Vladimir G Dubrovskii
email: dubrovskii@mail.ioffe.ru

Frank Glas
email: frank.glas@c2n.upsaclay.fr

Vapor-liquid-solid growth of semiconductor nanowires

Keywords: III-V nanowires, vapor-liquid-solid growth, nucleation, growth modeling

Vapor-liquid-solid growth of semiconductor nanowires

Introduction

In this chapter, we discuss the growth of semiconductor nanowires, wich an emphasis put on the vapor-liquid-solid growth of III-V nanowires. Particular attention is paid to theoretical understanding and the relevant models for the growth, morphology, crystal phase, composition and statistical properties within the naniowire ensembles.

The chapter is organized as follows. Section 2 gives a general description of the vaporliquid-solid growth by different epitaxy techniques, including the role of equilibrium phase diagrams, growth in reguar arrays on patterned substrates such as SiO x /Si(111), Au-catalyzed and self-catalyzed growth of III-V nanowires. We also present some recent examples of shaping GaAs nanowires in the self-catalyzed approach. In section 3, we review the basics for the nanowire growth modeling, including the kinetics of group III and V species, axial growth rate, the role of supersaturation, nucleation-limited and regular growth modes. Section 4 deals with the role of surface energetics in the vapor-liquid-solid growth of GaAs nanowires, which has an important impact on the nanowire morphology and crystal phase. Section 5 presents a brief decription of selective area epitaxy of III-V nanowires and a model for such growth. In section 6, we consider a detailed description of the nanowire growth kinetics, including the transportlimited growth, chemical potentials for the vapor-liquid-solid growth of III-V nanowires, nucleation and growth of two-dimensional islands at the liquid-solid interface, and some advanced self-consistent growth models combining the material transport equations with the Zeldovich nucleation rate. Section 7 desribes an interesting effect of radius self-equilibration in self-catalyzed III-V nanowires. Section 8 presents the recent results on the length distributions within the ensembles of different III-V nanowires, including the role of surface diffusion and delayed nucleation of nanowires on the substrate surface, and possibilities to obtain sub-Poissonian length distributions induced by nucleation antibunching. Section 9 deals with ternary alloys in nanowires, and describes some theoretical models for the nanowire composition. In section 10, we consider heterostructures within nanowires, in particular, relaxation of elastic stress induced by lattice mismatch, the vapor-liquid-solid growth of axial heterostructures in different III-V material systems, and sharpness of the nanowire heterointerfaces. Section 11 deals with the zincblendewurtzite polytypism in III-V nanowires and its theoretical understanding achieved to this day.

Vapor-liquid-solid growth of semiconductor nanowires

The vapor-liquid-solid (VLS) method was introduced by Wagner and Ellis in 1964 to grow sub-micrometer Si "whiskers" on Si(111) substrates with Au catalyst [START_REF] Wagner | Vapor-liquid-solid mechanism of single crystal growth[END_REF]. Many fundamental aspects of the VLS growth of elemental and compound semiconductors were later developed by Givargizov [START_REF] Givargizov | Highly Anisotropic Crystals[END_REF]. At the beginning of the 2000s, a rapid development of semiconductor nanowires (NWs) started, lead by the groups of Lieber [START_REF] Zhang | Nanowires: building blocks for nanoscience and nanotechnology[END_REF], Yang [START_REF] Yang | Semiconductor nanowires: What's next?[END_REF] and Samuelson [START_REF] Seifert | Growth of one-dimensional nanostructures in MOVPE[END_REF], soon followed by many others. This has resulted in tremendous progress in the NW synthesis, characterization and applications (see, for example, Refs. [START_REF] Zhang | Nanowires: building blocks for nanoscience and nanotechnology[END_REF][START_REF] Yang | Semiconductor nanowires: What's next?[END_REF][START_REF] Dubrovskii | Nucleation theory and growth of nanostructures[END_REF][START_REF] Dubrovskii | Theory of VLS growth of compound semiconductors[END_REF] for a review).

