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Cyclic cluster calculation on the electronic structure of α-quartz and of its vacancy
Vo1uii~88A, number S
PHYSICS LETTERS
15 March 1982
CYCLIC CLUSTER CALCULATION ON THE ELECTRONIC STRUCTURE OF~-QUARTZ AND OF ITS VACANCY P. DEAK and J. GIBER Physical Institute, Technical University of Budapest, H-1521, Hungary Received 2 October 1981
Calculations on c~-quartzwere carried out using a CNDO/S Fock matrix optimized for saturated molecules of 0 and Si (including d-orbitals). Characteristic features of the electronic structure are reproduced and the density of states agrees considerably with the X-ray data. The results on the V~center are also in accordance with various experimental data.
1. Introduction. The electronic structure of Si02 constitutes the subject of a vast amount of experimental and theoretical work. The comprehensive review of Griscom [1] treats the topic in 80 pages with 112 references while the proceedings of the International Conference [2] held in Yorktown Heights contained 16 articles about the electronic structure of the various Si02 forms. Despite the numerous investigations some problems have remained unresolved till now. Previous cluster calculations were generally not self-consistent and were applied to Si04 or Si20 units or to idealized structures. As exceptions, the Xa SCF study [3] on Si04 reproduced the X-ray fluorescence spectrum while the LCLO MO method [4] was successful in describing the ESR parameters of the V~center in a real a-quartz structure. The best band structure calculation using a pseudopotential technique was carried out for a-quartz by Chelikowsky and SchlUter [5]. Their results serve as a reference for tight binding parametrizations (eg. ref. [6,71), though comparing them with experiments some controversies still have remained. First, the theory predicts a gap being about 1 eV in the upper valence band while photoelectron spectra show a minimum but not zero intensity in this region even if corrected for experimental resolution [11. Theorists claim this latter is a consequence of the background contributions. Second, the lower valence band was predicted too narrow and that as lies too low without fitting [6]. On the other hand, none of the calculations could predict an XPS threshold up till now, while ex0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
perirnentalists discuss the possibility of either 10.4 or 6.8 eV [1] - Furthermore, the calculations [51reproducing many of the experimental spectra were not able to yield a correct Si L2,3 spectrum likely in connection with the d-orbital problem [81
2. Cyclic cluster calculations. Since all experimental spectra being in connection with the valence bands (VB) of a-quartz and vitreous silica show sufficient similarity [1,6], a cluster model applying a localized basis seems to be an effective tool to face the issue. The cyclic cluster model based on the LUC concept of Evarestov et a!. [9,10] eliminates the shortcomings of the previous cluster calculations. The details of the method are given elsewhere [11—13]. The starting point is the choice of a large unit cell (LUC). It must be done that way [101 , that in the corresponding Brillouin zone (BZ) the K = 0 state should be equivalent to a set of high-symmetry points of the original BZ, forming a special k-point set in the sense of ref. [14] - Then a cluster calculation substituting the atomic orbital basis {p~}by the lattice sums =
N’~
~II~ ~u7
(where i runs over the N LUCs of the crystal) is equivalent to a partial band structure calculation on the highsymmetry points which applies Chadi and Cohen’s special k-point theorem. Regarding the fact that the VB of Si02 is highly determined by the short-range order 237
Volume 88A, number 5
PHYSICS LETTERS
a cut off in the lattice sums is possible. Since the tight binding studies [2,6,7] operate only with Si—O and O—O interactions this restriction might be done in our cluster calculation too. In our calculations a CNDO/S Fock matrix [15] on an spd basis was used with consequent CI calculation for electronic transitions. The parameters have been determined on saturated molecules of Si and 0 [16] This approximation yielded ionization potentials and electronic transitions with an accuracy better than 0.6 eV and 0.3 eV respectively for the case of molecules.
15 March 1982 Si 5012
SiC
2
CNDO
0
—
-10 -3-
-
~ Lack of the other two causes the lack of the weighting factor in the k-space averaging.
238
seLicons non[oc.pseudop.
