Supplementary Information

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1 Supplementary Information Marco Govoni 1, Ivan Marri 2, and Stefano Ossicini 2,3 1 Department of Physics, University of Modena and Reggio Emilia, via Campi 213/A, Modena, Italy 2 Department of Science and Methods for Engineering (DISMI), via Amendola 2 Pad. Morselli, Reggio Emilia, Italy 3 CNR-NANO Research Center S3, Via Campi 213/A, Modena, Italy July 19, 2012 We report here details about the simulations presented in the paper. 1 Electronic structures Electronic structures of the silicon-based systems considered in this paper have been obtained from first-principles using density functional theory. The Kohn-Sham electronic structures have been calculated using the pseudopotential and plane-wave PWscf code contained in the QuantumEspresso package [1]. Details about all calculated structures are reported in Tab. 1. Silicon and hydrogen norm-conserving pseudopotentials have been taken from the QuantumEspresso pseudopotential database (Si.pz-vbc.UPF and H.pz-vbc.UPF, respectively). Local density approximation has been used to calculate the exchange-correlation functional. Quasiparticle corrections, obtained within the G 0 W 0 scheme, have been applied to the electronic structure of silicon bulk for all k-points and bands as implemented in the Yambo package [2]. The outcome of this step on carrier multiplication (CM) lifetimes was positively checked with previously theoretical calculations of Ref. [3] and Ref. [4] (see Sec. 3). marco.govoni@unimore.it ivan.marri@unimore.it NATURE PHOTONICS 1

2 System Num. electrons Num. bands Cell size (nm) Si bulk 8 50 bulk cell, relaxed Si 35 H ( ) Si 87 H ( ) Si 147 H ( ) Si 293 H ( ) Si 293 H 172 Si 35 H ( ) Si 293 H 172 Si 147 H ( ) Table 1: Electronic structure details. All simulations were performed integrating the Brillouin Zone at Γ, except for silicon bulk for which we found converged results adopting a ( ) uniform Monkhorst-Pack k-mesh. Convergence trends let us use an energy cut-off for the wavefunction expansion in plane waves of 30.0 Hartree for Sibulk and of 20.0 Hartree for the nanosystems. For both isolated and interacting systems, nanocrystals (NCs) are surrounded by vacuum. 2 Screened Coulomb potential The dielectric screening, at the basis of the calculation of the matrix elements of the screened Coulomb potential, was computed from first-principles following the procedure reported below (coded in the Yambo package [2]). 1. Calculation of the plane-wave expanded noninteracting static polarizability χ 0, see Eq. (8) of Ref. [2]. 2. Calculation of the plane-wave expanded interacting polarizability χ, obtained by solving the Dyson s equation in the random phase approximation (RPA), i.e. where v c is bare Coulomb potential. χ = (1 χ 0 v c ) 1 χ 0 (1) 3. Calculation of the screened Coulomb potential as a sum of two contributions [5]: W = v c + v c χv c. (2) The first (second) part of Eq. (2) is termed bare ( screened ). After convergence tests, we adopted an energy cutoff of about 0.5 Hartree for the screened term in all considered nanosystems. When using a supercell approach, an exact box-shaped Coulomb cut-off technique was adopted in order to remove the spurious Coulomb interaction among replicas (see Ref. [6]). 2 NATURE PHOTONICS

3 SUPPLEMENTARY INFORMATION 3 CM lifetimes In the general case where each Bloch s state is given by a band index n and a k-point index k (chosen inside the first Brillouin Zone, 1BZ), according to the Fermi s golden rule, electron- and hole-initiated CM lifetimes are obtained from the recombination rates Rn e a k a and Rn h a k a, i.e. R e n a k a = cond. val. 1BZ (3) n cn d n b k b k c k d 4π [ M D 2 + M E 2 + M D M E 2] δ(e a + E b E c E d ) R h n a k a = val. cond. 1BZ (4) n cn d n b k b k c k d 4π [ M D 2 + M E 2 + M D M E 2] δ(e a + E b E c E d ) where Hartree atomic units have been adopted. M D and M E of Eq. (3) and (4) are the direct and exchange screened Coulomb matrix elements (see Ref. [5]). They are obtained starting from the calculated Si bulk and Si-NCs electronic structures (Sec. 1) by adopting the screened Coulomb potential of Eq. (2). a labels the initial carrier igniting the CM recombination while b, c and d represent the final products of recombination (see Fig. 1). Energy conservation is guaranteed by the delta function [5]. The calculated CM lifetime τ nak a, associated to the initial state (labelled with a and of energy E in ) is obtained as the reciprocal of rate τ (e/h) n a k a = 1 R (e/h) n a k a. (5) This relation is used to compute CM lifetimes of silicon bulk, isolated and interacting Si-NCs as a function of the energy of the carrier igniting the transition, identified by n a k a, or shortly by in. CM can be considered to follow multiple exciton generation in devices where the excitons are separated, dissociated and the resultant charge carriers are transported as individual charge carriers through the device. Calculations of CM lifetimes are obtained using a recently developed code, where Eq.s (3), (4) and (5) have been implemented. For standard semiconductor materials, like for instance Si bulk, our results can be compared with previously published theoretical results [3, 4] (see Fig. 2), where a fair agreement is found. NATURE PHOTONICS 3

