# First-Principles Property Assessment of Hybrid Formate Perovskites

Abduljelili Popoola1 P. S. Ghosh2,3 Maggie Kingsland1 Ravi Kashikar1 D.
DeTellem1 Yixuan Xu4 S. Ma4 S. Witanachchi1 S. Lisenkov1 I. Ponomareva1 1\.
Department of Physics, University of South Florida, Tampa, Florida 33620, USA
2\. Glass & Advanced Materials Division, Bhabha Atomic Research Centre, Mumbai
400 085, India  3\. Homi Bhabha National Institute, Anushaktinagar, Mumbai 400
094, India  4\. Department of Chemistry, University of North Texas, CHEM 305D,
1508 W Mulberry Street, Denton, Texas 76201, USA

###### Abstract

Hybrid organic inorganic formate perovskites, AB(HCOO)3, is a large family of
compounds which exhibit variety of phase transitions and diverse properties.
Some examples include (anti)ferroelectricity, ferroelasticity,
(anti)ferromagnetism, and multiferroism. While many properties of these
materials have already been characterized, we are not aware of any study that
focuses on comprehensive property assessment of a large number of formate
perovskites. Comparison of the materials property within the family is
challenging due to systematic errors attributed to different techniques or the
lack of data. For example, complete piezoelectric, dielectric and elastic
tensors are not available. In this work, we utilize first-principles density
functional theory based simulations to overcome these challenges and to report
structural, mechanical, dielectric, piezoelectric, and ferroelectric
properties for 29 formate perovskites. We find that these materials exhibit
elastic stiffness in the range 0.5 to 127.0 GPa , highly anisotropic linear
compressibility, including zero and even negative values; dielectric constants
in the range 0.1 to 102.1; highly anisotropic piezoelectric response with the
longitudinal values in the range 1.18 to 21.12 pC/N, and spontaneous
polarizations in the range 0.2 to 7.8 $\mu$C/cm2. Furthermore, we propose and
computationally characterize a few formate perovskites, which have not been
reported yet.

Hybrid organic inorganic perovskites (HOIP) are receiving a lot of attention
presently owing to the rapid progress in their synthesis and characterization.
They have the chemical formula ABX3, where A is typically an organic molecule,
B is a metallic cation, while X site could be halogen or molecular linker.
They exhibit a variety of phase transitions and rich range of properties, such
as ferromagnetism, (diel/ferro)ectricity, non-linear optical properties,
caloric effects, ferroelasticity, multiferroicity, among others Li  _et al._
(2020); Ma and Sun (2020). Among chemically diverse HOIPs, AB(HCOO)3, is one
of the largest families, where one can find most of the aforementioned
properties. The structural phase transitions in these materials are primarily
driven by the hydrogen bond stabilization and often occur close or even above
room temperature, which is highly desirable feature Li  _et al._ (2020). For
example, most ethyl ammonium metal formate perovskites exhibit transition in
the range 293 to 400 K Shang  _et al._ (2014a, 2016); Ptak  _et al._ (2016a,
b). Magnetic properties of formates are mostly determined by weak magnetic
interactions mediated by formate linker causing them to exhibit magnetic
ordering at low temperatures only, typically below 50 K Li  _et al._ (2020).
Furthermore, [AZE][M(HCOO)3] (AZE = azetidinium; M = Mn2+, Cu2+ and Zn2+)
family was reported to have extraordinarily large dielectric constants higher
than 104 in the vicinity of room temperature Imai  _et al._ (2012); Zhou  _et
al._ (2011); Asaji  _et al._ (2012). Often times, the value exhibit strong
frequency dependence, which resemble behavior of ferroelectric relaxors
Shvartsman and Lupascu (2012). Many formates undergo transitions into polar
space groups and, therefore, are possible candidates for ferroelectricity,
which is defined by the presence of spontaneous electric polarization
reversible by electric field. However, the value of spontaneous polarization
is typically below 5 $\mu$C/cm2, which makes its experimental measurement very
challenging Asadi and van der Veen (2016). Table 2 provides polarization
values from the literature and conditions for which it was reported/computed.
The simultaneous realization of ferroelectricity, ferromagnetism and/or
ferroelasticity in some hybrid formates classify them as multiferroics. It was
shown that DMA-Zn(HCOO)3 becomes multiferroic on substitution of Zn with
transition metals such as Ni, Mn, Co and Fe Jain  _et al._ (2009, 2008a); Ma
and Sun (2020). DMA-Co(HCOO)3 is another hybrid in which multiferroicity has
been observed Fu  _et al._ (2011). All abbreviations for A sites used in this
study are listed in Table 1.

Table 1: Abbreviations for A sites used in the study, following Stroppas bookLi  _et al._ (2020) Material | NH2NH3 | C2H5NH3 | C(NH2)3 | (CH3)2NH2 | CH3NH3 | NH2CHNH2
---|---|---|---|---|---|---
abbreviation | HAZ | EA | Gua | DMA | MA | FA
Table 2: Polarizations from literature and those computed in this work. A = NH3CH2CH3, P = PH3CH2CH3, AF = NH3CH2CF3, PF = PH3CH2CF3. Material | Type | Literature | Conditions | P ($\mu$C/cm2)
---|---|---|---|---
A-Mn | Comp. | 1.80${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{ref84Stroppabook}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
P-Mn | Comp. | 1.00${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{ref84Stroppabook}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
AF-Mn | Comp. | 5.10${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{ref84Stroppabook}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
PF-Mn | Comp. | 5.90${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{ref84Stroppabook}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
DMA-Mn | Exp. | 0.30${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C8RA00799C}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 150 K; B = 9 T (during growth) | 7.52
| Exp. | 2.70 – 3.61${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{WangW2013Mcit, doi:10.1021/acs.jpcc.0c03916}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 150 K; B = 0 – 5 T; E = 5 kV/cm | –
| Exp. | 0.8-2.4${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{doi:10.1063/1.4989783}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 184 K; E = 3.1-7.7 kV/cm | –
DMA-Ni | Exp | 0.42 – 0.52${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Ma_2019}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 150 K; B = 0 – 10 T | –
DMA-Co | Exp. | 0.30${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{doi:10.1021/acs.jpclett.0c02943}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 125 K | 7.44
DMA-Zn | Exp. | 0.45${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C6CP03414D}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 125 K | 7.77
NH4Mn | Exp. | 0.97${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{doi:10.1021/ja206891q}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 140 K | 2.45
NH4Mg | Exp. | 1.15${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{10.1002/chem.201303425}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 93 K | –
NH4Zn | Exp. | 0.02 – 0.93${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{doi:10.1021/ic4020702}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 120 – 248 K | 2.43
| Exp. | 4.00${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C6CE01891B}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 273 K; P = 1.44 GPa | –
| Exp. | 1.03${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{doi:10.1021/ja104263m, doi:10.1021/ja206891q}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 163 K | –
NH4Sc | Comp. | 3.71${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C6RA24182D}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
NH4Ti | Comp. | 2.46${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C6RA24182D}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
NH4V | Comp. | 2.40${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C6RA24182D}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
NH4Cr | Comp. | 2.51${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C6RA24182D}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
NH4Mn | Comp. | 2.38${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C6RA24182D}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
NH4Fe | Comp. | 2.37${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C6RA24182D}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
NH4Co | Comp. | 2.36${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C6RA24182D}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
NH4Ni | Comp. | 2.17${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C6RA24182D}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | 2.27
NH4Cu | Comp. | 2.20${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C6RA24182D}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
NH4Zn | Comp. | 2.30${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C6RA24182D}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | 2.43
CH3NH2NH2Mn | Exp. | 0.14${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{doi:10.1021/acs.chemmater.6b05249}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 150 K; B = 10 T | –
NH3(CH2)4NH3Mg2 | Exp. | 1.51${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{10.1002/chem.201303425}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 93 K | –
HAZ-Mn | Exp. (estimated) | 3.58${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C3QI00034F}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 110 K | 2.17
HAZ-Co | Exp. (estimated) | 2.61${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C3QI00034F}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 405 K | 2.53
HAZ-Zn | Exp. (estimated) | 3.48${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C3QI00034F}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 110 K | 2.32
HAZ-Mg | Exp. (estimated) | 3.44${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{C3QI00034F}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 400 K | 2.53
| Comp. | 2.6${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{doi:10.1021/acs.jpcc.6b10714}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 150 – 375 K | 2.53
EA-Mg | Exp. (estimated) | 3.43${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{10.1002/chem.201303425}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 93 K | 1.26
Gua-Cr | Comp. | 0.22${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{StroppaA2013HIFi}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
Gua-Cu0.5Mn0.5 | Comp. | 9.90${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{PhysRevMaterials.2.014407}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | –
Gua-Cu | Comp. | 0.11 - 0.37${}^{\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{2015PSSRR962T, ref79Stroppabook}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | T = 0 K | 0.21

Mechanical properties have been investigated for several members of formate
families and are reviewed in Ref.Tan  _et al._ (2012). Some representative
data from the literature for Young’s and elastic moduli are compiled in Table
3. The exotic negative linear compressibility, defined as an increase in
lattice parameter(s) under hydrostatic pressure, has been computationally
predicted in HAZ-M(HCOO)3 (M = Mn,Fe,Co) Ghosh and Ponomareva (2021, 2022) and
NH4Zn(HCOO)3 Li  _et al._ (2012). Negative linear compressibility finds
applications in pressure sensors and actuators, and possibly in design of
artificial musclesCairns and Goodwin (2015).

