- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Open Access
  • Access by Xinjiang University

Extreme-ultraviolet optical response of atomically thin molybdenum disulfide

Giacomo Fiorentini1,*, Nicola Di Palo1,*, Giacomo Inzani1, Gian Luca Dolso1, Simone Bonetti1, Qiuyang Li2, Fang Liu2,3, Xiaoyang Zhu2, Angelo Giglia4,5 et al.

Nicola Mahne4,5, Luca Pasquali4,6,7, Marco D'Alessandro8, Mikhail Malakhov9, María Camarasa-Gómez10, Juan José Esteve-Paredes11, Juan José Palacios11, Rocío Borrego-Varillas12, Mauro Nisoli1,12, Antonio Picón13, Davide Sangalli14, and Matteo Lucchini1,12,†

  • *These authors contributed equally to this work
  • †Contact author: matteo.lucchini@polimi.it
PDF

Phys. Rev. B 114, 185419 – Published 16 September, 2026

DOI: https://doi.org/10.1103/xbcw-n8sc

Abstract

We report multiangle reflectivity measurements in the extreme-ultraviolet (XUV) range for mono- and bilayer MoS2 on a Si3N4 substrate. Using a single-sheet 2D conductivity model, we extract the complex optical response of the MoS2 bilayer between 25 and 90 eV and derive an effective refractive index by introducing a thickness equal to the interlayer spacing. The MoS2 monolayer response is consistently reproduced either by halving the 2D conductivity or the effective thickness, indicating a robust scaling with layer number. Conversely from what was previously observed around the energy gap, the resulting optical constants display a broad resonance at the Mo N2,3 edge with no clear core-exciton peaks below the absorption edge despite the reduced dimensionality. First-principles calculations reproduce the experimental results and show that local-field (Hartree) effects dominate the XUV response, while screened-exchange (SEX) contributions remain weak and mainly induce spectral shifts. Our analysis demonstrates that excitonic effects play a minor role in the XUV optical response of atomically thin MoS2, highlighting key differences with respect to the visible and infrared regimes, and calling for a reassessment of the use of Mo-based transition metal dichalcogenides in attosecond spectroscopy and XUV excitonics.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (103)

  1. G. Moody, J. Schaibley, and X. Xu, Exciton dynamics in monolayer transition metal dichalcogenides, J. Opt. Soc. Am. B 33, C39 (2016).
  2. G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand, and B. Urbaszek, Colloquium: Excitons in atomically thin transition metal dichalcogenides, Rev. Mod. Phys. 90, 021001 (2018).
  3. Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides: MoS2, MoSe2, WS2, and WSe2, Phys. Rev. B 90, 205422 (2014).
  4. S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev, and A. Kis, 2D transition metal dichalcogenides, Nat. Rev. Mater. 2, 17033 (2017).
  5. Y. Jing, B. Liu, X. Zhu, F. Ouyang, J. Sun, and Y. Zhou, Tunable electronic structure of two-dimensional transition metal chalcogenides for optoelectronic applications, Nanophotonics 9, 1675 (2020).
  6. C. Gong, Y. Zhang, W. Chen, J. Chu, T. Lei, J. Pu, L. Dai, C. Wu, Y. Cheng, T. Zhai, L. Li, and J. Xiong, Electronic and optoelectronic applications based on 2D novel anisotropic transition metal dichalcogenides, Adv. Sci. 4, 1700231 (2017).
  7. K. F. Mak and J. Shan, Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides, Nat. Photonics 10, 216 (2016).
  8. A. Ciarrocchi, F. Tagarelli, A. Avsar, and A. Kis, Excitonic devices with van der Waals heterostructures: Valleytronics meets twistronics, Nat. Rev. Mater. 7, 449 (2022).
  9. J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, Valleytronics in 2D materials, Nat. Rev. Mater. 1, 16055 (2016).
  10. T. Mueller and E. Malic, Exciton physics and device application of two-dimensional transition metal dichalcogenide semiconductors, npj 2D Mater. Appl. 2, 29 (2018).
  11. K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Atomically thin MoS2: A new direct-gap semiconductor, Phys. Rev. Lett. 105, 136805 (2010).
