Chemical Physics 393 (2012) 148–152

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Chemical Physics

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Ab initio calculation of chromium oxide containing Ti dopant

Frank Maldonado a, Corina Novillo b, Arvids Stashans a,⇑a Grupo de Fisicoquímica de Materiales, Universidad Técnica Particular de Loja, Apartado 11-01-608, Loja, Ecuadorb Escuela de Ingeniería Química, Universidad Técnica Particular de Loja, Apartado 11-01-608, Loja, Ecuador

a r t i c l e i n f o

Article history:Received 16 September 2011In final form 30 November 2011Available online 9 December 2011

Keywords:a-Cr2O3

DFT + U methodTi-dopingStructureElectronic propertiesElectrical conductivityMagnetism

0301-0104/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.chemphys.2011.11.038

⇑ Corresponding author.E-mail address: arvids@utpl.edu.ec (A. Stashans).

a b s t r a c t

First-principles computations based on the density functional theory within the generalised gradientapproximation and introduced intra-atomic interaction term for strongly correlated electrons (DFT + Umethod) has been used in this work. Ti impurity doping in the a-Cr2O3 crystal has been carried out con-sidering single defect model within the periodic crystalline structure. Atomic displacements, Badercharges on atoms have been computed showing that Ti dopant converts the chemical bonding in itsneighbourhood into more ionic one. The defect-local microstructure is such as there exist general ten-dency of atomic rearrangements away with respect to the Ti imperfection. It is found that defect incor-poration produces some local changes upon the band structure of the material and also induces a metallicstate. That implies n-type electrical conductivity in the Ti-doped a-Cr2O3 crystals and relates our workdirectly to a number of experimental studies in this area. Our results provide evidence over change inmagnetic moments in the vicinity of defect, which means that the chromium oxide doped with Ti impu-rity might not act as an antiferromagnetic substance.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Chromium oxide (a-Cr2O3) is rather interesting material andbelongs to the family of transition metal oxides. It is a wideband-gap semiconductor since its direct band-gap width is equalto 3.3 eV [1,2]. It has a wide range of applications, i.e. a-Cr2O3

materials are used as catalysts, solar thermal energy collectors,as well as in black matrix films, liquid crystal displays [3,4], protec-tive layers (corrosion and wear resistance of stainless steel) [5] andadhesion promoters [5]. The a-Cr2O3 has been also employed inspintronic devices such as non-volatile magnetoelectric memories[6,7]. Another attractive exploitation of the material is that the a-Cr2O3 thin films are being considered as electrochromic materials.There exist many crystalline modifications of the chromium oxide,such as Cr2O3 (corundum), CrO2 (rutile), Cr5O12 (three-dimensionalframework), Cr2O5 and CrO3 (unconnected strings of CrO4 tetrahe-dral). However, the only stable bulk oxide form is Cr2O3, which is amagnetic dielectric with the corundum structure [3]. In the a-Cr2O3 corundum structure, the O atoms form a hexagonal close-packing array. The metal atoms occupy two thirds of the octahe-dral interstices between two layers. The a-Cr2O3 might be de-scribed by the rhombohedral primitive unit cell, where Cr atomsare eight-coordinated with oxygens placed in two oxygen layers.

Ti-doped a-Cr2O3 has an extra significance since currently hasbeen used as a material for sensors to detect trace quantities of

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reducing gases in air such as CO and ethanol vapour by changingtheir resistance [8] at certain temperatures. The sensing mecha-nism involves the trapping of electrons by chemisorption of oxy-gen from air on the surface of a semiconducting oxide. It is of aprimary importance to understand the behaviour of the n-typeand p-type electrical conductivity in the material to generate a use-ful sensitivity for sensing trace quantities of reducing gases in air.

Up to now, chromium oxide has been investigated mainly by anumber of different experimental techniques. However, a couple ofattempts have been made to study this system theoretically. As anexample, we can mention density functional theory (DFT) calcula-tions of SO2 chemisorption on the a-Cr2O3 (0001) surfaces [9] aswell as study of chemical states of water on the a-Cr2O3 (0001)surfaces [10].

The present investigation has a purpose to find out the effects ofTi influence upon the structural, electronic, electrical and magneticproperties of corundum-type chromium oxide.

2. Methodology

The present calculations have been carried out using ab initioDFT approach as it is implemented in the Vienna ab initio Simula-tion Package (VASP) [11,12] and the generalised gradient approxi-mation (GGA) [13]. The projector augmented wave (PAW)pseudopotentials as proposed by Blöchl [14] and adapted by Kresseand Joubert [15] were utilised in our investigation.

