Abstract
The rate at which excited charge carriers relax to their equilibrium state affects many aspects of the performance of nanoscale devices, including switching speed, carrier mobility and luminescence efficiency. A better understanding of the processes that govern carrier relaxation therefore has important technological implications. A significant increase in carrier–carrier interactions caused by strong spatial confinement of electronic excitations in semiconductor nanostructures leads to a considerable enhancement of Auger effects, which can further result in unusual, Augerprocesscontrolled recombination and energy relaxation regimes. Here, we report the first experimental observation of efficient Auger heating in CdSe quantum rods at high pump intensities, leading to a strong reduction of carrier cooling rates. In this regime, the carrier temperature is determined by the balance between energy outflow through phonon emission and energy inflow because of Auger heating. This equilibrium results in peculiar carrier cooling dynamics that closely correlate with recombination dynamics, an effect never seen before in bulk or nanoscale semiconductors.
Main
Quantization of electronic and phonon energies and large surfacetovolume ratios significantly modify energy relaxation mechanisms in nanoscale semiconductors compared with bulk materials. In the case of semiconductor nanocrystals^{1}, strong quantum confinement leads to greatly enhanced carrier–carrier interactions that open new nanocrystalspecific energy relaxation and recombination channels^{2,3,4,5,6,7,8,9}. For example, highly efficient electron–hole (e–h) energy transfer (Fig. 1a) can lead to fast, subpicosecond electron intraband dynamics^{3} despite a wide energy separation between quantized states, which can exceed multiple longitudinal optical (LO) phonon energies, ħω_{0}. Furthermore, enhanced carrier–carrier interactions in nanocrystals result in large rates of nonradiative Auger recombination^{6,7,8,9}, in which the e–h recombination energy is not emitted as a photon but is transferred to an electron or a hole (Fig. 1b). Although e–h energy transfer does not change the total energy of the e–h pair, Auger recombination leads to heating of the electronic system (that is, an increase of the average e–h pair energy) that can, in principle, slow down carrier relaxation dynamics.
The role of Auger heating is not significant in bulk semiconductors^{10,11} because of the restrictions imposed by energy and momentum conservation. However, Auger processes can be much more efficient in semiconductor nanocrystals because of the relaxation in momentum conservation and a 'forced' overlap of carrier wavefunctions induced by strong quantum confinement. For example, in 2nmradius CdSe nanocrystals, the Auger decay time of a two e–h pair state (biexciton) is only 50 ps (ref. 7). The energy released in this process is approximately equal to the energy gap (E_{g}≈2 eV), and it is transferred to the remaining exciton, which corresponds to a heating rate of approximately 0.04 eV ps^{−1} per e–h pair. The latter value is close to the energy loss rate typically observed in bulk CdSe and CdS (∼0.1 eV ps^{−1}; ref. 12), indicating that in nanosized particles Auger heating can significantly alter carrier cooling dynamics at high pump intensities.
In this article, we analyse the effect of Auger heating on energy relaxation dynamics in elongated CdSe nanocrystals (quantum rods (QRs)). At high pump intensities (more than 2–3 e–h pairs per QR), we detect a dramatic, orderofmagnitude reduction in the energy relaxation rate resulting from efficient Auger heating. The peculiarity of this regime is that the energy relaxation directly correlates with recombination dynamics, which has never previously been observed in bulk or lowdimensional materials. Furthermore, we find that Auger heating differs in short and long QRs, which can be explained by the difference in the scaling of Auger rates with respect to carrier density in zerodimensional (0D) and 1D semiconductors^{8}.
In this work, we study highly monodisperse, colloidal CdSe QRs (Fig. 2a,b) prepared as hexane solutions^{13,14,15}. We use a series of samples with the same QR diameter of 4.6 nm and various lengths from 22 to 44 nm. The samples are excited at 400 nm using frequencydoubled, 100fs pulses from an amplified Ti:sapphire laser (100kHz repetition rate). Timeresolved photoluminescence (PL) measurements are carried out using a femtosecond PL upconversion (uPL) technique^{16} with a time resolution of approximately 300 fs. All measurements are conducted at room temperature. Further details of the experimental procedures can be found in the Methods section.
