Abstract

We develop a low-temperature theory of quasi-elastic secondary emission from a semiconductor quantum dot, the electronic subsystem of which is resonant with the confined longitudinal-optical (LO) phonon modes. Our theory employs a generalized model for renormalization of the quantum dot’s energy spectrum, which is induced by the polar electron-phonon interaction. The model takes into account the degeneration of electronic states and allows for several LO-phonon modes to be involved in the vibrational resonance. We give solutions to three fundamental problems of energy-spectrum renormalization—arising if one, two, or three LO-phonon modes resonantly couple a pair of electronic states—and discuss the most general problem of this kind that admits an analytical solution. With these results, we solve the generalized master equation for the reduced density matrix, in order to derive an expression for the differential cross section of secondary emission from a single quantum dot. The obtained expression is then analyzed to establish the basics of optical spectroscopy for measuring fundamental parameters of the quantum dot’s polaron-like states.

© 2011 OSA

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References

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  1. A. V. Fedorov, I. D. Rukhlenko, A. V. Baranov, and S. Y. Kruchinin, Optical Properties of Semiconductor Quantum Dots (Nauka, 2011).
  2. T. Takagahara, “Electron-phonon interactions in semiconductor quantum dots,” in Semiconductor Quantum Dots: Physics, Spectroscopy and Applications, Y. Masumoto and T. Takagahara, eds. (Springer, 2002).
  3. V. Cesari, W. Langbein, and P. Borri, “Dephasing of excitons and multiexcitons in undoped and p-doped InAs/GaAs quantum dots-in-a-well,” Phys. Rev. B82, 195314 (2010).
  4. E. A. Muljarov and R. Zimmermann, “Exciton dephasing in quantum dots due to LO-phonon coupling: an exactly solvable model,” Phys. Rev. Lett.98, 187401 (2007).
  5. K. Kojima and A. Tomita, “Influence of pure dephasing by phonons on exciton-photon interfaces: quantum microscopic theory,” Phys. Rev. B73, 195312 (2006).
  6. A. Vagov, V. M. Axt, T. Kuhn, W. Langbein, P. Borri, and U. Woggon, “Nonmonotonous temperature dependence of the initial decoherence in quantum dots,” Phys. Rev. B70, 201305 (2004).
  7. R. R. Cooney, S. L. Sewall, E. A. Dias, D. M. Sagar, K. E. H. Anderson, and P. Kambhampati, “Unified picture of electron and hole relaxation pathways in semiconductor quantum dots,” Phys. Rev. B75, 245311 (2007).
    [CrossRef]
  8. B. Patton, W. Langbein, U. Woggon, L. Maingault, and H. Mariette, “Time- and spectrally-resolved four-wave mixing in single CdTe/ZnTe quantum dots,” Phys. Rev. B73, 235354 (2006).
    [CrossRef]
  9. M. R. Salvador, M. W. Graham, and G. D. Scholes, “Exciton-phonon coupling and disorder in the excited states of CdSe colloidal quantum dots,” J. Chem. Phys.125, 184709 (2006).
  10. S. Sanguinetti, E. Poliani, M. Bonfanti, M. Guzzi, E. Grilli, M. Gurioli, and N. Koguchi, “Electron-phonon interaction in individual strain-free GaAs/Al0.3Ga0.7As quantum dots,” Phys. Rev. B73, 125342 (2006).
    [CrossRef]
  11. D. Valerini, A. Cretí, M. Lomascolo, L. Manna, R. Cingolani, and M. Anni, “Temperature dependence of the photoluminescence properties of colloidal CdSe/ZnS core/shell quantum dots embedded in a polystyrene matrix,” Phys. Rev. B71, 235409 (2005).
    [CrossRef]
  12. A. V. Fedorov and A. V. Baranov, “Exciton-vibrational interaction of the Fröhlich type in quasi-zero-size systems,” J. Exp. Theor. Phys.83, 610–618 (1996).
  13. T. Itoh, M. Nishijima, A. I. Ekimov, C. Gourdon, A. L. Efros, and M. Rosen, “Polaron and exciton-phonon complexes in CuCl nanocrystals,” Phys. Rev. Lett.74, 1645–1648 (1995).
    [CrossRef] [PubMed]
  14. I. D. Rukhlenko and A. V. Fedorov, “Propagation of electric fields induced by optical phonons in semiconductor heterostructures,” Opt. Spectrosc.100, 238–244 (2006).
    [CrossRef]
  15. I. D. Rukhlenko and A. V. Fedorov, “Penetration of electric fields induced by surface phonon modes into the layers of a semiconductor heterostructure,” Opt. Spectrosc.101, 253–264 (2006).
    [CrossRef]
  16. A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, and S. V. Gaponenko, “Enhanced intraband carrier relaxation in quantum dots due to the effect of plasmon–LO-phonon density of states in doped heterostructures,” Phys. Rev. B71, 195310 (2005).
  17. A. V. Baranov, A. V. Fedorov, I. D. Rukhlenko, and Y. Masumoto, “Intraband carrier relaxation in quantum dots embedded in doped heterostructures,” Phys. Rev. B68, 205318 (2003).
    [CrossRef]
  18. B. A. Carpenter, E. A. Zibik, M. L. Sadowski, L. R. Wilson, D. M. Whittaker, J. W. Cockburn, M. S. Skolnick, M. Potemski, M. J. Steer, and M. Hopkinson, “Intraband magnetospectroscopy of singly and doubly charged n-type self-assembled quantum dots,” Phys. Rev. B74, 161302 (2006).
    [CrossRef]
  19. V. Preisler, R. Ferreira, S. Hameau, L. A. de Vaulchier, Y. Guldner, M. L. Sadowski, and A. Lemaitre, “Hole–LO phonon interaction in InAs/GaAs quantum dots,” Phys. Rev. B72, 115309 (2005).
    [CrossRef]
  20. J. Zhao, A. Kanno, M. Ikezawa, and Y. Masumoto, “Longitudinal optical phonons in the excited state of CuBr quantum dots,” Phys. Rev. B68, 113305 (2003).