4 E in a d c b b d c E in a e-initiated CM h-initiated CM Figure 1: Schematics of CM recombinations. E in identifies the energy of the carrier igniting the CM processes. a labels the initial state, while b, c and d are the final states. In the case of two interacting Si-NCs, it is possible to extract from Eq. (5) the contribution of one- and two-site CM processes. When NC-NC interaction is turned on, each CM decay process can be split into a linear combination of one-site CM, CDCT and SSQC using the spill-out as weighting factor that measures the localization of the wavefunction. For instance, when both final and initial states are localized onto the same NC, CM is one-site; however if, as a consequence of the NC-NC interaction, the initial state delocalizes over both NCs, the same effect becomes a sum of one-site CM and CDCT. A general definition of one-site CM, CDCT and SSQC valid for all possible configurations (delocalized or not) is reported below 1 = 1 [s nak τ one-site τ a s nb k b s nckc s nd k d + (1 s naka )(1 s nb k b )(1 s nckc )(1 s nd k d )] naka (6) 1 τ SSQC = 1 τ na k a {[(1 s na k a )s nb k b + s na k a (1 s nb k b )] + (7) [s nck c (1 s nd k d )+s nd k d (1 s nck c )]} 1 = (8) τ CDCT τ na k a τ SSQC τ one-site where s a, s b, s c and s d are the spill-out of a, b, c and d carriers, respectively. For a couple of interacting NCs of different size, the spill-out gives the probability of having 4 NATURE PHOTONICS

5 SUPPLEMENTARY INFORMATION Kotani et al. PRB (2010) Sano et al. PRB (1992) This work Time (s) VBM CBM E gap Energy (ev) Figure 2: Calculated CM (by impact ionization) lifetimes in Si bulk as a function of the energy of the initial state. Red marks: results of our method. Blue marks: calculation reported by Sano et al. [3]. Black marks: calculations reported by Kotani et al. [4]. VBM (CBM) stands for valence (conduction) band maximum (minimum). The zero of the energy scale is set at half gap. A fair agreement with previously published e-initiated CM lifetimes is found. the state localized onto the small NC, i.e. it equals 0 (1) when the initial carrier is totally located onto the large (small) NC or can take any value in between. For example, considering the system Si 293 H 172 Si 35 H 36, s na k a is zero when the initial carrier is located onto Si 293 H 172 and equals 1 when the initial carrier is located onto Si 35 H Rate equations The time-evolution of the number of e-h pairs induced into the sample after pumping of above-cm-threshold photons (under low fluence conditions) can be simulated combining the lifetime hierarchy, calculated by first principles and discussed in the main NATURE PHOTONICS 5

