Signatures of Exciton Dynamics and Carrier Trapping in the Time

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J. Phys. Chem. C 2009, 113, 18632–18642

Signatures of Exciton Dynamics and Carrier Trapping in the Time-Resolved Photoluminescence of Colloidal CdSe Nanocrystals Marcus Jones,† Shun S. Lo, and Gregory D. Scholes* Department of Chemistry, 80 St. George Street, Institute for Optical Sciences, and Center for Quantum Information and Quantum Control, UniVersity of Toronto, Toronto, Ontario, M5S 3H6 Canada ReceiVed: August 14, 2009; ReVised Manuscript ReceiVed: September 12, 2009

Surface effects significantly affect the photoexcitation dynamics of colloidal nanocrystals, but their influence is hard to study because of sample complexity and the typically small extinction coefficient of trap states. Using temperature-dependent time-resolved photoluminescence (PL) measurements, we investigate the perturbations induced by surface-localized carrier traps on the exciton dynamics of the nanocrystals. We present a model of carrier trapping that is based on Marcus’ electron-transfer theory and use it to accurately reproduce PL dynamics over a wide temperature range in five core-shell CdSe/CdZnS nanocrystal samples. The resulting pictures of carrier dynamics are then used to identify features in the PL data that may be used in subsequent experiments to reveal information about the energy and distribution of surface-localized trap states. We find that in certain cases, the shape of the ensemble distribution of trap energies can be accurately determined from data recorded at a single temperature that is easily identified from plots of average PL lifetime. 1. Introduction Colloidal semiconductor nanocrystals (NCs) have attracted considerable attention in recent years due to their potential as an enabling technology in a variety of applications such as photovoltaics, photodetectors, or chemical sensors. Foremost among their properties is their widely tunable absorption thresholds: NC excited electronic states are examples of nanoscale excitons with diameters in the range 1-10 nm. In addition NCs represent the smallest materials to have a welldefined surface, and this inherent property is exhibited in the interplay between quantum-confined electronic properties of the core semiconductor versus those of atoms located at the interface between semiconductor material and insulating ligands and solvent. Electronic processes occurring in the core are reasonably well understood and can be modeled with ever-increasing degrees of accuracy, but the effect on these processes of proximate surface atoms and coordinating ligands is not accurately known. Successful applications of NCs in photovoltaic devices, for example, will rely on finding ways to optimize carrier dynamics. Time-dependent photoluminescence (PL) measurement could potentially be instrumental in that regard, but only if we are able to interpret the PL signals. A simple way to elucidate dynamical information about NC-containing materials from PL decays without the need for rigorous analysis would immensely aid application of this ultimately sensitive technique. In this paper we derive a model of NC excitation dynamics that is able to reproduce PL transients over the temperature range, 77-300 K. We then use the detailed information about our samples obtained from this model to identify features in the decay data that can directly inform us about processes occurring in our * To whom correspondence should be addressed. E-mail: gscholes@ chem.utoronto.ca. † Present address: Department of Chemistry, University of North Carolina, Charlotte, 9201 University City Blvd., Charlotte, NC 28223.

materials. Finally we outline several signatures of NC excitation dynamics that can be easily applied to the analysis of NC PL decays. In a 4 nm diameter spherical Wurtzite CdSe NC more than half of the Cd and Se atoms in the crystal lie within 5 Å of the surface: it is therefore not surprising that the chemistry and photophysics at the interface plays such a significant role in determining NC electronic properties. For example, timeresolved photoluminescence (PL) signals, which indicate the transient population of radiative states in the material, are found to be extremely sensitive to perturbations affecting interfacial electronic structure, such as ligand exchange, NC photooxidation, or temperature.1-6 Unfortunately, the inherent dispersion of NC size and morphology and the countless structural variations at NC surfaces, coupled with the very low optical activity of surface-localized electronic states, makes it difficult to directly probe interfacial dynamics and learn about the influence of surface atoms on NC electronic processes. As a result, our ability to understand the transport of photogenerated charge and energy in potential useful materials, combining, e.g., NCs and other molecular species, has been limited by effects intrinsic to individual NCs that we do not yet understand. Surface-localized NC states are able to trap electrons or holes forming metastable trap states that can recombine to core exciton states or decay (non)radiatively to the ground state. In an attempt to minimize the effect of these states on core electronic processes, colloidal NCs are typically coated with passivating ligands that both sustains their dispersion in solution and lowers the number of trap sites (atoms with coordination number less than four). Thicker shells typically result in higher PL quantum yields7,8 and have been recently associated with the termination of PL intermittency in single NCs:9,10 a process that is linked to interfacially trapped carrier dynamics.11 Thickly shelled CdSe NCs with high PL quantum yields and suppressed blinking can make ideal markers for bioimaging applications;12 however, thick shells may actually diminish the effectiveness of NCs in applications that require some degree of interaction between

10.1021/jp9078772 CCC: $40.75  2009 American Chemical Society Published on Web 09/30/2009

