Abstract
In light of directives around the world to eliminate toxic materials in various technologies, finding leadfree materials with high piezoelectric responses constitutes an important current scientific goal. As such, the recent discovery of a large electromechanical conversion near room temperature in (1−x)Ba(Zr_{0.2}Ti_{0.8})O_{3}−x(Ba_{0.7}Ca_{0.3})TiO_{3} compounds has directed attention to understanding its origin. Here, we report the development of a largescale atomistic scheme providing a microscopic insight into this technologically promising material. We find that its high piezoelectricity originates from the existence of large fluctuations of polarization in the orthorhombic state arising from the combination of a flat freeenergy landscape, a fragmented local structure, and the narrow temperature window around room temperature at which this orthorhombic phase is the equilibrium state. In addition to deepening the current knowledge on piezoelectricity, these findings have the potential to guide the design of other leadfree materials with large electromechanical responses.
Introduction
Piezoelectricity is a physical phenomenon that converts mechanical into electrical energy and viceversa (see ref. 1 and references therein). It has been used for devices such as sensors, actuators or ultrasonic motors^{2,3}. Up to now, the materials exhibiting the highest piezoelectric responses include lead ions, such as Pb(Zr,Ti)O_{3} or Pb(Mg,Nb,Ti)O_{3} (refs 4, 5, 6, 7, 8), that introduce toxicity concerns. As a result, the search for leadfree compounds exhibiting large roomtemperature piezoelectricity constitutes an important current research direction that is partially driven by regulations announced by several countries^{9,10}. In that regard, the discovery of a large electromechanical response found at room temperature in the (1−x)Ba(Zr_{0.2}Ti_{0.8})O_{3}−x(Ba_{0.7}Ca_{0.3})TiO_{3} solid solutions with x=0.50 (to be denoted as BCTZ0.5 in the following) and reported in ref. 11 is a major finding.
Interestingly, the microscopic origin of such large piezoelectricity in this leadfree system remains subject to debate. For instance, the experimental study of ref. 11 suggests that it arises from the proximity of a tricritical point, where two ferroelectric phases of rhombohedral and tetragonal symmetries meet with the paraelectric phase of cubic symmetry. In contrast, the combined theoretical and experimental investigation of ref. 12 proposes that the highest piezoelectric coefficients are reached at the boundary between ferroelectric phases of orthorhombic and tetragonal symmetries as a result of a combination of reduced anisotropy energy, high polarization and enhanced elastic softening. The experimental analyses of refs 13, 14 offer yet another explanation, pointing out the coexistence of tetragonal, orthorhombic and rhombohedral phases and the strong electricfielddependency of their relative contributions to the total system as the culprits responsible for the large observed electromechanical response. On the basis of the polarizationrotation mechanism proposed in leadbased materials^{4,15,16,17,18}, it is also legitimate to wonder if an overlooked lowsymmetry phase, inside which the spontaneous polarization easily rotates, may be responsible for large piezoelectricity in (Ba_{0.85}Ca_{0.15})(Zr_{0.10}Ti_{0.90})O_{3}.
A plausible explanation for the paucity of knowledge of BCTZ0.5 is that atomistic simulations, which have been particularly important to understand piezoelectricity in leadbased materials^{15,16,17,18}, are currently lacking for (1−x)Ba(Zr_{0.2}Ti_{0.8})O_{3}−x(Ba_{0.7}Ca_{0.3})TiO_{3} compounds. This lack of simulations is likely due to the difficulty in realistically mimicking these latter solid solutions, since not only do they possess chemical mixing at both the A and B sublattices of the ABO_{3} perovskite structure, but can also exhibit local inhomogeneities (especially if phase coexistence occurs as advocated in refs 13, 14). As a result, large supercells are most likely required to accurately model BCTZ0.5, which is typically problematic from the standpoint of memory and computational time.
Here, we build a largescale atomistic approach to tackle roomtemperature piezoelectricity in (Ba_{0.85}Ca_{0.15})(Zr_{0.10}Ti_{0.90})O_{3}. The use of such a scheme leads to a successful explanation of its origin that we find residing in the dyadic combination of the narrow temperature range of stability of the macroscopic orthorhombic phase near 300 K, and the flatness of the free energy associated with this orthorhombic phase, which allows large fluctuations of the polarization around its equilibrium value. Such macroscopic effects are also associated with specific characteristics of the local structures, including the existence of the socalled percolating cluster (which is of orthorhombic symmetry) while its strength (that is, volume per cent occupied in the material) is found here to directly correlate with piezoelectricity. It is also worth realizing that clusters of orthorhombic symmetry are not ingredients of the widely used Comes–Guinier–Lambert^{19}, which implies that this model ought to be generalized to be more realistic.