Figure 2c shows uPL spectra of 29nmlong QRs taken at different time delays (Δt) after excitation for the initial e–h density, n_{eh}, of ∼5.5×10^{18} cm^{−3} that corresponds to 2–3 e–h pairs per QR on average. In contrast to hot PL spectra of spherical nanocrystals^{17,18} (quantum dots) that show wellseparated emission peaks arising from distinct quantized states, the timeresolved spectra of QRs consist of a single peak with an extended, highenergy tail, which reflects the distribution of charge carriers, n(ħω), over the dense manifold of highenergy QR states. In addition to n(ħω), the shape of the PL spectrum is determined by spectral distributions of the interband transition oscillator strength, f(ħω), and the e–h joint density of states, ρ(ħω):I_{PL}(ħω)∝f(ħω)ρ(ħω)n(ħω). As the absorption coefficient, α_{0}(ħω), is proportional to f(ħω)ρ(ħω), we can extract the populationrelated term by dividing the PL spectra by α_{0}(ħω). Applying this procedure to measured uPL spectra, we find that the highenergy tail of n(ħω) can be well described by the exponential dependence n(ħω)∝exp(−ħω/k T_{e}) (inset in Fig. 2c; k is the Boltzmann's constant), implying that carrier distributions produced in our experiments are thermal. These observations further suggest that carrier thermalization occurs on timescales that are shorter than our time resolution (∼300 fs) and the carrier temperature, T_{e}, can be directly derived from highenergy tails of the uPL spectra.
A progressive increase of the highenergy slope of uPL spectra with time in Fig. 2c reflects carrier cooling dynamics, which is shown in Fig. 2d by filled squares. At pump levels below ∼5.5×10^{18} cm^{−3}, the temperature relaxation (time constant is 0.5 ps) does not show a significant dependence on either rod length or pump level, indicating that highcarrierdensity effects play a minor role in this intensity range. In Fig. 3a, we plot the energy loss rate (ɛ_{r}=d(1.5k T_{e})/dt) as a function of T_{e} (filled squares). The initial rate is approximately 0.2 eV ps^{−1}. It stays nearly constant until T_{e} reaches ∼700 K and then drops steeply by orders of magnitude as T_{e} approaches the sample temperature. We find that the final electron temperature derived from the uPL data is always higher than 300 K (the nominal sample temperature) even at low excitation densities. This divergence is primarily due to the size dispersion of the QRs leading to T_{e}independent broadening of the emission spectra. For example, for the QR sample in Fig. 2c, the slope of the highenergy tail of the PL spectrum measured using lowintensity, continuouswave excitation (spectroscopic lamp) formally corresponds to the temperature of 370 K, which is close to the final carrier temperature (∼400 K) measured for the same sample in the PL upconversion studies.
Both the initial value of the energy relaxation rate measured at low pump intensities and its qualitative behaviour with changing carrier temperature are similar to those observed in bulk II–VI semiconductors at comparable excitation densities. In bulk semiconductors this behaviour has been explained in terms of strong coupling between the e–h and the LOphonon subsystems that are in equilibrium with each other and cool together via interactions with acoustic phonons^{19,20}. The temperature dependence of the relaxation rate in this regime can be described by the following expression^{12}:
in which τ_{LO} is the characteristic LOphonon decay time, T_{a} is the acoustic phonon temperature and N_{LO}(T) is the LOphonon occupation number for temperature T.
We can use equation (1) to accurately explain our results for energy relaxation rates measured at low pump intensity assuming that T_{a}=410 K and τ_{LO}=0.5 ps (Fig. 3a, blue dotted line). This agreement provides strong evidence that at low pump levels, carrier intraband relaxation in QRs is bulklike. This behaviour results from the high density of electronic states in QRs and is different from that observed in quantum dots with sparse energy spectra^{3,4,18}.
