Cite this Article
Wang Dawei, Liu Laijun, Liu Jia, Zhang Nan, Wei Xiaoyong. Epitaxially strained SnTiO3 at finite temperatures . Chinese Physics B, 2018, 27(12): 127702  
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Epitaxially strained SnTiO3 at finite temperatures
Wang Dawei1, †, Liu Laijun2, ‡, Liu Jia3, Zhang Nan4, Wei Xiaoyong4
School of Microelectronics and State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, China
College of Materials Science and Engineering, Guilin University of Technology, Guilin 541004, China
State Key Laboratory for Mechanical Behavior of Materials & School of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, China
Electronic Materials Research Laboratory–Key Laboratory of the Ministry of Education and International Center for Dielectric Research, Xi’an Jiaotong University, Xi’an 710049, China

 

† Corresponding author. E-mail: dawei.wang@mail.xjtu.edu.edu 2009011@glut.edu.edu

Project supported by the National Natural Science Foundation of China (Grant Nos. 11574246, 51390472, U1537210, and 11564010), the National Basic Research Program of China (Grant No. 2015CB654903), the Natural Science Foundation of Guangxi Zhuang Autonomous Region (Grant Nos. GA139008 and AA138162), and the “111” Project of China (Grant No. B14040).

Abstract

By combining the effective Hamiltonian approach and direct ab initio computation, we obtain the phase diagram of SnTiO3 with respect to epitaxial strain and temperature. This demonstrates the complex features of the phase diagram and provides an insight into this system, which is a presumably simple perovskite. Two triple points, as shown in the phase diagram, may be exploited to achieve high-performance piezoelectric effects. Despite the inclusion of the degree of freedom related to oxygen octahedron tilting, the ferroelectric displacements dominate the structural phases over the whole misfit strain range. Finally, we show that SnTiO3 can change from hard to soft ferroelectrics with the epitaxial strain.

PACS: 77.80.-e;77.84.-s;81.30.Bx
Keyword:SnTiO3;phase diagram;epitaxial strain
1. Introduction

Piezoelectricity is a phenomenon in certain materials that strain and electric polarization can induce and/or influence each other. In response to an applied mechanical strain, ferroelectric materials, which are inherently piezoelectric, can produce an electric polarization proportional to the load. Similarly, these materials will produce a mechanical deformation (strain) in response to an applied voltage. Switchable polarization makes ferroelectrics a critical component in memories, actuators, electro-optic devices, and potential candidates for nanoelectronics.[1]

In recent years, it has been found that materials of high-performance piezoelectricity are often associated with morphortropic phase boundary (MPB), with examples including Pb(Zr,Ti)O3[2] and (K,Na)NbO3-LiTaO3-LiSbO3,[3] or triple points, such as in (Ba,Ca)(Zr,Ti)O3.[4] Engineering solid solutions to a certain composition can create phase boundaries and tricritical points, where the crystal structure changes abruptly, inducing maximal piezoelectric properties. Three important situations have been extensively studied: (i) MPB in pure perovksites that separates regions of the tetragonal from the rhombohedral symmetry;[5] (ii) MPB formed in perovskites dissoluted with a small amount of non-perovskite-structured materials that can cause lattice distortions,[3] and grain boundary effects;[6] and (iii) regions close to a triple point where cubic paraelectric phase (C), ferroelectric rhombohedral (R), and tetragonal (T) phases meet.[4] In addition to MPB, epitaxial thin-film growth, which introduces intrinsic lattice strain, has matured as another important method to design desired ferroelectric materials, which is important for highly integrated design and intelligent control technology.[7,8] Strain engineering is widely adopted and can tune the large 2p–3d charge hybridization between the strongly correlated 3d electrons in transition metal ions and the 2p electrons of oxygens.[9–12] For instance, both compressive and tensile strains increase the Ni 3d band width and favor the metallic phase in NdNiO3.[13] In addition, substrate clamping will force the temperature dependence of in-plane lattice constants to follow that of the substrates, which may lead to unexpected phase transitions and domain formation.[14,15] Therefore, strain constraint can introduce muti-phase coexistence in thin films, which makes it an attractive method to fine tuning film properties.

