Layered perovskite structure Sr₂M₂O₇, as a new type of functional material, has become a hot topic,^[1–4] for instance, Cai et al. studied the dielectric constant and loss of Sr₂Nb₂O₇ specimen sintered at 1200 °C, which were 57% and 1.6% at 10 kHz, and the piezoelectric constant d₃₃ reached 2.1 pC/N, respectively. In Chen’s work, by using the solid-state reaction method, the author investigated significant effects on the sinterability, microstructure, and electrical properties of Sr₂Nb₂O₇ ceramics prepared by CuO addition. Peng et al. studied the electronic structures of S and V/Nb mono- and (anionic–cationic) co-doped Sr₂Ta₂O₇, the band gap engineering and the shifts of the VBM and CBM for sufficiently utilizing visible light of the solar spectrum by generalized gradient approximation (GGA) and Perdew–Burke–Eruzerh (PBE) calculations. Teshima et al. prepared Sr₂Ta₂O₇ film growth under several different conditions, and found that the growth atmospheres were important in forming idiomorphic crystals with well-developed facets and characteristic shapes. The above results provide a comparison and research direction for our theoretical calculation.

At room temperature, Sr₂Nb₂O₇ and Sr₂Ta₂O₇ are orthogonal structures (see Fig. 1), which are composed of many “plates” that are parallel to (010) planes (seen Fig. 1), and every “plate” is composed of the MO₆ octahedrons. One of Sr occupies the boundary of the structure, another Sr in internal system, for M, there are also two types and all locate at the center of the octahedron. Sr₂Ta₂O₇ (space group Cmcm) and Sr₂Nb₂O₇ (space group Cmc2₁) are isomorphic heterogeneity, the symmetry of Sr₂Ta₂O₇ is higher than that of Sr₂Nb₂O₇ at room temperature.

Fig. 1.

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	Fig. 1. (color online) The perovskite-slab structures of Sr2M2O7 unit cell, the green and brown regions represent NbO6 and TaO6 octahedra, respectively.

Because of the unique structure and excellent physical properties, Sr₂M₂O₇ can also be widely used in various industrial fields. For Sr₂Nb₂O₇, it is one of the few high temperature ferroelectric materials with high Curie temperature (T_c = 1615 K), chemical and thermal stability, and significant physical properties.^[5,6] As a new photocatalyst, Sr₂Ta₂O₇ is the paraelectric orthorhombic phase at room temperature without any additives to split water into H₂ and O under UV light.^[7] The unique structure determines that Sr₂M₂O₇ has many advantages: (i) it is easy to adjust the band gap and absorption spectrum of the semiconductor by changing the particle size; (ii) the optical absorption of semiconductor particles present absorption bands, which is beneficial to the effective collection of sunlight; (iii) the photostability can be increased by the surface modification; (iv) using layers as a suitable reaction point to control the reverse reaction, the efficiency of the reaction can be improved, therefore, the semiconductors that enter layers are in the nanometer scale, it is beneficial to the separation of photogenerated electrons and holes; (v) when the semiconductor is irradiated by the light, it can generate the conduction electron and the corresponding hole. The excited state of electrons can recombine with the holes, and the light energy can be converted into thermal energy or other forms of energy in the process of recombinations. After the combination of semiconductors and layered compounds, there are effective migrations of photoelectrons from semiconductors to layered substrates, and the generated electron–hole pairs in the semiconductor surface cannot be quickly combined for other reasons; the semiconductor has activity, showing that the reason for the inhibition of electron–hole pairs recombination may be that the semiconductors are affected. In this case, electron–hole pairs recombination will be significantly suppressed, which also reduces the recombination rate of photogenerated electrons and holes, thereby improving the photocatalytic properties of semiconductor. On the side, due to the special layered structure, the metal, non-metal, organics, semiconductor oxide, and composite nanoparticles can be introduced into the interlayers, to improve the thermal stability and catalytic activity, because of the great concern.^[4,8]

