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

10.3 Point Defects in Perovskites

439

density within the skin region [735]. Oxygen vacancies play a noticeable role in the structural transformations in bulk SrTiO3, which possesses two relevant structures: the tetragonal antiferrodistortive (AFD) phase, and the cubic phase [726]. The latter (Fig. 10.3a) is stable at 105 K, whereas AFD (slightly distorted cubic phase) is stable at lower temperatures.

Fig. 10.3. Three types of equidistant crystalline cells with a centered O vacancy for the cubic phase of SrTiO3 perovskite: (a) simple cubic (sc); (b) face-centered cubic (fcc); (c) body-centered cubic (bcc). Sticks between the oxygen and titanium ions indicate the partly covalent bonds between them. To construct the di erence electron-density plots, each cell is sectioned along the Ti-O-Ti axis by the plane PP

When simulating a single point defect, the main problem is to understand changes induced by it in the atomic and electronic structure of a host crystal. A perfect SrTiO3 crystal has a mixed ionic-covalent nature of the chemical bonding, which is not in complete conformity with the formal charges on Sr2+, Ti4+, and O2− ions, see Chap. 9. The formalism of Wannier functions applied for the determination of e ective charges and bond populations in several perovskite crystals [736] calculated previously using the density-functional theory (DFT) method [606] gives the atomic charge of +2.55 on titanium and –1.55 on oxygen, whereas the Ti–O bond order is 0.35; only the charge on Sr (+1.95) is close to the nominal ionic value +2, confirming that strontium is ionically bonded in SrTiO3. Partly covalent chemical bonding makes the simulations of the structural defects in strontium titanate rather complicated, even in a cubic phase.

The simplest native defect may be described in terms of either a neutral oxygen vacancy (single O atom removed from the lattice site) or neutral F center (the oxygen ion O2− vacancy trapped 2 electrons remaining in a host crystal).

Theoretical studies mainly predict equal contributions of the 3d orbitals of the two nearest titanium ions (Fig. 10.3) into the wavefunction of the F center [729, 730, 734]. According to DFT calculations on a cubic phase of SrTiO3 perovskite [730], the Mulliken electron charge of 1.1–1.3 is localized in the neutral O vacancy (depending on the supercell size), and 0.6–0.8 are equally divided by the two nearest Ti ions if we consider the F center. This result does not confirm the formal conclusions that the F center is supposed to have released both electrons, whereas the nearest titanium

440 10 Modeling and LCAO Calculations of Point Defects in Crystals

ions change their valence from Ti4+ to Ti3+. The position of the F center level in the optical bandgap of SrTiO3 is also not completely clear: it is an open question as to whether it lies well below the conduction-band bottom or close to it. The latter is supported by the indirect experimental study on the conductivity of SrTiO3 ceramics, suggesting that the F center is a rather shallow defect [737].

Supercell ab-initio calculations on the SrTiO3 give the values of the formation energy for O vacancy within the range of 6.5–8.5 eV [726, 729, 730, 734], whereas in cluster models [738], removal of an O atom from the lattice has a higher energy cost (> 9 eV). However, this value cannot be directly measured experimentally. Conductivity in SrTiO3 at low partial oxygen pressures depends on the mobility of O vacancies [737]. Experiments performed at high temperatures suggest an energy barrier of 0.86 eV for the F center di usion in bulk, whereas semiempirical pair-potential calculations on migration on the empty O vacancy result in the barrier of 0.76 eV [739].

In [732] the results of both LCAO (made with the CRYSTAL code [23]) and PW (made with the VASP code [740]) DFT calculations on the F center in a cubic phase of SrTiO3 perovskite are analyzed, combining the advantages of LCAO and planewave formalisms. The lattice structural relaxation around an oxygen vacancy was optimized for supercells of di erent shapes and sizes. This is important as DFT PW calculations [726] show the strong dependence of the results on the supercell size (it was concluded that at least a 4 × 4 × 4 supercell of 320 atoms is needed to describe the structural parameters of an oxygen vacancy accurately). In [732] the supercells were obtained by a consequent equidistant extension of crystalline lattice vectors (Fig. 10.3) increasing the supercell size from 80up to 320-atom cells, in order to eliminate the interaction of periodically repeated point defects and to reach the limit of a single

