CHINESE JOURNAL OF PHYSICS VOL. 1, NO. 3 June 213 Calculation and Analysis of the Dielectric Functions for BaTiO 3, PbTiO 3, and PbZrO 3 Chao Zhang and Dashu Yu School of Physics & Electronic Information Science, Tianjin Normal University, Tianjin 3389, China (Received December 1, 211; Revised August, 212) The pseudo-potential plane wave method and the generalized gradient approximation (GGA) have been used to calculate the electronic structures and dielectric functions of cubic BaTiO 3 (BT), PbTiO 3 (PT), and PbZrO 3 (PZ). The dielectric constant imaginary parts (ε 2 (ω)) of these three materials have been analyzed with the optical transition theory. Furthermore, the ε 2 (ω) peaks have been fitted by the Lorentz formula, and the fitting results were discussed using the dielectric theory. We hope the results will be helpful for the dielectric spectrum measurements for these three perovskite oxides. DOI: 1.6122/CJP.1.32 PACS numbers: 78.2.Bh I. INTRODUCTION In dielectric physics, the dielectric properties are closely related to the electronic structures of the material, such as the energy band structure and the density of states [1]. The first-principle methods by computer simulation can obtain the relationship between the dielectric characterization and the parameters of the microscopic description. Also, firstprinciple methods have proved to be generally quite accurate in the theoretical prediction of the ground state structure type and structural parameters of perovskite oxides [2]. In this paper, BaTiO 3 (BT), PbTiO 3 (PT), and PbZrO 3 (PZ) have been chosen, both because of their scientific and technological importance, and because they allow us to investigate the effects on the dielectric features by substituting one cation with the other atoms unchanged. In many papers on the dielectric properties using the first principle method, see, e.g., [3 ], the contributions of the ε 2 peaks are always discussed with the electronic structures, while the relationship between the character of the ε 2 curve (for example, the ε 2 peak number, the ε 2 peak values, etc.) and the electronic structures has been less studied. In this paper, the character of the dielectric functions of the three materials have been discussed with relation to the density of states and Mulliken bond populations. We hope the results may be helpful for the dielectric spectrum analysis for these three perovskite oxides. Electronic address: zhangchao81831224@163.com http://psroc.phys.ntu.edu.tw/cjp 32 c 213 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA
VOL. 1 CHAO ZHANG AND DASHU YU 33 II. METHOD OF CALCULATION The Cambridge serial total energy (CASTEP) code was chosen, which uses a plane wave expansion technology in reciprocal space. The generalized gradient approximation (GGA) of the Perdew Wang (PW91) scheme [6] was used for the exchange-correlation energy calculations. The presence of tightly-bound core electrons was represented by ultrasoft pseudo-potentials of the Vanderbilt type. A special k point sampling method proposed by Monkhorst-Pack was used for the energy integration in the first irreducible Brillouin zone. In the calculation, the kinetic energy cut-off value was selected as 38 ev, which was sufficient to obtain reliable results. The k point mesh was selected as 6 6 6. All atoms were relaxed to their equilibrium positions when the energy change on each atom between successive steps converged to 1 6 ev/atom, the forces on each atom converged to.1 ev/å, the stress on each atom converged to.2 GPa, and the displacement converged to 1 4 Å. The space group of BT, PT, and PZ in the cubic phase is PM-3M, with the lattice parameters 4.1 Å, 3.969 Å, and 4.161 Å [7 9], respectively. Their calculated band gaps were corrected using the experimental data [1] (BT: 3.3 ev, PT: 3.4 ev, PZ: 3.7 ev) by using the scissors operator. III. RESULTS