Since Nanamatsu et al.^[9] first successfully synthesized Sr₂Nb₂O₇, Sr₂Ta₂O₇, and the corresponding solid solution by the floating zone method, several researchers have experimentally and theoretically investigated related properties of Sr₂M₂O₇. Kudo et al.^[10] studied the perovskite-tantalates Sr₂(Ta_{1 − x}Nb_x)₂O₇, band structure, and water splitting activity by partial substitution of Nb; the results indicated that the band gap decreases with the increase of Nb/Ta. After loading the NiO, the photolysis of water according to the stoichiometric produced H₂ and O, NiO/Sr₂Ta₂O₇ had the highest activity. Kato et al.^[11] researched the surface morphology and dielectric properties of Sr₂Ta₂O₇ and Sr₂(Ta_0.7Nb_0.3)₂O₇ films when the annealing temperature reached 750 °C, the dielectric constant was around 90, while dielectric loss was less than 0.05 at 100 kHz, which showed that these compounds were suitable for memory materials. Yoshioka et al.^[12] investigated the relationship between photocatalytic activity and crystal structure of different perovskite tantalate (Sr₂Ta₂O₇, Sr₂Ta₂O₆, and Sr₄Ta₂O₉). The research suggested that Sr₂Ta₂O₇ had the highest photocatalytic and great relationship with special structure. TaO₆ octahedrons in Sr₂Ta₂O₇ were connected with the common vertex, while the common surface for the TaO₆ in Sr₄Ta₂O₉. Thus, the conduction bands of Sr₄Ta₂O₉ were so narrow that it cannot provide effective carrier and the poor photocatalytic activity.

With the development of computers, the theoretical researches of materials have increased, especially based on the first principles.^[13–15] Nisar et al.,^[16] based on hybrid functionals (HSE06), surveyed the photocatalytic of Sr₂Nb₂O₇ from the band structure and bandgap. The results showed that the band gap in layered perovskite Sr₂Nb₂O₇ was significantly reduced by the cationic–anionic co-doping, and band position was excellent for the visible-light photocatalysis. Liu et al.^[17] performed hybrid functional calculations for the mono- and co-doping of Sr₂Ta₂O₇ by the controlled band gap engineering for an efficient solar-driven photocatalyst. The results showed that doping creates impurity states in the band gap and therefore reduces the band gap significantly, and the stability of the co-doped system is governed by the Coulomb interactions and charge compensation effects.

By reviewing the previous work, we find that the researches about Sr₂M₂O₇ have made great achievements both in experiment and theory, but these studies mainly focused on optical research under UV light and there have been few theoretical calculations that were not systematic. The present study, based on first-principles, will focus on the discussion of dielectric, piezoelectric properties, Born effective charge, and spontaneous polarization of Sr₂M₂O₇. As is known to all, the above properties have been widely applied in the field of photoelectricity, especially photocatalytic degradation of organic pollutants, photolysis of water to produce ammonia. More and more attention has been paid to the degradation of environmental pollutants and the decomposition of water by using solar semiconductor photocatalysis technology. It is expected to solve the problems of energy shortage and environmental pollution, therefore our preliminary research can provide important theoretical guidance for studying Sr₂M₂O₇ and expanding its application.

Because many experiments are difficult or costly in practice, the theoretical calculation is usually the first step in this experiment. Piezoelectricity is one of the most important practical properties of ferroelectric materials, so the theoretical investigation of dielectric and piezoelectric properties is taken as the first step to guide the practice.

We used the generalized gradient approximation (GGA)^[12] in the Perdew–Burke–Eruzerh (PBE)^[18] to study Sr₂M₂O₇, and the theoretical mechanisms are discussed. The calculations have been performed by using the Cambridge Sequential Total Energy Package (CASTEP) code.^[19,20] Using the plane wave pseuodopotential method, the ion potential is replaced and the electron wave function is carried out with the plane wave basis set. We use the GGA to correct the exchange interaction between the electrons and the correlation potential.

In the density functional theory (DFT), the Schrödinger equation of the single electron motion can be expressed as follows (in atomic unit): 1 2 where Z_q is the nuclear charge, r is the position vector of the nuclear motion, R_q is the distance between two cores, ρ(r) indicates the electron density, V(r) indicates the external potential field, Ψ_i(r) is a single electron wave function, ε_i is the energy of a single electron, and n_i indicates the number of electrons occupied by the intrinsic state. The first term is effective electron kinetic energy in the system of Eq. (1). The second represents the attractive Coulomb potential of the atoms in the system, which is expressed by the norm-conserving pseudopotential. The term is electron Coulomb potential. The fourth represents the exchange correlation energy, and the concrete form is expressed by the local density approximation (LDA) or the GGA.

