Letter pubs.acs.org/NanoLett
Nonradiative Auger Recombination in Semiconductor Nanocrystals Roman Vaxenburg,† Anna Rodina,‡ Andrew Shabaev,§ Efrat Lifshitz,† and Alexander L. Efros*,∥ †
Technion - Israel Institute of Technology, Haifa 32000, Israel Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia § George Mason University, Fairfax, Virginia 22030, United States ∥ Naval Research Laboratory, Washington, DC 20375, United States ‡
ABSTRACT: We calculate the rate of nonradiative Auger recombination in negatively charged CdSe nanocrystals (NCs). The rate is nonmonotonic, strongly oscillating with NC size, and sensitive to the NC surface. The oscillations result in nonexponential decay of carriers in NC ensembles. Using a standard single-exponential approximation of the decay dynamics, we determine the apparent size dependence of the Auger rate in an ensemble and derive CdSe surface parameters consistent with the experimental dependence on size. KEYWORDS: Auger recombination, CdSe, nanocrystal, quantum dot, trion, boundary conditions
A
shown recently20 that AR rate of negative trions, which consist of two electrons and a hole, in colloidal CdSe NCs follows the size dependence ⟨1/τA⟩ ∝ ⟨a⟩−4.3 and that this rate is slower than in biexcitons9 by about an order of magnitude. This suggests that the biexciton decay is dominated by positive trion AR that must be very efficient and can have different sizedependence due to the complex valence-band structure and different band offsets. A stronger dependence of the AR rate on the NC size was predicted theoretically.18 The calculations of AR in charged CdS NCs showed that the AR rate in individual NCs exhibits a highly nonmonotonic oscillatory dependence on NC size. Averaging this dependence over the NC size distribution gives ⟨1/τA⟩ ∝ ⟨a⟩−7 in the limit when the momentum, k, of the excited carrier is very large, such that ka ≫ 1. The averaging leads to a weaker size dependence of the AR rate, ⟨1/τA⟩ ∝ ⟨a⟩−5, when the energy of the excited carrier is just above the confining potential barrier. It was shown recently that the AR rate is intimately connected with properties of surfaces and interfaces.21,22 In the calculation of the AR rate in ref 18, however, the NC surface properties were not taken into account because the model utilized standard boundary conditions23 (BCs). To take the effect of the NC surface into account, the AR must be considered using general BCs,24 which contain surface-related parameters. In this Letter, we use the eight-band k·p model and general BC formalism to calculate the size dependence of the AR rate of negative trions in CdSe NCs embedded in a solid or liquid
fter more than 30 years of research on semiconductor nanocrystals (NCs), started by Ekimov1 and Brus2 in stained glasses and aqueous solutions, correspondingly, they became much more than just objects of scientific curiosity. The demonstration of tunable room-temperature lasing using NC quantum dot solids,3 the development of NC-based lightemitting diodes, and photovoltaic cells4,5 are just a few illustrations of the broad technological potential of these materials. The performance of these optical devices, however, is hindered by a significant enhancement of the nonradiative Auger recombination (AR) in NCs.6−8 In the AR process, an electron−hole pair recombines nonradiatively, transferring the recombination energy to a third carrier, to an electron, for example, as shown schematically in Figure 1a. Owing to its enhanced efficiency, AR becomes the central nonradiative relaxation channel affecting all aspects of carrier dynamics in NCs. In particular, AR dominates the decay dynamics of multiexcitons,9,10 induces fluorescence intermittency,11,12 causes gain decay in NC lasing,3,13 and triggers the droop phenomenon in quantum dot14 and quantum well15,16 lightemitting diodes. The high relevance of the AR processes in NC photophysics raises the question of the size-dependence of the AR rate. For Auger processes in which an extra electron is ejected out of the NC (see Figure 1a), the average rate of AR, ⟨1/τA⟩, measured in ensemble of CdS NCs embedded in glass matrix,17,18 is described as ⟨1/τA⟩ ∝ ⟨a⟩−4.5, where ⟨a⟩ is the average NC radius. Later it was found9,19 that the rate of the multiexcitonic AR in colloidal NCs is inversely proportional to the NC volume: ⟨1/τA⟩ ∝ ⟨a⟩−3. This so-called volume scaling was observed in both direct and indirect-gap NCs,19 suggesting that the ∝ ⟨a⟩−3 scaling is a universal trend among NCs. It was © 2015 American Chemical Society
Received: December 29, 2014 Revised: February 2, 2015 Published: February 19, 2015 2092
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Figure 1. Auger recombination of a negative trion in CdSe nanocrystals. (a) Schematic diagram of the Auger process leading to ejection of an electron into the continuous spectrum of the surrounding medium. Two 1S1/2 electrons, labeled by 1 and 2, and 1S3/2 hole, h, comprise the initial state, while the ejected nP3/2 electron, f, represents the final state. (b) Radial probability density, R2(r)r2, of the ejected Auger electron in CdSe nanocrystal with radius a = 2.5 nm at time t ≪ t0 ; (c) R2(r)r2 at time t ∼ t0 ; (d) R2(r)r2 at time t ≫ t0. Here, t0 is the onset time of disintegration of the wavepacket shown in (b).
