Effect of Auger Recombination on Lasing in Heterostructured Quantum

Sep 23, 2015 - Francesco Di Stasio , Anatolii Polovitsyn , Ilaria Angeloni , Iwan Moreels , and Roman Krahne. ACS Photonics ... Byeong Guk Jeong , You...
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Letter pubs.acs.org/NanoLett

Effect of Auger Recombination on Lasing in Heterostructured Quantum Dots with Engineered Core/Shell Interfaces Young-Shin Park,†,‡ Wan Ki Bae,§ Thomas Baker,† Jaehoon Lim,† and Victor I. Klimov*,† †

Chemistry Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States Center for High Technology Materials, University of New Mexico, Albuquerque, New Mexico 87131, United States § Photo-Electronic Hybrids Research Center, National Agenda Research Division, Korea Institute of Science and Technology, Seoul 02792, Korea ‡

S Supporting Information *

ABSTRACT: Nanocrystal quantum dots (QDs) are attractive materials for applications as laser media because of their bright, size-tunable emission and the flexibility afforded by colloidal synthesis. Nonradiative Auger recombination, however, hampers optical amplification in QDs by rapidly depleting the population of gain-active multiexciton states. In order to elucidate the role of Auger recombination in QD lasing and isolate its influence from other factors that might affect optical gain, we study two types of CdSe/CdS core/shell QDs with the same core radii and the same total sizes but different properties of the core/shell interface (“sharp” vs “smooth”). These samples exhibit distinctly different biexciton Auger lifetimes but are otherwise virtually identical. The suppression of Auger recombination in the sample with a smooth (alloyed) interface results in a notable improvement in the optical gain performance manifested in the reduction of the threshold for amplified spontaneous emission and the ability to produce dual-color lasing involving both the band-edge (1S) and the higherenergy (1P) electronic states. We develop a model, which explicitly accounts for the multiexciton nature of optical gain in QDs, and use it to analyze the competition between stimulated emission from multiexcitons and their decay via Auger recombination. These studies re-emphasize the importance of Auger recombination control for the realization of real-life QD-based lasing technologies and offer practical strategies for suppression of Auger recombination via “interface engineering” in core/shell structures. KEYWORDS: quantum dot, semiconductor nanocrystal, core/shell heterostructure, Auger recombination, optical gain, lasing ince the first demonstration of optical amplification in colloidal quantum dots (QDs),1 these structures have been extensively studied as prospective materials for the realization of solution-processed lasing media with broadly tunable emission wavelengths.2−19 Despite their favorable light emitting properties, application of QDs in lasing is complicated by extremely short optical gain lifetimes limited by nonradiative multicarrier Auger decay,20 the process wherein the energy released during electron−hole recombination is transferred to a third carrier. Indeed, because of the degenerate character of the band-edge states involved in light emission, population inversion required for lasing action can only be achieved if the QD is populated with more than one exciton; hence, multicarrier Auger recombination will have a strong effect on carrier dynamics (Figure 1).1 The gain threshold is defined by the condition G = 0 (G is gain magnitude). For the 2-fold degenerate band-edge levels, and the situation when the number of excitons (N) per QD is the same across the entire QD ensemble, the gain-threshold condition is met when N = Ng = 1.1 In a more realistic situation of Poisson distribution of

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© XXXX American Chemical Society

excitons across the QD ensemble, the onset of optical gain corresponds to the average QD occupancy ⟨N⟩ = ⟨Ng⟩ = 1.15.21 As was pointed out above, fast relaxation of optical gain resulting from Auger decay represents a major complication for both generating population inversion and maintaining it for time durations sufficient for the development of the lasing regime. One approach to outcompete Auger decay at the pumping stage is to use pulsed laser excitation with pulse duration shorter than the Auger lifetime.20 For the multiexciton gain mechanism, when the gain onset is defined by the condition ⟨Ng⟩ of ca. 1, the corresponding threshold per-pulse photon fluence (jg) can be estimated from ⟨Ng⟩ = jgσ ≈ 1 (σ is the QD absorption cross-section), which yields jg ≈ 1/σ−1. For standard CdSe QDs, σ is around 10−15 cm2, and hence jg is of the order of 1015 photons per cm2, which corresponds to the Received: July 1, 2015 Revised: September 15, 2015

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Figure 1. A three-state model of optical gain in QDs. (a) The ground state, |0⟩ (bottom), is nondegenerate and contains a spin-up and a spin-down electron in the valence band-edge state. The single-exciton state, |X⟩ (middle), is 4-fold degenerate and contains one electron in one of the spin sublevels of the conduction band-edge state and the other in one of the spin sublevels of the valence band-edge state; because of optical selection rules, only two of these states are optically active. The biexciton state, |XX⟩ (at the top), is nondegenerate and contains two electrons in different spin sublevels of the conduction-band-edge state. Arrows show allowed optical transitions. (b) The three states are connected by optical transitions leading to photon absorption (arrows facing up) or stimulated emission (arrows facing down). The rates of these transitions per unit photon density (ϕ = 1) are indicated in the figure; γ is the transition probability per single spin-allowed transition (shown by black arrows in the panel at left). As apparent from the figure, the net contribution from single excitons to optical gain is zero as stimulated emission resulting from the |X⟩−|0⟩ transition is exactly compensated by optical absorption due to the |X⟩−|XX⟩ transition. Therefore, optical gain depends only on the occupancies of the biexciton and the ground state levels that define, respectively, the net rates of stimulated emission (the |XX⟩−|X⟩ transition) and photon absorption (the |0⟩ − |X⟩ transition). As a result of these two contributions, optical gain is G0 ∝ 2γ(PXX − P0).

