Letter pubs.acs.org/NanoLett
Variation of Energy Density of States in Quantum Dot Arrays due to Interparticle Electronic Coupling Manca Logar,† Shicheng Xu,† Shinjita Acharya,† and Fritz B. Prinz*,†,‡ †
Department of Mechanical Engineering and ‡Department of Material Science and Engineering, Stanford University, Stanford, California 94305, United States S Supporting Information *
ABSTRACT: Subnanometer-resolved local electron energy structure was measured in PbS quantum dot superlattice arrays using valence electron energy loss spectroscopy with scanning transmission electron microscopy. We found smaller values of the lowest available transition energies and an increased density of electronic states in the space between quantum dots with shorter interparticle spacing, indicating extension of carrier wave functions as a result of interparticle electronic coupling. A quantum simulation verified both trends and illustrated the wave function extension effect.
KEYWORDS: Energy density of states, STEM-EELS, electronic coupling, quantum dot, superlattice interparticle distance decreases in thin films of monodisperse QDs.19−24 However, such red shifting can be caused by changes in the dielectric environment23 in addition to changes in electronic coupling24 between the dots. It is difficult to isolate these factors when examining the macroscopic optical transition energies of the arrays, especially considering the inhomogeneity in the dielectric environment introduced by the inevitable size and shape variation of the QDs in the sample. On the microscopic level, scanning tunneling microscopy and spectroscopy (STM-STS) have been used to investigate the electronic local density of states (LDOS) of single nanoparticles25−29 and their superlattice films. Steiner et al.30 studied arrays assembled by trioctylphosphine capped InAs QDs and measured the LDOS at different regions of the array. Band gap reduction was verified by comparing the density of states in isolated and arrayed QDs. Liljeroth et al.31 studied oleic acid capped PbSe QD arrays and found variations in LDOS due to local microscopic disorders. Reductions of the energy gap were only found in certain locations where the electron and hole wave functions were delocalized. Overgaag et al.32 investigated the LDOS of single PbSe QDs arrays stabilized by CdSe QDs. While no significant difference was observed in the zeroconductance gap between isolated and aggregated oleic acid capped QDs, the first conduction-level resonances were found to broaden as the number of particles in the molecular aggregate increased.
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rdered arrays of colloidal quantum dots (superlattices) exhibiting collective optical and electrical phenomena have gained considerable interest among the scientific community with promising applications in photovoltaics,1−3 optoelectronics,4,5 information technology,6 and catalysis.7−9 Mediated by capping ligands, the quantum dots (QDs) can be treated as artificial atoms and assembled into higher order nanostructures such as metamaterials and supraparticles. These structures have programmable physical and chemical properties,10−12 widening the array of available functional materials for relevant applications. Concomitant existence of quantum confinement and high packing density can be realized by using short capping ligands, a technique that has been found critical for performance in a variety of devices. Exchanging long chain ligands with shorter ones leads to successful manipulation of various properties of QD films, such as carrier mobility, which can be greatly improved.13−16 Shorter ligands also enhance exciton recombination in a QD film, which leads to efficiency improvement in infrared LEDs.5 Additionally, stronger interparticle coupling can facilitate dissociation of photogenerated excitons,17 which contributes to increased efficiency of solar cells fabricated from atomically passivated QDs.18 By reducing interparticle spacing in QD arrays, electronic coupling among QDs increases, which modifies the energy states and their corresponding density. Gaining in-depth insight into the electronic coupling phenomena among the QD arrays is of great importance for the precise engineering of superstructures. Most work in the area is presently focused on the effect of different capping ligands on electron energies. On the macroscopic level, the effect is seen as a red shift of the optical absorption peak as the © XXXX American Chemical Society
Received: December 3, 2014 Revised: January 26, 2015
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DOI: 10.1021/nl5046507 Nano Lett. XXXX, XXX, XXX−XXX
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Nano Letters
Figure 1. HRTEM images of hexagonal arrays of PbS quantum dots capped by (a) butylamine (denoted C4), (b) octylamine (C8), (c) dodecylamine (C12), and (d) oleic acid (C18). The diameter of the quantum dots is approximately 5.0 nm. (e) Schematic showing variation in spacing of QDs as a function of ligand length. Average values of d1, d2, d3, and d4 are 0.6, 1.0, 1.5, and 2.0 nm.
Figure 2. Variations in the lowest available electron transition energy (LATE) across neighboring PbS QDs with different interparticle spacings. EEL spectra taken at the positions of the red dots in the ADF images were used to extract LATE line profiles across QDs capped by (a) butylamine, (b) octylamine, (c) dodecylamine, and (d) oleic acid. The arrows denote EELS acquisition directions.
reproducing previously observed changes in the macroscopic band gap of the lattices, our results reveal changes in the local electronic structure in between QDs as their spacing is diminished, providing direct experimental observation of electronic coupling of the dots. To explore the theoretical basis for our observations, we present the results of a quantum mechanical finite element simulation of our samples. PbS QDs capped with various organic ligands self-assemble in close-packed arrays with hexagonal symmetry (Figure 1). In this study, the synthesis of PbS quantum dots was performed using a modification of the standard colloidal procedure described by Hines et al.,34 with lead acetate as the lead precursor and bis(trimethylsilyl)-sulfide as the sulfur precursor (details presented in the Supporting Information). Capping of the PbS QDs with organic ligands using a solution-based ligand exchange process4,35,36 not only suppresses surface oxidation, enabling the formation of chemically homogeneous arrays of dots with rock-salt crystal structure (Supporting Information Figure S2) but also allows for tailoring of the interparticle spacing, which is a parameter crucial to this study. Spacing
To date, no such microscopic studies have provided evidence correlating electronic coupling with changes in electronic structure in the region between individual coupled QDs. Moreover, the interpretation of tunneling spectra from STS on such samples is clouded by unknown shape effects from the tip and by the Schottky barrier in between the tip and sample. Given the limited aspect ratio of an STM probe and the threedimensional topologies of QDs, even resolving the geometric features of QDs is a challenging task, not to mention discerning the variation of local density of states within and between QDs. In this work, we present localized measurements of the electronic energy structure of lead sulfide (PbS) QD pairs, as well as spatial variation in the lowest available electron transition energy (LATE) in a set of monodisperse QD superlattices capped with different size ligands. The measurements were performed using electron energy-loss spectroscopy in a (scanning) transmission electron microscope [(S)TEMEELS].33 The spatial resolution of this technique allows independent measurement of the joint local density of states (JLDOS) inside of and between individual QDs. In addition to B
DOI: 10.1021/nl5046507 Nano Lett. XXXX, XXX, XXX−XXX
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Figure 3. The 2D maps of integrated EELS signal intensity over the energy range 0.9−2 eV (left panes) and EELS spectra taken at the locations indicated by the red dots in Figure 2 over PbS QDs (right panes) capped by (a) butylamine, (b) octylamine, (c) dodecylamine, and (d) oleic acid.
