Microscopic origin and quantitative theory of the resonant quantum magnetoelectric effect at resonant quantum tunneling of magnetization in multiferroic formate metal–organic frameworks

In the multiferroic metal–organic framework [(CH3)2NH2]Fe(HCOO)3, the dielectric permittivity exhibits sharp peaks exactly at the magnetic fields where resonant quantum tunneling of magnetization (RQTM) occurs — an effect termed the resonant quantum magnetoelectric (ME) effect. A decade after its observation, the microscopic origin of this spin-to-charge transduction remains unexplained quantitatively. The linear ME tensor cannot by itself produce field-localized peaks, and the candidate channels — among them higher-order static ME coupling tensors ($\beta_{ijk}$, $\gamma_{ijk}$), spin-driven polarization mechanisms such as the inverse Dzyaloshinskii–Moriya effect (a spin–orbit-coupled mechanism in which non-collinear spin textures induce electric polarization), spin–lattice (magnetostrictive) coupling, and dynamical Einstein–de Haas-type spin–phonon processes that transfer angular momentum and torque to the lattice — have not been discriminated, computed, or directly measured. The same phenomenology has since been reproduced in a second formate framework [CH3NH3]Co(HCOO)3, in dysprosium-based ferroelectric single-molecule magnets (molecular complexes whose high-spin metal centers show slow magnetic relaxation and hysteresis), and in the oxide spin-chain compound Ca3Co2O6, always with the same mechanistic gap. The problem is to identify the coupling channel(s) responsible for the dielectric peaks at the RQTM fields of [(CH3)2NH2]Fe(HCOO)3 and to supply the quantitative microscopic description that is still missing: derive or compute the relevant coupling coefficients and reproduce the measured magnitude, field positions, temperature dependence and orientation dependence of $\Delta\varepsilon$ at the tunneling fields, explaining how the magnetic metal–formate framework couples, through the hydrogen-bond network, to the dipolar degrees of freedom of the organic A-site cations. Direct experimental tests of the dynamical channels (e.g., detection of phonon angular momentum or transient lattice torque at tunneling events) and a numerically resolved A-site-versus-framework decomposition remain open corroborating directions.

Background

A metal–organic framework (MOF) is a crystalline solid built from metal ions or clusters bridged by organic linker molecules; formate MOFs of the perovskite-like ABX3 type adopt a ReO3-connected network of corner-sharing B-site metal–formate octahedra (B = Fe2+, Co2+, Mn2+, Ni2+; X = HCOO−) with A-site ammonium cations (dimethylammonium DMA+ or methylammonium MA+) occupying the cuboctahedral cavities. In [(CH3)2NH2]Fe(HCOO)3 (DMA-Fe), the DMA+ cations order their N–H···O hydrogen bonds below $T_{\mathrm{C}} \approx 164$ K, breaking inversion symmetry and producing ferroelectric order; the Fe2+ (S = 2) sublattice orders as a weakly canted antiferromagnet below $T_{\mathrm{N}} \approx 19$ K, and below a blocking temperature $T_{\mathrm{B}} \approx 8$ K the material is magnetically phase-separated: a canted-antiferromagnetic matrix coexists with blocked single-ion quantum magnets (nanophase regions with strong Fe2+ uniaxial anisotropy). In those regions, resonant quantum tunneling of magnetization (RQTM) — coherent tunneling of the magnetization through the anisotropy barrier at avoided level crossings — produces stair-shaped magnetic hysteresis with steps at the resonant fields, most prominently near 0 and ±2.2 T at 2 K. Magnetoelectric (ME) coupling denotes any coupling between magnetic and electric order; in the standard Landau expansion of the free energy F(E,H), the lowest-order coupling is the linear ME tensor $\alpha_{ij}$, with higher-order tensors $\beta_{ijk}$ (E·H·H-type) and $\gamma_{ijk}$ (E·E·H-type) beyond it. The linear ME effect varies smoothly with field and cannot by itself generate peaks localized at discrete resonant fields; a field-localized dielectric anomaly therefore requires a coupling contribution that changes abruptly at the tunneling fields. Mechanisms of that kind are not limited to higher-order static ME coupling and dynamical spin–phonon processes: spin-driven polarization of the inverse Dzyaloshinskii–Moriya type and spin–lattice (magnetostrictive) coupling are equally candidates, and extrinsic explanations (electrode-contact or Maxwell–Wagner-type interfacial charge accumulation, artifact magnetostriction) have to be excluded by measurement controls rather than by a priori argument. One specific dynamical channel is of the Einstein–de Haas type: a tunneling event changes the spin angular momentum abruptly, the lattice experiences the compensating torque, and the associated change in phonon angular momentum — the rotational angular momentum carried by lattice vibrations, shown by Zhang and Niu to be a well-defined and experimentally accessible quantity — can, through spin–phonon interaction, alter the dielectric permittivity. Prior density-functional theory (DFT) work on the same Fe-MOF had proposed a 'hybrid' ME mechanism in which non-polar metal–formate framework distortions drive polar displacements of the A-site cations via hydrogen bonds, but stated explicitly that a quantitative ME calculation for this compound was not yet available, blocked on the missing low-temperature crystal structure in the multiferroic state. In 2016 it was discovered that the dielectric permittivity of DMA-Fe shows sharp peaks precisely at the RQTM resonant fields below $T_{\mathrm{B}}$ (absent above it) — the resonant quantum ME effect. Subsequent work reproduced the analogous effect in MA-Co [CH3NH3]Co(HCOO)3 (there termed resonant quantum magnetodielectric, RQMD, attributed qualitatively to intrinsic ME coupling via the inverse Dzyaloshinskii–Moriya mechanism — a spin–orbit-coupled mechanism in which a canted or spiral spin texture induces electric polarization — and to spin–lattice coupling, with the magnetostriction, i.e. the field-induced lattice strain, tracking the dielectric response), in Dy-based ferroelectric single-molecule magnets (molecular complexes whose high-spin metal centers show slow magnetic relaxation and hysteresis; electrical detection and modulation of single-molecule-magnet magnetism), and in the oxide spin-chain compound Ca3Co2O6 (attributed to 'finite spin units'), while direct measurement of phonon angular momentum was demonstrated in non-magnetic systems and electric-field modulation of spin resonance was demonstrated in the same Fe-MOF — yet no work has delivered a microscopic quantitative account of the effect. Throughout this statement, $\Delta\varepsilon$ denotes the field-induced change of the dielectric permittivity measured along a fixed crystallographic direction at a fixed measurement frequency, referenced to its value at a field away from the tunneling resonances; the measured direction, frequency, sign convention and normalization are those of the original experiments on each compound (Tian et al., J. Am. Chem. Soc. 138, 782 (2016), for the Fe compound; Su et al., Small 21, 2409564 (2024), for the Co compound). Likewise, 'the relevant coupling coefficients' means the coefficients required by whichever channel is identified — the symmetry-allowed components of $\beta_{ijk}$, $\gamma_{ijk}$, or of the spin–phonon coupling, in the low-temperature multiferroic phase — not a prescribed set.

