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Electronic structure of the vanadyl oxygen vacancy in V2O5 : periodic vacancy single layer model
0038-lO98/8l/260257-05$02.00/O
Solid State Communications, Vo1.39, pp.257-261. Pergamon Press Ltd. 1981. Printed in Great Britain.
ELECTRONIC
STRUCTURE PERIODIC
OF THE VANADYL VACANCY
OXYGEN VACANCY
l
W. Lambrecht*,
IN V205
:
SINGLE LAYER MODEL.
B. Djafari-Rouhani
*
and J. Vennik
Laboratorium voor Kristallografie en Studie van de Vaste Stof, Rijksuniversiteit Gent, Krijgslaan 271, B-9000 Gent, Belgium.
A single layer model of V205 with one vanadyl oxygen vacancy per unit Two vacancy derived bands in the main band gap cell is considered. Their origin and nature are disare interpreted as localized states. Infrared data are discussed but a definite interpretation is cussed. The total spin of the groundstate of the two-elecnot yet possible. tron system corresponding to the vacancy is in agreement with the EPR experimental data.
1. Introduction. The important role of the vanadyl oxygen vacancy (W) in V 05 for electrical and catalytic behaviour has o P ten been stressed in the literature (ref.l and refs. therein). The vacancy is furthermore supposed to provide localized donor levels near the bottom of the conduction band, which may govern the electrical conductivity of Surfa e vanadyl oxygen vacanthis materia12. cies and the related vE+ centers are believed to be active sites for the catalytic oxidation of hydrocarbons. Furthermore the transitions of V205 into lower oxides (V409 and V6013) which may take place during the initial state of the catalytic reactions at a V205-surface, proceeds by vacancy formatior;, accompanied by cert in $ IR (inlattice distortions . Experimentally, frared) and EPR (Electron Paramagnetic ResonanIn the present conce) spectra were reported. tribution we present preliminary results of a first theoretical treatment of the electronic structure of the W.
indications of how localized states might be produced. Some deficiencies of the present model are the following. 1) The vacancies are quite close to one another in the b-direction and consequently there may be considerable interactions between them, which will give rise to band dispersion of the localized states of an isolated vacancy. This dispersion is expected to be largest along IY and will give an idea about the mutual interaction between vacancies. 2) The periodic arrangement of vacancies considered here, imposes symmetry restrictions, not present for the isolated vacancy for which only one mirror plane, namely the one passing through the vacancy, remains. 3) Atomic relaxation effects are not taken into account. Two types of relaxation might be considered : a shift of the nearest vanadium towards the il oxygen plane; ii/ relaxation of down pointing vanadyl oxygens of the neighbouring layer. A general discussion of structural aspects and figures of the crystal structure may be found in reference'. For a representation of the effect (ii) see reference 4b fig. 4.14. For the surface W, the latter effect is of course absent and even in the bulk it is by no means certain to occur. The former effect would probably only slightly change the interactions between the nearest vanadium neighbour of the W and its surrounding oxygens and in a first approximation both effects may consequently be ignored. Apart from the perturbation consisting in the removal of an oxygen and the breaking of the related bonds, a second important effect must be considered at least in a qualitative manner. This is the charge perturbation created by the removal of a negatively charged oxygen, which will give rise to an attractive potential on the neighbouring atoms. A calculation of this potential was not attempted at this stage, because it involves the delicate balance of two nearly equal large effects : the inter-atomic Coulomb (or Madelung) effect gnd the intra-atomic Coulomb repulsion effects . The most im-
2. Model. In previous work6 it was shown by compari on between a full 3-dimensional band calculation 7 and our 2-dimensional calculation, that a single layer approximation accounts quite well for the main features of the band structure of V205. In order to obtain a first indication of what happens to the bandstructure upon creation of W's, we consider the same single layer tight binding model as used previously, but now introduce one W per unit cell. The model obtained in this way is one with at the bottom side of the layer all vanadyl oxygens present and at the top side alternatingly rows in the b-direction with and without vanadyl oxygens. This model is fairly unrealistic for a study of isolated vacancies, but nevertheless may provide a few
