Interactions of charged domain walls and oxygen vacancies in BaTiO 3 : a ﬁ rst-principles study

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Interactions of charged domain walls and oxygen vacancies in BaTiO ₃ : a ﬁ rst-principles study

J.J. Gong

^a^,^b^,^*

, C.F. Li

^a

, Y. Zhang

^a

, Y.Q. Li

^a

, S.H. Zheng

^a

, K.L. Yang

^a

, R.S. Huang

^a

, L. Lin

^a

, Z.B. Yan

^a

, J.-M. Liu

^a^,^c

aLaboratory of Solid State Microstructure, Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

bDepartment of Applied Physics, Lanzhou University of Technology, Lanzhou 730050, China

cInstitute for Advanced Materials, South China Normal University, Guangzhou 510006, China

a r t i c l e i n f o

Article history:

Received 26 May 2018 Received in revised form 23 June 2018

Accepted 24 June 2018

Keywords:

Barium titanate Ferroelectricity Electrical conductivity Charged defects

a b s t r a c t

Ferroelectric domain walls have been promised for some potential applications due to their unique properties. In particular, the electrical conductivity of charged domain walls (DWs) allows a new dimension to ferroelectric functionalities. In this work, we construct two representative types of charged DWs, i.e. head-to-head (HH) wall and tail-to-tail (TT) wall, and employ theﬁrst-principles method to study the electronic structure of these charged walls in BaTiO3and the interactions between them and oxygen vacancies. It is revealed that the HH walls show then-type conductivity, but the TT walls show thep-type conductivity. While embedded oxygen vacancies attract the TT wall and repel the HH wall, the interaction between the walls and oxygen vacancies depends on the vacancy occupation. This interaction enhances the conductivity of HH walls and reduces the conductivity of TT walls, and in particular a TT wall in binding with oxygen vacancies will drive the transition ofp- type wall conductivity into n-type wall conductivity. The interaction of these walls with oxygen vacancies is discussed using the electrostatic model. This work represents a comprehensive understanding of electrical transport of charged DWs in ferroelectrics and possible roadmaps for manipulation.

1. Introduction

Ferroic domain structure has been a long-standing topic because it contains a characteristic scale essential for ferroic properties[1,2]. An understanding of domain structure in ferroics becomes highly necessary[3e5]. Attention is being paid to domain wall (DW) itself where the electronic structure is different from bulk[6,7]. It is known that ferroelectrics exhibit complex domain structures because of relaxation of depolarization energy together with other ingredients of physics[8]. A few of emergent phenomena associated with ferroelectric DW have been observed[9e16].

One concerned effect is the unusual electrical transport inside a wall. The enhanced wall electrical conductivity (hereafter abbreviated as wall conductivity)[17e20]mayﬁnd relevant potential applications in microelectronics [21] and energy-conversion

devices[22,23]. In addition, a wall can be multiferroic[24], further stimulating interest in electronic property inside a wall.

Although wall conduction has been observed in numerous ferroelectrics such as BiFeO3and PbTiO3[15,17,25e30], no clear correlation between wall conﬁguration and its conductivity is understood, noting that wall conductivity can be observed in both small band-gapped YMnO3 (~1.55 eV) and large gapped BaTiO3

(BTO) (3.2 eV)[31,32]. So far, most observations have been focused on charged wall across which the continuity of electric polarization is broken. The head-to-head (HH) wall and tail-to-tail (TT) wall are two simple cases and usually have better conductivity than others [17e19,27]. A schematic representation in Fig. 1(a) shows three domains separated by two walls. The electrons in the outer shell of cationic ion near a HH wall would have higher energy, likely to be excited into the conduction band. This is equivalent to then-type carrier doping. For a TT wall, the situation is just opposite, equivalent to thep-type carrier doping.

The above discussion hints potentials to mediate the wall conductivity by electric polarization [11,16,18,20]. The charge

*Corresponding author. Laboratory of Solid State Microstructure, Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China.

E-mail address:[email protected](J.J. Gong).

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doping via various approaches could be a good scheme to improve the wall conductivity. On the other hand, the role of lattice defects in influencing transport properties has been well recognized, suggesting the defects-oriented strategy to improve wall conductivity. For instance, wall conductivity in BiFeO3thin films can be remarkably mediated by Fe⁴^þions and Bi vacancies [9]. The interactions of walls with lattice defects may lead to DW variants in PbTiO3 [10]and other systems [33e37]. Theoretical calculations [38] revealed that point defects may have lower formation energy inside walls as sinks on one hand and are pinned by immobile defects on the other hand [15]. For ferroelectric oxides, the role of oxygen vacancies (OVs) has been often discussed, including the accumulation and pinning in walls [39e41]as well as structural variations[30,32]. Here it is noted that an implanted OV is bivalent unless otherwise stated. Owing to the electrostatic interaction of an OV with its neighbors, the electronic structure and spatial configuration of a wall will be modulated. For a concrete case, attraction between OV and TT wall and repulsion between vacancy and HH wall are expected.

Meanwhile, neighboring HH wall and TT wall also have electrostatic interaction. One can propose an interaction triangle con- necting OV, HH wall, and TT wall, as shown inFig. 1(b).

Nevertheless, such a triangle has not yet well understood [9,33,40]. First, the electronic structure and conductivity of a charged wall remain to be investigated in detail. Second, the correlation between wall conducting behaviors and charged defects deserves for attention. For simplifying consideration, one takes BTO as the object of present study. A charged wall can be generated by the super-bandgap illumination method at room temperature [13,42], and a metallic-like conduction with conductivitys^{~ 10}⁹ times larger than that inside domain is possible [16]. BTO is a ferroelectric in which the density of OVs can be mediated over many orders of magnitude, allowing the appearance of insulating, semiconducting, or metallic wall. Furthermore, earlier experiments [39,41]revealed that OVs can aggregate near some DWs, making their concentration much higher than other locations. This indicates the necessity for addressing the interactions between OVs and charged walls.

Owing to the complex interactions between charged wall and defects, a phenomenological study based on effective Hamiltonian is complex. Theﬁrst-principles approach allows us to proceed from the speciﬁc atomic structure to study these complex interactions. In this work, we mainly address the electronic structure of HH wall and TT wall and their interaction with charged defects such as OVs.

2. Model and computational details 2.1. Structural model

We start from a simple model based on a tetragonal BTO lattice withﬁxed symmetry (P4mm)[43]. The optimized lattice constants after a full relaxation area¼3.9851 Å,c¼4.1773 Å, in agreement with earlier results[44,45]. The lattice for the present calculation is a long strip supercell of 11Lin dimension with electric polarization along the c-axis, as shown in Fig. 2a. The periodic boundary conditions are applied. In this case, the ground state must be a monodomain[15,46].

Fig. 1.(a) A schematic representation of HH wall and TT wall as neighbors. The blue arrows indicate the polarization, the symbol‘þ’and‘’denote the charges. (b) The interaction triangle between HH wall, TT wall, and oxygen vacancy. HH, head-to-head; TT, tail-to-tail.

Fig. 2.(a) A schematic drawing of a wall zone. Each arrow indicates a BTO unit cell and also the polarization direction. The shaded areas at the two ends are the end zones of the lattice. (b) The distance of Ti atom and O atom near a domain wall along thec-axis, dz Ti-O, as a function ofLdm. The inserted picture in (b) is an enlargement of the unit cell near the HH wall plane. BTO, BaTiO3; HH, head-to-head.

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To construct a HH wall or TT wall, we divide the lattice strip into three zones. The two end zones as shaded inFig. 2a are set, and each consists ofLendunit cells whose atomic coordinates areﬁxed as the values of bulk BTO. The middle zone is called the wall zone withL_dmunit cells. In our calculations, only the wall zone is allowed to relax. Clearly, a HH wall or TT wall in the wall zone will be spontaneously generated if the polarizations in the two end zones are aligned inward or outward. The lattice shows lower energy if the wall central plane (hereafter abbreviated as wall plane) is on a Ba-O plane instead of a Ti-O plane, similar to earlier results for 180 wall[47,48].

Certainly, assigning the two end zones is an artifact in order to assure a HH wall or TT wall inside the wall zone. A sufficient wide wall zone is necessary for minimizing such artificial effects. We fixLend¼ 3 and vary Ldm to check the variation of local lattice

structure by monitoring the distance of Ti-O atom near the wall plane,d_{z Ti-O}, as a function ofL_dm, as shown inFig. 2b. It is found that dz Ti-O changes little when Ldm> 6. In our calculations, we takeLdm8, i.e. the wall zone is at least eight unit cells in width.

A largerL_dmis certainly better, but the computation cost increases hugely.

