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Electronic Properties of III-V Quantum Dots

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Multi-Band Effective Mass Approximations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 94))

Abstract

Electronic properties of quantum dots are reviewed based on eight-band k⋅ p theory. We will focus on the following interrelated subjects: First the role of crystallographic symmetry is evaluated. This includes the symmetry of the lattice of the substrate [wurtzite (wz) versus zinc blende (zb)] as well as different substrate orientations [zb-(001) versus zb-(111)]. Second, we discuss two different types of band alignment, type-I versus type-II, by comparing the common-anion system zb-InAs/GaAs to the common-cation system zb-GaSb/GaAs. Finally, the impact of large built-in fields resulting from piezo- and pyroelectric charges will be exemplified for the wz-GaN/AlN QD-system.

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References

  1. W.G. Cady Piezoelectricity: an Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals (McGraw-Hill, 1946)

    Google Scholar 

  2. O.Ambacher, M. Eickhoff, A. Link, M. Hermann, M. Stutzmann, F. Bernardini, V. Fiorentini, Y. Smorchkova, J. Speck, U. Mishra, W. Schaff, V. Tilak, L.F. Eastman, Electronics and sensors based on pyroelectric AlGaN/GaN heterostructures. Phys. Stat. Sol. (c) 0(6), 1878–1907 (2003)

    Google Scholar 

  3. O. Ambacher, J. Majewski, C. Miskys, A. Link, M. Hermann, M. Eickhoff, M. Stutzmann, F. Bernardini, V. Fiorentini, V. Tilak, Pyroelectric properties of Al (In) GaN/GaN hetero-and quantum well structures. J. Phys.: Condens. Matter 14(13), 3399 (2002)

    Google Scholar 

  4. A.D. Andreev, E.P. O’Reilly, Theory of the electronic structure of GaN/AlN hexagonal quantum dots. Phys. Rev. B 62(23), 851–870 (2000)

    Article  Google Scholar 

  5. A.D. Andreev, E.P. O’Reilly, Optical transitions and radiative lifetime in GaN/AlN self-organized quantum dots. Appl. Phys. Lett. 79(4), 521 (2001)

    Google Scholar 

  6. N. Baer, S. Schulz, P. Gartner, S. Schumacher, G. Czycholl, F. Jahnke, Influence of symmetry and Coulomb correlation effects on the optical properties of nitride quantum dots. Phys. Rev. B 76(7) (2007)

    Google Scholar 

  7. N. Baer, S. Schulz, S. Schumacher, P. Gartner, G. Czycholl, F. Jahnke, Optical properties of self-organized wurtzite InN/GaN quantum dots: A combined atomistic tight-binding and full configuration interaction calculation. Appl. Phys. Lett. 87(23), 114–231 (2005)

    Article  Google Scholar 

  8. T. Bahder, Eight-band k⋅ p model of strained zinc-blende crystals. Phys. Rev. B 41(17), 992–1012 (1990)

    Article  Google Scholar 

  9. F. Bernardini, V. Fiorentini, D. Vanderbilt, Spontaneous polarization and piezoelectric constants of III-V nitrides. Phys. Rev. B 56(16), 10,024–10,027 (1997)

    Article  Google Scholar 

  10. G. Bester, X. Wu, D. Vanderbilt, A. Zunger, Importance of second-order piezoelectric effects in zinc-blende semiconductors. Phys. Rev. Lett. 96(18) (2006)

    Google Scholar 

  11. G. Bester, A. Zunger, Cylindrically shaped zinc-blende semiconductor quantum dots do not have cylindrical symmetry: Atomistic symmetry, atomic relaxation, and piezoelectric effects. Phys. Rev. B 71(4) (2005)

    Google Scholar 

  12. G. Bester, A. Zunger, X. Wu, D. Vanderbilt, Effects of linear and nonlinear piezoelectricity on the electronic properties of InAs/GaAs quantum dots. Phys. Rev. B 74(8) (2006)

