Carbon Nanotube Far Infrared Detectors with High Responsivity and Superior Polarization Selectivity Based on Engineered Optical Antennas

Single-wall carbon nanotube (SWCNT) thin films are promising for sensitive uncooled infrared detection based on the photothermoelectric effect. The SWCNT film is usually shaped into a belt and diversely doped to form a p-n junction at the center. Under the illumination of a focused incident light, the temperature gradient from the junction to the contacts leads to photoresponse. When the SWCNTs are aligned in one direction, the photoresponse becomes polarization selective. Although a typical bowtie antenna can improve the responsivity and polarization extinction ratio by deep-subwavelength light focusing, the absolute absorptance of the junction region is only 0.6%. In this work, the antenna was engineered for a higher light coupling efficiency. By integrating a bottom metal plane at a specific distance from the SWCNT film and optimizing the antenna geometries, we achieved ultra-efficient impedance matching between the antenna and the SWCNTs, thus the absorptance of the junction region was further enhanced by 21.3 times and reached 13.5%, which is more than 3 orders of magnitude higher than that of the device without the engineered antenna. The peak responsivity was further enhanced by 19.9 times and responsivity reached 1500 V/W at 1 THz. The resonant frequency can be tuned by changing the size of the antenna. Over the frequency range of 0.5 THz to 1.5 THz, the peak responsivity was further enhanced by 8.1 to 19.9 times, and the polarization extinction ratio was enhanced by 2.7 to 22.3 times. The highest polarization extinction ratio reached 3.04 × 105 at 0.5 THz. The results are based on the numerical simulations of the light and the thermal fields.


Introduction
Far infrared (including THz) detectors are important to a wide range of applications in the fields of astronomy, sensing, spectroscopy, imaging, defense and communications [1][2][3]. Since the photon energy in this regime is quite low, photonic detectors require cryogenic cooling systems, which are expensive and bulky, to maintain the performance. In this respect, thermal detectors have an advantage as they can operate in an uncooled mode, although they are generally less sensitive and slower than photonic detectors [4]. In this situation, new materials with a prominent thermoelectric effect, low specific heat capacity, and high thermal stability become promising candidates for future infrared thermal detectors [5][6][7][8][9][10][11][12]. Among those materials, single-wall carbon nanotubes (SWCNT) have attracted great attention due to their unique electrical, optical and thermal properties [4,[8][9][10][11][12][13][14]. In particular, SWCNT thin films have been proposed and demonstrated as a potential infrared detection material by a lot of researchers, since they can be made highly uniform and can be diversely doped to form p-n junctions [8,13,14]. In addition, they are compatible (1) ε ⊥ denote the relative permittivity of an aligned SWCNT film for the light field polarized perpendicular to the SWCNTs, and ε denote the relative permittivity for the light field parallel to the SWCNTs. f represents the fill factor, ε SWCNT is the permittivity of a single SWCNT, which can be approximated as a solid rod [20,21], and ε e is the permittivity of the environment, which is air in this case. Based on the density of a SWCNT film [22,23], we assume f = 0.3. Since the SWCNT film contains both semiconductor nanotubes and metal nanotubes, ε CNT represents the average optical response. Based on the actual measurement results of SWCNT film under far-infrared conditions [24,25], ε CNT is obtained through effective medium theory and modeled in the Drude-Lorentz form [20]. Based on this, we obtained the permittivity ε ⊥ and ε of SWCNT. parallel to the SWCNTs. f represents the fill factor, is the permittivity of a single SWCNT, which can be approximated as a solid rod [20,21], and εe is the permittivity of the environment, which is air in this case. Based on the density of a SWCNT film [22,23], we assume f = 0.3. Since the SWCNT film contains both semiconductor nanotubes and metal nanotubes, represents the average optical response. Based on the actual measurement results of SWCNT film under far-infrared conditions [24,25], is obtained through effective medium theory and modeled in the Drude-Lorentz form [20]. Based on this, we obtained the permittivity and ∥ of SWCNT.