In EELS measurements, the sample is exposed to a beam of electrons with a narrow range of kinetic energies and the amount of energy lost by the transmitted electrons is measured by an EEL spectrometer. The spectra exhibit a characteristic zero-loss peak (ZLP) from electrons transmitted without inelastic scattering. After ZLP removal, information about the material band structure is obtained from the signal generated by inelastically scattered electrons with energy loss below 10 eV. The energy onset of inelastic scattering indicates the LATE at the beam position (Supporting Information Figures S3 and S5). Recently, the energy and spatial resolution of this technique has been greatly improved through the use of monochromated valence electron energy-loss spectroscopy (VEELS).37 It has been shown that for direct band gap semiconductors the correct values of the band gap can be obtained by VEELS acquisition in samples of thickness less than half the total mean free path for inelastic scattering, where the effect of Cerenkov radiation can be efficiently minimized.38,39 STEM-VEELS has also been applied to evaluate the size and shape dependence of the band gap of CdSe and PbS QDs,33,40 as well as the thickness dependence of the band gap of Si.41 Here, EELS line profiles across PbS QDs were obtained with a monochromated 80 kV electron beam focused to a 0.3 nm diameter. Figure 2 shows the LATE from Fourier deconvoluted EEL spectra taken at 0.5 nm intervals along lines crossing neighboring dots from hexagonal arrays with various ligand lengths (more details in Supporting Information, Figure S6). The line profiles reveal the spatial dependence of the allowed electronic transitions in the space within and between neighboring QDs. A gradual increase of the LATE from the center of the PbS QDs toward the edge is observed, which is in agreement with previous experiments.33 This effect is attributed to the reduced wave function density near the dot edges as a result of the confinement geometry. To further compare the LATE of QD samples with different interparticle spacings, two-dimensional maps of integrated EELS signal intensity from neighboring PbS QDs over the
Figure 4. Boxplot of the LATE among quantum dot pairs from hexagonal arrays, capped by C4, C8, C12, and C18. Twenty pairs were used in each case. The red line indicates the median of the sample, the black lines indicate the minimum and maximum, and blue lines indicate the 25 percentile and 75 percentile.
control was achieved by exchanging the as-synthesized oleic acid ligands (denoted C18) with three alkylamine ligands of variable lengths, namely, dodecylamine (C12), octylamine (C8), and butylamine (C4). The results of this process are illustrated in Figure 1. Average particle size and interparticle spacing in the assemblies of ligand-capped PbS QDs was measured from HRTEM images of 100 pairs of QDs. The average dot size is 5.0 ± 0.13 nm, and dots capped with C4, C8, C12, and C18 show spacings of 0.6 ± 0.2, 1.0 ± 0.3, 1.5 ± 0.2, and 2.0 ± 0.8 nm, respectively. STEM-EELS was used on the prepared samples to measure variations in the availability of low-energy electron transitions in the space between neighboring QDs as a function of interparticle spacing. Whereas the previously discussed STMSTS technique measures the energy density of individual electron states, the technique used here enables the direct measurement of interband electron transitions and avoids the tip geometry complications encountered with scanning probes. C
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Figure 5. Top view (a) and side view (b) of the simulation cell used to calculate JLDOS line-profiles across pairs of quantum dot pairs separated by (d) 0.5, (e) 1.0, (f) 1.5, and (g) 2.0 nm. (c) A side view (from the perspective in (b)) of the electron probability density for states near 2.1 eV for each value of separation.
edges.42 In analogy with theoretical treatment of the tunneling current in STM, we can treat the EELS signal as proportional to the JLDOS in the sample, which is the point-localized analogue of the standard joint density of states (JDOS)33
energy range 0.9−2 eV are presented in Figure 3. Significant integrated signal can be observed between the QDs with shorter interparticle spacing (C4 and C8), reducing noticeably as the interparticle spacing is increased (C12 and C18). Aligned to the right of each integrated heat map is a series of EEL spectra taken at the corresponding lateral positions over the red dots as indicated in Figure 2. These spectra reveal the nonzero EELS signal in the area between the closely spaced dots, which gradually disappears as the QD spacing increases. Brief reflection on the physical origin of the EELS signal suggests that the increased signal observed between the more proximal QDs is a result of their electronic coupling. In the low loss energy range of the EEL spectrum, the signal intensity can be approximated using Fermi’s golden rule for overlapping electron and hole states near the semiconductor band
JLDOS(r, ΔE) =
∑ |φhj(r)|2 |φei(r)|2 δ(Ehj − Eei , ΔE) i,j
(1)
where φe(h)i(j) is the ith (jth) electron (hole) wave function of the system and E is the corresponding energy of the state. Electron transitions that contribute to the EELS signal at a given location come from pairs of electron and hole states with the right energy difference that coexist there, in proportion to the probability density of the wave functions associated with those states. The sharp drop in the observed signal at the edges D
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states do not extend significantly outside the QD boundaries, explaining the higher onset of the JLDOS (and hence EELS signal) in between the dots than at the center of the dots. Close inspection of the QD-center spectra in Figure 5d,g reveals a reduction in the macroscopic LATE as the QDs are brought closer, in agreement with the experiment. The magnitude of the shift is smaller than measured (37 meV vs 40 meV from C18 to C4). Such a disagreement is to be expected from this model given the various approximations made. These approximations include an energy-independent value for the effective carrier mass that is only accurate near the band edges, an interface energy barrier independent of QD spacing, and perfect monodispersity of the QDs in the array. (For QDs with average size of 5 nm, a variation of 0.2 nm can change transition energies by 26 meV.) Despite the uncertainty in absolute value, this simple quantum mechanical model reproduces the trends observed in this study and provides an explanation of the observed EELS signal between the QDs based on electronic coupling as they are spaced more closely in arrays.
