Letter pubs.acs.org/NanoLett
Multiplicity of Steady Modes of Nanowire Growth K. W. Schwarz and J. Tersoff* IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, United States ABSTRACT: For nanowire growth by the vapor−liquid−solid process, we examine whether there is a unique steady-state growth morphology. Applying a continuum model for faceted nanowire evolution to a model crystal structure, we enumerate the possible growth morphologies and calculate their dynamical stability. We find that even for a single set of experimental conditions there can be multiple distinct modes of steady-state growth. The actual growth mode occurring in experiment thus depends on the initial conditions and growth history. Relevant experiments are discussed. KEYWORDS: Nanowire, VLS, growth, simulation
B
shape allows a number of distinct growth morphologies, as illustrated in Figure 1.
ecause of their technological promise and fundamental interest, nanowires have been the subject of extensive experimental and theoretical investigations. In the vapor− liquid−solid (VLS) process, a small catalyst droplet at the wire tip captures material from the vapor and deposits it at the catalyst−wire growth interface. For most technological applications, reproducibility and uniformity are essential. To achieve these, the growth should involve a unique dynamical steady state. Experimentally, the growth mode often appears to be nonunique, although it has been unclear whether this might result from small variations in local conditions across the sample.1−4 Here we consider VLS growth of a hexagonal crystal in two dimensions (2D), where one can enumerate and test all possible steady-state modes. Surprisingly, we find that even for this simple case the dynamical steady state of nanowire growth is not unique. For typical growth conditions we often find two or more modes being dynamically stable under identical conditions. In regimes where multiple modes are stable, the actual growth mode will depend on the precise initial conditions and growth history. This has important consequences for wire uniformityfor the same catalyst size, different modes result in wires of different diameter. These results help to clarify the requirements for growing identical wires in “bottom up” technologies and provide a new perspective on the diversity of wire morphologies that are observed experimentally. Our nanowire growth model has been described elsewhere.5,6 It uses a classical model of facet evolution based on wellestablished continuum physics.7 The liquid droplet captures material from the vapor at a rate proportional to the liquid− vapor interfacial area and deposits this material at the liquid− solid interface at a rate proportional to the difference between the chemical potentials of the liquid and the facet. New facets are introduced based on a linear stability analysis.6,7 To clearly illustrate the basic issues, we focus here on a single crystal structure in 2D with only 6 equivalent facets, so the equilibrium crystal shape is a regular hexagon. Even this simple crystal © 2012 American Chemical Society
Figure 1. Steady VLS growth modes for our 2D crystal. The stable modes found for free-standing wire growth are: single-facet (1f); symmetric two-facet (2fs); asymmetric two-facet (2fa); symmetric inverted two-facet (2fis); and asymmetric inverted two-facet (2fia). In addition we consider a crawling mode (c) which represents an alternative to free-standing wires. The modes shown were computed using γ̃vs = 1.05, γ̃ls = 0.5, and values of d̃ = 1.10, 1.90, 1.07, 1.70, 2.00, and 1.00, respectively.
Although our general focus is on free-standing wires, we have also included a crawling mode where the catalyst droplet moves along a surface, leaving behind a “lateral nanowire”.,4,8−11 This mode is of some technological interest both as a common problem when freestanding wires are desired and as an alternative mode compatible with traditional planar device structures.9,10 Depending on the values of γ, the lateral wire can rest on top of the original surface or the moving droplet can dissolve some substrate material as it moves, leading to a wire that is embedded into the surface. (The distinction becomes Received: November 2, 2011 Revised: December 26, 2011 Published: January 31, 2012 1329
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Our results for the fixed topology case are shown in Figure 2 as a “dynamical phase diagram”. The system is characterized by
