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石墨烯是一种由碳原子组成的单原子层六方晶格二维材料,其电导率可由Kubo方程推导出的表面电导率表示。当其化学势μC >> kBT时,其表面电导率可表示为[19]:
$$ \begin{split} {\sigma _{2D}} \approx & \frac{{i{e^2}{k_B}T}}{{{\textit{π}} {\hbar ^2}\left( {\omega} + i2 {{\textit{π}} \varGamma } \right)}}\left[ {\frac{{{\mu _C}}}{{{k_B}T}} + 2ln\left( {{e^{\frac{{{\mu _C}}}{{{k_B}T}}}} + 1} \right)} \right] + \frac{{i{e^2}}}{{4{\textit{π}} \hbar }}\\ & {\rm ln}\left[ {\frac{{2\left| {{\mu _C}} \right| - \left( {\omega + i2\varGamma } \right)\hbar }}{{2\left| {{\mu _C}} \right| + \left( {\omega + i2\varGamma } \right)\hbar }}} \right] \end{split}$$ (1) 式中:
$ \hbar $ 、kB与e分别为约化的普朗克常数、玻耳兹曼常数以及电子电量;kBT表示热能;$\varGamma $ 是带电粒子散射率,其值设定为0.43 eV;$\mu _C $ 为石墨烯化学势,可由载流子浓度决定[20]。$$ \begin{split} {n_0} = \frac{2}{{\pi {\hbar ^2}{v^2}}}\mathop \smallint \limits_0^{ + \infty } \varepsilon \left[ {{f_0}\left( {\varepsilon - {\mu _C}} \right) - {f_0}\left( {\varepsilon + {\mu _C}} \right)} \right]{\rm d}\varepsilon \end{split} $$ (2) 式中:
${f_0}=1+e^{\frac{{{(\varepsilon - \mu _C)}}}{{{k_B}T}}}$ 是费米-狄拉克分布函数,而费米速度v = 1.0×106 m/s。因为,载流子浓度可通过加载在石墨烯上的偏置电压控制。$$ \begin{split} {n_0} = \frac{{{\varepsilon _{dielectric}}{\varepsilon _0}{V_{bias}}}}{{ed}} \end{split}$$ (3) 式中:d为介质层厚度;
$\varepsilon _{dielectric} $ 为介质层相对介电常数。因此,可通过偏置电压实现对石墨烯化学势的有效调控。文中利用以下关系:$$ \begin{split} \sigma \left( \omega \right) = \frac{{{\sigma _{2D}}\left( \omega \right)}}{{{t_n}}} \end{split} $$ (4) 式中:tn是石墨烯薄膜的膜厚,在文中的仿真计算中设定为1 nm。化学势具体调控手段为:如图1(b)所示,将吸收器石墨烯层外接电源作为正极,底层金属层作为负极,通过增大外加电压的方式增大石墨烯层的化学势。
图 1 (a)宽带吸收器周期结构x-y平面示意图;(b)吸收器的三维单元示意图,其中P = 104 μm, R1= 11 μm, R2= 38 μm, R3= 26 μm, ts = 15 μm, tm = 2 μm
Figure 1. (a) x-y plane diagram of the periodic structure of the broadband absorber;(b) 3 D schematic diagram of the absorber, where P=104 μm, R1=11 μm, R2= 38 μm, R3=26 μm, ts =15 μm, tm =2 μm
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吸收器的结构示意图如图1所示,由下到上依次为连续金属接地层、PDMS(聚二甲基硅氧烷)薄膜、均匀单层石墨烯以及硅半球-半椭球复合结构层组成。该超材料结构单元的晶格周期P为104 μm,金属金作为接地层,其电导率为4.561×107 S/m,厚度tm为2 μm。其上为介质层,材料为低损耗的PDMS聚合物,相对介电常数ε2= 1.72,损耗角正切tanδ2= 0.15,介质厚度ts = 15 μm。介质层上为由半椭球与半圆球构成的硅周期结构,其单元结构分别由位于边缘的四个旋转对称半椭球和一个位于结构中心的半圆球组成。半椭球的长轴长R1 = 38 μm,短轴长R2 = 11 μm,且椭圆中心距离相邻侧壁距离分别为m = 40 μm和n = 12 μm,而中心半圆球半径R3= 26 μm。相对介电常数ε1和损耗角正切tanδ1分别为11.65和0.174。仿真基于有限元方法,利用商业计算软件CST Microwave Studio 2020中的频域求解器完成。仿真频率范围为0.1~10 THz,选择TE波作为入射波。其中,沿x和y方向单胞采用周期性边界条件,则吸收器的吸收率可通过公式(5)计算:
$$ A{\rm{ }} = 1{\rm{ }} - R-T$$ (5) 式中:R和T分别是反射率和透过率。这里,反射率R = | S11 | 2,其中S11是入射波的反射系数。由于底部金属反射层具有足够的厚度(ts = 2 μm),其远大于太赫兹波的趋肤深度而使得透过率T ≈ 0。因此,吸收率A仅与反射率有关,具体为:
$$ A \approx {\rm{ }}1{\rm{ }} - {\rm{ }}{\left| {{S_{11}}} \right|^2} $$ (6) 对于该结构上层的硅半椭球和半圆球,在TE波垂直入射时,波在半球内部沿x方向不断反射,该结构等效于几何轮廓尺寸不断随z和y变化的法
法布里-珀罗(FP)谐振腔,如图2(b)所示。所提出的吸收器中的激发石墨烯等离子体共振(GPRS)与具有波矢量的石墨烯等离子体的驻波模式的形成相关联。
图 2 不同石墨烯化学势μ(0~0.9 eV)条件下吸收器吸收率随频率变化图
Figure 2. Absorption spectra of the proposed absorber with various values of the graphene chemical potential μ from 0 to 0.9 eV
$$ \begin{split} {K_{GPRs}}\left( \omega \right) = \frac{{\pi {\hbar ^2}}}{{{e^2}{E_f}}}{\varepsilon _0}\left( {{\varepsilon _1} + {\varepsilon _2}} \right)\omega \left( {\omega + i{\tau ^{ - 1}}} \right) \end{split} $$ (7) 式中:ε0是真空介电常数;ε1是硅半球的相对介电常数;ε2是PDMS的相对介电常数。半椭球和半圆球谐振腔对应的共振角频率ω可以用公式(8)来计算:
$$ \begin{split} {\rm Re}({K_{GPRs}}\left( \omega \right)) = \frac{{\pi N + \varphi }}{{L\left( {y,z} \right)}} \end{split} $$ (8) 式中:
${\rm Re}({K_{GPRs}}\left( \omega \right))$ 是${K_{GPRs}}\left( \omega \right) $ 的实部。在某个位置的半椭球F-P腔长L(y,z)可以近似为$L\left( {{y},{{z}}} \right)=$ $2{R_2}\sqrt {1 - \dfrac{{{y^2} - {z^2}}}{{R_1^2}}} $ ,而半圆球的F-P腔长L(y,z)可以近似为$L\left( {{{y}},{{z}}} \right){\rm{ = }} 2{R_3}\sqrt {1 - \dfrac{{{y^2} - {z^2}}}{{R_3^2}}}$ N是正整数,是谐振模式的阶数;φ是FP腔中的反射系数的相位,文中研究可以忽略不计。因此,半椭球和半圆球第N阶模式的ωN1和ωN2可以分别简化为:$$ \begin{split} {\omega _{N1}} = \frac{e}{\hbar }\sqrt {\frac{{{E_f}\left( {N\pi + \varphi } \right){R_2}}}{{2\pi {\varepsilon _0}\left( {{\varepsilon _1} + {\varepsilon _2}} \right)\sqrt {{R_2}^2{R_1}^2 - {R_1}^2{y^2} - {R_2}^2{z^2}} }}} \end{split} $$ (9) $$ \begin{split} {\omega _{N2}} = \frac{e}{\hbar }\sqrt {\frac{{{E_f}\left( {N\pi + \varphi } \right){R_3}}}{{2\pi {\varepsilon _0}\left( {{\varepsilon _1} + {\varepsilon _2}} \right)\sqrt {{R_3}^4 - {R_3}^2({y^2} - {z^2})} }}} \end{split} $$ (10) 不同的整数N对应多个不同的离散GPR,然而,文中所示吸收器,半椭球和半圆球的几何轮廓不断变化,因此,通过连续改变L,相对于Y和Z形成一系列FP谐振腔;因此,将导出具有连续频率的谐振腔等离子体共振(FPRS)。连续FPRS不仅填充离散GPRS的频率间隔,还不会形成重叠的谐振频率。因此,该宽带吸收器的吸收谱较平滑[14]。
Wideband terahertz metamaterial absorber for composite graphene/silicon hemispheres
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摘要: 提出了一种性能可调的宽带、极化与入射角不敏感的超材料太赫兹吸收器,该吸收器自上而下分为四层结构,分别是:硅半椭球/半球体复合结构、连续石墨烯层、PDMS介质层和金属背板。通过在TE波垂直入射条件下仿真,在已知结果基础上,对不同石墨烯化学势和不同结构条件下的电场结果分析表明,在硅半椭球/半球体亚波长复合结构所形成的连续、多模法布里-珀罗共振,以及由石墨烯所激发的多个离散的等离子体共振的协同作用下,其吸收光谱得到平滑和扩展,使该结构可实现吸收率宽范围可调,以及接近100%吸收率的宽频带吸收特性。特别的,当石墨烯化学势分别为0.2与0.9 eV时,其分别可获得约5.7 THz与7 THz的宽带太赫兹波吸收(吸收率超过90%),且其最大吸收率接近完美吸收(约99.8%)。此外,该结构还具有360°极化不敏感和高于60°的入射角不敏感等优异特性,在以上角度范围内,吸收器吸收率仍可保持到90%以上。在太赫兹波探测、光谱成像以及隐身技术等方面具有潜在的应用前景。