Double integral regularization and calculation for the problem of anisotropic two-dimensional exciton

Ngoc-Tram D. Hoang; Dang-Khoa D. Le; Hoang-Viet Le

doi:10.15625/0868-3166/23911

Author affiliations

Authors

Ngoc-Tram D. Hoang Physics Department, Ho Chi Minh City University of Education https://orcid.org/0000-0001-8847-0610
Dang-Khoa D. Le Ho Chi Minh City University of Education https://orcid.org/0009-0009-8325-0964
Hoang-Viet Le Ho Chi Minh City University of Education https://orcid.org/0009-0008-0941-1478

DOI:

https://doi.org/10.15625/0868-3166/23911

Keywords:

exciton, regularized perturbation theory, anisotropic monolayer, anisotropic integral

Abstract

The development of a regularized perturbation theory method for solving the Schrödinger equation of an exciton in a black phosphorus monolayer—a two-dimensional anisotropic material—yields a novel form of double integrals that is substantially more complex than that in previous studies on isotropic excitons. Direct computation of these integrals requires a significant amount of computational time and resources. In this study, we propose an analytical calculation scheme for these integrals based on the Newton binomial theorem, combined with Taylor expansion and a regularization method that utilizes a free parameter. This approach expresses the integrals as linear combinations of simpler single integrals, which can be evaluated through recurrence relations, making them suitable for numerical implementation. Moreover, a free parameter is introduced and optimized to enhance calculation speed. With this scheme, the evaluation speed of the integrals is improved dramatically—by several orders of magnitude—resulting in exciton energy-level calculations that are accelerated by up to five orders of magnitude. These findings present a new approach for the numerical treatment of complicated integrals, thereby enhancing computational efficiency in similar problems.

Downloads

Download data is not yet available.

References

[1] J. Frenkel, On the transformation of light into heat in solids. {i}, Physical Review 37 (1931) 17.

[2] N. S. Rytova, Screened potential of a point charge in a thin film, {Moscow University Physics Bulletin 3 (1967) 18}.

[3] L. V. Keldysh, Coulomb interaction in thin semiconductor and semimetal films, {JETP Letters 29 (1979) 658}.

[4] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos Et al., Electric field effect in atomically thin carbon films, Science 306 (2004) 666.

[5] A. K. Geim and I. V. Grigorieva, Van der {waals heterostructures}, Nature 499 (2013) 419.

[6] J. Hur, J. Park, D. Kim and S. Jeon, Model for the operation of a monolayer {mo{S}₂ thin-film transistor with charges trapped near the channel interface}, Physical Review Applied 7 (2017) 044030.

[7] A. V. Stier, N. P. Wilson, K. A. Velizhanin, J. Kono, X. Xu and S. A. Crooker, Magnetooptics of exciton {rydberg states in a monolayer semiconductor}, Physical Review Letters 120 (2018) 057405.

[8] D.-N. Ly, D.-N. Le, N. H. Phan and V.-H. Le, Thermal effect on magnetoexciton energy spectra in monolayer transition metal dichalcogenides, Physical Review B 107 (2023) 155410.

[9] L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu Et al., Black phosphorus field-effect transistors, Nature Nanotechnology 9 (2014) 372.

[10] E. Prada, J. Alvarez, K. Narasimha-Acharya, F. Bailen and J. Palacios, Effective-mass theory for the anisotropic exciton in two-dimensional crystals: application to phosphorene, Physical Review B 91 (2015) 245421.

[11] Y. Gao, M. Zhang, X. Zhang and G. Lu, Decreasing exciton binding energy in two-dimensional halide perovskites by lead vacancies, The Journal of Physical Chemistry Letters 10 (2019) 3820.

[12] J. Li, J. Ma, X. Cheng, Z. Liu, Y. Chen and D. Li, Anisotropy of excitons in two-dimensional perovskite crystals, ACS Nano 14 (2020) 2156.

[13] L. Vannucci, U. Petralanda, A. Rasmussen, T. Olsen and K. S. Thygesen, Anisotropic properties of monolayer 2{d materials: An overview from the {C2DB} database}, Journal of Applied Physics 128 (2020) 105101.

[14] Y. Wang, F. Jia, X. Li, S. Teale, P. Xia, Y. Liu Et al., Self-assembled monolayer–based blue perovskite {__protected_0__s}, Science Advances 9 (2023) eadh2140.

[15] D. D.-K. Le, H.-V. Le, Q.-H. Le and V.-H. Le, Anisotropic exciton in two-dimentional black phosphorus in a uniform magnetic field: an algebra approach, {Ho Chi Minh City University of Education Journal of Science 21 (2024) 1229}.

[16] R. Y. Kezerashvili, A. Spiridonova and A. Andrew Dublin, Magnetoexcitons in phosphorene monolayers, bilayers, and {van der {W}aals heterostructures}, Physical Review Research 4 (2022) 013154.

[17] S. Wu, Anisotropic exciton states and excitonic absorption spectra in a freestanding monolayer black phosphorus, Physica E: Low-dimensional Systems and Nanostructures 141 (2022) 115238.

[18] A. Castellanos-Gomez, L. Vicarelli, E. Prada, J. O. Island, K. L. Narasimha-Acharya, S. I. Blanter Et al., Isolation and characterization of few-layer black phosphorus, 2D Materials 1 (2014) 025001.

[19] P. Cudazzo, I. V. Tokatly and A. Rubio, Dielectric screening in two-dimensional insulators: implications for excitonic and impurity states in graphane, Physical Review B 84 (2011) 085406.

[20] P. D.-A. Nguyen, D.-N. Ly, D.-N. Le, D. N.-T. Hoang and V.-H. Le, High-accuracy energy spectra of a two-dimensional exciton screened by reduced dimensionality with the presence of a constant magnetic field, Physica E: Low-dimensional Systems and Nanostructures 113 (2019) 152.

[21] R. Tian, R. Fei, S. Hu, T. Li, B. Zheng, Y. Shi Et al., Observation of excitonic series in monolayer and few-layer black phosphorus, Phys. Rev. B 101 (2020) 235407.

[22] X. Wang, A. M. Jones, K. L. Seyler, V. Tran, Y. Jia, H. Zhao Et al., Highly anisotropic and robust excitons in monolayer black phosphorus, Nature Nanotechnology 10 (2015) 517.

[23] G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand Et al., Colloquium: excitons in atomically thin transition metal dichalcogenides, Reviews of Modern Physics 90 (2018) 021001.

[24] I. D. Feranchuk, A. Ivanov, V. H. Le and A. Ulyanenkov, Non-perturbative description of quantum systems, Lecture Notes in Physics, volume 894. Springer, Cham / Heidelberg / New York / Dordrecht / London, 2015, 10.1007/978-3-319-13006-4.

[25] R. D. Santaella Forero, R. Mendoza Suárez and J. C. López Carreño, A new way to evaluate lobachevsky’s integrals, International Journal of Mathematics and Mathematical Sciences 2025 (2025) 3918563.

[26] I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey and D. Zwillinger, Table of integrals, series, and products. Elsevier / Academic Press, London / San Diego / etc., 7 ed., 2007.