6 text, with a set of rate equations that couples biexcition (or trion), below-cm-threshold exciton and above-cm-threshold exciton populations. The fraction of NCs with a biexciton [7] is called n XX, the fraction of NCs with a below-cm-threshold exciton, i.e. with an e-h pair that can only decay radiatively, is called n X and the fraction of NCs with an above-cm-threshold exciton, i.e. with an e-h pair that can decay via CM, is denoted with n X. After above-cm-threshold photon absorption, high-energy excitons (X ) are generated into the sample (initial time). This configuration can evolve according to several decay paths: via one-site CM leading to the formation of one biexciton located onto one nanostructure (XX), via SSQC leading to the formation of two separated e-h pairs (X + X) or via intraband thermalization leading to the formation of one e-h pair (X). While e-h pairs formed after the quantum cutting mechanism are long-lived (i.e. subjected to radiative decay only), biexcitons can quickly ( 1 ps) recycle back to X via Auger recycling, thus restoring an excited configuration that can again evolve via one-site CM, SSQC or thermalize. X can also decay via CDCT, albeit generating a configuration that is subjected to Auger recombination and hence very similar to the one produced via one-site CM. For this reason CDCT does not alter the conclusion of the model, which is kept simple neglecting this term. The implemented system of rate equations is reported below ( ) d n dt X = 1 τ one-site + 1 τ SSQC + 1 τ relax n X + 1 τ Auger n XX d n dt XX = 1 τ Auger n XX + 1 τ one-site n X ( ) d n 2 dt X = τ SSQC + 1 τ relax n X 1 τ radiative n X where τ one-site, τ SSQC and τ Auger are the calculated lifetimes (reciprocal of rates) discussed in the main text. τ relax and τ radiative are parameters introduced to simulate thermalization and radiative decay of excitons, respectively. A simple scheme of the cyclic procedure of CM and Auger recycling is depicted in Fig. 3. We report in Fig. 4 a solution of Eq. (9) obtained using the lifetimes τ one-site =0.01 ps, τ SSQC = 1 ps, τ Auger = 1 ps, τ relax = 1 ps (see Ref. [8]) and τ radiative = 1000 ps, and assuming an initial condition of 5% of NCs excited by the pump pulse. It is important to note that SSQC can benefit from experimental conditions where the embedding matrix or the presence of several interacting NCs (typical condition of three-dimensional realistic systems) favor the creation of delocalized orbitals. As a consequence, the most relevant SSQC lifetimes can decrease from few picoseconds to tenths of picoseconds in a system where each NC can interact with several nearby NCs (such as in the experimental samples studied by Timmerman et al. [9] and Trinh et al. [10]). In order to (9) 6 NATURE PHOTONICS

7 SUPPLEMENTARY INFORMATION Photon in Recycling... CM Threshold CM one-site CM one-site SSQC SSQC... Photons out Figure 3: A simple representation of e-h dynamics in interacting Si-NCs is reported. Horizontal black lines identify CM thresholds and not band edges. XX is a biexciton located on the same nanostructure, while X and X are an above-cm-threshold and a below-cm-threshold exciton, respectively. The cyclic procedure of fast one-site CM, Auger recycling and SSQC is sketched. CDCT processes are neglected for simplicity in the diagram because, being subjected to Auger recombination, they play the same role of one-site processes. After each cycle, the fraction of separated ( quantum cut ) e-h pairs is given by the ratio of SSQC rate and the total decay rate of the X configuration. take roughly into account the effect of many interacting NCs, we considered also the solution of Eq. (9) with a SSQC process being six times faster than the mechanism between just a couple of NCs (see Fig. 4). It is possible to see that an initial time regime appears where the population of biexcitons (n XX ) dominates. Then, single excitons located on separated NCs emerge thanks to the repetitive combination of fast one-site CM, Auger exciton-recycling and SSQC. After the crossing of red (n XX ) with green (n X ) line (few ps) it is more likely to have biexcitons located on separated NCs than biexcitons located onto the same NC. Few tens of picoseconds later, the population of biexcitons in the same nanostructure is reduced to zero. Anyway, the purple dashed line, that indicates the total fraction of excited e-h pairs (i.e. n X +n X +2n XX ), is close to 10% before and after the intersection point, showing that, since from the beginning, a double fraction of e-h pairs with respect to the number of absorbed above-threshold NATURE PHOTONICS 7

8 frac. (%) 6 4 frac. (%) 6 4 SSQC SSQC time (ps) one-to-one SSQC time (ps) one-to-many SSQC Figure 4: Solutions of Eq. (9) are reported here as a function of time. Lifetimes reported in the text are used for the one-to-one plot, while for the one-to-many plot we consider a six times faster SSQC process. According to Fig. 3 red, blue and green lines identify n XX, n X and n X inside the sample after above-cm-threshold photon absorption, respectively. n X is, for instance, given by the ratio of the number of X and the total number of nanocrystals. Solutions of the rate equations are obtained assuming an initial 5% population of X. The purple dashed line indicates the total fraction of excited e-h pairs, i.e. n X + n X +2n XX. The plot underlines the compatibility of our results with the experimental observations of Trinh et al. [10]. Radiative lifetimes are assumed to be 1 ns. This parameter slightly affects the solutions of rate equations introducing only a slow decay component into n X and, hence, into the total number of e-h pairs (purple dashed line). photons is generated, without being subjected to fast degradation even if Auger processes are involved. A time-step simulation of the evolution of the number of excited e-h pairs is recorded in the Movie 1 for a duration of 80 ps [11]. The exciton dynamics implemented in Eq. (9), schematized in Fig. 3 and recorded in Movie 1 result compatible with the recent experimental measurements of Timmerman et al. [9] and Trinh et al. [10]. The Auger recycling mechanism, depicted in Fig. 3 as a non-destructive process, represents an active procedure that is able to restore an active CM configuration. Auger recycling lifetimes have been calculated using effective Coulomb matrix elements and, for the largest NC, are 1 ps. Remarkably, the existence of exciton recycling mechanisms was already hypothesized by D. Navarro-Urris et al. [12] and A. Pitanti et al. [13] in order to interpret energy transfer processes in Er-doped Si-NCs systems 8 NATURE PHOTONICS

