CdS

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Determining the Complex Refractive Index of Neat CdSe/CdS Quantum Dot Films Dana B. Dement, Mayank Puri, and Vivian E. Ferry* Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Avenue SE, Minneapolis, Minnesota 55455, United States

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S Supporting Information *

ABSTRACT: Understanding how structural and synthetic factors influence the complex refractive index of quantum dot (QD) solids is crucial to tailoring the light−matter interactions of QD-containing photonic and optoelectronic devices. However, neat QD films are challenging to accurately model as they are a mixture of inorganic core/shell materials and surrounding organic ligands. Furthermore, both the QD film morphology and the complex refractive index vary due to particle size, ligand chain length, and the deposition process. Here, we study the complex refractive index of neat CdSe/CdS core/shell QD films by using variable-angle spectroscopic ellipsometry to derive the effective complex refractive index using Kramers−Kronig consistent dispersion models. We use this information in conjunction with intrinsic refractive index data of CdSe and CdSe/CdS QDs extracted from solution-state absorption data and effective medium approximations (EMA) to describe neat QD films. We find that EMAs can successfully be used as a tool to approximate the complex refractive index of QD films. This information allows us to better understand packing variations between QD films and more accurately predict the absorption in QD thin films, including those made with core/shell heterostructures.

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INTRODUCTION The unique optical properties of nanocrystal quantum dots (QDs) make them useful for many optoelectronic applications, including photovoltaic devices, LEDs, and sensors.1−6 With the increasing demand for solid-state QD devices also arises the need to accurately predict how light propagates through and interacts with these devices by characterizing the complex refractive index of the QD layer across the visible spectrum. When the complex refractive index of a QD solid is known, a variety of nanophotonic designs can be more accurately modeled and understood. Applications include the design of QD lasers and gain media,7 the prediction of photovoltaic device performance including calculations of exciton generation rates and quantum efficiencies,8,9 nanophotonic designs for absorption enhancement or light guiding,10−12 and interference engineering for improved photovoltaics and color filters.13 The ability to tune both the electronic and optical properties of the individual QDs, their matrix, and the QD spacing and organization in the solid state gives great freedom in device design. Unfortunately, while the many degrees of freedom within QD films grant them many unique features, they also make their complex refractive index challenging to accurately model as the resulting films depend on the properties of the individual QDs, their ligands, and the morphology of the resulting film. The QDs themselves can be made from single or © XXXX American Chemical Society

multicomponent materials in a variety of shapes, including dots, rods, and tetramers, depending on the structure of the underlying crystal lattice.14 Among these options, heterostructures such as core/shell QDs are particularly well-suited to nanophotonic applications due to the passivation of surface bonds and the added protection against photo-oxidation, leading, in many cases, to higher quantum yields and greater photoluminescence stability. The choice of core and shell material gives control over the confinement of the electron and hole wave functions, introducing additional device functionality.15,16 For example, the QD absorption and emission spectral overlap can be tuned to best suit the device of interest, helping to prevent reabsorption losses.17 In addition to the QDs themselves, the choice of ligands, as well as the way in which the QDs are deposited, can also be used to tune the properties of the collective quantum dot solid. While colloidal QDs are generally synthesized with long-chain organic ligands, these can be replaced with more compact, shorter-chain organic or inorganic ligands through ligand exchange methods.18 These shorter-chain ligands are often desirable because they can increase the optical density of the QD film and dramatically improve charge transport within the Received: May 12, 2018 Revised: August 17, 2018

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DOI: 10.1021/acs.jpcc.8b04522 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C solid.19 The QD film morphology also depends on the QD deposition method and size dispersity between the QDs; QD solids have been shown to exhibit short-range order, producing a glassy solid, or long-range order creating a QD superlattice.20 Both cases may be used within electronic and optoelectronic devices, with each case possessing its own unique properties. The dependence of solid-state properties on the QD synthesis and processing is further heightened in neat QD films as the individual particles are close enough that the strength of interparticle interactions may become significant.20,21 To date, estimates of the intrinsic complex refractive index of QDs have been performed on core-only materials, including PbS, PbSe, and PbTe nanocrystals, as well as CdSe and CdS nanocrystals.22,23 These studies use solution-state absorbance measurements with models that assume noninteracting, dilute particles. More recently, these methods were applied to find the complex refractive index of PbS QDs embedded in glass at volume fractions of less than 0.2% and in neat PbSe films with a volume fraction estimated between roughly between 30% and 40% by applying the Maxwell Garnett effective medium theory.24,25 To our knowledge, one study has utilized this method to describe core/shell QDs, but the calculated solution absorption coefficient values were not extended to predict the complex refractive index functions of films.26 While these methods can be very useful for understanding the intrinsic complex refractive index of isolated QDs, for high QD-loading fractions, it is unclear if these models are applicable.27 Variable-angle spectroscopic ellipsometry is an ideal experimental tool for obtaining the effective refractive index of solid-state QD films. Spectroscopic ellipsometry allows accurate measurement of the effective refractive index at high nanoparticle concentrations, when the QDs are in close proximity. This is accomplished by measuring the spectral reflectivity of a sample, which can then be used to construct a Kramers−Kronig consistent dispersion model. The dispersion model can be fit either by using an effective medium approximation (EMA), in cases where the individual material properties are known, or by using a series of oscillator models to describe the composite film. In both cases, the dispersion model should return the same spectral reflectivity as was experimentally measured. Currently, it is unclear what, if any, EMA models should be used for quantum dot films as previous studies on QD films have been either extended only to very dilute solids or applied at low frequencies.24,27 Effective complex refractive index measurements of QD films using oscillator models, however, have been made. Law et al. measured the effective complex refractive index of neat PbSe QD films to determine the quantum efficiency of QD solar cells.9 More recently, Diroll et al. used spectroscopic ellipsometry to characterize neat quantum dot films of CdSe and PbS and showed how different QD ligands and film annealing can lead to dramatic changes in the complex refractive index function.28 While spectroscopic ellipsometry measurement of the effective refractive index of neat QD films is a highly accurate way to characterize a specific QD film, it is difficult to extrapolate this information to describe QD films of different compositions made using varying deposition methods without performing additional spectroscopic ellipsometry measurements for each new film. Here we utilize solution-state absorbance data and variableangle spectroscopic ellipsometry to study both the intrinsic refractive index of QD nanocrystals as well as the effective refractive index of QD films. To carry out this study, a single

