Combined Ab Initio and Interatomic Potentials Based Assessment of

Article pubs.acs.org/JPCC

Combined Ab Initio and Interatomic Potentials Based Assessment of the Defect Structure of Mn-Doped SrTiO3 J. A. Dawson,*,† H. Chen,‡ and I. Tanaka† †

Department of Materials Science and Engineering, Kyoto University, Sakyo, Kyoto 606-8501, Japan Environmental Remediation Materials Unit, National Institute for Materials Sciences, Ibaraki 305-0044, Japan

‡

ABSTRACT: The energetics of doping SrTiO3 with Mn is investigated using density functional theory calculations within the GGA + U scheme and lattice statics simulations. Defect formation energies are derived for different Mn charge states and for occupation at both the Sr- and Ti-sites. For the GGA + U calculations, due consideration is given to finite-size effects and the treatment of the chemical potential of atomic oxygen to avoid errors associated with direct DFT calculations of the oxygen dimer. The lattice statics calculations have been completed using a recently developed potential model designed for the investigation of rare-earth and transition metal doping of SrTiO3. Intrinsic defects have also been considered in all possible charge states. The GGA + U calculations predict that for the three different thermodynamic environments examined both Mn2+ and Mn4+ ions occupy the Ti-site via either oxygen vacancy compensation (Mn2+) or no charge compensation (Mn4+). This is contradictory to the lattice statics calculations which predict Mn2+ doping at the Sr-site and Mn4+ at the Ti-site. Experiment suggests that Mn2+ doping at both the Sr- and Ti-sites is possible. (Sr1−xMnx)TiO3 are a result of the off-centering of Mn2+ ions caused by the large size difference between the dopant and host ions.2,11,13 This off-centering has been confirmed by both experiment13 and DFT calculations.2,16 Such displacements create dipoles, producing local polar clusters in the material.13 The doping of Mn2+ at the Sr-site has been confirmed in the recent years.11,13,16 Assessments and results in the literature for the magnetic properties of Mn-doped SrTiO3 vary considerably. It has been proposed that the lack of clarity on this topic is mostly a result of the poor understanding of the substitutional chemistry in the material.11 Valant et al.11 provide a review of recent studies on the magnetic properties of this complex system. Some findings suggest no magnetic ordering11,12 as a result of Mn-doping and that many of the magnetic anomalies associated with the system are caused by extrinsic sources. Alternatively, magnetic ordering has been suggested in samples with 3 mol % Mn concentration,12,17 whereas only paramagnetic behavior was observed for samples with 5 mol % Mn concentration. The potential existence of a Mn-doped SrTiO3 system with both a glassy magnetic state and glassy dielectric behavior (or a socalled “multiglass”15,18) is an intriguing prospect. In addition to the DFT studies on Mn off-centering2,16 in SrTiO3, many other ab initio studies have been completed on the defect chemistry19−22 and electrical properties23−25 of this technologically essential material. One previous DFT study of Mn-doping defect energies is available in the literature;

I. INTRODUCTION As a result of its ABO3 perovskite structure, SrTiO3 has a large number of technologically interesting physical properties resulting in it being a very well-studied and widely used electroceramic material. SrTiO3 is a diamagnetic material with a large band gap1,2 and has a variety of electrical properties including strain-induced ferroelectricity3 and a high relative permittivity.4,5 Therefore, it has many important applications including random access memories (RAMs),1,4 microwave devices,3 and thermoelectric devices.6 Although undoped SrTiO3 naturally has a variety of useful properties and applications, it is only when it is doped that the true potential of the material can be unlocked. It is therefore essential to have a detailed understanding of the defect chemistry of this material. Doping of SrTiO3 with rare-earth and transition metal elements is common and is often used to tailor electrical properties in particular. For example, La-doping has a dramatic effect on the permittivity of SrTiO3 and induces superconductivity.7,8 Transition metal doping has been used to produce a dilute magnetic semiconductor1,9,10 and also a dielectric material with long-range magnetic ordering.11,12 In recent years, Mn-doping of SrTiO3 has received considerable interest as a result of its unusual and dramatic effects on both electrical and magnetic properties. It has been reported that the electrical properties of Mn-doped SrTiO3 vary significantly depending upon whether the Mn substitutes at the A- or Bsite.2,13−15 For example, a dielectric relaxation2,13 and a coexistence of polar and spin glass behaviors12,13,16 are observed for (Sr1−xMnx)TiO3 but not for Sr(Ti1−xMnx)O3. It is thought that many of these interesting properties in © 2014 American Chemical Society

