Abstract

Optical methods, which allow the determination of the dominant channels of energy and phase relaxation, are the most universal techniques for the investigation of semiconductor quantum dots. In this paper, we employ the kinetic Pauli equation to develop the first generalized model of the pulse-induced photoluminescence from the lowest-energy eigenstates of a semiconductor quantum dot. Without specifying the shape of the excitation pulse and by assuming that the energy and phase relaxation in the quantum dot may be characterized by a set of phenomenological rates, we derive an expression for the observable photoluminescence cross section, valid for an arbitrary number of the quantum dot’s states decaying with the emission of secondary photons. Our treatment allows for thermal transitions occurring with both decrease and increase in energy between all the relevant eigenstates at room or higher temperature. We show that in the general case of N states coupled to each other through a bath, the photoluminescence kinetics from any of them is determined by the sum of N exponential functions, whose exponents are proportional to the respective decay rates. We illustrate the application of the developed model by considering the processes of resonant luminescence and thermalized luminescence from the quantum dot with two radiating eigenstates, and by assuming that the secondary emission is excited with either a Gaussian or exponential pulse. Analytic expressions describing the signals of secondary emission are analyzed, in order to elucidate experimental situations in which the relaxation constants may be reliably extracted from the photoluminescence spectra.

© 2012 OSA

1. Introduction

The effect of size quantization and the associated modification of the interaction between various types of elementary excitations inside semiconductor quantum dots change the dominant mechanisms of energy and phase relaxation of charge-carrier excitations (as compared to bulk materials), leading to a significant variation in the interband transition rates [13]. The unique optical properties [47] being acquired by quantum dots owing to this change—as well as the ability of altering them through the variation of size, shape, and chemical composition—makes quantum dots a key material for nanotechnology and an exceptionally interesting subject of fundamental investigation [810].

The performance of optical and optoelectronic devices based on semiconductor quantum dots essentially depends on the efficiency of energy and phase relaxation occurring upon their optical excitation [1114], thus making the development of accurate theoretical models of such processes a crucial step towards successful design strategies. The most frequently occurring mechanisms of energy relaxation are those involving interactions with elementary excitations inherent to either quantum dots or their environment. These types of mechanisms include relaxation mediated by the emission of one [1521], two [22], or several [23] phonons, as well as the decay of the excited state accompanied by the excitation of plasmon or plasmon-phonon modes [2429]. The relaxation may also occur via the Auger process [30], nonradiative energy transfer [3133], or through the interaction of the quantum dot’s electronic subsystem with surface defects [3436]. Finally, in the case of weak electron-phonon coupling, the relaxation in sufficiently small quantum dots with defect-free surfaces is accounted for by the radiative transitions [37]. In order to be able to give a definite answer to the question ‘which of these (or other possible) mechanisms is dominant in any particular situation,’ one relies on a number of optical methods.

The commonly used optical methods for studying the dynamics of spectroscopic transitions inside a semiconductor quantum dot include such well-known techniques as four-wave mixing [38], photon echo [39, 40], and the coherent control of secondary emission [4144]. These methods allow one to gain a valuable information on the total dephasing rates of quantum transitions between different pairs of electronic states. Among the nonstationary optical methods for studying the decay of the excited states, the transient pump–probe spectroscopy [4549] should be pointed out as a multipurpose one. Several experimental schemes [5052] that can be realized within the framework of this spectroscopic technique enable the measurement of the total rate of energy relaxation (lifetime) of a particular state of the electronic subsystem, as well as the determination of individual intraband relaxation rates of electrons and holes. Another valuable method for measuring the relaxation rates, for which the development of a theoretical treatment constitutes the primary aim of the present paper, is the spectroscopy of secondary emission induced upon a pulsed excitation of either a single quantum dot or the entire ensemble [5356].

Despite the fact that the relaxation mechanisms of quantum dots have been a subject of intense scholarly research over the past two decades [34,5762], there are still certain aspects that need to be clarified. One of the main reasons for this is insufficient elaboration of the theory of the quantum dot’s optical response on the continuous-wave, and especially pulsed, excitation. It is essential in this connection to develop advanced theoretical models of secondary emission, which would enable a reliable extraction of the information on the dynamics of quantum transitions from the time- and frequency-resolved photoluminescence spectra.

In this paper, we present a generalized theory of nonstationary secondary emission from the lowest-energy states of a semiconductor quantum dot. The generality of our treatment stems from the following facts. First, we do not specify the macroscopic mechanisms of energy and phase relaxation, but assume that they are determined by the interaction with a bath and may be described by the respective relaxation constants. Second, we allow for an arbitrary number of energy states whose decay contributes to the signal of secondary emission and, although we prefer to talk about electron–hole pairs, the developed formalism is well suited for the analysis of relaxation of just electrons or holes. Third, our theory takes into account all possible transitions (with either an increase or decrease in energy) between the excited states, which are induced by thermal fluctuations. Such transitions normally occur at room or higher temperature, where the thermal energy is comparable to, or above, the energy gap between a pair of states. We show that the allowing for the transitions with the increase in energy leads to a considerable modification of the photoluminescence kinetics; additional exponentially decaying terms arise in the response spectra of the quantum dot, while the luminescence decay rates change substantially. And fourth, the theory is applicable to an arbitrarily shaped pulse, provided its spectral width is well below the dephasing rates of the optical transitions.

Two schemes that are most commonly employed to excite a photoluminescence are considered: (i) resonant excitation of a particular radiating state of the quantum dot’s electronic subsystem; and (ii) excitation of some high-energy electronic state, which then decays to the lower-lying states that directly contribute to the secondary emission. The general theory is illustrated by the example of the quantum dot with two radiating energy states. In this case, the kinetic Pauli equation [63] admits a solution in quadratures, which enables us to derive analytic expressions for the time-dependent photoluminescence signal from a quantum dot excited by a pulse with either an exponential or Gaussian profile. A comprehensive analysis of the obtained expressions is performed and the conditions upon which the relaxation constants of the system can be extracted from the experimental data are established.

2. Quantum dot photoluminescence

If the electronic subsystem of a semiconductor quantum dot interacts with a laser pulse and vacuum radiation field, then—depending on the confinement regime inside the quantum dot—the interaction leads to the creation and annihilation of either electron–hole pairs or excitons. For the sake of definiteness, we assume the strong confinement regime throughout this paper, but keep in mind that the results being obtained are equally applicable to the case where the spatial confinement is weak. The quantum behavior of the system “field plus particles” is contained in its total Hamiltonian H(t) = H0 + HL(t)+ HV, which is the sum of the Hamiltonian H0 describing noninteracting electron–hole pairs and vacuum radiation field, and the Hamiltonians HL(t) and HV governing generation and annihilation of the pairs due to the interaction with classical excitation light and quantum radiation field. The Hamiltonian

H0=nh¯ωn|nn|+λh¯ωλcλ+cλ
is expressed through the energies h̄ωn and eigenvectors |n〉 of the electron–hole pairs, and through the energies h̄ωλ and creation ( cλ+) and annihilation (cλ) operators of photons. The summations in H0 extend over the finite set of relevant eigenstates (marked by the subscript n = 1, 2,...,N) and over all photon modes (marked by the wavelength λ). Here, by the term “relevant eigenstates” we mean the eigenstates that may decay with emission of a secondary photon, directly contributing to the photoluminescence signal, as well as the eigenstates whose populations may be transferred to the emitting states due to the thermal interaction with a bath. The interaction Hamiltonian HL(t) depends on the parameters of the laser pulse, including polarization eL, carrier frequency ωL, and envelope ϕ(t), and may be written in the rotation wave approximation as
HL(t)=ELnϕ(t)Vn,0(L)eiωLt|n0|+H.c.,
where EL is the electric field amplitude, Vn,0(L)=n|peL|0 is the matrix element of the dipole moment operator p = −er (−e is the charge of the electron), and |0〉 denotes the vacuum of electron–hole pairs. Similarly, the interaction of the quantum dot’s electronic subsystem with the radiation field is described by the Hamiltonian
HV=λnigλV0,n(λ)cλ+|0n|+H.c.,
which depends on the polarization eλ of emitted photons and the matrix elements V0,n(λ)=0|peλ|n; gλ=2πh¯ωλ/(ɛV), with ε and V being the high-frequency permittivity of the quantum dot and the normalization volume, respectively.

After the quantum dot is excited by the laser pulse, the evolution of its electronic subsystem may be conveniently described with the kinetic Pauli equation for the diagonal components of the density matrix [63]. Such a description is applicable where the spectral width σ of the pulse is much smaller than the dephasing rates γnn of optical transitions, so that the effects of coherence relaxation may be safely neglected. Since γnn ≳ 1013 s−1 at room (or higher) temperature [6466], we restrict ourself to the consideration of optical pulses with σ ≪ 1013 s−1. If ζnn is the rate of transitions from the state |n′〉 to the state |n〉 due to the thermal interaction with a bath and γnn is the inverse lifetime of the state |n〉, then the kinetic Pauli equation is of the form

ρnnt=γnnρnn+nnζnnρnn+fn(t),
where the function fn(t) explicitly takes into account the population buildup due to either generation of electron–hole pairs by the pulse or relaxation of the quantum dot’s carriers from the high-energy (non-relevant) eigenstates.

Suppose that there are no electron–hole pairs and emitted photons in the initial state |i〉〉 of our system (at t = −∞), while the final state |f〉〉 has one emitted photon in mode λ and zero electron–hole pairs, i.e., |i〉〉 = |0〉|0λ〉 and |f〉〉 = |0〉|1λ〉. During its evolution between these states, the system may be found in one of N intermediate states |n〉〉 = |n〉|0λ〉, with zero emitted photons and an electron–hole pair in the state |n〉. The filling rates of the intermediate states vary with the intensity of the pulse. By assuming that this intensity is relatively small and neglecting the nonlinear effects, we find with Eq. (2) the generation rate of electron–hole pairs in the state |n〉 to be fn(t) = Wnϕ2(t), where

Wn=(ELh¯)2|Vn,0(L)|22γ^ni(ωnωL)2+γni2,
γni = γnn/2 + γ̂ni is the total dephasing rate of the optical transition |i〉〉 → |n〉〉, and γ̂ni is the pure dephasing rate.

Owing to its linear nature, Eq. (4) may be solved in quadratures for an arbitrary pulse envelope and arbitrary number of relevant eigenstates of the quantum dot’s electronic subsystem (see Appendix). The resulting populations ρnn(ωL,t) determine the time evolution of photoluminescence, the intensity of which is characterized by the differential cross section [67]

Σ(ωL,ωλ,t)=h¯2ωλ4πc4EL2n=1Nρnn(ωL,t)W˜n(ωλ),
where
W˜n(ωλ)=|V0,n(λ)|2h¯22γfn(ωnωλ)2+γfn2
gives the probability of the transition |n〉〉 → |f〉〉, with the total dephasing rate γfnσ.

In order to obtain from Eq. (6) the observable quantity, one needs to take into account both the finite frequency resolution and the finite response time of the photon detector filtering the signal [6870]. For the sake of definiteness, we assume that the detector is a combination of a Fabry–Perot interferometer and a “white” photodetector. Then, the observable cross section of the photoluminescence is given by the double convolution [68, 69]

Σ¯(ωL,ωF,t)=0dτΓFeΓFτdωλΓF/(2π)(ωFωλ)2+(ΓF/2)2Σ(ωL,ωλ,tτ)=1π(ωFc)4n=1N|V0,n(λ)|22(γfn+ΓF/2)(ωnωF)2+(γfn+ΓF/2)2Rn(t)EL2,
where ΓF and ωF are the bandpass and central frequencies of the interferometer, and
Rn(t)=0ρnn(tτ)ΓFeΓFτdτ
is a dimensionless parameter determining the contribution of the nth energy level to the signal of photoluminescence. It may be noted that the ratio Rn(t)/EL2 is independent of the electric field amplitude due to the linearity of the photoluminescence phenomenon.

The developed theoretical formalism allows us to describe two important types of the photoluminescence: resonant luminescence (RL) and thermalized luminescence (TL), which are schematically shown in Fig. 1 for the case of three relevant eigenstates of the electron–hole pairs. During the process of resonant luminescence in Fig. 1(a), the laser pulse generates electron–hole pairs in one of the excited states (|1〉, |2〉, or |3〉) directly, whereas during the thermalized luminescence in Fig. 1(b) the pairs are first excited to some high-energy state |n〉 and then relax (at effective rates ζ1n, ζ2n, or ζ3n, which allow for all direct and step-by-step transitions, including those involving eigenstates located between states |n〉 and |3〉) to one of the relevant eigenstates due to the thermal interaction with the bath. After the relevant eigenstate |i〉 is populated, it either decays to another excited eigenstate |j〉, with the energy shift Eij = h̄(ωiωj), or decays to the ground state |0〉 with or without the emission of a photon in mode λ. If the temperature T of the system expressed in energy units is comparable to, or larger than, Eij, then transitions |i〉 ⇆ |j〉 may occur at sufficiently high rates ζji and ζij = ζji exp(−Eij/T). As a result of this, the signal of secondary emission generally contains information about the lifetimes of all relevant eigenstates of the quantum dot’s electronic subsystem, even where the laser pulse excites only one of them. The proof of this statement is given in Appendix, where we show that for N relevant eigenstates and t ≫ 1/σ the components of the density matrix are given by the expression ρnn(ωL,t)=n=1NAnexp(snt), where An (n = 1, 2,..., N) are the time-independent coefficients and sn are the modified energy relaxation rates.

 

Fig. 1 Optical (solid arrows) and relaxation (dashed arrows) transitions corresponding to the processes of (a) resonant luminescence and (b) thermalized luminescence from a semiconductor quantum dot with three energy levels. Ket vectors |0〉, |1〉, |2〉, and |3〉 denote the ground and excited states of electron–hole pairs; |n〉 is the high-energy state that does not directly contribute to the secondary emission; ωLk (ωL) and ωλk (k = 1, 2, 3) are the excitation frequencies and the frequencies of emitted photons; ζkk is the rate of transitions |k′〉 → |k〉 due to the thermal interaction with a bath. The spectral width of the excitation pulse is assumed to be much smaller than the dephasing rates of all interband optical transitions.

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In the rest of the paper, we analyze the effects of resonant luminescence and thermalized luminescence for the simplest case of two relevant states of the electron–hole pairs and two different envelopes of the excitation pulse, which enables us to establish the basic features of the photoluminescence kinetics and come up with experimental conditions required to reliably measure the relaxation parameters.

