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Computer Modelling & New Technologies, 2002, Volume 6, No.1,29-41 Transport and Telecommunication Institute, Lomonosov Str.1, Riga, LV-1019, Latvia

FIRST PRINCIPLES CALCULATIONS ON Fe4+ IMPURITY IN STRONTIUM TITANATE MONOCRYSTAL*.

S. PISKUNOV

Department of Theoretical Solid State Physics University of Osnabrück

Barbarastr. 7, D-49069 Osnabrück, Germany

The first ab initio Hartree-Fock investigation of Fe impurity in SrTiO3 by means of CRYSTAL computer code for periodic systems is presented. Using the Unrestricted Hartree-Fock (UHF) method and a supercells containing 80 atoms, the energy level positions in the gap and atomic geometry for the Fe3+ and Fe4+ impurities substituting for a host Ti atom in SrTiO3 are calculated. The energy level positions strongly depend on the asymmetric displacement mode of the six nearest O ions which is a combination of the Jahn-Teller and breathing modes. A considerable covalent bonding between the Fe ion and four nearest O ions takes place. The main points of seminar were:

• Why, where and for what we use perovskite ferroelectrics • Difference between pure SrTiO3 crystal (5 atoms) and ST LUC16 and ST LUC32 (80 and 160 atoms respectively) • Lattice relaxation near impurity • Crystal properties in ST with Fe impurity

Keywords: Ab initio Unrestricted Hartree-Fock calculations, CRYSTAL code, SrTiO3, Fe impurity, cyclic cluster model

1. Introduction

Perovskite ferroelectrics crystals are very important for many high-technological applications. The bulk ferroelectrics are used for sonar applications and the thin films are used for phase-array antennas, microwaves mixer, filters, phase shifters, as well as ABO3 perovskites ferroelectrics are used in electronics for DRAM, non-volatile RAM, planar waveguides, electro-optical materials and for substrates for high Tc cuprate superconductor growth [1-4]. Strontium titanate (STO) with Fe4+ impurity, where Fe ion substitutes for Ti4+ ion, is used in photochromic and photorefractive processes, STO with Fe3+ impurity is used for electrochemical electrodes and resistive oxygen sensor [5,6].

Figure 1. Devices, using ferroelectric films.

* The Seminar Report, Riga, Institute of Solid State Physics, CAMART’ 2001

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The unit cells of cubic SrTiO3 lattice sectioned by three different planes, (100), (110) and (111), are shown in Figs. 2a-c. The SrTiO3 primitive unit cell contains five atoms which is also the case for other ABO3 perovskites. According to structural studies of low-temperature strontium titanate [7], its unit cell possesses a simple cubic symmetry Pm3m and a lattice constant a0 ≈ 3.89 Å [8], which below 105 K transforms to the tetragonal structure [7]. Oxygen ions in a cubic unit cell of SrTiO3 form a perfect octahedron, thus internal titanium ion in its center is closer to O atoms (RTi−O =a0/2) than strontium ions outside oxygen octahedron (RSr−O = a0/√2). Therefore, chemical bonding along Ti−O bonds could be stronger than for Sr−O bonds. The color sketch of STO lattice is presented on Fig.3 and Fig.4 shows us the picture with the samples of SrTiO3.

Figure 2. The structural units of cubic SrTiO3 crystal cross-sectioned by three different planes (shaded): a) the (100) surface containing O2- and Sr2+ ions, b) the (110) surface containing Ti4+ , O2- and Sr2+, c) the (001) surface containing Ti4+ and O2- ions. a0 is the lattice constant.

Figure 3. SrTiO3 lattice.

Figure 4. Samples of SrTiO3 .

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1. Theoretical background

2.1. HARTREE-FOCK CALCULATIONS ON PERIODIC SYSTEMS

The crystalline-orbital (CO) method, upon which CRYSTAL treatment is based, uses a set of localized atom-centered GTFs, χµ (r ), for the ne-electron-containing unit reference cell (UC) of periodic lattice described by g translation vectors. The unknown COs are expanded as linear combinations of a set of m Bloch functions (BF) built from these GTFs (CO LCGTF):

∑ −⋅∑=ϕ= g

µm

1µµiei g)(rg)χexp(ik(k)cnr)(k, ,

(1)

where k is a general vector in the first Brillouin zone (BZ). Solution of one-electron Hartree-Fock (HF) equations for this crystal

