Theory and experiment of submonolayer quantum-dot metal-cavity surface-emitting microlasers

Pengfei Qiao; Chien-Yao Lu; Dieter Bimberg; Shun Lien Chuang

doi:10.1364/OE.21.030336

Theory and experiment of submonolayer quantum-dot metal-cavity surface-emitting microlasers

Pengfei Qiao, Chien-Yao Lu, Dieter Bimberg, and Shun Lien Chuang

Optics Express
Vol. 21,
Issue 25,
pp. 30336-30349
(2013)
•https://doi.org/10.1364/OE.21.030336

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Pengfei Qiao, Chien-Yao Lu, Dieter Bimberg, and Shun Lien Chuang, "Theory and experiment of submonolayer quantum-dot metal-cavity surface-emitting microlasers," Opt. Express 21, 30336-30349 (2013)

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Related Topics
Table of Contents Category
- Lasers and Laser Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Semiconductor lasers (140.5960)
- Microcavity devices (140.3948)
- Vertical cavity surface emitting lasers (140.7260)
- Quantum-well, -wire and -dot devices (250.5590)

About this Article
History
- Original Manuscript: October 1, 2013
- Revised Manuscript: November 19, 2013
- Manuscript Accepted: November 19, 2013
- Published: December 3, 2013

Accessible

Open Access

Abstract

We present a theoretical model for metal-cavity submonolayer quantum-dot surface-emitting microlasers, which operate at room temperature under electrical injection. Size-dependent lasing characteristics are investigated experimentally and theoretically with device radius ranging from 5 μm to 0.5 μm. The quantum dot emission and cavity optical properties are used in a rate-equation model to study the laser light output power vs. current behavior. Our theory explains the observed size-dependent physics and provides a guide for future device size reduction.

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References

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Author
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Publication

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18, 2329–2330 (1979).
[Crossref]
K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18, 8790–8799 (2010).
[Crossref] [PubMed]
M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[Crossref]
C.-Y. Lu, C.-Y. Ni, M. Zhang, S. L. Chuang, and D. Bimberg, “Metal-cavity surface-emitting microlasers with size reduction: theory and experiment,” IEEE J. Sel. Top. Quantum Electron. 19, 1701809 (2013).
K. Ding, Z. Liu, L. Yin, H. Wang, R. Liu, M. T. Hill, M. J. H. Marell, P. J. van Veldhoven, R. Ntzel, and C. Z. Ning, “Electrical injection, continuous wave operation of subwavelength-metallic-cavity lasers at 260 K,” Appl. Phys. Lett. 98, 231108 (2011).
[Crossref]
D. Bimberg, “Quantum dot based nanophotonics and nanoelectronics,” Electron. Lett. 44, 168–171 (2008).
[Crossref]
S. S. Mikhrin, A. E. Zhukov, A. R. Kovsh, N. A. Maleev, V. M. Ustinov, Y. M. Shernyakov, I. P. Soshnikov, D. A. Livshits, I. S. Tarasov, D. A. Bedarev, B. V. Volovik, M. V. Maximov, A. F. Tsatsulnikov, N. N. Ledentsov, P. S. Kopev, D. Bimberg, and Z. I. Alferov, “0.94 μm diode lasers based on Stranski-Krastanow and sub-monolayer quantum dots,” Semicond. Sci. Technol. 15, 1061 (2000).
[Crossref]
F. Hopfer, A. Mutig, G. Fiol, M. Kuntz, V. A. Shchukin, V. A. Haisler, T. Warming, E. Stock, S. S. Mikhrin, I. L. Krestnikov, D. A. Livshits, A. R. Kovsh, C. Bornholdt, A. Lenz, H. Eisele, M. Dahne, N. N. Ledentsov, and D. Bimberg, “20 Gb/s 85 °C error-free operation of vcsels based on submonolayer deposition of quantum dots,” IEEE J. Sel. Top. Quantum Electron. 13, 1302–1308 (2007).
[Crossref]
J. Kim and S. L. Chuang, “Theoretical and experimental study of optical gain, refractive index change, and linewidth enhancement factor of p-doped quantum-dot lasers,” IEEE J. Quantum Electron. 42, 942–952 (2006).
[Crossref]
T. Baba, T. Hamano, F. Koyama, and K. Iga, “Spontaneous emission factor of a microcavity DBR surface-emitting laser,” IEEE J. Quantum Electron. 27, 1347–1358 (1991).
[Crossref]
F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. A. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, A. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. 89, 141106 (2006).
[Crossref]
A. Schliwa, M. Winkelnkemper, and D. Bimberg, “Impact of size, shape, and composition on piezoelectric effects and electronic properties of In(Ga)As/GaAs quantum dots,” Phys. Rev. B 76, 205324 (2007).
[Crossref]
G. Bester, X. Wu, D. Vanderbilt, and A. Zunger, “Importance of second-order piezoelectric effects in zinc-blende semiconductors,” Phys. Rev. Lett. 96, 187602 (2006).
[Crossref] [PubMed]
S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009), Chap. 4, 9, and 11.
C. Pryor, “Eight-band calculations of strained InAs/GaAs quantum dots compared with one-, four-, and six-band approximations,” Phys. Rev. B 57, 7190–7195 (1998).
[Crossref]
G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, 1974).
I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815–5875 (2001).
[Crossref]
Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, 149–154 (1967).
[Crossref]
E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).
J. M. Gérard and B. Gayral, “InAs quantum dots: artificial atoms for solid-state cavity-quantum electrodynamics,” Physica E 9, 131–139 (2001).
[Crossref]
Y. Yamamoto, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44, 657–668 (1991).
[Crossref] [PubMed]
S.-W. Chang, C.-Y. A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 10580–10595 (2008).
[Crossref] [PubMed]
C.-Y. Lu, S. L. Chuang, and D. Bimberg, “Metal-cavity surface-emitting nanolasers,” IEEE J. Quantum Electron. 49, 114–121 (2013).
[Crossref]
S.-W. Chang, C.-Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE J. Sel. Top. Quantum Electron. 17, 1681–1692 (2011).
[Crossref]
P. V. Mena, J. J. Morikuni, S.-M. Kang, A. V. Harton, and K. W. Wyatt, “A comprehensive circuit-level model of vertical-cavity surface-emitting lasers,” J. Lightwave Technol. 17, 2612–2632 (1999).
[Crossref]
S.-W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 8, 1014–1023 (2009).
[Crossref]