F defect. These supercells are described by transformation matrices

n

0

0

0

n n

−n

n

n

0

n

0

,

n

0 n

,

n

−n

n

(10.14)

0

0

n

n n 0

n

n −n

where n is varied between 2 and 4. The corresponding equidistant supercells form sc, f cc and bcc lattices (cases a), b) and c) in Fig. 10.3, respectively). The ratio of volumes for the supercells extended from the primitive unit cell using matrices (10.14) with the same n is

Vbcc = 2Vf cc = 4Vsc

(10.15)

Supercells with extensions 3×3×3 (135 atoms) and 4×4×4 (320 atoms) form a simple

cubic lattice (Fig. 10.3a). In turn, supercells with the fcc extensions 2√

2

×2√

2

×2√

2

√

√

√

(80 atoms) and 3

2

× 3

2

× 3

2

(270 atoms) are rhombohedral with a 60◦ angle

between the lattice vectors, see Fig. 10.3b. Lastly, for a 160-atom rhombohedral bcc
√		√		√
supercell (extension 2	3 × 2		3 × 2 3), the angle is 109.47◦. For all three types of

equidistant cells shown in Fig. 10.3, their shapes correlate with the orientation of the –Ti–O–Ti– axis, it may be: (i) the rotation axis joining the centers of opposite faces (Fig. 10.3a), (ii) the diagonal joining the opposite apices of the rhombohedron (Fig. 10.3b), and (iii) the axis joining the centers of opposite rhombohedron edges (Fig. 10.3c).

The computational details (the optimization of LCAO basis and choice of cuto energy in PW calculations, k-meshes used for BZ integration, choice of pseudopotentials) can be found in [732]. The advantage of PW calculations is that the complete

10.3 Point Defects in Perovskites

441

optimization of lattice relaxation upon vacancy creation, even for large supercells, can be performed much faster than in LCAO calculations. Geometry optimizations in PW calculations have been carried out with an accuracy of 10−3 eV in the total energy. The electronic structure of defective SrTiO3 perovskite calculated with both the CRYSTAL and VASP codes has been studied for the diamagnetic closed electronic shell (singlet) state, since no lower state was found employing spin-polarized calculations [730].

The basic properties of a perfect cubic SrTiO3 crystal (lattice constant, bulk modulus, elastic constants) were obtained in reasonable agreement with experiment in LCAO calculations. The best agreement with experimental results for the energy gap was obtained for the hybrid HF-DFT technique (B3PW), see Chap. 9. Further improvement of the calculated optical gap was achieved by adding the d polarization orbital into the oxygen basis set: at the Γ point of BZ the calculated gap 3.65 eV is close to the experiment, 3.25 eV for the indirect bandgap. In DFT (PW91) PW calculations the structural parameters were again found to be quite reasonable but the optical bandgap of 2.59 eV is an evident underestimate, typical for the DFT method.

The F center was modeled by removing an oxygen atom in the supercell (see Fig. 10.3). For estmation of the formation energy Ef (F ) of the F center the energy for the spin-polarized isolated oxygen atom (3P state) was taken and (10.1) was used. A similar approach for the determination of Ef (F ) was used in [734]. In an alternative approach [726,729] the formation energies of the O vacancy were expressed via chemical potentials of O, Sr, and Ti atoms. However, the range of Ef (F ) calculated using both approaches for supercells of di erent shapes and sizes is not so large as to give a preference to one of them.

Table 10.10 shows the dependence of the vacancy-formation energy in SrTiO3 bulk on both supercell shape and size, which is accompanied by the large contribution coming from the lattice relaxation upon vacancy formation. The formation energy is reduced considerably (by 1.5–2.0 eV) when the positions of all atoms in the supercells are fully optimized. This demonstrates that the relaxation of even 14 nearest ions (neighboring Ti, O, and Sr coordination spheres directly shown in Fig. 10.3a) might be somewhat insu cient, and the inclusion of next-nearest coordination spheres is necessary, see Table 10.11.

