J. Phys. Chem. C 2007, 111, 6343-6349
6343
First-Principles Investigation of Stability and Structural Properties of the BaTiO3 (110) Polar Surface Ying Xie,†,‡ Hai-tao Yu,† Guo-xu Zhang,† Hong-gang Fu,*,†,‡,§ and Jia-zhong Sun†,§ School of Chemistry and Materials Science, Heilongjiang UniVersity, Harbin 150080, People’s Republic of China, Department of Applied Chemistry, Harbin Institute of Technology, Harbin 150001, People’s Republic of China, and State Key Laboratory of Theoretical and Computational Chemistry, Institute of Theoretical Chemistry, Jilin UniVersity, Changchun 130023, People’s Republic of China ReceiVed: September 10, 2006; In Final Form: February 27, 2007
The atomic and electronic structures of the BaTiO3 (110) polar surface were systematically investigated by first-principles density functional theory (DFT) calculations with use of slab models. The relaxations and rumplings of five different (1 × 1) terminations were considered. According to the results of the charge redistribution, the polarity compensation conditions can be achieved in both stoichiometric and nonstoichiometric terminations, but their compensation mechanisms are obviously different. For the BaTiO and O2 stoichiometric terminations, the intensive electronic structure changes with respect to the bulk crystal result in larger structural distortions and cleavage energies than the nonstoichiometric ones. For the TiO, Ba, and O nonstoichiometric terminations, whose electronic structures are qualitatively similar to that of the bulk crystal, their insulating characteristics are retained because no filling of surface states was found. Furthermore, the computation results of the surface grand potentials (SGPs), which were used to distinguish the relative stabilities of different terminations, clearly suggest the existence of four distinct stable (110) terminations, in which the BaTiO stoichiometric termination can only exist in a small region with O-poor condition.
1. Introduction Barium titanate is a material that has many potential applications. Its high dielectric constant makes it a very promising candidate in various electronic devices.1 In recent years, with the miniaturization trend in device size, lowering the dimensions of BaTiO3 crystal has become the focus of many research efforts, and the nanocrystal and single-nanowire BaTiO3 materials have been prepared and characterized successfully.2-4 At the same time, due to the applications of BaTiO3 in the fields of thin films and catalysts, there has been continuous interest in its surface structures and properties.5,6 Among all the orientations of the BaTiO3 crystal, the (100) surface was the most extensively studied owing to its nonpolar characteristic and high stability. On one hand, its surface step edges preferentially along [100] axes and the in-gap electronic states below Fermi energy have been characterized by scanning tunneling microscopy (STM) and ultraviolet photoelectron spectroscopy (UPS), respectively.7-9 On the other hand, larger surface relaxations, which can cause surface rumplings and polarizations, have been found in previous theoretical studies,10-12 and no deep-gap states were found.11,12 Relative to the wellknown (100) surface, however, the (110) and (111) surfaces were much less known. Such a scarcity may be attributed to their polar characteristics. Although a dense and smooth (110) surface has been successfully prepared by a two-step method including the pulsed laser deposition technique,13 our knowledge on this polar surface is still limited. To reveal the structure and polarization properties, Heifets and co-workers have investigated four nonstoichiometric (110) terminations by the semiempirical †
Heilongjiang University. Harbin Institute of Technology. § Jilin University. ‡