Table 3: Elastic properties from the literature. AZE = (CH2)3NH2 Material | Type | Young’s Moduli (GPa) | Elastic Moduli (GPa) | Ref.
---|---|---|---|---
DMA-Ni | Exp. | 24.5 |  | Tan  _et al._ (2012)
DMA-Mn | Exp. | 19.0 |  | Tan  _et al._ (2012)
DMA-Co | Exp. | 21.5 |  | Tan  _et al._ (2012)
DMA-Zn | Exp. | 19.0 |  | Tan  _et al._ (2012)
Gua-Cu | Exp. |  | 15.0 – 21.0 | Gui  _et al._ (2018)
Gua-Zn | Exp. |  | 24.0 – 29.0 | Gui  _et al._ (2018)
Gua-Mn | Exp. & Comp. |  | 23.5(6) – 28.6(4) | Li  _et al._ (2014)
AZE-Cu | Exp. & Comp. |  | 11.5(4) – 12.6(3) | Li  _et al._ (2014)
HAZ-Zn | Exp. |  | 24.5 – 26.5 | Kieslich  _et al._ (2016)
HAZ-Mn | Exp. |  | 24.5 – 28.6 | Kieslich  _et al._ (2016)
NH4Zn | Exp. & Comp. |  | 18.2 – 34.4 | Li  _et al._ (2012)

Evidences of pyroelectricity have reported in some hybrid formates
perovskites. For instance, pyroelectric coefficient was measured to exhibit a
maximum of 5.16$\times$10-2 C/m2 K under a poling electric field of 7.7 kV/cm
at 192 K for DMA-Mn(HCOO)3 Ma  _et al._ (2017). In another instance,
pyroelectric current was reported and used to study the order-disorder
transition under different pressures in DMA-Co(HCOO)3 Yu  _et al._ (2020).
Some other hybrid formates perovskites in which pyroelectric current has been
measured include DMA-Mg(HCOO)3 LoCicero  _et al._ (2020), DMA-Mn(HCOO)3, DMA-
Mn0.5Ni0.5(HCOO)3 Liu  _et al._ (2020a); Wang  _et al._ (2013), Gua-Cu(HCOO)3
Tian  _et al._ (2015), CH3NH2NH2Mn(HCOO)3 Maczka  _et al._ (2017) and DMA-
Zn(HCOO)3 Simenas  _et al._ (2016). The dependence of pyroelectric current on
applied magnetic field has also been demonstrated in DMA-Ni(HCOO)3 Ma  _et
al._ (2019).

Although, the aforementioned studies highlight the outstanding progress that
has been made in characterization of these materials, the survey also reveals
scarcity of such investigations, especially in the light of the fact that
formates subgroup hosts at least 64 known members Li  _et al._ (2020, 2017);
Wang  _et al._ (2010). It should also be recognized that many such
characterizations, spontaneous polarization for example, are rather
challenging experimentally. On the other hand, computational investigation is
an inexpensive, reliable and efficient tool to overcome these challenges and
achieve a comprehensive assessment of structural, piezoelectric, dielectric
and elastic properties for a wide range of materials in the formate family.

Therefore, in this study, we aim: (i) to predict structural parameters,
polarization, piezoelectric coefficients, dielectric constants and elastic
stiffness of 29 formate compounds using first-principles density functional
theory (DFT) based simulations; (ii) to provide a comprehensive comparative
assessment of the aforementioned properties; (iii) to catalog the properties
which could aid screening of promising materials.

##  I Computational Methodology

Table 4: Materials investigated in this study (upper part of the Table). The underscored compounds do not have experimental structure available from the literature. The bottom part of the Table gives other materials from the same family, which are not part of this investigation. Material | Space Group | T (K) | Experimental Ref.
---|---|---|---
Gua-M (M=Co, Fe, Mn, Ni, Zn) | Pnna | 293 | Hu  _et al._ (2009)
Gua-Cu | Pna21 | 293 | Hu  _et al._ (2009)
EA-Mg | Pna21 | 292 | Shang  _et al._ (2014a)
DMA-M (M = Co, Mn, Zn) | Cc | 285 | Liu  _et al._ (2020b)
MA-M (M = Mn, Zn, Co, Ni) | Pnma | 135, 290, 290, 100 | Shang  _et al._ (2012); Mazzuca  _et al._ (2018); Pato-Doldán  _et al._ (2016)
HONH3M (M=Co, Mg, Mn, Ni, Zn, Fe) | P212121 | 293 | Liu  _et al._ (2012)
FA-Mn | C2/c | 110 | Maczka  _et al._ (2014b)
HAZ-M (M = Mn, Zn, Co) | Pna21 | 110, 110, 298 | Chen  _et al._ (2014); Gunatilleke  _et al._ (2017)
HAZ-M (M = Mg) | P212121 | 110 | Chen  _et al._ (2014)
NH4M (M=Co, Fe, Mn, Zn) | P63 | 110 | Xu  _et al._ (2011)
Gua-M (M = Cd) | R$\overline{3}$c | 150 | Collings  _et al._ (2016b)
Gua-M (M = Zn) | Pnna | 120 | Asaji  _et al._ (2019)
NH4M (M = Cu) | P212121 | 120 | Peksa  _et al._ (2020)
NH4M (M = Mg) | P63 | 93 | Shang  _et al._ (2014b)

Table 4 lists the hybrid formates that we have investigated and the associated
experimental references from which the structures have been retrieved along
with temperatures at which the structures were recorded. The bottom part of
the panel lists some of the HOIPs in the same family, which did not become
part of this study. The experimental structures were used to initialize DFT
based computations as implemented in VASP package Kresse and Hafner (1993);
Kresse and Furthmuller (1996); Kresse and Furthmüller (1996); Kresse and
Joubert (1999). Technically, all experimental structures were first fully
relaxed using Perdew-Burke-Ernzerhof (PBE) version of the generalized gradient
approximation for exchange correlation functional Perdew  _et al._ (1996). In
order to model hydrogen bonds, we used dispersion corrections of zero-damping
D3 Grimme (2006); Grimme  _et al._ (2010), which was previously shown to
provide good agreement with experimental structures Ghosh  _et al._ (2021,
2022a, 2020); Kingsland  _et al._ (2021); Kashikar  _et al._ (2021). The
electron-ion interactions are treated with the projected augmented wave (PAW)
potentials Blöchl (1994). We used the plane wave cutoff energy in range
700-850 eV and non-Gamma centered k-point mesh which corresponds to k-point
densities in range 0.19 – 0.57 Å-1. Note that k-point density and cutoff
energy for each material are given in Table S1 in supplementary material. The
Hubbard correction as proposed by Dudarev et al. Dudarev  _et al._ (1998) is
introduced to account for the Coulomb repulsion between localized d-electrons
of transition metals. Unit cell parameters and atomic positions were relaxed
until stress and forces are less than 0.1 GPa and 1 meV/$\AA$, respectively.
The energy convergence criterion for self-consistent calculations was 10-6 eV.
The crystal polarization is evaluated by the Berry phase method developed by
King-Smith and Vanderbilt King-Smith and Vanderbilt (1993); Vanderbilt and
King-Smith (1993). We computed the intrinsic piezoelectric constants $e_{ij}$
and $d_{ij}$ (in matrix notations) defined as the linear response of the
polarization to the applied strain and stress, respectively. The $d_{ij}$
coefficients were obtained from $d_{ij}=e_{ik}(C^{-1})_{kj}$, where $C$ is the
single crystal elastic constant matrix. The constants $e_{ik}$ and $C_{ij}$
were computed using finite difference method as implemented in VASP Le Page
and Saxe (2002). Hubbard U were employed for transition metal atoms. The
following values computed using the linear response ansatz of Cococcioni et al
Cococcioni and de Gironcoli (2005) from PAW approach in VASP were utilized:
6.5 eV for Mn, 7.2 eV for Fe , 4.6 eV for Co and 5.1 eV for Ni.

