## Theory of quasi-elastic secondary emission from a quantum dot in the regime of vibrational resonance |

Optics Express, Vol. 19, Issue 16, pp. 15459-15482 (2011)

http://dx.doi.org/10.1364/OE.19.015459

Acrobat PDF (1712 KB)

### 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

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. 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]

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]

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. 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]

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. 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]

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]

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]

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]

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. 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]

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]

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]

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]

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]

## 2. Theory of vibrational resonance in a semiconductor quantum dot

### 2.1. Hamiltonian formalism

*e*) with phonons (

*ph*), excitation light (

*L*), and radiation field (

*λ*). The notations on the right-hand side of Eq. (2) are as follows:

*E*=

_{p}*E*+

_{g}*E*+

_{pe}*E*is the energy of electron-hole pair state |

_{ph}*p*〉 ≡ |

*p*;

_{e}*p*〉, which depends on the quantum numbers of electron (

_{h}*p*) and hole (

_{e}*p*), and the band gap,

_{h}*E*of the bulk semiconductor;

_{g}*h̄*Ω

*and*

_{q}*h̄ω*are the energies of LO-phonon mode

_{λ}*q*and photon mode

*λ*.

*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 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 in which –

*e*is the charge of the electron;

*q*;

*δ*

_{pe2, pe1}and

*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.

*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]

*ɛ*

**= (1/**

^{*}*ɛ*

_{0}– 1/

*ɛ*

_{∞})

^{–1},

*ɛ*

_{0}and

*ɛ*

_{∞}are the low- and high-frequency dielectric permittivities of the bulk semiconductor,

*j*(

_{l}*z*) is the spherical Bessel function of the first kind, with

*ξ*being its

_{nl}*n*th zero [i.e.,

*j*(

_{l}*ξ*) = 0]. Clebsch-Gordan coefficients

_{nl}*n*,

_{a}*l*, and

_{a}*m*(

_{a}*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}).

*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. 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]

*ϕ*(

*t*) is the complex envelope of classical optical field,

*V*is the normalization volume, and is the matrix element of the electric dipole moment operator –

*e*

**r**, calculated on Bloch functions

*u*and

_{c}*u*at the Brillouin zone center, for excitation light (

_{v}*η*=

*L*) or emitted photon (

*η*=

*λ*) of frequency

*ω*and polarization vector

_{η}**ê**

*. 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 T*

_{η}_{d}and O

_{h}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]

### 2.2. Formation of polaron-like states

*E*

_{p2}–

*E*

_{p1}≈

*h̄*Ω

*. 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].*

_{q}### 2.3. Vibrational resonance involving one, two, and three phonon modes

*H̃*that admit analytical solutions. The simplest problems of this type arise where two nondegenerate states |

*Ẽ*

_{p2}〉 and |

*Ẽ*

_{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,

*Ẽ*

_{p2}; 0

_{q1}〉 = |

*Ẽ*

_{p2}〉|0

_{q1}〉 and |

*Ẽ*

_{p1}; 1

_{q1}〉 =

**|**

*Ẽ*

_{p1}〉|1

_{q1}〉, where kets

**|**0

_{q1}〉 and |1

_{q1}〉 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 where

*Ẽ*

_{p2}; 0

_{q1}〉 and |

*Ẽ*

_{p1}; 1

_{q1}〉. The formation of new energy-shifted levels of the polaron-like states

*q*

_{1}and

*q*

_{2}are equal to the energy,

*h̄*Ω

*of the bulk LO phonon at the Brillouin zone center, we find the polaron-like states of the Hamiltonian*

_{LO}*→ Ω*

_{qk}*,*

_{LO}**q**= (

*q*

_{1},

*q*

_{2}). The matrix that puts

*h̄*Ω

_{q1}=

*h̄*Ω

_{q2}=

*h̄*Ω

_{q3}= Ω

*, we may represent the polaron-like states as where*

_{LO}**q**= (

*q*

_{1},

*q*

_{2},

*q*

_{3}). The diagonalization

*q*

_{1},

*q*

_{2}, . . . ,

*q*) are resonant with the quantum dot’s electronic subsystem. The resulting energies of the polaron-like states are of the form Thus, whenever the

_{k}*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

*H̃*

_{e}_{,}

*and*

_{L}*H̃*,

_{e}*given in Eqs. (8) through (10). In order to evaluate the efficiency of low-temperature secondary emission in the presence of vibrational resonance with one, two, or three LO-phonon modes, the following matrix elements of the transformed Hamiltonians*

