- the illumination wavelength λ (193nm for present advanced systems)
- the coherence parameter σ which specifies the directions from which the mask is illuminated
- the physical layout of the mask
- the numerical aperture (NA) of the projection lens which depends on the opening angle and the refractive index n of the material between the projection lens and the photoresist (air, vacuum, or immersion liquid)
- the position of the wafer with respect to the image plane (defocus)
- several other parameters such as polarization of the source, wave aberrations of the projector lens, flare (scattered light from rough surfaces), etc.
The aerial and resist images are computed by scalar and vector Fourier optics. Therefore, the light diffracted from the mask - the so called diffraction spectrum - is computed by a Fourier transformation of the mask layout or by more rigorous methods. The projector is described by a pupil function which specifies the amplitude transmission and phase delay for the diffracted light which passes through the lens. The aerial image is obtained by an inverse Fourier transformation of the product of the diffraction spectrum with the pupil function and by integration over the source (Figure 2). The computation of the bulk image requires the additional application of thin film transfer matrix techniques to describe the propagation of the light in the photoresist / wafer stack.
The following examples demonstrate some typical applications of aerial image modeling:
Verification and optimization of mask layout
Figure 3 shows a typical problem which occurs for the imaging of line-ends. The projector lens limits the spatial frequency (or angular) range for the diffracted light from the mask. This frequency limitation results in a rounding of the aerial image contours compared to the mask layout (black lines). The occurrence of light in the nominally dark areas of the image results in a line end shortening which, for example, may cause contact problems in electronic circuits. To avoid such a phenomenon, one has to block a certain amount of light. This is done by adding serifs or hammerheads to the end of the line on the mask. In Figure 3 it can be seen how the size of these hammerheads modifies the resulting image. Aerial image simulation can be very helpful to find critical positions in the layout of the mask, which produce similar phenomena, and to devise correction strategies. This is what lithographers call optical proximity correction (OPC).
Impact of wave aberrations of the projector lens on the aerial image.
Figure 4 demonstrates the simulation of the impact of wave aberrations on the aerial image quality. The mask layout is shown in the left upper position. The smallest linewidth in the layout is 90nm. The mask is imaged with a wavelength of 193nm and a numerical aperture of 1.35 under immersion conditions (refractive index of the immersion liquid 1.44). The "ideal" image without aberrations (lower, left) shows some pronounced proximity effects like corner rounding of the letters "I", pronounced deformation of the letter "B". Symmetrical aberrations such as defocus and spherical aberration produce a pronounced image blur. A combination of defocus and astigmatism (upper right) results in a strong orientation dependence of the image. Vertical features like the "I" letters are well resolved whereas horizontal features appear blurred. Coma aberration (lower right) produces other asymmetries and image artifacts. For demonstration purposes the degree of wave aberration was assumed to by much larger than the values of typical lithographic projection systems.
Impact of modeling approach and polarization of the illumination on the resulting image
At large numerical apertures, image formation becomes strongly polarization dependent. A corresponding simulation example can be seen in Figure 5. It shows aerial images of an array of 68nm wide contact holes with a period of 136nm (exposure wavelength = 193nm, NA=0.75, optimized geometry of the illuminator) with different polarizations of the illuminator. For unpolarized illumination the resulting image (left) shows a relatively poor contrast. This contrast becomes even worse for radial polarization of the source with respect to the optical axis (center). The best contrast is obtained with azimuthal polarization of the source (right).