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Determination of Exciton Diffusion Length by Transient Photoluminescence Quenching and Its Application to Quantum Dot Films Elizabeth M. Y. Lee and William A. Tisdale* Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States S Supporting Information *

ABSTRACT: We present a framework for analyzing transient photoluminescence interfacial quenching experiments to extract exciton diffusivity and diffusion length. Through analytical solutions and finite element simulations at the continuum level, we derive spatiotemporal exciton distributions in films with arbitrary optical thickness and under noninstantaneous photoexcitation, paying particular attention to the effects of imperfect quenching and time-dependent diffusivity (i.e., subdiffusive transport). We demonstrate the utility of our model by applying it to a colloidal quantum dot (QD) thin film interface found in a recently reported record-efficiency QD light-emitting device. We find the exciton diffusion length in these CdSe/CdS core/shell QD films to be in the range 19−24 nm, in agreement with recent measurements of similar materials. We discuss limitations of the continuum-level analysis due to the finite size of individual QDs and an apparent subpopulation of “stationary” excitons that do not diffuse.

1. INTRODUCTION Excitons are bound electron−hole pairs that are formed when light is absorbed in certain materials, such as inorganic and organic molecular semiconductors,1,2 colloidal quantum dots (QDs),3 carbon nanotubes,4 conjugated polymers,5,6 molecular aggregates,7,8 and transition-metal dichalcogenides.9 These excitonic materials have a wide variety of optoelectronic applications, including light-emitting diodes (LEDs),10−13 solar cells,14,15 photodetectors,16,17 and lasers.18 Exciton transport is at the heart of operation of many of these devices. In the case of solar cells and photodetectors, the farther an exciton travels within its lifetime, the higher the chance of that photogenerated exciton reaching a charge separation interface prior to recombination.15,19 On the other hand, a shorter exciton propagation distance can improve the performance of some LEDs, where the emission efficiency is limited by exciton quenching at charge injection interfaces.20−22 In molecular semiconductors and QDs, exciton diffusion often proceeds by incoherent site-to-site hopping.1,2,23−28 When there are many exciton hopping steps such that the exciton’s mean free path is much greater than the lattice spacing, exciton transport can be described by a diffusion formalism, in which the length of exciton migration is known as the diffusion length (LD).29 The diffusion formalism has also been used to model charge carrier transport in traditional semiconductors30 and more recently in organo-metal-halide perovskites.31−33 While microscopic models such as kinetic © 2015 American Chemical Society

Monte Carlo (KMC) can capture a wider array of dynamic phenomena,27,34,35 these models require a larger number of known parameters such as the homogeneous and inhomogeneous line width, orientation of neighboring transition dipole moments, and the exact arrangement of molecular sites within the assembly. In many situations, the phenomenological concepts of exciton diffusivity and diffusion length are sufficiently descriptive and alleviate the need for microscopic detail. A variety of methods have been developed to extract exciton diffusion lengths and diffusion coefficients (D’s) to understand and to control energy transport in semiconductor materials, particularly for organic semiconductor materials.6,19,23,26,28,29,36−39 One of the most common techniques to measure the exciton diffusion length is based on device measurements, including photocurrent-based response of Schottky diodes40,41 and lateral architectures,38 and analysis of the thickness-dependent performance of solar cells19 and LEDs.42 We note that, in such approaches, optical fields near metal electrodes must be treated carefully, and the possibility of enhanced exciton quenching at the metal interfaces should be considered. 43 These device measurements are typically performed under steady state illumination. Received: December 18, 2014 Revised: April 3, 2015 Published: April 7, 2015 9005

DOI: 10.1021/jp512634c J. Phys. Chem. C 2015, 119, 9005−9015

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The Journal of Physical Chemistry C An alternative way to measure LD is by transient exciton− exciton annihilation, wherein a sample is irradiated with a pump laser pulse. Then, changes in photoluminescence (PL) or absoprtion dynamics are monitored as a function of pump intensity.36,44,45 The advantage of this approach is that it can be performed in situ and does not require fabrication of a heterostructure; however, there is uncertainty in D since the capture radius or the annihilation radius is difficult to find experimentally. Recently, a more sophisticated technique was developed to record both spatial and temporal distributions of excitons via time-resolved optical microscopy.39 The power of this approach lies in the ability to observe the evolution of the exciton diffusivity D as a function of time, revealing dynamic behavior that was previously unobservable. However, the approach requires sophisticated instrumentation, and the data analysis can be involved for short LD due to deconvolution of spatial broadening from the imaging system.27 Perhaps the most popular method for studying exciton diffusion is the exciton quenching experiment, in which the photoluminescence of an excitonic material is compared in the presence and absence of quenching sites.23,29,37,46−51 These experiments can be performed under either steady state29,37 or pulsed excitation,37,46−48 and may include quenching sites that are distributed homogeneously within the excitonic medium23,50,51 or localized at a surface or interface.29,37,46,47,49,50 In addition to excitons, transient photoluminescence measurements with a quenching interface can be used to characterize the diffusion of free charges.31,32 With the possible exception of time-resolved imaging,27,39 each of these experimental techniques offers an indirect view of exciton diffusion; therefore, they rely heavily on theoretical models to extract technologically important information such as exciton diffusivity and propagation distance. In this paper, we expand upon the continuum model that facilitates the extraction of such parameters from transient PL quenching experiments. We present both analytical solutions and numerical simulations, illustrating the spatial and temporal exciton distributions in thin films with arbitrary optical thickness and noninstantaneous photoexcitation, and devote particular attention to the effects of incomplete exciton quenching and time-dependent diffusivity.27 To demonstrate the usefulness of our models, we analyze the quenching of quantum dot (QD) thin film photoluminescence next to an adjacent ZnO layer found in a recently reported recordefficiency QD LED.21,22 In section 2, we introduce the physical model, derive analytical solutions for limiting cases, and present finite element simulations of spatiotemporal exciton profiles under varying conditions. In section 3, we demonstrate the success of our model by applying it to the analysis of transient photoluminescence quenching from QD films in a technologically relevant device architecture.

