Electronic Origin of Defect States in Fe-Doped LiNbO3 Ferroelectrics

We investigate the role of Fe in the electronic structure of ferroelectric LiNbO 3 by density-functional theory calculations. We show that Fe on the Li site (Fe2+Li) features a displacement opposite to the direction of spontaneous polarization and acts as a trigger for the bulk photovoltaic (PV) effect. In contrast to Fe on the Li site that forms the defect states (1e, a, and 2e) below the conduction band minimum, the reduction from Fe to Fe accompanied by a lattice relaxation markedly lowers only the a state (d z 2 ) owing to a strong orbital hybridization with Nb-4d. The a state of Fe2+Li provides the highest electron-occupied defect state in the middle of the band gap. A reduction treatment of Fe-LN is expected to increase the concentration of Fe and therefore to enhance the PV effect under visible light illumination.

Recently, the PV effect in ferroelectrics has attracted renewed interests, because the working principle can be applied to the devices that generate a high voltage beyond the limitation of present semiconductor-based solar cells [16][17][18][19].The bulk PV effect in ferroelectrics arises from asymmetric carrier dynamics because of spatial symmetry breaking [20].The photoinduced feature of the generation and dissociation of charge carriers in Fe-LN under visible light make it a promising candidate for the ferroelectric material used in the PV devices.Although a large number of experimental studies on Fe-LN have been conducted for several decades [21,22], first-principles calculations of the electronic structures have been reported very recently [23][24][25][26].
Sanson et al. [25] have performed density-functional theory (DFT) calculations for Fe-LN in addition to local structure studies by an extended X-ray absorption fine structure spectroscopy.They reveal that Fe 2+ on the Li site (Fe 2+ Li ) produces defect states inside the band gap and that the  state in the minority spin band forms the Fermi energy (  ).According to Inoue et al. [26], a tensile strain from substrates markedly enhances the PV effect in Fe-LN epitaxial thin films whereas their DFT calculations show that neither tensile nor compressive strain plays a minor role in the electronic structure of Fe 2+ Li -LN.However, the following question remains unresolved: why the -wave states of Fe 2+ Li differ in the majority and minority spin bands.In addition, the orbital interaction between Fe 2+ Li and its neighbors should play an important role, but the bonding and antibonding interactions related to the Fe 2+ -d states have not been understood yet.

Advances in Condensed Matter Physics
To establish the materials design for enhanced PV effect in the LN system, the electronic origin of the defect states originating from Fe 2+ Li needs to be clarified.In this paper, we report the electronic structure of Fe 2+ Li -LN obtained by DFT calculations and discuss the orbital interactions of Fe 2+ Li with its neighbors.We demonstrate that the mixing between Fe-3d and Nb-4d plays a crucial role in the electronic structure.Our calculations show that the  state of Fe 2+ Li resulting from the hybridization of Fe-  2 and Nb-4d is the highest electron-occupied defect state in the middle of the band gap and acts as a trigger for the PV effect under visible light.

