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Role of Nonequilibrium Populations in Dark-Exciton Formation

Paul Werner1, Wiebke Bennecke1, Jan Philipp Bange1, Giuseppe Meneghini2,3, David Schmitt1, Marco Merboldt1, Anna M. Seiler1, AbdulAziz AlMutairi4,5, Kenji Watanabe6 et al.

Takashi Taniguchi7, G. S. Matthijs Jansen1, Junde Liu1, Daniel Steil1, Stephan Hofmann4, R. Thomas Weitz1,8, Ermin Malic2,3,9, Stefan Mathias1,8,*, and Marcel Reutzel1,2,3,†

  • *Contact author: smathias@uni-goettingen.de
  • †Contact author: marcel.reutzel@uni-marburg.de
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Phys. Rev. Lett. 136, 186903 – Published 8 May, 2026

DOI: https://doi.org/10.1103/w29j-z48v

Abstract

The optical excitation of a bright exciton may be followed by the formation of lower-energy dark states. In these formation and relaxation processes, nonequilibrium exciton and phonon populations play a dominant role but remain so far largely unexplored, as most states are inaccessible by regular spectroscopies. Here, on the example of homobilayer 2H−MoS2, we realize direct access to the full exciton relaxation cascade from experiment and theory. We find distinct changes in the time-, energy-, and in-plane momentum-resolved photoemission spectral function that can be explained only when considering the formation and subsequent thermalization of excitonic nonequilibrium occupation distributions. In agreement with microscopic many-particle calculations, we quantify the timescales for the formation of a nonequilibrium dark-excitonic occupation and its subsequent thermalization to 85 and 150 fs, respectively. Our results provide a previously inaccessible view of the complete exciton relaxation cascade, which is of importance for the future characterization of nonequilibrium excitonic phases and the efficient design of optoelectronic devices.

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References (73)

  1. S.-J. Liang, B. Cheng, X. Cui, and F. Miao, Van der Waals heterostructures for high-performance device applications: Challenges and opportunities, Adv. Mater. 32, 1903800 (2020).
  2. T. Mueller and E. Malic, Exciton physics and device application of two-dimensional transition metal dichalcogenide semiconductors, npj 2D Mater. Appl. 2, 29 (2018).
  3. 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).
  4. R. Perea-Causin, D. Erkensten, J. M. Fitzgerald, J. J. P. Thompson, R. Rosati, S. Brem, and E. Malic, Exciton optics, dynamics, and transport in atomically thin semiconductors, APL Mater. 10, 100701 (2022).
  5. D. A. Ruiz-Tijerina and V. I. Fal’ko, Interlayer hybridization and moiré superlattice minibands for electrons and excitons in heterobilayers of transition-metal dichalcogenides, Phys. Rev. B 99, 125424 (2019).
  6. Y. Wang, Z. Wang, W. Yao, G.-B. Liu, and H. Yu, Interlayer coupling in commensurate and incommensurate bilayer structures of transition-metal dichalcogenides, Phys. Rev. B 95, 115429 (2017).
  7. A. Chernikov, A. M. van der Zande, H. M. Hill, A. F. Rigosi, A. Velauthapillai, J. Hone, and T. F. Heinz, Electrical tuning of exciton binding energies in monolayer WS2, Phys. Rev. Lett. 115, 126802 (2015).
  8. A. Raja, A. Chaves, J. Yu, G. Arefe, H. M. Hill, A. F. Rigosi, T. C. Berkelbach, P. Nagler, C. Schüller, T. Korn, C. Nuckolls, J. Hone, L. E. Brus, T. F. Heinz, D. R. Reichman, and A. Chernikov, Coulomb engineering of the bandgap and excitons in two-dimensional materials, Nat. Commun. 8, 15251 (2017).
