Piezoelectric Response of Plastic Ionic Molecular Crystals: Role of Molecular Rotation

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I. INTRODUCTION
Piezoelectric, ferroelectric, and relaxor-ferroelectric materials are key components in a range of technologies including sonar, ultrasound, radio frequency (RF) filters, accelerometers, motion sensors, micropositioning systems, and mechanical energy harvesting devices [1][2][3][4][5][6][7][8][9][10][11][12].Characteristic for these materials is the coupling of mechanical and electrical properties.In the direct piezoelectric effect, an applied stress σ j induces a polarization P i at a fixed electrical field E, d i,j = (∂P i /∂σ j ) E .Here, d i,j are the piezoelectric coefficients; i = {x, y, z} are the field directions (by convention the polar direction is set to i = 3 = z); j are the elastic tensor directions, indicated with Voigt indicies runing from 1 to 6, i.e j = {xx, yy, zz, yz, xz, xy}.In the converse piezoelectric effect, d i,j = (∂ε i /∂E j ) σ , an electrical field induces a mechanical strain ϵ.Piezoelectric coefficients include the longitudional and transverse coefficients, d 33 , d 31 , and shear coefficients include d 15 , d 24 , d 16 , and the face-shear mode d 36 , some of which are equivalent or vanish depending on the crystal symmetry.
The desired piezoelectric material properties depend on the application.High-power applications typically require large d ij s but are tolerant to frequency and temperature instability [13].Typical piezoelectric coefficients for Pb(Zr,Ti)O 3 (PZT) can exceed d 33 = 600 pC/N [14,15].Low power applications, such as RF filters, on the other hand, require stable coefficients as a function of frequency.AlN, a material used for this purpose, has * elin.dypvik.sodahl@nmbu.noa small but stable d 33 of 3.4 pC/N [16].State-of-the-art polymer piezoelectric materials, attractive for flexible devices such as energy harvesting cantilevers, are based on the polymer polyvinylidene fluoride (PVDF).The piezoelectric coefficient of PVDF-co-trifluoroethylene(TrFE) is d 33 = −65.3pC/N [17,18].Although piezoelectrics like PZT, AlN, and PVDF have important commercial applications, there remain a large parameter spaces between these known materials where combinations of properties, such as soft or flexible materials with robust or stable piezoelectric responses, are yet to be found.Plastic ionic molecular crystals, which consist of inorganic anions bonded to organic cations, are an emerging class of ferroelectric and piezoelectrics that could fill many gaps in the material property space [19][20][21][22].These materials possess immense compositional tunability, low processing temperatures compared to inorganic materials, and compositions with toxic and scarce elements such as Pb, Ba, and Co [23][24][25][26][27][28] can be avoided.They exhibit a broad range of bonding properties -including van der Waals forces, hydrogen bonding, charge-transfer, and intermolecular orbital hybridization [29] -which gives rise to unique mechanical and functional properties.The key example of a unique property is the existence of a "plastic" high-temperature mesophase, where molecules retain lattice order but increase their degree of local orientational disorder, introducing molecular rotation into the structure.In the mesophase the material has weakened intramolecular bonding and ease of dislocation movement, causing it to become malleable and deformable into desired shapes [30,31].However, the molecular rotations do not only play a role in the plasticity and phase transitions of plastic crystals, they are also involved in the polarization switching of ferroelectric materials [32][33][34][35].
Molecular rotations that are triggered by external stimuli make plastic ionic crystals intriguing candidates for piezoelectricity, as the change in dipoles due to rotation could be a significant contribution to the piezoelectricity.Several recent papers on plastic ionic crystals emphasize the potential of this material class for piezoelectricity, but the role of molecular rotation is not well established.[33,[36][37][38][39].This paper provides computed dielectric constants, elastic moduli, spontaneous polarization, and piezoelectric coefficients of 11 representative plastic ionic crystal ferroelectrics.Comparing the different systems, we identified a high shear compliance, caused by molecular rotation in response to shear strain, as a mechanism that can give high shear piezoelectric response in molecular crystals.

