Effects of Disorder on Electronic Properties of Nanocrystal Assemblies

Article pubs.acs.org/JPCC

Effects of Disorder on Electronic Properties of Nanocrystal Assemblies Jun Yang* and Frank W. Wise School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14850, United States S Supporting Information *

ABSTRACT: Structural disorder is one of the major obstacles for band-like transport in semiconductor nanocrystal (NC) solids. However, systematic study of the effect of structural disorder on electronic properties is lacking. Here, we use a tightbinding model that explicitly incorporates energetic and positional disorders to evaluate quantitatively the effects of disorder on electron transport in semiconductor NC assemblies. The density of states and wave function delocalization are analyzed to identify the mobility edges, if they exist. We summarize the results in phase diagrams of the localization− delocalization transition for different superlattice structures. We find that the critical disorder at the transition point mainly depends on short-range order, not the long-range order of the superlattice. We also use an effective-mass model to calculate the coupling energy between adjacent nanocrystals. The calculation reveals that a red-shift of the excitonic peak in the optical absorption spectrum is not directly related to the magnitude of electronic coupling. For the specific case of PbSe nanocrystals coupled by ethanedithiol ligands, we conclude that the disorder is too large to allow delocalized states. Lastly, we analyze the effects of valley degeneracy and intervalley coupling, both within a NC and across adjacent NCs, using a phenomenological model. The impact of multiple valleys on charge transport depends on the exact nature of the coupling of distinct valleys, which is unknown. We estimate the magnitude of the coupling and conclude that the presence of multiple valleys has a minor impact on the wave function delocalization. Overall, we expect that band-like transport should be attainable with suitable ligands and proper surface passivation.

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INTRODUCTION Semiconductor nanocrystals continue to be the focus of intense research. Their nanoscale sizes lead to quantum confinement effects, which reduces the density of states into discrete energy levels. These “artificial atoms” can be the basis of “artificial solids”, which provide great scientific opportunities. On the other hand, their tunable energy gaps, easy access to surface modification, and relatively cheap cost of manufacturing, makes them attractive for a number of optoelectronic devices, such as solar cells,1−4 light-emitting diodes,5,6 and photodetectors.7,8 Currently, one of the major goals in this area is to observe coherent transport of charge carriers. In an ordinary crystalline solid, atoms line up periodically to form a lattice. Instead of scattering the electrons, the interaction of the atoms with the electron forms an energy band. Ideally, the wave function of an electron in the band extends over the entire crystal, and electrons transport though these extended states. Typical semiconductors have carrier mobilities around 103 cm2/(V s) to 104 cm2/(V s). For comparison, NC solids currently have mobilities that range from 10−3 cm2/(V s) to 101 cm2/(V s).9−14 The transport is usually attributed to nearest-neighbor or variablerange hopping.15 Several studies have reported band-like transport,16−18 which is based on the observation of increased © XXXX American Chemical Society

mobility with decreasing temperature. However, there has been controversy as to the real nature of transport. It was shown19 that even hopping transport can lead to a similar temperature dependence, and that alone cannot prove real transport through band-like states. The fundamental limit for true band-like transport can be traced to the inherent disorder in a NC solidthat is, the energy disorder due to NC size inhomogeneity and the coupling disorder due to imperfect alignment. Sufficient disorder leads to Anderson localization,20,21 the absence of delocalized states. Anderson localization has been extensively studied analytically,22,23 numerically,24−28 and experimentally.29−31 Noriega et al. studied transport in organic semiconductors and showed a strong correlation between paracrystallinity (structural disorder) in conjugated polymers and their charge transport efficiency.32 Remacle et al. studied the effects of charging energy and disorder on charge transport in metal NC solids.33 Artemyev et al. studied the existence of delocalized electron states in CdSe NC ensembles with different sizes and packing ratios.34 Received: September 29, 2014 Revised: January 7, 2015

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DOI: 10.1021/jp5098469 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Figure 1. Simulation results for a 3D simple-cubic lattice. Parameters: s/t0 = 2.0, κg = 0.2. (a) Density of states. Red line is an exponential fit to the tail states. (b) Logarithm of I2 (Inverse Participation Ratio) versus energy, different lines correspond to different system size L. (c) Scaling of I2 versus the system size L. Solid lines are power-law fits. The opposite of the fitted exponent corresponds to d2, which is the fractal dimension. (d) Summary of the fractal dimensions for all energies. Error bars are estimated from the fitting procedure. The red line is a parabolic fit of d2 near the mini-band center.

tij = t0e−κ Δdij

However, the modeling of structural disorder in this work is not realistic enough to allow comparison with experiments. Here, we study numerically the criterion for band-like transport (i.e., carriers conduct through delocalized electronic states) in widely investigated cadmium-salt and lead-salt NC solids. We start from a Hamiltonian that explicitly models the onsite energy disorder and positional disorder realistically. We then use scaling of the wave functions to the system size to identify whether a state is localized or delocalized. We find the disorder threshold for the existence of delocalized states in NC solids with different superlattice symmetries. To compare with the experimental status quo, we use an effective-mass model to calculate the coupling strength, which is then compared with the coherent transport criterion. As an example, we find that NC solids treated with short organic ligands commonly used in photovoltaic devices are unlikely to support delocalized states. Finally, we discuss the effect of the valley degeneracy present in lead-salt NCs.

