Quantum-chemical simulations of free and bound hole polarons in corundum crystal

COMPUTATIONAL MATERIALS SCIENCE

Computational Materials Science 7 (1997) 285-294

Quantum-chemical simulations of free and bound hole polarons corundum crystal

Yu.F. Zhukovskii a7bT*, E.A. Kotomin a, R.M. Nieminen ‘, A. Stashans aTc

a Insritufe of Solid Smre Physics, Uniuersify oflatviu, Kengnraga 8, LV-1063 Rigu, Laruiu b Laboratory of Physics, Helsinki Uniuersiiy of Technology, Ofakaari I, FIN-02150 Espoo, Finlund

’ Department of Quantum Chemistry, Uppsalu University, Box 518, S-751 20 Uppsala, Sweden

Received 1 July 1996; accepted 28 August 1996

in

Abstract

The semi-empirical method of the so-called intermediate neglect of differential overlap (INDO) has been applied to the calculations of the hole small-radius polarons in corundum crystals. Results for optimized atomic and electronic structure using two different approaches (the molecular cluster and periodic, supercell model) are critically compared. It is shown that the main results are similar in both cases.

1. Introduction

Corundum (cu-Al,O,) is an important ceramic

material [l] which is also used as a substrate for thin film growth and as an optical material. Of possible important applications, its use in single-crystal IR

fibers [2] is perhaps the principal one. The theoretical studies of pure corundum crystals started only re- cently [3-71 with emphasis mainly on surface prop- erties, interfaces and adsorption [8,9]. These studies were carded out using a number of quantum-mecha- nical techniques, including semi-empirical (extended Hiickel, complete neglect of the differential overlap, tight binding) and ab initio (Hartree-Fock, orthogo- nalized LCAO in the local density approximation), as well as both cluster and periodical models. These calculations are greatly complicated because of the

’ Corresponding author.

non-cubic structure of the corundum crystal (space group D& with 10 atoms per unit cell) and its partly covalent chemical bonding [lo]. Along with quan- tum-mechanical studies, the atom-atom potential ap- proach, as implemented in the Harwell HADES II [l I] and MIDAS codes, has been successfully ap- plied to pure corundum crystals, their surfaces and impurities therein [ 12,131.

Up to recently, electronic defect properties in corundum have not been studied satisfactorily, espe- cially theoretically [ 14,151. Along with corundum’s importance as a potential dosimeter material, such a study is of considerable fundamental importance. In particular, the question of whether free holes can be self-trapped in a regular oxide lattice, as they are in alkali halide crystals, has been debated for a long time [15-171.

This is why recently we used the semi-empirical method of the intermediate neglect of differential overlap (INDO) for the study of the electronic struc-

0927-0256/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SO927-0256(96)00090-O

286 Yu.F. Zhukovskii et al./Computational Materials Science 7 (1997) 285-294

ture of electron centers (F, F,) [ 181 as well as both self-trapped holes (SIX) [19] and holes trapped by defects [20] in the bulk corundum. Our main find- ings, achieved using large stoichiometric quantum clusters, are the following: * both one-site (atomic) and two-site (quasi-molec-

ular) free polarons are energetically favorable; the latter configuration has lower energy;

* the same is true for a hole polaron trapped by an Mg atom substituting one of the nearest Al atoms (the so-called V,, center). Our use of the cluster model has a potential threat

that obtained results are affected by the boundary conditions imposed on the finite quantum cluster (the electrostatic field of the rest of the infinite crystal). This is why in the present paper we repeat our previous INDO study using a ‘periodic, supercell’ model. With the help of this model, we study care- fully both STH and two different configurations of V Mg centers (only one of these configurations was considered by us earlier [20]).