##  II Results and Discussion

Table 5: Structural properties and polarizations. Ground state magnetic
ordering of magnetic materials is in parenthesis, where FM and AFM denote
ferromagnetic and antiferromagnetic orderings, respectively. The data in
paranthesis give experimental values taking from the reference given in the
last column.

material | space group | a(Å) | b(Å) | c(Å) |  $\beta$ (o) |  V (Å3) | $\mu_{B}$ |  P ($\mu$C/cm2) | ref
---|---|---|---|---|---|---|---|---|---
Gua-Mn (AFM) | Pnna | 8.42 (8.52) | 11.91 (11.98) | 9.12 (9.06) | 90 | 915 (925) | 4.7 | non-polar | Hu  _et al._ (2009)
Gua-Fe (AFM) | Pnna | 8.35 (8.42) | 11.78 (11.85) | 8.98 (8.95) | 90 | 883 (892) | 3.8 | non-polar | Hu  _et al._ (2009)
Gua-Co (AFM) | Pnna | 8.28 (8.33) | 11.64 (11.75) | 8.99 (8.91) | 90 | 866 (873) | 2.8 | non-polar | Hu  _et al._ (2009)
Gua-Ni (AFM) | Pnna | 8.24 (8.26) | 11.63 (11.64) | 8.91 (8.83) | 90 | 853 (850) | 1.8 | non-polar | Hu  _et al._ (2009)
Gua-Cu (AFM) |  Pna21 | 8.50 (8.52) | 9.07 (9.03) | 11.27 (11.35) | 90 | 869 (874) | 0.6 | (0, 0, 0.21) (0.11–0.37) | Hu  _et al._ (2009); Stroppa  _et al._ (2011); Tian  _et al._ (2015)
Gua-Zn | Pnna | 8.27 (8.35) | 11.66 (11.73) | 8.99 (8.91) | 90 | 868 (872) | 0.0 | non-polar | Hu  _et al._ (2009)
HONH3-Mn (AFM) |  P212121 | 7.80 (7.81) | 8.04 (7.96) | 13.06 (13.17) | 90 | 819 (819) | 4.5 | non-polar | Liu  _et al._ (2012)
HONH3-Co (AFM) |  P212121 | 7.67 (7.68) | 7.82 (7.76) | 13.00 (13.02) | 90 | 780 (776) | 2.7 | non-polar | Liu  _et al._ (2012)
HONH3-Ni (AFM) |  P212121 | 7.59 (7.62) | 7.98 (7.78) | 12.80 (12.73) | 90 | 775 (755) | 1.8 | non-polar | Liu  _et al._ (2012)
HONH3-Fe (AFM) |  P212121 | 7.70 | 8.00 | 13.05 | 90 | 802 | 3.6 | non-polar |
HONH3-Zn |  P212121 | 7.65 (7.69) | 7.83 (7.74) | 13.18 (13.02) | 90 | 790 (775) | 0.0 | non-polar | Liu  _et al._ (2012)
HONH3-Mg |  P212121 | 7.67 (7.69) | 7.89 (7.79) | 12.73 (12.86) | 90 | 770 (770) | 0.0 | non-polar | Liu  _et al._ (2012)
HAZ-Co (AFM) |  Pna21 | 8.63 (8.65) | 7.74 (7.76) | 11.46 (11.55) | 90 | 765 (776) | 2.7 | (0, 0, 2.81) (2.61 at 405 K) | Gunatilleke  _et al._ (2017)
HAZ-Mn (FM) |  Pna21 | 8.99 (8.93) | 7.83 (7.82) | 11.66 (11.69) | 90 | 820 (817) | 4.7 | (0, 0, 2.51) (3.58 at 110 K) | Chen  _et al._ (2014)
HAZ-Zn |  Pna21 | 8.65 (8.66) | 7.75 (7.72) | 11.49 (11.48) | 90 | 771 (768) | 0.0 | (0, 0, 2.59) (2.6–3.48 at 0–110 K) | Chen  _et al._ (2014); Svane and Walsh (2017)
HAZ-Mg |  Pna21 | 8.87 | 7.63 | 11.45 | 90 | 775 | 0.0 | (0, 0, 2.90) (3.44 at 400 K) | Chen  _et al._ (2014)
HAZ-Mg |  P212121 | 8.00 (7.89) | 13.90 (13.75) | 7.28 (7.38) | 90 | 809 (802) | 0.0 | non-polar | Chen  _et al._ (2014)
NH4-Co (AFM) |  P3 | 12.54 (12.59) | 12.54(12.59) | 8.34 (8.22) | 90 | 1136 (1128) | 2.6 | (0, 0, 2.35) | Xu  _et al._ (2011)
NH4-Fe (AFM) |  P63 | 12.58 (12.62) | 12.58 (12.62) | 8.57 (8.36) | 90 | 1174 (1153) | 3.8 | (0, 0, 2.41) | Xu  _et al._ (2011)
NH4-Mn (AFM) |  P63 | 12.55 (12.67) | 12.55 (12.67) | 8.71 (8.54) | 90 | 1189 (1187) | 4.5 | (0, 0, 2.45) | Xu  _et al._ (2011)
NH4-Zn |  P63 | 12.56 (12.59) | 12.56 (12.59) | 8.38 (8.20) | 90 | 1144 (1126) | 0.0 | (0, 0, 2.43) (0.93–1.03 at 120–163 K) | Xu  _et al._ (2011, 2010, 2011); Maczka  _et al._ (2014a)
MA-Co (AFM) | Pnma | 8.18 (8.28) | 11.67 (11.67) | 8.28 (8.15) | 90 | 790 (789) | 2.8 | non-polar | Mazzuca  _et al._ (2018)
MA-Co (AFM) |  P21/c | 8.25 (8.18) | 11.69 (11.67) | 8.20 (8.15) | 93.6 (91.9) | 789 (789) | 2.8 | non-polar | Mazzuca  _et al._ (2018)
MA-Mn (AFM) | Pnma | 8.39 (8.68) | 11.93 (11.95) | 8.42 (8.17) | 90 | 843 (847) | 4.7 | non-polar | Shang  _et al._ (2012)
MA-Mn (AFM) |  P21/c | 8.41 | 8.47 | 14.12 | 121.9 | 853 | 4.7 | non-polar |
MA-Ni (AFM) | Pnma | 8.15 (8.18) | 11.62 (11.52) | 8.24 (8.08) | 90 | 781 (762) | 1.8 | non-polar | Pato-Doldán  _et al._ (2016)
MA-Zn | Pnma | 8.31 (8.41) | 11.69(11.71) | 8.17 (8.10) | 90 | 794 (798) | 0.0 | non-polar | Shang  _et al._ (2012)
MA-Zn |  P21/c | 8.27 | 13.89 | 8.26 | 122.4 | 802 | 0.0 | non-polar |
DMA-Co (AFM) | Cc | 14.10 | 8.24 | 8.62 | 121 | 859 | 2.7 | (6.93, 0, 2.70)(0.3 at 125 K) | Yu  _et al._ (2020)
DMA-Mn (AFM) | Cc | 14.35 (14.35) | 8.39 (8.32) | 8.86 (8.88) | 120.3 | 922 (912) | 4.7 | (7.10, 0, 2.48) (0.3–2.7 at 150–184 K) | Liu  _et al._ (2020b, a); Ma  _et al._ (2017); Wang  _et al._ (2013); Hu  _et al._ (2018)
DMA-Zn | Cc | 14.13 | 8.24 | 8.64 | 120.71 | 865 | 0.0 | (7.30, 0, 2.66) (0.45 at 125 K) | Simenas  _et al._ (2016)
EA-Mg |  Pna21 | 8.82 (8.90) | 8.12 (8.12) | 11.73 (11.81) | 90 | 840 (853) | 0.0 | (0, 0, 1.45) (3.34 at 93 K) | Shang  _et al._ (2014a)
FA-Mn (AFM) | C2/c | 13.12(13.44) | 8.86(8.68) | 8.23(8.41) | 118.1 | 859(865) | 4.7 | non-polar | Maczka  _et al._ (2014b)

###  II.1 Structure

The ground state structural parameters are reported in Table 5. Comparison
with experimental data, where available, was also provided in the table. We
find that, in most cases, the lattice parameters are within 1% of experimental
(see Supplementary material, Table S1). The pictorial representation of how
experimental lattice parameters compare with computational ones is given in
Fig. 1. The figure reveals good agreement between experiment and computations.
We thus conclude that our computational approach provide reliable structural
predictions. The ground state structures are available from Ref.git (2023).