where t0 is the coupling strength at average distance, and Δdij is the deviation of the distance from the average distance. The exponential decay of coupling strength has been verified previously10,35 and will be discussed further in the next section. As a first approximation, the dependence of coupling strength on NC diameter is neglected. We use Monte Carlo simulation to model an ensemble of lattices and calculate the density of states as well as the wave functions. The onsite energy disorder s describes the inhomogeneous broadening of energy levels of the NCs, 2 2 which is modeled as P(Ei = E) ∝ e−E /(2s ). The positional disorder g describes the fluctuation of the position of each NC relative to a perfect superlattice, which is modeled as P(Δxi = 2 2 Δx) ∝ e−Δx /(2g ), and Δdij = Δxj(α) − Δxi(α), where α is the direction from lattice point i to lattice point j. To simplify the units, all energies will be given in terms of the dimensionless quantity E/t0, and all distances will be given in terms of the dimensionless quantity κx. The analysis mostly follows the approach of ref 28. Results. We first consider the simple-cubic lattice. Each NC has six nearest neighbors. The calculated density of states (DOS) for s/t0 = 2.0, κg = 0.2 is plotted in Figure 1a. The calculation predicts formation of a symmetric mini-band with bandwidth around 12t0, centered at E = 0 (the average energy of uncoupled NC electron states), consistent with expectation. The DOS outside the mini-band shows an exponential decay, which is common among amorphous materials.36−39 Since we are more interested in the wave function, we use the inverse participation ratio40

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EFFECT OF DISORDER ON WAVE FUNCTION LOCALIZATION Method. We use the tight binding Hamiltonian to model NC solids.28 The total Hamiltonian is H = Honsite + Hcoupling =∑ Eiai†ai + i

∑ tij(ai†aj + h. c. ) ⟨i , j⟩

Here Ei is the electronic state (HOMO or LUMO) of the NC at lattice point i. The sum over ⟨i,j⟩ includes all the nearestneighbor pairs. tij is the coupling energy, which depends critically on the edge-to-edge distance between neighboring NCs,

I2[Ψn] =

∑ |Ψn(r )|4 r

B


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Figure 2. Contour plots for the fractal dimension d2(E) at E = 0, versus onsite energy disorder s/t0 and positional disorder κg, for different crystal structures: (a) simple cubic (sc), (b) body-centered cubic (bcc), (c) face-centered cubic (fcc), (d) random close packing (rcp), (e) 2D honeycomb, (f) 2D square, and (g) 2D hexagonal. N is the number of nearest neighbors. N ∼ 7 for random close packing is estimated from the cumulative pair distribution function.

It was shown previously that at the localization−delocalization transition, the scaling exponent d2 = 1.33 ± 0.02.41 The scaling exponent is a universal quantity; it does not depend on the details of the coupling but only on the dimension and symmetry of the interaction. If we draw a line at d2(E) = 1.33, then statistically, states below this line are delocalized, whereas states above it are localized (Figure 1d). Here, the transition points are at E = ± 6.8t0, which also correspond to the mobility edges. There are two mobility edges, one on each side of the miniband (if the band does not overlap with bands formed from higher-energy states). States that are within the mini-band are delocalized; states outside are localized. One can change the transport regime by moving the Fermi level of the electrons, by electrostatic doping (via the gate voltage in a field effect transistor), or by electrochemical doping (changing the concentration of dopants).42,43 However, if the disorder is large enough, even the states at band center E = 0 are completely localized. In this case, all the states will be localized, and there is no coherent transport in the system. Figure 2a summarizes the fractal dimension at band center (d2(E = 0)) with varying onsite energy disorder s/t0 and positional disorder κg for the simple cubic lattice. The black line

to describe the degree of delocalization. Suppose the system is of size L and dimension d. If the wave function Ψn(r) is completely delocalized, then |Ψn(r)| has to be on the scale of L−d/2 for all lattice sites, in order to satisfy wave function normalization (∑r|Ψn(r)|2 = 1). This implies that I2[Ψn] ∼ Ld(L−(d/2))4 ∝ L−d. On the other hand, if the wave function is completely localized, then |Ψn(r)| ∼ 1 in the localized region, and |Ψn(r)| ∼ 0 in other regions. This gives I2[Ψn] ∼ constant ∝ L0. At the transition from being delocalized to localized, the wave function is multifractal,41 which gives I2[Ψn] ∝ L−d2, with d2 the fractal dimension (0 < d2 < d). Figure 1b shows the average value of I2 as a function of energy, for varying system size L. At fixed energy, we can fit I2 to a power law function of L (Figure 1c) and extract d2(E) as the exponent. Here E = 5.0t0 corresponds to states within the mini-band. As expected, d2(E) = 2.67, which is close to 3 and so indicates that states are mostly delocalized. E = 7.8t 0 corresponds to states deep in the exponential band tail. Here d2(E) = 0 within the uncertainty, which indicates that states at this energy are completely localized. At E = 7.0t0, d2(E) = 1.24, close to the critical point of transition, and the DOS has a fractal shape. d2(E) is plotted in Figure 1d. C


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than those of 3D lattices, even after considering their smaller number of nearest neighbors. This indicates that dimensionality does play an important role in localization.