Received: January 20, 2014 Revised: June 5, 2014 Published: June 11, 2014 14485

dx.doi.org/10.1021/jp5006428 | J. Phys. Chem. C 2014, 118, 14485−14494

The Journal of Physical Chemistry C

Article

perturbation in their local potential.35 This value has also been successfully applied to the study of Mn-doping in BaTiO3.36 Using the Zhang−Northrup formalism,37 the formation energy (ΔEf) of a defect in charge state q can be calculated with respect to chemical potentials and the Fermi level

however, it suffers from various shortcomings in the methodology. The calculations by Yang et al.19 showed that for all chemical environments tested Mn substitution at the Sr-site is preferred but that only Mn substituted at the Ti-site can cause the formation of impurity states in the band gap of SrTiO3. These calculations were completed using only a 40 atom supercell, and no consideration was given to finite-size effects. Furthermore, only single defect species were considered as binding pairs were ignored and a U parameter was not considered for the Mn ions. Another DFT study of Mn-doped SrTiO320 suggests that Mn2+ at the Ti-site is stabilized by a tilting deformation; unfortunately, Sr-site doping was not considered. To the best of our knowledge there exists only one previous lattice statics study on Mn-doping of SrTiO3. Akhtar et al.26 simulated the doping of Mn2+, Mn3+, and Mn4+ at both cation sites in SrTiO3. The preference for divalent Mn was for substitution at the Sr-site whereas for tetravalent Mn the preference was for Ti-site substitution in complete agreement with experiment. Trivalent Mn which is usually not considered experimentally and is known to be unstable in BaTiO327 is shown to dope via a so-called “self-compensation” mechanism where a dopant ion substitutes at both cation sites ensuring charge neutrality. More recent lattice statics studies report the use of our new titanate potential set28 in modeling rare-earth doping of SrTiO329 and Mn-doping of BaTiO3.30 In the present work, we carry out GGA + U and lattice statics calculations of Mn2+ and Mn4+ doping at both the Sr and Ti sites with the appropriate charge compensation considered when necessary. This work reports the first in detail ab initio study of Mn defect formation energies in SrTiO3 with consideration given to finite-size effects. Comparison is made with the findings from the lattice statics calculations to provide a complete computational assessment of Mn-doping site preference in SrTiO3. The energies of vacancy formation are also calculated to assess whether the charge-compensating defects are ionic or electronic. Comparison to previous computational and experimental findings is made wherever possible.

ΔEf = (E Tdef, q − E Tperf ) + q(E VBM + E F) −

∑ Δniμi i

(1)

Edef,q T

where is the total energy of the defective system and Eperf T is the total energy of the perfect reference cell. The dependence upon the Fermi level, EF, is given in the second term; here EVBM is the position of the valence band maximum (VBM). In this equation, EF takes a value between 0 and Eg, where Eg is the band gap of the material. The final term defines the contribution of the chemical potentials of each atomic species; Δni is the number of atoms of element i added to or removed; μi represents the chemical potential of element i and is the sum of the chemical potentials of the reference state and the chosen chemical environment (as calculated by eqs 4−6 below). The chemical potential of the reference state for an element is equivalent to its calculated total free energy per atom (see Section III A). A weakness of these types of DFT calculations is the spurious defect−defect interactions that occur between periodic images, with the magnitude being very much dependent on the charge of the defect and the size and shape of the supercell. One of the main sources of this error is elastic interactions. These interactions scale inversely to supercell volume, L−3, and can be corrected for by a finite-size scaling extrapolation process. This is done by calculating the defect energies for a variety of cells with different sizes so that the defect energy can be extrapolated to infinite dilution. This approach is adopted in this work. For charged defects, electrostatic interactions can be corrected by a multipole expansion as presented by Makov and Payne.38 The first term in this expression concerns monopole− monopole interactions and scales as L−1. This correction can be determined with prior knowledge of the static dielectric constant of the crystal, ε, and the Madelung constant of the supercell, α

