Interfacial Properties and Electronic Structure of Ag(001)/BaTiO₃(001): A First Principle Study

Surface energy (γ_s ) is the excess energy of surface particles relative to internal particles, which can be used for evaluation of surface stability. ³⁷ The γ_s was usually calculated by the following formula: ³⁸

$$\begin{eqnarray}&&{\gamma }_{{\rm{s}}}\approx \left[{E}_{{\rm{slab}}}\left(N\right)-N{E}_{{\rm{bulk}}}\right]/\left(2{A}_{{\rm{s}}}\right)\end{eqnarray} \tag{ 1 }$$

Where E_slab(N) is total energy of surface supercell slab, whose number of atoms or formulas is N. E_bulk is the total energy of each atom or formulas in the bulk phase. A_s is the area corresponding to E_slab(N), and the factor 2 is due to the fact that the slab for calculation has two surfaces.

In this section, the convergent surface energy of Ag(001) was obtained by using this formula, which was shown in Table I. When the surface energy of BaTiO₃(001) was calculated by this method, the number of atoms in the plate system with odd total number of layers did not match the atomic ratio of the bulk. And for plates with even number of layers, only the average surface energy of two kinds of atoms-terminated could be calculated. Therefore, the method used by Mastrikov ³⁹ to calculate the surface energy of LaMnO₃ with different atoms termination could be applied to calculate the surface energy of BaTiO₃(001). The surface energy was calculated by using 9-layer surface supercells with different atoms-terminated as shown in Figs. 1-(d-1,d-2). The surface energy is defined as the sum of the cleavage energy E_c and the relaxation energy E_r . As the ABO₃ crystal was cut along the (001) crystal plane direction, both AO and BO₂ surfaces appeared at the same time, so it could be considered that the cutoff energy of the two surfaces were the same. The cleavage energy E_c can be calculated according to the following formula:

$$\begin{eqnarray}&&{E}_{{\rm{c}}}=\displaystyle \frac{1}{4}\left[{E}_{{\rm{slab}}}^{{\rm{unr}}}\left({\rm{BaO}}\right)+{E}_{{\rm{slab}}}^{{\rm{unr}}}\left({{\rm{TiO}}}_{{\rm{2}}}\right)-9{E}_{{\rm{bulk}}}\left({{\rm{BaTiO}}}_{{\rm{3}}}\right)\right]\end{eqnarray} \tag{ 2 }$$

Where ${E}_{{\rm{slab}}}^{{\rm{unr}}}\left({\rm{BaO}}\right)$ and ${E}_{{\rm{slab}}}^{{\rm{unr}}}\left({{\rm{TiO}}}_{2}\right)\,$ are the energy before optimization of 9-layer BaO-terminated and TiO₂-terminated BaTiO₃(001) surface supercells respectively. E_bulk(BaTiO₃) is the total energy of each BaTiO₃ in the bulk phase (formulas = BaTiO₃). The factor 1/4 is due to the fact that the two surface supercells contain four surfaces. The factor 9 is due to the fact that the two surface supercells contain nine BaTiO₃.

The relaxation energy E_r is defined as the energy change of geometric plate before and after structural optimization:

$$\begin{eqnarray}&&{E}_{{\rm{r}}}=\displaystyle \frac{1}{2}\left[{E}_{{\rm{slab}}}^{{\rm{r}}}\left({\rm{I}}\right)-{E}_{{\rm{slab}}}^{{\rm{unr}}}\left({\rm{I}}\right)\right]\end{eqnarray} \tag{ 3 }$$

Where ${E}_{{\rm{slab}}}^{r}\left({\rm{I}}\right)$ and ${E}_{{\rm{slab}}}^{{\rm{unr}}}\left({\rm{I}}\right)$ are the energy of slab I before and after optimization respectively. The factor 1/2 is due to the fact that the surface supercell has two identical surfaces. From the calculation results, BaO-terminated BaTiO₃(001) has strong stability. The surface energy of BaTiO₃(001) with different layers could also be calculated by this method, which was shown in Table I. As shown in Table I the surface energies of Ag(001), BaO-terminated and TiO₂-terminated BaTiO₃(001) has converged to 1.28 J m⁻², 1.26 J m⁻² and 1.39 J m⁻², respectively, when the number of atomic layers is 7.

