NANO LETTERS
Phase Equilibrium and Nucleation in VLS-Grown Nanowires
2008 Vol. 8, No. 11 3739-3745
Edwin J. Schwalbach and Peter W. Voorhees* Department of Materials Science and Engineering, Northwestern UniVersity, EVanston, Illinois 60208 Received July 7, 2008; Revised Manuscript Received September 23, 2008
ABSTRACT Phase diagrams accounting for capillarity and surface stress in VLS-grown nanowires have been calculated, and linearized forms for the compositions of the solid and liquid are given. The solid-vapor interfacial energy causes a significant depression of the liquidus, and the impurity concentration in the wire decreases with decreasing wire diameter. Nucleation calculations give upper bounds on the nucleation temperature and liquid supersaturation during growth that are consistent with measurements in the Au-Ge system.
Silicon and germanium nanowires have been extensively studied due to their unique properties, and recent interest has been generated by a broad array of potential applications including gene delivery to cells,1 sensors for biological molecules,2 and improvements to Li-ion battery anodes.3,4 A common technique for growing these wires is the vapor-liquid-solid (VLS) mechanism originally described by Wagner and Ellis.5 In this method, a nanometer-sized metallic seed particle which forms a low temperature eutectic alloy with the wire material acts as a site of preferential, unidirectional crystal growth. This growth technique has been studied for a number of years, but it is not yet completely understood. While wire growth is a nonequilibrium process, the equilibrium phase diagram provides important guidance on the conditions, such as temperature and liquid composition, needed for nanowire fabrication. Unfortunately, these temperatures and compositions cannot be determined from the bulk phase diagram, even if the wire is growing very close to equilibrium, due to the major influence of capillarity and stress in these nanoscale objects. We shall focus on the effects of capillarity and stress generated by the surfaces on phase equilibrium in the important Au-Ge and Au-Si systems. In doing so, we outline a general methodology for computing phase diagrams relevant to nanowire growth that can be applied to other systems as well. There has been recent experimental and theoretical progress in understanding the role of capillarity in setting the phase boundaries in these systems. Sutter and Sutter have measured the equilibrium composition of a Au-Ge liquid alloy drop on the end of a Ge nanowire using in situ heating in a transmission electron microscope (TEM).6 These results indicate that phase boundaries in the nanowire system are shifted to lower temperatures compared to those of a bulk 10.1021/nl801987j CCC: $40.75 Published on Web 10/28/2008
2008 American Chemical Society
system. Theoretical guidance in this area exists for the Au-Ge system in the form of calculated phase diagrams.7 However, these calculations assume that all of the Au in the initial seed particle is present in the alloy droplet on the tip of the nanowire throughout growth. Thus as the liquid composition changes, both the radius of the droplet and the wire diameter must change. As a result there is an implicit dependence of wire diameter on liquid catalyst composition which is not valid under all experimental conditions. For example, surface migration of Au atoms on clean Si surfaces was observed in the Au-Si system under ultrahigh vacuum conditions, leading to coarsening of the initial Au particles prior to wire growth.8 Given the complexities of the wire nucleation process, we take a different approach. Here, we calculate phase diagrams that account for surface energies and stresses by considering constant wire diameters rather than initial seed particle diameters. The electrical properties of Si can be strongly affected by small quantities of Au impurity atoms, potentially limiting the performance of Si-nanowire-based electronic devices.9 Recent TEM experiments have confirmed that Au atoms are incorporated into Si nanowires during growth, although the