A. Density Functional Theory Calculations
The density functional theory (DFT) calculations were performed with the VASP software package [40][41][42][43] using projector augmented plane wave method (PAW) pseduopotentials [44,45].The planewave energy cutoff was set to 530 eV.[46] The systems were iteratively relaxed until forces fell below 0.01 eV/ Å.The Brillouin zone was sampled with a Γ-centered Monkhorst-Pack k-point grid with a k-spacing of (1/15) Å−1 , which converged the lattice constants of HdabcoReO 4 to below 0.02 Å.Half this -spacing was used for obtaining elastic and piezoelectric coefficients using density functional perturbation theory (DFPT) [47,48].DFPT calculations provide piezoelectric coefficients e ij which links polarization and strain, as follows The piezoelectric coefficients d ij was obtained with where c jk is the compliance tensor, the inverse of the elastic stiffness tensor Y jk .The bulk moduli B reported are Voigt-Reuss-Hill averages obtained with ELATE package [49].The spontaneous polarization was computed using the Berry phase method [50][51][52] using a slab method to determine the polarization branch [53].

A. Benchmarking Density Functionals
Piezoelectric properties can be very sensitive to lattice constants making the choice of exchange-correlation functional in DFT important.Figure 1 shows the mean (absolute) deviations, M(A)D of the computed lattice parameters for the materials studied here for a set of different functionals.Reference structures are provided by low-temperature experimental crystal structures [32,[54][55][56][57][58][59].The van der Waals (vdW) functionals, vdW-DFcx [60], vdW-DF2-B86R [61], and vdW-DF2, [62] all provide accurate lattice constants, so does the Taktchenko-Scheffler method (PBE-TS) [63].PBE [64], which lacks an explicit account of van der Waals forces, overestimates lattices constants.SCAN+rVV10, [65] on the other hand, underestimates the lattice constants.The largest deviations are found for the C[NH 2 ] 3 ClO 4 system, which is a supermolecular layered system, with intralayer hydrogen and intermolecular bonds in the layers, and weaker dispersion forces between the layers.PBE, SCAN+rVV10, and vdW-DF-cx resulted in c-axis lenghts of 9.48 Å, 8.17 Å, and 8.54 Å, compared to an experimental lattice constant is 8.83 Å.Further details on lattice constants can be found in the supplementary materials (SM).Based on this study, vdW-DF-cx was adopted for predicting material properties.

B. Dielectric, piezoelectric and ferroelectric properties
The following sections present the computed properties of 11 plastic ionic molecular crystals.Four different types of plastic crystals were considered: 1. Quinuclidinium compounds, 2. Dabco-compounds, 3. (R)-OH-Q-compounds, and 4. C[NH 2 ] 3 -compounds.They all consist of one organic cation per inorganic anion and are quite representative of the broad range of ferroelectric ionic plastic crystal compositions that have emerged in recent years [34,[66][67][68][69][70][71][72].The quinuclidinium and dabco compounds have globular organic cations and tetrahedral anions, where the central atom is coordinated by oxygen.The (R)-OH-Q-compounds also have globular cations, but halogen anions.Finally, the C[NH 2 ] 3 -compounds have planar organic cations and anions with tetrahedral geometry.