  12. D. Y. Qiu, F. H. da Jornada, and S. G. Louie, Optical spectrum of MoS2: Many-body effects and diversity of exciton states, Phys. Rev. Lett. 111, 216805 (2013).
  13. Alejandro Molina-Sánchez, M. Palummo, A. Marini, and L. Wirtz, Temperature-dependent excitonic effects in the optical properties of single-layer MoS2, Phys. Rev. B 93, 155435 (2016).
  14. Y. Liu, Y. Gao, S. Zhang, J. He, J. Yu, and Z. Liu, Valleytronics in transition metal dichalcogenides materials, Nano Res. 12, 2695 (2019).
  15. F. Liu, W. Wu, Y. Bai, S. H. Chae, Q. Li, J. Wang, J. Hone, and X.-Y. Zhu, Disassembling 2D van der Waals crystals into macroscopic monolayers and reassembling into artificial lattices, Science 367, 903 (2020).
  16. S. Carr, S. Fang, and E. Kaxiras, Electronic-structure methods for twisted moiré layers, Nat. Rev. Mater. 5, 748 (2020).
  17. E. C. Regan, D. Wang, E. Y. Paik, Y. Zeng, L. Zhang, J. Zhu, A. H. MacDonald, H. Deng, and F. Wang, Emerging exciton physics in transition metal dichalcogenide heterobilayers, Nat. Rev. Mater. 7, 778 (2022).
  18. S. Li, K. Wei, Q. Liu, Y. Tang, and T. Jiang, Twistronics and moiré excitonic physics in van der Waals heterostructures, Front. Phys. 19, 42501 (2024).
  19. Y. Li, J. Shi, Y. Mi, X. Sui, H. Xu, and X. Liu, Ultrafast carrier dynamics in two-dimensional transition metal dichalcogenides, J. Mater. Chem. C 7, 4304 (2019).
  20. C. Trovatello, F. Katsch, N. J. Borys, M. Selig, K. Yao, R. Borrego-Varillas, F. Scotognella, I. Kriegel, A. Yan, A. Zettl, et al., The ultrafast onset of exciton formation in 2D semiconductors, Nat. Commun. 11, 5277 (2020).
  21. M. Selig, F. Katsch, R. Schmidt, S. Michaelis de Vasconcellos, R. Bratschitsch, E. Malic, and A. Knorr, Ultrafast dynamics in monolayer transition metal dichalcogenides: Interplay of dark excitons, phonons, and intervalley exchange, Phys. Rev. Res. 1, 022007R (2019).
  22. L. Waldecker, R. Bertoni, H. Hübener, T. Brumme, T. Vasileiadis, D. Zahn, A. Rubio, and R. Ernstorfer, Momentum-resolved view of electron-phonon coupling in multilayer WSe2, Phys. Rev. Lett. 119, 036803 (2017).
  23. C. Heide, P. D. Keathley, and M. F. Kling, Petahertz electronics, Nat. Rev. Phys. 6, 648 (2024).
  24. S. M. Cavaletto, K. M. Kowalczyk, F. O. Navarrete, and J. Rivera-Dean, The attoscience of strong-field-driven solids, Nat. Rev. Phys. 7, 38 (2025).
  25. F. Gucci, E. B. Molinero, M. Russo, P. San-Jose, F. V. A. Camargo, M. Maiuri, M. Ivanov, Á. Jiménez-Galán, R. E. F. Silva, S. D. Conte, and G. Cerullo, Encoding and manipulating ultrafast coherent valleytronic information with lightwaves, Nat. Photonics 20, 266 (2026).
  26. A. O. Slobodeniuk, P. Koutenský, M. Bartoš, F. Trojánek, P. Malý, T. Novotný, and M. Kozák, Ultrafast valley-selective coherent optical manipulation with excitons in WSe2 and MoS2 monolayers, npj 2D Mater. Appl. 7, 17 (2023).
  27. J. Kim, X. Hong, C. Jin, S.-F. Shi, C.-Y. S. Chang, M.-H. Chiu, L.-J. Li, and F. Wang, Ultrafast generation of pseudo-magnetic field for valley excitons in WSe2 monolayers, Science 346, 1205 (2014).