  9. E. M. Alexeev, D. A. Ruiz-Tijerina, M. Danovich, M. J. Hamer, D. J. Terry, P. K. Nayak, S. Ahn, S. Pak, J. Lee, J. I. Sohn, M. R. Molas, M. Koperski, K. Watanabe, T. Taniguchi, K. S. Novoselov, R. V. Gorbachev, H. S. Shin, V. I. Fal’ko, and A. I. Tartakovskii, Resonantly hybridized excitons in moiré superlattices in van der Waals heterostructures, Nature (London) 567, 81 (2019).
  10. K. Tran et al., Evidence for moiré excitons in van der Waals heterostructures, Nature (London) 567, 71 (2019).
  11. K. L. Seyler, P. Rivera, H. Yu, N. P. Wilson, E. L. Ray, D. G. Mandrus, J. Yan, W. Yao, and X. Xu, Signatures of moiré-trapped valley excitons in MoSe2/WSe2 heterobilayers, Nature (London) 567, 66 (2019).
  12. Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, V. Elser, K. F. Mak, and J. Shan, Correlated insulating states at fractional fillings of moiré superlattices, Nature (London) 587, 214 (2020).
  13. C. Jin, E. Y. Ma, O. Karni, E. C. Regan, F. Wang, and T. F. Heinz, Ultrafast dynamics in van der Waals heterostructures, Nat. Nanotechnol. 13, 994 (2018).
  14. X.-X. Zhang, Y. You, S. Y. F. Zhao, and T. F. Heinz, Experimental evidence for dark excitons in monolayer WSe2, Phys. Rev. Lett. 115, 257403 (2015).
  15. A. Raja, M. Selig, G. Berghäuser, J. Yu, H. M. Hill, A. F. Rigosi, L. E. Brus, A. Knorr, T. F. Heinz, E. Malic, and A. Chernikov, Enhancement of exciton–phonon scattering from monolayer to bilayer WS2, Nano Lett. 18, 6135 (2018).
  16. X. Jiang, Q. Zheng, Z. Lan, W. A. Saidi, X. Ren, and J. Zhao, Real-time GW-BSE investigations on spin-valley exciton dynamics in monolayer transition metal dichalcogenide, Sci. Adv. 7, eabf3759 (2021).
  17. M. Selig, G. Berghäuser, M. Richter, R. Bratschitsch, A. Knorr, and E. Malic, Dark and bright exciton formation, thermalization, and photoluminescence in monolayer transition metal dichalcogenides, 2D Mater. 5, 035017 (2018).
  18. P. Merkl, F. Mooshammer, P. Steinleitner, A. Girnghuber, K. Q. Lin, P. Nagler, J. Holler, C. Schüller, J. M. Lupton, T. Korn, S. Ovesen, S. Brem, E. Malic, and R. Huber, Ultrafast transition between exciton phases in van der Waals heterostructures, Nat. Mater. 18, 691 (2019).
  19. J. Madéo, M. K. L. Man, C. Sahoo, M. Campbell, V. Pareek, E. L. Wong, A. Al-Mahboob, N. S. Chan, A. Karmakar, B. M. K. Mariserla, X. Li, T. F. Heinz, T. Cao, and K. M. Dani, Directly visualizing the momentum-forbidden dark excitons and their dynamics in atomically thin semiconductors, Science 370, 1199 (2020).
  20. R. Wallauer, R. Perea-Causin, L. Münster, S. Zajusch, S. Brem, J. Güdde, K. Tanimura, K.-Q. Lin, R. Huber, E. Malic, and U. Höfer, Momentum-resolved observation of exciton formation dynamics in monolayer WS2, Nano Lett. 21, 5867 (2021).
  21. J. P. Bange, P. Werner, D. Schmitt, W. Bennecke, G. Meneghini, A. AlMutairi, M. Merboldt, K. Watanabe, T. Taniguchi, S. Steil, D. Steil, R. T. Weitz, S. Hofmann, G. S. M. Jansen, S. Brem, E. Malic, M. Reutzel, and S. Mathias, Ultrafast dynamics of bright and dark excitons in monolayer WSe2 and heterobilayer WSe2/MoS2, 2D Mater. 10, 035039 (2023).