Quinuclidinium-compounds
The quinuclidinium-based compounds were among the first ionic plastic crystals with which multi-axial ferroelectric switching of the polarization vectors were observed.[32] The three ionic quinuclidinium-based compounds studied here are presented in Fig. 2. Two of them consists of quinuclidinium (HQ) cations and the inorganic anions ReO 4 and IO 4 , while the third consists of 4-fluoroquinuclodinium (F-Q) and ReO 4 .All three systems pack in similar side-centered crystal structures.The side-centered packing allows for monoclinic and orthorhombic Bravais lattices.HQReO 4 and HQIO 4 pack in the orthorhombic space group P mn2 1 , while HQIO 4 pack in the monoclinic P n space group.Note that while the F-Q cation in [F -Q]ReO 4 points in the c-axis direction, the direction of the HQ-cations is tilted in HQReO 4 , and likewise for HQIO 4 .
Table I lists the computed piezoelectric coefficients (of sizeable magnitude), relative permittivity (on the diagonal elements), computed and experimental spontaneous polarization and lattice parameters for the quinuclidinium-compounds.Full set of parameters are provided in the Supplementary material (SM).The computed polarization of [F -Q]ReO 4 is almost twice that of the polarization of the two other quinuclidiniumcompounds.The fact that polarization increases with a H/F substitution is in agreement with results for other ferroelectric molecular crystals [70,[74][75][76].For this system, the substitution also inverts the polarization direction.This inversion occurs because the H-F substitution is on the opposite side of the nitrogen atom involved in setting up the direction of polarization in the HQ compound.
With a shear coefficient of d 16 = −94 pC/N, the largest predicted piezoelectric coefficient obtained in this work is that of HQReO 14.9 14.2 a Ref. [56] b DFT-predicted.c Ref. [78] Table II provides the results for the dabco compounds.It shows that substitution of ReO 4 with IO 4 reduces the piezoelectric coefficients, which was also found for the quinuclidinium compounds.However, the d 24 value, which drops from 84 to 63 pC/N, remains sizeable.The substitution of ReO 4 with IO 4 increases the spontaneous polarization from 8.0 µC/cm 2 to 11.5 µC/cm 2 .Both the anisotropy and polarization decreased due to the same substitution for the quinuclidinium compounds.The different effect of the substitution on the piezoelectric anisotropy can be linked to the different crystal structures and the mechanisms for the shear piezoelectric response, where the rotational response is more pronounced for the quinuclidinium compounds than in the dabco case.
Figure 5 shows the structure of [Hdabco]ReO 4 in a shear strained unit cell.Remarkably, only one of the two ReO 4 tetrahedra in the unit cell have rotated significantly, while both the dabco-molecules have rotated compared to the unstrained crystal structure, one of the dabco-molecules are distorted.As the rotated ReO 4 anion and the distorted molecule are in the same plane, indicating alternating layers where the structure is distorted due to shear strain.This particular result obtained for a finite strain, unlike the DFPT results, hints at onset of a slip-like mechanism and might be sensitive to the size of the unit cell used in the computation.

(R)-OH-Q-compounds
The (R)-OH-Q-compounds consist of a (R)-(-)-3-hydroxlyquinuclidinium-molecule ((R)-OH-Q) and a halogen (Cl/Br/I) ion.The comparison of these compounds, allows for the investigation of the role of electronegativity of the anion and halogen size in the plastic organic crystals.Figure 6, left panel, shows the crystal structures of the compounds in this group.They all pack in the same P4 1 space group.While all three are reported as ferroelectrics [79], the experimental polarization has only been reported for (R)-OH-QCl.The right panel of Fig. 6 shows a strained structure.Unlike in the quinuclidinium-compounds, there is no clear molecular rotation in the strained unit cell, indicating that the piezoelectricity arises from an extender mechanism similar to the inorganic extender piezoelectrics, like PbTiO 3 and AlN.The anisotropies in the (R)-OH-Q compounds are quite comparable to that of LiTa which has a piezoelectric anisotropy of 3.3 [77].
The computed polarization of the (R)-OH-QCl is more than three times larger that of the experimentally measured.The discrepancy can have many origins, both 10.0 10.3 8.9 a Ref. [57] b Ref. [79] experimental and computational: Incomplete switching typically cause experimental polarization to fall below the predicted.One cause of incomplete switching is finite electrical conductivity causing the sample to experience a heterogeneity of the electric field.Ferroelectric domain wall motion can also be pinned by point defects, grain boundaries or interfaces during the switching process.Such defects can arise easily because of microstructural properties of polycrystalline materials and specific synthesis conditions.The discrepancy might be related to the use of polycrystalline samples.The presence of defects can also in some samples prevent complete polarization switching [80].Moreover, DFT calculations do not include temperature effects, such as thermal vibration.As a result, the microscopic dipoles tend to align to a larger degree than they would at room temperature, which causes overestimation of polarization.The predicted lattice constants are also smaller than the experimentally reported, with the largest deviation of 2.58% for the c-axis in the Br-compound.This underestimation can also contribute to an overestimation of the calculated polarization.

