Structural, electronic and magnetic properties of metal–organic-framework perovskites [AmH][Mn(HCOO)₃]: a first-principles study

Abstract

We calculate the structural, electronic and magnetic properties of the subgroup of Metal–Organic-Frameworks (MOFs) [AmH][M(HCOO)₃] (in which AmH⁺ = organic ammonium cation, M = divalent metal ion) using density functional theory with GGA+U approximation. The optimized structures and magnetic ground states are in good agreement with available experimental results. The electronic structures of these MOFs are obtained at their magnetic ground states. Using hybrid functional method (HSE06), the band gap is 4.33 eV, 4.12 eV, 4.15 eV and 4.78 eV for NH₂NH₃⁺, HONH₃⁺, CH₃CH₂NH₃⁺ and NH₄⁺ compounds, respectively. The band gap of NH₂NH₃⁺ varies from 2.63 eV (−5% compressive strain) to 3.50 eV (+5% tensile strain) at U_eff = 4 eV. It is demonstrated that the band gap of such MOFs can be easily tuned by applying external strain and the AmH⁺ ligand for the first time. These MOFs all show insulating properties. In addition, such strain engineering may also be useful for enhancing the Neel temperature by changing the distance of magnetic Mn ions. Interestingly, Bader charge analysis indicates that AmH⁺ is fully ionic suggesting that appropriate arrangement may give rise to polar order associated with the magnetic ordering, these MOFs materials can be considered as potential multiferroics. Finally, this work reveals that both strain and chemical modification are efficient approaches for designing improved and novel MOFs for future applications in photocatalytic, optoelectronic, ferroelectric or multiferroic and electronic device.

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1. Introduction

In past decades, metal–organic framework (MOF) materials, consisting of organic ligands, metal ions and the connecting formate, have attracted considerable attention because of their multifunctional properties and potential applications, such as gas storage,¹ catalysis,² photocatalysis,³ luminescence,^4,5 magnetism,⁶ sensors^7,8 and electronic devices.^9,10 [AmH][M(HCOO)₃] (with AmH⁺ = organic ammonium cation, M = divalent metal ion) is a classical subgroup of MOFs. They are formed into frameworks of metal ions and formate HCOO⁻, in which cation ligands are located in their channels. Many MOFs of this type have been synthesized, such as [(CH₃)₂NH₂][M(HCOO)₃] (M = Mn, Co, Ni, Fe and Zn),^11–14 [NH₄][M(HCOO)₃] (M = Mn, Fe, Co, Ni, Zn and Mg),^15–18 [HONH₃][M(HCOO)₃] (M = Mn, Co, Ni, Zn and Mg),¹⁹ [NH₂NH₃][M(HCOO)₃] (M = Mn, Zn, Co and Mg)²⁰ or [AmH][Mn(HCOO)₃] (AmH = CH₃NH₃⁺, CH₃CH₂NH₃⁺, (CH₃)₂NH₂⁺ and (CH₂)₃NH₂⁺).¹⁴ [AmH][M(HCOO)₃] have also been investigated experimentally as a new type of multiferroic materials because of their structural diversity and chemical versatility. Jain et al.¹³ have found that an antiferroelectric order and a weakly ferromagnetic order coexist below 8–36 K in [(CH₃)₂NH₂][M(HCOO)₃] with M = Mn, Fe, Co and Ni. Gao et al.^18,20 have also reported multiferroic properties in the [NH₂NH₃][M(HCOO)₃] and [NH₄][M(HCOO)₃] (M = Mn, Fe, Co and Ni) systems. In the NH₄⁺ compound, the ferroelectric or antiferroelectric order is triggered by the order-disorder transition at 191–254 K. Recently, this class of MOFs can be optimized in terms of magnetic, ferroelectric and dielectric properties through mixing ammonium species and/or substituting B-site metal ions. The introduction of nonpolar CH₃NH₃⁺ into [NH₂NH₃][Mn(HCOO)₃] decreases framework distortions and phase transition temperature from 355 K to 285 K, depending on CH₃NH₃⁺ content.²¹ On the other hand, the mixed-metal ions of MOFs have been studied mainly in terms of structural, thermal, dielectric properties as well as IR, Raman and luminescence properties by Maczka et al., such as trivalent cations doped-[(CH₃)₂NH₂][M(HCOO)₃] (M = Mg, Mn and Co),²² [C₂H₅NH₃][Na_0.5Fe_0.5(HCOO)₃],²³ [(CH₃)₂NH₂][Fe^IIIM^II(HCOO)₆] (M^II = Zn, Ni, Cu, Fe, Mg),^24,25 [(CH₃)₂NH₂][Na_0.5Cr_0.5(HCOO)₃].²⁶ Recently, luminescent properties of europium or chromium-doped in such multiferroic MOFs were identified,^22,26 which may open a new route for novel potential applications in sensors and solid-state lighting devices. While the spin-canted weak ferromagnetism may originate from antiferromagnetic or ferromagnetic couplings on the corresponding metal sites connected by formate bridge, HCOO⁻. Unfortunately, the Neel temperature (T_N) of these MOFs is often very low. For instance, the highest Neel temperature obtained in [(CH₃)₂NH₂][Ni(HCOO)₃] reported so far is only about 36 K, which hinders their multiferroic applications in the future.²⁷

