The optical properties of atomically thin transition metal dichalcogenide (TMDC) semiconductors are shaped by the emergence of correlated many-body complexes due to strong Coulomb interaction. Exceptional electron–hole exchange predestines TMDCs to be used to study fundamental and applied properties of Coulomb complexes such as valley depolarization of excitons and fine-structure splitting of trions. Biexcitons in these materials are less well understood and it has been established only recently that they are spectrally located between excitons and trions. Here we show that biexcitons in monolayer TMDCs exhibit a distinct and rich fine structure on the order of millielectronvolts due to electron–hole exchange. Ultrafast pump–probe experiments on monolayer WSe2 reveal decisive biexciton signatures and a fine structure in excellent agreement with a microscopic theory. We provide a pathway to understand the complex spectral structure of higher-order Coulomb complexes in TMDCs going beyond the usual classification scheme in terms of four-particle configurations.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


  1. 1.

    Onodera, Y. & Toyozawa, Y. Excitons in alkali halides. J. Phys. Soc. Jpn 22, 833–844 (1967).

  2. 2.

    Denisov, M. M. & Makarov, V. P. Longitudinal and transverse excitons in semiconductors. Phys. Status Solidi B 56, 9–59 (1973).

  3. 3.

    Warming, T. et al. Hole–hole and electron–hole exchange interactions in single InAs/GaAs quantum dots. Phys. Rev. B 79, 125316 (2009).

  4. 4.

    Kadantsev, E. & Hawrylak, P. Theory of exciton fine structure in semiconductor quantum dots: Quantum dot anisotropy and lateral electric field. Phys. Rev. B 81, 045311 (2010).

  5. 5.

    Sieh, C. et al. Coulomb memory signatures in the excitonic optical Stark effect. Phys. Rev. Lett. 82, 3112–3115 (1999).

  6. 6.

    Schäfer, W. & Wegener, M. Semiconductor Optics and Transport Phenomena (Springer, Berlin, New York, 2002).

  7. 7.

    Forney, J. J., Quattropani, A. & Bassani, F. Electron–hole exchange contributions to the biexciton states. Il Nuovo Cimento B 22, 153–178 (1974).

  8. 8.

    Quattropani, A., Forney, J. J. & Bassani, F. Biexcitons in indirect-gap semiconductors: applications to GaSe and AgBr. Phys. Status Solidi B 70, 497–504 (1975).

  9. 9.

    Ekardt, W. & Sheboul, M. I. The influence of the electron–hole exchange interaction on the biexciton ground state in CuCl and CuBr and related optical transitions. Physica Status Solidi B 73, 475–482 (1976).

  10. 10.

    Bassani, F. & Rovere, M. Biexciton binding energy in Cu2O. Solid State Commun. 19, 887–890 (1976).

  11. 11.

    Quattropani, A. & Forney, J. J. Theory of excitonic molecules. Il Nuovo Cimento B 39, 569–578 (1977).

  12. 12.

    Chung, S. G., Sanders, G. D. & Chang, Y.-C. Theory of excitonic molecules in thallous halide: The role of electron-hole exchange interaction. Solid State Commun. 45, 237–241 (1983).

  13. 13.

    Ungier, W. Electron–hole exchange interaction in a biexciton molecule. Solid State Commun. 69, 53–55 (1989).

  14. 14.

    Yu, T. & Wu, M. W. Valley depolarization due to intervalley and intravalley electron-hole exchange interactions in monolayer MoS2. Phys. Rev. B 89, 205303 (2014).

  15. 15.

    Yu, H., Liu, G.-B., Gong, P., Xu, X. & Yao, W. Dirac cones and Dirac saddle points of bright excitons in monolayer transition metal dichalcogenides. Nat. Commun. 5, 3876 (2014).

  16. 16.

    Glazov, M. M. et al. Spin and valley dynamics of excitons in transition metal dichalcogenide monolayers. Phys. Status Solidi B 252, 2349–2362 (2015).

  17. 17.

