Matthias
Ruppert
a,
Hanh
Bui‡
bc,
Laxmi Kishore
Sagar§
de,
Pieter
Geiregat
de,
Zeger
Hens
de,
Gabriel
Bester
bc and
Nils
Huse
*ac
aInstitute for Nanostructure and Solid-State Physics, Department of Physics, University of Hamburg and Center for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany. E-mail: nils.huse@uni-hamburg.de
bPhysical Chemistry and Physics departments, University of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
cThe Hamburg Centre for Ultrafast Imaging, University of Hamburg, Luruper Chaussee, 149, 22761 Hamburg, Germany
dPhysics and Chemistry of Nanostructures, Department of Chemistry, Ghent University, Krijgslaan 281 - S3, B-9000 Gent, Belgium
eCenter for Nano and Biophotonics, Ghent University, Technologiepark Zwijnaarde 15, B-9052 Gent, Belgium
First published on 7th December 2021
Femtosecond pump–probe spectroscopy reveals ultrafast carrier dynamics in mid-infrared (MIR) colloidal HgTe nanoparticles with a bandgap of 2.5 μm. We observe intraband relaxation processes after photoexcitation ranging from resonant excitation up to the multi-exciton generation (MEG) regime by identifying initially excited states from atomic effective pseudopotential calculations. Our study elucidates the earliest dynamics below 10 ps in this technologically relevant material. With increasing photon energy, we find carrier relaxation times as long as 2.1 ps in the MEG regime close to the ionization threshold of the particles. For all photon energies, we extract a constant mean carrier energy dissipation rate of 0.36 eV ps−1 from which we infer negligible impact of the density of states on carrier cooling.
In contrast to these well described quantum dot systems, the dynamics during the first 10 picoseconds after photo-excitation are largely unknown in MIR HgTe quantum dots, while the slower band edge dynamics are well understood.17,18 The possibility of tuning their band-gap through the entire infrared down to the Terahertz spectral range by varying the size of the crystallites19–21 makes these particles very promising for technological applications such as infrared sensing22–27 and emission.27–29 MEG has been demonstrated in these particles as well,18 meaning that a photon of sufficient energy may efficiently excite several electron hole pairs upon being absorbed, as also observed in other quantum dot systems.3,5,30
Here, we report the early dynamics in MIR HgTe quantum dots with a band-gap of 2.5 μm using ultrafast visible/infrared pump–probe spectroscopy. We observe the early state filling dynamics of the energetically lowest exciton state upon non-resonant excitation, which allow assigning timescales for intraband carrier relaxation mechanisms in different excitation regimes. By varying pump wavelengths from 2.4 μm up to 400 nm, we gain insight into the exciton cooling process from the lowest optical excitation up to the highly non-resonant MEG regime, where a single photon may excite up to four electron hole pairs in HgTe quantum dots.18
![]() | (1) |
Here, Ni,e(0) and Ni,b(0) denote the number of initially excited electron hole pairs in particles with a single exciton only and particles with two excitons, respectively. τe, τb and τc, denote the exciton recombination time, the bi-exciton recombination time and the intraband cooling time. The first term in Eqn (1) accounts for the finite width of the instrument response where erf denotes the Gaussian error function and IRF denotes the full width at half maximum (FWHM) of the instrument response function. All pump and probe pulses are close to transform limited with a pulse duration of 100 fs (FWHM), which results in a signal rise time (10%-to-90%) of ≤150 fs. IRF was therefore fixed to 150 fs so as not to overparameterize the fit.
