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黑磷是层状结构,如图 1a所示[26],图中标示出了4层黑磷,其中s=1代表第1层,s=4代表第4层, a2是晶格常数,对于s层黑磷而言,总厚度为a2s/2。同一层内的黑磷原子并非处于同一个平面,而是表现为较为立体的蜂巢型结构,如图 1所示,沿着x方向称为扶手椅(armchair,AC)方向,沿着y方向称为锯齿(zigzag,ZZ)方向。由于原子排列的各向异性,使得黑磷在入射偏振光沿AC方向和ZZ方向表现出高度的各向异性。
图 1 黑磷的原子结构图及介电常数与波长的关系
Figure 1. Schematic diagram of BP structure and relationship between effective permittivity and wavelength
单层黑磷的光学特性可以用半经典的Drude模型来描述[14]。薄膜的介电常数ε可以由下式计算:
$ \left\{\begin{array}{l} \boldsymbol{\varepsilon}=\left[\begin{array}{ccc} \varepsilon_x & 0 & 0 \\ 0 & \varepsilon_y & 0 \\ 0 & 0 & \varepsilon_z \end{array}\right] \\ \varepsilon_j=\varepsilon_{\mathrm{r}}+\frac{\mathrm{i} \sigma_j}{\varepsilon_0 \omega t}, (j=x, y, z) \end{array}\right. $
(1) 式中,ε0是自由空间的介电常数, ε0=8.854×10-12 F/m; 对于单层黑磷来说,其在高频时的相对介电常数εr=5.76;ω是入射光的角频率; t是单层黑磷的厚度,也即1 nm; σj是表面电导率。σj可以由下式计算:
$ \left\{\begin{array}{l} \sigma_j=\frac{\mathrm{i} D_j}{\pi\left(\omega+\mathrm{i} \frac{\eta}{\hbar}\right)} \\ D_j=\frac{\pi e^2 n_{\mathrm{d}}}{m_j} \end{array}\right. $
(2) 式中, 弛豫速率η=1 meV;约化普朗克常数$\hbar$-=1.05457266×10-34J·s;单个电子的电荷量e=1.6022×10-19C;nd为电子掺杂浓度,可以通过外加偏置电压来改变[13];mj为沿着不同方向(x方向或者y方向)的有效质量,可以由下式计算:
$ \left\{\begin{array}{l} m_x=\frac{\hbar^2}{\frac{2 \gamma^2}{\Delta}+\eta_1} \\ m_y=\frac{\hbar^2}{2 \nu} \end{array}\right. $
(3) 式中, γ=4a/π,π/a代表布里渊区的宽度,a是黑磷的特征尺寸,对于单层黑磷而言,a=0.223 nm;Δ为黑磷的带隙宽度,对于单层黑磷而言,Δ=2 eV;η1=$\hbar{^{2}}$/0.4m0,ν=$\hbar{^{2}}$/1.4m0,m0=9.10938×10-31 kg,是标准电子质量。
根据上述公式,可以计算得到在不同的电子掺杂浓度下,介电常数的实部εRe和虚部εIm与波长的关系,如图 1b和图 1c所示。由图可知,在相同的电子掺杂浓度下,介电常数的实部(或者虚部)在AC(x方向)和ZZ(y方向)的数值相差很大,说明黑磷对不同方向的偏振光表现出强烈的偏振敏感性。
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图 2为本文作者所设计的基于黑磷的双频带超材料吸收体的单元结构。图 2a为立体图; 图 2b为剖面图。最下面一层为厚度h1=100 nm的Ag,作为反射镜使用;上面是厚度h2=1200 nm的透明介质层Al2O3,其折射率为1.7[22];再上面为2条带状结构的单层黑磷,具有不同的电子掺杂浓度,用以实现不同的吸收波长。结构参数为:周期p=500 nm,黑磷的宽度w=150 nm,黑磷边缘到中心线的距离d=20 nm。
本文中利用仿真软件模拟此超材料吸收体的光谱特性。这是一款基于时域有限差分法(finite difference time-domain,FDTD)的光学模拟软件,被广泛应用于光电器件的模拟中。由于本文作者提出的超材料吸收体具有周期性排列的结构,因此将图 2的单元结构的x和y方向的边界条件设为周期性结构,z方向设为完美匹配层即可。入射光的偏振方向为x方向,即沿着黑磷的AC方向。首先研究电子掺杂浓度对器件性能的影响。已有的研究表明,黑磷的电子掺杂浓度nd在1013 cm-2这个量级[27],因此在模拟时将nd限制在这个数量级。当将右边黑磷的电子掺杂浓度nd, 2固定为9×1013 cm-2,左边黑磷的电子掺杂浓度nd, 1从6×1013 cm-2变化到9×1013 cm-2时,此器件的吸收光谱如图 3所示。由图可知,在光谱范围内有2个吸收峰,当nd, 1变大时,2个吸收峰都出现了蓝移,而且吸收率都是先变大后变小。当左边黑磷的电子掺杂浓度为nd, 1=7×1013 cm-2、右边黑磷的电子掺杂浓度为nd, 2=9×1013 cm-2时,2个峰都实现了完美吸收,分别在波长2863.55 nm和3566.05 nm处,吸收率分别为99.96%和99.94%,实现了双频带的完美吸收。