Results
Atomistic scheme
We adopt the virtual crystal approximation (VCA)^{20,21} mimic (1−x)Ba(Zr_{0.2}Ti_{0.8})O_{3}−x(Ba_{0.7}Ca_{0.3})TiO_{3} with x=0.50. We first model a virtual 〈A〉〈B〉O_{3} simple perovskite system, for which the 〈A〉 atom involves a compositional average of Ba and Ca potentials of 85 and 15% respective contributions, while the 〈B〉 atom is built from a mixing of the Zr and Ti potentials of 10 and 90% respective contributions. An effective Hamiltonian (H_{eff}) is then developed for this 〈A〉〈B〉O_{3} system. As in ref. 22, the degrees of freedom of this H_{eff} are, for each 5atom unit cell i, the local soft mode u_{i} that is proportional to the electric dipole moment of the cell, the η_{H} homogeneous strain tensor, and inhomogeneousstrainrelated dimensionless displacements {v_{i}}. Technically, the various {u_{i}} and {v_{i}} are, respectively, centred on the 〈B〉 and 〈A〉 sites. The analytical expression for the total internal energy of this effective Hamiltonian is the one provided in ref. 22 for pure BaTiO_{3}, and therefore contains a localmode selfenergy, a longrange dipole–dipole interaction, a shortrange interaction between soft modes, an elastic energy, and an interaction between the local modes and local strains. In particular, the localmode selfenergy is given by:
where the sum runs over all the 〈B〉 sites and where (u_{i,x}, u_{i,y}, u_{i,z}) are the Cartesian components of u_{i} in the orthonormal basis formed by the [100], [010] and [001] pseudocubic directions. In the first step, the parameters κ_{2}, α and γ are determined, along with the other coefficients of the effective Hamiltonian, by performing density functional theory calculations within the VCA approach^{21} on small 〈A〉〈B〉O_{3} cells (less than 20 atoms). In a second step, MonteCarlo (MC) simulations using E_{tot} are conducted on large supercells (typically of 12 × 12 × 12 or 18 × 18 × 18 dimensions) made of 〈A〉〈B〉O_{3}. During these simulations, κ_{2} is varied to fit the experimental value of the Curie temperature^{11,12,23} (since H_{eff} techniques can underestimate the paraelectric–ferroelectric transition temperature^{22,24}) and γ is slightly adjusted to reproduce the measured lowest transition temperature observed in refs 12, 23 for (1−x)Ba(Zr_{0.2}Ti_{0.8})O_{3}−x(Ba_{0.7}Ca_{0.3})TiO_{3} solid solutions having x=0.50. For comparison, we also computed finitetemperature properties of pure BaTiO_{3} (BTO), as arising from the use of the effective Hamiltonian of ref. 24.
Phase transitions
The results of these MC simulations for BCTZ0.5 and BTO are shown in Fig. 1a,b for the Cartesian components of the supercell average of the local modes, 〈u〉 (which is directly proportional to the spontaneous polarization), when averaging over 4 million MC sweeps and using 18 × 18 × 18 supercells. The x, y and z axes are chosen along the [100], [010] and [001] pseudocubic directions, respectively. The computations for BCTZ0.5 correctly qualitatively and even quantitatively reproduce the three observed successive (firstorder) transitions^{12,23} when cooling down the system: first, a paraelectric cubic Pmm to ferroelectric tetragonal P4mm transition at around 360 K, for which the z component of 〈u〉 becomes nonzero; second, a P4mm to ferroelectric orthorhombic Amm2 transition near 297 K (that is, very close to room temperature), for which the y component of the supercell average of the local modes suddenly becomes nonzero and equal to the z component; and third, a Amm2 to ferroelectric rhombohedral R3m transition at ≃270 K, for which all the Cartesian components of 〈u〉 are now nonzero and equal to each other. In Supplementary Note 2, it is also demsonstrated that the effective Hamiltonian scheme used within the VCA approach (with a rescaling of the κ_{2} and γ parameters) can correctly reproduce the temperature–compositional phase diagram of (1−x)Ba(Zr_{0.2}Ti_{0.8})O_{3}−x(Ba_{0.7}Ca_{0.3})TiO_{3} (BCTZx) for x ranging between 0.25 and 0.65, as well as other properties of BCTZx, which further attests to the validity of the presently used numerical method. Moreover, for pure BaTiO_{3}, Fig. 1b shows that the Pmm−P4mm, P4mm−Amm2 and Amm2−R3m transitions are predicted to be about 384, 283 and 226 K, which agree reasonably well with the corresponding experimental values of 400, 280 and 180 K (refs 25, 26).
