Quantum Chemistry of Solids / 23-Modeling and LCAO Calculations of Point Defects in Crystals

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centered dumbbell configurations. The excitation of this configuration to the relaxed face-centered interstitial requires an energy of 2.12 eV, which would therefore be the activation energy for di usion of f d. This energy may be compared with: 1.45 eV the activation energy for oxygen-atom di usion calculated by the supercell FPLMTO approach [708]; the 1.6 eV for some di usion-controlled process involving interstitial oxygen atom that has been observed experimentally in MgO.

10.2.2 Neutral and Charged Oxygen Vacancy in Al2O3 Crystal: Supercell and Cyclic-cluster Calculations

Aluminum oxide (α-Al2O3, corundum) has a large number of technological applications. Due to its hardness, its chemical and mechanical stability at high temperatures and its electronic properties as a widegap insulator it is used for the fabrication of abrasives, as a carrier for thin metal films in heterogeneous catalysis and in optical and electronic devices.

The structural and electronic properties of α-Al2O3 are altered by defects such as oxygen vacancies, which can be formed as a result of the bombardment of the oxide with high-energy particles. Neutral oxygen vacancies (F centers) and positively charged oxygen vacancies (F+ centers) in aluminum oxide have been investigated in a number of experimental and theoretical studies. Experimental and theoretical results agree that the two electrons of the F center are localized and remain in the vacancy vicinity indicating a singlet ground state. Despite numerous investigations, the defectformation energies of the F center, Ef (F), and the F+ center, Ef (F+), are rarely given in the literature. To our knowledge, no experimental data are available up to now.

In [709] supercell calculations based on the orthogonalized linear combination of atomic orbitals (OLCAO) method within the local density approximation (LDA) were performed. In this case, the oxygen vacancy was modeled by removing one oxygen atom from the cell. They obtained a defect-formation energy of Ef (F) = 38.36 eV for the neutral oxygen vacancy without relaxation of the geometry. When the four nearest-neighbor aluminum and 12 next-nearest-neighbor oxygen atoms around the vacancy (see Fig. 2.19) were allowed to relax the defect-formation energy decreased to Ef (F) = 5.83 eV. This corresponds to a relaxation energy of Erel = 32.53 eV. The distance of the neighboring aluminum atoms to the vacancy increased by about 16% and the distance of the oxygen neighbors by about 8% with respect to the unrelaxed structure. Similar values were obtained for the F+ center. The defect-formation energy for the F center was calculated with respect to an isolated neutral O atom, see (10.1), using the di erence of the total energy of the supercell with and without vacancy, and adding the energy of the free oxygen atom EO.

In [710] a 65-atom molecular cluster is used that was treated semiempirically at the INDO level and embedded in the electrostatic field generated by multipoles. To model the F and F+ centers in corundum additional 2s and 2p Slater-type atomic orbitals were placed in the vacancy after removing one O atom. The relaxation of the four aluminum atoms nearest to the vacancy resulted in an outward displacement by 2–3% and 5–6% for the F and F+ center, respectively. From the 12 next-nearest oxygen atoms only two were allowed to relax. In contrast to the results of [709] the two oxygen atoms moved toward the vacancy by about –2.2% and –4.6% for the F and F+ center, respectively. Defect-formation energies were not calculated.

430 10 Modeling and LCAO Calculations of Point Defects in Crystals

The neutral oxygen vacancy was investigated in [711] by supercell-DFT calculations within the generalized gradient approximation (GGA) using a plane-wave basis and ultrasoft Vanderbilt pseudopotentials. The calculated defect-formation energy is Ef (F) = 7.13 eV and Ef (F) = 7.08 eV for the F center without and with the relaxation of the nearest aluminum and next-nearest oxygen atoms, at the oxygen-rich limit. As a reference for the calculation of the defect-formation energy EO = 0.5EO2 was taken instead of a free oxygen atom energy EO. The aluminum atoms show an inward relaxation of –1.5%, the positions of the oxygen atoms are almost unchanged (–0.1%).

