Abstract

In long wavelength quantum well lasers the effective electron temperature (Te) is often a strong function of the pump current and hence the Te correlates with the carrier concentration n in the active region. On the other hand, the material gain g in the active layer depends on both variables, g=g(n,Te). We discuss a convenient way of analyzing this situation, based on considering the contours of constant gain g on the surface g (n,Te). 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.

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References

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  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
  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]
  3. 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]
  4. 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]
  5. 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]
  6. 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]
  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 (OaPm) quantum cascade lasers operating above room temperature, Appl. Phys. Lett. 68, 3680-3682 (1996).
    [CrossRef]
  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]
  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]

Other (9)

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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Figures (5)

Figure 1.
Figure 1.

Dependence of the modal gain (3) on the carrier concentration n and temperature Te . Device comprises N=10 quantum wells of width 15 nm, g 0 = 103 cm-1, the mode confinement factor Γ=0.01N =0.1, the effective energy transferred per carrier pair E eff = 1.5E GE G = 0.75 eV , where E G=0.3eV is the bandgap in the active region and ΔE G=0.3eV is the band discontinuity between the active region and the cladding; the energy relaxation time τε =10-12s, the Auger coefficient C A=10-26 cm6/s , and the effective carrier masses are m e=0.025 m 0 and m h=20 m e.

Figure 2.
Figure 2.

The contours of constant gain for the g (n,Te) of Fig.1. The isogain curves are shown in blue color for selected values of modal gain g=α indicated in units of cm-1. Red curves show the relation between T e and n from the energy balance (2).

Figure 3.
Figure 3.

Temperature dependence of the intersubband transition rate in a model quantum cascade laser.

Figure 4.
Figure 4.

The 2D surface g(n,T e ) for a model quantum cascade laser.

Figure 5.
Figure 5.

The contours of constant gain (blue) for the g(n,Te ) surface of Fig.4. Red lines indicate the carrier temperature as fixed by equations (5) and (6). In the present model of carrier heating the carrier temperature T e is independent of n.

Equations (7)

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g ( n , T e ) = α .
nk ( T e T ) τ ε = E eff C A n 3 .
g ( n , T e ) = Γ g 0 ( 1 f e f h ) ,
f e ( n , T e ) = 1 e π ħ 2 n m e k T e
f h ( n , T e ) = 1 e π ħ 2 n m h k T e
nk ( T e T ) τ ε = n 2 τ 21
n 1 n 2 = τ 1 out τ 21

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