    Schmidt, R. et al. Ultrafast Coulomb-induced intervalley coupling in atomically thin WS2. Nano Lett. 16, 2945–2950 (2016).

  18. 18.

    Hao, K. et al. Direct measurement of exciton valley coherence in monolayer WSe2. Nat. Phys. 12, 677–682 (2016).

  19. 19.

    Qiu, D. Y., Cao, T. & Louie, S. G. Nonanalyticity, valley quantum phases, and lightlike exciton dispersion in monolayer transition metal dichalcogenides: theory and first-principles calculations. Phys. Rev. Lett. 115, 176801 (2015).

  20. 20.

    Jones, A. M. et al. Excitonic luminescence upconversion in a two-dimensional semiconductor. Nat. Phys. 12, 323–327 (2016).

  21. 21.

    Plechinger, G. et al. Trion fine structure and coupled spin–valley dynamics in monolayer tungsten disulfide. Nat. Commun. 7, 12715–12723 (2016).

  22. 22.

    Singh, A. et al. Long-lived valley polarization of intravalley trions in monolayer WSe2. Phys. Rev. Lett. 117, 257402 (2016).

  23. 23.

    Voss, T., Rückmann, I., Gutowski, J., Axt, V. M. & Kuhn, T. Coherent control of the exciton and exciton–biexciton transitions in the generation of nonlinear wave-mixing signals in a semiconductor quantum well. Phys. Rev. B 73, 115311 (2006).

  24. 24.

    Gilliot, P. et al. Measurement of exciton spin coherence by nondegenerate four-wave mixing experiments in the χ (3) regime. Phys. Rev. B 75, 125209 (2007).

  25. 25.

    Mayers, M. Z., Berkelbach, T. C., Hybertsen, M. S. & Reichman, D. R. Binding energies and spatial structures of small carrier complexes in monolayer transition-metal dichalcogenides via diffusion Monte Carlo. Phys. Rev. B 92, 161404 (2015).

  26. 26.

    Zhang, D. K., Kidd, D. W. & Varga, K. Excited biexcitons in transition metal dichalcogenides. Nano Lett. 15, 7002–7005 (2015).

  27. 27.

    Kylänpää, I. & Komsa, H.-P. Binding energies of exciton complexes in transition metal dichalcogenide monolayers and effect of dielectric environment. Phys. Rev. B 92, 205418 (2015).

  28. 28.

    Szyniszewski, M., Mostaani, E., Drummond, N. D. & Fal’ko, V. I. Binding energies of trions and biexcitons in two-dimensional semiconductors from diffusion quantum Monte Carlo calculations. Phys. Rev. B 95, 081301 (2017).

  29. 29.

    Hao, K. et al. Neutral and charged inter-valley biexcitons in monolayer MoSe2. Nat. Commun. 8, 15552 (2017).

  30. 30.

    Mai, C. et al. Many-body effects in valleytronics: direct measurement of valley lifetimes in single-layer MoS2. Nano Lett. 14, 202–206 (2014).

  31. 31.

    You, Y. et al. Observation of biexcitons in monolayer WSe2. Nature Physics 11, 477–481 (2015).

  32. 32.

    Plechinger, G. et al. Identification of excitons, trions and biexcitons in single-layer WS2. Phys. Status Solidi Rapid Res. Lett. 9, 457–461 (2015).

  33. 33.

    Shang, J. et al. Observation of excitonic fine structure in a 2D transition-metal dichalcogenide semiconductor. ACS Nano 9, 647–655 (2015).

  34. 34.

    Sie, E. J., Frenzel, A. J., Lee, Y.-H., Kong, J. & Gedik, N. Intervalley biexcitons and many-body effects in monolayer MoS2. Phys. Rev. B 92, 125417 (2015).

  35. 35.

    Lee, H. S., Kim, M. S., Kim, H. & Lee, Y. H. Identifying multiexcitons in MoS2 monolayers at room temperature. Phys. Rev. B 93, 140409 (2016).

  36. 36.