* **
Research assistant of the National Fund for Scientific Research Belgium. Present address : Equipe de Physique des Solides du C.N.R.S., Institut Supdrieur d'electronique du Nord, 3 rue Franfois Bass, 59046 Lille Cedex, France. 257
vANADYL
258
OXYGEN
portant effect will be felt by the nearest vanadium and consequently we studied this charge transfer effect by applying a semi-empirical shift a to the V3d-level of this atom, which we allow to assume values between 0 and -0.03 Ryd. 3. Results. As the vanadyl oxygen contribution to the valence band is situated in the middle and bottom part of the band, one does not expect localized levels to be created in the main gap originating from the valence band. A calculation of the levels in T, the centre of the Brillouin Although some zone, confirms this statement. changes may occur in the valence band, this will In the not be discussed in the present study. figures we consequently only show the energy leIn figure 1 the vels of the conduction band. levels in r are presented as a function of the parameter a, introduced above, and are compared to the corresponding levels for the perfect crystal. The symmetry with respect to the mirto the b-direction, ror planes m , perpendicular is indicatedYby + or - and levels which are believed to the interrelated are joined by dashed lines. A few conclusions can be drawn from this figure. 1) Some energy levels are shifted downwards into regions where in the case of the perfect crystal no levels are present in T and where also the density of states is low, as may be seen
VACANCY
Vol. 39, No. 2
IN V2O5
in figure 6. These levels furthermore depend more strongly on a than the other levels which could be an indication that they are more or less localized on the first vanadium and thus could present resonance behaviour. 2) In the energy regions where in the case of the perfect crystal the levels are closely spaced, which corresponds to a high density of states, the levels of the vacancy on the contrary have become more widely spaced, corresponding to a decrease in density of states. This anti-correlation between the peaks of the density of states and the changes in density of states induced by the vaca;cy, has been observed previously by Pollmann , and was explained to be a quite general effect on the basis of aGreen's function treatment. As for the presence of localized states, we must consider the lowest three levels at the bottom of the conduction band. Figure 2 shows the dispersion of the corresponding bands along TY.
-O’@jr _o,6*; ;; c?
s w
-0.3
1
N
V
1
+ +
t----+ t----+ -0.4
+
:
L
3 -0.51-+ W
+ l
+
i__ I
1
+ + +
1 =I
+ z i
a(Ryd) Figure 1 : Energy levels at the point for the normal (N) and vacancy (V) unit cell of V205. The vanadyl oxygen vacancy levels are given as a function of the parameter a, discussed in the text.
------.
I -’
-0,701
1 0 r
I ky
n/b Y
Figure 2 : Energy bands along the b-direction (I'Y) for the three lowest vacancy bands (full lines) for u=O, and for the lowest two perfect crystal bands (dashed lines).
Only the lowest band is completely separated from the normal conduction band and may consequently, with some confidence, be considered to correspond to a localized level of an isolated vacancy. The second level, even upon enlarging lul, does not separate from the conduction band. Nevertheless its center of gravity is clearly moving out of the band. This level, moreover, depends strongly on a as can be concluded from figure 1. This suggests that a second localized state may be present if c1 is strong enough, say -0.005 at least. The third level is much less a-dependent, and one is thus inclined to consider it rather as a normal band state. The fourth and fifth level may eventually give rise to localized states within the small gap found at the bottom of the conduction band, which was discussed in previous work6. The interpretation of certain bands as localized states is however by no means straightforward; the number of localized states may be larger or smaller.
