.0004 on n and k from the exact value. For the TiO2/glass, the solution of interest is found to be n =2.6149 and k =0.0146, which makes a deviation of 0.0101 and 0.0004 on n and k from the exact value. For the silica/polysilicon/ silica/silicon, the algorithm yields to a slightly shifted value n=3.8531 and k=0.0341, which makes a deviation of 0.0169 and 0.0029 on n and k from the exact value. For the gold/alumina/aluminum, the solution of interest is found to be n=0.6619 and k =1.9109, which makes a deviation of 0.0031 and 0.0009 on n and k from the exact value. Deviation found in the different examples on the results of n and k from the exact values are low, which shows the relative insensitiveness of n and k to the parameters. As these errors depend on the sample and problem to be solved, they should be estimated to predict in more details the effects of these errors. The sensitivity of the n, k solutions to perturbation of input parameters of the contours method are estimated to the first order extent by linearization of the ellipsometric function around the point of interest. Considering successively a perturbation on the experimental ellipsometric values Ψ e or Δ e, which is a direct variation of level of contour lines, and then a perturbation on any other parameter x (angle of incidence, wavelength, optical constants of used materials, thicknesses), which does not change the contour level values but slightly changes the ellipsometric function, yields to: ∂n 1 ∂Δ ¼− D ∂k ∂Ψe ∂k 1 ∂Δ ¼ ∂Ψe D ∂n ∂n 1 ∂Ψ ¼ ∂Δe D ∂k ∂k 1 ∂Ψ ¼− ∂Δe  D ∂n  ∂n 1 ∂Ψ ∂Δ ∂Ψ ∂Δ ¼ − ∂x D  ∂x ∂k ∂k ∂x  ∂k 1 ∂Ψ ∂Δ ∂Ψ ∂Δ ¼ − ∂x D ∂n ∂x ∂x ∂n

ð10Þ

Output parameters Δ(∘)

d (nm)

n

k

29.3606

58.7186

100.0

29.2661

58.8939

99.4

1.1206 1.5300 2.4850 3.4841 1.1273 1.5349 2.4970 3.5027

1.1031 0.1369 0.0920 0.0558 1.1070 0.1376 0.0924 0.0561

Table 4 n and k solutions found for the silica/polysilicon/silica/silicon sample using exact values (top line) and erroneous values (bottom line) for Ψ, Δ, and d input parameters. Input parameters Ψ(∘) 21.0483 21.0026

Output parameters Δ(∘)

d (nm)

n

k

175.0763 174.9768

120.0 120.8

3.8700 3.8511

0.0370 0.0365

5574

M. Gilliot / Thin Solid Films 520 (2012) 5568–5574

Table 5 n and k solutions found for the gold/alumina/aluminum sample using exact values (top lines) and erroneous values (bottom lines) for Ψ, Δ, and d input parameters. Input parameters Ψ(∘)

Output parameters Δ(∘)

d (nm)

n

Table 6 Partial derivatives of n and k solutions for the SiN/Si sample at 5.8 eV relative to the different parameters: experimental ellipsometric angles (Ψe and Δe), thickness (d), angle of incidence (θ), and refractive and extinction indices of substrate (ns and ks). δn δx

x

k

e

40.1014

70.1856

20.0

40.0693

70.2733

20.2

0.4485 0.6650 0.4521 0.6619

0.2326 1.9100 0.2303 1.9109

Ψ Δe d θ ns ks

δk δx −3

2.64 × 10 − 3.04 × 10− 3 − 2.12 × 10− 2 − 9.57 × 10− 3 2.59 × 10− 2 4.80 × 10− 2

−1

(deg ) (deg− 1) (nm− 1) (deg− 1)

− 7.12 × 10− 3 − 1.13 × 10− 3 7.43 × 10− 5 − 6.45 × 10− 4 − 4.80 × 10− 2 2.59 × 10− 2

(deg− 1) (deg− 1) (nm− 1) (deg− 1)

where D¼

∂Ψ ∂Δ ∂Ψ ∂Δ − : ∂k ∂n ∂n ∂k

ð11Þ

Relative derivatives of n and k according to any input parameters for the solution of interest n = 2.485, k = 0.092 at 5.8 eV for the SiN/ Si sample are given in Table 6 using previous formulas. The appropriateness of the linearization for measurement error has been more extensively confirmed by comparison with the exact calculations on a larger number of values (not shown here). Especially errors according to the principal parameters Ψe, Δe and d can be considered. For instance an error of 1 nm on d would lead to partial error contributions of dn=2.12×10− 2(0.9%) and dk= 7.43×10− 5(0.1%). An error of 0.1° on Ψe would lead to partial error contributions of dn=2.64×10− 4(0.01%) and dk=7.12×10− 4(0.8%). An error of 0.1° on Δe would lead to partial error contributions of dn= 3.04×10− 4(0.01%) and dk=1.13×10− 4(0.1%). All these errors are less than 1%, and are in agreement with example results obtained using erroneous data. In fact errors on input parameters can hardly be precisely known and reasonable margins of errors for these parameters are considered. Corresponding margins of errors for final results depend on the sample parameters and measured data, and would be different for each particular case. The error formulas allow an easy implementation in standard procedures so that accuracy of the obtained solutions can be systematically calculated and checked for each set of new input data. 7. Conclusion A method to extract point by point n and k of an unknown layer in any stack of thin films from ellipsometric measurement has been presented. The method neither requires the use of a parametric dispersion dielectric function nor requires guess values as starting points of a minimization process. It provides accurately all possible solutions of the ellipsometric equations in a given (n, k) range that can be as large as desired. This inversion process is of particular interest when the materials is completely unknown and when estimate values for complex refractive index can not be provided. Efficiency of the method has been tested by different examples.

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