Photoluminescence of Colloidal CdSe Nanocrystals NCs and their environment, e.g., in a NC photovoltaic device, by physically isolating the core exciton from its surroundings. For uses such as these we must develop subtler ways of treating interfacial processes in NCs; an effective way to do this is to understand first the nature of the electronic processes that link exciton and surface states. Several studies have attempted to determine the nature of NC surfaces using direct structural probes. Magnetic resonance studies13-18 have explored the nature of NC-ligand binding interactions and found19,20 that surface ligand coverage is typically incomplete: remaining unbound surface atoms carry slight positive or negative charge after local distortions minimize surface free energy.21 These sites are capable of trapping electrons or holes at the surface. Ab initio quantum chemical calculations22 and positron annihilation spectroscopy on CdSe NCs19 suggest that Se atoms relax outward irrespective of passivation, indicating, as previously suggested,20,23-25 that hole traps are more prevalent than electron traps and this is supported by recent spectroscopic investigations.26,27 While the number of trap sites around a NC is likely to be both inhomogeneous and sample dependent, consideration of the number of atoms near the surface in a wurtzite NC gives us an approximate upper limit on that number.20 More recently, calculations on small CdSe NCs28 have found hybrid core-ligand states that may enhance intraband carrier relaxation and could increase the exposure of electrons and holes to surface defects. Since NC PL is particularly sensitive to surface and environmental changes, PL techniques can potentially be used as a delicate probe of surface structure and dynamics. In particular, PL transients contain dynamical information about processes that occur over a wide time scale range from ps to several µs. In spherical wurtzite CdSe NCs, PL rates depend strongly on the radiative recombination rates of the lowest exciton states. The relative oscillator strengths of these transitions have been the subject of theory29 and experiment.2,30,31 The corresponding exciton radiative lifetimes fall into two categories: “bright” states with ∼10 ns lifetimes32 and “dark” states with lifetimes of the order of 1 µs,2,30 leading to room temperature average PL lifetimes of ∼20-30 ns.31,33 Since the splitting between the lowest state (dark) and the next state (bright) is just a few meV,29 the average PL lifetime is strongly temperature dependent, converging toward the dark-state radiative lifetime at 4 K. Our analysis of NC time-dependent PL demonstrates that the shape of the PL decays cannot be entirely described by an exciton-only picture. Instead, we show that complex temperature-dependent variations of the decay shape and, hence, the average PL lifetimes are brought about via nonradiative transitions between exciton and trap states. In an attempt to understand these processes we develop a theoretical framework, based on classical Marcus electron transfer (ET) theory,34 that successfully reproduces PL decays over a wide temperature range. Data analysis using these methods is effective but very timeconsuming, so a central aim of this work is to link the results of our calculations with signature features in plots of average NC PL lifetimes. We show that there are several features that point to the average energy of occupied trap states in the NC ensemble and even, in some cases, to the precise energy dependence and shape of the trap state distribution. In this way we aim to suggest a general means of interpreting NC timeresolved PL data that allows rapid and straightforward characterization of photoexcitation dynamics. 2. Experimental Section We obtained NC samples from Evident Technologies. They consisted of a CdSe core overcoated with CdS then ZnS. For

J. Phys. Chem. C, Vol. 113, No. 43, 2009 18633 low-temperature studies, the NCs were dispersed in a glassforming solvent consisting of a 6:1 mixture of isopentane and methylcyclohexane. The solutions were subsequently injected between two sapphire plates separated with a Teflon spacer and mounted on the sample holder. To minimize decay distortions, the optical density of the solutions was kept less than 0.1 at the wavelength of the first exciton peak. PL dynamics were measured by time-correlated single-photon counting (TCSPC), using an IBH Datastation Hub system with an IBH 5000 M PL monochromator and an R3809U-50 cooled MCP PMT detector. The light source was a model 3950 ps Ti: sapphire Tsunami laser (Spectra-Physics), pumped by a Millenium X (Spectra-Physics) diode laser, pulse picked (Model 3980 Spectra-Physics), and frequency doubled using a GWU-23PL multiharmonic generator (Spectra-Physics). Pulse repetition rates were kept below 100 kHz. Sample temperatures were adjusted using an N2 flow cryostat coupled with a Lakeshore model 331 temperature controller. Excitation wavelengths for the photon-counting experiments were set 400 meV higher energy than the peak of the room temperature absorption in each of the NC samples. Data were recorded on each sample at 77 and 100 K and then at 20 K intervals until 300 K. Emission wavelengths were adjusted, at each temperature, to remain coincident with the peak of the steady-state PL. To ensure both the short- and long-decay components could be accurately extracted, PL decays were measured twice at each temperature: once with a 50 ns TAC ramp (27.6 ps channel width) and again with a 5 µs TAC (2.36 ns channel width). As recently discussed,35 long and short TCSPC data sets were analyzed simultaneously by least-squares iterative reconvolution of an N-component multiexponential decay function with two experimentally determined instrument response functions.36 The value of N was determined, for each pair of data sets, as the minimum required to yield a satisfactory reproduction of the measured decay curves. Criteria for an acceptable fit have been justified in a previous article.37 3. Photoluminescence Decay Analysis 3.1. Average PL Lifetimes. Time-resolved PL decays are typically assigned an average PL lifetime, 〈τ〉, defined as the average time taken for a photon to be emitted from the chromophore. For a multiexponential decay function, 〈τ〉 is given by

〈τ〉 )

∑ n

Rnτ2n

∑ Rmτm

(1)

m

Values of 〈τ〉 for the five CdSe/CdZnS samples were previously shown to exhibit a marked temperature dependence,35 indicative of complex trapping dynamics; however, inferring details about these processes from 〈τ〉 alone is impractical in these systems because they disclose no information regarding the decays’ time-dependent structure. A more general expression for the average lifetime that incorporates the time-dependent behavior of the decay is given by

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∑ ∫0

Rnτn exp(-t'/τn) dt'

∑ ∫0

Rm exp(-t''/τm) dt''

texp

〈τ(texp)〉 )

n

texp

Jones et al.