图 3 模拟得到的吸收光谱与电子掺杂浓度的关系
Figure 3. The simulated absorption spectra as a function of doping concentration
为了解释完美吸收的形成机制,采用了耦合模理论(coupled-mode theory,CMT)[28-29]。根据CMT,此超材料吸收体可以用下式表示:
$ \left\{\begin{array}{l} \frac{\mathrm{d} a_1}{\mathrm{~d} t}=\left(\mathrm{i} \omega_0-\delta-\gamma_1\right) a_1+\sqrt{2 \gamma_1} S_{+} \\ S_{-}=-S_{+}+\sqrt{2 \gamma_1} a_1 \end{array}\right. $
(4) 式中,a1是共振幅度,t是时间, S+和S-代表入射光和出射光的幅度,δ和γ1分别代表内部本征损耗和外部漏率,ω0是共振频率。
对于角频率ω不同的入射光,此系统的反射系数r可以用下式表示:
$ r=\frac{S_{-}}{S_{+}}=\frac{\mathrm{i}\left(\omega-\omega_0\right)+\delta-\gamma_1}{\mathrm{i}\left(\omega-\omega_0\right)+\delta+\gamma_1} $
(5) 由于银层足够厚,可以认为此系统的透过率为0,则吸收率为:
$ A=1-|r|^2=\frac{4 \delta \gamma_1}{\left(\omega-\omega_0\right)^2+\left(\delta+\gamma_1\right)^2} $
(6) 由上式可知,当δ=γ1时,在共振频率ω0处,临界耦合(critical coupling, CC)条件被满足,A=1,即实现了完美吸收。
共振峰处的Q值由下式计算:
$ Q_{\mathrm{CMT}}=\frac{Q_\delta \cdot Q_{\gamma_1}}{Q_\delta+Q_{\gamma_1}} $
(7) 式中, QCMT是指根据CMT理论计算得到的Q值,Qδ是指由内部本征损耗δ计算得到的Q值,Qδ=ω0/(2δ);Qγ1是指由外部漏率γ1计算得到的Q值,Qγ1=ω0/(2γ1)。
图 4为利用FDTD相关软件模拟得到的双频带完美吸收体的光谱图。为了与CMT理论的公式相匹配,横坐标为ω(rad/s),纵坐标为吸收率。由光谱图可以测得根据FDTD算法得到的2个吸收峰的Q值QFDTD,其中左边峰的QFDTD值为23.11,右边峰的QFDTD值为18.60。图中红色虚线为用洛伦兹曲线拟合这两个峰所得到的曲线。由拟合结果可知,左边峰的δ=γ1=5.66×1012 Hz, 根据CMT理论,由(7)式可以计算得到QCMT=23.42,与FDTD模拟得到的QFDTD非常接近;右边峰的δ=γ1=9.15×1012 Hz, 根据CMT理论,由(7)式可以计算得到QCMT=18.04,与FDTD模拟得到的QFDTD非常接近。2个峰的由软件模拟得到的QFDTD值与计算得到的QCMT值都是非常接近的,说明在这两个频率处的完美吸收是由共振频率处的临界耦合引起的。图 4还显示了在共振波长及非共振波长处的x-z平面内的电场强度的分布图,由图可知, 在高吸收率的波长处,入射光波主要被限制在单层黑磷的附近,形成了共振加强;在低吸收率的波长处,入射光波没有与器件产生共振,大部分都被反射回了自由空间。
图 4 双频带超材料吸收体的吸收光谱及洛伦兹拟合曲线
Figure 4. Absorption spectrum of BP based dual-band metamaterial and Lorenz fitting curves
下面将探索器件结构参数对吸收光谱(吸收波长、吸收率、半波宽等)的影响。图 5为带状黑磷的宽度w从120 nm增加到180 nm时,吸收光谱的变化。由图可知,当w增大时,2个吸收峰都发生了红移,这是因为谐振波长$\lambda \propto \sqrt{w / n_{\mathrm{d}}}$[19],因此当带状黑磷的宽度w变大时,谐振波长也会变大。
图 5 带状黑磷的宽度w从120 nm变化到180 nm时, 双频带超材料吸收体的吸收特性仿真曲线
Figure 5. The simulated absorption spectra as a function of mave lenght when BP ribbon with w changed from 120 nm to 180 nm
