Strain Engineered Band Gaps and Electronic Properties in PbPdO2 and PbPd0.75Co0.25O2 Slabs

Electronic structure and corresponding electrical properties of PbPdO2 and PbPd0.75Co0.25O2 ultrathin slabs with (002) preferred orientation were systematically investigated using first-principles calculations. The calculated results revealed the strain induced evidently the changes of band structure and carrier concentration in both slabs. It was also found that PbPdO2 and PbPd0.75Co0.25O2 ultrathin slabs exhibited evident differences in the external strain dependence of the band gap and charge carrier concentration, which was strongly dependent on bond angle and bond length induced by in-plane anisotropy strain. Interestingly, the carrier concentration of the PbPd0.75Co0.25O2 slab could increase up to 5–6 orders of magnitude with the help of external strain, which could explain the potential mechanism behind the observed colossal strain-induced electrical behaviors. This work demonstrated that the influence of the doping effect in the case of PbPdO2 could be a potentially fruitful approach for the development of promising piezoresistive materials.


Introduction
In the past decade, spin-gapless semiconductors (SGS) have attracted increasing interest because of their unique physical properties, leading to their potential application in electronic devices, such as field-effect transistors, optoelectronics, electronic sensors, and supercapacitors, amongst others [1][2][3][4][5][6]. Among them, the PbPdO 2 -based spin gapless semiconductor is considered as a promising candidate because of its non-toxicity, compatibility to the oxide semiconductor devices, and sensitivity to the doping metal elements, electric field, and operation current. Based on local density approximation calculations, the oxide-based PbPdO 2 gapless semiconductor was firstly discovered by Wang [4]. Following this, extensive investigations on the electric and magnetic properties of PbPdO 2 -based semiconductors were carried out theoretically and experimentally. Wang et al. studied the roles of both electrical current and magnetic field on the resistivity of PbPd 0.75 Co 0.25 O 2 thin films, and unusual colossal electroresistance and magnetoresistance were observed [5]. Moreover, the distinct different magnetoresistance effects were observed in PbPd 0.9 Cu 0.1 O 2 and PbPd 0.9 Zn 0.1 O 2 , which would be attributed to local structure deformation due to Pd/O deficiencies [7]. Based on the bound magnetic polaron (BMP) theory, the potential mechanism behind the observed ferromagnetic, paramagnetic, and antiferromagnetic properties coexisting in Co-doped PbPdO 2 film were suggested [8]. It was suggested that Pd-O hybridization in Co-doped PbPdO 2 thin films were responsible for the transition where ρ 0 is the resistance without strain and ∆ρ is change of resistance with strain ε. The resistivity (ρ) is in inverse proportion to carrier concentration (n). As ε = 0, let n = n 0 ; ε = 0, let n = n ε . The gauge factor can be re-expressed as follows, In PbPdO 2 -based composites, different preparation and processing methods result unavoidably in different microstructure and strain states, which consequently influences the band gap, carrier concentration, and corresponding electrical properties. Specifically, being similar to the layered MoS 2 , (002) preferred orientation layered PbPdO 2 has a small band gap [14], and the piezoresistive effect is expected.
In this work, based on the first-principle calculation method, a plane strain model was set up to obtain a deformed lattice with in-plane arbitrary uniaxial strain. In-plane anisotropy strain dependence of band-gap and carrier concentration were systematically investigated in the PbPdO 2 and PbPd 0.75 Co 0.25 O 2 slabs with preferred (002) orientation. These results can be well explained according to the p-d exchange interaction. Moreover, it is strongly suggested that the element-doping PbPdO 2 should become an important piezoresistance candidate material.

Methods
Based on Vienna ab initio simulation package (VASP), the self-consistent total energy was calculated and the geometry was optimized using the perdew-burke-ernzerhof (PBE) exchange-correlation functionals and the projector-augmented wave potentials [19,20]. The strain effect on the electronic properties of PbPdO 2 and PbPd 0.75 Co 0.25 O 2 slabs were simulated via standard DFT (Density functional theory) with generalized gradient approximation (GGA) method [21].
