Abstract

Active plasmonic waveguiding has become a key requirement for designing and implementing nanophotonic devices. We study theoretically the performance of an Au/GaSb-based, metal–insulator–semiconductor (MIS) structure acting as a hybrid electrically pumped waveguide with gain. The surface-plasmon polariton (SPP) mode supported by this configuration is analyzed in the third telecommunication window and discussed in detail. Changes in the effective mode index, confinement factor and effective mode area are illustrated for different core widths and layer thicknesses. Electrical behavior of the MIS junction is analyzed using a self-consistent numerical technique and used to study variations in the material and model gains within the semiconducting region of the device. Our results indicate the possibility of achieving low loss SPP propagation while maintaining a strong field confinement.

© 2014 Optical Society of America

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    [CrossRef]
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  50. M. Alavi, D. K. Reinhard, C. C. W. Yu, “Minority-carrier injection in PtSi Schottky-barrier diodes at high current densities,” IEEE Trans. Electron Dev. 34(5), 1134–1140 (1987).
    [CrossRef]
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    [CrossRef] [PubMed]

2013

T. M. Wijesinghe, M. Premaratne, “Surface plasmon polaritons propagation through a Schottky junction: influence of the inversion layer,” IEEE Photonics J. 5(2), 4800216 (2013).
[CrossRef]

C. Wang, H. J. Qu, W. X. Chen, G. Z. Ran, H. Y. Yu, B. Niu, J. Q. Pan, W. Wang, “Polarization of the edge emission from Ag/InGaAsP Schottky plasmonic diode,” Appl. Phys. Lett. 102(6), 061112 (2013).
[CrossRef]

S. Belan, S. Vergeles, P. Vorobev, “Adjustable subwavelength localization in a hybrid plasmonic waveguide,” Opt. Express 21, 7427–7438 (2013).
[CrossRef] [PubMed]

2012

T. Wijesinghe, M. Premaratne, “Dispersion relation for surface plasmon polaritons on a schottky junction,” Opt. Express 20(7), 7151–7164 (2012).
[CrossRef] [PubMed]

D. Y. Fedyanin, “Toward an electrically pumped spaser,” Opt. Lett. 37(3), 404–406 (2012).
[CrossRef] [PubMed]

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[CrossRef] [PubMed]

2011

J. Zhang, L. Cai, W. Bai, Y. Xu, G. Song, “Hybrid plasmonic waveguide with gain medium for lossless propagation with nanoscale confinement,” Opt. Lett. 36(12), 2312–2314 (2011).
[CrossRef] [PubMed]

D. Dai, Y. Shi, S. He, L. Wosinski, L. Thylen, “Gain enhancement in a hybrid plasmonic nano-waveguide with a low-index or high-index gain medium,” Opt. Express 19(14), 12925–12936 (2011).
[CrossRef] [PubMed]

I. B. Udagedara, I. D. Rukhlenko, M. Premaratne, “Complex–ω approach versus complex–k approach in description of gain–assisted SPP propagation along linear chains of metallic nano spheres,” Phys. Rev. B 83, 115451 (2011).
[CrossRef]

D. Y. Fedyanin, A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express 19, 12524–12531 (2011).
[CrossRef] [PubMed]

V. J. Sorger, Z. Ye, R. F. Oulton, Y. W. G. Bartal, X. Yin, X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

2010

2009

2008

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

R. F. Oulton, G. Bartal, D. F. P. Pile, X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10(10), 105018 (2008).
[CrossRef]

2007

R. Kirchain, L. Kimerling, “A roadmap for nanophotonics,” Nat. Photonics 1(6), 303–305 (2007).
[CrossRef]

T. Holmgaard, S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007).
[CrossRef]

A. V. Krasavin, A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 90(21), 211101 (2007).
[CrossRef]

2006

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006).
[CrossRef]

E. Dulkeith, F. Xia, L. Schares, W. Green, Y. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express 14, 3853–3863 (2006).
[CrossRef] [PubMed]

2005

J. Seidel, S. Grafstrom, L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94(17), 177401 (2005).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

2004

2003

J. Darabi, K. Ekula, “Development of a chip-integrated micro cooling device,” Microelectr. J. 34(11), 1067–1074 (2003).
[CrossRef]

1998

1995

Z. M. Li, “Two-dimensional numerical simulation of semiconductor lasers,” Prog. Electromagn. Res. 11, 301–344 (1995).

1992

O. J. Martin, G. L. Bona, P Wolf, “Thermal behavior of visible AlGaInP-GaInP ridge laser diodes,” IEEE J. Quantum Electron. 28(11), 2582–2588 (1992).
[CrossRef]

