Search

Article

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

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

Wang Fei Wei Bing

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

Wang Fei, Wei Bing
PDF
HTML
Get Citation
  • 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.
      Corresponding author: Wang Fei, wfei79@163.com
    [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

  • 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:
Metrics
  • Abstract views:  636
  • PDF Downloads:  26
  • Cited By: 0
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.

    • 石墨烯是由一层碳原子构成的二维蜂窝状晶体结构, 其厚度只有一个原子大小, 约为0.34 nm, 是世界上最薄的半导体材料. 石墨烯出色的力学特性和承载超大电流的能力, 使其成为未来微纳米技术中最有前途的材料之一[1-7], 并且在2004年通过微机械剥离法制备成功[8-11].

      相对于其他单原子金属薄膜, 石墨烯在热力学上很稳定, 并且在载流子浓度消失的情况下仍具有最小的导电性[9]; 同时, 石墨烯是一种零带隙半导体, 利用静电和/或静磁偏置的电场和霍尔效应可以调节其导电性[8,10]. 讨论分析石墨烯与电磁波相互作用问题时, 可以将其看成具有导电性的“无限薄”表面[12], 其导电性由表面导电率$\sigma $描述[13]; 微观量子动力学方法推导出了$\sigma $的数学模型, 其中还包括静电和/或静磁偏置情况下的重要影响—在存在偏压和偏磁的情况下, 石墨烯呈各向异性, 其表面导电率为张量${{\sigma}} $[14]. 此外应注意的是, 即使没有偏置场, 在太赫兹(terahertz, THz)频率以上, 石墨烯也具有空间色散效应, 这时亦呈现各向异性特性[15].

      本文主要研究在太赫兹频率以下, 存在偏置外静电和静磁场时, 各向异性的“无限薄”石墨烯与平面电磁波的相互作用: 利用电磁场边界条件和相位匹配条件[16], 推导出平面电磁波在各向异性石墨烯平面两侧的跨界面传播矩阵(propagation matrix, PM), 其中耦合了基本的横电(transverse electric, TE)波和横磁(transverse magnetic, TM)波极化, 同时包括偏置电、磁场的影响; 基于该传播矩阵, 进一步给出石墨烯界面的反透射系数解析解; 将跨石墨烯界面传播矩阵嵌入各向同性分层介质传播矩阵[17](TE和TM波合并), 获得的新传播矩阵可用于解析分析平面电磁波以任意角度入射含各向异性石墨烯界面层状介质时的传播和反透射特性. 本文中时谐因子取$\exp \left( { - {\rm{j}}\omega t} \right)$.

    2.   电、磁偏置各向异性石墨烯界面的传播矩阵
    • 设一“无限薄”石墨烯层(各向异性)平行于xoy平面, 如图1所示, 该石墨烯层将全空间分开为上、下两部分, 设其分别为均匀各向同性介质${\varepsilon _l}, {\mu _l}$${\varepsilon _{l + 1}}, {\mu _{l + 1}}$(介电系数$\varepsilon $和磁导系数$\mu $可以是复数, 代表有耗介质).

      Figure 1.  “Infinitely thin” graphene sheet.

      在不考虑太赫兹情形下石墨烯的空间色散效应[15], 石墨烯层的空间电磁特性可以通过垂直方向的外置偏置电、磁场${E_{{\rm{bias}}}}\hat{{z}}, {B_{{\rm{bias}}}}\hat{{z}}$调节, 一般地, 石墨烯层表面电导率可表示为如下张量形式[14,18-21]:

      其中

      其中e, $\hbar $是电子电量绝对值、约化普朗克常量, ${v_{\rm{F}}} \cong {10^6}\;{\rm{m}}/{\rm{s}}$为石墨烯中的费米速率, $\varGamma = {1 / {\left( {2\tau } \right)}}$是散射率($\tau $为电子弛豫时间), $\varDelta $表示激子能隙(室温下近似为0), 而费米-狄拉克分布${n_{\rm{F}}}$可表示为

      其中$\varepsilon $表示能量; ${k_{\rm{B}}}$是玻尔兹曼常数; ${\mu _{\rm{c}}}$是依赖于偏置电场${E_{{\rm{bias}}}}$的化学势, 关系为[15]

      将(5)式代入(6)式, 并通过数值求解可获得化学势与偏置电场关系${\mu _{\rm{c}}}\left( {{E_{{\rm{bias}}}}} \right)$, 如图2所示.