Figure 1. Schematic illustration of the transient photoluminescence quenching experiment in which the photoluminescence of the sample, a QD thin film with thickness L, is collected with and without an adjacent quenching layer at x = L.

constant. Using these approximations, we write a onedimensional diffusion equation ∂n(x , t ) ∂ 2n(x , t ) n(x , t ) = D(t ) − + G (x , t ) 2 ∂t τ ∂x

(1)

Because even the simplest functional forms for D(t) do not yield analytical solutions when inserted into eq 1, and because very often exciton diffusion closely mimics a random walk process,39 we assume constant diffusivity (D(t) = D) for the remainder of this section. We revisit the case of time-dependent diffusivity in section 3. 2.2. Exciton Distributions in One Dimension. Let us consider an initial exciton distribution that is generated instantaneously upon light absorption, following Beer’s law. For instantaneous excitation, the generation term G(x, t) in eq 1 is omitted, and we use instead an initial condition ⎛ αx ⎞ ⎟ n(x , 0) = G(x) = n0 exp⎜ − ⎝ cos θ ⎠

(2)

where α is the absorption coefficient at the excitation wavelength, θ is the incidence angle of the laser pulse with respect to the surface of the sample (see Figure 1), and n0 is dependent on the laser fluence. By adopting Beer’s law with a constant α at a given wavelength, we neglect any optical interference effects. It has been shown that transient quenching techniques are less sensitive to optical interference effects than steady state quenching experiments when using nonmetallic quenchers.37 Further, when the refractive index of the quenching layer at x = L is nearly the same as that of the corresponding blocking layer, optical effects can be neglected entirely. If the refractive indices are not similar, as is the case with metallic quenchers, then the Beer’s-law approach is insufficient, and the transfer-matrix method should be used.37,52 For the perfect quenching case, the effects of exciton blocking and quenching layers at x = 0 and x = L, respectively, appear as boundary conditions

2. THEORY 2.1. Model System and Governing Equations. The model system is illustrated in Figure 1. An excitonic material (a QD film or organic molecular film, for instance) with thickness L is deposited on a substrate, which can be coated with either an exciton blocking or quenching layer. If we assume dilute exciton density and a homogeneous material, diffusivity is practically independent of the exciton density and the location within the film; i.e., D(n, x, t) = D(t), where D is the diffusion

∂n(0, t ) =0 ∂x

(3)

n(L , t ) = 0

(4)

We have assumed that the film−air interface at x = 0 is inert (i.e., blocking or nonquenching). In the absence of any quenching, the boundary condition at x = L is replaced by (∂n(L, t))/(∂x) = 0. In both cases, the partial differential equation can be solved analytically using the finite Fourier transform method53 (see Supporting Information). 9006


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Figure 2. Finite element simulations of the spatiotemporal evolution of the exciton population subject to different boundary conditions at the quenching interface. (a) Spatial and temporal distribution of excitons with LD = 15 nm in an L = 30 nm thick film under perfect quenching (left), imperfect quenching with Da = 4 (middle), and no quenching (right) boundary conditions at x = L. (b) Simulated normalized photoluminescence decay profiles in an L = 30 nm thick film with diffusion length LD ranging from 0 to 60 nm are subject to perfect quenching (left) and imperfect quenching with Da = 4 (right). Insets show the same temporal distributions in log scale to emphasize near-single-exponential behavior. In both parts a and b, L = 30 nm, θ = 30°, τ = 5 ns, and α = 5.0 × 10−3 nm−1.

2.3. Generalized Boundary Conditions. A more general boundary condition that is valid for any type of interfacial process (perfect blocking, perfect quenching, or imperfect quenching) is written by equating the surface quenching rate per unit area with the diffusive flux of excitons to that interface D

∂n(L , t ) = kq n ∂x

Spatial profiles of exciton population density at different times for three types of quenching interfaces are shown in Figure 2a. Exciton distributions were obtained numerically using the finite element method as implemented in the COMSOL Multiphysics 4.3 package. For convenience, we let n0 = 1 so that n(x, t) is now the fraction of surviving exciton population. In this figure, we plotted n(x, t) for a 30-nm-thick film with an intrinsic exciton diffusion length of 15 nm. To be consistent with our transient PL quenching experiment that will be addressed in section 3, we let θ = 30°, τ = 5 ns, and α = 5.0 × 10−3 nm−1. The initial distribution of excitons is determined by both the quenching interface and the absorption coefficient as expected from eq 2, in which α creates an exciton population density gradient that drives excitons toward the quenching interface (for the experimental geometry indicated in Figure 1). As Da decreases from left to right in Figure 2a, the depletion rate of exciton population slows down; for example, at 3 ns, the spatially integrated fraction of surviving excitons (n(t)) is the ̅ highest for the smallest Da (no quenching case). In the perfect quenching case, there is a higher gradient of exciton population density leading to a faster diffusion process, following Fick’s law. 2.4. Relationship to Experimental Measurables. The experimentally measured instantaneous photoluminescence (PL) rate is proportional to the total exciton population within the film at a given time. To compare our model results to those from the transient PL quenching experiment, it is useful to define a spatially averaged exciton population density

