IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 8, NO. 4, NOVEMBER 1995 419

Rigorous Three-Dimensional Time-Domain Finite-Difference Electromagnetic Simulation

for Photolithographic Applications Alfred K. Wong, Member, IEEE, and Andrew R. Neureuther, Fellow, IEEE

Abstract-The parallel electromagnetic simulation program TEMPEST has been generalized to analyze three-dimensional problems in photolithography. TEMPEST, which has been made available on the National Center for Supercomputing Applica- tions, combines together techniques for rigorous Simulation of electromagnetic scattering and diffraction, a novel and effiaent absorbing boundary condition, and synthesis of partially coherent images including the effects of optical system aberrations. The electromagnetic solution is based on the time-domain finite- difference method, but exploits the power of parallel computer architectures. Equations suitable for parallel implementation are given. Simulation time is fifteen to twenty minutes with 64 M simulation nodes on a CM-5 with 512 nodes and 16 GBytes of memory. The usefulness and effectiveness of the program for photolithographic applications are demonstrated by considering problems in projection printing of polarization and transmission effects in contact holes, and reflective notching which causes undesired exposure in photoresists.

I. INTRODUCTION

OMPUTER-AIDED design (CAD) tools have come to C play an important role in integrated circuit design, device design, and process design. Circuit simulation programs such as SPICE [20] were among the first to be developed and gain popular acceptance. Device simulation programs such as MINWIOS [30] and PISCES [27] are helpful in understanding device performance. Process simulation tools such as DEPICT [35], SAMPLE [24], SOLID 1111, and SUPREM [15] are becoming more popular as the cost of performing experiments rises continually with time and the cost of computation de- creases dramatically from year to year. However, with the introduction of new technologies and the scaling of optical lithography to smaller feature sizes, greater demands are placed on the scope of the physical models and the accuracy of their implementation in algorithms. At the same time, these numerical models must be efficient in order to provide a rapid solution. These needs together have placed stringent requirements on the completeness, accuracy, and efficiency of process simulation tools.

Modeling of electromagnetic problems in photolithography are particularly computation-intensive because typical feature

Manuscript received Febraury 21, 1995. This research was supported by

A. K. Wong is with the Interuniversity Micro-Electronics Center (IMEC),

A. R. Neureuther is with the Department of Electrical Engineering and

JEEE Log Number 9414525.

SRUSEMATECH under Grant 94-MC-500.

Leuven, Belgium.

Computer Science, University of Califomia at Berkeley, CA 94720 USA.

sizes of interest are on the order of a wavelength. In this regime, neither geometric optics (assuming that the radii of curvature of the surfaces are much larger than the wavelength) nor Rayleigh’s method (assuming that the wavelength is much larger than the radii of curvature) suffices. Even with the most advanced numerical techniques, some of these problems are not tractable. For example, solving the problem with rigorous frequency-domain methods requires the solution to a system of millions of simultaneous equations. Direct solution of the matrix is not feasible with current numerical techniques. Iterative solutions such as the conjugate gradient method [12] or GMRES [29] are not attractive as the matrix is not positive definite and the condition number is large. With time-domain methods, typically 15 simulation nodes per wavelength are required to achieve the desired accuracy. This translates to 16 million simulation nodes for a problem size of (16X3). The size of the problem makes it difficult to solve even on the most advanced workstations.

The general problem of electromagnetic scattering from topography has been addressed in various ways. These tech- niques can be classified as either frequency-domain or time- domain methods. (This is not the only classification; Ye- ung [49] classified the different techniques as modal expan- sion, Green’s function-based, volume-based, and fast multipole methods.) For frequency-domain methods, the electromagnetic fields are usually expressed as a superposition of some basis functions. The steady-state electromagnetic solution is found by solving (directly or iteratively) a matrix. For time-domain methods, a time parameter is introduced and the steady- state electromagnetic fields are found by time-marching, i.e., electromagnetic interaction with matter is solved in time until the fields become time-harmonic.

Rayleigh made one of the first attempts to analyze the problem of electromagnetic scattering from a periodic grat- ing rigorously. He made the assumption that the fields can be ,expressed as a linear superposition of propagating and evanescent waves [28]. Gallatin et al. [7] as well as Bobroff and Rosenbluth [2] applied this method to model the images of alignment marks under photoresist. Nyyssonen and Kirk [23] developed the waveguide method and applied it in the examination of photolithographic issues such as edge detec- tion [22] and alignment mark signals [13]. Yuan et al. [50] extended the waveguide method to the transverse magnetic polarization and applied it in the study of wafer alignment and line width measurement [50]. Lucas et al. [16] also applied

.OO 0 1995 IEEE 0894-6507/95$04

420 IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 8, NO. 4, NOVEMBER 1995

the same technique in the study of two-dimensional phase- shifting mask structures. Matsuzawa et al. [18] solved the Helmholtz equation using the finite-element method together with the boundary-element method and applied it in the study of photoresist bleaching on a stepped perfectly conducting substrate. The approach was improved by Urbach and Bernard [37] with the extension to more general domains and partial coherence. Using the spectral element method in which the electromagnetic field is expanded with the Legendre polyno- mials as the basis functions, Barouch et al. [ 11 is able to study three-dimensional reflective notching on nonplanar substrates.

Yee [48] was one of the first to replace Maxwell's equa- tions by a set of finite difference equations and solve the electromagnetic problem via a staggered grid. Using this approach, Taflove and Brodwin [34] examined the steady- state electromagnetic field resulting from the scattering of a two-dimensional uniform and circular dielectric cylinder. Improvements of the conventional TDFD method to allow for more flexible geometries and more efficient memory us- age were also suggested. Mei et al. [19] demonstrated the feasibility of the conformal TDFD method which enables the finite-difference mesh to conform to the object surfaces. Zivanovic et al. [51] proposed a subgridding TDFD method which employs a variable step size.