VANADYL
Vol. 39, No. 2
OXYGEN
VACANCY
IN V205
259
In order to discuss the origin of the lowest two localized levels, which we will from now on call Ea for the level of symmetry - and Eb for the level of symmetry +, we consider the corresponding wavefunctions in table 1. Table
1
: Eigenvectors nadium
of localized contribution.
states
'a Y Vl v2 v3 v4
&I1 0.146 0.398 0.130
: va-
'b
Y -0.5:7 O.Z9 -0.094 -0.228 0.200 -0.171 0.084 I 0.012
3z2-r* 0.149 -0.030 -0.026 -0.006
x2_ 2 0.180 -0.142 -0.132 0.047
As the oxygen contribution is always an order of magnitude lower, only the vanadium contribution is presented. Due to symmetry restrictions, 'Pa contains only2xy2and yz2coyponents and Yb contains only xz, 32 -r and x -y . The numbering of the atoms is as indicated in figure 3.
Figure
3
: Numbering
of atoms in the unit cell.
Y, is primarily delocalized in the a-direction and Yb is fairly localized on the first vanadium Vl. Taking into account the signs on the coefficients, we conclude tha:. Y may be considered to be related to the perfectacrystal T3-state and Yb to the T7-state. As the coefficents on different atoms are different in the case of the vacancy, these levels cannot have these symmetries and thus will in fact be a mixture of several conduction band states, but at least their main constituent orbitals are oriented relative to each other in the same way as for I3 and T7. It is interesting to note that for both these symmetries the relative orientation of orbitals corresponds to indirect bonding of the vanadium orbitals across the bridge oxygen : for Y between xy-orbitals and for Yb between The case of Y is illustrated in xz-or e.itals. levels are lofigure 4. The reason why &ese wered in energy with respect to the case of the perfect crystal, may be understood as follows. Breaking the vanadiumvanadyl-oxygen pdn The interaction lowers the yz- and xz-levels. strongest effect is expected for Eb, which has primarily xz-character, whereas E has only partially yz-character and is actually rather an Application of the potential xy-derived level. a, further lowers the vanadium levels. However it is not clear in how far the present conclusions about the wavefunctions may be extrapoThe lated to the case of the single vacancy. removal of certain symmetry restrictions will
Figure 4 : Orbitals and interactions involved in Y the lowest localized state. The size of the irbitals is drawn in agreement with their contribution to the wave function. The full lines indicate direct, the dashed linges indirect interactions. The bonding or antibonding effect is indicated by B or AB respectively.
allow mixing of (xy,yz) with (xz, 3z2-r2, x2-y2) and will also not impose a fixed orientation and magnitude of orbitals in the neighbouring unit cells; further delocalization may thus occur. In the present case we can for instance conclude that delocalization in the vanadium chain via Vl will be small, but we cannot say anything about delocalization onto the Vl atoms of the adjacent unit cell. Furthermore, the mutual interaction of the close lying vacancies consistrongdered here, may perturb the wavefunctions ly. Charge accumulation could occur on an atom like V2 which is as close to the vacancy of its own unit cell as it is to that of the adjacent unit cell. 4. Comparison
with experimental
4.1. Infrared
Spectroscopy.
data.
The optical infrared band, corresponding to the W is reproduced in figure 5 (after Clauws The different features are labeled et al. 4). (A-E) for further reference. The transition energies involved and the closeness of the defect levels to the bottom of the conduction band, as obtained in the present calculations, indicate that the spectrum should result from transitions from the localized states towards the conduction band, and not from transitions between different defect levels. The general shape of the density of states in the conduction band, consisting in a large peak, followed by a second smaller one, which is also observed for the experimental spectrum, especially for polarization E//a, is a further indication for the validity of this interpretation. The defect levels of the isolated W should however lie at somewhat lower energies, in order to allow a straightforward interpretation of the spectra in this way. Another possible interpretation of the spectrum, which emphasizes the polarization dependence and consequently involves the selection rules for dipole transitions, is as indicated in figure 6. The peaks in the density of states arelabeled according to their critical points in the Brillouin zone. In this case however, eitner drastic changes in the density of states or transition probability effects must be assumed to explain why the first peak in the densi-
260
VANADYL
OXYGEN
VACANCY
hv.