shell model, and their results showed that the relaxations and surface energies of the (110) terminations are much larger than those of the (001) terminations.14-16 The results sufficiently extend our knowledge on this polar surface. Although some achievements have been realized, the stability problem that is very important to the applications of polar surface was still unsettled. Along the [110] orientation, the stack sequence of the charged layers (O2 and BaTiO) will result in a monotonic rise of the microscopic electric field, and thus, the polar surface is expected to be unstable. However, once the polarity perpendicular to the surface is suppressed, the polar surface is likely to be stabilized. As suggested by Noguera, two mechanisms can work for canceling the surface polarity.17,18 The first is to modify the surface composition, which can result in a nonstoichiometric and reconstructed surface. The second is to modify the surface electronic structure, which can lead to a total or partial filling of surface states. For the strongly ionic MgO compound, the former mechanism is clearly favored from both theoretical and experimental viewpoints,19,20 while the later one may work in the less ionic ZnO (0001) and (0001h) polar surfaces.21,22 According to a recent investigation, the surface polarity can also be compensated by a quite unusual mechanism, as an example, for the MgO (111) (2 × 2) surface in very O-poor conditions, the anomalous filling of surface states of the insulating nonstoichiometric termination leads to the polarity cancellation.23 But for BaTiO3 with ionic Ba-O and covalent Ti-O bonds, which mechanism is at work to cancel the surface polarity remains an open question. Even if the surface polarity is compensated by whichever mechanism, the stability of the surface should be discussed as a function of the chemical potential of the actual surface composition when considering the influence of the environment. Since the surface grand potential (SGP) is a measure of surface
10.1021/jp0658997 CCC: $37.00 © 2007 American Chemical Society Published on Web 04/06/2007
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stability and has been successfully applied to binary and ternary compounds,11,24-26 we introduce it to discuss the relative stabilities of different terminations of BaTiO3. To solve the structure and stability issues above, the present investigation is aimed at providing an application of firstprinciples approaches to the (110) polar surface of BaTiO3. In particular, three aspects are focused on (1) the atomic and electronic structures of the BaTiO3 (110) polar surface, (2) the mechanisms for canceling the macroscopic polarization, and (3) the relative stabilities of different terminations. 2. Computational Methods and Models 2.1. Methods. The calculations are performed within the framework of density functional theory (DFT)27 embedded in CASTEP code. The exchange-correlation energy is treated with generalized gradient approximation (GGA), using the PerdewWang parametrization.28 The electronic wave functions at each k-point are expanded in terms of a plane-wave basis set, and an energy cutoff of 380 eV is employed. Since there are some inherent limits on optimizing the convergence of norm-conserving pseudopotential (NCP),29 the ultrasoft pseudopotential (USP), which allows the calculations to be performed with the lowest possible energy cutoff for the plane-wave basis set, is used.30 The original valence configurations for pseudopotentials are 3s23p63d24s2 for Ti, 5s25p66s2 for Ba, and 2s22p4 for O. The sampling over the Brillouin zone is treated by a (6 × 4 × 1) Monkhorst-Pack mesh.31,32 The stable surface configuration is obtained by geometry optimization from the ideal unrelaxed surface. The whole optimization procedure is repeated until the average force on the atoms is less than 0.01 eV/Å and the energy change less than 5.0 × 10-6 eV/atom. To estimate the surface atom charges, the Mulliken population analysis is used in this study.33 To determine which termination is likely to emerge from the cleavage of the BaTiO3 crystal along the [110] orientation, the cleavage energy (Erel cl ) is calculated as follows,
Erel cl )
1 rel [E (A) + Erel slab(B) - nEbulk] 4S slab
(1)
where Erel slab is the total energy of relaxed termination, and S represents the surface area per unit cell. A and B denote the complementary terminations; they are BaTiO and O2 on one hand and TiO and Ba on the other. The O termination can be considered as self-complementary. To compare the relative stabilities of different terminations, the surface grand potential (SGP) is defined as34
