mathematical modelling of indirect measurements in scatterometry

Measurement 39 (2006) 782–794

www.elsevier.com/locate/measurement

Mathematical modelling of indirect measurementsin scatterometry

H. Gross a, R. Model a, M. Bar a,*, M. Wurm b,B. Bodermann b, A. Rathsfeld c

a Department 8.4 Mathematical Modelling and Data Analysis, Physikalisch-Techn. Bundesanstalt, Abbestr. 2-12, D-10587 Berlin, Germanyb Department 4.2 Imaging and Wave Optics, Physikalisch-Techn. Bundesanstalt, Bundesallee 100, D-38116 Braunschweig, Germany

c RG 4 Nonlinear Optimization and Inverse Problems, Weierstrass Institute for Applied Analysis and Stochastics,

Mohrenstr. 39, D-10117 Berlin, Germany

Received 9 February 2006; received in revised form 10 April 2006; accepted 13 April 2006Available online 4 May 2006

Abstract

In this work, we illustrate the benefits and problems of mathematical modelling and effective numerical algorithms todetermine the diffraction of light by periodic grating structures. Such models are required for reconstruction of the gratingstructure from the light diffraction patterns. With decreasing structure dimensions on lithography masks, increasingdemands on suitable metrology techniques arise. Methods like scatterometry as a non-imaging indirect optical methodoffer access to the geometrical parameters of periodic structures including pitch, side-wall angles, line heights, top and bot-tom widths. The mathematical model for scatterometry is based on the Helmholtz equation derived as a time-harmonicsolution of the Maxwell equations. It determines the incident and scattered electric and magnetic fields, which fully specifythe light propagation in a periodic two-dimensional grating structure. For numerical simulations of the diffraction pat-terns, a standard finite element method (FEM) or a generalized finite element method (GFEM) is used for solving the ellip-tic Helmholtz equation. In a first step, we performed systematic forward calculations for different varying structureparameters to evaluate the applicability and sensitivity of different scatterometric measurement methods. Furthermoreour programs include several iterative optimization methods for reconstructing the geometric parameters of the gratingstructure by the minimization of a functional. First reconstruction results for different test data sets are presented.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Indirect measurements; Mathematical modelling; Inverse methods; Diffractive optics

1. Introduction

The investigation of micro- or nano-structuredsurfaces regarding their structure geometries and

0263-2241/$ - see front matter � 2006 Elsevier Ltd. All rights reserved

doi:10.1016/j.measurement.2006.04.009

* Corresponding author. Tel.: +49 30 38417687; fax: +49 3034817030.

E-mail address: [email protected] (M. Bar).

dimensions can be performed in a rapid and non-destructive way by the measurement and analysisof light diffraction from the structured surfaces.Non-imaging metrology methods like scatterometryare in contrast to optical microscopy non diffractionlimited and they grant access to the geometricalparameters of periodic structures like structurewidth (CD), pitch, side-wall angle or line height

.

mailto:[email protected]

H. Gross et al. / Measurement 39 (2006) 782–794 783

[1,2]. An important application of scatterometricmetrology is the evaluation of structure dimensionson photo-masks and wafers in lithography [3,4]. Inparticular in the semiconductor industry both thefeature sizes and the required measurement uncer-tainty decrease continuously. On the other handconventional microscopical metrology techniqueslike atomic force, electron or optical microscopyunfortunately do not necessarily yield the sameresults. Therefore scatterometry is an importantindependent measurement technique for the charac-terization of such structures. However, scatteromet-ric methods are not necessarily unambiguous.Typically a-priori information is required, whichallows to confine the variation intervals of the struc-ture parameters to be determined. These measure-ment techniques depend crucially on a highprecision rigorous modelling of the light-structureinteraction, which includes the vectorial and the3D character of light and structure, respectively.Furthermore to determine the structure parametersfrom a measured diffraction pattern the inverse dif-fraction problem [5] has to be solved. Except for theproblem of unambiguity this is strongly related tooptimization problems for the development of dif-fractive optics (cf. [6]).

The mathematical modelling of scatterometryrequires the computation of the relation betweenthe input (the incoming wave) and output (diffrac-tion efficiencies, phase shifts). These quantities aredescribed by a model based on Maxwell’s equations[7–9]. For the numerical solution of the resultingHelmholtz equation there exists a whole variety ofdifferent methods. The most common ones areprobably the rigorous coupled wave analysis (cf.[10–14]) and the so-called C method (cf. [15]),whereas the integral equation method is the fastestfor profile gratings (cf. the references in [16]). Onthe other hand, the most flexible mathematicalmethod to solve boundary value problems for ellip-tic partial differential equations is the finite elementmethod (FEM). FEM has been applied to gratingse.g. by Urbach [17], Bao [18], and Elschner et al.[19]. Apart from the forward computations of theHelmholtz equation, the solution of the inverseproblem, i.e. the reconstruction of the grating pro-file from measured data, is desirable. Our approachhere employs a FEM-based optimization procedure[20].

