The research field of magnetic frustration is dominated by triangle based lattices but exotic phenomena can also be observed in pentagonal networks. The Fe³⁺ ions in Bi₂Fe₄O₉ materialize the first analogue of a magnetic pentagonal lattice [1]. The unit cell contains two different sites of four iron atoms each, which have different connectivities with the other irons (three or four neighbours for Fe₁ and Fe₂ respectively), and that form a lattice of pentagons. Because of its odd number of bonds per elemental brick, this lattice is prone to geometric frustration. The compound magnetically orders around 240 K: the resulting spin configuration on the two sites is the same, i.e. two orthogonal pairs of antiferromagnetic spins in a plane, with a global rotation between the two sites Fe₁ and Fe_2. This peculiar magnetic structure, which is the result of the complex connectivity and magnetic frustration, has opened new perspectives in the field of magnetic frustration.

In this original compound, we have measured the spin wave excitations in the magnetically ordered state by inelastic neutron scattering. The measurements have revealed an unconventional excited state related to local precession of pairs of spins. The confrontation of the experimental results with spinwave calculations allowed to determine the Hamiltonian of the system and shows a hierarchy of the interactions. This leads to a paramagnetic state constituted of strongly coupled antiferromagnetic pairs of spins (materializing isolated dimers) separated by much less correlated spins. This produces two types of response to an applied magnetic field associated with the two nonequivalent Fe sites, as observed in the magnetization density distributions obtained using polarized neutrons.

[1] E. Ressouche, V. Simonet, B. Canals, M. Gospodinov, V. Skumryev, Phys. Rev. Lett. 103, 267204 (2009)

Crystal electric fields play an essential role in shaping the local electronic density of ions. For ions with strong spin-orbit coupling such as rare earths, this also results in defining the single-ion magnetic moment properties. Therefore, the study of crystal electric field levels is a common approach to establish the fundamental building blocks of the magnetic Hamiltonian. Indeed, the moment anisotropy and the single-ion wavefunction provide crucial information to describe complex magnetic materials such as spin liquids and multipolar systems [1,2]. In the first part of my presentation, I will illustrate this approach on the frustrated magnet SrDy₂O₄. This material exhibits two inequivalent zigzag chains of magnetic ions. The combination of low dimensionality and frustration inhibits long range order and only short range magnetic correlations are observed down to 60 mK [3]. Domain walls in the chains decay slowly and interchain interactions ultimately lead to their freezing, leading to a weakly fluctuating short range order [4]. The understanding of this complex behaviour could only be achieved from the knowledge of the moment anisotropies, established from the analysis of crystal field electric levels.

Above, we took advantage of strong spin-orbit coupling to determine magnetic properties by studying electric ones, i.e. the effect of crystal electric fields. This strong coupling between the electric charge and magnetic moment can also be used in the other direction: using magnetism to learn more about electric effects beyond the single-ion properties. As spin waves are collective excitations of the magnetic moments, the local charge densities can also sustain collective modes. Taking again the magnetic insulator SrDy₂O₄ as an example, I will demonstrate that neutron spectroscopy can measure these electric waves and that this observation is facilitated by the material’s magnetism. Interestingly, our results indicate that electric interactions dominate the magnetic interactions in this case, although they remain hidden to most measurement techniques. This observation encourages a reassessment of the description of rare-earth based magnets with unconventional properties.

[1] P. Santini, S. Carretta, G. Amoretti, R. Caciuffo, N. Magnani, G. H. Lander, Rev. Mod. Phys. 81, 807 (2009).

[2] R. Sibille, N. Gauthier, E. Lhotel, V. Porée, V. Pomjakushin, R. A. Ewings, T. G. Perring, J. Ollivier, A. Wildes, C. Ritter, T. C. Hansen, D. A. Keen, G. J. Nilsen, L. Keller, S. Petit & T. Fennell, Nat. Phys. 16, 546 (2020).

[3] A. Fennell, V. Y. Pomjakushin, A. Uldry, B. Delley, B. Prévost, A. Désilets-Benoit, A. D. Bianchi, R. I. Bewley, B. R. Hansen, T. Klimczuk, R. J. Cava & M. Kenzelmann, Phys. Rev. B 89, 224511 (2014).

[4] N. Gauthier, A. Fennell, B. Prévost, A.-C. Uldry, B. Delley, R. Sibille, A. Désilets-Benoit, H. A. Dabkowska, G. J. Nilsen, L.-P. Regnault, J. S. White, C. Niedermayer, V. Pomjakushin, A. D. Bianchi & M. Kenzelmann, Phys. Rev. B 95, 134430 (2017).

Two-dimensional (2D) materials are of intense current fundamental and applied interest as a route to create novel fundamental phenomena beyond well-established classical behaviour within their topologically constrained layers. In this context 2D monolayer graphene, formed from the isolation of weakly connected van der Waals (vdW) bonded 2D layers in graphite by exfoliation, ignited widespread interest. Exotic quantum relativistic phenomena, such as Dirac semi-metals and quantum anomalous Hall insulators, have been predicted in graphene and related materials ranging from isolated 2D monolayers to quasi-2D bulk materials with vdW bonded layers. The focus has expanded to “beyond graphene” 2D vdW layered materials with intrinsic properties such as magnetism and semiconductivity not present in graphene, however the number of materials is limited and detailed understanding only just beginning.