IN V205
Vol. 39, No. 2
eV
Figure 5 : Infrared specumtrum of the vanadyl oxygen vacancy as taken from reference 4b. The labeling of the peaks is added to the original figure.
-0.55
r;; c -0,60 w
II
D
l
-0.701 0
I 100
200
300
LOO
I
500
DOS Figure 6 : Tentative interpretation of the infrared spectrum. The density of states corresponds to the perfect crystal and is obtained from diagonalization of the full tight binding matrix and not from an effective hamiltonian for the conduction band alone as in ref. 6b, which slightly modifies the higher part of the band. The energy region believed to be of interest here is shaded and the symmetry of peaks in the density of states and localized states is indicated.
ty of states would be suppressed. The only conelusion which can be drawn with a certain degree of confidence is that the conduction band states are involved in the optical transitions as final states. A definite assignment of the different features of the IR-band seems as yet impossible.
4.2. ---____________Electron paramagnetic
resonance.
With the present limited knowledge of the vacancy electronic structure, it is not possible to give 5" detailed discussion of all the EPR parameters ; we will in fact limit the discussion to the total effective spin S of the ground sta-
VANADYL
Vol. 39, No. 2 Table 2 S
OXYGEN
VACANCY
: Ground state determination eigenvalues
of interaction
1
Ea+Eb +Jab-K
0
E +E +Jab+Kab a b
c +E a
261
IN Vz05
of the two-electron hamiltonian
system.
numerical
value
12.34 eV
ab
13.25
b
and K stand for the direct and exchange ab ab and Yb,. similarly for Jaa and Jbb.
3
te.
In order to see whether an S=l ground state is possible for the system under consideration, as is required by experiment, we performed a simplified configuration interaction calculation, including only the lowest two localized states E and E . This approximation is by no means tritial hue as we are only interested in obtaining the spin of the groundstate, it is The reasonable to take it as a starting point. procedure followed is a diagonalization of the Coulomb-interaction matrix in the basis of all possible spin configurations on the two levels The Coulomb integrals between the E and Eb. w&e functions I and fb were determined by introducing their gxpansion in atomic orbitals, which are cut off after the nearest neighbours. The following supplementary approximations are introduced for the remaining Coulomb integrals : only one and two center terms are included and only for the one center terms exchange contriFor the twobutions are taken into account. center terms we furthermore neglect the diffeNo screerence between different orbitals. ning of the Coulomb interaction is considered. This latter effect could drastically change the involved energies, but would probably not change
integrals
between
Ya
the order of the levels. This, as may be seen in the equations included in table 2, would only be the case if the energy level difference E,-Eb would
References.
(1) L. Fiermans, P. Clauws, W. Lambrecht, L. Vandenbroucke and J. Vennik, Phys. Stat. Solidi (a) 59, 485, 1980. (2) J. Haemers, E. Baetens and J. Vennik, Phys. Stat. Solidi (a) 0, 381, 1973. (3) G. Grymonprez, L. Fiermans and J. Vennik, Acta Crystallogr. AZ, 833, 1977. (4) a. P. Clauws and J. Vennik, Phys. Stat. Solidi (b) 66, 553, 1974. b. P. Clauws, Studie van optische eigenschappen van vanadiumoxides in het infrarood, Koninklijke Academic voor Wetenschappen, Letteren en Schone Kunsten van Belgie, Brussel 1980.
(5) M. Van Haelst and P. Clauws, Phys. Stat. Solidi (b), I, 719, 1978. M. Lan(6) a. W. Lambrecht, B. Djafari-Rouhani, noo, P. Clauws, L. Fiermans and J. Vennik, J. Phys. C : Solid State Phys. E, 2503, 1980. b. W. Lambrecht, B. Djafari-Rouhani and J. Vennik, to be published. (7) D.W. Bullett, J. Phys. C : Solid State Phys. 13, L595, 1980. (8) x Lannoo and J.N. Decarpigny, Phys. Rev. B8, 5709, 1980. XX, Advances (9) J. Pollmann, FestkGrperprobleme in Solid State Physics, 117, 1980.