2. Theoretical background

The quantum-chemical INDO method has been used successfully in order to study oxide crystals and defects therein, including a-Al,O, [18,19], MgO [21], Li,O [22], and SiO, [23]. This method, modi- fied for defect studies in ionic and semi-ionic solids, was described in detail earlier [23-251. For calcula- tions of a pure crystal the periodic large unit cell model was used [26,27], whereas a cluster model [28] was used for defect calculations since the cen- ters under study are charged with respect to the perfect crystalline lattice. The use of the INDO method allowed us to perform self-consistent calcu- lations of the corundum electronic structure for both models: 65-atom cluster Al,,O,, and 80-atom 2 X 2 X 2 supercell Al,,O,,. For the simulation of V,, centers in these models one of the Al atoms was substituted by Mg (Figs. l-3). The valence basis set in both cases included 3s and 3p atomic orbitals (AO’s) on Al, 2s and 2p AO’s on 0 atoms as well as a 3s A0 on the Mg atom. The set of INDO parame- ters [24] used for these calculations is described elsewhere in detail [ 19,251.

0 0 atom q two Al atoms

Fig. 1. hojection of atoms in the stoichiometric Al,,O,, cluster

onto the (OG01) corundum plane (either Al(l) or Al(4) atoms were

replaced by Mg atoms, in order to simulate NNN or NN types of V Mg defects, respectively). The quantum cluster consists of three

basal 0 planes shown in white, light-grey (2.17 A below figure

plane) and dark-grey (2.17 A above one). For all the models of

defects the two-site hole is shared by the o(1) and O(2) atoms,

whereas the one-site hole is localized on the o(3) atom.

The stoichiometric molecular cluster (MC) con- tains 13 basic structural elements of the corundum, each containing a structural triangle of 0 atoms with two atoms of Al situated symmetrically above and below 0 plane (Fig. 1). This cluster is embedded into the electrostatic field of non-point ions. The populations of AO’s of these non-point ions outside cluster are ‘frozen’ and equal to those in the periodic AldO unit cell calculation of the perfect crystal. In the cluster calculations, the peripheral boundary atoms were allowed to relax to the equilibrium ge- ometry until the total energy minimum is reached, in order to take into account the effect of their broken bonds with atoms outside the cluster. We discussed this point in more detail in previous papers [19]; the boundary relaxations were found to be typically sev- eral percent of the corresponding crystallographic distances in perfect corundum. Then a hole was inserted into a cluster (perfect, or Mg-substitutional), and different ionic relaxations were imposed in order to find the local minima on the total energy surface

Yu.F. Zhukouskii et al./ Computational Materials Science 7 (1997) 285-294 287

without any a priori assumptions about the electron (hole) density distribution. Recently we have carried out a detailed analysis of various boundary condi- tions for both 35 and 65-atom MC models of STH defects in corundum using their INDO simulation

1291. The 80-atom corundum 2 X 2 X 2 supercell con-

sists of eight primitive unit cells (UC>. Each cell contains two Al,O, formula units rotated with re- spect to each other by 180” (upper part of Fig. 2a). Such a supercell (SC) may be formed by the exten- sion of a single lo-atom UC by a factor of two along the three translational vectors. Each Al atom in a perfect corundum structure is surrounded by a dis-

a.

torted octahedron of, 0 atoms forming two kinds of Al-O bond: 1.89 A (oxygen atom is the nearest neighbor to Al atom, NN) and 1.93 A (0 is the next-nearest neighbor to Al, NNN). In turn, each 0 atom is surrounded by four Al ones, each two of them at these distances. Obviously, every Al,O, unit contains only the NNN Al-O bonds; NN bonds are formed by nearest Al and 0 atoms belonging to the nearest, but not the same UC (Fig. 3a). For instance, O(l), O(2), and o(3) atoms of the basic oxygen triangle arranged in the center of 65atom fragment of the corundum lattice have six NN Al atoms from adjacent units (Fig. 1): three upper Al atoms belong- ing to the ‘light-grey’ units, including Al(4) atom,

b.

Fig. 2. 1%atom crystalline fragments of corundum containing three Al,O, formula units (two neighboring units form a primitive unit cell Al,O,) where a Mg atom replaces a regular Al atom simulating both NNN type of V,, defects (a) and STH (b). In these two models a hole

is either shared by the o(1) and o(2) atoms (quasi-molecular polaron) or localized on the o(3) atom (one-site polaron). The arrows indicate the main directions for the atom displacements during structure relaxation.

288 Yu.F. Zhukooskii et al./ Computational Materials Science 7 (1997) 285-294

and three lower Al atoms from ‘dark-grey’ units. At the same time, Al(l) and Al(2) atoms are the only NNN for these oxygen atoms.