One important advantage of NWs is that they allow for a very efficient relaxation of elastic stress induced by lattice mismatch on the lateral sidewalls, thus enabling dislocation-free growth on dissimilar substrates (such as Si for III-V NWs), and in NW heterostructures [START_REF] Glas | Critical dimensions for the plastic relaxation of strained axial heterostructures in freestanding nanowires[END_REF][START_REF] Zhang | Analytical study of elastic relaxation and plastic deformation in nanostructures on lattice mismatched substrates[END_REF][START_REF] Glas | Strain in nanowires and nanowire heterostructures[END_REF]. Peculiarities of the VLS growth from nanoscale catalyst droplets in largest measure determines the resulting morphology, crystal quality and phase, and statistical properties within the NW ensembles, which explains the importance of theoretical understanding and modeling [START_REF] Dubrovskii | Nucleation theory and growth of nanostructures[END_REF][START_REF] Dubrovskii | Theory of VLS growth of compound semiconductors[END_REF].

The VLS method makes use of a catalytic effect of a liquid metal droplet such as Au to fabricate semiconductor NWs which grow away from the substrate, with the position determined by the initial location of the droplet on the substrate. Usually, NWs grow in <111> direction perpendicular to (111) substrate. Parasitic growth between the NWs is either completely suppressed (for example, by using lithographically patterned substrates with a mask oxide layer) or proceeds at a lower rate than that of NWs. Radial growth rate of NWs by the vapor-solid (VS) mechanism on the sidewalls or due to non-stationary droplet swelling is much lower than the axial one. In chemical epitaxy techniques, a liquid metal acts as a chemical catalyst which enhances the cracking efficiency of semiconductor precursors at the droplet surface, so ideally NWs would grow only axially [START_REF] Wagner | Vapor-liquid-solid mechanism of single crystal growth[END_REF][START_REF] Givargizov | Highly Anisotropic Crystals[END_REF][START_REF] Seifert | Growth of one-dimensional nanostructures in MOVPE[END_REF]. In molecular beam epitaxy (MBE) [START_REF] Dubrovskii | Diffusion-induced growth of GaAs nanowhiskers: theory and experiment[END_REF][START_REF] Harmand | Analysis of vapor-liquid-solid mechanism in Au-assisted GaAs nanowire growth[END_REF], the droplet acts as a material collector (or a "physical" catalyst) which directs the material diffusion fluxes to the NW top. Whatever the epitaxy technique used to grow NWs, the VLS process necessarily involves two phase transitions of a semiconductor material: from vapor (for example, Ga and As2 beams in MBE) to liquid (Au-Ga-As liquid alloy in the droplet) and from liquid to solid (stoichiometric GaAs NW). This explains the term "vapor-liquid-solid" growth.

The nature of Au-catalyzed VLS growth, particularly for elemental semiconductors such as Si or Ge, can be understood through the equilibrium phase diagrams [START_REF] Wagner | Vapor-liquid-solid mechanism of single crystal growth[END_REF][START_REF] Givargizov | Highly Anisotropic Crystals[END_REF][START_REF] Zhang | Nanowires: building blocks for nanoscience and nanotechnology[END_REF]. Let us consider the phase diagram shown in Figure 2.1 for a binary alloy of Au with a NW material M. At least in the M-rich region of the diagram (for large enough M fractions x ), the solid state is immiscible and consists of pure GaAs and Au crystallites. In this region, above the eutectic temperature e T and below the liquidus, the system consists of a mixed M-Au liquid and pure M crystallites, with temperature-dependent fractions of liquid and the remaining solid M. Above the liquidus, the M-Au alloy becomes a single liquid melt. Therefore, any growth temperature T between e T and max T for a given composition x (determined, say, by material flux) will produce solid M NW, with the remaining liquid melt in the droplet seated on the NW top. In other words, no Au should incorporate to the NW according to the equilibrium phase diagram. This property of the binary Au-Si, Au-Ge or pseudo-binary Au-GaAs alloys explains why Au is the most common catalyst to grow NWs. According to the lever rule [START_REF] Smith | Foundations of materials science and engineering[END_REF], the fraction of liquid equals

) 1 /( ) 1 ( L x x  
, while the fraction of solid M equals ) 1 /( ) (

L L x x x  
, with L x the M content in liquid.

