Abstract

All-InSb film-based and spiral antenna-assisted Au-InSb-Au metal-semiconductor-metal detector is reported with dual-band photoresponse in the infrared (IR) and millimeter wave range. At IR, the detector exhibits a long wavelength 100% cut-off at 7.3 µm. Under an applied bias of 5 mA, the uncooled blackbody responsivity and specific detectivity are 3.5 A/W and 1×108 Jones, respectively. The f-3dB value measured at 2.94 µm is 75 KHz, corresponding to a detector rise speed of 4.7 µs. At millimeter wave range, the detector shows a narrowband response determined by the coupling of the antenna. A voltage responsivity of 25 V/W is achieved at 167 GHz (1.796 mm) under an applied bias of 25 mA, and the corresponding noise equivalent power (NEP) is 1.0×10−10 WHz−1/2, which can be improved to 1.8×10−12 WHz−1/2 if normalized to the real active semiconductor area. A f-3dB value of 17.5 KHz, corresponding to a detector rise speed of 20 µs is achieved in this range. A proof of principle for IR-modulated photoresponse for millimeter wave is achieved with a maximum modulation depth of 47.5%. This All-InSb film-based detector and the modulation are promising for future novel optoelectronic devices in IR and millimeter waves.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

J. Tong, Y. Qu, F. Suo, W. Zhou, Z. Huang, and D. H. Zhang, “Antenna-assisted subwavelength metal–InGaAs–metal structure for sensitive and direct photodetection of millimeter and terahertz waves,” Photonics Res. 7(1), 89 (2019).
[Crossref]

2018 (3)

D. Palaferri, Y. Todorov, A. Bigioli, A. Mottaghizadeh, D. Gacemi, A. Calabrese, A. Vasanelli, L. Li, A. G. Davies, E. H. Linfield, F. Kapsalidis, M. Beck, J. Faist, and C. Sirtori, “Room-temperature nine-µm-wavelength photodetectors and GHz-frequency heterodyne receivers,” Nature 556(7699), 85–88 (2018).
[Crossref]

A. Jollivet, B. Hinkov, S. Pirotta, H. Hoang, S. Derelle, J. Jaeck, M. Tchernycheva, R. Colombelli, A. Bousseksou, M. Hugues, N. Le Biavan, J. Tamayo-Arriola, M. Montes Bajo, L. Rigutti, A. Hierro, G. Strasser, J.-M. Chauveau, and F. H. Julien, “Short infrared wavelength quantum cascade detectors based on m-plane ZnO/ZnMgO quantum wells,” Appl. Phys. Lett. 113(25), 251104 (2018).
[Crossref]

N. Henry, D. Burghoff, Q. Hu, and J. B. Khurgin, “Temporal characteristics of quantum cascade laser frequency modulated combs in long wave infrared and THz regions,” Opt. Express 26(11), 14201 (2018).
[Crossref]

2017 (3)

A. Haddadi, A. Dehzangi, R. Chevallier, S. Adhikary, and M. Razeghi, “Bias–selectable nBn dual–band long–/very long–wavelength infrared photodetectors based on InAs/InAs1−xSbx/AlAs1−xSbx type–II superlattices,” Sci. Rep. 7(1), 3379 (2017).
[Crossref]

B. Schwarz, P. Reininger, A. Harrer, D. MacFarland, H. Detz, A. M. Andrews, W. Schrenk, and G. Strasser, “The limit of quantum cascade detectors: A single period device,” Appl. Phys. Lett. 111(6), 061107 (2017).
[Crossref]

J. Tong, W. Zhou, Y. Qu, Z. Xu, Z. Huang, and D. H. Zhang, “Surface plasmon induced direct detection of long wavelength photons,” Nat. Commun. 8(1), 1660 (2017).
[Crossref]

2016 (1)

2015 (2)

L. Zhang, K. Mu, Y. Zhou, H. Wang, C. Zhang, and X.-C. Zhang, “High-power THz to IR emission by femtosecond laser irradiation of random 2D metallic nanostructures,” Sci. Rep. 5(1), 12536 (2015).
[Crossref]

K. Peng, P. Parkinson, L. Fu, Q. Gao, N. Jiang, Y.-N. Guo, F. Wang, H. J. Joyce, J. L. Boland, H. H. Tan, C. Jagadish, and M. B. Johnston, “Single Nanowire Photoconductive Terahertz Detectors,” Nano Lett. 15(1), 206–210 (2015).
[Crossref]

