Abstract

A self-consistent theory for semiconductor lasers, in which plasma and lattice temperatures are treated as two independent variables, is presented. This theory consists of a set of coupled equations for the total carrier density, field amplitude, and plasma and lattice temperatures with the coupling that is due to phonon-carrier scattering and to the band gap’s dependence on lattice temperature. The self-consistent theory is then employed to study thermal effects in vertical-cavity surface-emitting lasers. We first investigate the plasma heating by solving the stationary (cw) solution of the set of equations with a fixed lattice temperature. The solution is studied systematically with respect to different parameters for both bulk and quantum-well media. Significant plasma-heating effects are found. These include the carrier-density dependence on pumping, decrease of input–output efficiency, dependence of the cw frequency shift on pumping, and a pronounced Pauli-blocking effect that is due to plasma heating. Furthermore, we solve the whole set of equations, including that for lattice temperature. We show that the output power is strongly saturated or switched off with an increase of pumping. Details of the saturation depend on the position of the cavity frequency in the gain spectrum and on the heat transfer rate from the lattice to the ambient.

© 1995 Optical Society of America

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  1. P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, and S. Sun, “Lasing threshold in quantum well surface-emitting lasers: many-body effects and temperature dependence,” Appl. Phys. Lett. 55, 2698 (1989).
    [Crossref]
  2. J. W. Scott, S. W. Corzine, D. B. Young, and L. A. Coldren, “Modeling the current to light characteristics of index-guided vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 62, 1050 (1993); “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295 (1993).
    [Crossref]
  3. B. Lu, P. Zhou, J. Cheng, and K. J. Malloy, “High temperature pulsed and cw operation and thermally stable threshold characteristics of vertical-cavity surface-emitting lasers grown by metal-organic chemical vapor deposition,” Appl. Phys. Lett. 65, 1337 (1994).
    [Crossref]
  4. W. Nakwaski and M. Osinski, “Thermal properties of the etched-well surface-emitting semiconductor lasers,” IEEE J. Quantum Electron. 27, 1391 (1991); “Self-consistent thermal electric modeling of photon-implanted top-surface-emitting semiconductor lasers,” in Physics and Simulation of Opto-electronic Devices II, W. W. Chow and M. A. Osinski, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2146, 365 (1994).
    [Crossref]
  5. F. Jahnke and S. W. Koch, “Theory of carrier heating through injection pumping and lasing in semiconductor microcavity lasers,” Opt. Lett. 18, 1438 (1993); F. Jahnke, K. Henneberger, W. Schäfer, and S. W. Koch, “Transient nonequilibrium and many-body effects in semiconductor microcavity lasers,” J. Opt. Soc. Am. B 10, 2396 (1993).
    [Crossref] [PubMed]
  6. We assume a quasi-thermal equilibrium between electrons and holes because this equilibration happens on a fast time scale because of the fast carrier–carrier scattering, so that the same temperature for the whole plasma can be used for the cw operation and for any evolution slower than the polarization dynamics.
  7. In terms of thermodynamics different parts of a composite system in a nonequilibrium stationary state can have different temperatures while maintaining a time-independent state by constant energy fluxes through different parts. This is different from the thermal equilibrium state in which the different parts maintain the same temperature and no net energy fluxes exist through different parts.
  8. T. L. Koch, L. C. Chiu, Ch. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982); R. F. Nabiev, A. Z. Obidin, A. N. Pechenov, and Y. Popov, “Spectral characteristics and kinetics of radiation emitted by a streamer semiconductor laser,” Sov. J. Quantum Electron. 17, 783 (1987).
    [Crossref]
  9. B. N. Gomatam and A. P. DeFonzo, “Theory of hot carrier effects on nonlinear gain in GaAs–GaAlAs lasers and amplifiers,” IEEE J. Quantum Electron. 26, 1689 (1990); G. P. Bava, P. Debernardi, and G. Osella, “Density matrix model for highly nondegenerate four-wave mixing in semiconductor laser devices,” IEEE Proc. Optoelectron. 141, 119 (1994).
    [Crossref]
  10. A. Uskov, J. Mork, and J. Mark, “Wave-mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769 (1994).
    [Crossref]
  11. W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor Laser Physics (Springer, New York, 1994), Chap. 3, p. 71.
    [Crossref]
  12. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 2nd. ed. (World Scientific, Singapore, 1993), Chap. 12, p. 218.
  13. Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, 149 (1967).
    [Crossref]
  14. This situation corresponds to the statistics of a grand canonical ensemble in which one has both material (number of particles) and energy contact with the environment. This justifies the use of plasma temperature as an independent macroscopic variable from the statistical physics point of view, because in the laser operation the plasma is constantly in both material and energy contact with the laser field and with the environment (heat baths).
  15. One could equally use hole energy density or the total energy density. Under the assumption that the holes and the electrons have already reached quasi-thermal equilibrium, the hole energy and the electron energy could be still different. But they are no longer independent; they both are related to the common temperature.
  16. R. Binder, Optical Sciences Center, University of Arizona, Tucson, Ariz. 85721 (personal communication, 1994).
  17. H. Haken, Light 2—Laser Light Dynamics (North-Holland, Amsterdam, 1985), Chap. 5, p. 98; Chap. 6, p. 123.
  18. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Sect. 9.7, p. 376.
  19. C. Z. Ning and J. V. Moloney, “Plasma heating induced intensity-dependent gain in semiconductor lasers,” Appl. Phys. Lett. 66, 559 (1995); see also, M. Willatzen, A. Uskov, J. Moerk, H. Olesen, B. Tromborg, and A. P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606 (1991).
    [Crossref]
  20. M. P. Kesler and E. P. Ippen, “Subpicosecond gain dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 51, 1765 (1987).
    [Crossref]
  21. C. Z. Ning and J. V. Moloney, “Thermal effects on threshold of vertical-cavity surface-emitting lasers: first- and second-order phase transitions,” Opt. Lett. 20, 1151 (1995).
    [Crossref] [PubMed]
  22. C. Z. Ning, R. Indik, J. V. Moloney, and S. W. Koch, “Effects of plasma and lattice heating in VCSEL’s, in Physics and Simulation of Optoelectronic Devices II, W. Chow and M. Osinski, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2399, 617 (1995).