Note, that we also augmented our list of HOIPs with the following structures
DMA-Zn, DMA-Co, HONH3-Fe and Pna21 phase of HAZ-Mg, which so far have not been
reported experimentally. Such structures were obtained by replacing Mn in DMA-
Mn with Zn or Co, Mn in HONH3-Mn with Fe, and Zn in HAZ-Zn with Mg, followed
by full structural relaxation. These hypothetical structures are underscored
in Table 4. Majority of the fully relaxed HOIPs structures retained their
experimental space groups. However, there were some exceptions. The
experimental structures of MA-Co are available in both Pnma and P21/c phases
whereas experimental structures of MA-(Mn,Zn) are available only in Pnma
phase. Our computations predicted that MA-(Mn,Zn,Co) are mechanically unstable
in Pnma phase while P21/c phase of MA-Co is mechanically stable. To ensure
mechanical stability of MA-(Mn,Zn) we deformed the Pnma structure along the
eigenvector associated with negative value of $C_{44}$ and subjected such
deformed structure to full structural relaxation, which resulted in P21/c
structure. It is therefore plausible, that these materials may undergo another
structural phase transition to P21/c phase at low temperatures. Both P21/c and
Pnma phases of MA-(Mn,Zn,Co) are reported in Table 5 and Ref. git (2023).
However, dielectric and mechanical properties were calculated from P21/c phase
of the structures.

Figure 1: Comparison between computational and experimental lattice
parameters. Only the structures where space group is the same for both
computations and experiment are compared.

It has been reported in previous experimental studies that at low temperature,
DMA-Zn crystallizes in space group Cc with no partial occupancy at N position
and possesses crystal structure similar to DMA-Mn Jain  _et al._ (2008b).
However, no structural file has been provided. Therefore, we have initiated
our calculation for DMA-Zn by replacing Mn with Zn in experimentally reported
DMA-Mn Liu  _et al._ (2020b). In case of HAZ-Mg, previous experimental study
reports a non-polar crystal structure P212121 Chen  _et al._ (2014), but a
recent DFT study Kieslich  _et al._ (2015) shows entropy driven effects are
responsible for stabilizing the structure in Pna21 space group. Therefore, we
have initiated our calculation for Pna21 phase of HAZ-Mg by replacing Zn with
Mg in experimentally reported Pna21 phase of HAZ-Zn Chen  _et al._ (2014).

NH4-Co experimentally is reported in P63 space group at low temperature.
However, relaxed structure in the same space group was found to be
mechanically unstable so further relaxation resulted in P3 space group.

For all structures with transition metal atoms we computed energies for
different magnetic orderings and selected the one with the lowest energy as
the ground state. It should be noted that in agreement with previous
studiesGhosh  _et al._ (2022b); Ghosh and Ponomareva (2021, 2022), we find
only very small differences in energy between structures with different
magnetic orderings. The magnetic orderings are given in Table 5.

###  II.2 Polarization

An inherent periodicity of crystal lattice makes polarization, $\mathbf{P}$, a
multivalued quantity. To overcome this challenge the polarization is typically
computed along a distortion path that connects polar structure to nonpolar
oneSpaldin (2012). However, for the case of HOIP the nonpolar high symmetry
structure is typically associated with partial occupancy and therefore cannot
be used as a reference point. One approach to construct nonpolar phase was
suggested in Ref.Hu  _et al._ (2017). Another approach is to model
experiments, where polarization is obtained from the measurement during its
reversal. Such an approach was used in Ref.Kingsland  _et al._ (2021); Ghosh
_et al._ (2022b), where the polarization reversal was achieved from creation
of inverted structure and generating a roto-distortion path between the
structure and its inversion. The inversion was applied with respect to the
inversion center of high symmetry experimental structure, where available, or
with respect to B-site. The roto-distortion path consists of distortion of the
framework and rotation of the A site molecule. We used same approach for EA-M,
HAZ-M, DMA-M as these compounds have inversion center in their high
temperature phase. Example of polarization evolution along such a path is
given in Fig. 2 (b) and Fig. 2 (c).

For Gua-Cu, rotations of the Gua molecules resulted in metallic structures,
which did not allow for polarization calculations. So we created a non-polar
structure using pseudosymmetry module of Bilbao Crystallographic
serverCapillas  _et al._ (2011) and generated a distortion path between the
polar and nonpolar structures. The polarization along such a path is given in
Fig. 2(e). For NH4-M family, the high temperature high symmetry structure is
P6322 and does not have an inversion center. In this case, we used U2 axis of
P6322 to generate the structure with reversed polarization direction.
Technically, we applied the following transformation $x\rightarrow y$,
$y\rightarrow x$ and $z\rightarrow-z$ on the Wykoff positions of NH4-M in P63
phase. Note that for NH4-Co, we report polarization for P63 phase, although it
was found to be mechanically unstable in calculations. An example of
polarization along such path is given in Fig. 2 (f).

Polarizations along the roto-distortion paths for all polar materials studied
are given in Fig. S1 of Supplementary material, while the associated
structures are given in Ref.git (2023). The Figures also report the energy
along the path. The energies are not likely to be physical as no optimization
has been performed. However, they do reveal two minima, that is double-well
potential. The typical barrier height is below 200 meV/atom which is
considered surmountableAcosta  _et al._ (2020). The comparison of our results
with experimental and computationally predicted values available from the
literature can be found in Table 2 and Fig. 3. We find excellent agreement
between our computational data and computational data from the literature.
However, there exist discrepancies with experimental data. This could be
attributed to the difference in temperature, and in some cases in phase, the
difference in the direction of measurement. In our case we report the value
along the polar direction. The data reveal that the polarization values for
the formate family is in the range of 0.2-7.8 $\mu$C/cm2 with largest values
found in DMA-M. The values are a factor of ten lower than the ones for
prototypical oxide ferroelectrics including BaTiO3 and PbTiO3 Smidt  _et al._
(2020).

Figure 2: Structural evolution along roto-distotion path schematically shown
by overlapping structures along the path (a). Variation of polarization and
energy along the path for a representative of each family, as given in the
legend (b)-(f) Figure 3: Comparison of our computational polarization values
with experimental and computational results from the literatureChen  _et al._
(2014); Hu  _et al._ (2018); Yu  _et al._ (2020); Simenas  _et al._ (2016); Xu
_et al._ (2011); Shang  _et al._ (2014a); Maczka  _et al._ (2014a); Collings
_et al._ (2016a); Shang  _et al._ (2014a). Note, ”est.” indicates that the
polarization was estimated from the separation between positive and negative
charge.

###  II.3 Piezoelectric response

The independent components of piezoelectric tensors, eij and dij, which are
allowed by symmetry are given in Table 6 and Table 7, respectively. Figure 4
provides comparative picture. For the formates with Pna21 space group, we
mostly find largest values for e15 and d15 components of the tensor. For
materials in Cc and P212121 space groups, the largest components are e35 (d35)
and e36 (d36) respectively, and can reach 0.26 C/m2 (25.36 pC/N) and 0.18 C/m2
(14.64 pC/N) in DMA-Zn and HAZ-Mg, respectively. The longitudinal coefficients
along the crystallographic directions, eii and dii, $i=$1, 2, 3, range from
0.01 to 0.14 C/m2 and 0.01 to 11.46 pC/N, respectively, with the largest of
these values belonging to DMA-Zn. The transverse coefficients eij and dij,
$i,j=$1, 2, 3 are in the range 0.00 to 0.20 C/m2 and 0.07 to 9.15 pC/N,
respectively, with the largest values belonging to DMA-Zn.

Figure 4: Comparative view of the components of the (a) piezoelectric stress
and (b) piezoelectric strain tensors

The directional dependence of the longitudinal piezoelectric stress and strain
responses was analyzed using MTexMainprice  _et al._ (2015) and is presented
in Fig. 5 and Fig. 6, respectively, for a representative material in each
family. For all materials we find response to be highly anisotropic. The
longitudinal piezoelectric stress coefficient can reach 0.22 C/m2 in DMA-Co in
$\langle\frac{1}{2},0,\frac{\sqrt{3}}{2}\rangle$ direction, while the strain
coefficient can reach 12.93 pC/N in the vicinity of $\langle 1/2,0,1\rangle$
direction. 3D visualizations of the piezoelectric stress/strain surfaces for
the rest of the materials are given in Fig. S2 and Fig. S3 in the
Supplementary material.

Thus, our data indicate that the intrinsic piezoelectric strain response in
the formate family can reach 26.7 pC/N (in HAZ-Mn) for the shear stress
component and 21.12 pC/N (in DMA-Zn along
$\langle\frac{1}{2},0,\frac{\sqrt{3}}{2}\rangle$ direction) for the
longitudinal one. DMA family exhibits the best values.

Figure 5: Piezoelectric stress surface for a representative from each family,
as indicated in the titles.

###  II.4 Dielectric response

The symmetry allowed components of the dielectric tensor are reported in Table
9. The typical value is 5. However, computations predict Gua-M to exhibit
distinctively high values, up to 100.00, comparable in order of magnitude to
dielectric constants of BaTiO3Yusoff  _et al._ (2020). The comparative view of
the dielectric constants is given in Fig. 7, which confirm that Gua-M family
exhibits largest response. The nature of such unusual response deserves
further investigation.

Figure 6: Piezoelectric strain surface for a representative from each family,
as indicated in the titles. Figure 7: Comparative view of the components of
the dielectric tensor.

