Table 1 The first eleven Zernike polynomials.

Z	Name	Equation for Z	Imaging consequence
Z	Piston	1	None
Z	Image translation X	2 cos()	Shift of image, independent of pattern
Z	Image translationY	2 sin()
Z	Defocus	3 (2²-1)	Image degradation
Z	Astigmatism ±45°	6 ² sin(2)	Orientation-dependent shift of focus
Z	Astig. Hor./Vert.	6 ² cos(2)
Z	Coma Y	8(3³ -2) sin()	Image asymmetry and pattern-dependent shift of image
Z	Coma X	8(3³ -2) cos()
Z	Three-leaf clover	8³ sin(3)	Imaging anomalies with threefold symmetry
Z	Three-leaf rotated 30°	8³ cos(3)
Z	Third-order spherical	5(6 -6²+1)	Pattern-dependent focus shifts

Image shifts--Z , Z

These two aberrations represent a simple tilt of the OPD surface, and the imaging consequence is a positional shift of the image in the plane of the wafer. The shift can be represented as a vector (X, Y) which is proportional to the Zernike coefficients as

(X, Y)= (a, a) ×2/NA,            (4)

Thus, an aberration of a =0.05 waves causes the image to shift in the X direction by 0.1 /NA, relative to that of an unaberrated lens. The amount of shift is completely independent of the complexity or feature size of the pattern. If a and a were constant across the entire lens field, a simple realignment of the wafer would correct the positional error. However, in most lithographic optics there is significant variation of a and a across the lens field, resulting in lens distortion which ultimately causes overlay errors. In an IC manufacturing line with many lithographic tools, lens distortion variations among the tools are one of the most important limits to overlay performance.

Defocus--Z

The Z aberration leads to a quadratic OPD surface which tends to degrade the image contrast, edge slope, pattern fidelity, and resolution relative to an aberration-free image. Under the paraxial assumption which is valid when NA « 1, a is directly proportional to the defocus Z through the relationship

           (5)

i.e., a = 0.072 waves is equivalent to one Rayleigh unit /(2NA²) of defocus. [At higher NA, when the paraxial assumption breaks down, small amounts of higher-order spherical aberrations are introduced by defocus, but Equation (5) still describes the dominant effect.]

Focus variation is ubiquitous in IC manufacturing because of the combined effects of many problems: wafer nonflatness, autofocus errors, leveling errors, lens heating, etc. A useful lithographic process must be able to print acceptable patterns in the presence of these unavoidable focus variations. Similarly, a useful lithographic process must be able to print acceptable patterns in the presence of variations in the exposure dose. To account for simultaneous variations of exposure dose and focus, it is useful to map out the "process space," i.e., the exposure-defocus space [10], within which acceptable lithographic quality occurs. Figure 3 shows an example of a process space calculated for a 350-nm line/space grating pattern printed by an aberration-free 0.5-NA projector with = 248 nm and partial coherence = 0.6, and assuming an approximate model for APEX-E resist [11]. The irregular area within the four curves represents the exposure-defocus space where the resist linewidth, or critical dimension (CD), prints within ±10% of the target 350-nm CD. Various rectangular process windows can be defined within this space, such as the one shown in Figure 3, which has an exposure latitude of 15% and a 1450-nm depth of focus (DOF). The process window is the standard measure of the robustness of the process to variations [10,12].

Figure 3

The effect of Z aberrations is to shift the process window along the focus axis according to Equation (5). Process windows of all pattern types, orientations, and feature sizes are shifted by the same amount. If a varies across the lens field, the surface of best imagery is not planar, and the usable depth of focus (UDOF) is correspondingly reduced [10]. For the example in Figure 3, if a varied by 0.036 waves over the lens field, the best focus position would vary by 250 nm across the lens. Since IC production requires that all parts of the lens print simultaneously, the UDOF at 15% exposure latitude is reduced from 1450 nm to 1200 nm, corresponding to the overlapped area of the two extreme process windows.

Astigmatism--Z , Z

These two Zernike polynomials are saddle-shaped OPD surfaces that are positively curved in one direction and negatively curved in an orthogonal direction. The effect of such aberrations on imagery is to cause lines with one orientation to be positively shifted in focus while lines of the orthogonal orientation are negatively shifted in focus. Pure Z introduces a focus difference Z between lines with horizontal and vertical orientations, given by

           (6)

In a similar way, pure Z causes focus differences between lines of +45° and -45° orientations. The proper combination of a and a can represent a more general astigmatism between two orthogonal lines of arbitrary orientation [13].

