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, Auger-process-controlled 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.
Quantization of electronic and phonon energies and large surface-to-volume ratios significantly modify energy relaxation mechanisms in nanoscale semiconductors compared with bulk materials. In the case of semiconductor nanocrystals1, strong quantum confinement leads to greatly enhanced carrier–carrier interactions that open new nanocrystal-specific energy relaxation and recombination channels2,3,4,5,6,7,8,9. For example, highly efficient electron–hole (e–h) energy transfer (Fig. 1a) can lead to fast, sub-picosecond electron intraband dynamics3 despite a wide energy separation between quantized states, which can exceed multiple longitudinal optical (LO) phonon energies, ħω0. Furthermore, enhanced carrier–carrier interactions in nanocrystals result in large rates of non-radiative Auger recombination6,7,8,9, 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 semiconductors10,11 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 2-nm-radius 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 (Eg≈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, order-of-magnitude 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 low-dimensional 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 zero-dimensional (0D) and 1D semiconductors8.
In this work, we study highly monodisperse, colloidal CdSe QRs (Fig. 2a,b) prepared as hexane solutions13,14,15. 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 frequency-doubled, 100-fs pulses from an amplified Ti:sapphire laser (100-kHz repetition rate). Time-resolved photoluminescence (PL) measurements are carried out using a femtosecond PL up-conversion (uPL) technique16 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 29-nm-long QRs taken at different time delays (Δt) after excitation for the initial e–h density, neh, of ∼5.5×1018 cm−3 that corresponds to 2–3 e–h pairs per QR on average. In contrast to hot PL spectra of spherical nanocrystals17,18 (quantum dots) that show well-separated emission peaks arising from distinct quantized states, the time-resolved spectra of QRs consist of a single peak with an extended, high-energy tail, which reflects the distribution of charge carriers, n(ħω), over the dense manifold of high-energy 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):IPL(ħω)∝f(ħω)ρ(ħω)n(ħω). As the absorption coefficient, α0(ħω), is proportional to f(ħω)ρ(ħω), we can extract the population-related term by dividing the PL spectra by α0(ħω). Applying this procedure to measured uPL spectra, we find that the high-energy tail of n(ħω) can be well described by the exponential dependence n(ħω)∝exp(−ħω/k Te) (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, Te, can be directly derived from high-energy tails of the uPL spectra.
A progressive increase of the high-energy 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 ∼5.5×1018 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 high-carrier-density effects play a minor role in this intensity range. In Fig. 3a, we plot the energy loss rate (ɛr=d(1.5k Te)/dt) as a function of Te (filled squares). The initial rate is approximately 0.2 eV ps−1. It stays nearly constant until Te reaches ∼700 K and then drops steeply by orders of magnitude as Te 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 Te-independent broadening of the emission spectra. For example, for the QR sample in Fig. 2c, the slope of the high-energy tail of the PL spectrum measured using low-intensity, continuous-wave 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 up-conversion 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 LO-phonon sub-systems that are in equilibrium with each other and cool together via interactions with acoustic phonons19,20. The temperature dependence of the relaxation rate in this regime can be described by the following expression12:
in which τLO is the characteristic LO-phonon decay time, Ta is the acoustic phonon temperature and NLO(T) is the LO-phonon 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 Ta=410 K and τLO=0.5 ps (Fig. 3a, blue dotted line). This agreement provides strong evidence that at low pump levels, carrier intraband relaxation in QRs is bulk-like. This behaviour results from the high density of electronic states in QRs and is different from that observed in quantum dots with sparse energy spectra3,4,18.
Despite this similarity with bulk semiconductors, carrier relaxation in QRs shows several distinct features arising from the nanoscale-size 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 neh=1018 cm−3 (ref. 12). In the bulk case, overheating occurs due to non-equilibrium filling of acoustic modes, which can eventually result in the situation for which Te=Ta and, hence, the cooling of the e–h system is controlled by the decay of the acoustic phonons (acoustic-phonon bottleneck12). The cooling of acoustic modes is controlled by relatively slow, heat-diffusion-limited interactions with the environment. The corresponding time constant (τa) 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 τa is on the sub-picosecond to picosecond timescale for QR dimensions studied in this paper. This timescale is comparable to that characteristic of the LO-phonon decay, which slows down the buildup of acoustic modes and hence significantly reduces the role of the acoustic-phonon 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 Te dynamics in Fig. 2d measured for neh=2.2×1019 cm−2 (open circles) and 5.5×1018 cm−2 (filled squares)). We also observe a dramatic change in the ɛr versus Te dependence (Fig. 3a, open circles). Instead of a sharp drop in the range of intermediate temperatures observed at low pump intensities, ɛr starts to decrease immediately with reducing Te and this reduction continues steadily until the final temperature is reached. Furthermore, we observe that for a given Te the magnitude of ɛr is greatly reduced compared with the low-intensity situation. For example, at high excitation intensity, ɛr 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 acoustic-phonon-bottleneck 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 non-radiative Auger recombination. The Auger recombination rate (RA) is a function of the carrier density and hence of the number of e–h pairs per nanocrystal, N: (dN/dt)|Auger=−RA(N) (refs 78). The Ne–h pair decay time constant is τN=N/R(N). Auger recombination leads to the heating of the electronic system by releasing energy that is approximately equal to the energy gap (Eg) in each recombination event. The Auger decay of the Ne–h pair state produces the (N−1) state and, therefore, the corresponding heating rate (ɛA) 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 temperature12 and is only weakly dependent on e–h pair density through the weak effect of the acoustic-phonon bottleneck: ɛph=ɛph(Te,N). The phonon-related relaxation rate decreases with decreasing Te (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=ΔtA) 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 negative-feedback mechanism, which operates in the following way. If ɛph drops below ɛA, the Auger heating takes over, which increases Te and hence ɛph (see equation (1)). In the opposite case of ɛph>ɛA, carrier cooling due to electron–phonon interactions dominates over Auger heating, which drives Te down until ɛph becomes equal to ɛA. The above considerations imply that during the Auger-process-controlled relaxation stage, Te is a function of carrier density and it can be determined from equation (3). The direct correspondence between carrier density and Te is clearly manifested in both measured cooling dynamics and pump-intensity-dependent data as discussed below.
