Issue 
Eur. Phys. J. Appl. Phys.
Volume 88, Number 1, October 2019



Article Number  10501  
Number of page(s)  6  
Section  Photonics  
DOI  https://doi.org/10.1051/epjap/2019190178  
Published online  17 December 2019 
https://doi.org/10.1051/epjap/2019190178
Regular Article
Band gap modulation effect on electronic and optical properties in PbTiO_{3} under stress: a DFT study
^{1}
Department of Physics, University of Gujrat, Gujrat 50700, Pakistan
^{2}
School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, P.R. China
^{3}
Department of Physics, Government College Women University, Sialkot, Pakistan
^{4}
Department of Basic Sciences and Humanities, University of Engineering and Technology Lahore (Narowal Campus), Lahore, Pakistan
^{5}
Department of Physics, Government College University Lahore, Lahore 54000, Pakistan
^{6}
Department of Physics, University of Education DG Khan Campus, DG Khan, Pakistan
^{7}
CHEP, University of the Punjab, Punjab 54590, Pakistan
^{*} email: dr.rizwan@uog.edu.pk
Received:
28
May
2019
Received in final form:
19
September
2019
Accepted:
14
October
2019
Published online: 17 December 2019
First principles calculation was implemented to explore the effect of band gap modulation under stress in cubic PbTiO_{3} by using generalized gradient approximation (GGA) and ultrasoft pseudopotential (USP). The band gaps at different external pressures show indirect band nature with significant reduction in band gapes. The density of states indicated that the upper valence states were influenced by O2p and in the meantime Tid states have main contribution in conduction band. The inclusion of external stress not only changes electronic structure but also significantly changes the optical properties such as complex dielectric function, absorption, energy loss function, refractive index and reflectivity. The shifting of absorption edges toward higher energies confirm the presence of blue shift and make this material an appealing candidate for optoelectronic devices.
© EDP Sciences, 2019
1 Introduction
The structure of perovskite which is ferroelectric in nature contain the generic formula ABO_{3}, where A and B are two cations, A is a univalent or bivalent with larger radius, B is the metal with the valency of four or five and its radius is smaller than A cation and O is oxygen. First principles calculation [1] gives the value of spontaneous polarization [88 C/cm^{2}] and value obtained from experiment is [75 C/cm^{2}] [2,3]. Lead titanate founds in different phases such as paraelectric and ferroelectric. In paraelectric phase, it exhibits the cubic structure and tetragonal structure in ferroelectric phase. It has been recognized that at critical temperature of 766 K, it transform from cubic to tetragonal structure. Detail discussion on its electronic structure has been done by various researchers [4–7]. Some possible energy ranges are occupied and some are not occupied by electrons. Information about these energy ranges can be obtained from electronic band structure. If we take the difference of valence band maximum (VBM) and the conduction band minimum (CBM) then it gives the electronic band gap in the semiconductors and insulators. Electronic band structure computations give detail of the feasible electronic transitions from valance band maximum (VBM) to conduction band maximum (CBM).
At extreme pressure, structural factors are needed. In the previous studies, the pressure range of 0.0–4.0 GPa was taken and Xray diffraction was used for the investigation of crystal structure of PbTiO_{3} [8]. PT material exhibits many applications as actuators, medical imaging, transducers, acoustic sensors and many more [9,10]. The electronic basis of ferroelectricity has been understood by the detailed study [11–15] of lead titanate. The ferroelectric distortion increased just because of the coupling of Ti–O and Pb–O but it cause to diminish the repulsions of short range. In the past studies it has been investigated that ferroelectricity is repressed by pressure and repulsion of short range are supported by constriction [13]. From last many years, different materials of the perovskite type was an interesting topic for the researchers but electronic properties of these materials are not described clearly up to now. The Local density approximation and different techniques such as fullpotential linearized augmentedplanewave was used by many authors [16–19]. PerdewBurkeErzernhof, generalized gradient approximation (GGA) computations were done for different perovskite oxide and local spin density approximation consequences were described by the Tinte and Stachiotti [20]. Waghhmare, Ghosez and workmates explained the atomic and electronic properties of PTO [21,14]. To examine the ferroelectricity in PbTiO_{3} and BaTiO_{3}, the FPLAPW technique was utilized as well as was given by Cohen and Krakauer [22]. At pressure of 11.0 GPa, the transformation from paraelectric to ferroelectric phase is noticed at room temperature [23] and it was described latterly the decrease of Curie temperature due to the influence of large pressure. By the deviation of pressure, the changes in structure were observed at the constant temperature.