large-scale batch of CdSe QD nanocrystals was synthesized to reduce variability between films. This same batch of CdSe core nanocrystals was used to synthesize a family of CdSe/CdS core/shell QDs with varying CdS shell-thicknesses, ranging from, approximately, two to six monolayers of CdS. We first find the complex permittivity of the core CdSe nanocrystals using the solution absorbance data and an iterative minimization method.22,23 We then modify this method to describe CdSe/CdS core/shell materials. This allows us to compare the intrinsic refractive index of a series of CdSe and CdSe/CdS nanocrystals with identical core sizes. In parallel, by utilizing spectroscopic ellipsometry and Kramers−Kronig consistent dispersion models, we find effective refractive index data for neat QD films. Unlike the intrinsic refractive index, the effective refractive index is sensitive to morphology, as it accounts for the QDs, ligands, and void space which make up the overall film. With information about both the intrinsic and effective refractive index of our QD materials, we show that the Maxwell Garnett and Bruggeman EMA models are able to describe the major features of our QD films. We then extend the use of these EMA models to estimate the refractive index of neat CdSe and CdSe/CdS QD films, when the packing is held constant at 64%, and use this refractive index information to simulate the absorption in theoretical QD thin films. We find that this model both assists in understanding the experimental measurements of the complex refractive index of QD films and provides insight into the tunability of the effective optical properties of QD films via differences in processing.

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EXPERIMENTAL SECTION Chemicals. Cadmium nitrate tetrahydrate (98%, SigmaAldrich), myristic acid (98%, Acros Organics), sodium hydroxide pellets (98.9%, Fisher Chemicals), selenium dioxide (99.8%, Acros Organics), oleic acid (technical grade 90%, Alfa Aesar), 1-octadecene (90%, Acros Organics), octane (98%, Sigma-Aldrich), acetonitrile (99.5%, Sigma-Aldrich), dodecane (99%, Sigma-Aldrich), oleylamine (99%, Sigma-Aldrich), cadmium chloride (99.9%, Sigma-Aldrich), cadmium acetate dehydrate (Cd(Ac)2·2H2O, 98%, Sigma-Aldrich), and sodium diethyldithiocarbamate trihydrate (NaDDTC·3H2O, ACS reagent grade, Sigma-Aldrich) were all purchased from the stated vendor and used without further purification. Cadmium myristate and cadmium oleate were prepared on a multigram scale according to the method reported by Chen and coworkers.29 Cadmium diethyldithiocarbamate (Cd(DDTC)2) was synthesized according to Nan et al.30 Toluene (anhydrous, Alfa Aesar) and methyl acetate (anhydrous, Sigma-Aldrich) were further treated with 3 Å molecule sieves in a nitrogenfilled glovebox for at least 24 h prior to use. Synthesis of Zinc-Blende CdSe QDs. Zinc-blende CdSe QDs with oleic acid capping ligands were synthesized on a 5 g scale following the procedure reported by Owen and coworkers,31 based on the original synthesis by Chen and coworkers.29 Large-scale purification of the QDs was carried out following the method of Owen and co-workers,32 except that the resulting CdSe nanoparticles were only precipitated with methyl acetate once prior to undergoing dispersion in toluene or octane. Synthesis of CdSe/CdS QDs. CdSe/CdS core/shell particles with various shell-thicknesses were prepared using the same batch of CdSe cores, following a synthetic method reported by Pu et al.29,33 The synthetic procedure was B


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The Journal of Physical Chemistry C

Figure 1. (a) Histogram of CdSe and CdSe/CdS QD sizes as determined by TEM micrographs as shown. TEM scale bar is 20 nm in all micrographs. (b) Normalized photoluminescence (PL) emission spectra of the QDs in solution. (c) Extinction measurements of the QDs in solution, normalized to the 1st excitonic peak. Inset is scaled so that the first excitonic features are more easily seen. Both PL data and extinction show a red-shift as the CdSe QDs (blue) are coated with additional layers of CdS, either 2 (green), 4 (yellow), or 6 (red) monolayers.

