Kinetics of pulse-induced photoluminescence from a semiconductor quantum dot

doi:10.1364/OE.20.027612

Kinetics of pulse-induced photoluminescence from a semiconductor quantum dot

Ivan D. Rukhlenko, Mikhail Yu. Leonov, Vadim K. Turkov, Aleksandr P. Litvin, Anvar S. Baimuratov, Alexander V. Baranov, and Anatoly V. Fedorov »View Author Affiliations

Optics Express, Vol. 20, Issue 25, pp. 27612-27635 (2012)
http://dx.doi.org/10.1364/OE.20.027612

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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.

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

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–3

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]. The unique optical properties [4

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] 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

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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

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], 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

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]

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]

X.-Q. Li, H. Nakayama, and Y. Arakawa, “Phonon bottleneck in quantum dots: Role of lifetime of the confined optical phonons,” Phys. Rev. B 59, 5069–5073 (1999). [CrossRef]

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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]

], two [22

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], or several [23

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] phonons, as well as the decay of the excited state accompanied by the excitation of plasmon or plasmon-phonon modes [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]

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).

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]

–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]

]. The relaxation may also occur via the Auger process [30

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. B 74, 075403 (2006). [CrossRef]

], nonradiative energy transfer [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]

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

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], or through the interaction of the quantum dot’s electronic subsystem with surface defects [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]

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]

]. 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

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]

]. 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

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]

], photon echo [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.

], and the coherent control of secondary emission [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]

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]

–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]

]. 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

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]

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]

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]

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]

] should be pointed out as a multipurpose one. Several experimental schemes [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]

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]

] 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

K. Rivoire, S. Buckley, Y. Song, M. L. Lee, and J. Vučković, “Photoluminescence from In_0.5Ga_0.5As/GaP quantum dots coupled to photonic crystal cavities,” Phys. Rev. B 85, 045319 (2012). [CrossRef]

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]

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]

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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]

Despite the fact that the relaxation mechanisms of quantum dots have been a subject of intense scholarly research over the past two decades [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]

,57

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

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]

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]

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]

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]

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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]

], 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

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

] 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 + H_L(t)+ H_V, which is the sum of the Hamiltonian H₀ describing noninteracting electron–hole pairs and vacuum radiation field, and the Hamiltonians H_L(t) and H_V governing generation and annihilation of the pairs due to the interaction with classical excitation light and quantum radiation field. The Hamiltonian

H_{0} = \sum_{n} \bar{h} ω_{n} | n 〉 〈 n | + \sum_{λ} \bar{h} ω_{λ} c_{λ}^{+} c_{λ}

(1)

is expressed through the energies h̄ω_n and eigenvectors |n〉 of the electron–hole pairs, and through the energies h̄ω_λ and creation (

c_{λ}^{+}

) and annihilation (c_λ) 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 H_L(t) depends on the parameters of the laser pulse, including polarization e_L, carrier frequency ω_L, and envelope ϕ(t), and may be written in the rotation wave approximation as

H_{L} (t) = E_{L} \sum_{n} ϕ (t) V_{n, 0}^{(L)} e^{- i ω_{L} t} | n 〉 〈 0 | + H . c .,

(2)

where E_L is the electric field amplitude,

V_{n, 0}^{(L)} = 〈 n | p e_{L} | 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

H_{V} = \sum_{λ} \sum_{n} i g_{λ} V_{0, n}^{(λ)} c_{λ}^{+} | 0 〉 〈 n | + H . c .,

(3)

which depends on the polarization e_λ of emitted photons and the matrix elements

V_{0, n}^{(λ)} = 〈 0 | p e_{λ} | n 〉

;

g_{λ} = \sqrt{2 π \bar{h} ω_{λ} / (ɛ_{\infty} V)}

, 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

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

]. 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_′ ≳ 10¹³ s⁻¹ at room (or higher) temperature [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]

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]

], we restrict ourself to the consideration of optical pulses with σ ≪ 10¹³ 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_nn is the inverse lifetime of the state |n〉, then the kinetic Pauli equation is of the form

\frac{\partial ρ_{n n}}{\partial t} = - γ_{n n} ρ_{n n} + \sum_{n^{'} \neq n} ζ_{n n^{'}} ρ_{n^{'} n^{'}} + f_{n} (t),

(4)

where the function 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_λ〉. During its evolution between these states, the system may be found in one of N intermediate states |n〉〉 = |n〉|0_λ〉, 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 f_n(t) = W_nϕ²(t), where

W_{n} = {(\frac{E_{L}}{\bar{h}})}^{2} {| V_{n, 0}^{(L)} |}^{2} \frac{2 {\hat{γ}}_{n i}}{{(ω_{n} - ω_{L})}^{2} + γ_{n i}^{2}},

(5)

γ_ni = γ_nn/2 + γ̂_ni is the total dephasing rate of the optical transition |i〉〉 → |n〉〉, and γ̂_ni 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 ρ_nn(ω_L,t) determine the time evolution of photoluminescence, the intensity of which is characterized by the differential cross section [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]

]

Σ (ω_{L}, ω_{λ}, t) = \frac{{\bar{h}}^{2} ω_{λ}^{4}}{π c^{4} E_{L}^{2}} \sum_{n = 1}^{N} ρ_{n n} (ω_{L}, t) {\tilde{W}}_{n} (ω_{λ}),

(6)

where

{\tilde{W}}_{n} (ω_{λ}) = \frac{{| V_{0, n}^{(λ)} |}^{2}}{{\bar{h}}^{2}} \frac{2 γ_{f n}}{{(ω_{n} - ω_{λ})}^{2} + γ_{f n}^{2}}

(7)

gives the probability of the transition |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

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]

G. Nienhuis, “Time and frequency dependence of nearly resonant light scattered from collisionally perturbed atoms,” Physica C 96, 391–409 (1979). [CrossRef]