Eqn (1) was used to simultaneously fit data for different excitation densities and wavelengths. For this global fit, the exciton and bi-exciton lifetime parameters, τe and τb, were shared across this large data set and only the amplitudes, Ni,e(0) and Ni,b(0), and the intraband cooling time, τc, vary between different time traces. An exemplary fitted time trace is also shown in Fig. 1 (panel A, red curve), demonstrating that the data can be adequately modelled using only 3 time-constants covering both, the initial rise and the long-lived interband recombination. We obtain τE = 623 ± 150 ps and τB = 31 ± 10 ps for the exciton recombination and biexciton Auger relaxation times. These lifetimes have already been reported for HgTe NCs with a similar bandgap.17 Our findings for the bi-exciton lifetime are identical within the margins of error. However, we find 5 times shorter exciton lifetimes, which may be caused by differences in synthesis and ligands used.
We now discuss the influence of the excitation wavelength on the band edge dynamics. Fig. 2 shows time traces for excitations at 2.4 μm, 2.0 μm, 1.2 μm, 800 nm and 400 nm. The excitation densities for the data shown correspond to less than 0.2 photons absorbed per quantum dot on average, meaning these measurements are not distorted by multi-photon processes (details of the excitation density analysis can be found in the ESI†). All time traces have been normalized to their peak signal for easier comparability.
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Fig. 2 Exemplary bleach signal at the band-gap of 0.5 eV (2.5 μm) for low excitation density (solid lines) and fits according to eqn (1) (dotted lines) for different pump wavelengths: 2.4 μm (red), 2.0 μm (orange), 1.2 μm (yellow), 800 nm (green) and 400 nm (blue). For 400 nm and 800 nm excitation, the long-lived dynamics are governed by multi-excitons generated through MEG. |
For 2.4 μm, 2.0 μm and 1.2 μm the long-lived dynamics are well described by a single exponential decay with a lifetime of 623 ± 150 ps, shown as dotted lines. For 800 nm and 400 nm excitation, the photon energy exceeds the MEG threshold.18 Consequently, the long-lived dynamics are governed by multi-excitons. These are no longer well described by a single exponential decay due to Auger recombination of multi-excitons as can be clearly seen in the inset. Apart from carrier multiplication, the excitation wavelength does not affect carrier dynamics past 5 ps after excitation in our data. This means that all intraband relaxation occurs within this time window. By comparing the early dynamics, a trend of longer rise times with higher photon energy can be observed. This trend can be expected because the carriers need to bridge increasingly larger energy differences between photon energy and particle band-gap. We also note that the nearly instantaneous rise of the bleach signal for resonant excitation at 2.4 μm within 111 ± 43 fs is faster than the instrument response function and therefore confirms a temporal resolution of ≲150 fs.
Fig. 3 shows the cooling time as a function of excitation density for all excitation wavelengths. For better comparison we normalize the fluence with respect to signal saturation (see ESI† for details). Below the MEG onset (for 2.4 μm, 2.0 μm and 1.2 μm excitation wavelength), Poissionian statistics can be employed to calculate the mean exciton number per quantum dot, 〈N〉, for which a direct correspondence to the normalized fluence, F/F0, is found in the linear regime as detailed in the ESI.† We do not observe a dependence of intraband cooling time on excitation density. I.e. a ratio of F/F0 = 0.5 corresponds to 25% of excited particles being doubly excited. Yet, no significant change in the cooling time is apparent within our experimental precision. Consequently, multi-particle scattering appears to play only a minor role in intraband relaxation in these quantum dots as also reported for much larger THz-gap HgTe crystallites.39
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Fig. 3 Exciton cooling time as a function of excitation fluence for different wavelengths: 2.4 μm (red), 2.0 μm (orange), 1.2 μm (yellow), 800 nm (green) and 400 nm (blue). The dotted lines correspond to the weighted average for each excitation wavelength (also see Table 1). |
In the following discussion, the mean value of the cooling times for each excitation wavelength in Fig. 3 is used as a more precise value. Here, the inverse uncertainty of the fit results (error bars in Fig. 3) has been used to calculate the weighted average. The uncertainties correspond to the standard deviation of the fit results for the cooling time. These values are summarized in Table 1. The top left panel of Fig. 4 shows the relation between photon energy and cooling time. This representation also reflects the increase of cooling time with higher photon energy as already apparent in the time traces in Fig. 2. Moreover, the increment of cooling time compared to photon energy appears to stagnate past 1200 nm excitation wavelength, suggesting much higher energy dissipation rates for 800 nm and 400 nm excitation. However, these wavelengths lie in the MEG regime, resulting in the generation of more than a single exciton from a single photon. The initial intraband cooling process is complete within less than 5 ps regardless of excitation wavelength (cf. Fig. 2) while Auger recombination of multi-excitons requires several tens of picoseconds. This means that the electronic configuration after intraband cooling consists of several excitons in the MEG regime.