Solid State Communications, Vo1.39, pp.257-261. Pergamon Press Ltd. 1981. Printed in Great Britain.
ELECTRONIC
STRUCTURE PERIODIC
OF THE VANADYL VACANCY
OXYGEN VACANCY
l
W. Lambrecht*,
IN V205
:
SINGLE LAYER MODEL.
B. Djafari-Rouhani
*
and J. Vennik
Laboratorium voor Kristallografie en Studie van de Vaste Stof, Rijksuniversiteit Gent, Krijgslaan 271, B-9000 Gent, Belgium.
A single layer model of V205 with one vanadyl oxygen vacancy per unit Two vacancy derived bands in the main band gap cell is considered. Their origin and nature are disare interpreted as localized states. Infrared data are discussed but a definite interpretation is cussed. The total spin of the groundstate of the two-elecnot yet possible. tron system corresponding to the vacancy is in agreement with the EPR experimental data.
1. Introduction. The important role of the vanadyl oxygen vacancy (W) in V 05 for electrical and catalytic behaviour has o P ten been stressed in the literature (ref.l and refs. therein). The vacancy is furthermore supposed to provide localized donor levels near the bottom of the conduction band, which may govern the electrical conductivity of Surfa e vanadyl oxygen vacanthis materia12. cies and the related vE+ centers are believed to be active sites for the catalytic oxidation of hydrocarbons. Furthermore the transitions of V205 into lower oxides (V409 and V6013) which may take place during the initial state of the catalytic reactions at a V205-surface, proceeds by vacancy formatior;, accompanied by cert in $ IR (inlattice distortions . Experimentally, frared) and EPR (Electron Paramagnetic ResonanIn the present conce) spectra were reported. tribution we present preliminary results of a first theoretical treatment of the electronic structure of the W.
indications of how localized states might be produced. Some deficiencies of the present model are the following. 1) The vacancies are quite close to one another in the b-direction and consequently there may be considerable interactions between them, which will give rise to band dispersion of the localized states of an isolated vacancy. This dispersion is expected to be largest along IY and will give an idea about the mutual interaction between vacancies. 2) The periodic arrangement of vacancies considered here, imposes symmetry restrictions, not present for the isolated vacancy for which only one mirror plane, namely the one passing through the vacancy, remains. 3) Atomic relaxation effects are not taken into account. Two types of relaxation might be considered : a shift of the nearest vanadium towards the il oxygen plane; ii/ relaxation of down pointing vanadyl oxygens of the neighbouring layer. A general discussion of structural aspects and figures of the crystal structure may be found in reference'. For a representation of the effect (ii) see reference 4b fig. 4.14. For the surface W, the latter effect is of course absent and even in the bulk it is by no means certain to occur. The former effect would probably only slightly change the interactions between the nearest vanadium neighbour of the W and its surrounding oxygens and in a first approximation both effects may consequently be ignored. Apart from the perturbation consisting in the removal of an oxygen and the breaking of the related bonds, a second important effect must be considered at least in a qualitative manner. This is the charge perturbation created by the removal of a negatively charged oxygen, which will give rise to an attractive potential on the neighbouring atoms. A calculation of this potential was not attempted at this stage, because it involves the delicate balance of two nearly equal large effects : the inter-atomic Coulomb (or Madelung) effect gnd the intra-atomic Coulomb repulsion effects . The most im-
2. Model. In previous work6 it was shown by compari on between a full 3-dimensional band calculation 7 and our 2-dimensional calculation, that a single layer approximation accounts quite well for the main features of the band structure of V205. In order to obtain a first indication of what happens to the bandstructure upon creation of W's, we consider the same single layer tight binding model as used previously, but now introduce one W per unit cell. The model obtained in this way is one with at the bottom side of the layer all vanadyl oxygens present and at the top side alternatingly rows in the b-direction with and without vanadyl oxygens. This model is fairly unrealistic for a study of isolated vacancies, but nevertheless may provide a few
