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Propagation matrix for electromagnetic interaction through electrostatically and magnetostatically biased graphene sheet

## Propagation matrix for electromagnetic interaction through electrostatically and magnetostatically biased graphene sheet

Wang Fei, Wei Bing
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• #### Abstract

The reflection and transmission of plane electromagnetic waves on monolayer graphene are studied theoretically in this paper. From an electromagnetic point of view, monolayer graphene is described as an “infinitely thin” graphene sheet characterized by a surface conductivity, and based on a microscopic quantum dynamical approach, the graphene sheet becomes anisotropic in the presence of both an electrostatic and a magnetic bias. In this work, starting from boundary conditions and phase-matching conditions, the propagation matrix for the analysis of the interaction between an electromagnetic field and thin graphene sheet which is biased electrostatically and magnetostatically, and then characterized by an anisotropic conductivity, is derived. Furthermore, the analytical solutions of co- and cross-polarization reflective and transmittance coefficients through an anisotropic graphene planar surface are obtained from the proposal matrix above, which couples the fundamental transverse electric (TE) polarization and transverse magnetic (TM) polarization and includes the possible effects of electrostatic and/or magnetostatic bias. In conclusion, the cross-polarization reflective coefficient of TE wave and that of TM wave are equal, and their cross-polarization transmittance coefficients have opposite phase. Finally, a new propagation matrix for stratified medium containing anisotropic graphene interfaces is deduced by embedding the matrix across graphene sheet mentioned above into the traditional propagation matrix for isotropic stratified medium. The proposed new matrix can be used to investigate the propagation properties of plane wave in a complex structure of layered medium and anisotropic conductivity interfaces (including graphene sheet) analytically and quickly, and represents a very simple tool for the relevant analysis and design.

#### References

 [1] Lovat G 2012 IEEE Trans. Electromagn. Compat. 54 101 [2] Geim A K 2009 Science 324 1530 [3] Das T, Sharma B K, Katiyar A K, Ahn J H 2018 J. Semicond. 39 011007 [4] Nimbalkar A, Kim H 2020 Nano-Micro Lett. 12 126 [5] Moshizi S A, Azadi S, Belford A, Razmjou A, Wu S, Han Z J, Asadnia M 2020 Nano-Micro Lett. 12 109 [6] Chen X Y, Tian Z, Li Q, Li S X, Zhang X Q, Ouyang C M, Gu J Q, Han J G, Zhang W L 2020 Chin. Phys. B 29 077803 [7] Chaea M S, Leea T H, Sona K R, Parka T H, Hwangb K S, Kim T G 2020 J Mater. Sci. Technol. 40 72 [8] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsove A A 2004 Science 306 666 [9] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsove A A 2005 Nature 438 197 [10] Zhang Y, Tan Y W, Stormer H L, Kim P 2005 Nature 438 201 [11] Zhang Y, Small J P, Pontius W V, Kim P 2005 Appl. Phys. Lett. 86 073104 [12] Slepyan G Y, Maksimenko S A, Lakhtakia A, Yevtushenko O, Gusakov A V 1999 Phys. Rev. B 60 17136 [13] Hanson G W 2008 J. Appl. Phys. 103 064302 [14] Gusynin V P, Sharapov S G, Carbotte J P 2007 J. Phys. Condens. Matter 19 026222 [15] Hanson G W 2008 IEEE Trans. Antennas Propag. 56 747 [16] 葛德彪, 魏兵 2011 电磁波理论 (北京: 科学出版社) 第5, 6, 39, 40, 56−73页 Ge D B, Wei B 2011 Electromagnetic wave theory (Beijing: Science Press) pp5, 6, 39, 40, 56−73 (in Chinese) [17] 王飞, 魏兵 2019 物理学报 68 244101 Wang F, Wei B 2019 Acta Phys. Sin. 68 244101 [18] Gusynin V P, Sharapov S G 2006 Phys. Rev. B 73 245411 [19] Gusynin V P, Sharapov S G, Carbotte J P 2006 Phys. Rev. Lett. 96 256802 [20] Peres N M R, Guiner F, Castro Neto A H 2006 Phys. Rev. B 73 125411 [21] Ziegler K 2007 Phys. Rev. B 75 233407 [22] George W H 2008 Journal of Applied Physics 103 064302 [23] Balanis C A 1989 Advanced Engineering Electromagnetics (New York: Wiley) pp168−170

#### Cited By

• 图 1  “无限薄”石墨烯层

Figure 1.  “Infinitely thin” graphene sheet.