The intrinsic defects in α-Al2O3 were also studied in [712] by the supercell DFT PW method (the supercell contained up to 120 atoms). The relaxation of the four nearest aluminum atoms, 12 next-nearest oxygen atoms and two next-next-nearest aluminum atoms around the neutral and positively charged oxygen vacancy was taken into account. The results given in [712] show an average inward relaxation of the nearest Al for the neutral oxygen vacancy of about –1.5% and a slight inward relaxation of the next-nearest O atoms by –0.4%. For the single positively charged oxygen vacancy they find an outward relaxation of the nearest Al atoms of 4.6%. The oxygen atoms relax by –1.4% toward the vacancy. The defect-formation energy of the neutral oxygen vacancy Ef (F)= 13 eV was calculated at the oxygen-rich limit.

It is seen that the results of the di erent theoretical simulations of oxygen vacancies in corundum are contradictory. An attempt to resolve this contradiction was made in [713] where the energies of the formation of an F center and an F+ center were calculated and relaxation e ects were investigated with a combination of HF, DFT and semiempirical methods in the framework of supercell and cyclic-cluster models. All supercell LCAO calculations were made using the CRYSTAL code [23], the details of the basis and computational parameters choice can be found in [713]. The semiempirical cyclic-cluster calculations were made using the MSINDO code developed in the Hannover theoretical chemistry research group [256], see Chap. 6. The cyclic cluster was embedded in an infinite field of point charges using the Ewald summation technique [714]. The charges for the embedding can either be calculated self-consistently from the atoms of the cyclic cluster using a Lowdin population analysis at each SCF cycle or can be kept fixed at the values of the perfect crystal.

The scheme of the SCM model realization, discussed in Sect. 10.1.3, requires in stage 1 the calculations of the perfect crystal. The perfect corundum crystal has a

¯ 6

rhombohedral structure with the space group R3c(D3d, see Chap. 2). The oxygen atoms form a hexagonal close-packed structure, the aluminum atoms occupy 2/3 of the octahedral vacancies so that all Al atoms have the same environment. Every O atom has two neighboring Al atoms at a distance of r1 = 1.86 ˚A and two others at a distance of r2 = 1.97 ˚A. The primitive rhombohedral unit cell consists of two Al2O3 formula units with experimental lattice parameters a = 5.128 ˚A and α = 55.333◦.

Using the transformation of basic rhombohedral lattice translation vectors with

the matrix	2n1 n1		0
−0		2n1	n2
l =	n1	n1	0	, L = 3n12n2	(10.12)

one obtains the hexagonal setting of the corundum structure structure. Its unit cell (n1 = n2 = 1) consists of L=3 primitive unit cells. The lattice parameters in this

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hexagonal setting are a = 4.763 ˚A and c = 13.003 ˚A. Furthermore, corundum has two internal structure parameters uAl = 0.352 and uO = 0.306, which define the Al and O positions, respectively (see Sect. 2.3.5). The experimental heat of atomization is ∆aH0 = 31.60 eV/Al2O3. This has been obtained by the experimental heat of formation of the solid (∆f H0(Al2O3) = −17.22 eV) and the heats of formation of the corresponding atoms in the gas phase (∆f H0(Al) = 3.35 eV, ∆f H0(O) = 2.56 eV).

As was noted in Sect. 10.1.3, the first stage of supercell-cyclic-cluster calculations of the defective crystal consists of the band-structure calculations of the perfect crystal to choose the supercell for its modeling. Table 10.5 shows the results for the HF and DFT LCAO band calculations of the perfect Al2O3.

