Study of correlations in the stacking sequence of a NW

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The vapor-liquid-solid mechanism of nanowire growth

The VLS mechanism was proposed by Wagner and Ellis in 1964 [140] to explain the growth of single-crystal silicon whiskers on Si (111), with diameters from 100 nm to hundreds of microns. Whisker growth was performed in a chemical vapor deposition reactor after the deposition of a thin film of gold followed by an annealing step at 950 °C. During the annealing step, Si from the substrate mixes with Au to form Au:Si droplets (Fig. 1.1). The Au-Si alloy has an eutectic point located at 363 °C for a Si atomic fraction of 18.6%. Therefore, the alloy may be liquid at temperatures considerably lower than the melting point of the constituents (Si: 1414 °C, Au: 1064 °C) and is liquid at standard growth temperatures.
Upon exposure to the gaseous reactants SiCl4 and H2, the liquid droplet becomes su-persaturated with Si atoms. This is possible, despite the low temperature, because the decomposition reaction is catalyzed by the gold droplet. For this reason, the droplet is referred to as the catalyst, and NWs obtained by the VLS method are often called metal-catalyzed or catalyst-assisted. This terminology is retained even when no chemical catalysis occurs, as in the case of molecular beam epitaxy (MBE). In MBE, the NW constituents are supplied in their elemental form and no complex precursors are used: the droplet acts as a collector and reservoir of material, which thus may be seen as a « physical » catalyst, in the sense that solid phase condensation via the droplet is more favorable than direct vapor-solid deposition.

Understanding NW growth with nucleation theory

We have seen that NW growth is described as the addition of individual monolayers. This occurs in a cyclic fashion and is limited by nucleation on the NW top surface. In order to understand which factors influence the rate at which the nuclei appear we need a theory of nucleation. However, the problem is not an easy one and nucleation theory is still an active field of research. Our discussion will be limited to the essential concepts of classical nucleation theory, presenting some formulas that will be used later in this work.

Calculation of the 2D nucleation barrier

Imagine having a perfectly flat surface surrounded by a supersaturated environment and let ∆µ be the difference of chemical potential between the liquid and solid. The nucleation process can be seen as the spontaneous (random) aggregation of the supersaturated species to form a cluster. The stability of this cluster is determined by the ability of the thermodynamic force for solidification ∆µ to overcome the energy cost of creating a new surface bounding the cluster.
We want to evaluate the energy barrier for the formation of a nucleus sitting on the top (111)B facet of a III-V nanowire. The height of the barrier is given by the maximum work needed to form the nucleus. The work ∆G is the sum of the energy cost of forming the nucleus surface and edges, minus the gain due to the thermodynamic drive for solidification ∆µ and to the destruction of the nucleus-NW interface. We suppose that the nucleus has the shape of an equilateral triangle of side length l. This is justified by the fact that nuclei on (111) surfaces have such shape. Other shapes are possible, affecting the final result only for a geometric pre-factor. Here we are taking the surface energies of the cluster to be the same as the surface energies of the equivalent macroscopic surfaces, an approach known as the capillary approximation. Introducing σ and Γ, with σ the energy cost per unit area of replacing the substrate surface with the nucleus top surface and Γ the energy cost per unit length of creating the nucleus edge, we write:
∆G = −Ah∆µ + Aσ + 3lhΓ (1.1) √ where A = 3 l2 and h are the area and height of the nucleus, respectively. Note that σ = 0 if 4 the substrate and nucleus top are of the same nature (e.g. in the case of homoepitaxy).
The sum of these volume and surface terms results in ∆G to be a concave function of l with a maximum ∆G∗ for some l∗, which is known as the critical size of the nucleus (Fig. 1.4a). Nuclei with a size smaller than the critical one will dissolve and nuclei bigger than the critical size will grow indefinitely.

