Abstract

The feasibility of population inversion, by impurity-scattering enhancement of the acoustic-phonon-limited lower-laser-level intersubband relaxation rate, is theoretically investigated in nonpolar three-level SiGe/Si systems. The dependence of the acoustic-phonon depopulating rate on the barrier thickness and the effect of the position of a δ-doped region on the impurity scattering are treated rigorously. A 10-Å-doped region with a 1010 or a 5×1010 cm-2 sheet carrier density enhances the acoustic-phonon-limited depopulating rate by more than one or two orders of magnitude, respectively. Thus for equal barrier widths between the depopulating and the lasing levels, the depopulating rate becomes at least an order of magnitude (1010 cm-2 doping) or a factor of 2–4 (5×1010 cm-2 doping) faster than the lasing transition's acoustic- or optical-phonon limit, respectively. This allows for the design of nonpolar intersubband lasers, in which population inversion between discrete valence-band states is achieved by impurity-scattering enhancement of the acoustic-phonon-limited depopulating rate.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Kastalsky, V. J. Goldman, and J. Abeles, “Possibility of infrared laser in a resonant tunneling structure,” Appl. Phys. Lett. 59, 2636 (1991); J. P. Loehr, J. Singh, R. K. Mains, and G. I. Haddad, “Theoretical studies of the applications of resonant tunneling diodes as intersubband laser and interband excitonic modulators,” Appl. Phys. Lett. 59, 2070 (1991); S. I. Borenstain and J. Katz, “Evaluation of the feasibility of a far-infrared laser based on intersubband transitions in GaAs quantum wells,” Appl. Phys. Lett. APPLAB 55, 654 (1989); J. Faist, F. Capasso, C. Sirtori, D. Sivco, A. L. Hutchinson, S. N. G. Chu, and A. Y. Cho, “Mid-infrared field-tunable intersubband electroluminescence at room temperature by photon-assisted tunneling in coupled-quantum wells,” Appl. Phys. Lett. APPLAB 64, 1144 (1994); J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, S. N. G. Chu, and A. Y. Cho, “Quantum-well intersubband electroluminescent diode at λ=5 μm,” Electron. Lett. ELLEAK 29, 2230 (1993).
    [CrossRef]
  2. G. Sun and J. B. Khurgin, “Optically pumped four-level infrared laser based on intersubband transitions in multiple quantum wells: feasibility study,” IEEE J. Quantum Electron. 29, 1104 (1993).
    [CrossRef]
  3. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553 (1994).
    [CrossRef] [PubMed]
  4. R. A. Soref, “Silicon based optoelectronics,” Proc. IEEE 81, 1687 (1993).
    [CrossRef]
  5. G. Sun, L. Friedman, and R. A. Soref, “Intersubband lasing lifetimes of SiGe/Si and GaAs/AlGaAs multiple quantum well structures,” Appl. Phys. Lett. 66, 3425 (1995).
    [CrossRef]
  6. J. V. D. Veliadis, J. B. Khurgin, and Y. J. Ding, “Engineering of the nonradiative transition rates in modulation doped multiple quantum wells,” IEEE J. Quantum Electron. 32, 1155 (1996).
    [CrossRef]
  7. B. K. Ridley, “The electron-phonon interaction in quasi-two-dimensional semiconductor quantum well structures,” J. Phys. C 15, 5899 (1982); P. J. Price, “Two-dimensional electron transport in semiconductor layers,” Ann. Phys. 133, 217 (1981).
    [CrossRef]
  8. R. Ferreira and G. Bastard, “Evaluation of some scattering times for electrons in unbiased and biased single- and multiple-quantum-well structures,” Phys. Rev. B 40, 1074 (1989).
    [CrossRef]
  9. The three-dimensional (3D) screening parameter λ is defined as λ2=4πn3De2/εkBT with n3D=n2D/d= kF2/2πd. Thus λ2=kF2 rexd 2e2εrex 1kBT=kF2 rexd 2EbindkBT≈kF2, where rex and Ebind are the exciton radius and the binding energy, respectively. To neglect screening, Δk≫λ(≅kF), where Δk is the change in the in-plane wave vector that is due to the impurity scattering. Δk≫kF=(2πn2D)1/2⇒ n2D≪Δk2/2π=9.2×1011 cm−2. In the impurity-scattering calculations, doping concentrations of 1010 and 5×1010 cm−2 were assumed. Therefore according to the rough calculation outlined above, we can neglect the screening effect.
  10. J. F. Young, D. J. Lockwood, J. M. Baribeau, and P. J. Kelly, in Light Scattering in Semiconductor Structures and Superlattices, D. J. Lockwood and J. F. Young, eds. (Plenum, New York, 1991), Vol. 41; A. Blancha, H. Presting, and M. Cardona, “Deformation potentials of k=0 states of tetrahedral semiconductors,” Phys. Status Solidi B 126, 11 (1984); B. K. Ridley, Quantum Processes in Semiconductors (Clarendon, Oxford, 1982).
    [CrossRef]
  11. G. Bastard, Wave Mechanics Applied to Semiconductors Heterostructures (Les Editions de Physique, Paris, 1990), Chap. 3.
  12. G. Sun and L. Friedman, “Heavy-hole scattering by confined nonpolar optical phonons in a single Si1−xGex/Si quantum well,” Phys. Rev. B 53, 3966 (1996).
    [CrossRef]