Nowadays, the commonly used high-performance piezoelectric materials, including PbTiO3 and Pb(Mn,Nb)O3, contain hazardous lead (Pb). Since Pb is harmful to environment and human health, lead-free ferroelectric materials are highly desired. Many lead-free materials are based on (Bi0.5,Na0.5)TiO3, (K,Na)NbO3 or BaTiO3; however, their performance is still sub-optimal compared to Pb-containing materials.[16] Since Sn and Pb belong to the same family, SnTiO3 is expected to achieve high-performance piezoelectricity with environmentally benign elements.[17,18] Indeed, SnTiO3 has large polarization and large axial ratio,[19,20] even larger than PbTiO3, making it a promising candidate. But for various reasons, SnTiO3 bulk material is hard to prepare because Sn2+ can easily become Sn4+, and Sn is prone to enter into the B site (where the Ti ion stays) due to its small ionic radius. However, researchers continued to look for opportunities to exploit the remarkable properties of SnTiO3. For instance, researchers have considered Sn-doped BaTiO3,[21,22] Bennett et al. considered Sn(Al0.5,Nb0.5)O3,[23] while Suzuki et al. obtained Sn-doped SrTiO3,[24] and Laurita investigated (Sr,Sn)TiO3 and (Ba,Ca,Sn)TiO3.[25] Recently, Agarwal et al.[26] obtained perovskite phase SnTiO3 with the atomic-layer deposition technique. This is an important breakthrough that will excite more work on SnTiO3.

In addition to experimental work, there are also many theoretical investigations of SnTiO3.[17–19,27–31] However, most of previous theoretical investigations are based on direct ab initio computation, which cannot provide information regarding SnTiO3 at finite temperatures. Therefore, many important questions remain unanswered. For instance, what are the conditions for the existence of different ferroelectric phases? What does the phase diagram of SnTiO3 look like? Can such a seemingly simple perovskite (SnTiO3 is not a solid solution and no doping is applied) possess complex features? In this work, we will focus on the finite temperature properties of epitaxially strained SnTiO3 and address these questions. We note that this information will be useful for the fabrication of SnTiO3 bulk, or the growth of SnTiO3 film, and engineering SnTiO3-containing ferroelectric materials. Since SnTiO3 bulk is not available, there is little a priori information at finite temperatures that can be used in this work. While our computational results are less convincing without the support from experiments, it demonstrates the value of theoretical and numerical work—i.e., their power to predict something unknown—and this is one of the reasons that motivated this investigation.

2. Method

To fully understand SnTiO3, it is necessary to know all of its possible phases under various conditions. However, it is not trivial to achieve this goal. For instance, the direct ab initio approach usually provides us with the structural phase of local energy minimum (in contrast to the most stable phase of global energy minimum), which is exacerbated by the 0-K assumption adopted. The application of this approach often requires a comparison of many different phases that may not be able to cover all possibilities. To address this problem, here we adopt the first-principles based effective Hamiltonian approach and Monte–Carlo (MC) simulations, which were developed exactly to address these challenges. To use this approach, it is necessary to compute tens of coefficients appearing in the effective Hamiltonian using direct ab initio computation before carrying out MC simulations.

The effective Hamiltonian used here was originally developed in Refs. [32] and [33], which incorporate the coupled dynamics of the soft mode, strain, and dipole. Its internal energy is given by

where ui is the local soft-mode in unit cell i, and is proportional to the local electric dipole moment in that cell when multiplied by Born effective charge. We note that ui here is located on the A-site (where Sn stays), and represents the collective motion of Sn, Ti, and O atoms inside one unit cell. The vi are Sn-centered local displacements related to the inhomogeneous strain inside each unit cell. ηH is the homogeneous strain tensor. The energy terms and associated parameters can be found in Refs. [32], [33], and [34]. In this work, we extended the effective Hamiltonian and expanded the local energy term to 8th order (similar to Ref. [35]) to describe the internal energy more precisely.[36] In addition, the antiferro-distortive (AFD) oxygen octahedron tilting is also considered with the energy term[37]
where the new variable ωi represents the oxygen tilting on the unit cell i, which centers on the B-site atom (i.e., Ti). The pristine expression of EAFD can be found in Ref. [37], which also includes the couplings between AFD, and all the other dynamical variables. The associated parameters in EAFD are obtained in a similar way as in Ref. [34]. The effective Hamiltonian approach is a well established methodology that has been developed since 1994[32,33,38] and similar to direct ab initio methods, it has been used in many investigations, e.g., to predict new structural phases of perovksites.[39,40] The most appealing features of this approach are the following: (i) it can often lead us to the most stable structural phase of global energy minimum; and (ii) it can produce finite temperature properties, which explains why we adopt this approach in our work.