Of course, using pseudo-binary diagrams for Au-III-V alloys (that is, assuming stoichiometric GaAs composition in the liquid phase) is not very relevant due to the known low solubility of highly volatile group V elements such as As and P in liquids [START_REF] Glas | Chemical potentials for Au-assisted vapor-liquid-solid growth of III-V nanowires[END_REF]. Therefore, Au-III-As or Au-III-P liquid melts contains very small fractions of the group V elements rather than being stoichiometric. This property considerably complicates theoretical treatment as will be explained in what follows. , while the fraction of solid M is ) 1 /( ) (

L L x x x   .
Specifically for III-V NWs, Au-catalyzed VLS growth with a high degree of control over the position, morphology, composition and crystal phase has been achieved in a wide range of epitaxy techniques, including metal organic vapor phase epitaxy (MOVPE, also known as metal organic chemical vapor deposition, MOCVD) [START_REF] Seifert | Growth of one-dimensional nanostructures in MOVPE[END_REF][START_REF] Joyce | Unexpected benefits of rapid growth rate for III-V nanowires[END_REF][START_REF] Chuang | Critical diameter for III-V nanowires grown on lattice-mismatched substrates[END_REF], chemical beam epitaxy (CBE, the same as molecular organic MBE, MO-MBE) [START_REF] Halder | Growth of wurtzite InP/GaP core-shell nanowires by metal-organic molecular beam epitaxy[END_REF][START_REF] Husanu | Growth of defect-free GaP nanowires[END_REF], and MBE [START_REF] Dubrovskii | Diffusion-induced growth of GaAs nanowhiskers: theory and experiment[END_REF][START_REF] Harmand | Analysis of vapor-liquid-solid mechanism in Au-assisted GaAs nanowire growth[END_REF][START_REF] Plante | Growth mechanisms of GaAs nanowires by gas source molecular beam epitaxy[END_REF]. In MOVPE, both group III and V elements are supplied in the form of chemical precursors, such as TMGa [Ga(CH 3 ) 3 ] for Ga and AsH 3 for As, in a carrier gas flow (H 2 ). CBE uses metal-organic precursors for the group III and either hydride gases or metal-organics TBAs (TBP) for the group V elements under high vacuum conditions. In MBE (either solid source or gas-source), III-V materials are deposited from an atomic beam of the group III element (Ga) and molecular beam of the group V element (As 4 or As 2 after pre-cracking). Hydride vapor phase epitaxy (HVPE), which uses chloride precursors for the group III element (GaCl) and hydride gases for the group V element in a vector from of H 2 with some additional HCl, yields a less controlled nucleation but on the other hand enables the highest aspect ratio and crystal quality in the zincblende (ZB) phase of GaAs NWs [START_REF] Gil | Record pure zincblende phase in GaAs nanowires down to 5 nm in radius[END_REF]. Despite this variety of techniques, the major difference between them is in the different mechanisms and kinetic pathways of delivering the material into the droplet and onto the substrate surface. These differences result in more or less regular nucleation of NWs (which also depends on the substrate preparation), parasitic growth and, most importantly, different group III and V contents in an Au-III-V droplet at the typical temperatures employed in a particular growth method. The latter manifests through different axial growth rates, which are typically on the order of 1 nm/s in MBE, 1-10 nm/s in MOVPE, and reach 40 nm/s in HVPE.

However, the liquid-solid phase transition can be understood from a general perspective, because the liquid alloy in any case consist of a mixture of group III and V atoms with Au.

Au-catalyzed VLS growth of III-V NWs usually consists of the following steps:

1) Preparation of the substrate surface with the Au catalyst particles;

2) Annealing the surface above the melting temperature of the alloy of Au with a given substrate material to produce liquid droplets.

3) Deposition of semiconductor material, where the properties of the NW ensembles (surface density, size uniformity, mean length, width, shape and even the crystal phase) can be tuned by the deposition conditions such as the growth time, temperature, group III and V fluxes.

In particular, Figure 2.2 illustrates the Au-catalyzed VLS growth of GaAs NWs by MBE on GaAs(111)B substrate in the simplest procedure where a thin solid Au layer is pre-deposited onto the substrate outside the MBE growth chamber. Then the substrate is transferred to the growth chamber and annealed above the eutectic temperature of Au-Ga alloy (the influence of

As can be ignored in this stage) to break the Au film into an ensemble of liquid droplets, with a broad Gaussian-like size distribution. Deposition of GaAs at a temperature of around 550 o C leads to rapid VLS growth of <111>B oriented NWs, and a much slower growth of a rough parasitic GaAs layer between the NWs. The resulting NW ensemble is irregular in terms of random positioning, broad length and radius distributions originating from the initial size inhomogeneity of the droplets and random character of the NW nucleation. In order to achieve more regular ensembles of NWs, Au particles can be organized before growth using electron beam [START_REF] Halder | Growth of wurtzite InP/GaP core-shell nanowires by metal-organic molecular beam epitaxy[END_REF] or nanoimprint [START_REF] Mårtensson | Nanowire arrays defined by nanoimprint lithography[END_REF] lithographical patterning of a substrate. In particular, Figure 2.3 presents the examples of InP/GaP core-shell NW arrays grown by MO-MBE from the organized Au seeds in the openings of a SiN x mask layer on InP(111)B [START_REF] Halder | Growth of wurtzite InP/GaP core-shell nanowires by metal-organic molecular beam epitaxy[END_REF].