    [CrossRef]
  21. S. Hameau, J. N. Isaia, Y. Guldner, E. Deleporte, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, and J. M. Gérard, “Far-infrared magnetospectroscopy of polaron states in self-assembled InAs/GaAs quantum dots,” Phys. Rev. B65, 085316 (2002).
    [CrossRef]
  22. A. V. Fedorov, A. V. Baranov, A. Itoh, and Y. Masumoto, “Renormalization of energy spectrum of quantum dots under vibrational resonance conditions,” Semiconductors35, 1390–1397 (2001).
    [CrossRef]
  23. S. Hameau, Y. Guldner, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, A. Lemaître, and J. M. Gérard, “Strong electron-phonon coupling regime in quantum dots: evidence for everlasting resonant polarons,” Phys. Rev. Lett.83, 4152–4155 (1999).
    [CrossRef]
  24. P. Palinginis, S. Tavenner, M. Lonergan, and H. Wang, “Spectral hole burning and zero phonon linewidth in semiconductor nanocrystals,” Phys. Rev. B67, 201307 (2003).
  25. E. A. Chekhovich, A. B. Krysa, M. S. Skolnick, and A. I. Tartakovskii, “Direct measurement of the hole-nuclear spin interaction in single InP/GaInP quantum dots using photoluminescence spectroscopy,” Phys. Rev. Lett.106, 027402 (2011).
    [CrossRef] [PubMed]
  26. P. Fallahi, S. T. Yilmaz, and A. Imamoğlu, “Measurement of a heavy-hole hyperfine interaction in InGaAs quantum dots using resonance fluorescence,” Phys. Rev. Lett.105, 257402 (2010).
    [CrossRef]
  27. S. Y. Kruchinin and A. V. Fedorov, “Renormalization of the energy spectrum of quantum dots under vibrational resonance conditions: persistent hole burning spectroscopy,” Opt. Spectrosc.100, 41–48 (2006).
    [CrossRef]
  28. O. Verzelen, R. Ferreira, and G. Bastard, “Excitonic polarons in semiconductor quantum dots,” Phys. Rev. Lett.88, 146803 (2002).
    [CrossRef] [PubMed]
  29. T. Stauber, R. Zimmermann, and H. Castella, “Electron-phonon interaction in quantum dots: a solvable model,” Phys. Rev. B62, 7336–7343 (2000).
    [CrossRef]
  30. T. Inoshita and H. Sakaki, “Density of states and phonon-induced relaxation of electrons in semiconductor quantum dots,” Phys. Rev. B56, R4355–R4358 (1997).
    [CrossRef]
  31. A. V. Fedorov, A. V. Baranov, and K. Inoue, “Exciton-phonon coupling in semiconductor quantum dots: resonant Raman scattering,” Phys. Rev. B56, 7491–7502 (1997).
    [CrossRef]
  32. A. V. Fedorov, A. V. Baranov, and K. Inoue, “Two-photon transitions in systems with semiconductor quantum dots,” Phys. Rev. B54, 8627–8632 (1996).
    [CrossRef]
  33. O. Madelung, M. Schultz, and H. Weiss, eds., Semiconductors. Physics of Group IV Elements and III–V Compounds, Landolt-Börnstein, New Series, Group III, Vol. 17, Pt. a (Springer-Verlag, 1982).
    [PubMed]
  34. A. I. Anselm, Introduction to Semiconductor Theory (Prentice-Hall, 1978).
  35. A. S. Davydov, Theory of Solid State (Nauka, 1976).
  36. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill Book Company, 1968).
  37. A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of the quasi-elastic resonant secondary emission: semiconductor quantum dots,” Opt. Spectrosc.92, 732–738 (2002).
    [CrossRef]
  38. A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of optical-phonon-assisted resonance secondary emission in semiconductor quantum dots,” Opt. Spectrosc.93, 52–60 (2002).
    [CrossRef]
  39. K. Blum, Density Matrix Theory and Its Applications (Plenum Press, 1981).
  40. R. W. Boyd, Nonlinear Optics (Academic Press, 2003).
  41. E. A. Zibik, T. Grange, B. A. Carpenter, R. Ferreira, G. Bastard, N. Q. Vinh, P. J. Phillips, M. J. Steer, M. Hopkinson, J. W. Cockburn, M. S. Skolnick, and L. R. Wilson, “Intersublevel polaron dephasing in self-assembled quantum dots,” Phys. Rev. B77, 041307 (2008).
    [CrossRef]
  42. A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Acoustic phonon problem in nanocrystaldielectric matrix systems,” Solid State Commun.122, 139–144 (2002).
    [CrossRef]
  43. A. V. Fedorov and S. Y. Kruchinin, “Acoustic phonons in a quantum dot-matrix system: hole-burning spectroscopy,” Opt. Spectrosc.97, 394–402 (2004).
    [CrossRef]
  44. I. D. Rukhlenko, D. Handapangoda, M. Premaratne, A. V. Fedorov, A. V. Baranov, and C. Jagadish, “Spontaneous emission of guided polaritons by quantum dot coupled to metallic nanowire: beyond the dipole approximation,” Opt. Express17, 17570–17581 (2009).
    [CrossRef] [PubMed]
  45. A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, T. S. Perova, and K. Berwick, “Quantum dot energy relaxation mediated by plasmon emission in doped covalent semiconductor heterostructures,” Phys. Rev. B76, 045332 (2007).
    [CrossRef]
  46. S. Y. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Double quantum dot photoluminescence mediated by incoherent reversible energy transport,” Phys. Rev. B81, 245303 (2010).
    [CrossRef]
  47. D. Gammon, N. H. Bonadeo, G. Chen, J. Erland, and D. G. Steel, “Optically probing and controlling single quantum dots,” Physica E (Amsterdam)9, 99–105 (2001).
    [CrossRef]
  48. M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B65, 041308 (2002).
    [CrossRef]

Other (48)

A. V. Fedorov, I. D. Rukhlenko, A. V. Baranov, and S. Y. Kruchinin, Optical Properties of Semiconductor Quantum Dots (Nauka, 2011).

