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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 16 — Aug. 1, 2011
  • pp: 15459–15482
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Theory of quasi-elastic secondary emission from a quantum dot in the regime of vibrational resonance

Ivan D. Rukhlenko, Anatoly V. Fedorov, Anvar S. Baymuratov, and Malin Premaratne  »View Author Affiliations


Optics Express, Vol. 19, Issue 16, pp. 15459-15482 (2011)
http://dx.doi.org/10.1364/OE.19.015459


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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

1. Introduction

The coupling between electrons and phonons is one of the most influential interactions occurring inside semiconductor quantum dots, which results in the scattering of electrons, holes, and excitons [1

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

, 2

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).

]. In the photoluminescence experiments performed on quantum dots, this coupling is often manifested in enhanced dephasing rates of spectroscopic transitions [3

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. B 82, 195314 (2010).

6

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. B 70, 201305 (2004).

] and reduced lifetimes of electronic excitations [7

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. B 75, 245311 (2007). [CrossRef]

11

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. B 71, 235409 (2005). [CrossRef]

]. Electron-phonon interaction may also cause a significant modification of the quantum dot’s energy spectrum, known to be especially prominent in the regime of vibrational resonance [12

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

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]

]. This regime is realized where the energy of a phonon coincides with the energy spacing between a pair of states of the quantum dot’s electronic subsystem. Renormalization of the electron-phonon energy spectrum in this situation leads to the formation of hybrid energy states characterizing a new class of polaron-like excitations, which are similar to the polarons in a bulk semiconductor.

Several physical theories of polaron-like excitations in semiconductor quantum dots were developed by different research groups [13

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]

, 22

22. A. V. Fedorov, A. V. Baranov, A. Itoh, and Y. Masumoto, “Renormalization of energy spectrum of quantum dots under vibrational resonance conditions,” Semiconductors 35, 1390–1397 (2001). [CrossRef]

, 23

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]

, 27

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]

30

30. T. Inoshita and H. Sakaki, “Density of states and phonon-induced relaxation of electrons in semiconductor quantum dots,” Phys. Rev. B 56, R4355–R4358 (1997). [CrossRef]

], and subsequently employed for the interpretation of experimental data. For example, the first theory proposed in Ref. [13

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]

] helped to explain the shape of the excitonic luminescence spectra obtained under the resonant size-selective excitation of CuCl-nanocrystals. The model of Ref. [23

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]

] was applied for the analysis of data on intraband transitions obtained with far-infrared magnetospectroscopy for self-assembled quantum dots made of InAs [18

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. B 74, 161302 (2006). [CrossRef]

, 19

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. B 72, 115309 (2005). [CrossRef]

, 21

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. B 65, 085316 (2002). [CrossRef]

, 23

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]

]. Another model, reported in Ref. [22

22. A. V. Fedorov, A. V. Baranov, A. Itoh, and Y. Masumoto, “Renormalization of energy spectrum of quantum dots under vibrational resonance conditions,” Semiconductors 35, 1390–1397 (2001). [CrossRef]

], proved useful in interpreting the effect of renormalization for the lowest excitonic states in CuCl-quantum dots embedded into NaCl-matrix, which were studied with the spectroscopy of secondary emission under two-photon resonance excitation. Unlike theoretical treatment of the vibrational resonance itself, the mathematical description of its manifestation in optical spectra is developed rather poorly, and the information these spectra are capable of providing is not perfectly understood. To the best of our knowledge, the only technique that has been placed on solid theoretical ground up to date is the persistent spectral hole burning in an inhomogeneously broadened light absorption profile [27

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]

]. Thus, a fundamental theory of quantum dot optical response allowing for the renormalization of the electron-LO-phonon energy spectrum is still in demand.

2. Theory of vibrational resonance in a semiconductor quantum dot

We start our theoretical treatment of vibrational resonance arising in a quantum dot due to the polar electron-phonon interaction, by introducing the Hamiltonian formalism for the physical system being of interest.

2.1. Hamiltonian formalism

The physical system to be analyzed is composed of a semiconductor quantum dot, classical excitation light, and vacuum (quantum electromagnetic) radiation field. It is convenient to take the total Hamiltonian of this system to be of the form
H(t)=H0+Hint(t),
(1)
where
H0=pEpap+ap+qħΩqbq+bq+λħωλcλ+cλ
(2)
gives the contribution from noninteracting electron-hole pairs, LO phonons, and photons, while
Hint(t)=He,ph+He,L(t)+He,λ
(3)
describes the interaction of electron-hole pairs (e) with phonons (ph), excitation light (L), and radiation field (λ). The notations on the right-hand side of Eq. (2) are as follows: ap+(ap), bq+(bq), and cλ+(cλ) are the creation (annihilation) operators of electron-hole pairs, LO phonons, and photons, respectively; Ep = Eg + Epe + Eph is the energy of electron-hole pair state |p〉 ≡ |pe; ph〉, which depends on the quantum numbers of electron (pe) and hole (ph), and the band gap, Eg of the bulk semiconductor; Ωq and h̄ωλ are the energies of LO-phonon mode q and photon mode λ.

For simplicity, we restrict our analysis to the quantum dot in the regime of strong confinement, and describe its electronic subsystem using the two-band approximation [1

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

, 2

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).

]. Although polar electron-phonon interaction in quantum dots is relatively strong, it does not excite electrons from the valence (v) band to the conduction (c) band, but induces only intraband transitions between the different states of electron-hole pairs. This fact allows electron-phonon Hamiltonian to be represented as
He,ph=qp1,p20(Vp2,p1(q)ap2+ap1bq+H.c.),
(4)
where the second summation does not extend over the quantum numbers p 1 = p 2 = 0, representing vacuum of electron-hole pairs, and the matrix element of polar electron-phonon interaction is given by the expression
Vp2,p1(q)=e(φpe2,pe1(q)δph2,ph1φph1,ph2(q)δpe2,pe1),
(5)
in which –e is the charge of the electron; φpe2,pe1(q) and φph1,ph2(q) are the matrix elements of the electric potential induced by the LO-phonon mode q; δ pe2, pe1and δ ph2, ph1 are the products of Kronecker deltas for the quantum numbers of electron and hole. In writing Eq. (5) in the form shown, we assumed that the spatial confinement provided by a quantum dot for electrons, holes, and phonons is approximated suitably enough by an infinitely deep potential well.