    Google Scholar 

  13. D. Bimberg, M. Grundmann, N.N. Ledentsov, Quantum Dot Heterostructures (John Wiley & Sons, New York, 1999)

    Google Scholar 

  14. H. Born, L. Müller-Kirsch, R. Heitz, A. Hoffmann, D. Bimberg, Rapid research notes-radiative recombination in type II GaSb/GaAs quantum dots. Phys. Stat. Sol. (b) 228(3), 4 (2001)

    Google Scholar 

  15. M. Braskén, M. Lindberg, D. Sundholm, J. Olsen, Full configuration interaction calculations of electron-hole correlation effects in strain-induced quantum dots. Phys. Rev. B 61(11), 7652–7655 (2000)

    Article  Google Scholar 

  16. M. Braskén, M. Lindberg, D. Sundholm, J. Olsen, Full configuration interaction calculations of electron-hole correlation effects in strain-induced quantum dots. Phys. Stat. Sol. (b) 224(3), 775–779 (2001)

    Google Scholar 

  17. T. Bretagnon, S. Kalliakos, P. Lefebvre, P. Valvin, B. Gil, N. Grandjean, A. Dussaigne, B. Damilano, J. Massies, Time dependence of the photoluminescence of GaN/AlN quantum dots under high photoexcitation. Phys. Rev. B 68(20), 205–301 (2003)

    Article  Google Scholar 

  18. T. Bretagnon, P. Lefebvre, P. Valvin, R. Bardoux, T. Guillet, T. Taliercio, B. Gil, N. Grandjean, F. Semond, B. Damilano, A. Dussaigne, J. Massies, Radiative lifetime of a single electron-hole pair in GaN/AlN quantum dots. Phys. Rev. B 73(11), 113–304 (2006)

    Article  Google Scholar 

  19. C.H. Chan, M.C. Chen, H.H. Lin, Y.F. Chen, G.J. Jan, Y.H. Chen, Characterization of piezoelectric (111)B InGaAs/GaAs p-i-n quantum well structures using photoreflectance spectroscopy. Appl. Phys. Lett. 72(10), 1208 (1998)

    Google Scholar 

  20. S. Cho, J. Kim, A. Sanz-Hervás, A. Majerfeld, G. Patriarche, B.W. Kim, Characterization of piezoelectric and pyroelectric properties of MOVPE-grown strained (111)A InGaAs/GaAs QW structures by modulation spectroscopy. Phys. Stat. Sol. (a) 195(1), 260–264 (2003)

    Google Scholar 

  21. M. Cusack, P. Briddon, Electronic structure of InAs/GaAs self-assembled quantum dots. Phys. Rev. B (1996)

    Google Scholar 

  22. B. Daudin, F. Widmann, G. Feuillet, Y. Samson, M. Arlery, J. Rouvière, Stranski-Krastanov growth mode during the molecular beam epitaxy of highly strained GaN. Phys. Rev. B 56(12), R7069–R7072 (1997)

    Article  Google Scholar 

  23. P. Enders, A. Bärwolff, M. Woerner, D. Suisky, k⋅ p theory of energy bands, wave functions, and optical selection rules in strained tetrahedral semiconductors. Phys. Rev. B 51(23), 16,695–16,704 (1995)

    Google Scholar 

  24. V.A. Fonoberov, A.A. Balandin, Excitonic properties of strained wurtzite and zinc-blende GaN/Al x Ga1−x N quantum dots. J. Appl. Phys. 94(11), 7178 (2003)

    Google Scholar 

  25. H. Fu, L.W. Wang, A. Zunger, Applicability of the k⋅ p method to the electronic structure of quantum dots. Phys. Rev. B 57(16), 9971–9987 (1998)

    Article  Google Scholar 

  26. M. Geller, A. Marent, T. Nowozin, D. Bimberg, N. Akçay, N. Öncan, A write time of 6 ns for quantum dot–based memory structures. Appl. Phys. Lett. 92(9), 092–108 (2008)

