Ab-initio Hartree–Fock study of Mg+2 as a substitutional impurity in CaF2

PERGAMON

Solid State Communications 118 (2001) 569±574

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Ab-initio Hartree±Fock study of Mg 12 as a substitutional impurity in CaF2 V.M. Bermudez* Naval Research Laboratory, 4555 Overlook Avenue, S.W., Washington, DC 20375-5347, USA Received 1 November 2000; received in revised form 11 January 2001; accepted 30 March 2001 by S.G. Louie

Abstract The energetics and electronic structure of Mg 12 as a substitutional impurity in CaF2 have been computed using ab-initio Hartree±Fock theory and the supercell approach. After correcting for electron correlation, a substitution energy (referenced to Ê , relative free Mg and Ca atoms) of 13.30 eV is found. At a Mg 12 site, the F 2 nearest-neighbor shell contracts by about 0.093 A Ê greater than in pure MgF2. No new valence to the ideal CaF2 shell radius; however, the Mg 12 ±F 2 distance remains about 0.32 A states occur outside the CaF2 valence band, but Mg 12 s-like states are found at the very bottom of the conduction band, lying just below the lowest empty Ca 12 states. The position of these states is sensitive to the extent of lattice relaxation around the impurity cation. It is also found that displacement of the Mg 12 from the ideal Ca 12 lattice site can occur easily. q 2001 Published by Elsevier Science Ltd. PACS: 61.72.Bb; 71.20.Ps; 78.20.Bh Keywords: A. Insulators; B. Crystal growth; D. Optical properties

1. Introduction As the technology of deep-ultraviolet (DUV) lithography advances toward shorter wavelengths for increased resolution, the properties of materials used in lenses, windows, etc. become more important. For lithography using the l ˆ 157 nm …hn ˆ 7:89 eV† line of the F2 laser [1±3], CaF2 is the material of choice for high transmission and durability. A critical issue in the manufacture of DUV components is the availability of large quantities of CaF2 stock with a reproducibly low bulk optical loss coef®cient 1 …a , 5 £ 1023 cm21 † at 157 nm. It has been found [1±3] that CaF2 obtained from different sources, and at different times from the same source, often exhibits considerable variation in both surface and bulk losses. While surface losses might result from polishing damage [4±6] or adsorbed contaminants [7], bulk losses quite probably derive from impurities introduced during growth. Identifying these impurities has been dif®cult due to the small * Corresponding author. Fax: 11-202-404-4071. E-mail address: [email protected] (V.M. Bermudez). 1 The loss coef®cient is de®ned by T ˆ 102ad ; where T is the transmittance of a sample of thickness d.

concentrations and the wide range of possible species. Work aimed at minimizing DUV losses in MgF2 [8] has suggested a number of possible causes of bulk contamination, and experience in the semiconductor industry illustrates the ease with which trace amounts of impurity can result from subtle errors in material processing and handling. Many species (e.g. transition metal ions, 2OH and O 22) will obviously degrade CaF2 DUV transmission due to excitation of the impurities themselves or to low-lying charge-transfer transitions, and others, such as Pb 12, are already known empirically to be problematic. The work described here focuses on a third, previously overlooked, class of impurities; namely, those which for chemical reasons are dif®cult to separate from CaF2 and for which the effects on DUV transmission in CaF2 have not yet been studied in detail. The a priori wide range of possible trace impurities in this group makes a purely empirical study rather prohibitive. Hence we have turned to numerical modeling as an ef®cient way to identify which of many possible species might be the most important and, therefore, worthy of further experimental effort. Here we are concerned with the electronic and structural properties in the vicinity of the impurity, an assessment of local states

0038-1098/01/$ - see front matter q 2001 Published by Elsevier Science Ltd. PII: S 0038-109 8(01)00194-6