2013 (2)

C.-Y. Lu, C.-Y. Ni, M. Zhang, S. L. Chuang, and D. Bimberg, “Metal-cavity surface-emitting microlasers with size reduction: theory and experiment,” IEEE J. Sel. Top. Quantum Electron. 19, 1701809 (2013).

C.-Y. Lu, S. L. Chuang, and D. Bimberg, “Metal-cavity surface-emitting nanolasers,” IEEE J. Quantum Electron. 49, 114–121 (2013).
[Crossref]

2011 (2)

S.-W. Chang, C.-Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE J. Sel. Top. Quantum Electron. 17, 1681–1692 (2011).
[Crossref]

K. Ding, Z. Liu, L. Yin, H. Wang, R. Liu, M. T. Hill, M. J. H. Marell, P. J. van Veldhoven, R. Ntzel, and C. Z. Ning, “Electrical injection, continuous wave operation of subwavelength-metallic-cavity lasers at 260 K,” Appl. Phys. Lett. 98, 231108 (2011).
[Crossref]

2010 (2)

K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18, 8790–8799 (2010).
[Crossref] [PubMed]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[Crossref]

2009 (1)

S.-W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 8, 1014–1023 (2009).
[Crossref]

2008 (2)

S.-W. Chang, C.-Y. A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 10580–10595 (2008).
[Crossref] [PubMed]

D. Bimberg, “Quantum dot based nanophotonics and nanoelectronics,” Electron. Lett. 44, 168–171 (2008).
[Crossref]

2007 (2)

F. Hopfer, A. Mutig, G. Fiol, M. Kuntz, V. A. Shchukin, V. A. Haisler, T. Warming, E. Stock, S. S. Mikhrin, I. L. Krestnikov, D. A. Livshits, A. R. Kovsh, C. Bornholdt, A. Lenz, H. Eisele, M. Dahne, N. N. Ledentsov, and D. Bimberg, “20 Gb/s 85 °C error-free operation of vcsels based on submonolayer deposition of quantum dots,” IEEE J. Sel. Top. Quantum Electron. 13, 1302–1308 (2007).
[Crossref]

A. Schliwa, M. Winkelnkemper, and D. Bimberg, “Impact of size, shape, and composition on piezoelectric effects and electronic properties of In(Ga)As/GaAs quantum dots,” Phys. Rev. B 76, 205324 (2007).
[Crossref]

2006 (3)

G. Bester, X. Wu, D. Vanderbilt, and A. Zunger, “Importance of second-order piezoelectric effects in zinc-blende semiconductors,” Phys. Rev. Lett. 96, 187602 (2006).
[Crossref] [PubMed]

F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. A. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, A. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. 89, 141106 (2006).
[Crossref]

J. Kim and S. L. Chuang, “Theoretical and experimental study of optical gain, refractive index change, and linewidth enhancement factor of p-doped quantum-dot lasers,” IEEE J. Quantum Electron. 42, 942–952 (2006).
[Crossref]