  22. F. Caruso, Nonequilibrium lattice dynamics in monolayer MoS2, J. Phys. Chem. Lett. 12, 1734 (2021).
  23. S. Ovesen, S. Brem, C. Linderälv, M. Kuisma, T. Korn, P. Erhart, M. Selig, and E. Malic, Interlayer exciton dynamics in van der Waals heterostructures, Commun. Phys. 2, 23 (2019).
  24. G. Meneghini, M. Reutzel, S. Mathias, S. Brem, and E. Malic, Hybrid exciton signatures in ARPES spectra of van der Waals materials, ACS Photonics 10, 3570 (2023).
  25. V. R. Policht, H. Mittenzwey, O. Dogadov, M. Katzer, A. Villa, Q. Li, B. Kaiser, A. M. Ross, F. Scotognella, X. Zhu, A. Knorr, M. Selig, G. Cerullo, and S. Dal Conte, Time-domain observation of interlayer exciton formation and thermalization in a MoSe2/WSe2 heterostructure, Nat. Commun. 14, 7273 (2023).
  26. R. Rosati, K. Wagner, S. Brem, R. Perea-Causín, E. Wietek, J. Zipfel, J. D. Ziegler, M. Selig, T. Taniguchi, K. Watanabe, A. Knorr, A. Chernikov, and E. Malic, Temporal evolution of low-temperature phonon sidebands in transition metal dichalcogenides, ACS Photonics 7, 2756 (2020).
  27. C. Poellmann, P. Steinleitner, U. Leierseder, P. Nagler, G. Plechinger, M. Porer, R. Bratschitsch, C. Schüller, T. Korn, and R. Huber, Resonant internal quantum transitions and femtosecond radiative decay of excitons in monolayer WSe2, Nat. Mater. 14, 889 (2015).
  28. M. Reutzel, G. S. M. Jansen, and S. Mathias, Probing excitons with time-resolved momentum microscopy, Adv. Phys. X 9, 2378722 (2024).
  29. S. Dong, M. Puppin, T. Pincelli, S. Beaulieu, D. Christiansen, H. Hübener, C. W. Nicholson, R. P. Xian, M. Dendzik, Y. Deng, Y. W. Windsor, M. Selig, E. Malic, A. Rubio, A. Knorr, M. Wolf, L. Rettig, and R. Ernstorfer, Direct measurement of key exciton properties: Energy, dynamics, and spatial distribution of the wave function, Nat. Sci. 1, e10010 (2021).
  30. M. K. L. Man, J. Madéo, C. Sahoo, K. Xie, M. Campbell, V. Pareek, A. Karmakar, E. L. Wong, A. Al-Mahboob, N. S. Chan, D. R. Bacon, X. Zhu, M. M. M. Abdelrasoul, X. Li, T. F. Heinz, F. H. da Jornada, T. Cao, and K. M. Dani, Experimental measurement of the intrinsic excitonic wave function, Sci. Adv. 7, eabg0192 (2021).
  31. D. Schmitt, J. P. Bange, W. Bennecke, A. AlMutairi, G. Meneghini, K. Watanabe, T. Taniguchi, D. Steil, D. R. Luke, R. T. Weitz, S. Steil, G. S. M. Jansen, S. Brem, E. Malic, S. Hofmann, M. Reutzel, and S. Mathias, Formation of moiré interlayer excitons in space and time, Nature (London) 608, 499 (2022).
  32. O. Karni et al., Structure of the moiré exciton captured by imaging its electron and hole, Nature (London) 603, 247 (2022).
  33. A. Kunin, S. Chernov, J. Bakalis, Z. Li, S. Cheng, Z. H. Withers, M. G. White, G. Schönhense, X. Du, R. K. Kawakami, and T. K. Allison, Momentum-resolved exciton coupling and valley polarization dynamics in monolayer WS2, Phys. Rev. Lett. 130, 046202 (2023).
  34. J. P. Bange, D. Schmitt, W. Bennecke, G. Meneghini, A. AlMutairi, K. Watanabe, T. Taniguchi, D. Steil, S. Steil, R. T. Weitz, G. S. M. Jansen, S. Hofmann, S. Brem, E. Malic, M. Reutzel, and S. Mathias, Probing electron-hole coulomb correlations in the exciton landscape of a twisted semiconductor heterostructure, Sci. Adv. 10, eadi1323 (2024).
  35. D. Schmitt, J. P. Bange, W. Bennecke, G. Meneghini, A. AlMutairi, M. Merboldt, J. Pöhls, K. Watanabe, T. Taniguchi, S. Steil, D. Steil, R. T. Weitz, S. Hofmann, S. Brem, G. S. M. Jansen, E. Malic, S. Mathias, and M. Reutzel, Ultrafast nano-imaging of dark excitons, Nat. Photonics 19, 187194 (2025).
  36. W. Bennecke, I. Gonzalez Oliva, J. P. Bange, P. Werner, D. Schmitt, M. Merboldt, A. M. Seiler, K. Watanabe, T. Taniguchi, D. Steil, R. T. Weitz, P. Puschnig, C. Draxl, G. S. M. Jansen, M. Reutzel, and S. Mathias, Hybrid Frenkel–Wannier excitons facilitate ultrafast energy transfer at a 2D–organic interface, Nat. Phys. 21, 1973 (2025).
  37. E. Perfetto, D. Sangalli, A. Marini, and G. Stefanucci, First-principles approach to excitons in time-resolved and angle-resolved photoemission spectra, Phys. Rev. B 94, 245303 (2016).
  38. A. Steinhoff, M. Florian, M. Rösner, G. Schönhoff, T. O. Wehling, and F. Jahnke, Exciton fission in monolayer transition metal dichalcogenide semiconductors, Nat. Commun. 8, 1166 (2017).
  39. A. Rustagi and A. F. Kemper, Photoemission signature of excitons, Phys. Rev. B 97, 235310 (2018).
  40. D. Christiansen, M. Selig, E. Malic, R. Ernstorfer, and A. Knorr, Theory of exciton dynamics in time-resolved ARPES: Intra- and intervalley scattering in two-dimensional semiconductors, Phys. Rev. B 100, 205401 (2019).
  41. G. Meneghini, M. Löwe, R. Perea-Causin, J. P. Bange, W. Bennecke, M. Reutzel, S. Mathias, and E. Malic, ARPES signatures of trions in van der Waals materials, arXiv:2511.11448.
  42. 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).
  43. N. Scheuschner, O. Ochedowski, A.-M. Kaulitz, R. Gillen, M. Schleberger, and J. Maultzsch, Photoluminescence of freestanding single- and few-layer MoS2, Phys. Rev. B 89, 125406 (2014).
  44. M. Keunecke, C. Möller, D. Schmitt, H. Nolte, G. S. M. Jansen, M. Reutzel, M. Gutberlet, G. Halasi, D. Steil, S. Steil, and S. Mathias, Time-resolved momentum microscopy with a 1 MHz high-harmonic extreme ultraviolet beamline, Rev. Sci. Instrum. 91, 063905 (2020).
  45. K. Medjanik, O. Fedchenko, S. Chernov, D. Kutnyakhov, M. Ellguth, A. Oelsner, B. Schönhense, T. R. F. Peixoto, P. Lutz, C.-H. Min, F. Reinert, S. Däster, Y. Acremann, J. Viefhaus, W. Wurth, H. J. Elmers, and G. Schönhense, Direct 3D mapping of the Fermi surface and Fermi velocity, Nat. Mater. 16, 615 (2017).