I
3. Results and discussion. The crucial point in a cluster calculation is the choice of the LUC. For Si02 this is rendered more difficult by the large size of the elementary cell (for a-quartz it is Si3O6, i.e. the basis consists of 51 orbitals). The special k-point set at the lowest number of orbitals are F and M of the hexagonal BZ. This corresponds to a Si12024 cluster, i.e. to 204 orbitals. However, since all of the band structure calculations predicted a highly dispersionless VB (supporting the importance of short-range order) one can expect reasonable results from a calculation including the F point alone which accounts for the point group symmetry. On the other hand, regarding the fact that the VB edge can be expected at the Brillouin zone boundary (at M [5] or at A [6]), it seems necessary to include one of these points into the calculation. To save the computer time we have carried out preliminary calculations on Si6012 clusters, i.e. on the two elementary cells along the c axis (F and A) and on the two cells in the hexagonal plane as well (F and one of the three M points *i). Justifying our expectations the one-electron energies in the VB at F differed by less than 0.1 eV in the two cases. So the results of the two-cluster calculations could be given together in fig. 1 being compared with results of ref. [5] . (By the previous band structure calculations the conduction band was predicted to be more dispersive. In our calculations the two different k-sets actually result in a difference of about 1 eV in the lowest conduction band levels.) The overall agreement with the results of ref. [5] is striking, especially in the lower part of the upper YB. On the other hand, position and width of the lower
/S
20
=~=~
2
=
— — .
—
—
30 —
—
[eV] A
=
..
—
r
NI
A
P
M
Fig. 1. Cluster electronic structure compared with the band 151. Dotted lines are plotted corresponding to compatibffity relations.
structure
VB is correct in our calculation. This can be seen in fig. 2 where XPS and UPS spectra are compared with the DOS from ref. [5] and with a histogram DOS from our calculation. The latter also denotes a broadening of the upper part of and no gap in the upper VB. The first orbitals (F3, F1) of the upper subband are almost exclusively of oxygen p character (up to 96%) while the lower ones (F2, F3, F3, F2, F3, F1) contain d admixtures of 9—19%. These values are not greater than those of Gilbert et al. [171 however, similarly to the calculation of ref. [3] , they are responsible for the broadening. So we think, the gap between nonbonding (0 2p) and strong bonding (Si 3s,.3p—O 2p) orbitals in the band calculations is a result of neglecting Si 3d-orbitals and so Si 3d—O 2p bonding. (None of the band calculations up to now has applied explicit d-orbitals for Si. In ref. [5] a plane wave expansion was used but no 1 = 2 terms in the potential.) The sequence of orbitals at the top of the YB dif-
Volume 88A, number S
PHYSICS LETTERS
15 March 1982
Si L2,3 spectrum also shows the importance of d-or/ ..
-~
UPS XPS
[1]
[1]
bitals. The presence of these in the upper region may account for the first peak as predicted in [31or as sug-
DOS
[5]
gested in refs. [4,8] - (The explanation of ref. [5] can
—
be ruled out [1,8].)
iiiL~4’ Fig.
4. The V~center of Si02. The E’ is the experimentally best documented center in SiO2. It could have been detected in both of vitreous silica and a-quartz
fers from that of ref. [5] however, this sequence could also not be reproduced in ref. [6] which was fitted to the results of ref. [5] The first ionization potential (Koopmans’ theorem) in our calculation is 11.61 eY. It agrees with the first peak in the UPS spectrum at 11.7 eV, and taking into consideration the line broadening of about 1 eY, it strongly supports the 10.4 eY suggestion for the XPS
by ESR and UY absorption experiments. Calculations of Yip and Fowler [4] regarding the ESR data strongly suggest the asymmetrically relaxed oxygen vacancy (Y~)as an origin. Collecting the ESR and UV results Greaves [2, p. 268] established a model electronic structure based on the trivalent Si atoms. As the E’ center however is also present in the more rigorous aquartz structure an alternative vacancy model might be also justified. Supplementing the data of Greaves with those of Griscom [2, p. 107] and with electrical
threshold.
