ARTICLES PUBLISHED ONLINE: 22 FEBRUARY 2016 | DOI: 10.1038/NMAT4576
Charge transport and localization in atomically coherent quantum dot solids Kevin Whitham1, Jun Yang2, Benjamin H. Savitzky3, Lena F. Kourkoutis2,4, Frank Wise2 and Tobias Hanrath5* Epitaxial attachment of quantum dots into ordered superlattices enables the synthesis of quasi-two-dimensional materials that theoretically exhibit features such as Dirac cones and topological states, and have major potential for unprecedented optoelectronic devices. Initial studies found that disorder in these structures causes localization of electrons within a few lattice constants, and highlight the critical need for precise structural characterization and systematic assessment of the effects of disorder on transport. Here we fabricated superlattices with the quantum dots registered to within a single atomic bond length (limited by the polydispersity of the quantum dot building blocks), but missing a fraction (20%) of the epitaxial connections. Calculations of the electronic structure including the measured disorder account for the electron localization inferred from transport measurements. The calculations also show that improvement of the epitaxial connections will lead to completely delocalized electrons and may enable the observation of the remarkable properties predicted for these materials.
T
he unique electronic properties of nanostructured materials inspire future applications and present intriguing scientific challenges. Recent advances in the controlled formation of superlattices by epitaxial connection of semiconductor nanocrystals (NCs), also known as quantum dots, have the potential for creating materials with properties by design. The ability to control the atomic structure of the quantum dot building block (that is, size, shape and composition) and the geometry of the superstructure creates opportunities to synthesize and study designer materials. Calculations of such quantum dot solids with a dimensionality less than 2 forecast a rich electronic structure that differs profoundly from the corresponding quantum well1 . Although the predicted emergence of electronic phenomena including topological states and Dirac cones intrigue scientists1–4 , experimental support is still lacking. Prerequisite to an experimental realization of predicted electronic band structure is a transition from localized to delocalized electrons, which requires highly ordered superlattices and strong coupling between NCs (ref. 5). Recent experimental efforts to improve the electronic properties of quantum dot solids have focused on increasing inter-dot coupling by replacing organic ligands with inorganic ligands6,7 or by ligand removal8 . However, this approach leads to band formation only when the coupling energy overcomes energetic and translational disorder5,9 . Oriented attachment is a simple route to simultaneously increase inter-dot coupling and enforce translational order, as demonstrated in several recent reports10–13 . Initial studies find that charges in these structures are localized to within a few lattice constants owing to disorder14,15 . It has been shown recently that charge transport in such structures occurs through incoherent hopping between epitaxial segments16 . These findings highlight the critical need for precise analysis of the structure of the quantum dot solid and a quantitative assessment of the effects of disorder on transport.
We address this challenge by combining advanced structure characterization techniques (aberration-corrected electron microscopy and X-ray scattering) to precisely determine structure and disorder parameters as inputs for tight-binding electronic structure calculations. We discovered that translational order in these structures is controlled down to about a single atomic bond length, a limit imposed by the 4% size dispersion of the NC building blocks. To probe the intrinsic charge transport in these structures we performed variable temperature field-effect measurements in epitaxially connected quantum dot superlattice devices. By integrating structure analysis, transport measurements and electronic structure calculation, we validate that carriers are localized to a few dots owing to the heterogeneity of the epitaxial inter-dot connections. Importantly, our analysis shows that complete delocalization is possible with optimized dot-to-dot bonding, thereby providing a path forward to create quantum dot solids in which theoretically predicted properties can be realized.
Disorder and carrier localization Oriented attachment of PbSe NCs can be controlled by ligand desorption from selected facets10 . In contrast to previous approaches, where oriented attachment and superlattice assembly occur simultaneously, the two processes are temporally decoupled in the approach used in this work. We first fabricate highly ordered superlattices of ligand-capped NCs, then chemically induce oriented attachment in a separate step11,17,18 . Decoupling assembly and attachment transfers the high degree of order in the unconnected superlattice to the final structure. A colloidal dispersion of 6.5-nm-diameter PbSe NCs in hexane was dispersed on ethylene glycol. Subsequently ethylene diamine was injected into the ethylene glycol subphase. A common metal chelation agent, ethylene diamine has been used to remove or replace oleic acid ligands on PbSe NCs (refs 19–21). We propose that
1 Department
of Materials Science and Engineering, Cornell University, Ithaca, New York 14853, USA. 2 School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA. 3 Department of Physics, Cornell University, Ithaca, New York 14853, USA. 4 Kavli Institute for Nanoscale Science, Cornell University, Ithaca, New York 14853, USA. 5 School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, New York 14853, USA. *e-mail: [email protected] NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
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NATURE MATERIALS DOI: 10.1038/NMAT4576
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Figure 1 | Structural characterization of a nanocrystal superlattice. a,b, Annular dark-field scanning transmission electron microscopy shows superlattice structure (a) and atomic structure (b). c, Measured radial distribution function and calculated radial distribution function of a paracrystalline square lattice with 3.4% translational disorder. d, Grazing incidence small-angle X-ray scattering (GISAXS) image shows reflections of a square superlattice (SL). e, Grazing incidence wide-angle X-ray scattering (GIWAXS) image shows alignment of the atomic lattices (AL) of the nanocrystals.
ethylene diamine, an L-type ligand, binds selectively to Pb atoms on {100} facets, desorbing as L-Pb(O2 CR)2 (ref. 19). After deprotection, NCs undergo oriented attachment to fuse with {100} facets of proximate NCs. Similar results have been obtained by heating the glycol subphase11 . Amines have been used previously to achieve oriented attachment of NC films deposited on a solid substrate with less long-range order than we observe using a liquid substrate12 . The nature of the epitaxial connections between NCs is revealed by aberration-corrected scanning transmission electron microscopy (STEM) of monolayer areas, shown in Fig. 1b. Atomicresolution images show that oriented attachment occurs in the h100i crystallographic directions. Both STEM and X-ray scattering show the superlattice is simple square and less than 10 NC layers thick (see Supplementary Fig. 10). The axes of the atomic lattice and the superlattice are coherent as a result of the oriented-attachment self-assembly mechanism. To quantify translational order in the superlattice structure, we calculated the radial distribution function (RDF) based on NC locations directly extracted from STEM images. Using a paracrystalline model22 , we fit the measured RDF with only one free parameter (see Supplementary Information) and find the standard deviation of the nearest-neighbour distance to be 0.22 nm, or approximately one Pb–Se bond length (0.306 nm). We also fit the paracrystal model to our GISAXS data and find the standard deviation of the nearest-neighbour distance to be 0.3 nm, consistent with TEM results (see Supplementary Fig. 3). Importantly, because the standard deviation of the diameter of the as-synthesized NCs is approximately one Pb–Se bond length (see Supplementary Figs 1 and 2), we conclude that the spatial 2
coherence of the superlattice is fundamentally limited by the inherent polydispersity of the NCs. Besides translational disorder we quantified two other forms of disorder: the size of the NC cores and the width of the epitaxial connections. We developed an image analysis algorithm to collect unbiased statistics of NC diameters from STEM images. An image with 1,693 NCs yields an average diameter of 5.75 nm ± 0.19 nm, whereas GISAXS analysis yields 6.1 ± 0.3 nm (see Supplementary Figs 5 and 3). Analysis of an image with 10,122 connections yields a connection width of 3.26 nm ± 0.48 nm (see Supplementary Figs 6 and 7). These two forms of disorder have significant impact on energetic coupling and delocalization, as we show with tight-binding calculations discussed below. The high fidelity of the quantum dot solid and the epitaxial nature of the inter-dot bond raises an intriguing question about delocalization of electrons and holes. We address this question by measuring temperature-dependent charge transport with a field-effect transistor (FET), the structure of which is shown in Fig. 2a. The FET is designed to modulate conductance by changing the average chemical potential of charge carriers. In disordered materials, carriers are conducted through a network of localized states distributed spatially and energetically. STEM and X-ray data can be used to model the density of states. Spatial density can be extracted from the radial distribution, and the distribution of states in energy can be inferred from measurement of the NC diameters23 . We used FET transport measurements to test the validity of this simple density-of-states model and to estimate the localization length.
