FR2849504A1 - Optical characterization method for chemical analysis of materials, involves determining error using initial wavelength, initial refractive index, and interpolation law until desired precision is obtained on spectrum - Google Patents

Abstract

The method involves selecting initial wavelengths, an interpolation law for refractive index of material, and an initial index for each initial wavelength. A theoretical spectrum and the error between measured spectrum and theoretical spectrum are determined using initial wavelength, initial refractive index, and the interpolation law. Selecting and determining steps are repeated for other initial wavelengths until obtaining desired precision on each obtained spectrum. The theoretical spectrum is determined according to selected reflectometry and ellipsometry calculation methods, and refractive index deducted by interpolation of index value in the initial wavelength. The error is determined between measured spectrum and theoretical spectrum, by varying the position of unknown indices values and/or depth of layer and/or values of refraction index to the initial value and then the spectrum is obtained.

Description

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PROCESS FOR OPTICAL CHARACTERIZATION OF MATERIALS WITHOUT
USE OF PHYSICAL MODEL
DESCRIPTION TECHNICAL FIELD
The present invention relates to a method for optical characterization of materials.

 This method makes it possible to characterize thin or thick layers of these materials, which are formed on substrates. The physical quantities, which this method makes it possible to determine, are: the thickness of a layer of a material, the refractive index of this material, and the absorption coefficient of this material.

 The optical characterization of materials is useful for the chemical analysis of these materials (in particular the study of absorption bands, densification properties and oxidation properties), in the fields of microelectronics, sensors, biology, medicine), or for the analysis of the deposit thicknesses of these materials.

 For examples of application, reference is made to document [1] which, like the other documents cited later, is mentioned at the end of the present description.

 Characterization of the optical properties of a material is also useful when the material is subsequently structured (to form, for example, etchings or roughnesses) and the properties of

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 Optical diffraction of the obtained structure should be calculated (see document [2]).

 Let us indicate from now on that the invention is particularly useful when the physical law followed by the complex refractive index of the material that one wants to characterize is, a priori, not known.

d'ondes [i,"" iM] . STATE OF THE PRIOR ART
It is recalled that the optical measurements can be of various types:
These may be reflectometric measurements. In this case, the intensity reflection coefficient of a structure is measured over a spectrum (i.e., range) of lengths.

of waves [i, "" iM].

 The angle of incidence of the illumination light may be non-zero. The reflection coefficient can be measured for several angles of incidence #. We will denote R (#, #, p) the reflectance spectrum, where p is the polarization of the incident beam and # the wavelength of the latter.

 Generally, the angle # is zero and the polarization p indeterminate. In the case where # is not zero, it is necessary to know this polarization p. In general, the latter is of type (S) where (P).

 It can also be ellipsometric measurements. The magnitudes measured are then the real and imaginary parts of the ratio of the polarization reflection coefficient (P) to the polarization reflection coefficient (S).

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On note généralement p=pexp(j) ce rapport complexe (avec j2=-1). Les grandeurs

généralement traitées sont 1 p que l'on note tan(yI), et cos(), ou des combinaisons des deux. Par souci de généralité, nous noterons Si et 82 ces grandeurs.

We generally note p = pexp (j) this complex ratio (with j2 = -1). The sizes

usually treated are 1 p that we denote tan (yI), and cos (), or combinations of both. For the sake of generality, we will note Si and 82 these quantities.

une plage de longueurs d'ondes [7,I", iM] . L'angle d'incidence peut être quelconque. Plusieurs spectres peuvent être mesurés à différents angles d'incidence afin d'obtenir un spectre plus riche. Nous noterons

s (6, î) =S1 (9, ,) , ,S2(S,À)} } le spectre ellipsométrique. The spectra Si, i # [1,2], are measured on

a range of wavelengths [7, I ", iM] The angle of incidence can be arbitrary, and several spectra can be measured at different angles of incidence in order to obtain a richer spectrum.

s (6, 1) = S1 (9,),, S2 (S, to)}} the ellipsometric spectrum.

 In a complementary manner, goniometric measurements (reflection coefficient as a function of the angle of incidence) may be added to the measurements used for the characterization, in order to determine the thickness of the various layers, for one or more wavelengths. These measurements are not sufficient in themselves since it is desired to determine the complex refractive index over a spectral range from #m to λM.

 To simplify the presentation, we will note # a set of spectrophotometric and / or ellipsometric spectra (s) (and possibly goniometric for some wavelengths).

 Without losing any generality, we will disclose, in the present description, the mode of use of the processes of the prior art and the present invention only in the case of a single thin layer of a material, formed on a known substrate.