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Fig. 4 Top left. Exciton cooling times as a function of photon energy. Colored data points represent mean values for the respective excitation wavelength, error bars correspond to the standard deviation of all measured cooling times in the linear regime. Arrows indicate the energy corresponding to the energetically lowest exciton (EX), bi-exciton (EXX) and quad-exciton (EXXXX). This assignment of exciton multiplicity to photon energies is based recent results on MEG in HgTe NCs.18 Subtracting the (multi-) exciton energies from the photon energies accordingly (grey symbols), reveals a linear relation between cooling time and energy to be dissipated from the electronic system at a rate of 0.36 eV ps−1, independent of excitation wavelength or density of states. Bottom left. Calculated absorption spectrum and experimental spectra for pump and probe pulses. The calculated spectrum has been shifted by +33 meV to match the experimental bandgap. Right. Calculated level scheme and optical transitions. |
λ exc (nm) | t cool (fs) | k diss (eV ps−1) |
---|---|---|
For 800 nm and 400 nm carrier multiplication was considered in the calculation of the excess energy. The rates for the resonant excitation at 2.4 μm are governed by the instrument response function. | ||
2400 | (111) (±43) | — |
2000 | 314 ± 136 | 0.39 ± 0.17 |
1200 | 1438 ± 141 | 0.37 ± 0.04 |
800 | 1552 ± 212 | 0.36 ± 0.05 |
400 | 2092 ± 152 | 0.34 ± 0.02 |
Given these different band-gap states for the different excitation regimes, we subtract the energy of the corresponding lowest (multi-) exciton energy from the photon energy in order to obtain a meaningful comparison for the energy dissipation during intraband cooling. Following results from photoconductivity measurements of HgTe quantum dots,18 we subtract the lowest bi-exciton energy, EXX, for 800 nm excitation and the energy of the lowest quadruple exciton, EXXXX, for 400 nm excitation. Our approach is illustrated in the top left panel of Fig. 4 by arrows that correspond to the energy of the respective (multi-)excitons. The grey data points then correspond to the excess of absorbed photon energy that is dissipated during intraband cooling. This representation of our data reveals an essentially linear relationship between dissipated energy and cooling time with a mean energy dissipation rate 0.36 eV ps−1. Individual cooling times and energy dissipation rates for all excitation wavelengths are summarized in Table 1.
Also shown in Fig. 4 is the single particle energy spectrum obtained from the pseudopotential calculations and the corresponding absorption spectrum. These calculations allow assigning the most probable electron and hole states after photo-excitation with 2.4 μm, 2.0 μm and 1.2 μm, shown as arrows in the level scheme. Most importantly, excitation at 2.0 μm creates only hot holes, allowing quantification of the 2Sh-to-1Sh relaxation time of 314 ± 136 fs, which is comparable to the rates found in CdSe NCs with similar 2sh–1sh energy spacings40 and roughly one order of magnitude faster than in PbSe quantum dots with comparable hole energy spacing.41 Excitation at 1.2 μm excites both, hot electrons and holes. Yet, the energy dissipation rates are identical within the margin of error for both excitations. We would like to note that our measurements are blind to the cooling rate associated to any states outside the band-gap, such as long-lived (surface)trap states as we do not measure depopulation of the hot state but population of the cold state. This means that these energy dissipation rates represent lower limits.