Theoretical simulations are mainly concentrated on ferroelectric polarization in this subclass of MOFs. Stroppa et al.^28,29 have investigated that multiferroic properties and strong magnetoelectric coupling do exist in [C(NH₂)₃][Cu/Cr(HCOO)₃] using Berry phase method based on space group analysis. It is shown that Jahn–Teller effect which is active for Cu²⁺ or Cr²⁺ ions plays an important role for getting strong magnetoelectric coupling in these MOFs. On the other hand, Sante et al.^30,31 have proposed that it is possible to tune the ferroelectric polarization by mediating the magnitude and/or the canting of the organic molecular dipoles as well as epitaxial strain. For instance, 4% compressive strain can enhance the ferroelectric polarization by more than 300% in [C(NH₂)₃]Cr(HCOO)₃. In this series of [NH₄][M(HCOO)₃], higher phase transition temperature of [NH₄][Mg(HCOO)₃] is attributed to stronger hydrogen-bonding interactions between NH₄⁺ ligands and oxygen atoms and hardly related to the local environment of the metal center.³² Considering this, the study of electronic structure of these MOFs offer some useful insights for polarization in ferroelectric materials, such as BaTiO₃ system.³³ The investigation of electronic structure may exploit their other applications in electronic device, sensor, photocatalysis. As a matter of fact, it is thus desirable to modulate the electronic structure of MOFs by modifying the organic linker, the metal center or adding guest molecule as well as external strain. Here, we consider four typical [AmH⁺][Mn(HCOO)₃] with (a) AmH⁺ = NH₂NH₃⁺, (b) AmH⁺ = HONH₃⁺, (c) AmH⁺ = CH₃CH₂NH₃⁺ and (d) AmH⁺ = NH₄⁺ and investigate their structural, electronic and magnetic properties using density functional theory. Our predictive results on these specific MOFs indicate that strain engineering can modulate the band gap and magnetic interaction which is related to Neel temperature. Bade charge analysis reveals that the polarization mechanism of these MOFs is originated from the ionic properties of the AmH⁺ ligands.

2. Computational details

All calculations are implemented using the Vienna Ab Initio Simulation Package (VASP)^34,35 with the Perdew–Burke–Ernzerhof (PBE) Gradient Generalized Approximation (GGA) functional³⁶ for the exchange–correlation. The cutoff energy is 400 eV and a 3 × 4 × 2 Monkhorst–Pack grid of k-points is used. Spin-polarized effects are considered in all simulations. Ions are relaxed towards their equilibrium positions until the Hellman–Feynman force is less than 1 meV Å⁻¹. For Mn (3d) MOFs, the strong on-site Coulomb repulsion is modified by considering the Hubbard U correction proposed by Dudarev et al.³⁷ In our simulations, an effective Hubbard U parameter U_eff (U_eff = U − J) is utilized for calculating electronic structure. We test the effects of Mn U_eff values with respect to lattice parameters and compare with the band gap of other MOFs. The total energies are obtained at different magnetic configurations in order to determine the magnetic ground states.