Table 10.10. Dependence of the nearest distance between F centers (dF−F), formation energy of a single oxygen vacancy Ef (F), and energy barrier Edif f (F) of its (011) di usion on both shape and size of supercell used for PW calculations, [732]

Supercell		Extension								Type of	dF−F	Ef (F)(eV)			Edi (F)
										lattice	˚	unrelaxed	relaxed		(eV)
	√		√				√				(A)
S80	2	2	× 2 2 ×			2	2			f cc	11.04	9.00	7.73		0.41
S135			3 × 3 × 3							scc	11.71	9.17	7.89		0.35
	√		√				√
S160	2√	3	× 2√	3	×	2√		3		bcc	13.52	8.98	7.35		0.50
S270	3	2	× 3 2		×	3	2			f cc	16.56	8.98	7.17		0.38

The lattice relaxation around the defect is periodically repeated in the supercell model, thus a ecting the calculated total energy per cell: the larger the supercell,

442 10 Modeling and LCAO Calculations of Point Defects in Crystals

Table 10.11. Dependence of lattice relaxation for the nearest equivalent atoms around a single F center in a cubic SrTiO3 crystal on shape and size of supercells in PW calculations [732]. Relative radial shifts from unrelaxed positions (%) are given

	Nearest	Distance			Number of	S80	S135	S160	S270	S320
	atoms	(in units of a)			equiv. atoms
	Ti	1/2			2	7.21	7.16	7.08	8.28	7.76
		√
	O		2/2		8	–7.59	–7.92	–7.98	–7.43	–7.79
	Sr	√
			2/2		4	3.51	3.48	3.45	3.42	3.94
	O	1.0			4	3.16	2.98	2.49	2.87	3.56
	O	1.0			2	–1.72	–1.56	–1.67	–1.05	–1.28

Positive shift corresponds to expansion of the atomic coordinate sphere, whereas negative sign means its compression

the smaller this artifact. The numerical results given in Tables 10.10 and 10.11 are obtained in PW calculations, as complete optimization of the lattice relaxation in the large supercells is extremely time consuming in the LCAO calculations.

The calculated defect-formation energies mainly decrease with the increase of the supercell size, but they also depend on the shape of the supercell (compare the corresponding values for fcc 80-atom and sc 135-atom cells). To calculate the energy barrier Edif f (F ) for oxygen-vacancy di usion a jump of the O atom from the eight possible sites nearest to the F center (Fig. 10.3a) toward the vacancy is considred. The saddle point energy has been estimated by fixing a hopping O atom at the middle of the Sr–Ti–Sr triangle (Fig. 10.3a) crossed by the oxygen migration trajectory, which has been found to be nonlinear, while the rest of the lattice has been allowed to relax to the minimum of the total energy. The Edif f (F ) (Table 10.10) is sensitive to both the shape and size of the SrTiO3 supercell; moreover, for optimized rhombohedral f cc and bcc supercells (Fig. 10.3, b and c), the migration trajectories are not completely equivalent. Nevertheless, migration energies mainly decrease with increase of the supercell size. The sensitivity of the calculated lattice relaxation around the defect to both supercell shape and size is also clearly seen in Table 10.11. For the same type of superlattice (sc, f cc, or bcc), expansion of the first coordination sphere (two Ti ions) is larger, whereas compression of the second sphere (eight O ions) is smaller with increasing size of the supercell. However, the convergence of the lattice relaxation is complex, and a very low concentration of F centers should be used to achieve it. For instance, fcc supercells are stretched along the z-axis and are compressed in the xy-plane (Fig. 10.3b). This causes the larger z-shifts of Ti ions nearest to the O vacancy in 80and 270-atomic f cc supercells, as compared with 135and 320-atomic cubic supercells (Table 10.11), whereas xy-shifts of the nearest O and Sr ions are smaller in the former case. Nevertheless, the range of δR for equivalently shifted atoms in equidistant supercells of di erent shapes and sizes is small enough (≤ 1.0%) to suggest the stabilization of a single O vacancy in a cubic SrTiO3 crystal when using large equidistant supercells containing 270 and 320 atoms.

When trying to use the LCAO basis for partial optimization of the total energy for the same supercells, a markedly smaller expansion of the first coordination sphere involving the two nearest titanium ions as compared with complete optimization in

10.3 Point Defects in Perovskites

443

PW calculations (1.5–2% vs. 7–8%) was obtained. The same is true for the nextnearest coordination spheres.