A cut-off kinetic energy of 500 eV is used by converging the to-tal energy to less than 1 meV/atom. C-centred Monkhorst–Pack

Fig. 1. DFT + U (U = 4 eV) computed total energies for each magnetic configurationas a function of volume. The vertical dotted line shows the optimum volume for 10-atom cell of the a-Cr2O3 crystal.

Fig. 2. DFT + U (U = 4 eV) computed total DOS for the AFM � + � + state of purechromium oxide crystal. The dotted line marks the Fermi level (EF).

F. Maldonado et al. / Chemical Physics 393 (2012) 148–152 149

(MP) grid with a 0.045 Å�1 separation is applied, which corre-sponds to a k-point mesh of 6 � 6 � 6 for the 10-atom primitiveunit cell of the rhombohedral a-Cr2O3. The previously mentionedparameters were obtained through the atomic relaxation until allthe forces are less than 0.008 eV/Å.

Some most common magnetic configurations (Table 1) havebeen taken into consideration in order to find out the equilibriumground state and the corresponding magnetic properties of pure a-Cr2O3 crystal. The considered magnetic states for the structure areas follows: antiferromagnetic (AFM) � � + +, AFM � + + �, AFM � +� + and ferromagnetic (FM) + ++ +. We found that the most favour-able case is the one corresponding to the AFM � + � + state. Fig. 1shows the computed total energy of the system for different mag-netic states as a function of the cell volume. It shows a possibletransition from the AFM � + + � state to the FM + ++ +configura-tion, when volume of the cell is around 99.5 Å3. However, neitherthe AFM � + + � state nor the FM + ++ + configuration are the min-imum-energy states, so we will focus further mainly on the AFM �+ � + case. The computed magnetic moment for the optimum vol-ume of the AFM � + � + configuration is found to be equal to 2.92lB/Cr, which is very close to the corresponding experimental valueof 2.76 lB/Cr [1].

The a-Cr2O3 has a classification as an intermediate-type insula-tor being placed in between the charge-transfer insulator and theMott–Hubbard insulator. This is supported by photoemission andelectron-energy-loss spectroscopy studies [16] as well as X-rayphotoemission experiments [17]. Thus it is necessary to take intoaccount that the DFT-GGA describes inappropriately the strongCoulomb repulsion between the d electrons localised on metalions. One way to correct this deficiency is through the use of an in-tra-atomic interaction for the strongly correlated electrons by anunrestricted Hartree–Fock (UHF) approximation, resulting in theso-called DFT + U method [18,19]. The corresponding equationsdescribing this approach can be found elsewhere [20]. One of thesignificant corrections being provided by the DFT + U is a partialimprovement for the band-gap width, which is underestimatedby the standard DFT method. That is done by shifting the valenceband (VB) states downwards and moving the conduction band(CB) states upwards, respectively. The important point in the appli-cation of this approximation is the choice of a proper value for theU parameter.

We tested a number of different values for U parameter corre-sponding to the d electrons of Cr atom and finally arrived at valueU = 4 eV as a proper value for pure a-Cr2O3 system (J = 0 has beenutilised throughout the study). Thus, our DFT + U modification pro-vides a band-gap width value being equal to 3.21 eV, which isreally very close to the experimental magnitude equal to 3.3 eV[1,2].

The equilibrium structural parameters have been computed byfitting the total energy of the system within a wide range of vol-umes. The optimum volume was found to be equal to 97.5 Å3,

Table 1Main results obtained for the ground-state of the a-Cr2O3 bulk: total energies E plattice parameter a. The corresponding experimental values are displayed in the

Magnetic state E (eV) Magnetic momen

FM ++++ �86.81 2.80AMF�++� �86.82 2.56AMF��++ �87.09 2.56AMF�+�+ �87.31 2.56Exp. – 2.76a

a Ref. [1].b Ref. [23].c Ref. [21].d Ref. [22].

which is close to the experimental findings of 96.5 Å3 [23]. In orderto study Ti impurity in the a-Cr2O3, 10-atom primitive unit cellwas expanded eight times (2 � 2 � 2 extension) resulting in 80-atom supercell, and the MP k-point mesh of 4 � 4 � 4 was appliedfor the new cell.

