Composition of Gold Alloy Seeded InGaAs Nanowires in the

Mar 13, 2017 - Synopsis. We use binary nucleation modeling to calculate the composition of gold particle seeded InGaAs nanowires in the nucleation lim...
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Composition of Gold Alloy Seeded InGaAs Nanowires in the Nucleation Limited Regime Jonas Johansson* and Masoomeh Ghasemi NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden ABSTRACT: We explain the composition of gold alloy particle seeded InGaAs nanowires in the nucleation limited regime. We use binary nucleation modeling to account for the nucleation of InGaAs from a supersaturated quaternary liquid alloy particle containing Au, In, Ga, and As. In our modeling we use realistic chemical potential differences between the seed particle and the nanowire. The chemical potentials are calculated from assessed thermodynamic parameters for Au, In, Ga, As, InAs, and GaAs in all relevant phases. Using binary nucleation theory we are able to link the composition of the seed particle to the composition of the nanowire under different conditions. We vary the seed particle concentrations of As, Ga, and In and the temperature. For each of these conditions, we calculate the Gibbs free energy landscape for nucleation. The size and composition of the critical nucleus is given by the location of the saddle point in this free energy landscape. We foresee that these results will be essential for understanding the limitations of composition control in gold alloy seeded InGaAs nanowires.

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INTRODUCTION III−V semiconductor nanowires are highly versatile nanoscale objects with great potential in several applications. Especially promising are the energy related applications in solid state lighting1,2 and photovoltaics,3 but other applications are also being considered. Some examples include life sciences,4 thermoelectrics,5 electronics,6 and photonics.7 Nanowires are also utilized as test systems for low dimensional quantum physics.8−10 Ternary materials combinations are especially promising for nanowires since they can allow for bandgap engineering along the nanowire, enabling, for instance, the fabrication of heterojunctions or quantum dots. This approach requires composition control, and important steps in that direction has been taken by Wu et al., who have demonstrated composition control in InxGa1−xAs nanowires grown by metal−organic vapor phase epitaxy (MOVPE).11 Caroff et al. have also demonstrated composition control in the same materials system, and in addition, they explained their findings using a mass transport model for the incorporation of InAs and GaAs.12 Moreover, Dubrovskii has developed an analytical model for the composition of ternary nanowires in the limit of irreversible growth.13 In the current investigation, the aim is to explain the composition of gold alloy particle seeded InxGa1−xAs nanowires in the nucleation limited regime and relate it to the composition of the particle. Here we use the recently developed knowledge of the chemical potential in the alloy particle, which is a quaternary liquid alloy containing the elements As, Au, Ga, and In.14 Moreover, we use binary, or two-component, nucleation modeling15−18 to link the composition of the solid nanowire to the composition of the seed particle under various conditions. © XXXX American Chemical Society

Binary nucleation theory is rarely used in modeling the growth of solid binary alloys, and this is the first time, to our knowledge, that binary nucleation theory is being used in nanowire growth modeling. Another kind of two-parameter nucleation theory has previously been used to model the nucleation of islands in lattice mismatched systems with size and aspect ratio as the two variables.19,20 The focus of this investigation is, however, to use binary nucleation modeling to characterize the free energy surface for nucleation of InxGa1−xAs from a supersaturated gold based quaternary liquid alloy and to link the composition of the solid nucleus to the composition of the liquid alloy.

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FREE ENERGY SURFACE Our approach generalizes the nucleation model by Dubrovskii et al.21 to the case of binary nucleation. That is, we consider a mixed nucleus containing two materials, in this case InAs and GaAs. Due to nucleation on a {111} interface, we consider a triangular nucleus, equilateral with the side length r and one bilayer thick (thickness h). The system we consider is schematically shown in Figure 1. The Gibbs free energy for this nucleation scenario can be written as ⎡ 2γ (ΩS − Ω L) sin β ⎤ ⎥(iInAs + iGaAs) ΔG = −⎢Δμ − LV R ⎣ ⎦ + 3rh Γ

(1)

Received: November 15, 2016 Revised: February 22, 2017 Published: March 13, 2017 A

DOI: 10.1021/acs.cgd.6b01653 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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the triangular nucleus. The side length of the equilaterally triangular nucleus is given by r=

Γ=

where iInAs and iGaAs are the numbers of InAs and GaAs pairs in the nucleus, respectively. The chemical potential difference between the quaternary liquid and the ternary nanowire is given by

⎞ Ω 2 1⎛ γSL + ⎜γSV − γLV L sin β ⎟ 3 3⎝ ΩS ⎠

(7)