per-pulse energy fluence, wg, of 0.1−1 mJ/cm2 depending on excitation wavelength. Experimental studies, however, indicate that the onsets of amplified spontaneous emission (ASE, a “precursor” of lasing), and the lasing regime in standard QDs are in the range of a few to a few hundreds of mJ/cm2, which is considerably higher than estimations based on the condition G = 0. This is not surprising given that in order to establish the lasing regime, optical gain has to reach a critical value (Gc) sufficient for compensating optical losses in the cavity. The presence of fast Auger recombination is expected to further raise the value of Gc as optical gain has to be sufficiently high to ensure that the buildup of photon population due to stimulated emission outcompetes deactivation of optical gain via the Auger process. Because Auger lifetimes scale in direct proportion to the nanocrystal volume,20,22 the use of larger QD is expected to lower the lasing threshold. Experimental studies indicate that it can indeed be reduced to ∼10−100 μJ/cm2 using, for example, thick-shell “giant” CdS/CdSe QDs or large-size nonspherical nanostructures such as nanorods,3,8,11,18,19,23 tetrapods,10 or nanoplatelets.13,14,24 In these reported examples, however, lasing onsets were usually evaluated in terms of the pump fluence but not the QD excitonic occupancy, therefore, the observed reduction in the threshold intensity did not have to be necessarily due to lengthening of the Auger lifetime but could be a simple consequence of the increased absorption crosssection. In this Letter, we conduct theoretical and experimental analysis of optical amplification in colloidal QDs with a goal to quantify the effect of Auger recombination on ASE and lasing thresholds. In our experimental studies, in order to decouple the effect of Auger recombination from other factors such as absorption cross sections, QD packing density, and linear optical losses, we perform a comparative study of ASE and lasing in two types of CdSe/CdS QD samples that are engineered to have the same volumes (and hence, same absorption cross sections) and similar emission wavelengths but different Auger lifetimes. The first (reference) sample

represents standard core/shell (C/S) QDs with a sharp (nonalloyed) core/shell interface realized using a recently developed “fast-shelling” procedure.25 In the second (alloyed) sample, the QDs comprise a CdSexS1−x alloy layer at the core/ shell interface (C/A/S QDs). As was reported previously, the C/A/S QDs feature strong suppression of Auger decay resulting from the grading (smoothing) of the confinement potential,25,26 which reduces the strength of the intraband transition involved in Auger recombination.27 For both samples, we observe the transition from spontaneous emission to ASE and then random lasing with increasing pump fluence. Importantly, the thresholds of both ASE and lasing are consistently lower for the alloyed QDs compared to the reference samples. Furthermore, the films of C/A/S QDs in addition to lasing at the band-edge (1S) transition show a regime of dual-color lasing involving both the band-edge and the higher-energy (1P) electronic states. This indicates the possibility of obtaining strong optical gain not only due to biexcitons but also higher-order multiexcitons. Because the C/S and C/A/S samples are characterized by identical absorption cross sections and QD packing densities, the observed improvements can be unambiguously attributed to suppression of Auger decay. Our observations are rationalized using a model, which explicitly accounts for the multiexciton nature of optical gain in colloidal QDs. This modeling suggests that a modest improvement in the lasing threshold observed for alloyed QDs in the regime of pulsed excitation should translate into orders of magnitude threshold reduction in the case of continuous wave (cw) pumping. These studies re-emphasize the importance of Auger-decay engineering for the realization of cw lasing with optical and potentially electrical excitation. Auger Recombination and QD Lasing. Theory. The onset of optical gain (corresponding to G = 0) does not require generation of multiexcitons and therefore is fairly easy to realize. While being a prerequisite for the lasing action, attaining the gain threshold does not guarantee the establishment of the lasing regime, which requires a sufficiently high (critical) value of G = Gc for the stimulated emission to B

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Figure 2. Modeling of the effect of Auger decay on QD lasing. (a) Illustration of a gain-switching model. The optical gain medium is excited with a δfunction-like pulse (red bar) at t = 0 (top), which leads to the instantaneous buildup of carrier populations in the QDs with average truncated QD occupancy ⟨N*0⟩. If ⟨N*0⟩ is above the lasing threshold, population decay via stimulated emission leads to a gradual buildup of the photon population in the cavity (bottom). (b) Normalized decay of the biexciton population, PXX(t), for varied biexciton lifetimes (50 ps to 2 ns) and a fixed initial QD occupancy ⟨N*0⟩ = 1.5. (c) Temporal evolution of the photon density in the cavity for the same conditions as in panel b (same color coding). The double arrow shows a delay (τD) in the buildup of photon population with respect to the excitation pulse. For example, for τXX = 2 ns, τD = 56 ps. (d) Time integrated photon density (Φ) as a function ⟨N*0⟩ for the same set of biexciton lifetimes as in panel b (same color coding). (e) Time delay τD (black solid squares) and τ1/e,XX/τXX (blue open circles) as a function of ⟨N*0⟩. The onset of lasing is indicated by a sharp drop in the τ1/e,XX/τXX vs ⟨N*0⟩ dependence (marked by the red arrow), which correlates with the position of the peak in the τD vs ⟨N*0⟩ dependence. (f) The 1/e biexciton lifetimes (τ1/e,XX) derived from the PXX(t) dynamics in panel b normalized by τXX and plotted as a function of ⟨N*0⟩; same set of biexciton lifetimes as in panel b (same color coding). Lasing thresholds, ⟨N*las⟩, are defined as the value of ⟨N*0⟩ at which τ1/e,XX drops by 5% from its maximum (dashed line).

These considerations indicate that the net contribution from single excitons to optical gain is zero as stimulated emission via the |X⟩−|0⟩ transition is exactly compensated by optical absorption associated with the |X⟩−|XX⟩ transition. As a result, optical gain is defined solely by the occupancies of the biexciton and the ground state-levels that are responsible, respectively, for the net stimulated emission (|XX⟩−|X⟩ transition) and the net photon absorption (|0⟩−|X⟩ transition); see Figure 1b. Using the probabilities introduced above, we can describe a coupled QD-light-field system using the following set of kinetic equations:

outcompete photon losses in the cavity and nonradiative carrier losses in the gain medium itself. As a result of these additional requirements the realization of ASE and lasing is considerably more difficult than the realization of optical gain. In QD-based gain media, a principle mechanism of nonradiative depletion of optical gain is multicarrier Auger recombination.1,20 To quantify its effect on QD optical gain performance and specifically lasing threshold, we carry out a simulation of a QD gain medium within a lasing cavity using a gain-switching model.28 We approximate a QD by a three-level system that comprises: a nondegenerate ground state (|0⟩), in which two electrons occupy spin-up and spin-down sublevels of the highest valence-band state; a 4-fold-degenerate singleexciton state (|X⟩) in which one electron is in one of the 2-folddegenerate valence-band states and the other in one of the 2fold-degenerate conduction-band states; and a nondegenerate biexciton state (|XX⟩) with both electrons in the lowest 2-fold spin-degenerate conduction-band state (Figure 1a). If we denote the probability of absorption or stimulated emission per spin-allowed transition as γ, then the total probability of the |0⟩−|X⟩ transition due to interaction with the resonant field at frequency ω is W01 = 2γϕ, where ϕ is the photon density in the cavity (Figure 1b). Optical selection rules dictate that only two of the four exciton states are optically active (Figure 1a); therefore, the probability of stimulated emission by the single exciton (|X⟩−|0⟩ transition) is W10 = (γ/2)ϕ. Based on the similar argument, the probability of absorption by a single exciton to produce a biexciton (|X⟩−|XX⟩ transition) is W12 = (γ/2)ϕ, whereas the probability of stimulated emission by the biexciton (|XX⟩−|X⟩ transition) is W21 = 2γϕ.

