Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials

Transcription

1 Convergence properties of critical imension measurements by spectroscopic ellipsometry on gratings mae of various materials Roman Antos a Research Institute of Electronics, Shizuoka University, Johoku 3-5-1, Hamamatsu , Japan an Frontier Research System, RIKEN, 2-1 Hirosawa, Wako, Saitama , Japan Jaromir Pistora Department of Physics, VSB-Technical University of Ostrava, 17 Listopau 15, Ostrava-Poruba, Czech Republic Jan Mistrik an Tomuo Yamaguchi Research Institute of Electronics, Shizuoka University, Johoku 3-5-1, Hamamatsu , Japan Shinji Yamaguchi an Masahiro Horie Dainippon Screen Mfg. Co., Lt., Kyoto , Japan JOURNAL OF APPLIED PHYSICS 100, Stefan Visnovsky Institute of Physics, Faculty of Mathematics an Physics, Charles University, Ke Karlovu 5, Prague 2, Czech Republic Yoshichika Otani Institute for Soli State Physics, University of Tokyo, Kashiwa, Chiba , Japan an Frontier Research System, RIKEN, 2-1 Hirosawa, Wako, Saitama , Japan Receive 13 February 2006; accepte 24 July 2006; publishe online 8 September 2006 Spectroscopic ellipsometry SE in the visible/near-uv spectral range is applie to monitor optical critical imensions of quartz, Si, an Ta gratings, namely, the epth, linewith, an perio. To analyze the SE measurements, the rigorous couple-wave theory is applie, whose implementation is escribe in etail, referre to as the Airy-like internal reflection series with the Fourier factorization rules taken into account. It is emonstrate that the Airy-like series implementation of the couple-wave theory with the factorization rules provies fast convergence of both the simulate SE parameters an the extracte imensions. The convergence properties are analyze with respect to the maximum Fourier harmonics retaine insie the perioic meia an also with respect to the fineness of slicing imperfect Ta wires with paraboloially curve eges American Institute of Physics. DOI: / I. INTRODUCTION Specular-moe spectroscopic ellipsometry SE has become perhaps the most important technique within optical scatterometry, especially for its precision, sensitivity, applicability for in situ, an other avantages over the conventional critical-imension measurement techniques such as scanning electron microscopy or atomic force microscopy. 1 In recent years scatterometry of iffraction gratings base on SE has wiely sprea owing to consierable progress in computation scopes that are necessary for the effective analysis of experimental iffraction-response ata, particularly for the real-time process monitoring. 2 5 Moreover, progress in theoretical algorithms with improving convergence properties enable researchers to refine the corresponence between surface-relief profiles monitore optically an profiles visible by conventional scanning probe methos. In particular, the introuction of Li s Fourier factorization rules for electromagnetic fiels propagating insie meia with perioic iscontinuous permittivity remarkably enhance the convergence of the algorithms. 6,7 a Author to whom corresponence shoul be aresse; electronic mail: antos@riken.jp Among many iffraction theories in literature, the couple-wave theory is the mostly use metho for scatterometry, wiely referre to as the rigorous couple-wave analysis. 8 Authors presenting various implementations of the theory usually focus their numerical analyses to singlewavelength numerical experiments on hypothetical gratings that are consiere perfectly known, an present convergence properties accoring to the number of Fourier harmonics retaine insie a perioic meium. 9 On the other han, authors of scatterometric analyses on real samples o not provie etails of algorithms they use, thereby not clarifying the effectiveness of their methos. For this reason, we evote this paper to the complete escription of the opticalscatterometric technique base on SE, incluing the etaile escription of our implementation of the couple-wave theory, its convergence properties with respect to extracting the imensions an topographic ata, an the comparison of results obtaine on gratings mae of transparent quartz, semiconuctive silicon, an metallic tantalum. After the appearance of original papers on couple waves initially implemente as the numerically unstable transfer-matrix approach, 8 the metho was revise several times to obtain algorithms usable for general structures, particularly with respect to higher epths of patterning an /2006/1005/054906/11/$ , American Institute of Physics

