A Parameterized Mask Model for Lithography Simulation

Transcription

1 A Parameterize Mask Moel for Lithography Simulation henhai hu Caence Research Labs, Berkeley, CA ABSTRACT e formulate the mask moeling as a parametric moel orer reuction problem base on the finite element iscretization of the Helmholtz equation. By using a new parametric mesh an a machine learning technique calle Kernel Metho, we convert the nonlinearly parameterize FEM matrices into affine forms. This allows the application of a well-unerstoo parametric reuction technique to generate compact mask moel. Since this moel is base on the first principle, it naturally inclues iffraction an couplings, important effects that are poorly hanle by the existing heuristic mask moels. Further more, the new mask moel offers the capability to make a smooth trae-off between accuracy an spee. Categories an Subject Descriptors B.7. Integrate Circuits: Design Aies Simulation General Terms Algorithms,Performance,Design Keywors Lithography, Mask Moel, Parameterize Moel Orer Reuction. INTRODUCTION From lithography simulation point of view, two classes of esigns pose particularly challenging problems. The first class is the memory esign. ith just six or seven transistors, each memory cell has small layout. The carefully esigne cell is uplicate hunre millions to billions of times. Since the manufacturing efects in one cell affect the entire row of cells, the memory esign is highly sensitive to such efects. Therefore, high accuracy lithography simulation is manatory for the memory esign. The secon class is the custom logic esign. ith a finite number of unique stanar cells as the builing blocks, the layout is typically large, in the orer of 1 1mm. Therefore, highly efficient lithography simulation is manatory for the custom logic esign. Since timing an power are the main concerns, the accuracy of lithography simulation has to be goo enough to moel the impact of manufacturing imperfection to the timing an power. Clearly, a goo lithography simulation tool shoul inherently have the capability to make smooth trae-off between accuracy an spee for various esigns. An it shoul be base on rigorous mathematical founation to ensure its robustness. Permission to make igital or har copies of all or part of this work for personal or classroom use is grante without fee provie that copies are not mae or istribute for profit or commercial avantage an that copies bear this notice an the full citation on the first page. To copy otherwise, to republish, to post on servers or to reistribute to lists, requires prior specific permission an/or a fee. DAC 009, July 6-31, 009, San Francisco, California, USA. Copyright 009 ACM ACM /08/ $5.00. There are three main steps in lithography simulation 1: photo mask moeling, aerial image simulation an photo resist simulation. This paper focuses on the photo mask moeling. The goal of this step is to compute the near fiel, the electric fiel at the bottom of the computational omain that contains the mask pattern. The existing approaches can be categorize into two groups: fiel solvers an heuristic moels. The fiel solver approach is to solve the Maxwell s equations using well-known numerical techniques such as finite ifference time omain (FDTD) or finite element metho (FEM) 3. This is the most accurate an robust approach. But experience inicates that even the state-of-the-art fiel solvers are too slow or memoryboune to hanle a meium-size mask pattern. In practice, the heuristic moels are use instea to obtain the approximate solution. A commonly use moel is base on the so-calle Kirchhoff approximation 4: if there is a mask opening, the light shines through it without any change in magnitue an phase; otherwise, light is completely blocke. This approximation neglects the effects such as iffraction, polarization an coupling. Attempts have been mae to obtain a moifie mask moel to improve the accuracy 5, 6. The piecewise constant curve fitting approach in 5, 6 is simple an efficient but not accurate an robust. For example, it has been shown in 7 that such moel can result in wrong wafer imaging preiction. The fiel solvers in, 3 an the heuristic moels in 5, 6 sit at the opposite corners in the accuracy-spee trae-off matrix an it is ifficult to smoothly trae accuracy with spee. In our early work 8, the mask moeling was formulate as a parametric moel orer reuction problem base on the finite element iscretization of the Helmholtz equation. The iscretization in 8 uses a uniform an rectangular mesh. This natually leas to the affine parametric form for the stiff an the mass matrices in FEM an hence the well-unerstoo parametric reuction technique 9, 10 can be irectly applie to generate the compact mask moel. However, the non-uniform an unstructure mesh is inispensable to hanle complicate mask patterns. Unfortunately, as will be shown in section 6, a irect application of the technique in 9 coul result in large mask moels. In this paper, we present a new approach to approximate the nonlinearly parameterize FEM matrices with affine forms. This allows the irect application of the parametric reuction in 9, 10 again. Though we emonstrate this new technique for the Helmholtz equation that governs the photo mask moeling, it shoul be straightforwar to apply the same approach to other partial ifferential equations in the context of the parametric moel orer reuction. 3. PROBLEM FORMULATION For the sake of simplicity, we present the problem formulation base on the D example shown in Fig 1. The extension to 3D cases is straightforwar. Assuming an S-polarization (TE) case, the governing D Helmholtz