AND DISCUSSION The equilibrium lattice constant calculated for BT, PT, and PZ are 4.23 Å, 3.97 Å, and 4.18 Å, the results are very close to the experiment lattice constant data mentioned. In Fig. 1, for BT and PT, there is strong hybridization between the Ti-d and O-p orbitals, which indicates that the Ti and O atoms form strong covalent bonds with each other. What s more, the Ba-p (or Pb-d) and O-s orbital have a weaker hybridization. The difference of the PDOS in BT and PT is that the Pb-s and O-p orbitals in PT have hybridization, but the Ba-s and O-p orbitals don t. For PZ, Zr, and O atoms form a covalent bond like the Ti-O bond in PT, and also there is hybridization between the Pb-s and O-s(or O-p) orbitals. The Mulliken bond populations have been calculated, the results are as follows: TABLE I: Calculated Mulliken bond populations, P (in e), for BT, PT, and PZ in their cubic phase. BT PT PZ P Ba/Pb O 1.12.3.18 P Ti/Zr O.26 1.11.98 P O O.28.13.9 According to the population values and the PDOS calculations, the Pb-O hybridiza-
34 CALCULATION AND ANALYSIS OF THE DIELECTRIC... VOL. 1 Density of States/(states ev -1 ) 1 1-2 -1 1 2 1-2 -1 1 2-2 -1 1 2 Pb-s Pb-p Pb-d Ti-s Ti-p Ti-d O-s O-p Pb-s Pb-p Pb-d Ti-s Ti-p Ti-d O-s O-p Ba-s Ba-p Ba-d Ti-s Ti-p Ti-d O-s O-p FIG. 1: The partial density of states (PDOS) of BT(a), PT(b), and PZ(c). tions in PT and PZ are both stronger than Ba-O s in BT. Pb-O hybridization leads to a larger strain in the crystal lattice, and makes PT more stable in the tetragonal phase than in the cubic phase. Apart from those, Pb-O hybridization in PZ is stronger than in PT, which promotes antiferroelectricty forming in PZ. The PDOS calculations and Mulliken bond population results are close to the article reports [11, 12]. The dielectric functions which can be obtained by experiments are determined and influenced by the PDOS and Mulliken bond populations, in other words, the electronic structure of the elements holds the key to understanding the relationship between the composition, atomic arrangement, and properties. When changing the ion constituent, the PDOS and Mulliken bond populations changes leading to a difference in the dielectric functions. The following part of the paper discussed how their corresponding PDOS and Mulliken bond populations determine and affect their dielectric functions. The real and the imaginary parts of the dielectric function are shown in Fig. 2 for BT, PT, and PZ in their cubic phase. In Fig. 2, there are normal and abnormal dispersions appearing in the dielectric functions, which is consistent with the dielectric theory [1]. So the calculations in the paper are reliable. Optical properties can be determined using the complex dielectric function as follows: ε(ω) = ε 1 (ω) + iε 2 (ω). (1) In Fig. 2, the positions of the ε 2 peaks are related to the density of states between the conduction and valence bands [1]. Combining the PDOS calculation results (Fig. 1), in
VOL. 1 CHAO ZHANG AND DASHU YU 3 1 1 1 BT PT PZ 2 1 1 O-p to Ti-d.7eV O-p to Pb-p.8eV O-p to Ti-d.44eV BT PT PZ - 4 8 12 16 4 8 12 16 (a) (b) FIG. 2: The real and the imaginary parts ε 2 (ω) of the dielectric functions of BT, PT, and PZ. the energy below 16 ev, the transitions have been taken into account from: (in BT): O-p Ti-d, (in PT): O-p Ti-d and O-p Pb-p, (in PZ): O-p Zr-d, O-p Pb-p, and Pb-s Pb-p. Furthermore, we can sure that the transitions related to the arrowed peaks of the ε 2 curves (in Fig. 2 (b)) have been taken into account from: (in BT) : O-p Ti-d, (in PT) : O-p Ti-d, (in PZ) : O-p Pb-p. In Fig. 2 (b), we checked the energy values of the arrowed peaks for BT, PT, and PZ. The values are.41 ev,.8 ev, and.72 ev, respectively, which shows that the transition in BT needs the least energy. This can be explained as follows: from Fig. 1, the band gap of BT is the smallest, so