An explicit form of the matrix elements in Eq. (5) depends on the specific shape of the quantum dot [1

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

, 2

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).

]. For a spherical quantum dot of radius R, which will be used to illustrate our results in Section 4, they can be expressed compactly as [31

31. A. V. Fedorov, A. V. Baranov, and K. Inoue, “Exciton-phonon coupling in semiconductor quantum dots: resonant Raman scattering,” Phys. Rev. B 56, 7491–7502 (1997). [CrossRef]

]
φpe2,pe1(q)=φph2,ph1(q)=4(2lq+1)(2lp1+1)ħΩq(2lp2+1)ɛ*R𝒥np2lp2,np1lp1nqlqClq0,lp10lp20Clqmq,lp1mp1lp2mp2,
(6)
where ɛ * = (1/ɛ 0 – 1/ɛ )–1, ɛ 0 and ɛ are the low- and high-frequency dielectric permittivities of the bulk semiconductor,
𝒥np2lp2,np1lp1nqlq=01dxx2jlq(ξnqlqx)jlp2(ξnp2lp2x)jlp1(ξnp1lp1x)ξnqlqjlq+1(ξnqlq)jlp2+1(ξnp2lp2)jlp1+1(ξnp1lp1),
and jl (z) is the spherical Bessel function of the first kind, with ξnl being its nth zero [i.e., jl(ξnl) = 0]. Clebsch-Gordan coefficients Clqmq,lp1mp1lp2mp2 appearing in this expression set the selection rules for electron-electron and hole-hole transitions. The quantities na, la, and ma (a = q, p 1, p 2) denote the principal quantum number, angular momentum, and its projection for LO phonon (a = q), electron (a = p e1, p e2), or hole (a = p h1, p h2).

The second and third terms in Eq. (3) account for the two mechanisms of interband transitions: (i) generation of electron-hole pairs through absorption of excitation light by the quantum dot’s electronic subsystem; and (ii) recombination of electron-hole pairs with emission of photons due to their interaction with the vacuum radiation field. The Hamiltonians describing these mechanisms can be represented in the forms [31

31. A. V. Fedorov, A. V. Baranov, and K. Inoue, “Exciton-phonon coupling in semiconductor quantum dots: resonant Raman scattering,” Phys. Rev. B 56, 7491–7502 (1997). [CrossRef]

, 32

32. A. V. Fedorov, A. V. Baranov, and K. Inoue, “Two-photon transitions in systems with semiconductor quantum dots,” Phys. Rev. B 54, 8627–8632 (1996). [CrossRef]

]
He,L(t)=p0(Vp,0(L)(t)ap++H.c.),
He,λ=λp0(iħgλVp,0(λ)ap+cλ+H.c.),
where Vp,0(L)(t)=Vp,0(L)ϕ(t)exp(iωLt), ϕ(t) is the complex envelope of classical optical field, gλ=2πωλ/(ɛħV) , V is the normalization volume, and
Vp,0(η)=euc|re^η|uvδpe,ph
(7)
is the matrix element of the electric dipole moment operator –e r, calculated on Bloch functions uc and uv at the Brillouin zone center, for excitation light (η = L) or emitted photon (η = λ) of frequency ωη and polarization vector ê η. Kronecker’s delta following the matrix element shows that the dipole-allowed generation and recombination of electron-hole pairs occur only if the quantum numbers of electron and hole are the same. In what follows, we shall consider only quantum dots made of semiconductors with either Td and Oh symmetry. In this case, matrix element in Eq. (7) is expressed through the Kane’s parameter, P of bulk material as [33

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

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

] uc|re^η|uv2Zcv=2P/Eg.

2.2. Formation of polaron-like states

In the case of no interaction occurring between electrons and phonons inside a quantum dot, a number of the total system “electrons plus phonons” energy states may be degenerate. This happens in the situation of vibrational resonance, where energy of a particular LO-phonon mode is close to the energy spacing between a pair of electron-hole states, i.e., when E p2E p1Ωq. The degeneracy is removed by the polar electron-phonon interaction, which results in the splitting of degenerate levels into two or more polaron-like states with different energies [35

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

].

2.3. Vibrational resonance involving one, two, and three phonon modes

Let us now consider those eigenvalue problems associated with Hamiltonian that admit analytical solutions. The simplest problems of this type arise where two nondegenerate states | p2〉 and | p1〉 are coupled via one (q 1), two (q 1, q 2), or three (q 1, q 2, q 3) LO-phonon modes. Using Eqs. (9a) and (9b) and assuming that the temperature of the system is low enough for the phonon modes to be unoccupied, we first draw the matrices that represent the electron-phonon part of the Hamiltonian (9) in these three cases,
H˜e,ph(1)=(E˜p2Vp2,p1(q1)Vp2,p1(q1)*E˜p1+ħΩq1),
(11a)
H˜e,ph(2)=(E˜p2Vp2,p1(q1)Vp2,p1(q2)Vp2,p1(q1)*E˜p1+ħΩq10Vp2,p1(q2)*0E˜p1+ħΩq2),
(11b)
H˜e,ph(3)=(E˜p2Vp2,p1(q1)Vp2,p1(q2)Vp2,p1(q3)Vp2,p1(q1)*E˜p1+ħΩq100Vp2,p1(q2)*0E˜p1+ħΩq20Vp2,p1(q3)*00E˜p1+ħΩq3).
(11c)

