Fast Lithography Image Simulation By Exploiting Symmetries in Lithography Systems

Transcription

1 Fat Lithograhy Image Simulation By Exloiting Symmetrie in Lithograhy Sytem Peng Yu, Weifeng Qiu David Z. Pan Abtract Lithograhy imulation ha been widely ued in many alication, uch a otical roximity correction, in emiconductor indutry. It i imortant to reduce the runtime of uch imulation. Dedicated hardware arallel comuatation have been ued to reduce the runtime. For full chi imulation, the imulation method, Otimal Coherent Aroximation (OCA ), i widely ued. But it ha not been imroved ince it firt incetion. In thi aer, we imrove it by conidering the ymmetric roertie of lithograhy ytem. The new method could eed u the runtime by without lo of accuracy. We demontrate the eedu i alicable to vectorial imaging model a well. In cae the ymmetric roertie do not hold trictly, the new method can be generalized uch that it could till be fater than the old method. I. INTRODUCTION Lithograhy image imulation a a te in lithograhy imulation i widely ued in alication uch a Otical Proximity Correction (OPC) []. The indutry ha uhed hard to reduce the imulation runtime by uing arallel comutation dedicated hardware. Mentor Grahic ha ued multiroceing multithreading on Linux worktation cluter []. Secific hardware-accelerated comutational lithograhy latform ha alo been ued in the indutry [3] IBM ha develoed oftware for IC deign DFM oftware with the IBM BlueGene uercomuter [4]. Thee effort are very imortant but require huge oftware hardware invetment. Otimal Coherent Aroximation (OCA ) [5], [6] i ued for the full-chi image imulation. In thi work, we imrove thi method by exloiting the ymmetric roertie that are commonly found in lithograhy ytem. The new method could give a eedu of without any lo in accuracy. Such imrovement can be eaily integrated with the other eedu technique mentioned above. The main contribution of thi aer are a follow: We derive that the well known Hokin equation can be reduced uch that the imaginary art of the tranmiion cro coefficient i not neceary. We derive an imroved image imulation formula, which could give a eedu without any accuracy lo by uing the ymmetric roertie in common lithograhy ytem. It work for both calar vectorial imaging model. The new method till imrove runtime even if lithograhy ytem are not erfectly ymmetric. Peng Yu David Z. Pan are with the Det. of Electrical Comuter Engineering, the Univerity of Texa at Autin. Weifeng Qiu i with the Intitute for Comutational Engineering Science, the Univerity of Texa at Autin. The ret of thi aer i organized a follow. In Section II, we review the calar imaging model the ymmetric roertie commonly found in lithograhy ytem. In Section III, we derive the roertie of the tranmiion cro coefficient (TCC) the reduced Hokin equation. In Section IV, we how the new imulation formula. The detailed derivation can be found in the Aendix. In Section V, we how that the new formula work for both vectorial model nonerfectly ymmetric lithograhy ytem. Section VI how the exeriment reult. Section VII conclude thi aer. II. LITHOGRAPHY SIMULATION REVIEW AND SYMMETRIES IN LITHOGRAPHY SYSTEM We review the calar lithograhy imaging model. The vectorial model will be dicued in Section V. We then oint out ome ymmetric roertie commonly found in lithograhy ytem, which be ued for the derivation of the new formula. The latent image intenity in the hotoreit i given by the Hokin equation [7], I(k) = T(k + k, k )F(k + k )F (k )d k. () F(k) i the mak tranmiion function F (r) in the frequency domain, where k denote a oint in the frequency domain r denote a oint in the atial domain. I(k) i the chemical latent image in the Fourier domain. T(k, k ) i the tranmiion cro coefficient (TCC) (including diffuion, ee below), given by T(k, k ) = G(k k ) J(k)K(k + k )K (k + k )d k. () The meaning of the ymbol are decribed below: G(k) i the diffuion kernel, written a G(k) = e π d k, (3) which correond the diffuion of the latent image during the ot-exoure-bake (PEB), where d i the diffuion length, k = k. J(k) i the illumination function. We illutrate ome commonly ued one in Figure. K(k) i the rojection ytem tranfer function. Auming a circular uil, it can be written a { K(k) = e i π z λ k in θ obj+iπφ(k) k < 0 otherwie, (4)