1990

K. Horio, H. Yanai, “Numerical modelling of heterojunctions including the thermionic emission mechanism at the heterojunction interface,” IEEE Trans. Electron. Dev. 37, 1093–1098 (1990).
[CrossRef]

1987

M. Alavi, D. K. Reinhard, C. C. W. Yu, “Minority-carrier injection in PtSi Schottky-barrier diodes at high current densities,” IEEE Trans. Electron Dev. 34(5), 1134–1140 (1987).
[CrossRef]

1984

W. Nakwaski, “Dynamical thermal properties of stripe-geometry laser diodes,” IEEE Proc. Electron Dev. 131(3), 94 (1984).

1983

M. S. Lundstrom, R. J. Schuelke, “Numerical analysis of heterostructure semiconductor devices,” IEEE Trans. Electron. Dev. 30, 1151–1159 (1983)
[CrossRef]

1973

S. Kameda, W. Carr, “Analysis of proposed MIS laser structures,” IEEE J. Quantum Electron 9(2), 374–378 (1973).
[CrossRef]

M. A. Green, J. Shewchun, “Minority carrier effects upon small signal and steady-state properties of schottky diodes,” Solid State Electron., 16(10), 1141–1150 (1973).
[CrossRef]

1972

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1971

H. C. Card, B. L. Smith, “Green injection luminescence from forward-biased Au/GaP Schottky barriers,” J. Appl. Phys. 42(13), 5863–5865 (1971).
[CrossRef]

J. N. Walpole, K. W. Nill, “Capacitance-voltage characteristics of metal barriers on p PbTe and p InAs: Effects of the inversion layer,” J. Appl. Phys. 42(13), 5609–5617 (1971).
[CrossRef]

1970

K. W. Nill, A. R. Calawa, T. C. Harman, J. N. Walpole, “Laser emission barriers on PbTe and PbSnTe,” Appl. Phys. Lett. 16(10), 375–377 (1970).
[CrossRef]

Adachi, S.

S. Adachi, Properties of Group-IV, III-V and II-VI Semiconductors (John Wiley, 1950).

Agrawal, G. P.

M. Premaratne, G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University, 2011).
[CrossRef]

Akbari, A.

Alavi, M.

M. Alavi, D. K. Reinhard, C. C. W. Yu, “Minority-carrier injection in PtSi Schottky-barrier diodes at high current densities,” IEEE Trans. Electron Dev. 34(5), 1134–1140 (1987).
[CrossRef]

Arsenin, A. V.

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[CrossRef] [PubMed]

D. Y. Fedyanin, A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express 19, 12524–12531 (2011).
[CrossRef] [PubMed]

Bai, W.

Banerjee, K.

K. Banerjee, S. C. Lin, N. Srivastava, “Electrothermal engineering in the nanometer era: from devices and interconnects to circuits and systems,” in Proceedings of ASP-DAC (IEEE, 2006), pp. 223–230.
[CrossRef]

Bartal, G.

R. F. Oulton, G. Bartal, D. F. P. Pile, X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10(10), 105018 (2008).
[CrossRef]

Bartal, Y. W. G.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. W. G. Bartal, X. Yin, X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

Belan, S.

Berini, P.

A. Akbari, R. N. Tait, P. Berini, “Surface plasmon waveguide Schottky detector,” Opt. Express 18(8), 8505–8514 (2010).
[CrossRef] [PubMed]

I. D. Leon, P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4, 382–388 (2010).
[CrossRef]

Bian, Y. S.

Bona, G. L.

O. J. Martin, G. L. Bona, P Wolf, “Thermal behavior of visible AlGaInP-GaInP ridge laser diodes,” IEEE J. Quantum Electron. 28(11), 2582–2588 (1992).
[CrossRef]

Bouhelier, A.

J. Grandidier, G. C. D. Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

T. Holmgaard, S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007).
[CrossRef]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Cai, L.

Calawa, A. R.

K. W. Nill, A. R. Calawa, T. C. Harman, J. N. Walpole, “Laser emission barriers on PbTe and PbSnTe,” Appl. Phys. Lett. 16(10), 375–377 (1970).
[CrossRef]

Capasso, F.

Card, H. C.

H. C. Card, B. L. Smith, “Green injection luminescence from forward-biased Au/GaP Schottky barriers,” J. Appl. Phys. 42(13), 5863–5865 (1971).
[CrossRef]

Carr, W.

