LETTER pubs.acs.org/NanoLett
Elementary Processes in Nanowire Growth K. W. Schwarz and J. Tersoff* IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, United States ABSTRACT: We propose that many of the complex morphological phenomena observed during nanowire growth arise from the interplay of just three elementary processes: facet growth, droplet statics, and the introduction of new facets. We incorporate these processes into an explicit model for the vaporliquid-solid growth of fully faceted nanowires. In numerical simulations with this model, different conditions can lead to either growth of a free-standing wire or lateral growth where the catalyst droplet crawls along the surface. An external perturbation can cause the wire to kink into a different direction. Different growth conditions can also change the shape of the growth tip. All of these phenomena have been observed, and the model behavior is consistent with the experimental observations. KEYWORDS: Nanowire, growth, simulation, semiconductor, liquid, epitaxy
T
he growth of semiconductor nanowires by the vaporliquid-solid (VLS) process offers unique opportunities for nanoscale device fabrication.1 However, nanowire growth is often complicated by undesired behavior, including kinking from one direction to another2,3 or crawling along a surface.4 Most applications require suppression of such unintended growth modes. On the other hand, control of kinking and crawling could enable growth of novel device structures.5,6 For either purpose, it is important to understand the underlying mechanisms that lead to the observed variety of growth morphologies. Here we propose that many of the observed growth modes can be understood as arising from the interplay of just three elementary processes, which we capture in an explicit model based on classic continuum physics. Several examples illustrate how some of the unexplained phenomena commonly observed in nanowire growth can arise naturally from the interplay of these three processes. Specifically, we address the kinking of a nanowire into a new orientation, as in Figure 1a,b; a competing growth mode in which the catalyst crawls along the surface, as in Figure 1c; and the occurrence of different wire-tip morphologies under different growth conditions, as in Figure 2. Previous work demonstrated that the basic features of nanowire growth could be described using an anisotropic continuum model.7 Extensive simulations using that model have taught us that crystalline anisotropy and faceting are crucial for describing the experimental behavior. Indeed, the most interesting aspects of nanowire growth were found to result from droplet pinning and depinning at facet edges and from the introduction of new facets at the trijunction (i.e., the line where vapor, liquid, and solid meet). Semiconductors are generally completely faceted at the relevant temperatures, and electron microscopy confirms that this is true of semiconductor nanowires. We therefore focus here on the fully faceted limit of our continuum model,7 distilling its key features into a simpler model where the nanowire is r 2010 American Chemical Society
composed entirely of strictly planar facets. (The transition from an anisotropic continuum model to a strictly faceted model has been discussed in detail by Carter et al.8) The new fully faceted model consists of the following three elements: (a) facet dynamics, the motion of a facet normal to itself due to crystal growth or dissolution; (b) droplet statics, in which capillary forces determine the droplet position and shape, and in particular the pinning and depinning of the catalyst droplet at facet edges; and (c) introduction of new facets where appropriate. As in any continuum model, we do not attempt to explicitly describe atomic-scale structure or thermally activated nucleation processes. These may enter implicitly, especially into the rules for (c) as discussed below. But processes that require explicit consideration of thermally activated nucleation9 are outside the scope of our model. While our approach is applicable to three dimensions (3D), our current implementation is restricted to two dimensions (2D). To avoid ambiguity, we use the language of 3D to describe our 2D geometry, referring to the “edge” where facets meet, the “volume” [dimensions (length)2] of the 2D catalyst and so forth. For the facet dynamics, we use standard linear attachmentlimited kinetics as in prior work.7,8 To lowest order in the supersaturation, each facet i at the liquid-solid interface then advances or retreats as vi ¼ Ri ðμl - μi Þ
ð1Þ
where vi is the normal velocity of the facet, μi is the chemical potential of that facet, and μl is the chemical potential of the wire material in the liquid. The rate constant Ri can in general be different for inequivalent facets. We assume that attachment Received: August 6, 2010 Revised: December 4, 2010 Published: December 28, 2010 316
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Figure 1. Different nanowire growth behaviors predicted by our model for a 12-fold crystal, that is, one having 12 equivalent facets (so the 2D equilibrium crystal shape is a regular dodecagon). Straight red lines show computed wire profiles at equal time intervals with the black dots indicating facet edge positions. The catalyst droplet (blue curve) is only shown for the final configuration. (a) Calculated wire growth for default conditions (see text for details). Note the transition from an initial tapering pedestal shape to a propagating steady-state shape, as seen in experiment. (b) Same conditions as (a), except that halfway through the calculation after the growth has already reached a steady state the sidewall energy is briefly perturbed (increased by 10%) on the right side, leading to kinking toward that side. (c) Same conditions as (a), except that we include an edge energy impeding step formation (εi = 0.4, see text for details). As in (a), the droplet eventually depins from the top of the tapering pedestal. But for sufficiently large εi, the droplet fails to introduce a step in the sidewall of the pedestal. Instead it rolls off the pedestal. This results in crawling and lateral wire growth.
