Introduction

The perovskite structures have been dubbed a “chameleon” due to their rich diversity of chemical compositions and physical properties.1,2,3 Searching for large piezoelectric perovskites has been the aspiration of science and engineering studies.4 Compared with inorganic perovskite, organic materials possess many advantages. For example, organic crystals can be synthesized at relatively low temperature.5 The easy reorientation inorganic cations make them promising candidates for order–disorder ferroelectrics.6,7 By replacing Pb and other toxic ions with inorganic cations, they can be designed to toxic-metal-free materials. Therefore, organic materials have become an attractive topic not only due to the remarkable structural variability and highly tunable properties, but also due to their potential applications in light sources, photovoltaics, and electronics.8

Recently, a breakthrough of piezoelectric material was reported by Yu-Meng You, Ren-Gen Xiong, and colleagues.9 Based on molecular design strategy and elaborate organic cation selection, they successfully synthesized a family of metal-free organic perovskite ferroelectrics. This family contains 23 new compounds, in which 21 compounds have cubic perovskite structure and the other two have hexagonal perovskite structure. Among these cubic perovskite compounds, 14 compounds show paraelectric-ferroelectric transition and their Curie temperatures range from 355 to 493 K. This means all of them are ferroelectric at room temperature. The general formula of this family is A(NH4)X3, in which A is a divalent organic cation and X is halogen anion. Despite their diversity, compounds of this family were obtained easily by a simple strategy: slow evaporation of solution containing organic ammines, inorganic ammoniums, and dilute hydrohalogen acid. Not only the facile synthesis method but also their desirable properties make them are possible for application in flexible devices.

Among this metal-free organic perovskite family, MDABCO-NH4-I3 (MDABCO is N-methyl-N’-diazabicyclo[2,2,2]octonium) is an outstanding one. Its ferroelectric Curie temperature (T0) is 448 K, spontaneous polarization (Ps) is 22 μC/cm2 and piezoelectric coefficient d33 (along [1 1 1] direction) is about 14 pC/N9. Furthermore, second-harmonic generation response and dielectric permittivity were also measured as a function of temperature. Although a detailed investigation of MDABCO-NH4-I3 was made in experiment, some important physical properties were not reported. For example, except for the longitude piezoelectric stress coefficient d33, other elements of piezoelectric stress tensor were not measured in experiment. Furthermore, the elastic tensor and piezoelectric strain tensor were not reported, too.

Results and discussion

In order to elucidate the piezoelectric and elastic properties of MDABCO-NH4-I3, we performed the density-functional theory (DFT) calculation using Perdew–Burke–Ernzerhof (PBE)10 and PBE-D311 methods as implemented in the Vienna ab initio simulation package (VASP).12,13,14

The primitive and conventional unit cells of MDABCO-NH4-I3 in ferroelectric phase are depicted in Fig. 1. Herein, the MDABCO-NH4-I3 crystal is composed of three building blocks: NH4, MDABCO, and I6 octahedron. The NH4 locates in the middle of I6 octahedron, which shares corners with its six neighboring octahedra to form a three-dimensional (3D) network. The MDABCO locates in the middle of a cage defined by 12 iodine ions. It has a trigonal structure with space group R3. The three-fold axis passes through the two nitrogen atoms of MDABCO and the spontaneous polarization direction is parallel to this axis (z-axis). Similarly with K3BO10Cl15,16,17 and trimethylchloromethyl ammonium trichloromanganese,5 it is the ordered cation MDABCO that breaks the inversion symmetry and permits the occurrence of ferroelectric, piezoelectricity, second-harmonic generation, and other interesting phenomena. When temperature increases above 448 K, it turns to a cubic paraelectric phase.9 Besides MDABCO-NH4-I3, MDABCO-NH4-Br3 is also successfully synthesized with similar method.9 It has a similar structure with MDABCO-NH4-I3 and exhibits paraelectric-ferroelectric transition at 390 K.