_{λ}*Ĥ*

_{e}_{,}

*(*

_{η}*η*=

*L*,

*λ*) are required: Here the rows from top to bottom, and columns from left to right, correspond to the states

^{(}

^{k}^{)}〉|0

*〉 for*

_{λ}*η*=

*L*, and

^{(}

^{k}^{)}〉|1

*〉 for*

_{λ}*η*=

*λ*; |0

^{(k)}〉 denotes the vacuum of polaron-like excitations, while |0

*〉 and |1*

_{λ}*〉 stand for zero and one photons in the mode*

_{λ}*λ*. Equations (17) are obtained with the same

*S*-matrices that were used to diagonalize Hamiltonians

*p*

_{e2}=

*p*

_{h2}).

### 2.5. Analytically solvable eigenvalue problems

*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

*S*-matrix of a similar block structure, Diagonalization of the Hamiltonian

_{e}_{,}

*with matrix 𝒮 can be performed analytically in the special case of any*

_{ph}*N*vibrational resonances involving no more than three phonon modes, i.e., if max (

*k*

_{1},

*k*

_{2}, . . . ,

*k*) ≤ 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

_{N}## 3. Quantum dot secondary emission

*T*≪

*h̄*Ω

*), in order to avoid turning the phonon subsystem out of its state of thermodynamic equilibrium. In this situation, it is convenient to describe the phenomenon of secondary emission within the density matrix formalism. The dynamics of spectroscopic transitions in the considered quantum system is then governed by the generalized master equation for the reduced density matrix,*

_{q}*ρ*(

*t*) [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]

*Ĥ*(

*t*) is the transformed Hamiltonian given in Eq. (10),

*γ*=

_{μν}*γ*= (

_{νμ}*γ*+

_{μμ}*γ*)/2

_{νν}**+**

*for*γ ^

_{μν}*μ*≠

*ν*is the coherence relaxation rate,

*μ*, and

*=*γ ^

_{μν}*is the pure dephasing rate. The last term on the right-hand side of the master equation accounts for the transitions |*γ ^

_{νμ}*ν*′〉 → |

*ν*〉 due to the thermal interaction with a bath through the relaxation parameters

*ζ*

_{νν}_{′}. We allow for this interaction in our subsequent discussion of quasi-elastic secondary emission, but neglect it in Section 4.

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

*k*phonon modes occurs between the nondegenerate states of the electron-hole pairs, we take the four eigenstates of the Hamiltonian

*H̃*

_{0}, 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.

*ω*due to the annihilation of the polaron-like excitations is characterized by the rate [37

_{λ}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]

*ρ*

_{24}(

*t*) and

*ρ*

_{34}(

*t*) are to be found from the master equation.

*ϕ*(

*t*) =

*E*, 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, d

_{L}*ω*:

_{λ}*c*is the speed of light in a vacuum, and

*γ*

_{0}is the inverse photon lifetime. The quantities

*, the relative importance of these terms is set by: (i) pure dephasing rates*

_{η}_{n}γ ^

_{12}and

γ ^

_{13}; (ii) lifetimes,

*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]

*γ*

_{12}and

*γ*

_{13}substantially increase if the transitions between states

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]

*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, 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. 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]

*γ*

_{0}≪

*γ*

_{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

*g*(

_{F}*ω*–

_{F}*ω*) centered at frequency

_{λ}*ω*. Following Ref. [46

_{F}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]

*being the detector’s bandpass. Using this expression for*

_{F}*g*and taking into account that Γ

_{F}*≫*

_{F}*γ*

_{0}in the majority of practical instances, we obtain after the convolution

*γ*

_{12}and

*γ*

_{13}, the bandpass is usually chosen to satisfy the inequalities in which case Eq. (21) acquires the form

*ω*and detection frequency

_{L}*ω*. The possibility to vary one of them while keeping the other constant results in two types of spectroscopic experiments. If

_{F}*ω*is fixed and

_{L}*ω*is varied, then an ordinary

_{F}*spectrum of secondary emission*is obtained. On the other hand, in the situation where

*ω*is fixed and

_{F}*ω*is altered, an

_{L}*excitation spectrum of secondary emission*is recorded.

*ω*, whereas the luminescence intensity peaks at frequencies

_{L}γ ^

_{12}/

*γ*

_{22}≪ 1 and 2

γ ^

_{13}/

*γ*

_{33}≪ 1 entering the luminescence terms, and because of Γ

*≪*

_{F}*γ*

_{12},

*γ*

_{13}.

