1. Introduction

ABO$_{3}$-type perovskite crystals have been widely used in high capacity computer memory cells, waveguides, laser frequency doubling and electro-optics, etc. One of the most popular examples is lead titanate (PbTiO$_{3})$. Due to its piezoelectric and dielectric properties, it can be used as a material to make non-volatile memories, IR sensors and actuation devices^[1]. It needs to be pointed out that lead titanate shows semiconductor characteristics in the crystalline form with a band gap of 3.0-4.5 eV^{[2, 3]}.

Lead titanate has attracted much attention^[3]. It has been proved that hydrogen forms a shallow donor in PbTiO$_{3}$ once H is implanted via an equilibrium process^[4]. The effects of different Ca concentrations on the structure and dielectric characteristics of PbTiO$_{3}$ polycrystals have also been investigated^[5]. Van Minh et al. prepared PbTi$_{1-x}$Fe$_{x}$O$_{3}$ ceramics by partly substituting the Ti site by Fe ions using a solid state reaction method; its structural, optical and magnetic properties were studied and it was shown that Fe-doping is the cause of ferromagnetism in PbTiO$_{3}$ perovskite^[6]. However, to date, the p-type conductivity of nonmetallic element doping PbTiO$_{3}$ has not been studied systematically. In this paper, we study the electronic structures of intrinsic and 5 at. % N-doped PbTiO$_{3}$ based on first-principles calculations, and the oxygen vacancy (V$_{\rm O})$ in N-doped PbTiO$_{3}$ is also discussed. In addition, as a comparison, the resistivity, carrier densities and mobility of pure, N-doped and N-V$_{\rm O}$ co-doped PbTiO$_{3}$ are studied through experiments.

2. Calculation details

First-principles calculations based on density functional theory (DFT), were performed with the CASTEP program, using the generalized gradient approximation (GGA) and the plane-wave pseudopotentials^[7]. The Perdew-Burke-Ernzerhof (PBE) electron-electron exchange and correlation effects were used in the simulation^[8]. In order to improve calculating efficiency, ultrasoft pseudopotentials were utilized for geometry optimization. In our calculations, a 3 $\times$ 3 $\times$ 3 k-point Monkhorst-Pack mesh in the Brillouin zone was used, and self-consistent calculations were carried out with a convergence criterion of 5.0 $\times$ 10$^{-6}$ eV/atom, the cutoff energy was 380 eV. All atoms were allowed to relax until the force on each atom was below 0.1 eV/nm and the displacement of each atom was below 5.0 $\times$ 10$^{-5}$ nm. In addition, for N-doped PbTiO$_{3}$ and N-doped PbTiO$_{3}$ with an oxygen vacancy, spin-polarized DFT (SP-DFT) calculations were used^[9]. The valence-electron configurations were 5d$^{10}$6s$^{2}$6p$^{2}$ for Pb, 3s$^{2}$3p$^{6}$3d$^{2}$4s$^{2}$ for Ti, 2s$^{2}$2p$^{4}$ for O and 2s$^{2}$2p$^{3}$ for N, respectively.

Paraelectric~phase PbTiO$_{3}$ has a cubic structure with the space group of Pm3m, the Pb and Ti cations are arranged on a simple cubic lattice, and the O ions lie on the face centers nearest to the Ti cations. The Ti cations are at the centers of the O octahedral, while the Pb cations lie at the larger 12-fold coordinated sites. The lattice parameters are $a = b = c =$ 0.398 nm, and the volume is 0.06163 nm$^{3}$ in the experiment^[10]. A 2 $\times$ 2 $\times$ 2 supercell of pure PbTiO$_{3}$ was used as the computational model (Figure 1), then two oxygen atoms were replaced by two nitrogen atoms to simulate 5 at. % N-doped PbTiO$_{3}$. In addition, an oxygen atom was removed from the N-doped PbTiO$_{3}$ to simulate the oxygen vacancy (the N$_{\rm O}$-V$_{\rm O}$ complex).