Because of its computation intensive nature, time-domain solutions of electromagnetic problems in photolithography was impractical until the advent of powerful supercomputers in the late 1980s. Wojcik et al., studied the time-domain finite- element (TDFE) method 1421 and applied it in the study of light scattering from silicon surfaces [39], alignment mark signals [40], and line width metrology [41]. Concurrently, Guerrieri et al. [9] formulated and Gamelin [8] implemented the TDFD approach on the connection machine CM-2 and applied it in the study of reflective notching [31], metrology of polysilicon gate structures [33], mask material and coating effects on image quality [5 ] , and alignment mark signal integrity [43].

This paper describes the major formulation extensions and algorithmic developments in the program TEMPEST [46] which have found wide utilities in modeling difficult problems in optical lithography [5 ] , [31], 1331, 1431, and extreme ultra- violet lithography [21]. Extension from two-dimensional anal- ysis to three-dimensional analysis is described. This extension increases the number of updating equations from three to six. Together with the increase in the number of simulation nodes for three-dimensional analysis, this increase in the number of updating equations (and hence the number of variables) makes the program memory limited rather than computation time limited. Accuracy of the TDFD scheme is discussed via the local truncation error as well as the eigenvector and eigenvalue of the set of difference equations. Stability of the numerical scheme is examined using the Fourier analysis. The problem of image synthesis, i.e., relating the electromagnetic fields calculated by TEMPEST to an aerial image, is tackled

Coherent Source of Arbitrary Amplitude Profile

fl Field V a l u e d Initially

Photomsist

si wafer

Fig. 1. A typical simulation domain in TEMPEST. The structure can rep- resent arbitrary three-dimensional nonplanar and inhomogeneous topography. The domain is excited at the top by a monochromatic plane wave.

in photolithography including mask transmission and reflective notching. The usefulness of the program is illustrated via the study of contact hole transmission and reflective notching.

11. FORMULATION A typical simulation domain for electromagnetic field calcu-

lation is shown in Fig. 1. The structure can represent arbitrary three-dimensional nonplanar and inhomogeneous topography. For lithographic applications, the interest is almost always the response of the structure at a particular frequency or at a narrow band of frequencies. The simulation domain is thus excited with monochromatic radiation at the top. The problem is to find the steady-state solution for Maxwell's equations:'

- aD - V x H = - + J at

- aB at V X E=--

supplemented with the constitutive relations:

$= E E

B=@ - - J=aE

where E, p, and a are, respectively, the permittivity, per- meability, and conductivity of the material. In general, the parameters E , p, and 0 are functions of the frequency of the electromagnetic wave. For the application in hand, however, they are assumed to be constant because of monochromatic excitation. Using Stokes' theorem, (1) and (2) can be re-written in the weak form

(4)

by integrating TEMPEST With another simulation Program called SPLAT [36]. Synthesis of the above techniques into the software package TEMPEST makes it a flexible and

where 4 F . d i and s, F . d S represent, respectively, the line integral and surface integral of a variable F".

rigorous topography simulator which finds broad applications In the MKS system.

WONG AND NEuREuTHER: RIGOROUS THREE-DIMENSIONAL TIME-DOMAIN FINITE-DIFFERENCE ELEClROMAGNETIC SIMULATION 421

Following the TDFD method proposed by Yee [48] in which the differential S t is replaced by At and Sx by Ax, (3) and (4) are solved using a cubic grid in which the field components are staggered and occupy distinct locations in space as shown in Fig. 2. The surface integral and line integral are thus evaluated on square surfaces as shown in Fig. 3. This discretization scheme leads to three scalar equations in two-dimensional analysis for the transverse electric (TE) or the transverse mag- netic (TM) polarization, but six equations in three-dimensional analysis for the field components E,, E,, E,, H,, Hy. and H , :

EF+'(i, j , k) = aE,"(i, j , k) + @[H,"+('/')(i, j - f, k) - H,"+('/2)(i, j + f , k) + H,"+(1/2)(i + f, j , k)

- H"+(1/2)(i Y - ;, j , k)] ( 5 ) E"+l(i y 73+5, . k + f )

+ @[H,"+'1/2'(i - f, j + f , k + f) - H,n+(1/2)(i + I . + f, k + f)

- H,"+(1/2)(2, j + f, k)] E,"+'(i + f , j , k + $)

+ @[H,"+(l/2)(1 + f , j , k) - H,"+(1/2)(i + f, j , k + 1)

- H,"+(1/2)(i + f , j - I, 2 k + f)]

- - H,"-Q/2)(2 + ;, j + f , k + f) - y[E,n(i + f , j , k + f) - E,"(i + ;, j + 1, k + f ) + E,"(i + 1 , j + 5 ' , k + f) - E;(i, j + ;, k + f)]

H"+(1/2)(i Y + f, j , k) k)

= aE:(i, j + a, k + f)

2 , 3 + H,"+(l12)(i, j + f , k + 1)

(6)

= aE,n(i + f , j , k + $)

+ H,"+(1/2)(i + f , j + f, k + f) (7)

H,"+(1/2)(i + f , j + f , k + 3)

(8)

- - H"-(1/2)(i + f, j , Y

- y[E,n(i, j , k) - E,"(i + 1, j , I C ) + E,"(i + f , j , k + $) - E,n(i + f, j , k - f)]

- - H,"+/2)(2, j + f, k) - y[Eyn(i, j + f , k - f) - Eyn(i, j + f , k + f) + E,"(i, j + 1, k) - EF(2, j , k)]

(9) H,n+(1/2)(i, j + ;, k)

(10)

where a = ( 2 ~ - a A t ) / ( 2 ~ + o a t ) , /3 = (At/Ax). [ 2 / ( 2 ~ + oa t ) ] , and y = (At/pAx). The superscript of the field variables stands for the time step (time = neat) , the subscript represents the direction of the field, and (i, j , k) signifies the node position at (iAx, jAx, kAx).

z IncidcotRadiation

X J L Absorbing Boundary Conditions

Fig. 2. The TEMPEST simulation domain. Maxwell's equations are solved over a cubic grid using the TDFD method. The field components are staggered over the grid.

