# First-Principles Study of Structure and Magnetism in Copper(II)-Containing Hybrid Perovskites

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{2}POO)

_{3}(A = triazolium (Trz) and guanidinium (Gua), and H

_{2}POO

^{−}= hypophosphite), and [HIm]Cu(HCO

_{2})

_{3}(HIm = imidazolium cation, ${\mathrm{HCO}}_{2}^{-}$ = formate). The triazolium hypophosphite and the formate have been suggested as possible ferroelectrics. We study the fully relaxed structures with different magnetic orderings and possible phonon instabilities. For the [Trz]Cu hypophosphite, the Trz cation is shown to induce large octahedral distortions due to the Jahn-Teller effect, with Cu-O long-bond ordering along two perpendicular directions, which is correlated with antiferromagnetic ordering and strongly one-dimensional. We find that the structure is dynamically stable with respect to zone-center distortions, but instabilities appear along high symmetry lines in the Brillouin zone. On the other hand, for the [HIm]Cu formate, large octahedral distortions are found, with large Cu-O bonds present in half of the octahedra, in this case along a single direction, and correspondingly, the magnetism is almost two-dimensional.

## 1. Introduction

_{3}perovskite topology, where the X

^{−}and/or A-site of simple perovskites are substituted by polyatomic ions, have shown fascinating ferroelectric, magnetic, and multiferroic properties [1,2,3], similar to their inorganic counterparts [4,5]. Theoretical studies of this family of compounds using density functional theory calculations [6] have matched well with experimental results, which highlights the predictive power of computation [7,8].

## 2. Methods

_{2}POO)

_{3}and 520 eV for the other two compounds; the full relaxations (atomic positions, cell shape, and volume) did not use symmetry, and stopped when the maximum residual force components were less than 0.005 eV/A. The FINDSYM program (Brigham Young University, Provo, UT, USA) [24,25] was used to find the symmetry of the optimized structure, the VESTA software for analysis and structural representation [26], and PHONOPY to post-process VASP results and calculate phonons.

## 3. Triazolium Cu Hypophosphite

#### 3.1. Magnetism

_{1}/c and subsequently replaces the Mn atoms by Cu. We performed a series of full structural optimizations with the ferromagnetic and the three antiferromagnetic orders obtained by the different combinations of up/down spins in the unit cell. These can be called A, C, and G-type antiferromagnetism (AFM) [27], considering that the Cu framework is cubic-like (even if the framework is distorted from ideal cubes by columnar shifts).

_{2}]3), and it could be understood on the basis of Jahn–Teller distortions in the CuO

_{6}octahedra. We will show in the following structural analysis that strong octahedral distortions are also present in our compound. Indications of low-dimensional magnetism were also seen in guanidinium Cu formate [29], which is consistent with first-principles calculations that have shown a dominant 1D interaction [7]. We also used the DFT-D3(BJ) method to check the influence of dispersion corrections, parametrized for GGA (U = 0), shown in the rightmost set of points of Figure 2. The FM/AFM out-of-plane difference corresponds to intermediate values of U, while the A/G and particularly C/F states are now more separated, indicating higher in-plane exchange, but dominant 1D magnetism is still seen.

_{B}, and the total moment per cell is 1 μ

_{B}/Cu for the FM state. The values are consistent with the Cu

^{2+}S = 1/2 Jahn-Teller state. Considering the cubic-like framework of Cu ions, we can use the same model as in Ref. [28] to map the density functional theory (DFT)total energies to a Heisenberg model and obtain three exchange interactions between spins: intra-chain nearest-neighbor (J

_{1}), interchain in-plane (J

_{2}), and interchain out-of-plane (J

_{3}). The interactions are shown in Table 1. (For comparison with the previous work, we follow the convention where AFM interactions are positive.) As it can be seen by comparison of the two rightmost columns, the exchange interactions in our compound are qualitatively similar to [DMA]Cu[HCO

_{2}]

_{3}for the same U value, with an AFM intrachain larger by far than the interchain interactions J

_{2}+ J

_{3}(overall weakly ferromagnetic). Therefore, similar experimental signatures of 1D magnetism should be found. The AFM intrachain spin exchange J

_{1}is slightly weaker in our case, but so is the interchain exchange (J

_{2}+ J

_{3}), such that the dimensionless ratio J

_{1}/(J

_{2}+ J

_{3}) is −38 for [Trz]Cu(H

_{2}POO)

_{3}and −35 for [DMA]Cu(HCO

_{2})

_{3}.