The cut-off energy was set to be 500 eV. The initial structure of PbPdO 2 and PbPd 0.75 Co 0.25 O 2 slabs were obtained from bulk PbPdO 2 and PbPd 0.75 Co 0.25 O 2 , respectively [10,20]. Then, to make the in-plane force reach up to the minimum, both initial structures were totally relaxed through the energy minimization method. Starting with the relaxed initial structures of PbPdO 2 and PbPd 0.75 Co 0.25 O 2 slabs, the effects on the atoms and band structures were studied systematically under the strain with the range of ±2%, which applied in the in-plane anisotropy uniaxial strain direction. The positive (negative) values of strain corresponded to the stretch and compression, respectively. The positions of all the atoms in the cell were relaxed by the optimizations of the strained structures with the Gaussians smearing method. After relaxation, each atom's convergence tolerance of force was smaller than 0.01 eV/Å. Meanwhile, 21 × 11 × 1 and 19 × 13 × 1 Monkhorst-Pack's meshes were used in the calculation of density of states (DOS) for PbPdO 2 and PbPd 0.75 Co 0.25 O 2 slabs, respectively. Figure 1 shows the relaxed crystal structures of PbPdO 2 and PbPd 0.75 Co 0.25 O 2 ultrathin slabs with (002) preferred orientation. As shown in Figure 1, the unstrained PbPdO 2 and ultrathin slab revealed an in-plane symmetric configuration, maintaining the important properties of PbPdO 2 . After Co-doping, the symmetric configuration of PbPd 0.75 Co 0.25 O 2 was broken. In comparison to the pristine PbPdO 2 , the broken-symmetry in PbPd 0.75 Co 0.25 O 2 was expected to bring different physical properties. It should be noted that the anisotropy of PbPd 0.75 Co 0.25 O 2 was greatly affected by the state of the Co-substitution. Moreover, its configurations were fairly complicated. Here, the configuration with the least number of atoms and only one Co doping atom was considered, as seen in Figure 1b the effects on the atoms and band structures were studied systematically under the strain with the range of ±2%, which applied in the in-plane anisotropy uniaxial strain direction. The positive (negative) values of strain corresponded to the stretch and compression, respectively. The positions of all the atoms in the cell were relaxed by the optimizations of the strained structures with the Gaussians smearing method. After relaxation, each atom's convergence tolerance of force was smaller than 0.01 eV/Å. Meanwhile, 21 ×11 × 1 and 19 ×13 × 1 Monkhorst-Pack's meshes were used in the calculation of density of states (DOS) for PbPdO2 and PbPd0.75Co0.25O2 slabs, respectively. Figure 1 shows the relaxed crystal structures of PbPdO2 and PbPd0.75Co0.25O2 ultrathin slabs with (002) preferred orientation. As shown in Figure 1, the unstrained PbPdO2 and ultrathin slab revealed an in-plane symmetric configuration, maintaining the important properties of PbPdO2. After Codoping, the symmetric configuration of PbPd0.75Co0.25O2 was broken. In comparison to the pristine PbPdO2, the broken-symmetry in PbPd0.75Co0.25O2 was expected to bring different physical properties. It should be noted that the anisotropy of PbPd0.75Co0.25O2 was greatly affected by the state of the Cosubstitution. Moreover, its configurations were fairly complicated. Here, the configuration with the least number of atoms and only one Co doping atom was considered, as seen in Figure 1b. Specifically, this Co atom forms square planar bonding with the nearest-neighbor four O atoms, which plays an important role on the electrical properties of PbPd0.75Co0.25O2.  .75Co0.25O2 slabs, respectively. It was found that PbPdO2 exhibits intrinsic characteristics of narrow band gap (0.051 eV), which was much smaller than that (0.4 eV) of the PbPd0.75Co0.25O2 slab. In our previous experimental work, PbPdO2 with (002) preferred orientation was prepared, and its band gap was found to be close to zero [22]. It is suggested that the localized Co would be responsible for the large band gap of 0.35 eV for the PbPd0.75Co0.25O2 bulk material. Interestingly, our calculated results were consistent with the reported results [23]. The band gap of PbPdO2 slab was slightly larger than that reported in our previous calculated work because of the different full in-plane structure relaxation [14]. From the density of electronic states in Figure 2b,d, it is found that DOSs of the pristine and Co-doped PbPdO2 slabs at minimum of the conduction band and maximum of the valence band were mainly composed of 4d(Pd) and 2p(O) states. These results were similar to those found by many research groups [6,10]. In contrast, the PbPd0.75Co0.25O2 slab had more distinct contribution of hybridization of 2p(O)-4d(Pd) states, where a Co 3d state added modified energy to the DOS at the Fermi energy level.  slabs, respectively. It was found that PbPdO 2 exhibits intrinsic characteristics of narrow band gap (0.051 eV), which was much smaller than that (0.4 eV) of the PbPd 0.75 Co 0.25 O 2 slab. In our previous experimental work, PbPdO 2 with (002) preferred orientation was prepared, and its band gap was found to be close to zero [22]. It is suggested that the localized Co would be responsible for the large band gap of 0.35 eV for the PbPd 0.75 Co 0.25 O 2 bulk material. Interestingly, our calculated results were consistent with the reported results [23]. The band gap of PbPdO 2 slab was slightly larger than that reported in our previous calculated work because of the different full in-plane structure relaxation [14]. From the density of electronic states in Figure 2b,d, it is found that DOSs of the pristine and Co-doped PbPdO 2 slabs at minimum of the conduction band and maximum of the valence band were mainly composed of 4d(Pd) and 2p(O) states. These results were similar to those found by many research groups [6,10]. In contrast, the PbPd 0.75 Co 0.25 O 2 slab had more distinct contribution of hybridization of 2p(O)-4d(Pd) states, where a Co 3d state added modified energy to the DOS at the Fermi energy level. To gain insight into the potential mechanism of strain-induced electronic properties in PbPdO2base composites, a plane-stress-strain model was set up. Figure 3a,b present the undeformed lattice and a deformed lattice with in-plane arbitrary uniaxial tensile strain (directional cosines (cos , cos )), respectively. The strain-related components could be obtained based on the coordinate transformation method. Assuming uniaxial strain is along the x′ direction in an unprimed coordinate system x-y, the strain tensor elements in the primed coordinate system are given as follows,

Results and Discussion
) and x y  ' ' are normal (tensile or compressive) and shear strains, respectively [24].
The directional cosines are where , , ′, ′ are arbitrary directions in the x-y and x′-y′ coordinate system, as seen in Figure  3.