(d2 = 1.33) is the boundary between delocalized and localized states. For systems in the left bottom corner, there are some states that are delocalized, permitting the possibility of coherent transport; for systems on the top right corner, all the states are localized. The threshold with only onsite energy disorder and no positional disorder (κg = 0, sth/t0 ∼ 6) is consistent with previous results.26 Results for the body-centered cubic (bcc) lattice, facecentered cubic (fcc) lattice, and random close packing (rcp) are calculated in the same way and summarized in Figure 2b−d. The random close-packed assembly is generated with forcebiased algorithms developed elsewhere.44 The packing ratio generated is 64%, close to the theoretical limit of maximum packing45 (see Supporting Information for detailed packing statistics). Results for the 2D honeycomb, square and hexagonal lattices are also shown for comparison. Strictly speaking, all 2D systems with time-reversal symmetry have localized wave functions.28 However, with low disorder, the localization length can be very long, much longer than the sample size, and therefore, the states are only weakly localized. Here we use finite sample dimension L in the range of 10 to 40. Also, because there is no true Anderson transition here, we use d2 = 1.0 as an approximate criterion. The actual value will be between 1.0 and 2.0. Discussion. The basic shapes of all the phase diagrams are very similar. To analyze further, we plot the number of nearest neighbors vs critical onsite energy disorder sth and critical positional disorder gth in Figure 3.

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EFFECTIVE MASS CALCULATION OF COUPLING ENERGY It is clear that the existence of coherent transport depends critically on the ratio of onsite energy disorder to coupling strength. There have been numerous attempts to estimate coupling strength in experiments. A large red shift in the peak position of the absorption spectrum is usually seen when quantum dots are deposited on a solid state film with short ligands, and the red shift is frequently attributed to the electronic coupling.10,47 However, further experiments48 showed that other factors, such as polarization effects, are more likely to be the cause. Application of the results above will require estimates of the coupling energy, and calculations will shed light on the relationship between coupling energy and optical spectra. Method. We use a single-band effective mass model to analyze the magnitude and distance dependence of internanocrystal coupling. The single-band model is only a rough approximation to the actual band structure. However, it is mathematically simple to evaluate, it gives a clear physical picture, and the energy levels it produces agree reasonably well with more sophisticated models and experiments, especially for large particles.49 We believe that the model is effective because the coupling energy depends more on the barrier height at the NC-host interface than on the electronic structure of the NC or host material. The Schrödinger Equation for the one-band model is50 ⎛ ℏ2 ⎞ −∇⎜ ∇ ⎟ φ (r ) + V (r ) φ (r ) = E 0 φ (r ) ⎝ 2meff (r ) ⎠

where [meff (r ), V (r )] = [meff (r ), V (r )] ⎧ ⎪[mNC , − Vwell ], r ≤ R =⎨ ⎪ ⎩[menv , 0], r > R

Figure 3. Threshold values for disorder at g = 0 and s = 0. N is the number of the nearest neighbors.

mNC is the effective mass of the electron (hole) for the bulk semiconductor of the nanocrystal; menv is the effective mass of the electron (hole) of the interstitial area; and Vwell is the potential-energy barrier. The lowest energy wave function has the form:

The results for 3D lattices lie on a straight line, which indicates a power law dependence of critical disorder on the number of nearest neighbors. Fitting the result with sth/t0 = ANα gives us the exponent α = 0.87 ± 0.08, which is close to 1. This can be understood as more neighbors introduce, on average, more coupling to each NC. The RMS bandwidth ⟨(E − E0)2⟩1/2 for a perfect lattice without disorder is Nt0. This can be viewed as the total coupling energy of a single NC, considering all its nearest neighbors. The fitting of threshold positional disorder κgth = BNβ yields the exponent β = 0.54 ± 0.09, close to 0.5. This reflects the model assumption that the positional disorder of each NC is independent. Thus, having more neighbors decreases the overall fluctuation of coupling energy of each NC, due to the central limit theorem. It is worth mentioning that the point for 3D random close packing is also fit well by the power law. This implies that the lack of long-range order does not affect delocalization. Indeed, extended states have been reported in disordered InGaAs/GaAs 2D quantum dot superlattices.46 On the other hand, the disorder thresholds for 2D lattices are all significantly lower

⎧ C1sin kr ,r≤R ⎪ ⎪ r φ (r ) = φ (r ) = ⎨ ⎪ C2e−αr ,r>R ⎪ ⎩ r

where k = (2mNC(E + Vwell))1/2/ℏ, α = (2menv(−E))1/2/ℏ. The wave function (Figure 4) is continuous at the boundary but has a discontinuous derivative due to the discontinuity of effective mass across the boundary. The Hamiltonian for a superlattice of coupled NCs is ⎛ ℏ2 ⎞ H(r ) = −∇⎜ ∇⎟ + V (r ) 2 m ( ) r ⎝ eff ⎠ D


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⎛

2

∫|r−R ||f 2p|, and therefore the total f 2 = f 2k + f 2p is small but positive. f 3k and f 3p have the same sign (their absolute phase depends on the phase of the wave across different NCs, but their relative phase is determined). Fits of the absolute values of the overlap integrals to exponential decays are shown in Figure 5b. It is clear that f 3 follows an exponential decay Ae−κd (with κ = 0.40 Å−1 ≈ β/2), whereas f 2 follows approximately an e−2κd decay.

Figure 4. Wave function of electron in a nanocrystal, calculated from a single-band effective mass model. Parameters used: mNC = mpbs = 0.09 me (me is the free electron mass), menv = mC = 0.28 me, Vwell 4.6 eV, R = 3 nm.