3. Results and discussions

3.1. Structural properties

The structures obtained experimentally using powder X-ray diffraction analysis are used as starting models for bulk [AmH⁺][Mn(HCOO)₃]^14,18–20 (8 Mn atoms in NH₂NH₃⁺; HONH₃⁺; CH₃CH₂NH₃⁺; 12 Mn atoms in NH₄⁺) and the unit cell volume and atomic positions are then fully optimized with the GGA+U method. The [NH₂NH₃⁺][Mn(HCOO)₃], [HONH₃⁺][Mn(HCOO)₃] and [CH₃CH₂NH₃⁺][Mn(HCOO)₃] compounds belong to an orthorhombic system. Both NH₂NH₃⁺ and CH₃CH₂NH₃⁺ compounds have a polar space group (Pna2₁), while space group (P2₁2₁2₁) of HONH₃⁺ compound is nonpolar. [NH₄⁺][Mn(HCOO)₃] compound crystallizes into a hexagonal symmetry, and belongs to a polar space group (P6₃). The four unit cells are shown in Fig. 1. NH₂NH₃⁺ and CH₃CH₂NH₃⁺ compounds crystallize into the same anionic perovskite-framework of [Mn(HCOO)₃] with the cubic cavities being occupied by NH₂NH₃⁺ and CH₃CH₂NH₃⁺cations, respectively. The optimized equilibrium lattice parameters of these four compounds are listed in Tables 1 and 2, along with the available experimental values published in the literature. The volume of AmH⁺ cations decreases as following: CH₃CH₂NH₃⁺ > NH₂NH₃⁺ > HONH₃⁺ > NH₄⁺. The calculated lattice constants of the three orthorhombic compounds are all smaller than their corresponding experimental values. The volume of NH₂NH₃⁺, HONH₃⁺ and CH₃CH₂NH₃⁺ are 7.5%, 10.5% and 4.2% smaller than their experimental values, respectively. However, in the hexagonal NH₄⁺ system, the calculated volume is 4.6% bigger than its experimental value. Although the relaxation process induces volume expansion or contraction, these compounds maintain its initial space group system.

Fig. 1 Schematic view of the structure of [AmH⁺][Mn(HCOO)₃] (AmH⁺ = (a) NH₂NH₃⁺; (b) HONH₃⁺; (c) CH₃CH₂NH₃⁺; (d) NH₄⁺).

Table 1 Calculated and experimental structural parameters for [AmH⁺][Mn(HCOO)₃] (AmH⁺ = NH₂NH₃⁺; HONH₃⁺; CH₃CH₂NH₃⁺; NH₄⁺)

AmH^+			a (Å)	b (Å)	c (Å)	c/a	α (°)	β (°)	γ (°)	V (Å^3)
a Ref. 20.b Ref. 19.c Ref. 14.d Ref. 18.
NH2NH3^+	Pna21	Cal.	17.24 (8.62)	7.72	11.36	1.32	90	90	90	1511.59 (755.80)
		Exp.^a	8.9319	7.82	11.70	1.31	90	90	90	816.61
HONH3^+	P212121	Cal.	15.30 (7.65)	7.74	12.40	1.62	90	90	90	1467.13 (733.57)
		Exp.^b	7.81	7.96	13.17	1.69	90	90	90	819.27
CH3CH2NH3^+	Pna21	Cal.	17.85 (8.93)	8.14	11.84	1.33	90	90	90	1720.05 (860.03)
		Exp.^c	9.09	8.22	12.03	1.32	90	90	90	897.92
NH4^+	P63	Cal.	25.68 (12.84)	12.84	8.70	0.68	90	90	120	2483.02 (1241.51)
		Exp.^d	12.67	12.67	8.54	0.67	90	90	120	1186.61

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Table 2 Calculated and experimental bond distance and bond angle for [AmH⁺][Mn(HCOO)₃] (AmH⁺ = NH₂NH₃⁺; HONH₃⁺; CH₃CH₂NH₃⁺; NH₄⁺)

		dMn–O (Å)	cis-O–Mn–O (°)	trans-O–Mn–O (°)	dMn–OCHO–Mn (Å)	dN–O (Å)	dN–H–O (Å)
a Ref. 20.b Ref. 19.c Ref. 14.d Ref. 18.
NH2NH3^+	Cal.	2.16–2.18	84.5–96.5	178.8	5.72–5.77	2.75–2.79	162.7–166.9
						2.91	106.3–113.0
	Exp.^a	2.178–2.196	86.37–94.43	176.1–179.1	5.908–5.951	2.876–2.926	164.9–173.4
						2.990–3.110	114.5–121.7
HONH3^+	Cal.	2.13–2.21	86.0–100.0	164.0–169.0	5.68–5.73	1.42	161.6–170.2
						2.72–2.85
	Exp.^b	2.15–2.24	79.87–97.17	167.58–173.80	5.945–6.071	1.405	159.1–168.2
						2.79–3.00
CH3CH2NH3^+	Cal.	2.17–2.21	87.1–95.7	175.1–178.2	5.95–6.00	2.81	121.0; 153.3; 139.1
	Exp.^c	2.177–2.218	87.68–95.07	175.05–178.22	6.037–6.155	—	175.6; 124.2; 170.7; 155.6; 139.9
NH4^+	Cal.	2.22–2.24	81.7–98.7	168.5–169.0	6.10	2.84–2.86	176.6
						3.17–3.24	100.2
	Exp.^d	2.168–2.194	81.62–97.95	169.26–169.44	6.005	2.824–2.867	171–173
						3.084–3.152	104.9–105.8