With the effective Hamiltonian, we perform MC simulations on a 12 × 12 × 12 supercell, containing 8640 atoms. The system is set to meet the periodic boundary condition along the (pseudo) x, y, z directions and subject to various epitaxial strains. In each simulation at a given epitaxial misfit strain s, we gradually cool down the system from 2500 K to 5 K. For each temperature, we carry out 1.6 × 105 MC steps to obtain averaged physical quantities, most importantly the supercell average of local mode, which can be used to determine the symmetry of the system and the approximate positions of each atom in it. The use of supercell is necessary to find various possible structural phases and accurate phase transition temperatures. For instance, to simulate antiferroelectric perovksites, at least two primitive unit cells are needed. Similarly, for AFD related phases, two or more primitive unit cells are necessary. In fact, the ability to simulate large supercells is an important feature and advantage of the effective Hamiltonian approach, which enables us to find the energy ground state.

In this approach, the symmetries or temperatures are not preset in the effective Hamiltonian. The starting point of the effective Hamiltonian is the cubic phase (), which is used as the reference state. The effective Hamiltonian quantifies the energy change when ion displacements and deformation (i.e., strain) happen with respect to this reference state. This energy change is then used in the MC simulations to find the averaged ion displacements and deformation at a given temperature (and epitaxial strain). Finally, the ion displacements and the deformation are used to determine the structural phase (and symmetry) of SnTiO3 at the given temperature. In this process, the temperature is a given parameter for each MC simulation, which employs the effective Hamiltonian and the symmetry of SnTiO3 is inferred after the MC simulation is done.

We have also performed direct ab initio computation to corroborate MC simulation results. For this purpose, the open source ABINIT software package[41] is used along with the local density approximation (LDA)[42] and the projector-augmented-wave (PAW) method.[43] We use the pseudo-potentials implemented in the GBRV package,[44] and the Sn 4d 5s 5p, Ti 3s 3p 4s 3d, and O 2s 2p orbitals are treated as valence orbitals. For convergence, we have chosen the cut-off energy (ecut) to be 25 Hartree (1 Hartree = 27.211 eV) for plane wave expansion, and the fine grid cut-off energy (pawcutdg) is selected to be 50 Hartree. In addition, k-point sampling of 6 × 6 × 6 Monkhorst–Pack grid[45] was used. The atomic coordinates are relaxed until all atomic-force components are smaller than 10−5 Hatree/Bohr, and the cell size and shape are varied until all stress components are below 10−7 Hatree/Bohr3.

Both the MC simulations and the direct ab initio computations deal with the strained bulks (in contrast to ultrathin two-dimensional films) given the periodic boundary condition used. However, many properties of the strained films (as long as they are not just a few atomic layers thick) can be inferred from these calculations. Comparing to our previous work,[34] the present investigation develops in three directions:

We now include the new degree of freedom, i.e., oxygen octahedron tilting (often called antiferrodistortive rotations, AFD), which is ignored in the previous work.[34]

To this end, we computed many new essential parameters for the effective Hamiltonian. While the current work build on the previous one, it has substantially extended the effort to fully simulate SnTiO3.

Here we consider the effects of epitaxial strain and show that SnTiO3 has a rather complex phase diagram, containing interesting phase boundaries and triple points.

3. Results

To obtain the phase diagram, we have performed MC simulations for misfit strain between s = 0 and s = 2% with a step size of Δs = 0.125%. For each misfit strain s, the system is gradually cooled down from ∼2500 K to 5 K and its evolution with temperature is observed. The simulation results are then summarized to form the phase diagram.

3.1. Phase transition and phase diagram

We first obtain the supercell averaged local mode versus temperature of SnTiO3 under various tensile epitaxial strains, which is expected to stabilize SnTiO3.[20] This information enables us to find the evolution of structural phases with respect to temperature and epitaxial strain, and more importantly, the phase transition temperatures of the system. Figure 1 shows the results with misfit strain s = 0.375% and s = 1%. Note that we have relaxed the cubic phase ( ) SnTiO3 using ABINIT (with settings specified in Section 2) to obtain the lattice parameter a = 7.312 Bohr, which is used as the reference value for specifying the misfit strain.

Fig. 1. (color online) Local mode versus temperature at two given strain misfit, s = 0.375% (a) and s = 1% (b).

