High Refractive Index Nanocomposite Fluids for Immersion Lithography

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High Refractive Index Nanocomposite Fluids for Immersion Lithography L. Bremer,† R. Tuinier,† and S. Jahromi*,‡ DSM Research, ACES, P.O. Box 18, 6160 MD Geleen, The Netherlands, and DSM Licensing, P.O. Box 53, 6160 AB Geleen, The Netherlands ReceiVed August 19, 2008 The concept of using dispersions of nanoparticles as high refractive index fluids in immersion lithography is examined both from a theoretical and experimental point of view. In the theoretical part we show that gelation and demixing can be controlled in high solid dispersions, needed to achieve a high (refractive) index, by using short stabilizing brushes. We considered both fluid-fluid demixing by using statistical thermodynamics and percolation, computed using liquid-state approaches. Whenever demixing or percolation takes place, the nanoparticle dispersion is unsuited for immersion lithography. The minimum thickness of the stabilizer layer of a stable suspension is estimated assuming particles plus steric stabilizer to act as hard spheres with van der Waals attraction between the cores. Since the van der Waals attraction can be related to the optical properties of the particles and dispersion medium, it is also possible to estimate the refractive index that can be attained with composite immersion fluids. Using materials that are known to be highly transparent in the bulk at a wavelength of 193 nm, indices above 1.8 can be attained. Other materials with higher indices are expected to be transparent at 193 nm due to a blue shift of the UV absorption and enable much higher indices. In the experiment, we show that it is possible to prepare suspensions with particles of about 4 nm diameter that increase the refractive index of the continuous phase with 0.2 at a wavelength of 193 nm. The refractive index and density of such dispersions are proportional to the volume fraction of the disperse phase, and it is shown that the refractive index of the composite fluid can be predicted very well from the optical properties of the components. Furthermore, successful imaging experiments were performed through a dispersion of silica nanoparticles. These findings lead to the conclusion that immersion lithography using nanoparticle dispersions is indeed possible.

I. Introduction Since the 1970s, advances in optical lithography in the semiconductor industry have made it possible to increase the level of integration on chips. This resulted, for example, in the production of microprocessors with ever increasing operational speeds. The critical dimension or the smallest half-pitch 1:1 feature size (HP) is related to the wavelength λ through the Rayleigh equation,1

HP )

k1λ NA

(1)

Here NA is the numerical aperture of the stepper’s lens, and k1 expresses the lithography process capability with a theoretical limit of 0.25. Moore’s law,2 claiming that the number of transistors on an integrated circuit increases exponentially with time, was shown to survive in the past decades mainly by reducing the wavelength of exposure light. In the early 1970s, the chip industry started using exposure light with λ ) 432 nm (G-line of Hg), printing micrometer size features, then, in the mid 1980s, shifted to 365 nm (I-line of Hg), and, 10 years later, λ ) 248 nm (KrF eximer laser) technology was introduced. Currently, chip manufacturers are using λ ) 193 nm ArF eximer lasers to print 65 nm lines. As the transition to smaller available laser light (i.e., λ ) 157 nm F2-He laser)3 proved to be very difficult, the * Corresponding author. E-mail: [email protected]. † DSM Research. ‡ DSM Licensing. (1) Owa, S.; Nagasaka, H. J. Microlithogr.: Microfabr. Microsyst. 2004, 3, 97. (2) (a) Peercy, P. S. Nature 2000, 406, 1023. (b) Lundstrom, M. Science 2003, 299, 210. (3) See, for example, Sewell, H.; McClay, J.; Jenkins, P.; Tirri, B.; Flageloo, D.; Mulkens, J. J. Photopolym. Sci. Tech. 2002, 15, 569.

community turned to an old concept, so-called immersion lithography,4 to push the roadmap to smaller HPs. The concept of immersion lithography can be understood by considering the definition of numerical aperture as