2 Proof: Figure. Commonly ued illumination cheme. The radii of the outer circle are. J i a contant over the gray region. T(k, k ) = G(k k ) J(k)K(k + k )K (k + k )d k ( ) = G(k k ) J(k)K (k + k )K(k + k )d k = T (k, k ) () where z denote the focu error, λ i the wavelength θ obj i the emi-aerture angle at the image lane [8] Φ(k) i the aberration term. The uercrit denote the comlex conjugation oeration. Baed on Eq. (3), Figure Eq. (4), it i obviou that G(k), J(k) K(k) have the following two roertie. Proerty. G(k) R, J(k) R K(k) C, (5) where R C denote the et of all real number all comlex number, reectively. Proerty. It i reaonable to aume that the diffuion i rotational invariant. Then we have G(k) = G( k). (6) Since ymmetrical illumination cheme are commonly ued (not neceary limited to the one in Figure ), we have J(k) = J( k). (7) Auming no odd order aberration, we have K(k) = K( k). (8) Remark. Thee roertie are true in a general ene are not necearily ecific only to the form in Eq. (3), Figure Eq. (4). The mak tranmiion function F (r) i real for commonly ued mak, uch a binary mak (BIM) or hae hift mak (PSM) with the hae of By the definition of the Fourier Tranform, we can eaily rove the following roerty of F (r) invere Fourier Tranform F(k). Proerty 3. F(k) = F ( k). (9) III. THE REDUCED HOPKINS EQUATION In thi ection, we rove a few lemma on TCC derive a reduced Hokin Equation. With Proerty, we have the following lemma. Lemma. T(k, k ) i Hermitian, that i, T(k, k ) = T (k, k ). (0) With Proerty, we have the following lemma. Lemma. T(k, k ) i ymmetric under the reflection oeration about the origin in the frequency domain, Proof: T(k, k ) = T( k, k ). () T(k, k ) = G(k k ) J(k)K(k + k )K (k + k )d k = G ( ( k ) ( k ) ) J( k)k( k k )K ( k k )d k = G ( ( k ) ( k ) ) J(k)K(k k )K (k k )d k = T( k, k ) (3) Remark. The roof of Lemma do not ue the articular function form of G, J K but only Proerty. Thee concluion are generally true for common lithograhy ytem. With Lemma, we immediately have the following corollary. Corollary. T(k, k ) = T ( k, k ). (4) Uing Proerty 3, we have the following lemma. Lemma 3. If T(k, k ) = T( k, k ), we have I(k) = 0. Proof: We rove the lemma by roving I(k) = I(k). (5) I(k) = T(k + k, k )F(k + k )F (k )d k = T( k, k k )F( k )F ( k k )d k = T(k + k, k )F(k + k )F (k )d k = I(k). (6) We relace k k by k to get the third integral.

3 3 T(k, k ), a a comlex function, can be earated into a real art (T real (k, k ) R) an imaginary art (T imag (k, k ) R), T(k, k ) = T real (k, k ) + it imag (k, k ). (7) I(k) can be earated accordingly a I(k) = I real (k) + ii imag (k), (8) where I real (k) = T real (k + k, k )F(k + k )F (k )d k (9) I imag (k) = T imag (k + k, k )F(k + k )F (k )d k. (0) By Corollary, we have T real (k, k ) T imag (k, k ) are ymmetric antiymmetric, reectively, T real (k, k ) = T real ( k, k ) () T imag (k, k ) = T imag ( k, k ). () Uing Lemma 3, we have I imag (k) = 0. Therefore, we have the following theorem. Theorem. The image can be comuted by the Reduced Hokin Equation, I(k) = T real (k + k, k )F(k + k )F (k )d k. (3) Uing the above theorem, we can derive the eedu imulation formula in the next ection. IV. IMPROVED OPTIMAL COHERENT APPROXIMATIONS Otimal Coherent Aroximation (OCA ) ha been derived in [5], which how that the image can be comuted by I(r) = σ Qn n F, (4) where F i the mak tranmiion function, i the convolution oerator Q n (called kernel) are comlex function. The real number σ n are ordered uch that σ 0 σ σ n. (5) An image can be aroximated by uing only the firt a few term, Ĩ (r) = σ Qn n F. (6) The error can be etimated a u I(r) Ĩ(r) σ F, (7) r where denote the L -norm. However, Theorem wa not ued in [5]. When we ue Theorem, we can rove in Aendix that the oerator i not needed. That i, intead of (4), we have I(r) = σ n(q n F ), (8) where F Q n are all real, σ 0 σ σ n. (9) Note that σ n σ n, Q n Q n may not be the ame. A an aroximation, we alo take the firt a few term for the image imulation Ĩ (r) = σ n(q n F ), (30) Similarly, the