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

      应注意, 石墨烯在偏置电场和磁场同时存在时为各向异性, 当偏置电场或/和磁场为0时, 石墨烯为各向同性, 即${\sigma _{\rm{O}}}{{ = }}0$, 而$\sigma = {\sigma _{\rm{D}}}$, 其表达式可参考文献[1].

    • 根据电磁场边界条件[16], 在石墨烯层两侧区域的电、磁场有如下关系[22]:

      其中${{J}}_{\rm{e}}^{\rm{s}}$${{J}}_{\rm{m}}^{\rm{s}}$为石墨烯表面电流和磁流, 由第二式得

      又因为

      代入(8)式得到

      而根据边界条件(7)式的第一式可知分界面(石墨烯层)上的电场切向分量是连续的, 即

      所以(11)式中等号右边的${E_x}$, ${E_y}$可以取介质l$l + 1$区域中电场的x, y分量.

    • 下来我们考虑平面电磁波入射石墨烯界面时, 界面两侧场分量间的关系. 根据电磁波理论, 平面电磁波斜入射界面垂直于z轴的层状各向同性介质时, 在介质的任意l层区域中, 波可按z轴分为TE和TM平面波, 波方程分别为[16]

      其中$A_l^{{\rm{TE/TM}}}$$B_l^{{\rm{TE/TM}}}$分别代表区域l中上行和下行TE/TM波的电/磁场幅值.

      设石墨烯层位于l$l + 1$区域分界面, 由边界条件(11)式和(12)式可解得

      其中

      称为各向异性石墨烯分界面的前向传播矩阵, 式中

      由于各向异性, 石墨烯层分界面两侧的TE和TM平面波出现耦合, 单纯TE/TM波入射会产生同时含TE和TM波分量的反射和透射波.

      ${\sigma _{\rm{O}}} = 0$时, 石墨烯为各向同性, (15)式可分解为TE, TM波对应的方程

      其中${{{V}}^{{\rm{TE}}}}$, ${{{V}}^{{\rm{TM}}}}$即为文献[17]中所给出的各向同性石墨烯分界面的前向传播矩阵.

    • 设石墨烯薄层位于$z = 0$处, 平面波自上往下斜入射, 下来分别讨论TE和TM入射波情形, 此时设入射区为介质1, 即$l = 1$, 则透射区$l + 1$为介质2.

    • 此时在1区没有下行的TM波, $B_1^{{\rm{TM}}} = 0$, 将下行即入射的TE波$B_1^{{\rm{TE}}}$看做已知参量, 是入射波电场幅值; 而2区为透射区, 所以没有上行波, $A_2^{{\rm{TE}}} = 0, A_2^{{\rm{TM}}} = 0$. 由(15)式可得

      其中

      定义TE波入射石墨烯单层时的反透射系数为

      其分别为反射TE波、TM波, 透射TE波、TM波与入射TE波的电场幅值比. 由于$A_1^{{\rm{TM}}}, \;B_2^{{\rm{TM}}}$表示的是TM波的磁场幅值, 所以(22)式中利用各自区域介质的波阻抗将其转换为电场幅值. 求解(20)式可得TE波入射石墨烯单层时的反透射系数