dP0 P γ = −2γϕP0 + ϕPX + X dt 2 τX

(1)

dPX P P γ = −2 ϕPX + 2γϕP0 + 2γϕPXX − X + XX dt 2 τX τXX

(2)

dPXX P γ = −2γϕPXX + ϕPX − XX dt 2 τXX

(3)

dϕ ϕ = 2γϕnQD(PXX − P0) − dt τc

(4)

where P0, PX, and PXX, are the probabilities of the QD to be in the ground, single-exciton, or biexciton state, respectively. These probabilities are constrained such that P0 + PX + PXX = 1. Time constants τX, τXX, and τc are the single-exciton, biexciton, and the cavity photon lifetimes, respectively. In eq 4, we account only for photons produced via stimulated emission that C

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multiplicity, which implies that τr,XX = τX/4.30 This is consistent with the ratio of the probabilities of the |X⟩−|0⟩ and the |XX⟩−| X⟩ transitions introduced earlier (see Figure 1b). In our calculations, where τX is fixed, τr,XX is also fixed (τr,XX = 12.5 ns) and the τXX value is tuned by changing its Auger-recombination component (τA,XX). To initiate the buildup of a photon population due to stimulated emission, we seed the cavity at time t = 0 with 1 photon per unit volume (ϕ0 = 1 cm−3) and then follow temporal changes in PXX = PXX(t) (Figure 2b) and ϕ = ϕ(t) (Figure 2c). We would like to note that the exact number of seed photons does not affect the final results of our modeling because after the lasing threshold is crossed, ϕ exceeds ϕ0 by many orders of magnitude. By integrating ϕ(t) from 0 to ∞, we calculate the total number of photons (Φ) emitted into the cavity per excitation pulse (displayed in Figure 2d). These calculations indicate that for subthreshold excitation, the photon number grows sublinearly with ⟨N*0⟩ as the rate of photon losses in this case exceeds the rate of stimulated emission. Above a certain threshold, Φ shows a transition to superlinear growth indicating the onset of lasing. Eventually, at even higher values of ⟨N*0⟩ the increase of Φ slows down until it becomes linear, which corresponds to the situation when the depletion of population inversion becomes dominated by stimulated emission. This behavior results in the S-shaped dependence of Φ on ⟨N*0⟩ typical of the lasing regime (Figure 2d). It is apparent from Figure 2d, that the transition to lasing becomes progressively steeper with increasing τA,XX, suggesting a reduction in the lasing threshold; however, because of the gradual character of this transition it is difficult to quantify the lasing threshold based on the Φ vs ⟨N*0⟩ characteristics. A more rigorous approach for evaluating the threshold is based on the analysis of population dynamics. Specifically, the onset of lasing can be inferred from the onset of the regime when population decay of gain-active biexciton states, PXX(t), becomes dominated by stimulated emission. When we examine the PXX dynamics in Figure 2b, we observe that for shorter Auger time constants [see, e.g., the trace for τA,XX ≈ τXX = 50 ps (magenta)], the biexciton population shows a single-exponential decay controlled by Auger recombination. However, when τXX is increased to 100 ps (light blue trace), the calculated dynamics exhibits a kink at ∼80−100 ps due to the influence of stimulated emission, which becomes strong enough to affect the PXX decay via the −2γϕPXX term in eq 3. This kink eventually evolves into a pronounced step (see traces shown by different colors), indicating that biexciton population is quickly depleted via stimulated emission rather than Auger recombination. The delayed onset of this fast decay indicates that the buildup of the photon population in the optical cavity is not instantaneous and depends on the stimulated emission rate (tuned here by ⟨N*0⟩) and the population inversion decay rate (tuned here by τXX). To illustrate these dependences, we model the delay (τD) in the buildup of the photon population (shown by the double arrow in Figure 2c) as a function of ⟨N*0⟩ for the fixed biexciton lifetime (τXX = 1 ns; black squares in Figure 2e) and as a function of τXX for the fixed value of ⟨N*0⟩ = 1.5 (black squares in Supporting Information Figure S1). As either of these two parameters is increased, τD first also increases, but then, above a certain critical value of τXX or ⟨N*0⟩, it sharply shortens, which provides a clear quantitative indicator of the transition to the lasing regime.

accumulate in the cavity mode. In principle, one can also account for photons produced by spontaneous emission using a separate rate equation. However, because these photons do not accumulate in the cavity, they do not affect carrier dynamics and, therefore, are disregarded here, which is a common approach when analyzing the lasing regime. On the basis of eq 4, the stimulated emission rate can be expressed as rSE = 2γnQD(PXX − P0) and by extension the optical gain as G = (c/n)rSE = 2(c/n)γnQDρ = G0ρ, where ρ = PXX − P0 is the per-QD population inversion in terms of traditional lasing theories, and G0 = 2(n/c)γnQD is the maximum gain value (saturated gain) achieved when ρ = 1. Gain threshold is still defined by the condition G = 0, or equivalently, ρ = 0. It is satisfied, for example, when PX = 1, and hence PXX = P0 = 0. As was mentioned in the introduction, this corresponds to the situation when all QDs in the ensemble are occupied with a single exciton. In our three-state model, the average QD occupancy is limited to 2 and can be expressed as ⟨N*⟩ = PX + 2PXX = 1 + ρ. We will refer to this quantity as a “truncated” average QD occupancy. In order to relate model parameters P0, PX, PXX, and ⟨N*⟩ to quantities measured experimentally, we will consider the situation of a Poisson distribution of carrier populations across the QD ensemble, which is typically realized for above band gap excitation.29 In this case, the probability of having N excitons in a QD is described by pN = ⟨N⟩N(N!)−1e−⟨N⟩, where 29 ⟨N⟩ = Σ∞ i=1 ipi is the average QD excitonic occupancy. In QDs with 2-fold degenerate emitting states, the contribution from a given multiexciton to the band-edge optical gain is independent of its order and is the same as for a biexciton. As a result, PXX can be calculated from PXX = Σ∞ i=2 pi, whereas PX = p1, P0 = p0, which ensures that the condition P0 + PX + PXX = 1 is met. Given these relationships between probabilities Pi (i = 0, X, XX) and pj (j = 0, 1, 2, ...), we can relate the true average QD occupancy to the truncated one by ⟨N *⟩ = 2 − e−⟨N ⟩(2 + ⟨N ⟩)