NA ) n sin θmax

(2)

where n is the refractive index (herein also denoted as the index) of the medium between the last lens element and the resist, and θ is the aperture angle. By changing the refractive index of the medium from air (n ) 1) to a liquid with a higher index, one can effectively decrease the exposure wavelength, thereby increasing the resolution. Water with an index of n ) 1.44 at λ ) 193 nm is now commercially accepted as the first generation fluid (Gen-1) to print features as small as 40 nm, and high-volume chip production using this technology is already in progress. When using water, NA is limited to 1.30-1.36, and further increase of the resolution by single exposure is only possible by replacing the water with fluids of higher index.5 With fluids of refractive indices between 1.60 and 1.65, so-called Gen-2 fluids, NA can be increased to values around 1.5, enabling printing features of 32 nm. Availability of fluids with n ) 1.8 (Gen-3) would even open the possibility to print 22 nm HP nodes because NA then increases to 1.7 (double patternning with k1 ) 0.19) and can thus compete with extreme UV. It must be noted that, apart from replacing the fluids, the index of the last lens element must also be increased. Recent studies have shown that garnets such as Lu3Al5O12 (LuAG) (4) (a) Switkes, M.; Kunz, R. R.; Sinta, R. F.; Rothschild, M.; GallagherWetmore, P. M.; Krukonis, V. J.; Williams, K. Proc. SPIE 2003, 5040, 690. (b) Owa, S.; Nagasaka, H. Proc. SPIE 2003, 5040, 724. (5) Sewell, H.; Mulkens, J.; McCafferty, D.; Markoya, L.; Streefkerk, B.; Graeupner, P. Proc. SPIE 2006, 6154, 6-1.

10.1021/la8026896 CCC: $40.75  2009 American Chemical Society Published on Web 01/14/2009

Immersion Lithography using Nanocomposite Fluids

combine the high transparency and high n at λ ) 193 nm to allow their use as high index final lens elements.6 For Gen-2 fluids, several approaches have been proposed such as mixing additives such as salts7 or nanoparticles in a fluid,8 or the use of (cyclic)alkanes.9 For Gen-3 fluids, however, the situation is different. So far, there have been numerous unsuccessful attempts to increase the refractive index without increasing the absorption, by, for example, changing the chemistry of organic compounds.10 It has now been widely recognized that the use of nanoparticles is the only remaining route to achieve refractive indices close to 1.8. Incorporation of inorganic domains is an effective way to fabricate high refractive index composite materials11-13 and composite materials with refractive indices above 2 have been prepared.14 However, the employed materials such as TiO2 or PbS absorb λ ) 193 nm light. Furthermore, transparency at 193 nm dictates the use of extremely small particles ( -6, a stable colloidal dispersion can be expected for all practical particle concentrations. Two-body interactions, reflected in B2, thus provide information on the many-body phase behavior. Here we do not account for fluid-solid phase coexistence, which is reasonable in the case where the particles are somewhat polydisperse, in which case crystallization is suppressed.30 The second osmotic virial coefficient is defined as

B2 ) 2π

∫0∞ r2(1 - exp[-βW(r)])dr

Figure 1. Stability diagram of a dispersion of spherical colloids with a hard core diameter s, van der Waals attraction characterized by the Hamaker value A, and brush thickness D. The diagram is calculated on the basis of the criterion B2* > -6 for stability.

(4)

with W(r) being the interaction potential between two spheres as a function of the distance between their centers r, and β ) 1/kT. It is convenient to introduce r˜ ) r/σ, so

B2* ) 12

∫0∞ r˜2(1 - exp[-βW(r˜)])dr˜

(5)

We describe the dispersed nanoparticles as core-shell particles with van der Waals attraction between the cores. The shell, with thickness ∆, consists of a layer with concentrated polymeric stabilizers. The particle interactions in the particle dispersion consequently consist of a steric repulsion whenever the stabilizing layers overlap plus a van der Waals attraction. The steric repulsion is assumed to be strong (at least a few kT as the brushes overlap), so the hard core plus brush layer can be considered as purely repulsive. Then the van der Waals attraction only operates for r > σ∆. The interaction potential within our model can thus be described as

W(r˜) )

{

∞ for r˜ e 1 + 2∆ ⁄ σ A - f(r˜) for r˜ > 1 + 2∆ ⁄ σ 6

(6)

with A being the Hamaker constant, and f(r˜) being the r-dependent part of the van der Waals interaction, -Af(r˜)/6:

f(r˜) )

( )

r˜2 - 1 1 1 + + ln 2(r˜2 - 1) 2r˜2 r˜2

(7)

