Fundamental Insights into Nanowire Diameter Modulation and the

Dec 20, 2012 - Sema Ermez , Eric J. Jones , Samuel C. Crawford , and Silvija Gradečak .... Cunjing Lv , Chao Chen , Yin-Chuan Chuang , Fan-Gang Tseng...
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Letter pubs.acs.org/NanoLett

Fundamental Insights into Nanowire Diameter Modulation and the Liquid/Solid Interface Sam Crawford, Sung Keun Lim, and Silvija Gradečak* Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02143, United States S Supporting Information *

ABSTRACT: Controlled modulation of diameter along the axis of nanowires can enhance nanowire-based device functionality, but the potential for achieving such structures with arbitrary diameter ratios has not been explored. Here, we use a theoretical approach that considers changes in the volume, wetting angle, and three-dimensional morphology of seed particles during nanowire growth to understand and guide nanowire diameter modulation. We use our experimental results from diameter-modulated InN and GaN nanowires and extend our analysis to consider the potential and limitations for diameter modulation in other materials systems. We show that significant diameter modulations can be promoted for seed materials that enable substantial compositional and surface energy changes. Furthermore, we apply our model to provide insights into the morphology of the liquid/solid interface. Our approach can be used to understand and guide nanowire diameter modulation, as well as probe fundamental phenomena during nanowire growth. KEYWORDS: Diameter, modulation, nanowires, nitrides, interface, wetting

S

the need for such templates, the diameter of nanowires can instead be modulated by altering the supersaturation of the seed particle during growth, producing changes in the seed volume and wetting angle, consequently yielding changes in nanowire diameter.13−15 Unintentional changes in nanowire diameter have commonly been reported in literature, but we have recently demonstrated intentional and well-controlled modulations.14 However, the exact mechanism of the nanowire diameter modulation remains unclear; for example, diameter fluctuations can be attributed to changes in the seed particle volume,13 wetting angle,16 or a combination of both. Understanding of the origin and relative contribution of these effects would open up possibilities to optimize the synthesis of nanowires with arbitrary and well-controlled diameter ratios. Here, we explore the potential and limitations of nanowire diameter modulation using template-free particle-mediated growth by first focusing on diameter-modulation of InN and GaN nanowires, which we have experimentally described previously,14 as examples in our investigation. We then extend our analysis beyond III−V nitrides to demonstrate the applicability of our synthesis technique and modeling approach to other materials systems and outline strategies and considerations for achieving significant diameter changes. Furthermore, we apply our model in analyzing the morphology of the liquid/solid interface during nanowire growth,

emiconductor nanowires offer a number of distinct advantages over planar materials for the development of photonic and electronic devices.1−3 In particular, due to efficient strain relaxation,4 nanowire geometry offers the ability to create unique composition-modulated architectures, including axial and radial heterojunctions,5,6 with low dimensionality and superior crystallinity in lattice-mismatched structures. In addition to the compositional variations, modulation of diameter along the nanowire axis can offer additional advantages, including improved light trapping for solar cells7 and decoupling of phonon and electron transport for thermoelectric devices.8 For nanowires with radial quantum confinement, changes in nanowire diameter could be used to modulate the bandgap of the nanowire,9 either as an alternative to or in conjunction with compositional control over bandgap. Furthermore, diameter-modulated nanowires with short, thindiameter regions could be used to promote preferential fracture, which could be useful in applications such as transfer printing,10 separation of free-standing nanowire-templated films,11 or as a unique platform for studying the mechanical properties of nanowires. Particle-mediated nanowire growth is one of the most common nanowire synthesis methods and involves the use of a foreign metal particle to define the size and location of the resulting nanowire.12 Diameter-modulation of nanowires using particle-mediated growth can be achieved either with or without a template. When a template such as anodized alumina is used, diameter modulation is produced via axial variations of the width of the pores in which the nanowires grow.7 To avoid © 2012 American Chemical Society