of the more largely spaced QDs (Figure 3c,d) indicates the spatial limit of the electron and hole wave functions in isolated dots at the energies shown. The appearance of a signal outside of those limits (between the QDs) as the QDs are brought closer together indicates an extension of the wave functions into that space as result of electronic coupling between the QDs. To verify our procedure against other measurements of the energetic effects of QD coupling, we collected statistics on the LATE from 20 QD pairs in each sample. Figure 4 shows the smallest values of the LATE found in the region of each dot pair (typically located at the center of a dot), corresponding to the absorption onset that would be observed in a macroscopic optical measurement of the array. Previous such measurements on dispersed PbS QDs have found an onset of approximately 0.9 eV for 5.0 nm diameter dots,43 similar to the value that we have measured. With smaller interparticle spacing, the macroscopic LATE decreases, in agreement with the optical absorption red shift observed in the previously mentioned studies. This is because the transition energy decreases with increasing QD coupling energy β as the electron wave functions overlap24 ΔE = ΔE0 − 6(|βe| + |βh |)
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ASSOCIATED CONTENT
S Supporting Information *
(2)
Supporting Information contains details of PbS QDs synthesis, ligand exchange, and their HRTEM characterizations, procedures of ZLP removal and LATE determination in STEMEELS, and simulation results on JLDOS as functions of ligand band gap and alignment. This material is available free of charge via the Internet at http://pubs.acs.org.
To gain further insight into our experimental results we performed a quantum finite element analysis of spherical QDs in 2D hexagonal arrays. Electronic structure was calculated with effective mass approximation,44,45 by solving the Schrödinger equation for this geometry using a k·p method, where Vr is the potential energy results from the physical position r, k is the superlattice vector (sampled 4 × 4), i is the imaginary unit, and φk and Ek are the corresponding wave function and energy
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected].
⎤ ⎡ ℏ2 ⎡ 1 2i·k ℏ2k 2 ⎤ − ∇·⎢ ∇φk (r) + φ(r)⎥ + ⎢V (r) + ⎥φ (r) = E k φk (r) 2 ⎣ meff meff 2meff ⎦ k ⎦ ⎣
Author Contributions
M.L. and S.X. contributed equally.
(3)
Notes
The calculation was performed using a hexagonal unit cell, as shown in Figure 5a,b. For the PbS QDs, the electron and hole effective mass were both approximated to be 0.080 m0, and the carrier effective mass within the barrier material was set to m0. Assuming an energy-symmetric interface band alignment and considering the 5 eV HOMO−LUMO energy gap of alkane chains and 0.4 eV bulk band gap of PbS, an energy barrier of 2.3 eV was used for both electrons and holes at the QD boundaries. (The effects of asymmetry in the band alignment are presented in the Supporting Information.) After solving for a sufficient set of low-energy eigenstates, the JLDOS was calculated at different points in space using eq 1. The resulting spectra at points along lines bisecting QD pairs are presented in Figure 5d,g as a function of QD spacing. (Delta functions from eq 1 are plotted as Lorentzian functions with room-temperature thermal energy broadening.) A similar trend to the experimental measurement is observed in which electronic transitions with lower energies become available between the QDs as they are brought closer. The wave function extension effect earlier suggested to be responsible for this can be seen in Figure 5c. Here we show the probability densities for a wave function at approximately 2.1 eV for several different values of QD separation. As the QDs move closer together, the wave function lobes extend further outside the atomic boundaries of the dots. They eventually connect, contributing to an increased JLDOS between the QDs for transitions involving that wave function. The wave functions corresponding to lower energy
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by Honda R&D and the Center on Nanostructuring for Efficient Energy Conversion (CNEEC) at Stanford University, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award DE-SC0001060. The authors also acknowledge helpful discussions with D. Thian, P. Schindler, A. L. Dadlani, and P. B. Van Stockum of the Nanoscale Prototyping Laboratory, Stanford University, and E. Shevchenko of Argonne National Laboratory.
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REFERENCES