Problem Statement

What is the microscopic origin of the resonant quantum magnetoelectric effect — the sharp dielectric permittivity peaks that appear exactly at the magnetic fields of resonant quantum tunneling of magnetization in [(CH3)2NH2]Fe(HCOO)3 below its blocking temperature, and are absent above it — and can it be described quantitatively? Specifically: (1) Which coupling channel(s) carry the observed field-localized signal? Candidates include, but are not limited to, higher-order static magnetoelectric coupling tensors ($\beta_{ijk}$, $\gamma_{ijk}$) — the coupling terms beyond the linear magnetoelectric tensor in the Landau free-energy expansion of the polarization in electric and magnetic fields; spin-driven polarization mechanisms of the inverse Dzyaloshinskii–Moriya type (spin–orbit-induced electric polarization arising from canted or spiral spin textures, which changes abruptly when the magnetization tunnels); spin–lattice (magnetostrictive) coupling, in which the magnetic energy released at a tunneling event is transferred to the lattice through damping and changes the permittivity; and the dynamical Einstein–de Haas channel, in which the abrupt spin angular-momentum change at a tunneling event transfers torque to the lattice, changing the phonon angular momentum (the rotational angular momentum of lattice vibrations) and, through spin–phonon interaction, the permittivity. May some other mechanism dominate instead, or several of these act together? Extrinsic explanations (electrode-contact and Maxwell–Wagner-type interfacial effects, or artifact magnetostriction) are to be ruled out by measurement controls, not by a priori argument. (2) Can a microscopic model with explicit coupling coefficients — derived from first principles, or calibrated against data other than the $\Delta\varepsilon$ peaks to be reproduced (for example magnetic anisotropy and tunneling-splitting or ESR data, structural, phonon or magnetostriction measurements) — reproduce the measured magnitude, field positions, temperature dependence (disappearance above the blocking temperature, $T_{\mathrm{B}} \approx 8$ K in the Fe compound) and orientation dependence of $\Delta\varepsilon$ at the tunneling fields, and specify the coupling pathway by which the framework couples, through the hydrogen-bond network, to the dipolar degrees of freedom of the A-site cations, so that the agreement at those fields is a prediction of the model rather than a fit to the very data it must explain? (3) Can the respective contributions of the A-site organic cations (dimethylammonium, ordered via hydrogen bonds) and the metal–formate (BX3) framework to the 'hybrid' coupling through the hydrogen-bond network expected from the DFT work be resolved quantitatively? (4) Can the dynamical channels be tested directly — e.g., by detecting phonon angular momentum or transient lattice torque at resonant tunneling events, now that direct phonon angular momentum measurement has been demonstrated in other systems? (5) What does the identified mechanism imply for using the dielectric readout as a probe of quantum tunneling of magnetization in this material? The required core of the task is sub-questions (1) and (2) — identifying the coupling channel(s) and supplying the quantitative microscopic description — while sub-questions (3)–(5) are optional corroborating and extension directions; a complete answer to (1) and (2) resolves the question.

The same phenomenology reported later in [CH3NH3]Co(HCOO)3, in Dy-based ferroelectric single-molecule magnets and in Ca3Co2O6 is recorded in the Background as evidence that the effect recurs; whether one and the same microscopic mechanism accounts for all of these classes is an open corollary of the question posed here, not part of it.

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