Table 10.5. HF and DFT cohesive energies EHF and EDFT (eV/Al2O3) obtained from band calculations using di erent sets (s1s2s3) of k-points, [713]*

Unit cell	(s1s2s3)	L	Ne cient	EHF	EDF T
rhombohedral	(222)	8	Al16O24	21.47	29.70
	(333)	27	Al108O162	21.50	29.72
hexagonal	(111)	3	Al12O18	—	29.84
	(221)	12	Al48O72	21.49	29.70
	(331)	27	Al108O162	21.50	29.71
	(662)	216	Al432O1296	21.50	29.72

*Eexp=31.60 eV

The geometrical parameters were fixed to the experimental values. The cohesive energy per formula unit was calculated as the di erence between the total energy per Al2O3 unit in the crystal and the sum of the energies of the corresponding free atoms. The calculations were made using Monkhorst–Pack sets (s1, s2, s3) of k points, which correspond to supercells of L-primitive unit cells consisting of Nef f icient atoms. Ecoh calculated corresponds to the experimental heat of atomization ∆aH0 except for the zero-point energy that is neglected in the calculation of Ecoh. The energies of the free Al and O atoms were obtained by increasing the basis sets successively with di use sp-functions until the energies of the free atoms were converged, see Chap. 8. The HF cohesive energy of 21.50 eV/Al2O3 is too low by 10.10 eV/Al2O3 compared with the experimental value of 31.60 eV/Al2O3. This discrepancy is mainly due to the missing correlation and the incompleteness of the basis set [715].

The DFT results, using the PWGGA-PWGGA exchange-correlation functional (29.72 eV/Al2O3), are closer to the experiment. But the cohesive energy is still underestimated by about 1.88 eV/Al2O3. The bandgap Eg = 6.28 eV from DFT calculation is about 0.4 eV larger than the value obtained in [712] but still too small compared with the experimental range of the bandgap between 8.5 and 9.9 eV. The bandgap of 17.51 eV calculated at the HF level is much too high in line with experience.

The results of the cyclic-cluster MSINDO calculations are shown in Table 10.6. Since the CCM does not involve a summation over the reciprocal space the

size of the cyclic cluster was increased in direct space to obtain convergence for the cohesive energy. Each cluster was embedded in an infinite Madelung field of

432 10 Modeling and LCAO Calculations of Point Defects in Crystals

Table 10.6. Cohesive energies EMSINDO (eV/Al2O3) for cyclic-cluster MSINDO calculations using di erent cyclic clusters, [713].

Cluster	L	n1	n2	EMSINDO
Al12O18	3	1	1	35.45
Al48O72	12	2	1	31.17
Al96O144	24	2	2	31.17
Al108O162	27	3	1	31.09
Al192O288	48	4	1	31.09

point charges. Again, the geometry was fixed to the experimental values to enable the comparison with the results of the supercell calculations. The MSINDOoptimized lattice parameters (a = 4.779 ˚A, c= 13.074 ˚A), internal structure parameters (uAl = 0.354, uO = 0.301) and the cohesive energy (Ecoh = 31.16 eV/Al2O3) [714] of the perfect α-Al2O3 are in very good agreement with the experimental data. A bandgap of Eg = 10.9 eV was obtained by a 25 × 21 configuration interaction (CI) calculation for the defect-free Al192O288 cluster. This value is at least 1 eV above the experimentally observed range of the bandgap of the perfect crystal but there are no forbidden transitions, which is an indication of a direct bandgap, in agreement with the generally accepted opinion. The valence-band maximum (VBM) consists of O 2p orbitals, while the conduction band minimum (CBM) consists mainly of Al 3s orbitals in agreement with other theoretical [712] and experimental results. The cohesive energies for the experimental geometry of α-Al2O3 given in Table 10.6 show a fast convergence with increasing cluster size. Already the energy value of the Al48O72 cluster that corresponds to L = 12 primitive unit cells is, with 31.17 eV/Al2O3, close to the calculated limit of 31.09 eV/Al2O3. Therefore, the experimental value of 31.60 eV/Al2O3 is reproduced with a deviation of about 0.50 eV/Al2O3.