Figure 3. The elementary processes considered here, schematically illustrated for a 6-fold crystal. (a) The chemical potential μi of facet i reflects all those contributions to the free energy that change when facet i moves normal to itself. These terms include the area (and hence interfacial energy) of i; the area of the facets j and j0 connected to i; and the surface area of the droplet. The trijunction where vapor, liquid, and solid meet is labeled t. (b) If the trijunction lies at an edge, there is a range of angles over which the trijunction will be stably pinned. This range is bounded by the equilibrium contact angles θc on the respective facets, as indicated by the dashed lines. (c) As the system evolves, an unpinned trijunction lying on facet i moves along the facet to satisfy the equilibrium contact angle. The facet i cannot advance, since it is partly dry. (d) A step can form, introducing a new facet, and splitting the facet i into a “wet” facet (liquid-solid interface) that can now grow, and a “dry” facet (vapor-solid interface) that remains immobile. (e) Trijunction pinned at an edge. Nonconvex perturbations introducing wet and dry facets are shown in (f) and (g), respectively. (h) Trijunction pinned at an edge having a missing facet. Perturbations introducing the missing facet as wet or dry are shown in (i) and (j), respectively.
covered part of the facet if it is energetically favorable to separate the facet into two distinct facets by introducing a step at the trijunction, as illustrated in Figure 3(c,d) and described below. The chemical potential μl in the liquid is approximated here as in previous work7 as ð2Þ μl ¼ βðc - c0 Þ þ Ωl γvl kl where c0 is the bulk liquidus composition, κl is the scalar curvature of the liquid surface, and γvl is the vapor-liquid interfacial free energy (i.e., the liquid surface tension). Ωl, the specific volume of the wire material in the liquid, is approximated as equal to the volume per atom in the crystal. (A more detailed treatment is given by Roper et al.10) The liquid composition c evolves as material is added from the vapor and captured by the wire. As in ref 7, the volume per unit time added from the vapor is rvlAvl, where rvl reflects the vapor pressure and probability of dissociative adsorption and Avl is the area of the vapor-liquid
Figure 2. Effect of growth rate on the calculated wire morphology. (a,b) A 6-fold crystal, that is, one having 6 equivalent facets (so the 2D equilibrium crystal shape is a regular hexagon). (c,d) A 12-fold crystal shape. Here (a,c) are computed for a low growth rate, rvl/Ri = 0.15 meV; (b,d) show the effects of a high growth rate, rvl/Ri = 50 meV. Note the differences in the geometry of the liquid-solid interface, reflecting the competition between the rate at which the facets advance and the rate at which the facets come to relative equilibrium.
