Electronic structure of SiO 2 : Charge redistribution contributions to the dynamic dipolesõeffective charges of the infrared active normal modes

Electronic structure of SiO 2 : Charge redistribution contributions to the dynamic dipolesõeffective charges of the infrared active normal modes J. L. Whitten Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695 Y. Zhang Department of Physics, North Carolina State University, Raleigh, North Carolina 27695 M. Menon Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695 G. Lucovsky a) Department of Physics, North Carolina State University, Raleigh, North Carolina 27695 Received 20 February 2002; accepted 16 April 2002 This article applies ab initio calculations at the i Hartree Fock self-consistent field single determinant and ii configuration interaction multideterminant expansion levels to study the diagonal components of the dynamic dipoles, i / q j with i j, or equivalently infrared effective charges, e i *, associated with asymmetric bond stretching y, symmetric bond stretching z, and out-of-plane bond rocking x, normal mode infrared active vibrations of noncrystalline SiO 2. The normal mode dynamic dipoles hereafter, /q i are decomposed into equilibrium charge density ionic and orbital variation charge redistribution contributions. The calculations are based on small clusters in which Si O Si groups are connected through O atoms to embedding Si atom terminators Si* that emulate the connectivity of these Si O Si groups to the SiO 2 continuous random network. Values of / q i have been determined as a function of the Si O Si bond angle at the bridging O-atom sites, and agree with values obtained from analysis of infrared spectra. Finally, the ab initio calculations are extended to noncrystalline silicon carbon alloys and silicon nitride, and values of / q i are determined for infrared active vibrations associated with N- and C-atom asymmetric stretching normal mode motions. The normalized equilibrium charge density or ionic contributions of the O, N, and C atoms follow trends expected on the basis of their relative Pauling electronegativities in bonds with Si. 2002 American Vacuum Society. DOI: 10.1116/1.1490382 a Electronic mail: gerry lucovsky@ncsu.edu I. INTRODUCTION The vibrational modes of noncrystalline SiO 2 hereafter SiO 2 in bulk fused silica, 1,2 and deposited and thermally grown thin films 3,4 have received considerable attention experimentally and in empirical force constant 1 and electronic structure 5 8 calculations. This article address the dynamic dipole moments / q i or equivalently infrared effective charges e i * associated with asymmetric bond stretching y, symmetric bond stretching z, and out-of-plane bond rocking hereafter bond rocking x infrared active normal mode vibrations of SiO 2. 1,2 The most important issues resolved in this article are the identification of microscopic contributions to / q i from: i equilibrium charge localization on the Si and O atoms, i.e., bond ionicity and ii orbital charge redistribution, or equivalently dynamic changes in the relative occupation of symmetry-determined bonding orbitals occurring during normal mode motions. The decomposition of dynamic dipoles infrared effective charges has previously been addressed for elemental and crystalline materials in Refs. 9, 10, and 11, where it has been shown that two qualitatively different mechanisms can be operative: one associated with the equilibrium charge localization or ionic bonding and a second that can occur in the absence of heteropolar bonding and is associated with a dynamic rehybridization of bonding orbitals occurring during normal mode atomic motions. There is no unique or unambiguous way to separate these two contributions experimentally when both are present; however, it is obvious that ionic contributions dominate in alkali halide crystals such as NaCl, KCl, etc., and that the second mechanism applies exclusively in crystalline Se and Te, where the bonding is identically homopolar. 9,12 In Ref. 10, an attempt was made to decompose the total infrared effective charge into local and nonlocal contributions. This was accomplished by using an empirical model to distinguish between two different contributions to the transverse optical phonon frequency: one determined by mechanical forces and a second with dipolar fields associated with charge localization on the atom sites. Using a model calculation to calculate the mechanical force contribution in terms of bulk elastic properties, this approach was applied to the eight-electron semiconducting and insulating crystals, NaCl, ZnS, GaAs, etc., and local effective charges were obtained. This model local charge, assumed to be equivalent to the 1710 J. Vac. Sci. Technol. B 20 4, JulÕAug 2002 1071-1023Õ2002Õ20 4 Õ1710Õ10Õ$19.00 2002 American Vacuum Society 1710

1711 Whitten et al.: Electronic structure of SiO 2 1711 equilibrium charge density of this article, was demonstrated to be proportional to an empirical bond ionicity determined from the spectroscopically based scale of Phillips and Van Vechten. 13,14 This article addresses the separation of the ionic and charge redistribution contributions to the dynamic dipoles/ infrared effective charges from a different perspective. Electronic structure calculations have been applied to: i equilibrium ground state structures and ii nonequilibrium bonding geometries defined by normal mode displacements of infrared active vibrations. The dynamic dipoles have been decomposed into two contributions: i one from the equilibrium charge density, equivalent to a local ionic bonding charge and ii the second from orbital changes that take place during normal mode motion, equivalent to a dynamic rehybridization of the equilibrium bonding orbitals. This approach has also been extended to noncrystalline Si 3 N 4, and to Si, C alloys, where values of / q i have been obtained for infrared active vibrations associated with asymmetric stretching N- and C-atom normal mode motions. Combined with the results for SiO 2, the equilibrium charge density contributions O, N, and C atoms follow trends expected on the basis of their relative Pauling electronegativity differences with respect to Si. This provides the incentive to expand this approach to other noncrystalline materials, including oxides such as GeO 2, 14 P 2 O 5, 15 and B 2 O 3, 16 and chalcogenide glasses and thin films such as As 2 S Se 3 Ref. 17 and GeS Se 2. 