Connecting the Dots: The Kinetics and Thermodynamics of Hot, Cold

Article pubs.acs.org/JPCC

Connecting the Dots: The Kinetics and Thermodynamics of Hot, Cold, and Surface-Trapped Excitons in Semiconductor Nanocrystals Jonathan Mooney, Michael M. Krause, and Patanjali Kambhampati* Department of Chemistry, McGill University, Montreal, QC, H3A 0B8, Canada ABSTRACT: The excitonics of semiconductor nanocrystals (NC) depend upon temperature in a complex manner due to the interplay between the kinetics of hot exciton relaxation/trapping and the thermodynamics leading to cold exciton recombination. We apply a semiclassical electron transfer model of surface trapping to temperature-dependent absorption and emission data to elucidate a microscopic picture of the factors which govern the fate of hot and cold excitons. The linear absorption spectra reveal a unique temperature-dependence to the energies of higher excitonic states, while oscillator strength is shown to be temperature invariant. We identify the phonon based origin to the anomalous low temperature peak energy trend in photoluminescence (PL) spectra. PL intensities, PL lifetimes, and absorption spectra are used to demonstrate that variation of quantum yield with temperatures arises from the thermally controlled fraction of NC which emit, rather than from an activated nonradiative pathway common to all NCs. Experimental quantum yield spectra are shown for several NCs and we perform a much-needed analysis of the role of surface PL in quantum yield. Finally, we show that a semiclassical electron transfer model including hot excitonic effects can explain experimental quantum yield spectra and suggests how to probe kinetic trapping processes via simple steady-state spectroscopy.

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INTRODUCTION Semiconductor nanocrystals (NCs) are of interest due to their potential application in devices such as light-emitting diodes,1 lasers,2 and solar cells.3 A proper understanding of the excitonics of these systems is critical to successful device design and fabrication. While a good understanding of the optical and electronic properties of the well-known quantized core states in NCs and their dependence on temperature has been established,4 the way in which hot excitons thermalize to surface traps remains poorly understood. Higher energy excitation results in hot excitons which may cool via standard relaxation pathways or alternatively experience hot exciton surface trapping, transport, or photoionization.5 Thermal aspects of NCs are particularly important because they affect design properties such as brightness and conductance. The effect of temperature on the intensity, lifetime, and energy of the band gap photoluminescence (PL) from semiconductor NCs has been well investigated.6 These studies have provided insight into the efficiency of electron−phonon coupling and the nonradiative relaxation pathways that compete with the desired radiative recombination,6a effects which are significant to the design of devices such as NC-based LEDs7 or field effect transistors (FET).8 Nevertheless, questions remain about NC temperature-dependence. For electronically coupled NC films, it is not obvious whether the conductance proceeds via coherent band-like transport or via incoherent (variable range) hopping mechanisms. As similar experiments9 suggest differing mechanistic pictures of a fundamental process, it is clear that our knowledge of the temperature-dependent electronic and optical properties of NCs is far from complete. © 2014 American Chemical Society

Our group has examined temperature-dependent PL to understand the processes behind surface trapping and white light production.10 It is well-known that at low temperatures NCs can emit broadened and red-shifted PL which is attributed to recombination involving a charge trapped on the surface of the NC. We have shown a simple thermodynamic connection between core and surface PL bands in which semiclassical electron-transfer theory can account for the coupling between these states.10a,c Our results focused upon the thermodynamics of relaxed or cold excitons, which are created when the band edge state is directly populated by a photon; here surface trapping occurs under thermodynamic equilibrium conditions which may be probed in continuous wave (CW) PL spectroscopy. But surface trapping may also proceed in nonequilibrium conditions via hot exciton trapping, a process known as photocharging or photoionization in the carrier multiplication (CM) literature.11 In this situation, a high-energy photon creates a high-energy hot (excited) exciton that may trap to the surface if this process is competitive with hot exciton relaxation.5a These hot exciton trapping effects have been explored in detail by our group5a,12 and others13 via pump/ probe transient absorption (TA) spectroscopy.5a,12,14 Our TA work has shown that time domain nonequilibrium measurements can directly monitor surface trapping of relaxed as well as hot excitons. Received: February 28, 2014 Revised: March 11, 2014 Published: March 12, 2014 7730

dx.doi.org/10.1021/jp502102a | J. Phys. Chem. C 2014, 118, 7730−7739

The Journal of Physical Chemistry C

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Lakeshore model 331 temperature controller. Steady-state absorption measurements were performed on a Varian Cary 5000 UV/visible spectrophotometer. Photoluminescence spectra were measured on a Spex Fluoromax-2 spectrofluorometer. Time-resolved data were obtained with use of a Picoquant Fluotime TSCPS setup employing an LDH 470 ps diode laser (Picoquant) with an excitation wavelength of 470 nm.