Fig. 1
figure 1

The ferroelectric structure of MDABCO-NH4-I3. a and b are NH4 and MDABCO groups, c and d are the primitive unit cell viewed along [0 0 1] and [1 1 1] directions, e and f are the conventional unit cell viewed along [1 1 0] and [0 0 1] directions

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So as to calculate the properties of these newly synthesized crystal, we relaxed their atomic positions and lattice parameters first. For MDABCO-NH4-I3 and MDABCO-NH4-Br3, we used the experimental data for their initial structures. Then, lattice parameters and atomic positions are relaxed without any symmetry constraint until calculated stresses and forces are less than the threshold. As MDABCO-NH4-Cl3 is not synthesized in experiment, we substitute the Br in MDABCO-NH4-Br3 to Cl as its initial structure.18,19 Then, we relaxed lattice parameters and atomic positions of MDABCO-NH4-Cl3 until they meet the threshold. So as to verify the calculated structure are stable, we calculated their zone center phonon frequencies and do not find unstable mode. Moreover, they all satisfy the mechanical stability criteria, namely, that the stiffness tensor is positive-definite.20 The calculated lattice parameters of primitive unit cell are listed in Table 1 and are compared with available experimental data.

Table 1 Experimental and calculated lattice parameters of ferroelectric MDABCO-NH4-X3
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Because MDABCO-NH4-X3 are molecular materials, we use the DFT-D3 method to take the van de Waals interaction into consideration.9 Unfortunately, PBE + D3 does not give better result compared with PBE. As far as the angle is concerned, PBE functional is slightly better than PBE + D3. However, when it comes to the lattice length, PBE functional performs much better than PBE + D3. The PBE result overestimates the lattice parameter of MDABCO-NH4-I3 1.5%, and PBE + D3 underestimates 1.9%. For MDABCO-NH4-Br3, PBE underestimates lattice parameter 2.1%, while PBE + D3 underestimates it as large as 5.2%. This means PBE is more suitable than PBE + D3 in MDABCO-NH4-X3 calculation. The poor performance of PBE-D3 is due to the hydrogen bond in MDABCO-NH4X3. The total energy of DFT-D3 is EKS-DFT − Edisp.11 The first term EKS-DFT is the usual self-consistent KS energy and the second term Edisp is the dispersion correction. This correction is a sum of two- and three-body energies, related with bond length, bond angle and etc. However, it has no relationship with the electronic band structure. Therefore, the developers of DFT-D3 says “the method is of molecular mechanics type in the sense that it is very fast and only geometric information is employed”.11 It is pointed in refs. 21,22,23,24 that the long-range part of the exchange contribution is overestimated in PBE. This overestimation compensates the missing vdW interaction to some extent. Consequently, directly adding additional dispersion correction to PBE would lead to an unphysical ‘double-counting’’ effect.22 As shown in ref. 11, this unphysical effect in hydrogen bond is severe and results into the poor performance of PBE-D3. Therefore, we use the atom position and lattice parameters of PBE and calculate other properties with PBE functional as well.

So as to investigate the origin of large spontaneous polarization (22 μC/cm2) of MDABCO-NH4-I3, we calculate its Born effective charge (BEC). BEC is a second-order tensor and the component along the dipole direction is related with ferroelectric polarization. The calculated BEC components along the dipole direction are marked in Fig. 2.

Fig. 2
figure 2

The calculated structure and Born effective charges of MDABCO-NH4-I3. a The Born effective charges of MDABCO. b The Born effective charges and relative positions of MDABCO, NH4, and I3 groups. The dashed lines show hydrogen bonds between H+ and I−

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As shown in Fig. 2a, MDABCO can be divide into three parts. The top hydrogen, the bottom CH3 and the middle N2C6H12 part. Their BECs are +0.84, +0.81, and +0.77, respectively. For MDABCO, all C and H show positive values, only N shows negative values. We calculate the positive and negative charge centers of MDABCO, and find the distance between the two centers is as small as 0.009 Å. They are almost overlap with each other. Therefore, the dipole moment of the MDABCO group is negligible. The total charge of MDABCO is +2.45 and the total charge center is marked as a red point in Fig. 2. The positive and negative charge centers of NH4 are almost overlap with each other, too. Therefore, the dipole moment of the NH4 group is negligible. The total charge of NH4 is +1.15, centered at the N position. The charge of I is −1.2, similar to its formal charge −1. The relative positions of MDABCO, NH4, and I3 along the dipole moment direction are depicted in Fig. 2b. It is clear that the charge center of MDABCO (A-site of perovskite) shifts upward with the distance dA = (5.37 – 3.84)/2 = 0.765 Å relative to the center of its nearest neighboring six I− ions. At the same time, the charge center of NH4 (B-site of perovskite) shifts upward, too, with the distance dB = (2.83 – 1.77)/2 = 0.53 Å relative to the center of I6 octahedron. As a comparison, the A-site and B-site displacements of rhombohedral BaTiO3 are 0.10 and 0.19 Å, respectively. They are much smaller than the corresponding off-center distance of MDABCO-NH4-I3. Furthermore, dA is larger than dB for MDABCO-NH4-I3, similarly to the condition of PbTiO3. This means that MDABCO-NH4-I3 is an A-site driven ferroelectricity. On the contrary, dA is smaller than dB for BaTiO3, indicating a B-site driven ferroelectricity. Therefore, MDABCO-NH4-I3 is an A-site driven order–disorder ferroelectricity and its large polarization results from the large off-center distances of MDABCO and NH4 relative to I.