(5)

Further, using Poisson probabilities in the expression for the optical gain onset (ρ = PXX − P0 = 0), we obtain that the gain threshold defined in terms of true average occupancy can be found from the equation ⟨Ng⟩ + 2 = e⟨Ng⟩, which yields ⟨Ng⟩ ≈ 1.15, as was found in ref 21. For lasing to occur, the average QD occupancy must exceed ⟨Ng⟩ (or ⟨N*g⟩ if evaluated in terms of truncated occupancy) in order for stimulated emission to outpace photon losses in the cavity and carrier losses resulting from Auger decay and other recombination processes. As a result, the lasing threshold (⟨Nlas⟩ or ⟨N*las⟩) is higher than the gain threshold. In order to quantify it and, specifically, evaluate the effect of Auger recombination on the onset of the lasing regime, we use eqs 1−4 to model temporal evolutions of P0, PX, PXX, and ϕ following excitation of a QD medium with a short δ-functionlike pulse (Figure 2a). The excitation intensity is characterized in terms of the average truncated QD occupancy ⟨N*0⟩ ≤ 2 created at time t = 0. On the basis of parameters of samples studied in the next section, in our calculations, we use fixed values of G0 = 100 cm−1 and τX = 50 ns, assuming that the single-exciton lifetime is purely radiative, that is, τX = τr,X. The biexciton decay is assumed to be due to both radiative recombination (τr,XX) and the nonradiative Auger process (τA,XX), and hence, the overall biexciton lifetime can be expressed as τXX = τr,XX τA,XX/(τr,XX + τA,XX). We further assume that the radiative rate exhibits a quadratic scaling with exciton D

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Nano Letters In order to gain a further insight into this behavior, we analyze the dependence of the 1/e lifetime of the biexciton population (τ1/e,XX) as a function of ⟨N*0⟩ shown in Figure 2f. We again observe a dramatic change in the behavior of τ1/e,XX as ⟨N*0⟩ is increased. First, τ1/e,XX stays nearly constant at the level τ1/e,XX = τXX (Figure 2f), but then, it sharply shortens once ⟨N*0⟩ reaches a certain value which is virtually identical to the critical value of ⟨N*0⟩ derived from the analysis of τD (compare data shown by the black squares and the blue circles in Figure 2e). This indicates that when the lasing regime is achieved the carrier decay in the gain medium becomes dominated not by intrinsic recombination processes but by stimulated emission. The above analyses show that sharp changes in carrier or photon population dynamics (Figure 2e) associated with the transition to lasing allow for a more accurate quantification of the lasing threshold compared to the analysis of the overall population of the cavity mode (Figure 2d). Therefore, here we define the lasing threshold, ⟨N*las⟩, as the value of ⟨N*0⟩ at which τ1/e,XX drops by 5% from its maximum and calculate the respective per-dot critical population inversion (ρlas) from ρlas = ⟨N*las⟩ − 1. In Figure 3a, we plot ρlas as a function of the overall biexciton lifetime (τXX) for different cavity time constants τc = 1 ps to 1 ns. We observe that the effect of τXX on ρlas increases with increasing τc and in the case of the lossless cavity ρlas becomes controlled solely by τXX. In fact, in the limit of τc → ∞, ρlas asymptotically approaches the dependence that can be approximated by the inverse square root scaling: ρlas ∝ (τXX)−1/2 (gray line in Figure 3a). If expressed in terms of the average QD occupancy the lasing threshold in this case can be presented as * ⟩ = 1 + (τ0/τXX)1/2 ⟨Nlas

(6)

where τ0 is a constant, which is directly related to the characteristic time of the stimulated emission buildup τSE = n/ (cG0). Here, we characterize τSE in terms of the inverse of the stimulated emission rate calculated for ρ = 1, which corresponds to gain saturation. If we tune τSE by varying G0, we observe that that τ0 scales almost linearly with τSE (the inset of Figure 3a) and can be expressed as τ0 = βτSE, where β is ca. 20 for the present parameters of the model. Because ⟨N*las⟩ in eq 6 is expressed in terms of the truncated QD occupancy, it cannot exceed 2, and hence, for lasing to occur the buildup of stimulated emission should be sufficiently fast to ensure that τ0 is shorter than the biexciton lifetime. As τSE ∝ 1/G0 ∝ 1/nQD, this explains, the existence of a critical density of QD emitters in the gain medium below which lasing becomes impossible as was first pointed out in ref 1. The effect of Auger decay on lasing threshold is even more dramatic in the case of cw excitation, as the population inversion sufficient to reach the lasing regime requires very high pumping rates in order to outcompete Auger decay. To consider the steady state situation, we add in eqs 1−3 the steady state carrier generation term Jσ (here J is the cw pump intensity expressed in terms of the photon flux) and set all time derivatives to zero [d(...)/dt = 0]. In the sublasing-threshold regime we can assume that ϕ = 0, which allows us to obtain the following relationship between ⟨N*⟩ and J: ⟨N *⟩ =

Figure 3. Effect of biexciton lifetimes on lasing threshold. (a) Lasing threshold in terms of ρlas plotted as a function of τXX for five different cavity photon lifetime: τc = 1 ns (black open squares), 0.1 ns (red solid circles), 0.01 ns (green open triangles), 0.005 ns (blue solid diamonds), 0.002 ns (magenta open pentagons), and 0.001 ns (brown solid hexagons); same labeling is used in the other two panels of this figure. In these calculations, τXX was varied by tuning the Auger decay time while keeping the biexciton radiative lifetime the same (τr,XX = 12.5 ns). The gray line is the lossless cavity limit, which can be approximately described by ρlas ∝ (τ0/τXX)1/2. Time constant τ0 scales linearly with the characteristic time of the stimulated emission buildup, τSE (inset). (b) The cw lasing threshold (Jlas) derived from data in panel a based on steady state solutions of eqs 1−3; Jlas is plotted as a function of τXX. (c) The plot of ρlas vs τXX for the case of purely radiative biexciton decay when τXX = τr,XX = τX/4; in these calculations, for each τXX the saturated gain was adjusted according to G0 ∝ 1/τXX.

