Engineering Auger recombination in colloidal quantum dots via dielectric screening

Auger recombination is the main non-radiative decay pathway for multi-carrier states of colloidal quantum dots, which affects performance of most of their optical and optoelectronic applications. Outstanding single-exciton properties of CdSe/CdS core/shell quantum dots enable us to simultaneously study the two basic types of Auger recombination channels—negative trion and positive trion channels. Though Auger rates of positive trion are regarded to be much faster than that of negative trion for II-VI quantum dots in literature, our experiments find the two rates can be inverted for certain core/shell geometries. This is confirmed by theoretical calculations as a result of geometry-dependent dielectric screening. By varying the core/shell geometry, both types of Auger rates can be independently tuned for ~ 1 order of magnitude. Experimental and theoretical findings shed new light on designing quantum dots with necessary Auger recombination characteristics for high-power light-emitting-diodes, lasers, single-molecular tracking, super-resolution microscope, and advanced quantum light sources.


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Supplementary Figure 6. PL saturation along with excitation power increase for CdSe650/6CdS micro-liquid film. Curves of total PL intensity (I) and the single-exciton-only PL intensity fraction ( ) versus 〈N〉 are fitted by a saturation function with biexciton QY as a fitting parameter (see below for details). For each excitation power, PL decay curve is measured in a fixed period of time. The single-exciton-only PL intensity ( ) is proportional to the total photon number of the long lifetime (single-exciton) component of the PL decay curve.  (2) traces (red) with fittings (black) of single CdSe/CdS QDs with different core sizes and 6 monolayers of shell. 0 (2) is the biexciton QY while single exciton QY is close to unity. b, Biexciton QYs of 50 single CdSe/CdS QDs with the first-exciton absorption peak of core at 650 nm and 6 monolayers of shell measured via second-order photon intensity correlation measurement (red circle) and ensemble biexciton QYs measured via PL saturation experiment with QD micro-liquid film (blue line). c, Ensemble biexciton QYs measured using QD micro-liquid film versus the single-dot values measured via second-order photon intensity correlation measurements for CdSe/CdS QDs with different core sizes and 6 monolayers of shell. The error bar is the standard deviation for 50 single QDs for each sample. The results show a great consistency of ensemble biexciton QYs measured using QD micro-liquid film and the single dot values measured via second-order photon intensity correlation measurements.
Supplementary Figure 8. Core radius and shell thickness dependent biexciton properties. a, Biexciton quantum yield and b, radiative recombination rates and nonradiative Auger recombination rates for CdSe/CdS core/shell QDs with different core radiuses and 6 monolayers of CdS shell. c, Biexciton quantum yield and d, radiative recombination rates and nonradiative Auger recombination rates for CdSe/CdS core/shell QDs with 2.1 nm core radius (the first abs peak for core at 590 nm) and different monolayers of CdS shell.

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Supplementary Figure 9. Recoverability of photochemical electron-doping of CdSe/CdS core/shell QDs. PL spectra, absorption spectra and PL decay curves of CdSe590/8CdS before photochemical doping (black) and after re-oxidation by exposure to air (red).The PL spectrum, absorption spectrum and PL decay dynamics of the recovered QDs are identical to the original ones. This indicated that the photochemical electron doping experiment is a reversible process that not harmful to the structure or surface passivation of the QDs. Figure 10. Auger rate of positive trion calculated via different methods. a, The Auger recombination rates of negative and positive trions for CdSe/CdS core/shell QDs with the first-exciton absorption peak at 630 nm and 6 monolayers of shell. The , + is calculated with , measured by QD micro-liquid film (hatched bar) or measured using single-dot photon statistics experiment for the same single dot (solid bar). Error bars are defined as s.d. b, The same for CdSe/CdS core/shell QDs with the first-exciton absorption peak at 630 nm and 8 monolayers of shell. S10 Supplementary Figure 11. Auger rates of negative and positive trions for CdSe/CdS core/shell QDs with hexahedral and spherical shape. TEM images of a, hexahedral and b, spherical CdSe/CdS core/shell QDs with the first-exciton absorption peak of core at 590 nm and 5 monolayers of shell. The spherical QDs were converted from the hexahedral counterpart. c, d, Auger recombination rates for negative and positive trions for hexahedral and spherical CdSe/CdS core/shell QDs with the first-exciton absorption peak of core at 590 nm and different shell thicknesses. Error bars are defined as s.d. These results suggest that the shape of QDs has limit influence on the Auger recombination rates. S11 Supplementary Figure 12. Charge-dependent volume scaling of Auger recombination. a, Log-log plot for Auger recombination lifetime of positive trion ( , + ) versus core radius and total radius (inset). b, Log-log plot for Auger recombination lifetime of negative trion ( , − ) versus total radius and core radius (inset). Dashed lines are power-law fits with exponents of 2.8 and 3.7 for , + to core radius and , − to total radius respectively.

Supplementary
Supplementary Figure 13. Atmosphere dependent experiment. Representative PL intensity trajectories of a single CdSe/CdS core/shell QDs with the first exciton absorption peak of core at 630 nm and 4 monolayers of shell during changing the surrounding atmosphere from air (black curve) to oxygen (yellow curve).   Table 3. Radiative rate of neural single exciton, and emission rate, emission quantum yield, radiative rate and Auger non-radiative rate of negative trion for CdSe/CdS QDs with different core sizes or shell thicknesses. The data is determined with single dot spectroscopy. PL spectra gradually shift to red and mono-exponential PL decay lifetime increases. In comparison, keeping the same shell thickness but increasing the core size, one would observe significant red-shift of the spectra but small change of the mono-exponential lifetime. Significant red-shift of the spectra suggests that both core and shell dimensions impact quantum confinement. Difference on influence of PL decay lifetime means that spatial distribution of electron and hole wavefunctions of single-exciton is affected differently by the dimensions of the core and shell. This is so because, with near unity PL quantum yield, PL decay lifetime is determined by the electron and hole envelop wavefunction overlapping and the corresponding transition dipole of single-exciton.