2 Antos et al. J. Appl. Phys. 100, FIG. 1. Diffraction of a polarize plane wave by a layer with perioic permittivity 1 y sanwiche between a superstrate with uniform permittivity 0 an a substrate with uniform permittivity 2. Owing to the symmetry properties of the planar iffraction mounting, all the wave vectors kˆ n r,t of the iffracte waves lie within the y-z plane an each of the two principal polarizations s an p is preserve. highly conuctive materials. There arose, for example the R-matrix propagation algorithm 10,11 borrowe from chemical-physics literature 12 an consecutively the scattering S matrix approach, 13,14 which has become the most common algorithm up to now. Within the theoretical part of this paper we escribe in etail a physically transparent implementation of the couple-wave approach, which we have applie in several recent scatterometric stuies an as a control moel to check the accuracy of more general simulations of iffraction on anisotropic gratings in limite cases We refer to our implementation as the Airy-like internal reflection series. It is worth noting two technical etails relate to an experimental metho applie here an a special sample uner investigation. First, application of a microscopic ellipsometer in the case of the Si gratings seems to be very avantageous. For the sake of prouction-quality control of lithography, the time-consuming patterning is preferable to perform on as small area as possible, to only make the grating bigger than or equal to the size of the incient beam an contain the sufficient number of perios to preserve the valiity of the Fraunhofer-iffraction approximation. Secon, we perform a topographical analysis on a high-quality Ta wire-gri polarizer fabricate with parameters to polarize visible light, which has so far not been carrie out by means of SE. Note that metallic gratings have generally been stuie by SE very rarely, 22 probably ue to ifficulties couple with the variability of material constants of metals an the high sensitivity to their surface properties. II. MATHEMATICAL CALCULATION OF THE ELLIPSOMETRIC PARAMETERS A. Formulation of the iffraction problem In orer to clearly emonstrate the Airy-like series implementation of the couple-wave theory, we evote this section to a etaile escription of iffraction of a polarize plane wave by a layer with perioic permittivity 1 y sanwiche between two semi-infinite ambient meia, a superstrate with uniform permittivity 0 an a substrate with uniform permittivity 2. The geometrical configuration of the problem is epicte in Fig. 1 incluing the enomination of the Cartesian coorinates, two principal polarizations, an irectional wave vectors of plane waves, kˆ i that is incient an kˆ n r an kˆ n t that are reflecte an transmitte into the nth iffraction orer, respectively. We assume the planariffraction mounting an the far-fiel Fraunhofer-iffraction approximation throughout this paper. Owing to the symmetry properties of the mounting, the iffraction response preserves the s polarization of a transverse-electric incient wave; the irection of the s polarization is here ientical to the irection of the x axis. We introuce unit vectors xˆ =ŝ, ŷ, an ẑ, parallel to the Cartesian coorinates. Similarly, in the case of a transverse-magnetic incient wave with p polarization, whose irection is perpenicular to both the ŝ an kˆ i vectors i.e., pˆ i=kˆ i ŝ, all the iffracte waves are p polarize as well provie that the normalize polarization vector of the nth iffraction orer is efine as pˆ r,t n =kˆ r,t n ŝ. For the sake of simplicity, we first limit our emonstration to the s polarization. Furthermore, we assume the e it time epenence throughout this paper, an the space coorinates scale by the 2/ factor, being the wavelength in vacuum. With these assumptions we can formulate the problem of fining the reflecte an transmitte amplitues of the electric fiel, enote f r n an f t n, respectively, corresponing to the nth iffraction orer. By means of Rayleigh s expansion, the incient, reflecte, an transmitte electric fiels are written E i i y,z = xˆ f 0 exp iq 0 y + s 0 0 z, 1a E r y,z = xˆ f r n exp iq n y s 0 n z, 1b E t t y,z = xˆ f n exp iq n y + s 2 n z, 1c respectively, where the x epenence is omitte owing to the planar-iffraction mounting an where the components of the wave vectors are efine as follows: q n = sin i + nq, 2a s n J = J q n 2, q =, 2b 2c where i an enote the angle of incience an the perio of the function 1 y, respectively, with J inexing here the uniform meia 0 an 2. B. Electromagnetic waves in a perioic meium Defining H =c 0 H, with c, 0, an H representing the light velocity in vacuum, the magnetic permeability in vacuum, an the magnetic fiel vector in the Système International SI units, respectively, we write Maxwell s equations insie perioic meium 1 in a concise, timeinepenent form E = ih, 3a

3 Antos et al. J. Appl. Phys. 100, k= 1 n k f 1 k q n 2 f 1 n = s 2 f 1 n, 8 which we transform to an eigenvalue equation of a matrix C 1 with eigenvalues j 1 an eigenvectors f j 1, FIG. 2. Rectangular grating patterne in a homogeneous layer ientifie by geometrical parameters, the epth, linewith W, an perio, an material parameters, the permittivity of wires w 1 an the permittivity of the meium between wires 1 b. C 1 f j 1 = j 1 f j 1, C 1 = 1 q 2, j 1 = s j 1 2, 9a 9b 9c H = i 1 ye. 3b In the case of a rectangular grating patterne in a homogeneous layer, isplaye in Fig. 2, we aopt the terminology of perioic wires commonly use for metallic gratings. Then the permittivity is a perioic function of only one lateral coorinate y with perioicity, assuming two values, = 1 y 1 w y 0,W 1 b y W,, where w 1 enotes the permittivity of wires, b 1 the permittivity of the meium between wires, an W the with of wires. Note that the bounary point between the wire an the space y=w as well as the limit points of the