Fig. 3. The electric field component Ez(z, j , k ) is calculated by summing up the magnetic field values of the four neighboring nodes. The magnetic field components are assumed to be constant along the line segments 1-2.2-3, 3-4, and 4-1, and the electric field component E, is assumed to be constant over the square surface bounded by 1-2-34

III. ACCURACY

As a rough estimation of the accuracy of the TDFD scheme, consider the wave equation resulting from Maxwell's equa- tions in nonconductive materials

- a2E

at2 V 2 E = pe -.

For time-harmonic electromagnetic waves, without loss of generality, the electric field can be assumed to be travelling in the z-direction and polarized in the x-direction with a magnitude EO, an angular frequency w and a wave number K. related to w by K = w/c , i.e., E ( F , t) = E,(z, t) = Eo sin ( w t - m ) . In continuous space, this sinusoidally varying electric field is an eigenvector of the operator V 2 with an eigenvalue of - K ~ , i.e.,

-

a 2

az V2E,(Z, t) = 2 E,(Z) t)

=--K2Ez(z, t)

For finite-difference schemes, the V 2 operator can be approx- imated by the discrete operator D:

v2 + D; =D: 1

Ax2 = - (K1- 2Ko + K-1)

where K,E(kAz) = E [ ( k + m) Ax]. Operation of 0: on the sinusoidally varying electric field E,(z, t) results in the

422 IEEE TRANSACllONS ON SEMICONDUCTOR MANUFACTURING, VOL. 8, NO. 4, NOVEMBER 1995

following discrete equation:

1 Ax2

D?Ez(z, t) = - { E [ ( k + l )Ax, t]

where 77 is the intrinsic impedance of the material. From stability considerations (which are discussed in Section IV), the relation between At and Ax is

Ax - 2E(kAx, t) + E[(k - l )Ax, t]} At 5 ~

c&. - -- Eo {sin (wt - & A X ( ~ + 111 Ax2 - 2 sin (wt - KAxk) + sin [ut - ~ A x ( k - l)]}

The relative error of the scheme is thus estimated to be

[ 1 - cos (&Ax)] sin (wt - K Z ) . - 2EO Ax2

Taylor series expansion of cos (KAx) for small &Ax gives the following result:

D?E&, t) = - K ~ ( 1 - - K2z2) EO sin (ut - nr)

K ~ A x ~ = -K2 (1 - T) EZ(Z, t). (11)

The discrete operator D: thus preserves the eigenvector but gives an error in the eigenvalue. The fractional error in the eigenvalue is ( ~ ~ A x ~ ) / 1 2 . In terms of the number of simulation nodes per wavelength d = (A/Ax), the relative error, which is defined as the total error in the value of the field divided by the magnitude of the incident field, is given by

To achieve a two percent accuracy thus requires about 15 simulation nodes per wavelength. This is consistent with the results shown by Guerrieri et al. [9] and from numerical experiments [46]. For conductive materials, the accuracy can be ascertained from the local truncation error T by Taylor series expansion of (5)-(10). For example, E,"(i, j , k) can be expanded at r'= (iAx, jAx, kAx) and t = [n + (1/2)]At as

E - - - + - At 8E 1 ( A t ) . % - 2 d t 2 2 1 At 6 2

- - ( -I3 + o(At4).

Using this procedure, the local truncation error for the z- component of the electric field is

.To estimate the magnitude of the local 'truncation error, approximations of the derivatives as well as the values of E

and U are necessary. The magnitudes of the derivatives can be estimated by

where d is the density of simulation nodes per wavelength, and E, and E; are, respectively, the real and imaginary parts of the relative permittivity of the material. Notice in (1 3) that the relative error is no longer second order accurate in the spatial discretization Ax when the material is lossy. In the limit when ~i >> E,, the relative error varies as d-' instead of d-2 and the TDFD scheme is only first order accurate. Such cases arise when the magnitudes of the real and imaginary parts of the refractive index are comparable because E~ = n: - n: and E; = 2nTn;. Only for lossless materials does the TDFD scheme retain the second order accurate behavior. For slightly lossy materials in which E, 2 ~ i , the TDFD scheme still possesses a pseudo second order behavior so long as d > 2 because the first term in the denominator ( E , ) dominates over the second term ( 1 . 7 7 ~ i l d ) . As an example, consider silicon at a wavelength of 0.365 pm. The refractive index is (6.522 - j2.705) [25]. In the MKS system of units, this gives a permittivity of 35.2~0 ( E , = 35.2) and a conductivity of 3 5 . 3 ~ ~ 0 ( E ; = 35.3). The estimated relative error is 2% with a node density of 15 per wavelength, agreeing with the estimation from the D," operator.

Iv. STABLITY

Engquist er al. [6] have shown that the TDFD scheme described in (5)-(10) is unstable unless the temporal discretiza- tion At and the spatial discretization Ax satisfy the following relation:

At 5 [ 2 Ax2 sin2 (i S,aS) + -112 + - 1 sin2 (f K ~ A Z ) ]

A22

for arbitrary K ~ , K ~ , and tsZ. Since 0 5 sin2 ( ~ ~ A x / 2 ) 5 1, and Ax = Ay = Az, equation reduces to

(14) Ax

At 5 ~

C G '

The factor dim in (14) represents the number of dimensions of the structure. For a planar structure, d i m = 1. For a structure which is uniform across one of the dimensions such as a photoresist line, d i m = 2. For a general structure, d i m = 3.

