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Electronic structure of defect centers in SiO2
J. Phys. Chem. Solids
Pergamon Press 1971. Vol. 32, pp. 1251-1261.
Printed in Great Britain.
E L E C T R O N I C S T R U C T U R E OF D E F E C T C E N T E R S
IN SiO2 A. J. BENNETT and L. M. ROTH General Electric Research and Development Center, P.O. Box 8, Schenectady, N.Y. 12301, U.S.A.
(Received 31 .4 ugust 1970) Abstract--Molecular orbital calculations based on the Extended Huckel Theory have been performed for clusters of two and eight SiO2 molecules. We report here the results for an ordered structure in which periodic boundary conditions are imposed to saturate the peripheral bonds which otherwise cause extraneous localized states to appear. We find an energy gap of - 13 eV as compared to the experimental value of - I 1 eV. The removal of an oxygen results in several levels in the energy gap which move upon displacement of the two neighboring silicons. The levels appear to account for the observed ultraviolet optical absorption band and are also a possible origin for the positively charged slow surface states present near a SiO2-Si interface. The addition of either atomic or molecular hydrogen to a perfect crystal is predicted to yield preferentially a double hydrogen-single oxygen center. If ambient oxygen atoms are present, a two hydroxl group center is then most favorable. 1. INTRODUCTION
THE PROPERTIES of various defects in SiO2 are of great technological interest. Vacancies and interstitials have been invoked to explain both 'fast' and 'slow' surface states at the semiconductor-oxide interface of a metaloxide-semiconductor field effect transistor (Mosfet)[1-3]. The 'fast' states which affect the voltage, frequency, and temperature dependence of the capacitance characteristics can i.nteract rapidly with the Si space charge and are known to be well localized at the interface. These states lie within the Si band gap. We consider here the 'slow' states which are responsible for translation of the C(V) curves along the V axis, and are thought to be localized at defect centers in the oxide (within 200 ,~ of the silicon). These defects are induced by the junction fabrication. The growth of the SiO2-Si interface may result in oxygen vacancies and accompanying trivalent Si near. the interface. This structure is indicated schematically in Fig. 1(b). If, as is often the case, hydrogen is present during fabrication, a variety of other defects may exist. Hydrogen additions to the perfect crystal are shown in Figs. l(c) and l(d); additions to the vacancy structure in Fig. l(e) and
l(f). A structure which may result if oxygen interstitials and/or water is also present, is shown in Fig. 1(g). Experiments by Bell, Hetherington and Jack [5] have shown that exposure of vitreous silica to hydrogen or to water vapor at high temperatures leads to the formation of O H groups in the SiO2 system. These are detected by a strong i.r. absorption at 2.75/z due to the stretching of the O - H bond. Defect states also result when pure amorphous or crystalline SiO2 is irradiated with neutrons, electrons or gamma rays [6]. In particular optical absorption [7] and Epr studies [8-10] have indicated the existence of a paramagnetic state associated with an oxygen vacancy. There have been no even semiquantitative calculations of the electronic properties of the various postulated defects states. We attempt such a treatment in this paper. The defect states may lie well within the SiO2 band gap ( - 10 eV) [11] and as a result can not be considered using the effective mass approximation[12]. We apply a simple molecular orbital scheme, the Extended Huckel Theory (EHT) developed by Hoffman and his collaborators [ 13-15], to a finite array of Si
1251
1252
A . J . B E N N E T T and L. M. R O T H
d ~,.~l e~ (o}
d , o.9,
(hi
s,
s,
(e}
o'
"
(d}
o
s,
..
Z
s,
d
~
H (e)
(f} H
Si
0
(g} Fig. 1. Atomic configurations in/3 crystobalite in the vicinity of (a) a normal oxygen site (b) an oxygen vacancy and (c-g) various centers containing hydrogen. The plane of the paper is the (I10) plane in which we have assumed the defect atoms to lie. The x and y axes are (111) and (112) respectively. The arrows in (b) show the extent of the Si displacements.
and O atoms chosen to represent the insulator. As is well known, the E H T yields quite useful semi-quantitative results when applied to small molecules. Interaction effects, not considered in single bond arguments, are naturally included in such calculations. Recently Messmer and Watkins [ 16] have successfully used the method in a study of the nitrogen defect in diamond. Although probably best
suited to totally covalent systems, the method will be applied here to SiO2 where the ionicity is not extremely large [ 17]. The SiO2 films present in Mosfets are, in general, amorphous. Locally, however, there exists strong short range order. X-ray and neutron scattering data indicate that even in SiOa glasses, the basic SiO4 tetrahedra are maintained[18]. Since the defect states are
D E F E C T C E N T E R S IN SiO2
probably well localized on an atomic scale, we use a crystalline model of the insulator, in particular a simplified cubic /3-crystobalite structure [19]. /3 crystobalite has a density similar to that of the amorphous phase and hence is a reasonable choice for a crystalline model. The structure can be visualized by noting that the silicons form a diamond lattice with oxygens midway between adjacent silicons. In the actual crystal, the oxygens are somewhat displaced from these positions. One disadvantage of the use of an ordered structure is that we are unable to consider certain network defects such as non-bridging oxygens. The edge atoms of our finite representation of the SiOz system have unsaturated bonds which are responsible for perimeter states, localized at the edge of the representation. In this study periodic boundary conditions will be applied to eliminate these states. In the next section we briefly describe the E H T , the representations used, and the application of periodic boundary conditions. Section 3 contains our results and comparison with experiment. We summarize in a final section. 2. FORMALISM
The Extended Huckel Theory has been discussed extensively in various publications [13-15]. It is a molecular orbital method in which the wave functions 9~ are taken as linear combinations of all the valence orbitals, Ih > (Slater functions) centered on the various atoms of the system. =
c,
lx > .
(2.1)
The SchriSdinger equation written in that non-orthogonal representation is given by IH-- ESI = 0
(2.2)
where Hxx = -- I~
(2.3)
HA,,- = 8 9
(2.4)
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Sx~ = (hltr)
(2.5)
Ix is the valence state ionization potential, K is a constant equal to 1.75, and (h[o-) is an overlap integral between atomic orbitals. Cohesive energies are calculated as the difference between the sum of the molecular one-electron energies and the sum of the one electron energies of the separated constituent atoms. The neglect of explicit electron-electron and core-core interaction is a distinct weakness of the approach. The results are, therefore, only semiquantitative. We find, for example, too much ionicity in the system which is a consequence of the lack of any provision for self consistency in the method. T w o representations of the SiO2 structure, consisting of six and twenty four atoms respectively, were used in our calculations. The smaller representation is shown in Fig. 2 by the solid circles. Extraneous perimeter states are eliminated by including Si-Si and Si-O bonds between an atom in the cluster and its nearest and nextnearest neighbors in translated versions of the representation. The cluster thus serves as one unit cell of a periodic system. The dotted circles in Fig. 2 represent atoms in the translated versions of the small cluster. The silicons la, lb, and lc, which are not in the cluster, are considered to be equivalent to silicon 1. To the overlap matrix element between silicon 1 and oxygen 2, we add the overlap between silicon la and oxygen 2. Similarly, we add overlaps of silicon lb with oxygen 3 and silicon l c with oxygen 4, to the silicon 1 - oxygen 3 and silicon 1 - oxygen 4 overlap integrals respectively. In addition, we add to the overlap matrix element between silicon 1 and silicon 2, three terms corresponding to overlaps between silicons 1a, 1b and 1c with silicon 2. We are able to use the small cluster due to the short range nature of the overlaps in SiO2. For some defects containing two hydrogens, however, it was found necessary to go to a larger cluster of 8 SiO~ molecules, which
1254
A . J . B E N N E T T and L. M. R O T H / ~ --'%
I [C t
/
/
./
.//
_i
.
