An atomistic study into the defect chemistry of hexagonal barium titanate J.A.Dawson,a)C.L.Freeman,L.-B.Ben,J.H.Harding,and D.C.Sinclair
Department of Materials Science and Engineering,University of Shef?eld,Sir Robert Had?eld Building,
Mappin Street,Shef?eld,S13JD,United Kingdom
(Received13December2010;accepted27January2011;published online19April2011)
Using a recently established BaTiO3potential model speci?cally designed for the calculation of defect
energetics,atomistic simulations have been carried out on the intrinsic defect chemistry and Rare
Earth(RE3t)doping of hexagonal barium titanate(h-BaTiO3).Five charge compensation schemes
have been considered as well as potential cluster binding energies.The results show that ion size
arguments are obeyed.In the dilute concentration limit,large RE3tcations dope at the Ba-site via a
titanium vacancy mechanism and mid sized RE3tcations dope at the Ba and Ti sites simultaneously
via a self compensation mechanism.In contrast,small RE3tcations dope exclusively on the Ti-site
via an oxygen vacancy compensation https://www.doczj.com/doc/b76748812.html,parisons between the hexagonal and cubic phases
of BaTiO3have also been drawn.It is suggested that Ba-site doping is less favorable and that Ti-site
doping is considerably more favorable in h-BaTiO3and that different defect con?gurations have a
signi?cant effect on the binding energies between such defects,leading to some mechanisms
becoming more or less energetically favorable as a result.V C2011American Institute of Physics.
[doi:10.1063/1.3560552]
I.INTRODUCTION
BaTiO3is renowned for its useful electrical properties which include ferroelectricity,piezoelectricity and a high rela-tive permittivity,1,2giving it a role in a wide range of applica-tions including multilayer ceramic capacitors(MLCC)and positive temperature coef?cient(PTC)ceramic thermistors.It is also an excellent example of a ferroelectric perovskite based on a d o cation occupying the B-site and is therefore a useful ‘model’system for fundamental studies related to structure-composition-property relationships.
There are four polymorphs and three polymorphic phase transition temperatures associated with the cubic close packed perovskite BaTiO3(c-BaTiO3).The ideal cubic polymorph exists from$1425down to$130 C where it transforms into a tetragonal polymorph.Further distortions occur at lower temperatures associated with transformations into orthorhom-bic and rhombohedral polymorphs at$0andà90 C,respec-tively.3In addition to these polymorphs,a high temperature hexagonal polymorph exists above$1425 C.The structure is based on a combination of cubic(c)and hexagonal(h),close packed BaO3layers based on a(cch)2sequence,and is classi-?ed as a6H-type hexagonal perovskite,Fig.1.This results in a combination of face-sharing Ti2O9dimers connected by a corner sharing TiO6unit.
By comparison with the well studied polymorphs of c-BaTiO3,the hexagonal polymorph(h-BaTiO3)has received much less attention,with a limited number of experimental studies into the defect chemistry and electrical properties of this polymorph.Undoped h-BaTiO3can be stabilized at room temperature by processing samples under inert or reducing con-ditions at high temperature.4,5This results in oxygen de?ciency and induces semiconductivity due to partial reduction of Ti4tto Ti3tions.4,5Partial re-oxidation can result in the develop-ment of‘colossal’or‘giant’permittivity effects associated with extrinsic effects such as interfacial layers and nonohmic electrode contacts.4Neutron Diffraction studies of h-BaTiO3àd powders reveal the oxygen de?ciency(d)to occur solely at the O1sites in the h-BaO3layers within the Ti2O9units,5Fig.1. Doping studies have concentrated mainly on3d transition metal ions such as Mn,Fe,Co,Ni and Cu;6–9however,there are also studies on Ga3t.10In general,these materials can be prepared in air and are insulators/modest semiconductors; oxygen de?ciency at the O1sites is again a key feature in their formation.8,10In some cases,e.g.,Ga-doped samples,they exhibit high room temperature permittivity(e r>60)and mod-est microwave dielectric resonance properties.11