(1) Kramer, I. J.; Sargent, E. H. Chem. Rev. 2014, 114, 863−882. (2) Sargent, E. H. Nat. Photonics 2012, 6, 133−135. (3) Ip, A. H.; Thon, S. M.; Hoogland, S.; Voznyy, O.; Zhitomirsky, D.; Debnath, R.; Levina, L.; Rollny, L. R.; Carey, G. H.; Fischer, A.; Kemp, K. W.; Kramer, I. J.; Ning, Z.; Labelle, A. J.; Chou, K. W.; Amassian, A.; Sargent, E. H. Nat. Nanotechnol. 2012, 7, 577−582. (4) Konstantatos, G.; Howard, I.; Fischer, A.; Hoogland, S.; Clifford, J.; Klem, E.; Levina, L.; Sargent, E. H. Nature 2006, 442, 180−183. (5) Sun, L.; Choi, J. J.; Stachnik, D.; Bartnik, A. C.; Hyun, B.-R.; Malliaras, G. G.; Hanrath, T.; Wise, F. W. Nat. Nanotechnol. 2012, 7, 369−373. (6) Kim, D. K.; Lai, Y.; Diroll, B. T.; Murray, C. B.; Kagan, C. R. Nat. Commun. 2012, 3, 1216. E
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Nano Letters (7) Kang, Y.; Li, M.; Cai, Y.; Cargnello, M.; Diaz, R. E.; Gordon, T. R.; Wieder, N. L.; Adzic, R. R.; Gorte, R. J.; Stach, E. A.; Murray, C. B. J. Am. Chem. Soc. 2013, 135, 2741−2747. (8) Chen, C.; Kang, Y.; Huo, Z.; Zhu, Z.; Huang, W.; Xin, H. L.; Snyder, J. D.; Li, D.; Herron, J. A.; Mavrikakis, M.; Chi, M.; More, K. L.; Li, Y.; Markovic, N. M.; Somorjai, G. A.; Yang, P.; Stamenkovic, V. R. Science 2014, 343, 1339−1343. (9) Sun, X.; Li, D.; Ding, Y.; Zhu, W.; Guo, S.; Wang, Z. L.; Sun, S. J. Am. Chem. Soc. 2014, 136, 5745−5749. (10) Murray, C. B.; Kagan, C. R.; Bawendi, M. G. Annu. Rev. Mater. Sci. 2000, 30, 545−610. (11) Talapin, D. V.; Lee, J.-S.; Kovalenko, M. V.; Shevchenko, E. V. Chem. Rev. 2010, 110, 389−458. (12) Hanrath, T. J. Vac. Sci. Technol., A 2012, 30, 030802. (13) Talapin, D. V.; Murray, C. B. Science 2005, 310, 86−89. (14) Kovalenko, M. V.; Scheele, M.; Talapin, D. V. Science 2009, 324, 1417−1420. (15) Zarghami, M. H.; Liu, Y.; Gibbs, M.; Gebremichael, E.; Webster, C.; Law, M. ACS Nano 2010, 4, 2475−2485. (16) Liu, Y.; Gibbs, M.; Puthussery, J.; Gaik, S.; Ihly, R.; Hillhouse, H. W.; Law, M. Nano Lett. 2010, 10, 1960−1969. (17) Choi, J. J.; Luria, J.; Hyun, B.-R.; Bartnik, A. C.; Sun, L.; Lim, Y.F.; Marohn, J. A.; Wise, F. W.; Hanrath, T. Nano Lett. 2010, 10, 1805− 1811. (18) Tang, J.; Kemp, K. W.; Hoogland, S.; Jeong, K. S.; Liu, H.; Levina, L.; Furukawa, M.; Wang, X.; Debnath, R.; Cha, D.; Chou, K. W.; Fischer, A.; Amassian, A.; Asbury, J. B.; Sargent, E. H. Nat. Mater. 2011, 10, 765−771. (19) Artemyev, M.; Bibik, A.; Gurinovich, L.; Gaponenko, S.; Woggon, U. Phys. Rev. B 1999, 60, 1504−1506. (20) Artemyev, M. V.; Woggon, U.; Jaschinski, H.; Gurinovich, L. I.; Gaponenko, S. V. J. Phys. Chem. B 2000, 104, 11617−11621. (21) Law, M.; Luther, J. M.; Song, Q.; Hughes, B. K.; Perkins, C. L.; Nozik, A. J. J. Am. Chem. Soc. 2008, 130, 5974−5985. (22) Luther, J. M.; Law, M.; Song, Q.; Perkins, C. L.; Beard, M. C.; Nozik, A. J. ACS Nano 2008, 2, 271−280. (23) Wolcott, A.; Doyeux, V.; Nelson, C. A.; Gearba, R.; Lei, K. W.; Yager, K. G.; Dolocan, A. D.; Williams, K.; Nguyen, D.; Zhu, X.-Y. J. Phys. Chem. Lett. 2011, 2, 795−800. (24) Williams, K. J.; Tisdale, W. A.; Leschkies, K. S.; Haugstad, G.; Norris, D. J.; Aydil, E. S.; Zhu, X.-Y. ACS Nano 2009, 3, 1532−1538. (25) Banin, U.; Cao, Y.; Katz, D.; Millo, O. Nature 1999, 400, 542− 544. (26) Katz, D.; Wizansky, T.; Millo, O.; Rothenberg, E.; Mokari, T.; Banin, U. Phys. Rev. Lett. 2002, 89, 086801. (27) Niquet, Y.; Delerue, C.; Lannoo, M.; Allan, G. Phys. Rev. B 2001, 64, 113305. (28) Bakkers, E. P. A. M.; Hens, Z.; Zunger, A.; Franceschetti, A.; Kouwenhoven, L. P.; Gurevich, L.; Vanmaekelbergh, D. Nano Lett. 2001, 1, 551−556. (29) Vanmaekelbergh, D.; Casavola, M. J. Phys. Chem. Lett. 2011, 2, 2024−2031. (30) Steiner, D.; Aharoni, A.; Banin, U.; Millo, O. Nano Lett. 2006, 6, 2201−2205. (31) Liljeroth, P.; Overgaag, K.; Urbieta, A.; Grandidier, B.; Hickey, S.; Vanmaekelbergh, D. Phys. Rev. Lett. 2006, 97, 096803. (32) Overgaag, K.; Liljeroth, P.; Grandidier, B.; Vanmaekelbergh, D. ACS Nano 2008, 2, 600−606. (33) Jung, H. J.; Dasgupta, N. P.; Van Stockum, P. B.; Koh, A. L.; Sinclair, R.; Prinz, F. B. Nano Lett. 2013, 13, 716−721. (34) Hines, M. A.; Scholes, G. D. Adv. Mater. 2003, 15, 1844−1849. (35) Moreels, I.; Justo, Y.; De Geyter, B.; Haustraete, K.; Martins, J. C.; Hens, Z. ACS Nano 2011, 5, 2004−2012. (36) Dong, A.; Ye, X.; Chen, J.; Kang, Y.; Gordon, T.; Kikkawa, J. M.; Murray, C. B. J. Am. Chem. Soc. 2011, 133, 998−1006. (37) Erni, R.; Browning, N. D. Ultramicroscopy 2005, 104, 176−192. (38) Kadkhodazadeh, S.; Ashwin, M. J.; Jones, T. S.; McComb, D. W. J. Phys. Conf. Ser. 2008, 126, 012049.
(39) Gu, L.; Srot, V.; Sigle, W.; Koch, C.; van Aken, P.; Scholz, F.; Thapa, S.; Kirchner, C.; Jetter, M.; Rühle, M. Phys. Rev. B 2007, 75, 195214. (40) Erni, R.; Browning, N. D. Ultramicroscopy 2007, 107, 267−273. (41) Stö ger-Pollach, M.; Hebert, C.; Zandbergen, H. W.; Schattschneider, P. Adv. Eng. Mater. 2004, 6, 826−828. (42) Rafferty, B.; Brown, L. Phys. Rev. B 1998, 58, 10326−10337. (43) Moreels, I.; Lambert, K.; Smeets, D.; De Muynck, D.; Nollet, T.; Martins, J. C.; Vanhaecke, F.; Vantomme, A.; Delerue, C.; Allan, G.; Hens, Z. ACS Nano 2009, 3, 3023−3030. (44) Efros, A. Phys. Rev. B 1992, 46, 7448−7458. (45) Lassnig, R. Phys. Rev. B 1985, 31, 8076−8086.
