1. Introduction

The secondary-emission spectroscopy is the powerful optical tool for studying both individual quantum dots (QDs) and their ensembles [1–8]. It enables one to determine the chemical composition of the QD material [9–14], measure the mechanical stress inside a QD [15–19], investigate the properties of the QD surface [20–23], and study various aspects of the electron–phonon interaction inside QDs [24–26]. Many of the unique physical properties [27, 28] of semiconductor QDs underlying their applications in photonics [29, 30] were determined using the experimental methods of the secondary-emission spectroscopy.

Among the most important interactions in semiconductor QDs is the electron–phonon coupling, which enhances the dephasing rates of spectroscopic transitions [31–34] and reduces the lifetimes of electronic excitations [35–39]. This interaction also leads to the formation of hybrid polaron-like states inside QDs with resonant electron and phonon subsystems [40–43], causing a significant modification of the QDs’ energy spectra. The vibrational resonance required for the modification to occur is realized, for instance, where the energy of a phonon mode (supported either by the QD or the host material [44, 45]) coincides with the energy spacing between a pair of the quantum-confined electron states of the QD. Often, these are the longitudinal optical (LO) phonons and the polar electron–phonon interaction that couple the states and induce the resonance [43,46–51]. The energy splitting of the resulting polaron-like excitations is typically in the millielectronvolt range and can therefore be experimentally resolved only at cryogenic temperatures.

The formation of the polaron-like excitations in the above circumstances have been theoretically described for a variety of semiconductors by several research groups [43, 50–58]. In particular, Hameau et al. [51] proposed a model of everlasting resonant polarons in InAs QDs to explain the experimental data on the QD intraband transitions obtained with the far-infrared magnetospectroscopy [46, 47, 49, 51]. The multiphonon Raman scattering and polaron effects in semiconductor nanocrystals were taken into account by Pokatilov et al. [53] and Miranda et al. [54, 57] to describe the Raman spectrum of spherical CdSe QDs. Itoh et al. [43] developed a theory of polaron and exciton-phonon complexes to explain the shape of the exciton luminescence spectra measured under the resonant size-selective excitation of CuCl nanocrystals. Another model of the polaron-like excitations, suggested by Fedorov et al. [50], was employed to interpret the renormalization of the lowest-energy exciton states in CuCl QDs embedded in the NaCl matrix, which were studied under the resonant two-photon excitation using the secondary-emission spectroscopy. This model was recently used to describe the transformation of the QD energy spectrum due to the resonant polar electron–phonon coupling and to study how this transformation is manifested in the stationary photoluminescence spectra [40].

This paper presents a theory of phonon-assisted secondary emission from a single QD in the regime of vibrational resonance. In contrast to previous works, we are considering a vibrational resonance in a QD with finite potential barriers for the confined carriers. Owing to the barriers finiteness, the selection rules of interband optical transitions are modified, and transitions to some electron–hole pair states that cannot be optically excited in the QDs with impenetrable boundaries become dipole allowed. Another important consequence of the QD model adopted is the emergence of a specific three-particle interaction between the optical radiation and the electron–phonon subsystem of the QD. This interaction leads to a multi-peak structure of the photoluminescence and excitation photoluminescence spectra, whose analysis allows one to estimate the strength of the electron–phonon interaction.

The paper is organized as follows. We begin in Section II by extending the Hamiltonian formalism employed earlier for the treatment of the vibrational resonance in the QD being an infinitely high potential well for the confined charge carriers [40,52] to the case of a spherical QD with a penetrable surface. This allows us to consider the three-particle interaction—involving electrons–hole pair, phonons, and photons—essential for the study. Important here is that this interaction is readily observable in the secondary emission spectra but absent in the model of infinitely high potential barriers. Section III presents a stationary solution to the generalized master equation for the reduced density matrix, which describes the process of photoluminescence mediated by the decay of the polaron-like excitations of the QD. The energies and wave functions of the excitations are calculated analytically by diagonalizing the Hamiltonian of the system with respect to the electron–phonon interaction in the Appendix. Section IV summarizes our results and concludes the paper.