This paper is organized as follows: A brief over-view on non-imaging methods and the measurementsystems will be given in Section 2. One of the most

common geometrical configurations for diffractiveoptical structures is a periodic pattern etched intothe surface of an optical substrate, such as the trap-ezoidal shaped grating with two layered materialcomponents shown in Section 2. The grating patternis often created with a sequential photolithographicmask-etch process, sometimes combined with thedeposition of additional material layers. In Sections3 and 4 we give a brief description of the mathemat-ical model and the numerical algorithms which areused to determine the diffraction of light by two-dimensional grating structures. The mathematicalmodels used are based on time-harmonic waves.Scattering of time-harmonic waves from infiniteperiodic structures is a classical problem, datingback to the Rayleigh expansion of the scatteredfield. In Section 5 we employ the program packageDIPOG (cf. [21]) to analyse the variability of thediffraction intensity pattern of trapezoidal shapedgratings in dependence on the different controlparameters. The latter include the frequency anddirection of the incident light and particularly thedimensions and optical constants of the gratinggeometry. DIPOG includes several iterative optimi-zation methods for the reconstruction of the gratingshape from a given set of the diffraction pattern. Incontrast to shape design problems we are interestedhere in the shape reconstruction from given diffrac-tion patterns. In Section 6 the reconstruction resultsfor several test data sets will be presented and dis-cussed. In particular, we are considering aspects ofthe validation such as the accuracy of reconstruc-tion in dependence on the given diffraction pattern.We will end up in a discussion of further researchand open questions.

2. Scatterometric measuring setups and test samples

Scatterometry is used as a collective term forseveral metrology methods, which may be generallydescribed as measurement techniques for a quanti-tative evaluation of surface properties by angle-resolved characterisation and analysis of lightscattered from a surface under test. Since no imag-ing optics is used, the surface and shape have tobe reconstructed from intensity and/or polarisationdata detected in the far field. Several measurementmodes can be classified as scatterometric techniques.In Fig. 1 four basic set-ups are sketched. These arestandard scatterometer, spectroscopic reflectometer,spectroscopic ellipsometer, and ellipsometric scatt-erometer.

Fig. 1. Schemes of different scatterometric setups: (a) standard scatterometry, (b) spectroscopic reflectrometry, (c) spectroscopicellipsometry and (d) ellipsometric scatterometry.

784 H. Gross et al. / Measurement 39 (2006) 782–794

A standard scatterometer (Fig. 1(a)) is used forone-dimensional periodically structured surfacesand the diffraction patterns are recorded. In the sim-plest case only planar diffraction is considered (clas-sical mount). A monochromatic light source with afixed state of polarisation is used. The polarisationis called transverse electric (TE) if the incident E-field is parallel to the grooves of the grating (alsocalled s-polarised) and transverse magnetic (TM) ifthe E-field is perpendicular to the grooves (alsocalled p-polarised). The angle of incidence h is aparameter of the experiment that is kept constantduring the measurement. Depending on the opticalconstants of the sample under test the measurementcan be performed not only in reflexion but also intransmission mode.

In addition, reflectometry measurements can berealised with such an instrument. Here the lightsource and the detector are moved simultaneouslyin such a way that the detector position is alwaysat an angle �hinc measured from the normal of thesurface. In spectral reflectometry (Fig. 1(b)) boththe angle of incidence hinc and the angular positionof the detector (at �hinc) are kept constant and themeasurement is done by varying the inspectionwavelength (e.g. by using a tuneable laser systemor a broad-band light source and a monochromatoras shown here). By adding polarisation optics thisset-up can be enhanced to a spectroscopic ellipsome-ter (Fig. 1(c)). For this a polariser in front of the lightsource and another one, working as an analyser,in front of the detector is needed. Optionally a

compensator/retarder (usually a quarter-wave plate,QWP) is placed in the optical path between bothpolarisers (either before or after the sample). Witha spectroscopic ellipsometer the influence of the sam-ple on a defined polarisation state of the incidentlight can be measured as a function of the inspectionwavelength.

In ellipsometric scatterometry (Fig. 1(d)) againthis influence on the polarisation state of the 0th dif-fraction order is measured but with respect to theangle of incidence and at a fixed wavelength. Theellipsometric measurands W and D describe the stateof polarisation. They are defined by the fundamen-tal equation of ellipsometry tan WeiD ¼ rp

rswith rp

and rs being the complex Fresnel reflexion coeffi-cients for p- and s-polarised light, respectively.When performing measurements in transmissionmode the coefficients have to be replaced accord-ingly by the Fresnel transmission coefficients.

At PTB, two scatterometric set-ups are available[22]: A standard scatterometer operating at a wave-length of 633 nm, and a spectroscopic reflectometerwhich especially can be used for measurements inthe EUV range (0.7–35 nm). The sample under testwe discuss here is a chrome on glass (CoG) mask.Their nominal geometrical parameters can be takenfrom Fig. 2 where the grating structure is shown. Inthe photolithographic imaging process CoG masksare used in transmission mode. To reduce interre-flexions, the chrome absorber is anti-reflexioncoated with CrO. Thus three optical efficaciousmaterials have to be taken into account: Cr, CrO,

Fig. 2. CoG1 grating – Chrome on glass mask used for forward calculations and reconstructions (d = 1120 nm, hSiO2 = 6.35 mm,hCr = 50 nm, hCrO = 23 nm, c = 73�).


and the fused silica substrate (SiO2). Their complexindices of refraction for wavelengths of 13.58, 193,and 632.8 nm are given in Table 1.