absolute concentrations are difficult to measure.10,11 We calculate the equilibrium solid solubility of Au in Si nanowires based on bulk thermodynamic data at high temperatures, estimate equilibrium Au concentrations in the range of growth temperatures, and compare these with experimental values. There is experimental evidence that nanowire growth can proceed at temperatures well below the bulk eutectic temperature with a liquid or solid seed particle in the Au-Ge system.7,12,13 The state of the seed particle influences the growth kinetics of the wire with solid seeds giving rise to significantly slower growth compared to liquid seeds at the
same temperature.12 There are also indications that subeutectic growth is possible in the Au-Si system, although the state of the seed particle is unknown.14,15 Far below the bulk eutectic temperature, the liquid seed will solidify, but calculations indicate that surface energy effects alone do not account for the degree of supercooling necessary for solidification.7,12 It has been proposed that a supersaturation of Ge in the seed stabilizes the liquid phase at these low temperatures. We examine this issue from the standpoint of nucleation of Au within the liquid droplet, explore the differences between nucleation in bulk and nanoscale systems, and estimate the magnitude of the Ge supersaturation. We approach this problem from the perspective of classical thermodynamics. The condition of chemical equilibrium requires that the diffusion potential of an atom of species Q in the solid be equal to the chemical potential of that same species in the liquid seed, where Q can be Au, Si, or Ge, MQs ) µQl
(1)
MQs is the diffusion potential of Q in the stressed crystalline wire and µQl is the chemical potential of Q in the liquid. We employ well-known results from the thermodynamics of elastically stressed solids; for a review see ref 16. These quantities are functions of the size, shape, and surface energies of the different phases as well as temperature and composition. The diffusion and chemical potentials for the various phases in the nanoscale system can be expressed in terms of the bulk thermodynamic data and corrections which account for the surface energy and stress effects. All compositions are expressed in terms of the atomic fraction of the semiconductor species unless otherwise noted. The pressure generated by surface energy of the liquid-vapor interface is given by the Laplace-Young equation, and an approach developed by Herring is employed to treat the solid-vapor and faceted solid-liquid interfaces.17,18 Stress effects are included according to the thermodynamics of elastically stressed crystals.16 The full derivation of the equations of chemical and mechanical equilibrium for the nanowire system is available in the Supporting Information. Our model assumes that the crystalline wire has a circular cross section and that the solid-liquid interfacial area does not change during growth. The latter assumption allows for wires that taper due to exchange of material between the solid and vapor directly but does not account for tapering due to a shrinking liquid droplet; this behavior will be addressed later. The liquid-vapor interface is spherical and the solid-liquid interface is planar (Figure 1a). The radius of the liquid seed Rl is 10% larger than the wire radius Rs. This is consistent with the surface energies and the force balance at the liquid-vapor-solid trijunction. Linear elasticity is assumed to hold in the solid. In the case of the Au-Si calculations, the Au atoms incorporated into the solid wire are assumed to be substitutional on the Si lattice, as observed for the majority of Au atoms in the results of Oh et al.11 The molar volume of this alloy is calculated assuming Vegard’s law using the diamond cubic semiconductor molar 3740
Figure 1. (a) Geometry of the wire used for the phase diagram calculations showing the solid wire in blue with radius Rs and the liquid cap of radius Rl as transparent. (b) Geometry used for the nucleation calculation showing the critical nucleus in yellow with radius R*.