  28. M. F. C. M. Quintela, M. Sá, A. J. Uría-Álvarez, M. Malakhov, G. Cistaro, J. Quereda, J. J. Palacios, and A. Picón, Quantum valley pseudospin controlled by strain, Phys. Rev. Res. 7, 043265 (2025).
  29. N. S. Kamath, A. Soni, and S. K. Pal, Valley properties and dynamics in 2D transition metal dichalcogenides: Prospects for valleytronic applications, J. Phys. Chem. Lett. 17, 25 (2026).
  30. R. Borrego-Varillas, M. Lucchini, and M. Nisoli, Attosecond spectroscopy for the investigation of ultrafast dynamics in atomic, molecular and solid-state physics, Rep. Prog. Phys. 85, 066401 (2022).
  31. A. Zong, B. R. Nebgen, S.-C. Lin, J. A. Spies, and M. Zuerch, Emerging ultrafast techniques for studying quantum materials, Nat. Rev. Mater. 8, 224 (2023).
  32. P. Agostini, Nobel lecture: Genesis and applications of attosecond pulse trains, Rev. Mod. Phys. 96, 030501 (2024).
  33. Anne L'Huillier, Nobel lecture: The route to attosecond pulses, Rev. Mod. Phys. 96, 030503 (2024).
  34. F. Krausz, Nobel lecture: Sub-atomic motions, Rev. Mod. Phys. 96, 030502 (2024).
  35. O. Alexander, D. Ayuso, M. Matthews, L. Rego, J. W. G. Tisch, B. Weaver, and J. P. Marangos, Attosecond physics and technology, Appl. Phys. Lett. 126, 170501 (2025).
  36. A. R. Attar, H.-T. Chang, and A. Britz, Simultaneous observation of carrier-specific redistribution and coherent lattice dynamics, ACS Nano, 14, 15829 (2020).
  37. H.-T. Chang, A. Guggenmos, C. T. Chen, J. Oh, R. Géneaux, Y.-D. Chuang, A. M. Schwartzberg, S. Aloni, D. M. Neumark, and S. R. Leone, Coupled valence carrier and core-exciton dynamics in WS2 probed by few-femtosecond extreme ultraviolet transient absorption spectroscopy, Phys. Rev. B 104, 064309 (2021).
  38. B. Buades, A. Picón, E. Berger, I. León, N. Di Palo, S. L. Cousin, C. Cocchi, E. Pellegrin, J. H. Martin, S. Mañas-Valero, E. Coronado, T. Danz, C. Draxl, M. Uemoto, K. Yabana, M. Schultze, S. Wall, M. Zürch, and J. Biegert, Attosecond state-resolved carrier motion in quantum materials probed by soft x-ray XANES, Appl. Phys. Rev. 8, 011408 (2021).
  39. Z. Schumacher, S. A. Sato, S. Neb, A. Niedermayr, L. Gallmann, A. Rubio, and U. Keller, Ultrafast electron localization and screening in a transition metal dichalcogenide, Proc. Natl. Acad. Sci. USA 120, e2221725120 (2023).
  40. J. Oh, H.-T. Chang, C. T. Chen, S. Aloni, A. Schwartzberg, and S. R. Leone, Carrier and phonon dynamics in multilayer WSe2 captured by extreme ultraviolet transient absorption spectroscopy, J. Phys. Chem. C 127, 5004 (2023).
  41. R. Geneaux, H. J. B. Marroux, A. Guggenmos, D. M. Neumark, and S. R. Leone, Transient absorption spectroscopy using high harmonic generation: A review of ultrafast x-ray dynamics in molecules and solids, Philos. Trans. R. Soc. A 377, 20170463 (2019).
  42. N. Di Palo, G. Inzani, G. L. Dolso, M. Talarico, S. Bonetti, and M. Lucchini, Attosecond absorption and reflection spectroscopy of solids, APL Photonics 9, 020901 (2024).
  43. G. Inzani and M. Lucchini, Attosecond electron dynamics in solid-state systems, J. Phys.: Photonics 7, 022001 (2025).
