Spin exchange Hamiltonian and topological degeneracies in elemental gadolinium

Abstract

We present a comprehensive study of the magnetic exchange Hamiltonian of elemental Gadolinium. We use neutron scattering to measure the magnon spectrum over the entire Brillouin zone, and fit the excitations to a spin wave model to extract the first 30 nearest neighbor magnetic exchange interactions with rigorously defined uncertainty. We find these exchange interactions to follow RKKY behavior, oscillating from ferromagnetic to antiferromagnetic as a function of distance. Finally, we discuss the topological features and degeneracies in Gd, and HCP ferromagnets in general. We show theoretically how, with asymmetric exchange, topological properties could be tuned with a magnetic field.

Dirac magnons, nodal lines, and nodal plane in elemental gadolinium

“…Like on the honeycomb lattice [12], it is symmetry-allowed on the HCP lattice second nearest neighbor bonds. Although the superexchange mechanism considered by Moriya [69] may seem prima facie incompatible with the Ruderman-Kittel-Kasuya-Yosida (RKKY)-like magnetism in rare-earths like Gd [52], RKKY-mediated DM exchanges have, in fact, been discussed in other contexts [70][71][72], which leads us to speculate that weak DM exchanges may be possible also in spin-orbit-coupled rare-earths. A more concrete reason to consider DM exchange is that it may be relevant to other HCP ferromagnets, or to potential devices where the interfacial DM exchange can be strong due to breaking of inversion symmetry.…”

“…Data were processed with Mantid software [50]; see the Supplemental Materials [51] and Ref. [52] for further details. The resulting full data set allows one to directly see topological features in the spectrum.…”

“…The resulting full data set allows one to directly see topological features in the spectrum. The data were thoroughly analyzed to determine an accurate spin exchange Hamiltonian: this is discussed in detail in a separate paper [52] focusing on the Gd magnetic interactions. Here we focus on the topological properties of the Gd magnon bands, showing the existence of a Dirac magnon at K, and nodal planes at half-integer .…”

“…To more firmly establish the topological nature of the nodal line, we turn to linear spin-wave theory [54,55]. Rather than using the full fitted Gd Hamiltonian [52], we here use a simplified J 1 − J 2 − J 3 model that qualitatively captures the main features of the full fitted model, including the band crossings,…”

“…(For the values of the exchange couplings, see Ref. [52].) J 1 and J 3 couple the two sublattices, whereas J 2 couple only sites within the same sublattice (within ab-planes).…”