2014 (1)

S. Tretyakov, “Maximizing Absorption and Scattering by Dipole Particles,” Plasmonics 9(4), 935–944 (2014).
[Crossref]

2013 (2)

C. W. Berry, N. Wang, M. R. Hashemi, M. Unlu, and M. Jarrahi, “Significant performance enhancement in photoconductive terahertz optoelectronics by incorporating plasmonic contact electrodes,” Nat. Commun. 4(1), 1622 (2013).
[Crossref]

L. Deng, J. Teng, H. Liu, Q. Y. Wu, J. Tang, X. Zhang, S. A. Maier, K. P. Lim, C. Y. Ngo, S. F. Yoon, and S. J. Chua, “Direct Optical Tuning of the Terahertz Plasmonic Response of InSb Subwavelength Gratings,” Adv. Opt. Mater. 1(2), 128–132 (2013).
[Crossref]

2012 (1)

S. M. Hanham, A. I. Fernández-Domínguez, J. H. Teng, S. S. Ang, K. P. Lim, S. F. Yoon, C. Y. Ngo, N. Klein, J. B. Pendry, and S. A. Maier, “Broadband Terahertz Plasmonic Response of Touching InSb Disks,” Adv. Mater. 24(35), OP226–OP230 (2012).
[Crossref]

2011 (1)

I. McKerracher, J. Wong-Leung, G. Jolley, L. Fu, H. H. Tan, and C. Jagadish, “Selective Intermixing of InGaAs/GaAs Quantum Dot Infrared Photodetectors,” IEEE J. Quantum Electron. 47(5), 577–590 (2011).
[Crossref]

2010 (5)

A. Semenov, O. Cojocari, H.-W. Hübers, F. Song, A. Klushin, and A.-S. Müller, “Application of Zero-Bias Quasi-Optical Schottky-Diode Detectors for Monitoring Short-Pulse and Weak Terahertz Radiation,” IEEE Electron Device Lett. 31(7), 674–676 (2010).
[Crossref]

B. Szentpáli, P. Basa, P. Fürjes, G. Battistig, I. Bársony, G. Károlyi, T. Berceli, V. Rymanov, and A. Stöhr, “Thermopile antennas for detection of millimeter waves,” Appl. Phys. Lett. 96(13), 133507 (2010).
[Crossref]

V. Giannini, A. Berrier, S. A. Maier, J. A. Sánchez-Gil, and J. G. Rivas, “Scattering efficiency and near field enhancement of active semiconductor plasmonic antennas at terahertz frequencies,” Opt. Express 18(3), 2797 (2010).
[Crossref]

X. Wang, A. A. Belyanin, S. A. Crooker, D. M. Mittleman, and J. Kono, “Interference-induced terahertz transparency in a semiconductor magneto-plasma,” Nat. Phys. 6(2), 126–130 (2010).
[Crossref]

F. Sizov and A. Rogalski, “THz detectors,” Prog. Quantum Electron. 34(5), 278–347 (2010).
[Crossref]

2009 (1)

2007 (2)

M. McFadden and W. R. Scott, “Analysis of the Equiangular Spiral Antenna on a Dielectric Substrate,” IEEE Trans. Antennas Propag. 55(11), 3163–3171 (2007).
[Crossref]

G. Jolley, L. Fu, H. H. Tan, and C. Jagadish, “Influence of quantum well and barrier composition on the spectral behavior of InGaAs quantum dots-in-a-well infrared photodetectors,” Appl. Phys. Lett. 91(17), 173508 (2007).
[Crossref]

2005 (1)

V. I. Shashkin, V. L. Vaks, V. M. Danil’tsev, A. V. Maslovsky, A. V. Murel, S. D. Nikiforov, O. I. Khrykin, and Y. I. Chechenin, “Microwave Detectors Based on Low-Barrier Planar Schottky Diodes and Their Characteristics,” Radiophys. Quantum Electron. 48(6), 485–490 (2005).
[Crossref]

2002 (1)

D. Hofstetter, M. Beck, and J. Faist, “Quantum-cascade-laser structures as photodetectors,” Appl. Phys. Lett. 81(15), 2683–2685 (2002).
[Crossref]

1959 (1)

J. Dyson, “The equiangular spiral antenna,” IRE Trans. Antennas Propag. 7(2), 181–187 (1959).
[Crossref]

1956 (1)

K. Wertheim, “Carrier Lifetime in Indium Antimonide,” Phys. Rev. 104(3), 662–664 (1956).
[Crossref]

Adhikary, S.