perioicity interval 0, are not important for evaluating the Fourier series of the function 4. Assuming s polarization, we may rewrite Eqs. 3 into a scalar wave equation for the single component E 1 =E x 1 as follows: 1 y + y 2 E 1 y,z = z 2 E 1 y,z, in which y an z represent partial erivatives with respect to the scale space coorinates. Next we transform Eq. 5 into a matrix eigenvalue problem by expaning the permittivity function an the electric-fiel function into the Fourier an pseuo-fourier series, respectively, i.e., 1 y = n 1 e inqy, E 1 y,0 = f 1 n e iq n y, 4 5 6a 6b where we have utilize Floquet s theorem. Looking for a propagation eigenmoe E 1s with the epenence E 1s y,z = E 1s y,0e isz assuming only propagation in the irection of the increasing z coorinate, we obtain from Eq. 5 using Eqs. 6 an 7 the following: 7 where = is a Toeplitz matrix compose of the Fourier coefficients n of the permittivity function 1 y truncate for the purpose of numerical evaluation, q = iag...q 2,q 1,q 0,q 1,q 2,..., 11 an with f j 1 being column vectors compose of the pseuo- Fourier coefficients f n 1. The subscript j inexes all the eigenvalue-eigenvector solutions of Eq. 9a. Let f 1 0 be a column vector of the pseuo-fourier coefficients from Eq. 6b in the plane z=0, an let f 1 be an analogous column vector corresponing to the plane z =. Then we can escribe the propagation of waves between the two planes as a matrix transform f 1 = P 1 f 1 0, in which the propagation matrix P 1 is efine as 12 1e is1 0 0 P 1 0 e is 2 0 = T 0 0 e is 3 13 ] ] ] T1 1, where T 1 is the transformation matrix between the Fourier an eigenmoe representations in meium 1; the columns of T 1 are the eigenvectors of C 1, an for brevity, we enote s 1 j =s j. Similar to Eq. 7, we can look for propagation moes in the opposite irection backwar moes E 1 s back y,z =E 1 s back y,0e +isz, which leas to an analogous matrix formula f back 1 =P f back 0. By inverting the propagation matrix we may conclue that the propagation in the opposite irection is escribe by the same matrix transform, i.e., 1 f back 0=P 1 1 f back.

4 Antos et al. J. Appl. Phys. 100, FIG. 3. Diffraction on the interface between meia 0 an 1. In general, incient, reflecte, an transmitte waves are ientifie with column vectors f i, f r, an f t, respectively, whose elements are the components of the pseuo- Fourier series of the corresponing electric fiels. The iffraction response is then escribe by linear transforms between those vectors, R 01 for reflection an T 01 for transmission. C. Bounary conitions The conitions on the bounary between two perioic meia or between a perioic an a homogeneous meium can be expresse as the requirement for the continuity of the tangential components of the electric an magnetic fiels. For the case of s polarization, we require the continuity of the components Ey,z an H yy,z, the latter of which follows from Eq. 3a for any of the Jth meium, H y J y,z = i z E J y,z. For the propagation moe proportional to e isz,weget H y Js y,z = se Js y,z which, after the pseuo-fourier expansion H y J y,0 = g n J e iq n y, becomes a matrix transform g J z 0 = D J f J z 0, where f J z 0 an g J z 0 are the column vectors of the pseuo-fourier coefficients of the fiels E J y,z 0 an H J y y,z 0 in any plane z=z 0. Here D J = T Js s s 3 ] ] ] TJ 1 18 represents the so-calle ynamical matrix of the Jth meium. For the illustration of applying the bounary conitions, we first calculate the iffraction response of the interface between meia 0 an 1, which is symbolically epicte in Fig. 3. Let f i an f r be the column vectors of the pseuo- Fourier coefficients of the incient an reflecte waves, respectively, an f t of the fiel transmitte into the perioic meium. All of the three vectors are evaluate in the plane z=0. Using the matrix formalism, we express the continuity of the electric an magnetic fiel at the interface as FIG. 4. Airy-like series as an implementation of the couple-wave theory base on superpositioning contributions from multiple reflections an propagations insie the perioic meium 1 sanwiche between the upper meium 0 an the lower meium 2. f i + f r = f t, D 0 f i f r = D 1 f t, 19a 19b respectively, from which we erive the reflection an transmission matrix transforms at the interface 0-1 as f r = R 01 f i, f t = T 01 f i. 20a 20b On the base of Eqs. 19, the reflection an transmission matrices can be efine as R 01 = 1 + D 1 1 D D 1 1 D 0, T 01 = 1 + R 01, respectively, where 1 enotes the unit matrix. D. Airy-like internal reflection series 21a 21b The essence of our implementation of the couple-wave theory is base on ecomposing the electromagnetic fiel insie the perioic meium into a series of multiple internal reflections an propagations, as epicte in Fig. 4. The total reflecte an transmitte fiel of the sanwich structure is thus expresse as a sum of contributions f r = f 0 r + f 1 r + f 2 r +, f t = f 1 t + f 2 t + f 3 t +. 22a 22b The corresponing reflection an transmission matrices then become R 02 = R 01 + T 10 P 1 R 12 P 1 T 01 + T 10 P 1 R 12 P 1 R 10 P 1 R 12 P 1 T 01 +, 23a