(5)

where kq is the surface quenching rate constant, which has units of velocity. When kq is very small, the diffusive flux at that interface is approximately zero, and we recover the perfect blocking condition. When kq is large, the exciton population at x = L is effectively zero, n(L, t) ≈ 0, and we recover the perfect quenching condition. Equation 5 suggests the introduction of another dimensionless number called the Damköhler number (Da) Da =

kq L D

(6)

which is the ratio between the interfacial quenching rate (kq) and the diffusive transport rate (D/L). The Damköhler number (Da) often appears in chemical reaction engineering, and indicates the extent to which interfacial processes are transport limited or reaction rate limited.54 For example, Da = 10 means that the rate of quenching excitons at the interface is 10 times faster than the rate of exciton transport to the interface, so the overall rate will be limited by transport to the interface. Since there is no analytical solution for the imperfect quenching model, eq 5 needs to be solved numerically. Through numerical simulations, we found that, for Da ≥ 50, an interface can be considered perfectly quenched with less than 5% error in the total exciton population.

n ̅ (t ) =

1 L

∫0

L

n(x , t ) dx

(7)

For the unquenched film, the spatially averaged exciton population density is 9007


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The Journal of Physical Chemistry C n ̅ (t ) = n 0

⎛ αL ⎞⎤ ⎛ t⎞ cos θ ⎡ ⎟ exp⎜ − ⎟ ⎢1 − exp⎝⎜ − ⎥ ⎝ τ⎠ αL ⎣ cos θ ⎠⎦

LD ≈

(8)

As expected, we see from eq 8 that the average lifetime in the unquenched film depends only on its native lifetime and not on the film thickness or exciton diffusion length. The total number of light-emitting excitons per unit volume per laser pulse, N, is related to the average exciton density n̅(t) through the relation N≡

∫0

(9)

such that n0 =

N τ

cos θ ⎡ 1 αL ⎣

αL − exp − cos θ ⎤⎦

(

)

(10)

Using eqs 8 and 10, we arrive at the familiar expression for the transient exciton population in an unquenched film n ̅ (t ) =

⎛ t⎞ N exp⎜ − ⎟ ⎝ τ⎠ τ

(11)

On the other hand, in the thickness-dependent transient quenching case, the average exciton lifetime depends on both the native lifetime and LD/L in the form of infinite series as derived in the Supporting Information. 2.5. Estimating LD Using the Quenched Film Lifetime. For experimental convenience, it would be useful to have an expression for quickly estimating the exciton diffusion length, LD, by comparing the photoluminescence lifetime in the presence and absence of the quenching layer. Such an expression can be obtained by considering the limiting behavior of the infinite series solution for the perfect quenching case (see Supporting Information). For the experimental geometry shown in Figure 1, the useful experiment is one in which the thickness of the film is less than the absorption length (αL < 1, where αL is the thickness of the film described in terms of the number of absorption lengths). Under such conditions, an average exciton lifetime in the presence of interfacial quenching τq can be estimated as 2 −1 ⎡ π 2 ⎛ LD ⎞ ⎤ ⎜ ⎟ ⎥ τq ≈ τ ⎢1 + 8⎝ L ⎠⎦ ⎣

(13)

In section 3, we will show that eq 13 provides an excellent guess for LD with an error of less than a factor of 2 for our system. 2.6. Effect of Imperfect Quenching. Intuitively, one would expect that imperfect quenching should lessen the change in exciton lifetime observed when moving from the unquenched film to the quenched film. Consequently, imperfect quenching would lead us to underestimate the exciton diffusion length, LD, as calculated by eq 13. To illustrate this effect, we compare theoretical photoluminescence decay curves for perfectly quenched and imperfectly quenched (Da = 4) films in Figure 2b, with values of LD/L ranging from 0 to 2. Insets show the same temporal exciton population decay on a semilog scale. As LD increases, the overall exciton population decays faster. From the insets, we note that the decay dynamics for all values of LD appear linear in log scale for both perfect quenching and imperfect quenching. For the experimental conditions simulated here (αL/cos θ = 0.2), the decay curves are well-described by a single-exponential function. The simulations shown in Figure 2b illustrate that a significant change in exciton lifetime will be observed even if the exciton diffusion length is only half the film thickness. When the sample is 10 times thicker than the diffusion length (LD/L = 0.1), then even in the presence of a perfectly quenching interface, it will be difficult to detect a significant change in the exciton lifetime. When excitons are only partially quenched at the interface, changes in the exciton dynamic are less noticeable as the film thickness is varied, compared to the perfectly quenched case. For example, the decay curve at LD = 15 nm with a perfect quenching interface is approximately the same as that with an imperfect quenching interface at LD = 60 nm for a 30-nm-thick film. Thus, quantitative information regarding LD may be more difficult to obtain with imperfectly quenched films.

