Defect clusters in titanium-substituted spinel-related lithium ferrite

H.M. Widatallaha,b, E.A. Mooreb,*

aDepartment of Physics, University of Khartoum, P.O. Box 321, Khartoum 1115, SudanbDepartment of Chemistry, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK

Received 19 January 2004; revised 12 April 2004; accepted 12 April 2004

Abstract

The cation distribution in spinel-related titanium-substituted lithium ferrite, Li0.50.5xFe2.521.5xTixO4 has been explored using interatomicpotential and ab initio calculations. The results suggest that the cation distribution with Ti4 substituting for Fe3 on octahedral B sites andexcess Li substituting for Fe3 on tetrahedral A sites is stabilised by the formation of clusters of two octahedrally coordinated Ti4 ions andone tetrahedrally coordinated Li ion linked through a common oxygen.q 2004 Elsevier Ltd. All rights reserved.

Keywords: A. Ceramics; A. Magnetic materials; C. Ab initio calculations; D. Defects

1. Introduction

The lithium ferrite Li0.5Fe2.5O4 has attractive electric and

magnetic properties for microwave and memory-core

applications [14]. These properties can be modified by

the substitution of Fe3 on either the tetrahedral A oroctahedral B sub-lattices thus allowing the material to be

tailored for specific applications [1,5].

At low temperatures, lithium ferrite adopts an inverse

spinel structure in which all the Li ions and 3/5 of the Fe3

ions occupy octahedral B sites whilst the remaining Fe3

ions occupy tetrahedral A sites and the space group is P4332

[6,7]. At temperatures above 735 755 8C, the siteoccupancy becomes disordered and the solid can be indexed

according to the space group Fd3m. For slowly cooled

samples of Ti-doped lithium ferrite with low concentrations

of Ti, it is widely accepted [810] that Ti4 ions replacesome of the Fe3 ions on the octahedral B sites, althoughsimilarity in the scattering factors for Ti and Fe mean that

XRD cannot unequivocally determine the cation

distribution [10]. These samples can be considered as

solid solutions in the Li2OTiO2 Fe2O3 system and

generally contain excess Li ions above the number in

Li0.5Fe2.5O4. Experimental data indicates that these excess

Li ions occupy tetrahedral A sites, giving a formula(Fe1.020.5xLi0.5x)A(Fe1.52xLi0.5Tix)BO4. XRD [10,11] shows

that substitution is ordered giving a space group P4332 for

0 # x # 0:4: The lattice constant increases only very

slightly over this range from 0.833 to 0.834 nm for powder

samples. 57Fe Mossbauer spectra of Ti-doped lithium ferrite

with the value of x in this range (11 and work in our own

laboratory) show two sextets corresponding to iron on the

A and B sites, respectively. The average hyperfine field for

both sextets is reduced relative to that for lithium ferrite

indicating substitution on both sub-lattices. Fe3 spins onthe B sites are ferromagnetically coupled to each other as

are the spins on the A sites. FeOFe exchange, however,

couples spins on A and B sites antiferromagnetically.

The overall magnetisation of undoped lithium ferrite is due

to the excess of Fe3 ions on octahedral sites over those ontetrahedral sites. Magnetisation of the Ti-doped lithium

ferrite is lower than that of the undoped ferrite suggesting

greater substitution on B sites than on A sites [8].

In this paper we use interatomic potential and ab initio

calculations to investigate the stability of the proposed

substitution pattern in Ti-doped lithium ferrite with low

concentrations of Ti relative to other possible defect

structures and gain insight into the factors determining

this distribution.

0022-3697/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jpcs.2004.04.002

Journal of Physics and Chemistry of Solids 65 (2004) 16631667

www.elsevier.com/locate/jpcs

* Corresponding author. Tel.: 44-1908-655028; fax: 44-1908-858327.

E-mail address: e.a.moore@open.ac.uk (E.A. Moore).

