Propagation of Structural Disorder in Epitaxially Connected Quantum

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Propagation of Structural Disorder in Epitaxially Connected Quantum Dot Solids from Atomic to Micron Scale Benjamin H Savitzky, Robert Hovden, Kevin Whitham, Jun Yang, Frank W. Wise, Tobias Hanrath, and Lena Fitting Kourkoutis Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b02382 • Publication Date (Web): 19 Aug 2016 Downloaded from http://pubs.acs.org on August 20, 2016

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Propagation of Structural Disorder in Epitaxially Connected Quantum Dot Solids from Atomic to Micron Scale Benjamin H. Savitzky1, Robert Hovden2, Kevin Whitham3, Jun Yang2, Frank Wise2, Tobias Hanrath4 and Lena F. Kourkoutis2,5,* 1.

2.

Department of Physics, Cornell University, Ithaca, NY 14853, USA.

School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA.3.

Department of Materials Science & Engineering, Cornell University, Ithaca, NY 14853, USA. 4.

School of Chemical & Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA. 5.

Kavli Institute for Nanoscale Science, Cornell University, Ithaca, NY 14853, USA. *.

Corresponding Author: [email protected]

ACS Paragon Plus Environment

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Abstract

Epitaxially connected superlattices of self-assembled colloidal quantum dots present a promising route towards exquisite control of electronic structure through precise hierarchical structuring across multiple length scales. Here we uncover propagation of disorder as an essential feature in these systems, which intimately connects order at the atomic, superlattice, and grain scales. Accessing theoretically predicted exotic electronic states and highly tunable minibands will therefore require detailed understanding of the subtle interplay between local and long range structure.

To that end, we developed analytical methods to quantitatively characterize the

propagating disorder in terms of a real paracrystal model, and directly observe the dramatic impact of angstrom scale translational disorder on structural correlations at hundreds of nanometers. Using this framework we discover improved order accompanies increasing sample thickness, and identify the substantial effect of small fractions of missing epitaxial bonds on statistical disorder. These results have significant experimental and theoretical implications for the elusive goals of long range carrier delocalization and true miniband formation.

Keywords: quantum dot solids, scanning transmission electron microscopy, paracrystal, structural disorder, PbSe, nanocrystals


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Superlattices (SLs) of epitaxially connected colloidal quantum dots (QDs), also called confined-but-connected structures (CBCs), hold tremendous potential for both optoelectronic applications and exploration of novel low-dimensional physics. This is due to the combination of highly tunable electronic structure via the QD size, shape, composition, spacing, and arrangement1–3; enhanced electronic coupling afforded by epitaxial connection4–6; and the scalability of modern colloidal processing methods. High mobility transistors and photovoltaic cells have been experimentally demonstrated7–11, and theoretical calculations of 2D QD assemblies reveal rich miniband structures including Dirac cones, nontrivial flat bands, and topological edge states12,13.

These tantalizing electronic properties are all fundamentally structurally mediated14–16, and prior work has explored local structure and its fascinating implications for the band structure13,17,18.

However, CBC’s considerable promise derives principally from rationally

designed hierarchical structuring across Ångströms to microns. Realizing that promise therefore requires understanding the precise nature of structural disorder spanning this sizable length range. Here, we reveal the vital influence of the propagation of disorder through the SL in square PbSe CBCs, and demonstrate that this is best described by a real paracrystalline model19,20. We show that order improves with increasing sample thickness, and use this result to illustrate experimentally how propagation of disorder allows atomic scale disorder to drastically alter long range structure. Finally, we examine the sources of disorder and find that although most adjacent QD pairs are connected, unconnected pairs contribute disproportionally to structural disorder. These results have important implications for optimization of future CBC growth, and the elusive goal of long range charge carrier delocalization14,21.


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We fabricated square PbSe CBCs with epitaxial connections across the facets, with SL spacing a = 6.5nm and grain sizes of several microns following the process we describe elsewhere10.

Using high resolution aberration corrected scanning transmission electron

microscopy (STEM), we obtained directly interpretable, Z-contrast micrographs spanning atomic to SL to single grain length scales22 (Fig 1a, i-iii).

See Supporting Information S1 for

experimental details.

Figure 1: STEM HAADF micrographs of epitaxially connected PbSe superlattices at the atomic (a,i), superlattice (a,ii) and grain sized (a,iii) length scales reveal the presence of cumulative


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disorder. Broadening peaks in the pair correlation function G(r) with increasing peak order clearly indicate propagation of disorder (b, c). The rate of broadening in the longitudinal and transverse directions (b, inset) is compared to several models of crystalline disorder, including a uniform model, the ideal paracrystalline model (e), and the real paracrystalline model. Over many datasets, the real paracrystalline model is found to describe the data best; a representative example in the longitudinal direction is shown in (d).