Ibm,.4
.
.
IB----'l
.
/ (~,k~
f-"~/
/ "~
~'~"
Ib I
II ~.;~'~.~.~ ~1~ v I.
.
.
.
L/"
.
.
.
j_ .
I i "
----
.
.
.
.
--t
I0
1
// /"
0
Fig. 2. A portion of our idealized beta crystobalite lattice showing the small cluster which consists of silicons 1 and 2, and oxygens 1, 2, 3, 4. Silicons la, lb, and lc are outside the cluster and are equivalent to silicon I. The dashed cube shown is ] of the cubic unit cell for the diamond lattice upon which the silicons occur and is drawn to show the tetrahedral arrangement of silicons. (It is not a unit cell-the neighboring cube has no Si in the center.)
then serves as the unit cell. Similar periodic boundary conditions were applied to this larger model. The above procedure for applying periodic boundary conditions is intuitively the obvious one to use. It is interesting to justify it formally by requiring the wave function for the system repeated periodically, to be periodic with unit cell given by our cluster. This is accomplished by adopting as basis functions the k --- 0 Bloch states. IX)' = ~ 1 xi Ih~)"
(2.6)
Here i labels the unit cell in which Ihi) is to be centered, and N is the number of cells. The overlap integral is then (~lh) ' = Z (/-*01h,) 9 i
(2.7)
T h e rule is therefore to add to a given overlap S~oxo within the cluster, all overlaps Sm~, between ~0 in the cluster and the h~ outside the cluster which are equivalent to h0. We have followed this rule in the procedure described above. However, the overlaps beyond a certain distance are quite small ( < 10 per cent of those included) and are ignored. In practice a difficulty could arise which should be noted. The matrix S associated with a finite molecule is readily shown to be positive definite [20]. If arbitrary additional overlaps are added to that matrix this property can be destroyed creating technical difficulties. The positive definite property holds also for a periodic array of molecules with all overlaps included. In practice, therefore, if the S matrix ceases to be positive definite, we must include more overlaps. Out system is quite tightly
1255
D E F E C T C E N T E R S IN SiOz
b o u n d and we have had relatively little difficulty in this regard. Our method is equivalent to a tight binding energy band calculation with nonorthogonality taken into account. The wave vector is, however, restricted to k = 0 for the Brillouin zone of the unit cell corresponding to the cluster. The energy values correspond to those of a selected set of k values for the Brillouin zone of the conventional primitive unit cell of the crystal. 3. RESULTS
The basis set consists of 2s and 2p orbitals on the oxygen and 3s, 3p, and 3d orbitals on the Si. The calculations were performed on a G E 635 computer using a modified version of an E H T Program used in chemisorption calculations[21]. The silicon and oxygen orbital exponents are taken as 1.38 and 2.27 respectively. The ionization potentials are
(a)
shown in Fig. 3. Because of computer time limitations we have selected reasonable configurations for various defects rather than extensively varying the configuration to obtain a minimum energy. (a) Perfect lattice We begin with the perfect fl crystobalite lattice [ 19]. Figures 3 (a) and 3 (b) show respectively the electron states associated with the 24 atom representation of the actual Beta crystobalite lattice and the simplified model of that lattice used in the following calculations. The band gap is changed somewhat by the simplification. In these figures, and those that follow, the levels which are due to the oxygen 2s state lie at about --32 eV and are not shown. Their position is relatively insensitive to the existence of the various defects. Notice that the present calculation gives a somewhat lower energy to the simplified
(b)
(c)
(d)
3d(Si)
-5
D
3p(Si) -IO hm
3s(Si) -15
2p(O)
-20 Fig. 3. Energy levels for SiO2 clusters. The longer lines correspond to two and threefold degenerate or near degenerate levels. The occupation of gap levels for electrical neutrality is shown by the arrows. Unless otherwise noted, the clusters are members of a periodic array (a) eight molecule cluster with actual/3 crystobalite structure (b) eight molecule cluster with model structure (c) two molecule cluster (d) isolated two molecule cluster.
1256
A.J.
BENNETT
a n d L. M. R O T H
structure. The total energy of various structures is given in the first column of Table 1; their cohesive e n e r g y - i n the second column. Figure 3(c) shows the energy levels in the small cluster. The main effect of the decrease in cluster size is the removal of energy levels. The calculation gives an energy gap o f - 13 eV as compared with - 10 eV obtained by Phillip experimentally[11]. This result was substantially unchanged upon going from the large cluster to the small cluster. Consideration of the cohesive energy as a function of the lattice constant shows that the maximum binding of 26-85 eV/molecule for the small lattice and 27.45 eV/molecule for the large lattice is obtained for the experimentally known Si-O separation of - 1 . 5 6 A . This value, as is usually the case for E H T calculations, is larger than the experimental value o f 18.8 eV/molecule [22]. The wave functions which characterize the lowest level shown consist of a linear combination of silicon s orbitals and oxygen p orbitals, the latter directed along the bond. The remaining valence levels correspond primarily to oxygen p orbitals. As mentioned above, there is too much ionicity. The calculation gives about 7.5 electrons on each oxygen and 1 on each silicon. The calculation shows very little d contribution to the valence band states. In fact, a calculation without d-states gave virtually the same results for the valence band
energies. The conduction band is however dominated by the d levels, and if they are not included, the lowest conduction band level appears at + 6 eV. Figure 3(d) shows the energy levels of the small cluster which result when periodic boundary conditions are not applied. States localized on Si 1 then lie in the gap. (b) O x y g e n vacancy Calculations on silicon and oxygen vacancies Show that approximately three times as much energy ( - 25 eV) is required to remove the tetrahedrally bonded Si. We may, therefore, restrict our attention to the oxygen vacancy. Figures 4(a), 4(b) and 4(c) show the energy levels of the small representation for the case of an oxygen vacancy (oxygen 1 of Fig. l) with and without distortion of the neighboring silicons. The three spectra correspond to unperturbed, reduced and increased Si-Si separations respectively. Figures 4(d) and 4(e) show the results for the unperturbed and reduced cases as calculated on the larger representation. Comparison of Figs. 4(a) and 4(d), as well as Figs. 4(b) and 4(e), shows that the smaller lattice is an adequate model of the defect. The reduced Si-Si separation was chosen to approximate that of pure silicon. The removal of one oxygen causes two levels to appear in the gap, the upper level being doubly degenerate. Upon displacing the
Table 1.