measurements on MOS devices reviewed by DiMaria
Comparing the histograms of the partial densities of states for various atomic orbitals with the XES spectra (fig. 3: the histograms are magnified in order to reach the main XES peak) it can be seen that the sequence of orbitals of various compositions is also correct. Naturally, the shape of a partial DOS curve can not be compared directly with XES but positions of the bands have to agree. The comparison with the
[2, p. 160] the following model can be suggested (fig. 4). The neutral relaxed vacancy provides a doubly occupied and an empty level in the gap. The excitation of them eventuates the well known E’ transitions (5.4 and 6.2 eY in a-quartz and 5.85 eV in vitreous silica). The vacancy, trapping a hole, relaxes asymmetrically forming the singly occupied Y+ center which shows the ESR signal. Transition from the YB edge to the occupied level has probably an energy about 8.2 eY. This model is supported by the results of ref. [4] and ref. [18] too. Our method allows the calculation of the electronic
-20
-10
0
2. The DOS histogram of the cluster and the DOS from ref. [51compared with XPS and UPS [1] spectra.
,
.
—
s states of Si d Si
S
—
—
L23 exp [1]
cdc [51-
structure of a vacancy including electronic transitions.
Si p K,, statesexpof [1]Si —Si Ka calc [5]
p O -20
—10
states
of 0 K~, cole exp [ii K~ [5J
0
Fig. 3. Histograms of the partial DOS compared with theoretical [51and experimental [1] XES spectra. The amplitudes of the histograms are fitted to the main peaks of the corresponding XES.
Fig. 4. Projection of the relaxed Si6 Oii V~cluster onto the hexagonal plane.
239
Volunoe 88A, number 5
PHYSICS LETTERS
__________________
5. C’onclusion. The cyclic cluster model has been applied ftr calculating the electronic structure of Si0
CB
2
j
V
and its V+ center. The proper parametrization of the
CNDO/S method on an spd basis was able to yield rca-
52
v°
sonable results regarding the orbital composition and energy relations of the valence band of a-quart!. as well as of the vacancy level. Further improvement of the results can be cx-
VB
Fig. 5. The vacancy model of the E’ center.
Removing an oxygen atom from the 5i6012 (F, M1) a-quartz cluster we fixed the Si atoms at the relaxed positions given in ref. [4] (fig. 5). Due to the strong perturbation at Si11 the change in the bond order matrix (a measure of convergence with cluster size) does not decay sufficiently in its surroundings (lower part of fig. 5). On the other hand, in the environment of Si1 (upper part of fig. 5) the change becomes negligible at the cluster boundary. Considering that the wavefunction of the singly occupied is localized 0~31p049d0~05 withstate a 2.22°tilt fromonthe Si1 in 85%(s vacancy direction), the energy results in connection with that level can be considered as nearly convergent. The ionization potential (Koopmans’ theorem) of the level is 2.02 eV being 9.6 eV above the YB. These 0~22p0~60 results with agree 9.2 welleV with those ref.The [4] parameters : an ~ hybrid above theofYB. of the strong hyperfine interaction are also in good agreement with experiments and other calculations (table 1). Furthermore, our calculation presented a minimum transition energy of 7.96 eV from the top of the valence band to the singly occupied level. It approximates the experimental 8.2 eV very well. Since the two unoccupied vacancy levels are localized almost totally on Si 11, the transitions calculated to these cannot be considered as convergent.
Table 1 Parameters of the strong hyperfme interaction as calculated in the same approximation as in ref. [19]. Data are in 1 o~ cm~. —
~-
Ref. [41 a b
—-------________
Ref. [191
_______
_______—-
303.2 20.2
337 16.5
——-——---—~--—————~
240
15 March 1982
—
Present
Exp 1201
425.1 16.5
384 20
±2 ±2
pected from applying a Si12024 cluster which corresponds to the use of a proper special k-point set. Calculations on the neutral vacancy are also under work. The authors are deeply grateful to the Institute of Theoretical Physics of the TU of Vienna for the possibility to use their computer and personally to Dr. H. Nowotny for his kind help.
References [1] S.T. D.L. Panteides, Griscom, J.ed., Noncryst. Solidsof24Si0 (1977) 155. [21 The physics 2, and its interfaces (Pergamon, New York, 1978). 131 G.A.D. Collins, D.W.J. Cruickshank and A. Breeze, J. Chem. Soc. Faraday Trans. 1168 (1972) 1189.
