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Simulation of GaAs Nanowire Growth and Crystal Structure Erik K. Mårtensson, Sebastian Lehmann, Kimberly A. Dick, and Jonas Johansson Nano Lett., Just Accepted Manuscript • Publication Date (Web): 08 Jan 2019 Downloaded from http://pubs.acs.org on January 8, 2019
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Nano Letters
Simulation of GaAs Nanowire Growth and Crystal Structure ∗,†
Erik K. Mårtensson,
†
Sebastian Lehmann,
Johansson
†Lund ‡Centre
†,‡
Kimberly A. Dick,
and Jonas
†
University, Solid State Physics, Box 118, 22100, Lund, Sweden
for Analysis and Synthesis, Lund University, Box 124, 22100, Lund, Sweden
E-mail:
[email protected] Phone: +46 (0)46 222 7692
1
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Abstract Growing GaAs nanowires with well de ned crystal structures is a challenging task, but may be required for the fabrication of future devices. In terms of crystal phase selection, the connection between theory and experiment is limited, leaving experimentalists with a trial and error approach to achieve the desired crystal structures. In this work, we present a modelling approach designed to provide the missing connection, combining classical nucleation theory, stochastic simulation and mass transport through the seed particle. The main input parameters for the model are the ows of the growth species and the temperature of the process, giving the simulations the same exibility as experimental growth. The output of the model can also be directly compared to experimental observables, such as crystal structure of each bilayer throughout the length of the nanowire and the composition of the seed particle. The model thus enables for observed experimental trends to be directly explored theoretically. Here, we use the model to simulate nanowire growth with varying As ows, and our results match experimental trends with good agreement. By analysing the data from our simulation, we nd theoretical explanations for these experimental results, providing new insights into how the crystal structure is aected by the experimental parameters available for growth.
Keywords Wurtzite, Zinc Blende, GaAs, Nanowire, Simulation When conventional III-V semiconducting materials are epitaxially grown in the form of nanowires, they often exhibit the metastable Wurtzite (WZ) phase. This increases the interest in III-V nanowires as the electronic properties of the WZ phase dier from the stable Zinc Blende (ZB) phase; for instance, GaP has an indirect band gap in the ZB phase 1 and a direct band gap in the WZ phase, 2 thus making WZ GaP interesting for optical devices. Further, for GaAs, InAs and InP there is a type II band alignment between the band gaps 2
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of WZ and ZB phases, which enables axial manipulation of the electronic band structure without changing the material. 35 WZ nanowires are not only interesting for their electronic properties, but also from the perspective of crystal growth, as the fact that they can be produced opens the question as to what makes a metastable phase form in nanowires grown using the particle-assisted vapourliquid-solid mechanism. 6 From a theoretical perspective, there is consensus that the crystal structure is determined by the nucleation event, and that once a WZ bilayer has formed it is unlikely for it to change back to the ZB phase. 7,8 Provided that the supersaturation is high enough, WZ formation could be promoted through nucleation at the Triple Phase Line (TPL), as proposed by Glas et al. 7 There, the higher cohesive energy of the WZ phase could be compensated by a lower surface energy compared to the ZB phase. Nucleation at the TPL is commonly used to address the experimentally observed formation of WZ. 911 The nucleation theory of nanowire growth has been developed futher in multiple works to include additional features such as the changes to the surface area of the particle, 8 next nearest neighbour interactions, 12 so-called Kashchiev renormalization, 13 depletion of the droplet 14 and polynucleation 15 to name a few. However, there still remain unresolved issues concerning the understanding of nanowire growth and crystal structure. The early theoretical models explicitly related the crystal phase selection to the supersaturation of the particle, with high supersaturations leading to growth of WZ. In experimental growth using MOVPE, the crystal structure is commonly controlled using the ow of group V precursors. Generally, a high ow of As precursors lead to growth in the ZB phase, and lowering the As ow leads to WZ formation. 10,16 At extremely low As ows and V/III ratios, it has been observed that an additional transition occurs in MOVPE where ZB again becomes dominant. 17 This matches the behaviour observed in growth using molecular beam epitaxy (MBE), 18 and while the 'MBE transition' has been heavily studied in theory, the ZB formation at high V/III ratio has received less attention. It has been suggested that the ZB formation using high ows of As and high nanowire growth rates can only be explained using 3
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polynucleation. 15 However, stacking faults and WZ segments can be observed in nanowires grown using the aerotaxy method, where growth rates of 1 µm/s are commonly observed, 19 which indicates that there are still uncertainties regarding the eect of high ows on the crystal structure. There are many examples where dierent uses of simulation have been applied to gain a deeper knowledge into the growth of nanowires. 