3. Mathematical modelling of scatterometry

In Section 2, a special structure, the CoG1 maskis described. Ideally, an optical grating is an infiniteplate consisting of different non-magnetic materialswith permeability l0 and dielectric constants �.The functional relation between the input (theincoming wave) and output (diffraction efficiencies,phase shifts) can be described by a mathematicalmodel based on Maxwell‘s equations [7,8]. Itdepends on the one hand on the parameters of theincoming wave, the wavelength k and the angle ofincidence h 2 (�p/2,p/2), and on the other handon the mask parameters, i.e. the refraction indicesand the geometric parameters of the grating. The

Table 1Optical constants of the material components of CoG1 (cf. [28]) used f

k 13.58 nm 193 nm

CrO 0.916358 + i0.044834 1.7452 + iCr 0.931609 + i0.039533 1.0549 + iSiO2 0.977821 + i0.010901 1.5608 + i

coordinate system is chosen as follows (cf. Fig. 2):the material distribution is supposed to be periodicwith period d in x direction and homogeneous par-allel to the grooves in z direction. Hence, � is invari-ant with respect to z, and the y-axis is perpendicularto the plate.

For simplicity, the upper cover material isassumed to be air and the incident wave is norma-lised to have unit amplitude. Unlike the generalized(conical) case, classical diffraction deals with inci-dent wave directions restricted to the x–y planeresulting in reflected and transmitted plane wavemodes in the x–y plane, too. Then in case of polar-ised incident light the wave is a superposition of TEand TM polarised light. Note that in the TM,respectively TE case the magnetic field H and theelectric field E remain parallel to the grooves so thatthe transverse component of H in the TM case andthe transverse component of E in the TE case can be

or the calculations and shape reconstructions

632.8 nm

1.3353 3.1185 + i0.38021.4269 3.7329 + i3.81130.000000035 1.4571 + i0.00000000058


determined from the two-dimensional Helmholtzequation

Duþ k2u ¼ 0; ð1Þwith the piecewise constant wave number functionk ¼ kðx; yÞ ¼ x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil0�ðx; yÞ

pand angular frequency x

of the incident light wave. On material interfacesthe solution u and its normal derivative onu for TEpolarisation, respectively the solution u and theproduct k�1onu for TM polarisation have to crossthe interface continuously. In the infinite regionsthe usual outgoing wave conditions for half spacesare required. Therefore, the domain X in the crosssection plane for the FEM solution of Eq. (1) canbe reduced to a rectangle with the x coordinatevarying between zero and the period d and withtwo artificial boundaries C± = {y = b±} (cf. Fig. 2).For (1), a boundary value problem on X is to besolved with quasi-periodic boundary conditionu(0,y) = u(d,y)exp(�iad) on the lateral boundarypart and with non-local boundary conditions onC±. For instance on C+, this non-local boundarycondition means that the trace @nujCþ on C+ of thenormal derivative onu must equal the y derivativeof the Rayleigh expansion (cf. Eq. (2)) of the traceujCþ of u from X. The Rayleigh expansion is a spe-cial case of the Fourier series expansion of u = Ez

at the horizontal boundaries (cf. [9]). E.g. in the caseof TE polarisation it is of the form

Ezðx; bþÞ ¼X1

n¼�1Aþn expðþibþn yÞ expðþianxÞ

þ Ainc0 expð�ibþ0 yÞ expðþiaxÞ; ð2Þ

Ezðx; b�Þ ¼X1

n¼�1A�n expðþibþn yÞ expðþianxÞ; ð3Þ

with k± = k(x,b±), an ¼ kþ sin hþ 2pd n and b�n ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðk�Þ2 � ðkþ sin hþ 2pd nÞ2

q, and Ainc

0 ¼ 1. The impor-

tant Rayleigh coefficients A�n are those with n 2 U±,

U� ¼ fn 2 Z : janj < k�g if Imk� ¼ 0

; if Imk� > 0:

(

Indeed, they describe magnitude and phase shift ofthe propagating plane waves. More precisely, themodulus jA�n j is the amplitude of the nth reflectedresp. transmitted wave mode and arg ðA�n =jA�n jÞ thephase shift. The terms with n 62 U± lead to evanes-cent waves only. The optical efficiencies of the grat-ing can be measured and are determined by

e�n ¼b�n A�n�� 2

bþ0 Ainc0

�� 2 ; ð4Þ

with (n,±) 2 {(n,+):n 2 U+} [ {(n,�):n 2 U�}. Thecoefficients of the Rayleigh expansion are com-puted from the FEM solution of Eq. (1). Then,the optical efficiencies are obtained from Eq. (4).Note that the efficiency of a transmitted or reflectedmode is nothing else but the portion of energy trans-ferred from the incoming light to this mode. Theefficiencies in case of TM polarisation are derivedanalogously.