The valence electron configuration for Sr, Nb, Ta, and O are 4p⁶5s², 4d⁶5s¹, 5p⁶5d³, and 2s²2p⁴, respectively. The ultrasoft pseudopotential is applied to describing the interaction energy between valence electron and ionic core. The Brillouin zone integration is performed over 4 × 4 × 8 grid sizes using the Monkhorst–Pack method for structure optimization. The convergence thresholds for total energy, maximum force, maximum stress, maximum displacement, and Hellmann–Feynman ionic force are less than 5 × 10⁻⁶ eV/atom, 0. 01 eV/Å, 0. 02 GPa, 5.0 × 10⁻⁴ Å, and 1 meV/Å. The linear response method of density functional perturbation theory (DFPT)^[21] is used to calculate the dielectric, piezoelectric properties, Born effective charge, and the calculation of spontaneous polarization using the Berry phase method.^[22]

From Table 1, it can be seen that the optimized structures are consistent with the experimental data,^[16,23] indicating that our calculations are reasonable. Nisar et al.^[16] simulated the structures of Sr₂Nb₂O₇ and Sr₂Ta₂O₇ using hybrid functionals and PBE function. The result shows that the error rate is greater than 1.2% for Sr₂Nb₂O₇, and greater than 1.6% for Sr₂Ta₂O₇. The PBE usually overestimates the lattice parameters of the equilibrium, so the PBEsol function is more suitable for our work.

Table 1.

Table 1.

Table 1. The space groups and lattice parameters (in unit Å) of the Sr2Nb2O7 and Sr2Ta2O7. . Sr2Nb2O7 Sr2Ta2O7 Cmc21 Present HSE06[16] Expt.[22] Cmcm Present PBE[23] Expt.[22] a/Å 3.9820 3.797 3.97 a/Å 3.9572 3.983 3.95 b/Å 26.9472 27.205 26.86 b/Å 27.3434 27.59 27.27 c/Å 5.7239 5.793 5.72 c/Å 5.6969 5.746 5.70

Table 1. The space groups and lattice parameters (in unit Å) of the Sr2Nb2O7 and Sr2Ta2O7. .

3.1. Lattice vibrations of Γ points for Sr₂M₂O₇

Sr₂Nb₂O₇ has 4 different phonon modes (A₁, A₂, B₁, and B₂), which are one-fold degeneracy. Phonon frequencies at the center (Γ) of the Brillouin zone are listed in Table 2. Based on the group theory, the irreducible representation of Sr₂Nb₂O₇ phonon modes is expressed as follows: In order to get the stability configuration of the local potential energy in the lowest point, we must ensure that no imaginary frequency appears. We further calculated the phonon dispersion of Sr₂M₂O₇, it can be seen that an imaginary frequency cannot be found from Fig. 2, which indicates that the Sr₂M₂O₇ structures are stable.

In terms of optical activity, except for the A₂, the other modes are both infrared and Raman. According to the Mulliken symbol, A₁ and A₂ for the main axis are symmetric, while B₁ and B₂ are antisymmetric. On the other hand, A₁ and B₁ are symmetric for twofold C₂ (perpendicular to the main axis), while A₂ and B₂ are antisymmetric for C₂. In addition, the phonon at the Γ does not have an imaginary frequency indicating that the optimized structure is stable. In the experiment, Ito et al.^[19] used Raman scattering and far-infrared scattering to study the phonon frequency of Sr₂Nb₂O₇. Because of the experimental conditions, they just observed eighteen A₁, six A₂, five B₁, and two B₂, which are less than expected. Compared with the experiment, the calculated A₁ frequencies are in good agreement with Ito’s results. The observed modes in the experiment are less than the calculation, which may be the reason why the unobserved oscillators are relatively weak.

Fig. 2.

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	Fig. 2. (color online) The phonon dispersion curves for Sr2M2O7, (a) Sr2Nb2O7 and (b) Sr2Ta2O7, different color curves represent different phonon modes, respectively.

Table 2.

Table 2.