*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

*p*

_{2}〉 = |

*n*1

_{e}

_{e}*m*;

_{e}*n*1

_{h}

_{h}*m*〉 and either of the 3-fold degenerate states |

_{h}*p*

_{1}〉 = |

*n*1

_{e}

_{e}*m*;

_{e}*n*0

_{h}*0*

_{h}*〉 or*

_{h}**|**

*n*0

_{e}*0*

_{e}*;*

_{e}*n*1

_{h}

_{h}*m*〉 (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 |

_{h}*p*

_{2}〉 are obtained as a direct product of the three degenerate eigenstates for electrons and the three degenerate eigenstates for holes, For the two lower states, the eigenvectors are and

*H̃*

_{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

*Ẽ*

_{p1},

*Ẽ*′

_{p1}, and

*Ẽ*″

_{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);

*V*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 |

_{k}*p*

_{2}〉, which resulted in

*Ẽ*″

_{p2}=

*Ẽ*′

_{p2}=

*Ẽ*

_{p2}=

*E*

_{p2}. The quantities

*d*in Eq. (24) are the constants describing renormalization of the electron-phonon spectrum, given by where

_{n}## 4. Examples and discussion

*ɛ*

_{0}= 15.15,

*ɛ*

_{∞}= 12.25,

*E*= 418 meV, and

_{g}*h̄*Ω

*= 29.5 meV [33*

_{LO}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]

*ζ*

_{νν}_{′}= 0. We also assume that the vibrational resonance takes place in the valence band, in which case Eq. (5) acquires the form where

*m*and

_{c}*m*are expressed through free-electron mass

_{v}*m*

_{0}as

*m*= 0.0219

_{c}*m*

_{0}and

*m*= 0.43

_{v}*m*

_{0}. It should be recorded that the energies of the electron-hole pairs are actually independent on the angular momentum projections

*m*and

_{e}*m*. As follows from Eqs. (20), (21), and (23), only the states with

_{h}*n*=

_{e}*n*≡

_{h}*n*,

*l*=

_{e}*l*≡

_{h}*l*, and

*m*=

_{e}*m*≡

_{h}*m*appear in the spectra of the quasi-elastic secondary emission. The last expression then reduces to where

*μ*=

*m*

_{c}*m*/(

_{v}*m*+

_{c}*m*).

_{v}### 4.1. Resonance with nondegenerate electronic states

*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

*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 where and

*β*≈ 0.1346.

*R*=

*R*

_{res}(where green and orange lines meet), the initially 3-fold degenerate state of energy

*E*

_{200,200}=

*Ẽ*

_{200,100}

**+**

*h̄*Ω

*splits into three nondegenerate states separated from each other in energy by*

_{LO}*V*

_{2}. Figure 2(b) shows the size dependance of probability amplitudes whose values directly affect the intensity of the quasi-elastic secondary emission.

*ω*

_{2}=

*V*

_{2}/

*h̄*≈ 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]

*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.

*resonant*excitation of the polaron-like states at frequencies

*nonresonantly*, at frequencies

*nonresonant*excitation of the quantum dot at frequency

*= 0.04 meV. Such a small value of Γ*

_{F}*is typical for optical experiments on a single quantum dot [48*

_{F}48. M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In_{0.60}Ga_{0.40}As/GaAs self-assembled quantum dots,” Phys. Rev. B **65**, 041308 (2002). [CrossRef]

*γ*

_{12}and

*γ*

_{13}, because they are ten times larger than the parameter Γ

*/2. The widths of the scattering peaks are set by the bandwidth Γ*

_{F}*.*

_{F}*= 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.*

_{F}*R*

_{res}, and the same set of detection frequencies as the one we used for the purpose of excitation,

_{F}**=**0.4 meV and employ the last two sets of the relaxation parameters in Table 1.

γ ^

_{12}=

γ ^

_{13}= 0). The green curve is obtained with a detector tuned to frequency

*. 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*

_{F}*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

*. The central dips present in all five spectra are due to the destructive interference of the scattered waves.*

_{F}### 4.2. Resonance with degenerate electronic states

*n*=

_{e}*n*= 1. Their energies are given by where

_{h}*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, and where

*α*≈ 0.0513 and

*α*′ ≈ 0.0974. These states exhibit the exact vibrational resonances (through phonons of energy

*h̄*Ω

*) with the upper electronic state inside the quantum dots of radii and The detunings from the two resonances are:*

_{LO}*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

*β*≈ 0.1265. With this result, we can write the energies of the six optically allowed polaron-like states in the following form: where

*d*in Eq. (24) are explicitly given by

_{n}*d*are plotted in Figs. 5(a) and 5(b). Magnifying insets reveal the two pairs of closely spaced polaron-like states. The splitting,

_{n}*R*

_{res}≈

*R*′

_{res}≈ 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

*μ*eV. For the upper pair of states, the maximal splitting of approximately 137

*μ*eV 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.