5 Antos et al. J. Appl. Phys. 100, T 0,J+1 = T J,J+1 1 Q J 1 P J T 0,J, 25b with Q J =P J R J,0 P J R J,J+1, an similarly for R J+1,0 an T J+1,0, until we obtain the matrices with J=N, corresponing to the total structure. FIG. 5. Recursive algorithm for applying Airy-like series to a general nonrectangular profile of wires that must be slice into N sublayers. Within each of the recursive iterations, a partial multilayer consisting of the first J 1 slices is regare as a pseuointerface whose reflection transmission matrices RT 0,J+1, RT J+1,0 are known. The Airy-like series is then applie to the Jth layer to obtain the matrices RT 0,J, RT J,0, until we reach the substrate. The slicing shoul be performe so that artificial roughness thereby prouce is of a scale below the wavelength. T 02 = T 12 P 1 T 01 + T 12 P 1 R 10 P 1 R 12 P 1 T 01 +, 23b where the brackete term is repeate with an increasing exponent. If we apply the formula for the geometric series j=0 Q j =1 Q 1, the result will be expresse in a concise form R 02 = R 01 + T 10 P 1 R 12 1 Q 1 P 1 T 01, 24a T 02 = T 12 1 Q 1 P 1 T 01, 24b where Q=P 1 R 10 P 1 R 12 is the matrix coefficient of the geometric series. E. Recursive algorithm for a slice nonrectangular relief So far we have assume only a rectangular-relief grating, ientifie by epth an one perioic permittivity function 1 y corresponing to the first layer. In orer to escribe more general relief profiles by means of perioic permittivity, we must slice the relief into N sufficiently thin layers so that the artificial roughness thereby prouce can be neglecte, as epicte in Fig. 5. The relief is then ientifie by a set of epths J an a corresponing set of perioic permittivity functions J y, each of those with the same perio ; j inexes the sublayers from 1 to N an the ambient meia numbere 0 an N+1. To evaluate the optical response of the multilayer, we utilize Eqs.24 recursively. Assuming we have evaluate the reflection an transmission matrices R,0j, R J,0, T 0,j, an T J,0 of a partial system of first J layers for both inciences from the top an bottom, we may simply regar this partial system as a pseuointerface fully escribe by these matrices. Then we apply the Airy-like series to the J+ 1st layer cf. Eqs. 23 an 24, i.e., R 0,J+1 = R 0,J + T J,0 P J R J,J+1 1 Q J 1 P J T 0,J, 25a F. Transverse-magnetic p polarization The evaluation of the iffracte amplitues for the p polarization follows the lines analogous to the above case of s polarization. We apply the uality transformation an replace the electric an magnetic fiels with each other to obtain formulas formally ientical to the case of s polarization, but with ifferent meaning of iniviual components. Now the magnetic fiel is the funamental quantity treate insie a perioic meium, while the electric fiel being the erive quantity helps us to etermine the bounary conitions. The wave equation for the single component of the magnetic fiel H =H x, following from Eqs. 3, becomes J y + J y y 1 J y yh y,z = z 2 H y,z, 26 where the superscript J inicates that we are treating the Jth perioic layer. Instea of Eqs. 1, 6b, 7, 8, 9b, 14 16, an 18 we now write H i i y,z = xˆ f 0 exp iq 0 y + s 0 0 z, H r y,z = xˆ f r n exp iq n y s 0 n z, H t t y,z = xˆ f n exp iq n y + s 2 n z, H J y,0 = f J n e iq n y, H Js y,z = H Js y,0e isz, k= J n k m= C J = J J q J 1 q, E y J y,z = E y Js y,z = J n m q m J 1 mk q kf J k = s 2 f J n, i J zh J y,z, E y J y,0 = s J y H Js y,z, g n J e iq n y, 27a 27b 27c

6 Antos et al. J. Appl. Phys. 100, D J = J 1 T Js s s 3 ] ] ] TJ Otherwise, the remaining formulas for etermining the iffracte amplitues f r t n an f n stay same. TABLE I. Nominal grating parameters of the samples uner measurement. Sample Structure lamella/substrate Depth Linewith Perio A SiO 2 /SiO B Si/ Si C Ta/SiO G. Fourier factorization Appropriate transformation of Eq. 26 into a matrix formula requires particular treatment of iscontinuous perioic functions, known as the Fourier factorization rules introuce by Li. 6 Accoring to his three factorization theorems, instea of Eqs. 30 an 31 we obtain 1 1 q m J 1 mk q kf J k = s 2 f J n, k= 1 1 Jnk m= C J = J J q J 1 q, Jnm where we have efine J =1/ J 1. For an arbitrary perioic function Fy, the symbol F, like in Eq. 10, enotes the Toeplitz matrix compose of its Fourier coefficients. Several authors escribe more general rules usable for relief gratings with large slopes of eges of wires, referre to as the fast Fourier factorization metho. 7 In this paper, however, we limit our analyses to rectangular an quasirectangular gratings, so such generalization woul not be necessary. H. Ellipsometric parameters For the purposes of this paper, we evaluate the two relevant ellipsometric parameters an in the specular moe, i.e., in the zeroth iffraction orer of reflection. We assume each of the incient s- an p-polarize waves, respectively, with the epenence exp iq 0 y + 0 q 0 2 z, 38 which can be expresse in a matrix form of the Fourier representation at z=0 as 0 0 f =] i ]. 