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DOI: 10.1021/nl5046507 Nano Lett. XXXX, XXX, XXX−XXX
Variation of Energy Density of States in Quantum Dot Arrays due to Interparticle Electronic Coupling Manca Logar,† Shicheng Xu,† Shinjita Acharya,† and Fritz B. Prinz*,†,‡ †
Department of Mechanical Engineering and ‡Department of Material Science and Engineering, Stanford University, Stanford, California 94305, United States S Supporting Information *
ABSTRACT: Subnanometer-resolved local electron energy structure was measured in PbS quantum dot superlattice arrays using valence electron energy loss spectroscopy with scanning transmission electron microscopy. We found smaller values of the lowest available transition energies and an increased density of electronic states in the space between quantum dots with shorter interparticle spacing, indicating extension of carrier wave functions as a result of interparticle electronic coupling. A quantum simulation verified both trends and illustrated the wave function extension effect.
KEYWORDS: Energy density of states, STEM-EELS, electronic coupling, quantum dot, superlattice interparticle distance decreases in thin films of monodisperse QDs.19−24 However, such red shifting can be caused by changes in the dielectric environment23 in addition to changes in electronic coupling24 between the dots. It is difficult to isolate these factors when examining the macroscopic optical transition energies of the arrays, especially considering the inhomogeneity in the dielectric environment introduced by the inevitable size and shape variation of the QDs in the sample. On the microscopic level, scanning tunneling microscopy and spectroscopy (STM-STS) have been used to investigate the electronic local density of states (LDOS) of single nanoparticles25−29 and their superlattice films. Steiner et al.30 studied arrays assembled by trioctylphosphine capped InAs QDs and measured the LDOS at different regions of the array. Band gap reduction was verified by comparing the density of states in isolated and arrayed QDs. Liljeroth et al.31 studied oleic acid capped PbSe QD arrays and found variations in LDOS due to local microscopic disorders. Reductions of the energy gap were only found in certain locations where the electron and hole wave functions were delocalized. Overgaag et al.32 investigated the LDOS of single PbSe QDs arrays stabilized by CdSe QDs. While no significant difference was observed in the zeroconductance gap between isolated and aggregated oleic acid capped QDs, the first conduction-level resonances were found to broaden as the number of particles in the molecular aggregate increased.
O
rdered arrays of colloidal quantum dots (superlattices) exhibiting collective optical and electrical phenomena have gained considerable interest among the scientific community with promising applications in photovoltaics,1−3 optoelectronics,4,5 information technology,6 and catalysis.7−9 Mediated by capping ligands, the quantum dots (QDs) can be treated as artificial atoms and assembled into higher order nanostructures such as metamaterials and supraparticles. These structures have programmable physical and chemical properties,10−12 widening the array of available functional materials for relevant applications. Concomitant existence of quantum confinement and high packing density can be realized by using short capping ligands, a technique that has been found critical for performance in a variety of devices. Exchanging long chain ligands with shorter ones leads to successful manipulation of various properties of QD films, such as carrier mobility, which can be greatly improved.13−16 Shorter ligands also enhance exciton recombination in a QD film, which leads to efficiency improvement in infrared LEDs.5 Additionally, stronger interparticle coupling can facilitate dissociation of photogenerated excitons,17 which contributes to increased efficiency of solar cells fabricated from atomically passivated QDs.18 By reducing interparticle spacing in QD arrays, electronic coupling among QDs increases, which modifies the energy states and their corresponding density. Gaining in-depth insight into the electronic coupling phenomena among the QD arrays is of great importance for the precise engineering of superstructures. Most work in the area is presently focused on the effect of different capping ligands on electron energies. On the macroscopic level, the effect is seen as a red shift of the optical absorption peak as the © XXXX American Chemical Society
Received: December 3, 2014 Revised: January 26, 2015
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DOI: 10.1021/nl5046507 Nano Lett. XXXX, XXX, XXX−XXX
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Figure 1. HRTEM images of hexagonal arrays of PbS quantum dots capped by (a) butylamine (denoted C4), (b) octylamine (C8), (c) dodecylamine (C12), and (d) oleic acid (C18). The diameter of the quantum dots is approximately 5.0 nm. (e) Schematic showing variation in spacing of QDs as a function of ligand length. Average values of d1, d2, d3, and d4 are 0.6, 1.0, 1.5, and 2.0 nm.
Figure 2. Variations in the lowest available electron transition energy (LATE) across neighboring PbS QDs with different interparticle spacings. EEL spectra taken at the positions of the red dots in the ADF images were used to extract LATE line profiles across QDs capped by (a) butylamine, (b) octylamine, (c) dodecylamine, and (d) oleic acid. The arrows denote EELS acquisition directions.
reproducing previously observed changes in the macroscopic band gap of the lattices, our results reveal changes in the local electronic structure in between QDs as their spacing is diminished, providing direct experimental observation of electronic coupling of the dots. To explore the theoretical basis for our observations, we present the results of a quantum mechanical finite element simulation of our samples. PbS QDs capped with various organic ligands self-assemble in close-packed arrays with hexagonal symmetry (Figure 1). In this study, the synthesis of PbS quantum dots was performed using a modification of the standard colloidal procedure described by Hines et al.,34 with lead acetate as the lead precursor and bis(trimethylsilyl)-sulfide as the sulfur precursor (details presented in the Supporting Information). Capping of the PbS QDs with organic ligands using a solution-based ligand exchange process4,35,36 not only suppresses surface oxidation, enabling the formation of chemically homogeneous arrays of dots with rock-salt crystal structure (Supporting Information Figure S2) but also allows for tailoring of the interparticle spacing, which is a parameter crucial to this study. Spacing
To date, no such microscopic studies have provided evidence correlating electronic coupling with changes in electronic structure in the region between individual coupled QDs. Moreover, the interpretation of tunneling spectra from STS on such samples is clouded by unknown shape effects from the tip and by the Schottky barrier in between the tip and sample. Given the limited aspect ratio of an STM probe and the threedimensional topologies of QDs, even resolving the geometric features of QDs is a challenging task, not to mention discerning the variation of local density of states within and between QDs. In this work, we present localized measurements of the electronic energy structure of lead sulfide (PbS) QD pairs, as well as spatial variation in the lowest available electron transition energy (LATE) in a set of monodisperse QD superlattices capped with different size ligands. The measurements were performed using electron energy-loss spectroscopy in a (scanning) transmission electron microscope [(S)TEMEELS].33 The spatial resolution of this technique allows independent measurement of the joint local density of states (JLDOS) inside of and between individual QDs. In addition to B
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Figure 3. The 2D maps of integrated EELS signal intensity over the energy range 0.9−2 eV (left panes) and EELS spectra taken at the locations indicated by the red dots in Figure 2 over PbS QDs (right panes) capped by (a) butylamine, (b) octylamine, (c) dodecylamine, and (d) oleic acid.