Because it is generally necessary to print both line orientations simultaneously, astigmatism has a degrading effect on the process window. Figure 4 illustrates a simulated process window similar to that of Figure 3, but with the addition of a = 0.05 waves of astigmatism. The process window for horizontal lines is identical in size and shape to the aberration-free calculation, but shifted by +243 nm in focus. Vertical lines are similarly shifted by -243 nm, resulting in a relative shift Z = 486 nm. The common process window which can print both orientations is substantially reduced, so that the UDOF at 15% exposure latitude is reduced from 1450 nm to 964 nm. The amount of focus shift for long line structures is dependent only on the line orientation, and not on the linewidth or proximity to other lines.

Figure 4

Coma--Z , Z

Coma occurs when image contributions from different pupil radii shift relative to one another, in contrast to the Z , Z image shift, where all image contributions shift by the same amount. The definition of Y coma in Table 1 can be rewritten as

Z = 2 (3²-2)× Z,            (7)

illustrating that the shift for on-axis rays with near 0 is different, and of the opposite sign, from the shift for off-axis rays with near 1. As with image shift, two numbers, a and a, are required to characterize both X and Y coma. It should also be noted that Z and Z represent only the lowest-order coma terms, normally called third-order coma.

A consequence of coma is that symmetric patterns may print asymmetrically. Let us consider the example of a three-bar pattern, where three 250-nm-wide dark lines are separated by 250-nm spaces, imaged by a 0.5-NA projector with = 248 nm. Figure 5 shows resist profiles simulated by the PROLITH/2* simulation program² under various imaging assumptions. Figure 5(a) illustrates the aberration-free case, using an ordinary chrome (binary) mask and a partial coherence = 0.6. The image is symmetric about the center of the pattern, and the outside lines are slightly wider than the interior line. Figure 5(b) shows the addition of coma, a = 0.035 waves, which causes the left line to be roughly 50 nm narrower than the right line. (Such a linewidth variation would be considered a major problem in an advanced CMOS gate-level process.) Figure 5(c) shows that for the case of = 0.3, the linewidth asymmetry is significantly increased, and the height of the left and right resist patterns is also different. Figure 5(d) shows annular off-axis illumination with = 0.7 and = 0.6. The direction of asymmetry has changed such that the left line is now wider than the right line. This can be understood through the different sign of the image shift for on-axis and off-axis light in Equation (7).

Figure 5

Not only can the linewidth of patterns change because of coma; the center position of the patterns can also change. This shift is highly dependent on the details of the mask pattern as well as illumination. For example, a small isolated contact hole with a feature size of 0.5 /NA shifts less with coma than a large contact hole with a feature size of 1.5 /NA, using a = 0.3 projector. Therefore, the presence of coma destroys the concept of a single "lens distortion" map that can be unambiguously measured and applied to any mask pattern. Since relatively large patterns, e.g., "box-in-box," are almost universally used to measure overlay errors in IC processing, coma can cause a relative overlay shift between the measured overlay patterns and the actual device patterns. The current overlay error control scheme in IC processing is based on the assumption that only simple image shifts, i.e., Z and Z, are present.

Three-leaf clover--Z , Z

The next two Zernike terms represent OPD surfaces with threefold symmetry. Z is identical to Z, except that it is rotated by 30° so that the proper combination of a and a can represent a surface of any desired orientation. The main effect of three-leaf-clover aberration on lithography is to cause undesirable imaging artifacts. One pattern which has high sensitivity to this aberration is the attenuated phase-shift mask (PSM) contact hole. Figure 6 shows aerial image calculations of an isolated 350-nm contact hole using a 0.5-NA projector with = 248 nm and = 0.3. Figure 6(a) shows the image from a chrome (i.e., binary) mask with a = 0.045 waves. While the peak intensity is down a few percent compared with that of an unaberrated image, the image contours show little evidence of an aberration. Figure 6(b) shows the image from a 13% attenuated PSM with the same three-leaf-clover aberration. Surrounding the main contact hole image are three secondary peaks with intensities of more than 0.3, and with the characteristic symmetry of the three-leaf-clover aberration. Figure 6(c) shows the image from the same PSM with no aberration, showing a secondary ring with intensity of roughly 0.22 surrounding the main contact image. Comparison of Figures 6(b)and Figure 6(c) illustrates that the three-leaf-clover aberration breaks up the circular ring of an aberration-free image and concentrates the energy into three spots. Resist processes with insufficient contrast may partially print the secondary peaks [14], causing problems in the final etched contact hole structure.

