DFT Calculations on Refractive Index Dispersion of Fluoro ... · implemented in Gaussian-03 with “POLAR” and “CPHF= Rdfreq” keywords. In this study, the polarizability (α
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351
DFT Calculations on Refractive Index Dispersion of
Fluoro-compounds in the DUV-UV-Visible Region
Shinji ANDO*
Department of Organic & Polymeric Materials, Tokyo Institute of Technology,Ookayama 2-12-1-S1-21, Meguro-ku, Tokyo, 152-8552, Japan
Density functional theory (DFT) calculations using the B3LYP hybrid functional have been performed to predict the refractive indices and their dispersion for fluoring-containing compounds which are expected to show high transparency in the vacuum UV (VUV) and the deep UV (DUV) region. The linear polarizabilities at wavelengths of 157, 193, 248, 300, 350, 434, 486, 540, 589, 656, 730 and 800 nm were calculated, and the corresponding refractive indices were estimated by assuming the molecular packing coefficient (Kp) as 0.56. The refractive indices at 193 nm are linearly proportitional to those at 589 nm for most of the compounds. In addition, the calclulated Abbe numbers representing the refractive index dispersion in the visible region are in linear relationships with the calculated refractive indices at 589 nm. The DFT calculations predict that alicyclic compounds, lactones, siloxanes, nitriles, sulfonylfluorides, and SO3-esters exbihit relatively high refractive indices and good transparency in the DUV region.Keywords / Density functional theory / Refractive Index / Dispersion / Abbe number / Fluoro-compounds / DUV region / Immersion fluid /
1. IntroductionThe wavelength dispersion of refractive indices
is an important property of optical materials. In particular, high refractive index and high transparency at wavelengths of 157 and 193 nm are required for fluids used for immersion lithography. This lithographic method uses transparent fluids to fill the lens-to-wafer gap. Although a convenient and transparent fluid at 193 nm is water [1], the development of an inexpensive, transparent fluid exhibiting higher refractive indices is required for the next-generation immersion lithography. Kunz et al.[2] have measured the transparencies of more than 50 fluorocarbon liquids over the wavelength range of 150−200 nm for use in the 157 nm immersion lithography. Very recently, Kaplan et al.[3] have measured the values of the refractive index, thermo-optic coefficient, and absorption coefficient of a number of organic solvents and aqueous inorganic solutions that may be used as immersion fluids at 193 or 248 nm.
We have reported that the time-dependent density functional theory (TD-DFT) calculations
can reproduce the observed absorption spectra of model compounds for optical polymers without incorporation of empirical corrections [4-8]. In this study, we calculated wavelengh-dependent linear polarizabilities for more than 100 organic compounds which are expected to show high transparency, and we predicted the refractive indices and their wavelength dispersions in the deep UV (DUV)-UV-visible region using the DFT.
2. Theory and MethodsThe DFT level of theory with the three-
parameter Becke-style hybrid functional (B3LYP), which employs the Becke exchange and LYP correlation functionals, was adopted in conjunction. The Gaussian basis set of 6-311G(d) was used for geometry optimizations under no constraints, and the 6-311++G(d,p) was used for calculations of wavelength-dependent linear polarizabilities. All the calculations were performed using the software package of Gaussian-03 (Rev.C02 and D01)[9] installed on a Compaq Alpha server of the computer center of Tokyo Inst. Tech.
Refractive index and its temperature dependence (thermo-optical (TO) coefficients) of molecular materials can be calculated using the Lorentz-Lorenz equation :
€
nλ2 −1
nλ2 + 2
= 4π3
ρ ⋅ NAMw
αλ = 4π3
αλ
Vmol (1)
€
dnλ
dT= − (nλ
2 − 1)(nλ2 + 2)
6nλβ (2)
where n is the refractive index, ρ the density, NA the Abogadoro number, Mw the molecular weight, α
λ the linear molecular polarizability, Vmol the
molecular volume, and β the volume expansion coefficient. The wavelength dependence of α
λ is
the origin of the dispersion of refractive indices and TO-coefficients. The αλ can be calculated by solving the coupled perturbed Hartree-Fock equations, and this procedure was recently implemented in Gaussian-03 with “POLAR” and “CPHF= Rdfreq” keywords. In this study, the polarizability (α
λ) of each molecule were calculated
at wavelengths of 157, 193, 248, 300, 350, 434, 486, 540, 589, 656, 730, and 800 nm.
