## 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 [1–3]. The unique optical properties [4–7] 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 [8–10].

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 [11–14], 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 [15–21], 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 [24–29]. The relaxation may also occur *via* the Auger process [30], nonradiative energy transfer [31–33], or through the interaction of the quantum dot’s electronic subsystem with surface defects [34–36]. 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 [41–44]. 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 [45–49] should be pointed out as a multipurpose one. Several experimental schemes [50–52] 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 [53–56].

Despite the fact that the relaxation mechanisms of quantum dots have been a subject of intense scholarly research over the past two decades [34,57–62], 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*) = *H*_{0} + *H _{L}*(

*t*)+

*H*, which is the sum of the Hamiltonian

_{V}*H*

_{0}describing noninteracting electron–hole pairs and vacuum radiation field, and the Hamiltonians

*H*(

_{L}*t*) and

*H*governing generation and annihilation of the pairs due to the interaction with classical excitation light and quantum radiation field. The Hamiltonian

_{V}*h*̄

*ω*and eigenvectors |

_{n}*n*〉 of the electron–hole pairs, and through the energies

*h*̄

*ω*and creation ( ${c}_{\lambda}^{+}$) and annihilation (

_{λ}*c*) operators of photons. The summations in

_{λ}*H*

_{0}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

*H*(

_{L}*t*) depends on the parameters of the laser pulse, including polarization

**e**

*, carrier frequency*

_{L}*ω*, and envelope

_{L}*ϕ*(

*t*), and may be written in the rotation wave approximation as

*E*is the electric field amplitude, ${V}_{n,0}^{\left(L\right)}=\u3008n\left|\mathbf{p}{\mathbf{e}}_{L}\right|0\u3009$ is the matrix element of the dipole moment operator

_{L}**p**= −

*e*

**r**(−

*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

**e**

_{λ}of emitted photons and the matrix elements ${V}_{0,n}^{\left(\lambda \right)}=\u30080\left|\mathbf{p}{\mathbf{e}}_{\lambda}\right|n\u3009$; ${g}_{\lambda}=\sqrt{2\pi \overline{h}{\omega}_{\lambda}/\left({\varepsilon}_{\infty}V\right)}$, 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 *γ _{nn}*

_{′}of optical transitions, so that the effects of coherence relaxation may be safely neglected. Since

*γ*

_{nn}_{′}≳ 10

^{13}s

^{−1}at room (or higher) temperature [64–66], we restrict ourself to the consideration of optical pulses with

*σ*≪ 10

^{13}s

^{−1}. If

*ζ*

_{nn′}is the rate of transitions from the state |

*n*′〉 to the state |

*n*〉 due to the thermal interaction with a bath and

*γ*is the inverse lifetime of the state |

_{nn}*n*〉, then the kinetic Pauli equation is of the form

*f*(

_{n}*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〉|0* _{λ}*〉 and |

*f*〉〉 = |0〉|1

*〉. During its evolution between these states, the system may be found in one of*

_{λ}*N*intermediate states |

*n*〉〉 = |

*n*〉|0

*〉, with zero emitted photons and an electron–hole pair in the state |*

_{λ}*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

*f*(

_{n}*t*) =

*W*

_{n}ϕ^{2}(

*t*), where

*γ*=

_{ni}*γ*/2 +

_{nn}*γ*̂

*is the total dephasing rate of the optical transition |*

_{ni}*i*〉〉 → |

*n*〉〉, and

*γ*̂

*is the pure dephasing rate.*

_{ni}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 *ρ _{nn}*(

*ω*,

_{L}*t*) determine the time evolution of photoluminescence, the intensity of which is characterized by the differential cross section [67]

*n*〉〉 → |

*f*〉〉, with the total dephasing rate

*γ*≫

_{fn}*σ*.

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 [68–70]. 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]

*and*

_{F}*ω*are the bandpass and central frequencies of the interferometer, and

_{F}*n*th energy level to the signal of photoluminescence. It may be noted that the ratio ${R}_{n}\left(t\right)/{E}_{L}^{2}$ 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 *ζ*_{1n}, *ζ*_{2n}, or *ζ*_{3n}, 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 *E _{ij}* =

*h*̄(

*ω*−

_{i}*ω*), or decays to the ground state |0〉 with or without the emission of a photon in mode

_{j}*λ*. If the temperature

*T*of the system expressed in energy units is comparable to, or larger than,

*E*, then transitions |

_{ij}*i*〉 ⇆ |

*j*〉 may occur at sufficiently high rates

*ζ*and

_{ji}*ζ*=

_{ij}*ζ*exp(−

_{ji}*E*/

_{ij}*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 ${\rho}_{nn}\left({\omega}_{L},t\right)={\sum}_{n=1}^{N}{A}_{n}\text{exp}\left(-{s}_{n}t\right)$, where

*A*(

_{n}*n*= 1, 2,...,

*N*) are the time-independent coefficients and

*s*are the modified energy relaxation rates.

