Development of the photoemission simulator
SPADExp and its application to an ARPES study ofnodal line materials
Hiroaki Tanaka(ISSP, Univ. of Tokyo)

2Angle-resolved photoemission
spectroscopy (ARPES)

Observation of the band dispersion using the photoelectric effectPhotoelectron analyzer

Photoelectron
UV, X-ray
X-ray→
Solid crystal
Schematic of the photoemission process
J. A. Sobota et al., Rev. Mod. Phys. 93, 025006 (2021).

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Synchrotron ARPES endstation@SPring-8 BL25SU

3Photoemission 3-step model In-plane momentum (𝐤 // ) is conserved in the whole process Out-of-plane momentum (𝑘𝑧 ) is conserved in the crystal

 Photoelectron energy 𝐸kin = Ground state energy + ℎ𝜈

ℏ2 𝑘 2ℏ2 𝐾 2𝐸kin =+ 𝑉0 =determines 𝑘𝑧 (𝑉0 < 0 is a parameter)
2𝑚
2𝑚

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4Starting point of our study Bulk band degeneration at the L point was not observed.
 Surface sensitivity (a few nm)→ Momentum selection rule along z may not be rigorous?

ARPES spectra of the transition metal dichalcogenide 2H-NbS2H. Tanaka, S. Okazaki et al., Phys. Rev. B 105, L121102 (2022).

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5Overview of this talk

 Development of the photoemission simulator SPADExp[H. Tanaka et al., J. Electron Spectrosc. 264, 147297 (2023)]

 Tight-binding model (previous studies) → DFT with localized basis sets
 Application to an ARPES study of nodal line materials[H. Tanaka et al., under review. (arXiv:2308.00999)]
 Line node at the AL path protected by the 21 screw rotation symmetry

 Photoemission calculations beyond the free-electron approximation[H. Tanaka et al., in preparation.] Wave function calculation using the Kohn-Sham potential

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Development of the photoemission
simulator SPADExp
H. Tanaka et al., J. Electron Spectrosc. 264, 147297 (2023)
https://github.com/Hiroaki-Tanaka-0606/SPADExp

7Photoemission probability
Electric fieldFermi’s golden rule
Energy conservation
Final state

Initial state

Final state: Free-electron approximation is applied

Bulk system: Plane wave

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Slab system: Plane wave × Decay

8Calculation formulae

Initial state: Linear combination of (pseudo-)atomic orbitals

Rn=lattice vector, τi=atom position, 𝜙=orbital wave function, 𝛼=(plm)
Final state: Partial wave expansion at each atom position

Electric field: Dipole approximation

e=electric field vector
Integration over angles → excitation selection rule𝑙𝑓 = 𝑙𝑖 ± 1, 𝑚𝑓 = 𝑚𝑖 + 𝑗
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9Previous study of graphene A tight-binding model (carbon pz orbital) is used for initial states Graphene Dirac cones are excited only within the Brillouin zoneARPESSimulation

50 eV polarization not specified, EF-2.1 eV[E. L. Shirley et al.,Phys. Rev. B 51, 13614 (1995)]

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ARPESSimulation

54 eV horizontal linear polarization,
EF-1.7 eV [H. Nishimoto et al., Solid State
Commn. 98, 671 (1996)]

10

Our software

Simulator of Photoemission Angular Distribution for Experiments (SPADExp)

Our contribution

 Use MO.fileout / MO.Nkpoint / <MO.kpoints> to extract LCAO coefficients C++ / OpenMP parallelization / Lapack / HDF5
 Slab calculations were performed on ISSP Supercomputer System B

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11

Calculation example

30°-twisted bilayer graphene: ARPES spectra with 12-fold symmetry
 Approximating the quasicrystal by a supercell (224 carbon atoms)
Band dispersion Photoemission

ARPES

21 eV no polarization

Ahn et al., Science 361, 782 (2018).
21 eV, horizontal polarization
constant-energy maps at the Fermi level

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Application of SPADExp to an ARPES studyof nodal line materials
Hiroaki Tanaka et al., under review.
(arXiv:2308.00999)

13

Bulk nodal lines in 2H-NbS2
wo/ SOC

2H-NbS2

Hexagonal Brillouin zone

 Without SOC, a nodal plane on 𝑘𝑧 =𝜋𝑐 Due to 21 screw and time-reversal

wo/ SOCw/ SOC With SOC, a nodal line remains on AL
Bulk band dispersion of 2H-NbS2