2 Glass Figure 1: The structure of the D phase-shift mask. Variables w 1, w an s are the parameters in the reuce mask moel. equation is Air u ω εµu= u+k u = 0 (1) where u(x,y) is the z-component of the total electric fiel, ω is the frequency, ε an µ are respectively the ieletric constant an permeability. Following the stanar FEM proceure 11, we obtain the weak form s v u ω Ω Ω sεµvu lvdtn(u) = Ω Ω lvdtn(u in ), () where Ω an Ω are respectively the computational omain an its bounary, v is the testing function, u in is the known incience plane wave an the DtN( ) operator efines the transparent bounary conition 1. In this paper, we assume the perioic bounary conition at the east an the west sie of the computational omain an transparent or non-reflecting bounary conition on the north an the south sie of the computational omain. Using the stanar FEM piecewise polynomial basis functions 11 to iscretize (), we arrive at the parameterize system equation S( w, s) M( w, s) Bū = r, (3) where vector w an s respectively contain the withs an the spacings in the layout, the stiff matrix S( w, s) correspons to the first term in (), the mass matrix M( w, s) correspons to the secon term in (), the matrix B correspons to the thir term in (), an vector r correspons to the right-han-sie term in (). For mask patterns with fixe topology but ifferent with an spacing values, computing the near fiel involves solving equation (3) for ifferent w an s. This kin of multiple-inquery scenario is precisely what the parametric moel orer reuction approach is esigne for. 4. PARAMETRIC MODEL ORDER REDUC- TION A parametric moel orer reuction technique calle Reuce Basis metho has been evelope by the finite element research community 9. A similar iea has also been inepenently propose in the area of parameterize moel orer reuction for circuit simulation 10. As will be shown later, the Reuce Basis is a very powerful iea on top of which we can buil our new mask moel to achieve the esirable accuracy an spee trae-off. Here we summarize its key steps. Please refer to 9 for more etails. Suppose the parameterize governing equation for the problem at han is A( σ)ū = A 0 + f i ( σ)a i ū = r, (4) i where the scalar function f i ( σ) can be arbitrary an the size of vector ū, r an constant matrix A i is N 1, N 1 an N N, respectively. In practical applications, N can be as large as a few millions for a meium-size 3D structure. The Reuce Basis metho in 9 has two stages: the off-line pre-characterization an the on-line evaluation. Off-line pre-characterization stage. e ranomly generate a set σ k = {σ k 1,σk,...} using the given ranges of σ i an solve (4) for ū k. After a few sampling solves, we collect all solutions into the projection matrix an perform projection P = ū 1,ū,...,ū M (5)  i = P T A i P; ˆr = P T r. (6) Now we arrive at the reuce governing equation Aˆ 0 + f i ( σ)â i û = ˆr, (7) i where the size of matrix  i is M M an M is the number of sampling solves. Similar to the stanar proceure in the Moel Orer Reuction 10, the columns in the projection matrix P are orthogonalize using techniques such as incremental QR ecomposition. This makes the matrix P well conitione. In aition, both theoretical an practical ways to estimate the error of the reuce moel are reaily available 9, 10. Hence the off-line moel generation can be mae incremental. On-line evaluation stage. e substitute the given set σ into (7) an solve for û. The approximate solution to equation (4) is obtaine from u = Pû. (8) The key observation here is that the CPU time of the on-line stage is only relate to M, not to N in (4). An M is typically many orers of magnitue smaller than N, as shown by the extensive experiments in 9, 10. Hence equation (7) is a much more efficient but still accurate reuce moel than the original moel in (4). However, this ramatic efficiency gain critically epens on the fact that matrix A i in (4) is not a function of σ. Otherwise, the projection step in (6) involves calculating A i ( σ) at a particular value σ k. This essentially means that the CPU time use by the reuce moel in (7) is relate to the original problem size N an hence we have gaine no efficiency at all 9, 13. This issue of representing the potentially arbitrary nonlinearity in A( σ) in the form amenable to the projection framework in (6) is one of the main challenges in the nonlinear Moel Orer Reuction 14. The main contribution in this paper is to show how to convert S( w, s) an M( w, s) into the appropriate form similar to that in equation (4) so that we can apply the congruence projection in (6). This is one in two steps: parameterization of mesh an parameterization of the FEM matrices. 5. PARAMETERIE MESH In this section, we show an effective technique to parameterize the unstructure triangular mesh in an affine form of the geometry parameters w an s. This is an important first step towar parameterizing the stiff matrix S( w, s) an the mass matrix M( w, s) in (3). hen the size of a geometric feature changes, say w in Fig 1, the mesh points surrouning the feature will move as well. However, to capture the changes in the resulting electric fiel, not all mesh points in the computational omain have to be move. Only mesh points within a certain istance from the changing geometric feature nee to be move. To measure such a istance, we borrow a well-establishe concept calle istance function from the celebrate Level Set Metho 15.