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longueur d' onde est noté n* (,) . The thickness of this layer is denoted e and the complex refractive index of the material at the

wavelength is denoted n * (,).

It is recalled in this connection that the real (respectively imaginary) part of this complex refractive index is denoted n (#) (respectively k (#)) and called "refractive index" (respectively "extinction coefficient").

deux spectres (1) et tp (2) . Moreover, Er ((1), vii / (2>) denotes an error function (for example the mean squared difference) between

two spectra (1) and tp (2).

angles 6;,, le {l..n}, et d'un spectre réflectométrique : E.(T',')=## J 1!= L \ l l e'' I ll !vi m Àm 1=1 + [Rm(Â)-Ri2)(Â)f\i (1)

avec (1) (A.) ={S{1) (Oi E 11.. nl, 1) , ,S2(1) (6El..n,,) R (1) (,) et pl2)(1)=ISJ(2)(0,.etl..nl,X),S2 (2) (eE{1..n},À.) ,R(2J (À.)}
Des facteurs de pondération peuvent être apportés à l'intégrale de façon à ce que la fonction d'erreur puisse tenir compte de variations sur la présicion de mesure des spectres. For example, we can take, when we have ellipsometric spectra on several

angles 6, ,, the {l..n}, and a reflectance spectrum: E. (T ',') = ## J 1! = L \ ll e '' I ll! vi mAm 1 = 1 + [Rm (Â) -Ri2) (Â) f \ i (1)

with (1) (A) = {S {1) (Oi E 11 .. nl, 1),, S2 (1) (6E1..n1) R (1) (,) and p12) (1) ) = ISJ (2) (0, .etl..nl, X), S2 (2) (eE {1..n}, to.), R (2J (To)}
Weighting factors can be applied to the integral so that the error function can take into account variations in the measurement presiction of the spectra.

The optical characterization of layers of materials generally revolves around two applications:
The first application is the dimensional control of the deposition of thin layers used in microelectronics.

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 Generally, the deposited material is well known, that is to say that the complex refractive index of this material is well known at the wavelengths of the light used for the characterization.

 The laws followed by the complex refractive index are either tabulated or approximated by known physical laws such as, for example, the Cauchy model, the Sellmeier model (see document [3]), the Forouhi laws ( see the document [4]), and the laws laws of harmonic oscillators (see the document [5]). These laws are defined by a finite number of parameters.

Lorsque l'on est sûr de la valeur des

coefficients ai de {1,2}) mais que l'on ne connaît pas l'épaisseur, un algorithme de recherche est utilisé afin de trouver l'épaisseur qui minimise l'erreur entre

la mesure *P et la réponse théorique Y compte tenu de l'indice modélisé. For example, a Cauchy type law without two-parameter absorption is defined as follows:

When one is sure of the value of

coefficients ai of {1,2}) but that we do not know the thickness, a search algorithm is used to find the thickness which minimizes the error between

the measure * P and the theoretical response Y taking into account the modeled index.

 The search algorithm may be, for example, the Simplex method, the Tabou search, the Levendt-Marquart method or the simulated annealing method (see Chapter 10 of document [6]).

 When the refractive index is approximate, the coefficients ai are integrated in the adjustment procedure of # and #. The search for

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 coefficients ai constitutes a method of characterization of the refractive index.

 However, when the law followed by this index of refraction is unknown (it happens that the material is unknown or is not well described by a known physical law), this method remains approximate and the thickness may be wrong.

 The second application is the characterization of materials.

 The process used remains the same, except that the material is not well known. It is precisely the complex refractive index function closest to reality that is targeted.

Dans l'expression ci-dessus, j2=-1 et l'indice de réfraction et le coefficient d'extinction sont exprimés non pas en fonction de # mais de E, avec E= 1240/À en nm) . The type of law can be chosen by analogy with other materials. However, the law followed by the complex refractive index can be complicated, which is for example the case of a law of harmonic oscillators:

In the above expression, j2 = -1 and the refractive index and the extinction coefficient are expressed not in terms of # but of E, with E = 1240 / λ in nm).

 In this case, the coefficients of the oscillators are difficult to find if we do not have their order of magnitude. Research is difficult to automate, search algorithms can give wrong answers and the time lost can be considerable.

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 There is an alternative to finding coefficients: the point-to-point method (PAP).