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In these framework structures, each Mn ion connects to six adjacent Mn ions, forming an octahedral spatial arrangement, through six anti–anti formates, thus constructing a topological metal–formate framework, while the ligand cations accommodate the channel sites. For NH₂NH₃⁺, the Mn–O distances are equal to 2.16–2.18 Å and the cis-O–Mn–O angles are ranging within 84.5–96.5° interval while trans-O–Mn–O angles are about 178.8°. The framework grid or cavity has Mn–OCHO–Mn edges of 5.72–5.77 Å. For CH₃CH₂NH₃⁺, the Mn–O distances are in a large range of 2.17–2.21 Å. The longer Mn–OCHO–Mn distances of 5.95–6.00 Å originate from the fact that the CH₃CH₂NH₃⁺ cation is larger than that of NH₂NH₃⁺ cation. The cation forms N–H–O and weak C–H–O hydrogen bonds with the framework. The Mn–O–CH–O–Mn linkages in the framework are parallel to the c axis as well as the two ab diagonal directions in the unit cell. After relaxation, HONH₃⁺ structure remains nonpolar with orthorhombic space group P2₁2₁2₁. It possesses a 3D anionic [Mn(HCOO)₃] chiral metal–organic framework of rare 4⁹.6⁶ topology, in which HONH₃⁺ cations are arranged, head to tail, in a zigzag manner along the b directional channel. Each octahedral Mn ion is linked to six neighboring Mn ions through the anti–anti formate, forming a trigonal prism. That is different from the octahedral spatial arrangement around Mn metals in polar NH₂NH₃⁺ and CH₃CH₂NH₃⁺ compounds. The MnO₆ octahedron has Mn–O distances of 2.13–2.21 Å with cis-O–Mn–O angles and trans-O–Mn–O angles in the interval 86.0–100.0° and 164.0–169.0°, respectively. The channel's wall consists of a bidirectional triple helix of ⋯Mn–OCHO–Mn–OCHO⋯. In the HONH₃⁺ cations, the NH₃ end points toward the HO end along the slight tilted b-axis direction with the N–O, H–N and H–O distances being equal to 1.42 Å, 1.05 Å and 1.02 Å, respectively. Each H–O (N–H) donor forms a pair of H–O (N–H)⋯O_HCOO, one short and one long distance, to the anionic metal formate framework. The N_HONH3–O_HCOO and O_HONH3–O_HCOO distances are 2.89 Å (short) and 3.14 Å (long), and 3.18 Å and 4.09 Å, respectively. The formate-bridged Mn⋯Mn distances are equal to 5.68–5.73 Å. For NH₄⁺ compound, the ab-plane is rhombus, with the NH⁴⁺ cations located at three different channels at (0,0,z), (2/3,1/3,z) and (1/3,2/3,z). Each NH⁴⁺ cation contains one apical and three basal N–H groups. Each basal N–H group forms stronger H bond to the metal formate framework, with N–O distances of 2.84–2.86 Å and N–H–O angles of 176.6°. The apical N–H group, with the N–H directing to the c axis, still forms a trifurcated acceptor-type H bond, with N–O distances of 3.17–3.24 Å and N–H–O angles of 100.2°. Compared to experimental values, our calculated structures (Tables 1 and 2) show that a good agreement is obtained to support our modeling approach.