The diagonal elements of matrix H˜e,ph(1) give the energies of states | p2; 0q1〉 = | p2〉|0q1〉 and | p1; 1q1〉 = | p1〉|1q1〉, where kets |0q1〉 and |1q1〉 stand for the vacuum of the phonon modes and one LO phonon in the mode q 1 [see Figs. 1(a) and 1(b)]. Being perturbed by electron-phonon interaction, these states change their energies and wave functions according to the relations
±(1)=12(E˜p2+E˜p1+ħΩq1±δ1),
(12a)
|+(1)=c1(1)|E˜p2;0q1+c3(1)*|E˜p1;1q1,|(1)=c2(1)|E˜p2;0q1c4(1)*c1(1)|E˜p1;1q1,
(12b)
where c3(1)=2Vp2,p1(q1)/Δ1, c4(1)=Vp2,p1(q1)/V1, and the rest of parameters are given by the generic expressions
c1(k)=E˜p2E˜p1ħΩqk+δkΔk,c2(k)=2VkΔk,δk=(E˜p2E˜p1ħΩqk)2+4Vk2,
(13a)
Δk=(E˜p2E˜p1ħΩqk+δk)2+4Vk2,Vk=(i=1k|Vp2,p1(qi)|2)1/2.
(13b)
Here, c1(k) and c2(k) give the probability amplitudes that the polaron-like states |+(1) and |(1) do not contain LO phonons. The Hamiltonian in Eq. (11a) is reduced to the diagonal form
H^e,ph(1)=(+(1)00(1))
by the matrix
S1=(c1(1)c2(1)c3(1)*c4(1)*c1(1)).
We see that the electron-phonon interaction completely removes the degeneracy of states | p2; 0q1〉 and | p1; 1q1〉. The formation of new energy-shifted levels of the polaron-like states |+(1) and |+(1) is illustrated in Figs. 1(a)–1(c).

Fig. 1 Formation of polaron-like states from [(a), (b)] a pair of states p2 and p1 coupled via LO phonon. (c) States +(1) and (1) arise due to the coupling of nondegenerate states p2 and p1 via a single phonon mode. (d) States +(k), i(k) (i = 3, . . . , k + 1), and (k) originate from two nondegenerate states p2 and p1 coupled via k phonon modes, or from either k-fold degenerate state p2 and nondegenerate state p1 or nondegenerate state p2 and k-fold degenerate state p1 coupled via a single phonon mode.

Closed expressions for energies and wave functions of the polaron-like states corresponding to matrices in Eqs. (11b) and (11c) can also be found. Generally, these expressions are rather cumbersome, because their derivation amounts to solving cubic and quartic equations [36

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

]. We, therefore, restrict ourselves to a specific situation of dispersionless phonon modes, in which simplified solutions to both eigenvalue problems can be obtained.

Assuming that the energies of phonons in the modes q 1 and q 2 are equal to the energy, ΩLO of the bulk LO phonon at the Brillouin zone center, we find the polaron-like states of the Hamiltonian H˜e,ph(2) to be
±(2)=12(E˜p2+E˜p1+ħΩLO±δ2),3(2)=E˜p1+ħΩLO,
(14a)
|+(2)=c1(2)|E˜p2;0q+c6(2)*|E˜p1;1q1+c3(2)*|E˜p1;1q2,
(14b)
|3(2)=-c5(2)|E˜p1;1q1+c4(2)|E˜p1;1q2,
(14c)
|(2)=c2(2)|E˜p2;0qc4(2)*c1(2)|E˜p1;1q1c5(2)*c1(2)|E˜p1;1q2,
(14d)
where c1(2) and c2(2) are given by Eq. (13) after the replacement Ωqk → ΩLO, c3(2)=2Vp2,p1(q2)/Δ2, c4(2)=Vp2,p1(q1)/V2, c5(2)=Vp2,p2(q2)/V2, c6(2)=2Vp2,p1(q1)/Δ2, and q = (q 1, q 2). The matrix that puts H˜e,ph(2) to the diagonal form H^e,ph(2)=diag(+(2),3(2),(2)) is
S2=(c1(2)0c2(2)c6(2)*c5(2)c1(2)c4(2)*c3(2)*c4(2)c1(2)c5(2)*).
Figure 1(d) shows how the energy level that was initially triply degenerate splits into three nondegenerate levels given in Eq. (14a).

Finally, by setting the energies of all phonon modes in Eq. (11c) alike, Ωq1 = Ωq2 = Ωq3 = ΩLO, we may represent the polaron-like states as
±(3)=12(E˜p2+E˜p1+ħΩLO±δ3),3(3)=4(3)=E˜p1+ħΩLO,
(15a)
|+(3)=c1(3)|E˜p2;0q+c9(3)*|E˜p1;1q1+c3(3)*|E˜p1;1q2+c4(3)*|E˜p1;1q3,
(15b)
|3(3)=c5(3)|E˜p1;1q1+c6(3)|E˜p1;1q2,
(15c)
|4(3)=c7(3)c6(3)*|E˜p1;1q1c7(3)c5(3)*|E˜p1;1q2+c8(3)|E˜p1;1q3,
(15d)
|(3)=c2(3)|E˜p2;0qc10(3)*c1(3)|E˜p1;1q1c11(3)*c1(3)|E˜p1;1q2c7(3)*c1(3)|E˜p1;1q3,
(15e)
where c1(2) and c2(2) are given by Eq. (13) , c3(3)=2Vp2,p1(q2)/Δ3, c4(3)=2Vp2,p1(q3)/Δ3, c5(3)=Vp2,p1(q2)/V2, c6(3)=Vp2,p1(q1)/V2, c7(3)=Vp2,p1(q3)/V3, c8(3)=V2/V3, c9(3)=2Vp2,p2(q1)/Δ3, c10(3)=Vp2,p1(q1)/V3, c11(3)=Vp2,p1(q2)/V3, and q = (q 1, q 2, q 3). The diagonalization H˜e,ph(3)H^e,ph(3)=diag(+(3),3(3),4(3),(3)) is realized by the matrix
S3=(c1(3)00c2(3)c9(3)*c5(3)c7(3)c6(3)*c1(3)c10(3)*c3(3)*c6(3)c7(3)c5(3)*c1(3)c11(3)*c4(3)*0c8(3)c1(3)c7(3)*).
It is seen that electron-phonon interaction is unable to completely remove the degeneracy of the four states in the present case. This is a consequence of the initial triple degeneracy of the LO-phonon mode.

By looking at Eqs. (12a), (14a), and (15a), it may be noted that they admit a straightforward generalization to the situation in which an arbitrary number of degenerate phonon modes (q 1, q 2, . . . , qk) are resonant with the quantum dot’s electronic subsystem. The resulting energies of the polaron-like states are of the form
±(k)=12(E˜p2+E˜p1+ħΩLO±δk),3(k)=4(k)==k+1(k)=E˜p1+ħΩLO.
(16)
Thus, whenever the k degenerate LO-phonon modes couple a pair of quantum-dot electronic states, it results in the formation of two nondegenerate and one (k – 1)-degenerate polaron-like states.

2.4. Polaron-photon interaction

By analogy, the matrices representing transformed electron-photon Hamiltonians for an arbitrary number of phonon modes can be obtained,
H^e,η(k)=(0000c1(k)Vp2,0(η)00000000000000c2(k)Vp2,0(η)c1(k)Vp2,0(η)*00c2(k)Vp2,0(η)*0).
(18)
Notice that these matrices are Hermitian.

2.5. Analytically solvable eigenvalue problems

Since the quantum dot usually has a large number of electronic states and confined LO-phonon modes, multiple vibrational resonances may occur between them. Suppose that N vibrational resonances simultaneously occur for the different electron-hole states of a single quantum dot, such that the first pair of states is resonant to k 1 phonon modes, the second pair is resonant to k 2 modes, and so on. By combining interaction matrices H˜e,ph(kj) describing the individual resonances, we can build an electron-phonon Hamiltonian of the entire system,
˜e,ph=(H˜e,ph(k1)000H˜e,ph(k2)000H˜e,ph(kN)).
This quasi-diagonal block Hamiltonian allows for various types of degeneration of the electron-hole pairs’ states and LO-phonon modes. Its diagonalization is realized with an S-matrix of a similar block structure,
𝒮=(Sk1000Sk2000SkN).
Diagonalization of the Hamiltonian ˜ e , ph with matrix 𝒮 can be performed analytically in the special case of any N vibrational resonances involving no more than three phonon modes, i.e., if max (k 1, k 2, . . . , kN) ≤ 3. It becomes particularly simple if, in addition, all modes are of the same frequency. In this case, renormalization of electron-hole spectra is described by Eqs. (12)(16). The matrix e,η(k) of the polaron-photon interaction is self-adjoint and similar in form to Eq. (18); its last row can be written as
(c1(k1)Vp2,0(η)*,0,,0,k11c2(k1)Vp2,0(η)*,c1(k2)Vp2,0(η)*,0,,0,k21c2(k2)Vp2,0(η)*,,c1(kN)Vp2,0(η)*,0,,0,KN1c2(kN)Vp2,0(η)*,0).

3. Quantum dot secondary emission

3.1. Secondary emission in the case of resonance with nondegenerate electronic states

In order to calculate the intensity of secondary emission for the case in which vibrational resonance of the k phonon modes occurs between the nondegenerate states of the electron-hole pairs, we take the four eigenstates of the Hamiltonian 0,
|1=|0(k)|0q|0λ,|2=|+(k)|0q|0λ,|3=|(k)|0q|0λ,|4=|0(k)|0q|1λ.
These eigenstates form the minimal sufficient basis for our system, since the rest of the polaron-like states given in Eq. (16) are not involved in the direct dipole-allowed optical transitions. The absence of the degeneration with reference to the spherical quantum dot adopted in our model, implies that the angular momenta of the electrons and holes must be zero, l e1 = l h1 = l e2 = l h2 = 0.

Through use of Eq. (18), it is easy to show that the emission of photons of frequency ωλ due to the annihilation of the polaron-like excitations is characterized by the rate [37

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

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]

]
W=1iħ[H^(t),ρ(t)]44=gλ(c1(k)Vp2,0(λ)*ρ24(t)+c2(k)Vp2,0(λ)*ρ34(t)+c.c.),
where c1(k) and c2(k) are given in Eq. (13), while the density matrix elements ρ 24(t) and ρ 34(t) are to be found from the master equation.

We now solve Eq. (19) for the case of stationary excitation, ϕ(t) = EL, using the method of perturbation theory and assuming the inverse lifetime of the ground state |1〉 to be zero. The solution we obtain leads to the following expression for the differential cross section of secondary emission per unit solid angle, dΘ and unit frequency interval, dωλ:
d2σdΘdωλ=Vɛ3/2ħωλ34(πc)3ILW=δpe2,ph2C(ωλ)(n=232γ^1nγnn(cn1(k))4ΔLn2+γ1n22γ1nΔλn2+γ1n2+2ζ32γ12γ22γ33(c1(k)c2(k))2ΔL22+γ1222γ13Δλ32+γ132+|n=23(cn1(k))2ΔLniγ1n|2γ0(ωLωλ)2+(γ0/2)2),
(20)
where IL=ɛcEL2/(2π) is the intensity of the excitation light, C(ωλ)=4(Zcvωλe)4/(πc4ħ2), c is the speed of light in a vacuum, and γ 0 is the inverse photon lifetime. The quantities Δηn=ωn(k)ωη denote detunings from the frequencies, ωn(k)=Σn(k)/ħ of the polaron-like energy states Σ2(k)=+(k) and Σ3(k)=(k).

The first, second, and third terms in the parenthesis of Eq. (20) describe the process of luminescence, whereas the fourth term corresponds to the resonant quasi-elastic scattering. For the given frequency detunings Δηn, the relative importance of these terms is set by: (i) pure dephasing rates γ^ 12 and γ^ 13; (ii) lifetimes, γ221 and γ331, of the electron-hole pairs in states |2〉 and |3〉; (iii) coherence relaxation rates, γ 12 = γ 22/2 + γ^ 12 and γ 13 = γ 33/2 + γ^ 13, for optical transitions with generation and recombination of the electron-hole pairs; (iv) relaxation rate ζ 32; and (v) photon lifetime. The values of these parameters depend on many factors, which makes their precise determination a challenging experimental task that has not still been accomplished [41

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. B 77, 041307 (2008). [CrossRef]

]. For example, γ 12 and γ 13 substantially increase if the transitions between states |+(k) and |(k) are intensified through the emission of acoustic phonons with a continuous energy spectrum [42

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

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]

]. The quantity ζ 32 is determined by the relaxation processes with the emission of local excitations of the quantum dot, and elementary excitations of the environment [16

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. B 71, 195310 (2005).