#### 3.2. Structure

^{3}/Mn, for the Cu compound, our calculations (after full relaxation and without use of symmetry) result in 279.29 to 285.91 Å

^{3}/Cu, depending on U

_{Cu}, with smaller volumes for larger U. For G/A-AFM orders, the changes in volume are similar and come mostly from the change of U. There is a small expansion in any case, as could be expected on the basis of a slightly larger Cu

^{2+}ionic radius relative to Mn

^{2+}[30]. Concerning the transition metal–oxygen bonds, in the synthesized Mn compound, the Mn-O bonds vary between 2.11 and 2.24 Å, with an average bond length of 2.18 Å, while for the Cu compound, our predictions for U = 0 (U = 5) show a much more distorted octahedron, consisting of a Cu-O square with “equatorial” bonds 1.95–1.99 (1.96–1.99) Å, while the other two “axial” distances are much larger, 2.62 (2.42) and 2.94 (2.76) Å. These distances are almost the same for A and G-type orders. The strong AFM coupling between ab-planes can be related to the Cu-O bonds along the c direction connecting planes, which are all the smaller equatorial ones, while in the ab-plane, half of the bonds are the long axial bonds. The very large Cu-O 2.94 (2.76) Å distance in particular, alternating direction in half of the in-plane framework edges, must considerably weaken the Cu-Cu super-exchange path through the O atoms, disrupting the formation of long-range magnetic order in the plane. For increasing U values, the two axial distances decrease, so the distortion of the Cu-O octahedra is not so large; nevertheless, the in-plane magnetic coupling is lower due to the enhanced localization.

_{1}/c, which is centrosymmetric. Using the structure found at U = 0 with the A-AFM order, we searched for possible dynamical instabilities using the VASP software by calculating the Hessian matrix (with central differences and a step size of 0.015 Å) and corresponding zone-center (Γ point) vibrational frequencies. The 297 (3N−3, where N = 100 is the number of atoms) non-translational modes are all real, showing that the P2

_{1}/c structure is stable with respect to zone-center distortions, and thus, this compound will likely not be a proper ferroelectric. For example, the five lowest mode frequencies are 34, 46, 47, 48, and 51 cm

^{−1}. However, this does not exclude other possibilities such as hybrid improper ferroelectricity [31], where the polarization would be induced by other modes. Therefore, we also calculated phonons in other regions of the reciprocal space, using density functional perturbation theory to calculate the force constants [32] and then post-processing using the PHONOPY code [33] to calculate the phonon dispersion in high symmetry lines of the Brillouin zone. The resulting phonon band structure plot is shown in Figure 3. Instabilities are seen in many lines in the Brillouin zone, which are shown as negative frequencies in the plot, but with small magnitudes (≈33 cm

^{−1}). Although a more extensive check is in order here, the calculations are already too computationally expensive using the original cell.

_{2}(Z–C

_{2}and C

_{2}–Y

_{2}directions) there is one more, with six total imaginary modes.

_{2}), and constructed modulated supercell structures with the displacements given by weighted eigenvectors. Then, these structures were optimized, as before, and the symmetry and energies of the optimized structures were analyzed. For the Z point, the modulated structure, as well as the optimized final structure, have P1 symmetry, although it remains P2

_{1}/c within tolerances of 10

^{−3}Å in lattice and 10

^{−2}Å in atomic positions or larger, which shows that the distortions are small. The energy lowering with respect to the initial P2

_{1}/c structure is only a 0.30 meV/formula unit. For the D and E points, the initial modulated and optimized structures have $\mathrm{P}\overline{1}$ symmetry (for D, the structure remains P2

_{1}/c symmetry for tolerances 10

^{−3}Å in lattice and 10

^{−2}Å in atomic positions or larger; for E, the structure remains P2

_{1}/c symmetry for tolerances 10

^{−4}Å in lattice and 10

^{−2}Å in atomic positions or larger), and the energies are lower by 1.00 and 0.82 meV/Cu, respectively. These energy and structural differences are indeed small but due to the computational cost of the calculations, we did not explore this point further.