3.2. Magnetic properties

The different magnetic configurations are schematized in Fig. 2. We calculate energy differences with respect to the ferromagnetic configuration as well as magnetic exchange interaction energies of Mn atoms for all four compounds at both Mn U_eff = 0 and 4 eV, listed in Table 3. By comparing the total energy of each compound in the four different magnetic configurations considered (i.e. antiferromagnetic (AFM) states with G-type, C-type and A-type arrangements as well as ferromagnetic configuration), it is found that NH₂NH₃⁺ and CH₃CH₂NH₃⁺ compounds are both stable in G-type AFM configuration, while A-type AFM configuration is favored in HONH₃⁺ and NH₄⁺ compounds. All the four compounds keep the same magnetic ground states at Mn U_eff = 0 and 4 eV, although the magnetic interaction strength becomes weaker with increasing of the Mn U_eff value. Previous results using density functional theory calculations obtained by Sante et al.³¹ have also suggested that CH₃CH₂NH₃⁺ compounds displays a G-type AFM configuration, which is in good agreement with our finding. Our simulations are also supported by previous experimental results showing that all the compounds exhibit antiferromagnetic properties.^14,18–20 Considering the magnetic moment for Mn ion, in case of CH₃CH₂NH₃⁺ compound, the calculated value is about 4.5 μ_B using GGA functional with Mn U_eff being zero. Within GGA+U method, the magnetic moment increases linearly with increasing of U_eff value towards the high-spin magnetic moment of Mn²⁺ ions that is 5.0 μ_B/Mn, displayed in Fig. 3(b). Similar phenomena^38,39 have also been observed in various systems like FeAs or Sr₂FeOsO₆, indicating that the GGA+U method leads to an enhancement of the magnetic moment through coulomb interactions. In contrast, Kanungo et al.³⁹ have preferred to use GGA functional instead of GGA+U to study the exchange coupling of transition metal ions arguing that it is a better choice for determining the magnetic exchange. In our work, the magnetic ground states at U_eff = 0 and 4 eV are consistent with all the four investigated compounds, see Table 3. Neutron diffraction and further magnetic measurements are desirable to verify the detailed antiferromagnetic configurations.

Fig. 2 Magnetic configurations of [AmH⁺][Mn(HCOO)₃] ((a) NH₂NH₃⁺; (b) HONH₃⁺; (c) CH₃CH₂NH₃⁺; (d) NH₄⁺).

Table 3 Calculated energy differences with respect to the ferromagnetic configuration and magnetic interaction strength of these four compounds at Mn U_eff = 0 and 4 eV as well as [NH₂NH₃][Mn(HCOO)₃] under 5% compressive and tensile strain at Mn U_eff = 0 eV. The energy unit is meV

	Mn Ueff	G-type	C-type	A-type	FM	J1	J2
NH2NH3^+	0	−118.674	−76.385	−26.890	0	19.0966	6.7225
	4	−24.820	−17.405	−8.829	0	4.3513	−2.2073
Comp005	0	−14.057	16.740	152.032	0	−4.185	−38.008
Ten005	0	−106.811	−46.537	−42.889	0	11.634	10.722
HONH3^+	0	−56.591	−42.112	−96.159	0	10.5280	24.0398
	4	−19.935	−18.315	−31.554	0	4.5788	7.8885
CH3CH2NH3^+	0	−136.265	−105.133	−55.257	0	26.2833	13.8143
	4	−24.551	−16.671	−12.985	0	4.1678	3.2463
NH4^+	0	−50.825	−78.584	−108.824	0	19.6460	27.2060
	4	−1.931	−16.689	−18.150	0	4.1723	4.5375

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Fig. 3 (a) The total density of states of [NH₃NH₂⁺][Mn(HCOO)₃] in G-type antiferromagnetic configuration at Mn U_eff = 0 eV and 4 eV; the relationship between (b) magnetic moment; (c) band gap; (d) lattice constant with increasing of Mn U_eff.

According to Goodenough–Kanamori–Anderson rules,⁴⁰ superexchange interaction of neighboring magnetic ions (Mn) via HCOO and ligand cation plays a leading role in determining their magnetic order. In NH₂NH₃⁺, the distance of Mn–HCOO–Mn in ab-plane and c-axis are nearly equal to 5.72–5.77 Å (and weakly affected, if any, by U_eff values, see Fig. 3(d)). The Mn atoms are coupled antiferromagnetically in both the ab-plane and along the c axis. Therefore, for NH₂NH₃⁺ and CH₃CH₂NH₃⁺ compounds, the ground states display a G-type antiferromagnetic (AFM) spin configuration. For NH₄⁺ compound, the Mn–HCOO–Mn along the c-axis direction has a distance of 6.10 Å, while the nearest distance of Mn–Mn in the ab-plane is 7.35 Å. So the superexchange interaction along the c-axis is of AFM type and the magnetic interaction in the ab-plane is ferromagnetic in nature, which satisfies A-type AFM configuration and similarly in HONH₃⁺ compound.

