Abstract

A novel simulation strategy is proposed for searching for semiconductor quantum devices that are optimized with respect to required performances. Based on evolutionary programming, a technique that implements the paradigm of genetic algorithms in more-complex data structures than strings of bits, the proposed algorithm is able to deal with quantum devices with preset nontrivial constraints (e.g., transition energies, geometric requirements). Therefore our approach allows for automatic design, thus avoiding costly by-hand optimizations. We demonstrate the advantages of the proposed algorithm through a relevant and nontrivial application, the optimization of a second-harmonic-generation device working in resonance conditions.

© 2000 Optical Society of America

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  1. See, e.g., H. L. Stormer, D. C. Tsui, and A. C. Gossard, Rev. Mod. Phys. 71, S298 (1999).
    [CrossRef]
  2. F. Capasso, C. Gmachl, D. L. Sivco, and A. Y. Cho, Phys. World12(6), 27 (1999); F. Capasso, ed., Physics of Quantum Electron Devices (Springer-Verlag, Berlin, 1990).
  3. J. Faist, F. Capasso, and D. L. Sivco, Science 264, 553 (1994); G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, Science 276, 773 (1997); A. Tredicucci, C. Gmachl, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, Nature 396, 350 (1998); C. Sirtori, P. Kruck, S. Barbieri, P. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, Appl. Phys. Lett. 73, 3486 (1998).
    [CrossRef] [PubMed]
  4. F. Capasso, W. T. Tsang, and G. F. Williams, IEEE Trans. Electron Devices ED-30, 381 (1983); B. F. Levine, J. Appl. Phys. 74, R1 (1993); S. Barbieri, F. Mango, F. Beltram, M. Lazzarino, and L. Sorba, Appl. Phys. Lett. 67, 250 (1995).
    [CrossRef]
  5. Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs (Springer-Verlag, Berlin, 1992).
  6. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989); M. Gen and R. Cheng, Genetic Algorithms & Engineering Design (Wiley, New York, 1997).
  7. D. M. Deaven and K. M. Ho, Phys. Rev. Lett. 75, 288 (1995).
    [CrossRef] [PubMed]
  8. Generalizations to more-complex structures such as ternary alloys and doped layers is straightforward.
  9. For example, for a type I AlxGa1-xAs heterostructure we typically require that 0<xi<0.4.
  10. In the proposed implementation, we were inspired by the GENOCOP system described in Ref. 5, but additional operators have also been introduced that are suited to the specific problem.
  11. One can, at least in principle, write any property of the quantum structure in terms of the orthonormal basis set obtained from the single-particle wave functions.
  12. The parameters are as follows: The AlxGa1-xAs bandgap Egx is obtained from Egx=EgGaAs+1.36x+0.22x2 [C. Bosio, J. L. Staelhi, M. Guzzi, G. Burri, and R. A. Logan, Phys. Rev. B 38, 3263 (1998)]. The valence-band contribution is ΔEvx=0.48x [E. T. Yu, J. O. McCaldin, and T. C. McGill, Solid State Phys. 46, 2 (1992)]. The space-dependent effective mass is mex=0.067+0.083x [O. Madelung, ed., Semiconductors: Physics of Group IV Elements and III–IV Compounds, Vol. 17 of Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1982)].
    [CrossRef]
  13. E. Rosencher and Ph. Bois, Phys. Rev. B 44, 11315 (1991).
    [CrossRef]
  14. S. Tomić, V. Milanović, and Z. Ikonić, Phys. Rev. B 56, 1033 (1997); J. Phys. Condens. Matter 10, 6523 (1998).
    [CrossRef]
  15. The optimized potential in Ref. 14 spans more than 0.5 eV and cannot be implemented as a strictly type I structure. A distinct advantage of the present method is the easy implementation of physical limitations, such as alloy concentrations, on the parameter space.

1999

See, e.g., H. L. Stormer, D. C. Tsui, and A. C. Gossard, Rev. Mod. Phys. 71, S298 (1999).
[CrossRef]

1998

The parameters are as follows: The AlxGa1-xAs bandgap Egx is obtained from Egx=EgGaAs+1.36x+0.22x2 [C. Bosio, J. L. Staelhi, M. Guzzi, G. Burri, and R. A. Logan, Phys. Rev. B 38, 3263 (1998)]. The valence-band contribution is ΔEvx=0.48x [E. T. Yu, J. O. McCaldin, and T. C. McGill, Solid State Phys. 46, 2 (1992)]. The space-dependent effective mass is mex=0.067+0.083x [O. Madelung, ed., Semiconductors: Physics of Group IV Elements and III–IV Compounds, Vol. 17 of Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1982)].
[CrossRef]

