Egor D.
Leshchenko
^{a},
Masoomeh
Ghasemi
^{ab},
Vladimir G.
Dubrovskii
^{c} and
Jonas
Johansson
*^{a}
^{a}Solid State Physics and NanoLund, Lund University, Box 118, 221 00 Lund, Sweden. E-mail: jonas.johansson@ftf.lth.se
^{b}Physics Department, Persian Gulf University, Box 7513613817, Booshehr, Iran
^{c}ITMO University, Kronverkskiy prospekt 49, 197101 St. Petersburg, Russia

Received
21st December 2017
, Accepted 16th February 2018

First published on 19th February 2018

We derive an analytic expression for the composition of a ternary solid material nucleating from a quaternary liquid melt. The calculations are based on the two-component nucleation theory with realistic descriptions of the liquid and solid phases. We apply this theory to gold-catalyzed, nucleation limited vapor–liquid–solid growth of ternary III–V nanowires. We consider ternary gallium, indium, and aluminum arsenides and antimonides and discuss growth conditions for optimum composition control in these materials. Furthermore, we compare our calculations with the results of an equilibrium thermodynamic model.

The critical step in NW-based device design is the ability to control the composition of ternary III–V NWs, which involves the bandgap tunability by variation of the compound concentrations in the ternary alloy. In recent years, enormous progress has been made toward improving the growth technologies by broadening the spectrum of the NW materials. Particularly, growth of AlGaAs and InGaAs NWs using molecular beam epitaxy (MBE)^{15} and metal–organic vapor phase epitaxy (MOVPE)^{16} have been studied. Sb-based NWs such as GaAsSb,^{17} InAsSb (ref. 18) and InGaSb (ref. 19) are nowadays of particular interest in terms of mid-infrared band gap engineering. In these investigations, growth of ternary NWs over a broad range of compositions has been demonstrated. However, the liquid droplet composition has not been systematically investigated which impedes a direct comparison between experimental and theoretical results. NW synthesis resulting in compositions within the miscibility gap can, for systems where this is relevant, be explained in terms of growth kinetics,^{20} whereas the present work is devoted to thermodynamically stable compositions.

Analytical models which link the solid–liquid and solid–vapor compositions have been reported by Dubrovskii et al.^{21,22} and Glas.^{23} However, a full description of ternary NWs that form from quaternary alloys (three NW constituencies and gold) is still lacking because of unknown thermodynamic and kinetic constants such as chemical potentials and crystallization rates of different III–V pairs.

The current investigation is aimed at improving the fundamental understanding of the ternary NW formation from quaternary alloys and is based on realistic thermodynamic descriptions of all the considered phases. In our approach, binary and ternary interactions are taken into account. We calculate and analyze compositional diagrams for highly relevant III–V materials including the In–Ga–As–Au, Al–Ga–As–Au, In–Ga–Sb–Au and In–Sb–As–Au systems. Furthermore, a comparison between self-catalyzed and Au-catalyzed NW growth in these materials systems is presented. The obtained results provide a basis for understanding the ternary NW formation and can be useful for choosing the growth conditions to tune the NW composition to the desired values.

(1) |

(2) |

It has been argued, however, that the surface energy of the critical nucleus is at a minimum due to surface segregation effects.^{25} This property corresponds to the condition da/dx = 0, which considerably simplifies the calculations and leads to the expression

(3) |

We will consider the formation of a ternary alloy, A_{x}B_{1−x}D, from a quaternary solution mother phase containing A, B, D, and U, where U can be thought of as a solvent, often gold. For this situation, the chemical potential difference of the ternary alloy in the liquid and solid can be expressed as^{26}

Δμ = x(μ^{L}_{A} + μ^{L}_{D} − μ^{S}_{AD}) + (1 − x)(μ^{L}_{B} + μ^{L}_{D} − μ^{S}_{BD}), | (4) |

(5) |

The chemical potentials of the species in the liquid are expressed by differentiation of G^{L} in eqn (5) according to

(6) |

The zincblende (ZB) solid phase, A_{x}B_{1−x}D, is modeled as a binary regular solid solution with chemical potentials given by

(7a) |

(7b) |

By explicitly expressing the chemical potentials of the species in the liquid, we construct the chemical potential difference according to eqn (4). Differentiating this expression according to eqn (3) yields an analytical expression for the composition of the critical nucleus, which serves as a first approximation of the composition of the solid phase,

(8) |

Here, we introduce the φ(y) function given by

(9) |

(10) |

In the above equations, we have made the substitutions c_{tot} = c_{A} + c_{B} and y = c_{A}/c_{tot}. In some relevant cases (see the discussion in ref. 20 for self-catalyzed growth), there is a closed form approximation for eqn (8) given by

(11) |

Here,

(12) |

b = α | (13) |

From the thermodynamic point of view, the selected states (c_{tot}, y, c_{D} and T) of the homogeneous liquid particle are not equilibrium states but rather represent a supersaturation with respect to the ZB solid phase. This does not guarantee that there is no thermodynamic driving force for other solid phases to form. According to the Gibbs phase rule, as many as six phases can coexist in a quaternary system. However, we assume that our system is at a non-equilibrium steady state where only the ZB phase forms from the supersaturated liquid and the formation of other solid phases (if they can exist at the given conditions) is kinetically inhibited. As far as possible, we will choose conditions based on the reported experimental results for the respective systems.