As shown in the dashed lines of Fig. 2b, there are three hydrogen bonds between three hydrogens of CH3 and three I− anions. And there are also three hydrogen bonds between three hydrogens of NH4 and its surrounding three I− anions.9 When MDABCO-NH4-I3 turns from ordered ferroelectric phase to disordered paraelectric phase, much energy is needed to break these six hydrogen bonds so as to make MDABCO and NH4 rotating freely. In paraelectric phase, the rotating MDABCO must overcome a barrier, in which the two N atoms of MDABCO lines along the [1 1 0] direction. So as to get the barrier value, we calculate the total energy of two configurations. In the first configuration, lattice parameters are set to the experimental value of paraelectric phase and I ions forms a cubic network. Then we fix the positions of I and set the two N and two C atoms of MDABCO along the C3 axis to move only along the [1 1 1] direction. Other atoms can move freely. In the second configuration, lattice parameters and positions of I ions are the same as configuration one. Then the selected two N and two C atoms of MDABCO in the first configuration are set to move only along the [1 1 0] direction. Other atoms can move freely. The energy difference between the above two configurations is ΔE = 0.156 eV per primitive unit cell. As MDABCO is polyatomic molecules with six freedom (three translations and three rotations), the energy barrier corresponds to the temperature T = ΔE/(3 × kB) = 398 K, which agrees with the experimental value 448 K.

The structure evolution with X ions is presented in four parts. Firstly, the structure parameters inner MDABCO are shown in Fig. 3a. The bond lengths (solid red symbol) refer to the left axis, while the bond angles (open blue symbol) correspond to the right axis. With halogen changing from Cl to I, the bond lengths change less than 0.01 Å and the bond angles change <0.2° Therefore, the MDABCO group performs like a rigid body in MDABCO-NH4-X3 crystals.

Fig. 3
figure 3

The structure parameters a of inner MDABCO and b between MDABCO and its nearest neighboring X ions. The solid red symbols refer to the left vertical axis, while the open blue ones refer to the right vertical axis. The inset structures depict the bond lengths and angles

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Secondly, the structure parameters between MDABCO and its surrounding X ions are shown in Fig. 3b. There are two kinds of bond length between H and X ions. The first one is dHX_w at the top of Fig. 3b inset, while the second one is dHX_s at the bottom of Fig. 3b inset (the subscript ‘w’’ and ‘s’’ represent ‘relatively weak’’, and ‘relatively strong’’). The first bond length is larger than the second one, indicating the first hydrogen–halogen interaction is weaker than the second one. With the radius of X ion increases from Cl to I, the two hydrogen-X bond lengths increase. We also give the angles between H–X bond and the X3 plane (inset of Fig. 3b). With halogen changing from Cl to I, the angle θHX3_w changes from 26.4° to 31.1°, increasing 4.7°. However, the angle θHX3_s changes from 46.4° to 44.4°, decreasing 2.2°. The opposite trends of θHX3_w and θHX3_s originate from the A-site space changing of perovskite. With ion changing from Cl− to I−, the lattice parameter of MDABCO-NH4-X3 increases from 6.84 to 7.37 Å. Then, the corresponding A-site space of perovskite increases. In MDABCO-NH4-Cl3, the MDABCO group is a little compressed as there is not enough room for it. When it comes to MDABCO-NH4-I3, there is spare room for MDABCO group. Because the hydrogen–halogen interaction at the bottom of Fig. 3b inset is stronger than the top one, the bottom hydrogen approaches its neighboring I, decreasing θHX3_s, while the top hydrogen leaves its neighboring I, increasing θHX3_w.