II. METHOD A. Density Functional Theory Calculations. The density functional theory calculations in this work were performed using the Vienna ab initio simulation package (VASP)31 with the generalized gradient approximation (GGA) according to Perdew, Burke, and Ernzerhof32 and the projector-augmented wave method.33,34 For greater accuracy, the 3s and 3p electrons of the Ti atoms were included in the valence electrons for all calculations. A plane-wave cutoff energy of 500 eV was applied for all defect calculations. Unless explicitly stated otherwise, calculations are not spin-polarized. Defect formation energies were calculated using cubic SrTiO3 supercells of 40 atoms (8 unit cells), 60 atoms (12 unit cells), 90 atoms (18 unit cells), 135 atoms (27 unit cells), and 180 atoms (36 unit cells). A Γpoint centered 2 × 2 × 2 k-point mesh was used for Brillouin zone integration. The structural relaxations were completed when the residual force of each atom was less than 0.01 eV/Å. The Hubbard U parameter was employed to adjust the electron configuration contribution (on-site Coulombic effects). A value of 5.04 eV was chosen for the Mn 3d electrons. This value was calculated for Mn using a method based on calculating the response in the occupation of transition metal states to a

ΔEmp = −

q 2α 2Lε

(2)

Given the very high static dielectric constant of SrTiO3, this error is generally small. The next term in the multipole expansion relates to monopole−dipole interactions. As for the elastic interactions, this scales with L−3 and therefore is accounted for by the finite-size scaling procedure. Recent work on oxygen vacancies in cubic SrTiO3 by Choi et al.21 also shows the importance of monopole−monopole interactions which scale with L−1. The results show that for very large cell sizes (625- and 1080-atom cells) the formation energies increase, contrary to what was observed for smaller cell sizes. Unfortunately, the testing of this complex behavior and the computational expense of the cell sizes required are beyond the scope of this study. The finite-size effects for Mn-doping defect energies are illustrated in Figure 1. An additional weakness of this type of electronic structure calculation is the underestimation of the band gap.39 To account for this problem, we use an approach previously applied to BaTiO3.40,41 By shifting the valence band (ΔEVB) and the conduction band (ΔECB) with respect to each other the correction energy can be obtained from 14486



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stability range of SrTiO3. In this study, we consider three different combinations of chemical potentials. The first is the metal-rich limit, i.e., ΔHf[SrTiO3] = 3ΔμO. The other two combinations are for the O-rich limit, one with excess TiO2 where ΔHf[TiO2] = ΔμTi and ΔμSr = ΔHf[SrTiO3] − ΔHf[TiO2] and one with excess SrO where ΔHf[SrO] = ΔμSr and ΔμTi = ΔHf[SrTiO3] − ΔHf[SrO]. The actual chemical potentials used are listed in Table 1. Table 1. Chemical Potential Values (eV) for the Three Chemical Environments Used in the Calculation of the Defect Formation Energies at T = 1400 K and pO2 = 1 atm metal-rich O-rich (excess TiO2) O-rich (excess SrO)

(3)

where ne and nh represent the number of electrons occupying conduction band states and the number of holes occupying valence band states, respectively. An experimental band gap of 3.2 eV42 is used to correct for the underestimation of the band gap while assuming the offset of the calculated band structure is restricted to the conduction band, i.e., ΔEVB = 0 and ΔECB = 3.2 eV − Ecalc G . This correction only affects oxygen vacancies and in-charge states below their full ionic values, i.e., V1+,0 O . These defects already have significantly higher formation energies than the equivalent defect in its nominal charge state over the entire band gap (see Section III B), and the band gap corrections succeed in only increasing these defect formation energies further. To assess the defect formation energies for a range of chemical environments, the chemical potentials must be calculated to establish the thermodynamic boundaries of the system. The entropic and pressure/volume dependence of the chemical potential per unit cell of a solid (e.g., Sr, Ti) is small and therefore can be neglected. This allows the use of T = 0 K total energy calculations for these atomic species and the bulk crystal (as well as its formation energy) to define thermodynamic reservoirs where the chemical potentials of the individual atoms cannot exceed the formation energy of the crystal ΔμSr + ΔμTi + 3ΔμO = ΔHf [SrTiO3]