2. Computational procedure

2.1. Interatomic potential calculations

Interatomic potential calculations were carried out using

the program GULP [12] in which the force field used consists

of a pairwise interaction energy that is composed of a

Buckingham potential to model the short-range Pauli

repulsion and the leading term of any dispersion energy

and the Coulomb interaction, such that

Eij A exp2rijr

2 Cr26ij qiqi

rij

where Eij is the interatomic potential, A; r and C areempirical constants, q is the charge of the particles and rijthe interatomic spacing. GULP uses the MottLittleton

approach to point defect calculation with the crystal

surrounding the defect divided into three spherical regions.

In the first region, all the ions are treated exactly and are

allowed to relax their positions in response to the defect.

The radius of this region was taken to be 8 A. In a second

spherical region, individual ions were displaced, and the

ions in a harmonic potential well were assumed to

approximate the response to the electrostatic force of the

defect. The radius of the second region was set to 14 A.

In the third region, a similar approximation is used except

that only polarisation of sub-lattices is considered.

The interatomic potential parameters for Li, Fe3 andO22 used are similar to those used [13] to investigate

quaternary LiMnFeO spinels. The OO potential

parameters are identical to those in an earlier paper [14]

and therefore the Ti4 parameters given there are used.The parameters used are given in Table 1.

2.2. Ab initio calculations

Ab initio calculations were carried out using the program

CRYSTAL98 [15]. This code is based on the ab initio periodic

linear combination of atomic orbitals. An unrestricted

HartreeFock treatment was employed. Because of the

large size of the unit cell (the unit cell of lithium ferrite is

Li4Fe20O32), a basis set in which the core electrons were

described by Poples 3-21G basis [16] was used. For the

valence electrons a larger set of functions was used based on

the 8-6-411G(Fe, Ti) or 8-5G (O) basis sets used in CRYSTAL

calculations on iron and titanium oxides by other workers

[17,18]. These valence basis sets were reoptimised for the

3-21G description of the inner electrons by varying the

coefficients using GAMESS [19] calculations on Fe3 andTi4. For Li we used Poples 3-21G basis set, but with theouter s/p shell replaced by a p function of coefficient 0.6.

This approach has been used successfully for calculating

electric field gradients [20]. The Fe and O basis sets were

tested on a-Fe2O3. The experimental order of energies of thepossible spin states of this oxide (ferromagnetic and the three

possible antiferromagnetic states) were reproduced correctly

for the experimental geometry. We also performed runs on

a-Fe2O3 for different values of the lattice parameter c; but afixed c=a ratio and for several values of c=a for a fixed value of

a: For the experimental c=a ratio of 2.7336, our calculations

gave a minimum around c 1:381 nm compared to theexperimental value 1.3772 nm. For a 0:5069 nm, ourminimum ratio, c=a was 2.727. The agreement with

experiment is reasonable and comparable to that obtained

using large basis sets and so gives us confidence in our basis

set. The basis sets are given in Table 2.

3. Results

3.1. Interatomic potential calculations

As Ti4 ions are the minority species, we consideredmodels in which these ions were present as defects in

Li0.5Fe2.5O4. Both models in which Ti4 substitutes for an

Fe3 ion or Li ion and those in which Ti4 ions are presentin interstitial sites were considered. Li and Fe3 vacanciesand excess Li substituting on Fe3 sites were considered asbalancing defects. Both substitution and vacancy formation

will reduce the number of magnetic ions in agreement with

the experimental results. Reduction of Fe3 to Fe2 was notconsidered as there is no experimental evidence for the

presence of Fe2. The defect reactions considered are givenbelow using the KrogerVink notation.

TiO2s FexFe LixLi! TizFe V0Li 1=2Li2Os 1=2Fe2O3s 1

TiO2s 4=3FexFe! TizFe 1=3V000Fe 2=3Fe2O3s 2TiO2s 3=2FexFe 1=4Li2Os! TizFe 1=2Li00Fe

3=4Fe2O3s 3TiO2s 2FexFe OxO! TizFe V000Fe VzzO Fe2O3s 4TiO2s FexFe LixLi! TizzzLi V000Fe 1=2Li2Os