Vectors connecting QD centroids quantitatively describe the SL structure in real space. The cumulative disorder is revealed by calculating the two-dimensional pair correlation function G(r) directly from HAADF STEM images, using an in house algorithm (see Supporting Information S2). Large field of view, high pixel density micrographs combined with sub-pixel centroid fitting allowed us to obtain good statistics over ~1µm2 areas while preserving subatomic spatial resolution. A representative G(r) plot is shown in Fig 1b. The central white X indicates the origin, while the four adjacent peaks can be interpreted as the probability distributions of nearestneighbor (NN) QD displacement vectors (Fig 1b, inset). It is immediately evident that peaks corresponding to larger lattice vectors become progressively broader, indicating propagation of disorder through the lattice. We quantified this broadening by fitting 2D Gaussians to the peaks centered at na, where a is any of the SL lattice vectors and n is an integer, then subsequently extracting the standard deviations longitudinal and transverse to a, σL and σT (Figs 1b,c).

In all datasets σT > σL, corresponding to a greater shearing disorder than

compressive/tensile disorder. Physically, this may reflect the degree of QD misalignment during the oriented attachment process, or a greater susceptibility to shearing than compression/tension under post assembly mechanical stresses. Representative experimental values of σL versus n are


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shown for a single dataset in a single direction in Fig 1d, green, illustrating the G(r) peak broadening as σL increases monotonically with n. Trends are identical for σT, as discussed below. Note that we have ruled out the effects of distortions due to the imaging process by varying acquisition parameters and observing identical behavior in both the slow and fast scan directions.

Theoretically, this behavior can be understood by contrasting three models of crystalline disorder: uniform, ideal paracrystalline, and real paracrystalline disorder19,23,24.

Their

fundamental premises and qualitative differences are sketched here in the simple 1D case; mathematical details, and the extension to two- and three-dimensions, are found in Supporting Information S3. By uniform disorder, we refer to the case of a perfect lattice in which each particle is displaced from its lattice site by a random vector sampled from some distribution. This is uniform in the sense that the disorder is non-cumulative, so that σ is constant with increasing n.

In contrast, the ideal paracrystal model describes the effects of cumulative

disorder. Its primary assumptions are that there exists some probability density function for NN lattice vectors p(x), and that p(x) between any pair of NN particles is independent of all other particle positions. The immediate result is that the probability density function between particles n lattice spacings apart pn(x) is the n-fold self-convolution of p(x), or ‫݌‬௡ ሺ࢞ሻ = ‫∗݌‬௡ ሺ࢞ሻ = ‫݌‬ሺ࢞ሻ ∗ ‫݌‬ሺ࢞ሻ ∗ ⋯ ∗ ‫݌‬ሺ࢞ሻ It follows that under an ideal paracrystal model the standard deviation of pn(x) grows as √n. This behavior is illustrated schematically in Fig. 1e. While disorder in the uniform model does not propagate, the ideal paracrystalline model may be thought of intuitively as describing ‘perfect’ propagation of disorder. The real paracrystalline model effectively allows for


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‘imperfect’ propagation of disorder, by combining an ideal disorder parameter σideal which is cumulative, and a real disorder parameter σreal which is non-cumulative.

Under a real

paracrystalline model, pn(x) increases monotonically with n no faster than √n.

The real paracrystal model, which includes both σideal and σreal, best captures the data with good fidelity. In Fig 1d, all three models are fit to the experimental σL(n) values (green) up to and including n=8. Unsurprisingly, the uniform model fits the data poorly (black), and the ideal paracrystal model accurately captures the trend of a broadening σ but overestimates rate of increase (blue). Comparison of the root mean square deviation of fits from each model for both σL and σT in each direction over many datasets confirms that the disorder is best described under a real paracrystalline model (see Supporting Information S4).

A priori, the paracrystalline nature of CBCs is unsurprising. Though the polydispersity of QD building blocks used here was only ~6%, which represents approximately a single lattice plane spacing, any polydispersity should in principle result in paracrystallinity (see Supporting Information S3), and recent GISAXS studies have concluded that unconnected colloidal crystal thin films are paracrystalline25. However, the superiority of the real paracrystal model has not yet been demonstrated in either unconnected or connected colloidal crystals, and is essential to understanding the relationship between short and long range SL order.