, 17

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. B 68, 205318 (2003). [CrossRef]

, 44

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. Express 17, 17570–17581 (2009). [CrossRef] [PubMed]

, 45

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. B 76, 045332 (2007). [CrossRef]

]. These processes are sensitive to the design and fabrication quality of the sample, as well as excitation conditions. Although in most practical situations the population relaxation rates satisfy the inequality γ 0γ 22, γ 33, their absolute values also vary in a broad range. It is reasonable, therefore, to treat parameters appearing in Eq. (20) as adjustable phenomenological constants, which should be determined (separately for each individual case) from the experimental data.

3.2. Spectral filtration of secondary emission

For experimental measurement of the coherence relaxation rates γ 12 and γ 13, the bandpass is usually chosen to satisfy the inequalities
ΓFγ12,γ13,
(22)
in which case Eq. (21) acquires the form
d2σdΘdωλ=δpe2,ph2C(ωF)(n=232γ^1nγnn(cn1(k))4ΔLn2+γ1n22γ1nΔFn2+γ1n2)+2ζ32γ12γ22γ33(c1(k)c2(k))2ΔL22+γ1222γ13ΔF32+γ132+|n=23(cn1(k))2ΔLniγ1n|2ΓF(ωLωF)2+(ΓF/2)2).
(23)
This expression is a function of the excitation frequency ωL and detection frequency ωF. The possibility to vary one of them while keeping the other constant results in two types of spectroscopic experiments. If ωL is fixed and ωF is varied, then an ordinary spectrum of secondary emission is obtained. On the other hand, in the situation where ωF is fixed and ωL is altered, an excitation spectrum of secondary emission is recorded.

Comparison of luminescence and scattering contributions to the total signal of secondary emission given in Eqs. (20), (21), and (23) reveals several important differences between their spectra. First, if the frequency of excitation is out of resonance with either of the electronic transitions, then the position of the scattering maximum coincides with ωL, whereas the luminescence intensity peaks at frequencies ωF=ω±(k)±(k)/ħ. Second, should conditions in Eq. (22) be satisfied, the scattering linewidth is narrower than the linewidths of the luminescence. Third, scattering strongly masks luminescence, as the peak intensity of the scattering signal is by far greater than the peak luminescence intensity. The domination of scattering over luminescence in the spectra of secondary emission is due to the factors 2γ^ 12/γ 22 ≪ 1 and 2γ^ 13/γ 33 ≪ 1 entering the luminescence terms, and because of ΓFγ 12, γ 13.

For the same reasons, scattering prevails in the excitation spectra of the secondary emission. The excitation spectrum of scattering consists of two peaks at ωL=ω±(k), whose widths are determined by the coherence relaxation rates. This fact allows parameters γ 12 and γ 13 to be calculated through fitting the experimentally measured excitation spectra with Eq. (23).

3.3. Secondary emission in the case of resonance with degenerate electronic states

The above approach allows the secondary emission to be treated analytically in the event that the degenerate states (with nonzero angular momenta) of the electron-hole pairs are involved in the vibrational resonance. In principle, this can be done for the situation discussed in Subsection 2.5; however, such a general treatment results in rather cumbersome mathematical formulas not suitable for this paper. The simplest problem that can be solved exactly is the case of vibrational resonance between the 9-fold degenerate state |p 2〉 = |ne1e me;nh1h mh〉 and either of the 3-fold degenerate states |p 1〉 = |ne1e me;nh0h0h〉 or | ne0e0e;nh1h mh〉 (which correspond to resonance in the valence or conduction band, respectively). Clearly, the 9-fold degeneracy of the upper state is the lowest possible degeneracy for optically excitable states with equal quantum numbers of electrons and holes [see Eq. (7)]. Eigenvectors for the state |p 2〉 are obtained as a direct product of the three degenerate eigenstates for electrons and the three degenerate eigenstates for holes,
{|ne1eme;nh1nmh}={|ne,1e,1e|ne,1e,0e|ne,1e,+1e}{|nh,1h,1h|nh,1h,0h|nh,1h,+1h}.
For the two lower states, the eigenvectors are
|nh,0h,0h{|ne,1e,1e|ne,1e,0e|ne,1e,+1e}
and
|ne,0e,0e{|nh,1h,1h|nh,1h,0h|nh,1h,+1h}.

Employing the results of Section 2 for the above pair of degenerate states, we solve Eq. (19) with a new minimal basis for the Hamiltonian 0 based on the renormalized electron-phonon spectrum. The obtained density matrix elements are then introduced into the expression similar to Eq. (20) to find that the differential cross section of the secondary emission is given by
d2σdΘdωλ=δpe2,ph2C(ωF)(n=272γ^1nγnndn4ΔLn2+γ1n22(γ1n+ΓF/2ΔFn2+(γ1n+ΓF/2)2)+r=37n=2r12ζrnγ1nγnnγrrdn2dr2ΔLn2+γ1n22(γ1r+ΓF/2)ΔFr2+(γ1r+ΓF/2)2+|n=27dn2ΔLniγ1n|2ΓF(ωLωF)2+(ΓF/2)2).
(24)
In deriving this expression, we took into account six optically allowed polaron-like states of energies
Σ2(k)=12(Ep2+E˜p1+ħΩLO+δk),Σ5(k)=12(Ep2+E˜p1+ħΩLOδk),Σ3(k)=12(Ep2+E˜p1+ħΩLO=δk),Σ6(k)=12(Ep2+E˜p1+ħΩLOδk),Σ4(k)=12(Ep2+E˜p1+ħΩLO+δk),Σ7(k)=12(Ep2+E˜p1+ħΩLOδk),
where p1, p1, and p1 are the energies of the electron-hole pairs shifted due to the interaction with k, k′, and k″ LO-phonon modes according to Eqs. (5) and (9a);
δk=χ2+4Vk2,δk=χ2+4Vk2,δk=χ2+4Vk2,χ=Ep2E˜p1ħΩLO,χ=Ep2E˜p1ħΩLO,χ=Ep2E˜p1ħΩLO,
and the parameters Vk are given by Eq. (13b). It was also taken into account that the diagonal part of the electron-phonon interaction does not change the energy of the state |p 2〉, which resulted in p2 = p2 = p2 = E p2. The quantities dn in Eq. (24) are the constants describing renormalization of the electron-phonon spectrum, given by
d2=(χ+δk)/Δk,d3=(χ+δk)/Δk,d4=(χ+δk)/Δk,d5=2Vk/Δk,d6=2Vk/Δk,d7=2Vk/Δk,
where
Δk=(χ+δk)2+4Vk2,Δk=(χ+δk)2+4Vk2,Δk=(χ+δk)2+4Vk2.