C[NH2]3-compounds
The group of C[NH 2 ] 3 -compounds consists of three plastic ionic molecular crystals built up of a C[NH 2 ] 3 -    Most theoretical studies on piezoelectricity have focused on inorganic materials or polymers.While piezoelectric properties of some molecular crystals have been investigated [82][83][84][85][86][87][88][89], microscopic insight into piezoelec-tric mechanisms for these materials is vital for designing new and better materials.Plastic crystals are molecular crystals characterized by the existence of orientationally disordered mesophases, triggered by the onset of rotational motion.In this paper, we investigated how the tendency of molecules to rotate manifest as a large shear compliance and hence the potential for high shear piezoelectric response and piezoelectric anisotropy.Molecules rotate in response to mechanical or electrical stimulus, which induce polarization or unit cell deformations.
Figure 8 plots the computed longitudinal piezoelectric coefficients d 33 against shear piezoelectric coefficients d 16 , d 24 , d 25 for the plastic ionic molecular crystals and makes a comparison with data for the five inorganic piezoelectrics for five inorganic materials: PbTi and BaTiO 3 in the tetragonal P4mm phase, LiNbO 3 in the rhombohedral R3c phase, wurtzite AlN and ZnO, and HfO 2 in the ortorhombic Pca2 1 phase.Data for the first 4 extracted from the Materials Project [90,91] databases of computed material properties, while HfO 2 data is obtained from Dutta et al. [92].The comparison highlights how the plastic ionic molecular crystals studied here have generally smaller d 33 values than the inorganics, but the piezoelectric shear coefficients d 16 , d 24 ,d 25 have similar or larger magnitude, hence large piezoelectric anisotropy factors.We note that the d 33 of the inorganic materials within the materials project are for 0 K conditions in phase-pure single crystalline form and can be significantly lower than experimental values for polycrystalline samples with extrinsic contributions from mechanisms such as domain wall movement.Experiments made phase pure high-quality samples show better agreement with theory.[93] Figure 8 plots the shear piezoelectric coefficients d 16 , d 24 , d 25 against the corresponding shear compliance coefficients c 44 , c 55 , c 66 for the materials studied.For the plastic ionic crystals, high shear compliance coefficients typically coincide with high shear piezoelectric response.This trend supports the key role of molecular rotation in the piezoelectric response from most of plastic ionic crystals.The material with the largest shear piezoelectric response, HQReO Many factors influence the ease of molecular rotation.First, the crystal packing density affects the steric hindrance of rotation.The shapes of the constituent molecules also affect the molecule's ability to rotate as it changes the moment of inertia, small globular molecules rotate more easily than large elongated molecules, in accordance with Timmermans [30] characterization of the plastic crystal material class.In this study, the plastic ionic crystals containing globular organic cations and tetrahedral inorganic anions, the dabco and the quinuclidinium compounds, have both the largest anisotropies and shear piezoelectric response.Finally, strong intermolecular bonds can hinder rotations of globular molecules.This effect explains the lower piezoelectric response of [F -Q]ReO 4 compared to HQReO 4 .
The analysis of the molecular rotations in HQReO 4 showed that the primary cause of the large shear piezoelectric response of this material is the rotation of the non-polar inorganic tetrahedra, rather than the polar HQ molecule.This finding highlights the importance of intermolecular charge transfer.In fact, the importance of the rotation of rigid molecules in plastic crystals is reminiscent of tilting of the corner-sharing oxygen octahedra in the perovskite piezoelectrics.[94,95] The rotation mechanism in the inorganic piezoelectrics is often linked to a morphotropic phase boundary (MPB), i.e, a compositionally engineered phase boundary where the free energy barrier between two structural symmetries is low.The polarization vector is allowed to rotate between its position in each phase at this boundary due to small energy difference from one phase to the other [77,96].In metal oxide perovskite ferroelectrics, engineering solid solutions with compositions close to a MPB is the state-of-the-art approach for enhancing piezoelectric coefficients.The fact that rotation of nonpolar molecules in both cases are critical for the piezoelectric shear response, suggest that an alloy-like design approach can also be suitable for plastic ionic crystals, leveraging greater ability to enhance molecular rotations.Combined with the immense possibilities offered by the different combinations of cation and anions, these design tools brings vast opportunities for designing novel plastic ionic molecular piezoelectrics with record-breaking properties.
All materials in the (R)-OH-Q, Hdabco, and C[NH 2 ] 3 compounds have negative d 33 values.Negative longitudinal piezoelectric coefficients are rather untypical and corresponds to a contraction in the direction of the applied electric field for the converse piezoelectric effect.Negative d 33 values have been reported for semicrystalline ferroelectric polymers such as two-phase P(VDF-TrFE) and ferroelectric liquid crystals BTAs (trialkylbenzene-1,3,5-tricarboxamides) [85,97,98].For these materials, the negative piezoelectric response have been attributed to extrinsic contributions, such as defects, and DFT computations of phase-pure materials have predicted positive piezoelectric coefficients [85].However, compounds with negative d 33 can be found in other first principles studies [99] and was recently measured for inorganic HfO 2 thin films [92], with DFT calculations attributing it to an intrinsic effect [100][101][102].
The substitution of ReO 4 for IO 4 for both the HQ and dabco compounds resulted in a decrease in the piezoelec-tric response, see Sec.II B. Still, in both cases, the shear piezoelectric responses are among the highest predicted responses in this work.Rhenium is a high-cost element, and this work shows that substituting ReO 4 for IO 4 can be used to reduce the price of a molecular piezoelectric if a reduction of the piezoelectric coefficients is acceptable.