While use of the PW basis is more e cient for optimization of the lattice relaxation around a vacancy, the LCAO basis possesses a noticeable advantage when describing the electronic properties of defective crystals. To gain deeper insight into the defective SrTiO3 bulk from LCAO calculations on di erent supercells in [732] the electron density due to O-vacancy formation was analyzed. Redistribution of the electron density due to O-vacancy formation is shown in Fig. 10.4(a and b) as calculated for the equidistant fcc supercells with di erent extensions.

Fig. 10.4. Two-dimensional (2D) di erence electron-density maps (the total density in the perfect SrTiO3 bulk minus the sum of electron densities of both isolated oxygen atoms and defective SrTiO3) projected onto the (110) section plane PP for 80-atom (a) and 270-atom

(b) fcc supercells containing a single oxygen vacancy [732]. Dash-dot isolines correspond to the zero level. Solid and dashed isolines describe positive and negative values of electron density, respectively. The isodensity increment is 0.002 e/A−3.

In both plots, the Mulliken electron charge (1.1–1.3 e) is localized within a neutral O vacancy; in other words, 0.6–0.8 e is equally divided by the two Ti ions nearest to the neutral F center and mainly localized on their 3dz2 orbitals, making the largest contribution to the defect bands shown in Fig. 10.5(a and b). Figure 10.4 clearly demonstrates the e ect of size of the f cc-type supercell (80 atoms and 270 atoms) on localization of the charge redistribution. For the 80-atom supercell, mutual interaction of the neighboring O vacancies is clearly seen, especially along the –Ti–O–Ti– axes, whereas for the 270-atom supercell, the more-or-less visible redistribution of the electron density is limited by a region of 1.5–2.0 lattice constants around a vacancy, in the z-direction.

From Fig. 10.5 it is seen that the defect-band dispersion becomes very small for a 270-atom supercell demonstrating decreasing the defect–defect interaction. The latter is also manifested in LCAO calculations through the finite defect bandwidth. As noted earlier, a pure DFT (PW91) functional strongly underestimates the bandgap. As is seen from Table 10.12 (the results of LCAO calculations are given), the F-center

444 10 Modeling and LCAO Calculations of Point Defects in Crystals

Fig. 10.5. Band structure of unrelaxed SrTiO3 crystal with a single F center per fcc supercell containing either 80 atoms (a) or 270 atoms (b), [732]. Energy bands corresponding to the F center are split o the conduction bands. Their depth (gap) is shown relative to the bottom of the conduction band at the Γ point

energy level, remaining in the bandgap, approaches the conduction-band bottom, moving from 0.69 eV for the 80-atom supercell (with a bandwidth of 0.15 eV), down to 0.57 eV (0.08 eV) for 160 atoms, and finally reaching the optical ionization energy of 0.49 eV (almost neglecting the dispersion of 0.02–0.03 eV) for supercells of 270 and 320 atoms where the distance between the nearest defects is close to four lattice constants.

The results presented in Table 10.12 confirm that the 135-atom supercell is not big enough to reduce defect–defect interactions; its dispersion δε is even larger than for an 80-atom supercell with a di erent shape, i.e. δε is sensitive to both the shape and size of the equidistant supercell, similar to other properties described before. At the same time, the defect band for the 270-atom supercell (Fig. 10.5b) is almost a straight

10.3 Point Defects in Perovskites

445

Table 10.12. Dependence of the F-center energy-level position with respect to the conduction-band bottom of unrelaxed SrTiO3 crystal with periodically distributed oxygen vacancies ( ), its dispersion (δ ) and distance between the nearest F centers (dF−F) as a function of the supercell size used in LCAO B3PW calculations [732] with 2 ×2 ×2 k-mesh

Supercell		Extension								Type of lattice	˚		δ
											dF−F (A)
	√		√				√
S80	2	2	× 2 2 ×			2	2			f cc	11.04	0.69	0.15
S135			3 × 3 × 3							scc	11.71	0.72	0.23
	√		√				√
S160	2√	3	× 2√	3	×	2√		3		bcc	13.52	0.57	0.09
S270	3	2	× 3 2		×	3	2			f cc	16.56	0.49	0.02
S320			4 × 4 × 4							sc	15.61	0.49	0.03

The defect-level position is calculated at the Γ point of the BZ

line. Table 10.12 clearly shows that the use of both supercells of 270 and 320 atoms practically eliminates the interaction between periodically distributed point defects.