∞

n ̅ (t ) d t

⎞ 2L ⎛⎜ τ 2⎜ − 1⎟⎟ π ⎝ τq ⎠

3. APPLICATION TO EXCITON DIFFUSION IN A QUANTUM DOT FILM We apply our theoretical framework to the analysis of exciton diffusion in a colloidal quantum dot (QD) solid film. Specifically, we study CdSe/CdS core/shell QD thin films situated adjacent to a nanocrystalline ZnO layer, an interface found in a recently reported record-efficiency colloidal QD light-emitting device.21 Previous studies confirmed that exciton dissociation can occur at this type-II ZnO/QD interface, and interfacial charge dynamics were believed to play a central role in device operation.21,22 3.1. Experimental Methods. CdSe/CdS QD films on ZnO were provided by QD Vision, Inc., and used as received. Detailed sample preparation procedures can be found in the previous publication by Mashford et al.21 Briefly, QDs were made by traditional hot-injection synthesis of CdSe cores followed by slow growth of a CdS shell from solution. The resulting nanocrystals, which were approximately 6 nm in diameter and capped with a long-chain aliphatic ligand, exhibited bright and stable photoluminescence centered at λ = 613 nm (32 nm fwhm) with 80% photoluminescence quantum efficiency. Nanocrystalline ZnO films were prepared by spin-coating colloidal ZnO nanocrystals from solution onto a glass substrate, followed by a vacuum oven bake at 100 °C. The resulting ZnO films had an RMS roughness of 1.2 nm, as

(12)

Equation 12 is most accurate when the film thickness is much less than the optical absorption length, L ≪ 1/α. If an interface is known to be perfectly quenching, the error in eq 12 is less than 30% for αL < 0.2 and is greater than 80% for αL ≥ 1. As expected, τq is less than τ as long as LD > 0. In the presence of interfacial quenching, the spatial gradient of exciton density will be high, leading to a large net flux of excitons to the quenching interface. Therefore, the rate of decay for the exciton population, which is inversely proportional to LD/L, will be higher (or the lifetime will be shorter) than that with a nonquenching interface. Equation 12 provides a very useful expression to easily approximate LD from the transient PL quenching data, especially if the data fits well to a single-exponential function. Using the fitted τq and the known film thickness L, LD can be estimated by 9008


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decay curves for two different thickness films overlap completely (red and blue circles), indicating that neither the QD−glass interface nor the QD−nitrogen interface acts as a quenching interface. In contrast, for the quenched (i.e., QDson-ZnO) films shown in Figure 3b, the PL decays faster as the film thickness is reduced, and in all cases the QDs-on-ZnO PL decays faster than that of QDs-on-glass. Comparing the PL decay curves of QDs in unquenched films and in solution (Figure 3a), we observe that the PL intensity decays faster in solid film than in solution. This phenomenon was accompanied by a reduction in quantum yield going from solution to film. Previous authors have attributed this behavior to efficient exciton transport via nonradiative energy transfer (e.g., FRET) within the inhomogeneous size distribution of QDs, in contrast to that of the noninteracting QDs present in solution.11,24,25,27,35 It has been hypothesized that there could be “self-quenching” of excitons in QD films that occurs when an exciton hops to a nonluminescent site, such as a QD in its blinking “off” state.11,35 Additionally, the PL decay curves from the unquenched QD films appear biexponential (Figure 3a). Such behavior has previously been interpreted to indicate the presence of a subset of QDs that do not participate in energy transfer, likely due to the absence of proximal resonant acceptor QDs.24 This can be explained in the context of FRET; for example, when an exciton is created on a QD whose dipole moment is not aligned with its neighboring QDs, there would be a smaller probability of energy transfer or exciton hopping from the donor to the acceptor QDs. On the basis of this evidence, we fit our data using a twopopulation model, where the total exciton population is composed of two independent populations: excitons that participate in diffusion and those that are stationary. The two-population model can then be written mathematically as

determined by atomic force microscopy (AFM). QD films were prepared by spin-coating from colloidal suspension. Four different thicknesses of QD films on ZnO were prepared for the study: 15, 30, 45, and 60 nm. As a control group, two different thickness QD films (15 and 30 nm) were deposited onto a clean glass substrate. The QD films had an RMS roughness of 2.0 nm, as determined by AFM. All samples were prepared inside a nitrogen glovebox and sealed with a glass cover plate and UV-curable epoxy before exposure to ambient conditions for photoluminescence testing. QD films were excited with 100 fs duration pulses of λ = 525 nm laser light from an optical parametric amplifier (Coherent OPerA Solo), which was pumped by a 1 kHz Ti:sapphire regenerative amplifier (Coherent Libra). A 300 μm diameter spot on the sample surface was irradiated at an average fluence of 5.0 μJ/cm2, which was determined from power-dependent measurements to be below the threshold for multiexciton recombination or exciton−exciton annihilation (see Supporting Information). Photoluminescence was collected from the sample and reimaged onto the entrance slit of a streak camera (Hamamatsu) for time-resolved and spectrally resolved detection. The timing resolution varied depending on the temporal width of the streak scan window, but had an overall lower limit of ∼5 ps. 3.2. Transient Photoluminescence from Quenched and Unquenched Films. Photoluminescence (PL) decay traces for the quenched and unquenched films are shown in Figure 3. In Figure 3a, the unquenched (i.e., QDs-on-glass) PL

ntotal(t ) = (1 − w0)n ̅ (t ) + w0n0̅ (t )