Method of DFT Calculations
We performed DFT calculations within the local density approximation (LDA) [27] by the PAW (projector-augmented wave) method [28] using a plane wave basis set as implemented in the Vienna ab initio simulation package (VASP) [29].A plane wave cut-off energy was set to be 520 eV, and an electronic energy was converged to less than 10 −5 eV in all calculations.First, the hexagonal unit cell of LN in space group R3c was structurally optimized until the Hellmann-Feynman force on each atom was less than 0.1 eV/nm.A kmesh of 3 × 3 × 3 centered on the Γ point was used in the geometrical optimization for lattice parameters and fractional coordinates.The lattice parameters of the optimized LN cell agree well with the experimental values [30] within 0.7%.
We employed a simplified local spin density approximation (LSDA + U approach) [31] as a correction for localized and strongly correlated electrons within on-site Coulomb terms of  −  = 3 eV for Nb-4d [32] and  −  = 4 eV for Fe-3d [23,30].The details of the influence of the  −  terms on the electronic structures are discussed later.The occupation sites of Fe in LN have been reported as Fe on the Li site (Fe Li ) [33][34][35][36][37][38][39][40] and Fe on the Nb site (Fe Nb ) [38,39].Here, we adopt the model of Fe 2+ on the Li site (Fe 2+ Li ), according to the reports [23,41], because we focus our attention on the defect states arising from Fe 2+ Li that absorbs visible light.To make a supercell of Fe-LN, the setting of the optimized LN cell (hexagonal) was transformed to the rhombohedral cell (Li 16 Nb 16 O 48 ) in space group P1.In this cell one Li atom was replaced by one Fe atom, leading to Li 15 FeNb 16 O 48 .This rhombohedral setting was changed again to the hexagonal one in space group R3.This Fe-LN cell [Li 15 FeNb 16 O 48 ( = 3)] was geometrically optimized (with a k-mesh of 3 × 3 × 3 centered on the Γ point).
In the model of Fe 2+ Li , the charge neutrality is supposed to be satisfied by the following two possible compensations: one is the formation of the small polaron, Fe 2+ Li -Nb 4+ Nb , [41,42] and the other is the creation of Li vacancy (  Li ) [23], where the prime denotes one negative charge.First, we tested the small polaron model of Fe 2+ Li -Nb 4+ Nb .In the Fe-LN cell, there are eight different kinds of the Nb sites in view of the symmetry.Prior to the geometrical optimizations, magnetic moment of +1   (where   denotes Bohr magneton) or −1   was put to each of the Nb atom, and that of Fe was set to be +4   assuming the high spin configuration.After the geometrical optimizations for all of the supercells, the total magnetic moment was converged to be +4   and the   was located inside the conduction band (the Nb-4d band).This implies that Fe-LN crystals with the Fe 2+ Li -Nb 4+ Nb polaron exhibit a metallic behavior having a low resistivity, which cannot explain the experimental result of the high resistivity of Fe-LN crystals [43].These results force us to consider that the small polaron model of Fe 2+ Li -Nb 4+ Nb is not suitable for the charge compensation in the ground state.
Next we examined the   Li model [23]  Li exhibited a small total energy by 30-90 meV compared with the other cells.These results suggest that   Li is not necessarily stabilized adjacent to Fe 2+ Li and lead to a structural picture of the Fe 2+ Li -LN lattice with a random distribution of   Li .

In reality, Fe 2+
Li -LN should have an averaged structure similar to the LN host in R3c space group.This symmetry is, however, completely lost in the   Li model owing to the presence of   Li on the specific site.In order to take into account the threefold rotation axis along the  axis, we adopted the Fe 2+ Li -LN cell without where the valence state of Fe was controlled to be Fe 2+ .This structural model is the same as that reported by Sanson et al. [25].The calculations of the Fe 3+ Li -LN cell were also conducted in a similar manner for comparison.The electronic band structures and density of states (DOS) of the geometrically optimized cells were calculated with a k-mesh of 3 × 3 × 3 centered on the Γ point.While the calculations with a denser k-mesh were tested, we confirmed that the results of the electronic structures are essentially the same, as reported in [25].Li -LN cell after the geometrical optimization.In the LN cell, Nb atoms are displaced along the  axis by approximately 0.02 nm, and this cooperative displacement yields a spontaneous polarization (  ).In the Fe 2+

