Visualizing Current Flow at the Mesoscale in Disordered Assemblies

Jul 12, 2017 - The SCS model predicts that a single network within the nanocrystal assembly, composed of sites connected by small resistances, dominat...
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Visualizing Current Flow at the Mesoscale in Disordered Assemblies of Touching Semiconductor Nanocrystals Qinyi Chen,† Jeffrey R. Guest,§ and Elijah Thimsen*,†,‡ †

Institute of Materials Science and Engineering and ‡Department of Energy, Environmental and Chemical Engineering, Washington University in Saint Louis, St. Louis, Missouri 63130, United States § Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, United States S Supporting Information *

ABSTRACT: The transport of electrons through assemblies of nanocrystals is important to performance in optoelectronic applications for these materials. Previous work has primarily focused on single nanocrystals or transitions between pairs of nanocrystals. There is a gap in knowledge of how large numbers of nanocrystals in an assembly behave collectively and how this collective behavior manifests at the mesoscale. In this work, the variable range hopping (VRH) transport of electrons in disordered assemblies of touching, heavily doped ZnO nanocrystals was visualized at the mesoscale as a function of temperature both theoretically, using the model of Skinner, Chen, and Shklovskii (SCS), and experimentally, with conductive atomic force microscopy on ultrathin films only a few particle layers thick. Agreement was obtained between the model and experiments, with a few notable exceptions. The SCS model predicts that a single network within the nanocrystal assembly, composed of sites connected by small resistances, dominates conduction, namely, the optimum band from variable range hopping theory. However, our experiments revealed that in addition to the optimum band there are subnetworks that appear as additional peaks in the resistance histogram of conductive atomic force microscopy (CAFM) maps. Furthermore, the connections of these subnetworks to the optimum band change in time, such that some subnetworks become connected to the optimum band while others become disconnected and isolated from the optimum band; this observation appears to be an experimental manifestation of the “blinking” phenomenon in our images of mesoscale transport.

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INTRODUCTION Thin films composed of semiconductor nanocrystals are being actively explored for optoelectronic applications such as photovoltaic solar cells,1−9 light-emitting diodes,10−12 transparent conductive electrodes, 13−17 electrochromic windows,18−20 and field effect transistors.21,22 Majority carrier transport (e.g., electrons in an n-type semiconductor) is important for performance, and recent studies have endeavored to unravel these dynamics.23−26 For example, as nanocrystals are brought closer together, the longitudinal mobility through the assembly increases,27 and thus, recent work has focused on semiconductor nanocrystals that are separated by very short ligands or abutted against one another.14,28−31 For example, we have recently demonstrated experimentally that the transport mechanism in a film composed of touching nanocrystals can be successfully controlled to be either Efros−Shklovskii variable range hopping (ES-VRH) or diffusive transport by adjusting the interparticle contact radius,32 which is consistent with a recently proposed theory.30 In general, research has focused on studying the properties of single nanocrystals,33 or transitions between pairs of nanocrystals,30,34 and then this information has been used to draw conclusions about macroscopic properties of an assembly such as conductivity. Such an analysis involves © 2017 American Chemical Society

connecting widely disparate length scales on the order of 10−6 cm for single nanocrystals to the much larger scale of experimental thin film samples on the order of 1 cm. Further complicating the situation, experimental thin film samples (10−4 × 1 × 1 cm) are expected to contain as many as 1014 nanocrystals. Distributions are unavoidable in parameters such as size, composition, energy levels, and carrier density. The multiscale nature of the system and the observation that the ensemble is made up of an extremely large population composed of diverse individuals raise a question: what is the behavior at intermediate length scales? While a few experimental studies have been conducted,35,36 there remains a gap in knowledge of the collective behavior of nanocrystals at the mesoscale in general and for majority carrier transport specifically. Mesoscale is defined here as length scale from 100 nm to 10 μm. Two known phenomena that are expected to produce interesting collective behavior at the mesoscale are (i) long-range tunneling involved in a variable range hopping (VRH) transport mechanism and (ii) current Received: May 22, 2017 Revised: June 30, 2017 Published: July 12, 2017 15619

DOI: 10.1021/acs.jpcc.7b04949 J. Phys. Chem. C 2017, 121, 15619−15629

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The Journal of Physical Chemistry C

Shklovskii (SCS),39 with a few key modifications (vide inf ra). The resistor network calculated using the SCS model was used to construct simulated conductive atomic force microscope (CAFM) current maps as a function of temperature in the range from 38 to 296 K. These simulated CAFM current maps were then compared to experimentally measured CAFM current maps measured in the same temperature range. Qualitative agreement was found between the model and experiments with a few notable differences. In addition to the optimum band predicted by the theoretical calculations, in the experiments we also observed subnetworks that appeared as additional peaks in the resistance histogram. Furthermore, we observed that large portions of the current map displayed dynamic behavior. Some portions of the current map became more conductive, abruptly appearing to join the optimum band, while other portions of the current map appeared to abruptly disconnect from the network. The results of these experiments indicate that the constituency of the optimum band is dynamic.

blinking. These two phenomena will be discussed in sequence. If the contact resistance between nanocrystals is greater than the quantum resistance, then transport in an assembly of nanocrystals is via an ES-VRH mechanism.30,32,34 In an assembly that exhibits an ES-VRH transport mechanism, electrons do not necessarily hop to adjacent nanocrystals since every nanocrystal is believed to be connected to every other nanocrystal, as described by the Miller−Abrahams network.37,38 The nondimensional equation for the resistance between nanocrystal i and j, Rij, can be written as39 ⎛ 2rij εij ⎞ + R ij = R 0 exp⎜ ⎟ kBT ⎠ ⎝ ξ

(1)

where R0 is a prefactor that is often assumed to be a constant with respect to changes in temperature,38,39 rij is the separation between nanocrystals i and j, ξ is the electron localization length, which is a sample-dependent parameter that is typically on the order of 31−64 nm in our materials (when the transport mechanism is ES-VRH),32 and εij is an activation energy. The energy εij depends on the type of VRH. For ES-VRH, Coulombic interactions are considered, and for energy levels on opposite sides of the Fermi level, εij = |εi − εj| − e2/κrij, where κ is the effective dielectric constant of the medium.39 The probability of finding a pair of nanocrystals that produces a small εij increases with increasing rij. Thus, statistically, there is a separation between nanocrystals for which the pairwise resistance (eq 1) is minimized. Interestingly, this distance can be relatively large; in some cases, this distance can be up to 100 nm apart at low temperature.32 Long-range hopping may give rise to interesting features in a visualization of current flow at the mesoscale for quasi-two-dimensional samples, for example, current hotspots that are spatially disconnected while being electronically connected according to eq 1. Another recently observed phenomenon that may give rise to interesting collective behavior is current blinking.40−42 The basic idea is that as time proceeds, individual nanocrystals randomly switch between an “on” state and an “off” state. For a given applied potential, in the on state the current flow is relatively large, and in the off state the current flow is relatively small. The governing behavior is not well understood but is believed to be related to the ubiquitous photoluminescence blinking that has been reported by many authors.43 It is interesting to think about current blinking in the context of ESVRH. If several nearby nanocrystals in the optimum band all switched off, it may result in large portions of the optimum band becoming partially or fully disconnected from the conduction backbone, or vice versa. Such behavior should be readily observed experimentally by scanning probe measurements. In this work, we report the results of a study focused on visualizing current flow at the mesoscale in assemblies of heavily doped touching ZnO nanocrystals that exhibited an ESVRH transport mechanism. The current flow was visualized using both computations and experiments. The assemblies were ultrathin films composed of nanocrystals deposited on insulating substrates. For the samples studied here, the areal density of nanocrystals on the substrate was approximately 5 × 1012 particles/cm2, such that a square area on the substrate with a side equal to the particle diameter contained three particles on average. In other words, the ultrathin film was only a few particle layers thick on average. The computational model was constructed using the theory outlined by Skinner, Chen, and

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METHODS Computational Model. The model consists of four subroutines. The first subroutine deposits the film composed of nanocrystals by a simulated ballistic impaction process using procedures that have been previously reported.44,45 This model deposition method was chosen since it is a fair representation of the physical process by which the experimental films were deposited.14,46,47 The (x, y) position of the particles was randomly initialized on the 150 nm × 150 nm domain at a height far away from the surface of the substrate. A scan was carried out to determine if any particles were already deposited in the projected area on the substrate beneath the new particle. If there were no particles deposited on the substrate in the projected area beneath the new particle, then the center of the new particle was placed on the substrate at the initialized (x, y) position, at a height of one particle radius. If there were particles deposited on the substrate in the projected area of the new particle, then the one at the greatest height was identified. The new particle was then deposited on top of the particle of greatest height at the initialized (x, y) position and a height determined by a vector connecting the two particle centers that was one diameter in length. This procedure was carried out iteratively until 1000 nanocrystals had been deposited. After the film had been deposited, the spacing between all particles in the film was checked to ensure that there were no overlaps. The number of donors in these particles was then initialized to be a random integer variable described by the Poisson distribution with a mean of 8. The donor distribution in the ensemble is plotted in Figure S1a. The donors are considered to be a fixed positive charge. The number of electrons in the particles was then initialized to be equal to the number of donors. A ground state charge distribution in the film was determined using a method similar to that described by SCS. In this procedure, the number of donors in the particles remains constant, but the electrons are redistributed to minimize the total energy of the system. In Gaussian units, the Hamiltonian of the system can be written as39 all particles

H=

∑ i=1

+

15620

⎡ e 2(N − n )2 i i ⎢ + ⎢⎣ 2κrp

ni

⎤

k=0

⎥⎦

∑ E Q (k )⎥

⎡ e 2(N − n )(N − n ) ⎤ i i j j ⎢ ⎥ ⎢ ⎥⎦ κ r ij ⟨i , j⟩, i ≠ j ⎣

∑

(2)

DOI: 10.1021/acs.jpcc.7b04949 J. Phys. Chem. C 2017, 121, 15619−15629

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The Journal of Physical Chemistry C The first term in eq 2 describes the electrostatic self-energy, the second term is energy associated with filling up the quantum confined energy levels with electrons, and the third term describes the electrostatic energy associated with the coulomb interaction between nanocrystals i and j. The number of donors in particle i is Ni, the number of electrons is ni, e is the unit charge, and κ is a dielectric constant that is approximately 8.8 for our ZnO−Al2O3 nanocomposites,32 rp is the nanocrystal radius, and rij is the separation between nanocrystals i and j. In the calculation of the function EQ, we assumed that the energy level spacing between the quantum confined levels was constant and approximately equal to Δ = 5ℏ2/merp2,32,48 where me is the effective mass of an electron in ZnO (∼0.3m0).49 The degeneracies of the 1Se, 1Pe, 1De, 2Se, and 1Fe levels are 2, 6, 10, 2, and 14, respectively. Thus, EQ(n = 0) = 0, EQ(n = 1, 2) ≈ Δ, EQ(3 ≤ n ≤ 8) ≈ 2Δ, EQ(9 ≤ n ≤ 18) ≈ 3Δ, EQ(n = 19, 20) ≈ 4Δ, and EQ(21 ≤ n ≤ 34) ≈ 5Δ. Equation 2 was minimized by randomly moving an electron from one nanocrystal to another nanocrystal. If the move decreased H, it was accepted; and if the move increased H, it was rejected and the electron was moved back to its initial location. The procedure was carried out iteratively until the two ES ground state criteria were satisfied for every particle pair i and j in the ensemble:32 εj(e) − εi(f) −

e2 >0 κrij

εi(e) − εj(f) −

e2 >0 κrij

in Origin 2015 and smoothed by increasing the number of points by a factor of 100 and using a smoothing parameter of 0.001. Experiment. Samples were synthesized using a previously reported procedure using the same apparatuses.32 Prior to deposition, substrates were clean using a two-step sonication in first acetone and then isopropanol. The substrates were made of silicon with approximately 800 nm of thermal SiO2 (MTI Corp.). The process parameters were as follows. The flow rates of the dilution Ar and O2 were 300 standard cubic centimeters per minute (sccm) and 30 sccm, respectively. The flow rate of Ar through the diethylzinc (DEZ) bubbler was 30 sccm. The bubbler was maintained at ambient temperature and a pressure of 94 Torr using a metering valve. The plasma was generated by applying 33 W of RF power through an impedance matching network at a frequency of 13.56 MHz. The pressure in the nanocrystal synthesis stage was 12 Torr. The ZnO nanocrystal aerosol was expanded to supersonic velocity through a slit nozzle that had a cross section of 0.8 × 20 mm and was 67 mm long in the direction of flow. The measured pressure in the impaction stage was 0.55 Torr. The substrates were passed back and forth through the particle beam six times to deposit samples for CAFM characterization. Subsequent to deposition in the plasma reactor, the nanocrystals were coated with ZnO and Al2O3 by ALD. The ALD reactor was maintained at 180 °C and a pressure of 0.6 Torr when no precursors were being pulsed. The carrier gas was argon. The VRH sample was coated with 1 cycle of ZnO, while the diffusive sample was coated with 10 cycles of ZnO. The ZnO was deposited using a DEZ pulse time of 1 s and a purge time of 60 s, which was followed by an H2O pulse time of 1 s and a purge time of 60 s. Following ZnO deposition, both samples were coated with the same amount of Al2O3; in fact, they were coated in the same run. Specifically, the samples were coated with 12 cycles of Al2O3 by pulsing trimethylaluminum for 1 s, followed by a 60 s purge, which was then followed by 1 s H2O pulse and a 60 s purge. The growth rate of Al2O3 under these conditions was 0.11 nm per cycle. The samples were characterized using a variety of techniques to determine the structure. Transmission electron microscopy was performed using a JEOL JEM-2100 field emission TEM operating at an accelerating voltage of 200 kV to determine the nanocrystal size. The specimen supports were ultrathin carbon films approximately 3 nm in thickness supported on a lacey carbon network, which was supported on a copper grid. Scanning electron microscopy was performed using a JEOL JSM-7001 LVF field emission SEM operating at an accelerating voltage of 15 kV. X-ray florescence (XRF) was used to determine the mass-effective thickness of the ultrathin nanocrystal film. This was straightforward since all films studied here were in the thin-film limit where the mass-effective thickness was much less than the absorption depth at the photon energy of the analytical line. The mass-effective thickness is the thickness of the film if it were present as a compact, nonporous layer. XRF was performed using a Spectro Midex operating at a tube voltage of 50 kV. The Zn Kα line was used for analysis. The calibration factor, which relates the analytical line peak area to the mass-effective thickness, was determined using compact ZnO films deposited by ALD. The thicknesses of the compact films used for calibration were measured by spectroscopic ellipsometry using a J.A. Woolam α-SE. The mass-effective thickness of the nanocrystal samples used in this study was 12 nm before ALD coating.