important for heteroepitaxial growth, which lies outside the scope of our current analysis.) In order for a given wire morphology to represent a steady growth mode within our VLS model, it must satisfy three dynamical stability requirements: (1) It must be stable against droplet depinning from the sidewall edges; (2) it must be stable against introduction of a new facet; and (3) if the growth front includes more than one facet, all facets must advance with the same axial velocity, and this uniform advance must be stable against small perturbations. The key inputs into the model are the facet orientations and energies and the surface tension of the catalyst.6 For our hexagonal crystal, this reduces to three parameters: γvs, γls, and γvl for the vapor−solid, liquid−solid, and vapor−liquid interfacial energies, respectively. In addition, the behavior depends on the growth rate and on the volume of the catalyst droplet. The droplet is typically a eutectic solution of a metal catalyst with the nanowire material, e.g., Au and Si. We make the approximation (valid for not-too-fast growth and not-toonarrow wires) that the droplet composition is close to the bulk liquidus composition. Then for a fixed amount of the metal catalyst, the droplet is characterized by a fixed volume Vl. Increasing growth rate increases the droplet supersaturation, which in general has two effects: It increases the droplet volume and composition, an effect that we can ignore for not-too-fast growth rates; and if there are multiple growing facets at the liquid−solid interface, it changes the relative velocity of these facets if they are at different angles to the wire axis. For our hexagonal geometry, all growing facets are at the same angle to the wire axis, so we can neglect growth rate in our analysis. For a given set of parameters (interfacial energies γ and droplet volume Vl), we determine the steady growth modes as follows. We consider each of the candidate wire “topologies” illustrated in Figure 1. For each topology, we consider all possible geometries. The geometry is characterized by the wire diameter, and if there is more than one growth facet, by the relative sizes of these facets. (In our 2D model, there are at most two growth facets, so this introduces one additional variable besides diameter.) For each wire geometry, we calculate the droplet geometry and then test the three stability criteria listed above. In determining the steady growth modes, we are not concerned with the overall size or energy scales of the system. If we take γvl and Vl as setting these overall scales, we need only consider two dimensionless interfacial energy parameters, γ̃vs = γvs/γvl and γ̃ls = γls/γvl. Also, the wire diameter d only matters relative to the overall size scale set by the droplet, so we need only consider the dimensionless diameter d̃ = d/Vl1/2. [In 3D this would be d̃ = d/Vl1/3.] Our growth model includes an additional parameter ε characterizing the energetics for introducing a new facet edge.6,12 For simplicity, here we only address two limits. One limit is where the topology is fixed, i.e., a prohibitively large value of ε suppresses creation of new facet edges. Then we can simply omit the test for stability against introducing new facets. For cases where there is only a single facet at the liquid−solid interface, stability is then limited only by droplet depinning, which has been addressed by Roper et al.13 for a different geometry. The other limit is where edge-energy effects are negligible (ε = 0 in the model of ref 6), so introduction of new facets is controlled by the γ̃ values and the geometry. We expect that real systems lie somewhere between these limits.
Figure 2. (a) Regions in the γ̃vs−γ̃ls plane where the various modes can occur, in the limit that new facets cannot be introduced. The cross marks the location γ̃vs = 1.05 and γ̃ls = 0.5 used to compute Figure 1. Dotted line is region discussed in text where solid “wets” liquid. (b) The range of diameter d̃ over which each mode occurs for γ̃vs = 1.05 and γ̃ls = 0.5.
the two dimensionless interfacial energies γ̃vs and γ̃ls, and these energies will be different for different materials (e.g., Si vs Ge, different catalysts) and even for the same system under different experimental conditions (temperature, source-gas pressure, etc.4,14,17). Thus different possible growth systems correspond to different points in the γ̃vs−γ̃ls plane. We label the regions of the plane according to which modes are dynamically stable. There are two regions in Figure 2 occupying two corners of the “phase diagram”, where no nanowire growth is possible.13 The upper left corner corresponds to complete wetting of the solid by the liquid. In this regime there cannot be a localized droplet. Instead the entire solid is wetted by liquid, and growth occurs everywhere indiscriminately. The lower right corner corresponds to the completely nonwetting regime, with the liquid−solid contact area shrinking to zero. In addition, the region in the lower left corner is generally unfavorable for VLS growththe energy is lowered if the liquid becomes encapsulated by solid, which would prevent further growth. We have verified that for an isotropic crystal model5 wires cannot grow for this range of parameters. Nevertheless, with the limited set of facets included here, a growing nanowire can remain dynamically stable in the region. The most striking feature of Figure 2 is that typically there are multiple steady growth modes. An extreme instance is the region containing the point indicated by a cross, corresponding 1330