(4)

ΔμTi + 2ΔμO ≤ ΔHf [TiO2 ]

ΔμO

0 −6.10 −4.91

0 −10.44 −11.63

−5.51 0 0

The temperature contribution Δμ(T) is represented by

Further constraints come from the formation of competing compounds ΔμSr + ΔμO ≤ ΔHf [SrO]

ΔμTi

In addition to the bulk chemical potentials, the chemical potential of Mn must also be considered. Manganese oxide exists in a number of different structures including MnO, Mn2O3, Mn3O4, and MnO2. From these phases, it is Mn3O4 that exists over the largest oxygen partial pressure range at the typical sintering temperatures of multilayer ceramic capactiors (MLCCs).36 Therefore, the chemical potential of Mn is determined on the assumption of equilibrium with Mn3O4. While the use of T = 0 K total energy calculations is justified for defining thermodynamic boundaries for solids, the chemical potential of gaseous oxygen has a far stronger dependence upon temperature and pressure. The choice of functional and pseudopotentials can also cause significant error in the DFT calculation of the oxygen dimer.43 In this work, the effects of temperature and pressure are taken into account via the use of a method developed by Finnis et al.44 By combining the formation energy of SrTiO3 and T = 0 K total energy calculations for Sr and Ti, the oxygen chemical potential at standard pressure and temperature, μO(pO0 2 ,T0), can be determined without the need for direct DFT calculation. Through the use of ideal gas relations and μO(p0O2,T0), the oxygen chemical potential can be derived for a specific temperature and pressure, μO(pO2,T). The reliability of this approach has been confirmed by comparison to thermodynamic data.45 The value of μO(pO2,T) for a specific temperature and pressure is obtained from the ideal gas expression and formula for an ideal gas of rigid dumbbells 1 μO(pO , T ) = (μSrTiO − μSr − μTi − ΔGfSrTiO3) 3 2 3 ⎛p ⎞ 1 O + Δμ(T ) + kBT log⎜⎜ 02 ⎟⎟ 2 p ⎝ O2 ⎠ (7)

Figure 1. Scaling behavior for the formation energies of charge-neutral Mn defects. These data represent only the energy difference between the defective and perfect supercells; chemical potentials and other corrections are not applied here.

ΔEg = neΔECB + nhΔE VB

ΔμSr

Δμ(T ) = −

(5)

⎛T ⎞ 1 0 1 (SO2 − C p0)(T − T 0) + C p0T log⎜ 0 ⎟ ⎝T ⎠ 2 2 (8)

S0O2

where is the molecular entropy of oxygen gas (0.0021 eV/ K46) and C0p is its constant pressure heat capacity (7kB/246). Unless explicitly stated otherwise, a temperature of 1400 K and an oxygen partial pressure of 1 atm have been used to

(6)

The combination of these three equations confines the values of the chemical potentials of the atomic species within the 14487



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calculate μO(pO2,T) in accordance with a typical sintering temperature of SrTiO3. B. Lattice Statics Calculations. For the lattice statics calculations, the energy of the system is modeled from contributions of the long-range and short-range forces with respect to the atomic positions in the lattice. The long-range interactions are Coulombic, and the short-range repulsive interactions are represented by interatomic potentials such as the Buckingham potential n