T. Takagahara, “Electron-phonon interactions in semiconductor quantum dots,” in Semiconductor Quantum Dots: Physics, Spectroscopy and Applications, Y. Masumoto and T. Takagahara, eds. (Springer, 2002).

V. Cesari, W. Langbein, and P. Borri, “Dephasing of excitons and multiexcitons in undoped and p-doped InAs/GaAs quantum dots-in-a-well,” Phys. Rev. B82, 195314 (2010).

E. A. Muljarov and R. Zimmermann, “Exciton dephasing in quantum dots due to LO-phonon coupling: an exactly solvable model,” Phys. Rev. Lett.98, 187401 (2007).

K. Kojima and A. Tomita, “Influence of pure dephasing by phonons on exciton-photon interfaces: quantum microscopic theory,” Phys. Rev. B73, 195312 (2006).

A. Vagov, V. M. Axt, T. Kuhn, W. Langbein, P. Borri, and U. Woggon, “Nonmonotonous temperature dependence of the initial decoherence in quantum dots,” Phys. Rev. B70, 201305 (2004).

R. R. Cooney, S. L. Sewall, E. A. Dias, D. M. Sagar, K. E. H. Anderson, and P. Kambhampati, “Unified picture of electron and hole relaxation pathways in semiconductor quantum dots,” Phys. Rev. B75, 245311 (2007).
[CrossRef]

B. Patton, W. Langbein, U. Woggon, L. Maingault, and H. Mariette, “Time- and spectrally-resolved four-wave mixing in single CdTe/ZnTe quantum dots,” Phys. Rev. B73, 235354 (2006).
[CrossRef]

M. R. Salvador, M. W. Graham, and G. D. Scholes, “Exciton-phonon coupling and disorder in the excited states of CdSe colloidal quantum dots,” J. Chem. Phys.125, 184709 (2006).

S. Sanguinetti, E. Poliani, M. Bonfanti, M. Guzzi, E. Grilli, M. Gurioli, and N. Koguchi, “Electron-phonon interaction in individual strain-free GaAs/Al0.3Ga0.7As quantum dots,” Phys. Rev. B73, 125342 (2006).
[CrossRef]

D. Valerini, A. Cretí, M. Lomascolo, L. Manna, R. Cingolani, and M. Anni, “Temperature dependence of the photoluminescence properties of colloidal CdSe/ZnS core/shell quantum dots embedded in a polystyrene matrix,” Phys. Rev. B71, 235409 (2005).
[CrossRef]

A. V. Fedorov and A. V. Baranov, “Exciton-vibrational interaction of the Fröhlich type in quasi-zero-size systems,” J. Exp. Theor. Phys.83, 610–618 (1996).

T. Itoh, M. Nishijima, A. I. Ekimov, C. Gourdon, A. L. Efros, and M. Rosen, “Polaron and exciton-phonon complexes in CuCl nanocrystals,” Phys. Rev. Lett.74, 1645–1648 (1995).
[CrossRef] [PubMed]

I. D. Rukhlenko and A. V. Fedorov, “Propagation of electric fields induced by optical phonons in semiconductor heterostructures,” Opt. Spectrosc.100, 238–244 (2006).
[CrossRef]

I. D. Rukhlenko and A. V. Fedorov, “Penetration of electric fields induced by surface phonon modes into the layers of a semiconductor heterostructure,” Opt. Spectrosc.101, 253–264 (2006).
[CrossRef]

A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, and S. V. Gaponenko, “Enhanced intraband carrier relaxation in quantum dots due to the effect of plasmon–LO-phonon density of states in doped heterostructures,” Phys. Rev. B71, 195310 (2005).

A. V. Baranov, A. V. Fedorov, I. D. Rukhlenko, and Y. Masumoto, “Intraband carrier relaxation in quantum dots embedded in doped heterostructures,” Phys. Rev. B68, 205318 (2003).
[CrossRef]

B. A. Carpenter, E. A. Zibik, M. L. Sadowski, L. R. Wilson, D. M. Whittaker, J. W. Cockburn, M. S. Skolnick, M. Potemski, M. J. Steer, and M. Hopkinson, “Intraband magnetospectroscopy of singly and doubly charged n-type self-assembled quantum dots,” Phys. Rev. B74, 161302 (2006).
[CrossRef]

V. Preisler, R. Ferreira, S. Hameau, L. A. de Vaulchier, Y. Guldner, M. L. Sadowski, and A. Lemaitre, “Hole–LO phonon interaction in InAs/GaAs quantum dots,” Phys. Rev. B72, 115309 (2005).
[CrossRef]

J. Zhao, A. Kanno, M. Ikezawa, and Y. Masumoto, “Longitudinal optical phonons in the excited state of CuBr quantum dots,” Phys. Rev. B68, 113305 (2003).
[CrossRef]

S. Hameau, J. N. Isaia, Y. Guldner, E. Deleporte, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, and J. M. Gérard, “Far-infrared magnetospectroscopy of polaron states in self-assembled InAs/GaAs quantum dots,” Phys. Rev. B65, 085316 (2002).
[CrossRef]

A. V. Fedorov, A. V. Baranov, A. Itoh, and Y. Masumoto, “Renormalization of energy spectrum of quantum dots under vibrational resonance conditions,” Semiconductors35, 1390–1397 (2001).
[CrossRef]

S. Hameau, Y. Guldner, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, A. Lemaître, and J. M. Gérard, “Strong electron-phonon coupling regime in quantum dots: evidence for everlasting resonant polarons,” Phys. Rev. Lett.83, 4152–4155 (1999).
[CrossRef]

P. Palinginis, S. Tavenner, M. Lonergan, and H. Wang, “Spectral hole burning and zero phonon linewidth in semiconductor nanocrystals,” Phys. Rev. B67, 201307 (2003).