(2)

m

where texp is a finite variable that can be thought of as a time window for the PL decay experiment. Note that the usual average PL lifetime expression, eq 1, is recovered when texp ) ∞. The contribution to 〈τ(texp)〉 of each component in the multiexponential decay is weighted by a factor (1 - exp(-texp/ τn)), causing faster processes to dominate at early times relative to 〈τ〉. Values of 〈τ(texp)〉 therefore increase with texp and converge on 〈τ〉 when texp is several times larger than the longest τn. As shown in Figure 1, 〈τ(texp)〉 varies markedly in each of our CdSe/CdZnS samples and the data start to reveal some of the time and temperature dependence of important processes occurring after photoexcitation. Several features in the 〈τ(texp)〉 landscape are common to all of the NC samples: the shortest PL lifetimes occur when the experimental window, texp, is smallest; one or more lifetime peaks appear when texp is long; and both hot and cold temperature limits see an upswing in 〈τ(texp)〉, especially at small texp. When the lifetime window is short, 〈τ(texp)〉 emphasizes processes that occur with a decay time that is shorter than texp, so transitions that occur rapidly on the time scale of the experiment, such as radiative recombination and exciton trapping will dominate the shape of 〈τ(texp)〉 when texp is just a few tens of nanoseconds. At much longer times after photoexcitation detrapping and other low-frequency transitions have a growing effect on 〈τ(texp)〉, which indicates that the maxima in Figure 1 at long texp times arise, at least in part, from slow reformation of excitons from confined trap states. In this article our goal is to start unraveling the excitation dynamics that give rise to the features in Figure 1, with our ultimate aim to be able to infer details about excited-state processes directly from TRPL measurements. In order to do this we develop a model of NC carrier trapping dynamics based on classical Marcus theory, demonstrate its ability to reproduce NC TRPL data, and illustrate how its predictions can help us draw some general conclusions about the fate of photogenerated excitons in CdSe NCs. 3.2. Kinetic Model of NC Photoexcitation Dynamics. The influence of trap states on CdSe NC PL dynamics has recently been the subject of several studies;38-45 however, to date, little consensus has emerged on the nature of exciton-trap interactions or on how the trap sites might be distributed either spatially about the quantum dots or energetically relative to exciton states. This is largely because of the inherent inhomogeneity of NC size, shape, and composition, which enforces a choice upon those who try to understand their photoexcitation dynamics: we can either use a microscopic approach in which we infer ensemble properties from a representative number of individual NCs; or a macroscopic approach, which requires us to deduce individual NC properties from a knowledge of ensemble photophysics. The microscopic approach has been exclusively used to study NC PL intermittency46 or to measure single-NC PL transients,3 and recent developments have even enabled the tracking of individual NCs in solution;47 however, the propensity for NC photo-oxidation4 and the dynamic nature of surface ligand binding48,49 coupled with the complexity of the experiments makes it impractical to measure a sufficient number of high resolution PL decays using microscopic techniques. The challenge, addressed in the present work, is to elucidate photophysical information at the single NC level

by analysis of ensemble PL decay data. To that end, a model needs to be developed that relates to the properties of individual NCs but allows for systematic variation of these properties when reproducing ensemble measurements. A comprehensive model of NC photoexcitation dynamics must account for both exciton relaxation dynamics and the way exciton population is depleted, and possibly sensitized some time later, by surface traps. As already mentioned, exciton structure and radiative recombination rates have been the subject of several studies and their calculated dependence on NC diameter accurately reflect experimental predictions. We can therefore utilize the wurtzite CdSe exciton model of Efros et al.29 as a foundation onto which the dynamical effects of trap states can be appended. Incorporating the interaction among excitons and surface trap states presents the main challenge because each NC in the ensemble has a distinct trap state number and energy distribution. Even at the single NC level, the trap state distribution can change as a function of time. This probably happens on a time scale of approximately ms and longer according to NMR50 and blinking studies.51 Figure 2a depicts a general kinetic scheme for a single NC that describes nonradiative transitions between a manifold of exciton states (Xi) and a few discrete trap states (Sj) located in or around the NC. The total exciton radiative recombination rate is kR. This scheme is the basis for all data analysis described in this article; however, in order for it to be useful we have to convert the single NC scheme into a scheme capable of reproducing photoexcitation dynamics in an ensemble of NCs. We do this by making two general assumptions: i. The likelihood of finding a NC possessing a trap state at energy, ε, at any given time within an ensemble of NCs is given by one or more probability distribution functions, PK(ε), where K labels each trap distribution. ii. Carrier trapping is an ET reaction that that converts a highly delocalized exciton state into a state in which one charge carrier is chiefly localized at the surface or at another interface and may be described in terms of classical Marcus ET theory.34 These assumptions greatly reduce the complexity of the problem since they allow us to make a connection between individual trapping events that could occur in a single NC and the collective dynamics that emerge when signals from a large number of NCs are averaged. Although there is growing evidence that trap states can be accessed by hot electrons and holes26,52 formed immediately after photoexcitation, by far the dominant trapping processes that occur after ∼1 ps are due to depopulation of NC exciton states near the band edge. We will consider the effects of hot carrier trapping in a future article; however, for now we base our study on band-edge trapping transitions only. The depopulation and repopulation of exciton states by interaction with surface trap states occurs by ET and recombination reactions, respectively. In the case of NCs, the ET is often specified as hole transfer because there is good evidence to suggest that the electron is transferred from a trap state to the unoccupied valence state of the NC. Figure 2b illustrates a graphical representation of the ET model we developed to model NC photoexcitation dynamics. It depicts the ET reaction between exciton fine-structure states and the trap state SK(ε) with energy ε in the Kth trap state distribution. Here we use the electronic state representation, not the electron-hole representation. Therefore the trap state distribution equally describes electron or hole transfer from the exciton to the surface. The electron

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Figure 1. Maps of weighted average PL lifetime, 〈τ(texp)〉, from five CdSe/CdZnS NC samples, plotted as a function of the time window, texp, and sample temperature (77-300 K) for each of the five NC samples. The horizontal dotted lines indicates the location of a slice taken from each of the data sets at 2 µs after photoexcitation and depicted in Figure 7. The contour labels are in units of ns.