Figure 6

Third-order spherical--Z

Just as coma can be viewed as an image shift which depends on pupil radius , so spherical aberration can be thought of as a focus shift which depends on . By rewriting Z in terms of Z,

Z = (15/16) × [(4² - 2) Z - 2/ 3],            (8)

it is evident that the focus shift depends on such that, for on-axis rays with near 0, the shift is of opposite sign from that for off-axis rays with near 1. As with coma, the effect of this aberration is highly dependent on the mask pattern and the method of illumination, since these determine which part of the aperture is used in image formation. By changing the pitch of an alternating PSM, different parts of the imaging aperture can be chosen. Figure 7 displays a plot of focus shift versus feature size of a 1:1 alternating PSM grating, using a 0.5-NA projector with = 248 nm, = 0.3, and a = 0.045 waves of spherical aberration. The small feature sizes diffract light into the outer parts of the aperture, resulting in a focus shift with sign opposite to that of the larger feature sizes, in accord with Equation (8). Ordinary binary mask patterns also exhibit shifts of best focus with different feature sizes and feature types, though the focus shifts are smaller than in Figure 7.

Figure 7

Another aspect of spherical aberration is that the image displays asymmetric behavior through focus. Figure 8 shows results for a 350-nm line/space grating mask imaged with the 0.5-NA, = 248 nm, = 0.3 projector. Aerial image contours in the X- Z plane are shown for an aberration-free projector in Figure 8(a); imagery is symmetric about best focus Z = 0. At roughly Z = ±1600 nm, the image is observed to reverse, with the opaque portion of the mask having higher intensity than the clear portion. Figure 8(b) shows similar image contours for a = 0.045 waves of spherical aberration; best focus is shifted upward by several hundred nm, and the image reversal is weakened at defocus Z = 1900 nm but strengthened at defocus Z = -1300 nm. Figures 8(c) and Figure 8(d) respectively display the process window for the aberration-free and aberrated cases. The imaging asymmetry induced by spherical aberration cuts off the process window for negative values of defocus. For the process windows with 15% exposure range, the unaberrated DOF of 1370 nm is reduced by spherical aberration to approximately 1000 nm.

Figure 8

Measuring aberrations with resist patterns

Pattern Placement

Wide-line patterns (e.g., with k > 2) are normally used to measure lens distortion, and it is assumed that all patterns are shifted the same, as in Z , Z image shifts. However, coma can cause pattern-dependent shifts. Figure 9 shows the image shift of an isolated clear line, with an opaque background, as a function of linewidth, for a 0.5-NA projector with = 248 nm, = 0.3, and a = 0.035 waves coma. It is apparent that, under these conditions, the narrowest line is shifted less by coma than wider lines. This observation can lead to a coma measurement pattern. A "box-in-box" pattern can be designed with the inner box made from a narrow (250-nm) linewidth and the outer box made from a wide (500-nm) linewidth. Coma aberrations would then induce a shift of the center of the inner box with respect to the center of the outer box, a measurement that can be made with a few nm precision by modern optical overlay metrology tools. Since both X shifts and Y shifts are measured, information about both a and a can be obtained. Illumination that is not properly centered in the aperture could also be detected with this pattern by observing the slope of the overlay shift versus focus [16]. Increasing to 0.6 results in smaller shifts that depend less on feature size, as shown by the second curve in Figure 9.

Figure 9

Pattern symmetry

The three-leaf-clover aberration is most clearly observed through the breaking of symmetry. A 13% attenuated PSM imaging small contacts with low , as in Figure 6, is a sensitive indicator of lens asymmetry. Sensitivity can be increased by deliberately increasing the exposure dose, i.e., overexposing, so as to bring out relatively small imaging artifacts. The presence of a symmetrical ring around the main contact image is a good indication that asymmetric aberrations are small. Three-leaf-clover aberrations, a and a , break the ring into three spots. Coma aberrations, a and a, cause one side of the ring to be more prominent than the other side.

