Inheritance of the exciton geometric structure from Bloch electrons in two-dimensional layered semiconductors

Abstract

We theoretically studied the exciton geometric structure in layered semiconducting transition metal dichalcogenides. Based on a three-orbital tight-binding model for Bloch electrons which incorporates their geometric structures, an effective exciton Hamiltonian is constructed and solved perturbatively to reveal the relation between the exciton and its electron/hole constituent. We show that the electron−hole Coulomb interaction gives rise to a non-trivial inheritance of the exciton geometric structure from Bloch electrons, which manifests as a valley-dependent center-of-mass anomalous Hall velocity of the exciton when two external fields are applied on the electron and hole constituents, respectively. The obtained center-of-mass anomalous velocity is found to exhibit a non-trivial dependence on the fields, as well as the wave function and valley index of the exciton. These findings can serve as a general guide for the field-control of the valley-dependent exciton transport, enabling the design of novel quantum optoelectronic and valleytronic devices.

Graphical abstract

Keywords

transition metal dichalcogenides / exciton / geometric structure / Berry curvature / van der Waals stacking

References

[1] A. Ciarrocchi, F. Tagarelli, A. Avsar, A. Kis. Excitonic devices with van der Waals heterostructures: Valleytronics meets twistronics. Nat. Rev. Mater., 2022, 7(6): 449

[2] K. F. Mak, D. Xiao, J. Shan. Light-valley interactions in 2D semiconductors. Nat. Photonics, 2018, 12(8): 451

[3] X. D. Xu, W. Yao, D. Xiao, T. F. Heinz. Spin and pseudospins in layered transition metal dichalcogenides. Nat. Phys., 2014, 10(5): 343

[4] K. F. Mak, J. Shan. Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides. Nat. Photonics, 2016, 10(4): 216

[5] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, M. S. Strano. Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nat. Nanotechnol., 2012, 7(11): 699

[6] A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi, Y. Li, B. Aslan, D. R. Reichman, M. S. Hybertsen, T. F. Heinz. Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS₂. Phys. Rev. Lett., 2014, 113(7): 076802

[7] D. Y. Qiu, F. H. da Jornada, S. G. Louie. Optical spectrum of MoS₂: Many-body effects and diversity of exciton states. Phys. Rev. Lett., 2013, 111(21): 216805

[8] X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, W. Y. Ching. Analytic solution of a two-dimensional hydrogen atom (I): Nonrelativistic theory. Phys. Rev. A, 1991, 43(3): 1186

[9] T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu, P. Tan, E. Wang, B. Liu, J. Feng. Valley-selective circular dichroism of monolayer molybdenum disulphide. Nat. Commun., 2012, 3(1): 887

[10] H. L. Zeng, J. Dai, W. Yao, D. Xiao, X. Cui. Valley polarization in MoS₂ monolayers by optical pumping. Nat. Nanotechnol., 2012, 7(8): 490

[11] K. F. Mak, K. He, J. Shan, T. F. Heinz. Control of valley polarization in monolayer MoS₂ by optical helicity. Nat. Nanotechnol., 2012, 7(8): 494

[12] A. M. Jones, H. Yu, N. J. Ghimire, S. Wu, G. Aivazian, J. S. Ross, B. Zhao, J. Yan, D. G. Mandrus, D. Xiao, W. Yao, X. Xu. Optical generation of excitonic valley coherence in monolayer WSe. Nat. Nanotechnol., 2013, 8(9): 634

[13] P. Rivera, H. Yu, K. L. Seyler, N. P. Wilson, W. Yao, X. Xu. Interlayer valley excitons in heterobilayers of transition metal dichalcogenides. Nat. Nanotechnol., 2018, 13(11): 1004

[14] D. Xiao, M. C. Chang, Q. Niu. Berry phase effects on electronic properties. Rev. Mod. Phys., 2010, 82(3): 1959

[15] D. Xiao, G. B. Liu, W. Feng, X. Xu, W. Yao. Coupled spin and valley physics in monolayers of MoS₂ and other group-VI dichalcogenides. Phys. Rev. Lett., 2012, 108(19): 196802

[16] G. Aivazian, Z. Gong, A. M. Jones, R. L. Chu, J. Yan, D. G. Mandrus, C. Zhang, D. Cobden, W. Yao, X. Xu. Magnetic control of valley pseudospin in monolayer WSe₂. Nat. Phys., 2015, 11(2): 148

