The present note gives a brief account of the new results. Shoenberg, Nature , , Verkin, Lazarev and Rudenko, J. CAS Google Scholar. See appendix to Shoenberg, Proc.
Shoenberg and Zaki Uddin, Proc. Marcus, J. Andrew, Proc. Goodman private communication of preliminary results. Mackinnon, Proc. Download references. Wohlfarth Elsevier, Amster- quadrupole moment. The quadrupole moment couples dam, p. Andres, P. Wang, Y. Wong, B. Lfithi and H. The reduc- [3] R. Ahmet, K.
Motoki, N. Kimura, tion of the dHvA amplitude in the metamagnetic H. Ebihara, H. Sugawara, K. Sugiyama and transition is thus due to this crystal distortion. The Y. Onuki, J. Japan 64 Haen and Louis-Pierre Regnault. Cornelius and P. B 8 , Lee, Phys. Myron, Solid State Commun. Download references. You can also search for this author in PubMed Google Scholar.
Reprints and Permissions. Dunsworth, A. The de Haas-van Alphen effect in PdTe 2. J Low Temp Phys 19, 51—57 Download citation. Received : 30 September Despite the vast number of materials identified as topological in nature, design principles of electronic topology besides numerical identification involving extensive band details are much less established. This particularly impedes the quest for correlated topological materials 6 , 7 , where calculation is known to be challenging to yield precise band information.
The theoretical development of the topological band theories itself has relied heavily on conceptual lattice models 8 ; to what degree such models are relevant for real materials remains an open question. The two-dimensional 2D kagome lattice is a system of corner-sharing triangles assembled in a hexagonal fashion analogous to the graphene lattice 9. These triangular and hexagonal structural features are a test ground to access the physics of magnetic frustration 10 and of lattice-driven Dirac fermions 11 , respectively.
In the context of electronic hopping models, the kagome network also gives rise to a flat band together with a pair of Dirac bands that potentially support exotic phases such as interaction-driven ferromagnetism 12 , 13 and chiral superconductivity Compared with the graphene lattice, the nearest-neighbor bonds in the kagome lattice are not contained in mirror planes perpendicular to the basal lattice, and can therefore experience an electric field orthogonal to the nearest-neighbor bonds 15 , introducing explicitly spin—orbit effects into the band structure and pathways to topologically nontrivial electronic bands 16 , In terms of material realizations, a number of recent efforts have focused on metallic kagome lattice materials that potentially connect to the electronic hopping behavior expected for the 2D lattice.
In particular, the kagome lattice has been realized in a series of hexagonal 3 d transition metal stannides and germannides 18 , The importance in this construction can also be seen by comparing with kagome lattice-containing Co 3 Sn 2 S 2 —there, despite a relatively large layer spacing, the electronic structure is 3D in nature as the additional sulfur network bridges the kagome layers 24 , The use of the 3 d transition elements allows the introduction of magnetism; in the case of Fe 3 Sn 2 , this is a soft ferromagnetic order 26 , which along with atomic spin—orbit coupling, gives rise to substantial intrinsic anomalous Hall conductivity from the massive Dirac bands extending above room temperature Given the softness of this magnetic order, a natural question that arises is how a general positioning of the magnetic moment m affects the electronic structure and topology of this system.
For Kane—Mele-type spin—orbit coupling, the orthogonality between the local electric field and magnetic moment orientation can selectively open the gap; such control of electronic topology with magnetic moment orientation is uniquely enabled in ferromagnetic systems Pulsed field torque magnetometry and de Haas—van Alphen oscillations in Fe 3 Sn 2.
The black arrows correspond to the eighth and ninth oscillation of the slow frequency at each angle. Here we report a torque magnetometry study that captures the evolution of the quasi-2D Dirac bands via the de Haas—van Alphen effect dHvA while at the same time monitoring changes in the magnetic order.
These observations together demonstrate a systematic development of the massive Dirac states consistent with a Kane—Mele spin—orbit coupling 8 with a relativistic energy shift comparable to those observed in elemental ferromagnets 28 , 29 , 30 , 31 , This is indicative of the lower energy scale for Landau quantization compared with the magnetic order and the associated anisotropy see Supplementary Note 1. Angular dependence of dHvA oscillation frequencies.
Empty circles are collected from pulsed field experiments, while solid circles are from DC field experiments. Data taken from different samples are represented with different colors. The black curves are guides to the eye. This dependence is shown as a dashed line in Fig. A deviation of this type is observed in systems with local hyperboloid geometries as depicted in Fig.
Applying such a scenario to the present case of a quasi-2D surface is shown in Fig. As we describe below, this is well described by a massive Dirac model with systematically evolving band parameters constructed for the inner Dirac band and extended to capture the outer Dirac band solid lines in Fig.
We note from these observations that the size of the Fermi surface can be used as a caliper to probe the orientation of m. By converting this to an angular projected area, we find that it corresponds to the c -normal Fermi surface in Fig.
The overall dHvA oscillation amplitude of multiple oscillations in magnetization can be written as The inset shows a schematic of the double Dirac spectrum. Error bars correspond to standard errors in least-squares fitting. By extending this analysis across the dHvA spectrum see Supplementary Fig. Based on previous observations of the double massive Dirac spectrum in this system see schematic in Fig.
We note that the outer Dirac pocket has been observed to have substantial warping near the Fermi level E F illustrated in the inset in Fig. In analogy to the spin—orbit models of Ni 30 , we consider that the Dirac band parameters are modulated by m. By assuming a Kane—Mele spin—orbit coupling with massive Dirac fermions 8 , the intrinsic anomalous Hall conductivity per kagome bilayer provides a further constraint to these parameters.
Here t is the thickness of a structural unit that contains a single kagome bilayer. With these smooth functions we obtain the solid fits to Fig. We can use the band parameters to reconstruct the trends observed in experiment for the inner Dirac surface solid curves in Fig. These observations suggest that in the presence of spin—obit coupling, the Dirac bands have a considerable response to changes in the intrinsic ferromagnetism 22 where the spin—orbit coupling is likely of Kane—Mele type.
The spin—orbit coupling energy scale in the present system is substantially enhanced compared with that in graphene 8 and provides a model system for studying the topological phases associated with Kane—Mele term. Extending the angular range of these measurements, as well as more sophisticated modeling of this behavior, including the role of the other electronic bands as charge reservoirs, are of considerable interest.
While modeling the evolution of the three-dimensional bands is more challenging owing to their angular evolution from intrinsic ellipticity, further theoretical efforts in understanding the electronic structure may help to elucidate the spin—orbit-induced changes exiting therein.
Finally, we return to the overall magnetic torque behavior with H. In Fig. This trend is clearer when extended to high field: Fig. Quantitatively, from the angular dependence of the torque, a moderate easy-plane anisotropy can be inferred see Supplementary Note 1 , similar to previous reports in which shape anisotropy plays an important role Further study of the interplay of bulk, surface, and shape anisotropies with the electronic structure of this system 43 is an important area for future work; as the Dirac mass itself can influence magnetic order in similar systems 44 , 45 , an exciting prospect is that the Dirac fermions themselves along with spin—orbit coupling play a role in determining evolution of the magnetic order.
Torque response from the soft ferromagnetism in Fe 3 Sn 2. At low angles, the torque response exhibits an initial increase that gradually transforms to a broad shoulder at high angles, consistent with the observation at high fields with piezoresistive cantilevers. The inset shows the transverse magnetization extracted for each torque curve. The dHvA results presented here are a thermodynamic probe of the ground state in the presence of a strong polarizing magnetic field of the correlated, topological bands of Fe 3 Sn 2.
The magnetoquantum oscillations confirm the bulk nature of the quasi-2D massive Dirac bands arising from the kagome network previously observed spectroscopically 21 , and provide guidance for theoretical models of this system. Viewed more broadly, the results here demonstrate how lattice-derived topological electronic bands can be wed with the robust ferromagnetism in correlated electron systems.
Given the widespread use of 3 d ferromagnets in spintronics, this provides the exciting prospect that topologically nontrivial analogs of the workhorse materials for spintronics may be developed, allowing direct integration of topological electronic states into such architectures 46 , The development of such materials where the charge, spin, and heat transport properties are dominated by the topological bands and controllable with spintronic techniques will be an important direction in realizing the promise of topological electronic states to impact the next generation of electronic devices.
Single crystals were grown with an I 2 -catalyzed reaction starting from stoichiometric Fe and Sn powders Hexagonally shaped crystals were formed near the hot side.
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