2.2. First-principles calculations

The electronic structure is calculated using the Vienna Ab- initio Simulation Package [49], and the exchange correlation functional is described by the generalized gradient approximation with the Perdew-Bruke-Eruzerhof functional [50]. The pro- jector-augmented wave pseudo-potentials [51] with 5s²5p⁶6s², 3s²3p⁶3d²4s², and 2s²2p⁴ valence electron conﬁgurations are

Fig. 3.The lattice model for a HH wall and TT wall in absence of oxygen vacancy. The atomic coordinates of the two end zones areﬁxed to be identical to bulk BTO, and the lattice between the two end zones is assigned as the wall zone. (a) A HH wall zone with non-relaxed lattice and relaxed lattice, and (b) a TT wall zone with non-relaxed lattice and relaxed lattice. The coarse solid arrows represent the local polarization. Parametersdandd⁰stand for the separation between Ti and O atoms in a Ti-O plane along thez-axis. (c) The number labeling of lattice atoms in the domain wall zone. BTO, BaTiO3; HH, head-to-head; TT, tail-to-tail.

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used for Ba, Ti, and O atoms, respectively, noting that we use the semicore pseudo-potentials to include the contributions from inner electrons of Ba and Ti atoms. The cutoff energy (Ecut) for the plane wave basis is set at 550 eV. A Monkhorst-Pack k-point sampling with 888k-mesh for structural optimization and self-consistent calculation for BTO unit cell is used. For the supercell containing a wall, the 881k-mesh for structural optimization and 14 14 1 k-mesh for self-consistent and electron density of states (DOS) calculation are used. The convergence thresholds are set at 1.0 10⁶ eV in energy and 0.005 eV/Å in force. Here, for data reliability, we further increase the value of Ecut and the number of k-points and ﬁnd no remarkable variations of the total energy, atomic force, and atomic coordinates. Therefore, E_cut¼550 eV and the chosenk- points are sufﬁcient for the data reliability.

3. Results and discussion

3.1. Structural and electronic properties

Weﬁrst look at the lattice without defect. For either a HH wall or a TT wall, initially assigned wall may remain stationary during the relaxation, but the local polarization in the wall zone is position-dependent. The polarization becomes weaker, and the deviations of O and Ti atoms from the high symmetry positions become smaller as the cell position is closer to the wall plane, as shown in Fig. 3. Usually, the ferroelectric polarization is calculated using the Berry phase method in ﬁrst-principles calculations. This method cannot be applied to the present case because the polarization is non-uniform over the whole lattice because of the existence of imposed DWs. As a compromise and also for an approximation treatment, we measure the interdistance d_z _Ti-O between Ti atom and centroid of its surrounding six oxygen atoms, and this distance should be proportional to the local electric dipole moment of each unit cell. In Fig. 3, the dipole moments are scaled using the coarse solid arrows with the length proportional to the dipole moment.

The local atomic coordinates and lattice distortions inside the wall zone before and after the lattice full relaxations are plotted in Fig. 3, and details can be seen by zoom-out looking.

For understanding the electrical conductivity inside the wall zone, we mainly look at the DOS. In this case, one labels the atomic position numerically in the wall zone, as shown inFig. 3(c) where Ba^2þ, Ti^4þ, and O^2-are indexed using various color integer numbers. The calculated DOS obtained by summing the atomic projected DOS (PDOS) of each atom in the wall zone is plotted in Fig. 4. For a HH wall, the Fermi level (Ef) moves into the conduction band with a remarkable non-zero DOS atE_f, indicating the n-type conductivity. For a TT wall, the Fermi level (Ef) shifts into the valence band with a remarkable non-zero DOS too, exhibiting the p-type conductivity. The calculated electronic structure is qualitatively consistent with the prediction from an earlier Landau theory[52]. These non-zero DOS values indicate that the wall zone is no longer an insulator[53].

These effects can be understood using a simple electrostatic charge scenario. Taking one unit cell in the end zone as a reference, one canﬁnd the variation of atomic coordinates in the unit cell at the wall plane. For a HH wall, the cell at the wall plane is charged positively since the distance of Ti atoms from the wall plane becomes smaller. Here, the Ti atoms become Ti^4þ by releasing electrons that cannot be completely collected by surrounding O atoms, allowing the wall more free electrons than

holes. This scenario is expressed as/(q))inFig. 4(a), whereq represents the amount of free charges. This effect is similar to electron doping by other methods. Similarly, a TT wall can be expressed as)(þq)/and considered as the hole doped zone, as shown inFig. 4(b). Therefore, both HH wall and TT wall can be viewed as charged and conducting walls. They also attract or repel opposite charges, suggesting the appearance of interaction between walls and charged defects.