18 These dielectrics have continuous random network structures and predominantly covalent bonding that obeys the 8-N rule. 1 II. THEORETICAL APPROACH The electronic structure calculations are ab initio in character, and employ variational methods in which an exact Hamiltonian is used. No core potential or exchange approximations are assumed. 19,20 The Hamiltonian H is given in Eq. 1 : H i 2 i k Z k /r ik i j 1/r ij, 1 2 containing kinetic energy i nuclear-electron attraction ( Z k /r ik ), and electron electron repulsion (1/r ij ) contributions. Equation 1 is in atomic units a.u : 1 a.u. (distance) 1 bohr a 0 0.5292 Å; 1 a.u. (energy) 1 hartree e 2 /a 0 27.21 ev. The calculations are done at two levels, initially through a self-consistent field SCF Hartree Fock calculation with a single determinant wave function of the general form C det a 1 b 2 c 3 d 4,..., x N, where functions are molecular orbitals containing spatial and spin components. This approach does not include effects due to electron correlation. Following this, there is a configuration interaction Cl refinement of the bonding orbitals that is based on a multideterminant expansion wave function i c i i, 2 3 FIG. 1. Schematic representation of the Si O Si terminated cluster used for the ab initio calculations of this article. The Si O Si bond angle is 180 in this diagram, and will be varied from 120 to 150 for the calculations. The Si* represent an embedding potential that assures Si core eigenvalues are correct. where the i denote determinants of the form specified in Eq. 2 with different choices of occupied molecular orbitals. 21 Expansions of this type include electron correlation effects. The expansion coefficients c i are determined by energy minimization. Since the Hamiltonian is exact through electrostatic interactions, the energy variational theorem is satisfied. The SCF and Cl calculations have been applied to the clusters in Figs. 1 and 2. The local cluster in Fig. 1 is embedded mathematically in a larger network structure through a one-electron embedding potential V(r) and basis functions S1 and S2, which are represented by Si* in the diagram. For Si bonded to O, V(r) 1/r, relative to an origin 1.94 a.u. from the O nucleus, which is close to an OH bond length. Functions S1 and S2 are Gaussian expansions of long and short range functions representing a sp 3 hybrid orbital of Si. The mixing parameter in S1 S2, is set such that: i all Si core energy levels are correct this assures a correct overall charge distribution in the cluster and ii the structure in Fig. 2 has a zero dipole moment. The approach used in this article is similar in spirit to a cluster Bethe lattice CBL 6 in that both approaches seek to terminate the local cluster to represent a continuation of the noncrystalline network. Otherwise, the approaches are completely different. The present work models the system as a finite cluster of atoms in which the terminating atoms Si* represent the next shell of SiO n atoms. The Si* are chosen such that the correct dipole moment of the system and uniformity in the charge distribution, as measured by core eigenvalues, are achieved. As noted above, the Si* terminator approach is conceptually similar to the CBL approach of Ref. 6; however, for our FIG. 2. Schematic representation of a second Si O Si cluster that establishes the validity of the embedding potentials Si*. JVST B-Microelectronics and Nanometer Structures

1712 Whitten et al.: Electronic structure of SiO 2 1712 FIG. 3. Calculated energy in hartree a.u. 27.2 ev as a function of the Si O Si bond angle a for SCF with d polarization, SCF Cl no d polarization, and SCF Cl d polarization. calculations, the Hamiltonian is exact and the calculation is ab initio rather than empirical. The CBL and the embedding potential approach of this article are used to emulate the properties of the continuous random network amorphous morphology of SiO 2 1 e.g., the CBL provides a featureless density of states, etc., and Si* termination removes any dipole fields, or induced moments that come from the termination of the small cluster. Achieving a dipole moment of zero on replacing the (Si* O) 3 Si group by Si* while maintaining correct core energy levels provides a sensitive test of this effective potential. A flexible basis set is used for Si 13 s-type, 9 p-type Gaussians along with two sets of d-type Gaussians representing: i short range polarization and correlation effect exponent 0.5 and ii a3d state of the Si atom exponent 0.015. Oxygen atoms are also described by a basis of comparable quality 10 s-type, 5 p-type Gaussians plus an additional set of more diffuse p-type functions exponent 0.06 to improve the description of the polarization of the electron distribution on displacement of the nuclei. III. ELECTRONIC STRUCTURE CALCULATIONS A. Total energy The total energy for the cluster in Fig. 1 has been calculated as a function of the Si O Si bond angle see Fig. 3 at three different computational levels: i SCF with d polarization on the Si atoms, ii SCF Cl with no d polarization, and iii SCF Cl with d polarization. The Si O distances and angles for the central Si O Si were optimized, 22,23 and the optimum values are very close to the distances used for the next shell of atoms. The important point is that the energy difference between the linear structure and the optimum structure at an Si O Si angle of 150 is quite small as shown in Fig. 3. As a consequence, the internuclear angle will likely be more sensitive to randomness of bond angles in the next shell of atoms that reorient bond dipoles than to small distance variations. Since the Hamiltonian is exact, any increase in the total binding energy through a variational process represents an improved solution to the total energy calculation, and additionally to the dynamic dipole calculations that use the equilibrium cluster geometry as a point of reference. This variational approach to binding energy optimization unequivocally establishes that the SCF Cl with d polarizations is the best approach for calculating the total binding energy, and therefore for other properties of the cluster to be addressed in this article as well. It is significant to note that the minimum in binding energy occurs at a Si O Si bond angle of approximately 145 150, very close the average bond angle determined experimentally for SiO 2. 