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NATURE MATERIALS DOI: 10.1038/NMAT4576
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Figure 2 | Charge transport measurement by field-effect transistor. a, Cross-sectional schematic of the field-effect transistor. The gate is formed by highly doped silicon (Si++) with a 200 nm dielectric layer of SiO2 . The source and drain electrodes were patterned on the SiO2 to form a channel 100 µm × 3 mm. A layer of photocured polymer served as a barrier between the NC layer and ambient water and oxygen. b, Drain current (ID )-Gate voltage (VG ) transfer curves show ambipolar transport with electron current at positive VG and hole current at negative VG . The source–drain bias was 1 V. The conductance was measured every 5 K from 85 K to 245 K. c, Arrhenius plot of electron conductance at a gate bias of 22 V with a source–drain bias of 1 V. The solid line is a linear fit to the higher temperature data. d, Log–log plot of the data in c. Straight lines illustrate that the data are not a single power law over this temperature range.
By constructing the density-of-states model from the physical structure of the superlattice, we assumed defect states to be negligible. However, removal of ligands could expose the NCs to adventitious water or oxygen, creating electronic defects8,24–26 . The sensitivity of PbSe NCs to adsorbed molecules has been well documented27–30 . To achieve negligible defect density we encapsulated devices with a photocurable polymer (see Methods and Supplementary Fig. 12). With encapsulation we observed negligible hysteresis in the FET characteristics at temperatures between 85 K and 245 K (see Supplementary Fig. 11). Above 245 K we did observe hysteresis, and therefore limit our analysis to data below 245 K. Transfer curves (current versus gate voltage) in Fig. 2b show ambipolar transport (see also Supplementary Fig. 11). We find that electron and hole transport are thermally activated (hopping), with conductance decreasing exponentially with decreasing temperature, a clear sign of carrier localization. Hopping conductance is p commonly interpreted using the expression G = G0 e−(T0 /T ) , where p depends on the type of hopping and T0 depends on material properties31 . A plot of ln(G) versus ln(T −1 ) in Fig. 2d shows that a single value of p fails to describe the data over the full temperature range. Instead, the data suggests that there is a transition between two regimes. In all types of hopping transport, conductance is described by G = G00 e−ξ , where ξ = (2r/a) + (ε/kT ), r is the hopping distance, a is the electron or hole localization length, ε is the hopping energy32 , with k the Boltzmann constant. In the limit of 2r/a ε/kT , the hopping distance is the nearest-neighbour distance; called
nearest-neighbour hopping (NNH). In NNH the term 2r/a is independent of temperature, therefore conductance is given by G = G000 e−εNN /kT , where εNN is the NNH energy. Figure 2c shows this Arrhenius-type behaviour at high temperatures. Conductance at low temperature deviates from Arrhenius behaviour. In this regime the hopping distance varies with temperature; called variable-range hopping (VRH). The transition temperature between NNH and VRH can be determined by fitting two linear regimes to ln(G) versus ln(T −1 ), as in Fig. 2d. Analysis of transport data in Fig. 3 clearly shows the transition temperature (Tc ) and NNH energy (εNN ) directly relate to the applied gate voltage (VG ). Increasing the gate bias moves the chemical potential to a higher density of states. The average hopping energy decreases as the density of states increases. This is the physical reason for the relation between εNN and VG . Below, we will show that the localization length can be estimated from the transition temperature Tc . At high temperature, hopping occurs between nearest neighbours. As the temperature is decreased, the density of states accessible to a charge carrier within a few kT of the chemical potential becomes increasingly sparse. At a certain temperature, hopping longer distances becomes more favourable to minimize the energy between initial and final states32 . This transition occurs at a higher temperature for more delocalized states because these carriers can more easily ‘reach’ distant states. Changing the gate voltage shifts the chemical potential, thereby probing different states, whereas changing the temperature varies the energy window available to carriers near the chemical potential. By varying both, we determined transition temperatures for a range of chemical
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NATURE MATERIALS DOI: 10.1038/NMAT4576
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potentials, and from the transition temperatures estimated the localization lengths. Our procedure follows the analysis of hopping transport in doped semiconductors32 . We adapted the analysis for the density of states inferred from the physical structure of the superlattice (see Supplementary Information). The result in Fig. 4d shows the localization lengths calculated from FET transport measurements. We find that charge carriers are localized to just a few NCs at a gate bias of ±40 V. We note that Coulomb blockade should not be an important factor in the measured temperature range (see Supplementary Information). We measure mobilities of 0.54 cm2 V−1 s−1 for holes and 0.2 cm2 V−1 s−1 for electrons at 245 K (see Supplementary Fig. 11), Despite relatively good mobilities compared to ligand-exchanged quantum dot solids28,33–35 , the localization length reveals that transport is still far from band-like. Before discussing the theoretical prediction of the localization length in the next section, it is important to note that the localization lengths shown in Fig. 4d are not the localization lengths at the chemical potential, but the smallest localization lengths within the energy window accessible to carriers near the chemical potential32 . Recently reported field-effect mobilities of 3–13 cm2 V−1 s−1 were measured in single superlattice grains16 . The difference between 3 and 13 cm2 V−1 s−1 reported recently and 0.2 cm2 V−1 s−1 in this work may be due to several factors. First, the values of 3–13 cm2 V−1 s−1 were measured with unintentionally n-doped samples, whereas we fabricated ambipolar samples, preventing unintentional doping via encapsulation. Second, we measured 4
0.2 cm2 V−1 s−1 at 245 K to avoid hysteresis, whereas values of 3–13 cm2 V−1 s−1 were measured at 290 K without accounting for hysteresis. Finally, we estimate the effect of grain boundaries on mobility. If coherent transport occurs over a few NCs (20 nm), then transport across 0.1 mm requires at least 5 × 103 hops. Introducing grain boundaries every 100 nm increases the number of hops to 6 × 103 , and effectively decreases the coherence length to 16.7 nm. Because the mobility µ is related to the diffusion coefficient D by the Einstein relation D = µkT /q, and D ∝ r 2 , where the distance between hops r is the coherence length, this reduction in coherence length reduces the mobility by 30%.