Thirdly, we give the structure parameters inner NH4 in Fig. 4a. There are two N–H bond length: lNH_up and lNH_dn. The first bond almost points to the neighboring X ion, while the second bond points to the middle of three X ions. Therefore, the first N–H bond is enlarged by X ions relative to the second one, resulting into lNH_up > lNH_dn. Because NH4 forms a tetrahedron, we calculate the Baur’s distortion index D with the following equation17,25:

$$D = \frac{1}{n}\mathop {\sum}\limits_{i = 1}^n {\frac{{|l_i - l_{{\mathrm{av}}}|}}{{l_{{\mathrm{av}}}}}}$$
(1)

where li is the distance from the central atom to the ith coordinating atom, and lav is the average bond length. For a regular tetrahedron, the distortion index equals zero. For NH4 tetrahedron, the distortion index is very little, indicating that NH4 performs like a rigid body in MDABCO-NH4-X3.

Fig. 4
figure 4

The structure parameters a of inner NH4 and b between NH4 and its nearest neighboring X ions. The solid red symbols refer to the left vertical axis, while the open blue ones refer to the right vertical axis. The inset structures depict the bond lengths and angles

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Fourthly, we show the structure parameters between NH4 and its surrounding X ions in Fig. 4b. For a regular octahedron, N atom should locate at the center and φXNX should be 180°. Actually, the angle φXNX is about 157°, obviously deviating from the ideal value of 180°. With halogen changing from Cl to I, φXNX changes from 154.6° to 158.7°, increasing 4.1°. Similarly with MDABCO, we also give the angles between H–X bond and the X3 plane (inset of Fig. 4b). With halogen changing from Cl to I, the angle φHX3_w changes from 36.0° to 33.8°, decreasing 2.2°. However, the angle φHX3_s changes from 30.6° to 33.2°, increasing 2.6°. With halogen changing from Cl to I, the evolutions of φHX3_s (increasing) and φHX3_w (decreasing) are opposite to those of θHX3_s (decreasing) and θHX3_w (increasing). It originates from the B-site space changing of perovskite. With X ion changing from I− to Cl− (not Cl− to I−), the B-site space of perovskite increases. In MDABCO-NH4-I3, the NH4 group is a little compressed as there is not enough room for it. When it comes to MDABCO-NH4-Cl3, there is spare room for NH4 group. Because the hydrogen–halogen interaction at the top of Fig. 4b inset is stronger than the bottom one, the top hydrogen approaches its neighboring Cl, decreasing θHX_s, while the bottom hydrogen leaves its neighboring Cl, increasing θHX_w.

The calculated electronic density of state (DOS) of MDABCO-NH4-X3 are exhibited in Fig. 5. It can be clearly seen that these DOS are divided into three parts. The low energy part (below −2 eV) is mainly composed of occupied N-2p, C-2p, and H-1s states, the middle energy part (from −2 to 0 eV) is mainly composed of occupied Cl-3p, Br-4p, and I-5p states, and the high energy parts (above 3 eV) is unoccupied states of N, C, H, and halogen. By comparing Fig. 5a–c, we can see two obvious evolutions with anion changes from Cl− to I−. Firstly, the width of the occupied halogen p orbital increases from 0.8 (Cl−) to 1.1 (I−) eV. This width broadening originates from the X-N-X angle changing in Fig. 4b. With X ion changing from Cl− to I−, the angle of X-N-X angle (φXNX) increases toward 180°, resulting into more electron interaction along the X-N-X line.26 Secondly, there are obvious gap between the occupied halogen p state and the occupied states of C, N, and H. This characters indicate the halogen interactions with MDABCO and NH4 are weak.16,27 It agrees with the analysis of Figs. 3a and 4a.

Fig. 5
figure 5

The calculated electronic density of state of MDABCO-NH4-X3. Fermi level is set to zero. Some special points are label with arrows of different color

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Piezoelectric effect allows interconversion between electric field and mechanical deformation. This property is important for applications of sensors, actuators, vibration reducer, and so on. Conventional perovskite ferroelectrics usually require high processing temperatures,18 have little mechanical flexibility, and even contain potential toxic elements.4 These drawbacks make scientists pay much attention to organic materials.5,9 Experiment shows that MDABCO-NH4-I3 is an organic piezoelectrics without metal element. The piezoelectric strain coefficient dz3 of MDABCO-NH4-I3 is measure to be ~14 pC/N at room temperature.9 However, other components of piezoelectric tensor are still unknown. Therefore, we calculated the piezoelectric strain and stress tensors of MDABCO-NH4-X3.