Received: October 23, 2012 Revised: December 11, 2012 Published: December 20, 2012 226

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nanowire cross-sectional geometry. The values of d and V can be determined via transmission electron microscopy (TEM) and energy dispersive X-ray spectroscopy (EDS), allowing calculation of f(β) and therefore β. From eq 1, f(β) for cylindrical nanowires is defined as

demonstrating its ability for probing fundamental aspects of particle-mediated nanowire synthesis. Details of our experimental approach for producing diameter-modulated III−V nitride nanowires via template-free vapor−liquid−solid growth have been described previously.14 In short, controlled diameter modulations in Au-seeded GaN and InN nanowires were produced by varying the fluxes of III and V sources, respectively, during nanowire growth. Ex situ measurements of nanowire diameter and the composition of the III source in the seed particle indicated that the seed volume was greater in the thicker versus thinner nanowire segments. For these binary nanowires, an increase in the III flux leads to an increase in nanowire diameter due to an increased rate of incorporation of III source into the seed particle. On the other hand, an increase in the V flux leads to a decrease in nanowire diameter due to an increased rate of extraction of III source out of the seed particle. In this work, we develop a model to evaluate the relative contributions of changes in both the seed volume and wetting angle to observed changes in nanowire diameter, apply the model to the experimental results from our diametermodulated InN and GaN nanowires, and then extend the model to consider the potential and limitations for nanowire diameter modulation in other materials systems. We first consider cylindrical nanowires in which the relationship between the seed particle volume (V), wetting angle (β) between the vapor/liquid and liquid/solid surfaces at the triplephase boundary, and nanowire diameter (d) can be described as follows:17 ⎛ 3V ⎞ d = 2⎜ ⎟ ⎝ π ⎠

1/3

f (β ) =

⎛ V ⎞1/3 f (β2) d2 = ⎜ 2⎟ d1 ⎝ V1 ⎠ f (β1)

(4)

The volume of the seed particle depends on both the initial volume of the pristine metal seed particle and the amount of alloying precursor material incorporated during growth. In the case of III−V nitride nanowires synthetized using Au seed particles, nitrogen is insoluble in Au,18 so only the composition of the III element is considered when calculating the volume of the Au−III seed particles x III V = nAu ΩAu + nAu Ω III 1 − x III (5) Here, n is the number of atoms, Ω is the atomic volume, and xIII is the composition of the III element (either In for InN nanowires or Ga for GaN nanowires) in the seed (see Supporting Information for more details). The amount of gold (nAu) remains constant throughout growth and is determined for InN from known Au colloid sizes and, for GaN grown using Au films, from a combination of TEM and EDS measurements.19 Role of Cross-Sectional Geometry in DiameterModulated InN Nanowires. We first consider diametermodulated InN nanowires. Here, we use measured values of the thin diameter (d1) and corresponding seed composition (x1) from our studies on diameter-modulated InN nanowires14 and determine V1 and f(β1) from eqs 5 and 2, respectively (see Supporting Information, Table S1). We then calculate the changes in wetting angle and composition that are required to achieve different diameter ratios between thick and thin segments (Figure 2) and compare these with experimental results obtained from measured values of the thick diameter (d2) and corresponding seed composition (x2). To calculate β, an expression for f(β) that defines the relationship between seed volume and nanowire diameter as a function of wetting angle is needed. As mentioned before, eq 3 assumes a cylindrical nanowire and perfectly spherical seed particle, so it cannot be applied to most semiconductor nanowires, which generally have noncircular cross sections due to the formation of low-surface-energy facets. Our InN nanowires grow in the [0001] c-direction and [101̅0] mdirection with hexagonal and triangular cross sections, respectively.14 To determine f(β) for noncylindrical geometries, we employ the program Surface Evolver, which minimizes the energy of a surface according to given constraints.20 (In our case, the key constraints are the shape of the liquid/solid

(1 + cos(β ))

(1)

Figure 1. Diameter-modulated nanowire growth and relevant parameters, shown here for cylindrical nanowires. (a) Thin-diameter (d1) nanowire segment (gray) and corresponding seed particle (yellow) with volume V1 and wetting angle β1. (b) Diametermodulated nanowire with thick-diameter region (d2) and corresponding seed particle on top of thin-diameter region.