S. Kameda, W. Carr, “Analysis of proposed MIS laser structures,” IEEE J. Quantum Electron 9(2), 374–378 (1973).
[CrossRef]

Chen, W. X.

C. Wang, H. J. Qu, W. X. Chen, G. Z. Ran, H. Y. Yu, B. Niu, J. Q. Pan, W. Wang, “Polarization of the edge emission from Ag/InGaAsP Schottky plasmonic diode,” Appl. Phys. Lett. 102(6), 061112 (2013).
[CrossRef]

Cho, A. Y.

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Chuang, S. L.

S. L. Chuang, Physics of Photonic Devices, 2nd ed. (John Wiley, 2009).

Dai, D.

Dai, D. X.

Darabi, J.

J. Darabi, K. Ekula, “Development of a chip-integrated micro cooling device,” Microelectr. J. 34(11), 1067–1074 (2003).
[CrossRef]

Dereux, A.

J. Grandidier, G. C. D. Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009).
[CrossRef] [PubMed]

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Dulkeith, E.

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Ekula, K.

J. Darabi, K. Ekula, “Development of a chip-integrated micro cooling device,” Microelectr. J. 34(11), 1067–1074 (2003).
[CrossRef]

Eng, L.

J. Seidel, S. Grafstrom, L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94(17), 177401 (2005).
[CrossRef] [PubMed]

Fedyanin, D. Y.

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[CrossRef] [PubMed]

D. Y. Fedyanin, “Toward an electrically pumped spaser,” Opt. Lett. 37(3), 404–406 (2012).
[CrossRef] [PubMed]

D. Y. Fedyanin, A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express 19, 12524–12531 (2011).
[CrossRef] [PubMed]

Feng, C.

Finot, C.

J. Grandidier, G. C. D. Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009).
[CrossRef] [PubMed]

Francs, G. C. D.

J. Grandidier, G. C. D. Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009).
[CrossRef] [PubMed]

Genov, D. A.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

Gmachl, C.

Grafstrom, S.

J. Seidel, S. Grafstrom, L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94(17), 177401 (2005).
[CrossRef] [PubMed]

Gramotnev, D. K.

Grandidier, J.

J. Grandidier, G. C. D. Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009).
[CrossRef] [PubMed]

Green, M. A.

M. A. Green, J. Shewchun, “Minority carrier effects upon small signal and steady-state properties of schottky diodes,” Solid State Electron., 16(10), 1141–1150 (1973).
[CrossRef]

Green, W.

Hall, T.

R. Millett, J. Wheeldon, T. Hall, H. Schriemer, “Towards modelling semiconductor heterojunctions,” in Proceedings of the COMSOL Users Conference, Boston, 2006.

Handapangoda, D.

Harman, T. C.

K. W. Nill, A. R. Calawa, T. C. Harman, J. N. Walpole, “Laser emission barriers on PbTe and PbSnTe,” Appl. Phys. Lett. 16(10), 375–377 (1970).
[CrossRef]

He, S.

He, S. L.

Holmgaard, T.

T. Holmgaard, S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007).
[CrossRef]

Horio, K.

K. Horio, H. Yanai, “Numerical modelling of heterojunctions including the thermionic emission mechanism at the heterojunction interface,” IEEE Trans. Electron. Dev. 37, 1093–1098 (1990).
[CrossRef]

Hutchinson, A. L.

Jagadish, C.

Johnson, P. B.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Kameda, S.

S. Kameda, W. Carr, “Analysis of proposed MIS laser structures,” IEEE J. Quantum Electron 9(2), 374–378 (1973).
[CrossRef]

Kimerling, L.

R. Kirchain, L. Kimerling, “A roadmap for nanophotonics,” Nat. Photonics 1(6), 303–305 (2007).
[CrossRef]

Kirchain, R.

R. Kirchain, L. Kimerling, “A roadmap for nanophotonics,” Nat. Photonics 1(6), 303–305 (2007).
[CrossRef]

Krasavin, A. V.

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[CrossRef] [PubMed]

A. V. Krasavin, A. V. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express 18(11), 11791–11799 (2010).
[CrossRef] [PubMed]

A. V. Krasavin, A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 90(21), 211101 (2007).
[CrossRef]

Lee, S. Y.

Leon, I. D.

I. D. Leon, P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4, 382–388 (2010).
[CrossRef]

Li, T.

Li, Y.

Li, Z. M.

Z. M. Li, “Two-dimensional numerical simulation of semiconductor lasers,” Prog. Electromagn. Res. 11, 301–344 (1995).

Lin, S. C.