Responsivity Obtained by Optical and Thermal Field Simulation
As shown in Figure 1a, the aligned SWCNT film (1 mm long, 5 μm wide, and 2 μm thick) is supported by a transparent dielectric substrate (teflon or SU8), and connected by two metal contacts. By diversely doping the two halves of the SWCNT film, a p-n junction is formed at the center. The Seebeck coefficient is positive in the p-doped region and negative in the n-doped region. In this way, when the incident light is focused at the p-n junction, a distributed temperature rise is created. The temperature at the junction is the highest and gradually decreases from the center to the two ends. Then, a photoresponse is created due to the photothermoelectric effect. The SWCNT film can be formed by first growing vertically and rolling down to form a macroscopic film [13,14,22], and then patterning the film by photolithography and etching [26,27]. The photoresponse can be estimated by numerical simulations of the optical field and the thermal field. The light field and the thermal field are described by the following equations (Equations (1)-(3)): where k0 represents the wave vector in vacuum, is the relative permittivity tensor of the aligned SWCNT film, is the thermal conductivity tensor of the aligned SWCNT film.
is the vacuum permittivity, and Qe denotes the light power absorption density. The absorbed light power becomes heat, so Qe is the distributed heat source term in Equation (4). When a far infrared Gaussian beam (waist around 800 μm) is incident on the aligned SWCNT film (1 mm long, 5 μm wide, and 2 μm thick, as shown in Figure 1a), the light absorption distribution and the temperature rise distribution are simulated as the plots in Figure 1b,c. The frequency of the incident light is 1 THz and the beam waist is 400 μm. The simulations are based on the finite element method and through the software COMSOL. After the temperature distribution over the SWCNT film is worked out, the photovoltage (Δ ) due to the PTE effect (Δ = ( − )Δ ) is obtained. Sp and

Responsivity Obtained by Optical and Thermal Field Simulation
As shown in Figure 1a, the aligned SWCNT film (1 mm long, 5 µm wide, and 2 µm thick) is supported by a transparent dielectric substrate (teflon or SU8), and connected by two metal contacts. By diversely doping the two halves of the SWCNT film, a p-n junction is formed at the center. The Seebeck coefficient is positive in the p-doped region and negative in the n-doped region. In this way, when the incident light is focused at the p-n junction, a distributed temperature rise is created. The temperature at the junction is the highest and gradually decreases from the center to the two ends. Then, a photoresponse is created due to the photothermoelectric effect. The SWCNT film can be formed by first growing vertically and rolling down to form a macroscopic film [13,14,22], and then patterning the film by photolithography and etching [26,27]. The photoresponse can be estimated by numerical simulations of the optical field and the thermal field. The light field and the thermal field are described by the following equations (Equations (1)-(3)): parallel to the SWCNTs. f represents the fill factor, is the permittivity of a single SWCNT, which can be approximated as a solid rod [20,21], and εe is the permittivity of the environment, which is air in this case. Based on the density of a SWCNT film [22,23], we assume f = 0.3. Since the SWCNT film contains both semiconductor nanotubes and metal nanotubes, represents the average optical response. Based on the actual measurement results of SWCNT film under far-infrared conditions [24,25], is obtained through effective medium theory and modeled in the Drude-Lorentz form [20]. Based on this, we obtained the permittivity and ∥ of SWCNT.

Responsivity Obtained by Optical and Thermal Field Simulation
As shown in Figure 1a, the aligned SWCNT film (1 mm long, 5 μm wide, and 2 μm thick) is supported by a transparent dielectric substrate (teflon or SU8), and connected by two metal contacts. By diversely doping the two halves of the SWCNT film, a p-n junction is formed at the center. The Seebeck coefficient is positive in the p-doped region and negative in the n-doped region. In this way, when the incident light is focused at the p-n junction, a distributed temperature rise is created. The temperature at the junction is the highest and gradually decreases from the center to the two ends. Then, a photoresponse is created due to the photothermoelectric effect. The SWCNT film can be formed by first growing vertically and rolling down to form a macroscopic film [13,14,22], and then patterning the film by photolithography and etching [26,27]. The photoresponse can be estimated by numerical simulations of the optical field and the thermal field. The light field and the thermal field are described by the following equations (Equations (1)-(3)): where k0 represents the wave vector in vacuum, is the relative permittivity tensor of the aligned SWCNT film, is the thermal conductivity tensor of the aligned SWCNT film.