_{n}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

*γ*

_{11}and

*γ*

_{22}are always larger than the respective thermal relaxation rates

*ζ*

_{21}and

*ζ*

_{12}, 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

*ζ*

_{12}

*ζ*

_{21}<

*γ*

_{11}

*γ*

_{22}, which implies that the parameters

*s*

_{1}and

*s*

_{2}are positive. Also noteworthy is the inequality

*s*

_{2}>

*γ*

_{22}>

*γ*

_{11}>

*s*

_{1}, 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
$\varphi \left(t\right)=\sqrt{A}\text{exp}\left(-{\varsigma}^{2}{t}^{2}\right)$. The constants
$A=2/\sqrt{\pi \text{ln}2}$ and
$\varsigma =\sigma \sqrt{2/\text{ln}2}$ 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

*H*(

*t*) is the Heaviside step function,

*Q*(

*α*,

*t*) may be approximated by the exponential (Γ

*/*

_{F}*σ*)exp[

*α*

^{2}/(8

*ς*

^{2})]

*e*

^{−αt}for a time $t\ge \alpha /{\left(2\varsigma \right)}^{2}+\sqrt{2}/\varsigma $.

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 (*e*^{−s1t} and *e*^{−s2t}) 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

*Q*(Γ

*,*

_{F}*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 *h*̄*ω _{n}* of the state |

*n*〉 is such that

*h*̄(

*ω*−

_{n}*ω*) ≫

_{j}*T*for

*j*= 1, 2, we arrive at the following equations governing the populations of the three states:

*w*= (

_{j}*ζ*/

_{jn}*γ*)

_{nn}*W*and

_{n}*e*

^{−γnnt}and ∝

*Q*(

*γ*,

_{nn}*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) *σ* ≪ *γ*, adiabatic excitation; (ii) *σ* ≫ *γ*, instantaneous excitation; and (iii) *σ* ∼ *γ*, 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, *γ* ∼ *s*_{1} for regime (i) and *γ* ∼ *s*_{2} 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).

#### 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
$\mathrm{\Xi}={\overline{h}}^{2}{\omega}_{F}^{4}/\left(\pi {c}^{4}{E}_{L}^{2}\right)$ and

*γ*

_{1i},

*γ*

_{2i}, and

*γ*), while the relative peak intensities contain information about the relaxation constants.

_{ni}The probability of transition |1〉 → |2〉 becomes vanishingly small at low temperatures (*T* ≪ *E*_{21}), for which Eqs. (24) and (25) reduce to the well-known formulas [16, 31, 67]

By closely looking at Eq. (24), it is seen that the spectrum of stationary resonant luminescence allows one to reliably determine the ratio *ζ*_{12}*ζ*_{21}/(*γ*_{11}*γ*_{22}), 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 *E*_{21} ≫ max(*γ*_{1i}, *γ*_{2i}, *γ*_{f1}, *γ*_{f2}), 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

*ω*≈

_{L}*ω*, 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

_{F}*W*

_{1}and

*W*

_{2}), and for the

*j*th eigenstate lie within the range Γ

*≪ |*

_{F}*ω*−

_{L}*ω*| ≲

_{j}*γ*. This inequality sets the upper limit for the filter’s bandpass in stationary experiments, Γ

_{ji}*≪ min(*

_{F}*γ*

_{1i},

*γ*

_{2i}).

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

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 *ζ*_{21}/*ζ*_{12} = *e*^{−E21/T}. By denoting the recombination rates as *ζ*_{01} and *ζ*_{02} and introducing a size-dependent parameter *x* = *E*_{21}/*T*, we may write

*x*,

*x*

_{2}>

*x*

_{1}, gives in accordance with Eq. (24)

*ζ*

_{01}and

*ζ*

_{02}, we arrive at the following relations between the relaxation rates:

*ζ*

_{1n}/

*ζ*

_{2n}.