Ω)
1 [E - NTiµTi - NBaµBa - NOµO] 2S slab
(2)
where NTi, NBa, and NO are the numbers of Ti, Ba, and O atoms in the slab, respectively, and the PV term and vibrational contributions in the equation are neglected.35,36 The term µ is the chemical potential of species. As the atoms in the bulk and on surfaces are in thermodynamical equilibrium, therefore, the chemical potentials of different atoms are not independent any more but are related to the chemical potential of the bulk crystal,35
µBaTiO3 ) µBa + µTi + 3µO
(3)
When neglecting the PV term and vibrational contributions to the bulk Gibbs free energy of the perovskite, the bulk chemical
potential is then replaced by the total energy of the bulk crystal (µBaTiO3 ≈ Ebulk). When µTi is eliminated by eqs 2 and 3 and the variations, ∆µO ) µO - (1/2)EOgas2 and ∆µBa ) µBa - µbulk Ba , are introduced, the SGP can be determined as
Ω)φ-
1 [∆µO(NO - 3NTi) + ∆µBa(NBa - NTi)] (4) 2S
where the constant φ is defined as
[
gas
EO 2 1 Eslab - NTiEbulk φ) (NO - 3NTi) 2S 2
]
Ebulk Ba (NBa - NTi) (5) bulk In eq 5, Ebulk Ba is the approximation of µBa , to which the vibrational contributions and volume effects are neglected.35 Furthermore, because the surface atoms do not precipitate out of the slabs, the chemical potential of each species must be lower than the Gibbs free energy of the corresponding atoms in the stable phase.35,36 Therefore, by replacing the Gibbs free energy with the total energy,35 the upper limits of each species are determined as34
1 ∆µO ) µO - EOgas2 < 0 2
(6)
∆µBa ) µBa - Ebulk Ba < 0
(7)
∆µTi ) µTi - Ebulk Ti < 0
(8)
By combining eqs 3 and 8, the lower limit of ∆µO and ∆µBa can be determined as follows,34 f ∆µBa + 3∆µO < -EBaTiO 3
3 gas f bulk bulk ) EBaTiO - Ebulk -EBaTiO Ti - EBa - EO2 3 3 2
(9) (10)
f where EBaTiO is the formation energy of BaTiO3 bulk crystal 3 with respect to the Ti and Ba atoms in their bulk phase and the O atom in the gas phase. Once the effective intervals of ∆µO and ∆µBa are determined, the accessible values of SGP are thus obtained. 2.2. Models. To accurately describe the surface charge distribution, the slab model is used in our calculations. The slab has to be thick enough to display bulk characteristics in its central layer and should be symmetric with zero total dipole moment. When a plane wave basis set is used for self-consistent calculations, the slab is repeated in the direction perpendicular to the surface. To get rid of the spurious interactions between these images, a vacuum region is needed. According to our tests, the vacuum region thickness should be 11 Å at least. The slabs should contain not less than 11 atomic layers for stoichiometric terminations and 9 for nonstoichiometric ones. As precise experimental results are lacking, we cannot clearly understand which terminations are availably stable under different thermodynamic conditions. Even if numerous terminations, from theoretical and practical viewpoints, may be obtained under different thermodynamic conditions, only five terminations, two stoichiometric (O2 and BaTiO) and three nonsto-
Investigation of the BaTiO3 (110) Polar Surface
J. Phys. Chem. C, Vol. 111, No. 17, 2007 6345
Figure 1. Slab model of the BaTiO3 (110) polar surface. The smallest white spheres represent oxygen atoms, the medium green spheres represent barium atoms, and the largest dark spheres represent titanium atoms.
TABLE 1: Relaxations (Å) and Rumplings (R, Å) of Different Terminations along Surface Normala termination layer 1 Ba Ti O R layer 2 Ba Ti O R layer 3 Ba Ti O R layer 4 Ba Ti O R
TiO
BaTiO
O2
-0.07 -0.04 0.40 0.47
-0.25 (-0.28)b 0.24 0.22 (0.26)b 0.47 (0.54)b
0.38 -0.09 0.23 -0.16 -0.01 (0.00)c 0.54 0.01 -0.08 -0.06 0.11 0.19
-0.03 0.05 0.16 -0.03 0.08
0.04 0.15 0.08 -0.02 0.17
0.05 (-0.02)c 0.07
Ba
O
-0.37 (-0.54)b -0.25 (-0.44)b
0.03 -0.01 0.01 -0.01 0.02
0.00
0.10 -0.04 0.06 0.14
0.06 (-0.08)c 0.14 -0.01 0.00 0.02 0.03
a Positive (negative) displacement means the direction outward (inward) from (to) the surface. b The values are the results derived from the shell mode.15 c The atomic displacements of O atoms in the same layer are no longer equivalent.