Calculation of electronic structure The strong localization of electrons and holes may seem surprising in view of the high fidelity of the quantum dot solid shown in Fig. 1. To gain a deeper understanding of the electronic structure of the quantum dot solid and the sensitivity to disorder, we performed tight-binding calculations that directly incorporate disorder parameters determined from TEM and GISAXS. A recent calculation of the electronic band structure of a square superlattice of PbSe NCs predicts the evolution of delocalized mini-bands from quantized NC states1 . We reproduced the band structure of a perfect superlattice, and then impressed disorder on the structure to determine the localization length in a realistic sample. Inclusion of disorder in an atomistic calculation of a structure with a length scale of hundreds of nanometres is computationally prohibitive. Conveniently, the mini-band structure of a 2D square superlattice of PbSe NCs has been calculated recently by an
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NATURE MATERIALS DOI: 10.1038/NMAT4576
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Figure 4 | Extent of delocalization in a disordered superlattice. a, Model density of 1Se or 1Sh states based on experimental measurements of the superlattice structure. b, Relationship between the NNH energy and chemical potential for the density-of-states model in a. Points were calculated numerically, the solid line is a power law approximation. Inset arrows show the ranges of hopping energies measured for electrons and holes at gate voltages between 0 and ±40 V. Open circles mark the measured hopping energies of electrons (holes) for a gate bias of 40 V (−40 V). c, Theoretical localization length of electrons (holes) in 1Se (1Sh ) states for a range of connectivity values. Zero energy refers to the average 1Se or 1Sh energy level. The lowest energy states accessible to electrons (holes) at a gate bias of 40 V (−40 V) are indicated. Error bars represent the standard deviation of multiple numerical calculations using the Monte Carlo method. Localization lengths are given in nanometres on the left-hand scale and by dimensionless units of a0 on the right-hand scale, where a0 represents the localization length of a charge carrier in a completely isolated NC, equal to the NC radius. d, Localization lengths of electrons and holes calculated from the measured temperature dependence of the conductance. Error bars reflect the uncertainty of the transition temperature.
atomistic tight-binding approach1,3 . We fit the band structure from the atomistic calculation using an effective Hamiltonian (see Supplementary Information for details) and then use the effective Hamiltonian to calculate the localization length in a large disordered system. We modelled the two sources of disorder observed in the experiment: fluctuation of the NC size, and fluctuation of the width of the epitaxial connections between NCs. Size fluctuation is modelled as a Gaussian distribution of the energy levels, where the size distribution is related to the energy distribution because of quantum confinement (see Supplementary Information). The fluctuation of the width of the epitaxial connections is modelled as a distribution of coupling energies estimated from the atomistic calculation (see Supplementary Fig. 14 for the calculated densities of states). Analysis of STEM images gives accurate distributions for the NC diameter and epitaxial connection width. Approximately 80% of neighbouring NCs in our STEM images are epitaxially connected; we include the missing bonds as zero-width connections in our calculation. In Fig. 4c we show the localization lengths predicted considering the disorder measured directly from our samples. The maximum localization length is 50 nm for states near the average NC energy level. In domains larger than ∼100 nm, hopping transport will remain dominant. Furthermore, the sites with the largest localization lengths do not determine conductance through a disordered system; conductance is limited by the smallest localization lengths in the percolation network.
We now compare the experimental result with the theoretical calculation. With the highest applied gate voltage (40 V), we find the NNH energy for electrons is 32 meV. This corresponds to a chemical potential 8.5 meV below the average NC energy level. The lowest energy state accessible to an electron is therefore 41 meV from the average NC energy level. Hopping transitions to NCs at this lowest energy level limit the overall conductance. The theoretically calculated localization length at 41 meV is 4 nm to 9 nm for networks with 80%–95% connectivity, respectively (Fig. 4c). This stands in good qualitative agreement with the 9.8 nm inferred from measured conductance data (Fig. 4d). Now that we understand the limitations on electron transport that arise from disorder in our system, we can identify a realistic path towards more extended delocalization in epitaxial quantum dot solids. Despite great progress in semiconductor NC synthesis over the past 20 years, control of NC size dispersion to better than one atomic layer (roughly 3% of the diameter) remains elusive36–38 . Epitaxial connection of NCs into atomically coherent superstructures, however, is a relatively unexplored area, and we see many opportunities to enhance the structural fidelity. In particular, we see controlling the connection width disorder as the most promising path towards quantum dot solids that enable delocalized transport. To illustrate this point, we calculated the effect of three major sources of disorder on the localization length (see Supplementary
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1Se (1Sh ) levels. We confirmed experimentally that localization length does increase, and we have shown qualitative agreement with a theoretical calculation. Despite the high degree of order compared to common NC solids prepared with organic ligands39 , we find that the combined effect of multiple sources of disorder limit the electron wavefunction to just a few superlattice unit cells. Although size disorder of the NCs is unlikely to improve significantly, we show this limitation may be overcome through more uniform epitaxial connections. This inspires further study of the oriented-attachment mechanism, and we expect optimization will lead to quantum dot solids with desired properties. The results presented here therefore chart a course for future research towards realizing the exciting predicted properties of quasi-2D quantum dot solids.
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Figure 5 | Complete delocalization in a quantum dot solid. a, Effect of reducing disorder relative to experimental values on the maximum localization length. b, Calculated localization lengths for a quantum dot solid with no epitaxial connection disorder and near 100% connectivity while retaining the NC size dispersion. The dashed line separates localized from delocalized states. Error bars represent the standard deviation of multiple numerical calculations using the Monte Carlo method.
Information for details). In Fig. 5 we show that in our current sample the coupling energy disorder (due to variation of connection width) and NC size disorder are almost equally important and have more influence than missing connections (connectivity) with respect to delocalization. Reducing either the coupling disorder or the connectivity disorder alone to less than 1% of current values does not create delocalized states. Reducing both simultaneously, but without reducing NC size disorder, can lead to delocalized states, as we show in Fig. 5b. Importantly, this analysis illustrates the path forward to realize quasi-two-dimensional quantum dot solids in which the theoretically predicted properties can be realized. Although it has been shown that the average connection width strongly influences energetic coupling, reducing connection width disorder is clearly important for creating delocalized states and should be explored1,12 .