1996 (2)

J. V. D. Veliadis, J. B. Khurgin, and Y. J. Ding, “Engineering of the nonradiative transition rates in modulation doped multiple quantum wells,” IEEE J. Quantum Electron. 32, 1155 (1996).
[CrossRef]

G. Sun and L. Friedman, “Heavy-hole scattering by confined nonpolar optical phonons in a single Si1−xGex/Si quantum well,” Phys. Rev. B 53, 3966 (1996).
[CrossRef]

1995 (1)

G. Sun, L. Friedman, and R. A. Soref, “Intersubband lasing lifetimes of SiGe/Si and GaAs/AlGaAs multiple quantum well structures,” Appl. Phys. Lett. 66, 3425 (1995).
[CrossRef]

1994 (1)

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553 (1994).
[CrossRef] [PubMed]

1993 (2)

R. A. Soref, “Silicon based optoelectronics,” Proc. IEEE 81, 1687 (1993).
[CrossRef]

G. Sun and J. B. Khurgin, “Optically pumped four-level infrared laser based on intersubband transitions in multiple quantum wells: feasibility study,” IEEE J. Quantum Electron. 29, 1104 (1993).
[CrossRef]

1989 (1)

R. Ferreira and G. Bastard, “Evaluation of some scattering times for electrons in unbiased and biased single- and multiple-quantum-well structures,” Phys. Rev. B 40, 1074 (1989).
[CrossRef]

Bastard, G.

R. Ferreira and G. Bastard, “Evaluation of some scattering times for electrons in unbiased and biased single- and multiple-quantum-well structures,” Phys. Rev. B 40, 1074 (1989).
[CrossRef]

Capasso, F.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553 (1994).
[CrossRef] [PubMed]

Cho, A. Y.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553 (1994).
[CrossRef] [PubMed]

Ding, Y. J.

J. V. D. Veliadis, J. B. Khurgin, and Y. J. Ding, “Engineering of the nonradiative transition rates in modulation doped multiple quantum wells,” IEEE J. Quantum Electron. 32, 1155 (1996).
[CrossRef]

Faist, J.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553 (1994).
[CrossRef] [PubMed]

Ferreira, R.

R. Ferreira and G. Bastard, “Evaluation of some scattering times for electrons in unbiased and biased single- and multiple-quantum-well structures,” Phys. Rev. B 40, 1074 (1989).
[CrossRef]

Friedman, L.

G. Sun and L. Friedman, “Heavy-hole scattering by confined nonpolar optical phonons in a single Si1−xGex/Si quantum well,” Phys. Rev. B 53, 3966 (1996).
[CrossRef]

G. Sun, L. Friedman, and R. A. Soref, “Intersubband lasing lifetimes of SiGe/Si and GaAs/AlGaAs multiple quantum well structures,” Appl. Phys. Lett. 66, 3425 (1995).
[CrossRef]

Hutchinson, A. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553 (1994).
[CrossRef] [PubMed]

Khurgin, J. B.

J. V. D. Veliadis, J. B. Khurgin, and Y. J. Ding, “Engineering of the nonradiative transition rates in modulation doped multiple quantum wells,” IEEE J. Quantum Electron. 32, 1155 (1996).
[CrossRef]

G. Sun and J. B. Khurgin, “Optically pumped four-level infrared laser based on intersubband transitions in multiple quantum wells: feasibility study,” IEEE J. Quantum Electron. 29, 1104 (1993).
[CrossRef]

Sirtori, C.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553 (1994).
[CrossRef] [PubMed]

Sivco, D. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553 (1994).
[CrossRef] [PubMed]

Soref, R. A.