Moreover, either molecular volume or density is required to calculate the refractive index. However, such parameters are not easy to predict because of the difficulty in estimating the degrees of molecular
packing. In addition, the high sensitivity of density to temperature makes the estimation complicated. In contrast, van der Waals volumes (Vvdw) can be easily calculated from the optimized geometries using the Slonimski’s method [10,11] with the van der Waals radii of atoms.[12] The molecular packing constant Kp, which is used as a measure of molecular packing, is defined as
€
Kp = VvdwVmol
= ρ ⋅ NAMw
Vvdw (3)
The Vmol of a certain molecule is the summation of Vvdw and intermolecular free spaces. For instance, dense molecular packing caused by intermolecular hydrogen bondings or charge transfer interactions increases Kp due to the decrease in free space.
Abbe number, ν, has been used as a measure of the wavelength dispersion of refractive indices in the visible region. This value is defined as
€
ν = nD −1nF − nC
(4)
where nD, nF and nC are the refractive indices at the wavelengths of Fraunhofer D-, F- and C- spectral lines (589.2 nm, 486.1 nm and 656.3 nm respectively). Note that low dispersion materials have high values of ν. We have recently calculated the linear polarizabilities for 86 fundamental organic compounds using DFT, and demonstrated that the calculated Abbe numbers are linearly proportional to the experimental values as shown in Fig.1 when we assume Kp as 0.60. Fig.1 clearly indicates that the experimental dispersions in the Compounds Mw Vvdw ρ Kp
cyc((CF2)3CF(CF3)CF2CF(CF3)) 400.1 213.1 1.828 0.586 average - - - 0.556
Table 1. Molecular weight, Van der Waals volumes (A3), density, and packing coefficients of fluorocompounds.
10
20
30
40
50
60
70
10 20 30 40 50 60 70
Cal
cula
ted
Ab
be
Nu
mb
er (K
p=
0.6)
Experimental Abbe Number
Figure 1. Calcuated Abbe numbers assuming Kp=0.6 vs. Experimental Abbe numbers for 86 fun-damental organic compounds.[13]
353
visible region can be well reproduced by the DFT calculations. In addition, we tentatively define the second Abbe number which expresses the dispersion in the DUV-UV region as follows.
€
ν2 = n248 −1n193 − n300
(5)
where n248, n193 and n300 are the refractive indices at wavelengths of 248. 193, and 300 nm, respectively.
The objectives of this study is to calculate the wavelength-dependent linear polarizabilities and predict the refractive indices and their dispersions for fluorine-containing compounds which are expected to show high refractive indices and high transparency in the DUV region. The systematic examination of calculated refractive indices and their dispersions will be useful for screening immersion fluids for next generation lithography.
Figure 2. Calculated linear polarizability vs. van der Waals volumes with the optimized geometries for fundamental fluoro-compounds (Group I).
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.20 1.25 1.30 1.35
c-C4F
4H
4CO
Calculated Ref. Index @ 589 nm
Cal
cula
ted
Ref
. In
dex
@ 1
93 n
m
5
10
15
20
50 100 150 200 250 300
Van der Waals Volume / A3
Po
lari
zab
ility
@ 5
89 n
m /
A3
Figure 3. Calculated refractive indices at 589 nm vs. those at 193 nm. Most of the points are located on the correlation : n193=1.588∙n589 −0.696.
Figure 4. Calculated refractive indices at 589 nm vs. calculated Abbe numbers for fundamental fluoro-compounds (Group I).
50 60 70 80 90 100 110 1201.20
1.25
1.30
1.35
Calculated Abbe Number
Cal
cula
ted
Ref
ract
ive
Ind
ex a
t 58
9 n
m
2
4
6
8
10
12
14
60 70 80 90 100 110 120
DU
V A
bb
e N
um
ber
, ν2
Visible Abbe Number, ν
c-C4F
4H
4CO
Figure 5. Calculated Abbe number in the visible region (ν) vs. calculated Abbe numbers in the DUV region (ν2).
3. Results & Discussion3.1 Packing coefficients of fluoro-compounds.
At first, we need to determine the molecular packing constant using experimental densities because experimental data for the wavelength dispersion of refractive indices for fluoro-compounds are very limited. Table 1 shows the values of molecular weight, van der Waals volumes, experimental density together with the packing coefficients (Kp) estimated using Eq.(2). The average value of Kp is ~0.56, which is slightly smaller than that for non-fluorinated compounds (~0.60).[12] This is due to the low polarizability of fluorines which significantly reduces the inter-molecular attracting interactions. We used this value (Kp=0.56) throughout this study. Table 2 lists the values of molecular weight, van der Waals volume, calculated polarizability, refractive index
354
(nλ) at five wavelengths, and Abbe parameter for the fundamental fluorocarbons called Group I. This group does not contain neither aromatic rings nor polar structures, such as -CN, -COO-, -SO2F, and -SO3-. As shown in Fig. 3, the values of α at 589 nm are linearity proportional to Vvdw. Since n is determined by the ratio of αλ/Vmol as expressed in Eq.(1), the variations in n are caused by the distributions in Kp and αλ/Vvdw. As seen in Table 1, the refractive indices of Group I fall in a small range (1.224−1.319). Note that the calculated values of n at 193 nm are in linear relationships with those at 589 nm except for c-C4F4H4CO. The relation is expressed as n193=1.5879 n589 −0.69584, which means that the wavelength dispersion disappears at n193=n589=1.184. Figure 4 shows that the values of n589 are also in linear relationships with the estimated Abbe numbers (ν) despite the scatter of points. The estimated ν at n589=1.184 goes beyond 130, which can be regarded as almost no dispersion. The linear relationships observed in Figs.3−5 indicate that the dispersion curves for Group I compounds can be well expressed by a common equation; the Cauchy’s formula
Figure 8. Calculated dispersions of refractive indices for fluoroalkylethers. The smaller fluorine contents, the higher refractive indices.