–70

J. H. Eberly and K. Wódkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am. 67, 1252–1261 (1977). [CrossRef]

]. 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

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]

, 69

G. Nienhuis, “Time and frequency dependence of nearly resonant light scattered from collisionally perturbed atoms,” Physica C 96, 391–409 (1979). [CrossRef]

]

\begin{array}{l} \bar{Σ} (ω_{L}, ω_{F}, t) = \int_{0}^{\infty} d τ Γ_{F} e^{- Γ_{F} τ} \int_{- \infty}^{\infty} d ω_{λ} \frac{Γ_{F} / (2 π)}{{(ω_{F} - ω_{λ})}^{2} + {(Γ_{F} / 2)}^{2}} Σ (ω_{L}, ω_{λ}, t - τ) \\ = \frac{1}{π} {(\frac{ω_{F}}{c})}^{4} \sum_{n = 1}^{N} {| V_{0, n}^{(λ)} |}^{2} \frac{2 (γ_{f n} + Γ_{F} / 2)}{{(ω_{n} - ω_{F})}^{2} + {(γ_{f n} + Γ_{F} / 2)}^{2}} \frac{R_{n} (t)}{E_{L}^{2}}, \end{array}

(8)

where Γ_F and ω_F are the bandpass and central frequencies of the interferometer, and

R_{n} (t) = \int_{0}^{\infty} ρ_{n n} (t - τ) Γ_{F} e^{- Γ_{F} τ} d τ

(9)

is a dimensionless parameter determining the contribution of the nth energy level to the signal of photoluminescence. It may be noted that the ratio

R_{n} (t) / 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 = h̄(ω_i − ω_j), 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, E_ij, then transitions |i〉 ⇆ |j〉 may occur at sufficiently high rates ζ_ji and ζ_ij = ζ_ji exp(−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

ρ_{n n} (ω_{L}, t) = \sum_{n = 1}^{N} A_{n} exp (- s_{n} t)

, where A_n (n = 1, 2,..., N) are the time-independent coefficients and s_n 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; ω_{L_k} (ω_L) and ω_{λ_k} (k = 1, 2, 3) are the excitation frequencies and the frequencies of emitted photons; ζ_kk_′ 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.

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

\frac{\partial ρ_{11}}{\partial t} = - γ_{11} ρ_{11} + ζ_{12} ρ_{22} + W_{1} ϕ^{2},

(10a)

\frac{\partial ρ_{22}}{\partial t} = - γ_{22} ρ_{22} + ζ_{21} ρ_{11} + W_{2} ϕ^{2} .

(10b)

The solution to this system may be written in the form (see Appendix)

ρ_{11} (t) = \int_{- \infty}^{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^{'},

(11a)

ρ_{22} (t) = \int_{- \infty}^{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^{'},

(11b)

where

p = \frac{γ_{22} - s_{1}}{D}, q = \frac{s_{2} - γ_{22}}{D}, ϑ_{i j} = \frac{ζ_{i j}}{D},

(12a)

D = \sqrt{{(γ_{11} - γ_{22})}^{2} + 4 ζ_{12} ζ_{21}},

(12b)

s_{1} = \frac{1}{2} (γ_{11} + γ_{22} - D), s_{2} = \frac{1}{2} (γ_{11} + γ_{22} + D) .

(12c)

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) = \sqrt{A} exp (- ς^{2} t^{2})

. The constants

A = 2 / \sqrt{π ln 2}

and

ς = σ \sqrt{2 / 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

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),

(13a)

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),

(13b)

where H(t) is the Heaviside step function,

η_{i j}^{\pm} = \frac{(γ_{i j} \pm 2 σ) W_{i} + ζ_{i j} W_{j}}{(s_{1} \pm 2 σ) (s_{2} \pm 2 σ)} \frac{Γ_{F}}{Γ_{F} \pm 2 σ},

(14a)

A_{i j}^{(r)} = \frac{4 σ}{4 σ^{2} - s_{1}^{2}} \frac{Γ_{F}}{Γ_{F} - s_{1}} (r W_{i} + ϑ_{i j} W_{j}),

(14b)

B_{i j}^{(r)} = \frac{4 σ}{4 σ^{2} - s_{2}^{2}} \frac{Γ_{F}}{Γ_{F} - s_{2}} (r W_{i} - ϑ_{i j} W_{j}) .

(14c)

For the Gaussian pulse, the same equations give

R_{1} (t) = ξ_{12} Q (Γ_{F}, t) + C_{12}^{(p)} Q (s_{1}, t) + D_{12}^{(q)} Q (s_{2}, t),

(15a)

R_{2} (t) = ξ_{21} Q (Γ_{F}, t) + C_{21}^{(q)} Q (s_{1}, t) + D_{21}^{(p)} Q (s_{2}, t),

(15b)

where

ξ_{i j} = \frac{(γ_{j j} - Γ_{F}) W_{i} + ζ_{i j} W_{j}}{(Γ_{F} - s_{1}) (Γ_{F} - s_{2})},

(16a)

C_{i j}^{(r)} = \frac{r W_{i} + ϑ_{i j} W_{j}}{Γ_{F} - s_{1}}, D_{i j}^{(r)} = \frac{r W_{i} - ϑ_{i j} W_{j}}{Γ_{F} - s_{2}},

(16b)

and

Q (α, t) = \frac{Γ_{F}}{2 σ} exp (\frac{α^{2}}{8 ς^{2}}) erfc (\frac{α - 4 ς^{2} t}{2 \sqrt{2} ς}) e^{- α t} .