Here, we study the effective exchange interaction among Mn ions with the aim of providing insights about how to increase Neel temperature (T_N) of this class of MOFs. Based on the Heisenberg model, the magnetic configurations are usually determined by two parameters J₁ and J₂; which are the exchange interaction strengths of nearest neighbors and second nearest neighbors, respectively. J₁ and J₂ are either positive or negative depending on ferromagnetic or antiferromagnetic interactions, respectively. Molecular field theory is utilized to qualitatively estimate Neel temperature of this subgroup of MOFs where T_N is approximately given by , here n is the number of atoms per unit volume; κ is Boltzmann's constant; μ_B is Bohr magnetron; g is Lande factor; J is the total angular momentum. According to molecular field theory, T_N is related to the difference between J₁ and J₂. However, whatever the compounds considered with either G-type or A-type magnetic arrangement, T_N remains rather modest suggesting that (i) whatever the nature (antiferromagnetic or ferromagnetic) of the interaction for the second nearest neighbors this does not affect T_N and thus J₂ may be considered as negligible and (ii) J₁ remains small in agreement with the low T_N temperature reported experimentally. The Mn–OCHO–Mn superexchange magnetic interactions are related to the angles and distances between magnetic Mn sites. In case of NH₂NH₃⁺ compound, the distance of nearest neighbor Mn–Mn is 5.7 Å while that of second nearest neighbor is 7.7 Å, and the angle of Mn–OCHO–Mn is 140°. The magnetic interaction strength of NH₂NH₃⁺ compound under 5% compressive and tensile strain is displayed in Table 3. It is found that 5% compressive strain enhance the difference of exchange interaction strength of nearest neighbors J₁ and second nearest neighbors J₂ to get an increase of Néel temperature. This challenge might be addressed by applying a compressive stress by clamping mechanically the MOFs onto a substrate through strain engineering to reinforce the magnetic interactions and finally increase Neel temperature. The Neel temperature of these four MOFs is about 8 K,^14,18–20 which indicates that A-ligands offer a slight effect on Neel temperature. In contrast, the introduction of mixed metal atoms may enhance the Neel temperature. For instance, the Neel temperature of [(CH₃)₂NH₂][Fe^IIINi^II(HCOO)₆] is up to 42 K.⁴¹ As we shall see later on, such strain engineering approach is also powerful for tuning the band gap.

3.3. Electronic properties

3.3.1 Density of states. We firstly study the effect of the different A-site ligands (NH₂NH₃⁺, HONH₃⁺, CH₃CH₂NH₃⁺ and NH₄⁺) on the electronic structure of these MOFs. As traditional GGA functional neglects the on-site Coulomb interactions of transition metal (Mn) and underestimates seriously the band gap, the GGA+U method is performed in this work. As the band gap of these four MOFs are experimentally unavailable, we considered other classic MOFs (i.e. MOF-5) to obtain a suitable Mn U_eff correction for our calculations. Taking U_eff = 4 eV, the band gap of the four MOFs ranges from 3.12 eV to 3.56 eV (see Table 4) which are close to those of MOF-5 references.^42–44 Let us take the NH₂NH₃⁺ compound as an example to describe the relationship between band gap, magnetic moment, lattice constants and the effective Hubbard Mn U_eff correction, as shown in Fig. 3. In Fig. 3(b)–(d), it is shown that both band gap and magnetic moment of NH₂NH₃⁺ compound increase almost linearly with increasing of the Mn U_eff value. For examples, the band gap increases from 1.64 eV (with U_eff = 0 eV) to 3.12 eV (with U_eff = 4 eV) and to 3.84 eV (with U_eff = 9 eV); the magnetic moment reaches 4.84 μ_B with Mn U_eff = 9 eV while it is 4.66 μ_B with U_eff = 4 eV and 4.47 μ_B with U_eff = 0 eV. Similar magnetic features were observed in MnO by Franchini et al.⁴⁵ the magnetic moment of MnO is 4.67 μ_B with Mn U_eff = 6 eV, which is comparable with our results. Interestingly, the lattice constants are only slightly, if any, affected with increasing of Mn U_eff value, presented in Fig. 3(d). Fig. 3(a) displays the total density of states (TDOS) of NH₂NH₃⁺ compound at Mn U_eff = 0 eV and 4 eV, respectively. The bottom of conduction band is pushed away from the Fermi level in the TDOS when Mn (U_eff) is increased from 0 eV to 4 eV, which induces a larger band gap. In case of Mn U_eff = 0 eV, the conduction band consists of Mn 3d, C 2p and O 2p orbitals, while the valance band mainly originates from Mn 3d and O 2p orbitals. For Mn U_eff = 4 eV, the big difference in TDOS shows that Mn 3d electrons become more localized and are pushed away from the bottom of the conduction band.