A deformation of the unit cell is created by changing the Bravais lattice vectors R of the undeformed unit cell to R′ using a strain matrix as follows: To gain insight into the potential mechanism of strain-induced electronic properties in PbPdO 2 -base composites, a plane-stress-strain model was set up. Figure 3a,b present the undeformed lattice and a deformed lattice with in-plane arbitrary uniaxial tensile strain (directional cosines (cos α, cos β)), respectively. The strain-related components could be obtained based on the coordinate transformation method. Assuming uniaxial strain ε is along the x direction in an unprimed coordinate system x-y, the strain tensor elements in the primed coordinate system are given as follows, where ε x (ε y ) and γ x y are normal (tensile or compressive) and shear strains, respectively [24]. The directional cosines are where α, β, α , β are arbitrary directions in the x-y and x -y coordinate system, as seen in Figure 3. A deformation of the unit cell is created by changing the Bravais lattice vectors R of the undeformed unit cell to R using a strain matrix as follows: where R is the Bravais lattice vectors with strain, and R is the Bravais lattice vectors without strain, ε x (ε y ) and γ xy (γ yx ) are the normal (tensile or compressive) and shear strain-related components, respectively. Where R is the Bravais lattice vectors with strain, and R′ is the Bravais lattice vectors without strain, x  ( y  ) and xy  ( yx  ) are the normal (tensile or compressive) and shear strain-related components, respectively.      ) and y axis ( = 90   ), respectively. As shown in Figure 5a, it is found that the band-gap value increases with increasing strain along the y-axis, whilst the band-gap value decreases with increasing strain along the x-axis. It is interesting that the band gap of the PbPdO2 slab would widen when a compressive stress is applied closely to the x-axis or a tensile stress is applied closely to the  Where R is the Bravais lattice vectors with strain, and R′ is the Bravais lattice vectors without strain, x  ( y  ) and xy  ( yx  ) are the normal (tensile or compressive) and shear strain-related components, respectively.     Figure 5a, it is found that the band-gap value increases with increasing strain along the y-axis, whilst the band-gap value decreases with increasing strain along the x-axis. It is interesting that the band gap of the PbPdO2 slab would widen when a compressive stress is applied closely to the x-axis or a tensile stress is applied closely to the   Figure 5a, it is found that the band-gap value increases with increasing strain along the y-axis, whilst the band-gap value decreases with increasing strain along the x-axis. It is interesting that the band gap of the PbPdO 2 slab would widen when a compressive stress is applied closely to the x-axis or a tensile stress is applied closely to the y-axis, as seen in Figure 4a. These calculated results can be explained according to the interaction of Pd-O bonding. It is expected that a compressive stress along the x-axis or tensile stress along the y-axis pulls O atoms apart from Pd atoms, which weakens the interaction of Pd and O. On the other hand, a tensile stress along the x-axis or compressive stress along the y-axis would push O atoms closely to Pd atoms and strengthen the interaction of Pd and O. As a result, the band gap is decreased. A similar result has also been reported in MoS 2 and black phosphorus [25,26].
Materials 2018, 11, x FOR PEER REVIEW 6 of 11 y-axis, as seen in Figure 4a. These calculated results can be explained according to the interaction of Pd-O bonding. It is expected that a compressive stress along the x-axis or tensile stress along the yaxis pulls O atoms apart from Pd atoms, which weakens the interaction of Pd and O. On the other hand, a tensile stress along the x-axis or compressive stress along the y-axis would push O atoms closely to Pd atoms and strengthen the interaction of Pd and O. As a result, the band gap is decreased.
A similar result has also been reported in MoS2 and black phosphorus [25,26].  that Co atom should act as the source of the localized magnetic moment, and the coupling between the p-state from O and d-state from Co could induce a strong exchange interaction (named as p-d exchange interaction) in PbPd0.75Co0.25O2. Moreover, the p-d exchange interaction was found to be nearly inversely proportional to the unit cell volume [27]. Therefore, p-d exchange interaction mediated by strain should be responsible for variation of the plane averaged electron density difference, leading to a clear change of the band gap.    Figure 7, it is concluded that Co atom should act as the source of the localized magnetic moment, and the coupling between the p-state from O and d-state from Co could induce a strong exchange interaction (named as p-d exchange interaction) in PbPd 0.75 Co 0.25 O 2 . Moreover, the p-d exchange interaction was found to be nearly inversely proportional to the unit cell volume [27]. Therefore, p-d exchange interaction mediated by strain should be responsible for variation of the plane averaged electron density difference, leading to a clear change of the band gap. y-axis, as seen in Figure 4a. These calculated results can be explained according to the interaction of Pd-O bonding. It is expected that a compressive stress along the x-axis or tensile stress along the yaxis pulls O atoms apart from Pd atoms, which weakens the interaction of Pd and O. On the other hand, a tensile stress along the x-axis or compressive stress along the y-axis would push O atoms closely to Pd atoms and strengthen the interaction of Pd and O. As a result, the band gap is decreased.