4. Numerical algorithms for solving the Helmholtz

equations and the reconstruction problem

For the numerical solution of the boundary valueproblem from the last section, there exists a wholevariety of different methods. The most popular areprobably the rigorous coupled wave analysis (cf.[10]) and the so-called C method (cf. [15]), whereasthe integral equation method is probably the fastestfor profile gratings (cf. the references in [16]). On theother hand, the most general mathematical methodto solve boundary value problems for elliptic partialdifferential equations is the finite element method(FEM). The advantage of FEM is that it is capableto simulate the diffraction by gratings with generalgeometry due to existing general triangulation pro-grams. No approximation of the domain X by aunion of rectangular subdomains is needed, andno critical Rayleigh expansion is used. FEM hasbeen applied to gratings e.g. by Urbach [17], Bao[18], and Elschner et al. [19].

To set up the numerical scheme of finite elements,the first step is to derive a variational reformulationof the boundary value problem for the Helmholtzequation (1). Formally, a test function v is intro-duced over X. Multiplying (1) by the complex con-jugation of v and integrating over X yieldsR

XfDuþ k2ug�v ¼ 0 for all v. Applying the Gaußintegration formula and taking into account theboundary and transmission conditions, the varia-tional formulation equivalent to the boundary valueproblem of (1) over X is obtained. For instance, inthe case of the classical TM polarization this yieldsZ

X

1

k2ru � rv�

ZX

uvþ 1

ðkþÞ2Z

CþðTþa uÞvþ 1

ðk�Þ2Z

C�ðT�a uÞv

¼ � 1

ðkþÞ2Z

Cþð2ibeifax�bygÞv; v 2 H 1

pðXÞ:

ð5Þ


Here H 1pðXÞ stands for the first order Sobolev space

of quasiperiodic functions and T�a is the non-localoperator mapping a trace function u on C± to they-derivative of its Rayleigh expansion (2) resp. (3).

Having derived a variational equation like (5), theFEM is obtained by approximating u with uh from afinite element space Sh � H 1

pðXÞ and by restrictingalso the test functions v = vh to Sh. More precisely,the FEM seeks the approximate solution uh solving

ZX

1

k2ruh � rvh �

ZX

uhvh þ1

ðkþÞ2Z

CþðTþa uhÞvh

þ 1

ðk�Þ2Z

C�ðT�a uhÞvh

¼ � 1

ðkþÞ2Z

Cþð2ibeifax�bygÞvh; vh 2 Sh: ð6Þ

The space Sh is the collection of all continuous andpiecewise linear functions subordinate to a given tri-angulation of X. By h we denote the mesh size of thetriangulation, i.e. the maximal diameter of the sub-triangles of X. Clearly, the smaller h is, the better thefunction u can be approximated by uh. However,smaller h results in a larger number of subtrianglesand in a higher dimension of the vector space Sh

which increases the computational effort to solve(6). If uh is determined, then instead of u the approx-imation uh is expanded into the Rayleigh series onthe right-hand sides of (2) and (3). This way approx-imate values of the Rayleigh coefficients are gener-ated and, using (4), the efficiencies can be computed.

The FEM works well if the ratio period overwave length is moderate. However, if the last ratiois large, the solution u is oscillatory and a tiny meshsize h together with an unacceptably long comput-ing time is needed to produce a reasonable approx-imation. Better results are possible with thegeneralized FEM (GFEM). Here the only differencebetween FEM and GFEM is a different trial spaceSh of approximate functions. In the case of GFEMthe trial space consists of continuous functions therestrictions of which to the subtriangles of the trian-gulations are special solutions of the Helmholtzequation. More precisely, they are piecewiseapproximate solutions since there are no explicitHelmholtz solutions with prescribed boundary dataeven for simple domains as triangles.

For the optimal design of gratings (compare e.g.[20,8]), the starting point is a functional f defined onthe set of gratings and expressing the objective ofthe optimization task. This can be any user defined

linear or quadratic expression of the grating efficien-cies and phase shifts. For example

f ðgratingÞ :¼X

ðn;�Þ:n2U�

fa�n e�n þ b�n ½e�n � c�n �2g; ð7Þ

where the constants a�n , b�n , and c�n are fixed by theuser. Choosing aþn ¼ �1 for index n = n0 and settingall the other constants a�n , b�n and c�n to zero, theminimization of f amounts to searching for a gratingwith minimal resp. maximal efficiency eþn0

. Settingaþn ¼ 1 for all reflected efficiencies and all other con-stants to zero results in finding a grating with mini-mal reflected energy, i.e. in an antireflection grating.Values a�n different from ±1 correspond to an opti-mization of the e�n with different priorities, high pri-ority for a large modulus ja�n j and low for a small. Inthe quadratic terms b�n ½e�n � c�n �

2 the c�n are desiredvalues for the efficiencies e�n which are to be approx-imated by the real efficiencies of the optimized grat-ing. Again, the non-negative weight factors b�ncontrol the priority of the approximation task. Ofcourse, the terms like those in (7) may depend onseveral prescribed wave lengths or on several pre-scribed angles of incidence. The optimization ofthe objective functional in the class of all gratingsis, from the mathematical point of view, a so-calledseverely ill-posed problem and even the best numer-ical algorithms cannot be very accurate. Therefore,the class of admissible gratings should be restrictedto a subclass of gratings described by a small num-ber of real parameters rj, j = 1, . . . ,J. Box conditionsare supposed, i.e. the admissible parameters arerestricted to user prescribed intervals.