1,24 The broad minimum in binding energy is consistent with radial distribution functions studies on fused silica that indicate that the bond angle distribution is broad, extending from approximately 120 to 180. 24 It is also consistent with the unusually small value of the empirical Si O Si three-body bond-bending force constant. 1 The spread in bond angle also contributes significantly to the configurational entropy that promotes the high metastability of SiO 2. The calculated dependence of the total energy on the Si O Si bond angle adds additional validity of this local cluster approach for other properties which depend primarily on the local Si O Si bonding geometry, and in which in the terminal O-atom groups and embedding terminal atom Si* emulate their connectivity to the SiO 2 continuous random network. B. Dynamic dipoles The normal mode dynamic dipoles / q i, q i x, y, and z have been calculated for the asymmetric bondstretching vibration y, symmetric bond-stretching z, and bond rocking x infrared active vibrations of noncrystalline SiO 2. 1 These values of the dynamic dipoles should be compared with the experiment, and are essentially the same as the values of the Born effective charges that are calculated for the infrared active modes in a crystalline system. The Born effective charge for crystals is a tensor that includes off-diagonal as well as diagonal terms. 25 For each of the materials systems studied in this article, off-diagonal dipole moment components, where v x, y, z, and w x, y, z, and v w, were also obtained. These components, which cannot be probed by the experiment, are significantly smaller than the normal mode diagonal components with v w, which can be compared directly with experiment. Therefore the offdiagonal elements v w will not be addressed. The displacements of the O atoms for the three normal mode SiO 2 vibrations are defined with respect to the coordinate system displayed in Fig. 4. For the asymmetric bond-stretching vibration y, the normal coordinate is y and the relative motion of the O atom is parallel to a line joining the two Si atom neighbors. For the symmetric bond-stretching vibration J. Vac. Sci. Technol. B, Vol. 20, No. 4, JulÕAug 2002

1713 Whitten et al.: Electronic structure of SiO 2 1713 FIG. 4. Atomic displacement directions for normal mode vibrations. The normal mode coordinates y and z shown in the diagram are for the asymmetric and symmetric bond-stretching modes, respectively. The normal coordinate for the out-of-plane bond rocking mode is the out-of-plane coordinate x. FIG. 6. Changes in the dynamic dipoles for the x, y, and z normal mode motions at 150 as function of changes in the Si O Si bond angles of the terminating Si O Si* groups. z, the normal coordinate is z, and the relative motion of the O atom is in the direction of the bisector of the Si O Si bond angle. Finally, for the bond-rocking vibration x the normal coordinate is x, which is perpendicular to the plane of the Si O Si bonding group in Fig. 4. The dynamic dipoles / y, / z, and / x have been calculated from SCF and Cl calculations, with and without d polarizations, and as a function of. The displacements x, y, and z for the calculations were 5% of a nominal Si O bond length of 0.16 nm, or 3 a.u. (1 a.u. 0.053 nm). The dynamic dipoles in Fig. 5 are in units of the electronic charge e to facilitate comparisons with other theoretical and experimental studies. 5,7 10,25 The inclusion of the d polarizations increases the magnitude of the dynamic dipole for the asymmetric stretching mode with the normal coordinate displacement y. The dynamic dipoles for the other two normal modes are each significantly smaller than that of the asymmetric stretching mode, and are decreased in magnitude when the d polarizations are included. For the calculations including the d polarizations, / x is essentially independent of, while / y and / z display opposite dependencies on. / y decreases becomes more negative with increasing, whereas / z increases, but with a total positive change in / z that is about a factor of 7 smaller than the total negative change in / y, 0.1 e for / z as compared to 0.7 e for / y. The influence of next nearest neighbor Si O Si* bond angles on the dynamic dipoles for 150 is displayed in Fig. 6. Changes in the dynamic dipoles are about 5% of their 150 values for 30 excursions of the Si O Si* bond angle. This demonstrates that the dynamic dipoles are determined primarily by nearest neighbor bonding within the Si O Si portion of the cluster, and are effectively insensitive to the details of the next-nearest neighbor O-atom bonding arrangements with embedding Si* terminations, establishing that the dynamic dipoles for the infrared active normal modes of SiO 2 are a local property of the continuous random network amorphous morphology. FIG. 5. Dynamic dipoles in units of electronic charge as a function of Si O Si bond angle for x, y, and z normal mode displacements. The solid lines connecting the calculated values for 120, 140, 150, 160, and 180 are for SCF Cl d polarization calculations, and the dashed lines are for SCF Cl. C. Decomposition of dynamic dipoles These calculations compare the dynamic dipoles for two different conditions. The total dynamic dipoles for the normal mode displacements have been determined by comparing the equilibrium structure x, y, and z dipole moments for a given bond angle with the dipole moments of dynamic structures that include, respectively, the x, y, and z displacements of the normal modes. This was done at: i SCF with d polarizations, ii SCF Cl without d polarizations, and iii SCF Cl with d polarization levels. For the determination of JVST B-Microelectronics and Nanometer Structures