39 The vectors of reflecte an transmitte waves can then be calculate as f sr s = R 0,N+1 f si, 40a f pr p = R 0,N+1 f pi, 40b f st s = T 0,N+1 f si, 40c f pt p = T 0,N+1 f pi, 40 where the superscripts s an p represent inepenent calculations corresponing to the s an p polarizations, respectively. Finally, the ellipsometric parameters follow from the complex-number ratio tan exp i = R p 0,N+1;00 s, 41 R 0,N+1;00 s p where R 0,N+1;00 an R 0,N+1;00 enote the 0,0 element of the s p matrices R 0,N+1 an R 0,N+1 accoring to the pseuo-fourier expansions Eqs. 1 an 27. The minus sign on the right p han sie of Eq. 41 is because R 0,N+1 represents the transformation matrix for the magnetic fiel, whereas in ellipsometry we evaluate the electric one. III. SAMPLES AND EXPERIMENTAL PROCEDURES Three sets of samples were analyze, mae as A rectangular-relief gratings patterne on a fuse-quartz substrate, B rectangular-relief gratings patterne on a Si substrate, an C rectangular-relief gratings patterne in a Ta thin film eposite on a fuse-quartz substrate. For the purposes of this paper we have chosen one sample from each set, with their fabrication parameters summarize in Table I. The Ta film of sample C was eposite by means of rf sputtering. The patterning was mae by means of x-ray lithography. The SE measurements on samples A an C were performe on a four-zone null spectroscopic ellipsometer with the polarizer-sample-compensator-analyzer configuration, 23 covering the wavelength range of nm, an with an ajustable angle of incience i. Here we report on measurements at i =70. To avoi the influence of incoherent backreflections from the transparent quartz substrates for which the specular-moe SE is highly sensitive, we utilize the liqui solution proceure whose etails are escribe in Ref. 17. For the present samples we use the same glycerin solution. The SE measurements on sample B were performe on a rotating-analyzer ellipsometer in the polarizer-sampleanalyzer configuration with a measurement spot iameter of 30 m, covering the wavelength range of nm, an with a fixe angle of incience i = The technical etails of the apparatus are escribe in Ref. 24. Since the measurement on an ellipsometer in the polarizer-sample-

7 Antos et al. J. Appl. Phys. 100, analyzer configuration oes not provie the imaginary component of tan exp i the value in Eq. 41, we present, in this case, the values of an cos. IV. DATA PROCESSING Optical scatterometry is a measurement technique base on analyzing the experimental optical response of laterally patterne structures in orer to etermine their material an/or geometrical properties, particularly optical critical imensions. The principle of etermining unknown grating parameters, known as fitting, is minimizing the ifference between simulate an measure values. This error is here evaluate as the angular istance between the measure an simulate points plotte on Poincaré s sphere. This istance is specifie by the azimuthal angle 2 an the polar angle, i.e., cos E j = S e,j S m,j, 42 where E j represents the error of the jth value, while S e,j an S m,j enote the three-imensional normalize Stokes vectors, = sin 2 e,j cos e,j S e,j sin 2 e,j sin e,j, 43a cos 2 = e,j sin 2 m,j cos m,j S m,j sin 2 m,j sin m,j, 43b cos 2 m,j of the jth experimental Eq. 43a an moele Eq. 43b ellipsometric values, respectively, in an ieal nonepolarizing system. The ot on the right han sie of Eq. 42 enotes the scalar prouct. In the case of fitting, the sum of the squares of ifferences E 2 LS = j E 2 j is minimize. In tables, however, the average error over M experimental values E=1/M M j=1 E j is liste. To evaluate errors of values measure by rotatinganalyzer ellipsometry sample B, which only provies Ree i e,j=cos e,j instea of the complete information on phase e,j, we simply complete that information by efining sin e,j = sinarccoscos e,j, 44a sin m,j = sinarccoscos m,j, 44b to substitute in Eqs. 43. In orer to perform a topographical analysis on the samples, we parametrize the geometrical relief profile by expaning the z epenence of the filling factor into a Taylor series up to the secon orer, i.e., wz = w 1 + w 2 2z 1 w 3 2z 1 2, 45 where is the epth of the relief structure an w j represents parameters of the relief profile. The function wz=wz/ epens on the normal coorinate z an has its omain of efinition between 0 an. For rectangular gratings, only w 1 =W/ is nonzero, where W is the linewith constant within epth. Trapezoial-wire gratings are characterize by w 1 an w 2 that are relate to the top linewith by W0 =w 1 w 2 an to the average linewith W/2=w 1. In a special case of wires with paraboloial eges, the grating is characterize by w 1 an w 3 that are relate to the top linewith by W0=w 1 w 3 an to the maximum linewith W/2=w 1. Finally, a general profile is characterize by all the three w j parameters in Eq. 45, or more when assuming a polynomial approximation of higher orer. For the purpose of the application of the Airy-like series, we slice the relief profile into N sublayers an corresponingly evaluate N iscrete values of Eq. 45 as 1 w J = w 1 + w 2 2J 1/2 N w 3 2J 1/2 12 N 46 for each Jth sublayer. In the case of sample C, the top layer correspons to a native Ta 2 O 5 overlayer assume present on the top of wires, an hence the relief slices are inexe from 2toN+1. The convergence properties are in this paper presente either accoring to the increasing number of slices N or accoring to the maximum iffraction orer n max retaine in the Rayleigh, Floquet, an Fourier expansions in Eqs. 1, 6, 27, an 28, where the infinite sum is replace with +n max n= nmax. This means that the fiel in the perioic meium as well as in the sanwiching meia is being escribe by the so-calle 2n max +1-wave approximation, the value of which correspons to the imension of the column vectors an to the orer of the matrices of the linear-algebraic equations in Sec. II. The epenences of the optical critical imensions on N an n max are obtaine by applying the fitting proceure escribe above for each pair of N an n max that are fixe uring fitting. In the calculations we use optical constants of SiO 2, Si, an Ta 2 O 5 publishe in references. 