Electric-field switching of magnetic ground states in single pi-extended metalloporphyrin molecules via tunable superexchange: reliable critical fields and experimental realization

Lu et al. (Physical Review B 106, 064405, 2022) predicted that a static electric field applied along the edge-metal-edge axis of single pi-extended metalloporphyrin molecules (denoted TM-Pc) shifts the relative energies of the d and pi orbitals and thereby tunes the d-pi and pi-pi superexchange couplings $J_1$ and $J_0$, switching the molecular magnetic ground state between ferromagnetic, ferrimagnetic, and antiferromagnetic configurations at critical fields of roughly 0.015-0.65 V/Angstrom depending on the molecule and the electronic-structure method. The prediction is quantitatively unstable - DFT+U and hybrid HSE calculations disagree even on the zero-field ground state of V-Pc - and it has never been tested experimentally. The open problem is to make the mechanism reliable and real: determine method-robust ground states and critical fields, experimentally demonstrate reversible electric-field switching of a single molecule's magnetic ground state via superexchange tuning, and put the source's predicted critical-field-reducing molecular designs on a method-robust and experimentally validated footing. Later work has achieved electric control of molecular spins on the same phthalocyanine/porphyrin platforms by other mechanisms (bistable spin switches, transport-mediated tip exchange fields), leaving the superexchange mechanism itself open.

Background

Single magnetic molecules are among the smallest conceivable magnetic units, and controlling their spin states with electric rather than magnetic fields is a central goal of molecular spintronics: electric fields can be confined to nanometer scales, switched fast, and applied at low energy cost, whereas magnetic fields cannot be localized comparably. An isolated spin does not couple directly to an electric field, so spin-electric coupling - any mechanism by which an electric field shifts the relative energies of spin states - must arise indirectly. The mechanisms available before 2022 relied on spin-orbit coupling (the interaction between a particle's spin and its orbital motion, strong in lanthanide ions) or on the magnetic dipole moments of intrinsically polar molecules; both can modulate fine spin structure but are too weak to flip a molecular spin configuration, and extrinsic alternatives based on charge transfer between molecule and substrate are difficult to integrate into devices.

Lu et al. proposed an intrinsic mechanism strong enough to flip spin configurations: tune the magnetic superexchange couplings themselves. Superexchange is the indirect magnetic interaction between two localized spins mediated by virtual hopping of an electron through intermediate orbitals; in second-order perturbation theory its strength scales as the inverse square of the energy gap between the occupied and virtual orbitals of the hopping path. The molecules they studied - TM-Pc, the pi-extended metalloporphyrins synthesized and characterized on surfaces by Sun et al. - carry two kinds of magnetism: a central transition-metal ion with a partially filled d shell (d magnetism) and, at each zigzag molecular edge, a pi magnetism of about one Bohr magneton that had been verified by scanning tunneling spectroscopy. The magnetic states are described by the Heisenberg model

$H = J_0 \mathbf{S}_{\pi L} \cdot \mathbf{S}_{\pi R} + J_1 \mathbf{S}_{d} \cdot \mathbf{S}_{\pi L} + J_1 \mathbf{S}_{d} \cdot \mathbf{S}_{\pi R} + A_x (S_{dx})^2 + A_y (S_{dy})^2,$