ichiometric (O, Ba, and TiO), as shown in Figure 1, are reasonably considered in the present study. 3. Results and Discussion 3.1. Surface Relaxations and Rumplings. In this simulation, all the atoms are allowed to relax. The calculated results of the atomic relaxations and rumplings of each termination along the surface normal are presented in Table 1. Because the layers below the central layer are mirror images of the upper ones, as plotted in Figure 1, only the results of the upper layers are listed. As shown in Table 1, in the BaTiO termination, the O atoms in the first layer intensively move outward by 0.40 Å (9.9% of the bulk lattice constant, cubic phase, a0 ) 4.008 Å), whereas the displacement decreases to 0.11 Å in the third layer and almost disappears in the inner layer. On the contrary, the Ba
and Ti atoms move inward and their displacements are relatively small. In the O2 termination, the larger outward displacements are computed to be 0.24 and 0.38 Å for O and Ba atoms in the first and second layers, respectively. Due to larger relaxations in the outermost two layers, the top bilayers of the BaTiO and O2 stoichiometric terminations are contracted severely. Moreover, the rumpling caused by surface relaxation decreases from 0.47 Å (the first layer) to 0.19 Å (the third layer) for the BaTiO termination, and from 0.54 Å (the second layer) to 0.17 Å (the fourth layer) for the O2 termination, indicating that the relaxation equilibrium can be established. For the three nonstoichiometric terminations, the atomic relaxations are somewhat moderate. In the first layer of the TiO termination, the Ti atoms move inward by 0.25 Å and the O atoms outward by 0.22 Å. As regards the Ba and O terminations, the largest relaxations are also found in the first layers and are predicted to be -0.37 and -0.25 Å, respectively. The abovecalculated results for the three nonstoichiometric terminations are in good agreement with the prediction derived from the shell model,15 as shown in Table 1. As the layer number increases, the atomic relaxations of the three nonstoichiometric terminations are quickly damped. Furthermore, Table 1 also shows a trend that the rumplings of stoichiometric terminations are much larger than those of nonstoichiometric ones (0.47-0.54 Å vs 0.14-0.47 Å in the first two layers, 0.17-0.19 Å vs 0.020.14 Å in the second two layers). Furthermore, it was observed that the atomic displacements of two kinds of O atoms in the same inner O2 layer of the TiO and O terminations are no longer equivalent, as shown in Table 1. A similar result was also found for the O termination of the SrTiO3 (110) surface.34 Bottin and co-workers have investigated the atomic relaxations of the SrTiO3 (110) surface by firstprinciples calculations and suggested that such a behavior corresponds to the anti-ferroelectric distortion (AFD). But no experiments suggested the BaTiO3 material with this behavior possesses the property of AFD. Therefore, the nature of the inequivalent atomic displacements in the two materials is very different. As can be noted in the study by Bottin and co-workers the rumplings of the inner O2 layers are not damped as a function of slab thickness.34 But our results, shown in Table 1, clearly indicate that the rumplings of the inner O2 layer are small and decrease quickly going inward through the slab. Recently, Heifets and co-workers investigated two O terminations of the BaTiO3 (110) surface with and without mirror symmetry along the [1h10] direction. They found that the symmetric configuration is higher in energy than the asymmetric one, and that only the asymmetric O termination reveals on-plane displacement.15 According to our calculated results, we found that the symmetric O termination converges to the asymmetric one after the full optimization procedure was performed, which agrees well with Heifets’ results. As the mirror symmetry along the [1h10] direction does not exist anymore, the rumpling in the O2 planes becomes possible. 3.2. Charge Redistribution. In general, most of the polar surfaces are expected to be unstable due to the existence of surface dipole moment. However, specific charge density modifications in the outer layers may cancel out the macroscopic component of the dipole moment and the polarity.17 In the polar surfaces constructed by equidistant layers, the condition for the cancellation of macroscopic dipole moment reads,18 m
∑ j)1
σj ) (-1)m+1
σm+1 ; 2
|σj| * σ, 1 e j e m, |σm+1| ) σ (11)
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TABLE 2: Atomic Charges (e) and Layer Excess Charges (e) for Stoichiometric Terminations BaTiO termination
O2 termination
layer
Ba
Ti
O
∆ea
1 2 3 4 5 6 bulkb
0.87
1.10
-1.00 -1.04 -1.08 -1.07 -1.09 -1.08 -1.07
-1.17 0.06 0.03 0.00 0.02 -0.02
1.37
1.88
1.48
1.77
1.48
1.71
Ba
Ti
1.34
1.68
1.40
1.65
1.51
1.79
O
∆e
-0.53 -0.98 -1.02 -1.00 -1.05 -1.06
1.08 -0.10 0.10 -0.09 0.04 0.10
a The layer excess charge relative to the bulk plane (elayer - ebulk-plane). The layer charges of BaTiO and O2 plane in bulk crystal are (2.14 e, respectively.