),()(),(F iii rkkrk ϕε=ϕ (2)

defines the energy spectrum of eigenvalues ε i (k). BF representation of the Fockian becomes )(F)exp()(F ggkk

gµνµν ∑ ⋅= i , (3)

where )(gµνF is the matrix element of the Fock operator between the µ-th AO located in the zero UC (0) and the ν-th AO located in the g cell (the row index can be limited to the 0 cell for translation symmetry) and may be presented as a sum of four different contributions:

)(K)(J)(V)(T)(F ggggg µνµνµνµνµν +++= , (4)

where matrix elements of kinetic ( T ), electron-nuclei (V ), Coulomb ( J ), and non-local exchange ( K ) operators are defined throughout the first BZ:

∫ −∇= rgrrg r dT 2 )()()( νµµν χχ , (5a)

rgrgRr

rgg

d'

ZV

j

jN

1j ')()()( −

−−∑ ∑ ∫==

νµµν χχ , (5b)

∑ ∫∫ −−−−−

−∑ ∑=''

m

''dd'''''''

''''PJ

ggrrggrgr

grrgrrgg )()(

1)()()()( σλνµ

λσλσµν χχχχ ,

(5c)

∑∫∫ −−−−−

−∑∑−=''

m

''dd''''

'''''''PK

ggrrggrgr

grrgrrgg )()(

1)()()(

2

1)( σνλµ

λσµν χχχχλσ ,

(5d)

where ∇r is a Laplace operator, N, Rj, and Zj are number of atoms per UC, their radii-vectors and charges, respectively, g′ and g′′ determine lattice summations, whereas elements of density matrix are defined as:

∫

∑ ε−εθ⋅= σλλσ

zoneBrillouin

n

iiFi

*i d)}({)(c)(c)'exp()'(P

bkkkkgkg i2 ,

(6)

where θ is the so-called Heaviside "zero-temperature" occupation function, εF is the Fermi energy which determines the occupied manifold in k-space, whereas cµi(k) and εi(k) are eigenvectors and eigenvalues of Fock matrix ||F(k)||, respectively. In matrix form, HF equations for equations for closed shell systems may be presented as: ||F(k)|| × ||C(k)|| = ||S(k)|| × ||C(k)|| × ||E(k)|| , (7) where ||C(k)|| and ||E(k)|| are matrix of eigenvectors and diagonalized matrix of eigenvalues, respectively, whereas ||S(k)|| contains Fourier transformations of pair-wise overlap integrals:

rgrrgkkg

d)()()exp()(S −χ∑ ∫χ⋅= νµµν i . (8)

An expression for the total electron energy per UC, which is obtained using variational principle and taking into account representation of the trial HF wave function as a single determinantal function constructed from antisymmetrized spin-orbitals, can be written as follows:

[ ])(V)(T)(F)(PEm

tot ggggg

µνµνµνµν

µν ++∑∑= (9)

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In the case of open shell periodic systems, the so-called unrestricted HF method (UHF) is usually applied, where two matrix equations must be solved self-consistently, for both α and β electrons: ||Fα(k)|| × ||Cα(k)|| = ||S(k)|| × ||Cα(k)|| × ||Eα(k)|| , (10a) ||Fβ(k)|| × ||Cβ(k)|| = ||S(k)|| × ||Cβ(k)|| × ||Eβ(k)|| . (10b) The total density and spin density matrices are defined in direct space computational scheme using UHF, as: ||P(g)|| = ||Pα(g)|| + ||Pβ(g)|| , (11a) ||Pspin(g)|| = ||Pα(g)|| − ||Pβ(g)|| , (11b) where ||Pα(g)|| and ||Pβ(g)|| are obtained as in Eq. (6) by using the eigenvectors ||Cα(k)|| and ||Cβ(k)|| obtained from Eqs. (10a,b), respectively. Elements of Fockians are defined as follows:

)(K)(F)(F spin ggg µνµναµν −= , (12a)

)(K)(F)(F spin ggg µνµν

βµν += , (12b)

where ||F(g)|| matrix is defined as in Eq. (4), the total density matrix ||P(g)|| from Eq. (11a) is used in the Coulomb and exchange terms written in Eqs. (5c,d), whereas )(gspinKµν is defined as )(gµνK in Eq. (5d),

however, elements of spin density matrix )(gspinPµν are used there instead of Pµν(g) for total density matrix. An expression for the total electron energy per UC according to the UHF procedure is slightly more complicated than that in Eq. (9):