2001 (2)

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815–5875 (2001).
[Crossref]

J. M. Gérard and B. Gayral, “InAs quantum dots: artificial atoms for solid-state cavity-quantum electrodynamics,” Physica E 9, 131–139 (2001).
[Crossref]

2000 (1)

S. S. Mikhrin, A. E. Zhukov, A. R. Kovsh, N. A. Maleev, V. M. Ustinov, Y. M. Shernyakov, I. P. Soshnikov, D. A. Livshits, I. S. Tarasov, D. A. Bedarev, B. V. Volovik, M. V. Maximov, A. F. Tsatsulnikov, N. N. Ledentsov, P. S. Kopev, D. Bimberg, and Z. I. Alferov, “0.94 μm diode lasers based on Stranski-Krastanow and sub-monolayer quantum dots,” Semicond. Sci. Technol. 15, 1061 (2000).
[Crossref]

1999 (1)

P. V. Mena, J. J. Morikuni, S.-M. Kang, A. V. Harton, and K. W. Wyatt, “A comprehensive circuit-level model of vertical-cavity surface-emitting lasers,” J. Lightwave Technol. 17, 2612–2632 (1999).
[Crossref]

1998 (1)

C. Pryor, “Eight-band calculations of strained InAs/GaAs quantum dots compared with one-, four-, and six-band approximations,” Phys. Rev. B 57, 7190–7195 (1998).
[Crossref]

1991 (2)

Y. Yamamoto, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44, 657–668 (1991).
[Crossref] [PubMed]

T. Baba, T. Hamano, F. Koyama, and K. Iga, “Spontaneous emission factor of a microcavity DBR surface-emitting laser,” IEEE J. Quantum Electron. 27, 1347–1358 (1991).
[Crossref]

1979 (1)

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18, 2329–2330 (1979).
[Crossref]

1967 (1)

Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, 149–154 (1967).
[Crossref]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Alferov, Z. I.

Baba, T.

T. Baba, T. Hamano, F. Koyama, and K. Iga, “Spontaneous emission factor of a microcavity DBR surface-emitting laser,” IEEE J. Quantum Electron. 27, 1347–1358 (1991).
[Crossref]

Bedarev, D. A.

Bester, G.

G. Bester, X. Wu, D. Vanderbilt, and A. Zunger, “Importance of second-order piezoelectric effects in zinc-blende semiconductors,” Phys. Rev. Lett. 96, 187602 (2006).
[Crossref] [PubMed]

Bimberg, D.

C.-Y. Lu, S. L. Chuang, and D. Bimberg, “Metal-cavity surface-emitting nanolasers,” IEEE J. Quantum Electron. 49, 114–121 (2013).
[Crossref]

D. Bimberg, “Quantum dot based nanophotonics and nanoelectronics,” Electron. Lett. 44, 168–171 (2008).
[Crossref]

Bir, G. L.

G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, 1974).

Bondarenko, O.

Bornholdt, C.

Chang, S.-W.

S.-W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 8, 1014–1023 (2009).
[Crossref]

S.-W. Chang, C.-Y. A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 10580–10595 (2008).
[Crossref] [PubMed]

Chuang, S. L.

C.-Y. Lu, S. L. Chuang, and D. Bimberg, “Metal-cavity surface-emitting nanolasers,” IEEE J. Quantum Electron. 49, 114–121 (2013).
[Crossref]

S.-W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 8, 1014–1023 (2009).
[Crossref]

S.-W. Chang, C.-Y. A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 10580–10595 (2008).
[Crossref] [PubMed]

S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009), Chap. 4, 9, and 11.

Coldren, L.

L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Corzine, S.

L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Dahne, M.

Ding, K.

Eisele, H.

Fainman, Y.

Feng, L.

Fiol, G.

Gayral, B.

J. M. Gérard and B. Gayral, “InAs quantum dots: artificial atoms for solid-state cavity-quantum electrodynamics,” Physica E 9, 131–139 (2001).
[Crossref]

Gérard, J. M.

J. M. Gérard and B. Gayral, “InAs quantum dots: artificial atoms for solid-state cavity-quantum electrodynamics,” Physica E 9, 131–139 (2001).
[Crossref]

Germann, T. D.

Haisler, V. A.

Hamano, T.

T. Baba, T. Hamano, F. Koyama, and K. Iga, “Spontaneous emission factor of a microcavity DBR surface-emitting laser,” IEEE J. Quantum Electron. 27, 1347–1358 (1991).
[Crossref]

Harton, A. V.

Hill, M. T.

Hopfer, F.

Iga, K.