[1...n], avec ,1=7,a, et 2=Xm. This PAP method proposes not to choose a physical law and to look for the complex refractive index of the material for each wavelength #i, iE

[1 ... n], with, 1 = 7, a, and 2 = Xm.

entre la mesure IF (li) et la réponse théorique Y ( %1,i , n ( %l,i ) . k ( %li ) F-) Un tel procédé pose un problème parce que les divers points (,i, , n (,;,) , k (,;,) ) ne sont pas forcément physiquement compatibles entre eux : par exemple, l'épaisseur trouvée peut varier en fonction de la longueur d'onde et la loi suivie par l'indice de réfraction complexe, plus simplement appelée loi d'indice, peut présenter des discontinuités. For each #i, a search algorithm tries to find the thickness, the index n (#i) and the extinction coefficient k (#i) that minimize the error

between the measurement IF (li) and the theoretical response Y (% 1, i, n (% l, i). k (% li) F-) Such a method is problematic because the various points (, i,, n (,;,), k (,;,)) are not necessarily physically compatible with each other: for example, the thickness found may vary according to the wavelength and the law followed by the refractive index complex, more simply called index law, may have discontinuities.

 This method is generally valid only when the thickness is very well known and that the measurements are of very good quality.

STATEMENT OF THE INVENTION
The present invention aims to overcome the above disadvantages.

 The method which is the subject of the invention makes it possible to characterize a material without using a physical model, that is to say without using a physical law followed by the complex refractive index of the material.

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 studied. It is therefore particularly useful when such a law is not known.

coordonnées (Ài n; ) , où nl est la valeur prise par l'indice de réfraction complexe à la longueur d'onde #i et i prend un nombre limité de valeurs (entières). This process constitutes an alternative to the known characterization methods mentioned above. It can be called the "node method" because it uses "nodes", that is, points of

coordinates (Ài n;), where nl is the value taken by the complex refractive index at the wavelength #i and i takes a limited number of (integer) values.

 Specifically, the subject of the present invention is a process for the optical characterization of at least one layer of a material in a wavelength range [# min,, max], this layer being formed on a substrate, this method being characterized in that: - a set of measurements of reflectometry and / or ellipsometry, this set of measurements leading to a measured spectrum, denoted #, - m wavelengths initial # 1 are selected ... #m belonging to this interval, m being an integer at least equal to 1, at each wavelength is associated with a refractive index, - at least one interpolating law is chosen for the refractive index of the material, for the wavelengths between the initial wavelengths # 1 ... # m, - M is chosen initial parameters, M being at least equal to m, namely an initial refractive index and for each length of initial wave #i, 1 # im, the initial wavelengths were chosen from

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i allant de 1 à m, sur le spectre [À,min, À,max] , - on détermine l'erreur Er (\fi, \fi), entre le spectre mesuré et le spectre théorique, - on minimise cette erreur en faisant varier la position des valeurs des indices inconnus et/ou l'épaisseur de la couche et/ou les valeurs des indices de réfraction aux longueurs d'ondes initiales, et l'on obtient un spectre, - on ajoute des longueurs d'ondes aux longueurs d'ondes initiales #1 ... #m, les longueurs d'ondes ajoutées constituant de nouveaux n#uds, - on répète le procédé en choisissant un nombre m' de longueurs d'ondes initiales, m' étant supérieur à m, so that at least the refractive index can be determined by interpolation for any wavelength of the interval [# min,, max], the pairs (#i, ni) being called n # uds, - methods are chosen for calculating reflectometry and ellipsometry, an error function Er, representative of the difference between two spectra # 1 and # 2, is also chosen, the spectra # 1 and 2 being calculated or measured on a higher number of points. to the number m of nodes, - using the initial m wavelengths, M initial parameters and the interpolation law, the following optimization process is implemented: - a spectrum is determined theoretical, denoted #, according to the chosen calculation methods, and of the index deduced by interpolation of its value in #i,

i ranging from 1 to m, on the spectrum [À, min, À, max], the error Er (\ fi, \ fi), between the measured spectrum and the theoretical spectrum is determined, - this error is minimized by varying the position of the unknown index values and / or the thickness of the layer and / or the values of the refractive indices at the initial wavelengths, and a spectrum is obtained, - wavelengths are added at the initial wavelengths # 1 ... #m, the added wavelengths constituting new nodes, - the process is repeated by choosing a number m 'of initial wavelengths, m' being greater than m,

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 and M 'initial parameters, M' being greater than M, until the accuracy on each spectrum thus represented at best is equal to a predefined precision.

d'ondes X, en l'occurrence l'intervalle [Xmin, kax] , mais on ne sortirait pas du cadre de l'invention en effectuant cette caractérisation optique - sur un intervalle de longueurs d'ondes inverses 1/#, en l'occurrence sur un intervalle