An automated procedure of atomic relaxation within both MC and SC models has been used for the simulation of various configurations of free and

bound hole polarons in a corundum crystal. STH defects have been simulated either by embedding of a charged Al,,O$ cluster into perfect corundum

lattice (MC model, Fig. 1) or by a substitution of a remote Al atom by Mg in a neutral crystalline frag- ment (SC model, Fig. 2b) keeping the supercell neutral. (The distance between the STH region within supercell and the remote Mg atom is large enough to avoid its direct influence to the properties of STH, in particular, the hole density distribution in the defec- tive region.) The first stage of this relaxation proce- dure included both a structural optimization of the corresponding Al,O, formula unit for both models

a.

as well as a simultaneous variation of Mg position for the SC model. As the next step a relaxation of the adjacent formula units was also carried out. A simi- lar procedure has been used for simulation of NNN V,, defects (Fig. 2a). The first stage of a simulation of NN V,, defect included the geometry optimiza-

tion for atoms O(l), O(2), O(3) forming a basic oxygen triangle as well as for both NN Mg and NNN Al(l), Al(2) atoms (Fig. 3a). Continuation and com- pletion of this procedure included also relaxation of all the surrounding Al,O, units (shown in ‘grey’ in Fig. 1). Such a way of defect simulation corresponds to the hypothetical two-stage process of the hole self-trapping [29]: localization of a free hole on one or two atoms in a perfect crystal region at the first stage, and the subsequent relaxation of the lattice with accompanying electron redistribution in this local region at the second stage, until the self-trapped (ST) state is formed.

b.

Fig. 3. IO-atom crystalline fragment of corundum containing two AI,O, formula units from the adjacent UC where an Al atom from a

neighboring (lower) unit is replaced by a Mg atom, in order to simulate NN type of V,, defects (a - initial, unrelaxed crystalline structure,

b - relaxed structure). The two-site hole is shared by the o(1) and o(2) atoms. The shaded region corresponds to = 90% spin density

localization. The arrows indicate the main directions for atom displacements during structure relaxation.

Yu.F. Zhukovskii et al./Computatioml Materids Science 7 (1997) 285-294

Table 1

Results of calculations using both cluster (MC) and supercell (SC) models of STH and V,, defects in corundum

Feature Model Two-center STH Various types of V,, defects

two-center NN one-center NNN two-center NNN

IEJ (eV) MC 5.30 3.80 0.92 4.3 1 SC 3.41 7.46 1.61 4.30

spin density (e) MC 0.92fO.46 + 0.46) 0.93co.43 + 0.50) 0.66 0.91cO.45 + .0.46) SC 0.92cO.45 + 0.47) 0.91(0.44 + 0.47) 0.70 0.91fO.45 + 0.46)

“Since all relaxation energies are negative (i.e. defects are stable) their absolute values are given here.

289

The stability of various hole defects in a crystal may be characterized by the sign and magnitude of the relaxation energy, E,,. This is a gain in energy due to displacement of the lattice atoms induced by the net charge of the localized hole:

&I = E&f - qxrf ’

where Eder is the total energy of cluster or supercell for the fully relaxed hole state, EFrf is the total energy of the relevant unrelaxed fragment of a lat- tice. Values of the relaxation energies and spin densi- ties for stable configurations of both STH and V,, defects, which correspond to the local minima on the total energy surface, are shown in Table 1. The initial (unrelaxed) and relaxed atomic coordinates of central fragments of all these configurations for both

models are presented in Tables 2 and 3. The origin of coordinates for all these defect configurations coincides with a center of the basic oxygen triangle

0(1)0(2)0(3) (Fig. 1).

3. Results and discussion

In our INDO calculation of a perfect corundum we found the effective atomic charges to be 2.34 e (MC)/2.36 e (SC) on Al as well as - 1.56 e (MC)/ - 1.57 e (SC) on 0 atoms. These charges are in relatively good agreement with data of recent ab initio calculations (2.02 e on Al and - 1.35 e on 0) [6]. The Mulliken population analysis shows that

Table 2 Optimized coordinates (in A) of atoms nearest to the two-center STH obtained for both MC and SC models

Atomic coordinatesa Perfect structure Optimized structure

MC SC MC SC

Al(l) x

Y ‘

AK21 X

Y L

o(l) x

Y Z

o(2) x

Y L

o(3) X

Y Z

0. 0.