b
where σ is the charge of BaTiO or O2 layer in bulk crystal and equals the charge of layer m + 1 in the surface, while σj represents the charge of layer j, which has been modified and no longer equals the bulk value. In other words, the compensation criterion can be fulfilled when the total charge of the modified layers equals half of the bulk layer charge. Our calculated atomic charge and layer excess charge for stoichiometric and nonstoichiometric terminations are listed in Tables 2 and 3, respectively. The charges of Ba, Ti, and O atoms in bulk crystal are 1.48, 1.71, and -1.07 e, respectively. Then, the bulk BaTiO and O2 layers bear charges of 2.14 and -2.14 e, as presented in Table 2. In the BaTiO termination, the charges of the metal atoms in the outermost layer are reduced, while the topmost O atoms in the O2 termination are oxidized with respect to the bulk crystal. As the layer number increases, the charges of all atoms in the two terminations gradually approach the bulk. Since the total charges of six layers are -1.08 and 1.13 e, which are nearly half of the bulk value ((2.14 e), for the BaTiO and O2 terminations, respectively, the compensation criterion for polarity is thus fulfilled. As can be noted in Table 3, for the three nonstoichiometric terminations, the charge redistribution in their outer layers is also observed, and the total charges of the five layers are 1.21, 1.12, and -1.06 e for the nonstoichiometric Ba, TiO, and O terminations, respectively. Therefore, the condition for polarity compensation can also be achieved. Although the compensation criterion can be achieved for the considered five terminations, the compensation mechanisms are very different. It is often thought that surface relaxations and change of covalency in the surface layers are helpful in the stabilization process of the polar surface. But in an investigation by Noguera,17 in which a bond-transfer model for describing the charge distribution in oxide surfaces was proposed and used, the surface relaxation and change of covalency in the MgO (111) and SrTiO3 (110) polar surfaces were found to be invalid for canceling the polarity, and only the filling of surface states and change of surface stoichiometry can work to cancel the polarity. In BaTiO3, if the formal charges of +2, +4, and -2 e are assigned to Ba, Ti, and O atoms, respectively, then the Ba, TiO, and O terminations bear formal charges of (2 e, being half of the bulk layer charges ((4 e). Therefore, all three nonstoichio-
Figure 2. TDOSs of the first and second layers in stoichiometric terminations (smearing width: 0.10 eV).
metric terminations are self-compensated. On the contrary, the compensation criterion in stoichiometric terminations can be achieved only through the change of surface electronic structure. Such a change in electronic structure will lead to larger structural distortions of the stoichiometric terminations than the nonstoichiometric ones, which has been confirmed by the previously calculated relaxations and rumplings. Furthermore, the electronic structure change can also lead to larger cleavage energies of the stoichiometric terminations than those of the nonstoichiometric ones, which is confirmed in the next discussion. 3.3. Density of States. The total densities of states (TDOSs) of the first and second layers in the five different terminations are depicted in Figures 2 and 3. The Fermi energy is taken as the zero point (E - Ef ) 0.0 eV). Since the relative position of the conduction to valence bands is erroneous when the KohnSham eigenvalues are used,37 a scissor operator was introduced to fix the problem.38 The TDOSs of BaTiO and O2 layers in bulk crystal, whose band gap is corrected to the experimental value of 3.3 eV with the mentioned scissor technique,39 are also displayed in Figure 2b as a reference. The valence bands (VBs) located at about -5.0 to 0.0 eV are assigned to O2p orbitals, while the conduction bands (CBs) at about 3.5 to 6.0 eV are mainly composed of Ti3d orbitals.40 For the BaTiO stoichiometric termination, there is no change in its surface stoichiometry, and therefore, the electronic structure should change considerably for satisfying the compensation condition, and it can be noted from above discussion that the compensation condition is realized by charge redistribution. Our results identified the electronic structure changes as shown in Figure 2a. The Fermi level is located across the CBs and some CB states are filled by additional electrons. With the filling of the surface states, some surface atoms are reduced, which is very consistent with the atomic charges calculated in the preceding section. Thus, eq 11 can be fulfilled once the
TABLE 3: Atomic Charges (e) and Layer Excess Charges (e) for Nonstoichiometric Terminations Ba termination layer
Ba
1 2 3 4 5
0.40
Ti
1.42
2.43
1.52
1.95
TiO termination O
∆e
-1.12 -1.06 -1.08 -1.05
-1.74 -0.10 0.65 -0.02 0.28
Ba
O termination
Ti
O
∆e
1.34
-0.78 -0.85 -0.92 -1.06 -1.07
-1.58 0.44 0.03 0.02 0.07
1.51
1.58
1.49
1.79
Ba
Ti
1.21
1.59
1.46
1.88
O
∆e
-0.90 -1.03 -1.04 -1.03 -1.08
1.24 -0.37 0.06 0.17 -0.02
Investigation of the BaTiO3 (110) Polar Surface
J. Phys. Chem. C, Vol. 111, No. 17, 2007 6347 TABLE 4: Cleavage Energies (J/m2) of Different Terminations BaTiO Erel cl φa a
Figure 3. TDOSs of the first and second layers in nonstoichiometric terminations (smearing width: 0.10 eV).