is the vacuum permittivity, and Qe denotes the light power absorption density. The absorbed light power becomes heat, so Qe is the distributed heat source term in Equation (4). When a far infrared Gaussian beam (waist around 800 μm) is incident on the aligned SWCNT film (1 mm long, 5 μm wide, and 2 μm thick, as shown in Figure 1a), the light absorption distribution and the temperature rise distribution are simulated as the plots in Figure 1b,c. The frequency of the incident light is 1 THz and the beam waist is 400 μm. The simulations are based on the finite element method and through the software COMSOL. After the temperature distribution over the SWCNT film is worked out, the photovoltage (Δ ) due to the PTE effect (Δ = ( − )Δ ) is obtained. Sp and − Sensors 2021, 21, 5221 3 of 9 parallel to the SWCNTs. f represents the fill factor, is the permittivity of a single SWCNT, which can be approximated as a solid rod [20,21], and εe is the permittivity of the environment, which is air in this case. Based on the density of a SWCNT film [22,23], we assume f = 0.3. Since the SWCNT film contains both semiconductor nanotubes and metal nanotubes, represents the average optical response. Based on the actual measurement results of SWCNT film under far-infrared conditions [24,25], is obtained through effective medium theory and modeled in the Drude-Lorentz form [20]. Based on this, we obtained the permittivity and ∥ of SWCNT.

Responsivity Obtained by Optical and Thermal Field Simulation
As shown in Figure 1a, the aligned SWCNT film (1 mm long, 5 μm wide, and 2 μm thick) is supported by a transparent dielectric substrate (teflon or SU8), and connected by two metal contacts. By diversely doping the two halves of the SWCNT film, a p-n junction is formed at the center. The Seebeck coefficient is positive in the p-doped region and negative in the n-doped region. In this way, when the incident light is focused at the p-n junction, a distributed temperature rise is created. The temperature at the junction is the highest and gradually decreases from the center to the two ends. Then, a photoresponse is created due to the photothermoelectric effect. The SWCNT film can be formed by first growing vertically and rolling down to form a macroscopic film [13,14,22], and then patterning the film by photolithography and etching [26,27]. The photoresponse can be estimated by numerical simulations of the optical field and the thermal field. The light field and the thermal field are described by the following equations (Equations (1)-(3)): where k0 represents the wave vector in vacuum, is the relative permittivity tensor of the aligned SWCNT film, is the thermal conductivity tensor of the aligned SWCNT film.
is the vacuum permittivity, and Qe denotes the light power absorption density. The absorbed light power becomes heat, so Qe is the distributed heat source term in Equation (4). When a far infrared Gaussian beam (waist around 800 μm) is incident on the aligned SWCNT film (1 mm long, 5 μm wide, and 2 μm thick, as shown in Figure 1a parallel to the SWCNTs. f represents the fill factor, is the permittivity of a single SWCNT, which can be approximated as a solid rod [20,21], and εe is the permittivity of the environment, which is air in this case. Based on the density of a SWCNT film [22,23], we assume f = 0.3. Since the SWCNT film contains both semiconductor nanotubes and metal nanotubes, represents the average optical response. Based on the actual measurement results of SWCNT film under far-infrared conditions [24,25], is ob-tained through effective medium theory and modeled in the Drude-Lorentz form [20]. Based on this, we obtained the permittivity and ∥ of SWCNT.

Responsivity Obtained by Optical and Thermal Field Simulation
As shown in Figure 1a, the aligned SWCNT film (1 mm long, 5 μm wide, and 2 μm thick) is supported by a transparent dielectric substrate (teflon or SU8), and connected by two metal contacts. By diversely doping the two halves of the SWCNT film, a p-n junc-tion is formed at the center. The Seebeck coefficient is positive in the p-doped region and negative in the n-doped region. In this way, when the incident light is focused at the p-n junction, a distributed temperature rise is created. The temperature at the junction is the highest and gradually decreases from the center to the two ends. Then, a photoresponse is created due to the photothermoelectric effect. The SWCNT film can be formed by first growing vertically and rolling down to form a macroscopic film [13,14,22], and then patterning the film by photolithography and etching [26,27]. The photoresponse can be estimated by numerical simulations of the optical field and the thermal field. The light field and the thermal field are described by the following equations (Equations (1)-(3)): where k0 represents the wave vector in vacuum, is the relative permittivity tensor of the aligned SWCNT film, is the thermal conductivity tensor of the aligned SWCNT film.