    • 此时在1区没有下行的TE波, $B_1^{{\rm{TE}}} = 0$, 将下行即入射的TM波$B_1^{{\rm{TM}}}$看做已知参量, 是入射波磁场幅值; 而2区为透射区, 所以没有上行波, $A_2^{{\rm{TE}}} = 0, A_2^{{\rm{TM}}} = 0$. 按照和2.4.1小节类似讨论方法可得TM波入射石墨烯单层时的反透射系数:

      其中

      分别是反射TM波、TE波, 透射TM波、TE波与入射TM波的磁场幅值比, 由于$A_1^{{\rm{TE}}}, \;B_2^{{\rm{TE}}}$表示的是TE波的电场幅值, 所以(25)式中利用各自区域介质的波阻抗将其转换为磁场幅值.

      (23)式和(24)式即为平面波入射石墨烯单层时的反透射系数公式(含同极化和交叉极化). 另, 由二式可知, 石墨烯单层的交叉反射系数相同, ${R^{{\rm{TM}} \text{-} {\rm{TE}}}} = {R^{{\rm{TE \text{-} TM}}}}$.

      当石墨烯界面两侧为同种介质, 在均匀平面波入射时, 有

      代入(23)式和(24)式得到

      可见, 均匀各向同性介质中石墨烯薄层对入射TE和TM波的交叉极化反射系数以及入射TM波的交叉极化透射系数是相等的, 且与入射TE波的交叉极化透射系数模值相等、相位相差${\text{π}}$. 令上式中$\eta = {\eta _0}$, 即得真空中石墨烯薄层的反透射系数.

    3.   含各向异性石墨烯界面的层状介质传播矩阵
    • 考虑如图3所示各向同性分层介质, 假设其中某一个或多个界面被嵌入“无限薄”石墨烯, 这时由于各向异性石墨烯的交叉极化效应, 使得该层状结构对入射平面波产生交叉极化方向的反射和透射, 所以我们把透射区t和入射区0的场值关系表示为

      Figure 3.  Stratified medium.

      上式等号左边和右边的$4 \times 1$矩阵各行依次分别表示透射区t和入射区0中的上行TE波电场、下行TE波电场、上行TM波电场、下行TM波电场. 矩阵${{{V}}_{t0}}$为从入射点${M_0}$到透射点${N_t}$的前向传播矩阵.

      当不存在石墨烯界面时, 各向同性分层介质的前向传播矩阵有两组${{V}}_{t0}^{{\rm{TE}}}$${{V}}_{t0}^{{\rm{TM}}}$, 分别对应TE和TM两种线极化波情形[16,17]. 考虑到石墨烯界面各向异性特性产生的交叉极化效应, 我们首先将这两组传播矩阵整合为一组, 如下

      其中$V_l^{{\rm{TE/TM}}}$, ${{U}}_l^{{\rm{TE/TM}}}$分别是l层介质中、跨越区域l$l + 1$界面的传播矩阵.

      假设在区域l$l + 1$的分界面$z = {z_l}$处嵌入石墨烯, 则将(30)式中的矩阵${{{U}}_l}$替换为(15)式和(16)式中的${{{U}}_G}$.

      如果是纯粹的各向同性分层介质, (28)式就退化为TE模和TM模两组传播矩阵, 此时${R^{{\rm{TM}} \text{-} {\rm{TE}}}}, \;{T^{{\rm{TM}}\text{-} {\rm{TE}}}}, \;{R^{{\rm{TE \text{-} TM}}}}, \;{T^{{\rm{TE \text{-} TM}}}}$都是0.

      下面我们分别考虑TE和TM线极化波入射时的情形:

    • 此时入射TE波电场幅值设为1, 而入射TM波磁场幅值为0, 即$B_0^{{\rm{TE}}} = 1$$B_0^{{\rm{TM}}} = 0$, 则(28)式成为

      可求得

      其中${v_{ij}}$${{{V}}_{t0}}$的元素. 当没有石墨烯界面时

      代入(32)式得到

      即为TE波入射一般分层介质反透射计算式[16,17].