However, (14) provides only a necessary condition for the TDFD scheme to be stable. To guarantee stability, it turns out that in addition to the condition in (14), the magnitude of the real part of the refractive index n, must be greater than the

WONG AND NEUREUTHER RIGOROUS THREE-DIMENSIONAL TLMEWMAIN FINITE-DIFFERENCE ELECTROMAGNETIC SIMLTLAnON

~

423

TABLE I REFRACTIVE INDEXES OF COMMONLY USED MATW IN THE FABRICMION

OF INTEGRATED CIRCUITS [25]. *KODAK KTFR P H ~ T ~ R E S I ~ T [38]

imaginary part ni. In other words, E, = (np - n:) > 0. The Appendix demonstrates this result by Fourier analysis of the numerical scheme.

The condition E,. > 0 puts a constraint on the allowable material properties. Since E, = np - n:, this condition means that only materials with n,. > ni can be studied. This is a severe restriction as the refractive indices of a lot of commonly used materials in the fabrication of integrated circuits shown in Table I possess the quality that ni > n,. The problem of simulation of highly dispersive materials, i.e., materials in which ni > n,, has been tackled by Wong el al. [46] following the approach by Luebbers et al. [17]. The basic idea is to model the time-domain convolution relation between the electric displacement D and the electric field E.

V. IMAGE SYNTHESIS

A. Motivation

TEMPEST calculates the steady-state electromagnetic fields throughout the volume of a structure under the excitation of a monochromatic harmonic field. In many instances in photolithography, however, the interest of the user is not in the electromagnetic fields themselves but the image of the structure as viewed through an optical imaging system. Furthermore, typically only the intensity distribution resulting from the fields can be observed to validate the simulation results. There thus exists a need to relate the steady-state fields calculated by TEMPEST into an intensity profile. The basic idea is to Fourier transform the electric and magnetic fields across a horizontal plane into the diffraction harmonics from which imaging information can be found.

There are two physical phenomena to consider in extending TEMPEST for use in image formation. First, the imaging systems used in photolithography are (spatially) partially co- herent to reduce ringing at the dark-bright transitions whereas illumination is assumed to be coherent in TEMPEST. One way to model partial coherent effects is to superpose simulation results for a lot of different angles of illumination. This approach, although accurate and feasible, generally requires a lot of simulation runs and is inefficient. Hopkins’ approx- imation [3] can be used to simplify the problem. The key assumption is that the magnitudes of the diffraction efficiencies are independent of the incident angle. This approximation is adequate even for 1X projection systems with a numerical aperture (NA) of less than 0.5 [45].

The second physical phenomenon to consider is aberration of the imaging system. For nonaberrated systems, the calcula- tion of the optical image is a straightforward integral. For aberrated systems, however, the calculation becomes com- plicated. This calculation can again be simplified by using Hopkins approximation to separate mask and imaging system parameters. With this assumption, the calculation becomes primarily that of computation of the transmission cross coeffi- cients [3] (TCC’s). Routines for the calculation of these TCC’s are available, for example, in a simulation program called SPLAT [36]. However, SPLAT, ,which is based on the scalar diffraction theory and assumes infinitesimally thin masks, must be modified to link in the vector diffracted field information from TEMPEST.

B. TEMPEST-SPLAT Inte$ace

SPLAT simulates aerial images of photomasks with the assumption that the masks are infinitesimally thin and have ideal transitions. The optical image profile of any mask at any spatial point is thus given by [3]

. M(%a, K E & ) M * ( K : , K ; )

. , - J - - r [ ( K : - n a ) z + ( K : - K b : ) Y ] d K; dK& dtc: drci

(15)

where M ( K % , tcY) represents the Fourier coefficients of the mask and the asterisk represents complex conjugation. In SPLAT simulations, the values of M(K,, tcY) are calculated by Fourier transformation of the ideal mask. Thus, in order to link SPLAT and TEMPEST, the values of M(tcZ, tsy) must be modified.

The quantity MM’ for the scalar fields in (15) can be interpreted as the energy transmitted through the ideal mask. n u s , it is analogous to the Poynting vector S =E x H* for the vector fields. Assuming that the mask lies in the zy-plane, the quantity of interest is then the energy travelling in the z- direction. Thus, only the z-component of the Poynting vector S, = (EzHG - EyH,*) is of interest. The spatial intensity distribution can thus be expressed by modifying (15) as

r r r r

With this connection, rigorous electromagnetic simulation is combined with arbitrary lens aberrations, allowing the study of mask topography effects, partial coherent effects as well as the influence of lens aberrations simultaneously. Different illumination schemes such as annular and quadruple illumi- nation as well as resolution enhancement techniques such as spatial filtering can be modeled. Application is not limited

-

424 IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 8, NO. 4, NOVEMBER 1995

to mask transmission studies, however, because SPLAT can be perceived as imitating the functions of a microscope. Therefore, many problems such as a dark-field or bright-field alignment system can be studied.

C. Implementation

An interface to SPLAT requires information on the diffrac- tion harmonics of the electric and the magnetic fields in both the 2- and y-directions as discussed above. To calculate the diffraction harmonics, Fourier transforms of the instantaneous fields are taken on either the top (for the reflected diffraction harmonics) or the bottom (for the transmitted diffraction harmonics) zy-plane. Since the quantity (ExH; - EyHz) is needed, Fourier transformation must be performed on each of the four field variables E,, Ey, H,, and H V . However, since these variables are displaced from one another in the FDTD grid, a correction factor must be multiplied to the Fourier transformed variables to take into account the staggering of the numerical grid. For example, if the reference position is chosen to be (i + 1/4, j + 1/4, k + 1/4), the correction factor for the E, component can be determined as follows:

Ex(.,, .y)

VI. MEMORY REQUIREMENT AND PERFOFWANCE

The computation time requirement as a function of problem size has been discussed by Guerrieri et al. [9]. It was shown that the computation time is constant when the number of processors increases proportionally to the problem size. When the number of processors is fixed, the computation time increases linearly with the problem size.