These structural

distinctions may have dramatic implications for electronic transport in CBCs. Recent work demonstrated that delocalization and transport in high mobility conjugated polymer networks is dominated by paracrystallinity, which introduces localized bandtail states26,27.

Further, the


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paracrystalline framework enables exploration of precisely which factors contribute to the disorder, and how.

Interestingly, the thickness of the CBC film has a direct impact on disorder, with order improving with film thickness. We quantify this effect at short ranges by plotting both σL (blue) and σT (red) against film thickness, measured in number of QD layers, Fig. 2a. Representative samples 1, 3, and 4 QD layers thick are shown in Figs. 2b-d. Thickness was determined directly from the HAADF intensities (see Supporting Information S5).

Paracrystalline disorder

decreased from monolayer to 4 layer samples by 33% (34%) in the longitudinal (transverse) directions. In combination with reported approaches to controlling SL thickness28, this result provides a straightforward method to tune paracrystalline disorder.


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Figure 2: Increasing sample thickness reduces both longitudinal (blue) and transverse (red) paracrystalline disorder (a). Solid circles indicate mean σ values at each thickness and error bars indicate the 95% confidence interval. The values shown represent the disorder in a single twodimensional layer even in thicker samples, calculated by correcting for STEM projection effects (see Supporting Information S6). Representative samples 1, 3, and 4 QD layers thick are shown (b-d).

Comparing disorder across differing sample thicknesses requires care, as STEM provides projection images. The centroids identified in micrographs of multilayer samples thus represent


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the average position of the QD centroids in a single out-of-plane column. Fortunately, the 2D projection of a 3D real paracrystal is again a real paracrystal, thus the 3D parameters can be then extracted from the calculated 2D parameters and the sample thickness (see Supporting Information S6). The σ values plotted in Fig. 2 represent the disorder of a single 2D layer of the CBC structure even for thicker samples, enabling direct comparison.

The improvements in paracrystalline disorder with thickness shown in Fig. 2 occur on subAngstrom length scales, however cumulative disorder magnifies their effects over large distances, leading to dramatic structural differences between thin and thick samples at length scales of hundreds of nanometers.

To achieve good signal-to-noise at long distances we

averaged G(r) azimuthally to obtain the radial distribution function g(r), shown in Fig. 3a for samples 2-3 layers thick (blue) and 6 layers thick (red). After calculating g(r) analytically for the 2D isotropic ideal paracrystal model (see Supporting Information S7), we fit the experimental curves (Fig. 3b) and find σ2-3 layer = 3.5Å, and σ6 layer = 2.2Å, in good agreement with the short range results of Fig. 2. The ideal paracrystal model is sufficient here as the central limit theorem ensures that the real and ideal models converge at large distance (see Supporting Information S3). Although the fitted disorder parameters differ by just over an Ångström, the effect on long range order is pronounced, evidenced by the rapid decay to unity of g2-3 layer(r)

relative to g6 layer(r). Quantitative interpretation of the long range decay of g(r) is

complicated by the interesting and non-trivial behavior of g(r) for a perfect lattice and the effect of slight SL anisotropies on g(r) at large length scales (see Supporting Information S8 and S9). Nonetheless, the qualitative picture is striking: increasing the paracrystalline


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disorder by a fraction of an atomic lattice plane spacing reduces the length scale over which QD positions are correlated by hundreds of nanometers.

Figure 3: Angstrom-scale variations in nearest neighbor translational disorder have dramatic effects on long range structure in the presence of cumulative disorder. While g(r) converges to unity in thin, 2-3 layer samples (blue) at ~100 nm, QD positions remain correlated in thicker, 6 layer samples (red) at ~300 nm (a). Ideal paracrystal fits (b) confirm that the origin of this long range structural disparity is sub-atomic differences in the paracrystalline disorder parameter σ, from σ2-3 layer = 3.5Å, to σ6 layer = 2.2Å.


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Next we examine the origins of paracrystalline disorder. Atomic resolution analysis of the epitaxial connections reveals that the presence or absence of epitaxial bonds is a significant source of paracrystalline disorder, augmenting the expected effect of polydispersity.