Outlook Oriented attachment is a generic method to strongly couple NC building blocks. In this work we have controlled the attachment process to self-assemble completely inorganic arrays of NCs with a high degree of order. We have fabricated square superlattices of PbSe NCs by a very simple, room-temperature, solution-based process. We show that epitaxial superlattices can be made with small translational disorder (near 3%) from NC building blocks with small size disorder (also near 3%). Transport through quantized states can be measured after mitigation of trap states by encapsulation. Of great interest is the expansion of the electron (hole) wavefunction as the chemical potential moves deeper into the 6
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NATURE MATERIALS DOI: 10.1038/NMAT4576 19. Anderson, N. C., Hendricks, M. P., Choi, J. J. & Owen, J. S. Ligand exchange and the stoichiometry of metal chalcogenide nanocrystals: spectroscopic observation of facile metal-carboxylate displacement and binding. J. Am. Chem. Soc. 135, 18536–18548 (2013). 20. Kitada, S., Kikuchi, E., Ohno, A., Aramaki, S. & Maenosono, S. Effect of diamine treatment on the conversion efficiency of PbSe colloidal quantum dot solar cells. Solid State Commun. 149, 1853–1855 (2009). 21. Murphy, J. E., Beard, M. C. & Nozik, A. J. Time-resolved photoconductivity of PbSe nanocrystal arrays. J. Phys. Chem. B 110, 25455–25461 (2006). 22. Vogel, W. & Hosemann, R. Evaluation of paracrystalline distortions from line broadening. Acta Crystallogr. A 26, 272–277 (1970). 23. Moreels, I. et al. Composition and size-dependent extinction coefficient of colloidal PbSe quantum dots. Chem. Mater. 19, 6101–6106 (2007). 24. Kim, D., Kim, D.-H., Lee, J.-H. & Grossman, J. C. Impact of stoichiometry on the electronic structure of PbS quantum dots. Phys. Rev. Lett. 110, 196802 (2013). 25. Oh, S. J. et al. Stoichiometric control of lead chalcogenide nanocrystal solids to enhance their electronic and optoelectronic device performance. ACS Nano 7, 2413–2421 (2013). 26. Voznyy, O. et al. A charge-orbital balance picture of doping in colloidal quantum dot solids. ACS Nano 6, 8448–8455 (2012). 27. Dai, Q. et al. Stability study of PbSe semiconductor nanocrystals over concentration, size, atmosphere, and light exposure. Langmuir 25, 12320–12324 (2009). 28. Law, M. et al. Structural, optical, and electrical properties of PbSe nanocrystal solids treated thermally or with simple amines. J. Am. Chem. Soc. 130, 5974–5985 (2008). 29. Leschkies, K. S., Kang, M. S., Aydil, E. S. & Norris, D. J. Influence of atmospheric gases on the electrical properties of PbSe quantum-dot films. J. Phys. Chem. C 114, 9988–9996 (2010). 30. Sykora, M. et al. Effect of air exposure on surface properties, electronic structure, and carrier relaxation in PbSe nanocrystals. ACS Nano 4, 2021–2034 (2010). 31. Guyot-Sionnest, P. Electrical transport in colloidal quantum dot films. J. Phys. Chem. Lett. 3, 1169–1175 (2012). 32. Shklovski, B. I. & Efros, A. L. Percolation theory and conductivity of strongly inhomogeneous media. Sov. Phys. Usp. 18, 845–862 (1975). 33. Luther, J. M. et al. Structural, optical, and electrical properties of self-assembled films of PbSe nanocrystals treated with 1, 2-ethanedithiol. ACS Nano 2, 271–280 (2008). 34. Sandeep, C. S. S. et al. High charge-carrier mobility enables exploitation of carrier multiplication in quantum-dot films. Nature Commun. 4, 2360 (2013).
ARTICLES 35. Liu, Y. et al. PbSe quantum dot field-effect transistors with air-stable electron mobilities above 7 cm2 /Vs. Nano Lett. 13, 1578–1587 (2013). 36. Cademartiri, L. et al. Multigram scale, solventless, and diffusion-controlled route to highly monodisperse PbS nanocrystals. J. Phys. Chem. B 110, 671–673 (2006). 37. Hendricks, M. P., Campos, M. P., Cleveland, G. T., Jen-LaPlante, I. & Owen, J. S. A tunable library of substituted thiourea precursors to metal sulfide nanocrystals. Science 348, 1226–1230 (2015). 38. Yu, W. W., Falkner, J. C., Shih, B. S. & Colvin, V. L. Preparation and characterization of monodisperse PbSe semiconductor nanocrystals in a noncoordinating solvent. Chem. Mater. 16, 3318–3322 (2004). 39. Weidman, M. C., Yager, K. G. & Tisdale, W. A. Interparticle spacing and structural ordering in superlattice PbS nanocrystal solids undergoing ligand exchange. Chem. Mater. 27, 474–482 (2014).
Acknowledgements This research was supported by the Cornell Center for Materials Research with funding from the NSF MRSEC program (DMR-1120296). 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’. B.H.S. was supported by the NSF IGERT grant DGE-0903653 and NSF GRFP grant DGE-1144153. This work was based on research conducted at the Cornell High Energy Synchrotron Source (CHESS), which is supported by the National Science Foundation and the National Institutes of Health/National Institute of General Medical Sciences under NSF award DMR-1332208. Charge transport measurements were performed in a facility supported by Award No. KUS-C1-018-02, made by King Abdullah University of Science and Technology (KAUST). The authors wish to thank CHESS staff scientist Detlef Smilgies for assistance with X-ray scattering experiments.
Author contributions K.W. prepared samples for electron microscopy and X-ray scattering, fabricated FET devices, performed electrical transport and X-ray scattering measurements, acquired bright-field TEM images and analysed the electron micrographs, X-ray data and electrical transport data. J.Y. performed calculations of electronic structure and localization length. B.H.S. acquired and analysed STEM micrographs. All authors contributed to the interpretation of results and preparation of the manuscript.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to T.H.
Competing financial interests The authors declare no competing financial interests.