In EELS measurements, the sample is exposed to a beam of electrons with a narrow range of kinetic energies and the amount of energy lost by the transmitted electrons is measured by an EEL spectrometer. The spectra exhibit a characteristic zero-loss peak (ZLP) from electrons transmitted without inelastic scattering. After ZLP removal, information about the material band structure is obtained from the signal generated by inelastically scattered electrons with energy loss below 10 eV. The energy onset of inelastic scattering indicates the LATE at the beam position (Supporting Information Figures S3 and S5). Recently, the energy and spatial resolution of this technique has been greatly improved through the use of monochromated valence electron energy-loss spectroscopy (VEELS).37 It has been shown that for direct band gap semiconductors the correct values of the band gap can be obtained by VEELS acquisition in samples of thickness less than half the total mean free path for inelastic scattering, where the effect of Cerenkov radiation can be efficiently minimized.38,39 STEM-VEELS has also been applied to evaluate the size and shape dependence of the band gap of CdSe and PbS QDs,33,40 as well as the thickness dependence of the band gap of Si.41 Here, EELS line profiles across PbS QDs were obtained with a monochromated 80 kV electron beam focused to a 0.3 nm diameter. Figure 2 shows the LATE from Fourier deconvoluted EEL spectra taken at 0.5 nm intervals along lines crossing neighboring dots from hexagonal arrays with various ligand lengths (more details in Supporting Information, Figure S6). The line profiles reveal the spatial dependence of the allowed electronic transitions in the space within and between neighboring QDs. A gradual increase of the LATE from the center of the PbS QDs toward the edge is observed, which is in agreement with previous experiments.33 This effect is attributed to the reduced wave function density near the dot edges as a result of the confinement geometry. To further compare the LATE of QD samples with different interparticle spacings, two-dimensional maps of integrated EELS signal intensity from neighboring PbS QDs over the
Figure 4. Boxplot of the LATE among quantum dot pairs from hexagonal arrays, capped by C4, C8, C12, and C18. Twenty pairs were used in each case. The red line indicates the median of the sample, the black lines indicate the minimum and maximum, and blue lines indicate the 25 percentile and 75 percentile.
control was achieved by exchanging the as-synthesized oleic acid ligands (denoted C18) with three alkylamine ligands of variable lengths, namely, dodecylamine (C12), octylamine (C8), and butylamine (C4). The results of this process are illustrated in Figure 1. Average particle size and interparticle spacing in the assemblies of ligand-capped PbS QDs was measured from HRTEM images of 100 pairs of QDs. The average dot size is 5.0 ± 0.13 nm, and dots capped with C4, C8, C12, and C18 show spacings of 0.6 ± 0.2, 1.0 ± 0.3, 1.5 ± 0.2, and 2.0 ± 0.8 nm, respectively. STEM-EELS was used on the prepared samples to measure variations in the availability of low-energy electron transitions in the space between neighboring QDs as a function of interparticle spacing. Whereas the previously discussed STMSTS technique measures the energy density of individual electron states, the technique used here enables the direct measurement of interband electron transitions and avoids the tip geometry complications encountered with scanning probes. C
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Figure 5. Top view (a) and side view (b) of the simulation cell used to calculate JLDOS line-profiles across pairs of quantum dot pairs separated by (d) 0.5, (e) 1.0, (f) 1.5, and (g) 2.0 nm. (c) A side view (from the perspective in (b)) of the electron probability density for states near 2.1 eV for each value of separation.
edges.42 In analogy with theoretical treatment of the tunneling current in STM, we can treat the EELS signal as proportional to the JLDOS in the sample, which is the point-localized analogue of the standard joint density of states (JDOS)33
energy range 0.9−2 eV are presented in Figure 3. Significant integrated signal can be observed between the QDs with shorter interparticle spacing (C4 and C8), reducing noticeably as the interparticle spacing is increased (C12 and C18). Aligned to the right of each integrated heat map is a series of EEL spectra taken at the corresponding lateral positions over the red dots as indicated in Figure 2. These spectra reveal the nonzero EELS signal in the area between the closely spaced dots, which gradually disappears as the QD spacing increases. Brief reflection on the physical origin of the EELS signal suggests that the increased signal observed between the more proximal QDs is a result of their electronic coupling. In the low loss energy range of the EEL spectrum, the signal intensity can be approximated using Fermi’s golden rule for overlapping electron and hole states near the semiconductor band
JLDOS(r, ΔE) =
∑ |φhj(r)|2 |φei(r)|2 δ(Ehj − Eei , ΔE) i,j
(1)
where φe(h)i(j) is the ith (jth) electron (hole) wave function of the system and E is the corresponding energy of the state. Electron transitions that contribute to the EELS signal at a given location come from pairs of electron and hole states with the right energy difference that coexist there, in proportion to the probability density of the wave functions associated with those states. The sharp drop in the observed signal at the edges D
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states do not extend significantly outside the QD boundaries, explaining the higher onset of the JLDOS (and hence EELS signal) in between the dots than at the center of the dots. Close inspection of the QD-center spectra in Figure 5d,g reveals a reduction in the macroscopic LATE as the QDs are brought closer, in agreement with the experiment. The magnitude of the shift is smaller than measured (37 meV vs 40 meV from C18 to C4). Such a disagreement is to be expected from this model given the various approximations made. These approximations include an energy-independent value for the effective carrier mass that is only accurate near the band edges, an interface energy barrier independent of QD spacing, and perfect monodispersity of the QDs in the array. (For QDs with average size of 5 nm, a variation of 0.2 nm can change transition energies by 26 meV.) Despite the uncertainty in absolute value, this simple quantum mechanical model reproduces the trends observed in this study and provides an explanation of the observed EELS signal between the QDs based on electronic coupling as they are spaced more closely in arrays.