the transition will happen most easily, needing the least energy. Such a result implies that the quantity relations of the band gaps in BT, PT, PZ determine the quantity relations of the energies corresponding to the first peak in the ε 2 curves in the three materials. Apart from that, there are the most peaks in the ε 2 curve for PZ (six peaks) and the least peaks for BT (four peaks). This phenomenon may be explained as follows: Below 16eV, there are only transitions from the O-p to the Ti-d state in BT. In PT, the transitions from the O-p to the Pb-p state also exist besides the transitions from the O-p to the Ti-d state. For PZ, the PDOS curve for Pb-p is closer to the Fermi level than that of PT; as a result, more transitions from the O-p to the Pb-p state occur in the energy below 16eV. Apart from that, the transition from the Pb-s to the Pb-p state participates, so the number
36 CALCULATION AND ANALYSIS OF THE DIELECTRIC... VOL. 1 of peaks in the ε 2 curve for PZ is the most of all. This part of the results shows that the distribution of the densities of states in BT, PT, PZ lead to the difference of the ε 2 peak amount in these materials. Furthermore, we considered the formula [1] N(ω) = 2mε e 2 π ω ωε 2 (ω)dω, (2) where N is the density of electrons participating in the transition, m is the electronic mass, ε is the vacuum dielectric constant; we calculated N(ω) of BT, PT, and PZ, as shown in Fig. 3. 3.x1 29 2.x1 29 BT PT PZ 2.x1 29 N 1.x1 29 1.x1 29.x1 28. 4 8 12 16 FIG. 3: the density of electrons participating in the transition for the energy. In Fig. 3, the N value in PT is the largest in general, and the value in BT is the smallest. So there are the most electrons taking part in the transition, which indicates that the transition probability is the largest in PT. Because the value of ε 2 is proportional to the integration of the transition probability in the whole Brillouin zone [1], the value of ε 2 is the largest in PT, and the smallest in BT. This can be explained with the PDOS curves of the three materials, firstly, the transitions mentioned above all involve the O-p state in valence bands to another state. The O-p state density curve in PT spans the vastest energy range (the three material s quantities of energy range crossing were shown in Fig. 4). And the band gap of BT is small, and also there are the transitions of the O-p state to the Ti-d state and the Pb-p state. Comparing the same situations in BT and PZ, the transition probability in PT is the largest, leading to the largest value of the ε 2 curve of PT as a
VOL. 1 CHAO ZHANG AND DASHU YU 37 whole. The result says that the value of the ε 2 curve is influenced by the energy range which the state density curve spans. Density of States/(states ev -1 ) 1 1-1 1 2.64eV 6.1eV PZ 1-1 1 9.1eV PT 6.2eV -1 1 BT FIG. 4: The quantities of the energy range crossing of the O-p state density curve in the valence bands for the three materials (in the remarked field, the state density of the O-p state is not zero). According to the dielectric theory, the Lorentz theory can give the characterization of the dielectric function in the energy range of Fig. 3 [1, 13]. The imaginary part of the dielectric function is described by the formula as follows: ε 2 (ω) = i Nf i e 2 ε m γ i ω (ωi 2 ω2 ) 2 + γi 2, (3) ω2 where the subscript i stands for the electrons with the particular resonance frequency ω i ; f i is called the oscillator strength, it is a measure of relative probability of the transition; γ i /ω i reflects the loss of the energy in the transition. When the value is small, the ε 2 curve shows the resonance image, meaning a small energy loss, whereas the ε 2 curve turns into the Debye relaxation curve showing no abnormal dispersion and this means a large energy loss [14]. In the paper, the ε 2 curves have been fitted by multi-peak fitting for obtaining the parameters ω i, γ i. Because the peaks which are marked by arrows in Fig. 2 (b) only refer to the transitions from one kind of electron state to the other one, it is convenient for analysis and obtaining the obvious conclusion. The results of fitting the peak marked by arrows are