2. Theory of vibrational resonance

Our approach to the theoretical treatment of vibrational resonance in a semiconductor QD is based on the use of the Hamiltonian formalism that was initially developed by Levinson and Rashba [59] for bulk materials and later extended by Fedorov et al. [40, 50, 52] for low-dimensional structures. Without loss of generality, we restrict our consideration to the polar electron–phonon interaction between the charge carriers and longitudinal optical (LO) phonons confined inside a semiconductor QD with a high degree of ionicity. This interaction is relatively easy to study experimentally due to its greater strength as compared to the strengths of the other kinds of electron–phonon coupling [24, 26, 41, 43, 49, 51]. The following theory, with the exception of Eqs. (2)–(9), is applicable to any kind of electron–phonon interaction leading to the formation of polaron-like states in quantum nanostructures.

2.1. Hamiltonian formalism

The phonon-assisted photoluminescence from a semiconductor QD can occur according to four different physical scenarios, in which phonons are either absorbed or emitted upon the optical excitation or radiative relaxation of the QD’s electronic subsystem [40]. We describe the electron (e) and phonon (ph) subsystems of the QD, their excitation with light (L), the radiation field (r), and the interactions between the electromagnetic fields and the QD using six Hamiltonians: (i) the free electron–hole pair Hamiltonian H_e expressed through energies, E_p, creation operators, $a_p^{+}$, and annihilation operators, a_p, of the pairs; (ii) the Hamiltonian H_ph of the dispersionless LO phonon modes of energy h̄Ω described by operators $b_q^{+}$ and b_q; (iii) the radiation-field Hamiltonian H_r, which is expressed through the energies h̄ω_λ of the electromagnetic field modes λ and the modes’ creation, $c_{\lambda }^{+}$, and annihilation, c_λ, operators; (iv) the polar electron–phonon coupling Hamiltonian H_e,ph; (v) the Hamiltonian H_e,L of the QD interaction with the classical excitation field; and (vi) the Hamiltonian H_e,r of the interaction between the QD and the quantum radiation field. The energy of an electron–hole pair in state |p〉 is comprised of three terms, E_g + E_pe + E_ph, which are the band gap of bulk semiconductor and the quantum-confinement energies of electrons and holes in the states characterized by the sets of quantum numbers p_e and p_h.

Let us focus on the QD in the strong confinement regime and describe its electron subsystem using a simple two-band model of semiconductor material [24–26]. Using the parabolic model of the valence band and neglecting the Coulomb interaction allow us to derive simple formulas for rapid plotting of photoluminescence spectra and checking the simulation results. Since E_g ≫ h̄Ω for most semiconductors with a high degree of ionicity, the electron–phonon coupling induces only intraband transitions of the confined charge carriers, whereas the interaction of the QD’s electronic subsystem with electromagnetic fields leads solely to interband transitions. The Hamiltonian operators describing these interactions are of the forms [40]

The strength of the polar electron–phonon interaction is determined by the matrix element [24] $V_{p_2;p_1}^{(q)}=e\left({\phi }_{p_{h_1},p_{h_2}}^{(q)}{\delta }_{p_{e_2},p_{e_1}}-{\phi }_{p_{e_2},p_{e_1}}^{(q)}{\delta }_{p_{h_2},p_{h_1}}\right)$, where −e is the charge of the electron, ${\phi }_{p_{h_1},p_{h_2}}^{(q)}$ and ${\phi }_{p_{e_2},p_{e_1}}^{(q)}$ are the matrix elements of the electric field induced by the LO phonons of mode q; and δ_ph2,ph1 and δ_pe2,pe1 are the products of Kronecker deltas for the respective pairs of quantum numbers from sets p_h1 and p_h2 for holes and p_e1 and p_e2 for electrons.

The direct creation of an electron–hole pair by the classical excitation field and the subsequent annihilation of the pair due to its interaction with the vacuum radiation field are described with the matrix elements $V_{p;0}^{(L)}=-e{E}_L{D}_{cv}{F}_p\varphi (t)\text{exp}(-i{\omega }_{L}t)$ and $V_{p;0}^{(\lambda )}=-ie\sqrt{2\pi \overline{h}{\omega }_{\lambda }/({\epsilon }_{\infty }V)}{D}_{cv}{F}_p$, where ϕ(t) is the complex-valued envelope of the excitation pulse of the peak field strength E_L, D_cv is the dipole matrix element calculated on the Bloch wave functions at the Brillouin zone center, F_p is the overlap integral calculated on the envelope wave functions of electrons and holes, ε_∞ is the high-frequency permittivity of the bulk semiconductor, and V is the normalization volume.

