Enhancement of Optical Gain in Semiconductor Nanocrystals through

Jul 1, 2008 - The development of optical gain in semiconductor nanocrystals requires a high concentration of electronic excitation energy. Generating ...
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J. Phys. Chem. C 2008, 112, 10622–10631

Enhancement of Optical Gain in Semiconductor Nanocrystals through Energy Transfer P. Gregory Van Patten* Department of Chemistry & Biochemistry, Clippinger Laboratories, Ohio UniVersity, Athens, Ohio 45701 and Nanoscale & Quantum Phenomena Institute, Ohio UniVersity, Athens, Ohio 45701 ReceiVed: August 29, 2007

The development of optical gain in semiconductor nanocrystals requires a high concentration of electronic excitation energy. Generating and maintaining such a high excitation density is difficult because of fast Auger relaxation channels that operate under such conditions. In this article, the use of energy transfer within nanocrystal clusters is proposed as a means for concentrating excitation energy in a subset of optically active particles. Numerical simulations of the dynamic evolution of the system under moderate pumping intensity have been carried out to evaluate the potential efficacy of this strategy. Using realistic values for relevant rate constants, these simulations predict that energy transfer in nanocrystal clusters can reduce the effective gain threshold by 3- to 4-fold and can increase the gain lifetime by an order of magnitude as compared with those of samples of noninteracting nanocrystals. Strong performance enhancements are realized even if the energytransfer rate is slower than the Auger relaxation process that normally hinders the development of gain. The proposed scheme is entirely compatible with other nanocrystal lasing strategies, such as the use of type II heteronanocrystals and electrical pumping, and should contribute substantially to the ultimate goal of nanocrystal lasing under practical pumping conditions. Introduction Colloidal semiconductor nanocrystals are promising candidates for optical gain media in new types of lasers. High-quality colloidal nanocrystal samples may exhibit very high luminescence efficiencies with quantum yields approaching unity, and their emission wavelength can easily be tuned over a wide range by controlling the particle size during synthesis. In addition to their light-emitting properties, colloidal nanocrystals have some unusual characteristics that make them uniquely suited for certain types of applications. For example, they can be prepared in bulk solution in large quantities (compared to epitaxial quantum dots) and can be manipulated and processed like molecular species using solution-based methods. At the same time, they can be readily integrated into solid-state devices by evaporating the solvent to produce glassy or ordered nanocrystal solids. Most molecular species, if dried to a solid, lose their luminescence due to various quenching mechanisms; nanocrystals, however, have organic and/or inorganic coatings that insulate them from their environment and make them less susceptible to such quenching. Nanocrystal-based lasers produced by wet methods would be highly flexible in terms of geometry, miniaturization, and integration with other materials. For this reason they could find use in a variety of device platforms such as microelectronic, microfluidic, and microelectro-mechanical systems. Despite these favorable characteristics, it has been difficult to realize lasing in samples of colloidal nanocrystals. The practical difficulty arises as a result of some of the photophysical idiosyncrasies of these systems. The lowest energy excitonic state in semiconductor nanocrystals generally has a degeneracy of at least two, and in most systems studied to date, even higher degeneracies exist. For the case of colloidal CdSe nanocrystals, which have been studied extensively, the lowest-energy exciton states consist of a doubly degenerate, dark exciton state and a * Corresponding author. E-mail: [email protected].

doubly degenerate bright exciton state that are separated in energy by less than 20 meV.1 At room temperature, these states are strongly mixed, and the electronic structure can be modeled effectively as a 4-fold-degenerate [valence band] hole energy level and a pair of doubly degenerate [conduction band] electron energy levels.1,2 This model gives excellent agreement with measurements of transient bleaching and gain in pump-probe measurements on CdSe nanocrystals. In order to achieve a population inversion in such a system, it is necessary to produce multiple excitons per nanocrystal across the entire system. In practice, optical inversion of such a system requires an excitation density of more than 1.5 excitons per nanocrystal,2–4 so multiply excited nanocrystals are generally a requirement for lasing in ordinary nanocrystal systems. Unfortunately, the existence of multiple excitons on a single nanocrystal is a highly unstable situation. The instability of biexcitons and multiexcitons arises due to the spatial confinement of these particles within a very small volume. This confinement results in a greatly increased probability (relative to the bulk material) for Auger recombination, whereby one exciton may recombine nonradiatively by transferring its energy to another carrier residing on the same nanocrystal. Multiexciton decay occurs in a stepwise fashion, with one exciton being eliminated in each step. Furthermore, the Auger rate increases rapidly with the number of excitons in the system, so that even under strong pumping conditions, biexcitons are likely to represent the main contribution to optical gain. In strongly confined nanocrystals, the biexciton Auger rate constant is typically on the order of 1010-1011 s-1, so any biexcitons present in the system disappear within tens of picoseconds along with the associated optical gain.5 This short gain lifetime has been a major obstacle to producing a practical lasing device from nanocrystals. In order to maintain a steady-state population inversion, it would be necessary to pump the system faster than the biexcitons disappear, and this is simply impractical with conventional continuous wave pumping.

10.1021/jp802720u CCC: $40.75  2008 American Chemical Society Published on Web 07/01/2008