QD Films. All QD films were spin-coated onto an Al2O3coated Si substrate. A 20 nm Al2O3 layer was deposited on Si via atomic layer deposition at 150 °C and was found to improve the wetting of the QD solution. Spin-coating and buildup of QD film thickness was achieved by using a solidstate ligand exchange procedure, where CdSe and CdSe/CdS quantum dots were dispersed in small amounts of octane and filtered using a 0.2 μm polytetrafluoroethylene (PTFE) syringe filter. The QDs were first deposited onto the substrate and then spun at 2000 rpm for 30 s. Next, 3-mercaptopropionic acid (MPA), at a concentration of 1% v/v in acetonitrile, was deposited onto the film and allowed to sit for 15 s to better facilitate the solid-state exchange. After this time, the sample was spun again at 2000 rpm for 15 s. Finally, the film was rinsed twice with pure acetonitrile by spinning at 2000 rpm for 10 s each. This process was repeated 3−5 times until the desired QD film thickness was achieved. Spectroscopic Ellipsometry. Spectroscopic ellipsometry measurements were all taken in reflection mode on a J.A. Woollam Vase ellipsometer. Data was collected at a minimum of four angles (60°, 65°, 70°, and 75°) from 350 to 1000 nm using dynamic averaging and zone average polarization measurement settings. Characterization of the bare Al2O3coated Si substrate was performed first, followed by characterization of the QD film. Three or more different locations were sampled on each QD film. Reflectance data was described with Kramers−Kronig consistent dispersion models, as described in the main text using J.A. Woollam WVase software.

modified to include a 4:1 molar ratio of cadmium diethyldithiocarbamate/cadmium oleate for core/shell QDs of all shell-thicknesses, rather than just for those with eight or more monolayers of CdS as stated in the original procedure. Additionally, a final surface treatment of cadmium chloride, rather than cadmium formate, was used to passivate the QD surface. To carry out this treatment, cadmium chloride was dissolved in octylamine and injected via syringe into the reaction mixture at 50 °C under nitrogen. The QD solution was then allowed to stir for 5 min at which point it was cooled to room temperature, transferred to a glovebox for purification, and dispersed in octane. UV−Vis Spectroscopy Procedure. UV−vis spectroscopy experiments were carried out using a 1.0 cm quartz cuvette containing either a dilute solution of CdSe in toluene or CdSe/ CdS QDs in octane. The UV−vis spectroscopy measurements were carried out with either an Agilent Cary 5000 or Cary 7000 UV−vis−NIR spectrometer at room temperature. Transmission Electron Microscopy Procedure. TEM samples were prepared by drop-casting a dilute solution of CdSe or CdSe/CdS QDs dispersed in toluene or octane onto a 300 mesh copper TEM grid supplied by Ted Pella, Inc. TEM experiments were run on a Tecnai T12 microscope at an operating voltage of 120 kV. Sizing of the nanoparticles was carried out using the PEBBLES software34, and images were produced using ImageJ software. For CdSe/4CdS and CdSe/ 6CdS, the nanocrystal shapes took on slightly prolate characteristics. To size these particles, TEM micrographs were fit to ellipsoidal shapes and the long axis was used to estimate the CdS monolayer thickness. C


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RESULTS AND DISCUSSION In order to accurately compare the characteristics of both CdSe QD films and various CdSe/CdS core/shell QD films, we synthesized a single, 5 g batch of CdSe core nanocrystals using a noninjection method.29,33 This single batch of CdSe core nanocrystals was then subsequently used for preparing CdSe/CdS QDs of varying CdS shell thickness, where CdSe/ XCdS represents a batch of CdSe/CdS core/shell QDs with a X monolayer CdS shell thickness. On the basis of UV−vis absorption and TEM size fitting using PEBBLES software,34,35 the diameters of the QD nanocrystals were determined. It was found that the CdS shell thickness ranged from approximately two to six monolayers surrounding 4 nm diameter CdSe cores (Figure 1a). Extinction and photoluminescence (PL) emission measurements were taken for CdSe and CdSe/XCdS nanocrystals, confirming that the QDs all show a narrow size dispersity, consistent with TEM data (Figure 1b,c). As expected, a red-shift is observed between the CdSe core’s first excitonic feature at 572 nm and the CdSe/XCdS core/ shell first excitonic features at 591, 613, and 618 nm for CdSe/ 2CdS, CdSe/4CdS, and CdSe/6CdS nanocrystals, respectively, due to increasing delocalization of the CdSe electron wave function.15 By measuring the absorbance spectra of dilute solutions of QD nanocrystals using a UV−vis spectrometer, we can obtain the intrinsic refractive index of isolated QD particles in a manner similar to those reported elsewhere.7,8 In bulk materials, the absorption coefficient, α, is defined as α(λ ) =

4πk(λ) λ

Figure 2. (a) Absorption coefficients as a function of wavelength for bulk zinc-blende CdSe (gray) and for CdSe QDs in toluene as determined by the Maxwell Garnett mixing rule and accounting for field screening (blue). Also shown is a comparison of our effective CdSe QD absorption coefficient values to those of zinc-blende CdSe QDs reported elsewhere (yellow).39 (b) The real (red) and imaginary (blue) portions of the CdSe QD permittivities found after extracting absorption coefficient data from solution measurements and performing a Kramers−Kronig iterative procedure are compared to those of bulk CdSe (gray). Bulk values were extracted from Ninomiya et al.50

(1)

where k is the imaginary portion of the refractive index of the nanocrystal. However, in a colloidal dispersion, the electric field within the inclusion will change on the basis of the refractive index of its surroundings. For QD systems, the QD inclusion will generally have a higher refractive index than the surroundings. This leads to dielectric screening and a reduction in the local electric field inside the QD relative to the external electric field. As shown by Ricard et al., this screening can be accounted for by using the Maxwell Garnett mixing rule and defining the absorption coefficient, μ, of spherical inclusions in solution over the wavelengths of interest, λ, as36,37 μ(λ) =

findings.39 The calculated function μ(λ) is dependent on both the real and imaginary permittivities of CdSe QDs, and therefore, it is not straightforward to solve for each component separately. However, the real and imaginary permittivities must also uphold the Kramers−Kronig (KK) relation which is derived from the requirement that a material cannot respond to an applied electric field prior to the application of the field. The relation is given by

