Letter pubs.acs.org/NanoLett
Measuring Ligand-Dependent Transport in Nanopatterned PbS Colloidal Quantum Dot Arrays Using Charge Sensing Nirat Ray,*,† Neal E. Staley,† Darcy D. W. Grinolds,‡ Moungi G. Bawendi,‡ and Marc A. Kastner† †
Department of Physics and ‡Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States S Supporting Information *
ABSTRACT: Colloidal quantum dot arrays with long organic ligands have better packing order than those with short ligands but are highly resistive, making low-bias conductance measurements impossible with conventional two-probe techniques. We use an integrated charge sensor to study transport in weakly coupled arrays in the low-bias regime, and we nanopattern the arrays to minimize packing disorder. We present the temperature and field dependence of the resistance for nanopatterned oleic-acid and n-butylamine-capped PbS arrays, measuring resistances as high as 1018 Ω. We find that the conduction mechanism changes from nearest neighbor hopping in oleic-acid-capped PbS dots to Mott’s variable range hopping in n-butylamine capped PbS dots. Our results can be understood in terms of a change in the interdot coupling strength or a change in density of trap states and highlight the importance of the capping ligand on charge transport through colloidal quantum dot arrays. KEYWORDS: PbS nanocrystals, electrical transport, nanoscale charge sensor, nanopatterning
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level9,25 have been investigated in some detail, very little has been done to determine the influence of the ligand on the conduction mechanism.26 In particular, transport in weakly coupled arrays, expected to have minimum packing disorder as well as a lower density of sub band gap states, has not been investigated, and a technique to measure low conductivity is crucial to investigation of such transport measurements. In this Letter, we introduce a novel technique to study transport in weakly coupled PbS QD arrays, based on a timeresolved measurement of charge using a nanoscale charge sensor. This method has been previously used to study transport in highly resistive nanopatterned films of amorphous silicon27 and amorphous germanium.28 The method has also been shown to be insensitive to contact effects such as blocking contacts.28 We use a nanopatterning technique developed for colloidal dots29 to integrate a charge sensor with the array of dots and fabricate the necessary devices. Nanopatterning also helps to minimize cracks and clusters in the array resulting in more conducting arrays and can be used to make samples whose dimensions approach the short ordering length scales of the dots. We study charge transport in PbS QD arrays with two capping ligands: the native oleic acid (OA) ligand and the ligand-exchanged n-butylamine (NBA), and from the evolution of the resistance as a function of temperature and electric field, we find a different conduction mechanism for these two ligands.
olloidal quantum dots (QDs) have emerged as promising candidates for applications in a variety of photovoltaic and optoelectronic devices,1−4 owing to their tunable optical and electronic properties. It has recently been shown that, in complement to the tunability achieved by a modification of the dot size, the electronic properties of coupled QD arrays could also be tuned with the surface ligands.5 There has also been considerable interest in using dots as building blocks for artificial solids to gain insight into correlated electron physics,6 and interesting theoretical predictions exist for the observation of charge ordering and other novel aspects of charge transport in dot arrays.7−9 These effects are difficult to observe experimentally because of the finite size distribution of the dots and packing disorder in the array. Long-range ordering of dots in the array is stabilized by van der Waal’s interactions10−12 and leads to highly resistive arrays, due to the insulating nature of the organic ligands and correspondingly large tunnel barriers between dots. Techniques to decrease the spacing between dots, such as vacuum annealing or ligand exchange, typically lead to a larger packing disorder in the array,13,14 and the incompleteness of the ligand exchange process can further lead to the creation of sub-band gap states.15,16 Many different experimental configurations, such as the field effect transistor (FET) geometry,8,17−19 interdigitated electrodes,20,21 and doping through chemical22,23 or electrochemical gating methods24 have been used to study transport in a variety of colloidal dot assemblies. However, a relatively large variability exists in the conduction mechanisms reported in literature, and while the influence of dot size20,21 and doping © XXXX American Chemical Society
Received: February 17, 2015 Revised: May 23, 2015
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DOI: 10.1021/acs.nanolett.5b00659 Nano Lett. XXXX, XXX, XXX−XXX
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electrode connected to the nanopattern, we observe an instantaneous jump in the MOSFET conductance followed by a further decrease slowly with time (which we call the charge transient). The time dependence of the charging transient from the nanopattern has been shown to be accurately exponential,27 and the resistance of the nanopattern can be extracted from an exponential fit to this transient response. The voltage step and the transient response of the MOSFET are shown in Figure 1b. The solid lines are fits to GM = G∞ + G0 exp(−Γt); the conductance, GPbS, of the nanopattern is given by GPbS = wLCΓ/π2, where w and L are the width and length of the sample, respectively, and C is the capacitance per unit area of the quantum dot film. It is important to note here that the technique measures the differential conductance, g = dI/dV, rather than just the conductance, G, of the material. For ohmic regimes, near zero bias, they are identical; however, they may be significantly different for nonohmic regimes. To measure the differential conductance, GPbS, at finite source drain bias, in order to study the field dependence of the conductance, we apply a static field across the nanopattern, rapidly step the voltage on one of the electrodes and obtain the conductance from the observed charge transient. For example, to measure the differential conductance, gPbS, at an applied bias voltage of VPbS = −3.25 V, we apply a fixed voltage VPbS = −3 V to one electrode and rapidly step the voltage to VPbS = −3.5 V. Our device structure allows us to measure the current− voltage characteristics in addition to transients resulting from the spreading of charge. At high temperatures, we are able to measure the current and extract the differential conductance gPbS = dI/dV, directly. As we move to lower temperatures, our current measurements are limited by instrument noise from the amplifiers. At these temperatures, we extract the conductance from exponential fits to charge transients as discussed above. Figure 2a shows the measured conductance of the NBA capped PbS dots, GPbS−NBA at a bias of 0.5 V, from room temperature to 22 K, plotted as a function of inverse temperature. At this low bias, the differential conductance is expected to be the same as
Our measurements also reveal the highest resistances ever measured for colloidal QD arrays. We use an n-channel MOSFET as the charge sensor, where the MOS (metal−oxide−semiconductor) structure is obtained by growing 100 nm of silicon dioxide (SiO2) on top of a p-type silicon substrate and depositing a layer of metal (aluminum). The aluminum MOSFET gate tapers down to a narrow width of almost 80 nm so that the MOSFET is highly sensitive to its electrostatic environment. The structure includes two additional terminals (drain and source), each connected to individual highly doped n-type regions on the p-type substrate. Two gold electrodes spaced 1 μm apart are fabricated ∼100 nm away from the sensor for contacts to the nanopatterned dot arrays. We use PbS dots with a mean diameter of approximately 4.5 nm. The dots are synthesized air-free using a Schlenk line by high temperature pyrolysis of Pb and S precursors in an oleic acid/octadecene mixture.30,31 We process the growth solution to remove remaining products and, when desired, perform a solution exchange of the native oleic acid capping ligand for the smaller butylamine ligand. We create nanopatterns of PbS dots by dropcasting a dilute solution of dots into a 200-nm-wide trench created in the electron beam resist that is aligned with the patterned gold electrodes.29 Figures 1a and 1b show a
Figure 1. (a) Schematic of the device structure used for measurements. (b) False color scanning electron micrograph of the MOSFET gate (blue), adjacent to a nanopatterned array of PbS dots (red). The nanopattern is 200 nm wide and 1 μm long, and we make electrical contact to the nanopattern with Ti/Au electrodes (yellow). (b) Transient response of the MOSFET conductance, GM, as a function of time after the voltage VAu on one of the gold electrodes is stepped from 0 to −1 (red trace) at 50 K (blue markers) and 60 K (green markers). The solid lines are fits to GM = G∞ + G0 exp(−Γt).