## 4. Guanidinium Cu Hypophosphite

_{2})

_{3}

^{±}) at the A site. For this system, we also started with the structure from the Mn analogue synthesized by Wu et al. [9]. In this case, we used the low temperature $P\overline{1}$ structure [9]. Due to the low symmetry, the only possible symmetry-lowering distortion is the break of inversion symmetry. We considered the FM and the three AFM states supported by the four Cu atoms in the triclinic unit cell. In this case, the + + − − state (where + (−) refers to the sign of projection of the spin moment on a given Cu atom along the quantization axis, considered as a z-axis, and the order corresponds to Cu atoms with z coordinates approximately 0.25, 0.75, 0.5, and 0), corresponds to the G-AFM alternating state for the pseudo-cubic framework, while the + − + − and − + + − cases correspond to double-layered AFM states for ferromagnetic (001) triclinic planes (corresponding to (111) pseudo-cubic planes), as shown in Figure 4. Furthermore, the initial calculations in some cases converged to a ferrimagnetic state of the type + + − +, suggesting that this is also a competing state; therefore, we also included it in our analysis.

## 5. Imidazolium Cu Formate

#### 5.1. Magnetism

_{1}/c) as [Trz]Mn(H

_{2}PO

_{2})

_{3}, but the HIm cations order such that there are now two crystallographically inequivalent cations and metal centers in 4e Wyckoff positions, and thus, the cell is approximately doubled with respect to the previous hypophosphites.

_{2}is the in-plane interaction, while J

_{1}is the vertical out-of-plane interaction, and J

_{3}is the diagonal one. Table 3 shows the calculated exchange interactions. As seen for [Trz]Cu(H

_{2}PO

_{2})

_{3}, the main exchange interaction decreases with U. In the dimethylammonium Cu formate case, the DFT values were much closer to experiment for the higher U values calculations [28], which indicates that for this Cu compound, the U = 5 eV results, with overall ≈75 K AF interaction, are also more reasonable than U = 0 eV. In this case, the degree of 2D magnetism can be estimated by J

_{2}/(J

_{1}+ J

_{3}) = −22 for U = 5 eV, i.e., the AFM in-plane interaction is much larger than the out-of-plane ones (overall weakly ferromagnetic). Note that there are two times more octahedral in-plane than out-of-plane bonds, so the overall exchange interactions, here dominated by J

_{2}(in-plane), are of similar magnitude to the ones in the hypophosphite (where the dominant out-of-plane J

_{1}= 780 K for U = 0 eV and 141 K for U = 5 eV, approximately double the in-plane interactions here), and to the ones in the previously studied DMA Cu formate (J

_{1}= 214 K for U = 5 eV) [28].

#### 5.2. Structure

_{1}polar space group for the ferromagnetic calculation, U = 4 eV for small tolerances (10

^{−3}Å for lattice and position parameters), while increasing the positions tolerance (to 10

^{−2}Å), the centrosymmetric P2

_{1}/c space group is found instead. Nevertheless, this indicated the possible stability of the polar phase. In order to quantify a possible energy lowering, we performed relaxations with the symmetry constraints of both P2

_{1}and P2

_{1}/c space groups (using the initial structures found with different tolerances) to well converged values for the more stable magnetic orders (C and G cases), with U = 4 eV. However, after converging the calculations to more stringent parameters (energy cutoff of 600 eV and forces threshold of 2 meV/fu), the energy difference is found to be 0.03 meV/fu for the C magnetic order, meaning that P2

_{1}and P2

_{1}/c are almost degenerate in energy, and it is also almost vanishing for the G order, −0.006 meV/fu. In this case, we could not try to calculate phonons, due to the larger size of the system.

## 6. Conclusions

_{1}/c space group of the Mn analogues, which suggests these compounds are not ferroelectric. We have shown that [Trz]Cu hypophosphite and [HIm]Cu formate exhibit strong, almost ideal 1D and 2D magnetism, respectively, highlighting the role of strong octahedral distortions in combination with different periodic orderings of the organic cations. We hope that our study motivates the synthesis and characterization of these compounds.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Unit cells of the considered materials. (

**a**) [Trz]Cu(H

_{2}POO)

_{3}, with space group P2

_{1}/c. (

**b**) [Gua]Cu(H

_{2}POO)

_{3}, with space group $P\overline{1}$. (

**c**) [HIm]Cu(HCO

_{2})

_{3}, with space group P2

_{1}/c. Color code: The orange octahedra are Cu centered, with oxygen (black) at the vertices; P—gray; C—brown; H—white; N—blue.