surface atomic charges are reduced to half of the bulk value. As a result, in the BaTiO stoichiometric termination, the compensation criterion is actually achieved through the filling of surface states. Recently, Bottin and co-workers have investigated the stability of the SrTiO3 (110) polar surface by firstprinciples calculations.34 They confirmed that the filling of surface states was the only feasible mechanism for the polarity compensation in the SrTiO stoichiometric termination.34 Because of the lack of relevant data, a direct comparison with the experimental results on the BaTiO3 (110) polar surface is impossible at present. But for the SrTiO3 polar surface, the result probed by STM/S shows the presence of rather flat regions with metallic characteristics,41 which is very consistent with the typical Ti3+ and Ti2+ features in the XPS spectra and the metallic surface state with a Ti3d characteristic in the UPS spectra.42,43 On the basis of these experimental observations, Bando and coworkers proposed a SrTiO termination for the unreconstructed SrTiO3 (110) surface. Considering the similarity of bonding characteristics among ATiO3 perovskites, we can reasonably suppose that the above compensation mechanism may be very common in the ATiO stoichiometric terminations. For the O2 stoichiometric termination, the two O atoms in the first layer move close to each other to form a bond with a length of 1.49 Å, which indicates the formation of a peroxo group. This result can also be confirmed by the calculated TDOS. As shown in Figure 2c, the TDOS of the topmost O2 plane in the range of [-6.00 eV, 0.00 eV] changes extensively with respect to the bulk crystal. The new TDOS peaks at about -5.60, -4.59, and -4.00 eV correspond to σ and π bonding states formed by the O2p orbitals, while the peaks at about -1.11 and -0.18 eV denote π* antibonding states. The Fermi level is located at the top of the antibonding π* state and does not cross the energy bands. Consequently, the O2 termination still retains its insulating characteristic. Moreover, due to the emptying of antibonding σ* states, the surface charges are reduced in magnitude. According to the calculated atomic charges listed in Table 2, the charge of the O2 plane in the first layer is predicted to be -1.06 e, which equals half of the bulk layer charge. Therefore, the forming of a peroxo group will provide the charges that are required to compensate the surface dipole moment and lead to the fulfillment of eq 11. The above results agree well with those found in the O2 termination of the SrTiO3 (110) surface.34
2.90 4.81
O2
TiO
Ba
O
2.90 0.99
1.55 6.19
1.55 -3.10
0.98 0.97
Constants of eq 5 for different terminations.