Another useful symmetry test uses three-line patterns to search for coma, as in Figure 5. Linewidth differences between the two outer lines are an indication of coma. By orienting such patterns in both horizontal and vertical orientations, one can determine both a and a. It is useful to adjust to as low a value as possible, resulting in the most sensitive detection of aberrations.

Imagery through focus

Many techniques to measure astigmatism and focal plane nonflatness by tracking image performance through focus are well established. In the pin bar technique [13], the best focus is picked out by visual observation of a "microstepped" focus matrix. By measuring lines of different orientation at many locations across the lens field, astigmatism (a, a) and focus plane nonflatness (a) can be accurately measured.

The determination of spherical aberration is a more challenging problem. One approach is to look for a dependence of best focus on feature size. Figure 7 showed such a case using alternating PSM gratings of various sizes, imaged with small . Such a PSM is not commonly available, and the linewidths are extremely small. Similar results, albeit with reduced sensitivity, can be obtained with ordinary binary mask gratings. Figure 10 shows the best focus as a function of feature size for imaging with spherical aberration a = 0.03 waves, at two values of partial coherence . The = 0.6 imagery is considerably less sensitive to spherical aberration than the = 0.3 imagery as a result of the greater averaging across the aperture. Unfortunately, real experimental data may also contain effects due to imaging into the relatively thick (e.g., 1000 nm) resist layer. Perhaps ultrathin imaging layers (e.g., 50 nm) might be used to circumvent these difficulties.

Figure 10

90° Phase-shift mask patterns--"Focus monitor"

An alternating PSM with phase near 90° possesses unusual optical properties that can be exploited to measure focus errors [17,18]. It is possible to design a "box-in-box" pattern, termed the focus monitor, in which the measured overlay error is proportional to the focus error. Focal plane nonflatness can be assessed by measuring focus monitor patterns across the lens field. Astigmatism information appears as differences between the X overlay error and the Y overlay error measurement. This technology has proven to be particularly useful for assessing variations in focus across the wafer due to lens heating, misfocusing near the edge of the wafer, and wafer chuck flatness.

The focus monitor pattern is also sensitive to spherical aberration. Full resist simulations were performed to determine the calibration curve (i.e., the overlay shift versus focus offset) of a focus monitor pattern consisting of a 200-nm-wide chrome line with 90° phase shifter to the left and no phase shifter to the right. Figure 11 plots such curves both with and without spherical aberration, and for two different values of partial coherence. The solid square points, representing an aberration-free projector with = 0.5, are less strongly dependent on focus than the open square points with = 0.3. The two curves cross at approximately zero overlay and focus offset of -250 nm, a focus very close to that for optimum resist imagery. Similar simulations, with an aberration of a = 0.045 waves, are shown in Figure 11 as the triangle data points. The aberrated curves are shifted relative to the aberration-free curves, with a significantly larger shift for = 0.3 than for = 0.5. The aberration has a huge impact on the crossing point of the = 0.3 and = 0.5 curves, moving it to a focus offset of more than 800 nm and an overlay shift of about 30 nm. For lithographic tools with variable , measuring the overlay error of the crossing point may provide a sensitive measurement of spherical aberration.

Figure 11

Conclusions

The increasing use of advanced imaging techniques such as off-axis illumination or phase-shift masks will motivate tighter control of aberration. Such techniques can put more energy into the outer parts of the aperture, which can cause a greater sensitivity to aberrations. For example, in Figure 6 the ordinary chrome-on-glass contact hole was relatively insensitive to three-leaf-clover aberration compared with the attenuated PSM contact hole. Both coma and spherical aberration were found to cause larger image deviations when was small. This is probably due to increased averaging across the aperture when is large. In situations where is adjustable, one can choose small to measure aberrations and large for production use, though this certainly oversimplifies the trade-offs. It is hoped that new aberration-measurement techniques, in addition to the ones presented here, will be developed on the basis of simulations of aberrated test patterns.

Acknowledgments

*PROLITH/2 is a trademark of FINLE Technologies.

¹The interested reader is referred to the extensive literature on lens aberration theory, e.g., V. N. Mahajan, Aberration Theory Made Simple, SPIE Press, Bellingham, WA, 1991.

²PROLITH/2 Version 5.0 is a product of FINLE Technologies, Austin, TX.

References and note

Received February 9, 1996; accepted for publication August 9, 1996