The crystallographic symmetry plays an important role in the piezoelectric phenomena.28 According to the definition of the piezoelectric effect, in the matrix notation (with two indices), the piezoelectric strain tensor dij of MDABCO-NH4-X3 (space group R3) is

$$\left[ {\begin{array}{*{20}{c}} {d_{x1}} & { - d_{x1}} & 0 & {d_{x4}} & {d_{x5}} & { - 2d_{y2}} \\ { - d_{y2}} & {d_{y2}} & 0 & {d_{x5}} & { - d_{x4}} & { - 2d_{x1}} \\ {d_{z1}} & {d_{z1}} & {d_{z3}} & 0 & 0 & 0 \end{array}} \right]$$

The corresponding piezoelectric stress tensor eij is

$$\left[ {\begin{array}{*{20}{c}} {e_{x1}} & { - e_{x1}} & 0 & {e_{x4}} & {e_{x5}} & { - e_{y2}} \\ { - e_{y2}} & {e_{y2}} & 0 & {e_{x5}} & { - e_{x4}} & { - e_{x1}} \\ {e_{z1}} & {e_{z1}} & {e_{z3}} & 0 & 0 & 0 \end{array}} \right]$$

There are six independent coefficients in piezoelectric strain (stress) tensor.

The piezoelectric strain tensor is defined as the ratio of developed strain to the applied electric field.28 Our calculated piezoelectric strain tensors are shown in Fig. 6a. The calculated dz3 of MDABCO-NH4-I3 is 14.4 pC/N, which agrees very well with the experimental value ~14 pC/N9. This agreement proves the validity of our calculation.

Fig. 6
figure 6

The calculated piezoelectric strain (a) and stress (b) tensor of MDABCO-NH4-X3. The experimental measured dz3 of MDABCO-NH4-I3 is also given

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The calculated dx4 and dx5 are much larger than dz3. The largest components of piezoelectric strain tensor for the three crystals are all dx5, which is 119 pC/N for MDABCO-NH4-Cl3, 248 pC/N for MDABCO-NH4-Br3, and 179 pC/N for MDABCO-NH4-I3. They are much larger than the largest component of inorganic crystals, such as lithium niobate (dx5 = 68 pC/N),29 lithium tantalite (dx5 = 26 pC/N),29 cadmium sulfide (dx5 = -14 pC/N),30 barium sodium niobate (dy4 = 52 pC/N),28 hexagonal zinc oxide (dz3 = 12.4 pC/N),31 and hexagonal zinc sulfide (dz3 = 3.2 pC/N).32,33 They are even comparable with that of barium titanate (dx5 = 242 pC/N)34 and are almost one third of that of PZT-5H (dx5 = 741 pC/N).34 The above comparison shows that MDABCO-NH4-X3 have large piezoelectric responds and are good candidate materials for a variety of piezoelectric applications.

Similarly with piezoelectric strain tensor, piezoelectric stress tensor is defined as the ratio of developed stress to the applied electric field.28 Our calculated piezoelectric stress tensors are shown in Fig. 6b. Except for ex1 and ez3, the other components change little with X anion. Furthermore, differently with the condition of dx5, ex5 is not the largest component of piezoelectric stress tensor. It is well known that the piezoelectric strain tensor [d], piezoelectric stress tensor [e], and elastic compliance tensor [s] satisfy the relation [d] = [e][s].28 This indicates that the large value of dx5 may originates from the large elastic compliance. In the matrix notation (with two indices), the elastic compliance tensor of MDABCO-NH4-X3 (space group R3) is

$$\left[ {\begin{array}{*{20}{c}} {s_{11}} & {s_{12}} & {s_{13}} & {s_{14}} & { - s_{25}} & 0 \\ {s_{12}} & {s_{11}} & {s_{13}} & { - s_{14}} & {s_{25}} & 0 \\ {s_{13}} & {s_{13}} & {s_{33}} & 0 & 0 & 0 \\ 0 & 0 & 0 & {s_{44}} & 0 & {2s_{25}} \\ { - s_{25}} & {s_{25}} & 0 & 0 & {s_{44}} & {2s_{14}} \\ 0 & 0 & 0 & {2s_{25}} & {2s_{14}} & {2(s_{11} - s_{12})} \end{array}} \right]$$

There are seven independent coefficients in elastic compliance tensor. The calculated values are shown in Fig. 7.