(3)

ε(e) i

where is the lowest empty electron energy level of nanocrystal i and ε(f) i is the highest filled electron energy level. These energy levels can be calculated by the following equations:39 εi(f) = EQ (ni) + −

∑ j≠i

e 2[(Ni − ni)2 − (Ni − ni + 1)2 ] 2κrp

e(Nj − nj)2 2κrij

εi(e) = EQ (ni + 1) + −

∑ j≠i

(4)

e 2[(Ni − ni − 1)2 − (Ni − ni)2 ] 2κrp

e(Nj − nj)2 2κrij

(5)

The ground state charge distribution of the ensemble for the calculations presented in this paper is presented in Figure S1b. From the ground state of the system, the resistor network can be calculated. The localization length was assumed to be 64 nm, as we have previously measured.32 We calculated the resistor network using an identical procedure to SCS,39 and thus it will not be repeated here. To generate the simulated CAFM current maps, a given particle was biased with a unit potential, and all nanocrystals within one diameter of the edge of the sample were grounded. The resulting current flow was calculated by formulating the system as a set of linear equations in matrix notation and then solved using Matlab. A three-dimensional data set was produced in this way, which consisted of a series of (x, y, current) triplets. These triplets were plotted as color maps 15621

DOI: 10.1021/acs.jpcc.7b04949 J. Phys. Chem. C 2017, 121, 15619−15629

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Figure 1. Computational model. (a) Three-dimensional perspective of the assembly of nanocrystals used for model calculations. (b) Density of states in the assembly of touching ZnO nanocrystals. A soft Coulomb gap is clearly observed at the Fermi level, as expected for a material that exhibits ES-VRH. (c) Network plot showing all of the connections that have a normalized resistance value of Rij/R0 < 80 at T = 38 K. The smallest normalized resistance in the network at 38 K was approximately Min⟨Rij/R0⟩ ≈ 0.1. The resistors in (c) are color coded according to their value.

nm in diameter; a total number of 1000 were deposited into a square area that was 150 × 150 nm. The nanocrystal diameter of 5.6 nm was chosen because it is the largest size at which ZnO can reasonably be considered to be quantum confined,50 which is an assumption of the SCS model.39 The ZnO nanocrystals were assumed to be n-type with an average donor concentration of n̅d = 9 × 1019 cm−3, in accordance with our previous experiments.32 The number of donors in each particle was initialized to an integer value according to the Poisson distribution with a mean of nd̅ νp = 8.3, where vp is the nanocrystal volume. The assumption of monodispersed size distribution and Poisson-distributed donor concentration implies that Coulomb disorder is more important than size disorder in determining the energy levels of the system; this assumption has been justified elsewhere.48 The dispersion in donor number, and the resulting dispersion in occupancy of electron energy levels, is the source of disorder in the system that controls the charge distribution. In the second subroutine, a ground state of the system was identified. The Hamiltonian was minimized by keeping the donor number constant in each particle, but allowing the electrons to redistribute (see Methods section). Some particles became charged by this process, and some remained neutral, depending on the number of donors and the location in the film. The ground state of the system resulting from the Hamiltonian minimization is a set of locations, number of donors, and number of electrons for each nanocrystal in the film. Using the ground state, the energy levels of the nanocrystals were calculated. The resulting density of states, which is simply a histogram of the energy levels, is plotted in Figure 1b. The density of states clearly exhibits a Coulomb gap at the Fermi level, as expected.39 The third subroutine uses the energy levels of the particles to calculate the resistance connecting each particle i and j using eq 1. To illustrate the highly parallel nature of the resulting resistor network, in Figure 1c is plotted all connections that had a resistance value of Rij/R0 < 80 at a temperature of 38 K (note the minimum value in the set of Rij/R0 was approximately ≈0.1 at 38 K). The resistances are color coded according to their value. Many particles were connected to other particles that

CAFM measurements were performed in the Center for Nanoscale Materials at Argonne National Laboratory using the variable temperature ultrahigh vacuum (VT-UHV) AFM system (Omicron, controlled by Nanonis electronics) operated in contact mode. The AFM cantilevers were fabricated by BudgetSensors. The cantilevers were silicon, coated with approximately 5 nm of Cr and 25 nm of Pt to impart electrical conductivity. SEM images of the tips can be found in Figure S2. The measured cantilever resonance frequency was 62.4 kHz, and the spring constant reported by the manufacturer was 3 N/ m. The cantilevers were mounted using EPOTEK H21D conductive epoxy. The samples and cantilevers were degassed at a pressure of 1.3 × 10−10 mbar before insertion into the analysis chamber. The pressure in the analysis chamber was approximately 2 × 10−12 mbar during measurement. The sample temperature was controlled using a liquid He cryostat. Note that in the VT-UHV system the sample is actively cooled; however, the AFM tip is not. The system was allowed to stabilize for at least 20 min after reaching the temperature set point before starting measurements. Approach−retract curves were used to determine the applied force, which was kept constant at a value of approximately 0.2 μN. Images were acquired at a scan speed of 1 μm/s. During map acquisition, the applied potential to the tip was 30 mV with respect to ground. Grid spectroscopy was performed on a 200 × 200 nm square that had 32 × 32 points. At each point, 120 current−voltage pairs were acquired at a sweep rate of approximately 300 mV/s. The voltage was scanned in both the forward and reverse directions to check for hysteresis, but only the forward scan was recorded. The data were processed using Gwyddion, Matlab, and Origin.

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RESULTS AND DISCUSSION Computations were carried out by first depositing the nanocrystals, then minimizing the energy of the system by redistributing electrons while keeping donor number constant, and finally calculating the simulated CAFM map. The film used for modeling was porous and disordered (Figure 1a). The particles were assumed to be spherical, monodispersed, and 5.6 15622

DOI: 10.1021/acs.jpcc.7b04949 J. Phys. Chem. C 2017, 121, 15619−15629

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The Journal of Physical Chemistry C were distant but had small values of εij. A number of particles illustrated in Figure 1c had low resistance connections to many more particles than they physically touched. Given the fully connected resistor network provided by the third subroutine, finally the fourth subroutine calculated a simulated CAFM current map. All nanocrystals within one diameter from the edge of the square sample were grounded (i.e., voltage = 0). The map was generated by biasing each of the particles in the interior with a unit potential (i.e., voltage = 1) and then calculating the resulting current flow to the grounded edges through the fully connected resistor network (where we take R0 to have a value of 1). All of the pairwise resistances (eq 1) were considered in the calculation of the current flow. The resistor network with external constraints was formulated as a system of linear equations using Kirchoff’s law and solved in matrix form using Gaussian elimination. The CAFM map is a plot of the current that results from biasing each of the uppermost nanocrystals as a function of each particle’s (x, y) position (i.e., the nanocrystals that would be probed by a CAFM experiment). The results of the computational model are plotted in Figure 2. Plotted in Figure 2a is the topography of the model film composed of nanocrystals, and Figure 2b shows a schematic of a single nanocrystal being biased as an example of how one point on the simulated CAFM current map was generated. Topography over a larger area is presented in Figure S3a. While the average film thickness was approximately 3 particles, some regions contained no particles and other regions contained many. This inhomogeneity, in addition to the porosity of the film, results in thickness variations on the order of 10 nanocrystal diameters in some locations (Figure S3a). The simulated CAFM maps, generated by sequentially biasing all nanocrystals on the surface of the film and calculating the resulting current flow to the grounded edges for each, are plotted as a function of temperature in Figures 2c1 to 2e1. The histogram of resistance values for the simulated CAFM maps is plotted in Figures 2c2 to 2e2. Each resistance value was determined by dividing the unit bias applied to a given nanocrystal by the current that flowed through the resistor network to the sample edges. The number of values used to generate the histogram was the number of nanocrystals in the film minus the number of particles within one diameter of the sample edge (861 for the calculations presented in this work). There are several notable features in the computational results. At low temperature, e.g. 38 K, there appears in the CAFM current map current hotspots that are separated from one another by low current regions (Figures 2c1 and 2d1). In other words, there were nanocrystals in the interior of the sample that were connected to the edges by long-range tunneling contacts due to the ES-VRH transport mechanism. One might argue against the idea of long-range tunneling by the possibility that current could flow through the layers of nanocrystals beneath the topmost surface that is plotted in Figure 2c1. However, the spatially separated current hotspots were also observed if the simulated CAFM map was generated using all of the nanocrystals in the film (Figure S4). As temperature increased, the conducting hotspots become more numerous and closer together until eventually at room temperature there was nearly a continuous path of conducting nanocrystals to the edge of the substrate (Figure 2e1). There are some difficulties in comparing the simulated CAFM maps to experiments. The distance between the current hotspots, which was approximately 20 nm at 38 K and decreased with

Figure 2. Computational CAFM simulations (a, c1−e1) and resistance histograms (c2−e2). (a) Topography and (b) schematic of the method by which the simulated CAFM current maps were generated. Simulated CAFM current maps for different temperatures: (c1) 38 K, (d1) 50 K, (e1) 100 K, (d1) 200 K, and (e1) 296 K. The range of the color maps was chosen such that red was the maximum value and blue was the minimum value. Resistance histograms for the CAFM current maps: (c2) 38 K, (d2) 50 K, (e2) 100 K, (d2) 200 K, and (e2) 296 K.

increasing temperature, was less than the experimental CAFM tip diameter (Figure S2). It would be difficult, if not impossible, to clearly reproduce the features in the simulated maps of Figure 2 experimentally by CAFM measurements. Thus, a different method must be used for presenting the data to make it comparable to the experimental measurements. If one assumes an ohmic contact with negligible contact resistance, which is obviously valid for the idealized model calculations, then the simulated CAFM current map can be simply treated as a random sample of conductance values between the AFM tip and the sample edges. The reciprocals of these conductance values are resistances. There is a single peak in the resistance histogram at the small resistance end of the distribution (Figures 2c2 to 2e2) for all temperatures 15623