occurs only at the liquid-solid interface. Therefore if the droplet only partially covers a facet, growth can only occur on the 317
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interface.11 The rate of capture by the wire is simply the growth velocity, eq 1, integrated over the liquid-solid interface. The chemical potential μi of a facet is8 μi = (Ωs/Ai)dF/dni, where Ωs is the atomic volume of the solid, Ai is the facet area, F is the free energy of the solid, and ni is a displacement of the facet normal to itself by attachment of material. For convenience we refer to a facet at the liquid-solid interface as a wet facet, and one at the vapor-solid interface as a dry facet. For our three-phase system and for a wet facet,12 the chemical potential becomes 0 1 Ωs @ dAi X A μi ¼ ð3Þ pAi þ γi þ fj Ai dni j
introduction of facets of every possible orientation at every point on the liquid-solid interface. However, both continuum modeling7 and experimental observations2,3,15 suggest that new facets form only at the trijunction, greatly simplifying the numerical simulation. The specific instabilities considered here are illustrated schematically in Figure 3c-j. If the trijunction is unpinned, as in Figure 3c, a new facet can only be introduced as a step in the existing facet, as in Figure 3d. This breaks the partially wetted facet into a wet facet and a dry facet, allowing the wet facet to participate in the wire growth. At pinned trijunctions, we have two distinct classes of perturbations. In one class, introducing a new facet into Figure 3e makes the solid nonconvex. The new facet could be either wet as in Figure 3f or dry as in Figure 3g. The other class occurs when there is a “missing facet” at the trijunction, as for a solid having an acute internal angle in the 6-fold crystal, Figure 3h. (Missing facets do not occur in two-phase systems,8 they become possible here because of the force exerted by the liquid-vapor interface at the trijunction.) In this case, as the system evolves, it may become favorable to introduce the missing facet, as in Figure 3i,j. In addition to the interfacial energies γi, there is in general an extra energy associated with the edge where two facets meet. Thus, there is an extra energy cost associated with introducing new facets, since this creates additional edges. Little is known about the structure and energetics of semiconductor facet edges, or the microscopic processes by which new facets are introduced. Nevertheless, it is useful to include edge energetics, at least in a crude way, to study what is the qualitative effect on nanowire growth. To this end, when introducing a new facet i, we imagine that the energy per unit area is higher for a very small facet than for a macroscopic one, by a factor of 1 þ εi. This effect extends over a characteristic facet width wi. Therefore, for a facet of width λ (initially infinitesmal), we take the energy (per unit edge length in 3D) to be γi(1 þ εi)λ up to the characteristic facet width wi, saturating to γiλ þ γiεiwi for facet widths greater than wi. Roughly speaking, wi is the spatial extent of the edge, and εiγi is the extra tension needed to introduce a new facet i, playing a role analogous to the parameter Γc in ref 15. In the macroscopic limit, γiεiwi is the extra energy per unit length associated with adding two new edges. Thus the dimensionless parameter εi characterizes the extra difficulty of introducing new facets due to edge effects. In general, εi may depend on whether facet i is wet or dry, and on whether it represents a step (Figure 3d), a nonconvex corner perturbation (Figure 3e-g), or a missing facet (Figure 3h-j). For systems having inequivalent facets, the possibilities multiply rapidly. The choice of εi reflects the microscopic physics of the edges, which unfortunately is not well understood for the systems of interest here. We therefore assume for the present that εi = 0 for missing facets (since we could imagine εi being either positive or negative in this case), while εi has a single non-negative value for all other perturbations. For these, we take εi = 0 unless otherwise stated. In all simulations presented here, we take wi to be sufficiently small that εi plays no role except in determining the initial stability of a given perturbation. At this point, we have all of the elements needed to simulate nanowire growth. To illustrate the potential usefulness of the model, we consider several examples of how phenomena commonly observed in nanowire growth can be reproduced and understood in terms of this model.