In general,Rare Earth(RE3t)doping of c-BaTiO3has received considerable attention due to the possibility of a variety of substitution mechanisms dependent on the size of the RE ion and the Ba/Ti ratio.12,13Examples include,A-site doping com-pensated by electrons([Ba1-x RE x]TiO3);A-site doping compen-sated by either A-or B-site vacancies(e.g.,[Ba1-3x RE2x] TiO3and[Ba1-x RE x]Ti1-x/4O3,respectively);B-site doping compensated by O-site vacancies(Ba[Ti1-x RE x]O3-x/2);and A-and B-site doping or‘self compensation’([Ba1-x RE x][Ti1-x RE x]O3).These mechanisms can have a signi?cant in?uence on the polymorphic phase transition temperatures and dielec-tric properties of BaTiO3.In particular,A-site doping by large RE cations such as La has been a controversial topic for many years14and can result in signi?cant levels of electrical con-ductivity for samples processed in air or reducing atmos-pheres but can also result in insulating materials for samples processed in?owing O2.Recently,A-site La3tdoping of h-BaTiO3has also been reported to occur by an electronic com-pensation mechanism15and this permitted a giant permittivity effect to be obtained without having to induce(or control) oxygen de?ciency.15Ho3tdoping at the Ti-site via an oxygen vacancy charge compensation mechanism of h-BaTiO3 has also been explored;16Ho3tincorporation was found to
a)Electronic mail:mtp09jd@shef?https://www.doczj.com/doc/b76748812.html,.
JOURNAL OF APPLIED PHYSICS109,084102(2011)
destabilize the hexagonal phase with the cubic/hexagonal phase transition temperature increasing with increasing Ho concentration.Other types of dopants such as Nb have also been incorporated into h-BaTiO3and the effects on properties monitored.17
We have found very little simulation work on h-BaTiO3 in the literature.The most detailed study was completed by Colson et al.18where Density Functional Theory(DFT)was used to study both c-and h-BaTiO3.Good agreement with ex-perimental structures was obtained,although an increased error in lattice parameters was observed for h-BaTiO3,suggesting an increased dif?culty in modeling this polymorph.Ru doping at various concentrations and substitutional sites was also con-sidered and it was reported that O1(face sharing)vacancies were energetically more favored than O2(corner sharing) vacancies.To the best of our knowledge there exists no classi-cal(potential-based)study of h-BaTiO3.There have been two studies on c-BaTiO3,19,20based on a potential set developed by Lewis and Catlow;19however,their model was based exclusively on ionic bonding and there were recognized de?-ciencies,for example,negligible defect association energy between highly charged defects such as titanium vacancies with RE-ions occupying the A-site.20We have recently devel-oped a new force?eld for BaTiO3that includes covalency associated with the Ti-O bonding and we report the results for c-BaTiO3elsewhere.21This paper reports on the use of this potential set to model undoped and RE3tdoped h-BaTiO3and also compares the results obtained for c-and h-BaTiO3.
II.SIMULATION METHODS AND POTENTIALS All results presented in this work were obtained using the General Lattice Utility Program(GULP)Version3.1.22 The BaTiO3potential parameters used in this work are given in the supplementary data(Table I).23A detailed description of their development and?tting is available in Ref.21A cutoff distance of12A?was used for both the Buckingham potentials and the Lennard-Jones potential.covalent character of the Ti-O bond.All RE short-range potentials and the Ti3tàO2àpotential were taken from pre-vious work on BaTiO3by Lewis and Catlow.19
In these calculations,the energy of the system is mod-eled from contributions of the long-range and short-range forces with respect to the atomic positions in the lattice.For ionic materials such as BaTiO3,the long-range interactions are Coulombic:
V ij?
X n
i?j
q i q j
4p e0r ij
(1)
where q i,q j are the full ionic charges corresponding to the formal valence state of the ions.
Short-range interactions including electron repulsion and Van der Waals attractions are represented in this work by the Buckingham(2)and Lennard Jones7-6potentials(3):
V ij?
X n
i?j
A expàr ij=q
àá
à
C
r6ij
(2)
V ij?
X n
i?j
4e ij
r ij
r
7
à
r ij
r
6!
;(3)
where the symbols have their usual meanings.