1997

S. Tomić, V. Milanović, and Z. Ikonić, Phys. Rev. B 56, 1033 (1997); J. Phys. Condens. Matter 10, 6523 (1998).
[CrossRef]

1995

D. M. Deaven and K. M. Ho, Phys. Rev. Lett. 75, 288 (1995).
[CrossRef] [PubMed]

1994

J. Faist, F. Capasso, and D. L. Sivco, Science 264, 553 (1994); G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, Science 276, 773 (1997); A. Tredicucci, C. Gmachl, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, Nature 396, 350 (1998); C. Sirtori, P. Kruck, S. Barbieri, P. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, Appl. Phys. Lett. 73, 3486 (1998).
[CrossRef] [PubMed]

1991

E. Rosencher and Ph. Bois, Phys. Rev. B 44, 11315 (1991).
[CrossRef]

1983

F. Capasso, W. T. Tsang, and G. F. Williams, IEEE Trans. Electron Devices ED-30, 381 (1983); B. F. Levine, J. Appl. Phys. 74, R1 (1993); S. Barbieri, F. Mango, F. Beltram, M. Lazzarino, and L. Sorba, Appl. Phys. Lett. 67, 250 (1995).
[CrossRef]

Bois, Ph.

E. Rosencher and Ph. Bois, Phys. Rev. B 44, 11315 (1991).
[CrossRef]

Bosio, C.

The parameters are as follows: The AlxGa1-xAs bandgap Egx is obtained from Egx=EgGaAs+1.36x+0.22x2 [C. Bosio, J. L. Staelhi, M. Guzzi, G. Burri, and R. A. Logan, Phys. Rev. B 38, 3263 (1998)]. The valence-band contribution is ΔEvx=0.48x [E. T. Yu, J. O. McCaldin, and T. C. McGill, Solid State Phys. 46, 2 (1992)]. The space-dependent effective mass is mex=0.067+0.083x [O. Madelung, ed., Semiconductors: Physics of Group IV Elements and III–IV Compounds, Vol. 17 of Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1982)].
[CrossRef]

Burri, G.

The parameters are as follows: The AlxGa1-xAs bandgap Egx is obtained from Egx=EgGaAs+1.36x+0.22x2 [C. Bosio, J. L. Staelhi, M. Guzzi, G. Burri, and R. A. Logan, Phys. Rev. B 38, 3263 (1998)]. The valence-band contribution is ΔEvx=0.48x [E. T. Yu, J. O. McCaldin, and T. C. McGill, Solid State Phys. 46, 2 (1992)]. The space-dependent effective mass is mex=0.067+0.083x [O. Madelung, ed., Semiconductors: Physics of Group IV Elements and III–IV Compounds, Vol. 17 of Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1982)].
[CrossRef]

Capasso, F.

J. Faist, F. Capasso, and D. L. Sivco, Science 264, 553 (1994); G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, Science 276, 773 (1997); A. Tredicucci, C. Gmachl, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, Nature 396, 350 (1998); C. Sirtori, P. Kruck, S. Barbieri, P. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, Appl. Phys. Lett. 73, 3486 (1998).
[CrossRef] [PubMed]

F. Capasso, W. T. Tsang, and G. F. Williams, IEEE Trans. Electron Devices ED-30, 381 (1983); B. F. Levine, J. Appl. Phys. 74, R1 (1993); S. Barbieri, F. Mango, F. Beltram, M. Lazzarino, and L. Sorba, Appl. Phys. Lett. 67, 250 (1995).
[CrossRef]

F. Capasso, C. Gmachl, D. L. Sivco, and A. Y. Cho, Phys. World12(6), 27 (1999); F. Capasso, ed., Physics of Quantum Electron Devices (Springer-Verlag, Berlin, 1990).

Cho, A. Y.

F. Capasso, C. Gmachl, D. L. Sivco, and A. Y. Cho, Phys. World12(6), 27 (1999); F. Capasso, ed., Physics of Quantum Electron Devices (Springer-Verlag, Berlin, 1990).

Deaven, D. M.

D. M. Deaven and K. M. Ho, Phys. Rev. Lett. 75, 288 (1995).
[CrossRef] [PubMed]

Faist, J.

J. Faist, F. Capasso, and D. L. Sivco, Science 264, 553 (1994); G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, Science 276, 773 (1997); A. Tredicucci, C. Gmachl, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, Nature 396, 350 (1998); C. Sirtori, P. Kruck, S. Barbieri, P. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, Appl. Phys. Lett. 73, 3486 (1998).
[CrossRef] [PubMed]

Gmachl, C.

F. Capasso, C. Gmachl, D. L. Sivco, and A. Y. Cho, Phys. World12(6), 27 (1999); F. Capasso, ed., Physics of Quantum Electron Devices (Springer-Verlag, Berlin, 1990).