Figure 2. (a) Ohmic conductance, G, as a function of inverse temperature for n-butylamine (NBA)-capped PbS dots near zero bias. (b) Differential conductance, g, as a function of inverse temperature for oleic acid (OA)-capped PbS dots. Empty markers represent conductance extracted from charge sensing measurements and filled markers represent values extracted from current measurements. The conductance using both methods has been found to agree within a factor of 4, thereby validating our method for obtaining the conductance from the charge transient (see Supporting Information, Figure S4). For OA-capped dots, we were unable to probe the resistance in the ohmic regime, and so the data presented actually represent the differential conductance (gPbS−OA) at a bias of 3.5 V. In the main figure of (a) the dashed line is a fit to the sum of two simply activated components; the inset in (a) shows the conductance and fit to Motts variable range hopping. In (b) the dashed line is a fit to a single activated component.
schematic and an electron micrograph of the final device structure. For all of the measurements reported here, a positive voltage, Vg, is applied to the gate so that the MOSFET is in inversion, and the nanopattern as well as the gold electrodes are capacitively coupled to the inversion region. See Supporting Information (Figure S2) for more details on the measurement setup and device structure. The measurement consists of rapidly stepping the voltage on one of the gold electrodes, while the other is held at 0 V relative to the substrate. As we apply a negative voltage step to the gold B
DOI: 10.1021/acs.nanolett.5b00659 Nano Lett. XXXX, XXX, XXX−XXX
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markers represent the differential conductances extracted from charge sensing measurements. We find that, for all values of the applied field, the conductance is simply activated. A simple model of nearest neighbor hopping has previously been used to describe charge transport in large scale arrays of annealed PbSe dots.17 In this model of hole dominated transport, it is assumed that, at low electric fields, a hole must overcome two contributions to the activation energy in order to contribute to the current. First it must overcome the binding energy of the acceptor state, and second it must overcome the disorder broadening of the 1Sh states. We assume that most of the Coulomb interaction between the hole and the acceptor that provides it is overcome when the hole is removed from the acceptor and placed on the nearest dot. This would be the case if the acceptor is inside the cap layer or if the acceptor is positively charged when occupied. As the strength of the electric field, F, is increased, the activation energy from disorder is reduced by eFd, where d is the distance between neighboring dots, while the activation energy from the acceptor binding is reduced by eFx, where x is the distance from the acceptor state to the 1S state. This gives rise to two length scales in the problem, and two slopes of the activation energy as a function of field. At a critical field, Fc, the activation energy from disorder is completely overcome by the field, F, and only the activation energy for excitation from the acceptor state remains. The conductance in the presence of a field may be written as G = G0 exp(−(Ea − eF(x + d))/kBT). Figure 3b shows the extracted activation energies as a function of applied electric field across the nanopattern, where we can see the decrease in activation energy with increasing field. If we fit to this model, at the highest fields (11−14 × 106 V/m), the slope is given by −ex, from which we extract x = 0.8 ± 0.1 nm. In the low field region, the slope gives us x + d = 6.2 ± 0.6 nm. Using x = 0.8 nm, we extract d = 5.4 ± 0.6 nm, which is within error of the expected interdot separation of approximately 5.5 nm, given a dot diameter between 4.3 and 4.6 nm and ligand size of roughly 1 nm (see Supporting Information). From optical absorption of measurements of the dots in solution,32 we extract roughly the disorder in site energies to be approximately 100 meV. The change in the slope at ∼10 × 107 V/m indicates that this is approximately the critical field. At this field the activation energy has been reduced by approximately 45 meV, which is roughly half the variation in energy measured optically. Nearest neighbor hopping therefore provides a consistent explanation for transport in OA-capped PbS dots. For NBA-capped PbS dots, Figure 4a shows the differential conductance at different fields, as a function of inverse temperature. We fit the data shown to a sum of two simply activated processes, as one would expect for a case of two conducting channels in parallel, and extract both the activation energies. We plot the activation energies as a function of field, as shown in Figure 4b. From the evolution of activation energies with field, at low fields we find length scales of 9 and 3.1 nm for the larger and smaller activation energy, respectively. These are difficult to understand because neither of them corresponds to the interdot spacing. If we consider the electric field dependence of variable range hopping, for large electric fields, F, such that F > 2kBT/ea, where a is the localization length, the electron can move by downward hops only, emitting a phonon at each hop. The conductance at high field is then temperature independent and only depends on the field as, G ∝ exp(F*/F)1/4.33 However,
the ohmic conductance. Filled circles represent conductances extracted from the current−voltage characteristics, and empty circles represent the conductances from transient measurements. From Figure 2a, we see that the conductance deviates from a simple linear relationship at temperatures below ∼50 K. We would not have observed this deviation near zero bias, without our charge-sensing measurements. While the data can be fit to Mott’s variable range hopping, G = G0 exp(−(T*/ T)1/4), with T*= 1.2 × 107 K, over the whole temperature range as shown in the inset of Figure 2a, it can also be fit to a sum of two simply activated processes with activation energies of 55 and 13 meV. We are also able to measure current in nanopatterned arrays of OA-capped PbS dots. Although the change in the height of the tunnel barrier from OA-capped to NBA-capped PbS dots is difficult to estimate, the interdot separation changes from 0.5 nm in NBA-capped dots to approximately 1 nm in OA-capped dots, as determined from TEM measurements. For OA-capped dots, we are unable to measure the conductance in the ohmic regime, so we measure the differential conductance, gPbS−OA = dI/dV at the lowest voltage we can. Figure 2b shows the differential conductance at a bias of 3.5 V for an OA-capped PbS nanopattern. In this case we find simply activated behavior over the entire temperature range; thus, we find a clear difference in the observed temperature dependence of the conductance depending on the capping ligand. It is difficult to make any strong claims about the transport mechanism simply from the observed temperature dependence; however, the simply activated behavior for OA-capped dots suggests nearest neighbor hopping. The data for NBA-capped dots can be understood in terms of variable range hopping or as two conduction channels in parallel, giving rise to two different activation energies in the system. We therefore study the electric field dependence or bias dependence of the conductance and find that the extracted parameters, such as the localization length and hopping distances, are consistent with Mott’s variable range hopping for NBA-capped PbS arrays. Figure 3a shows the electric field dependence of the differential conductance for OA-capped PbS dots. Filled markers represent differential conductance extracted from derivatives of the current−voltage characteristics, and empty
Figure 3. Electric field dependence of the differential conductance in OA-capped PbS dots. (a) Differential conductance of OA-capped PbS dots, gPbS−OA, as a function of inverse temperature, for different values of bias voltage across the dot array. Filled markers represent differential conductance extracted from derivatives of the current− voltage characteristics, and empty markers represent the differential conductances extracted from charge sensing measurements. Black dashed lines are fits to gPbS−OA ∼ exp(−Ea/kBT), where Ea is the activation energy. (b) Activation energy, Ea, as a function of the applied electric field, F. Blue markers indicate the data points, and the black dashed lines represent straight line fits to the data. C
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range hopping,36 the hopping distance, rm, is weakly temperature-dependent and is given by
rm =
(2)
where a is the localization length. Figure 4d shows the hopping distance as a function of 1/T1/4, and from a fit to eq 2, represented by the black dashed line, we extract a localization length, a = 1.75 ± 0.2 nm, which is comparable to the dot radius. At low temperatures, as we move to higher fields, we expect a transition out of the low field regime for hopping conduction. The flattening of the conductance at low temperatures and high fields likely represents the onset of the expected high field F1/4 behavior. At 22 and 40 K, the quantity eFa/2kBT starts to become comparable to 1 at field strengths of 2 and 3 V/m, respectively, which matches the field at which we start to observe the flattening of the curves. Variable range hopping therefore seems to provide a consistent explanation for transport in NBA-capped PbS dots. While the hopping distance decreases with increasing temperature, we find a hopping distance of ∼9 nm even at room temperature. It is interesting to note that, while this value is comparable to the length-scale extracted from fits to simply activated conductance at high temperatures, it is still much larger than the nearest neighbor separation between the dots. We therefore have no clear evidence for a crossover from variable-range to nearestneighbor hopping. We also see no evidence for a crossover to the form exp(−(T*/T) 1/2), expected for ES-VRH,37 and elastic and inelastic cotunneling between electronic states for several granular metals and metallic nanocrystal arrays.38−42 The change of the conduction mechanism when the ligand is changed can be caused by a change in the wave function overlap between dots or by a change in the density of trap states. We assume that hopping occurs in the band of occupied acceptor states, which is partially occupied because of compensation by donors or defect states.17 In Mott’s picture of hopping between localized states, a smaller wave function overlap, which can result from a larger interdot separation in our case, can favor hopping to the nearest neighbor site over variable range hopping.36 However, shorter ligands are also associated with a higher density of trap states due to the incompleteness of the ligand exchange process, which could also favor variable range hopping between trap sites. A complete understanding of the ligand dependence of the conductance would require taking into consideration changes in the dot-ligand surface dipole moments and changes in the local dielectric environment among other factors. Our results here show the impact of capping ligand on the transport through the array and could also explain the variability in transport measurements on QD arrays reported in literature. The observation of conduction mechanisms similar to those observed in large area films provides strong evidence to suggest that size disorder rather than packing disorder or charge disorder, dominates transport through the nanopatterned PbS array for the ligands studied here. Our measurements are made possible only by a combination of nanopatterning and subsequent integration of a charge sensor to measure high resistances. The measurement technique is fairly general and can further be applied to a range of highly resistive samples. The ability to approach ordering length scales of dots and measure high resistances opens up new possibilities
Figure 4. Electric field dependence of the current in NBA-capped PbS dots. (a) Differential conductance, gPbS, of NBA-capped PbS dots as a function of inverse temperature for different values of bias voltage across the dot array. (b) Activation energies, Ea1 and Ea2, extracted by fitting the conductance to a sum of two simply activated processes, as functions of the bias voltage (c) Bias dependence of the differential conductance at various temperatures. The dashed lines are fits to gPbS ∼ exp(0.17 eVPbSrm/LkBT), where rm is the variable range hopping distance. (d) Variable range hopping distance, rm, extracted from exponential fits in (c) as a function of 1/T1/4.
not a lot of work has been done to understand the dependence in the moderate or low electric field regime, F < 2kBT/ea. The most significant experimental results on the field dependence of the conductance are from Elliot et al.34 on amorphous germanium, because it is the amorphous semiconductor that clearly exhibits variable range hopping. They find an exponential dependence of the conductance on the electric field, which has since been supported by theory from Pollak and Reiss.35 Within the percolation model proposed by Pollak and Reiss,35 the field is expected to change the percolation path in such a way that the critical impedance is lowered, leading to an exponential increase in the conductance with field. The current, I, in the low field regime is given by ⎛ ⎛ T * ⎞1/4 ⎞ ⎛ 3 eFrm ⎞ I ∼ exp⎜⎜ −⎜ ⎟ ⎟⎟ exp⎜ ⎟ ⎝ 16 kBT ⎠ ⎝ ⎝T ⎠ ⎠
1/4 3 ⎛⎜ T * ⎞⎟ a 8 ⎝T ⎠
(1)
where rm is the variable range hopping distance. Figure 4c shows the differential conductance as a function of applied field across the NBA-capped PbS nanopattern, for different temperatures. The solid symbols are data obtained from current measurements, and open symbols are from charge sensing measurements. We find that the differential conductance increases exponentially with applied field for low fields at all temperatures. However, at lower temperatures, we observe a rollover of the differential conductance beyond a certain field strength. We extract the hopping distance, rm, from exponential fits to the conductance measured as a function of field as given by eq 1. The fits are shown by black dashed lines. At 22 K, we extract a hopping distance of approximately 20 nm. Given an interdot spacing of approximately 5 nm, this would be hopping roughly across 4 dots. In Mott’s picture of variable D
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(16) Kalyuzhny, G.; Murray, R. W. J. Phys. Chem. B 2005, 109, 7012−7021. (17) Mentzel, T. S.; Porter, V. J.; Geyer, S.; MacLean, K.; Bawendi, M. G.; Kastner, M. A. Phys. Rev. B 2008, 77, 075316. (18) Geyer, S.; Porter, V. J.; Halpert, J. E.; Mentzel, T. S.; Kastner, M. A.; Bawendi, M. G. Phys. Rev. B 2010, 82, 155201. (19) Romero, H. E.; Drndic, M. Phys. Rev. Lett. 2005, 95, 156801. (20) Kang, M. S.; Sahu, A.; Norris, D. J.; Frisbie, C. D. Nano Lett. 2010, 10, 3727−3732. (21) Kang, M. S.; Sahu, A.; Norris, D. J.; Frisbie, C. D. Nano Lett. 2011, 11, 3887−3892. (22) Yu, D.; Wang, C.; Guyot-Sionnest, P. Science 2003, 300, 1277− 1280. (23) Shim, M.; Guyot-Sionnest, P. Nature 2000, 407, 981−983. (24) Vanmaekelbergh, D.; Houtepen, A. J.; Kelly, J. J. Electrochim. Acta 2007, 53, 1140−1149. (25) Bekenstein, Y.; Vinokurov, K.; Keren-Zur, S.; Hadar, I.; Schilt, Y.; Raviv, U.; Millo, O.; Banin, U. Nano Lett. 2014, 14, 1349−1353. (26) Liu, Y.; Gibbs, M.; Puthussery, J.; Gaik, S.; Ihly, R.; Hillhouse, H. W.; Law, M. Nano Lett. 2010, 10, 1960−1969. (27) MacLean, K.; Mentzel, T. S.; Kastner, M. A. Nano Lett. 2010, 10, 1037−1040. (28) Mentzel, T. S.; MacLean, K.; Kastner, M. A. Nano Lett. 2011, 11, 4102−4106. (29) Mentzel, T. S.; Wanger, D. D.; Ray, N.; Walker, B. J.; Strasfeld, D.; Bawendi, M. G.; Kastner, M. A. Nano Lett. 2012, 12, 4404−4408. (30) Steckel, J. S.; Coe-Sullivan, S.; Bulović, V.; Bawendi, M. G. Adv. Mater. 2003, 15, 1862−1866. (31) Zhao, N.; Osedach, T. P.; Chang, L.-Y.; Geyer, S. M.; Wanger, D.; Binda, M. T.; Arango, A. C.; Bawendi, M. G.; Bulovic, V. ACS Nano 2010, 4, 3743−3752. (32) Grinolds, D. D. W.; Brown, P. R.; Harris, D. K.; Bulovic, V.; Bawendi, M. G. Nano Lett. 2015, 15, 21−26. (33) Shklovskii, B. I. Sov. Phys.-Semicond. 1973, 6, 1964. (34) Elliott, P. J.; Yoffe, A. D.; Davis, E. A. Hopping conduction in amorphous germanium. Tetrahedrally Bonded Amorphous Semicond. 1974, 311−319. (35) Pollak, M.; Riess, I. J. Phys.: Condens. Matter 1976, 9, 2339. (36) Mott, N. F.; Davis, E. A. Electronic processes in non-crystalline materials; Oxford University Press: New York, 2012. (37) Shklovskii, B. I.; Efros, A. L. Electronic Properties of Doped Semiconductors; Springer Series in Solid-State Sciences; Springer: Berlin, 1984; Vol. 45; pp 202−227. (38) Beloborodov, I. S.; Lopatin, A. V.; Vinokur, V. M.; Efetov, K. B. Rev. Mod. Phys. 2007, 79, 469−518. (39) Tran, T. B.; Beloborodov, I. S.; Lin, X. M.; Bigioni, T. P.; Vinokur, V. M.; Jaeger, H. M. Phys. Rev. Lett. 2005, 95, 076806. (40) Beloborodov, I. S.; Lopatin, A. V.; Vinokur, V. M. Phys. Rev. B 2005, 72, 125121. (41) Tran, T. B.; Beloborodov, I. S.; Hu, J.; Lin, X. M.; Rosenbaum, T. F.; Jaeger, H. M. Phys. Rev. B 2008, 78, 075437. (42) Beloborodov, I.; Glatz, A.; Vinokur, V. Phys. Rev. B 2007, 75, 052302.
for transport studies in colloidal systems, particularly in the emerging field of single and multicomponent superlattices.
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ASSOCIATED CONTENT
* Supporting Information S
Transmission electron micrographs of PbS assemblies. Charge sensor characterization and instrumentation. This material is available free of charge via the Internet at http://pubs.acs.org/ .The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b00659.
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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 We acknowledge useful discussions with Prof. B. I. Shklovskii. This work was carried out in part through the use of MITs Microsystems Technology Laboratories. We are grateful to Mark Mondol and RLE SEBL facility for help with electronbeam lithography. This work was supported in part by the U.S. Army Research Laboratory and the U.S. Army Research Office through the Institute for Soldier Nanotechnologies, under contract number W911NF-13-D-0001 (synthesis and nanopatterning of colloidal QDs), by the Department of Energy under award number DE-FG02-08ER46515 (charge sensing on QD arrays, including MOSFET sensor fabrication) and by Samsung SAIT. N.R. acknowledges support from the Schlumberger Foundation through the Faculty for the Future Fellowship Program. D.D.W.G. acknowledges support from the Fannie and John Hertz Foundation Fellowship.