A. Haddadi, A. Dehzangi, R. Chevallier, S. Adhikary, and M. Razeghi, “Bias–selectable nBn dual–band long–/very long–wavelength infrared photodetectors based on InAs/InAs1−xSbx/AlAs1−xSbx type–II superlattices,” Sci. Rep. 7(1), 3379 (2017).
[Crossref]

Andrews, A. M.

B. Schwarz, P. Reininger, A. Harrer, D. MacFarland, H. Detz, A. M. Andrews, W. Schrenk, and G. Strasser, “The limit of quantum cascade detectors: A single period device,” Appl. Phys. Lett. 111(6), 061107 (2017).
[Crossref]

Ang, S. S.

S. M. Hanham, A. I. Fernández-Domínguez, J. H. Teng, S. S. Ang, K. P. Lim, S. F. Yoon, C. Y. Ngo, N. Klein, J. B. Pendry, and S. A. Maier, “Broadband Terahertz Plasmonic Response of Touching InSb Disks,” Adv. Mater. 24(35), OP226–OP230 (2012).
[Crossref]

Balanis, C. A.

C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. (John Wiley & Sons: New York, 2005).

Bársony, I.

B. Szentpáli, P. Basa, P. Fürjes, G. Battistig, I. Bársony, G. Károlyi, T. Berceli, V. Rymanov, and A. Stöhr, “Thermopile antennas for detection of millimeter waves,” Appl. Phys. Lett. 96(13), 133507 (2010).
[Crossref]

Basa, P.

B. Szentpáli, P. Basa, P. Fürjes, G. Battistig, I. Bársony, G. Károlyi, T. Berceli, V. Rymanov, and A. Stöhr, “Thermopile antennas for detection of millimeter waves,” Appl. Phys. Lett. 96(13), 133507 (2010).
[Crossref]

Battistig, G.

B. Szentpáli, P. Basa, P. Fürjes, G. Battistig, I. Bársony, G. Károlyi, T. Berceli, V. Rymanov, and A. Stöhr, “Thermopile antennas for detection of millimeter waves,” Appl. Phys. Lett. 96(13), 133507 (2010).
[Crossref]

Beck, M.

D. Palaferri, Y. Todorov, A. Bigioli, A. Mottaghizadeh, D. Gacemi, A. Calabrese, A. Vasanelli, L. Li, A. G. Davies, E. H. Linfield, F. Kapsalidis, M. Beck, J. Faist, and C. Sirtori, “Room-temperature nine-µm-wavelength photodetectors and GHz-frequency heterodyne receivers,” Nature 556(7699), 85–88 (2018).
[Crossref]

D. Hofstetter, M. Beck, and J. Faist, “Quantum-cascade-laser structures as photodetectors,” Appl. Phys. Lett. 81(15), 2683–2685 (2002).
[Crossref]

Bello, K. D.

P. Tcheg, K. D. Bello, and D. Pouhe, “A planar equiangular spiral antenna array for the V-/W-band,” in2017 11th European Conference on Antennas and Propagation (EUCAP) (IEEE, 2017), pp. 1148–1152.

Belyanin, A. A.

X. Wang, A. A. Belyanin, S. A. Crooker, D. M. Mittleman, and J. Kono, “Interference-induced terahertz transparency in a semiconductor magneto-plasma,” Nat. Phys. 6(2), 126–130 (2010).
[Crossref]

Berceli, T.

B. Szentpáli, P. Basa, P. Fürjes, G. Battistig, I. Bársony, G. Károlyi, T. Berceli, V. Rymanov, and A. Stöhr, “Thermopile antennas for detection of millimeter waves,” Appl. Phys. Lett. 96(13), 133507 (2010).
[Crossref]

Berrier, A.

Berry, C. W.