volume and an estimated diamond cubic Au molar volume based on the atomic radius of Au. The surface energy for the liquid-vapor interface is taken from the literature for the liquid semiconductor at its melting temperature.19 The actual surface energy at the liquid-vapor interface can depend on the liquid composition, potentially in a nonlinear way.7 However, the energy is bounded between the surface energies of pure Au and pure semiconductor. Because the surface energies of the semiconductor materials considered are both lower than Au, they are a lower bound on the composition-dependent surface energy. The solid-vapor surface energy is taken to be that of a [111] plane of the semiconductor material, which is the lowest energy surface for a diamond cubic solid.20,21 The use of these low estimates for the liquid-vapor and liquid-solid surface energies results in a calculation which is an overall lower bound on the effect of surfaces on the system. The solid-liquid interfacial energy is estimated based on a balance of surface tensions at the solid-liquid-vapor interface in the plane of the liquid-solid interface. This energy is approximately 10% of the liquid-vapor interfacial energy. The thermodynamic data needed to determine the chemical potentials is available in the literature.22-24 Unfortunately, these data are not complete and assume that the solubilities of Au in diamond cubic Ge and of Si in FCC Au are zero. The phase boundaries are strongly dependent on the morphology of the phases in a nanoscale system because of the large surface area to volume ratios. To calculate the equilibrium compositions of the system, the size and shape of the liquid and solid phases must be known. For the semiconductor-rich side of the phase diagram this information is clear, the liquid is a spherical cap on the end of the cylindrical nanowire (Figure 1a). However, for the Au-rich side of the phase diagram, the morphology of the appropriate solid is not clear. Therefore, we do not calculate the Aurich liquidus or define a eutectic temperature for the nanowire system. The eutectic temperature is thought to be important since it defines the temperature below which solid Au can nucleate from a liquid rich in Si. Instead, we will deal with Nano Lett., Vol. 8, No. 11, 2008
the issue of the temperature at which the liquid seed solidifies by considering the nucleation of a Au particle within the seed. The assumption that the solid-liquid interfacial area remains constant leads to the result that neither the liquidus nor the solidus is dependent on the solid-liquid interfacial energy. Thus, the solid-vapor surface energy changes the chemical potential of the solid proportional to 2/Rs. The geometric factor of 2 follows from Herring’s formulation for a cylinder growing along its axis; see the Supporting Information and ref 18. This result holds for wires which taper due to homogeneous gas decomposition on the solid-vapor surface or etching of the solid by reactive gas moledules because the solid-liquid interfacial area does not change as the wire length increases.25 However, if the diameter of the catalyst droplet changes during growth, which has been observed in the Au-Si system,8 the contribution to the capillarity terms from the solid-liquid interfacial energy will not be zero. Because the solid-liquid interfacial area decreases during this type of tapered growth, this will cause a decrease in the magnitude of the capillarity term. For long wires, this effect will be small for the majority of the wire where tapering is only slight. The liquid-solid interfacial energy could play a larger role in the latest stages of growth, where tapering has been observed to accelerate in the Au-Si system.26 An estimate of the magnitude of this effect is included in the Supporting Information. For wires with noncircular cross sections, the factor of 2/Rs must be modified to reflect the appropriate ratio of perimeter to cross sectional area. Since a circle has the lowest perimeter for a given area, the capillarity effects will generally be larger for noncircular cross sections. If the wire has an equilateral triangular cross section, for example, the effect of capillarity is increased by approximately 30% (assuming equal energy on all solid-vapor facets). Again, our calculation reflects a lower bound on the effects of surfaces on the equilibrium of the system. If, for a given temperature, the compositions of the solid and liquid in the nanowire do not differ too much from those of the bulk, linearized equations for the solid and liquid compositions can be derived. The accuracy of these predictions decreases as the radius of the wire decreases. However, the equations are useful for determining the relative importance of the effects included in the analysis and can provide very good estimates of the phase compositions in many cases. We present the results in the limit where γ/(RK) , 1 where γ is a representative surface energy of the system, K is an elastic constant of the solid such as the bulk or shear modulus, and R is a representative radius of curvature. For typical materials, this parameter is much less than 1 for values of R on the order of 1 nm or larger. Also, we take the solid to be a dilute solution, as is the case for the Au-Ge and Au-Si systems, and thus the second derivative of the molar Gibbs free energy of the solid evaluated at the bulk equilibrium solid composition is very large and positive. Under these conditions, the solid composition can be approximated as Nano Lett., Vol. 8, No. 11, 2008