In the present work, theoretical model of the influence of pressure on the optical and electronic properties are studied. Many publications have been done on the ferroelectricity, phase transition and doping in PbTiO_{3} but no proper study has been done on the outcomes of pressure on the optical and electronic properties in lead titanate up to now. The features of density of state (DOS), energy band gap and lattice parameters are published by many researchers [24]. On the basis of Density Functional theory (DFT), the electronic and optical properties of PbTiO_{3} (Pm3m) are examined under the influence of pressure (0.0–24.0 GPa). The paper is arranged in following manners: In segment 2 some technical explanation is given for our computation. In Section 3.1 we describe the electronic properties and in Section 3.2; optical properties of material are described. In Section 4, the conclusion is given.
2 Computational detail
At high temperatures, lead titanite exhibits a cubic perovskite structure with spce group Pm3m. It contain two phases, in phaseI (paraelectric) it has cubic structure with space group Pm3m as well as tetragonal structure in phaseII (ferroelectric) with space group P4mm in which Pb atom sits at edges, at body centered Ti is placed and oxygen at face centered and axial ratio of c/a 1.063 is possessed by the lead titanate [25]. In our computations, stress is applied from 0.0 to 24.0 GPa for a 2 × 2 × 1 supercell as shown in Figure 1. The calculations are done with the help of DFT with generalized gradient approximation. These calculations involve PBE exchange relationship parameterization. With this approach, calculations are done with improved efficiency. Bonding of the atoms was supposed to be done by the valance electrons not by the bonded electrons which are closer to the nucleus, here the electrons are iced up on nucleus, and this bonding process is done by the pseudo potential perspective. The electronic configuration of lead and titanium are 4f^{14} 5d ^{10} 6s ^{2} 6p ^{2} and 3d ^{2} 4s ^{2} respectively. The representation of electron ion interaction by ultrasoft pseudo potential as well as for the plane wave basis set, the achieved cutoff energy for our computation is 489.8 eV. MonkhorstPack kpoint grid of 2 × 2 × 1 for the purpose of Brillouin zone integration is used. Moreover, the structure which is under study needs its geometry to be optimized. During this process, the atoms possess the residual forces is less than 0.001 V/Å. Displacement of the atom during geometry optimization is less than 0.00001 Å.
Fig. 1 Super cell of lead titanate (PbTiO_{3}). 
3 Result and discussion
3.1 Electronic properties
3.1.1 Band structure
The electronic band structure of PbTiO_{3} in ideal cubic phase along the symmetry lines of the Brillouin zone under the stress of 0.0 GPa is shown in Figure 2a whereas different lattice constants of lead titanate at various pressures are given in Table 1. After geometry optimization, the optical and electronic properties are calculated. On the energy scale, the peak of valance band was positioned on zero on given energy ranges. There are nine valence bands in the region of energy that lies from −7.0 eV to 0.0 eV underneath the Fermi level and they are generally formed from coupling between Ti3d and O2p orbitals which will be discussed in later subsection.