0. 0.

1.2550 1.2413

0. 0.

0. 0.

- 1.2550 - 1.2413 - 1.1001 - 1.0881

0.635 1 0.6282 0. 0. 1.1001 1.088 1 0.635 1 0.6282

0. 0. 0. 0.

- 1.2702 - 1.2564

0. 0.

0.0748

- 0.0578

1.4288

- 0.0748

- 0.0578

- 1.4288

- 0.6365 0.7112

0.0447

0.6365

0.7112

- 0.0447 0.

- 1.1602

0.

0.0349

0.0087

1.4094

0.0280

0.0157

- 1.4096 - 0.593 1

0.6982

0.05 12

0.6405

0.6027 - 0.0221

-0.1212

- 1.1929

-0.0115

aAtomic numbering corresponds to Figs. 1 and 2b.

290

Table 3

Yu.F. Zhukmskii et al./ Computational Materials Science 7 (1997) 285-294

Optimized coordinates (in A) of atoms nearest to the V,, defect obtained for both MC and SC models

Type of defect Atomic coordinates’ Perfect structure Optimized structure

MC SC MC SC

NN Mg

NNN one-center

NNN two-center

two-center

Al(l)

Al(2)

o(l)

00)

o(3)

Mg

Al(2)

o(l)

o(2)

o(3)

Mg

Al(2)

o(1)

o(2)

o(3)

x

Y z x

Y Z x

Y Z x

Y Z x

Y 2

x

Y 2

x

Y Z x

Y Z x

Y Z x

Y Z x

Y Z

x

Y

Z x

Y Z x

Y

Z x

Y Z x

Y Z

- 1.2606 - 1.2469 2.1834 2.1597

- 0.737 1 - 0.7291

0. 0. 0. 0. 1.2550 1.2413 0. 0.

0. 0. - 1.2550 - 1.2413

- 1.1001 - 1.0881 0.635 1 0.6282 0. 0. 1.1001 1.0081 0.635 1 0.6282

0. 0. 0. 0.

- 1.2702 - 1.2564 0. 0.

- 1.1909 - 1.1293

2.1210 2.0994

- 0.6732 - 0.6366

0.0151 0.035 1 0.0126 0.0276

1.4104 1.4277

0.0452 0.0533

- 0.0063 0.0181

- 1.4137 - 1.3192

- 0.7523 -0.8212

0.4457 0.3715

-0.1980 - 0.0735

0.665 1 0.5912

0.6238 0.685 1

0.0196 0.0279

0.2913 0.397 1

- 1.2674 - 1.2237

- 0.0469 -0.0185

0. 0. 0. 0. 1.2550 1.2413 0. 0. 0. 0.

- 1.2550 - 1.2413 - 1.1001 - 1.0881

0.635 I 0.6282 0. 0. 1.1001 1.0881 0.635 I 0.6282 0. 0. 0. 0.

- 1.2702 - 1.2564 0. 0.

0.0308

0.0119

1.4229

0.0085

0.0041 1.3221

- 1.0238

- 0.0178

1.0478

0.5296

- 0.0054

- 0.0325

- 1.0302

- 0.0102

0.0444

0.0499

1.4576

0.0392

0.0575 - 1.2934

- 1.0266

0.6543

- 0.0953

1.1030

0.5860

- 0.0423

-0.0810

- 1.0088

- 0.0529

0. 0. 0.0104 - 0.0204 0. 0. 0.0398 0.0590 1.2550 1.2413 1.3762 1.3548 0. 0. 0.0170 -0.0113 0. 0. 0.0601 0.0889

- 1.2550 - 1.2413 - 1.396 - 1.4282 - 1.1001 - 1.0881 - 0.7068 - 0.6975

0.635 1 0.6282 0.7513 0.8474 0. 0. - 0.0523 -0.1601 1.1001 1.0881 0.5725 0.7335 0.635 1 0.6282 0.8765 - 0.7274 0. 0. -0.1569 -0.1205 0. 0. 0.1452 -0.1736

- 1.2702 - 1.2564 - 1.3335 - 1.3081 0. 0. - 0.2097 -0.1385

aAtomic numbering corresponds to Fig. 2a (NNN defects) and Fig. 3 (NN defect).

