Abstract
Berry phases and the related concept of Berry curvature can give rise to many unconventional phenomena in solids. Here, we discover a colossal orbital Zeeman effect of topological origin in a bilayer kagome metal, TbV6Sn6. Using spectroscopic imaging scanning tunnelling microscopy, we reveal that the magnetic field leads to a splitting of the gapped Dirac dispersion into two branches with enhanced momentum-dependent g factors, resulting in a substantial renormalization of the Dirac band. These measurements provide a direct observation of a magnetic field-controlled orbital Zeeman coupling to the orbital magnetic moments of up to 200 Bohr magnetons near the gapped Dirac points. Our work provides direct insight into the momentum-dependent nature of topological orbital moments and their tunability via the magnetic field, concomitant with the evolution of the spin Berry curvature. These results can be extended to explore large orbital magnetic moments driven by the Berry curvature governed by other quantum numbers beyond spin, such as the valley in certain graphene-based structures.
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Data availability
The data supporting the findings of this study are available via Zenodo at https://zenodo.org/records/10800955 (ref. 63) and upon request from the corresponding author. Source data are provided with this paper.
Code availability
The computer code used for data analysis is available upon request from the corresponding authors.
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Acknowledgements
We thank X. Hu and X. Li for useful discussions. I.Z. gratefully acknowledges the support from DOE Early Career Award DE-SC0020130 for STM measurements. Z.W. acknowledges the support of US Department of Energy, Basic Energy Sciences grant no. DE-FG02-99ER45747 and Cottrell SEED award no. 27856 from the Research Corporation for Science Advancement. F.M. greatly acknowledges the SoE action of pnrr, number SOE_0000068. S.D.W. and G.P. gratefully acknowledge support via the UC Santa Barbara NSF Quantum Foundry funded via the Q-AMASE-i programme under award DMR-1906325. P.E. was supported by the SFB 1170 ToCoTronics, funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project ID 258499086. G.S. acknowledges financial support from the DFG through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter-ct.qmat (EXC 2147, project-id 390858490) and through “QUAST” FOR 5249-449872909 (project P05). D.D.S. has received funding from the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement no. 897276. P.E. gratefully acknowledges the Gauss Center for Supercomputing e.V. (www.gauss-center.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Center (www.lrz.de).
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H.L. performed the STM experiments. H.L. analysed the STM data with help from S.C. G.P. synthesized and characterized the samples under the supervision of S.D.W. P.E., G.S. and D.D.S. performed density functional theory calculations. Z.W. provided theoretical input on the interpretation of STM data. C.B. and F.M. provided insights on angle-resolved photoemission spectroscopy data in connection to STM data. H.L., G.S., D.D.S., S.D.W., Z.W. and I.Z. wrote the paper with input from all authors. I.Z. supervised the project.
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Extended data
Extended Data Fig. 1 Temperature-dependent dI/dV spectra.
Average dI/dV spectra acquired over the same region of the Sn surface at about 2 K and at 4.2 K. The two spectra appear nearly identical. STM setup conditions: tunneling current Iset = 1 nA, sample bias Vsample = 300 mV, bias excitation Vexc = 2 mV; 0 T magnetic field.
Extended Data Fig. 2 Evolution of the Dirac band as a function of magnetic field direction.
Radial linecut of Fourier transforms of dI/dV maps (which we refer to as \(S(q,E)\) in the text) for (a-d) positive magnetic fields and (e-h) negative magnetic fields (the minus sign denotes the reversal of the magnetic field direction from parallel to antiparallel to the c-axis). The band dispersion appears to be the same for positive and negative magnetic fields of the same magnitude. STM setup conditions: tunneling current Iset = 1.5 nA, sample bias Vsample = 300 mV, bias excitation Vexc = 2 mV.
Extended Data Fig. 3 Determination of wave vector dispersion from Fourier transform (FT) linecuts.
(a) The second derivative of radially averaged Fourier transform linecuts, \({\partial }_{q}{\partial }_{E}S(q,E)\), at \(E\)=260 mV from 0 T (blue circles) to 6 T (red circles). Black lines are the Gaussian fits to the data to locate the wave vector q-space positions. (b-f) The second derivative of radially averaged FT linecuts \({\partial }_{q}{\partial }_{E}S(q,E)\). All Fourier transforms are applied to dI/dV maps, which are processed by subtracting the average value from each dI/dV map to artificially suppress the bright center in the resulting Fourier transform and smoothed on 1 pixel length scale prior to the Fourier transform. Dark blue points are extracted by row fits method as shown in (a), while the white dots are extracted by column fits, the same as was done in Fig. 4. The data points obtained by the two extraction methods match reasonably well with one another.
Extended Data Fig. 4 Theoretical model.
(a,b) Electronic band structure of TbV6Sn6 in the vicinity of the K point (a) without and (b) with spin-orbit coupling included. The bands’ colors in (b) highlights the strength of the orbital magnetic moment, with red and blue being positive and negative contributions, respectively. The blue arrow in (a) denotes the Dirac point that is gapped by magnitude Δ in (b). (c) Evolution of the orbital moment of the topmost Dirac cone upon a continuous change of the Dirac gap Δ. The gap is artificially tuned by acting on the strength of the local spin-orbit coupling in our first-principles calculations. For the calculation of the Berry curvature and orbital magnetic moments, the Kohn–Sham wave functions were projected onto a Tb d, V d and Sn s, p-type basis [Ref. 60], and the quantities were computed by using our in-house post-wan library [https://github.com/philipp-eck/post_wan], consistent with [Ref. 62].
Extended Data Fig. 5 Band extraction using different fitting methods.
White and blue points are the same second derivative fits as described in Extended Data Fig. 3. The red points with error bars are extracted peaks (E(q) ± δE) for each q point by fitting a Gaussian function to the vertical/column linecuts. The error bar δE represents standard error of peak position in energy axis from Gaussian peak fittings of QPI vertical linecuts (number of data points for each vertical linecut is 95). It can be seen that the extracted band does not vary substantially by using different fitting methods.
Supplementary information
Supplementary Information
Supplementary Notes 1 and 2 and Figs. 1–6.
Source data
Source Data Fig. 1
Band structure plots.
Source Data Fig. 2
Temperature-dependent spectral peak evolution.
Source Data Fig. 4
Field-dependent evolution of band features.
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Li, H., Cheng, S., Pokharel, G. et al. Spin Berry curvature-enhanced orbital Zeeman effect in a kagome metal. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02487-z
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DOI: https://doi.org/10.1038/s41567-024-02487-z