Results and Discussion
Li -LN cell the bond length of Fe-O1 (  = 0.200 nm) is shorter by 0.02 nm than that of Fe-O2 (  = 0.221 nm), showing that Fe 2+ is displaced in the direction opposite to   .The bond valence sum (BVS) of Fe 2+ Li is estimated to be +2.3, which is slightly larger than the nominal valence (+2).Figures 1(b)-1(d) display the electronic band structures.The total density of states (DOS) and their partial DOS (PDOS) are also shown in Figures 2(a)-2(c).For the LN (R3c) cell (Figures 1(b) and 2(a)), the   is set to be 0 eV.In these figures, the valence band maximum (VBM) mainly composed of the O-2p band is aligned to the same level.Our calculations show that the band gap is indirect: the VBM is located at the Λ point (1/2 1/2 1/2) and the conduction band minimum (CBM) is positioned at the Γ point.Because the maximum of the valence band at the Γ point is very  In point group 3, the  states are divided into 1e, a, and 2e (Figure 3(b)).Fe 2+ has a 3d 6 electron configuration, which is described as 1e 2 a 1 2e 2 in the majority spin (↑) band and a 1 1e 0 2e 0 in the minority spin (↓) band [41,46].The detailed analysis of the Fe 2+ -3d states leads to the order of 1e, a, and 2e with increasing energy in the ↑ band, as is illustrated in the energy diagram of the  states in the Fe 2+ Li -LN cell (Figure 3 the antibonding state.In the ↑ band, all of these states are occupied by electrons and the  * and 2 * states form defect states above the VBM (Figure 1(c)).
In the ↓ band, we note that the  state is markedly lowered in energy whereas the 1e and 2e are located above the CBM, as can be seen in Figures 1(c), 2(b), and 3(f).The detailed analysis reveals that the lowering in energy of the  state (↓) stems from the two orbital interactions as depicted in Figure 3: one is the direct hybridization with Nb1-4d (the line I) and the other is the mixing with the antibonding state of the Nb2-4d (O-2p) states (the line II).As a result, Fe 2+ possesses three localized defect states inside the band gap, so-called gap states:  * and 2 * in the ↑ band and  in the ↓ band.The gap states of  * and 2 * exist near the VBM, and the  state (↓) forms the   in the middle of the band gap.Due to Fe 2+ Li with the  state (↓), the apparent gap between the occupied state and the CBM narrows to 1.6 eV, which agrees well with the onset of the optical absorption of Fe-LN crystals [42].This result shows that visible light with ℏ ≥ 1.6 eV leads to the electron excitation from Fe 2+ to Nb 5+ .This intervalence transfer transition oxidizes Fe 2+ to Fe 3+ and reduces Nb 5+ to Nb 4+ , which forms the Fe-Nb dipole.The total optical absorption essentially results from a superposition of the electron transfers from the  state (↓) to the CB [41].
Next, we discuss the results of the Fe 3+ Li -LN cell shown in Figures 1(d) and 2(c).Fe 3+ has a 3d 5 electron configuration with 1e 2 a 1 2e 2 in the ↑ band and 1e 0 a 0 2e 0 in the ↓ band.The analysis of the Fe 3+ -d sates shows the orders of 1e, a, and 2e in the ↑ band and a, 1e, and 2e in the ↓ band with increasing energy.Compared with the Fe 2+ Li -LN cell, the Fe 3+ -3d states are lowered in energy regardless of the ↑ and ↓ bands.Only the 2 * state (↑) constitutes the occupied gap state and forms the   .The unoccupied states of a, 1e, and 2e in the ↓ band are present below the CBM.The energy gap between the occupied 2 * state and the CBM on the Γ point is as large as 3 eV, indicating that visible light with a photon energy less than 3 eV (ℏ < 3 eV) cannot cause a charge transfer from Fe 3+ to Nb 5+ .Thus, Fe 3+ Li in the ground state plays a minor role in the PV effect under visible light.
For the bulk PV effect in Fe-LN under visible light, the defect states inside the band gap are of particular importance.We depict the wave function of the occupied  state (↓) of the Fe 2+ Li -LN cell in Figure 4(a) and its two-dimensional projections in Figures 4(b)-4(d).This  state is composed mainly of the   2 orbital of Fe 2+ and has a small contribution of O-2p.In Figures 4(b) and 4(d) we display the above-described two orbital interactions, that is, the direct hybridization of Fe-  2 with Nb1-4d and the mixing with the Nb2-4d (O-2p) antibonding state, respectively.It is interesting to note that the  state (↓) is indeed localized around Fe 2+ Li .In contrast, the wave function of the CBM on the Γ point exhibits the Nb-4d character and is delocalized throughout the crystal, as shown in Figure 5.