Yu.F. Zhukovskii et al./Computational Materials Science 7 (1997) 285-294 291

O-O bonds are practically non-populated by elec- trons whereas both NN and NNN kinds of Al-O bonds reveal rather high electron populations (0.258 e and 0.170 e, respectively). That is, the Al-O bonds are directly responsible for the partly covalent nature of the chemical bonding in corundum.

The SO-atom supercell calculation gives the total energy minimum at theoretical lattice constant being 10% less than the experimental value. This differ- ence is twice as large as the corresponding value found earlier [19] for a small AldO supercell. These data indicate the necessity of further improvement of INDO parameters for corundum. However, careful atomic relaxation within the 80-atom supercell shows that the unit cell symmetry remains unchanged.

3.1. STH centers

We calculated MC and SC models of both ‘one’- site and ‘two’-site free polarons. The optimized STH geometry for a free quasi-molecular polaron shows that two oxygen atoms O(1) and O(2) are strongly relaxed inwards (= 40% of the O-O distance in a regular lattice) which is accompanied by an outward = 20% displacement of both NNN Al(l) and Al(2) atoms on each side of the 0 triangle (Fig. 2b). The displacement vectors of these two 0 atoms form an angle of = 20” with respect to the straight line connecting their perfect-lattice sites. (The same con- figuration is predicted for a free three-atom molecule by the general Jahn-Teller theory.) More detailed information on the structure of the basic Al,O, unit containing STH centers is presented in Table 2. The supercell STH geometry is close to that obtained in our previous MC calculations [19,29]. The quasi- molecular model of a free polaron obtained in our calculations (for both MC and SC approaches) is very similar to the one known for a long time as the V, center in alkali halides [30]. Similar diatomic centers were also discovered in EPR experiments on fused SiO, glasses [31].

For the MC model of quasi-molecular STH the total energy gain due to hole self-trapping is 5.3 eV [29]; its main contribution comes from the linear relaxation of the two oxygen atoms O(1) and O(2) (3.5 eV). These 0 atoms share more than 90% of the hole density. A closer analysis of the origin of this strong localization shows that it arises from the

chemical bonding between the two mentioned oxy- gen atoms: the population of their bond is close to that one for the Al-O bond in a perfect crystal. A value of 1 E,,) for SC model (3.4 eV> is essentially less than for the MC model. Moreover, an automated atomic relaxation done in the supercell model shows rotation by = 6” of the basic oxygen triangle O(1) o(2) O(3) with respect to the Al(l)-Al(2) axis. Such a rotation is accompanied by a simultaneous Mg atom displacement from that axis and is analogous to what is observed for the so-called off-center impuri- ties in ionic solids.

The supercell model indicates that one-site STH center localized on the O(3) atom (Fig. 2b) is unsta- ble with respect to the transformation into a quasi- molecular configuration. In turn, the cluster model predicts the existence of a local minimum at the potential energy surface corresponding to such a polaron, with E,, = 1.9 eV. However, we were un- able to find the barrier height of its transformation into a quasi-molecular form. Possible reasons for the disagreements considered could be attributed to the difference between both models. The results obtained for SC model could be affected by both the interac- tion of periodically arranged hole defects from dif- ferent supercells and possible interaction between holes and Mg atoms within each supercell. On the other hand, the MC model neglects bond breaking of atoms at the cluster boundary.

3.2. VMg centers

By analogy to STH simulations, two spatial con- figurations of V,, centers (Figs. 2 and 3) were carefully studied and compared for MC and SC models (Tables 1 and 3): atomic polaron with a hole localized on a single O(3) atom and quasi-molecular polaron where a hole is shared by the two atoms, o(1) and O(2). Both cluster and supercell models agree that these bound polarons could co-exist (E,, is negative) but give a preference for the two-site configurations as the energetically most favorable. In

*e vMg centers a hole is trapped by an Mg atom having a smaller charge than an Al atom which it substitutes for, effectively Mg has a negative charge with respect to the perfect corundum lattice. In a purely ionic model its charge would be - 1 e. In our INDO calculations we obtained qMg = 1.90 e