    Article  Google Scholar 

  27. D. Gershoni, C.H. Henry, G.A. Baraff, Calculating the optical properties of multidimensional heterostructures: Application to the modeling of quaternary quantum well lasers. IEEE J. Quantum Electron. 29(9), 2433–2450 (1993)

    Article  Google Scholar 

  28. K. Gradkowski, T.J. Ochalski, D.P. Williams, J. Tatebayashi, A. Khoshakhlagh, G. Balakrishnan, E.P. O’Reilly, G. Huyet, L.R. Dawson, D.L. Huffaker, Optical transition pathways in type-II Ga(As)Sb quantum dots. J. Luminescence 129(5), 456–460 (2009)

    Article  Google Scholar 

  29. M. Grundmann, O. Stier, D. Bimberg, InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure. Phys. Rev. B 52(16), 11,969 (1995)

    Google Scholar 

  30. T. Hammerschmidt, Role of strain relaxation during different stages of InAs quantum dot growth. In: Physics of Semiconductors: 27th International Conference on the Physics of Semiconductors - ICPS-27, pp. 601–602. AIP (2005)

    Google Scholar 

  31. F. Hatami, M. Grundmann, N. Ledentsov, F. Heinrichsdorff, R. Heitz, J. Böhrer, D. Bimberg, S. Ruvimov, P. Werner, V. Ustinov, P. Kop’ev, Z. Alferov, Carrier dynamics in type-II GaSb/GaAs quantum dots. Phys. Rev. B 57(8), 4635–4641 (1998)

    Google Scholar 

  32. M. Hayne, J. Maes, S. Bersier, V.V. Moshchalkov, A. Schliwa, L. Müller-Kirsch, C. Kapteyn, R. Heitz, D. Bimberg, Electron localization by self-assembled GaSb/GaAs quantum dots. Appl. Phys. Lett. 82(24), 4355–4357 (2003)

    Article  Google Scholar 

  33. R. Hogg, T. Fisher, A. Willcox, D. Whittaker, M. Skolnick, D. Mowbray, J. David, A. Pabla, G. Rees, R. Grey, J. Woodhead, J. Sanchez-Rojas, G. Hill, M. Pate, P. Robson, Piezoelectric-field effects on transition energies, oscillator strengths, and level widths in (111)B-grown (In,Ga)As/GaAs multiple quantum wells. Phys. Rev. B 48(11), 8491–8494 (1993)

    Article  Google Scholar 

  34. K. Hoshino, S. Kako, Y. Arakawa, Formation and optical properties of stacked GaN self-assembled quantum dots grown by metalorganic chemical vapor deposition. Appl. Phys. Lett. 85(7), 1262–1264 (2004)

    Article  Google Scholar 

  35. H. Jiang, J. Singh, Strain distribution and electronic spectra of InAs/GaAs self-assembled dots: An eight-band study. Phys. Rev. B 56(8), 4696–4701 (1997)

    Article  Google Scholar 

  36. S. Kako, C. Santori, K. Hoshino, S. Götzinger, Y. Yamamoto, Y. Arakawa, A gallium nitride single-photon source operating at 200K. Nat Mater 5(11), 887–892 (2006)

    Article  Google Scholar 

  37. E. Kane, Energy Band Theory (Handbook on Semiconductors, 1982)

    Google Scholar 

  38. E.Kane, Phonon spectra of diamond and zinc-blende semiconductors. Phys. Rev. B 31(12), 7865–7876 (1985)

    Article  MathSciNet  Google Scholar 

  39. J. Kim, L.W. Wang, A. Zunger, Comparison of the electronic structure of InAs/GaAs pyramidal quantum dots with different facet orientations. Phys. Rev. B 57(16), R9408–R9411 (1998)

    Article  Google Scholar 

  40. N. Kirstaedter, N.N. Ledentsov, M. Grundmann, D. Bimberg, V.M. Ustinov, S.S. Ruvimov, M.V. Maximov, P.S. Kop’ev, Z.I. Alferov, U. Richter, P. Werner, U. Gosele, J. Heydenreich, Low threshold, large To injection laser emission from (InGa)As quantum dots. Electron. Lett. 30(17), 1416–1417 (1994)