For the three nonstoichiometric terminations, the compensation criterion can be achieved easily. With the adsorption or desorption of Ba and O atoms from ideal surfaces, the charges of the three terminations compensate for each other. In principle, no change of surface electronic structure is expected, and the surface may retain its insulating characteristics. To testify to this, the computed TDOSs of the first and second layers in different nonstoichiometric terminations are drawn in Figure 3. The results clearly indicate that the electronic structures of the Ba, TiO, and O terminations are qualitatively similar to that of bulk crystal, and no filling of surface states is observed. Therefore, in the three nonstoichiometric terminations, the polarity compensation can be achieved by the change of surface stoichiometry. Another problem that should be pointed out is the infinite charge issue. When the surface is terminated with the same type of crystalline planes (O2-O2, BaTiO-BaTiO), as depicted in Figure 1, the models obtained from a perfect-crystal cleavage will bear an infinite charge, which is known as another origin of the surface instability.17,44 According to our results, when the charges of all layers, including the mirror images below the central layer, are added, the sum is nearly zero. Consequently, the charge neutrality is achieved. Again considering the surface polarity is compensated for as discussed above, the polar surface is likely to be stabilized. 3.4. Thermodynamic Stability. To determine which termination is likely to emerge from the cleavage of the BaTiO3 crystal along the [110] orientation, we have performed relevant calculations, and the results are listed in Table 4. Our numerical results indicate that the O termination has the lowest cleavage energy (0.98 J/m2) among all terminations, in good agreement with the value (1.05 J/m2) derived from the shell model,15 whereas the cleavage energies of the O2 and BaTiO terminations are the highest (2.90 J/m2). Because the three nonstoichiometric terminations have lower cleavage energies than the two stoichometric terminations, and considering the fact that the polarity compensation of the nonstoichiometric terminations is achieved by the change of surface stoichiometry, it seems that the change of surface stoichiometry is more effective for the polarity compensation of the (110) polar surface than the changes of surface electronic structures. The mechanism is in good agreement with the stabilization of the SrTiO3 (110) polar surface.34 Although the nonstoichiometric terminations predominate over the cleavage or growth of the BaTiO3 crystal along the [110] orientation according to the calculated cleavage energies, the stoichiometric terminations should also be considered if they can be stabilized. The stoichiometric termination can be stabilized when the influence of environment is taken into consideration. For illuminating the effects of environment, the surface grand potential (SGP) is introduced. To determine the accessible values of the SGP, one should first calculate the effective intervals of ∆µBa and ∆µO according to the formation energy of BaTiO3 bulk crystal, as defined in ref 34. In our calculation, the obtained formation energy of BaTiO3 is -17.70 eV. On the basis of eq 9, we can deduce the lower limits of ∆µBa and ∆µO to be -17.70 and -5.90 eV, respectively. Thus ∆µBa and ∆µO are restricted within the intervals of [-17.70
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Figure 4. Stability diagram of the BaTiO3 (110) (1 × 1) surface. The surface grand potential (SGP) is expressed as functions of the excess Ba and O chemical potentials (∆µBa and ∆µO).
Figure 6. SGPs of different terminations in the condition of ∆µBa ) -1.96 eV.
Figure 5. SGPs of different terminations in the condition of ∆µBa ) 0.00 eV.
eV, 0.00 eV] and [-5.90 eV, 0.00 eV], respectively. Furthermore, the calculated constants (φ) in eq 5 are also listed in Table 4. The accessible values of the SGP within the allowed region are displayed in Figure 4, and they describe the stability diagram of the BaTiO3 (110) (1 × 1) surface in the O and Ba external environment. According to Figure 4, only four out of five possible terminations are found to be stable. Indeed, the O2 termination cannot be stabilized even if it is in a very O-rich chemical environment because its SGP is always larger than that of the O termination within the whole region. The calculated results indicate that the Ba termination is the most stable one in the O- and Ba-rich environment (Section II) while its complementary, TiO termination is in the O- and Ba-poor environment (Section IV). Furthermore, the O termination shows a stability domain in the moderate O and Ba environment (Section III). And finally, the BaTiO stoichiometric termination is found to be stable in a small region with O-poor and Ba-rich conditions (Section I). Figure 5 shows the SGPs of different terminations in the condition of ∆µBa ) 0.00 eV. For the complementary terminations, TiO and Ba or BaTiO and O2, the sum of their SGPs is independent of the chemical potential of ∆µO but corresponds to the sum of their cleavage energies. The O termination can be considered as self-complementary, and its SGP is a constant of 0.98 J/m2. The results indicate that only two stable configurations can be obtained under the condition of ∆µBa ) 0.00 eV,