We start our investigation with the analysis of gold-catalyzed In_{x}Ga_{1–x}As NWs. The InAs–GaAs pseudobinary interaction parameter is large and gives rise to a significant miscibility gap where the formation of the homogeneous solid solution is thermodynamically forbidden at the relevant growth temperatures. From the GaAs–InAs vertical section of the phase diagram of the In–Ga–As–Au system presented in Fig. 1 (c_{tot} = 0.5 and c_{As} = 0.02), it is seen that the In_{x}Ga_{1−x}As phase exists over the entire range of relevant temperatures. Fig. 2 shows the liquid droplet composition versus the In_{x}Ga_{1–x}As solid composition at T = 477 °C and c_{As} = 0.02 in a wide range of the total concentrations of group III elements from c_{tot} = 0.98 (corresponding to self-catalyzed growth without gold) to c_{tot} = 0.3, where gold dominates in the catalyst droplet. It should be noted that, whereas the NW composition can be measured accurately using high-resolution electron microscopy techniques,^{17} the composition of the droplet during growth is generally unknown. However, an experimental value of group III elements concentration in the Au–III droplet has been estimated to c_{tot} = 0.5 by Harmand et al.^{28} The amount of group V elements is very small due to the low solubility of arsenic in Au–III alloys. In spite of its negligible amount, the concentration of group V elements has drastic influence on the chemical potentials and, consequently, on the growth rates and crystal structures of ternary NWs.^{29} As seen from Fig. 2, reasonable agreement between our analytical model and a purely thermodynamic phase segregation model (represented by the open circles) is observed. This model is based on segregation of a hypothetical, homogeneous supersaturated liquid with composition y into a solid phase with composition x, and a remaining liquid in equilibrium with the solid phase. The phase segregation data were evaluated using the Thermo-Calc software.

Fig. 2 Analytical calculations (solid curves) for the liquid–solid composition dependence for different c_{tot} at T = 477 °C and c_{As} = 0.02. The dashed parts of the curves correspond to the miscibility gap. The small open circles correspond to phase segregation, evaluated using the Thermo-Calc software. The squares represent results from ref. 24. |

The slope of the y(x) curve is practically vertical over almost the entire y range and changes only for y > 0.97. Therefore, very high y values are needed to tune the composition of In_{x}Ga_{1−x}As NWs. As noticed above, a direct comparison between the experimental and theoretical liquid–solid composition dependences is seriously hampered because of in most cases unknown compositions of the catalyst particles. However, it is possible to describe the main observed trends. So, for example, to synthesize self-catalyzed In_{x}Ga_{1−x}As NWs with x = 0.03–0.05 as obtained in ref. 30, y ≈ 0.96–0.98 is required. Moreover, the use of Au catalyst droplets allows one to increase the In fraction up to x = 0.22.^{31} This is consistent with the obtained theoretical results: the In fraction in the nanowire decreases with increasing c_{tot} at constant y and T (Fig. 2).

The squares are the numerical results obtained using the previous two-component nucleation model including surface energies and a specific VLS term offsetting the chemical potential difference.^{24} The main difference between that nucleation model and the one presented here is that the previous model that includes surface energies admits NW compositions within the miscibility gap. This is because the composition dependent surface energies influence the saddle point coordinates in such a way that it turns the two minima that would correspond to the miscibility gap compositions into one shallow minimum between these two minima.

The relationship between the solid and liquid compositions for different growth temperatures at the fixed c_{tot} = 0.5 is presented in Fig. 3. Clearly, the width of the miscibility gap shrinks with increased temperature and disappears completely at T = 543 °C. This would enable the thermodynamically stable growth of ternary In_{x}Ga_{1−x}As NWs with any x value. However, such high temperatures can be unfeasible for MBE and MOVPE growth due to increased desorption and potential decomposition of III–V materials. Thus, as an example, thermodynamically stable growth of In_{x}Ga_{1−x}As NWs is not possible in the range 0.22 < x < 0.78 at T = 477 °C (see Fig. 2).

To sum up the results of this section, the required y values for obtaining reasonable InAs fractions in ternary InGaAs NWs increase with the total concentration of group III elements, whereas the III/Au ratio does not change the miscibility gap region which is entirely determined by the temperature-dependent pseudobinary interaction parameter.