1995 (2)

C. Z. Ning and J. V. Moloney, “Plasma heating induced intensity-dependent gain in semiconductor lasers,” Appl. Phys. Lett. 66, 559 (1995); see also, M. Willatzen, A. Uskov, J. Moerk, H. Olesen, B. Tromborg, and A. P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606 (1991).
[Crossref]

C. Z. Ning and J. V. Moloney, “Thermal effects on threshold of vertical-cavity surface-emitting lasers: first- and second-order phase transitions,” Opt. Lett. 20, 1151 (1995).
[Crossref] [PubMed]

1994 (2)

B. Lu, P. Zhou, J. Cheng, and K. J. Malloy, “High temperature pulsed and cw operation and thermally stable threshold characteristics of vertical-cavity surface-emitting lasers grown by metal-organic chemical vapor deposition,” Appl. Phys. Lett. 65, 1337 (1994).
[Crossref]

A. Uskov, J. Mork, and J. Mark, “Wave-mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769 (1994).
[Crossref]

1993 (2)

F. Jahnke and S. W. Koch, “Theory of carrier heating through injection pumping and lasing in semiconductor microcavity lasers,” Opt. Lett. 18, 1438 (1993); F. Jahnke, K. Henneberger, W. Schäfer, and S. W. Koch, “Transient nonequilibrium and many-body effects in semiconductor microcavity lasers,” J. Opt. Soc. Am. B 10, 2396 (1993).
[Crossref] [PubMed]

J. W. Scott, S. W. Corzine, D. B. Young, and L. A. Coldren, “Modeling the current to light characteristics of index-guided vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 62, 1050 (1993); “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295 (1993).
[Crossref]

1991 (1)

W. Nakwaski and M. Osinski, “Thermal properties of the etched-well surface-emitting semiconductor lasers,” IEEE J. Quantum Electron. 27, 1391 (1991); “Self-consistent thermal electric modeling of photon-implanted top-surface-emitting semiconductor lasers,” in Physics and Simulation of Opto-electronic Devices II, W. W. Chow and M. A. Osinski, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2146, 365 (1994).
[Crossref]