However,these short-range potentials do not take into account the electronic polarization of the atoms.In this work, this is overcome by using the shell model developed by Dick and Overhauser.24The model works by splitting atoms into negatively charged massless shells and positively charged cores with the relevant atomic mass.Cores and shells belong-ing to the same ion interact via a harmonic spring and the spring constant k replaces the conventional electrostatic interaction.The shell model parameters used in this work are also supplied in the supplementary data(Table II).23 The defect calculations were performed using the Mott-Littleton method.25This approach divides the lattice around the defect into two spherical regions;an inner region and an outer region.Interactions are calculated explicitly in the inner region and the ions are relaxed to positions of zero force,whereas in the outer region the polarization energy and ionic positions are approximated using a dielectric con-tinuum method.The interactions between the ions of the inner region with the ions of the outer region are calculated explicitly to ensure that the inner region is properly embed-TABLE I.Vacancy energies for(a)h-BaTiO3and(b)c-BaTiO3.
(a)Ion Site
Vacancy Energy
(eV)(b)Ion Site
Vacancy Energy
(eV)a
Ba120.54Ba19.50
Ba220.25
Ti1105.52Ti98.96
Ti2101.39
O123.99O24.66
O226.55
a Reference21
. FIG.1.(Color online)h-BaTiO3unit cell.
The methods described here have been used in a range of studies on BaTiO3(Refs.19and20)and for other perov-skite compounds.26–28
III.RESULTS AND DISCUSSION
A.Structural analysis
The structure of h-BaTiO3,space group P63/mmc(num-ber194),29is given in Fig.1.The structure contains two differ-ent sites for each of the Ba2t,Ti4tand O2àions.The oxygen ions are associated with the Ti2O9units and are subdivided into face-sharing ions(O1)and corner-sharing ions(O2).
The cohesive energies as well as the lattice parameters for the structure were calculated to ensure the force?eld accurately replicated the observed structure.The results are provided in detail in our previous work with the values obtained being compared to those calculated from the Lewis and Catlow original force?eld.21Experimental cohesive energies were calculated from Born-Haber cycles.21 The results are in good agreement with the experimental values and the cohesive energies are far more accurate with the new force?eld than those for the Lewis and Catlow force ?eld.For example,our simulations for h-BaTiO3give a co-hesive energy ofà159.94eV in comparison to the Lewis and Catlow value ofà147.69eV and an experimental value ofà159.73eV.This is important as these values are essen-tial in obtaining accurate dopant energetics.
B.Intrinsic defects
Vacancy energies were calculated for each type of ion at each site in h-BaTiO3and the results are presented in Table I(a).Values obtained for c-BaTiO3from Ref.21are included for comparison in Table I(b).
In comparison with the c-BaTiO3,21the energies are generally higher as a result of the more complex coordina-tion environments.However,one defect where the energy is lower is the O1vacancy.This result supports experimental ?ndings from Neutron Diffraction studies for undoped,5Ga-doped10and3d transition-metal doped8,9h-BaTiO3that show face-sharing oxygen vacancies(O1)are more likely than corner-sharing oxygen vacancies(O2).Similar to c-BaTiO3,Ti vacancies are very unfavourable(at both sites). Ti2vacancies are considerably more favorable than Ti1Ti2sites within the Ti2O9dimers.This is a result of destabi-lisation from the shorter interatomic cation distance between the Ti2sites($2.9387A?)within the face sharing dimers compared to the closest Ti1/Ti2distance($4.0730A?)asso-ciated with corner shared octahedra and hence increased Coulombic repulsion.This result is consistent with the obser-vation that cubic close packed perovskites based exclusively on corner shared BO6octahedra are much more abundant than hexagonal close packed perovskites where face sharing octahedra are present.
C.Dopant substitution and solution energy calculations
Substitution energies(e.g.E M3t
sub;Ba
)represent the energy required to bring a defect ion into the lattice from in?nity and remove the ion it is replacing to in?nity.These values are strongly dependent on the charge and size of the incom-ing ion and the ion that is being substituted.The calculated substitution energies are shown in Table II.
The substitution energies obey ion-size arguments,i.e. as the ion size decreases from La to Lu and the charge den-sity increases,the substitution becomes energetically more favorable.For Ba-substitution,the Ba1site is preferred whereas for Ti-substitution the Ti2site is more favorable (with the exception of Ti3tsubstitution),although the energy difference is small compared to the difference between Ti vacancies.However,these energies take no account of charge compensation and therefore solution ener-gies were calculated.