In stage 2 (the supercell calculation of a defective crystal, see Sect. 10.1.3), for the simulation of the F center in α-Al2O3 a supercell was created by using the transformation matrix (10.12) with n1 = 2 and n2 = 1 (L=12). One oxygen atom was removed from the supercell and the defect-formation energy for the F center was calculated for di erent k-point grids (s1s2s3) keeping the geometry fixed to the experimental values. For the calculation of the defect-formation energy EO = 0.5EO2 was used. Table 10.7 shows the results using the HF and the DFT (PWGGA-PWGGA) approach.

Table 10.7. HF and DFT defect-formation energies EHF(F) and EDFT(F) (eV) for the neutral oxygen vacancy in Al2O3 obtained from supercell calculations using di erent sets (s1, s2, s3) of k-points, [713].

(s1, s2, s3) EHF (F)		EDFT(F)
(1 1 1)	16.55	13.08
(2 2 2)	16.38	12.92
(3 3 3)	16.38	12.24

PW-DFT GGA [712]

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The defect-formation energy shows fast convergence with the increase of the number of sampling k points. Already the (2 × 2 × 2) grid gives the converged HF energy of EfHF (F ) = 16.38 eV. For the DFT calculations the same set of k points was used to yield a defect-formation energy of EfDF T (F ) = 12.92 eV, which is about 3.46 eV lower than the HF result.

In stage 3 relaxation e ects were included in the simulation of the F and the F+ centers in α-Al2O3 using the MSINDO cyclic-cluster calculations (Table 10.8).

Table 10.8. Cyclic-cluster MSINDO				defect-formation energies EMSINDO (F) and
E	MSINDO	+	and r2	˚
		(F ) (eV), distances r1		(A) of nearest-neighbor (NN) and next-nearest

neighbor (NNN) atoms from the defect and changes of the distances ∆r(%) for the unrelaxed and relaxed F and F+ center in Al2O3 using the Al192O288 cyclic cluster, [713]

		Unrelaxed	Relaxed (NN+NNN)
F center	EMSINDO(F) = 12.32,		EMSINDO(F) = 11.81,
	r1	= 1.86, r2 = 1.97	r1	= 1.872 (∆r = 0.5),
			r2	= 2.053 (∆r = 4.0)
F+ center	EMSINDO(F+) = 12.22,		EMSINDO(F+) = 10.33,
	r1	= 1.86, r2 = 1.97	r1	= 1.971 (∆r = 6.0),
			r2	= 2.089 (∆r = 6.0)

The Al192O288 cyclic cluster (n1 = 4, n2 = 1, L=48) was chosen for the simulations. One oxygen atom was removed from the cyclic cluster to create the F center, keeping the system neutral. To model the F+ center the charge of the system was set to +1 after removing one oxygen atom. In both cases the cyclic cluster was embedded in the Madelung field of the perfect crystal. The charges for this embedding were the L¨owdin charges calculated from the simulation of the corresponding perfect cyclic cluster. In contrast to the standard supercell model of a periodic point defect, no artificial neutralizing background and additional charge corrections are needed in the cyclic-cluster model as the point defect is not periodically repeated. The four nearestneighbor (NN) aluminum atoms and the 16 next-nearest-neighbor (NNN) oxygen atoms close to the vacancy were allowed to relax using a quasi-Newton method until the residual forces had converged to less than 0.02 eV/˚A. The rest of the cyclic cluster was fixed in the experimental geometry. The defect-formation energy of the F+ center was calculated using as the reference a single oxygen atom energy EO = 0.5EO2 as in the case of the neutral oxygen vacancy. Additionally, one electron is removed from the cluster to create a positive charge and the chemical potential of the electron µe is assumed to be zero, di erent from the models of the charged defects where the electron is assumed to remain in the lattice. The CCM value for the defect-formation energy of the unrelaxed F center EfMSINDO(F ) = 12.32 eV is close to the DFT supercell value of EfDF T (F ) = 12.92 eV and the 13 eV obtained in [712], despite substantial di erences in the defect-level positions above the VBM. The latter are 8.1 eV for MSINDO, 4.9 eV for DFT supercell calculation and 2.3 eV as obtained in [712]. The SCM calculation using the HF method gives a larger defect-formation energy (EfHF (F ) = 16.38 eV). The defect-formation energy of 7.13 eV calculated