Optical Gain in Semiconductor Nanocrystals Klimov and co-workers have recently demonstrated that it is possible to achieve a population inversion without biexciton formation in nanocrystal ensembles containing so-called typeII heteronanocrystals.4,6 The theoretical minimum threshold for gain in the type II system is 〈N0〉 ) 0.67, where 〈N0〉 represents the average number of excitons per nanocrystal in the system. For an ensemble of ordinary nanocrystals, the threshold lies near 〈N0〉 ) 1.8, as mentioned previously, so the threshold reduction in the type II systems is nearly 3-fold. While this threshold reduction seems minor, it is important because it means that biexcitons are not necessary in the system to achieve lasing. Accordingly, the pump intensity required to produce gain may be greatly reduced (by at least 1 to 2 orders of magnitude) because it is no longer necessary for optical pumping to compete with the fast Auger decay process. The present work discusses an alternative strategy for lowering the optical gain threshold in nanocrystal ensembles through excitation energy transfer. This article presents and discusses a series of numerical simulations designed to quantitatively estimate improvements in lasing performance that are possible through long-range energy transfer. The results suggest that large increases in optical gain can be realized by using energy transfer within donor-acceptor nanocrystal clusters to concentrate excitons within the [spectroscopically distinct] acceptor subset of nanocrystals. These structures are analogous to antenna complexes used in biological and biomimetic light harvesting systems aimed at solar energy conversion.7 In the nanocrystal clusters described here, the energy-transfer process enhances biexciton production and allows for substantial reductions in the excitation density required to achieve gain. Importantly, it is found that these gain enhancements are possible even with realistic energy-transfer rates that might be somewhat slower than the Auger recombination rate. In addition to increasing the gain magnitude, the nanocrystal antenna clusters described here also produce a significant increase in the gain lifetime of the system. As mentioned above, increasing the gain lifetime is a crucial part of reducing the pump intensity required to maintain gain. The gain lifetime is also important because once gain is attained, a certain time is required for stimulated emission to build up in the system. If the gain lifetime is not longer than this stimulated emission buildup time, then lasing cannot occur. This antenna-based strategy for concentrating electronic energy in the acceptor subset of nanocrystals is shown to be effective in reducing the gain threshold in conventional (type I) nanocrystal systems, where multiexcitons are required for lasing. It is important to realize, however, that this strategy can also be used to advantage in ensembles of type II nanocrystals. While multiexcitons are not required for lasing in those systems, the ability to collect energy over a large volume and concentrate that energy at the active gain centers can significantly reduce gain threshold and improve performance even in those systems. Theory Energy Transfer in Semiconductor Nanocrystals. Energy transfer between nanocrystals has been widely studied both experimentally and theoretically.8–18 The operative mechanism is generally assumed to be identical to that which was originally explained by Fo¨rster for molecular systems.19,20 In that mechanism, the transfer of energy is mediated through Coulombic interactions between an excited donor chromophore and a nearby, unexcited, acceptor chromophore. Most frequently it is assumed that the dominant term in the Coulomb interaction expression is the dipole-dipole interaction and that the chro-

J. Phys. Chem. C, Vol. 112, No. 29, 2008 10623 mophores are of small size relative to their separation so that they can be modeled as point dipoles. This assumption leads directly to a simple set of criteria that can be used to predict the probability (rate) of energy transfer between a given donor and acceptor. In order for the process to occur, the acceptor chromophore must have an electronic transition that is in resonance with the donor relaxation process and that fulfills appropriate selection rules. Because only dipole-dipole interactions are considered, the applicable selection rules are the same as those which govern absorption of dipole radiation. Consequently, the probability for energy transfer can be accurately gauged by measuring spectroscopically the overlap between the absorption spectrum of the acceptor and the luminescence spectrum of the donor. Intensities of the absorption and luminescence peaks are directly linked to the strength of the Coulombic coupling and thus to the rate of energy transfer that can be expected. Because nanocrystals tend to have broad absorption peaks, it is generally a sufficient criterion for energy transfer that the band gap energy of the donor be larger than the band gap energy of the acceptor. The point dipole approximation predicts that the energytransfer rate should scale with the inverse sixth power of the distance between the centers of the chromophores involved. Precise, controlled measurements of energy-transfer rates between nanocrystals are difficult due to difficulties in preparing perfectly monodisperse nanocrystals and to difficulties in linking them together in precise geometries. Nevertheless, the body of experimental data suggests that Fo¨rster-type energy transfer between nanocrystals with surface-to-surface distances of a few nanometers or less may be expected to occur on a subnanosecond time scale. Although precise numbers are not yet available, some of the best measurements suggest that first order rate constants of approximately 1010 s-1 may be expected for nanocrystals in close proximity (i.e., with only small molecule ligands separating them).8,9,11–16 Biexcitons can be formed in semiconductor nanocrystals under intense irradiation through sequential absorption of photons. The band gap of a singly excited nanocrystal is nominally the same as the band gap of the unexcited nanocrystal (a small shift of a few tens of meV may be observed, owing to Coulombic interactions between excitons).21,22 Consequently, an excited nanocrystal should be able to accept energy from a donor just like a ground-state nanocrystal can. The transfer of excitation energy to a singly excited nanocrystal would produce a biexciton. Thus, it should be possible to produce a biexciton on an acceptor nanocrystal either through absorption of one photon followed by energy transfer from an excited donor or through two energy-transfer events occurring in rapid succession from multiple excited donors. Production of higher order multiexcitons through energy transfer should also be possible. Photophysical Model. The electronic energy spectrum of an individual nanocrystal consists of numerous states that give rise to a large number of transitions above the band gap energy. The homogeneous line width of each transition is generally assumed to be dominated by thermal broadening at room temperature, and is thus usually taken to be approximately 30 meV. In reality, measured room temperature emission line widths from single quantum dots are more typically in the 50-80 meV range.23 The difference between these values may be mainly due to rapid spectral diffusion.24 [It is noteworthy that this single nanocrystal line width is not much different from inhomogeneously broadened line widths reported for the best colloidal samples being prepared today (60-80 meV).25] As a result of their relatively narrow line widths, the two or three

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Van Patten SCHEME 1: Nanocrystal Antenna Clusters Comprising Multiple Energy Donors (Green) Surrounding a Central Energy Acceptor (Red)

Figure 1. Electronic energy level diagram for donor and acceptor nanocrystals under consideration in this study. Energy values on the vertical axis are relative to the nanocrystal ground states, which are shown at the bottom. The donor band gap is approximately 2.5 eV, and the acceptor band gap is approximately 2.0 eV. This allows energy transfer from the lowest excited donor state (D*) to higher-lying excited states on the acceptor. Because the acceptor’s electronic spectrum is relatively dense at 2.5 eV (quasi-continuum), the resonance matching condition for energy transfer is not too stringent in this system.

