The Role of Surface Energies and Chemical Potential during

LETTER pubs.acs.org/NanoLett

The Role of Surface Energies and Chemical Potential during Nanowire Growth Rienk E. Algra,†,‡,§ Marcel A. Verheijen,‡,|| Lou-F
e Feiner,‡,|| George G. W. Immink,‡ Willem J. P. van Enckevort,§ Elias Vlieg,§ and Erik P. A. M. Bakkers*,‡,|| †

Materials innovation institute (M2i), 2628CD Delft, The Netherlands Philips Research Laboratories Eindhoven, High Tech Campus 11, 5656AE Eindhoven, The Netherlands § Institute for Molecules and Materials, Radboud University Nijmegen, Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands Photonics and Semiconductor Nanophysics, Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands

)

‡

bS Supporting Information ABSTRACT: We present an approach to quantitatively determine the magnitudes and the variation of the chemical potential in the droplet (Δμ), the solid-liquid (γSL) and the liquid-vapor (γLV) interface energies upon variation of the group III partial pressure during vapor-liquid-solid-growth of nanowires. For this study, we use GaP twinning superlattice nanowires. We show that γLV is the quantity that is most sensitive to the Ga partial pressure (pGa), its dependence on pGa being three to four times as strong as that of γSL or Δμ, and that as a consequence the surface energies are as important in determining the twin density as the chemical potential. This unexpected result implies that surfactants could be used during nanowire growth to engineer the nanowire defect structure and crystal structure.

KEYWORDS: Nanowire, VLS mechanism, chemical potential, superlattice, twin, gallium phosphide

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or all applications and to fully use the potential of nanowires it is important to understand and control their structural properties.1-3 The defect density and crystal structure of nanowires grown by the vapor-liquid-solid (VLS) growth mechanism can be tuned by additive atoms,4 temperature,5,6 diameter,7 III-V ratio8 or by a combination of these external parameters.9 It has been argued that these parameters can affect the chemical potential in the droplet, Δμ, the nanowire surface energy, γSV, and the solid-liquid or the liquid-vapor interface energies, γSL or γLV,4-13 which in turn drive the formation of planar defects and/or specific crystal structures. However, these energies have not been determined experimentally and their effect on nanowire growth is not clear. The formation of a twinning superlattice in nanowires, in which the distance between neighboring twin planes is approximately constant, has been quantitatively explained with a model based on two-dimensional nucleation and droplet deformation.4,14 This model gives the particular possibility to reveal the contribution of individual parameters to the nanowire growth r 2011 American Chemical Society

from the combination of experiment and theory. In this study, we use GaP twinning superlattice nanowires and investigate the effect of the trimethyl-gallium (TMG) vapor pressure, pGa, on the defect plane density for a range of nanowire diameters. From model fits to the experimental data we calculate Δμ, γSL, and γLV. We have synthesized twin superlattices in GaP nanowires via the VLS growth mechanism on (111)B GaP substrates at a temperature of 550 °C (see ref 15 for more details). Wires grown without additives have the wurtzite crystal structure containing many defect planes.15 Following the strategy to make twinning superlattices (TSLs) in InP,4 we have added diethylzinc (DEZn). With a DEZn partial pressure (PDEZn) above ∼7
10-3 mbar, the wires crystallize in the zinc-blende structure with randomly spaced defect planes, and for PDEZn >1.4
10-2 mbar the wires exhibit a periodic twinning structure. Below 550 °C, no Received: December 7, 2010 Revised: January 24, 2011 Published: February 18, 2011 1259

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Figure 1. Fabrication of GaP twinning superlattice nanowires. (a) Different types of morphology of GaP wires as a function of the III-V ratio and temperature as reported in ref 14. Here type 1 and 2 represent morphologies with the zinc blende crystal structure. Type 3 represents the wurtzite crystal structure. By adding sufficient diethylzinc at 550 °C wires with a twin superlattice were formed (type 4). We note that the samples with DEZn have been grown on a GaP substrate, whereas the samples in the III-V versus temperature graph were grown on SiO2. (b) SEM image of a 100 nm diameter GaP twin superlattice nanowire. The characteristic regular {111} faceting can be seen, which induces the formation of the twin superlattice. (c) TEM image revealing the twin superlattice in GaP. (d) High-resolution TEM revealing the zinc blende crystal structure and a {111} facet.