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989); M. Gen and R. Cheng, Genetic Algorithms & Engineering Design (Wiley, New York, 1997).

Gossard, A. C.

See, e.g., H. L. Stormer, D. C. Tsui, and A. C. Gossard, Rev. Mod. Phys. 71, S298 (1999).
[CrossRef]

Guzzi, M.

The parameters are as follows: The AlxGa1-xAs bandgap Egx is obtained from Egx=EgGaAs+1.36x+0.22x2 [C. Bosio, J. L. Staelhi, M. Guzzi, G. Burri, and R. A. Logan, Phys. Rev. B 38, 3263 (1998)]. The valence-band contribution is ΔEvx=0.48x [E. T. Yu, J. O. McCaldin, and T. C. McGill, Solid State Phys. 46, 2 (1992)]. The space-dependent effective mass is mex=0.067+0.083x [O. Madelung, ed., Semiconductors: Physics of Group IV Elements and III–IV Compounds, Vol. 17 of Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1982)].
[CrossRef]

Ho, K. M.

D. M. Deaven and K. M. Ho, Phys. Rev. Lett. 75, 288 (1995).
[CrossRef] [PubMed]

Ikonic, Z.

S. Tomić, V. Milanović, and Z. Ikonić, Phys. Rev. B 56, 1033 (1997); J. Phys. Condens. Matter 10, 6523 (1998).
[CrossRef]

Logan, R. A.

The parameters are as follows: The AlxGa1-xAs bandgap Egx is obtained from Egx=EgGaAs+1.36x+0.22x2 [C. Bosio, J. L. Staelhi, M. Guzzi, G. Burri, and R. A. Logan, Phys. Rev. B 38, 3263 (1998)]. The valence-band contribution is ΔEvx=0.48x [E. T. Yu, J. O. McCaldin, and T. C. McGill, Solid State Phys. 46, 2 (1992)]. The space-dependent effective mass is mex=0.067+0.083x [O. Madelung, ed., Semiconductors: Physics of Group IV Elements and III–IV Compounds, Vol. 17 of Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1982)].
[CrossRef]

Michalewicz, Z.

Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs (Springer-Verlag, Berlin, 1992).

Milanovic, V.

S. Tomić, V. Milanović, and Z. Ikonić, Phys. Rev. B 56, 1033 (1997); J. Phys. Condens. Matter 10, 6523 (1998).
[CrossRef]

Rosencher, E.

E. Rosencher and Ph. Bois, Phys. Rev. B 44, 11315 (1991).
[CrossRef]

Sivco, D. L.

J. Faist, F. Capasso, and D. L. Sivco, Science 264, 553 (1994); G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, Science 276, 773 (1997); A. Tredicucci, C. Gmachl, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, Nature 396, 350 (1998); C. Sirtori, P. Kruck, S. Barbieri, P. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, Appl. Phys. Lett. 73, 3486 (1998).
[CrossRef] [PubMed]

F. Capasso, C. Gmachl, D. L. Sivco, and A. Y. Cho, Phys. World12(6), 27 (1999); F. Capasso, ed., Physics of Quantum Electron Devices (Springer-Verlag, Berlin, 1990).

Staelhi, J. L.

The parameters are as follows: The AlxGa1-xAs bandgap Egx is obtained from Egx=EgGaAs+1.36x+0.22x2 [C. Bosio, J. L. Staelhi, M. Guzzi, G. Burri, and R. A. Logan, Phys. Rev. B 38, 3263 (1998)]. The valence-band contribution is ΔEvx=0.48x [E. T. Yu, J. O. McCaldin, and T. C. McGill, Solid State Phys. 46, 2 (1992)]. The space-dependent effective mass is mex=0.067+0.083x [O. Madelung, ed., Semiconductors: Physics of Group IV Elements and III–IV Compounds, Vol. 17 of Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1982)].
[CrossRef]

Stormer, H. L.

See, e.g., H. L. Stormer, D. C. Tsui, and A. C. Gossard, Rev. Mod. Phys. 71, S298 (1999).
[CrossRef]

Tomic, S.

S. Tomić, V. Milanović, and Z. Ikonić, Phys. Rev. B 56, 1033 (1997); J. Phys. Condens. Matter 10, 6523 (1998).
[CrossRef]

Tsang, W. T.

F. Capasso, W. T. Tsang, and G. F. Williams, IEEE Trans. Electron Devices ED-30, 381 (1983); B. F. Levine, J. Appl. Phys. 74, R1 (1993); S. Barbieri, F. Mango, F. Beltram, M. Lazzarino, and L. Sorba, Appl. Phys. Lett. 67, 250 (1995).
[CrossRef]

Tsui, D. C.

