comprehensive 3-d notching simulator with non-planar ......2. formulation the exposure model is...

Comprehensive 3-D Notching SimulatorWith Non-Planar Substrates

Eytan Barouch, Brian BradieDepartment of Mathematics and Computer Science

Clarkson University, Potsdam, NY 13676

Uwe Hollerbach, George Karniadakis, and Steven OrszagApplied and Computational Mathematics

Princeton University, Princeton, NJ 08544-1000

ABSTRACT

A comprehensive three-dimensional simulation model for non-planar substrate lithographyis presented. Matching substrate as well as standing wave effects are examined. The projectionprinting is simulated using Hopkins' results and the exposure model is solved using spectral elementdiscretizations of the nonlinear wave equation coupled with the rate equation for the photoactivecompound concentrate evolution. The dissolution algorithm describing moving fronts has beenmodified to handle various topographies, thus yielding the final profiles. Results are presented forseveral test problems.

1. INTRODUCTIONThe design of high resolution VLSI devices imposes significant demands on resist and

exposure tool performance. Optimization of photoresist performance can be greatly facilitated bycomplementing experimental results with numerical simulation of image characteristics. However,as pattern dimensions shrink, the influence of three-dimensional effects on image definition becomesignificant, so new classes of high resolution, accurate numerical simulators must be developed.To date, available simulators have been developed for planar substrates and have focused on two-dimensional effects. Future progress will be paced by the ability to handle non-planar problemsin three dimensions as well as problems involving oblique incidence.

In this paper, we present results from a new class of simulator that we are developing tohandle non-planar substrates in three dimensions, as well as oblique exposures. The model equa-tions used in the simulator involve three basic steps: projection printing, exposure, and dissolution.In future development of our simulator, we plan to include the effects of post-exposure bake byconvolving the results of the photoactive compound (PAC) concentrate with a heat (diffusion)operator following C. Mack (unpublished).

In Sec. 2 we present the mathematical formulation of the problems solved here. In Sec.3 we present some details of the numerical procedures used to solve these equations and theboundary conditions used in this formulation. In Sec. 4, we present results for model problemsinvolving both rectangular and square masks, integrating the results of the projection printing,exposure, and dissolution algorithms. We also discuss future enhancements of our simulator.

334 / SPIE Vol. 1264 Optical/Laser Microlithography 111(1990)

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2. FORMULATIONThe exposure model is composed of a coupled set of two non-linear partial differential

equations. The first equation is Maxwell's equation for standing waves of wavelength ) in a non-isotropic medium with complex refractive index N. The second equation is Dill's rate equationfor the photoactive compound (FAG) concentrate evolution. Explicitly we obtain1:

EE +2

E + V(E . Vlog(N2)) = 0 (1)

M=_CME.E* (2)

N=n-i_[AM+B] (3)

where E(x) is the electric field, M(x, t) is the PAC concentrate, ) is the wavelength of the incidentlight, n is the (real) index of refraction of the substrate, and the parameters A, B, C are thestandard Dill parameters.

Asymptotic (WKB) solutions of these equations for planar substrates have been studiedelsewhere2. In this paper, we report numerical solutions of the full equations for non-planarsubstrates.

3. NUMERICAL METHODS3. 1 Spectral element methods

A typical elliptic equation for a field variable c5 [see Eq. (1) above] can be put in the formof a Helmholtz equation with variable coefficients

—— (L/-f') _ A2 = f in 1 (4)19xj \ t9XjJ

In addition, let us assume for simplicity of presentation that there are homogeneous boundaryconditions on the boundary of the domain l. Let H be the space of differentiable functionswhich satisfy these homogeneous boundary conditions. Introducing test functions 'çb E H ,we canthen write the equivalent variational statement of (4) as,

fv_±ds + A2fid3 = _jcbfd3 (5)

A spectral element discretization3 corresponds to numerical quadrature of the variationalform (5) restricted to a subspace Xh C H . The discrete space X1, is defined in terms of thespectral element discretization parameters (K, N1 ,N2, N3), where K is the number of "spectralelements," and N1, N2, N3 are the degrees of piecewise high-order polynomials in the threedirections respectively that fill the domain ft By selecting appropriate Gauss-Lobatto quadraturepoints and corresponding weights Ppqr = PpPqP?, equation (5) can be replaced by its discrete