    • 此时入射TM波磁场场幅值设为1, 而入射TE波磁场幅值为0, 即$B_0^{{\rm{TM}}} = 1$$B_0^{{\rm{TE}}} = 0$, 则(28)式成为

      可求得

      其中${v_{ij}}$${{{V}}_{t0}}$的元素. 当没有石墨烯界面时, 将(33)式代入(36)式得到

      即为TM波入射一般分层介质反透射计算式[16,17].

      此外, 还可以证明(32)式中第三式和(36)式中第二式是相等的(限于篇幅, 证明略), 即含各向异性石墨烯界面的各向同性层状介质对入射TE波和TM波的交叉极化反射系数相等, ${R^{{\rm{TM}} \text{-} {\rm{TE}}}} = {R^{{\rm{TE \text{-}TM}}}}$.

    4.   算 例
    • 以下算例皆考虑室温条件下$T = 300\;{\rm{K}}$以及石墨烯电子弛豫时间为$\tau = 3\;{\rm{ps}}$.

      例 1 电、磁偏置下石墨烯薄层的表面电导率及其屏蔽效率、透射波极化旋转

      图4(a)是偏置电场${E_{{\rm{bias}}}} = 0.5$V/nm, 频率$f = 1\;{\rm{GHz}}$时, 石墨烯电导率张量元素${\sigma _{\rm{D}}}, {\sigma _{\rm{O}}}$的实虚部和偏置磁场${B_{{\rm{bias}}}}$($- 0.2\text{—} + 0.2\;{\rm{T}}$)的关系. 图4(a)中清楚体现了${\sigma _{\rm{D}}}, {\sigma _{\rm{O}}}$分别为偏置磁场${B_{{\rm{bias}}}}$的偶函数和奇函数.

      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.

      采用本文给出的传播矩阵, 计算了$f = 1\;{\rm{GHz}}$时石墨烯层的反透射并得到其屏蔽效率[1] (shielding effectiveness, SE)如图4(b)所示. 图4(b)中给出偏置电场${E_{{\rm{bias}}}}$分别为$0, \;0.5, \;2, \;5\;{\rm V/nm}$时, SE与偏置磁场${B_{{\rm{bias}}}}$($- 0.2 \text{—} + 0.2\;{\rm{T}}$)的关系.

      当偏置电磁场都不为0时, 石墨烯呈各向异性, 即使入射波为线极化, 在反射和透射波中也会产生交叉极化分量, 从而导致反射、透射波成为椭圆极化波. 图5给出了$f = 1\;{\rm{GHz}}$线极化波垂直入射石墨烯层时, 透射椭圆极化波的极化状态参量$\tau, \;\varepsilon $(在庞加莱球上的参数)[23]与偏置电磁场的关系, 其中图5(a)图5(b)子图分别是偏置电场${E_{{\rm{bias}}}} = 1\;{\rm{V}}/{\rm{n}}{{\rm{m}}}$、偏置磁场${B_{{\rm{bias}}}} = 0.5\;{\rm{T}}$时极化状态参量与偏置磁场${B_{{\rm{bias}}}}$($- 0.2\text{—} + 0.2\;{\rm{T}}$)、偏置电场${E_{{\rm{bias}}}}$($- 4\text{—} + 4\;{\rm V/nm}$)的关系.

      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) ${B_{{\rm{bias}}}} = 0.5\;{\rm{T}}$.

      该算例为验证算例, 采用和文献[1]相同参数, 计算结果也和文献一致, 表明本文推导正确, 方法有效.