is determined by a superposition of eigenstates from a certain range of energies, allowing formation of short-lived wavepackets. Radial wave functions of the ejected Auger electron calculated at different times in CdSe NC with radius a = 2.5 nm are shown in Figure 1b−d (calculation details are given below). We find three distinct stages in the course of the Auger electron time-evolution. At short times, a stable spherical wavepacket localized in the vicinity of the NC surface is formed (Figure 1b). Later, it begins to disintegrate and gradually becomes delocalized (Figure 1c). Finally, in the long-time limit, the full delocalization is reached and a final free-particle state is recovered (Figure 1d). Analysis of the perturbative expansion coefficients allows us to estimate the time scales of this process. It can be shown that the onset of the wavepacket disintegration process, t0, is on the order t0 ∼ a(8m0/Ep)1/2, where m0 is the free electron mass and Ep = 2m0P2 where P is the semiconductor’s Kane parameter. For CdSe NC with a = 2.5 nm, this gives t0 ∼ 4 fs. Time-evolution of the Auger electron wave function starts off with a wavepacket localized at the NC surface at short times t ≪ t0. This supports the conjecture that the AR occurs right at the NC surface. In our calculations, we assume that CdSe NCs have spherical shape and zinc-blende lattice structure. Under this assumption, the electron and hole wave functions inside the NC can be described by the eight-band k·p model27 in the spherical-band approximation.28 Within this model, each electron and hole state is characterized by the magnitude, j, and projection, m, of the total angular momentum, J = F + L, and by parity, π = ±1. Here, F and L are the Bloch and envelope angular momenta, respectively.29 The full wave function of the electron and hole inside the NC can be written, using vector notation,30 as
matrix. The calculations show strong, orders-of-magnitude, oscillations of the single NC AR rate, 1/τA, with the NC radius, a, which behaves after averaging as ⟨1/τA⟩ ∝ a−6.5. The oscillatory behavior of 1/τA results in multiexponential photoluminescence (PL) decay dynamics in a NC ensemble. Approximation of the calculated PL decay by a singleexponential function, which mimics the experimental procedure of the AR lifetime measurements, results in apparently weaker size-dependence of the ensemble AR rate, which is in good agreement with experimental data. Why is the nonradiative AR so efficient in NCs in comparison with bulk materials? In bulk wide-gap semiconductors, such as CdSe and CdS, the AR is suppressed due to the kinematic threshold that limits the AR efficiency.7,17,18,25 The threshold originates from the requirement of energy and momentum conservation during the AR process in an effectively infinite bulk crystal. In NCs, however, the strong confining potential, which localizes electrons and holes, introduces high momentum components into their wave functions. This lifts the requirement of momentum conservation, rendering the AR an efficient thresholdless process. Semiclassically, one would expect that the electron acquires the high momentum components in the regions where its local momentum has high uncertanty.7,17 In NCs, this should occur in the neighborhood of abrupt interfaces, suggesting that the NC surface is where the AR and generation of the energetic Auger electrons are most likely to take place. To verify this semiclassical expectation we investigate the time-evolution of the wave function of the electron excited during the AR process and ejected into the continuous spectrum of the surrounding material (see Figure 1a). The time-evolution can be described by standard time-dependent perturbation theory26 in which the wave function is expanded in energy eigenstates with time-dependent coefficients. In the long time limit, the perturbative expansion reduces to Fermi’s Golden Rule that dictates strict energy conservation, such that the initial state wave function evolves into a specific final state with definite energy (a free-particle state in our case). At shorter times, however, the time-evolution of the wave function
Ψ(r )Ωc
(2)
⎛ R (a + λ / 2) ⎞ ⎜ c ⎟ = T̂ ⎜ c ⎟ ⎜ < ⎟ ⎜ > ⎟ ⎝ iV τ (a − λ / 2)⎠ ⎝ iVτ (a + λ / 2)⎠