The program package DIPOG [21] can treatthree classes. The first class is that of polygonal pro-file gratings, i.e. of gratings consisting of only twodifferent materials separated by an interface whichis described by a polygonal curve in the cross-sec-tion. In this class the x and y coordinates of the cor-ners of the polygonal profile curve are optimized.The second class is that of stacks of a finite numberof trapezoids placed one on each other (cf. Fig. 2)and the parameters of optimization are the refrac-tive indices of the trapezoid materials and the geo-metry parameters of the trapezoids. The last classis that of a general fixed grating geometry with avarying polygonal interface curve. Here the param-eters of optimization are the refractive indices of thegrating materials (except those of the substrate andcover material) and the coordinates of the corners ofthe polygonal interface curve.


To solve the optimization, DIPOG offers threegradient based algorithms, namely a method of aug-mented Lagrangian, a conjugate gradient method,and an interior point method [24–26]. The algorith-mic parameters of these methods must be adaptedto the optimization problem by the user. Unfortu-nately, the gradient based algorithms are localmethods, i.e. the solution of these optimizationmethods render local solutions only. Recall that alocal minimum of an optimization problem is a setof parameters ~rj; j ¼ 1; . . . ; J such that there is asmall threshold e with

f ðf~rj; j ¼ 1; . . . ; JgÞ 6 f ðfrj; j ¼ 1; . . . ; JgÞ;for all {rj, j = 1, . . . ,J} with jrj � ~rjj < e; j ¼ 1 ; . . . ; J .The optimization problems for gratings usually havea lot of local minima different from global minima. Insuch a case the determination of the global optimumis difficult. A restart of the optimization algorithmfrom various initial solutions and choosing the min-imum of the local solutions can be a helpful strategy.Alternatively, DIPOG includes a global optimizationalgorithm of simulated annealing [27]. This stochas-tic algorithm, however, can be extremely time con-suming and the determination of a global solutionis not guaranteed either.

An important application for the optimizationtools is the reconstruction problem (cf. [8,23]). Herea grating of a certain class is given and the corre-sponding material, respectively geometry parame-ters should be reconstructed from the measuredefficiencies and phase shifts of the grating. This isdone by solving an optimization problem, e.g. withthe objective functional

f ðfrj; j ¼ 1; . . . ; JgÞ :¼XL

l¼1

XM

m¼1

Xðn;�Þ:n2U�

b�n ðkl; hmÞ

� ½e�n ðkl; hmÞ � c�n ðkl; hmÞ�2;ð8Þ

where kl, l = 1, . . . ,L and hm, m = 1, . . . ,M are thewave lengths resp. angles of incidence of the mea-surement, the c�n ðkl; hmÞ are the measured efficien-cies, and the b�n ðkl; hmÞ > 0 some weight factors.Clearly, the parameter set of the sought grating isa global minimum of the non-negative objectivefunctional. Though there is no theoretical resultclaiming the uniqueness of the minimal solution,one should be optimistic that the solution is uniqueat least for sufficiently many measured efficienciesand phase shifts. Then there is a good chance thatthe optimization tools of DIPOG will reconstruct

the parameters of the sought grating. Of course,the accuracy depends on the measurement uncer-tainty and on the behavior of the mapping assigningthe efficiency data to each parameter set.

5. Variation of model parameters – analysing

the dynamics of the diffraction pattern

To analyse the variability of the diffraction patternof trapezoidal shaped gratings such as CoG1,described in Section 2, we employ DIPOG by callingits FEM or GFEM executables. The most importantparameters are the optical indices of all materialcomponents and the geometrical dimensions of thegrating structure as well as the wavelength of the inci-dent light and its angle relative to the normal of thegrating surface. The period or pitch of the gratingstructure is part of the geometrical dimensions.

The scattering of the incoming light dependsstrongly on the chosen parameters. In Eq. (4) theratio of the energy of a propagating mode of thescattered waves to the energy of the incoming waveis called the efficiency of the mode. With respect tothe shape reconstruction by measured efficiencies,one of the key features of any grating is the distribu-tion of its efficiencies over the propagating modes independence from the model parameters. To supportthe ‘metrological’ expectations about the measur-able efficiencies of photo-masks such as CoG1(Chrome on Glass) we have performed systematicforward calculations for different control parame-ters. In the following we present some examples.

The Fig. 3(a) and (b) show the simulation resultsfor the reflected efficiency distribution of the polar-ization type TE if the angle of the incoming wave ischanged. Here we have used the GFEM algorithmsof DIPOG in order to get a sufficient accuracy alsofor the example with small wavelength. In Fig. 3(a)the wavelength is fixed to 13.58 nm and in Fig. 3(b)to 632.8 nm. The same trapezoidal grating (CoG1,see Section 2) is used for all calculations: TheChrome-on-glass mask is assumed with a bottomlength of 624.6 nm, an trapezoidal angle of 73�, aheight of the Cr layer of 50.0 nm and a height ofthe CrO layer of 18 nm. The period of the gratingis 1120 nm and the SiO2 substrate has a thicknessof 6.35 mm. The used optical indices for the differ-ent wavelengths are given in Table 1. In order tosupport a better recognition of the dynamicalchanges in the distribution we have chosen a 3d-pre-sentation of the calculated results formed by apseudo-surface of efficiencies as a function of the

Fig. 3. Efficiency patterns calculated for CoG1 for a variation of incidence angle h with: (a) k = 13.58 nm and (b) k = 632.8 nm.


order of diffraction and the control parameter,which is the incidence angle in the Fig. 3(a) and(b). In Fig. 3(b) one can observe increasing efficien-cies with an increasing angle of the incident light.However, for the small wavelength (Fig. 3(a)),which is comparable to the critical dimensions ofthe grating structure, the angular dependence ofthe efficiency distribution becomes more complex:The increasing efficiencies of all diffraction orderslook like a modulated periodic function. The peri-odicity is w.r.t. h and the period is about 8�.