For more details, we evaluate the contribution of each atom to the DOS, i.e. the atomic PDOS nearEf.Fig. 5(a) presents the DOS nearEffor the case of HH wall and the valance band is at ~1.8 eV belowE_f. The PDOS nearE_ffrom atoms at different layers are plotted inFig. 5(b) and (c), and it is seen that the PDOS is mainly contributed from Ti atoms, and those atoms at smaller distances to the wall plane contribute more to the PDOS. These contributions can be also plotted in the histograms, as shown inFig. 5(e)e(g). The DOS(E_f) from Ba and Ti atoms decreases rapidly with the distance from the wall plane. As shown inFig. 5(g), the O atoms in the Ti-O layers (O positions:3,1, 1, 3) have larger DOS(E_f) than those in the Ba-O layers (O positions:4,2, 0, 2, 4).

For the TT wall, the calculated results are summarized in Fig. 6. In this case,E_fshifts into the valence band, and the conduction band is ~2.0 eV above Ef. The PDOS near Efis mainly contributed by O atoms while Ba and Ti atoms contribute little. It is thus suggested that OVs, if properly doped into the DW region, would change remarkably the electrical conductivity of TT wall instead of HH wall. This suggestion can be further biased by the DOS(Ef) histogram for the TT wall, as shown in Fig. 6(e)e(g).

While the DOS(E_f) from Ba and Ti atoms is negligible, the O atoms Fig. 4.Calculated density of states (DOS) near the Fermi levelEffor (a) a HH wall and (b) a TT wall. Here,Efis set to zero. The HH wall can be treated as a zone doped with negative charges, showing then-type conductivity. The electrostatic interaction is expressed by/(q)). The TT wall can be seen as a zone doped with positive charges, showing the p-type conductivity. The electrostatic interaction is expressed by )(þq)/. HH, head-to-head; TT, tail-to-tail.

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at positions 0 and±2 have the largest contributions to the PDOS atEf, a reasonable fact. In the other words, for a TT wall, an OV exactly at the wall plane has the most remarkable inﬂuence on wall conductivity.

3.2. Interactions between wall and oxygen vacancy

The more attractive issue is the interaction between DW and OVs. Again, we consider two speciﬁc cases without losing the generality: one is that the polarizations at the two end zones point inward (I) and the other is that they point outward (II). As discussed earlier, there would appear an HH wall and a TT wall in the wall zone, respectively, if no OV is embedded.

We calculate the total lattice energy for these lattices before and after the lattice relaxation, respectively. For the case I, the results for the non-relaxed lattices are plotted inFig. 7(a) in the histogram format. It is seen that the energy for vacancy at Ba-O layers is higher than that for vacancy at Ti-O layers. The highest energy appears when the vacancy is at the middle Ba-O layer, a reasonable result because the wall plane is located at this layer. However, these

results are physically insigniﬁcant since the lattice is non-relaxed and the wall plane is ﬁxed at the middle Ba-O layer. When the lattices are fully relaxed, the results are very different and plotted in Fig. 7(b) in the histogram format too. It is clearly shown that the energy for vacancy at Ba-O layers becomes much lower than that for vacancy at Ti-O layers, and the lowest energy occurs when vacancy is at the middle Ba-O layer. It is thus demonstrated that a vacancy prefers to occupy Ba-O layer instead of Ti-O layer in the wall zone.

For the case II, we also calculate the lattice total energy given different oxygen atoms in the wall zone are one by one replaced with a vacancy. The energies for the non-relaxed lattices are plotted inFig. 7(c). It is seen that the lowest energy occurs when the vacancy is at the middle Ba-O layer. The energy increases when the vacancy occupies other sites, and the higher the energy the more the distance of the vacancy from the middle line. Very differently too, when the wall zone is allowed to fully relax, the results are summarized inFig. 7(d). For all cases, the energies for vacancy at different Ba-O layers are identical but much lower than those for vacancy at different Ti-O layers.

