In the simulation of high frequency nanoscale semiconductor devices in which electromagnetic (EM) fields and carrier transport are coupled, and optoelectronic devices in which strong interactions between EM fields and charged particles exist, both the Maxwell's equations and the time-dependent Schrödinger equation (TDSE) need to be solved to capture the interactions between EM and quantum mechanics (QM). One of the numerical simulation methods for solving these equations is the finite difference time domain (FDTD) method. In this review paper, the development of FDTD method applied in EM and QM simulation is discussed. Several widely used FDTD techniques, i.e., explicit, implicit, explicit staggered-time, and Chebyshev methods, for solving the TDSE are introduced and compared. The hybrid approaches based on FDTD method, which are used to solve the Poisson-TDSE and Maxwell-TDSE coupled equations for EM-QM simulation, are also discussed. Furthermore, the applications of these simulation methods for nanoscale semiconductor devices and optoelectronic devices are introduced. Finally, a conclusion is given.
2. Chen, Y. P., W. E. I. Sha, W. C. H. Choy, L. Jiang, and W. C. Chew, "Study on spontaneous emission in complex multilayered plasmonic system via surface integral equation approach with layered medium Green’s function," Optics Express, Vol. 20, No. 18, 20210, 2012.
3. Capua, A., O. Karni, and G. Eisenstein, "A finite-difference time-domain model for quantum-dot lasers and amplifiers in the Maxwell-Schrodinger framework," IEEE Journal of Selected Topics in Quantum Electronics, Vol. 19, No. 5, 1-10, 2013.
4. Yankwich, P. E., "Introduction to quantum mechanics," Journal of the American Chemical Society, Vol. 82, No. 14, 3803-3803, 1960.
5. Rae, A. I. M., "The picture book of quantum mechanics," Physics Today, Vol. 49, No. 1, 65-66, 1996.
6. Chan, T. F., D. Lee, and L. Shen, "Stable explicit schemes for equations of the Schrodinger type," SIAM Journal on Numerical Analysis, Vol. 23, No. 2, 274-281, 1986.
7. Chen, J. B. and M. Z. Qinz, "Multi-symplectic Fourier pseudospectral method for the nonlinear Schrodinger equation," Electronic Transactions on Numerical Analysis Etna, Vol. 12, 193-204, 2001.
8. Chang, Q. and G. Wang, "Multigrid and adaptive algorithm for solving the nonlinear Schrodinger equation," Journal of Computational Physics, Vol. 85, No. 2, 504, 1989.
9. Dai, W. Z. and R. Nassar, "A finite difference scheme for the generalized nonlinear Schrodinger equation with variable coefficients," Journal of Computational Mathematics, Vol. 18, No. 2, 123-132, 2000.
10. Delfour, M., M. Fortin, and G. Payr, "Finite-difference solutions of a non-linear Schrodinger equation," Journal of Computational Physics, Vol. 44, No. 2, 277-288, 1981.
11. Herbst, B. M., J. Ll Morris, and A. R. Mitchell, "Numerical experience with the nonlinear Schrodinger equation," Journal of Computational Physics, Vol. 60, No. 2, 282-305, 1985.
12. Taflove, A. and S. C. Hagness, Computational Electrodynamics (The Finite-difference Time-domain Method), 3rd Ed., Artech House, 2001.
13. Sullivan, D. M., Electromagnetic Simulation Using the FDTD Method, 2nd Ed., Chapters 1–11, Wiley-IEEE Press, 2000.
14. Sullivan, D. and D. S. Citrin, "Time-domain simulation of two electrons in a quantum dot," Journal of Applied Physics, Vol. 89, No. 7, 3841-3846, 2001.
15. Sullivan, D. M. and D. S. Citrin, "Determination of the eigenfunctions of arbitrary nanostructures using time domain simulation," Journal of Applied Physics, Vol. 91, No. 5, 3219-3226, 2002.
16. Soriano, A., E. A. Navarro, J. A. Porti, and V. Such, "Analysis of the finite difference time domain technique to solve the Schrodinger equation for quantum devices," Journal of Applied Physics, Vol. 95, No. 12, 8011-8011, 2004.
17. Sudiarta, I. W. and D. J. W. Geldart, "Solving the Schrodinger equation using the finite difference time domain method," Journal of Physics A: Mathematical and Theoretical, Vol. 40, No. 8, 1885-1896, 2007.
18. Moxley, F. I., D. T. Chuss, and W. Dai, "A generalized finite-difference time-domain scheme for solving nonlinear Schrodinger equations," Computer Physics Communications, Vol. 184, No. 8, 1834-1841, 2013.
19. Tay, W. C. and E. L. Tan, "Pentadiagonal alternating-direction-implicit finite-difference time-domain method for two-dimensional Schrodinger equation," Computer Physics Communications, Vol. 185, No. 7, 1886-1892, 2014.
20. Wilson, J. P. and W. Dai, "Generalized finite-difference time-domain method with absorbing boundary conditions for solving the nonlinear Schrodinger equation on a GPU," Computer Physics Communications, Vol. 235, 279-292, 2019.
21. Dai, W., G. Li, R. Nassar, and S. Su, "On the stability of the FDTD method for solving a time-dependent Schrodinger equation," Numerical Methods for Partial Differential Equations, Vol. 21, No. 6, 1140-1154, 2010.
22. Adamowski, J., "A numerical solution of the Poisson-Schr¨odinger problem for a vertical gated quantum dot," TASK Quarterly, Vol. 8, 603, 2004.
23. Fiori, G. and G. Iannaccone, "The effect of quantum confinement and discrete dopants in nanoscale 50 nm n-MOSFETs: A three-dimensional simulation," IEEE Transactions on Nanotechnology, Vol. 13, No. 3, 294, 2002.
24. Guo, J., et al., "Assessment of high-frequency performance potential of carbon nanotube transistors," IEEE Transactions on Nanotechnology, Vol. 4, No. 6, 715-721, 2005.
25. Stefanucci, G., S. Kurth, A. Rubio, and E. K. U. Gross, "Time-dependent approach to electron pumping in open quantum systems," Physical Review B, Vol. 77, 75339, 2008.
26. Chen, Z.-D., J.-Y. Zhang, and Z.-P. Yu, "Time-dependent transport in nanoscale devices," Chinese Physics Letters, Vol. 26, No. 3, 37303-37306(4), 2009.
27. Yang, J. and W. Sui, "Solving Maxwell-Schrodinger equations for analyses of nano-scale devices," European Microwave Conference, 2007.
28. Ahmed, I., E. H. Khoo, E. Li, and R. Mittra, "A hybrid approach for solving coupled Maxwell and Schrodinger equations arising in the simulation of nano-devices," IEEE Antennas and Wireless Propagation Letters, Vol. 9, 914-917, 2010.
29. Shibayama, J., M. Muraki, J. Yamauchi, and H. Nakano, "Efficient implicit FDTD algorithm based on locally one-dimensional scheme," Electronics Letters, Vol. 41, No. 19, 1046-1047, 2006.
30. Ahmed, I., E. K. Chua, E. P. Li, and Z. Chen, "Development of the three-dimensional unconditionally stable LOD-FDTD method," IEEE Transactions on Antennas and Propagation, Vol. 58, No. 11, 832-837, 2010.
31. Pierantoni, L., D. Mencarelli, and T. Rozzi, "A new 3-D transmission line matrix scheme for the combined Schrodinger-Maxwell problem in the electronic/electromagnetic characterization of nanodevices," IEEE Transactions on Microwave Theory and Techniques, Vol. 56, No. 3, 654-662, 2008.
32. Xiang, C., F. Kong, K. Li, and M. Liu, "A high-order symplectic FDTD scheme for the Maxwell-Schrodinger system," IEEE Journal of Quantum Electronics, Vol. 54, No. 1, 1-8, 2018.
33. Lee, K. H., I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, "Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications," Progress In Electromagnetics Research, Vol. 116, 441-456, 2011.
34. Chen, Y. P., Y. M.Wu, and W. E. I. Sha, "Modeling Rabi oscillation by rigorously solving Maxwell-Schrodinger equation," IEEE International Symposium on Microwave, 2016.
35. Hatori, N., M. Sugawara, T. Akiyama, and Y. Nakata, "Low frequency chirp self-assembled InGaAs/GaAs quantum dot lasers," Lasers & Electro-optics Society, Leos the Meeting of the IEEE, 2001.
36. Yang, Z. D., L. Zhang, H. Zeng, D. Z. Ding, and R. S. Chen, "Multi-quantum state control of nano-tube by the Maxwell-Schrodinger hybrid method," 2018 Cross Strait Quad-Regional Radio Science and Wireless Technology Conference (CSQRWC), 2018.
37. Takeuchi, T., S. Ohnuki, and T. Sako, "Maxwell-Schrodinger hybrid simulation for optically controlling quantum states: A scheme for designing control pulses," Physical Review A, Vol. 91, No. 3, 033401, 2015.
38. Meshulach, D. and Y. Silberberg, "Coherent quantum control of two-photon transitions by a femtosecond laser pulse," Nature, Vol. 396, No. 6708, 239-242, 1998.