(17)

Notice that Q(α, t) may be approximated by the exponential (Γ_F/σ)exp[α²/(8ς²)]e^−αt for a time

t \geq α / {(2 ς)}^{2} + \sqrt{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 (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 h̄ω_n of the state |n〉 is such that h̄(ω_n − ω_j) ≫ T for j = 1, 2, we arrive at the following equations governing the populations of the three states:

\frac{\partial ρ_{11}}{\partial t} = - γ_{11} ρ_{11} + ζ_{12} ρ_{22} + ζ_{1 n} ρ_{n n},

(18a)

\frac{\partial ρ_{22}}{\partial t} = - γ_{22} ρ_{22} + ζ_{21} ρ_{11} + ζ_{2 n} ρ_{n n},

(18b)

\frac{\partial ρ_{n n}}{\partial t} = - γ_{n n} ρ_{n n} + W_{n} ϕ^{2} .

(18c)

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)

ρ_{11} (t) = \int_{- \infty}^{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^{'},

(19a)

ρ_{22} (t) = \int_{- \infty}^{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^{'},

(19b)

where w_j = (ζ_jn/γ_nn)W_n and

g (t) = γ_{n n} \int_{- \infty}^{t} ϕ^{2} (t^{'}) e^{- γ_{n n} (t - t^{'})} d t^{'} .

Using this result in Eq. (9), gives for the case of the exponential pulse

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),

(20a)

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),

(20b)

where

χ_{i j}^{\pm} = \frac{γ_{n n}}{γ_{n n} \pm 2 σ} \frac{(γ_{i j} \pm 2 σ) w_{i} + ζ_{i j} w_{j}}{(s_{1} \pm 2 σ) (s_{2} \pm 2 σ)} \frac{Γ_{F}}{Γ_{F} \pm 2 σ},

(21a)

E_{i j}^{(r)} = \frac{γ_{n n}}{γ_{n n} - s_{1}} \frac{4 σ}{4 σ^{2} - s_{1}^{2}} \frac{Γ_{F}}{Γ_{F} - s_{1}} (r w_{i} + ϑ_{i j} w_{j}),

(21b)

F_{i j}^{(r)} = \frac{γ_{n n}}{γ_{n n} - s_{2}} \frac{4 σ}{4 σ^{2} - s_{2}^{2}} \frac{Γ_{F}}{Γ_{F} - s_{2}} (r w_{i} - ϑ_{i j} w_{j}),

(21c)

G_{i j} = \frac{4 σ γ_{n n}}{γ_{n n}^{2} - 4 σ^{2}} \frac{(γ_{n n} - γ_{j j}) w_{i} - ζ_{i j} w_{j}}{(γ_{n n} - s_{1}) (γ_{n n} - s_{2})} \frac{Γ_{F}}{Γ_{F} - γ_{n n}} .

(21d)

In the case of the Gaussian pulse, we obtain

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),

(22a)

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),

(22b)

where

η_{i j} = \frac{γ_{n n}}{γ_{n n} - Γ_{F}} \frac{(γ_{i j} - Γ_{F}) w_{i} + ζ_{i j} w_{j}}{(Γ_{F} - s_{1}) (Γ_{F} - s_{2})},

(23a)

K_{i j}^{(r)} = \frac{γ_{n n}}{γ_{n n} - s_{1}} \frac{r w_{i} + ϑ_{i j} w_{j}}{Γ_{F} - s_{1}}, L_{i j}^{(r)} = \frac{γ_{n n}}{γ_{n n} - s_{2}} \frac{r w_{i} - ϑ_{i j} w_{j}}{Γ_{F} - s_{2}},

(23b)

M_{i j} = \frac{γ_{n n}}{Γ_{F} - γ_{n n}} \frac{(γ_{j j} - γ_{n n}) w_{i} + ζ_{i j} w_{j}}{(γ_{n n} - s_{1}) (γ_{n n} - s_{2})} .

(23c)

The structure of Eqs. (20) and (22) is quite similar to that of Eqs. (13) and (15), except for additional terms ∝ e^−γ_nnt 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 ∼ s₁ for regime (i) and γ ∼ s₂ 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[(2σ/Γ_F)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.

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 = {\bar{h}}^{2} ω_{F}^{4} / (π c^{4} E_{L}^{2})

and

{\bar{W}}_{n} (ω_{F}) = \frac{{| V_{0, n}^{(λ)} |}^{2}}{{\bar{h}}^{2}} \frac{2 (γ_{f n} + Γ_{F} / 2)}{{(ω_{n} - ω_{F})}^{2} + {(γ_{f n} + Γ_{F} / 2)}^{2}},

we find

{\bar{Σ}}_{RL}^{(0)} (ω_{L}, ω_{F}) = Ξ ({\bar{W}}_{1} \frac{γ_{22} W_{1} + ζ_{12} W_{2}}{γ_{11} γ_{22} - ζ_{12} ζ_{21}} + {\bar{W}}_{2} \frac{γ_{11} W_{2} + ζ_{21} W_{1}}{γ_{11} γ_{22} - ζ_{12} ζ_{21}})

(24)

for the stationary signal of resonant luminescence and

{\bar{Σ}}_{TL}^{(0)} (ω_{L}, ω_{F}) = Ξ ({\bar{W}}_{1} \frac{γ_{22} ζ_{1 n} + ζ_{12} ζ_{2 n}}{γ_{11} γ_{22} - ζ_{12} ζ_{21}} + {\bar{W}}_{2} \frac{γ_{11} ζ_{2 n} + ζ_{21} ζ_{1 n}}{γ_{11} γ_{22} - ζ_{12} ζ_{21}}) \frac{W_{n}}{γ_{n n}}

(25)

for the stationary signal of thermalized luminescence. We see that the two-dimensional spectra

{\bar{Σ}}_{RL}^{(0)} (ω_{L}, ω_{F})

and

{\bar{Σ}}_{TL}^{(0)} (ω_{L}, ω_{F})

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 (γ_1i, γ_2i, and γ_ni), 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.