B. Assessment of Energy Harvesting Potential
Piezoelectric energy harvesting devices scavenge waste kinetic energy, transforming it to electrical energy.While power outputs are typically modest, on the order of µW, the small size of piezoelectric energy harvesting devices make the technology attractive for powering microwireless electronics [103][104][105][106].The electromechanical coupling factor gives a measure of how efficiently a piezoelectric converts input mechanical energy to electric energy [1], The suitability of a piezoelectric in an energy harvesting device can be assessed by the following figure of merit [1]: An ideal piezoelectric material for energy harvesting applications have a high piezoelectric coefficient relevant for the mode of operation.The currently realized plastic ionic molecular crystals may not have piezoelectric coefficients that compete with the best commercial metaloxide piezoelectrics, such as PZT and lead magnesium niobate-lead titanate (PMN-PT).Still, the plastic crystals have inherent potential for energy harvesting application due to their low dielectric permittivity.Comparing with commercial inorganics, PZT-5H has a d 33 of 593 pC/N and a relative permittivity of 3400 [7], while all the plastic crystals studied in this work have dielectric constants below 5.In addition, the plastic crystals have other advantages such as low-temperature synthesis and plasticity in the mesophases which could enable simplified co-fabrication with flexible polymeric substrates used for energy harvesting devices.This process is currently technically challenging for metal oxides, involving etching of sacrificial layers and a process of transfer of the piezoelectric from silicon to polymer substrates [107].The predicted figures of merits indicate that the quinuclidinium and the dabco compounds have potential for application in shear mode energy harvesting devices.Most energy harvesting devices rely on the longitudinal piezoelectric response, but shear devices have also been studied [8][9][10][110][111][112][113][114].

