Direct 3D mapping of the Fermi surface and Fermi velocity


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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

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.


  1. 1

    Felser, C., Fecher, G. H. & Balke, B. Spintronics: a challenge for materials science and solid-state chemistry. Angew. Chem. Int. Ed. 46, 668–699 (2007).

  2. 2

    Hwang, H. Y. et al. Emergent phenomena at oxide interfaces. Nat. Mater. 11, 103–113 (2012).

  3. 3

    Hsieh, D. et al. Observation of unconventional quantum spin textures in topological insulators. Science 323, 919–922 (2009).

  4. 4

    Riley, J. M. et al. Direct observation of spin-polarized bulk bands in an inversion-symmetric semiconductor. Nat. Phys. 10, 835–839 (2014).

  5. 5

    Roushan, P. et al. Topological surface states protected from backscattering by chiral spin texture. Nature 460, 1106–1109 (2009).

  6. 6

    Sakano, M. et al. Topologically protected surface states in a centrosymmetric superconductor β-PdBi2. Nat. Commun. 6, 8595 (2015).

  7. 7

    Pickel, M. et al. Spin-orbit hybridization points in the face-centered-cubic cobalt band structure. Phys. Rev. Lett. 101, 066402 (2008).

  8. 8

    Hüfner, S. Photoelectron Spectroscopy: Principles and Applications (Springer, 2003).

  9. 9

    Suga, S. & Sekiyama, A. Photoelectron Spectroscopy (Springer, 2014).

  10. 10

    Miyamoto, K. et al. Spin-polarized Dirac-cone-like surface state with d character at W(110). Phys. Rev. Lett. 108, 066808 (2012).

  11. 11

    Miyamoto, K. et al. Massless or heavy due to two-fold symmetry: surface-state electrons at W(110). Phys. Rev. B 86, 161411 (2012).

  12. 12

    Mirhosseini, H., Giebels, F., Gollisch, H., Henk, J. & Feder, R. Ab initio spin-resolved photoemission and electron pair emission from a Dirac-type surface state in W (110). New J. Phys. 15, 095017 (2013).

  13. 13

    Braun, J. et al. Exceptional behavior of d-like surface resonances on W(110): the one-step model in its density matrix formulation. New J. Phys. 16, 015005 (2014).

  14. 14

    Engelkamp, B. et al. Spin-polarized surface electronic structure of Ta(110): similarities and differences to W(110). Phys. Rev. B 92, 085401 (2015).

  15. 15

    Chernov, S. et al. Anomalous d-like surface resonances on Mo(110) analyzed by time-of-flight momentum microscopy. Ultramicroscopy 159, 453–463 (2015).

  16. 16

    Tusche, C., Krasyuk, A. & Kirschner, J. Spin resolved band structure imaging with a high resolution momentum microscope. Ultramicroscopy 159, 520–529 (2015).

  17. 17

    Tanuma, S., Powell, C. J. & Penn, D. R. Calculations of electron inelastic mean free paths (IMFPs) VI. analysis of the gries inelastic scattering model and predictive IMFP equation. Surf. Interface Anal. 25, 25–35 (1997).

  18. 18

    Gray, A. X. et al. Probing bulk electronic structure with hard X-ray angle-resolved photoemission. Nat. Mater. 10, 759–764 (2011).

  19. 19

    Xu, S.-Y. et al. Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide. Nat. Phys. 11, 748–754 (2015).

  20. 20

    Papp, C. et al. Band mapping in x-ray photoelectron spectroscopy: an experimental and theoretical study of W(110) with 1.25 keV excitation. Phys. Rev. B 84, 045433 (2011).

  21. 21

    Strocov, V. N., Cirlin, T. G. E., Sadowski, J., Kanski, J. & Claessen, R. GaSb/GaAs quantum dot systems: in situ synchrotron radiation X-ray photoelectron spectroscopy study. Nanotechnology 16, 1326–1334 (2005).

  22. 22

    Fadley, C. S. X-ray photoelectron spectroscopy: progress and perspectives. J. Electron Spectrosc. Relat. Phenom. 2, 178–179 (2010).

  23. 23

    Lv, B. Q. et al. Observation of Weyl nodes in TaAs. Nat. Phys. 11, 724–727 (2015).

  24. 24

    Kobata, M. et al. Electronic structure of EuAl4 studied by photoelectron spectroscopy. J. Phys. Soc. Jpn 85, 094703 (2016).

  25. 25

    Mattheis, L. F. Fermi surface in tungsten. Phys. Rev. 139, A1893 (1965).