Self-catalyzed GaAs NWs: a model system

We saw that gold is commonly used to induce the growth of both elemental and III-V NWs. In fact, the vast majority of works on VLS NWs use Au, which is quite surprising given the variety of materials and growth techniques, but means that Au is suitable to seed NW growth in a very broad material and temperature window ranges. An interesting discussion of why gold is probably the most effective metal to induce NW growth can be found in Ref. [98].
On the other hand, the presence of Au in Si is extremely detrimental even at dopant concentrations. It has long been known that Au introduces both acceptor and donor states in the band gap, causing a significant reduction of carrier lifetimes, an effect detectable at concentrations as low as 1012 cm−3 [26].
While III-V NWs grown with gold still exhibit excellent electronic properties, use of Au must be avoided if integration with silicon is sought. Because of its low chemical reactivity, removal of Au surface contamination is difficult. This incompatibility with the Si platform stimulated the research for alternatives to gold. One alternative is to avoid the catalyst droplet altogether, which usually involves selective area growth, with long and expensive substrate preparation procedures. Furthermore, compared to VLS growth, the temperature/flux rate growth window is reduced and high aspect ratios are more difficult to obtain (with some notable exceptions [100]). Growth of III-V NWs using alternative foreign catalysts was explored, with examples including noble metals (Pd, Pt, Ag) and other metals (Ni, Mn, Cu and Fe) [32] each of them with unique advantages and drawbacks. For example, Mn has been exploited to grow spontaneously Mn-doped GaAs NWs [94]. The interested reader may refer to a recent review by Dick and Caroff on gold-free NW growth [32]. Finally, a third option is to exploit the VLS mechanism using a group III metal as catalyst e.g. In for InAs [57]. In this case we speak of a self-catalyzed growth mechanism. Gallium-assisted GaAs NWs is, by far, the most investigated system for self-catalyzed growth. The first report of GaAs whiskers grown using liquid gallium dates back to 1965 [11] and the first example of Ga-catalyzed GaAs growth by MBE was reported by Arthur and LePore in 1969 [8] when MBE was still in its infancy. In the attempt of achieving perfect two-dimensional growth, the authors described whisker growth as a parasitic effect induced by the presence of scratches and other surface defects. Since MBE was developed with the objective of realizing high-quality epitaxial films, the formation of such peculiar structures did not stimulate much interest.
The interest revamped in recent years, when the synthesis of epitaxial high purity Ga-seeded GaAs NWs was achieved [43, 71]. Ga-assisted growth has been reported first on GaAs [4, 43] and shortly afterwards on cleaved Si(100) [71] and Si(111) [103]. In both cases, a thin layer of Si oxide was deposited or spontaneously formed on the substrate. However, the role of the oxide is not yet completely clarified: Plissard et al. [114] found that NW growth can occur on both oxidized and oxide-free Si, although with different growth conditions. On one hand, the presence of oxide enhances the NW yield and reduces the formation of parasitic crystallites, but it reduces NW verticality [114].

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Heterostructure formation in nanowires: advantages and challenges

Epitaxy refers to the growth of a single crystal on top of another (usually semiconductors) with the same or fixed relative crystal orientation. We speak of homoepitaxy if the two crystals are of the same material and of heteroepitaxy if they are not. Epitaxial layers of two different materials are referred to as heterostructures. Heterostructures widely expand the range of application of semiconductor materials beyond simple doping. High quality heterostructures allowed for major developments, such as quantum wells and superlattices. Nevertheless, as mentioned previously, the quality of NW heterostructures obtained by top-down methods may be limited by the degradation of the starting bulk material (or thin film) upon etching. Actually, the advantage of bottom-up NWs is to grow structures that cannot be obtained in planar form. In the two-dimensional case (except for homoepitaxy), the epilayer and its substrate have generally different lattice parameters. Let as and ae be the lattice parameters Fig. 1.7 The simplest nanowire heterostructures. a) NW-substrate; b) core-shell; c) axial. of the relaxed substrate and epilayer, respectively, the lattice mismatch is: f = ae −as . (1.10).

Absolute calibration of the vapor fluxes

Knowing the direction and magnitude of the atomic fluxes is important for NW growth modeling, in order to quantify the amount of material reaching the NW growth front and to identify the possible material pathways. The orientation of the vapor sources with respect to the substrate is known and fixed by the geometry of the growth chamber. We calibrated the fluxes of Ga, Al and As by performing RHEED oscillations on GaAs(001) substrates at 590 °C. In presence of an excess of As, thus with an atomic V/III ratio greater than 1, growth speed is limited by the flux of group III elements. By opening the shutter of Ga and measuring the oscillation period, the deposition rate in ML/s is obtained. Since the surface atomic density of GaAs(001) is known (2/a20, where a0 is the GaAs lattice parameter) the number of atoms per unit surface and unit time is easily obtained. Similarly, the Al flux is measured by growing an AlGaAs layer on GaAs(001) using the same Ga BEP used for the Ga calibration, so that the Al deposition rate is obtained by subtracting the Ga rate from the total growth rate. Alternatively, one can measure the AlAs growth rate directly.

Table of contents :

Thesis overview
List of figures
List of tables
1 Vapor-liquid-solid growth of III-V nanowires 
1.1 What are nanowires?
1.2 The vapor-liquid-solid mechanism of nanowire growth
1.2.1 Growth directions
1.2.2 Nanowire growth dynamics
1.3 Understanding NW growth with nucleation theory
1.3.1 Calculation of the 2D nucleation barrier
1.3.2 The classical nucleation rate
1.3.3 Polytypism in III-V nanowires
1.4 Self-catalyzed GaAs NWs: a model system
1.4.1 Crystal structure
1.5 Heterostructure formation in nanowires: advantages and challenges
1.5.1 The reservoir effect
2 Experimental methods 
2.1 Molecular beam epitaxy
2.1.1 Effusion cells and flux measurements
2.1.2 RHEED
2.1.3 Absolute calibration of the vapor fluxes
2.2 Electron microscopy characterization
2.2.1 Scanning electron microscopy
2.2.2 Transmission electron microscopy
2.2.3 Quantification of composition using medium and high-resolution
HAADF contrast in scanning transmission microscopy
3 Study of correlations in the stacking sequence of a NW 
3.1 Introduction
3.2 NW synthesis and data collection
3.3 Probabilistic analysis of the stacking sequence: conditional probabilities .
3.4 Distribution of cubic and hexagonal segments
3.5 The pair correlation function
3.6 Including correlations in the classical nucleation theory
3.6.1 Standard choice
3.6.2 ANNNI for the nucleus interface energy
3.6.3 Extending the ANNNI model to the step energy
3.7 Conclusions
4 Development of self-catalyzed Ga(As,P) axial heterostructures 
4.1 Introduction
4.2 Growth of pure, self-catalyzed GaP nanowires
4.3 Growth of GaAs (GaP) insertions in GaP (GaAs)
4.4 Behavior of As and P fluxes
4.5 Morphology of the growth front
4.6 Conclusions
5 (Al,Ga)As axial heterostructures 
5.1 (Al,Ga)As insertions in self-catalyzed GaAs NWs
5.2 Experimental details
5.3 Abruptness of interfaces
5.3.1 Estimation of Al fraction in the liquid
5.3.2 The effect of As flux and Al diffusion
5.3.3 Improving interface abruptness with droplet pre-filling
5.4 Modeling the interface composition
5.4.1 Liquid-solid equilibrium of the Al-Ga-As alloy
5.4.2 Predicting the composition profile
5.4.3 An analytical solution for the interface profile
5.5 Conclusions
6 On thin nanowires and the ultimate control of nucleation events 
6.1 Nucleation statistics in time
6.1.1 Growth rate diagram of self-catalyzed GaAs nanowires
6.1.2 Depleting the droplet: growth versus evaporation
6.1.3 Difficulties of the steady-state approach
6.1.4 Pulsing the As supply
6.1.5 First experimental demonstrations
6.1.6 Conclusions
6.2 Growth of thin NWs
General conclusions and perspectives
References 

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