1714 Whitten et al.: Electronic structure of SiO 2 1714 TABLE I. Decomposition of total dynamic dipole. a Mode Equilibrium charge Dynamic dipoles e orbital change Total a 150 x, rocking 1.0 0.1 0.9 z, symmetric stretch 1.0 0.1 0.9 y, asymmetric stretch 1.3 1.2 2.5 a 180 x, rocking 1.0 0.1 0.9 z, symmetric stretch 1.0 0.1 0.9 y, asymmetric stretch 1.4 1.3 2.7 a The calculations have precision of at least one part in a hundred; however, a realistic assessment of the model is consistent with uncertainty of the order of at least 0.1 e. the equilibrium charge density contribution, / q i ] eq, the change in the dipole moment was calculated for normal mode displacements of the O-atom localized orbitals, with all other orbitals fixed. The contribution due to the orbital displacements, i.e., the dynamic rehybridization of the bonding orbitals, / q i ] oc, is then the difference between the total dynamic dipole / q i ] total and the contribution from the equilibrium charge density shift / q i ] eq as in Eq. 4 : / q i ] total / q i ] eq / q i ] oc. 4 These calculations have been performed for two different Si O Si bond angles, 150, the optimum bond angle from Fig. 3, and for 180. The results of these calculations are also included in Table I. The average value of the equilibrium charge density contribution is about 35% larger ( 150 ) for the asymmetric stretching mode with the normal coordinate y than the symmetric stretching mode and rocking modes with respective normal mode coordinates z and x. The equilibrium charge density contributions are negative, consistent with the Pauling electronegativities of 1.90 for Si and 3.44 for O. 26 Finally the orbital change contributions for the symmetric stretching and rocking modes are relatively small and positive, and decrease the magnitude of the total dynamic dipole, while the orbital change contribution for the asymmetric mode is more than 1 order of magnitude larger, and is negative, so that it increases the magnitude of the total dynamic dipole. There is also an increase of about 10% in the orbital change contribution when the Si O Si angle is increased from 150 to 180, while the equilibrium density contribution increases about 7%. D. Extension to Si C and Si N vibrations Since the Pauling electronegativity differences between Si and N atoms ( X 1.14) and Si and C atoms ( X 0.65) are smaller than the corresponding electronegativity difference between Si and O atoms ( X 1.54) 26 it is reasonable to assume that the equilibrium density contributions to the respective dynamic dipoles for Si N and Si C asymmetric bond-stretching infrared active vibrations would be smaller than for the asymmetric bond stretching vibration of Si O. FIG.7. a and b Clusters used for the Si C calculation and c cluster used for the Si N calculations. Terminating groups of all three clusters are H atoms. This suggests that the orbital change contributions should become increasingly more important in noncrystalline Si N and Si C, making proportionally larger contributions to the total dynamic dipole of the asymmetric bond-stretching vibrations. To test this hypothesis, calculations have been performed on the Si N and Si C containing clusters displayed in Figs. 7 a and 7 b. These clusters are terminated with next-nearest neighbors that emulate the network bonding; however, they do not include a Si-atom embedding potential Si* as in Fig. 1 for the Si O Si cluster. The Si N and Si C bonding arrangements of interest for the noncrystalline network vibrational properties are based on 8-N bonding, i.e., threefold coordinated planar bonding between Si and N atoms in Si 3 N 4, 27,28 and tetrahedral bonding of Si and C atoms in Si, C alloys. 29 Since the Si and C atoms are each in tetrahedral bonding arrangements, a calculation has also been performed for both C- and Si-atom normal mode displacements to confirm that the sign of the calculated equilibrium charge, positive for Si and negative for C, was consistent with their electronegativity difference. Table II gives the results of these calculations for the clusters in Figs. 7 a and 7 b, but does not include the confirmatory calculation for Si and C described above. The total dynamic dipole charges decrease approximately linearly as the coordination of the central atom of the cluster increases Fig. 8. This scaling applies when the Si O Si bond angle is 180 ; however the Si O point for a coordination of two is de- J. Vac. Sci. Technol. B, Vol. 20, No. 4, JulÕAug 2002

1715 Whitten et al.: Electronic structure of SiO 2 1715 TABLE II. Dynamic dipoles for Si O, Si N, and Si C asymmetric stretching vibrations, and local contributions for out-of-plane modes. Bond Cluster Dynamic dipole e Equilibrium charge from out-of-plane modes e Si O Fig. 1 150 2.5 O atom, bond rocking 1.0 Si O Fig. 1 180 2.7 O atom, bond rocking 1.0 Si N Fig. 7 a 2.2 N atom, out-of-plane stretch 1.1 Si C Fig. 7 b 1.5 C atom, degenerate stretch a 0.5 a Due to tetrahedral symmetry of Si C bonding, this vibration is threefold degenerate and does not have the same out-of plane character as the Si O rocking and Si N modes. creased when the Si O Si bond angle is reduced to the average value of 150. The behavior of the equilibrium charge contribution for the asymmetric stretching modes displays a similar behavior but with a larger departure from linearity when plotted as a function of bond coordination in Fig. 8. However, as discussed in Sec. IV, this ionic charge displays a linear behavior when normalized to coordination, and plotted as function of bond ionicity. Finally, the orbital change charge shows a much smaller variation with bond ionicity that is also a monotonically decreasing function of bond coordination. As a consequence of this smaller dependence on bond coordination, the contribution of the orbital change component becomes increasingly more important as the bond coordination increases from two for O, to three for N, and finally to four for C. IV. DISCUSSION A. Applicability of local structure calculations The total energy ab initio calculations of this article represent an improvement over the more empirical approaches in previous publications. The calculations of Pantelides and Harrison, 5 based on empirical tight-binding theory, gave an optimum of 90, which is a significant departure from the experimentally determined average Si O Si bond angle of approximately 145 150. 