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to γ̃vs= 1.05 and γ̃ls = 0.50. For these values, which fall within the plausible range for Au-catalyzed Si and Ge nanowires, every growth topology in Figure 1 is dynamically stable. It is important to note that for a given point in the γ̃vs−γ̃ls plane and for a given catalyst volume, each steady mode is actually stable over a range of wire diameters. For the point marked by a cross in Figure 2a, the stable diameter range for each mode is shown in Figure 2b. (The 2 fs mode exhibits two separate ranges of d̃, apparently reflecting an instability against going asymmetric at intermediate diameters.) Thus not only does the growth mode depend on the initial conditions and growth history but also for each mode the wire diameter can depend on these as well. In experiment, the initial conditions include the substrate orientation (including miscut) and the degree of annealing to react the catalyst with the substrate prior to growth. Sometimes patterning or masks are employed to give additional control. Because there is a transient period after growth begins, during which new facets form,5,6 the connection between initial conditions and final steady state is not simple or one-to-one. The connection between initial conditions and final steady state is even more complex for heteroepitaxial or nonepitaxial growth. And if conditions change during growth, the final steady state may in general depend on the entire history prior to the point where conditions stabilized. In principle one could give a more complete picture with a 3D diagram in γ̃vs−γ̃ls−d̃, showing the regions in which the respective modes can occur. However, γ̃vs and γ̃ls are fundamental system parameters, while d̃ values represent possible outcomes depending on the initial conditions. We therefore consider it more useful to project the intractable 3D diagram onto the γ̃vs−γ̃ls plane, to simply show which modes can occur for a given pair of values of the material parameters γ̃vs and γ̃ls. Another way to look at this projection is to consider slices from the 3D diagram corresponding to fixed values of d̃, as shown in Figure 3. Figure 4 shows our results for the opposite limit, where edge energies are negligible and the introduction of new facets is limited only by whether this increases or decreases the total of the interfacial energies.6,7 The requirement of stability against new facets shrinks the range of dynamical stability of the various modes. In particular the 1f mode does not occur for any γ values in this limit. Nevertheless, there remain large regions where multiple modes are possible under identical conditions. We note in particular that the modes 2fa and 2fs coexist for a wide range of γ’s in both limits and also with 1f in the fixededge limit of Figure 2. This is intriguingly reminiscent of the experimental situation for Si ⟨110⟩ nanowires. These have been clearly observed in symmetric geometries that are close 3D analogs of our 2D 2fs structure.1,15 But ⟨110⟩ nanowires have been equally observed in a highly asymmetric structure analogous to our 2fa.16 Nevertheless, we must emphasize that the two modes have never been observed under identical conditions, so at this time there is no direct connection between such experiments and our theoretical results. Indeed, for two different structures to occur under identical experimental conditions would require some difference in initial conditions or subsequent perturbations between the respective wires. Lateral nanowire growth via the crawling mode has been observed by numerous authors,9,4,10,8,11 despite a geometry unfavorable for transmission electron microscopy. This is less surprising in view of the mode’s uniquely broad stability range.
Figure 3. Connection between d̃ and the mode stability regions in Figure 2, illustrated for the 2fs mode. Red lines indicate the stability regions for the 2fs mode for specific values of d̃. (For clarity we only show a limited range of values, ranging from 0.6 down in increments of 0.1, except that zero is replaced by 0.02.) Green dots indicate the corners of these regions, over the entire range of d̃ (i.e., including values larger than 0.6). Superposing all the regions that are stable for different values of d̃ leads to the outer envelope indicated by a blue line.
Figure 4. Regions in the γ̃vs-γ̃ls plane where the various modes can occur, in the limit that edge energies can be neglected (ε = 0, the case where new facets are most easily introduced). The cross marks the location γ̃vs = 1.05 and γ̃ls = 0.5. Dotted line is region discussed in text where solid “wets” liquid.