Here p = γvlκl is the capillary pressure of the liquid (and hence the pressure applied to the solid at the liquid-solid interface). The next term reflects the change in area of the facet as it moves, with γi being the interface energy for facet i. The final term represents the capillary force exerted on facet i by its neighboring facets, and also by the droplet surface tension if the trijunction lies at the facet edge. (The capillary force exerted by a neighboring facet j is simply the change in energy due to changing length of facet j per unit length of displacement of facet i.) This is illustrated in Figure 3a. We emphasize that the capillary force exerted by the liquid at the trijunction is comparable in magnitude to the other terms, and therefore plays a crucial role in determining the morphology and in controlling the introduction of new facets. We now turn to the droplet statics, which have been well understood since the time of Gibbs.13 As in ref 7, we assume that droplet motion and internal equilibration are rapid on the time scale of nanowire growth. The droplet then has uniform composition and chemical potential and is in mechanical equilibrium, with a uniform scalar surface curvature. (This is consistent with in situ microscopy, where the trijunctions are typically pinned at facet edges, occasionally jumping from one static position to another too quickly to resolve the jump or the subsequent relaxation in video-rate imaging.) For a given configuration of the solid, under these assumptions, specification of the droplet volume and of the trijunction position uniquely determines the droplet geometry, including the contact angles. In equilibrium, the trijunction position adjusts so that the contact angles satisfy the condition of zero force tangential to the surface (Young's equation). For faceted systems, however, the tangent is not defined at the edges where facets meet. Instead, the trijunction is pinned at an edge if a displacement onto either facet leads to a restoring force toward the edge.13 Typically, there will be a range of angles satisfying this condition, bounded by the equilibrium contact angles for the respective facets, as illustrated in Figure 3b. As a result, a trijunction will remain pinned at the facet edge as the system evolves, until the contact angle passes outside the stable range. At this point, the trijunction depins and moves abruptly along the facet until it either satisfies Young's equation, or pins on an adjacent facet edge, or creates a new facet edge on which it can pin, as described below. The third and final elementary process considered here is the introduction of new facets. In the spirit of previous work,7,8 we introduce a new facet of infinitesmal size whenever the system is linearly unstable against such a perturbation. To test linear stability, we introduce an infinitesmal facet and ask whether it grows or shrinks, using the same dynamical rules that govern the wire growth.14 In principle, at every moment we should test the 318
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For simplicity, we consider highly symmetric crystals in 2D, consisting of 6 equivalent facets having 60 external angles, or 12 equivalent facets having 30 external angles. Thus the equilibrium crystal shape is a regular hexagon or dodecagon, as in ref 7. In these systems, there are only three distinct interfacial energies: γvl, γvs, and γls for vapor-liquid, vapor-solid, and liquid-solid interfaces, respectively. These are chosen as in ref 7 to facilitate comparison with the prior results: Ωsγvs = 0.16 eV nm, Ωsγls=0.07 eV nm, Ωsγvl = 0.14 eV nm. In all simulations here, we take a 2D metal-catalyst volume of 5000 nm2, diluted to 80% metal at c0. Steady addition of material from the vapor leads to the growth modes shown in Figure 2. As expected, these closely resemble the results of our earlier continuum simulations.7 For crystals having 6 and 12-fold symmetry, the resulting wire shapes resemble those seen experimentally for Au-catalyzed Si nanowires growing in Æ110æ and Æ111æ directions, respectively.11,16,17 “Top-down” device applications typically require a uniform wire growing normal to the substrate, as in Figure 1a. However, in experiment there is often a mixture of morphologies with undesired shapes occurring side by side with the desired shapes in an uncontrolled manner. While the causes are generally unknown, Wagner reported an intriguing instance of systematic kinking from one growth direction into another: a thermal gradient in the growth chamber led to kinking of wires toward the high-temperature side.2 This suggests that the apparently random kinking frequently observed might occur in response to small variations in local environment. We find that similar effects occur naturally within our model. This is illustrated in Figure 1b. We begin with growth as in Figure 1a. The growth soon reaches a steady state, which would continue indefinitely if left undisturbed. We then perturb the growth by briefly increasing the vapor-solid interfacial energy on the right sidewall by 10%. As shown in Figure 1b, the wire then kinks toward the side with increased energy. We speculate that increased solid-vapor interfacial energy could occur on the hightemperature side due to desorption of passivating molecules from the sidewall, perhaps increasing as the wire grows away from the shielding substrate. An asymmetric incident flux of disilane or other precursor molecules could also lead to differences in passivation and surface energy between the two sides. The behavior in Figure 1b is strikingly similar to that reported by Wagner2 and, under rather different conditions, by Madras et al.3 with the wire pivoting about one edge during the kinking process. We observe similar kinking for 6-fold and 8-fold crystals when we perturb the growth in the same way. The results in Figure 1a,b are obtained without including any edge effects (i.e., εi = 0). It is far from clear how the facet edges are affected by absorbed gases, temperature, impurities, and so forth. Therefore we investigate whether variations in εi could account for any of the unexplained behaviors observed experimentally. In the early stages of growth, the droplet sits on a tapering pedestal, which must introduce new facets to evolve into steadystate nanowire growth. This process is seen in Figure 1a and Figure 2 and has been discussed in ref 7. Increasing εi impedes the introduction of new facets, and so affects the transition from a tapering pedestal to a uniform wire. For small εi, the effect is only a slight change in the shape of the wire base and the width of the wire. However, if εi is too large we find a qualitatively different growth mode, shown in Figure 1c. The initial base continues to taper until the catalyst droplet rolls off. The catalyst then crawls along the surface, growing a lateral wire.