Using this expression, we can translate values of ρlas into cw lasing thresholds (Jlas) and then plot the results as Jlas vs τXX (Figure 3b). This plot indicates a rapid increase in the

JστX + 2(Jσ )2 τXτXX 1 + JστX + (Jσ )2 τXτXX

(7) E

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Figure 4. Optical characterization of the C/S and C/A/S QD samples. (a) PL (solid line) and absorption (dashed line) spectra of ensemble samples of the C/S (black) and C/A/S (red) QDs. (b) Ensemble PL dynamics for the C/S (black) and C/A/S (red) samples indicating similar excition lifetimes of ∼50 ns. (c) Representative biexciton dynamics measured from single C/A/S (red) and C/S (black) QDs. These measurements indicate a substantial difference in τXX between the two samples. (d) Histogram of biexciton lifetimes for 37 individual C/S (black) and C/A/S (red) QDs.

Auger recombination and is close to the fundamental limit defined by the optical gain onset. This is an expected observation, as in both cases the biexciton decay is dominated by radiative recombination and its rate is sufficiently high to ensure that stimulated emission outcompetes fairly slow photon losses immediately above the gain threshold. The situation however changes when τXX is reduced. In Figure 3a, the biexciton decay rate is increased (τXX is decreased) by shortening the Auger decay time while preserving its radiative component. In this case, in order to compensate for shortening of the gain lifetime due to shorter τXX, one must increase the rate of stimulated emission by increasing ρ, which raises the lasing threshold according to ρlas ∝ (τXX)−1/2, as discussed earlier. In the purely radiative case, the shortening of the gain lifetime is accompanied by the proportional increase in the rate of stimulated emission, and as a result, the lasing threshold stays almost unchanged. Interestingly, the difference in trends between the “nonradiative” and “radiative” cases becomes especially pronounced for short cavity lifetimes (τc = 10 ps and less) when the lasing threshold calculated for the radiative limit increases with increasing τXX, as opposed to the situation of Auger recombination when ρlas drops for longer τXX. In the radiative case, the increase in τXX implies that the strength of the radiative transition is decreased, and hence, ρlas must be increased in order to keep the stimulated emission rate at the level, which is sufficiently high to compete with fast photon losses from the cavity. In the presence of Auger decay, the increase in τXX leads to longer gain lifetimes for the same rate of stimulated emission, which favors the lasing regime and leads to the drop in the lasing threshold. This analysis indicates the

threshold intensity with decreasing biexciton lifetime. The growth of Jlas is almost linear for larger values of τXX (10−1 ns) but becomes progressively steeper for shorter biexciton lifetimes. For example, in the case of low cavity losses (τc = 1 ns), the change of τXX from 1 ns to 50 ps results in the increase of Jlas by 3 orders of magnitude. This emphasizes the importance of Auger recombination control for the realization of cw lasing. Specifically, on the basis of these calculations, one might expect that extension of Auger lifetimes to 1 ns or slightly longer will already have very significant benefits for cw lasing. An interesting question is whether the observed strong effect of τXX on the lasing threshold is a consequence of the nonradiative character of Auger recombination or is a general feature of any processes (radiative or nonradiative) limiting the optical gain lifetime. To answer this question, we examine the situation where the nonradiative Auger decay rate is zero and biexciton relaxation is instead controlled solely by radiative recombination. In this purely radiative limit, τXX is directly linked to the single exciton lifetime by τXX = τr,XX = τX/4.30 This implies that τXX is controlled by the oscillator strength of the same transition as stimulated emission and further suggests that when we tune τXX, we must simultaneously adjust the gain magnitude according to G0 ∝ 1/τXX. Assuming that G0 = 100 cm−1 for τXX = 12.5 ns, we simulate the τXX dependence of ρlas for several cavity lifetimes and display the results in Figure 3c. A comparison of Figure 3a and c indicates a dramatic difference in the trends between the “radiative” and “nonradiative” cases. For example, in the case of low cavity losses (τc = 1 ns) and the longest biexciton lifetime (τXX = 12.5 ns), the lasing threshold (ρlas ≈ 0.035) is the same for the situations with (Figure 3a) and without (Figure 3c) F

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Figure 5. ASE and random lasing in films of C/A/S and C/S QDs. (a) Spectrally integrated PL intensity as a function of per-pulse pump fluence for the C/A/S QD film; arrows mark onsets of the ASE and lasing regimes. Inset: pump-fluence dependent PL spectra. (b) Same for the C/S QD film. (c) Spectra of random lasing observed for the C/A/S QD film at w = 13.8 μJ/cm2. Vertical blue lines mark spectral positions of four prominent peaks separated by equal intervals Δλ = 2.4 nm. (d) Power Fourier transform of the spectrum in panel c. Peak-to-peak separation Δd = 27 μm yields the cavity round-trip length s = 95.4 μm. The inset shows schematically a “random cavity” created by the local inhomogeneities in the QD film due to, for example, refractive index or thickness variations.