Piezoelectricity
Piezoelectric coefficients, d_{ij}, of BCTZ0.5 and BTO are calculated using the correlationfunction approach of ref. 27, that is:
where T is the temperature, k_{B} the Boltzmann constant, N_{s} the total number of 5atom cells composing the supercell, Z* the Born effective charge associated with the soft mode and a the 5atom lattice constant. u_{i} is the icomponent of the supercell average of the local mode at a given MC sweep, and η_{H,j} is the j component of the homogeneous strain tensor (in Voigt notation) at this MC sweep. The 〈〉 symbol denotes statistical averages over the different MC sweeps. Figure 1c,d report an averaged computed piezoelectric coefficient, 〈d_{ave}〉, for BCTZ0.5 and BaTiO_{3}, respectively. More precisely, 〈d_{ave}〉 is equal to (for the aforementioned (x, y, z) basis) in the R3m phase, since d_{11}, d_{22} and d_{33} coefficients are all nonzero and equal to each other in this phase. On the other hand, 〈d_{ave}〉 is chosen to be (respectively, d_{11}+d_{22}+d_{33}) in the Amm2 (P4mm) phase because only two (one) of these three coefficients are (is) nonzero there. We also practically choose 〈d_{ave}〉 to be equal to d_{11}+d_{22}+d_{33} in the paraelectric phase, as in the tetragonal ferroelectric state (note that all three aforementioned choices provide the same 〈d_{ave}〉 in the Pmm phase since it has no piezoelectricity).
In BCTZ0.5, large values of this piezoelectric coefficient exist in the temperature range associated with the stability of the Amm2 phase, as consistent with the experimental findings of refs 11, 12. In particular, the computed averaged 〈d_{ave}〉 coefficient is always bigger than ≃225 pC/N and can be as high as 525 pC/N in the macroscopic Amm2 phase of BCTZ0.5 (note that piezoelectric coefficients larger than 525 pC/N shown in Fig. 1c correspond to frequent fluctuations between different macroscopic phases, such as Amm2 and P4mm, and are therefore inherently linked to phase transitions). In contrast, the piezoelectric coefficient can be as small as ≃150 pC/N and does not exceed values of about ≃330 pc/N in the macroscopic Amm2 state of BaTiO_{3}. Interestingly and unlike in leadbased Pb(Zr,Ti)O_{3} and Pb(Mg,Nb,Ti)O_{3} solid solutions near their morphotropic phase boundary^{4,15,16}, these large piezoelectric responses in BCTZ0.5 are not due to the existence of a lowsymmetry (that is, monoclinic) phase that is associated with the ease of rotating the polarization, since our calculations reported in Fig. 1a (as well as corresponding data related to strain tensors that are not shown here) indicate that they occur within a macroscopic orthorhombic phase.
It is also worthwhile to realize that Fig. 1c further predicts that BCTZ0.5 should also have large piezoelectric coefficients (larger than 200 pC/N) in the rhombohedral R3m phase in the vicinity of the Amm2–R3m phase transition, namely for temperature varying between 260 and 266 K, as well as close to the Curie temperature of ≃360 K, which is consistent with the experimental data of ref. 12.
Fluctuations
To better understand the piezoelectric responses within the Amm2 phases of BCTZ0.5 and BTO, as well as their differences and origins, Fig. 2a reports 〈d_{ave}〉 versus in the ferroelectric orthorhombic state of these two materials, where δu_{y} and δu_{z} are the error bars that are associated with the y and z components of the supercell average of the local mode that are shown in Fig. 1a,b. These error bars quantify the fluctuation of the polarization about its macroscopic average, which is oriented along the [011] direction. What is remarkable is that not only 〈d_{ave}〉 is found to nearly linearly increase with , but also that this linear relationship is rather similar between BCTZ0.5 and BaTiO_{3} (Supplementary Note 3 provides an understanding of the linear relationship between 〈d_{ave}〉 and . We did not include δu_{x} in the computation of the fluctuations reported in the horizontal axis of Fig. 2a because the polarization in the orthorhombic phase has a vanishing x component). One can thus assert that BCTZ0.5, in contrast with BTO, can adopt values of 〈d_{ave}〉 larger than 330 pc/N in its Amm2 state because it can sustain larger fluctuations of its polarization. Moreover, unlike BaTiO_{3}, BCTZ0.5 is prevented from having piezoelectric coefficients smaller than 225 pC/N because the narrow temperature stability of its macroscopic Amm2 phase (about 30 K in BCTZ0.5 versus 60 K in BTO, see Fig. 1a,b) prevents the orthorhombic symmetry from reaching lower temperatures where thermal fluctuations are restricted. Note that the inherent relation between a limited temperature range of stability of the orthorhombic phase and large piezoelectricity is further demonstrated in the Supplementary Note 2, where we also examined BCTZ0.4. As a matter of fact, in this latter system, the range of stability of its orthorhombic state further contracted (about 15 K), which results in even larger piezoelectric coefficients. However, such range of stability in BCTZ0.4 occurs for temperatures higher than 300 K, which is detrimental to generate high roomtemperature electromechanical response.