For the smaller epitaxial strain s = 0.375%, the system adopts P4mm phase for temperature T ≲ 1000 K, (Px = Py = 0, Pz > 0), as the temperature increases it undergoes a phase transition to become the Amm2 phase (Px = Py > 0, Pz = 0), and eventually the cubic phase for T ≳ 1500 K. The whole process is somewhat similar to that of BaTiO3 and (Na0.5,K0.5)NbO3, where several structural phases are involved in phase transitions. Here, however, a rather drastic change happens at around 1000 K as the polarization rotates from out-of-plane to the in-plane configuration. For the larger epitaxial strain (s = 1%), there are also two phase transitions. At low temperature (T ≲ 197 K), it adopts the Cm (Px = Py > 0, Pz > 0) phase, which changes to the Amm2 phase as the temperature increases, and finally becomes paraelectric at T ≃ 1750 K. This phase transition temperature is high comparing to other typical ferroelectric materials, e.g., PbTiO3 (763 K) and BiFeO3 (1100 K).[46] This is likely due to the strong dipole–dipole interaction inside SnTiO3, noting that the spontaneous polarization of SnTiO3 (1.32 C/m2)[34] is even larger than the strained BiFeO3 (1.30 C/m2).[47] Given the high phase transition temperatures, we note that in the MC simulations SnTiO3 is assumed to be always in the solid state (i.e., it is not melted) when the structural phases are obtained. While the melting point of SnTiO3 is not yet available experimentally, it is likely lower than that of PbTiO3, which is 1554 K.

No correlated AFD was observed for the epitaxial strains investigated here, which is consistent with previous known results[20] (also see Section 4). Figure 1 indicates that four phases (paraelectric , orthorhombic Amm2, tetragonal P4mm, and monoclinic Cm) can all exist in SnTiO3 at proper temperature and epitaxial strain. Without strain constraint or at tiny strain (s < 0.5%, see Fig. 2), the system adopts the P4mm phase (Pz > 0, Px,y = 0) at low temperature, and only has one phase transition (P4mm to paraelectric) as temperature increases. Interestingly, figure 1(b) shows a region (T ≤ 200 K) that corresponds to the MB phase (belong to space group Cm),[36,38] and for smaller s (e.g., at s = 0.75%), we have observed that Pz > Px = Py, which is also Cm, but corresponds to the MA phase.[48] Moreover, in Fig. 1(a), both the MA and MB phases exist between the T (P4mm) and O (Amm2) phases. The phase transition sequence resembles the local structure evolution of Pb(Ti1−x, Zrx)O3 with increasing x.[49] This monoclinic region has been shown to play critical role in high-performance piezoelectric materials.[50]

Fig. 2. Phase diagram of epitaxially strained SnTiO3.

We now turn to the phase diagram of SnTiO3 with respect to temperature and epitaxial strain. First, for all the investigated misfit strains (up to 2%), figure 2 shows that SnTiO3 undergoes a paraelectric to ferroelectric phase transition at rather high temperature (TC > 1400 K). The high TC is likely due to the large intrinsic spontaneous polarization discussed previously.[34] Second, TC initially decreases with s, reaching a minimum, and then increases again.[51] The lowest point corresponds to a triple point separating the paraelectric phase and the other two ferroelectric phases (tetragonal P4mm and orthorhombic Amm2), similar to what happens in ultrathin PbTiO3 films.[15] We note that the existence of the Cm phase is consistent with results obtained in Ref. [20]. The abrupt transition from the P4mm to the Cm phase (with respect to the misfit strain) happens in a slender region (< 0.125%) represented by a straight line in Fig. 2 due to the limit of Δs used in simulations.

The first triple point in Fig. 2, where one paraelectric cubic phase and two ferroelectric phases converge, contains potential large piezoelectric effects. For instance, a triple point was found and exploited in binary compounds including (Ba0.7,Ca0.3)TiO3-Ba(Zr0.2,Ti0.8)O3 (BCZT),[4] Ba(Sn0.12,Ti0.88)O3−x(Ba0.7,Ca0.3)O3,[52] and other BaTiO3-derived systems.[53] More importantly, the three ferroelectric phases (P4mm, Amm2, and Cm) converge to a second triple point. There are three boundaries around this point. The boundary at s = 0.5% separates the P4mm and the Cm phase, resembling the MPB seen in Pb(Zr1−x,Tix)O3, in which two different structural phases exist with a buffer region at x ≃ 0.48. In Pb(Zr1−x,Tix)O3 the monoclinic phase serves as a bridge between the higher symmetry tetragonal phase (with [001] polarization) and the rhombohedral phases (with [111] polarization).[37] In this connecting phase, the polarization can align anywhere on the {110} mirror plane between the pesudocubic [111] and [001] directions, giving rise to the high piezoelectric response.[37,54,55]