K. Banerjee, S. C. Lin, N. Srivastava, “Electrothermal engineering in the nanometer era: from devices and interconnects to circuits and systems,” in Proceedings of ASP-DAC (IEEE, 2006), pp. 223–230.
[CrossRef]

Lundstrom, M. S.

M. S. Lundstrom, R. J. Schuelke, “Numerical analysis of heterostructure semiconductor devices,” IEEE Trans. Electron. Dev. 30, 1151–1159 (1983)
[CrossRef]

Maier, S. A.

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006).
[CrossRef]

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Markey, L.

J. Grandidier, G. C. D. Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009).
[CrossRef] [PubMed]

Martin, O. J.

O. J. Martin, G. L. Bona, P Wolf, “Thermal behavior of visible AlGaInP-GaInP ridge laser diodes,” IEEE J. Quantum Electron. 28(11), 2582–2588 (1992).
[CrossRef]

Massenot, S.

J. Grandidier, G. C. D. Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009).
[CrossRef] [PubMed]

Mei, T.

Millett, R.

R. Millett, J. Wheeldon, T. Hall, H. Schriemer, “Towards modelling semiconductor heterojunctions,” in Proceedings of the COMSOL Users Conference, Boston, 2006.

Nakwaski, W.

W. Nakwaski, “Dynamical thermal properties of stripe-geometry laser diodes,” IEEE Proc. Electron Dev. 131(3), 94 (1984).

Ng, K. K.

S. M. Sze, K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (John Wiley, 2006).

Nill, K. W.

J. N. Walpole, K. W. Nill, “Capacitance-voltage characteristics of metal barriers on p PbTe and p InAs: Effects of the inversion layer,” J. Appl. Phys. 42(13), 5609–5617 (1971).
[CrossRef]

K. W. Nill, A. R. Calawa, T. C. Harman, J. N. Walpole, “Laser emission barriers on PbTe and PbSnTe,” Appl. Phys. Lett. 16(10), 375–377 (1970).
[CrossRef]

Niu, B.

C. Wang, H. J. Qu, W. X. Chen, G. Z. Ran, H. Y. Yu, B. Niu, J. Q. Pan, W. Wang, “Polarization of the edge emission from Ag/InGaAsP Schottky plasmonic diode,” Appl. Phys. Lett. 102(6), 061112 (2013).
[CrossRef]

Oulton, R. F.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. W. G. Bartal, X. Yin, X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

R. F. Oulton, G. Bartal, D. F. P. Pile, X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10(10), 105018 (2008).
[CrossRef]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

Ozbay, E.

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

Pan, J. Q.

C. Wang, H. J. Qu, W. X. Chen, G. Z. Ran, H. Y. Yu, B. Niu, J. Q. Pan, W. Wang, “Polarization of the edge emission from Ag/InGaAsP Schottky plasmonic diode,” Appl. Phys. Lett. 102(6), 061112 (2013).
[CrossRef]

Pile, D. F. P.

R. F. Oulton, G. Bartal, D. F. P. Pile, X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10(10), 105018 (2008).
[CrossRef]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

D. F. P. Pile, D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29(10), 1069–1071 (2004).
[CrossRef] [PubMed]

Premaratne, M.

T. M. Wijesinghe, M. Premaratne, “Surface plasmon polaritons propagation through a Schottky junction: influence of the inversion layer,” IEEE Photonics J. 5(2), 4800216 (2013).
[CrossRef]

T. Wijesinghe, M. Premaratne, “Dispersion relation for surface plasmon polaritons on a schottky junction,” Opt. Express 20(7), 7151–7164 (2012).
[CrossRef] [PubMed]

I. B. Udagedara, I. D. Rukhlenko, M. Premaratne, “Complex–ω approach versus complex–k approach in description of gain–assisted SPP propagation along linear chains of metallic nano spheres,” Phys. Rev. B 83, 115451 (2011).
[CrossRef]

D. Handapangoda, I. D. Rukhlenko, M. Premaratne, C. Jagadish, “Optimization of gain–assisted waveguiding in metal–dielectric nanowires,” Opt. Lett. 35(24), 4190–4192 (2010).
[CrossRef] [PubMed]

M. Premaratne, G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University, 2011).
[CrossRef]

Qu, H. J.

C. Wang, H. J. Qu, W. X. Chen, G. Z. Ran, H. Y. Yu, B. Niu, J. Q. Pan, W. Wang, “Polarization of the edge emission from Ag/InGaAsP Schottky plasmonic diode,” Appl. Phys. Lett. 102(6), 061112 (2013).
[CrossRef]

Ran, G. Z.