#### 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 ∝ *ϕ*^{2}(*t*). Indeed, assuming in Eqs. (13), (14), (20), and (21) that Γ* _{F}* ≫

*s*

_{2}and

*s*

_{1}≫

*σ*, yields

*α*≫

*ς*, we obtain from Eqs. (15), (16), (22) and (23) for the Gaussian pulse

*γ*

_{11}and

*γ*

_{22}, 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 *γ*_{22} −*γ*_{11} ≫ 2*ζ*_{12} exp[−*E*_{21}/(2*T*)]. This is because in this case

To illustrate the dependence of parameters *s*_{1} and *s*_{2} 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: *n*_{1} = *n*_{e1} = *n*_{h1} = 1, *l*_{1} = *l*_{e1} = *l*_{h1} =0, *n*_{2} = *n*_{e2} = *n*_{h2} = 2, *l*_{2} = *l*_{e2} = *l*_{h2} = 0. Then the energy gap between them is given by the expression [1, 67]

*μ*=

*m*/(

_{e}m_{h}*m*+

_{e}*m*) is the reduced mass of electron and hole. Suppose also that the rates

_{h}*γ*

_{11}and

*γ*

_{22}are in the form of Eq. (30). Figure 4 shows how relaxation parameters

*s*

_{1}and

*s*

_{2}change with

*R*and the ratio

*ζ*

_{02}/

*ζ*

_{01}for a PbS quantum dot with

*m*=

_{e}*m*= 0.25

_{h}*m*

_{0}(

*m*

_{0}is the free-electron mass) at room temperature. It is seen that if

*ζ*

_{01}≫

*ζ*

_{02}, then

*s*

_{1}≈ min(

*ζ*

_{12},

*ζ*

_{01}) and

*s*

_{2}≈ max(

*ζ*

_{12},

*ζ*

_{01}) for

*R*< 7 nm, while for

*R*> 10 nm the values of

*s*

_{1}and

*s*

_{2}steeply diverge with the radius. In the case of equal recombination rates,

*ζ*

_{01}=

*ζ*

_{02}, we find that

*s*

_{1}=

*ζ*

_{01}is independent of

*R*and

*s*

_{2}=

*γ*

_{22}+

*ζ*

_{12}

*e*

^{−x}increases from

*γ*

_{22}in small quantum dots to

*γ*

_{22}+

*ζ*

_{12}in large ones. If

*ζ*

_{02}≫

*ζ*

_{01}, then

*s*

_{1}≈ min(

*ζ*

_{12},

*ζ*

_{02}) and

*s*

_{2}≈ max(

*ζ*

_{12},

*ζ*

_{02}) for quantum dots with

*R*< 7 nm. Since

*ζ*

_{12}∼

*ζ*

_{01}for our example in Fig. 4, this result implies that

*s*

_{2}≫

*s*

_{1}. 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.

### 5.3.1. Kinetics of resonant luminescence

Suppose now that Γ* _{F}* ≫

*σ*≫

*s*

_{2}. 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

*W*(

_{j}*ω*

_{1}) ∼

*W*(

_{j}*ω*

_{2}) or

*W̄*(

_{j}*ω*

_{1}) ∼

*W̄*(

_{j}*ω*

_{2}) for

*j*= 1, 2]. Specifically, the decay of luminescence is governed by the functions

*ω*≠

_{L}*ω*, 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 (

_{F}*t*) at which the signal of luminescence peaks, we obtain a relation (

_{m}*s*

_{2}−

*s*

_{1})

*t*= ln(

_{m}*s*

_{2}/

*s*

_{1}). This relation may be used to calculate

*s*

_{2}in the event that the temporal resolution of the detector is insufficient to determine

*s*

_{2}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

*σ*≪

*t*≲ 1/

*s*

_{2}and

*t*≳ 1/

*s*

_{1}. This allows one to experimentally measure

*s*

_{1}and

*s*

_{2}and—if the transition rate

*ζ*

_{12}is known—to calculate the relaxation constants

*ζ*

_{12}, one may determine parameter

*ψ*in the regime of stationary excitation using Eq. (28), and then find

*s*

_{1}and

*s*

_{2}from any single time dependency in Eq. (33) is only possible when

*p*∼

*q*. It should be also recognized that

*s*

_{2}is always much larger than

*s*

_{1}in the regime of dominant thermal interaction (1 −

*ψ*≪ 1), however, the inequality

*s*

_{2}≫

*s*

_{1}itself does not necessarily imply that the interaction with the bath gives major contribution to the eigenstates’ lifetimes. For instance,

*s*

_{2}≫

*s*

_{1}for

*ψ*≪ 1 and

*γ*

_{22}≫

*γ*

_{11}.

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 *γ _{jj}* when the state is excited resonantly (

*i.e.*,

*ω*≈

_{L}*ω*). 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 (

_{j}*t ∼*1/

*σ*≪ 1/

*γ*

_{11}, 1/

*γ*

_{22}).