is the vacuum permittivity, and Qe denotes the light power absorption density. The absorbed light power becomes heat, so Qe is the distributed heat source term in Equation (4). When a far infrared Gaussian beam (waist around 800 μm) is incident on the aligned SWCNT film (1 mm long, 5 μm wide, and 2 μm thick, as shown in Figure 1a), the light absorption distribution and the temperature rise distribution are simulated as the plots in Figure 1b,c. The frequency of the incident light is 1 THz and the beam waist is 400 μm. The simulations are based on the finite element method and through the software COMSOL. After the temperature distribution over the SWCNT film is worked out, the photovoltage (Δ ) due to the PTE effect (Δ = ( − )Δ ) is obtained. Sp and where k 0 represents the wave vector in vacuum, ↔ ε r is the relative permittivity tensor of the aligned SWCNT film, ↔ k is the thermal conductivity tensor of the aligned SWCNT film. ε 0 is the vacuum permittivity, and Q e denotes the light power absorption density. The absorbed light power becomes heat, so Q e is the distributed heat source term in Equation (4). When a far infrared Gaussian beam (waist around 800 µm) is incident on the aligned SWCNT film (1 mm long, 5 µm wide, and 2 µm thick, as shown in Figure 1a), the light absorption distribution and the temperature rise distribution are simulated as the plots in Figure 1b,c. The frequency of the incident light is 1 THz and the beam waist is 400 µm. The simulations are based on the finite element method and through the software COMSOL. After the temperature distribution over the SWCNT film is worked out, the photovoltage (∆V) due to the PTE effect ( ∆V = S p − S n ∆T) is obtained. S p and S n denote the Seebeck coefficients of the p-doped and the n-doped regions, respectively. The details of simulation settings can be found in a previous study [18]. Since the real part of the parallel permittivity of the aligned SWCNT film (Re(ε )) is negative in the frequency range from 0 to 2.8 THz [18], the film with a limited width (i.e., the length of the SWCNTs) can intrinsically support a localized surface plasmon (LSPR) mode. The LSPR frequency depends on the W film . Although the LSPR of the aligned SWCNT film could also enhance the light absorption, the light coupling efficiency is low and the light field is not concentrated at the junction region. Thus, the responsivity enhancement and the polarization extinction ratio are 1 to 2 orders of magnitudes smaller than those induced by an optical antenna [18].

Results
In order to increase the responsivity and the polarization extinction ratio of the aligned SWCNT film based detector, it is natural to add an antenna to concentrate the incident light at the junction region of the SWCNT film, as shown in Figure 2a. The SWCNT film is 1 mm long, 5 µm wide, and 2 µm thick, as shown in Figure 1a. The antenna has a bowtie shape. The length L a and the width W a are both 150 µm. The feed point overlaps the junction region. There is an extension bar (20 µm long and 2 µm wide) at the tip of each wing. It is also 2 µm away from the SWCNT film. Although an ordinary bowtie antenna can enhance the local Q e at the junction by 240 times and thus enhance the responsivity by 23 times [18], the absolute absorptance of the junction region (A j ) is only 0.6%. In order to further enhance A j and the responsivity, we propose to integrate a bottom metal plane at a specific distance from the SWCNT film and the antenna, as shown in Figure 2b. As a result, the Q e at the junction is further increased by 18 times (Figure 2c-e), and ∆T is further enhanced by 17.5 times (Figure 2f). A j is enhanced from 0.6% to 11.7%, which is more than 3 orders of magnitude higher than that of the device without any antenna. The antenna dimensions W a and L a are adjusted from 150 µm to 171 µm after the bottom metal plane is integrated to keep the resonant frequency at 1 THz. When the thickness of the dielectric layer is as small as 13 µm, each wing of the bowtie antenna couples with its image due to the deep subwavelength thickness of the dielectric spacer and then forms a magnetic dipole, as exhibited in Figure 2c,e. At the magnetic dipole resonance, the incident light is efficiently coupled into the system, leading to an enhanced local field that significantly improves the responsivity of the carbon nanotube detector. The resonant frequency of the engineered bowtie antenna can be estimated by the antenna dimensions based on a patch antenna model [28]. The performance enhancement by the antenna engineering is attributed to the improved light coupling efficiency.