[41 K.L. Yip and W.B. Fowler, Phys. Rev. Bil (1975) 2327. [5] (1977) J.R. Chelikowsky and M. Schluter, Phys. Rev. B15 4020. [6] R.B. Laughlin, D. Joannopoulos and D.J. Chadi, Phys. Rev. B20 (1979) 5228. [7] R.N. Nucho and A. Madhukar, Phys. Rev. B21 (1980) 1576. [8] F. Calabrese and W.B. Fowler, Phys. Rev. B18 (1978) 2888.
191 R.A. Evarestov, MI. Petrashen and E.M. Ledovskaya,
Phys. Stat. Sol. 68b (1975) 453. [101 R.A. Evarestov and V.A. Lovchikov, Phys. Stat. Sol. 79b (1977) 743. [11] P. Deák, J. Kazsoki and J. Giber, Phys. Lett. 66A (1978) 395.
[12] P. Deák, Lecture Notes in Physics 122 (1980) 253.
[13] P. Deák Phys. Lett. 83A (1981) 39; Acta Phys. Hung. 50(1981)139. [141 D.J. Chadi and ML. Cohen, Phys. Rev. B7 (1973) 692. [151 J. Del Bene and H.H. Jaffe, J. Chem. Phys. 48 (1968) 1807; R.C. Ellis, G. Kuehnlenz and H.H. Jaffé, Theoret. Chim. Acta 26 (1972)131. 1161 P.Deák, tobe published. [17] T.L. Gilbert, W.J. Stevens, H. Schrenk, M. Yoshimine and P.S. Bagus, Phys. Rev. B8 (1973) 5977. [181 A.J. Bennett and L.M. Roth, J. Phys. Chem. Solids 32 (1971) 1251. [19] C. Gobsch, M. Haberlandt, J. Wechner and J. Reinhold, Phys. Stat. Sol. 90b (1978) 309. [20] RH. Silsbee, J. Appi. Phys. 32 (1961) 1459.
PHYSICS LETTERS
15 March 1982
CYCLIC CLUSTER CALCULATION ON THE ELECTRONIC STRUCTURE OF~-QUARTZ AND OF ITS VACANCY P. DEAK and J. GIBER Physical Institute, Technical University of Budapest, H-1521, Hungary Received 2 October 1981
Calculations on c~-quartzwere carried out using a CNDO/S Fock matrix optimized for saturated molecules of 0 and Si (including d-orbitals). Characteristic features of the electronic structure are reproduced and the density of states agrees considerably with the X-ray data. The results on the V~center are also in accordance with various experimental data.
1. Introduction. The electronic structure of Si02 constitutes the subject of a vast amount of experimental and theoretical work. The comprehensive review of Griscom [1] treats the topic in 80 pages with 112 references while the proceedings of the International Conference [2] held in Yorktown Heights contained 16 articles about the electronic structure of the various Si02 forms. Despite the numerous investigations some problems have remained unresolved till now. Previous cluster calculations were generally not self-consistent and were applied to Si04 or Si20 units or to idealized structures. As exceptions, the Xa SCF study [3] on Si04 reproduced the X-ray fluorescence spectrum while the LCLO MO method [4] was successful in describing the ESR parameters of the V~center in a real a-quartz structure. The best band structure calculation using a pseudopotential technique was carried out for a-quartz by Chelikowsky and SchlUter [5]. Their results serve as a reference for tight binding parametrizations (eg. ref. [6,71), though comparing them with experiments some controversies still have remained. First, the theory predicts a gap being about 1 eV in the upper valence band while photoelectron spectra show a minimum but not zero intensity in this region even if corrected for experimental resolution [11. Theorists claim this latter is a consequence of the background contributions. Second, the lower valence band was predicted too narrow and that as lies too low without fitting [6]. On the other hand, none of the calculations could predict an XPS threshold up till now, while ex0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
perirnentalists discuss the possibility of either 10.4 or 6.8 eV [1] - Furthermore, the calculations [51reproducing many of the experimental spectra were not able to yield a correct Si L2,3 spectrum likely in connection with the d-orbital problem [81
2. Cyclic cluster calculations. Since all experimental spectra being in connection with the valence bands (VB) of a-quartz and vitreous silica show sufficient similarity [1,6], a cluster model applying a localized basis seems to be an effective tool to face the issue. The cyclic cluster model based on the LUC concept of Evarestov et a!. [9,10] eliminates the shortcomings of the previous cluster calculations. The details of the method are given elsewhere [11—13]. The starting point is the choice of a large unit cell (LUC). It must be done that way [101 , that in the corresponding Brillouin zone (BZ) the K = 0 state should be equivalent to a set of high-symmetry points of the original BZ, forming a special k-point set in the sense of ref. [14] - Then a cluster calculation substituting the atomic orbital basis {p~}by the lattice sums =