[ ]{ }∑∑ +++=µν

µνµνµνµνµνµν

m spinspintot )(K)(P)(V)(T)(F)(PE

ggggggg

(13)

To correct Etot for the standard HF method by a posteriori estimating the electron correlation energy, CRYSTAL code provides an integration of the applying DFT functional:

[ ] rrr d)()(Ecellunit c

)DFT(c ∫ ρερ=

(14)

where εc is a correlation energy per particle, whereas HF electron density ρ(r) is defined as:

∫ ∑ ε−εθϕ=ρzoneBrillouin

statesoccupied

iiF

2i d)}({|),(|)( kkrkr

(15)

Cyclic procedure of the Self-Consistent-Field (SCF) solution for matrix Eqs. (7) and (10a,b) in the framework of CRYSTAL computational scheme is illustrated in Figure below. All the relevant quantities (mono- and bi-electronic integrals, overlap and Fock matrices) are computed in the direct configuration space. Just before diagonalization step ||F(g)|| matrix is Fourier transformed to reciprocal space of BF according to Eq. (3), then both eigenvalues ε i (k) and eigenvectors cµ i (k) are combined to generate direct space matrix ||P(g)|| using Eq. (6) for the next SCF cycle. Each of them is completed by total energy calculation using either Eq. (9) or Eq. (13). Two sections of the UHF scheme, corresponding to α and β electrons, are independent until the Fermi energy εF calculation and ||F(g)|| matrix reconstruction. It is then possible to force the system into a state with a particular total spin value (Sz), by imposing the corresponding value when the crystalline one-electron energy levels ε i are populated at each cycle of the

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SCF step. The whole procedure is completed after n-th SCF cycle when either set tolerance of the total energy per UC is achieved, i.e. E|EE| )1n(

tot)n(

tot δ<− − , or n exceeds set limit of cycles [9].

Figure 5. Scheme of implemenation of the standardHF method in CRYSTAL code. 2.2. PROPERTIES

To construct the difference 2D plots of the electron charge distribution we have sectioned bulk

SrTiO3 cells by main crystallographic planes. The difference density ∆ρ(r) represent the total electron density minus the superposition of atomic (or ionic) densities. When constructing a rectangle 2D section of ∆ρ(r), its area limited by a set frame is divided on equal elemental squares ∆sij (i and j being indices for their enumeration along length and height of rectangle, respectively). Electron density ρ(rij) concentrated on ∆sij includes contribution from each crystalline atom (ion) and may be described if the Gaussian-type function )( ij

A rµχ localized on arbitrary A atom is decomposed on the radial and angular components

)( ijAR rµ and )( ij

AY rµ , respectively. In the direct space, representation of CO LCGTF (for the closed electronic shell) using algorithm, which includes one- and two-center components of electron density (and neglects three- and four center contributions), the charge distribution in each elemental square ∆sij of the plot may be defined as:

,)(Y)(R)(RP)(Y)(Y)(R)(RP

)()(P)(

ijAB

ijB

ijA

BA

ijA

ijA

ijA

ijA

AA

ijij

atomsall

ij

rrrrrrr

rrr

µννµµν

<

νµλµλµµλ

λ<µ

∗νµµν

νµ

∑∑+∑∑

≈χχ∑∑=ρ

(16)

Fourier transformation

of ||F(g)|| to ||F(k)||

SCF

Geometry and basis set input; symmetry analysis;

integrals classification; computation of mono- and

bi-electronic integrals

Reconstruction of Fock matrix ||F(g)||

Diagonalization of ||F(k)||

Calculation of Fermi energy εF(k)

as well as reconstruction of

density matrix ||P(g)||

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where µνP is an element of the density matrix described above, )( ijAY rµ a spherical harmonic of one-

center component density distribution and )( ijABY rµ the same for two-center component which describes

the angular overlap of any pair of atomic orbital with s-, p- or d-configurations. To construct the density of electronic states (DOS) for bulk strontium titanate we have used energy levels in the interval of both valence and bottom of conduction bands. The total and projected DOS (s- and p- and d- levels of Sr2+, Ti4+ and O2- ions) have been calculated to estimate their contribution to a chemical bonding in SrTiO3 crystal. To smooth the DOS curves we have used Gaussian-like broadening of the discrete energy levels ε k defined as:

( ) ( )∑

σε−ε

−πσ

=εk

2

2k

2total

2exp

81N ,

(17)

( ) ( )∑ ∑∑

σε−ε

−

πσ=ε

µ νµν

k2

2k

statesprojected

2projected

2expP

81N ,

(18)

where σ is a Gaussian-like broadening parameter modelling the effect of electron-phonon interaction [10]. 3. Main results

The ab initio periodic Unrestricted Hartree-Fock (UHF) calculations were performed using the

CRYSTAL-98 computer code [11]. The crystalline orbitals used as the basis set for the wavefunction expansion were constructed from a linear combination of atom-centered Gaussian orbitals (HF- LCGO approximation). To reduce the computational time Durand-Barthelat [12] for Ti and O atoms and Hay-Wadt small-core pseudopotentials for Sr atoms [13] were used. The impurity iron atoms were considered including all electrons and full-electron basis set was used for Fe [11]. For Ti and O basis sets were taken from [14] and for Sr from [15]. To avoid boundary effects typical for the molecular cluster model of partly-covalent solids a periodic supercell model was used. Increasing the supercell size allows us to decrease inter-defect interactions caused by an artificial periodicity of defects. In the limit of large supercells the results correspond to Large Unit Cell (LUC) or cyclic cluster model [16,17]. The transformation of simple cubic STO basic translation vectors using the matrices

200020002

,

220202022

,

−−

−

222222222

,

(19)

generates supercells of 8, 16 and 32 primitive cells corresponding to simple, face-centered and body-centred cubic lattices, respectively. The same supercells were used for defect calculations, substituting one Ti atom by an Fe atom. Calculations of the lattice constant a0 and bulk modulus B for the pure STO crystal (Fig. 6) give a0 = 3.92 Å and 3.84 Å, and B=222 and 242 GPa for the pure HF and HF-PWGGA methods, respectively. The pure HF calculations give quite good agreements with experimental data. It allows suggest that the optimisations of geometry and basis sets were made correctly. The table 1 shows the principal differences between LUC with 40, 80 and 160 atoms. It is evident that the characteristics of

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Figure 6. The equilibrium lattice constant calculated using the HF and HF-GGA methods a0 = 3.92 Å and 3.84 Å, and B=222 and 242 GPa for the pure HF and HF-PWGGA methods, respectively

TABLE 1. Positions of one-electron energies for the valence band top and the width of the Fe impurity band, Ew, (in eV) calculated for the large unit cells of three different sizes LUC No atoms VB top, eV Fe-Fe distance (Å) Ew, eV L8 40 -6.85 7.81 1.42 L16 80 -6.89 11.04 0.23 L32 160 -6.90 13.53 0.14 the LUC 16 and LUC 32 are quite similar and LUC 16 with 80 atoms was chosen for the basic calculations. The effective charges q of atoms collected in Table 2 demonstrate considerable covalence effects, well known for ABO3 perovskites. The difference electron density maps for STO cubic lattice 5 atoms with shrinking factor of IS=6 (6 6 6 Pack-Monhorst k-mesh) and for LUC 16 cyclic cluster of 80 atoms (IS=1) are compared on Fig. 7-9. On the plots, both titanium and oxygen atoms clearly demonstrate formation of covalent bonding as for simple cubic lattice as for LUC 16 cyclic cluster, pictures for both cases are similar and it shows us one more time that the cyclic cluster model is an excellent instrument to describe the solids, both pure and with defects. The Fig. 10 shows the difference electron density plots and spin density plots for STO with Fe impurity. The four alpha electrons are well localized on Fe impurity atom and six nearest oxygen atoms. The Fig. 11,12 show the distributions of the covalent bonding after Fe4+ impurity adding. These pictures are very interested from the chemical point of view. TABLE 2. The effective Mulliken charges of atoms q and bond orders P (in e) calculated for unrelaxed and relaxed STO:Fe lattices