434 10 Modeling and LCAO Calculations of Point Defects in Crystals

in [711] is considerably lower than the values obtained in [713]. The defect-formation energy of Ef (F ) = 38.36 eV calculated in [709] was obtained using the total energy of the oxygen atom. If one usees the same reference [709] and not 1/2EO2 one finds a defect-formation energy of 15.06 eV for the MSINDO cyclic-cluster simulation of the unrelaxed vacancy.

An absorption band of about 6 eV has been assigned experimentally to the F center, indicating a location of a doubly occupied defect level roughly 3 eV above the VBM. The corresponding excitation energies are 1.8 eV and 8.8 eV for DFT and HF calculations [713], respectively. The defect level obtained using the MSINDO-CCM method consists mainly of Al 3s orbitals of the aluminum atoms next to the vacancy. Using a 40 × 30 CI the first allowed excitation was found to occur at 3.1 eV for the relaxed structure. Therefore, the excitation energy is underestimated by about 3.0 eV. In comparison the excitation energy in [712] is 3.52 eV which is also too small.

Concerning the relaxation, the CCM simulations of the F center show that the NN Al atoms increase their distance from the vacancy by 1% to 4% which agrees well with the corresponding data of 2% to 6% of [710]. The value from [709] is larger (16%) but the NN Al atoms also show a movement away from the vacancy. This is reasonable, since the NN Al atoms are positively charged and should therefore repel each other. Under these circumstances the inward relaxation to the vacancy (1.5%) of the aluminum neighbors found in [711, 712] is di cult to understand. Their choice of pseudopotentials and of basis set, restriction only by Γ point in BZ might be the reason for the discrepancy. Table 10.7 shows that a Monkhorst–Pack net of at least (222) must be used for the defective supercell to reach convergence with respect to the energy. Using only the Γ point might also influence the relaxation.

Considering the NNN O atoms, an average relaxation of –0.4% toward the vacancy and the total relaxation energy Erel =0.51 eV was found in the LCAO calculations [713] of the F center. This agrees well with the values –0.1% and –0.4% of [711, 712] indicating that the positions of the oxygen atoms are almost unchanged. An inward relaxation was also found in [710]. In contrast to these results, a large outward relaxation of 8% and Erel=32.53 eV were found in [709]. The value of 32.53 eV for the relaxation energy appears to be much too high, because this value is larger than the experimental heat of formation for the α-Al2O3 (31.60 eV).

A singly occupied defect level occurs in the bandgap in the case of the F+ center. The results of the CCM calculations for the F+ center (Table 10.8) indicate a larger relaxation of the NN Al atoms (6.0%) and of the NNN oxygen atoms (-1.2%) compared to the values obtained for the neutral oxygen vacancy. This is reasonable, since the aluminum atoms repel each other more strongly due to the missing electron in the vacancy. In consequence, the relaxation energy (Erel = 1.90 eV) is also larger than the corresponding value for the F center. This trend is confirmed by results found in DFT PW calculations [712]: an outward relaxation of 4.6% for the Al neighbors and an inward relaxation of –1.4% of the oxygen atoms close to the vacancy, which agrees well with the MSINDO CCM results. The movements found in semiempirical LCAO calculations [710] are slightly larger (5–6% for the Al atoms, –4.6% for the oxygen atoms) but they also confirm the tendency.

The defect-formation energy of the F+ center is smaller than that of the F center in both the relaxed and the unrelaxed structures using µe = 0 for the chemical potential of the electrons. This agrees qualitatively with the results of [712] when compared to

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435

their values at k = 0. It was noted that no experimental data for the defect-formation energy are available in the literature for the F and F+ center in α-Al2O3. The detailed consideration of the results of application of di erent models and calculation schemes to F and F+ centers in corundum demonstrate that LCAO methods reproduce well the results of PW calculations for the neutral defect. Furthermore, the LCAO methods are more appropriate for the charged-defect modeling as the artificial defect-periodicity is escaped in the LCAO cyclic-cluster approach.