Because the heights of the material layers andtheir top and bottom widths are crucial controlparameters, the sensitivity of the efficiency patternw.r.t. these parameters is important. Fig. 4(a) pre-sents the calculated efficiency pattern for a variationof the thickness of the chrome layer: Here we have

chosen an intermediate wavelength of 193 nm andan incidence angle of 10�. The height hCr in Fig. 2is varied in the range from 30 to 150 nm by stepsof 5 nm. A distinctive diffraction pattern with devi-ations in the reflected efficiencies of about one evenover the step size of 5 nm can be observed.

The sensitivity of the diffraction efficiencies w.r.t.different top and bottom widths of CoG1 is alsostrong as can be seen in Fig. 4(b). At the same side-wall angle of 73� we have changed the bottomlength of the trapezoid grating CoG1 from 200 nmto 900 nm in steps of 25 nm (incidence angle andwavelength: 10�, respectively 632.8 nm).

In contrast to the strong sensitivity of the efficien-cies in the previous examples a variation of the realpart of the refractive index of Cr leads to almostconstant values.

Fig. 4. Efficiency patterns calculated for CoG1 for: (a) variation of hCr in CoG1 at k = 193 nm and (b) variation of bottom length inCoG1 at k = 632.8 nm.


6. Optimization results – reconstruction of

the model parameters from simulated data sets

The shape reconstruction from a given efficiencyis an inverse problem and equivalent to the minimi-zation of an objective functional f describing the dif-ference between a calculated and a given efficiencypattern. The examples in the previous section givesome insight into the sensitivity w.r.t. the differentmodel parameters which specify the grating struc-ture and the inspecting light. The additive terms ofthe functional f in Eq. (8) depend on the efficienciesrepresenting the known data. The number M ofthese terms has always to be greater or equal tothe number N of the model parameters which are

sought. Only if this condition is fulfilled a meaning-ful reconstruction can be expected.

In order to make the reconstruction feasible, wehave reduced the inverse problem to the reconstruc-tion of a finite set of geometrical parameters specify-ing the CoG1-mask. We fix the inspectingwavelength to 632.8 nm and calculate several effi-ciency patterns for different incidence angles of theinspecting light. The efficiencies were calculatedwith the GFEM algorithm of DIPOG at a higherror level and over a triangulation different fromthose used in the reconstruction algorithm. More-over we fix the optical parameters of the trapezoidalgrating and those of the superstrate and substratematerials (air resp. SiO2) to the values given in


Table 1. Then the following seven model parametersremain for the optimization: the bottom length ofthe Cr layer, the height of the Cr layer, the x coor-dinates of the upper left and the upper right cornersof this layer, the height of the CrO layer and the x

coordinates of the upper left and the upper rightcorners of the CrO layer. The admissible domainof these parameters has to be restricted by addinglower and upper bounds and furthermore initial val-ues have to be provided. All the settings are declaredin an ASCII-formatted input file for the optimiza-tion algorithms of the DIPOG software. At leastM different values from the simulated measurementdata, i.e. from the known efficiency pattern for pos-sible different angles of incidence, have to beinserted into this input file, too. They are neededto specify the functional f. Remember, the numberof measurement data should exceed the number ofunknown parameters, i.e. M should satisfy the con-straint M P 7.

By M we denote the set of all possible efficienciesfor all possible wavelengths, incidence angles, anddiffraction orders. Clearly, for a reconstruction weuse a subset of M with M entities of the efficiencyvalues of the simulated measurement data. We havetested different subsets which satisfy the constraintM P 7. If we choose the reflected and transmittedmodes of diffraction orders 0 and �1 for the polar-ization types TE and TM at the four incidenceangles of 10�, 15�, 30�, and 35� (M = 32), a goodreconstruction result is found (cf. Fig. 5(a)). Thesmaller subset of the same reflected and transmittedmodes (order 0 and �1, polarization TE and TM) at

Fig. 5. Optimization results for simulated CoG1 test sets: (a) right recon50.04 nm (right value: 50.00 nm) and for hCrO: 23.08 nm (right value:with M = 8. For details see text.

a single incidence angle h of 80� (M = 8) leads to asurprisingly good result for the shape reconstruc-tion. Nevertheless, many other examined subsetsdid not produce acceptable results for the shape ofthe sought CoG1 grating and an example for sucha case is shown in Fig. 5(b) (reflected and transmit-ted modes of diffraction orders 0 and �1 and polar-isation type TE and TM at incidence angle h = 75�).

If we create new sets of known data by changingthe height of the CrO layer from 23 nm to 18 nm, wefind a very similar behavior. Good reconstructionresults are achieved by the same subsets and partic-ularly the new height of the CrO layer is foundproperly. In many cases of reconstruction all theseven parameters specifying the CoG1 mask arefound with an accuracy less than 1%.