Despite this similarity with bulk semiconductors, carrier relaxation in QRs shows several distinct features arising from the nanoscalesize regime. One interesting observation is that the final carrier temperature in QRs (corrected for spectral broadening associated with sample polydispersity) is close to room temperature, whereas in bulk semiconductors, significant overheating is already observed at n_{eh}=10^{18} cm^{−3} (ref. 12). In the bulk case, overheating occurs due to nonequilibrium filling of acoustic modes, which can eventually result in the situation for which T_{e}=T_{a} and, hence, the cooling of the e–h system is controlled by the decay of the acoustic phonons (acousticphonon bottleneck^{12}). The cooling of acoustic modes is controlled by relatively slow, heatdiffusionlimited interactions with the environment. The corresponding time constant (τ_{a}) is determined by the diameter of the excitation spot and is typically in the microsecond to millisecond time range. In the case of QR samples, the efficiency of cooling of acoustic modes is increased by orders of magnitude because heat exchange is controlled by the dimensions of the individual nanoscale particle. We estimate that τ_{a} is on the subpicosecond to picosecond timescale for QR dimensions studied in this paper. This timescale is comparable to that characteristic of the LOphonon decay, which slows down the buildup of acoustic modes and hence significantly reduces the role of the acousticphonon bottleneck in nanosized rods.
At high excitation densities, energy relaxation changes significantly compared with that at low pump powers. Specifically, we find that in this case the cooling process is not terminated after a few picoseconds (as it does at low pump intensities) but persists up to tens of picoseconds (compare T_{e} dynamics in Fig. 2d measured for n_{eh}=2.2×10^{19} cm^{−2} (open circles) and 5.5×10^{18} cm^{−2} (filled squares)). We also observe a dramatic change in the ɛ_{r} versus T_{e} dependence (Fig. 3a, open circles). Instead of a sharp drop in the range of intermediate temperatures observed at low pump intensities, ɛ_{r} starts to decrease immediately with reducing T_{e} and this reduction continues steadily until the final temperature is reached. Furthermore, we observe that for a given T_{e} the magnitude of ɛ_{r} is greatly reduced compared with the lowintensity situation. For example, at high excitation intensity, ɛ_{r} at 700 K is 0.008 eV ps^{−1} whereas at low pump levels it is approximately 0.1 eV ps^{−1}. These observations cannot be explained by the acousticphononbottleneck model alone (black dashed line in Fig. 3a) and indicate the existence of other mechanisms that become active at high excitation densities. Below we show that the observed behaviour can be accurately described if we account for Auger heating.
At the pump levels used in these measurements, carrier decay is dominated by nonradiative Auger recombination. The Auger recombination rate (R_{A}) is a function of the carrier density and hence of the number of e–h pairs per nanocrystal, N: (dN/dt)_{Auger}=−R_{A}(N) (refs 78). The Ne–h pair decay time constant is τ_{N}=N/R(N). Auger recombination leads to the heating of the electronic system by releasing energy that is approximately equal to the energy gap (E_{g}) in each recombination event. The Auger decay of the Ne–h pair state produces the (N−1) state and, therefore, the corresponding heating rate (ɛ_{A}) per e–h pair is given by:
During the initial fast cooling (Δt<1 ps), energy relaxation is dominated by interactions with phonons with a rate that depends primarily on carrier temperature^{12} and is only weakly dependent on e–h pair density through the weak effect of the acousticphonon bottleneck: ɛ_{ph}=ɛ_{ph}(T_{e},N). The phononrelated relaxation rate decreases with decreasing T_{e} (see equation (1)) and eventually becomes equal to the Auger heating rate, which marks the onset of the relaxation stage in which carrier intraband dynamics are controlled by the Auger process (inset of Fig. 3b). Starting from this point (t=Δt_{A}) the e–h temperature is determined by the equilibrium between the energy outflow through interactions with phonons and the energy inflow because of Auger heating:
This equilibrium is maintained by the negativefeedback mechanism, which operates in the following way. If ɛ_{ph} drops below ɛ_{A}, the Auger heating takes over, which increases T_{e} and hence ɛ_{ph} (see equation (1)). In the opposite case of ɛ_{ph}>ɛ_{A}, carrier cooling due to electron–phonon interactions dominates over Auger heating, which drives T_{e} down until ɛ_{ph} becomes equal to ɛ_{A}. The above considerations imply that during the Augerprocesscontrolled relaxation stage, T_{e} is a function of carrier density and it can be determined from equation (3). The direct correspondence between carrier density and T_{e} is clearly manifested in both measured cooling dynamics and pumpintensitydependent data as discussed below.