The calculations [732] clearly demonstrate the advantage of combining DFT PW and DFT LCAO calculations. The former is necessary for the complete optimization of both lattice relaxation upon vacancy creation (especially for large supercells) and its migration, whereas the latter allows one to study in more detail the electronic structure for both unrelaxed and relaxed lattices. Such a combined study of the oxygen vacancy in SrTiO3 crystal shows that achieving the convergence with the supercell increases up to 270–320 atoms, the defect–defect interaction becomes negligible, thus approaching a realistic model of a single F center. A similar conclusion follows from the study of the Fe impurities in SrTiO3, considered in next section.

10.3.2 Supercell Model of Fe-doped SrTiO3

The properties of transition-metal impurities, especially iron, in ABO3 perovskite ferroelectrics are of considerable interest due to their photochromic, photorefractive and other applications.

There were several theoretical calculations for ion impurities substituting for B atoms. Molecular-cluster calculations were made to study charged point defects in SrTiO3–3d3 ions Cr3+, Mn4+, Fe5+ [741] and Fe3+, Fe4+, Fe5+ ions [742–744]. The formal charges used in designations of impurity ions mean the metal-atom oxidation state. As was already noted the calculated charges on transition-metal atoms in perovskites di er essentially from the formal charges due to strong covalent interaction with oxygen atoms. Bearing in mind that in perfect SrTiO3 crystal Ti4+ means the titanium-atom oxidation state 4, Fe3+, Fe4+, Fe5+ ions can be considered as impurity centers with charges –1, 0, +1, respectively. The supercell calculations are known for Fe impurities in KNbO3 [745] in the LDA+U approximation, Li-doped KTaO3 (semiempirical INDO simulations [746] and comparative DFT, PW and INDO calculations [747]).

The supercell DFT PW calculations of di erent donor and acceptor centers in SrTiO3 [748] demonstrated the dependence of the formation energy on doping levels in the bulk crystal. In the majority of calculations cited no lattice relaxation around the impurity was studied, the calculated density of states depends on the parameters, used in the LDA+U or INDO approximations for the Hamiltonian.

446 10 Modeling and LCAO Calculations of Point Defects in Crystals

A combination of first-principles supercell HF and DFT calculations that include lattice relaxations have been used in [680] to investigate the energy levels of neutral Fe4+ substitution on the Ti site. Here we consider these calculations in more detail. A consistent supercell–cyclic-cluster approach was applied for defective solids (see Sect. 10.1.3) with a focus on a detailed treatment of lattice relaxation around a single defect.

For a perfect crystal results of periodic HF and DFT calculations based on LCAO approximation are compared. Despite the fact that the supercell approach is widely used in defect calculations, very little attention is paid to the supercell shape and size optimization and the e ect of periodically repeated defect interaction. Following the method accepted for the above-considered supercell calculations of native point defects (interstitial oxygen in MgO, F center in Al2O3 and SrTiO3) a study of the convergence of results to the limit of a single defect is one of the main emphases of the calculations [680] on Fe-doped SrTiO3. The transformation matrices for the supercells used are given in (10.14) in general form and define the equidistant supercells, forming sc, fcc and bcc lattices. In the footnote to Table 10.13 actual transformation matrices used in Fe-doped SrTiO3 calculations are given [680].

For the perfect crystal the special kq points sets corresponding to these supercells satisfy the Chadi–Cohen condition, see (4.83):

Wq

exp(ikqAj ) = 0, m = 0, 1, 2, 3, . . .

(10.16)

kq |Aj |=Rm

where the second sum is over lattice vectors of the same length equal with the mth neighbor distance Rm, the first sum is over a set of these special kq points, and Wq are weighting factors equal to the number of rays in their stars. The larger the number m, the better is the electronic-density approximation for the perfect crystal. The numbers M (m = 0, 1, 2, . . .), defined according to (10.16) the accuracies of the corresponding sets, are given in Table 10.13.