Vij(r ) =

∑ A exp(−rij/ρ) − i≠j

C rij6

Table 3. Comparison of the Lattice Energy and Cell Parameters Obtained from Lattice Statics Calculations with Experimental Data for SrTiO3 and the Relevant Metal Oxides lattice energy (eV)

lattice constants (Å)

crystal

potentials

experiment

potentials

experiment

SrTiO3 SrO TiO2

−162.25 −34.47 −127.64

−161.2929 −33.4029 −125.5029

a = 3.96 a = 5.04 a = 4.49, c = 3.14

a = 3.9154 a = 5.1656 a = 4.59, c = 2.9541

(9)

where the symbols have their usual meanings. In the static limit, lattice vibrations are ignored, and the structure is determined by the static contribution to the internal energy. Ionic polarization is taken into account by the Dick and Overhauser shell model.47 In this model atoms are divided into negatively charged massless shells and positively charged cores with the appropriate atomic mass. The usual electrostatic interaction between the cores and shells is replaced by a harmonic one determined by the spring constant, k. The potential model used in this work has been fitted using ab initio calculations and reproduces experimental structures (see Section III A) and results very accurately.28,29 A detailed description of the SrTiO3 potential model used here and the fitting procedure is given in refs 28 and 29. A cutoff of 12 Å was applied to all of the potentials. The defect calculations are performed using the Mott− Littleton approximation.48 In this method defects are simulated at the infinitely dilute concentration limit. The lattice surrounding the defect is divided into two spherical regions: an inner region and outer region. In the inner region the interactions are calculated explicitly and ions are relaxed to positions of zero force. In the outer region, where the interactions are weaker, the polarization energy and ionic positions are approximated using a dielectric continuum method. To ensure the inner region is properly bedded in the crystal, the interactions between ions of the inner region and the ions of the outer regions are calculated explicitly. All the lattice statics calculations in this work were completed using the General Utility Lattice Program (GULP).49 Comprehensive reviews of the methodology briefly described here are available elsewhere.50,51

III. RESULTS AND DISCUSSION A. SrTiO3 Bulk Properties. For both the DFT and lattice statics calculations it is essential that not only the SrTiO3 structure and energy are reproduced correctly but also its end member phases, SrO and TiO2. The values calculated using both techniques for the standard states of the materials are Table 2. Comparison of the Formation Energies and Cell Parameters Obtained from GGA Calculations with Experimental Data for SrTiO3 and the Relevant Metal Oxides formation energy (eV)

Figure 2. GGA + U vacancy formation energies for SrTiO3 under (a) metal-rich, (b) O-rich (TiO2 excess), and (c) O-rich (SrO excess) conditions at T = 1400 K and pO2 = 1 atm.

lattice constants (Å)

crystal

GGA

experiment

GGA

experiment

SrTiO3 SrO TiO2

−16.54 −4.91 −10.44

−17.1453 −6.1455 −9.7841

a = 3.94 a = 5.20 a = 4.65, c = 2.97

a = 3.9154 a = 5.1656 a = 4.59, c = 2.9541

compared with the experimental equivalents in Tables 2 and 3. Such calculations are also required to define thermodynamic boundaries of SrTiO3 for the DFT calculations (as described in Section II B). All bulk property calculations were completed using a Γ-point centered 13 × 13 × 13 k-point mesh and were 14488



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Figure 3. Density of states (DOS) from spin-polarized GGA + U calculations for (a) Mn2+ doped at the Sr-site, (b) Mn4+ doped at the Ti-site, (c) Mn2+ doped at the Ti-site with oxygen vacancy compensation, and (d) Mn4+ doped at the Sr-site with Sr vacancy compensation.

majority of the band gap. At high Fermi level energies V1+ O and V0O do become more stable than the fully ionic oxygen vacancy, and a deep defect level associated with the reduction of two Ti ions is formed. For low Fermi energies, oxygen vacancies are clearly the dominant defect in the system. Under metal-rich conditions the formation energy for an oxygen vacancy is positive only for high Fermi energies. This suggests that under these highly reducing conditions the formation of stoichiometric SrTiO 3 is impossible. This is supported by experiment where grossly nonstoichiometric SrTiO3 samples with up to 9.5% oxygen deficiency are possible under reduction.57 This is also true for BaTiO3 as confirmed by both experiment58 and simulation.41 Low defect formation energies for the oxygen vacancy have also been calculated using screened hybrid density functional theory59 and GGA + U calculations.21 Sr vacancies are only dominant for part of the O-rich (TiO2 excess) environment, whereas Ti vacancies are clearly the dominant defect at high Fermi energies as a result of their large negative charge. Hybrid functional calculations suggest that Ti vacancies are very high in energy; however this is to be expected given that they only consider neutral vacancies.59 Our results are in good agreement with previous calculations and are similar to those calculated for BaTiO3 also.40,41 C. Mn-Doped SrTiO3 Electronic Structure. Figure 3 shows the densities of states (DOSs) of Mn-doped SrTiO3 calculated using the GGA + U method with spin polarization. In SrTiO3 the valence band (VB) is mostly made up of O p states as well as hybridized bonding states containing d orbitals of Ti and p states of O. The conduction band (CB) is made up

fully converged to an accuracy greater than 1 meV per unit cell. A band gap of 1.79 eV is obtained for undoped SrTiO3. The value of 1.79 eV is significantly lower than the experimental value of 3.2 eV,42 as a result of the discussed band gap underestimation associated with DFT calculations. This value is, however, in very good agreement with previously calculated values.52 The data obtained from both the GGA and lattice statics calculations are in good agreement with the experimental values, and the use of the GGA avoids the common problem associated with LDA calculations of overbinding in solids as reported previously for Ti.40,41 The lattice constants calculated with the GGA functional are slightly larger than the experimental values for the oxide materials; this is a result of typical overestimation of the lattice constant by the GGA functional. For the potential calculations, the lattice constants of SrO and TiO2 are slightly underestimated, while for SrTiO3, the lattice constant is slightly overestimated. Despite these generally smaller errors, both GGA and lattice statics calculations can accurately replicate not only the structure of interest but also its end member phases and constituent elements. B. DFT Vacancy Formation Energies. In addition to being of fundamental importance in their own right, it is also important to assess the charge states of the compensating defects associated with Mn-doping. The vacancy formation energies are plotted for three chemical potential limits: O-rich (SrO and TiO2 rich) and metal-rich conditions in Figure 2. 2− 4− Only energies in their nominal charge states (V2+ O , VSr , VTi ) are plotted, as these are the dominant charge states over the 14489



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Figure 4. Isosurfaces of the spin density for (a) Mn2+ doped at the Sr-site, (b) Mn4+ doped at the Ti-site, (c) Mn2+ doped at the Ti-site with oxygen vacancy compensation, and (d) Mn4+ doped at the Sr-site with Sr vacancy compensation. Sr ions have been omitted for clarity. Yellow refers to spinup density, and blue is spin-down density. The isosurface is set to 0.01 μBÅ−3.

Table 4. GGA + U Defect Formation Energies for MnDoping of SrTiO3 under Metal-Rich and O-Rich Conditions at T = 1400 K and pO2 = 1 atma

a

defect

Ef (eV) metal-rich

Ef (eV) O-rich (TiO2 excess)

Ef (eV) O-rich (SrO excess)

Mn0Sr 2− (Mn2+ Sr −VSr ) Mn0Ti 2+ Mn2− Ti −VO

7.39 13.62 7.98 4.25

1.29 1.42 −2.46 −0.68

2.50 3.80 −3.65 −1.82

The energies are calculated using finite-size extrapolation.

of empty d orbitals from the Ti and Sr atoms. For Figure 3(a) and (b) the band gap remains unchanged with the addition of a Mn dopant ion. Previous hybrid exchange calculations also show no change to the band gap after the introduction of Mn2+ to the Sr-site; however, they report a narrowing of 0.72 eV of the band gap with the introduction of Mn4+ at a Ti-site.19 Direct comparison between these two electronic structure methods can be difficult as hybrid functional calculations generally produce much larger, more accurate band gaps. The strong peaks produced by Mn are also observed in the hybrid functional calculations. For Mn-doping at the Ti-site with oxygen vacancy compensation (Figure 3(c)), the band gap is significantly reduced with the introduction of occupied states above the oxygen 2p states as a result of the Mn dopant ion. For Mn-doping at the Sr-site with Sr vacancy compensation (Figure 3(d)), the band gap is actually increased to 2.1 eV. It must be noted that given the difficulty in accurately calculating

Figure 5. GGA + U defect formation energies of Mn-doped SrTiO3 under metal-rich and O-rich conditions at T = 1400 K and pO2 = 1 atm.