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REFERENCES
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DOI: 10.1021/acs.nanolett.5b00659 Nano Lett. XXXX, XXX, XXX−XXX
Measuring Ligand-Dependent Transport in Nanopatterned PbS Colloidal Quantum Dot Arrays Using Charge Sensing Nirat Ray,*,† Neal E. Staley,† Darcy D. W. Grinolds,‡ Moungi G. Bawendi,‡ and Marc A. Kastner† †
Department of Physics and ‡Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States S Supporting Information *
ABSTRACT: Colloidal quantum dot arrays with long organic ligands have better packing order than those with short ligands but are highly resistive, making low-bias conductance measurements impossible with conventional two-probe techniques. We use an integrated charge sensor to study transport in weakly coupled arrays in the low-bias regime, and we nanopattern the arrays to minimize packing disorder. We present the temperature and field dependence of the resistance for nanopatterned oleic-acid and n-butylamine-capped PbS arrays, measuring resistances as high as 1018 Ω. We find that the conduction mechanism changes from nearest neighbor hopping in oleic-acid-capped PbS dots to Mott’s variable range hopping in n-butylamine capped PbS dots. Our results can be understood in terms of a change in the interdot coupling strength or a change in density of trap states and highlight the importance of the capping ligand on charge transport through colloidal quantum dot arrays. KEYWORDS: PbS nanocrystals, electrical transport, nanoscale charge sensor, nanopatterning
C
level9,25 have been investigated in some detail, very little has been done to determine the influence of the ligand on the conduction mechanism.26 In particular, transport in weakly coupled arrays, expected to have minimum packing disorder as well as a lower density of sub band gap states, has not been investigated, and a technique to measure low conductivity is crucial to investigation of such transport measurements. In this Letter, we introduce a novel technique to study transport in weakly coupled PbS QD arrays, based on a timeresolved measurement of charge using a nanoscale charge sensor. This method has been previously used to study transport in highly resistive nanopatterned films of amorphous silicon27 and amorphous germanium.28 The method has also been shown to be insensitive to contact effects such as blocking contacts.28 We use a nanopatterning technique developed for colloidal dots29 to integrate a charge sensor with the array of dots and fabricate the necessary devices. Nanopatterning also helps to minimize cracks and clusters in the array resulting in more conducting arrays and can be used to make samples whose dimensions approach the short ordering length scales of the dots. We study charge transport in PbS QD arrays with two capping ligands: the native oleic acid (OA) ligand and the ligand-exchanged n-butylamine (NBA), and from the evolution of the resistance as a function of temperature and electric field, we find a different conduction mechanism for these two ligands.
olloidal quantum dots (QDs) have emerged as promising candidates for applications in a variety of photovoltaic and optoelectronic devices,1−4 owing to their tunable optical and electronic properties. It has recently been shown that, in complement to the tunability achieved by a modification of the dot size, the electronic properties of coupled QD arrays could also be tuned with the surface ligands.5 There has also been considerable interest in using dots as building blocks for artificial solids to gain insight into correlated electron physics,6 and interesting theoretical predictions exist for the observation of charge ordering and other novel aspects of charge transport in dot arrays.7−9 These effects are difficult to observe experimentally because of the finite size distribution of the dots and packing disorder in the array. Long-range ordering of dots in the array is stabilized by van der Waal’s interactions10−12 and leads to highly resistive arrays, due to the insulating nature of the organic ligands and correspondingly large tunnel barriers between dots. Techniques to decrease the spacing between dots, such as vacuum annealing or ligand exchange, typically lead to a larger packing disorder in the array,13,14 and the incompleteness of the ligand exchange process can further lead to the creation of sub-band gap states.15,16 Many different experimental configurations, such as the field effect transistor (FET) geometry,8,17−19 interdigitated electrodes,20,21 and doping through chemical22,23 or electrochemical gating methods24 have been used to study transport in a variety of colloidal dot assemblies. However, a relatively large variability exists in the conduction mechanisms reported in literature, and while the influence of dot size20,21 and doping © XXXX American Chemical Society
Received: February 17, 2015 Revised: May 23, 2015
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electrode connected to the nanopattern, we observe an instantaneous jump in the MOSFET conductance followed by a further decrease slowly with time (which we call the charge transient). The time dependence of the charging transient from the nanopattern has been shown to be accurately exponential,27 and the resistance of the nanopattern can be extracted from an exponential fit to this transient response. The voltage step and the transient response of the MOSFET are shown in Figure 1b. The solid lines are fits to GM = G∞ + G0 exp(−Γt); the conductance, GPbS, of the nanopattern is given by GPbS = wLCΓ/π2, where w and L are the width and length of the sample, respectively, and C is the capacitance per unit area of the quantum dot film. It is important to note here that the technique measures the differential conductance, g = dI/dV, rather than just the conductance, G, of the material. For ohmic regimes, near zero bias, they are identical; however, they may be significantly different for nonohmic regimes. To measure the differential conductance, GPbS, at finite source drain bias, in order to study the field dependence of the conductance, we apply a static field across the nanopattern, rapidly step the voltage on one of the electrodes and obtain the conductance from the observed charge transient. For example, to measure the differential conductance, gPbS, at an applied bias voltage of VPbS = −3.25 V, we apply a fixed voltage VPbS = −3 V to one electrode and rapidly step the voltage to VPbS = −3.5 V. Our device structure allows us to measure the current− voltage characteristics in addition to transients resulting from the spreading of charge. At high temperatures, we are able to measure the current and extract the differential conductance gPbS = dI/dV, directly. As we move to lower temperatures, our current measurements are limited by instrument noise from the amplifiers. At these temperatures, we extract the conductance from exponential fits to charge transients as discussed above. Figure 2a shows the measured conductance of the NBA capped PbS dots, GPbS−NBA at a bias of 0.5 V, from room temperature to 22 K, plotted as a function of inverse temperature. At this low bias, the differential conductance is expected to be the same as
Our measurements also reveal the highest resistances ever measured for colloidal QD arrays. We use an n-channel MOSFET as the charge sensor, where the MOS (metal−oxide−semiconductor) structure is obtained by growing 100 nm of silicon dioxide (SiO2) on top of a p-type silicon substrate and depositing a layer of metal (aluminum). The aluminum MOSFET gate tapers down to a narrow width of almost 80 nm so that the MOSFET is highly sensitive to its electrostatic environment. The structure includes two additional terminals (drain and source), each connected to individual highly doped n-type regions on the p-type substrate. Two gold electrodes spaced 1 μm apart are fabricated ∼100 nm away from the sensor for contacts to the nanopatterned dot arrays. We use PbS dots with a mean diameter of approximately 4.5 nm. The dots are synthesized air-free using a Schlenk line by high temperature pyrolysis of Pb and S precursors in an oleic acid/octadecene mixture.30,31 We process the growth solution to remove remaining products and, when desired, perform a solution exchange of the native oleic acid capping ligand for the smaller butylamine ligand. We create nanopatterns of PbS dots by dropcasting a dilute solution of dots into a 200-nm-wide trench created in the electron beam resist that is aligned with the patterned gold electrodes.29 Figures 1a and 1b show a
Figure 1. (a) Schematic of the device structure used for measurements. (b) False color scanning electron micrograph of the MOSFET gate (blue), adjacent to a nanopatterned array of PbS dots (red). The nanopattern is 200 nm wide and 1 μm long, and we make electrical contact to the nanopattern with Ti/Au electrodes (yellow). (b) Transient response of the MOSFET conductance, GM, as a function of time after the voltage VAu on one of the gold electrodes is stepped from 0 to −1 (red trace) at 50 K (blue markers) and 60 K (green markers). The solid lines are fits to GM = G∞ + G0 exp(−Γt).
Figure 2. (a) Ohmic conductance, G, as a function of inverse temperature for n-butylamine (NBA)-capped PbS dots near zero bias. (b) Differential conductance, g, as a function of inverse temperature for oleic acid (OA)-capped PbS dots. Empty markers represent conductance extracted from charge sensing measurements and filled markers represent values extracted from current measurements. The conductance using both methods has been found to agree within a factor of 4, thereby validating our method for obtaining the conductance from the charge transient (see Supporting Information, Figure S4). For OA-capped dots, we were unable to probe the resistance in the ohmic regime, and so the data presented actually represent the differential conductance (gPbS−OA) at a bias of 3.5 V. In the main figure of (a) the dashed line is a fit to the sum of two simply activated components; the inset in (a) shows the conductance and fit to Motts variable range hopping. In (b) the dashed line is a fit to a single activated component.
schematic and an electron micrograph of the final device structure. For all of the measurements reported here, a positive voltage, Vg, is applied to the gate so that the MOSFET is in inversion, and the nanopattern as well as the gold electrodes are capacitively coupled to the inversion region. See Supporting Information (Figure S2) for more details on the measurement setup and device structure. The measurement consists of rapidly stepping the voltage on one of the gold electrodes, while the other is held at 0 V relative to the substrate. As we apply a negative voltage step to the gold B
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markers represent the differential conductances extracted from charge sensing measurements. We find that, for all values of the applied field, the conductance is simply activated. A simple model of nearest neighbor hopping has previously been used to describe charge transport in large scale arrays of annealed PbSe dots.17 In this model of hole dominated transport, it is assumed that, at low electric fields, a hole must overcome two contributions to the activation energy in order to contribute to the current. First it must overcome the binding energy of the acceptor state, and second it must overcome the disorder broadening of the 1Sh states. We assume that most of the Coulomb interaction between the hole and the acceptor that provides it is overcome when the hole is removed from the acceptor and placed on the nearest dot. This would be the case if the acceptor is inside the cap layer or if the acceptor is positively charged when occupied. As the strength of the electric field, F, is increased, the activation energy from disorder is reduced by eFd, where d is the distance between neighboring dots, while the activation energy from the acceptor binding is reduced by eFx, where x is the distance from the acceptor state to the 1S state. This gives rise to two length scales in the problem, and two slopes of the activation energy as a function of field. At a critical field, Fc, the activation energy from disorder is completely overcome by the field, F, and only the activation energy for excitation from the acceptor state remains. The conductance in the presence of a field may be written as G = G0 exp(−(Ea − eF(x + d))/kBT). Figure 3b shows the extracted activation energies as a function of applied electric field across the nanopattern, where we can see the decrease in activation energy with increasing field. If we fit to this model, at the highest fields (11−14 × 106 V/m), the slope is given by −ex, from which we extract x = 0.8 ± 0.1 nm. In the low field region, the slope gives us x + d = 6.2 ± 0.6 nm. Using x = 0.8 nm, we extract d = 5.4 ± 0.6 nm, which is within error of the expected interdot separation of approximately 5.5 nm, given a dot diameter between 4.3 and 4.6 nm and ligand size of roughly 1 nm (see Supporting Information). From optical absorption of measurements of the dots in solution,32 we extract roughly the disorder in site energies to be approximately 100 meV. The change in the slope at ∼10 × 107 V/m indicates that this is approximately the critical field. At this field the activation energy has been reduced by approximately 45 meV, which is roughly half the variation in energy measured optically. Nearest neighbor hopping therefore provides a consistent explanation for transport in OA-capped PbS dots. For NBA-capped PbS dots, Figure 4a shows the differential conductance at different fields, as a function of inverse temperature. We fit the data shown to a sum of two simply activated processes, as one would expect for a case of two conducting channels in parallel, and extract both the activation energies. We plot the activation energies as a function of field, as shown in Figure 4b. From the evolution of activation energies with field, at low fields we find length scales of 9 and 3.1 nm for the larger and smaller activation energy, respectively. These are difficult to understand because neither of them corresponds to the interdot spacing. If we consider the electric field dependence of variable range hopping, for large electric fields, F, such that F > 2kBT/ea, where a is the localization length, the electron can move by downward hops only, emitting a phonon at each hop. The conductance at high field is then temperature independent and only depends on the field as, G ∝ exp(F*/F)1/4.33 However,
the ohmic conductance. Filled circles represent conductances extracted from the current−voltage characteristics, and empty circles represent the conductances from transient measurements. From Figure 2a, we see that the conductance deviates from a simple linear relationship at temperatures below ∼50 K. We would not have observed this deviation near zero bias, without our charge-sensing measurements. While the data can be fit to Mott’s variable range hopping, G = G0 exp(−(T*/ T)1/4), with T*= 1.2 × 107 K, over the whole temperature range as shown in the inset of Figure 2a, it can also be fit to a sum of two simply activated processes with activation energies of 55 and 13 meV. We are also able to measure current in nanopatterned arrays of OA-capped PbS dots. Although the change in the height of the tunnel barrier from OA-capped to NBA-capped PbS dots is difficult to estimate, the interdot separation changes from 0.5 nm in NBA-capped dots to approximately 1 nm in OA-capped dots, as determined from TEM measurements. For OA-capped dots, we are unable to measure the conductance in the ohmic regime, so we measure the differential conductance, gPbS−OA = dI/dV at the lowest voltage we can. Figure 2b shows the differential conductance at a bias of 3.5 V for an OA-capped PbS nanopattern. In this case we find simply activated behavior over the entire temperature range; thus, we find a clear difference in the observed temperature dependence of the conductance depending on the capping ligand. It is difficult to make any strong claims about the transport mechanism simply from the observed temperature dependence; however, the simply activated behavior for OA-capped dots suggests nearest neighbor hopping. The data for NBA-capped dots can be understood in terms of variable range hopping or as two conduction channels in parallel, giving rise to two different activation energies in the system. We therefore study the electric field dependence or bias dependence of the conductance and find that the extracted parameters, such as the localization length and hopping distances, are consistent with Mott’s variable range hopping for NBA-capped PbS arrays. Figure 3a shows the electric field dependence of the differential conductance for OA-capped PbS dots. Filled markers represent differential conductance extracted from derivatives of the current−voltage characteristics, and empty
Figure 3. Electric field dependence of the differential conductance in OA-capped PbS dots. (a) Differential conductance of OA-capped PbS dots, gPbS−OA, as a function of inverse temperature, for different values of bias voltage across the dot array. Filled markers represent differential conductance extracted from derivatives of the current− voltage characteristics, and empty markers represent the differential conductances extracted from charge sensing measurements. Black dashed lines are fits to gPbS−OA ∼ exp(−Ea/kBT), where Ea is the activation energy. (b) Activation energy, Ea, as a function of the applied electric field, F. Blue markers indicate the data points, and the black dashed lines represent straight line fits to the data. C
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range hopping,36 the hopping distance, rm, is weakly temperature-dependent and is given by
rm =
(2)
where a is the localization length. Figure 4d shows the hopping distance as a function of 1/T1/4, and from a fit to eq 2, represented by the black dashed line, we extract a localization length, a = 1.75 ± 0.2 nm, which is comparable to the dot radius. At low temperatures, as we move to higher fields, we expect a transition out of the low field regime for hopping conduction. The flattening of the conductance at low temperatures and high fields likely represents the onset of the expected high field F1/4 behavior. At 22 and 40 K, the quantity eFa/2kBT starts to become comparable to 1 at field strengths of 2 and 3 V/m, respectively, which matches the field at which we start to observe the flattening of the curves. Variable range hopping therefore seems to provide a consistent explanation for transport in NBA-capped PbS dots. While the hopping distance decreases with increasing temperature, we find a hopping distance of ∼9 nm even at room temperature. It is interesting to note that, while this value is comparable to the length-scale extracted from fits to simply activated conductance at high temperatures, it is still much larger than the nearest neighbor separation between the dots. We therefore have no clear evidence for a crossover from variable-range to nearestneighbor hopping. We also see no evidence for a crossover to the form exp(−(T*/T) 1/2), expected for ES-VRH,37 and elastic and inelastic cotunneling between electronic states for several granular metals and metallic nanocrystal arrays.38−42 The change of the conduction mechanism when the ligand is changed can be caused by a change in the wave function overlap between dots or by a change in the density of trap states. We assume that hopping occurs in the band of occupied acceptor states, which is partially occupied because of compensation by donors or defect states.17 In Mott’s picture of hopping between localized states, a smaller wave function overlap, which can result from a larger interdot separation in our case, can favor hopping to the nearest neighbor site over variable range hopping.36 However, shorter ligands are also associated with a higher density of trap states due to the incompleteness of the ligand exchange process, which could also favor variable range hopping between trap sites. A complete understanding of the ligand dependence of the conductance would require taking into consideration changes in the dot-ligand surface dipole moments and changes in the local dielectric environment among other factors. Our results here show the impact of capping ligand on the transport through the array and could also explain the variability in transport measurements on QD arrays reported in literature. The observation of conduction mechanisms similar to those observed in large area films provides strong evidence to suggest that size disorder rather than packing disorder or charge disorder, dominates transport through the nanopatterned PbS array for the ligands studied here. Our measurements are made possible only by a combination of nanopatterning and subsequent integration of a charge sensor to measure high resistances. The measurement technique is fairly general and can further be applied to a range of highly resistive samples. The ability to approach ordering length scales of dots and measure high resistances opens up new possibilities
Figure 4. Electric field dependence of the current in NBA-capped PbS dots. (a) Differential conductance, gPbS, of NBA-capped PbS dots as a function of inverse temperature for different values of bias voltage across the dot array. (b) Activation energies, Ea1 and Ea2, extracted by fitting the conductance to a sum of two simply activated processes, as functions of the bias voltage (c) Bias dependence of the differential conductance at various temperatures. The dashed lines are fits to gPbS ∼ exp(0.17 eVPbSrm/LkBT), where rm is the variable range hopping distance. (d) Variable range hopping distance, rm, extracted from exponential fits in (c) as a function of 1/T1/4.
not a lot of work has been done to understand the dependence in the moderate or low electric field regime, F < 2kBT/ea. The most significant experimental results on the field dependence of the conductance are from Elliot et al.34 on amorphous germanium, because it is the amorphous semiconductor that clearly exhibits variable range hopping. They find an exponential dependence of the conductance on the electric field, which has since been supported by theory from Pollak and Reiss.35 Within the percolation model proposed by Pollak and Reiss,35 the field is expected to change the percolation path in such a way that the critical impedance is lowered, leading to an exponential increase in the conductance with field. The current, I, in the low field regime is given by ⎛ ⎛ T * ⎞1/4 ⎞ ⎛ 3 eFrm ⎞ I ∼ exp⎜⎜ −⎜ ⎟ ⎟⎟ exp⎜ ⎟ ⎝ 16 kBT ⎠ ⎝ ⎝T ⎠ ⎠
1/4 3 ⎛⎜ T * ⎞⎟ a 8 ⎝T ⎠
(1)
where rm is the variable range hopping distance. Figure 4c shows the differential conductance as a function of applied field across the NBA-capped PbS nanopattern, for different temperatures. The solid symbols are data obtained from current measurements, and open symbols are from charge sensing measurements. We find that the differential conductance increases exponentially with applied field for low fields at all temperatures. However, at lower temperatures, we observe a rollover of the differential conductance beyond a certain field strength. We extract the hopping distance, rm, from exponential fits to the conductance measured as a function of field as given by eq 1. The fits are shown by black dashed lines. At 22 K, we extract a hopping distance of approximately 20 nm. Given an interdot spacing of approximately 5 nm, this would be hopping roughly across 4 dots. In Mott’s picture of variable D
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(16) Kalyuzhny, G.; Murray, R. W. J. Phys. Chem. B 2005, 109, 7012−7021. (17) Mentzel, T. S.; Porter, V. J.; Geyer, S.; MacLean, K.; Bawendi, M. G.; Kastner, M. A. Phys. Rev. B 2008, 77, 075316. (18) Geyer, S.; Porter, V. J.; Halpert, J. E.; Mentzel, T. S.; Kastner, M. A.; Bawendi, M. G. Phys. Rev. B 2010, 82, 155201. (19) Romero, H. E.; Drndic, M. Phys. Rev. Lett. 2005, 95, 156801. (20) Kang, M. S.; Sahu, A.; Norris, D. J.; Frisbie, C. D. Nano Lett. 2010, 10, 3727−3732. (21) Kang, M. S.; Sahu, A.; Norris, D. J.; Frisbie, C. D. Nano Lett. 2011, 11, 3887−3892. (22) Yu, D.; Wang, C.; Guyot-Sionnest, P. Science 2003, 300, 1277− 1280. (23) Shim, M.; Guyot-Sionnest, P. Nature 2000, 407, 981−983. (24) Vanmaekelbergh, D.; Houtepen, A. J.; Kelly, J. J. Electrochim. Acta 2007, 53, 1140−1149. (25) Bekenstein, Y.; Vinokurov, K.; Keren-Zur, S.; Hadar, I.; Schilt, Y.; Raviv, U.; Millo, O.; Banin, U. Nano Lett. 2014, 14, 1349−1353. (26) Liu, Y.; Gibbs, M.; Puthussery, J.; Gaik, S.; Ihly, R.; Hillhouse, H. W.; Law, M. Nano Lett. 2010, 10, 1960−1969. (27) MacLean, K.; Mentzel, T. S.; Kastner, M. A. Nano Lett. 2010, 10, 1037−1040. (28) Mentzel, T. S.; MacLean, K.; Kastner, M. A. Nano Lett. 2011, 11, 4102−4106. (29) Mentzel, T. S.; Wanger, D. D.; Ray, N.; Walker, B. J.; Strasfeld, D.; Bawendi, M. G.; Kastner, M. A. Nano Lett. 2012, 12, 4404−4408. (30) Steckel, J. S.; Coe-Sullivan, S.; Bulović, V.; Bawendi, M. G. Adv. Mater. 2003, 15, 1862−1866. (31) Zhao, N.; Osedach, T. P.; Chang, L.-Y.; Geyer, S. M.; Wanger, D.; Binda, M. T.; Arango, A. C.; Bawendi, M. G.; Bulovic, V. ACS Nano 2010, 4, 3743−3752. (32) Grinolds, D. D. W.; Brown, P. R.; Harris, D. K.; Bulovic, V.; Bawendi, M. G. Nano Lett. 2015, 15, 21−26. (33) Shklovskii, B. I. Sov. Phys.-Semicond. 1973, 6, 1964. (34) Elliott, P. J.; Yoffe, A. D.; Davis, E. A. Hopping conduction in amorphous germanium. Tetrahedrally Bonded Amorphous Semicond. 1974, 311−319. (35) Pollak, M.; Riess, I. J. Phys.: Condens. Matter 1976, 9, 2339. (36) Mott, N. F.; Davis, E. A. Electronic processes in non-crystalline materials; Oxford University Press: New York, 2012. (37) Shklovskii, B. I.; Efros, A. L. Electronic Properties of Doped Semiconductors; Springer Series in Solid-State Sciences; Springer: Berlin, 1984; Vol. 45; pp 202−227. (38) Beloborodov, I. S.; Lopatin, A. V.; Vinokur, V. M.; Efetov, K. B. Rev. Mod. Phys. 2007, 79, 469−518. (39) Tran, T. B.; Beloborodov, I. S.; Lin, X. M.; Bigioni, T. P.; Vinokur, V. M.; Jaeger, H. M. Phys. Rev. Lett. 2005, 95, 076806. (40) Beloborodov, I. S.; Lopatin, A. V.; Vinokur, V. M. Phys. Rev. B 2005, 72, 125121. (41) Tran, T. B.; Beloborodov, I. S.; Hu, J.; Lin, X. M.; Rosenbaum, T. F.; Jaeger, H. M. Phys. Rev. B 2008, 78, 075437. (42) Beloborodov, I.; Glatz, A.; Vinokur, V. Phys. Rev. B 2007, 75, 052302.
for transport studies in colloidal systems, particularly in the emerging field of single and multicomponent superlattices.
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ASSOCIATED CONTENT
* Supporting Information S
Transmission electron micrographs of PbS assemblies. Charge sensor characterization and instrumentation. This material is available free of charge via the Internet at http://pubs.acs.org/ .The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b00659.
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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 We acknowledge useful discussions with Prof. B. I. Shklovskii. This work was carried out in part through the use of MITs Microsystems Technology Laboratories. We are grateful to Mark Mondol and RLE SEBL facility for help with electronbeam lithography. This work was supported in part by the U.S. Army Research Laboratory and the U.S. Army Research Office through the Institute for Soldier Nanotechnologies, under contract number W911NF-13-D-0001 (synthesis and nanopatterning of colloidal QDs), by the Department of Energy under award number DE-FG02-08ER46515 (charge sensing on QD arrays, including MOSFET sensor fabrication) and by Samsung SAIT. N.R. acknowledges support from the Schlumberger Foundation through the Faculty for the Future Fellowship Program. D.D.W.G. acknowledges support from the Fannie and John Hertz Foundation Fellowship.
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REFERENCES
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