10.2.3 Supercell Modeling of Metal-doped Rutile TiO2

Titanium dioxide (TiO2) is a wide-bandgap material useful in many practical applications (electrochemistry, sun lotions, nanostructured electrodes). The electronic stucture of TiO2 can be strongly influenced by 3d transition-metal dopants that introduce local electronic levels into the bulk energy bandgap and thus shift the absorption edge to the visible region.

The e ects of metal doping on the properties of TiO2 have been a frequent topic of experimental investigation. In particular, according to a systematic study on the photoreactivity and absorption spectra of TiO2 doped with 21 di erent metals [716] the energy level and d-electron configuration of the dopants were found to govern the photoelectrochemical process in TiO2. The resonance photoemission study [717] was applied to investigate the nature of bandgap states in V-doped TiO2 in rutile structure.

The computer simulation has been employed to clarify in detail impurity-doping e ects. A molecular-cluster approach was extensively adopted, see references in [718]. These cluster calculations have never led to unifying conclusions regarding the e ects of doping on the electronic states due to the influence of dangling bonds.

The supercell approach demonstrated its capacity to deal with metal-doped TiO2 defective crystals in DFT FPLAPW supercell electronic-structure calculations [718, 719]. It was found that when TiO2 is doped with V, Cr, Mn, Fe or Co, an electronoccupied localized level in the bandgap occurs and the electrons are localized around each dopant. As the atomic number of the dopant increases the localized level shifts to the lower energy. The energy of the localized level due to Co is su ciently low to lie at the top of the valence band, while the other metals produce midgap states. In contrast, the electrons from the Ni dopant are somewhat delocalized, thus significantly contributing to the formation of the valence band with the O 2p and Ti 3d electrons. These conclusions were made in [718] on the basis of the DFT FPLAPW calculations on a relatively small supercell consisting of four primitive unit cells stacked along the b- and c-axis of the simple tetragonal lattice. The experimental values of the lattice constants a=b=4.594 ˚A, c=2.959 ˚A were taken, so that the translation vectors lengths a, b, c of the orthorhombic supercell equal 4.594 ˚A, 9.188 ˚A and 5.918 ˚A, respectively. Such a choice of supercell is not optimal as the defect–defect distance Rd−d is not the same along the three supercell translation vectors and the defect– defect interaction corresponds in fact to the smallest distance Rd−d = 4.594 ˚A (the length of the primitive tetragonal unit cell translation in the perfect rutile crystal).

As in the FPLAPW method the further increasing of the supercell is su ciently di cult that the dependence of the results on the supercell size was not studied in [718]. Such a study was made in semiempirical cyclic-cluster calculations of Li and

436 10 Modeling and LCAO Calculations of Point Defects in Crystals

Cl, Br impurities in TiO2 [720, 721] without preliminary symmetry analysis of the cyclic clusters chosen. Such an analysis was made in [683] and allowed the convergence of the supercell UHF LCAO calculations of V-doped rutile with the supercell size to be investigated. This first nonempirical supercell LCAO calculation of the defective rutile crystal is discussed here in more detail.

The impurity vanadium atom substitutes for a titanium atom introducing an unpaired electron in the unit cell. The symmetry of the supercell model is described by any of four orthorhombic space groups D21h, D219h, D223h, D225h. Analyzing the period of the defect for the same supercell size (given L-value) in these groups it was found that for rutile structure the best choice would be D225h corresponding to the symmetric transformation of the simple tetragonal lattice translation vectors with matrix (n1, n2, n3 - integers, n1 =n2):

l =

−n1 −n2

n3

L = 2 n

2

−

2

3

(10.13)

n2

n1

−n3

,

1

n

2

n1

n2

n3

n

The space group D225h corresponds to the body-centered orthorhombic lattice of supercells with the translation vectors of equal length in all three translation directions. This means that the point-defect period is also the same in these directions (equidistant configuration of defects). This choice of defective crystal space group ensures the largest defect period for a chosen supercell size (fixed value of L).