⎛ ℏ2 ⎞ = −∇⎜ ∇⎟ + ⎝ 2menv ⎠

f2 (dmn))am +

d3rφ*(r − R i)H′(r − R j)φ(r − R k)

Figure 5. (a) Values of the overlap integrals as a function of edge-to-edge distance between NCs. Integral due to kinetic energy and potential energy are plotted individually. Vwell 4.0 eV, menv = meff,C = 0.162m0, mNC = mPbSe = 0.047m0, d = 6 nm. (b) Exponential fit of the overlap integrals. E


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Figure 6. Summary of calculated coupling energy t0 for EDT coupled nanocrystals, and onsite energy disorder s. t0 is calculated assuming surface to surface distance between NCs is 0.4 nm; s is calculated assuming standard deviation in NC diameters is 0.3 nm for all sizes.

To study the size dependence of coupling and disorder, we use ethanedithiol (EDT), a short linker molecule commonly used in NC devices. The molecule length is about 0.4 nm.10,35 The coupling energy t0 for NCs connected by EDT is plotted as a function of diameter in Figure 6a. The onsite energy disorder s is estimated using a four-band model for lead salt NCs49 and a one-band model for cadmium salt NCs. The NC diameter is assumed to vary with a standard deviation of 0.3 nm,10,52 which corresponds to one layer of crystal lattice. The ratio of disorder to coupling is plotted in Figure 6b. Smaller NCs are more favorable for delocalized transport, due to their larger coupling strength. For fixed size, PbSe NCs are generally more favorable than CdSe NCs, then CdS, then PbS NCs. Discussion. The electronic coupling itself does not lead to a red shift in absorption. We saw previously that f 2 is a diagonal term, which only shifts the overall energy and does not contribute to the actual delocalization of the wave function, and vice versa for f 3. We also saw that f 2 is always orders of magnitude smaller than f 3 in absolute value. Furthermore, f 2 is actually positive. When the wave function of one NC leaks into an adjacent NC, the kinetic energy it picks up is larger in magnitude than the decrease in potential energy, so the overall energy increases slightly. As a result, if only electronic coupling is considered, the absorption peak position should shift to shorter rather than longer wavelengths. Therefore, the observed red-shift must arise from other processes. We would like to point out, however, that electronic coupling does red-shift the absorption edge (as opposed to the peak) because of the much larger f 3. Broadening of the peak is a direct measure of electronic coupling. The effect of dielectric contrast between the NC and host medium is not included in the calculation. In most cases, the dielectric constant of the NC is much larger than that of the medium. For example, ε0 (PbS) = 169, ε0 (PbSe) = 210, while ε0 (air) = 1, ε0 (polyethylene) = 2.25. In that case, the image charge at the interface has the same sign as the original charge, which repels the charge from the interface, causing “dielectric confinement”. Inclusion of dielectric contrast would therefore reduce the coupling strength. Figure 7 shows the critical values of the disorder for 3 nm diameter PbSe quantum dots with different linker lengths, plotted in units of the measurable quantities. For an EDTcoupled NC solid with diameter fluctuations of 0.3 nm and edge-to-edge distance fluctuations of 0.3 nm, the point lies outside the delocalization threshold, which implies that there should be no delocalized states.

Figure 7. Summary of critical disorder for diameter fluctuation σdiameter and edge-to-edge distance fluctuation σe−e distance for 3 nm PbSe NCs, for average edge-to-edge distances of 0, 0.2, 0.4, and 0.6 nm. The red dot is for EDT-coupled dots. It lies to the right of the yellow line (0.4), so there are no delocalized states.

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EFFECTS OF VALLEY DEGENERACY IN LEAD-SALT NANOCRYSTALS

One aspect that is missing from the previous analysis is the 8fold degeneracy in the HOMO and LUMO levels of PbS or PbSe NCs, due to the four equivalent L valleys in the bulk band structure, and the 2-fold spin degeneracy. Coupling between the valleys due to quantum confinement then splits this degeneracy. The exact nature of the coupling and the resulting splitting of the energy levels are currently unknown. As far as we know, there are no experiments that directly measure this valley coupling strength, although there are numerous observations of the splitting or broadening of lowest exciton states that suggest its existence.53−56 Nevertheless, first principle calculations estimate the splitting due to intervalley coupling to be 10−80 meV, for PbSe dots ranging from 3 to 8 nm, which shows oscillatory behavior and is sensitive to size.57,58 This energy is on the same order of magnitude as the coupling energy between adjacent dots and the onsite energy disorder and therefore suggests that it could appreciably affect the delocalization of electron states. The simplest intuition might be that splitting of energy levels comparable to the coupling energy should lead to enhanced coupling and transport. We use a phenomenological model to study 4 nmdiameter PbSe NC simple-square-lattice and simple-cubiclattice as examples to evaluate the impact of intervalley coupling on delocalization. Method. To describe the degeneracy and coupling for four valleys, each state in the original one-band model is expanded into four states, with intervalley coupling Vint between them: F


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Figure 8. Fractal dimension d2(E) for different valley coupling model. (Noise in the data is due to limited calculation times and has no physical meaning.) Parameters: s/t0 = 1.5, κg = 0.6, ϕ = 0.