4. Examples and discussion

In the strong confinement regime, the energy spectrum of noninteracting electrons and holes inside a spherical quantum dot is given by [1

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

, 2

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).

]
Eneleme,nhlhmh=Eg+Eneleme+Enhlhmh=Eg+ħ2ξnele22mcR2+ħ2ξnhlh22mvR2,
where effective masses mc and mv are expressed through free-electron mass m 0 as mc = 0.0219 m 0 and mv = 0.43 m 0. It should be recorded that the energies of the electron-hole pairs are actually independent on the angular momentum projections me and mh. As follows from Eqs. (20), (21), and (23), only the states with ne = nhn, le = lhl, and me = mhm appear in the spectra of the quasi-elastic secondary emission. The last expression then reduces to
Enlm,nlm=Eg+ħ2ξnl22μR2,
where μ = mc mv/(mc + mv).

4.1. Resonance with nondegenerate electronic states

As a first application of the developed theory, we consider the vibrational resonance coupling a pair of orbitally nondegenerate electronic states of the lowest possible energies
Ep2E200,200=Eg+ħ2(2π)22μR2
and
Ep1E200,100=Eg+ħ2π22R2(4mc+1mv).
The detuning from the resonance in this instance is characterized by the parameter
χ=Ep2E˜p1ħΩLO=3ħ2π22mvR2+αe2ɛ*RħΩLO,
in which α ≈ 0.0549 is calculated using Eqs. (9a) and (25). For the exact resonance, equation χ = 0 is readily solved to obtain the radius of the quantum dot wherein the resonance occurs
Rres=αe22ɛ*ħΩLO+12[(αe2ɛ*ħΩLO)2+6ħπ2mvΩLO]1/29.5nm.

Evaluation of different matrix elements of the polar electron-phonon interaction in Eq. (6) shows that only two LO-phonon modes with quantum numbers q 1 = {1, 0, 0} and q 2 = {2, 0, 0} materially couple the states |p 2〉 = |200; 200〉 and |p 1〉 = |200; 100〉, and hence we deal with the second problem considered in Subsection 2.3. The energies of the polaron-like states in this situation are given according to Eq. (14) by
±(2)=12[2Eg+ħ2π22R2(8mc+5mv)αe2ɛ*R+ħΩLO±δ2],
(26a)
3(2)=Eg+ħ2π22R2(4mc+1mv)αe2ɛ*R+ħΩLO,
(26b)
where
δ2=χ2+4V22,V2=(βe2ħΩLOɛ*R)1/2,
and β ≈ 0.1346.

The anticrossing behavior of the polaron-like states given by Eq. (26) is shown in Fig. 2(a). We see that at the exact resonance, R = R res (where green and orange lines meet), the initially 3-fold degenerate state of energy E 200,200 = 200,100 + ΩLO splits into three nondegenerate states separated from each other in energy by V 2. Figure 2(b) shows the size dependance of probability amplitudes
c1(2)=χ+δ2(χ+δ2)2+4V22,c2(2)=2V2(χ+δ2)2+4V22,
whose values directly affect the intensity of the quasi-elastic secondary emission.

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; ΩLO = 29.5 meV. For other parameters refer to text.

In Fig. 3, we plot the spectra of the secondary emission given by Eq. (21) for the following five excitation frequencies in the vicinity of, or coinciding with, the electron-hole states exhibiting resonant coupling:
ωL=ω(2)ω2,ω(2),ω(2)+ω2,ω+(2),ω+(2)+ω2,
where ω 2 = V 2/ ≈ 3.1 meV. In the calculations, we assume that the quantum dot has radius R res and use the relaxation parameters of Table 1, chosen in accordance with the experimental data of Ref. [47

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]

]. The spectra corresponding to the set of parameters S 1 [panel (a)] represent the intensity of the resonant quasi-elastic scattering, while the spectra for the parameters of the set S 2 [panels (b)–(d)] show the total signal of the secondary emission, which contains contributions from both scattering and luminescence. The structure of Eq. (21) enables clear interpretation for the peculiarities of these spectra.

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 ΓF = 0.04 meV. Total secondary emission for relaxation parameters S 2, (c) ΓF = 0.04 meV, and (d) ΓF = 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.

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

table-icon
View This Table

Different curves in Fig. 3(a) originate as follows. The upper two peaks correspond to the scattering upon resonant excitation of the polaron-like states at frequencies ω±(2)1223 and 1229.1 meV. The left and right spectra in the middle represent the scattering excited nonresonantly, at frequencies ω±(2)±ω21219.9 and 1232.2 meV. Finally, the lower peak shows the signal of scattering emerging upon nonresonant excitation of the quantum dot at frequency ω(2)+ω21226meV, which lies right in the middle between the two polaron-like states exhibiting the vibrational resonance. This peak is much less intense than the other four, due to the destructive interference between the different terms under the modulus in Eq. (21). The widths of all peaks in Fig. 3(a) are equal to the bandwidth of the filter function, ΓF = 0.04 meV. Such a small value of ΓF is typical for optical experiments on a single quantum dot [48

48. M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65, 041308 (2002). [CrossRef]

].