图 6为带状黑磷的边缘到中心线的距离d从20 nm增加到40 nm时,吸收光谱的变化。内嵌小图为吸收波峰处的放大图。由图 6可知,除了吸收波长有微小的偏移之外,吸收率和半波宽几乎没有变化。这是因为本文中提出的双频带吸收体是周期性重复的结构,虽然在一个单元格内,带状黑磷偏离了中心位置,但是相邻单元格之间,同样电子掺杂浓度的带状黑磷的距离是没有变化的,因此不会造成吸收率的改变。吸收率和半波宽对d不敏感的这一特性,是所提出的双频带完美吸收体的一个优势,说明在实际的器件制作过程中,即使由于工艺等的问题造成带状黑磷与单元格中心线之间的间距偏离设计尺寸,也不会影响完美吸收体对红外光的吸收性能。
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由于此超材料吸收体具有完美的吸收特性,可以与物质发生强烈的相互作用,因此可以作为传感器使用。当不同折射率的待测物覆盖在此传感器上时,吸收光谱(吸收波长、吸收率、半波宽等)会发生变化,从而可以推测出待测物的折射率。本文中提出的双频带超材料吸收体作为传感器,具有独特的优势,因为它有2个吸收峰,可以与待测物质的特征频率实现多点匹配, 减少由于实验条件的改变、人员的误操作等带来的附加误差,从而提高传感器的可靠性和准确性[30]。图 7是待测物的折射率n从1增加到1.6时,吸收光谱的变化。可以看出,当待测物的折射率增加时,2个吸收峰都出现了红移,而且红移的幅度是不一样的。
图 7 待测覆盖物折射率n从1变化到1.6时,双频带超材料吸收体的吸收特性仿真曲线
Figure 7. The simulated absorption spectra as a function of wavelength when refractive index of the cladding n changed from 1 to 1.6
图 8为吸收波长的变化量Δλ与折射率的变化量Δn的关系。方点是在不同的折射率的情况下,较小的吸收波长λ1的偏移量Δλ1的模拟结果;圆点是在不同的折射率的情况下,较大的吸收波长λ2的偏移量Δλ2的模拟结果。黑线和红线分别对应的是它们的线性拟合。由直线的斜率可知,在吸收波长λ1处的折射率灵敏度S(λ1)=629.1 nm/RIU, 在吸收波长λ2处的折射率灵敏度S(λ2)=666.2 nm/RIU,RIU为单位折射率(reflective index unit)。由于折射率传感器的灵敏度与谐振波长有很强的相关性,因此通常用归一化灵敏度系数S′来比较不同传感器之间的性能。S′由下式计算:S′= S/λresonant, 其中λresonant是谐振波长[31]。计算可知,本文中的传感器在2个谐振波长处的归一化灵敏度系数S′分别为0.219/RIU和0.187/RIU。另外一个评价折射率传感器性能的重要参数为品质因数(figure of merit, FOM),FOM是折射率灵敏度与谐振峰的半峰全宽(full width at half maximum,FWHM)的比值。计算可知, 传感器在2个谐振波长处的品质因数FOM分别为3.12/RIU和4.34/RIU。
图 8 谐振波长的偏移量Δλ与折射率的变化量Δn的关系
Figure 8. Resonant wavelength shift Δλ as a function of refractive index variation Δn
表 1中列出了近期文献中报道的采用不同材料或者不同结构制成的折射率传感器的谐振波长、灵敏度S、归一化灵敏度系数S′和品质因数FOM[24]。由表 1可知,本文中设计的传感器在性能方面与其它的传感器具有可比性,并且其谐振波长正好在染料激光器的频谱范围内,在检测仪器的可靠性和价格方面有一定的优势。本文中传感器还有结构简单、对制作工艺要求不高等优势,是实用的折射率传感器。
表 1 折射率传感器在性能方面的对比
Table 1. Comparison between our refractive index sensor and other reported refractive index sensors
sensor material resonant wavelength/frequency S S′/RIU-1 FOM/RIU-1 reference black phosphorus 4.16 μm 1.4 μm/RIU 0.34 4 borophene 1.585 μm 560 nm/RIU 0.35 5.5 black phosphorus left peak 8.802 μm 140 nm/RIU — 125 black phosphorus right peak 8.807 μm 180 nm/RIU — 261 black phosphorus 7.7 μm 2 μm/RIU 0.26 0.29 black phosphorus 19.06 THz 7.62 THz/RIU 0.38 — black phosphorus peak 1 7.6 μm 2.4 μm/RIU 0.32 4.8 black phosphorus peak 2 8.3 μm 3.0 μm/RIU 0.36 4.2 graphene peak 1 23.5 μm 3.98 μm/RIU 0.17 16.6 graphene peak 2 24.3 μm 4.13 μm/RIU 