Emergent Electronic and Dielectric Properties of Interacting

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Emergent electronic and dielectric properties of interacting nanoparticles at finite temperature Arin R Greenwood, Márton Vörös, Federico Giberti, and Giulia Galli Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b04047 • Publication Date (Web): 11 Dec 2017 Downloaded from http://pubs.acs.org on December 14, 2017

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Emergent electronic and dielectric properties of interacting nanoparticles at finite temperature Arin R. Greenwood,† Márton Vörös,‡ Federico Giberti,† and Giulia Galli∗,†,‡ †Institute for Molecular Engineering, The University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA ‡Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA E-mail: [email protected] Abstract Lead chalcogenide nanoparticle solids have been successfully integrated into certified solar cells, and represent promising platforms for the design of novel photo-absorbers for photo-electrochemical cells. While much attention has been drawn to improving efficiency and device performance through altering the character of the individual nanoparticles, the role of interactions between nanoparticles is not yet well understood. Using first principles molecular dynamics and electronic structure calculations, we investigated the combined effect of temperature and interaction on functionalized lead chalcogenide nanoparticles (NPs). Here we show that at finite temperature, interacting NPs are dynamical dipolar systems, with average values of dipole moments and polarizabilities substantially increased with respect to those of the isolated building blocks. In addition, we show that the interacting NPs exhibit slightly smaller fundamental gaps that decrease as a function of temperature, and that the radiative lifetimes of both the isolated NPs and the solids are greatly reduced at finite temperature compared to T=0. Finally, we present a critical discussion of various results reported in the literature for the values of dipole moments of nanoparticles.

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Keywords Chalcogenide quantum dots, Density Functional Theory, Photovoltaic Devices, Molecular Dynamics

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Semiconducting colloidal nanoparticles (NPs) are promising candidates for next generation photovoltaic devices due to their tunable electronic and optical properties. 1–5 In particular, lead chalcogenide quantum dots have a large exciton Bohr radius (18 nm for PbS), leading to strong quantum confinement and hence a band gap that can be tuned over the solar spectrum by changing the NP size. Recently, halide-capped PbS NPs 6–9 have been successfully integrated into certified solar cell devices with efficiency in excess of 12%, 9,10 which could be further improved and potentially even pushed beyond the Shockley-Queisser limit by exploiting the efficient multiple exciton generation observed in PbS NPs. 11–18 In addition to tunable band gaps, the frontier orbital energies (or band edges) of these NPs may be tuned by choosing appropriate ligands; 19,20 for example they could be tuned with respect to the redox potentials of water, CO2 or H2 S, thus allowing for the design of novel and efficient photoabsorbers for photoelectrochemical cells. 21 In NP based solar cells, quantum dots are assembled into films 22–24 where charge transport usually occurs via hopping, possibly leading to low mobilities. 25–28 It is often assumed that mobilities can be enhanced by precisely arranging the dots into ordered solids, or by decreasing the distance between NPs. 29,30 For example, epitaxially-connected NP superlattices 31,32 and ultrathin 2D sheets of PbS NPs 33 have been synthesized, and the formation mechanisms of 2D and 3D NP superstructures have been studied using X-ray scattering experiments, electron microscopy, and Monte Carlo simulations. 34,35 Treml et al used successive ionic layer adsorption and reaction (SILAR) to enhance interdot coupling 36 in similar epitaxially-connected NP assemblies, while Dolzhnikov et al 37,38 used composition-matched molecular “solders” to increase carrier mobilities between NPs. Despite the importance of NP interactions to understand and predict the properties of energy conversion devices composed of quantum dots, most theoretical and computational studies to date have addressed isolated NPs at zero temperature (T). Only a few finite temperature simulations of isolated Ge and Si NPs 39,40 and of PbSe and CdSe 41,42 NPs have appeared in the literature. Recent studies on the interactions of 2D PbSe nanoplatelets 43 and

