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Abstract
We theoretically propose a nonplasmonic optical refractive index sensor based on black phosphorus (BP) and other dielectric materials in the infrared band. Due to the anisotropic property of BP, the proposed sensor can achieve alternative sensitivity and figure of merit (FOM) in its different crystal directions. The high sensitivity and FOM are attributed to the strong magnetic resonance in the all-dielectric configuration. The coupled-mode theory (CMT) is used to verify the simulation results and reveal the physical mechanism. Furthermore, influences of the sample and the incident angle on the performance of the sensor are also discussed. Our design utilizes a simple dielectric structure with a BP monolayer, which exhibits great potential for the future high-performance sensor with low cost.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As a leading technology with great research significance in the field of nanophotonics, optical refractive index sensors are widely used in medical detection [1], environmental monitoring [2] and other fields due to their better detection performance and more compact size, non-destructive measurement, stronger anti-interference capacity, more convenient and faster detection speed compared with conventional sensors. Currently, many studies are focusing on improving their sensitivity and performance. Over the past few years, optical refractive index sensors using various plasmonic structures have been extensively studied as a result of their high sensitivity [3]. Theoretical research on a plasmonic sensing system has shown that the sensitivity and figure of merit (FOM) (at the wavelength of 1 µm) can reach as high as 13000 nm/RIU and 138 [4], respectively. Shen et al. proposed a plasmonic gold mushroom array with sensitivity and FOM (at the wavelength of 0.9 µm) up to 1010 nm/RIU and 108, respectively [5].
Recently, two-dimensional (2D) materials have attracted significant attention due to their unique optical and electrical features [6–9] and have been considered to have the ability to improve the performance of sensors [10–13]. Many researchers have used graphene [14–16], transition metal dichalcogenide [17] and other materials to improve the sensitivity of plasmonic biosensing system. For example, Tobias Wenger et al. proposed the graphene-based plasmonic sensor at an infrared band with sensitivity and FOM (at the wavelength of 7.3 µm) up to 2500 nm/RIU and 10.7, respectively [18]. Dash et al. proposed the graphene-based surface plasmonic optical fiber sensing system based D-shaped with sensitivity and FOM (at the wavelength of 0.5 µm) up to 3700 nm/RIU and 216, respectively [19].
Since being separated from massive crystals in 2014, black phosphorus (BP) has attracted significant attention [20–22]. As a new 2D material, BP has unique photoelectric features compared with other common 2D materials, for example, direct band gap [23,24], higher carrier effective quality, higher carrier mobility, and anisotropy [25,26]. It has attracted great attention in the fields of optoelectronic and sensor application [27]. In addition, BP is more applicable to sensing system due to its higher adsorption energy than graphene and MoS2 [28]. Sarika Pal’s study showed that the sensitivity of BP-based plasmonic sensor can be 1.40 times higher than that of conventional graphene-based plasmonic sensor [29]. Although many studies have been made with the plasmonic sensing system of 2D materials and performances have been improved to a certain extent, this sensing system requires special detection methods and complex manufacturing technology. Moreover, its FWHM will easily result in extension which further reduces the sensing resolution [30]. Besides, metals are usually contained to bring high consumption and the precious metals also result in relatively higher costs.
In this work, a BP-based all-dielectric optical refractive index sensor (BPAORIS) with high sensitivity has been proposed, in which the monolayer BP is placed at the top of the Al2O3 layer and the conventional perfect electric conductor (PEC) is replaced with the cold mirror [31] to avoid metal loss and reduce costs and processing challenges. The absorption is enhanced with magnetic resonance and coupled-mode theory (CMT) is used to verify the results of numerical simulation. Through optimizing the sensing system, this study achieves the sensitivity up to 4950 nm/RIU and 5000 nm/RIU and FOM up to 1395 and 682 respectively in different directions of crystal. It verifies that BP can be effectively combined with refractive index sensors and indicates that the proposed structure has great potential in the field of sensing devices.