  44. A. Moulet, J. B. Bertrand, T. Klostermann, A. Guggenmos, N. Karpowicz, and E. Goulielmakis, Soft x-ray excitonics, Science 357, 1134 (2017).
  45. R. Géneaux, C. J. Kaplan, L. Yue, A. D. Ross, J. E. Bækhøj, P. M. Kraus, H.-t. Chang, A. Guggenmos, M.-Y. Huang, M. Zürch, K. J. Schafer, D. M. Neumark, M. B. Gaarde, and S. R. Leone, Attosecond time-domain measurement of core-level-exciton decay in magnesium oxide, Phys. Rev. Lett. 124, 207401 (2020).
  46. M. Lucchini, S. A. Sato, G. D. Lucarelli, B. Moio, G. Inzani, R. Borrego-Varillas, F. Frassetto, L. Poletto, H. Hübener, U. De Giovannini, A. Rubio, and M. Nisoli, Unravelling the intertwined atomic and bulk nature of localised excitons by attosecond spectroscopy, Nat. Commun. 12, 1021 (2021).
  47. A. Geondzhian, A. Rubio, and M. Altarelli, Valley selectivity of soft x-ray excitations of core electrons in two-dimensional transition metal dichalcogenides, Phys. Rev. B 106, 115433 (2022).
  48. K. J. Gannan, L. B. Drescher, R. Quintero-Bermudez, N. Rana, C. Huang, K. Schafer, M. B. Gaarde, and S. R. Leone, Polarization-resolved core exciton dynamics in LiF using attosecond transient absorption spectroscopy, Phys. Rev. B 111, 134306 (2025).
  49. A. C. Thompson, D. T. Attwood, E. M. Gullikson, M. R. Howells, J. B. Kortright, A. L. Robinson, J. H. Underwood, K.-J. Kim, J. Kirz, I. Lindau, P. Pianetta, H. Winick, G. P. Williams, and D. Vaughan, X-ray Data Booklet (Lawrence Berkeley National Laboratory, 2009).
  50. A. Parija, Y.-H. Choi, Z. Liu, J. L. Andrews, L. R. De Jesus, S. C. Fakra, M. Al-Hashimi, J. D. Batteas, D. Prendergast, and S. Banerjee, Mapping catalytically relevant edge electronic states of MoS2, ACS Cent. Sci. 4, 493 (2018).
  51. S. W. Han, H. Kwon, S. K. Kim, S. Ryu, W. S. Yun, D. H. Kim, J. H. Hwang, J.-S. Kang, J. Baik, H. J. Shin, and S. C. Hong, Band-gap transition induced by interlayer van der Waals interaction in MoS2, Phys. Rev. B 84, 045409 (2011).
  52. S. Nannarone, F. Borgatti, A. DeLuisa, B. Doyle, G. Gazzadi, A. Giglia, P. Finetti, N. Mahne, L. Pasquali, M. Pedio, et al., The BEAR beamline at ELETTRA, in AIP Conference Proceedings (American Institute of Physics, 2004), Vol. 705, pp. 450–453.
  53. L. Pasquali, A. De Luisa, and S. Nannarone, The UHV experimental chamber for optical measurements (reflectivity and absorption) and angle resolved photoemission of the BEAR beamline at ELETTRA, in AIP Conference Proceedings (American Institute of Physics, 2004), Vol. 705, pp. 1142–1145.
  54. J. C. Fuggle and N. Mårtensson, Core-level binding energies in metals, J. Electron Spectrosc. Relat. Phenom. 21, 275 (1980).
  55. N. Kumar, A. T. Kozakov, A. V. Nezhdanov, R. M. Smertin, V. N. Polkovnikov, N. I. Chkhalo, A. I. Mashin, A. N. Nikolskii, A. A. Scrjabin, and S. Y. Zuev, Quantum confinement effect in a nanoscale Mo/Si multilayer structure, J. Phys. Chem. C 124, 17795 (2020).
  56. D. J. Trainer, A. V. Putilov, C. D. Giorgio, T. Saari, B. Wang, M. Wolak, R. U. Chandrasena, C. Lane, T.-R. Chang, H.-T. Jeng, H. Lin, F. Kronast, A. X. Gray, X. Xi, J. Nieminen, A. Bansil, and M. Iavarone, Inter-layer coupling induced valence band edge shift in mono- to few-layer MoS2, Sci. Rep. 7, 40559 (2017).