**Figure 2.**[Trz]Cu(H

_{2}POO)

_{3}GGA+U total energies, as a function of U, and GGA+D3(BJ) total energies, relative to the ferromagnetic state.

**Figure 3.**[Trz]Cu(H

_{2}POO)

_{3}phonon band structure (THz) for high-symmetry directions of the Brillouin zone.

**Figure 4.**Description of the antiferromagnetism (AFM) orders supported by the triclinic unit cell (dark lines). (

**a**) + + − − (G-AFM type) order. (

**b**) + − + −, and (

**c**) − + + −: double-layered AFM orders, with (001) FM triclinic planes (equivalent to (111) pseudocubic planes).

**Figure 5.**[Gua]Cu(H

_{2}POO)

_{3}GGA+U total energies relative to the ferromagnetic state as a function of U.

**Figure 6.**[HIm]Cu(HCO

_{2})

_{3}GGA+U total energies as a function of U, relative to the ferromagnetic state.

**Table 1.**Spin exchange interactions (in K) for H = 1/2∑

_{ij}J

_{ij}S

_{i}S

_{j}, with S = 1/2, calculated with GGA, GGA+U, GGA+D3(BJ), and comparison with another quasi-1D Cu MOF.

U = 0 eV | D3 | U = 5 eV | [DMA]Cu(HCO_{2})_{3} ^{1} | |
---|---|---|---|---|

J_{1} | 780 | 533 | 141 | 214 |

J_{2} | −18.6 | −36 | −6.8 | −8.8 |

J_{3} | 7.8 | 5.9 | 3.1 | 2.7 |

^{1}U = 5 eV [27].

**Table 2.**Structural parameters a, b, c (Å), β (°), Volume V (Å

^{3}/fu), and Cu–O equatorial and axial distances (Å) for different approximations: GGA, GGA+D3(BJ), and GGA+U (U

_{Cu}= 5 eV).

Structural Parameter | GGA | D3 | U = 5 eV |
---|---|---|---|

a | 9.88 | 9.58 | 9.86 |

b | 9.18 | 8.54 | 9.04 |

c | 12.68 | 12.66 | 12.73 |

β | 99.47 | 99.14 | 99.30 |

V | 285.91 | 255.45 | 279.29 |

Cu-O eq. | 1.95–1.99 | 1.95–1.99 | 1.96–1.99 |

Cu-O ax. | 2.56 & 2.95 | 2.39 & 2.63 | 2.49 & 2.76 |

**Table 3.**Spin exchange interactions (in K) for H = 1/2∑

_{ij}J

_{ij}S

_{i}S

_{j}, with S = 1/2, calculated with GGA and GGA+U.

U = 0 eV | U = 5 eV | |
---|---|---|

J_{1} (out-of-plane) | −11.7 | −1.5 |

J_{2} (in-plane) | +307.3 | +77.9 |

J_{3} (out-of-plane) | 2.5 | −1.9 |

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**MDPI and ACS Style**

Gonçalves, J.N.; Phillips, A.E.; Li, W.; Stroppa, A. First-Principles Study of Structure and Magnetism in Copper(II)-Containing Hybrid Perovskites. *Crystals* **2020**, *10*, 1129.
https://doi.org/10.3390/cryst10121129

**AMA Style**

Gonçalves JN, Phillips AE, Li W, Stroppa A. First-Principles Study of Structure and Magnetism in Copper(II)-Containing Hybrid Perovskites. *Crystals*. 2020; 10(12):1129.
https://doi.org/10.3390/cryst10121129

**Chicago/Turabian Style**

Gonçalves, João N., Anthony E. Phillips, Wei Li, and Alessandro Stroppa. 2020. "First-Principles Study of Structure and Magnetism in Copper(II)-Containing Hybrid Perovskites" *Crystals* 10, no. 12: 1129.
https://doi.org/10.3390/cryst10121129