SPIE Vol. 1264 Optical/Laser Microfithography 111(1990) / 335


formK N1 N2 N3 K Ni N2 N3

i: : .ii Ppqr Jqr [1 3Jpqr + x2 > i: i: :i: Ppqr 4qrk=1 p=O q=O ?0 ' k=1 p=O q=O r=O

K Ni N2 N3= — i: : i i: Ppqrqr [1I;fIeqr (6)

k=1 p=O q=O r0

Here pqr S the Jacobian of the transformation from global to local coordinates (x, y, z) = (r, 3, t)for the three-dimensional element k. The Jacobian is easily calculated from the partial derivativesof the geometry transformation V(r, 3, t). The next step in implementing (6) is the selection ofa basis which reflects the structure of the piecewise smooth space Xh. We choose an interpolantbasis with components defined in terms of Legendre-Lagrangian interpolants, h(r1) = 5,. Here,r' represents a local coordinate and Sjj is the Kronecker-delta symbol. It may be shown that sucha spectral element implementation converges spectrally fast4 [convergence faster than any powerof 1/N] to the exact solution for a fixed number of elements K and N —+ oo, for smooth data andsolution, even in non-rectilinear geometries.

Having selected the basis we proceed by writing spectral element approximations for (or',b) within element k [denoted q5C (or çbIc)] as follows,

k 7nphm(r)hn(s)hp(t) Vm,np e (0, . . . , N1), (0, . . . , N2), (0, . . . , N3) (7a)

where is the local nodal value of . The geometry is also represented via similar typetensorial products with same-order polynomial degree, i.e.,

(:i;,y,z)k (xLv, znp)hm(r)hn(3)hp(t) Vm, n, p E (0, . . . , N1), (0, . . . , N2), (0, . . . , N3)

(7b)Here z;t are the global physical coordinates of the node mnp in the kth element.We now insert (7) into (6) and choose test functions i/'mnp which are non-vanishing at only oneglobal node to arrive at the final discrete matrix system.

The natural choice of solution algorithm for this latter system for a space-dependent co-efficient A(x, y, z) [see (4)] is an iterative procedure; to date direct solvers as well as conjugategradient techniques and multigrid methods have been implemented.

Next we discuss the convergence in space of the spectral element method. The maincontributions of the error in the spatial discretization come from the fact that the test functions'p that are employed for obtaining the discrete equations belong to a restricted subspace of thespace H1 . Secondly, errors may be introduced by inexact representation of the integrals involvedand the insufficiency of the quadrature rules used. As regards the latter, it is the choice of thecollocation points that greatly influence this error-component. The set of polynomials4 used inspectral element methods (Chebyshev or Legendre) allows for exact quadrature evaluation witha straightforward implementation. As regards the first error-component, there are twodifferent

approaches to project more closely the restricted subspace onto the space H1 , and thus attainconvergence: either the order of interpolants, N1 ,N2, N3, can be kept constant and increase thenumber K of spectral elements, or the number of spectral elements can be kept constant and the



number of collocation points per element can be increased. These two different approaches reflectthe difference in the convergence philosophy between finite element and spectral methods.

We conclude this section by commenting on the flexibility of the spectral element methodon representing not only curvilinear smooth geometries, but also on treating singular sharp-edge..like geometries. To demonstrate this we solve the Poisson equation in the domain shown inFig. 1 (A,C) using two different discretization approaches. In both cases a standard isoparametricmapping is employed, according to which the geometry is represented in terms of the same basisas the solution itself. The following Poisson equation,

is solved, which has the exact (smooth) solution

U = sinxeFirst, the solution is obtained for discretization (A); in this case the geometrical irregularity istreated as in global spectral methods. This results in poor convergence, as is indicated in Fig. 1,where we plot the error as a function of the total number of degrees of freedom in the x-direction,N. It is seen that the geometrical singularity results in a loss of the exponential convergence.The spectral element method, however, offers other alternatives by discretizing the geometry as in

(C). The spectral accuracy is recovered if an elemental interface passes exactly through the sharpcorner. For reference, we also include the results of the solution of the same equation solved inthe curvilinear domain (B). Exponential convergence is recovered again for this smooth geometry.