While these direct time-domain measurements of hot excitonic processes, e.g., femtosecond TA measurements, are information rich,5a,14 simpler continuous-wave (CW) techniques to measure surface trapping and/or hot exciton processes are desired. We5a and others15 have recently suggested that the quantum yield (QY) depends on the initially excited core state and that it should be possible to measure a quantum yield spectrum by varying excitation energy. Thus hot exciton effects may be present even in simple continuous-wave PL experiments. Previously, Rumbles and Nozik argued that there was a strong excitation energy-dependence to quantum yield,16 while Chergui and co-workers indicated no intrinsic energy dependence to quantum yield.17 More recently our group has suggested the presence of a QY spectrum in CdSe core5a as well as core/barrier/shell systems,18 while Loomis and coworkers showed an unusually strong excitation energy dependence to the PL quantum yield, i.e., a quantum yield spectrum.19 While the presence or absence of a quantum yield spectrum has been discussed for some time,20 it is clear that a microscopic understanding of the processes which govern the fate of hot excitons is far from well understood. Such an understanding may enable simpler continuous-wave (CW) measurements of hot exciton processes in lieu of the far more complex femtosecond transient absorption experiments. Here we report on the temperature dependence of excitonic processes in CdSe semiconductor nanocrystals which, in conjunction with theory, reveals the first microscopic basis for understanding hot exciton surface trapping. We perform temperature-dependent continuous-wave spectroscopy as well as photoluminescence lifetime measurements. Absorption spectra show a strong state-dependence to changes in peak energy with temperature, while the oscillator strengths are temperature invariant. Analysis of absorption spectra, PL spectra, and PL lifetimes reveals that the well-known temperature-dependence to the quantum yield in NCs arises due to the fraction of NC which emit changing, rather than the expected changes in radiative or nonradiative decay rates. The PLE spectra are used to explore the possibility of measuring a photoaction spectrum for surface trappinga quantum yield spectrum for PL. We provide a microscopic theory that rationalizes both the presence and the absence of a QY spectrum in all NC systems based upon the thermodynamics of excitons.

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RESULTS AND DISCUSSION Lattice Expansion and Phonon Progressions Explain Temperature Dependence of Absorption and Photoluminescence Spectra. To begin the exploration of exciton thermodynamics, steady-state absorption measurements were obtained on several samples of CdSe and CdS NCs at temperatures between 5 and 300 K. Figure 1a shows the

Figure 1. Linear absorption (a) and photoluminescence (b) spectra of the same sample of CdSe NCs. The energy of the band edge exciton, X1, is 2.13 eV at 300 K.

absorption spectra of a characteristic sample of CdSe NCs (R = 1.93 nm) over this temperature range. The spectra display several expected features, which are assigned to excitonic states denoted X1 (1Se-1S3/2 in the effective mass approximation), X2 (1Se−3S3/2), and X3 (1Pe−1P3/2).22 In this NC, these features are located at 2.13, 2.25, and 2.57 eV at 300 K, respectively. As temperature decreases, these features blueshift. Figure 1b shows the photoluminescence (PL) spectra of the same CdSe NCs over the same temperature range, presented in energy units and corrected by using the Jacobian transformation.23 Two bands are visible in the emission spectra: an intense, narrow, high-energy band (∼2.1 eV at 295 K) that arises from the band edge excitonic PL from the NC core, and a weak, broad, low-energy band (∼1.65 eV at 295 K) that arises from surface-related emission in which a charge carrier is localized on the surface of the nanocrystal.24 As with the absorption spectrum features, the band-edge PL from the core blueshifts as temperature is decreased, although unlike in the absorption spectra, this trend reverses below 50 K. The temperature-dependence of the X1 feature of the absorption spectrum for several different samples of NCs is shown in Figure 2a. This temperature-dependence can be modeled by using the empirical Varshni equation:25