0.17 20.7 graphene peak 3 27.8 μm 5.06 μm/RIU 0.18 18.1 graphene 31.11 μm 13.67 μm/RIU 0.44 6 MoS2, peak 1 583 nm 500 μm/RIU 0.86 — MoS2, peak 2 770 nm 200 nm/RIU 0.26 — black phosphorus left peak 2.863 μm 629.1 nm/RIU 0.219 3.12 our work black phosphorus right peak 3.566 μm 666.2 nm/RIU 0.187 4.34 our work 由上述的分析可知,当某种未知的待测物覆盖在本文中的传感器的表面时,只要测得吸收峰相比于空气的折射率。n0=1时的偏移量Δλ,就可以由下述公式计算得到该待测物的折射率n:
$ n=\frac{\Delta \lambda_j}{S\left(\lambda_j\right)}+n_0 $
(8) 式中, Δλj是有覆盖物时的吸收峰相比于没有覆盖物时的吸收峰的偏移量(j=1或者2,对应的是双吸收峰中的较小波长或者较大波长),S(λj)是在λj处的折射率灵敏度。
为了验证本文中设计的传感器在检测待测物折射率方面的性能,模拟了待测物折射率分别为1.45和1.55的吸收谱,如图 9所示。由图可知,n=1.45时,对应的Δλ1和Δλ2分别为276.81 nm和297.37 nm,由(8)式可以计算得到n=1.4400和n=1.4464;n=1.55时,对应的Δλ1和Δλ2分别为344.07 nm和365.09 nm,由(8)式可以计算得到n=1.5469和n=1.5480。该传感器计算的折射率与实际折射率的偏差如表 2所示。误差都在1%以内,可见该传感器可以较为精确地测量待测物的折射率。
图 9 覆盖物折射率分别为1, 1.45和1.55时的吸收光谱
Figure 9. Absorption spectra as a function of wavelength when refractive index of the cladding n is 1, 1.45 and 1.55 respectively
表 2 传感器计算的折射率与实际折射率的对比
Table 2. Comparison of the calculated refractive index and actual refractive index
actual
nthe calculated n from λ1 error/% the calculated n from λ2 error/% 1.45 1.4400 -0.69 1.4464 -0.25 1.55 1.5469 -0.20 1.5480 -0.13
基于黑磷的双频带超材料吸收体及其传感特性
Dual-band metamaterial absorber based on black phosphorus and its sensing characteristics
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摘要: 为了实现红外波段的双频带完美吸收, 采用将不同电子掺杂浓度的单层带状黑磷在同一平面内交错排列的方法, 进行了理论分析和仿真模拟, 得到了此器件在红外波段的吸收光谱和传感性能。结果表明, 此吸收体可以在波长2 μm~5 μm的红外波段范围内实现双频带的完美吸收(吸收率大于99.9%), 此高吸收率是由于入射光波与器件满足了临界耦合条件而形成了共振加强; 在共振波长处, 光波被限制在黑磷附近; 此超材料吸收体的双频带特性在其作为传感器使用时具有独特的优势, 可以提高传感器的可靠性和准确性; 吸收波峰的偏移量与覆盖在此器件上的未知物质的折射率基本呈线性关系, 用此器件测得的未知物质的折射率与实际的折射率的误差在1%以内。该超材料吸收体结构简单, 对制作工艺的尺寸精确度要求不高, 在红外波段的多频带吸收和传感检测方面将会有广泛的应用。Abstract: In order to realize dual-band perfect absorption in the infrared wavelength range, single-layer black phosphorus ribbons were arranged in parallel with alternating carrier doping concentration. Theoretical analysis and optical simulation were performed to get absorption spectra and sensing characteristics of the device in the infrared wavelength range. The results show the proposed device can achieve dual-band perfect absorption (>99.9% absorption efficiency) in the 2 μm~5 