[17] A. Srivastava, M. Sidler, A. V. Allain, D. S. Lembke, A. Kis, A. Imamoğlu. Valley Zeeman effect in elementary optical excitations of monolayer WSe. Nat. Phys., 2015, 11(2): 141

[18] D. MacNeill, C. Heikes, K. F. Mak, Z. Anderson, A. Kormányos, V. Zólyomi, J. Park, D. C. Ralph. Breaking of valley degeneracy by magnetic field in monolayer MoSe₂. Phys. Rev. Lett., 2015, 114(3): 037401

[19] A. Kormányos, V. Zólyomi, V. I. Fal’ko, G. Burkard. Tunable Berry curvature and valley and spin Hall effect in bilayer MoS₂. Phys. Rev. B, 2018, 98(3): 035408

[20] K. F. Mak, K. L. McGill, J. Park, P. L. McEuen. The valley Hall effect in MoS₂ transistors. Science, 2014, 344(6191): 1489

[21] T. Yu, M. W. Wu. Valley depolarization dynamics and valley Hall effect of excitons in monolayer and bilayer MoS₂. Phys. Rev. B, 2016, 93(4): 045414

[22] W. Yao, D. Xiao, Q. Niu. Valley-dependent optoelectronics from inversion symmetry breaking. Phys. Rev. B, 2008, 77(23): 235406

[23] Q. Z. Zhu, M. W. Y. Tu, Q. Tong, W. Yao. Gate tuning from exciton superfluid to quantum anomalous Hall in van der Waals heterobilayer. Sci. Adv., 2019, 5(1): eaau6120

[24] C.Y. JiangA. RasmitaH.MaQ.TanZ.Zhang Z.HuangS. LaiN.WangS.LiuX.Liu T.YuQ.Xiong W.Gao, A room-temperature gate-tunable bipolar valley Hall effect in molybdenum disulfide/tungsten diselenide heterostructures, Nat. Electron. 5(1), 23 (2021)

[25] M. Onga, Y. Zhang, T. Ideue, Y. Iwasa. Exciton Hall effect in monolayer MoS₂. Nat. Mater., 2017, 16(12): 1193

[26] Z. Huang, Y. Liu, K. Dini, Q. Tan, Z. Liu, H. Fang, J. Liu, T. Liew, W. Gao. Robust room temperature valley Hall effect of interlayer excitons. Nano Lett., 2020, 20(2): 1345

[27] W. Yao, Q. Niu. Berry phase effect on the exciton transport and on the exciton Bose‒Einstein condensate. Phys. Rev. Lett., 2008, 101(10): 106401

[28] H. Y. Yu, W. Yao. Electrically tunable topological transport of moire polaritons. Sci. Bull. (Beijing), 2020, 65(18): 1555

[29] N. Ubrig, S. Jo, M. Philippi, D. Costanzo, H. Berger, A. B. Kuzmenko, A. F. Morpurgo. Microscopic origin of the valley Hall effect in transition metal dichalcogenides revealed by wavelength-dependent mapping. Nano Lett., 2017, 17(9): 5719

[30] M. Trushin, M. O. Goerbig, W. Belzig. Model prediction of self-rotating excitons in two-dimensional transition-metal dichalcogenides. Phys. Rev. Lett., 2018, 120(18): 187401

[31] A. Hichri, S. Jaziri, M. O. Goerbig. Charged excitons in two-dimensional transition metal dichalcogenides: Semiclassical calculation of Berry curvature effects. Phys. Rev. B, 2019, 100(11): 115426

[32] A. Srivastava, A. Imamoğlu. Signatures of Bloch-band geometry on excitons: Nonhydrogenic spectra in transition-metal dichalcogenides. Phys. Rev. Lett., 2015, 115(16): 166802

[33] J. H. Zhou, W. Y. Shan, W. Yao, D. Xiao. Berry phase modification to the energy spectrum of excitons. Phys. Rev. Lett., 2015, 115(16): 166803

[34] P. Gong, H. Yu, Y. Wang, W. Yao. Optical selection rules for excitonic Rydberg series in the massive Dirac cones of hexagonal two-dimensional materials. Phys. Rev. B, 2017, 95(12): 125420

[35] T. Cao, M. Wu, S. G. Louie. Unifying optical selection rules for excitons in two dimensions: Band topology and winding numbers. Phys. Rev. Lett., 2018, 120(8): 087402