    Okada, M. et al. Observation of biexcitonic emission at extremely low power density in tungsten disulfide atomic layers grown on hexagonal boron nitride. Sci. Rep. 7, 322 (2017).

  37. 37.

    Van Tuan, D., Scharf, B., Žutić, I. & Dery, H. Marrying excitons and plasmons in monolayer transition-metal dichalcogenides. Phys. Rev. X 7, 041040 (2017).

  38. 38.

    Xu, X., Yao, W., Xiao, D. & Heinz, T. F. Spin and pseudospins in layered transition metal dichalcogenides. Nat. Phys. 10, 343–350 (2014).

  39. 39.

    Hulin, D. & Joffre, M. Excitonic optical stark redshift: the biexciton signature. Phys. Rev. Lett. 65, 3425–3428 (1990).

  40. 40.

    Tran, K. et al. Disorder-dependent valley properties in monolayer WSe2. Phys. Rev. B 96, 041302 (2017).

  41. 41.

    Kern, J. et al. Nanoscale positioning of single-photon emitters in atomically thin WSe2. Adv. Mater. 28, 7101–7105 (2016).

  42. 42.

    Branny, A., Kumar, S., Proux, R. & Gerardot, B. D. Deterministic strain-induced arrays of quantum emitters in a two-dimensional semiconductor. Nat. Commun. 8, 15053 (2017).

  43. 43.

    Palacios-Berraquero, C. et al. Large-scale quantum-emitter arrays in atomically thin semiconductors. Nat. Commun. 8, 15093 (2017).

  44. 44.

    He, Y.-M. et al. Cascaded emission of single photons from the biexciton in monolayered WSe2. Nat. Commun. 7, 13409 (2016).

  45. 45.

    Axt, V. M. & Stahl, A. A dynamics-controlled truncation scheme for the hierarchy of density matrices in semiconductor optics. Z. Phys. B 93, 195–204 (1994).

  46. 46.

    Lindberg, M., Hu, Y. Z., Binder, R. & Koch, S. W. Χ (3) formalism in optically excited semiconductors and its applications in four-wave-mixing spectroscopy. Phys. Rev. B 50, 18060 (1994).

  47. 47.

    Sham, L. J. & Rice, T. M. Many-particle derivation of the effective-mass equation for the Wannier exciton. Phys. Rev. 144, 708–714 (1966).

  48. 48.

    Hernandez, V., Roman, J. E. & Vidal, V. SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31, 351–362 (2005).

  49. 49.

    Balay, S. et al. Portable, extensible toolkit for scientific computation. Argonne National Laboratory http://www.mcs.anl.gov/petsc/ (2018).

  50. 50.

    Steinhoff, A., Rösner, M., Jahnke, F., Wehling, T. O. & Gies, C. Influence of excited carriers on the optical and electronic properties of MoS2. Nano Lett. 14, 3743–3748 (2014).

  51. 51.

    Liu, G.-B., Shan, W.-Y., Yao, Y., Yao, W. & Xiao, D. Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides. Phys. Rev. B 88, 085433 (2013).

  52. 52.

    FLEUR: The Jülich FLAPW code family. FLEUR http://www.flapw.de/pm/index.php (2018).

  53. 53.

    Friedrich, C., Blügel, S. & Schindlmayr, A. Efficient implementation of the GW approximation within the all-electron FLAPW method. Phys. Rev. B 81, 125102 (2010).

  54. 54.

    Rösner, M., Şaşıoğlu, E., Friedrich, C., Blügel, S. & Wehling, T. O. Wannier function approach to realistic Coulomb interactions in layered materials and heterostructures. Phys. Rev. B 92, 085102 (2015).

  55. 55.

    Florian, M. et al. The dielectric impact of layer distances on exciton and trion binding energies in van der Waals heterostructures. Nano Lett. 18, 2725–2732 (2018).

  56. 56.

    Steinhoff, A. et al. Exciton fission in monolayer transition metal dichalcogenide semiconductors. Nat. Commun. 8, 1166 (2017).