Density of states (DOS) for pure chromium oxide have beencomputed using U = 4 eV for the AFM � + � + magnetic state(Fig. 2). The obtained value for the band-gap width equals to3.21 eV, which contrasts greatly to the value of 1.4 eV, which isthe computed band-gap width if value U = 0 eV is utilised. Our

er 10-atom cell, magnetic moments per Cr atom, optimum volumes V0 andlast line.

t (lB) V0 (Å3) a (Å)

99.50 5.3398.50 5.3097.50 5.4597.50 5.4096.50b 5.359b 5.362c 5.350d

150 F. Maldonado et al. / Chemical Physics 393 (2012) 148–152

DOS pattern shows that the upper VB is dominated by the Cr 3dstates, with some admixture of the O 2p states, whereas the bot-tom of the CB is almost completely formed by the Cr 3d states.These results are in complete accordance to another available re-port [24] implying that the a-Cr2O3 crystal is an intermediate-typeinsulator between the charge-transfer and Mott–Hubbardinsulators.

Parameter U for Ti 3d electrons were fitted before by Morganand Watson [25] and later slightly readjusted by Çelic et al. [26]to obtain a correct electron description in TiO2 crystals [26].According to our experience [27], any semi-empirical parameterremains practically constant for a given chemical element indepen-dently of the compound. That is why we used value U = 4.5 eV asan additional Hubbard term for electron on-site repulsion for theimpurity states.

Fig. 3. Frontal view of atoms in the vicinity of Ti impurity.

Fig. 4. Atomic displacements in the a-Cr2O3 crystalline lattice in the neighbour-hood of the Ti dopant. Both Cr and O atoms have a tendency to move outwards withrespect to the defect.

3. Results and discussion

3.1. Geometry of Ti-doped Cr2O3

Ti doping was done by replacing one of the lattice central hostCr atoms by a Ti impurity. It was found that the total energy ofthe system increases as a result of such a substitution by approxi-mately 0.22 eV. This value can be considered as the formation en-ergy for a single Ti impurity in the chromium oxide crystal. Thetotal energy augmentation implies that Ti presence in the a-Cr2O3 is energetically unfavourable. However, due to very small va-lue of this energy, the process of Ti doping might be athermic one.In any case, if necessary this energy amount could be easily com-pensated by intrinsic defects like Cr vacancies leading to a favour-able formation energy [28,29].

As a result of Ti atom inclusion in the otherwise pure a-Cr2O3

structure, the atoms in its neighbourhood have tendency to dis-place themselves in order to find their new equilibrium positions.Although we allowed both atomic and cell relaxation, only defect-nearest atomic shifts were encountered while the lattice parame-ters as well as cell volume remained unchanged due to the Ti intro-duction. It is necessary to state that each Cr atom has six O atomsin its vicinity, three at 1.97 Å distance, and the other three at 2.03 Ådistance (Table 2, Figs. 3 and 4). There are two types of Cr atomscharacterised by their magnetic moment, i.e. the first type of Cratoms has a positive magnetic moment (case A) while the secondone has a negative magnetic moment (case B). So, two separatecases have to be taken into consideration in order to study impu-rity doping in the a-Cr2O3 crystalline lattice.

In case A of the supercell, i.e. when the Ti atom replaces for a Cratom possessing a positive magnetic moment, the O atoms have a

Table 2Charges on atoms obtained by the Bader population analysis for the perfect (Q1) andTi-doped (Q2) Cr2O3 crystals. The initial distance to the impurity (R1) and the atomicdisplacements (DR) regarding the impurity for atoms within the defective region arealso shown. Positive atomics displacements stand for the defect-outward movements.The atomic numeration corresponds to the one indicated in Fig. 4.

Atom Q1 (e) Q2 (e) R1(Å) DR(Å)

Ti (1) – 2.02 – –O (2) �1.12 �1.23 1.97 0.01O (3) �1.12 �1.22 1.97 0.03O (4) �1.12 �1.22 1.97 0.04O (5) �1.12 �1.22 2.03 0.06O (6) �1.12 �1.22 2.03 0.06O (7) �1.12 �1.21 2.03 0.07Cr (8) 1.69 1.78 2.68 0.07Cr (9) 1.69 1.78 2.90 0.02Cr (10) 1.69 1.78 2.90 0.02Cr (11) 1.69 1.78 2.90 0.03

