## Hot electrons and curves of constant gain in long wavelength quantum well lasers

Optics Express, Vol. 2, Issue 4, pp. 125-130 (1998)

http://dx.doi.org/10.1364/OE.2.000125

Acrobat PDF (315 KB)

### Abstract

In long wavelength quantum well lasers the effective electron temperature (*T _{e}
*) is often a strong function of the pump current and hence the

*T*correlates with the carrier concentration

_{e}*n*in the active region. On the other hand, the material gain

*g*in the active layer depends on both variables,

*g*=

*g*(

*n*,

*T*. We discuss a convenient way of analyzing this situation, based on considering the contours of constant gain

_{e})*g*on the surface

*g*

*(n*,

*T*. This is qualitatively illustrated with two model examples involving quantum well lasers, the long-wavelength quantum well laser with current dominated by the Auger recombination and the unipolar quantum cascade laser.

_{e})© Optical Society of America

## 1. Introduction

1. S. Luryi, “Hot electrons in semiconductor devices”, in *Hot Electrons in Semiconductors*, N. Balkan, ed. (Oxford University Press,1998) pp. 385–427; http://www.ee.sunysb.edu/~serge/152.dir/152.html

*g*on the carrier temperature

*T*. In near infrared lasers, hot electron effects are relatively small and arise mainly from heterostructure barrier injection and free-carrier absorption of cavity radiation. Nevertheless, even these small effects are not entirely benign: they are responsible for a substantial intermodulation distortion limiting the number of channels in optical communication systems [2

_{e}2. V. B. Gorfinkel and S. Luryi, “Fundamental limits for linearity of CATV lasers”, J. Lightwave Technol. **13**, 252–260 (1995); http://www.ee.sunysb.edu/–~serge/133.html [CrossRef]

*et al*. estimated [3

3. M. Silver, E. P. O’Reilly, and A. R. Adams, “Determination of the wavelength dependence of Auger recombination in long-wavelength quantum-well semiconductor lasers using hydrostatic pressure”, IEEE J. Quantum Electron. **33**, 1557–1566 (1997). [CrossRef]

4. Z. Shi, M. Tacke, A. Lambrecht, and H. Böttner, “Midinfrared lead salt multi-quantum-well diode lasers with 282 K operation”, Appl. Phys. Lett. **66**, 2537–2539 (1995). [CrossRef]

*I*, this may lead to a substantial carrier heating. The increased carrier temperature

*T*suppresses the optical gain

_{e}*g*and may even lead to the appearance of a maximum

*g*

_{max}in the dependence

*g*

*(I)*. If the total losses in the laser cavity exceed

*g*

_{max}then the structure will not lase at any current. Note that for a constant

*T*, the “isotherm” dependence

_{e}*g*

*(I)*is always monotonic. If the losses do not exceed

*g*

_{max}, then the generation regime can be reached, but the negative slope of the

*g*

*(I)*characteristic may result in peculiar instabilities for currents exceeding

*I*=

_{cr}*I*

*(g*. For the same value of gain one would have two regimes that differ in the carrier concentration and the temperature and, of course, in the output radiation power. The higher

_{max})*T*regime would correspond to higher concentration and lower power. Such a regime should be metastable. It could be switched into the stable lower-

_{e}*T*regime by a sufficiently powerful external illumination pulse that would temporarily suppress Auger recombination processes.

_{e}7. J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “High power mid-infrared (λ~5μm) quantum cascade lasers operating above room temperature”, Appl. Phys. Lett. **68**, 3680–3682 (1996). [CrossRef]

*T*arises owing to the de-phasing of inter-subband transition by scattering processes whose rate depends on the electron energy and also from the non-parabolicity of the conduction band. The resultant non-monotonic

_{e}*g*

*(I)*is responsible [8

8. Vera Gorfinkel, Serge Luryi, and Boris Gelmont, “Theory of gain spectra for quantum cascade lasers and temperature dependence of their characteristics at low and moderate carrier concentrations”, IEEE J. Quantum Electron. **32**, 1995–2003 (1996); http://www.ee.sunysb.edu/~serge/145.html [CrossRef]

*T*

_{e}*(n)*. These curves, referred to below as “isogains”, provide a “phase portrait” of the laser. For a given value of the total loss α, the intersection of a corresponding isogain with the

*T*

_{e}*(n)*curve that results from the energy balance equation, defines the operating point of the laser. If for a particular α there is no intersection, such a device will not generate at any pumping.