Table 2. Calculated frequencies (in unit cm−1) of phonon modes at the Γ point of Sr2Nb2O7. . Irrep ωλ/(cm−1) Irrep ωλ/(cm−1) Present Previous[22] Present Previous[22] 838.5 850 839.5 834 820.6 848 783.8 776 665.7 673 699.7 605.2 599 669.5 572.2 585 554.1 428.1 460 511.5 398.5 427 412.4 359.1 376 378.8 322.2 342 353.5 306.5 329 345.6 275.8 281 289.1 A1 259.7 B2 279.9 (IR & Raman) 225.3 242 (IR & Raman) 234.6 191.8 223.1 161.7 162 185.1 156.1 151 170.7 129.8 124 141.7 124.1 124 123.7 93.1 99 107.2 81.5 76.1 55.5 59 55.3 542.3 541 564.9 510.6 511.5 352.3 340 355.8 342 310.2 315.2 A2 295.5 B1 288.0 285 (Raman) 250.1 (IR & Raman) 234.6 258 210.9 220 178.1 179 130.1 132 130.5 120 111.4 115 100.7 90.9 67.9 58.6 60

Table 2. Calculated frequencies (in unit cm−1) of phonon modes at the Γ point of Sr2Nb2O7. .

For Sr₂Ta₂O₇, there are 8 different phonon modes (A_g, B_1g, B_2g, B_3g, A_u, B_1u, B_2u, and B_3u), and all modes are also onefold degeneracy. The irreducible representation at the Γ is expressed as:

Table 3.

Table 3.

Table 3. Calculated frequencies (in unit of cm−1) of IR modes at the Γ of Sr2Ta2O7. . Irrep ωλ/cm−1 Irrep ωλ/cm−1 Irrep ωλ/cm−1 676.8 863.9 612.6 609.3 800.2 561.3 575.3 625.7 332.4 291.4 422.4 273.0 233.4 423.3 193.9 B1u 222.1 B2u 357.7 B3u 114.0 201.1 308.9 102.0 121.9 243.1 66.3 92.2 179.9 73.8 127.2 0.56 115.6

Table 3. Calculated frequencies (in unit of cm−1) of IR modes at the Γ of Sr2Ta2O7. .

Table 4.

Table 4.

Table 4. Calculated frequencies (in unit of cm−1) of Raman and static modes at the Γ of Sr2Ta2O7. . Irrep ωλ/cm−1 Irrep ωλ/cm−1 Irrep ωλ/cm−1 856.7 593.0 695.9 812.0 560.8 624.9 424.1 342.7 324.5 411.3 240.3 286.6 Ag 347.4 B1g 137.1 B3g 235.2 (Raman) 315.0 (Raman) 112.9 (Raman) 202.1 197.6 84.3 159.2 143.4 57.7 142.4 119.1 88.7 101.6 60.6 321.6 343.0 Au 300.6 B2g 266.9 (Static) 232.4 (Raman)

Table 4. Calculated frequencies (in unit of cm−1) of Raman and static modes at the Γ of Sr2Ta2O7. .

From the point of view of optical activity, B_1g, B_3g, and A_u are Raman activity, while B_1u, B_2u, and B_3u are infrared, the remaining A_g is static. According to the Mulliken symbol, A_g and A_u for the main axis are symmetric, the others, namely B_1g, B_2g, B_3g, B_1u, B_2u, and B_3u, are antisymmetric. On the other hand, A_g, B_1g, B_2g, and B_3g are symmetric for the inversion center, while A_u, B_2u, and B_3u are antisymmetric. In addition, different from Sr₂Nb₂O₇, Sr₂Ta₂O₇ has an imaginary frequency (infrared B_1u mode, −0.56 cm⁻¹) at the Γ point. The infrared frequencies at the center Γ of the Brillouin zone are listed in Table 3, and the Raman and static frequencies are shown in Table 4. Unfortunately, there is no experimental data to compare.

In the present work, we investigated the atom displacement, in order to better understand the lattice vibration. The M2 atoms are located at the center of the orthogonal Sr₂M₂O₇ as shown in Fig. 3. Therefore, with the orderly rotation of the octahedron, the atomic displacement of M2 are always 0. The M1 atoms are symmetrically distributed in the body center of the MO₆ octahedron in the orthogonal Sr₂M₂O₇, and clockwise rotation occurs with the rotation of the central axis. All the O atoms on the octahedron move orderly with the rotation of the M1 atom.