Here in SnTiO3, the pronounced existence of the Cm phase and the second triple point, which bridges the tetragonal phase (with [001] polarization) and the orthorhombic phase (with [110] polarization), may also enable high performance, following the pattern of the universal phase diagram discussed in Refs. [54] and [55]. The reason is similar to that of Pb(Zr1−x,Tix)O3: in this region, polarization anisotropy nearly vanishes and thus polarization rotations are easy.[48] Many alkaline niobate perovskites show a polymorphic phase transition[56] between the tetragonal phase and the orthorhombic phase, similar to what happens in Fig. 2. The polymorphic behavior can also lead to high piezoelectricity due to the instability with respect to polarization rotation.[57] However, the temperature stability of their piezoelectric properties for alkaline niobate perovskites is not as good as Pb(Zr1−x,Tix)O3. In addition, the large anisotropy along the whole polymorphic boundary line leads to a larger energy barrier between the two polarization states (tetragonal and orthorhombic), preventing possible polarization rotations because the phase coexistence results from the diffusive tetragonal to orthorhombic phase transformation.[4] In SnTiO3, unlike the polymorphic phase transition, the Cm phase is associated with a triple point, where a low energy barrier between two ferroelectric phases (P4mm and Amm2) may exist that facilitates the polarization rotation and lattice distortion, leading to high performance. Unfortunately, this triple point is not at room temperature for pure SnTiO3 (which is at ∼750 K as shown in Fig. 2), and may need to be tuned (e.g. by doping). For instance, following the lessons from BCZT, it may be possible to use Pb to substitute Sn and/or Zr to substitute Ti. In this way, SnTiO3 can be taken as the matrix material for designing high performance piezoelectric materials.

To show that the polarization in the Cm phase can easily rotate, we also obtained the hysteresis loop at different epitaxial strains. As figure 3 shows, the Cm phase has a strong effect on the hysteresis loop of SnTiO3. When the Cm phase exists [Fig. 3(a)] at s = 0.75%, the coercive field is ∼1.6 × 107 V/m, reduced by a factor of 10 compared to the result obtained when s = 0% (> 2 × 108 V/m, see Fig. 3(b)). Analogous to magnetic materials, by applying a proper strain (e.g., 0.75%), SnTiO3 becomes “soft” ferroelectrics. Interestingly, in the whole process, the magnitude of the polarization P is approximately a constant [dark green line in Fig. 3], making the whole process a rotation as well as switching of polarization (albeit a sudden rotation), consistent with previous computations.[30,31]

Fig. 3. (color online) Hysteresis loop of SnTiO3 at s = 0.75% (a) and s = 0% (b) at 300 K. The electric field is applied along the z axis.
3.2. Direct ab initio computation

To corroborate the results obtained from effective Hamiltonian-based computations, we also obtained numerical results from direct ab initio computation.

Figure 4(a) shows the energy versus strain for the P4mm, Amm2, and Cm phases. These phases all appear in the phase diagram of SnTiO3 (Fig. 2). For a large range of epitaxial strain, the Cm phase has the lowest energy. At s = 1%, the energy of the Cm phase is approximately 29.4 meV lower than that of P4mm or Amm2. These results indicate that the Cm phase can be the ground state for a restricted range of strain, which supports our effective Hamiltonian results. The Cm phase appears around s = 0.5% is likely because at this point the three phases (Cm, P4mm, and Amm2) have similar energies, and macroscopically the average of P4mm and Amm2 also give rise to the Cm phase. On the other hand, when s > 1%, P4mm is no longer an option as its energy becomes much higher than the other two. Because the energy difference between Amm2 and Cm continues to be smaller with the misfit strain, the Cm phase can only exist at low temperatures, and eventually disappears. The Cm phase rarely appear in pure perovskites, whether it is epitaxially strained or not, and SnTiO3 seems to be an important exception. Figure 4(b) plots the polarization versus the misfit strain for the Cm phase, which is similar to the results obtained in Ref. [9], showing that Pz increases while Px,y decreases with increasing epitaxial strain. We finally note that, despite many attempts, no structural phases involving AFD tilting were found to be at the ground state. For 0% < s < 5%, structural phases without AFD are consistently more stable in terms of energy.