*γ*=

_{nn}*γ*

_{1}

*= 2*

_{n}γ ^

_{1}

*= 20*

_{n}*μ*eV for

*n*= 2, 3, . . . , 7. In order to be able to resolve the four energy levels of the polaron-like spectrum, the condition Γ

*≤ Δ*

_{F}_{±}needs to be satisfied. We shall meet this condition in the quantum dot of radius

*R*

_{res}, by setting Γ

*= 40*

_{F}*μ*eV. Figures 6(a)–6(c) show the five excitation spectra of the secondary emission detected at frequencies

*ω*

_{1}=

*V*

_{1}/

*h̄*. 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

*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

*μ*eV. 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

## Acknowledgments

## References and links

1. | A. V. Fedorov, I. D. Rukhlenko, A. V. Baranov, and S. Y. Kruchinin, |

2. | T. Takagahara, “Electron-phonon interactions in semiconductor quantum dots,” in |

3. | V. Cesari, W. Langbein, and P. Borri, “Dephasing of excitons and multiexcitons in undoped and |

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

5. | K. Kojima and A. Tomita, “Influence of pure dephasing by phonons on exciton-photon interfaces: quantum microscopic theory,” Phys. Rev. B |

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 |

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 |

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 |

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. |

10. | S. Sanguinetti, E. Poliani, M. Bonfanti, M. Guzzi, E. Grilli, M. Gurioli, and N. Koguchi, “Electron-phonon interaction in individual strain-free GaAs/Al |

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 |

12. | A. V. Fedorov and A. V. Baranov, “Exciton-vibrational interaction of the Fröhlich type in quasi-zero-size systems,” J. Exp. Theor. Phys. |

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. |

14. | I. D. Rukhlenko and A. V. Fedorov, “Propagation of electric fields induced by optical phonons in semiconductor heterostructures,” Opt. Spectrosc. |

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. |

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 |

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 |

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 |

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 |

20. | J. Zhao, A. Kanno, M. Ikezawa, and Y. Masumoto, “Longitudinal optical phonons in the excited state of CuBr quantum dots,” Phys. Rev. B |

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 |

22. | A. V. Fedorov, A. V. Baranov, A. Itoh, and Y. Masumoto, “Renormalization of energy spectrum of quantum dots under vibrational resonance conditions,” Semiconductors |

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. |

24. | P. Palinginis, S. Tavenner, M. Lonergan, and H. Wang, “Spectral hole burning and zero phonon linewidth in semiconductor nanocrystals,” Phys. Rev. B |

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. |

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. |

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. |

28. | O. Verzelen, R. Ferreira, and G. Bastard, “Excitonic polarons in semiconductor quantum dots,” Phys. Rev. Lett. |

29. | T. Stauber, R. Zimmermann, and H. Castella, “Electron-phonon interaction in quantum dots: a solvable model,” Phys. Rev. B |

30. | T. Inoshita and H. Sakaki, “Density of states and phonon-induced relaxation of electrons in semiconductor quantum dots,” Phys. Rev. B |

31. | A. V. Fedorov, A. V. Baranov, and K. Inoue, “Exciton-phonon coupling in semiconductor quantum dots: resonant Raman scattering,” Phys. Rev. B |

32. | A. V. Fedorov, A. V. Baranov, and K. Inoue, “Two-photon transitions in systems with semiconductor quantum dots,” Phys. Rev. B |

33. | O. Madelung, M. Schultz, and H. Weiss, eds., |

34. | A. I. Anselm, |

35. | A. S. Davydov, |

36. | G. A. Korn and T. M. Korn, |

37. | A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of the quasi-elastic resonant secondary emission: semiconductor quantum dots,” Opt. Spectrosc. |

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. |

39. | K. Blum, |

40. | R. W. Boyd, |

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 |

42. | A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Acoustic phonon problem in nanocrystaldielectric matrix systems,” Solid State Commun. |

43. | A. V. Fedorov and S. Y. Kruchinin, “Acoustic phonons in a quantum dot-matrix system: hole-burning spectroscopy,” Opt. Spectrosc. |

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 |

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 |

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 |

47. | D. Gammon, N. H. Bonadeo, G. Chen, J. Erland, and D. G. Steel, “Optically probing and controlling single quantum dots,” Physica E (Amsterdam) |

48. | M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In |

**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

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