E. A. Chekhovich, A. B. Krysa, M. S. Skolnick, and A. I. Tartakovskii, “Direct measurement of the hole-nuclear spin interaction in single InP/GaInP quantum dots using photoluminescence spectroscopy,” Phys. Rev. Lett.106, 027402 (2011).
[CrossRef] [PubMed]

P. Fallahi, S. T. Yilmaz, and A. Imamoğlu, “Measurement of a heavy-hole hyperfine interaction in InGaAs quantum dots using resonance fluorescence,” Phys. Rev. Lett.105, 257402 (2010).
[CrossRef]

S. Y. Kruchinin and A. V. Fedorov, “Renormalization of the energy spectrum of quantum dots under vibrational resonance conditions: persistent hole burning spectroscopy,” Opt. Spectrosc.100, 41–48 (2006).
[CrossRef]

O. Verzelen, R. Ferreira, and G. Bastard, “Excitonic polarons in semiconductor quantum dots,” Phys. Rev. Lett.88, 146803 (2002).
[CrossRef] [PubMed]

T. Stauber, R. Zimmermann, and H. Castella, “Electron-phonon interaction in quantum dots: a solvable model,” Phys. Rev. B62, 7336–7343 (2000).
[CrossRef]

T. Inoshita and H. Sakaki, “Density of states and phonon-induced relaxation of electrons in semiconductor quantum dots,” Phys. Rev. B56, R4355–R4358 (1997).
[CrossRef]

A. V. Fedorov, A. V. Baranov, and K. Inoue, “Exciton-phonon coupling in semiconductor quantum dots: resonant Raman scattering,” Phys. Rev. B56, 7491–7502 (1997).
[CrossRef]

A. V. Fedorov, A. V. Baranov, and K. Inoue, “Two-photon transitions in systems with semiconductor quantum dots,” Phys. Rev. B54, 8627–8632 (1996).
[CrossRef]

O. Madelung, M. Schultz, and H. Weiss, eds., Semiconductors. Physics of Group IV Elements and III–V Compounds, Landolt-Börnstein, New Series, Group III, Vol. 17, Pt. a (Springer-Verlag, 1982).
[PubMed]

A. I. Anselm, Introduction to Semiconductor Theory (Prentice-Hall, 1978).

A. S. Davydov, Theory of Solid State (Nauka, 1976).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill Book Company, 1968).

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of the quasi-elastic resonant secondary emission: semiconductor quantum dots,” Opt. Spectrosc.92, 732–738 (2002).
[CrossRef]

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of optical-phonon-assisted resonance secondary emission in semiconductor quantum dots,” Opt. Spectrosc.93, 52–60 (2002).
[CrossRef]

K. Blum, Density Matrix Theory and Its Applications (Plenum Press, 1981).

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

E. A. Zibik, T. Grange, B. A. Carpenter, R. Ferreira, G. Bastard, N. Q. Vinh, P. J. Phillips, M. J. Steer, M. Hopkinson, J. W. Cockburn, M. S. Skolnick, and L. R. Wilson, “Intersublevel polaron dephasing in self-assembled quantum dots,” Phys. Rev. B77, 041307 (2008).
[CrossRef]

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Acoustic phonon problem in nanocrystaldielectric matrix systems,” Solid State Commun.122, 139–144 (2002).
[CrossRef]

A. V. Fedorov and S. Y. Kruchinin, “Acoustic phonons in a quantum dot-matrix system: hole-burning spectroscopy,” Opt. Spectrosc.97, 394–402 (2004).
[CrossRef]

I. D. Rukhlenko, D. Handapangoda, M. Premaratne, A. V. Fedorov, A. V. Baranov, and C. Jagadish, “Spontaneous emission of guided polaritons by quantum dot coupled to metallic nanowire: beyond the dipole approximation,” Opt. Express17, 17570–17581 (2009).
[CrossRef] [PubMed]

A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, T. S. Perova, and K. Berwick, “Quantum dot energy relaxation mediated by plasmon emission in doped covalent semiconductor heterostructures,” Phys. Rev. B76, 045332 (2007).
[CrossRef]

S. Y. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Double quantum dot photoluminescence mediated by incoherent reversible energy transport,” Phys. Rev. B81, 245303 (2010).
[CrossRef]

D. Gammon, N. H. Bonadeo, G. Chen, J. Erland, and D. G. Steel, “Optically probing and controlling single quantum dots,” Physica E (Amsterdam)9, 99–105 (2001).
[CrossRef]

M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B65, 041308 (2002).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Formation of polaron-like states from [(a), (b)] a pair of states p 2 and p 1 coupled via LO phonon. (c) States + ( 1 ) and ( 1 ) arise due to the coupling of nondegenerate states p 2 and p 1 via a single phonon mode. (d) States + ( k ) , i ( k ) (i = 3, . . . , k + 1), and ( k ) originate from two nondegenerate states p 2 and p 1 coupled via k phonon modes, or from either k-fold degenerate state p 2 and nondegenerate state p 1 or nondegenerate state p 2 and k-fold degenerate state p 1 coupled via a single phonon mode.

Fig. 2
Fig. 2

Size dependence of (a) polaron-like energy levels and (b) renormalization coefficients for vibrational resonance between nondegenerate electron-hole states |p 2〉 = |200; 200〉 and |p 1〉 = |200; 100〉 of a spherical quantum dot in the regime of strong confinement. The exact resonance occurs at the intersection of the green and orange lines in panel (a) and blue and red curves in panel (b) for R ≈ 9.5 nm. The energies are reckoned from the energy of the state |p 1〉 shifted by electron-phonon interaction; Ω LO = 29.5 meV. For other parameters refer to text.

Fig. 3
Fig. 3

Spectra of the quasi-elastic secondary emission from a single quantum dot of radius R res ≈ 9.5 nm (see Fig. 2) for the excitation frequencies ωL ≈ 1219.9, 1223, 1226, 1229.1, and 1232.2 meV shown by vertical ticks. (a) Resonant scattering and (b) total secondary emission for relaxation parameters S 1 and S 2 (see Table 1), respectively, and Γ F = 0.04 meV. Total secondary emission for relaxation parameters S 2, (c) Γ F = 0.04 meV, and (d) Γ F = 0.4 meV. Legends in panels (c) and (d) show excitation frequencies for all spectra of the same color; ω 2 = V 2/. For other parameters refer to text.