T. Baba, T. Hamano, F. Koyama, and K. Iga, “Spontaneous emission factor of a microcavity DBR surface-emitting laser,” IEEE J. Quantum Electron. 27, 1347–1358 (1991).
[Crossref]

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18, 2329–2330 (1979).
[Crossref]

Kang, S.-M.

Kim, J.

Kitahara, C.

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18, 2329–2330 (1979).
[Crossref]

Kopev, P. S.

Kovsh, A. R.

Koyama, F.

T. Baba, T. Hamano, F. Koyama, and K. Iga, “Spontaneous emission factor of a microcavity DBR surface-emitting laser,” IEEE J. Quantum Electron. 27, 1347–1358 (1991).
[Crossref]

Krestnikov, I. L.

Kuntz, M.

Lakhani, A.

K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18, 8790–8799 (2010).
[Crossref] [PubMed]

Ledentsov, N. N.

Lenz, A.

Liu, R.

Liu, Z.

Livshits, D. A.

Livshits, D. L.

Lomakin, V.

Lu, C.-Y.

C.-Y. Lu, S. L. Chuang, and D. Bimberg, “Metal-cavity surface-emitting nanolasers,” IEEE J. Quantum Electron. 49, 114–121 (2013).
[Crossref]

Maleev, N. A.

Marell, M. J. H.

Maximov, M. V.

Mena, P. V.

Meyer, J. R.

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815–5875 (2001).
[Crossref]

Mikhrin, S. S.

Mizrahi, A.

Morikuni, J. J.

Mutig, A.

Nezhad, M. P.

Ni, C.-Y.

Ni, C.-Y. A.

S.-W. Chang, C.-Y. A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 10580–10595 (2008).
[Crossref] [PubMed]

Ning, C. Z.

Ntzel, R.

Pikus, G. E.

G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, 1974).

Pohl, U. W.

Pryor, C.

C. Pryor, “Eight-band calculations of strained InAs/GaAs quantum dots compared with one-, four-, and six-band approximations,” Phys. Rev. B 57, 7190–7195 (1998).
[Crossref]

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Ram-Mohan, L. R.

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815–5875 (2001).
[Crossref]

Schliwa, A.

Shchukin, V. A.

Shernyakov, Y. M.

Simic, A.

Slutsky, B.

Soda, H.

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18, 2329–2330 (1979).
[Crossref]

Soshnikov, I. P.

Stock, E.

Suematsu, Y.

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18, 2329–2330 (1979).
[Crossref]

Tarasov, I. S.

Tsatsulnikov, A. F.

Ustinov, V. M.

van Veldhoven, P. J.

Vanderbilt, D.

G. Bester, X. Wu, D. Vanderbilt, and A. Zunger, “Importance of second-order piezoelectric effects in zinc-blende semiconductors,” Phys. Rev. Lett. 96, 187602 (2006).
[Crossref] [PubMed]

Varshni, Y. P.

Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, 149–154 (1967).
[Crossref]

Volovik, B. V.

Vurgaftman, I.

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815–5875 (2001).
[Crossref]

Wang, H.

Warming, T.

Werner, P.

Winkelnkemper, M.

Wu, M. C.

K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18, 8790–8799 (2010).
[Crossref] [PubMed]

Wu, X.

G. Bester, X. Wu, D. Vanderbilt, and A. Zunger, “Importance of second-order piezoelectric effects in zinc-blende semiconductors,” Phys. Rev. Lett. 96, 187602 (2006).
[Crossref] [PubMed]

Wyatt, K. W.

Yamamoto, Y.

Y. Yamamoto, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44, 657–668 (1991).
[Crossref] [PubMed]

Yin, L.

Yu, K.

K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18, 8790–8799 (2010).
[Crossref] [PubMed]

Zakharov, N. D.

Zhang, M.

Zhukov, A. E.

Zunger, A.

G. Bester, X. Wu, D. Vanderbilt, and A. Zunger, “Importance of second-order piezoelectric effects in zinc-blende semiconductors,” Phys. Rev. Lett. 96, 187602 (2006).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

Electron. Lett. (1)

D. Bimberg, “Quantum dot based nanophotonics and nanoelectronics,” Electron. Lett. 44, 168–171 (2008).
[Crossref]

IEEE J. Quantum Electron. (4)

T. Baba, T. Hamano, F. Koyama, and K. Iga, “Spontaneous emission factor of a microcavity DBR surface-emitting laser,” IEEE J. Quantum Electron. 27, 1347–1358 (1991).
[Crossref]

S.-W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 8, 1014–1023 (2009).
[Crossref]

C.-Y. Lu, S. L. Chuang, and D. Bimberg, “Metal-cavity surface-emitting nanolasers,” IEEE J. Quantum Electron. 49, 114–121 (2013).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (3)