2. Design and modeling
Figure 1 shows the proposed BPAORIS which consists of periodical SiO2 nanoribbon, monolayer BP, Al2O3 layer, cold mirror constituted by alternative high refractive index layer and low refractive index layer, and transparent substrate. Performances of the sensor are studied with the commercial software COMSOL based on finite element method (FEM).In our simulation, the interaction is investigated using COMSOL Multiphysics, which solves Maxwell Equations with finite element method (FEM) in the wavelength domain. Periodic boundary conditions are applied in the x-axis and y-axis directions. User-controlled tetrahedral meshes are applied in the whole computational domain. The infrared wave illuminates upon the top of the structure vertically, and interacts with the BP metasurface. In the simulation, it is assumed that refractive indexes (RI) of SiO2, Al2O3, high refractive index layer, low refractive index layer, and substrate are 1.35, 1.7, 3.9, 1.2 and 1.48 respectively. The thickness of each layer and the number of layers are optimized to greatly enhance the performance of the optical sensor. Due to its ultra-thin thickness compared with the incidence wavelength, BP can be assumed as 2D conductive surfaces with zero thickness. The optical properties of BP can be described with a simplified Drude model, where the conductivity can be expressed as below [32]:
where, j represents x-direction or y-direction; ω is the angular frequency, Dj is the Drude weight, the relaxation rate η is set as 10 meV, n is electron doping concentration, ħ is the reduced Planck’s constant, e is the charge of an electron. The effective mass along the x-direction and y-direction can be expressed as below:
Fig. 1. Schematic drawing of the proposed BPAORIS. The symbols w, p, h and t indicate width of SiO2 nanoribbon, period of nanoribbon, thickness of SiO2 and thickness of Al2O3, respectively.
For the monolayer BP, we have γ=${{4} \over \pi }eVm$, Δ=2 eV, ηc=$\frac{{\mathrm{\hbar} ^{2}}}{{0.4{m_0}}}$, νc=$\frac{{{\mathrm{\hbar}^2}}}{{1.4{m_0}}}$, the electron doping n=1013 cm-2. In addition, α is the scale length of BP; $\frac{\pi }{a}$ is the width of Brillouin Zone. The dielectric function with 2D conductivity can be expressed as below [32]:
where ɛ0 is the permittivity of free space, the relative permittivity ɛr of monolayer BP is 5.76. BP has two internal crystal arrangement modes, i.e., armchair along x-axis arrangement (“AX arrangement”) and armchair along z-axis arrangement (“AZ arrangement”), which determine the anisotropic property of BP.3. Results and discussions
Figure 2(a) compares the reflection spectrum of all-dielectric cold mirror and PEC. The role and design method of all-dielectric cold mirror will be discussed firstly. The all-dielectric cold mirror consists of stacked all-dielectric materials without metal loss. Through design and optimization, it is possible to reflect infrared waves at the expected frequency band. This study uses a group of dielectric layers with quarter wavelength that are stacked alternatively with high and low refractive index. Under normal incidence, the effective admittance and reflectivity of reflecting mirror can be calculated as below [33]:
where, nH, nL and ns represent the refractive index of dielectric layer with high refractive index, dielectric layer with low refractive index, and substrate respectively. The odd number m represents the total number of dielectric layers. Equations (6) and (7) indicate that the more the number of dielectric layers is, the larger the reflectivity of cold mirror will be. The bandwidth of high-reflectivity Δλ=λmax - λmin can be calculated as below [33]:Fig. 2. (a) Comparison of reflectance spectrum under TE polarization between PEC and cold mirror within 13.000 µm and 16.800 µm; (b) Comparison of absorption spectrum under TE and TM polarization in different arrangement directions of BP based on FEM, and comparison of absorption spectrum under TE and TM polarization in different arrangement directions of BP based on CMT analysis, where w=8.0 µm, p=12.0 µm, nb=4.2 µm, h=0.5 µm and t=3.0 µm
After showing the functions of cold mirror, we next compare the absorption spectrum of the sensor arranged along different directions under different polarizations, as shown in Fig. 2(b). The incident wave under TM polarization does not induce any absorption in both AX arrangement and AZ arrangement. However, under TE polarization, resonance occurs at wavelength λ=15.210 µm in AX arrangement, when the absorption rate is 78.9% and FWHM of related absorption peak is Δλ=3.5 nm, so that the quality factor can be defined to Qx=λ/Δλ=4330.25. On the other hand, resonance occurs at wavelength λ=15.195 µm in AZ arrangement, when the absorption rate is 46.5% and FWHM of related absorption peak is Δλ=7.3 nm, so that the quality factor Qy=2073.17 can be achieved. Therefore, the following simulations are conducted based on TE incidence because effective absorption cannot be observed under TM incidence.
In order to verify the accuracy of simulation results, CMT is used to calculate the absorption spectrum of the optical sensor, as shown in Fig. 2(b). Based on CMT analysis, the optical behavior of the system can be expressed as below [34–36]:
where, α is resonance amplitude; w0 is resonance frequency; S+ and S- are input and output wave amplitude; γ and δ are external leakage rate and intrinsic loss. The reflection coefficient of the system is:In the CMT analysis, for AX arrangement, the intrinsic loss and external leakage rate of BPAORIS are δ=0.62 × 109 Hz and γ=1.66 × 109 Hz, respectively. The quality factor which is defined as QCMTx=QδQγ/(Qδ + Qγ) can be calculated to be 4319.28, which is close to Qx . Qδ= w0/(2δ) is defined as intrinsic loss and Qγ= w0/(2γ) is defined as external leakage rate. For AZ arrangement, the intrinsic loss and external leakage rate are δ=0.66 × 109 Hz and γ=4.34 × 109 Hz, respectively. QCMTy is calculated to be 1972.63, which is close to Qy. Therefore, the analytical results are consistent with simulation results, thus providing theoretical support for the analysis of the proposed structure.
In order to illustrate the sensing mechanism of BPAORIS, Fig. 3 shows the amplitude distributions of magnetic field and electric field inside the sensor in action. From top to bottom, it is air, liquid sample, periodical SiO2 nanoribbon, Al2O3 layer, first high refractive index layer, and first low refractive index layer successively, while other all-dielectric layers and substrates with especially low amplitude of magnetic field and electric field are neglected. As shown in Fig. 3(a), compared with the magnetic field at non-resonant point λ=15.000 µm in AX arrangement, the proposed sensor has a significant magnetic dipole resonance at resonant point λ=15.210 µm in AX arrangement and traps infrared magnetic field between SiO2 nanoribbon and cold mirror. Compared with non-resonant state at λ=15.000 µm, the induced magnetic dipole contributes to the concentration and enhancement of magnetic field. The concentrated magnetic field is mainly distributed in the gap between the layer of liquid sample and the cold mirror. Moreover, magnetic dipole resonance also occurs at resonant point λ=15.195 µm in AZ arrangement, as shown in the middle of Fig. 3(a). The electric field at resonant point in AX arrangement is confined and enhanced in the area between Al2O3 layer and the liquid sample, which is guided along the direction perpendicular to incident plane, as shown in Fig. 3(b). The light absorption depends on the intensity of the electric field on the surface of BP. Therefore, under the resonant state at λ=15.210 µm in AX arrangement, the intensively strengthened electric field localized on BP layer results in significantly enhanced absorption. In addition, it can be seen that the distribution area of the electric field at resonant point λ=15.195 µm in AZ arrangement is the same as the distribution area at resonant point λ=15.210 µm in AX arrangement, but only the intensity of electric field changes to a certain extent.
Fig. 3. (a) Amplitude distributions of magnetic field at non-resonant wavelength in AX arrangement and at resonant wavelengths in AZ arrangement and AX arrangement; (b) Amplitude distributions of electric field at non-resonant wavelength in AX arrangement and at resonant wavelengths in AZ arrangement and AX arrangement
Next, the sensing performance of BPAORIS is mainly studied, including sensitivity (S) and figure of merit (FOM). The sensitivity can be expressed as below:
where Δλ represents the change of resonant wavelength of the sensor; Δn represents the change of refractive index of the sample under detection. FOM can be expressed as below: where, FWHM is the full width at half maximum of the resonance peak.In order to study the sensitivity and FOM of BPAORIS for different samples under detection, a series of simulations are conducted on the related absorption features when the refractive index of sample has any change. In the simulation, the fixed thickness of the sample is nb = 4.2 µm and the range of the refractive index is set within 1.50 and 1.55.
As shown in Fig. 4(a), if the refractive index of sample under detection is n=1.50, the resonant wavelength of the proposed sensor in AX arrangement and AZ arrangement is λ=15.210 µm and λ=15.195 µm, and the corresponding absorbance is 78.9% and 46.5%, respectively. However, as the refractive index of sample increases to n=1.55, the resonance peaks of BPAORIS are red-shifted to λ=15.457 µm and λ=15.444 µm, and the corresponding absorbance are 99.6% and 78.7%, while the wavelength offsets Δλ are 0.247 µm and 0.249 µm respectively. Besides, it can be observed that optical loss at related resonant wavelength in AZ arrangement is larger. Figure 4(b) shows the sensitivity and FOM of the sensor. In AX arrangement, sensitivity is 4900 nm/RIU and FOM is 1395 when n=1.50. Then, sensitivity increases to 4950 nm/RIU and FOM declines to 1209 within the range between n=1.51 and n=1.54. In AZ arrangement, sensitivity is 5000 nm/RIU and FOM is 679.6 when n=1.50. Then, sensitivity maintains at about 5000 nm/RIU and FOM declines to 662.5 within the range between n=1.51 and n=1.54. Distinguishing sensitivities and FOMs can be seen in different directions of BP crystal, which reflects the anisotropic property of BP-based sensor.
Fig. 4. (a) Absorption spectra for various samples in different directions of BP crystal, where nb=4.2 µm; (b) Sensitivity and FOM for various samples in different directions of BP crystal, where nb=4.2 µm
Subsequently, in order to further study the influence of thickness of the sample on the sensitivity and FOM of the proposed sensor, we plot the absorption spectra for different thicknesses of sample (nb) when the refractive index maintains at n=1.50. In AX arrangement as shown in Fig. 5(a), when the refractive index of sample is fixed to be n=1.50, the resonant wavelength exhibits an obvious redshift from λ=15.210 µm to λ=15.753 µm as nb changes from 4.2 µm to 6.2 µm. The absorptivity is 78.9% when nb=4.2 µm and it declines to 70.7% when nb increases to 6.2 µm. As shown in Fig. 5(b), the sensitivity reaches the maximum of 6250 nm/RIU when nb=4.6 µm and FOM reaches up to 1935. When the thickness of the sample exceeds 5.0 µm, both sensitivity and FOM of the proposed sensor increase with the increase of thickness nb.
Fig. 5. (a) Absorption spectra for different samples under detection in AX arrangement, where n=1.50; (b) Sensitivity and FOM for different samples under detection in AX arrangement, where n=1.50
Many optical sensors are sensitive to the incident angle, which may be boundedness in practical sensing applications. Therefore, we finally investigate the influence of incident angle on the proposed structure, as shown in Fig. 6(a). When n=1.50 and the angle of incidence increases from 0° to 5°, the resonance peak at λ=15.210 µm is divided into two separated peaks with low absorptivity at resonant wavelength λ=14.585 µm and λ=15.769 µm respectively. Meanwhile, its FWHM changes from 3.5 nm to 2.7 nm and 3.0 nm. Accordingly, based on Eqs. (18) and (19), sensitivity at left resonance peak is 5100 nm/RIU and FOM is 1653; while sensitivity at right resonance peak is 4450 nm/RIU and FOM is 1719. As shown in Fig. 6(b), when θ2=5°, the electric field is concentrated on the sample under detection, distribution mode of the electric field at left and right resonance point is similar, but the intensity is different. It can be seen that when the angle of incidence is θ2=5°, the amplitude of the electric field distributed on the sensor declines. This indicates that the absorption of infrared incidence inside BP also declines. In spite of this, its sensitivity at θ2=5° is still higher than conventional optical sensors in the infrared range, which verifies the feasibility of the proposed sensor in practical sensing application.
Fig. 6. (a) Absorption spectra of BPAORIS for θ=0° and θ=5° under TE polarization; (b) Electric field distributions at resonance λ=14.585 µm and λ=15.769 µm and non-resonance λ=15.000 µm under TE polarization, where θ=5°, nb=4.2 µm
Table 1 summarizes the performance comparison between BPAORIS and some other latest optical refractive index sensors. As shown in Table 1, compared with previous sensors based on plasmonic structures, 2D materials and other optical solutions, the proposed sensor is greatly improved in sensitivity and FOM. Moreover, the existence of BP monolayer brings the anisotropic characteristic to this sensor. This indicates that the proposed structure is greatly promising in the design of refractive index sensor.
Table 1. Performance comparison between latest optical refractive index sensor and BPAORIS
4. Conclusions
In conclusion, this paper proposes a BP-based all-dielectric refractive index sensor whose costs are greatly reduced due to avoidance of noble metal materials. Due to the existence of BP, the sensitivity has been achieved up to 4950 nm/RIU in AX arrangement and 5000 nm/RIU in AZ arrangement, and FOM has been achieved up to 1395 and 682 respectively. This paper also verifies the practical potential of BP monolayer in optical refractive index sensors and creates a new way for realizing various refractive index sensors with BP material.
Funding
Opening Project of Key Laboratory of Microelectronic Devices & Integrated Technology, Institute of Microelectronics, Chinese Academy of Sciences; Young and Middle-aged Teachers Education and Scientific Research Foundation of Fujian Province (JAT190674); Xiamen Youth Innovation Fund Project (3502Z20206074); Natural Science Foundation of Fujian Province (2020J01294, 2020J01295); National Natural Science Foundation of China (11804071, 62005232).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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