large energy gradient to a smoother potential has a dramatic effect on the probability of the intraband transition involved in the Auger recombination event. Specifically, one-dimensional model calculations demonstrate that the replacement of the rectangular confinement potential with a parabolic profile may result in orders of magnitude suppression of Auger decay.27 These theoretical ideas were initially confirmed by the studies of thick-shell “giant” CdSe/CdS QDs,34,35 where, for example, unintentional alloying was observed to lead to a broad spread of biexciton lifetimes and their emission efficiencies across nominally identical QDs within the same sample.35 Even more convincing evidence of the effect of the core/shell interface on Auger recombination has been provided by recent studies of CdSe/CdS core/shell samples grown by a novel technique of “fast shelling”, which allowed for the preparation and comparative studies of samples with “sharp” core/shell interfaces versus samples, in which a well-defined CdSexS1−x alloy layer was introduced between the core and the shell.25 Here, we use similar samples to elucidate the effect of interfacial alloying on lasing properties of the QDs. The C/S and C/A/S QDs synthesized for the present study exhibit similar photoluminescence (PL) wavelengths (630−650 nm; Figure 4a), emission quantum yields (∼45%), and PL lifetimes measured in the single-exciton regime (τX ≈ 50 ns; Figure 4b). These observations are consistent with those from

existence of a principal difference between the effects of nonradiative and radiative recombination on optical gain performance of the QDs and emphasizes again the detrimental role of nonradiative Auger decay for lasing applications with these materials. Effect of Auger Recombination on ASE Thresholds. Experiment. To experimentally evaluate the role of Auger recombination in QD lasing, we synthesize two types of core/ shell CdSe/CdS QDs using the procedures from our earlier publications (see Supporting Information Figure S2).25,31,32 Both samples have the same CdSe core radius (r = 1.5 nm) and the same total radius (R = 7 nm). However, the core−shell interface is different between the two samples. In one sample (C/S), the interface is “sharp” (i.e., does not contain an alloyed layer), whereas in the other (C/A/S sample), the CdSe core is separated from the CdS shell by an intermediate 1.5 nm-thick layer of CdSexS1−x alloy with x ≈ 0.5. As a result of this difference, the two samples exhibit distinct rates of Auger recombination with time constants that are considerably longer in the alloyed sample.25,31,32 As was discussed in ref 23, lengthening of the biexciton lifetime in the C/A/S QDs arises from the suppression of Auger decay due to smoothing of the interfacial potential. According to original calculations27 and follow-up theoretical studies,33 the transition from a sharp step-like potential with a G

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Nano Letters our earlier studies of analogous QDs.25,31,32 To characterize biexciton dynamics, we use a single-QD micro-PL technique with time-tagged, time-correlated single photon counting (TCSPC), as detailed, for example, in ref 31. We derive biexciton lifetimes by measuring the arrival time of the first photon during excitations cycles where two PL photons are detected from the same pump pulse. Based on these measurements, τXX is longer in the C/A/S samples compared to the C/S samples. In Figure 4c, we plot representative biexciton dynamics measured for two single C/S and C/A/S QDs. Based on single-exponential fits, τXX is 1.8 ns for the C/ A/S QD and 1 ns, indicating a significant suppression of Auger decay. In fact, according to our calculations (Figure 3a and c), the biexciton lifetimes realized in the C/A/S samples are sufficiently long to allow for appreciable improvements in lasing performance. For the ASE/lasing measurements, we fabricated closepacked films of the C/S and C/A/S QDs by spin coating. The film thickness and surface roughness were characterized using atomic force microscopy (AFM). A frequency doubled output (400 nm wavelength) of an amplified femtosecond Ti:sapphire laser operating at the 1 kHz repetition rate was focused with a cylindrical lens (5 cm focal length) into a narrow stripe onto the QD film (see Supporting Information Figure S3). The PL signal was collected from one end of the stripe in the direction normal to the laser beam and sent to a spectrometer coupled to a liquid-nitrogen-cooled charge coupled device. In Figure 5a and b, we display spectrally integrated PL intensities and emission spectra (insets) as a function of perpulse pump fluence (w) for the best performing C/A/S and C/ S QD films selected out of 10 samples. As the pump fluence is increased, the PL intensity for both samples, first increases sublinearly and then exhibits a sharp transition to superlinear growth, which is accompanied by the development of a narrower peak on the blue side of a broad spontaneous emission band. This is a signature of the transition to the ASE regime due to stimulated emission of biexcitons. As pump fluence is increased further, we observe the development of multiple sharp peaks signifying a transition to lasing. To determine the modal gain of these films, we conduct variable stripe length (VSL) measurements (see Supporting Information Figure S4), where the intensity of PL detected at the edge of the sample is measured as a function of the length of the excited stripe varied with an adjustable slit. According to VSL measurements, the biexciton modal gain is ∼85 cm−1 and ∼110 cm−1 for the C/A/S and C/S QD films, respectively (Supporting Information Figure S4). These values are close to the magnitude of the saturated gain (100 cm−1) used in the modeling discussed in the previous section. First, we consider the spectral positions of the ASE and lasing peaks. Their blue shift with regard to the peak of spontaneous emission is consistent with the biexciton nature of optical gain in these structures. Indeed, because of a small conduction band energy offset, the electron wave function in CdSe/CdS QDs is