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〉 = |*n*_{2} = 2, *l*_{2} = 0〉 to state |1〉 = |*n*_{1} = 1, *l*_{1} = 0〉, by setting *ζ*_{01} = *ζ*_{02} = *ζ*_{12}/10 = 10 *μ*eV. Owing to this choice, the lifetime of the low-energy state and ratio *q* = *e*^{−}* ^{x}*/(1 +

*e*

^{−x}) 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 (

*E*

_{21}≈ 361 meV) to 4.95 for

*R*= 30 nm (

*E*

_{21}≈ 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
$\left|{V}_{1,0}^{L}\right|=\left|{V}_{2,0}^{L}\right|=\left|{V}_{0,1}^{\lambda}\right|=\left|{V}_{0,2}^{\lambda}\right|=1$ and is normalized by a factor of
${\omega}_{F}^{4}/\left(\pi {\overline{h}}^{2}{c}^{4}\right)$. Even though *s*_{2} ≫ *s*_{1} and *p* ∼ *q* 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 *τ*_{1} = 1/*s*_{1} = 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 *E*_{21} ∼ *γ _{ji}*,

*γ*(

_{fj}*j*= 1, 2). When

*R*is reduced, the energy gap

*E*

_{21}grows larger, resulting in: (i) luminescence being described by Eq. (33); (ii)

*p*≫

*q*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

*τ*

_{1}when

*ω*=

_{L}*ω*=

_{F}*ω*

_{1}[see Fig. 5(a)]. For

*ω*=

_{L}*ω*=

_{F}*ω*

_{2}, the decrease in

*q*makes the decay ∝

*qe*

^{−γ11t}+

*pe*

^{−γ22t}being characterized by the smaller time constant,

*τ*

_{2}= 1/

*γ*

_{22}≈ 9 meV

^{−1}[see Fig. 5(b)]. If the pulse resonantly excites the high-energy eigenstate (

*ω*=

_{L}*ω*

_{2}) and the signal is detected at the frequency of the low-energy eigenstate (

*ω*=

_{F}*ω*

_{1}), then the luminescence from small quantum dots peaks at a time

*t*≈ (1/

_{m}*γ*

_{22}) ln(

*γ*

_{22}/

*γ*

_{11}) ≫ 1/

*σ*. This is clearly seen from Fig. 5(c), where the buildup of the luminescence intensity, governed by the function 1 −

*e*

^{−γ22t}(red curve), is overcome by the decay, proportional to

*e*

^{−γ11t}(blue curve), at

*t*≈ 24 meV

_{m}^{−1}. Since transitions |1〉 → |2〉 become extremely improbable in small quantum dots, the luminescence from state |2〉 excited at frequency

*ω*=

_{L}*ω*

_{1}steeply diminishes with the reduction of

*R*below 15 nm, as shown in Fig. 5(d). Hence, in the example considered,

*γ*

_{11}and

*γ*

_{22}may be calculated precisely by analyzing the kinetics of the resonant luminescence.

### 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 Γ* _{F}* ≫

*σ*≫

*γ*(

_{nn}*γ*>

_{nn}*γ*

_{22}), these equations for

*t*≫ 1/

*σ*are reduced to

*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 *σ* ≫ *γ _{nn}* ≫

*s*

_{2}, as in this case the relaxation constant

*γ*may be reliably determined from the luminescence spectrum for a registration time 1/

_{nn}*σ*≪

*t*≲ 1/

*γ*, whereas for a time

_{nn}*t*≫ 1/

*γ*only two exponentials are contributing to the overall signal, namely,

_{nn}*μ*=

_{j}*ζ*/

_{jn}*γ*. Again, if the thermal interaction between the two relevant eigenstates is so strong that

_{nn}*s*

_{2}≫

*s*

_{1}, then

*s*

_{1}and

*s*

_{2}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

*s*

_{1}∼

*s*

_{2}), but only if the rates

*ζ*

_{12}and

*ζ*

_{21}are known. The signal of thermalized luminescence from eigenstate |2〉 decays in this regime according to

*γ*

_{22}[see Eq. (38) with

*q*,

*ϑ*

_{21}≪ 1], and the remaining relaxation constant is readily found form Eq. (28),

*γ*

_{11}= (

*ζ*

_{12}

*ζ*

_{21})/(

*γ*

_{22}

*ψ*). Furthermore, since the parameter

*χ*is measurable with the stationary excitation [see Eq. (29)], knowing

*γ*

_{11}and

*γ*

_{22}enables the calculation of the relaxation rates ratio

*ζ*

_{1n}/

*ζ*

_{2n}.