J. Appl. Phys. (1)

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815–5875 (2001).
[Crossref]

J. Lightwave Technol. (1)

Jpn. J. Appl. Phys. (1)

H. Soda, K. Iga, C. Kitahara, and Y. Suematsu, “GaInAsP/InP surface emitting injection lasers,” Jpn. J. Appl. Phys. 18, 2329–2330 (1979).
[Crossref]

Nat. Photonics (1)

Opt. Express (2)

S.-W. Chang, C.-Y. A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 10580–10595 (2008).
[Crossref] [PubMed]

K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18, 8790–8799 (2010).
[Crossref] [PubMed]

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Phys. Rev. A (1)

Y. Yamamoto, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44, 657–668 (1991).
[Crossref] [PubMed]

Phys. Rev. B (2)

C. Pryor, “Eight-band calculations of strained InAs/GaAs quantum dots compared with one-, four-, and six-band approximations,” Phys. Rev. B 57, 7190–7195 (1998).
[Crossref]

Phys. Rev. Lett. (1)

G. Bester, X. Wu, D. Vanderbilt, and A. Zunger, “Importance of second-order piezoelectric effects in zinc-blende semiconductors,” Phys. Rev. Lett. 96, 187602 (2006).
[Crossref] [PubMed]

Physica (1)

Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, 149–154 (1967).
[Crossref]

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[Crossref]

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Fig. 1 (a) Schematic of a submonolayer (SML) quantum-dot (QD) metal-cavity surface-emitting laser. The active region contains 3 groups of SML QDs. (b) Schematic of each group of SML QDs, consisting of 10 stacks of 0.5 ML InAs QDs separated by 2.2 ML thick GaAs spacers. [8] (c) A scanning electron micrograph of a 0.5-μm-radius microlaser before SiN_x passivation and metal coating.

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Fig. 2 (a) Wavefunctions of the first two conduction band states (CB1 and CB2) and the first two heavy-hole states (HH1 and HH2). The wavefunctions are shown at the isosurface of

{| Ψ |}^{2} = 0.5 {| Ψ |}_{max}^{2}

. The blue disks are the 10-fold vertically-correlated submonolayer quantum dots, assuming no lateral coupling. (b) Wavefunction of the conduction band ground state (CB1), considering lateral coupling. The wavefunction is shown at the isosurface of

{| Ψ |}^{2} = 0.16 {| Ψ |}_{max}^{2}

. In this example, the quantum-dot 2D fill factor is 62.8% and the 2D dot density is 2 × 10¹¹ cm⁻².

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Fig. 3 (a) Ground state transition energy as a function of the temperature for different effective dot sizes. The measured photoluminescence peak is shown as the star. [8] (b) The quasi-Fermi level F_c for conduction band (blue circles) as a function of injected carrier density. Horizontal lines show 50 conduction band states with bound states circled. (c) The quasi-Fermi level F_v for valence band (red circles) as a function of injected carrier density. Horizontal lines show 50 heavy-hole and 20 light-hole states.

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Fig. 4 (a) TE-polarized material gain and (b) TE-polarized spontaneous emission rate calculated for the submonolayer quantum dots at T = 300 K, with carrier densities from n = 6 × 10¹⁷ cm⁻³ to n = 5.4 × 10¹⁸ cm⁻³.

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Fig. 5 (a) Calculated lasing wavelength and the photon lifetime as a function of the cavity diameter for the fundamental HE₁₁ transverse mode. (b) Calculated cavity quality factor and effective mode volume as a function of the cavity diameter for the HE₁₁ transverse mode.

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Fig. 6 Theoretical and measured light output power vs. current (L-I) curves for submono-layer quantum-dot metal-cavity surface-emitting microlasers with different device diameters at T = 300 K. The turn-on behavior below lasing threshold is explained by the increasing β_sp factor with carrier injection, i.e., increasing amount of spontaneous emission coupled into the cavity mode.

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Fig. 7 (a) Calculated Purcell factor for the fundamental mode in the metal-cavity micro-lasers with different cavity diameters. (b) Extracted spontaneous emission coupling factor β_sp from the fitting of device light output power vs. current (L-I) curves with the rate-equation model.

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Tables (2)

Table 1 Size-dependent laser characteristics extracted from the rate-equation model.

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Table 2 Parameters used in our theoretical model.