Fig. 4. (color online) (a) Energy and polarization versus strain for three different phases Amm2, P4mm, and Cm. At s = 0%; (b) The polarization (Px = Py and Pz) versus strain for the Cm phase.
4. Discussion

In Section 3 we have shown that SnTiO3 has P4mm, Amm2, and Cm phases under different conditions. However, it is hard to prove that SnTiO3 can only has these three phases for the misfit strain we have investigated. This point is further discussed below in Subsection 4.1. Moreover, we will compare the MC results to direct ab initio results and focus on the Cm phase in Subsection 4.2.

4.1. Seeking new phases

Phonon calculations have shown that for SnTiO3 the AFD-related modes are also unstable.[20] Considering this fact, it is rather surprising that the MC simulation results are unable to identify new phases involving oxygen octahedron tilting, which is a pity and an important lesson.

We had intentionally played with the coefficients in the effective Hamiltonian and performed additional MC simulations to suggest new phases involving AFD. The simulations results indeed generated a few AFD-related candidates (e.g., the Ima2, Imma, and I4cm phases). However, direct ab initio calculations do not validate them as energy ground states, which is consistent with the fact that Parker et al. did not propose any AFD related phases although their calculations have shown strong AFD instability.[20] Therefore, our results strongly suggest that SnTiO3 may not have AFD-related phases as ground state for the misfit strain of s < 5%.

The fact that AFD-related phases do not appear is most likely to be due to the strong competition from the polar local mode, which is responsible for the polarization in SnTiO3. For ferroelectric materials, AFD is usually adverse to the development of polarization. One term in the effective Hamiltonian specifically represents such an effect, which is Dω2u2,[58] where the sign of the coupling coefficient (D) will determine how strong the competition (or in rare cases the cooperation) between the local mode u (related to polarization) and ω (related to AFD). In SnTiO3, the ferroelectric local mode (which is responsible for the P4mm, Amm2, and Cm phases) apparently dominates the system.

4.2. The Cm phase

The results from ab initio computations in Fig. 4 show that the Cm phase has the lowest energy beyond s = 0 among the P4mm, Cm, and Amm2 phases. Meanwhile, the phase diagram obtained from MC simulations shows that SnTiO3 exhibits the Cm phase in a much smaller misfit strain range. The most likely reason of this discrepancy is that the lone pair on Sn2+ can cause an extra out-of-plane polarization that is not well-accounted for in the effective Hamiltonian, where the polarization is closely related to the Γ-point (of the cubic phase) unstable polar mode.[32,33]

For SnTiO3 that mode is ux,y,z = (ξSn,ξTi,ξO||,ξO⊥,ξO⊥) = (0.534,0.169,−0.411,−0.508,−0.508) (normalized), which approximately represents the ion displacements in SnTiO3 and specifies how spontaneous polarization develops. However, some anomaly happens for SnTiO3 as can be seen from the direct ab initio computation. For instance, at s = 1%, along the z direction the ion displacements are while along the x, y directions, . It is important to note that u′ has a much larger weight on Sn2+, which is not reflected in u. In fact, this is a known issue during the development of the effective Hamiltonian approach as discussed in Ref. [38] where it was pointed out that for KNbO3 and PbTiO3 there is large difference between the experimental and theoretical local modes. Here, SnTiO3 has the same problem while the difference between PbTiO3 and SnTiO3 is discussed in detail in Ref. [59]. In principle, for SnTiO3 it is possible to tune the values of u (or adding an extra polar mode) in the effective Hamiltonian to alleviate or fix this issue. However, this move will involve many more cumbersome calculations of coupling coefficients, making the approach more complicated. This issue again shows the complexity of SnTiO3 despite its simple composition.

5. Conclusion

Using effective Hamiltonian-based MC simulations, we have investigated epitaxially strained SnTiO3, found their structural phases at finite temperatures, and obtained its phase diagram with respect to temperature and misfit strain. The phase diagram of SnTiO3 turns out to be rather complicated, containing two triple points and boundaries that separate ferroelectric phases. These special features provide unique opportunities to design novel high-performance ferroelectric materials containing SnTiO3, in which the rarely seen Cm phase can exist. In addition, while phonon calculation has shown that AFD related modes are unstable,[20] no AFD-related structural phases were found in our simulations, which is likely to be due to the strong competition from unstable polar modes.

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