1,24 However, attempts to improve the tight-binding empirical calculations by Harrison through the addition of an empirical Si Si repulsion term resulted in a small improvement, increasing to approximately 114. 7 Revesz and Gibbs performed ab initio calculations on a pyrosilicate acid molecule, (HO) 3 SiOSi(OH) 3 or equivalently H 6 Si 2 O 7, at the SCF level. 30 This molecule terminates the Si O Si arrangement of interest with terminal OH groups. The calculation of Ref. 30 specifically included Si 3d states in the basis set. They obtained results that are qualitatively similar to those presented in this article in Fig. 3, displaying a shallow minimum in binding energy at approximately 140. The energy of this minimum of about 5% is lower than the value of the binding energy at 180, while in Fig. 3, it is lower by 1.5% for the SCF Cl with d polarization calculations, and is at angle of 150. The calculations of Lucovsky and Yang in Ref. 8 at the SCF Cl level, but without the Si d polarizations, give essentially the same results as in the corresponding curve of Fig. 3. These calculations show a continuous increase in binding energy between 120 and 180, but with no shallow minimum in the vicinity of 150 or elsewhere. As evident in Fig. 3, the inclusion of the Si d polarization gives a large 2.7 ev increase in the total energy, and also produces a shallow minimum in binding energy close to the experimentally determined Si O Si value of 145 150. From a comparison between the calculations of Lucovsky and Yang, 8 Revesz and Gibbs, 30 and those presented in this article, it can be concluded that the inclusion of Si 3d states as in Ref. 30, and Si d polarizations as in this article, are most likely the determinant factor in generating the shallow minimum in the binding energy curve. FIG. 8. Calculated dynamic dipoles, equilibrium charges, and orbital change charges for asymmetric Si O, Si N, and Si C bond-stretching vibrations as a function of bond coordination. The plot includes values of the dynamic dipole for the asymmetric stretching vibration of the Si O Si group for Si O Si bond angles of 150 and 180. B. Calculated dynamic dipoles for the Si O Si group There are several issues to be addressed relative to the differences between the calculated dynamic dipoles of this article, and those of Refs. 7 and 8 as presented in Table III. All of the calculations give similar results, with the asymmetric bond-stretching frequency having the largest dynamic dipole, and the dynamic dipoles of the symmetric bondstretching and out-of-plane bond-rocking vibrations being smaller by a factors ranging from 1.6 to 2.8. Since the calculations of Ref. 7 were based on an entirely different electronic structure calculation than those of this work and Ref. 8, it is difficult to make specific comparisons JVST B-Microelectronics and Nanometer Structures

1716 Whitten et al.: Electronic structure of SiO 2 1716 TABLE III. Calculated dynamic dipoles for a 150 for the Si O Si bonding group. Vibrational mode Wave number 5 cm 1 This work SCF Cl This work SCF Cl d polarizations Harrison a Lucovsky and Yang b SCF Cl Asymmetric stretching Symmetric stretching Out-of-plane rocking 1072 2.1 2.5 2.6 2.4 810 1.2 0.9 1.3 1.0 455 1.3 0.9 1.3 0.9 a See Ref. 7. b See Ref. 8. other than to note that the factor of 1.6 2.8 between the magnitude of the asymmetric bond-stretching dynamic dipole and the other two dynamic dipoles is simply a manifestation of a qualitatively important difference in their normal mode atomic displacements. For the asymmetric bondstretching vibration, the Si O bond lengths change relative to one another, while for the other two vibrations, they change by equal amounts, and in the same sense during the normal mode symmetric-stretching and out-of-plane rocking displacements. An asymmetric bond stretching normal mode motion then invariably leads to a larger dynamic dipole. The differences between the SCF Cl and SCF Cl with d polarizations, calculations of this work, and the differences between each of these and the calculations of Ref. 8 are now addressed in more detail. Consider first the SCF Cl calculations of this work and of Ref. 8. The differences between these calculations are attributed to the differences in the clusters used to perform the calculations. The calculations of Ref. 8 were based on the same 3 (HO)SiOSi(OH) 3 molecule as in Ref. 30, but without inclusion of Si d states or d polarizations. This means that the differences between the SCF Cl calculations in this work and Ref. 8 are due to the d polarization effects and in the way the Si O Si group was terminated, by OH groups in Ref. 8, and O Si* groups in this work. Since these terminations are qualitatively different, the differences in: i binding energy versus the Si O Si bond angle and ii the dynamic dipoles are assumed to reflect these differences in cluster termination. Finally, the differences between the calculations of this work for SCF Cl, and SCF Cl with d polarizations are then related to qualitatively different dynamic effects associated with the d polarizations. This is evident in the comparisons presented in