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near the CaF2 band edges which result from such impurities and the mechanisms whereby these states are formed. To our knowledge, the only previous theoretical work addressing the issue of impurity effects on CaF2 DUV transmission is that of Medvedeva et al. [9] which reported cluster calculations showing gap states introduced by an O 22 impurity. Magnesium, the impurity selected for initial study, is a prototype for this third class of impurity. It substitutes, without charge compensation, at a high-symmetry (Ca 12) site and exhibits no low-lying intraionic or charge-transfer excitations. This allows the effects of local lattice relaxation on near-edge impurity states to be observed, and it will be shown here that it is important to consider such factors in analyzing the effects of impurities on DUV transmission losses for this and for other species. Recent experimental work [10] for CaF2:Mg has shown a small but ®nite effect of the impurity on near-edge absorption; however, it is dif®cult to determine from the data alone whether this is due to gap states or to perturbation of the exciton structure. The results of the present work will suggest that the former is an important factor. 2. Computational details Ab-initio restricted Hartree±Fock (HF) calculations were performed using the crystal98 code [11,12] which was written for periodic structures. For Ca 12 and F 2 previously reported [13] all-electron basis sets, optimized for CaF2, were used which are constructed from contracted Gaussiantype functions and designated 86-511G p and 7-311G, respectively. 2 For Mg 12, the 8-511G basis set employed in a previous study 3 of MgF2 [14] was used here. Substitutional Mg 12 was treated using the supercell approach [15], and convergence was checked by performing calculations for supercells consisting of 4, 8 and 16 primitive unit cells. 3. Results Initial calculations were done for pure CaF2 using a lattice Ê (vs the measured value of 5.445 A Ê, a constant of 5.538 A 1.7% difference). This was found to minimize the total energy (Etot), in accord with previous work [13] using the same procedure and basis set. An Etot of 2875.946525 Hartree was found, vs the previous result [13] of 2875.946307 Hartree. The difference of ,6 meV is not considered signi®cant for the present work. Fig. 1 shows the calculated valence band (VB) density of states 2

A typographical error was found in the Ca basis set given in Table 1 of Ref. [13]. The exponent of the last 1s Gaussian function should be 2.509(11), i.e. 25.09. See http://www.ch.unito.it/ifm/ teorica/crystal.html for a tabulation of atomic basis sets. 3 A typographical error was found in the Mg basis set given in Table 1 of Ref. [14]. The p-coef®cient of the ®rst 2sp Gaussian function should be 7.72(23), i.e. 0.00772.

Fig. 1. Computed VB DOS for bulk CaF2 (solid line) compared with the angle-integrated UPS (points) (Ref. [7]). The spectra have been aligned at the bottom of the VB (see text). The bottom of the experimental VB has been located by linear extrapolation to the baseline as shown by the dashed line. The binding energy zero is taken as the calculated VBM. The computed VB DOS has not been broadened to account for the ,0.40 eV instrumental resolution. The UPS data were obtained for an epitaxial CaF2 ®lm, grown in ultrahigh vacuum on a Si(111)-(7 £ 7) surface, using He-II excitation …hn ˆ 40:8 eV†:

(DOS) for pure CaF2 together with the experimental [7] angle-integrated ultraviolet photoemission spectrum (UPS). As expected, the VB is almost purely F 2p with only a few percent of the DOS coming from Ca orbitals. The UPS data in Fig. 1 were obtained under conditions of high surface sensitivity …hn ˆ 40:8 eV† for the atomicallyclean (111) surface of a CaF2 ®lm grown on a Si(111)(7 £ 7) surface in ultra-high vacuum. Hence, there may be unresolved surface state emission near the VB maximum (VBM) which will not appear in the bulk calculation. The CaF2 surface used in the experiment is known [16] to be terminated in a layer of F 2 ions, and a surface state would consist of an F 2p level near the bulk VBM which is shifted to lower binding energy by the reduction in Madelung potential at the surface [17]. Therefore, to avoid possible complications related to surface states, the observed and calculated spectra have been aligned at the bottom of the VB. The width and shape of the computed DOS are in reasonable agreement with the data. The position of the conduction band minimum (CBM, see below) indicates a gap of Eg ˆ 20:0 eV: An experimental value for Eg is dif®cult to obtain for materials (such as CaF2) with strong excitonic structure in the near-edge absorption spectrum. An estimate of Eg < 12:1 eV has been given [18,19] for CaF2; although, a recent analysis [20] indicates a value of about 11.8 eV. Although HF calculations typically give an Eg which is too large by a factor of ,2, the widths and shapes of the valence and