  46. G. Meneghini, S. Brem, and E. Malic, Ultrafast phonon-driven charge transfer in van der Waals heterostructures, Nat. Sci. 2, e20220014 (2022).
  47. I. C. Gerber, E. Courtade, S. Shree, C. Robert, T. Taniguchi, K. Watanabe, A. Balocchi, P. Renucci, D. Lagarde, X. Marie, and B. Urbaszek, Interlayer excitons in bilayer MoS2 with strong oscillator strength up to room temperature, Phys. Rev. B 99, 035443 (2019).
  48. I. Paradisanos, S. Shree, A. George, N. Leisgang, C. Robert, K. Watanabe, T. Taniguchi, R. J. Warburton, A. Turchanin, X. Marie, I. C. Gerber, and B. Urbaszek, Controlling interlayer excitons in MoS2 layers grown by chemical vapor deposition, Nat. Commun. 11, 2391 (2020).
  49. X. Zhu, D. R. Bacon, V. Pareek, J. Madéo, T. Taniguchi, K. Watanabe, M. K. L. Man, and K. M. Dani, A holistic view of the dynamics of long-lived valley polarized dark excitonic states in monolayer WS2, Nat. Commun. 16, 6385 (2025).
  50. S. Kusaba, K. Watanabe, T. Taniguchi, K. Yanagi, and K. Tanaka, Role of dark exciton states in the relaxation dynamics of bright 1s excitons in monolayer WSe2, Appl. Phys. Lett. 119, 093101 (2021).
  51. Z. Jin, X. Li, J. T. Mullen, and K. W. Kim, Intrinsic transport properties of electrons and holes in monolayer transition-metal dichalcogenides, Phys. Rev. B 90, 045422 (2014).
  52. W. Jin, P.-C. Yeh, N. Zaki, D. Zhang, J. T. Sadowski, A. Al-Mahboob, A. M. van der Zande, D. A. Chenet, J. I. Dadap, I. P. Herman, P. Sutter, J. Hone, and R. M. Osgood, Direct measurement of the thickness-dependent electronic band structure of MoS2 using angle-resolved photoemission spectroscopy, Phys. Rev. Lett. 111, 106801 (2013).
  53. W. Bennecke, A. Windischbacher, D. Schmitt, J. P. Bange, R. Hemm, C. S. Kern, G. D’Avino, X. Blase, D. Steil, S. Steil, M. Aeschlimann, B. Stadtmüller, M. Reutzel, P. Puschnig, G. S. M. Jansen, and S. Mathias, Disentangling the multiorbital contributions of excitons by photoemission exciton tomography, Nat. Commun. 15, 1804 (2024).
  54. See Supplemental Material at http://link.aps.org/supplemental/10.1103/w29j-z48v for a description of the theoretical modeling, which includes Refs. [55–62].
  55. S. Brem, K.-Q. Lin, R. Gillen, J. M. Bauer, J. Maultzsch, J. M. Lupton, and E. Malic, Hybridized intervalley moiré excitons and flat bands in twisted WSe2 bilayers, Nanoscale 12, 11088 (2020).
  56. J. Hagel, S. Brem, C. Linderälv, P. Erhart, and E. Malic, Exciton landscape in van der Waals heterostructures, Phys. Rev. Res. 3, 043217 (2021).
  57. S. Brem, M. Selig, G. Berghaeuser, and E. Malic, Exciton relaxation cascade in two-dimensional transition metal dichalcogenides, Sci. Rep. 8, 8238 (2018).
  58. G. Meneghini, S. Brem, and E. Malic, Ultrafast phonon-driven charge transfer in van der Waals heterostructures, Nat. Sci. 2, e20220014 (2022).
  59. Z. Jin, X. Li, J. T. Mullen, and K. W. Kim, Intrinsic transport properties of electrons and holes in monolayer transition-metal dichalcogenides, Phys. Rev. B 90, 045422 (2014).