In our pump-intensity-dependence 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 carrier-density dependences of Auger recombination rates8. 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: RA(N)∝N3. In the case of long, 1D rods, electrons and holes are bound into 1D excitons that recombine in the bimolecular fashion with RA(N)∝N2. This difference in scaling of RA should have a pronounced effect on the pump dependence of carrier overheating ΔTe=Te−TL (where TL is the nominal lattice temperature), if it is indeed because of the Auger process.
If we neglect a weak dependence of ɛph on carrier density, we find that at moderate overheating (ΔTeħωLO/(k TeTa)<1, Ta≈TL), the phonon-related relaxation rate is approximately proportional to ΔTe. From the condition of the equilibrium between ɛph and ɛA (equation (3)), we obtain ΔTe(N)∝ɛA(N). Finally using equation (2), we find that ΔTe(N) is proportional to RA(N) and, hence, the pump-power dependence of the ratio of the overheating measured in 0D and 1D rods scales as N: ΔTe(N)|0D/ΔTe(N)|1D∝N. To account for the dependence of ɛph on the carrier density, we need to take into consideration the acoustic-phonon bottleneck. Because of extremely efficient heat exchange between nanoscale QRs and the outside medium (solvent in this case), the non-equilibrium heating of acoustic modes in QRs is not significant and, therefore, it can be accounted for in the linear approximation: Ta−TL∝N (ref. 12). The use of a corrected expression for ɛph in equation (3) modifies the dependence of ΔTe on N; however, it still preserves an approximate linearity of the ratio ΔTe(N)|0D/ΔTe(N)|1D with respect to N.
To experimentally compare carrier heating behaviour in 0D and 1D cases, we analyse the carrier density dependence of ΔTe measured at Δt=2 ps (that is, after the equilibrium between ɛph and ɛA is established; see below) for 22- and 44-nm-long QRs that are characterized by cubic and quadratic Auger recombination rates, respectively8. As a measure of carrier density, we use an average number of e–h pairs per QR, 〈N〉, which is calculated as 〈N〉=jpσ(ħωp), where jp is the pump per-pulse fluence (in photons per cm2) and σ(ħωp)is the QR absorption cross-section at the pump photon energy, ħωp. We clearly observe that shorter rods show a faster increase of the temperature with increasing pump level (ΔTe∝〈N〉1.8) than longer rods (ΔTe∝〈N〉0.9) (Fig. 4). Furthermore, the ratio of ΔTe measured for these two samples scales as 〈N〉0.9, which is in good agreement with the prediction of the Auger-heating model.
These results are also consistent with observed cooling dynamics. In Fig. 3b, we compare relaxation of ΔTe to dynamics of PL intensity squared, IPL2, and observe that both transients show nearly identical behaviour at delay times Δt>2 ps. As IPL is proportional to the carrier density (IPL∝nehτr−1;τr is the radiative decay time constant), the above observation indicates that ΔTe is proportional to 〈N〉2, which agrees with results of the pump-intensity-dependence studies (short-rod case in Fig. 4a). The dynamical data also provide information on the onset (ΔtA) of the Auger-controlled cooling stage, which corresponds to the time delay, after which ΔTe and IPL show 'correlated' relaxation (∼2 ps in Fig. 3b).
Finally, we demonstrate that the Auger-heating model also allows us to accurately describe the temperature dependence of the energy loss rates measured at high pump fluences. In the Auger-heating regime, ɛr∝[d(ΔTe)/dN](dN/dt)=[d(ΔTe)/dN]RA. Our experimental data in Fig. 4 indicate that ΔTe is approximately proportional to 〈N〉2 and 〈N〉 in short and long rods, respectively, which yields ɛr∝(ΔTe)2 for both cases. The measured temperature dynamics indicate that the Auger-heating regime establishes at temperatures below approximately 700 K. In this range the quadratic dependence on ΔTe 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.
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 nanocrystals13, we used a combination of hexylphosphonic acid (HPA) and tetradecylphosphonic acid (TDPA) to promote formation of elongated, rod-shaped 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.
Steady-state 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 wide-gap semiconductor. These relatively low quantum yields imply a significant role of non-radiative recombination processes in our samples. On the basis of typical room-temperature radiative lifetimes for QRs (several nanoseconds to tens of nanoseconds), we can estimate that the average non-radiative 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 (sub-picosecond to tens of picosecond time constants), the results of our relaxation studies are not significantly affected by non-radiative decay channels and, specifically, by exact values of steady-state emission quantum yields.
In the PL up-conversion experiment16, the QR samples were excited at 400 nm with 100-fs pulses from a frequency-doubled, amplified Ti:sapphire laser (100-kHz repetition rate). An elliptical mirror was used to collect and to refocus the QR PL onto a nonlinear-optical β-barium borate crystal. The sample emission was frequency-mixed (gated) in the β-barium borate crystal with variably delayed 100-fs pulses at a fundamental frequency of the Ti-sapphire laser. The up-converted, sum-frequency 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 time-resolved PL data. The spectral resolution in these measurements was ∼5 nm and the temporal resolution was ∼300 fs.
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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Nature Physics (2006)