C. Wang, H. J. Qu, W. X. Chen, G. Z. Ran, H. Y. Yu, B. Niu, J. Q. Pan, W. Wang, “Polarization of the edge emission from Ag/InGaAsP Schottky plasmonic diode,” Appl. Phys. Lett. 102(6), 061112 (2013).
[CrossRef]

Reinhard, D. K.

M. Alavi, D. K. Reinhard, C. C. W. Yu, “Minority-carrier injection in PtSi Schottky-barrier diodes at high current densities,” IEEE Trans. Electron Dev. 34(5), 1134–1140 (1987).
[CrossRef]

Rukhlenko, I. D.

I. B. Udagedara, I. D. Rukhlenko, M. Premaratne, “Complex–ω approach versus complex–k approach in description of gain–assisted SPP propagation along linear chains of metallic nano spheres,” Phys. Rev. B 83, 115451 (2011).
[CrossRef]

D. Handapangoda, I. D. Rukhlenko, M. Premaratne, C. Jagadish, “Optimization of gain–assisted waveguiding in metal–dielectric nanowires,” Opt. Lett. 35(24), 4190–4192 (2010).
[CrossRef] [PubMed]

Schares, L.

Schriemer, H.

R. Millett, J. Wheeldon, T. Hall, H. Schriemer, “Towards modelling semiconductor heterojunctions,” in Proceedings of the COMSOL Users Conference, Boston, 2006.

Schuelke, R. J.

M. S. Lundstrom, R. J. Schuelke, “Numerical analysis of heterostructure semiconductor devices,” IEEE Trans. Electron. Dev. 30, 1151–1159 (1983)
[CrossRef]

Seidel, J.

J. Seidel, S. Grafstrom, L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94(17), 177401 (2005).
[CrossRef] [PubMed]

Shewchun, J.

M. A. Green, J. Shewchun, “Minority carrier effects upon small signal and steady-state properties of schottky diodes,” Solid State Electron., 16(10), 1141–1150 (1973).
[CrossRef]

Shi, Y.

Sirtori, C.

Sivco, D. L.

Smith, B. L.

H. C. Card, B. L. Smith, “Green injection luminescence from forward-biased Au/GaP Schottky barriers,” J. Appl. Phys. 42(13), 5863–5865 (1971).
[CrossRef]

Snowden, C. M.

C. M. Snowden, Introduction to Semiconductor Device Modelling (World Scientific, 1998).
[CrossRef]

Song, G.

Sorger, V. J.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. W. G. Bartal, X. Yin, X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

Srivastava, N.

K. Banerjee, S. C. Lin, N. Srivastava, “Electrothermal engineering in the nanometer era: from devices and interconnects to circuits and systems,” in Proceedings of ASP-DAC (IEEE, 2006), pp. 223–230.
[CrossRef]

Sze, S. M.

S. M. Sze, K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (John Wiley, 2006).

Tait, R. N.

Thylen, L.

Udagedara, I. B.

I. B. Udagedara, I. D. Rukhlenko, M. Premaratne, “Complex–ω approach versus complex–k approach in description of gain–assisted SPP propagation along linear chains of metallic nano spheres,” Phys. Rev. B 83, 115451 (2011).
[CrossRef]

Vergeles, S.

Vlasov, Y.

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Vorobev, P.

Walpole, J. N.

J. N. Walpole, K. W. Nill, “Capacitance-voltage characteristics of metal barriers on p PbTe and p InAs: Effects of the inversion layer,” J. Appl. Phys. 42(13), 5609–5617 (1971).
[CrossRef]

K. W. Nill, A. R. Calawa, T. C. Harman, J. N. Walpole, “Laser emission barriers on PbTe and PbSnTe,” Appl. Phys. Lett. 16(10), 375–377 (1970).
[CrossRef]

Wang, B.

Wang, C.

C. Wang, H. J. Qu, W. X. Chen, G. Z. Ran, H. Y. Yu, B. Niu, J. Q. Pan, W. Wang, “Polarization of the edge emission from Ag/InGaAsP Schottky plasmonic diode,” Appl. Phys. Lett. 102(6), 061112 (2013).
[CrossRef]

Wang, G. P.

Wang, L.

Wang, W.

C. Wang, H. J. Qu, W. X. Chen, G. Z. Ran, H. Y. Yu, B. Niu, J. Q. Pan, W. Wang, “Polarization of the edge emission from Ag/InGaAsP Schottky plasmonic diode,” Appl. Phys. Lett. 102(6), 061112 (2013).
[CrossRef]

Weeber, J. C.

