Lithographic image simulations

Lithographic image simulations

Microelectronic North-Holland Engineering 3 (1985) 355-362 355 Lithographic image simulations Douglas S. Goodman IBM T.J.Watson Research Center Yor...

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Microelectronic North-Holland

Engineering 3 (1985) 355-362

355

Lithographic image simulations Douglas S. Goodman IBM T.J.Watson Research Center Yorktown Heights, New York 10598

Lithographic aerial images have been studied by digital image simulation. The aspects considered include the variation of numerical aperture, defocus, aberrations, source size, source shape, and source centration.

The physics of image formation is well understood and the equations governing image formation are known, at least in the scalar wave approximation.i Nevertheless, the situation is sufficiently complicated that practical questions about lithographic imaging cannot be conveniently answered by analytical methods; numerical modelling is required. Over the past years we have done a number of twodimensional image simulations to investigate the effects of various parameters of imaging, such as numerical aperture, defocus, and aberrations, as well as source size, shape, and position. The details of the program are described elsewhere. 2 The distribution of irradiance (power/area) in the aerial image is computed; photoresist and processing effects are not included. A typical image is shown in Figure 1. The dashed line is the ideal geometrical image. The solid lines are the irradiance contours for I = 0.1,0.3,0.5, &c. The normalization used throughout is that the mask is illuminated with unit irradiance, so the ideal geometric image has unit irradiance (assuming unit magnification). Thus, the brightness of the source decreases as its size increases. On the upper left corner of the figures are arrows showing the diameter of the first zero of the Airy pattern, 1.2h/NA, and the cutoff period h/2NA, the inverse of the highest spatial frequency present in the image irradiance. Here h is the wavelength and NA is the numerical aperture. Information about the conditions of the image formation are given to the right of the contour plot. The linewidths here are 0.75X/NA, as they are in the other images shown. The imaging system has a circular pupil and no aberrations. The source for the projector is a centered circular disk. The ratio of the diameter of the source image to that of the pupil is denoted by 0, which here equals 0.75. Throughput is defined as the fraction of the power transmitted by the object that reaches the image, the losses here being due to light diffracted by the object beyond the collection cone of the imaging lens. The maximum value of the irradiance in the image is I,,,,,. In this image the contour Z - 0.3 corresponds closely to the edge of the straight portions of the nominal image. This result is found empirically over a wide range of conditions. Defocus: The simulations confirm that for an aberration-free lens, the image is degraded very little fora quarter wave of defocus, giving a depth of focus in air of f X/2(NA)z. Figure 2 shows how badly degraded the image is at twice this distance from focus, i.e. for W,, = h/2. Aberrations: Computations for third-order aberrations show that the Raleigh quarter-wave criterion holds very well. That is, wavefront aberrations of less than h/4 have little effect on images, and greater amounts of aberration degrade images significantly. Coma is a special case, because the comatic wavefront is equivalent to a tilted wavefront plus a residual. If there is not too much coma, then the residual is sufficiently small that the principal effect is a lateral shift of the image, that is, an effective distortion. Figure 3 shows a comatic image for which W,, = h/2. The irradiance contours are shifted with respect to the nominal image by nearly half the linewidth. A comparison with Figure 1, shows that the edges are quite sharp. The relative sharpness is in part due to the fact that here u = 0.5 rather than 0.75.

0167-93 17/85/$3.30

0 1985, Elsevier Science Publishers B.V. (North-Holland)

356

D.S. Goodman / Lithographic

image simulations

Source size and image form: It is well known that images formed by projectors with very small sources are characterized by coherent ringing and that, as (Tincreases, irradiance profiles become more rounded, like those of noncoherent images. There is also a change in the shapes of the irradiance contours, which are quite wiggly for small sources. Figure 4 shows an image formed with a centered point source (coherent illumination). The irradiance has numerous overshoots, the greatest reaching the value of 1.72. Figure 5 shows slices through images formed with centered disk sources with o = 0,0.25,0.5, 0.75, 1.0, 1.25. The first source is a centered point, and the last overfills the pupil. These curves show the well-know phenomenon of increasing slope with decreasing (T. Source size and power efficiency: Some of the light that passes the entrance pupil of the imaging lens, so it does not reach the reaching the image plane, decreases as the source size increases, of the pupil is more easily diffracted beyond it. The throughputs 88%, 87%, 83%, 78%, 72%, 53%. (Note that the throughput the particular slice shown.)

through the mask is diffracted beyond image plane. The fraction of the light since light directed closer to the edge for the images shown in Figure 5 are is defined for the entire image, not for

Ring-shaped sources: A method of using coherent sources for projector illumination without introducing speckle is to image a spot of light originating from the source into the entrance pupil of the lens, and to rotate the spot about the lens axis. The time-averaged equivalent source is then a narrow ring. The images formed using ring-shaped sources are quite similar to those formed with disk-shaped sources of the same diameter for values of a up to about 0.75. For greater ring diameters, the image undergoes a transition to a darkfield image, which obtains when the ring overfills the pupil. For a given diameter, the throughput with a ring source is less than that a disk, the ratio decreasing as a increases. Figure 6 shows an image formed with a ring whose diameter is 0.75 that of the pupil. Except for total power, this image is little different from that due to a disk source of the same diameter, Figure 1. Source symmetry and centration: A practical question in projectors is that of the extent to which the source image must be symmetrical about the center of the pupil. Image simulations have shown that this is not critical. For example, if a small source is moved from the center of the pupil to a point halfway to the edge of the pupil in any direction, the irradiance contours that would be seen in photoresist show little change. (We assume here that illumination uniformity is maintained in the mask plane; in actual projectors, misalignment inthe pupil may be correlated to nonuniformity at the mask.) Source centration and defocus: If the source is not centered in the pupil, then geometrical optics predicts that with a telecentric lens defocus of the image plane results in a lateral shift of the image. This translation occurs because the effective chief ray is defined by the source rather than the pupil. Simulations have been done to study the magnitude of this effect in the diffraction theory of image formation. The case used was that of a single point source imaged halfway between the lens axis and the edge of its pupil--an extreme decentration. For tolerable amounts of defocus there is a small distortion effect. In particular, for linewidths of 0.75h/NA and defocus W,, = X/4, the shift is about l/6 of the linewidth, half that predicted geometrically. Source size and depth of focus: The claim sometimes appears in the literature that depth of focus increases as ~7decreases. This may be true in some cases, but it does not hold in general. Simulations were done for linewidths of 0.75h/NA and a defocus of W,, = X/2, with e ranging from 0 to 1. For 0 < 0.25, the image is completely jumbled, while for (T2 0.5, the image is blurred, but recognizable. (With coherent illumination it is possible to have excellent contrast, but no fidelity. For instance, a random speckle pattern has 100% contrast.) 1.

H.H.Hopkins, “On the diffraction pp 408-432 (1953)

2.

Marc D. Levenson, Douglas S. Goodman, Scott Lindsey, Paul W. Bayer and Hugo A. E. Santini, “The Phase-Shifting Mask II: Imaging Simulations and Submicrometer Resist Exposures,” IEEE Transactions on Electron Devices, v ED-3 1, pp 753-763 (1984)

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