Here, β can be considered constant along the circumference of the boundary (Figure 1). More generally, this equation can be written as ⎛ d3 ⎞1/3 ⎜ ⎟ = f (β ) ⎝V ⎠

(3)

However, for noncylindrical nanowires, the shape of the seed particle is not spherical, and eq 3 is no longer valid. Instead, f(β) has a different form and sometimes cannot be expressed analytically, which we discuss later in the text. In the case of diameter modulation (see Figure 1), the ratio of diameters between the thick (d2) and thin (d1) nanowire segments can be expressed as

1/2

(1 − cos(β ))1/6 (2 + cos(β ))1/3

⎛ 24 ⎞1/3 (1 + cos(β ))1/2 ⎜ ⎟ ⎝ π ⎠ (1 − cos(β ))1/6 (2 + cos(β ))1/3

(2)

Simply put, f(β) describes the relationship between nanowire diameter and seed particle volume and is dependent upon the 227

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Figure 2. Comparison of seed morphology and diameter modulation for InN nanowires. (a) Cross-sectional shape and seed particle morphology for circular, hexagonal, and triangular nanowires. (b) f(β) as a function of wetting angle (β) for the nanowire cross sections described in (a). Black solid line was calculated using eq 3, whereas dashed lines were fitted to the data using quadratic functions. (c) Comparison of actual wetting angle (β*) at edges (black) and corners (red) as measured from Surface Evolver images to nominal wetting angle (β). The inset illustrates β (solid gray lines), as well as β* at both the edges (dashed black lines) and corners (dashed red lines) for a seed particle on a triangular cross-section. (d−f) Diameter modulation maps for circular (d), hexagonal (e), and triangular (f) cross sections. Points on the map indicate experimentally measured values from the thin-diameter (small symbols) and thick-diameter (large symbols) segments of InN nanowires. Top axis indicates the change in wetting angle (Δβ) from the thin to the thick segment.

interface and a surface tension that corresponds to an input wetting angle.) Figure 2a illustrates seed particle shapes obtained using this approach for three different cross-sectional geometries: circular, hexagonal, and triangular. For each geometry, we determine the ratio d/V1/3 for different values of β (Figure 2b). The Surface Evolver model fits accurately the analytical predictions from eq 3 for cylindrical nanowires, but f(β) for the hexagonal and triangular nanowires expectedly deviates from the cylindrical case and is instead determined empirically by fitting the data in Figure 2b with a simple quadratic function. Interestingly, while the nominal wetting angle (β) used as an input in the Surface Evolver model is assumed to be constant, the actual wetting angle output from the model (β*) for noncircular cross sections differs from β and varies across vapor/liquid/solid interface. Most notably, β* > β at the nanowire edges and β* < β at corners (Figure 2c). Details of the measurements are described in Supporting Information, Figure S1. A similar prediction of variation in wetting angle around the seed particle has been described previously in analyzing changes in seed particle shape during the growth of twinning superlattices in InP nanowires.21 Both our InN and GaN nanowire growths and the growths of the latter InP nanowires reported by others were observed ex situ, thus preventing a direct comparison of theory and experiment. However, asymmetrical wetting angles have been observed in situ during the growth of Si nanowires with asymmetrical cross sections.22,23 As shown in Figure 3, our Surface Evolver model accurately matches the observed morphology of the seed

Figure 3. Comparison of Surface Evolver model with in situ TEM studies. (a) Si nanowire seeded by a AuSi seed particle. Adapted from ref 23. Used with permission from the American Physical Society. Copyright (2012). The dashed line indicates the boundary of the seed particle, produced using the cross-sectional geometry illustrated in (b), consistent with the geometry reported by the authors. (b) Seed particle morphology (top) and corresponding cross-sectional geometry (bottom).