The probability of transition |1〉 → |2〉 becomes vanishingly small at low temperatures (T ≪ E₂₁), for which Eqs. (24) and (25) reduce to the well-known formulas [16

, 31

, 67

]

{\bar{Σ}}_{RL}^{(0)} = Ξ (\frac{{\bar{W}}_{1} W_{1}}{γ_{11}} + \frac{{\bar{W}}_{2} W_{2}}{γ_{22}} + \frac{{\bar{W}}_{1} ζ_{12} W_{2}}{γ_{11} γ_{22}})

(26)

and

{\bar{Σ}}_{TL}^{(0)} = Ξ (\frac{{\bar{W}}_{1} ζ_{1 n}}{γ_{11}} + \frac{{\bar{W}}_{2} ζ_{2 n}}{γ_{22}} + \frac{{\bar{W}}_{1} ζ_{12} ζ_{2 n}}{γ_{11} γ_{22}}) \frac{W_{n}}{γ_{n n}} .

(27)

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(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

ψ = \frac{ζ_{12} ζ_{21}}{γ_{11} γ_{22}} = \frac{{\bar{Σ}}_{RL}^{(0)} (ω_{1}, ω_{2})}{{\bar{Σ}}_{RL}^{(0)} (ω_{1}, ω_{1})} \frac{{\bar{Σ}}_{RL}^{(0)} (ω_{2}, ω_{1})}{{\bar{Σ}}_{RL}^{(0)} (ω_{2}, ω_{2})} .

(28)

In order to make it possible to filter out the signal of resonant scattering in the experiments with ω_L ≈ ω_F, 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 Γ_F ≪ |ω_L −ω_j| ≲ γ_ji. This inequality sets the upper limit for the filter’s bandpass in stationary experiments, Γ_F ≪ min(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

χ = \frac{ζ_{1 n}}{ζ_{2 n}} \frac{γ_{22}}{ζ_{12}} = \frac{φ - 1}{1 - φ ψ},

(29)

where

φ = \frac{{\bar{Σ}}_{TL}^{(0)} (ω_{n}, ω_{1})}{{\bar{Σ}}_{TL}^{(0)} (ω_{n}, ω_{2})} \frac{{\bar{Σ}}_{RL}^{(0)} (ω_{2}, ω_{2})}{{\bar{Σ}}_{RL}^{(0)} (ω_{2}, ω_{1})} .

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 = e^−E₂₁/T. By denoting the recombination rates as ζ₀₁ and ζ₀₂ and introducing a size-dependent parameter x = E₂₁/T, we may write

γ_{11} (x) = ζ_{01} + ζ_{12} e^{- x}, γ_{22} = ζ_{02} + ζ_{12} .

(30)

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 = \frac{{\bar{Σ}}_{RL}^{(0)} (ω_{1}, ω_{1}, x_{2})}{{\bar{Σ}}_{RL}^{(0)} (ω_{1}, ω_{1}, x_{1})} = \frac{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 = \frac{{\bar{Σ}}_{RL}^{(0)} (ω_{2}, ω_{2}, x_{1})}{{\bar{Σ}}_{RL}^{(0)} (ω_{1}, ω_{1}, x_{1})} \frac{{\bar{Σ}}_{RL}^{(0)} (ω_{1}, ω_{1}, x_{2})}{{\bar{Σ}}_{RL}^{(0)} (ω_{2}, ω_{2}, x_{2})} = \frac{γ_{11} (x_{1})}{γ_{11} (x_{2})} .

By solving these equations with respect to the unknowns ζ₀₁ and ζ₀₂, we arrive at the following relations between the relaxation rates:

γ_{11} (x) = (e^{- x} + \frac{e^{- x_{1}} - b e^{- x_{2}}}{b - 1}) ζ_{12},

γ_{22} = \frac{b - 1}{b - a} \frac{e^{- x_{1}} - a e^{- x_{2}}}{e^{- x_{1}} - e^{- x_{2}}} ζ_{12} .

The second of these relations, being employed in Eq. (29), allows one to find the ratio 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 ∝ ϕ²(t). Indeed, assuming in Eqs. (13), (14), (20), and (21) that Γ_F ≫ s₂ and s₁ ≫ σ, yields

{\bar{Σ}}_{RL}^{(i)} \approx {\bar{Σ}}_{RL}^{(0)} e^{- 2 σ | t |}, {\bar{Σ}}_{TL}^{(i)} \approx {\bar{Σ}}_{TL}^{(0)} e^{- 2 σ | t |},

where

{\bar{Σ}}_{RL}^{(0)}

and

{\bar{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 (\frac{α - 4 ς^{2} t}{2 \sqrt{2} ς}) = exp [- \frac{{(α - 4 ς^{2} t)}^{2}}{8 ς^{2}}] (\frac{2 ς}{α} \sqrt{\frac{2}{π}} + O [{(ς / α)}^{2}]),

which is valid for α ≫ ς, we obtain from Eqs. (15), (16), (22) and (23) for the Gaussian pulse

{\bar{Σ}}_{RL}^{(i)} \approx {\bar{Σ}}_{RL}^{(0)} A e^{- 2 ς^{2} t^{2}}, {\bar{Σ}}_{TL}^{(i)} \approx {\bar{Σ}}_{TL}^{(0)} A e^{- 2 ς^{2} t^{2}} .