Next, we consider nucleation of gold-catalyzed Al_{x}Ga_{1−x}As NWs. Fig. 4a shows a vertical section of the quaternary Al–Ga–As–Au system with c_{tot} = 0.5 and c_{As} = 0.02. While many different phases can be present in the case of Au-assisted NW growth, the Al_{x}Ga_{1–x}As is a dominating solid phase at almost any Au–III ratios and in a wide range of NW growth temperatures. Fig. 4b shows a vertical section of the ternary Al–Ga–As system which is relevant for self-catalyzed growth of Al_{x}Ga_{1−x}As NWs. Fig. 4a reveals that the minimum temperature for Au-catalyzed growth of Al_{x}Ga_{1−x}As NWs should be higher than the eutectic point of the system (T = 353 °C). The minimum VLS growth temperature decreases as the Au concentration decreases. For the Al_{x}Ga_{1−x}As system, the pseudobinary interaction parameter has a low enough value leading to the small dome height in comparison with the In_{x}Ga_{1−x}As system: the critical temperature at which the miscibility gap disappears equals −142°. This enables the fabrication of Al_{x}Ga_{1−x}As NWs with any solid composition at relevant MBE and MOVPE growth temperatures. The relationship between the solid and liquid compositions for different c_{tot} values at T = 610 °C is presented in Fig. 5. For obtaining self-catalyzed Al_{x}Ga_{1−x}As NWs with non-zero x, very low y values are necessary, while an almost horizontal slope cumbers precise compositional control. Indeed, a small addition of Al to the liquid results in a tremendous increase of the Al concentration in the solid. However, with decreasing c_{tot}, the slope of y(x) changes drastically and the required y values increase by several orders at high enough Au concentrations (c_{tot} = 0.3). The change of temperature also influences y(x) but not so significant as c_{tot} does. As seen in Fig. 5, the results are well fitted by the one-parametric (ε) Langmuir expression given by x = εy/(1 + y(ε − 1)),^{15} whereas tremendous discrepancy in comparison to the phase segregation approach (dotted curves) is observed.

Fig. 4 a. Vertical section of the Al–Ga–As–Au system at c_{tot} = 0.5 and c_{As} = 0.02. b. Vertical section of the Al–Ga–As system at c_{tot} = 0.98 and c_{As} = 0.02. |

Further, we investigate the nucleation of In_{x}Ga_{1−x}Sb NWs from quaternary In–Ga–Sb–Au alloy. As seen from the vertical section of the In–Ga–As–Au system at c_{tot} = 0.64 and c_{Sb} = 0.06 presented in Fig. 6, the compositional tuning of In_{x}Ga_{1−x}Sb throughout the entire compositional range is possible at T < 430 °C, which is the lowest temperature of the liquid single phase region. In comparison to As-based system, a relatively high Sb concentration c_{Sb} = 0.06 is needed to ensure positive difference of chemical potentials and, consequently, a positive supersaturation. Fig. 7 shows the liquid composition at c_{tot} = 0.64 and c_{Sb} = 0.06 plotted against the solid composition for different growth temperatures. The blue dashed curve at T = 450 °C indicates the absence of the solid phase and corresponds to the In molar fraction range within the 0.31–0.54 range on the GaSb–InSb vertical section of the In–Ga–Sb–Au phase diagram. When y decreases below 0.47, the solid phase occurs again (not shown in Fig. 7). The black dotted curve corresponds to the phase segregation model and is obtained using the Thermo-Calc software at T = 430 °C, c_{tot} = 0.64, and c_{Sb} = 0.06. It is seen that the analytical model given by eqn (8) agrees well with the results of the phase segregation model. Fig. 8 shows the comparison of exact [given by eqn (8)] and approximate [eqn (11) and (12)] solutions for different total concentrations of group III elements and demonstrates a good agreement between them, especially for high x values. However, a significant divergence between exact and approximate solutions is observed for self-catalyzed growth.

Finally, we analyze nucleation of InSb_{x}As_{1−x} NWs from the In–Sb–As–Au melt. According to the phase diagram for the In–Sb–As–Au system presented in Fig. 9, the VLS growth of InSb_{x}As_{1−x} NWs is possible with any solid composition at T = 450 °C. The main feature of this system is the compositional-dependent pseudobinary interaction parameter. This results in a composition-dependent, asymmetric miscibility gap. Fig. 10 shows the liquid composition versus the InSb_{x}As_{1−x} solid composition for different temperatures in the cases of self-catalyzed (c_{tot} = 0.05 and c_{In} = 0.95) and gold-catalyzed growth at T = 450 °C (c_{tot} = 0.05 and c_{In} = 0.45). Defect-free, self-catalyzed NWs with an Sb fraction greater than x = 0.35 have been obtained at T = 520 °C, as reported in ref. 32. Such a relatively high value corresponds to y ≈ 0.975. Clearly, it is more difficult to control the NW composition in the case of Au-catalyzed growth because of the much steeper slope of the y(x) curve. Interestingly, in the case of Al_{x}Ga_{1−x}As, the influence of the total concentration is reversed so that composition control would be easier in the Au-catalyzed case, as seen from comparing Fig. 5 and 10.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7ce02201h |

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