xAs ) ˆxAs +
[ ( )
ˆ s ˆ l 2γsvV γlv V m m 1 + ξ 1 ′′ s ˆ γ ˆ sv R ∆xˆ G V s A
s
]
m
ˆ s∆xˆ ∆V A (ξγlv + 2fsv) ˆ sγ 3V m
sv
and for the liquid composition,
[ (
ˆ s ˆ 2γsvV γlv Vml(xˆAs) m xA ) ˆxA + s 1+ξ 1ˆ ′′ γsv ˆ s R ∆xˆ G V l
l
A
l
m
)]
(2)
(3)
where xˆAi is the composition of the bulk equilibrium phase i, ∆xˆA ) xˆAs - xˆAl, Vˆmi is the molar volume of the equilibrium phase i, Vˆml(xˆAs) is the molar volume of a liquid with the js ) V j As composition of the bulk equilibrium solid, ∆V s s j j VB , VQ is the partial molar volume of species Q in the solid, ˆ i′′ is the second derivative with respect to composition of G the molar Gibbs free energy of the of phase i evaluated at composition xˆAi, γsv is the solid-vapor surface energy, γlv is the liquid vapor surface energy, fsv is the solid-vapor surface stress, and ξ ) Rs/Rl. The Supporting Information contains more general versions of these equations which can be applied to systems where the above assumptions are not valid. The prefactor 2γsv/Rs in both composition equations indicates that the shifts in composition from their bulk values are driven primarily by the curvature of the solid-vapor interface, with larger shifts for smaller radii wires. Both equations have terms which are related to ratios of molar volumes, surface energies, and surface forces. While these terms are not negligible, they tend to be small for the systems in question. Also, the absence of the quantity fsv in eq 3 indicates that the effect of the solid-vapor surface stress on the liquidus is negligible. However, the surface stress can have a weak effect on the solid composition as indicated by the last term in eq 2. The linearized results for the solid composition lie within 5% of the full calculation for wires 30 nm or larger in diameter, and the linearized liquid composition is typically even closer to the nonlinear solution. Figure 2 shows the bulk Au-Ge phase diagram as well as liquidus curves determined by solving the full nonlinear equations for chemical equilibrium for several nanowire diameters. The liquidus for the 100 nm diameter wires is roughly equal to that of bulk material whereas it is displaced approximately 40 K below the bulk value for the 30 nm diameter wire and more than 100 K below for the 10 nm diameter wire. Similar effects are visible for the Au-Si system in Figure 3. The calculated Au-Ge liquidus exhibits a roughly constant downward shift across a wide composition range that depends on the diameter of the wire, whereas the experimental data suggest that the shift is relatively small near the eutectic composition, increases for higher Ge compositions in the liquid, and is nearly independent of wire diameter.6 The cause of the differences between the predictions and experimental results could result from the volumebased technique used for the experimental composition measurement. Figure 4 shows the solidus for 30 and 10 nm diameter Si nanowires. Unlike the surface energy which is always a positive quantity, the surface stress acting at the solid-vapor 3741
Figure 2. Phase diagram for the Au-Ge system showing the bulk phase boundaries as well as liquidus curves calculated for 100 nm (green), 30 nm (blue), 10 nm (red), and 5 nm (magenta) diameter nanowires. Because no morphology is assumed for the Au-rich solid phase, the liquidus curves are extended below the bulk eutectic temperature and could represent metastable states at low temperatures. The composition of the solid wire is assumed to be pure Ge due to limited thermodynamic data.
Figure 3. Phase diagram for the Au-Si system showing the bulk phase boundaries as well as liquidus curves calculated for 100 nm (green), 30 nm (blue), 10 nm (red), and 5 nm (magenta) diameter nanowires.
interface can be positive or negative. Positive fsv causes compressive stresses in the solid, and conversely negative fsv results in tensile stresses. The magnitude of the surface stress in these calculations is set equal to the solid-vapor surface energy, and the calculations are repeated with both positive and negative values of fsv. Figure 4 shows that the solubility of Au in diamond cubic Si decreases with decreasing diameter. For a fixed radius wire, the solubility is lower for positive values of fsv. Positive surface stresses compress the lattice, making it less accommodating to the presence of the large substitutional Au atoms. A similar trend might be expected for interstitial Au atoms as well. The maximum solubility of Au at elevated temperatures for the 30 nm wire is approximately XAu ) 1 × 10-6, roughly half the bulk solubility. However, Si exhibits a retrograde solidus, and nanowire growth is typically carried out at temperatures much closer to the bulk eutectic temperature of 636 K. At a typical growth temperature of 723 K, the Au solubility is calculated to be XAu ) 1.2 × 10-8 for the bulk 3742
Figure 4. Solidus for the bulk (black), 10 nm (red), and 30 nm (blue) diameter wires. The surface force fsv at the solid-vapor interface can be positive (solid) or negative (dashed). There is no surface stress acting in the bulk system. The eutectic line of the bulk diagram at 636 K and liquidus curves are not shown for clarity.