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EXPERIMENTAL METHODS Solutions of colloidal CdSe and CdSe/ZnS NCs passivated with octadecylamine (ODA) ligands were used as received from NN Laboratories. Solutions of colloidal CdS NCs passivated with oleic acid ligands were used as received from NN Laboratories. Cellulose triacetate was purchased from Sigma Aldrich. Polymer solutions of 5.5 wt %/wt cellulose triacetate (CTA) in 1:9 methanol:dicholoromethane were prepared. Samples were prepared by thoroughly mixing NC solutions (original concentration: 0.5 g NC/100 mL toluene) in a 1:20 proportion with the CTA solutions.21 These samples were drop-cast in Petri dishes which were covered and the solvent was allowed to evaporate over a period of at least 48 h, producing films of NCs dispersed in hard polymer matrices. Films were mounted in a Janis (STVP-100) flow cryostat, the vacuum chamber of which was evacuated continuously during the experiment by using a turbomolecular pump. Sample temperatures were adjusted via N2 or LHe flow with a 7731



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Figure 2. (a) Temperature dependence of the energy of the X1 feature of several samples and sizes of NCs. The X1 feature at 300 K is listed in the figure for each sample. (b) Temperature dependence of the energy of the X1, X2, and X3 features of samples of CdSe NCs possessing an X1 feature at 2.26 eV at 300 K. The linear constant from the Varshni equation provides slope values of 5.15 × 10−4, 5.77 × 10−4, and 6.16 × 10−4 eV/K for X1, X2, and X3, respectively. (c) Temperature dependence of the energy of band edge (blue) and surface (red) photoluminescence bands of NCs. The energy of the X1 absorption feature is shown in black for comparison. The Varshni equation can successfully reproduce the trend above 50 K for the band edge emission, but cannot account for the behavior of band-edge emission below 50 K nor for that of surface emission at any temperature.

Eg = E0 −

αT 2 T+β

observed dE/dT for the higher states does not exactly match this simple calculation, one can anticipate more complex electronic structural effects are at play. Next, we examine the temperature dependence of the PL spectra. The temperature dependence of the peak energies of the NC emission bands is shown in Figure 2c. The Varshni equation can be used to successfully model the temperature dependence of the energy of band-edge emission between 50 an 300 K; however, below ∼50 K the value dE/dT becomes positive, in contradiction to the Varshni equation. Such anomalous behavior has been observed previously in some cases28 but not in others.6a,26 With regard to surface emission, there is no clear trend of energy with temperature among the different samples studied. Scholes and co-workers have previously posited that the “turnover temperature” for the band edge PL is related to the lowest longitudinal acoustic phonon frequency and population of disorder states at intermediate temperatures.28a To explore the behavior of dE/ dT below the turnover temperature, we modeled photoluminescence spectra as a function of temperature, assuming one emitting state and including Franck−Condon phonon progressions. The phonon progressions are simulated in the time domain by using the wavepacket approach of Heller et al.29 but explicitly incorporating the thermal occupation of phonon modes:30

(1)

Clearly, as temperature decreases, the energy of the X1 feature increases consistently in all samples, albeit at a much slower rate below 50 K than above 50 K. These observations are consistent with prior works.6a,26 While the temperature dependence of the band edge excitonic peak energy is well-known, the response of the higher excitonic states is not. Figure 2b shows the temperature dependence of the X1, X2, and X3 features for NCs with an X1 feature at 2.26 eV at 300 K. The temperature dependence of the energy of each of these features can be fit by using the Varshni equation, in which the parameter α is linearly proportional to the magnitude of temperature dependence. For the NCs shown in Figure 2, the X1, X2, and X3 features exhibit values for the α parameter of 5.15 ± 0.51, 5.77 ± 0.83, and 6.16 ± 2.32 (×10−4 eV/K), respectively. This overall trend of increasing values of α for higher energy features is reproduced generally by other NC samples. The primary factors contributing to the band gap temperature dependence are lattice expansion with temperature and change in electron−phonon interaction with temperature.25,27 The effect of lattice expansion with temperature is modeled in Figure 2a. Since lattice expansion with temperature plays a major role in dE/dT behavior, the greater temperature dependence observed for higher quantum states can be qualitatively analyzed by examining the effect of lattice expansion on higher energy eigenstates. The particles in sphere energy eigenvalues are

∞

IT(ω) =

2 2

∫−∞ ⟨0|0(t )⟩eiωt−(1/2)Θ t −Γ|t|

(4)