[ ( 1/,) min, (1/1)max], où ( 1/,) min est égal à 1/ (Xmax) et (1/À.)max à 1/ (,min) , - ou sur un intervalle d'énergies E (avec E = hv = hc/# où h est la constante de Planck, c la vitesse de la lumière dans le vide et v la fréquence correspondant à #), en l'occurrence sur un intervalle [Emin, Emax] , où Emin est égal à hc/(#max) et Emax à hc/ (#min), ou, plus généralement, sur un intervalle [amin, amax] de valeurs d'une variable a fonction de #. It is a question, above, to perform optical characterization over a range of lengths

X-wave, in this case the interval [Xmin, kax], but it would not be outside the scope of the invention to carry out this optical characterization - over a range of inverse wavelengths 1 / #, in occurrence on an interval

[(1 /,) min, (1/1) max], where (1 /,) min is equal to 1 / (Xmax) and (1 / to.) Max to 1 / (, min), - or an interval of energies E (with E = hv = hc / # where h is the Planck constant, c the speed of light in vacuum and v the frequency corresponding to #), in this case over an interval [Emin , Emax], where Emin is equal to hc / (# max) and Emax to hc / (#min), or, more generally, to an interval [amin, amax] of values of a variable with function of #.

 Those skilled in the art will readily adapt the definition of the process which is the subject of the invention which has just been given as a function of the variable, as well as the particular embodiments and the examples which are given thereafter, at the same time. one or other of the variables 1 / # and E and, more generally, to the variable [alpha] dependent on #.

 For example, if one carries out the optical characterization on the interval [(1 / #) min,

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(1/,)1...(1/,)m (respectivement des énergies initiales Ei...Em) et, plus généralement, par une étape où l'on choisit des valeurs initiales [alpha]1...[alpha]m de [alpha]. (1 / #) max] (respectively on the interval [Emin, Emax]), we will replace the step where we choose the initial wavelengths # 1 ... # m by a step where we chooses initial inverse wavelengths

(1 /,) 1 ... (1 /,) m (respectively initial energies Ei ... Em) and, more generally, by a step where initial values [alpha] 1 are chosen ... [ alpha] m of [alpha].

 In addition, the invention applies to characterize a spectrum or part of the spectrum.

 According to a first particular embodiment of the method which is the subject of the invention, m is at least equal to 2.

 According to a second particular embodiment of the method which is the subject of the invention, m is equal to 1 and initial initial refractive indexes are chosen.

 According to a particular embodiment of the invention, the material is non-absorbent and the number M is equal to m, the extinction coefficient of the material being taken equal to 0.

 According to another particular embodiment of the invention, - M is at least equal to 2 m, - one further chooses an interpolation law for the extinction coefficient of the material, - for each initial wavelength # i, 1 # im, an initial extinction coefficient ki is also chosen, the initial wavelengths being furthermore chosen so as to be able to determine by

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 interpolation the extinction coefficient for any wavelength of the interval [# min, λ max], - in the optimization process, the error is minimized by additionally varying the values of the extinction coefficients at Initial wavelengths and added wavelengths are further placed to best represent the spectrum of the extinction coefficient of the material.

 In this case, according to a particular embodiment of the invention, m is equal to 1 and initial equal refractive indices and initial initial extinction coefficients are chosen.

 According to a particular embodiment of the method that is the subject of the invention, the layer of material is thin, that is to say has a thickness less than the coherence length of the light used for the measurements. an additional initial parameter, namely an initial layer thickness, and in the optimization process the error is minimized by further varying the value of the layer thickness.

 According to another particular embodiment of the method which is the subject of the invention, the layer of material is thick, that is to say is not thin, and M is at most equal to 2 m.

 According to another particular embodiment, the thickness of the layer of material is known with sufficient accuracy and M is at most equal to 2 m.

 Each interpolation law can be chosen from linear interpolation laws,

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 cubic interpolation laws, polynomial interpolation laws, and spline function interpolation laws.

 In the invention, the distribution of nodes can be homogeneous.

BRIEF DESCRIPTION OF THE DRAWINGS
The present invention will be better understood on reading the description of exemplary embodiments given below, purely by way of indication and in no way limiting, with reference to the appended drawings, in which: FIG. 1 is a schematic view of devices for characterizing a layer according to the invention, - Figure 2 shows the variations of the refractive index as a function of the wavelength, for a material according to a Cauchy law (curve I) and for a material characterized according to the invention (curve II), and - FIG. 3A (respectively 3B) shows the variations of the refractive index (respectively of the extinction coefficient) as a function of the wavelength, for a material following a two oscillator law (curve I) and for a material characterized according to the invention.