3. Calculation results and discussion

3.1 Intrinsic PbTiO$_{3}$

The Broyden-Fletcher-Goldfarb-Shanno minimization algorithm was used to conduct geometry optimization. The equilibrium structure was obtained after the cell geometry and volume were fully relaxed by minimizing the total energy and forces. The optimized geometry parameters $a,$ $b$ and $c$ are 0.3973 nm, which is nearly equal to the experimental values (0.3969 nm)^[11]. The volume of every unit cell is 0.062728 nm$^{3}$. The formation enthalpy is a key quantity to determine the crystal structural stability. Thus, the formation enthalpies of the PbTiO$_{3}$ with and without N impurities are calculated using Equation (1):

$\begin{split} \label{eq1} \Delta H_{\rm f} [{\rm PbTiO}_{(24-x)/8} {\rm N}_{x/8}] = {}& \frac{1}{8}[E_{\rm total} ({\rm Pb}_8 {\rm Ti}_8 {\rm O}_{24-x} {\rm N}_x ) \\& -8E_{\rm Pb} -8E_{\rm Ti} -(24-x)E_{\rm O} -xE_{\rm N}], \end{split}$

(1)

where $x$ is the number of N impurities in the supercell, $\Delta H_{\rm f} [{\rm PbTiO}_{(24-x)/8} {\rm N}_{x/8}]$ is the formation enthalpy of the PbTiO$_{3}$ with N impurity, $E_{\rm total} ({\rm Pb}_8 {\rm Ti}_8 {\rm O}_{24-x} {\rm N}_x )$ is the total energy of a supercell with N impurities, and $E_{\rm Pb}$ and $E_{\rm Ti}$ are calculated from the energies of the bulk Pb and Ti crystal, respectively. $E_{\rm O}$ and $E_{\rm N}$ are the energies of the O and N atoms taken from the energies of molecular O$_{2}$ and N$_{2}$, respectively. Calculated using Equation (1), the value of the formation enthalpy of the intrinsic PbTiO$_{3}$ is $-13.19$ eV, this result is smaller than that of other compounds calculated under the same conditions, so we can certify that the intrinsic PbTiO$_{3}$ is thermodynamically stable^[12].

Figure 2 shows contour plots of the charge density difference of intrinsic PbTiO$_{3}$. The charge density difference distributions are used to analyze the bonding characteristics in the crystals^[13]. The charge distribution is in the middle of the Ti and O atoms, but the charges near Ti are fewer than those near O, so the Ti-O bonds primarily show covalence characteristics with fewer ionic characteristics. The charge densities distribute spherically around Pb and O, so Pb-O bonds show ionic characteristics, which is in accordance with Reference~^[14].

The band structure of the intrinsic PbTiO$_{3}$ is shown in Figure 3(a). The Fermi energy level is set to 0 eV, the calculated band gap is 1.67 eV. The result is consistent with the value calculated by Lv et al. using different DFT functions and various plane-wave pseudopotential methods^[15], the value of the calculated band gap above is smaller than the value of the experimental band gap (3.4 eV)^[16]. This is because DFT is based on ground state theory and it underestimates the exchange-correlation potential of the excited electrons, but it should not affect the study of electronic structures under the same computing conditions^[7]. In addition, both the conduction band minimum (CBM) and the valence band maximum (VBM) are located at the same Gamma point in the Brillouin zone, this would imply that the intrinsic PbTiO$_{3}$ is a direct-gap semiconductor.

There is a close relationship between the conductivity and the effective mass. The hole effective mass $m_{\rm h}^*$ is expressed by the following equation^[17]:

$m_{\rm h}^\ast =\frac{h^2}{4\pi ^2}\left( {\frac{{\rm d}^2E}{{\rm d}k^2}} \right)^{-1},$

(2)

where $h$ represents the Planck constant, and $\frac{{\rm d}^2E}{{\rm d}k^2}$ represents the second derivative of the top of the valance bands. The effective mass of intrinsic PbTiO$_{3}$ is $-1.42m_{\rm e}$.

Figure 3(b) shows the total density of states (TDOS) and partial density of states (PDOS) of intrinsic PbTiO$_{3}$. It indicates that the valence bands of pure PbTiO$_{3}$ are fully occupied by electrons and are mainly attributed to the O 2p state, partially hybridized with Pb 6s and Ti 3d states. In the calculations, the number of empty bands is 12. The conduction bands from 1.5 to 3.5 eV contain Pb 6p and Ti 3d states, the width of the conduction bands is 2 eV, which is limited by the number of empty bands selected in the calculations.

3.2 N-doped PbTiO$_{3}$

The optimized geometry parameters of N-doped PbTiO$_{3}$ are $a =$ 0.4065 nm, $b =$ 0.3964 nm, $c =$ 0.3964 nm. The volume is 0.063885 nm$^{3}$, which is larger than that of the intrinsic bulk, this is because the ionic radius of N$^{3- }$ (0.146 nm) is larger than the radius of O$^{2- }$ (0.132 nm)^[18]. $\Delta H_{\rm f}$ for N-doped PbTiO$_{3}$ is -11.16 eV, indicating that the N-doped PbTiO$_{3}$ is also thermodynamically stable.