The ground-state hole on the other hand, which is strongly confined (mostly due to the heavy-hole mass), can be approximated as being surrounded by an infinite barrier, implying vanishing of the hole wave function outside the NC, Ψ>h ≡ 0. To find the energies and the wave functions of the electrons and holes, one needs to solve a system of differential equations for the radial functions inside and outside the NC and impose appropriate BCs on the solutions at the NC surface. The electron and hole radial functions in eq 1 satisfy the system of differential equations given in eq 14 in ref 28, while the electron radial function in eq 2 satisfies the free-electron Schrodinger equation, [(ℏ2/2m0)(−∇2r + ( + 1)/r2) + U0 − E]R>c (r) = 0. Here, ∇2r is the radial Laplacian,26 is the angular momentum, U0 is the height of the potential barrier acting on electrons, and E is the electron energy measured from the bottom of the conduction band. We use CdSe material parameters from ref 31, and U0 = 1.75 eV. In the eight-band model, utilization of standard BCs32 is known to admix the wave functions with unphysical (spurious) components with wavevectors lying outside the first Brillouin zone.24,33 We eliminate these unphysical contributions using the procedure proposed in refs 24 and 33. When solving eq 14 of ref 28 for electrons (holes) inside the NC, we set to zero the remote band contribution to the valence (conduction) band effective masses, γ1 = γ = 0 (α = 0). This procedure results in a reduction of the order of the differential system in eq 14 and reduces the number of required BCs, thereby removing the spurious components. Applying the BCs that connect the solutions inside and outside the NC, we should note that the confining potential
(3)
t t where T̂ = t11 t12 is the energy and state independent 21 22 transfer matrix characterizing the NC surface within the k·p model. The off-diagonal element t21 describes an interfacial δfunction potential,35 which is absent in our model, therefore t21 = 0. In addition, the requirement24 det T̂ = 1 implies t22 = 1/t11. The surface parameter Ta0 = −0.6 Å introduced in ref 34 can be shown to satisfy the relation Ta0 = (ℏ/m0)t11t12 in the case of infinite potential barrier, U0 → ∞. Assuming that this relation holds also for a finite barrier, the matrix T̂ contains only one independent parameter, which we choose to be t11. As a result, the NC surface in our model is described by two parameters, t11 and λ. The radial envelope velocity, Vτ, of the 1S1/2 and nP3/2 electron states participating in the trion AR is given by34
(
)
Here, the energy-dependent conduction band effective mass is m0/m(E) = α + (Ep/3)[2/(Eg+E) + 1/(Eg + Δ + E)], Δ is the spin−orbit splitting energy, and Eg is the bulk energy gap. For holes, the wave function outside the NC is identically zero and we require vanishing of the hole radial functions Rh1, Rh2, and Rso at r = a − λ/2. 2094
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well as the overall shape of the oscillatory pattern, are affected by the parameter t11, as shown in Figure 2b,c. In particular, the dependencies ⟨1/τA⟩ ∝ a−7.5 and ⟨1/τA⟩ ∝ a−6.4 for t11 = 0.5 and 1.5 are obtained. On the other hand, we find that the increase of λ produces no significant effect, apart from shifting the calculated 1/τA toward larger sizes by ≈ λ/2 (not shown). The nonmonotonic behavior of the total trion decay rate, 1/τT, and its dependence on the choice of the BC parameters are also evident in Figure 2. At larger NCs, the AR decay becomes slower than the trion radiative decay, and the oscillations of 1/ τT become significantly suppressed. Experimentally, the rate of AR is usually extracted by probing the excitation decay dynamics in an ensemble of NCs.9,20 One of the methods to probe such dynamics is by measuring PL decay.20 The AR rate is then obtained from the best-fitting of the PL decay time-dependence by a single-exponential function. However, the strongly nonmonotonic oscillatory behavior of the AR rate with NC radius, as shown in Figure 2, should result in a highly nonhomogenuous distribution of the AR rates across a size-dispersed ensemble of NCs. Consequently, the PL decay in such an ensemble should exhibit complex multiexponential behavior. Indeed, the time-dependent PL intensity, I(t), can be written in this case as I(t) ∝ ∫ ∞ 0 G(a)exp[−(1/τA(a) + 1/τR) t]da, where G(a) describes the NC size-distribution. To determine an apparent AR rate of the NC ensemble, which we label by 1/τens A , we mimic the experimental procedure by calculating the PL decay and