25,26 To obtain the constants of Ta, however, we analyze optical experiments carrie out on a thin-film reference sample prepare with the same conitions as a nonpatterne version of sample C but with a shorter eposition time to facilitate measurement in the transmission moe. A joint analysis of the SE an energy transmittance measurements yiele the refractive inex an extinction coefficient of the Ta Fig. 6 as well as the thickness t Ta2 O 5 =4.5 nm of the native oxie overlayer assume present on the top of the Ta film. V. RESULTS AND DISCUSSION In the case of samples A an B, the simulation corresponing to rectangular gratings, that assume a single slice N=1, aequately interprets the SE experimental ata. The epenences of extracte imensions on increasing n max are summarize in Table II, together with imensions assuming an ieal rectangular-wire grating profile of sample C as well. The corresponing fitte SE parameters, an, are isplaye for a few emonstrative examples of n max in Fig. 7 for sample A, in Fig. 8 for sample B, an in Fig. 9 for sample C.

8 Antos et al. J. Appl. Phys. 100, FIG. 6. Refractive inex n an extinction coefficient k specifie by an analysis of SE an energy transmittance measurement carrie out on a nonpatterne reference Ta/quartz sample. As clearly visible from Fig. 7, the SE response of the transparent sample in the spectral range of interest is perfectly escribe by the three-wave approximation since the curves corresponing to n max =1 are almost overlappe with those of the 21-wave approximation corresponing to n max =10. It is nearly surprising that the three-wave approximation is capable of simulating with high accuracy not only the two interference oscillations below 500 nm of wavelength but also Woo s anomaly appearing near Rayleigh s wavelength of 504 nm at which the minus first iffraction orer of reflection passes off. 15 What has been state about sample A can analogously be repeate for the case of sample B except that instea of three waves we nee the five-wave approximation to interpret the SE measurement correctly. This is evient in Fig. 8 where we compare two fits, with n max equal to 2 an 10. In particular, the three-wave approximation woul be insufficient in the near-uv spectral range where Si is more absorbing. Slight iscrepancies between the measure an fitte SE values in the peripheral parts of the spectral range can be explaine by the presence of a TECH SPEC heat absorbing glass filter within the rotating-analyzer ellipsometer. The filter cuts off the intensity of light out of the range nm an hence increases the noise an systematic errors of SE measurement near the limit points. Contrary to samples A an B, the SE response of sample C in Fig. 9 cannot be correctly interprete as a response measure on an ieal rectangular-wire grating, as is obvious FIG. 7. Experimental marks an fitte four-zone null SE parameters, an, on a rectangular-relief quartz grating sample A in incience of 70 for two examples of maximum Fourier harmonics, n max =1 soli curves an n max =10 otte curves. Obviously, the three-wave approximation corresponing to n max =1 is sufficient to fit the measurement almost perfectly. from the iscrepancy between the experimental ata an the fits using any value of n max. However, carrying out the same proceure is useful to fin out that the nine-wave approximation corresponing to n max =4 is necessary in the case of a similar but rectangular metallic grating. Rather than a rectangular-wire grating, we ientify the geometrical profile of sample C with a profile whose filling factor epens on the z coorinate as escribe by Eq. 45. Accoring to various numerical experiments performe, we consier the profile as symmetric wires with paraboloial eges, i.e., with the linear element in Eq. 45 omitte. Thus four imensions are use to ientify the profile, i.e., epth, top linewith, maximum linewith in half-epth, an perio. Two kins of analyses on the convergence properties of extracte imensions were carrie out: accoring to increasing n max while the number of slices N was fixe to 20 Table III an accoring to increasing N while n max was fixe to 20 Table IV. TABLE II. Fitte critical imensions epth, linewith W, an perio an errors of the fits of rectangular gratings accoring to increasing n max. Sample A Sample B Sample C n max W Error eg W Error eg W Error eg