    Google Scholar 

  41. R.B. Laghumavarapu, A. Moscho, A. Khoshakhlagh, M. El-Emawy, L.F. Lester, D.L. Huffaker, GaSb/GaAs type II quantum dot solar cells for enhanced infrared spectral response. Appl. Phys. Lett. 90(17), 173,125–173,125–3 (2007)

    Google Scholar 

  42. O.L. Lazarenkova, P. von Allmen, F. Oyafuso, S. Lee, G. Klimeck, Effect of anharmonicity of the strain energy on band offsets in semiconductor nanostructures. Appl. Phys. Lett. 85(18), 4193 (2004)

    Google Scholar 

  43. S. Lee, F. Oyafuso, P. von Allmen, G. Klimeck, Boundary conditions for the electronic structure of finite-extent embedded semiconductor nanostructures. Phys. Rev. B 69(4), 045,316 (2004)

    Google Scholar 

  44. J.A. Majewski, S. Birner, A. Trellakis, M. Sabathil, P. Vogl, Advances in the theory of electronic structure of semiconductors. Phys. Stat. Sol. (c) 1(8), 2003–2027 (2004)

    Google Scholar 

  45. G. Malpuech, A. Di Carlo, A. Kavokin, J.J. Baumberg, M. Zamfirescu, P. Lugli, Room-temperature polariton lasers based on GaN microcavities. Appl. Phys. Lett. 81(3), 412–414 (2002)

    Article  Google Scholar 

  46. A. Marent, M. Geller, A. Schliwa, D. Feise, K. Pötschke, D. Bimberg, N. Akçay, N. Öncan, 106 years extrapolated hole storage time in GaSb/AlAs quantum dots. Appl. Phys. Lett. 91(24), 242,109 (2007)

    Google Scholar 

  47. R. McWeeny, B.T. Pickup, Quantum theory of molecular electronic structure. Rep. Prog. Phys. 43(9), 1065–1144 (2000)

    Article  Google Scholar 

  48. P. Michler, A quantum dot single-photon turnstile device. Science 290(5500), 2282–2285 (2000)

    Article  Google Scholar 

  49. M. Migliorato, A. Cullis, M. Fearn, J. Jefferson, Atomistic simulation of strain relaxation in In x Ga1−x As/GaAs quantum dots with nonuniform composition. Phys. Rev. B 65(11), 115,316 (2002)

    Google Scholar 

  50. V. Mlinar, F.M. Peeters, Optical properties of (In,Ga)As capped InAs quantum dots grown on [11k] substrates. Appl. Phys. Lett. 91(2), 021,910 (2007)

    Google Scholar 

  51. V. Mlinar, A. Zunger, Spectral barcoding of quantum dots: deciphering structural motifs from the excitonic spectra. Phys. Rev. B 80(3), 035,328 (2009)

    Google Scholar 

  52. F.H. Pollak, Effects of homogeneous strain on the electronic and vibrational levels in semiconductors. In: T.P. Pearsall (ed.) Semiconductors and Semimetals, p. 17. Strained-Layer Superlattices: Physics (1990)

    Google Scholar 

  53. M. Povolotskyi, A. Di Carlo, S. Birner, Electronic and optical properties of [N11] grown nanostructures. Phys. Stat. Sol. (c) 1(6), 1511–1521 (2004)

    Google Scholar 

  54. C. Pryor, Eight-band calculations of strained InAs/GaAs quantum dots compared with one-, four-, and six-band approximations. Phys. Rev. B 57(12), 7190–7195 (1998)

    Article  Google Scholar 

  55. C. Pryor, J. Kim, L.W. Wang, A.J. Williamson, A. Zunger, Comparison of two methods for describing the strain profiles in quantum dots. J. Appl. Phys. 83(5), 2548 (1998)

    Google Scholar 

  56. V. Ranjan, G. Allan, C. Priester, C. Delerue, Self-consistent calculations of the optical properties of GaN quantum dots. Phys. Rev. B 68(11) (2003)