⎛ E Vinteiϕ Vinteiϕ Vinteiϕ ⎞ 0 ⎜ ⎟ ⎜ V e−iϕ iϕ iϕ ⎟ E V e V e int 0 int int ⎟ E0 → ⎜ ⎜ ⎟ −iϕ −iϕ Vint e E0 Vinteiϕ ⎟ ⎜ Vint e ⎜ ⎟ −iϕ Vint e−iϕ Vint e−iϕ E0 ⎠ ⎝ Vint e

stays the same. When Vint is increased, the maximum fractal dimension stays the same, while the overall bandwidth is increased. Table S3 summarizes the results for three different cases of disorder. The trend observed previously is consistent across all cases. Next, we consider the effect of coupling selectivity. Figure 9 shows the fractal dimension for different α values. Most

Here ϕ is the phase of the coupling. Depending on the exact value of ϕ, the splitting of the energy levels will be different. Nevertheless, the total energy splitting is consistently 4 times Vint (see Supporting Information for details). From the total energy splitting previously reported,57,58 we estimate Vint ≅ 15 meV for 4 nm PbSe NCs. Each coupling term between adjacent NCs in a one-band model is also expanded into a 4 × 4 matrix that describes coupling across different valleys:

⎛α ⎜ ⎜β t → t *⎜ ⎜β ⎜ ⎝β

β β β⎞ ⎟ α β β⎟ ⎟ β α β⎟ ⎟ β β α⎠

Figure 9. Fractal dimension d2(E) for different coupling selectivity α. Parameters: s/t0 = 1.5, κg = 0.6, ϕ = 0, Vint = 2.0t0.

where t is the coupling strength in the one-band model, which has an exponential dependence on the distance between dot surfaces, α is the parameter that describes the valley selectivity of the coupling, α2 + 3β2 = 1 so as to preserve the total tunneling probability from one valley in a dot to the adjacent dot. α = 1 (β = 0) corresponds to the case where an electron in one valley is only coupled to the electron state in the adjacent dot in the same valley; α = 0.5 (β = 0.5) corresponds to the case where an electron in one valley is equally coupled to all four valleys in the adjacent dot. Results. First we consider the simple case where α = 1. Figure 8 compares the fractal dimensions of the original oneband model and the four-band model with different Vint. In the case of Vint = 0, and since β = 0, all four bands in the NC solid are completely decoupled. As expected, the fractal dimension

noticeable is when α = 0.5, where a “localization band” appears from −5t0 to 0: the states are almost completely localized. This is caused by the destructive interference of coupling to the next NC from different valleys: the coupling matrix has an eigenvalue of 0 at α = 0.5; the corresponding eigenstate represents a state which has no coupling to its adjacent NCs and is therefore completely localized. Away from the localization band, the system shows improved delocalization compared with the one-band model. Intermediate values of α produce the best potential for transport. Table S4 summarizes the results for different disorders for α = 0.8. The increase of delocalization is largest when the onsite energy disorder is large. Compared with the G


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phase diagrams that show directly the critical disorder that eliminates delocalized transport, for several crystal structures. We used a one-band effective mass model to calculate the coupling energy. We found that in EDT-treated nanocrystal solids, the disorder is large enough to completely eliminate coherent transport. For lead salt NCs, we also studied the effect of intervalley coupling, which does generally improve delocalization, but not enough to make a real difference with current synthetic capabilities. However, we believe that with the right ligand for electronic coupling, infilling of the void space with smaller-bandgap semiconductors, and proper passivation of surfaces, band-like transport is attainable.

one-band model, the increase of maximum is equivalent to the reduction of onsite energy disorder by 0.9−1.4t0. Calculations for 3D lattices are listed in the Supporting Information and show similar results. The intervalley coupling increases the bandwidth of delocalization by approximately 4Vint. The localization band is also present and is even more prominent than in the 2D case. The net result is that in some cases, the maximum delocalization is slightly decreased instead of increased. We estimate that the increase in wave function delocalization due to intervalley coupling corresponds to the increase that would result from a reduction of onsite energy disorder by about t0. Discussion. There is currently no way to identify which model is more accurate than the other. What we conclude from the analysis is that intervalley coupling does change the delocalization spectrum appreciably. For α = 1.0, valley coupling increases the bandwidth; for α = 0.5, it creates a localization band; for intermediate α, it effectively decreases onsite energy disorder by about t0. We expect that firstprinciple calculations will provide new data that can help identify the true value of α. Also, low-temperature charge transport experiments over a large doping range might provide experimental evidence for the localization band. On the other hand, the ratio of onsite energy disorder to coupling energy is generally very large (≳ 8) for current synthesis capabilities, and a change of 1 is not going to significantly impact the delocalization. To achieve coherent transport, the system has to be well within the delocalization threshold to have a considerable energy band. As seen from Figure 7, if the edge-to-edge distance is less than 0.2 nm, it will be on the threshold of having delocalized states. Metal chalcogenide complex18 and atomic ligands12 both show much improved carrier mobility. To decrease the positional disorder κg, much progress has been made to improve g in terms of superlattice formation. Recently, atomically coherent PbSe superlattices have been reported.59−61 Due to the constraints of the superlattice structures, their positional disorder will be significantly improved. In these systems, the onsite energy disorder will be the major obstacle for delocalized transport. Another way to decrease positional disorder is to decrease κ by infilling the void space of the solid with a small-bandgap semiconductor. The use of atomic layer deposition12 proves to be an excellent way to passivate the nanocrystal surface while decreasing the energy barrier between nanocrystals. In order to observe coherent transport through delocalized states, all the incoherent scattering should be reduced so that the mean free path due to incoherent scattering is larger than or at least comparable to the sample size. These include scattering from domain boundaries, scattering from thermal fluctuation of the lattice, and charge trapping in defects. Also, the Fermi level of carriers has to be within the mini-band for the transport to benefit from these delocalized states. If the mini-band is very narrow, it requires a significant amount of charge doping (more than one electron per dot). This will introduce strong electron−electron scattering. This is not captured in the current single particle model and will lower the mobility further.