Total energy Configuration SisO~6 (actual crystal) SisO16
Si204 Si2Oz (dss-sl = 2"40 A ) Si8015 (dsi-si = 2"40/~) Si204H SisOlaH2 SisO3H SisOlsH2 SisO17H2
Cohesive energy
(eV)
(eV)
-- 2600.73 -- 2604.58 -- 649.94 --512.92 -- 2466.57 --658.01 -- 2630"78 -- 523'06 -- 2495"35 -- 2766"28
--215.77 --219.62 --'53 "70 - - 44.84 -- 209.77 --48.17 --218-62 --41"38 --211-35 -- 225"96
Total energy relative to p e r f e c t crystal
T o t a l energy relative to vacancy structure
(eV)
8.86 9.85 5.53 1-00 12.32 8.27 3.51
(7.90) (5.75) (14.69) (13-02) (8.26)3.41
(eV)
3.46 (5.83) -
1.58
(3.17)
- 6 . 3 4 ( - 1.59)
D E F E C T C E N T E R S IN SiOe
(o)
(b)
Ic)
(d)
1257
re)
-5
H H
,.=,-io Z I,IJ
N m
"~
~:~ ILl
-509.65 -512.92 -498.95 -499.90 - 488.24 -486.88
m
-505~5 -2465~5 -2466.57 -497.56 -2455.44 -2454.38 -489.27 -2445.43 -2442.19
Fig. 4. Energy levels for SiO2 clusters containing oxygen vacancies. The longer lines correspond to two and threefold degenerate or near degenerate levels. The occupation of gap levels for electrical neutrality is shown by the arrows. The clusters are members of a periodic array. The total energies (eV) listed beneath the figures are for the (neutral) doubly occupied, singly occupied, and unoccupied centers. (a) two molecule cluster containin.g an oxygen vacancy with silicons undisplaced, ds~-sl = 3.10 A (b) two molecule cluster containing an oxygen vacancy with silicons displaced inward, dsi-si = 2.41 A (c) two molecule cluster containing an oxygen vacancy with silicons displaced outward, dsi-s~ = 3.79 A (d) eight molecule cluster containin~g an oxygen vacancy with silicons undisplaced, dsi-si = 3.10 A (e) eight molecule cluster containing an oxygen vacancy with silicons displaced inward, dsi-si = 2.41 ,~.
neighboring silicon ions toward each other, the levels move down while if the silicons are pulled apart, the levels move up and the upper one merges with the continuum. At the bottom of each column in Fig. 4 is the total energy of each configuration with the lower gap level doubly occupied (the neutral configuration), singly occupied, and unoccupied. If the lower gap level is doubly or singly occupied, it is energetically favorable for the neighboring silicons to come together; while if it is unoccupied, they tend to move apart. The neutral oxygen vacancy with neighboring silicons brought together, for both small
and large clusters, is described in Table 1. The energy of formation, defined here as the difference between the total energy of the particular center plus isolated oxygen atoms and the total energy of the comparable unperturbed lattice is given in the third column of the table. An examination of the eigenfunctions connected with the gap levels shows that the lower one is composed of a bonding combination of s and p orbitals (with z along the bond direction between the two silicons) whereas the upper level corresponds to Px and pu orbitals also in a bonding combination. The energy separation between those levels, as
t258
A . J . B E N N E T T and L. M. R O T H
obtained in the calculation on the large cluster, is 6-6 eV, and there is a dipole moment for the optical transition between them. These results suggest strongly that this pair of levels is to be identified with an u.v. absorption band at ~ 6 eV observed in neutron irradiated quartz and amorphous silica by Nelson and Weeks [7]. This band was denoted as E~, and was assigned by the authors to the singly occupied oxygen vacancy. They also describe a possible precurser, E~, which would be the neutral oxygen vacancy. A similar ultraviolet band is observed in certain synthetic samples before oxidizing. Weeks and Nelson[8] and Silsbee[9] have observed electron spin resonance signals which are correlated with the E'1 absorption. In addition Nishi[10] has observed electron spin resonance in amorphous SiO2 films. Silsbee [9] has examined the anisotropy of the gfactor and hyperfine interaction in detail for the case of quartz, and finds three principle pairs of weak satellites, with maximum hyperfine splitting of 400, 9, and 8 G with intensities of - 289per cent of the main line. These splittings are due to interaction with 29Si nuclei, whose abundance is 4-7 per cent. The magnitude of the largest splitting and its intensity lead Silsbee to argue that the electron involved must belong to a single silicon, and interact weakly with two other silicon atoms. Since the g and hyperfine tensors bear no relationship to Si-O or Si-Si directions this model is still far from certain. Our result indicates that an electron trapped on an oxygen vacancy would be shared between two neighboring silicons which are pulled together. It is possible that this could give a large enough hyperfine interaction since an electron shared between two silicons in irradiated silicon, the A center[23], has a maximum hyperfine splitting of 153 G and, in our case, the silicons are closer. The intensity predicted by the two silicon model is not, however, correct. Clearly further work remains before this center is understood. At a Si-SiO2 junction, the 1.2 eV silicon
energy gap is in the middle of the large SiO2 gap. Levels in the SiO2 below the middle of the gap would be filled and those a b o v e empty. The oxygen vacancy with silicons moved together would probably be uncharged and not affect the electronic properties of the interface. The doubly charged center, in which the silicons tend to separate, has its emptied level above the Si band gap and would thus remain charged. It could serve, therefore, as a positively charged 'slow' surface state. We should reiterate, however, that there are a number of relevant defects which we cannot consider in the context of our present m o d e l such as the non bridging oxygen and defects on the interface. (c) Hydrogen centers We now consider the hydrogen centers shown in Figs. 1(c-g). The crystobalite structure is quite open and calculation indicates that an interstitial hydrogen is relatively uninfluenced by the lattice. It is, in fact, known that atomic and molecular hydrogen freely diffuse through fused silica [24]. In the following cases, some primitive attempts have been made to minimize the configuration energy by varying the atom positions. Again the total and cohesive energies of the various centers are summarized in the first and second columns of the Table. The third column gives the total energy relative to that of suitable reference structures consisting of the appropriate perfect large or small SiOz cluster and isolated hydrogen molecules and atoms. These numbers indicate the thermodynamic favorability of the center, but its formation rate depends on potential surfaces not calculated here. Figure 5(a) shows the energy levels resulting from the addition of a single hydrogen .to a normal bond (Fig. 1(c)). The wavefunction of the highest occupied level is concentrated on the O H group. The third column of the Table shows that this configuration has a rather high energy compared to that of the unperturbed lattice plus hydrogen interstitial. Here the
DEFECT
CENTERS
(b)
(c)
(o)
-E--
IN
SiO2
1259
(d)
(e)
4__
-f-
q,l
ILl Z I.,I.I
H -15
N c-t
m m
m
m
-2~ Fig, 5. Energy levels for SiO2 clusters containing hydrogen. The longer lines correspond to two and threefold degenerate or near degenerate levels. T h e occupation of gap levels for electrical neutrality is shown by the arrows. The clusters are members of a periodic army (a) two molecule cluster containing one hydrogen at a normal bond (Fig. l(c)) (b) eight molecule cluster containing two hydrogens at a normal bond (Fig. 1(d)) (c) two molecule cluster containing one hydrogen at an oxygen vacancy (Fig. 1(e)) (d) eight molecule cluster containing two hydrogcns at an oxygen vacancy (Fig. l(f)) (e) eight molecule cluster containing two hydrogens and an oxygen at a normal bond (Fig. 1(g)).