2π |f (λ)|2 εIm(λ) λns LF 4

9ns 2π = εIm(λ) λns (εRe(λ) + 2ns 2)2 + εIm(λ)2

2 π

∫0

∞

E′εIm(E′)

dE ′ (3) E′2 − E2 Here, E is energy, and ε∞ is the value of εIm(∞). By using a gradient minimization process, the discrete form of the KK relation,40 and fixing the imaginary permittivity values to those of bulk CdSe at wavelengths shorter than 350 nm, we can obtain the real and imaginary values of the permittivity through an iterative process adapted from those used elsewhere.7,8 Further details about the iterative process can be found in the Supporting Information. The resulting intrinsic permittivity values for our CdSe QDs after seven iterative steps (steps shown in Figure S1) are shown in Figure 2b along with values of bulk CdSe. Our permittivity values are consistent with those of zinc-blende CdSe QDs previously reported.23 Having derived the intrinsic refractive index of the CdSe QDs from solution measurements, we next prepared neat thin films of CdSe QDs to find solid-state experimental values of the effective refractive index values from spectroscopic εRe = ε∞ +

(2)

Here, f LF is the local field factor, ns is the refractive index of the surrounding, and εRe and εIm are the real and imaginary permittivities of the suspended material. For CdSe QDs, it can be shown that, at wavelengths ≤350 nm, quantum confinement is weak and μ of the CdSe QDs in solution approaches that of bulk-like inclusions of CdSe in solution (i.e., the value of μ found by using the permittivity of bulk CdSe). This allows the absorbance of the CdSe QD solution to be appropriately scaled to give μ of the CdSe QDs in solution as a function of wavelength. Figure 2a compares μ(λ) of bulk CdSe found using eq 1 to that of CdSe QDs dispersed in toluene using eq 2 with ns of toluene taken as 1.5.38 Absorption coefficient values reported for use in zinc-blende CdSe QDs sizing are also shown in Figure 2a and are in close agreement with our D


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Figure 3. (a) Plot of raw spectroscopic ellipsometry reflectance measurements of Ψ (dashed) and Δ (solid) for a neat CdSe QD thin film taken at 60° (blue), 65° (green), 70° (yellow), and 75° (red). (b) Breakdown of dispersion models used to fit raw ellipsometry data for the CdSe film. The red, yellow, and green curves are Gaussian oscillators, while the blue curve is a Tauc−Lorentz oscillator, added in the UV portion of the spectrum to account for other high-energy transitions. The sum of these oscillators gives the final imaginary permittivity function of the film, shown in black. The real (c) and imaginary (d) effective refractive index of the CdSe QD films. The dashed line shows values extracted from the raw ellipsometry data shown in part a while the values in blue and red were found by combining intrinsic refractive index values for isolated QDs in solution with a Bruggeman or Maxwell-Garnet EMA, respectively, using the packing fractions listed.

ellipsometry. To prepare the neat QD films, concentrated solutions of QDs dispersed in octane were spun-cast onto Al2O3-coated Si substrates and subjected to a solid-state ligand exchange procedure adapted from Labelle et al.41 The ligand exchange step allowed the QD film thickness to be built up with successive spin-coating steps, as the oleate ligands are replaced with short (0.6 nm), polar, 3-mercaptopropionic acid (3-MPA) ligands. The use of 3-MPA also led to closer QD packing in the QD films, a quality that is desirable in many device designs.2 Fourier transform infrared spectroscopy (FTIR) results before and after solid-state ligand exchange support the loss of the native oleate ligands and the introduction of the new 3-MPA ligands onto the surface of the QDs (Figure S2). After deposition, film thickness was measured using either profilometry or atomic force microscopy (AFM) and used in the dispersion model for spectroscopic ellipsometry fitting. The thickness of the films ranged from 25 to 60 nm, with each individual film having a percent deviation of 5% or less between the regions measured by spectroscopic ellipsometry (Figure S3). For spectroscopic ellipsometry measurements, data was collected at angles between 60° and 75° over a wavelength range 350−1000 nm in reflection mode. The raw ellipsometry data measured from the CdSe film is given in Figure 3a. To model the experimental reflectivity data and extract the effective complex refractive index, the spectral data was split into two regimes. At low energies, where absorption is negligible (750−1000 nm), a Sellmeier fit was initially used. This allowed the QD film thickness, with an initial guess given

by AFM or profilometry data, to be refined as the imaginary portion of the permittivity was negligible in the region. Thickness variation between the initial guess and the final ellipsometry model generally varied by 5 nm or less. After the transparent region was fit, the QD film thickness was fixed and the wavelength range was slowly expanded so that spectral features above the band gap energy could be fit using Kramers−Kronig consistent dispersion models. Gaussian oscillators were chosen to describe the absorption features in the visible region, as they better describe the slight polydispersity present in the QD ensemble and any resulting inhomogeneous line-broadening when compared to Lorentz oscillators. In the UV region of the spectrum, past the measured range, further high-energy features were grouped together and described by a Tauc−Lorentz oscillator. A Tauc− Lorentz oscillator was chosen as it accounts for the band gap of the QDs and prevents absorption contributions past the band gap. The final dispersion model for the neat CdSe QD film consisted of the sum of four individual oscillators as shown in Figure 3b. For this dispersion model, the resulting meansquared error (MSE) of the standard deviation between the measured and generated reflectivity data was 1.2 (Figure S4). Once the effective refractive index values of the CdSe QD film were obtained from the experimental spectroscopic ellipsometry measurements, we studied how well the solution-extracted intrinsic optical constants, when used in conjunction with an EMA, could recreate the measured effective refractive index values across the visible. We chose to compare results from the two most common EMAs, the E