3. Projection printingThe classical techniques5 of Hopkins and of the optical Fourier transform are used to corn-

pute the image intensity distribution incident upon the resist surface. Let (x, y) be an orthogonalcoordinate system in the image plane, (c',9) an orthogonal system in the pupil plane and (,i)an orthogonal system in the object plane. Additionally, let F denote the complex transmissionfunction of the mask, K denote the coherent transfer function of the optical system and 'y12 denote

the phase-coherence factor as defined by Hopkins. Assuming illumination of the object plane bymonochromatic radiation with uniform intensity Io, the total intensity as a function of positionin the image plane is

I(x, y) = 1of f f f Y12( — , — ')F(, )F*(, i')K(x — , y —

xK*(x _ _ (1)

The object transmission function F is taken to be zero in opaque regions of the mask andunity in transparent regions. Assuming a circular source which radiates uniformly in all directions,the phase-coherence factor has the form

'Y12(' — c,7/ — i) = 2J1(z)/z (2)

SPIEVol. 1264 Optical/Laser Microlithography 111(1990) / 337


where Ji (z) is the Bessel function of the first kind of order 1 and

z = : _ 1)2 + (, ,I)2] un sin a ()nsin a is the numerical aperture of the imaging lens at the side of the image and o is the degreeof partial coherence. Finally, if the lens law is satisfied, K is given by

K(x - , y - ) = f I P(C, O)exp { _i [(x - + (y - )O]} dCdO (4)

where P(C, 9) is the pupil function. In the absence of aberration P(C, 9) is taken to be unity insidethe opening of the imaging aperture and zero outside. Following Hopkins the effect of defocus caneasily be incorporated into the pupil function.

For illustration purposes we also employ coherent illumination. In this case the phase-coherence factor reduces to unity and the six-fold integration becomes a four-fold integration.Furthermore, this quadruple integral can be decomposed into an inner double integral takenover the object plane (this is the Fraunhofer diffraction integral or equivalently the optical Fouriertransform of the object transmission function) and an outer double integral taken over the pupil ofthe imaging system. The necessary integrations are performed numerically using Gauss-Legendrequadrature. We thus have a working code for computing both coherent and partially coherentimages for masks of arbitrary shapes.

3.3 Dissolution ofphotoresistThe least-action-principle dissolution algorithm6 is based on the local mathematical equiv-

alence of the underlying developer solvent diffusion system to the curvature-dependent eikonaldescription of front propagation. The local nature of this equivalence implies that the solventtrajectories must be dealt with infinitesimally and not globally. Moreover, analysis of the eikonalsystem provides a system of coupled nonlinear ordinary differential equations for the coordinatesof solvent trajectories along with the essential initial conditions.

In its current implementation the dissolution algorithm is a time marching procedure. Theundeveloped resist surface is taken as the initial profile. During each time step the most recentprofile is divided in the two lateral coordinate directions into segments of equal and fixed arclength. The length of these segments controls the resolution of the final profile. Each arclength isallowed to develop separately for a small time interval Lit. A special fourth-order Runge-Kutta isemployed to solve the system of seven ordinary differential equations which control the trajectoryof the developer solvent. After each time step, a tensor product B-spline interpolating functionis used to obtain a smooth representation of the evolving surface. In essence, this is sufficientregularization that the formation of non-physical loops and shock waves is avoided. Also, thecurvature control can provide the extra necessary protection and, in addition, it contains globalproperties of the surface. The latter information, which is missing from ray-tracing equations,provides a more general methodology for profile dissolution.



4. RESULTSIn Figs. 2-9, we display various cases of matching substrates and standing waves for recti-

linear substrate geometries with square and rectangular masks. The projection printing employspartial coherence a = 0.5 and coherent optics as well. Since the geometries used in this paperhave one symmetry axis, it is sufficient to run the exposure model for 2D cases. The 3D PACconcentrate is thus obtained from scaling with Io(x, y), while the dissolution algorithm runs in3D . For more complex topographies, the full 3-D exposure model has been implemented. Resultsfor these more complex topographies will be presented elsewhere.