      例 2 电、磁偏置下单石墨烯层的反射、透射特性

      应用本文方法在偏置电、磁场为${E_{{\rm{bias}}}} = 0.5\;{\rm V/nm}$, ${B_{{\rm{bias}}}} = 0.1\;{\rm{T}}$条件下, 计算了入射波频率$f = 0-1000\;{\rm{GHz}}$频段内石墨烯的反射、透射系数, 如图6所示. 图6(a)为同极化反、透射系数模值, 随着入射角增大, TE波的同极化反射(黑色线图例)会增强而透射(灰色线图例)减弱, 而TM波则刚好相反(黑色、灰色散点图例); 图6(b)为交叉极化反、透射系数模值, 如前所述((27)式), 石墨烯薄层的交叉极化反射系数和交叉极化透射系数的模值都是相等的, 并由图知, 它们随着入射角增大而减小; 图6(c)给出了TE波入射时交叉极化反、透射系数的相位, 和前面的理论一致, 它们相差$\pi $; 图6(d)是将两种透射系数整合并按照定义计算得到的石墨烯的屏蔽效率SE, 可见对TE波的屏蔽效率要高于对TM波的, 而且入射角越大, 对TE波的屏蔽效率越高, TM波则相反.

      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.

      例 3 含各向异性石墨烯界面的分层介质的反射与透射

      设有一Si/SiO2周期层状结构如图7所示, 第一层为厚度${d_s}$的SiO2层, 第2—10层交替为厚度${d_{\rm{a}}}$${d_{\rm{b}}}$的Si和SiO2层, 它们的相对复介电系数${\varepsilon _{{\rm{ra}}}}$${\varepsilon _{r{\rm{b}}}}$分别为

      Figure 7.  Si/SiO2 periodic layered structure.

      介质层厚度分别为${d_{\rm{a}}} = 0.25\dfrac{{\lambda _0}} {\sqrt {{{\varepsilon '}_{{\rm{ra}}}}} }, {d_{\rm{b}}} = 0.25\dfrac{{\lambda _0}} {\sqrt {{{\varepsilon '}_{{\rm{rb}}}}} }, {d_{\rm{s}}} = 2{d_{\rm{b}}}$, 其中${\lambda _0} = 0.5\;{\rm{mm}}$.

      图8中给出当石墨烯(偏置电、磁场为$E_{\rm bias} = 0.5~\rm V/nm$, ${B_{{\rm{bias}}}} = 0.1\;{\rm{T}}$)位于该层状结构表面、正中间(第5和6层界面处)和底层时对垂直入射平面波的反射和透射. 由于石墨烯的各向异性, 反透射场中出现交叉极化成分, 但相对较弱; 作为对比, 图中用灰线给出在没有石墨烯时该层状结构的反透射, 可以看出, 该结构在大约500—700 GHz 频段有一禁带, 石墨烯界面的加入会使禁带向高频方向偏移, 同时会增强非禁带频段的反射, 削弱非禁带频段的透射; 此外, 在禁带频段, 反透射场中的交叉极化成分几乎为0.

      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是石墨烯位于该层状结构表面时对$\theta = {60^ \circ }$斜入射线极化平面波的反射和透射. 可以明显地看到, TE波的同极化反射要强于TM波的, 而同极化透射比TM波弱; 同时, 两种交叉极化反射系数相同, 和前面理论讨论一致.

      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.

    5.   结 论
    • 基于边界条件和相位匹配推导出在电、磁偏置情形下具有各向异性特征的石墨烯界面的传播矩阵, 并且进一步将其嵌入传统各向同性层状介质传播矩阵序列, 使其能够解析计算含石墨烯界面的分层介质对平面电磁波的反透射. 偏置静电、静磁场影响调节石墨烯表面电导率张量, 从而实现对石墨烯反透射特性及屏蔽性能的控制, 并可能使用偏压的石墨烯作为极化旋转器. 本文方法同样适用于石墨烯为各向同性时情形(偏置静电或/和静磁场为零时), 其可看作一种特例. 此外, 在太赫兹及以上更高频率, 需要考虑空间色散效应引起的各向异性, 此时可采用文中相似的处理方法进行理论上的推导及进一步分析讨论.

Reference (23)

Catalog

    /

    返回文章
    返回