N’~
~II~ ~u7
(where i runs over the N LUCs of the crystal) is equivalent to a partial band structure calculation on the highsymmetry points which applies Chadi and Cohen’s special k-point theorem. Regarding the fact that the VB of Si02 is highly determined by the short-range order 237
Volume 88A, number 5
PHYSICS LETTERS
a cut off in the lattice sums is possible. Since the tight binding studies [2,6,7] operate only with Si—O and O—O interactions this restriction might be done in our cluster calculation too. In our calculations a CNDO/S Fock matrix [15] on an spd basis was used with consequent CI calculation for electronic transitions. The parameters have been determined on saturated molecules of Si and 0 [16] This approximation yielded ionization potentials and electronic transitions with an accuracy better than 0.6 eV and 0.3 eV respectively for the case of molecules.
15 March 1982 Si 5012
SiC
2
CNDO
0
—
-10 -3-
-
~ Lack of the other two causes the lack of the weighting factor in the k-space averaging.
238
seLicons non[oc.pseudop.
I
3. Results and discussion. The crucial point in a cluster calculation is the choice of the LUC. For Si02 this is rendered more difficult by the large size of the elementary cell (for a-quartz it is Si3O6, i.e. the basis consists of 51 orbitals). The special k-point set at the lowest number of orbitals are F and M of the hexagonal BZ. This corresponds to a Si12024 cluster, i.e. to 204 orbitals. However, since all of the band structure calculations predicted a highly dispersionless VB (supporting the importance of short-range order) one can expect reasonable results from a calculation including the F point alone which accounts for the point group symmetry. On the other hand, regarding the fact that the VB edge can be expected at the Brillouin zone boundary (at M [5] or at A [6]), it seems necessary to include one of these points into the calculation. To save the computer time we have carried out preliminary calculations on Si6012 clusters, i.e. on the two elementary cells along the c axis (F and A) and on the two cells in the hexagonal plane as well (F and one of the three M points *i). Justifying our expectations the one-electron energies in the VB at F differed by less than 0.1 eV in the two cases. So the results of the two-cluster calculations could be given together in fig. 1 being compared with results of ref. [5] . (By the previous band structure calculations the conduction band was predicted to be more dispersive. In our calculations the two different k-sets actually result in a difference of about 1 eV in the lowest conduction band levels.) The overall agreement with the results of ref. [5] is striking, especially in the lower part of the upper YB. On the other hand, position and width of the lower
/S
20
=~=~
2
=
— — .
—
—
30 —
—
[eV] A
=
..
—
r
NI
A
P
M
Fig. 1. Cluster electronic structure compared with the band 151. Dotted lines are plotted corresponding to compatibffity relations.
structure
VB is correct in our calculation. This can be seen in fig. 2 where XPS and UPS spectra are compared with the DOS from ref. [5] and with a histogram DOS from our calculation. The latter also denotes a broadening of the upper part of and no gap in the upper VB. The first orbitals (F3, F1) of the upper subband are almost exclusively of oxygen p character (up to 96%) while the lower ones (F2, F3, F3, F2, F3, F1) contain d admixtures of 9—19%. These values are not greater than those of Gilbert et al. [171 however, similarly to the calculation of ref. [3] , they are responsible for the broadening. So we think, the gap between nonbonding (0 2p) and strong bonding (Si 3s,.3p—O 2p) orbitals in the band calculations is a result of neglecting Si 3d-orbitals and so Si 3d—O 2p bonding. (None of the band calculations up to now has applied explicit d-orbitals for Si. In ref. [5] a plane wave expansion was used but no 1 = 2 terms in the potential.) The sequence of orbitals at the top of the YB dif-
Volume 88A, number S
PHYSICS LETTERS
15 March 1982
Si L2,3 spectrum also shows the importance of d-or/ ..