Besides, the number of convergence layers and surface energy of Ag surface are consistent with other experimental ⁴⁰ and computational results. At the same time, in order to better analyze the surface structure, we have made statistics and comparisons of the atomic positions before and after relaxation of different atoms in different layers (refers to Table II). After undergoing surface relaxation, each atomic layer moved to the bulk phase in different degrees, which reduced the distance between the atomic layers. And the closer the distance between the atoms to the surface, the greater the degree of relaxation. As the atomic layer was further away from the surface, there was decreasing degree of displacement compared with the closer one. The phenomenon was due to the atom surface effect, which was caused by the unsaturation of surface atoms. In general, the atoms of solid phase were in a state of mutual attraction, which made the solid atoms stack on top of each other and keep a certain volume and shape instead of scattering away. The above phenomenon of surface atoms relaxation towards bulk phase originated from the asymmetric arrangement of atoms on both sides of the surface layer. Due to the lack of attraction of some atoms on the side of the atomic layer near the surface, the atoms were subjected to the resultant force pointing to the bulk phase. And since the closer to the surface, the greater the degree of this asymmetry, its tendency to move was more obvious. In this BaO-terminated BaTiO₃(001) surface calculation, after structural optimization, the position changes of metal atoms in the same number layer were far greater than that of non-metallic atoms in the same layer, which phenomenon was also found in TiO₂-terminated BaTiO₃(001) surface. In other words, the relaxation of surface led a considerable rumpling that was also found by Padilla, Vanderbilt ⁴¹ and Eglitis ⁴² for BaTiO₃(001) surface rumpling.

Adhesion work (W_adh) is one of the most important characteristic in interface particularities, which is reversible work to separate a complete interface into two free surfaces. ^43,44 In calculation of slab interface system, W_adh can be calculated by the following formula:

$$\begin{eqnarray}&&{W}_{{\rm{adh}}}=\left({E}_{\mathrm{slab},\mathrm{Ag}}+{E}_{{\mathrm{slab},\mathrm{BaTiO}}_{{\rm{3}}}}-{E}_{{\rm{Ag}}/{{\rm{BaTiO}}}_{{\rm{3}}}}\right)/A\end{eqnarray} \tag{ 4 }$$

where W_adh is the adhesion work of interface Ag/BaTiO₃, E_slab,Ag and ${E}_{{\mathrm{slab},\mathrm{BaTiO}}_{3}}$ are the the total energy of Ag and BaTiO₃ surface after complete relaxation, ${E}_{{\mathrm{Ag}/\mathrm{BaTiO}}_{3}}$ is the total energy of interface Ag/BaTiO₃. A is the interface area corresponding to interface Ag/BaTiO₃.

Two methods were applied to calculate W_adh, first of which was UBER method. ^45,46 For each interface model with a series of different interfacial distances (D₀), W_adh was calculated respectively. A series of W_adh of different models were combined with their corresponding interfacial distances and W_adh-D₀ curves were got (shown as Fig. 3). With the increase of D₀, the value of W_adh for the six models increased at first and then converged or decreased. The extreme value of W_adh were obtained at a certain D₀ as shown in Fig. 3. And the extreme value of W_adh were maximum value for adhesion work and their corresponding value of D₀ was optimal interface distance simultaneously (The calculation results were listed in the UBER column of Tables III, IV). What needs to be explained here was that the stability of the interface whose UBER curve had no extremum did have none extreme value with the change of the interface spacing. The interface distance corresponding to the convergence of adhesion work was relative equilibrium spacing. ⁴⁷ The reason why the optimal interface spacing of these interfaces was selected as the minimum interface spacing corresponding to the convergence of the adhesion work was to save computing resources. For the other method, the interface models with the optimal interface distance obtained by UBER method were completely optimized, and the new equilibrium interface distance D_eq and adhesion work were obtained, which were listed in the fully optimized column of Tables III, IV.

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After observing and analyzing Tables III and IV, it was found that the relative values of adhesion work (W_adh) and interface distance (D) obtained by the two methods were coincident. At the same time, the difference of interface distance calculated by the two methods was not worth mentioning, But there was a certain gap in adhesion work. The calculated results after complete relaxation could be regarded as equilibrium state. Surface capping and surface atoms had important influence on the interfacial distance and adhesion work. Equilibrium atomic distance showed that the interface composed of TiO₂-terminated BaTiO₃ surface had smaller interface distance and larger adhesion work than that with BaO as the terminal. Model M 22 had the smallest interface distance and the largest adhesion work, followed by M 23 and interface M 21. These three models were composed of TiO₂-terminated BaTiO₃ surface and Ag surface. Simultaneously, it was also found that the adhesion work of M 12 and M13 were negative number, which indicated that those two interfaces were unstable. In addition, as long as surface Ag atoms were not at the top of O atoms, its relative position had little effect on the interfacial distance and adhesion work.