In our pumpintensitydependence studies, we concentrate on two QR samples with two different QR lengths of 22 nm (short rods) and 44 nm (long rods). According to our previous studies, these two samples show distinctly different carrierdensity dependences of Auger recombination rates^{8}. In short rods that can be considered as 0D objects, carriers are present in the form of unbound e–h pairs and, therefore, Auger rates are cubic with respect to carrier density: R_{A}(N)∝N^{3}. In the case of long, 1D rods, electrons and holes are bound into 1D excitons that recombine in the bimolecular fashion with R_{A}(N)∝N^{2}. This difference in scaling of R_{A} should have a pronounced effect on the pump dependence of carrier overheating ΔT_{e}=T_{e}−T_{L} (where T_{L} is the nominal lattice temperature), if it is indeed because of the Auger process.
If we neglect a weak dependence of ɛ_{ph} on carrier density, we find that at moderate overheating (ΔT_{e}ħω_{LO}/(k T_{e}T_{a})<1, T_{a}≈T_{L}), the phononrelated relaxation rate is approximately proportional to ΔT_{e}. From the condition of the equilibrium between ɛ_{ph} and ɛ_{A} (equation (3)), we obtain ΔT_{e}(N)∝ɛ_{A}(N). Finally using equation (2), we find that ΔT_{e}(N) is proportional to R_{A}(N) and, hence, the pumppower dependence of the ratio of the overheating measured in 0D and 1D rods scales as N: ΔT_{e}(N)_{0D}/ΔT_{e}(N)_{1D}∝N. To account for the dependence of ɛ_{ph} on the carrier density, we need to take into consideration the acousticphonon bottleneck. Because of extremely efficient heat exchange between nanoscale QRs and the outside medium (solvent in this case), the nonequilibrium heating of acoustic modes in QRs is not significant and, therefore, it can be accounted for in the linear approximation: T_{a}−T_{L}∝N (ref. 12). The use of a corrected expression for ɛ_{ph} in equation (3) modifies the dependence of ΔT_{e} on N; however, it still preserves an approximate linearity of the ratio ΔT_{e}(N)_{0D}/ΔT_{e}(N)_{1D} with respect to N.
To experimentally compare carrier heating behaviour in 0D and 1D cases, we analyse the carrier density dependence of ΔT_{e} measured at Δt=2 ps (that is, after the equilibrium between ɛ_{ph} and ɛ_{A} is established; see below) for 22 and 44nmlong QRs that are characterized by cubic and quadratic Auger recombination rates, respectively^{8}. As a measure of carrier density, we use an average number of e–h pairs per QR, 〈N〉, which is calculated as 〈N〉=j_{p}σ(ħω_{p}), where j_{p} is the pump perpulse fluence (in photons per cm^{2}) and σ(ħω_{p})is the QR absorption crosssection at the pump photon energy, ħω_{p}. We clearly observe that shorter rods show a faster increase of the temperature with increasing pump level (ΔT_{e}∝〈N〉^{1.8}) than longer rods (ΔT_{e}∝〈N〉^{0.9}) (Fig. 4). Furthermore, the ratio of ΔT_{e} measured for these two samples scales as 〈N〉^{0.9}, which is in good agreement with the prediction of the Augerheating model.
These results are also consistent with observed cooling dynamics. In Fig. 3b, we compare relaxation of ΔT_{e} to dynamics of PL intensity squared, I_{PL}^{2}, and observe that both transients show nearly identical behaviour at delay times Δt>2 ps. As I_{PL} is proportional to the carrier density (I_{PL}∝n_{eh}τ_{r}^{−1};τ_{r} is the radiative decay time constant), the above observation indicates that ΔT_{e} is proportional to 〈N〉^{2}, which agrees with results of the pumpintensitydependence studies (shortrod case in Fig. 4a). The dynamical data also provide information on the onset (Δt_{A}) of the Augercontrolled cooling stage, which corresponds to the time delay, after which ΔT_{e} and I_{PL} show 'correlated' relaxation (∼2 ps in Fig. 3b).