Compared to Figure 2.2, these NW structures (i) are positioned in regular arrays, with the inter-NW distance and arrangement determined by the lithographically defined pitch; (ii) have no parasitic layer between the NWs, because semiconductor material does not stick onto the inert mask layer and (iii) are more regular in terms of both radius and length. The latter feature is explained by the fact that the initial growth seeds have a narrower size distribution compared to the case of thermally de-wetted Au droplets. It will be shown later that the axial NW growth rate depends on the droplet size and hence more uniform droplets should yield less variation in terms of length. It can be anticipated that the NW nucleation is better synchronized in time in the case of the VLS growth in regular templates, which also contributes into the resulting size uniformity. Using Au as the VLS growth catalyst for III-V NWs is not desirable for integration with Si electronics, because Au can contaminate the NWs due to kinetic factors. Consequently, selfcatalyzed (or self-assisted) VLS growth has emerged as a promising Au-free alternative, starting from the pioneering works by the group of Fontcuberta i Morral [START_REF] Colombo | Gaassisted catalyst-free growth mechanism of GaAs nanowires by molecular beam epitaxy[END_REF] and Jabeen et al. [START_REF] Jabeen | Self-catalyzed growth of GaAs nanowires on cleaved Si by molecular beam epitaxy[END_REF], and quickly adapted by other groups [START_REF] Jabeen | Self-catalyzed growth of GaAs nanowires on cleaved Si by molecular beam epitaxy[END_REF][START_REF] Gibson | Opportunities and pitfalls in patterned selfcatalyzed GaAs nanowire growth on silicon[END_REF][START_REF] Priante | Stopping and resuming at will the growth of GaAs nanowires[END_REF][START_REF] Ramdani | Arsenic pathways in self-catalyzed growth of GaAs nanowires[END_REF][START_REF] Dubrovskii | New mode of vapor-liquid-solid nanowire growth[END_REF][START_REF] Dubrovskii | Self-equilibration of the diameter of Ga-catalyzed GaAs nanowires[END_REF][START_REF] Bastiman | Growth map for Ga-assisted growth of GaAs nanowires on Si(111) substrates by molecular beam epitaxy[END_REF][START_REF] Koivusalo | Sub-Poissonian narrowing of length distributions realized in Ga-catalyzed GaAs nanowires[END_REF][START_REF] Tauchnitz | A simple route to synchronized nucleation of self-catalyzed GaAs nanowires on silicon for sub-Poissonian length distributions[END_REF][START_REF] Dong | Self-catalyzed GaAs nanowires on silicon by hydride vapor phase epitaxy[END_REF]. In this method, Au is replaced by the group III metal, that is, one of the NW constituents. Self-catalyzed VLS growth of III-V NWs allows one not only to safely avoid the unwanted Au contamination, but also to achieve excellent phase purity [START_REF] Dubrovskii | New mode of vapor-liquid-solid nanowire growth[END_REF] and high degree of size homogeneity within the NW ensembles in terms of both radius [START_REF] Dubrovskii | Self-equilibration of the diameter of Ga-catalyzed GaAs nanowires[END_REF] and length [START_REF] Koivusalo | Sub-Poissonian narrowing of length distributions realized in Ga-catalyzed GaAs nanowires[END_REF][START_REF] Tauchnitz | A simple route to synchronized nucleation of self-catalyzed GaAs nanowires on silicon for sub-Poissonian length distributions[END_REF]. It should be noted that the self-catalyzed growth of III-V NWs is more easily achieved in MBE. Chemical epitaxy techniques such as MOVPE, CBE or HVPE are less suited for the self-catalyzed approach, most probably due to the difficulty in maintaining Ga droplets under effectively group V rich conditions typically employed in these growth methods (see Ref.