If we compare the shape of the functional f forthe two subsets of the known data which aresuitable for a good reconstruction, we observe thedifferent shapes shown in Fig. 6. ParticularlyFig. 6(a) exhibits a steeper descent toward the min-imum in the direction of the hCr axis. In Fig. 6 thefunctional f is calculated varying the heights hCrand hCrO of the two trapezoidal layers. The remain-ing 5 parameters are fixed to their values used forthe simulation of the input data. For the selectedadmissible range of the 2 heights the coordinatesof the minimum values of the objective functionalare very near to the expected values of 50 nm forhCr and 18 nm for hCrO. However, they were onlyfound if the initial value for hCrO is set to a valuesmaller than 60 nm where the functional f has aridge parallel to the hCr axis. Otherwise the (second)

struction by a suitable subset (M = 32); optimized values for hCr:23.00 nm) and (b) wrong reconstruction by an unsuitable subset

Fig. 6. Objective functions for test case CoG1: hCr versus hCrO for the two suitable subsets with: (a) M = 32 (4 h 0s: 10�,15�,30�,35�) and(b) M = 8 (1 h: 80�; see text for details).


local minimum of f on the other side of this maxi-mum is found. Because we use gradient based opti-mization methods the admissible range of modelparameters and their initial values can have a stronginfluence on the accuracy of the reconstructionresult.

Besides there are further strong determinants forthe 1% accuracy of the presented reconstructionresults, namely the scaling factors of the modelparameters. The performance of the local optimiza-tion depends on a proper scaling of the parameterswhich controls the size of the partial derivativesof/ori of the functional f in Eq. (8), where ri withi = 1, . . . , 7 are sought-after model parameters. Forinstance, suppose the partial derivate of/or1 is muchlarger than the other partial derivatives. Then it may

be that the small ones are not used for a correctionof the iterative solution towards the local minimum.To prevent such a situation the model parametersand their lower and upper bound values have tobe replaced by scaled parameters (r0i ¼ siri). Choos-ing the right scaling factors, the partial derivativescan be made to be almost of the same size, andthe iteration converges well. The above mentionedcases of good reconstruction results can only befound with properly selected scaling factors.

7. Discussion

We have demonstrated that the proposed gener-alized finite element method (GFEM), implementedin the DIPOG software enables the numerical com-


putation of diffraction efficiencies, even if the ratioof inspecting wavelength (k = 13.58 nm) and gratingperiod (d = 1.12 lm) is in the range of 1/100. Thevariability of the efficiency patterns depends onthe parameters of the inspecting light and on thegeometry of the scattering probe. Here we havedetermined these quantities for examples of trape-zoidal shaped gratings typically found on CoGphotolithography masks. Forward computationswith systematic parameter changes have been per-formed in order to achieve an estimate of the sensi-tivity of the scattering efficiencies on the parametersspecifying a ‘‘virtual experimental set-up’’.

In a second step, the diffraction modes computedfrom forward calculations have been used as inputof an optimization routine aimed at reconstructingthe profile of the grating. This amounts to a solu-tion of the inverse problem that might fail if it isbased upon insufficient input data. Recall that thelast is always a subset chosen from the set of allavailable efficiencies. All together, we have testeddifferent subsets of the modelled diffraction patternswith quite different reconstruction results. Goodresults, i.e. an accuracy less than 1% for the inputparameters of modelled CoG mask, were only pro-duced for special subsets such as the scattering effi-ciencies of the TE and TM modes at four differentangles of incidence. Subsets with a single angle ofincidence in most cases are not sufficient to solvethe inverse problem. Here a systematic methodto find suitable subsets is still missing. Clearly,the optimal subsets of this method would alsoreduce the expense for the sampling of the requiredscatterometric measurement data. Also using addi-tional measurands as input data like e.g. the rela-tive phase information between the diffracted TEand TM modes (cf. ellipsometric scatterometry)could be helpful to decrease the reconstructionuncertainty.

In general, simulations are well suited to test theaccuracy of optimization algorithms and inverseproblem solvers designed for processing measure-ment data. Clearly, the determination of the uncer-tainty in indirect measurements relying upon inversemethods or related parametric fitting routines are achallenge in metrology, which will require a lot offurther mathematical analysis and computationalwork. Simulations like the ones described above willbe crucial to resolve these questions. In an impor-tant next step, we plan to reconstruct profiles ofthe gratings from experimentally measured dataalong the lines described above. This will provide

a crucial test for the accuracy of the model andthe used algorithm and may require further consi-derations regarding measurement noise.

References

[1] B.K. Minhas, S.A. Coulombe, S. Sohail, H. Naqvi, J.R.McNeil, Ellipsometric scatterometry for metrology of sub-0.1 lm-linewidth structures, Appl. Opt. 37 (22) (1998) 5112–5115.

[2] H.T. Huang, F.L. Terry, Erratum to ‘‘Spectroscopic ellips-ometry and reflectometry from gratings (Scatterometry) forcritical dimension measurement and in situ, real-time processmonitoring’’ [Thin Solid Films 455–456, 2004, 828–836],Thin Solid Films 468 (2004) 1155–1169.