number in parenthesis gives the formation energy/center when two centers are formed frommolecular hydrogen (binding energy 4.75 eV). The total energy of the singly ionized version of this center (653-51 eV) is lower than that of the interstitial proton. The energy levels resulting from the addition of two hydrogen atoms to the normal bond (Fig. l(d)) are shown in Fig. 5(b). The wave function of the highest occupied gap level is primarily localized on the hydrogen member of the HSi unit, though it spreads onto the three oxygens nearest to the hydrogen. The doubly occupied (neutral) center is comparable in energy to the normal lattice plus two hydrogen interstitials (column 3). The energy for formation from a hydrogen molecule is shown in parenthesis. As mentioned in the Introduction, Bell,
JPCS VoL 32 No. 6
K
Hetherington and Jack have heated samples of amorphous silica in hydrogen and found that O H groups are formed. These groups are the origin of a strong 2.75/z i.r. absorption band which is associated with the stretching of their bond. One or the other of the above centers was postulated as the site of the hydroxyl ion. Our calculation indicates the two hydrogen center is more favorable than two one hydrogen centers. We now consider the addition of one and two hydrogens to the oxygen vacancy. A single hydrogen equidistant from the two silicons (Fig. l(e))'results in the energy diagram shown as Fig. 5(c). Assymetrichydrogen positions were energetically less favorable. The highest singly occupied level consists of silicon p orbitals. The level most localized on the hydrogen is more strongly bound. The addi-
1260
A . J . B E N N E T T and L. M. R O T H
tion of the atom apparently splits the vacancy levels further apart (cf. Fig. 4). The total energy of the center plus interstitial oxygen compared to that of the perfect lattice plus interstitial hydrogen is given in the third column of the Table. The formation energy center for two centers formed from a hydrogen molecule is given in parenthesis. Analogous results for the case when an oxygen vacancy is initially present are given in the fourth column. The ionized defect is energetically more advantageous than the ionized interstitial plus vacancy. The energy spectrum characteristic of two hydrogens bound to the oxygen vacancy (Fig. l(f) is shown in Fig. 5(d). As in the case of two hydrogen atoms at the normal site, the configuration is a more favorable one. That is, a comparison of the total energy of the c e n t e r plus oxygen interstitial with that of the relevant perfect crystal plus hydrogen interstitials shows that the formation of a center conraining two hydrogens requires less energy than the formation of two centers containing one hydrogen. The energy for formation from a molecule is again given in parenthesis. Similar results for the case when an oxygen vacancy is initially present are given in the fourth column. The formation is then quite favorable. The final configuration to be considered is the center containing two hydrogens and two oxygens (Fig. l(g)) whose energy levels are shown in Fig. 5(e). This could occur if interstitial oxygen due to vacancy formation and hydrogen interstitials are present, or upon the addition of water to the perfect lattice. Experiment indicates that O H stretching modes are also observed with the latter addition [5]. The occupied gap level just above the valence band edge consists mainly of prr orbitals on the oxygens. The total energy of this center plus a vacancy center (taken as the source of the oxygen) is compared in the third column with the energy of two perfect lattices plus the energy of two interstitial hydrogens. The comparison with an interstitial hydrogen molecule present is given in parenthesis. Similar
comparisons assuming the presence of ambient O are given in the fourth column. The energy of formation from a water molecule is the underlined number in the third column. It is interesting to consider the combination of a S i O H - S i O H center and an oxygen vacancy. The Table shows that the total energy of these two configurations is 3-5 eV less favorable than that of two normal bonds and hydrogen interstitials. Hetherington and Jack observed that certain samples exhibiting both O H stretching modes and u.v. modes, associated presumably with optical absorption at oxygen vacancies, when heated lost both spectra. The combination of the two centers appears to be a likely cause of this loss. Glasses which exhibit u.v. but not i.r. absorption require an oxygen atmosphere to remove the u.v. absorption. 4. SUMMARY
We have found that the E H T gives a reasonable semiquantitative description of the electronic states of ordered SiO2 and of defects in that material involving the oxygen vacancy and the addition of hydrogen. It appears that the oxygen vacancy is responsible for certain ultraviolet absorption bands and electron spin resonance signals, although some evidence suggests that we should examine other possible defects to explain the resonance examined in detail by Silsbee. The doubly ionized oxygen vacancy may account for some of the slow states observed in Mosfet systems. Our examination of defects involving hydrogen enables us to compare the energy for formation, of various configurations as given in Table 1. The results in the third and fourth columns of the Table idicate that the most likely defect formed by the addition of either atomic or molecular hydrogen to the perfect glass is the S i O H - S i H center. If, however, oxygen vacancies with accompanying interstitial oxygen or ambient atomic oxygen are present, the S i O H - S i O H denter is then most favorable. A comparison of our results with single
D E F E C T C E N T E R S IN SiO~
bond argumeats based on standard bond energies[25] shows that the energy cost we find for center production is in general larger than that pledicted by single bonds arguments. This is often the case withEHT calculations, which we emphasize aresemiquantitative. Our conclusions as to which centers are lowest in energy are in agreement with single bond arguments. E H T calculations include interaction between various bonds in the system; in the cases studied these interactions are apparently not of great importance. We emphasize that computer time restrictions have limited our efforts to minimize the various configuration energies by considering many possible atomic arrangements. The E H T also yields information about the electronic states associated with various defects, as we have reported in the text and figures. In future work we shall examine less ordered structures by distorting the present mode in various ways. We hope to examine defects which are not characteristic of the ordered structure, and to obtain some information on the electronic density of states in an amorphous material. REFERENCES 1. G R A Y P. V.,Proc. IEEE57, 1543 (1969). 2. R E V E S Z A. G. IEEE Trans. Electron Dev. ED-12, 97 "(1965). 3. KOOI E., Phillips Res. Rept. 21,477 (1966).
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4. D E A L B. Z . , S K L A R M . , G R O V E A . S. andSNOW E. H., J. electrochem. Soc. 114, 266 (1967). 5. BELL T., H E T H E R I N G T O N G. and J A C K K. H. Phys. Chem. Glasses 3, 141 (1962). 6. L E L L E., K R E I D L N. J. and H E N S L E R J. R. in Progress in Ceramic Science (Edited by J. Burke), Vol. 4, Pergamon Press, New York (1966). 7. N E L S O N C. M. and W E E K S R. A., J. Am. c e r a m . Soc. 43,396 (1960). 8. W E E K S R. A. and N E L S O N C. M., J. Am. ceram. Soc. 43,399 (1960). 9. SILSBEE R. H.,J. appl. Phys. 32, 1459 (1961). 10. N I S H I Y.,Japan J. appl. Phys. 5, 333 (1966). 11. P H I L L I P P H. R., SolidState Commun. 4, 73 (1966). 12. K O H N W., in Solid State Physics (Edited by F. Seitz and D. Turnbull) Vol. 5, p. 257, Academic Press, New York (1957). 13. H O F F M A N R. and L I P S C O M B W. N., J. Chem. Phys. 36, 2179 (1962). 14. H O F F M A N R.,J. Chem. Phys. 39, 1397 (1963). 15. H O F F M A N R.,J. Chem. Phys. 40, 2474 (1964). 16. M E S S M E R R. P. and W A T K I N S G. Phys. Rev. Lett. 26,656 (1970). 17. P A U L I N G L.,J. Phys. Chem. 56, 361 (1952). 18. BELL R. J. and D E A N P., Nature 212, 1354 (1966). 19. W Y C K O F F R. W. G., Cl~stal Structures, Vol 1, I nterscience, New York (1963). 20. This is readily proven by considering the sum, for any vector c~; ~,~ct~ = f ( ~ ~,l,k))* ( ~ c~lo-) ) d r ~>0 ( M E S S M E R R. P., private communication). 21. B E N N E T T A. J, M c C A R R O L B., and M E S S M E R R. Surface Sci. to be published. 22. Handbook of Chemistry and Physics 49th ed. pp. D-38, F-158, Chem. Rubber Co., Cleveland (1968). 23. W A T K I N S G. D. and C O R B E T T J. W., Phys. Rev. 121, 1001 (1961). 24. N O R T O N F. J., J. Am. ceram. Soc. 36, 90 (1953). 25. P A U L I N G L., The Nature of the Chemical Bond, Cornell University Press, Ithaca (1960).