2.2. A spherical QD with a penetrable surface

We shall eventually be interested in calculating the intensity of the QD secondary emission. This will require explicit expressions for the matrix elements entering Eq. (1). To find the forms of these elements, we now specify the model of the QD.

We first assume that the QD semiconductor has T_d or O_h symmetry. This enables us to relate the dipole matrix element to Kane’s parameter P of the bulk material through the bandgap energy via $\left|D_{cv}\right|=\sqrt{2}P/E_g$ [40]. We also consider our QD to be a sphere of radius R embedded in a dielectric host. Then the states of electrons, holes and phonons are characterized by sets {n, l, m} of three quantum numbers: the principal quantum number n, the angular momentum l, and its projection m. The overlap integral in this case is of the form [60]

Using the continual model of the dispersionless LO phonons, we next analytically calculate the electric field potential induced by the phonons inside and outside the QD [40]. The matrix elements of this field calculated on the envelope wave functions of electrons and holes are given by

The forms of the radial functions R_nl(x) and ℛ_nqlq (x) are determined by the profiles and heights of the potential barriers that need to be overcome by electrons, holes and phonons to leave the QD. These barriers are usually finite and unequal for electrons and holes confined by a semiconductor QD in a dielectric host. By describing them with the step functions of heights V_e and V_h, we obtain the following functional dependency [60]:

Since our QD is embedded in the dielectric that does not support optical phonon modes, its surface may be considered impenetrable for the confined LO phonons. The radial dependency of the phonon potential in this case is given by [61–63]

The overlap integral and the radial part of the matrix element ${\phi }_{p_{k_2},p_{k_1}}^{(q)}$ can now be calculated by substituting Eqs. (5) and (8) into Eqs. (2) and (4) to obtain

Equation (9a) shows that if a QD provides finite potential barriers for the confined carriers then the overlap integrals of the product of the envelope wave functions of electrons and holes with different principal quantum numbers (n_e ≠ n_h) are nonzero. In the case of infinite barriers this integral is given by the triple product δ_ne,nhδ_le,lhδ_me,mh, vanishing for n_e ≠ n_h. Hence, the finiteness of the potential barriers drastically modifies the selection rules of interband optical transitions, allowing some transitions that are dipole-forbidden in QDs with impenetrable boundaries.

2.3. Formation of polaron-like states

The vibrational resonance occurs where the energy of one or several LO phonon modes is close to the energy gap between a pair of the electron–hole states, i.e. where h̄Ω ≈ E_p2 − E_p1. The energy of the QD with one phonon and an electron in the lower state is equal in this situation to the energy of the QD without phonons and an electron in the upper resonant state. The degeneracy of the QD states is removed by the polar electron–phonon interaction, which leads to their splitting into several hybrid polaron-like states of different energies [40, 50, 52]. The number and degeneracy of the resulting states depend on the degeneracy orders of the electron, hole and phonon states involved in the vibrational resonance. The splitting is schematically illustrated by Fig. 1.

 

Fig. 1 Formation of polaron-like states from a pair of nondegenerate electron–hole states coupled via one, two, three or four degenerate LO phonon modes of energy h̄Ω. The electron–phonon interaction transforms the degenerate state of the QD into two, three, four or five nondegenerate polaron-like states.

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The calculation of the energies and wave functions of the polaron-like states requires diagonalization of the Hamiltonians in Eq. (1) with respect to the electron–phonon interaction. This diagonalization is carried out in two steps. With the first step, we exclude the diagonal part of the electron–phonon coupling using the transformation [40]

The transformed Hamiltonians responsible for the coupling of the polaron-like states to the excitation light [Eq. (11c)] and radiation field [Eq. (11d)] are seen to contain additional terms describing three-particle interactions. These interactions lead to the creation or annihilation of a polaron-like quasiparticle with the simultaneous absorption and/or emission of one photon and one phonon, as illustrated by the Feynman diagrams in Fig. 2. The four processes represented by the diagrams mediate the phonon-assisted secondary emission, which can be observed in the QD photoluminescence spectra.