Inserting eq 7 into eq 6 for B2* yields the form

B2 * ) 4 + 12

2∆ 2 2∆ 3 2∆ +4 + + 12 σ σ σ

( ) ( ) 12

∞ A r˜2(1 - exp[ f(r˜)])dr˜ (8) ∫1+∆⁄σ 6

where 4 is the hard core part, and the terms that depend on 2∆/σ are contributions due to the repulsive layer. The van der Waals contribution in the last term reduces the value for B2*. A state diagram is plotted in Figure 1. From this diagram it follows that (27) Van Helden, A. K.; Jansen, J. W.; Vrij, A. J. Colloid Interface Sci. 1981, 81, 354. (28) Pusey, P. N.; Van Megen, W. Nature 1986, 320, 340. (29) Vliegenthart, G. A.; Lekkerkerker, H. N. W. J. Chem. Phys. 2000, 112, 5364. (30) Bolhuis, P. G.; Kofke, D. A. Phys. ReV.E, 1996, 54, 634.

Figure 2. Percolation line calculated for the adhesive hard-sphere model and PY approximation following Chiew and Glandt.

∆/σ sets the range of Hamaker constants for which a dispersion is stable against colloidal gas-liquid (or fluid-fluid) phase separation. In the case of short-ranged attractions, percolation can lead to instability of nanoparticle dispersions. At the percolation point, the dispersion faces a liquid-to-gel crossover. In the case of sticky hard-sphere interactions, as described in this work, it is possible to estimate the relation between B2* and the effective percolation volume fraction using the Ornstein-Zernike equation in the Percus-Yevick (PY) approximation.31 The result is shown in Figure 2, and it turns out that gelation will occur already at about 10% effective volume fraction if the thinnest possible stabilizer layer is chosen. At higher volume fractions, larger values of B2* and thus thicker stabilizer layers are needed. Above this volume fraction, the thickness of the stabilizer layer needed for a stable liquid is thus dependent on both the Hamaker constant and the volume fraction. The effective volume fraction in the PY approximation we used to calculate the percolation threshold is the core volume fraction of the particles including the stabilizer layer, i.e., φPY ) φ(1 + 2∆/σ)3. The percolation threshold at given φPY can again be used in eq 8 in order to obtain the stabilizer thickness that is needed for stability at various Hamaker constants. For φPY ) 0.2, 0.3, and 0.4, the corresponding B2* values are -1.647, 1.213, and 2.667, respectively. Values for the percolation threshold are plotted in Figure 3. (31) (a) Chiew, Y. C.; Glandt, E. D. J. Phys. A: Math. Gen. 1983, 16, 2599. (b) Ornstein, L. S.; Zernike, F. Proc. K. Ned. Akad. Wet. 1914, 17, 793. (c) Percus, J. K.; Yevick, G. J. Phys. ReV. 1958, 110, 1.

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The fraction of core material in a spherical particle with core diameter σ and shell thickness ∆ is

(

1+

2∆ σ

of transparent materials by two parameters is the so-called ‘‘Cauchy equation”:

-3

)

n2(ω) - 1 )

(9)

The maximum effective volume fraction in a liquid is that of random closed packed spheres, 0.64 for monodisperse spheres. In practice, the viscosities of such highly concentrated dispersions are too large to handle. Therefore, realistic values of φPY ) 0.3 or 0.4, leading to low viscous dispersions, should be considered in order to examine the feasibility of high index nanocomposite liquids. The value of 2∆/σ can be related to the Hamaker constant, as shown in Figure 3. Therefore, it is possible to relate the maximum volume fraction that can be achieved to the Hamaker constant as shown in Figure 4. The Hamaker constant can be calculated from the dielectric properties of the particles and the continuous phase liquid using Lifshitz theory.32 An exact calculation using this theory is complicated due to the limited availability of data over the whole frequency range from static contributions to microwave range, infrared, and UV. Therefore approximations have been developed that enable estimation of the Hamaker constant based on a reduced amount of spectral information.33 For transparent materials it has been shown that the UV-vis contributions dominate the Hamaker constant.34 Below we compute the Hamaker constant from the optical properties of the medium and disperse phase. A useful estimation that describes the frequency dependence of the refractive index

n(ω) )

(n2(ω) - 1)ω2 + CUV

(10)



CUV 1 - ω2 ⁄ ωUV2

+1

(11)

If the particles and the medium are allowed to have different absorption frequencies (ωp ) ωUV of the particles and ωm of the medium) and refractive index, np ) (CUVp + 1) and nm ) (CUVm + 1), the Hamaker constant then follows as

Apmp )

(

)

3kT εp - εm 2 + 4 εp + εm 3h√ωpωm (X2ε + 2X∆ ε ε 1 ⁄ 2 + ∆ε2(3 + 2Y)) (12) 128πε 7⁄4 ((Y - √Y2 - 1)1 ⁄ 2 + (Y + √Y2 - 1)1 ⁄ 2)3

with

np2 + nm2 2

Y)

Figure 4. The volume fraction of nanoparticle “cores” sterically stabilized with the minimum stabilizer thickness that can lead to stability.