Such lateral wire growth has been observed by numerous authors3,4,18 but the cause was unknown. Our simulations suggest a natural explanation. Under some growth conditions, the difficulty of introducing new facets prevents formation of freestanding wires, and lateral crawling predominates. Growers naturally tend to avoid these conditions. Under borderline conditions, small variations in the local environment of neighboring wires could lead to variations in the difficulty of introducing new facets, so that some cross the threshold at which crawling growth occurs. While changing the growth conditions will naturally change parameters such as ε and γ, little is known at present about the energetics of facet edges, and how facet and edge energies depend on temperature and the partial pressures of different gases. Figure 2 shows that the growth rate can have a striking effect on the wire morphology. This may explain a longstanding puzzle regarding Æ110æ-oriented Si nanowires. Early marker-layer experiments showed that the wire tip has a symmetric shape,2 as in Figure 2a. In contrast, more recent in situ studies clearly see a highly asymmetric shape,16 as in Figure 2b. This apparent discrepancy has never been resolved. In the simulations shown in Figure 2a,b, the reason for the difference is clear. The growth rate of each facet is given by eq 1. In the asymmetric geometry, the smaller facet has a higher weighted curvature8 and thus a higher μi, giving it a smaller velocity than the larger facet. Thus the two facet sizes tend to equalize over time. However, for fast growth the droplet supersaturation dominates, that is, the difference between μi of the liquid and μ of either facet is much larger than the different between the μ values of the two facets. Therefore the two facets propagate at nearly equal velocities despite their different size. For the 6-fold crystal, this tends to preserve the initial asymmetric shape. Nevertheless, the wire is gradually evolving toward a symmetric geometry; we choose a growth rate high enough that this evolution is barely visible in Figure 2b. For slow growth, Figure 2a, we have the opposite situation. The difference between μ of the two facets is comparable to the difference between these and μl, so the evolution to a symmetric geometry is largely completed before the wire has grown far. With these results in mind, the origin of the different experimental observations becomes intelligible. Here “slow” and “fast” growth really refer to the supersaturation of the liquid, relative to the difference in facet chemical potentials.7 Thus it depends on the ratio rvl/Ri of our kinetic parameters. The early work used a mixture of H2 and SiCl4 at high temperature. This leads to a balance between growth and etching which makes it possible to grow very near equilibrium,19 leading to the symmetric structure. More recent in situ microscopy16 used disilane, giving an appreciable growth rate at temperatures so low that the rate constant Ri in eq 1 is extremely small. This leads to large supersaturations.11,20,21 Because the (111) facets are symmetric with respect to the Æ110æ growth direction, the resulting behavior is analogous to our 6-fold crystal. A different effect is seen in Figure 2c,d. For the 12-fold crystal, the interface has a large central facet and two smaller side facets, symmetric about the growth direction. Increasing the growth rate (and hence the liquid supersaturation) adds a constant normal facet velocity to all three facets. From geometry alone, increasing growth rate tends to drive out the side facets, as in Figure 2d. (There are actually still tiny facets present. As the side facets shrink, their μi increases, so that they resist being driven out entirely.) In contrast to the transient effect of Figure 2a,b, this is a 319
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change in the actual steady-state interface structure. Other factors, such as small changes in the surface energies, can also lead to a variety of different steady-state configurations. In our illustrations, we focused in Figure 1 on the role of facet energetics, and in Figure 2 on the role of kinetics. However, in extensive simulations we find that there is actually a complex interplay, which we have only begun to explore. Growth rate can strongly affect the occurrence of kinking and crawling in our simulations (and in experiment3), and changes in facet energy can lead to different steady-state morphologies of the wire tip. Therefore, the examples here, while illustrating real effects, do not represent a unique determination of the detailed mechanism in specific experiments. Rather, they demonstrate how the observed complex behavior can grow naturally out of the interplay of a few simple elements. In conclusion, we have shown that a surprisingly simple model with only three elementary processes can capture the striking variety of behaviors observed in nanowire growth. In this way, we provide the first clear indication of the processes leading to such important behaviors as kinking, crawling, and different wire-tip morphologies. The complex atomic-scale physics of growth enters only implicitly, via the model parameters, especially the εi controlling facet introduction. The simplicity and power of this approach promises to make it a useful tool for further research in the area of VLS nanowire growth.
(16) Kodambaka, S.; Hannon, J. B.; Tromp, R. M.; Ross, F. M. Nano Lett. 2006, 6, 1292. (17) Wu, Y.; Cui, Y.; Huynh, L.; Barrelet, C. J.; Bell, D. C.; Lieber, C. M. Nano Lett. 2004, 4, 433. (18) Fortuna, S. A.; Wen, J.; Chun, I. S.; Li, X. Nano Lett. 2008, 8, 4421. (19) Givargizov, E. I. J. Cryst. Growth 1975, 31, 20. (20) Kodambaka, S.; Tersoff, J.; Reuter, M. C.; Ross, F. M. Science 2007, 316, 729. (21) Adhikari, H.; Marshall, A. F.; Goldthorpe, I. A.; Chidsey, C. E. D.; McIntyre, P. C. ACS Nano 2007, 1, 415.
’ AUTHOR INFORMATION Corresponding Author
*E-mail: tersoff@us.ibm.com.
’ REFERENCES (1) (a) Li, Y.; Qian, F.; Xiang, J.; Lieber, C. M. Mater. Today 2006, 9, 18. (b) Thelander, C.; et al. Mater. Today 2006, 9, 28. (2) Wagner, R. S. In Whisker Technology; Levitt, A.P., Ed.; Wiley Interscience: New York, 1970; p 47. (3) Madras, P.; Dailey, E.; Drucker, J. Nano Lett. 2009, 9, 3826. (4) Wen, C.-Y.; Reuter, M. C.; Tersoff, J.; Stach, E. A.; Ross, F. M. Nano Lett. 2010, 10, 514. (5) Tian, B.; Xie, P.; Kempa, T. J.; Bell, D. C.; Lieber, C. M. Nat. Nanotechnol. 2009, 4, 824–829. (6) Quitoriano, N. J.; Wu, W.; Kamins, T. I. Nanotechnology 2009, 20, No. 145303. (7) Schwarz, K. W.; Tersoff, J. Phys. Rev. Lett. 2009, 102, 206101. (8) (a) Carter, W. C.; Roosen, A. R.; Cahn, J. W.; Taylor, J. E. Acta Metall. Mater. 1995, 43, 4309. (b) Taylor, J. E.; Cahn, J. W.; Handwerker, C. A. Acta Metall. Mater. 1992, 40, 1443. (9) Glas, F.; Harmand, J.-C.; Patriarche, G. Phys. Rev. Lett. 2007, 99, No. 146101. (10) Roper, S. M.; Davis, S. H.; Norris, S. A.; Golovin, A. A.; Voorhees, P. W.; Weiss, M. J. Appl. Phys. 2007, 102, No. 034304. (11) Kodambaka, S.; Tersoff, J.; Reuter, M. C.; Ross, F. M. Phys. Rev. Lett. 2006, 96, No. 096105. (12) The droplet applies a pressure p = γvlκl only to facets covered by the droplet, not to dry facets. (13) This classic result is discussed in detail and applied to nanowire growth by Roper, S. M.; Anderson, A. M.; Davis, S. H.; Voorhees, P. W. J. Appl. Phys. 2010, 107, No. 114320. (14) Strictly speaking, here we assume finite attachment rate R within an arbitrarily small but finite neighborhood of the liquid. Thus it is possible to introduce an infinitesmal dry facet at the trijunction; but once introduced, it cannot move further, it can only grow or shrink in length due to motion of the adjacent wet facet. (15) Ross, F. M.; Tersoff, J.; Reuter, M. C. Phys. Rev. Lett. 2005, 95, No. 146104. 320
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