C. Temperature effects and DFT Calculations
In this study, we use standard DFT to study plastic ionic molecular crystals and do not take temperature effects into account.At finite temperature intra-and intermolecular vibration arise.In plastic ionic molecular crystals, liberation motion of the molecular constituents is likely to be particularly pronounced.In addition, some plastic crystals have an intermediate plastic phases, in which the orientational order is only partially lost [26].
The intermediate phase can itself host attractive material properties; in the case of HQReO 4 , the ferroelectric coercive field is reduced from 340 kV/cm 2 to below 5 kV/cm 2 in the intermediate temperature phase [73].Yoneya et al. describe the molecular motion of HQReO 4 in the low, intermediate and high-temperature phases, where the difference between phases is characterized by the number of positions the molecules are allowed to liberate around, as well as the rate of liberation [73].The liberation of the molecules in the plastic ionic crystals will likely affect the piezoelectric molecular rotation mechanism.Due to the soft nature of the intermolecular bonding, temperature is likely to have a stronger influence on the piezoelectric response of plastic ionic molecular crystals than for inorganic materials.
The relationship between molecular rotation, structure, material properties, and temperature in plastic crystals is not well understood, meriting further studies.In particular, molecular dynamic (MD) simulations can yield valuable insight for this material class.[73] However, the interplay of short-and long-range interactions, and the structure dependency of piezoelectric properties, calls for the usage of MD based on accurate DFT calculations to capture the temperature-dependent piezoelectric properties and the coupling of molecular rotations of the plastic ionic molecular crystals.

V. CONCLUSIONS
We presented the computed dielectric, ferroelectric and piezoelectric properties of four classes of plastic ionic molecular crystals.Several of the materials display large shear piezoelectric responses.Our analysis shows that molecular rotation in response to shear stress is key to understanding the piezoelectric properties of plastic ionic crystals.On one hand, molecular rotations can allow for the unit cell to easily deform, i.e. provide large compliance coefficients.On the other hand, molecular rotation can both rotate molecular dipoles, but also induces related to intermolecular-charge transfer.This finding highlights the role of intermolecular bonding nature.The molecular rotation mechanism causing large shear responses reminiscent of the polarization vector rotation in inorganic perovskites, but while this mechanism is related to rotation and twisting of linked octahedra in the inorganics, the rotational freedom in the plastic inorganic crystals are linked to the dominantly non-covalent bonding nature.The fact that piezoelectric shear coefficients are comparable to phase pure inorganic piezoelectrics coupling with low dielectric functions make plastic ionic crystals promising for application in piezoelectric devices, such as energy harvesting and actuators.In particular, [Hdabco]ReO 4 and HQReO 4 have large energy harvesting figures of merits in their phase pure form, heralding great potential of this materials class with further material optimization.
VI. ACKNOWLEDGEMENT S. Seyedraoufi, C.H. Gørbitz, and O. Nilsen are acknowledged for valuable discussions.The computations of this work were carried out on UNINETT Sigma2 high performance computing resources (grant NN9650K).This work is supported by the Research Council of Norway as a part of the Young Research Talent project FOX (302362).The crystal structures of the plastic ionic molecular crystals in this work were retrieved from the Cambridge Structural Database [115].Crystal structure figures have been made with the VESTA software [116].

VII. SUPPORTING INFORMATION
(1) Predicted lattice parameters for all 11 plastic ionic crystals for 6 density functionals, (2) the full tensors of all predicted properties: dielectric permittivity, piezoelectric constants, stiffness and compliance tensors, and figures of merit.

FIG. 1 .
FIG. 1.The mean absolute deviation (MAD) (bars) and mean deviation (MD) (lines) between the experimental and the predicted lattice parameters.The polar axes are along the c-axes.