1990 (1)

B. N. Gomatam and A. P. DeFonzo, “Theory of hot carrier effects on nonlinear gain in GaAs–GaAlAs lasers and amplifiers,” IEEE J. Quantum Electron. 26, 1689 (1990); G. P. Bava, P. Debernardi, and G. Osella, “Density matrix model for highly nondegenerate four-wave mixing in semiconductor laser devices,” IEEE Proc. Optoelectron. 141, 119 (1994).
[Crossref]

1989 (1)

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, and S. Sun, “Lasing threshold in quantum well surface-emitting lasers: many-body effects and temperature dependence,” Appl. Phys. Lett. 55, 2698 (1989).
[Crossref]

1987 (1)

M. P. Kesler and E. P. Ippen, “Subpicosecond gain dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 51, 1765 (1987).
[Crossref]

1982 (1)

T. L. Koch, L. C. Chiu, Ch. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982); R. F. Nabiev, A. Z. Obidin, A. N. Pechenov, and Y. Popov, “Spectral characteristics and kinetics of radiation emitted by a streamer semiconductor laser,” Sov. J. Quantum Electron. 17, 783 (1987).
[Crossref]

1967 (1)

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

Binder, R.

R. Binder, Optical Sciences Center, University of Arizona, Tucson, Ariz. 85721 (personal communication, 1994).

Brennan, T. M.

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, and S. Sun, “Lasing threshold in quantum well surface-emitting lasers: many-body effects and temperature dependence,” Appl. Phys. Lett. 55, 2698 (1989).
[Crossref]

Cheng, J.

B. Lu, P. Zhou, J. Cheng, and K. J. Malloy, “High temperature pulsed and cw operation and thermally stable threshold characteristics of vertical-cavity surface-emitting lasers grown by metal-organic chemical vapor deposition,” Appl. Phys. Lett. 65, 1337 (1994).
[Crossref]

Chiu, L. C.

T. L. Koch, L. C. Chiu, Ch. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982); R. F. Nabiev, A. Z. Obidin, A. N. Pechenov, and Y. Popov, “Spectral characteristics and kinetics of radiation emitted by a streamer semiconductor laser,” Sov. J. Quantum Electron. 17, 783 (1987).
[Crossref]

Chow, W. W.

W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor Laser Physics (Springer, New York, 1994), Chap. 3, p. 71.
[Crossref]

Coldren, L. A.

J. W. Scott, S. W. Corzine, D. B. Young, and L. A. Coldren, “Modeling the current to light characteristics of index-guided vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 62, 1050 (1993); “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295 (1993).
[Crossref]

Corzine, S. W.

J. W. Scott, S. W. Corzine, D. B. Young, and L. A. Coldren, “Modeling the current to light characteristics of index-guided vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 62, 1050 (1993); “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295 (1993).
[Crossref]

DeFonzo, A. P.

B. N. Gomatam and A. P. DeFonzo, “Theory of hot carrier effects on nonlinear gain in GaAs–GaAlAs lasers and amplifiers,” IEEE J. Quantum Electron. 26, 1689 (1990); G. P. Bava, P. Debernardi, and G. Osella, “Density matrix model for highly nondegenerate four-wave mixing in semiconductor laser devices,” IEEE Proc. Optoelectron. 141, 119 (1994).
[Crossref]

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Sect. 9.7, p. 376.

Gomatam, B. N.

B. N. Gomatam and A. P. DeFonzo, “Theory of hot carrier effects on nonlinear gain in GaAs–GaAlAs lasers and amplifiers,” IEEE J. Quantum Electron. 26, 1689 (1990); G. P. Bava, P. Debernardi, and G. Osella, “Density matrix model for highly nondegenerate four-wave mixing in semiconductor laser devices,” IEEE Proc. Optoelectron. 141, 119 (1994).
[Crossref]

Gourley, P. L.

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, and S. Sun, “Lasing threshold in quantum well surface-emitting lasers: many-body effects and temperature dependence,” Appl. Phys. Lett. 55, 2698 (1989).
[Crossref]

Haken, H.

H. Haken, Light 2—Laser Light Dynamics (North-Holland, Amsterdam, 1985), Chap. 5, p. 98; Chap. 6, p. 123.

Hammons, B. E.

