Efficient Energy Level Calculations in InP 2D-Quantum Box with Two Distinct Potentials Using the Sparse Numerov Method

Öz

In this study, energy level calculations for an InP 2D quantum box structure with two distinct (infinite potential power-exponential) potential potentials have been conducted using the sparse Numerov method. The 2D Schrödinger equation has been transformed in accordance with the sparse Numerov approach, followed by the creation of the solution matrix employing appropriate finite difference expressions. A comparative analysis of calculation results has been performed with respect to CPU time, memory usage, and ground state energy for both O(h^4) and O(h^6) accuracy. The suitability of the sparse Numerov method for 2D nanostructures has been thoroughly discussed. The results revealed that the sparse Numerov approach yields physically meaningful and rational outcomes in the InP 2D quantum box structure. Importantly, it demands significantly lower CPU time and memory resources compared to the classical Numerov method, emphasizing its practical applicability in this context.

Anahtar Kelimeler

• Sparse Numerov
• approach 2D quantum box
• Nanostructure
• 2D-Stencil

Kaynakça

1. Chang, J., Gao, J., Esmaeil Zadeh, I., Elshaari, A. W., & Zwiller, V. (2023). Nanowire-based integrated photonics for quantum information and quantum sensing. Nanophotonics, 12(3), 339-358.
2. Chen, B., Li, D., & Wang, F. (2020). InP quantum dots: synthesis and lighting applications. Small, 16(32), 2002454.
3. Ciurla, M., Adamowski, J., Szafran, B., & Bednarek, S. (2002). Modelling of confinement potentials in quantum dots. Physica E: Low-dimensional Systems and Nanostructures, 15(4), 261-268.
4. Dongjiao, T., Ye, Y., & Dewanto, M. A. (2014). Generalized Matrix Numerov Solutions to the Schrödinger Equation. Bachelor’ s thesis, National University of Singapore, Singapore.
5. Ed-Dahmouny, A., Zeiri, N., Fakkahi, A., Arraoui, R., Jaouane, M., Sali, A., ... & Duque, C. A. (2023). Impurity photo-ionization cross section and stark shift of ground and two low-lying excited electron-states in a core/shell ellipsoidal quantum dot. Chemical Physics Letters, 812, 140251.
6. Gamper, J., Kluibenschedl, F., Weiss, A. K., & Hofer, T. S. (2023). Accessing Position Space Wave Functions in Band Structure Calculations of Periodic Systems─ A Generalized, Adapted Numerov Implementation for One-, Two-, and Three-Dimensional Quantum Problems. The Journal of Physical Chemistry Letters, 14(33), 7395-7403.
7. Graen, T., & Grubmüller, H. (2016). NuSol—Numerical solver for the 3D stationary nuclear Schrödinger equation. Computer Physics Communications, 198, 169-178.
8. Hu, L., & Mandelis, A. (2021). Advanced characterization methods of carrier transport in quantum dot photovoltaic solar cells. Journal of Applied Physics, 129(9).

9. Jiang, W., Low, B. Q. L., Long, R., Low, J., Loh, H., Tang, K. Y., ... & Ye, E. (2023). Active site engineering on plasmonic nanostructures for efficient photocatalysis. ACS nano, 17(5), 4193-4229.
10. Killingbeck, J. (1987). Shooting methods for the Schrodinger equation. Journal of Physics A: Mathematical and General, 20(6), 1411.
11. Kuenzer, U., Sorarù, J. A., & Hofer, T. S. (2016). Pushing the limit for the grid-based treatment of Schrödinger's equation: a sparse Numerov approach for one, two and three dimensional quantum problems. Physical Chemistry Chemical Physics, 18(46), 31521-31533.
12. Koch, O., Kreuzer, W., & Scrinzi, A. (2006). Approximation of the time-dependent electronic Schrödinger equation by MCTDHF. Applied mathematics and computation, 173(2), 960-976.
13. Li, J., Liu, X., Wan, L., Qin, X., Hu, W., & Yang, J. (2022). Mixed magnetic edge states in graphene quantum dots. Multifunctional Materials, 5(1), 014001. Liang, K., Wang, R., Huo, B., Ren, H., Li, D., Wang, Y., ... & Zhu, B. (2022). Fully printed optoelectronic synaptic transistors based on quantum dot–metal oxide semiconductor heterojunctions. ACS nano, 16(6), 8651-8661.
14. Liu, G., Poole, P. J., Lu, Z., Liu, J., Song, C. Y., Mao, Y., & Barrios, P. (2023). Mode-Locking and Noise Characteristics of InAs/InP Quantum Dash/Dot Lasers. Journal of Lightwave Technology, 41(13), 4262-4270.
15. Lu, P., Lu, M., Zhang, F., Qin, F., Sun, S., Zhang, Y., ... & Bai, X. (2023). Bright and spectrally stable pure-red CsPb (Br/I) 3 quantum dot LEDs realized by synchronous device structure and ligand engineering. Nano Energy, 108, 108208.
16. Pillai, M., Goglio, J., & Walker, T. G. (2012). Matrix Numerov method for solving Schrödinger’s equation. American Journal of Physics, 80(11), 1017-1019.
17. Rafailov, E. U., Cataluna, M. A., & Sibbett, W. (2007). Mode-locked quantum-dot lasers. Nature photonics, 1(7), 395-401.
18. Sanderson, C., & Curtin, R. (2016). Armadillo: a template-based C++ library for linear algebra. Journal of Open Source Software, 1(2), 26.
19. Sanderson, C., & Curtin, R. (2018). A user-friendly hybrid sparse matrix class in C++. In Mathematical Software–ICMS 2018: 6th International Conference, South Bend, IN, USA, July 24-27, 2018, Proceedings 6 (pp. 422-430). Springer International Publishing.
20. Terada, S., Ueda, H., Ono, T., & Saitow, K. I. (2022). Orange–red Si quantum dot LEDs from recycled rice husks. ACS Sustainable Chemistry & Engineering, 10(5), 1765-1776.
21. Wang, K., Xu, G., Gao, F., Liu, H., Ma, R. L., Zhang, X., ... & Guo, G. P. (2022). Ultrafast coherent control of a hole spin qubit in a germanium quantum dot. Nature Communications, 13(1), 206.
22. Wang, X., Xu, L., Ge, S., Foong, S. Y., Liew, R. K., Chong, W. W. F., ... & Huang, R. (2023). Biomass-based carbon quantum dots for polycrystalline silicon solar cells with enhanced photovoltaic performance. Energy, 274, 127354.
23. Won, Y. H., Cho, O., Kim, T., Chung, D. Y., Kim, T., Chung, H., ... & Jang, E. (2019). Highly efficient and stable InP/ZnSe/ZnS quantum dot light-emitting diodes. Nature, 575(7784), 634-638.
24. Yadav, A., Chichkov, N. B., Avrutin, E. A., Gorodetsky, A., & Rafailov, E. U. (2023). Edge emitting mode-locked quantum dot lasers. Progress in Quantum Electronics, 100451.
25. Zhang, H., Hu, N., Zeng, Z., Lin, Q., Zhang, F., Tang, A., ... & Du, Z. (2019). High‐efficiency green InP quantum dot‐based electroluminescent device comprising thick‐shell quantum dots. Advanced Optical Materials, 7(7), 1801602.