Several conclusions can be drawn from these observations. Most striking is the almost universal energy dissipation rate independent of the initially excited state and hence, the density of states. By comparing the theoretically assigned initial states for excitation at 2.0 μm and 1.2 μm (as indicated in the right-hand side of Fig. 4), the presence of an excitation in the conduction band does not seem to influence the cooling rate of the system. This suggests mainly two options for intraband carrier relaxation. Firstly, the cooling rate in the conduction band could be identical to the cooling rate in the valence band. This seems unlikely given the large difference of level spacings in the conduction and valence bands, and much slower electron intraband relaxation times reported for doped MIR HgSe quantum dots in the range of tens to hundreds of picoseconds9 and intraband lifetimes of more than 4 picoseconds reported for self-doped Terahertz-gap HgTe nanoparticles with roughly ten-fold smaller intraband gaps.39 Instead, these findings strongly point to Auger cooling as the dominant decay channel for conduction band relaxation in HgTe quantum dots, similar to CdSe quantum dots12,13 where the process is well understood: electrons rapidly transfer energy to holes in the valence band through Auger coupling. The hole subsequently relaxes back to the band-gap by dissipating energy to phonons and/or ligands. Assuming Auger cooling to be much quicker than the relaxation of the hole, the overall energy dissipation rate would be equal to the dissipation rate of holes alone, which would explain the observation of similar energy dissipation rates for 2.0 μm and 1.2 μm excitation. In this scenario, electron intraband cooling via an Auger mechanism in HgTe quantum dots might be substantially slowed down by spatial separation of electrons and holes in heterostructured core–shell quantum dots.8
This still leaves the question of a universal hole cooling rate unanswered, as the higher density of states for hole excitations larger than 0.1 eV (see Fig. 4, right hand side) should lead to more efficient energy dissipation through coupling to phonons and, consequently, larger cooling rates for higher photon energies. This is clearly not what we observe, not even in the MEG regime close to the ionization threshold of the particles. A possible explanation could be nonadiabatic coupling of states as already discussed in carrier relaxation for CdSe and PbSe quantum dots11,41 where it was argued that the transition rate for two states is scaling with the inverse of their energy spacing.11,40 Such a scaling law would explain the observed constant energy loss rate independent of excitation energy.
Confirmation of this proposed mechanism requires further experiments. These could include similar experiments on MIR HgTe quantum dots of various sizes which will give insight into the influence of intraband energy level spacings and localization effects on carrier cooling. The Auger cooling processes could be addressed by using type II heterostructured quantum dots. Here, spatial separation of electrons and holes should slow down carrier cooling in the picture that we propose.8 Further information on the role of ligands in carrier cooling can be accessed by ligand exchange studies or the use of type I core shell particles.
The cooling time found for 800 nm excitation also imposes an upper limit for the MEG process in MIR HgTe quantum dots which is at most 1.5 ps. Taking intraband-cooling into account as well, the process is likely much quicker than our temporal resolution of ≲150 fs. By estimating a mean carrier cooling time on the order of 1.4 ps after inverse Auger-recombination at an excess energy of one band-gap (see Fig. 4, excitation at 1.2 μm), a time-scale for electron–electron scattering during inverse Auger-recombination of 30 ps is retrieved in these particles. The time-scales obtained for the electron–electron interactions match theoretical calculations for inverse Auger-recombination and MEG in PbSe quantum dots42 where the mismatch between the ultrafast sub-100 fs Auger process during MEG and inverse Auger recombination was attributed to the differences in the density of final states. If this reasoning also holds for HgTe quantum dots will have to be addressed in future theoretical studies.
Footnotes |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/d1nr07007j |
‡ Current address: Faculty of Fundamental Science, Phenikaa University, Yen Nghia, Ha-Dong District, Hanoi, 10000, Viet Nam. |
§ Current address: Department of Electrical and Computer Engineering, University of Toronto, 10 King's College Road, Toronto, Ontario M5S 3G4, Canada. |
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