G. Sun, L. Friedman, and R. A. Soref, “Intersubband lasing lifetimes of SiGe/Si and GaAs/AlGaAs multiple quantum well structures,” Appl. Phys. Lett. 66, 3425 (1995).
[CrossRef]

R. A. Soref, “Silicon based optoelectronics,” Proc. IEEE 81, 1687 (1993).
[CrossRef]

Sun, G.

G. Sun and L. Friedman, “Heavy-hole scattering by confined nonpolar optical phonons in a single Si1−xGex/Si quantum well,” Phys. Rev. B 53, 3966 (1996).
[CrossRef]

G. Sun, L. Friedman, and R. A. Soref, “Intersubband lasing lifetimes of SiGe/Si and GaAs/AlGaAs multiple quantum well structures,” Appl. Phys. Lett. 66, 3425 (1995).
[CrossRef]

G. Sun and J. B. Khurgin, “Optically pumped four-level infrared laser based on intersubband transitions in multiple quantum wells: feasibility study,” IEEE J. Quantum Electron. 29, 1104 (1993).
[CrossRef]

Veliadis, J. V. D.

J. V. D. Veliadis, J. B. Khurgin, and Y. J. Ding, “Engineering of the nonradiative transition rates in modulation doped multiple quantum wells,” IEEE J. Quantum Electron. 32, 1155 (1996).
[CrossRef]

Appl. Phys. Lett. (1)

G. Sun, L. Friedman, and R. A. Soref, “Intersubband lasing lifetimes of SiGe/Si and GaAs/AlGaAs multiple quantum well structures,” Appl. Phys. Lett. 66, 3425 (1995).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. V. D. Veliadis, J. B. Khurgin, and Y. J. Ding, “Engineering of the nonradiative transition rates in modulation doped multiple quantum wells,” IEEE J. Quantum Electron. 32, 1155 (1996).
[CrossRef]

G. Sun and J. B. Khurgin, “Optically pumped four-level infrared laser based on intersubband transitions in multiple quantum wells: feasibility study,” IEEE J. Quantum Electron. 29, 1104 (1993).
[CrossRef]

Phys. Rev. B (2)

R. Ferreira and G. Bastard, “Evaluation of some scattering times for electrons in unbiased and biased single- and multiple-quantum-well structures,” Phys. Rev. B 40, 1074 (1989).
[CrossRef]

G. Sun and L. Friedman, “Heavy-hole scattering by confined nonpolar optical phonons in a single Si1−xGex/Si quantum well,” Phys. Rev. B 53, 3966 (1996).
[CrossRef]

Proc. IEEE (1)

R. A. Soref, “Silicon based optoelectronics,” Proc. IEEE 81, 1687 (1993).
[CrossRef]

Science (1)

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553 (1994).
[CrossRef] [PubMed]

Other (5)

A. Kastalsky, V. J. Goldman, and J. Abeles, “Possibility of infrared laser in a resonant tunneling structure,” Appl. Phys. Lett. 59, 2636 (1991); J. P. Loehr, J. Singh, R. K. Mains, and G. I. Haddad, “Theoretical studies of the applications of resonant tunneling diodes as intersubband laser and interband excitonic modulators,” Appl. Phys. Lett. 59, 2070 (1991); S. I. Borenstain and J. Katz, “Evaluation of the feasibility of a far-infrared laser based on intersubband transitions in GaAs quantum wells,” Appl. Phys. Lett. APPLAB 55, 654 (1989); J. Faist, F. Capasso, C. Sirtori, D. Sivco, A. L. Hutchinson, S. N. G. Chu, and A. Y. Cho, “Mid-infrared field-tunable intersubband electroluminescence at room temperature by photon-assisted tunneling in coupled-quantum wells,” Appl. Phys. Lett. APPLAB 64, 1144 (1994); J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, S. N. G. Chu, and A. Y. Cho, “Quantum-well intersubband electroluminescent diode at λ=5 μm,” Electron. Lett. ELLEAK 29, 2230 (1993).
[CrossRef]

The three-dimensional (3D) screening parameter λ is defined as λ2=4πn3De2/εkBT with n3D=n2D/d= kF2/2πd. Thus λ2=kF2 rexd 2e2εrex 1kBT=kF2 rexd 2EbindkBT≈kF2, where rex and Ebind are the exciton radius and the binding energy, respectively. To neglect screening, Δk≫λ(≅kF), where Δk is the change in the in-plane wave vector that is due to the impurity scattering. Δk≫kF=(2πn2D)1/2⇒ n2D≪Δk2/2π=9.2×1011 cm−2. In the impurity-scattering calculations, doping concentrations of 1010 and 5×1010 cm−2 were assumed. Therefore according to the rough calculation outlined above, we can neglect the screening effect.