Fig. 3.

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	Fig. 3. (color online) The atomic displacement for Sr2M2O7, (a) Sr2Nb2O7 and (b) Sr2Ta2O7, the blue arrows represent the vibration direction of atom O.

3.2. Sr₂M₂O₇ electrical properties

The dielectric constants, contributions of electrons and phonons to the dielectric constants, are shown in table 5. According to the symmetry, the dielectric constant for an orthogonal structure has three independent components: ε₁₁, ε₂₂, and ε₃₃, the expression is as follows: 3

The contributions of electrons to dielectric constants from Sr₂Nb₂O₇ are between 4.6–5.2, and the average is 4.9. The total dielectric constants are ε₁₁ = 65.8, ε₂₂ = 31.4, and ε₃₃ = 43, and the results of the experiment^[7] were ε₁₁ = 75, ε₂₂ = 46, and ε₃₃ = 43. Obviously, ε₃₃ is very approximate to the experimental value, while ε₁₁ and ε₂₂ are less than the results in the experiment. The dielectric constants of Sr₂Ta₂O₇ were ε₁₁ = 39.4, ε₂₂ = 18.1, and ε₃₃ = 74.5, respectively, while the previous results^[7] are ε₁₁ = 37, ε₂₂ = 22, ε₃₃ = 644, because of the limited experimental conditions at that time, it was impossible to determine whether the space group of Sr₂Ta₂O₇ was Pbcn or Cmcm.

Table 5.

Table 5.

Table 5. Calculated electronic (ε∞, ij) and phonon contributions (εph, ij), average dielectric constants ( ), and total dielectric constants (εij = ε∞, ij + εph, ij) for Sr2Nb2O7 and Sr2Ta2O7. . Sr2Nb2O7 Sr2Ta2O7 11 22 33 11 22 33 ε∞, ij 5.2 4.6 5.0 4.9 ε∞, ij 4.6 4.3 4.7 4.5 Cal. εph, ij 60.6 26.8 38.0 41.8 εph, ij 34.8 13.8 69.8 39.5 εij 65.8 31.4 43.0 46.7 εij 39.4 18.1 74.5 44.0 Expt.[5] εij 75 46 43.1 54.7 εij 37 22 644 234.3

Table 5. Calculated electronic (ε∞, ij) and phonon contributions (εph, ij), average dielectric constants ( ), and total dielectric constants (εij = ε∞, ij + εph, ij) for Sr2Nb2O7 and Sr2Ta2O7. .

The dielectric constants of Sr₂Nb₂O₇ are not high, so is the measured dielectric loss in the experiment. However, materials with ferroelectric properties at high temperature can be used for ferroelectric random storage materials.^[24] Table 6 lists the phonon frequency and oscillator strength of Sr₂Nb₂O₇ with major contributions to the dielectric constant. We can see that greater contributions are the frequency of 124.1, 55.5, 178.1, 100.7, and 55.3 cm⁻¹. Table 7 lists the contribution of ions to dielectric constants in Sr₂Nb₂O₇. The main contribution to the dielectric constants are Nb₁ and O₃–O₇ ions, in other words, the covalent structure of NbO₆ is formed by these ions, which contribute greatly to the dielectric constants.

Table 6.

Table 6.

Table 6. The calculated mode frequencies ωλ (in unit of cm−1), the oscillator strength, make dominant contributions to the dielectric intensity for Sr2Nb2O7. . ωλ (A1) Sλ, 33 ωλ (B1) Sλ, 11 ωλ (B2) Sλ, 22 124.1 9.3 178.1 33.4 76.1 4.8 93.1 2.7 100.7 16.4 55.3 9.9 81.5 5.7 67.9 4.6 55.5 8.4

Table 6. The calculated mode frequencies ωλ (in unit of cm−1), the oscillator strength, make dominant contributions to the dielectric intensity for Sr2Nb2O7. .

Table 7.

Table 7.