periodic structures were obtained even with high PDEZn (see Figure 1a). Although it is not clear what the effect of the addition of Zn is, it appears to have the general effect to change the crystal structure from wurtzite16 to zinc-blende and induce a TSL above a certain concentration. All data in this paper has been collected from wires grown with a PDEZn of 1.4
10-2 mbar. In Figure 1b, a scanning electron microscopy (SEM) image is shown of a GaP nanowire that reveals a periodic structure, typical for a twinning superlattice. The TSL can be seen in more detail in the bright-field transmission electron microscopy (TEM) images (Figure 1c) where the twin planes are at a constant distance. These planar defects can be recognized by the alternating change in contrast. From high-resolution TEM (HRTEM) in Figure 1d, the crystal structure is determined to be pure zinc-blende. Additionally, the characteristic {111} side facets, which make an angle of 19.5° with respect to the growth direction, are clearly visible. During growth, these nonparallel side facets result in a continuously changing shape of the nanowire top facet, which is in contact with the liquid metal particle. This leads to a distortion of the liquid particle, changing the contact angles with the nanowire. At a specific level of distortion a twin plane is formed, inverting the structure, and the process starts over again. The repetition of this process results in the formation of periodic twin superlattices.4 We note that the present work is the first realization of a twinning superlattice

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Figure 2. Effect of pGa on segment length. (a) TEM images of GaP TSL nanowires for different pGa. All images are taken at the same magnification for direct comparison. (b) Segment length distribution for different pGa. The data are collected by counting 56-253 segments in a single wire with diameters between 51 and 61 nm.

structure with long-range order in a semiconductor with an indirect bandgap,17 which allows to test the predicted effects on optical properties. For prospective optical studies, it is important to be able to tune the periodicity of the superlattice structure. We exploit the Ga partial pressure,18 pGa, as a new parameter, which allows us to tailor the defect plane density without affecting other important nanowire properties, such as diameter, and dopant concentration. In Figure 2a, TEM images, which are all taken at the same magnification, are shown of GaP TSL nanowires grown with pGa = 9.2
10-5 (top) to 9.2
10-3 (bottom) mbar. Note that the diameters of these wires are all within a small range (51 nm < D < 61 nm). Importantly, the segment length increases with decreasing pGa. In Figure 2b, the segment length distributions, obtained from high-resolution TEM studies, are plotted for the different Ga pressures. The narrow distributions in the histograms demonstrate the quality of the periodicity of the superlattice structures. The absolute width of the distribution does not significantly change with pGa (except for the highest pGa, which is discussed in Supporting Information S1). The range in which a TSL can be controllably grown is limited to 2.3
10-4 < pGa < 4.6
10-3 mbar (see Supporting Information S1). We have studied the dependence of the segment length on the wire diameter (Figure 3a). Clearly, the segment length increases with diameter (for a given pGa) following the same trend as observed for InP wires (Supporting Information S2).4 In the following, we analyze the data in Figure 3a with the help of 1260

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Figure 3. Fits to experimental data. (a) Segment length of a twin superlattice as a function of diameter for different pGa ranging from 9.2
10-5 to 9.2
10-3 mbar. The solid lines are curves fitted by the model according to eq 1. The gray box indicates the regime where the distributions shown in Figure 2b have been determined. (b) The fit parameters A and Δ as determined from panel a versus pGa. (c) The width of the segment length distribution as measured with HRTEM (red triangles) and calculated from the fit parameters A and Δ (black curve) as a function of ln(pGa/p0) with reference pressure ~ and Δ1/2, respectively, plotted versus ln(pGa/p0). p0 = 9.2
10-4 mbar. (d) Values for γLV sin(δ0) and Γ/Δμ as obtained from Δ1/2/A

the growth model presented in ref 4, using only the data for the middle five Ga partial pressure values, for which the nanowires are well within the TSL regime (compare Figure 2). To elucidate the effect of pGa on the growth of the TSL, we have first fitted for each pGa the segment length (Ns) as a function of the nanowire diameter, D, by   1 -Δ=Nc Þ ð1Þ Ns ¼ 2Nc 1 þ lnð1 - e Δ Here Nc = AD is half the critical segment length at which twin formation becomes the more favorable process, and Δ = δΔG*0/ kBT with δΔG*0 being the difference between the free energy of formation of a critical 2D nucleus that induces a twin, and a critical nucleus that leads to continuation of the side facets. The good agreement between the data and the fits shows that the formation of the GaP TSL is accurately described by the model. The two fitting parameters A and Δ are plotted versus Ga partial pressure in Figure 3b. Here the relative errors in the

individual values of both parameters are estimated to be smaller than 5%. Since we now have quantitative values for Δ and A, we can calculate the width of the distribution of segment lengths as a function of pGa for the particular case where we have detailed experimental data, that is D ≈ 55 nm and at various pGa (see Figure 2b). According to the model, the halfwidth, δNs, is given by δNs ¼