Figure 2. (a) Simplified general kinetic scheme for a single NC showing radiative decay with rate, kR, from a manifold of exciton states (Xi) together with many possible transitions of rate kij(kji) to (from) a series of discrete of trap states (Sj). (b) A graphical representation of the Marcus ET model linking the free energy curves of NC exciton states (labeled by their total angular momentum) and that of a trap state, SK(ε), from a continuous distribution PK(ε). For clarity, we only illustrate ET parameters arising from the (2 fine-structure state and we identify the reorganization energy, λK, and the free energy change, ∆G((2,SK(ε)), of the transition between this state and SK(ε). The curves represent free energy because they include the perturbation of all the nuclear and solvent polarization degrees of freedom.

(hole) transfer is described by three parameters: the free energy difference, ∆G(i,SK(ε)), between minima in the nuclear potential energy curves; the reorganization energy, λK, required to distort the NC and its local environment until it can accept the transferred charge distribution; and the electronic coupling matrix element, VK, that reflects the strength of the interaction between reactants and products at the nuclear configuration of the transition state. Note that trapping is pictured, in Figure 2b, as an endergonic process: this is not meant to imply that ∆G(i,SK(ε)) must be positive. We depict just one of many trap states and ∆G can be both positive and negative for different traps in a single NC. A single reorganization energy and coupling strength is assigned to each trap distribution because we assume that traps mirror the spherical symmetry of the NCs and exist predominantly in narrow radial bands at NC interfaces, with the consequence that trap states within these “shells” experience roughly the same local environment. These parameters are required to calculate the ET rate from each fine-structure state to each trap state encompassed by PK(ε). To construct a kinetic model we need to estimate the PK(ε) trap distributions. In organic photoconductors, trap site energies are typically taken from a Gaussian distribution53 whereas the surface traps in bulk semiconductors are exponentially distrib-

uted below the band edge. Unlike bulk semiconductors, quantum dots have discrete exciton states so a Gaussian distribution may be appropriate; however, this approximation in organic materials relies on the assumption that the self-energies of adjacent sites are uncorrelated, which may not be true in NCs, because all the traps sampled by charge carriers on a single dot are structurally connected and the number of traps visited by carriers is limited to the number of traps on the dot. We therefore found that a probability distribution based on the convolution of Gaussian and exponential functions was able to model our data more effectively than Gaussian or exponential distributions alone. This function is shown in eq 3,

PK(ε) )

∫ R σ √2π

∞

1

K K

ε

(

1 exp 2RK

(

exp -

σK2

) ( )( (

(E - εK)2 2σK2

+ 2RK(ε - εK) 2RK2

exp

)

ε-E dE ) RK

1 - erf

σK2

+ RK(ε - εK)

√2RKσK

))

(3)

where εK is the peak energy of the Gaussian distribution, σK is its width, and RK is the exponential decay constant. Trapping rates from Xi to trap states in the Kth trap distribution follow from Marcus theory:34

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2π|VK | 2 kiK ) gKPK(ε) p



(

Jones et al.

)

-(∆GiK(ε) + λK)2 1 exp 4πλKkBT 4λKkBT (4)

dFi(t) ) dt

∑ (FK(t) ∫0

∞

kKi(ε) dε) -

K

Fi(t)(

∑ ( ∫0

∞

kKi(ε) dε) + kiR)

(6)

K

and reverse rates are given by

kKi )

(

gi ∆GiK(ε) k exp gKPK(ε) iK kBT

)

(5)

where gi and gK are, respectively, the degeneracy of Xi and the average number of trap sites in the Kth trap distribution. We account for all eight states within the ground exciton manifold and model their energies and radiative transition dipole strengths following Efros et al.29 An adjustable exchange parameter, η, and a fixed crystal field parameter, ∆ ) 19.4 meV, are used to determine the energies of all eight exciton states in the wurtzite quantum dots. A single adjustable-rate parameter, kR, is used, with η, to calculate the radiative rates of the (1L, (1U, and 0U states. The dark states, (2 and 0L, are assumed to have zero transition dipole strength. All nonradiative processes linking exciton or trap states to the ground state are disregarded for simplicity. Finally a fixed fast ∼1 ps rate was chosen for transitions descending the exciton manifold,54 variation of this rate had negligible effect on the PL dynamics at the temporal resolution of our experiments. All reverse rates were calculated via detailed balance. We generated a kinetic scheme based on the framework in Figure 2b. The number of states, N, in the scheme is given by N ) 9 + ∑KNK, where NK is the number of traps used to approximate the Kth “continuous” ensemble trap distribution. It was adequate to set NK such that the spacing between consecutive trap states was e5 meV. To model TRPL decays, we are interested in the timedependent population of the bright excitons, therefore PL dynamics are formulated in terms of probabilities, Fi(t), that Xi is populated at time t after excitation. Since N was typically >100, we determined Fi(t) by finding numerical solution to a series of N rate equations:

where FK(t) is the total population of the Kth trap distribution and kiR is the radiative transition rate from Xi. The multiexponential PL decay functions, I(t,T), which had been extracted directly from TCSPC data at each temperature, were then modeled by minimizing the value, χ, given by