[32] and a discussion therein). https://pubs.acs.org/doi/10.1021/acs.nanolett.7b00842, with permission from the American Chemical Society, to which further permissions related to the material excerpted should be directed. Figure 2.4 shows the initial stages of self-catalyzed GaAs NW growth in regular arrays on patterned Si [START_REF] Vukajlovic-Plestina | Engineering the size distributions of ordered GaAs nanowires on silicon[END_REF]. MBE growth was performed on a Si(111) substrate covered with a 10 nm thick thermal oxide, in which nanoscale holes were etched with reactive ion etching. The yield of vertical NWs was higher than 85%. SEM images in Figures 2.4 (a) to (i) show the GaAs NW arrays obtained on 45 nm wide holes as a function of time. For this series of samples, the Ga flux corresponded to a GaAs planar growth rate of 0.1 nm/s, the As 4 beam equivalent pressure (BEP) was 2 ×10 6 Torr, and the substrate temperature was 635 °C. Morphological analysis [START_REF] Vukajlovic-Plestina | Engineering the size distributions of ordered GaAs nanowires on silicon[END_REF] shows that the NWs elongate axially and extend radially, as shown in Figures 2.4 (j) and (k). Radial growth occurs due to effectively Ga-rich growth conditions, whereby more Ga atoms arrive to the droplet than used for NW growth [START_REF] Priante | Stopping and resuming at will the growth of GaAs nanowires[END_REF]. As a consequence, the droplets inflate, first by increasing their contact angle and then by extending the radius of the NW top. Cylindrical geometry is maintained by the step flow growth on the NW sidewalls, following the radial extension of the tip. Very importantly, one can observe a long delay of the NW nucleation, about 4 min, and a wide spread of the nucleation times for different NWs, which explains the broad length distribution within the array. Minimizing the nucleation randomness is crucial for improving the length uniformity [START_REF] Koivusalo | Sub-Poissonian narrowing of length distributions realized in Ga-catalyzed GaAs nanowires[END_REF][START_REF] Tauchnitz | A simple route to synchronized nucleation of self-catalyzed GaAs nanowires on silicon for sub-Poissonian length distributions[END_REF] as will be discussed in Section 8.

One of the most interesting features of self-catalyzed VLS growth is that the droplet serves as a non-stationary reservoir of a group III metal, which can either swell or shrink depending on the V/III flux ratio [START_REF] Priante | Stopping and resuming at will the growth of GaAs nanowires[END_REF][START_REF] Dubrovskii | Self-equilibration of the diameter of Ga-catalyzed GaAs nanowires[END_REF][START_REF] Tersoff | Stable self-catalyzed growth of III-V nanowires[END_REF][START_REF] Kim | Bistability of contact angle and its role in achieving quantum-thin self-assisted GaAs nanowires[END_REF]. Indeed, Ga is always present in the droplet for crystallization of GaAs with the arriving As. Hence, the axial NW growth rate is determined by the As influx and temperature-dependent desorption. Then the droplet will swell under negative and shrink under positive V/III influx imbalance, depending on whether more or less As atoms arrive to the droplet compared to Ga, respectively. The Ga influx usually includes surface diffusion flux of Ga adatoms from the NW sidewalls to the droplet, which is radius-dependent and increases for smaller radii. This may result in an interesting self-equilibration effect for the NW radius [START_REF] Dubrovskii | Self-equilibration of the diameter of Ga-catalyzed GaAs nanowires[END_REF][START_REF] Tersoff | Stable self-catalyzed growth of III-V nanowires[END_REF]. The effect requires that the influx of As atoms from vapor (minus desorption) is larger than that of Ga. The missing Ga is brought by surface diffusion along the sidewalls. This contribution scales with NW radius whereas Ga consumption (an As input) scales with its square. As a consequence, the droplet size, and consequently the radius of the NW top, will increase with time for thin NWs, and decrease with time for thick NWs. As a result, the NW radii will converge to a stationary value, corresponding to the stable growth at this constant radius. Therefore, the radius distribution of NWs will self-regulate to an almost delta-like function regardless of the initial size distribution of Ga droplets. Details of this process will be discussed in Section 7. The V/III influx imbalance can easily be regulated by the V/III flux ratio. This can be used for the fine tuning of the morphology of self-catalyzed III-V NWs. Besides engineering the radius of vertical NWs, growth interruptions can be used for changing the NW growth direction. Controllable switching of the NW growth direction opens up new horizons in the bottom-up engineering of self-assembled nanostructures, for example, to fabricate interconnected nanowires used for quantum transport measurements. In Ref. [START_REF] Koivusalo | Deterministic switching of the growth direction of self-catalyzed GaAs nanowires[END_REF], a robust and highly controllable method was proposed for deterministic switching of the growth direction of self-catalyzed GaAs NWs. The method is based on the modification of the droplet-NW interface in the annealing stage without any fluxes and subsequent growth in the horizontal direction by a twin-mediated mechanism with indications of a novel type of interface oscillations. By systematically optimizing the growth parameters, 100% yield of switching the NW growth direction from vertical to horizontal was achieved, and occurring in the same horizontal plane.