-~
UPS XPS
[1]
[1]
bitals. The presence of these in the upper region may account for the first peak as predicted in [31or as sug-
DOS
[5]
gested in refs. [4,8] - (The explanation of ref. [5] can
—
be ruled out [1,8].)
iiiL~4’ Fig.
4. The V~center of Si02. The E’ is the experimentally best documented center in SiO2. It could have been detected in both of vitreous silica and a-quartz
fers from that of ref. [5] however, this sequence could also not be reproduced in ref. [6] which was fitted to the results of ref. [5] The first ionization potential (Koopmans’ theorem) in our calculation is 11.61 eY. It agrees with the first peak in the UPS spectrum at 11.7 eV, and taking into consideration the line broadening of about 1 eY, it strongly supports the 10.4 eY suggestion for the XPS
by ESR and UY absorption experiments. Calculations of Yip and Fowler [4] regarding the ESR data strongly suggest the asymmetrically relaxed oxygen vacancy (Y~)as an origin. Collecting the ESR and UV results Greaves [2, p. 268] established a model electronic structure based on the trivalent Si atoms. As the E’ center however is also present in the more rigorous aquartz structure an alternative vacancy model might be also justified. Supplementing the data of Greaves with those of Griscom [2, p. 107] and with electrical
threshold.
measurements on MOS devices reviewed by DiMaria
Comparing the histograms of the partial densities of states for various atomic orbitals with the XES spectra (fig. 3: the histograms are magnified in order to reach the main XES peak) it can be seen that the sequence of orbitals of various compositions is also correct. Naturally, the shape of a partial DOS curve can not be compared directly with XES but positions of the bands have to agree. The comparison with the
[2, p. 160] the following model can be suggested (fig. 4). The neutral relaxed vacancy provides a doubly occupied and an empty level in the gap. The excitation of them eventuates the well known E’ transitions (5.4 and 6.2 eY in a-quartz and 5.85 eV in vitreous silica). The vacancy, trapping a hole, relaxes asymmetrically forming the singly occupied Y+ center which shows the ESR signal. Transition from the YB edge to the occupied level has probably an energy about 8.2 eY. This model is supported by the results of ref. [4] and ref. [18] too. Our method allows the calculation of the electronic
-20
-10
0
2. The DOS histogram of the cluster and the DOS from ref. [51compared with XPS and UPS [1] spectra.
,
.
—
s states of Si d Si
S
—
—
L23 exp [1]
cdc [51-
structure of a vacancy including electronic transitions.
Si p K,, statesexpof [1]Si —Si Ka calc [5]
p O -20
—10
states
of 0 K~, cole exp [ii K~ [5J
0
Fig. 3. Histograms of the partial DOS compared with theoretical [51and experimental [1] XES spectra. The amplitudes of the histograms are fitted to the main peaks of the corresponding XES.
Fig. 4. Projection of the relaxed Si6 Oii V~cluster onto the hexagonal plane.
239
Volunoe 88A, number 5
PHYSICS LETTERS
__________________
5. C’onclusion. The cyclic cluster model has been applied ftr calculating the electronic structure of Si0
CB
2
j
V
and its V+ center. The proper parametrization of the
CNDO/S method on an spd basis was able to yield rca-
52
v°
sonable results regarding the orbital composition and energy relations of the valence band of a-quart!. as well as of the vacancy level. Further improvement of the results can be cx-
VB
Fig. 5. The vacancy model of the E’ center.