(14)

where ntotal(t) is the total exciton population, n(t) is the ̅ population of diffusing excitons, n̅0(t) is the population of stationary excitons, and w0 is the fraction of of the total exciton population that is stationary. n̅(t) follows eq 8 for unquenched films and a more complicated infinite series form for the quenched films (see Supporting Information). We note that the two populations should have different effective lifetimes, as the diffusing population is subject to self-quenching by encountering “dark” sites in the film or undergoing nonradiative decay to trap states, whereas the stationary population should have a lifetime similar to that of QDs in solution. To determine the lifetimes, τ and τ0, of both diffusing excitons and stationary excitons, respectively, we fitted the normalized PL measurements of unquenched films to eq 14, which was in turn convolved with the measured Gaussian IRF. We minimized the sum of residuals from data sets with two different QD film thicknesses to solve for three unknown parameters: τ, τ0, and w0. We note that, for unquenched films, values of α and L do not contribute to the fitting quality because they only affect the magnitude of the intensity, and thus are canceled when the PL intensity is normalized to its maximum value (see Supporting Information). The fitted curves are shown in Figure 4. The solution data set had a lifetime of about 23 ns. From the unquenched films, we calculated that 1 − w0 = 65 ± 2% of the total exciton population is mobile, with an average lifetime of τ = 4.9 ± 0.2 ns. The remaining stationary excitons have an average lifetime of τ0 = 17.8 ± 0.7 ns, which is comparable to the solution

Figure 3. Normalized photoluminescence decay of the QD samples plotted in linear (left) and log (right) scales for (a) solution and unquenched films (15 and 30 nm thick), and (b) quenched films (15, 30, 45, and 60 nm thick). In part b, the data from the unquenched 15 nm thick film on glass is included for reference. Solid black line denotes the measured instrument response function. 9009


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respectively. A value of α = 5.0 × 10−3 nm−1 was used for the absorption coefficient in the QD film, assuming that 50% of the volume of the QD film is occupied by the semiconductor cores and with a bulk CdSe value56 at λ = 525 nm of α ≈ 1.0 × 10−2 nm−1. The spatially averaged exciton density n(t) was then ̅ found by simultaneously solving eqs 1, 2, 3, and 5 using COMSOL. A total of six parameters were solved for: four w0’s (one for each film) and two common parameters, D and kq. The fitted curves are plotted alongside the experimental data in Figure 5a. The diffusion coefficient was found to be D = (3.54 ± 1.46) × 10−4 cm2/s, and the quenching velocity at the QD− ZnO interface was kq = 6.5 ± 2.5 m/s, yielding Da ranging from 3 (at L = 15 nm) to 11 (at L = 60 nm). The corresponding exciton diffusion length was LD = (2Dτ)1/2 = 19 nm. 3.4. Effect of Time-Dependent Diffusivity. In Figure 5a, it can be seen that the fitted PL curves tend to overestimate the experimental data between t = 0 and 5 ns but underestimate the data between t = 5 and 15 ns. (The fitted curves continue to overestimate the data beyond t = 15 ns, but this time range is less important because it captures only a small surviving fraction of the initially created exciton population.) During the first 15 ns, the rate of decay appears to decrease rapidly, resembling subdiffusive behavior.57 Subdiffusive transport of excitons has been examined in both theoretical (conjugated polymers)34 and experimental (tetracene crystals and CdSe QD solids)27,39 studies, and is characterized by a decrease in exciton diffusivity over time, usually arising from orientation or energetic disorder within the system. We observed evidence of energetic disorder in our QD thin films via transient red-shift of the peak emission energy in time (not shown), which indicates that there is a fast downhill migration of energy transport to lower energy sites.25,27,58 In QD films, the exciton diffusivity is expected to be high initially and then decrease over time until the diffusivity saturates at a limiting value characteristic of a thermalized population. To incorporate this phenomenon, we introduce a time-dependent diffusivity

Figure 4. Comparison of PL dynamics in solution and unquenched film. Normalized PL intensities plotted in linear (left) and log (right) scales for solution and an unquenched film (30 nm). Open circles represent experimental measurements; solid curves represent a fit to the two-population model expressed in eq 14.

lifetime. The shorter lifetime of the diffusing excitons compared to that of to the static excitons is consistent with the underlying motivation for the two-population model: that the faster decay rate and lower quantum yield observed in QD solids are due to exciton diffusion to nonluminescent sites,11,35 and that the longer lifetime component of thin film PL decay is due to excitons that do not participate in energy transfer to neighboring QDs.25,35 3.3. Constant Diffusivity Analysis. Before considering more complicated models of exciton diffusivity,27 we first analyzed the transient PL quenching data assuming that D is constant, i.e., normal diffusion. As a starting point, we estimated LD using eq 13 and considering only the 1/e lifetime (τ = 8.4 ns for the unquenched film) and the thickness of each film. The effective lifetime τq and LD are listed in Table 1. The average Table 1. Diffusion Length Calculated on the Basis of Equation 13 and 1/e Lifetime of the Unquenched Film (τ = 8.4 ns)

a

L (nm)

τq (ns)

LD (nm)a

15 30 45 60

3.2 4.1 5.3 6.5

17 27 31 29

⎡ ⎛ t ⎞⎤ D(t ) = D0⎢1 + R exp⎜ − ⎟⎥ ⎢⎣ ⎝ τR ⎠⎥⎦

(15)