  57. M. Lucchini, L. Castiglioni, L. Kasmi, P. Kliuiev, A. Ludwig, M. Greif, J. Osterwalder, M. Hengsberger, L. Gallmann, and U. Keller, Light-matter interaction at surfaces in the spatiotemporal limit of macroscopic models, Phys. Rev. Lett. 115, 137401 (2015).
  58. S. Yoo and Q. Park, Spectroscopic ellipsometry for low‐dimensional materials and heterostructures, Nanophotonics 11, 2811 (2022).
  59. L. Pasquali, S. Mukherjee, F. Terzi, A. Giglia, N. Mahne, K. Koshmak, V. Esaulov, C. Toccafondi, M. Canepa, and S. Nannarone, Structural and electronic properties of anisotropic ultrathin organic films from dichroic resonant soft x-ray reflectivity, Phys. Rev. B 89, 045401 (2014).
  60. R. Capelli, N. Mahne, K. Koshmak, A. Giglia, B. P. Doyle, S. Mukherjee, S. Nannarone, and L. Pasquali, Quantitative resonant soft x-ray reflectivity of ultrathin anisotropic organic layers: Simulation and experiment of PTCDA on Au, J. Chem. Phys. 145, 024201 (2016).
  61. L. Pasquali, N. Mahne, A. Giglia, A. Verna, L. Sponza, R. Capelli, M. Bonfatti, F. Mezzadri, E. Galligani, and S. Nannarone, Analysis of resonant soft x-ray reflectivity of anisotropic layered materials, Surfaces 4, 18 (2021).
  62. Bruno Majérus, E. Dremetsika, M. Lobet, L. Henrard, and P. Kockaert, Electrodynamics of two-dimensional materials: Role of anisotropy, Phys. Rev. B 98, 125419 (2018).
  63. C. J. Kaplan, P. M. Kraus, E. M. Gullikson, L. Borja, S. K. Cushing, M. Zürch, H.-T. Chang, D. M. Neumark, and S. R. Leone, Retrieval of the complex-valued refractive index of germanium near the M4,5 absorption edge, J. Opt. Soc. Am. B 36, 1716 (2019).
  64. B. L. Henke, E. M. Gullikson, and J. C. Davis, X-ray interactions: Photoabsorption, scattering, transmission, and reflection at E=50−30,000eV, Z=1−92, At. Data Nucl. Data Tables 54, 181 (1993).
  65. R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. Peres, and A. K. Geim, Fine structure constant defines visual transparency of graphene, Science 320, 1308 (2008).
  66. M. Merano, Fresnel coefficients of a two-dimensional atomic crystal, Phys. Rev. A 93, 013832 (2016).
  67. Y. Li and T. F. Heinz, Two-dimensional models for the optical response of thin films, 2D Mater. 5, 025021 (2018).
  68. P. Wang, W. Gou, T. Jiang, W. Zhao, K. Ding, H. Sheng, X. Liu, Q. Xu, and Q. Fan, An interlayer spacing design approach for efficient sodium ion storage in N-doped MoS2, Nanoscale Horiz. 8, 473 (2023).
  69. J. Xiao, M. Long, X. Li, Q. Zhang, H. Xu, and K. S. Chan, Effects of van der Waals interaction and electric field on the electronic structure of bilayer MoS2, J. Phys.: Condens. Matter 26, 405302 (2014).
  70. L. G. Parratt, Surface studies of solids by total reflection of x-rays, Phys. Rev. 95, 359 (1954).
  71. G. Ulian and G. Valdre, Structural, elastic, electronic, optical and vibrational properties of single-layered, bilayered and bulk molybdenite MoS2-2H, J. Appl. Crystallogr. 56, 611 (2023).
  72. S. A. Yamusa, A. Shaari, N. A. Alsaif, I. M. Alsalamah, I. Isah, and N. Rekik, Elucidating the structural, electronic, elastic, and optical properties of bulk and monolayer MoS2 transition-metal dichalcogenides: A DFT approach, ACS Omega 7, 45719 (2022).
  73. T. J. Whitcher, A. D. Fauzi, D. Caozheng, X. Chi, A. Syahroni, T. C. Asmara, M. B. H. Breese, A. H. C. Neto, A. T. S. Wee, M. A. Majidi, and A. Rusydi, Unravelling strong electronic interlayer and intralayer correlations in a transition metal dichalcogenide, Nat. Commun. 12, 6980 (2021).
  74. A. Molina-Sánchez, D. Sangalli, K. Hummer, A. Marini, and L. Wirtz, Effect of spin-orbit interaction on the optical spectra of single-layer, double-layer, and bulk MoS2, Phys. Rev. B 88, 045412 (2013).
  75. G. Onida, L. Reining, and A. Rubio, Electronic excitations: Density-functional versus many-body Green's-function approaches, Rev. Mod. Phys. 74, 601 (2002).
  76. C. Attaccalite, M. Grüning, and A. Marini, Real-time approach to the optical properties of solids and nanostructures: Time-dependent Bethe-Salpeter equation, Phys. Rev. B 84, 245110 (2011).
  77. Y.-H. Chan, D. Y. Qiu, F. H. da Jornada, and S. G. Louie, Giant exciton-enhanced shift currents and direct current conduction with subbandgap photo excitations produced by many-electron interactions, Proc. Natl. Acad. Sci. USA 118, e1906938118 (2021).
  78. G. Cistaro, M. Malakhov, J. J. Esteve-Paredes, A. J. Uría-Álvarez, R. E. F. Silva, F. Martín, J. J. Palacios, and A. Picón, Theoretical approach for electron dynamics and ultrafast spectroscopy (EDUS), J. Chem. Theory Comput. 19, 333 (2023).
  79. M. Malakhov, G. Cistaro, F. Martín, and A. Picón, Exciton migration in two-dimensional materials, Commun. Phys. 7, 196 (2024).
  80. J. F. P. Mosquera, G. Cistaro, M. Malakhov, E. Pisanty, A. Dauphin, L. Plaja, A. Chacón, M. Lewenstein, and A. Picón, Topological phase transitions via attosecond x-ray absorption spectroscopy, Rep. Prog. Phys. 87, 117901 (2024).
  81. D. Sangalli, R. Reho, I. M. Alliati, J. Cervantes-Villanueva, A. Esquembre Kučukalić, T. Geirsson, D. A. Leon Valido, B. Mellado Pinto, P. Milev, M. Nalabothula, F. Paleari, A. Romani, L. Bellentani, J. M. Castelo Ares, M. D'Alessandro, S. Laricchia, E. M. Montagna, S. Orlandini, J. E. Roman, F. Affinito, et al., Lumen, Zenodo, 2026, 10.5281/zenodo.20121198.
  82. D. Sangalli, A. Ferretti, H. Miranda, C. Attaccalite, I. Marri, E. Cannuccia, P. Melo, M. Marsili, F. Paleari, A. Marrazzo, G. Prandini, P. Bonfà, M. O. Atambo, F. Affinito, M. Palummo, A. Molina-Sánchez, C. Hogan, M. Grüning, D. Varsano, and A. Marini, Many-body perturbation theory calculations using the Yambo code, J. Phys.: Condens. Matter 31, 325902 (2019).
  83. M. Camarasa-Gómez, A. Ramasubramaniam, J. B. Neaton, and L. Kronik, Transferable screened range-separated hybrid functionals for electronic and optical properties of van der Waals materials, Phys. Rev. Mater. 7, 104001 (2023).
  84. P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. Dal Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, et al., Advanced capabilities for materials modelling with Quantum ESPRESSO, J. Phys.: Condens. Matter 29, 465901 (2017).
  85. G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Ibañez-Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, et al., Wannier90 as a community code: New features and applications, J. Phys.: Condens. Matter 32, 165902 (2020).
  86. P. Milev, B. Mellado-Pinto, M. Nalabothula, A. Esquembre-Kučukalić, F. Alvarruiz, E. Ramos, F. Filippone, A. Molina-Sanchez, L. Wirtz, J. E. Roman, and D. Sangalli, Solvers for the Hermitian and the pseudo-Hermitian Bethe–Salpeter equation in the Yambo code: Implementation and performance, Comput. Cond. Matter 47, e01279 (2026).