lowest energy transitions in the electronic spectrum of semiconductor nanocrystals appear as resolved peaks that show relatively little overlap with adjacent transitions. At higher energies, however, the spectrum becomes quite dense, so that with thermal broadening, the spectrum above the first few transitions can effectively be considered to be a continuum. Figure 1 depicts the electronic energy levels of hypothetical donor and acceptor nanocrystals. States are plotted versus energy relative to the ground state. Adjacent to the energy level diagrams and plotted on the same energy scale are absorption spectra from actual CdSe nanocrystal samples prepared in our laboratory. This figure can be used to understand the model used for our simulations. Following excitation to a high-lying energy level (into the quasi-continuum, for example), nanocrystals relax to the lowest excitonic state (D* or A*) within 1 ps.26 Compared with other excited-state processes, this intraband relaxation of the carriers is so fast that it may be considered to be instantaneous for the purposes of our dynamic simulations. At room temperature, the lifetime of the lowest exciton state in high-quality II-VI nanocrystals is typically in the range of a few tens of nanoseconds.27 If the band gap energy of the donor nanocrystal is significantly higher than that of the acceptor (∼0.5 eV for the case depicted in Figure 1), then the donor exciton energy will be in resonance with the quasi-continuum portion of the acceptor spectrum. If these two nanocrystals are placed in close proximity to one another, excitation energy may be transferred from the donor to the acceptor nanocrystal as depicted by the arrow in Figure 1. Following energy transfer, ultrafast intraband relaxation within the acceptor occurs to produce the A* state, so that the excitation energy is now effectively trapped on the acceptor. From the A* state, reverse energy transfer to the D* state requires a significant input of thermal energy, so that the thermally activated reverse transfer process is effectively shut off. In the case shown, the difference in band gaps is 0.5 eV, or 20 times larger than kT at room temperature, so that the reverse transfer rate is predicted to be approximately 5 × 108 times slower than the forward process. Because of its small rate constant, this reverse transfer process has been neglected in the simulations presented here. In a nanocrystal cluster consisting of multiple donor nanocrystals surrounding a central acceptor (Scheme 1), energy

transfer gives rise to processes that cannot occur in isolated nanocrystals. These clusters are similar to starburst antenna multichromophoric arrays studied previously for light-harvesting applications.7 When a short pulse of light excites a random selection of donors and acceptors in a collection of such clusters, the following equations represent the set of possible photophysical processes: 4kAuger

A**** 98 A*** 2.25kAuger

A*** 98 A** kAuger

A** 98 A* k1

A* 98 A kAuger

D** 98 D* k1

D* 98 D kDA

D* + An* 98 D + A(n+1)* 2kDA

D** + An* 98 D* + A(n+1)* kDD

D* + D* 98 D + D** kDD

D* + D 98 D + D* 2kDD

D** + D 98 D* + D*

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

Equations 1–6 represent all of the possible de-excitation processes intrinsic to excited nanocrystals in the cluster. The rate constant k1, assumed to apply to both donors and acceptors, includes both radiative and nonradiative decay components present in these singly excited nanocrystals. For the purposes of this work it is not necessary to distinguish between the relative contributions of radiative and nonradiative decay because we are ultimately concerned only with the relative concentrations of acceptor nanocrystals with varying degrees of excitation in the sample. The rate constant kAuger represents the rate constant for Auger relaxation in biexcitonic nanocrystals. Donors and

Optical Gain in Semiconductor Nanocrystals

J. Phys. Chem. C, Vol. 112, No. 29, 2008 10625

TABLE 1: Relative Distances and Energy-Transfer Rates between Donor Nanocrystals in ADx Clusters donor/acceptor molar ratio, x nearest neighbor (NN) second NN third NN fourth NN

dDD/dDA kDD/kDA no. of NNs dDD/dDA kDD/kDA no. of second NNs dDD/dDA kDD/kDA no. of third NNs dDD/dDA kDD/kDA no. of fourth NNs

2

3

4

6

8

12

2 0.0156 1 n/a n/a n/a n/a n/a n/a n/a n/a n/a

1.732 0.0370 2 n/a n/a n/a n/a n/a n/a n/a n/a n/a

1.633 0.0527 3 n/a n/a n/a n/a n/a n/a n/a n/a n/a

1.414 0.125 4 2 0.0156 1 n/a n/a n/a n/a n/a n/a

1.155 0.4219 3 1.633 0.0527 3 2 0.0156 1 n/a n/a n/a

1 1 4 1.414 0.125 2 1.732 0.0370 4 2 0.0156 1

acceptors in the system are assumed to have identical Auger relaxation rate constants. In the above equations, this Auger rate constant has been scaled by a prefactor that depends on the number of excitons on the nanocrystal, in accordance with experimental data.5 Donors excited beyond the biexciton level are not considered in these simulations. Effectively, this means that all triply and quadruply excited donors are considered to instantaneously relax to the biexcitonic state. This approximation leads to a very minor reduction in predicted gain in the samples because some of the donor excitons are simply “lost” at the outset; however, it greatly simplifies the treatment of donor-donor energy transfer in the clusters. Equations 7 and 8 represent energy transfer from a singly or doubly excited donor to an acceptor nanocrystal with n excitons (n ) 0, 1, 2, or 3). Equations 9–11 represent donor-to-donor energy transfer. The production of donor biexcitons through energy transfer (eq 9) is important because this opens the possibility for rapid Auger relaxation within the donor subset before the excitons reach the acceptors. The process represented by eq 10 can be ignored in the calculations of the present study because the products are indistinguishable from the reactants. For these simulations, eq 11 has also been neglected, with further minor impact on the results. Prediction of the rate constant for this nominally uphill process is problematic, and elimination of this process from consideration produces the most conservative estimate of gain production from these clusters. It has been assumed in these simulations that the rate constant for energy transfer between two donors is the same as that for energy transfer between a donor and an acceptor separated by an equivalent distance. The rate constant for donor-to-acceptor energy transfer is given by a function of the form kDA(d) ) Ad-6 where A is a constant and d is the center-to-center distance between the nanocrystals. The rate constant for donor-to-donor transfer is given by the same function with the same value of A, but donor-donor distances within the clusters may be different from the donor-acceptor distance (see following section and Table 1). The donor nanocrystals in these model systems have been assumed to be exactly identical so that energy can migrate randomly between multiple donors within a given cluster. In a real system, size heterogeneity would give rise to non-negligible differences between individual donor nanocrystals and would thus give rise to nonrandom energy migration between donor nanocrystals. This inaccuracy should have only a minor impact on our modeling of these clusters. The model does account for directionality in donor-acceptor transfer. Because the donor band gap energy is assumed to be greater than that of the