9 SUPPLEMENTARY INFORMATION and by W.D.A.M de Boer et al. [8] in order to explain the observed enhancement of photoluminescence intensity in Si-NCs. A direct experimental estimation of the Auger recycling lifetime is, at today, missing. Our calculated Auger recycling lifetime is in agreement with a rough estimation of this value made on the basis of measured biexciton Auger decay lifetimes reported in literature, that is referred to the irreversible dynamics XX X. Even if the latter can be considered only an upper bound to the actual Auger recycling lifetime, Ref. [14] reported a linear scaling of the Auger decay lifetime as a function of the Si-NC volume and, for NCs of 2.4 nm of diameter it corresponds to few ps, supporting the validity of our calculations. 5 Importance of counting wavefunction delocalization Relying on a multipole expansion of the Coulomb potential, Förster resonant energy transfer (FRET) has been frequently adopted to estimate energy transfer rates. When only the dipole term is counted, the lifetimes of energy transfer mechanisms between two NCs predicted by FRET are R 6 (R is the interdot distance [15]). Remarkably, we clearly see from Fig. 5 of the main text that the SSQC lifetimes calculated by first principles do not follow this power-law and can drop of orders of magnitude when wavefunction delocalization is present (not counted in FRET). As discussed in the main text, SSQC shows optimal efficiency in the wavefunction sharing regime. For this reason wavefunction delocalization should be counted when SSQC lifetimes in dense arrays of NCs are calculated. References [1] Giannozzi, P. et al. Quantum Espresso: a modular and open-source software project for quantum simulations of materials. J. Phys.-Cond. Mat. 21, (2009). [2] Marini, A., Hogan, C., Grüning, M & Varsano, D. Yambo: an ab initio tool for excited state calculations. Comput. Phys. Commun. 180, (2009). [3] Sano, N. & Yoshi, A. Impact-ionization theory consistent with a realistic band structure of silicon. Phys. Rev. B 45, 4171 (1992). [4] Kotani, T. & van Schilfgaarde, M. Impact ionization rates for Si, GaAs, InAs, ZnS, and GaN in the GW approximation. Phys. Rev. B 81, (2010). 9 NATURE PHOTONICS 9

10 [5] Govoni, M., Marri, I. & Ossicini, S. Auger recombination in Si and GaAs semiconductors: ab initio results. Phys. Rev. B 84, (2011). [6] Rozzi, C.A. et al. Exact coulomb cutoff technique for supercell calculations. Phys. Rev. B 73, (2006). [7] The fraction of NCs containing one biexciton is given by the ratio of the number of biexcitons localized in the same nanostructure and the total number of nanocrystals. [8] de Boer, W.D.A.M. et al. Red spectral shift and enhanced quantum efficiency in phonon-free photoluminescence from silicon nanocrystals. Nature Nanotech. 5, (2010). [9] Timmerman, D., Izeddin, I., Stallinga, P., Yassievich, I.N. & Gregorkiewicz, T. Space-separated quantum cutting with silicon nanocrystals for photovoltaic applications. Nature Photon. 2, (2008). [10] Trinh, M.T. et al. Direct generation of multiple excitons in adjacent silicon nanocrystals revealed by induced absorption. Nat. Photonics 6, (2012). [11] A cubic cell of 20 3 close-packed NCs with periodic boundary conditions is adopted to record the movie where, at each time-step (1 fs), transition probabilities are calculated according to the lifetimes used to get the results shown in Fig. 4, and the next snapshot is generated. [12] Navarro-Urrios, D. et al. Energy transfer between amorphous Si nanoclusters and Er 3+ ions in a SiO2 matrix. Phys. Rev. B 79, (2009). [13] Pitanti, A. et al. Energy transfer mechanism and Auger effect in Er 3+ coupled silicon nanoparticle samples. J. Appl. Phys. 108, (2010). [14] Beard, M.C. et al. Multiple Exciton Generation in Colloidal Silicon Nanocrystals. Nano Lett. 7, (2007). [15] Dexter, D.L. A theory of sensitized luminescence in solids. J. Chem. Phys. 21, (1953). 10 NATURE PHOTONICS