To understand the larger fluctuations occurring in the orthorhombic phase of BCTZ0.5 with respect to the case of BaTiO_{3}, Fig. 3 displays a quantity related to free energy–internal energy of both systems at their P4mm–Amm2 transition, as obtained using the WangLandau algorithm of ref. 28 within the effective Hamiltonians presently developed and/or used here. More precisely, this quantity related to free energy corresponds to the logarithm of the canonical probability function^{28}. Figure 3 demonstrates the existence of two minima of similar free energy in both systems. The right minimum with larger internal energies corresponds to the macroscopic P4mm state while the left minimum with smaller internal energies is associated with Amm2. The existence of these two minima demonstrates the firstorder character of the P4mm–Amm2 transition. Figure 3 also indicates that the energetic barrier between these two minima is smaller in BCTZ0.5 than in BTO, therefore making the exploration of different orthorhombic states of different polarization direction, via an intermediate tetragonal state, easier of access close to the P4mm–Amm2 transition in (Ba_{0.85}Ca_{0.15})(Zr_{0.10}Ti_{0.90})O_{3} than in BaTiO_{3}. This finding is fully consistent with the suggestion of ref. 11 that BCTZ0.5 possesses a low energetic barrier between different ferroelectric states that allows its polarization to easily rotate and that results in large piezoelectric responses very near the P4mm–Amm2 transition. Furthermore and understand Figs 2a and 3 also reveals that the free energy–internal energy curve is much flatter around the orthorhombic minimum in BCTZ0.5 than in BTO. In other words, there is a wider range of orthorhombic states having different internal energies (and thus different magnitudes of the polarization) that hold a similar freeenergy in BCTZ0.5. As a result, BCTZ0.5 can exhibit larger fluctuations of its polarization within the macroscopic Amm2 phase. Note also that the existence of a low freeenergy barrier and of flat minima revealed in Fig. 3 for BCTZ0.5 is consistent with the proximity of a tricritical point in the phase diagram of (1−x)Ba(Zr_{0.2}Ti_{0.8})O_{3}−x(Ba_{0.7}Ca_{0.3})TiO_{3}, as discussed in ref. 11 (see Supplementary Note 2 for our predictions and discussion about tricritical point in this solid solution).
Cluster analysis
Let us now determine whether the piezoelectric response and polarization’s fluctuations of BCTZ0.5 and BTO correlate with microscopic features. For that, we identified clusters of tetragonal (T), Orthorhombic (O) and Rhombohedral (R) symmetry (within which dipoles nearly all lie along a 〈001〉, 〈110〉 and 〈111〉 pseudocubic direction, respectively) in both BCTZ0.5 and BaTiO_{3}, using a modified version of the Hoshen–Kopelman algorithm^{29,30}. Interestingly, we numerically found (not shown here) that both systems support T, O and R clusters in their macroscopic P4mm state; this is reminiscent of the coexistence of P4mm, Amm2 and R3m phases close to the P4mm–Amm2 transition, reported in ref. 13 based on a Rietveld analysis of Xray powder diffraction for a (Ba_{0.85}Ca_{0.15})(Zr_{0.10}Ti_{0.90})O_{3} sample. Moreover, while our T clusters fully vanish in the Amm2 state of these two materials, the O and R clusters remain, which bears resemblance with the decrease of the per cent of P4mm phase experimentally reported in refs 13, 14 when subjecting BCTZ0.5 to electric field or stress near room temperature. These changes in microstructures are associated with the huge piezoelectric response found at the P4mm–Amm2 transition occurring between 296 and 298 K. Furthermore, our findings about local clusters demonstrate that the microscopic and macroscopic symmetries of BaTiO_{3}, but also of BCTZ0.5, can be quite different, which is in line with the celebrated Comes–Guinier–Lambert model^{19}. On the other hand, our predictions of the existence of T clusters (in the P4mm phases) and O clusters (in both the P4mm and Amm2 phases) in addition to R clusters are in line with the findings of ref. 31 and go beyond the Comes–Guinier–Lambert model, since this latter model only expects R clusters (with different 〈111〉 directions) to occur in the macroscopic P4mm and Amm2 phases.