χ)

∑ [ ∫t T

texp

1

(I(t, T) - C

∑ kiRFi(t, T))2 dt]

(7)

i

where C is a variable scaling factor, t1 ) 500 ps and texp ) 2 µs: a temporal range within which we had good confidence in the accuracy of our data. As indicated in eq 7, for each NC sample we analyzed all 12 data sets simultaneously using just one set of parameters. We found that the constraints imposed by this type of global analysis were essential to help extract physically valid dynamics. 4. Results and Discussion It was found that two trap state distributions were required to model the PL decays adequately in each of the five NC samples. An illustration of typical exciton and ensemble state distributions that were recovered from analysis of the 2.8 nm diameter NC PL data are presented in Figure 3a. Zero energy, on the ordinate, is defined to coincide with the exciton state energy in the absence of crystal field and exchange interactions. We found that the two trap distributions were distinct in their contributions to the NC excitation dynamics. Trap distribution 1 (K ) 1) is typically broad and centered more than 100 meV higher in energy than the exciton states. Trap distribution 2 is roughly the same width as the total exciton splitting and is centered close to 0 eV. The broad K ) 1 distributions are associated with reorganization energies that range from 150 meV in sample I up to 275 meV in sample IV. As discussed in our previous article,35 the

Figure 3. (a) Exciton states (left) and two trap distributions (K ) 1,2) required to acceptably fit PL decay data in the 2.8 nm NCs (sample III). (b) Trap 1 probability distribution functions of each of the five NC samples plotted vertically on a semilog axis. The exponential tail of each distribution is highlighted and extrapolated for clarity by dotted lines.

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Figure 4. Experimental and calculated maps of weighted average lifetime, 〈τ(texp)〉, for two representative NC samples: (a) 2.1 nm core diameter (sample I) and (b) 4.3 nm core diameter (sample V). Experimental values are on the left in (a) and (b) and calculated values from the results of our analysis are shown on the right.

magnitude of these reorganization energies together with their sensitivity to solvent polarizability means this distribution is likely to represent surface-localized trap states. Conversely, the narrower K ) 2 distributions have reorganization energies that are just a few tens of meV and were assigned to trap sites localized between core and shell, although it is possible that this distribution is needed to make up for inadequacies in our exciton model. The existence of a broad distribution of nonradiative surface-localized states that are predominantly higher in energy than the excitons is largely consistent with recent theoretical work from Kilina, et al.28,55 Considering fully ligated CdSe clusters, they predict substantial charge redistribution and polarization interactions on the surface that are due to surface reconstruction and strong bonding interactions between the surface and ligands. Optically dark hybridized states result, where electron density is at least partially localized on the ligand atoms. Removal of one of the more weakly coordinating ligands was found to have little effect on the electronic structure; however, removal of a strongly bound ligand introduced trap states within the NC band gap. These theoretical results therefore point to an ensemble distribution of dark, or at least semibright, incompletely passivated states with energies in the vicinity of the exciton state that arise due to incomplete ligand coverage. Figure 3b depicts each of the K ) 1 trap distributions plotted on a semilog axis, which emphasizes the exponential tails (highlighted by dotted lines) in the trap distributions. Each of these tails extend the distributions well below the exciton energies; in fact, it was found that the quality of the fits were most sensitive to the precise shape of the low-energy tails and far less sensitive to the distributions’ profiles at their peak energies. It turns out that the exponential tails are also very important in determining the shape of the average PL lifetime curves, 〈τ(texp)〉, as we shall discuss below. An illustration of the excellent correspondence between our model and the experimental decays has been previously reported.35 Here we plot calculated values of 〈τ(texp)〉 from parameters that were obtained from the analysis of each sample and we compare them to experimental data for the smallest and largest (samples I and V) NCs, Figure 4. There is good correspondence between calculated and experimental values of 〈τ(texp)〉. Each of the major features are well reproduced, although there are some small discrepancies in the maximum and minimum lifetime values, especially in sample V. Figure 4 demonstrates that such a relatively simple

model can capture most of the NC dynamics; however, despite its simplicity, application of the model to fit large quantities of PL data is not trivial. In order for our results to be of more general use we should consider how to connect information about trap-state dynamics gleaned from our model directly with trends in the PL data. 4.1. Elementary Kinetic Schemes: the Influence of the Experimental Time Window. The average PL lifetime data, highlighted in Figure 1, represents a good foundation from which we can construct links with the calculated dynamics, because these data are straightforward to calculate directly from PL decays and because they contain temperature-dependent structure. As a starting point we highlight two simple kinetic schemes, which can be solved analytically. These will enable us to identify similar trends in the dynamics in the full scheme that was depicted in Figure 2b. The simplest scheme that we can envisage as arising from Figure 2a consists of just three states: X (exciton), S (trap), and G (ground). We can write differential equations describing the populations of each of these states:

dFX ) FSkSX - FX(kXS + kR) dt dFS ) FXkXS - FSkSX dt dFG ) FXkR dt

(8)

which can be solved to give an expression for the population of X:

FX ) A1 exp(k1t) + A2 exp(k2t)

(9)

where k1(k2) )

( 21 )(-(k

XS

+ kSX + kR) ( √(kXS + kSX + kR)2 - 4kSXkR) A1 )

kXS + kR + k2 k 2 - k1 A2 ) 1 +

kXS + kR + k2 k 1 - k2

(10)

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Jones et al.