DETAILED PRESENTATION OF PARTICULAR EMBODIMENTS
The invention provides an alternative to the conventional methods mentioned above. It makes it possible to combine the coherence of a layer model (corresponding to a continuous index law and to

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 constant physical thickness), the generality concerning the index law to be found (as in the PAP method). In addition, the resolution is limited only by the resolution of the measured spectrum.

coordonnées (Xi, ni,ki) , avec ni = n (li) et ki = k (li) et - une loi d'interpolation entre les n#uds, qui peut être, par exemple, linéaire, cubique, de type spline ou polynômiale (de degré quelconque). In the method which is the subject of the invention, the spectrum of index n * (#) is characterized by: a reduced number of "nodes", which are points of

coordinates (Xi, ni, ki), with ni = n (li) and ki = k (li) and - a law of interpolation between the nodes, which can be, for example, linear, cubic, of spline type or polynomial (of any degree).

 This interpolation law makes it possible to calculate, from the nodes, the refractive indices and the extinction coefficients for the wavelengths located between the nodes.

na)~(4+1-A) (4) + (A-W4+1) #l+1 - #t avec #i<#<#i+1
On peut faire de même pour le coefficient d'extinction. For example, when the refractive index is characterized by a set of values for wavelengths # 1 ... #m, we can use a linear interpolation between two wavelengths #i and # i + 1 for calculate the index n at the wavelength (see document [6] chapter 3):

na) ~ (4 + 1-A) (4) + (A-W4 + 1) # 1 + 1 - #t with #i <# <# i + 1
The same can be done for the extinction coefficient.

 When the number of nodes allows it, more complex interpolation formulas, involving neighboring nodes, can be used (see document [6] chapter 3).

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 A layer model is therefore characterized by a thickness e and a family of nodes.

 An example of the process which is the subject of the invention is given below. In this example, the measurements consist of a reflectometric measurement R (#) and an ellipsometric measurement SI, 2 (#, #) where # is the angle of incidence of the light beam that is sent on the layer to be studied when measuring ellipsometry.

 This layer is a thin layer so that the thickness of this layer is also a variable of the problem. In addition, it is assumed that only one layer is unknown, this layer being formed on a known substrate.

 Let's first briefly explain this example, which uses an algorithm.

 From information supposed on the thickness # of the studied layer, on the refractive index n (#) and on the extinction coefficient k (X), of the material of this layer, one builds nodes starting point (in low numbers) and starting thickness #.

 We thus have m nodes and, by interpolation, we can know n (#) and k (#) outside the values of the wavelengths associated with the nodes.

 From the starting thickness and these starting values n (#) and k (#), the theoretical spectrum # is determined using ellipsometric and reflectrometric calculations.

 In addition, by means of ellipsometry and reflectometry devices and a

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 spectrometer, we obtain SI, 2 (#, #) and R (#) and we deduce the measures noted #.

 Then compare # and * using an error function Er and optimize the value of the refractive index and the value of the extinction coefficient at the different nodes, as well as the value of the thickness, by trying to minimize Er (#, #).

 When these values are optimized and if the precision on the spectrum n (#), the spectrum k (#) and the thickness # are not sufficient, we add new nodes, we vary the thickness #, and we repeat the determination of #, the comparison of # and and the optimization mentioned above, and so on.

 The loop thus defined is stopped when the precision on each of the spectra n (#) and k (#) and on the thickness e is considered sufficient.

 The spectra n (A,), k (#) and the thickness # are thus characterized.

permettant de caractériser n (,) , k(À) et E en fonction des informations fournies par le spectromètre 10 et conformément au procédé de l'invention. Ces moyens 12 sont munis de moyens d'affichage 14. FIG. 1 schematically shows the studied layer 2 formed on a substrate 4. The ellipsometry device 5.6, the reflectrometry device 8 and the spectrometer 10 are seen. In addition, electronic processing means 12 are seen.

to characterize n (,), k (A) and E according to the information provided by the spectrometer 10 and in accordance with the method of the invention. These means 12 are provided with display means 14.

 Let's go back in more detail to the example given.

 Phase 1

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The method of this example comprises first an initialization step.

 The algorithm starts with a reduced number of nodes (2 or more than 2). However, one can start with a single node by imposing a refractive index and an extinction coefficient that remain constant when the wavelength changes.

 Node positions are chosen so that, from this family of nodes, the whole spectrum can be deduced by interpolation. The layer model is therefore 3 or more parameters, since the thickness is also a variable to be determined. The index table over the entire spectrum is therefore deduced from the nodes by interpolation.

 This is the case when the thickness of the layers is of the order of magnitude of the wavelength (thin layers). But when the thickness of the layer is very important, the thickness intervenes almost no longer in the calculation of the response of the layer, and is no longer a variable of the problem.