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Equations (23)

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\begin{array}{l} n = 2 \frac{N_{dot}^{2 D}}{L_{z}} \sum_{i} \int d E [\frac{1}{\sqrt{2 π} σ_{c}} e^{- {(E - E_{c}^{i})}^{2} / 2 σ_{c}^{2}}] f_{c} (E, F_{c}), \\ p = 2 \frac{N_{dot}^{2 D}}{L_{z}} \sum_{j} \int d E [\frac{1}{\sqrt{2 π} σ_{v}} e^{- {(E - E_{v}^{j})}^{2} / 2 σ_{v}^{2}}] f_{v} (E, F_{v}) \end{array}

\begin{array}{l} g (\bar{h} ω) & = \frac{2 N_{dot}^{2 D}}{L_{z}} C_{0} \sum_{i, j} \int d E {| M_{env}^{i j} |}^{2} {| \hat{e} \cdot p_{c v} |}^{2} D (E, E_{c v}^{i j}) L (E, \bar{h} ω) (f_{c, i} - f_{v, j}), \\ r_{spon} (\bar{h} ω) & = \frac{2 N_{dot}^{2 D}}{L_{z}} B_{0} C_{0} \sum_{i, j} \int d E {| M_{env}^{i j} |}^{2} {| \hat{e} \cdot p_{c v} |}^{2} D (E, E_{c v}^{i j}) L (E, \bar{h} ω) f_{c, i} (1 - f_{v, j}) \end{array}

\begin{array}{l} D (E, E_{c v}^{i j}) = \frac{1}{\sqrt{2 π (σ_{c}^{2} + σ_{v}^{2})}} exp [- {(E - E_{c v}^{i j})}^{2} / 2 (σ_{c}^{2} + σ_{v}^{2})], \\ L (E, \bar{h} ω) = \frac{Γ_{c v}}{π} \frac{1}{Γ_{c v}^{2} + {(E - \bar{h} ω)}^{2}}, B_{0} = \frac{n_{a}^{2} ω^{2}}{π^{2} \bar{h} c^{2}}, C_{0} = \frac{π e^{2}}{n_{a} c ε_{0} m_{0}^{2} ω} \end{array}

F_{p} = \frac{3 Q}{4 π^{2} V_{eff}} {(\frac{λ}{n_{r}})}^{3}

β = \frac{F_{p} / 3}{g F_{p} / 3 + 1}

β = \frac{λ^{4}}{4 π^{2} V Δ λ ε^{3 / 2}}

\begin{matrix} R_{sp, m} = \int d (\bar{h} ω) {[\frac{2 N_{dot}^{2 D}}{L_{z}} \frac{c}{V_{eff} n_{a}} C_{0} \sum_{i j} {| M_{env}^{i j} |}^{2} {| \hat{e} \cdot p_{c v} |}^{2} f_{c, i} (1 - f_{v, j}) \frac{Γ_{c v} / π}{Γ_{c v}^{2} + {(E_{c v}^{i j} - \bar{h} ω)}^{2}}] \\ \times \frac{Γ_{m} / π}{Γ_{m}^{2} + {(\bar{h} ω_{m} - \bar{h} ω)}^{2}}} \end{matrix}

\begin{array}{l} R_{sp, m} & = \int d (\bar{h} ω) \frac{c / (V_{eff} n_{a})}{B_{0}} r_{spon} (\bar{h} ω) \frac{Γ_{m} / π}{Γ_{m}^{2} + {(\bar{h} ω - \bar{h} ω_{m})}^{2}} \\ \approx D_{cav} (\bar{h} ω_{m}) \int d (\bar{h} ω) r_{spon} (\bar{h} ω) \frac{Γ_{m} / π}{Γ_{m}^{2} + {(\bar{h} ω - \bar{h} ω_{m})}^{2}} \end{array}

\begin{array}{l} D_{cav} (\bar{h} ω_{m}) & = \frac{c / (V_{eff} n_{a})}{B_{0}} = \frac{π^{2} \bar{h} c^{3}}{V_{eff} n_{a}^{3} ω_{m}^{2}} = \frac{\bar{h} ω_{m}}{8 π V_{eff}} {(\frac{λ_{m}}{n_{a}})}^{3} = \frac{2 Γ_{m} Q}{8 π V_{eff}} {(\frac{λ_{m}}{n_{a}})}^{3} \\ = π Γ_{m} [\frac{Q}{4 π^{2} V_{eff}} {(\frac{λ_{m}}{n_{a}})}^{3}] = π Γ_{m} \frac{F_{p}}{3} \end{array}

\begin{array}{l} β_{sp, m} = \frac{R_{sp, m}}{R_{sp}} = \frac{R_{sp, m}}{g R_{sp, m} + R_{sp, cont}} \\ = \frac{D_{cav} (\bar{h} ω_{m}) \int d (\bar{h} ω) r_{spon} (\bar{h} ω) \frac{Γ_{m} / π}{Γ_{m}^{2} + {(\bar{h} ω - \bar{h} ω_{m})}^{2}}}{g D_{c a v} (\bar{h} ω_{m}) \int d (\bar{h} ω) r_{spon} (\bar{h} ω) \frac{Γ_{m} / π}{Γ_{m}^{2} + {(\bar{h} ω - \bar{h} ω_{m})}^{2}} + R_{spon, cont}} \end{array}