Removing an oxygen atom from the 5i6012 (F, M1) a-quartz cluster we fixed the Si atoms at the relaxed positions given in ref. [4] (fig. 5). Due to the strong perturbation at Si11 the change in the bond order matrix (a measure of convergence with cluster size) does not decay sufficiently in its surroundings (lower part of fig. 5). On the other hand, in the environment of Si1 (upper part of fig. 5) the change becomes negligible at the cluster boundary. Considering that the wavefunction of the singly occupied is localized 0~31p049d0~05 withstate a 2.22°tilt fromonthe Si1 in 85%(s vacancy direction), the energy results in connection with that level can be considered as nearly convergent. The ionization potential (Koopmans’ theorem) of the level is 2.02 eV being 9.6 eV above the YB. These 0~22p0~60 results with agree 9.2 welleV with those ref.The [4] parameters : an ~ hybrid above theofYB. of the strong hyperfine interaction are also in good agreement with experiments and other calculations (table 1). Furthermore, our calculation presented a minimum transition energy of 7.96 eV from the top of the valence band to the singly occupied level. It approximates the experimental 8.2 eV very well. Since the two unoccupied vacancy levels are localized almost totally on Si 11, the transitions calculated to these cannot be considered as convergent.
Table 1 Parameters of the strong hyperfme interaction as calculated in the same approximation as in ref. [19]. Data are in 1 o~ cm~. —
~-
Ref. [41 a b
—-------________
Ref. [191
_______
_______—-
303.2 20.2
337 16.5
——-——---—~--—————~
240
15 March 1982
—
Present
Exp 1201
425.1 16.5
384 20
±2 ±2
pected from applying a Si12024 cluster which corresponds to the use of a proper special k-point set. Calculations on the neutral vacancy are also under work. The authors are deeply grateful to the Institute of Theoretical Physics of the TU of Vienna for the possibility to use their computer and personally to Dr. H. Nowotny for his kind help.
References [1] S.T. D.L. Panteides, Griscom, J.ed., Noncryst. Solidsof24Si0 (1977) 155. [21 The physics 2, and its interfaces (Pergamon, New York, 1978). 131 G.A.D. Collins, D.W.J. Cruickshank and A. Breeze, J. Chem. Soc. Faraday Trans. 1168 (1972) 1189.
[41 K.L. Yip and W.B. Fowler, Phys. Rev. Bil (1975) 2327. [5] (1977) J.R. Chelikowsky and M. Schluter, Phys. Rev. B15 4020. [6] R.B. Laughlin, D. Joannopoulos and D.J. Chadi, Phys. Rev. B20 (1979) 5228. [7] R.N. Nucho and A. Madhukar, Phys. Rev. B21 (1980) 1576. [8] F. Calabrese and W.B. Fowler, Phys. Rev. B18 (1978) 2888.
191 R.A. Evarestov, MI. Petrashen and E.M. Ledovskaya,
Phys. Stat. Sol. 68b (1975) 453. [101 R.A. Evarestov and V.A. Lovchikov, Phys. Stat. Sol. 79b (1977) 743. [11] P. Deák, J. Kazsoki and J. Giber, Phys. Lett. 66A (1978) 395.
[12] P. Deák, Lecture Notes in Physics 122 (1980) 253.
[13] P. Deák Phys. Lett. 83A (1981) 39; Acta Phys. Hung. 50(1981)139. [141 D.J. Chadi and ML. Cohen, Phys. Rev. B7 (1973) 692. [151 J. Del Bene and H.H. Jaffe, J. Chem. Phys. 48 (1968) 1807; R.C. Ellis, G. Kuehnlenz and H.H. Jaffé, Theoret. Chim. Acta 26 (1972)131. 1161 P.Deák, tobe published. [17] T.L. Gilbert, W.J. Stevens, H. Schrenk, M. Yoshimine and P.S. Bagus, Phys. Rev. B8 (1973) 5977. [181 A.J. Bennett and L.M. Roth, J. Phys. Chem. Solids 32 (1971) 1251. [19] C. Gobsch, M. Haberlandt, J. Wechner and J. Reinhold, Phys. Stat. Sol. 90b (1978) 309. [20] RH. Silsbee, J. Appi. Phys. 32 (1961) 1459.