Figure 5. Simulated average PL lifetime curves: (a) 〈τ3(texp)〉 × kR and (b) 〈τ4(texp)〉 × kR calculated for the two schemes shown on the left. The red lines in plots (a) and (b) follow eqs 11 and 13 (texp ) ∞), while the blue dashed lines, (texp ) 10 µs), were calculated numerically with λXS ) 200 meV and VXS ) 10 µeV. The overlaid numbers are the ∆G values (in meV) used in each calculation.

From this we can obtain an expression for the three-state average PL lifetime, 〈τ3〉 (cf. eq 1):

〈τ3〉 )

(

( ))

kXS + kSX 1 ∆G ) 1 + exp kRkSX kR kBT

(11)

This expression is notably lacking in any dependence on the reorganization energy or electronic coupling matrix element for the trapping transition: the average three-state PL lifetime is determined solely by ∆G and kR. Equation 11 is valid when texp ) ∞, but if - 1/k1 or - 1/k2 are of the order of, or greater than, the time window of the experiment, a more complex expression for 〈τ3(texp)〉 results (eq 12) that is no longer independent of the electron transfer parameters and can be markedly different to 〈τ3〉.

〈τ3(texp)〉 ) A1k22(1 - exp(k1t2)) + A2k21(1 - exp(k2t2)) k1k2(a1k2(1 - exp(k1t2)) + a2k1(1 - exp(k2t2)))

initially rise and then start to fall again as they converge with 〈τ3〉, thereby giving rise to the peaks in 〈τ3(texp)〉 that are observed, in the three-state model, when ∆G < 0. With a slight change, we can make the simple three-state model more closely resemble the electronic structure of CdSe NCs. We do this by splitting the exciton state, X, by energy, δX, into a bright and a dark state, XB and XD. If we assume that the population of XB and XD can be deduced from Boltzmann statistics we find, analogously to eq 11:

〈τ4(texp)〉 )

Both average lifetime functions are shown in Figure 5a, in which the red and blue dashed lines represent 〈τ3〉 and 〈τ3(texp)〉, respectively, calculated for a series of ∆G values from -100 to 100 meV. At low temperatures 〈τ3〉 approaches (1/kR) when ∆G > 0 and the excited population resides almost completely in the X state. Conversely, when ∆G < 0, 〈τ3〉 increases dramatically at low temperatures, as the majority of the population ends up in the S state. In both cases 〈τ3〉 converges toward (2/kR) at high temperatures when the X and S populations are approximately equal. While 〈τ3〉 has no turning points, we observe that 〈τ3(texp)〉 with ∆G < 0 has a peak that resembles the peaks observed at long texp times in our 〈τ(texp)〉 NC data in Figure 1 (see also the slice taken from 〈τ(texp)〉 at texp ) 2 µs and shown in Figure 7). This peak arises because 1/k1 is longer than the time window, texp, at low temperatures, which, as mentioned earlier, causes these rates to be de-emphasized by a factor (1 - exp (k1texp)) in the calculation of 〈τ3(texp)〉. This results in 〈τ3(texp)〉 being dominated at low temperatures by the contribution of k2 and converging toward the radiative lifetime, 1/kR. As the temperature rises, however, the XTS transition rates increase and start to occur within the time window, texp, causing the 〈τ3(texp)〉 to

( ))

(13)

where:

φ)

(12)

(

1 1 ∆G + exp kR φ kBT

∆G ) ES - EXB exp(-EXB - EXD /kBT)

(14)

(1 + exp(-EXB - EXD /kBT))

Values of 〈τ4〉 and 〈τ4(texp ) 10 µs)〉, computed using eq 12, are plotted in Figure 5b for δX ) 2 meV (red and blue dashed curves, respectively). The shapes of the 〈τ4〉 functions are slightly different to 〈τ3〉. As temperature drops, 〈τ4〉 approaches a Boltzmann-weighted radiative lifetime (1/φkR) when ∆G > 0 and increases even more dramatically than 〈τ3〉 when ∆G < 0. In all cases 〈τ4〉 converges toward (3/kR) at high temperatures when the XB, XD, and S populations are approximately equal. This slightly different behavior of 〈τ4〉 versus 〈τ3〉 produces a turning point in 〈τ4〉 when ∆G > 0; however, when the observation window is reduced to 10 µs, the average PL lifetimes, as given by 〈τ4(texp ) 10 µs)〉, clearly show two turning points when ∆G < 0. The low-temperature turning point is a minimum that is caused by 1/k2 approaching a radiative lifetime, (1/φkR), that grows as the temperature drops. The second, a maximum at higher temperatures, has the same origins as the one observed in the 〈τ3(texp ) 10 µs)〉 data. Since these two turning points bear a resemblance to the two features highlighted in Figure 1b and despite the greater complexity of our full model of NC dynamics, we suggest that the trends we observe in PL data have the same general origins as the trends in these simple models. We can point to several

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Figure 6. Temperature-dependent population of trap distribution K ) 1 in each of the five NC samples with texp ) 2 µs. Saddle points (SPs) in the population maps are indicated by a red dot and are located at the intersection of the dashed lines. The horizontal dotted lines denote the energy that the gradient of P1(ε) reaches a maximum (i.e., where ∂2P1(ε)/∂ε2 ) 0); while the vertical dotted lines marks the temperature, TR, of the ∆G < 0 population peak, which is denoted by a white dot. The red lines that wind across each image locate the approximate temperature-dependent peak population energies.