ωUV2

The frequency, ω (rad/s), and wavelength, λ (m), are related through ω ) 2πc/λ, where c is the speed of light in vacuum. The Cauchy equation implies that n2(ω) - 1 depends linearly on (n2(ω) - 1)ω2, with slope 1/ωUV2 and intercept CUV. Values of ωUV and CUV can be obtained from the frequency dependence of the refractive index and have been reported in the literature34 for various materials. At any wavelength where the system is transparent, the refractive index can then be estimated with these values:

ε)

Figure 3. Stabilizer thickness needed to prevent gelation of concentrated dispersions of nanoparticles at various effective volume fractions φPY.

1

∆ ε ) np2 - nm2

(

X)

ωp 2 (n - 1) ωm p ωm 2 (n - 1) ωp m

)

ωm 2 ωp 2 (np + 1) + (n + 1) ωp m 4√ε ωm 1

All input for the calculation can be obtained from the Cauchy plots of the solvent and the dispersed phase. Decalin is one of the candidate liquids for immersion lithography because it has a relatively high refractive index and is transparent at λ ) 193 nm after thorough purification. We studied two decalin mixtures: the heavy fraction of a distillation, containing 90% cis-decalin and 10% trans-decalin, and the light fraction with about 90% trans- and 10% cis-decalin. The Cauchy plots of both fractions are shown in Figure 5, demonstrating the Cauchy equation can be used to describe the data. The parameters for Cauchy fits of relevant materials for immersion liquids are compiled in Table 1. The data for decalin and water are obtained from Cauchy plots of ellipsometry data over the wavelength range between 190 and 1700 nm. The data on R-Al2O3, MgO, MgAl2O4, and ZrO2 are obtained from Bergstro¨m,34 those on LuAG and HfO2 follow from plotting literature data35-38 for several wavelengths and the value of np ) 2.14 for LuAG at λ ) 193 nm in Cauchy plots. The values on thin films38 showed a smaller value for ωUV than reported for bulk materials by Bergstro¨m and the value we use for HfO2 is (32) Lifshitz, E. M. SoViet Phys. JETP 1956, 2, 73. (33) (a) IsraelachviliJ. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1992. (b) Horn, R. G.; Israelachvili, J. N. J. Chem. Phys. 1981, 75, 1400. (34) Bergstro¨m, L. J. Am. Chem. Soc. 1996, 79, 339; AdV. Colloid Interface Sci. 1997, 70, 125.

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Figure 5. The refractive index of decalin mixtures as a function of the wavelength. The upper plot is the deep-UV part of the measurements. The lower plot is the Cauchy plot of the same data. The linear regression (dashed) only deviates somewhat in the high frequency/deep UV part. Table 1. Optical Parameters Used for Estimating the Hamaker Constant Using Eq 12 ωUV (10 rad/s)

CUV

n (ω ) 0)

F (g/ml)

1.734 1.722 1.781 2.033 2.00 1.71 1.87 1.34 1.61 1.628 1.5

1.103 1.135 0.751 1.098 2.07 1.95 1.89 3.743 4.642 2.29 3.2

1.4501 1.4612 1.3233 1.4484 1.752 1.718 1.700 2.178 2.375 1.815 2.05

0.870 0.896 0.998 2.2 4.0 3.6 3.7 5.6 3.5 6.7 9.7

16

trans-decalin cis-decalin water silica R-Al2O3 MgO MgAl2O4 ZrO2 diamond LuAG HfO2