2023 While these articles provide insights into the growth rate and shape evolution of nanowires, none of these focused on the polytypism of nanowires. In this paper, we investigate the WZ-ZB polytypism in Au-seeded GaAs nanowires by combining existing nucleation models with stochastic simulations. We rely on the idea that nanowire growth is generally assumed to be a steady state process, meaning that although the composition of the particle changes over the course of a single nucleation cycle, the average composition does not change on a longer time scale. This suggests that at a speci c set of growth parameters, a constant ow of III- and V-atoms reach the seed particle. By looking at nanowire growth solely from the perspective of the seed particle, we use the concentrations of Ga and As to compare the formation of WZ and ZB nuclei. The timedependent composition aects the supersaturation, the surface energy, the surface area and the contact angle of the particle, which will aect the thermodynamic and kinetic dierences between WZ and ZB formation. The concentrations in turn depend on the formation of new bilayers, desorption and, importantly, the ows of III- and V atoms to the seed. By controlling the ows and allowing the other factors to evolve dynamically, we create a model that can bridge the experimental and the theoretical sides of the nanowire growth. From our simulations, we gain an understanding into how the composition of the seed particle and the supersaturation can be aected by the V/III ratio during growth and how this in turn can control the crystal structure. We show that the model can simulate two ZB regimes, where one is due to center nucleation and one is due to TPL nucleation. We compare the results of our model with experimental knowledge, and discuss the eects of 4
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experimentally accessible growth parameters. a)
b)
Nucleation Barrier ∆G*WZ , ∆G*ZB, ∆G*ZB,C
Nucleation Rate RWZ , RZB
Update Seed Properties ∆µ, γLV, β, Asurface
Nucleation Probability PWZ , PZB
Update Concentrations cIII , cV
Randomize Outcome WZ, ZB or No Growth
JAs+ Supply ZB Nuclei
JGa+ Supply Seed Particle
JAs- Evaporation WZ Nuclei
JAs-,Ga- Nucleation GaAs Nanowire
Figure 1: A schematic representation of the steps involved in the model (a), together with a schematic gure of a nanowire with the material balance used in the simulations(b). Our model is described by the schematics presented in Fig. 1, where a full cycle containing six steps is completed at each time step. Here each step in the cycle is presented and discussed. Further details on both the expressions and the variables used can be found in the Supplementary Information. The rst step is to calculate the nucleation barriers for WZ and ZB, for growth in the
h1 1 1i/h0 0 0 1i direction. As mentioned previously, there are many dierent adjustments and additions to the classical nucleation model for nanowire growth which have been presented over the years. However, no attempt to unify these alterations has been proposed, meaning that there is no agreement on which of these additions should be considered. For this reason we chose to work with the simplest model available which was still capable of explaining the experimentally observed trends, while leaving the possibility of including additional eects open. It is worth mentioning, that the concept of truncating corners 24 has recently gained interest in terms of crystal phase selection. 25 This theory describes that above a critical angle, the interface between the particle and the nanowire becomes truncated, which suppresses 5
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nucleation at the TPL. In our model the contact angle is an accessible output, meaning that the possible implications from truncated corners can easily be extrapolated to our results. In addition, for growth below this critical contact angle, our model would still be applicable within the framework of truncated corners. We include nucleation at the TPL for both polytypes, as well as center nucleation for ZB. Following the nomenclature of Dubrovskii et al., 8 our expression for the nucleation barriers, where variables with the subscript i depends on polytype, are given by:
Γ2i c22 hΩS 4c1 ∆µ − ψi 2 2 c2,c hΩS γSL = 4c1,c ∆µ
∆G∗i,T P L =
(1)
∆G∗ZB,C
(2)
Γi = (1 − x)γSL + x(γSV,i − γLV sin(β))
(3)
Here, cj are geometrical constants depending on the shape of the nuclei. For nucleation at the TPL, it is favourable for the nuclei to have a large part of the perimeter in contact with the vapour, and for this reason c1 and c2 were chosen to correspond to a triangular nucleus. For center nucleation, it is instead favourable to have a small perimeter/area ratio, and therefore a hexagonal shape was used for center nucleation. Furthermore, h is the height of a bilayer, ΩS is the volume of a Ga-As pair in the solid ZB phase, ∆µ is the dierence in chemical potential per mole between Ga-As pairs in the liqud phase with respect to the solid ZB phase, hence referred to as the supersaturation, and ψi is the dierence in cohesive energy between the WZ and the ZB phase. Γi is the average surface energy of the nuclei, which for TPL nucleation is given in Eq. 3 following the motivation by Glas et al. 7 This surface energy depends on the fraction of the nuclei in contact with the vapour, x, the contact angle, β , and the surface energies of the solid-liquid, γSL , solid-vapour (for the vertical side facets, {1 1 0} and {1 0 1 0}), γSV,i , and liquid-vapour interfaces ,γLV . Some of these variables will depend on the composition of the seed particle (e.g. γLV ). These variables are updated in the last 6
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step of the simulation cycle and will be discussed further when this step is addressed. From Eq. 1 we also de ne a critical supersaturation for nucleation at the TPL where both polytypes have nucleation barriers of the same height, ∆µ∗ZB=W Z . This is done by setting
∆G∗ZB,T P L = ∆G∗W Z and solving for ∆µ. Above this critical supersaturation, the barrier for forming WZ is lower than the barrier for forming ZB at the TPL. As mentioned previously, many dierent variations of the nucleation barrier can be found in literature, and can easily be implemented into the model if desired. The second step is to calculate the nucleation rates for the two polytypes, using the nucleation barriers from Eq. 1. Here we use the expression from classical nucleation theory which is a common approach to calculate nucleation rates. 26 For the pre-exponential factors, we use the following equation: 27
Ri = VN,i
NV ∗ ω Zi exp [−∆G∗i /(kB T )] VSeed
(4)
The rst factor VN,i corresponds to the volume in the seed where nucleation is possible. The second factor NV /VSeed describes the number of As atoms divided by the volume of the seed particle. This assumes that the nucleation is limited by the availability of As atoms, which was deemed reasonable as it is generally believed that cAs