A similar result has also been reported in MoS2 and black phosphorus [25,26].   Figure 7, it is concluded that Co atom should act as the source of the localized magnetic moment, and the coupling between the p-state from O and d-state from Co could induce a strong exchange interaction (named as p-d exchange interaction) in PbPd0.75Co0.25O2. Moreover, the p-d exchange interaction was found to be nearly inversely proportional to the unit cell volume [27]. Therefore, p-d exchange interaction mediated by strain should be responsible for variation of the plane averaged electron density difference, leading to a clear change of the band gap.         Unique electrical properties are highly desirable for practical application, and charge carrier concentration is a key parameter for the intrinsic semiconductor. For the intrinsic semiconductor, the charge charier concentration can be estimated as follows [28], where K B , E g are the Boltzmann constant and band gap, respectively. In this paper, all the temperatures in carrier concentration were calculated at T = 100 K. As ε = 0, let n = n 0 ; ε = 0, let n = n ε . Combined with the results presented in Figure 4a,b, the external strain dependence of charge carrier concentration ratio (n ε /n 0 ) for PbPdO 2 and PbPd 0.75 Co 0.25 O 2 slabs were evaluated, respectively. Figure 9a,b shows the orientation distribution curves of the intrinsic charge carrier concentration ratio n/n 0 for PbPdO 2 and PbPd 0.75 Co 0.25 O 2 slabs with ε = −0.02, −0.01, 0.00, 0.01, 0.02, respectively. Similar to the dependence of band gap E g on strain orientation, PbPd 0.75 Co 0.25 O 2 exhibits more distinct anisotropy in carrier concentration with strain direction, especially along the x-axis. As shown in Figure 9a, the pristine PbPdO 2 slab demonstrates the symmetrical and olive-like (n ε /n 0 ) −α curves, and having its largest and smallest band-gap values along the x-axis or y-axis, respectively. For the PbPd 0.75 Co 0.25 O 2 slab, the carrier concentration is sensitive to the application direction of strain. When the compressive stress applies along a direction of 75 • (α ≈ 75 • ) and x-axis, the remarkable variety in carrier concentration appears, as shown in Figure 9b. Figure 10c,d shows the intrinsic charge carrier concentration ratio (n ε − n 0 )/n 0 as a function of strain for PbPdO 2 and PbPd 0.75 Co 0.25 O 2 slabs along the x and y-axes, respectively. For the PbPdO 2 slab, the carrier concentration increases monotonically with increasing compressive stress, but decreasing with increasing tensile stress along the x-axis. On the contrary, the carrier concentration decreases with increasing compressive stress, while increasing with increasing tensile stress along the y-axis. In contrast, the carrier concentration of PbPd 0.75 Co 0.25 O 2 slab increases with increasing compressive and tensile stresses along both the x and y-axes. As the compressive stress increases beyond 0.015 along the x-axis, carrier concentration of the PbPd 0.75 Co 0.25 O 2 slab increases rapidly. It was found that the carrier concentration of PbPd 0.75 Co 0.25 O 2 could sharply increase up to 5-6 orders of magnitude with the help of external strain with ε = 0.02. The calculated results suggest strongly that the element-doping PbPdO 2 should become an important piezoresistance candidate material.
Unique electrical properties are highly desirable for practical application, and charge carrier concentration is a key parameter for the intrinsic semiconductor. For the intrinsic semiconductor, the charge charier concentration can be estimated as follows [28], where KB, Eg are the Boltzmann constant and band gap, respectively. In this paper, all the temperatures in carrier concentration were calculated at T = 100 K. As  = 0, let n = n0;  ≠ 0, let n = n  .