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, and S. Sun, “Lasing threshold in quantum well surface-emitting lasers: many-body effects and temperature dependence,” Appl. Phys. Lett. 55, 2698 (1989).
[Crossref]

Harder, Ch.

T. L. Koch, L. C. Chiu, Ch. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982); R. F. Nabiev, A. Z. Obidin, A. N. Pechenov, and Y. Popov, “Spectral characteristics and kinetics of radiation emitted by a streamer semiconductor laser,” Sov. J. Quantum Electron. 17, 783 (1987).
[Crossref]

Haug, H.

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 2nd. ed. (World Scientific, Singapore, 1993), Chap. 12, p. 218.

Indik, R.

C. Z. Ning, R. Indik, J. V. Moloney, and S. W. Koch, “Effects of plasma and lattice heating in VCSEL’s, in Physics and Simulation of Optoelectronic Devices II, W. Chow and M. Osinski, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2399, 617 (1995).

Ippen, E. P.

M. P. Kesler and E. P. Ippen, “Subpicosecond gain dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 51, 1765 (1987).
[Crossref]

Jahnke, F.

Kesler, M. P.

M. P. Kesler and E. P. Ippen, “Subpicosecond gain dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 51, 1765 (1987).
[Crossref]

Koch, S. W.

F. Jahnke and S. W. Koch, “Theory of carrier heating through injection pumping and lasing in semiconductor microcavity lasers,” Opt. Lett. 18, 1438 (1993); F. Jahnke, K. Henneberger, W. Schäfer, and S. W. Koch, “Transient nonequilibrium and many-body effects in semiconductor microcavity lasers,” J. Opt. Soc. Am. B 10, 2396 (1993).
[Crossref] [PubMed]

C. Z. Ning, R. Indik, J. V. Moloney, and S. W. Koch, “Effects of plasma and lattice heating in VCSEL’s, in Physics and Simulation of Optoelectronic Devices II, W. Chow and M. Osinski, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2399, 617 (1995).

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 2nd. ed. (World Scientific, Singapore, 1993), Chap. 12, p. 218.

W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor Laser Physics (Springer, New York, 1994), Chap. 3, p. 71.
[Crossref]

Koch, T. L.

T. L. Koch, L. C. Chiu, Ch. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982); R. F. Nabiev, A. Z. Obidin, A. N. Pechenov, and Y. Popov, “Spectral characteristics and kinetics of radiation emitted by a streamer semiconductor laser,” Sov. J. Quantum Electron. 17, 783 (1987).
[Crossref]

Lu, B.

B. Lu, P. Zhou, J. Cheng, and K. J. Malloy, “High temperature pulsed and cw operation and thermally stable threshold characteristics of vertical-cavity surface-emitting lasers grown by metal-organic chemical vapor deposition,” Appl. Phys. Lett. 65, 1337 (1994).
[Crossref]

Lyo, S. K.

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, and S. Sun, “Lasing threshold in quantum well surface-emitting lasers: many-body effects and temperature dependence,” Appl. Phys. Lett. 55, 2698 (1989).
[Crossref]

Malloy, K. J.

B. Lu, P. Zhou, J. Cheng, and K. J. Malloy, “High temperature pulsed and cw operation and thermally stable threshold characteristics of vertical-cavity surface-emitting lasers grown by metal-organic chemical vapor deposition,” Appl. Phys. Lett. 65, 1337 (1994).
[Crossref]

Mark, J.

A. Uskov, J. Mork, and J. Mark, “Wave-mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769 (1994).
[Crossref]

Moloney, J. V.

C. Z. Ning and J. V. Moloney, “Plasma heating induced intensity-dependent gain in semiconductor lasers,” Appl. Phys. Lett. 66, 559 (1995); see also, M. Willatzen, A. Uskov, J. Moerk, H. Olesen, B. Tromborg, and A. P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606 (1991).
[Crossref]

C. Z. Ning and J. V. Moloney, “Thermal effects on threshold of vertical-cavity surface-emitting lasers: first- and second-order phase transitions,” Opt. Lett. 20, 1151 (1995).
[Crossref] [PubMed]

C. Z. Ning, R. Indik, J. V. Moloney, and S. W. Koch, “Effects of plasma and lattice heating in VCSEL’s, in Physics and Simulation of Optoelectronic Devices II, W. Chow and M. Osinski, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2399, 617 (1995).