The green spectrum in Fig. 3(b) is excited at frequency ω(2)ω2. Its left peak represents scattering (which masks the weak signal of the resonant luminescence), while the middle and right peaks correspond to the luminescence from the lower and upper polaron-like states. The situation is somewhat similar for the wine-color spectrum, in which case both excitation and scattering occur at frequency ω+(2)+ω2, whereas the two peaks of luminescence arise on the left, at frequencies ω±(2). The orange curve is the spectrum of secondary emission excited at frequency ω(2)+ω2. Again, it has a strong scattering peak centered at the excitation frequency, and the two luminescence maxima located symmetrically at both its sides and marking the energies of the polaron-like excitations. The widths of the luminescence peaks are determined by the parameters γ 12 and γ 13, because they are ten times larger than the parameter ΓF/2. The widths of the scattering peaks are set by the bandwidth ΓF.

Figure 3(c) shows the two spectra of the resonant secondary emission. We see that, in addition to the main maxima whose intensities are determined by the processes of scattering and luminescence, these spectra contain two small peaks owing to the luminescence from the polaron-like states. The spectra of secondary emission plotted in Fig. 3(d) are obtained for the same excitation conditions as the spectra in Fig. 3(b), but with a smaller spectral resolution of the detection system, ΓF = 0.4 meV. Therefore the luminescence peaks in Fig. 3(d) are twice as wide as in Fig. 3(b), and the scatting peaks are wider by a factor of ten. Yet they are all still rather distinct to be used for studying the quantum dot’s polaron-like spectrum. It should also be noted that the decrease in spectral resolution leads to the increase of the luminescence contribution to the spectrum of the total secondary emission, and redistribution of the energy within the spectra.

The excitation spectra of the secondary emission are plotted in Figure 4. As before, the spectra are modeled for the quantum dot of radius R res, and the same set of detection frequencies as the one we used for the purpose of excitation, ωF=ω(2)±ω2, ω±(2), and ω+(2)+ω2. We also assume that ΓF = 0.4 meV and employ the last two sets of the relaxation parameters in Table 1.

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); ΓF = 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.

Figures 4(a) and 4(b) show the excitation spectra of the quasi-elastic scattering (γ^ 12 = γ^ 13 = 0). The green curve is obtained with a detector tuned to frequency ω(2)ω2. Its left peak represents nonresonant scattering, since the excitation light in this case is out of resonance with any of the polaron-like energy levels. The full width at half maximum (FWHM) of the peak is determined by the bandwidth of the filter function, ΓF. By contrast, the second and third peaks of the green spectrum correspond to the resonant excitation of the lower and upper polaron-like states. Their widths are equal to the coherence relaxation rates γ 12 and γ 13. The same features are seen for the orange and wine-color spectra in Fig. 4(a), which are detected at frequencies ω±(2)+ω2, as well as for the blue and red spectra in Fig. 4(b), detected at frequencies ω±(2). The peaks of resonant scattering at ω±(2) have widths set by the relaxation parameters, while the width of the nonresonant scattering peak is equal to ΓF. The central dips present in all five spectra are due to the destructive interference of the scattered waves.

The excitation spectra of the total secondary emission are presented in Figs. 4(c) and 4(d).

The origins of different peaks and their spectral widths are evident from the above discussion. However, at this time, resonant peaks are formed by both scattering and luminescence. Note that, in general, the developed stationary theory does not allow us to separate the contributions of these two processes to the intensity of the total secondary emission. The regime of nonstationary excitation of the secondary emission from a quantum dot exhibiting vibrational resonance will be analyzed in a subsequent paper.

4.2. Resonance with degenerate electronic states

Let us now consider the situation discussed in Subsection 3.3, in which LO phonons couple a pair of orbitally degenerate electronic states. Without loss of generality, we shall look at the lowest states of such kind, with quantum numbers ne = nh = 1. Their energies are given by
Ep2E11m,11m=Eg+ħ2ξ1122μR2,Ep1E11m,100=Eg+ħ22R2(ξ112mc+π2mv),
where m = −1, 0, +1 and ξ 11 ≈ 4.4934. As stated in connection with Eq. (24), the diagonal part of the polar electron-phonon interaction does not affect the upper electronic state, but modifies the lower one. As a result, the lower state splits into two,
E˜11±1,100=Eg+ħ22R2(ξ112mc+π2mv)αe2ɛ*R,
and
E˜110,100=Eg+ħ22R2(ξ112mc+π2mv)αe2ɛ*R,
where α ≈ 0.0513 and α′ ≈ 0.0974. These states exhibit the exact vibrational resonances (through phonons of energy ΩLO) with the upper electronic state inside the quantum dots of radii
Rresαe22ɛ*ħΩLO+[ħ(ξ112π2)2mvΩLO]1/25.59nm
and
Rresαe22ɛ*ħΩLO+[ħ(ξ112π2)2mvΩLO]1/25.61nm.
The detunings from the two resonances are:
χ=ħ2(ξ112π2)2mvR2ħΩLO+αe2ɛ*R,χ=ħ2(ξ112π2)2mvR2ħΩLO+αe2ɛ*R.

Estimates show that only one LO phonon at a time contributes predominantly to the coupling of any pair of the electronic states. Specifically, states |11+1; 11+1〉 and |11+1; 100〉 are mainly coupled via the phonon mode with quantum numbers q 1 = {1, 1, −1}, states |110; 100〉 and |110; 100〉 are coupled via the phonon mode q 2 = {1, 1, 0}, and states |11−1; 11−1〉 and |11−1; 100〉 via the phonon mode q 3 = {1, 1, +1}. The relevant off-diagonal matrix elements of the electron-phonon interaction are equal to each other and are given by
|V11+1,11+1;11+1,100(111)|=|V110,110;110,100(110)|=|V111,111;111,100(11+1)|V1=(βe2ħΩLOɛ*R)1/2,
where β ≈ 0.1265. With this result, we can write the energies of the six optically allowed polaron-like states in the following form:
Σ2(1)=Σ3(1)=+,Σ5(1)=Σ6(1)=,
(27a)
Σ4(1)=+,Σ7(1)=,
(27b)
±=12[2Eg+ħ2ξ1122μR2+ħ22R2(ξ112mc+π2mv)αe2ɛ*R+ħΩLO±δ1],
(27c)
±=12[2Eg+ħ2ξ1122μR2+ħ22R2(ξ112mc+π2mv)αe2ɛ*R+ħΩLO±δ1],
(27d)
where δ1=(χ2+4V12)1/2 and δ1=(χ2+4V12)1/2. The parameters dn in Eq. (24) are explicitly given by
d2=d3=χ+δ1(χ+δ1)2+4V12,d5=d6=2V1(χ+δ1)2+4V12,d4=χ+δ1(χ+δ1)2+4V12,d7=2V1(χ+δ1)2+4V12.

The energies defined in Eq. (27) and parameters dn are plotted in Figs. 5(a) and 5(b). Magnifying insets reveal the two pairs of closely spaced polaron-like states. The splitting, Δ±=±± of the fine-structure levels is shown in Fig. 5(c). One can see that the quantum dot of radius R resRres ≈ 5.6 nm is the best for the simultaneous observation of the upper and lower energy doublets. The splitting of the doublets near the exact vibrational resonance is about 92 μeV. For the upper pair of states, the maximal splitting of approximately 137 μeV is achieved in the quantum dot of radius ≈ 6.9 nm. In contrast, the splitting of the lower doublet grows monotonously with the decrease of R, as the constituent electronic states keep diverging due to the nonresonant electron-phonon interaction.

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.

In Fig. 6, we plot the spectra of the secondary emission given by Eq. (24), assuming that γnn = γ 1 n = 2γ^ 1 n = 20 μeV for n = 2, 3, . . . , 7. In order to be able to resolve the four energy levels of the polaron-like spectrum, the condition ΓF ≤ Δ± needs to be satisfied. We shall meet this condition in the quantum dot of radius R res, by setting ΓF = 40 μeV. Figures 6(a)–6(c) show the five excitation spectra of the secondary emission detected at frequencies ω(1)ω1, ω(1)/ħ, ω(1)+ω1, ω+(1)+/ħ, and ω+(1)+ω1, where ω 1 = V 1/. The large-scale structure of these spectra and the spectra in Figs. 4(c) and 4(d) are similar. However, the spectra maxima that previously corresponded to the resonant excitation of the lower and upper polaron-like states are now distinctly split into two peaks each (see magnified portions of the spectra in the insets). The first and second low-energy peaks can be interpreted as resonant scattering and luminescence from the polaron-like states of the lower doublet; they appear at frequencies /ħ and ω(1). Analogously, the first and second high-energy peaks are located at frequencies +/ħ and ω+(1) and represent the secondary emission mediated by the scattering and luminescence from the upper doublet. The difference in the intensities of the first and second fine-structure peaks in Figs. 4(a)–4(c) is explained by the fact that the exact vibrational resonance occurs only for the higher-energy polaron-like states in each doublet, whereas the lower-energy states are slightly off resonance with the phonon modes.

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); ΓF = 0.04 meV. For other parameters refer to text.

Figure 6(d) shows how the spectra of the secondary emission in Fig. 6(a) are modified for a quantum dot of radius R ≈ 3.5 nm (R −2 = 0.08 nm−2). As the spectra suggest, and as is seen from Fig. 5(c), the level splitting in the lower doublet of the polaron-like states increases up to ≈ 290 μeV. At the same time, the splitting of the upper doublet becomes negligible and cannot be observed in the real spectra. The relative intensities of the two fine-structure peaks are now determined by the proximity of the corresponding polaron-like states to the vibrational resonance. Notice also a strong scattering peak, which occurs where the detection and excitation frequencies coincide. It is not suppressed by the destructive interference as in Fig. 6(a), since the contributions from the polaron-like states are not symmetrical.

5. Conclusions

This paper is intended to lay the foundation for the optical spectroscopy of the semiconductor quantum dots with electron and phonon subsystems strongly coupled due to the resonant polar electron-phonon interaction. Based on the generalized model for renormalization of the quantum dot’s energy spectrum and using the density matrix technique, we have developed a new theory of the quasi-elastic secondary emission from a single quantum dot, while assuming that several LO-phonon modes may resonantly couple a pair of electron-hole states. We considered two examples of special simplicity and practical interest, namely, where the vibrational resonance involves either orbitally nondegenerate electron-hole states and two degenerate LO-phonon modes, or orbitally degenerate electron-hole states and three degenerate phonon modes. The derived analytical expressions for the differential cross section of the secondary emission provide an intuitive, yet accurate description of the quantum dot’s optical response, and may prove useful in retrieving the coherence relaxation rates of the polaron-like states from the experimentally measured spectra.

Acknowledgments

References and links

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. B 82, 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. B 73, 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. B 70, 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. B 75, 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. B 73, 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. B 73, 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. B 71, 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. B 71, 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. B 68, 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. B 74, 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. B 72, 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. B 68, 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. B 65, 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,” Semiconductors 35, 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. B 67, 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. B 62, 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. B 56, 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. B 56, 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. B 54, 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. B 77, 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. Express 17, 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. B 76, 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. B 81, 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. B 65, 041308 (2002). [CrossRef]

OCIS Codes
(290.5870) Scattering : Scattering, Rayleigh
(300.3700) Spectroscopy : Linewidth
(300.6280) Spectroscopy : Spectroscopy, fluorescence and luminescence
(300.6470) Spectroscopy : Spectroscopy, semiconductors
(160.4236) Materials : Nanomaterials
(250.5590) Optoelectronics : Quantum-well, -wire and -dot devices

ToC Category:
Spectroscopy

History
Original Manuscript: April 26, 2011
Revised Manuscript: June 24, 2011
Manuscript Accepted: July 11, 2011
Published: July 28, 2011

Citation
Ivan D. Rukhlenko, Anatoly V. Fedorov, Anvar S. Baymuratov, and Malin Premaratne, "Theory of quasi-elastic secondary emission from a quantum dot in the regime of vibrational resonance," Opt. Express 19, 15459-15482 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-15459


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References

  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]

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