Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Biexciton fine structure in monolayer transition metal dichalcogenides


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.

This is a preview of subscription content

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Pump–probe selection rules of biexcitons in monolayer TMDC.
Fig. 2: Configuration picture of biexcitons in monolayer WSe2.
Fig. 3: Differential absorption of monolayer WSe2 on sapphire from theory and experiment.

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.


  1. 1.

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

    ADS  Article  Google Scholar 

  2. 2.

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  5. 5.

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

    ADS  Article  Google Scholar 

  6. 6.

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

    Book  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  10. 10.

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

    ADS  Article  Google Scholar 

  11. 11.

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  13. 13.

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  17. 17.

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

    ADS  Article  Google Scholar 

  18. 18.

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

    Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  20. 20.

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

    Article  Google Scholar 

  21. 21.

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

    ADS  Article  Google Scholar 

  22. 22.

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  26. 26.

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  29. 29.

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  31. 31.

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  33. 33.

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

    Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    Google Scholar 

  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).

    Article  Google Scholar 

  39. 39.

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

    ADS  Article  Google Scholar 

  40. 40.

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

    ADS  Article  Google Scholar 

  41. 41.

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

    Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  43. 43.

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

    ADS  Article  Google Scholar 

  44. 44.

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    MathSciNet  Article  Google Scholar 

  49. 49.

    Balay, S. et al. Portable, extensible toolkit for scientific computation. Argonne National Laboratory (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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  52. 52.

    FLEUR: The Jülich FLAPW code family. FLEUR (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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  56. 56.

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

    ADS  Article  Google Scholar 

  57. 57.

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

    ADS  Article  Google Scholar 

  58. 58.

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

    ADS  Article  Google Scholar 

  59. 59.

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

    ADS  Article  Google Scholar 

  60. 60.

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

  62. 62.

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

    ADS  Article  Google Scholar 

  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).

    ADS  Article  Google Scholar 

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




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.

Corresponding author

Correspondence to Alexander Steinhoff.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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

Supplementary information

Supplementary Information

7 chapters, 6 figures, 1 table, 23 references

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Steinhoff, A., Florian, M., Singh, A. et al. Biexciton fine structure in monolayer transition metal dichalcogenides. Nature Phys 14, 1199–1204 (2018).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing