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Sub-micron pixel polarization-sensitive photodetector based on silicon nanowire

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Abstract

Silicon nanowire is a potential candidate to be used as polarization-sensitive material, but the relative mechanism of polarization response must be carried out. Herein, a sub-micron metal-single silicon nanowire-metal photodetector exhibits polarization-sensitive characteristics with an anisotropic photocurrent ratio of 1.59 at 780 nm, an excellent responsivity of 24.58 mA/W, and a high detectivity of 8.88 × 109 Jones at 980 nm. The underlying principle of optical anisotropy in silicon nanowire is attributed to resonance enhancement verified by polarizing light microscopy and simulation. Furthermore, Stokes parameter measurements and imaging are all demonstrated by detecting the characteristics of linearly polarized light and imaging the polarizer array, respectively. Given the maturity of silicon processing, the sub-micron linearly polarized light detection proposed in this study lays the groundwork for achieving highly integrated, simplified processes, and cost-effective on-chip polarization-sensitive optical chips in the future.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Polarization, as one of the basic optical properties, carries indispensable optical information. It plays a pivotal and ubiquitous role in various fields, ranging from analyzing material composition [1,2] to underwater navigation [3], remote sensing, atmospheric science, machine vision, and even autonomous driving [4]. Traditional methods of acquiring polarization information primarily involve division-of-time polarization detection, division-of-aperture polarization detection, division-of-amplitude polarization detection, and division-of-focal-plane polarization imaging [5,6], where optical components, such as polarizers, half-wave plates, linear retarders, and quarter-wave plates, are all needed, which definitely results in a bulky and intricate system structures. Also, the large size structure of the polarized selection components not only induces pixel crosstalk in the detector [7], but also prevents the miniaturization of pixels into the sub-wavelength [8]. Consequently, much work needs to be made to focus on polarization-sensitive materials to realize polarization detection without polarizers.

So far, to the best of our knowledge, non-polarizer polarization-sensitive detection relies on two primary approaches. One way is by integrating polarization structures before detectors. Many researchers utilize optical structures with polarization effects covering the detector surface to selectively control the polarization state of incident light [9]. Polarization selection using plasmonic cavities and optical transitions in quantum wells or quantum cascade detectors were all proposed to realize polarization-sensitive detectors [10,11]. Additionally, metasurface structures (such as rectangular gratings [4], nano-antennas [8], plasmonic nanostructures [12], and metasurface polarization filter arrays [13]) have been proven to serve as polarization selectors on the surface of the detector. While these detectors exhibit outstanding polarization extinction ratios, a considerable amount of light is lost, and integrating intricately designed microstructures with detectors poses certain challenges [5]. Another method is directly employing polarization-sensitive materials, which are different from polarization-sensitive structures. Many scholars have utilized 2D and quasi-2D materials [14] with in-plane structural anisotropy for polarization detection. Heterojunctions, such as black phosphorus/MoS2 [15], AsP/MoS2 [16], MoS2/Ta2NiSe5 [17], InSe/1T-MoTe2 [18], and GeSe/Ge [19], were all introduced to enhance the device's polarization sensitivity. However, the intricate fabrication steps, complex processes, and lower integration levels of heterojunctions pose obstacles to their applications. Nanowires of ReS2 [20], SnIISnIVS3 [21], Bi2S3 [22], ZrSe3 [23], and SbBiS3 [24] in binary and ternary materials have gained widespread attention owing to their polarization detection capabilities due to material anisotropic absorption from the asymmetry of crystal lattices and exhibit polarization-sensitive photoelectric responses. However, these binary and ternary materials still face formidable challenges in acquisition, complex preparation processes, suboptimal crystallinity, and incompatibility with complementary metal-oxide-semiconductor (CMOS) technology [9].

Compared to the above multicomponent materials, silicon with mature processing technology and high integration shows distinct advantages and excellent prospects. Many scholars have conducted research on silicon nanowire (SiNW) [2528]. Some work has used germanium to produce nanowire and explained the polarization mechanism [29]. However, less work has been made on polarization detection using SiNW, and the related polarization mechanisms need to be clarified. Recently, the polarization-sensitive structure of vertical SiNW has been demonstrated for acquiring polarization information and performing imaging [26], where an array of four elliptical SiNW oriented in different directions was prepared by using electron beam lithography. Later, a horizontal SiNW device was fabricated and demonstrated its polarization characteristics [28]. Despite having a more compact quasi-1D structure than vertical SiNW, unfortunately, this work did not undergo application validation. Very recently, researchers integrated a silicon metasurface on the surface of an InGaAs/InP photodetector to enable the detection of polarized light [30]. The silicon metasurface was only utilized for polarization selection and not to respond to light.

Herein, the polarization-sensitive of SiNW with a sub-micro pixel is determined using polarization detection experiments, simulations of light absorption, photoelectric response, and imaging applications. SiNW arrays were prepared through the metal-assisted chemical etching (MACE) method. Metal-semiconductor-metal (MSM) devices were fabricated by using SiNW, and photoelectric response tests demonstrated the devices’ polarization-sensitive characteristics. Subsequently, polarizing light microscopy confirmed the anisotropic optical properties of SiNW, and simulations demonstrated resonance phenomena between the nanowire and polarized light in different directions. Our work lays the groundwork and theoretical foundation for the miniaturization and high integration of on-chip polarization detectors in the future.

2. Experimental details

SiNW arrays were fabricated by a metal-assisted chemical etching method, as shown in Fig. 1(a). The N-type silicon wafer (100, from Shunsheng Electronic Technology) was cut into an appropriate size and sequentially cleaned in acetone, ethanol, and deionized water to ensure the cleanliness of the wafer. To remove the oxide layer from the wafer surface, the wafer was immersed in a 5% HF solution. Subsequently, the silicon wafer was immersed in a mixed solution of AgNO3 (0.005 mol/L) and HF (4.8 mol/L) to facilitate the attachment of Ag nanoparticles to the silicon surface. Then, at room temperature, the silicon was etched for 30 minutes using a solution ratio of 5:12:37 of H2O2, HF, and H2O to generate the SiNW array. The obtained SiNW array was rinsed with deionized water and dried with nitrogen. Finally, the SiNW array was sonicated in ethanol, resulting in the detachment of SiNW from the silicon substrate and obtaining individual SiNW dispersed in the ethanol. Using ethanol solution as a carrier, the SiNWs were transferred onto a prefabricated interdigitated electrode (Ti/Au: 30 nm/100 nm, substrate: SiO2, spacing: 5 µm). The device underwent a 20-minute annealing process (vacuum at 200 °C) to ensure stable connections with the electrode.

 figure: Fig. 1.

Fig. 1. (a) Schematic of the SiNW array and device fabrication process. (b) SEM image of SiNW. (c) Schematic of MSM device, inset: device diagram under an optical microscope. (d) SiNW connected to Au electrodes under an optical microscope.

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The diameter of SiNW was measured using scanning electron microscopy (SEM), as shown in Fig. 1(b), with a diameter of ∼450 nm. To further investigate the optoelectronic properties of SiNW, electrodes were fabricated at both ends to form a photodetector, as depicted in the schematic diagram in Fig. 1(c) along with an optical microscope image of the device. Figure 1(d) shows the SiNW connected to the Au electrode observed under an optical microscope, with an electrode spacing of 5 µm. The contact between SiNW and Au electrodes forms two back-to-back Schottky junctions, constituting a MSM device.

3. Result and discussion

3.1 Polarization-sensitive photoresponse and its mechanism

In order to verify the ability of the device to detect linearly polarized light, a setup depicted in Fig. 2(a) was established for testing. Unpolarized light emitted from the source becomes linearly polarized after passing through the polarizer. By rotating a half-wave plate, the direction of the linearly polarized light can be adjusted while maintaining light intensity. Moreover, when the half-wave plate is rotated by an angle $\alpha $, the polarization direction of the linearly polarized light rotates by $2\alpha $. The modulated linearly polarized light is directed onto the device, and the KEITHLEY 2450 digital source meter records the device's optoelectronic response, allowing the measurement of the anisotropic photocurrent ratio (Imax/Imin), also referred to as polarization sensitivity [31]. Figures 2(b) and 2(c)-(e) show the angle-photocurrent variation induced by linearly polarized light at 650 nm, 780 nm, and 808 nm at bias of -1 V, presented in Cartesian coordinates and polar plots (where the angle represents the rotated angle of polarized light). The measurements indicate an anisotropic photocurrent ratio of 1.47 at 650 nm, 1.59 at 780 nm, and 1.44 at 808 nm, respectively. The ability of the polarized light detector to detect anisotropy directly relies on the material's capacity to absorb polarized light of different directions. Stronger absorption of polarized light in a particular direction correlates with a larger responsive photocurrent [9,32].

 figure: Fig. 2.

Fig. 2. (a) Schematic diagram of the polarization-sensitive test structure. (b) Cartesian coordinate for anisotropic photocurrent under 650, 780, and 808 nm, respectively. (c)-(e) Polar plot for polarization-sensitive photocurrent under 650, 780, and 808 nm, respectively.

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To further investigate the anisotropic optical refraction of SiNW, polarizing light microscopy (PLM) was used to acquire absorption properties. The simplified testing principle is illustrated in Figs. 3(a) and 3(b): initially setting both polarizers in the cross mode in a completely dark field (Fig. 3(a)). Placing the sample between the two polarizers allows the polarized light passing through the sample to split into two beams with different polarization directions. The second polarizer permits the passage of the altered polarization direction beams, thus forming an image of the sample (Fig. 3(b)). The optical anisotropy of anisotropic low-dimensional materials can be directly characterized by optical contrast using PLM [24]. A polar plot depicting the grayscale values of SiNW at different angles is depicted in Fig. 3(c), which better illustrates the anisotropic birefringence phenomenon exhibited by the prepared SiNW. Figure 3(d) displays SiNW at different rotation angles in the cross mode, showcasing noticeable changes in optical contrast upon sample rotation. The sample darkens every 90° rotation, resulting in four extinction positions after a 360° rotation. Additionally, we discussed the anisotropic optical refraction of SiNW in the wavelength ranges of 600-700 nm and 700-800 nm in Supplement 1.

 figure: Fig. 3.

Fig. 3. (a) Schematic diagram of the light in the cross mode of PLM without the sample. (b) Schematic diagram of the light and testing principle after adding the sample in the cross mode of the PLM. (c) Polar plot of grayscale values of SiNW at different angles under PLM (fitting with the sine function). (d) SiNW at different rotation angles in the cross mode of the PLM.

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To deeply clear the polarization mechanism, simulation analysis was conducted using the electromagnetic wave frequency domain simulation. Figure 4(a) illustrates the schematic of the simulation structure, consisting of a single SiNW (diameter: 450 nm) placed in air, positioned above SiO2. By introducing TE and TM modes fields to the entire system and employing the finite element method to solve Maxwell's equations, the electromagnetic responses were computed, and the distribution of electric fields within and around the nanowire was revealed [33]. Figure 4(b) shows the absorption characteristics of SiNW ranging from 400 to 1000 nm in TE/TM modes and when the polarization direction of the light at an angle of -45° and 45° with the long axis of the nanowire. Notably, both TE and TM modes exhibit resonance peaks at different wavelengths. The absorptance is roughly the same when angles are -45° and 45°, and this value lies between the absorptance in TE and TM modes. In this case, quantum confinement influences the characteristics of electrons, in that the physical dimensions of electronic devices reach subwavelength scales [34]. The nanowire can be regarded as a scaled-down microcylindrical resonator, capturing light in circulating orbits through multiple total internal reflections. However, due to the small size of the nanowire, the resonant modes leak out, allowing for more efficient interaction with the surroundings and achieving antenna functionality [29]. In fact, this absorption caused by the optical antenna effect is closely related to the diameter of the SiNW [35]. The calculated absorption curves for different diameters can be found in Supplement 1. Due to factors such as the actual preparation resulting in a non-smooth surface of the NW and the accuracy of measurements affecting the assessment of the actual diameter, the actual diameter of the NW cannot be perfectly consistent with the simulated diameter. Therefore, there are discrepancies between the measured anisotropic photocurrent ratio and the calculated absorption rate. Specific spectral light absorption of NWs is enhanced when incident light couples effectively with leaky-mode resonances. When the polarization direction of the incident light is perpendicular to the nanowire, the amplitude inside the NW varies according to the material's dielectric constant and its diameter; when parallel, the internal field amplitude within the nanowire is similar to the incident field amplitude in air [28]. Figures 4(d) and 4(f) show the distribution of the electric field in TE/TM mode at 650 nm (The bands of the calculated absorption cross section are shown in the inset of Fig. 4(b), it can be observed that even though the difference in the absorption graph is small, polarization-sensitive characteristics still exist when observing the electric field distribution). With variations in the polarization direction, evident changes occur in the internal electric fields of the nanowire. Notably, the maximum intensity of the electric field distribution indicates that the internal electric field is higher in TM mode compared to TE mode, signifying enhanced absorption within the material in TM mode. Additionally, the electric field distribution results are shown in Figs. 4(c) and 4(e) by introducing light with polarization directions of -45° and 45°. Overall, as the angle between the polarization direction and the nanowire changes from -45° to 90°, the internal electric field distribution within the nanowire demonstrates a trend of medium-large-medium-small changes, which is consistent with the aforementioned findings. In Supplement 1, the cross-sectional plot of the simulated magnetic field distribution at 650 nm is provided. Simulation results illustrate that light waves with different polarization directions interact with SiNW structure, generating corresponding resonance phenomena and imparting SiNW with polarization-sensitive characteristics.

 figure: Fig. 4.

Fig. 4. (a) Schematic of a single SiNW in the simulation structure. (b) Absorbance of NW in the wavelength range from 400 to 1000 nm under light with different polarization directions, inset: the enlarged view of the absorption rate between 620 and 680 nm. (c)-(f) Distribution of electric fields inside and around the NW for light waves with polarization angles of -45°, 0°, 45°, and 90° with respect to the long axis of the NW at 650 nm. (In which the black arrow indicates the long axis direction of the nanowire, while the red arrow indicates the direction of the electric field.)

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3.2 Optoelectronic response mechanism

In order to investigate the optoelectronic response performance and mechanisms of the device, the same equipment mentioned above was used to conduct optoelectronic performance tests at room temperature. A schematic diagram of the test structure and photocurrent with a voltage ranging from -3 to 3 V is shown in Figs. 5(a) and 5(b), and the inset illustrates the current-voltage (I-V) characteristics under dark and illuminated conditions. It can be observed that the increase in photogenerated carriers leads to a corresponding increase in photocurrent when light is illuminated on SiNW. Additionally, the non-linearity of the I-V characteristics indicates the presence of a back-to-back Schottky structure between the metal electrode and the SiNW. Comparing responses under positive bias, the photocurrent increases more rapidly under negative bias, which is consistent with the I-V curve results under illumination in the inset. It is important to note that the device exhibits a minor negative photocurrent at zero bias, indicating the presence of an electric field within the MSM device at zero bias, likely due to uneven contact between the metal Au and the semiconductor NW, causing non-uniform barrier heights [33]. The energy band diagrams of the MSM device are depicted in Figs. 5(c) and 5(d), where applying bias reduces the barrier height, thereby enhancing device current. Figure 5(e) displays the optical response of the device under -1 V for light sources at 650 (optical power density: 9.26 mW cm-2), 780 (optical power density: 40.99 mW cm-2), 808 (optical power density: 470 mW cm-2), and 980 nm (optical power density: 14.56 mW cm-2), respectively. Photo responsivity (R) and detectivity (D*) are two important parameters for evaluating the performance of photodetectors. Photo responsivity, representing the photoelectric conversion performance of an optical component, is defined as the following equation:

$${R = }\frac{{{{I}_{{light}}}{ - }{{I}_{{dark}}}}}{{{{P}_{{in}}}{A}}}$$
where ${I_{light}}$ is the photocurrent response, ${I_{dark}}$ is the noise current (assuming it is primarily caused by dark current), ${P_{in}}$ is the incident light power, and A is the effective area of illumination, respectively. The detectivity is used to characterize the sensitivity of a photodetector, defined as the following equation:
$${D^\ast } = \frac{{R \cdot {A^{{\raise0.7ex\hbox{$1$} \!\mathord{/ {\vphantom {1 2}}}\!\lower0.7ex\hbox{$2$}}}}}}{{{{({2q{I_{dark}}} )}^{{\raise0.7ex\hbox{$1$} \!\mathord{/ {\vphantom {1 2}}}\!\lower0.7ex\hbox{$2$}}}}}}$$
where q is the absolute value of electron charge. Calculate the R and D* at different wavelengths, as shown in Fig. 5(f). Clearly, the maximum responsivity and detectivity can be up to 24.58 mA/W and 8.88 × 109 Jones at 980 nm. As mentioned above, the diameter of the nanowires directly affects the spectral absorption. The main reason for the difference between the calculated absorption and the measured responsivity and detectivity is that the surface of SiNW is not smooth, which leads to the shape and diameter of the prepared nanowires not being completely consistent with the simulated dimensions.

 figure: Fig. 5.

Fig. 5. (a) Schematic diagram of the optoelectronic characteristic testing structure of SiNW devices. (b) Optical response within the range of -3 to 3 V; inset: I-V characteristic curves with and without illumination. (c) Schematic energy-band diagrams of the MSM device for zero bias. (d) Schematic energy-band diagrams of the MSM device for an applied bias condition. (e) Optical response under different semiconductor lasers (650, 780, 808, and 980 nm, linearly polarized light:50:1) at bias of -1 v. (f) Responsivity and detectivity of the device at 650, 780, 808, and 980 nm, respectively.

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3.3 Application verification

Generally, a polarization imaging system obtains the polarization characteristics of a target by measuring its Stokes vector. To verify that our device can detect the characteristics of linearly polarized light, we carried out testing of the characteristics of linearly polarized light with the setup depicted in Fig. 6(a). Light emitted from the semiconductor laser (650 nm) passes through a polarizer, transforming into linearly polarized light. To determine the polarization state of this beam, the device is rotated at -45°, 0°, 45°, and 90°. These responses allow calculation of the angle of polarization (AOP) and degree of linear polarization (DOLP) through the ${S_0}$, ${S_1}$ and ${S_2}$(here, the three parameters are all Stokes parameters, calculated respectively by Eqs. (3), (4), and (5)), using the following formulas:

$${S_0} = {I_{0^\circ }} + {I_{90^\circ }}$$
$${S_1} = {I_{0^\circ }} - {I_{90^\circ }}$$
$${S_2} = {I_{45^\circ }} - {I_{ - 45^\circ }}$$

 figure: Fig. 6.

Fig. 6. (a) Schematic diagram of the testing structure. (b) Deviation plot between the testing result and the theoretical value of AOP. (c) Schematic diagram of the imaging structure. (d) Results of the four simulated imaging sessions.

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Here, ${I_{0^\circ }}$, ${I_{45^\circ }}$, ${I_{90^\circ }}$, and ${I_{ - 45^\circ }}$ are the photocurrents of the 0-degree, 45-degree, 90-degree and -45-degree polarization photodetector, the values are 30, 25, 20, and 23 pA, respectively. And then, the DOLP and AOP could be deduced in the equations as follows:

$$DOLP = \frac{{\sqrt {{S_1}^2 + {S_2}^2} }}{{{S_0}}}$$
$$AOP = \frac{1}{2}{\tan ^{ - 1}}\frac{{{S_2}}}{{{S_1}}}$$

After calculation, the detected DOLP for the linearly polarized light is 0.2, with an AOP of 5°39′. The deviation plot in Fig. 6(b) shows a comparison between the AOP of theory and the actual detected. The difference between the testing result and the theoretical value is largely attributed to the inaccurate experimental layout of the imprecise input light polarization.

Imaging for the polarizer array, one of the primary devices for obtaining a polarized image, was also conducted. As shown in Fig. 6(c) for the schematic diagram of the imaging structure, the computer sends commands to control the microcontroller, driving the motor to enable scanning by the detector. This scanning process captures images of polarizer arrays (each group has polarization directions of -45°, 0°, 45°, and 90°, respectively, totaling 4 groups). The detector performs a point-by-point scan 4 × 4 times, the results of these four scans are illustrated in Fig. 6(d). The scanning results indicate that imaging with polarization directions of 0° and 90° yields the highest and lowest signal values, respectively. Imaging with polarization directions of -45° and 45° results in medium signal values. It reveals distinct response values for different polarized light directions and exhibits good stability.

3.4 Comparison of different polarization detectors

Table 1 presents a comparison of the performance parameters of polarization detectors made from different materials. It's evident that, compared with the listed 1D nanowires, our fabricated SiNW exhibits superior detectivity and polarization sensitivity. Particularly noteworthy is that other materials present relatively complex fabrication and processing challenges, so these multicomponent materials do not possess the advantage of easy integration as compared to silicon material in the long term.

Tables Icon

Table 1. Performance comparison of polarization detectors made from different materials

4. Conclusion

In summary, we have employed metal-assisted chemical etching to fabricate a SiNW array from n-type silicon. The SiNW was transferred onto metal interdigital electrodes using ethanol, resulting in the fabrication of MSM devices. Through optoelectronic testing, this device not only exhibits a responsivity of ∼24.58 mA/W and a specific detectivity of ∼8.88 × 109 Jones at 980 nm but also shows significant polarization-sensitive characteristics (photocurrent anisotropy ratio of ∼1.47 at 650 nm, ∼1.59 at 780 nm, and ∼1.44 at 808 nm, respectively). We characterized the optical anisotropy of the SiNW using polarizing light microscopy, followed by structural simulations to explain this phenomenon theoretically. Finally, in order to demonstrate its capability and potential in detecting the AOP and the DOLP, the device was used for measuring Stokes parameters and simulating imaging. Compared to other quasi-1D materials, considering aspects such as integration and processability, SiNW possesses sufficient competitive advantages for future compact polarization-sensitive detectors which are sub-wavelength in size.

Funding

Industrial Chain Project of Shaanxi Provincial Science and the Technology Department (2021ZDLGY12-05); Key Research and Development Plan General Project (2022GY-262); Xi'an Innovation Capability Plan Project (21XJZZ0028).

Disclosures

The authors declare no conflicts of interest.

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.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

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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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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic of the SiNW array and device fabrication process. (b) SEM image of SiNW. (c) Schematic of MSM device, inset: device diagram under an optical microscope. (d) SiNW connected to Au electrodes under an optical microscope.
Fig. 2.
Fig. 2. (a) Schematic diagram of the polarization-sensitive test structure. (b) Cartesian coordinate for anisotropic photocurrent under 650, 780, and 808 nm, respectively. (c)-(e) Polar plot for polarization-sensitive photocurrent under 650, 780, and 808 nm, respectively.
Fig. 3.
Fig. 3. (a) Schematic diagram of the light in the cross mode of PLM without the sample. (b) Schematic diagram of the light and testing principle after adding the sample in the cross mode of the PLM. (c) Polar plot of grayscale values of SiNW at different angles under PLM (fitting with the sine function). (d) SiNW at different rotation angles in the cross mode of the PLM.
Fig. 4.
Fig. 4. (a) Schematic of a single SiNW in the simulation structure. (b) Absorbance of NW in the wavelength range from 400 to 1000 nm under light with different polarization directions, inset: the enlarged view of the absorption rate between 620 and 680 nm. (c)-(f) Distribution of electric fields inside and around the NW for light waves with polarization angles of -45°, 0°, 45°, and 90° with respect to the long axis of the NW at 650 nm. (In which the black arrow indicates the long axis direction of the nanowire, while the red arrow indicates the direction of the electric field.)
Fig. 5.
Fig. 5. (a) Schematic diagram of the optoelectronic characteristic testing structure of SiNW devices. (b) Optical response within the range of -3 to 3 V; inset: I-V characteristic curves with and without illumination. (c) Schematic energy-band diagrams of the MSM device for zero bias. (d) Schematic energy-band diagrams of the MSM device for an applied bias condition. (e) Optical response under different semiconductor lasers (650, 780, 808, and 980 nm, linearly polarized light:50:1) at bias of -1 v. (f) Responsivity and detectivity of the device at 650, 780, 808, and 980 nm, respectively.
Fig. 6.
Fig. 6. (a) Schematic diagram of the testing structure. (b) Deviation plot between the testing result and the theoretical value of AOP. (c) Schematic diagram of the imaging structure. (d) Results of the four simulated imaging sessions.

Tables (1)

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Table 1. Performance comparison of polarization detectors made from different materials

Equations (7)

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R = I l i g h t I d a r k P i n A
D = R A 1 / 1 2 2 ( 2 q I d a r k ) 1 / 1 2 2
S 0 = I 0 + I 90
S 1 = I 0 I 90
S 2 = I 45 I 45
D O L P = S 1 2 + S 2 2 S 0
A O P = 1 2 tan 1 S 2 S 1
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