in which $\mathbf{S}_{\pi L}$ and $\mathbf{S}_{\pi R}$ are the two edge pi spins, $\mathbf{S}_{d}$ is the central metal d spin, $J_0$ is the exchange constant between the two edge pi spins, $J_1$ the exchange constant between the metal d spin and each pi edge, and $A_x$ and $A_y$ are the single-ion magnetic anisotropy energies of the 3d ion for spin along the $x$ and $y$ directions, arising from spin-orbit interaction. According to the signs and magnitudes of $J_0$ and $J_1$, the model admits five ground-state types, named FM, FRI, AFM, AFM0 and FM0 in the source: ferromagnetic (FM, all spins aligned), ferrimagnetic (FRI, the two edge pi spins parallel to each other and antiparallel to the d spin, so the net moment is $|m_d - 2 \mu_B|$), and antiferromagnetic (AFM, the two edge pi spins antiparallel to each other, so that they cancel and the uncompensated central d moment $m_d$ is the net moment), plus the two states AFM0 and FM0, which the source associates with a vanishing transition-metal moment. With the source's extracted d moments ($m_d = 3 \mu_B$ for V-Pc, $m_d = 1 \mu_B$ for Co-Pc), this is why the predicted V-Pc FRI-to-AFM switch raises the total moment from 1 to 3 Bohr magnetons while the Co-Pc FM-to-AFM switch lowers it from 3 to 1. Because the d and pi orbitals sit at different heights along the molecular axis, a static electric field applied along the pi-d-pi (edge-metal-edge) direction - the source takes the field sign as positive from the porphyrin unit carrying the transition metal toward the one without it - shifts their relative orbital energies, changes the superexchange energy denominators, and thereby tunes $J_0$ and $J_1$. First-principles calculations based on density-functional theory (DFT) with the Hubbard-$U$ correction for localized d states, carried out for isolated molecules in vacuum supercells separated by more than 12 Angstrom and with spin-orbit coupling included except in the partial-density-of-states calculations, predicted ground-state switching: for V-Pc a FRI-to-AFM transition at about 0.17 V/Angstrom with the total moment changing from 1 to 3 Bohr magnetons, for Co-Pc an FM-to-AFM transition at about 0.12 V/Angstrom with the total moment changing from 3 to 1 Bohr magnetons, and much smaller critical fields in the longer porphyrin-nanotape molecules (TM-2Pc), for example about 0.015 V/Angstrom for Cr-2Pc (AFM-to-FM, total moment 6 to 4 Bohr magnetons) and about -0.05 V/Angstrom for Fe-2Pc (FM-to-AFM, total moment 4 to 2 Bohr magnetons). However, the same paper showed that the quantitative predictions are method-sensitive: calculations with the HSE hybrid functional, which mixes a fraction of exact exchange into DFT and cures part of its self-interaction error, change the zero-field ground state of V-Pc from FRI to FM and raise its critical field to about 0.65 V/Angstrom - the small orbital-energy gaps that control the superexchange are precisely where the approximations differ most. As a supporting check of its switching predictions the source also reported that $A_x$ and $A_y$ are nearly constant under the electric field within a reasonable range (details in its Supplemental Material S7), and it noted that the required field amplitudes are within current experimental capability, pointing to Supplemental Material S4 for potential tip and gate materials.

Problem Statement

The open problem is to turn the predicted superexchange-based spin-electric switching mechanism into a reliable and experimentally validated capability for single molecules. Here superexchange is the indirect magnetic interaction between localized spins, mediated by virtual hopping of electrons through intermediate orbitals; in the molecules of the source prediction it couples the central transition-metal d spin to the pi spins at the two molecular edges and the two edge spins to each other, and an electric field applied along the pi-d-pi (edge-metal-edge) molecular axis tunes these couplings by shifting the relative orbital energies. The components below serve one common objective - establishing electric-field switching of a single molecule's magnetic ground state through field-tuned superexchange as a quantitative, designable, and realizable effect. Specifically: (1) Quantitative reliability - determine the magnetic ground states and critical electric fields of the pi-extended metalloporphyrin molecules of the source prediction (the TM-Pc family, together with the longer porphyrin-nanotape molecules TM-2Pc and, as reported in the source's Supplemental Material S5, TM-3Pc) under a static electric field applied along the pi-d-pi (edge-metal-edge) direction, with controlled, quantified dependence on the electronic-structure method, replacing the current discrepancy in which density-functional theory with the Hubbard-$U$ correction (DFT+U) gives a ferrimagnetic ground state for V-Pc switching at about 0.17 V/Angstrom while the HSE hybrid functional (density-functional theory mixed with a fraction of exact exchange) gives a ferromagnetic ground state with a switching field of about 0.65 V/Angstrom. (2) Experimental realization - demonstrate for an individual molecule, in a geometry that delivers the required static field (the source suggests candidate tip and gate materials in its Supplemental Material S4), that the field reversibly switches the magnetic ground state between the ferromagnetic (FM), ferrimagnetic (FRI, net moment $|m_d - 2 \mu_B|$) and antiferromagnetic (AFM, net moment $m_d$) configurations defined in the Background, with the predicted accompanying change of the total magnetic moment, and establish that the switching is produced by field-tuned superexchange between the central d spin and the pi edge spins, the mechanism proposed in the source. (3) Design rules - put the source's predicted critical-field-reducing designs (size extension from TM-Pc to the TM-2Pc and TM-3Pc nanotapes, transition-metal substitution across V, Cr, Fe, Co, Ni, Cu and Zn, and multi-metal architectures such as the computed Cr-Fe TM1-TM2-2Pc example) on a method-robust footing and test them experimentally; the source prescribes no target value for an experimentally convenient critical field, so no particular threshold is required here, and any further reduction obtained by such re-evaluation or by new molecular designs belongs to this component. A rigorous negative resolution - a well-supported demonstration that the mechanism cannot operate in any realistic molecule-device geometry - would equally settle the problem. Progress on individual components counts as partial progress toward the common objective.

References

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