3 5.1 The Distance Function e use a simple example to explain the basic ieas in the istance function. Let (x 1,y 1 ) an (x,y ) be the lower-left an upperright corner of a rectangle, respectively. The istance from a point (x, y) to such a rectangle is efine as (x,y) = min(min(y y 1,y y),min(x x 1,x x)), (9) where function min(a, b) returns the smaller value of the two variables. Fig. shows the contour plot of the istance function for a square where x 1 = y 1 = 5 an x = y = 10. It shoul be note that the zero level set correspons to the bounary of the square. All points outsie of the square have positive istance an all points insie the square have negative istance. 5. The Blening Function The movement of mesh points ue to geometry changes epen on the istance of the mesh points to the changing geometry. Intuitively, the function that characterizes such movement shoul satisfy the following constrains b(η) = 1, i f η = 0 (10) b η < 0, i f 0 < η < 1 (11) b(η) = 0, i f η 1 (1) b η = 0, i f η = 1 (13) where η is the normalize istance efine as η = (x,y) B, (14) B is a user-efine raius of influence, an (x,y) is the istance function like the one efine in (9). Equation (10) means that the mesh points on the geometry move by the same amount as that of the moving geometry features. Equation (11) means that the movement magnitue ecreases monotonically as the mesh points are away from the geometry. Equation (1) means that the movement magnitue is zero if the mesh point is not insie the influence region. Equation (13) ensures a smooth transition of the movement at the bounary of the influence region. In spirit, function b(η) is similar to the so-calle blening function in 16 use to generate the mesh for the computational omain with moving bounaries. In this paper we have chosen the following function as the blening function b(η) = (1 p( η + 0.5)), η 0,1 (15) where p(η) is a thir-orer polynomial p 3 (η) = 3η η 3. (16) 5.3 The Parametric Form of Mesh Arme with the blening function in (15) base on the istance function like the one efine in (9), we are reay to parameterize the movement of the mesh points ue to the change of geometric features. ithout loss of generality, we use the parameter w in Fig. 1 as an example. Suppose w is change by δw. Further more, again without loss of generality, suppose that the center of the opening oes not move, only the left an the right wall move by δw an δw, respectively. ithin the influence region aroun the left an the right sie wall, the location (x,y) of a mesh point can be expresse as x = x 0 + b R b L δw = x 0 + b δw, (17) y = y 0 (18) Figure : Contour plot of the istance to a square. Figure 3: Triangular mesh for the D mask in Fig. 1. Notice the eformation of the 90-egree triangles ue to the change in w 1 an w. where b L = b( L(x 0,y 0 ) B ), b R = b( R(x 0,y 0 ) ) (19) B (x 0,y 0 ) is the initial location, an L (x 0,y 0 ) an R (x 0,y 0 ) are its istance to the left an the right sie wall, respectively. Using linear superposition, one can easily generalize (17) to the multiple parameter cases x = x 0 + < b, σ > (0) where <, > is the inner prouct, an b i is the blening term for parameter σ i which represents the change of w 1, w or s in Figure 1. Clearly the parametric form in (0) is an affine function of the perturbation in each geometry parameter. It shoul be pointe out that the simple parametric form in (17) an (18) can be easily extene to hanle parameter changes along arbitrary irections. But for the purpose of moeling masks, it is probably sufficient to consier just the cases where the parameters are changing only along the horizontal irection (x-irection for D). This will be the case in the remaining part of this paper. Figure 3 shows the eformation in the triangular mesh for the mask shown in Fig. 1 ue to the change in w 1 an w. 6. PARAMETERIE THE FEM MATRICES For the isoparametric linear triangular elements efine on the triangle element with vertexes (x i,y i ),i = 1,,3, the element stiff