The composition dependent interface energies γSL and γSV, solid−liquid and solid−vapor, respectively, are calculated using the Butler equations23 as described in Ghasemi et al.24 For the pure InAs and GaAs we choose the solid−vapor surface energies of the {111}B surfaces since we model the nucleation of zinc blende nanowires. These wires are generally highly twinned, and the twin lamellae are octahedron shaped with inclined {111}A and {111}B side facets.25 Calculations indicate that the {111}B surface energy is slightly smaller than the {111}A;26,27 therefore, we consider nucleation at the {111}B facets and choose the solid−vapor surface energies, γSV InAs and γSV GaAs, for {111}B. For the solid−liquid interface energy in the case of pure GaAs, we use the value calculated by Sakong et al.28 The corresponding interface energy for pure InAs, γInAs SL , is unknown, and we estimate this based on the assumption that the ratio, α, between the pure component interface energies is SV composition independent, that is, we assume α = γSL InAs/γInAs = SV γSL /γ . The composition-dependent interface energies as GaAs GaAs calculated using the Butler equations at 750 K are shown in Figure 2. In this investigation, we consider the nanowire to be in the zinc blende crystal phase. For the wurtzite crystal phase, the thermodynamic parameters and the surface energies are less known, but the surface energies are generally assumed to be

(2)

where x is the indium composition of the solid nucleus, given by iInAs x= iInAs + iGaAs (3) The chemical potentials of the liquid, μLX (X = In, Ga, and As), all depend on temperature, T, and the composition of the liquid, given by y, c3, and cAs, where c3 is the sum of the atomic fractions of In and Ga, c3 = cIn + cGa, cAs is the atomic fraction of arsenic, and y is the indium part of the total group III atomic fraction, y = cIn/c3. The chemical potentials of the liquid and the solid phase, μS, in this materials system are outlined and discussed to some detail in ref 14. The second term in eq 1 accounts for the energetics due to the change in volume of the seed particle during nucleation. Here, γLV is the surface energy of the liquid alloy particle, and we estimate it using a linear interpolation of the surface energies of the constituting species γLV = yc3γIn + (1 − y)c3γGa + (1 − c3)γAu

(6)

where h is the thickness of the nucleus, calculated as h = xhInAs + (1 − x)hGaAs, where hX is the thickness of a layer of material X (InAs and GaAs) in the ⟨111⟩ direction, hX = aX/√3, with aX the lattice constant. The volumes per pair are calculated as ΩX = aX3/4 with X as InAs and GaAs. The final factor in the last term in eq 1 is the surface energy term. For our system, where one-third of the nucleus perimeter lies along the triple phase line22 and two-thirds are inside, the surface energy term can be expressed as

Figure 1. Schematic illustration of the system. The nucleus is modeled as an equilateral triangle with one of the sides along the triple phase line and the other sides buried under the liquid alloy particle.

L L Δμ = xμInL + (1 − x)μGa + μAs − μ S (x , T )

4 (iInAs Ω InAs + iGaAs ΩGaAs) h 3

(4)

where we have replaced the surface energy of As with the one for Au, due to lack of surface energy data for As. We do not believe that we introduce any significant errors due to this since the As concentration is small and the term in eq 1 where γLV is included is generally small compared to Δμ anyway. The volume per III−V pair in the solid is estimated using Vegard’s law ΩS = x

aInAs 3 a 3 + (1 − x) GaAs 4 4

(5)

where aInAs and aGaAs are the lattice constants of InAs and GaAs, respectively. The volume per III−V pair in the liquid is also estimated using Vegard’s law. Since the molar volume of As in a liquid is unknown, these volumes are approximated as ΩLInAs ≈ 2ΩLIn, and similarly for GaAs (where the molar volumes are calculated from the respective densities). The parameters β and R in eq 1 are the wetting angle and the radius of the nanowire, respectively. The last term in eq 1 accounts for the increase in surface energy as the nucleus grows. Here, 3r is the total edge length of

Figure 2. Interface energies, γSV and γSL, calculated at 750 K using the Butler equations. B

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nucleus will be unary. We use a graphical algorithm to solve eq 8 to find x as a function of y for different conditions. In Figure 4 we show x as a function of y at T = 750 K and cAs = 0.02 for varying values of c3, ranging from c3 = 0.3 up to pure

lower than for the zinc blende phase. Nucleation modeling of nanowires in different crystal phases is, for example, presented in refs 21 and 29−32. In Figure 3 we show the Gibbs free energy surface for nucleation of InxGa1−xAs as given by eq 1 at the conditions, T =