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 γ₂₂ −γ₁₁ ≫ 2ζ₁₂ exp[−E₂₁/(2T)]. This is because in this case

q ≪ p \approx 1, s_{1} \approx γ_{11}, s_{2} \approx γ_{22},

(31)

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 = n_e₁ = n_h₁ = 1, l₁ = l_e₁ = l_h₁ =0, n₂ = n_e₂ = n_h₂ = 2, l₂ = l_e₂ = l_h₂ = 0. Then the energy gap between them is given by the expression [1

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

, 67

]

E_{21} = \frac{3 π^{2} {\bar{h}}^{2}}{2 μ R^{2}},

where μ = m_em_h/(m_e + m_h) 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 m_e = m_h = 0.25 m₀ (m₀ is the free-electron mass) at room temperature. It is seen that if ζ₀₁ ≫ ζ₀₂, 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 + ζ₁₂e^−x increases from γ₂₂ in small quantum dots to γ₂₂ + ζ₁₂ in large ones. If ζ₀₂ ≫ ζ₀₁, then s₁ ≈ min(ζ₁₂, ζ₀₂) and s₂ ≈ max(ζ₁₂,ζ₀₂) for quantum dots with R < 7 nm. Since ζ₁₂ ∼ ζ₀₁ for our example in Fig. 4, this result implies that s₂ ≫ s₁. 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)]/(γ₂₂ −γ₁₁) > 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.

5.3.1. Kinetics of resonant luminescence

Suppose now that Γ_F ≫ σ ≫ s₂. 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

\begin{array}{l} {\bar{Σ}}_{RL}^{(ii)} \approx \frac{Ξ}{σ} {[{\bar{W}}_{1} (p W_{1} + ϑ_{12} W_{2}) + {\bar{W}}_{2} (q W_{2} + ϑ_{21} W_{1})] e^{- s_{1} t} \\ + [{\bar{W}}_{1} (q W_{1} - ϑ_{12} W_{2}) + {\bar{W}}_{2} (p W_{2} - ϑ_{21} W_{1})] e^{- s_{2} t}} . \end{array}

(32)

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 W_j(ω₁) ∼ W_j(ω₂) or W̄_j(ω₁) ∼ W̄_j(ω₂) for j = 1, 2]. Specifically, the decay of luminescence is governed by the functions

{\bar{Σ}}_{RL}^{(ii)} \propto {\begin{array}{l} p e^{- s_{1} t} + q e^{- s_{2} t} & for ω_{L}, ω_{F} \approx ω_{1}, \\ q e^{- s_{1} t} + p e^{- s_{2} t} & for ω_{L}, ω_{F} \approx ω_{2}, \\ e^{- s_{1} t} - e^{- s_{2} t} & for ω_{L} \approx ω_{1 (2)}, ω_{F} \approx ω_{2 (1)}, \end{array}

(33)

provided the energy gap between the two eigenstates is much larger than the total dephasing rate of optical transitions. In the case of ω_L ≠ ω_F, 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 (t_m) at which the signal of luminescence peaks, we obtain a relation (s₂ − s₁)t_m = 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

s_{1} \approx \frac{1 - ψ}{1 / γ_{11} + 1 / γ_{22}} ≪ s_{2} \approx γ_{11} + γ_{22} .

(34)

The luminescence from either of the eigenstates is described in this situation by the monoexponential signals within the time domains 1/σ ≪ t ≲ 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

γ_{11} = \frac{1}{2} (s_{1} + s_{2} - \sqrt{{(s_{1} - s_{2})}^{2} - 4 ζ_{12} ζ_{21}}),

(35a)

γ_{22} = \frac{1}{2} (s_{1} + s_{2} + \sqrt{{(s_{1} - s_{2})}^{2} - 4 ζ_{12} ζ_{21}}) .

(35b)

In case of unknown ζ₁₂, one may determine parameter ψ in the regime of stationary excitation using Eq. (28), and then find

γ_{11} = \frac{1}{2} (s_{1} + s_{2} - \sqrt{{(s_{1} + s_{2})}^{2} - 4 s_{1} s_{2} / (1 - ψ)}),

(36a)

γ_{22} = \frac{1}{2} (s_{1} + s_{2} + \sqrt{{(s_{1} + s_{2})}^{2} - 4 s_{1} s_{2} / (1 - ψ)}) .

(36b)

Of significance is that the simultaneous measurement of s₁ and s₂ from any single time dependency in Eq. (33) is only possible when p ∼ q. 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 s₂ ≫ s₁ itself does not necessarily imply that the interaction with the bath gives major contribution to the eigenstates’ lifetimes. For instance, s₂ ≫ s₁ for ψ ≪ 1 and γ₂₂ ≫ γ₁₁.

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_j). 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 = 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 (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

| V_{1, 0}^{L} | = | V_{2, 0}^{L} | = | V_{0, 1}^{λ} | = | V_{0, 2}^{λ} | = 1

and is normalized by a factor of

ω_{F}^{4} / (π {\bar{h}}^{2} c^{4})

. Even though s₂ ≫ s₁ 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 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 E₂₁ ∼ γ_ji, γ_fj (j = 1, 2). When R is reduced, the energy gap E₂₁ 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 τ₁ when ω_L = ω_F = ω₁ [see Fig. 5(a)]. For ω_L = ω_F = ω₂, the decrease in q makes the decay ∝ qe^−γ₁₁t + pe^−γ₂₂t 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 (ω_L = ω₂) and the signal is detected at the frequency of the low-energy eigenstate (ω_F = ω₁), then the luminescence from small quantum dots peaks at a time t_m ≈ (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 − e^−γ₂₂t (red curve), is overcome by the decay, proportional to e^−γ₁₁t (blue curve), at t_m ≈ 24 meV⁻¹. Since transitions |1〉 → |2〉 become extremely improbable in small quantum dots, the luminescence from state |2〉 excited at frequency ω_L = 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) ω_L = ω_F = ω₁; (b) ω_L = ω_F = ω₂; (c) ω_L = ω₂, ω_F = ω₁; (d) ω_L = ω₁, ω_F = 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 γ̂_1i = γ̂_2i = γ̂_f ₁ = γ̂_f ₂ = 20 meV, Γ_F = 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.

