Direct 3D mapping of the Fermi surface and Fermi velocity

Abstract

We performed a full mapping of the bulk electronic structure including the Fermi surface and Fermi-velocity distribution vF(kF) of tungsten. The 4D spectral function ρ(EB; k) in the entire bulk Brillouin zone and 6 eV binding-energy (EB) interval was acquired in 3 h thanks to a new multidimensional photoemission data-recording technique (combining full-field k-microscopy with time-of-flight parallel energy recording) and the high brilliance of the soft X-rays used. A direct comparison of bulk and surface spectral functions (taken at low photon energies) reveals a time-reversal-invariant surface state in a local bandgap in the (110)-projected bulk band structure. The surface state connects hole and electron pockets that would otherwise be separated by an indirect local bandgap. We confirmed its Dirac-like spin texture by spin-filtered momentum imaging. The measured 4D data array enables extraction of the 3D dispersion of all bands, all energy isosurfaces, electron velocities, hole or electron conductivity, effective mass and inner potential by simple algorithms without approximations. The high-Z bcc metals with large spin–orbit-induced bandgaps are discussed as candidates for topologically non-trivial surface states.

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Figure 1: Cross-section of the ToF k-microscope (lenses schematic) with calculated trajectories for 300 eV initial kinetic energy and 30 eV drift energy.
Figure 2: Brillouin zone, calculated Fermi surface and direct transitions.
Figure 3: Momentum sections I(EF; kx, ky) at the Fermi energy.
Figure 4: Experimentally determined Fermi surface and Fermi velocity.
Figure 5: Band dispersions between the Fermi energy and EB = 6 eV.
Figure 6: Comparison of bulk and surface spectral functions and spin signature revealing a Dirac-like state in a partial bandgap.

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Acknowledgements

Excellent support by staff members of PETRA III and BESSY II is gratefully acknowledged. Sincere thanks are due to C. Tusche (Forschungszentrum Juelich, Germany) and J. Kirschner (MPI fuer Mikrostrukturphysik, Halle, Germany) for very fruitful cooperation. The project is funded by BMBF (05K13UM1, 05K13UM2, 05K13GU3, 05K12UM2), DFG through SFB 1170 (project C06) and Transregio SFB TRR 173 (Spin+X).

Author information

K.M. and G.S. wrote the paper. K.M., O.F., S.C., D.K., M.E., J.V., H.J.E. and G.S. set up and carried out the experiment and prepared the samples. B.S., H.J.E. and K.M. performed the data evaluation. A.O., S.D. and Y.A. helped with the microscopy control unit. G.S. and H.J.E. coordinated the project. All authors discussed the results and contributed to the writing of the manuscript.

Correspondence to K. Medjanik.

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Medjanik, K., Fedchenko, O., Chernov, S. et al. Direct 3D mapping of the Fermi surface and Fermi velocity. Nature Mater 16, 615–621 (2017). https://doi.org/10.1038/nmat4875

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