Michael S. Yeung

Single integral equation for electromagnetic scattering by three-dimensional homogeneous dielectric objects

A single integral equation formulation for electromagnetic scattering by three-dimensional (3-D) homogeneous dielectric objects is developed. In this formulation, a single effective electric current on the surface S of a dielectric object is used to generate the scattered fields in the interior region. The equivalent electric and magnetic currents for the exterior region are obtained by enforcing the continuity of the tangential fields across S. A single integral equation for the effective electric current is obtained by enforcing the vanishing of the total field due to the exterior equivalent currents inside S. The single integral equation is solved by the method of moments. Numerical results for a dielectric sphere obtained with this method are in good agreement with the exact results. Furthermore, the convergence speed of the iterative solution of the matrix equation in this formulation is significantly greater than that of the coupled integral equations formulation.

Modeling High Numerical Aperture Optical Lithography

Electromagnetic diffraction theory is applied to obtain a rigorous and comprehensive description of the imaging and exposure process in a projection optical system imaging a one-dimensional, periodic object in a planar layer of photoresist. The method is applicable to high numerical aperture and thick-photoresist systems, and accounts for the exposure dependent absorption characteristics of positive photoresists. It is used with the development simulator in SAMPLE to simulate the physical profile of the developed image. Theoretical and experimental results are given, which show asymmetrical variation of the developed image with focus. This asymmetry is found to depend on photoresist thickness, and the dependence is shown to be incompatible with the usual approximation of normal ray propagation in the photoresist.

Extension of the Hopkins theory of partially coherent imaging to include thin-film interference effects

In this paper, we extend the Hopkins formulation to take into account high numerical aperture and thin-film interference effects by introducing a new TCC function for each depth inside the photoresist, which completely characterizes the lens/thin-film system with respect to partial coherence, aberrations, defocus and interference effects at the given depth within the photoresist. The basis of the new formulation lies in the fact that, in the presence of the thin- film stack, each point on the exit pupil of the optical system maps linearly not into a single plane wave, but into a family of multiply reflected and generally obliquely propagating plane waves, when bleaching induced scattering effects are neglected. The response within the photoresist due to each incident plane wave is calculated by the method of thin-film optics. The results are then used in the calculation of a new, matrix pupil function of the lens/thin- film system for each depth within the photoresist. Obliquity factors appropriate to high-NA systems are included in the new pupil function. For the Koehler illumination commonly used in reduction projection systems, it is shown that the total irradiance at each depth within the photoresist is expressible in terms of a matrix TCC in the limit when the rays incident on the mask are all nearly vertical, as is the case in a 5X reduction system.

Limitation of the Kirchhoff boundary conditions for aerial image simulation in 157-nm optical lithography

The aerial images of half-wavelength features with 0/spl deg/ and 180/spl deg/ phases obtained by using the Kirchhoff boundary conditions are compared with those obtained by using rigorous electromagnetic field computation for 248-nm lithography and 157-nm lithography. The discrepancies between the aerial images computed by the two methods are large at both wavelengths, but they are much larger for TM polarization at the wavelength /spl lambda/=157 nm. These discrepancies are due to diffraction effects in the aperture regions, which are more pronounced at /spl lambda/=157 nm because of the larger ratio of the thickness of the chromium absorber to the wavelength required at /spl lambda/=157 nm for a given attenuation factor. This shows that diffraction effects in the aperture regions must be included when simulating aerial images in 157-nm lithography.

Illuminator optimization for projection printing

In this paper we report a new algorithm designed to enable printability and enhanced defocus budget at half and sub- half wavelength feature sizes. An integral part of this algorithm is the optimization of aerial image contrast, performed in stages, for an algorithmically determined set of contrast cost functions. The optimization is performed on the geometric shape of the condenser filter, herein referred to as the illuminator. Combining (1) illuminator optimization, (2) reticle proximity correction, and (3) attenuated phase shift masks allows one to perform corrections to aggressive SRAM mask designs with features sizes as small as 140 nm, when employing 248 nm illumination, as well as 125 nm feature sizes of lines and spaces. We also present optimizations for 80 nm lines, with 120 nm spaces using 193 nm illumination.

Three-dimensional nonplanar lithography simulation using a periodic fast multipole method

This paper discusses an extension of the fast multipole method to electromagnetic scattering from doubly periodic, multilayer wafer topography. The novelty of our approach lies in the use of a pseudo-periodic translation operator which can be computed efficiently using fast Fourier transform. Results obtained using the rigorous boundary conditions for dielectric surfaces are compared with those obtained using the approximate impedance boundary condition. The latter is shown to give good results for the type of topography usually encountered in lithography simulation. Results of reflective-notching simulation using the IBC method are presented.

Three-dimensional mask transmission simulation using a single integral equation method

A single integral equation formulation for electromagnetic scattering from three-dimensional dielectric objects is discussed. The new formulation converges significantly faster than the traditional, coupled integral equation formulation. The new formulation is extended to incorporate the exact boundary conditions for isolated mask features by using dyadic Greens functions for the stratified medium background. Results of three-dimensional phase-shifting mask simulation are presented.

Simulation study of process latitude for liquid immersion lithography

A simulation package has been developed for predicting the influence of immersion, i.e. the presence of a uniform liquid layer between the last objective lens and the photoresist, on optical projection lithography. This technology has engendered considerable interest in the microlithography community during the past year, as it enables the real part of the index of refraction in the image space, and thus the numerical aperture of the projection system, to be greater than unity. The simulation program described here involves a Maxwell vector solution approach, including polarization effects and arbitrary thin film multilayers. We examine here the improvement in process window afforded by immersion under a variety of conditions, including λ = 193 nm and 157 nm, annular illumination, and the use of alternating phase shift mask technology. Immersion allows printing of dense lines and spaces as small as 45 nm with acceptable process window. We also examine the effect of variations in liquid index on the process window and conclude that the index of the liquid must be known to and maintained within a few parts-per-million. This has important implications for the temperature control required in future liquid immersion projection systems.

Measurement of wave-front aberrations in high-resolution optical lithographic systems from printed photoresist patterns

A new method of measurement of wave-front aberrations in high-resolution optical lithographic systems used for semiconductor manufacturing is proposed. The method is based on the measurement of the positional shifts and focus offsets of a set of printed photoresist grating patterns with different periods and orientations, produced under nearly fully coherent illumination condition. The proposed experimental procedures are described in detail and various sources of systematic and random errors are discussed. It is estimated that a measurement precision of /spl lambda//50 for the wave-front aberrations at selected points on the lens pupil can be achieved with this method.

Photolithography simulation on nonplanar substrates

A iod1 of oJ. )tica. I hthograpliy suitable for certain types of onedimensional 1)eriOdic tOpogra)hy including birds beaks afl(L reflowed BISC4 structures is described. it is based on a formalism of diffraction grating theory which uses a. coordinate t. raiisforna. tiou to iap all the nonpla. uar surfaces oiit. o parallel planes a. iid it ca. ii be used for the rigorous simulation of photoresist. latent images of oiiedimensional mask patterns with periodicity fuller j)a. ra. lleI or I)(rI)e11(Iicu1a. r to that of the topogra)hy. Effects of t. opogra)11y scattering 1)111k illiaging and )11otob1eachu1Ig are fully ta. keii into a. ccounl. for both types of mask patterns. Simulation results illustrate the combined effects of topography scattering and I) ulk iiiaging in iioiipla. na. r pliotoli tliogra. phiy using high iiuinerical aperture optics.