Pergamon Press 1971. Vol. 32, pp. 1251-1261.
Printed in Great Britain.
E L E C T R O N I C S T R U C T U R E OF D E F E C T C E N T E R S
IN SiO2 A. J. BENNETT and L. M. ROTH General Electric Research and Development Center, P.O. Box 8, Schenectady, N.Y. 12301, U.S.A.
(Received 31 .4 ugust 1970) Abstract--Molecular orbital calculations based on the Extended Huckel Theory have been performed for clusters of two and eight SiO2 molecules. We report here the results for an ordered structure in which periodic boundary conditions are imposed to saturate the peripheral bonds which otherwise cause extraneous localized states to appear. We find an energy gap of - 13 eV as compared to the experimental value of - I 1 eV. The removal of an oxygen results in several levels in the energy gap which move upon displacement of the two neighboring silicons. The levels appear to account for the observed ultraviolet optical absorption band and are also a possible origin for the positively charged slow surface states present near a SiO2-Si interface. The addition of either atomic or molecular hydrogen to a perfect crystal is predicted to yield preferentially a double hydrogen-single oxygen center. If ambient oxygen atoms are present, a two hydroxl group center is then most favorable. 1. INTRODUCTION
THE PROPERTIES of various defects in SiO2 are of great technological interest. Vacancies and interstitials have been invoked to explain both 'fast' and 'slow' surface states at the semiconductor-oxide interface of a metaloxide-semiconductor field effect transistor (Mosfet)[1-3]. The 'fast' states which affect the voltage, frequency, and temperature dependence of the capacitance characteristics can i.nteract rapidly with the Si space charge and are known to be well localized at the interface. These states lie within the Si band gap. We consider here the 'slow' states which are responsible for translation of the C(V) curves along the V axis, and are thought to be localized at defect centers in the oxide (within 200 ,~ of the silicon). These defects are induced by the junction fabrication. The growth of the SiO2-Si interface may result in oxygen vacancies and accompanying trivalent Si near. the interface. This structure is indicated schematically in Fig. 1(b). If, as is often the case, hydrogen is present during fabrication, a variety of other defects may exist. Hydrogen additions to the perfect crystal are shown in Figs. l(c) and l(d); additions to the vacancy structure in Fig. l(e) and
l(f). A structure which may result if oxygen interstitials and/or water is also present, is shown in Fig. 1(g). Experiments by Bell, Hetherington and Jack [5] have shown that exposure of vitreous silica to hydrogen or to water vapor at high temperatures leads to the formation of O H groups in the SiO2 system. These are detected by a strong i.r. absorption at 2.75/z due to the stretching of the O - H bond. Defect states also result when pure amorphous or crystalline SiO2 is irradiated with neutrons, electrons or gamma rays [6]. In particular optical absorption [7] and Epr studies [8-10] have indicated the existence of a paramagnetic state associated with an oxygen vacancy. There have been no even semiquantitative calculations of the electronic properties of the various postulated defects states. We attempt such a treatment in this paper. The defect states may lie well within the SiO2 band gap ( - 10 eV) [11] and as a result can not be considered using the effective mass approximation[12]. We apply a simple molecular orbital scheme, the Extended Huckel Theory (EHT) developed by Hoffman and his collaborators [ 13-15], to a finite array of Si
1251
1252
A . J . B E N N E T T and L. M. R O T H
d ~,.~l e~ (o}
d , o.9,
(hi
s,
s,
(e}
o'
"
(d}
o
s,
..
Z
s,
d
~
H (e)
(f} H
Si
0
(g} Fig. 1. Atomic configurations in/3 crystobalite in the vicinity of (a) a normal oxygen site (b) an oxygen vacancy and (c-g) various centers containing hydrogen. The plane of the paper is the (I10) plane in which we have assumed the defect atoms to lie. The x and y axes are (111) and (112) respectively. The arrows in (b) show the extent of the Si displacements.
and O atoms chosen to represent the insulator. As is well known, the E H T yields quite useful semi-quantitative results when applied to small molecules. Interaction effects, not considered in single bond arguments, are naturally included in such calculations. Recently Messmer and Watkins [ 16] have successfully used the method in a study of the nitrogen defect in diamond. Although probably best
suited to totally covalent systems, the method will be applied here to SiO2 where the ionicity is not extremely large [ 17]. The SiO2 films present in Mosfets are, in general, amorphous. Locally, however, there exists strong short range order. X-ray and neutron scattering data indicate that even in SiOa glasses, the basic SiO4 tetrahedra are maintained[18]. Since the defect states are
D E F E C T C E N T E R S IN SiO2
probably well localized on an atomic scale, we use a crystalline model of the insulator, in particular a simplified cubic /3-crystobalite structure [19]. /3 crystobalite has a density similar to that of the amorphous phase and hence is a reasonable choice for a crystalline model. The structure can be visualized by noting that the silicons form a diamond lattice with oxygens midway between adjacent silicons. In the actual crystal, the oxygens are somewhat displaced from these positions. One disadvantage of the use of an ordered structure is that we are unable to consider certain network defects such as non-bridging oxygens. The edge atoms of our finite representation of the SiOz system have unsaturated bonds which are responsible for perimeter states, localized at the edge of the representation. In this study periodic boundary conditions will be applied to eliminate these states. In the next section we briefly describe the E H T , the representations used, and the application of periodic boundary conditions. Section 3 contains our results and comparison with experiment. We summarize in a final section. 2. FORMALISM
The Extended Huckel Theory has been discussed extensively in various publications [13-15]. It is a molecular orbital method in which the wave functions 9~ are taken as linear combinations of all the valence orbitals, Ih > (Slater functions) centered on the various atoms of the system. =
c,
lx > .
(2.1)
The SchriSdinger equation written in that non-orthogonal representation is given by IH-- ESI = 0
(2.2)
where Hxx = -- I~
(2.3)
HA,,- = 8 9
(2.4)
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Sx~ = (hltr)
(2.5)
Ix is the valence state ionization potential, K is a constant equal to 1.75, and (h[o-) is an overlap integral between atomic orbitals. Cohesive energies are calculated as the difference between the sum of the molecular one-electron energies and the sum of the one electron energies of the separated constituent atoms. The neglect of explicit electron-electron and core-core interaction is a distinct weakness of the approach. The results are, therefore, only semiquantitative. We find, for example, too much ionicity in the system which is a consequence of the lack of any provision for self consistency in the method. T w o representations of the SiO2 structure, consisting of six and twenty four atoms respectively, were used in our calculations. The smaller representation is shown in Fig. 2 by the solid circles. Extraneous perimeter states are eliminated by including Si-Si and Si-O bonds between an atom in the cluster and its nearest and nextnearest neighbors in translated versions of the representation. The cluster thus serves as one unit cell of a periodic system. The dotted circles in Fig. 2 represent atoms in the translated versions of the small cluster. The silicons la, lb, and lc, which are not in the cluster, are considered to be equivalent to silicon 1. To the overlap matrix element between silicon 1 and oxygen 2, we add the overlap between silicon la and oxygen 2. Similarly, we add overlaps of silicon lb with oxygen 3 and silicon l c with oxygen 4, to the silicon 1 - oxygen 3 and silicon 1 - oxygen 4 overlap integrals respectively. In addition, we add to the overlap matrix element between silicon 1 and silicon 2, three terms corresponding to overlaps between silicons 1a, 1b and 1c with silicon 2. We are able to use the small cluster due to the short range nature of the overlaps in SiO2. For some defects containing two hydrogens, however, it was found necessary to go to a larger cluster of 8 SiO~ molecules, which
1254
A . J . B E N N E T T and L. M. R O T H / ~ --'%
I [C t
/
/
./
.//
_i
.