Here, we discuss the generation and recombination of carriers photoexcited by visible light illumination.Figure 6 exhibits the schematic illustration of the configuration coordinate diagram representing the optical absorption ( opt ), Frank-Condon relaxations ( relax1 and  relax2 ), and photoluminescence ( PL ).The energy curve of Fe 2+ Li -LN varies with respect to the configuration coordinate (q) where the energy minimum (  ) achieved at q − represents a thermal ionization energy of acceptor (1.8 eV for Fe 2+ in LN) [23,47].The curve of Fe 3+ Li -LN exhibits an excited state caused by light illumination.We adopt the experimental value of the   of 3.8 eV.The optical absorption in Fe 2+ Li -LN ( opt ) has been reported to show a peak at around 2.5 eV, which changes the valence state from Fe 2+ to Fe 3+ [42].During the photoinduced electron excitation with a time scale of ∼0.1 ps [48,49], the valence state of Fe changes at a fixed coordinate of q − while the Fe 2+ Li -LN lattice remains unchanged.This excited state loses its surplus energy by displacing the neighboring ions (thermalization) and is relaxed to the coordinate of  0 along the Fe 3+ Li -LN curve, leading to the formation of free small polaron composed of an electron (  ) and Nb 5+ [41,42].The recombination time of the small polaron has been reported as ∼40 ns in MgO(5%)doped LN [49].The Fe valence state returns to Fe 2+ at the coordinate of  0 accompanied by a photoluminescence with an energy of  PL ∼ 1.6 eV [50].Then, the ground state coordinate  − is reached by a relaxation along the curve of Fe 2+ Li -LN.From our DFT calculations, we estimate the two Frank-Condon relaxations ( relax1 and  relax2 ) as 0.5 eV and 0.4 eV, respectively, which satisfy the following energy relations: Finally, we consider the influences of the  −  terms on the electronic structures and the gap state of Fe-3d in the Fe 2+ Li -LN cell.In the simplified LSDA + U approach, the − term is treated as an empirical parameter for representing localized, strongly correlated electrons.Nahm and Park [32] have investigated the crystal and electronic structures of LiNbO 3 with an antisite defect of Nb on the Li site in the range of − = 3-5 eV and reported that the electronic structure is well described with  −  = 4 eV for Nb-4d.Sanson et al. [25] have studied the structural distortion and electronic states of Fe 2+ Li -LN in the range of  −  = 3-5 eV for the  orbitals and concluded that the  −  value of 4 eV is suitable for both Nb-4d and Fe-3d for describing the local structure around Fe and the gap state of Fe-3d.Here, we investigate the influence of the  −  terms (2-8 eV) on the   and the a↓ gap state, the details of which are summarized in Tables 1 and 2. In Figure 7, we plot the CBM and the a↓ state with respect to the VBM as a function of  − .Because the VBM is set to be 0 eV, the value of the CBM is equal to   and the energy of the a↓ state indicates the energy gap from the VBM.With the fixed − of 3 eV for Nb-4d (Figure 7(a)), the   does not depend on − of Fe-3d and remains unchanged at 3.5 eV.This result shows that the   is essentially independent of the character of Fe-3d.The energy level of the a↓ state is lowered with increasing  −  of Fe-3d.As for the results with the fixed  −  of 4 eV for Fe-3d (Figure 7(b)), the   widens with increasing  −  of Nb-4d while the a↓ level remains constant, revealing that the  −  term of Nb-4d does not lead to a marked change in the character and depth of the gap states of Fe-3d in LN.
To verify the calculation results, we need to take account of the two experimental types of data: one is the   of 3.8 eV [44] and the other is the onset energy of the optical absorption in Fe 2+ Li -LN of 1.6 eV [42].Because the highest occupied electron is located at the a↓ state, the difference of Figure 7: Energy levels at the Γ point of the a↓ gap sate of Fe-3d and the CBM with respect to the VBM in the Fe 2+ Li -LiNbO 3 cell calculated by the LSDA + U approach with various  −  terms: (a) the  −  value of Nb-4d is fixed to be 3 eV and that of Fe-3d varies and (b) the  −  value of Fe-3d is fixed to be 4 eV and that of Nb-4d varies.The value of band gap (  ) accords with the energy difference between the CBM and the VBM.
Table 2: Parameters of the electronic band structures at the Γ point calculated for the Fe 2+ Li -LiNbO 3 cell within the LSDA +  approach with various  −  terms of Nb-4d (the  −  value of Fe-3d is fixed to be 4 eV).The value of band gap (  ) accords with the energy difference between the CBM and the VBM.[42].
the CBM and the a↓ state (CBM − a↓) can be considered to indicate the onset energy of the optical absorption.The CBM − a↓ is overestimated for the  −  values larger than 6 eV and underestimated for those smaller than 2 eV.We conclude that the  −  terms of 4 eV (Fe-3d) and 3 eV (Nb-4d) are appropriate for representing the electronic feature of the gap state originating from Fe-3d in Fe 2+ Li -LN, which is consistent with [23,25,30,32].We confirmed that the orbital interactions shown in Figures 3-5 play an essential role in the calculation results regardless of the  −  values.
One straightforward way to enhance the bulk PV effect in Fe 2+ Li -LN is to increase the concentration of Fe 2+ .Investigations with Mössbauer spectroscopy have shown that the dominant valence state of Fe in LN sintered in air is Fe 3+ and that the concentration of Fe 2+ [Fe 2+ ] is lower than the detection limit [51].This result suggests that [Fe 2+ ] is much lower than that of [Fe 3+ ] in Fe-LN sintered or annealed in air.A reduction treatment of Fe-LN is expected to increase [Fe 2+ ] and therefore to enhance the bulk PV effect in either bulk or thin-film form.