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can be seen that the shapes of the first excitonic peak and higher-order peaks are well-defined, and both the spectroscopic ellipsometry dispersion models and the solutionextracted EMA models have similar feature shapes. A slight 2 nm blue-shift of the excitonic features is observed in the spectroscopic ellipsometry data as compared to the features in the EMA models. We attribute this to slight etching of our particles after exposure to 3-MPA, as confirmed by UV−vis measurements after solution-state ligand exchange (Figure S5). For the limiting case we studied, the error in k, when using the above EMA models, was never greater than 20% when using the Bruggeman model or 13% in the Maxwell Garnett case for any given wavelength. In contrast, absorbance enhancement by as much as 400% has been claimed previously for CdSe monolayers due to dipolar coupling.46 More recent spectroscopic ellipsometry studies of CdSe thin films predict more modest enhancements in absorbance of 10% with an inorganic CdSe QD packing fraction of 27%.28 Absorption enhancement depends strongly on the film packing fraction, details of the ligand structure interdigitation, and QD polydispersity.25 Our findings of at most a 20% increase in QD film absorption support only modest absorption enhancements in our films as compared to solution measurements. We note that, in our neat films, there is no significant spectral shift of the absorption features when going from solution to the solid state. In cases where dipole coupling is significant, coupled dipole models can be used to describe the spectral changes that may be present.46,47 Further understanding of ligand density between QDs may lead to more accurate descriptions of the QD surroundings, which could provide improved descriptions of the extinction coefficient of the ligand and refine the model. For example, we also chose to explore whether or not a global minimum in the difference between the ellipsometry data and the EMAs could be found when both the packing fractions of QDs and the surrounding refractive index (ranging from 1 to 1.49) were allowed to vary. For the Maxwell Garnett model, the overall best agreement was found at a packing fraction of 41% and a surrounding refractive index of 1.42 and very closely replicates the ellipsometry results (Figure S7). The Bruggeman model produced no such ideal case where both n and k closely followed ellispsometry data. As a comparison case for the packing fraction results from the EMA models, a random-close-packed hard-spheres model (which assumes a 64% packing) was used. QD spacing due to the bound ligand shell was taken as 1.2 nm as measured via TEM. The random-close-packed model was chosen because it has been shown to be a reasonable packing estimate for glassy films of CdSe QDs.48 It was found that the EMA predictions matched closely the results of the inorganic fraction given by the hard-sphere model, which predicted 29% CdSe in the QD film. The EMA model also predicted packing fractions similar to those found for glassy QD films estimated elsewhere.25 Next, neat core/shell CdSe/CdS films were studied using a method similar to that used for the CdSe films. In contrast to the CdSe QDs, the absorption coefficient of CdSe/CdS core/ shell particles is dependent not only on the refractive index of the surroundings, core, and shell materials but also on the volume ratio of the CdSe to CdS. We compared two methods to solve for the absorption coefficient of the core/shell nanocrystals. In the first case, we used an expression for core/ shell inclusions proposed by Neeves et al.49

Maxwell Garnett approximation and the Bruggeman approximation. In the Maxwell Garnett model, it is assumed that spherical inclusions are well-separated within a host matrix, as is generally used to describe composites with low inclusion volume fractions. The Maxwell Garnett model is given by42 εQD ̃ − εs̃ ε ̃ − εs̃ = fQD ε ̃ + 2εs̃ εQD ̃ + 2εs̃

(4a)

where f QD is the volume fraction of QDs and ε̃QD, ε̃s, and ε̃ are the complex permittivity functions for the QDs, the surroundings (i.e., ligands and void space), and the overall composite, respectively. The second model followed the two component Bruggeman form given by43,44 fQD

εQD ̃ − ε̃ εQD ̃ + 2ε ̃

+ fs

εs̃ − ε ̃ =0 εs̃ + 2ε ̃

(4b)

where fs is the volume fraction of the surroundings. The Bruggeman model makes no distinction between the inclusions and the surroundings and instead treats them symmetrically. It is often used when the inclusion volume fraction is high. For neat QD films, we chose to explore both models because, while neat QDs films do have a relatively high packing fraction, they also have inclusions that are well-separated and, therefore, not fully interchangeable with the surroundings. Further, for randomly packed materials, there is evidence that the Maxwell Garnett model remains accurate up to packing fractions of 40%.45 In both models, rather than solving for ε̃, the effective permittivity of the composite, as is commonly done, we instead used the effective value of the complex permittivity found from spectroscopic ellipsometry (ε̃) and the solution-extracted intrinsic permittivity for ε̃QD to solve for the packing fraction of quantum dots in the film. As there was uncertainty in both the exact packing fractions of the QDs and in the refractive index of the surrounding material (ranging between 1, for the case of complete void space, and 1.49 in the case of being fully surrounded by 3-MPA ligands), we chose to examine the case where the extinction coefficient (and resulting absorption coefficient) would exhibit the greatest underestimation. To do this, the theoretical minimum packing density of QDs was extracted by assuming that the surrounding medium was made up of entirely free 3-MPA ligands with a real refractive index, ñs, of approximately 1.49, rather than void. This assumption produces a model where the values of the imaginary portion of the refractive index, to which only the inorganic QDs contribute, are at their lowest point relative to those of the real portion of the refractive index. The packing fraction of quantum dots in the film is obtained by fitting each EMA model to the nonabsorbing portion of the effective permittivity found from spectroscopic ellipsometry, between 700 and 1000 nm. Using this method, the Maxwell Garnett case predicted a 35% inorganic QD fraction while the Bruggeman model predicted a 32% fraction. Both results are similar to those predicted for glassy QD films as discussed below. The packing fraction found by applying this assumption is a lower bound that allows any absorption enhancement in the film to be most easily seen. Once the estimated packing fraction of QDs is known on the basis of the fit in the nonabsorbing region, the EMA model is then applied across the entire spectral range, and the resulting EMA functions for n and k are compared to those from spectroscopic ellipsometry measurements. From Figure 3c,d, it F