particle in this case and thus confirms validity of our approach (see also Supporting Information, Figure S2). The seed composition and wetting angle in the thin-diameter regions of diameter-modulated InN nanowires (Supporting Information, Table S1 and ref 14), along with the obtained expressions for f(β), were used in eq 4 to produce the diameter modulation maps in Figure 2d−f. These maps illustrate the 228

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diameter ratio as a function of both composition and wetting angle for the circular, hexagonal, and triangular geometries. The calculated wetting angles are similar for the hexagonal and triangular geometries (127 and 128°, respectively, in the thindiameter region), as would be expected from the similarity between their surface energies. However, the wetting angle is notably different for the circular geometry (121° in the thindiameter region), indicating the importance of accounting for cross-sectional geometry. We measure a slightly larger average diameter ratio (d2/d1) for triangular InN nanowires (1.40 ± 0.09) than for hexagonal ones (1.30 ± 0.06). Because the measured seed composition and wetting angle are similar for these geometries, the larger diameter ratio for triangular nanowires can be understood by the correspondingly larger change in the value of f(β) (Figure 2b). In both cases, we find that the observed increase in nanowire diameter is driven primarily by an increase in seed volume (25 and 24% for triangular and hexagonal InN nanowires, respectively), although there is a lesser but notable contribution from a reduction in wetting angle (12 and 5% for triangular and hexagonal InN nanowires, respectively). Note that the changes in seed composition are responsible for changes in both the seed particle volume and the wetting angle, which is determined by composition-dependent surface energies. While there may be some degree of error in the ex situ measurement of seed particle composition, we find that it would be unreasonable to attribute the diameter changes to variations in wetting angle alone, as significant changes in seed particle composition, and therefore volume, are necessary to produce the observed diameter modulation (Supporting Information, Table S2). Role of Seed Composition in Diameter-Modulated GaN Nanowires. In the case of InN nanowires, the maximum observed diameter ratio was 1.5, although it may be possible to achieve higher diameter ratios. In fact, we have produced diameter ratios in excess of 2 within GaN nanowires, which we grew at 850 °C and varied the Ga flux to change the seed particle composition (more experimental details in Supporting Information). Figure 4a shows an example of a GaN nanowire with a diameter ratio of 2.2. The measured Au−Ga seed particle composition at the thin diameter region was 28 atom. % Ga; measurements of seed particles in the thick-diameter regions were difficult to obtain reliably due to the greater Ga composition which led to significant extraction of Ga from the seed particle during the substantially longer cooling period for GaN versus InN (growth temperatures of 850 versus 560 °C, respectively). Instead, the diameter modulation map (Figure 4b) must be inspected to extract values for the composition and wetting angle in the thick-diameter regions of GaN nanowires. Greater modulations in diameter were achieved for GaN nanowires than for InN nanowires in part because of the lower group-III composition (xIII) in the thin-diameter regions (28 atom. % Ga and 50 atom. % In, respectively; see Figure 4c). Nonetheless, to achieve a diameter ratio of 2.2 with a constant wetting angle, the seed composition in the thick diameter region would need to be roughly 93 atom. % Ga. Such a high Ga composition is unlikely because of the significant amount of Ga that must be incorporated into the seed particle (Figure 4c, inset) despite the fact that the extraction rate of Ga out of the seed particle increases as its composition within the seed increases.14 Thus a substantial change in wetting angle likely contributes to the large diameter ratio in this case. For example,

Figure 4. Large diameter-ratio GaN nanowires. (a) TEM image of a GaN nanowire with a diameter ratio in excess of 2. (b) Diameter modulation map for GaN nanowires based on measurements from thin-diameter region. Points on map indicate measured conditions in thin-diameter region (small triangles), as well conditions in the thickdiameter region (large triangles) for a diameter ratio of 2.2 assuming a change in wetting angle (Δβ) of 0, −10, and −20°. (c) Volume changes (V2/V1)1/3 as a function of thick-diameter composition for different thin-diameter compositions assuming only volume changes, without any contribution from changes in wetting angle. Inset shows molar ratio of Ga to Au in the seed particle as a function of Ga composition.