图 2  石墨烯化学势${\mu _{\rm{c}}}\left( {{E_{{\rm{bias}}}}} \right)$与偏置电场${E_{{\rm{bias}}}}$的关系

Figure 2.  Graphical representation of the relation between the chemical potential ${\mu _{\rm{c}}}\left( {{E_{{\rm{bias}}}}} \right)$ and the electrostatic bias field ${E_{{\rm{bias}}}}$.

图 3  层状介质

Figure 3.  Stratified medium.

图 4  石墨烯电导率张量元素及屏蔽效率随偏置磁场变化　(a) 电导率张量元素; (b) 屏蔽效率

Figure 4.  Diagonal and off-diagonal components of the graphene conductivity tensor and SE as a function of the applied magnetostatic bias: (a) Components of the tensor; (b) SE.

图 5  透射波极化状态参量$\tau, \varepsilon$与偏置磁场、偏置电场关系　(a) 偏置电场${E_{{\rm{bias}}}} = 1\;{\rm{V}} \cdot {\rm{n}}{{\rm{m}}^{{{ - }}1}}$; (b) 偏置磁场${B_{{\rm{bias}}}} = 0.5\;{\rm{T}}$

Figure 5.  Angles $\tau, \;\varepsilon$ describing the polarization state of the transmitted wave as functions of the magnetostatic bias and the electrostatic bias: (a) ${E_{{\rm{bias}}}} = 1\;{\rm{V}} \cdot {\rm{n}}{{\rm{m}}^{{{ - }}1}}$; (b) ${B_{{\rm{bias}}}} = 0.5\;{\rm{T}}$.

图 6  各向异性石墨烯界面的反透射及屏蔽效率随频率变化关系　(a) 同极化反透射系数模值; (b) 交叉极化反透射系数模值; (c) 交叉极化反透射系数相位; (d) 屏蔽效率

Figure 6.  Reflection/transmission and the SE of an isotropic graphene sheet as functions of frequency: (a) Modulus of co-polarization reflective and transmittance coefficients; (b) modulus of cross-polarization reflective and transmittance coefficients; (c) phase of cross-polarization reflective and transmittance coefficients; (d) SE.

图 7  Si/SiO2周期层状结构

Figure 7.  Si/SiO2 periodic layered structure.

图 8  各向异性石墨烯界面位于不同位置时层状结构对垂直入射波的反透射随频率变化关系　(a) 反射系数模值; (b) 透射系数模值

Figure 8.  Reflection/transmission of normally incident waves from a periodic layered structure with an isotropic graphene at different interface as functions of frequency: (a) Modulus of reflective coefficient; (b) modulus of transmittance coefficient.

图 9  各向异性石墨烯界面位于上表面时层状结构对斜入射波的反透射随频率变化关系　(a) 反射系数模值; (b) 透射系数模值

Figure 9.  Reflection/transmission of oblique incident waves from a periodic layered structure with an isotropic graphene surface as functions of frequency: (a) Modulus of reflective coefficient; (b) modulus of transmittance coefficient.