4 matrix an mass matrix are respectively 11 an S e = δ x δt x + δ y δt y 4A T (1) M e = 4AM c () where A is the area of the triangle an can be written as A = 1 et( δ x, δ y ), (3) δ x = x 1 x 3,x x 1 T, (4) δ y = y 3 y 1,y 1 y T, (5) 1 1 = 1 0, (6) 0 1 M c = (7) The Parametric Element FEM Matrices Suppose a triangle element is within the region of the influence efine in section 5, in view of (0) an (18), (4) an (5) become x1 x δ x = 3 x1,0 x = 3,0 + < 13, σ > x x 1 x,0 x 1,0 + < (8) 1, σ > δ y = y3,0 y 1,0 y 1,0 y,0 (9) where (x i,0,y i,0 ) is the nominal position for the noe-i, (k) i j = b (k) i b (k) j an b (k) i is the k-th component of the vector b in (0) for noei. For the sake of clarity, we assume that the triangle is a 90-egree triangle that satisfies x1,0 x 3, y3,0 y =, 1,0 hy =, (30) x,0 x 1,0 hx y 1,0 y, then we have < 13 δ x = h x, σ > 1+ < = h 1 x, σ > δ y = h y 1 0 Y 1+ (31), (3) A = 0.5 h y (1+) (33) where Y =< 13, σ > an =< 1, σ >. Substituting (31), (3) an (33) into (1), we obtain the parametric element stiff matrix S e = = + where Y 1+ Y 1+ Y Y 1+ ( = S e 0 + S e 0 = T + h y T ) 0 Y T Y T h y 1 0 T T σ k S e k + Y 1+ Se Se + (34) 0 1 T, Sk e = 1 0 (k) 13 (k) 13 (k) 1 T (35) SN e p +1 = h x 1 0 T, SN e p + = h y 1 0 T. (36) In view of (33) an (), the parametric element mass matrix is M e ( σ) = h y (1+)M c = M0 e + σ k Mk e (37) where M e 0 = M c an M e k = M c (k) The Assemble FEM Matrices The stanar way of generating matrices S( w, s) an M( w, s) is by a proceure calle stamping. The 3 3 element stiff matrix S e ( w, s) an the 3 3 element mass matrix M e ( w, s) are first generate for each element. Using the local-to-global vertex inex map, the entries of S e an M e are irectly ae (stampe) to the large matrices S( w, s) an M( w, s), respectively. The parametric stiff matrix is S( σ) = S 0 + σ k S (k) N f i ( σ)s (i) N + g j ( σ)s ( j) 3 (38) i=1 j=1 where matrices S 0, S (k) 1, S(i) an S ( j) 3 are respectively the reults of the stamping of the element matrices S0 e, Se k, Se +1 an Se + in (34), f i ( σ) = Y i, g j ( σ) = 1, (39) 1+ i 1+ j Y i =< (i) 13, σ >, i =< (i) 1, σ >, (40) N 1 an N are respectively the number of unique f i ( σ) an g j ( σ) for all the triangle elements in the computation omain, is the number of parameters in the vector σ. Similarly, the parametric mass matrix is M( σ) = M 0 + σ k M (k) 1, (41) where matrices M 0 an M (k) 1 are respectively the reults of the stamping of the element matrices M0 e an Me k in (37). 6.3 Regression Using the Kernel Metho The parametric forms in (38) an (41) are very similar to that in (4). But since N 1 an N are relate to the number of elements in the computational omain, so is the size of the reuce moel. Hence we have not obtaine a mask moel that is inepenent of the original problem size yet. In fact, this is the main reason why we can not irectly apply the technique in 9 to generate the reuce mask moel. The next critical task is to approximately represent the large number of f i ( σ) an g j ( σ) by the linear combination of a small number of the so-calle Kernel functions that are inepenent of the original problem size. Essentially we seek to approximate a high-imension space where the nonlinear function f i an g j resie with a much lower imension space. This kin of approximation has been well-stuie in the Machine Learning literatures an the so-calle Kernel Metho has been shown to be effective 17. The gist of the Kernel Metho is the following. Consier representing a scalar function f( x) : R n R of multi-variable argument. An interesting class of approximations are of the form N k f( x) = α k K( x, x k ) (4)