The UHF LCAO calculations of V-doped rutile were made using the CRYSTAL code [23] and Durand–Barthelat pseudopotentials [484]. The atomic basis functions of Ti and O atoms were taken from [323] that were fitted to reproduce the band structure and bonding properties of perfect rutile crystal. The vanadium atom functions were found by fitting these properties of VO2 crystal in the rutile modification. The accuracy of the calculation was ensured by the choice of the computational parameters of the CRYSTAL code ensuring a good accuracy of the bielectronic series, the convergence thresholds of the eigenvalues and total energy and k-point sampling in BZ.

Table 10.9 shows the results obtained for the supercells of increasing size. The Monkhorst–Pack shrinking factor was taken to be 6 for supercells with L=2, 4, 6 and 4 for supercells with L=10, 14. Table 10.9 gives the integers n1, n2, n3 defined according to (10.13) the supercell of the defective crystal consisting of L primitive unit cells and corresponding doping per cent, the number of atoms NA and atomic basis functions NB in the supercell, vanadium–vanadium distance Rd−d (defect period) in the supercell model. The convergence of the absolute value of the one-electron energy Ed occupied by the unpaired electron is seen. It appears that already the supercell for L =6 is enough to reproduce the vanadium atom charge QV , covalence CV (the sum of the bond orders of the impurity atom with all the crystal atoms)

and spin-density NVα−β				. Using the calculated QV and CV the total valence Vtot =
0.5 CV	4			is 3.88, i.e. the impurity atom is in oxidation state IV,
	+	4Q2	+ C2
		V	V

and the calculated spin density corresponds to the fifth vanadium electron not taking part in the chemical bonding. This result is in accordance with the experimentally found [717] ESR signal for V-doped rutile.

The Fermi energy EF in the perfect rutile TiO2 (EF =–0.3029 a.u.) was calculated using the same pseudopotential and basis functions of titanium and oxygen atoms

10.2 Point Defects in Binary Oxides

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Table 10.9. The convergence of the results of the unrestricted Hartree–Fock LCAO cal-

culations [683] of the V-doped rutile (defective-crystal space group D225h = Immm; L =

2(n21 − n22)n3)

	(n1, n2, n3)	(1,0,1)	(1,0,2)			(2,1,1)		(3,2,1)		(4,3,1)
	L	2		4			6		10	14
	doping (%)	0.25	0.125				0.083	0.050		0.036
	NA	12		24			36		60	84
	NB	120	240				360	600		840
	˚	5.45	7.53				10.62	16.69		22.97
	Rd−d (A)
	Ed (a.u.)	–0.2955	–0.2991			–0.3004		–0.3012		–0.3016
	QV	2.761	2.760				2.759	2.759		2.759
	CV	1.921	1.924				1.927	1.927		1.927
	NVα−β	1.115	1.114				1.099	1.099		1.099
				−n1 −n2				n3
The supercell transformation matrix l =					n2		n1 −n3
					n1		n2	n3

that were used for the defective-crystal calculation. Thus, the V-atom impurity level (occupied by an electron) is relatively close to the perfect-crystal valence-band edge (∆E=–0.3016 + 0.3029 = 0.0013 a.u. or 0.04 eV). Experimental data [717] predict larger values of ∆E (0.03 a.u. or 0.81 eV). The di erence may be explained both by the necessity to take into account the atomic relaxation around the impurity atom and experimental ambiguity in the assignment of the gap states to electrons localized on the V dopant. Bearing in mind the energy gap of 3.0 eV for the perfect crystal and midgap position of the localized level in FPLAPW calculations [718] for V-doped rutile, we conclude that both LCAO (0.04 eV) and FPLAPW (1.5 eV) results for the defect-level position in the bandgap di er from the experimental estimation (0.81 eV).