  26. 26

    Choi, D. et al. Electron mean free path of tungsten and the electrical resistivity of epitaxial (110) tungsten films. Phys. Rev. B 86, 045432 (2012).

  27. 27

    Christensen, N. E. & Feuerbacher, B. Volume and surface photoemission from tungsten. I. Calculation of band structure and emission spectra. Phys. Rev. B 10, 2349–2357 (1974).

  28. 28

    Schönhense, G. Circular dichroism and spin polarization in photoemission from adsorbates and non-magnetic solids. Phys. Scr. T31, 255–275 (1990).

  29. 29

    Weinelt, M., Schmidt, A. B., Pickel, M. & Donath, M. in Dynamics at Solid State Surfaces and Interfaces (eds Bovensiepen, U., Petek, H. & Wolf, M.) Part 1 (Wiley, 2010).

  30. 30

    Kessler, J. The ‘perfect’ photoionization experiment. Comments At. Mol. Phys. 10, 47–56 (1981).

  31. 31

    Bauer, E. LEEM and UHV-PEEM: a retrospective. Ultramicroscopy 119, 18–23 (2012).

  32. 32

    Schönhense, G. et al. Correction of the deterministic part of space-charge interaction in momentum microscopy of charged particles. Ultramicroscopy 159, 488–496 (2015).

  33. 33

    Oelsner, A. et al. Time- and energy resolved photoemission electron microscopy-imaging of photoelectron time-of-flight analysis by means of pulsed excitations. J. Electron Spectrosc. Relat. Phenom. 178, 317–330 (2010).

  34. 34

    Viefhaus, J. et al. The Variable Polarization XUV Beamline P04 at PETRA III: optics, mechanics and their performance. Nucl. Instrum. Methods 710, 151–154 (2013).

  35. 35

    Smith, N. V., Thiry, P. & Petroff, Y. Photoemission linewidths and quasiparticle lifetimes. Phys. Rev. B 47, 15451–15476 (1993).

  36. 36

    Plucinski, L. et al. Band mapping in higher-energy X-ray photoemission: phonon effects and comparison to one-step theory. Phys. Rev. B 78, 035108 (2008).

  37. 37

    Takata, Y. et al. Recoil effect of photoelectrons in the Fermi edge of simple metals. Phys. Rev. Lett. 101, 137601 (2008).

  38. 38

    Kutnyakhov, D. et al. Spin texture of time-reversal symmetry invariant surface states on W(110). Sci. Rep. 6, 229394 (2016).

  39. 39

    Elmers, H. J. et al. Spin mapping of surface and bulk Rashba states in ferroelectric a-GeTe(111) films. Phys. Rev. B 94, 201403(R) (2016).

  40. 40

    Ovsyannikov, R. et al. Principles and operation of a new type of electron spectrometer—ArTOF. J. Electron Spectrosc. Relat. Phenom. 191, 92–103 (2013).

  41. 41

    Berntsen, M. H., Götber, O. & Tjernberg, O. An experimental setup for high resolution 10.5 eV laser-based angle-resolved photoelectron spectroscopy using a time-of-flight electron analyser. Rev. Sci. Instrum. 82, 095113 (2011).

  42. 42

    Liang, A. et al. Electronic evidence for type II Weyl semimetal state in MoTe2. Preprint at (2016).

  43. 43

    Vollmer, A. et al. Two dimensional band structure mapping of organic single crystals using the new generation electron energy analyzer ARTOF. J. Electron Spectrosc. Relat. Phenom. 185, 55–60 (2012).

  44. 44

    Hilbert, S. A., Barwick, B., Fabrikant, M., Uiterwaal, C. J. G. J. & Batelaan, H. A high repetition rate time-of-flight electron energy analyzer. Appl. Phys. Lett. 91, 173506 (2007).

  45. 45

    Eastman, D. E., Donelon, J. J., Hien, N. C. & Himpsel, F. J. An ellipsoidal mirror display analyzer system for electron energy and angular measurements. Nucl. Instrum. Methods 172, 327–336 (1980).

  46. 46

    Daimon, H. New display-type analyzer for the energy and the angular distribution of charged particles. Rev. Sci. Instrum. 59, 545–549 (1988).

  47. 47

    Matsuda, H. et al. Development of display-type ellipsoidal mesh analyzer: computational evaluation and experimental validation. J. Electron Spectrosc. Relat. Phenom. 195, 382–398 (2014).

Download references


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.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 187 kb)

Supplementary Information

Supplementary movie 1 (AVI 1999 kb)

Supplementary Information

Supplementary movie 2 (AVI 811 kb)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Further reading