Fig. 4
Fig. 4

Excitation spectra of the quasi-elastic secondary emission from a single quantum dot of radius R res ≈ 9.5 nm (see Fig. 2) for relaxation parameters [(a) and (b)] S 3 and [(c) and (d)] S 4 (see Table 1); Γ F = 0.4 meV. Detection frequencies ωF ≈ 1219.9, 1223, 1226, 1229.1, and 1232.2 meV, labeled in panels (a) and (b), are shown by vertical ticks; ω 2 = V 2/. For other parameters refer to text.

Fig. 5
Fig. 5

Size dependence of (a) polaron-like energy levels, (b) renormalization coefficients, and (c) level splitting Δ± = ±± for vibrational resonance between the degenerate electron-hole states |p 2〉 = |11m; 11m〉 and |p 1〉 = |11m; 100〉 (m = 0, ±1) of a spherical quantum dot in the regime of strong confinement. In panels (a) and (b), the energies are reckoned from the lower state in the electron-hole doublet. For other parameters refer to text.

Fig. 6
Fig. 6

Excitation spectra of the quasi-elastic secondary emission from a single quantum dot of radius [(a)–(c)] R res ≈ 5.59 nm and (d) R ≈ 3.54 nm [see Fig. 5(c)], for the vibrational resonance between a pair of orbitally degenerate electronic states. Detection frequencies are: ωF ≈ 1592.7, 1596.6, 1600.5, 1604.4, and 1608.2 meV in panels (a)–(c); and 3322.6, 3332.3, and 3377.4 meV in panel (d); Γ F = 0.04 meV. For other parameters refer to text.

Tables (1)

Tables Icon

Table 1 Relaxation Parameters (in μeV) used in Figs. 3 and 4

Equations (85)