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Figure 4. (a) Plot comparing the real and imaginary CdSe refractive index (dashed blue line) to that of CdSe/CdS QDs (red, yellow, and green lines) found after extracting absorption coefficient data from solution measurements and performing a Kramers−Kronig iterative procedure. (b) Plot comparing the real and imaginary effective CdSe thin film refractive index (dashed blue line) to that of CdSe/CdS QD films (red, yellow and green lines) measured via spectroscopic ellipsometry.

μ(λ) =

2π Im(3εsβ ) λns

β (λ ) =

εsh̃ (λ)εã − εsεb̃ εsh̃ (λ)εã + 2εsεb̃

Figure S8. Because the refractive index contrast is low, a Bruggeman EMA can accurately describe the bulk CdSe/CdS composite and also takes into account dielectric screening of the CdSe at high energies. The core/shell particles were dispersed in octane for absorbance measurements, and ns of octane was assumed to be 1.40 in both methods. The absorption coefficient values for the CdSe/CdS QDs found using either of these two methods, with QD absorbance scaled to 325 nm, were nearly identical (Figure S9), with a difference of less than 2.5%. We also compared the Bruggeman method to a linear combination method as used elsewhere for describing core/shell materials and found close agreement.26 After obtaining the behavior of a bulk CdSe/CdS composite, the intrinsic optical constants for the CdSe/CdS QDs were treated as those of a composite material and were found in the same manner as that of CdSe, again using eqs 2 and 3. By describing the core/shell heterostructure with a single composite complex refractive index term, we extracted the effective real and imaginary permittivities using the same method as that for CdSe QD cores. The iterative permittivity values are shown in Figure S10. For the effective refractive index of core/shell QD films, spectroscopic ellipsometry was again utilized with dispersion models found in the same manner as that of CdSe films. Because of the increased number of energy transitions at higher energies from CdS, the CdSe/ CdS films generally required four to five Gaussian oscillators across the visible in contrast to three for those of CdSe. The MSE was below 6 for all neat core/shell films reported here. Figure 4a compares the solution-state extracted intrinsic optical constants of the CdSe QDs to those of the CdSe/CdS QDs. Several interesting features are seen. As with the CdSe cores, the excitonic features remain clearly visible, though the

(5)

ij V yz V εã (λ) = εc̃ jjjj3 − 2 sh zzzz + 2εsh̃ sh j z V V QD { QD k ij V yz V εb̃ (λ) = εsh̃ jjjj3 − sh zzzz + εc̃ sh j VQD z{ VQD k

where ε̃c, ε̃sh, and ε̃s are the complex permittivities of the core, shell, and solvent materials and Vsh is the ratio of the QD shell VQD

volume to the total QD volume, as determined by TEM sizing. This method takes into account dielectric screening of the CdSe core and makes solving for the absorption coefficient of the QDs more straightforward but adds additional complexity to the iterative extraction method required to solve for the QD permittivity values. Because of the low refractive index contrast between the materials, we also treated the QDs as a single effective QD composite by applying eq 2 as we had in the core QD case. For the core/shell materials, because both CdSe and CdS contribute to absorption at high energies, the absorbance cannot be scaled to a pure bulk component as was done with CdSe. Instead, the high-energy CdSe/CdS composite was described using a Bruggeman two component model with relative volume fractions, found by taking the volume of a 2, 4, or 6 monolayer CdS hollow sphere relative to the volume of a 4 nm diameter CdSe sphere, and bulk permittivities shown in G


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Figure 5. Comparison of the real and imaginary refractive index values of QD films found using solution absorbance data and a Kramers−Kronig iterative procedure with either a Bruggeman model (blue) or Maxwell Garnett model (red) to approximate spectroscopic ellipsometry data (black dashed) of neat QD films.

