The recent effort to extend optical lithography below 193 nm to 157 nm faces a number of new challenges. One of the most troublesome is that the 157-nm wavelength requires the use of crystalline materials like calcium fluoride (CaF2), rather than the fused silica commonly used for 193-nm wavelengths. Single-crystal CaF2 is produced very slowly and is difficult to grow in large sizes with low birefringence.
Birefringence refers to the dependence of refractive index on polarization direction. Crystalline materials are in general naturally birefringent and have anisotropic optical properties, but the prevailing belief used to be that crystalline materials with cubic crystal structure, such as CaF2, were constrained by their high symmetry to be isotropic with no intrinsic birefringence. Temperature gradients introduced during the crystal growth process commonly resulted in high grown-in stresses, however, and the resultant stress-induced birefringence often exceeded tolerances. Though stress-induced birefringence presents engineers with a challenge, the community believed that it could be reduced by improvements in growth techniques. Indeed, by the spring of 2001, CaF2 growth efforts began to yield large single crystals that seemed to meet the 157-nm lithography birefringence target specification of 1 nm/cm (or equivalently Δn = 1 x 10-7). This critical materials issue seemed under control.
Figure 1. A plot of directional dependence of intrinsic birefringence in one octant shows maxima and zeros, with maximum scaled to one. A VRML version may be found in reference 5.
That was until May of 2001, when our group at the National Institute of Standards and Technology (NIST; Gaithersburg, MD) announced experimental and theoretical evidence of a large birefringence in CaF2 that was intrinsic to the material. The disturbing aspect of this result is that the value is far larger than the industry specifications, and worse, being intrinsic, cannot be reduced by any material improvements. Further, the effect is highly anisotropic and results in significant average refractive index anisotropies (see figure 1). The 157-nm lithography systems that were designed up to this point simply would not work. The new challenge was whether and how these systems could be redesigned to correct for this effect. cubic birefringence
Though symmetry arguments demand that crystals with cubic symmetry have isotropic linear optical properties, including birefringence, H.A. Lorentz pointed out more than a century ago that the photon wavevector q introduces a small symmetry-breaking term roughly proportional to 1/λ2, which can produce birefringence in cubic crystals.1 This term becomes significant when the wavelength of light approaches the size of a unit cell of the crystal, or equivalently when q approaches the size of the Brillouin zone. Analysis of this effect on the dielectric tensor that governs the optical properties of the crystal shows that only one free-scaling parameter determines the complete angular dependence of the intrinsic birefringence for crystals with the structure of CaF2.2
The angular dependence for propagation directions in one octant is shown in figure 1, which represents the birefringence in a given direction as a point on a surface whose distance from the origin gives the magnitude of the birefringence in that direction. The figure shows that there are maxima in the , , and  crystal directions and zeros in the , , , and  directions. The other seven octants of directions are equivalent because of the cubic symmetry of the crystal, with the result that there are 12 birefringent maxima in the equivalent <110> directions (out to the cube edge centers), and 14 birefringent zeros, six along the equivalent <100> directions (cube edge directions), and eight along the equivalent <111> directions (cube diagonals).3
Figure 2. Measured values of the intrinsic birefringence in CaF2 and BaF2 as functions of wavelength (symbols) show the roughly 1/λ2 wavelength dependence predicted from theory (curves).
This effect of finite q (spatial dispersion) was first measured in semiconductors in the 1970s4,5, but it seemed to be too small to have any practical consequences; its implications for UV optics were not explored until recently.2 Our group measured the wavelength dependence of the intrinsic birefringence of CaF2 and barium fluoride (BaF2), a similar UV material (see figure 2 and table). The magnitude of the intrinsic birefringence depends strongly on the direction of light propagation in the crystal, and the results shown are for propagation in a direction with the strongest effect. For light propagation in the crystal  direction, these results give the index for polarization in the [-110] direction minus the index for polarization in the  direction. These results also hold for the other propagation directions in the crystal equivalent by symmetry. Measurements of the angular dependence are consistent with that predicted by theory.
The values for CaF2 and BaF2 are of opposite sign at short wavelength, and the magnitudes decrease rapidly, approximately as 1/λ2 as the wavelength increases toward the visible, which explains why the effect was not observed previously in visible measurements. For CaF2 at 157 nm, the magnitude of the intrinsic birefringence is more than 10 times that of the target specification. lithography implications
For lenses made of crystalline material, the intrinsic birefringence has two main effects. The first is a different refraction for the two polarization components at the lens surfaces, which causes a ray bifurication at each lens. The second is that each polarization component accumulates a different phase as it traverses the crystal, resulting in a phase-front distortion. Both effects contribute to blurring of the image, which limits the achievable resolution. The second effect is considered more troublesome for lithography and has been included in a recent modification of CODE V lens-design software (Optical Research Association; Pasadena, CA). The second effect also depolarizes linearly polarized light.