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hydrogenated Si NP dimers 44,45 and CdSe NP solids 46 have neglected temperature effects altogether. To our knowledge, no atomistic studies have been conducted yet on nanostructured lead chalcogenide solids that include at the same time the effects of temperature, NP interactions and that consider realistic capping ligands. Here we present a first principles study of the finite temperature properties of PbS NPs assembled into solids. We used ab-initio molecular dynamics to investigate the modification of the electronic and dielectric properties of 1.6 nm iodine-capped lead sulfide NPs (Pb32 S28 I8 ) upon formation of thin films. We chose iodide ligands, as they are employed in experiments to prevent oxidative attacks and thus improve air-stability. 47 Although smaller than those usually considered experimentally (often 3-10 nm in diameter), the NPs studied here are representative models of experimental quantum dots, inclusive of a core and a surface with cation excess, whose charge is compensated for by an appropriate number of anionic ligands. We show that when the NPs interact to form a solid, the band gap slightly decreases by up to approximately 0.2 eV, the radiative lifetimes can decrease by about two orders of magnitude, and intra-gap states emerge in the NP films at finite T. We also show that, at and above room T, interacting NPs behave as oscillating dipoles with instantaneous values of their dipole moments increasing by an order of magnitude compared to that of the isolated NPs. Polarizabilities also increase in condensed phases, although much less than dipole moments. Finally our results are used to shed light on some of the controversial findings reported in the literature of the last two decades regarding dipole moments of NP assemblies. 48–55 In order to assess the effect of interaction strength, we compared the properties of isolated NPs to those of interacting ones in two regimes: the first regime represents an assembly in which the NPs are kept apart by the ligands (similar to what is observed in spin-cast or drop-cast experimental films 3,56 ), and where the NPs are interacting but not covalently bonded; the second regime is representative of chemically bonded NP solids, or “connected NP solids.” 31,32,37,57 The models built in this work are illustrated in Fig. 1a and are labeled “necked” and “fused” solids. The subunits for the necked solid remain largely intact, with

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the bonds between NPs occurring at the edges (or “neck”) of the unit, whereas the NPs in the fused solid are strongly bound and they arrange into a periodic sheet of units fused together. Formation of these two solids is illustrated with a movie in the SI. Although these models represent only two possible bonding geometries, they are expected to be representative of epitaxially-connected structures synthesized by several groups using techniques designed to enhance electronic coupling. 31,36,58,59

a)

c)

1.9

τ/τ0 ( s)

b) Eg (eV)

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1.7

1.5

1.3 0

isolated NP necked solid fused solid 200

400

Temperature (K)

101

100

10-1

10-2

600

isolated NP necked solid fused solid

0

200

400

Temperature (K)

600

Figure 1: a) Ball and stick representation of the nanoparticles (NPs) considered in this work; from left to right: isolated Pb32 S28 I8 NP; solid of weakly interacting NPs; necked and fused solids of NPs forming bonds (simulation cell shown as inner black box). Black, yellow and purple spheres indicate lead, sulfur, and iodine atoms, respectively. b) Fundamental gaps as a function of temperature, computed as differences between the energies of the lowest unoccupied and highest occupied single particle states using DFT and the PBE functional. c) Ratio of radiative lifetimes (τ ) as a function of temperature, where τ0 is the value computed at T=0; calculations were carried out at the DFT-PBE level of theory (see text).

We carried out ab initio molecular dynamics (MD) simulations at the DFT-PBE 60 level of theory with the Qbox code, 61 focusing first on the isolated Pb32 S28 I8 NP at approximately 250 K and 510 K, respectively (see SI for details). At both temperatures, simulations started from the 0 K relaxed geometry. The NP solids were generated with the following procedure: two NVT simulations were performed at a constant T (targeting about 600 K) starting from 5