approximating it by the best-fit single-exponential function. We assume Gaussian distribution of the NC sizes with standard deviation σ = 8% (this value can be obtained by analysis of the absorption spectra in ref 20 by the procedure described in ref 38). Figure 3a shows calculated PL decay traces and the corresponding single-exponential fits in CdSe NC ensembles with mean radii ⟨a⟩ = 1.85, 2.25, 2.75, and 3.5 nm. For each one of the ensembles, the inset shows the fitted apparent AR rate, 1/τens A (symbols). Additionally, the inset illustrates how the individual NC AR rates, 1/τA, (shown by curve) are distributed within the full width at half-maximum (fwhm) of the 8% Gaussian size-dispersion. One can see that orders-of-magnitude dispersion of 1/τA can exist in a typical NC ensemble. The multiexponential behavior of I(t) can be observed, especially in the ⟨a⟩ = 1.85 nm ensemble in which the AR lifetimes are significantly shorter than τR. In addition, it can be seen in Figure 3a that the PL decay curves, as well as the corresponding apparent ensemble AR rates shown in the inset, exhibit nonmonotonic behavior with mean NC size, the effect caused by the oscillations of 1/τA as well. The behavior of the negative trion PL decays in NC ensembles with mean radii in the entire range ⟨a⟩ = 1.6−5 nm is shown in Figure 3b. Here, the nonmonotonic behavior of I(t) with ⟨a⟩ can be seen in full, suggesting that even with σ = 8%, the apparent singleexponential AR lifetimes can be nonmonotonic with NC size. Calculating the NC ensemble PL decays and fitting them to a single-exponential function provides us with the apparent ensemble AR rates, 1/τens A . We find that, generally, this apparent rate decreases as a lower power of the NC radius than the AR rate of the underlying single NCs. In addition, the apparent AR rate depends on the choice of the BC parameters, t11 and λ, as well as on the value of the effective dielectric constant, ε, (see eq 6) whose value should generally be smaller than the 39 corresponding bulk constant, ε∞ bulk. In our model, this effective constant accounts for the complex electrostatic interaction between charge densities both inside and outside the NC. In
We describe the nonradiative AR process of a negative ground-state trion, shown schematically in Figure 1a, that leads to ejection of a bound electron into the continuous spectrum above the NC barrier. The rate of this process is given by Fermi’s Golden Rule integrated over the continuum of final states18,22,26 ⎛ 1 2π 2 ⎜ ∂E(k) = |M | ⎜ ∂k τA ℏ ⎝
⎞−1 ⎟ ⎟ k = qf ⎠
(5)
where M is the Coulomb matrix element, and E(k) and qf are the dispersion law and momentum of the ejected electron outside the NC. The wave functions of the initial and final states entering the matrix element M are constructed in the form of antisymmetrized products of the participating singleparticle states written in the electron−electron representation. This allows expression of the Coulomb matrix element as M = M1 − M2, where M1 =
∬
d3r1 d3r2 Ψ†h(r1)Ψ1(r1)
e2 † Ψf (r2)Ψ2(r2) εr12
(6)
Here, ε is the dielectric constant, and M2 is obtained from M1 by exchanging the indices Ψ1 ↔ Ψ2. The initial state, comprised of two 1S1/2 electrons and a 1S3/2 hole, is represented by Ψ1,2 and Ψh, respectively (see Figure 1a). With this initial configuration, the requirements of angular momentum and parity conservation allows Auger electron excitation only to the nP3/2 state, represented here by Ψf (Figure 1a). The integral in eq 6 is solved using the standard technique36 of multipole expansion of the Coulomb term, 1/r12 ≡ 1/|r1 − r2| . Upon the multipole expansion, M can be expressed in terms of up to 36 integrals of the form:
where r< = min(r1, r2), r> = max(r1, r2), = 0,1,2, ... is summation index,36 and Rv[β](r) are the radial envelope functions of the four states (β = h, 1, f, 2) involved in the negative trion AR. The subscripts A,B,C,D = c, h1, h2, so specify the radial components of each state (see eq 1). To investigate the size-dependence of AR, we perform a full numerical calculation of the negative trion AR rate (eq 5) in single CdSe NCs with radii a = 1.2−8 nm. Figure 2a shows the calculated AR rates, 1/τA, and total trion decay rates, 1/τT = 1/ τA + 1/τR, in individual NCs using standard BC parameters (t11 = 1, λ = 0) and ε = ε∞ bulk. Here, τR ∼ 3 ns is the negative trion radiative lifetime,37 and ε∞ bulk = 6.25 is the bulk