Abstract: A tunable broadband, polarization insensitive and incident angle insensitive metamaterial terahertz absorber is proposed, which consists of silicon semi-ellipsoid/semi-spherical structure, graphene, dielectric layer and metal back plate. Based on the known results, the electric field results under different chemical potentials of graphene and different structural conditions were analyzed by simulation under the condition of vertical incident TE wave show that under the synergism of continuous and multimode Fabry-Perot resonances formed by silicon semi-ellipsoid/semi-spherical subwavelength structure and multiple discrete plasma resonances excited by graphene the absorption spectrum is smoothed and expanded so that the structure can achieve a wide range of adjustable absorptivity and a broadband absorption characteristic of nearly 100% absorptivity. When the chemical potential of graphene is around 0.2 eV and 0.9 eV, it can obtain about 5.7 THz and 7 THz wideband terahertz wave absorption (the absorption rate is more than 90%), respectively, and its maximum absorption rate is close to perfect absorption (about 99.8%). In addition, the structure is insensitive to 360° polarization and incident angle higher than 60°. In the above angle range, the absorptivity of the absorber can still be maintained to more than 90%. and the structure has potential applications in terahertz wave detection, spectral imaging and stealth technology.
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Key words:
- terahertz /
- silicon hemispheric layer /
- chemical potential /
- wideband
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图 1 (a)宽带吸收器周期结构x-y平面示意图;(b)吸收器的三维单元示意图,其中P = 104 μm, R1= 11 μm, R2= 38 μm, R3= 26 μm, ts = 15 μm, tm = 2 μm
Figure 1. (a) x-y plane diagram of the periodic structure of the broadband absorber;(b) 3 D schematic diagram of the absorber, where P=104 μm, R1=11 μm, R2= 38 μm, R3=26 μm, ts =15 μm, tm =2 μm
图 4 μ = 0.2 eV时五种不同结构的吸收谱:(1)无石墨烯无谐振结构;(2)有石墨烯且只有半球结构(蓝线);(3)有石墨烯且只有半椭球结构(绿线);(4)无石墨烯但具有半椭球/半球谐振结构(粉线);(5)有石墨烯与半椭球/半球谐振结构(红线)
Figure 4. (a) Simulated absorption spectra of the proposed G-SemiEllip/SemiSphere absorber (red line) and four other absorbers with NN (black line), G-SemiSphere (blue line), G-SemiEllip (green line) and N-SemiEllip/SemiSphere (pink line)
图 6 (a) TE波斜入射时吸收率随极化角ϕ(电场方向与y轴的夹角)的变化率; (b)~(c) TE/TM波斜入射时吸收率随入射角θ(入射方向与z轴的夹角)的变化率
Figure 6. (a) The change rate of absorptivity with polarization angle (angle between electric field direction and y-axis) at oblique incident of TE wave; (b)-(c) The rate at which the absorptivity varies with the incidence angle θ (the angle between the incident direction and the z-axis) for oblique incidence of TE/TM waves
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