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ASSOCIATED CONTENT

S Supporting Information *

Detailed simulation method, detailed statistics on random close packing, band parameters used in the calculation and discussion on the dependence of coupling energy on value of energy barrier potential and effective mass. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Division of Materials Sciences and Engineering, under Award No. DE-SC0006647. The authors would also like to thank T. Hanrath, K. Whitham, and H. Zhang for helpful discussions.

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REFERENCES

(1) Luther, J. M.; Law, M.; Beard, M. C.; Song, Q.; Reese, M. O.; Ellingson, R. J.; Nozik, A. J. Schottky Solar Cells Based on Colloidal Nanocrystal Films. Nano Lett. 2008, 8, 3488−3492. (2) Choi, J. J.; Lim, Y.-F.; Santiago-Berrios, M. B.; Oh, M.; Hyun, B.R.; Sun, L.; Bartnik, A. C.; Goedhart, A.; Malliaras, G. G.; Abruña, H. D.; et al. PbSe Nanocrystal Excitonic Solar Cells. Nano Lett. 2009, 9, 3749−3755. (3) Pattantyus-Abraham, A. G.; Kramer, I. J.; Barkhouse, A. R.; Wang, X.; Konstantatos, G.; Debnath, R.; Levina, L.; Raabe, I.; Nazeeruddin, M. K.; Grätzel, M.; et al. Depleted-Heterojunction Colloidal Quantum Dot Solar Cells. ACS Nano 2010, 4, 3374−3380. (4) Ip, A. H.; Thon, S. M.; Hoogland, S.; Voznyy, O.; Zhitomirsky, D.; Debnath, R.; Levina, L.; Rollny, L. R.; Carey, G. H.; Fischer, A.; et al. Hybrid Passivated Colloidal Quantum Dot Solids. Nat. Nanotechnol. 2012, 7, 577−582. (5) Wood, V.; Panzer, M. J.; Caruge, J.-M.; Halpert, J. E.; Bawendi, M. G.; Bulović, V. Air-Stable Operation of Transparent, Colloidal Quantum Dot Based LEDs with a Unipolar Device Architecture. Nano Lett. 2010, 10, 24−29. (6) Sun, L.; Choi, J. J.; Stachnik, D.; Bartnik, A. C.; Hyun, B.-R.; Malliaras, G. G.; Hanrath, T.; Wise, F. W. Bright Infrared QuantumDot Light-Emitting Diodes through Inter-Dot Spacing Control. Nat. Nanotechnol. 2012, 7, 369−373. (7) Prins, F.; Buscema, M.; Seldenthuis, J. S.; Etaki, S.; Buchs, G.; Barkelid, M.; Zwiller, V.; Gao, Y.; Houtepen, A. J.; Siebbeles, L. D. a; et al. Fast and Efficient Photodetection in Nanoscale Quantum-Dot Junctions. Nano Lett. 2012, 12, 5740−5743. (8) Keuleyan, S.; Lhuillier, E.; Brajuskovic, V.; Guyot-Sionnest, P. Mid-Infrared HgTe Colloidal Quantum Dot Photodetectors. Nat. Photonics 2011, 5, 489−493.

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CONCLUSION We used a tight binding model with explicit modeling of onsite energy disorder and positional disorder to evaluate the existence of delocalized states in a nanocrystal solid. We obtain H