•  [1] Lovat G 2012 IEEE Trans. Electromagn. Compat. 54 101 [2] Geim A K 2009 Science 324 1530 [3] Das T, Sharma B K, Katiyar A K, Ahn J H 2018 J. Semicond. 39 011007 [4] Nimbalkar A, Kim H 2020 Nano-Micro Lett. 12 126 [5] Moshizi S A, Azadi S, Belford A, Razmjou A, Wu S, Han Z J, Asadnia M 2020 Nano-Micro Lett. 12 109 [6] Chen X Y, Tian Z, Li Q, Li S X, Zhang X Q, Ouyang C M, Gu J Q, Han J G, Zhang W L 2020 Chin. Phys. B 29 077803 [7] Chaea M S, Leea T H, Sona K R, Parka T H, Hwangb K S, Kim T G 2020 J Mater. Sci. Technol. 40 72 [8] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsove A A 2004 Science 306 666 [9] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsove A A 2005 Nature 438 197 [10] Zhang Y, Tan Y W, Stormer H L, Kim P 2005 Nature 438 201 [11] Zhang Y, Small J P, Pontius W V, Kim P 2005 Appl. Phys. Lett. 86 073104 [12] Slepyan G Y, Maksimenko S A, Lakhtakia A, Yevtushenko O, Gusakov A V 1999 Phys. Rev. B 60 17136 [13] Hanson G W 2008 J. Appl. Phys. 103 064302 [14] Gusynin V P, Sharapov S G, Carbotte J P 2007 J. Phys. Condens. Matter 19 026222 [15] Hanson G W 2008 IEEE Trans. Antennas Propag. 56 747 [16] 葛德彪, 魏兵 2011 电磁波理论 (北京: 科学出版社) 第5, 6, 39, 40, 56−73页 Ge D B, Wei B 2011 Electromagnetic wave theory (Beijing: Science Press) pp5, 6, 39, 40, 56−73 (in Chinese) [17] 王飞, 魏兵 2019 物理学报 68 244101 Wang F, Wei B 2019 Acta Phys. Sin. 68 244101 [18] Gusynin V P, Sharapov S G 2006 Phys. Rev. B 73 245411 [19] Gusynin V P, Sharapov S G, Carbotte J P 2006 Phys. Rev. Lett. 96 256802 [20] Peres N M R, Guiner F, Castro Neto A H 2006 Phys. Rev. B 73 125411 [21] Ziegler K 2007 Phys. Rev. B 75 233407 [22] George W H 2008 Journal of Applied Physics 103 064302 [23] Balanis C A 1989 Advanced Engineering Electromagnetics (New York: Wiley) pp168−170
•  Citation:
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• Abstract views:  637
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##### Publishing process
• Received Date:  09 July 2020
• Accepted Date:  03 September 2020
• Available Online:  20 December 2020
• Published Online:  05 January 2021

## Propagation matrix for electromagnetic interaction through electrostatically and magnetostatically biased graphene sheet

###### Corresponding author: Wang Fei, wfei79@163.com;
• 1. School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China
• 2. Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi’an 710071, China

Abstract: The reflection and transmission of plane electromagnetic waves on monolayer graphene are studied theoretically in this paper. From an electromagnetic point of view, monolayer graphene is described as an “infinitely thin” graphene sheet characterized by a surface conductivity, and based on a microscopic quantum dynamical approach, the graphene sheet becomes anisotropic in the presence of both an electrostatic and a magnetic bias. In this work, starting from boundary conditions and phase-matching conditions, the propagation matrix for the analysis of the interaction between an electromagnetic field and thin graphene sheet which is biased electrostatically and magnetostatically, and then characterized by an anisotropic conductivity, is derived. Furthermore, the analytical solutions of co- and cross-polarization reflective and transmittance coefficients through an anisotropic graphene planar surface are obtained from the proposal matrix above, which couples the fundamental transverse electric (TE) polarization and transverse magnetic (TM) polarization and includes the possible effects of electrostatic and/or magnetostatic bias. In conclusion, the cross-polarization reflective coefficient of TE wave and that of TM wave are equal, and their cross-polarization transmittance coefficients have opposite phase. Finally, a new propagation matrix for stratified medium containing anisotropic graphene interfaces is deduced by embedding the matrix across graphene sheet mentioned above into the traditional propagation matrix for isotropic stratified medium. The proposed new matrix can be used to investigate the propagation properties of plane wave in a complex structure of layered medium and anisotropic conductivity interfaces (including graphene sheet) analytically and quickly, and represents a very simple tool for the relevant analysis and design.

Reference (23)

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