39. Chen, Z., J. Zhang, and Z. Yu, "Solution of the time-dependent Schrodinger equation with absorbing boundary conditions," Journal of Semiconductors, Vol. 30, No. 1, 1-6, 2009.
40. Subasi, M., "On the finite-differences schemes for the numerical solution of two dimensional Schrodinger equation," Numerical Methods for Partial Differential Equations, Vol. 18, No. 6, 752-758, 2002.
41. Burden, R. L. and J. D. Faires, Numerical Analysis, 5th Ed., PWS Publishing Co., 1988.
42. Visscher, P. B., "A fast explicit algorithm for the time-dependent Schrodinger equation," Computers in Physics, Vol. 5, No. 6, 596-598, 1991.
43. Tal-Ezer, H. and R. Kosloff, "An accurate and efficient scheme for propagating the time dependent Schrodinger equation," Journal of Chemical Physics, Vol. 81, No. 9, 3967-3971, 1984.
44. Leforestier, C., et al., "A comparison of different propagation schemes for the time dependent Schrodinger equation," Journal of Computational Physics, Vol. 94, No. 1, 59-80, 1991.
45. Leforestier, C., R. H. Bisseling, C. Cerjan, M. D. Feit, and R. Kosloff, "A comparison of different propagation schemes for the time dependent Schrodinger equation," Journal of Computational Physics, Vol. 89, No. 1, 490-491, 1991.
46. De Raedt, H., K. Michielsen, J. S. Kole, and M. T. Figge, "One-step finite-difference time-domain algorithm to solve the Maxwell equations," Physical Review E Statal Nonlinear & Soft Matter Physics, Vol. 67, No. 5, Pt. 2, 056706, 2003.
47. Bar-On, I. and M. Leoncini, "Stable solution of tridiagonal systems," Numerical Algorithms, Vol. 18, No. 3, 361-388, 1998.
48. Zhang, Y., J. Cohen, A. A. Davidson, and J. D. Owens, "A hybrid method for solving tridiagonal systems on the GPU," GPU Computing Gems Jade Edition, 117-132, 2012.
49. Chen, Y. C. and C. R. Lee, Augmented Block Cimmino Distributed Algorithm for Solving Tridiagonal Systems on GPU, Chapter 9, Advances in GPU Research and Practice, 2017.
50. Chen, Y., Finite element method modeling of advanced electronic devices, Electronic Theses and Dissertations, 2006.
51. Alsunaidi, M. A., S. M. S. Imtiaz, and S. M. El-Ghazaly, "Electromagnetic wave effects on microwave transistors using a full-wave time-domain model," IEEE Transactions on Microwave Theory and Techniques, Vol. 44, No. 6, 799-808, 1996.
52. Grondin, R. O., S. M. El-Ghazaly, and S. M. Goodnick, "A review of global modeling of charge transport in semiconductors and full-wave electromagnetics," IEEE Transactions on Microwave Theory and Techniques, Vol. 47, No. 11, 2167-2167, 2002.
53. Naeemi, A., R. Sarvari, and J. D. Meindl, "Performance comparison between carbon nanotube and copper interconnects for GSI," IEEE International Electron Devices Meeting, 2005.
54. Kim, G., E. Arvas, V. Demir, and A. Z. Elsherbeni, "A novel nonuniform subgridding scheme for FDTD using an optimal interpolation technique," Progress In Electromagnetics Research B, Vol. 44, 137-161, 2012.
55. Mailloux, R., "Theory of electromagnetic waves," IEEE Antennas & Propagation Society Newsletter, Vol. 26, No. 2, 13-14, 1984.
56. Ahmed, I. and E. Li, "A hybrid FDTD and ADI-FDTD technique for coupled Maxwell’s and Schrodinger’s equations," IEEE Antennas & Propagation Society International Symposium, 2010.
57. Ren, X., et al., "High-order unified symplectic FDTD scheme for the metamaterials," Computer Physics Communications, Vol. 183, No. 6, 1192-1200, 2012.
58. Ryu, C. J., A. Liu, W. E. I. Sha, and W. C. Chew, "Finite-difference time-domain simulation of the Maxwell-Schrodinger system," IEEE Journal on Multiscale & Multiphysics Computational Techniques, Vol. 1, 40-47, 2016.
59. Turati, P., "FDTD modelling of nanostructures at microwave frequency," Surface & Coatings Technology, Vol. 254, No. 10, 402-409, 2014.