  57. 57.

    Rooney, A. P. et al. Observing imperfection in atomic interfaces for van der Waals heterostructures. Nano Lett. 17, 5222–5228 (2017).

  58. 58.

    Xiao, D., Yao, W. & Niu, Q. Valley-contrasting physics in graphene: magnetic moment and topological transport. Phys. Rev. Lett. 99, 236809 (2007).

  59. 59.

    Singh, A. et al. Trion formation dynamics in monolayer transition metal dichalcogenides. Phys. Rev. B 93, 041401 (2016).

  60. 60.

    Combescot, M. & Combescot, R. Excitonic Stark shift: a coupling to semivirtual biexcitons. Phys. Rev. Lett. 61, 117–120 (1988).

  61. 61.

    Wang, G. et al. In-plane propagation of light in transition metal dichalcogenide monolayers: optical selection rules. Phys. Rev. Lett. 119, 047401 (2017).

  62. 62.

    Zhang, X.-X. et al. Magnetic brightening and control of dark excitons in monolayer WSe2. Nat. Nanotech. 12, 883–888 (2017).

  63. 63.

    Danovich, M., Zólyomi, V. & Fal’ko, V. I. Dark trions and biexcitons in WS2 and WSe2 made bright by e–e scattering. Sci. Rep. 7, 45998 (2017).

Download references


This work was supported by Deutsche Forschungsgemeinschaft (DFG) within CRC 1558 and RTG 2247. A.W.A. acknowledges funding from DFG via grant no. AC290-2/1. The spectroscopic experiments were jointly supported by NSF DMR1306878 (A. Singh) and the NSF MRSEC program DMR-1720595 (K.T.). X.L. gratefully acknowledges support from the Welch foundation (F-1662) and the Alexander von Humboldt Foundation, which facilitated the collaboration with TU-Berlin. A. Steinhoff and M.F. would like to acknowledge P. Gartner for fruitful discussions. We thank G. Schönhoff, M. Rösner and T. Wehling for providing material-realistic band structures and bare as well as screened Coulomb matrix elements.

Author information

Author notes

    • Akshay Singh

    Present address: Department of Material Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA


  1. Institut für Theoretische Physik, Universität Bremen, Bremen, Germany

    • Alexander Steinhoff
    • , Matthias Florian
    •  & Frank Jahnke
  2. University of Texas at Austin, Austin, TX, USA

    • Akshay Singh
    • , Kha Tran
    •  & Xiaoqin Li
  3. Institut für Optik und Atomare Physik, Technische Universität Berlin, Berlin, Germany

    • Mirco Kolarczik
    • , Sophia Helmrich
    • , Alexander W. Achtstein
    • , Ulrike Woggon
    •  & Nina Owschimikow
  4. MAPEX Center for Materials and Processes, Universität Bremen, Bremen, Germany

    • Frank Jahnke


  1. Search for Alexander Steinhoff in:

  2. Search for Matthias Florian in:

  3. Search for Akshay Singh in:

  4. Search for Kha Tran in:

  5. Search for Mirco Kolarczik in:

  6. Search for Sophia Helmrich in:

  7. Search for Alexander W. Achtstein in:

  8. Search for Ulrike Woggon in:

  9. Search for Nina Owschimikow in:

  10. Search for Frank Jahnke in:

  11. Search for Xiaoqin Li in:


A. Steinhoff and M.F. performed analytical and numerical calculations of the biexciton spectra. A. Singh, K.T. and M.K. designed and performed the experiments. K.T. exfoliated the sample. N.O., A. Singh, A.W.A. and S.H. analysed the data. N.O., A. Steinhoff, M.F. and A. Singh prepared the manuscript. U.W., F.J. and X.L. initiated and coordinated the project. All authors contributed to the discussion and the writing of the manuscript.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to Alexander Steinhoff.

Supplementary information

  1. Supplementary Information

    7 chapters, 6 figures, 1 table, 23 references

About this article

Publication history




Issue Date