of the more largely spaced QDs (Figure 3c,d) indicates the spatial limit of the electron and hole wave functions in isolated dots at the energies shown. The appearance of a signal outside of those limits (between the QDs) as the QDs are brought closer together indicates an extension of the wave functions into that space as result of electronic coupling between the QDs. To verify our procedure against other measurements of the energetic effects of QD coupling, we collected statistics on the LATE from 20 QD pairs in each sample. Figure 4 shows the smallest values of the LATE found in the region of each dot pair (typically located at the center of a dot), corresponding to the absorption onset that would be observed in a macroscopic optical measurement of the array. Previous such measurements on dispersed PbS QDs have found an onset of approximately 0.9 eV for 5.0 nm diameter dots,43 similar to the value that we have measured. With smaller interparticle spacing, the macroscopic LATE decreases, in agreement with the optical absorption red shift observed in the previously mentioned studies. This is because the transition energy decreases with increasing QD coupling energy β as the electron wave functions overlap24 ΔE = ΔE0 − 6(|βe| + |βh |)
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ASSOCIATED CONTENT
S Supporting Information *
(2)
Supporting Information contains details of PbS QDs synthesis, ligand exchange, and their HRTEM characterizations, procedures of ZLP removal and LATE determination in STEMEELS, and simulation results on JLDOS as functions of ligand band gap and alignment. This material is available free of charge via the Internet at http://pubs.acs.org.
To gain further insight into our experimental results we performed a quantum finite element analysis of spherical QDs in 2D hexagonal arrays. Electronic structure was calculated with effective mass approximation,44,45 by solving the Schrödinger equation for this geometry using a k·p method, where Vr is the potential energy results from the physical position r, k is the superlattice vector (sampled 4 × 4), i is the imaginary unit, and φk and Ek are the corresponding wave function and energy
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected].
⎤ ⎡ ℏ2 ⎡ 1 2i·k ℏ2k 2 ⎤ − ∇·⎢ ∇φk (r) + φ(r)⎥ + ⎢V (r) + ⎥φ (r) = E k φk (r) 2 ⎣ meff meff 2meff ⎦ k ⎦ ⎣
Author Contributions
M.L. and S.X. contributed equally.
(3)
Notes
The calculation was performed using a hexagonal unit cell, as shown in Figure 5a,b. For the PbS QDs, the electron and hole effective mass were both approximated to be 0.080 m0, and the carrier effective mass within the barrier material was set to m0. Assuming an energy-symmetric interface band alignment and considering the 5 eV HOMO−LUMO energy gap of alkane chains and 0.4 eV bulk band gap of PbS, an energy barrier of 2.3 eV was used for both electrons and holes at the QD boundaries. (The effects of asymmetry in the band alignment are presented in the Supporting Information.) After solving for a sufficient set of low-energy eigenstates, the JLDOS was calculated at different points in space using eq 1. The resulting spectra at points along lines bisecting QD pairs are presented in Figure 5d,g as a function of QD spacing. (Delta functions from eq 1 are plotted as Lorentzian functions with room-temperature thermal energy broadening.) A similar trend to the experimental measurement is observed in which electronic transitions with lower energies become available between the QDs as they are brought closer. The wave function extension effect earlier suggested to be responsible for this can be seen in Figure 5c. Here we show the probability densities for a wave function at approximately 2.1 eV for several different values of QD separation. As the QDs move closer together, the wave function lobes extend further outside the atomic boundaries of the dots. They eventually connect, contributing to an increased JLDOS between the QDs for transitions involving that wave function. The wave functions corresponding to lower energy
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by Honda R&D and the Center on Nanostructuring for Efficient Energy Conversion (CNEEC) at Stanford University, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award DE-SC0001060. The authors also acknowledge helpful discussions with D. Thian, P. Schindler, A. L. Dadlani, and P. B. Van Stockum of the Nanoscale Prototyping Laboratory, Stanford University, and E. Shevchenko of Argonne National Laboratory.
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REFERENCES