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(30) Hu, H.; Strybulevych, A.; Page, J. H.; Skipetrov, S. E.; van Tiggelen, B. A. Localization of Ultrasound in a Three-Dimensional Elastic Network. Nat. Phys. 2008, 4, 945−948. (31) Billy, J.; Josse, V.; Zuo, Z.; Bernard, A.; Hambrecht, B.; Lugan, P.; Clément, D.; Sanchez-Palencia, L.; Bouyer, P.; Aspect, A. Direct Observation of Anderson Localization of Matter Waves in a Controlled Disorder. Nature 2008, 453, 891−894. (32) Noriega, R.; Rivnay, J.; Vandewal, K.; Koch, F. P. V; Stingelin, N.; Smith, P.; Toney, M. F.; Salleo, A. A General Relationship between Disorder, Aggregation and Charge Transport in Conjugated Polymers. Nat. Mater. 2013, 12, 1−7. (33) Remacle, F.; Levine, R. D. Quantum Dots as Chemical Building Blocks: Elementary Theoretical Considerations. ChemPhysChem 2001, 2, 20−36. (34) Artemyev, M.; Bibik, A.; Gurinovich, L.; Gaponenko, S.; Woggon, U. Evolution from Individual to Collective Electron States in a Dense Quantum Dot Ensemble. Phys. Rev. B 1999, 60, 1504−1506. (35) Gao, Y.; Aerts, M.; Sandeep, C. S. S.; Talgorn, E.; Savenije, T. J.; Kinge, S.; Siebbeles, L. D. a; Houtepen, A. J. Photoconductivity of PbSe Quantum-Dot Solids: Dependence on Ligand Anchor Group and Length. ACS Nano 2012, 6, 9606−9614. (36) Soukoulis, C. M.; Cohen, M. H.; Economou, E. N. Exponential Band Tails in Random Systems. Phys. Rev. Lett. 1984, 53, 616−619. (37) Werner, J.; Peisl, M. Exponential Band Tails in Polycrystalline Semiconductor Films. Phys. Rev. B 1985, 31, 6881−6883. (38) Tiedje, T.; Cebulka, J.; Morel, D.; Abeles, B. Evidence for Exponential Band Tails in Amorphous Silicon Hydride. Phys. Rev. Lett. 1981, 46, 1425−1428. (39) Schiff, E. Trap-Controlled Dispersive Transport and Exponential Band Tails in Amorphous Silicon. Phys. Rev. B 1981, 24, 6189− 6192. (40) Mirlin, A. Statistics of Energy Levels and Eigenfunctions in Disordered Systems. Phys. Rep. 2000, 326, 259−382. (41) Evangelou, S. N. Multifractal Wavefunctions at the Mobility Edge. J. Phys. A. Math. Gen. 1990, 23, L317−L323. (42) Vanmaekelbergh, D.; Liljeroth, P. Electron-Conducting Quantum Dot Solids: Novel Materials Based on Colloidal Semiconductor Nanocrystals. Chem. Soc. Rev. 2005, 34, 299−312. (43) Wang, S.; Ha, M.; Manno, M.; Daniel Frisbie, C.; Leighton, C. Hopping Transport and the Hall Effect near the Insulator-Metal Transition in Electrochemically Gated poly(3-Hexylthiophene) Transistors. Nat. Commun. 2012, 3, 1210. (44) Baranau, V.; Hlushkou, D.; Khirevich, S.; Tallarek, U. Pore-Size Entropy of Random Hard-Sphere Packings. Soft Matter 2013, 9, 3361−3372. (45) Song, C.; Wang, P.; Makse, H. A. A Phase Diagram for Jammed Matter. Nature 2008, 453, 629−632. (46) Lan, S.; Akahane, K.; Song, H.-Z.; Okada, Y.; Kawabe, M.; Nishimura, T.; Nishikawa, S.; Wada, O. Formation of Extended States in Disordered Two-Dimensional In[sub 0.4]Ga[sub 0.6]As/ GaAs(311)B Quantum Dot Superlattices. J. Appl. Phys. 2000, 88, 227. (47) Williams, K. J.; Tisdale, W. A.; Leschkies, K. S.; Haugstad, G.; Norris, D. J.; Aydil, E. S.; Zhu, X.-Y. Strong Electronic Coupling in Two-Dimensional Assemblies of Colloidal PbSe Quantum Dots. ACS Nano 2009, 3, 1532−1538. (48) 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. Anomalously Large Polarization Effect Responsible for Excitonic Red Shifts in PbSe Quantum Dot Solids. J. Phys. Chem. Lett. 2011, 2, 795− 800. (49) Kang, I.; Wise, F. W. Electronic Structure and Optical Properties of PbS and PbSe Quantum Dots. J. Opt. Soc. Am. B 1997, 14, 1632. (50) Glutsch, S. Excitons in Low-Dimensional Semiconductors: Theory, Numerical Methods, Applications; Springer-Verlag: Berlin, Heidelberg, 2004. (51) Tomfohr, J.; Sankey, O. Complex Band Structure, Decay Lengths, and Fermi Level Alignment in Simple Molecular Electronic Systems. Phys. Rev. B 2002, 65, 1−12.