system, and XAu ) 0.9 × 10-8 for the 30 nm diameter nanowire. The Au-Si solid solution thermodynamic data are based on experimental measurements for alloys in the range 1173-1598 K.22,24 Therefore, the solid composition is not expected to be quantitatively accurate because of the extrapolation to lower temperatures; however the general result of decreased impurity solubility due to capillarity is consistent with self-purification effects in other nanocyrstal systems.27 A rough experimental estimate of the AU composition in a Si nanowire grown at 723 K is 4 × 10-6.10 This is 2 orders of magnitude higher than the calculated nanowire value at the same temperature and is even high compared to the bulk material at elevated temperatures. The authors point out that kinetic factors are likely responsible for the Au concentrations in excess of the equilibrium predictions. Recent measurements using secondary ion mass spectrometry indicate a solubility of roughly XAu ) 4 × 10-7 for 2 µm diameter wires grown at 1273 K.28 Our calculations indicate that for wires of such large diameters, the solubility of Au would be roughly equivalent to that of a bulk system, or approximately 1 × 10-6, agreeing reasonably well with the experimental data. Next, we explore the ability to grow nanowires below the bulk eutectic temperature. It is clear that sufficiently below the bulk eutectic temperature solid Au will nucleate in the liquid droplet which will then solidify. This nucleation process could be heterogeneous, with the nucleus forming at either the liquid-vapor or liquid-solid interfaces or solid-liquid-vapor contact line. For simplicity we treat the problem of homogeneous nucleation of a solid spherical Au particle in contact with only liquid. Also, we assume that the molar volumes of the various phases are equal and ignore stress in the nucleus. Nucleation in a nanosized droplet differs significantly from the standard nucleation problem. In the standard approach, the composition of the liquid matrix with which the nucleus is in equilibrium is assumed to be constant, or similarly, the Nano Lett., Vol. 8, No. 11, 2008
chemical potentials of the species in the liquid are fixed. This is equivalent to assuming that there is a source of Au atoms which replaces those atoms that leave the liquid to form the nucleus, resulting in an open system. This assumption is valid for bulk systems where the nucleus is small with respect to the liquid, but might not apply for a nanoscale system where the volumes of the nucleus and liquid are similar. For this reason we allow the liquid composition to be a variable and introduce the constraint that the number of Au atoms is constant. Under the simplifying assumptions that all phases have the same molar volume Vm, that the total volume of the system remains constant, that there is only one nucleus present, and that the system is closed, conservation of mass requires 4 4 V0x0 ) V0 - πR* 3 xl + πR* 3x* 3 3
(
)
(4)
where V0 is the volume of the liquid and nucleus combined, x0 is the composition of the liquid seed, xl is the composition of the liquid in equilibrium with the nucleus, R* is the radius of the spherical nucleus, and x* is the composition of the nucleus. The critical nucleus and liquid are in equilibrium. We assume that over the time scale of the nucleation process, the liquid does not exchange material with the wire or vapor phase. The equations of chemical equilibrium are then µQ* + V m
(
2γlv Rl
+
)
2γls 2V mγlv ) µQl + R* Rl
(5)
where µQ* and µQl are the bulk chemical potentials of species Q in the face-centered cubic nucleus and liquid, respectively, and γls is the liquid-solid interfacial energy. As an order of magnitude estimate, we assume that this quantity is equal to the Si-liquid interfacial energy. Solving eq 4 simultaneously with eqs 5 gives the composition of the liquid, nucleus, and critical radius at a given temperature and alloy composition. The available thermodynamic data for Au-Si assumes that x* ) 0 or that the nucleus is pure Au. This is not the case for the Au-Ge system, and the nucleus composition must be determined in the calculation. For comparison, classical nucleation calculations