Fig. 5.The atomic projected density of states (PDOS) near Fermi levelEffor a Ba, Ti, and O atoms in a HH wall zone. The numbers in (aed) and the number on thex-axis in (eeg) indicate the atomic positions marked inFig. 3(c). HH, head-to-head; DOS, density of states.

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The substantial difference in energy between the non-relaxed and fully relaxed lattices is due to reconfiguration of domain structure in the wall zone. For case I, we start from an initial OV-free lattice with a HH wall (Fig. 8(a)). A replacement of one oxygen atom in the wall zone leads to reconfiguration of the lattice, as shown in Fig. 8(b)e(g), and the lattice energies for these configurations are plotted inFig. 7(b). Interestingly, if the vacancy occupies a Ba-O layer, three walls will be generated: one TT wall bound with the vacancy and two HH walls that take the positions self-consistently so that the total energy is minimized. If the vacancy appears at a Ti- O layer, only one HH wall is formed which appears at either Ba-O layer or Ti-O layer far from the vacancy position. For case II, we also start from an initial lattice with a TT wall (Fig. 9(a)), the fully relaxed configurations are shown inFig. 9(b)e(g), and the lattice energies for these configurations are plotted inFig. 7(d). It is seen that the wall plane is always tightly bound with the vacancy no matter where the vacancy occupies.

The above phenomena can be explained by the electrostatic interaction in the straightforward way. Certainly, a HH wall is positively charged, repelling OV, and leading to a separation of the

vacancy from the wall, as shown inFig. 8(h). Therefore, a combination of a vacancy at Ba-O layer and a HH wall always generates two HH walls separated by a TT wall bound with the vacancy. A TT wall is negatively charged, and it certainly attracts the vacancy, and thus the vacancy is always bound with the wall plane, as shown in Fig. 9(h).

Here, an unsolved issue is why a combination of a HH wall and a vacancy at Ba-O layer results in two additional HH walls. If the vacancy is at Ba-O layer, it means missing of the O atom between two neighboring Ti atoms. We look at the case ofFig. 8(b) without losing the generality. The distance between the two neighboring Ti atoms will increase due to the strong repulsive force, and the distance between the next-neighboring Ti atoms will increase too.

Consequently, local polarizations pointing away from the vacancy are generated. Noting that the polarizations in the two end zones are inwardﬁxed, two additional HH walls appear.

The inﬂuences of OV on the domain structure, as discussed above, can be illustrated by the simple diagram inTable 1. If the polarizations in the two end zones are inward-aligned, as shown in the upper part of Table 1, there may appear three domain Fig. 6.The atomic projected density of states (PDOS) near Fermi levelEffor a Ba, Ti, and O atoms in a TT wall zone. The numbers in (aed) and the number on thex-axis in (eeg) indicate the atomic positions marked inFig. 3(c). TT, tail-to-tail; DOS, density of states.

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conﬁgurations (modes) in the wall zone: mode 1, mode 2, and mode 3, where mode 1 is the reference free of vacancy. If the two end zones have their polarizations outward-aligned, as shown in the bottom part ofTable 1, there may appear also three domain conﬁgurations (modes) in the wall zone: mode 4, mode 5, and mode 6, where mode 4 is the reference free of vacancy. The six modes have different electronic structures and thus different wall conductivities.

3.3. Wall conductivity

Now we calculate the DOS of the wall zone for the six modes listed inTable 1. Without losing the generality, inFig. 10(a) and (b) we present the calculated DOS data. They correspond respectively to the results for modes 1, 2, 3 and modes 4, 5, 6. As shown in Fig. 10(a), the three modes have similar band structures, and then- type conduction behavior remains unchanged when a vacancy occupies a site at either Ba-O layer or Ti-O layer. Looking at the DOS atEf, one sees that all the cases have larger DOS(Ef) with respect to mode 1. Mode 2 where the vacancy is at Ba-O layers exhibits the largest DOS(E_f) with a larger slope in conduction band edge, suggesting the best electrical conductivity, as shown inFig. 10(c) as a locally exempliﬁed plot. Nevertheless, for modes 5 and 6 where the

TT wall is always bound with the vacancy, the situations become quite different. While mode 4 shows the p-type conduction behavior, as shown inFig. 10(b), mode 5 and mode 6 exhibit then- type conduction behaviors. The DOS(Ef) is slightly suppressed by the vacancy. In particular, for mode 6 where the vacancy occupies a site at Ti-O layer, DOS(Ef) becomes nearly zero with a smaller slope in conduction band edge, suggesting that the wall becomes insulating.