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NATURE MATERIALS DOI: 10.1038/NMAT4576
ARTICLES Methods Note. Unless stated otherwise, all steps were performed in a nitrogen glovebox with less than 1 ppm oxygen and 0.1 ppm water. Nanocrystal synthesis. PbSe nanocrystals were synthesized by modification of the method reported by Yu and colleagues38 . Pb-oleate was prepared by heating 892 mg of PbO (Sigma, 99%) with 3.2 ml oleic acid (Sigma, tech. grade) in 15.5 ml 1-octadecene (Sigma, tech. grade) at 120 ◦ C under 100 mtorr vacuum for 1 h. The Se precursor (TOPSe) was prepared by dissolving 2.95 g Se pellets (Sigma, 99.99%) in 22 ml tri-n-octylphosphine (Strem, 97%) (1.7 M). To 883 µl of the TOPSe solution was added 15 µl diphenylphosphine (DPP) (Sigma, 98%). The Pb-oleate solution (2.36 ml) was heated to 160 ◦ C before injection of TOPSe:DPP under vigorous stirring, held at 160 ◦ C for 4 min, and quenched by addition of 4 ml toluene. Nanocrystals were purified three times by centrifugation with toluene and acetonitrile as solvent and antisolvent, respectively. Nanocrystal size, dispersion and concentration were measured in tetrachloroethylene (Sigma, 99.9%) by absorption spectroscopy using a Cary-5000 spectrometer23 . Field-effect transistor fabrication. Highly doped Si wafers (95%), 1,3,5-triallyl-1,3,5-triazine-2,4,6(1H,3H,5H)-trione (Sigma, 98%), and acetonitrile (Sigma, anhydrous 99.8%) (1:1:2 vol.). The encapsulation layer was deposited on the nanocrystal film by spin-casting at 700 r.p.m. for 30 s, then the speed was increased to 2,000 r.p.m. for 30 s. Solvent was removed by annealing the device on a hotplate at 50 ◦ C for 10 min. Finally, the encapsulant was photocured using a 4 W, 256 nm wavelength lamp at a distance of 3 cm for 10 min. Mixing, deposition and curing were performed in a nitrogen glovebox. Electrical characterization. Electrical measurements were performed in a Desert Cryogenics TTP4 probe station. After loading the sample, the chamber was pumped to 10−5 torr for at least 12 h at 295 K. Current and voltage were measured using an Agilent B1500A semiconductor parameter analyser. For temperature-dependent measurements, the temperature was scanned in both directions between 85 K and 350 K at intervals of 5 K or 10 K. The temperature was scanned repeatedly to ensure the sample had equilibrated and the measurement did not depend on temperature scan rate. Devices were stable over the duration of the measurement (see Supplementary Fig. 17). Electron microscopy. Samples were prepared for electron microscopy by transferring NC films to a carbon-coated grid using the Langmuir–Schaefer method. Excess ethylene glycol was removed by rinsing in anhydrous methanol and acetonitrile, before drying under 100 mtorr vacuum. All sample preparation was completed in a nitrogen glovebox. STEM images were acquired on a NION superSTEM at an accelerating voltage of 60 keV. Lens aberrations were corrected up to and including fifth order, and an aperture size of 30 mrad was used. Bright-field TEM images were acquired on an FEI T12 at an accelerating voltage of 120 keV. X-ray scattering. GISAXS and GIWAXS data were acquired at the Cornell High Energy Synchrotron Source, station D1. The X-ray beam was positioned in the channel of a transistor device to collect scattering from the superlattice film in the active region of the transistor. Radiation from a hard bend magnet source was filtered to produce a 1.157 Å wavelength beam with typical flux of 1012 photons s−1 mm−2 . The sample was positioned for grazing incidence at an angle of 0.25◦ . Small-angle and wide-angle scattering images were collected simultaneously by Pilatus 200k and 100k detectors, respectively.
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Charge transport and localization in atomically coherent quantum dot solids Kevin Whitham1, Jun Yang2, Benjamin H. Savitzky3, Lena F. Kourkoutis2,4, Frank Wise2 and Tobias Hanrath5* Epitaxial attachment of quantum dots into ordered superlattices enables the synthesis of quasi-two-dimensional materials that theoretically exhibit features such as Dirac cones and topological states, and have major potential for unprecedented optoelectronic devices. Initial studies found that disorder in these structures causes localization of electrons within a few lattice constants, and highlight the critical need for precise structural characterization and systematic assessment of the effects of disorder on transport. Here we fabricated superlattices with the quantum dots registered to within a single atomic bond length (limited by the polydispersity of the quantum dot building blocks), but missing a fraction (20%) of the epitaxial connections. Calculations of the electronic structure including the measured disorder account for the electron localization inferred from transport measurements. The calculations also show that improvement of the epitaxial connections will lead to completely delocalized electrons and may enable the observation of the remarkable properties predicted for these materials.
T
he unique electronic properties of nanostructured materials inspire future applications and present intriguing scientific challenges. Recent advances in the controlled formation of superlattices by epitaxial connection of semiconductor nanocrystals (NCs), also known as quantum dots, have the potential for creating materials with properties by design. The ability to control the atomic structure of the quantum dot building block (that is, size, shape and composition) and the geometry of the superstructure creates opportunities to synthesize and study designer materials. Calculations of such quantum dot solids with a dimensionality less than 2 forecast a rich electronic structure that differs profoundly from the corresponding quantum well1 . Although the predicted emergence of electronic phenomena including topological states and Dirac cones intrigue scientists1–4 , experimental support is still lacking. Prerequisite to an experimental realization of predicted electronic band structure is a transition from localized to delocalized electrons, which requires highly ordered superlattices and strong coupling between NCs (ref. 5). Recent experimental efforts to improve the electronic properties of quantum dot solids have focused on increasing inter-dot coupling by replacing organic ligands with inorganic ligands6,7 or by ligand removal8 . However, this approach leads to band formation only when the coupling energy overcomes energetic and translational disorder5,9 . Oriented attachment is a simple route to simultaneously increase inter-dot coupling and enforce translational order, as demonstrated in several recent reports10–13 . Initial studies find that charges in these structures are localized to within a few lattice constants owing to disorder14,15 . It has been shown recently that charge transport in such structures occurs through incoherent hopping between epitaxial segments16 . These findings highlight the critical need for precise analysis of the structure of the quantum dot solid and a quantitative assessment of the effects of disorder on transport.
We address this challenge by combining advanced structure characterization techniques (aberration-corrected electron microscopy and X-ray scattering) to precisely determine structure and disorder parameters as inputs for tight-binding electronic structure calculations. We discovered that translational order in these structures is controlled down to about a single atomic bond length, a limit imposed by the 4% size dispersion of the NC building blocks. To probe the intrinsic charge transport in these structures we performed variable temperature field-effect measurements in epitaxially connected quantum dot superlattice devices. By integrating structure analysis, transport measurements and electronic structure calculation, we validate that carriers are localized to a few dots owing to the heterogeneity of the epitaxial inter-dot connections. Importantly, our analysis shows that complete delocalization is possible with optimized dot-to-dot bonding, thereby providing a path forward to create quantum dot solids in which theoretically predicted properties can be realized.
Disorder and carrier localization Oriented attachment of PbSe NCs can be controlled by ligand desorption from selected facets10 . In contrast to previous approaches, where oriented attachment and superlattice assembly occur simultaneously, the two processes are temporally decoupled in the approach used in this work. We first fabricate highly ordered superlattices of ligand-capped NCs, then chemically induce oriented attachment in a separate step11,17,18 . Decoupling assembly and attachment transfers the high degree of order in the unconnected superlattice to the final structure. A colloidal dispersion of 6.5-nm-diameter PbSe NCs in hexane was dispersed on ethylene glycol. Subsequently ethylene diamine was injected into the ethylene glycol subphase. A common metal chelation agent, ethylene diamine has been used to remove or replace oleic acid ligands on PbSe NCs (refs 19–21). We propose that
1 Department
of Materials Science and Engineering, Cornell University, Ithaca, New York 14853, USA. 2 School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA. 3 Department of Physics, Cornell University, Ithaca, New York 14853, USA. 4 Kavli Institute for Nanoscale Science, Cornell University, Ithaca, New York 14853, USA. 5 School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, New York 14853, USA. *e-mail: [email protected] NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
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NATURE MATERIALS DOI: 10.1038/NMAT4576
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Figure 1 | Structural characterization of a nanocrystal superlattice. a,b, Annular dark-field scanning transmission electron microscopy shows superlattice structure (a) and atomic structure (b). c, Measured radial distribution function and calculated radial distribution function of a paracrystalline square lattice with 3.4% translational disorder. d, Grazing incidence small-angle X-ray scattering (GISAXS) image shows reflections of a square superlattice (SL). e, Grazing incidence wide-angle X-ray scattering (GIWAXS) image shows alignment of the atomic lattices (AL) of the nanocrystals.