Fig. 7
figure 7

The calculated elastic compliance tensor of MDABCO-NH4-X3, in which s44 is the largest component

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It is clear in Fig. 7 that the s44 is the largest component in MDABCO-NH4-X3. With X anion changes from Cl− to I−, it firstly increases from 449 to 658 and then decreases to 473 × 10−12 m2/N. This evolution is similar with that of dx5 in Fig. 6a. For MDABCO-NH4-X3, the piezoelectric strain tensor component dx5 = ex5 × s44 − 2ex1 × s25 − 2ey2 × s14. The last term 2ey2 × s14 is about two orders smaller than the first term, and the second term 2ex1 × s25 is about one-third of the first term. Because the ex5 difference for MDABCO-NH4-X3 is rather small, the dx5 evolution with halogen is dominated by s44.

For MDABCO-NH4-X3, s44 equals (c12 − c11)/(2c214 + 2c225 − c11c44 + c12c44), related with five parameters c12, c11, c14, c25, and c44. No one dominates the others. Therefore, s44 is affected by many structure parameters. So as to give an intuitive explanation, we propose a model based on the tolerance factor of perovskite. For an inorganic perovskite ABX3, the Goldschmidt tolerance factor (TF) is defined as below35

$${\mathrm{TF}} = \frac{{r_{\mathrm{A}} + r_{\mathrm{X}}}}{{\sqrt 2 \left( {r_{\mathrm{B}} + r_{\mathrm{X}}} \right)}}$$
(2)

where rA, rB, and rX are the ionic radii of the ions A, B, and X, respectively. In this model, all ions are assumed to be hard spheres. TF is widely used to analysis the structure of inorganic perovskite.36 TF = 1 corresponds to an ideal cubic perovskite with A, B and X ions touching each other perfectly. TF > 1 implies that A cation is too large or B cation too small, while TF < 1 implies A cation is too small or B cation too large.

This concept is generalized to hybrid perovskites containing organic cations by Kieslich and Becker et al.37,38,39 It is found that the TF value of most hybrid perovskites lies in the range from 0.8 to 1. Hybrid perovskite structures with TF > 1 or TF < 0.8 are rarely observed.39 So as to get the TF values of MDABCO-NH4-X3, ions radii are needed. The Shannon radius of Cl−, Br−, and I− are 1.81, 1.96, and 2.2 Å, respectively.40 The radius of (NH4)+ is proposed to be 1.46 Å39. But the radius of MDABCO is difficult to define because it is not a hard sphere. Based on the available radii of X and NH4, we calculate the TF value evolution with MDABCO radius (rA). As shown in Fig. 8a, the TF value of MDABCO-NH4-Cl3 is the largest, while that of MDABCO-NH4-I3 is the smallest. In experiment, MDABCO-NH4-Br3 and MDABCO-NH4-I3 are successful synthesized, while MDABCO-NH4-Cl3 is not. Therefore, it is reasonable to assume that the TF values of MDABCO-NH4-Br3 and MDABCO-NH4-I3 is <1, while that of MDABCO-NH4-Cl3 is >1 (the middle light blue region of Fig. 8a). This assumption agrees with the angles analysis of Figs. 3b and 4b. We also give the sketch of AX plane and BX plane for different TF values in Fig. 8b–d. For TF > 1, the A cation is too large, resulting into over compression between A and X ions. It is the condition of MDABCO-NH4-Cl3. For TF < 1, the B cation is too large, resulting into over compression between B and X ions. It is the condition of MDABCO-NH4-I3. The over compressions of MDABCO-NH4-Cl3 and MDABCO-NH4-I3 increase their shear resistance, reducing their s44 values. For MDABCO-NH4-Br3, its TF value is closer to 1 than that of MDABCO-NH4-I3. It implies there is less over compression between A and X ions or B and X ions in MDABCO-NH4-Br3. Consequently, its shear resistance is less than that of MDABCO-NH4-Cl3 and MDABCO-NH4-I3, resulting in large s44 value.