Mixing of the bands occurs and also they are very near to each other. The Ti3dt_{2g} triplet orbital have main contribution in the formation of few bands of conduction band which lies above the Fermi level. Pb6s orbital plays an important role in the formation of the top of valance band and valance band has significant contribution in the band structure. In the formation of valance band, O2p also takes part and it is also the possibility that valance band is due to the coupling of these two orbitals. An obtained band gap is indirect with the value of 1.666 eV. The maxima of valance band lies at Xdirection and minima lies at Gdirection. The previously calculated values for direct band gap (Γ–Γ) of cubic PbTiO_{3} are 2.65 eV and for indirect band gap (X–Γ) is 1.63 eV [24]. These band gaps were calculated by using PerdewBurkeErnzerhof PBE exchange relationship parameterization. It is truly noted that phonons are responsible for the transformation of electron from valance band to conduction band in the indirect band gap and these transformations cause the loss of energy. This loss of energy influences the capability of optoelectronic materials based on PbTiO_{3}.
As we increased the pressure from 0.0 GPa to 6.0 GPa, the value of band gap reduce from 1.666 eV to 1.643 eV. It means there is inverse relation between pressure and band gap. In the case of 6.0 GPa, the band gap is indirect because maxima lies at X point and minima lie at G point and it is clearly described in Figure 2b. It can be noticed clearly from the Figure 2b that as pressure increases, the valance band shifted downward. Further, at 15 GPa (Fig. 2c) and 24 GPa (Fig. 2d), obtained values for the band gaps are 1.576 eV, 1.511 eV respectively. In all the above cases, we obtained the indirect band gaps.
Fig. 2 Band structures of PbTiO_{3} at different ranges of the pressure. 
Different cell volumes and lattice constants of PbTiO_{3} at various pressures.
3.1.2 Density of state
Density of state basically explains the electron distribution in energy spectrum. The density of state of lead titanate is shown in Figure 3a. At Fermi level, this zero energy is fixed at the top of the valence band. The two states by which the lower valance band is formed are O2s and Pb5d states. At 0.0 GPa, the formation of the upper valence band is predominantly due to the O2p and Pb6s states. The range of upper valance band is from −10.0 eV to 0.0 eV. The Ti3d states are responsible for the formation of the conduction band between the energy range of 0.0 eV to 5.0 eV and Pb6p orbitals plays an important role in the formation of conduction band between to energy ranges from 5.0 eV to 10.0 eV. It can be clearly seen from graphs that the Ti3d and O2p orbital peaks are approximately equivalent, these two orbitals are coupled with each other in definite manners. So these sates have almost equal contribution in TDOS. Moreover, due to Pb5p, the conduction band (CB) has little participation and this due to the reason of coupling between Pb5p and O2p.
In partial density of state pstates are sharper than sstate and dstates. So in the valance band there is a main contribution of p states whereas in conduction band, dstates are sharper than sstates and pstates, it means that dstates have main contribution in the conduction band.
Elemental partial densities of states (PDOS) at different pressures (0.0–24.0 GPa) are described in Figure 3b–d. These graphs show the major difference between the conduction band and valance band after applying the pressure. It is cleared from our results that for the PDOS of Pb, the top of the valance band have main contribution from sstates, while in the conduction band; the pstates are dominant. The effect of pressure is mainly on the lead. Further if we precede the PDOS of titanium (Ti), it can be easily examined that peaks of the dstates are sharper than the others. In the case of oxygen (O), we obtained the sharp peaks of pstates in both valance and conduction bands. In the case of oxygen, the conduction band has small contribution.
Now we describe the effect of pressure on the TDOS. Figure 3e describes that top of valance band is from −10.0 eV to 0.0 eV due to the effect of pressure and conduction band peaks are from 2.0 eV to 8.0 eV. The valance band shifted downward due to the pstates of conduction band in lead due to the effect of pressure. The upper valance band and conduction band are not at same points so we obtained the indirect band gap again. We concluded from our results that band gap decreases as pressure increase and secondly, there is a shifting of Fermi level in the downward direction.
Fig. 3 (a) PDOS of PbTiO_{3}. (b) PDOS of Pb. (c) PDOS of Ti. (d) PDOS of O. (e) TDOS of PbTiO_{3}. 