In brief, the initial self-catalyzed GaAs NWs were grown by solid source MBE on lithography-free oxide pattern templates fabricated on p-Si(111) substrates via droplet epitaxy, with remarkably narrow length distribution [START_REF] Koivusalo | Sub-Poissonian narrowing of length distributions realized in Ga-catalyzed GaAs nanowires[END_REF]. After growth of the vertical part, the NWs were annealed for 20 to 70 s at the growth temperature of 640 °C, without any fluxes in order to reshape the droplet-NW interface. After the annealing, the NW growth was resumed by It was found that horizontal type 1 NWs have a better crystal quality, exhibiting pure ZB structure throughout the whole NW, with a single twin plane extending along the horizontal section. For type 2 NWs, the downward section is dominated by periodic twinning. Therefore, the yield of type 1 horizontal NWs was carefully optimized by the growth parameter tuning, and reached 100% for smaller diameters of the initial NWs (75 nm), higher number density (2.5  10 8 cm -2 ), or longer annealing times (70 s).

Basics of the nanowire growth modeling

These examples demonstrate almost unlimited possibilities for bottom-up design of III-V NWs. Let us now discuss the most important ingredients of the NW growth modeling, leaving the details for the next sections. We start from the kinetic equations describing the time evolution of the total number of group V ( 5 N ) and III ( 3 N ) atoms in a catalyst droplet with base radius R, which equals the radius of the NW top [START_REF] Dubrovskii | Influence of the group V element on the chemical potential and crystal structure of Au-catalyzed III-V nanowires[END_REF]

dt dL R R I R I dt dN S des r       2 2 5 2 5 5 5 ) cos 1 ( 2 ) 1 (       , ( 3.1
)  (with the same meaning of the parameters as for group V), and the net diffusion current through the triple phase line (TPL) of cylindrical NW (in s -1 ). The latter is given by In the Au-catalyzed case under group V rich conditions, the excess group V atoms will desorb from the droplet. The axial NW growth rate is then obtained from the condition 0 /  ) and group III adatoms on the NW sidewalls at large enough distance from the droplet, and may be sensitive to the vapor flux of As atoms and their concentration in the droplet. The simplest approximation

dt dL R j R I dt dN S diff     2 , 3 2 3 3 3    . ( 3
L z diff z n RD j      ) / ( 2
3 3 3 3 2 I R j     (where 3
 summarizes the geometrical effect in MBE or the cracking efficiency at the NW sidewalls in MOVPE and the collection length of Ga atoms 3  can depend on the group V flux) leads to the typical inverse radius dependence of the NW axial growth rate [START_REF] Seifert | Growth of one-dimensional nanostructures in MOVPE[END_REF][START_REF] Dubrovskii | Diffusion-induced growth of GaAs nanowhiskers: theory and experiment[END_REF][START_REF] Dubrovskii | Influence of the group V element on the chemical potential and crystal structure of Au-catalyzed III-V nanowires[END_REF]

         R I dt dL S 3 3 3 3 2   . (3.4)
In this transport-limited regime, the NW elongation is independent of the droplet composition.

Similar equation is often used to describe the length-radius correlation of elemental semiconductor NWs such as Si [START_REF] Schubert | Silicon nanowhiskers grown on <111> Si substrates by molecular beam epitaxy[END_REF].