are performed for both systems. For the classical, open system calculation, eq 4 is ignored. Figure 5 shows R* as a function of T calculated in the Au-Ge system for a liquid seed composition of x0 ) 0.30 using both the open and closed system approaches. The open system calculation results in R* approaching infinity as the temperature increases toward a metastable equilibrium for liquid of composition x0, which is given by an extension of the bulk Au-rich liquidus below the eutectic temperature and is 589 K with x0 ) 0.30. This behavior is not physically allowable for the nanoscale system in question because there is a finite amount of Au present from which to form the nucleus. The closed system result exhibits fundamentally different behavior. The radius does not grow unbounded but rather increases toward a finite maximum which is set by the liquid seed composition. There is a temperature T* above which the closed system does not have a critical nucleus and below which there are two possible values of R*. This temperature is below the temperature given by the extension of the bulk Nano Lett., Vol. 8, No. 11, 2008
Figure 5. R* as a function of T calculated for the Au-Ge system with x0 ) 0.3 using the open (black) and closed (blue) system calculations. The maximum temperature at which the closed system has a critical radius is T* ) 535 K (dashed) which is below the temperature at which the open system critical radius diverges to infinity of 589 K. Also, the closed system critical nucleus is always smaller than the theoretical maximum size of 13.5 nm assuming that all of the Au initially in the liquid droplet forms a pure Au nucleus.
Au liquidus, indicating that the closed nanoscale system requires a finite undercooling for a nucleus to form. In other words, the maximum temperature at which nucleation can occur in the nanodroplet is less than the temperature at which nucleation is possible in a bulk system. At temperatures well below T* the closed system nucleus with a smaller R* approaches the open system R* value, and the liquid composition in equilibrium with the nucleus approaches x0. The reversible work of forming the nucleus, WRev*, also approaches the classical result, although this is not shown here. This indicates that the classical nucleation theory is valid in the nanoscale system for undercoolings where the nuclei are small. However, as the temperature increases the equilibrium nuclei become larger and the open and closed system results diverge. At T* there is a significant difference between the open and closed system R* values even though the volume of the nucleus is only approximately 2% of the volume of the total system. These nuclei sizes near T* are clearly too large to likely result from spontaneous composition fluctuations. The importance of T* is that it is the temperature below which the droplet must be cooled for homogeneous nucleation to occur. In this sense it is similar to the eutectic temperature in bulk materials. The reversible work of nucleation indicates that nuclei in the open system are always unstable, as is the smaller of the two nuclei calculated with the closed-system approach. An unstable nucleus tends to grow if extra material is added or shrink if material is removed. The larger nucleus in the closed system for T < T* is a stable particle but is not likely to be present due to kinetic considerations as it requires a large change in the liquid composition. The value of T* is a function of the liquid seed composition and the nanowire diameter. Figure 6 shows the eutectic region of the Au-Ge phase diagram with the calculated liquidus curves and T* calculated over a range of x0. T* is 3743
Figure 6. Eutectic region of the Au-Ge phase diagram showing the 10 nm (red) and 30 nm (blue) liquidus curves. The dashed curves represent T* which is a maximum temperature for nucleation of Au in the closed nanoscale droplet. The solid green curve at 528 K is the experimental solidification temperature of a Au-Ge alloy seed particle on a growing ≈30 nm diameter nanowire.12 The seed composition during growth at this temperature must lie between xGe ) 0.24 and 0.31 given by the two vertical green lines.