3. Resonant luminescence

Suppose first that optical pulses directly generate electron–hole pairs in either state |1〉 or state |2〉, which then relax to the ground state |0〉 with or without the emission of photons in mode λ. Then the populations of the excited states may be calculated by solving the system in Eq. (4), which contains only two equations

ρ11t=γ11ρ11+ζ12ρ22+W1ϕ2,
ρ22t=γ22ρ22+ζ21ρ11+W2ϕ2.
The solution to this system may be written in the form (see Appendix)
ρ11(t)=tϕ2(t)[(ρW1+ϑ12W2)es1(tt)+(qW1ϑ12W2)es2(tt)]dt,
ρ22(t)=tϕ2(t)[(qW2+ϑ21W1)es1(tt)+(pW2ϑ21W1)es2(tt)]dt,
where
p=γ22s1D,q=s2γ22D,ϑij=ζijD,
D=(γ11γ22)2+4ζ12ζ21,
s1=12(γ11+γ22D),s2=12(γ11+γ22+D).
It is significant that the inverse lifetimes γ11 and γ22 are always larger than the respective thermal relaxation rates ζ21 and ζ12, because they are determined by all possible processes contributing to the decay of the two states, including annihilation of electron–hole pairs and any types of relaxation processes. This leads to the inequality ζ12ζ21 < γ11γ22, which implies that the parameters s1 and s2 are positive. Also noteworthy is the inequality s2 > γ22 > γ11 > s1, which is always valid because the higher the energy of the electron–hole pair, the more relaxation channels exist for it.

In order to evaluate the integrals in Eq. (11), one needs to specify the pulse envelope. We restrict our further analysis to the pulses characterized by the exponential profile ϕ(t) = exp(−σ|t|) and Gaussian profile ϕ(t)=Aexp(ς2t2). The constants A=2/πln2 and ς=σ2/ln2 are chosen such as to make the power and full width at half maximum (FWHM) of both pulses equal, and thus ensure a fair comparison of the corresponding results. A little algebra shows that Eqs. (9) and (11) for the exponential pulse yield

R1(t)=η12+e2σtH(t)+(η12e2σt+A12(p)es1t+B12(q)es2t)H(t),
R2(t)=η21+e2σtH(t)+(η21e2σt+A21(q)es1t+B21(p)es2t)H(t),
where H(t) is the Heaviside step function,
ηij±=(γij±2σ)Wi+ζijWj(s1±2σ)(s2±2σ)ΓFΓF±2σ,
Aij(r)=4σ4σ2s12ΓFΓFs1(rWi+ϑijWj),
Bij(r)=4σ4σ2s22ΓFΓFs2(rWiϑijWj).
For the Gaussian pulse, the same equations give
R1(t)=ξ12Q(ΓF,t)+C12(p)Q(s1,t)+D12(q)Q(s2,t),
R2(t)=ξ21Q(ΓF,t)+C21(q)Q(s1,t)+D21(p)Q(s2,t),
where
ξij=(γjjΓF)Wi+ζijWj(ΓFs1)(ΓFs2),
Cij(r)=rWi+ϑijWjΓFs1,Dij(r)=rWiϑijWjΓFs2,
and
Q(α,t)=ΓF2σexp(α28ς2)erfc(α4ς2t22ς)eαt.
Notice that Q(α, t) may be approximated by the exponential (ΓF/σ)exp[α2/(8ς2)]eαt for a time tα/(2ς)2+2/ς.

According to Eqs. (8) and (13) to (17), the kinetics of resonant luminescence is described by four exponentials in the case of a pulse with an exponential envelope, and by three functions Q(α,t) if the excitation pulse has a Gaussian profile. In the first case, two exponentials (es1t and es2t) appear due to the energy relaxation of the excited states, and the other two (e±2σt) describe the part of the signal that follows the shape of the excitation pulse. An important point is that the detecting system does not affect the time variation of the resonant luminescence induced by an exponential pulse. The filtering of secondary emission excited by a Gaussian pulse, contrastingly, always leaves an imprint of the detector, represented by the dependence QF, t) in Eq. (15). This difference is associated with the specific form of the filter function in Eq. (9).

4. Thermalized luminescence

Now consider the process of thermalized luminescence in which the laser field excites an electron–hole pair in some high-energy state |n〉. This state then decays to one of the lower-lying states (|1〉 or |2〉), whose populations are interrelated via thermal fluctuations, and which further decay with or without the emission of photons in mode λ. By assuming that the energy h̄ωn of the state |n〉 is such that h̄(ωnωj) ≫ T for j = 1, 2, we arrive at the following equations governing the populations of the three states:

ρ11t=γ11ρ11+ζ12ρ22+ζ1nρnn,
ρ22t=γ22ρ22+ζ21ρ11+ζ2nρnn,
ρnnt=γnnρnn+Wnϕ2.
Since the last equation in this system is decoupled from the first two, the elements of the density matrix may be written in the form similar to Eq. (11)
ρ11(t)=tg(t)[(pw1+ϑ12w2)es1(tt)+(qw1ϑ12w2)es2(tt)]dt,
ρ22(t)=tg(t)[(qw2+ϑ21w1)es1(tt)+(pw2ϑ21w1)es2(tt)]dt,
where wj = (ζjn/γnn)Wn and
g(t)=γnntϕ2(t)eγnn(tt)dt.
Using this result in Eq. (9), gives for the case of the exponential pulse
R1(t)=χ12+e2σtH(t)+(χ12e2σt+E12(p)es1t+F12(q)es2t+G12eγnnt)H(t),
R2(t)=χ21+e2σtH(t)+(χ21e2σt+E21(q)es1t+F21(p)es2t+G21eγnnt)H(t),
where
χij±=γnnγnn±2σ(γij±2σ)wi+ζijwj(s1±2σ)(s2±2σ)ΓFΓF±2σ,
Eij(r)=γnnγnns14σ4σ2s12ΓFΓFs1(rwi+ϑijwj),
Fij(r)=γnnγnns24σ4σ2s22ΓFΓFs2(rwiϑijwj),
Gij=4σγnnγnn24σ2(γnnγjj)wiζijwj(γnns1)(γnns2)ΓFΓFγnn.
In the case of the Gaussian pulse, we obtain
R1(t)=η12Q(ΓF,t)+K12(p)Q(s1,t)+L12(q)Q(s2,t)+M12Q(γnn,t),
R2(t)=η21Q(ΓF,t)+K21(q)Q(s1,t)+L21(p)Q(s2,t)+M21Q(γnn,t),
where
ηij=γnnγnnΓF(γijΓF)wi+ζijwj(ΓFs1)(ΓFs2),
Kij(r)=γnnγnns1rwi+ϑijwjΓFs1,Lij(r)=γnnγnns2rwiϑijwjΓFs2,
Mij=γnnΓFγnn(γjjγnn)wi+ζijwj(γnns1)(γnns2).
The structure of Eqs. (20) and (22) is quite similar to that of Eqs. (13) and (15), except for additional terms ∝ eγnnt and ∝ Q(γnn,t), which arise due to the decay of the excited state |n〉. The impact of these terms on the kinetics of thermalized luminescence is analyzed in the next section.

5. Analysis of photoluminescence kinetics

The main objective of performing the time analysis of the photoluminescence signal is to recover the relaxation parameters of the emitting system. To make such a recovery feasible, one needs to know the major factors contributing to the secondary emission and the timescales on which these factors are dominant. With this in mind, we now work out the time dependence of the quantum dot’s luminescence under different assumptions on the excitation, relaxation, and detection parameters. Depending on the values of the pulse bandwidth σ and the characteristic relaxation constant γ, it is convenient to distinguish between the following four excitation regimes: (0) σ = 0, stationary excitation; (i) σγ, adiabatic excitation; (ii) σγ, instantaneous excitation; and (iii) σγ, pulse-sensitive excitation. For such a distinction to be meaningful, the parameter γ should have different values for different regimes, unless all relaxation constants are of the same order of magnitude. Specifically, for the scenario of two relevant eigenstates considered earlier, γs1 for regime (i) and γs2 for regime (ii). Figure 2 illustrates different excitation regimes by the example of a Gaussian pulse, for which the system’s response is governed by the function Q(α, t) in Eq. (17).

 

Fig. 2 Three-dimensional plot of log[(2σF)Q(α, t)] as a function of the dimensionless parameters αt and α/σ [see Eq. (17)]. Function values are shown by labels of contour levels. Three regimes of excitation by a Gaussian pulse are clearly seen: (i) adiabatic excitation for log(α/σ) ≳ 1; (ii) instantaneous excitation for log(α/σ) ≲ −1; and (iii) pulse-sensitive excitation for |log(α/σ)| ≲ 0.5.

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5.1. Stationary excitation

In the situation of stationary excitation of the quantum dot’s electronic subsystem, the signal of secondary emission may be calculated simply using Eqs. (14a) and (21a) with σ = 0. By introducing new functions Ξ=h¯2ωF4/(πc4EL2) and

W¯n(ωF)=|V0,n(λ)|2h¯22(γfn+ΓF/2)(ωnωF)2+(γfn+ΓF/2)2,
we find
Σ¯RL(0)(ωL,ωF)=Ξ(W¯1γ22W1+ζ12W2γ11γ22ζ12ζ21+W¯2γ11W2+ζ21W1γ11γ22ζ12ζ21)
for the stationary signal of resonant luminescence and
Σ¯TL(0)(ωL,ωF)=Ξ(W¯1γ22ζ1n+ζ12ζ2nγ11γ22ζ12ζ21+W¯2γ11ζ2n+ζ21ζ1nγ11γ22ζ12ζ21)Wnγnn
for the stationary signal of thermalized luminescence. We see that the two-dimensional spectra Σ¯RL(0)(ωL,ωF) and Σ¯TL(0)(ωL,ωF) generally comprise of, respectively, four and two peaks schematically shown in Fig. 3. The spectral widths of the peaks determine the dephasing rates of optical transitions (γ1i, γ2i, and γni), while the relative peak intensities contain information about the relaxation constants.

 

Fig. 3 Two-dimensional spectra of (a) resonant luminescence and (b) thermalized luminescence for two relevant eigenstates of the quantum dot’s electronic subsystem. The widths of the peaks in the spectra are determined by the dephasing rates of optical transitions, while the relative peak intensities depend on the interband matrix elements [see Eqs. (5) and (7)], relaxation constants, and temperature of the system. Peak 4 vanishes in the limit of small temperatures.

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The probability of transition |1〉 → |2〉 becomes vanishingly small at low temperatures (TE21), for which Eqs. (24) and (25) reduce to the well-known formulas [16, 31, 67]

Σ¯RL(0)=Ξ(W¯1W1γ11+W¯2W2γ22+W¯1ζ12W2γ11γ22)
and
Σ¯TL(0)=Ξ(W¯1ζ1nγ11+W¯2ζ2nγ22+W¯1ζ12ζ2nγ11γ22)Wnγnn.
Here, the first two terms in the parentheses represent luminescence from the first and second quantum dot states that are either directly excited by light (RL) or populated by transitions from the high-energy state (TL). The last terms describe luminescence from state |1〉 populated upon the decay of state |2〉.

By closely looking at Eq. (24), it is seen that the spectrum of stationary resonant luminescence allows one to reliably determine the ratio ζ12ζ21/(γ11γ22), provided that the relevant eigenstates are excited independently of one another and the luminescence is measured from each of them separately. These requirements are met when E21 ≫ max(γ1i, γ2i, γf1, γf2), which is assumed to hold true in the following discussion. The desired ratio may be found by measuring the signal of luminescence for four combinations of resonant excitation and resonant detection frequencies, and by taking the ratio

ψ=ζ12ζ21γ11γ22=Σ¯RL(0)(ω1,ω2)Σ¯RL(0)(ω1,ω1)Σ¯RL(0)(ω2,ω1)Σ¯RL(0)(ω2,ω2).
In order to make it possible to filter out the signal of resonant scattering in the experiments with ωLωF, the excitation frequency needs to be slightly detuned from the exact resonance with the eigenstate transition. The detuning should be fixed for each eigenstate in all four experiments (to ensure equal excitation probabilities W1 and W2), and for the jth eigenstate lie within the range ΓF ≪ |ωLωj| ≲ γji. This inequality sets the upper limit for the filter’s bandpass in stationary experiments, ΓF ≪ min(γ1i, γ2i).

The evaluation analogous to that in Eq. (28) may be performed (with the same result) using a signal of thermalized luminescence. By comparing the spectra in Eq. (24) and (25) we may also get additional information on the relaxation constants in the form of the relationship

χ=ζ1nζ2nγ22ζ12=φ11φψ,
where
φ=Σ¯TL(0)(ωn,ω1)Σ¯TL(0)(ωn,ω2)Σ¯RL(0)(ω2,ω2)Σ¯RL(0)(ω2,ω1).
This relationship will prove useful in the analysis of photoluminescence kinetics.

Up until now, it has been assumed that we are dealing with the luminescence from a single quantum dot of a fixed size. If there is a capability of analyzing the signal of luminescence emitted by a number of similar quantum dots of different sizes or a number of different quantum-dot ensembles with relatively narrow size distributions, then further information on the relaxation constants may be obtained. We illustrate this scenario by supposing that the decay of the two relevant eigenstates is determined by the recombination of electron–hole pairs, whose rate is independent of the quantum dot’s size, as well as by thermal transitions, whose relative efficiency varies with the energy gap between the states as ζ21/ζ12 = eE21/T. By denoting the recombination rates as ζ01 and ζ02 and introducing a size-dependent parameter x = E21/T, we may write

γ11(x)=ζ01+ζ12ex,γ22=ζ02+ζ12.
The measurement of signals of the resonant luminescence from both eigenstates for two values of x, x2 > x1, gives in accordance with Eq. (24)
a=Σ¯RL(0)(ω1,ω1,x2)Σ¯RL(0)(ω1,ω1,x1)=γ11(x1)γ22ζ122ex1γ11(x2)γ22ζ122ex2,
b=Σ¯RL(0)(ω2,ω2,x1)Σ¯RL(0)(ω1,ω1,x1)Σ¯RL(0)(ω1,ω1,x2)Σ¯RL(0)(ω2,ω2,x2)=γ11(x1)γ11(x2).
By solving these equations with respect to the unknowns ζ01 and ζ02, we arrive at the following relations between the relaxation rates:
γ11(x)=(ex+ex1bex2b1)ζ12,
γ22=b1baex1aex2ex1ex2ζ12.
The second of these relations, being employed in Eq. (29), allows one to find the ratio ζ1n/ζ2n.