Table I, where the total dynamic dipoles are analyzed in terms of equilibrium charge densities and orbital changes occurring in the normal mode motion. The equilibrium charge densities for all three modes lie between 1.0 and 1.4 for the bond angles of 150 and 180. The orbital change contributions are small for the symmetric bondstretching and out-of-plane bond-rocking vibrations. However, the orbital change component for the asymmetric bondstretching vibration is 20 times larger, and additionally, there is a 10% increase in the orbital change component in going from a 150 to a 180 Si O Si bond angle. This is consistent with the d polarization effect being more important for vibrations in which the two Si O bond lengths of the Si O Si group change in opposite directions during a normal mode motion, rather than change in the same sense and magnitude. O Keefe 22 has shown that the Si Si distances in essentially all silicate bonding arrangements are approximately the same (0.305 0.005 nm) so that Si O bond length is reduced by at least 0.05 nm when increases from 150 to 180. This means that d polarization effects increase with increasing Si O Si bond angle and decreasing Si O bond length. More significantly, the decrease in the Si O bond length associated with this change in bond angle does not change in any significant way the equilibrium charge density on the O atom. In is important to note that the quantity that is measured experimentally is the square of the dynamic dipole or effective charge. Therefore it is only the relative sign of the equilibrium charge density and dynamic change contributions that is important. The absolute sign of the equilibrium charge component is an important aspect of the theory and the agreement with empirical bond ionicity criteria based on electronegativity difference is then significant, but unfortunately is not susceptible to direct experimental verification. C. Comparisons with experiment Comparisons with the experiment are at best difficult to make for several reasons. First noncrystalline SiO 2 is a metastable material, and its local atomic bonding and infrared properties depend on the method of preparation, e.g., bulkquenched glass, deposited thin film, or thermally grown oxide, and heat treatments or annealing cycles after preparation. 31 The as-deposited properties of thin film SiO 2, prepared by plasma chemical vapor deposition depend on the details of the deposition process and deposition process conditions; 3,4 however, the properties of these films after a 900 C, 30 s 1 min anneal in a nonoxidizing ambient such as Ar are essentially independent of the deposition process variations, and similar to those of fused silica that has been annealed at a fictive temperature close to the viscoelastic relaxation, or softening temperature of approximately 1000 C. 3 Analysis of the infrared spectra of fused silica in Ref. 2, and thin film SiO 2 produced by remote plasma-enhanced chemical vapor deposition and annealed at 900 C for 1 min J. Vac. Sci. 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1717 Whitten et al.: Electronic structure of SiO 2 1717 in Ar, 32 gives values of the dynamic dipoles/infrared effective charges that are nearly equal to the calculated values. The results are also given as positive charges, since the analysis experimental results always yields the square of the dynamic dipole/infrared effective charge. The uncertainties in the values of these measured dynamic dipoles (2.5 0.15)e, (0.8 0.15)e, and (1.0 0.15)e for the respective asymmetric stretching, symmetric stretching, and rocking vibrations, are in excellent agreement with the SCF Cl with d polarization calculations of this article. Even though the agreement is equally good with the results of Ref. 8, the observation that the calculations of this article show a shallow minimum close to the experimentally determined average bond angle of SiO 2 favors the approach of this article and the increased reliability of the results. D. Scaling of equilibrium charge with empirical bond ionicity and coordination As noted above, the equilibrium charge density component of the dynamic dipole is expected to scale with bond ionicity. Consider ideal ionic bonding in a diatomic molecule, A B, with an ionic charge of e A on A, e B on B. The magnitudes of e A and e B are equal. The dynamic dipole / x is then identically equal to either the charge on A or B, depending on the way the molecular coordinate axis is chosen; in particular, the sign of the dynamic dipoles depends on the sign of the charge on the A or B atom. This is a mute point, since the experimentally determined quantity is always the square of the dynamic dipole or effective charge. This simple calculation provides the basis for the comparisons between the magnitude of the equilibrium charge density and an empirical bond ionicity f i as defined by Pauling. 33 For example, if X is the difference in electronegativity between a pair of bonding partners such as Si and O, N or C, then the Pauling bond ionicity is given by Eq. 5 : f i 1 exp 0.25 X 2. 5 Figure 9 a is based on Table IV and includes a plot of calculated values of the total dynamic dipole for the asymmetric bond stretching vibration for C, N, and O atoms in Si C, Si N, and Si O bonds, respectively, as a function of the Pauling bond ionicity f i. The calculated value of the total dynamic dipole for the Si O Si group is for a 180 bond angle. The table also includes calculated values of the equilibrium charge, as presented in Table II, which have been normalized by the formal chemical valences of C, N, and O, i.e., n 4, 3, and 2, respectively. Table IV includes the equilibrium charges calculated for the asymmetric stretching vibrations, and as noted above in Sec. III C, these are not equal to the equilibrium charges of Table II. The dynamic dipole and normalized equilibrium charges of Table II increase approximately linearly with f i, but with different slopes obtained from a least squares fit to the data points: 