 For example, in the case of optical disks, on which a very large layer is deposited (its thickness is of the order of 1 millimeter), the reflection coefficient of such a layer is no longer a function of the thickness of the film. the layer, but only the refractive index thereof.

 For example, we choose to place the first two nodes at the 1 min and 1 max ends of the spectrum. The values of the complex index at these ends are chosen according to the type of material studied. For example, on a spectrum

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 ellipsometric between 300nm and 800nm of a thin layer of photoresist, n (300nm) = n (800nm) = 1.5 and k (300nm) = k (800nm) = 0.

considéré, on a donc n (,) =1, 5 et k (,) =0 pour appartenant à [300nm, 800nm] . When the spectrum is characterized by only two nodes, the index between the extremes is determined by linear interpolation. In the case

considered, we therefore have n (,) = 1, 5 and k (,) = 0 for belonging to [300 nm, 800 nm].

 From three nodes, a cubic interpolation is chosen instead to obtain softer index forms than those obtained by linear interpolation.

 The starting thickness is, for its part, chosen as close as possible to the actual thickness.

Phase 2
The values of the refractive index and the extinction coefficient on the nodes and the value of the thickness are then optimally determined.

 To do this, the # (#) spectra are calculated using the layer model used, resulting from the choice of nodes, the interpolation law and the thickness of the layer.

 The physical model used for the calculation of # is of course a function of the measurement method used, that is to say in particular the angle of incidence of the light, the spectrum used and the model of thin layers or thick layers where appropriate (see for example stacked layer models in document [3]).

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 The spectra being constituted by a set of measurements of various natures (for example ellipsometric and reflectometric measurements), a reflectometry (ellipsometry) calculation method is used for the reflectometry (respectively ellipsometry) measurements.

 Reflectometric and ellipsometric measurements are combined via an error function Er (#, #) which is for example of the kind defined by equation (1).

 Thanks to a search function such as Simplex for example, the difference between # (#) and # (#) is minimized, by varying the value of the refractive index and the value of the extinction coefficient at the position of each of the nodes as well as the layer thickness (if this thickness is an important factor in the calculation of #). When the deviation is minimum, ie when Er (#, #) is minimum, it means that the reflectometry and ellipsometric measurements coincide at best (for a given number of nodes).

 At this stage, for a known number of nodes and a known spectral position for each of these nodes, we obtain the layer model (refractive index, extinction coefficient and thickness) which best corresponds to the actual layer.

 The validity of the model found is even more assured that the number of measurements is large. To have a large number of measurements, it is possible for example to use several angles of incidence of the light #i,

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 1 i #, for ellipsometric spectra, make a reflectometric measurement and make complementary goniometric measurements.

Phase 3
Then we add a finite number of nodes.

spectres n (,) et k (,) . A titre d'exemple, on place ces n#uds supplémentaires aux endroits où l'écart entre et # est maximum ou aux endroits en lesquels les n#uds sont les plus espacés. Et l'on retourne à la phase 2 tant que la précision sur chacun des spectres n(#) et k(#) et sur l'épaisseur e n'est pas suffisante, c'est-à-dire n'est pas égale à une précision prédéfinie. In a first embodiment, the added nodes are positioned in such a way as to best represent the

spectra n (,) and k (,). As an example, these additional nodes are placed where the distance between and # is maximum or at the places in which the nodes are the most spaced apart. And we go back to phase 2 as long as the precision on each of the spectra n (#) and k (#) and on the thickness e is not sufficient, that is to say is not equal to a predefined precision.

 In a second embodiment, new nodes are inserted between two nodes of the set of previously selected nodes that will be homogeneously distributed over the spectrum.

 In what follows, we give two common examples of application of the invention. These examples involve two types of materials that follow different laws.

From ellipsometric and reflectometric measurements, we propose to find the physical laws followed by these materials. We proceed as follows:
A fictitious material is created, this material following a known theoretical law (a Cauchy law or a law of harmonic oscillators), with parameters that we arbitrarily set. The

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 variations of the complex refractive index as a function of the wavelength are thus perfectly known.

 We also impose the material a thickness of 200.00nm, on a silicon substrate, the latter being very well known.

 Fictitious measurements (ellipsometric measurements, reflectometry measurements) are calculated and then noised so as to introduce a defective apparatus.

 Everything happens as if we had real measurements made on the material. But, contrary to reality, we know perfectly the complex refractive index since we fixed it, just as we fixed the thickness of the layer of material.