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a periodically repeated NP array in a 15.6 ˚ A unit cell. The lattice constant was chosen as that corresponding to the minimum of the binding energy curve for a simple cubic NP solid at T=0, where the geometry at each lattice constant was optimized until forces were less than 10−5 Ry Bohr−1 (see SI for simulation details, including the effect of spin orbit coupling). After short simulation times of about 3 ps, we observed the NP building blocks merge into connected units to form either the “necked” or the “fused” solid. For both simulations, the temperature resulted to be approximately 510 K and configurations extracted from the initial 3 ps were considered as representative of weakly interacting NPs. We then continued the simulations of the solids at 510 K (for approximately 80 ps) to investigate whether additional transformations would occur. After the initial, fast formation of the fused solid, we observed major structural changes. In particular, after 30 ps of simulation time, the 2D NP film initially obtained formed a 3D network bridged by ligands between the fused sheets, which then remained intact for the remainder of the simulation. In contrast, after formation of the 2D necked solid, only minor structural reorganization occurred with ligands migrating to and from the “neck” between NPs. Configurations of the solid at 250 K and 0 K were generated by lowering the temperature from 510 K to target temperatures of around 300 K and then to 150, 50, and 0 K, while allowing for the systems to equilibrate at each temperature. We begin by describing the electronic properties of isolated and interacting NPs. The electronic energy levels and wave functions were calculated in Qbox 61 using snapshots of the MD trajectories over 10 ps taken every 0.1 ps, for a total of 100 finite temperature structures per simulation (results obtained with 200 snapshots by decreasing the spacing between snapshots yielded the same average values). Fig. 1b shows the calculated band gap (Eg ) for the isolated NP and the two solid structures. We found that the band gap decreases linearly as a function of temperature for all three systems. For the isolated NP, we observed a band gap decrease from 1.93 eV at 0 K to 1.75 ± 0.06 eV at 250 K and 1.54 ± 0.11 eV at 510 K. The calculated slope was found to be -0.8 meV/K, and was approximately the same for

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the NP solids as for the isolated NP. We remark that calculations were performed using the PBE exchange correlation functional, which is known to underestimate the magnitude of the fundamental band gap, but it is expected to accurately reproduce qualitative trends; 62,63 we also note that the gap of the small NPs considered here correspond to an experimental gap of larger NPs. The decrease in band gap with increasing temperature is in contrast to what is observed in bulk lead chalcogenides 64–66 and for larger NPs with diameter of approximately 4 nm, 67 but it is consistent with some experimental reports 64,68 of a negative

dEg dT

for PbS

NPs below approximately 4 nm in diameter. However, other experimental studies reported a positive

dEg dT

for PbS NPs with diameter larger than 3 nm, with the magnitude of the slope

decreasing as a function of diameter. 69 To investigate the effect of NP size, we also calculated the band gap of a larger NP (Pb62 S55 I14 , 2.1 nm in diameter) as an average value of results for configurations during a 540 K simulation as well as at T=0; we found a similar trend in band gap as a function of temperature for this system, with a slope of approximately -0.6 meV/K compared to -0.8 meV/K for the isolated Pb32 S28 I8 NP (see SI for further details). Due to the interest in PbS NPs for photovoltaic applications, we calculated Boltzmannaveraged radiative lifetimes of the charge carriers, which include the transitions accessible at the simulation temperatures: P

if hτ i = P

τif e−ωif /kB T

if

(1)

e−ωif /kB T

where τif was calculated in the dipole approximation 70,71 with the Quantum Espresso code: 72

τif =

3 4nαF S ωif 1 X |hψi |r|ψf i|2 2 3c 3 r∈x,y,z

!−1 (2)

Here, τif is the radiative lifetime of the transition from the occupied state i to the unoccupied state f with energy ωif = f − i (i is the energy of state i ); n is the index of refraction; αF S is the fine-structure constant; the matrix element is the transition dipole moment between states i and f computed using density functional perturbation theory techniques. 73 The ratio

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of radiative lifetimes τ to the value at T=0 are reported in Fig.

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1c. For the Boltzmann

average, 100 empty states (up to about 3 eV above the LUMO) and 2000 transitions were considered. For the T=0 case, only the HOMO-LUMO transition was used for the calculation of the lifetime. Calculations for the radiative lifetimes were performed using 50 snapshots over the 10 ps trajectory (results obtained with 100 snapshots by decreasing the spacing between snapshots yielded the same average values). The lifetimes are inversely proportional to the rate of radiative carrier recombination and are an important property to characterize optoelectronic devices built from NP solids. We found that for isolated NPs, the radiative lifetime of the transition between the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO, respectively) decreases by about two orders of magnitude from 0 K to 250 K. This is due to symmetry breaking at finite T: the high symmetry of the optimized structure at T=0 leads to degenerate, nearly forbidden transitions that become allowed when the symmetry is broken due to thermal motion. When analyzing the single particle wave functions we found that at 0 K both the HOMO and LUMO are delocalized over the NP; at finite temperature the LUMO remains delocalized but the HOMO localizes in different regions of the NP over the course of the simulation, leading to an increase of the transition dipole moment and thus to a decrease of the radiative lifetime. We observed similar decreases in lifetimes for the NP solids (green and red dotted lines in Fig. 1b and c) at finite T, though to a lesser extent, with the lifetimes about 5 times smaller than at T=0. Similarly, we calculated the lifetimes for a few select snapshots of a 2.1 nm NP (Pb62 S55 I14 ) and found a decrease in lifetime at finite temperature similar to that of the solids with τ540K /τ0 = 0.25 for Pb62 S55 I14 and τ510K /τ0 = 0.19 and τ510K /τ0 = 0.52 for the fused and necked solid, respectively. The trend in the lifetimes observed here is similar to that reported in the experiments by Maikov et al 68 for PbSe NPs of comparable size, and highlights the importance of including finite-temperature effects in predictive models of NPs. In addition to a decrease of electronic gaps and lifetimes, in the case of the NP solids, we

