volume | PIER Journals

Abstract

In this paper, we model the interband conductivity of a graphene sheet using a rational-function expansion based on complex-conjugate pole-residue (CCPR) pairs via an efficient vector-fitting approach. We then incorporate the developed model into a time-domain transmission-line matrix (TLM) method with a voltage source and normalized admittance, representing graphene as an equivalent surface current density via an auxiliary differential equation (ADE) scheme. Reflection and transmission coefficients for electromagnetic (EM) wave interaction with an infinite graphene sheet are computed. Numerical results show excellent agreement with available theoretical data, demonstrating that interband conductivity contributions significantly modify graphene's electromagnetic response in the mid- and near-infrared regimes and cannot be neglected. Furthermore, the tunability of the transmission coefficient under varying temperatures and chemical potentials is investigated. The proposed method provides an alternative TLM-based approach for modeling graphene's full Kubo conductivity, extending the application domain of the TLM method from the far-infrared regime to the mid- and near-infrared regimes, and offers a new approach for the design and analysis of tunable graphene-based electromagnetic devices.

1. Novoselov, K. S., A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, "Electric field effect in atomically thin carbon films," Science, Vol. 306, No. 5696, 666-669, October 2004.
doi:10.1126/science.1102896 Google Scholar

2. Novoselov, K. S., V. I. Fal′ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, "A roadmap for graphene," Nature, Vol. 490, No. 7419, 192-200, October 2012.
doi:10.1038/nature11458 Google Scholar

3. Wu, Jiayang, Han Lin, David J. Moss, Kian Ping Loh, and Baohua Jia, "Graphene oxide for photonics, electronics and optoelectronics," Nature Reviews Chemistry, Vol. 7, No. 3, 162-183, January 2023.
doi:10.1038/s41570-022-00458-7 Google Scholar

4. Samaha, Anna-Christina, Jacques Doumani, T. Elijah Kritzell, Hongjing Xu, Andrey Baydin, Pulickel M. Ajayan, Mario El Tahchi, and Junichiro Kono, "Graphene terahertz devices for sensing and communication," Small, Vol. 21, No. 28, 2401151, August 2024.
doi:10.1002/smll.202401151 Google Scholar

5. Liu, Jihong, Siyu Bao, and Xinzhe Wang, "Applications of graphene-based materials in sensors: A review," Micromachines, Vol. 13, No. 2, 184, January 2022.
doi:10.3390/mi13020184 Google Scholar

6. Rodriguez-Lopez, Pablo, Jian-Sheng Wang, and Mauro Antezza, "Electric conductivity in graphene: Kubo model versus a nonlocal quantum field theory model," Physical Review B, Vol. 111, No. 11, 115428, March 2025.
doi:10.1103/physrevb.111.115428 Google Scholar

7. Taflove, A. and S. C. Hagness, Computational Electromagnetics: The Finite-Difference Time-Domain Method, Artech House, 2005.

8. He, Guo-Qiang, Feng-Xiang Li, Zi-Xuan Yi, Mei-Ling Li, Xue-Xia Yang, and Johan H. Stiens, "Modeling graphene-based THz devices with an improved surface boundary condition in Leapfrog FDTD scheme," IEEE Transactions on Plasma Science, Vol. 50, No. 5, 1369-1377, May 2022.
doi:10.1109/tps.2022.3166672 Google Scholar

9. Bouzianas, Georgios D., Nikolaos V. Kantartzis, Christos S. Antonopoulos, and Theodoros D. Tsiboukis, "Optimal modeling of infinite graphene sheets via a class of generalized FDTD schemes," IEEE Transactions on Magnetics, Vol. 48, No. 2, 379-382, February 2012.
doi:10.1109/tmag.2011.2172778 Google Scholar

10. Chen, Wei-Jun, Qi-Wen Liang, Shi-Yu Long, and Min Zhao, "Modeling thin graphene sheets in the WLP-FDTD algorithm with surface boundary condition," Progress In Electromagnetics Research Letters, Vol. 91, 93-98, 2020.
doi:10.2528/pierl20041503 Google Scholar

11. Sarker, Pejush Chandra, Md. Masud Rana, and Ajay Krishno Sarkar, "A simple FDTD approach for the analysis and design of graphene based optical devices," Optik, Vol. 144, 1-8, September 2017.
doi:10.1016/j.ijleo.2017.06.054 Google Scholar

12. Baker, G. A. and P. Graves-Morris, Padé Approximants, Cambridge University Press, Cambridge, UK, 1996.

13. Chen, Juan, Jianxing Li, and Qing Huo Liu, "Designing graphene-based absorber by using HIE-FDTD method," IEEE Transactions on Antennas and Propagation, Vol. 65, No. 4, 1896-1902, April 2017.
doi:10.1109/tap.2017.2670610 Google Scholar