J. F. Young, D. J. Lockwood, J. M. Baribeau, and P. J. Kelly, in Light Scattering in Semiconductor Structures and Superlattices, D. J. Lockwood and J. F. Young, eds. (Plenum, New York, 1991), Vol. 41; A. Blancha, H. Presting, and M. Cardona, “Deformation potentials of k=0 states of tetrahedral semiconductors,” Phys. Status Solidi B 126, 11 (1984); B. K. Ridley, Quantum Processes in Semiconductors (Clarendon, Oxford, 1982).
[CrossRef]

G. Bastard, Wave Mechanics Applied to Semiconductors Heterostructures (Les Editions de Physique, Paris, 1990), Chap. 3.

B. K. Ridley, “The electron-phonon interaction in quasi-two-dimensional semiconductor quantum well structures,” J. Phys. C 15, 5899 (1982); P. J. Price, “Two-dimensional electron transport in semiconductor layers,” Ann. Phys. 133, 217 (1981).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

One period of heavy-hole valence-band diagram of the proposed three-level intersubband laser. Hole energy increases in the upward direction. The lasing transition is between levels 3 and 2. The 21 acoustic-phonon-transition limit is enhanced through impurity scattering. VHH is the heavy-hole valence-band offset. Indicated are the well and the barrier width symbols used in Eq. (7), and the subband levels and transitions.

Fig. 2
Fig. 2

Heavy-hole valence-band diagram of the DACQW structure where the lower-laser (2) and the ground (1) states are mainly localized. Hole energy increases in the upward direction.

Fig. 3
Fig. 3

Filled circles represent the acoustic-phonon intersubband transition rate as a function of the barrier width separating the lower-laser from the ground states. The well sizes are 20 Å and 30 Å. The solid curve fit represents the exponential decay of the acoustic-phonon rate with increasing barrier width.

Fig. 4
Fig. 4

Solid curve represents the 21 impurity intersubband transition rate Wimp as a function of the position of the 10-Å-doped region in the DACQW shown in the inset. The dotted-dashed line represents the 21 acoustic-phonon intersubband transition rate Wac. The dashed curve represents the total 21 intersubband transition rate Wtot.

Fig. 5
Fig. 5

32 phonon intersubband transition rate (filled circles) and the E3E2 subband-energy difference (triangles, solid curve) as a function of the upper-laser-level well width L. The upper- and the lower-laser-level wells and their separating barrier are shown in the inset.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

Wifac=Ξ2kBTm*4πcL3|Gif(qz)|2dqz,
Wifop=m*[n(ω0)+1/2±1/2]×D02/4πρ2ω0|Gif(qz)|2dqz,
Gif(qz)=f|exp(iqzz)|i.
Wifimp=2πNimp(Ei-Ef)e2r2|F(zimp, kf)|2
Ei-Ef=2kf22m*.
F(zimp, kf)=z=- χi(z)exp(-kf|z-zimp|)χf*(z)dz
Wifimp=Γ|{exp[-kf(zimp-w2)][M2+A exp(kfw2)+B exp(kfb1)]+exp[kf(w3-w1)]M1+M3}/1+exp(-kfw3)[C+DTf+ETi+FTfTi+G exp(kfb1)]/2+{H exp(-kfzimp)[1-exp(-3b2)]+I exp[kf(w3-w1)]×[1-exp(-3bper)]}/3|2,
Γ=2πNimpEi-Efe2r2,
1=(kzf2+kzi2+kf2)2-4kzf2kzi2,
2=[(qzf-qzi)2-kf2][(qzf+qzi)2-kf2],
3=qzf+qzi+kf,
Tf(i)=[exp(qzf(i)b1)]2,
Mj=lj cos(k- wj)+mj cos(k+ wj)+nj sin(k- wj)+oj sin(k+ wj),j=1,2,3,
λ2=kF2rexd2e2rex1kBT=kF2rexd2EbindkBTkF2,

Metrics