Figure 3. Normalized calculated magnitudes of the directional dependence of the intrinsic birefringence (red) and average index shift (green) for propagation in directions in the cube-diagonal plane containing the , , and  directions, represented by the angle measured from the  direction. The birefringence plotted is proportional to the index of refraction for the polarization normal to the - plane, minus the index for the polarization in the - plane.
The implications for refractive optics can be seen from the angular dependence of the intrinsic birefringence and the resulting index shift (see figure 3). Most CaF2 lenses used in lithography objectives are made with the annular axis of the lens near the  crystal direction. Fortunately, as figure 3 shows, this is a direction of zero birefringence and an extremum in average index. This means that with -oriented lenses, for rays traveling along or close to the central lens axis, the birefringence and average index variation is zero or small. Note, however, that for this lens orientation, the birefringence increases in magnitude rapidly as rays deviate from the optical axis (the  direction).
In the  direction, CaF2 has another birefringence zero that in this case is a local extremum, meaning that the magnitude of the birefringence increases relatively slowly for angles off the  direction. This means that for sufficiently small angles, lenses with the annular axis oriented along the  direction have smaller birefringence than those with the annular axis oriented along the  direction. Thus, for low numerical aperture optics, which feature rays traveling at small angles off the central axis,  orientation of lenses is preferred.
Unfortunately, high numerical apertures, which introduce large ray angles in the optical system, are key to achieving high resolution in lithographic projection systems. Even for -oriented lenses, the birefringence exceeds acceptable values for some of the non-paraxial rays in the large numerical aperture designs. Clearly there is a problem with CaF2 optics at 157 nm, and some correction approaches are required. designing solutions
Most 157-nm lithography system designs are catadioptric, i.e., incorporating both mirrors and lenses in the optics to minimize the chromatic aberrations. Catadioptric designs can also help the birefringence problem by putting most of the focusing power in the mirrors, maintaining the high numerical aperture while keeping the angles in the lenses relatively small.
Figure 4. 3-D directional dependences of the intrinsic birefringence show low-symmetry in a general direction (a), four-fold symmetry along the  direction (b), three-fold symmetry along the  direction (c), and two-fold symmetry in the  direction (d).
Further birefringence reduction can be achieved by exploiting the high symmetry of the effect. In a general propagation direction, the effect has low symmetry, whereas along the  direction, for example, a four-fold symmetry exists (see figure 4). This means that for a given angle Θ off the  direction, as a function of azimuthal angle Φ, there is a birefringence maximum and minimum every 90°. By coupling two lenses with their maxima and minima staggered by 45°, the net birefringence through the two lenses is reduced compared to that for a comparable single lens. Coupling  lenses with transverse crystal axes staggered by 60°, and coupling  lenses with their transverse axes staggered by 90°, give similar reductions.
Even further reductions can be achieved by coupling these three combinations in such a way that their respective residuals partially cancel. The average index variation can be compensated for as well. The present consensus of 157-nm system designers is that implementing these correction approaches succeeds in reducing the distortion due to the intrinsic birefringence in CaF2 to acceptable levels.
This comes at a cost, however. The freedom to relatively rotate (clock) the system of 20 or so lenses is routinely exploited to compensate for lens-surface-figure error and index variations. Using the clocking freedom to minimize the intrinsic birefringences necessarily gives up some of the freedom to clock for the other aberrations, with the consequence that the approach may require more stringent surface-figure and index tolerances.
Given the complexity of the correction approaches required to minimize the intrinsic birefringence in CaF2, we need to find UV-transmitting materials with much less intrinsic birefringence at 157 nm. All crystals should exhibit the effect. Figure 2 shows that BaF2 has an opposite sign of the effect to that of CaF2 at 157 nm, however. Mixed crystals of Ca1-xBaxF2 can in principle be made of these two materials, and the properties, including intrinsic birefringence, would be expected to be intermediate. Thus an x can be chosen that eliminates the intrinsic birefringence at 157 nm. Though this is an appealing approach, it should be pointed out that there might be significant difficulties in maintaining the other optical specifications such as index homogeneity in these mixed crystals. Nevertheless, we are exploring this approach.
The recent addition of intrinsic birefringence in CaF2 to the 157-nm optical-design picture caught the lithography industry by surprise. The present consensus is that the design-based solutions described will work, though the designs will be more complex and the fabrication process more challenging. All future 157-nm designs will take careful account of the lens crystal orientations to manage the effects of the intrinsic birefringence. oe
1. V. Agranovich and V. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons, Second Ed., Springer-Verlag, New York, NY (1984).
2. J. Burnett, Z. Levine, et al., Phys. Rev. B 64, 241102 (2001).
4. J. Pastrnak and K. Vedam, Phys. Rev. B 3, 2567 (1971).
5. P.Y. Yu and M. Cardona, Solid State Commun. 9, 1421 (1971).
John Burnett, Zachary Levine, Eric Shirley
John Burnett, Zachary Levine, and Eric Shirley are staff scientists in the Physics Laboratory at NIST, Gaithersburg, MD.