We identified adjacent QD pairs with (green) and without (blue) a

continuous adjoining atomic lattice in a monolayer sample, and plotted a histogram of QD spacings color-coded by these designations, revealing a clear increase in spacing between unconnected, adjacent QD pairs (Fig. 4). More importantly, unconnected QDs significantly increased the disorder in the NN spacing in spite of representing only 20% of adjacent pairs; while σconnected = 3.9 Å, including both connected and unconnected pairs yields σadjacent = 5.3 Å (see Supporting Information S10). As noted above, these changes can have a significant impact on the long range decay in CBC order. Additionally, this result suggests that a possible cause for the observed thickness-order correlation is increased epitaxial bond number due to bond formation in the out-of plane direction, however confirmation is complicated by the difficulty in experimentally ascertaining out of plane connectivity. Interestingly, while disorder in the width of epitaxial bonds between connected QDs does not appear to affect the NN spacing, previous work has shown bondwidth variations significantly impact carrier localization6,10. Details, including mapping and characterization of bridge width variations, are in Supporting Information S11.


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Figure 4: Missing epitaxial bonds significantly increase the variance in nearest neighbor spacing. The increase in spacing between adjacent, unconnected QDs (blue) relative to connected QDs (green) broadens σ from σconnected = 3.9 Å to σall adjacent = 5.3 Å. Due to cumulative disorder the relatively small fraction (20%) of missing epitaxial bonds thus disproportionately effects structure, particularly at large length scales.

We have observed the propagation of disorder through CBC SLs, elucidated its real paracrystalline nature, demonstrated its considerable impact on long range structure, and discovered correlations between paracrystalline disorder and both thickness and epitaxial bonding. Improved understanding of the origins and nature of CBC structural disorder is particularly important in light of the central role disorder plays in determining electronic structure and transport properties. Theoretical investigations into this connection have successfully accounted for several key experimental observations in QD SLs, such as Anderson-type transitions and photoconductivity measurements, and guide continued efforts towards incorporation into devices.14,21,29,30 However, transport in paracrystalline


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QD SLs has not been considered, implying the need to evaluate the role of cumulative structural disorder in colloidal SLs in general. High mobility conjugated polymers, in which charge transport occurs through π-stacks exhibiting both local, on-site energy disorder as well as cumulative, paracrystalline disorder, provide an illuminating comparison26,27. In these systems paracrystallinity leads to bandtails, increased carrier localization, and a clear mobility edge, however this effect dominates local disorder only above a critical value of the paracrystalline disorder. Although quantitative comparison between these systems is not meaningful, it is qualitatively striking to note that the range of CBC paracrystalline disorders reported here span precisely this crossover region in conjugated polymers, highlighting the critical need to understand electronic structure in paracrystalline colloidal QD SLs.

To address this need, we performed tight binding calculations of carrier localization lengths, varying σ and holding all other parameters constant. The localization length increases by a factor of ~10 as σ is decreased over the range of values observed experimentally in monolayers to 6 layer thick samples (see Supporting Information S12). Given our observed twofold improvement in ordering with thickness, we might expect a increase in photoconductivity due to translational disorder by a factor of ~4 according to calculations by Shabaev et al, which showed a square dependence of the scattering rate on disorder. Our estimates are consistent with recent reports of delocalization across several QD diameters10, suggesting the feasibility of efficient transport over the length scales of modern transistor gates, or along the direction of charge separation in thin film photovoltaic cells31. These are also precisely the length scales at which cumulative and


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non-cumulative disorder models become structurally distinct, providing some physical indication that paracrystallinity may be electronically important. Experimentally, our results emphasize the need to optimize epitaxial bonding so as to minimize disorder, not only in the energetic coupling8,10, but also in short and long range translational order. Further, we establish a new, straightforward approach to controlling disorder by tuning sample thickness. This suggests that miniband formation and tailored optoelectronic properties in CBCs may be more readily achievable in thicker films where order is improved.

Associated Content Supporting Information Available: Experimental and image processing details, more complete analytical description of the models used, and additional supporting analysis may be found in the supporting information. This material is available free of charge via the Internet at http://pubs.acs.org

Acknowledgements This research was supported by the Cornell Center for Materials Research with funding from the NSF MRSEC program (DMR-1120296). B.H.S. was supported by NSF GRFP grant DGE1144153.

K.W. and J.Y. were supported by the Basic Energy Sciences Division of the

Department of Energy through Grant DE-SC0006647 ‘Charge Transfer Across the Boundary of Photon-Harvesting Nanocrystals’. The authors wish to thank CCMR staff John Grazul and Mick Thomas for experimental support.


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Nano Letters


Nano Letters

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Nano Letters


Propagation of Structural Disorder in Epitaxially Connected Quantum

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