Let us focus on the macroscopic orthorhombic Amm2 phase of both BCTZ0.5 and BTO since we are interested in relating its large piezoelectric coefficients displayed in Fig. 1c with atomistic characteristics. In this phase, we numerically found that the R clusters are dynamical in nature^{32}, since they can change of location within the supercell and can also jump from one 〈111〉 direction to another 〈111〉 direction between different MC sweeps. Note that jump of polarization is typically associated with the socalled central mode, as demonstrated in ref. 33 for pure BaTiO_{3}. On the other hand, we further discovered that there are two different types of O clusters in the macroscopic orthorhombic Amm2 phase of BCTZ0.5 and BaTiO_{3}. One type has a strong dynamical character as a result of the different 〈110〉 directions and locations within the supercell it can adopt during the MC simulations at fixed temperature. On the other hand, the second type of O clusters has a pronounced static character in the sense that its polarization is always oriented along the spontaneous polarization. However, this second type of cluster also possesses some dynamics, that is, it breathes rather than change location during these simulations, which may be related to the softmode that has been predicted and observed to exist (in addition to the central mode) in BTO^{33,34}. As a result, such second type of O cluster can be referred to as quasistatic (it is worthwhile to realize that our simulations thus show that both quasistatic and dynamical clusters can coexist inside a pure system, such as BaTiO_{3}, and not only in complex solid solutions such as relaxor ferroelectrics^{35}). Interestingly, this quasistatic type of O cluster is, in fact, the socalled percolating cluster that spreads from one side of the supercell to its opposite side along the [100], [010] or [001] pseudocubic directions^{36}.
To corroborate these observations, we have conducted additional molecular dynamics simulations so as to estimate the relative timescale of cluster dynamics in BCTZ0.5 within the orthorhombic phase, at 280 K. We found that the polarization of the percolating O cluster does not change orientation throughout the 400 ps total simulation time, thus being indeed quasistatic at these time scales accessible to molecular dynamics simulations. Moreover, we found that the autocorrelation time of the volume of the nonpercolating O clusters is of ∼1.4 ps, while for the percolating O cluster, it is of ∼2.3 ps, hence indicating that the latter, while featuring significantly slower dynamics nevertheless has a breathing component to its time evolution, essentially due to volume fluctuations. Examples of R clusters, as well as the two types of O clusters, are shown in Fig. 4e,f for both BCTZ0.5 and BaTiO_{3}. Figure 4a–d indicate the relative evolution with temperature of different types of clusters in BCTZ0.5 and BaTiO_{3}, and show that the priorly evidenced enhancement of piezoelectric response in the Amm2 phase of the former is associated with a more fragmented local structure. Specifically, Fig. 4a reports the ratio between the number of sites belonging to all R clusters over the total number of sites in the whole supercell in the macroscopic orthorhombic Amm2 phase of these two systems and as a function of temperature, while Fig. 4b displays the same ratio but for all O clusters. These two ratios are denoted as r_{R} and r_{O}, respectively. Figure 4c shows the socalled strength of the percolating O cluster, , corresponding to the per cent of sites belonging to the (infinite) percolating O cluster. Furthermore, Fig. 4d reports the difference between r_{O} and , that is it represents the per cent of total volume occupied by the aforementioned first type of O clusters (that is, by the dynamical O clusters). Figure 4a,b demonstrate that the R and O clusters occupy a significant amount of the whole supercell in the macroscopic Amm2 state of both BCTZ0.5 and BTO. For instance, for BaTiO_{3} at 240 K, r_{R} and r_{O} are both close to 40%. Interestingly, comparing Fig. 4b–d also tells us that most of the space occupied by the O clusters originates from the percolating O cluster, as demonstrated by the fact that r_{O}– is always smaller than ≃6% for any temperature and decreases down to ≃1% when decreasing the temperature to 230 K. Other important information provided by Fig. 4 is that r_{O} and thus are rather sensitive to temperature in the Amm2 state of both BCTZ0.5 and BTO, unlike r_{R}. For instance, increases from 25 to 41% when decreasing temperature from 296 to 230 K, while r_{R} remains close to 40% in both the studied materials in that temperature range. Recalling that the range of stability of the Amm2 state typically occurs for higher temperatures in BCTZ0.5 than in BaTiO_{3}, one can therefore conclude that the local structure of the Amm2 phase of BCTZ0.5 is more disordered/fragmented than that of BaTiO_{3}. Such enhanced disordering allows for easier fluctuations of the polarization, and thus according to Fig. 2a to larger piezoelectricity. Figure 2b confirms the correlation between large piezoelectricity and enhancement of disordering of the local structure, as well as further sheds light into the strong connection between large electromechanical responses and percolating clusters, since 〈d_{ave}〉 is found to typically increase when decreases in both BCTZ0.5 and BTO.