⟨0|0(t )⟩ = Πke−Sk[(2⟨nj⟩+ 1)(1 − cos ωk) + i sin ωk] − iω0t

(5)

⟨nj⟩ = (e ωk / kbT − 1)−1

(6)

2 2

E=

hk 8mr 2

(2)

Thus

Here ω is the excitation frequency, Θ is an inhomogeneous broadening parameter, Γ is the homogeneous line width, S is the Huang−Rhys parameter, ωk is the phonon energy, and ω0 represents the energy difference between the ground and excited state potential energy surface minima. The homogeneous line width was taken to be 130 cm−1 at 10 K based on ref 31. The value of S was taken to be 0.32, consistent with our recent time-domain measurements,32 which provide the most precise measure of the intrinsic exciton−phonon coupling.5a,14,33 The inhomogeneous broadening was taken to be 40 cm−1. Details of spectral line shape modeling may be found elsewhere.10c,33

2 2

dE dE hk ≈ =− dT dr 4mr 3

(3)

The zeroes of the spherical Bessel function, k, increase as π, 4.14, and 2π for the states X1, X2, and X3, respectively. Therefore, dE/dT, which due to lattice expansion is proportional to dE/dr and hence the Bessel function zeros, should also increase from X1 to X3. This increase is observed experimentally (Figure 2b), showing that a basic understanding of the state dependence of dE/dT can be understood with reference to the lattice expansion with temperature. Since the 7732



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lowered.6a Quantum yields of CdSe NCs at 300 K are commonly 5−10%, but can increase to 50% below 100 K. For bright CdSe/ZnS and related core/shell structures, the QY can start at 50% at 300 K and increase to 90% below 100 K. Trivially one expects some thermally activated nonradiative process to be responsible, but the specific origin of the observation has remained elusive. To better connect the PL intensities to an understanding of the temperature-dependence of the quantum yield, we measure the PL lifetimes as a function of temperature. The intensityweighted average lifetime was determined from the luminescence decay profiles. Within the emission band, substantially different lifetimes may be obtained, depending on the energy of emission monitored (Figure 4a). Specifically, it was observed

The results of this modeling can be observed in Figure 1b (gray line). The experiments and simulations suggest a simple possible explanation for the anomalous low-temperature trend in PL peak energy. The high-temperature PL line shapes are more symmetric than the low-temperature spectra. With weak LO phonon coupling, individual phonon lines only become visible for T < 50 K. At higher temperatures, the effects of dephasing and Boltzmann statistics alter the line shape and symmetrize it. The emergence of individual phonon lines at low temperature leads to a shift in the spectral peak to lower energy below the turnover temperature, suggesting another possible explanation for this anomalous low-temperature PL behavior. As a final point in examining the temperature dependence of the linear spectra, we investigate whether there is any temperature dependence to the oscillator strength of the X1 absorption feature. Figure 3a shows the temperature depend-

Figure 4. (a) The observed lifetime of the band-edge luminescence demonstrates a strong spectral dependence. (b) Correcting the measurement for changes in the emission peak energy with temperature yields no consistent lifetime trend with temperature, while monitoring at the same wavelength yields an apparent increase in lifetime as temperature decreases.

Figure 3. Temperature dependence of absorption peak spectral properties: (a) while peak area of the X1 feature does not change with temperature, peak height decreases with increasing temperature; and (b) in contrast, the full-with at half-maximum of the peak increases with temperature, resulting in a roughly constant peak area for the X1 feature.

that the average lifetime obtained was inversely proportional to the energy of emission monitored. These observations are consistent with previous reports.34 In particular, the smaller NCs within the ensemble have substantially shorter lifetimes, a fact that has been previously ascribed to Förster Resonance Energy Transfer from smaller to larger NCs, which introduces an additional relaxation pathway for smaller NCs that shortens their relative lifetime.34 Moreover, the PL spectra have contributions from size distributions as well as a Boltzmann distribution of emitting states within the fine structure of the band edge exciton. Since the lifetime will not change much within the 5% size variation, only the distribution within the fine structure will affect the lifetime.35 Since the long-lived dark exciton state is lower in energy within the electronic fine structure, the lifetime will be shorter at higher energy. Together, these explanations account for the decrease in lifetime at higher energy within the emission band.