Table 4 Band gap of [AmH⁺][Mn(HCOO)₃] (AmH⁺ = (a) NH₂NH₃⁺; (b) HONH₃⁺; (c) CH₃CH₂NH₃⁺; (d) NH₄⁺) at Mn U_eff = 0 eV and 4 eV as well as hybrid functional (HSE06)

Mn Ueff	NH2NH3^+	HONH3^+	CH3CH2NH3^+	NH4^+
0	1.64	1.23	1.80	1.71
4	3.12	3.03	3.27	3.56
HSE06	4.33	4.12	4.15	4.78

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Total density of states (TDOS) and partial density of states (PDOS) of the four MOFs in their antiferromagnetic configuration with U_eff = 4 eV are displayed in Fig. 4 and 5, respectively. The band gap is 3.12 eV, 3.03 eV, 3.27 eV and 3.56 eV for NH₂NH₃⁺, HONH₃⁺, CH₃CH₂NH₃⁺ and NH₄⁺ compounds, respectively. In order to obtain accurate band gap, hybrid functional method (HSE06) is performed, as shown in Table 4. In NH₂NH₃⁺, the band gap is 4.33 eV. For the case of IRMOF-1,⁴⁶ the band gap using HSE06 is 4.66 eV, larger than the experimental band gap of 3.4 eV. For MOFs, hybrid functional may overestimate the band gap. In summary, these MOFs are good insulators. Observing partial density of states, a small bandwidth in the lower part of the conduction band is observed for all MOFs, except NH₄⁺ compound. At Mn U_eff = 4 eV, these four compounds exhibit similar density of states. The Mn 3d and O 2p orbitals contribute mainly to the valence band, while the conduction band consists of C 3d and N 2p orbitals. In Fig. 5(d) of NH₄⁺, the H 1s and N 2p orbitals are weakly hybridized with the C 2p and O 2p orbitals near Fermi level, which indicates that ammonium exist weak interaction with the Mn–OCHO framework. It is clear that the C 2p orbitals have stronger hybridization of O 2p orbitals at the bottom of valance band. So HCOO⁻ is more stable than ammonium like NH₄⁺, which is easily rotated, distorted and moved under external stimulus such as high temperature, high pressure. Such results offer some insights to the mechanism of polarization and phase transitions originating from A-site ligands. Two possible methods are suggested to modulate the band gap of these MOFs: (1) our results show that the A-site ligands induce the change of electronic structure, suggesting that it is also possible to modifying the band gap through shifting the conduction band through implantation of foreign molecule.⁴⁷ (2) Doping with metal atoms may shift the valence band to the Fermi level.

Fig. 4 Total density of states of [AmH⁺][Mn(HCOO)₃] at U_eff = 4 eV ((a) NH₂NH₃⁺; (b) HONH₃⁺; (c) CH₃CH₂NH₃⁺; (d) NH₄⁺).

Fig. 5 Partial density of states of [AmH⁺][Mn(HCOO)₃] at U_eff = 4 eV ((a) NH₂NH₃⁺; (b) HONH₃⁺; (c) CH₃CH₂NH₃⁺; (d) NH₄⁺).

3.3.2 Band gap modulation with strain engineering. Next, we investigate another strategy to tune the electronic properties of NH₂NH₃⁺ compound, i.e. the application of compressive or tensile strain. Recently, thin films of MOFs^48–53 have been fabricated to study their multifunctional properties. The mismatch of the lattices between MOFs and substrates can indeed affect the mechanical, electronic, and magnetic properties of the materials. In this work, in-plane strain is applied biaxially on the orthorhombic and hexagonal unit cell of NH₂NH₃⁺ compound. The in-plane strain is defined as ε = Δa/a₀, where a₀ and Δa + a₀ are the unit cell parameters without and with external strain, respectively (ε < 0 stands for compressive strain, ε > 0 represents tensile strain). The effect of strain (from −5% compressive strain to +5% tensile strain) on the band gap at U_eff = 4 eV is plotted in Fig. 6. The Fermi energy level is marked at zero. The band gap of NH₂NH₃⁺ compound increases from 2.63 eV under −5% compressive strain to 3.50 eV under +5% tensile strain at U_eff = 4 eV. The modulation of band gap arises mainly from the shift of the conduction band (CBM), the valence band (VBM) is influenced slightly. Compared to previous studies,^54,55 the band gap of MOFs such as MIL-53, MOF-5 can be modulated through hydrostatic pressure or internal (external) pressure. In MIL-53-Al materials,⁵⁵ the bang gap increases from 3.6 eV to 4.6 eV under tensile pressure, which also indicates that tensile strain increases the band gap. Therefore, strain engineering is revealed to be an powerful tool for modifying the electronic structure by shifting the conduction band in these MOFs.