1.06 for the normalized equilibrium charge and 3.43 for the dynamic dipole charge. The orbital change charge has been calculated by the method described in Sec. III D for the asymmetric stretching modes, and also displays a monotonic increase FIG. 9. Dynamic dipole charge, orbital change charge, and normalized equilibrium for Si O, Si N, and Si C bonds as functions of: a the Pauling bond ionicity f i and b the coordination of O, N, and C. with increasing f i. These data points show a more significant departure from linearity than the dynamic dipole and normalized equilibrium charges as indicated by the second order polynomial fit. Figure 9 b, in which bond coordination of the O, N, and C atoms is the x-axis variable, shows complementary behaviors to those in Fig. 9 a. The linear fits to the dynamic dipole and normalized equilibrium charges result from an empirical linear scaling relationship between coordination of the O, N, and C atoms and the Pauling bond ionicity f i for their bonding with Si, which also applies to Si F bonds with a coordination of one. The slope of the normalized equilibrium charge versus Pauling bond-ionicity plot in Fig. 9 a displays the linear scaling anticipated by the definition of the equilibrium charge. The slope of this plot is remarkably close to one; however, the agreement cannot be taken as a measure of inherent accuracy of these calculations since: i the definition of electronegativity is based on a decomposition of empirical bond energies in homo-and heteropolar contributions and ii the relationship between electronegativity differences and bond ionicity in Eq. 5 is not derived from a first principles calculation, but rather reflects an emprical and intui- JVST B-Microelectronics and Nanometer Structures

1718 Whitten et al.: Electronic structure of SiO 2 1718 TABLE IV. Magnitudes of equilibrium orbital charge and total dynamic dipole charges. Atom, coordination Pauling bond ionicity f i Normalized equilibrium charge a e Orbital change charge e Equilibrium charge b e Total dynamic dipole e O, 2 0.45 ( 1.0/2) 0.50 1.27 1.4 2.7 N, 3 0.28 ( 1.1/3) 0.37 1.12 1.1 2.2 C, 4 0.10 ( 0.5/4) 0.13 1.04 1.1 1.5 a Normalization of the equilibrium charges of Table II in parentheses with respect to primary chemical valence of 2, 3, and 4, respectively. b Equilibrium charges calculated for the asymmetric stretching modes in Sec. III D. tive approach to chemical bonding. The approach of this article in identifying the equilibrium charge is complementary, and also less arbitrary than the empirical model of Ref. 10, in which the decomposition of the infrared effective charge into local and nonlocal charges is based on application of macroscopic and linearized equations of motion for the infrared active vibrations. V. CONCLUSIONS The calculations of this paper have established that: a The concept of an embedding potential Si* adds a dimension to the applicability of using small cluster calculations to provide important information about the electronic structure of noncrystalline solids with a continuous random network structure in which the local bonding is predominantly covalent, and the coordinations of the constituent atoms obey the 8-N rule. 34 b Si d polarization functions make significant contributions to the total electronic energy and the dynamic dipoles of the infrared active network vibrations. c Inclusion of the d polarizations in the total electronic energy yields a shallow minimum very close to the experimentally determined average Si O Si bond angle of approximately 150. d Changes in the Si O Si bond angle produce relatively small changes in the total energy between about 130 and 180, and with increased destablization occurring for bond angles less than about 125, consistent with incorporation of OH and F into thin film SiO 2. 8,35 e Calculated values of respective dynamic dipoles for the asymmetric bond-stretching, symmetric bondstretching, and out-of-plane bond-rocking vibrations are in good agreement with the experiment. f The calculated dynamic dipoles show very small changes for next-nearest-neighbor Si O Si* bond angle variations as large as 30. g The insensitivity to next-nearest-neighbor bond angle variations in f establishes that the dynamic dipoles are local properties of the Si O Si groups of the SiO 2 continuous random network structure. h Decomposition of the total dynamic dipoles into contributions from the equilibrium charge density, and orbital changes during the normal mode motions, establishes that these are significantly larger for the asymmetric stretching vibration, wherein the Si O bond length changes are oppositely directed, than for the symmetric stretching and rocking vibrations where the changes are in the same sense. i Extension of the Si N and Si C bond stretching vibrations in nc-si 3 N 4 and nc-si,c infrared active vibrations indicates that normalized values of the equilibrium charge scale linearly with the Pauling bond ionicity with a slope, 1.05, approximately to one. j The total dynamic dipoles for asymmetric stretching vibrations show a stronger linear dependence on Pauling bond ionicity with a slope 3.43 more than three times that of the normalized equilibrium charges. k The plots in Figs. 8 and 9 indicate that the orbital change contribution increases in relative importance as the bond ionicity decreases, and the coordination increases. Note Added in Proof: Infrared effective charges for noncrystalline As 2 S 3 and As 2 Se 3 have previously been estimated from analysis of IR reflectance data, and it had been suggested that the large values of these charges for asymmetric bond-stretching vibrations, 1 e, were the result of a charge redistribution mechanism 36 similar to what has been discussed in this article for SiO 2. Studies were also performed on GeS 2 and Ge S alloy glasses, 37,38 and the analysis of reflectance data also indicated a large IR effective charge for the asymmetric bond-stretching modes. Based on these observations, the calculations of this