To confirm this observation, we have additionally estimated the contribution to piezoelectricity stemming from each type of clusters in the Amm2 phase of BCTZ0.5 (at 280 K) and BTO (at 250 K), by first determining at each MC sweep which local modes belong to which type of clusters and then using equation (2) for the local modes associated with each type of clusters. We found that, in the case of BCTZ0.5 (BTO), the percolating O cluster, which occupies 31% (38.3%) of the supercell, has an individual piezoelectric response of 70.5 pC/N (48.9 pC/N), while the dynamical R and nonpercolating O clusters, occupying, respectively, 38.4% (37.8%) and 3.3% (1.2%) of the supercell, have piezoelectric contributions of 68.8 pC/N (27.3 pC/N) and 29.5 pC/N (11.5 pC/N). These results consistently indicate that the larger the volume of the percolating O cluster, the lower is its contribution to the piezoelectric response (see Supplementary Note 1 for additional information about BCTZ0.5). In light of these results, the trend line to achieve enhanced piezoelectricty appears to rest upon the relative fragmentation of local order. The latter can be tuned via x (see Supplementary Note 2 for additional information about BCTZx), via the application of a small electric field along one of the equivalent 〈111〉 directions that would depopulate the percolating O cluster, or alternatively, given the interplay between epitaxial strain and the orientation and morphology of local order^{37}, via the application of epitaxial strain. Note, however, that these levers that would allow the tuning of the ratio of R and nonpercolating O clusters to percolating O clusters in favour of the former ones are in interplay with temperature, a parameter that is intrinsically related to the studied phenomenon via thermal fluctuations.
Discussion
In summary, atomistic simulations within an effective Hamiltonian scheme predict that BCTZ0.5 undergoes a P4mm–Amm2 transition that occurs near room temperature, and that yields an orthorhombic state that has a rather flat freeenergy landscape as well as a small temperature range of stability. A a result, larger fluctuations of the polarization occur in the Amm2 state of BCTZ0.5 with respect to BaTiO_{3}, thereby inducing higher piezoelectric responses near 300 K. Moreover, our study further reveals that this larger piezoelectric response is intrinsically linked to a specific feature of the local structure, namely the smaller strength of the percolating cluster. Interestingly, such cluster is of orthorhombic rather than rhombohedral local symmetry, and, as result, is missing in the famous Comes–Guinier–Lambert model^{19}. In other words, such latter model ought to be generalized (by including O clusters in the Amm2 phase) be able to capture the microscopic origins of physical properties of BCTZ0.5 and BTO.
Note that our study focuses on single phases. However, we also expect that the formation and coexistence of several phases inside BCTZ0.5 will contribute to further enhancing piezoelectricity. This expectation stems from the fact that the barrier height of the free energy corresponds to the interface tension or, in other words, to the energy of the domain wall between phases of different symmetry^{38}, and, from this perspective, the reduction of the barrier height revealed in Fig. 3 when going from BaTiO_{3} to BCTZ0.5 enables domain wall fluctuations that should further strengthen the electromechanical response^{39}.
We hope that such findings not only provide a better understanding of BCTZx systems but also can be used in the quest for other leadfree systems with high electromechanical conversion. For instance, our results suggest that one possibility for generating large piezoelectricity is to mix one system having a singlephase transition from cubic paraelectric to ferroelectric rhombohedral at a temperature to be denoted by T_{c1}, with another material having another singlephase transition but from cubic paraelectric to ferroelectric tetragonal at a temperature to be denoted T_{c2}. This mixing can then result in the emergence of a ferroelectric orthorhombic state having flat freeenergy minimum, within a narrow temperature region that is located inbetween the temperature stability regions of the ferroelectric tetragonal and ferroelectric rhombohedral states. The key factor to then obtain large roomtemperature piezoelectric response is to make the stability of the orthorhombic state occurring in a region comprising room temperature, which can happen by finding the right amount of mixing between the two compounds and the right T_{c1} and T_{c2} temperatures. Note that, in this scenario, the existence of a tricritical point, where the tetragonal and rhombohedral ferroelectric states meet with the paraelectric phase, can occur at a specific composition and temperature but the highest roomtemperature piezoelectric response can correspond to other compositions, namely the ones for which the orthorhombic state is stable within a small temperature region comprising 280–300 K. These are precisely the conditions encountered in BCTZx (refs 12, 23). Note also that this scenario is different from, that is, the mixing of rhombohedral Pb(Zr,Ti)O_{3} of lower Ti compositions with tetragonal Pb(Zr,Ti)O_{3} of larger Ti concentrations for which the coexistence of rhombohedral and tetragonal domains^{40}, or the occurrence of a compositionally induced bridging monoclinic phase^{4,15}, can yield large electromechanical response, since our scenario consists in creating a macroscopic orthorhombic phase of small temperature range of stability inbetween the temperature ranges of the tetragonal and rhombohedral states.
Data availbility
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Additional information
How to cite this article: Nahas, Y. et al. Microscopic origins of the large piezoelectricity of leadfree (Ba,Ca)(Zr,Ti)O_{3}. Nat. Commun. 8, 15944 doi: 10.1038/ncomms15944 (2017).
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Change history
25 October 2017
A correction has been published and is appended to both the HTML and PDF versions of this paper. The error has not been fixed in the paper.