 We test here the method "blind", that is to say that we start from a false thickness (220nm) and false complex refractive indices, since they are supposed to be unknown.

 We apply the method that is the subject of the invention and then compare the complex refractive index found with the theoretical complex refractive index. We find the same laws, very precisely, as well as the same layer thickness.

Take as a first example a material whose complex refractive index follows a Cauchy law such that:

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This index law is typical of photosensitive resins (in the 300-800 nm spectral range).

 In order to find, by means of the node method (that is to say of the method which is the subject of the invention) the index law mentioned above, we carry out two measurements, namely an ellipsometric measurement. at an angle of 70 and a reflectometric measurement.

The treatment conditions are as follows: the spectrum treated is between 300 nm and 800 nm, the nodes are initially at positions (300 nm, 1.6) and (800 nm, 1.6), that is, to say that the index is considered to vary linearly between 300nm and
800nm, and that its value is constant (equal to 1.6), - the number of nodes is increased iteratively by adding at least one node to each iteration (for example, we add 2 nodes), - the interpolation law is a cubic law, when the number of nodes is greater than 2, otherwise it is linear, and the minimization algorithm used is a Simplex type algorithm.

composé d'une mesure ellipsométrique S1 (,) , S2 (,) à 70 et d'une mesure réflectométrique R(#). FIG. 2 makes it possible to compare the refractive index corresponding to the fictitious material which perfectly follows a Cauchy law (curve I) to the refractive index which we find by the node method (curve II), to the using 6 nodes (represented by circles in Figure 2). We used a *?

composed of an ellipsometric measurement S1 (,), S2 (,) at 70 and a reflectometric measurement R (#).

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Take as a second example a material whose complex refractive index follows a two-oscillator law, such that:

The node method is applied to a set of ellipsometric measurements made between 250nm and 800nm at 75, 70, 60, and 45, plus a reflectometric measurement. The actual thickness of the material being 200 nm, we find a thickness of 200.25 nm with the node method. The fit on the index law is very good.

 Figures 3A and 3B respectively show the reconstructions of the curves n (#) and k (#) of the material using the node method. The reconstruction is performed using four ellipsometric spectra and a reflectometric spectrum. The actual absorption peaks are very well represented by the curve obtained by cubic interpolation between the nodes (represented by circles in FIGS. 3A and 3B).

 In FIG. 3A, the curve I (respectively II) corresponds to n perfectly following the chosen law (respectively to n found by the node method).

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 In FIG. 3B, the curve I (respectively II) corresponds to k perfectly following the chosen law (respectively at k found by the node method).

valeurs, avec X= {nx, n2 , ... , n , ...nm , kl7 k2,..., ki, ..., km,e}, où ni est la valeur de l'indice de réfraction au n#ud #i, i#{l...m}, m étant le nombre de noeuds ki est la valeur du coefficient d'absorption

au n#ud ,i, i e {l...m} , e est l'épaisseur de la couche étudiée. Examples of the invention have been described. It will be noticed in a more general way that, in the latter, we consider a set X of

values, with X = {nx, n2, ..., n, ... nm, kl7 k2, ..., ki, ..., km, e}, where ni is the value of the refractive index at n # ud #i, i # {l ... m}, where m is the number of nodes ki is the value of the absorption coefficient

at n # ud, i, ie {l ... m}, e is the thickness of the studied layer.

 The error minimization operation Er (#, #) amounts to finding the set X such that Er is minimum.

 When no particular constraint is imposed, minimization is minimization to 2xm + l parameters. We can of course add constraints to reduce the number of variables.

de {l...m} et X devient : X={nl,n2...,ni,...nm, E . In particular, if we know that the material is non-absorbent, we impose ki = 0 for all i

from {l ... m} and X becomes: X = {nl, n2 ..., ni, ... nm, E.

 If, by a complementary measurement (for example a goniometry measurement or a non-optical direct measurement), the thickness of the layer considered is known with a sufficient precision, the thickness e is no longer a variable and : X = {nl, n2 ...., ni, ... nm, ki, k2, ..., ki, ..., km}.

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 The two previous options can of course be combined.

 In addition, the present invention is not limited to the characterization of thin layers. It also applies to the characterization of thick layers.

 In addition, the present invention is not limited to the characterization of a single layer formed on a substrate. It also applies to the characterization of two or more layers formed on a substrate.

 The documents cited in this specification are: [1] R. M.A.Azzam and N. M. Bashara, Ellipsometry and Polarized Light, North Holland Physics Publishing, 1997, Chapter 6.

for the metrology of sub-0.1-.m-linewidth structures, Applied Optics, 37(22) : 5112-5115,1998. [2] BK Minhas, Coulombe SA, S. Sohail, H. Naqvi and JR McNeil, Ellipsometric scatterometry

for the metrology of sub-0.1-.m-linewidth structures, Applied Optics, 37 (22): 5112-5115,1998.