MnPSe₃ and CrPS₄ are such layered vdW materials that are both magnetic and semiconducting, with magnetic ions forming hexagonal and rectangular 2D motifs. To access their low dimensional behaviour we probe bulk powder and single crystal samples with neutron scattering measurements [1,2]. Through magnetic symmetry analysis and spin wave analysis we are able to isolate and explore the 2D structural and magnetic behaviour in these bulk materials. Interactions shown in Fig. 1. The data highlights subtle competing interactions in both materials that leads to the stabilization of the determined magnetic ground states. These magnetic ground states were further tuned with small applied perturbations of field and temperature and found to undergo both subtle spin alterations and more dramatic metamagnetic transitions. The determination of the intralayer and interlayer exchange interactions and anisotropy within model spin Hamiltonians allowed the underlying observed exotic bulk behaviour to be explored.

The results show that for MnPSe₃ the Se ion drives unexpectedly strong magnetic interactions between the 2D layers, which forms a contrast to the wider studies S analogue MnPS₃. While for CrPS₄ a further lowering of interaction dimensionality to 1D-chains is shown to be of significance. Collectively, these results highlight the subtle role of the crystalline structure on the emergent behaviour and show the powerful insights neutron scattering can supply to studies of low dimensional materials.

[1] S. Calder, A. Haglund, Y. Liu, D. M. Pajerowski, H. B. Cao, T. J. Williams, O. V. Garlea, D. Mandrus, “Magnetic structure and exchange interactions in the layered semiconductor CrPS₄”, Physical Review B, Phys. Rev. B 102, 024408 (2020).

[2] S. Calder, A. Haglund, A. I. Kolesnikov, D. Mandrus, “Magnetic exchange interactions in the van der Waals layered antiferromagnet MnPSe₃”, Physical Review B 103, 024414 (2021).

In the dense metal-organic framework Na[Mn(HCOO)₃], Mn²⁺ ions (S = 5/2) occupy the nodes of a ‘trillium’ net. We show that this material exhibits a variety of behaviour characteristic of geometric frustration: the Néel transition is suppressed well below the characteristic magnetic interaction strength; neutron scattering indicates that short-range magnetic order persists far above the Néel temperature; and the magnetic susceptibility exhibits a pseudo-plateau at 1/3-saturation magnetisation. We demonstrate that a simple nearest-neighbour Heisenberg antiferromagnet model accounts quantitatively for each observation, and hence Na[Mn(HCOO)₃] is the first experimental realisation of this model on the trillium net. We demonstrate how both link geometric frustration within the classical spin liquid regime to a strong magnetocaloric response at low fields.

The transverse-field Ising model on a triangular lattice is predicted to support a topological Kosterlitz-Thouless (KT) phase at nonzero temperature through a mapping of the Ising spins to a complex order parameter defined for each triangular unit. Recently, the triangular lattice antiferromagnet TmMgGaO₄ has emerged as a candidate material to realize this theoretical scenario. Through the complementary use of neutron diffraction and magnetic pair distribution function (mPDF), we have quantitatively investigated the spin correlations in TmMgGaO₄ in the temperature region of interest, tracking their evolution across the proposed transitions into and out of the KT phase. We confirm the presence of the three-sublattice order predicted for the ground state and show that the local magnetic structure undergoes distinct changes in the temperature range expected for the KT phase. Modeling the real-space mPDF reveals a preferential tendency for the system to form bound vortex-antivortex pairs, the hallmark of the KT phase, precisely in the expected temperature range. These findings constitute promising evidence for the KT phase, potentially establishing TmMgGaO₄ as a rare platform for studying KT physics in a dense spin system.

Geometrically frustrated magnets, such as triangular networks of antiferromagnetically coupled spins, can display incredibly rich physical properties that may have potential applications in quantum information science and other technologies. Determining if and how magnetic order emerges from competing magnetic tendencies is an important objective in this field. Here, we discuss the Jahn-Teller active triangular AMnO₂ (A= Na, Cu; Fig. 1) antiferromagnets [1] to highlight that the degree of frustration, mediated by residual disorder, contributes to the rather differing pathways towards a single, stable magnetic ground state, albeit with varying ordering temperatures. For these insulating sister compounds, complementary high-resolution synchrotron XRD, local-probe muon-spin relaxation (μ⁺SR) studies, corroborate that the layered NaMnO₂ adopts a remarkable magnetostructurally inhomogeneous ground state. [2] In view of this peculiarity, we employ powerful neutron total scattering and magnetic pair distribution function (PDF) analysis to uncover that although CuMnO₂ undergoes a conventional symmetry-lowering lattice distortion driven by Néel order, in the Na-derivative a short-range triclinic distortion (Fig. 2) lifts the degeneracy of the isosceles triangular network on the nanoscale, thereby enabling long-range magnetism to develop with enhanced magnetic correlations above the transition. [3] More generally, the work illuminates the cooperative intertwining of the local atomic and magnetic structures that permits ground state selection when spatial inhomogeneity meets geometrical frustration, a mechanism that may also be operative in other frustrated materials with electronically active transition metal cations.

[1] M. Giot et al., Phys. Rev. Lett. 99, 247211 (2007).

[2] A. Zorko et al., Sci. Rep. 5, 9272 (2015).

[3] B. A. Frandsen et al., Phys. Rev. B 101, 024423 (2020).