14. Sarker, Pejush Chandra, Md. Masud Rana, and Ajay Krishno Sarkar, "Modeling of graphene conductivity using FDTD in the near infrared frequency," 2016 2nd International Conference on Electrical, Computer & Telecommunication Engineering (ICECTE), 1-4, Rajshahi, Bangladesh, December 2016.
doi:10.1109/icecte.2016.7879581

15. Mock, Adam, "Padé approximant spectral fit for FDTD simulation of graphene in the near infrared," Optical Materials Express, Vol. 2, No. 6, 771, May 2012.
doi:10.1364/ome.2.000771 Google Scholar

16. Gustavsen, B. and A. Semlyen, "Rational approximation of frequency domain responses by vector fitting," IEEE Transactions on Power Delivery, Vol. 14, No. 3, 1052-1061, July 1999.
doi:10.1109/61.772353 Google Scholar

17. Choi, Hongjin, Yeon-Hwa Kim, Jae-Woo Baek, and Kyung-Young Jung, "Accurate and efficient finite-difference time-domain simulation compared with CCPR model for complex dispersive media," IEEE Access, Vol. 7, 160498-160505, 2019.
doi:10.1109/access.2019.2951173 Google Scholar

18. Lin, Hai, Mario F. Pantoja, Luis D. Angulo, Jesus Alvarez, Rafael G. Martin, and Salvador G. Garcia, "FDTD modeling of graphene devices using complex conjugate dispersion material model," IEEE Microwave and Wireless Components Letters, Vol. 22, No. 12, 612-614, December 2012.
doi:10.1109/lmwc.2012.2227466 Google Scholar

19. Nayyeri, Vahid, Mohammad Soleimani, and Omar M. Ramahi, "Wideband modeling of graphene using the finite-difference time-domain method," IEEE Transactions on Antennas and Propagation, Vol. 61, No. 12, 6107-6114, December 2013.
doi:10.1109/tap.2013.2282535 Google Scholar

20. Wang, Da-Wei, Wen-Sheng Zhao, Xiao-Qiang Gu, Wenchao Chen, and Wen-Yan Yin, "Wideband modeling of graphene-based structures at different temperatures using hybrid FDTD method," IEEE Transactions on Nanotechnology, Vol. 14, No. 2, 250-258, March 2015.
doi:10.1109/tnano.2014.2387576 Google Scholar

21. Xu, Di, Cheng Deng, Hai Lin, and Helin Yang, "A versatile material model for the FDTD simulation of graphene," 2013 IEEE International Conference on Microwave Technology & Computational Electromagnetics, 345-348, August 2013.
doi:10.1109/icmtce.2013.6812488

22. Wang, Xiang-Hua, Jian-Yun Gao, and Fernando L. Teixeira, "Stability-improved ADE-FDTD method for wideband modeling of graphene structures," IEEE Antennas and Wireless Propagation Letters, Vol. 18, No. 1, 212-216, January 2019.
doi:10.1109/lawp.2018.2886335 Google Scholar

23. Johns, P. B., "A symmetrical condensed node for the TLM method," IEEE Transactions on Microwave Theory and Techniques, Vol. 35, No. 4, 370-377, April 1987.
doi:10.1109/tmtt.1987.1133658 Google Scholar

24. Johns, P. B. and R. L. Beurle, "Numerical solution of 2-dimensional scattering problems using a transmission-line matrix," Proceedings of the Institution of Electrical Engineers, Vol. 118, No. 9, 1203, 1971.
doi:10.1049/piee.1971.0217 Google Scholar

25. Christopoulos, C., The Transmission-Line Modeling (TLM) Method in Electromagnetics, Springer International Publishing, Cham, 2022.

26. El Adraoui, Soufiane, Khalid Mounirh, Mohamed Iben Yaich, and Mohsine Khalladi, "Shift operator-TLM method for modeling gyroelectric media," Progress In Electromagnetics Research M, Vol. 87, 189-197, 2019.
doi:10.2528/pierm19081803 Google Scholar

27. Mounirh, Khalid, Soufiane El Adraoui, Yasser Ekdiha, Mohamed Iben Yaich, and Mohsine Khalladi, "Modeling of dispersive chiral media using the ADE-TLM method," Progress In Electromagnetics Research M, Vol. 64, 157-166, 2018.
doi:10.2528/pierm17110103 Google Scholar

28. Kanjaa, Mohammed, Otman El Mrabet, and Mohsine Khalladi, "A TLM formulation based on fractional derivatives for dispersive Cole-Cole media," Electromagnetic Wave Propagation for Industry and Biomedical Applications, L. Wang (ed.), IntechOpen, 2021.
doi:10.5772/intechopen.96378 Google Scholar