Table 7. Contributions of individual ions (εk, ij) to the dielectric constants for Sr2Nb2O7. . Atom WP εk, 11 εk, 22 εk, 33 Sr1 4a −0.593 3.711 3.072 Sr2 4a 0.011 0.123 0.478 Nb1 4a 10.805 −0307 1.649 Nb2 4a 0.89 −1.972 3.139 O1 4a 1.394 1.695 −0.848 O2 4a 0.582 0.665 −0.693 O3 4a 3.191 2.835 6.143 O4 4a 1.597 1.718 3.175 O5 4a 3.847 0.843 1.013 O6 4a 6.825 0.982 2.988 O7 4a 2.344 3.178 5.147

Table 7. Contributions of individual ions (εk, ij) to the dielectric constants for Sr2Nb2O7. .

In order to analyze the origin of the components in the dielectric tensor, the oscillators strength of greater frequencies and the effective charge of different frequencies are shown in Table 8. According to symmetry, the oscillator strength of B_1u has a non-zero component S_{λ, 33}, and B_2u only contributes to S_{λ, 22}. The major contribution to S_{λ, 22} is the infrared phonons of 308.9 cm⁻¹ and 115.6 cm⁻¹, the oscillator strength is 4.9 and 5.2, and the effective charges are 7.5 and 5.1, respectively. The dielectric component ε_{ph, 11} is mainly contributed by B_3u of 193.9 cm⁻¹ and 102.9 cm⁻¹, and the ε_{ph, 33} is derived from the B_1u, 121.9 cm⁻¹ and 92.2 cm⁻¹.

Table 8.

Table 8.

Table 8. The calculated mode frequencies ωλ (in unit of cm−1), oscillator strength, and mode effective charge (e) make dominant contributions to the dielectric intensity for Sr2Ta2O7. . ωλ B1u ωλ B3u ωλ B2u Sλ, 33 Sλ, 11 Sλ, 22 121.9 3.4 3.2 193.9 21.0 10.1 308.9 4.9 7.5 92.2 17.1 5.8 102.9 7.66 6.1 115.6 5.2 5.1 73.8 37.3 5.3

Table 8. The calculated mode frequencies ωλ (in unit of cm−1), oscillator strength, and mode effective charge (e) make dominant contributions to the dielectric intensity for Sr2Ta2O7. .

Phonon’s contribution to the dielectric constants can be decomposed into the contributions of each ion (Table 9). It can be seen that contributions from O1, O2, and O5 to ε_{ph, 11} is major. The greatest contribution to ε_{ph, 22} is Sr₁ and O₅. The main contribution to ε_{ph, 33} is Sr₂, O1, and O4, and O1 with a value of about 5.596.

Table 9.

Table 9.

Table 9. The contribution (εk, ij) of ions in Sr2Ta2O7 to the phonon dielectric constants. . Atom WP εk, 11 εk, 22 εk, 33 Sr1 4c 0.683 1.423 1.994 Sr2 4c 1.409 0.109 3.433 Ta1 4c 2.011 0.366 1.230 Ta2 4c 0.419 0.625 1.871 O1 4c 4.193 0.516 5.596 O2 4c 2.953 0.504 1.426 O3 8f 0.893 0.524 0.007 O4 8f 1.403 1.132 4.686 O5 4a 2.424 1.304 3.243

Table 9. The contribution (εk, ij) of ions in Sr2Ta2O7 to the phonon dielectric constants. .

Orthogonal structure Sr₂Nb₂O₇ exhibits the ferroelectric state, and the piezoelectric constant has five independent components (Table 10), of which, the larger values are e₃₁ (1.276) and e₃₃ (−2.932), respectively, while Sr₂Ta₂O₇ has no piezoelectric constant.

Generally known, the piezoelectric strain coefficient can be expressed as, 4 where S represents the elastic compliance constant, and C represents the elastic stiffness constant. The obtained elastic constants from the calculation are C₁₁ = 113.8 GPa, C₂₂ = 120.1 GPa, and C₃₃ = 196.8 GPa, which are in agreement with the existing experimental data.^[25] The piezoelectric constant is obtained by further calculation, d₃₃ = 24 pC/N, while the measured d₃₃ = 2.8 pC/N in the experiment.^[26]

Table 10.

Table 10.