for a wetting angle change of −10 or −20°, the composition of the seed would only need to reach 85 or 73 atom. % Ga, respectively (Figure 4b and Supporting Information, Table S3). Thus the low composition in the thin-diameter region not only allows large changes in volume with composition but also greater changes in wetting angle due the changes in the composition-dependent surface energies that define it. Opportunities, Limitations, and Role of Materials Selection. On the basis of our studies of diameter-modulated InN and GaN nanowires, we now extend this analysis to consider general opportunities and limitations for template-free modulation of nanowire diameter. Most importantly, greater diameter ratios are possible for materials that can grow over a large range of supersaturation. However, as described above, adding large amounts of alloying elements into an already highly supersaturated seed particle is challenging. Consequently, achieving large diameter ratios in excess of 2 would require significant changes in wetting angle. Changes in wetting angle are driven by changes in surface energy, which can be altered by changes in seed particle composition.24 In addition, the seed material can be selected such that larger changes in wetting angle are possible. For example, if we assume that that the liquid/solid surface energy (σls) does not change substantially with the seed composition, choosing a seed metal with a higher vapor/liquid surface energy 229

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(σvl) relative to that of the alloying element would result in greater changes in σvl, and therefore β, as seed composition changes. Furthermore, it may be possible to employ other foreign elements to alter interfacial energies.19,21 Different materials systems may require different strategies and mechanisms for achieving diameter modulation. The III−V nitride system offers a unique ability to separately tune the rates of incorporation of alloying elements into the seed particle and extraction out of the seed particle by modulating the flux of III and V precursors, respectively.14 Although the solubility of the V element in the seed may not be negligible for other III−V materials, the III source is generally the predominant alloying precursor,25 so modulation of incorporation and extraction rates with precursor fluxes should still be possible. However, in elemental nanowires such as Si other strategies may need to be employed to modulate the diameter, for example, adjusting the supersaturation of Si in the seed particle by adjusting the temperature.26 In some materials systems, defects such as twins may preferentially form under certain conditions, which may place some limits on the range of conditions that can be used without diminishing material performance.27 We note that all of the discussion thus far has centered on liquid seed particles, but the seed particle remains solid in many cases.13,28,29 For example, Caroff et al.13 reported InAs/InSb axial heterostructures with a 40% greater diameter for the InSb segment than the InAs segment, which had seed compositions of 67% and 30 atom. % In, respectively. The calculated volume changes alone accounted for a 35% increase in diameter, similar to the observed 40% increase, indicating that the diameter changes were driven primarily by changes in seed particle volume. For solid seed particles with compound-forming alloys, the seed composition and nanowire diameter may be limited to a discrete range of values, but continuous diameter modulation should be possible for solid solutions. However, in order to model the diameter modulation potential via eq 4, different models that account for changes in the three-dimensional geometry of solid seed particles would be necessary to determine f(β). Liquid/Solid Interface Energy and Morphology. We now extend our model beyond diameter modulation and apply it to an investigation of the liquid/solid interface morphology, which has often been assumed to be flat with sharp corners,17,19,21 but recent theoretical30−33 and experimental22,34,35 reports have challenged this assumption. It is possible to extract the liquid/solid interfacial energy (σls) by knowing the nominal wetting angle (β), the energies of the vapor/solid (σvs) and vapor/liquid (σvl) interfaces, and the morphology at the triple-phase boundary. We consider three different potential morphologies (Figure 5), calculate the corresponding liquid/ solid surface energy, and compare those values with expected values to provide insight into the structure of liquid/solid interface at the triple-phase boundary. We determine the values of β and σvs from our experimental data and literature,36,37 respectively, for our InN and GaN nanowires. To calculate σvl, we account for the migration of low-surface-energy elements to the liquid surface by solving the following equation