obtained by interpolating data obtained for SiO2 and ZrO2. One must be careful when treating these data, because the values on thin films of HfO2 may differ from data obtained from bulk material. In colloidal systems the materials are in many cases less dense than in the bulk state. The data in Table 1 are therefore only indicative and used to estimate the maximum refractive index that can be achieved using various materials. Hamaker constants for various materials are shown in Figure 6 and are calculated using eq 12 with εm ) 2 and 80 for decalin and water, respectively, and εp ) 10 for the particles. Diamond nanoparticles are particularly interesting because of the extremely high refractive index. At λ ) 193 nm the index can be estimated with eqn. 11 to be 2.90. However, the Hamaker (35) Thielsch, R.; Gatto, A.; Heber, J.; Kaiser, N. Thin Solid Films 2002, 410, 86. (36) Landolt-Bo¨rnstein Part 8, Springer-Verlag: Berlin, 1962. (37) Kuwano, Y.; Suda, K.; Ishizawa, N.; Yamada, T. J. Cryst. Growth 2004, 260, 159. (38) Jerman, M.; Qiao, Z.; Mergel, D. Appl. Opt. 2005, 44, 3006.

Figure 6. Estimation of Hamaker constants of materials that are of interest for the preparation of composite immersion fluids. Continuous phase is water (a) or trans-decalin (b).

constant is also very high and is estimated to be 37 kT and 30 kT in water and decalin, respectively. With eq 8, the value of ∆/σ needed for a stable dispersion can be calculated, which is 0.18 for water and 0.15 for decalin, limiting the attainable volume fraction to 25% in water and 29% in decalin. The optimum materials for composite fluids have a low Hamaker constant and a high refractive index. The high refractive index can be obtained by densification of the material leading to an increase of np, but this leads also to an increase of the Hamaker constant. It can also be achieved by an increase of the polarizability, leading to a decrease of ωUV and a decrease of the Hamaker constant. However, a decrease of ωUV will shift the UV absorption to higher wavelengths (Kramers-Kronig relation). Best candidate materials for composite immersion fluids have ωUV of about 1.6 × 1016 rad/s, and this value is employed to estimate the limiting refractive index that can be achieved in an immersion fluid. However, it cannot be excluded that sufficiently transparent materials can be found with still higher polarizability (lower ωUV), which would enable the production of liquids with still higher index. Assuming the limit for sufficient transparency at λ ) 193 nm would be for materials with ωUV ) 1.6 × 1016 rad/s, i.e., such as LuAG, the Hamaker constants can be estimated as a function of the refractive index of the particles at large wavelength (1.82 in the case of LuAG). From the Hamaker constant, the maximum volume fraction and the maximum refractive index for an immersion liquid can be estimated. This is shown in Figure 7, assuming the effective volume fractions of 0.3 and 0.4 for low viscous dispersions, and the limiting value of 0.64 for a very high viscosity (see Figure 4). Important consequences of this approach are that the maximum volume fraction of composite fluid with nanopar-

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Figure 8. Turbidity as a function of the volume fraction of particles with refractive index 2 in a medium with refractive index 1.64 and λ ) 193 nm, calculated using the Rayleigh equation with Percus-Yevick approximation of the structure factor.

indices of the medium and particles, respectively. It is also possible to relate the turbidity to the refractive index of the system, n, and that of the solvent, n0:

(n2 - n02)2 4 τ ) π4σ3 9 φλ4

Figure 7. Maximum achievable refractive index at λ ) 193 nm of dispersions in water (a) and decalin (b), as a function of index of particles. The values are plotted for different volume fractions with 0.3 and 0.4 for low viscous dispersions.

ticles is expected to be independent of the particle size and that the molar mass of the stabilizer (setting the thickness) needed depends on the size of the particles to be stabilized. The latter implies a need of control during particle preparation. For immersion lithography, this analysis indicates that the production of low viscous composite fluids, with for example decalin as dispersion medium, is indeed possible, reaching indices above 1.8 at λ ) 193 nm. II.b. Particle Size Requirements for Scattering. The nanoparticle dispersion should be sufficiently transparent at the wavelength of 193 nm. This means that both the absorption of light and the scattering should be very low. The extinction, E, is

E ) -log

I I0

(13)

I is the intensity of the light through the medium, and I0 is the incident light intensity. If both the particles and the medium does not absorb light, any extinction is caused by scattering. In this case, the term turbidity, τ, is employed:

τ)

ln(I0 ⁄ I) ln 10 × -log(I ⁄ I0) ) ) 2.3E ⁄ x x x

(14)

where x is the path length of the cell. For very small particles, the turbidity can be calculated using Rayleigh theory:

( )(

n0 τ ) 24π NV λ 3

2

4

np2 - n02 np2 + 2n02

)

2

( )(

n0 ) 4π σ φ λ 4 3

4

np2 - n02

)