the electronic states of transition metals these results must be treated with some caution. Figure 4 shows the spin density isosurfaces for different types of Mn defects in SrTiO3. These illustrate the difference between the up and down spin density. For all examples it is the Mn ions that carry the majority of the spin density as a result of the five unpaired electrons of Mn2+ (t2g3eg2) and three unpaired electrons of Mn4+ (t2g3eg0). All magnetic moments calculated for the doped cells are also in agreement with the expected oxidation states of Mn. The different oxidation states 14490



Article 2− 2− 2+ −1.60 and −1.00 eV for (Mn2+ Sr −VO ) and (MnTi −VO ), respectively. For the O-rich environments, again both divalent and tetravalent Mn ions dope at the Ti-site. The defect formation energies for these environments are all lower than those for the metal-rich environment as now the removal of metal ions is energetically more favorable. In agreement with experiment,60,63 the lowest energy incorporation mechanism for both conditions is that of the direct replacement of Ti4+ with Mn4+. For both O-rich environments tested, the defect formation energy for this direct replacement is strongly negative suggesting an exothermic intake of Mn into the system. This is supported by experiment where up to 50 mol % Mn solubility has been observed for this type of incorporation.64 Beyond 50 mol % Mn concentrations, hexagonal SrMnO3 was observed as well as nonreacted Mn2O3 and SrCO3. It is interesting that for Mn2+ doping the Ti-site is favored over the Sr-site. This is even true for the TiO2 excess example where Sr-site doping is energetically favored as a result of the chemical potentials. Experiment shows that Mn2+ doping of both the Sr- and Ti-sites is possible, and this is supported by our results as the energy for Mn2+ doping of the Sr-site is also low. In addition to the charge-neutral single defects and defect pairs, we also consider the dependence of the valence state of Mn on the Fermi level. These energies are plotted against the Fermi energy along with the charge-neutral equivalent in Figure 6. For Ti-site substitution, Mn4+ is dominant under p-type conditions where the Fermi level is located near the top of the valence band. However, at n-type conditions (higher Fermi energies), Mn 2+ becomes more favorable. For Sr-site substitution, the neutral defect is dominant over the entire band gap. This further supports the preference of Mn2+ substitution at the Ti-site over the Sr-site. E. Lattice Statics Mn-Doping Defect Energies. The same four dopant incorporation mechanisms used for the DFT calculations have also been used for the potential-based calculations (see Schemes 1− 4). The defect energy (solution energy, Es) is calculated using the relevant oxide lattice energy, EL (Section III A), the Mn substitution energy, and the vacancy energy (if charge compensation is required). For the two schemes where defect pairs are present, the substitution and vacancy energies are calculated together in one cell so that the binding energy is taken into account, as discussed previously for the DFT calculations. All types of vacancy in SrTiO3 have been calculated previously using this potential model.29 Comparison of these values with the DFT values calculated in this work is very difficult and arguably meaningless as the lattice statics values do not take into account the chemical potential environment and the contribution from the Fermi energy. The lattice energies of MnO and MnO2 are calculated to be −38.90 and −114.41 eV, respectively. The incorporation schemes are provided here for the convenience of the reader. The calculated solution energies for the four incorporation schemes are given in Table 5. Again, the direct substitution of Ti4+ with Mn4+ is the lowest energy mechanism in agreement

Figure 6. Fermi level dependence of Mn defect formation energy for substitution on (a) a Sr-site under O-rich (TiO2 excess) and (b) a Tisite under O-rich (SrO excess) conditions in different valence states at T = 1400 K and pO2 = 1 atm.