Table 10.9 shows that the increase of the supercell size moves the defect energy level closer to the valence-band edge. Therefore, DFT FPLAPW-correlated calculations with the larger supercells would make the agreement with the experimental data better. When using HF LCAO approach for the calculations it is important to estimate the possible influence of the electron correlation on the obtained results. We have seen that the highest-occupied level position moves to higher energies when the correlation is taken into account and this movement may appear di erent for the perfect and defective crystal, so that the defect-level position in the bandgap also changes. This requires further study.

Nevertheless, the calculation [683] demonstrates: (1) the e ciency of the more detailed symmetry analysis for the supercell choice when the periodic defect calculations are made in the complicated crystalline structures with the symmetry of a nonsymmorphic space group and noncubic lattice; (2) the reality of the supercell model for the nonempirical LCAO calculations of the point defects in such a complicated crystalline structure as a rutile structure; 3) higher e ciency of LCAO basis compared with LAPW basis in the supercecell calculations of defective crystals. Moreover, the supercell model allows the dependence of the electronic properties of doped crys-

438 10 Modeling and LCAO Calculations of Point Defects in Crystals

tals on the doping level to be investigated. In the next section we consider supercell calculations of defective crystals with perovskite structure.

10.3 Point Defects in Perovskites

10.3.1 Oxygen Vacancy in SrTiO3

Ternary ABO3 perovskite materials have numerous technological applications as these materials display a wide range of useful physical and chemical properties. They are important as catalysts, as ceramics, ferroelectrics, superconductors, as materials for fuel cells, fusion reactors, and optical and piezoelectric devices.

The properties of perovskite materials are heavily dominated by their oxygen content, as well as by donorand acceptor-type impurities. An essential contribution to the knowledge of the structural and electronic properties of point defects in these materials comes from theoretical approaches. The results of large-scale computer semiempirical and first-principles modeling of point defects, polarons and perovskite solid solutions can be found in [722], focusing mostly on KNbO3 and KTaO3.

First-principles calculations of formation energies of point defects were made on BaTiO3 [723, 724] and NaNbO3 [725] crystals. Among the various fundamentally and technologically important oxides, SrTiO3 is a simple structural prototype for many perovskites, in which the detailed investigation of native and dopant defects can lay the theoretical groundwork that can be applied to structurally and chemically more complex perovskite materials [726].

The Green-function method appeared to be very useful for displaying the chemical trends in defect energy levels [727, 728]. However, the calculation of other defectivecrystal properties (defect-formation energy, lattice relaxation, local-states localization) requires approaches based on molecular cluster or supercell models. Only recently have these models been used in the first-principles calculations to study point defects in SrTiO3.

The supercell calculations are made mostly by the DFT PW method [726, 729]. The comparative PW (supercell) and LCAO (cluster and supercell) calculations are known for the oxygen vacancy in SrTiO3 crystal [730–732].

Oxygen vacancies in SrTiO3 act as e ective donors and are important defects in SrTiO3. Theoretical simulation of defective SrTiO3 is very important since modern scanning transmission electron microscopy (STEM) and reflection high-energy electron di raction (RHEED) combined with atomic-scale electron-energy-loss spectroscopy (EELS) are able to detect the presence of even single impurities and vacancies in SrTiO3. A deliberate deviation of oxygen content from the ideal stoichiometry of perovskites, is relevant for their numerous high-tech applications as sensors, fuel cells, microelectronic devices. At low oxygen partial pressure, the electrical conductivity of SrTiO3 perovskite is controlled by both the concentration and mobility of oxygen vacancies, which act as e ective donors; therefore, this material becomes n- type conductive. Increasing the partial pressure reduces the carrier concentration, and at high pressures the conductivity goes through a minimum: the material becomes p-type [733, 734]. Consequent thermal reduction of oxygen in single-crystalline strontium titanate results in the insulator-to-metal transition, up to a possible superconducting state, accompanied by intensive formation of vacancies and their high