 

Fig. 2 Feynman diagrams describing [(a) and (b)] creation and [(c) and (d)] annihilation of a polaron-like quasiparticle of energy Ẽ_p with the simultaneous (a) absorption of a photon, h̄ω_L, and a phonon, h̄Ω, (b) absorption of a photon and emission of a phonon, (c) emission of a photon, h̄ω_λ, and absorption of a phonon, and (d) emission of a photon and a phonon.

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In Fig. 3 we plot the absolute values of matrix elements Φ_p,p and overlap integrals F_p as functions of potential barrier heights for electrons and holes in a spherical InAs QD with R = 9.14 nm (the barrier heights are assumed to be related as V_e = 1.35V_h). The plots correspond to the vibrational resonance between the orbitally nondegenerate states |p₁〉 = |1, 0, 0;1, 0, 0〉 and |p₂〉 = |1, 0, 0; 2, 0, 0〉 of the valence band and a phonon mode of quantum numbers q = {1, 0, 0}. The Hamiltonian terms of the form $F_p\left(1+{\mathrm{\Phi }}_{p;p}^{(q)}{b}_q-{\mathrm{\Phi }}_{p;p}^{(q)*}{b}_q^{+}\right)a_p^{+}$, which describe the interaction of the polaron-like states with excitation light, are seen to be all nonzero for finite potential barriers. Since Φ_p1,p1 and F_p2 (shown by dashed curves) vanish when the barriers are infinitely high, the phonon-assisted photoluminescence in this case is absent.

 

Fig. 3 Matrix elements and overlap integrals as functions of potential barrier heights for electrons and holes in a spherical InAs QD with R = 9.14 nm. The material parameters are: ε₀ = 15.15, ε_∞ = 12.25, E_g = 418 meV, h̄Ω = 29.5 meV, m_e = 0.0219m₀, and m_h = 0.43m₀ [64] (m₀ is the free-electron mass). For other parameter refer to the text.

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With the second step, we exclude from Eqs. (11b)–(11d) the nondiagonal part of the electron–phonon coupling corresponding to the states in the vibrational resonance using the transformation Ĥ_e,ph = S⁺H̃_e,phS. In order to construct the unitary operator S performing the transformation, we solve the standard eigenvalue problem Ĥ_e,ph|ψ̂〉 = ℰ|ψ̂〉, where eigenstates |ψ̂〉 are related to the eigenstates |ψ̃〉 of the Hamiltonian H̃_e,ph through the Hermitian conjugate of S via |ψ̂〉 = S⁺|ψ̃〉. For the general case in which the resonance condition is simultaneously satisfied for k phonon modes and the polaron-like states of degeneracies k₁ and k₂ (subscripts 1 and 2 designate the lower and upper states, respectively), the eigenvalue problem leads to the algebraic equation of order s = kk₁ + k₂. This equation can be solved exactly only for s ≤ 4, while also admitting analytical solutions in many situations of practical interest for s > 4.

The spherical symmetry of the QD enables analytical treatment of the formation of the polaron-like states in the quite general case of arbitrary degeneracies of the electron and hole states with respect to the projections of the angular momentum. In the Appendix, we use the selection rules of the electron–phonon coupling to construct the Hamiltonian ${\tilde{H}}_{e,\mathit{ph}}^{(k)}$, carry out its diagonalization, and obtain the energies and wave functions of the polaron-like excitations [see Eq. (18)–(23)] for a pair of arbitrarily degenerate hole states resonant to an arbitrary number of dispersionless phonon modes. We also show that the diagonalization of the electron–phonon Hamiltonian in the case of degenerate electron–hole-pair states can be reduced to a number of identical diagonalization problems for the pairs of nondegenerate states coupled via eligible phonon modes (with nonzero matrix elements). In Section III, we consider an example leading to such a basic problem, in which the vibrational resonance occurs between a pair of orbitally nondegenerate hole states and two phonons of zero angular momentum.