1=2Fe2O3s 5

Table 1

Interatomic potentials used in this work

A /eV r /A C/eV A6 Spring

constant

k /eV A22

Shell

charge/e

Li core-O22 shel 479.837 0.3 0Fe3oct shel-O

22 shel 1342.754 0.3069 0

Fe3tet shel-O22 shel 1240.232 0.3069 0

Ti4 shel-O22 shel 2088.107 0.2888 0O22 shel-O22 shel 22.410 0.6937 32.32

Fe3 core-Fe3 shel 10082.50 1.029O22 core-O22 shel 20.53 2 2.513

H.M. Widatallah, E.A. Moore / Journal of Physics and Chemistry of Solids 65 (2004) 166316671664

TiO2s 3=2FexFe LixLi 1=4Li2Os! TizzzLi 3=2Li00Fe 3=4Fe2O3s 6

TiO2s 4LixLi! TizzzLi 3V0Li 2Li2Os 7TiO2s FexFe LixLi!Tizzzzint V000Fe V0Li 1=2Li2Os 1=2Fe2O3s 8

TiO2s 4=3FexFe! Tizzzzint 4=3V000Fe 2=3Fe2O3s 9TiO2s 2FexFe Li2Os! Tizzzzint 2Li00Fe Fe2O3s 10

The results are given in Table 3.

Defect reactions involving interstitial Ti4 were not veryfavourable. Defect reactions involving substitution of Ti for

Fe were more favourable. Substitution of Ti4 on anoctahedral Fe3 site was much more favourable thansubstitution on a tetrahedral Fe3 site. However, Li alsoshowed a preference for octahedral sites, although the

energy difference between substitution on octahedral and

tetrahedral sites was smaller than that for Ti4. The lowestenergy defect reaction was reaction (3) with both Ti and Li

replacing Fe on octahedral B sites. This would lead to a

large drop in magnetisation and would not agree with

the observed changes in the Mossbauer spectrum on both

Table 2

Exponents and coefficients of the contracted Gaussian-type basis functions

used in the ab initio calculations

Type Exponent Coefficient

s p d

Fe

s 3299.184 0.06358590

499.0886 0.3762016

109.1614 0.6817845

sp 143.4652 20.1105517 0.1411006

31.16858 0.0968468 0.5603874

9.483612 0.9587974 0.4676444

sp 9.464565 20.2920555 0.02376201

3.100373 0.3375236 0.4689113

0.986493 0.8519416 0.6083113

sp 1.314 1.0 1.0

sp 0.532 1.0 1.0

d 30.48 0.0583

8.692 0.2591

3.101 0.5162

1.171 0.5656

d 0.4298 1.0

Ti

s 2335.020 0.06421661

353.0442 0.378412

77.05846 0.67968

sp 99.57387 20.109471 0.1372966

21.54671 0.1019427 0.5458753

6.413965 0.9546224 0.4890681

sp 6.238379 20.2861372 0.01923665

1.996108 0.3218278 0.4404422

0.06464899 0.8595511 0.6356195

sp 1.017186 1.0 1.0

sp 0.758559 1.0 1.0

d 6.141248 0.048967

2.453525 0.146523

0.952059 0.302947

d 0.496983 1.0

Li

s 36.8382 0.0696686

5.48172 0.381346

1.11327 0.681702

sp 0.540205 20.263127 0.161546

p 0.102255 1.14339 0.915663

0.6 1.0

O

s 322.037 0.0592394

48.4308 0.3515

10.4206 0.707658

sp 7.40294 20.404453 0.244586

1.5762 1.22156 0.853955

sp 0.373684 1.0 1.0

d 0.8 1.0

Table 3

Defect energies for reactions (1)(10)

Defect reaction Defect energy (eV)

Reaction (1) Ti4 on B site 21.56Ti4 on A site 0.52

Reaction (2) Ti4 on B site, vacancy on B site 21.84Ti4 on B site, vacancy on A site 21.25Ti4 on A site, vacancy on B site 0.23Ti4 on A site, vacancy on A site 0.82

Reaction (3) Ti4 on B site, Li on B site 22.44Ti4 on B site, Li on A site 22.36Ti4 on A site, Li on B site 20.37Ti4 on A site, Li on A site 20.29

Reaction (4) Ti4 on B site, vacancy on B site 1.21Ti4 on B site, vacancy on A site 2.99Ti4 on A site, vacancy on B site 3.28Ti4 on A site, vacancy on A site 5.06