β_{sp, m} = \frac{γ F_{p} / 3}{γ g F_{p} / 3 + 1}

\begin{array}{l} γ = \frac{1}{R_{sp, cont}} \int d (\bar{h} ω) r_{spon} (\bar{h} ω) \frac{Γ_{m}^{2}}{Γ_{m}^{2} + {(\bar{h} ω - \bar{h} ω_{m})}^{2}} \\ = τ_{sp, cont} V_{a} \int d (\bar{h} ω) r_{spon} (\bar{h} ω) \frac{Γ_{m}^{2}}{Γ_{m}^{2} + {(\bar{h} ω - \bar{h} ω_{m})}^{2}} \end{array}

[\nabla_{t}^{2} + n^{2} (ρ) k_{0}^{2}] [\begin{matrix} E_{z} \\ H_{z} \end{matrix}] = k_{z}^{2} [\begin{matrix} E_{z} \\ H_{z} \end{matrix}] = n_{eff}^{2} k_{0}^{2} [\begin{matrix} E_{z} \\ H_{z} \end{matrix}]

\begin{array}{l} \frac{d n}{d t} = η_{i} \frac{I - I_{l} (n)}{q V_{a}} - (A n + C n^{3}) - R_{sp} (n) - v_{g} g (n) S, \\ \frac{d S}{d t} = Γ_{E} v_{g} g (n) S - \frac{S}{τ_{p}} + Γ_{E} β_{sp, m} (n) R_{sp} (n) \end{array}

A = \frac{A_{a}}{V_{a}} v_{s} = \frac{4}{D} v_{s} (cylindrical)

I_{l} (n) = I_{l 0} \cdot exp (\frac{E_{g, barrier} - [F_{c} (n) - F_{v} (n)]}{k T})

P = β_{c 1} \bar{h} ω \frac{V_{a}}{Γ_{E}} v_{g} α_{m} S + β_{c 2} \bar{h} ω R_{sp} V_{a}

\begin{array}{l} R_{sp, m} = \frac{2 N_{dot}^{2 D}}{L_{z}} \frac{2 π}{\bar{h}} \sum_{i, j} {| 〈 ψ_{i}^{c} | μ \cdot \frac{E_{m}}{2} | ψ_{v}^{j} 〉 |}^{2} \frac{Γ_{c v} + Γ_{m}}{π} \frac{f_{c, i} (1 - f_{v, j})}{{(E_{c v}^{i j} - \bar{h} ω_{m})}^{2} + {(Γ_{c v} + Γ_{m})}^{2}} \\ \approx \frac{2 N_{dot}^{2 D}}{L_{z}} \frac{2 π}{\bar{h}} [\int_{V_{a}} \frac{d^{3} r}{V_{a}} \frac{{| E_{m} (r) |}^{2}}{4}] \sum_{i, j} {| 〈 ψ_{c}^{i} | μ \cdot \hat{e} | ψ_{v}^{j} 〉 |}^{2} \frac{Γ_{c v} + Γ_{m}}{π} \frac{f_{c, i} (1 - f_{v, j})}{{(E_{c v}^{i j} - \bar{h} ω_{m})}^{2} + {(Γ_{c v} + Γ_{m})}^{2}} \end{array}

\frac{V_{a}}{V_{eff}} = Γ_{E} = \frac{\int_{V_{a}} d^{3} r ε_{0} n_{a}^{2} {| E_{m} (r) |}^{2} / 2}{\int_{V} d^{3} r ε_{0} n^{2} (r) {| E_{m} (r) |}^{2} / 2} = \frac{\int_{V_{a}} d^{3} r ε_{0} n_{a}^{2} {| E_{m} (r) |}^{2} / 2}{\bar{h} ω_{m}}

n^{2} \to \frac{1}{2} [{\frac{\partial [ω^{'} n^{2} (ω^{'})]}{\partial ω^{'}} |}_{ω^{'} = ω} + n^{2} (ω)]

{| 〈 ψ_{c}^{i} | μ \cdot \hat{e} | ψ_{v}^{j} 〉 |}^{2} = {| M_{env}^{i j} |}^{2} {| μ_{c v} \cdot \hat{e} |}^{2} = {| M_{env}^{i j} |}^{2} \frac{e^{2}}{m_{0}^{2} ω_{i j}^{2}} {| \hat{e} \cdot p_{c v}^{i j} |}^{2}