In Figs. 2-4, we display a series of rectilinear substrate topographies for a rectangular maskshape of 0.8 x 41Lwith incident light at 365 nm. These color representations of the PAC concentrateclearly display standing wave patterns. The blue regions represent maximum exposure while thered represent minimal exposure. The effect of geometry is illustrated in Fig. 5 by results obtainedby projecting a square mask of dimension 0.8 x O.8j on a sloped substrate. Notice the interestinginterference pattern.

In Figs. 6-9, we present 3D dissolution profiles for matched substrates. The rectangularand square masks for various dissolution times for coherent incident light.

For all these examples we see the clear cut appearance of notching. With our quantita-tive, robust and versatile model for notching, we can also undertake a quantitative study of thecombined effects of post-exposure bake, addition of dyes and non-reflective layers. An optimiza-tion algorithm is being developed to mazimize performance and line resolution, while minimizingoperator effort. With these additional developments, we hope that research as well as productionlithography systems will benefit significantly from the versatility of our computer codes.

5. ACKNOWLEDGEMENTSThis work was supported in part by AFOSR, DARPA, and NIST.

6. REFERENCES1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergammon, New York, 1980).2. S. V. Babu and E. Barouch, J. Opt. Soc. Am. 5 1460 (1988).3. G. E. Karniadakis, Spectral Element Simulations of Laminar and Turbulent Flows in Com-

plex Geometries, Appi. Num. Math., to appear (1990).4. D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods, NSF-CBMS Mono-

graph No. 26, Soc. md. Appl. Math., Philadelphia (1977).5. B. J. Lin, in Introduction to Microlithography, L. F. Thompson, C. G. Wilson, and M. J.

Bowden, Eds., American Chemical Society, Washington, D.C., 1983.6. E. Barouch, B. Bradie, and S. V. Babu, J. Vac. Sci. Technol. B 6, 2234 (1988); also ibid

B 7, 1766 (1989).

SPIE Vol. 1264 Optical/Laser Microllthography 111(1990) / 339


FIGURE CAPTIONS

1 . Error versus number of degrees of freedom in the x-direction. Exponential convergence is

recovered for appropriate spectral element discretization.

2. PAC concentration for rectangular mask (coherent illumination) and substrate with slope

f.J 300 Here A = O.6/j, B = O.1/j, C = O.02cm2/mJ and n = 1.65. Concentration

increases from blue (M 0.36) to red (M 0.90)

3. PAC concentration for rectangular mask (coherent illumination) and substrate with 45°

slope. Concentration increases from blue (M 0.32) to red (M 0.90). Other parameters

as in Fig 1.

4. PAC concentration for rectangular mask (coherent illumination) and substrate with slope

r...J 60°. Concentration increases from blue (M 0.25) to red (M 0.88). Other parameters

as in Fig. 1.

5. PAC concentration for square mask (coherent illumination) and substrate with equilateral

triangle depression at center of mask. Concentration increases from blue (M 0.34) to red

(M 1.00). Other parameters as in Fig. 1.

6. 10 sec dissolution profile for square mask (coherent illumination) and matched substrate

with 45° slope. Here A = 1.0/ti, B = 0.106/j, C = 0.019cm2/mJ and n = 1.69.

7. 40 sec dissolution profile for square mask (coherent illumination) and matched substrate

with 45° slope. Other parameters as in Fig. 5.

8. 5 sec dissolution profile for rectangular mask (o = 0.5) and matched substrate with 75°

slope. Other parameters as in Fig 5.

9. 20 sec dissolution profile for rectangular mask (o = 0.5) and matched substrate with 75°

slope. Other parameters as in Fig. 5.



Fig. 1 Error versus number of degrees of freedom in the x.direction. Exponential convergenceis recovered for appropriate spectral element discretization,

SPIE Vol. 1264 Optical/Laser Microlithography 111(1990) / 341

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comprehensive 3-d notching simulator with non-planar ......2. formulation the exposure model is...

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