The local field intensity at the junction region can be derived from the coupled mode theory as [29] |E loc | 2 E loc is the amplitude of the local field, E 0 the amplitude of the incident light, A c is the effective aperture, λ res the resonant wavelength, V eff the effective mode volume. Q as the total quality factor is related to the radiation quality factor (Q rad ) and the absorption quality factor (Q abs ) through the equation Q −1 = Q rad −1 + Q abs −1 . Based on the derivative of |E loc | 2 /|E 0 | 2 with respect to Q rad , it is revealed that |E loc | 2 /|E 0 | 2 reaches a maxima when Q rad = Q abs , which is called a critical coupling condition or an impedance matching condition. When the system reaches a critical coupling status, the light coupling efficiency becomes a maxima. Q abs is decided by the absorptance of the SWCNT film and the metal. In the absence of the bottom metal plane, the light mode is not very well confined (Figure 3a,b), and Q rad is much smaller than Q abs , so the system is far from a critical coupling status.
In the presence of the bottom metal plane, the radiation from the antenna and that from the mirror image of the induced charges will constructively or destructively interfere with each other as the thickness of the dielectric layer varies (Figure 3c,d), so Q rad is tuned and the light coupling efficiency changes accordingly (Figure 3e). At a specific dielectric thickness, Q rad matches Q abs so the system reaches the critical coupling status and the light coupling efficiency is maximized. As revealed in Figure 3b,d, the local field induced by the optimized metal plane integrated antenna is several tens of times higher than that induced by the ordinary antenna. fere with each other as the thickness of the dielectric layer varies (Figure 3c,d), so Qrad is tuned and the light coupling efficiency changes accordingly (Figure 3e). At a specific dielectric thickness, Qrad matches Qabs so the system reaches the critical coupling status and the light coupling efficiency is maximized. As revealed in Figure 3b,d, the local field induced by the optimized metal plane integrated antenna is several tens of times higher than that induced by the ordinary antenna.  The length of the extension bar at each tip of the two wings of the bowtie antenna is an important parameter that has a big impact on the coupling efficiency. As shown in Figure 4a, A j varies non-monotonously with the length of the extension bar (L bar ). The highest junction region absorptance occurs at L bar = 40 µm. Correspondingly, the ∆T at the junction and the responsivity of the device also reaches the maxima at L bar = 40 µm (Figure 4b, red line). This effect is attributed to the fine tuning of the antenna efficiency. According to the antenna theory [30], the antenna efficiency gets maximized when the radiation resistance (R rad ) equals to the load resistance (R load ). Concerning the bowtie antenna without the bottom metal plane, R rad is around 73 Ω [30]. R load is decided by the portion of the SWCNT film within the feed point region (defined by L bar and marked out by the purple frame in Figure 4a). As L bar increases, there are more and more SWCNTs bridging the feed point region in parallel, so R load decreases. R load = (1/σ)(1/L bar ) can be calculated through the conductivity of the SWCNT film (σ = iωε). ω is the angular frequency, ε is the dielectric constant. Based on this expression, R load is calculated to be 158 Ω at L bar = 20 µm, and 78 Ω at L bar = 60 µm. R load is close to R rad at L bar = 60 µm, so the antenna efficiency and thus the junction region absorptance reach a maxima in this condition (Figure 4a black line). When the antenna is integrated with a metal plane, the length of the bowtie antenna (L a ) needs to be enlarged to maintain the resonant wavelength. Since R load increases with L a , the situation R rad = R load occurs at a smaller L bar for the metal plane integrated antenna. As confirmed by the red line in Figure 4b,c, the optimized A j appears around L bar = 40 µm. In this case, the engineered antenna with L bar = 40 µm   The length of the extension bar at each tip of the two wings of the bowtie antenna is an important parameter that has a big impact on the coupling efficiency. As shown in Figure 4a, Aj varies non-monotonously with the length of the extension bar (Lbar). The highest junction region absorptance occurs at Lbar = 40 μm. Correspondingly, the ∆T at the junction and the responsivity of the device also reaches the maxima at Lbar = 40 μm (Figure 4b, red line). This effect is attributed to the fine tuning of the antenna efficiency. According to the antenna theory [30], the antenna