References
Bellaiche, L. Piezoelectricity of ferroelectric perovskites from first principles. Curr. Opin. Solid State Mater. Sci. 6, 19–25 (2002).
Lines, M. E. & Glass, A. M. Principles and Applications of Ferroelectrics and Related Materials Oxford university press (1977).
Uchino, K. Piezoelectric, Actuators and Ultrasonic Motors Kluwer Academic Publishers (1996).
Noheda, B. et al. A monoclinic ferroelectric phase in the Pb(Zr1−xTix)O3 solid solution. Appl. Phys. Lett. 74, 2059–2061 (1999).
Lee, C.K. & Moon, F. C. Modal sensors/actuators. J. Appl. Mech. 57, 434–441 (1990).
Zhang, Q. M., Wang, H., Kim, N. & Cross, L. E. Direct evaluation of domainwall and intrinsic contributions to the dielectric and piezoelectric response and their temperature dependence on lead zirconatetitante ceramics. J. Appl. Phys. 75, 454–459 (1994).
Guo, R. et al. Origin of the high piezoelectric response in PbZr1−xTixO3 . Phys. Rev. Lett. 84, 5423–5426 (2000).
Park, S.E. & Shrout, T. E. Ultrahigh strain and piezoelectric behavior in relaxor based ferroelectric single crystals. J. Appl. Phys. 82, 1804–1811 (1997).
European Commission. Directive 2002/95/EC of the European Parliament and of the Council of 27 January 2003 on the restriction of the use of certain hazardous substances in electrical and electronic equipment. Off. J. Eur. Union L 37, 19–23 (2003).
European Commission. Directive 2011/65/EU of the European Parliament and of the Council of 8 June 2011 on the restriction of the use of certain hazardous substances in electrical and electronic equipment. Off. J. Eur. Union L 174, 88–110 (2011).
Liu, W. & Ren, Z. Large piezoelectric effect in Pbfree ceramics. Phys. Rev. Lett. 103, 257602 (2009).
Acosta, M. et al. Origin of the large piezoelectric activity in ((1−x)Ba(Zr0.2Ti0.8)O3−x(Ba0.7Ca0.3)TiO3 ceramics. Phys. Rev. B 91, 104108 (2015).
Brajesh, K., Tanwar, K., Abebe, M. & Ranjan, R. Relaxor ferroelectricity and electricfielddriven structural transformation in the giant leadfree piezoelectric (Ba,Ca)(Ti,Zr)O3 . Phys. Rev. B 92, 224112 (2015).
Brajesh, K., Abebe, M. & Ranjan, R. Structural transformations in morphotropicphaseboundary composition of the leadfree piezoelectric system Ba(Zr0.2Ti0.8)O3−(Ba0.7Ca0.3)TiO3 . Phys. Rev. B 94, 104108 (2016).
Bellaiche, L., A. García, L. & Vanderbilt, D. Finitetemperature properties of Pb(Zr1−xTix)O3 alloys from firstprinciples. Phys. Rev. Lett. 84, 5427–5430 (2000).
Bellaiche, L., García, A. & Vanderbilt, D. Lowtemperature properties of Pb(Zr1−xTix)O3 solid solutions near the morphotropic phase boundary. Ferroelectrics 266, 41–56 (2002).
Fu, H. & Cohen, R. E. Polarization rotation mechanism for ultrahigh electromechanical response in singlecrystal piezoelectrics. Nature 403, 281–283 (2000).
Grinberg, I., Cooper, V. R. & Rappe, A. M. Relationship between local structure and phase transitions of a disordered solid solution. Nature 419, 909–911 (2002).
Comes, R., Lambert, M. & Guinier, A. The chain structure of BaTiO3 and KNbO3 . Solid State Commun. 6, 715–719 (1968).
Van Vechten, J. A. Quantum dielectric theory of electronegativity in covalent systems. I. Electronic dielectric constant. Phys. Rev. A 182, 891–905 (1969).
Bellaiche, L. & Vanderbilt, D. Virtual crystal approximation revisited: application to dielectric and piezoelectric properties of perovskites. Phys. Rev. B 61, 7877–7882 (2000).
Zhong, W., Vanderbilt, D. & Rabe, K. M. Firstprinciples theory of ferroelectric phase transitions for perovskites: the case of BaTiO3 . Phys. Rev. B 52, 6301–6312 (1995).
Keeble, D. S., Benabdallah, F., Thomas, P. A., Maglione, M. & Kreisel, J. Revised structural phase diagram of Ba(Zr0.2Ti0.8)O3−(Ba0.7Ca0.3)TiO3 . Appl. Phys. Lett. 102, 092903 (2013).
Walizer, L., Lisenkov, S. & Bellaiche, L. Finitetemperature properties of (Ba,Sr)TiO3 systems from atomistic simulations. Phys. Rev. B 73, 144105 (2006).