Mork, J.

A. Uskov, J. Mork, and J. Mark, “Wave-mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769 (1994).
[Crossref]

Nakwaski, W.

W. Nakwaski and M. Osinski, “Thermal properties of the etched-well surface-emitting semiconductor lasers,” IEEE J. Quantum Electron. 27, 1391 (1991); “Self-consistent thermal electric modeling of photon-implanted top-surface-emitting semiconductor lasers,” in Physics and Simulation of Opto-electronic Devices II, W. W. Chow and M. A. Osinski, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2146, 365 (1994).
[Crossref]

Ning, C. Z.

C. Z. Ning and J. V. Moloney, “Plasma heating induced intensity-dependent gain in semiconductor lasers,” Appl. Phys. Lett. 66, 559 (1995); see also, M. Willatzen, A. Uskov, J. Moerk, H. Olesen, B. Tromborg, and A. P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606 (1991).
[Crossref]

C. Z. Ning and J. V. Moloney, “Thermal effects on threshold of vertical-cavity surface-emitting lasers: first- and second-order phase transitions,” Opt. Lett. 20, 1151 (1995).
[Crossref] [PubMed]

C. Z. Ning, R. Indik, J. V. Moloney, and S. W. Koch, “Effects of plasma and lattice heating in VCSEL’s, in Physics and Simulation of Optoelectronic Devices II, W. Chow and M. Osinski, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2399, 617 (1995).

Osinski, M.

W. Nakwaski and M. Osinski, “Thermal properties of the etched-well surface-emitting semiconductor lasers,” IEEE J. Quantum Electron. 27, 1391 (1991); “Self-consistent thermal electric modeling of photon-implanted top-surface-emitting semiconductor lasers,” in Physics and Simulation of Opto-electronic Devices II, W. W. Chow and M. A. Osinski, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2146, 365 (1994).
[Crossref]

Press, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Sect. 9.7, p. 376.

Sargent, M.

W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor Laser Physics (Springer, New York, 1994), Chap. 3, p. 71.
[Crossref]

Schaus, C. F.

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, and S. Sun, “Lasing threshold in quantum well surface-emitting lasers: many-body effects and temperature dependence,” Appl. Phys. Lett. 55, 2698 (1989).
[Crossref]

Scott, J. W.

J. W. Scott, S. W. Corzine, D. B. Young, and L. A. Coldren, “Modeling the current to light characteristics of index-guided vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 62, 1050 (1993); “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295 (1993).
[Crossref]

Sun, S.

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, and S. Sun, “Lasing threshold in quantum well surface-emitting lasers: many-body effects and temperature dependence,” Appl. Phys. Lett. 55, 2698 (1989).
[Crossref]

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Sect. 9.7, p. 376.

Uskov, A.

A. Uskov, J. Mork, and J. Mark, “Wave-mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769 (1994).
[Crossref]

Varshni, Y. P.

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

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Sect. 9.7, p. 376.

Yariv, A.

T. L. Koch, L. C. Chiu, Ch. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982); R. F. Nabiev, A. Z. Obidin, A. N. Pechenov, and Y. Popov, “Spectral characteristics and kinetics of radiation emitted by a streamer semiconductor laser,” Sov. J. Quantum Electron. 17, 783 (1987).
[Crossref]

Young, D. B.

J. W. Scott, S. W. Corzine, D. B. Young, and L. A. Coldren, “Modeling the current to light characteristics of index-guided vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 62, 1050 (1993); “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295 (1993).
[Crossref]

Zhou, P.