    Google Scholar 

  57. P. Rinke, M. Scheffler, A. Qteish, M. Winkelnkemper, D. Bimberg, J. Neugebauer, Band gap and band parameters of InN and GaN from quasiparticle energy calculations based on exact-exchange density-functional theory. Appl. Phys. Lett. 89(16), 161,919 (2006)

    Google Scholar 

  58. A. Rodina, M. Dietrich, A. Göldner, L. Eckey, A. Hoffmann, A. Efros, M. Rosen, B. Meyer, Free excitons in wurtzite GaN. Phys. Rev. B 64(11), 115,204 (2001)

    Google Scholar 

  59. T. Saito, Y. Arakawa, Electronic structure of piezoelectric In0. 2Ga0. 8N quantum dots in GaN calculated using a tight-binding method. Physica E 15(3), 169–181 (2002)

    Google Scholar 

  60. J.L. Sanchez Rojas, A. Sacedon, F. Gonzalez Sanz, E. Calleja, E. Munoz, Dependence on the In concentration of the piezoelectric field in (111)B InGaAs/GaAs strained heterostructures. Appl. Phys. Lett. 65(16), 2042–2044 (1994)

    Article  Google Scholar 

  61. R. Santoprete, B. Koiller, R. Capaz, P. Kratzer, Q. Liu, M. Scheffler, Tight-binding study of the influence of the strain on the electronic properties of InAs/GaAs quantum dots. Phys. Rev. B 68(23) (2003)

    Google Scholar 

  62. C. Santori, S. Götzinger, Y. Yamamoto, S. Kako, K. Hoshino, Y.Arakawa, Photon correlation studies of single GaN quantum dots. Appl. Phys. Lett. 87(5), 051,916–051,916–3 (2005)

    Google Scholar 

  63. A. Schliwa, M. Winkelnkemper, D. Bimberg, Impact of size, shape, and composition on piezoelectric effects and electronic properties of In(Ga)As/GaAs quantum dots. Phys. Rev. B 76(20) (2007)

    Google Scholar 

  64. A. Schliwa, M. Winkelnkemper, D. Bimberg, Few-particle energies versus geometry and composition of In x Ga1−x As/GaAs self-organized quantum dots. Phys. Rev. B 79(7) (2009)

    Google Scholar 

  65. A. Schliwa, M. Winkelnkemper, A. Lochmann, E. Stock, D. Bimberg, In(Ga)As/GaAs quantum dots grown on a (111) surface as ideal sources of entangled photon pairs. Phys. Rev. B 80(16) (2009)

    Google Scholar 

  66. S. Schulz, M.A. Caro, E.P. O’Reilly, O. Marquardt, Symmetry-adapted calculations of strain and polarization fields in (111)-oriented zinc-blende quantum dots. Phys. Rev. B 84(12), 125,312 (2011)

    Google Scholar 

  67. R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. Pohl, D. Bimberg, Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots. Phys. Rev. Lett. 95(25) (2005)

    Google Scholar 

  68. V. Shchukin, D. Bimberg, Spontaneous ordering of nanostructures on crystal surfaces. Rev. Mod. Phys. 71(4), 1125–1171 (1999)

    Article  Google Scholar 

  69. W. Sheng, J.P. Leburton, Electron-hole alignment in InAs/GaAs self-assembled quantum dots: effects of chemical composition and dot shape. Phys. Rev. B 63(16) (2001)

    Google Scholar 

  70. E. Siebert, T. Warming, A. Schliwa, E. Stock, M. Winkelnkemper, S. Rodt, D. Bimberg, Spectroscopic access to single-hole energies in InAs/GaAs quantum dots. Phys. Rev. B 79(20) (2009)

    Google Scholar 

  71. R. Singh, G. Bester, Nanowire quantum dots as an ideal source of entangled photon pairs. Phys. Rev. Lett. 103(6) (2009)

    Google Scholar 

  72. J.C. Slater, G.F. Koster, Simplified LCAO method for the periodic potential problem. Phys. Rev. 94(6), 1498–1524 (1954)