Fig. 8
figure 8

The tolerance factor MDABCO-NH4-X3. a The evolution of tolerance factor with A ion radius (rA). b–d The sketches of AX and BX planes for different tolerance factors. The arrows in b represent shear stress

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The large elastic compliance tensor in Fig. 7 suggests that MDABCO-NH4-X3 are soft materials.32 So as to investigate their elastic properties, Table 2 gives their bulk and shear moduli with Voigt-Reuss-Hill(VRH) approximation,41 which is a good estimation for mechanical properties of polycrystalline materials.16,20,42,43,44 Bulk modulus describes the resistance of a material to a change in volume and shear modulus describes the resistance to a change in shape. Although the crystal structures of MDABCO-NH4-X3 are similar with inorganic ABO3 perovskite, their bulk moduli are only around 10 GPa. They are much smaller than that of inorganic perovskite CaTiO3 (170.9 GPa),45 BaTiO3 (162 GPa),46,47 and PbTiO3(144 GPa).46,47 The Young’s moduli of MDABCO-NH4-X3 are also around 10 GPa, even smaller than that of flexible metal-organic frameworks (MOFs)43: MIL-53(Al) (25.5 GPa), MIL-53(Ga) (19.2 GPa), MIL-47 (25.3 GPa), DMOF-1(lozenge) (11.2 GPa), and DMOF-1(square) (12.5 GPa).

Table 2 Calculated elastic stiffness tensor cij, bulk modulus B, shear modulus G, and Young’s modulus E (GPa) of MDABCO-NH4-X3
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Bulk modulus is an average value of all directions. Instead, directional Young’s modulus characterizes uniaxial stiffness in different directions. As a reference, the directional Young’s modulus of rhombohedral BaTiO3 based on the calculated elastic stiffness using PBE functional48 are given in Fig. 9a, b. Similarly with MDABCO-NH4-X3, BaTiO3 takes rhombohedral structure as its lowest energy phase. We set its polar direction along z-axis. The crystal structure and directional Young’s modulus are shown in the same view point in Fig. 9a. It is clear that the directional Young’s modulus has six local minimums and eight high value lobes. The local minimum clearly seen in Fig. 9a is along the a1 direction (a1, a2, and a3 are the lattice vectors of the primitive unit cell). The other five local minimums are along -a1, ± a2, and ± a3 directions. They correspond to compression or stretching of O–Ti–O bonds. Owing to the Ti displacement from the center of oxygen octahedron, these six directions show low rigidity. The directional Young’s modulus viewed along the polar direction is shown in Fig. 9b. It is clear that the view direction is perpendicular to the facet of oxygen octahedron. Similarly, the other seven high value lobes are perpendicular to the other seven facets of oxygen octahedron.

Fig. 9
figure 9

The calculated directional Young’s modulus. a, b are directional Young’s modulus and crystal structure of BaTiO3 viewed from different directions. c–e are directional Young’s modulus of MDABCO-NH4-Cl3, MDABCO-NH4-Br3, and MDABCO-NH4-I3, respectively. They are represented as three-dimensional (3D) surfaces with axes tick labels in GPa

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Similarly with BaTiO3, the polar direction of MDABCO-NH4-X3 is parallel to z-axis in Fig. 9c–e. There are six local minimums and eight high-value lobes in the directional Young’s modulus of MDABCO-NH4-X3. These local minimums are along the lattice vectors of their primitive unit cells. They correspond to compression or stretching of X–N–X bonds in octahedrons. The directions of these high value lobes are perpendicular to the facets of X6 octahedron. Their directional Young’s modulus can be described in the following equation42:

$$\begin{array}{*{20}{l}} E \hfill & = \hfill & {1/\left[ {s_{11}sin^4\varphi + s_{33}cos^4\varphi + \left( {2s_{13} + s_{44}} \right)sin^2\varphi {\mathrm{cos}}^2\varphi } \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { + 2\left( {s_{14}{\mathrm{sin}}3\theta - s_{25}{\mathrm{cos}}3\theta } \right){\mathrm{cos}}\,\varphi \,{\mathrm{sin}}^3\varphi } \right]} \hfill \end{array}$$
(3)