5 where x R n is the evaluation point, K( x,ȳ) : R n R n R is the Kernel, an the N k vectors x k R n are enote as the support vectors. The basic kernel methos 17 use simple function forms for the kernels an pick the coefficient α k base on certain loss functions. In this paper, we choose the kernel form to be the same as the nonlinear functions to be approximate. So the kernel regression form is where N k f i ( σ) α (i) N k k K 1(ξ k ), g j ( σ) β ( j) k K (ξ k ), (43) K 1 (ξ k ) = ξ k, K (ξ 1+ξ k ) = 1, (44) k 1+ξ k an ξ k =< k, σ >, N k is the number of kernels necessary to achieve the esire accuracy. As shown in section 8, 10 to 15 kernels can achieve 3 to 4 igits of accuracy. The coefficients α (i) k an β ( j) k are obtaine by the simple Least Square Fitting, the simplest loss function form 17. The support vectors k are selecte from all possible values of (i) 13 an (i) 1 in (40) by the K-mean Clustering Methos 17. It shoul be note that ue to the locality of the parametric mesh, each mesh noe is influence by a small number of parameters, typically in the D cases an 4 in the 3D cases. This means that the length of the vector (i) 13 an (i) 1 is for D cases an 4 for 3D cases, respectively. Essentially the number of clusters is inepenent of the number of the parameters in the mask structures. This is a key benefit from using the parametric mesh in section 5. Covering a relatively low-imensional space with the K-mean clustering usually means small cluster number N k in (43). Hence the mask moel size is largely etermine by the number of sampling solves in (5). Substituting (43) into (38), we obtain the final parametric form of the stiff matrix S( σ) S 0 + σ k S (k) 1 + N k K 1 (ξ k )S (k) + N k K (ξ k )S (k) 3 (45) where S (k) an S (k) 3 are the results of the stamping α (i) k Se +1 an β ( j) k Se +, respectively. 7. PARAMETRIC MASK MODEL Now we are reay to put everything together to present the final mask moel. Substituting (45) an (41) into (3), we obtain A 0 + σ k A (k) N k 1 + K 1 (ξ k )A (k) N k + K (ξ k )A (k) 3 ū = r, (46) where A 0 = S 0 M 0 B, A (k) 1 = S (k) 1 M(k) 1, A(k) = S (k), an A(k) 3 = S (k) 3. Using the congruence projection in (6), we obtain the reuce moel  0 + σ k  (k) 1 + N k K 1 (ξ k ) (k) N k + K (ξ k ) (k) 3 û = ˆr. (47) The usage of this mask moel is similar to that escribe in section 4. But there is a key ifference. In lithography simulation flow, we only care about the near fiel at the bottom of the mask. In view of (8), this can be accomplishe by using ū n = Λū = ΛPû = P n û, (48) where ū n is the portion of ū that correspons to the near fiel at the bottom of the mask, an the sparse matrix Λ selects those entries from ū. An important benefit from (48) is that we only nee to store P n = ΛP in the mask moel. Matrix P n is much smaller than matrix P. The algorithmic etails for the offline an the online stage are shown in Algorithm 1 an Algorithm, respectively. e want to emphasize again that the cost of online stage is O(N 3 s + ( + N k )N s ). This is clearly inepenent of the orginal problem size N. Algorithm 1: Offline Stage to generate mask moel Input: Range of parameters in σ; mesh; N s : number of samplings; tol: truncation tolerance for rank revealing QR Output:  0,  (k) 1,Â(k),Â(k) 3 ; ˆr; P n (1) Form matrices A 0,A (k) 1,A(i) j),a( 3 an r in (46) () foreach k = 1 : N s (3) Ranomly sample σ (k) using the given ranges (4) Solve (46) for ū (k) (5) P = P ū (k) (6) Run rank-revealing incremental QR to obtain rank r an the fullrank columns P 1,P,...,P r using the given truncation tolerance tol (7) if r < k, i.e., P is rank eficient (8) P = P 1,P,...,P r (9) Exit the sampling loop (10) Use P as projector an compute  0, (k) 1,Â(i) j),â( 3, ˆr as shown in (6) (11) P n = ΛP as shown in (48) Algorithm : Online Stage to evaluate mask moel Input: σ*;  0, (k) 1,Â(k),Â(k) 3 ; ˆr;P n Output: ū n : the near fiel at the bottom of the mask (1) Instantiate the compact mask moel in (47) using σ () Solve (47) for û (3) ū n = P n û as shown in (48) 8. EXPERIMENTAL EVIDENCE e use the mask in Figure 1 to valiate our ieas. For the 3nm noe, the nominal values of w 1, w an s are assume to be 18nm. They