2.4. Polaron–photon interaction

The formation of the polaron-like states modifies interactions of the QD with excitation light and emitted photons. This modification is described by the same transformation of the Hamiltonians H̃_e,L and H̃_e,r that was used earlier to eliminate the nondiagonal part of the electron–phonon coupling. The results of the Appendix show that the matrix elements of the transformed Hamiltonian describing the optical excitation of the polaron-like states are given by

The matrix representation of the modified interaction with the emitted photons is

The rows from top to bottom and columns from left to right in Eqs. (12a) and (13a) correspond to states $|{ℰ}_1^{(k)}〉|0_{\lambda }〉|0_q〉$, $|{ℰ}_2^{(k)}〉|0_{\lambda }〉|0_q〉,\dots ,|{ℰ}_{k+1}^{(k)}〉|0_{\lambda }〉|0_q〉$, and |0^(k)〉|0_λ〉|0_q〉 for the interaction with the excitation light, and to states $|{ℰ}_1^{(k)}〉|0_{\lambda }〉|0_q〉,\dots ,|{ℰ}_{k+1}^{(k)}〉|0_{\lambda }〉|0_q〉$, |0^(k)〉|1_λ〉 |1_q1〉,..., |0^(k)〉|1_λ〉|1_qk〉 for the interaction with the emitted photons. Here $|{ℰ}_n^{(k)}〉\equiv |{\widehat{\psi }}_n〉$ are the polaron-like states of energies ${ℰ}_n^{(k)}$ given in Eq. (23); |0^(k)〉 denotes the vacuum of the polaron-like excitations; and |N_λ〉 and |N_q〉 represent the states with N photons in mode λ and N phonons in mode q.

It should be noted that the use of the infinite potential well model of the QD in our previous work [40] results in the possibility of generation and annihilation of the polaron-like excitations only in states $|{ℰ}_1^{(k)}〉$ and $|{ℰ}_2^{(k)}〉$, whereas the three-particle processes involving phonons enable all polaron-like states to directly interact with light. Indeed, the terms describing the three-particle interaction in Eqs. (11c) and (11d) are proportional to the product $F_p{V}_{p;p}^{(q)}$, where the set of quantum numbers p = {n_e, l_e, m_e; n_h, l_h, m_h} describes the state of the electron–hole pair. If the QD is modeled by an infinitely deep potential well, then the first factor in this product (equal to δ_ne,nhδ_le,lhδ_me,mh) ensures that only electrons and holes of equal quantum numbers are generated or can recombine. The second factor in this case is identically zero, implying that the triple interaction is absent. On the other hand, $F_p{V}_{p;p}^{(q)}\ne 0$ for n_e ≠ n_h in the QD with a penetrable surface, in which case Eq. (3) shows that the simultaneous emission or absorption of a photon and a phonon by an electron–hole pair is possible for l_q = 0, 2, 4,..., 2l and m_q = 0.

It should be noted that the above calculations can be performed in a similar fashion for nonspherical QDs. The lowering of the QD symmetry would reduce the degeneracy degree of electron and hole states, and modify the selection rules of interband optical transitions. With the reduction of the states degeneracy, the treatment of vibrational resonance becomes simpler but the overall calculation scheme remains the same: the renormalization of energy spectrum induced by the electron–phonon interaction is followed by the modification of the Hamiltonian describing the interaction of phonon-polaritons with light.

3. Phonon-assisted secondary emission

We now use the results of the previous section to calculate the differential cross section of the phonon-assisted photoluminescence within the frame of the density matrix formalism [65]. To do so, we assume that the optical excitation of the QD is sufficiently weak for the depletion of the ground electron–hole-pair state and that the temperature of the system is low enough (i.e. k_BT ≪ h̄Ω) for the spontaneous generation of the optical phonons. The first assumption enables the employment of the perturbation theory whereas the second allows us to neglect the processes of secondary emission accompanied by the emission of the phonons [see the diagrams in Figs. 2(b) and 2(d)].