Finally, we demonstrate that the Augerheating model also allows us to accurately describe the temperature dependence of the energy loss rates measured at high pump fluences. In the Augerheating regime, ɛ_{r}∝[d(ΔT_{e})/dN](dN/dt)=[d(ΔT_{e})/dN]R_{A}. Our experimental data in Fig. 4 indicate that ΔT_{e} is approximately proportional to 〈N〉^{2} and 〈N〉 in short and long rods, respectively, which yields ɛ_{r}∝(ΔT_{e})^{2} for both cases. The measured temperature dynamics indicate that the Augerheating regime establishes at temperatures below approximately 700 K. In this range the quadratic dependence on ΔT_{e} describes the behaviour of the energy loss rate (solid red line in Fig. 3a) remarkably well.
In conclusion, we study carrier cooling dynamics in semiconductor QRs as a function of rod length and pump intensity. We observe a significant reduction of the energy loss rate (by more than one order of magnitude) at high pump levels resulting from highly efficient Auger heating. In this regime, the carrier temperature is determined by the balance between energy outflow through interactions with phonons and energy inflow produced by Auger recombination. This approximate equilibrium results in a cooling dynamics that is controlled by the carrier recombination process. Because of the direct correlation between energy relaxation and recombination dynamics, the carrier cooling behaviour is significantly different in short and long QRs due to the difference in scaling of Auger recombination rates with respect to carrier density in 0D and 1D semiconductors.
Methods
We fabricated QR samples using colloidal synthetic procedures adapted from refs 13–15. In addition to trioctylphosphine (TOP) and trioctylphosphine oxide (TOPO), standard coordinating ligands used in the preparation of spherical CdSe nanocrystals^{13}, we used a combination of hexylphosphonic acid (HPA) and tetradecylphosphonic acid (TDPA) to promote formation of elongated, rodshaped particles. Typically, a particular HPA:TDPA ratio (for example, 1:1 to 1:3) was chosen to achieve a desired rod diameter, and different rod lengths were obtained by selecting aliquots from the reaction mixture over time. Longer reaction times yielded longer rods, with little impact on rod diameter. This procedure produced highly monodisperse QR samples that had 5%–7% and 10% standard deviations for QR radii and lengths, respectively.
Steadystate PL quantum yields of QRs are a few per cent, which is typical for samples fabricated without subsequent overcoating with an inorganic layer of a widegap semiconductor. These relatively low quantum yields imply a significant role of nonradiative recombination processes in our samples. On the basis of typical roomtemperature radiative lifetimes for QRs (several nanoseconds to tens of nanoseconds), we can estimate that the average nonradiative time constants in QR samples are on the hundreds of picoseconds timescales. As these timescales are significantly longer than those of the cooling processes studied in this work (subpicosecond to tens of picosecond time constants), the results of our relaxation studies are not significantly affected by nonradiative decay channels and, specifically, by exact values of steadystate emission quantum yields.
In the PL upconversion experiment^{16}, the QR samples were excited at 400 nm with 100fs pulses from a frequencydoubled, amplified Ti:sapphire laser (100kHz repetition rate). An elliptical mirror was used to collect and to refocus the QR PL onto a nonlinearoptical βbarium borate crystal. The sample emission was frequencymixed (gated) in the βbarium borate crystal with variably delayed 100fs pulses at a fundamental frequency of the Tisapphire laser. The upconverted, sumfrequency signal was spectrally filtered by a monochromator and detected with a cooled photomultiplier tube coupled to a photon counting system. By scanning the monochromator and the time delay between excitation and gating pulses we were able to obtain spectrally and timeresolved PL data. The spectral resolution in these measurements was ∼5 nm and the temporal resolution was ∼300 fs.
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Acknowledgements
The authors thank Dr. H. Htoon for useful discussions. This work was supported by Los Alamos LDRD Funds and the Chemical Sciences, Biosciences, and Geosciences Division of the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy.
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Chemistry Division, CPCS, MSJ567, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
 Marc Achermann
 , Andrew P. Bartko
 , Jennifer A. Hollingsworth
 & Victor I. Klimov
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Correspondence to Victor I. Klimov.
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