Ibm,.4
.
.
IB----'l
.
/ (~,k~
f-"~/
/ "~
~'~"
Ib I
II ~.;~'~.~.~ ~1~ v I.
.
.
.
L/"
.
.
.
j_ .
I i "
----
.
.
.
.
--t
I0
1
// /"
0
Fig. 2. A portion of our idealized beta crystobalite lattice showing the small cluster which consists of silicons 1 and 2, and oxygens 1, 2, 3, 4. Silicons la, lb, and lc are outside the cluster and are equivalent to silicon I. The dashed cube shown is ] of the cubic unit cell for the diamond lattice upon which the silicons occur and is drawn to show the tetrahedral arrangement of silicons. (It is not a unit cell-the neighboring cube has no Si in the center.)
then serves as the unit cell. Similar periodic boundary conditions were applied to this larger model. The above procedure for applying periodic boundary conditions is intuitively the obvious one to use. It is interesting to justify it formally by requiring the wave function for the system repeated periodically, to be periodic with unit cell given by our cluster. This is accomplished by adopting as basis functions the k --- 0 Bloch states. IX)' = ~ 1 xi Ih~)"
(2.6)
Here i labels the unit cell in which Ihi) is to be centered, and N is the number of cells. The overlap integral is then (~lh) ' = Z (/-*01h,) 9 i
(2.7)
T h e rule is therefore to add to a given overlap S~oxo within the cluster, all overlaps Sm~, between ~0 in the cluster and the h~ outside the cluster which are equivalent to h0. We have followed this rule in the procedure described above. However, the overlaps beyond a certain distance are quite small ( < 10 per cent of those included) and are ignored. In practice a difficulty could arise which should be noted. The matrix S associated with a finite molecule is readily shown to be positive definite [20]. If arbitrary additional overlaps are added to that matrix this property can be destroyed creating technical difficulties. The positive definite property holds also for a periodic array of molecules with all overlaps included. In practice, therefore, if the S matrix ceases to be positive definite, we must include more overlaps. Out system is quite tightly
1255
D E F E C T C E N T E R S IN SiOz
b o u n d and we have had relatively little difficulty in this regard. Our method is equivalent to a tight binding energy band calculation with nonorthogonality taken into account. The wave vector is, however, restricted to k = 0 for the Brillouin zone of the unit cell corresponding to the cluster. The energy values correspond to those of a selected set of k values for the Brillouin zone of the conventional primitive unit cell of the crystal. 3. RESULTS
The basis set consists of 2s and 2p orbitals on the oxygen and 3s, 3p, and 3d orbitals on the Si. The calculations were performed on a G E 635 computer using a modified version of an E H T Program used in chemisorption calculations[21]. The silicon and oxygen orbital exponents are taken as 1.38 and 2.27 respectively. The ionization potentials are
(a)
shown in Fig. 3. Because of computer time limitations we have selected reasonable configurations for various defects rather than extensively varying the configuration to obtain a minimum energy. (a) Perfect lattice We begin with the perfect fl crystobalite lattice [ 19]. Figures 3 (a) and 3 (b) show respectively the electron states associated with the 24 atom representation of the actual Beta crystobalite lattice and the simplified model of that lattice used in the following calculations. The band gap is changed somewhat by the simplification. In these figures, and those that follow, the levels which are due to the oxygen 2s state lie at about --32 eV and are not shown. Their position is relatively insensitive to the existence of the various defects. Notice that the present calculation gives a somewhat lower energy to the simplified
(b)
(c)
(d)
3d(Si)
-5
D
3p(Si) -IO hm
3s(Si) -15
2p(O)
-20 Fig. 3. Energy levels for SiO2 clusters. The longer lines correspond to two and threefold degenerate or near degenerate levels. The occupation of gap levels for electrical neutrality is shown by the arrows. Unless otherwise noted, the clusters are members of a periodic array (a) eight molecule cluster with actual/3 crystobalite structure (b) eight molecule cluster with model structure (c) two molecule cluster (d) isolated two molecule cluster.
1256
A.J.
BENNETT
a n d L. M. R O T H
structure. The total energy of various structures is given in the first column of Table 1; their cohesive e n e r g y - i n the second column. Figure 3(c) shows the energy levels in the small cluster. The main effect of the decrease in cluster size is the removal of energy levels. The calculation gives an energy gap o f - 13 eV as compared with - 10 eV obtained by Phillip experimentally[11]. This result was substantially unchanged upon going from the large cluster to the small cluster. Consideration of the cohesive energy as a function of the lattice constant shows that the maximum binding of 26-85 eV/molecule for the small lattice and 27.45 eV/molecule for the large lattice is obtained for the experimentally known Si-O separation of - 1 . 5 6 A . This value, as is usually the case for E H T calculations, is larger than the experimental value o f 18.8 eV/molecule [22]. The wave functions which characterize the lowest level shown consist of a linear combination of silicon s orbitals and oxygen p orbitals, the latter directed along the bond. The remaining valence levels correspond primarily to oxygen p orbitals. As mentioned above, there is too much ionicity. The calculation gives about 7.5 electrons on each oxygen and 1 on each silicon. The calculation shows very little d contribution to the valence band states. In fact, a calculation without d-states gave virtually the same results for the valence band
energies. The conduction band is however dominated by the d levels, and if they are not included, the lowest conduction band level appears at + 6 eV. Figure 3(d) shows the energy levels of the small cluster which result when periodic boundary conditions are not applied. States localized on Si 1 then lie in the gap. (b) O x y g e n vacancy Calculations on silicon and oxygen vacancies Show that approximately three times as much energy ( - 25 eV) is required to remove the tetrahedrally bonded Si. We may, therefore, restrict our attention to the oxygen vacancy. Figures 4(a), 4(b) and 4(c) show the energy levels of the small representation for the case of an oxygen vacancy (oxygen 1 of Fig. l) with and without distortion of the neighboring silicons. The three spectra correspond to unperturbed, reduced and increased Si-Si separations respectively. Figures 4(d) and 4(e) show the results for the unperturbed and reduced cases as calculated on the larger representation. Comparison of Figs. 4(a) and 4(d), as well as Figs. 4(b) and 4(e), shows that the smaller lattice is an adequate model of the defect. The reduced Si-Si separation was chosen to approximate that of pure silicon. The removal of one oxygen causes two levels to appear in the gap, the upper level being doubly degenerate. Upon displacing the
Table 1.