μm infrared wavelength range. The high absorption is caused by the critical coupling of incident light to the device, and instructive resonance is formed; the on-resonance light is restricted around black phosphorus; the dual-band absorption characteristic of the metamaterial makes it an ideal sensor with high reliability and accuracy; shift of absorption peaks is almost in a linear relationship with change of refractive index of cladding material. The margin of error between the calculated and actual refractive index is within 1%. The simple structure and reasonable tolerance in dimension deviation make the proposed metamaterial a good candidate for applications such as multiple-band absorption and sensing in the infrared wavelength range.
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Key words:
- spectroscopy /
- black phosphorus /
- dual-band /
- metamaterial /
- perfect absorption /
- sensing
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表 1 折射率传感器在性能方面的对比
Table 1. Comparison between our refractive index sensor and other reported refractive index sensors
sensor material resonant wavelength/frequency S S′/RIU-1 FOM/RIU-1 reference black phosphorus 4.16 μm 1.4 μm/RIU 0.34 4 borophene 1.585 μm 560 nm/RIU 0.35 5.5 black phosphorus left peak 8.802 μm 140 nm/RIU — 125 black phosphorus right peak 8.807 μm 180 nm/RIU — 261 black phosphorus 7.7 μm 2 μm/RIU 0.26 0.29 black phosphorus 19.06 THz 7.62 THz/RIU 0.38 — black phosphorus peak 1 7.6 μm 2.4 μm/RIU 0.32 4.8 black phosphorus peak 2 8.3 μm 3.0 μm/RIU 0.36 4.2 graphene peak 1 23.5 μm 3.98 μm/RIU 0.17 16.6 graphene peak 2 24.3 μm 4.13 μm/RIU 0.17 20.7 graphene peak 3 27.8 μm 5.06 μm/RIU 0.18 18.1 graphene 31.11 μm 13.67 μm/RIU 0.44 6 MoS2, peak 1 583 nm 500 μm/RIU 0.86 — MoS2, peak 2 770 nm 200 nm/RIU 0.26 — black phosphorus left peak 2.863 μm 629.1 nm/RIU 0.219 3.12 our work black phosphorus right peak 3.566 μm 666.2 nm/RIU 0.187 4.34 our work 表 2 传感器计算的折射率与实际折射率的对比
Table 2. Comparison of the calculated refractive index and actual refractive index
actual
nthe calculated n from λ1 error/% the calculated n from λ2 error/% 1.45 1.4400 -0.69 1.4464 -0.25 1.55 1.5469 -0.20 1.5480 -0.13 -
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