The ab-initio periodic restricted and unrestricted HF (RHF,UHF) calculations were performed for perfect and defective SrTiO3 crystals, respectively, using the CRYSTAL computer code [23]. This code has the option to perform both HF and DFT calculations on equal grounds, for a large number of implicitly or a posteriori used exchange-correlation functionals that permits one to analyze directly the relevant electron-correlation e ects keeping other computational conditions the same. The same LCAO basis set was used in the HF and DFT (PWGGA) calculations. Large-core Durand–Barthelat [484] for Ti and O atoms and Hay–Wadt small-core pseudopotentials [483] for Sr atoms were used. The impurity iron atoms were treated as all-electron atoms. The “standard” basis for Ti and O was taken from previous TiO2 LCAO calculations [574] whereas that for Fe and Sr from [629] and the CRYSTAL code site [23], respectively. The outer Ti, Fe and oxygen O basis functions were reoptimized. To characterize the chemical bonding and covalency e ects for both defective and perfect crystals, a standard Mulliken population analysis was used for the e ective atomic charges q and other local properties of electronic-structure – bond orders, atomic covalencies and full valencies (these local properties of electronic structure are defined in Sect. 9.1.1).

Since the lattice-relaxation calculations around a point defect was one of the aims, the band-structure calculations with the lattice constant a optimization were made

10.3 Point Defects in Perovskites

447

Table 10.13. Convergence of results for pure SrTiO3 (a=3.905 ˚A) obtained for pure HF

(a) and DFT-PWGGA (b) LCAO band calculations corresponding to cyclic clusters of an increasing size [680]. All energies in eV, total energies are presented with respect to the reference point of 80 a.u.=2176.80 eV. q and V are e ective atomic charges and valencies (in e), respectively. RM and M are explained in the text

	Supercell		Matrix		M		˚		Etot (eV)		V			C		q(Ti)	q(O)		q(Sr)		V (Ti)	V (O)			V (Sr)
						RM(A)
										a) HF LCAO calculations
	S8, sc			A	4		7.81			–24.288	–6.838			3.393		2.39	21.41		1.84		3.94		2.07			2.01
	S16, f cc			B	7	11.04				–24.818	–6.895			3.766		2.54	21.46		1.84		3.97		2.06			2.01
	S32, bcc			C	11	13.53				–24.873	–6.906			3.725		2.54	21.46		1.84		3.98		2.06			2.01
	S64, sc			D	14	15.62				–24.873	–6.906			3.720		2.54	21.46		1.84		3.98		2.06			2.01
	S108, bcc			E	24	20.29				–24.883	–6.895			3.744		2.54	21.46		1.84		3.98		2.06			2.01
										b) DFT-PWGGA LCAO calculations
	S8, sc			A	4		7.81			–73.059	–4.647			4.367		3.43	21.74		1.79		3.98		2.06			2.04
	S16, f cc			B	7	11.04				–73.024	–2.735			2.169		2.82	21.51		1.71		3.97		2.09			2.04
	S32, bcc			C	11	13.53				–70.874	–2.737			1.219		2.55	21.42		1.70		3.99		2.11			2.04
	S64, sc			D	14	15.62				–66.101	–2.414			0.027		1.69	21.13		1.70		4.12		2.18			2.04
	S108, bcc			E	24	20.29				–66.134	–2.443			–1.025		1.69	21.13		1.70		4.12		2.18			2.04
Transformation matrices
		2 0 0					2 2 0				2	2		2				4 0 0					3	3		3
		0 0 2					0 2 2				−2	−2 2						0 0 4					−3	−3 3
A =		0 2 0		, B =			2 0 2			, C =	2 2			−2	, D =			0 4 0		, E =			3 3			−3

for the perfect crystal. The values obtained were a=3.92, 3.84, and 3.92 ˚A, for HF, HF-PWGGA and DFT-PWGGA, respectively. The experimental value is a0=3.905 ˚A. The bulk modulii are B=222, 242, and 195 GPa, respectively, to be compared with the experimental value (extrapolated to 0 K) B=180 GPa. That is, pure HF gives an error of 0.5% only for the lattice constant, and by 20% overestimates the bulk modulus. Its a posteriori electron-correlation correction, HF-PWGGA, gives too small a value, and B even larger than the pure HF. Use of an optimized basis set results in a=3.93, 3.85, and 3.93 ˚A for HF, HF-PWGGA, and DFT-PWGGA, respectively. The relevant bulk moduli are B=220, 249, and 191 GPa, respectively. That is, the basis optimization only slightly a ected the calculated a and B values.