60. Pierantoni, L., D. Mencarelli, and T. Rozzi, "The combined Schrodinger-Maxwell problem in the electronic/electromagnetic characterization of nanodevices," Time Domain Methods in Electrodynamics, 105-133, 2008.
61. Xie, G., Z. Huang, M. Fang, and W. Sha, "Simulating Maxwell-Schrodinger equations by high-order symplectic FDTD algorithm," IEEE Journal on Multiscale and Multiphysics Computational Techniques, Vol. 4, 143-151, 2019.
62. Zheng, F. and Z. Chen, "A finite-difference time-domain method without the Courant stability conditions," IEEE Microw. Guided Wave Lett., Vol. 9, No. 11, 441-443, 1999.
63. Ravi, K., Y. Huang, and S. Ho, "A computationally efficient, non-equilibrium, carrier temperature dependent semiconductor gain model for FDTD simulation of optoelectronic devices," 2011 Numerical Simulation of Optoelectronic Devices, 113-114, Sep. 5–8, 2011.
64. Bhardwaj, S., "Electronic-electromagnetic multiphysics modeling for terahertz plasmonics: A review," IEEE Journal on Multiscale and Multiphysics Computational Techniques, Vol. 4, 307-316, 2019.
65. Wang, G., et al., "The numerical modeling of 3D microfiber couplers and resonators," IEEE Photonics Technology Letters, Vol. 28, No. 15, 1707-1710, 2016.
66. Tan, E. L. and D. Y. Heh, "Multiple 1-D fundamental ADI-FDTD method for coupled transmission lines on mobile devices," IEEE Journal on Multiscale and Multiphysics Computational Techniques, Vol. 4, 198-206, 2019.
67. Zhai, M., H. Peng, J. Mao, and W. Yin, "Modeling tunable graphene-based filters using leapfrog ADI-FDTD method," 2015 IEEE MTT-S International Microwave Workshop Series on Advanced Materials and Processes for RF and THz Applications (IMWS-AMP), 1-3, Jul. 1–3, 2015.
68. Bahl, M., et al., "Mixed-level simulation of opto-electronic devices," 2016 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), 101-102, Jul. 11–15, 2016.
69. Bandrauk, E. L. C., "A numerical Maxwell-Schrodinger model for intense laser-matter interaction and propagation," Computer Physics Communications, 2007.
70. Navarro, D., "A carrier-transit-delay-based nonquasi-static MOSFET model for circuit simulation and its application to harmonic distortion analysis," IEEE Transactions on Electron Devices, Vol. 53, No. 9, 2025-2034, 2006.
71. Chen, Y. P., W. E. I. Sha, L. Jiang, M. Meng, Y. M. Wu, and W. C. Chew, "A unified Hamiltonian solution to Maxwell-Schrodinger equations for modeling electromagnetic field-particle interaction," Computer Physics Communications, Vol. 215, 63-70, 2017.
72. Takeuchi, T., S. Ohnuki, and T. Sako, "A simple formula to predict the influence of the near-field in the optical control of confined electron systems," Journal of Physics B Atomic Molecular & Optical Physics, Vol. 50, No. 4, 045002, 2017.
73. Gerry, C., Introductory Quantum Optics, 1st Ed., Cambridge University Press, London, 2004.
74. Rabitz, H., "Whither the future of controlling quantum phenomena?," Science, Vol. 288, No. 5467, 824-828, 2000.
75. Townsend, D., et al., "A Stark future for quantum control," The Journal of Physical Chemistry A, Vol. 4, No. 115, 357-373, 2011.
76. Rangan, C. and P. H. Bucksbaum, "Optimally shaped terahertz pulses for phase retrieval in a Rydberg-atom data register," Physical Review A, Vol. 64, No. 3, 033417, 2001.
77. Palao, J. P. and R. Kosloff, "Quantum computing by an optimal control algorithm for unitary transformations," Physical Review Letters, Vol. 89, No. 18, 188301, 2002.
78. Nunn, J., et al., "Mapping broadband single-photon wave packets into an atomic memory," Physical Review A, Vol. 75, No. 1, 011401, 2007.
79. Lewis, A. and K. Lieberman, "Near-field optical imaging with a non-evanescently excited high-brightness light source of sub-wavelength dimensions," Nature, Vol. 354, No. 6350, 214-216, 1991.
80. Zenhausern, F., "Apertureless near-field optical microscope," Applied Physics Letters, Vol. 65, No. 13, 1623-1625, 1994.
81. Choi, S., et al., "Active tailoring of nanoantenna plasmonic fields using few-cycle laser pulses," Physical Review A, Vol. 93, No. 2, 021405, 2016.