If the vibrational resonance occurs between k phonon modes and a pair of nondegenerate electron–hole-pair states, then the minimal sufficient basis of our system is formed by the following 2k + 2 eigenstates of Hamiltonian Ĥ:

The process of the QD photoluminescence involves several transitions between the states of the basis. The excitation light first transfers the QD from the ground state to a superposition of the excited states by creating one or several polaron-like excitations. As we mentioned earlier, the creation can occur either directly (i.e., purely by light) in states $|{ℰ}_1^{(k)}〉$ and/or $|{ℰ}_2^{(k)}〉$ or in any of the excited states with the emission of a phonon [see Eqs. (22a) and (23c)]. The excited polaron-like states can then nonradiatively convert to one another due to the thermal interaction with a bath at rates which we denote as ζ_νμ. The polaron-like excitations finally annihilate with the emission of photons h̄ω_λ, and the system transits to one of the final states. Unlike the creation, the annihilation can directly produce a phonon which is already present in any of the excited states, as well as occur indirectly in the first two polaron-like states with the simultaneous emission of secondary radiation and creation of a phonon.

By perturbatively solving the generalized master equation [40, 65]

As an illustration of the developed theory, we consider the vibrational resonance coupling the orbitally nondegenerate states |p₁〉 = |1, 0, 0;1, 0, 0〉 and |p₂〉 = |1, 0, 0;2, 0, 0〉 in the valence band. In this case only phonons with zero angular momentum are involved in the resonance, and we restrict our analysis to a pair of them (s = 2) characterized by the quantum numbers q₁ = {1, 0, 0} and q₂ = {2, 0, 0} (see Fig. 1). We consider InAs QD with V_e = 2.7 eV and V_h = 2 eV, and the rest of material parameters the same as in Fig. 3. We also neglect transitions between the polaron-like states by setting ζ_νμ = 0 for simplicity. The rest of the relaxation parameters depends on many factors (including the geometry, fabrication quality, and excitation conditions of the sample) and is taken to be γ̂_iμ = 10 μeV, γ_μμ = 20 μeV, γ_fμ = 50 μeV and γ_ph = 60 μeV.

Figure 4 shows five photoluminescence spectra of InAs QD in the spectral range of the one-phonon Stokes sideband. The upper three curves represent the case of resonant excitation at the energies of 530.6, 533.7, and 536.8 meV, whereas the lower two curves correspond to the nonresonant excitation at 527.6 and 539.9 meV. All five photoluminescence spectra are seen to have three peaks each, corresponding to the radiative decay of the polaron-like QD states $|{ℰ}_1^{(2)}〉$, $|{ℰ}_2^{(2)}〉$, and $|{ℰ}_3^{(2)}〉$. It is seen that the blue, orange and red spectra have main peaks at resonant energies $\overline{h}\left({\omega }_2^{(2)}-\mathrm{\Omega }\right)=501.1\hspace{0.17em}\text{meV}$, $\overline{h}\left({\omega }_3^{(2)}-\mathrm{\Omega }\right)=504.2\hspace{0.17em}\text{meV}$ and $\overline{h}\left({\omega }_1^{(2)}-\mathrm{\Omega }\right)=507.3\hspace{0.17em}\text{meV}$. The amplitudes of the additional two peaks of the spectra are much smaller than the amplitudes of the main peaks. The nonresonant excitation, represented by the wine-color and green spectra, significantly reduces the intensity of secondary emission while making the amplitudes of the three peaks comparable. The widths of the phonon-assisted photoluminescence peaks are all set by the relaxation parameters γ_fμ and equal to 100 μeV.

 

Fig. 4 Photoluminescence spectra of InAs QD with a pair of nondegenerate electronic states coupled through two phonon modes. The excitation frequencies are: $\overline{h}{\omega }_2^{(2)}=530.6\hspace{0.17em}\text{meV}$, $\overline{h}{\omega }_3^{(2)}=533.7\hspace{0.17em}\text{meV}$, and $\overline{h}{\omega }_1^{(2)}=536.8\hspace{0.17em}\text{meV}$; V⁽²⁾ = 3.1 meV. For material parameters refer to the text.