ethylene diamine, an L-type ligand, binds selectively to Pb atoms on {100} facets, desorbing as L-Pb(O2 CR)2 (ref. 19). After deprotection, NCs undergo oriented attachment to fuse with {100} facets of proximate NCs. Similar results have been obtained by heating the glycol subphase11 . Amines have been used previously to achieve oriented attachment of NC films deposited on a solid substrate with less long-range order than we observe using a liquid substrate12 . The nature of the epitaxial connections between NCs is revealed by aberration-corrected scanning transmission electron microscopy (STEM) of monolayer areas, shown in Fig. 1b. Atomicresolution images show that oriented attachment occurs in the h100i crystallographic directions. Both STEM and X-ray scattering show the superlattice is simple square and less than 10 NC layers thick (see Supplementary Fig. 10). The axes of the atomic lattice and the superlattice are coherent as a result of the oriented-attachment self-assembly mechanism. To quantify translational order in the superlattice structure, we calculated the radial distribution function (RDF) based on NC locations directly extracted from STEM images. Using a paracrystalline model22 , we fit the measured RDF with only one free parameter (see Supplementary Information) and find the standard deviation of the nearest-neighbour distance to be 0.22 nm, or approximately one Pb–Se bond length (0.306 nm). We also fit the paracrystal model to our GISAXS data and find the standard deviation of the nearest-neighbour distance to be 0.3 nm, consistent with TEM results (see Supplementary Fig. 3). Importantly, because the standard deviation of the diameter of the as-synthesized NCs is approximately one Pb–Se bond length (see Supplementary Figs 1 and 2), we conclude that the spatial 2
coherence of the superlattice is fundamentally limited by the inherent polydispersity of the NCs. Besides translational disorder we quantified two other forms of disorder: the size of the NC cores and the width of the epitaxial connections. We developed an image analysis algorithm to collect unbiased statistics of NC diameters from STEM images. An image with 1,693 NCs yields an average diameter of 5.75 nm ± 0.19 nm, whereas GISAXS analysis yields 6.1 ± 0.3 nm (see Supplementary Figs 5 and 3). Analysis of an image with 10,122 connections yields a connection width of 3.26 nm ± 0.48 nm (see Supplementary Figs 6 and 7). These two forms of disorder have significant impact on energetic coupling and delocalization, as we show with tight-binding calculations discussed below. The high fidelity of the quantum dot solid and the epitaxial nature of the inter-dot bond raises an intriguing question about delocalization of electrons and holes. We address this question by measuring temperature-dependent charge transport with a field-effect transistor (FET), the structure of which is shown in Fig. 2a. The FET is designed to modulate conductance by changing the average chemical potential of charge carriers. In disordered materials, carriers are conducted through a network of localized states distributed spatially and energetically. STEM and X-ray data can be used to model the density of states. Spatial density can be extracted from the radial distribution, and the distribution of states in energy can be inferred from measurement of the NC diameters23 . We used FET transport measurements to test the validity of this simple density-of-states model and to estimate the localization length.
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NATURE MATERIALS DOI: 10.1038/NMAT4576
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Figure 2 | Charge transport measurement by field-effect transistor. a, Cross-sectional schematic of the field-effect transistor. The gate is formed by highly doped silicon (Si++) with a 200 nm dielectric layer of SiO2 . The source and drain electrodes were patterned on the SiO2 to form a channel 100 µm × 3 mm. A layer of photocured polymer served as a barrier between the NC layer and ambient water and oxygen. b, Drain current (ID )-Gate voltage (VG ) transfer curves show ambipolar transport with electron current at positive VG and hole current at negative VG . The source–drain bias was 1 V. The conductance was measured every 5 K from 85 K to 245 K. c, Arrhenius plot of electron conductance at a gate bias of 22 V with a source–drain bias of 1 V. The solid line is a linear fit to the higher temperature data. d, Log–log plot of the data in c. Straight lines illustrate that the data are not a single power law over this temperature range.
By constructing the density-of-states model from the physical structure of the superlattice, we assumed defect states to be negligible. However, removal of ligands could expose the NCs to adventitious water or oxygen, creating electronic defects8,24–26 . The sensitivity of PbSe NCs to adsorbed molecules has been well documented27–30 . To achieve negligible defect density we encapsulated devices with a photocurable polymer (see Methods and Supplementary Fig. 12). With encapsulation we observed negligible hysteresis in the FET characteristics at temperatures between 85 K and 245 K (see Supplementary Fig. 11). Above 245 K we did observe hysteresis, and therefore limit our analysis to data below 245 K. Transfer curves (current versus gate voltage) in Fig. 2b show ambipolar transport (see also Supplementary Fig. 11). We find that electron and hole transport are thermally activated (hopping), with conductance decreasing exponentially with decreasing temperature, a clear sign of carrier localization. Hopping conductance is p commonly interpreted using the expression G = G0 e−(T0 /T ) , where p depends on the type of hopping and T0 depends on material properties31 . A plot of ln(G) versus ln(T −1 ) in Fig. 2d shows that a single value of p fails to describe the data over the full temperature range. Instead, the data suggests that there is a transition between two regimes. In all types of hopping transport, conductance is described by G = G00 e−ξ , where ξ = (2r/a) + (ε/kT ), r is the hopping distance, a is the electron or hole localization length, ε is the hopping energy32 , with k the Boltzmann constant. In the limit of 2r/a ε/kT , the hopping distance is the nearest-neighbour distance; called
nearest-neighbour hopping (NNH). In NNH the term 2r/a is independent of temperature, therefore conductance is given by G = G000 e−εNN /kT , where εNN is the NNH energy. Figure 2c shows this Arrhenius-type behaviour at high temperatures. Conductance at low temperature deviates from Arrhenius behaviour. In this regime the hopping distance varies with temperature; called variable-range hopping (VRH). The transition temperature between NNH and VRH can be determined by fitting two linear regimes to ln(G) versus ln(T −1 ), as in Fig. 2d. Analysis of transport data in Fig. 3 clearly shows the transition temperature (Tc ) and NNH energy (εNN ) directly relate to the applied gate voltage (VG ). Increasing the gate bias moves the chemical potential to a higher density of states. The average hopping energy decreases as the density of states increases. This is the physical reason for the relation between εNN and VG . Below, we will show that the localization length can be estimated from the transition temperature Tc . At high temperature, hopping occurs between nearest neighbours. As the temperature is decreased, the density of states accessible to a charge carrier within a few kT of the chemical potential becomes increasingly sparse. At a certain temperature, hopping longer distances becomes more favourable to minimize the energy between initial and final states32 . This transition occurs at a higher temperature for more delocalized states because these carriers can more easily ‘reach’ distant states. Changing the gate voltage shifts the chemical potential, thereby probing different states, whereas changing the temperature varies the energy window available to carriers near the chemical potential. By varying both, we determined transition temperatures for a range of chemical
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NATURE MATERIALS DOI: 10.1038/NMAT4576