3.2 Optical properties
Lead titanate belongs to the family of electrooptic devices; hence we explored the impact of stress in PbTiO_{3} compound. Optical properties, like energy loss function, refractive index, dielectric function, absorption coefficient and reflectivity, explains how light interacts with matter. These optical properties are computed for cubic phase PbTiO_{3} under different pressures (0.0 GPa, 6.0 GPa, 15.0 GPa, 24.0 GPa) as depicted in Figure 4a–f. These properties are interlinked with each other and these are dependent on frequency. If we calculate the complex dielectric function from Maxwell's equations, then we can derive all other properties.
The real (ε_{1}) and imaginary (ε_{2}) part of complex dielectric function are given in Figure 4a, b. Real part of dielectric function describes the polarization and imaginary part give information about how much energy is wasted in the system as well as describes the absorption. From our computed results, the imaginary part of epsilon ε_{2}(ω) gives two major peaks at approximately 3.0 eV and 8.0 eV and other peaks are about 12.0 eV and 38.0 eV. The interband transition to conduction band from valance band is associated with these points. The basis of the various peaks in dielectric functions is from interband transitions. The peak of ε_{2}(ω) at energy level of 3.0 eV is responsible for the conversion from the 6s to 6p state of Pb. The static dielectric function ε_{1}(ω) is obtained by the ε_{1} of dielectric function and it can be seen in Figure 4a and its value is 8.9. Further, at higher pressures (6.0 GPa, 15.0 GPa, 24.0 GPa), different graphs are obtained in dielectric function.
The spectrum of absorption for lead titanate is given in Figure 4c. It revealed that significant absorption is possessed by this compound in the ultraviolet region of high energy spectrum. The threshold point in the absorption is a particular energy which is absorbed fastly by material in the electromagnetic radiations. This point is appears at approximately 9.0 eV. The level of absorption increases as soon as energy increases. At 0.0 GPa, absorption edge sets at 0.0 eV. As we increase the pressure from 0.0 GPa to 24.0 GPa, the absorption edge shifted toward the higher energies, peaks becomes sharper and peaks are shifted toward the right it mean that blue shift appears in the absorption spectrum. The maximum I(ω) is around 38.50 eV for PbTiO_{3}. The optical transitions between 2p state of oxygen and 3d states of titanium helps to describe the maximum absorption. The nature of absorption of this material is in correspondence with the previous studies [28].
Figure 4d gives the information about the electron energy loss function L(ω) which explains typical plasmonic oscillations. The macroscopic and microscopic properties of the compound are explained by the fast electrons that traverses the compound. In this process, energy loss appears and this whole process is directly associated with electron energy loss function L(ω). In energy loss function spectrum, the most significant peak that founds is plasmon peak. These peaks and irregular peaks of R(ω) are related with each other. The volume plasmon (ℏωp) is taken by the energy of major maximum and is equal to 25 eV.
In term of energy, the optical reflectivity of PbTiO_{3} was computed and is shown in Figure 4e. The major peak about 9.0 eV was observed. This spectrum is owing to the involvement between p state of oxygen and d state of titanium in the valance and conduction band respectively. At 0.0 GPa, the higher peaks of reflectivity are observed. But with the increase of pressure up to 24.0 GPa, the reflectivity goes to decrease. Moreover, the R(ω) is initiated to be lesser than 10% in energy ranges 11.0–17.7 eV and 28.0–33.5 eV for PbTiO_{3}. Our studied compound is extremely transparent in low energy regions for example in ultraviolet, visible and infrared region present in energy spectrum in these given energy regions. At about 9.0 eV, the reflectivity goes to its high value of 97.3% significantly. This material exhibits best reflecting characteristics at these energies. Absorption is inversely proportional to reflectivity. It can be clearly seen from the graphs that where the absorption edge is at 0.0 eV, the reflectivity has some value.