Five charge compensation schemes are generally con-sidered for RE3tdoping of BaTiO3.Four of these schemes are detailed in Buscaglia et al.;20the schemes are reproduced here with the addition of a RE3tsubstitution at the Ba2tsite with a Ba2tvacancy scheme.
1.Substitution of M31at Ba21with conduction electron compensation
M2O3t2Ba Ba!2M_Bat
1
2
O2t2e0t2BaO(4)
E s?
1
2
2E M3t
sub;Ba
t2E Ti3t
sub
t2E e0à
1
2
D O
2
tE O2à
A
BaO M2O3
i
TABLE II.Substitution energies for h-BaTiO3.
Defect Ion(3t)
Ba-site substitution,E M3t
sub;Ba
(eV)Ti-site substitution,E M3t
sub;Ti
(eV)
Ba1site Ba2site Ti1site Ti2site
Laà21.66à21.1658.3458.19 Ndà22.74à22.1856.0055.85 Euà23.49à22.8954.3454.19 Gdà23.65à23.0553.9153.76 Hoà24.43à23.8252.1952.01 Ybà24.90à24.2550.9750.76 Luà25.13à24.4550.5150.28 Ti——49.4249.52
2.Substitution of M31at Ba21with Ti41vacancy compensation
2M2O3t4Ba BatTi Ti!4M_BatV0000Tit3BaOtBaTiO3
(6)
E s?1
4
4E M3t
sub;Ba
tE Ti4t
vac
tE BaTiO3
L
t3E BaO
L
à2E M2O3
L
h i
:(7)
3.Substitution of M31at Ti41with O22vacancy compensation
M2O3t2Ti TitO o!2M0TitV€
O
t2TiO2(8)
E s?1
2
2E M3t
sub;Ti
tE O2à
vac
t2E TiO2
L
àE M2O3
L
h i
:(9)
4.Substitution of M31at Ba21and M31at Ti41leading to self compensation
M2O3tBa BatTi Ti!M_BatM0TitBaTiO3(10)
E s?1
2
E M3t
sub;Ba
tE M3t
sub;Ti
tE BaTiO3
L
àE M2O3
L
h i
:(11)
5.Substitution of M31at Ba21with Ba21vacancy compensation
M2O3t3Ba Ba!2M_BatV00Bat3BaO(12)
E s?
1
2
2E M3t
sub;Ba
tE Ba2t
vac
t3E BaO
L
àE M2O3
L
h i
:(13)
The energy of solution(E s)for each dopant species and for each scheme can then be calculated using the relevant cohe-sive energies,substitution energies and vacancy energies (see above)(note that the fraction present in each E s equation ensures the energy calculated is per RE3tion substituting). Other terms used in these calculations are given in the previ-ous paper on the cubic polymorph.21
The solution energies for the lowest energy con?gura-tions(i.e.the structure with the lowest substitution and va-cancy energies)are displayed in Table III.All other solution energy data for the numerous con?gurations can be found in the online supplementary data.23
RE substitution via self compensation is preferred at the Ba1and Ti2sites and vacancies are preferred at the O1,Ba2 and Ti2sites for other various con?gurations.The self com-pensation mechanism is signi?cantly lower in energy than the other competing mechanisms;however,these solution energies are for isolated defects and so only display trends due to ion size and do not take into account the binding energy between the charged defects.
D.Binding Energies
It has been illustrated for c-BaTiO3that binding energies between defect clusters have a dramatic effect on the overall
TABLE III.Solution energies of the lowest energy con?gurations for RE3tdoping of h-BaTiO3.
Dopant Ion Scheme Ionic Radii(A?)a
Solution Energy,E S(eV)
12345
RE Ba1te Ti1RE Ba1tV Ti2RE Ti2tV O1RE Ba1tRE Ti2RE Ba1tV Ba2
La 1.0329.96 2.37 5.62 1.41 2.67 Nd0.98310.83 2.85 4.84 1.26 3.15 Eu0.94711.96 3.69 4.78 1.65 4.00 Gd0.93811.58 3.83 4.65 1.66 4.14 Ho0.90112.27 4.41 4.26 1.76 4.72 Yb0.86812.86 4.94 4.01 1.90 5.25 Lu0.86112.59 5.13 3.95 1.96 5.44
a Reference30.