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 > γ₂₂), these equations for t ≫ 1/σ are reduced to

\begin{array}{l} {\bar{Σ}}_{TL}^{(ii)} \approx \frac{Ξ}{σ} {\frac{{\bar{W}}_{1} (p ζ_{1 n} + ϑ_{12} ζ_{2 n}) + {\bar{W}}_{2} (q ζ_{2 n} + ϑ_{21} ζ_{1 n})}{γ_{n n} - s_{1}} e^{- s_{1} t} \\ + \frac{{\bar{W}}_{1} (q ζ_{1 n} - ϑ_{12} ζ_{2 n}) + {\bar{W}}_{2} (p ζ_{2 n} - ϑ_{21} ζ_{1 n})}{γ_{n n} - s_{2}} e^{- s_{2} t} \\ + \frac{{\bar{W}}_{1} [(γ_{22} - γ_{n n}) ζ_{1 n} + ζ_{12} ζ_{2 n}] + {\bar{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} . \end{array}

(37)

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 σ ≫ γ_nn ≫ s₂, as in this case the relaxation constant γ_nn may be reliably determined from the luminescence spectrum for a registration time 1/σ ≪ t ≲ 1/γ_nn, whereas for a time t ≫ 1/γ_nn only two exponentials are contributing to the overall signal, namely,

{\bar{Σ}}_{TL}^{(ii)} \propto {\begin{array}{l} (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} \approx 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} \approx ω_{2}, \end{array}

(38)

where μ_j = ζ_jn/γ_nn. Again, if the thermal interaction between the two relevant eigenstates is so strong that s₂ ≫ s₁, 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 s₁ ∼ s₂), 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 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 ≈ ω₁ when ϑ₁₂ζ₂_n > qζ_1n and for ω_F ≈ ω₂ when ϑ₂₁ζ_1n > pζ₂_n. Using Eq. (37), we find the time corresponding to the luminescence peak to be

t_{m} = \frac{1}{s_{2} - s_{1}} \times {\begin{array}{l} ln (\frac{s_{2}}{s_{1}} \frac{ϑ_{12} ζ_{2 n} - q ζ_{1 n}}{ϑ_{12} ζ_{2 n} + p ζ_{1 n}}) & for ω_{F} \approx ω_{1}, \\ ln (\frac{s_{2}}{s_{1}} \frac{ϑ_{21} ζ_{1 n} - q ζ_{2 n}}{ϑ_{21} ζ_{1 n} + p ζ_{2 n}}) & for ω_{F} \approx ω_{2} . \end{array}

(39)

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 ζ₁_n = ζ₂_n.

The value of γ_nn normally exceeds 10¹¹ s⁻¹ at room temperature [58

–60

], so that the condition σ ≫ γ_nn is satisfied for pulses shorter than 100 ps. For much longer pulses, an opposite relationship holds (γ_nn ≫ σ ≫ s₂) and the signal of thermalized luminescence is seen to be described by Eq. (38) for t ≫ 1/σ if γ_nn ≪ Γ_F. Hence, only the quantities 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_{e_n} = n_{h_n} = 9 and l_n = l_{e_n} = l_{h_n} = 0, in which case [1

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

, 67

] E_n₁ ≈ E_n₂ ≈ 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 γ_nn/ζ₁_n = 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) ω_L = ω_n, ω_F = ω₁; (b) ω_L = ω_n, ω_F = ω₂. The relevant eigenstates are weakly coupled via thermal interaction for R ≲ 8 nm. It was assumed that γ̂_ni = 20 meV, γ_nn = 2 meV, and ζ_1n = ζ_2n = 500 μeV. The other parameters are the same as in Fig. 5.

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 e^−γ₁₁t and may be used to find γ₁₁. The described situation is illustrated by Fig. 7(a), where it is assumed that ζ_2n = 10ζ_1n = 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 ζ_1n and ζ_2n (i.e., ζ_1n = 10ζ_2n = 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) ζ₁_n = 50 μeV, ζ_2n = 500 μeV and (b) ζ_1n = 500 μeV, ζ_2n = 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.

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

{\bar{Σ}}_{RL}^{(iii)} \approx {\bar{Σ}}_{RL}^{(0)} e^{- Γ_{F} t}, {\bar{Σ}}_{TL}^{(iii)} \approx {\bar{Σ}}_{TL}^{(0)} e^{- Γ_{F} t} .

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.

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.

Excitation Regime	Time Dependence	Retrievable Parameters
σ = 0	—	χ, ψ
Γ_F ≫ s₂, s₁ ≫ σ	ϕ²(t)	χ, ψ
Γ_F ≫ σ ≫ s₂ ≫ s₁	e^−s₂t, 1/σ ≪ t ≲ 1/s₂ e^−s₁t, t ≳ 1/s₁	γ₁₁, γ₂₂
Γ_F ≫ σ ≫ s₂ ∼ s₁ (weak thermal coupling)	$\begin{array}{l} e^{- γ_{11} t}, ω_{L}, ω_{F} \approx ω_{1} \\ e^{- γ_{22} t}, ω_{L}, ω_{F} \approx ω_{2} \end{array}} t ≫ \frac{1}{σ}$	γ₁₁, γ₂₂
Γ_F ≫ σ ≫ γ_nn ≫ s₂ ≫ s₁	e^−γ_nnt, 1/σ ≫ t ≲ 1/γ_nn e^−s₂t, 1/γ_nn ≪ t ≲ 1/s₂ e^−s₁t, t ≳ 1/s₁	γ_nn, ζ₁_n/ζ₂_n γ₁₁, γ₂₂
Γ_F ≫ σ ≫ γ_nn ≫ s₂ ∼ s₁ (weak thermal coupling)	e^−γ_nnt, 1/σ ≫ t ≲ 1/γ_nn e^−γ₂₂t, t ≳ 1/γ₂₂, ω_L ≈ ω₂	γ_nn, γ₂₂ (ζ₁_n/ζ₂_n, γ₁₁ for known ζ₁₂)
Γ_F ≫ γ_nn ≫ σ ≫ s₂ ∼ s₁ (weak thermal coupling)	e^−γ₂₂t, ω_L ≈ ω₂, $t ≫ \frac{1}{σ}$	γ₂₂ (ζ₁_n/ζ₂_n, γ₁₁ for known ζ₁₂)
Γ_F ≪ σ ∼ γ₁₁, γ₂₂, γ_nn	e^−Γ_Ft, t ≫ 1/σ	χ, ψ

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

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]

]. 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

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]

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]

]. 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.