wavelengths, the n and k values of the core/shell QDs start to converge. Interestingly, the effective n and k functions show that the overall magnitude of the refractive index of the CdSe QD film is much lower than expected compared to that of the CdSe/ CdS QD films as predicted from both the intrinsic refractive index values and bulk values. This can be explained by accounting for the change in the size of the nanocrystals, as the fraction of the film that consists of inorganic QD material depends on the size of the nanocrystal. To study this further, we again utilize EMA models to predict packing fractions in the QD films and compare them to those of the random-close packing models while also taking into account the slightly elongated nature of the thickest core/shell structures. Using the Maxwell Garnett model as before and treating the CdSe/ CdS as a single composite material we found that, when assuming the surroundings of the QDs were free 3-MPA ligands (the point at which inorganic QD packing should be at a minimum), the lower packing fraction limits for CdSe/2CdS, CdSe/4CdS, and CdSe/6CdS films were 43%, 40%, and 40%, respectively. Likewise, using the Bruggeman EMA, we found the packing fractions were approximately 2% lower than those predicted by the Maxwell Garnett model, at 40%, 38%, and 38% for CdSe/2CdS, CdSe/4CdS, and CdSe/6CdS films, respectively. The resulting fits of the EMA models and the spectroscopic ellipsometry data are shown in Figure 5. These estimated core/shell packing fractions are significantly higher than the 32−35% limiting value predicted for the CdSe film. These packing values are again compared to those of a theoretical random-close-packed model. The random-closepacked model is a simple approximation that assumes hard spheres and does not account for ligand interdigitation, differences in processing conditions, film cracking, particle attachment, or polydispersity in the QD’s size. Nevertheless, it does give a reasonable estimate for the packing we should expect in our films and the general trends that occur as we go from cores to larger and more elongated core/shell QDs. On the basis of TEM analysis of the QD shapes, 64% random-close

magnitude of the refractive index for core/shell QDs is lower than that of CdSe QDs. Throughout the visible spectrum, bulk CdSe has higher values of n compared to those of bulk CdS, which is consistent with the higher values of the intrinsic refractive index for the CdSe cores compared to those of the core/shell particles.50,51 Additionally, the magnitude of the extinction coefficient near the first excitonic feature dramatically decreases as the shell thickness increases from the CdSe cores to those of CdSe/2CdS and CdSe/4CdS. However, the magnitude of the extinction coefficients of CdSe/4CdS QDs compared to those of CdSe/6CdS particles stays relatively constant. This trend can again be explained by looking at bulk extinction coefficient values for zinc-blende CdS, whose band gap is approximately 2.39 eV (∼518 nm).52 For the core/shell QDs, at wavelengths longer than the CdS band gap position, the refractive index features are dominated by the CdSe features (Figure S8). Initially, CdSe makes up a large fraction of the QD film and the excitonic features are prominent, but as the volume fraction of CdSe is decreased relative to that of CdS, these features become much weaker. In contrast, at short wavelengths, both CdSe and CdS contribute to the QD extinction coefficient, meaning there is a less dramatic change as the QD composition transitions from pure CdSe to mostly CdS. We next compared the effective complex refractive index of the neat CdSe/CdS films found using spectroscopic ellipsometry on deposited films, as shown in Figure 4b. Compared to the intrinsic refractive index values, which include only the inorganic portion of the QDs, the magnitude of the effective refractive index for all QD films studied is much lower. This is because the effective values include the inorganic portion of the QDs, their ligands, and the surroundings within the films. As found in the intrinsic refractive index values, there is a red-shift in the excitonic features as the shell thickness increases from CdSe/2CdS to CdSe/6CdS, along with a decrease in the magnitude of the refractive index at energies lower than the band gap of CdS. We also see that, at short H


Article


Table 1. Theoretical Percent Volume of Materials in Neat QD Films Compared to Those Found Using EMAs and Effective Ellipsometry Data MLs CdS

QD aspect ratio

packing fraction

bound ligand

CdSe

CdS

random-close-packedinorganic

Bruggeman EMA inorganic

Maxwell Garnett EMA inorganic

0 2 4 6

1 1 1.2 1.3

64 64 69 70

35 28 26 22

29 13 8.1 5.2

0.0 22 35 37

29 36 43 45

32 40 38 38

35 43 40 40

Figure 6. (a) Plot comparing the simulated absorption of 100 nm thick QD films using effective refractive index values measured via spectroscopic ellipsometry. Inorganic QD content is estimated to be 32%, 40%, 38%, and 38% for films of CdSe, CdSe/2CdS, CdSe/4CdS, and CdSe/6CdS QDs. (b) Plot comparing the simulated absorption of 100 nm thick QD films using theoretical effective refractive index values found using a Bruggeman model and solution-extracted intrinsic QD refractive index data. The inorganic QD content is set at 33%, 38%, 42%, and 45% for films of CdSe, CdSe/2CdS, CdSe/4CdS, and CdSe/6CdS QDs, respectively.

higher packing fraction than the other films. However, we believe this highlights the advantage of using EMA models to understand film packing as differences in processing conditions and any resulting film variability are otherwise difficult to track. In the limiting case, taking the surroundings again as only ligands where QD packing is lowest, we see that the EMA models are able to closely describe the real portion of the refractive index throughout most of the visible spectrum. However, in our limiting case of high index surroundings, EMA models consistently underestimate the extinction coefficient of the QD films near the peaks of CdSe features. For the CdSe film, the maximum underestimation was 12% while in the core/shell films it is 38% at some wavelengths. Given the complexity of the core/shell system it is expected that greater error is seen when it is compared to the core-only case. However, we believe the insight these EMA models give into QD film packing and optical behavior shows that they can indeed be useful tools to interpret QD film behavior. To show how these EMAs can be utilized to design and understand absorption in quantum dot devices, we simulated the absorption in 100 nm thick neat films of QDs on Al2O3coated Si. The absorption was calculated using finite difference time domain simulations. We first calculated the absorption using the experimentally measured effective refractive index values obtained using spectroscopic ellipsometry. The absorption, A, of a plane wave source was isolated to the QD layer only and calculated after solving for the electric field distribution in the film via

