Carrier multiplication detected through transient photocurrent in device-grade films of lead selenide quantum dots

In carrier multiplication, the absorption of a single photon results in two or more electron–hole pairs. Quantum dots are promising materials for implementing carrier multiplication principles in real-life technologies. So far, however, most of research in this area has focused on optical studies of solution samples with yet to be proven relevance to practical devices. Here we report ultrafast electro-optical studies of device-grade films of electronically coupled quantum dots that allow us to observe multiplication directly in the photocurrent. Our studies help rationalize previous results from both optical spectroscopy and steady-state photocurrent measurements and also provide new insights into effects of electric field and ligand treatments on multiexciton yields. Importantly, we demonstrate that using appropriate chemical treatments of the films, extra charges produced by carrier multiplication can be extracted from the quantum dots before they are lost to Auger recombination and hence can contribute to photocurrent of practical devices.


Supplementary
. TPC traces as a function of applied voltage (V) normalized by V; these measurements were conducted on the EDT treated film of PbSe QDs with E g = 0.69 eV using 1.55 eV excitation with <N abs > = 0.02. The fact the V-normalized traces are virtually indistinguishable indicates the photoconductance of the QD film is electric-field independent, that is, Ohmic.

Supplementary Notes
Supplementary Note 1: Modeling of photocurrent transients. Transient photocurrent (TPC) traces could be accurately modeled by the sum of two exponentials and an offset, convoluted with a Gaussian to account for the finite temporal resolution as is typical for experiments with slower temporal resolution 2 . Example fits can be seen in Fig. 3A in the main text. Prior to fitting the baseline signal at times before the pulse arrival, was subtracted. In some cases, the data was also filtered through a Fourier analysis to remove GHz noise arising from impedance mismatches in the device.
In typical optical experiments, the low fluence transients taken below the carrier multiplication (CM) threshold display a nearly constant signal for several nanoseconds. In this circumstance, the contribution of the multiexciton populations to the signal can be easily extracted by simply taking a ratio of the early time peak signal to the late-time "single-excitonic" signal measured at a few nanoseconds after completion of Auger decay 3 . However, in most of the measurements the recorded photocurrent transients exhibit a prominent nanosecond decay at low fluences which does not permit the usual data analysis strategy. Below, we briefly discuss different models which help elucidate the measured TPC dynamics. To explain the multiexponential decay, we consider a model case of a QD film containing two sub-ensembles of dots (fractions f 1 and f 2 ) with differing decay times of single excitons, arising from, e.g., different numbers of surface defects. We further assume that both sub-ensembles have the same biexcitons lifetime. In this situation, we can describe the QD population dynamics with the following set of rate equations: Here we take into consideration that biexcitons contribute twice the signal of a single exciton and further that the QD fractions in the two sub-ensembles are normalized to unity: Next we assume that the initial single-exciton and biexciton QD occupancies are identical for both subensembles as defined by excitation fluence: Equations (1)-(4) can be analytically solved and the resulting photocurrent can be presented as the sum of three exponential terms: In the limit that and are both much smaller than A,XX k (as observed in our experiments), we find: Using the latter expressions, we can re-write Eq. (9) as: where t is a functional form of the solution in the case when p 2 =0, that is, the photoexcited system is purely single-excitonic, the situation experimentally realized at low excitation fluences. From the experimental perspective, this implies that when there is a clear separation of biexciton and single-exciton timescales, a purely biexcitonic signal can be extracted from the measured high-pumpintensity time transients by subtracting a "tail-normalized" single-exciton trace measured at low fluences.
This subtractive procedure introduced in ref. 4, has been frequently used in transient absorption (TA) spectroscopy for isolating multiexciton dynamics. Another useful implication of this analysis is that the early time amplitudes of tail-normalized single-exciton and multiexciton traces can be used to directly evaluate the exciton multiplicity, <N X >, and hence quantum efficiency of photon-to-exciton conversion when <N X > is measured in the limit of low fluences. Indeed, according to Eqs. (13) and (14), the t = 0 total signal amplitude is (p 1 + 2p 2 ), while the amplitude of the single-exciton component is (p 1 + p 2 ). The ratio of these amplitudes is (p 1 + 2p 2 )/(p 1 + p 2 ), which is the definition of <N X >. Again, this procedure is similar to one used in CM studies utilizing ultrafast TA, where the exciton multiplicity is evaluated from the ratio of the early-to late-time TA signal 3 .
To evaluate the range of validity of analytical solutions given by Eqs. (13) and (14) we have compared them to numerical solutions of Eqs. (1)-(4). We have found that for carrier relaxation parameters indicated by our measurements, the correction factor is less than 5%. However, when the k i /k A ratio becomes greater than 0.15 the correction can become significant.
We have additionally tested two other models, where we consider only one population and the single exciton rate equation was altered to one of the following forms: , Where α is an arbitrary positive number. These rate equations were numerically solved and the resulting solutions were incorporate into Eq. (5) which then was used to fit TPC data to find initial QD occupancies. Shown in Supplementary Figure 2A we compare the fits of the three different models along with the resulting exciton multiplicity derived from the fits. Here, "diff model 1" refers to Eq. (15), "diff model 2" refers to Eq. (16), and "exp model" corresponds to Eqs. (1) to (4). On this timescale all three models accurately reproduce the data. The extracted exciton multiplicity derived from the fits in Supplementary Figure 2A are shown in Supplementary Figure 2B. We find that the extracted exciton multiplicity is virtually independent on the specific model used to describe the low fluence data. We conclude that the specific details of the decay of the photocurrent does not strongly influence our measurement, chiefly because there still remains a clear separation of timescales.  (6). With our limited time resolution we are not able to resolve Auger decay components due to triexcitons and multiexcitons of higher orders. However, as an ultimate product of decay of these species is a biexciton, they all contribute to probability p 2 , which hence can be calculated from 2 2 i i pk     . Probabilities p 0 and p 1 are simply equal to k 0 and k 1 , respectively, which preserves the required normalization p 0 + p 1 + p 2 =1. Using this truncated version of the Poisson distribution, we can simultaneously fit pump-intensity dependence of p 1 and p 2 derived from the TPC traces using a QD absorption cross-section and a shared amplitude factor as two adjustable parameters (see Fig. 3B; main text). Using a truncated version of Poisson distribution we can also obtain the following expression for the exciton multiplicity: <N X > = (2p 2 + p 1 )/(p 1 + p 2 ) = 2 -<N abs >e -<Nabs> (1 -e -<Nabs> ) -1 . (17) This expression is used in the main text to model experimentally derived multiplicities in Fig. 3C, using the same absorption cross-section as in Fig 3B.