Appendices

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

\frac{\partial ρ_{n n}}{\partial t} + \sum_{n^{'} = 1}^{N} a_{n n^{'}} ρ_{n^{'} n^{'}} = f_{n} (t),

(A.1)

where a_nn_′ = γ_nnδ_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_n and adding equations corresponding to different n termwise, yield

\sum_{n = 1}^{N} b_{n} \frac{\partial ρ_{n n}}{\partial t} + \sum_{n^{'} = 1}^{N} ρ_{n^{'} n^{'}} \sum_{n = 1}^{N} b_{n} a_{n n^{'}} = \sum_{n = 1}^{N} b_{n} f_{n} (t) .

(A.2)

Suppose that the coefficients b_n are such that

\sum_{n^{'} = 1}^{N} b_{n^{'}} a_{n^{'} n} = s_{n} b_{n} .

(A.3)

It is crucial for the subsequent discussion that all N roots s_n of the characteristic polynomial of Eq. (A.3) are positive. The positiveness of the roots follows from two observations. First, the coefficient matrix

{\hat{a}}_{n n^{'}} = (\begin{matrix} γ_{11} & - ζ_{12} e^{- E_{21} / T} & - ζ_{13} e^{- E_{31} / T} & \dots & - ζ_{1 N} e^{- E_{N 1} / T} \\ - ζ_{12} & γ_{22} & - ζ_{23} e^{- E_{32} / T} & \dots & - ζ_{2 N} e^{- E_{N 2} / T} \\ - ζ_{13} & - ζ_{23} & γ_{33} & \dots & - ζ_{3 N} e^{- E_{N 3} / T} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ - ζ_{1 N} & - ζ_{2 N} & - ζ_{3 N} & \dots & γ_{N N} \end{matrix})

is strictly diagonally dominant [75

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

], as its elements satisfy the condition

| {\hat{a}}_{n n} | > \sum_{n^{'} \neq n} | {\hat{a}}_{n n^{'}} | .

This condition holds true since the inverse lifetime γ_nn 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 ζ_1n + ζ_2n + ... + ζ_n_−1,n + ζ_n_+1,n + ... + ζ_Nn of relaxation rates within the channels occurring due solely to thermal fluctuations. Second, the matrix â_nn_′ can be rewritten in the symmetric form by simply multiplying its jth row (j ≥ 2) by the factor e^−E_j1/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

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

] 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

x_{n} = \sum_{n^{'} = 1}^{N} b_{n^{'}}^{(n)} ρ_{n^{'} n^{'}},

(A.4)

where

{b_{n^{'}}^{(n)}}

is the set of coefficients b_n corresponding to the root s_n. As a result, Eq. (A.2) takes the form

\frac{\partial x_{n}}{\partial t} + s_{n} x_{n} = \sum_{n^{'} = 1}^{N} b_{n^{'}}^{(n)} f_{n^{'}} (t) .

(A.5)

The solution to this equation vanishing for t → −∞ is found to be

x_{n} (t) = \sum_{n^{'} = 1}^{N} b_{n^{'}}^{(n)} \int_{- \infty}^{t} f_{n^{'}} (τ) e^{- s_{n} (t - τ)} d τ .

(A.6)

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

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, New York, 2000), 2nd ed.

]. 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).

References and links

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OCIS Codes
(300.3700) Spectroscopy : Linewidth
(300.6280) Spectroscopy : Spectroscopy, fluorescence and luminescence
(300.6470) Spectroscopy : Spectroscopy, semiconductors
(160.4236) Materials : Nanomaterials

ToC Category:
Spectroscopy

History
Original Manuscript: October 11, 2012
Revised Manuscript: November 16, 2012
Manuscript Accepted: November 16, 2012
Published: November 28, 2012

Citation
Ivan D. Rukhlenko, Mikhail Yu. Leonov, Vadim K. Turkov, Aleksandr P. Litvin, Anvar S. Baimuratov, Alexander V. Baranov, and Anatoly V. Fedorov, "Kinetics of pulse-induced photoluminescence from a semiconductor quantum dot," Opt. Express 20, 27612-27635 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-25-27612