CdSe highfrequency dielectric constant. The effect of the general BCs on calculated 1/τA and 1/τT is shown in Figure 2b,c. One can see that in all cases, 1/τA is a rapidly decreasing function of the NC radius with strong, orders-of-magnitude, oscillatory behavior. These oscillations are caused by the fact that the integrand in eq 6 is itself an oscillating function of coordinates, such that during the integration the positive and negative contributions exactly cancel each other at certain NC sizes. Qualitative analysis of the integral shows that the oscillation period can be estimated as Δa ∼ (πℏ/Eg)(Ep/2m0)1/2, which is in good agreement with the numerical calculations. In the case of standard BCs (Figure 2a), averaging over the oscillations results in the power-law dependence ⟨1/τA⟩ ∝ a−6.6. In the case of general BCs, this averaged size-dependence, as 2095
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Figure 4. Comparison of the size dependences of Auger recombination rate in individual nanocrystals and nanocrystal ensembles. Calculated Auger rates of negative trions in single CdSe nanocrystals, 1/τA (blue curve), and apparent Auger rates in nanocrystal ensembles, 1/τens A (red curve). The latter were obtained by approximating the calculated multiexponential ensemble photoluminescence decays by a single-exponential function. Experimental data from ref 20 are shown by symbols. The straight dashed lines indicate averaged power-law behavior of each one of the curves. The orange strip shows the range of the negative trion radiative rates. The calculation was performed using ε = 0.4ε∞ bulk and boundary condition parameters t11 = 1.6,λ = 0.25a0. Inset shows the same calculation but with ε = ε∞ bulk.
Figure 3. Time dependence of the photoluminescence (PL) intensity decay calculated for nanocrystal ensembles. (a) Calculated PL decay, I(t), (solid curves) in ensembles of CdSe nanocrystal having Gaussian size-distribution with mean radius ⟨a⟩ and standard deviation σ = 8%. The corresponding single-exponential fits are shown by dashed curves. Inset: Calculated negative trion Auger rates in individual CdSe nanocrystals, 1/τA (blue curve), and fitted apparent Auger rates of the ensembles in the panel, 1/τens A (color-matched symbols). Gray strips indicate how single-nanocrystal Auger recombination rates are distributed within the fwhm of the Gaussian size-dispersion in each of the ensembles. (b) Contour plot of I(t) in nanocrystal ensembles with ⟨a⟩ = 1.6−5 nm. Locations of the PL decay traces of the ensembles in (a) are indicated by dotted lines.
monotonic behavior of I(t) with NC radius, as mentioned above. To obtain the power-law size-dependence of the calculated apparent ensemble AR rate, the residual oscillations of 1/τens A should be averaged over. This averaging, however, is not unique, as it depends on the choice of the end-points of the curve. Accounting for this uncertainty, we find that the averaged scaling of the apparent ensemble AR rate is given −4.3±0.4 by ⟨1/τens , which is in agreement with the A ⟩ ∝ ⟨a⟩ experimental scaling of ∝ ⟨a⟩−4.3. It should be mentioned that the upper and lower limits of the ⟨1/τens A ⟩ scaling obtained by exploring the entire physically meaningful parameter range are ∝ ⟨a⟩−6 and ∝ ⟨a⟩−3.2, respectively. To the best of our knowledge, the general BC parameters for CdSe NCs with finite barrier height have not been determined experimentally or calculated before. The magnitude of the extracted interface parameters, however, is consistent with the parameters calculated in ref 40 for GaAs/AlGaAs heterostructure. It is seen in Figure 4 that the averaged AR rate in single NCs follows the size dependence 1/τA ∝ a−6.5, which is stronger than the experimental and theoretical AR size-dependence ∝ a−4.3 in NC ensembles. This apparent disagreement stems from the single-exponential approximation of the multiexponential AR dynamics in size-dispersed NC ensembles in which the AR rates of individual NCs can vary widely and nonmonotonically. Indeed, single NC spectroscopic studies41,42 showed a large spread in biexciton lifetimes, which could not be explained by monotonic dependence of the AR rate on NC size. The oscillatory behavior of 1/τA results in vanishingly small AR rate at certain NC sizes. This suggests a possibility to fabricate NCs in which AR is intrinsically suppressed simply by a careful choice of the NC size.