2

np2 + 2n02 (15)

where N is the particle concentration (m-3), V is the volume of a particle (m3), λ is the wavelength, and n0 and np are the refractive

(16)

This can be more practical because the refractive index of the dispersion can be easily measured.39 For optical lithography, there are two principle criteria for transparency and scattering. The first criterion considers the socalled pupil apodization, depending on the ratio between the transmission of the axial ray that travels the shortest distance between the lens and the resist, x, and the marginal ray that travels the longest possible distance, x/cosθmax, and is thus dependent on NA. To enable imaging, the pupil apodization (further also denoted as PA), should be below 0.1, and this limits the extinction of the axial ray to E < cos θmax/(cos θmax - 1) log(1 - PA). For a liquid with refractive index 1.8 and a lens with NA ) 1.7, the aperture angle is 70.8°, and the extinction should thus be below 0.0224. If the extinction was due to turbidity alone, this would limit the turbidity to 0.0516/x (eq 14). Even in the case of a path length, (the distance between the last lens element and the photoresist), on the order of x ) 1 mm the required particle size must be below 1.3 nm according to the Rayleigh equation for turbidity (ref 17 or 18). The same liquid in a setup with NA ) 1.5 would have an aperture angle of 56.4° and thus an extinction of the axial ray that should be below 0.0566 and turbidity below 0.13/x. Here the size requirement would be below 1.8 nm according to Rayleigh. Concentrated systems (φ > 0.2) of repulsive spheres have, in practice, a lower turbidity than dilute systems (0.05 < φ < 0.2) due to structure formation (see Figure 8). Pure Rayleigh scattering (ignoring structural contributions) leads to an overestimation of the scattering of several orders of magnitude at high particle volume fractions. In order to obtain large effects of the structure, rather monodisperse particles and strong repulsion are required. However, this approach is useless because the viscosity will increase to high levels. For practical systems, high concentrations and thus short-range repulsion is required, and the hard-sphere approximation of Percus and Yevick will give a very good estimation of the structure factor for small particle size that can be achieved in practice:40,41

S)

(1 - φPY)4 (1 + 2φPY)2

(17)

where φPY is the volume fraction of the particles including the protective barrier. At volume fractions of about 40%, this would

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lead to a turbidity of 4% of the value expected by pure Rayleigh scattering. The particle diameter can thus be bigger by a factor of about 3 than expected by Rayleigh and about 4 nm in the case of NA ) 1.7 and 6 nm for NA ) 1.5. If the path length could be reduced further, x , 1 mm, even larger particles are suitable. The other criterion is more difficult to quantify and has to do with the scattered light that reaches the resist, thereby reducing the contrast. If both the particles and the carrier liquid are completely transparent, then almost 50% of the scattered light will reach the resist. The effect will be dependent on the resist dose and on the sensitivity of the resist. In the case of the relatively high NA systems that we consider, the extinction of the marginal ray will be less than 0.1 in order to fulfill the pupil apodization criterion. From eq 13 it follows that less than 20% of the light is scattered, and that less than 10% will reach the resist. Therefore, at high NA, the pupil apodization will limit the particle size. It is interesting to note that some absorption can strongly reduce the amount of scattered light that reaches the resist because only light that has to travel over a short distance will reach the resist without being absorbed.