C. W. Berry, N. Wang, M. R. Hashemi, M. Unlu, and M. Jarrahi, “Significant performance enhancement in photoconductive terahertz optoelectronics by incorporating plasmonic contact electrodes,” Nat. Commun. 4(1), 1622 (2013).
[Crossref]

Bhattarai, K.

Bigioli, A.

D. Palaferri, Y. Todorov, A. Bigioli, A. Mottaghizadeh, D. Gacemi, A. Calabrese, A. Vasanelli, L. Li, A. G. Davies, E. H. Linfield, F. Kapsalidis, M. Beck, J. Faist, and C. Sirtori, “Room-temperature nine-µm-wavelength photodetectors and GHz-frequency heterodyne receivers,” Nature 556(7699), 85–88 (2018).
[Crossref]

Boland, J. L.

K. Peng, P. Parkinson, L. Fu, Q. Gao, N. Jiang, Y.-N. Guo, F. Wang, H. J. Joyce, J. L. Boland, H. H. Tan, C. Jagadish, and M. B. Johnston, “Single Nanowire Photoconductive Terahertz Detectors,” Nano Lett. 15(1), 206–210 (2015).
[Crossref]

Bousseksou, A.

A. Jollivet, B. Hinkov, S. Pirotta, H. Hoang, S. Derelle, J. Jaeck, M. Tchernycheva, R. Colombelli, A. Bousseksou, M. Hugues, N. Le Biavan, J. Tamayo-Arriola, M. Montes Bajo, L. Rigutti, A. Hierro, G. Strasser, J.-M. Chauveau, and F. H. Julien, “Short infrared wavelength quantum cascade detectors based on m-plane ZnO/ZnMgO quantum wells,” Appl. Phys. Lett. 113(25), 251104 (2018).
[Crossref]

Burghoff, D.

Calabrese, A.

D. Palaferri, Y. Todorov, A. Bigioli, A. Mottaghizadeh, D. Gacemi, A. Calabrese, A. Vasanelli, L. Li, A. G. Davies, E. H. Linfield, F. Kapsalidis, M. Beck, J. Faist, and C. Sirtori, “Room-temperature nine-µm-wavelength photodetectors and GHz-frequency heterodyne receivers,” Nature 556(7699), 85–88 (2018).
[Crossref]

Chauveau, J.-M.

A. Jollivet, B. Hinkov, S. Pirotta, H. Hoang, S. Derelle, J. Jaeck, M. Tchernycheva, R. Colombelli, A. Bousseksou, M. Hugues, N. Le Biavan, J. Tamayo-Arriola, M. Montes Bajo, L. Rigutti, A. Hierro, G. Strasser, J.-M. Chauveau, and F. H. Julien, “Short infrared wavelength quantum cascade detectors based on m-plane ZnO/ZnMgO quantum wells,” Appl. Phys. Lett. 113(25), 251104 (2018).
[Crossref]

Chechenin, Y. I.

V. I. Shashkin, V. L. Vaks, V. M. Danil’tsev, A. V. Maslovsky, A. V. Murel, S. D. Nikiforov, O. I. Khrykin, and Y. I. Chechenin, “Microwave Detectors Based on Low-Barrier Planar Schottky Diodes and Their Characteristics,” Radiophys. Quantum Electron. 48(6), 485–490 (2005).
[Crossref]

Chevallier, R.

A. Haddadi, A. Dehzangi, R. Chevallier, S. Adhikary, and M. Razeghi, “Bias–selectable nBn dual–band long–/very long–wavelength infrared photodetectors based on InAs/InAs1−xSbx/AlAs1−xSbx type–II superlattices,” Sci. Rep. 7(1), 3379 (2017).
[Crossref]

Chu, J.

J. Chu and A. Sher, Device Physics of Narrow Gap Semiconductors (Springer New York, 2010).

Chua, S. J.

L. Deng, J. Teng, H. Liu, Q. Y. Wu, J. Tang, X. Zhang, S. A. Maier, K. P. Lim, C. Y. Ngo, S. F. Yoon, and S. J. Chua, “Direct Optical Tuning of the Terahertz Plasmonic Response of InSb Subwavelength Gratings,” Adv. Opt. Mater. 1(2), 128–132 (2013).
[Crossref]

Cojocari, O.