38 CALCULATION AND ANALYSIS OF THE DIELECTRIC... VOL. 1 as follows: (peak arrowed in BT): γ i = 2.47 1 14, ω i = 1.3 1 1, γ i /ω i =.191; (peak arrowed in PT): γ i = 2.2 1 14, ω i = 1.3 1 1, γ i /ω i =.167; (peak arrowed in PT): γ i = 2.41 1 14, ω i = 1.37 1 1, γ i /ω i =.17. Considered from the discussion above, the two peaks fitted from the transitions are between the O-p and the Ti-d state in BT and PT, and another peak is from the transition between the O-p and the Pb-p state in PZ. It can be known that: for BT and PT, the replacement of a Ba atom by a Pb atom slightly reduced the energy loss in the transition from the Op to the Ti-d state below 6eV, which may be due to the difference of the Ti-O coupling strength between Ti and O. This may be explained by means of the Mulliken bond populations results (PT: Ti-O 1.11; BT: Ti-O.26), thus the intensity of the Ti-O in PT is stronger than in BT. The electrons which belong to the Ti and O atom are bound around the Ti and O atom by the strong Ti-O bond. When the electrons get the energy for a transition, they are likely to move intensively from an O atom to a Ti atom, which can reduce the energy loss by the inter-collision with an electron of the Pb atoms. The result illustrates that the Mulliken bond populations value affects the energy loss of the transition. IV. CONCLUSION The PDOS and Mulliken bond populations have been calculated for BT, PT, and PZ, which shows that: the distributions of the PDOS are similar among the three; there is no obvious hybridization between the Ba-s and the O-p orbitals in BT, but hybridization exists between the Pb-s and the O-p orbitals in PT and PZ; and the coupling of Pb-O in PZ is stronger than in PT. The dielectric functions have been calculated, which indicates that: the calculations are in accordance with the dielectric theory with the appearance of normal and abnormal dispersions; there is an internal relation between the imaginary part ε 2 (ω) of the curve and the distribution of the PDOS curve and Mulliken bond populations, in other words, we can obtain much information about the electronic structure from the macroscopic ε 2 (ω) curve obtained from a dielectric experiment. We hope these results will be helpful for the dielectric measurement and analysis of these three perovskite oxides. References [1] R. C. Fang, Solid spectroscopy (University of Science and Technology of China Press, Hefei, 23) p. 1 98 (in Chinese). [2] D. Vanderbilt, Curr. Opin. Solid State Mater. Sci. 2, 71 (1997). [3] H. Salehi, Chinese J. Phys. 48, 829 (21). [4] J. Baedi, S. M. Hosseini, A. Kompany, and E. Attaran Kakhki, Phys. Stat. Sol. (b) 24, 272 (28).
VOL. 1 CHAO ZHANG AND DASHU YU 39 [] M. Q. Cai, Z. Yin, and M. S. Zhang, Appl. Phys. Lett. 83, 28 (23). [6] J. P. Perdew and Y. Wang, Phys. Rev. B 4, 13244 (1992). [7] Z. X. Chen, Y. Chen, and Y. S. Jiang, J. Phys. Chem. B 1, 766 (21). [8] T. Mitsui et al., Landolt-Bornstein Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1981) NS, III/16. [9] S. Aoyagi, Y. Kuroiwa, A. Sawada, H. Tanaka, J. Harada, et al., J. Phys. Soc. Jpn. 71, 233 (22). [1] P. W. Peacocka and J. Robertson, J. Appl. Phys. 92, 4712 (22). [11] P. Ghosez, E. Cockayne, U. V. Waghmare, and K. M. Rabe, Phys. Rev. B 6, 836 (1999). [12] S. Piskunov, A. Gopeyenko, E. A. Kotomin, Yu. F. Zhukovskii, and D. E. Ellis, Comput. Mater. Sci 41, 19 (27). [13] F. Wooten, Optical properties of solids (Academic Press, New York, 1972) p 1 182. [14] A. K. Jonscher, Dielectric Relaxation in Solids (Xi an Jiaotong University Press, Xi an, 28) p 1 12.