acceptor, energy can be transferred from donor to acceptor, but the reverse has been strictly forbidden in these calculations (i.e., kAD ) 0). The rate constant for process 7 is assumed to be twice the value for process 8. The factor of 2 comes from the number of potential donor “oscillators” in these processes. Because the donor’s exciton energy is far above the acceptor band gap, the probability, or rate constant, for transfer does not depend, to a good approximation, on the excitation state of the acceptor. These assumptions provide a reasonable starting point for estimating the effects of energy-transfer processes on biexciton yield and gain development in such a material. This is particularly true because the three relevant rate constants, k1, kDA, and kAuger are not expected to have similar values. For example, slight differences that might be expected between the Auger processes in real donors and acceptors are overshadowed by the comparatively large differences between k1, kDA, and kAuger. In real nanocrystals, the single exciton decay generally occurs on a time scale of several nanoseconds or longer, whereas Auger occurs on a time scale of tens of picoseconds.5 The expected rate for energy transfer falls in between and may vary significantly depending on the energetics of the nanocrystals and the distances separating them.8,9,11,12,14–16 For the simulations described in this paper, the rate constants have been chosen in observance of these general guidelines. For some of the simulations, the energy-transfer rates have been varied in order to assess the effect of energy-transfer rate on the photophysical behavior of the system. Dependence of Gain on Excited Nanocrystal Populations. Immediately following excitation by a short laser pulse, the distribution of excitons among a collection of nanocrystals can be described by Poisson statistics. This statistical model holds if the excitation density is very low or if the probability of absorbing a photon does not depend significantly on whether a particular nanocrystal is already excited. This latter criterion applies if the photon energy is much higher than the nanocrystal band gap, as assumed in this work. The Poisson statistics can then be used to calculate the numbers of nanocrystals containing given numbers of excitons as a function of overall excitation density, expressed in terms of 〈N0〉, the ratio of excitons to nanocrystals in the system, which is directly proportional to the pump fluence. In such a case, the Poisson distribution gives directly the initial concentrations of nanocrystals with a given exciton multiplicity

[A] ) [NA]e-〈N0〉

(12)

[A*] ) [NA]〈N0 〉e-〈N0〉

(13)

10626 J. Phys. Chem. C, Vol. 112, No. 29, 2008

Van Patten

〈N0 〉2 -〈N0〉 [A**] ) [NA] e 2

(14)

〈N0 〉3 -〈N0〉 (15) e [A***] ) [NA] 6 [A****] ) [NA] - {[A***] + [A**] + [A*] + [A]} (16) where [NA] is the total concentration of acceptor nanocrystals in the sample. Previous work2 has shown that at room temperature the lowest energy states for excited electrons and holes may effectively be considered to be doubly and quadruply degenerate in hexagonal II-VI nanocrystals such as CdSe. As a result, the fractional occupation of excited electron and hole levels by photoexcited carriers can be calculated according to

ne ) 1 nh ) 1 -

{

{ (

1 [A*] [A] + [NA] 2

)}

1 ([A] + 0.75[A*] + 0.5[A**] + [NA]

(17)

}

0.25[A***]) (18) Klimov and co-workers have made extensive measurements of photoinduced bleaching and optical gain in nanocrystal ensembles as a function of excitation density.3–5,28,29 Their results indicate that photoinduced bleaching at the lowest energy absorption peak is dominated by state-filling of conduction-band states by excited electrons. However, gain observed at the peak gain wavelength follows a different dependence due to Coulombic interactions between multiple excitons on the same nanocrystal. By fitting Klimov’s photobleaching measurements as a function of 〈N0〉 (which are collected in Figure 17 of ref 2) to a linear combination of [An/], under the constraints of eqs 12–18, it is possible to determine the dependence of the normalized gain coefficient, γnorm on the relative populations of nanocrystals with particular exciton multiplicities. This normalized gain coefficient varies from a value near -1.0 in a system with zero excitons to a maximum value of +1.0. The gain threshold is at γnorm ) 0. This normalized gain coefficient can be scaled to an actual gain value by multiplying by the gain cross section, the concentration of nanocrystals in the gain medium, and optical path length of the gain medium. The functional dependence of gain on the various subpopulations of ground-state and excited nanocrystals is found to be

γnorm ) ne + nh - 1.15[A*] + 0.75[A**] - 1.0

(19)

The third and fourth terms in this expression are a direct result of the Coulombic interactions, which shift the gain band away from the lowest energy absorption peak. Equation 19 can be adjusted to represent relative photobleaching by eliminating the final term (-1.0); the resulting function is an excellent fit to the data compiled by Klimov and co-workers (see Supporting Information).2 Equation 19 gives the normalized relative gain as a function of the exciton distribution within a collection of acceptor nanocrystals. Of course, the simulations here assume that there are donor nanocrystals as well as acceptors in the sample. However, because the band gap energies of the donors are assumed to be greater than that of the acceptors, the donors cannot absorb light emitted by the acceptors and therefore cannot contribute to losses at the acceptor emission wavelength. This point is critical for understanding the enhancement of gain