###  II.5 Mechanical Properties

Mechanical properties describe the materials response to external mechanical
stimuli, such as pressure, stress or strain. The independent components of
stiffness tensors are given in Table 8. They satisfy the Bohn conditions for
the elastic stability Born (1940); Mouhat and Coudert (2014) as checked by
VASPKITWang  _et al._ (2021). The typical diagonal elements are in the range
3.3 to 127.0. Comparative view of the stiffness tensor components among all
the formates is given in Fig. 8. We computed average Bulk modulus ($B$), Young
modulus ($E$), shear modulus ($G$), Poisson’s ratio ($\nu$) and Cauchy’s
pressure (CP) for bulk polycrystals within the Hills’ approximation as
implemented in VASPKITWang  _et al._ (2021); Woldemar (1928); Hill (1952);
Reuss (1929); Anderson (1963); VAS (2011); Tian  _et al._ (2012) and reported
them in Table 10. The values compare well with the experimental results,
listed in Table 3.

Figure 8: Comparative view of the components of the stiffness tensor (Cij).

Poisson’s ratio, defined as the ratio of transverse compressive strain to
longitudinal tensile strain, and Pugh’s ratio, commonly expressed as $B/G$
ratio, can be used to characterize ductility or brittleness of crystals. The
former one typically ranges from 0.0 to 0.5. Ductility-brittleness border line
is usually drawn at Poisson ratio of 0.26 and at Pugh ratio of 1.75 Islam and
Hossain (2020); Hossain  _et al._ (2021). As shown in Fig. 9, most of the
formates studied in this work are ductile and therefore are able to withstand
large stresses and exhibit malleability.

Figure 10 shows the directional dependence of linear compressibility, defined
as linear expansion or compression of materials upon application of isotropic
pressure. Interestingly, the data predict that a few formates have negative
values (indicated by red color) and some exhibit nearly zero values. For
example, HONH3-Ni, MA-Co and NH4-Mn exhibit negative values along $\langle
0,1,0\rangle$, $\langle$-$0.6447,0,$-$0.7644\rangle$ and $\langle
0,0,1\rangle$ directions, respectively. Directional dependence of linear
compressibility for other materials are presented in supplementary material
Fig. S4. Previously negative linear compressibility was predicted for HAZ-Co,
HAZ-Mn, HAZ-Fe and NH4-Zn Ghosh and Ponomareva (2021, 2022); Li  _et al._
(2012) and explained on the basis of strut-hinge modelViswanathan (2019);
Ogborn  _et al._ (2012).