5.2. Adiabatic excitation

If the intensity of the excitation pulse changes noticeably on a time scale that is much larger than the time required for the photoluminescence to reach its steady state, then the excitation is adiabatic. This type of excitation is characterized by the signal of the luminescence following the intensity of the pulse, which varies in time ∝ ϕ2(t). Indeed, assuming in Eqs. (13), (14), (20), and (21) that ΓFs2 and s1σ, yields

Σ¯RL(i)Σ¯RL(0)e2σ|t|,Σ¯TL(i)Σ¯TL(0)e2σ|t|,
where Σ¯RL(0) and Σ¯TL(0) are given in Eqs. (24) and (25). Similarly, using in Eq. (17) the Taylor series expansion of the complementary error function
erfc(α4ς2t22ς)=exp[(α4ς2t)28ς2](2ςα2π+O[(ς/α)2]),
which is valid for ας, we obtain from Eqs. (15), (16), (22) and (23) for the Gaussian pulse
Σ¯RL(i)Σ¯RL(0)Ae2ς2t2,Σ¯TL(i)Σ¯TL(0)Ae2ς2t2.
It is seen that the regime of adiabatic excitation is analogous from the experimental viewpoint to the stationary excitation regime, as they both convey the same information on the quantum dot’s electronic subsystem. More importantly, these regimes do not permit determination of γ11 and γ22, which are to be found via excitation of the quantum dot with subpicosecond pulses.

5.3. Instantaneous excitation

In the regime of instantaneous excitation, the pulse duration is much smaller than the characteristic photoluminescence decay time, so that most of the time the system evolves in the absence of the excitation field. This regime is the most attractive from the experimental viewpoint, as it allows one to reliably measure the parameters of the quantum dot’s eigenstates. By looking at Eqs. (12), (13), and (15) one may notice that the favorable experimental conditions in the instantaneous excitation regime arise where a pair of relevant eigenstates are coupled via thermal interaction so weakly that their relaxation parameters satisfy the inequality γ22γ11 ≫ 2ζ12 exp[−E21/(2T)]. This is because in this case

qp1,s1γ11,s2γ22,
and the secondary emission decays according to the eigenstates’ lifetimes. The regime of weak thermal coupling may be realized by either reducing the temperature of the system (but keeping it high enough for the coherence relaxation effects to be negligible) or by reducing the size of the quantum dot and increasing the energy gap between the two states.

To illustrate the dependence of parameters s1 and s2 on the quantum dot’s size, we model the quantum dot using a spherically symmetric, infinitely deep potential well of radius R, and consider its first two orbitally nondegenerate states of electron–hole pairs. Suppose that these states have equal quantum numbers for electrons and holes: n1 = ne1 = nh1 = 1, l1 = le1 = lh1 =0, n2 = ne2 = nh2 = 2, l2 = le2 = lh2 = 0. Then the energy gap between them is given by the expression [1, 67]

E21=3π2h¯22μR2,
where μ = memh/(me + mh) is the reduced mass of electron and hole. Suppose also that the rates γ11 and γ22 are in the form of Eq. (30). Figure 4 shows how relaxation parameters s1 and s2 change with R and the ratio ζ02/ζ01 for a PbS quantum dot with me = mh = 0.25 m0 (m0 is the free-electron mass) at room temperature. It is seen that if ζ01ζ02, then s1 ≈ min(ζ12, ζ01) and s2 ≈ max(ζ12, ζ01) for R < 7 nm, while for R > 10 nm the values of s1 and s2 steeply diverge with the radius. In the case of equal recombination rates, ζ01 = ζ02, we find that s1 = ζ01 is independent of R and s2 = γ22 + ζ12ex increases from γ22 in small quantum dots to γ22 + ζ12 in large ones. If ζ02ζ01, then s1 ≈ min(ζ12, ζ02) and s2 ≈ max(ζ12,ζ02) for quantum dots with R < 7 nm. Since ζ12ζ01 for our example in Fig. 4, this result implies that s2s1. The red curves in Fig. 4 bound the domain of strong thermal coupling between the eigenstates. The approximation in Eq. (31) is applicable outside of this domain.

 

Fig. 4 Variation of (a) s1 and (b) log s2 with radius R of PbS quantum dot and ratio y = ζ02/ζ01 of radiative recombination rates for two relevant eigenstates of electron–hole pairs (s1 and s2 are in meV). The quantum numbers of the eigenstates are n1 = 1, l1 = 0 and n2 = 2, l2 = 0. Bound by red curves are the domains of strong thermal coupling between the eigenstates, where 2ζ12 exp[−E21/(2T)]/(γ22γ11) > 0.1. Quantum dot’s boundary is assumed to be impenetrable for both electrons and holes; ζ01 = 200 μeV, ζ12 = 150 μeV, and T = 25 meV. For other parameters, refer to the text.

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5.3.1. Kinetics of resonant luminescence

Suppose now that ΓFσs2. Then the shape of the excitation pulse is insignificant for the registration time t ≫ 1/σ, and from Eqs. (8) and (13) to (16) we obtain

Σ¯RL(ii)Ξσ{[W¯1(pW1+ϑ12W2)+W¯2(qW2+ϑ21W1)]es1t+[W¯1(qW1ϑ12W2)+W¯2(pW2ϑ21W1)]es2t}.
The signal of resonant luminescence from either of the quantum dot’s eigenstates is seen to be described by the biexponential function of time, which is due to the transitions |1〉 ⇆ |2〉 caused by the interaction with the bath and/or nonresonant excitation/detection of the signal [resulting in Wj(ω1) ∼ Wj(ω2) or j(ω1) ∼ j(ω2) for j = 1, 2]. Specifically, the decay of luminescence is governed by the functions
Σ¯RL(ii){pes1t+qes2tforωL,ωFω1,qes1t+pes2tforωL,ωFω2,es1tes2tforωLω1(2),ωFω2(1),
provided the energy gap between the two eigenstates is much larger than the total dephasing rate of optical transitions. In the case of ωLωF, the decay is seen to be always preceded by a buildup, due to the time delay associated with the transitions between the relevant eigenstates. By measuring the time (tm) at which the signal of luminescence peaks, we obtain a relation (s2s1)tm = ln(s2/s1). This relation may be used to calculate s2 in the event that the temporal resolution of the detector is insufficient to determine s2 directly from the experimental data.

It may turn out that the spectral widths of the two eigenstates are predominantly dictated by the thermal interaction between them. If this is the case, then parameter ψ is close to unity and Eqs. (12b) and (12c) yield

s11ψ1/γ11+1/γ22s2γ11+γ22.
The luminescence from either of the eigenstates is described in this situation by the monoexponential signals within the time domains 1/σt ≲ 1/s2 and t ≳ 1/s1. This allows one to experimentally measure s1 and s2 and—if the transition rate ζ12 is known—to calculate the relaxation constants
γ11=12(s1+s2(s1s2)24ζ12ζ21),
γ22=12(s1+s2+(s1s2)24ζ12ζ21).
In case of unknown ζ12, one may determine parameter ψ in the regime of stationary excitation using Eq. (28), and then find
γ11=12(s1+s2(s1+s2)24s1s2/(1ψ)),
γ22=12(s1+s2+(s1+s2)24s1s2/(1ψ)).
Of significance is that the simultaneous measurement of s1 and s2 from any single time dependency in Eq. (33) is only possible when pq. It should be also recognized that s2 is always much larger than s1 in the regime of dominant thermal interaction (1 −ψ ≪ 1), however, the inequality s2s1 itself does not necessarily imply that the interaction with the bath gives major contribution to the eigenstates’ lifetimes. For instance, s2s1 for ψ ≪ 1 and γ22γ11.

The relaxation constants may be also found directly from the luminescence spectra in the regime of weak thermal coupling between the two relevant eigenstates. This is evidenced by Eqs. (31) and (33), which show that the luminescence from eigenstate |j〉 decays according to the relaxation rate γjj when the state is excited resonantly (i.e., ωLωj). An important point is that the scattering of the excitation pulse does not mask the signal of resonant luminescence, as it occurs on a time scale that is much smaller than the lifetimes of the relevant eigenstates (t ∼ 1/σ ≪ 1/γ11, 1/γ22).

To illustrate the discussed features of the photoluminescence kinetics, we consider the same pair of eigenstates of a PbS quantum dot as before and assume that they are populated by a Gaussian pulse with σ = 1 meV. In order to leave room for the regime of strong thermal coupling, we ensure the radiative relaxation rates are much smaller than the rate of intraband transitions from state |2〉 = |n2 = 2, l2 = 0〉 to state |1〉 = |n1 = 1, l1 = 0〉, by setting ζ01 = ζ02 = ζ12/10 = 10 μeV. Owing to this choice, the lifetime of the low-energy state and ratio q = ex/(1 + ex) become strongly dependent on the quantum dot’s size [see Eqs. (12a) and (30)]. So too does the function sech(x) in determining the strength of thermal coupling; it increases at room temperature of 25 meV from 14.6 × 10−3 for R = 5 nm (E21 ≈ 361 meV) to 4.95 for R = 30 nm (E21 ≈ 10 meV).

Figure 5 shows the signal of resonant luminescence calculated using Eqs. (15) to (17) as a function of time and quantum dot radius, for four combinations of excitation and detection frequencies corresponding to peaks 1–4 in Fig. 3. For simplicity, the signal is evaluated under the assumption that |V1,0L|=|V2,0L|=|V0,1λ|=|V0,2λ|=1 and is normalized by a factor of ωF4/(πh¯2c4). Even though s2s1 and pq for quantum dots with R ≳ 30 nm, the decay of resonant luminescence from both eigenstates of large quantum dots is seen to be governed by a single time constant τ1 = 1/s1 = 100 meV−1 regardless of the excitation frequency. This behavior is explained by Eq. (32), where the pre-exponential factors differ by two orders of magnitude, due to the fact that E21γji, γfj (j = 1, 2). When R is reduced, the energy gap E21 grows larger, resulting in: (i) luminescence being described by Eq. (33); (ii) pq for R ≲ 10 nm; and (iii) weak thermal coupling of the eigenstates for R ≲ 8 nm. Therefore, the luminescence from small quantum dots, much like from the large ones, decays according to τ1 when ωL = ωF = ω1 [see Fig. 5(a)]. For ωL = ωF = ω2, the decrease in q makes the decay ∝ qeγ11t + peγ22t being characterized by the smaller time constant, τ2 = 1/γ22 ≈ 9 meV−1 [see Fig. 5(b)]. If the pulse resonantly excites the high-energy eigenstate (ωL = ω2) and the signal is detected at the frequency of the low-energy eigenstate (ωF = ω1), then the luminescence from small quantum dots peaks at a time tm ≈ (1/γ22) ln(γ22/γ11) ≫ 1/σ. This is clearly seen from Fig. 5(c), where the buildup of the luminescence intensity, governed by the function 1 − eγ22t (red curve), is overcome by the decay, proportional to eγ11t (blue curve), at tm ≈ 24 meV−1. Since transitions |1〉 → |2〉 become extremely improbable in small quantum dots, the luminescence from state |2〉 excited at frequency ωL = ω1 steeply diminishes with the reduction of R below 15 nm, as shown in Fig. 5(d). Hence, in the example considered, γ11 and γ22 may be calculated precisely by analyzing the kinetics of the resonant luminescence.

 

Fig. 5 Temporal evolution of resonant luminescence from PbS quantum dots of different sizes excited by a Gaussian pulse. Excitation and detection frequencies correspond to the peaks in Fig. 3(a): (a) ωL = ωF = ω1; (b) ωL = ωF = ω2; (c) ωL = ω2, ωF = ω1; (d) ωL = ω1, ωF = ω2. Blue and red curves indicate decay/buildup of luminescence with characteristic times 1/s1 and 1/s2, respectively. The relevant eigenstates are weakly coupled via thermal interaction with the bath for R ≲ 8 nm. Simulation parameters are γ̂1i = γ̂2i = γ̂f 1 = γ̂f 2 = 20 meV, ΓF = 10 meV, σ = 1 meV, ζ12 = 100 μeV, and ζ01 = ζ02 = 10 μeV. Parameters of the quantum dot are the same as in 4. For other parameters, refer to the text.

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5.3.2. Kinetics of thermalized luminescence

The signal of thermalized luminescence evolves according to Eqs. (8) and (20) to (23). When the quantum dot is ‘instantaneously’ excited by the pulse whose spectral width satisfies the inequalities ΓFσγnn (γnn > γ22), these equations for t ≫ 1/σ are reduced to

Σ¯TL(ii)Ξσ{W¯1(pζ1n+ϑ12ζ2n)+W¯2(qζ2n+ϑ21ζ1n)γnns1es1t+W¯1(qζ1nϑ12ζ2n)+W¯2(pζ2nϑ21ζ1n)γnns2es2t+W¯1[(γ22γnn)ζ1n+ζ12ζ2n]+W¯2[(γ11γnn)ζ2n+ζ21ζ1n](γnns1)(γnns2)eγnnt}Wn.
The major difference of this experimental situation from the previous one [cf. Eq. (33)] is that the relevant eigenstates cannot be excited independently of one another, since the excitation is mediated by the transition to the high-energy eigenstate |n〉. As a consequence, the thermalized luminescence from each eigenstate is generally governed by three exponentials in Eq. (37).

Of particular interest is the situation where σγnns2, as in this case the relaxation constant γnn may be reliably determined from the luminescence spectrum for a registration time 1/σt ≲ 1/γnn, whereas for a time t ≫ 1/γnn only two exponentials are contributing to the overall signal, namely,

Σ¯TL(ii){(pμ1+ϑ12μ2)es1t+(qμ1ϑ12μ2)es2tforωFω1,(qμ2+ϑ21μ1)es1t+(pμ2ϑ21μ1)es2tforωFω2,
where μj = ζjn/γnn. Again, if the thermal interaction between the two relevant eigenstates is so strong that s2s1, then s1 and s2 are extractable from the photoluminescence spectra and the corresponding relaxation constants may be calculated using either Eq. (35) or Eq. (36). The calculation of these constants is also feasible in the regime of weak thermal coupling between the two eigenstates (for which s1s2), but only if the rates ζ12 and ζ21 are known. The signal of thermalized luminescence from eigenstate |2〉 decays in this regime according to γ22 [see Eq. (38) with q, ϑ21 ≪ 1], and the remaining relaxation constant is readily found form Eq. (28), γ11 = (ζ12ζ21)/(γ22ψ). Furthermore, since the parameter χ is measurable with the stationary excitation [see Eq. (29)], knowing γ11 and γ22 enables the calculation of the relaxation rates ratio ζ1n/ζ2n.