FIG. 2 .
FIG. 2. The structure of the Quinuclidinium-compunds.The top figures show the structures of HQReO4 and HQIO4, the bottom figures show the structures of [F -Q]ReO4.The packing of HQReO4 and [F -Q]ReO4 differ slightly, the fluorideside group of the latter aligns with the c-axis, the former does not.
FIG. 3. The structure of HQReO4 (top) and [F -Q]ReO4 (bottom) relaxed in unit cell with a shear strain of 0.12%

Figure 3
shows a unit cell with 0.12 % shear strain for HQReO 4 and [F -Q]ReO 4 in the ZX-plane.This strain corresponds to the d 16 shear coefficient.Fig.4clearly demonstrates the rotation of the individual molecules.For [F -Q]ReO 4 , the polarization vector rotates by 2.6 • .For both compounds, the rotation is more pronounced for the tetrahedral anions than for the organic cations.The two symmetry inequivalent tetrahedra in HQRe 4 O rotate 6.0 • and 3.4 • within the ZX plane, while in [F -Q]ReO 4 the rotations are of 5.8 • and 2.6 • .The organic cation rotates more in the [F -Q]-compound compared to the HQcompound, with rotations of 2.0 • and 1.6 • for [F -Q]ReO 4 and 0.5 • and 0.02 • for HQReO 4 .The fact that the cation in the compound with the largest piezoelectric response barely rotate, while the anion, which does not have an intrinsic dipole, rotate significantly is interesting and provides a critical piece in the puzzle of explaining the origin of piezoelectric properties in ionic plastic crystals.It underlines how the polarization rotation, while driven by molecular rotation, can not be understood as arising primarily from the rotation of polar molecules.This finding points to the role of intermolecular charge transfer and rotation-driven lattice displacement as key to understanding and thus engineering of functional properties.The rotation of ReO 4 -tetrahedra under shear strain makes HQReO 4 a clear example of polarization vector rotation-like mechanism.Compared to the inorganic systems exhibiting polarization vector rotation, the rotation of the anion is facilitated by the non-covalent bonding in these systems.

FIG. 6 .
FIG. 6.The crystal structure of the (R)-OH-Q-compounds (left) and the crystal structure of R-OH-Cl under 0.12% shear strain (right).

FIG. 7 .
FIG. 7. C[NH2]3ClO4 and C[NH2]3BF4 have similar structures, as illustrated in the unit cells to the left, while C[NH2]3C2H5SO4 packs in a different crystal structure, as illustrated in the right top figure.
molecule and an inorganic or hybrid anion based on a tetrahedral geometry.C[NH 2 ] 3 ClO 4 and C[NH 2 ] 3 BF 4 both have the space group R3m.

Figure 7
shows that the planar C[NH 2 ] 3 -molecules in these two structures stack in layers, where one C[NH 2 ] 3 is shifted compared to the layer below.The tetrahedral anions stack similarly, where each of the tetrahedra is located directly above the center carbon atom in a C[NH 2 ] 3unit.The SO 4 -tetrahedra in C[NH 2 ] 3 C 2 H 5 SO 4 is bonded to an organic tail of C 2 H 5 , giving the anion an elongated shape compared to the tetrahedral anions in C[NH 2 ] 3 ClO 4 and C[NH 2 ] 3 BF 4 .This more complex anion causes a different, but still somewhat layered packing of C[NH 2 ] 3 C 2 H 5 SO 4 .Table IV lists the results and lattice parameters for the three C[NH 2 ] 3 -materials.C[NH 2 ] 3 ClO 4 has a predicted d 33 of −11.2 pC/N which has a similar magnitude, but opposite sign as the experimentally measured d 33 of 15 pC/N for a single crystal sample [81].This is the only such discrepancy in this work, as such it is difficult to determine if the origin of the difference in the values is related to the computation or the experimental methods used in the literature.This is a topic for further investigation.C[NH 2 ] 3 ClO 4 have in general larger piezoelectric coefficients than C[NH 2 ] 3 BF 4 .C[NH 2 ] 3 C 2 H 5 SO 4 has both smaller polarization and piezoelectric coefficients than the two other compounds in the C[NH 2 ] 3 -group.The large C 2 H 5 SO 4 anion results in a different crystal structure with a larger unit cell, per anion-cation pair, partly explaining its lower polarizaiton.The piezoelectric anisotropy factor |d 25 /d 33 | is 29 for C[NH 2 ] 3 C 2 H 5 SO 4 , more than ten times that of the two other compounds.In line with the prior discussion, this larger anisotropy indicates that rotation of the molecules is a more pronounced part of the