through energy transfer. All nanocrystals in the ensemble are involved in absorbing the incident pump radiation, but the excitation energy is then concentrated within the acceptors. Because only acceptors contribute to gain and losses at the emission wavelength, only acceptor nanocrystals need to be factored into the calculation of the relative gain coefficient, so eq 19 applies to the donor-acceptor mixtures discussed in this article. Calculations Numerical simulations were run using custom C code on a laptop computer running an Intel Core Duo processor with a clock speed of 1.83 GHz. Virtual nanocrystal clusters were created with several different donor/acceptor molar ratios, x, ranging from 0 to 12. The clusters, which are designated as ADx with x ) 0, 1, 2, 3, 4, 6, 8, or 12, were assumed to have a central acceptor with uniformly distributed donors on the periphery. The distances between donor nanocrystals were calculated in terms of the donor-acceptor distance, and these distances were used to scale the donor-donor energy-transfer rate constant, kDD, relative to kDA assuming a d-6 distance dependence in accord with Fo¨rster theory. Table 1 gives donor-donor distances and kDD values relative to donor-acceptor values for the various clusters considered. A set of differential rate equations was generated from eqs 1–11. Initial populations of A, A*, A**, A***, and A**** were calculated assuming a given initial excitation density, 〈N0〉 according to eqs 12–16. Excitation densities were varied from 〈N0〉 ) 0.1 to 2.5 excitons per nanocrystal. For donor populations, these equations were modified slightly. For the donor populations, [ND] ) x[NA] replaced [NA] in eqs 12 and 13, and [D**] was calculated as {[ND] - ([D] + [D*])}. An optical pulse of infinitesimally short duration and high photon energy was assumed to provide excitation. The absorption cross sections of the donors and acceptors at the excitation wavelength were assumed to be equal so that excitons were initially distributed randomly across the nanocrystal clusters. The concentrations of species at time t + dt were evaluated by numerically solving the differential rate equations according to the Euler method. In this work, the finite time interval used was dt ) 1 fs, and it was found that this small time interval allowed for truncation after only two terms (i.e., [D](t + dt) ) [D](t) + (d[D]/dt) dt) without noticeable effect on the results. The normalized relative gain coefficient, γnorm, was computed for each time t from the concentrations of the species present using eqs 17–19. Typical values of rate constants used in these simulations were k1 ) (25 ns)-1, kAuger ) (50 ps)-1, and kDA ) (100 ps)-1. In some cases, values other than these may have been used; such cases are explicitly noted. Calculations of relative gain under steady state conditions were performed using MathCad 2000 Professional. To the set of differential equations generated from eqs 1–11 was added another which described steady state photoexcitation in the system (i.e., (d[A(n+1)/]/dt) ) σcΦ[An/], where σ is the nanocrystal absorption cross section, c is the speed of light, and Φ is the photon density in m-3). These differential equations were then combined and the steady state assumption was made (i.e., (d[Q]/dt) ) 0 for all species Q, which include all donor and acceptor nanocrystals present in the system). These equations were solved simultaneously for the respective [Q] variables over a range of Φ values, and the relative gain for each case was computed using eqs 17–19.

Optical Gain in Semiconductor Nanocrystals

Figure 2. (A) Relative populations of the ground state (A), exciton state (A*), and multiexciton states (A**, A***, A****) versus time in a collection of uncoupled semiconductor nanocrystals following optical excitation. The initial excitation density is 〈Nx〉 ) 1.60 excitons per nanocrystal, and the rate constants for excited-state relaxation processes are k1 ) (25 ns)-1 and kAuger ) (50 ps)-1. No energy transfer between nanocrystals is considered for this case. (B) Relative gain versus time for the same collection of nanocrystals. At the pump intensity assumed here, the system cannot achieve positive gain.

Results and Discussion To appreciate the behavior of the ADx nanocrystal clusters, it is first necessary to examine the behavior of noninteracting nanocrystals under moderate optical pumping. Figure 2A shows the relative concentrations of ground state ([A]) and excited ([A*], [A**], [A***], and [A****]) nanocrystals as a function of time after a short laser pulse. No donors were present in this simulation. The pulse energy is assumed to be sufficient to achieve an initial 〈N0〉 value of 1.60 excitons per nanocrystal. Immediately after excitation, the sample contains a mixture of ground-state nanocrystals, single excitons, and multiexcitons. Multiply excited nanocrystals make up 80% of the total population at t ) 0, but these multiexcitons disappear rapidly due to the fast Auger decay process that is assumed here to have a first order rate constant of (50 ps)-1 for biexcitons, (22 ps)-1 for triexcitons, and (12.5 ps)-1 for quadexcitons. Since the excitation density is below the gain threshold for this system (〈Nx〉 ≈ 1.8), the normalized relative gain, γrel (shown in Figure 2B), does not reach a positive value. At t ) 0, γrel ) -0.16, and it rapidly decays to its minimum value as multiexcitons are destroyed through the Auger process.