Total energy Configuration SisO~6 (actual crystal) SisO16
Si204 Si2Oz (dss-sl = 2"40 A ) Si8015 (dsi-si = 2"40/~) Si204H SisOlaH2 SisO3H SisOlsH2 SisO17H2
Cohesive energy
(eV)
(eV)
-- 2600.73 -- 2604.58 -- 649.94 --512.92 -- 2466.57 --658.01 -- 2630"78 -- 523'06 -- 2495"35 -- 2766"28
--215.77 --219.62 --'53 "70 - - 44.84 -- 209.77 --48.17 --218-62 --41"38 --211-35 -- 225"96
Total energy relative to p e r f e c t crystal
T o t a l energy relative to vacancy structure
(eV)
8.86 9.85 5.53 1-00 12.32 8.27 3.51
(7.90) (5.75) (14.69) (13-02) (8.26)3.41
(eV)
3.46 (5.83) -
1.58
(3.17)
- 6 . 3 4 ( - 1.59)
D E F E C T C E N T E R S IN SiOe
(o)
(b)
Ic)
(d)
1257
re)
-5
H H
,.=,-io Z I,IJ
N m
"~
~:~ ILl
-509.65 -512.92 -498.95 -499.90 - 488.24 -486.88
m
-505~5 -2465~5 -2466.57 -497.56 -2455.44 -2454.38 -489.27 -2445.43 -2442.19
Fig. 4. Energy levels for SiO2 clusters containing oxygen vacancies. The longer lines correspond to two and threefold degenerate or near degenerate levels. The occupation of gap levels for electrical neutrality is shown by the arrows. The clusters are members of a periodic array. The total energies (eV) listed beneath the figures are for the (neutral) doubly occupied, singly occupied, and unoccupied centers. (a) two molecule cluster containin.g an oxygen vacancy with silicons undisplaced, ds~-sl = 3.10 A (b) two molecule cluster containing an oxygen vacancy with silicons displaced inward, dsi-si = 2.41 A (c) two molecule cluster containing an oxygen vacancy with silicons displaced outward, dsi-s~ = 3.79 A (d) eight molecule cluster containin~g an oxygen vacancy with silicons undisplaced, dsi-si = 3.10 A (e) eight molecule cluster containing an oxygen vacancy with silicons displaced inward, dsi-si = 2.41 ,~.
neighboring silicon ions toward each other, the levels move down while if the silicons are pulled apart, the levels move up and the upper one merges with the continuum. At the bottom of each column in Fig. 4 is the total energy of each configuration with the lower gap level doubly occupied (the neutral configuration), singly occupied, and unoccupied. If the lower gap level is doubly or singly occupied, it is energetically favorable for the neighboring silicons to come together; while if it is unoccupied, they tend to move apart. The neutral oxygen vacancy with neighboring silicons brought together, for both small
and large clusters, is described in Table 1. The energy of formation, defined here as the difference between the total energy of the particular center plus isolated oxygen atoms and the total energy of the comparable unperturbed lattice is given in the third column of the table. An examination of the eigenfunctions connected with the gap levels shows that the lower one is composed of a bonding combination of s and p orbitals (with z along the bond direction between the two silicons) whereas the upper level corresponds to Px and pu orbitals also in a bonding combination. The energy separation between those levels, as
t258
A . J . B E N N E T T and L. M. R O T H
obtained in the calculation on the large cluster, is 6-6 eV, and there is a dipole moment for the optical transition between them. These results suggest strongly that this pair of levels is to be identified with an u.v. absorption band at ~ 6 eV observed in neutron irradiated quartz and amorphous silica by Nelson and Weeks [7]. This band was denoted as E~, and was assigned by the authors to the singly occupied oxygen vacancy. They also describe a possible precurser, E~, which would be the neutral oxygen vacancy. A similar ultraviolet band is observed in certain synthetic samples before oxidizing. Weeks and Nelson[8] and Silsbee[9] have observed electron spin resonance signals which are correlated with the E'1 absorption. In addition Nishi[10] has observed electron spin resonance in amorphous SiO2 films. Silsbee [9] has examined the anisotropy of the gfactor and hyperfine interaction in detail for the case of quartz, and finds three principle pairs of weak satellites, with maximum hyperfine splitting of 400, 9, and 8 G with intensities of - 289per cent of the main line. These splittings are due to interaction with 29Si nuclei, whose abundance is 4-7 per cent. The magnitude of the largest splitting and its intensity lead Silsbee to argue that the electron involved must belong to a single silicon, and interact weakly with two other silicon atoms. Since the g and hyperfine tensors bear no relationship to Si-O or Si-Si directions this model is still far from certain. Our result indicates that an electron trapped on an oxygen vacancy would be shared between two neighboring silicons which are pulled together. It is possible that this could give a large enough hyperfine interaction since an electron shared between two silicons in irradiated silicon, the A center[23], has a maximum hyperfine splitting of 153 G and, in our case, the silicons are closer. The intensity predicted by the two silicon model is not, however, correct. Clearly further work remains before this center is understood. At a Si-SiO2 junction, the 1.2 eV silicon
energy gap is in the middle of the large SiO2 gap. Levels in the SiO2 below the middle of the gap would be filled and those a b o v e empty. The oxygen vacancy with silicons moved together would probably be uncharged and not affect the electronic properties of the interface. The doubly charged center, in which the silicons tend to separate, has its emptied level above the Si band gap and would thus remain charged. It could serve, therefore, as a positively charged 'slow' surface state. We should reiterate, however, that there are a number of relevant defects which we cannot consider in the context of our present m o d e l such as the non bridging oxygen and defects on the interface. (c) Hydrogen centers We now consider the hydrogen centers shown in Figs. 1(c-g). The crystobalite structure is quite open and calculation indicates that an interstitial hydrogen is relatively uninfluenced by the lattice. It is, in fact, known that atomic and molecular hydrogen freely diffuse through fused silica [24]. In the following cases, some primitive attempts have been made to minimize the configuration energy by varying the atom positions. Again the total and cohesive energies of the various centers are summarized in the first and second columns of the Table. The third column gives the total energy relative to that of suitable reference structures consisting of the appropriate perfect large or small SiOz cluster and isolated hydrogen molecules and atoms. These numbers indicate the thermodynamic favorability of the center, but its formation rate depends on potential surfaces not calculated here. Figure 5(a) shows the energy levels resulting from the addition of a single hydrogen .to a normal bond (Fig. 1(c)). The wavefunction of the highest occupied level is concentrated on the O H group. The third column of the Table shows that this configuration has a rather high energy compared to that of the unperturbed lattice plus hydrogen interstitial. Here the
DEFECT
CENTERS
(b)
(c)
(o)
-E--
IN
SiO2
1259
(d)
(e)
4__
-f-
q,l
ILl Z I.,I.I
H -15
N c-t
m m
m
m
-2~ Fig, 5. Energy levels for SiO2 clusters containing hydrogen. The longer lines correspond to two and threefold degenerate or near degenerate levels. T h e occupation of gap levels for electrical neutrality is shown by the arrows. The clusters are members of a periodic army (a) two molecule cluster containing one hydrogen at a normal bond (Fig. l(c)) (b) eight molecule cluster containing two hydrogens at a normal bond (Fig. 1(d)) (c) two molecule cluster containing one hydrogen at an oxygen vacancy (Fig. 1(e)) (d) eight molecule cluster containing two hydrogcns at an oxygen vacancy (Fig. l(f)) (e) eight molecule cluster containing two hydrogens and an oxygen at a normal bond (Fig. 1(g)).