  87. J. Stöhler, S. Blügel, and C. Friedrich, Efficient all-electron Bethe-Salpeter implementation using crystal symmetries, arXiv:2603.24860.
  88. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, et al., Quantum ESPRESSO: A modular and open-source software project for quantum simulations of materials, J. Phys.: Condens. Matter 21, 395502 (2009).
  89. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  90. D. R. Hamann, Optimized norm-conserving Vanderbilt pseudopotentials, Phys. Rev. B 88, 085117 (2013).
  91. A. Guandalini, P. D'Amico, A. Ferretti, and D. Varsano, Efficient GW calculations in two dimensional materials through a stochastic integration of the screened potential, npj Comput. Mater. 9, 44 (2023).
  92. P. Cudazzo, I. V. Tokatly, and A. Rubio, Dielectric screening in two-dimensional insulators: Implications for excitonic and impurity states in graphane, Phys. Rev. B 84, 085406 (2011).
  93. L. X. Benedict, Screening in the exchange term of the electron-hole interaction of the Bethe-Salpeter equation, Phys. Rev. B 66, 193105 (2002).
  94. T. Deilmann and K. S. Thygesen, Important role of screening the electron-hole exchange interaction for the optical properties of molecules near metal surfaces, Phys. Rev. B 99, 045133 (2019).
  95. F. Hüser, T. Olsen, and K. S. Thygesen, How dielectric screening in two-dimensional crystals affects the convergence of excited-state calculations: Monolayer MoS2, Phys. Rev. B 88, 245309 (2013).
  96. D. Y. Qiu, F. H. da Jornada, and S. G. Louie, Screening and many-body effects in two-dimensional crystals: Monolayer MoS2, Phys. Rev. B 93, 235435 (2016).
  97. T. Tian, D. Scullion, D. Hughes, L. H. Li, C.-J. Shih, J. Coleman, M. Chhowalla, and E. J. G. Santos, Electronic polarizability as the fundamental variable in the dielectric properties of two-dimensional materials, Nano Lett. 20, 841 (2020).
  98. F. Sottile, Response functions of semiconductors and insulators: From the Bethe-Salpeter equation to time-dependent density functional theory, Ph.D. thesis, Ecole Polytechnique X, 2003.
  99. M. Bruno, M. Palummo, A. Marini, R. Del Sole, V. Olevano, A. N. Kholod, and S. Ossicini, Excitons in germanium nanowires: Quantum confinement, orientation, and anisotropy effects within a first-principles approach, Phys. Rev. B 72, 153310 (2005).
  100. E. Chang, G. Bussi, A. Ruini, and E. Molinari, Excitons in carbon nanotubes: An ab initio symmetry-based approach, Phys. Rev. Lett. 92, 196401 (2004).
  101. J. Hong, K. Li, C. Jin, X. Zhang, Z. Zhang, and J. Yuan, Layer-dependent anisotropic electronic structure of freestanding quasi-two-dimensional MoS2, Phys. Rev. B 93, 075440 (2016).
  102. G. A. Ermolaev, D. V. Grudinin, Y. V. Stebunov, K. V. Voronin, V. G. Kravets, J. Duan, A. B. Mazitov, G. I. Tselikov, A. Bylinkin, D. I. Yakubovsky, S. M. Novikov, D. G. Baranov, A. Y. Nikitin, I. A. Kruglov, T. Shegai, P. Alonso-González, A. N. Grigorenko, A. V. Arsenin, K. S. Novoselov, and V. S. Volkov, Giant optical anisotropy in transition metal dichalcogenides for next-generation photonics, Nat. Commun. 12, 854 (2021).
  103. G. Fiorentini, Figure data for “Extreme-ultraviolet optical response of atomically-thin molybdenum disulfide” [Data set], Zenodo, 2026, https://doi.org/10.5281/zenodo.21162549.

Outline

Information

Phys. Rev. B 114, 185419– Published 16 September, 2026
  • Received 15 June 2026
  • Revised 19 August 2026
  • Accepted 20 August 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation