DOI: 10.1002/smll.200600334 ... · Robert J. Barsotti, Jr., Michael D. Vahey, Ryan Wartena, Yet-Ming Chiang, Joel Voldman, and Francesco Stellacci* T he directed assembly ofnanoparticles

Nanoparticle assembly

DOI: 10.1002/smll.200600334

Assembly of Metal Nanoparticles into NanogapsRobert J. Barsotti, Jr., Michael D. Vahey, Ryan Wartena, Yet-Ming Chiang,Joel Voldman, and Francesco Stellacci*

The directed assembly of nanoparticles and nanoscale materials ontospecific locations of a surface is one of the major challenges in nanotech-nology. Here we present a simple and scalable method and model for theassembly of nanoparticles in between electrical leads. Gold nanoparticles,20 nm in diameter, were assembled inside electrical gaps ranging from 15to 150 nm with the use of positive ac dielectrophoresis. In this method, analternating current is used to create a gradient of electrical field thatattracts particles in between the two leads used to create the potential.Assembly is achieved when dielectrophoretic forces exceed thermal andelectrostatic forces; the use of anchoring molecules, present in the gap,improves the final assembly stability. We demonstrate with both experi-ment and theory that nanoparticle assembly inside the gap is controlledby the applied voltage and the gap size. Experimental evidence andmodeling suggest that a gap-size-dependent threshold voltage must beovercome before particle assembly is realized. Assembly results as afunction of frequency and time are also presented. Assembly of fewerthan 10 isolated particles in a gap is demonstrated. Preliminary electricalcharacterization reveals that stable conductance of the assembledparticles can be achieved.

Keywords:· dielectrophoresis· nanoelectronics· nanoparticles· nanotechnology· self-assembly

1. Introduction

Dielectrophoresis (DEP) is the force felt by a polariza-ble object when it is placed in a nonuniform electricfield.[1–3] In vacuum, when a polarizable object is placed intoan electric field, polarization charge is induced at the sur-face of the object; if the applied electric field is nonuniform,it interacts with this polarization charge to attract the objecttowards the point of strongest field. This also happens in so-lution, when an object with a larger polarizability than thesolvent molecules interacts with a nonuniform field (positiveDEP). When objects of lower polarizability interact with afield, the polarization charge aligns such that the object ispushed away from the region of highest field (negativeDEP). DEP has been demonstrated to manipulate microme-ter-sized objects such as cells for over three decades.[4–9]

One of the major challenges for nanotechnology is themanufacturing of devices with easily scalable approaches.While there are many obstacles to be overcome in nanoma-

[*] R. J. Barsotti, Jr., Dr. R. Wartena, Prof. Y.-M. Chiang,Prof. F. StellacciDepartment of Materials Science and EngineeringMassachusetts Institute of Technology77 Massachusetts Avenue, 13-5043Cambridge, MA 02139 (US)Fax: (+1)617-324-2500E-mail: [email protected]

M. D. Vahey, Prof. J. VoldmanDepartment of Electrical Engineering and Computer ScienceResearch Laboratory of ElectronicsMassachusetts Institute of Technology77 Massachusetts Avenue, 36-824Cambridge, MA 02139 (US)

Supporting information for this article is available on the WWWunder http://www.small-journal.com or from the author.

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full papers F. Stellacci et al.

nufacturing, here we focus on the placement of nanoscalematerials in specific locations on a patterned substrate. DEPoffers the unique opportunity of selectively placing nano-ob-jects inside electrically defined gaps with several potentialadvantages, mostly versatility and quick assemblytime.[2,3, 10–34] One could envision DEP as a technologicallyscaleable approach to assemble many nanoscale objects inparallel.[26] The forces and experimental variables affectingDEP-based assemblies have been well-studied in the case ofassembling nanoparticles into microwires.[12,13] However,much less is known regarding the manipulation of nanoma-terials into nanoscale gaps. The technological understandingof DEP on the nanoscale could lead to the scalable integra-tion of isolated nano-objects into electronic devices. Elec-tronic devices with assembled nano-objects could havemajor application in low-power logics, highly parallel bio-logical and chemical sensing, and possibly distributedenergy systems.

Here we present a combined theoretical and experimen-tal study to investigate DEP assembly of metal nanoparti-cles inside gaps of widths comparable to or slightly largerthan (1–7.55) the particle size. We investigate experimen-tally the key factors in the controlled assembly of nanoparti-cles, that is, ac voltage, ac frequency, time, nanoparticle con-centration, and gap width. The process is accurately mod-eled with consideration of thermal motion and electrostaticinteractions. Novel aspects of this work include: a) experi-mental evidence of the existence of a gap-size-dependentthreshold voltage in the DEP assembly of nanoparticles intonanogaps; b) theoretical modeling of the assembly of nano-particles into nanogaps with consideration of DEP forces,thermal motion and electrostatic effects; c) a quantitativecomparison of theoretical predictions of the number of par-ticles assembled in a gap with experimental data as a func-tion of voltage and gap size with a qualitative discussion ofresults with consideration of the dominant forces in the as-sembly.