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found that instantaneous structural distortions at finite temperature lead to the emergence of localized occupied orbitals within the gap. In Fig. 2a we plot the near-gap energy levels as a function of time for a short (0.2 ps) portion of the 510 K simulation for the fused solid, during which one can observe the periodic emergence of an occupied gap state, shown in red in Fig.

2a. We found a similar electronic structure for the necked solid at 510 K

(See Fig. S5 in the SI). At 250 K, such a state was still present, although positioned closer to the valence band maximum (VBM) and it did not occur as frequently throughout the simulations. On average, these dynamic gap states occurred on a timescale of approximately 50 fs during the entire simulation, and they were localized at times on the NP core and at times on the ligands (wave functions are shown in the inset of Fig. 2a). This emergent gap state may contribute to the decrease of Eg at finite T, although to a small extent: over a 10 ps portion of the trajectory, we found that the intra-gap state is on average only 0.1 eV above the state immediately lower in energy. It is expected that localized states could give rise to trapping, 70,74 thus we speculate that these emergent localized occupied states may be detrimental to hole transport in NP films. We have not observed any localized unoccupied states, indicating the electron transport may not be affected by trap states. Following our electronic structure calculations, we characterized the dielectric properties of the NPs at different temperatures, which have been widely disputed in the literature and are of great importance to understand assembly of NPs into solids. The magnitude of the dipole for lead (and cadmium) chalcogenide NPs in particular is highly controversial in the literature, i.e. with Klokkenburg et al, 51 Sashchiuk et al, 48 and Bertolotti et al 75 reporting dipole moments between 300 and 600 D for NPs above 2.5 nm in diameter, and other groups 49,50,53 reporting values between 0 - 150 D for the same sized systems using different experimental techniques and analytical expressions (a summary of the numerous dipole values reported in literature can be found in SI). To shed light on the sensitivity of the dipole moment to the interactions between NPs, we compared the dipole moment computed for the isolated Pb32 S28 I8 NP at 0, 250, and 510

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a) 3 LUMO HOMO HOMO 1 HOMO 2

2

10.4

b) Frequency

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10.5

10.6

Simulation Time (ps)

0.06

0.04

0.02

0.00 0

20

40

HOMO IPR

60

80

Figure 2: a) Emergence of intra-gap states (red dots, corresponding to the highest occupied level (HOMO)) during a small (0.2 ps) portion of the 510 K simulation for the fused solid. The size of R dots indicates the value of the inverse participation ratio (IPR, defined as:

|ψi |4 d3 r

2 ; larger IPR ( |ψi |2 d3 r) indicated more localized wave functions). Ball and stick representations of the NP structures show where the HOMO is localized at specific snapshots (from left to right, corresponding to #16, 29, 47, 61, and 72 of the 100 snapshots spaced evenly over the 0.2 ps window). b) Distribution of IPR for the HOMO wave function (intra-gap state) over a 10 ps simulation for the fused solid. Dotted line represents the average value.