Figure 5. Surface energies and force balances at the triple-phase boundary for different liquid/solid interface morphologies. (a) Macroscopic view of a cylindrical nanowire and seed particle. (b−d) Models for surface energies for different nanoscopic morphology of the vapor/liquid/solid interface, zoomed in at red box in (a): flat with a sharp corner (b), faceted with a sharp corner (c), or inclined along a rounded corner (d). Shaded areas between the nanowire and seed particle indicate the portions of the nanowire that are encapsulated within the seed particle. Figures show the steady-state case in which α = 90°.

σAu − i = σAu + = σi +

⎛ 1 − xS ⎞ RT i ⎜ ⎟ ln 1/3 2/3 − 1 x 1.091NA V̅Au ⎝ i ⎠

⎛ xS ⎞ RT ⎜ i ⎟ ln 2/3 1.091NA1/3VIn ⎝ xi ⎠ ̅

(6)

where σ is the vapor/liquid surface energy, R is the gas constant, T is temperature, i is either In or Ga, NA is Avogadro’s number, V̅ is molar volume,38 and xSi is the concentration of element i at the liquid surface.24 Finally, to calculate σls, the morphology of the liquid/solid interface must be known. Figure 5 shows three different potential morphologies, described below. Note that differences in these models affect the calculated liquid/solid interfacial energy, but do not affect the rest of the analysis in this work as the nominal value of β remains the same in all cases. It has generally been assumed that the liquid/solid interface is flat and meets at a sharp corner (Figure 5b).17,19,21 In this case, the liquid/solid energy is calculated from the following force balance 38

σls = σvscos(α) − σvl cos(β)

(7a)

where α is the inclination angle between the vapor/solid and liquid/solid interfaces.17 At steady-state, when the diameter remains constant, α = 90°, and eq 7a reduces to:

σls = −σvl cos(β)

(7b)

This condition can only be met when β ≥ 90° and σvl ≥ σls. Furthermore, eq 7a represents only the force balance in the direction along the liquid/solid interface. The force balance along the vapor/solid interface can be written as: σvs = σvl cos(β − α) + σls cos(α)

(8a)

When α = 90°, the condition is only met when σvl ≥ σvs.

σvs = σvl sin(β)

(8b)

For InN and GaN nanowires seeded by Au seed particles, σvl < σvs, indicating that eq 8b cannot be satisfied. Thus, either the 230

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force balance along the vapor/solid interface does not have to be satisfied because the seed is not in contact with the sidewall, or the liquid/solid interface is not flat with a sharp corner, contrary to the common assumption for nanowire growth models. Liquid/solid interfaces with faceted edges have been observed during in situ growth studies.22,34,35 If we assume that all edges are faceted at some length scale39 with an inclination angle of θf (Figure 5c), the force balances along the liquid/solid interface and the vapor/solid interface are, respectively σls = σvs cos(α − θf ) − σvl cos(β − θf )

(9)

σvs = σvl cos(β − α) + σls cos(α − θf )

(10)

The values for θf and σls can be determined by solving eqs 9 and 10 together such that both conditions are met. It has also been suggested that, because high facet curvature is unfavorable, the corners at the triple-phase boundary should be rounded at some length scale (Figure 5d).30 In this case, the edge of the seed particle would intersect the rounded corner at some inclination angle (θi) such that 0° ≤ θi ≤ 90°. The force balance takes the standard form of Young’s equation σls = σvs − σvl cos(β − θi)

(11)