For a fixed amount of physical resource (number of proces- sor nodes), there corresponds a maximum problem size over which larger problems cannot be solved because of memory limitations. For the CM-5 at the National Center for Supercom- puting Applications (NCSA) with 512 processor nodes and a total memory of 16 GBytes (each node has 32 MBytes of memory), the maximum number of simulation nodes is 64M (due to memory limitation on each of the processor node), corresponding to a simulation domain with 512 x 512 x 256 nodes. From the accuracy requirement discussed in Section 111, about 15 simulation nodes per wavelength is required to achieve a 2% accuracy in the fields. The largest simulation domain is thus approximately 16384X3. Assuming a deep-UV wavelength of 0.248 pm and a refractive index of 2, the largest simulation domain is 4 pm x 4 pm x 2 pm, or 32 pm3. The time required for simulation of such a problem is typically 15-20 minutes. For example, the smaller contact hole problem shown in Fig. 5 (256 x 256 x 64 nodes) requires 271 seconds with 128 processor nodes. (This simulation time corresponds

to 14.6 MFlops per processor node, 11.4% of the theoretical maximum of 128 MFlops.) Scaling to the maximum problem size of 512 x 512 x, 256 nodes with 512 processor nodes gives a simulation time of 18 min. Larger problems must be simulated with either more processor nodes or more memory per processor node. The computation time in the former scenario will not increase as demonstrated by Guerrieri et al. [9]. In the latter scenario, the computation time only increases linearly with the amount of memory. Thus, a twofold increase in memory per processor node would only result in a 15- to 20-min increase in simulation time. The algorithm is therefore memory-limited rather than computation time-limited on the CM-5.

VU. APPLICATION EXAMPLES The ability to analyze three-dimensional structures is very

important in photolithography as it allows many complicated issues to be examined. Effects in two-dimensional features may be exaggerated or diminished in a three-dimensional structure. For example, results from two-dimensional rigor- ous simulation of a 0.5-MNA wide isolated space feature on a binary mask show polarization effects of 3% and a transmission loss of 10% as the light passes through the opening [42]. Reflective notching is another important three- dimensional issue as nonplanar topography on the wafer can cause undesired exposure in the photoresist and a change in the critical dimension of the feature being formed.

The existence of these effects and the ability of TEMPEST as a prediction tool has been verified in two-dimensional struc- tures. For instance, two-dimensional TEMPEST has been used to predict effects of mask topography and glass edge shape on the performance of alternating phase-shifting masks [26]. The difference in transmission between the 0' (unetched) and 180" (etched) openings on alternating phase-shifting masks was predicted by TEMPEST as shown in Fig. 4 in which the peak intensity at the 0' opening is 10% higher than that at the 180' opening. This result is confirmed by the SEM picture of the imaged 0.25 pm linespaces (negative) photoresist shown in the same figure. There is a 0.1-pm linewidth difference between the 180' and the 0' openings. This difference in intensity can be reduced by undercutting the glass beneath the opaque chromium layer via isotropic wet etching as shown in Fig. 4(b). However, the intensity of the light going through the etched portion of the mask in this case becomes higher than that going through the unetched portion. Another possible solution is to etch the glass first by anisotropic dry etch and then slightly undercutting both the shifted and nonshifted regions on the mask using an isotropic wet etch as shown in Fig. 4(c) and (d). Depending on the amount of glass etched during the second wet etch, the intensity of light going through the etched portion of the mask is higher or lower compared to the unetched portion of the mask. In this case, an undercut of 60 nm shown in Fig. 4(c) produces equal peak intensities for both openings as indicated by the simulated images and the photoresist lines. An undercut of 120 nm tends to make the peak intensity at the etched opening higher as shown in Fig. 4(d).

WONG AND N E U R E W R RIGOROUS THREE-DIMENSIONAL TIME-DOMAIN FINITE-DIFFERENCE ELECTROMAGNETIC SIMULATION 425

Fig. 4. Simulated images and exposure results of 0.25-pm lines and spaces created by different alternating PSM's. (a) A mask with vertical glass edges creates an intensity imbalance which is manifested as a 0.1-pm linewidth difference in the photoresist. (b) Isotropic etching of the 180' glass opening overcompensates the problem and makes the peak intensity at the 180' region higher and the photoresist line wider. (c) Anisotropic etching followed by 60 nm of isotropic etching produces images of equal peak intensity and photoresist lines with equal widths. (d) The same mask as in case (c) except that the amount of isotropic etch is 120 nm tends to overcompensate the problem. (Photoresist pictures courtesy of Christophe Pierrat at AT&T Bell Laboratories.)

A. I X Contact Hole Example

In this section, the suitability and usefulness of TEMPEST in analyzing three-dimensional photolithographic problems are demonstrated via the study of 1X contact hole printing. The structure of a 0.5 pm x 0.5 pm contact hole is shown in Fig. 5 . The domain of 4 pm x 4 pm x 1 pm (length btimes width x height) is divided into a cubic grid of 256 x 256 x 64 simulation nodes. This translates to about one simulation node per 16 nm. The incident radiation is assumed to be a normally incident plane wave with the electric field polarized in the %-direction (the length direction). A prototypical 1X deep-W projection printing system with a numerical aperture (NA) of 0.5 and a partial coherence factor (a) of 0.4 is chosen for imaging.