A. Semenov, O. Cojocari, H.-W. Hübers, F. Song, A. Klushin, and A.-S. Müller, “Application of Zero-Bias Quasi-Optical Schottky-Diode Detectors for Monitoring Short-Pulse and Weak Terahertz Radiation,” IEEE Electron Device Lett. 31(7), 674–676 (2010).
[Crossref]

Colombelli, R.

A. Jollivet, B. Hinkov, S. Pirotta, H. Hoang, S. Derelle, J. Jaeck, M. Tchernycheva, R. Colombelli, A. Bousseksou, M. Hugues, N. Le Biavan, J. Tamayo-Arriola, M. Montes Bajo, L. Rigutti, A. Hierro, G. Strasser, J.-M. Chauveau, and F. H. Julien, “Short infrared wavelength quantum cascade detectors based on m-plane ZnO/ZnMgO quantum wells,” Appl. Phys. Lett. 113(25), 251104 (2018).
[Crossref]

Crooker, S. A.

X. Wang, A. A. Belyanin, S. A. Crooker, D. M. Mittleman, and J. Kono, “Interference-induced terahertz transparency in a semiconductor magneto-plasma,” Nat. Phys. 6(2), 126–130 (2010).
[Crossref]

Danil’tsev, V. M.

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K. Peng, P. Parkinson, L. Fu, Q. Gao, N. Jiang, Y.-N. Guo, F. Wang, H. J. Joyce, J. L. Boland, H. H. Tan, C. Jagadish, and M. B. Johnston, “Single Nanowire Photoconductive Terahertz Detectors,” Nano Lett. 15(1), 206–210 (2015).
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K. Peng, P. Parkinson, L. Fu, Q. Gao, N. Jiang, Y.-N. Guo, F. Wang, H. J. Joyce, J. L. Boland, H. H. Tan, C. Jagadish, and M. B. Johnston, “Single Nanowire Photoconductive Terahertz Detectors,” Nano Lett. 15(1), 206–210 (2015).
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A. Haddadi, A. Dehzangi, R. Chevallier, S. Adhikary, and M. Razeghi, “Bias–selectable nBn dual–band long–/very long–wavelength infrared photodetectors based on InAs/InAs1−xSbx/AlAs1−xSbx type–II superlattices,” Sci. Rep. 7(1), 3379 (2017).
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A. Jollivet, B. Hinkov, S. Pirotta, H. Hoang, S. Derelle, J. Jaeck, M. Tchernycheva, R. Colombelli, A. Bousseksou, M. Hugues, N. Le Biavan, J. Tamayo-Arriola, M. Montes Bajo, L. Rigutti, A. Hierro, G. Strasser, J.-M. Chauveau, and F. H. Julien, “Short infrared wavelength quantum cascade detectors based on m-plane ZnO/ZnMgO quantum wells,” Appl. Phys. Lett. 113(25), 251104 (2018).
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K. Peng, P. Parkinson, L. Fu, Q. Gao, N. Jiang, Y.-N. Guo, F. Wang, H. J. Joyce, J. L. Boland, H. H. Tan, C. Jagadish, and M. B. Johnston, “Single Nanowire Photoconductive Terahertz Detectors,” Nano Lett. 15(1), 206–210 (2015).
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I. McKerracher, J. Wong-Leung, G. Jolley, L. Fu, H. H. Tan, and C. Jagadish, “Selective Intermixing of InGaAs/GaAs Quantum Dot Infrared Photodetectors,” IEEE J. Quantum Electron. 47(5), 577–590 (2011).
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K. Peng, P. Parkinson, L. Fu, Q. Gao, N. Jiang, Y.-N. Guo, F. Wang, H. J. Joyce, J. L. Boland, H. H. Tan, C. Jagadish, and M. B. Johnston, “Single Nanowire Photoconductive Terahertz Detectors,” Nano Lett. 15(1), 206–210 (2015).
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I. McKerracher, J. Wong-Leung, G. Jolley, L. Fu, H. H. Tan, and C. Jagadish, “Selective Intermixing of InGaAs/GaAs Quantum Dot Infrared Photodetectors,” IEEE J. Quantum Electron. 47(5), 577–590 (2011).
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Lau, K.