2AD ¼ Δ

Ns   1 Δ 1 þ lnð1 - e-Δ=NC Þ Δ

ð2Þ

which is plotted in Figure 3c together with the data. Comparison shows that the overall dependency on ln(pGa/p0) is reproduced well, demonstrating the consistency of the model description. The absolute width, however, is off by a factor of about 3 for reasons still not understood, like in the case of InP.4 ~ = According to the model,4 after transformation of A to A A/(1 - dhA/Δ) where d = 4.46 (a geometrical constant), the 1261

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Figure 4. Chemical potential and surface energies during nanowire growth. (a) Growth velocity v versus TMGa partial pressure, displayed as ln(v/v0) versus ln(pGa/p0) with reference velocity v0 = 1.4 nm/s and reference pressure p0 = 9.2
10-4 mbar. The growth velocities refer to nanowires with a diameter of 51-61 nm. The data points are measured values, and the curve is the fit by eq 5 for Q0 = 90. (b) Chemical potential in the liquid, Δμ = μL - μS, and interface energies γLV and γSL, and Γ = γSL þ x (γSV - γSL - γLV cos δ0) as a function of ln(pGa/p0). (c) Segment length versus ln(pGa/p0). The data points are measured values for nanowires with diameter of 51-61 nm (same as in the gray box in Figure 3a), the curves are calculated using eq 1 for D = 54 nm with (black curve)Δμ, γSL, and γLV all depending on pGa and parameters as quoted in the text, with (blue curve) only Δμ depending on pGa and with (green curve) only γSL and γLV depending on pGa. The middle five values of pGa pertain to nanowires in the TSL regime (see Figure 2).

fitting parameters are related to the physically relevant quantities by   γT Γ ~ ¼ b A ð3Þ h2 γLV sin δ0 Δμ and

 2 c2 Γ γT Δ¼ Δμ kB T

ð4Þ

Here Γ = γSL þ x (γSV - γSL - γLV cos δ0), which is the effective sidewall energy (per unit area) of the nucleus, which takes into account that one of the edges is in contact with the vapor instead of the liquid.10,11 Further, γT is the twinning energy (0.021 J/m2),19 h is the height of a monomolecular growth layer, δ0 is the contact angle between the droplet and the {111}B side facet, Δμ is the difference in chemical potential (per unit volume) of GaP in the liquid droplet and the solid, and x, b, and c are

geometrical constants (for a nucleus of hexagonal shape, x = 1/6, b = 1.16, and c = 1.98). From the relations 3 and 4, we directly ~ and Δ1/2, reobtain γLV sin δ0 and Γ/Δμ (from Δ1/2/A spectively), which have been plotted versus ln(pGa/p0) in Figure 3d, where p0 = 9.2
10-4 mbar is chosen as the reference pressure. In the remaining part of this letter, the subscript 0 refers to this standard pressure condition. Because the externally varied physical quantity is the chemical potential of the vapor, μV = μV,0 þ kBT ln(pGa/p0), the natural independent variable is ln(pGa/p0), and one expects that the pressure dependence of any physical quantity of interest can be represented by an expansion of the form X = X0 [1 þ uX ln(pGa/ p0) þ wX ln2(pGa/p0) ] with small quadratic coefficients. This is corroborated by the linear behavior of γLV sin δ0 shown in Figure 3d, where uLV = 0.266 while wLV = 0 within experimental accuracy (See Supporting Information S3 for other physical quantities). The offset value Γ0/Δμ0 = 0.88 nm (=R*0, the critical radius of the 2D nucleus without twinning at p0) is consistent 1262

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with our observation that at pGa = 4.6
10-4 mbar nanowire growth still occurs at D = 5 nm, but is suppressed at D = 2 nm owing to the Gibbs-Thomson effect.20 Whereas the offset value γLV,0 sin δ0 = 0.201 J/m2 is determined highly accurately by the experimental data (relative error less than 2%; see Figure 3d), the accuracy of γLV,0 is limited by the uncertainty in δ0. We assume that the contact angle does not depend on pGa, and in accordance with previous estimates4,13 we take δ0 = 7°, which gives γLV,0 = 1.65 J/m2. From Figure 3d, it is clear that pGa affects γLV much stronger than Γ/Δμ. Remarkably, γLV increases with pGa, which corresponds to an increasing Ga concentration in the Au-Ga droplet. The increase of γLV can be understood by a decrease of the atomic volume when Ga is added to Au (see Supporting Information S4).21 As a result, the number of dangling bonds per unit area at the liquid-vapor interface increases and with this the surface energy increases. This shows that it is not permitted to calculate the liquid-vapor surface energy of the Au-Ga alloy by simple interpolation of the surface energies of Ga and Au, in which replacement of Au atoms by Ga atoms is assumed.22 Separate values for Γ and Δμ cannot be determined from the segment length data, because all measurable quantities depend only on (a power of) their ratio. To disentangle these energies, we use the nanowire growth rate as a function of pGa as measured for the nanowires in the small diameter range 51 nm < D < 61 nm (see Figure 4a). We find that the growth is not limited by diffusion or decomposition of reactants22-24 (see Supporting Information S5), but by the nucleation kinetics at the dropletnanowire interface. The growth rate, v, is then given by25 (see Supporting Information S6) -ΔG=kB T