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potentials, and from the transition temperatures estimated the localization lengths. Our procedure follows the analysis of hopping transport in doped semiconductors32 . We adapted the analysis for the density of states inferred from the physical structure of the superlattice (see Supplementary Information). The result in Fig. 4d shows the localization lengths calculated from FET transport measurements. We find that charge carriers are localized to just a few NCs at a gate bias of ±40 V. We note that Coulomb blockade should not be an important factor in the measured temperature range (see Supplementary Information). We measure mobilities of 0.54 cm2 V−1 s−1 for holes and 0.2 cm2 V−1 s−1 for electrons at 245 K (see Supplementary Fig. 11), Despite relatively good mobilities compared to ligand-exchanged quantum dot solids28,33–35 , the localization length reveals that transport is still far from band-like. Before discussing the theoretical prediction of the localization length in the next section, it is important to note that the localization lengths shown in Fig. 4d are not the localization lengths at the chemical potential, but the smallest localization lengths within the energy window accessible to carriers near the chemical potential32 . Recently reported field-effect mobilities of 3–13 cm2 V−1 s−1 were measured in single superlattice grains16 . The difference between 3 and 13 cm2 V−1 s−1 reported recently and 0.2 cm2 V−1 s−1 in this work may be due to several factors. First, the values of 3–13 cm2 V−1 s−1 were measured with unintentionally n-doped samples, whereas we fabricated ambipolar samples, preventing unintentional doping via encapsulation. Second, we measured 4
0.2 cm2 V−1 s−1 at 245 K to avoid hysteresis, whereas values of 3–13 cm2 V−1 s−1 were measured at 290 K without accounting for hysteresis. Finally, we estimate the effect of grain boundaries on mobility. If coherent transport occurs over a few NCs (20 nm), then transport across 0.1 mm requires at least 5 × 103 hops. Introducing grain boundaries every 100 nm increases the number of hops to 6 × 103 , and effectively decreases the coherence length to 16.7 nm. Because the mobility µ is related to the diffusion coefficient D by the Einstein relation D = µkT /q, and D ∝ r 2 , where the distance between hops r is the coherence length, this reduction in coherence length reduces the mobility by 30%.
Calculation of electronic structure The strong localization of electrons and holes may seem surprising in view of the high fidelity of the quantum dot solid shown in Fig. 1. To gain a deeper understanding of the electronic structure of the quantum dot solid and the sensitivity to disorder, we performed tight-binding calculations that directly incorporate disorder parameters determined from TEM and GISAXS. A recent calculation of the electronic band structure of a square superlattice of PbSe NCs predicts the evolution of delocalized mini-bands from quantized NC states1 . We reproduced the band structure of a perfect superlattice, and then impressed disorder on the structure to determine the localization length in a realistic sample. Inclusion of disorder in an atomistic calculation of a structure with a length scale of hundreds of nanometres is computationally prohibitive. Conveniently, the mini-band structure of a 2D square superlattice of PbSe NCs has been calculated recently by an
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NATURE MATERIALS DOI: 10.1038/NMAT4576
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Figure 4 | Extent of delocalization in a disordered superlattice. a, Model density of 1Se or 1Sh states based on experimental measurements of the superlattice structure. b, Relationship between the NNH energy and chemical potential for the density-of-states model in a. Points were calculated numerically, the solid line is a power law approximation. Inset arrows show the ranges of hopping energies measured for electrons and holes at gate voltages between 0 and ±40 V. Open circles mark the measured hopping energies of electrons (holes) for a gate bias of 40 V (−40 V). c, Theoretical localization length of electrons (holes) in 1Se (1Sh ) states for a range of connectivity values. Zero energy refers to the average 1Se or 1Sh energy level. The lowest energy states accessible to electrons (holes) at a gate bias of 40 V (−40 V) are indicated. Error bars represent the standard deviation of multiple numerical calculations using the Monte Carlo method. Localization lengths are given in nanometres on the left-hand scale and by dimensionless units of a0 on the right-hand scale, where a0 represents the localization length of a charge carrier in a completely isolated NC, equal to the NC radius. d, Localization lengths of electrons and holes calculated from the measured temperature dependence of the conductance. Error bars reflect the uncertainty of the transition temperature.
atomistic tight-binding approach1,3 . We fit the band structure from the atomistic calculation using an effective Hamiltonian (see Supplementary Information for details) and then use the effective Hamiltonian to calculate the localization length in a large disordered system. We modelled the two sources of disorder observed in the experiment: fluctuation of the NC size, and fluctuation of the width of the epitaxial connections between NCs. Size fluctuation is modelled as a Gaussian distribution of the energy levels, where the size distribution is related to the energy distribution because of quantum confinement (see Supplementary Information). The fluctuation of the width of the epitaxial connections is modelled as a distribution of coupling energies estimated from the atomistic calculation (see Supplementary Fig. 14 for the calculated densities of states). Analysis of STEM images gives accurate distributions for the NC diameter and epitaxial connection width. Approximately 80% of neighbouring NCs in our STEM images are epitaxially connected; we include the missing bonds as zero-width connections in our calculation. In Fig. 4c we show the localization lengths predicted considering the disorder measured directly from our samples. The maximum localization length is 50 nm for states near the average NC energy level. In domains larger than ∼100 nm, hopping transport will remain dominant. Furthermore, the sites with the largest localization lengths do not determine conductance through a disordered system; conductance is limited by the smallest localization lengths in the percolation network.
We now compare the experimental result with the theoretical calculation. With the highest applied gate voltage (40 V), we find the NNH energy for electrons is 32 meV. This corresponds to a chemical potential 8.5 meV below the average NC energy level. The lowest energy state accessible to an electron is therefore 41 meV from the average NC energy level. Hopping transitions to NCs at this lowest energy level limit the overall conductance. The theoretically calculated localization length at 41 meV is 4 nm to 9 nm for networks with 80%–95% connectivity, respectively (Fig. 4c). This stands in good qualitative agreement with the 9.8 nm inferred from measured conductance data (Fig. 4d). Now that we understand the limitations on electron transport that arise from disorder in our system, we can identify a realistic path towards more extended delocalization in epitaxial quantum dot solids. Despite great progress in semiconductor NC synthesis over the past 20 years, control of NC size dispersion to better than one atomic layer (roughly 3% of the diameter) remains elusive36–38 . Epitaxial connection of NCs into atomically coherent superstructures, however, is a relatively unexplored area, and we see many opportunities to enhance the structural fidelity. In particular, we see controlling the connection width disorder as the most promising path towards quantum dot solids that enable delocalized transport. To illustrate this point, we calculated the effect of three major sources of disorder on the localization length (see Supplementary
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NATURE MATERIALS DOI: 10.1038/NMAT4576
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1Se (1Sh ) levels. We confirmed experimentally that localization length does increase, and we have shown qualitative agreement with a theoretical calculation. Despite the high degree of order compared to common NC solids prepared with organic ligands39 , we find that the combined effect of multiple sources of disorder limit the electron wavefunction to just a few superlattice unit cells. Although size disorder of the NCs is unlikely to improve significantly, we show this limitation may be overcome through more uniform epitaxial connections. This inspires further study of the oriented-attachment mechanism, and we expect optimization will lead to quantum dot solids with desired properties. The results presented here therefore chart a course for future research towards realizing the exciting predicted properties of quasi-2D quantum dot solids.