Similar to the case of resonant luminescence, the signal of thermalized luminescence may peak long after the excitation pulse dies away. This occurs for *ω _{F}* ≈

*ω*

_{1}when

*ϑ*

_{12}

*ζ*

_{2}

*>*

_{n}*qζ*

_{1n}and for

*ω*≈

_{F}*ω*

_{2}when

*ϑ*

_{21}

*ζ*

_{1n}>

*pζ*

_{2}

*. Using Eq. (37), we find the time corresponding to the luminescence peak to be*

_{n}It is readily seen that *ϑ*_{12} > *q* and *p* > *ϑ*_{21} for relaxation constants in the form of Eq. (30) and equal rates of radiative recombination (*ζ*_{01} = *ζ*_{02}). The signal of thermalized luminescence from the lower eigenstate will therefore always peak in this situation (for *t* ≫ 1/*σ*), should *ζ*_{1}* _{n}* =

*ζ*

_{2}

*.*

_{n}The value of *γ _{nn}* normally exceeds 10

^{11}s

^{−1}at room temperature [58–60], so that the condition

*σ*≫

*γ*is satisfied for pulses shorter than 100 ps. For much longer pulses, an opposite relationship holds (

_{nn}*γ*≫

_{nn}*σ*≫

*s*

_{2}) and the signal of thermalized luminescence is seen to be described by Eq. (38) for

*t*≫ 1/

*σ*if

*γ*≪ Γ

_{nn}*. Hence, only the quantities*

_{F}*ψ*and

*χ*may be found from the stationary experiment, and

*γ*(

_{jj}*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 *n _{n}* =

*n*

_{en}=

*n*

_{hn}= 9 and

*l*=

_{n}*l*

_{en}=

*l*

_{hn}= 0, in which case [1, 67]

*E*

_{n}_{1}≈

*E*

_{n}_{2}≈ 26

*E*

_{21}. 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 (

*τ*

_{1}), but the intensity of thermalized luminescence is reduced by a factor of

*γ*/

_{nn}*ζ*

_{1}

*= 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*

_{n}^{−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/

*γ*

_{11}≈ 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/

*γ*

_{22}≈ 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.

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 *γ*_{22}. The decay of luminescence from the lower state is then governed by the exponential *e*^{−γ11t} and may be used to find *γ*_{11}. The described situation is illustrated by Fig. 7(a), where it is assumed that *ζ*_{2n} = 10*ζ*_{1n} = 500 *μ*eV. 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 *ζ*_{1n} and *ζ*_{2n} (*i.e.*, *ζ*_{1n} = 10*ζ*_{2n} = 500 *μ*eV), 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).

#### 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* ≫ Γ* _{F}*). In this case, Eqs. (15) to (17), (22), and (23) for

*t*≫ 1/

*σ*give

## 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.

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 [72–74]. 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

*a*

_{nn}_{′}=

*γ*

_{nn}δ_{nn}_{′}−

*ζ*

_{nn}_{′}(1 −

*δ*

_{nn}_{′}),

*n*= 1, 2,...,

*N*, and the function

*f*(

_{n}*t*) is assumed to tend to zero sufficiently fast at minus infinity in time. Multiplying Eq. (A.1) by an arbitrary constant

*b*and adding equations corresponding to different

_{n}*n*termwise, yield

*b*are such that It is crucial for the subsequent discussion that all

_{n}*N*roots

*s*of the characteristic polynomial of Eq. (A.3) are positive. The positiveness of the roots follows from two observations. First, the coefficient matrix

_{n}*γ*of electron–hole state |

_{nn}*n*〉 is determined by all possible relaxation channels (including those leading to the annihilation of electron–hole pairs) and, therefore, always exceeds the sum

*ζ*

_{1n}+

*ζ*

_{2n}+ ... +

*ζ*

_{n}_{−1,n}+

*ζ*

_{n}_{+1,n}+ ... +

*ζ*of relaxation rates within the channels occurring due solely to thermal fluctuations. Second, the matrix

_{Nn}*â*

_{nn}_{′}can be rewritten in the symmetric form by simply multiplying its

*j*th row (

*j*≥ 2) by the factor

*e*

^{−Ej1/T}[note that a similar transformation of Eq. (A.3) does not change its roots]. Being essentially symmetric, the matrix

*â*

_{nn}_{′}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

*b*corresponding to the root

_{n}*s*. As a result, Eq. (A.2) takes the form

_{n}*t*→ −∞ is found to be

## 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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