The electron density difference shown in Figure 4 indicates that the charge density mainly distributes near the N atoms. Between Ti and N, the charge density, which is near nitrogen, is increased in comparison with the charge densities near oxygen between Ti and O, but the interaction of charge density between the N atom and Ti atom is weakened. Therefore, the ionic characteristics of the Ti-N bonds become weakened as compared with those of the Ti-O bonds. This results from the fact that the electronegativity difference between N (3.04) and Ti (1.54) atoms is less than that between O (3.44) and Ti (1.54) atoms.

Figure 5(a) is the calculated band structure of N-doped PbTiO$_{3}$, it shows that the N-doped PbTiO$_{3}$ system is spin polarized, which is caused by the unpaired electrons in N at the Fermi level. The band gap is 1.20 eV, which is 0.47 eV smaller than that of intrinsic PbTiO$_{3}$. The narrowing trend of the band gap is the same as those of N-doped TiO$_{2}$^[19]. There are impurity states (N 2p) appearing on the upper edge of the valence bands and partially mixing with the intrinsic valance bands. The impurity states caused by spin-down bands above the Fermi energy level represent the acceptor levels, it is confirmed that the N-doped PbTiO$_{3}$ is a typical p-type semiconductor. From Equation (2), we obtain the value of the hole effective mass $m_{\rm h}^*$ of N-doped PbTiO$_{3}$, it is -1.63$m_{\rm e}$. Reference ^[20] states that if, $\vert m_{\rm h}^* \vert > 2 m_{\rm e}$, the acceptor levels form heavy holes and show poor p-type semiconducting characteristics. While, if $\vert m_{\rm h}^* \vert < 2 m_{\rm e}$, the acceptor levels form light holes and the p-type semiconducting characteristics are good. The results reveal that the p-type conductivity of 5 at. % N-doped PbTiO$_{3}$ may be excellent.

The calculated spin-polarized TDOS and PDOS of N-doped PbTiO$_{3}$ are plotted in Figures 5(b)-5(f). There are three conclusions: (1) the spin-down states of Pb 6s, Pb 6p, Ti 3d, O 2p and N 2p have contributions to the acceptor states, (2) there are spin polarized phenomena near the Fermi energy, and (3) the N 2p is the major contributor to the acceptor states. The third conclusion is similar to that for N-doped TiO$_{2}$, ^[21].

From the spin-down TDOS of N-doped PbTiO$_{3}$, it can be seen that the acceptor states are in the top of the valance bands. As we know, there is a positive correlation between the values of the hole carrier density and the relative hole number. For the sake of simplicity, we define $p$ as the relative hole number in the N-doped PbTiO$_{3}$. As is shown in the TDOS of Figure 5(b), $p$ is determined by the area under the curve of the spin-down DOS, which is between the Fermi energy level and the top of the valance band. By integrating based on Origin 8.0, $p$ $=$ 2.02 is obtained.

3.3 The N-doped PbTiO$_{3}$ with an oxygen vacancy

Due to the charge imbalance in the doped crystal, it is likely to form an oxygen vacancy in N-doped PbTiO$_{3}$. When an oxygen vacancy occurs, the value of the lattice parameters are $a =$ 0.4088 nm, $b =$ 0.3971 nm, $c =$ 0.3969 nm. The unit cell volume is 0.064434 nm$^{3}$, larger than that of N-doped PbTiO$_{3}$. The value of $\Delta H_{\rm f}$ is -10.96 eV, indicating that the N-doped PbTiO$_{3}$ with oxygen vacancy is thermodynamically stable.

The electron density difference of the atoms around the oxygen vacancy (Figure 6) shows that there is a charge accumulation near Ti, this indicates that only parts of the electrons are transferred from Ti to the surrounding atoms and there is no obvious impact on the other bonds in N-doped PbTiO$_{3}$.

The band structure and density of states of N-doped PbTiO$_{3}$ with an oxygen vacancy are shown in Figures 7(a)-7(f). There is no spin-polarized phenomenon for this solid~system. The conduction bands move to low energy levels and the value of the band gap is reduced to 1.18 eV. The shrinking of the band gap is mainly derived from the Urbach band tail caused by the many-body effect^[22]. There are almost no acceptors near the top of the valance bands, this illustrates that the p-type conductivity is fully compensated by the oxygen vacancy. The effective mass is -1.63$m_{\rm e}$. The shift of Ti 3d states leads to the conduction band moving to the low energy level. Although the N 2p states make a contribution to the hole levels, the hole levels are fully compensated by the oxygen vacancy.