not a phase boundary but rather a necessary condition for homogeneous Au nucleation in the liquid droplet and therefore solidification of the seed particle, and we will refer to it as the nucleation boundary. Since the solid-liquid interface is faceted, local equilibrium at the interface is likely not present and the growth is therefore driven by a supersaturation of Ge in the liquid. At a temperature of 528 K the nonequilibrium liquid composition must therefore satisfy xGel > 0.24 for a 30 nm wire according to Figure 6. However, solidification of the seed during growth has been experimentally observed to occur at this temperature and diameter.12 Nucleation of solid Au clearly occurs under these conditions, and therefore the liquid composition must be to the left of the nucleation boundary. Consistent with our calculation, we find that this is possible if xGel < 0.31. This is an upper bound on the liquid composition from which it is possible to nucleate solid Au within the liquid droplet and is therefore the maximum supersaturation possible in the liquid at the solidification temperature. The intersection of the liquidus with the nucleation boundary is an upper bound on the temperature at which a seed of any composition can solidify, but it is not a eutectic point. At a eutectic, three phases are in equilibrium, but at the intersection of the nucleation boundary and liquidus, the only phases in equilibrium are the nucleus and liquid. Because of the composition change due to nucleation, the liquid that is in equilibrium with the nucleus is no longer in equilibrium with the nanowire. Figure 7 indicates that the Au-Si system exhibits similar behavior to the Au-Ge system. There is little data available on subeutectic growth of Si from Au seeds, but based purely on the thermodynamic data, it appears that it should be possible to obtain low temperature growth similar to that in the Au-Ge system. A methodology for computing phase diagrams for VLS grown nanowires has been developed that accounts for 3744
Figure 7. Eutectic region of the Au-Si phase diagram showing the 10 nm (red) and 30 nm (blue) liquidus curves. The dashed curves represent T*, which is a maximum temperature for nucleation in the closed nanoscale system.
capillarity and surface stress. Analytical forms for the compositions of the solid and liquid are given. These show that the solid-vapor interfacial energy plays a crucial role in setting the phase diagram of a nanowire system. Through nonlinear calculations in both the Au-Ge and Au-Si systems, we find that the liquidus temperatures can be significantly depressed from their bulk values and that surface stress has little effect on the magnitudes of the depressions. The small size of the nanowire acts to purify the solid, with the degree of purification dependent on the solid-vapor surface stress. Since the eutectic temperature provides no information on the conditions needed to solidify the liquid droplet, the conditions for nucleation of a solid in a nanodroplet are considered. The size of the liquid droplet in which the solid forms places constraints on the nucleation process that yields different conditions for nucleation than in the classical open-system case. This gives a temperature below which the liquid must be cooled to initiate solidification. This temperature is consistent with measurements in the Au-Ge system and places an upper bound on the supersaturation present in the experiments. Acknowledgment. This research is supported by a National Science Foundation Grant CMMI-0507053, and E.J.S. acknowledges support from a National Defense Science and Engineering Graduate Fellowship. Supporting Information Available: A derivation of the full equations of chemical and mechanical equilibrium including linearized approximations of the nanowire and liquid compositions. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Kim, W.; Ng, J. K.; Kunitake, M. E.; Conklin, B. R.; Yang, P. D. J. Am. Chem. Soc. 2007, 129, 7228–7229. (2) Zheng, G. F.; Patolsky, F.; Cui, Y.; Wang, W. U.; Lieber, C. M. Nat. Biotechnol. 2005, 23, 1294–1301. (3) Chan, C. K.; Peng, H. L.; Liu, G.; McIlwrath, K.; Zhang, X. F.; Huggins, R. A.; Cui, Y. Nat. Nanotechnol. 2008, 3, 31–35. (4) Chan, C. K.; Zhang, X. F.; Cui, Y. Nano Lett. 2008, 8, 307–309. (5) Wagner, R. S.; Ellis, W. C. Appl. Phys. Lett. 1964, 4, 89–90. Nano Lett., Vol. 8, No. 11, 2008
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