error can be etimated a u I(r) Ĩ (r) σ F. (3) r Baed on Eq. (7) Eq. (3), we can chooe the number of term that are needed ( ) for a given error requirement. It i obviou that the TCC T i real when there are no aberration (z = 0 Φ(k) = 0). In thi cae, we have Q n = Q n σ n = σ n (3) for any n. Therefore, the ame number of term ( = ) are needed for the ame error requirement. Since the convolution of a comlex function (Q) with a real function (F ) i lower than the convolution of two real function (Q F ), we get a eedu. We can eaily ee that the aroximated image are alway the ame Ĩ (r) = Ĩ (r). Therefore, there i no lo in the accuracy from (6) to (30). We have the following corollary. Corollary. When there are no aberration, (8) give a eedu without lo of accuracy. Otherwie the eedu i =. (33) In Section VI, we how exerimentally the eedu for ome other cae. V. EXTENSIONS TO VECTORIAL IMAGING AND NON-PERFECT SYMMETRIES The above imrovement wa hown for calar image modeling with erfect ymmetrie. We will how below it work for vectorial image modeling [8] non-erfect ymmetric lithograhy ytem.

4 4 A. Vectorial Imaging According to [8], the TCC in the calar model become a TCC matrix in the vectorial model, which can be written a T ij (k, k ) = J(k)K(k + k )K (k + k ) M ki (k + k )M kj(k + k )d k, where = M 0xx M 0xy M 0xz k={x,y,z} M 0 (k) = M 0 (f, g) M 0yx M 0yy M 0yz = α = f in θ obj, β +α γ γ αβ +γ α β = g in θ obj, γ = (f + g ) in θ obj. αβ +γ α +β γ γ β For unolarized illumination, an equivalent TCC can be written a [9], [0] T = T xx + T yy. (34) Similar to Lemma, it i eay to check that T in Eq. (34) i Hermitian T(k, k ) = T (k, k ) ymmetric under the reflection oeration about the origin T(k, k ) = T( k, k ). Therefore, Eq. (30) i valid for vectorial imaging a well. B. Non-Perfect Symmetrie Practically, lithograhy ytem may not be erfectly ymmetric due to ome error (Proerty may not hold erfectly). But we can earate T(k, k ) into two art a where T(k, k ) = T ym (k, k ) + T anti (k, k ), (35) T ym (k, k ) = T(k, k ) + T ( k, k ) (36) T anti (k, k ) = T(k, k ) T ( k, k ). (37) It i eay to check that T ym i ymmetric T anti i antiymmetric T ym (k, k ) = T ym( k, k ) (38) T anti (k, k ) = T anti( k, k ). (39) Similar to the deduction of Theorem, we only need the real art of T ym (k, k ) the imaginary art of T anti (k, k ) for the comutation of I(k). Aume term are needed to decomoe T, q term are needed to decomoe T ym,real q term are needed to decomoe T anti,imag. Therefore, the runtime eedu i = q + q. (40) If lithograhy ytem are cloe to ymmetric, we have that T ym i cloe to T T anti i mall. Therefore, i cloe to q q i much maller than. In thi cae, the eedu i cloe to. VI. EXPERIMENTAL RESULTS In thi ection, we numerically validate our reviou tatement. The imlementation were in C++ [], imulation were on a.8 GHz Pentinum-4 Linux machine. We ued the conventional artially coherent illumination with σ = 0.7, the numerical aerture NA = 0.8, the wavelength λ = 93 nm the defocu z = 00 nm unle otherwie noted. A. Validation of TCC ymmetrical roerty For the roertie of TCC, we only numerically validate the ymmetric roerty (Lemma ), becaue the Hermitian roerty (Lemma ) i well known. We denote T(k, k ) a T(f, g, f, g ), where (f, g) = k (f, g ) = k to imlify the dicuion below. It i eay to check that T(f, g, f, g ) = 0, for (f, g, f, g ) / B, (4) where B i a 4-dimenional box B = ( σ, + σ) 4. (4) We numerically imulate the TCC on the all oint (i, j, i, j ) B, (43) where the grid ize in the frequency domain = 0., the number i, j, i j are integer in the interval [ N, N], where N = +σ = 8. T(f, g, f, g ) i a 4-dimenional function, which need to be reindexed to draw -dimenionally. We denote T(i, j, i, j ) a T ij, where { i = (i + N) + (N + )(i + N), j = (j + N) + (N + )(j + N) (44) to hel viualize the TCC []. The indexe i j are in [0, (N + ) ] = [0, 88]. We alo denote T( i, j, i, j ) a T ij We ue R I to denote the real art the imaginary art. Figure 3 how R(T ij ), I(T ij ), R( T ij ) I( T ij ) for both the calar model the vectorial model. From thee two figure, it i clear that T ij = T ij for both model. Therefore, Lemma i validated.