The value of σls is calculated such that the minimum total surface energy for σls occurs at the measured value of β. Notably, for all conditions here, total surface energy is minimized when θi = 90°. However, θi may be less than 90° as σvl approaches σvs. On the basis of our analysis, as well as the results of other experimental22,34,35 and theoretical30−33 studies, we conclude that the edges of the liquid/solid surface in contact with the triple-phase boundary may be either faceted (Figure 5c) or rounded (Figure 5d), but it is unlikely that they are flat (Figure 5b). First, our results indicate that the force balance along the vapor/solid interface is not upheld for the flat interface model. Additionally, the faceted and rounded models produce values for σls that, relative to σvs, are more consistent with values for elemental liquids on compound semiconductors40,41 than what is calculated for a perfectly flat interface (see Supporting Information, Table S1), suggesting an interface structure that differs from the flat one that is generally assumed. However, for both the faceted and rounded interface models, the extracted values for σls are similar and remain nearly constant from the thin to thick regions, so we cannot distinguish between these two models. Further in situ TEM investigation, paired with the models that we have developed here, should provide clarity into the nanoscopic structure of the liquid/solid interface at the triple-phase boundary, which has significant implications for particle-mediated nanowires, including the fundamental growth mechanism, the sharpness of heterointerfaces, and the mechanism of twinning.22 On the basis of these liquid/solid interface models, we now consider materials selection for achieving larger diameter changes through changes in the wetting angle. According to the flat interface model in eq 7b, a larger thin-diameter wetting angle resulting from a lower ratio of σvl/σls would yield a larger wetting angle and make the nanowire diameter more sensitive to changes in surface energy (Figure 6a). In this case, selecting seed materials with lower surface energy (σvl) would facilitate greater changes in nanowire diameter. However, by assuming constant σls the faceted and rounded interface models instead

Figure 6. Impact of changes in surface energy and wetting angle on changes in nanowire diameter. (a) Changes in nanowire diameter as a function of relative surface energy for different thin-diameter wetting angles, assuming a flat liquid/solid interface. f(β2)/f(β1) indicates diameter ratio that would be expected from changes in surface energy alone, without any contribution from changes in seed volume. (b) Changes in nanowire diameter as a function of σvl for triangular InN and GaN nanowires, assuming a faceted or rounded interface with constant σls. Inset shows the change in the surface energy of the seed particle as a function of composition for Au−In and Au−Ga alloys, according to eq 6. Dots correspond to values in thin-diameter region.

suggest that greater reductions in σ vl with increasing composition of the alloying precursor element will yield greater changes in nanowire diameter (Figure 6b). Hence, for either of these models, selection of metals with high σvl that form alloys in which σvl decreases more linearly with composition would produce greater changes in nanowire diameter. In summary, based on our modeling of experimental results with diameter-modulated InN and GaN nanowires, we have revealed fundamental strategies, opportunities, and limitations for diameter modulation of nanowires via template-free particle-mediated growth. We found that the potential for diameter modulation is greatest for seed alloys that can promote nanowire growth both at low and high compositions of the precursor element(s). Additionally, materials can be selected such that changes in wetting angle have a greater contribution to changes in nanowire diameter. With significant changes in both volume and wetting angle, we were able to fabricate GaN nanowires with diameter ratios greater than 2. The model is consistent with in situ TEM results and not only allows assessment of diameter modulation but can also be used to probe fundamental aspects of nanowire growth, including the three-dimensional shape of the seed particle and the liquid/ solid interfacial energy. We find that the structure of the liquid/ solid interface may be more complex than the simple flat surface that is commonly assumed, which can have a variety of implications for nanowire growth. In sum, we have developed a 231