1.2
1.3
1.4
0.0
0.2
0.4
0.6
0.8
1.0
150 200 250 300 350 400
C3F8C4F10C5F12C6F14C7F16
C3F8C4F10C5F12C6F14C7F16
Wavelength (nm)
Cal
cula
ted
ab
sorp
tio
n /
arb
.un
it
Cal
cula
ted
Ref
ract
ive
Ind
ex
Figure 6. Calculated absorption spectra and refrac-tive index dispersions for perfluoroalkanes in the VUV-DUV region.
€
nλ = A + B
λ2 + C
λ4 (6)
where A, B and C are intrinsic coefficients for each material (C is generally very small).
Figures 6−8 show the wavelength dispersion of refractive indices for perfluoroalkanes, cyclicfluoroethers, cyclicfluoroalkanones, and fluoroalkylethers which are included in Group I. Fig. 6 shows that perfluoroalkanes exhibit the lowest n and the smallest dispersion with the largest ν and ν2 in all the compounds. Note that the values of n increase as the molecular size
355
5
10
15
20
25
50 100 150 200 250 300
Van der Waals Volume / A3
Po
lari
zab
ility
@ 5
89 n
m /
A3
1.3
1.4
1.5
0.0
0.5
1.0
1.5
2.0
150 200 250 300 350 400
CH3SO2FC2H5SO2FCF3CH2SO2FCF3SO2F
CH3SO2FC2H5SO2FCF3CH2SO2FCF3SO2F
Cal
cula
ted
ab
sorp
tio
n /
arb
.un
it
Wavelength (nm)
Cal
cula
ted
Ref
ract
ive
Ind
ex
1.3
1.4
1.5
0.0
0.5
1.0
1.5
2.0
150 200 250 300 350 400
CH3SO3CH3C2H5SO3CH3CF3CH2SO3CH3CF3SO3CH3
CH3SO3CH3C2H5SO3CH3CF3CH2SO3CH3CF3SO3CH3
Cal
cula
ted
ab
sorp
tio
n /
arb
.un
it
Wavelength (nm)
Cal
cula
ted
Ref
ract
ive
Ind
ex
Figure 11. Calculated dispersions of sulfonylfluorides and sulfonylesters which were firstly reported as highly ransparent maerials at 157 nm by the authors.
and/or Mw increases. This trend is also observed for other types compounds and agrees well with the experimental results for hydrocarbons.[14] Fig. 7 shows that fluorocycloalkanones show characteristic absorptions in the VUV region, which should cause anomalous dispersion at longer wavelengths. These absorptions are assigned to the transitions from the lone-pair (n-) orbitals at the carbonyl oxygen to spacially spreading Rydberg orbitals. The deviations from the liearity of c-C4F4H4CO in Figs. 3 and 5 are attributable to the anomalous dispersion at 193 nm as seen in Fig. 7. Fig. 8 shows the calculated dispersion for fluoroalkylethers. Since there are no absorption peaks over 150 nm, these dispersion curves are well fitted by the Cauchy’s formula. The dispersion behaviors are similar to those of perfluorocarbons and can be expressed as 1) the higher fluorine content, the lower n, and 2) the larger molecular size and/or Mw, the higher n.
Table 3 lists the calculated parameters for fluorocarbons having polar groups, which are included in Group II. In addition, Figs. 9−11 show the dispersions of fluorocarbonylesters, sulfonylfluorides, and sulfonylesters, which are expected to show good transparency in the VUV region.[5,7] Fluoro-carbonylesters show complicated shapes of dispersion as seen in Fig. 9. These are due to the characteristic absorptions appearing at 220−230 nm, and these peaks can be assigned to the n→π* transitions at the carbonyl group. These esters are not suitable for immersion fluids because of the lower refractive indices and significant absorption. On the other hand, the lactones containing C=O group show characteristic absorptions at 165−175 nm as shown in Fig. 10, which are originated both from n→Rydberg* and π→π* transitions. Since the oscillator strenghs of n→π* absorptions are much smaller than those
Figure 12. Calculated linear polarizability vs. van der Waals volumes with the optimized geometries for fundamental fluoro-compounds (Group I & II).