5 5 (a) Uing Hokin Equation (b) Uing Reduced Hokin Equation (a) R(T ij ) (b) I(T ij ) Figure 4. The imulated image for a five-via attern. Each via i of ize 00 nm. The ditance between the center via any other via i 00 nm. (c) R( e T ij ) (d) I( e T ij ) Figure. Viualization of T(k, k ) T( k, k ) of the calar model (z = 00 nm). Subfigure (a) (c) are the ame, Subfigure (b) (d) are the ame. Therefore, T(k, k ) = T( k, k ). B. Validation of the Reduced Hokin Equation Figure 4 how the imulated image uing the calar model for a five-via attern uing Hokin Equation Reduced Hokin Equation. The maximum image difference between thee two image i , which i numerically zero. Therefore, we verified the Reduced Hokin Equation for the calar model. The Reduced Hokin Equation for the vectorial model can alo be verified. C. Runtime Seedu Figure 5 how the number of term, their ratio a a function of the error requirement for z = 00 nm z = 00 nm, reectively (the calar model). The exeriment how that the runtime eedu i for z = 0 nm. When z = 00 nm z = 00 nm, The eedu can be bigger than for ome. In the wort, the eedu i aroximately.. D. Non-Perfect Symmetrie When there are odd aberration, Eq. (8) doe not hold. Let u conider a mall x-coma aberration (z = 0 nm). The x-coma aberration term Φ(k) i [8] (a) R(T ij ) (b) I(T ij ) Φ(k) = Φ(f, g) = c (3k )f where c i a coefficient (we take it a a mall number, 0.0) Figure 6 how the number of term, their ratio a a function of the error requirement (the calar model). The eedu varie a the error requirement change. But in the wort cae, the eedu i aroximately.. The eedu can be bigger than. VII. CONCLUSIONS (c) R( e T ij ) (d) I( e T ij ) Figure 3. Viualization of T(k, k ) T( k, k ) of the vectorial model (z = 00 nm). Subfigure (a) (c) are the ame, Subfigure (b) (d) are the ame. Therefore, T(k, k ) = T( k, k ). In thi aer, we derive a new method for the lithograhy imulation, which eed u the widely ued method (OCA ) uing the ymmetric roertie of the lithograhy imaging ytem. It can give eedu if there are no aberration. It work for both the calar the vectorial model. The new method till give eedu when lithograhy imaging ytem are not erfectly ymmetric.

6 6 or or or (a) z = 0 nm. The curve for are the ame (b) z = 00 nm (c) z = 00 nm Figure 5. Number of term ( ) the runtime eed (uing Eq. (33)) v. the error requirement (). or Figure 6. Imrovement for x-coma with c = APPENDIX In thi ection, we will rove that the eigenfunction a real Hermitian oerator under certain condition can be made either ymmetric or antiymmetric. Thi reult will be alied to our lithograhy image imulation roblem at the end of thi aendix. Define the oerator A baed on a real function A(k, k ) a Aφ(k) = A(k, k )φ(k )dk, where A(k, k ) = A(k, k). Define the arity oerator P a P φ(k) = φ( k). (45) Theorem. The arity oerator P ha only eigenvalue. If ψ i the eigenfunction aociated with the eigenvalue, then ψ(k) = ψ( k). If ψ i the eigenfunction aociated with the eigenvalue, then ψ(k) = ψ( k). Proof: Aume ψ i an eigenfunction of P uch that P ψ = λψ. We ut ψ(k) + ψ( k) ψ (k) =, ψ(k) ψ( k) ψ (k) =. Then P ψ = P ψ +P ψ = ψ ψ. Since P ψ = λψ, we have ψ ψ = λψ + λψ. Then we have (λ )ψ = (λ + )ψ. Since ψ i an even function ψ i a odd function, we have λ = or λ =. So we can ay that the arity oerator P ha only eigenvalue. Obviouly, if λ =, then ψ = 0. So we have ψ(k) = ψ( k). If λ =, then ψ = 0. So we have ψ(k) = ψ( k). Theorem 3. If A(k, k ) i real A(k, k ) = A( k, k), A(k, k ) can be exed in term of orthonormal real function ψ i (k) a A(k, k ) = σ i ψ i (k)ψ i (k ), where ψ i (k) atifie i= ψ i (k) = ψ i ( k) or ψ i (k) = ψ i ( k). Proof: Aume {σ i } are eigenvalue {φ i } are normalized eigenfucntion. Since A i real ymmetric, σ i φ i are real with φi (k)φ j (k)dk = δ ij for any i, j N. It i eay to ee that A(k, k )φ i ( k )dk = σ i φ i ( k). Thi imlie that σ i (P φ i ) = A(P φ i ). for any i N. So P φ i i till an eigenfunction of A aociated with engenvalue σ i.