Figure 3. Three-dimensional (3D) plot and corresponding twodimensional (2D) contour plot of ΔG as a function of iInAs and iGaAs at T = 750 K, cAs = 0.02, c3 = 0.5, and y = 0.979. The saddle point is * = 18 and iGaAs * = 19. located at iInAs

750 K, cAs = 0.02, c3 = 0.5, y = 0.979, R = 25 nm, and β = 125°. The exact value of β is not critical but is chosen so that it agrees with experimental observations29 and previous investigations.21,28 From the 2D contour plot in Figure 3, it is clear that the free energy surface has a saddle point at these conditions, and it is located at i*InAs = 18 and i*GaAs = 19. In the next section, we calculate the loci of the saddle points at different concentrations and temperatures. Numerical values of the parameters used in the calculations are given in Table 1.

Figure 4. InAs composition of the solid nucleus, x, shown as a function of the In fraction of the group III content of the liquid, y, at T = 750 K and an arsenic atomic fraction of 0.02. The total group III content of the liquid phase was varied, and the following molar fractions were investigated: 0.3 (blue), 0.5 (black), 0.7 (red), and 0.98 (green). The upper inset is a zoom-in at 0.95 < y < 1.00. The solid lines connecting the circles are guides for the eye. The two bottom insets show the location of the saddle point, that is, the size and composition of the critical nucleus (the same color code as in the main * -axis correspond to figure) for increasing y. The values close to the iGaAs low y values, and the values close to the i*InAs-axis correspond to y values close to one.

Table 1. Materials Parameters Used for the Calculation of the Free Energy Surfacea property

a

unit

value

γIn

J/m2

γGa γAu ρLIn ρLGa γSL InAs

J/m2 J/m2 kg/m3 kg/m3 J/m2

0.568 − 4 × 10−5(T − 273) − 7.0 × 10−8(T − 273)2 0.708 − 6.6 × 10−5(T − 302.8) 1.15 − 1.64 × 10−4(T − 1337) 7022−0.762(T − 429.7) 6077−0.611(T − 302.9) 0.63

γSL GaAs γSV InAs γSV GaAs

J/m2 J/m2 J/m2

0.73 1.19 1.36

ref

group III-assisted growth, c3 = 0.98. The inset in the upper right corner is a zoom-in for high values of y, whereas the two bottom insets show the size of the critical nucleus for the different values of c3. The first clear trend that we notice is that the In ratio in the seed particle is always higher than the InAs concentration of the solid nucleus, y > x. Moreover, a quite high In ratio in the seed particle is needed to get any InAs in the nucleus. This critical y value depends on c3 and is highest for the two intermediate values of c3. Concerning the flat parts of the curves, the higher the c3, the closer to one is the curve. This trend is clearly shown in the upper right inset. In the bottom insets we note an interesting nonlinear trend, namely, that the critical nucleus is very large for the lowest and the highest c3. At c3 = 0.3, the supersaturation is low due to the low group III content, only slightly above the equilibrium solubility. At the gold free conditions (group III- or self-seeded), c3 = 0.98, the supersaturation is low because As has a higher solubility in the group III liquid compared to the gold containing liquid alloy. In this case, for nucleus compositions, x, approximately above 0.05, the liquid composition, y, is highest. In Figure 5 we show x as a function of y at T = 750 K and c3 = 0.5 for varying values of cAs, from 0.005 to 0.10. Here we see the same general trends as in Figure 4. The upper inset shows, however, that the composition curves are more converging for y values close to one, in this case, than in the previous case when c3 was varied. The lower inset shows that the size of the critical

43 44 45 46 46 this work 28 27 26

All the temperatures, T, have the unit K.

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RESULTS AND DISCUSSION We calculate the composition of the critical nucleus, x, as a function of the composition of the liquid alloy, y. The composition of the critical nucleus is given by eq 3 with iInAs = i*InAs and iGaAs = i*GaAs, where i*InAs and i*GaAs are solutions to ∂ΔG ∂ΔG = 0, =0 ∂iInAs ∂iGaAs (8) so that the locus iInAs * , iGaAs * represents a saddle point, see Figure 3. In case only one of the equalities in eq 8 is satisfied, the C

DOI: 10.1021/acs.cgd.6b01653 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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Figure 5. InAs composition of the solid nucleus, x, shown as a function of the In fraction of the group III content of the liquid, y, at T = 750 K and a total group III molar fraction of the liquid of 0.5. The arsenic atomic fraction of the liquid phase was varied and the following values were investigated: 0.005 (green), 0.01 (blue), 0.02 (black), 0.05 (red), and 0.10 (pink). The upper inset is a zoom-in at 0.95 < y < 1.00. The solid lines connecting the circles are guides for the eye. The bottom inset shows the location of the saddle point, that is, the size and composition of the critical nucleus (the same color code as in the * -axis main figure) for increasing y. The values close to the iGaAs correspond to low y values, and the values close to the i*InAs-axis correspond to y values close to one.