ence of peak height and peak area of the X1 feature in the absorption spectrum. While integrated peak area remains the same over the temperature range studied, the peak height decreases with temperature. In contrast to the peak height, the peak width increases as temperature rises (Figure 3b), which results in the nearly constant value for total peak area shown in Figure 3a. Thus, we can conclude that there is no temperature dependence to the oscillator strength over this range, a point that will be revisited in the context of PL quantum yield. PL Intensity Is Dictated by a Thermally Controlled Fraction of Emissive Particles. PL intensities and lifetimes were measured as a function of temperature in order to understand the physical origin of the well-known observation that NC emit more brightly at low temperature. It is wellknown that the PL intensity of NCs increases as temperature is 7733



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monotonically as temperature is lowerednot the band edge excitonic PL from the core state of the NC. In the case of larger CdSe, these effects are not large. But for other systems like smaller CdSe and CdS, these surface contributions can be significant. Having considered different metrics of PL spectral intensity with which to compare the total decay rate, we are now poised to explore the relationship between decay rate and PL quantum yield. The experimental observation of increasing total PL quantum yield at low temperatures is simple to explain by some thermally activated decay process common to all NCs. This process would experimentally manifest itself as a temperature dependence to the total decay rate. For a single emitter or a homogeneous system, one has the PL quantum yield as

This energy-dependence to the observed lifetime is significant when experimentally determining the temperaturedependence of the PL lifetimes. Either the entire PL band should be monitored, or the detection wavelength should be chosen based on the temperature-dependent PL peak energy. When photoluminescence decays were monitored at the peak of the emission feature, adjusted for each temperature, no clear temperature-dependent trend was observed in the range 80− 300 K (Figure 4b). However, monitoring at a fixed wavelength and failing to adjust for the change in peak energy yields an apparent linear change in lifetime with temperature (Figure 4b). Several prior investigations of lifetime as a function of temperature28b,36 have demonstrated that while the lifetime exhibits a strong temperature dependence at very low temperature due to thermal population of the dark exciton state, the lifetime shows little or no temperature-dependent trend above ∼70 K, in accord with our results. In several other cases, the near temperature independence of the lifetime has been reported alongside significant temperature dependence to the quantum yield over the same temperature range.28b,36a Jones, Lo, and Scholes6b,37 have reported on changes in lifetime with temperature above 77 K, but these variances are on the order of 10% and cannot alone account for quantum yield changes of over an order of magnitude. This near temperature invariance to the PL decay rate contrasts strongly with the strong temperature dependence to the PL intensity. Figure 5 shows integrated PL intensities of

Φ(T ) =

k rad(T ) kD(T)

(7)

and total decay rate as kD(T ) = k rad(T ) + k nr(T )

(8)

where krad is the intrinsic or radiative decay rate, kD is the measured decay rate, and knr is the rate of nonradiative decay. The temperature dependencies of the radiative and nonradiative rates will not necessarily be the same since they reflect different physical processes. The radiative rate is commonly assumed to be temperature independent. There may in principle be some temperature-dependence due to some Boltzmann distribution of electronic states or via non-Condon effects.38 The experimental temperature invariance of the oscillator strength of the absorptive transitions suggests the radiative rate should be mostly temperature invariant. Since the experimental total decay rate is also constant yet the PL quantum yield decreases monotonically with temperature, there is a basic inconsistency. This inconsistency can be resolved by considering that the system of NCs is not homogeneous. Considerable recent work has suggested that the NCs within an ensemble are far from homogeneous in their decay pathways.28b,39 Thus, we propose that the percentage of NCs within the ensemble which emit in the first place is a temperature-dependent quantity, n(T). Hence the ensemble quantum yield has a temperature dependence that now includes the fraction of NC which emit Φ(T ) =

k rad n(T ) k rad + k nr

(9)

Thus, the loss of QY at higher temperature does not result from a thermally activated nonradiative decay pathway common to all NCs, but rather from a fraction of NCs within the ensemble ceasing to emit at higher temperatures. Relating the Kinetics of Hot Exciton Surface Trapping to Steady State Photoluminescence Excitation Spectra and Quantum Yield. Our recent work on low-temperature CW PL spectroscopy of NC has provided insights into the thermodynamics of cold exciton emission and trapping. Excitation with high-energy photons can create excited or hot excitons. Hot excitons normally undergo a relaxation or cooling process to the X1 state.5a,14,40 But these hot excitons may also experience direct trapping to surface states, which competes with the process of relaxation. Hot exciton processes, such as surface trapping,12,41 are fundamentally kinetic processes governed by ultrafast transition rates; their analysis typically demands time domain measure-

Figure 5. The temperature-dependent PL intensity of CdSe (a) and CdS (b) NC. CdSe has X1 = 2.13 eV and R = 1.94 nm. CdS has X1 = 2.95 eV and R = 2.04 nm. In all cases, there is both PL from the core of the NC as well as the surface. The surface PL increases at low temperatures in a complex manner. Similarly the total PL intensity does not follow a simple temperature response. In contrast, the total PL monotonically increases as temperature is lowered.