τ−1 R

Here, D(t) decays exponentially at a rate of and approaches an asymptotic value of D0 as t → ∞. The variable R is the relative enhancement of diffusivity at early time, R = [D(t = 0) − D0]/D0. For example, a value of R = 2 corresponds to a 200% increase in diffusivity initially compared its value at steady state, D0. Sometimes anomalous diffusion is explained as a power law,59 ⟨x2⟩ = Atβ, where ⟨x2⟩ is the mean square displacement, and A is a prefactor related to the magnitude of the displacement. β is the exponent that indicates the type of diffusion: β = 1 is normal diffusion, β < 1 is subdiffusion, and β > 1 is superdiffusion.60 For normal diffusion, ⟨x2⟩ = (LD)2 = 2Dτ. Thus, D(t) can also be expressed in a power-law form

av LD (nm) = 26.

value of LD estimated from all four films is 26 nm, which leads to D ∼ 10−4 cm2/s. Table 1 also shows that the calculated value of LD increases as the film thickness increases. Since LD is an intrinsic material property, this unphysical result is an indication that our assumption of a perfectly quenching interface is likely invalid. Others have previously shown that ZnO is a poor quencher of excitons generated in CdSe QDs,55 reinforcing our conclusion that an imperfect quenching model would be better suited to describe the CdSe QD−ZnO interface. To obtain a more rigorous estimate of the exciton diffusivity, we performed a global fitting analysis of all four data sets (L = 15, 30, 45, and 60 nm; Figure 3b) allowing for imperfect exciton quenching at the QD−ZnO interface (see Supporting Information for detailed fitting procedure). From the results of the unquenched film analysis, lifetimes of diffusing and static excitons were set as constants, τ = 4.9 ns and τ0 = 17.8 ns,

D(t ) ∝ t β− 1

(16) 34

However, as others have noted, the power-law form fails to capture the finite value of the diffusivity at long times. To perform the fitting, two additional parameters, R and τR, were allowed to vary, in addition to the six parameters that were allowed to vary in the case of constant D. Again, a global fitting procedure was used wherein the qualities of all four best-fit curves were assessed simultaneously to obtain self-consistent 9010


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Figure 5. Results of the full global fitting analysis assuming either (a) constant diffusivity, or (b) time-dependent diffusivity as expressed in eq 15. Best-fit parameters are listed in Table 2. In both parts a and b, L = 15, 30, 45, and 60 nm; θ = 30°; τ = 4.9 ns; τ0 = 17.8 ns; σ = 1.14 ns; and α = 5.0 × 10−3 nm−1.

Table 2. Summary of Results of the Full Global Fitting Analysis Assuming Either Constant Exciton Diffusivity (Left) or TimeDependent Diffusivity (Right)a D = constant

D(t) = D0[1 + R exp(−t/τR)]

L (nm)

D (10−4 cm2/s)

kq (m/s)

Da

15 30 45 60

3.54 ± 1.46

6.5 ± 2.5

2.8 5.5 8.3 11.0

w0 0.14 0.16 0.23 0.35

± ± ± ±

D0 (10−4 cm2/s)

R

τR (ns)

kq (m/s)

Da

1.34 ± 0.7

9.0 ± 3.4

2.0 ± 1.9

7.0 ± 3.3

1.0 2.1 3.1 4.1

0.01 0.01 0.01 0.01

w0 0.12 0.16 0.26 0.35

± ± ± ±

0.01 0.01 0.01 0.01

L = film thickness; D = exciton diffusivity; kq = interfacial quenching rate; Da = Damköhler number (ratio of diffusion time scale to quenching time scale); Da = time-averaged Damköhler number; w0 = fraction of exciton population that is stationary, or nondiffusing; D0 = limiting diffusivity (t → ∞); R = ratio of initial diffusivity to limiting diffusivity; τR = diffusivity decay time constant. a

values for D, R, kq, and τR. The best-fit curves are shown in Figure 5b, and the values of the fitted parameters are listed Table 2. D(t) begins at an initial value of 1.3 × 10−3 cm2/s and then decays with an exponential time constant of ∼2 ns to a saturation value of 1.3 × 10−4 cm2/s, an order of magnitude smaller than the initial diffusivity. Additionally, we fitted the data using the power-law expression for D(t) presented in eq 16. We obtained a bestfit exponent of β = 0.60, which is slightly smaller than those reported recently for CdSe/ZnCdS QDs via time-resolved microscopy (β = 0.67−0.73).27 For the case of constant diffusivity, the dimensionless Damköhler number is simply Da = kqL/D. For the case of time-varying diffusivity, the instantaneous Damköhler number is a function of time, Da = Da(t), so it is useful to compute a time-averaged Damköhler number over the exciton lifetime, τ Da =

1 τ

∫0

contribution of larger values of D at early times to the net displacement of the exciton at t = τ. Accordingly, the square root of the variance of the exciton distribution in one dimension (i.e., exciton propagation distance) at any time t is given by l(t ) =

∫0

t

D(t )̃ dt ̃

(18)

The diffusion length is defined as the propagation distance achieved when the population falls to 1/e of its starting value. For a single-exponential decay, the time at which the population falls to 1/e of its initial population is τ. Using the best-fit parameters for D0, R, and τR shown in Table 2, and the freely diffusing exciton lifetime of τ = 4.9 ns, the corresponding exciton diffusion length in the case of time-dependent diffusivity is LD = (2 ∫ τ0D(t) dt)1/2 = 24 nm. 3.5. Discussion. Exciton diffusivities for both the constant and time-dependent cases are compared graphically in the upper panel of Figure 6. As demonstrated previously,27 timedependent diffusivity gives a more accurate representation of the exciton mobility over time. This is evident in the quality of the fits to experimental data shown in Figure 5. While the constant diffusivity value of D = 3.5 × 10−4 cm2/s can be