III. Experiment III.A. Materials. Silica dispersions DIL0 and DIL1 are prepared using Snowtex 50 and Snowtex C of Nissan Corporation. The dispersions are concentrated by cross-flow filtration over a 20 kDa polyethersulphone membrane until the increase of the viscosity was too high to allow further filtration (about 100 mPa s). Diluted samples have been prepared by adding MilliQ water to the concentrated dispersion. The ZrO2 dispersion is obtained from Nyacol Nanotechnologies, Inc. The product contains 20% ZrO2 and 15% acetic acid according to the information of Nyacol. The particles are amorphous. The dispersion is first diluted in water by adding equal amounts of MilliQ water and dispersion, subsequently centrifuged for 4 h at 10.000 g in order to remove larger agglomerates, and concentrated as described above. Alumina dispersions are obtained from various sources. Dispersions of R-Al2O3 are prepared as described by Stenger.42 A γ-Al2O3 suspension has been obtained from Sasol Corporation and another dispersion of this material is prepared by grinding Degussa C powder in water using a stirred media mill.42 Boehmite is prepared using the method of Yoldas43 with Al-isopropoxide as precursor and HClO4 for peptization. III.b. Methods. The density of the suspensions is measured using a Mettler/Paar vibrating tube densimeter DMA55 at room temperature. Before each measurement, the densimeter is calibrated using MilliQ water and air using the corresponding densities at room temperature. All samples are degassed by agitating under vacuum for 5 min prior to measurements. The refractive index at λ ) 598 nm, nD, is measured using a Mettler Toledo RA-510 M refractometer operating at 22.00 °C. Indices at other wavelengths are measured using a spectroscopic ellipsometer at J.A. Woolam, Inc. The extinctions of the dispersions are measured with a ThermoSpectronic Aquamate UV-vis spectrophotometer relative to MilliQ water in Hellma Quartz cuvettes with path lengths of 1, 2, or 10 mm. (39) Stauff, J. Kolloidchemie; Springer-Verlag: Berlin, 1960. (40) (a) Wertheim, M. S. Phys. ReV. Lett. 1963, 10, 321. (b) Thiele, E. J. Chem. Phys. 1963, 39, 474. (41) Vrij, A.; Tuinier, R. In Fundamentals of Interface and Colloid Science; Lyklema, J., Ed.; Elsevier: New York, 2005; Vol 4. (42) Stenger, F.; Mende, S.; Schwedes, J.; Peukert, W. Powder Technol. 2005, 156, 103. (43) Yoldas, B. E. J. Mater. Sci. 1975, 10, 1856. (44) Yaminsky, V. V.; Ninham, B. W.; Pashley, R. M. Langmuir 1998, 14, 3223. (45) Vigil, G.; Xu, Z. H.; Steinberg, S. J. Colloid Interface Sci. 1994, 165, 367.

Figure 9. TEMs of silica dispersions DIL 1/120804 and DIL 0/190804.

The hydrodynamic diameter of the suspended particles is measured with a Malvern Zetasizer Nano. Samples are diluted in 1 mM KCl and filtered through a 0.22 µm filter before measuring. The solid content of the dispersions is determined by measuring the weight loss during drying at 180 °C in an oven under nitrogen. The dried powders are immediately weighed after removal from the oven since they are often very hygroscopic. The samples are replaced in the oven afterward and weighed again in order to determine whether weight loss is completed. The composition of the solids of the ZrO2 suspension after drying is determined using X-ray fluorescence (XRF). The maximum solid content is obtained by slowly concentrating the dispersion until the viscosity increases to such a degree that further filtration is impossible (about 0.1 Pa s). Imaging experiments have been performed at ASML with a twobeam interferometer operating at λ ) 248 nm, yielding 130 nm lines/spaces. The resist consists of three layers: 62 nm DUV 30 bottom antireflective coating, 166 nm Sumitomo KX 923 S95 resist, and 38 nm TOK TSP-3A top coat.

IV. Experimental Results IV.A. CharacterizationofModelSystemswithNanoparticles. Here we will examine the physical properties of a series of silica dispersions, mainly in terms of particle size and distribution. Silica dispersions in water are chosen because silica is known to be transparent in deep UV and also because it is easy to prepare small particles. Transmission electron micrographs (TEMs) of two silica dispersions with different particle diameters are shown in Figure 9. The particle size distributions are shown in Figure 10. From these distributions several average diameters have been calculated. Relevant diameters are the Sauter diameter d32, and the numberaverage diameter d10. The relevant diameter for turbidity is d63, and that for PCS measurements is d65.46 (46) Finsy, R.; De Jaeger, N. Part. Part. Syst. Charact. 1991, 8, 187.

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F ) φFp + (1 - φ)F0

Figure 10. Particle size distribution of the dispersions of Figure 9. All particles in the picture were classified in size classes. The first class ranges from 5 nm to 1.05 × 105 nm and subsequent classes range from 1.05i × 105 nm to 1.05i+1 × 105 nm, where i is the class number.

a

In Figure 11, measurements (upper graph; data points) of the extinction of several concentrations of the two silica dispersions measured in a 1 cm quartz cell are compiled. The extinction has been calculated and is given in Figure 11 (lower graph; curves) using Rayleigh scattering for diameters, σ, of 16.4 and 24.6 nm for the DIL1 and DIL0 samples, respectively. These values have been chosen because they lead to a good fit between Rayleigh calculations and experimental data at low concentration (