CdSe and CdS NCs as a function of temperature. The integrated area from the PL is comprised primarily of core PL in CdSe but has significant contributions from the surface at low temperature in CdSe and at all temperatures in CdS. We have recently shown that these effects are general to NC and can be described via a simple semiclassical electron transfer approach.10 Notably, it is the total PL that increases almost 7734



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ments such as time-resolved PL, pump/probe TA spectroscopy,12,41 or two-dimensional electronic spectroscopy.42 In contrast, continuous wave (CW) PL spectroscopy is done at steady-state under conditions of thermodynamic equilibrium. Hence it is not self-evident that kinetic processes such as hot exciton surface trapping will manifest themselves in CW spectroscopies such as PL or PLE. Nonetheless, there appears to be experimental evidence of a QY spectrum (excitation energy dependence to the QY),5a,16−19 including CW studies.17,19 Hence it is essential to be able to rationalize via kinetic and thermodynamic arguments whether one can experimentally observe a CW QY spectrum reflecting kinetic processes. As we have recently shown, PL from the surface becomes significant at low temperature10a,c and may be controlled via ligand chemistry.10b The surface of the NC has traditionally been viewed as a source of unwanted nonradiative processes. Hence measures of hot exciton effects like quantum yield spectra or photoaction spectra for photocharging13b typically consider that an exciton (whether hot or cold) gets trapped to the surface at which point it is no longer emissiveor at least relevant to the spectroscopy. Such a view is incorrect. The NC itself is fundamentally characterized by two emissive states: the core and the surface. Since there are two coupled measurable quantities, a quantum yield or photoaction spectral analysis of surface effects should clearly delineate between core and surface PL. As noted before, this point is particularly important in CdS and small CdSe. With this understanding in mind, an investigation of the question of hot carrier trapping was undertaken. In each case, PLE spectra were obtained for both the surface and core bands. Also obtained were absorption spectra which were then converted to absorbed light (1 − T) as discussed by Chergui.17 Conversion (1 − T) is useful to correct for nonlinearities in absorbance vs absorbed light. Figure 6 shows representative photoaction spectra for two observables for a range of NC samples. In this case, the spectra was computed by including weighted contributions to the PLE spectrum from both core and surface emission and dividing by the spectrum of absorbed light. We focus upon the total quantum yield spectrum as well as the PLE ratios of surface relative to core emission to analyze the system. Figure 6a shows the QY spectra for two samples of CdSe NC. At low energy there are spectral oscillations which likely arise from the fact that the PLE spectra provide some size selection thereby narrowing the lines relative to those in the linear absorption spectra. At higher energy the spectrum is relatively free of these artifactual signals and is more robust for quantitative analysis. In these samples, we see either no QY spectra in the high-energy continuum region, or a small QY variation in the case of 1.48 nm CdSe at 80 K. The absence of a significant excitation energy dependence to the PLE spectra suggests that hot carriers experience the same quantum yield as carriers excited at the X1 feature. Our findings are in contrast to recent work by the Loomis group, which found evidence of a variation of quantum yield with temperature,19 but consistent with prior reports by Tonti et al.17 The variance in results, assuming identical analysis, is difficult to explain. We suggest that hot exciton trapping rates, which compete with relaxation rates, depend upon sample quality, passivation, and other important material parameters, analgous to the strong ligand dependence we have previously shown for cold exciton trapping rates in ultrasmall NCs.10b

Figure 6. Photoluminescence excitation (PLE) spectra of several NC samples when monitored at the band edge luminescence and the surface luminescence. The spectra show negligible differences as a function of excitation energy, suggesting that hot exciton trapping does not occur in these samples.