R

K to those of the weakly interacting solids (See SI for details of dipole moment calculations). At T=0, we found a non-zero dipole moment (0.98 D) for the isolated NP, due to the asymmetric ligand arrangement on the surface. As a function of temperature, the dipole moment increased to 4.16 ± 1.9 D at 250 K and 5.68 ± 3.0 D at 510 K. Importantly, we observed that the dipole moment fluctuates on the order of 10 D with a maximum magnitude of 16 D at 510 K, as shown in Fig. 3b. The increase in magnitude and the fluctuations are due to structural distortions at finite temperature, which increase with increasing temperature, as well as to ligand migration at the NP surface, for example from one facet to another. We also calculated the dipole moment of Pb62 S55 I14 . We found that the fluctuations of 10


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the dipole moment are similar to those found for the smaller NP, and that the band gaps and lifetimes follow the same trends. In particular, the dipole moments of Pb62 S55 I14 and Pb32 S28 I8 averaged over 8 ps are 4.18 ± 1.6 D and 6.06 ± 3.2 D, respectively (without including the corrections for the induced electric field described in the SI); the maximum dipole moment calculated for Pb62 S55 I14 was 9.0 D, with a minimum value of 1.0 D, compared to 17.0 and 1.0 D for the maximum and minimum calculated values for Pb32 S28 I8 . For the necked and fused solids, the absolute dipole moment is ill-defined as these systems are periodic. Hence analysis of the dipole moments was limited to the weakly interacting systems (Fig.

3c). We found that weak interactions lead to an increase in the dipole

moment even in the absence of bond formation between neighboring NPs. Here, the dipole fluctuations were again significant, with values up to 77.5 D and as low as 5 D at 510 K.

Figure 3: Instantaneous dipole moment (µ) as a function of time for a) an isolated Pb32 S28 I8 NP over a 10 ps trajectory at T=510 K and b) weakly interacting NPs over two short trajectories of 2 and 3 ps at T=510 K. c) Difference in polarization with respect to the origin of the trajectories for two interacting NP solids (necked and fused; see text) at T=510 K; d) Computed dipole moments at T=0 for the lead selenide and lead sulfide NPs with a variety of ligands (L=halides (F,Cl,Br,I), ethane dithiol (EDT), and cinnamic acids)

For the necked and fused solids, in Fig. 3d we plot the difference in polarization, with respect to the initial configurations where bonds were not yet present, as a function of time, showing again a fluctuating behavior. These results indicate that for all solid structures studied here, the NP building blocks acted as dynamic, fluctuating dipolar systems rather than static point-dipoles. 11


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The comparison between the different values of dipole moments reported in the literature is not straightforward, as experiments used different ligands as well as NP shape. We calculated the dipole moment for a number of PbS and PbSe NPs at 0 K with different ligands, including replacement of the iodine on Pb32 S28 I8 with ethane dithiol (EDT), and halide ligands (F, Cl, Br and I) and we considered NP cores ranging from 1.5 - 2.1 nm. We also computed the dipole moments of NPs previously investigated by Kroupa et al, 20 capped with cinnamic acid ligands. Our results are reported in Fig. 3a. We found that the magnitude of the dipole can be tuned by ligand choice for a given NP core, an observation consistent with the claims of Rabani 55 and Shanbhag 54 that the dipole moment of a NP does not necessarily increase linearly as a function of NP size. In our calculations, we did not observe that the longest ligands (cinnamic acids) always provided the largest dipole. However, we did find that the choice of ligand can have a significant effect on the magnitude of the dipole, despite the ligands being often ignored in the literature. We caution that it may be necessary to revisit some of the approximations made to fit and interpret experimental data, as NP structure, ligand selection, and interactions between tightly packed NPs all contribute to the value of the dipole moment, with even residual interactions between NPs playing an important role in the interpretation of experimental data. a)

b)

Figure 4: Computed polarizabilities as a function of simulation time at T=510 K for the dilute, weakly interacting, and strongly interacting solids of Pb32 S28 I8 NPs (see text)

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Finally, we computed the electronic polarizability, α of the isolated NP and the solids using finite differences of the polarization under application of an electric field, with the polarizability tensor defined as:

αij = Ω

Pi (+∆Ej ) − Pi (−∆Ej ) 2∆Ej

(3)

where Pi (∆Ej ) denotes the polarization in the i th direction in response to an applied field ∆Ej in the j th direction. The ionic coordinates were not allowed to relax upon application of the applied field. For the isolated NP, we found a surprisingly constant, nearly temperature-independent 3 3 polarizability of 398 ± 3 ˚ A at 250 K and 401.5 ± 2.5 ˚ A at 510 K, where we defined α to be

the trace of the polarizability tensor. In contrast to the dipole moments, the fluctuations in the polarizability were small, however increasing in magnitude with the interaction strength. 3 While the increase in α from 700 to 800 ˚ A between the weakly interacting systems and

the solids is in part due to the increased number of bonds upon formation of the solid, the 3 increase from 400 to 700 ˚ A from the isolated NP to the weakly interacting solid corresponds

solely to non-bonding interactions. These results indicate that it may not be sufficient to use a fixed polarizability value (such as the polarizability of bulk PbS) for models of self-assembly of PbS NP solids, as the electronic part varies substantially. Furthermore, we compared our computed polarizabilities with that calculated from the commonly-used Clausius-Mossotti relation:

αCM =

3Ω (∞ − 1) 4π (∞ + 2)

(4)

where the dielectric constant of the solid, ∞ , was obtained by applying a finite electric field to the NP solids and using the Qbox code. We found a value of approximately 3.2 at 0 K for the weakly interacting solid (increasing to 3.6 and 3.7 at 510 K for the fused and necked solids, respectively, which is much smaller than the PbS bulk dielectric constant of

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22). We found that the Clausius-Mossotti relation underestimated the polarizability of the 3

3

interacting solids by 60%, providing values of only 376 ˚ A instead of 700 ˚ A , due to inherent approximations of a homogeneous non-dipolar material underlying the derivation of Eq. 4. In conclusion, we investigated the combined effect of temperature and NP-NP interaction on functionalized PbS nanoparticles, showing the emergence of electronic and dielectric properties different from those of isolated NPs. Using first principles molecular dynamics and electronic structure calculations, we showed that the fundamental gap of the NP slightly decreases with T and as a function of interaction, while the radiative lifetimes exhibited a much stronger dependence on interaction, becoming an order of magnitude less sensitive to temperature compared to those of isolated NPs. Our calculations also showed that weakly interacting NPs behave as fluctuating dipoles, with amplitudes on the order of 300% with respect to the average dipole moment, similar to those that we observed for the polarization of strongly interacting NP solids. We found that electronic polarizabilities, similar to dipoles, are affected by the interaction between NPs, increasing by about a factor of 2 in strongly interacting solids; however, their fluctuations at finite T are much smaller than those of the polarization. Hence, at finite T the NP solids behave as dynamical, polarizable dipolar systems, and dispersion interactions, e.g. as Casimir forces, are expected to contribute to their bonding and stability. Finally, in view of our findings, we revisited and discussed several results present in the literature for the dipole moments of lead chalcogenide NPs terminated with various ligands.

Acknowledgement The authors thank Nicholas Brawand, Fran¸cois Gygi, Sergio Mazzotti and Kirill Velizhanin for insightful discussions. AG was supported by NSF-CCI, under grant NSF-CHE-1305124; FG and GG were supported by MICCoM, as part of the Computational Materials Sciences Program funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences,

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Materials Science and Engineering Division through Argonne National Laboratory, under Contract No. DE-AC02-06CH11357. MV was supported by Laboratory Directed Research and Development (LDRD) funding from Argonne National Laboratory, provided by the Director, Office of Science, of the U.S. Department of Energy under Contract No. DEAC02-06CH11357. This research used computational resources provided by the Los Alamos National Laboratory Institutional Computing Program, which is supported by the U.S. DOE National Nuclear Security Administration under Contract No. DE-AC52-06NA25396, and the National Energy Research Scientific Computing Center (NERSC) which is supported by the Office of Science of the U.S. DOE under Contract No. DE-AC02-05CH11231.

Supporting Information Available This material is available free of charge via the Internet at http://pubs.acs.org: • Additional computational details • Video showing the change in atomic structure upon formation of both the fused and necked solids

References (1) Murray, C. B.; Noms, D. J.; Bawendi, M. G. J. Am. Chem. Soc 1993, 115, 8706–8715. (2) Rossetti, R.; Nakahara, S.; Brus, L. E. The Journal of Chemical Physics 1983, 79, 1086–1088. (3) Luther, J. M.; Law, M.; Beard, M. C.; Song, Q.; Reese, M. O.; Ellingson, R. J.; Nozik, A. J. Nano Letters 2008, 8, 3488–3492. (4) Shirasaki, Y.; Supran, G. J.; Bawendi, M. G.; Bulović, V. Nature Photonics 2012, 7, 13–23. 15


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Emergent Electronic and Dielectric Properties of Interacting

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