[3] C.J. Raymond, M.R. Murnane, S. Sohail, H. Naqvi, J.R.McNeil, Metrology of subwavelength photoresist gratingsusing optical scatterometry, J. Vac. Sci. Technol. 13 (4)(1995) 1484.

[4] C.J. Raymond, M.R. Murnane, S.L. Prins, S. Sohail, H.Naqvi, J.R. McNeil, Multiparameter grating metrologyusing optical scatterometry, J. Vac. Sci. Technol. 15 (2)(1997) 361–368.

[5] I. Kallioniemi, J. Saarinen, E. Oja, Optical scatterometry ofsubwavelength diffraction gratings: neural-networkapproach, Appl. Opt. 37 (25) (1998) 5830.

[6] J. Turunen, F. Wyrowski (Eds.), Diffractive Optics forIndustrial and Commercial Applications, Akademie Verlag,Berlin, 1997.

[7] R. Petit (Ed.), Electromagnetic Theory of Gratings,Springer-Verlag, Berlin, 1980.

[8] G. Bao, D.C. Dobson, Modeling and optimal design ofdiffractive optical structures, Surveys on Mathematics forIndustry 8 (1998) 37–62.

[9] X. Chen, A. Friedmann, Maxwell’s equation in a periodicstructure, Trans. Amer. Math Soc. 323 (1991) 811–818.

[10] M.G. Moharam, T.K. Gaylord, Rigorous coupled waveanalysis of planar grating diffraction, J. Opt. Soc. Amer. 71(1981) 811–818.

[11] M.G. Moharam, E.B. Grann, D.A. Pommet, T.K. Gaylord,Formulation for stable and efficient implementation of therigorous coupled-wave analysis of binary gratings, JOSA A12 (1995) 1068–1076.

[12] M.G. Moharam, E.B. Grann, D.A. Pommet, T.K. Gaylord,Stable implementation of the rigorous coupled-wave analysisfor surface-relief gratings: enhanced transmittance matrixapproach, JOSA A 12 (1995) 1077–1086.

[13] P. Lalanne, G.M. Morris, Highly improved convergence ofthe coupled-wave method for TM-polarization, JOSA A 13(1996) 779–784.

[14] L. Li, New formulation of the Fourier modal method forcrossed surface-relief gratings, JOSA A 14 (1997) 2758–2767.

[15] J. Chandezon, D. Maystre, G. Raoult, A new theoreticalmethod for diffraction gratings and its applications, J. Opt.(Paris) 11 (1980) 235–241.

[16] B.H. Kleemann, Elektromagnetische Analyse von Oberfla-chengittern von IR bis XUV mittels einer parametrisiertenRandintegralmethode: Theorie, Vergleich und Anwendun-gen. Dissertation, TU Ilmenau, 2002, Mensch und BuchVerlag Berlin, 2003.


[17] H.P. Urbach, Convergence of the Galerkin method for two-dimensional electromagnetic problems, SIAM J. Numer.Anal. 28 (1991) 697–710.

[18] G. Bao, Finite element approximation of time harmonicwaves in periodic structures, SIAM J. Numer. Anal. 32(1995) 1155–1169.

[19] J. Elschner, R. Hinder, G. Schmidt, Finite element solutionof conical diffraction problems, Adv. Comp. Math. 16 (2002)139–156.

[20] J. Elschner, G. Schmidt, Numerical solution of optimaldesign problems for binary gratings, J. Comput. Physics 146(1998) 603–626.

[21] J. Elschner, R. Hinder, A. Rathsfeld, and G. Schmidt,DIPOG Homepage. Available from: <http://www.wias-berlin.de/software/DIPOG>.

[22] M. Wurm, B. Bodermann, W. Mirande, Investigation andevaluation of scatterometric CD metrology methods, Proc.SPIE 5858 (2005) 585813.

[23] J. Chandezon, A.Ye. Poyedinchuk, N.P. Yashina, Recon-struction of periodic boundary between dielectric media, in:Mathematical Methods in Electromagnetic Theory, Confer-ence Proceedings of MMET*02, Kiev, 2002, pp. 416–419.

[24] Ch. Großmann, J. Terno, Numerik der Optimierung, Teub-ner Studienbucher der Mathematik, Teubner Stuttgart, 1997.

[25] F. Jarre, J. Stoer, Optimierung, Springer Verlag, New YorkBerlin Heidelberg, 2004.

[26] J. Nocedal, S.J. Wright, Numerical optimization, SpringerVerlag, New York, Berlin Heidelberg, Springer Series inOperation Research, 2000.

[27] P.J.M. van Laarhoven, E.H.L. Aarts, Simulated annealing:Theory and Applications, D. Reidel Publishing Company,Mathematics and Applications, Member of Kluwer Aca-demic Publishing Group, Dodrecht Boston LancasterTokyo, 1988.

[28] E.M. Gullikson, X-ray Interactions With Matter. Availablefrom: <http://www-cxro.lbl.gov/optical_constants>.

http://www.wias-berlin.de/software/DIPOG

http://www.wias-berlin.de/software/DIPOG

http://www-cxro.lbl.gov/optical_constants

mathematical modelling of indirect measurements in scatterometry

Documents