Figure 9:  Pugh (a) and Poisson (b) ratios of formates studied in this work. Figure 10: 3D plots of linear compressibility for a representative material in each family as given in the titles. Green and red colors correspond to positive and negative values, respectively. Table 6: Piezoelectric stress constants $e_{ij}$ in C/m2. Materials which do not have experimentally reported structure are underscored. | e15 | e24 | e31 | e32 | e33 |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---
Gua-Cu |  $-$0.011 | 0.069 |  $-$0.017 | 0.018 | 0.051 |  |  |  |  |
| e15 | e24 | e31 | e32 | e33 |  |  |  |  |
EA-Mg |  $-$0.261 |  $-$0.014 |  $-$0.035 |  $-$0.045 |  $-$0.015 |  |  |  |  |
| e11 | e12 | e13 | e15 | e24 | e26 | e31 | e32 | e33 | e35
DMA-Co | 0.065 | 0.071 | 0.160 | 0.077 | 0.012 |  $-$0.009 |  $-$0.020 | 0.025 | 0.107 | 0.175
DMA-Mn | 0.124 | 0.129 | 0.165 | 0.058 | 0.041 | 0.006 |  $-$0.004 | 0.073 | 0.122 | 0.187
DMA-Zn | 0.112 | 0.062 | 0.196 | 0.114 | 0.019 |  $-$0.017 | 0.020 | 0.001 | 0.141 | 0.258
| e14 | e25 | e36 |  |  |  |  |  |  |
HONH3-Mn |  $-$0.124 | 0.003 |  $-$0.211 |  |  |  |  |  |  |
HONH3-Co |  $-$0.102 |  $-$0.027 |  $-$0.031 |  |  |  |  |  |  |
HONH3-Ni |  $-$0.226 | 0.103 | 0.247 |  |  |  |  |  |  |
HONH3-Fe |  $-$0.065 |  $-$0.004 |  $-$0.135 |  |  |  |  |  |  |
HONH3-Zn |  $-$0.097 |  $-$0.130 |  $-$0.218 |  |  |  |  |  |  |
HONH3-Mg |  $-$0.005 |  $-$0.002 |  $-$0.193 |  |  |  |  |  |  |
| e15 | e24 | e31 | e32 | e33 |  |  |  |  |
HAZ-Co |  $-$0.185 | 0.138 |  $-$0.012 | 0.035 |  $-$0.054 |  |  |  |  |
HAZ-Mn |  $-$0.143 | 0.060 | 0.004 | 0.037 |  $-$0.067 |  |  |  |  |
HAZ-Zn |  $-$0.194 | 0.104 |  $-$0.021 | 0.028 |  $-$0.050 |  |  |  |  |
HAZ-Mg (Pna21) |  $-$1.172 | 0.090 |  $-$0.032 | 0.022 |  $-$0.088 |  |  |  |  |
| e14 | e25 | e36 |  |  |  |  |  |  |
HAZ-Mg (P212121) |  $-$0.190 |  $-$0.074 |  $-$0.177 |  |  |  |  |  |  |
| e14 | e15 | e31 | e33 |  |  |  |  |  |
NH4-Co | 0.001 |  $-$0.057 | 0.017 | 0.011 |  |  |  |  |  |
NH4-Fe | 0.078 |  $-$0.046 | 0.037 | 0.023 |  |  |  |  |  |
NH4-Zn | 0.055 |  $-$0.049 | 0.031 | 0.019 |  |  |  |  |  |
NH4-Mn | 0.034 |  $-$0.069 |  $-$0.013 |  $-$0.015 |  |  |  |  |  |
Table 7: Piezoelectric stress constants $d_{ij}$ in pC/N. Materials which do not have experimentally reported structure are underscored. | d15 | d24 | d31 | d32 | d33 |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---
Gua-Cu |  $-$1.05 | 7.36 |  $-$1.41 | 0.39 | 1.23 |  |  |  |  |
| d15 | d24 | d31 | d32 | d33 |  |  |  |  |
EA-Mg |  $-$40.55 |  $-$1.24 |  $-$0.14 |  $-$0.84 | 0.01 |  |  |  |  |
| d11 | d12 | d13 | d15 | d24 | d26 | d31 | d32 | d33 | d35
DMA-Co |  $-$2.77 | 0.18 | 6.52 | 7.21 | 0.59 |  $-$0.57 |  $-$6.19 | 1.19 | 7.37 | 15.05
DMA-Mn |  $-$1.29 | 1.58 | 7.11 | 8.16 | 4.04 | 2.19 |  $-$8.56 | 2.81 | 11.29 | 23.16
DMA-Zn |  $-$2.34 |  $-$1.39 | 9.15 | 11.85 | 1.05 |  $-$1.32 |  $-$7.10 |  $-$0.50 | 11.46 | 25.36
| d14 | d25 | d36 |  |  |  |  |  |  |
HONH3-Mn |  $-$8.06 | 0.33 |  $-$11.01 |  |  |  |  |  |  |
HONH3-Co |  $-$5.47 |  $-$2.35 |  $-$1.25 |  |  |  |  |  |  |
HONH3-Ni |  $-$9.65 | 9.97 | 11.38 |  |  |  |  |  |  |
HONH3-Fe |  $-$3.66 |  $-$0.75 |  $-$5.85 |  |  |  |  |  |  |
HONH3-Zn |  $-$6.03 |  $-$13.16 |  $-$10.86 |  |  |  |  |  |  |
HONH3-Mg |  $-$0.22 |  $-$0.16 |  $-$6.47 |  |  |  |  |  |  |
| d15 | d24 | d31 | d32 | d33 |  |  |  |  |
HAZ-Co |  $-$20.73 | 14.08 |  $-$1.01 | 1.54 |  $-$1.32 |  |  |  |  |
HAZ-Mn |  $-$26.72 | 6.90 |  $-$0.07 | 1.71 |  $-$2.33 |  |  |  |  |
HAZ-Zn |  $-$22.48 | 10.71 |  $-$1.41 | 1.62 |  $-$1.24 |  |  |  |  |
HAZ-Mg (Pna21) |  $-$22.40 | 8.86 |  $-$1.53 | 1.73 |  $-$1.74 |  |  |  |  |
| d14 | d25 | d36 |  |  |  |  |  |  |
HAZ-Mg (P212121) |  $-$8.91 |  $-$3.62 |  $-$14.64 |  |  |  |  |  |  |
| d14 | d15 | d31 | d33 |  |  |  |  |  |
NH4-Co | 0.49 |  $-$2.90 | 0.37 |  $-$0.20 |  |  |  |  |  |
NH4-Mn | 1.99 |  $-$4.09 |  $-$3.55 | 4.00 |  |  |  |  |  |
NH4-Zn | 3.07 |  $-$2.73 | 0.71 |  $-$0.42 |  |  |  |  |  |
NH4-Fe | 4.30 |  $-$2.47 | 1.05 |  $-$0.63 |  |  |  |  |  |
Table 8: Components of stiffness tensor $C_{ij}$ in GPa. Materials which do not have experimentally reported structure are underscored. | C11 | C12 | C13 | C22 | C23 | C33 | C44 | C55 | C66 |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Gua-Mn | 28.6 | 9.5 | 22.3 | 42.6 | 13.2 | 68.2 | 7.94 | 10.6 | 8.3 |  |  |  |
Gua-Fe | 34.6 | 11.2 | 26.5 | 46.9 | 15.3 | 80.0 | 9.3 | 11.6 | 6.4 |  |  |  |
Gua-Co | 34.5 | 12.3 | 26.7 | 49.9 | 17.0 | 82.9 | 10.3 | 13.0 | 11.9 |  |  |  |
Gua-Ni | 68.4 | 46.7 | 64.2 | 84.6 | 53.6 | 127.0 | 11.0 | 14.4 | 13.0 |  |  |  |
Gua-Cu | 29.7 | 19.7 | 14.3 | 63.6 | 17.4 | 52.8 | 9.4 | 10.6 | 8.9 |  |  |  |
Gua-Zn | 36.6 | 15.3 | 27.0 | 49.4 | 19.5 | 83.1 | 10.3 | 10.9 | 12.0 |  |  |  |
| C11 | C12 | C13 | C22 | C23 | C33 | C44 | C55 | C66 |  |  |  |
EA-Mg | 55.3 | 32.3 | 20.9 | 48.5 | 15.9 | 63.6 | 11.2 | 6.4 | 21.2 |  |  |  |
| C11 | C12 | C13 | C15 | C22 | C23 | C25 | C33 | C35 | C44 | C46 | C55 | C66
DMA-Co | 50.6 | 27.7 | 25.2 | 4.9 | 54.8 | 24.8 |  $-$3.4 | 37.8 |  $-$3.0 | 16.3 |  $-$4.8 | 15.4 | 10.1
DMA-Mn | 34.9 | 18.3 | 17.1 | 2.2 | 39.6 | 15.2 |  $-$2.3 | 26.7 |  $-$3.3 | 12.0 |  $-$3.3 | 10.8 | 8.8
DMA-Zn | 46.4 | 29.2 | 24.2 | 3.4 | 54.2 | 25.4 |  $-$2.2 | 35.7 |  $-$3.3 | 13.9 |  $-$3.3 | 12.6 | 9.9
| C11 | C12 | C13 | C22 | C23 | C33 | C44 | C55 | C66 |  |  |  |
HONH3-Mn | 36.9 | 15.2 | 7.8 | 29.3 | 8.0 | 18.6 | 15.4 | 9.0 | 19.2 |  |  |  |
HONH3-Co | 46.9 | 25.6 | 17.9 | 50.9 | 18.0 | 31.1 | 18.6 | 11.3 | 25.0 |  |  |  |
HONH3-Ni | 16.1 | 15.0 |  $-$6.6 | 59.8 | 15.3 | 16.1 | 23.4 | 10.4 | 21.7 |  |  |  |
HONH3-Fe | 46.0 | 21.8 | 13.7 | 39.1 | 12.9 | 26.2 | 17.9 | 10.8 | 23.1 |  |  |  |
HONH3-Zn | 45.6 | 25.1 | 21.4 | 44.8 | 16.5 | 38.4 | 16.0 | 9.0 | 20.1 |  |  |  |
HONH3-Mg | 50.2 | 22.5 | 14.2 | 45.6 | 14.4 | 31.2 | 23.2 | 12.6 | 29.8 |  |  |  |
| C11 | C12 | C13 | C22 | C23 | C33 | C44 | C55 | C66 |  |  |  |
MA-Ni (Pnma) | 29.7 | 12.2 | 22.2 | 56.7 | 21.2 | 46.3 | 3.6 | 17.3 | 4.9 |  |  |  |
| C11 | C12 | C13 | C15 | C22 | C23 | C25 | C33 | C35 | C44 | C46 | C55 | C66
MA-Co (P21/c) | 39.1 | 18.6 | 16.3 | 6.0 | 30.8 | 8.6 | 5.2 | 52.1 |  $-$0.8 | 6.2 | 0.5 | 3.3 | 18.5
MA-Zn (P21/c) | 55.9 | 27.1 | 24.3 | 1.8 | 33.8 | 16.4 |  $-$1.4 | 55.2 | 0.6 | 6.2 |  $-$0.7 | 8.2 | 16.1
MA-Mn (P21/c) | 46.4 | 22.7 | 19.7 | 1.3 | 26.4 | 11.6 |  $-$1.2 | 47.8 | 0.8 | 4.9 |  $-$0.3 | 7.0 | 15.5
| C11 | C12 | C13 | C22 | C23 | C33 | C44 | C55 | C66 |  |  |  |
HAZ-Co | 35.0 | 28.4 | 15.2 | 61.2 | 23.3 | 56.0 | 9.8 | 9.0 | 15.8 |  |  |  |
HAZ-Mn | 25.0 | 24.4 | 15.5 | 52.0 | 21.5 | 43.9 | 8.7 | 5.3 | 13.0 |  |  |  |
HAZ-Zn | 30.4 | 26.6 | 16.7 | 58.8 | 23.9 | 52.4 | 9.7 | 8.7 | 13.7 |  |  |  |
HAZ-Mg (Pna21) | 36.8 | 29.7 | 15.3 | 58.3 | 19.5 | 56.6 | 10.1 | 7.7 | 16.9 |  |  |  |
HAZ-Mg (P212121) | 53.3 | 28.0 | 35.1 | 49.1 | 28.5 | 42.7 | 21.3 | 20.5 | 12.1 |  |  |  |
| C11 | C12 | C13 | C15 | C22 | C23 | C25 | C33 | C35 | C44 | C46 | C55 | C66
FA-Mn | 19.6 | 11.9 | 14.1 | 5.7 | 67.2 | 35.8 |  $-$0.9 | 54.4 | 7.1 | 18.0 |  $-$2.1 | 12.5 | 3.4
| C11 | C12 | C13 | C33 | C44 |  |  |  |  |  |  |  |
NH4-Co | 38.4 | 24.1 | 33.2 | 71.5 | 20.2 |  |  |  |  |  |  |  |
NH4-Fe | 30.8 | 20.6 | 30.4 | 65.6 | 18.1 |  |  |  |  |  |  |  |
NH4-Mn | 28.3 | 17.7 | 37.7 | 63.1 | 17.0 |  |  |  |  |  |  |  |
NH4-Zn | 39.9 | 24.2 | 34.3 | 71.4 | 18.0 |  |  |  |  |  |  |  |
Table 9: Dielectric constants. Materials which do not have experimentally reported structure are underscored. | $\epsilon_{11}$ | $\epsilon_{22}$ | $\epsilon_{33}$ |  |  |
---|---|---|---|---|---|---
Gua-Mn | 5.31 | 79.19 | 34.42 |  |  |
Gua-Fe | 4.62 | 102.12 | 30.26 |  |  |
Gua-Co | 5.26 | 67.72 | 30.55 |  |  |
Gua-Ni | 4.94 | 73.59 | 35.51 |  |  |
Gua-Cu | 6.79 | 6.85 | 6.26 |  |  |
Gua-Zn | 5.33 | 5.20 | 5.68 |  |  |
| $\epsilon_{11}$ | $\epsilon_{22}$ | $\epsilon_{33}$ |  |  |
EA-Mg | 4.82 | 4.63 | 4.67 |  |  |
| $\epsilon_{11}$ | $\epsilon_{22}$ | $\epsilon_{33}$ | $\epsilon_{13}$ | Expt | Ref.
DMA-Co | 4.92 | 4.61 | 5.35 | 0.33 |  |
DMA-Mn | 4.93 | 4.53 | 5.53 | 0.41 | 3 – 6 | Sánchez-Andújar  _et al._ (2010)
DMA-Zn | 5.50 | 4.98 | 6.00 | 0.41 | 8 – 10 | Abhyankar  _et al._ (2017)
| $\epsilon_{11}$ | $\epsilon_{22}$ | $\epsilon_{33}$ |  |  |
HONH3-Mn | 5.90 | 6.04 | 5.21 |  |  |
HONH3-Co | 5.84 | 5.93 | 4.95 |  |  |
HONH3-Ni | 5.56 | 6.48 | 5.08 |  |  |
HONH3-Fe | 5.14 | 5.29 | 4.46 |  |  |
HONH3-Zn | 6.26 | 6.04 | 5.15 |  |  |
HONH3-Mg | 4.84 | 5.01 | 4.43 |  |  |
| $\epsilon_{11}$ | $\epsilon_{22}$ | $\epsilon_{33}$ | $\epsilon_{13}$ |  |
MA-Co (P21/c) | 5.21 | 5.34 | 5.89 | 0.22 |  |
MA-Zn (P21/c) | 5.17 | 5.83 | 6.06 |  $-$0.20 |  |
MA-Mn (P21/c) | 4.62 | 4.85 | 5.29 |  $-$0.13 |  |
MA-Ni (Pnma) | 5.21 | 13.69 | 5.11 |  |  |
| $\epsilon_{11}$ | $\epsilon_{22}$ | $\epsilon_{33}$ |  |  |
HAZ-Co | 4.87 | 5.05 | 5.03 |  |  |
HAZ-Mn | 4.66 | 4.84 | 4.83 |  |  |
HAZ-Zn | 5.30 | 5.41 | 5.49 |  |  |
HAZ-Mg (Pna21) | 4.31 | 4.65 | 4.50 |  |  |
HAZ-Mg (P212121) | 5.17 | 4.75 | 9.29 |  |  |
| $\epsilon_{11}$ | $\epsilon_{22}$ | $\epsilon_{33}$ | $\epsilon_{13}$ |  |
FA-Mn | 4.36 | 4.72 | 5.07 | 0.27 |  |
| $\epsilon_{11}$ | $\epsilon_{22}$ | $\epsilon_{33}$ |  |  |
NH4-Co | 5.38 | 5.38 | 6.04 |  |  |
NH4-Fe | 4.77 | 4.77 | 5.28 |  |  |
NH4-Mn | 5.51 | 5.51 | 5.94 |  |  |
NH4-Zn | 5.28 | 5.28 | 6.17 |  |  |
Table 10: Mechanical properties of formates. $B$, $E$ and $G$ are average bulk, Young and shear moduli of bulk polycrystal, respectively. B/G, $\nu$, CP and LCmin are Pugh’s ratio, Poisson’s ratio, Cauchy’s pressure and minimum linear compressibility, respectively. Materials which do not have experimentally reported structure are underscored | B (GPa) | E (GPa) | G (GPa) | B/G | $\nu$ | CP (GPa) | LCmin (TPa-1)
---|---|---|---|---|---|---|---
Gua-Mn | 23.5 | 28.2 | 10.8 | 2.2 | 0.30 | 1.5 | 2.41
Gua-Fe | 27.4 | 30.2 | 11.5 | 2.4 | 0.32 | 1.9 | 2.18
Gua-Co | 29.0 | 35.7 | 13.8 | 2.1 | 0.30 | 2.0 | 2.45
Gua-Ni | 64.1 | 40.7 | 14.6 | 4.4 | 0.4 | 35.6 | 0.05
Gua-Cu | 26.1 | 30.0 | 11.4 | 2.3 | 0.3 | 10.4 | 5.03
Gua-Zn | 30.3 | 34.2 | 13.0 | 2.3 | 0.3 | 5.0 | 2.46
EA-Mg | 33.7 | 34.2 | 12.9 | 2.6 | 0.33 | 21.1 | 6.72
DMA-Co | 32.5 | 31.3 | 11.7 | 2.8 | 0.34 | 11.4 | 6.85
DMA-Mn | 20.9 | 23.8 | 9.0 | 2.3 | 0.32 | 6.3 | 8.51
DMA-Zn | 31.7 | 28.2 | 10.4 | 3.1 | 0.35 | 15.3 | 5.36
HONH3-Mn | 15.3 | 27.5 | 11.5 | 1.3 | 0.20 |  $-$0.2 | 11.33
HONH3-Co | 26.8 | 36.9 | 14.5 | 1.8 | 0.27 | 7.0 | 7.18
HONH3-Ni | 8.5 | 21.7 | 10.1 | 0.8 | 0.08 |  $-$8.4 |  $-$188.08
HONH3-Fe | 22.0 | 34.0 | 13.6 | 1.6 | 0.24 | 3.9 | 7.63
HONH3-Zn | 28.0 | 33.8 | 13.0 | 2.2 | 0.30 | 9.1 | 7.51
HONH3-Mg | 24.5 | 41.2 | 16.9 | 1.5 | 0.22 |  $-$0.7 | 8.72
MA-Co (P21/c) | 19.9 | 21.1 | 8.0 | 2.5 | 0.32 | 12.4 |  $-$32.24
MA-Mn (P21/c) | 23.7 | 23.4 | 8.8 | 2.7 | 0.34 | 17.8 |  $-$0.15
MA-Ni (Pnma) | 27.8 | 23.1 | 8.5 | 3.3 | 0.35 | 8.6 | 1.96
MA-Zn (P21/c) | 29.7 | 27.5 | 10.2 | 2.9 | 0.35 | 20.9 | 4.22
HAZ-Co | 30.1 | 31.5 | 11.9 | 2.5 | 0.33 | 18.6 | 1.77
HAZ-Mn | 24.8 | 23.1 | 8.6 | 2.9 | 0.35 | 15.6 |  $-$1.73
HAZ-Zn | 28.6 | 28.8 | 10.8 | 2.6 | 0.33 | 16.9 | 0.82
HAZ-Mg (Pna21) | 30.2 | 31.3 | 11.8 | 2.6 | 0.33 | 19.6 | 3.28
HAZ-Mg (P212121) | 36.2 | 34.7 | 12.9 | 2.8 | 0.34 | 6.6 | 5.28
FA-Mn | 22.8 | 25.4 | 9.6 | 2.4 | 0.32 |  $-$6.1 | 1.07
NH4-Co | 33.9 | 32.0 | 11.9 | 2.8 | 0.34 | 3.9 |  $-$1.73
NH4-Fe | 29.5 | 27.8 | 10.3 | 2.9 | 0.34 | 2.5 |  $-$4.50
NH4-Mn | 18.5 | 15.1 | 5.5 | 3.4 | 0.36 | 0.7 |  $-$439.63
NH4-Zn | 34.7 | 31.5 | 11.7 | 3.0 | 0.35 | 6.2 |  $-$2.04