where θ is the azimuthal angle in the xy plane from the x-axis (0 ≤ θ < 2π), and φ is the zenith angle from the positive z axis (0 ≤ φ ≤ π). It can be derived from Eq. (3) that E equals 1/s33 with φ = 0. The s33 value of MDABCO-NH4-X3 (X = Cl, Br, I) is 52.3, 60.1, and 73.9 × 10−12m2/N, respectively. Correspondingly, the highest value of the top lobe in Fig. 9c–e is 19.1, 16.6, and 13.5 GPa, respectively. When it comes to xy plane, φ = π/2 and E = 1/s11. Therefore, E value in the xy plane has no relationship with θ and becomes a circle with radius 1/s11. The s11 value of MDABCO-NH4-X3 (X = Cl, Br, I) is 110.8, 107.3, and 88.9 × 10−12 m2/N, respectively. Their corresponding circle radii in the xy plane are 9.0, 9.3, and 11.2 GPa. In Eq. (3), the only term related with angle θ is (s14sin3θ – s25cos3θ), which means z-axis is a threefold axis. The maximum and minimum values of this term are \(\sqrt {(s_{14}^2 + s_{25}^2)}\) and \(- \sqrt {(s_{14}^2 + s_{25}^2)}\), respectively. Therefore, the anisotropy with angle θ is only related with \(g = \sqrt {(s_{14}^2 + s_{25}^2)}\). The g value of MDABCO-NH4-X3 (X = Cl, Br, I) is 112.9, 134.1, and 46.1 × 10−12 m2/N, respectively. MDABCO-NH4-Br3 has the largest g value, and the g value of MDABCO-NH4-I3 is much lower than the others. That is why Fig. 9e looks much smoother than Fig. 9c, d.

In summary, we calculated the structure, electronic density of state, piezoelectric response, and elastic properties of metal-free organic perovskite MDABCO-NH4-X3. The MDABCO and NH4 groups performs like a rigid body in MDABCO-NH4-X3, and they are linked to the I6 octahedron network by H–X interaction. The high ferroelectric Curie temperature is closely related with the break of this H–X interaction. With halogen changing from Cl to I, the tolerance factor of MDABCO-NH4-X3 increases from lower than 1 to higher than 1. Based on Born effective charges, the calculated dipole moments of the MDABCO and NH4 groups are negligible. Therefore, the large spontaneous polarization of MDABCO-NH4-I3 mainly results from the off-centers of MDABCO and NH4 relative to X ions. Our calculated piezoelectric strain component dz3 (14.4 pC/N) of MDABCO-NH4-I3 agrees very well with experimental value (~14pC/N). Although most organic materials have a very weak piezoelectric response, we found that some piezoelectric strain components of MDABCO-NH4-X3 have large values. Among the six independent components, the largest one is dx5. It is 119 pC/N for MDABCO-NH4-Cl3, 248 pC/N for MDABCO-NH4-Br3, and 179 pC/N for MDABCO-NH4-I3. The large value of dx5 is closely related with the large value of s44. This means that the large piezoelectric strain response of MDABCO-NH4-X3 is mainly originated from their softness. These unique properties of MDABCO-NH4-X3 make them good candidates as soft, low-density and bio-friendly piezoelectrics for various important applications, such as flexible devices, implantable systems, and micro robots.

Methods

we performed the density-functional calculation using PBE10 and PBE-D311 methods as implemented in the Vienna ab initio simulation package (VASP).12,13,14 The projector-augmented wave method was used to represent the electron–ion interaction.49 The valence-electron configurations were 2s22p2 for C, 2s22p3 for N, and ns2np5 for halogen. The wave function was expanded in a plane-wave basis set with an energy cutoff of 650 eV and the first Brillouin zone of primitive unit cell was sampled on a 3 × 3 × 3 mesh. The lattice parameters and atomic positions were fully relaxed until stress and forces were <0.05 GPa and 10 meV/Å, respectively. The elastic stiffness is calculated with finite difference method.50 It applies six finite strains (three normal and three shearing strains) to an optimized structure. All these six deformations correspond to strains of ±0.015, so as to keep strains in the elastic region. Then, the corresponding stress are calculated and the elastic stiffness constants are obtained by the strain-stress relationship. After that, the elastic compliance is obtained by inverting the elastic stiffness matrix directly. The piezoelectric constants, Born effective charges and zone center phonon frequencies are calculated by VASP package using density-functional perturbation theory.51 In order to obtain these values, three kinds of perturbations are applied to a crystal. They are atomic displacement away from equilibrium position ui, strain Sj and electric field Ek. The corresponding responses to these three perturbations are force Fl, stress Tm, and polarization Pn. Based on these perturbations and their responses, one can obtain force-constant matrix dFl/dui, Born effective charge dPn/dui and piezoelectric response dPn/dSj.