J. Phys. Chem. C, Vol. 112, No. 29, 2008 10627 Nanocrystal clusters containing multiple donors surrounding a central acceptor exhibit behavior that is qualitatively different from that of a collection of isolated nanocrystals. Figure 3 shows the excited-state population trajectories for a sample consisting of AD8 clusters under the same excitation conditions as were considered above (〈Nx〉 ) 1.60). In addition to [An*], Figure 3 shows the relative concentrations of the ground-state donors, [D], and excited donors, [D*] and [D**]. At t ) 0, the distribution of excitons among acceptor nanocrystals is the same in the AD6 clusters as in the individual acceptor nanocrystals described in Figure 2. Thus, the value of γrel at t ) 0 is -0.16 as before. In the case of the AD8 cluster, energy is transferred from excited donors to the central acceptor nanocrystal on a subnanosecond time scale (kDA ) (100 ps)-1 for this case). This energy-transfer process leads to a dramatic decrease in groundstate and singly excited acceptors within the first nanosecond and a strong rise in multiply excited acceptors. The energytransfer process also substantially slows the observed decrease of multiexcitonic acceptors so that these multiexcitonic acceptors outnumber ground-state acceptors for 660 ps. As a consequence of the energy-transfer pumping of the acceptor nanocrystals, the AD8 system achieves positive gain for the case where the initial excitation density 〈N0〉 ) 1.60 excitons per nanocrystal. Figure 4 shows the time dependent behavior of γrel for the AD8 cluster after optical pumping to 〈N0〉 ) 1.60 excitons/ nanocrystal. As explained above, energy transfer leads to rapid development of optical gain after excitation. The gain reaches a maximum at a value of 0.72 and remains positive for a period of 230 ps. This extended gain duration in the cluster is a consequence of the ability of the donors to store single excitons for relatively long periods and to continually transfer this stored excited-state energy to the acceptors as long as the excited donors remain. The significance of the increased gain duration is discussed in a later section of this article. The most obvious effect of energy transfer on the optical gain in the ADx nanocrystal clusters is to enhance the maximum relative gain that can be attained at a given pump fluence. Figure 5 shows the maximum relative gain attained following an ultrashort optical excitation pulse for various ADx clusters. For these simulations, the relevant rate constants were assigned the values given previously: k1 ) (25 ns)-1, kAuger ) (50 ps)-1, and kDA ) (100 ps)-1. From the graph it can be seen that positive gain can be produced in AD12 clusters even when 〈N0〉 < 0.40. For clusters with lower coordination number, such as AD6 or AD8, the threshold lies between 〈N0〉 ) 0.50 and 0.80. Gain thresholds are significantly reduced for all ADx clusters where x g 3. For cases in which x g 8, the gain threshold predicted here is lower than the threshold that theoretically can be achieved with type II heteronanocrystals.4 Gain Duration and Time-Integrated Gain. The gain duration is an important parameter because it determines the time window available for stimulated emission to build up in the sample. Figure 6 shows the gain duration calculated for ADx clusters under the same conditions described for Figure 5. Once the optical pumping threshold for gain is exceeded, the gain duration increases rapidly with pump intensity and then begins to level off. Gain durations exceeding 100 ps are readily attained for the ADx clusters when x g 4. The extended gain duration relative to the free acceptor nanocrystals is due to the storage of exciton energy in the donors prior to transfer to the acceptors. The maximum possible storage time depends on the rate of deexcitation of D*, which is dominated by the energy-transfer rate

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Van Patten

Figure 4. Normalized relative gain versus time for a collection of AD8 nanocrystal clusters for the same system and excitation conditions depicted in Figure 3. Energy transfer from donors to acceptors concentrates excitation energy in the acceptors and produces positive gain under conditions that could not produce gain in a collection of noninteracting nanocrystals (cf. Figure 2). The gain remains positive for more than 200 ps. This gain duration is significantly longer than the Auger recombination rate that limits the gain duration in collections of conventional semiconductor nanocrystals.

Figure 5. Maximum relative gain attained for ADx nanocrystal clusters as a function of excitation pump intensity. For clusters with x g 4, the gain threshold is significantly reduced relative to collections of noninteracting nanocrystals.

Figure 3. Ground-state and excited-state populations versus time for semiconductor nanocrystals in AD8 clusters following optical excitation. The initial excitation density is 〈Nx〉 ) 1.60 excitons per nanocrystal, and rate constants for excited-state relaxation are the same as in Figure 2. Donor-to-acceptor energy transfer is operative with a rate constant of kDA ) (0.10 ns)-1, and donor-to-donor energy transfer occurs with the rate constants given in Table 1.

in most of the cases studied (kDA ) (100 ps)-1 > k1). The value of kDA, then, dictates where the gain duration ultimately must level off. In ref 4 the characteristic decay time of the photoinduced bleach for transient absorption measurements in the optical gain regime is given for type II heteronanocrystals as 1.7 ns. This number does not represent the actual gain duration, but rather represents the rate constant for decay of 〈Nx〉 from its initial value back to zero (or to the value corresponding to thermal population). The 1.7 ns decay time can be used to estimate the gain duration, though, if the initial value of 〈Nx〉 is known. The transient absorption measurement compares the absorbance of the sample both with (R) and without (R0) an excitation pump pulse. The criterion for optical gain, expressed in terms of quantities from the transient absorption measurement, is -∆R/ R0 > 1.0, where ∆R ) R - R0. A negative value of ∆R implies

Optical Gain in Semiconductor Nanocrystals

J. Phys. Chem. C, Vol. 112, No. 29, 2008 10629

Figure 6. Gain duration versus pump intensity for collections of ADx nanocrystal clusters.

photoinduced bleaching, whereas a positive value is associated with photoinduced absorption. When ∆R becomes sufficiently negative that -∆R > R0, then the absorption of the sample after the arrival of the pump pulse, (R0 + ∆R), becomes negative (positive gain). If we consider the system as a set of two-level oscillators as described above, the maximum value of -∆R/R0 is 2 (corresponding to a normalized relative gain value of +1). Reference 1 reported the attainment of -∆R/R0 ≈ 1.5 for optimally pumped, type II heteronanocrystals (Figure 5d in that work). Starting with this value and assuming an exponential decay of -∆R with a lifetime of 1.7 ns, it can be shown that the sample would exit the gain regime (-∆R/R0 > 1.0) after less than 0.7 ns. This analysis shows that the gain duration values predicted in the present work for ADx clusters are somewhat shorter than those attained in type II heteronanocrystals. Time-Integrated Gain. As has been discussed by Klimov et al.,3 the rate constant for stimulated emission buildup, ks () 1/τs), is proportional to the gain of the medium and is inversely proportional to the refractive index of the medium:

ks )

c 1 c ) γ r ) (ntotσγrel) τs n nr

(20)