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References

A. V. Fedorov, I. D. Rukhlenko, A. V. Baranov, and S. Yu. Kruchinin, Optical Properties of Semiconductor Quantum Dots (Nauka, St. Petersburg, 2011).
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. 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]
J. H. Davies, The Physics of Low-Dimensional Semiconductors: An Introduction (Cambridge University Press, Cambridge, 1998), 1st ed.
S. V. Gaponenko, Optical Properties of Semiconductor Nanocrystals (Cambridge University Press, Cambridge, 1998), 1st ed. [CrossRef]
U. Woggon, Optical Properties of Semiconductor Quantum Dots (Springer, Berlin, 1996), 1st ed.
A. P. Alivisatos, “Semiconductor clusters, nanocrystals, and quantum dots,” Science271, 933–937 (1996). [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]
C. Weisbuch and B. Vinter, Quantum Semiconductor Structures: Fundamentals and Applications (Academic Press, Boston, 1991), 1st ed.
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]
N. Koshida, ed., Device Applications of Silicon Nanocrystals and Nanostructures (Springer Science+Business Media, LLC, New York, 2009). [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]
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 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]
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]
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]
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]
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]
T. Inoshita and H. Sakaki, “Electron relaxation in a quantum dot: Significance of multiphonon processes,” Phys. Rev. B46, 7260–7263 (1992). [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]
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, 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 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]
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]
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]
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]
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]
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]
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]
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]
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]
W. Demtröder, Laser Spectroscopy: Basic Concepts and Instrumentation (Springer, Berlin, 2002), 3rd ed.
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]
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]
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]
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]
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]
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]
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]
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]
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]
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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]
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Abbarchi, M.
Akahane, K.
Al Salman, A.
Albrecht, A. C.
Alén, B.
Alivisatos, A. P.
Anda, E. V.
Anderson, K. E. H.
Arakawa, Y.
Artemyev, M.
Auer, T.
Axt, V. M.
Bandelow, U.
Baranov, A. V.
Bastard, G.
Bawendi, M. G.
Bayer, M.
Baymuratov, A. S.
Benisty, H.
Berstermann, T.
Berwick, K.
Bester, G.
Bhattacharya, P. K.
Bimberg, D.
Blum, K.
Bockelmann, U.
Bonati, C.
Bonn, M.
Borri, P.
Buckley, S.
Cannizzo, A.
Carmele, A.
Chergui, M.
Cooney, R. R.
Dachner, M.-R.
Dai, Q.-F.
Davies, J. H.
Davydov, V.
de Mello Donegá, C.
Demtröder, W.
Dias, E. A.
Dou, X. M.
Eberly, J. H.
Ema, K.
Empedocles, S. A.
Fedorov, A. V.
Forchel, A.
Franchi, S.
Frigeri, P.
Gaponenko, S. V.
Gartner, P.
Gies, C.
Gomis-Bresco, J.
Goupalov, S. V.
Gradshteyn, I. S.
Griffiths, D. J.
Grundmann, M.
Guo, Q.
Gurioli, M.
Guyot-Sionnest, P.
Hartmann, G.
Hebling, J.
Hendry, E.
Hoffmann, M. C.
Hu, W.
Hwang, H. Y.
Inoshita, T.
Ishi-Hayase, J.
Ivanov, S. A.
Jahnke, F.
Jiang, D. S.
Kabuss, J.
Kambhampati, P.
Kavousanaki, E. G.
Kim, J.-E.
Kim, K.
Klimov, V. I.
Knorr, A.
Koeberg, M.
Korn, G. A.
Korn, T. M.
Kruchinin, S. Yu.
Kuhn, T.
Kujiraoka, M.
Kuroda, T.
Kurtze, H.
Lan, S.
Langbein, W.
Lee, M. L.
Leonov, M. Yu.
Li, X.-Q.
Litvin, A. P.
Liu, H.-Y.
Liu, S.-H.
Maingault, L.
Makler, S. S.
Malic, E.
Mano, T.
Mariette, H.
Martínez-Pastor, J.
Mastrandrea, C.
Masumoto, Y.
Melinger, J. S.
Meng, Z.-M.
Mohamed, M. B.
Mukamel, S.
Muñoz-Matutano, G.
Nakayama, H.
Narvaez, G. A.
Nelson, K. A.
Ni, H. Q.
Nienhuis, G.
Niu, Z. C.
Norris, D. J.
Norris, T. B.
Ouyang, D.
Pandey, A.
Parfenov, P. S.
Patton, B.
Perova, T. S.
Petruska, M. A.
Pietryga, J. M.
Premaratne, M.
Prudnikau, A. V.
Ren, H.-W.
Reuter, D.
Richter, M.
Rivoire, K.
Roslyak, O.
Rukhlenko, I. D.
Ryzhik, I. M.
Sakaki, H.
Sakoda, K.
Sasaki, M.
Schaller, R. D.
Schneider, S.
Schoth, M.
Schroeter, D. F.
Schulze, F.
Schwab, M.
Seebeck, J.
Sellin, R. L.
Seravalli, L.
Sercel, P. C.
Sewall, S. L.
Singh, J.
Song, Y.
Sotomayor-Torrès, C. M.
Stavarache, V.
Strauf, S.
Sugou, S.
Sun, B. Q.
Tonti, D.
Tortschanoff, A.
Trevisi, G.
Turkov, V. K.
Urayama, J.
Ushakova, E. V.
van Mourik, F.
Vanmaekelbergh, D.
Vasilevskiy, M. I.
Vinattieri, A.
Vinter, B.
Vuckovic, J.
Wang, F.
Weber, C.
Wehrenberg, B.
Weisbuch, C.
Wieck, A. D.
Wiersig, J.
Wilms, A.
Wódkiewicz, K.
Woggon, U.
Wolters, J.
Wu, L.-J.
Wu, Z. K.
Yakovlev, D. R.
Yamamoto, N.
Yang, T.
Yeh, K.-L.
Yu, D.
Zhang, H.
Zunger, A.

ACS Nano

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]

Appl. Phys. Lett.

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]
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]

J. Appl. Phys.

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]
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]

J. Chem. Phys.

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]
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]
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]

J. Opt. Soc. Am.

J. H. Eberly and K. Wódkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am.67, 1252–1261 (1977). [CrossRef]

J. Phys. D: Appl. Phys.

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]

Laser Photonics Rev.

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

New J. Phys.

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]

Opt. Express

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]

Opt. Spectrosc.

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

Phys. Rev. B

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).
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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]
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]
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]
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. 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]
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]
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]
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]
M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B65, 041308 (2002). [CrossRef]
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X.-Q. Li and Y. Arakawa, “Anharmonic decay of confined optical phonons in quantum dots,” Phys. Rev. B57, 12285–12290 (1998). [CrossRef]
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]
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]
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]
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]
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]
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]
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]
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