Conclusion
We have investigated the electronic structures of Fe 2+ Li -LN and Fe 3+ Li -LN by DFT calculations and discussed the electronic origin of the defect states inside the band gap.Our calculations show that Fe 2+  Li is displaced in the direction opposite to spontaneous polarization and acts as a trigger for the bulk PV effect under visible light.The  state, which is mainly composed of the Fe 2+ Li -  2 orbital, forms a localized defect state in the middle of the band gap.The following two orbital interactions lead to a noticeable lowering in energy of the  state in the minority spin band that forms the   inside the band gap: the direct hybridization of Fe-  2 with Nb-4d and the orbital mixing of Fe 2+ Li -  2 with the antibonding state of Nb-4d (O-2p).It is proposed that a reduction treatment of Fe-LN increases the concentration of Fe 2+ and therefore enhances the PV effect in either bulk or thin-film form.

Figure 1 (
Figure 1(a) displays the crystal structure around iron in the Fe 2+Li -LN cell after the geometrical optimization.In the LN cell, Nb atoms are displaced along the  axis by approximately 0.02 nm, and this cooperative displacement yields a spontaneous polarization (  ).In the Fe 2+Li -LN cell the bond length of Fe-O1 (  = 0.200 nm) is shorter by 0.02 nm than that of Fe-O2 (  = 0.221 nm), showing that Fe 2+ is displaced in the direction opposite to   .The bond valence sum (BVS) of Fe 2+Li is estimated to be +2.3, which is slightly larger than the nominal valence (+2).Figures1(b)-1(d) display the electronic band structures.The total density of states (DOS) and their partial DOS (PDOS) are also shown in Figures2(a)-2(c).For the LN (R3c) cell (Figures1(b) and 2(a)), the   is set to be 0 eV.In these figures, the valence band maximum (VBM) mainly composed of the O-2p band is aligned to the same level.Our calculations show that the band gap is indirect: the VBM is located at the Λ point (1/2 1/2 1/2) and the conduction band minimum (CBM) is positioned at the Γ point.Because the maximum of the valence band at the Γ point is very

Figure 1 :
Figure 1: (a) Crystal structure around Fe 2+ in the Fe 2+ Li -LN cell after the geometrical optimization obtained by the DFT calculations and the electronic band structures of the (b) LN, (c) Fe 2+ Li -LN, and (d) Fe 3+ Li -LN cells.

Figure 4 :
Figure 4: Wave functions of the occupied  state in the minority spin (↓) band of the Fe 2+ -LN cell (a) in a three-dimensional view and its two-dimensional projections on the (b) Nb1-Fe-Nb3 (c) Nb1-Fe-O2 and (d) O2-Fe-O1 planes.

Figure 5 :Figure 6 :
Figure 5: Wave function of the conduction band minimum (CBM) on the Γ point representing the Nb-4d character, which is delocalized throughout the crystal.

Table 1 :a
Parameters of the electronic band structures at the Γ point calculated for the Fe 2+Li -LiNbO 3 cell within the LSDA +  approach with various  −  terms of Fe-3d (the  −  value of Nb-4d is fixed to be 3 eV).The value of band gap (  ) accords with the energy difference between the CBM and the VBM. −  (eV) [Fe-3]   (eV) CBM a −  ↓ (eV)  ↓ − VBM b (eV) Conduction band maximum.b Valence band maximum.c Reference [44].d The onset energy of the optical absorption in Fe 2+Li -LiNbO 3[42].
by using the supercell containing   Li and Fe 2+ Li , that is, Li 14 FeNb 16 O 48 (the   Li cell) in space group P1.The Fe-LN cell has seven different kinds of Li sites.We performed the geometrical optimizations for all of the supercells with   Li on each of the Li sites.The cells with   Li on the first nearest neighbor (NN) and seventh NN sites with respect to Fe 2+