Reaction (5) Vacancy on B site 3.47Vacancy on A site 5.25

Reaction (6) Li on B site 1.68Li on A site 1.92

Reaction (7) 4.33Reaction (8) Vacancy on B site 8.57

Vacancy on A site 10.35Reaction (9) Vacancy on B site 8.28

Vacancy on A site 10.65Reaction (10) Li on B site 5.90

Li on A site 6.22

H.M. Widatallah, E.A. Moore / Journal of Physics and Chemistry of Solids 65 (2004) 16631667 1665

A and B sites. For defect reaction (3), we also ran

calculations on supercells of formula Li5.25Fe16.25Ti2.5O32corresponding to x 0:3125 in Li0.50.5xFe2.521.5xTixO4with both Fe A and B sites partially occupied by Fe and by

Ti and/or Li to represent random occupation according to

the possible patterns of substitution over A and B sites.

The supercells were optimised. With Fe and Li only on B

sites the lattice parameter, a; was reduced from 0.8314 nm

in pure lithium ferrite to 0.8263 nm. With Ti on B sites and

Li on A sites, the lattice parameter suffered a smaller

reduction (0.8299 nm) and the optimised cell had lower

energy. Experimentally, the Ti-doped lithium ferrites show

a small increase in lattice parameter from 0.833 to 0.834 nm

for x 0! 0:4: Thus, the supercell with Ti on A sites andLi on B sites produced closer agreement with experiment.

Our initial calculations were for isolated defects, but the

supercells include contributions from defects in close

proximity as well as isolated defects. We therefore

considered clusters of defects corresponding to reaction

(3). Particularly, an energetically favourable cluster

(defect energy 25.76 eV) was found with two Ti4substituting for Fe3 on B sites and one Li substituting forFe3 on an A site and with all three substituting ions linkedto a common oxygen. The position of this defect in the unit

cell is shown in Fig. 1.

A cluster with all substitution on B sites had a defect

energy of 23.53 eV. These results suggest that substitutionof Li for Fe on a tetrahedral A site is favoured by the

formation of defect clusters in which the Li is linkedthrough a common oxygen to two Ti4 on octahedral B sites.

3.2. Ab initio calculations

Calculations were performed on supercells Li5Fe17Ti2O32 corresponding to x 0:25 in the formula Li0.50.5x

Fe2.521.5xTixO4 formed by substituting for three Fe3 ions in

the unit cell of lithium ferrite by two Ti4 ions and one Li

ion. The ordered structure of lithium ferrite [20] was used.

In case the results were affected by changes in lattice

parameter, we ran all substitution patterns at two values of

the lattice parameter, a; 0.8314 and 0.833 nm,

corresponding to the values found for single crystal and

powder X-ray diffraction determinations of the structure.

The results for various patterns of substitution on the

A and B sites are given in Table 4.

The order of the energies is the same for both lattice

parameters. In contrast to the interatomic potential calcu-

lations, the supercell in which both Ti4 and Li substituteon B sites is the least favourable. This may be due to the

reduced spin interactions as spinspin interactions enter

into these calculations, but are ignored in interatomic

potential calculations. However, despite maximising the

unpaired spin giving an advantage when the defects are

relatively isolated (see highest energy for each spin state),

the formation of the cluster in which Li on an A site andtwo Ti4 on B sites are linked through a common oxygenagain produces the lowest energy.

4. Discussion

Interatomic potential calculations indicated that as

isolated defects, Ti4 and Li substitute for Fe3 ions onoctahedral B sites. This is not surprising as Ti4 is known tofavour octahedral sites in other doped iron oxides and Li

occupies octahedral sites in lithium ferrite itself. However, a

cluster of two Ti4 ions on B sites and one Li ion on atetrahedral A site was found to stabilise the structure of the

Ti-doped lithium ferrite. This concentrating of the defects in

one part of the unit cell reduces disruption in the cell,

for example giving only a small change in lattice parameter.

The ab initio calculations took into account not only

structural factors as in the interatomic potential calculations,

but also changes in spin interactions on substitution. In the

ferrites, spins on Fe3 ions on the A sub-lattice areferromagnetically coupled to each other as are the spins on

the B sub-lattice. Ions on the A and B sub-lattice are

antiferromagnetically coupled via exchange FeAOFeB.