* **
Research assistant of the National Fund for Scientific Research Belgium. Present address : Equipe de Physique des Solides du C.N.R.S., Institut Supdrieur d'electronique du Nord, 3 rue Franfois Bass, 59046 Lille Cedex, France. 257
vANADYL
258
OXYGEN
portant effect will be felt by the nearest vanadium and consequently we studied this charge transfer effect by applying a semi-empirical shift a to the V3d-level of this atom, which we allow to assume values between 0 and -0.03 Ryd. 3. Results. As the vanadyl oxygen contribution to the valence band is situated in the middle and bottom part of the band, one does not expect localized levels to be created in the main gap originating from the valence band. A calculation of the levels in T, the centre of the Brillouin Although some zone, confirms this statement. changes may occur in the valence band, this will In the not be discussed in the present study. figures we consequently only show the energy leIn figure 1 the vels of the conduction band. levels in r are presented as a function of the parameter a, introduced above, and are compared to the corresponding levels for the perfect crystal. The symmetry with respect to the mirto the b-direction, ror planes m , perpendicular is indicatedYby + or - and levels which are believed to the interrelated are joined by dashed lines. A few conclusions can be drawn from this figure. 1) Some energy levels are shifted downwards into regions where in the case of the perfect crystal no levels are present in T and where also the density of states is low, as may be seen
VACANCY
Vol. 39, No. 2
IN V2O5
in figure 6. These levels furthermore depend more strongly on a than the other levels which could be an indication that they are more or less localized on the first vanadium and thus could present resonance behaviour. 2) In the energy regions where in the case of the perfect crystal the levels are closely spaced, which corresponds to a high density of states, the levels of the vacancy on the contrary have become more widely spaced, corresponding to a decrease in density of states. This anti-correlation between the peaks of the density of states and the changes in density of states induced by the vaca;cy, has been observed previously by Pollmann , and was explained to be a quite general effect on the basis of aGreen's function treatment. As for the presence of localized states, we must consider the lowest three levels at the bottom of the conduction band. Figure 2 shows the dispersion of the corresponding bands along TY.
-O’@jr _o,6*; ;; c?
s w
-0.3
1
N
V
1
+ +
t----+ t----+ -0.4
+
:
L
3 -0.51-+ W
+ l
+
i__ I
1
+ + +
1 =I
+ z i
a(Ryd) Figure 1 : Energy levels at the point for the normal (N) and vacancy (V) unit cell of V205. The vanadyl oxygen vacancy levels are given as a function of the parameter a, discussed in the text.
------.
I -’
-0,701
1 0 r
I ky
n/b Y
Figure 2 : Energy bands along the b-direction (I'Y) for the three lowest vacancy bands (full lines) for u=O, and for the lowest two perfect crystal bands (dashed lines).
Only the lowest band is completely separated from the normal conduction band and may consequently, with some confidence, be considered to correspond to a localized level of an isolated vacancy. The second level, even upon enlarging lul, does not separate from the conduction band. Nevertheless its center of gravity is clearly moving out of the band. This level, moreover, depends strongly on a as can be concluded from figure 1. This suggests that a second localized state may be present if c1 is strong enough, say -0.005 at least. The third level is much less a-dependent, and one is thus inclined to consider it rather as a normal band state. The fourth and fifth level may eventually give rise to localized states within the small gap found at the bottom of the conduction band, which was discussed in previous work6. The interpretation of certain bands as localized states is however by no means straightforward; the number of localized states may be larger or smaller.