Figure 6. InAs composition of the solid nucleus, x, shown as a function of the In fraction of the group III content of the liquid, y, at a total group III molar fraction of the liquid of 0.5 and an arsenic atomic fraction of 0.02. The temperature, T, was varied, and the following values were investigated: 650 (green), 700 (blue), 750 (black), and 800 (red). The upper inset is a zoom in at 0.95 < y < 1.00. The solid lines connecting the circles are guides for the eye. The bottom inset shows the location of the saddle point, that is, the size and composition of the critical nucleus (the same color code as in the * -axis main figure) for increasing y. The values close to the iGaAs correspond to low y values, and the values close to the i*InAs-axis correspond to y values close to one.

nucleus decreases with increasing cAs. This is because the supersaturation increases with increasing cAs. Our last set of saddle point-based x−y-curves are shown in Figure 6. Here the total group III concentration and the arsenic concentration in the liquid particle are kept constant (c3 = 0.5 and cAs = 0.02), while the temperature is varied from T = 650 to 800 K. Also here, we observe the same general trend as in the two previous figures. Specifically, when the temperature is increasing, the size of the critical nucleus increases. This is because the supersaturation decreases with increasing temperature since the solubilities of both the group III species and As are increasing with temperature. A common trend that can be observed in Figures 4−6 is that the lower the supersaturation, the smaller the In ratio, y, which is required to get any InAs in the solid nucleus (x > 0). For the saddle point to exist, there must be a solution to the two equations in eq 8. If only one of these equations is satisfied, the nucleation is unary and one of the pure components form, that is x = 0 or x = 1. In order to investigate the generality of our results we perform a robustness analysis where we vary four parameters, and test if the saddle point still exists. The parameters we vary are related to the solid phase and three of them are the rather uncertain surface energy parameters, γSV InAs, γSV GaAs, and α, the ratio between the solid−liquid and solid− vapor energies. The fourth parameter we vary is the regular solution parameter, ω, describing the interaction between InAs and GaAs in the solid solution.14 This is an assessed thermodynamic parameter, but we still vary it to get a general understanding of its impact on the existence of a saddle point. The conditions we choose are, T = 750 K, cAs = 0.02, c3 = 0.5, and y = 0.979, and the standard values for our parameters are

2 SV 2 γSV InAs = 1.19 J/m , γGaAs = 1.36 J/m , α = 0.53, and ω = 14.06 kJ/ mol. For easier calculations we here estimate the composition dependence of the surface energy with a linear relation, Vegard’s law, instead of the Butler equations. Already this has some impact on the position of the saddle point; it is moved from (iInAs * ,iGaAs * ) = (18, 19) to (17, 10). First, by changing γSV InAs and keeping the other parameters constant we see that the saddle point exists for the approximate SV inequalities 1 < γSV InAs < 1.8. For smaller or larger γInAs, pure InAs or GaAs nucleates, respectively. By instead changing γSV GaAs and keeping the other parameters constant, the saddle point exists if SV γSV GaAs lies in the same interval, that is, if 1 < γGaAs < 1.8. SV Correspondingly, for smaller or larger γGaAs, pure GaAs or InAs nucleates, respectively. Next we vary α, the ratio of the solid− liquid and solid−vapor surface energies, and we find that when α < 0.1, pure InAs nucleates. For α larger than this, there is a saddle point, and both i*InAs and i*GaAs increase with increasing α. Care should however be taken when varying the surface energy parameters independently since this could result in unphysical situations or situations where the stability criteria for nanowire growth33,34 are not met. As a final test, we vary ω. For positive ω, as is the case for the InAs−GaAs materials system, the attraction is stronger between like than unlike species. At lower temperatures such systems have a miscibility gap, but at higher temperature the species are fully miscible due to entropic effects. We find that as ω is increased, iGaAs * decreases and iInAs * increases until the saddle point vanishes and the formation of pure InAs is favorable. When ω instead is decreased, both i*GaAs and i*InAs decrease, and their ratio approaches one. From these tests we conclude that the existence of a saddle point should not be a general feature for ternary nanowire materials systems