V.M. Bermudez / Solid State Communications 118 (2001) 569±574 Table 1 Energy of substitution, DEN …Mg† (eV), for Mg in CaF2 as a function of relaxation and of the number (N) of primitive cells in the supercell. DEN …Mg† is referenced to the free Ca and Mg atoms (see text) N

No relaxation a

Relaxed (dRmin) b

4 8 16

1 2.90 1 2.90 1 2.90

1 2.69 (20.073) 1 2.64 (20.093) c 1 2.63 (20.093)

a

All atoms ®xed in the optimized positions for pure CaF2. Relaxation of ®rst-nearest-neighbor F 2-shell is allowed. dRmin Ê ) needed to minimize DEN …Mg†. gives the change in shell radius (A c Including electron correlation (see text) increases DEN …Mg† to 13.30 eV. b

lower conduction bands are well described [12]. This is illustrated in a recent study [21] comparing HF and density functional theory results for SiO2 and has been invoked [22] as justi®cation for treating the error in Eg as a constant offset in excitation energies. Here we are concerned mainly with

Fig. 2. The N ˆ 8 supercell used in the calculations. The large (small) dark spheres are Ca 12 (F 2), and the shaded spheres at the vertices are Mg 12.

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detecting impurity-induced states near the band edges, not with computing an accurate joint DOS. The energy of substitution, DEN …Mg†; for a supercell consisting of N primitive unit cells is given by [15] DEN …Mg† ˆ EN …CaF2 : Mg† 2 N £ E…CaF2 † 1 E…Ca† 2 E…Mg†

…1†

where EN …CaF2 : Mg† is the energy of the supercell and E(CaF2) the energy per primitive cell of pure CaF2. The energies of the isolated atoms, E(Ca) and E(Mg), were calculated using the isolated-atom basis sets given in Refs. [13] and [14], respectively. A substitution energy, DEN1 …Mg†; referenced to the free ions can be obtained by using E(Ca 12) in Eq. (1) in place of E(Ca) and similarly for Mg. The energies of the isolated ions were calculated using 8-61G Mg 12 and 86-5111G p Ca 12 basis sets, optimized for the free ions [15], after correcting typographical errors in the Mg set 4 and in the Ca set (see above). The ionic radii of eight-fold coordinated Ca 12 and sixfold coordinated Mg 12 (as in MgF2) are given [23] as 1.12 Ê , respectively. Replacement of Ca 12 by the and 0.72 A smaller Mg 12 will lead to lattice contraction in the vicinity of the impurity. Table 1 gives DEN …Mg† vs N for the case of no relaxation and for relaxation of the ®rst-nearest-neighbor (NN) shell of F 2 ions. The energies were obtained by expanding or contracting the NN-shell radially about the Mg 12 in increments of dR < 0:02 A and computing EN …CaF2 : Mg† at each step. The minimum in DEN …Mg† vs R was then found by ®tting with a quadratic in R. Table 1 gives dRmin, the displacement required to minimize DEN …Mg† at the value given. With the ®rst-NN shell set at dRmin, the effect of displacements of the second-NN Ca 12 shell was considered for the N ˆ 16 supercell. A smaller supercell does not permit separation of the second-NN shells of different Mg 12 sites. A radial displacement of the Ca 12 Ê toward Mg 12 gives a further ,0.06 eV shell by ,0.03 A reduction in the minimum DEN …Mg†. This is a small effect, and second-NN relaxation was subsequently neglected. The possibility of off-center displacement of the Mg 12, i.e. away from the ideal Ca 12 position in CaF2, was also considered. Although the potential energy surface was not thoroughly mapped, it was found that displacements as large Ê changed the energy over a range of only about as 0.10 A ^5 meV (vs a calculational precision of 0.3 meV) from that of the undisplaced ion. Calculations for displacements in the (100), (110) and (111) directions showed the energy surface Ê. to be roughly isotropic for a radial displacement of 0.10 A The existence of a low-energy Mg 12 displacement mode in CaF2:Mg might be revealed in measurements of the thermal or acoustic properties of this system which, to our knowledge, have not yet been reported. The effect of electron correlation was estimated a 4 In the Mg basis of Ref. [15] the p-coef®cient of the third 2sp function should be 0.2104.