In contrast, the “inverted” modes 2fis and 2fia in Figure 1 have not to our knowledge ever been observed experimentally. They require a relatively high droplet surface tension, relative to the solid surface and interface energies, and it is not clear whether there exists suitable materials within the required range of γ. In addition to the steady modes discussed so far, we have also identified a quasi-steady sawtooth mode shown in Figure 5. Strictly speaking, this mode is oscillatory rather than steady, 1331
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REFERENCES
(1) Wagner, R. S. In Whisker Technology; Levitt, A. P., Ed.; Wiley Interscience: New York, 1970. (2) Schmidt, V.; Senz, S.; Gösele, U. Nano Lett. 2005, 5, 931. (3) Jagannathan, H.; Deal, M.; Nishi, Y.; Woodruff, J.; Chidsey, C.; McIntyre, P. C. J. Appl. Phys. 2006, 100, 024318. (4) Madras, P.; Dailey, E.; Drucker, J. Nano Lett. 2009, 9, 3826. (5) Schwarz, K. W.; Tersoff, J. Phys. Rev. Lett. 2009, 102, 206101. (6) Schwarz, K. W.; Tersoff, J. Nano Lett. 2011, 11, 316. (7) Carter, W. C.; Roosen, A. R.; Cahn, J. W.; Taylor, J. E. Acta Metall. Mater. 1995, 43, 4309. (8) Wen, C.-Y.; Reuter, M. C.; Tersoff, J.; Stach, E. A.; Ross, F. M. Nano Lett. 2010, 10, 514. (9) Fortuna, S. A.; Wen, J.; Chun, I. S.; Li, X. Nano Lett. 2008, 8, 4421. (10) Quitoriano, N. J.; Wu, W.; Kamins, T. I. Nanotechnology 2009, 20, 145303. (11) Zhang, G.; Tateno, K.; Gotoh, H.; Nakano, H. Nanotechnology 2010, 21, 095607. (12) Ross, F. M.; Tersoff, J.; Reuter, M. C. Phys. Rev. Lett. 2005, 95, 146104. (13) Roper, S. M.; Anderson, A. M.; Davis, S. H.; Voorhees, P. W. J. Appl. Phys. 2010, 107, 114320. (14) Dailey, E.; Madras, P.; Drucker, J. Appl. Phys. Lett. 2010, 97, 143106. (15) Wu, Y.; Cui, Y.; Huynh, L.; Barrelet, C. J.; Bell, D. C.; Lieber, C. M. Nano Lett. 2004, 4, 433. (16) Kodambaka, S.; Hannon, J. B.; Tromp, R. M.; Ross, F. M. Nano Lett. 2006, 6, 1292. (17) Schwarz, K. W.; Tersoff, J.; Kodambaka, S.; Chou, Y.-C.; Ross, F. M. Phys. Rev. Lett. 2011, 107, 265502.
Figure 5. (a) Sawtooth wire, calculated for γvs0 = 0.95 and γls0 = 0.5. Structure shown is for ε = 0.05 to give sawtooth facets that are easily visible; for ε → 0, the facets become infinitesmal. (b) Stability range for this mode (in the limit of small ε) is indicated by red dots connected by solid lines. This range is superposed on a simplified copy of Figure 4. For points inside the region, the sawtooth mode is stable in addition to the modes indicated in Figure 4.
with the continual introduction of new facets. The size of the facets scales with the edge-energy parameter ε. We call this mode quasi-steady because in the small-ε limit of Figure 4, the sawtooth facets become infinitesmal in size, so the wire appears smooth, and the growth appears continuous. The range over which this growth mode occurs is indicated in Figure 5b. This mode bears a striking resemblance to the Au-catalyzed Si ⟨111⟩ nanowires observed experimentally.12 Both here and in experiment, the wire sidewalls on average run normal to the (111) growth interface, but they are actually composed of a sawtooth of tiny angled facets. Moreover, in experiment as here, this mode competes16,17 with angled modes resembling 2fs and 2fa. Despite these intriguing similarities with experiments, in the context of the present 2D hexagonal model, we do not consider the sawtooth mode strictly realistic. While it is stable against low levels of random noise, it is not stable against a systematic bias where the sidewall energies are very slightly higher on one side of the wire than the other. Therefore we expect that under realistic conditions, any small environmental asymmetry associated with the flux of source gas or the temperature would destabilize this mode. In contrast, the other modes discussed here are all robust against such bias (as well as being robust against noise). In experiment, the sawtooth mode has both {113} and {111} facets in the sidewall, and in the absence of {113} facets, no sawtooth occurs. As discussed in ref 17, the sawtooth mode can similarly be stabilized within the present model by including an additional set of facets. In conclusion, we have identified the stability regime for the various VLS growth modes of a simple crystal model. This provides a new perspective on several aspects of nanowire growth. In particular, we find that for a specific set of conditions, there are typically multiple steady growth modes that are dynamically stable. Moreover, for the same catalyst size they yield wires with different diameters. Even a single mode can grow stably over a range of diameters. In such a situation, to guarantee arrays of identical wires may require that the starting configuration for each wire be the same.
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