 [3] Born and E. Wolf, Principle of Optics, Cambridge University Press.

 [4] R. Forouhi and I. Bloomer, Optical Dispersal Relationships for amorphous semiconductors and amorphous dielectrics, Physical Review B, 34 (10) 7018-7026, November 1986.

 [5] F. L. Terry, Jr., A modified harmonic oscillator approximation scheme for the dielectric constants of AlxG1-xAs, Journal of Applied Physics, 70 (1), 1991, pp. 409-417.

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 [6] W. H. Press, S. A. Teukolsky, W. T.

Vetterling and B. P. Flannery, Numerical Recipes in C, Cambridge University Press, 1992, Chapters 3 and 10.

Claims (11)

A method for the optical characterization of at least one layer of a material in a wavelength range [# min, max], said layer being formed on a substrate, said method being characterized in that: - a set of measurements of reflectometry and / or ellipsometry, this set of measurements leading to a measured spectrum, denoted #, - m wavelengths # 1 ... # m belonging to this interval are chosen m, m being a an integer at least equal to 1, a refractive index is associated with each wavelength, - at least one interpolation law is chosen for the refractive index of the material, for the wavelengths between the initial wavelengths # 1 ... # m, - we choose M initial parameters, M being at least equal to m, namely an initial index of refraction nor for each initial wavelength #i, 1 # im , the initial wavelengths being chosen so as to be able to determine by interpol at least the refractive index for any wavelength of the interval [# min,, max], the pairs (# i, ni) being called nodes, - methods for calculating reflectometry and of ellipsometry, - we also choose an error function Er, representative of the difference between two spectra # 1 and # 2, the spectra # 1 and # 2 being calculated or measured <Desc / Clms Page number 28>  ranging from 1 to m, on the spectrum [, min, tanax], the error Er (#, #), between the measured spectrum and the theoretical spectrum is determined - this error is minimized by varying the position of the values of the unknown indices and / or layer thickness and / or refractive index values at initial wavelengths, and a spectrum is obtained - wavelengths are added at initial wavelengths 1 ... #m, the added wavelengths constituting new nodes, - the process is repeated by choosing a number m 'of initial wavelengths, m' being greater than m, and M 'initial parameters , M 'being greater than M, until the accuracy on each spectrum thus represented at best is equal to a predefined precision.

 on a number of points greater than the number m of nodes, - using the initial m wavelengths, M initial parameters and the interpolation law, the following optimization process is implemented. : - a theoretical spectrum, denoted #, is determined according to the chosen calculation methods, and the index deduced by interpolation of its value in #i, i  2. The method of claim 1, wherein m is at least 2.  The method of claim 1, wherein m is 1 and initial initial refractive indices are selected. <Desc / Clms Page number 29>  4. Method according to any one of claims 1 to 3, wherein the material is non-absorbent and the number M is equal to m, the extinction coefficient of the material being taken equal to 0.  5. Method according to any one of claims 1 to 3, wherein: - M is at least equal to 2 m, - one further chooses an interpolation law for the extinction coefficient of the material, - for each length initial wavelength # i, 1 # im, an initial extinction coefficient ki is also chosen, the initial wavelengths being furthermore chosen so as to be able to determine by interpolation the extinction coefficient for any length of wavelength. interval wave [A. min, max], - in the optimization process, the error is minimized by additionally varying the values of the extinction coefficients at the initial wavelengths and the added wavelengths are furthermore placed so as to best represent the spectrum of the extinction coefficient of the material.  The method of claim 5, wherein m is 1 and initial initial refractive indices and initial initial extinction coefficients are selected.  The method of any one of claims 1 to 6, wherein the material layer is thin, i.e., less than the coherence length of the light used for the measurements, a additional initial parameter, namely an initial thickness of <Desc / Clms Page number 30>  layer, and in the optimization process the error is minimized by further varying the value of the layer thickness.  The method of any one of claims 1 to 6, wherein the material layer is thick, i.e. is not thin, and M is at most 2 m.  9. Method according to any one of claims 1 to 6, wherein the thickness of the material layer is known with a predefined precision and M is at most equal to 2 m.  The method of any one of claims 1 to 9, wherein each interpolation law is selected from linear interpolation laws, cubic interpolation laws, polynomial interpolation laws, and interpolation laws. Spline function type.  11. The method of any one of claims 1 to 10, wherein the distribution of nodes is homogeneous.