subtle differences caused by the size and shape of the trap state distributions, but as a guide to NC photoexcitation dynamics, the three- and four-state schemes pictured in Figure 5 do surprisingly well. These simple models serve to underscore a general effect that applies to NC PL, namely: the temporal window through which we observe PL decays affects the calculation of average PL lifetimes and must be taken into account before 〈τ(texp)〉 can be considered a useful indicator of photoexcitation dynamics. If we examine Figure 1a we see that the values of 〈τ(texp)〉 appear to plateau around 1 µs. This does not fit with recent reports56,57 that link PL decay dynamics with PL intermittency that occurs over millisecond and longer time scales and would ensure a steady increase in 〈τ(texp)〉 for texp longer than 1 µs. In reality, the plateau is a result of our experimental window (texp ) 5 µs) that sets an upper limit on the time scale of processes that can be recorded accurately. 4.2. Connection between Trap Population and PL Lifetime. We now discuss the NC 〈τ(texp)〉 data in more detail and, in combination with our detailed models, start to unravel what they can tell us about NC photoexcitation dynamics. In Figure 6 we illustrate the calculated average population located within the K ) 1 distribution functions (cf. Figure 3) during the first 2 µs after photoexcitation. The maps from each NC sample except sample III are split into quadrants defined by a horizontal line at the mean exciton energy (0 eV) and a vertical line that crosses the abscissa at a temperature, TSP, where “SP” denotes the saddle point on the population landscape. The maps show that traps in the distribution that are lower in energy than the exciton states (∆G < 0), are predominantly populated when T < TSP (in the lower left quadrant); while traps with ∆G > 0 are populated when T > TSP (upper right quadrant). This behavior is strongly dependent on P1(ε), the shape of trap 1’s probability distribution function, which describes the density of states that are available for trapping. Low temperatures favor transitions with small activation energies, causing traps with low ∆G to evolve most of the population; however, as shown in Figure 3, P1(ε) has an exponential tail that stretches well below the exciton states and as the temperature rises, the number of accessible traps also grows. This has the effect of generating a population peak at a temperature, TR, for traps with ∆G < 0, as shown in Figure 6. As T continues to rise, rapidly escalating values of P1(ε) causes the trapping rates to the higher traps to grow ever more

quickly, which results in a broadening and flattening of the trap 1 population distribution along the energy axis until a saddle point is reached, in all but one of our NC samples, at a temperature, TSP. Around TSP the peak population energy (red line in Figure 6) swings to higher energy by up to 100 meV (in sample II) within a ∼10 K temperature range and then continues to rise as more and more population is forced into higher energy traps by the rapidly rising trap density. At temperatures greater than TSP the majority of the trap population is situated in traps that are higher in energy than the exciton states. Trap population energies continue to rise with temperature in each of our samples, but the peak trap population energy appears to converge upon the energy at which the trap state distribution gradient is a maximum, i.e., where ∂2P1(ε)/ ∂ε2 ) 0. This point is denoted in Figure 6 by horizontal dotted lines. We can link several features in Figure 6 to trends in 〈τ(texp ) 2 µs)〉, the 2 µs average PL lifetimes. In Figure 7 we plot 〈τ(texp ) 2 µs)〉 data extracted from the PL decays of each of the five NC samples; note that these data correspond to horizontal slices of Figure 1a at texp ) 2 µs. Onto these data we have highlighted three trends in blue, green, and red, and we can identify each of these with specific processes occurring in our NCs. i. Blue. As illustrated in Figure 5b, when texp is of the order of a few µs, a four-state system consisting of a bright and a dark exciton state that are both coupled to a trap state results in a steeply decaying average PL lifetime, 〈τ(texp)〉, at low temperatures. We therefore identify the components of Figure 7, highlighted in blue, as arising primarily from the splitting between “dark” (2 and “bright” (1L exciton states. These components are most important in samples I and II because, at 7.0 and 5.7 meV, respectively, the (2 - (1L splitting is largest in the smallest diameter NCs. The importance of this effect on 〈τ(texp ) 2 µs)〉 dwindles almost to nothing by sample III, which has a calculated splitting of 3.6 meV. ii. Green. We know from Figure 6 that when T < TSP, in the first 2 µs, populated traps are predominantly lower energy than the exciton states, i.e., ∆G < 0. In the three- and four-state schemes discussed in Figure 5 we see that when ∆G < 0, there is a peak in 〈τ(texp)〉 that arises as trapping and detrapping transitions occur with increasing frequency within the experimental time window and 〈τ(texp ) 2 µs)〉 approaches 〈τ〉. We

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Figure 7. Plots of 〈τ(texp ) 2 µs)〉 for each of our five NC samples. Trends in the data are highlighted by red, green, and blue curves, and their meaning is discussed in the text. The two vertical dashed lines denote the temperatures TR and TSP.

PL decays of four of the samples, but what does the value of TSP tell us about the NC photoexcitation dynamics? To reveal the importance of TSP we plot in Figure 8 the ratio, kiS/kSi, of trapping to detrapping rates for the transition (1LfSK)1(ε), where SK(ε) is the Kth trap distribution. When T < TSP the maximum value of kiS/kSi occurs for traps that are lower energy than the exciton state, whereas when T > TSP the maximum value of kiS/kSi occurs for higher energy traps. At T ) TSP the maximum value of kiS/kSi occurs at the exciton state energy (ε ) 0), i.e.,

∂(kiS /kSi) )0 ∂ε

(15)

From eqs 4 and 5 we can derive the relation: Figure 8. Ratio of trapping to detrapping rates kiS/kSi in sample I for the transition (1LfSK)1(ε) plotted as a function of energy (ε) at temperatures from 100 to 300 K. Note that ε ) 0 corresponds to the mean energy of the exciton states and is therefore slightly higher than the energy of the (1L state. When T < TSP we observe that kiS/kSi is larger for traps lower in energy than the (1L state, but when T > TSP the maximum value of kiS/kSi involves traps higher in energy than (1L.