## 2. Examples

### 2.1 Quantum-well laser with Auger heating

*E*

_{eff}

*C*

_{A}

*n*, where

^{3}*E*

_{eff}is an effective energy transferred into the carrier system per each act of pair recombination:

*f*and

_{e}*f*are the Fermi functions, describing the occupation of electrons and holes, respectively, at the bottom of the quantum well, viz.

_{h}### 2.2 Quantum cascade laser

8. Vera Gorfinkel, Serge Luryi, and Boris Gelmont, “Theory of gain spectra for quantum cascade lasers and temperature dependence of their characteristics at low and moderate carrier concentrations”, IEEE J. Quantum Electron. **32**, 1995–2003 (1996); http://www.ee.sunysb.edu/~serge/145.html [CrossRef]

*n*is the total sheet carrier concentration in both subbands,

*n*

_{2}is the electron concentration in the upper subband, and τ

_{21}is the intersubband transition rate. The latter depends on the carrier temperature, being mainly determined by the emission of polar optic phonons. Equation (5 ) assumes that all carriers in each cascade period of the QCL have the same temperature, which is a reasonable approximation if

*n*is not too low [8

8. Vera Gorfinkel, Serge Luryi, and Boris Gelmont, “Theory of gain spectra for quantum cascade lasers and temperature dependence of their characteristics at low and moderate carrier concentrations”, IEEE J. Quantum Electron. **32**, 1995–2003 (1996); http://www.ee.sunysb.edu/~serge/145.html [CrossRef]

_{21}affects the QCL operation not only through the energy balance (5) but primarily because it controls the subband population ratio,

_{21}= τ

_{21}(

*T*

_{e}) was calculated in our earlier work [9

9. M. V. Kisin, V. B. Gorfinkel, M. A. Stroscio, G. Belenky, and S. Luryi, “Influence of complex phonon spectra on intersubband optical gain”, J. Appl. Phys. **82**, 2031–2038 (1997); http://www.ee.sunysb.edu/~serge/148.pdf [CrossRef]

_{1out}describing the escape of electrons from the quantum well was assumed independent of the carrier temperature and equal 0.5 ps.

*g(n*,

*T*) calculated in the model [8

_{e}**32**, 1995–2003 (1996); http://www.ee.sunysb.edu/~serge/145.html [CrossRef]

9. M. V. Kisin, V. B. Gorfinkel, M. A. Stroscio, G. Belenky, and S. Luryi, “Influence of complex phonon spectra on intersubband optical gain”, J. Appl. Phys. **82**, 2031–2038 (1997); http://www.ee.sunysb.edu/~serge/148.pdf [CrossRef]

*g*=

*α*, corresponding to the surface

*g(n*,

*T*) are plotted in Fig. 5 in blue with the values of α indicated. The red lines correspond to the

_{e}*T*versus

_{e}*n*relation as given by the energy balance equation (5). Evidently, in the present model the temperature does not vary with the overall concentration

*n*=

*n*

_{1}+

*n*

_{2}so long as the ratio

*n*

_{1}/

*n*

_{2}is fixed and that depends on temperature only.

*g*in this model is fixed by the ratio of subband concentrations. If the gain is positive, it increases with

*n*, if it is negative it decreases with

*n*. Evidently, at the transparency value, which is attained when

*n*

_{1}=

*n*

_{2}, i.e., at the temperature when τ

_{21}(

*T*)=τ

_{e}_{1out}, the gain is independent of the overall sheet carrier density in the quantum well.

## Conclusion.

*g*=

*a*on the surface

*g(n*,

*T*

_{e}) contains valuable information and offers a unique view of the highly nonlinear device.