TABLE IV.Binding energies of the lowest energy full cluster con?gurations for RE3tdoping of h-BaTiO3.a
Dopant Ion Scheme Ionic Radii(A?)b
Binding Energy(eV)
12345
RE Ba1te Ti1RE Ba1tV Ti2RE Ti2tV O1RE Ba1tRE Ti2RE Ba1tV Ba2
La 1.032à0.56à5.29à6.69à0.68à1.54 Nd0.983à0.56à5.58à6.57à0.71à1.59 Eu0.947à0.55à5.90à6.55à0.74à1.70 Gd0.938à0.55à6.04à6.56à0.78à1.76 Ho0.901à0.55à6.59à6.57à0.89à2.03 Yb0.868à0.55à7.24à6.66à1.02à2.39 Lu0.861à0.54à7.60à6.64à1.08à2.54
a Negative energies imply binding behavior.
energy for the doping process.Within the hexagonal phase there are numerous possible con?gurations of defects and hence numerous binding energies with varying magnitudes.All potential con?gurations were calculated and the most favorable binding energies are given in Table IV .Again binding energy data for other con?gurations are provided in the supplementary data.23
There is clearly a signi?cant disparity between the size of the binding energies for different defect clusters;this is primarily due to the difference in charge on the defects.
Scheme 1(RE Ba telectron compensation)shows a rela-tively weak series of binding energies that remain constant with an increase in dopant ion charge density.These small negative energies are the result of the interaction between a (t1)charged defect site (RE Ba )and a (à1)charged defect site (Ti 0Ti )(simulated by calculating the energy for the substi-tution of a Ti 3tion on a Ti 4tsite via a Ti 3tàO 2àpoten-tial 19).A possible explanation for why no clear trend is observed is the large distance between the Ba1and Ti1sites compared to other con?gurations.
Scheme 2(RE Ba tV Ti )has a much larger set of binding energies that also show a large increase from La to Lu.The larger energies are a result of the strong interaction between the Ti vacancy (charge à4)and RE Ba (charge t1)defects.Also,as the size of the dopant ion decreases,the RE-O inter-action increases and the negatively charged V Ti further pushes the O anions toward the RE cation.
Scheme 3(RE Ti tV O )again has large binding energies,however,these values are less affected by the increase in charge density from La to Lu.The interaction between the oppositely charged O vacancy and RE Ti sites is the cause of
these large values.For this particular con?guration (RE Ti2tV O1)a unique pattern is observed.With other con-?gurations a similar pattern to the cubic phase was observed,i.e.,with decreasing dopant ion size the binding energy became less negative.This con?guration,however,shows a small decrease in negativity with decreasing ion size only up to Eu,which is then followed by an increase in binding energy up to Yb;this is proposed to be a strain effect.Our simulations show that larger RE ions such as La cause other O1ions to be marginally displaced as the La-O bonds are longer than,for example,shorter and stronger Lu-O bonds.In this particular con?guration the separation between the O1ions and the RE Ti2sites is signi?cantly smaller compared to the rest of the lattice which restricts the displacement of the O1ions as illustrated by Fig.2.This suggests possible competition between the energetically favored larger RE 3tcations and the strain that these large cations cause.
Scheme 4shows similar results to that of scheme 1as again the binding energy is between a singly charged posi-tive defect and a singly charged negative defect.The same trend is also observed with decreasing dopant ion size caus-ing a more negative binding energy.This is as a result of stronger RE-O interactions as the charge density of the RE 3tion increases.
Scheme 5illustrates the same pattern as scheme 2;how-ever,the binding energies are not as favorable as the vacancy now has a charge of à2as opposed to the à4charge of V Ti in scheme 2.