delocalized over the entire nanostructure while the hole wave function is tightly confined to the core.34 This type of carrier localization, usually referenced to as quasi-type-II,36 is characterized by significant asymmetry in spatial distributions of positive and negative charge densities, which leads to strong exciton−exciton repulsion.6,37 As a result, biexciton emission is shifted to higher energies from the single-exciton band, which explains the spectral position of ASE and lasing peaks detected in our measurements. Next, it is instructive to compare ASE and lasing threshold between the C/A/S and C/S samples. The threshold intensity for the development of ASE in the C/S QD film is ∼19 μJ/cm2 or ⟨N⟩ = 4.6 in terms of the average QD occupancy. This corresponds to ⟨N*⟩ = 1.93, which is very close to the maximum value of 2, suggesting the need for almost complete inversion of the QD population (ρ = 0.93) to achieve the regime of optical amplification. On the other hand, use of the C/A/S decreases the ASE threshold to ∼6 μJ/cm2, which establishes a new record for QD samples and is on a par with the lowest thresholds published for large 2D colloidal platelets.24 This threshold fluence corresponds to ⟨N⟩ = 1.45, which is closer to the ultimate limit defined by the gain threshold (⟨Ng⟩ = 1.15). The corresponding value in terms of the truncated average QD occupancy is ⟨N*⟩ = 1.19, indicating that the per-dot population inversion (ρ) is reduced to 0.19. On the basis of our earlier theoretical analysis, these improvements in the optical-gain performance can be ascribed to the lengthening of Auger lifetimes. The improvement in the ASE/lasing thresholds due to suppression of Auger decay in C/A/S samples is systematically observed across all studied samples with the average improvement factor of 1.6−1.7 (Supporting Information Figure S5). It might seem to be a small difference; however, one should realize that even small improvements observed in the case of pulsed excitation will translate into much greater benefits in the case of the cw pumping (compare Figure 3a and c). Given present results for the pulsed excitation and our theoretical modeling, with C/A/S samples, we should be able to realize cw lasing after we resolve the problem of film thermal stability and improve optical quality of our samples. These efforts are currently in progress. Random Lasing in QD Films. In the last part of this article, we analyze the lasing regime, which develops at the highest excitation intensities. For the films in Figure 5, the onset of this regime is at ∼17 and ∼21 μJ/cm2 for the C/A/S and C/S samples, respectively. The lower lasing threshold measured for the C/A/S QDs is systematically observed for other studied samples (Supporting Information Figure S5), which again illustrates a beneficial effect of interfacial alloying on lasing performance of the QDs. Above the threshold, the PL intensity grows superlinearly with a log−log slope of ∼3−4 in both cases. After the transition to lasing the ASE band splits into multiple sharp peaks (insets of Figure 5a and b) that strongly fluctuate in time. These behaviors are typical signatures of random lasing previously observed, for example, for ZnO powder, organic polymer, and QD films.15,38,39 We have been able to excite random lasing in all studied samples (both C/A/S and C/S) with threshold fluences in the range of ∼10 to ∼60 μJ/cm2. In Figure 5c, we provide a more detailed analysis of random lasing in the C/A/S QD film. In general, the lasing spectrum consists of several sharp peaks (line widths as narrow as 0.2 nm) with heights and spectral positions fluctuating in time and showing strong dependence on the exact position of the H

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develops at w = 18 μJ/cm2 and quickly grows in intensity as w is increased further (inset of Figure 6a). In addition to “single-color” lasing, using C/A/S samples we have been able to realize “dual-color” lasing due to optical transitions involving both band-edge and excited electronic states. An example of these data is shown in Figure 6b. At lower pump intensities, this sample exhibits single-band lasing at ∼645 nm due to the band-edge (1S) transition. As the pump intensity is increased, the intensity of band-edge band begins to saturate (at w ∼30 μJ/cm2) and simultaneously a new lasing band comprising multiple sharp peaks emerges at around 590 nm (Figure 6b and c), which is consistent with the energy of the 1Pe−1Ph transition (see Supporting Information Figure S7). The threshold pump intensity observed for 1P lasing is ∼40 μJ/cm2, which corresponds to ⟨N⟩ of ∼10. Considering the 6-fold degeneracy of 1P state, the optical amplification on the 1P transition is only possible when the average QD occupancy is at least 5, which corresponds to the nominal 1P gain threshold. Because Auger decay rates quickly increase with exciton multiplicity,30 the ASE/lasing threshold can be considerably higher than the nominal gain threshold to ensure rapid buildup of photon population by stimulated emission which outcompetes very fast Auger recombination of highorder multiexcitons. Therefore, it is not surprising that the onset of 1P lasing requires QD occupancies that are twice as high as those required for the onset of 1P optical gain. Importantly, we have been able to obtain the two-color lasing regime with all of the C/A/S QDs, whereas none of the C/S samples showed lasing at the 1P transition because of extremely fast depletion of the 1P gain by Auger recombination. This result again highlights the importance of Auger decay suppression for enhancing lasing performance of QD materials and demonstrates that the proposed strategy of interfacial alloying is highly effective in extending lifetimes of not only biexcitons but also higher-order multiexciton states. To summarize, we observe that two types of CdSe/CdS core/shell samples (C/S vs C/A/S) characterized by similar single-exciton properties but distinct biexciton lifetimes also exhibit different optical-gain properties. Specifically, the ASE thresholds observed for the C/A/S samples are systematically lower compared to those in the C/S samples. Furthermore, although both types of the QDs exhibit random lasing, the C/ A/S QDs are able to lase simultaneously at two wavelengths due to participation of both the band-edge (1S) and excited (1P) states. As indicated by our modeling, these improvements in optical-gain performance are derived from the suppression of Auger recombination in the C/A/S samples due to the presence of the interfacial alloy. The resulting increase of the multiexciton lifetimes lowers the critical population inversion (ρlas) required for the lasing action at the band-edge transition and allows for population-inversion buildup at the higherenergy states responsible for lasing at the 1P transition. According to our calculations, in the limit of the lossless cavity, ρlas scales as (τXX)−1/2. Although under pulsed excitation, this scaling leads to only modest dependence of the lasing threshold on the rate of Auger decay; the effect of τXX becomes much more dramatic in the case of cw excitation, when carrier generation directly competes with Auger recombination at the stage of preparation of critical population inversion. For example, for the parameters of our model and in the ideal situation of the lossless cavity, the increase of the biexciton Auger lifetimes from 50 ps to 1 ns leads to the reduction of the cw lasing threshold by 3 orders of magnitude. Interestingly,

excitation spot on the QD film. Random lasing is due to the recurrent scattering of light in randomly formed cavities within the optical gain medium.38 For thin QD films, this scattering can originate from nonuniformities of the refractive index or film-thickness variation.15,40 Indeed, AFM images of both the C/A/S and the C/S QD films show a nonuniform corrugation on film surfaces with about 20−30% height modulation (see Supporting Information Figure S6). To estimate the dimensions of these randomly formed cavities within our films, we analyze the random lasing spectrum shown in Figure 5c. This particular spectrum exhibits four dominant lasing peaks (indicated by the blue lines in Figure 5c) that are nearly equally spaced by Δλ = 2.4 nm, which yields the cavity round-trip length s = λ2/(nΔλ) = 95.6 μm. In Figure 5d, we show a power Fourier transform of the lasing spectrum from Figure 5c. It also allows us to estimate a round-trip path length from s = 2πΔd/n, where Δd is the peak-to-peak spacing.40 On the basis of a series of equally spaced diminished peaks at 27 μm, 55 μm, 83 μm, and 110 μm (primary microcavity), we find Δd ≅ 27 μm. This corresponds to s = 94.2 μm, which is in agreement with the value inferred directly from the lasing spectrum. In the spectrum of Figure 5d, we can also discern peaks at 43 μm and 129 μm that can be interpreted in terms of the first and the third overtones of the secondary cavity with Δd of 43 μm; the second overtone expected at 86 μm is likely unresolved due to its close proximity to the 83 μm peak of the primary cavity. Random lasing spectra normally consist of several sharp peaks originating from multiple resonances of one cavity or several cavities residing within the excitation spot. Interestingly, for our films we can locate regions that produce single-mode lasing as illustrated in Figure 6a. The displayed spectra feature a single narrow peak (line width of ∼0.2 nm) at 644 nm which