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conditions, provided that the vapor−liquid kinetics and thermodynamics are known. However, several of the experimental parameters and materials parameters are interdependent, and generally, one cannot expect to vary one parameter at the time as in Figures 4−6.

at arbitrary conditions. It is easy to imagine a materials system where, for instance, the surface energies of the pure semiconductors and the chemical potentials are so mutually dissimilar that a saddle point does not exist at experimental conditions. While the composition of the critical nucleus is given by the location of the saddle point, the composition of the stable, growing nucleus is still given by eq 3, but iInAs and iGaAs are ⎡ ∂ΔG / ∂iInAs ⎤ determined by diInAs/dt = WInAs⎢1 − exp ⎥⎦ and kBT ⎣ ⎡ ∂ΔG / ∂iGaAs ⎤ 17 diGaAs/dt = WGaAs⎢1 − exp ⎥⎦. The coefficients kBT ⎣ WInAs and WGaAs are mainly determined by mass transport kinetics and have been expressed for unary nucleation in metal alloy seeded nanowires.31 In the following we use the composition of the critical nucleus as an estimate for the composition of the solid material, InxGa1−xAs. This might be a crude approximation for large or infinite layers if the composition based on the incorporation kinetics deviates from the composition of the critical nucleus. However, we believe that this is an acceptable approximation for understanding the main features of composition control in this materials system. There are a few experimental reports on growth of gold alloy particle seeded InGaAs nanowires with specified composition. Some recent examples, excluding approaches to form InAs− GaAs heterostructures, are given by refs 11, 12, and 35−38. In addition, Koblmüller and Abstreiter have written a review focusing on growth of InGaAs nanowires on Si.39 Of these investigations, Jung et al. report compositional tuning of InGaAs throughout the entire compositional range.36 They used a vapor phase growth approach utilizing evaporation of InAs and GaAs powders. Wu et al. grew their InGaAs nanowires with MOVPE, and they report composition control by tuning the composition of the vapor phase.11 They found that the InAs fraction in the nanowires is larger than the In part of the group III molar fraction of the vapor, which was experimentally confirmed by Ameruddin et al.12 and shown by Dubrovskii.13 However, no detailed measurements on the composition of the particle were made in any of these investigations. It is interesting to note that the vapor phase composition of In is generally smaller than the solid InAs composition, while according to our calculations the seed particle composition of In is significantly higher than the solid InAs composition. This was experimentally demonstrated by Jabeen et al, who reported results on Au-catalyzed InGaAs nanowires grown in molecular beam epitaxy.35 Following earlier work, this effect can be explained by the higher affinity of Au for In than for Ga.40−42 This could make the In incorporation in the alloy particle much more efficient than the Ga incorporation and lead to In-rich particles even when the vapor is In-poor. We believe that the current investigation will be of highest importance for future research on compositional control of gold alloy seeded InGaAs nanowires. Our theory could be used to directly compare and correlate the seed particle composition to the nanowire composition. The surface energies of the solid phase are the most uncertain parameters and could be used as fitting parameters. As soon as these parameters are fitted to experiments one could use the model to predict solid compositions as a function of the composition of the seed particle and ultimately as a function of the experimental

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CONCLUSIONS We have characterized the Gibbs free energy surface for InxGa1−xAs nanowire nucleation from a gold based alloy seed particle using realistic chemical potentials. According to binary nucleation theory, the saddle point of this energy landscape gives the size and composition of the critical nucleus, which in turn is an estimate of the composition of the solid nanowire. We have calculated the composition of the critical nucleus for various temperatures and compositions of the seed particle. In addition we have tested the robustness of the existence of the saddle point. For this materials system, we find that the surface energies of the solids and the solid−liquid interface energies, the most uncertain parameters, can be varied to some extent with the saddle point still existing, while a too large variation leads to unary nucleation. Finally, we foresee that this model will be important for guiding and explaining future experimental research on compositional control of gold alloy seeded InGaAs nanowires.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +46 46 2221472. Fax: +46 46 2223637. Web: http://www.nano.lu.se/jonas. johansson. ORCID

Jonas Johansson: 0000-0002-2730-7550 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge financial support from NanoLund (the Center for Nanoscience at Lund University), the Swedish Research Council (VR), and the Knut and Alice Wallenberg Foundation (KAW). We thank K. Deppert, R. C. Flagan, K. A. Dick, and M. Selleby for stimulating discussions.

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DOI: 10.1021/acs.cgd.6b01653 Cryst. Growth Des. XXXX, XXX, XXX−XXX