of Mn are also reflected by the shape of the spin density isosurfaces. D. DFT Mn-Doping Defect Formation Energies. The defect formation energies for Mn-doping calculated by the GGA + U scheme for three different chemical environments at a temperature of 1400 K and an oxygen partial pressure of 1 atm are given in Table 4. The energies for the neutral defects are plotted in Figure 5. Only dopants and defects in their 4− nominal charge states (e.g., Mn2+, Mn4+, V2− Sr , and VTi ) have been considered as these are by far the most likely dominant species as confirmed by our own calculations and experiment.2,13−15 For defect pairs, the formation energy is calculated with the two defects neighboring each other to maximize the binding energy between the oppositely charged species. It is clear from Figure 5 that there is significant variation in formation energy depending on not only the defect type but also the chemical environment, with energies ranging from −3.65 to 13.62 eV. The vast majority of the larger defect formation energies is for the metal-rich limit. This is logical given that such conditions strongly favor the formation of oxygen vacancies over metal vacancies. This is illustrated by the fact that the lowest defect energy for metal-rich conditions is 2+ for the (Mn2− Ti −VO ) defect pair. In terms of site preference, the Ti-site is strongly preferred for both divalent and tetravalent Mn dopant ions, with the lowest overall energy being for Mn2+ doping with oxygen vacancy compensation, as discussed. Experimental findings confirm that Mn2+ can exist at both the Sr- and Ti-site2,11,13,60 and also that under reducing conditions Mn4+ at the Ti-site becomes Mn2+ with the formation of associated oxygen vacancies.60−62 DFT calculations of Mndoped BaTiO3 also confirm the energetic preference of this incorporation mechanism at the reducing limit.36 It is noteworthy to add, however, that generally Mn4+ substituting naturally for Ti4+ is the usual mechanism in most samples.60,63 Binding energies for the two defect pairs were calculated as Scheme 1. Substitution of Mn2+ at Sr2+ Sites

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Scheme 2. Substitution of Mn4+ at Sr2+ Sites with Sr Vacancy Compensation

Scheme 3. Substitution of Mn4+ at Ti4+ Sites

Scheme 4. Substitution of Mn2+ at Ti4+ Sites with O Vacancy Compensation

energies for all monovacancies in SrTiO3 in all possible charge states. Vacancies are shown to be fully ionic as they exist in their nominal charge states over the entire band gap. Comparisons between the results from the two methods have been made, and the influence the chemical environment has on the DFT results (when compared to the lattice statics results where no environment is applied) has been assessed. Excellent agreement with experiment is also observed.

Table 5. Solution Energies Calculated for Mn-Doped SrTiO3 defect

solution energy (eV)

Mn0sr 2− (Mn2+ Sr −VSr ) 0 MnTi 2+ Mn2− Ti −VO

1.55 7.18 −0.56 5.50

■

with experiment.60,63 The negative solution energy for this scheme also suggests an actual stabilization of the material. A result is also observed for the Mn4+ doping of cubic and hexagonal BaTiO330 where negative solution energies are used to explain the lowering of the cubic to hexagonal phase transition temperature. For Mn2+ doping, Sr-site substitution is strongly preferred over the formation of the dopant−defect pair. As discussed, divalent Mn is known to exist at both cation sites experimentally. It is noteworthy that lattice statics calculations predict only direct substitutions at sites with equivalent charges and not defect pairs where the strength of the binding is undoubtedly strong. On the basis of ion size arguments alone it is clear that Mn4+ (0.53 Å)65 should dope at the Ti4+ (0.61 Å)65 site; however it is not clear that Mn2+ (0.83 Å)65 should dope at the Sr2+ (1.44 Å)65 site as its ionic radius is actually closer to that of Ti4+. Our results completely agree with the previous lattice statics calculations which also predicted Mn2+ doping at Sr-sites and Mn4+ doping at Ti-sites.26

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +8175-753-5435. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Japan Society for the Promotion of Science (JSPS) for funding through Grants-in-Aid for (a) Scientific Research on Innovative Areas “Nano Informatics” (Grant No. 25106005) and for (b) JSPS fellows (Grant No. 2503370).

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IV. CONCLUSIONS We have calculated the defect formation energies for the Mndoping of SrTiO3 for a variety of incorporation methods using the GGA + U scheme. In addition, defect energies are also calculated using lattice statics methods. Both divalent and tetravalent Mn ions have been tested with both direct substitutions and any necessary charge compensation considered. We found that in agreement with experiment direct substitution of Ti4+ with Mn4+ is the most energetically preferred mechanism for GGA + U calculations when O-rich conditions are applied and for lattice statics calculations. GGA + U calculations suggest that Mn2+ doping occurs at the Ti-site also, whereas the lattice statics results strongly suggest Sr-site substitution. We have also calculated the defect formation 14492



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