τ

Da(t ) dt

2

(17)

where Da(t) ≡ kqL/D(t). These values are listed in Table 2. Similarly, the exciton diffusion length cannot simply be calculated as LD = (2Dτ)1/2 because the metric must properly take into account the history of the diffusivity and the 9011


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and time-dependent diffusivity analyses. This parameter can be cast into more intuitively accessible terms by converting it to a probability of quenching per visit by an exciton to a QD at the ZnO interface. The quenching rate per interfacial QD is kq′ = kq

kq (number of QDs per volume) ≈ (number of QDs per cross‐sectional area) d (19)

where d is the center-to-center distance between any two QDs. For an average d = 8 nm used in this study, kq′ = 0.88 ns−1. For comparison, the average exciton hopping rate along the xdirection (Figure 1) is

k hop,x =

L D2 d 2τ

(20)

where d is the distance traveled per hop (approximately equal to the center-to-center distance), and τ is the exciton lifetime (see the Supporting Information for a detailed discussion of dimensionality and the effect of homogeneous quenching sites). For LD = 24 nm and τ = 4.9 ns, khop,x = 1.8 ns−1. Because the hopping rate is faster than the interfacial quenching rate (kq′ = 0.88 ns−1), an exciton is more likely to hop to an adjacent QD than to be quenched when it visits the ZnO interface. Accordingly, the probability of quenching per visit to the QD layer nearest the ZnO interface is η = (kq′ )/(kq′ + khop,x + 1/τ) = 0.33 (see the Supporting Information for further discussion). Similarly, the fraction of stationary excitons, w0, was found to be the same for both the constant and time-dependent analyses at each film thickness, L. However, in both analyses the best-fit value of w0 monotonically increased as the film thickness increased. As discussed earlier in the context of eq 14, these “stationary” excitons likely represent a subset of QDs that do not participate in energy transfer, probably due to the absence of proximal resonant acceptor QDs.24 One possible explanation for the correlation between w0 and L is that, as L decreases, the surface area to volume ratio of the film increases, so a larger percentage of QDs are already at the quenching interface. For example, since the average center-to-center spacing is approximately 8 nm, half of the QDs in the 15 nm thick film are already in contact with the underlying ZnO film. Since all quenched excitons are considered as diffusing excitons in our two-population model, this leads to smaller values of w0 as L decreases. Consequently, the L = 60 nm value of w0 = 0.35 is probably closest to the true intrinsic value for the fraction of stationary excitons in this particular QD system. The scaling of w0 with film thickness highlights an inherent limitation of the continuum approach to modeling exciton transport in QD films: the finite size of the QDs themselves. In the classical Förster regime, exciton diffusion in QD solids most likely proceeds by site-to-site hopping.24,25,27,35 Since the hopping distance is equal to the QD diameter (plus ligand spacing), the continuum approximation is only sensible when both the film thickness L and the exciton diffusion length LD are much larger than the QD diameter. More physically representative models may be implemented via coarse-grained simulations, such as kinetic Monte Carlo.27,34,35 However, such methods present their own challenges since a larger number of known parameters are required, including the homogeneous and inhomogeneous line width, orientation of neighboring transition dipole moments, and the exact arrangement of molecular sites within the assembly. Despite the limitations of the continuum approach, the results are remarkably consistent

Figure 6. Comparison of exciton diffusivity and diffusion length under conditions of constant (blue) and time-dependent (red) diffusivity. (a) Diffusivity vs time. (b) Fraction of excitons surviving to a given propagation distance. The dashed line indicates 1/e of the initial exciton population, revealing the exciton diffusion length.

considered a time-averaged D(t), the importance of the initial large diffusivity (D(t = 0) = 13.4 × 10−4 cm2/s) is revealed in a plot of the fraction of excitons surviving to a given propagation distance, as shown in the lower panel of Figure 6. All excitons experience the large early time diffusivity, whereas only a small fraction live long enough to be subject to the small limiting value as t → ∞. Consequently, the calculated exciton diffusion length, LD = 24 nm, is larger for the case of time-dependent diffusivity than it is for the constant diffusivity case where LD = 19 nm. We note that these exciton diffusion lengths are comparable to those reported recently for CdSe/ZnCdS QDs via time-resolved optical microscopy (LD = 21−32 nm) and are slightly longer than exciton diffusion lengths observed in polymers (LD = 6−18 nm)61 and organic semiconductors (LD = 7−31 nm),29,61using the definition LD = (2Dτ)1/2 for these comparisons. However, these diffusion lengths are 2−3 orders of magnitude smaller than those reported for triplet excitons in molecular crystals such as tetracene39 and rubrene.38 While the fits to experimental data were sensitive to changes in D, L, kq, τ, and w0, they were not strongly affected by experimental parameters such as α (absorption length) and θ (angle of incidence). For instance, doubling the value of α from 0.005 to 0.010 nm−1 did not change the best-fit value of LD. These reassuring observations underscore the robustness of transient quenching techniques, and are consistent with previous authors’ claims that accurate knowledge of the optical absorption profile is not essential to the success of transient quenching analyses, whereas steady state quenching experiments are strongly affected by the optical absorption profile due to its effect on total light absorbed.37 Another reassuring observation is that the best-fit interfacial quenching rate, kq = 7 ± 3 m/s, was the same (within the margin of error, see Table 2) for both the constant diffusivity 9012