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(19) Wallentin, J.; Ek, M.; Wallenberg, L. R.; Samuelson, L.; Deppert, K.; Borgström,, M. T. Nano Lett. 2010, 10, 4807−4812. (20) Brakke, K. A. Exp. Math. 1992, 1, 141−165. (21) Algra, R. E.; Verheijen, M. A.; Borgström, M. T.; Feiner, L. F.; Immink, G.; van Enckevort, W. J. P.; Vlieg, E.; Bakkers, E. P. A. M. Nature 2008, 456, 369−372. (22) Wen, C. Y.; Tersoff, J.; Hillerich, K.; Reuter, M. C.; Park, J. H.; Kodambaka, S.; Stach, E. A.; Ross, F. M. Phys. Rev. Lett. 2011, 107, 025503. (23) Ross, F. M.; Tersoff, J.; Reuter, M. C. Phys. Rev. Lett. 2005, 95, 146104. (24) Egry, I.; Brillo, J. J. Chem. Eng. Data 2009, 54, 2347−2352. (25) Dick, K. A.; Bolinsson, J.; Borg, B. M.; Johansson, J. Nano Lett. 2012, 12, 3200−3206. (26) Sutter, E.; Sutter, P. Nano Lett. 2008, 8, 411−414. (27) Joyce, H. J.; Wong-Leung, J.; Gao, Q.; Tan, H. H.; Jagadish, C. Nano Lett. 2010, 10, 908−915. (28) Persson, A. I.; Larsson, M. W.; Stenström, S.; Ohlsson, B. J.; Samuelson, L.; Wallenberg, L. R. Nat. Mater. 2004, 3, 677−681. (29) Dick, K. A.; Deppert, K.; Martensson, T.; Mandl, B.; Samuelson, L.; Seifert, W. Nano Lett. 2005, 5, 761−764. (30) Schwarz, K. W.; Tersoff, J. Phys. Rev. Lett. 2009, 102, 206101. (31) Schwarz, K. W.; Tersoff, J. Nano Lett. 2011, 11, 316−320. (32) Haxhimali, T.; Buta, D.; Asta, M.; Voorhees, P. W.; Hoyt, J. J. Phys. Rev. E 2009, 80, 050601(R). (33) Ryu, S.; Cai, W. J. Mater. Res. 2011, 26, 2199−2206. (34) Oh, S. H.; Chisholm, M. F.; Kauffmann, Y.; Kaplan, W. D.; Luo, W. D.; Rühle, M.; Scheu, C. Science 2010, 330, 489−493. (35) Gamalski, A. D.; Ducati, C.; Hofmann, S. J. Phys. Chem. C 2011, 115, 4413−4417. (36) Filippetti, A.; Fiorentini, V.; Cappellini, G.; Bosin, A. Phys. Rev. B 1999, 59, 8026−8031. (37) Northrup, J. E.; Neugebauer, J. Phys. Rev. B 1999, 60, R8473− R8476. (38) CRC Handbook of Chemistry and Physics, 92nd ed.; Taylor and Francis Group: Boca Raton, FL, 2011. (39) While faceting at all edges has been observed, most reports only detect faceting at one edge. We note that we investigated the potential impact on our diameter-modulation models of introducing a small tilted facet at the edge of the liquid/solid interface and found it to have minimal impact on the analysis (Supporting Information, Figure S3). (40) Nikolopoulos, P.; Agathopoulos, S.; Angelopoulos, G. N.; Naoumidis, A.; Grübmeier, H. J. Mater. Sci. 1992, 27, 139−145. (41) König, U.; Keck, W. J. Electrochem. Soc. 1983, 130, 685−686.

theoretical foundation to guide template-free nanowire diameter modulation and probe fundamental aspects of nanowire growth for a broad range of materials systems, including other III−V materials and elemental materials such as Si.

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ASSOCIATED CONTENT

S Supporting Information *

Experimental data and calculations for InN nanowires, details about Surface Evolver models, calculations regarding the origin of diameter changes in InN nanowires, data and calculations for GaN nanowires, and evaluation of the impact of liquid/solid interface morphology on modeling. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by NSF CAREER award no. DMR0745555. The authors acknowledge access to Shared Experimental Facilities provided by the MIT Center for Materials Science Engineering supported in part by MRSEC Program of National Science Foundation under award number DMR-0213282. We thank Jordan Chesin for helpful discussion.

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REFERENCES

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dx.doi.org/10.1021/nl3039019 | Nano Lett. 2013, 13, 226−232