R_{sp, m} = \frac{2 N_{dot}^{2 D}}{L_{z}} \frac{2 π}{\bar{h}} (\frac{Γ_{E} \bar{h} ω_{m}}{2 ε_{0} n_{a}^{2} V_{a}}) \sum_{i, j} \frac{e^{2}}{m_{0}^{2} ω_{i j}^{2}} {| M_{env}^{i j} |}^{2} {| \hat{e} \cdot p_{c v} |}^{2} \frac{Γ_{c v} + Γ_{m}}{π} \frac{f_{c, i} (1 - f_{v, j})}{{(E_{c v}^{i j} - \bar{h} ω_{m})}^{2} + {(Γ_{c v} + Γ_{m})}^{2}}

\begin{array}{l} R_{sp, m} = \frac{2 N_{dot}^{2 D}}{L_{z}} \frac{2 π}{\bar{h}} \sum_{i, j} {\frac{\bar{h} e^{2}}{2 ε_{0} n_{a}^{2} V_{eff} m_{0}^{2} ω_{m}} {| M_{env}^{i j} |}^{2} {| \hat{e} \cdot p_{c v} |}^{2} f_{c, i} (1 - f_{v, j}) \\ \int d (\bar{h} ω) \frac{Γ_{c v} / π}{{(E_{c v}^{i j} - \bar{h} ω)}^{2} + Γ_{c v}^{2}} \frac{Γ_{m} / π}{{(\bar{h} ω - \bar{h} ω_{m})}^{2} + Γ_{m}^{2}}} \\ = \int d (\bar{h} ω) [\frac{2 N_{dot}^{2 D}}{L_{z}} \frac{c C_{0}}{V_{eff} n_{a}} \sum_{i, j} {| M_{env}^{i j} |}^{2} {| \hat{e} \cdot p_{c v} |}^{2} f_{c, i} (1 - f_{v, j}) \frac{Γ_{c v} / π}{{(E_{c v}^{i j} - \bar{h} ω)}^{2} + Γ_{c v}^{2}}] \frac{Γ_{m} / π}{{(\bar{h} ω - \bar{h} ω_{m})}^{2} + Γ_{m}^{2}} \end{array}

Device diameter (μm)	Threshold carrier density n_th (×10¹⁸ cm⁻³)	Threshold material gain g_th (cm⁻¹)	Leakage current density at threshold J_l_,th (kA/cm²)	Threshold current density J_th (kA/cm²)

10	1.96	160.72	1.488	5.730
5	1.97	163.01	6.305	15.83
4	1.98	165.96	14.21	24.89
3	2.29	175.02	25.96	39.82
2	2.77	216.83	128.7	150.2
1	3.58	685.68	411.4	452.1

Name and symbol	Value	Name and symbol	Value

Surface recombination velocity v_s	6 × 10⁵ cm/s	Injection efficiency η_i	0.4∼0.7
Auger coefficient C	1×10⁻²⁹ cm⁶/s	Quantum dot diameter	20 μm
Coupling efficiency β_c₁	0.05∼0.15	Quantum dot 2D density	1×10¹¹ cm⁻²
Coupling efficiency β_c₂	0.005∼0.05	Series leakage current parameter I_l₀	100∼300 mA
Homogeneous broadening linewdith Γ_cv	15 meV	Inhomogeneous broadening linewidth for conduction band σ_c	15 meV
Inhomogeneous broadening linewdith for valence band σ_v	2 meV

Device diameter (μm)	Threshold carrier density n_th (×10¹⁸ cm⁻³)	Threshold material gain g_th (cm⁻¹)	Leakage current density at threshold J_l_,th (kA/cm²)	Threshold current density J_th (kA/cm²)

10	1.96	160.72	1.488	5.730
5	1.97	163.01	6.305	15.83
4	1.98	165.96	14.21	24.89
3	2.29	175.02	25.96	39.82
2	2.77	216.83	128.7	150.2
1	3.58	685.68	411.4	452.1

Name and symbol	Value	Name and symbol	Value

Surface recombination velocity v_s	6 × 10⁵ cm/s	Injection efficiency η_i	0.4∼0.7
Auger coefficient C	1×10⁻²⁹ cm⁶/s	Quantum dot diameter	20 μm
Coupling efficiency β_c₁	0.05∼0.15	Quantum dot 2D density	1×10¹¹ cm⁻²
Coupling efficiency β_c₂	0.005∼0.05	Series leakage current parameter I_l₀	100∼300 mA
Homogeneous broadening linewdith Γ_cv	15 meV	Inhomogeneous broadening linewidth for conduction band σ_c	15 meV
Inhomogeneous broadening linewdith for valence band σ_v	2 meV