    Article  MATH  Google Scholar 

  73. O. Stier, D. Bimberg, Modeling of strained quantum wires using eight-band k⋅ p theory. Phys. Rev. B 55(12), 7726–7732 (1997)

    Article  Google Scholar 

  74. O. Stier, M. Grundmann, D. Bimberg, Electronic and optical properties of strained quantum dots modeled by 8-band k⋅ p theory. Phys. Rev. B 59(8), 5688 (1999)

    Google Scholar 

  75. C.K. Sun, G. Wang, J.E. Bowers, B. Brar, H.R. Blank, H. Kroemer, M.H. Pilkuhn, Optical investigations of the dynamic behavior of GaSb/GaAs quantum dots. Appl. Phys. Lett. 68(11), 1543–1545 (1996)

    Article  Google Scholar 

  76. J. Tersoff, Modeling solid-state chemistry: interatomic potentials for multicomponent systems. Phys. Rev. B 39(8), 5566–5568 (1989)

    Article  Google Scholar 

  77. I. Vurgaftman, J.R. Meyer, Band parameters for nitrogen-containing semiconductors. J. Appl. Phys. 94(6), 3675 (2003)

    Google Scholar 

  78. F. Widmann, J. Simon, B. Daudin, G. Feuillet, J. Rouvière, N. Pelekanos, G. Fishman, Blue-light emission from GaN self-assembled quantum dots due to giant piezoelectric effect. Phys. Rev. B 58(24), R15,989–R15,992 (1998)

    Article  Google Scholar 

  79. A.J. Williamson, A. Franceschetti, A. Zunger, Multi-excitons in self-assembled InAs/GaAs quantum dots: a pseudopotential, many-body approach. Europhys. Lett. 53(1), 59–65 (2007)

    Article  Google Scholar 

  80. A.J. Williamson, L. Wang, A. Zunger, Theoretical interpretation of the experimental electronic structure of lens-shaped self-assembled InAs/GaAs quantum dots. Phys. Rev. B 62(19), 12,963–12,977 (2000)

    Article  Google Scholar 

  81. M. Winkelnkemper, A. Schliwa, D. Bimberg, Interrelation of structural and electronic properties in In x Ga1−x N/GaN quantum dots using an eight-band k⋅ p model. Phys. Rev. B 74(15) (2006)

    Google Scholar 

  82. M. Winkelnkemper, R. Seguin, S. Rodt, A. Hoffmann, D. Bimberg, GaN/AlN quantum dots for single qubit emitters. J. Phys.: Condens. Matter 20(45), 454,211 (2008)

    Google Scholar 

  83. M. Winkelnkemper, R. Seguin, S. Rodt, A. Schliwa, L. Reissmann, A. Strittmatter, A. Hoffmann, D. Bimberg, Polarized emission lines from A- and B-type excitonic complexes in single InGaN/GaN quantum dots. J. Appl. Phys. 101(11), – (2007)

    Google Scholar 

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Acknowledgements

Many colleagues contributed to the success of this work. However, three people stand out who contributed most over the past 20 years: We feel deeply indebted to Oliver Stier, Momme Winkelnkemper, and Marius Grundmann.

The work was funded by DFG in the frame of SFBs 296 and 787.

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  1. Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstr. 36, 10623, Berlin, Germany

    Andrei Schliwa, Gerald Hönig & Dieter Bimberg

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  1. Andrei Schliwa
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  2. Gerald Hönig
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  1. & Numerische Mathematik, Bergische Universität Wuppertal Lehrstuhl für Angewandte Mathematik, Wuppertal, Germany

    Matthias Ehrhardt

  2. Forschungsgruppe “Partielle Differentialgleichungen", Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Germany

    Thomas Koprucki

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Schliwa, A., Hönig, G., Bimberg, D. (2014). Electronic Properties of III-V Quantum Dots. In: Ehrhardt, M., Koprucki, T. (eds) Multi-Band Effective Mass Approximations. Lecture Notes in Computational Science and Engineering, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-01427-2_2

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