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posteriori (Ref. [24] and works cited) using the HF electron density and the Perdew±Wang GGA density functional. The correction was computed for each term in Eq. (1), leading to a net increase by 10.66 eV (i.e. 25%) in DEN …Mg† for the relaxed N ˆ 8 supercell. The correlation-corrected result is then DEN …Mg† ˆ 13:30 eV: The entire calculation was also performed with referencing to the free ions, as mentioned above, giving DEN1 …Mg† ˆ 21:69 eV after the correlation correction. To our knowledge there is no comparable experimental value for the substitution energy. Table 1 shows that DEN …Mg† is independent of N until relaxation is allowed, in which case DEN …Mg† decreases (and dR increases) slightly with N. This indicates that the impurities do not interact directly, i.e. that the unrelaxed DEN …Mg† is independent of impurity concentration. However, increasing the distance between impurities allows a larger contraction of the NN shell leading to a lower DEN …Mg†. The dependence of DEN …Mg† on N is weak, and the rest of the discussion will focus on results for the NN-optimized N ˆ 8 supercell shown in Fig. 2. The ionic charges of 11.87 (Ca) and 20.937 (F) in CaF2:Mg are essentially identical to those of pure CaF2. However, the Mg charge of 11.88 is slightly higher than in MgF2 [14] (11.80). Likewise, the F 2 charge is slightly higher than in MgF2 (20.901). Fig. 3 shows the CaF2:Mg charge density map for the supercell (100) plane. The Ca 12 and F 2 valence densities exhibit slight distortions toward each other Ð i.e. the weak hexagonal (trigonal) shape of the Ca 12 (F 2) map Ð indicating a slight covalency. However, no similar effect is evident for Mg 12. This suggests that the NN F 2-shell cannot contract suf®ciently to reproduce the local Mg 12 environment in MgF2. This is Ê, consistent with the Mg 12 ±F 2 NN distance of 2.305 A corresponding to minimum DEN …Mg†; which is larger than Ê [25] in bulk MgF2. the value of 1.98 A Figs. 4 and 5, respectively, show the VB and CB partial densities of states (PDOS) for pure CaF2 and for CaF2:Mg. For a meaningful comparison, the pure-CaF2 results were

Fig. 3. Charge density map for CaF2:Mg showing the (100) plane of the relaxed N ˆ 8 supercell (cf. Fig. 2). The Mg 12 and Ca 12 ions lie in the plane, and the F 2 ions lie above and below. The separation between isodensity curves is 0.01 e/Bohr 3.