4. Experiment and results

PbTiO$_{3}$ thin films were prepared using RF-magnetron sputtering (JGP-450H). The substrates were (100)-oriented MgO and they were heated to 550 C. The vacuum chamber was pumped down to 3 $\times$ 10$^{-4}$ Pa before introducing the sputtering gas Ar, the reactive gas O$_{2}$ and N$_{2}$. The flow rates of these gases were 40 sccm, 4 sccm and 2 sccm, respectively. The sputtering pressure was maintained at 0.5 Pa, the mixed gases Ar $+$ O$_{2}$, Ar $+$ O$_{2}$ $+$ N$_{2}$ and Ar $+$ N$_{2}$ were used for the deposition of intrinsic PbTiO$_{3}$, N-doped PbTiO$_{3}$ and N-doped PbTiO$_{3}$ with an oxygen vacancy, respectively. The RF power density was 2.8 W/cm$^{2}$, the thickness of all the samples was 200 nm. The optical absorption was measured using a double beam spectrophotometer (TU1901), the electrical properties were determined by a Hall effect measurement system (HL5500PC).

The electrical properties of the PbTiO$_{3}$ films deposited under different conditions are summarized in Table 1. The p-type N-doped PbTiO$_{3}$ films are deposited under Ar $+$ O$_{2}$ $+$ N$_{2}$ ambient. The values of resistivity (1.34 $\times$ 10$^{2}$ $\Omega$$\cdot$cm), carrier density (6.2 $\times$ 10$^{15}$ cm$^{-3}$) and Hall mobility (7.8 cm$^{2}$/(V$\cdot$s)) are obtained. When the films are prepared under the Ar $+$ N$_{2}$ conditions, the carrier type is changed from p-type to n-type. The oxygen vacancy is easily formed in an O-poor environment and the p-type characteristics are compensated, all of the experimental results are almost in accordance with the calculation results (Section 3). The mobility of N-doped PbTiO$_{3}$ with an oxygen vacancy is lower than that of N-doped PbTiO$_{3}$, which could be explained by the larger effective mass of the N-doped PbTiO$_{3}$ with an oxygen vacancy calculated above (Section 3.3).

The dependence of the absorption coefficient ($\alpha$) on the photon energy $(h \nu)$ is analyzed using the following equation for near edge optical absorption of direct band gap semiconductors^[23].

$\alpha ^2=A(h\nu -E_{\rm g} ).$

(3)

Here, $E_{\rm g}$ is the optical band gap, and $A$ is constant. The variations of the squared adsorption coefficient $\alpha ^{2}$ versus the photon energy in the absorption region are plotted in Figure 8. The $E_{\rm g}$ values are given by the extrapolation of the linear portion to the energy axis at $\alpha^{2}$ $=$ 0. The band gap is determined to be about 3.40 eV for the undoped PbTiO$_{3}$, which is larger than the calculation value 1.67 eV. Therefore a scissor operator of 1.73 eV is introduced to modify the calculation band-gap. The modified values of the calculation band gaps of N-doped PbTiO$_{3}$ and N-doped PbTiO$_{3}$ with an oxygen vacancy are 2.93~eV and 2.91 eV, which are consistent with the experimental values of 2.97 eV and 2.93 eV.

5. Conclusions

Using first-principles calculations, the charge density differences, band structures and density of states of N-doped PbTiO$_{3}$ and N-doped PbTiO$_{3}$ with an oxygen vacancy are studied. Compared with intrinsic PbTiO$_{3}$, the Fermi energy level of N-doped PbTiO$_{3}$ enters the top of the valance bands, which shows the p-type conductive characteristics. The effective mass of N-doped PbTiO$_{3}$ is -1.63$m_{\rm e}$, and the relative hole number is 2.02. For N-doped PbTiO$_{3}$ with an oxygen vacancy, the value of the band gap decreases and the acceptor states are fully compensated. The calculation results are almost consistent with the experimental results. Therefore we can make the conclusion that it is necessary to fabricate p-type N-doped PbTiO$_{3}$ under suitable O-rich conditions to prevent the formation of an oxygen vacancy.