J. Grandidier, G. C. D. Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009).
[CrossRef] [PubMed]

Wheeldon, J.

R. Millett, J. Wheeldon, T. Hall, H. Schriemer, “Towards modelling semiconductor heterojunctions,” in Proceedings of the COMSOL Users Conference, Boston, 2006.

Wijesinghe, T.

Wijesinghe, T. M.

T. M. Wijesinghe, M. Premaratne, “Surface plasmon polaritons propagation through a Schottky junction: influence of the inversion layer,” IEEE Photonics J. 5(2), 4800216 (2013).
[CrossRef]

Wolf, P

O. J. Martin, G. L. Bona, P Wolf, “Thermal behavior of visible AlGaInP-GaInP ridge laser diodes,” IEEE J. Quantum Electron. 28(11), 2582–2588 (1992).
[CrossRef]

Wosinski, L.

Xia, F.

Xu, Y.

Yanai, H.

K. Horio, H. Yanai, “Numerical modelling of heterojunctions including the thermionic emission mechanism at the heterojunction interface,” IEEE Trans. Electron. Dev. 37, 1093–1098 (1990).
[CrossRef]

Ye, Z.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. W. G. Bartal, X. Yin, X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

Yin, X.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. W. G. Bartal, X. Yin, X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

Yu, C. C. W.

M. Alavi, D. K. Reinhard, C. C. W. Yu, “Minority-carrier injection in PtSi Schottky-barrier diodes at high current densities,” IEEE Trans. Electron Dev. 34(5), 1134–1140 (1987).
[CrossRef]

Yu, H. Y.

C. Wang, H. J. Qu, W. X. Chen, G. Z. Ran, H. Y. Yu, B. Niu, J. Q. Pan, W. Wang, “Polarization of the edge emission from Ag/InGaAsP Schottky plasmonic diode,” Appl. Phys. Lett. 102(6), 061112 (2013).
[CrossRef]

Zayats, A. V.

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[CrossRef] [PubMed]

A. V. Krasavin, A. V. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express 18(11), 11791–11799 (2010).
[CrossRef] [PubMed]

A. V. Krasavin, A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 90(21), 211101 (2007).
[CrossRef]

Zhang, J.

Zhang, X.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. W. G. Bartal, X. Yin, X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

X. Zhang, Y. Li, T. Li, S. Y. Lee, C. Feng, L. Wang, T. Mei, “Gain-assisted propagation of surface plasmon polaritons via electrically pumped quantum wells,” Opt. Lett. 35(18), 3075–3077 (2010).
[CrossRef] [PubMed]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

R. F. Oulton, G. Bartal, D. F. P. Pile, X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10(10), 105018 (2008).
[CrossRef]

Zhao, X.

Zheng, Z.

Zhou, T.

Zhu, J. S.

Appl. Phys. Lett.

A. V. Krasavin, A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 90(21), 211101 (2007).
[CrossRef]

K. W. Nill, A. R. Calawa, T. C. Harman, J. N. Walpole, “Laser emission barriers on PbTe and PbSnTe,” Appl. Phys. Lett. 16(10), 375–377 (1970).
[CrossRef]

C. Wang, H. J. Qu, W. X. Chen, G. Z. Ran, H. Y. Yu, B. Niu, J. Q. Pan, W. Wang, “Polarization of the edge emission from Ag/InGaAsP Schottky plasmonic diode,” Appl. Phys. Lett. 102(6), 061112 (2013).
[CrossRef]

IEEE J. Quantum Electron

S. Kameda, W. Carr, “Analysis of proposed MIS laser structures,” IEEE J. Quantum Electron 9(2), 374–378 (1973).
[CrossRef]

IEEE J. Quantum Electron.

O. J. Martin, G. L. Bona, P Wolf, “Thermal behavior of visible AlGaInP-GaInP ridge laser diodes,” IEEE J. Quantum Electron. 28(11), 2582–2588 (1992).
[CrossRef]

IEEE Photonics J.

T. M. Wijesinghe, M. Premaratne, “Surface plasmon polaritons propagation through a Schottky junction: influence of the inversion layer,” IEEE Photonics J. 5(2), 4800216 (2013).
[CrossRef]

IEEE Proc. Electron Dev.

W. Nakwaski, “Dynamical thermal properties of stripe-geometry laser diodes,” IEEE Proc. Electron Dev. 131(3), 94 (1984).

IEEE Trans. Electron Dev.