B. Lu, P. Zhou, J. Cheng, and K. J. Malloy, “High temperature pulsed and cw operation and thermally stable threshold characteristics of vertical-cavity surface-emitting lasers grown by metal-organic chemical vapor deposition,” Appl. Phys. Lett. 65, 1337 (1994).
[Crossref]

Appl. Phys. Lett. (6)

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, and S. Sun, “Lasing threshold in quantum well surface-emitting lasers: many-body effects and temperature dependence,” Appl. Phys. Lett. 55, 2698 (1989).
[Crossref]

J. W. Scott, S. W. Corzine, D. B. Young, and L. A. Coldren, “Modeling the current to light characteristics of index-guided vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 62, 1050 (1993); “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295 (1993).
[Crossref]

B. Lu, P. Zhou, J. Cheng, and K. J. Malloy, “High temperature pulsed and cw operation and thermally stable threshold characteristics of vertical-cavity surface-emitting lasers grown by metal-organic chemical vapor deposition,” Appl. Phys. Lett. 65, 1337 (1994).
[Crossref]

T. L. Koch, L. C. Chiu, Ch. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982); R. F. Nabiev, A. Z. Obidin, A. N. Pechenov, and Y. Popov, “Spectral characteristics and kinetics of radiation emitted by a streamer semiconductor laser,” Sov. J. Quantum Electron. 17, 783 (1987).
[Crossref]

C. Z. Ning and J. V. Moloney, “Plasma heating induced intensity-dependent gain in semiconductor lasers,” Appl. Phys. Lett. 66, 559 (1995); see also, M. Willatzen, A. Uskov, J. Moerk, H. Olesen, B. Tromborg, and A. P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606 (1991).
[Crossref]

M. P. Kesler and E. P. Ippen, “Subpicosecond gain dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 51, 1765 (1987).
[Crossref]

IEEE J. Quantum Electron. (3)

B. N. Gomatam and A. P. DeFonzo, “Theory of hot carrier effects on nonlinear gain in GaAs–GaAlAs lasers and amplifiers,” IEEE J. Quantum Electron. 26, 1689 (1990); G. P. Bava, P. Debernardi, and G. Osella, “Density matrix model for highly nondegenerate four-wave mixing in semiconductor laser devices,” IEEE Proc. Optoelectron. 141, 119 (1994).
[Crossref]

A. Uskov, J. Mork, and J. Mark, “Wave-mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769 (1994).
[Crossref]

W. Nakwaski and M. Osinski, “Thermal properties of the etched-well surface-emitting semiconductor lasers,” IEEE J. Quantum Electron. 27, 1391 (1991); “Self-consistent thermal electric modeling of photon-implanted top-surface-emitting semiconductor lasers,” in Physics and Simulation of Opto-electronic Devices II, W. W. Chow and M. A. Osinski, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2146, 365 (1994).
[Crossref]

Opt. Lett. (2)

Physica (1)

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

Other (10)

This situation corresponds to the statistics of a grand canonical ensemble in which one has both material (number of particles) and energy contact with the environment. This justifies the use of plasma temperature as an independent macroscopic variable from the statistical physics point of view, because in the laser operation the plasma is constantly in both material and energy contact with the laser field and with the environment (heat baths).

One could equally use hole energy density or the total energy density. Under the assumption that the holes and the electrons have already reached quasi-thermal equilibrium, the hole energy and the electron energy could be still different. But they are no longer independent; they both are related to the common temperature.

R. Binder, Optical Sciences Center, University of Arizona, Tucson, Ariz. 85721 (personal communication, 1994).

H. Haken, Light 2—Laser Light Dynamics (North-Holland, Amsterdam, 1985), Chap. 5, p. 98; Chap. 6, p. 123.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Sect. 9.7, p. 376.

C. Z. Ning, R. Indik, J. V. Moloney, and S. W. Koch, “Effects of plasma and lattice heating in VCSEL’s, in Physics and Simulation of Optoelectronic Devices II, W. Chow and M. Osinski, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2399, 617 (1995).

We assume a quasi-thermal equilibrium between electrons and holes because this equilibration happens on a fast time scale because of the fast carrier–carrier scattering, so that the same temperature for the whole plasma can be used for the cw operation and for any evolution slower than the polarization dynamics.

In terms of thermodynamics different parts of a composite system in a nonequilibrium stationary state can have different temperatures while maintaining a time-independent state by constant energy fluxes through different parts. This is different from the thermal equilibrium state in which the different parts maintain the same temperature and no net energy fluxes exist through different parts.

W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor Laser Physics (Springer, New York, 1994), Chap. 3, p. 71.
[Crossref]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 2nd. ed. (World Scientific, Singapore, 1993), Chap. 12, p. 218.

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Figures (10)

Fig. 1
Fig. 1

Pumping dependence of the cw quantities: carrier density N, plasma temperature Tp, output power P, frequency shift δω, and carrier-density pumping efficiency ηN. Parameters: γT/γ|| = 1 × 103; mj(=L/λ0) = 7; Lm/L = 0.1; γ = 1 × 1013/s; γ|| = 1 × 109/s; δ0ħ = 0.02 eV; μ0/e = 5 Å; mh = 0.45 m0; me = 0.0665m0, where m0 is the free-electron mass. Throughout this paper, the values of γ, γT, γ||, μ0, mh, and me are always the same as in this figure.

Fig. 2
Fig. 2

Dependence of the cw quantities on γT/γ||. Other parameters are the same as in Fig. 1, except that pumping is now fixed at Np = 6.0 × 1018/cm3.

Fig. 3
Fig. 3

Dependence of the cw quantities on detuning δ0ħ. Other parameters are the same as in Fig. 1, except that pumping is now fixed at Np = 8.0 × 1018/cm3.

Fig. 4
Fig. 4

Dependence of the cw quantities on Lm/L. Other parameters are the same as in Fig. 3, except that detuning is now fixed at δ0ħ = 0.01 eV.

Fig. 5
Fig. 5

Dependence of the cw quantities on pumping Np. Parameters: mj = 7, Lm/L = 0.05, δ0ħ = 0.01 eV.

Fig. 6
Fig. 6

Dependence of the cw quantities on γT/γ||. Np = 7 × 1018/cm3, Lm/L = 0.07, δ0ħ = 0.005 eV. Other parameters are the same as in Fig. 5.

Fig. 7
Fig. 7

Dependence of the cw quantities on the detuning (δ0ħ). Np = 6 × 1018/cm3. Other parameters are the same as in Fig. 6.

Fig. 8
Fig. 8

Dependence of the cw quantities on Lm/L. Np = 8 × 1018/cm3, mj = 7, δ0ħ = 0.08 eV.

Fig. 9
Fig. 9

Continuous-wave output power versus pumping density for the case with the self-consistently determined lattice temperature for different given ambient temperatures, as marked at each curve. Parameters: γa/γ|| = 50.0. The cavity frequency is adjusted to the band-gap frequency at Tl = 300 K. Other parameters are the same as in Fig. 5. Additional parameters that appeared in the lattice-temperature equation are cq = 1.862 × 106 J/K m3, S = 100 μm2, R = 1000 Ω, γnr = γ||.

Fig. 10
Fig. 10

Continuous-wave output power versus pumping density for different values of γa/γ|| as marked at the top of each curve. Ta = 300 K. Other parameters are the same as in Fig. 9.

Equations (49)