The images of the 0.5 pm x 0.5 pm (1 ANA x 1 ANA) opening predicted by TEMPEST and SPLAT (thin mask and scalar approximations) simulations are shown in Fig. 5. Notice in the figure that the SPLAT image shows a peak intensity which is 25% higher than that of the TEMPEST image. This loss in peak intensity is also observed for space patterns [44], and is apparently due to a combination of propagation through a small aperture and energy dissipation in the chromium layer due to its finite thickness. In fact, a plot of the peak intensity as a function of opening size for TEMPEST and SPLAT simulations shown in Fig. 6 indicates that the SPLAT image

shows higher peak intensity for opening sizes of less than 1 ANA. The SPLAT image can give a peak intensity which is almost twice that predicted by TEMPEST for the case of a 0.4 pm x 0.4 pm (0.8 ANA x 0.8 "A) opening. The difference in the peak intensities between SPLAT and TEMPEST is greater for contact hole structures than for a space pattern. An exact relationship of transmission loss between an open space and a square contact hole is complicated. However, for feature sizes smaller than 1 ANA, transmission loss for a square contact hole is about three times that of an isolated space. This increase is due to the simultaneous presence of the north/south and easdwest edge effects in a contact hole structure.

Since the transmission loss of a square contact hole is about three times that of an isolated space, the effects in a contact hole cannot be approximated by summing the effects of two perpendicular space openings. Nonlinear effects are present. These effects can also be ascertained from Fig. 6 as the transmission loss is not constant over any range of the feature size, implying that a different bias must be used for features of different sizes. This is not a serious restriction in IC fabrication, however, because all the contact holes usually have the same dimensions.

The transmission loss phenomenon discussed above is im- portant for 1X steppers (such as the Ultrastep steppers).

-

426 IEEE TRANSACI'IONS ON SEMICONDUCTOR MANUFACTURING, VOL. 8, NO. 4, NOVEMBER 1995

. .. z Y

w

+- lPm ~ silica

chromium

4Crm 4 P plane view cross-sectional view

Fig. 5. Structure of a 0.5 pm x 0.5 pm contact hole used in TEMPEST simulation (bottom). The simulation domain of 4 pm x 4 pm x 1 pm is divided into a cubic grid of 256 x 256 x 64 simulation nodes. Images of the 1 X/NA x 1 X/NA contact hole predicted by TEMPEST (top left) and SPLAT (top right) simulations show differences in transmission and polarization effects.

Square Opening Size (UNA per side) - Fig. 6. Peak intensity of square openings as a function of size. For openings smaller than 1 X/NA, the SPLAT images always show higher peak intensity than the TEMPEST images.

However, for reduction printing on 4X or 5X steppers, the transmission loss is much reduced and the aerial image pre- dictions from SPLAT are adequate. Nevertheless, the finite transmission loss in reduction masks can render photoresist exposure time predictions from the thin-mask approximation in error by a few percent [IO].

Another interesting feature in Fig. 5 is that the image predicted by SPLAT shows a four-fold symmetry (with respect to the x-axis, the y-axis, the x = y line and the s = -y line) whereas the TEMPEST image exhibits only a two-fold

symmetry (with respect to the x-axis and the y-axis). Since the four-fold symmetry shown by the SPLAT image is also present in the square mask pattern, lack of symmetry with respect to the x = y line and the x = - y line in the TEMPEST image indicates that the electromagnetic fields interact with the metallic chromium layer differently depending on the relative orientation between the incident polarization and the metal surface. In fact, the TEMPEST image in Fig. 5 shows an elliptical shape which is elongated in the s-direction when the incident electric field is polarized in the %-direction. The eccentricity (defined as the ratio of the image width in the y- direction to the width in the 2-direction) as a function of the opening size is shown as the open squares in Fig. 7. In the absence of polarization effects, a square contact with IC1 = 0.8 prints slightly rounded by 6% as shown by the SPLAT image in Fig. 5. The polarization effect is about 3 times larger as the eccentricity reaches a maximum of 1.17 when the opening size is 1NNA. This suggests that there is a critical size at which polarization effects are the most important. For sizes smaller than this critical size, polarization effects are relatively less important because of the difficulty in transmission through a small opening, and the contact hole becomes more difficult to resolve. For contact holes larger than this critical size, the relative importance of the metal edges decreases as the clear area increases.

This polarization effect is expected to be more pronounced for rectangular patterns because a narrow slit may transmit different amounts of energy depending on the incident polar- ization. Fig. 8 shows the TEMPEST images of a 0.4 pm x

WONG AND NELJREUTHER: RIGOROUS THREE-DIMENSIONAL TIME-DOMAIN FINITE-DIFFERENCE ELECTROMAGNETIC SlMULATION 427

1 : lo ~ 1 2

Square Opening Size (UNA per side)

Fig. 7. Eccentricity as a measure of lack of symmetry. The value should be one in the absence of any polarization effects. For a 1X chromium mask, the eccentricity can be as large as 17%.

0.5 pm mask and a 0.5 pm x 0.4 pm mask. It can be seen from the figure that for the 0.4 x 0.5 mask, the image is longer in the x-direction than in the y-direction although the mask opening is longer in the y-direction. This indicates that polarization effects are even stronger than the orientation of the rectangular mask feature. For the 0.5 x 0.4 mask, polarization shortening of the y dimension causes the structure to appear more elliptical as indicated by the oval shape of the image. The image predicted by SPLAT shown in the same figure displays a less elliptical image, indicating the effects of polarization in the TEMPEST images.