Figure 6b shows the relative PLE spectra, PLEsurface/PLEcore, normalized at high energy. In the presence of hot exciton surface trapping, there is the potential for an initial nonequilibrium population of surface trapped excitons. Hence provided these surface trapped excitons undergo radiative recombination prior to thermalization with the band edge exciton, there should be a spectral dependence to this PLE ratio. As in the QY spectra, there are large-amplitude spectral oscillations at low energy where states are resolved. These arise due to small differences in the peaks of the PLE spectra when monitoring at the surface. As in the QY spectra, the analysis is clearer at higher energy where there is a contribution from a continuum of states. Again, there is no strong trend in this region. These results also indicate that there is no evidence of hot exciton trapping in these samples. As the equilibrium process of core/surface redistribution is thermally activated,10a,c one might expect a thermal response for these QY spectra. Figure 6c shows the QY spectra computed by using core PL for one sample of CdSe NCs over a range of temperatures, normalized at the X1 feature. In this sample, there is no clear trend for quantum yield as a function of energy of excitation. In addition, the trend of an 7735



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Figure 7. (a) One-dimensional schematic illustration of the semiclassical electron transfer model of hot exciton surface trapping. Shown is the classical transfer path. The quantum path from phonons is not shownsee text for details. (b) Semiempirical simulation of the total PL quantum yield spectrum at various temperatures. (c) Calculated trapping rates using a microscopic model of surface charge trapping using semiclassical Marcus−Jortner electron transfer theory, assuming λ = 300 meV and ΔG(X1) = 50 meV. (d) Microscopic simulation of total PL quantum yield spectrum at various temperatures.

absence of variation in quantum yield as a function of excitation energy is maintained at all temperatures, suggesting that hot carrier trapping is not thermally activated. Thus there is no evidence of hot carrier trapping in these NCs. The experimental spectra shown here and elsewhere may be understood in terms of a simple semiclassical electron transfer theory. We have recently produced the first microscopic model that accounts for the thermodynamics of equilibration between cold (band edge) excitons and surface trapped excitons.10a,c In those works, the X1 band edge exciton is coupled to the surface via a thermally activated classical configuration coordinate that corresponds to the bath, and a quantum tunneling configuration coordinate corresponding to the LO phonon. The coupling strengths for each degree of freedom dictate the thermodynamics of PL from cold excitons. Here, we draw from the same theoretical approach to explain the effect of hot exciton trapping to the surface as a kinetic, nonequilibrium process. We explore the conditions under which these nonequilibrium processes may manifest themselves in equilibrium CW measurements like PL and PLE. Figure 7a shows a configuration coordinate diagram of NC states including several quantized exciton states (X1, X2, etc.) and a surface state to which carrier trapping processes can occur. The LO phonon coordinate is not shown here but is discussed in detail elsewhere.10a,c In such a system, we have previously shown that X1 exciton trapping can be modeled as an electron-transfer process: ⎞ 1⎛ π ⎟ ⎜ 2 τf ⎝ ℏ λmkbT ⎠

1/2

kET =

e −S ∑ n

Here, we allow exciton trapping to take place not only from the band edge (X1) state as in our previous works but also from higher excited states. The difference in trapping rate between states is dictated by the activation barrier to trapping ΔG‡ =

(λ + ΔG 0)2 4λ

(11)

Higher energy states will have different activation barriers to trapping and hence different trapping rates than the band edge (X1) feature. To extend the cold electron transfer model to include hot exciton effects, the rates of hot exciton relaxation and hot exciton surface trapping must be known. Our prior works have produced detailed real time measurements of both the cooling rate as a function of energy,5a,14,40 krelax(E), as well as the surface trapping rate as a function of energy, ktrap(E).5a,12,41a Based upon experimentally determined values, we input these rates to perform a semiempirical calculation, Figure 7b. The ratio of trapping to cooling rates was taken to increase from 1:1 for the X2 state to 10:1 for the continuum. Within the framework of this approach, the quantum yield is now a function of both energy and temperature, QY(E,T) due to the possibility of thermally repopulating the X1 state. This semiempirical approach predicts the existence of a QY spectrum. The extent of this effect is shown to be temperature dependent; lower temperatures enable the effect to be seen more readily due to the inefficient detrapping from the surface state. In addition to the semiempirical approach, a fully microscopic model offers some insight. In the case of a fully microscopic model, the transition rates and trapping rates need