9 Antos et al. J. Appl. Phys. 100, TABLE III. Fitte critical imensions epth, top linewith W top, maximum linewith W max, an perio an errors of the fits of the Ta paraboloial-wire grating accoring to increasing n max with a fixe number of slices N=20. n max W top W max Error eg FIG. 8. Experimental marks an fitte rotating-analyzer SE parameters, an cos, on a rectangular-relief Si grating sample B in incience of for two examples of maximum Fourier harmonics, n max =2 soli curves an n max =10 otte curves. In this case the five-wave approximation corresponing to n max =2 is sufficient to fit the measurement. The fitte SE parameters corresponing to the former case for three emonstrative examples of n max are isplaye in Fig. 10. In most of the spectral range, the 17-wave approximation corresponing to n max =8 is sufficient to perform a reliable fit. Higher value of n max can slightly improve the accuracy of simulation in the short-wavelength range an at the extreme points near Rayleigh s wavelengths, one at 388 nm where the minus first iffraction orer of reflection passes off an another at 480 nm where the minus first iffraction orer of transmission passes off. The fitte SE parameters with n max fixe to 20 for four emonstrative examples of N are isplaye in Fig. 11. Here the agreement between experiment an fitting is improving graually while N is increasing. Again, the short spectral ranges near Rayleigh s wavelengths are most sensitive to insufficient slicing. We can conclue that the fit with N=10 is satisfactory accoring to the error of the SE parameters efine by Eq. 42. Finally, in Fig. 12 we present the convergence epenences of relative imensions, i.e., parameters efine as pn max / pn max, where pn max enotes a imension whose convergence properties accoring to increasing n max are stuie, while pn max represents a reference value obtaine from a fit that assumes sufficiently high n max, chosen 50 for both the rectangular-grating profiles an Ta paraboloial-wire profiles. Thus, the subfigures epict how the imensions are relatively approaching their approximate-limit values with high n max. In Figs. 12a 12c, the convergences of the relative epth, linewith, an perio are isplaye as obtaine on samples A, B, an C, respectively. The errors of all the imensions of the transparent grating are less than 1% for n max 2 an are rapily ecreasing for higher n max Fig. 12a. Note that the perio converges fastest, while the line- TABLE IV. Fitte critical imensions epth, top linewith W top, maximum linewith W max, an perio an errors of the fits of the Ta paraboloial-wire grating accoring to increasing number of slices N with fixe n max =20. FIG. 9. Experimental marks an fitte four-zone null SE parameters, an, on a rectangular-wire-assume Ta grating sample C in incience of 70 for three examples of maximum Fourier harmonics, n max =2 ashe curves, n max =4 soli curves, an n max =10 otte curves. Since the artificially assume rectangular profile oes not correspon to reality, none of the approximations fits the measurement correctly. Nevertheless, the ninewave approximation corresponing to n max =4 can obviously be consiere sufficient to simulate a rectangular-wire metallic grating with comparable imensions. N W top W max Error eg

10 Antos et al. J. Appl. Phys. 100, FIG. 10. Experimental marks an fitte four-zone null SE parameters, an, on a paraboloial-wire Ta grating sample C in incience of 70 for three examples of maximum Fourier harmonics, n max =4 ashe curves, n max =8 soli curves, an n max =20 otte curves, slice into N=20 sublayers. In this case, higher-orer approximations fit the measurement reasonably, especially in the longer-wavelength spectral range. with oes slowest. Similarly, the errors of all the imensions of the Si grating are less than 1% for n max 4 Fig. 12b. In the case of the Ta grating with its wires assume rectangular, the errors of the epth an perio are less than 1% for n max 3, whereas the linewith converges much slower. This can be explaine by the inhomogeneity of the linewith within epth or, in other wors, by the incorrectness of the moel applie for fitting which oes not take into account all the aspects of the reality. FIG. 11. Experimental marks an fitte four-zone null SE parameters, an, on a paraboloial-wire Ta grating sample C in incience of 70 for four examples of the number of slices, N=3 ashe curves, N=5 ashotte curves, N=10 soli curves, an N=20 otte curves, in each case with the maximum Fourier harmonics n max =20. Here the quality of the fit is almost linearly improving by increasing N. In Figs. 12 an 12e, the convergences of the relative epth, top linewith, maximum linewith, an perio are isplaye as obtaine on sample C with the number of slices N assume 3 an 20, respectively. Here the convergences of all the imensions are slower than in the case assuming the rectangular profile. The most probable explanation follows from a partial inaequacy of the Fourier factorization rules applie here to wires with slightly curve eges, though the rules FIG. 12. Convergence properties of the relative critical imensions accoring to increasing n max a e an accoring to increasing N f, obtaine on samples A an B a an b, on the rectangular-wire-assume sample C c, an on the paraboloial-wire-assume sample C with fixe N=3, with fixe N=20 e, an with fixe n max =20 f.