The complex refractive index contains two parts; refractive index (n) and extinction coefficient, these both depend on energy. At zero photon energy, the highest value of refractive index is related to the smaller absorption energy. The refractive index goes on decreasing as the absorption goes to its higher values which can be easily seen in Figure 4f. The absorption spectrum and extinction coefficient also correlates with each other and it gives the information about the annihilation of energy present in system when the light having particular frequency falls on it. So it measures how much light absorbed in PbTiO_{3}. The extinction coefficient has zero value at energy of approximately 1.6 eV which shows that band gap is indirect. It means process of annihilation of energy in the material up to approximately 1.6 eV does not occur. Due to this reason absorption was also zero. So there is direct relation between absorption and extinction coefficient. As the extinction coefficient peaks become higher the absorption also increases and then with the decrease of extinction coefficient, there is also decrease in absorption. The computed extinction coefficient and refractive index are plotted in Figure 4f. The features of n(ω) spectrum are related to ε_{1}(ω) spectrum. The value of the static refractive index obtained is n_{a} (0) = = = 2.93 and previously calculated refractive index is 2.83 [29]. The refractive index starts to rise to approach highest point of 3.8 at the energy of approximately 4.0 eV. The local maxima value of extinction coefficient associates to the zero value of real part of dielectric function. It results in the value of 2.42 for extinction coefficient at around 8.0 eV.
Fig. 4 (a) Real part of dielectric function. (b) Imaginary part of complex dielectric function. (c) The spectrum of absorption. (d) Energy loss function. (e) Reflectivity. (f) Refrective index and extinction coefficient. 
4 Conclusion
The density functional theory with generalized gradient approximation and PBE functional is used for the calculations of the optical and electronic properties of lead titanate under application of stress. The obtained band gap is 1.511 eV at 24.0 GPa in which the maxima of valance band lies at X point and minima of conduction band lies at G point. Moreover, valance band shifts downward which shows that material becomes ptype semiconductor. It is predicted that increase of pressure results a noticeable reduction in cubic PbTiO_{3} band gap. The computed band gaps are in inverse relation with the pressure. The density of states revealed that 2p states of oxygen are responsible for the formation of upper valance band and d states of titanium cause to form conduction band. To explain the band structure in detail, the TDOS and PDOS graphs at different pressures (0.0–24.0 GPa) are given. The results of the optical properties were also discussed in detail in cubic phase (Pm3m) which shows the shifting of absorption peaks toward higher energies. The maximum absorption (I(ω)) is around 38.50 eV. The calculations show a static dielectric function 8.9, energy loss spectrum of 25 eV and a refractive index of 2.93. Thus, this material is suitable for the use in electrooptic devices.
Author contribution statement
Muhammad Rizwan and Tariq Mahmood conceived of the presented idea. Rabia Bibi performed the computational work as a student. Imran Aslam and Zahid Usman verified the analytical methods. Hai Bao Jin and Chuan Bao Cao encouraged the authors to investigate the effect of stress on subjected properties of PbTiO_{3} and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.
Acknowledgments
The research supported by Higher Education Commission of Pakistan (HEC) (Grant No. 1541). This work is partially performed under Humboldt postdoctoral research fellowship.
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Cite this article as: Muhammad Rizwan, Rabia Bibi, Tariq Mahmood, Imran Aslam, Syed Sajid Ali Gillani, Hai Boa Jin, Chuan Bao Cao, Zahid Usman, A. Maqsood, Band gap modulation effect on electronic and optical properties in PbTiO_{3} under stress: a DFT study, Eur. Phys. J. Appl. Phys. 88, 10501 (2019)
All Tables
All Figures
Fig. 1 Super cell of lead titanate (PbTiO_{3}). 

In the text 
Fig. 2 Band structures of PbTiO_{3} at different ranges of the pressure. 

In the text 
Fig. 3 (a) PDOS of PbTiO_{3}. (b) PDOS of Pb. (c) PDOS of Ti. (d) PDOS of O. (e) TDOS of PbTiO_{3}. 

In the text 
Fig. 4 (a) Real part of dielectric function. (b) Imaginary part of complex dielectric function. (c) The spectrum of absorption. (d) Energy loss function. (e) Reflectivity. (f) Refrective index and extinction coefficient. 

In the text 
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