VANADYL
Vol. 39, No. 2
OXYGEN
VACANCY
IN V205
259
In order to discuss the origin of the lowest two localized levels, which we will from now on call Ea for the level of symmetry - and Eb for the level of symmetry +, we consider the corresponding wavefunctions in table 1. Table
1
: Eigenvectors nadium
of localized contribution.
states
'a Y Vl v2 v3 v4
&I1 0.146 0.398 0.130
: va-
'b
Y -0.5:7 O.Z9 -0.094 -0.228 0.200 -0.171 0.084 I 0.012
3z2-r* 0.149 -0.030 -0.026 -0.006
x2_ 2 0.180 -0.142 -0.132 0.047
As the oxygen contribution is always an order of magnitude lower, only the vanadium contribution is presented. Due to symmetry restrictions, 'Pa contains only2xy2and yz2coyponents and Yb contains only xz, 32 -r and x -y . The numbering of the atoms is as indicated in figure 3.
Figure
3
: Numbering
of atoms in the unit cell.
Y, is primarily delocalized in the a-direction and Yb is fairly localized on the first vanadium Vl. Taking into account the signs on the coefficients, we conclude tha:. Y may be considered to be related to the perfectacrystal T3-state and Yb to the T7-state. As the coefficents on different atoms are different in the case of the vacancy, these levels cannot have these symmetries and thus will in fact be a mixture of several conduction band states, but at least their main constituent orbitals are oriented relative to each other in the same way as for I3 and T7. It is interesting to note that for both these symmetries the relative orientation of orbitals corresponds to indirect bonding of the vanadium orbitals across the bridge oxygen : for Y between xy-orbitals and for Yb between The case of Y is illustrated in xz-or e.itals. levels are lofigure 4. The reason why &ese wered in energy with respect to the case of the perfect crystal, may be understood as follows. Breaking the vanadiumvanadyl-oxygen pdn The interaction lowers the yz- and xz-levels. strongest effect is expected for Eb, which has primarily xz-character, whereas E has only partially yz-character and is actually rather an Application of the potential xy-derived level. a, further lowers the vanadium levels. However it is not clear in how far the present conclusions about the wavefunctions may be extrapoThe lated to the case of the single vacancy. removal of certain symmetry restrictions will
Figure 4 : Orbitals and interactions involved in Y the lowest localized state. The size of the irbitals is drawn in agreement with their contribution to the wave function. The full lines indicate direct, the dashed linges indirect interactions. The bonding or antibonding effect is indicated by B or AB respectively.
allow mixing of (xy,yz) with (xz, 3z2-r2, x2-y2) and will also not impose a fixed orientation and magnitude of orbitals in the neighbouring unit cells; further delocalization may thus occur. In the present case we can for instance conclude that delocalization in the vanadium chain via Vl will be small, but we cannot say anything about delocalization onto the Vl atoms of the adjacent unit cell. Furthermore, the mutual interaction of the close lying vacancies consistrongdered here, may perturb the wavefunctions ly. Charge accumulation could occur on an atom like V2 which is as close to the vacancy of its own unit cell as it is to that of the adjacent unit cell. 4. Comparison
with experimental
4.1. Infrared
Spectroscopy.
data.
The optical infrared band, corresponding to the W is reproduced in figure 5 (after Clauws The different features are labeled et al. 4). (A-E) for further reference. The transition energies involved and the closeness of the defect levels to the bottom of the conduction band, as obtained in the present calculations, indicate that the spectrum should result from transitions from the localized states towards the conduction band, and not from transitions between different defect levels. The general shape of the density of states in the conduction band, consisting in a large peak, followed by a second smaller one, which is also observed for the experimental spectrum, especially for polarization E//a, is a further indication for the validity of this interpretation. The defect levels of the isolated W should however lie at somewhat lower energies, in order to allow a straightforward interpretation of the spectra in this way. Another possible interpretation of the spectrum, which emphasizes the polarization dependence and consequently involves the selection rules for dipole transitions, is as indicated in figure 6. The peaks in the density of states arelabeled according to their critical points in the Brillouin zone. In this case however, eitner drastic changes in the density of states or transition probability effects must be assumed to explain why the first peak in the densi-
260
VANADYL
OXYGEN
VACANCY
hv.
IN V205
Vol. 39, No. 2
eV
Figure 5 : Infrared specumtrum of the vanadyl oxygen vacancy as taken from reference 4b. The labeling of the peaks is added to the original figure.