11 Antos et al. J. Appl. Phys. 100, have been erive for rectangular-relief gratings. Nevertheless, the slopes of the wires of sample C are so small that the factorization rules shoul be approximately vali. In Fig. 12f, the convergences of the relative epth, top linewith, maximum linewith, an perio accoring to increasing N are isplaye as obtaine on sample C with n max fixe to 20. The relative parameters can now be written as pn/ pn, where pn is a parameter whose convergence properties accoring to increasing N are stuie, while pn represents an approximate-limit value obtaine by assuming N=50 slices. Just as the SE parameters o in Fig. 11, the relative imensions graually converge while the number of slices is increasing. We can conclue from Figs f that the perio converges most rapily, whereas the maximum linewith most slowly in the case of a slice nonrectangular profile. VI. CONCLUSIONS A couple-wave implementation referre to as Airy-like series was here escribe in etail to increase the clarity of monitoring critical imensions of gratings by means of optical scatterometry, particularly with respect to the convergence properties of the imensions accoring to the level of approximation such as the truncation orer of Fourier series, n max, or the number of slices N of a nonrectangular-wire grating. It was emonstrate that applying the Fourier factorization rules increibly enhance the convergence properties of both the simulate SE parameters an the extracte imensions, though an improvement base on generalizing the factorization rules woul provie better results for wires with curve eges. In the case of the gratings mae of transparent quartz an semiconucting Si, the agreement between the experimental ata an the fitte parameters was nearly perfect, obviously inicating high relevance of the moel assuming a rectangular-relief structure of samples A an B. In the case of the Ta grating, however, slight iscrepancies were observe in the short-wavelength part of the range of interest, even though a top native Ta 2 O 5 overlayer an a nonieal shape of wires with curve eges were consiere. The metho coul be improve by generalizing the moel incluing more aspects of reality. The most important aspects are the optical constants of Ta to be monitore more precisely in a broaer spectral range with an energy reflectance measurement inclue, the surface roughness of Ta for which SE is sensitive, the surface oxie on the eges of wires, an/or a more general shape of the Ta wires with higher-orer parameters of the Taylor series in Eq. 45. ACKNOWLEDGMENTS This work was partially supporte by the Grant Agency of the Czech Republic Project No. 202/06/0531, Grant Agency of Charles University Project No. 314/2004/B-FYZ/ MFF, an Ministry of Eucation, Youth an Sport of the Czech Republic Project No. MSM One of the authors R.A. is thankful to Professor Ivan Ohlial Masaryk University, Brno, Czech Republic for a number of useful iscussions. 1 D. E. Aspnes, Thin Soli Films 455, B. S. Stutzman, H.-T. Huang, an F. L. Terry, Jr., J. Vac. Sci. Technol. B 18, H.-T. Huang, W. Kong, an F. L. Terry, Jr., Appl. Phys. Lett. 78, X. Niu, N. Jakatar, J. Bao, an C. J. Spanos, IEEE Trans. Semicon. Manuf. 14, H.-T. Huang an F. L. Terry, Jr., Thin Soli Films 455&456, L. Li, J. Opt. Soc. Am. A 13, E. Popov an M. Neviere, J. Opt. Soc. Am. A 17, M. G. Moharam an T. K. Gaylor, J. Opt. Soc. Am. 71, M. G. Moharam, D. A. Pommet, E. B. Grann, an T. K. Gaylor, J. Opt. Soc. Am. A 12, L. Li, J. Opt. Soc. Am. A 10, F. Montiel an M. Neviere, J. Opt. Soc. Am. A 11, J. C. Light an R. B. Walker, J. Chem. Phys. 65, N. P. K. Cotter, T. W. Preist, an J. R. Sambles, J. Opt. Soc. Am. A 12, L. Li, J. Opt. Soc. Am. A 11, R. Antos, I. Ohlial, J. Mistrik, K. Murakami, T. Yamaguchi, J. Pistora, M. Horie, an S. Visnovsky, Appl. Surf. Sci. 244, R. Antos, I. Ohlial, D. Franta, P. Klapetek, J. Mistrik, T. Yamaguchi, an S. Visnovsky, Appl. Surf. Sci. 244, R. Antos, J. Pistora, I. Ohlial, K. Postava, J. Mistrik, T. Yamaguchi, S. Visnovsky, an M. Horie, J. Appl. Phys. 97, R. Antos, J. Mistrik, M. Aoyama, T. Yamaguchi, S. Visnovsky, an B. Hillebrans, J. Magn. Magn. Mater , R. Antos, J. Mistrik, S. Visnovsky, M. Aoyama, T. Yamaguchi, an B. Hillebrans, Trans. Magn. Soc. Jpn. 4, R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokritov, an B. Hillebrans, Appl. Phys. Lett. 86, R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokritov, an B. Hillebrans, Opt. Express 13, J. Bao, X. Niu, N. Jakatar, C. Spanos, an J. J. Benik, Proc. SPIE 3998, R. M. A. Azzam an N. M. Bashara, Ellipsometry an Polarize Light North-Hollan, Amsteram, A. Kinoshita, F. Kitamura, M. Horie, an T. Yoshia, Proc. SPIE 5752, D. F. Ewars, in Hanbook of Optical Constants of Solis, eite by E. D. Palik Acaemic, New York, 1998, Vol. 1, pp ; H. R. Philipp, ibi., Vol. 1, pp E. Franke, C. L. Trimble, M. J. DeVries, J. A. Woollam, M. Schubert, an F. Frost, J. Appl. Phys. 88,