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where N has been previously defined in eq 9. In an excitonic solar cell, this number represents the fraction of photogenerated excitons that have successfully dissociated at a heterointerface. In an LED, this number reflects unwanted quenching of excitons at a charge injection layer. For the purpose of discussion, we consider an exciton population with constant diffusivity. The shape of the curves shown in Figure 7 depends heavily on the quenching efficiency and the exciton diffusion length. For a perfect quenching boundary condition, when the film thickness is equal to the exciton diffusion length, the fraction of excitons captured is 1 − 1/e ≈ 63%. As expected, a larger value of LD leads to a greater fraction of excitons captured at a given film thickness (Figure 7a). In fact, in order to collect ∼90% of the charges, L ≈ 0.4LD. Additionally, the fraction of excitons captured at the interface becomes smaller as the quenching efficiency at the interface decreases (Figure 7b).We note that these plots are applicable to other material systems as any differences in the radiative and nonradiative recombination rates are implicit to LD. For the CdSe QD/ZnO interface studied here, we would expect as much as 60% exciton quenching in the shortest film (L = 15 nm), which would severely limit light-emitting device efficiency. To combat these effects, recent improvements in QD LEDs have included the addition of an inert buffer layer such as poly(methyl methacrylate) between the ZnO injection layer and the luminescent QD layer to prevent interfacial exciton quenching.22

with results obtained via more realistic experimental and computational approaches.27 With the large number of fitting parameters (see Table 2) and the lack of analytical solutions to the diffusion equation for realistic quenching boundary conditions and diffusivity models (see section 2), it is worth noting the remarkable success of eq 13 in predicting the exciton diffusion length. Simply using the 1/e lifetime for the quenched and unquenched films and averaging over the four film thicknesses yielded LD = 26 nm, whereas a full parallel and computationally expensive global fitting to our time-dependent diffusivity model yielded LD = 24 nm. This agreement is, to some extent, coincidental since the effects of imperfect interfacial quenching are offset by the longer lifetime of the nondiffusing exciton subpopulation in this specific material system. Nonetheless, eq 13 is expected to be particularly accurate when the film thickness is comparable to the diffusion length and when interfacial quenching efficiency is near 100%. 3.6. Implications for Optoelectronic Devices. To understand the role of exciton diffusion in optoelectronic devices, particularly those that involve the CdSe QD/ZnO interface,21,22 we calculate the fraction of excitons ( f) that are ultimately quenched at the interface as a function of the film thicknesses and the diffusion length as shown in Figure 7. This

4. CONCLUSIONS In conclusion, we find that continuum modeling can provide reasonable analysis of transient photoluminescence quenching in quantum dot films. The advantages of this approach are (1) direct applicability to experiments, (2) the analytic predictions regarding the time scale and the length scale of diffusion versus quenching, and (3) relatively short computational time compared to the time needed for ab initio simulations and stochastic models. In particular, eq 13 can provide a quick and reasonably accurate estimate of the exciton diffusion length. Despite such successes, our model overlooks other physical phenomena that can be important to exciton transport in nanostructured thin films. For quantum dots, these phenomena include energetic disorder, spatial disorder of QDs, transition dipole orientation disorder, fluctuations in PL intensities and lifetimes known as blinking,62 and charging of QDs with free carriers. These effects can be studied in stochastic simulations such as kinetic Monte Carlo,27,34 and are the subject of further investigation.

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Figure 7. Fraction of excitons captured at the interface as a function of film thickness (a) under perfect quenching for LD ranging from 20 to 100 nm, and (b) in a comparison of perfectly quenched and imperfectly quenched conditions (kq = 6.5 m/s) for LD = 19 nm. The graphs were generated using best-fit parameters with constant D, θ = 30°, τ = 4.9 ns, and α = 5.0 × 10−3 nm−1.

* Supporting Information Full derivations, laser power dependence, details of the global fitting procedure, and further discussion of the exciton hopping rate and the effect of homogeneous quenching sites. This material is available free of charge via the Internet at http:// pubs.acs.org.

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quantity is related to the difference in the total number of excitons (N) that radiatively decay in the unquenched and quenched cases f=1−

AUTHOR INFORMATION

Corresponding Author

*Phone: +1 (617) 253-4975. E-mail: [email protected].

Nquenched Nunquenched

ASSOCIATED CONTENT

S

Notes

The authors declare no competing financial interest.

(21) 9013


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ACKNOWLEDGMENTS The authors thank B. Mashford, J. Steckel, and S. Coe-Sullivan of QD Vision, Inc., for preparation of QD samples, and V. Bulović for use of laser spectroscopy equipment and insightful discussions. This work was supported as part of the Center for Excitonics, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award DE-SC0001088 (MIT). E.M.Y.L. gratefully acknowledges fellowship support from the National Science Foundation Graduate Research Fellowship Program under Grant 1122374. Transient photoluminescence quenching measurements were performed in the MIT Nanostructured Materials Metrology Laboratory within the MIT Center for Materials Science and Engineering on equipment provided by the Eni-MIT Solar Frontiers Center.

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