Supplementary Note 2. Calculation of Auger recombination rates of biexciton and trions.
As the PL QY of single-exciton is near unity (Figure 1b and Supplementary Note 1), non-radiative recombination of single-exciton is negligible and the mono-exponential S17 decay rate measured should be the radiative recombination rate of single-exciton ( , ).
Furthermore, for a multi-carrier state, Auger recombination is the only non-radiative decay pathway and its rate can be determined as = (1 − ) × by measuring its emission QY and rate (k).
For biexciton, with biexciton QY ( ) and decay rate (  . The total emission intensity saturation curve is essential a function S19 of 2 and 〈 〉. Fitting the total emission saturation curve in Supplementary Figure 6 with the model above, the biexciton quantum yield for CdSe650/6CdS QDs was determined as 2 = 0.2.
Supplementary Note 5. Core radius and shell thickness dependence of biexciton radiative rates, Auger recombination rates and quantum yield for CdSe/CdS core/shell QDs.
One can see that the biexciton quantum yield increases with core radius whereas almost independent on shell thickness for CdSe/CdS core/shell QDs. Given single-exciton QY being unity, biexciton quantum yield can be written as where krad,XX and kAuger,XX are the biexciton radiative recombination rate and nonradiative Auger recombination rate respectively. We found that the core radius dependent biexciton quantum yield is due to a faster decline of The Auger recombination rates of negative trion determined with single-dot spectroscopy is almost identical to the values of the negatively doped QDs by photochemical doping (Fig. 2f). These results are identical evidence for the most prone to occur trion state in single QD is a negative trion state. In addition, radiative recombination rates should be different for QDs within different local environments due to the local field effect 3 (the S20 first column in Supplementary Table 3 and the first column in Fig. 2f). While changing the surrounding atmosphere from air to oxygen, the discharging rate was highly increased and the dim state was almost eliminated 5 . This result implies that the dim state is a negatively charged state that can readily be oxidized (as shown in Supplementary Figure 13).

Supplementary
As demonstrated by previous work 6 , the charge sign of a dim state can be determined by combining PL blinking trace, PL decay dynamics and (2) trace of the state. In detail,

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for a high quality QD sample, the PL QY of a dim state can be calculated from the blinking trace considering the bright state has unity PL QY and nonradiative recombination is negligible. Combining with PL decay dynamics, radiative recombination rate ( , * ) and nonradiative Auger recombination rate ( , * ) can be figured out (as shown in Supplementary Table 4 and Supplementary Table 5 where , is the intrinsic nonradiative decay rate which is negligible for high quality QDs. From Eq. S4 and S5, one notes that the sign of charge has a direct influence on * * ⁄ which can be used to identify the sign. Take CdSe630/4CdS as an example, we firstly assume that the dim state with 20% PL QY ('Dim 1' in Figure 4 in the main text) is a negative trion emission state, that is   Figure 14). These results confirm that the dim state with 20% QY is a negatively charged state for CdSe630/4CdS.

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To further confirmation, we also do the similar calculation with the assumption of the dim state with 20% QY being a positively charged state. In this case with , − = 0.093 ns -1 , , + = 0.32 ns -1 and , = 0.044 ns -1 , + + is 49.7% which is much higher than the 0 (2) value measured for the state. Therefore the dim state with 20% QY for CdSe630/4CdS should not be a positively charged state.
However second-order photon intensity correlation measurement is hard to perform on 'Dim 2' for its rare occurrence. As the dim state with lower QY for CdSe630/4CdS is deciphered as a negatively charged state, the dim state with higher QY can only be a positively charged state. This is because a double-charged negative state would have a QY value even lower than the single-charged negative state.
Similar to above, 'Dim 1' for CdSe630/8CdS is also deciphered as a negatively charged state which is in agreement with literatures [6][7][8][9] . Dim 2 with lower QY than 'Dim 1' could be either a positively charged state (X + ) or a double-charged negative state (X 2-). By counting the radiative recombination pathways for X -, X + and X 2states, the relation between the rates of the radiative recombination rates of the states are  where | ⟩ and | ⟩ are the initial and final Auger electronic states, and are their eigenenergies, and Δ is the Coulomb interaction. Γ is a broadening parameter that accounts for the finite lifetime of the Auger final states due to electron-phonon coupling.
By using a single Slater determinant to represent | ⟩ and | ⟩, the Auger rate for negative trion ( ) and positive trion ( ℎ ) can be calculated by: where m(r) is a mask function that changes smoothly from 1, when r is inside the QD, to 0, when r is outside. in ( , ′ ) is the screening inside the QD, and we have used the proposed model in Ref.10 which includes the G-space dependence.
In our calculation, the surfaces of the constructed QDs are passivated by pseudo H atoms, and the structures are relaxed using the valence force field (VFF) method 11 . The wavefunctions of the QDs are obtained using the charge patching method (CPM) and folded spectrum method (FSM). The motif based CPM produces ab initio quality charge S24 densities for large systems without actually doing self-consistent calculations 12