##  III Conclusion and Outlook

In summary, we have used DFT computations to assess structural, electric,
piezoelectric, and mechanical properties of 29 hybrid formate perovskites. We
predict that the ground state phase of most MA-M (M = Co, Mn, Zn) formates is
different from the low temperature phase reported experimentally, which
suggests additional phase transitions at very low temperatures. The
spontaneous polarizations range from 0.2 to 7.8 $\mu$C/cm2 with the largest
values being in DMA-M family. They are expected to be reversible by the
electric field as the upper estimate for the energy barrier is 200 meV/atom.
We also find polarization values often exceeding experimentally reported ones,
which we attribute to the difference in the direction of measurement. Thus,
our study could guide towards optimization of materials performance. Typical
dielectric constants are typically 5.0. Nevertheless, Gua family exhibits
outstandingly large values in range 4.6 – 102.1, which, however, need to be
further validated. Intrinsic piezoelectric strain and stress constants are in
the range 0.1 – 25.8 $\mu$C/cm2 and 0.1 – 26.7 pC/N, respectively. The
responses were also found to be highly anisotropic. Components of elastic
stiffness tensor range from 0.3 to 127.0 GPa. On the basis of Pugh and Poisson
ratio we found most of the materials to be ductile. Computations predict that
linear compressibility is highly anisotropic and many materials (e.g.
HONH3-Ni, NH4-Mn, Gua-Ni and MA-Co) exhibit either zero or even negative
values. All computational data are available from Ref. git (2023).

Our study reveals that additional investigations are needed to validate and
explain outstanding dielectric response of Gua-M formates, and large
piezoelectric response of DMA-M formates, along with the large negative
compressibility values for HONH3-Ni and NH4-Mn. Investigation on the origin of
negative and/or nearly zero values of compressibility is also required.

##  IV Acknowledgment

The work is supported by the National Science Foundation under the grant
EPMD-2029800.

##  V Data availability

The data that support the findings of this study are available from our github
repositorygit (2023) as well as from the corresponding author upon reasonable
request.