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n ˙ α , k = γ f ¯ α , k ( 1 - n α , k ) - γ n α , k - 2 E ¯ 2 γ 2 μ k 2 γ 2 + δ k 2 × ( n e , k + n h , k - 1 ) ,
E = E ¯ exp ( - i ω 0 t ) + c . c .
ω k = E g ( T l ) + k 2 2 m r 1 [ E g ( T l ) + r ( k ) ] ,
E g ( T l ) = E g ( 0 ) - v 1 T l 2 / ( T l + v 2 ) .
N ˙ = γ N 0 ( T p , T l ) - γ N - G N ( T p , T l ) E ¯ 2 ,
N 0 = 1 V k f ¯ α , k ( 1 - f α , k ) = η N N p ,
η N = k f ¯ α , k ( T l ) [ 1 - f α , k ( T p ) ] k f ¯ a , k 1.
γ N 0 = γ N p - γ V k f ¯ α , k f α , k j e L m - γ V k f ¯ α , k f α , k ,
G N ( T p , T l ) = 2 γ V k μ k 2 ( γ ) 2 + ( δ k ) ( f e , k + f h , k - 1 ) ,
f α , k = { exp [ β p ( α , k - μ α ) ] + 1 } - 1 ,             ( α = e , h ) ,
W = 1 V k 2 k 2 2 m e n k 1 V k e ( k ) n k ,
W ˙ = γ W 0 ( T l , T p ) - γ W - E ¯ 2 G W ( T p , T l ) ,
G W ( T p , T l ) = 2 γ V k e ( k ) μ k 2 ( γ ) 2 + ( δ k ) 2 ( f e , k + f h , k - 1 ) .
W 0 = 1 V k e ( k ) f ¯ α , k ( 1 - f α , k ) W p η E ,
η E = k e ( k ) f ¯ k ( T l ) [ 1 - f k ( T p ) ] k e ( k ) f ¯ k ( T l ) 1.
W p = 1 V f ¯ k e ( k ) .
T ˙ = ( N μ e W ˙ - W μ e N ˙ ) / ( W T p N μ e - W μ e N T p ) J W W ˙ - J N N ˙ ,
T ˙ p = γ [ J W ( W 0 - W ) - J N ( N 0 - N ) ] - E ¯ 2 G T - γ T ( T p - T l ) ,
G T = J W G W - J N G N .
d T l d t = - γ a ( T l - T a ) + γ T ( T p - T l ) + ω 0 γ nr N c q + J 2 R c q V t .
2 E z 2 - n b 2 c 2 2 E t 2 = μ 0 2 P t 2 + μ 0 σ E t ,
P = 0 χ E .
c 2 2 i ω 0 n b 2 ( 2 z 2 + k 0 2 ) E ¯ + E ¯ t = i ω 0 2 n b 2 χ E ¯ - σ 2 n b 2 0 E ¯ ,
k 0 = n b ( ω 0 / c ) ,
χ = 1 V k ( δ k + i γ ) 0 μ k 2 ( γ ) 2 + ( δ k ) 2 ( 1 - f e , k - f h , k ) .
E ¯ ( z , t ) = A ( t ) sin ( k 0 z ) .
β = 2 L m z m - L m / 2 z m + L m / 2 sin 2 ( k 0 z ) d z = 1 - λ 0 2 π L m sin ( 2 π L m λ 0 ) ,
A ˙ = i ω 0 L m 2 n b 2 L β χ A - σ 2 n b 2 0 A
N ˙ = γ ( N 0 - N ) - ½ β G N A 2 ,
T ˙ p = γ [ J W ( W 0 - W ) - J N ( N 0 - N ) ] - ½ β G T p A 2 - γ T ( T p - T l ) ,
T ˙ l = - γ a ( T l - T a ) + γ T ( T p - T l ) + γ nr ω 0 c q N + S 2 R c q V t j 2 .
A = A ¯ exp ( - i δ ω t ) ,
δ ω ω 0 = - β L m 2 n b 2 L χ ,
κ + ω 0 β L m 2 n b 2 L χ = 0 ,
β 2 A ¯ 2 - γ ( N 0 - N ) / G N = 0 ,
J W [ ( W 0 - W ) - G W G N ( N 0 - N ) ] - γ T γ ( T p - T l ) = 0 ,
κ = σ 2 n b 2 0 c 2 L n b ln ( 1 r m ) ,
1 V k D i d e ,
D 2 = m e π L m 2 ,
D 3 = ( 2 m e 3 ) 1 / 2 π 2 3 e .
F ( e ) = μ k 2 0 e m e m r - δ 0 + i γ 0 ( γ ) 2 + ( e m e m r - δ 0 ) 2 ( f e + f h - 1 ) .
χ = - d e D i F ( e ) .
L = m j λ 0 = m j 2 π c ω 0 n b .
κ = ω 0 4 π m j ln ( 1 r m ) .
F 1 1 + 2 m j β π L m n b 2 ln ( 1 r m ) L χ = 0 ,
F 2 ( T p T l ) 2 [ ( W 0 N 0 - W N 0 ) G N + ( N N 0 - 1 ) G W ] - γ T γ G N M K B T l ( T p T l - 1 ) = 0 ,
M = e 2 ( 1 - f e ) ( 1 - f e ) - e ( 1 - f e ) 2 η N ( 1 - f e ) f ¯ ,
g ( e ) = D i g ( e ) d e .
P = ( 1 - r m 2 ) 0 c n b 2 β N 0 - N G N .

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