4 with d 16 =
−94 pC/N has c 66 of 505 TPa −1 , while [F -Q]ReO 4 with similar packing and composition has d 16 = −33 pC/N with c 66 = 187 TPa −1 .While the HQ compound has an anisotropy of 118, [F -Q] has an anisotropy of 7, underlining the more important role of molecular rotation for the former.While there is a clear trend, the exact relationship between the shear piezoelectric coefficients and the corresponding compliance values varies, especially across material class.For instance, C[NH 2 ] 3 BF 4 has c 55 value of 230 TPa −1 and d 25 of 10.9 pC/N.So while the shear compliance values of C[NH 2 ] 3 BF 4 are larger than those of [F -Q]ReO 4 , the piezoelectric response is lower.

FIG. 8 .
FIG. 8. Piezoelectric shear coefficients (vertical axis) plotted against longitudinal piezoelectric coefficients d33 (left panel, horizontal axis) and diagonal shear compliance coefficients cjj, j = 4, 5, 6 (right panel, horizontal axis) The piezoelectric coefficients for the inorganics are obtained from the Materials Project.Piezoelectric coefficients vanishing due to crystal symmetry are omitted.

TABLE II .
Computed (and experimental) properties of the dabco-compounds.

TABLE III .
Computed (and experimental) properties of the (R)-OH-Q-compounds.
NH 2 ] 3 C 2 H 5 SO 4 than for C[NH 2 ] 3 ClO 4 and C[NH 2 ] 3 BF 4 .The less dense packing of C[NH 2 ] 3 C 2 H 5 SO 4 facilitates molecular rotation.Further, the orientation of the C[NH 2 ] 3 relative to the polarization axis is different between C[NH 2 ] 3 C 2 H 5 SO 4 .The orientation of the C[NH 2 ] 3 molecules will also affect the piezoelectric response.The largest spontaneous polarization in this group of materials is predicted for C[NH 2 ] 3 BF 4 at 7.1 µC/cm 2 .Both the calculated and measured polarization of C[NH 2 ] 3 BF 4 is larger than that of C[NH 2 ] 3 ClO 4 , which relates to the fact that the smaller ion, BF 4 , results in a larger dipole density in the material than the larger ClO 4 -ion.
[81]f.[58]bRef.[59]cRef.[81]piezoelectricmechanism in C[ Table V lists the computed figures of merits and electromechanical coupling factors k 2 for all the materials investigated here.The k 2 values vary greatly even among systems with comparable crystal structures.In particular, there is a high sensitivity of k 2 to the composition of the plastic ionic molecular crystals.While some systems have low k 2 values, the predicted values of HQReO 4 of 0.75 and [Hdabco]ReO 4 of 0.79, exceeds those of PZT-5H, AlN, and PVDF-TrFE with values 0.76, 0.23 and

TABLE V .
[92]largest energy harvesting figures of merits and electromechanical coupling coefficients of all materials in this work.The large compliance, as well as the low dielectric constants, of the ionic plastic molecular crystals explains the large figure of merits even though the piezoelectric strain coefficients e i,j are far lower than that of engineered inorganic piezoelectrics.For instance, where HQReO 4 has a c 66 = 505 TPa −1 , PbTi has a c 33 = 49 TPa −1[92].