##  VI References

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Supplementary Material

Table S1: Computational details and errors associated with estimation of structural parameters. KEYS: ENCUT = Cut-Off Energy and KD = k-points Density | Error-a (%) | Error-b (%) | Error-c (%) | Error-V (%) | ENCUT (eV) | KD (Å-1)
---|---|---|---|---|---|---
Gua-Mn |  $-$1.2 |  $-$0.6 | 0.7 |  $-$1.1 | 708 | 0.33
Gua-Fe |  $-$0.8 |  $-$0.6 | 0.3 |  $-$1.1 | 850 | 0.27
Gua-Co |  $-$0.6 |  $-$0.9 | 0.9 |  $-$0.6 | 708 | 0.39
Gua-Ni |  $-$0.2 |  $-$0.1 | 0.9 |  $-$0.6 | 850 | 0.33
Gua-Cu |  $-$0.2 | 0.4 |  $-$0.7 |  $-$0.5 | 700 | 0.33
Gua-Zn |  $-$1.0 |  $-$0.6 |  $-$0.9 | 0.7 | 708 | 0.27
EA-Mg |  $-$0.9 | 0.0 |  $-$0.7 |  $-$1.6 | 700 | 0.42
DMA-Mn | 0.0 | 0.9 | -0.2 | 0.6 | 700 | 0.33
HONH3-Mn |  $-$0.1 | 1.0 |  $-$0.8 | 0.0 | 850 | 0.28
HONH3-Co |  $-$0.1 | 0.8 |  $-$0.2 | 0.5 | 850 | 0.28
HONH3-Ni |  $-$0.4 | 2.6 | 0.5 | 2.7 | 850 | 0.29
HONH3-Zn |  $-$0.6 | 1.2 | 1.2 | 1.9 | 850 | 0.57
HONH3-Mg |  $-$0.3 | 1.3 |  $-$1.0 | 0.0 | 708 | 0.29
MA-Co(Pnma) |  $-$1.2 | 0.0 | 1.6 | 0.4 | 708 | 0.34
MA-Co(P21/c) | 0.9 | 0.2 | 0.6 | 1.6 | 708 | 0.34
MA-Mn | -3.3 |  $-$0.2 | 3.1 |  $-$0.5 | 708 | 0.34
MA-Ni |  $-$0.4 | 0.9 | 2.0 | 2.5 | 708 | 0.34
MA-Zn(Pnma) |  $-$1.2 |  $-$0.2 | 0.9 |  $-$0.5 | 708 | 0.28
FA-Mn |  $-$2.4 | 2.1 |  $-$2.1 |  $-$2.5 | 700 | 0.33
HAZ-Co |  $-$0.2 |  $-$0.3 |  $-$0.8 |  $-$1.3 | 700 | 0.40
HAZ-Mn | 0.7 | 0.1 |  $-$0.3 | 0.5 | 700 | 0.34
HAZ-Zn |  $-$0.1 | 0.4 | 0.1 | 0.3 | 700 | 0.35
HAZ-Mg (P212121) | 1.4 | 1.1 |  $-$1.4 | 1.1 | 700 | 0.40
NH4-Co |  $-$0.4 | 0$-$0.4 | 0.2 | 0.5 | 850 | 0.22
NH4-Fe |  $-$0.3 |  $-$0.3 | 2.5 | 1.9 | 850 | 0.19
NH4-Mn |  $-$0.9 |  $-$0.9 | 2.0 | 0.0 | 850 | 0.22
NH4-Zn |  $-$0.2 |  $-$0.2 | 2.2 | 1.7 | 708 | 0.22
Table S2: Mechanical properties of formates. $B$, $E$ and $G$ are average bulk, Young and shear moduli of bulk polycrystal, respectively. B/G, $\nu$, CP and LCmin are Pugh’s ratio, Poisson’s ratio, Cauchy’s pressure and minimum linear compressibility, respectively. Materials which do not have experimentally reported structure are underscored | B(GPa) | E(GPa) | G(GPa) | B/G | v | CP(GPa) | ALC | LCmin(TPa-1) | LCmax(TPa-1)
---|---|---|---|---|---|---|---|---|---
Gua-Mn | 23.5 | 28.2 | 10.8 | 2.2 | 0.30 | 1.5 | 11.5 | 2.41 | 27.04
Gua-Fe | 27.4 | 30.2 | 11.5 | 2.4 | 0.32 | 1.9 | 10.2 | 2.18 | 22.32
Gua-Co | 29.0 | 35.7 | 13.8 | 2.1 | 0.30 | 2.0 | 8.5 | 2.45 | 20.87
Gua-Ni | 64.1 | 40.7 | 14.6 | 4.4 | 0.4 | 35.6 | 215.6 | 0.05 | 10.47
Gua-Cu | 26.1 | 30.0 | 11.4 | 2.3 | 0.3 | 10.4 | 5.0 | 5.03 | 25.32
Gua-Zn | 30.3 | 34.2 | 13.0 | 2.3 | 0.3 | 5.0 | 8.2 | 2.46 | 20.04
EA-Mg | 33.7 | 34.2 | 12.9 | 2.6 | 0.33 | 21.1 | 1.9 | 6.72 | 12.76
DMA-Co | 32.5 | 31.3 | 11.7 | 2.8 | 0.34 | 11.4 | 2.6 | 6.85 | 17.54
DMA-Mn | 20.9 | 23.8 | 9.0 | 2.3 | 0.32 | 6.3 | 3.2 | 8.51 | 26.90
DMA-Zn | 31.7 | 28.2 | 10.4 | 3.1 | 0.35 | 15.3 | 3.6 | 5.36 | 19.55
HONH3-Mn | 15.3 | 27.5 | 11.5 | 1.3 | 0.20 |  $-$0.2 | 3.7 | 11.33 | 41.90
HONH3-Co | 26.8 | 36.9 | 14.5 | 1.8 | 0.27 | 7.0 | 3.2 | 7.18 | 23.08
HONH3-Ni | 8.5 | 21.7 | 10.1 | 0.8 | 0.08 |  $-$8.4 |  $-$2.2 |  $-$188.08 | 405.43
HONH3-Fe | 22.0 | 34.0 | 13.6 | 1.6 | 0.24 | 3.9 | 3.7 | 7.63 | 28.34
HONH3-Zn | 28.0 | 33.8 | 13.0 | 2.2 | 0.30 | 9.1 | 2.2 | 7.51 | 16.72
HONH3-Mg | 24.5 | 41.2 | 16.9 | 1.5 | 0.22 |  $-$0.7 | 2.7 | 8.72 | 23.35
MA-Co(P21/c) | 19.9 | 21.1 | 8.0 | 2.5 | 0.32 | 12.4 |  $-$1.8 |  $-$32.24 | 59.26
MA-Mn(P21/c) | 23.7 | 23.4 | 8.8 | 2.7 | 0.34 | 17.8 |  $-$210.5 |  $-$0.15 | 32.22
MA-Ni(Pnma) | 27.8 | 23.1 | 8.5 | 3.3 | 0.35 | 8.6 | 6.3 | 1.96 | 22.48
MA-Zn(P21/c) | 29.7 | 27.5 | 10.2 | 2.9 | 0.35 | 20.9 | 11.7 | 4.22 | 26.36
HAZ-Co | 30.1 | 31.5 | 11.9 | 2.5 | 0.33 | 18.6 | 12.6 | 1.77 | 22.30
HAZ-Mn | 24.8 | 23.1 | 8.6 | 2.9 | 0.35 | 15.6 |  $-$20.0 |  $-$1.73 | 34.68
HAZ-Zn | 28.6 | 28.8 | 10.8 | 2.6 | 0.33 | 16.9 | 32.3 | 0.82 | 26.56
HAZ-Mg(Pna21) | 30.2 | 31.3 | 11.8 | 2.6 | 0.33 | 19.6 | 6.1 | 3.28 | 19.88
HAZ-Mg(P212121) | 36.2 | 34.7 | 12.9 | 2.8 | 0.34 | 6.6 | 2.3 | 5.28 | 13.56
FA-Mn | 22.8 | 25.4 | 9.6 | 2.4 | 0.32 |  $-$6.1 | 53.6 | 1.07 | 58.19
NH4-Co | 33.9 | 32.0 | 11.9 | 2.8 | 0.34 | 3.9 |  $-$9.8 |  $-$1.73 | 16.92
NH4-Fe | 29.5 | 27.8 | 10.3 | 2.9 | 0.34 | 2.5 |  $-$4.7 |  $-$4.50 | 21.31
NH4-Mn | 18.5 | 15.1 | 5.5 | 3.4 | 0.36 | 0.7 |  $-$0.9 |  $-$439.63 | 381.82
NH4-Zn | 34.7 | 31.5 | 11.7 | 3.0 | 0.35 | 6.2 |  $-$8.2 |  $-$2.04 | 16.69
Figure S1: Variation of polarization and energy along the path for (b) HAZ-Co
(c) HAZ-Mg; (d) HAZ-Zn (e) DMA-Zn (f) DMA-Mn (g) NH4-Zn (h) NH4-Mn (i) NH4-Fe
Figure S2: Piezoelectric stress surface for some of the formates studied in
this work, as indicated in the titles. Figure S3: Piezoelectric strain surface
for some of the formates studied in this work, as indicated in the titles.
Figure S4: 3D plots of linear compressibility for some of the formates studied
in this work, as given in the titles. Green and red colors correspond to
positive and negative values, respectively.