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The Journal of Physical Chemistry C investigated. The peak is composed of particles that are connected by low resistance contacts and is tentatively assigned as the conduction band, since it occurs at low resistance values and is expected to dominate conduction in the network. The remainder of the resistance distribution at higher values does not display any distinguishing features and is nominally flat. These higher resistance values correspond to particles that are not a part of the optimum band. As temperature increased, the peak position of the conduction band shifted to lower resistance values and the entire distribution become narrower. The shift and narrowing of the resistance distribution with increasing temperature are consistent with eq 1. Given these computational results, in the experimental data we expect to observe a single peak in the resistance histogram at the small resistance end of the distribution, and that peak is expected to shift to smaller resistance values and become narrower as temperature increases. Experimental samples were synthesized using a procedure similar to the one previously reported32 and then characterized by CAFM as a function of temperature. The sample synthesis consisted of three steps: (1) nanocrystal deposition, (2) atomic layer deposition (ALD) of ZnO to control the contact radius and therefore transport mechanism,32 and (3) capping with Al2O3 also by ALD to impart conductivity.13 In brief, ZnO nanocrystals were synthesized in a low-temperature argon plasma using O2 and diethylzinc as chemical precursors. The ZnO nanocrystals were nominally free of organic ligands on their surfaces at all times during the synthesis. These nanocrystals were accelerated through a nozzle to supersonic velocity to form a particle beam. Substrates were passed back and forth under the particle beam to deposit the nanocrystals. One pass consisted of moving the substrate under the beam and then pulling it back again under the beam using a pushrod in a reciprocating motion. Presented in Figure 3a are transmission electron microscope (TEM) images of particles deposited using 1 pass. The particles had a mean diameter of 7.3 nm and a standard deviation of 1.5 nm and were nominally single crystal. The measured size distribution agreed with previous characterization at the same experimental conditions.32 Using six passes, thin films composed of ZnO nanocrystals were deposited onto a substrate that consisted of an 800 nm thermal SiO2 film on Si (Figures 3b and 3d). The number of passes was chosen to obtain the thinnest film composed of nanocrystals that had a finite sheet resistance, as measured by a source meter with an input resistance of >10 GΩ. The mass effective thickness of the resulting film composed of ZnO nanocrystals was determined to be 12 nm by X-ray fluorescence spectroscopy, which corresponds to an average of approximately 2.9 nanocrystals between the substrate surface and the top of the filmsimilar to the model calculations. A plan view SEM image of the sample used for CAFM measurements is presented in Figure 3b. After the ZnO nanocrystals were deposited onto the SiO2/Si substrate, they were coated with a small amount of material by ALD. Specifically, one type of sample was coated with a sufficient number of ZnO ALD cycles to impart a diffusive transport mechanism to the network by increasing the contact radius, as described in a previous publication.32 The other type of sample had a sufficiently small contact radius that the transport mechanism was ES-VRH.32 Both the diffusive and VRH samples were then coated with the same amount of Al2O3 (12 cycles, approximately 1.3 nm) by ALD to render them conductive and protect from air exposure between synthesis

Figure 3. Experimental setup. (a) Transmission electron micrographs of ZnO nanocrystals deposited using 1 pass. (b) Scanning electron micrograph of the VRH sample that was deposited using 6 passes. (c) Digital photograph of the experimental sample mounted to the CAFM sample holder using conductive epoxy. (d) Schematic of the CAFM experiment. Current flowed from the tip, laterally through the network composed of nanocrystals, to the ground electrodes at the edges. Examples of a topographical map and a current map squired at 296 K are presented in (e) and (f), respectively. The z range for the color maps goes from the minimum value to the maximum value in the region of interest.

and measurement.13,14 The Al2O3 coating was thin enough that the resulting contact resistance in the CAFM experiment was small with respect to the resistance of the nanocrystal network (vide inf ra). These ultrathin films were then imaged using CAFM in contact mode by placing ground electrodes at the edges of the samples and biasing the AFM tip (32 nm tip diameter, Cr/Pt coated silicon probe, Figure S3). Current and topographic maps were recorded simultaneously using the variable temperature ultrahigh vacuum (VT-UHV) AFM system in the Center for Nanoscale Materials at Argonne National Laboratory. The current flowed laterally from the AFM tip through the nanocrystal network to the ground electrodes at the edges (Figures 3c and 3d). Examples of topographical and current maps obtained at 296 K are provided in Figures 3e and 3f. The variations in the experimental topography were consistent with the simulated topography using the same nanocrystal diameter and domain size (Figure S3). To generate a resistance histogram from experimental CAFM maps, several assumptions must be made. Our interest here is the VRH sample, which ought to be comparable to the model calculations. First, it must be assumed that the entire circuit, including the contacts between the conductive AFM tip and 15624

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Figure 4. IV curves measured by grid spectroscopy for a VRH sample at different temperatures: (a) 38 K, (b) 100 K, (c) 200 K, and (d) 296 K. The plots are 2D histograms of IV curves measured from a grid of approximately 1024 spatial locations at each temperature. In this experiment, the tip was biased at the indicated potential and the edges of the sample were grounded. A schematic of the configuration is presented in Figure 3d. At each temperature, the curves were nominally linear, which indicates ohmic behavior.

transport mechanism, and the nanocrystal−ground contact resistance, all connected in series. Therefore, the upper limit of the contact resistance between the tip and the nanocrystals, which includes the thin Al2O3 coating, can be estimated to be 4.3 kΩ. This resistance value is several orders of magnitude smaller than the smallest resistance measured in the CAFM current maps for the VRH sample (Figures 4 and 5). Therefore, the assumptions are accepted, and resistance values can be calculated as the applied voltage to the CAFM tip divided by the measured current flow for each position in the map. The data were corrected for the slight temperature-dependent voltage offset of the zero current point (Figure 4a−d and Figure S5c,d). Qualitatively, the experimental CAFM maps showed a similar trend to the computational model. The measured CAFM maps were acquired from different locations (a notable difference compared to the theoretical calculations) of the same sample as a function of temperaturethe results are plotted in Figures 5a2 to 5e2 for the VRH case. At low temperature, it appeared as though there are small regions of high current in the middle of larger regions of lower current. As the temperature increased, the current hotspots grew, until they become quite large at room temperature. However, such observations are somewhat subjective, and as mentioned previously, experimental observation of spatially separated current hotspots is frustrated by the fact that the tip diameter is similar to the spacing between the hotspots. Thus, as before, we turn to resistance histograms to aid in an objective comparison to the model predictions. The experimental resistance histograms plotted in Figures 5a3 to 5e3 qualitatively resemble the model predictions, with a striking exception. Similar to the model predictions, a large peak was observed at low resistance values. Similar to the model calculations, as temperature increased, the large peak at low resistance shifted to smaller values and became narrower (Figure S6). The striking difference is that in many of the

ZnO nanocrystals, as well as the contacts between the ZnO nanocrystals and the ground electrodes (Figure 3), is ohmic. To verify this assumption, we performed grid spectroscopy and measured current as a function of voltage (IV curves) for 1024 points on the VRH sample as a function temperature. Twodimensional histograms of these IV measurements for the VRH sample are plotted in Figure 4. It can be seen from Figure 4 that in the temperature range of interest the measured IV curves are linear. Second, it must be assumed that the contact resistance between the conductive AFM tip and the ZnO nanocrystal network is negligible with respect to the nanocrystal network itself. Recall that the ZnO nanocrystals were coated with approximately 1.3 nm of Al2O3, so this assumption is important to verify. The contact resistance between CAFM tip and ZnO nanocrystals was estimated using the diffusive sample. The diffusive sample was coated with a sufficient number of ZnO ALD cycles to increase the contact radius between the nanocrystals to impart a diffusive transport mechanism.32 Following the ALD step to increase the contact radius, the diffusive sample was coated with the same amount of Al2O3 as the VRH sample (approximately 1.3 nm). In the CAFM current map, the diffusive sample displayed binary type behavior (Figure S5a2 and S5b2). The conductive regions saturated the current preamplifier, and the insulating regions gave very little current. The two-dimensional IV histograms for the diffusive sample are presented in Figures S5c and S5d. At most applied voltages, the current flowing was sufficiently large to saturate the preamplifier. However, there was a narrow voltage range where the current transitioned from the negative saturated value to the positive saturated value. This voltage range, which was only a few millivolts wide, contained a sufficient number of points that the resistance could be estimated to be 4.3 kΩ. The resistance value of 4.3 kΩ contains the contact resistance between the AFM tip and nanocrystals, the lateral resistance of the film composed of ZnO nanocrystals with a diffusive 15625

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Figure 5. Experimental CAFM data for the same VRH sample as Figures 3 and 4. Topography measured at different temperatures: (a1) 38 K, (b1) 50 K, (c1) 100 K, (d1) 200 K, and (e1) 296 K. Current maps measured at an applied potential of 30 mV to the tip: (a2) 38 K, (b2) 50 K, (c2) 100 K, (d2) 200 K, and (e2) 296 K. Topography and current were measured simultaneously. For a given color map, the range was determined by the maximum and minimum values. Resistance histograms generated from the current maps by assuming ohmic behavior at different temperatures: (a3) 38 K, (b3) 50 K, (c3) 100 K, (d3) 200 K, and (e3) 296 K. The vertical lines in the histograms are the upper and lower limits of the measurement.

second scan; meanwhile, a region in dashed circle γ became disconnected from the optimum band. The observation can be explained by current blinking.40−42 For example, a connection between α or β and the optimum band changed from an off state to an on state; meanwhile, the connection between γ and the optimum band blinked from an on state to an off state. Further work is necessary to support such a hypothesis, but nevertheless, the evidence suggests that in thin films composed of nanocrystals that display an ES-VRH electron transport mechanism the population of particles comprising the optimum band is dynamic. These unexpected experimental observations could be incorporated into the SCS model. To our knowledge, the SCS model is the state-of-the-art numerical technique for simulating experiments such as the ones reported here, but nevertheless, there are a few notable inconsistencies between the assumptions of the model and the experiments. In the model, the edges of the image frame were assumed to be at a constant potential, while in experiments this is not the case. A more realistic study could be carried out using either a smaller experimental domain or a larger model domain. As it is impractical to have a model domain on the order of millimeters,

CAFM current maps, only some of which are shown here, there were large regions that had a uniformly higher resistance (i.e., lower current at the same applied voltage), for example, Figures 5b2 and 5d2. These regions appeared as additional peaks in the resistance histograms (Figures 5b3 and 5d3). Such peaks were not observed in the diffusive sample (Figures S5a3 and S5b3). It appeared as though the subnetworks were connected to the optimum band by a tenuous high resistance link, which produced the shifted peak in the resistance histogram compared to the optimum band. Subnetworks are not predicted by the calculations we have performed using the SCS model, and to our knowledge, this report is the first experimental observation of them. Interestingly, the connections of portions of the nanocrystal network to the optimum band appear to be dynamic. Plotted in Figure 6 are two consecutive CAFM scans of the same region of the VRH sample at a temperature of 38 K. Regions of interest are outlined by the dashed circles α, β, and γ. The topography in these regions did not appear to change between the first and second scan. However, the current maps did. Regions in dashed circles α and β became connected to the optimum band at some point in time between the first and 15626

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according to an experimentally observed blinking probability distribution40−42 and then calculate the CAFM map.

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CONCLUSIONS In this work, we have presented visualizations of current flow at the mesoscale in quasi-two-dimensional films of ZnO nanocrystals that exhibited an ES-VRH transport mechanism. A computational model was presented to predict CAFM maps. The model predicted a peak in the resistance histogram at low resistance values, which was attributed to the well-known optimum band. This peak shifted to smaller resistances and became narrower as temperature increased. A similar trend was observed in the experimental CAFM measurements, with a notable exception. In the experiments, additional peaks in the resistance histogram were observed. These peaks were assigned as subnetworks, which are hypothesized to be connected to the optimum band by tenuous links. Additionally, in the CAFM maps acquired at low temperature, different regions of the nanocrystal network appeared to both join and leave the optimum band dynamically. The result indicates that the subset of nanocrystals that comprise the optimum band is dynamic. Suggestions were made for future work to incorporate the results of the experiments into the model.

Figure 6. Dynamic optimum band illustrated by subsequent scans on the same location at a constant temperature of 38 K. Topography for the (a1) first scan and (b1) second scan. Topography and current were measured simultaneously. Current maps at an applied bias of 30 mV for the (a2) first scan and (b2) second scan. For a given color map, the range was determined by the maximum and minimum values. Several regions of interest are outlined by white dashed circles.

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the more realistic strategy is to shrink the experimental domain using nanofabrication techniques. Another important discrepancy between the model and the experiments is the particle size. In the SCS model, the large spacing between quantum confined energy levels is what allows electrons to overcome the charging energy and be redistributed in the assembly during energy minimization to the ground state charge distribution. In the model calculations, the ZnO nanocrystals were assumed to have the maximum diameter for which ZnO is clearly in the quantum confined regime, which is 5.6 nm. In the experiments, the nanocrystals were slightly larger and had an average diameter of 7.3 nm and thus were at the border of the quantum confined regime. Films composed of these 7.3 nm diameter particles are known to exhibit an ES-VRH transport mechanism if the contact radius between particles is sufficiently small,32 which suggests that there is a charge distribution in the film and the energy levels are distributed in a qualitatively similar manner to the SCS model. The key difference is the cause of the redistribution of electrons. In the model, the cause is the large spacing between quantum confined energy levels and the Poisson distribution of donors. In the experimental sample, the cause is uncertain, but the effect appears to be similar. We note that other authors have used a generic random coulomb potential applied to the particles, which allows for nananocrystals that are not quantum confined to display a transport mechanism similar to the SCS model.51 Furthermore, in the SCS model, the nanocrystals were assumed to be placed on an ideal lattice.39 In our materials, the particles have a large degree of spatial disorder as a result of the supersonic impact deposition process (Figures 1 and 3). While spatial disorder does not explicitly affect the calculation of the pairwise resistance (eq 1), it is a notable difference if one considers cotunneling as a possible process by which electrons move between sites.52 Finally, it would be useful to extend the SCS model to incorporate current blinking, although that would require a considerable increase in computational resources. The expectation is that current blinking would cause the appearance and disappearance of subnetworks. One could introduce a time step and set each nanocrystal to be in either an on or off state

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b04949. Figure S1: the model ground state charge distribution; Figure S2: SEM images of the type of AFM tip used in this work; Figure S3: a comparison of model topography to the experimentally measured topography; Figure S4: simulated CAFM maps at 38 K generated using the topmost particles as well as all of the particles in the ensemble showing that current hotspots are observed in both; Figure S5: CAFM characterization of a sample prepared at the same conditions as the one in the text, except that it was coated with a sufficient amount of ZnO by ALD to induce a diffusive electron transport mechanism; Figure S6: trends in the peak at the low end of the resistance distribution as a function of temperature for both model and experiments (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Phone 314-935-6103 (E.T.). ORCID

Jeffrey R. Guest: 0000-0002-9756-8801 Elijah Thimsen: 0000-0002-7619-0926 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Use of the Center for Nanoscale Materials, an Office of Science user facility, was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract DE-AC02-06CH11357. The authors acknowledge financial support from Washington University and the Institute of Materials Science and Engineering for the use of instruments and Dr. H. Li for assistance in performing electron microscopy. The authors also thank Prof. J. Catalano for use of an ambient 15627

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