Figure 6. Single mode and dual-color lasing regimes in the C/A/S QD film. (a) Single-mode lasing with a threshold of 18 μJ/cm2. (b) Dualcolor lasing due to transitions involving the band-edge (1S) and excited (1P) QD states. I

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(15) Wang, Y.; Ta, V. D.; Gao, Y.; He, T. C.; Chen, R.; Mutlugun, E.; Demir, H. V.; Sun, H. D. Adv. Mater. 2014, 26, 2954−2961. (16) Cooney, R. R.; Sewall, S. L.; Sagar, D. M.; Kambhampati, P. Phys. Rev. Lett. 2009, 102, 127404. (17) Cooney, R. R.; Sewall, S. L.; Sagar, D. M.; Kambhampati, P. J. Chem. Phys. 2009, 131, 164706. (18) Zavelani-Rossi, M.; Krahne, R.; Della Valle, G.; Longhi, S.; Franchini, I. R.; Girardo, S.; Scotognella, F.; Pisignano, D.; Manna, L.; Lanzani, G.; Tassone, F. Laser Photonics Rev. 2012, 6, 678−683. (19) Di Stasio, F.; Grim, J. Q.; Lesnyak, V.; Rastogi, P.; Manna, L.; Moreels, I.; Krahne, R. Small 2015, 11, 1328−1334. (20) Klimov, V. I.; Mikhailovsky, A. A.; McBranch, D. W.; Leatherdale, C. A.; Bawendi, M. G. Science 2000, 287, 1011−1013. (21) Klimov, V. I., Ch.5. In Semiconductor and Metal Nanocrystals: Synthesis and Electronic and Optical Properties, 1st ed.; Klimov, V. I., Ed.; Marcel Dekker: New York, 2004; pp 159−214. (22) Robel, I.; Gresback, R.; Kortshagen, U.; Schaller, R. D.; Klimov, V. I. Phys. Rev. Lett. 2009, 102, 177404. (23) Htoon, H.; Hollingsworth, J. A.; Dickerson, R.; Klimov, V. I. Phys. Rev. Lett. 2003, 91, 227401. (24) She, C.; Fedin, I.; Dolzhnikov, D. S.; Demortière, A.; Schaller, R. D.; Pelton, M.; Talapin, D. V. Nano Lett. 2014, 14, 2772−2777. (25) Bae, W. K.; Padilha, L. A.; Park, Y.-S.; McDaniel, H.; Robel, I.; Pietryga, J. M.; Klimov, V. I. ACS Nano 2013, 7, 3411−3419. (26) Bae, W. K.; Park, Y.-S.; Lim, J.; Lee, D.; Padilha, L. A.; McDaniel, H.; Robel, I.; Lee, C.; Pietryga, J. M.; Klimov, V. I. Nat. Commun. 2013, 4, 3661. (27) Cragg, G. E.; Efros, A. L. Nano Lett. 2010, 10, 313−317. (28) Svelto, O. Principles of Lasers. Springer: New York, 2010. (29) Klimov, V. I. J. Phys. Chem. B 2000, 104, 6112−6123. (30) Klimov, V. I. Annu. Rev. Condens. Matter Phys. 2014, 5, 285− 316. (31) Park, Y.-S.; Bae, W. K.; Padilha, L. A.; Pietryga, J. M.; Klimov, V. I. Nano Lett. 2014, 14, 396−402. (32) Park, Y.-S.; Bae, W. K.; Pietryga, J. M.; Klimov, V. I. ACS Nano 2014, 8, 7288−7296. (33) Climente, J. I.; Movilla, J. L.; Planelles, J. Small 2012, 8, 754− 759. (34) García-Santamaría, F.; Chen, Y.; Vela, J.; Schaller, R. D.; Hollingsworth, J. A.; Klimov, V. I. Nano Lett. 2009, 9, 3482−3488. (35) Park, Y.-S.; Malko, A. V.; Vela, J.; Chen, Y.; Ghosh, Y.; GarciaSantamar, F.; Hollingsworth, J. A.; Klimov, V. I.; Htoon, H. Phys. Rev. Lett. 2011, 106, 187401. (36) Piryatinski, A.; Ivanov, S. A.; Tretiak, S.; Klimov, V. I. Nano Lett. 2007, 7, 108−115. (37) Ivanov, S. A.; Nanda, J.; Piryatinski, A.; Achermann, M.; Balet, L. P.; Bezel, I. V.; Anikeeva, P. O.; Tretiak, S.; Klimov, V. I. J. Phys. Chem. B 2004, 108, 10625−10630. (38) Cao, H.; Zhao, Y. G.; Ho, S. T.; Seelig, E. W.; Wang, Q. H.; Chang, R. P. H. Phys. Rev. Lett. 1999, 82, 2278. (39) Frolov, S. V.; Vardeny, Z. V.; Yoshino, K.; Zakhidov, A.; Baughman, R. H. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59, R5284. (40) Polson, R. C.; Raikh, M. E.; Vardeny, Z. V. C. R. Phys. 2002, 3, 509−521.

when we analyze a purely radiative scenario of multiexciton decay, we do not see any dependence of lasing threshold on τXX (considering again the case of a lossless cavity), because the reduction of the gain lifetime due to shorter τXX is compensated by the proportional increase of the stimulated emission rate. These results re-emphasize the detrimental role of Auger recombination in QD lasing and highlight the importance of the development of effective approaches for its suppression.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b02595. Additional information on calculations of lasing thresholds, structural characteristics of reference and alloyed quantum dot samples, amplified spontaneous emission and optical gain measurements, quantum dot film morphology, and calculations of energies of the 1S and 1P optical transitions. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS These studies were supported by the Chemical Sciences, Biosciences and Geosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy.

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REFERENCES

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