(1) Kramer, I. J.; Sargent, E. H. Chem. Rev. 2014, 114, 863−882. (2) Sargent, E. H. Nat. Photonics 2012, 6, 133−135. (3) Ip, A. H.; Thon, S. M.; Hoogland, S.; Voznyy, O.; Zhitomirsky, D.; Debnath, R.; Levina, L.; Rollny, L. R.; Carey, G. H.; Fischer, A.; Kemp, K. W.; Kramer, I. J.; Ning, Z.; Labelle, A. J.; Chou, K. W.; Amassian, A.; Sargent, E. H. Nat. Nanotechnol. 2012, 7, 577−582. (4) Konstantatos, G.; Howard, I.; Fischer, A.; Hoogland, S.; Clifford, J.; Klem, E.; Levina, L.; Sargent, E. H. Nature 2006, 442, 180−183. (5) Sun, L.; Choi, J. J.; Stachnik, D.; Bartnik, A. C.; Hyun, B.-R.; Malliaras, G. G.; Hanrath, T.; Wise, F. W. Nat. Nanotechnol. 2012, 7, 369−373. (6) Kim, D. K.; Lai, Y.; Diroll, B. T.; Murray, C. B.; Kagan, C. R. Nat. Commun. 2012, 3, 1216. E
DOI: 10.1021/nl5046507 Nano Lett. XXXX, XXX, XXX−XXX
Letter
Nano Letters (7) Kang, Y.; Li, M.; Cai, Y.; Cargnello, M.; Diaz, R. E.; Gordon, T. R.; Wieder, N. L.; Adzic, R. R.; Gorte, R. J.; Stach, E. A.; Murray, C. B. J. Am. Chem. Soc. 2013, 135, 2741−2747. (8) Chen, C.; Kang, Y.; Huo, Z.; Zhu, Z.; Huang, W.; Xin, H. L.; Snyder, J. D.; Li, D.; Herron, J. A.; Mavrikakis, M.; Chi, M.; More, K. L.; Li, Y.; Markovic, N. M.; Somorjai, G. A.; Yang, P.; Stamenkovic, V. R. Science 2014, 343, 1339−1343. (9) Sun, X.; Li, D.; Ding, Y.; Zhu, W.; Guo, S.; Wang, Z. L.; Sun, S. J. Am. Chem. Soc. 2014, 136, 5745−5749. (10) Murray, C. B.; Kagan, C. R.; Bawendi, M. G. Annu. Rev. Mater. Sci. 2000, 30, 545−610. (11) Talapin, D. V.; Lee, J.-S.; Kovalenko, M. V.; Shevchenko, E. V. Chem. Rev. 2010, 110, 389−458. (12) Hanrath, T. J. Vac. Sci. Technol., A 2012, 30, 030802. (13) Talapin, D. V.; Murray, C. B. Science 2005, 310, 86−89. (14) Kovalenko, M. V.; Scheele, M.; Talapin, D. V. Science 2009, 324, 1417−1420. (15) Zarghami, M. H.; Liu, Y.; Gibbs, M.; Gebremichael, E.; Webster, C.; Law, M. ACS Nano 2010, 4, 2475−2485. (16) Liu, Y.; Gibbs, M.; Puthussery, J.; Gaik, S.; Ihly, R.; Hillhouse, H. W.; Law, M. Nano Lett. 2010, 10, 1960−1969. (17) Choi, J. J.; Luria, J.; Hyun, B.-R.; Bartnik, A. C.; Sun, L.; Lim, Y.F.; Marohn, J. A.; Wise, F. W.; Hanrath, T. Nano Lett. 2010, 10, 1805− 1811. (18) Tang, J.; Kemp, K. W.; Hoogland, S.; Jeong, K. S.; Liu, H.; Levina, L.; Furukawa, M.; Wang, X.; Debnath, R.; Cha, D.; Chou, K. W.; Fischer, A.; Amassian, A.; Asbury, J. B.; Sargent, E. H. Nat. Mater. 2011, 10, 765−771. (19) Artemyev, M.; Bibik, A.; Gurinovich, L.; Gaponenko, S.; Woggon, U. Phys. Rev. B 1999, 60, 1504−1506. (20) Artemyev, M. V.; Woggon, U.; Jaschinski, H.; Gurinovich, L. I.; Gaponenko, S. V. J. Phys. Chem. B 2000, 104, 11617−11621. (21) Law, M.; Luther, J. M.; Song, Q.; Hughes, B. K.; Perkins, C. L.; Nozik, A. J. J. Am. Chem. Soc. 2008, 130, 5974−5985. (22) Luther, J. M.; Law, M.; Song, Q.; Perkins, C. L.; Beard, M. C.; Nozik, A. J. ACS Nano 2008, 2, 271−280. (23) Wolcott, A.; Doyeux, V.; Nelson, C. A.; Gearba, R.; Lei, K. W.; Yager, K. G.; Dolocan, A. D.; Williams, K.; Nguyen, D.; Zhu, X.-Y. J. Phys. Chem. Lett. 2011, 2, 795−800. (24) Williams, K. J.; Tisdale, W. A.; Leschkies, K. S.; Haugstad, G.; Norris, D. J.; Aydil, E. S.; Zhu, X.-Y. ACS Nano 2009, 3, 1532−1538. (25) Banin, U.; Cao, Y.; Katz, D.; Millo, O. Nature 1999, 400, 542− 544. (26) Katz, D.; Wizansky, T.; Millo, O.; Rothenberg, E.; Mokari, T.; Banin, U. Phys. Rev. Lett. 2002, 89, 086801. (27) Niquet, Y.; Delerue, C.; Lannoo, M.; Allan, G. Phys. Rev. B 2001, 64, 113305. (28) Bakkers, E. P. A. M.; Hens, Z.; Zunger, A.; Franceschetti, A.; Kouwenhoven, L. P.; Gurevich, L.; Vanmaekelbergh, D. Nano Lett. 2001, 1, 551−556. (29) Vanmaekelbergh, D.; Casavola, M. J. Phys. Chem. Lett. 2011, 2, 2024−2031. (30) Steiner, D.; Aharoni, A.; Banin, U.; Millo, O. Nano Lett. 2006, 6, 2201−2205. (31) Liljeroth, P.; Overgaag, K.; Urbieta, A.; Grandidier, B.; Hickey, S.; Vanmaekelbergh, D. Phys. Rev. Lett. 2006, 97, 096803. (32) Overgaag, K.; Liljeroth, P.; Grandidier, B.; Vanmaekelbergh, D. ACS Nano 2008, 2, 600−606. (33) Jung, H. J.; Dasgupta, N. P.; Van Stockum, P. B.; Koh, A. L.; Sinclair, R.; Prinz, F. B. Nano Lett. 2013, 13, 716−721. (34) Hines, M. A.; Scholes, G. D. Adv. Mater. 2003, 15, 1844−1849. (35) Moreels, I.; Justo, Y.; De Geyter, B.; Haustraete, K.; Martins, J. C.; Hens, Z. ACS Nano 2011, 5, 2004−2012. (36) Dong, A.; Ye, X.; Chen, J.; Kang, Y.; Gordon, T.; Kikkawa, J. M.; Murray, C. B. J. Am. Chem. Soc. 2011, 133, 998−1006. (37) Erni, R.; Browning, N. D. Ultramicroscopy 2005, 104, 176−192. (38) Kadkhodazadeh, S.; Ashwin, M. J.; Jones, T. S.; McComb, D. W. J. Phys. Conf. Ser. 2008, 126, 012049.
(39) Gu, L.; Srot, V.; Sigle, W.; Koch, C.; van Aken, P.; Scholz, F.; Thapa, S.; Kirchner, C.; Jetter, M.; Rühle, M. Phys. Rev. B 2007, 75, 195214. (40) Erni, R.; Browning, N. D. Ultramicroscopy 2007, 107, 267−273. (41) Stö ger-Pollach, M.; Hebert, C.; Zandbergen, H. W.; Schattschneider, P. Adv. Eng. Mater. 2004, 6, 826−828. (42) Rafferty, B.; Brown, L. Phys. Rev. B 1998, 58, 10326−10337. (43) Moreels, I.; Lambert, K.; Smeets, D.; De Muynck, D.; Nollet, T.; Martins, J. C.; Vanhaecke, F.; Vantomme, A.; Delerue, C.; Allan, G.; Hens, Z. ACS Nano 2009, 3, 3023−3030. (44) Efros, A. Phys. Rev. B 1992, 46, 7448−7458. (45) Lassnig, R. Phys. Rev. B 1985, 31, 8076−8086.
F
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