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

H ( t ) = H 0 + H int ( t ) ,
H 0 = p E p a p + a p + q ħ Ω q b q + b q + λ ħ ω λ c λ + c λ
H int ( t ) = H e , p h + H e , L ( t ) + H e , λ
H e , p h = q p 1 , p 2 0 ( V p 2 , p 1 ( q ) a p 2 + a p 1 b q + H . c . ) ,
V p 2 , p 1 ( q ) = e ( φ p e 2 , p e 1 ( q ) δ p h 2 , p h 1 φ p h 1 , p h 2 ( q ) δ p e 2 , p e 1 ) ,
φ p e 2 , p e 1 ( q ) = φ p h 2 , p h 1 ( q ) = 4 ( 2 l q + 1 ) ( 2 l p 1 + 1 ) ħ Ω q ( 2 l p 2 + 1 ) ɛ * R 𝒥 n p 2 l p 2 , n p 1 l p 1 n q l q C l q 0 , l p 1 0 l p 2 0 C l q m q , l p 1 m p 1 l p 2 m p 2 ,
𝒥 n p 2 l p 2 , n p 1 l p 1 n q l q = 0 1 d x x 2 j l q ( ξ n q l q x ) j l p 2 ( ξ n p 2 l p 2 x ) j l p 1 ( ξ n p 1 l p 1 x ) ξ n q l q j l q + 1 ( ξ n q l q ) j l p 2 + 1 ( ξ n p 2 l p 2 ) j l p 1 + 1 ( ξ n p 1 l p 1 ) ,
H e , L ( t ) = p 0 ( V p , 0 ( L ) ( t ) a p + + H . c . ) ,
H e , λ = λ p 0 ( i ħ g λ V p , 0 ( λ ) a p + c λ + H . c . ) ,
V p , 0 ( η ) = e u c | r e ^ η | u v δ p e , p h
U = exp [ p , q ( Φ p , p ( q ) b q H . c . ) a p + a p ]
H ˜ ( t ) = U + H ( t ) U = H ˜ 0 + H ˜ e , p h + H ˜ e , L ( t ) + H ˜ e , λ ,
H ˜ 0 = p E ˜ p a p + a p + q ħ Ω q b q + b q + λ ħ ω λ c λ + c λ , E ˜ p = E p q ħ Ω q | Φ p , p ( q ) | 2 ,
H ˜ e , p h = q p 1 p 2 0 ( V p 2 , p 1 ( q ) a p 2 + a p 1 b q + H . c . ) ,
H ˜ e , L ( t ) = p 0 { V p , 0 ( L ) ( t ) [ 1 + q ( Φ p , p ( q ) b q H . c . ) ] a p + + H . c . } ,
H ˜ e , λ = λ p 0 { i ħ g λ V p , 0 ( λ ) [ 1 + q ( Φ p , p ( q ) b q H . c . ) ] a p + c λ + H . c . } .
H ^ ( t ) = S + H ˜ ( t ) S .
H ˜ e , p h ( 1 ) = ( E ˜ p 2 V p 2 , p 1 ( q 1 ) V p 2 , p 1 ( q 1 ) * E ˜ p 1 + ħ Ω q 1 ) ,
H ˜ e , p h ( 2 ) = ( E ˜ p 2 V p 2 , p 1 ( q 1 ) V p 2 , p 1 ( q 2 ) V p 2 , p 1 ( q 1 ) * E ˜ p 1 + ħ Ω q 1 0 V p 2 , p 1 ( q 2 ) * 0 E ˜ p 1 + ħ Ω q 2 ) ,
H ˜ e , p h ( 3 ) = ( E ˜ p 2 V p 2 , p 1 ( q 1 ) V p 2 , p 1 ( q 2 ) V p 2 , p 1 ( q 3 ) V p 2 , p 1 ( q 1 ) * E ˜ p 1 + ħ Ω q 1 0 0 V p 2 , p 1 ( q 2 ) * 0 E ˜ p 1 + ħ Ω q 2 0 V p 2 , p 1 ( q 3 ) * 0 0 E ˜ p 1 + ħ Ω q 3 ) .
± ( 1 ) = 1 2 ( E ˜ p 2 + E ˜ p 1 + ħ Ω q 1 ± δ 1 ) ,
| + ( 1 ) = c 1 ( 1 ) | E ˜ p 2 ; 0 q 1 + c 3 ( 1 ) * | E ˜ p 1 ; 1 q 1 , | ( 1 ) = c 2 ( 1 ) | E ˜ p 2 ; 0 q 1 c 4 ( 1 ) * c 1 ( 1 ) | E ˜ p 1 ; 1 q 1 ,
c 1 ( k ) = E ˜ p 2 E ˜ p 1 ħ Ω q k + δ k Δ k , c 2 ( k ) = 2 V k Δ k , δ k = ( E ˜ p 2 E ˜ p 1 ħ Ω q k ) 2 + 4 V k 2 ,
Δ k = ( E ˜ p 2 E ˜ p 1 ħ Ω q k + δ k ) 2 + 4 V k 2 , V k = ( i = 1 k | V p 2 , p 1 ( q i ) | 2 ) 1 / 2 .
H ^ e , p h ( 1 ) = ( + ( 1 ) 0 0 ( 1 ) )
S 1 = ( c 1 ( 1 ) c 2 ( 1 ) c 3 ( 1 ) * c 4 ( 1 ) * c 1 ( 1 ) ) .
± ( 2 ) = 1 2 ( E ˜ p 2 + E ˜ p 1 + ħ Ω L O ± δ 2 ) , 3 ( 2 ) = E ˜ p 1 + ħ Ω L O ,
| + ( 2 ) = c 1 ( 2 ) | E ˜ p 2 ; 0 q + c 6 ( 2 ) * | E ˜ p 1 ; 1 q 1 + c 3 ( 2 ) * | E ˜ p 1 ; 1 q 2 ,
| 3 ( 2 ) = - c 5 ( 2 ) | E ˜ p 1 ; 1 q 1 + c 4 ( 2 ) | E ˜ p 1 ; 1 q 2 ,
| ( 2 ) = c 2 ( 2 ) | E ˜ p 2 ; 0 q c 4 ( 2 ) * c 1 ( 2 ) | E ˜ p 1 ; 1 q 1 c 5 ( 2 ) * c 1 ( 2 ) | E ˜ p 1 ; 1 q 2 ,
S 2 = ( c 1 ( 2 ) 0 c 2 ( 2 ) c 6 ( 2 ) * c 5 ( 2 ) c 1 ( 2 ) c 4 ( 2 ) * c 3 ( 2 ) * c 4 ( 2 ) c 1 ( 2 ) c 5 ( 2 ) * ) .
± ( 3 ) = 1 2 ( E ˜ p 2 + E ˜ p 1 + ħ Ω L O ± δ 3 ) , 3 ( 3 ) = 4 ( 3 ) = E ˜ p 1 + ħ Ω L O ,
| + ( 3 ) = c 1 ( 3 ) | E ˜ p 2 ; 0 q + c 9 ( 3 ) * | E ˜ p 1 ; 1 q 1 + c 3 ( 3 ) * | E ˜ p 1 ; 1 q 2 + c 4 ( 3 ) * | E ˜ p 1 ; 1 q 3 ,
| 3 ( 3 ) = c 5 ( 3 ) | E ˜ p 1 ; 1 q 1 + c 6 ( 3 ) | E ˜ p 1 ; 1 q 2 ,