When comparing the binding energies of RE-doped c -BaTiO 3there are a few notable differences,although it should be noted that direct comparison is dif?cult due to the very dif-ferent crystal structures.Generally,the magnitude of the bind-ing energies is greater in h -BaTiO 3;this is especially the case with schemes 2and 5.For these schemes in the cubic phase,the four RE Ba ions are located in the cubic plane,21which causes an increase in repulsion in the system,however in the hexagonal phase the separation between the Ba-sites is gener-ally much larger,meaning the repulsion is minimized and results in a more negative binding energy.This is also observed for the RE Ba1tV Ti2/RE Ba1tV Ba2con?gurations.The ion size trends are the same with the exception of scheme 3,which has been previously
discussed.
FIG.2.(Color online)Representation of the strongest binding energy defect cluster for scheme 3.
TABLE V.Final solution energies of the lowest energy con?gurations for RE 3tdoping of h àBaTiO 3.
Dopant Ion Scheme Ionic Radii (A
?)a Final Solution Energy (eV)
1
2
3
4
5
RE Ba1te Ti1
RE Ba1tV Ti2
RE Ti2tV O1
RE Ba1tRE Ti2
RE Ba1tV Ba2
La 1.032
9.40 1.04 2.28 1.07 1.90Nd 0.98310.27 1.45 1.560.91 2.36Eu 0.94711.41 2.21 1.50 1.28 3.15Gd 0.93811.03 2.32 1.37 1.27 3.26Ho 0.90111.72 2.770.98 1.31 3.71Yb 0.86812.31 3.130.68 1.39 4.06Lu
0.861
12.05
3.23
0.63
1.42
4.17
E.Final solution energies
To gain an accurate representation of the energies and trends in these dopant mechanisms,solution and binding energies must be https://www.doczj.com/doc/b76748812.html,ing the appropriate fraction of binding energy (determined from the charge compensa-tion schemes,e.g.,[1/4]for scheme 2),the ?nal solution energies were calculated and results for the lowest energy con?guration are presented in Table V and graphically in Fig.3.
Similar to the cubic phase,the electron compensation mechanism is signi?cantly unfavourable,a considerable 7–8eV above the nearest other compensation scheme.Scheme 5is also high in energy;this correlates well with experimental ?ndings as this scheme is considered less plausible than the other three mechanisms in h -BaTiO 3,31at least for samples prepared in air.
The large range present for the cohesive energies of RE 2O 3results in all schemes excluding scheme 3becoming more unfavourable with smaller defect ions,as a result of the RE defect ion being moved from its stable oxide to the per-ovskite structure,a phenomenon explained previously.21
It is also notable that Nd seems to be energetically more favorable than expected based only on its ion size,
although this effect is not as pronounced as in the cubic phase.This feature is visible in schemes 2,3,4and 5but particularly in 3and 4.For scheme 3there is a rather small increase in energy from Eu to Nd,however
the size difference is rather substantial (Eu—0.947A
?,Nd—0.983A
?)and hence a larger energy increase would be expected.In scheme 4,Nd is the lowest energy dop-ant regardless of the trend of decreasing energy with increasing ion size.This is a feature of Nd 2O 3having a more positive cohesive energy than would be expected for its ion size,meaning there is less of an energy pen-alty when removing Nd from its oxide and into the h -BaTiO 3structure.It is interesting to note that the larg-est solid solution for the self-compensation mechanism found from experimental studies is for Nd.31
Another interesting feature of the results is that for scheme 2,there are two competing defect con?gurations,where one is favored over the other based on the defect ion,as illustrated by Table VI and Figs.4and 5.
Figure 4clearly shows the importance of the binding energy in such calculations.For the RE Ba1tV Ti2con?gura-tion (also featured in Fig.3),the solution energies are signi?-cantly lower for La and Nd doping when compared to the RE Ba2tV Ti2con?guration.However,the binding energy for the RE Ba2tV Ti2con?guration becomes far more
signi?cant
FIG.3.(Color online)Final solution energies (including binding energies)versus ionic radius for RE 3t-doped h -BaTiO 3.Code;Yellow (^)Scheme 1RE Ba te à;Blue (*)Scheme 2RE Ba tV Ti ;Orange (?)Scheme 3RE Ti tV O ;Red (h )Scheme 4RE Ba tRE Ti ;Black (D )Scheme 5RE Ba tV Ba
.