In lithium ferrite Li0.5Fe2.5O4, the calculatedab populationis higher on the B sites (4.697) than the A sites (24.660) andthe overlap FeO population is positive for both sites with

stronger bonding of the A sites to a electrons and the B sitesto b electrons. In the Ti-doped lithium ferrite, the averagecalculated ab population on the B sites shows a small dropto 4.688 and that on the A sites a small increase to 24.663.This would produce a small drop in the A-site magnetic field

and a small increase in the B-site magnetic field, but the effect

of substituting non-magnetic nuclei on both sites will cause a

much larger reduction in both fields. The larger hyperfine

field and isomer shift for Fe in octahedral sites than in

tetrahedral sites in the undoped lithium ferrite has been

Fig. 1. Two unit cells of Ti-doped lithium ferrite showing position of cluster

of two Ti and one Li atoms. The black spheres are Fe ions, the largest grey

spheres are Ti ions and the small grey spheres with no bonds shown are Li

ions. The cluster consists of the two Ti ions in the centre of the figure and

the Li ion on the right above these two. The c axis of the cell is vertical.

H.M. Widatallah, E.A. Moore / Journal of Physics and Chemistry of Solids 65 (2004) 166316671666

explained as due to the more covalent character of the FeO

interaction at the tetrahedral sites. Our calculated overlaps

show that this difference in covalency is maintained in the

doped ferrite with the FeO overlap population remaining

similar to that in the undoped ferrite, but with a slight increase

for Fe atoms close to the defect region. The LiO overlap for

both a and b electrons is markedly larger than the FeOoverlap. Spin interactions appear to play a role in stabilising

substitution on tetrahedral sites by favouring states with

higher numbers of unpaired spins. However, the determining

factor in the cation distribution appears to be structural. Ab

initio calculations agree with the interatomic potential

calculations in finding that a defect cluster of two Ti4 ionson octahedral sites and an Li ion on a tetrahedral sitestabilises the Ti-doped structure.

It is interesting to note that the formation of clusters has

been put forward to explain the Mossbauer spectra of

Li0.50.5xFe2.521.5xTixO4 with x 0:5 and 0.7 [11] andthe temperature-dependence of the spectra of MgTi

ferrites [21].

5. Conclusions

Both interatomic potential calculations and ab initio

calculations indicate that Ti-doped lithium ferrite of formula

Li0.50.5xFe2.521.5xTixO4 with low concentrations of Ti isstabilised by the formation of defect clusters of two Ti4

ions substituting for Fe3 on B sites and one Li ionsubstituting for Fe3 on an A site. In these clusters both Ti4

ions and the Li ion are linked through a common oxygen. Itis suggested that such a cluster is stable because it causes

little disruption to the structure of the unit cell.

Acknowledgements

We thank The Abdus Salam International Centre for

Theoretical Physics (ICTP) Trieste for support (HMW) and

the UK computational chemistry working party for a grant

of time on the EPSRC superscalar facility at the Rutherford

Appleton Laboratories.

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Table 4

Ab initio calculated energies of Li5Fe17Ti2O32 for different distributions of substituted Ti4 and Li

Li site Ti4 site A/B linkage Unpaired spinsper unit cell

Energy (104 eV)

a 0:8314 nm a 0:833 nm

A A 35 22.54781005 22.54780779

B A Both A linked to B 25 22.54780962 22.54780731

A B A linked to one B 15 22.54781444 22.54781213

A B A and B linked to common O 15 22.54781834 22.54781603

A A/B Both A linked to B 25 22.54781510 22.54781282

B A/B Both B linked to A 15 22.54780690 22.54780458

B A/B A and B linked to common O 15 22.54781092 22.54780860

B B 5 22.54780096 22.54779929

H.M. Widatallah, E.A. Moore / Journal of Physics and Chemistry of Solids 65 (2004) 16631667 1667

Defect clusters in titanium-substituted spinel-related lithium ferriteIntroductionComputational procedureInteratomic potential calculationsAb initio calculations

ResultsInteratomic potential calculationsAb initio calculations

DiscussionConclusionsAcknowledgementsReferences