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The excitation spectra of photoluminescence are plotted in Fig. 5. The upper spectra correspond to the resonant detection at the energies of 501.1, 504.2 and 507.3 meV, whereas the lower spectra are for the case of nonresonant detection at 498.1 and 510.4 meV. One can see that each of the resonant-detection spectra has one pronounced main peak and two smaller additional peaks. The main peaks are centered at the resonance frequencies ${\omega }_1^{(2)}$, ${\omega }_2^{(2)}$ and ${\omega }_3^{(2)}$ and characterize the decay of the respective polaron-like states, whereas the additional peaks appear due to the photoluminescence mediated by the decay of the states detected nonresonantly.

 

Fig. 5 Excitation photoluminescence spectra of InAs QD with a pair of nondegenerate electronic states coupled through two phonon modes. The detection frequencies are shown near the spectra. The parameter values and resonant frequencies ${\omega }_j^{(2)}$ (j = 1, 2, 3) are the same as in Fig. 4.

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The nonresonant-detection peaks are all of approximately equal intensities, and the widths of all peaks in the figure are 40 μeV. Notice also that had we taken into account the transitions between different polaron-like states (described by factors ζ_νμ), the intensity of the peak corresponding to the lowest-energy polaron-like state $|{ℰ}_2^{(k)}〉$ would have increased.

The analysis of Eq. (16), facilitated by the above examples, reveals that the spectra of the low-temperature photoluminescence from a semiconductor QD with resonant electron and phonon subsystems is the sum of Lorentzians centered at frequencies ${\omega }_{\mu }^{(k)}$ and ${\omega }_{\mu }^{(k)}-\mathrm{\Omega }$. The full widths at half maxima of the Lorentzians are equal to 2γ_fμ for the photoluminescence spectra and 2γ_iμ for the excitation photoluminescence spectra. By extracting the relaxation rates γ_fμ and γ_iμ from the experimental spectra, one can calculate the phonon lifetime as [2(γ_fμ − γ_iμ)]⁻¹. The nonstationary secondary emission from a QD in the regime of vibrational resonance provides more information on the relaxation parameters and will be theoretically analyzed in our subsequent papers.

Our theory shows that photoluminescence and excitation photoluminescence spectra measured in the regime of vibrational resonance significantly differ from the ordinary spectra of phonon-assisted secondary emission due to the formation of the polaron-like quasiparticles. Indeed, in the absence of vibrational resonance, there would be a single luminescence peak red-shifted from the excitation frequency by the LO-phonon energy. There would also be a similar red-shifted peak in the excitation photoluminescence spectrum corresponding to the optical transition to a certain resonant electron-hole pair state. In sharp contrast to this, in the case of vibrational resonance, both spectra have three peaks each in the spectral range of the one-phonon sideband even at cryogenic temperatures. The energy spacing between the adjacent peaks in the situation of the exact resonance equals V⁽²⁾ [see Eqs. (23a) and (23b)], i.e. directly related to the strength of the electron–phonon coupling.

4. Conclusions

We have theoretically studied the process of QD photoluminescence involving the emission of the dispersionless optical phonons that are confined by the QD and resonant to a pair of its electron states. To describe this process, we had to abandon the infinite-potential-well model of the QD and assume that its surface is penetrable for the confined charge carriers. This assumption changes the standard selection rules and allows the polaron-like states arising from the resonant electron–phonon coupling to decay with the emission of phonons in spherical QDs. The most important consequence of modelling the QD as finite potential barriers for electrons and holes is the emergence of the three-particle interaction between photons, phonons and electrons (holes). Allowing for this interaction drastically modifies the photoluminescence and excitation photoluminescence spectra, resulting in a fine structure of the one-phonon Stokes sideband. The analysis of this structure allows one to estimate the strength of the electron–phonon interaction inside the QD. Using the density matrix formalism, we then derived an exact expression for the differential cross section of the stationary, low-temperature photoluminescence. Being applicable to the cases of arbitrarily degenerate electron states and an arbitrary number of phonons, our expression describes a wide variety of practical situations. It may prove useful to experimentalists in the analysis of the frequency-resolved photoluminescence spectra and determination of the fundamental parameters of the polaron-like excitations in semiconductor QDs.