Fig. 6 Band gap, valence band (VBM), conduction band (CBM) of [NH₂NH₃][Mn(HCOO)₃] at U_eff = 4 eV under external strain.

3.4. Bader charge analysis

The investigation of Bader charges help us obtain information on electrons arrange of the compounds by revealing charge transfer, chemical state and so on. Bader charges are calculated using Bader's atoms-in-molecules theory.⁵⁶ For the four compounds (see Table 5), both the Bader charges of Mn atoms and strength of the bond are nearly same whatever the AmH⁺ ligands, which is consistent with the close distances of Mn–O bond in these four compounds. The Bader charges of H in HCOO⁻ and AmH⁺ are completely different for all the four compounds. The Bader charge of H is 0.07–0.26 e, 0.074–0.088 e, −0.03–0.21 e and 0.02–0.175 e in HCOO⁻ for NH₂NH₃⁺, HONH₃⁺, CH₃CH₂NH₃⁺ and NH₄⁺ compounds, respectively. As a result, the H atom transfers its charge to the C atom in HCOO⁻, regardless of the type of the compounds giving rise to a more covalent C–H bond. However, all the charge of H atoms in AmH⁺ are 1 e, implying that there is no charge transfer. The complete ionic bond in AmH⁺ indicates that the AmH⁺ cation is very soft and suggesting that the cations can be ordered and/or disordered using external parameters like temperature, pressure and electric field. As a consequence, the AmH⁺ cations by creating electrical dipoles may arrange into a polar state that might exhibit ferroelectric and/or antiferroelectric behavior associated with magnetic ordering, which makes these MOFs become novel multiferroic materials.

Table 5 Bader charges of [AmH⁺][Mn(HCOO)₃] (AmH⁺ = (a) NH₂NH₃⁺; (b) HONH₃⁺; (c) CH₃CH₂NH₃⁺; (d) NH₄⁺)

	HCOO^−				AmH^+
	Mn	H	C	O	N	C	O	H
NH2NH3^+	1.64	0.07–0.26	2.52–2.73	−1.79 to −1.82	−1.88	—	—	1.00
					−2.33
HONH3^+	1.66	0.074–0.088	2.70–2.74	−1.77 to −1.82	−2.12	—	−1.12	1.00
CH3CH2NH3^+	1.64	0.12, 0.21, 0.04, −0.05, −0.03	2.66–2.71	−1.77 to −1.85	−2.96	0.12	—	1.00
						0.60
NH4^+	1.63	0.02–0.175	2.67–2.78	−1.79 to −1.86	−3.178	—	—	1.00
					−3.183

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4. Conclusions

In summary, a comprehensive study is conducted to investigate structural, electronic and magnetic properties of a subgroup of MOFs using density functional theory with GGA+U method. The calculated lattice constants are in good agreement with previous results. The magnetic ground states are determined through comparing the energy difference of different magnetic configurations. Using hybrid functional method (HSE06), the band gap is 4.33 eV in NH₂NH₃⁺, 4.12 eV in HONH₃⁺, 4.15 eV in CH₃CH₂NH₃⁺ and 4.78 eV in NH₄⁺, respectively. The band gap of NH₂NH₃⁺ at U_eff = 4 eV decreases from 2.63 eV to 3.50 eV with increasing external strain (from −5% compressive to +5% tensile strain). The band gap in these four insulating MOFs is modulated via changing the AmH⁺ ligands or external strain. Bader charge analysis shows that the AmH⁺ ligands are purely ionic, which is helpful to elucidate the physical mechanism of ferroelectric polarization. We analyze magnetic interaction of Mn ions, it is expected that the low Neel temperature of these MOFs may be increased by mediating the distance of magnetic metal atoms using external strain or changing the metal atoms. Our results suggest thus that appropriate design of these MOFs may make them potential applications in novel multifunctional devices.

Acknowledgements

This work was supported by the National Science Foundation of China (NSFC No. 51372195, 41372055, 11204230 and 51390472), the Ministry of Science and Technology of China through a 973-Project (No. 2012CB619401, No. 2012CB619402, No. 2014CB644003 and No. 2015CB654903), the Fundamental Research Funds for the Central Universities (2013JDGZ03), and Grant No. IRT13034. X. J. Lou would like to thank the “One Thousand Youth Talents” program for support. Y. Liu and B. Dkhil acknowledge the China Scholarship Council (CSC) for funding Y. Liu's stay in France.

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