ABO3 perovskites such as PbZrO3, PbTiO3 and PbZrxTi1-xO3 exhibit important antiferroelectric, ferroelectric, piezoelectric and photoluminescent properties that make them useful in a wide range of technical applications. PZT materials are used in the fabrication of dynamic random-access memory (DRAM), non-volatile ferroelectrics random-access memory (NFRAM) devices, light emitting devices for displays and communication, piezoelectric sensors actuators and transducers and MEMS devices. The unique properties arise from the various crystal structures of the different compounds that effect domain wall formation and orientation (ferroelectric vs. antiferroelectric), polarization and piezoelectric response. Numerous characterization techniques like powder neutron diffraction, X-ray diffraction, photoluminescence spectroscopy can be used to determine the atomic structure and give insight to the electronic structure of these different materials. However, it is beneficial to combine such characterization techniques with ab initio calculations. Ab initio studies not only act as a compliment experimental work, confirming observations, but they allow researchers to investigate properties even further. With ab initio calculations researchers can isolate specific parameters and determine specific electronic events. Defects such as vacancies can be studied in an isolated system, allowing the researcher to investigate the effects of which atomic species vacancy has the most significant impact. Ab initio methods can also give insight as to how the electronic structure is affected by the presence of such vacancies or how domain wall formation/motion is governed by the electronic structure. Ab initio studies open the possibility for investigating and designing new materials based on specific properties of interest, giving insight as to how such properties can be exploited either through processing conditions, composition, heat treatments etc.
Vacancy Formation in PbTiO3
PbTiO3 (lead titanate) is a simple, yet important member of the perovskite family having a high curie temperature of 493°C and a high spontaneous polarization experimentally determined as 75µC/cm2 and 88µC/cm2 from first-principles calculations. It has been observed that PZT and PbTiO3 in both thin film and bulk form suffer from Pb evaporation during a variety of processing methods including sintering and in-air deposition. The vacancies that Pb leave behind result in an increase in acceptor levels, described by the reaction below:
PbPb ↔Pb + VPb∙∙ + 2h+
PbPb represents Pb in the lattice, while VPb is a vacancy. Loss of Pb has been shown to contribute to reduction in remnant polarization as well as oxygen vacancies playing a role in polarization fatigue by domain pinning. The experiment carried out by Z. Zhang et al.(2008) sought to investigate the formation of different types of neutral vacancies in PbTiO3. The calculations were based on the generalized gradient approximation (GGA) to the DFT using the projector-augmented wave (PAW) method and Perdew-Burke-Ernzerhof (PBE) exchange-correlation parameterization. The pseudo=potential approach they used regarded the Pb 5d, 6s, 6p, Ti 3d, 4s and O 2s, 2p orbitals as valence orbitals, totaling 36 valence electrons in one unit cell. Calculations were carried out using VASP. Four types of 2x2x2 supercells were established in order to investigate the four types of vacancies (Pb, Ti, O1 and O2). Converging tests were performed before any geometry relaxation or property calculation in order to determine the cutoff energy, which was 400eV for each supercell. 8 k-points in the Brillouin zone were used. Taking note from previous work they pointed out that a fully relaxed geometry optimization is unable to reliably determine lattice parameters of perovskite crystals, therefore the experimental value of 63.28Å3/unit cell was used as a fixed value for the volume of the lead titanate. The four supercells used are depicted in figure 1.
Figure1: Models of the four supercells used. Vacancies are located in the center of each supercell. (a) Pb vacancy, (b) Ti vacancy, (c) O1 vacancy, (d) O2 vacancy.
During the calculations for the system energy of a supercell with a point defect, only the first and second nearest ions near the vacancy were able to be rearranged. The lattice parameters and other ions were kept fixed. In calculations where all ions were allowed to relax, the energy difference was less than 0.2%. The chemical potentials were calculated for each of the reference materials, lead, titanium, PbO and TiO2 with high cutoff energy and k meshes. The chemical potential of oxygen was in calculated in the form of an isolated molecule in a 20Åx20Åx20Å box. The equation for the formation energy (Ef) of a neutral vacancy in the PbTiO3 system can be expressed as:
Ef = (Edefect – Eperfect) + µ
Where Edefect is the total energy of the supercell with a neutral vacancy, and Eperfect is the total energy of a defect free supercell. µ is the chemical potential of the atom removed from the perfect system. Figure 2 shows two tables, one comparing the theoretical lattice parameters with experimental values and the other representing the five different thermo-chemistry conditions impose on the systems.