Similar to the case of resonant luminescence, the signal of thermalized luminescence may peak long after the excitation pulse dies away. This occurs for ωFω1 when ϑ12ζ2n > 1n and for ωFω2 when ϑ21ζ1n > 2n. Using Eq. (37), we find the time corresponding to the luminescence peak to be

tm=1s2s1×{ln(s2s1ϑ12ζ2nqζ1nϑ12ζ2n+pζ1n)forωFω1,ln(s2s1ϑ21ζ1nqζ2nϑ21ζ1n+pζ2n)forωFω2.

It is readily seen that ϑ12 > q and p > ϑ21 for relaxation constants in the form of Eq. (30) and equal rates of radiative recombination (ζ01 = ζ02). The signal of thermalized luminescence from the lower eigenstate will therefore always peak in this situation (for t ≫ 1/σ), should ζ1n = ζ2n.

The value of γnn normally exceeds 1011 s−1 at room temperature [5860], so that the condition σγnn is satisfied for pulses shorter than 100 ps. For much longer pulses, an opposite relationship holds (γnnσs2) and the signal of thermalized luminescence is seen to be described by Eq. (38) for t ≫ 1/σ if γnn ≪ ΓF. Hence, only the quantities ψ and χ may be found from the stationary experiment, and γjj (j = 1, 2) are to be calculated as is described above.

Suppose that state |n〉, involved in the process of thermalized luminescence, is one of the optically accessible states of a spherical quantum dot made of PbS. Its energy should be high enough to prevent thermal transitions |n〉 ⇆ |1〉 and |n〉 ⇆ |2〉 in the quantum dot of interest. If the radius of the quantum dot does not exceed 30 nm, then it is sufficient for |n〉 to have quantum numbers nn = nen = nhn = 9 and ln = len = lhn = 0, in which case [1, 67] En1En2 ≈ 26E21. In Fig. 6, we plot the normalized cross section of thermalized luminescence, calculated using Eqs. (22) and (23)—as before—with matrix elements of all interband transitions being equal to unity. When the quantum dot is relatively large, the evolution of the secondary emission from states |1〉 and |2〉 is similar to that in Figs. 5(c) and 5(d). The decay occurs on the same timescale (τ1), but the intensity of thermalized luminescence is reduced by a factor of γnn/ζ1n = 4 with respect to the intensity of resonant luminescence. As the size of the quantum dot is reduced below 8 nm, the coupling between the eigenstates becomes weak and their radiative decay starts to follow the dependency in Eq. (38). For detection in resonance with the lower state [see Fig. 6(a)], the signal of luminescence has a pronounced peak at 17 meV−1, given in Eq. (39). The peak is preceded by an inhomogeneous growth due to the simultaneous decay of states |2〉 and |n〉, and is followed by the decay of the lower state with the characteristic time constant 1/γ11 ≈ 100 meV−1. When the detector is tuned to the resonance frequency of the upper state [see Fig. 6(b)], the luminescence decay is governed by the lifetime of this state (1/γ22 ≈ 9 meV−1), owing to the negligible probability of transitions |1〉 → |2〉 in small quantum dots. This same reason explains the reduction in the luminescence intensity with the decrease in the quantum dots’ radius.

 

Fig. 6 Temporal evolution of thermalized luminescence from PbS quantum dots of different sizes excited by a Gaussian pulse. Excitation and detection frequencies correspond to the peaks in Fig. 3(b): (a) ωL = ωn, ωF = ω1; (b) ωL = ωn, ωF = ω2. The relevant eigenstates are weakly coupled via thermal interaction for R ≲ 8 nm. It was assumed that γ̂ni = 20 meV, γnn = 2 meV, and ζ1n = ζ2n = 500 μeV. The other parameters are the same as in Fig. 5.

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In order to be able to retrieve the relaxation parameters of both eigenstates through analysis of the kinetics of thermalized luminescence from state |1〉, one needs to suppress direct transitions |n〉 → |1〉. Doing so in the regime of weak thermal coupling ensures that the lower state is populated due solely to the decay of state |2〉, thus allowing one to measure γ22. The decay of luminescence from the lower state is then governed by the exponential eγ11t and may be used to find γ11. The described situation is illustrated by Fig. 7(a), where it is assumed that ζ2n = 10ζ1n = 500 μeV. Comparing this figure with Fig. 6(a), we see the desired suppression of the intermediate state decay for R < 15 nm. The position of the luminescence peak therewith shifts to 23 meV−1. If we now assume an opposite relation between ζ1n and ζ2n (i.e., ζ1n = 10ζ2n = 500 μeV), then the upper state of the electron–hole pairs will be populated predominantly through the decay of the lower state in large quantum dots and via transitions from state |n〉 in small ones. As a result of this, the intensity of secondary emission from the upper state is drastically reduced for R > 15 nm, as may be seen from Fig. 6(b).

 

Fig. 7 The same as in Fig. 6 but for (a) ζ1n = 50 μeV, ζ2n = 500 μeV and (b) ζ1n = 500 μeV, ζ2n = 50 μeV. The relevant eigenstates are weakly coupled via thermal interaction for R ≲ 8 nm. The buildup and decay of luminescence from a 5-nm quantum dot in (a) allow one to determine relaxation constants γ22 and γ11, respectively.

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5.4. Pulse-sensitive excitation

The analysis of the photoluminescence signal obtained upon pulse-sensitive excitation is the most challenging, in comparison to those in other excitation regimes, as the signal evolves in time according to not only the relaxation constants of interest, but also the shape of the excitation pulse. Therefore, this regime is generally to be avoided in practice.

As a concluding remark, we would like to note that the information about relaxation parameters contained in Eqs. (28) and (29) may also be retrieved from the photoluminescence spectra in the regime of pulse sensitive excitation, provided that the quantum dot is excited by the Gaussian pulse and its bandwidth is chosen to be much larger than the filter’s bandpass (σ ≫ ΓF). In this case, Eqs. (15) to (17), (22), and (23) for t ≫ 1/σ give

Σ¯RL(iii)Σ¯RL(0)eΓFt,Σ¯TL(iii)Σ¯TL(0)eΓFt.

6. Conclusions

Using the kinetic Pauli equation, we have developed a unified theoretical treatment of photoluminescence from an arbitrary number of the lowest-energy states of a semiconductor quantum dot excited by a laser pulse. A generic expression for the photoluminescence cross section was derived by assuming that the processes of energy and phase relaxation in the quantum dot are characterized by a set of phenomenological constants, and without specifying the shape of the excitation pulse. Our treatment takes into account thermal transitions (with both decrease and increase in energy) between all eigenstates whose decay results in the emission of secondary photons. The transitions with the increase in energy lead to the additional exponentially decaying terms in the photoluminescence spectra, and to the renormalization of the decay rates. The application of the derived expression was illustrated by the examples of resonant luminescence and thermalized luminescence from the quantum dot with two radiating eigenstates, which are excited with either a Gaussian or exponential pulse. Specifically, we gave analytic expressions for the time-dependent cross section in the case of the two photoluminescence regimes and two types of pulses, and analyzed them for the purpose of elucidating experimental situations that are permitting retrieval of the relaxation constants from the secondary emission spectra. The results of our analysis for different excitation regimes are summarized in Table 1.

Tables Icon

Table 1. Time dependence of photoluminescence signal and experimentally measurable relaxation constants in different excitation regimes. It is assumed that experiments are conducted with either a single quantum dot or a quantum dot ensemble with a narrow size distribution, and that the quantum dot has two eigenstates decaying with the emission of secondary photons. The eigenstates are assumed to be excited independently of one another and the photoluminescence from each of them is meant to be measured separately.

The developed formalism is equally applicable to the quantum dots in the strong and weak confinement regimes, and can be used for various types of elementary excitations, including electron–hole pairs, excitons, hybrid excitations consisting of a trapped electron and hole in the conduction band, and even electron states of unknown nature [71]. However, its applicability is limited to weak excitation pulses, whose intensities are low enough for ignoring such nonlinear effects as generation of biexcitons and trions inside the quantum dot [7274]. The nonlinear treatment of excitonic complexes requires going beyond the conventional rate equations and using essentially different photoluminescence models based on the random population theory.

Appendix: Solution of Pauli equation

To solve Pauli equation for an arbitrary number N of the electron–hole states, we rewrite Eq. (4) in the form

ρnnt+n=1Nannρnn=fn(t),
where ann = γnnδnnζnn (1 −δnn), n = 1, 2,...,N, and the function fn(t) is assumed to tend to zero sufficiently fast at minus infinity in time. Multiplying Eq. (A.1) by an arbitrary constant bn and adding equations corresponding to different n termwise, yield
n=1Nbnρnnt+n=1Nρnnn=1Nbnann=n=1Nbnfn(t).
Suppose that the coefficients bn are such that
n=1Nbnann=snbn.
It is crucial for the subsequent discussion that all N roots sn of the characteristic polynomial of Eq. (A.3) are positive. The positiveness of the roots follows from two observations. First, the coefficient matrix
a^nn=(γ11ζ12eE21/Tζ13eE31/Tζ1NeEN1/Tζ12γ22ζ23eE32/Tζ2NeEN2/Tζ13ζ23γ33ζ3NeEN3/Tζ1Nζ2Nζ3NγNN)
is strictly diagonally dominant [75], as its elements satisfy the condition
|a^nn|>nn|a^nn|.
This condition holds true since the inverse lifetime γnn of electron–hole state |n〉 is determined by all possible relaxation channels (including those leading to the annihilation of electron–hole pairs) and, therefore, always exceeds the sum ζ1n + ζ2n + ... + ζn−1,n + ζn+1,n + ... + ζNn of relaxation rates within the channels occurring due solely to thermal fluctuations. Second, the matrix ânn can be rewritten in the symmetric form by simply multiplying its jth row (j ≥ 2) by the factor eEj1/T [note that a similar transformation of Eq. (A.3) does not change its roots]. Being essentially symmetric, the matrix ânn have real eigenvectors, which are all positive according to the Gershgorin’s theorem [75] for a strictly diagonally dominant matrix with positive diagonal elements.

Assuming that the roots of the characteristic polynomial are all different (it is extremely unlikely that some roots are multiple for a real quantum dot), we introduce new variables

xn=n=1Nbn(n)ρnn,
where {bn(n)} is the set of coefficients bn corresponding to the root sn. As a result, Eq. (A.2) takes the form
xnt+snxn=n=1Nbn(n)fn(t).
The solution to this equation vanishing for t → −∞ is found to be
xn(t)=n=1Nbn(n)tfn(τ)esn(tτ)dτ.
The Pauli equation is now readily solved by inverting the linear system of equations in Eq. (A.4) through the utilization of Cramer’s rule [76]. Hence, Eqs. (A.3), (A.4), and (A.6) give the general solution to Pauli equation.

Acknowledgments

The work of I.D.R was supported by the Australian Research Council, through its Discovery Early Career Researcher Award DE120100055. Six of the authors (M.Yu.L., V.K.T., A.P.L., A.S.B., A.V.B., and A.V.F.) gratefully acknowledge the financial support from the Ministry of Education and Science of the Russian Federation (Projects Nos. 11.519.11.3020, 11.519.11.3026, and 14.740.11.1366).

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21. T. Inoshita and H. Sakaki, “Electron-phonon interaction and the so-called phonon bottleneck effect in semiconductor quantum dots,” Physica B: Cond. Matt. 227, 373–377 (1996). [CrossRef]  

22. T. Inoshita and H. Sakaki, “Electron relaxation in a quantum dot: Significance of multiphonon processes,” Phys. Rev. B 46, 7260–7263 (1992). [CrossRef]  

23. M. I. Vasilevskiy, E. V. Anda, and S. S. Makler, “Electron-phonon interaction effects in semiconductor quantum dots: A nonperturbative approach,” Phys. Rev. B 70, 035318 (2004). [CrossRef]  

24. A. V. Fedorov and I. D. Rukhlenko, “Study of electronic dynamics of quantum dots using resonant photoluminescence technique,” Opt. Spectrosc. 100, 716–723 (2006). [CrossRef]  

25. 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).

26. A. V. Fedorov and A. V. Baranov, “Intraband carrier relaxation in quantum dots mediated by surface plasmon-phonon excitations,” Opt. Spectrosc. 97, 56–67 (2004). [CrossRef]  

27. A. V. Fedorov and A. V. Baranov, “Relaxation of charge carriers in quantum dots with the involvement of plasmon-phonon modes,” Semicond. 38, 1065–1073 (2004). [CrossRef]  

28. A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, and Y. Masumoto, “New many-body mechanism of intraband carrier relaxation in quantum dots embedded in doped heterostructures,” Solid State Commun. 128, 219–223 (2003). [CrossRef]  

29. 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]  

30. G. A. Narvaez, G. Bester, and A. Zunger, “Carrier relaxation mechanisms in self-assembled (In,Ga)As/GaAs quantum dots: Efficient PS Auger relaxation of electrons,” Phys. Rev. B 74, 075403 (2006). [CrossRef]  

31. S. Yu. 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]  

32. S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Electron-electron scattering in a double quantum dot: Effective mass approach,” J. Chem. Phys. 133, 104704 (2010). [CrossRef]   [PubMed]  

33. S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Resonant energy transfer in quantum dots: Frequency-domain luminescent spectroscopy,” Phys. Rev. B 78, 125311 (2008). [CrossRef]  

34. P. Guyot-Sionnest, B. Wehrenberg, and D. Yu, “Intraband relaxation in cdse nanocrystals and the strong influence of the surface ligands,” J. Chem. Phys. 123, 074709 (2005). [CrossRef]   [PubMed]  

35. D. F. Schroeter, D. J. Griffiths, and P. C. Sercel, “Defect-assisted relaxation in quantum dots at low temperature,” Phys. Rev. B 54, 1486–1489 (1996). [CrossRef]  

36. P. C. Sercel, “Multiphonon-assisted tunneling through deep levels: A rapid energy-relaxation mechanism in non-ideal quantum-dot heterostructures,” Phys. Rev. B 51, 14532–14541 (1995). [CrossRef]  

37. V. K. Turkov, S. Yu. Kruchinin, and A. V. Fedorov, “Intraband optical transitions in semiconductor quantum dots: Radiative electronic-excitation lifetime,” Opt. Spectrosc. 110, 740–747 (2011). [CrossRef]  

38. 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]  

39. E. G. Kavousanaki, O. Roslyak, and S. Mukamel, “Probing excitons and biexcitons in coupled quantum dots by coherent two-dimensional optical spectroscopy,” Phys. Rev. B 79, 155324 (2009). [CrossRef]  

40. W. Demtröder, Laser Spectroscopy: Basic Concepts and Instrumentation (Springer, Berlin, 2002), 3rd ed.

41. 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]  

42. 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]  

43. A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of thermalized luminescence in semiconductor quantum dots,” Opt. Spectrosc. 93, 555–558 (2002). [CrossRef]  

44. A. V. Baranov, V. Davydov, A. V. Fedorov, H.-W. Ren, S. Sugou, and Y. Masumoto, “Coherent control of stress-induced InGaAs quantum dots by means of phonon-assisted resonant photoluminescence,” Physica Status Solidi (b) 224, 461–464 (2001). [CrossRef]  

45. H. Kurtze, J. Seebeck, P. Gartner, D. R. Yakovlev, D. Reuter, A. D. Wieck, M. Bayer, and F. Jahnke, “Carrier relaxation dynamics in self-assembled semiconductor quantum dots,” Phys. Rev. B 80, 235319 (2009). [CrossRef]  

46. M. C. Hoffmann, J. Hebling, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Impact ionization in InSb probed by terahertz pump—terahertz probe spectroscopy,” Phys. Rev. B 79, 161201 (2009). [CrossRef]  

47. H.-Y. Liu, Z.-M. Meng, Q.-F. Dai, L.-J. Wu, Q. Guo, W. Hu, S.-H. Liu, S. Lan, and T. Yang, “Ultrafast carrier dynamics in undoped and p-doped InAs/GaAs quantum dots characterized by pump-probe reflection measurements,” J. Appl. Phys. 103, 083121 (2008). [CrossRef]  

48. T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007). [CrossRef]  

49. S. L. Sewall, R. R. Cooney, K. E. H. Anderson, E. A. Dias, and P. Kambhampati, “State-to-state exciton dynamics in semiconductor quantum dots,” Phys. Rev. B 74, 235328 (2006). [CrossRef]  

50. M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient intraband light absorption by quantum dots: Pump-probe spectroscopy,” Opt. Spectrosc. 111, 798–807 (2011). [CrossRef]  

51. M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Non-degenerate case of pump-probe spectroscopy,” Opt. Spectrosc. 110, 24–32 (2011). [CrossRef]  

52. M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Degenerate pump-probe spectroscopy,” Opt. Spectrosc. 109, 358–365 (2010). [CrossRef]  

53. K. Rivoire, S. Buckley, Y. Song, M. L. Lee, and J. Vučković, “Photoluminescence from In0.5Ga0.5As/GaP quantum dots coupled to photonic crystal cavities,” Phys. Rev. B 85, 045319 (2012). [CrossRef]  

54. X. M. Dou, B. Q. Sun, D. S. Jiang, H. Q. Ni, and Z. C. Niu, “Electron spin relaxation in a single InAs quantum dot measured by tunable nuclear spins,” Phys. Rev. B 84, 033302 (2011). [CrossRef]  

55. V. M. Axt and T. Kuhn, “Femtosecond spectroscopy in semiconductors: A key to coherences, correlations and quantum kinetics,” Rep. Prog. Phys. 67, 433 (2004). [CrossRef]  

56. S. A. Empedocles, D. J. Norris, and M. G. Bawendi, “Photoluminescence spectroscopy of single CdSe nanocrystallite quantum dots,” Phys. Rev. Lett. 77, 3873–3876 (1996). [CrossRef]   [PubMed]  

57. A. Pandey and P. Guyot-Sionnest, “Slow electron cooling in colloidal quantum dots,” Science 322, 929–932 (2008). [CrossRef]   [PubMed]  

58. C. Bonati, A. Cannizzo, D. Tonti, A. Tortschanoff, F. van Mourik, and M. Chergui, “Subpicosecond near-infrared fluorescence upconversion study of relaxation processes in PbSe quantum dots,” Phys. Rev. B 76, 033304 (2007). [CrossRef]  

59. E. Hendry, M. Koeberg, F. Wang, H. Zhang, C. de Mello Donegá, D. Vanmaekelbergh, and M. Bonn, “Direct observation of electron-to-hole energy transfer in CdSe quantum dots,” Phys. Rev. Lett. 96, 057408 (2006). [CrossRef]   [PubMed]  

60. R. D. Schaller, J. M. Pietryga, S. V. Goupalov, M. A. Petruska, S. A. Ivanov, and V. I. Klimov, “Breaking the phonon bottleneck in semiconductor nanocrystals via multiphonon emission induced by intrinsic nonadiabatic interactions,” Phys. Rev. Lett. 95, 196401 (2005). [CrossRef]   [PubMed]  

61. H. Benisty, C. M. Sotomayor-Torrès, and C. Weisbuch, “Intrinsic mechanism for the poor luminescence properties of quantum-box systems,” Phys. Rev. B 44, 10945–10948 (1991). [CrossRef]  

62. U. Bockelmann and G. Bastard, “Phonon scattering and energy relaxation in two-, one-, and zero-dimensional electron gases,” Phys. Rev. B 42, 8947–8951 (1990). [CrossRef]  

63. K. Blum, Density Matrix Theory and Applications (Springer, Berlin, 2012), 3rd ed. [CrossRef]  

64. A. Al Salman, A. Tortschanoff, M. B. Mohamed, D. Tonti, F. van Mourik, and M. Chergui, “Temperature effects on the spectral properties of colloidal CdSe nanodots, nanorods, and tetrapods,” Appl. Phys. Lett. 90, 093104 (2007). [CrossRef]  

65. P. Borri, W. Langbein, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Exciton dephasing via phonon interactions in InAs quantum dots: Dependence on quantum confinement,” Phys. Rev. B 71, 115328 (2005). [CrossRef]  

66. P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Relaxation and dephasing of multiexcitons in semiconductor quantum dots,” Phys. Rev. Lett. 89, 187401 (2002). [CrossRef]   [PubMed]  

67. I. D. Rukhlenko, A. V. Fedorov, A. S. Baymuratov, and M. Premaratne, “Theory of quasi-elastic secondary emission from a quantum dot in the regime of vibrational resonance,” Opt. Express 19, 15459–15482 (2011). [CrossRef]   [PubMed]  

68. J. S. Melinger and A. C. Albrecht, “Theory of time and frequency resolved resonance secondary radiation from a three-level system,” J. Chem. Phys. 84, 1247–1258 (1986). [CrossRef]  

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71. E. V. Ushakova, A. P. Litvin, P. S. Parfenov, A. V. Fedorov, M. Artemyev, A. V. Prudnikau, I. D. Rukhlenko, and A. V. Baranov, “Anomalous size-dependent decay of low-energy luminescence from PbS quantum dots in colloidal solution,” ACS Nano 6, 8913–8921 (2012). [CrossRef]   [PubMed]  

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73. M. Abbarchi, C. Mastrandrea, T. Kuroda, T. Mano, A. Vinattieri, K. Sakoda, and M. Gurioli, “Poissonian statistics of excitonic complexes in quantum dots,” J. Appl. Phys. 106, 053504 (2009). [CrossRef]  

74. M. Grundmann and D. Bimberg, “Theory of random population for quantum dots,” Phys. Rev. B 55, 9740–9745 (1997). [CrossRef]  

75. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, San Diego, CA, 1994), 5th ed.

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    [CrossRef]
  45. H. Kurtze, J. Seebeck, P. Gartner, D. R. Yakovlev, D. Reuter, A. D. Wieck, M. Bayer, and F. Jahnke, “Carrier relaxation dynamics in self-assembled semiconductor quantum dots,” Phys. Rev. B80, 235319 (2009).
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    [CrossRef]
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    [CrossRef]
  50. M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient intraband light absorption by quantum dots: Pump-probe spectroscopy,” Opt. Spectrosc.111, 798–807 (2011).
    [CrossRef]
  51. M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Non-degenerate case of pump-probe spectroscopy,” Opt. Spectrosc.110, 24–32 (2011).
    [CrossRef]
  52. M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Degenerate pump-probe spectroscopy,” Opt. Spectrosc.109, 358–365 (2010).
    [CrossRef]
  53. K. Rivoire, S. Buckley, Y. Song, M. L. Lee, and J. Vučković, “Photoluminescence from In0.5Ga0.5As/GaP quantum dots coupled to photonic crystal cavities,” Phys. Rev. B85, 045319 (2012).
    [CrossRef]
  54. X. M. Dou, B. Q. Sun, D. S. Jiang, H. Q. Ni, and Z. C. Niu, “Electron spin relaxation in a single InAs quantum dot measured by tunable nuclear spins,” Phys. Rev. B84, 033302 (2011).
    [CrossRef]
  55. V. M. Axt and T. Kuhn, “Femtosecond spectroscopy in semiconductors: A key to coherences, correlations and quantum kinetics,” Rep. Prog. Phys.67, 433 (2004).
    [CrossRef]
  56. S. A. Empedocles, D. J. Norris, and M. G. Bawendi, “Photoluminescence spectroscopy of single CdSe nanocrystallite quantum dots,” Phys. Rev. Lett.77, 3873–3876 (1996).
    [CrossRef] [PubMed]
  57. A. Pandey and P. Guyot-Sionnest, “Slow electron cooling in colloidal quantum dots,” Science322, 929–932 (2008).
    [CrossRef] [PubMed]
  58. C. Bonati, A. Cannizzo, D. Tonti, A. Tortschanoff, F. van Mourik, and M. Chergui, “Subpicosecond near-infrared fluorescence upconversion study of relaxation processes in PbSe quantum dots,” Phys. Rev. B76, 033304 (2007).
    [CrossRef]
  59. E. Hendry, M. Koeberg, F. Wang, H. Zhang, C. de Mello Donegá, D. Vanmaekelbergh, and M. Bonn, “Direct observation of electron-to-hole energy transfer in CdSe quantum dots,” Phys. Rev. Lett.96, 057408 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2012 (2)

K. Rivoire, S. Buckley, Y. Song, M. L. Lee, and J. Vučković, “Photoluminescence from In0.5Ga0.5As/GaP quantum dots coupled to photonic crystal cavities,” Phys. Rev. B85, 045319 (2012).
[CrossRef]

E. V. Ushakova, A. P. Litvin, P. S. Parfenov, A. V. Fedorov, M. Artemyev, A. V. Prudnikau, I. D. Rukhlenko, and A. V. Baranov, “Anomalous size-dependent decay of low-energy luminescence from PbS quantum dots in colloidal solution,” ACS Nano6, 8913–8921 (2012).
[CrossRef] [PubMed]

2011 (8)

J. Gomis-Bresco, G. Muñoz-Matutano, J. Martínez-Pastor, B. Alén, L. Seravalli, P. Frigeri, G. Trevisi, and S. Franchi, “Random population model to explain the recombination dynamics in single InAs/GaAs quantum dots under selective optical pumping,” New J. Phys.13, 023022 (2011).
[CrossRef]

I. D. Rukhlenko, A. V. Fedorov, A. S. Baymuratov, and M. Premaratne, “Theory of quasi-elastic secondary emission from a quantum dot in the regime of vibrational resonance,” Opt. Express19, 15459–15482 (2011).
[CrossRef] [PubMed]

X. M. Dou, B. Q. Sun, D. S. Jiang, H. Q. Ni, and Z. C. Niu, “Electron spin relaxation in a single InAs quantum dot measured by tunable nuclear spins,” Phys. Rev. B84, 033302 (2011).
[CrossRef]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient intraband light absorption by quantum dots: Pump-probe spectroscopy,” Opt. Spectrosc.111, 798–807 (2011).
[CrossRef]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Non-degenerate case of pump-probe spectroscopy,” Opt. Spectrosc.110, 24–32 (2011).
[CrossRef]

S. Strauf and F. Jahnke, “Single quantum dot nanolaser,” Laser Photonics Rev.5, 607–633 (2011).

F. Schulze, M. Schoth, U. Woggon, A. Knorr, and C. Weber, “Ultrafast dynamics of carrier multiplication in quantum dots,” Phys. Rev. B84, 125318 (2011).
[CrossRef]

V. K. Turkov, S. Yu. Kruchinin, and A. V. Fedorov, “Intraband optical transitions in semiconductor quantum dots: Radiative electronic-excitation lifetime,” Opt. Spectrosc.110, 740–747 (2011).
[CrossRef]

2010 (4)

S. Yu. 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]

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Electron-electron scattering in a double quantum dot: Effective mass approach,” J. Chem. Phys.133, 104704 (2010).
[CrossRef] [PubMed]

M.-R. Dachner, E. Malic, M. Richter, A. Carmele, J. Kabuss, A. Wilms, J.-E. Kim, G. Hartmann, J. Wolters, U. Bandelow, and A. Knorr, “Theory of carrier and photon dynamics in quantum dot light emitters,” Phys. Stat. Solidi (b)247, 809–828 (2010).
[CrossRef]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Degenerate pump-probe spectroscopy,” Opt. Spectrosc.109, 358–365 (2010).
[CrossRef]

2009 (4)

H. Kurtze, J. Seebeck, P. Gartner, D. R. Yakovlev, D. Reuter, A. D. Wieck, M. Bayer, and F. Jahnke, “Carrier relaxation dynamics in self-assembled semiconductor quantum dots,” Phys. Rev. B80, 235319 (2009).
[CrossRef]

M. C. Hoffmann, J. Hebling, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Impact ionization in InSb probed by terahertz pump—terahertz probe spectroscopy,” Phys. Rev. B79, 161201 (2009).
[CrossRef]

M. Abbarchi, C. Mastrandrea, T. Kuroda, T. Mano, A. Vinattieri, K. Sakoda, and M. Gurioli, “Poissonian statistics of excitonic complexes in quantum dots,” J. Appl. Phys.106, 053504 (2009).
[CrossRef]

E. G. Kavousanaki, O. Roslyak, and S. Mukamel, “Probing excitons and biexcitons in coupled quantum dots by coherent two-dimensional optical spectroscopy,” Phys. Rev. B79, 155324 (2009).
[CrossRef]

2008 (3)

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Resonant energy transfer in quantum dots: Frequency-domain luminescent spectroscopy,” Phys. Rev. B78, 125311 (2008).
[CrossRef]

H.-Y. Liu, Z.-M. Meng, Q.-F. Dai, L.-J. Wu, Q. Guo, W. Hu, S.-H. Liu, S. Lan, and T. Yang, “Ultrafast carrier dynamics in undoped and p-doped InAs/GaAs quantum dots characterized by pump-probe reflection measurements,” J. Appl. Phys.103, 083121 (2008).
[CrossRef]

A. Pandey and P. Guyot-Sionnest, “Slow electron cooling in colloidal quantum dots,” Science322, 929–932 (2008).
[CrossRef] [PubMed]

2007 (3)

C. Bonati, A. Cannizzo, D. Tonti, A. Tortschanoff, F. van Mourik, and M. Chergui, “Subpicosecond near-infrared fluorescence upconversion study of relaxation processes in PbSe quantum dots,” Phys. Rev. B76, 033304 (2007).
[CrossRef]

A. Al Salman, A. Tortschanoff, M. B. Mohamed, D. Tonti, F. van Mourik, and M. Chergui, “Temperature effects on the spectral properties of colloidal CdSe nanodots, nanorods, and tetrapods,” Appl. Phys. Lett.90, 093104 (2007).
[CrossRef]

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B76, 165318 (2007).
[CrossRef]

2006 (8)

S. L. Sewall, R. R. Cooney, K. E. H. Anderson, E. A. Dias, and P. Kambhampati, “State-to-state exciton dynamics in semiconductor quantum dots,” Phys. Rev. B74, 235328 (2006).
[CrossRef]

E. Hendry, M. Koeberg, F. Wang, H. Zhang, C. de Mello Donegá, D. Vanmaekelbergh, and M. Bonn, “Direct observation of electron-to-hole energy transfer in CdSe quantum dots,” Phys. Rev. Lett.96, 057408 (2006).
[CrossRef] [PubMed]

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]

G. A. Narvaez, G. Bester, and A. Zunger, “Carrier relaxation mechanisms in self-assembled (In,Ga)As/GaAs quantum dots: Efficient P → S Auger relaxation of electrons,” Phys. Rev. B74, 075403 (2006).
[CrossRef]

J. Ishi-Hayase, K. Akahane, N. Yamamoto, M. Sasaki, M. Kujiraoka, and K. Ema, “Long dephasing time in self-assembled InAs quantum dots at over 1.3 μm wavelength,” Appl. Phys. Lett.88, 261907 (2006).
[CrossRef]

A. V. Fedorov and I. D. Rukhlenko, “Study of electronic dynamics of quantum dots using resonant photoluminescence technique,” Opt. Spectrosc.100, 716–723 (2006).
[CrossRef]

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

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

2005 (4)

T. B. Norris, K. Kim, J. Urayama, Z. K. Wu, J. Singh, and P. K. Bhattacharya, “Density and temperature dependence of carrier dynamics in self-organized InGaAs quantum dots,” J. Phys. D: Appl. Phys.38, 2077 (2005).
[CrossRef]

P. Guyot-Sionnest, B. Wehrenberg, and D. Yu, “Intraband relaxation in cdse nanocrystals and the strong influence of the surface ligands,” J. Chem. Phys.123, 074709 (2005).
[CrossRef] [PubMed]

R. D. Schaller, J. M. Pietryga, S. V. Goupalov, M. A. Petruska, S. A. Ivanov, and V. I. Klimov, “Breaking the phonon bottleneck in semiconductor nanocrystals via multiphonon emission induced by intrinsic nonadiabatic interactions,” Phys. Rev. Lett.95, 196401 (2005).
[CrossRef] [PubMed]

P. Borri, W. Langbein, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Exciton dephasing via phonon interactions in InAs quantum dots: Dependence on quantum confinement,” Phys. Rev. B71, 115328 (2005).
[CrossRef]

2004 (4)

V. M. Axt and T. Kuhn, “Femtosecond spectroscopy in semiconductors: A key to coherences, correlations and quantum kinetics,” Rep. Prog. Phys.67, 433 (2004).
[CrossRef]

A. V. Fedorov and A. V. Baranov, “Intraband carrier relaxation in quantum dots mediated by surface plasmon-phonon excitations,” Opt. Spectrosc.97, 56–67 (2004).
[CrossRef]

A. V. Fedorov and A. V. Baranov, “Relaxation of charge carriers in quantum dots with the involvement of plasmon-phonon modes,” Semicond.38, 1065–1073 (2004).
[CrossRef]

M. I. Vasilevskiy, E. V. Anda, and S. S. Makler, “Electron-phonon interaction effects in semiconductor quantum dots: A nonperturbative approach,” Phys. Rev. B70, 035318 (2004).
[CrossRef]

2003 (2)

A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, and Y. Masumoto, “New many-body mechanism of intraband carrier relaxation in quantum dots embedded in doped heterostructures,” Solid State Commun.128, 219–223 (2003).
[CrossRef]

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]

2002 (6)

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]

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

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of thermalized luminescence in semiconductor quantum dots,” Opt. Spectrosc.93, 555–558 (2002).
[CrossRef]

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

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

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Relaxation and dephasing of multiexcitons in semiconductor quantum dots,” Phys. Rev. Lett.89, 187401 (2002).
[CrossRef] [PubMed]

2001 (2)

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett.87, 157401 (2001).
[CrossRef] [PubMed]

A. V. Baranov, V. Davydov, A. V. Fedorov, H.-W. Ren, S. Sugou, and Y. Masumoto, “Coherent control of stress-induced InGaAs quantum dots by means of phonon-assisted resonant photoluminescence,” Physica Status Solidi (b)224, 461–464 (2001).
[CrossRef]

1999 (1)

X.-Q. Li, H. Nakayama, and Y. Arakawa, “Phonon bottleneck in quantum dots: Role of lifetime of the confined optical phonons,” Phys. Rev. B59, 5069–5073 (1999).
[CrossRef]

1998 (1)

X.-Q. Li and Y. Arakawa, “Anharmonic decay of confined optical phonons in quantum dots,” Phys. Rev. B57, 12285–12290 (1998).
[CrossRef]

1997 (2)

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]

M. Grundmann and D. Bimberg, “Theory of random population for quantum dots,” Phys. Rev. B55, 9740–9745 (1997).
[CrossRef]

1996 (4)

S. A. Empedocles, D. J. Norris, and M. G. Bawendi, “Photoluminescence spectroscopy of single CdSe nanocrystallite quantum dots,” Phys. Rev. Lett.77, 3873–3876 (1996).
[CrossRef] [PubMed]

T. Inoshita and H. Sakaki, “Electron-phonon interaction and the so-called phonon bottleneck effect in semiconductor quantum dots,” Physica B: Cond. Matt.227, 373–377 (1996).
[CrossRef]

A. P. Alivisatos, “Semiconductor clusters, nanocrystals, and quantum dots,” Science271, 933–937 (1996).
[CrossRef]

D. F. Schroeter, D. J. Griffiths, and P. C. Sercel, “Defect-assisted relaxation in quantum dots at low temperature,” Phys. Rev. B54, 1486–1489 (1996).
[CrossRef]

1995 (1)

P. C. Sercel, “Multiphonon-assisted tunneling through deep levels: A rapid energy-relaxation mechanism in non-ideal quantum-dot heterostructures,” Phys. Rev. B51, 14532–14541 (1995).
[CrossRef]

1992 (1)

T. Inoshita and H. Sakaki, “Electron relaxation in a quantum dot: Significance of multiphonon processes,” Phys. Rev. B46, 7260–7263 (1992).
[CrossRef]

1991 (1)

H. Benisty, C. M. Sotomayor-Torrès, and C. Weisbuch, “Intrinsic mechanism for the poor luminescence properties of quantum-box systems,” Phys. Rev. B44, 10945–10948 (1991).
[CrossRef]

1990 (1)

U. Bockelmann and G. Bastard, “Phonon scattering and energy relaxation in two-, one-, and zero-dimensional electron gases,” Phys. Rev. B42, 8947–8951 (1990).
[CrossRef]

1986 (1)

J. S. Melinger and A. C. Albrecht, “Theory of time and frequency resolved resonance secondary radiation from a three-level system,” J. Chem. Phys.84, 1247–1258 (1986).
[CrossRef]

1979 (1)

G. Nienhuis, “Time and frequency dependence of nearly resonant light scattered from collisionally perturbed atoms,” Physica C96, 391–409 (1979).
[CrossRef]

1977 (1)

1953 (1)

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).

Abbarchi, M.

M. Abbarchi, C. Mastrandrea, T. Kuroda, T. Mano, A. Vinattieri, K. Sakoda, and M. Gurioli, “Poissonian statistics of excitonic complexes in quantum dots,” J. Appl. Phys.106, 053504 (2009).
[CrossRef]

Akahane, K.

J. Ishi-Hayase, K. Akahane, N. Yamamoto, M. Sasaki, M. Kujiraoka, and K. Ema, “Long dephasing time in self-assembled InAs quantum dots at over 1.3 μm wavelength,” Appl. Phys. Lett.88, 261907 (2006).
[CrossRef]

Al Salman, A.

A. Al Salman, A. Tortschanoff, M. B. Mohamed, D. Tonti, F. van Mourik, and M. Chergui, “Temperature effects on the spectral properties of colloidal CdSe nanodots, nanorods, and tetrapods,” Appl. Phys. Lett.90, 093104 (2007).
[CrossRef]

Albrecht, A. C.

J. S. Melinger and A. C. Albrecht, “Theory of time and frequency resolved resonance secondary radiation from a three-level system,” J. Chem. Phys.84, 1247–1258 (1986).
[CrossRef]

Alén, B.

J. Gomis-Bresco, G. Muñoz-Matutano, J. Martínez-Pastor, B. Alén, L. Seravalli, P. Frigeri, G. Trevisi, and S. Franchi, “Random population model to explain the recombination dynamics in single InAs/GaAs quantum dots under selective optical pumping,” New J. Phys.13, 023022 (2011).
[CrossRef]

Alivisatos, A. P.

A. P. Alivisatos, “Semiconductor clusters, nanocrystals, and quantum dots,” Science271, 933–937 (1996).
[CrossRef]

Anda, E. V.

M. I. Vasilevskiy, E. V. Anda, and S. S. Makler, “Electron-phonon interaction effects in semiconductor quantum dots: A nonperturbative approach,” Phys. Rev. B70, 035318 (2004).
[CrossRef]

Anderson, K. E. H.

S. L. Sewall, R. R. Cooney, K. E. H. Anderson, E. A. Dias, and P. Kambhampati, “State-to-state exciton dynamics in semiconductor quantum dots,” Phys. Rev. B74, 235328 (2006).
[CrossRef]

Arakawa, Y.

X.-Q. Li, H. Nakayama, and Y. Arakawa, “Phonon bottleneck in quantum dots: Role of lifetime of the confined optical phonons,” Phys. Rev. B59, 5069–5073 (1999).
[CrossRef]

X.-Q. Li and Y. Arakawa, “Anharmonic decay of confined optical phonons in quantum dots,” Phys. Rev. B57, 12285–12290 (1998).
[CrossRef]

Artemyev, M.

E. V. Ushakova, A. P. Litvin, P. S. Parfenov, A. V. Fedorov, M. Artemyev, A. V. Prudnikau, I. D. Rukhlenko, and A. V. Baranov, “Anomalous size-dependent decay of low-energy luminescence from PbS quantum dots in colloidal solution,” ACS Nano6, 8913–8921 (2012).
[CrossRef] [PubMed]

Auer, T.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B76, 165318 (2007).
[CrossRef]

Axt, V. M.

V. M. Axt and T. Kuhn, “Femtosecond spectroscopy in semiconductors: A key to coherences, correlations and quantum kinetics,” Rep. Prog. Phys.67, 433 (2004).
[CrossRef]

Bandelow, U.

M.-R. Dachner, E. Malic, M. Richter, A. Carmele, J. Kabuss, A. Wilms, J.-E. Kim, G. Hartmann, J. Wolters, U. Bandelow, and A. Knorr, “Theory of carrier and photon dynamics in quantum dot light emitters,” Phys. Stat. Solidi (b)247, 809–828 (2010).
[CrossRef]

Baranov, A. V.

E. V. Ushakova, A. P. Litvin, P. S. Parfenov, A. V. Fedorov, M. Artemyev, A. V. Prudnikau, I. D. Rukhlenko, and A. V. Baranov, “Anomalous size-dependent decay of low-energy luminescence from PbS quantum dots in colloidal solution,” ACS Nano6, 8913–8921 (2012).
[CrossRef] [PubMed]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient intraband light absorption by quantum dots: Pump-probe spectroscopy,” Opt. Spectrosc.111, 798–807 (2011).
[CrossRef]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Non-degenerate case of pump-probe spectroscopy,” Opt. Spectrosc.110, 24–32 (2011).
[CrossRef]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Degenerate pump-probe spectroscopy,” Opt. Spectrosc.109, 358–365 (2010).
[CrossRef]

S. Yu. 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]

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Electron-electron scattering in a double quantum dot: Effective mass approach,” J. Chem. Phys.133, 104704 (2010).
[CrossRef] [PubMed]

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Resonant energy transfer in quantum dots: Frequency-domain luminescent spectroscopy,” Phys. Rev. B78, 125311 (2008).
[CrossRef]

A. V. Fedorov and A. V. Baranov, “Intraband carrier relaxation in quantum dots mediated by surface plasmon-phonon excitations,” Opt. Spectrosc.97, 56–67 (2004).
[CrossRef]

A. V. Fedorov and A. V. Baranov, “Relaxation of charge carriers in quantum dots with the involvement of plasmon-phonon modes,” Semicond.38, 1065–1073 (2004).
[CrossRef]

A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, and Y. Masumoto, “New many-body mechanism of intraband carrier relaxation in quantum dots embedded in doped heterostructures,” Solid State Commun.128, 219–223 (2003).
[CrossRef]

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]

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

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of thermalized luminescence in semiconductor quantum dots,” Opt. Spectrosc.93, 555–558 (2002).
[CrossRef]

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

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

A. V. Baranov, V. Davydov, A. V. Fedorov, H.-W. Ren, S. Sugou, and Y. Masumoto, “Coherent control of stress-induced InGaAs quantum dots by means of phonon-assisted resonant photoluminescence,” Physica Status Solidi (b)224, 461–464 (2001).
[CrossRef]

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

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

Bastard, G.

U. Bockelmann and G. Bastard, “Phonon scattering and energy relaxation in two-, one-, and zero-dimensional electron gases,” Phys. Rev. B42, 8947–8951 (1990).
[CrossRef]

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S. Strauf and F. Jahnke, “Single quantum dot nanolaser,” Laser Photonics Rev.5, 607–633 (2011).

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A. V. Baranov, V. Davydov, A. V. Fedorov, H.-W. Ren, S. Sugou, and Y. Masumoto, “Coherent control of stress-induced InGaAs quantum dots by means of phonon-assisted resonant photoluminescence,” Physica Status Solidi (b)224, 461–464 (2001).
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C. Bonati, A. Cannizzo, D. Tonti, A. Tortschanoff, F. van Mourik, and M. Chergui, “Subpicosecond near-infrared fluorescence upconversion study of relaxation processes in PbSe quantum dots,” Phys. Rev. B76, 033304 (2007).
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H.-Y. Liu, Z.-M. Meng, Q.-F. Dai, L.-J. Wu, Q. Guo, W. Hu, S.-H. Liu, S. Lan, and T. Yang, “Ultrafast carrier dynamics in undoped and p-doped InAs/GaAs quantum dots characterized by pump-probe reflection measurements,” J. Appl. Phys.103, 083121 (2008).
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T. B. Norris, K. Kim, J. Urayama, Z. K. Wu, J. Singh, and P. K. Bhattacharya, “Density and temperature dependence of carrier dynamics in self-organized InGaAs quantum dots,” J. Phys. D: Appl. Phys.38, 2077 (2005).
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H. Kurtze, J. Seebeck, P. Gartner, D. R. Yakovlev, D. Reuter, A. D. Wieck, M. Bayer, and F. Jahnke, “Carrier relaxation dynamics in self-assembled semiconductor quantum dots,” Phys. Rev. B80, 235319 (2009).
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J. Ishi-Hayase, K. Akahane, N. Yamamoto, M. Sasaki, M. Kujiraoka, and K. Ema, “Long dephasing time in self-assembled InAs quantum dots at over 1.3 μm wavelength,” Appl. Phys. Lett.88, 261907 (2006).
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M. C. Hoffmann, J. Hebling, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Impact ionization in InSb probed by terahertz pump—terahertz probe spectroscopy,” Phys. Rev. B79, 161201 (2009).
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Yu, D.

P. Guyot-Sionnest, B. Wehrenberg, and D. Yu, “Intraband relaxation in cdse nanocrystals and the strong influence of the surface ligands,” J. Chem. Phys.123, 074709 (2005).
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Zhang, H.

E. Hendry, M. Koeberg, F. Wang, H. Zhang, C. de Mello Donegá, D. Vanmaekelbergh, and M. Bonn, “Direct observation of electron-to-hole energy transfer in CdSe quantum dots,” Phys. Rev. Lett.96, 057408 (2006).
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ACS Nano (1)

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Appl. Phys. Lett. (2)

A. Al Salman, A. Tortschanoff, M. B. Mohamed, D. Tonti, F. van Mourik, and M. Chergui, “Temperature effects on the spectral properties of colloidal CdSe nanodots, nanorods, and tetrapods,” Appl. Phys. Lett.90, 093104 (2007).
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J. Ishi-Hayase, K. Akahane, N. Yamamoto, M. Sasaki, M. Kujiraoka, and K. Ema, “Long dephasing time in self-assembled InAs quantum dots at over 1.3 μm wavelength,” Appl. Phys. Lett.88, 261907 (2006).
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J. Appl. Phys. (2)

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H.-Y. Liu, Z.-M. Meng, Q.-F. Dai, L.-J. Wu, Q. Guo, W. Hu, S.-H. Liu, S. Lan, and T. Yang, “Ultrafast carrier dynamics in undoped and p-doped InAs/GaAs quantum dots characterized by pump-probe reflection measurements,” J. Appl. Phys.103, 083121 (2008).
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J. Chem. Phys. (3)

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J. Opt. Soc. Am. (1)

J. Phys. D: Appl. Phys. (1)

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Laser Photonics Rev. (1)

S. Strauf and F. Jahnke, “Single quantum dot nanolaser,” Laser Photonics Rev.5, 607–633 (2011).

New J. Phys. (1)

J. Gomis-Bresco, G. Muñoz-Matutano, J. Martínez-Pastor, B. Alén, L. Seravalli, P. Frigeri, G. Trevisi, and S. Franchi, “Random population model to explain the recombination dynamics in single InAs/GaAs quantum dots under selective optical pumping,” New J. Phys.13, 023022 (2011).
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Opt. Express (1)

Opt. Spectrosc. (11)

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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).
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A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of thermalized luminescence in semiconductor quantum dots,” Opt. Spectrosc.93, 555–558 (2002).
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M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient intraband light absorption by quantum dots: Pump-probe spectroscopy,” Opt. Spectrosc.111, 798–807 (2011).
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M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Non-degenerate case of pump-probe spectroscopy,” Opt. Spectrosc.110, 24–32 (2011).
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M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Degenerate pump-probe spectroscopy,” Opt. Spectrosc.109, 358–365 (2010).
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V. K. Turkov, S. Yu. Kruchinin, and A. V. Fedorov, “Intraband optical transitions in semiconductor quantum dots: Radiative electronic-excitation lifetime,” Opt. Spectrosc.110, 740–747 (2011).
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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).
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A. V. Fedorov and I. D. Rukhlenko, “Study of electronic dynamics of quantum dots using resonant photoluminescence technique,” Opt. Spectrosc.100, 716–723 (2006).
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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).
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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).
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G. A. Narvaez, G. Bester, and A. Zunger, “Carrier relaxation mechanisms in self-assembled (In,Ga)As/GaAs quantum dots: Efficient P → S Auger relaxation of electrons,” Phys. Rev. B74, 075403 (2006).
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S. Yu. 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).
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F. Schulze, M. Schoth, U. Woggon, A. Knorr, and C. Weber, “Ultrafast dynamics of carrier multiplication in quantum dots,” Phys. Rev. B84, 125318 (2011).
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X. M. Dou, B. Q. Sun, D. S. Jiang, H. Q. Ni, and Z. C. Niu, “Electron spin relaxation in a single InAs quantum dot measured by tunable nuclear spins,” Phys. Rev. B84, 033302 (2011).
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T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B76, 165318 (2007).
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M. C. Hoffmann, J. Hebling, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Impact ionization in InSb probed by terahertz pump—terahertz probe spectroscopy,” Phys. Rev. B79, 161201 (2009).
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C. Bonati, A. Cannizzo, D. Tonti, A. Tortschanoff, F. van Mourik, and M. Chergui, “Subpicosecond near-infrared fluorescence upconversion study of relaxation processes in PbSe quantum dots,” Phys. Rev. B76, 033304 (2007).
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Phys. Rev. Lett. (5)

E. Hendry, M. Koeberg, F. Wang, H. Zhang, C. de Mello Donegá, D. Vanmaekelbergh, and M. Bonn, “Direct observation of electron-to-hole energy transfer in CdSe quantum dots,” Phys. Rev. Lett.96, 057408 (2006).
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Phys. Stat. Solidi (b) (1)

M.-R. Dachner, E. Malic, M. Richter, A. Carmele, J. Kabuss, A. Wilms, J.-E. Kim, G. Hartmann, J. Wolters, U. Bandelow, and A. Knorr, “Theory of carrier and photon dynamics in quantum dot light emitters,” Phys. Stat. Solidi (b)247, 809–828 (2010).
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Physica B: Cond. Matt. (1)

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Physica C (1)

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A. V. Baranov, V. Davydov, A. V. Fedorov, H.-W. Ren, S. Sugou, and Y. Masumoto, “Coherent control of stress-induced InGaAs quantum dots by means of phonon-assisted resonant photoluminescence,” Physica Status Solidi (b)224, 461–464 (2001).
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Rep. Prog. Phys. (1)

V. M. Axt and T. Kuhn, “Femtosecond spectroscopy in semiconductors: A key to coherences, correlations and quantum kinetics,” Rep. Prog. Phys.67, 433 (2004).
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Science (2)

A. Pandey and P. Guyot-Sionnest, “Slow electron cooling in colloidal quantum dots,” Science322, 929–932 (2008).
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A. P. Alivisatos, “Semiconductor clusters, nanocrystals, and quantum dots,” Science271, 933–937 (1996).
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Semicond. (1)

A. V. Fedorov and A. V. Baranov, “Relaxation of charge carriers in quantum dots with the involvement of plasmon-phonon modes,” Semicond.38, 1065–1073 (2004).
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Solid State Commun. (2)

A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, and Y. Masumoto, “New many-body mechanism of intraband carrier relaxation in quantum dots embedded in doped heterostructures,” Solid State Commun.128, 219–223 (2003).
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A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Acoustic phonon problem in nanocrystal–dielectric matrix systems,” Solid State Commun.122, 139–144 (2002).
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A. V. Fedorov, I. D. Rukhlenko, A. V. Baranov, and S. Yu. Kruchinin, Optical Properties of Semiconductor Quantum Dots (Nauka, St. Petersburg, 2011).

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Figures (7)

Fig. 1
Fig. 1

Optical (solid arrows) and relaxation (dashed arrows) transitions corresponding to the processes of (a) resonant luminescence and (b) thermalized luminescence from a semiconductor quantum dot with three energy levels. Ket vectors |0〉, |1〉, |2〉, and |3〉 denote the ground and excited states of electron–hole pairs; |n〉 is the high-energy state that does not directly contribute to the secondary emission; ωLk (ωL) and ωλk (k = 1, 2, 3) are the excitation frequencies and the frequencies of emitted photons; ζkk is the rate of transitions |k′〉 → |k〉 due to the thermal interaction with a bath. The spectral width of the excitation pulse is assumed to be much smaller than the dephasing rates of all interband optical transitions.

Fig. 2
Fig. 2

Three-dimensional plot of log[(2σF)Q(α, t)] as a function of the dimensionless parameters αt and α/σ [see Eq. (17)]. Function values are shown by labels of contour levels. Three regimes of excitation by a Gaussian pulse are clearly seen: (i) adiabatic excitation for log(α/σ) ≳ 1; (ii) instantaneous excitation for log(α/σ) ≲ −1; and (iii) pulse-sensitive excitation for |log(α/σ)| ≲ 0.5.

Fig. 3
Fig. 3

Two-dimensional spectra of (a) resonant luminescence and (b) thermalized luminescence for two relevant eigenstates of the quantum dot’s electronic subsystem. The widths of the peaks in the spectra are determined by the dephasing rates of optical transitions, while the relative peak intensities depend on the interband matrix elements [see Eqs. (5) and (7)], relaxation constants, and temperature of the system. Peak 4 vanishes in the limit of small temperatures.

Fig. 4
Fig. 4

Variation of (a) s1 and (b) log s2 with radius R of PbS quantum dot and ratio y = ζ02/ζ01 of radiative recombination rates for two relevant eigenstates of electron–hole pairs (s1 and s2 are in meV). The quantum numbers of the eigenstates are n1 = 1, l1 = 0 and n2 = 2, l2 = 0. Bound by red curves are the domains of strong thermal coupling between the eigenstates, where 2ζ12 exp[−E21/(2T)]/(γ22γ11) > 0.1. Quantum dot’s boundary is assumed to be impenetrable for both electrons and holes; ζ01 = 200 μeV, ζ12 = 150 μeV, and T = 25 meV. For other parameters, refer to the text.

Fig. 5
Fig. 5

Temporal evolution of resonant luminescence from PbS quantum dots of different sizes excited by a Gaussian pulse. Excitation and detection frequencies correspond to the peaks in Fig. 3(a): (a) ωL = ωF = ω1; (b) ωL = ωF = ω2; (c) ωL = ω2, ωF = ω1; (d) ωL = ω1, ωF = ω2. Blue and red curves indicate decay/buildup of luminescence with characteristic times 1/s1 and 1/s2, respectively. The relevant eigenstates are weakly coupled via thermal interaction with the bath for R ≲ 8 nm. Simulation parameters are γ̂1i = γ̂2i = γ̂f 1 = γ̂f 2 = 20 meV, ΓF = 10 meV, σ = 1 meV, ζ12 = 100 μeV, and ζ01 = ζ02 = 10 μeV. Parameters of the quantum dot are the same as in 4. For other parameters, refer to the text.

Fig. 6
Fig. 6

Temporal evolution of thermalized luminescence from PbS quantum dots of different sizes excited by a Gaussian pulse. Excitation and detection frequencies correspond to the peaks in Fig. 3(b): (a) ωL = ωn, ωF = ω1; (b) ωL = ωn, ωF = ω2. The relevant eigenstates are weakly coupled via thermal interaction for R ≲ 8 nm. It was assumed that γ̂ni = 20 meV, γnn = 2 meV, and ζ1n = ζ2n = 500 μeV. The other parameters are the same as in Fig. 5.

Fig. 7
Fig. 7

The same as in Fig. 6 but for (a) ζ1n = 50 μeV, ζ2n = 500 μeV and (b) ζ1n = 500 μeV, ζ2n = 50 μeV. The relevant eigenstates are weakly coupled via thermal interaction for R ≲ 8 nm. The buildup and decay of luminescence from a 5-nm quantum dot in (a) allow one to determine relaxation constants γ22 and γ11, respectively.

Tables (1)

Tables Icon

Table 1 Time dependence of photoluminescence signal and experimentally measurable relaxation constants in different excitation regimes. It is assumed that experiments are conducted with either a single quantum dot or a quantum dot ensemble with a narrow size distribution, and that the quantum dot has two eigenstates decaying with the emission of secondary photons. The eigenstates are assumed to be excited independently of one another and the photoluminescence from each of them is meant to be measured separately.

Equations (80)

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H 0 = n h ¯ ω n | n n | + λ h ¯ ω λ c λ + c λ
H L ( t ) = E L n ϕ ( t ) V n , 0 ( L ) e i ω L t | n 0 | + H . c . ,
H V = λ n i g λ V 0 , n ( λ ) c λ + | 0 n | + H . c . ,
ρ n n t = γ n n ρ n n + n n ζ n n ρ n n + f n ( t ) ,
W n = ( E L h ¯ ) 2 | V n , 0 ( L ) | 2 2 γ ^ n i ( ω n ω L ) 2 + γ n i 2 ,
Σ ( ω L , ω λ , t ) = h ¯ 2 ω λ 4 π c 4 E L 2 n = 1 N ρ n n ( ω L , t ) W ˜ n ( ω λ ) ,
W ˜ n ( ω λ ) = | V 0 , n ( λ ) | 2 h ¯ 2 2 γ f n ( ω n ω λ ) 2 + γ f n 2
Σ ¯ ( ω L , ω F , t ) = 0 d τ Γ F e Γ F τ d ω λ Γ F / ( 2 π ) ( ω F ω λ ) 2 + ( Γ F / 2 ) 2 Σ ( ω L , ω λ , t τ ) = 1 π ( ω F c ) 4 n = 1 N | V 0 , n ( λ ) | 2 2 ( γ f n + Γ F / 2 ) ( ω n ω F ) 2 + ( γ f n + Γ F / 2 ) 2 R n ( t ) E L 2 ,
R n ( t ) = 0 ρ n n ( t τ ) Γ F e Γ F τ d τ
ρ 11 t = γ 11 ρ 11 + ζ 12 ρ 22 + W 1 ϕ 2 ,
ρ 22 t = γ 22 ρ 22 + ζ 21 ρ 11 + W 2 ϕ 2 .
ρ 11 ( t ) = t ϕ 2 ( t ) [ ( ρ W 1 + ϑ 12 W 2 ) e s 1 ( t t ) + ( q W 1 ϑ 12 W 2 ) e s 2 ( t t ) ] d t ,
ρ 22 ( t ) = t ϕ 2 ( t ) [ ( q W 2 + ϑ 21 W 1 ) e s 1 ( t t ) + ( p W 2 ϑ 21 W 1 ) e s 2 ( t t ) ] d t ,
p = γ 22 s 1 D , q = s 2 γ 22 D , ϑ i j = ζ i j D ,
D = ( γ 11 γ 22 ) 2 + 4 ζ 12 ζ 21 ,
s 1 = 1 2 ( γ 11 + γ 22 D ) , s 2 = 1 2 ( γ 11 + γ 22 + D ) .
R 1 ( t ) = η 12 + e 2 σ t H ( t ) + ( η 12 e 2 σ t + A 12 ( p ) e s 1 t + B 12 ( q ) e s 2 t ) H ( t ) ,
R 2 ( t ) = η 21 + e 2 σ t H ( t ) + ( η 21 e 2 σ t + A 21 ( q ) e s 1 t + B 21 ( p ) e s 2 t ) H ( t ) ,
η i j ± = ( γ i j ± 2 σ ) W i + ζ i j W j ( s 1 ± 2 σ ) ( s 2 ± 2 σ ) Γ F Γ F ± 2 σ ,
A i j ( r ) = 4 σ 4 σ 2 s 1 2 Γ F Γ F s 1 ( r W i + ϑ i j W j ) ,
B i j ( r ) = 4 σ 4 σ 2 s 2 2 Γ F Γ F s 2 ( r W i ϑ i j W j ) .
R 1 ( t ) = ξ 12 Q ( Γ F , t ) + C 12 ( p ) Q ( s 1 , t ) + D 12 ( q ) Q ( s 2 , t ) ,
R 2 ( t ) = ξ 21 Q ( Γ F , t ) + C 21 ( q ) Q ( s 1 , t ) + D 21 ( p ) Q ( s 2 , t ) ,
ξ i j = ( γ j j Γ F ) W i + ζ i j W j ( Γ F s 1 ) ( Γ F s 2 ) ,
C i j ( r ) = r W i + ϑ i j W j Γ F s 1 , D i j ( r ) = r W i ϑ i j W j Γ F s 2 ,
Q ( α , t ) = Γ F 2 σ exp ( α 2 8 ς 2 ) erfc ( α 4 ς 2 t 2 2 ς ) e α t .
ρ 11 t = γ 11 ρ 11 + ζ 12 ρ 22 + ζ 1 n ρ n n ,
ρ 22 t = γ 22 ρ 22 + ζ 21 ρ 11 + ζ 2 n ρ n n ,
ρ n n t = γ n n ρ n n + W n ϕ 2 .
ρ 11 ( t ) = t g ( t ) [ ( p w 1 + ϑ 12 w 2 ) e s 1 ( t t ) + ( q w 1 ϑ 12 w 2 ) e s 2 ( t t ) ] d t ,
ρ 22 ( t ) = t g ( t ) [ ( q w 2 + ϑ 21 w 1 ) e s 1 ( t t ) + ( p w 2 ϑ 21 w 1 ) e s 2 ( t t ) ] d t ,
g ( t ) = γ n n t ϕ 2 ( t ) e γ n n ( t t ) d t .
R 1 ( t ) = χ 12 + e 2 σ t H ( t ) + ( χ 12 e 2 σ t + E 12 ( p ) e s 1 t + F 12 ( q ) e s 2 t + G 12 e γ n n t ) H ( t ) ,
R 2 ( t ) = χ 21 + e 2 σ t H ( t ) + ( χ 21 e 2 σ t + E 21 ( q ) e s 1 t + F 21 ( p ) e s 2 t + G 21 e γ n n t ) H ( t ) ,
χ i j ± = γ n n γ n n ± 2 σ ( γ i j ± 2 σ ) w i + ζ i j w j ( s 1 ± 2 σ ) ( s 2 ± 2 σ ) Γ F Γ F ± 2 σ ,
E i j ( r ) = γ n n γ n n s 1 4 σ 4 σ 2 s 1 2 Γ F Γ F s 1 ( r w i + ϑ i j w j ) ,
F i j ( r ) = γ n n γ n n s 2 4 σ 4 σ 2 s 2 2 Γ F Γ F s 2 ( r w i ϑ i j w j ) ,
G i j = 4 σ γ n n γ n n 2 4 σ 2 ( γ n n γ j j ) w i ζ i j w j ( γ n n s 1 ) ( γ n n s 2 ) Γ F Γ F γ n n .
R 1 ( t ) = η 12 Q ( Γ F , t ) + K 12 ( p ) Q ( s 1 , t ) + L 12 ( q ) Q ( s 2 , t ) + M 12 Q ( γ n n , t ) ,
R 2 ( t ) = η 21 Q ( Γ F , t ) + K 21 ( q ) Q ( s 1 , t ) + L 21 ( p ) Q ( s 2 , t ) + M 21 Q ( γ n n , t ) ,
η i j = γ n n γ n n Γ F ( γ i j Γ F ) w i + ζ i j w j ( Γ F s 1 ) ( Γ F s 2 ) ,
K i j ( r ) = γ n n γ n n s 1 r w i + ϑ i j w j Γ F s 1 , L i j ( r ) = γ n n γ n n s 2 r w i ϑ i j w j Γ F s 2 ,
M i j = γ n n Γ F γ n n ( γ j j γ n n ) w i + ζ i j w j ( γ n n s 1 ) ( γ n n s 2 ) .
W ¯ n ( ω F ) = | V 0 , n ( λ ) | 2 h ¯ 2 2 ( γ f n + Γ F / 2 ) ( ω n ω F ) 2 + ( γ f n + Γ F / 2 ) 2 ,
Σ ¯ RL ( 0 ) ( ω L , ω F ) = Ξ ( W ¯ 1 γ 22 W 1 + ζ 12 W 2 γ 11 γ 22 ζ 12 ζ 21 + W ¯ 2 γ 11 W 2 + ζ 21 W 1 γ 11 γ 22 ζ 12 ζ 21 )
Σ ¯ TL ( 0 ) ( ω L , ω F ) = Ξ ( W ¯ 1 γ 22 ζ 1 n + ζ 12 ζ 2 n γ 11 γ 22 ζ 12 ζ 21 + W ¯ 2 γ 11 ζ 2 n + ζ 21 ζ 1 n γ 11 γ 22 ζ 12 ζ 21 ) W n γ n n
Σ ¯ RL ( 0 ) = Ξ ( W ¯ 1 W 1 γ 11 + W ¯ 2 W 2 γ 22 + W ¯ 1 ζ 12 W 2 γ 11 γ 22 )
Σ ¯ TL ( 0 ) = Ξ ( W ¯ 1 ζ 1 n γ 11 + W ¯ 2 ζ 2 n γ 22 + W ¯ 1 ζ 12 ζ 2 n γ 11 γ 22 ) W n γ n n .
ψ = ζ 12 ζ 21 γ 11 γ 22 = Σ ¯ RL ( 0 ) ( ω 1 , ω 2 ) Σ ¯ RL ( 0 ) ( ω 1 , ω 1 ) Σ ¯ RL ( 0 ) ( ω 2 , ω 1 ) Σ ¯ RL ( 0 ) ( ω 2 , ω 2 ) .
χ = ζ 1 n ζ 2 n γ 22 ζ 12 = φ 1 1 φ ψ ,
φ = Σ ¯ TL ( 0 ) ( ω n , ω 1 ) Σ ¯ TL ( 0 ) ( ω n , ω 2 ) Σ ¯ RL ( 0 ) ( ω 2 , ω 2 ) Σ ¯ RL ( 0 ) ( ω 2 , ω 1 ) .
γ 11 ( x ) = ζ 01 + ζ 12 e x , γ 22 = ζ 02 + ζ 12 .
a = Σ ¯ RL ( 0 ) ( ω 1 , ω 1 , x 2 ) Σ ¯ RL ( 0 ) ( ω 1 , ω 1 , x 1 ) = γ 11 ( x 1 ) γ 22 ζ 12 2 e x 1 γ 11 ( x 2 ) γ 22 ζ 12 2 e x 2 ,
b = Σ ¯ RL ( 0 ) ( ω 2 , ω 2 , x 1 ) Σ ¯ RL ( 0 ) ( ω 1 , ω 1 , x 1 ) Σ ¯ RL ( 0 ) ( ω 1 , ω 1 , x 2 ) Σ ¯ RL ( 0 ) ( ω 2 , ω 2 , x 2 ) = γ 11 ( x 1 ) γ 11 ( x 2 ) .
γ 11 ( x ) = ( e x + e x 1 b e x 2 b 1 ) ζ 12 ,
γ 22 = b 1 b a e x 1 a e x 2 e x 1 e x 2 ζ 12 .
Σ ¯ RL ( i ) Σ ¯ RL ( 0 ) e 2 σ | t | , Σ ¯ TL ( i ) Σ ¯ TL ( 0 ) e 2 σ | t | ,
erfc ( α 4 ς 2 t 2 2 ς ) = exp [ ( α 4 ς 2 t ) 2 8 ς 2 ] ( 2 ς α 2 π + O [ ( ς / α ) 2 ] ) ,
Σ ¯ RL ( i ) Σ ¯ RL ( 0 ) A e 2 ς 2 t 2 , Σ ¯ TL ( i ) Σ ¯ TL ( 0 ) A e 2 ς 2 t 2 .
q p 1 , s 1 γ 11 , s 2 γ 22 ,
E 21 = 3 π 2 h ¯ 2 2 μ R 2 ,
Σ ¯ RL ( ii ) Ξ σ { [ W ¯ 1 ( p W 1 + ϑ 12 W 2 ) + W ¯ 2 ( q W 2 + ϑ 21 W 1 ) ] e s 1 t + [ W ¯ 1 ( q W 1 ϑ 12 W 2 ) + W ¯ 2 ( p W 2 ϑ 21 W 1 ) ] e s 2 t } .
Σ ¯ RL ( ii ) { p e s 1 t + q e s 2 t for ω L , ω F ω 1 , q e s 1 t + p e s 2 t for ω L , ω F ω 2 , e s 1 t e s 2 t for ω L ω 1 ( 2 ) , ω F ω 2 ( 1 ) ,
s 1 1 ψ 1 / γ 11 + 1 / γ 22 s 2 γ 11 + γ 22 .
γ 11 = 1 2 ( s 1 + s 2 ( s 1 s 2 ) 2 4 ζ 12 ζ 21 ) ,
γ 22 = 1 2 ( s 1 + s 2 + ( s 1 s 2 ) 2 4 ζ 12 ζ 21 ) .
γ 11 = 1 2 ( s 1 + s 2 ( s 1 + s 2 ) 2 4 s 1 s 2 / ( 1 ψ ) ) ,
γ 22 = 1 2 ( s 1 + s 2 + ( s 1 + s 2 ) 2 4 s 1 s 2 / ( 1 ψ ) ) .
Σ ¯ TL ( ii ) Ξ σ { W ¯ 1 ( p ζ 1 n + ϑ 12 ζ 2 n ) + W ¯ 2 ( q ζ 2 n + ϑ 21 ζ 1 n ) γ n n s 1 e s 1 t + W ¯ 1 ( q ζ 1 n ϑ 12 ζ 2 n ) + W ¯ 2 ( p ζ 2 n ϑ 21 ζ 1 n ) γ n n s 2 e s 2 t + W ¯ 1 [ ( γ 22 γ n n ) ζ 1 n + ζ 12 ζ 2 n ] + W ¯ 2 [ ( γ 11 γ n n ) ζ 2 n + ζ 21 ζ 1 n ] ( γ n n s 1 ) ( γ n n s 2 ) e γ n n t } W n .
Σ ¯ TL ( ii ) { ( p μ 1 + ϑ 12 μ 2 ) e s 1 t + ( q μ 1 ϑ 12 μ 2 ) e s 2 t for ω F ω 1 , ( q μ 2 + ϑ 21 μ 1 ) e s 1 t + ( p μ 2 ϑ 21 μ 1 ) e s 2 t for ω F ω 2 ,
t m = 1 s 2 s 1 × { ln ( s 2 s 1 ϑ 12 ζ 2 n q ζ 1 n ϑ 12 ζ 2 n + p ζ 1 n ) for ω F ω 1 , ln ( s 2 s 1 ϑ 21 ζ 1 n q ζ 2 n ϑ 21 ζ 1 n + p ζ 2 n ) for ω F ω 2 .
Σ ¯ RL ( iii ) Σ ¯ RL ( 0 ) e Γ F t , Σ ¯ TL ( iii ) Σ ¯ TL ( 0 ) e Γ F t .
ρ n n t + n = 1 N a n n ρ n n = f n ( t ) ,
n = 1 N b n ρ n n t + n = 1 N ρ n n n = 1 N b n a n n = n = 1 N b n f n ( t ) .
n = 1 N b n a n n = s n b n .
a ^ n n = ( γ 11 ζ 12 e E 21 / T ζ 13 e E 31 / T ζ 1 N e E N 1 / T ζ 12 γ 22 ζ 23 e E 32 / T ζ 2 N e E N 2 / T ζ 13 ζ 23 γ 33 ζ 3 N e E N 3 / T ζ 1 N ζ 2 N ζ 3 N γ N N )
| a ^ n n | > n n | a ^ n n | .
x n = n = 1 N b n ( n ) ρ n n ,
x n t + s n x n = n = 1 N b n ( n ) f n ( t ) .
x n ( t ) = n = 1 N b n ( n ) t f n ( τ ) e s n ( t τ ) d τ .

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