Combined with the results presented in Figure 4a PbPd0.75Co0.25O2 slabs along the x and y-axes, respectively. For the PbPdO2 slab, the carrier concentration increases monotonically with increasing compressive stress, but decreasing with increasing tensile stress along the x-axis. On the contrary, the carrier concentration decreases with increasing compressive stress, while increasing with increasing tensile stress along the y-axis. In contrast, the carrier concentration of PbPd0.75Co0.25O2 slab increases with increasing compressive and tensile stresses along both the x and y-axes. As the compressive stress increases beyond 0.015 along the x-axis, carrier concentration of the PbPd0.75Co0.25O2 slab increases rapidly. It was found that the carrier concentration of PbPd0.75Co0.25O2 could sharply increase up to 5-6 orders of magnitude with the help of external strain with  = 0.02. The calculated results suggest strongly that the elementdoping PbPdO2 should become an important piezoresistance candidate material.   Figure 11 presents the strain dependence of gauge factor for the PbPdO 2 and PbPd 0.75 Co 0.25 O 2 slab along the x and y axes. Table 1 gives some typical gauge factor values. When the tensile strain is 0.02, the piezoresistive gauge factors for the PbPdO 2 and PbPd 0.75 Co 0.25 O 2 slab along the x-axis are calculated to be respectively 62.8 and −43.3, which is comparable to trilayer MoS 2 and much higher than suspended-graphene-based strain sensor [29]. As Pd-O (Co-O) polar covalent bond is different from C-C bond, PbPdO 2 -based semiconductors can exhibit a higher piezoresistive gauge factor than graphene-based strain sensors. The predicted large gauge factors in our work implies that the element-doping PbPdO 2 may have promising opportunities to be used as strain sensors.  Figure 11 presents the strain dependence of gauge factor for the PbPdO2 and PbPd0.75Co0.25O2 slab along the x and y axes. Table 1 gives some typical gauge factor values. When the tensile strain is 0.02, the piezoresistive gauge factors for the PbPdO2 and PbPd0.75Co0.25O2 slab along the x-axis are calculated to be respectively 62.8 and −43.3, which is comparable to trilayer MoS2 and much higher than suspended-graphene-based strain sensor [29]. As Pd-O (Co-O) polar covalent bond is different from C-C bond, PbPdO2-based semiconductors can exhibit a higher piezoresistive gauge factor than graphene-based strain sensors. The predicted large gauge factors in our work implies that the element-doping PbPdO2 may have promising opportunities to be used as strain sensors.

Conclusions
Based on first-principles calculations, the electronic structures and electrical properties of  Figure 11 presents the strain dependence of gauge factor for the PbPdO2 and PbPd0.75Co0.25O2 slab along the x and y axes. Table 1 gives some typical gauge factor values. When the tensile strain is 0.02, the piezoresistive gauge factors for the PbPdO2 and PbPd0.75Co0.25O2 slab along the x-axis are calculated to be respectively 62.8 and −43.3, which is comparable to trilayer MoS2 and much higher than suspended-graphene-based strain sensor [29]. As Pd-O (Co-O) polar covalent bond is different from C-C bond, PbPdO2-based semiconductors can exhibit a higher piezoresistive gauge factor than graphene-based strain sensors. The predicted large gauge factors in our work implies that the element-doping PbPdO2 may have promising opportunities to be used as strain sensors.

Conclusions
Based on first-principles calculations, the electronic structures and electrical properties of PbPdO2 and PbPd0.75Co0.25O2 ultrathin slabs were systematically investigated. The calculated results indicated that the strain induces changes of band structure and carrier concentration in both slabs.

Conclusions
Based on first-principles calculations, the electronic structures and electrical properties of PbPdO 2 and PbPd 0.75 Co 0.25 O 2 ultrathin slabs were systematically investigated. The calculated results indicated that the strain induces changes of band structure and carrier concentration in both slabs. Specifically, the carrier concentration of the PbPd 0.75 Co 0.25 O 2 slab could be modulated with 5-6 orders externally induced strain, which renders the Co-doped pristine PbPdO 2 phase a potentially promising piezoresistive material. Moreover, the above evident external strain modulation of the band gap and carrier concentration can be well explained by spin-splitting theory.