and that the Ba termination is very stable at a large interval (-5.61 eV < ∆µO < 0.00 eV). But the stoichiometric BaTiO termination still exists in a small interval with the O-poor condition. As ∆µBa decreases, the relative stabilities of the five terminations undergo obvious changes. Figure 6 shows the SGPs of the five terminations under the condition of ∆µBa ) -1.96 eV. It can be noted that the allowed region is reduced from [-5.90 eV, 0.00 V] to [-5.24 eV, 0.00 eV], and that the SGP lines of the TiO and Ba terminations move rightward. But the SGP lines of BaTiO, O2, and O terminations remain unchanged because their SGPs are independent of ∆µBa. As a result, the O and Ba terminations now become the most stable configurations. As displayed in Figure 4, when ∆µBa is restricted to [-2.25 eV, -5.78 eV], three stable nonstoichiometric terminations, Ba, TiO, and O terminations, are thus obtained. Finally, when ∆µBa is smaller than -7.40 eV, only a stable TiO termination can be obtained. From theoretical viewpoints, the polarity compensation through the changes of surface electronic structures is always expected. But actually the mechanism is not often feasible. For example, for the strongly ionic MgO compound, the stoichiometric (111) polar surface cannot be stabilized even if the filling of surface states exists because its cleavage energy (Erel cl ) 7.2 J/m2) is much higher than that of the nonstoichiometric 2 19 According to our reconstructed surface (Erel cl ) 2.2 J/m ). calculations, the larger SGP, which is connected to the cleavage energy inherently, results in the disappearance of the stable area of the terminations in the stability diagram. However, for BaTiO3, the situation is very different as it was noted that the stoichiometric BaTiO polar termination can be stabilized in a small region with O-poor condition. In this context, Bottin suggested that such a peculiar behavior might be due to the presence of Ti-O covalent bonds and a not too large fundamental gap.34 Although the relationship between the covalent effect and stability of the polar surface is complex and still unclear, our calculations indicate at least that the relatively small cleavage energy of BaTiO termination is very crucial for its stabilization. 4. Conclusions We have performed a systematic study on the atomic and electronic structures of the BaTiO3 (110) polar surface by first-
Investigation of the BaTiO3 (110) Polar Surface principles density functional theory (DFT-GGA) calculations relying on a two-dimension periodic slab model for understanding its surface structure and stability. In particular, two different kinds of terminations are considered, the stoichiometric terminations with BaTiO and O2 compositions and the nonstoichiometric terminations with Ba, TiO, and O compositions. For all the terminations, the largest relaxations are found in the topmost two layers and the relaxation equilibrium can be achieved in the inner plane. The calculated results of the atomic charge and layer excess charge of each termination indicate that the specific charge modifications take place in the outer layers and the condition of polarity compensation is thus fulfilled. Although the compensation criterion can be achieved for the two kinds of terminations, the compensation mechanisms are different. For the stoichiometric terminations, the compensation criterion is achieved through the change of surface electronic structures. In the BaTiO stoichiometric termination, the Fermi level is located at the bottom of the conduction bands, which leads to a considerable filling of surface states. In the O2 stoichiometric termination, the TDOS of the O2 plane in the first layer changes extensively with respect to the bulk crystal, indicating the forming of a peroxo group. Due to such changes in the electronic structure, the structural distortions and cleavage energies of the stoichiometric terminations are found to be larger than those of the nonstoichiometric ones. For the nonstoichiometric terminations, the compensation criterion is achieved actually through the change of the surface stoichiometry, and their electronic structures are qualitatively similar to that of the bulk crystal, which allows them to retain their insulating characteristics. By calculating surface grand potentials (SGPs), we obtained four distinct stable (110) terminations. A quite large domain is found for the three nonstoichiometric terminations, which indicates that the compensation criterion realized through the change of surface electronic structures is more expensive than that by the change of surface stoichiometry. However, the stoichiometric BaTiO termination can still exist in a small region with O-poor and Ba-rich conditions, and such a peculiar behavior should be attributed to its relatively small cleavage energy. In conclusion, due to the scarcity of experimental investigations and the observation complexity, the detailed determination of the atomic and electronic structures of the BaTiO3 (110) surface remains a delicate task. However, our first-principles calculations suggest that the stable surface configuration is very sensitive to the environment and the chemical conditions should be carefully controlled to get a comprehensive understanding of the polar surface. Acknowledgment. This work was supported by the National Natural Science Foundation of China (Nos. 20301006 and 20431030) and the Research Project of Education Department of Heilongjiang Province of China (No. 10551234) References and Notes (1) Jia, Q. X.; Shi, Z. Q.; Yi, J.; Anderson, W. A. J. Electron. Mater. 1994, 23, 53.
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