292 Yu.F. Zhukouskii et al./Computational Materials Science 7 (1997) 285-294

whereas for an Al atom in perfect corundum qA, = 2.34 e. Therefore the effective Mg charge is only -0.44 e. Thus, v,, center may be considered as a perturbed STH. I

The only stable one-site V,, configuration is found there where the Mg impurity is in the NNN cation position with respect to an O--ion. The stable structural configuration of this defect is character-

ized, first of all, by a considerable displacement of an O(3) atom (Figs. 1 and 2a) towards the center of the basic oxygen triangle (Table 3). Hole localization on this atom has mainly an electrostatic (Madelung) origin since it is essentially closer to the negatively charged Mg ion than at the regular site. Both models indicate the instability of the one-site conformation for the NN configuration of V,, center (Fig. 3).

As to the quasi-molecular configurations of the V,, defect, the supercell model favors the Mg ion in the NN position whereas the cluster model gives a preference for its NNN position.

A comparison of data presented in Tables 2 and 3 shows that the two-site NNN type of the V,, centers has a similar geometry with the corresponding STH defect described above. In particular, the upward shift of O(1) and O(2) atoms from the basic oxygen triangle plane is negligible (Table 3). Bond popula- tion for this pair of 0 atoms sharing a hole in a cluster model is = 0.3 e, i.e. slightly higher than in the STH case (0.267 e). In turn, Al-O bond popula- tion (0.228 e) is reduced as compared with that in perfect corundum (0.258 e). The values of E,, calculated for this NNN center coincide for both models (Table 1). Moreover, an automated atom relaxation in cluster and supercell models for this configuration indicates also rotation by = 7” of the basic oxygen triangle with respect to the Al( 1 )-Al(2) axis. A gain in the relaxation energy caused by both the rotation of this oxygen triangle and the displace- ment of all the atoms of the basic Al,O, unit from positions of symmetrical quasi-molecular configura- tion of NNN VMs defect is found to be 0.5 eV.

The most interesting results are obtained for the two-site NN V,, center. Its equilibrium configura- tion is shown in Fig. 3b. During the optimization procedure atoms of the oxygen triangle 0( 1)0(2)0(3) are synchronously displaced along the two trajecto- ries: all of them rotate by = 15” with respect to the Al(l)-Al(2) axis, and simultaneously O(1) and O(2)

Table 4

Experimental coordinates (in A) of atoms in corundum primitive

UC for both perfect structure [33] and V,, center [32]

Atomic coordinate? Perfect corundum Corundum doped with Mg*’

AItl)/Mg x 0. 0.

Y 0. 0. Z 1.3245 1.3338

Al(2) X 0. 0.

Y 0. 0.

Z - 1.3245 - 1.3245

o(1) X - 1.2620 - 1.2620 Y 0.7286 0.7286

Z 0. 0.

o(2) X 1.2620 1.2620 Y 0.7286 0.7286

Z 0. 0.

o(3) x 0. 0.0608

Y - 1.4572 - 1.6116

2 0. -0.1667

‘Atomic numbering corresponds to Fig. 2.

atoms relax towards each other (as in the case of two-site centers of both STH and NNN type of V,, defects). At the same time, the Mg atom shifts towards the basic oxygen triangle but the Al(l) and Al(2) atoms are displaced outwards from it. Such a careful optimization for NN configuration of V,, center we carried out for the first time. The relaxed geometry of all these atoms for both models is summarized in Table 3. Due to a complicated lattice structure of corundum we were unable to estimate the transition barrier between the NN and NNN configurations of V,, defects on a total energy surface.