| 4 ( 3 ) = c 7 ( 3 ) c 6 ( 3 ) * | E ˜ p 1 ; 1 q 1 c 7 ( 3 ) c 5 ( 3 ) * | E ˜ p 1 ; 1 q 2 + c 8 ( 3 ) | E ˜ p 1 ; 1 q 3 ,
| ( 3 ) = c 2 ( 3 ) | E ˜ p 2 ; 0 q c 10 ( 3 ) * c 1 ( 3 ) | E ˜ p 1 ; 1 q 1 c 11 ( 3 ) * c 1 ( 3 ) | E ˜ p 1 ; 1 q 2 c 7 ( 3 ) * c 1 ( 3 ) | E ˜ p 1 ; 1 q 3 ,
S 3 = ( c 1 ( 3 ) 0 0 c 2 ( 3 ) c 9 ( 3 ) * c 5 ( 3 ) c 7 ( 3 ) c 6 ( 3 ) * c 1 ( 3 ) c 10 ( 3 ) * c 3 ( 3 ) * c 6 ( 3 ) c 7 ( 3 ) c 5 ( 3 ) * c 1 ( 3 ) c 11 ( 3 ) * c 4 ( 3 ) * 0 c 8 ( 3 ) c 1 ( 3 ) c 7 ( 3 ) * ) .
± ( k ) = 1 2 ( E ˜ p 2 + E ˜ p 1 + ħ Ω L O ± δ k ) , 3 ( k ) = 4 ( k ) = = k + 1 ( k ) = E ˜ p 1 + ħ Ω L O .
H ^ e , η ( 1 ) = ( 0 0 c 1 ( 1 ) V p 2 , 0 ( η ) 0 0 c 2 ( 1 ) V p 2 , 0 ( η ) c 1 ( 1 ) V p 2 , 0 ( η ) * c 2 ( 1 ) V p 2 , 0 ( η ) * 0 ) ,
H ^ e , η ( 2 ) = ( 0 0 0 c 1 ( 2 ) V p 2 , 0 ( η ) 0 0 0 0 0 0 0 c 2 ( 2 ) V p 2 , 0 ( η ) c 1 ( 2 ) V p 2 , 0 ( η ) * 0 c 2 ( 2 ) V p 2 , 0 ( η ) * 0 ) ,
H ^ e , η ( 3 ) = ( 0 0 0 0 c 1 ( 3 ) V p 2 , 0 ( η ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 2 ( 3 ) V p 2 , 0 ( η ) c 1 ( 3 ) V p 2 , 0 ( η ) * 0 0 c 2 ( 3 ) V p 2 , 0 ( η ) * 0 ) .
H ^ e , η ( k ) = ( 0 0 0 0 c 1 ( k ) V p 2 , 0 ( η ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 2 ( k ) V p 2 , 0 ( η ) c 1 ( k ) V p 2 , 0 ( η ) * 0 0 c 2 ( k ) V p 2 , 0 ( η ) * 0 ) .
˜ e , p h = ( H ˜ e , p h ( k 1 ) 0 0 0 H ˜ e , p h ( k 2 ) 0 0 0 H ˜ e , p h ( k N ) ) .
𝒮 = ( S k 1 0 0 0 S k 2 0 0 0 S k N ) .
( c 1 ( k 1 ) V p 2 , 0 ( η ) * , 0 , , 0 , k 1 1 c 2 ( k 1 ) V p 2 , 0 ( η ) * , c 1 ( k 2 ) V p 2 , 0 ( η ) * , 0 , , 0 , k 2 1 c 2 ( k 2 ) V p 2 , 0 ( η ) * , , c 1 ( k N ) V p 2 , 0 ( η ) * , 0 , , 0 , K N 1 c 2 ( k N ) V p 2 , 0 ( η ) * , 0 ) .
ρ μ ν ( t ) t = 1 i ħ [ H ^ ( t ) , ρ ( t ) ] μ ν γ μ ν ρ μ ν ( t ) + δ μ ν ν ν ζ ν ν ρ ν ν ( t ) ,
| 1 = | 0 ( k ) | 0 q | 0 λ , | 2 = | + ( k ) | 0 q | 0 λ , | 3 = | ( k ) | 0 q | 0 λ , | 4 = | 0 ( k ) | 0 q | 1 λ .
W = 1 i ħ [ H ^ ( t ) , ρ ( t ) ] 44 = g λ ( c 1 ( k ) V p 2 , 0 ( λ ) * ρ 24 ( t ) + c 2 ( k ) V p 2 , 0 ( λ ) * ρ 34 ( t ) + c . c . ) ,
d 2 σ d Θ d ω λ = V ɛ 3 / 2 ħ ω λ 3 4 ( π c ) 3 I L W = δ p e 2 , p h 2 C ( ω λ ) ( n = 2 3 2 γ ^ 1 n γ n n ( c n 1 ( k ) ) 4 Δ L n 2 + γ 1 n 2 2 γ 1 n Δ λ n 2 + γ 1 n 2 + 2 ζ 32 γ 12 γ 22 γ 33 ( c 1 ( k ) c 2 ( k ) ) 2 Δ L 2 2 + γ 12 2 2 γ 13 Δ λ 3 2 + γ 13 2 + | n = 2 3 ( c n 1 ( k ) ) 2 Δ L n i γ 1 n | 2 γ 0 ( ω L ω λ ) 2 + ( γ 0 / 2 ) 2 ) ,
g F ( ω F ω λ ) = 1 π Γ F / 2 ( ω F ω λ ) 2 + ( Γ F / 2 ) 2 ,
d 2 σ d Θ d ω λ = δ p e 2 , p h 2 C ( ω F ) ( n = 2 3 2 γ ^ 1 n γ n n ( c n 1 ( k ) ) 4 Δ L n 2 + γ 1 n 2 2 ( γ 1 n + Γ F / 2 Δ F n 2 + ( γ 1 n + Γ F / 2 ) 2 ) + 2 ζ 32 γ 12 γ 22 γ 33 ( c 1 ( k ) c 2 ( k ) ) 2 Δ L 2 2 + γ 12 2 2 ( γ 13 + Γ F / 2 ) Δ F 3 2 + ( γ 13 + Γ F / 2 ) 2 + | n = 2 3 ( c n 1 ( k ) ) 2 Δ L n i γ 1 n | 2 Γ F ( ω L ω F ) 2 + ( Γ F / 2 ) 2 ) ,
Γ F γ 12 , γ 13 ,
d 2 σ d Θ d ω λ = δ p e 2 , p h 2 C ( ω F ) ( n = 2 3 2 γ ^ 1 n γ n n ( c n 1 ( k ) ) 4 Δ L n 2 + γ 1 n 2 2 γ 1 n Δ F n 2 + γ 1 n 2 ) + 2 ζ 32 γ 12 γ 22 γ 33 ( c 1 ( k ) c 2 ( k ) ) 2 Δ L 2 2 + γ 12 2 2 γ 13 Δ F 3 2 + γ 13 2 + | n = 2 3 ( c n 1 ( k ) ) 2 Δ L n i γ 1 n | 2 Γ F ( ω L ω F ) 2 + ( Γ F / 2 ) 2 ) .
{ | n e 1 e m e ; n h 1 n m h } = { | n e , 1 e , 1 e | n e , 1 e , 0 e | n e , 1 e , + 1 e } { | n h , 1 h , 1 h | n h , 1 h , 0 h | n h , 1 h , + 1 h } .