obtained for an N ˆ 8 supercell under the same computational conditions as for CaF2:Mg. Fig. 2 shows two inequivalent F 2 sites in the CaF2:Mg supercell, i.e. those which are NN's of Mg 12 sites and those which are not. The F 2 PDOS's in Figs. 4b and 5b are for the former. For Mg 12 the VB PDOS, Fig. 4b, essentially matches that of Ca 12, and the impurity produces no new states above the VBM of pure CaF2. The situation is different for the CB, Fig. 5b. Here Mg 12 introduces states near the CBM which lie below the minimum of the Ca 12 PDOS (cf. Fig. 5a). Orbital decomposition of the PDOS shows these states to be mainly s-like. A calculation of the CB DOS was also done for a Ê off-center displacement of the Mg 12 in the (111) 0.20 A direction, as discussed above, which showed no signi®cant difference in the Mg 12 states from the results in Fig. 5b. The presence of these states at lower energy than the corresponding Ca 12 states is at ®rst surprising since Eg of MgF2 is reported to be larger than that of CaF2. Detailed comparisons of optical data and band structure calculations [26,27], for the purpose of distinguishing excitons from interband transitions, indicate Eg $ 13 eV for MgF2. The impurity Mg 12 states can be understood qualitatively in terms of the ,16% larger Mg 12 ±F 2 NN distance in

Fig. 4. Atom-resolved VB PDOS for (a) pure CaF2 and (b) CaF2:Mg (relaxed N ˆ 8 supercell). The relative scale factors are given, and the zero of energy is the VB maximum. Different traces have been displaced vertically for clarity. Dashed, solid and dotted lines show results for Ca 12, F 2 and Mg 12, respectively.

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Fig. 6. Atom-resolved CB PDOS for CaF2:Mg showing results for the unrelaxed N ˆ 8 supercell near the CBM. All plots are on the same (arbitrary) vertical scale. `F(1)' and `F(3)' refer to inequivalent F 2 ions which are ®rst- and third-nearest-neighbors, respectively, of a Mg 12. Note the different energy ranges in Figs. 5 and 6.

Fig. 5. Similar to Fig. 4 but showing the corresponding CB PDOS. The region near the CB minimum is shown on an expanded vertical scale, displaced vertically for clarity.

CaF2:Mg vs bulk MgF2 which leads to a corresponding reduction in the NN contribution to the Madelung energy at the Mg 12 site. The importance of relaxation was shown by computing the CB DOS for the unrelaxed N ˆ 8 supercell, Fig. 6. In this case, the Mg 12 states appear even farther below the bottom of the Ca 12 PDOS (cf. Fig. 5b), and corresponding states on NN F 2 ions are similarly affected. For the relaxed supercell, however, the lowest Mg 12 states fall within the tail of F 2 states at the CBM. Hence, it is concluded that Mg 12 contamination will increase the optical transmission loss near the interband absorption threshold but will not directly affect transmission at the 7.89 eV F2 laser line. There are, however, possible indirect effects (such as perturbation of excitons by the lattice relaxation at the impurity site [10]) which are dif®cult to model and are beyond the scope of the present work. In summary, the effect of substitutional Mg 12 impurities on near-edge DUV optical losses in CaF2 have been modeled using ab-initio HF calculations. The results indicate that Mg 12, a dif®cult impurity to remove from CaF2,

will increase transmission losses near the interband absorption edge but will not directly affect transmission at the 7.89 eV DUV line of the F2 laser. The extent of lattice relaxation around the impurity is found to have a signi®cant effect on the energetic position of impurity-related nearedge states. After correcting for electron correlation, a substitution energy (referenced to free Mg and Ca atoms) of 13.30 eV is found. At a Mg 12 site, the F 2 nearestÊ , relative to the neighbor shell contracts by about 0.093 A ideal CaF2 shell radius. It is also found that displacement of the Mg 12 away from the ideal Ca 12 site can occur easily.

Acknowledgements This work was supported by the Of®ce of Naval Research (ONR) and the Defense Advanced Research Projects Agency (DARPA). This work was also supported by a grant of computer time from the DOD High-Performance Computing Modernization Program at the ASC-MSRC, Wright-Patterson AFB. J.P. Blaudeau (ASC-MSRC) and members of the Torino University Theoretical Chemistry Group are thanked for generously helping with the operation of the crystal98 program. D.E. Ramaker is thanked for a critical reading of the manuscript, and R.T. Williams and W.E. Carlos are gratefully acknowledged for helpful discussions.

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