-0.55
r;; c -0,60 w
II
D
l
-0.701 0
I 100
200
300
LOO
I
500
DOS Figure 6 : Tentative interpretation of the infrared spectrum. The density of states corresponds to the perfect crystal and is obtained from diagonalization of the full tight binding matrix and not from an effective hamiltonian for the conduction band alone as in ref. 6b, which slightly modifies the higher part of the band. The energy region believed to be of interest here is shaded and the symmetry of peaks in the density of states and localized states is indicated.
ty of states would be suppressed. The only conelusion which can be drawn with a certain degree of confidence is that the conduction band states are involved in the optical transitions as final states. A definite assignment of the different features of the IR-band seems as yet impossible.
4.2. ---____________Electron paramagnetic
resonance.
With the present limited knowledge of the vacancy electronic structure, it is not possible to give 5" detailed discussion of all the EPR parameters ; we will in fact limit the discussion to the total effective spin S of the ground sta-
VANADYL
Vol. 39, No. 2 Table 2 S
OXYGEN
VACANCY
: Ground state determination eigenvalues
of interaction
1
Ea+Eb +Jab-K
0
E +E +Jab+Kab a b
c +E a
261
IN Vz05
of the two-electron hamiltonian
system.
numerical
value
12.34 eV
ab
13.25
b
and K stand for the direct and exchange ab ab and Yb,. similarly for Jaa and Jbb.
3
te.
In order to see whether an S=l ground state is possible for the system under consideration, as is required by experiment, we performed a simplified configuration interaction calculation, including only the lowest two localized states E and E . This approximation is by no means tritial hue as we are only interested in obtaining the spin of the groundstate, it is The reasonable to take it as a starting point. procedure followed is a diagonalization of the Coulomb-interaction matrix in the basis of all possible spin configurations on the two levels The Coulomb integrals between the E and Eb. w&e functions I and fb were determined by introducing their gxpansion in atomic orbitals, which are cut off after the nearest neighbours. The following supplementary approximations are introduced for the remaining Coulomb integrals : only one and two center terms are included and only for the one center terms exchange contriFor the twobutions are taken into account. center terms we furthermore neglect the diffeNo screerence between different orbitals. ning of the Coulomb interaction is considered. This latter effect could drastically change the involved energies, but would probably not change
integrals
between
Ya
the order of the levels. This, as may be seen in the equations included in table 2, would only be the case if the energy level difference E,-Eb would
References.
(1) L. Fiermans, P. Clauws, W. Lambrecht, L. Vandenbroucke and J. Vennik, Phys. Stat. Solidi (a) 59, 485, 1980. (2) J. Haemers, E. Baetens and J. Vennik, Phys. Stat. Solidi (a) 0, 381, 1973. (3) G. Grymonprez, L. Fiermans and J. Vennik, Acta Crystallogr. AZ, 833, 1977. (4) a. P. Clauws and J. Vennik, Phys. Stat. Solidi (b) 66, 553, 1974. b. P. Clauws, Studie van optische eigenschappen van vanadiumoxides in het infrarood, Koninklijke Academic voor Wetenschappen, Letteren en Schone Kunsten van Belgie, Brussel 1980.
(5) M. Van Haelst and P. Clauws, Phys. Stat. Solidi (b), I, 719, 1978. M. Lan(6) a. W. Lambrecht, B. Djafari-Rouhani, noo, P. Clauws, L. Fiermans and J. Vennik, J. Phys. C : Solid State Phys. E, 2503, 1980. b. W. Lambrecht, B. Djafari-Rouhani and J. Vennik, to be published. (7) D.W. Bullett, J. Phys. C : Solid State Phys. 13, L595, 1980. (8) x Lannoo and J.N. Decarpigny, Phys. Rev. B8, 5709, 1980. XX, Advances (9) J. Pollmann, FestkGrperprobleme in Solid State Physics, 117, 1980.