M. Alavi, D. K. Reinhard, C. C. W. Yu, “Minority-carrier injection in PtSi Schottky-barrier diodes at high current densities,” IEEE Trans. Electron Dev. 34(5), 1134–1140 (1987).
[CrossRef]

IEEE Trans. Electron. Dev.

K. Horio, H. Yanai, “Numerical modelling of heterojunctions including the thermionic emission mechanism at the heterojunction interface,” IEEE Trans. Electron. Dev. 37, 1093–1098 (1990).
[CrossRef]

M. S. Lundstrom, R. J. Schuelke, “Numerical analysis of heterostructure semiconductor devices,” IEEE Trans. Electron. Dev. 30, 1151–1159 (1983)
[CrossRef]

J. Appl. Phys.

J. N. Walpole, K. W. Nill, “Capacitance-voltage characteristics of metal barriers on p PbTe and p InAs: Effects of the inversion layer,” J. Appl. Phys. 42(13), 5609–5617 (1971).
[CrossRef]

H. C. Card, B. L. Smith, “Green injection luminescence from forward-biased Au/GaP Schottky barriers,” J. Appl. Phys. 42(13), 5863–5865 (1971).
[CrossRef]

Microelectr. J.

J. Darabi, K. Ekula, “Development of a chip-integrated micro cooling device,” Microelectr. J. 34(11), 1067–1074 (2003).
[CrossRef]

Nano Lett.

J. Grandidier, G. C. D. Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009).
[CrossRef] [PubMed]

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[CrossRef] [PubMed]

Nat. Commun.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. W. G. Bartal, X. Yin, X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

Nat. Photonics

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

R. Kirchain, L. Kimerling, “A roadmap for nanophotonics,” Nat. Photonics 1(6), 303–305 (2007).
[CrossRef]

I. D. Leon, P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4, 382–388 (2010).
[CrossRef]

New J. Phys.

R. F. Oulton, G. Bartal, D. F. P. Pile, X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10(10), 105018 (2008).
[CrossRef]

Opt. Commun.

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006).
[CrossRef]

Opt. Express

D. X. Dai, S. L. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express 17(19), 16646–16653 (2009).
[CrossRef] [PubMed]

Y. S. Bian, Z. Zheng, X. Zhao, J. S. Zhu, T. Zhou, “Symmetric hybrid surface plasmon polariton waveguides for 3D photonic integration,” Opt. Express 17(23), 21320–21325 (2009).
[CrossRef] [PubMed]

A. V. Krasavin, A. V. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express 18(11), 11791–11799 (2010).
[CrossRef] [PubMed]

D. Dai, Y. Shi, S. He, L. Wosinski, L. Thylen, “Gain enhancement in a hybrid plasmonic nano-waveguide with a low-index or high-index gain medium,” Opt. Express 19(14), 12925–12936 (2011).
[CrossRef] [PubMed]

T. Wijesinghe, M. Premaratne, “Dispersion relation for surface plasmon polaritons on a schottky junction,” Opt. Express 20(7), 7151–7164 (2012).
[CrossRef] [PubMed]

D. Y. Fedyanin, A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express 19, 12524–12531 (2011).
[CrossRef] [PubMed]

S. Belan, S. Vergeles, P. Vorobev, “Adjustable subwavelength localization in a hybrid plasmonic waveguide,” Opt. Express 21, 7427–7438 (2013).
[CrossRef] [PubMed]

E. Dulkeith, F. Xia, L. Schares, W. Green, Y. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express 14, 3853–3863 (2006).
[CrossRef] [PubMed]

A. Akbari, R. N. Tait, P. Berini, “Surface plasmon waveguide Schottky detector,” Opt. Express 18(8), 8505–8514 (2010).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. B

I. B. Udagedara, I. D. Rukhlenko, M. Premaratne, “Complex–ω approach versus complex–k approach in description of gain–assisted SPP propagation along linear chains of metallic nano spheres,” Phys. Rev. B 83, 115451 (2011).
[CrossRef]

T. Holmgaard, S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007).
[CrossRef]

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Phys. Rev. Lett.

J. Seidel, S. Grafstrom, L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94(17), 177401 (2005).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Prog. Electromagn. Res.

Z. M. Li, “Two-dimensional numerical simulation of semiconductor lasers,” Prog. Electromagn. Res. 11, 301–344 (1995).

Science

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

Solid State Electron.

M. A. Green, J. Shewchun, “Minority carrier effects upon small signal and steady-state properties of schottky diodes,” Solid State Electron., 16(10), 1141–1150 (1973).
[CrossRef]

Other

M. Premaratne, G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University, 2011).
[CrossRef]

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

C. M. Snowden, Introduction to Semiconductor Device Modelling (World Scientific, 1998).
[CrossRef]

R. Millett, J. Wheeldon, T. Hall, H. Schriemer, “Towards modelling semiconductor heterojunctions,” in Proceedings of the COMSOL Users Conference, Boston, 2006.