number in parenthesis gives the formation energy/center when two centers are formed frommolecular hydrogen (binding energy 4.75 eV). The total energy of the singly ionized version of this center (653-51 eV) is lower than that of the interstitial proton. The energy levels resulting from the addition of two hydrogen atoms to the normal bond (Fig. l(d)) are shown in Fig. 5(b). The wave function of the highest occupied gap level is primarily localized on the hydrogen member of the HSi unit, though it spreads onto the three oxygens nearest to the hydrogen. The doubly occupied (neutral) center is comparable in energy to the normal lattice plus two hydrogen interstitials (column 3). The energy for formation from a hydrogen molecule is shown in parenthesis. As mentioned in the Introduction, Bell,
JPCS VoL 32 No. 6
K
Hetherington and Jack have heated samples of amorphous silica in hydrogen and found that O H groups are formed. These groups are the origin of a strong 2.75/z i.r. absorption band which is associated with the stretching of their bond. One or the other of the above centers was postulated as the site of the hydroxyl ion. Our calculation indicates the two hydrogen center is more favorable than two one hydrogen centers. We now consider the addition of one and two hydrogens to the oxygen vacancy. A single hydrogen equidistant from the two silicons (Fig. l(e))'results in the energy diagram shown as Fig. 5(c). Assymetrichydrogen positions were energetically less favorable. The highest singly occupied level consists of silicon p orbitals. The level most localized on the hydrogen is more strongly bound. The addi-
1260
A . J . B E N N E T T and L. M. R O T H
tion of the atom apparently splits the vacancy levels further apart (cf. Fig. 4). The total energy of the center plus interstitial oxygen compared to that of the perfect lattice plus interstitial hydrogen is given in the third column of the Table. The formation energy center for two centers formed from a hydrogen molecule is given in parenthesis. Analogous results for the case when an oxygen vacancy is initially present are given in the fourth column. The ionized defect is energetically more advantageous than the ionized interstitial plus vacancy. The energy spectrum characteristic of two hydrogens bound to the oxygen vacancy (Fig. l(f) is shown in Fig. 5(d). As in the case of two hydrogen atoms at the normal site, the configuration is a more favorable one. That is, a comparison of the total energy of the c e n t e r plus oxygen interstitial with that of the relevant perfect crystal plus hydrogen interstitials shows that the formation of a center conraining two hydrogens requires less energy than the formation of two centers containing one hydrogen. The energy for formation from a molecule is again given in parenthesis. Similar results for the case when an oxygen vacancy is initially present are given in the fourth column. The formation is then quite favorable. The final configuration to be considered is the center containing two hydrogens and two oxygens (Fig. l(g)) whose energy levels are shown in Fig. 5(e). This could occur if interstitial oxygen due to vacancy formation and hydrogen interstitials are present, or upon the addition of water to the perfect lattice. Experiment indicates that O H stretching modes are also observed with the latter addition [5]. The occupied gap level just above the valence band edge consists mainly of prr orbitals on the oxygens. The total energy of this center plus a vacancy center (taken as the source of the oxygen) is compared in the third column with the energy of two perfect lattices plus the energy of two interstitial hydrogens. The comparison with an interstitial hydrogen molecule present is given in parenthesis. Similar
comparisons assuming the presence of ambient O are given in the fourth column. The energy of formation from a water molecule is the underlined number in the third column. It is interesting to consider the combination of a S i O H - S i O H center and an oxygen vacancy. The Table shows that the total energy of these two configurations is 3-5 eV less favorable than that of two normal bonds and hydrogen interstitials. Hetherington and Jack observed that certain samples exhibiting both O H stretching modes and u.v. modes, associated presumably with optical absorption at oxygen vacancies, when heated lost both spectra. The combination of the two centers appears to be a likely cause of this loss. Glasses which exhibit u.v. but not i.r. absorption require an oxygen atmosphere to remove the u.v. absorption. 4. SUMMARY
We have found that the E H T gives a reasonable semiquantitative description of the electronic states of ordered SiO2 and of defects in that material involving the oxygen vacancy and the addition of hydrogen. It appears that the oxygen vacancy is responsible for certain ultraviolet absorption bands and electron spin resonance signals, although some evidence suggests that we should examine other possible defects to explain the resonance examined in detail by Silsbee. The doubly ionized oxygen vacancy may account for some of the slow states observed in Mosfet systems. Our examination of defects involving hydrogen enables us to compare the energy for formation, of various configurations as given in Table 1. The results in the third and fourth columns of the Table idicate that the most likely defect formed by the addition of either atomic or molecular hydrogen to the perfect glass is the S i O H - S i H center. If, however, oxygen vacancies with accompanying interstitial oxygen or ambient atomic oxygen are present, the S i O H - S i O H denter is then most favorable. A comparison of our results with single
D E F E C T C E N T E R S IN SiO~
bond argumeats based on standard bond energies[25] shows that the energy cost we find for center production is in general larger than that pledicted by single bonds arguments. This is often the case withEHT calculations, which we emphasize aresemiquantitative. Our conclusions as to which centers are lowest in energy are in agreement with single bond arguments. E H T calculations include interaction between various bonds in the system; in the cases studied these interactions are apparently not of great importance. We emphasize that computer time restrictions have limited our efforts to minimize the various configuration energies by considering many possible atomic arrangements. The E H T also yields information about the electronic states associated with various defects, as we have reported in the text and figures. In future work we shall examine less ordered structures by distorting the present mode in various ways. We hope to examine defects which are not characteristic of the ordered structure, and to obtain some information on the electronic density of states in an amorphous material. REFERENCES 1. G R A Y P. V.,Proc. IEEE57, 1543 (1969). 2. R E V E S Z A. G. IEEE Trans. Electron Dev. ED-12, 97 "(1965). 3. KOOI E., Phillips Res. Rept. 21,477 (1966).
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4. D E A L B. Z . , S K L A R M . , G R O V E A . S. andSNOW E. H., J. electrochem. Soc. 114, 266 (1967). 5. BELL T., H E T H E R I N G T O N G. and J A C K K. H. Phys. Chem. Glasses 3, 141 (1962). 6. L E L L E., K R E I D L N. J. and H E N S L E R J. R. in Progress in Ceramic Science (Edited by J. Burke), Vol. 4, Pergamon Press, New York (1966). 7. N E L S O N C. M. and W E E K S R. A., J. Am. c e r a m . Soc. 43,396 (1960). 8. W E E K S R. A. and N E L S O N C. M., J. Am. ceram. Soc. 43,399 (1960). 9. SILSBEE R. H.,J. appl. Phys. 32, 1459 (1961). 10. N I S H I Y.,Japan J. appl. Phys. 5, 333 (1966). 11. P H I L L I P P H. R., SolidState Commun. 4, 73 (1966). 12. K O H N W., in Solid State Physics (Edited by F. Seitz and D. Turnbull) Vol. 5, p. 257, Academic Press, New York (1957). 13. H O F F M A N R. and L I P S C O M B W. N., J. Chem. Phys. 36, 2179 (1962). 14. H O F F M A N R.,J. Chem. Phys. 39, 1397 (1963). 15. H O F F M A N R.,J. Chem. Phys. 40, 2474 (1964). 16. M E S S M E R R. P. and W A T K I N S G. Phys. Rev. Lett. 26,656 (1970). 17. P A U L I N G L.,J. Phys. Chem. 56, 361 (1952). 18. BELL R. J. and D E A N P., Nature 212, 1354 (1966). 19. W Y C K O F F R. W. G., Cl~stal Structures, Vol 1, I nterscience, New York (1963). 20. This is readily proven by considering the sum, for any vector c~; ~,~ct~ = f ( ~ ~,l,k))* ( ~ c~lo-) ) d r ~>0 ( M E S S M E R R. P., private communication). 21. B E N N E T T A. J, M c C A R R O L B., and M E S S M E R R. Surface Sci. to be published. 22. Handbook of Chemistry and Physics 49th ed. pp. D-38, F-158, Chem. Rubber Co., Cleveland (1968). 23. W A T K I N S G. D. and C O R B E T T J. W., Phys. Rev. 121, 1001 (1961). 24. N O R T O N F. J., J. Am. ceram. Soc. 36, 90 (1953). 25. P A U L I N G L., The Nature of the Chemical Bond, Cornell University Press, Ithaca (1960).