slight tendency to increase their initial distances with respect tothe impurity as it is shown in Table 2 and Fig. 4. In order to explainthe reason of these motions, the Bader charge analysis [30] hasbeen carried out. Through the employment of this procedure it ispossible to obtain a charge on a particular atom in the crystallinelattice thanks to the use of a robust algorithm [31–33]. However,to understand better the atomic relaxation we also need to con-sider size of the atoms. It is known that atomic radius of the Tiatom, 1.47 Å, is bigger than the corresponding number of replacedCr atom, 1.27 Å. Consequently the impurity requires more spacethan the replaced Cr atom producing that the defect-closest Oatoms move outwards with respect to the impurity. These move-ments practically are compensated by the Coulomb electrostaticattraction between the same O atom and the Ti atom. That is truesince the Ti atom owns a bigger positive charge, +2.02 e, comparedto that of the replaced Cr atom, +1.69 e, and one also has to con-sider the charge increase on the O atoms (Table 2). That explainswhy the rearrangement of O atoms are rather small.

It is important to note that not all of the O atoms displace them-selves by the same distance (Table 2). The defect-farthest O atoms,O (5), O (6) and O (7) atoms, move by a larger quantity than theclosest ones. This behaviour could be explained again by the in-crease of electrostatic attractive interaction between the men-

F. Maldonado et al. / Chemical Physics 393 (2012) 148–152 151

tioned defect-closest O atoms and the impurity itself preventingthe O (2), O (3) and O (4) atoms from larger shifts.

In the case of Cr atoms we obtain defect-outward displacementsdue to the increase in the Coulomb electrostatic repulsion. Besides,Cr atoms are closer to the O atoms than to the impurity. So, theyare attracted stronger to the O atoms, which are moving outwardswith respect to the impurity. That is why the Cr atoms are trying topreserve their initial distances with their neighbouring O atomsand displace themselves outwards from the Ti atom. One can no-tice that the Cr (8) atom has the largest shift from its initial placing,but it is also worth mentioning that this particular Cr atom is sur-rounded by the O atoms, O (5), O (6), and O (7) atoms (Table 2 andFig. 4), which also display largest shifts. Thus this particular Cratom behaves according to our explanation.

The detailed description of local microstructure due to the Tidoping also points out to the fact that atomic displacements arerather diminutive (Table 2). That is in concordance with otheravailable reports [28,34]. Our outcomes indicate a general patternof atomic charge increase (Table 2). This implies that crystallinestructure rearrangement is accompanied by the increase of the io-nic nature in the chemical bonding in the material.

We would like to state that all results described above refer tothe case A. The data of atomic relaxation and atomic charges for thecase B is practically identical and therefore have been omitted fromour discussion.

3.2. Electronic and electrical properties of Ti-doped Cr2O3

In order to study effects that the substitution of Ti atom pro-duces upon the electronic properties of a-Cr2O3 material, the totaland partial DOS have been calculated and compared with the cor-responding results obtained for pure chromium oxide crystal(Fig. 2).

Fig. 5 displays the total DOS for the case A, i.e. when Ti atomsubstitutes for a Cr atom possessing a positive magnetic moment.Comparison of Figs. 2 and 5 implies that both pictures look verysimilar. Thus the introduction of Ti atom has not produced any lo-cal energy levels within the band-gap of material. However, it ispossible to notice that the defect produces some local-characterperturbations upon the upper and lower VB as well as on the CBof the material. Additionally, and more important, it is possibleto observe that the Fermi level shifts to the bottom of the CB as aresult of Ti doping. This implies that there is free electron in theCB. In order to be completely sure that we observe delocalizedelectron state (metallic state) in the bottom of the CB, partialDOS have been computed for the Ti-doped a-Cr2O3 material. Theoutcomes are shown in Fig. 6. As one can note the contribution to-

Fig. 5. DFT + U (U = 4 eV) computed total DOS for the AFM � + � + state consideringcase A, i.e. Ti impurity replacing for a Cr atom possessing positive magneticmoment. The dotted line marks the Fermi level (EF).

wards the Fermi level is due to the Cr 3d states wholly. That en-sures free electron states in the material and also it contributesto the n-type electrical conductivity in agreement with the avail-able experimental results [28,34].

Total DOS for Ti-doped a-Cr2O3 if Ti dopant is situated in the Crsublattice of a negative magnetic moment (Fig. 7) yields a similarpicture. However, a local one-electron energy level occurs withinthe band-gap of material. The major contribution to this state isdue to the Ti 3d atomic orbitals. This localised occupied state doesnot change, however, electrical features of the system. The Fermi

Fig. 6. DFT + U (U = 4 eV) computed partial DOS for the AFM � + � + stateconsidering case A for: (a) Cr atom, (b) O atom, and (c) Ti atom. The dotted linesmark the Fermi level (EF).