packing is assumed for the spherical CdSe/2CdS QDs, 69% random-close packing is assumed for the CdSe/4CdS QDs with an aspect ratio of 1.2, and 70% random-close packing is assumed for the CdSe/6CdS QDs with an aspect ratio of 1.3.53,54 In our model, these packing fractions are made up of the inorganic QDs along with a 0.6 nm 3-MPA bound ligand shell (Figure S11), and so the inorganic portion in the closepacked model is smaller. The predicted packing fractions as compared to those of the EMA models are summarized in Table 1. Using the random-close-packed estimate, it becomes clear that the differences in the inorganic percentages of CdSe and CdSe/CdS QD films help explain why the CdSe film’s refractive index values are lower than might be expected from the intrinsic nanocrystal properties alone. As shown in Table 1, for small CdSe cores with 0.6 nm 3-MPA ligands, a theoretical packing fraction of 64% QD and attached ligand would give an inorganic CdSe percentage of 29% in the film. However, as the QD diameter is increased with additional CdS shell thickness, the bound ligand surroundings take up less and less space compared to the inorganic portion of the quantum dot. This, in addition to the increasing aspect ratio and associated packing, leads to an increase in the inorganic packing fraction as QDs with the thicker shells are used, predicting up to 45% packing for the CdSe/6CdS QDs. While CdSe QDs have a higher extinction coefficient intrinsically than CdSe/CdS particles, because of their smaller diameter and overall lower inorganic volume fraction in neat films, the effective extinction coefficient in the film is lower than that in the core/shell QD films. The random-close-packed model does not fully explain why the CdSe/2CdS film has a significantly

A (λ ) = I

1 ω |E(λ)|2 Im(ε (̃ λ)) 2

(6a) DOI: 10.1021/acs.jpcc.8b04522 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C where E is the electric field, Im(ε̃) is the imaginary portion of the effective refractive index of the QD layer, and ω is the angular excitation frequency. This equation, equivalent to the divergence of the Poynting vector, must be integrated over the entire simulation volume and can be normalized relative to source power Ps, by A (λ ) =

∭

ω |E(λ)|2 Im(ε (̃ λ)) dx dy dz 2Ps

demonstrating that these methods improve the understanding of the optical behavior of neat QD films with varying shell thickness. This finding can be extended to full-wave simulations and other refractive index dependent calculations as well as the prediction of the response of ordered and disordered quantum dot solids.

■

ASSOCIATED CONTENT

S Supporting Information *

(6b)

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b04522. Additional information on the intrinsic refractive index extraction process for both core and core/shell particles, raw ellipsometry data, and additional experimental characterization techniques including TEM and FTIR (PDF)

The resulting absorption fraction for a 100 nm neat CdSe film with effective refractive index values as measured by spectroscopic ellipsometry is shown in Figure 6a and closely follows the trends seen in the extinction coefficient function shown in Figure 4b, which it depends on. This absorption data, however, is based on films that do not have constant packing, making it difficult to extract information on how core/shell structure may affect absorption, independent of packing difference, within the different QD films. To better extract packing-independent absorption changes, we use the Bruggeman EMA described above and solve for the effective refractive index of QD films when the fraction of inorganic QDs and their attached ligands is held constant at 64%, assuming perfectly spherical particles. Ligand length was chosen to be 0.5 nm, accounting for possible interdigitation between the MPA ligands, unlike in the fully hard-sphere limiting case discussed above. The predicted inorganic packing fractions were calculated as 33%, 38%, 42%, and 45%, respectively, for the CdSe, CdSe/2CdS, CdSe/4CdS, and CdSe/6CdS films in this model. The theoretical effective refractive index information for this structure is shown in Figure S12, and the results of the absorption calculated for this case are shown in Figure 6b. In this case, it is much easier to see that additional monolayers of CdS on CdSe core QDs lead to moderate absorption enhancement at short wavelengths, but dramatic absorption decreases at wavelengths longer than the CdS band gap where CdSe features dominate. With the tools shown here we believe that the solid-state behavior of QD devices can be better predicted to understand and improve QD-based optoelectronic devices.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Vivian E. Ferry: 0000-0002-9676-6056 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported partially by the National Science Foundation through the University of Minnesota MRSEC under Award Number DMR-1420013. A portion of this work was carried out in the Minnesota Nano Center which receives partial support from the National Science Foundation through the NNCI program. Part of this work was carried out in the College of Science and Engineering Characterization Facility, University of Minnesota, which has received capital funding from the NSF through the UMN MRSEC program under Award Number DMR-1420013.

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REFERENCES

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CONCLUSIONS In conclusion, variations in ligand packing and nanocrystal size can obscure systematic trends in the complex refractive index of colloidal QD films. The method we show here, to study the complex refractive index of core/shell QD films, unravels the differences in effective refractive index for films with varying shell thickness. Spectroscopic ellipsometry is a highly accurate way to measure the effective refractive index of neat QD films using Kramers−Kronig consistent dispersion models and can be extended to films made of core/shell heterostructures. However, on its own, the scope is limited as differences in QD size and deposition can lead to dramatic changes in film morphology. By combining the effective refractive index of neat QD films with Kramers−Kronig consistent intrinsic refractive index values extracted from solution-state absorption measurements, we show that the Maxwell Garnett and the two component Bruggeman EMAs closely describe many of the trends and features seen in neat core and core/shell QD films and predict packing fractions similar to those of hard-sphere packing models. The applicability of this tool is exhibited through the calculation of absorption in CdSe core and CdSe/ CdS core/shell QD films independent of packing fraction, J


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