S. Adachi, Properties of Group-IV, III-V and II-VI Semiconductors (John Wiley, 1950).

K. Banerjee, S. C. Lin, N. Srivastava, “Electrothermal engineering in the nanometer era: from devices and interconnects to circuits and systems,” in Proceedings of ASP-DAC (IEEE, 2006), pp. 223–230.
[CrossRef]

S. L. Chuang, Physics of Photonic Devices, 2nd ed. (John Wiley, 2009).

S. M. Sze, K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (John Wiley, 2006).

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Figures (9)

Fig. 1
Fig. 1

(a) Cross section of an hybrid waveguide with MIS contact in the x–y plane. External bias is applied across the terminals marked by an electrical contact on the metallic side and an ohmic contact on the semiconductor side. (b) Three-dimensional view of the structure showing notations and dimensions. Directions of the propagation vector associated with the SPP field are marked in blue.

Fig. 2
Fig. 2

Field components (a) Hx and (b) Ey of the SPP mode plotted in the x–y plane for wc = 50 nm, hs = 200 nm, and hm = 90 nm. (c) Hx and (d) Ey under the same conditions except for a wider nano-slot (wc = 200 nm). In both cases, the nano-slot is only 10 nm thick.

Fig. 3
Fig. 3

Left panel: Variations of the (a) real and (b) imaginary parts of effective mode index with core width wc for several choices of device parameters. Right panel: Confinement factor and effective mode area under the same operating conditions. In all cases curves for t = 0 show the behavior in the absence of a nano-slot.

Fig. 4
Fig. 4

(a) Schematic of the energy bands across an MIS Schottky junction. The symbols χs, ϕm, Evac, Fm, ϕB denote electron affinity, built-in potential, vacuum energy level, metal work function, and Fermi energy in equilibrium. (b) Energy band diagram of the structure in the vertical y direction under a forward bias of 1.05 V.

Fig. 5
Fig. 5

(a) Hole density, (b) electron density, (c) electron and hole current densities and (d) material gain distributions in the MIS junction along the y axis at a fixed plane x = 0 μm. Material parameters used are given in Table 1.

Fig. 6
Fig. 6

(a) The material gain spectrum of GaSb for different donor densities. (b) The variation of modal loss, modal gain and net gain with core width for Va = 1.05 V, hs = 200 nm and t = 10 nm.

Fig. 7
Fig. 7

The temperature distribution in the waveguide cross section (a) at 0.01 μs and (b) after 0.3 μs. The material parameters are listed in Table 2

Fig. 8
Fig. 8

(a) The time evolution of the internal temperature of the device for duration of 0.3 μs (the temperature rise of the device during the pulse operation of the device if assumes the pulse train width of 0.3 μs). (b) The temperature distribution along y-plane for different time durations. (c) The temperature distribution in the waveguide cross section after 0.3 μs after using heat sink materials along the sides that maintains T=293 K.

Fig. 9
Fig. 9

The frequency and group velocity dispersion of the MIS waveguide.

Tables (2)

Tables Icon

Table 1 The material parameters of GaSb and Au at T = 300 K and λ = 1.55 μm.

Tables Icon

Table 2 The material parameters for the thermal analysis.

Equations (13)

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2 E + ( k 0 2 ε β 2 ) E = 0 ,
Γ g = s E y H x * d x d y E y H x * d x d y ,
A eff = ( W d x d y ) 2 W 2 d x d y ,
W = 1 2 Re [ d ( ω ε ) d ω ] | E | 2 + 1 2 μ 0 | H | 2 ,
. ( ε φ ) = q ( p n + N d ) ,
. ( J n ) = q R n , . ( J p ) = q R p ,
J n = q μ n n φ + q D n n ,
J p = q μ p p φ q D p p ,
φ o = ( q φ B E g ) , n o = N c F 1 2 [ q ϕ B K T ] , p o = N v F 1 2 [ q ϕ B E g K T ] ,
J n | x = 0 = q v n ( n | x = 0 n o ) , J p | x = 0 = q v p ( p | x = 0 p o ) ,
R S R H = ( n p n i 2 ) τ p ( n + n i ) + τ n ( p + n i ) ,
g M = s g m E y H x * d x d y E y H x * d x d y .
T t s = 1 ρ d C p ( κ 2 T + Q )

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