One may be more interested in comparing the DOS(Ef) for the six modes. The results are summarized inFig. 11(a)e(c) for modes 1, 2, 3, andFig. 11(d)e(f) for modes 4, 5, 6. For the former three modes, the occupation of vacancy at Ba-O layer allows a remarkable enhancement of the Ti atomic contributions to the DOS(Ef), while the Ba and O atomic contributions are relatively weak. When the vacancy occupies the Ti-O layer, the major contributions are also from the Ti atoms. The Ti atoms at the wall plane make the largest contributions. Because OV and HH wall are both positively charged, the enhanced charge aggregation at HH wall prompts more electrons into the conduction band. On the other hand, for latter three modes (4, 5, 6), the wall zone contains only one TT wall that is bound with the vacancy for modes 5 and 6. For mode 4 free of vacancy, the contribution of DOS(Ef) from the O atoms at the TT wall is the largest, while those from the Ba and Ti atoms are weak.

Fig. 7.The lattice total energy histogram when an oxygen vacancy occupies different sites as number label indicated. For a HH wall zone, the data in the non-relaxed state (a) and in the relaxed state (b) are plotted respectively. For a TT wall zone, the data in the non-relaxed state (c) and in the relaxed state (d) are plotted respectively. The oxygen atom number labeling is shown in (e). HH, head-to-head; TT, tail-to-tail; OV, oxygen vacancy.

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However, when a vacancy is induced, the DOS(Ef) from the O atoms becomes small, and the major contributions are from the Ti atoms near the wall plane. The conductivity of the wall zone is thus suppressed as a whole, in particular for mode 6 where only the Ti atom near the wall plane makes some contribution and the others contribute little. In this case, the conductivity of wall zone becomes negligible.

To understand the reason for the difference of DOS(Ef) in modes 1 to 6, one may discuss the band edges and relative position of the

Fermi level in these modes, as shown inFig. 12 for a schematic illustration. For the case of HH wall (mode 1), the conduction band bottom (E_c) is lower than E_f, and thus electrons are inside the conduction band, leading to then-type conduction. Thep-type conduction makes sense for the TT wall (mode 4) where the valence band top (E_v) is higher thanE_f. For mode 2, more electrons are allowed in the conduction band, resulting in the best conductivity.

In modes 5 and 6, the DOS(Ef) becomes quite small, indicating low conductivity. The twon-type conduction channels are possible for Fig. 8.The fully relaxed lattice structure in the wall zone with a HH wall, given the oxygen vacancy taking different sites (a)e(g), where the arrows indicate the polarization. A HH wall is marked using/), and a TT wall is marked using with)/. (h) A schematic drawing of the electrostatic interaction between an oxygen vacancy and a charged domain wall. HH, head-to-head; TT, tail-to-tail; OV, oxygen vacancy.

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mode 3 because of the repulsion between HH wall and OV. It is also noted that the band sloping increases if OV and HH wall coexist, giving rise to larger DOS. A combined TT wall and OV reduce the band sloping, and thus smaller DOS is expected.

Furthermore, another issue is why a vacancy in the TT wall changes the carriers from thep-type to then-type. Since the TT wall and vacancy carry opposite charges, it is reasonable that the charge aggregation at the wall plane will be partially suppressed when the vacancy is absorbed into the wall plane. In addition, the vacancy as

the pinning center will hinder the motion of wall. Owing to the fact that OV carries more charges than the wall, a reversal of carriers from thep-type to then-type is a natural consequence.

3.4. Discussions

We would like to add some more discussion on the interactions between charged defect and charged wall in ferroelectrics. It has been revealed that an OV is always the sink of a TT wall plane, but it Fig. 9.The fully relaxed lattice structure in the wall zone with a TT wall, given the oxygen vacancy taking different sites (a)e(g), where the arrows indicate the polarization. A HH wall is marked using/), and a TT wall is marked using with)/. (h) A schematic drawing of the electrostatic interaction between an oxygen vacancy and a charged domain wall. HH, head-to-head; TT, tail-to-tail; OV, oxygen vacancy.