Fig. 7. DFT + U (U = 4 eV) computed total DOS for the AFM � + � + state consideringcase B, i.e. Ti impurity replacing for a Cr atom possessing negative magneticmoment. The dotted line marks the Fermi level (EF).

152 F. Maldonado et al. / Chemical Physics 393 (2012) 148–152

level is situated at the bottom of the CB signalising a free electronpresence in the CB. Thus, this metallic state will obviously contrib-ute towards the n-type electrical conductivity in the Ti-doped a-Cr2O3 crystal.

3.3. Magnetic properties of Ti-doped Cr2O3

Ti incorporation leads to the occurrence of a local magnetic mo-ment in the chromium oxide lattice. In pure a�Cr2O3 crystal, eachCr atom has a magnetic moment of 2.92 lB. The pure system isessentially AFM with a total magnetic moment being equal to zero.The introduction of Ti impurity produces some magnetic perturba-tions for each case, A and B, respectively.

In the case A, the extracted Cr atom has a positive magnetic mo-ment. The impurity incorporation into the material affects the totalmagnetisation of supercell since Ti atom possesses a smaller mag-netic moment, 0.88 lB, compared to that of the host Cr atom.Please note that the value of 0.88 lB for Ti atom was obtained inour DFT + U computations since initially we assigned value 0 forthe magnetic moment of impurity atom. In fact, the supercell mag-netic moment after the doping is found to be equal to �1.99 lB.The contribution to this value comes predominantly from the Cratoms. Nevertheless, some non-negligible admixture towards thesupercell magnetic moment is due to the O atoms.

Results obtained for the case B are almost identical. The onlydifference is that the replaced Cr atom corresponds to a B-typesublattice, which originally has a negative magnetic moment.Hence, magnetisation of the Ti-doped supercell is equivalent to1.99 lB, i.e. being positive. The O atoms have the same behaviourand contributions towards the total magnetic moment of thesupercell as described before for the case A.

Calculated total energy of the supercell is found to be the samefor both cases, A and B, thus we can conclude that both magneticstates are energetically equivalent and can occur in the nature.

4. Conclusion

A quantum–mechanical study of pure and Ti-doped antiferro-magnetic a-Cr2O3 crystals has been carried out using first princi-ples DFT within the GGA + U approximation. The work allows toget a better idea of effects produced by a Ti dopant upon the struc-tural, electronic, electrical and magnetic features of the material.

Calculated magnetic states as a function of volume for pure a-Cr2O3 material demonstrate the necessity of using an intra-atomicinteraction for the strongly correlated d electrons, i.e. DFT + U ap-proach, to give a reasonable description of such a system. TheAFM � + � + configuration has been found to be the ground state

magnetic arrangement. The electronic band structure and theband-gap width being equal to 3.21 eV imply that the a-Cr2O3

crystal is an intermediate-type insulator between the charge-transfer and Mott–Hubbard insulators. These obtained featuresare in close agreement with the available experimental data.

The revealed microstructure around the Ti defect consists in de-fect-closest atom displacements away with respect to the Ti impu-rity. The character of chemical bonding tends to become more ionicaccording to the obtained Bader population analysis.

Ti impurity incorporation produces a local energy level withinthe band-gap of the material if Ti replaces for a Cr atom initiallypossessing a negative magnetic moment. This state is found forthe b spin sub-system and is composed mainly of the Ti 3d atomicorbitals. Ti substitution for a Cr atom having either positive or neg-ative magnetic moment, i.e. for both cases, yields a metallic statewith the Fermi level position at the bottom of the CB. Free electronin the CB contributes towards the n-type electrical conductivity inthe material.

Ti atom introduction leads to the occurrence of the local mag-netic moment being equal to �1.99 lB and 1.99 lB for the A-typeand B-type Ti Cr substitution, respectively. In both cases the maincontribution towards the magnetic moment is because of the de-fect-neighbouring Cr atoms. Nevertheless, admixture of the near-est O atoms is not negligible and Ti defect itself contributesaround 0.88 lB towards the total magnetic moment of the super-cell. It is also discovered within our DFT + U modelling that both,A-type and B-type, magnetic configurations are equally favourable.

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