Since the effective length of the opening in the y-direction is smaller when polarization effects are considered, the eccen- tricity of the images predicted by TEMPEST is smaller than the eccentricity of the images predicted by SPLAT. Table 11 shows the eccentricity of the images predicted by TEMPEST and SPLAT for different sized rectangular openings. The eccentricity is determined by the x and y extent of the image at the 30% intensity level (with respect to clear field). The difference between the eccentricities can be as much as 3 1 % for an opening of 0.5 pm x 0.6 pm. Notice also that for certain structures, the eccentricity of the TEMPEST images may be less than 1 while the mask aspect ratio is greater than 1 . This effect may complicate the design of masks as different bias values are needed for the two different directions if polarized illumination is used. From Table 11, a bias of about 0.05 pm in the y-direction is needed to correct for polarization effects assuming that no bias is applied to the 2-direction. In most projection systems, however, the use of unpolarized light sources would reduce almost all asymmetries due to polarization effects in square contact holes, but it would still be important to bias the contact hole according to Fig. 6 to increase the peak intensity of the image.

'

B. Reflective Notching

Besides its applications in the study of mask problems, three-dimensional TEMPEST is also useful in the study of

TABLE II ECcE"l?UClTY OF THE TEMPEST AND SPLAT IMAGES FOR MASK OPENINGS OF

DIFFERENT DIMENSIONS. FOR ANY CONTACT HOLE, THE ECCENTRICITY OF THE TEMPEST IMAGE Is LESS THAN THE SPLAT IMAGE. THE

DIFFERENCE CAN BE AS LARGE AS 31% FOR A 0.5 ,Um X 0.6 pm OPENING

0.2 x 0.5 0.4 x 0.5 0.6 x 0.5 0.8 x 0.5 1.0 x 0.5 0.5 x 0.2 0.5 x 0.4 0.5 x 0.6 0.5 x 0.8 0.5 x 1.0

2.50 1.25 0.83 0.63 0.50 0.40 0.80 1.25 1 .a 2.00

N/A 0.97 0.75 0.57 0.45 NIA 0.85 0.91 1.32 1.88

1.66 1.15 0.84 0.58 0.45 0.60 0.87 1.19 1 .n 2.24

NIA -18.6% -12.0% -1.8% 0.0% NIA -24% -30.8% -30.3% -19.1%

formation of latent images in photoresist over nonplanar topography. This is especially important in the patterning of the polysilicon gate. Topography on the wafer may cause uneven exposure and hence nonuniform critical dimension of the polysilicon gate as it transverses over the active region and field oxide. As an example, consider the printing of a polysilicon line on a negative photoresist as shown in Fig. 9. (The effect of using a light-field mask and positive photoresist will be more pronounced.) A conformal layer of polysilicon with a thickness of 0.25 pm is coated on the oxide. The polysilicon runs from the field oxide (0.5-pm thick) through the bird's beak (lateral dimension of 0.4 pm) onto the active region with a gate oxide thickness of 90 A. A latent image is formed on the planar photoresist layer with Dill's ABC parameters [4] given as (0.95, 0.083, 0.016). The image is formed by SPLAT simulation of a 1.75-pm wide opening on a 5X reduction mask at a wavelength of 0.365 pm. A total dose of 100 mJ is delivered. Dynamic bleaching of the photoresist is modeled by dividing the total dose into five steps.

Fig. 10 shows the latent image within the photoresist (PAC concentration) of a vertical plane along the line aa' in Fig. 9. Far away from the step, the latent image shows the effects of standing waves caused by the high reflectance of the polysilicon layer. At the bird's beak, the exposure energy is re-directed by the polysilicon topography, resulting in a region of low exposure. The effect of the step can also be seen in Fig. 11, in which the PAC concentration of vertical planes along bb', cc', and dd' are shown. The picture on the top (bb') shows the standing wave within the photoresist in the field oxide region. The picture on the bottom (dd') shows the PAC concentration in the photoresist in the active region. These latent images show wider critical dimensions than the one in the middle picture (cc') in which the PAC concentration of a vertical plane at the location of the bird's beak is plotted. This reduction of polysilicon critical dimension due to the slope of the bird's beak has also been observed experimentally [14]. The efficacy of remedial techniques such as anti-reflective coatings and dyed photoresists in reducing the reflective notching problem has been examined by Socha et al. [31] via experimentation and TEMPEST. The study shows a good match between simulation and experimental results.

428 IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 8, NO. 4, NOVEh4BER 1995

I Plane View

C

. . . . . . . . . . . . . 12 . . . . . . . . .

s i I

d

’4 -I J 1J

a 1.35 pm

1’ ; :

. . . . . .

. . . . . . . .

...

......

...... .......

.... ...... ....

. . - . *

. . ’ % . ’ . 2J

--

Y (4 Y (4 Y

Fig. 8. Images of a 0.4 pm x 0.5 pm opening by TEMPEST (left), a 0.5 pm x 0.4 f rm opening by TEMPEST (middle), and a 0.5 p m x 0.4 pm opening by SPLAT (right). Polarization effects make the effective by dimension of the opening smaller in TEMPEST simulations.

Side View

0.7 pm

0.25 pm 0.171 pm 0.009 pm 0.329 pm

Photoresist (1.69, -jO.Ol)

Polysilkon (6.187, -j2.453)

Gate Oxide (1.4745, -jO.O) Field Oxide (1.4745, -jO.O)

Silicon (6.551, -j2.648)

t I

Fig. 9. Printing of a polysilicon gate. The 0.25-fim thick conformal polysil- icon is coated on the oxide; it passes from the field oxide through the bird’s beak onto the active region.

VIII. CONCLUSION

Generalization of the time-domain finite-difference method to three-dimensional analysis has enabled the study of difficult technological issues in photolithography. Image synthesis from the calculated electromagnetic fields with considerations of partial coherence effects and the influence of optical system aberrations is also demonstrated. A novel boundary condition has been implemented which improves the efficiency of the algorithm [47]. These improvements have been incorporated into the parallel TEMPEST program and made available for use on the CM-5 machine at the National Center for Supercomputing Applications in Illinois. To achieve a 2% accuracy requires 15 nodes per wavelength with a simulation time of 15-20 min.