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Received 26 October 2015; accepted 21 January 2016; published online 22 February 2016
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Figure 5 | Complete delocalization in a quantum dot solid. a, Effect of reducing disorder relative to experimental values on the maximum localization length. b, Calculated localization lengths for a quantum dot solid with no epitaxial connection disorder and near 100% connectivity while retaining the NC size dispersion. The dashed line separates localized from delocalized states. Error bars represent the standard deviation of multiple numerical calculations using the Monte Carlo method.
Information for details). In Fig. 5 we show that in our current sample the coupling energy disorder (due to variation of connection width) and NC size disorder are almost equally important and have more influence than missing connections (connectivity) with respect to delocalization. Reducing either the coupling disorder or the connectivity disorder alone to less than 1% of current values does not create delocalized states. Reducing both simultaneously, but without reducing NC size disorder, can lead to delocalized states, as we show in Fig. 5b. Importantly, this analysis illustrates the path forward to realize quasi-two-dimensional quantum dot solids in which the theoretically predicted properties can be realized. Although it has been shown that the average connection width strongly influences energetic coupling, reducing connection width disorder is clearly important for creating delocalized states and should be explored1,12 .
Outlook Oriented attachment is a generic method to strongly couple NC building blocks. In this work we have controlled the attachment process to self-assemble completely inorganic arrays of NCs with a high degree of order. We have fabricated square superlattices of PbSe NCs by a very simple, room-temperature, solution-based process. We show that epitaxial superlattices can be made with small translational disorder (near 3%) from NC building blocks with small size disorder (also near 3%). Transport through quantized states can be measured after mitigation of trap states by encapsulation. Of great interest is the expansion of the electron (hole) wavefunction as the chemical potential moves deeper into the 6
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Acknowledgements This research was supported by the Cornell Center for Materials Research with funding from the NSF MRSEC program (DMR-1120296). 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’. B.H.S. was supported by the NSF IGERT grant DGE-0903653 and NSF GRFP grant DGE-1144153. This work was based on research conducted at the Cornell High Energy Synchrotron Source (CHESS), which is supported by the National Science Foundation and the National Institutes of Health/National Institute of General Medical Sciences under NSF award DMR-1332208. Charge transport measurements were performed in a facility supported by Award No. KUS-C1-018-02, made by King Abdullah University of Science and Technology (KAUST). The authors wish to thank CHESS staff scientist Detlef Smilgies for assistance with X-ray scattering experiments.
Author contributions K.W. prepared samples for electron microscopy and X-ray scattering, fabricated FET devices, performed electrical transport and X-ray scattering measurements, acquired bright-field TEM images and analysed the electron micrographs, X-ray data and electrical transport data. J.Y. performed calculations of electronic structure and localization length. B.H.S. acquired and analysed STEM micrographs. All authors contributed to the interpretation of results and preparation of the manuscript.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to T.H.
Competing financial interests The authors declare no competing financial interests.
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NATURE MATERIALS DOI: 10.1038/NMAT4576
ARTICLES Methods Note. Unless stated otherwise, all steps were performed in a nitrogen glovebox with less than 1 ppm oxygen and 0.1 ppm water. Nanocrystal synthesis. PbSe nanocrystals were synthesized by modification of the method reported by Yu and colleagues38 . Pb-oleate was prepared by heating 892 mg of PbO (Sigma, 99%) with 3.2 ml oleic acid (Sigma, tech. grade) in 15.5 ml 1-octadecene (Sigma, tech. grade) at 120 ◦ C under 100 mtorr vacuum for 1 h. The Se precursor (TOPSe) was prepared by dissolving 2.95 g Se pellets (Sigma, 99.99%) in 22 ml tri-n-octylphosphine (Strem, 97%) (1.7 M). To 883 µl of the TOPSe solution was added 15 µl diphenylphosphine (DPP) (Sigma, 98%). The Pb-oleate solution (2.36 ml) was heated to 160 ◦ C before injection of TOPSe:DPP under vigorous stirring, held at 160 ◦ C for 4 min, and quenched by addition of 4 ml toluene. Nanocrystals were purified three times by centrifugation with toluene and acetonitrile as solvent and antisolvent, respectively. Nanocrystal size, dispersion and concentration were measured in tetrachloroethylene (Sigma, 99.9%) by absorption spectroscopy using a Cary-5000 spectrometer23 . Field-effect transistor fabrication. Highly doped Si wafers (95%), 1,3,5-triallyl-1,3,5-triazine-2,4,6(1H,3H,5H)-trione (Sigma, 98%), and acetonitrile (Sigma, anhydrous 99.8%) (1:1:2 vol.). The encapsulation layer was deposited on the nanocrystal film by spin-casting at 700 r.p.m. for 30 s, then the speed was increased to 2,000 r.p.m. for 30 s. Solvent was removed by annealing the device on a hotplate at 50 ◦ C for 10 min. Finally, the encapsulant was photocured using a 4 W, 256 nm wavelength lamp at a distance of 3 cm for 10 min. Mixing, deposition and curing were performed in a nitrogen glovebox. Electrical characterization. Electrical measurements were performed in a Desert Cryogenics TTP4 probe station. After loading the sample, the chamber was pumped to 10−5 torr for at least 12 h at 295 K. Current and voltage were measured using an Agilent B1500A semiconductor parameter analyser. For temperature-dependent measurements, the temperature was scanned in both directions between 85 K and 350 K at intervals of 5 K or 10 K. The temperature was scanned repeatedly to ensure the sample had equilibrated and the measurement did not depend on temperature scan rate. Devices were stable over the duration of the measurement (see Supplementary Fig. 17). Electron microscopy. Samples were prepared for electron microscopy by transferring NC films to a carbon-coated grid using the Langmuir–Schaefer method. Excess ethylene glycol was removed by rinsing in anhydrous methanol and acetonitrile, before drying under 100 mtorr vacuum. All sample preparation was completed in a nitrogen glovebox. STEM images were acquired on a NION superSTEM at an accelerating voltage of 60 keV. Lens aberrations were corrected up to and including fifth order, and an aperture size of 30 mrad was used. Bright-field TEM images were acquired on an FEI T12 at an accelerating voltage of 120 keV. X-ray scattering. GISAXS and GIWAXS data were acquired at the Cornell High Energy Synchrotron Source, station D1. The X-ray beam was positioned in the channel of a transistor device to collect scattering from the superlattice film in the active region of the transistor. Radiation from a hard bend magnet source was filtered to produce a 1.157 Å wavelength beam with typical flux of 1012 photons s−1 mm−2 . The sample was positioned for grazing incidence at an angle of 0.25◦ . Small-angle and wide-angle scattering images were collected simultaneously by Pilatus 200k and 100k detectors, respectively.
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