the present work, however, we do not attempt to calculate ε, but rather leave it as a free parameter. To describe the negative trion AR rates measured in CdSe NC ensembles in ref 20 (shown in Figure 4), we systematically explore the dependence of the calculated apparent AR rate of the ensembles on the three parameters, t11, λ, ε. For each such triad, we calculate the size dependence of 1/τA in individual CdSe NCs within radii interval a = 1.2−8 nm (the resultant dependences are similar to those shown in Figure 2). Then, for all the ensembles in that size range, we calculate the PL decay curves, I(t), from which we extract the apparent ensemble AR rates, 1/τens A , by single-exponential fits (the procedure is similar to that shown in Figure 3). Finally, we compare the calculated data with the experimental values from ref 20 and minimize the difference between the two (in the least-squares sense) by varying the BC parameters and ε. As a result, we find that the best agreement between theory and experiment is obtained when the BC parameters are in the range t11 = 1.6 ± 0.4, 0 < λ < 0.25a0, and the effective dielectric constant is ε = 0.4ε∞ bulk, where a0 is the CdSe lattice parameter. Figure 4 shows the calculated AR rates of negative trions in single CdSe NCs and in NC ensembles, obtained using t11 = 1.6, λ = 0.25a0, and ε = 0.4ε∞ bulk. The calculations performed with ε = ε∞ bulk are shown in inset for comparison. One can see that although the apparent AR rate, 1/τens A , calculated with these parameters describes the experimental data from ref 20 very well, it still shows residual oscillatory behavior in NC ensembles with 8% size-dispersion. These oscillations reflect the non2096
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The present theoretical analysis does not take into account several additional factors that should further average out the residual oscillations of the calculated ensemble AR rate shown in Figure 4. Variation in the NC shape, fluctuations in potential barrier height, and lifetime-broadening of the ejected electron state can contribute to the further averaging. In addition, we neglect phonon contributions to the AR process as in our present study they would produce only small second-order corrections. The phonon contributions could become important, however, in very small CdSe clusters not considered in this paper, as it was shown in ref 43. In summary, we calculated the rate of AR of negative trions in single CdSe NCs and in NC ensembles. We found that in individual NCs the AR rate has nonmonotonic, strongly oscillating, dependence on NC size. This oscillatory behavior implies high dispersion of the AR rates across a NC ensemble, which in turn results in multiexponential dynamics of the ensemble PL decay. We determine the range of the general BC parameters and the effective dielectric constant that allow quantitative description of the experimental data in the framework of our model. The single NC AR rate, calculated with these parameters and averaged over the oscillations, scales as ⟨1/τA⟩ ∝ a−6.5. On the other hand, the corresponding −4.3±0.4 apparent AR rate of an ensemble scales as ⟨1/τens , A ⟩ ∝ ⟨a⟩ which is in agreement with experiment. We show that the apparently weaker dependence of the average ensemble AR rate on the NC size is a result of the single-exponential approximation used for description of the multiexponential AR dynamics of the NC ensemble.
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AUTHOR INFORMATION
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[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS
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REFERENCES
The authors thank C. Stephen Hellberg for critical reading of the manuscript. R.V. and E.L. acknowledge the support of the Israel Science Foundation (Project No. 1009/07 and 1425/04), the U.S.A.-Israel Binational Science Foundation (No. 2006225), the Israel Council for High Education - Focal Area Technology (No. 872967), the Volkswagen Stiftung, (No. 88116), Russell Berrie Nanotechnology Institute, and Schulich Faculty of Chemistry, Technion. A.S. acknowledges the support of the Center for Advanced Solar Photophysics (CASP), an Energy Frontier Research Center (EFRC) funded by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES); A.L.E. acknowledges the financial support of the Office of Naval Research (ONR) through the Naval Research Laboratory Basic Research Program.
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DOI: 10.1021/nl504987h Nano Lett. 2015, 15, 2092−2098
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DOI: 10.1021/nl504987h Nano Lett. 2015, 15, 2092−2098