(9) Talapin, D. V.; Murray, C. B. PbSe Nanocrystal Solids for N- and P-Channel Thin Film Field-Effect Transistors. Science 2005, 310, 86− 89. (10) Liu, Y.; Gibbs, M.; Puthussery, J.; Gaik, S.; Ihly, R.; Hillhouse, H. W.; Law, M. Dependence of Carrier Mobility on Nanocrystal Size and Ligand Length in PbSe Nanocrystal Solids. Nano Lett. 2010, 10, 1960−1969. (11) Sakanoue, T.; Sirringhaus, H. Band-like Temperature Dependence of Mobility in a Solution-Processed Organic Semiconductor. Nat. Mater. 2010, 9, 736−740. (12) Liu, Y.; Tolentino, J.; Gibbs, M.; Ihly, R.; Perkins, C. L.; Liu, Y.; Crawford, N.; Hemminger, J. C.; Law, M. PbSe Quantum Dot FieldEffect Transistors with Air-Stable Electron Mobilities above 7 cm2 V(−1) S(−1). Nano Lett. 2013, 13, 1578−1587. (13) Fafarman, A. T.; Koh, W.; Diroll, B. T.; Kim, D. K.; Ko, D.; Oh, S. J.; Ye, X.; Doan-Nguyen, V.; Crump, M. R.; Reifsnyder, D. C.; et al. Thiocyanate-Capped Nanocrystal Colloids: Vibrational Reporter of Surface Chemistry and Solution-Based Route to Enhanced Coupling in Nanocrystal Solids. J. Am. Chem. Soc. 2011, 133, 15753−15761. (14) Kovalenko, M. V.; Scheele, M.; Talapin, D. V. Colloidal Nanocrystals with Molecular Metal Chalcogenide Surface Ligands. Science 2009, 324, 1417−1420. (15) Houtepen, A. J.; Kockmann, D.; Vanmaekelbergh, D. Reappraisal of Variable-Range Hopping in Quantum-Dot Solids. Nano Lett. 2008, 8, 3516−3520. (16) Choi, J.; Fafarman, A. T.; Oh, S. J.; Ko, D.; Kim, D. K.; Diroll, B. T.; Muramoto, S.; Gillen, J. G.; Murray, C. B.; Kagan, C. R. Bandlike Transport in Strongly Coupled and Doped Quantum Dot Solids: A Route to High-Performance Thin-Film Electronics. Nano Lett. 2012, 12, 2631−2638. (17) Talgorn, E.; Gao, Y.; Aerts, M.; Kunneman, L. T.; Schins, J. M.; Savenije, T. J.; van Huis, M. a.; van der Zant, H. S. J.; Houtepen, A. J.; Siebbeles, L. D. a. Unity Quantum Yield of Photogenerated Charges and Band-like Transport in Quantum-Dot Solids. Nat. Nanotechnol. 2011, 6, 733−739. (18) Lee, J.-S.; Kovalenko, M. V.; Huang, J.; Chung, D. S.; Talapin, D. V. Band-like Transport, High Electron Mobility and High Photoconductivity in All-Inorganic Nanocrystal Arrays. Nat. Nanotechnol. 2011, 6, 348−352. (19) Guyot-Sionnest, P. Electrical Transport in Colloidal Quantum Dot Films. J. Phys. Chem. Lett. 2012, 3, 1169−1175. (20) Anderson, P. W. Absence of Diffusion in Certain Random Lattices. Phys. Rev. 1958, 109, 1492−1505. (21) Lagendijk, A.; van Tiggelen, B.; Wiersma, D. S. Fifty Years of Anderson Localization. Phys. Today 2009, 62, 24. (22) Abrahams, E.; Anderson, P. W.; Licciardello, D. C.; Ramakrishnan, T. V. Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions. Phys. Rev. Lett. 1979, 42, 673−676. (23) Wegner, F. J. Electrons in Disordered Systems. Scaling near the Mobility Edge. Zeitschrift für Phys. B Condens. Matter Quanta 1976, 25, 327−337. (24) Edwards, J. T.; Thouless, D. J. Numerical Studies of Localization in Disordered Systems. J. Phys. C Solid State Phys. 1972, 5, 807−820. (25) Bulka, B. R.; Kramer, B.; MacKinnon, A. Mobility Edge in the Three Dimensional Anderson Model. Zeitschrift für Phys. B Condens. Matter 1985, 60, 13−17. (26) Bulka, B.; Schreiber, M.; Kramer, B. Localization, Quantum Interference, and the Metal-Insulator Transition. Zeitschrift für Phys. B Condens. Matter 1987, 66, 21−30. (27) Sepehrinia, R.; Sheikhan, A. Numerical Simulation of Anderson Localization. Comput. Sci. Eng. 2011, 13, 74−83. (28) Markoš, P. Numerical Analysis of the Anderson Localization. Acta Phys. Slovaca. Rev. Tutorials 2006, 56, 561−685. (29) Wiersma, D.; Bartolini, P.; Lagendijk, A.; Righini, R. Localization of Light in a Disordered Medium. Nature 1997, 390, 671−673. I


Article

The Journal of Physical Chemistry C (52) Yang, J.; Ling, T.; Wu, W.-T.; Liu, H.; Gao, M.-R.; Ling, C.; Li, L.; Du, X.-W. A Top-down Strategy towards Monodisperse Colloidal Lead Sulphide Quantum Dots. Nat. Commun. 2013, 4, 1695. (53) Overgaag, K.; Vanmaekelbergh, D.; Liljeroth, P.; Mahieu, G.; Grandidier, B.; Delerue, C.; Allan, G. Electron−Phonon Coupling and Intervalley Splitting Determine the Linewidth of Single-Electron Transport through PbSe Nanocrystals. J. Chem. Phys. 2009, 131, 224510. (54) Harbold, J. M.; Wise, F. W. Photoluminescence Spectroscopy of PbSe Nanocrystals. Phys. Rev. B 2007, 76, 125304. (55) Kigel, A.; Brumer, M.; Maikov, G. I.; Sashchiuk, A.; Lifshitz, E. Thermally Activated Photoluminescence in Lead Selenide Colloidal Quantum Dots. Small 2009, 5, 1675−1681. (56) Masia, F.; Langbein, W.; Moreels, I.; Hens, Z.; Borri, P. Exciton Dephasing in Lead Sulfide Quantum Dots by X-Point Phonons. Phys. Rev. B 2011, 83, 3−6. (57) Allan, G.; Delerue, C. Confinement Effects in PbSe Quantum Wells and Nanocrystals. Phys. Rev. B 2004, 70, 1−9. (58) An, J. M.; Franceschetti, A.; Zunger, A. The Excitonic Exchange Splitting and Radiative Lifetime in PbSe Quantum Dots. Nano Lett. 2007, 7, 2129−2135. (59) Evers, W. H.; Goris, B.; Bals, S.; Casavola, M.; de Graaf, J.; van Roij, R.; Dijkstra, M.; Vanmaekelbergh, D. Low-Dimensional Semiconductor Superlattices Formed by Geometric Control over Nanocrystal Attachment. Nano Lett. 2013, 13, 2317−2323. (60) Baumgardner, W. J.; Whitham, K.; Hanrath, T. Confined-butConnected Quantum Solids via Controlled Ligand Displacement. Nano Lett. 2013, 13, 3225−3231. (61) Boneschanscher, M. P.; Evers, W. H.; Geuchies, J. J.; Altantzis, T.; Goris, B.; Rabouw, F. T.; van Rossum, S. A. P.; van der Zant, H. S. J.; Siebbeles, L. D. a.; Van Tendeloo, G.; et al. Long-Range Orientation and Atomic Attachment of Nanocrystals in 2D Honeycomb Superlattices. Science 2014, 344, 1377−1380.

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