V,, centers were studied experimentally by means of ESR/ENDOR technique [32]. In these works the one-site hole polaron was discovered, and its pro- posed geometry is given in Table 4. As follows from these data, the experimental estimate of the defect geometry greatly differs from the results obtained by us (Table 3). Our calculations show that the ‘experi- mental’ geometry is even less energetically favorable than the perfect crystal geometry: the relevant E,, 1 0.3 eV. There are several reasons why such a geome- try could be inadequate. In fact, only the relative distances between atoms could be extracted from

Yu.F. Zhukouskii et al./Cmnputational Maferials Science 7 (1997) 285-294 293

ESR/ENDOR measurements, e.g. it is not clear from [32] whether the Mg atom is displaced from a regular Al site and the O--atom is positioned in the plane containing three nearest Al atoms (two of them lie below the basal 0 plane whereas the third one lies above it). Moreover, it is also not clear if Mg existed at all in samples studied [32] or an Al vacancy led to the same effect. Secondly, in these estimates the considerable covalency of the conm- dum crystal was neglected. Therefore, the experi- mental search of the theoretically predicted two-site V,, configuration could be of great interest. Proba- bly, its observation by ENDOR is masked by a great number of lines in the relevant experimental spec- trum for such a low-symmetry defect. In this respect, Raman scattering spectroscopy seems to be a more promising tool.

4. Conclusions

We have demonstrated in this paper that both cluster and supercell models are in favor of the two-site (quasi-molecular) hole polaron in corundum as lowest in energy. This is true for both the free, small-radius polaron (self-trapped hole) and the po- laron bound at Mg impurity. For the latter the peri- odic supercell model gives a preference for the con- figuration where the Mg ion substitutes a nearest (NN) Al atom. Further ab initio calculations of these defects with electron correlation effects included are of great interest.

Acknowledgements

E.K. was supported by the EC HCM Network Project on “Polarons, bipolarons and excitons. Prop- erties and occurrence in new materials” (contract No. ERBCIPDCT94-0031). Y.Z. is greatly indebted to the CIMO Foundation (Finland) for the financial support and members of the Laboratory of Physics at Helsinki University of Technology for warm hospi- tality during his stay there. These calculations were performed using computer facilities of the Center for Scientific Computing (CSC) whom we also grate- fully acknowledge.

References

[l] W.D. Kingery, ed., Structure and Properties of MgO and AlaO, Ceramics. Advances in Ceramics, Vol. IO (American

Ceramic Society, Columbus, OH, 1983).

[2] G. Meaberg and J.W. Hanington, in: Infrared Fibre Optics III. Proc. SPIE, 1591 (SPIE, Bellingham, WA, 1991) p. 13.

[3] I.P. Batra, J. Phys. C 15 (1982) 5399; S. Ciraci and L.P.

Batra, Phys. Rev. B 28 (1983) 982.

[4] S. Nagel, J. Phys. C 18 (1985) 3673.

[5] R.A. Evarestov, A.N. Ennoshkin and V.A. Lovchikov. Phys. Status Solidi B 99 (1980) 387.

[6] M. Cat&, R. Dovesi, C. Roetti, E.A. Kotomin and V.R. Saunders, Chem. Phys. Lett. 140 (1987) 120; R. Dovesi, C.

Roetti, M. Causa and C. Pisani, in: Structure and Reactivity

of Surfaces, eds. C. Moreterra, A. Zecchina and G. Costa

(Elsevier, Amsterdam, 1989) p. 385; L. Salasco, R. Dovesi,

R. Orlando, M. Causa and V.R. Saunders, Mol. Phys. 72

(1991) 267.

[7] Y.N. Xu and W.Y. Ching, Phys. Rev. B 43 (1991) 4461; W.Y. Ching and Y.N. Xu, J. Am. Ceram. Sot. 77 (1994)

404.

[8] C. Pisani, M. Causa, R. Dovesi and C. Rcetti, Prog. Surf. Sci. 25 (1987) 119; M. Causa, R. Dovesi, C. Pisani and C.

Roetti, Surf. Sci. 215 (1989) 259.

[9] M. Kohyama, S. Kose, M. Kinoshita and R. Yamamoto, J. Phys. Chem. Solids 53 (1992) 345.

[lo] J. Lewis, D. Schwarzenbach and H.D. Flack, Acta Crystal- logr. A 38 (1982) 733.

[l I] C.R.A. Catlow, R. James, W.C. Mackrodt and R.F. Stewart, Phys. Rev. B 25 (1982) 1006.

[12] C.R.A. Catlow and W.C. Mackrodt, eds., Computer Simula- tion of Solids (Springer, Berlin, 1982).