29. Yaich, M. I., M. Khalladi, and M. Essaaidi, "Efficient modeling of chiral media using SCN-TLM method," Serbian Journal of Electrical Engineering, Vol. 1, No. 2, 249-254, 2004. Google Scholar

30. Moumou, Mohamed, Soufiane El Adraoui, Khalid Mounirh, Mohammed Kanjaa, and Mohsine Khalladi, "Efficient ADE-TLM scheme for modeling Drude based graphene in terahertz spectrum," Progress In Electromagnetics Research Letters, Vol. 112, 119-126, 2023.
doi:10.2528/pierl23060904 Google Scholar

31. Hanson, George W., "Dyadic Green's functions and guided surface waves for a surface conductivity model of graphene," Journal of Applied Physics, Vol. 103, No. 6, 064302, March 2008.
doi:10.1063/1.2891452 Google Scholar

32. Falkovsky, L. A. and S. S. Pershoguba, "Optical far-infrared properties of a graphene monolayer and multilayer," Physical Review B, Vol. 76, No. 15, 153410, October 2007.
doi:10.1103/physrevb.76.153410 Google Scholar

33. Gong, Yansheng and Na Liu, "Advanced numerical methods for graphene simulation with equivalent boundary conditions: A review," Photonics, Vol. 10, No. 7, 712, 2023.
doi:10.20944/preprints202305.0697.v1 Google Scholar

34. Valizadeh, Mahsa, Leila Yousefi, and MirFaez Miri, "Analytical formulation of spatiotemporal modulated graphene-based waveguides using Floquet-Bloch theory," Scientific Reports, Vol. 14, No. 1, 7332, March 2024.
doi:10.1038/s41598-024-56815-7 Google Scholar

35. Du, Xuemei, Fengping Yan, Wei Wang, Luna Zhang, Zhuoya Bai, Hong Zhou, and Yafei Hou, "A broadband switchable metamaterial absorber/reflector based on multi-laps graphene sheets in the terahertz band," IEEE Photonics Journal, Vol. 13, No. 5, 1-8, October 2021.
doi:10.1109/jphot.2021.3109045 Google Scholar

36. Khalladi, Mohsine, M. I. Yaich, N. Aknin, and M. C. Carrión, "Modeling of electromagnetic waves propagation in nonlinear optical media using HSCN-TLM method," IEICE Electronics Express, Vol. 2, No. 13, 384-391, 2005.
doi:10.1587/elex.2.384 Google Scholar

37. Odeyemi, Jomiloju, Chris Smartt, Ana Vukovic, Trevor M. Benson, and Phillip Sewell, "PML effectiveness in the transmission line modelling method for radiation and scattering applications," 2020 14th European Conference on Antennas and Propagation (EuCAP), 1-4, Copenhagen, Denmark, March 2020.
doi:10.23919/eucap48036.2020.9135802

38. Peres, N. M. R., T. Stauber, and A. H. Castro Neto, "The infrared conductivity of graphene on top of silicon oxide," EPL (Europhysics Letters), Vol. 84, No. 3, 38002, October 2008.
doi:10.1209/0295-5075/84/38002 Google Scholar

39. Herbut, Igor F. and Vieri Mastropietro, "Universal conductivity of graphene in the ultrarelativistic regime," Physical Review B, Vol. 87, No. 20, 205445, May 2013.
doi:10.1103/physrevb.87.205445 Google Scholar

40. Mak, Kin Fai, Matthew Y. Sfeir, Yang Wu, Chun Hung Lui, James A. Misewich, and Tony F. Heinz, "Measurement of the optical conductivity of graphene," Physical Review Letters, Vol. 101, No. 19, 196405, November 2008.
doi:10.1103/physrevlett.101.196405 Google Scholar

41. Nair, R. R., P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, "Fine structure constant defines visual transparency of graphene," Science, Vol. 320, No. 5881, 1308, June 2008.
doi:10.1126/science.1156965 Google Scholar

42. Gusynin, V. P., S. G. Sharapov, and J. P. Carbotte, "On the universal ac optical background in graphene," New Journal of Physics, Vol. 11, No. 9, 095013, September 2009.
doi:10.1088/1367-2630/11/9/095013 Google Scholar

43. Tuckerman, M. E., Statistical Mechanics: Theory and Molecular Simulation, Oxford University Press, 2023.

Mr. Mohamed Moumou

Dr. Khalid Mounirh

Dr. Soufiane El Adraoui

Dr. Mohammed Kanjaa

Prof. Mohsine Khalladi