## Acknowledgement.

## References and links

1. | S. Luryi, “Hot electrons in semiconductor devices”, in |

2. | V. B. Gorfinkel and S. Luryi, “Fundamental limits for linearity of CATV lasers”, J. Lightwave Technol. |

3. | M. Silver, E. P. O’Reilly, and A. R. Adams, “Determination of the wavelength dependence of Auger recombination in long-wavelength quantum-well semiconductor lasers using hydrostatic pressure”, IEEE J. Quantum Electron. |

4. | Z. Shi, M. Tacke, A. Lambrecht, and H. Böttner, “Midinfrared lead salt multi-quantum-well diode lasers with 282 K operation”, Appl. Phys. Lett. |

5. | H. K. Choi, G. W. Turner, and H. Q. Le, “InAsSb/InAlAs strained quantum-well lasers emitting at 4.5 μm ”, Appl. Phys. Lett. |

6. | J. R. Meyer, I. Vurgaftman, R. Q. Yang, and L. R. Ram-Mohan, “Type-II and Type-I interband cascade lasers”, Electron Lett. |

7. | J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, “High power mid-infrared (λ~5μm) quantum cascade lasers operating above room temperature”, Appl. Phys. Lett. |

8. | Vera Gorfinkel, Serge Luryi, and Boris Gelmont, “Theory of gain spectra for quantum cascade lasers and temperature dependence of their characteristics at low and moderate carrier concentrations”, IEEE J. Quantum Electron. |

9. | M. V. Kisin, V. B. Gorfinkel, M. A. Stroscio, G. Belenky, and S. Luryi, “Influence of complex phonon spectra on intersubband optical gain”, J. Appl. Phys. |

**OCIS Codes**

(140.5960) Lasers and laser optics : Semiconductor lasers

(230.5590) Optical devices : Quantum-well, -wire and -dot devices

**ToC Category:**

Focus Issue: Quantum well laser design

**History**

Original Manuscript: November 4, 1997

Published: February 16, 1998

**Citation**

Vera Gorfinkel, Mikhail Kisin, and Serge Luryi, "Hot electrons and curves of constant gain
in long wavelength quantum well lasers," Opt. Express **2**, 125-130 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-4-125

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### References

- S. Luryi, "Hot electrons in semiconductor devices", in Hot Electrons in Semiconductors, N. Balkan, ed. (Oxford University Press, 1998) pp. 385-427; http://www.ee.sunysb.edu/~serge/152.dir/152.html
- V. B. Gorfinkel and S. Luryi, "Fundamental limits for linearity of CATV lasers", J. Lightwave Technol. 13, 252-260 (1995); http://www.ee.sunysb.edu/~serge/133.html [CrossRef]
- M. Silver, E. P. OReilly, and A. R. Adams, Determination of the wavelength dependence of Auger recombination in long-wavelength quantum-well semiconductor lasers using hydrostatic pressure, IEEE J. Quantum Electron. 33, 1557-1566 (1997). [CrossRef]
- Z. Shi, M. Tacke, A. Lambrecht, and H. Bttner, Midinfrared lead salt multi-quantum-well diode lasers with 282 K operation, Appl. Phys. Lett. 66, 2537-2539 (1995). [CrossRef]
- H. K. Choi, G. W. Turner, and H. Q. Le, InAsSb/InAlAs strained quantum-well lasers emitting at 4.5 Pm, Appl. Phys. Lett. 66, 3543-3545 (1995). [CrossRef]
- J. R. Meyer, I. Vurgaftman, R. Q. Yang, and L. R. Ram-Mohan, Type-II and Type-I interband cascade lasers, Electron Lett. 32, 45-46 (1996). [CrossRef]
- J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, High power mid-infrared (OaPm) quantum cascade lasers operating above room temperature, Appl. Phys. Lett. 68, 3680-3682 (1996). [CrossRef]
- Vera Gorfinkel, Serge Luryi, and Boris Gelmont, "Theory of gain spectra for quantum cascade lasers and temperature dependence of their characteristics at low and moderate carrier concentrations", IEEE J. Quantum Electron. 32, 1995-2003 (1996); http://www.ee.sunysb.edu/~serge/145.html [CrossRef]
- M. V. Kisin, V. B. Gorfinkel, M. A. Stroscio, G. Belenky, and S. Luryi, Influence of complex phonon spectra on intersubband optical gain, J. Appl. Phys. 82, 2031-2038 (1997); http://www.ee.sunysb.edu/~serge/148.pdf [CrossRef]

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