article have been extended to noncrystalline GeS 2 and As 2 S 3, 39 which like SiO 2 have continuous random network structures with twofold coordinated bridging S atoms. The results of these calculations give IR effective charges for the three S atom normal mode vibrations that are in excellent agreement with experiment, 39 and consistent with an empirical force constant and effective charge model that demonstrated that an approach based on the normal mode motions of the twofold coordinated atoms was equally applicable to both oxides and chalogenides, 40 in spite of large differences in average bond angles at the bridging O-atom and S-atom sites, 125 150 for the oxides compared to 95 100 for the sulfides. The new calculations of Ref. 39 also confirm the importance of the charge redistribution contribution for the asymmetric bondstretching vibrations of As 2 S 3 and GeS 2. In addition, the same calculations have identified significant quantitative differences between electronic structure calculations for noncrystalline Si-, Ge-, and As-atom oxides and chalcogenides. J. Vac. Sci. Technol. B, Vol. 20, No. 4, JulÕAug 2002

1719 Whitten et al.: Electronic structure of SiO 2 1719 Even though the bonding in these continuous random networks obeys the same 8-N rule for atomic coordination, the atomic basis sets necessary to give an adequate description of the electronic structure are quantitatively different for the O and S atoms. For example, as discussed in Ref. 39, the basis sets for the As, Ge, Si, O, and S atoms include the core levels and the respective valence bonding states, 4s and 4p for As and Ge, 3s and 3p for Si and S, and 2s and 2p for O. In addition, the basis sets for the larger Si, S, As, and Ge atoms must also include additional functions with d-state symmetries in order to optimize the binding energy of the system. This reflects the importance of the so-called p d bonding discussed in Cotton and Wilkenson, 41 and more importantly, the increased size of the Si, S, As, and Ge atoms with respect to first row atoms such as O, and also C and N. The extended basis sets suggest that the linear scaling in Fig. 9 a of this article between the normalized equilibrium charge and the Pauling bond ionicity which applies to first row elements bonded to Si, should then not be expected to apply to the bonding of As and Ge to S as indeed has been demonstrated in Ref. 39. ACKNOWLEDGMENTS The authors acknowledge support for this research from ONR, AFSOR, DOE, SRC, and the SEMATECH/SRC Front End Processes FEP Center. 1 R. J. Bell and P. Dean, Discuss. Faraday Soc. 50, 55 1970 ; inamorphous Materials, edited by R. W. Douglas Wiley-Interscience, London, 1972, p.443. 2 F. L. Galeener and G. Lucovsky, Phys. Rev. Lett. 37, 1474 1976. 3 J. T. Fitch, Ph.D. thesis, North Carolina State University, Raleigh, 1990. 4 S. V. Hattangady, R. G. Alley, G. G. Fountain, R. J. Markunas, G. Lucovsky, and D. Temple, J. Appl. Phys. 73, 7635 1993. 5 S. T. Pantelides and W. A. Harrison, Phys. Rev. B 13, 2667 1976. 6 R. B. Laughlin and J. D. Joannopoulos, Phys. Rev. B 16, 2942 1977. 7 W. A. Harrison, Elementary Electronic Structure World Scientific, Singapore, 1999, p.419. 8 G. Lucovsky and H. Yang, J. Vac. Sci. Technol. A 15, 836 1997. 9 E. Burstein, M. H. Brodsky, and G. Lucovsky, Int. J. Quantum Chem. 1s, 759 1967. 10 G. Lucovsky, R. M. Martin, and E. Burstein, Phys. Rev. B 4, 1367 1971. 11 G. Lucovsky and R. M. White, Phys. Rev. B 8, 660 1973. 12 G. Lucovsky, Phys. Status Solidi B 49, 633 1972. 13 J. C. Phillips, Phys. Rev. Lett. 19, 415 1967 ; Phys. Rev. 168, 905 1968. 14 J. C. Phillips and J. A. Van Vechten, Phys. Rev. Lett. 22, 705 1969. 15 J. Wong, AIP Conf. Proc. 31, 237 1976. 16 F. L. Galeener, G. Lucovsky, and J. C. Mikkelsen, Jr., Phys. Rev. B 22, 2983 1980. 17 G. Lucovsky, Phys. Rev. B 6, 1480 1972. 18 G. Lucovsky, J. P. deneufville, and F. L. Galeener, Phys. Rev. B 9, 1591 1974. 19 J. L. Whitten and H. Yang, Int. J. Quantum Chem., Quantum Chem. Symp. 29, 41 1995. 20 J. L. Whitten and H. Yang, Surf. Sci. Rep. 24, 55 1996. 21 J. L. Whitten and M. Hackmeyer, J. Chem. Phys. 51, 5584 1969. 22 M. O Keeffe and B. G. Hyde, Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem. 34, 27 1978. 23 A. Demkov, Phys. Rev. B 52, 1618 1995. 24 R. L. Mozzi and B. E. Warren, J. Appl. Crystallogr. 2, 164 1969. 25 A. Pasqurello and R. Car, Phys. Rev. Lett. 79, 1766 1997. 26 J. E. Huheey, Inorganic Chemistry, 2nd ed. Harper and Row, New York, 1978, p. 162. 27 F. L. Galeener, W. Stutius, and G. T. McKinley, in The Physics of MOS Insulators, edited by G. Lucovsky, S. T. Pantelides, and F. L. Galeener Pergamon, New York, 1980, p.77. 28 G. Lucovsky, J. Yang, S. S. Chao, J. E. Tyler, and W. Czubatyj, Phys. Rev. B 28, 3234 1983. 29 Y. Katayama, K. Usami, and T. Shimada, Philos. Mag. B 43, 283 1981. 30 A. G. Revesz and G. V. Gibbs, in The Physics of MOS Insulators, edited by G. Lucovsky, S. T. Pantelides, and F. L. Galeener Pergamon, New York, 1980, p.92. 31 F. L. Galeener, A. J. Leadbetter, and M. W. Stingfellow, Phys. Rev. B 27, 1052 1983. 32 D. V. Tsu, S. S. Kim, J. A. Theil, and G. Lucovsky, J. Vac. Sci. Technol. A 8, 1430 1990. 33 L. Pauling, The Nature of the Chemical Bond, 3rd ed. Cornell University Press, Ithaca, NY, 1948, Chap. 2. 34 R. Zallen, The Physics of Amorphous Solids Wiley, New York, 1983, Chap. 2. 35 J. A. Theil, D. V. Tsu, S. S. Kim, and G. Lucovsky, J. Vac. Sci. Technol. A 8, 1374 1990. 36 G. Lucovsky, Phys. Rev. B 6, 1480 1972. 37 G. Lucovsky, J. P. deneufville, and F. L. Galeener, Phys. Rev. B 9, 1591 1974. 38 G. Lucovsky, F. L. Gallener, R. C. Keezer, R. H. Geils, and H. A. Six, Phys. Rev. B 10, 5134 1974. 39 G. Lucovsky, L. Sremaniak, Y. Zhang, J. L. Whitten, and T. M. Mower, J. Non-Cryst. Solids in press. 40 G. Lucovsky, C. K. Wong, and W. B. Pollard, J. Non-Cryst. Solids 59&60, 839 1983. 41 F. A. Cotton and G. Wilkinson, Advanced Inorganic Chemistry, 3rd ed. Interscience, New York, 1972, Chap. 4. JVST B-Microelectronics and Nanometer Structures