In eq 20, γ is the sample gain coefficient in cm-1, c is the speed of light in cm/s, nr is the refractive index of the gain medium, ntot is the number density of nanocrystals in the sample (cm-3), and σ is the gain cross section per nanocrystal (cm2). To construct a working laser, it is necessary to maintain positive gain for a period of time longer than the stimulated emission buildup time, τs. If the sample gain is maintained at a constant, positive value for a period of time ∆t, and the stimulated emission buildup time associated with that constant, positive gain level is τs, then the quantity z ) ∆t/τs ) ks∆t gives the ratio of the gain lifetime to the stimulated emission buildup time. The quantity z measures the extent to which stimulated emission is able to build up in the system. If the gain coefficient is not constant, but rather is a function of time, γ(t), then ks is also a function of time, and the equation for z changes to

z)

c nr

∫t t γ(t) dt ) 2

1

( )∫ cntotσ nr

t2

t1

γrel(t) dt

(21)

where t1 to t2 represents the time interval during which the gain

Figure 7. Time-integrated gain versus pump intensity for collections of ADx nanocrystal clusters.

coefficient remains positive. The parameter z, termed the timeintegrated gain, is a useful figure of merit for comparing the predicted lasing performance of these nanocrystal clusters under various excitation intensities. The shaded area under the curve in Figure 4 represents the integral at the right side of eq 21, and this integral is related to the time integrated gain through the prefactor shown, (cntotσ/nr). The simulations presented here give values for γrel as a function of time. To scale these values to predict performance of an actual nanocrystal sample or to evaluate the time-integrated gain, it is necessary to estimate the value of the optical cross section of a single nanocrystal, the number density of nanocrystals in the sample, and the refractive index of the sample. In order to assign a value for the number density of acceptor nanocrystals in a sample, the acceptor nanocrystals have been assumed to reside at the corners of a primitive cubic lattice with lattice constant of 8.04 nm. This cluster-to-cluster distance gives a volume fraction of acceptor nanocrystals in the sample of 8% (assumes nanocrystal radius of with 2.15 nm). Furthermore, this cluster-to-cluster separation is sufficient to ensure that interactions between donors on neighboring clusters are much weaker than the interactions between donors on the same cluster. Using these assumptions, the number density of nanocrystals in the sample is 1.92 × 1018 cm-3. The optical cross section used here (σ ) 1.0 × 10-16 cm2) was based on values reported previously.3,28 The refractive index was set at 2.5, a value that might correspond to a polymer or glass with embedded semiconductor material. From the above considerations, it is possible to calculate time integrated gain for the various clusters under various excitation conditions. The results of these calculations are shown in Figure 7. From the earlier discussion, it is apparent that a value of unity for this quantity represents the barest minimum value required for build up of stimulated emission. For clusters with x g 6, time-integrated gain values greater than 50 may be attained under moderate excitation conditions. Effect of Energy-Transfer Rate on Performance. Simulations have been executed using different values of the energytransfer rate constant, kDA. The results show that, as expected, a faster energy-transfer process results in higher maximum relative gain and shorter gain duration. Figure 8 shows how performance changes for the AD6 cluster as kDA is increased or decreased by a factor of 5. For systems with an energy-transfer rate constant an order of magnitude slower than the Auger rate

10630 J. Phys. Chem. C, Vol. 112, No. 29, 2008

Figure 8. Effect of energy-transfer rate constant on the performance of AD6 clusters at various pump intensities. If the energy transfer is much slower than 10 ns-1, the gain threshold is not significantly reduced; however, there still can be meaningful increases in gain duration and the time-integrated gain for such systems.

constant, the gain threshold is not significantly reduced by the energy-transfer process. However, energy transfer may still produce noticeable improvements in the gain duration and timeintegrated gain when these systems are pumped above the gain threshold. Prospects for Continuous Lasing. Calculations have been performed to estimate the gain threshold reductions that may be obtained through energy transfer under steady state excitation

Van Patten

Figure 9. Relative gain enhancement in nanocrystal donor/acceptor clusters under steady state pumping. (A) Normalized relative gain plotted versus relative pump intensity for isolated nanocrystals and donor/acceptor clusters. (B) Relative pump threshold required to achieve optical gain for the isolated nanocrystals and clusters considered in (A). (C) Gain threshold reduction factor (red bars) compared with numbers of absorbing nanocrystals in each cluster type (black bars) for the isolated nanocrystals and clusters considered in (A). The red bars in (C) are proportional to the reciprocal of the values plotted in (B) for each cluster type.

conditions. Figure 9A shows the relative gain for free acceptors and for clusters with 6, 8, and 12 donors under steady-state excitation. The abscissa is given in relative units that are

Optical Gain in Semiconductor Nanocrystals proportional to photons per cubic meter. The scaling factor is dependent on the value of the absorption cross section for the nanocrystals under consideration. Although the plot is not given in absolute units, it still shows the gain enhancement effects of the donor nanocrystals in the clusters. Figure 9B shows the relative reduction in steady-state gain threshold in terms of pump intensity for the free acceptors and for the clusters with 6, 8, and 12 donors. The threshold reduction factor for these cases is shown in Figure 9C, and it is evident that this enhancement factor is directly proportional to the number of donors in the cluster. At first glance, this result may seem intuitively obvious and almost trivial, but it is important to recognize that this result is a fortuitous result of the relationships between the relevant rate constansts, k1, kAuger, and kDA. To clearly understand this, one may consider the effect of significantly reducing kDA. In the case of a very small kDA value, it is obvious that the gain threshold would not scale inversely with the number of donors in the cluster. In principle, it should be possible to construct multilayered antenna clusters containing a single, central acceptor surrounded by successive, graded layers of donors. In such a cluster, each layer of nanocrystals would preferentially transfer energy inward toward the ultimate acceptor. Additional performance enhancements might be realized in this way with a second layer or even a third layer of donor nanocrystals in the cluster. Conclusions Results from numerical simulations of optical gain in nanocrystal clusters demonstrate that energy transfer can significantly enhance the lasing performance in nanocrystal samples. In these clusters, donor nanocrystals store energy on a nanosecond time scale in the form of single excitons before transferring these excitons to acceptors in the center of the clusters. The donors act as a light collection antenna, absorbing photons and funneling them inward to the acceptor nanocrystal. With this scheme, it is possible to collect photons under conditions of moderate pump intensity and to sufficiently concentrate the excitons in the acceptor nanocrystals so that the gain threshold may be exceeded. Strong gain enhancement can be realized even when the energy-transfer rate constant is somewhat slower than the rate constants that govern multiexciton relaxation within the system. Realistic estimates of energy-transfer rate constants in clusters with more than 4 donors predict reductions of more than 50% in the pump intensity required to achieve gain. More importantly, the storage of excitation energy as single excitons on peripheral donors can significantly extend the gain duration, thus further enhancing the lasing performance of the system. The reduction in gain threshold and extension of the gain lifetime predicted for these clusters are comparable to the values that have been realized in type II heteronanocrystals that permit single exciton lasing. It is not yet certain whether the predicted performance can be realized in actual nanocrystal samples or whether the performance of these samples will be able to match that of type II heteronanocrystals. However, this energy pumping