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A. Jollivet, B. Hinkov, S. Pirotta, H. Hoang, S. Derelle, J. Jaeck, M. Tchernycheva, R. Colombelli, A. Bousseksou, M. Hugues, N. Le Biavan, J. Tamayo-Arriola, M. Montes Bajo, L. Rigutti, A. Hierro, G. Strasser, J.-M. Chauveau, and F. H. Julien, “Short infrared wavelength quantum cascade detectors based on m-plane ZnO/ZnMgO quantum wells,” Appl. Phys. Lett. 113(25), 251104 (2018).
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Linfield, E. H.

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B. Schwarz, P. Reininger, A. Harrer, D. MacFarland, H. Detz, A. M. Andrews, W. Schrenk, and G. Strasser, “The limit of quantum cascade detectors: A single period device,” Appl. Phys. Lett. 111(6), 061107 (2017).
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I. McKerracher, J. Wong-Leung, G. Jolley, L. Fu, H. H. Tan, and C. Jagadish, “Selective Intermixing of InGaAs/GaAs Quantum Dot Infrared Photodetectors,” IEEE J. Quantum Electron. 47(5), 577–590 (2011).
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X. Wang, A. A. Belyanin, S. A. Crooker, D. M. Mittleman, and J. Kono, “Interference-induced terahertz transparency in a semiconductor magneto-plasma,” Nat. Phys. 6(2), 126–130 (2010).
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A. Jollivet, B. Hinkov, S. Pirotta, H. Hoang, S. Derelle, J. Jaeck, M. Tchernycheva, R. Colombelli, A. Bousseksou, M. Hugues, N. Le Biavan, J. Tamayo-Arriola, M. Montes Bajo, L. Rigutti, A. Hierro, G. Strasser, J.-M. Chauveau, and F. H. Julien, “Short infrared wavelength quantum cascade detectors based on m-plane ZnO/ZnMgO quantum wells,” Appl. Phys. Lett. 113(25), 251104 (2018).
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D. Palaferri, Y. Todorov, A. Bigioli, A. Mottaghizadeh, D. Gacemi, A. Calabrese, A. Vasanelli, L. Li, A. G. Davies, E. H. Linfield, F. Kapsalidis, M. Beck, J. Faist, and C. Sirtori, “Room-temperature nine-µm-wavelength photodetectors and GHz-frequency heterodyne receivers,” Nature 556(7699), 85–88 (2018).
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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic of the InSb-based dual-band photodetector on sapphire substrate. (b) Cross-section of the central Au-InSb-Au MSM. (c) The SEM of the detector. The scale bar represents 200 µm.
Fig. 2.
Fig. 2. Schematic of measurement set-up for millimeter wave.
Fig. 3.
Fig. 3. (a) Spectral response of the detector at IR range. The inset is the typical current-voltage characteristic curve of the detector. (b-c) Blackbody responsivity (b) and Detectivity (c) of the device at different bias current. (d) Amplitude-frequency response of the detector in IR range. (e) The typical response waveform of the detector at 2.94 µm. (f) Rise speed of the detector at 2.94 µm.
Fig. 4.
Fig. 4. Responsivity (a) and NEP (b) of the detector as functions of bias current under 167 GHz radiation. Spectral responsivity (c) and NEP (d) of the detector under radiations from 164 GHz to 174 GHz under a bias current of 10 mA.
Fig. 5.
Fig. 5. (a) Amplitude-frequency response of the detector at 167 GHz. The inset is the Amplitude-frequency response of the Golay cell detector under the same radiation. (b) Typical response waveform of the detector to the 167 GHz radiations with modulation frequency of 10 KHz. (c) Rise time of the detector.
Fig. 6.
Fig. 6. (a) Spectral response of the detector from 164 GHz to 174 GHz under pumping of 1064 nm IR laser with different pumping power (different laser current). (b) The photovoltage signal of the detector at 167 GHz under different pumping power. (c) The simulated plasmonic intensity distribution at different plasma frequency. (d) Schematic of IR-millimeter switch by using this detector.

Equations (6)

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R = V / ( p A ) = V A G R G / ( V G A ) ,
N E P = v n / R ,
v n = ( v t 2 + v b 2 ) 1 / 2 = ( 4 k B T r + 2 q I d r 2 ) 1 / 2 ,
ε ( ω ) = ε ε 0 [ 1 ω p 2 / ( ω 2 + i ω ω τ ) ] ,
ω p 2 = q 2 n / ( m ε ε 0 ) ,
y = A e x p ( x / B )   + C ,