  60. M. Kira and S. W. Koch, Many-body correlations and excitonic effects in semiconductor spectroscopy, Prog. Quantum Electron. 30, 155 (2006).
  61. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 2009).
  62. E. Malic and A. Knorr, Graphene and Carbon Nanotubes: Ultrafast Optics and Relaxation Dynamics (John Wiley & Sons, New York, 2013).
  63. Z. Fei, M. E. Scott, D. J. Gosztola, J. J. Foley, J. Yan, D. G. Mandrus, H. Wen, P. Zhou, D. W. Zhang, Y. Sun, J. R. Guest, S. K. Gray, W. Bao, G. P. Wiederrecht, and X. Xu, Nano-optical imaging of WSe2 waveguide modes revealing light-exciton interactions, Phys. Rev. B 94, 081402(R) (2016).
  64. T. Byrnes, N. Y. Kim, and Y. Yamamoto, Exciton–polariton condensates, Nat. Phys. 10, 803 (2014).
  65. P. Werner, W. Bennecke, J. P. Bange, G. Meneghini, D. Schmitt, M. Merboldt, A. M. Seiler, A. AlMutairi, K. Watanabe, T. Taniguchi, G. S. M. Jansen, J. Liu, D. Steil, S. Hofmann, R. T. Weitz, E. Malic, S. Mathias, and M. Reutzel, Replication data for: The role of non-equilibrium populations in dark exciton formation (2026), 10.25625/EKTPZZ.
  66. A. M. Seiler, Y. Zhumagulov, K. Zollner, C. Yoon, D. Urbaniak, F. R. Geisenhof, K. Watanabe, T. Taniguchi, J. Fabian, F. Zhang, and R. T. Weitz, Layer-selective spin-orbit coupling and strong correlation in bilayer graphene, 2D Mater. 12, 035009 (2025).
  67. M. Düvel, M. Merboldt, J. P. Bange, H. Strauch, M. Stellbrink, K. Pierz, H. W. Schumacher, D. Momeni, D. Steil, G. S. M. Jansen, S. Steil, D. Novko, S. Mathias, and M. Reutzel, Far-from-equilibrium electron–phonon interactions in optically excited graphene, Nano Lett. 22, 4897 (2022).
  68. M. Merboldt, M. Schüler, D. Schmitt, J. P. Bange, W. Bennecke, K. Gadge, K. Pierz, H. W. Schumacher, D. Momeni, D. Steil, S. R. Manmana, M. A. Sentef, M. Reutzel, and S. Mathias, Observation of Floquet states in graphene, Nat. Phys. 21, 10931099 (2025).
  69. L. Miaja-Avila, C. Lei, M. Aeschlimann, J. L. Gland, M. M. Murnane, H. C. Kapteyn, and G. Saathoff, Laser-assisted photoelectric effect from surfaces, Phys. Rev. Lett. 97, 113604 (2006).
  70. B. Schönhense, K. Medjanik, O. Fedchenko, S. Chernov, M. Ellguth, D. Vasilyev, A. Oelsner, J. Viefhaus, D. Kutnyakhov, W. Wurth, H. J. Elmers, and G. Schönhense, Multidimensional photoemission spectroscopy—The space-charge limit, New J. Phys. 20, 033004 (2018).
  71. F. Roth, J. Mahl, M. Borgwardt, L. Wenthaus, F. Brausse, V. Garbe, O. Gessner, and W. Eberhardt, Dynamical nonlinear inversion of the surface photovoltage at Si(100), Phys. Rev. Lett. 132, 146201 (2024).
  72. 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).
  73. M. Weinelt, M. Kutschera, T. Fauster, and M. Rohlfing, Dynamics of exciton formation at the Si(100)  c(4×2) surface, Phys. Rev. Lett. 92, 126801 (2004).

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Phys. Rev. Lett. 136, 186903– Published 8 May, 2026
  • Received 17 June 2025
  • Accepted 25 March 2026
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