Figure 2: (Top)Theoretical lattice parameters compared with experimental values. Calculated chemical potentials are also included.(Bottom) External atmospheres and chemical potential of lead under the different thermo-chemistry conditions.
The formation energies of neutral vacancies at different thermo-chemistry conditions that were calculated indicate that under an oxidizing atmosphere, form during high temperature heating. A reducing atmosphere results in oxygen vacancies being more stable than the others, and that under both conditions the oxygen bonded with a Ti in the z-direction are more easily formed than those bonded with Ti in the x-y plane. It was also determined that a Ti vacancy is difficult to form under all conditions. Al of the results from the theoretical calculations were in agreement with experimentally determined values.
Oxygen Vacancies in PbZrO3
Lead zirconate (PbZrO3) is antiferroelectric, making it important for applications involving actuators and high energy storage devices. Like PbTiO3, PbZrO3 is a parent compound of PZT. PbZrO3 thin film heterostructures have been found to be radiation resistant, making them good candidates for diagnostic devices used in thermonuclear reactors. S. Piskunov et al. (2007) utilized ab initio hybrid density functional calculations to simulate cubic and orthorhombic PbZrO3 along with cubic PbZrO3 containing single neutral oxygen vacancies to observe the effects of the vacancy on the ferroelectric properties. They utilized CRYSTAL computer code to carry out hybrid B3PW DFT calculations. The inner core electrons of Pb and Zr were described by Hay-Wadt effective core potentials taking into account the relativistic effect. The hybrid referred to is a combination of Fock exact exchange and exchange potentials constructed using LDA and non-localized GGA, combined with the GGA correlation potential of Perdew and Wang. Though LDA alone is known to underestimate the band gap and lattice constant for perovskites, the hybrid functionals yield more reliable results. Isolated defects were simulated by extending a simple cubic cell into a 3x3x3 supercell containing 135 atoms. In the modeling of the oxygen vacancy they employed a “ghost” basis set, where when an O atom is removed, its basis set is left behind. This method reduces computational time and aids in the accurate description of electron distribution in the vacancy. Reciprocal space integrations sampled Brillouin zones with 8x8x8 Monkhorst-Pack meshes for the cubic unit cell, 6x6x6 for the orthorhombic unit cell and 4x4x4 for the SC cell with a defect. Figure 3 shows a table of results comparing fractional coordinates of atoms and lattice parameters for the orthorhombic structure of PbZrO3 calculated using the DFT-B3PW method to values obtained from powder neutron diffraction at 10K and 20K.
Figure 3: Fractional coordinates of atoms and lattice parameters for the orthorhombic structure of PbZrO3 calculated using the DFT-B3PW method compared to values obtained by powder neutron diffraction at 10K and 20K.
To obtain the equilibrium geometry of PbZrO3 with an oxygen vacancy, all atoms in the first eight coordination spheres around the vacancy were relaxed along allowed cubic symmetry directions. It was noted thet the magnitude of displacement of Pb atoms induced by the defect is close to that observed for the Pb atoms in the transition from cubic to orthorhombic, thus having the possibility to affect its ferroelectric-antiferroelectric properties. The formation energy of an oxygen vacancy in cubic PbZrO3 was calculated using:
Eform(F) = Etot(F/PZ) + Etotground state (O) – Etot(PZ)
Where Etotground state(O) is the energy of an isolated oxygen atom in its ground state (spin-polarized), Etot(F/PZ) and Etot(PZ) are the energies of the defective and perfect crystal respectively. This equation yields an oxygen vacancy formation energy of 7.25eV. The calculated electron density distribution shown in figure 4 gives further insight into the influence of the oxygen vacancy on the properties of PbZrO3. The vacancy retains -0.68e from a missing oxygen anion. The four Pb atoms nearest the defect have a charge decrease of +1.3e to +1.17e, indicating that these neighbors captured -0.5e from the missing oxygen. These values are confirmed by vacancy-induced electron charge difference density plot also shown in figure 4. This plot indicates a presence of trapped electron density in the vacancy site.
Figure 4: (Left) Predicted displacements and effective Mulliken charges of atoms positioned in the atomic sites of the first eight coordinate spheres around the F-center (oxygen vacancy). (Right) Calculated (110) crossection of difference electron density maps Δρ(r) for PbZrO3 containing and oxygen vacancy. Black lines indicate zero level, blue lines indicate a decrease in Δρ(r) and red for an increase in Δρ(r).
The formation of an oxygen vacancy in the cubic PbZrO3 is accompanied by a 0.25Å shift of the nearest Pb atoms towards the vacancy and traps 0.68e forming a defect level in the middle of the band gap. The remaining 0.5 from the missing oxygen anion are localized on the four nearest neighbor Pb atoms, thus the presence of point defects will affect the atomic polarization in PbZrO3, influencing its ferroelectric properties.
Photoluminescence in PZT
Photoluminescence (PL) is the emission of a photon due to an electron being excited to a higher energy state and subsequently returning to a lower state. Since the first observation of PL in silicon, studies have focused on the fabrication of new, disordered or nanostructured, materials. These materials have potential technological applications such as dynamic random-access memory (DRAM), non-volatile ferroelectrics random-access memory (NFRAM) devices, light emitting devices for displays and communication and piezo-electric sensors.
The ABO3 perovskites are a class of materials that exhibit the non-linear electro-optical properties associated with PL. Many studies have been done on crystalline PZT at low temperatures however, few have studied the room temperature photoluminescence of PZT. M.S. Silva et al. (2005) have devised a method to process PZT in its amorphous phase by high-energy mechanical milling and study room temperature PL, based on the findings of two other groups that had initially observed room temperature PL in amorphous PZT films.
Quantum mechanical characterizations were carried out using Density Functional Theory with the gradient-corrected correlation functional combined with the Becke exchange functional, B3LYP. Calculations were carried out using the crystalline orbital program, CRYSTAL98. They chose a k-point sampling of 40 points within the irreducible part of the Brillouin zone. They also utilized a special option within the program to simulate the displacement of the Ti atom called ATOMDISP. XRD analysis and PL spectra revealed a spectral dependence of the crystalline and amorphous powders. These results suggested that a radiative recombination between trapped electrons and trapped holes occurs between localized states situated inside the band gap and must be responsible for the PL of the amorphized powder. The presence of the localized states was confirmed with quantum mechanical calculations using 2 periodic models representing the crystalline and amorphous PZT powders (PZT-a and PZT-c respectively), shown in figure 5.
Figure5: a) Crystalline PZT (PZT-c) periodic model as a 1x1x2 supercell of the (Pb,Zr)TiO3 unit cell. b) Asymmetric unit of PZT (PZT-a) in which Ti10 is shifted by a (0.0 0.0 0.5)Å vector.
The optimized a and c parameters were 3.966Å and 4.163Å respectively, in close agreement with the experimental values of 3.952Å and 4.163Å. The translation vector was chosen as (0.0 0.0 0.5)Å because that is the value sufficient to break the bond between Ti and O. From the calculated band structures, the PZT-c model had the top of the valence band at the X point and bottom of the conduction band at Γ. The indirect gap between X and Γ is 3.03eV, close to the experimental value of 3.16eV. For the PZT-a model, the minimal gap was the direct gap at X of 1.87eV compared with the optical gap found to be 2.91eV. The calculated band structures for the PZT-a and PZT-c models is shown in figure 5. From the band structures it is observed that the breaking of the Ti10-O4 bond causes a broadening of the valence and conduction bands, leading to a reduction of the gap, and a creation of new states in the original band gap before deformation. Those states are identified in an orbital projection of DOS shown in figure 6.
Figure 6:(Left) Band structures calculated for the PZT-c (a), and PZT-a (b) models. (Right) Orbital projected DOS. PZT-c (top), PZT-a (bottom).
Charge gradient calculations reveal that the displacement of the metallic center from PZT-c to PZT-a causes a permanent charge gradient between ZrO6 and TiO5 to compensate for the breaking of the Ti10and O4 bond. This charge gradient, in the presence of new localized states provide the conditions for the trapping of electrons and holes, confirming the radiative recombination responsible for PL in amorphous PZT.
In a different experiment, M. Anicete-Santos et al.(2007) employed the same CRYSTAL98 package using the gradient corrected correlation functional and Becke exchange functional as M.S. Silva et al.. Though the XRD experiments were carried out using PbZr0.53Ti0.47O3, a composition of PbZr0.5Ti0.5O3 was assumed in order to facilitate calculations. The periodic model created represented the space group P4mm. Disorder in one type of periodic model was similar to M.S. Silva et al. in that the translation vector for Zr (as opposed to Ti) was (0.0 0.0 0.5)Å. To increase the amount of disorder in the system a third model was created where the Ti atom is displaced by a (0.0 0.5 0.0)Å vector, seen in figure 7. Geometry optimization was not performed on the PZT-Zr and PZT-Zr-Ti models to avoid including additional changes that would mask the effect of the atoms displacement.
Figure 7: Disordered structure where both Zr and Ti atoms are displaced by 0.5Å in z and y directions respectively (PZT-Zr-Ti). The other two models used, one ordered(PZT-c) and the other a single local disordered structure (PZT-Zr) are identical to the models represented in figure 3 with the exception that in this experiment the Zr atom is displaced in the single local disorder structure not Ti.
Figure 8 shows a table of Mulliken charges for each cluster of the PZT-c, PZT-Zr and PZT-Zr-Ti periodic models and their gap energies. From the data they were able to determine that the local hole is greater in the structure containing one local disorder (PZT-Zr) than in the case with two local disorders (PZT-Zr-Ti), suggesting that structures containing a lesser degree of structural disorder have greater local holes, making them more favorable for intense PL emission. Furthermore, the first principles calculations revealed that the localized electronic states in the band gap of the order-disordered PZT models were in agreement with the experimentally determined spectral dependence of PbZr0.53Ti0.47O3 powders on the degree of structural disorder in the lattice.
Figure 8: Mulliken charges for each of the PZT models and their gap energies.
Dipolar Defects and 180° Domain Walls in PbTiO3
One of the most important materials related problems with the use of ferroelectric meterials is fatigue. Fatigue in FE materials is indicated by a sharp decrease in the magnitude of switchable polarization after repeated polarization reversals. 180° domain walls are often present in a non-poled crystal, due to their presence reducing charge accumulation at the surface, thus lowering the free energy of the crystal. Previous experiments have shown that defects contribute to fatigue and imprint in FE materials, such as increased fatigue with elevated Ti content, due to enhanced oxygen vacancy concentration. It has also been shown that in films, oxygen vacancies are involved in voltage offsets that lead to imprint failure and that defect dipoles also play an important role in imprint. S. Pöykkö et al. have studied oxygen vacancies and domain walls (DW) in PbTiO3. They have chosen PbTiO3 because of its simple phase diagram.
They utilized plane-wave pseudopotential calculations based on DFT with LGA. They had 5d states of Pb, and 3s and 3p sates of Ti treated as valence electrons. A sufficient energy cutoff of 39Ry was determined for the structure. For the calculations of vacancy-impurity complexes, BZ sampling was done by a 2x2x1 Monkhorst-Pack k-point mesh for an 80 atom supercell. They had noticed the same effect of nearest neighbor defects as a polarization pinning center as M. Anicete-Santos et al.(2007) and S. Piskunov et al. (2007).
Domain walls were studied using three-dimensional periodic supercells. The (Rxax1ax1c) shaped supercells have two domain walls, and Rx takes on values of 6, 8 or 10. In order to reduce the effect of systematic errors the atomic structure for the both the bulk and domain cases for each supercell were optimized for size, BZ sampling and energy cutoff. Structures were optimized until the Hellman-Feynman forces acting on an atom were less than 0.003 eV/Å. The amount of FE lattice distortion near DWs were compared to distortion in the bulk using a parameter Pz, defined by the relation:
Where b1 and bs are the long and short Ti-O distances along the c-axis in the bulk, bave=(bs+b1)/2 and bs' is the short Ti-O bond at a distance d from a DW. The results show that a DW can be either Ti or Pb- centered, and that only Pb-centered DWs are stable while Ti-centered DWs are unstable towards a transition to Pb-centered. The domain walls studied were found to be extremely narrow and the DW energy density for PbTiO3 is calculated to be ~150erg/cm2. Figure 9 shows the domain wall structure indicating a polarization change at a 180° domain and a table of calculated domain wall energy densities.
Figure9: (Left) Polarization change at 180° domain. The numbers are Pz values calculated from the equation above, representing the magnitude of ferroelectric relaxation with respect to the bulk value. (Right) Calculated domain wall energy densities corresponding to the domain wall changes in the figure on the left. Rx is the number of primitive cells in the x-direction, Nat is the number of atoms in the supercell.
The studies presented above represent a variety of ab initio methods used to study specific defects in PZT materials and how those defects affect the domain wall and electronic structure. All of the theoretical studies were confirmed with actual experiments proving the accuracy of the calculations. It has been shown that ab initio studies are a useful tool for investigating specific properties of materials related to the atomic and electronic structure. This allows researchers to look more specifically at certain parameters and aids them in developing processing conditions of developing new materials.