Density functional simulation of carbon at the titanium site in perovskite barium titanate

The perovskite family includes many titanates which have been used in various applications. BaTiO3 is interesting due to its room-temperature ferroelectric properties and relative low toxicity. Organic precursors present during growth make carbon a potentially key impurity, which would subsequently impact upon the BaTiO3 properties. This paper presents a density function study of the structural and electronic properties of carbon substituting for Ti in rhombohedral BaTiO3. The local vibrational modes of the defect centre has been calculated and suggested as a possible route to experimental identification.


Introduction
BaTiO 3 is a ferroelectric material with a high dielectric constant and low loss characteristics, widely utilised in the manufacture of electronic components such as multilayer dielectric capacitors and high refractive index thin films [1]. Its piezoelectric and pyroelectric properties lend it to passive infrared detectors [2], piezoelectric actuators and sonar devices [3], and its positive temperature coefficient of resistivity allows it to be used in sensor applications [3].
Given the dependence of the BaTiO 3 properties upon both stoichiometry and purity, the crystal growth techniques used are highly significant. Various different techniques have been used to synthesis thin films, including solid-state reaction [22], the sol-gel method [23,24,25,16,19], mechanochemical synthesis [26,27], solvothermal synthesis methods [28], sputter deposition [29], pulsed-laser deposition [30], chemical vapour deposition [31] and atomic layer deposition [31,32], in order to eliminate problems such as heterogeneity, impurities, inaccurate thickness control and the poor electrical properties of sintered ceramics. In several of these methods organic precursors are used for Ba sources, and subsequent carbon contamination has been reported in BaTiO 3 thin films [33,25,26,32]. For example, sharp peaks at 1425, 1050, 860, and 695 cm −1 were observed and attributed to the carbonate ions (CO −2 3 ) [34]. In this paper, we present the results of first principles calculations on the geometry, electronic structure, vibrational modes and reorientation barrier of carbon substituting for Ti in BaTiO 3 .

Computational details
The properties of carbon substituting for Ti in rhombohedral BaTiO 3 are calculated using first-principles density functional theory within the local density approximation [35]. The selfconsistence total energy calculations were performed using the AIMPRO computer code [36,37] based on the stable and efficient preconditioned conjugate gradients method with the optimised structures having forces of < 10 −3 atomic units and the final structural optimisation step is required to result in a reduction in the total energy of less than 10 −5 Ha.
Atoms are modelled using norm-conserving, separable pseudo potentials [38] (valance sets of Ba, Ti, O and C are 5s 2 5p 6 6s 2 , 3s 2 3p 6 3d 2 4s 2 , 2s 2 2p 4 and 2s 2 2p 2 , respectively), with the Kohn-Sham eigen-functions expanded using atom centred Gaussian basis sets [39], comprised from four independent sets of s, p and d type functions. Matrix elements of the Hamiltonian are determined using a plane-wave expansion of the density and Kohn-Sham potential [40] with a cut-off of 150 Ha, resulting in absolute convergence of the total energy with respect to the expansion of the charge density to better than 0.03 meV.
For bulk rhombohedral BaTiO 3 symmetry, our computational approach yields an equilibrium lattice parameters of 3.97Å and 89.85 • , within 1% of experiment [41,42]. The calculated bandgap is 2.3 eV, reflecting the well documented underestimate arising from the underpinning methodology. Our estimate is consistent with previous comparable calculations [43].
To model defects, we use the same type of cell detailed previously [44], consisting of 160 host atoms. The Brillouin zone is sampled using a uniform 2 × 2 × 2 Monkhorst-Pack mesh [45].
Vibrational modes have been calculated by obtaining the second derivatives of the energy with respect to atomic positions, which are then assembled into the dynamical matrix. The derivatives are determined using a finite-difference approximation based upon analytical forces obtained with atomic displacement of 0.2 atomic units.
Finally, reaction pathways and activation energies are determined using the climbing nudgedelastic-band (NEB) method [46,47], and the convergence of the saddle point energy with respect to the number of images and the image-forces has been established to within a few meV.

Defect geometry
We start with the geometry of carbon substitution of titanium, C Ti , which may be anticipated on the grounds of both C and Ti being group-IV elements [48]. The first structure analysed was created from the octahedral symmetry, with a Ti ion simply replaced with carbon. In the absence of any symmetry constraint, the optimised structure of C Ti exhibits significant structural change from the initial geometry. The carbon is displaced from the centre where it was initially co-ordinated with six oxygen ions, toward the centre of a triangular group composed of three oxygen ions, as shown in figure 1(b). The final structure has three C-O bonds 35% shorter than the host Ti-O distances. The equilibrium geometry can be clearly identified as a carbonate group ion. To further support this view it is noted that the calculated C-O bond-lengths of 1.29Å are close to those of the carbonate group [49,50] and those in witherite [51].
We now turn to the impact that the formation of the carbonate group has upon the electronic properties of the material.

Electronic structure
The band structure for C Ti is shown in figure 1(c). The inclusion of a carbon atom at a Ti site does not have any significant effect on the band-gap, which we note is in contrast to the situation with C Ti in cubic SrTiO 3 [48]. Carbon substitution for titanium in BaTiO 3 is therefore expected to lead to electrically inactive defects, i.e. these centres are not expected to introduce any states in the band-gap that would lead to a change of charge state.

Vibrational modes
Turning to vibrational characteristics, we find that the calculated frequencies are consistent with C Ti resulting in the formation of carbonate ions. A set of characteristic local vibrational modes lies above the one-phonon maximum, which could prove highly effective in the experimental identification of this centre: with a nearly degenerate C-O stretch mode at 1383 cm −1 and 1379 cm −1 , and a breathing stretch-mode is at 1061 cm −1 . These are close to the experimental values of CO −2 3 gas phase [52] which have E ′ and A ′ 1 stretch modes lying around 1415 cm −1 and 1064cm, respectively, and also agree both with the modes detected in BaTiO 3 [34], and with those modes assigned to the carbonate groups in BaCO 3 [53]. Lower frequency modes of the carbonate group are also present for C Ti , but are resonant with the one-phonon density of states. All three local modes listed here are, by symmetry, infra-red and Raman active, and given sufficient concentrations one might seek to confirm the presence of this form of carbon centre directly via either IR or Raman spectroscopy.

Carbonate group reorientation
Finally, the energetics for carbonate group reorientation have been predicted. The initial and final structures are shown in figure 2(a) and (b), and the energy profile along the minimum energy path between them in figure 2(c). The energy difference between initial and final structure has been calculated to be about 50 meV, which is related to non-equivalent in the defect with respect to the rhombohedral distortion. The activation energy for reorientation process has been calculated to be around 2.5 eV. The barrier is found to be high in energy compared to, for example, diffusion of the oxygen vacancy, since the reorientation requires the breaking and formation of strong covalent C-O bonds. It is noted, however, that this activation energy is similar to the equivalent path in cubic SrTiO 3 [48]. The high barrier is expected to lead to slow reorientation at room temperature, which will impact upon the hysteresis seen in the material under an applied electric field.

Summary and conclusions
In summary, based on first-principles density-functional calculations, the characteristics of carbon impurities substituting for titanium in rhombohedral BaTiO 3 have been examined, including the structural configuration, electronic properties, vibrational modes and reorientation barrier. The calculations reveal that carbon energetically favours the formation of carbonate groups and suggest that the carbon substituting of titanium is electrically passive, so that C Ti is predicted to have a minimal impact upon the electrical conductivity of BaTiO 3 . The vibrational frequencies associated with defect centre have been determined and compared with available theoretical and experiment values. These spectroscopic properties are a clear route to experimental identification of this defect. Finally, the reorientation barrier for the carbonate group for C Ti has been estimated, and because the process involves the breaking and forming of C-O covalent bonds, the barrier has a relatively high value which will in general impact upon ferroelectric hysteresis.