Menoret, C. et al. Structural evolution and polar order in Sr1−xBaxTiO3 . Phys. Rev. B 65, 224104 (2002).
Lemanov, V. V., Smirnova, E. P., Syrnikov, P. P. & Tarakanov, E. A. Phase transitions and glasslike behavior in Sr1−xBaxTiO3 . Phys. Rev. B 54, 3151–3157 (1996).
Garcìa, A. & Vanderbilt, D. Electromechanical behavior of BaTiO3 from first principles. Appl. Phys. Lett. 72, 2981–2983 (1998).
BinOmran, S., Kornev, I. A. & Bellaiche, L. WangLandau Monte Carlo formalism applied to ferroelectrics. Phys. Rev. B 93, 014104 (2016).
Hoshen, J. & Kopelman, R. Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm. Phys. Rev. B 14, 3438–3445 (1976).
Nahas, Y. Gauge Theory for Relaxor Ferroelectrics (PhD thesis, Ecole Centrale, 2013).
Prokhorenko, S., Nahas, Y. & Bellaiche, L. Fluctuations and topological defects in proper ferroelectric crystals. Phys. Rev. Lett. 118, 147601 (2017).
Nahas, Y., Prokhorenko, S., Kornev, I. & Bellaiche, L. Topological point defects in relaxor ferroelectrics. Phys. Rev. Lett. 116, 127601 (2016).
Hlinka, J. et al. Coexistence of the phonon and relaxation soft modes in the terahertz dielectric response of tetragonal BaTiO3 . Phys. Rev. Lett. 101, 167402 (2008).
Salje, E. et al. Elastic excitations in BaTiO3 single crystals and ceramics: mobile domain boundaries and polar nanoregions observed by resonant ultrasonic spectroscopy. Phys. Rev. B 87, 014106 (2013).
Akbarzadeh, A. R., Prosandeev, S., Walter, E. J., AlBarakaty, A. & Bellaiche, L. Finitetemperature properties of Ba(Zr,Ti)O3 relaxors from first principles. Phys. Rev. Lett. 108, 257601 (2012).
Stauffer, D. & Aharony, A. Introduction to Percolation Theory Taylor & Francis (1994).
Prosandeev, S., Wang, D. & Bellaiche, L. Properties of epitaxial films made of relaxor ferroelectrics. Phys. Rev. Lett. 111, 247602 (2013).
Meyer, B. & Vanderbilt, D. Ab initio study of ferroelectric domain walls in PbTiO3 . Phys. Rev. B 65, 104111 (2002).
Brierley, R. T. & Littlewood, P. B. Domain wall fluctuations in ferroelectrics coupled to strain. Phys. Rev. B 89, 184104 (2014).
Topolov, V. Y. Heterogeneous Ferroelectric Solid Solutions: Phases and Domain States Springer (2011).
Acknowledgements
We thank Drs R. Ranjan and J. Kreisel for insightful discussions. Y.N. and L.B. acknowledge the ARO grant W911NF1610227. A.A. and S.Prok. thank the DARPA grant HR00111520038 (MATRIX programme). S.Pros. appreciates the ONR Grant N000141211034 and grants 3.1649.2017/PP from RME (Russian Ministry of Education) and 16520072 Bel_a from RFBR (Russian Foundation for Basic Research). R.W. thanks the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE0957325 and the University of Arkansas Graduate School Distinguished Doctoral Fellowship. We also acknowledge the support of the Luxembourg National Research Fund through the PEARL (Grant FNR/P12/4853155/Kreisel COFERMAT, J.Í.) and intermobility (Grant FNR/INTER/MOBILITY/15/9890527 GREENOX, L.B. and J.Í.) programs. S.Pros. also appreciates Russian Ministry of Education grant 3.1649.2017/PP.
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Y.N. performed effective Hamiltonian simulations and conducted WangLandau calculations, cluster and fluctuation analysis. A.A. computed the phase diagram displayed in the Supplementary Fig. 4. S.Prok. and Y.N. performed molecular dynamics simulations for assessing clusters dynamics, and computed their individual piezoelectric responses. S.Pros., R.W. and L.B. extracted the effective Hamiltonian parameters. I.K. developed the WangLandau code that was used to obtain Fig. 3, and provided discussions along with J.Í., S.Prosandeev. and S.Prokhorenko. L.B. wrote the original version of the manuscript, which was then extensively modified due to the feedback and suggestions of all authors.
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Nahas, Y., Akbarzadeh, A., Prokhorenko, S. et al. Microscopic origins of the large piezoelectricity of leadfree (Ba,Ca)(Zr,Ti)O_{3}. Nat Commun 8, 15944 (2017). https://doi.org/10.1038/ncomms15944
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DOI: https://doi.org/10.1038/ncomms15944
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