Table 10. Homogeneous strain contribution (eiv, hom) and internal-strain contribution (eiv, int) to the total piezoelectric stress constants element eiv, tot (in units C/m2). We use the Voigt notion, the Latin index i runs from 1 to 3, and the Greek index v from 1 to 6. . e15 e24 e31 e32 e33 eiv, hom 0.212 .081 −0.122 .201 0.110 eiv, int 0.783 0.909 1.398 .602 2.813 eiv, tot 0.995 .990 1.276 .803 2.932

Table 10. Homogeneous strain contribution (eiv, hom) and internal-strain contribution (eiv, int) to the total piezoelectric stress constants element eiv, tot (in units C/m2). We use the Voigt notion, the Latin index i runs from 1 to 3, and the Greek index v from 1 to 6. .

The contributions of different phonon modes to the internal-strain piezoelectric constants in Sr₂Ta₂O₇ (Table 11) are further analyzed. Based on symmetry, we can know that the contributions of e₃₁ and e₃₃ to the piezoelectric components come from the A₁. For e₃₁, it can be seen that the main contribution is A₁ of 81.5 cm⁻¹ and 55.5 cm⁻¹, while e₃₃, the contributions come from 124.1 cm⁻¹ and 82.5 cm⁻¹. Compared with the phonon contribution to the dielectric, it can be found that the phonon contributions to the piezoelectric and dielectric components are consistent. Dielectric and piezoelectric responses are polarizable properties, which can be expressed as polarization about the applied electric field and macroscopic strain. Both of them can be understood as the contribution of electron and the relative displacement of ionic lattice.

Table 11.

Table 11.

Table 11. Contribution of A1 to the internal-strain piezoelectric constants eiv, int (in units C/m2). . ωλ e31 e33 124.1 0.177 −0.465 93.1 0.123 −0.392 81.5 0.603 −0.698 55.5 0.426 −0.277

Table 11. Contribution of A1 to the internal-strain piezoelectric constants eiv, int (in units C/m2). .

Table 12 shows the ion contribution to Sr₂Ta₂O₇ piezoelectric components. The greater contribution to e₃₁ is from O1, besides O2, and O5. The calculated e₃₃ is −2.813 C/m², O1, O2, O3, and Sr₂ have a relatively large contribution, with the greatest contribution from O1. In addition, the contribution of ion to dielectric (Table 9) shows that the contribution of ions with large dielectric components to the piezoelectric components are also significant.

Table 12.

Table 12.

Table 12. Contribution of individual ions to the internal-strain piezoelectric constants eiv, int (in units C/m2) for Sr2Ta2O7. . Atom WP e31, int e33, int Sr1 4c 0.085 –0.113 Sr2 4c 0.005 –0.129 Ta1 4c 0.090 –0.085 Ta2 4c –0.092 0.183 O1 4c 0.324 –0.594 O2 4c 0.182 –0.227 O3 8f 0.028 –0232 O4 8f 0.77 –0.049 O5 4a 0.142 –0.115

Table 12. Contribution of individual ions to the internal-strain piezoelectric constants eiv, int (in units C/m2) for Sr2Ta2O7. .

3.3. The Born effective charge and spontaneous polarization of Sr₂M₂O₇

The Born effective charge of Sr₂Nb₂O₇ are shown in Table 13, because of symmetry, only , , and are listed. This is similar to the ABO₃ perovskite structure compound,^[22] and the Born effective charge of each atom is deviated from the valence, this is particularly evident for the Nb and O in Sr₂Nb₂O₇. As seen from the table, the Born effective charge of Nb₁ and Nb₂ along the x axis are 7.864 and 7.788, respectively. Meanwhile, the Born effective charge of O₄ and O₅ are –5.966 and –6.040, which are almost triple as much as O⁻². The root reasons are the covalent properties and the hybridization between O-2p and Nb-4d. The Nb–O covalent bond is stronger than the Sr–O ionic bond. It is clear that the Born effective charge of O along the Nb–O is greater than the others. Therefore, the anisotropy is determined by the covalence of Nb–O.

Table 13.

Table 13.

Table 13. The Born effective charge for each inequivalent atom in the Cmc21 sapce group of the Sr2Nb2O7 structure. . Atom Sr1 (4a) 2.611 2.992 2.840 Sr2 (4a) 2.608 2.115 2.643 Nb1 (4a) 7.864 6.349 6.041 Nb2 (4a) 7.788 5.631 7.515 O1 (4a) –1.999 –2.895 –3.454 O2 (4a) –1.366 –2.809 –3.159 O3 (4a) –1.865 –3.088 –2.639 O4 (4a) –5.966 –1.288 –1.711 O5 (4a) –6.040 –1.569 –1.292 O6 (4a) –1.541 –2.259 –2.957 O7 (4a) –1.947 –3.197 –3.833

Table 13. The Born effective charge for each inequivalent atom in the Cmc21 sapce group of the Sr2Nb2O7 structure. .