efficiency gets maximized when the radiation resistance (Rrad) equals to the load resistance (Rload). Concerning the bowtie antenna without the bottom metal plane, Rrad is around 73 Ω [30]. Rload is decided by the portion of the SWCNT film within the feed point region (defined by Lbar and marked out by the purple frame in Figure 4a). As Lbar increases, there are more and more SWCNTs bridging the feed point region in parallel, so Rload decreases. Rload = (1/σ)(1/Lbar) can be calculated through the conductivity of the SWCNT film (σ = iωε). ω is the angular frequency, ε is the dielectric constant. Based on this expression, Rload is calculated to be 158 Ω at Lbar = 20 μm, and 78 Ω at Lbar = 60 μm. Rload is close to Rrad at Lbar = 60 μm, so the antenna efficiency and thus the junction region absorptance reach a maxima in this condition (Figure 4a black line). When the antenna is integrated with a metal plane, the length of the bowtie antenna (La) needs to be enlarged to maintain the resonant wavelength. Since Rload increases with La, the situation Rrad = Rload occurs at a smaller Lbar for the metal plane integrated antenna. As confirmed by the red line in Figure 4b    The spectra of A j , ∆T, and R V for the SWCNT infrared detector integrated with the ordinary antenna and that with the engineered antenna are presented in Figure 5a,b. As confirmed by the spectra in Figure 5a,b, the engineered antenna enhances the peak absorptance and ∆T (R V ) significantly at the resonant frequency for y-polarized incident light. The peak R V reaches 1500 V/W, which is 19.9 times higher than that achieved by an ordinary antenna. For the x-polariztion, the engineered antenna induces an R V of 0.045 V/W at 1 THz and the ordinary antenna induces an R V of 0.043 V/W. Therefore, the engineered antenna enhances the PER by 19.1 times to 33,440. By altering the dimension of the antenna, the peak responsivity can be set at different frequencies. As shown in Figure 5c, as the antenna dimension rising from 95 µm to 379 µm, the peak responsivity shifts from 1.5 THz to 0.5 THz in the frequency domain. Concerning the antenna as large as 379 µm, it also supports higher order modes in this frequency range, as shown by the pink line in Figure 5c. The peak R v , peak PER, and the enhancement times of these two quantities compared to that of the ordinary antenna integrated devices are presented in Table 1.
The spectra of Aj, ∆T, and RV for the SWCNT infrared detector integrated with the ordinary antenna and that with the engineered antenna are presented in Figure 5a,b. As confirmed by the spectra in Figure 5a,b, the engineered antenna enhances the peak absorptance and ∆T (RV) significantly at the resonant frequency for y-polarized incident light. The peak RV reaches 1500 V/W, which is 19.9 times higher than that achieved by an ordinary antenna. For the x-polariztion, the engineered antenna induces an RV of 0.045 V/W at 1 THz and the ordinary antenna induces an RV of 0.043 V/W. Therefore, the engineered antenna enhances the PER by 19.1 times to 33,440. By altering the dimension of the antenna, the peak responsivity can be set at different frequencies. As shown in Figure  5c, as the antenna dimension rising from 95 μm to 379 μm, the peak responsivity shifts from 1.5 THz to 0.5 THz in the frequency domain. Concerning the antenna as large as 379 μm, it also supports higher order modes in this frequency range, as shown by the pink line in Figure 5c. The peak Rv, peak PER, and the enhancement times of these two quantities compared to that of the ordinary antenna integrated devices are presented in Table 1.

Conclusions
In conclusion, we numerically demonstrated that light coupling engineering of an optical antenna integrated aligned SWCNT film based far infrared detector can further significantly enhance the performance. By integrating a bottom metal plane at a specific distance from the antenna and tuning the length of the extension bar at the tip of each wing of the antenna, the peak responsivity is further enhanced by 8.1 to 19.9 times over the frequency range from 0.5 THz to 1.5 THz, and the polarization extinction ratio is enhanced by 2.7 to 22.3 times. The highest peak responsivity in this range reaches 1500 V/W at 1 THz, 19.9 times higher than that induced by the ordinary antenna. The peak polarization extinction ratio is higher than 1481 over the range from 0.5 to 1.5 THz and reaches an extremely high value of 3.04 × 10 5 at 0.5 THz. The mechanism of the enhancement is attributed to critical coupling and impedance matching, and this principle can be applied to a variety of systems to improve the light-matter interaction in subwavelength scales.