See, e.g., H. L. Stormer, D. C. Tsui, and A. C. Gossard, Rev. Mod. Phys. 71, S298 (1999).
[CrossRef]

Williams, G. F.

F. Capasso, W. T. Tsang, and G. F. Williams, IEEE Trans. Electron Devices ED-30, 381 (1983); B. F. Levine, J. Appl. Phys. 74, R1 (1993); S. Barbieri, F. Mango, F. Beltram, M. Lazzarino, and L. Sorba, Appl. Phys. Lett. 67, 250 (1995).
[CrossRef]

IEEE Trans. Electron Devices

F. Capasso, W. T. Tsang, and G. F. Williams, IEEE Trans. Electron Devices ED-30, 381 (1983); B. F. Levine, J. Appl. Phys. 74, R1 (1993); S. Barbieri, F. Mango, F. Beltram, M. Lazzarino, and L. Sorba, Appl. Phys. Lett. 67, 250 (1995).
[CrossRef]

Phys. Rev. B

The parameters are as follows: The AlxGa1-xAs bandgap Egx is obtained from Egx=EgGaAs+1.36x+0.22x2 [C. Bosio, J. L. Staelhi, M. Guzzi, G. Burri, and R. A. Logan, Phys. Rev. B 38, 3263 (1998)]. The valence-band contribution is ΔEvx=0.48x [E. T. Yu, J. O. McCaldin, and T. C. McGill, Solid State Phys. 46, 2 (1992)]. The space-dependent effective mass is mex=0.067+0.083x [O. Madelung, ed., Semiconductors: Physics of Group IV Elements and III–IV Compounds, Vol. 17 of Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1982)].
[CrossRef]

E. Rosencher and Ph. Bois, Phys. Rev. B 44, 11315 (1991).
[CrossRef]

S. Tomić, V. Milanović, and Z. Ikonić, Phys. Rev. B 56, 1033 (1997); J. Phys. Condens. Matter 10, 6523 (1998).
[CrossRef]

Phys. Rev. Lett.

D. M. Deaven and K. M. Ho, Phys. Rev. Lett. 75, 288 (1995).
[CrossRef] [PubMed]

Rev. Mod. Phys.

See, e.g., H. L. Stormer, D. C. Tsui, and A. C. Gossard, Rev. Mod. Phys. 71, S298 (1999).
[CrossRef]

Science

J. Faist, F. Capasso, and D. L. Sivco, Science 264, 553 (1994); G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, Science 276, 773 (1997); A. Tredicucci, C. Gmachl, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, Nature 396, 350 (1998); C. Sirtori, P. Kruck, S. Barbieri, P. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, Appl. Phys. Lett. 73, 3486 (1998).
[CrossRef] [PubMed]

Other

The optimized potential in Ref. 14 spans more than 0.5 eV and cannot be implemented as a strictly type I structure. A distinct advantage of the present method is the easy implementation of physical limitations, such as alloy concentrations, on the parameter space.

F. Capasso, C. Gmachl, D. L. Sivco, and A. Y. Cho, Phys. World12(6), 27 (1999); F. Capasso, ed., Physics of Quantum Electron Devices (Springer-Verlag, Berlin, 1990).

Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs (Springer-Verlag, Berlin, 1992).

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989); M. Gen and R. Cheng, Genetic Algorithms & Engineering Design (Wiley, New York, 1997).

Generalizations to more-complex structures such as ternary alloys and doped layers is straightforward.

For example, for a type I AlxGa1-xAs heterostructure we typically require that 0<xi<0.4.

In the proposed implementation, we were inspired by the GENOCOP system described in Ref. 5, but additional operators have also been introduced that are suited to the specific problem.

One can, at least in principle, write any property of the quantum structure in terms of the orthonormal basis set obtained from the single-particle wave functions.

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Figures (2)

Fig. 1
Fig. 1

Potential profile and ψiz2 for the lowest three confined states, shifted vertically by the confinement energy. The simulation was performed with Nl=30 and Np=100. Probabilities pc and pm were in the range 0.07,0.1. Resonance with pumping radiation ω=116 meV was enforced. The estimated μ=4.10 nm3 corresponds to μ12=1.725 nm, μ23=2.609 nm, and μ31=0.9104 nm.

Fig. 2
Fig. 2

Fitness-function values, expression (1) (top right-hand axis), and SHG intensity, μ (bottom left-hand axis), for a typical run (not the same run leading to the results shown in Fig. 1 but with same parameters). The two axes are vertically shifted for clarity. The solid curve in the SHG intensity plot traces the fittest chromosome at each iteration.

Tables (1)

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Table 1 Description of Crossover (C) and Mutation (M) Mechanisms

Equations (1)

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Vψnexp±Pψn/δpt;

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