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14

ARPES experiments

Hexagonal Brillouin zone

Band dispersion around L (ℎ𝜈 = 525 eV)ML

 Two bands were gapped (0.5 eV)
 All the observed L points hadgapped dispersions
 21 screw rotation symmetry isbroken in the near-surface
electronic structure?
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Band dispersion along ML(ℎ𝜈 = 370 ~ 750 eV)M15

Slab calculations

 20 bilayers (120 atoms, 1000 electrons) for no spin calculations
 10 bilayers for noncollinear calculations
 Reproduced gapped spectra by photoemission intensity calculations
 Probing depth parameter ~ 1 nm

Slab band dispersion(spin off)

ARPES spectra

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Simulation spectra

20 bilayers
2H-NbS2

Future work and summary

17

Beyond the free-electron approximation

Three-step model / Free-electron approximation

Precise description of photoelectron wave function→ Photoelectron wave function from the local Kohn-Sham potential

Local Kohn-Sham potential of 1T-TiS2 slab
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18Summary

Photoemission simulator SPADExp
 Directly use the ground state wave function exported from OpenMX
 Able to handle large slab systems and surface electronic structure
ARPES study of nodal line materials
 Bulk band degeneracy due to nonsymmorphic symmetry was not absent
 Photoemission intensity calculations considering surface sensitivitysuccessfully reproduced gapped spectra

ARPES
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Simulation

References

20References

Photoelectric effect, photoemission spectroscopy H. Hertz, Annalen der Physik 267, 983 (1887). (Discovery of the photoelectric effect) A. Einstein, Annalen der Physik 322, 132 (1905). (Light quantum hypothesis)
 S. Hüfner, Photoelectron Spectroscopy (Springer, 2003). (Popular textbook) J. A. Sobota et al., Rev. Mod. Phys. 93, 025006 (2021). (Recent review paper)Band degeneration due to symmetry 犬井, 田辺, 小野寺, 「応用群論」 (裳華房, 2003). (Textbook in Japanese) C. J. Bradley and A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Oxford, 2010).
(Textbook in English) Supplementary Section E in G. Chang et al., Nat. Mater. 17, 978 (2018). (Example of 21 screw
rotation symmetry) Bilbao Crystallographic Server (https://www.cryst.ehu.es/). (Utility webpage)

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21References

Photoemission angular distribution calculations (theory) T. Matsushita et al., Phys. Rev. B 56, 7687 (1997). (tight-binding model)

 A. F. Starace, Phys. Rev. A 3, 1242 (1971); A. J. Read and R. J. Needs, Phys. Rev. B 44, 13071(1991). (position operator vs. momentum operator when nonlocal potential exists)

Photoemission intensity calculations (tight-binding model)
 E. L. Shirley et al., Phys. Rev. B 51, 13614 (1995); H. Nishimoto et al., Solid State Commn. 98,671 (1996). (graphene)

First-principles calculations
 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964); W. Kohn and L. J. Sham, Phys. Rev.
140, A1133 (1965). (Density functional theory) T. Ozaki, Phys. Rev. B 67, 155108 (2003). (Theory for OpenMX)
 https://www.openmx-square.org/ (OpenMX software)

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22ReferencesSPADExp
 H. Tanaka, K. Kuroda, and T. Matsushita, J. Electron Spectrosc. 264, 147297(2023). (Paper about the software development) https://github.com/Hiroaki-Tanaka-0606/SPADExp (Repository)

Inelastic mean free path
 H. Kanter, Phys. Rev. B 1, 522 (1970). (Electron beam transmittance) M. Klasson et al., Physica Scripta 5, 93 (1972). (XPS) M. P. Seah and W. A. Dench, Surf. Interf. Anal. 1, 2 (1979). (so-called the universal curve) W. F. Krolikowski and W. E. Spicer, Phys. Rev. 185, 882 (1969) and Phys. Rev. B 1, 478 (1970).
(theory)

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23

References

ARPES
 H. Tanaka, S. Okazaki et al., Phys. Rev. B 105, L121102 (2022). (NbS2 & 3d-intercalated system) D. Huang et al., Phys. Rev. B 105, 245145 (2022) (NbS2)
 H. Tanaka et al., under review. (arXiv:2308.00999) (NbS2 & hBN)

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