Sn −(ΔG0 + λm + nℏω)2 /4λmkbT e n! (10) 7736



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to be explicitly obtained via the relevant state-to-state transition paths by using eq 10. Here, the transition path from the core states to the surface state proceeds via semiclassical electron transfer. Hence one needs to compute the temperaturedependent ET rates from each initial excitonic state. Figure 7c shows the results of this microscopic calculation, a simulation of the expected rates of hot carrier trapping for a given reorganization energy, λ, and X1 free energy difference, ΔG. The trapping rate increases with higher excitation energy (X2, X3) as the barrier to trapping is decreased, but decreases at very high energy as one enters the inverted region of electron transfer (continuum). Figure 7d shows the result of the microscopic calculation of QY(E,T) using trapping rates from the microscopic model. The trends from the semiempirical calculation are qualitatively reproduced here. Specifically, for some higher lying states at lower temperatures, carriers will be trapped at the surface and incapable of detrapping back to the band edge state, resulting in a lower quantum yield for the core state. But deviations arise at very high energy because the microscopic model predicts a decrease in trapping rates far in the inverted region where activation energy increases. It is worth noting that this simple model does not consider any energy dependence to the overall matrix elements. Most likely higher states are more delocalized and more strongly coupled to the surface thereby increasing the rate with excitonic energyas observed experimentally. One of the main effects of temperature upon the excitonic populations and spectra of NC is the perceived color of the PL. NC are commonly used for lighting and display applications due to their narrow core PL which results in saturated colors on the Commission internationale de l’éclairage (CIE) scale. In NCs, the effect of temperature on the color perception demonstrated by the CIE scale is somewhat complex. Lowering temperature results in blueshifting the core PL. However, lowering temperature also results in more surface PL. Figure 8a shows a schematic of these two effects in ultrasmall nanocrystals. These two effects in concert result in a change in the color of the NC as perceived by the eye. Figure 8b shows a CIE trajectory for PL from the core/surface/total bands of an ultrasmall NC system. The surface PL peak energy is roughly temperature invariant. The core PL follows empirical dE/dT response described previously. The ratio between the two is dictated by the thermodynamics of the ET model. The main point is that the total CIE trajectory for the total emissive system follows a complex path due to the interplay between two distinct processes. Figure 8c shows representative data from CdS NC. The complex interplay between the temperature response of the core emission peak and the thermally determined populations of core and surface emitting states allows a degree of temperature control over the perceived light emitted from NCs.

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Figure 8. (a) PL spectra from ultrasmall NCs at high and low temperatures illustrate blueshifting of the core PL and increase in the surface PL at low temperature. (b) CIE coordinate trajectories as a function of temperature for the NCs in part a. (c) CIE coordinate trajectories of CdS NCs from 10 to 300 K.

CONCLUSIONS We identify the role of temperature on the basic excitonics of light absorption and emission in semiconductor nanocrystals. Absorption of high-energy light creates electronically hot excitons. These energetic excitons could undergo hot carrier surface trapping (photocharging) thereby decreasing quantum yield for emission from the core or increasing the emission from the surface trap states. We experimentally report on the excitation energy and temperature dependence of the PL quantum yield for the core and surface PL. We show that a semiclassical electron transfer model can account not only for

cold exciton surface trapping but also the kinetics of hot exciton surface trapping and therefore could explain the presence of a quantum yield spectrum. Careful analysis of the temperaturedependent PL intensities, lifetimes, and absorption cross sections reveals that the temperature-controlled quantum yield arises from a decrease in the population of emitting NC rather than changes to the relevant decay rates. Finally, the 7737



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temperature dependence of the absorption and PL energies reveals a distinct temperature dependence for each excitonic state, and the origin of the low-temperature PL energy anomaly. Temperature dictates the excitonics so as to control the visual appearance of light emission from the nanocrystal via thermodynamically controlled brightness, emission energies, and contributions from surface states. The nonequilibrium kinetics of these hot exciton surface trapping processes may be probed in simple CW spectroscopy under certain conditions.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the CFI, NSERC, FQRNT, and McGill University is acknowledged. The authors thank the McGill University Center for Self-Assembled Chemical Structures for use of their facilities and Fred Kluck, J. P. Guay, Rick Rossi, Olivia Dinica, Klaudia Jumaa, and Jonathan Saari for support with experiments. J.M. acknowledges support from FQRNT.

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Connecting the Dots: The Kinetics and Thermodynamics of Hot, Cold

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