7 7 Since A i a comact oerator from L to L, then λ R, if λ 0, there are at mot finitely many i N uch that σ i = λ. Without loing generality, we aume σ = = σ n σ n+. Put V = an{φ,, φ n }, then P (V ) V. So there i an orthonormal bai {ψ,, ψ n } of V uch that P ψ i = ψ i or P ψ i = ψ i for i n. It i eay to ee that φ (k)φ (k ) + + φ n (k)φ n (k ) =ψ (k)ψ (k ) + + ψ n (k)ψ n (k ). Then we can conclude that A(k, k ) = σ i ψ i (k)ψ i (k ) i= with ψ i (k) = ψ i ( k) or ψ i (k) = ψ i ( k) for i N ψ i (k) i real. Since T real atifie the condition of Theorem 3, therefore we have T real (k, k ) = n σ n Q n (k )Q n (k ), (46) where each σ n i a real number, Q n (k) i real, Q n (k) atifie Q n (k) = Q n ( k) or Q n (k) = Q n ( k) (47) Therefore, Eq. (4) can be derived [5], where Q n i the invere Fourier tranform of Q n. Uing Eq. (47), it i eay to ee the invere Fourier tranform Q n of Q n i either real or ure imaginary. So the magnitude oerator in Eq. (6) i not neceary, becaue even it i ure imaginary, we can make it real by multilying the imaginary unit i, iq n Q n. Therefore, taking only term a an aroximation, we can rove Eq. (30). REFERENCES [] N. B. Cobb, Fat Otical Proce Proximity Correction Algorithm for Integrated Circuit Manufacturing, Ph.D. diertation, Univerity of California at Berkeley, 998. [] N. Cobb Y. Granik, New concet in OPC, in Proc. SPIE 5377, 004, [3] J. Ye, Y.-W. Lu, Y. Cao, L. Chen, X. Chen, Sytem method for lithograhy imulation, Patent US 7,7,478 B, Jan. 8, 005. [4] G. A. Gomba, Collaborative Innovation: IBM Immerion Lithograhy Strategy for 65 nm 45 nm Half-itch Node & Beyond, in Proc. SPIE 65, 007. [5] Y. C. Pati T. Kailath, Phae-hifting mak for microlithograhy: automated deign mak requirement, Journal of the Otical Society of America A, vol., , Se [6] Y. Pati, A. Ghazanfarian, R. Peae, Exloiting tructure in fat aerial image comutation for integrated circuit attern, IEEE Tran. on Semiconductor Manufacturing, vol. 0, no.,. 6 74, Feb [7] M. Born E. Wolf, Princile of Otic : Electromagnetic Theory of Proagation, Interference Diffraction of Light, 7th ed. [8] A. K.-K. Wong, Otical Imaging in Projection Microlithograhy. SPIE Publication, Mar [9] K. Adam, Y. Granik, A. Torre, N. B. Cobb, Imroved modeling erformance with an adated vectorial formulation of the Hokin imaging equation, in Proc. of SPIE 5040, Jun. 003, [0] K. Adam W. Maurer, Polarization effect in immerion lithograhy, Journal of Micro/Nanolithograhy, MEMS MOEMS, vol. 4, no. 3,. 0306, 005. [] P. Yu D. Z. Pan, ELIAS: An Extenible Lithograhy Aerial Image Simulator with Imroved Numerical Algorithm, 008, in rearation.