ν ¼ rðΔμ, ΤÞe

ð5Þ

where the by far dominant dependence on Δμ is in the exponentional term, where Γ2 Γ2  ξkB T  kB TQ ΔG ¼ c2 h Δμ Δμ

ð6Þ

which is the formation energy of a 2D nucleus without twinning and is seen to depend on different powers of Γ and Δμ. The parameter Q0 = ξΓ02/Δμ0 has been determined (for details, see Supporting Information S6) by fitting eq 5, in which v depends on pGa through Γ and Δμ, to the five central data points in Figure 4a, which correspond to the growth of nanonowires with well-developed TSLs. A good fit is obtained for a rather wide range of Q0, viz. 65 < Q0 < 110. Figure 4a shows the fit to ln(v/v0) for Q0 = 90. From this we obtain the offset values Δμ0 = 6.77 eV/ nm3, that is, Δμ0 Ω = 275 meV per molecule (Ω = 0.0405 nm3 is the molecular volume), and Γ0 = 0.95 J/m2, from which one obtains using14,26 γSV = 0.96 J/m2 and the value for γLV,0 found above, γSL,0 = 1.28 J/m2. We note here that there are substantial (∼25%) error bars on the offset values of these parameters (due to the uncertainty of Q0), but their variation with pGa shown in Figure 4b (and described by uΔμ = 0.065, wΔμ = 0.006, uSL = 0.092, and wSL = 0.001) is accurate within 10%. Surprisingly, Δμ increases by less than 20% for an increase of pGa by almost 2 orders of magnitude, while γLV has more than tripled in this pressure range. Note that the dependence of γSL on pGa is much weaker, and therefore the effects of γSL and γLV on Γ nearly compensate one another, such that Γ hardly changes with pGa. These results show that the physical quantity that is most affected by a change of the group III pressure is the liquid-vapor

interface energy, γLV, and not the chemical potential, Δμ, as has been assumed. Finally, since we now know how the chemical potential and the surface energies depend on pGa, we can reveal their separate effect on the twin density. In Figure 4c, we have plotted the observed segment lengths for wires with diameters around 55 nm (i.e., those in the gray area in Figure 3a) as a function of pGa. The black curve shows the description by the model (eq 1) when we use the offset values and coefficients for Δμ, γSL, and γLV as determined above, and, as expected, this gives an excellent agreement (note that only the middle five values of pGa are relevant here, being in the pressure range where the nanowires show well-defined TSLs (compare Figure 2)). In the other curves, we show what happens if we allow either only Δμ or only γLV and γSL to depend on pGa, by putting the appropriate coefficients u and w equal to zero while leaving all offset values untouched. It shows that Δμ alone does not reproduce the behavior at small pGa, while the surface energies by themselves underestimate the pressure dependence in the whole range, which also demonstrates that their respective contributions are not simply additive. Thus, we can conclude that the dependencies of both chemical potential and surface energies on the group III partial pressure are crucial in controlling the behavior of the twin density. These general insights are essential for a better understanding and proper modeling of nanowire growth.

’ ASSOCIATED CONTENT

bS

Supporting Information. Limits of controlled fabrication of twinning superlattices in GaP nanowires, comparison with InP, pressure dependence, atomic volumes in the crystalline Au-Ga system, rate-limiting step for GaP nanowire growth, analysis of growth velocity measurements, and additional figures, table, and references. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This research was carried out under project number MC3.05243 in the framework of the strategic research program of the Materials Innovation Institute (M2i) (www.M2i.nl), the former Netherlands Institute of Metals Research, the FP6 NODE (015783) project, the ministry of economic affairs in The Netherlands (NanoNed) and the European Marie Curie program. ’ REFERENCES (1) Mattila, M.; Hakkarainen, T.; Mulot, M.; Lipsanen, H. Crystalstructure-dependent photoluminescence from InP nanowires. Nanotechnology 2006, 17, 1580–1583. (2) Akopian, N.; Patriarche, G.; Liu, L.; Harmand, J.-C.; Zwiller, V. Crystal Phase Quantum Dots. Nano Lett. 2010, 10, 1198–1201. (3) Ikoni
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