At room temperature, the spontaneous polarization of Sr₂Nb₂O₇ was reported in the direction [001], which is 9 μC/cm².^[5] According to Berry-phase polarization theory, we calculated the spontaneous polarization of Sr₂Nb₂O₇ in the same direction, but 25 μC/cm². Because of the temperature, it is reasonable to make a difference in the spontaneous polarization. Sr₂Ta₂O₇ has no spontaneous polarization in the paraelectric state.

Sr₂Nb₂O₇ is an environmently-friendly piezoelectric material, and due to its high Curie point, it can be applied to high temperature conditions. PbTiO₃^[27] as one of the earliest discovered piezoelectric materials with good performance, and the piezoelectric constants were e₁₅ = 4.03 C/m², e₃₁ = 1.81 C/m², and e₃₃ = 3.23 C/m², respectively. The piezoelectric properties of Sr₂Nb₂O₇ need to be improved by comparison with PbTiO₃.^[28] It is well known that stress has an effect on improving the physical properties of ferroelectric materials.^[29] Therefore, we tried to apply a pressure on Sr₂Nb₂O₇, and the pressure was 0–4.5 GPa on the premise that the material did not have structural phase transition.

Fig. 4.

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	Fig. 4. (color online) The charge relation of volume, polarization, and bond length with pressure increase.

The changes of volume V, polarization P_s, and bond length with pressure increases are shown in Fig. 4, and it can be seen that parameters are all decreased in the pressure range. When the pressure reaches 4.5 GPa, the volume reduces to 588.19 ų, and the polarization is just 1.34 μC/cm². Because the volume decreases and the atomic position changes, the average bond lengths of the Sr–O and Nb–O shorten as well. This can also be attributed to the chain reaction in the crystal lattice along the crystal axis. Moreover, the long-range Coulomb force weakens and the short-range repulsion increases, which will restrain the ferroelectric properties of the material.

The results of further analysis of the relations among pressure, volume, and polarization (Fig. 4(a)) are summarized as follows. Firstly, the volume decreases to 599.23 ų when the pressure increases to about 2.5 GPa, the polarization is about 10 μC/cm², which are in good agreement with the experimental values.^[30] Therefore, we review the previous polarization results are greater than the experiment, perhaps because of the selection of function and limitations of simulation calculation. What is more, the negative pressure on the material may improve the spontaneous polarization and ferroelectric properties. Finally, in order to increase the tensile stress on the surface, by adding a substrate to Sr₂Nb₂O₇ so that lattice constants are slightly greater in the epitaxial-coating can also reach the goal of improving the properties.

First-principles calculations have been performed to study Sr₂M₂O₇, the optimized structures are consistent with previous experimental and theoretical data, which can indicate that our calculation method is reasonable. By comparing the dielectric constants in the experiment, the calculated results of Sr₂Nb₂O₇ and Sr₂Ta₂O₇ are within the limits of error. In addition, from the viewpoint of optical modes and phonons, the contribution and influence of the phonon modes at different frequencies and the various ions to the dielectric and piezoelectric properties were analyzed. For Sr₂Ta₂O₇, the phonon frequencies were also investigated, but there was no corresponding experimental data to compare with.

The larger values of Sr₂Nb₂O₇ piezoelectric components are e₃₁ and e₃₃, 1.276 C/m² and –2.932 C/m², respectively. The contribution to the piezoelectric component mainly comes from the A₁. The spontaneous polarization of Sr₂Nb₂O₇ is 25 μC/cm² in the direction [001], which is slightly larger than the experimental value. After pressure, the polarization of Sr₂Nb₂O₇ decreases linearly with the increasing pressure. Therefore, the application of tensile stress on the surface or negative pressure can enhance the spontaneous polarization and ferroelectric properties of the material.

Throughout the article, the study of lattice vibration is the key point, which relates to the properties of materials such as electromagnetism, specific heat, thermal conductivity, and conductance; our theoretical results will provide a guide to further investigate Sr₂M₂O₇.