Tables 10.13a and b demonstrate the e ect of the cyclic-cluster increase for both HF and DFT-PWGGA methods, respectively. The main calculated properties are: the total energy Etot (per primitive unit cell), one-electron band-edge energies of the valence-band top and conduction-band bottom εv and εc, Mulliken e ective atomic charges q and full atomic valencies V . As is seen, the result convergence, as the supercell size increases, is quite di erent for the HF and DFT. We explain the much slower DFT convergence by a more covalent calculated electron-charge distribution, as compared to the HF case. For both methods, the convergence of local properties of the electronic structure is faster than that for the total and one-electron energies.

Based on the results of Table 10.13 the conclusion can be drawn that in the HF LCAO calculations of a perfect crystal, the electronic structure is reasonably well

448 10 Modeling and LCAO Calculations of Point Defects in Crystals

reproduced by the supercell of 80 atoms (L=16). This is confirmed by the bandstructure analysis. The results of the standard band-structure calculations for the SrTiO3 primitive unit cell with Monkhorst–Pack k set 6 ×6 ×6 and the cyclic cluster of 80 atoms (only at the Γ point of the BZ) are very similar. It appears that the most important features of the electronic structure of a perfect crystal (valenceand conductionbandwidths, local properties of electronic structure) are well reproduced by the cyclic cluster of 80 atoms. The corresponding one-electron energies do not practically change along all the symmetry directions in the narrowed BZ for supercell band calculations.

Analysis of the di erence electron-density plots, calculated for the band and the 80atom cyclic-cluster calculations confirms that the latter well reproduces the electrondensity distribution in a perfect crystal. Lastly, the total and projected density of states for a perfect crystal show that the upper valence band consists of O 2p atomic orbitals with admixture of Ti 3d orbitals, whereas the Sr states contribute mainly to the energies close to the conduction-band bottom, in agreement with previous studies.

However, as follows from Table 10.13, very accurate modeling of pure SrTiO3 by means of DFT-PWGGA needs use of cyclic clusters as large as 320 atoms (L=64). This conclusion agrees with the results of DFT PW supercell calculations on SrTiO3 discussed in the preceding subsection.

Table 10.13 has shown that an increase of the cyclic cluster from S16 to S32 does not change the HF-calculated top of the valence band. However, the calculated width of the defect impurity band EW found using a standard Monkhorst–Pack set 6×6×6 for three di erent supercells (Table 10.14) demonstrates clearly a considerable dispersion of defect energies across the supercell BZ.

Table 10.14. The width of the Fe impurity band EW calculated for the relevant supercells

Supercell	Number of atoms		˚	EW (eV)
		Fe–Fe distance (A)
S8	40	7.81		1.42
S16	80	11.04		0.23
S32	160	13.53		0.14

Indeed, the EW decreases rapidly, from 1.42 eV (S8) down to 0.23 eV (S16), and further down to 0.14 eV (S32), when the Fe–Fe distance increases only by a factor of about 2, from 7.81 to 13.53 ˚A, since an overlap of the impurity atomic functions decreases exponentially. This is why only S32 (160-atom cyclic cluster) is suitable for a careful modeling of the single Fe impurity and lattice relaxation around it. This is in contrast to many previous supercell calculations of defects in perovskites where S8 supercells were often used without any convergence analysis.

Mulliken e ective charges calculated for ions at di erent positions in supercells modeling pure and Fe-doped SrTiO3 are summarized in Table 10.15.

Table 10.15 demonstrates that the standard band-structure calculation and the S64 cyclic cluster give essentially identical charges. The more so, charges of the same ions in a 320-atom supercell are the same, irrespective of the ion position inside the cyclic cluster. Next, in the defective-crystal calculations, say, for the S32 cyclic