Applications of TEMPEST have shown its usefulness in the study of difficult problems in different areas of photolithog- raphy. In its two-dimensional form, TEMPEST was shown to be useful as a prediction tool whose results were verified via experimentation. Three-dimensional TEMPEST is currently being heavily used in the modeling of metrology, photomask fabrication and projection printing problems to understand

J? P I 4

x oun) +.l

Fig 10 PAC concentration of a vemcal plane along the h e aa‘ in Fig 9 Far away from the bird’s beak, the latent image shows the effects of standing waves At the polysihcon step, the exposure energy is re-directed by the topography, resulting in a region of low exposure

topography scattering effects and to assess the effectiveness of various technologies.

APPENDIX STABILITY OF THE NUMERICAL

SCHEME VIA THE FOURIER ANALYSIS

Consider a two-dimensional structure in the TE polarization. (The TE polarization is defined in such a way that the electric field is parallel to all surfaces, i.e., the electric field is polarized such that it oscillates perpendicularly to the plane of the structure.) Under this special case, only three out of the six equations of the TDFD scheme remain

where Q, /3, and y have the same definitions as those in Section 11. To determine stability for this scheme, the Fourier analysis

WONG AND NEUREUTHER. RlGOROUS THREE-DIMENSLONAL TIME-DOMAIN FINITE-DIFFERENCE ELECTROMAGNETJC SIMULATION 429

grid. The two-norms of Ez(il j ) and f i , (Q~, 0 2 ) are related by Parseval's theorem

i j

Fig. 11. beak (middle picture) is different from the other two

PAC concentration of vertical planes along bb' (top), cc' (middle), and dd' (bottom). The critical dimension at the bird's

is used. Consider the discrete Fourier transform of E, The above equations can be written in terms of a system of

r

1 0

-227 sin - 0 1

02

01 i2y sin -

2

2 -

The "energy" in the time-domain variable E, is therefore equivalent to the integrated spectral energy.

Fourier transformation of (18)-(20) results in the following equations in the frequency space of 01 and 82

k:+l(e,, 02) = 2; [a + 4py (sin2 + sin2 -

+ it: (i2p sin :) + it; (-i2p sin 3 ) 2

H : + ' ( o ~ , 0 2 ) =it: + E ;

it;+'(01, 02) = 12; + E;

Parseval's theorem then gives

L I [An+' I IC0 I Pol 12. (22)

Assuming that A can be diagonalized, i.e., A = XDX-l, (22) then becomes

- < IIXD"+1X-1IIC0IJFo/12. (23)

The TDFD scheme is thus two-norm stable if llDllCO 5 1 and if X is bounded. The former condition means that the magnitude of the eigenvalues of the matrix A must be bounded

~

430 IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACIZTRING, VOL. 8, NO. 4, NOVEMBEK 1995

by 1; the latter condition implies that the eigenvectors of A must not diverge when the spatial and temporal discretizations Ax and At approach zero.

To verify the former condition, the characteristic equation for the matrix is

(A - 1) + x 4/37 sin2 - + sin2 2 - a - 1 + a { [ ( : 1 3 ) 2 I } = 0.

Denoting sin2 (61 f 2) + sin2 (132 f 2) by A, the eigenvalues are A 1 = 1, and

f J16f127’A2 - 8/37A(~y + 1) + (CY - l)’]. (24)

The magnitudes of X2,3 in (24) is less than or equal to 1 if and only if E > 0 and cAtfAx < 1/& (The factor under the square root sign corresponds to the number of dimensions and is 2 in this case.) The latter requirement corresponds to that derived by Engquist [6], but the former is also necessary for the TDFD scheme to be stable.

To verify the boundedness of the eigenvectors of A, notice that the eigenvectors

where i = 1, 2, 3 are bounded when Ax and At approach zero since they are related by (14).

ACKNOWLEDGMENT The authors would like to thank the National Center for

Supercomputing Applications at UIUC for use of their Con- nection Machine CM-5. They would also like to thank the reviewers for helpful comments and detailed review of the paper-

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Alfred K. Wong (S’94-M’95) was bom in Hong Kong, on November 19, 1969. He received the B.S., M.S. and Ph.D. degrees from the University of Califomia at Berkeley, all in electrical engineering, in 1990, 1992, and 1994, respectively.

His thesis research was camed out under the guidance of Professor Neureuther, and was focused on the development of a vector electromagnetic simulation algonthm on the connection machine and applications of ngorous electromagnetic simulation in photolithography. In 1994, he joined the Interum-

versity Micro-Electronics Center (IMEC) as a post-doctoral fellow, where he is conducting research in phase-shiftmg masks and opka l proximity correction.

Andrew R. Neureuther (S’62-M’67-SM’S7-F’89) was bom in Decatur, E, on July 30, 1941. He received the B.S., M.S., and Ph.D. degrees from the University of Illinois, Urbana, all in electrical engineering, in 1963, 1964, and 1966, respectively.

In 1966, he joined the Department of Electri- cal Engineering and Computer Sciences, University of California at Berkeley, as a faculty member, where he is currently a Professor. On industrial leave he has also worked for IBM at the T. J. Watson Research Center and the San Jose Research

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Laboratory on modeling optical and e-beam lithography and etching and deposition. He has worked on electromagnetic theory, diffraction gratings, and is currently engaged in the study of integrated circuit process technology and its simulation. His current research emphasis is on the collaborative development of the user-oriented computer programs for SIMulation from Layout (SIML).

Dr. Neureuther is a member of the National Academy of Engineering.