FIG.4.(Color online)Final solution energies of two defect con?gurations for Scheme 2versus ionic radius for RE 3t-doped h -BaTiO 3.Code:*RE Ba1tV Ti2;^RE Ba2tV Ti2.
TABLE https://www.doczj.com/doc/b76748812.html,parison of two Scheme 2defect con?guration energetics.
Dopant Ion Solution Energy (eV)
Binding Energy (eV)Final Solution Energy (eV)RE Ba1tV Ti2
RE Ba2tV Ti2
RE Ba1tV Ti2
RE Ba2tV Ti2
RE Ba1tV Ti2
RE Ba2tV Ti2
La 3.19 3.75à6.94à8.30 1.46 1.67Nd 3.83 4.44à7.36à9.53 1.99 2.06Eu 4.80 5.43à7.83à10.63 2.84 2.77Gd 4.96 5.60à7.99à10.96 2.97 2.86Ho 5.67 6.35à8.91à12.52 3.44 3.22Yb 6.28 6.98à9.14à13.92 4.00 3.50
as the ion size decreases,meaning all ions smaller than Nd prefer this con?guration.The reason why this particular con-?guration has such a strong binding energy is due to the close proximity of the RE Ba2sites and the Ti2vacancy,as shown in Fig.5(b).This proximity results in a strong interac-tion between theà4charged vacancy and the surrounding t1charged RE substitution defects.
A similar‘crossover’effect is observed for the RE Ba1te Ti1and RE Ba1te Ti2con?gurations for the electron com-pensation scheme.The RE Ba1te Ti1con?guration has the lowest solution energies,however the RE Ba1te Ti2con?gu-ration has stronger binding energies resulting in larger dopant ions preferring the former con?guration whereas mid-sized and smaller ions prefer the latter con?guration (data available in the supplementary data23).
https://www.doczj.com/doc/b76748812.html,parison with c-BaTiO3
Some comparisons have already been drawn between the two polymorphs such as the unusual behavior of the Nd ion,but other common features and contrasts are discussed below.
The same ion size trends are observed and also the same compensation schemes are preferred for certain ions,for example large ions like La prefer A-site occupation.Mid-sized ions such as Nd also prefer self compensation and small ions like Yb prefer substitution at the Ti-site with an oxygen vacancy mechanism,all in agreement with experi-mental?ndings.31,32
It is noteworthy that the position of the schemes rela-tive to each other in terms of energy is different in the hex-agonal phase.For instance the results suggest that Ba-site doping is generally less favored in the hexagonal phase meaning schemes1,2,5and to some extent4are all higher in energy compared to the cubic phase.This is a result of the signi?cantly higher energy required for Ba and Ti vacancies in the hexagonal phase and also that the Ba-site substitution energies are not as favorable as in the cubic phase.Alternatively,the lowest energy con?guration for scheme3is lower in energy for mid and small sized ions due to the strong binding energy and low energy cost for V O1relative to V O in the cubic phase.Such effects mean certain ions in the hexagonal phase.Note that these com-parisons are only for the lowest energy(and hence theoreti-cally most likely)con?gurations of h-BaTiO3.Other less favorable con?gurations produce signi?cantly different results albeit with the same general trends.
IV.CONCLUSIONS
Through the use of a recently designed BaTiO3potential set,classical simulations have been applied to the hexagonal phase of BaTiO3.Both the defect structure and energetics of this polymorph have been studied using potential based meth-ods.The structure shows excellent agreement with experiment and the calculation of intrinsic defect energetics also supports experimental?ndings,e.g.,the preference of O1face-sharing vacancies over O2corner-sharing vacancies.RE3tpoint defects in a large range of con?gurations have also been stud-ied and detailed comparisons with the cubic phase have been drawn,most notably the energetic penalty required for Ba-site doping in h-BaTiO3and the dramatic consequence of different defect locations on these energies.The comparison between the?nal solution energies(combined solution and binding energies)and experiments are also good.Experimentally, small RE ions like Ho and Yb dope at the Ti-site,16,31which is corroborated by our results.It is expected that mid-sized ions such as Nd dope at both Ba and Ti sites via the self compensa-tion mechanism,also suggested by experimental results.31,32 ACKNOWLEDGMENTS
We thank EPSRC for funding(Grant number-EP/ G005001/1).
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