for the esters in Fig. 9, these lactones can be candidates for immersion fluids. Fig. 11 show the dispersions of sulfonylfluorides and sulfonylesters which were firstly proposed by the authors as novel resist platforms for 157 nm lithography. Due to the appropriate absorptions in VUV region and the sulfur atoms possesing high atomic polarizability, these compounds show high refractive indices and good transparency at 193 nm. In particular,
1.3
1.4
1.5
1.6
0.0
0.5
1.0
1.5
2.0
150 200 250 300 350 400
Lactone-aLactone-bLactone-cLactone-c
Lactone-aLactone-bLactone-cLactone-d
Cal
cula
ted
ab
sorp
tio
n /
arb
.un
it
Wavelength (nm)
Cal
cula
ted
Ref
ract
ive
Ind
ex
Figure 10. Calculated absorption spectra and refractive index dispersions of lactones.
356
sulfonylfluorides can be promising as 157 nm immersion fluids having refractive indices higher than 1.6. In addition, higher refractive indices can be expected for siloxanes, nitriles, and alicyclic compounds such as norbornaes and bicyclo[2,2,2]octanes as listed in Table 3.
Figures 12-15 show the relationships among the calculated parameters for all compounds (Groups I and II). Linear relationships are observed between α and Vvdw (Fig. 12), n589 and n193 (Fig. 13), n589 and ν (Fig. 14), and ν and ν2 (Fig. 15) as well as in Figs. 2−5, though points are much more scattered. The wider distributions in the calculated parameters should be originated by the anomalous dispersion caused by absorptions in the VUV-DUV region. However, these relationships will be useful to infer and predict the optical properties of fluorocompounds in the VUV-DUV region from the data measured in the visible region. In addition, if we carefully choose compounds exhibiting high refractive indices and high transparency with the aid of DFT and TD-DFT calculations, wider range of fluoro-compounds can be taken as candidates for immersion fluids for the next-generation lithography.
References [1] S. Owa and H. Nagasaka, Proc. SPIE., 5040, 724
(2003). [2] R. Kunz, M Switkes, R. Sinta, J.E. Curtin, R.H.
French, R.C. Wheland, C.P. Chai Kao, M.P. Mawn, P. Wetmore, V. Krukonis, and K. Williams, J. Microlith., Microfab., Microsyst., 3, 73 (2004).
[3] S. G. Kaplan and J. H. Burnett, Appl. Opt., 45, 1721 (2006).
[4] S. Ando, T. Fujigaya, and M. Ueda, Jpn. J. Appl. Phys., 41, L105 (2002).
[5] S. Ando, T. Fujigaya, and M. Ueda, J. Photopolym. Sci. Technol., 15, 559 (2002).
[6] T. Fujigaya, S. Ando, Y. Shibasaki, M. Ueda, S. Kishimura, M. Endo, and M. Sasago, J. Photopolym. Sci. Technol., 15, 643 (2002).
[7] S. Ando and M. Ueda, J. Photopolym. Sci. Technol., 16, 537 (2003).
[8] T. Fujigaya, Y. Shibasaki, S. Ando, S. Kishimura, M. Endo, M. Sasago, and M. Ueda, Chem. Mater., 15, 1512 (2003).
[9] M. J. Frisch, and J. A. Pople et al., Gaussian 03, Rev.D.01, Gaussian, Inc., Pittsburgh PA, 2005.
[10] G. Slonimskii, A. Askadskii, and A. Kitai-godorodskii, Polym. Sci. USSR., A12, 556 (1970).
[11] S. Ando, Kobinshi-Ronbunshu, 51, 251 (1994).[12] A. Bondi, J. Phys. Chem., 68, 441 (1964).[13] S. Ando, S. Azami, and T. Takiue, submitted.[14] J. L. Lauer, J. Chem. Phys., 16, 612 (1983).
1.2
1.3
1.4
1.5
1.6
1.2 1.3 1.4
Calculated Ref. Index @ 589 nm
Cal
cula
ted
Ref
. In
dex
@ 1
93 n
m
Figure 13. Calculated refractive indices at 589 nm vs. those at 193 nm. Most of the points are located on the correlation : n193 = −0.690 + 1.5833*n589.
Figure 14. Calculated refractive indices at 589 nm vs. calculated Abbe numbers for fundamental fluoro-compounds (Group I & II).
Figure 15. Calculated Abbe numbers in the visible region vs. calculated Abbe numbers in the DUV re-gion (Group I & II).
Table 3. Molecular weights, van der Waals volumes (angstrom3), calculated polarizability (angstrom3) andrefractive indices at the five wavelengths (nm), and calculated Abbe numbers for Group II compounds.