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is not for a HH wall plane. As shown by mode 2 and mode 3 in Table 1, the HH wall always repels OV. This allows an opportunity to modulate the HH wall positions by charged defect engineering. It is noted that the electrical conductivity of a charge wall would always be damaged if it is absorbed by a charged defect. Aﬁnite distance

between a‘high-quality’and defect-free charged wall would have the best conductivity. One may explore charged defect engineering approaches to‘generate’a conductive wall at an assigned position for speciﬁc applications. In this sense, the present prediction of the mode 2 and mode 3 is of practical signiﬁcance.

Table 1

Several domain modes (patterns) in the domain wall region in presence of oxygen vacancy. Blue arrow indicates the polarization orientation near a wall and pink open cycle indicates the vacancy. The domain modes are illustrated in variousﬁgures (e.g.Fig. 8(b), (d), (f)).

No vacancy One vacancy

at Ba-O layer at Ti-O layer

Mode 1, Fig. 8(a) Mode 2, Fig. 8(b, d, f) Mode 3, Fig. 8(c, e, g)

No vacancy One vacancy

at Ba-O layer at Ti-O layer

Mode 4, Fig. 9(a) Mode 5, Fig. 9(b, d, f) Mode 6, Fig. 9(c, e, g)

Fig. 10.The calculated density of states (DOS) for different domain modes in the presence of one oxygen vacancy. (a) Modes 1~3, (b) modes 4~6. The enlarged view near the Fermi level (c) for modes 1~3, and (d) for modes 4~6.

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On the other hand, it is noted again that the conduction of both HH wall and TT wall can be explained as the doping effect (self- doping). A HH wall allows then-type conduction due to the electron aggregation, while a TT wall favors thep-type conduction due to the hole aggregation. In this sense, the interactions between charged walls and OVs can be described by the electrostatic interaction model. As a result, the effect of OV on charged wall conductance can be predicted by the charge superposition.

Although this scenario is obtained by considering the special cases here, i.e. the interaction between one OV and a HH wall or a TT wall, the physical nature for other more complicated situations remains the same. Therefore, this model can be reasonably extended to more extensive systems.

Finally, we have to mention that the present calculations are majorly based on a simpliﬁed model lattice where the periodic boundary conditions are employed, making a quantitative calculation of the electrical conductivity in the DW region chal- lenging if not impossible. Even though the DOS data can be

trusted in quantitative sense, the carrier mobility would be an issue either.

4. Conclusion

In this work, the electronic structure and atomic configurations of HH wall and TT wall in BTO and their interactions with OVs have been investigated using thefirst-principles method. It is found that the HH wall shows the n-type conductivity while the TT walls exhibit thep-type conductivity in the absence of OV. Significant interactions between the charged DWs and OVs have been found.

The repulsive interaction between a HH wall and a vacancy may make the domain structure reorganize, and the wall plane prefers to move away the vacancy. This repulsive force also allows an opportunity for charged defect engineering of conductive DW conﬁguration. As commonly known, the attractive interaction between a TT wall and a vacancy always drive the wall plane to be absorbed into the vacancy site and pinned by the vacancy. As a Fig. 11.The calculated DOS(Ef) histograms of the wall zone for the six domain modes listed inTable 1, (a), (b), (c), (d), (e) and (f) for modes 1, 2, 3, 4, 5 and 6, respectively. The blue, green, and red bars represent the contributions from Ba, Ti, and O atoms, respectively. Thex-axis represents the position of atomic layer in the model, consistent with the representation of oxygen atoms inFig. 3(c). The horizontal long blue arrows indicate the polarizations, marking the position of domain wall plane. DOS, density of states; OV, oxygen vacancy.

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consequence, the presence of OV can signiﬁcantly increase the conductivity of the HH wall but seriously reduce the conductivity of the TT wall. The interactions between the charged DWs and defects can be described by the electrostatic interaction. It is believed that the present work represents a comprehensive understanding of the electronic structure and electrical conductivity of charged DWs and their control by defect engineering although the model lattice for our computation may be slightly over simpliﬁed.

Acknowledgments

This work was supported by the National Key Research Program of China (Grant Nos. 2016YFA0300101 and 2015CB654602) and the National Natural Science Foundation of China (Grant Nos.

51431006, 51332006).

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Fig. 12.Schematic diagrams of band edge conﬁgurations for the six modes discussed inFig. 11, (a), (b), (c), (d), (e) and (f) for modes 1, 2, 3, 4, 5 and 6, respectively. The red vertical line indicates the position of wall plane or OV. The upper and lower black lines indicate the positions for bottom of conduction band (Ec) and top of valence band (Ev), respectively.

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(13)

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