J. Phys. Chem. C, Vol. 112, No. 29, 2008 10631 strategy could easily be applied even in systems containing the type II nanocrystals and could produce significant improvements there as well. Acknowledgment. For R.J.D. I gratefully acknowledge funding support from Ohio University’s Biomimetic Nanoscience & Nanotechnology Initiative. References and Notes (1) Efros, A. L.; Rosen, M.; Kuno, M.; Nirmal, M.; Norris, D. J.; Bawendi, M. G. Phys. ReV. B 1996, 54, 4843. (2) Klimov, V. I. Charge Carrier Dynamics and Optical Gain in Nanocrystal Quantum Dots: From Fundamental Photophysics to QuantumDot Lasing. In Semiconductor and Metal Nanocrystals: Synthesis and Electronic and Optical Properties; Klimov, V. I., Ed.; Marcel Dekker: New York, 2004; Vol. 87; p 159. (3) Klimov, V. I.; Mikhailovsky, A. A.; Xu, S.; Malko, A.; Hollingsworth, J. A.; Leatherdale, C. A.; Eisler, H. J.; Bawendi, M. G. Science 2000, 290, 314. (4) Klimov, V. I.; Ivanov, S. A.; Nanda, J.; Achermann, M.; Bezel, I.; McGuire, J. A.; Piryatinski, A. Nature 2007, 447, 441. (5) Klimov, V. I.; Mikhailovsky, A. A.; McBranch, D. W.; Leatherdale, C. A.; Bawendi, M. G. Science 2000, 287, 1011. (6) Piryatinski, A.; Ivanov, S. A.; Tretiak, S.; Klimov, V. I. Nano Lett. 2007, 7, 108. (7) Van Patten, P. G.; Shreve, A. P.; Lindsey, J. S.; Donohoe, R. J. J. Phys. Chem. B 1998, 102, 4209. (8) Franzl, T.; Klar, T. A.; Schietinger, S.; Rogach, A. L.; Feldmann, J. Nano Lett. 2004, 4, 1599. (9) Franzl, T.; Koktysh, D. S.; Klar, T. A.; Rogach, A. L.; Feldmann, J.; Gaponik, N. Appl. Phys. Lett. 2004, 84, 2904. (10) Govorov, A. O. Phys. ReV. B 2003, 68, 075315. (11) Achermann, M.; Petruska, M. A.; Crooker, S. A.; Klimov, V. I. J. Phys. Chem. B 2003, 107, 13782. (12) Crooker, S. A.; Hollingsworth, J. A.; Tretiak, S.; Klimov, V. I. Phys. ReV. Lett. 2002, 89, 186802. (13) Mamedova, N. N.; Kotov, N. A.; Rogach, A. L.; Studer, J. Nano Lett. 2001, 1, 281. (14) Kagan, C. R.; Murray, C. B.; Bawendi, M. G. Phys. ReV. B 1996, 54, 8633. (15) Kagan, C. R.; Murray, C. B.; Nirmal, M.; Bawendi, M. G. Phys. ReV. B 1996, 76, 1517. (16) Koole, R.; Liljeroth, P.; Donega, C. D.; Vanmaekelbergh, D.; Meijerink, A. J. Am. Chem. Soc. 2006, 128, 10436. (17) Al-Ahmadi, A. N.; Ulloa, S. E. Appl. Phys. Lett. 2006, 88, 043110. (18) Al-Ahmadi, A. N.; Ulloa, S. E. Phys. ReV. B 2004, 70, 201302(R) (19) Fo¨rster, T. Ann. Phys. 1948, 437, 55. (20) Lamola, A. A.; Turro, N. J. Energy Transfer and Organic Photochemistry; John Wiley & Sons: New York, 1970. (21) Achermann, M.; Hollingsworth, J. A.; Klimov, V. I. Phys. ReV. B 2003, 68, 245302. (22) Caruge, J. M.; Chan, Y. T.; Sundar, V.; Eisler, H. J.; Bawendi, M. G. Phys. ReV. B 2004, 70, 085316. (23) Shimizu, K. T.; Bawendi, M. G. Optical Dynamics in Single Semiconductor Quantum Dots In Semiconductor and Metal Nanocrystals: Synthesis and Electronic Optical Properties; Klimov, V. I., Ed.; Marcel Dekker: New York, 2004; Vol. 87; p 215. (24) Empedocles, S. A.; Bawendi, M. G. J. Phys. Chem. B 1999, 103, 1826. (25) Qu, L. H.; Peng, X. G. J. Am. Chem. Soc. 2002, 124, 2049. (26) Klimov, V. I.; Mikhailovsky, A. A.; McBranch, D. W.; Leatherdale, C. A.; Bawendi, M. G. Phys. ReV. B 2000, 61, R13349. (27) Fisher, B. R.; Eisler, H. J.; Stott, N. E.; Bawendi, M. G. J. Phys. Chem. B 2004, 108, 143. (28) Klimov, V. I. J. Phys. Chem. B 2000, 104, 6112. (29) Mikhailovsky, A. A.; Malko, A. V.; Hollingsworth, J. A.; Bawendi, M. G.; Klimov, V. I. Appl. Phys. Lett. 2002, 80, 2380.

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