In this paper, we first describe the fabrication of wireswith nanogaps formed by localized melting (a scalable tech-nique). We then demonstrate the controlled assembly of20-nm-diameter gold nanoparticles into these nanogaps.Preliminary electrical characterization of devices is shown.Finally, a theoretical model is used to predict the number ofnanoparticles assembled from solution using DEP for agiven set of experimental parameters. The dependence ofparticle assembly on ac voltage, gap size, frequency, andtime is discussed with the important consideration of elec-trostatic interactions between the particle and the substrate.

2. Results

2.1. Gap Formation

Gaps used in this work were formed either by direct pat-terning with electron-beam lithography (EBL) or by a local-ized melting technique. Gaps formed by direct patterninghad widths of 30 to 150 nm (controlled via the electron doselevel during EBL) and the wires were 300 nm wide with

almost hemispherical ends. Gaps were also formed using alocalized melting technique introduced by Richter et al.[35]

Gold wires that overlap in a �100-nm-wide by 30-nm-longregion (Figure 1) were subjected to a voltage sweep overtime. As the voltage increases, greater current densities passthrough the junction of the two wires, causing large increas-es in temperature. Once the melting point of gold isreached, a sharp decrease in current level is observed, indi-cating gap formation. The gap resistance after the breakusually exceeds 351012 W when measured at 300 mV. Scan-ning electron microscopy (SEM) imaging revealed gaps withwidths ranging from 10–70 nm. A similar strategy was em-ployed using an alternating voltage at 1 MHz, gradually in-creasing the ac voltage until wire failure occurred. It is theauthorsA opinion that localized melting for fabricating nano-gaps is simple, requiring no complicated lithographic techni-ques, and due to its reliance on only electrical current couldbe technologically valuable.

Gold particles,[36] stabilized by a proprietary coating thathas an overall negative surface charge (probably similar tocitrate molecules), were chosen for this study partly becauseof their unique optical and electronic properties, but mainlybecause of their solubility in water, monodispersity, andcommercial availability.

Prior to DEP assembly of nanoparticles, the substrateswere immersed in a toluene solution of hexanethiol andnonanedithiol. Both molecules bind (through their thiolgroup) to the metallic leads. The dithiol molecules chemical-ly bind mostly orthogonally to the surface with only onethiol group.[37] The second thiol group stays on the surfaceon the self-assembled monolayer (SAM) ready to bind tothe nanoparticles once they are brought into the gap by theDEP force.[17–19,23,34] The SAM, therefore, serves as a molec-ular glue, binding the nanoparticles to the wires. Control ex-periments where the dithiol molecules were not used result-ed in inconsistent nanoparticle assembly.

The experimental set-up used for dielectrophorectic as-sembly is shown in Scheme 1 with details provided in theExperimental Section. Briefly, a 0.1–0.25 mL drop of aque-ous nanoparticle solution was placed over the sample, cover-ing all devices on the chip. An alternating voltage of a givenmagnitude and frequency was applied for a set amount oftime. An oscilloscope was used to monitor the changes inconductivity of the nanoparticle bridge when testing for theconditions of fused versus isolated particle assembly (seebelow). After these conditions had been established, experi-ments were performed without the oscilloscope; the secondelectrode was connected directly to ground.

2.2. Assembly Versus Voltage and Concentration—FusedVersus Isolated Particle Assembly

The assembly of nanoparticles into nanogaps has beenshown for isolated individual particles[15,17–19,23,34] and forfused metallic particles[20,21] using similar conditions. Usingour experimental set-up it was possible to achieve bothtypes of assembly (Figure 2) and to detect the creation of afused metallic wire bridging the gap in real time. When a

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Directed Assembly of Nanoparticles

gap is present, the circuit is open and no voltage signal isseen on the oscilloscope. When a metallic bridge forms, ahighly conductive pathway forms across the gap, and the ap-plied peak-to-peak voltage (Vpp) waveform is seen on theoscilloscope. Fusing of particles occurred anywhere from 2–60 seconds after the voltage was applied. The post-assemblydc characteristics of the fused nanoparticle bridges showlinear IV curves and low resistances (150–600W) similar tothose reported previously.[20,21]

The key controlling factors differentiating the two typesof assembly are found to be the applied peak-to-peak volt-age and the nanoparticle concentration in solution. At ananoparticle concentration of 751011 particlesmL�1, 15 outof the 17 (15/17) gaps (�88%) assembled at voltages of3 Vpp and higher were found to be filled with fused particles.When Vpp was below 3 V, fused wire assembly was not ob-served (0/6) and isolated particles could be seen in subse-quent SEM images. At a more dilute nanoparticle concen-tration (751010 particlesmL�1), the voltage required forfusing increases. Voltages larger than 4.5 Vpp were neededto observe fused particles in the gap (�69%, 9/13) while atvoltages equal to or lower than 4 Vpp almost all gaps (28/29)showed isolated particle assembly. As our goal was to studythe assembly of isolated particles, we performed experi-ments with these latter conditions (with nanoparticle con-centrations of 751010 particlesmL�1 and lower voltages).

2.2.1. Assembly Versus Voltage and Gap Width

To investigate assembly as a function of voltage and gapwidth, assemblies were carried out for 120 s at a given alter-nating Vpp and a frequency of 1 MHz. After the assembly ofisolated particles, the samples were imaged by SEM and thenumber of particles captured in each gap was counted. A400-nm-diameter circle centered on the middle of the gapwas drawn and only nanoparticles within this circle wereconsidered as captured. SEM images were also used to mea-sure the gap width. Gap widths were measured at their nar-rowest points.

Figure 3 demonstrates the effect of voltage and gapwidth on nanoparticle assembly. A threshold voltage mustbe reached before any nanoparticle assembly is realized, asseen in previous work.[14,24,25] As the gap width is increased,the threshold voltage required for nanoparticle assembly in-

Figure 1. Top: Scanning electron microscopy (SEM) images of a goldwire patterned by electron-beam lithography with a predefined weakjunction point. Bottom: SEM image of such a wire after localizedmelting to form a �11-nm gap. Scale bars=100 nm.

Scheme 1. Schematic image of the experimental set-up used for thedielectrophoresis of nanoparticles.

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creases. At an average gap width of 30 nm (widths rangingfrom 15 to 50 nm) and voltages below 2 Vpp, no assemblywas observed. For slightly larger gaps (85 nmaverage ; widthsfrom 50–110 nm) assembly was not observed until 3 Vpp, andfor the largest gaps tested (125 nmaverage, widths from 110–150 nm), assembly was not observed until approximately4 Vpp on eight out of nine gaps tested. The use of largergaps was also observed to increase the voltage necessary forfusing. Isolated nanoparticle assembly was achieved on gaps>70 nm at 4.5 Vpp and on gaps >105 nm at voltages of 5and 6 Vpp. The number of assembled particles captured inthe gap increased with increasing voltage for all gap widthswhen voltages larger than the threshold voltage were ap-plied.

2.2.2. Assembly Versus Time

To achieve the assembly of a very few (<10) isolatedparticles in the gap, it was necessary to limit the assemblytime. Operating at a frequency of 1 MHz and at voltagesnear the threshold for a given gap size resulted in the cap-ture of a few nanoparticles. For small electrode gaps, �15–40 nm, 2 Vpp assemblies were performed for 5, 10, 30, 60,and 120 s, while on larger electrode gaps, �55–90 nm, 3 Vpp

assemblies were performed for similar durations. As illus-trated in Figure 4, the shortest assembly times resulted inthe assembly of fewer than 10 particles in the gap. Increasedassembly time resulted in more nanoparticles in the gap forall gap widths at voltages above the threshold value.

2.2.3. Assembly Versus Frequency

To study the effect of frequency on nanoparticle assem-bly, peak-to-peak voltages near the threshold (2.5–4 Vpp)were applied. Assemblies were carried out for 120 s at fre-quencies of 10 kHz, 100 kHz, and 1 MHz. Figure 5 shows astrong dependence of nanoparticle assembly on frequen-cy.[12, 14,24,25] At lower frequencies more nanoparticle assem-bly is seen and fusing occurs more readily. As frequency isincreased, fewer nanoparticles assemble and the particlesremain isolated.

2.3. Post-Assembly Electrical Characterization of Nanoparti-cle Bridges

The assembly of nanoparticles in a gap was character-ized in the dry state at room temperature with dc I–V mea-ACHTUNGTRENNUNGsurements between 0 and 300 mV prior to SEM imaging(Figure 6). Isolated particles generally showed a high resist-ance (2.5 MW to 60 GW).[17,18] Using higher applied voltagesled to more nanoparticle assembly and lower-resistance de-vices. These devices were measured in the dry state 16 hafter assembly. The current levels of the devices were stableup to 20 electrical sweeps.

2.4. Modeling

2.4.1. Nanoparticle Transport

In order to predict how experimental variables wouldaffect the assembly of nanoparticles into nanogaps, we havemodeled the forces that govern the DEP assembly of parti-cles. Specifically we have modeled the dielectrophorecticforce, the particlesA thermal motion, and the electrostatic re-pulsion between the substrate and the particles. The prob-lem was solved by determining the regions of space overwhich each force becomes dominant.

The small size of the nanoparticles and the associatedsignificance of thermal fluctuations precludes the use of de-terministic models[5, 38] for particle assembly. Instead, weadopt a continuum model formulated in terms of thenumber density of nanoparticles, c ACHTUNGTRENNUNG(r,t), where r is the dis-

Figure 2. SEM images of the two modes of nanoparticle assembly at1 MHz and a nanoparticle concentration of 7'1011 particlesmL�1.Top: Assembly of isolated particles at 2 Vpp for 60 s. Bottom: Assem-bly of a fused nanoparticle bridge occurs at 4 Vpp in 3 s. Assemblyappears to be out of the surface plane. This type of assembly wasdescribed by Velev for assemblies on much larger gaps.[12] Scalebars=100 nm.



tance from the origin, takento be on the substrate in thecenter of the electrode gap.The quantity c ACHTUNGTRENNUNG(r=r0, t) multi-plied by the incrementalvolume dV may be thoughtof as giving the expectednumber of nanoparticles in avolume dV, centered aroundr= r0, at time t. Because thenanoparticle solution isdilute and the length scaleswe are interested in aresmall, the relevant volumeelement dV will generallycontain exceptionally fewnanoparticles; at the concen-trations adopted, a cubemeasuring a few micrometersper side would be necessaryto encompass a single parti-cle with reasonably highprobability. The consequenceof modeling a discrete systemusing smooth functions ofspace and time is that themodel will not capture thestatistical fluctuations inher-ent in the assembly process.Since we are primarily inter-ested in how the number ofassembled particles dependson different operating param-eters, a relatively simple wayto reintroduce statistical fluc-tuations into our model is totreat assembly as a series ofevents following a Poissondistribution. Accordingly,after we have determined thenumber of particles collectedat time t,n(t), we can approxi-mate the standard deviationas [n(t)]1/2 Inherent in anyprocess with a distribution ofthis form is that the variabili-ty in the number of particlesassembled becomes larger (ina fractional sense) when oneis interested in smaller num-bers of particles, since n1/2/ndiverges as n!0. As a result,we must emphasize that themodel predicts assembly onlyin a statistical sense, and isexpected to be more accuratein predicting larger numbersof assembled particles.

Figure 3. a) SEM images of assembly as a function of voltage and gap width after assembly for 120 s at1 MHz and a nanoparticle concentration of 7'1010 particlesmL�1. Top row: Small gaps at Vpp=1.5 V(left), 2 V (center), and 3 V (right). Middle row: Medium gaps at Vpp=2 V (left), 3 V (center), and 4 V(right). Bottom row: Large gaps at Vpp=2 V (left), 3 V (center), and 4 V (right). Scale bars=200 nm.b) Plot of experimental data and theoretical model for nanoparticle assembly using DEP as a function ofvoltage and gap width. Data is plotted as a function of voltage for three different gap-width ranges: 15–50 nm, average=30 nm; 50–110 nm, average=85 nm; 110–150 nm, average=125 nm. The theoreti-cally predicted number of particles is shown at the average gap widths.



Within this continuum framework, we describe nanopar-ticle transport in terms of the conservation equation:

@c@t

þr � J ¼ 0 ð1Þ

where the nanoparticle flux is represented by J and is givenby:

J ¼ �Drc þ cFtotal

6pRmð2Þ

Here, D denotes the nanoparticle diffusivity, Ftotal denotesthe net (deterministic) force acting on a nanoparticle, and cis the concentration of nanoparticles. We have taken themobility as the coefficient of drag on a sphere from StokesAlaw, where m denotes viscosity and R denotes the radius ofthe nanoparticle.

To specify the equation governing c ACHTUNGTRENNUNG(r,t), we define thetotal force as the sum of the contributions of an electropho-retic (EP), dielectrophoretic (DEP), and drag term:

Ftotal ¼ FEF þ FDEF þ FDrag ð3Þ

To complete the problem formulation, we constrain thenanoparticle flux normal to any phase boundary, S, (e.g., thesubstrate, electrodes, or droplet) to vanish:

n � Jjs¼ 0 ! n � cFtotalð Þjs¼ 6pRmD n � rcð Þ ð4Þ

Given a description of the total force, it is possible to solvefor the number density of suspended nanoparticles. Ratherthan proceeding to solve this partial differential equation di-rectly, however, we pursue a simpler framework for describ-

ing nanoparticle assembly.Specifically, we rely upon thedisparate magnitudes andlength scales over which eachcontribution of the totalforce acts to reduce the prob-lem to a much simpler form.

2.4.2. Forces and Scaling

Dielectrophoresis refersto the force on a polarizableobject in the presence ofnonuniform electric fields. Inthe special case where theelectric field across a particledeviates only slightly from itsvalue at the center of thatparticle, a dipole approxima-tion of the DEP force is ap-propriate. For a sphericalparticle of radius R, in mediaof permittivity em, actedupon by applied field E0, thedipole component of theDEP force is given by:

F ð1ÞDEP ¼ 2pemR3Re CMðwÞrE2

0ðr;wÞ� �

ð5Þ

Here, CM denotes the Clausius–Mossotti factor, in this casegiven by:

CMðwÞ ¼ep � emep þ 2em

ð6Þ

where ep and em are the complex permittivities of the parti-cle and media, respectively. For a particle with permittivitye and conductivity s, the complex permittivity at frequencyw is of the form e=e + s/jw. For the special case of a per-fectly conducting particle (sp!1), the CM factor reducesto unity, the value used in our model. In addition to thedipole contribution to the DEP force (Eq. (5)), we take intoconsideration the effects of higher-order polarization mo-ments and forces.[39] These higher-order corrections to theDEP force are of particular importance when the field ischanging rapidly as is the case close to the electrode gap.

Our objective is to calculate the DEP force associatedwith idealized gap geometries. Even in the case of rectangu-lar electrodes, determining the spatial dependence of theDEP force produced requires numerical analysis. Some pre-liminary insight can be obtained, however, by consideringthe asymptotic scaling of the electric field at a few differentdistances: those larger than the gap width but smaller thanthe electrode width, and those much greater than the elec-trode width. At length scales larger than the gap width (15–150 nm) but smaller than the electrode width (�300 nm),the electrodes can be approximated by a long, narrow slit,for which the field scales as r�1 and the DEP force as r�3. Atdistances larger than the electrode width, the electrodes

Figure 4. SEM images of assembly as a function of time. The top row shows assembly on small (15–40 nm) gaps at 2 Vpp, 1 MHz, and 7'10

10 particlesmL�1 for 5 seconds (left), 30 seconds (center), and120 seconds (right). The bottom row shows that assembly of only a few particles can also be achievedwith the use of medium (55–90 nm) gaps at 3 Vpp, 7'10

10 particlesmL�1 for short times: 5 seconds(left), 30 seconds, (center), and 120 seconds (right). Scale bars=200 nm.



produce a dipole field, scaling as r�3, with an associatedDEP force scaling as r�7. Since we are primarily interestedin distances on the order of hundreds of nanometers, weexpect the DEP force to decay slightly faster than r�3. Thisprediction is supported by our numerical models, which sug-

gest a scaling of � r�4 up to distances of a few hundrednanometers.

In contrast to DEP, the electrophoretic (EP) force isproportional to the electric field, and is given by:

FEP ¼ mEPCDE ð7Þ

where mEP denotes the electrophoretic mobility, and CD de-notes the drag coefficient. Choosing CD according to StokesALaw gives the correct expression for the electrophoretic ve-locity in Equation (2). Because the EP force depends uponthe polarity of the field, when the gap is driven with acfields of sufficiently high frequency, the average contribu-tion of this force is negligible compared to that of DEP.However, when the length scales of interest are of the orderof 10 nm, our calculations show oscillation of the particlesby the EP force may be present at frequencies as high as1 MHz. This particle oscillation could have a range of ef-fects, including inducing fluid flow in the surroundingmedium, which could alter the assembly process.

In addition to the oscillatory EP force associated withthe applied field, the negatively charged nanoparticles willbe electrostatically repulsed by the negatively charged sub-strate when they are within a few Debye lengths. Assuminga very low to zero salt concentration in the aqueous dilutenanoparticle solution, we used values from literature at ourmeasured pH value of 8 for the zeta potentials of the nano-particles (�40 mV),[40] silicon dioxide substrate(�105 mV),[41] and gold elecACHTUNGTRENNUNGtrodes coated primarily with amethyl terminated SAM (�80 mV).[42] Measurements of theelectrical conductivity of the nanoparticle suspension sug-gest a Debye length of �10 nm. Accordingly, we expectelectrostatic repulsion to play a significant role for particleswithin a few tens of nanometers of the substrate. This is incontrast to DEP, which is generally significant for distancesof a few hundreds of nanometers. Although this force willgenerally be more localized than DEP, it will nonethelesshinder or block nanoparticle assembly in cases where theapplied voltage is not sufficiently high for it to be overcome.

In addition to EP and DEP, the nanoparticles may alsobe acted upon by convection of the surrounding fluid. Thisconvection could arise from numerous sources, includingthermally induced flows (from localized changes in thefluidAs density and/or electrical properties), Marangoniflows, or induced-charge electro-osmosis (ICEO).[43, 44] Be-cause we expect thermal gradients in our system to besmall, we suspect that ICEO is the most significant sourceof induced convection. Briefly, as the polarity of the electro-des changes under the applied voltage, counterions from thesurrounding fluid form a double layer at the electrode sur-face, resulting in a net space charge density. The appliedelectric field then acts on this charge, pushing fluid awayfrom the gap along the axis of the electrodes. The fluid ex-pelled from the gap is balanced by a net inward flow fromabove and to the sides, which could potentially entrainnanoparticles, carrying them closer to the gap for assembly.The significance of ICEO depends on the extent of the elec-trode screening, and would thus be expected to be greatestat lower frequencies, provided the screening is not complete.

Figure 5. SEM images of assembly as a function of frequency. Assem-bly increases drastically with decreasing frequency. Assembly condi-tions were 120 s, 3.5 Vpp, �100-nm gap size, 7'1010 particlesmL�1,1 MHz (top), 100 kHz (center), 10 kHz (bottom). Scale bars (top andcenter)=200 nm. Scale bar (bottom)=500 nm.



We must also note that, although we expect electrodescreening to be greatly diminished at frequencies of 1 MHzor greater, the fact that the Debye screening length is com-parable to the size of the gap presents the possibility thatICEO persists even to rather high frequencies, greatly in-creasing the complexity of the problem. To proceed, wemodel the fluid as static and restrict our analysis to frequen-cies (1 MHz) where ICEO is not expected to be dominant.

2.4.3. Determining the Region of Influence

The primary forces acting on a nanoparticle are thus theelectrostatic repulsion associated with the fixed surface

charge on the substrate andelectrodes, the DEP force,and thermal motion, which ismanifested by the diffusiveterm in Equation (2). Addi-tionally, these forces exhibitvery different spatial depend-encies. While the electrostat-ic force is typically confinedto within �10 nm of the sub-strate, the DEP force extends�100 nm from the electrodegap, and thermal motion isindependent of location rela-tive to the gap. It followsthat at sufficiently high vol-tages, there will exist aregion surrounding the gapin which DEP overwhelmsboth thermal motion and therepulsion of the substrate.Any particle entering thisregion will experience nobarrier to its assembly in thegap. We refer to this as theregion of influence (ROI).An illustration of the regionswhere different forces domi-nate assembly is given in

Figure 7. At lower voltages, we find that DEP still over-whelms thermal motion within a particular region, but is nolonger sufficient to overcome the electrostatic repulsion.For this case, nanoparticles may be held in a stable equilib-rium above the electrodes without contacting any surface;accordingly, no particles are assembled.

To determine the size and shape of the ROI, we use thenumerical results from the force calculations to determinethe trajectory of particles initialized at various locationswith respect to the origin, taken as the center of the elec-trode gap. To conservatively approximate the effects of ther-mal motion, the net force is calculated by the sum of theelectrostatic repulsion, DEP, and an outward radial force

Figure 6. SEM images and corresponding I–V characteristics of devices assembled using DEP. Stable cur-rent levels are seen through 20 electrical sweeps. Scale bars=200 nm.

Figure 7. Schematic image showing regions where different forces dominate during DEP assembly of nanoparticles into nanogaps at voltageslower (left) and higher (right) than the threshold voltage.



with a magnitude given by the approximate rms value ofthermal motion on the nanoparticles. If a particle contactsthe electrode or substrate surface, its initial position isadded to the locus of points defining the ROI. Note that incases where the electrostatic repulsion prevents the particlesfrom contacting the surface, the ROI will not exist; this isdespite the fact that DEP may overwhelm thermal motionover a large region.

2.4.4. Simplifying the Equations for Nanoparticle Assembly

Once a description of the ROI is obtained, we may usethe concept to simplify the problem defined by Equa-tions (1–4). We do this by recognizing that we have con-structed the ROI such that particle transport inside thisregion is determined predominantly from the migrationterm in Equation (2), whereas outside of it, transport is pri-marily diffusive. Because the dynamics inside the ROI areessentially deterministic, particles that enter this region bydiffusion are quickly and irreversibly assembled in the gap,or otherwise held in equilibrium against the electrostatic re-pulsion. In either case, the probability of finding a freelysuspended particle in the ROI can be approximated to zero;for a finite particle flux, as defined in Equation (2), largeforces constrain the free-particle density to be small. Assem-bly is thus limited primarily by the rate at which particlesdiffuse into the ROI, independent of the details of theirtransport once inside this region.

This presents a very useful simplification; once the sizeof the ROI has been determined, we need only solve a dif-fusion equation in the external region, subject to the bound-ACHTUNGTRENNUNGary condition that the density of free particles in the ROI isuniformly zero. Furthermore, although the ROI has a fairlycomplex shape, we may, to leading order, approximate it asa hemisphere. This approximation can be thought of astaking the first term in a series expansion of the actualshape of the surface, and simplifies the analysis considera-bly, by constraining the nanoparticle probability density tohave spherical symmetry. We estimate the upper bound forthe error associated with this approximation by calculatingthe standard deviation of the points on the actual ROI fromthe mean: typically about 30%. By modeling the ROI ashemispherical with radius a, we obtain:

@c@t

¼ Dr2

@

@rr2@c@r

� �

c r ¼ a; tð Þ ¼ 0; cðr ! 1; tÞ ¼ c0

ð8Þ

Here, c0 denotes the original, bulk concentration of nano-particles in the droplet, in units of particles per m3. In forc-ing c to approach c0 at large r, we are assuming that the de-pletion of particles does not penetrate a distance compara-ble to the size of the droplet for all times of interest. Thisproblem permits an analytic solution:

cðr; tÞ ¼ c0 1� arerfc

r � a

2ffiffiffiffiffiffiDt

p� ��

ð9Þ

To calculate the expected number of particles in the gap, werecognize that the time rate of change of the assembled par-ticles is equal to the net flux of particles into the ROI. Sincewe have modeled the problem as purely diffusive outside ofthe ROI, the migration term in Equation (2) vanishes fromthe calculation, and the surface integral of the inward fluxover the ROI becomes:

@n@t

¼ �Zr¼a

ðn � JÞdS ¼ c0

Zr¼a

Dðn � rcÞdS ¼2pc0D r2@c@r

� �r¼a

ð10Þ

Evaluating and integrating with respect to time, we obtain:

nðtÞ ¼ 2pc0aDt 1þ 2affiffiffiffiffiffiffiffiffipDt

p þ a2

3Dt

� �ð11Þ

As an initial condition, we have used that, at t=0+ (im-mediately after the electrodes have been turned on), all ofthe particles initially inside the ROI are instantaneously as-sembled, so that nACHTUNGTRENNUNG(t=0+) is given by the bulk concentrationof nanoparticles (c0) times the volume of the ROI. This ini-tial condition is represented by the third term, proportionalto a3, and has a value of �10�3 particles for typical condi-tions in our experiments. Accordingly, it may be ignored forall practical purposes. Additionally, at sufficiently long times(i.e., t@a2/D), the second term, proportional to a2, may alsobe neglected to an excellent approximation. The expressionfor n(t) then reduces to a linear relationship with time, anddepends on voltage and electrode geometry through the sizeof the ROI, a.

At this point, we note that since this derivation has ne-glected the effects of assembled particles on the electricfield and DEP force, it approximates the assembly dynamicsonly when sufficiently few particles have accumulated in theelectrode gap to not significantly effect the electric fields, sothat the parameter a remains constant. In reality, particlesthat assemble in the gap will be polarized by the appliedelectric field. For highly conductive particles, the field in-duced by this polarization will tend to oppose the appliedfield, resulting in a decrease in the DEP force and thus thesize of the region of influence. We may introduce a first cor-rection to Equation (11) by including the approximate de-pendence of a on the number of assembled particles, n.With this modified description of a, we find the first-ordercorrection for n(t) to be given by:

nðtÞ ¼ 2pc0a0Dt1� 2pc0DtðDa=DnÞ ð12Þ

This has the advantage of remaining finite as t!1. Thenew parameter, Da/Dn, is determined numerically, and typi-cally has a value of approximately �6510�10 m. Completedetails of the methods used to model nanoparticle assemblyand calculate the first-order correction are given in the sup-porting information



3. Discussion

3.1 Assembly as a Function of Voltage and Concentration—Fused Versus Isolated Particles

The observation of fusing as a function of voltage andnanoparticle concentration suggests that nanoparticle melt-ing is due to Joule heating. Once a critical number of nano-particles is reached, high currents are able to pass throughthe assembly. The high current passing through these parti-cles causes localized melting and the formation of conduc-tive metallic bridges. At lower Vpp, the current is not largeenough to cause melting. At lower nanoparticle concentra-tions, a larger DEP force (higher peak-to-peak voltages) isneeded to assemble the number of nanoparticles necessaryfor the current density required for melting.

3.1.2. Assembly as a Function of Voltage and Gap Width

Figure 3 shows experimental data points for the numberof particles assembled as a function of Vpp and gap widthalong with the predictions of the model at the average gapwidth of each experimental range (30 nm, 85 nm, and125 nm). Given the many differences between the set-up as-sumed in the model (idealized gap geometries as comparedwith the irregular gap shapes used in the experiment, no in-terparticle interactions, induced flows, or electrochemical re-actions) and that used in experiments, good agreement isachieved mostly in predicting the threshold voltages re-quired for assembly.

The appearance of a voltage threshold for particle as-sembly in both the experimental and theoretical work showsthe significance of electrostatic repulsion between the parti-cles and the electrode/substrate surfaces. In the absence ofany electrostatic force, nanoparticle assembly would occurdown to very low voltages, since the DEP force only needsto overcome the random thermal motion of the particles.Also, in the absence of electrostatic forces our model pre-dicts that the number of particles assembled should be inde-pendent of gap size at this length scale (see discussionbelow). The dependence of this voltage threshold on thesize of the gap supports the idea that the DEP force mustreach a critical value to overcome the electrostatic repulsionbetween charged particles and surfaces.

For voltages exceeding the threshold, the number of par-ticles assembled appears to be largely independent of thegap size. In terms of the model we propose, this representsthe relatively weak dependence of the size of the ROI onthe size of the gap; while smaller gaps produce a stronger,more confined DEP force for a given voltage, larger gapsproduce a weaker DEP force which acts over a larger area.Simulations show that this independence of gap size, afterthe threshold voltage is reached, is approximately true forgaps up to 150 nm, and is corroborated by experimental ob-servation.

The model slightly over-predicts the observed thresholdfor the smallest gap sizes. Gap sizes were measured at theirnarrowest width. In many cases, this smallest gap size is notthe average gap size. The model assumes a uniform gap at

the smallest size. This would overestimate the electrostaticrepulsion felt by the particles from the electrode surface ascompared with the experimental reality, causing the shiftseen in the threshold. This effect would be expected to begreatest for the narrowest gaps, as small changes in gapwidth result in large changes in the electrostatic repulsiveforce.

The model also seems to slightly under predict thenumber of assembled particles at 4.5 Vpp and above. Athigher voltages, irreversible water dissociation may becomea factor, leading to the generation of protons, which weak-ens the electrostatic repulsion term.[45]

3.1.3. Assembly as a Function of Time

Previous work has used a resistor in series with the gapto make DEP assembly self-limiting.[15, 17,18,24] In these works,the assembled particles were significantly larger than thegaps onto which they were assembled, allowing for a singleparticle to bridge the gap. This creates a low-resistancebridge. Preliminary experiments using a 1 kW to 1 MW resis-tor in series with our gaps have shown no self-limiting be-havior. This is likely due to the ratio of gap size to nanopar-ticle radius in our system. Our gaps are larger or the samesize as the diameter of our particle, making it difficult for asingle particle to bridge the gap. The nanoparticle bridgeswe formed have higher resistances (2.5 MW to 60 GW) thanthe resistors we placed in series with the gap. Future experi-ments will use smaller gap sizes to enable single-particle as-sembly.

To achieve assembly of a small number of particles wedecided to operate near the threshold voltage. As seen inFigure 3, there is a rather steep increase in the number ofparticles assembled when operating for 120 s, hence it wasnecessary to limit the assembly time to achieve the assemblyof <10 nanoparticles.

3.1.4. Assembly as a Function of Frequency

At low frequencies, assembly was found to drastically in-crease. A counterion cloud surrounds the particles, which atlow frequencies can respond to changes in the electricalfield.[12,14] This counterion cloud increases the effectiveradius of the particle, thereby increasing the DEP force,which scales as r3.