therefore identify the green-highlighted peaks in all but sample I with the peaks calculated for our simple models in Figure 5. It is notable that the green-highlighted peak is nonexistent in sample I; while this feature is present in sample II, it is much smaller compared to the other three larger samples. The reason for this is directly related to the rate of growth of P1(ε) with energy. As shown in Figure 3b, the trap state distribution functions, g1P1, in samples III, IV, and V are typically larger than in samples I and II at energies lower than the exciton states, so for transitions with ∆G < 0, the surface traps play a smaller role in the dynamics of I and II than of the other samples. iii. Red. The transition from green to red in Figure 7 occurs very close to the values of TSP that we extracted from Figure 6, which indicates that red-highlighted sections arise predominantly due to population of traps that have a higher energy than exciton states. In Figure 5, the states with ∆G > 0 generate 〈τ(texp)〉 curves that rise with temperature as trapping rates increase, and this fits with what we observe in our data: as the population of higher energy traps grow (Figure 6) the rate at which exciton states are available for radiative recombination diminishes, thereby increasing the average PL lifetimes. It is therefore fairly straightforward to identify TSP in the 〈τ(texp)〉 data extracted from

kBTSP

∂PK ) PK ∂ε

(16)

and using eq 3 this becomes

(

1 kBTSP

)

EK2 σK2 EK - 2 RK 2σK 2RK2 1 ) + EK RK σK √πσK erf -1 √2RK √2σK

√2 exp

((

) )

(17)

While this appears a somewhat complex expression it simplifies to

kBTSP ≈ RK

(18)

when PK is an approximately exponential function close to ε ) 0. Referring to Figure 3b we see that this is the case for samples I and II and we find that eq 18 holds very well for these two samples: TSP(I) ) 194 K and TSP(II) ) 250 K, yielding, RK(I) ) 0.017 eV and RK(II) ) 0.022, which are virtually identical to the values obtained using our detailed model and previously reported.35 In these cases therefore, identifying TSP from average PL lifetime data, as shown in Figure 7, allows us to determine the rate constant of the exponential trap distribution tail.

Photoluminescence of Colloidal CdSe Nanocrystals 5. Conclusions Understanding NC photoexcitation dynamics requires careful interpretation of PL data. We have developed a physically realistic model to do this, and we have demonstrated its ability to reproduce PL decays and average lifetime data with good accuracy. This has allowed us to start uncovering some of the kinetic complexities induced by the presence of traps that are able to interact with core excitons; however, in this article we have tried to emphasize signatures of the carrier dynamics that can be interpreted by direct examination of PL lifetime data. We found that what can be learned from average lifetimes depends strongly on the dynamic range of the time axis. Despite being an extremely common measurement in the experimentalist’s arsenal, our ability to extract NC dynamics from time-resolved PL remains limited. For this reason, the ultimate aim of this work has been to develop a set of guidelines or rules to aid the interpretation of PL data. To a limited degree, we can now examine the shape of temperature-dependent average lifetime data and potentially determine at which temperatures the occupied trap states are predominantly lower or higher energy than the exciton states. In certain cases, we have also demonstrated that the identification of a single temperature from the PL lifetime curves can enable us to unravel the functional form of the trap distribution’s tail. What does this mean for the researcher who might want to use time-dependent PL to study NCs? All our analysis supports the fact that the lowest exciton states dominate PL decay rates in NCs with high quantum yields and in these materials average PL lifetimes are a reasonable approximation of the radiative lifetime of the lowest exciton manifold in the absence of any other states. CdSe NCs have a lowest exciton state that is dark. This means that at very low temperatures the average PL lifetime is long, but it is also a good approximation of the exciton radiative lifetime because trap states are rarely formed. Approaching room temperature the perturbations imposed on the PL decays by the occupation of trap states can modify average lifetimes by more than 30% from their exciton-only values and this makes predictions of exciton dynamics from single decays unreliable. Time-dependent PL is potentially an extremely useful way to delve into the complexities of NC photoexcitation dynamics, but it is an experiment that should not be performed in isolation. At present our understanding of NCs is such that an accurate picture can only be obtained when experiments are performed in parallel with an additional perturbation, e.g., varying the temperature, dielectric constant, electric field, etc. Indeed, such experiments could add significantly to our understanding and, coupled with careful data analysis, could yield important contributions to the field of NC photophysics. Acknowledgment. The Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged for support of this research. G.D.S. acknowledges the support of an EWR Steacie Memorial Fellowship. References and Notes (1) Al Salman, A.; Tortschanoff, A.; Van Der Zwan, G.; Van Mourik, F.; Chergui, M. Chem. Phys. 2009, 357 (1-3), 96–101. (2) Donega, C.d. M.; Bode, M.; Meijerink, A. Phys. ReV. B 2006, 74, 085320. (3) Fomenko, V.; Nesbitt, D. J. Nano Lett. 2008, 8, 287–293. (4) Jones, M.; Nedeljkovic, J.; Ellingson, R. J.; Nozik, A. J.; Rumbles, G. J. Phys. Chem. B 2003, 107, 11346–11352. (5) Kippeny, T. C.; Bowers, M. J.; Dukes, A. D.; McBride, J. R.; Orndorff, R. L.; Garrett, M. D.; Rosenthal, S. J. J. Chem. Phys. 2008, 128 (8), 084713.

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