SrTiO3 q (Ti) q( Ox,y) q(Oz) P (Ti-Ox) P(Ti-Oz)

pure 2.540 -1.459 -1.459 0.375 0.375

Fe –doped q(Fe) q(Ox,y) q(Oz) P(Fe-Ox) P(Fe-Oz)

unrelaxed 2.570 -1.464 -1.464 0.164 0.164

relaxed 2.594 -1.440 -1.534 0.235 0.154

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Figure 7. The difference electron density plots for the (001) surface of pure SrTiO3 plotted for a) primitive unit cell of 5 atoms with shrinking factor of IS=6 (6 6 6 Pack-Monhorst k-mesh), b) a cyclic cluster of 80 atoms (IS=1). Isodensity curves are drawn from –0.8 to +0.8 e au-3 with an increment of 0.0022 e au-3. The difference is with respect to the spherical atoms

Figure 8. The difference electron density plots for the (100) surface

Figure 9. The difference electron density plots for the (110) surface

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Figure10. The difference electron density plots and the spin density plots for the SrTiO3 cyclic cluster of 80 atoms with a single Fe impurity: a) the (100) surface, b) the (110) surface, c) the (001) surface. Isodensity curves are drawn from 0.8 to +0.8 e au-3 with an increment of 0.0022 e au-3

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Figure11. The difference electron density plots for the SrTiO3 cyclic cluster of 80 atoms with a single Fe impurity for the (001) surface. Isodensity curves are drawn from -0.1 to +0.3 e au-3 with an increment of 0.008 e au-3

Figure12. The difference electron density plots for the SrTiO3 cyclic cluster of 80 atoms with a single Fe impurity for the (110) surface. Isodensity curves are drawn from -0.1 to +0.3 e au-3 with an increment of 0.008 e au-3

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Figure13. (a) Schematic view of the Fe impurity with asymmetric eg relaxation of six nearest O atoms, (b) The relevant energy levels before and after relaxation

In the perovskite crystalline field a 5-fold degenerate Fe 3d state splits into eg and t2g states (Fig. 13.b). In the high spin state with S=2 the upper eg level is occupied by one α-electron and three other α-electrons occupy t2g states. In that situation an Eg⊗eg Jahn Teller effect takes place [18,19]. This means that an orbital degeneracy is lifted by an asymmetrical eg displacement of six O ions, as shown in Fig. 13.a: four equatorial O atoms lying in the x-y plane relax towards the impurity, whereas the two other O ions relax outwards along the z axis. This results in two a1g and b1g levels close to the valence band top, eg level 0.4 eV above the band, and virtual b2g level lying much higher (Fig. 13.b). Fig. 14 shows the calculated lattice energy gain due to asymmetric oxygen ion displacements. The calculations give practically the same magnitude of the six oxygen displacements, δ=0.04 Å, a quite flat minimum and energy gain of 1.4 eV.

Figure14. Calculated lattice energy gain due to asymmetric O ion displacements. Both, pure HF and HF-PWGGA calculations give magnitude of the six O displacements, δ=0.04 Å, a quite flat minimum and an energy gain of 1.4 eV

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The following two pictures contain the total and projected densities of states (DOS) for pure STO 5 atoms and for LUC 16 80 atoms with Fe impurity. The total DOS for both cases qualitatively correlate with those published earlier on [10]. The 3d Fe PDOS shows the interested main Fe levels near bottom of valence band, probably, prediction of the Fe level positions in the ground state could be checked by means of UPS experiments.

Figure15. Total and projected densities of states for SrTiO3 with unit cell 5 atoms

Figure16. Total and projected densities of states for SrTiO3:Fe with LUC 16 80 atoms

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4 Conclusions

The present large scale ab initio calculations for periodical systems of 80 atoms have demonstrated strong covalent bonding between unpaired electrons of Fe impurity and four nearest O ions relaxed towards an impurity. Positions of Fe energy levels in an SrTiO3 gap are very sensitive to the lattice relaxation. The predicted positions of the Fe energy levels with respect to the valence band could be checked by means of the UPS spectroscopy.

Acknowledgements

I am very thankful to Dr. Yuri F. Zhukovskii (Institute of Solid State Physics, the University of Latvia), Prof. Dr. Eugene Kotomin (Max Planck Institut für Festkörperforschung (Stuttgart, Germany), Institute of Solid State Physics, the University of Latvia), Prof. Dr. Robert A. Evarestov (Department of Quantum Chemistry, St. Petersburg University, Russia) for a many fruitful discussions during my work in Excellence Center in Riga where the seminar was prepared.

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Received on the 29th of December 2001