Another potential factor contributing to the frequencydependence is the existence of ICEO flows. At low frequen-cies, where greater amounts of charge accumulate in thedouble layer around the electrodes, ICEO flows would beexpected to play a larger role. These flows potentiallychange the system dynamics and increase nanoparticle as-sembly.

4. Conclusions

We have modeled and demonstrated the effect of volt-age and gap size on the assembly of 20-nm-diameter goldnanoparticles into nanogaps. The model accurately describes



our experimental results explaining the observed thresholdvoltage required for nanoparticle assembly as a consequenceof the presence of repulsive electrostatic interactions be-tween nanoparticles and surfaces. The model introduces aspatial region of influence (ROI) where the DEP forceovercomes Brownian motion. The ROI is independent ofgap size when considering only DEP forces and Brownianmotion; the gap-size dependence of particle assembly is dueto the electrostatic interactions. Assembly was observed toincrease with increased assembly time and to increase withdecreasing assembly frequency. The fabrication of smallergaps will allow for the assembly of single particles. Futureimprovements to the theoretical study will include consider-ation of induced flows, interparticle repulsions and electro-chemical reactions. Future experimental studies on con-ACHTUNGTRENNUNGtrolled assembly will analyze parameters such as tempera-ture, electrode geometry, and particle functionalization.

5. Experimental Section

Si wafers with 700 nm of thermally grown SiO2 were used asthe underlying substrate for the devices. Gold wires were pat-terned by EBL. PMMA (2%) in Anisole was spin coated onto theSiO2 substrates at a speed of 1000 rotations per minute for90 s. The resist was then baked at 135 8C for 90 min. EBL wasperformed on a IBM VS-26 with a LaB6 source at 50 KV, atdoses ranging from 400–1100 mCcm�2. Development was thencarried out in a 2:1 solution of Ispopropyl Alcohol (IPA): MethylIsobutyl Ketone (MIBK) for 80 s, followed by immersion in an IPAbath. An adhesion layer of 4–7 nm of Cr or Ti followed by 15–25 nm of Au was deposited using electron-beam evaporation ata rate of �0.1 nms�1. Lift-off was performed by immersing thesubstrate in N-methylpyrrolidone at 120 8C for 30 min followedby low power sonication for 30–60 s. Optical lithography wasused to define the large optical pads and to align them to thepatterned gold wires. An adhesion layer of 10 nm of Cr followedby 500 nm of Au was deposited to form the pads using electron-beam evaporation followed by lift-off, as described above. Thedevices were imaged by SEM on a Raith 150 EBL tool. The goldelectrodes were immersed into a 95:5 10 mMol toluene solutionof Hexane Thiol:Nonanedithiol overnight, followed by washing intoluene for 2 h.

Gold wires with overlapping leads were used to form narrowgaps via a localized melting technique, described by Richteret al.[35] Using a microprobe station (Cascade Microtech Model11000) the voltage was swept across the wire from 0 to 2 V inincrements of 2.5 mV until a drastic decrease in current levelwas seen, which is indicative of gap formation. The resistance ofwires after the break was then measured using an Agilent 4155Csemiconductor parameter analyzer.

Gold particles, 20 nm in diameter, were purchased from Brit-ish Bio Cell International and used at the as-purchased concen-tration (7'1011 particlesmL�1) or diluted with deionized water toa concentration of 7'1010 particlesmL�1.

Scheme 1 shows the experimental set-up used for dielectro-phoresis of nanoparticles. A function generator (Agilent 33250A)

was connected to one side of the device through the microprobestation. The second side of the device was connected to ground.Nanoparticles were introduced to the system in one of two meth-ods. The sample was placed on the probe station and a �0.1–0.25 mL drop of aqueous nanoparticle solution was placed onthe sample, completely covering all devices (this method wasused for 90% of the results presented here). Alternatively, thesample was placed in a small polyethylene tray (2 cm by 2 cmwith a 5-mm lip) which was then filled with 2.5 mL of aqueousnanoparticle solution. This method was found to give compara-ble results to the drop method. After assembly onto several de-ACHTUNGTRENNUNGvices on the same substrate, the drop was absorbed with a Kim-Wipe, or the sample was removed from the tray and blown drywith compressed air. Resistance of the nanoparticle bridge wasmeasured at 300 mV. SEM imaging of the nanoparticle assem-blies was then performed.

Acknowledgements

The authors would like to acknowledge Mark Mondol, KurtBroderick, Tan Mau Wu, Jeffrey Kuna, and Andy Fan. This workmade use of MIT’s shared scanning electron-beam lithogra-phy facility in the Research Laboratory of Electronics (SEBL atRLE) and the shared facilities of the MRSEC Program of theNational Science Foundation under award number DMR 02-13282. Support from the Singapore-MIT Alliance and fromthe MARCO Interconnect Focus Center is gratefully acknowl-edged. R.B. is grateful to the Department of Defense NDSEGfellowship. M.V is grateful to the CSBi/Merck graduate fellow-ship. R.W.’s research was supported by an IC PostdoctoralFellowship.

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DOI: 10.1002/smll.200600334 ... · Robert J. Barsotti, Jr., Michael D. Vahey, Ryan Wartena, Yet-Ming Chiang, Joel Voldman, and Francesco Stellacci* T he directed assembly ofnanoparticles