have ientically inepenent istribution with the variance being 10%. This 0% variation range is probably more than sufficient for practical consieration. It shoul be note that if we a correlation among these parameters, it will only change the sampling but not the main steps in algorithm 1 an. ith a ranomly generate parameter set σ, we substitute (38) an (41) into (3) an solve (3) for ū 1. This is treate as the accurate solution. This way, we can clearly see the error introuce separately by the Kernelbase fitting in (43) an the moel orer reuction in Algorithm 1. A) Kernel-base Regression Accuracy ith the same parameter set σ mentione above, we solve (46) for ū an then compute the relative L norm error ū 1 ū ū 1. The relative error vs. the number of kernels is shown in Fig. 4. B) Moel Orer Reuction Accuracy In this experiment, N k = 0 kernels are use in (47). The corresponing small error in kernel fitting ensures that the MoR error in (47) ominates the overall error, as is clearly seen in Figure 5. ith the increase of the number

6 Overall Fitting MoR Relative error Relative error Number of Kernels Figure 4: Relative error in regression. of samples N s in Algorithm 1, the relative error ue to moel orer reuction ecreases rapily in Figure 5. C) Mask Moel Accuracy Figure 5 also shows the overall error in the approximate solution provie by the new mask moel. ith about 8 kernels an 10 samples, the mask moel can achieve a 1% overall relative error or -igits of accuracy. D) Mask Moel Spee The main cost of using the reuce mask moel is to instantiate the small ense matrices in (47) an then invert one small ense matrix. Instantiating a few ense matrices an then inverting a ense matrix takes about 1e 4 secon. For the simple two-imensional phase-shift mask shown in Fig. 1, irect use of FEM approach woul result in a sparse matrix A( σ) in (46) with N being aroun The CPU time use by a irect sparse solver for such a system woul be in the orer of a few secons on a esktop PC. So we see a 4-orer of magnitue improvement in spee. In aition, since the CPU time at the online stage irectly relates to the number of samples in Algorithm 1, the smooth trae-off between the accuracy an the CPU time is clearly emonstrate in Figure 5. Further more, our empirical stuies inicate that number of samples is a weak function of the original problem size or the number of parameters. This is certainly a very attractive feature of the new mask moel propose in this paper. 9. CONCLUSIONS In this paper, we propose two key new ieas to improve our work on the parametric mask moel in 8: 1) Parametric unstructure mesh using the istance function an the blening function; ) Kernel-base regression to significantly reuce the number of parametric terms in the final system. Numerical experiments emonstrate that the new mask moel offers smooth accuracy-spee traeoff an hence can be tune to ifferent esign applications. The moel generation in Algorithm 1 is inherently incremental an both theoretical an practical ways to estimate the moel error are reaily available. In aition, the parametric mesh allows the ecoupling between the mesh generation an the parametrization of the FEM matrices, a consierable simplification from implementation point of view. 10. ACKNOLEDGMENTS e want to thank Dr. Joel Phillips from Caence Research Labs for bringing the Kernel Methos in 14, 17 to our attention. e want to thank Prof. Per-olof Persson from University of California at Berkeley for bringing to our attention the use of blening function in 16. An finally, we want to thank Dr. Apo Sezginer from Caence Design Systems for the valuable iscussions Number of Samples Figure 5: Convergence of the L norm relative error in fiel istribution for the mask shown in Figure 1. Each parameter varies between 10% an +10% from its nominal value of 18nm. MoR refers to the error cause by the moel reuction stage. 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