[36] X. O. Zhang, W. Y. Shan, D. Xiao. Optical selection rule of excitons in gapped chiral fermion systems. Phys. Rev. Lett., 2018, 120(7): 077401

[37] G. B. Liu, W. Y. Shan, Y. Yao, W. Yao, D. Xiao. Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides. Phys. Rev. B, 2013, 88(8): 085433

[38] F. C. Wu, F. Y. Qu, A. H. MacDonald. Exciton band structure of monolayer MoS₂. Phys. Rev. B, 2015, 91(7): 075310

[39] Z. L. Ye, T. Cao, K. O’Brien, H. Zhu, X. Yin, Y. Wang, S. G. Louie, X. Zhang. Probing excitonic dark states in single-layer tungsten disulphide. Nature, 2014, 513(7517): 214

[40] D. Y. Qiu, T. Cao, S. G. Louie. Nonanalyticity, valley quantum phases, and lightlike exciton dispersion in monolayer transition metal dichalcogenides: Theory and first-principles calculations. Phys. Rev. Lett., 2015, 115(17): 176801

[41] C. K. Yong, M. I. B. Utama, C. S. Ong, T. Cao, E. C. Regan, J. Horng, Y. Shen, H. Cai, K. Watanabe, T. Taniguchi, S. Tongay, H. Deng, A. Zettl, S. G. Louie, F. Wang. Valley-dependent exciton fine structure and Autler‒Townes doublets from Berry phases in monolayer MoSe₂. Nat. Mater., 2019, 18(10): 1065

[42] S. Chaudhary, C. Knapp, G. Refael. Anomalous exciton transport in response to a uniform in-plane electric field. Phys. Rev. B, 2021, 103(16): 165119

[43] J. L. Cao, H. A. Fertig, L. Brey. Quantum geometric exciton drift velocity. Phys. Rev. B, 2021, 103(11): 115422

[44] M. Q. Sui, G. Chen, L. Ma, W. Y. Shan, D. Tian, K. Watanabe, T. Taniguchi, X. Jin, W. Yao, D. Xiao, Y. Zhang. Gate-tunable topological valley transport in bilayer graphene. Nat. Phys., 2015, 11(12): 1027

[45] Y. Shimazaki, M. Yamamoto, I. V. Borzenets, K. Watanabe, T. Taniguchi, S. Tarucha. Generation and detection of pure valley current by electrically induced Berry curvature in bilayer graphene. Nat. Phys., 2015, 11(12): 1032

[46] L. Ju, L. Wang, T. Cao, T. Taniguchi, K. Watanabe, S. G. Louie, F. Rana, J. Park, J. Hone, F. Wang, P. L. McEuen. Tunable excitons in bilayer graphene. Science, 2017, 358(6365): 907

[47] H. Y. Yu, W. Yao. Luminescence anomaly of dipolar valley excitons in homobilayer semiconductor moire superlattices. Phys. Rev. X, 2021, 11(2): 021042

[48] H. Y. Yu, G. B. Liu, P. Gong, X. Xu, W. Yao. Dirac cones and Dirac saddle points of bright excitons in monolayer transition metal dichalcogenides. Nat. Commun., 2014, 5(1): 3876

[49] M. H. He, P. Rivera, D. Van Tuan, N. P. Wilson, M. Yang, T. Taniguchi, K. Watanabe, J. Yan, D. G. Mandrus, H. Yu, H. Dery, W. Yao, X. Xu. Valley phonons and exciton complexes in a monolayer semiconductor. Nat. Commun., 2020, 11(1): 618

[50] P. Cudazzo, I. V. Tokatly, A. Rubio. Dielectric screening in two-dimensional insulators: Implications for excitonic and impurity states in graphane. Phys. Rev. B, 2011, 84(8): 085406

[51] M. Danovich, D. A. Ruiz-Tijerina, R. J. Hunt, M. Szyniszewski, N. D. Drummond, V. I. Fal’ko. Localized interlayer complexes in heterobilayer transition metal dichalcogenides. Phys. Rev. B, 2018, 97(19): 195452

Declarations

The authors declare that they have no competing interests and there are no conflicts.

Acknowledgements

H. Y. acknowledges the support by the National Natural Science Foundation of China (Grant No. 12274477) and the Department of Science and Technology of Guangdong Province (No. 2019QN01X061).

RIGHTS & PERMISSIONS

2024 Higher Education Press