There were more oxygen atoms on the surface of TiO₂-terminated BaTiO₃. Compared with other interfaces, the Ag atoms were all located at the top of O atom in the M 22 interface model. And after optimization, the distance between Ag and O atoms at interfaces decreased, which was more obvious on the Ag–O distance in M 11. It was speculated that the interaction between Ag and O atoms makes M 22 the optimal interface, and the TiO₂-terminated BaTiO₃ surface had lower interface distance because of the large number of oxygen atoms on the surface.

Based on the above analysis, it could be reasonably inferred that the interaction between the atoms at the interface composed of TiO₂ capped BaTiO₃ surface was stronger and more stable in thermodynamics. Among these 6 models, M 22 had the smallest interfacial distance and the largest adhesion work, which was probably due to the stronger interaction between Ag atoms and O atoms than Ag atoms and other atoms.

Interface energy (γ_i ) is the other quantity often used to describe the thermodynamic stability of the interface, which is defined as the difference between the free energy of interface atoms and that of bulk atoms. ⁴⁸ Interface energy is rarely involved in the literature related to experiments for the reason that it is of great difficulty to obtain accurate interface energy results by means of experiment experiments. And the value of interface energy of solid-solid interface is usually ignored. If there are substances with similar properties and microstructure on both sides of the interface, seeing that the absolute value of the interface energy is very small, it is tolerable to ignore it. However, it is obviously unreasonable to ignore the interfacial energy directly for the interface composed of heterogeneous materials with large difference in properties and microstructure. The two solid materials that make up the interface are completely different, with the interface energy positive, which is mainly due to defects such as lattice distortion at the interface, which increases the instability of the interface. At the same time, the interface atoms with negative interfacial energy are more likely to diffuse, resulting in interface alloying and even forming new interface phases.

$$\begin{eqnarray}&&{\gamma }_{i}=\displaystyle \frac{{E}_{{\mathrm{Ag}/\mathrm{BaTiO}}_{{\rm{3}}}}\left(M,N\right)-M{E}_{{\rm{Ag}}}^{{\rm{bulk}}}-N{E}_{{{\rm{BaTiO}}}_{{\rm{3}}}}^{{\rm{bulk}}}}{{A}_{i}}-{\gamma }_{{\rm{Ag}}}-{\gamma }_{{{\rm{BaTiO}}}_{{\rm{3}}}}\end{eqnarray} \tag{ 5 }$$

Where ${E}_{{\mathrm{Ag}/\mathrm{BaTiO}}_{3}\left({\rm{M}},{\rm{N}}\right)}$ is the total energy of fully optimized Ag/BaTiO₃ interface model. ${E}_{{\rm{Ag}}}^{{\rm{bulk}}}$ and ${E}_{{{\rm{BaTiO}}}_{3}}^{{\rm{bulk}}}$ are respectively the total energy of each atom in the Ag bulk phase and each formulas in the BaTiO₃ bulk phase. M and N are respectively the number of Ag atoms and BaTiO₃ in Ag/BaTiO₃ interface model. ${\gamma }_{{\rm{Ag}}}$ and ${\gamma }_{{{\rm{BaTiO}}}_{3}}$ are the surface energy of Ag and the BaTiO₃ surface slabs, A_i is the interface area.

The interface energy (γ_i ) results calculated by formula (5) were listed in Table V. It could be seen from Table V that the interface model with the lowest interface energy was M 22, which was −0.312 J m⁻². The order of interface energy from low to high was M 22 < M 23 < M 21 < M 13 < M 12 < M 11. Except for M 22 and M 23 interface energy calculation results were negative, the interface energy of the other interface models were all positive values. And the three interface models with the lowest interface energy were all interfaces composed of Ag and TiO₂-terminated BaTiO₃, while the BaO-terminated interface had larger interface energy and all of them were positive values.

The type and relative position of interface atoms were considered in the analysis. The three interface models with low interface energy were TiO₂-terminated, which contained more oxygen atoms than the BaO-terminated interface. According to Table IV, the Ag-O atomic distance at the M 22 and M 23 interface with the interface energy negative were the shortest two of the six models, and the Ag-O atomic distance all decreased after complete relaxation. And M 22, had the lowest interface energy of the six interfaces, where the interface Ag atoms were located at the top position of the O atom, in which the Ag–O atom distance was the lowest among all the interface models. Besides, after complete relaxation, the Ag–O atom distance of M 22 model decreased to a greater degree.

In the calculation results of adhesion work (W_adh) and interface energy (γ_i ) and subsequent analysis, the same conclusion was obtained: interface oxygen atoms played a vital role in adhesion work and interface energy, and the relative positions of Ag–O atoms as well as their interaction would strongly affect the stability of the interface. The correlation between Ag, O atoms was stronger than that of other atoms in the interface. Among the six interface models, the M 22 interface with the smallest interface energy was most likely to form a new interface phase due to atomic diffusion.

The interface is the only way in the process of stress transmission from the matrix to the reinforcing phase, which plays a vital role in the overall strength of the cermet composite. And the fracture performance plays a key role in the strength of the cermet. Ag and BaTiO₃ differ greatly in toughness and strength, which makes the interface the location where the properties change the most in the material. When the stress and temperature are constantly changing, the interface is the most vulnerable position, and interface fracture is the main reason for the failure of cermet composites.

Fracture toughness is the impedance value of the material when the unstable fracture starts from cracks or crack-like defects in the specimen or component. The fracture failure of the interface is the process of dividing the interface into two parts, which is similar to the definition of interface adhesion work W_adh, so that W_adh is often used to describe the fracture toughness of the interface. The relationship between W_adh and the critical stress σ_F required for crack growth can be described by Griffith equation: ⁴⁹

$$\begin{eqnarray}&&{\sigma }_{F}=\sqrt{\displaystyle \frac{{W}_{{\rm{adh}}}E}{\pi c}}\end{eqnarray} \tag{ 6 }$$

Where E is Young's modulus, c is crack length. It can be seen that σ_F increases with the increase of W_adh, so the larger the W_adh, the better the interface fracture toughness of the material. In the study of Chen and Bielawski, it was found that the interfacial fracture toughness along a specific crystal orientation could be calculated by the following formula: ⁵⁰

$$\begin{eqnarray}&&{K}_{{\rm{Ic}}}^{\mathrm{int}}=\sqrt{4{W}_{{\rm{adh}}}{E}_{\left[hkl\right]}}\end{eqnarray} \tag{ 7 }$$

Where E_[hkl] is the Young's modulus of a material along the crystal orientation with index [hkl]. According to the adhesion work W_adh and Young's modulus E_[hkl], ${K}_{{\rm{I}}c}^{{\rm{int}}}$ could be calculated by this formula. Since the materials and crystal orientations of the given interface models were determined, the ${K}_{{\rm{I}}c}^{{\rm{int}}}$ of the interface models could be analyzed qualitatively.

According to the above two formulas, the σ_F and ${K}_{{\rm{I}}c}^{{\rm{int}}}$ of the interfaces could be analyzed based on the W_adh (as shown in Table III) and E_[hkl] of different interface models to analyze the interface fracture toughness. For six different interface models, the order of σ_F and ${K}_{{\rm{I}}c}^{{\rm{int}}}$ were: M 22 > M 23 > M 21 > M 11 > M 13 > M 12. The Ag and TiO₂-terminated BaTiO₃ had the largest σ_F and ${K}_{{\rm{I}}c}^{{\rm{int}}}$ in all interfaces, as well as had better interfacial fracture toughness than others. The calculated values of Young's modes E_[hkl] of Ag and BaTiO₃ along the [001] direction were E_Ag,[001] = 15.18 GPa, ${E}_{{{\rm{BaTiO}}}_{3},\left[{\rm{001}}\right]}$ = 227.08 GPa. Therefore, the value range of ${K}_{{\rm{I}}c}^{{\rm{int}}}$ for the O top stacking of TiO₂-terminated BaTiO₃ was 0.26 to 1.04 MPam^1/2.

Electron density difference refers to the difference of the charge density before and after the interface formed by the completely relaxed surface combination. It is usually used to describe the electron transfer between the interfaces and the interaction between interfacial atoms. In order to further analyze the interaction between the interfaces, the electron density difference was calculated. The electron density difference could be expressed as:

$$\begin{eqnarray}&&{\rm{\Delta }}\rho ={\rho }_{{\mathrm{Ag}/\mathrm{BaTiO}}_{{\rm{3}}}}-{\rho }_{{\rm{Ag}}}-{\rho }_{{{\rm{BaTiO}}}_{{\rm{3}}}}\end{eqnarray} \tag{ 8 }$$

Where ${\rho }_{{\mathrm{Ag}/\mathrm{BaTiO}}_{3}}$ is the electron density after the complete relaxation of the interface, ${\rho }_{Ag}$ and ${\rho }_{{{\rm{BaTiO}}}_{3}}$ are the electron densities of Ag(100) and BaTiO₃(100) surface respectively.

As graphically shown in Fig. 4, the charge transfer degree of M 22 and M 23 with lower interface energy was significantly greater than that of other interfaces. The electron transfer around the Ag and O atoms at the M 22 interface and the Ag atoms at the M23 interface were high, which was consistent with the conclusion of the interface energy analysis. At the interface of M 22, electron density decreased around Ag atoms and oxygen atoms near the interface decreases, with the electron density between Ag and O atoms increasing, which was consistent with the formation characteristics of covalent bond. The electron density transfer at the M23 interface also had a similar phenomenon, while the degree of electron transfer was obviously lower than that at the M22 interface. The relatively low change of electron density at M12 and M13 indicated that the interaction between the two solid phases composing the interface was weak and the composed interface was unstable.

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The electron transfer between Ag–O atoms at the interface was much stronger than between Ag–Ba and Ag–Ti atoms, indicating that the interaction between Ag–O atoms was stronger than that between other atoms on both sides of the interface. During the formation of the interface, the contribution between Ag–O atoms was greater than the contribution produced by the interaction between other atoms, which was probably due to the covalent interaction between Ag and O atoms.

In order to further analyze the bonding mechanism, partial density of states (PDOSs) of different atoms with most stable configuration M 22 at the interface were calculated. The part of PDOSs was shown in Fig. 5. The density of states of the first layer atoms at the interface could clearly reflect the situation of the atoms participating in the interface bonding, and could accurately analyze the hybridization of the electronic orbits of the bonding atoms at the interface. Compared with the first layer atoms, the third layer atoms at the interface were relatively less affected by the interface, and its density of states were closer to the bulk atoms. By comparing it with the first layer of atoms, influence on the atoms by the interface could be obtained, making it easier to analyze the interface bonding. Therefore, in addition to the main figure in Fig. 5 describing the entire PDOSs near the Fermi energy level, the attached figures of density of states of Ag and O atoms were added to illustrating the information of PDOS completely. The enlarged part of PDOSs were pictured in Fig. 5-(1,2).

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As shown in Figs. 5a, 5d, there was a significant overlap between the Ag-5 s orbitals of the first layer of Ag atoms and the O-2 s orbitals of the first layer of oxygen atoms from −6 eV to −3 eV, which was mainly around 3 eV and 6.5 eV. At the same time, comparing Figs. 5a, 5d with Figs. 5b, 5e, it was found that a new peak appeared near the o-2p orbital-4ev of the oxygen atom in the first layer of the interface, which overlapped with the Ag-5 s orbital. In addition, comparing their attached figures, new peaks appeared of Ag atoms in the first layer at −12.5 eV, −19 eV, etc., which were corresponding to the peak of oxygen atoms at the interface. In addition, the orbitals of the Ag and O atoms in the interface layer moved to the Fermi level, which indicated that the activity of Ag and O was enhanced due to the influence of interface and made greater contribution to the interfacial bonding.

Owing to the high content of oxygen atoms on the surface of TiO₂-terminated BaTiO₃(001), the thermodynamic stability of the stacking interface was higher. The Ag atoms in the surface layer were all located at the top of the O atom in M 22 interface. Due to the high activity of the O atoms at the interface, it was easier to achieve orbital hybridization between Ag–O atoms. It was indicated that Ag–O bonds were formed at the interface, and the bonds had some covalent properties. There was no obvious difference in peak position and peak intensity between the first layer of Ti atoms and bulk atoms, which indicated that Ti atoms did not participate in the bonding of interface and made little contribution to the formation of interface.