Emergence is the phenomenon of collective behaviour that cannot predicted from considerations of the individual constituents but rather through complex interactions resulting in novel and diverse states of matter. A well-known example is the benzene ring, a loop of six carbon atoms, that can be further interconnected to create carbon nanotubes. Emergent behaviour in magnetic materials is often found in geometrically frustrated compounds. The crystalline structure and exchange interactions thwarts the magnetic ordering, hence the term frustration. It is believed that an understanding of these novel states of matter in magnetic materials will result in technological breakthroughs beneficial to our spintronic society.
Quintessential frustrated magnetic materials are those based on a triangular hyperkagome crystalline lattice and include the isostructural compounds Gd3Ga5O12 (GGG) and Yb3Ga5O12 (YGG). Our previous studies (Science 350, 6257 (2015)) uncovered a most unusual long range emergent director state in GGG derived from magnetic spins arranged on a ten-ion loop despite a disordered state at the local level. In this most recent work, we show that the director state is also found in YGG, albeit correlated over a much reduced distance, revealing the ubiquity of this unusual state of matter. We derive the magnitude of the near neighbour exchange and the dipolar exchange interaction required to develop the director state. Our work thus provides a strong basis for the understanding of the complex Hamiltonian required to fully describe and thus manipulate the magnetic state of YbGG.
Neutron scattering experiments were performed on single crystal YGG, see Figure 1(a) top, using the polarized diffuse scattering spectrometer, D7, at the ILL, the cold neutron scattering spectrometer, CNCS at the SNS and the thermal chopper spectrometer, IN4 at the ILL.
The quantum nature of the Yb3+ spins was verified via extraction of the crystal field parameters using IN4. Magnetic susceptibility measurements determined the relevant exchange energies while neutron scattering probed the magnetic structure factor, S(Q), across a wide region of reciprocal space provided by the various incident energies of D7 and CNCS. Reverse Monte Carlo of S(Q) provided the spin structure and determined the director state of YGG, see Figure 1(b). Monte Carlo (MC) theoretical analysis for two differing models, based on the specific heat, extracts a range of relevant exchange interactions necessary to understand the complex Hamiltonian of YGG. In order to verify the exchange interactions determined by MC, the resultant exchange interactions are used to recalculate S(Q), see Figure 1(b), which are largely comparable to the experimental data.
Figure 1 A (top): Photo of the single crystal studied. (bottom): Spin structure: a 10-spin-loop together with a single ion from the opposite hyperkagome lattice (central, red). The blue spheres depict Yb3+ ions while the red sphere can be considered as the net average magnetic moment of the ten-ion loop, the director (red arrow). Local spin distributions peak along the local z-direction (grey arrows)
Figure 1 B. Comparison of experimentally determined magnetic neutron scattering structure factor and Reverse Monte Carlo fit (left and right respectively)
A more complex Hamiltonian is required to fully describe the magnetic state of YbGG and will be the focus of further studies. Recently, further inelastic neutron scattering experiments have been performed on the magnetically polarised state to determine the magnetic exchange interactions and provide a further handle on the Hamiltonian. It has been suggested that these directors are inherent in geometrically frustrated systems, and that they provide an organizing principle in which emergent clusters form out of a manifold of ground states with the low-temperature dynamics governed by the director state. As such a further step is to characterise the low energy dynamics.
Figure 2: PhD students on experiment celebrating their experimental success. Left to right: R.Edberg (KTH), L: Sandberg (KU) and I.M. Bakke (Oslo University).
The students involved in this work were part of the Nordforsk funded project (2017-2021): Magnetic Frustration under Pressure Project id:82248 with a student based at Oslo University (I.M.Bakke (student), H. Fjellvåg (PI)) for single crystal synthesis, at Copenhagen University for bulk characterization and neutron scattering (S. Sandberg (student), P. Deen (PI) & K. Lefmann (PI)), and theoretical support at KTH Royal Institute of Technology, Stockholm (R. Edberg (student), P. Henelius (PI)), see Figure 2.
The crystals were grown using the floating zone technique at Warwick University by M. Ciomaga Hatnean and G. Balakrishnan. Kasper S. Pedersen of the Technical University of Denmark provided support with susceptibility measurements. Neutron scattering experiments were possible with the support of the neutron insturment scientists: A. Wildes (D7), B. Fåk (IN4), G. Ehlers (CNS) and G. Sala (CNCS).