| n h , 0 h , 0 h { | n e , 1 e , 1 e | n e , 1 e , 0 e | n e , 1 e , + 1 e }
| n e , 0 e , 0 e { | n h , 1 h , 1 h | n h , 1 h , 0 h | n h , 1 h , + 1 h } .
d 2 σ d Θ d ω λ = δ p e 2 , p h 2 C ( ω F ) ( n = 2 7 2 γ ^ 1 n γ n n d n 4 Δ L n 2 + γ 1 n 2 2 ( γ 1 n + Γ F / 2 Δ F n 2 + ( γ 1 n + Γ F / 2 ) 2 ) + r = 3 7 n = 2 r 1 2 ζ r n γ 1 n γ n n γ r r d n 2 d r 2 Δ L n 2 + γ 1 n 2 2 ( γ 1 r + Γ F / 2 ) Δ F r 2 + ( γ 1 r + Γ F / 2 ) 2 + | n = 2 7 d n 2 Δ L n i γ 1 n | 2 Γ F ( ω L ω F ) 2 + ( Γ F / 2 ) 2 ) .
Σ 2 ( k ) = 1 2 ( E p 2 + E ˜ p 1 + ħ Ω L O + δ k ) , Σ 5 ( k ) = 1 2 ( E p 2 + E ˜ p 1 + ħ Ω L O δ k ) , Σ 3 ( k ) = 1 2 ( E p 2 + E ˜ p 1 + ħ Ω L O = δ k ) , Σ 6 ( k ) = 1 2 ( E p 2 + E ˜ p 1 + ħ Ω L O δ k ) , Σ 4 ( k ) = 1 2 ( E p 2 + E ˜ p 1 + ħ Ω L O + δ k ) , Σ 7 ( k ) = 1 2 ( E p 2 + E ˜ p 1 + ħ Ω L O δ k ) ,
δ k = χ 2 + 4 V k 2 , δ k = χ 2 + 4 V k 2 , δ k = χ 2 + 4 V k 2 , χ = E p 2 E ˜ p 1 ħ Ω L O , χ = E p 2 E ˜ p 1 ħ Ω L O , χ = E p 2 E ˜ p 1 ħ Ω L O ,
d 2 = ( χ + δ k ) / Δ k , d 3 = ( χ + δ k ) / Δ k , d 4 = ( χ + δ k ) / Δ k , d 5 = 2 V k / Δ k , d 6 = 2 V k / Δ k , d 7 = 2 V k / Δ k ,
Δ k = ( χ + δ k ) 2 + 4 V k 2 , Δ k = ( χ + δ k ) 2 + 4 V k 2 , Δ k = ( χ + δ k ) 2 + 4 V k 2 .
V p 2 , p 1 ( q ) = e φ n h 1 l h 1 m h 1 , n h 2 l h 2 m h 2 ( q ) δ n e 2 , n e 1 δ l e 2 , l e 1 δ m e 2 , m e 1 ,
E n e l e m e , n h l h m h = E g + E n e l e m e + E n h l h m h = E g + ħ 2 ξ n e l e 2 2 m c R 2 + ħ 2 ξ n h l h 2 2 m v R 2 ,
E n l m , n l m = E g + ħ 2 ξ n l 2 2 μ R 2 ,
E p 2 E 200 , 200 = E g + ħ 2 ( 2 π ) 2 2 μ R 2
E p 1 E 200 , 100 = E g + ħ 2 π 2 2 R 2 ( 4 m c + 1 m v ) .
χ = E p 2 E ˜ p 1 ħ Ω L O = 3 ħ 2 π 2 2 m v R 2 + α e 2 ɛ * R ħ Ω L O ,
R res = α e 2 2 ɛ * ħ Ω L O + 1 2 [ ( α e 2 ɛ * ħ Ω L O ) 2 + 6 ħ π 2 m v Ω L O ] 1 / 2 9.5 nm .
± ( 2 ) = 1 2 [ 2 E g + ħ 2 π 2 2 R 2 ( 8 m c + 5 m v ) α e 2 ɛ * R + ħ Ω L O ± δ 2 ] ,
3 ( 2 ) = E g + ħ 2 π 2 2 R 2 ( 4 m c + 1 m v ) α e 2 ɛ * R + ħ Ω L O ,
δ 2 = χ 2 + 4 V 2 2 , V 2 = ( β e 2 ħ Ω L O ɛ * R ) 1 / 2 ,
c 1 ( 2 ) = χ + δ 2 ( χ + δ 2 ) 2 + 4 V 2 2 , c 2 ( 2 ) = 2 V 2 ( χ + δ 2 ) 2 + 4 V 2 2 ,
ω L = ω ( 2 ) ω 2 , ω ( 2 ) , ω ( 2 ) + ω 2 , ω + ( 2 ) , ω + ( 2 ) + ω 2 ,
E p 2 E 11 m , 11 m = E g + ħ 2 ξ 11 2 2 μ R 2 , E p 1 E 11 m , 100 = E g + ħ 2 2 R 2 ( ξ 11 2 m c + π 2 m v ) ,
E ˜ 11 ± 1 , 100 = E g + ħ 2 2 R 2 ( ξ 11 2 m c + π 2 m v ) α e 2 ɛ * R ,
E ˜ 110 , 100 = E g + ħ 2 2 R 2 ( ξ 11 2 m c + π 2 m v ) α e 2 ɛ * R ,
R res α e 2 2 ɛ * ħ Ω L O + [ ħ ( ξ 11 2 π 2 ) 2 m v Ω L O ] 1 / 2 5.59 nm
R res α e 2 2 ɛ * ħ Ω L O + [ ħ ( ξ 11 2 π 2 ) 2 m v Ω L O ] 1 / 2 5.61 nm .
χ = ħ 2 ( ξ 11 2 π 2 ) 2 m v R 2 ħ Ω L O + α e 2 ɛ * R , χ = ħ 2 ( ξ 11 2 π 2 ) 2 m v R 2 ħ Ω L O + α e 2 ɛ * R .
| V 11 + 1 , 11 + 1 ; 11 + 1 , 100 ( 11 1 ) | = | V 110 , 110 ; 110 , 100 ( 110 ) | = | V 11 1 , 11 1 ; 11 1 , 100 ( 11 + 1 ) | V 1 = ( β e 2 ħ Ω L O ɛ * R ) 1 / 2 ,
Σ 2 ( 1 ) = Σ 3 ( 1 ) = + , Σ 5 ( 1 ) = Σ 6 ( 1 ) = ,
Σ 4 ( 1 ) = + , Σ 7 ( 1 ) = ,
± = 1 2 [ 2 E g + ħ 2 ξ 11 2 2 μ R 2 + ħ 2 2 R 2 ( ξ 11 2 m c + π 2 m v ) α e 2 ɛ * R + ħ Ω L O ± δ 1 ] ,
± = 1 2 [ 2 E g + ħ 2 ξ 11 2 2 μ R 2 + ħ 2 2 R 2 ( ξ 11 2 m c + π 2 m v ) α e 2 ɛ * R + ħ Ω L O ± δ 1 ] ,
d 2 = d 3 = χ + δ 1 ( χ + δ 1 ) 2 + 4 V 1 2 , d 5 = d 6 = 2 V 1 ( χ + δ 1 ) 2 + 4 V 1 2 , d 4 = χ + δ 1 ( χ + δ 1 ) 2 + 4 V 1 2 , d 7 = 2 V 1 ( χ + δ 1 ) 2 + 4 V 1 2 .

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