[13] P.W. Tasker, in: Structure and Properties of MgO and Al,O, Ceramics. Advances in Ceramics, ed. WD. Kingery, Vol. 10

(American Ceramic Society, Columbus, OH, 1983) p. 176.

[14] J.H. Crawford, Jr., Semicond. Insulat. 5 (1983) 599.

[15] J. Valbis and N. Itoh, Radiat. Eff. Def. Solids 1 16 (1991) 171.

[16] I. Tale, P. Kulis, Z. Rachko, M. Springis and 3. Jansons, Radiat. Eff. Def. Solids 119-121 (1991) 963.

[17] P.W.M. Jacobs and E.A. Kotomin, Philos. Mag. A 68 (1993) 695.

[ 181 A. Stashans, E.A. Kotomin and J.-L. Calais, Phys. Rev. B 49 (1994) 14854; E.A. Kotomin, A. Stashans, L.N. Kantorovich,

AI. Lifshitz, AI. Popov, I.A. Tale and J.-L. Calais, Phys.

Rev. B 51 (1995) 8770.

[I91 P.W.M. Jacobs and E.A. Kotomin, Phys. Rev. Lett. 69 (1992) 1411; P.W.M. Jacobs, E.A. Kotomin, A. Stashans and

I. Tale, Philos. Mag. B 67 (1993) 557; P.W.M. Jacobs, E.A.

Kotomin, A. Stashans, E.V. Stefanovich and I. Tale, J. Phys.:

Condens. Matter 4 ( 1992) 753 1.

[20] P.W.M. Jacobs and E.A. Kotomin, Nucl. Instr. Methods Phys. Res. B 91 (1994) 52; P.W.M. Jacobs, E.A. Kotomin,

A. Stashans and 1. Tale, Model. Simul. Mater. Sci. Eng. 2

(1994) 109.

294 Yu.F. Zhukouskii et al./ Computational Materials Science 7 (1997) 285-294

[21] A.L. Shluger, M. Georgiev and N. Itoh, Philos. Mag. B 63

(1991) 955; A.L. Shluger, R.W. Grimes, C.R.A. Catlow and

N. Itoh, J. Phys.: Condens. Matter 3 (1991) 8027; R.I.

Eglitis, M.M. Kuklja, E.A. Kotomin, A. Stashans and AI.

Popov, Comp. Mater. Sci. 5 (1996) 298.

[22] A.L. Shluger, N. Itoh and K. Noda, J. Phys.: Condens.

Matter 3 (1991) 9895.

1231 A.L. Shluger and E.V. Stefanovich, Phys. Rev. B 42 (1990)

%46.

1241 A.L. Shluger, Theor. Chim. Acta 66 (1985) 355;

[25] E.V. Stefanovich, E.K. Shidlovskaya, A.L. Shluger and M.A.

Zakharov, Phys. Stat. Sol. B 160 (1990) 529.

[26] R.A. Evarestov and V.A. Lovchikov, Phys. Status Solidi B

79 (1977) 743. 93 (1979) 469.

[27] R.A. Evarcstov and V.P. Smimov, Use of Group Theoretical

Methods in Quantum Chemistry of Solids (Leningrad Uni-

versity Press, Leningrad, 1987).

[28] R.W. Grimes, C.R.A. Catlow and A.L. Shluger, eds., Quan-

tum Mechanical Cluster Calculations in Solid State Studies

(World Scientific, Singapore, 1992).

[29] L.N. Kantorovich, A. Stashans, E.A. Kotomin and P.W.M.

Jacobs, Int. J. Quantum Chem. 52 (1994) 1177.

[30] M.N. Kabler, Point Defects in Solids, eds. J.H. Crawford and

L.M. Slifkm (Plenum, New York, 1972) p. 372.

[31] D.L. Griscom, Phys. Rev. B 40 (1989) 4224.

1321 R.C. Du Vamey, J.R. Niklas and J.-M. Spaeth, Phys. Status

Solidi B 128 (1985) 673; F.J. Adrian, A.N. Nette and J.-M.

Spaeth, Phys. Rev. B 31 (1985) 3923.

[33] R.W.G. Wyckoff, Crystal Structures, Vol. 2, 2nd ed. (Inter-

science, New York, London, Sydney, 1964) p. 6.