For self-catalyzed VLS growth, or Au-catalyzed VLS growth under As-poor conditions (where the Ga fraction in liquid approaches unity), the axial growth rate is limited by the kinetics of group V species, while dt dN /

) 1 (          . (3.5)
However, this equation is not self-consistent because the desorption rate depends on the atomic concentration of group V atoms in the droplet 5 c (here and below, we use the normalization for

X c with  X 3, 5, Au such that 1 5 3    Au c c c
, which is reduced to 1 5 3

  c c

for self-catalyzed NWs). According to Refs. [START_REF] Glas | Predictive modeling of selfcatalyzed III-V nanowire growth[END_REF][START_REF] Dubrovskii | Group V sensitive vapor-liquid-solid growth of Au-catalyzed and selfcatalyzed III-V nanowires[END_REF], if group V species desorb in the form of dimers, we have by the crystallization rate at the liquid-solid interface under the droplet [START_REF] Glas | Predictive modeling of selfcatalyzed III-V nanowire growth[END_REF][START_REF] Dubrovskii | Group V sensitive vapor-liquid-solid growth of Au-catalyzed and selfcatalyzed III-V nanowires[END_REF].

Theoretical considerations [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF][START_REF] Dubrovskii | Growth rate of a crystal facet of arbitrary size and the growth kinetics of vertical nanowires[END_REF][START_REF] Dubrovskii | Growth kinetics and crystal structure of semiconductor nanowires[END_REF][START_REF] Kashchiev | Dependence of the growth rate of nanowires on the nanowire diameter[END_REF][START_REF] Mårtensson | Simulation of GaAs nanowire growth and crystal structure[END_REF] and in situ growth monitoring inside a transmission electron microscope (TEM) [START_REF] Wen | Periodically changing morphology of the growth interface in Si, Ge, and GaP nanowires[END_REF][START_REF] Jacobsson | Interface dynamics and crystal phase switching in GaAs nanowires[END_REF][START_REF] Harmand | Atomic step flow on a nanofacet[END_REF] suggest that NWs often grow in the so-called mononuclear regime whereby only one 2D island succeeds in nucleating for each NW monolayer (ML). The probability to nucleate a 2D island on the top facet of cylindrical NW of radius R per unit time

equals J R 2



, where J is the nucleation rate. On the other hand, the island fills the complete ML slice of a NW after the time period v R / , where

dt dr v / 
is the growth rate of a linear size of the island. The ratio of these characteristic times naturally yields the control parameter

v J R / 3  [43]. Mononuclear growth occurs at 1 / 3  v J R



, which requires small NW radii, slow nucleation rates and rapid island growth rates. Similar considerations apply when nucleation of 2D islands takes place at the TPL rather than on the whole top facet, as has been proposed [START_REF] Glas | Predictive modeling of selfcatalyzed III-V nanowire growth[END_REF] to explain polytypism in NWs (see section 10.2), in which case the nucleation probability equals

J Rr c  2
, with c r as the radius of the critical 2D island [START_REF] Dubrovskii | Group V sensitive vapor-liquid-solid growth of Au-catalyzed and selfcatalyzed III-V nanowires[END_REF][START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF].

Slow nucleation rates correspond to low chemical potentials of III-V pairs in the liquid with respect to the reference solid state, defined as [START_REF] Glas | Chemical potentials for Au-assisted vapor-liquid-solid growth of III-V nanowires[END_REF] ) ( ) , , ( ) , , ( ) , , (

T T c c T c c T c c S L L         . ( 3
with ) /( T k B     
the chemical potential in thermal units. The energetic constant a is proportional to the effective surface (or edge) energy of 2D island eff  . For regular triangle shape of the island, it is given by [7]

2 / 1 4 / 3 ) ( 3 2 h T k a S B eff     , ( 3.9)
with h the ML height. Using CNT requires that the critical size is macroscopic ( 1  c i

). In the mononuclear growth regime, the axial NW growth rate equals The Zeldovich nucleation rate of CNT was studied in Ref. [START_REF] Glas | Predictive modeling of selfcatalyzed III-V nanowire growth[END_REF] for self-catalyzed and in Ref. [START_REF] Dubrovskii | Zeldovich nucleation rate, self-consistency renormalization, and crystal phase of Au-catalyzed GaAs nanowires[END_REF] for Au-catalyzed III-V NWs. It was argued that the island formation energy ) (i F should include the so-called self-consistency renormalization to ensure that ) (i F equals zero for the island of size one [START_REF] Ameruddin | Understanding the growth and composition evolution of gold-seeded ternary InGaAs nanowires[END_REF]. The details of calculations can be found in Ref. [START_REF] Dubrovskii | Zeldovich nucleation rate, self-consistency renormalization, and crystal phase of Au-catalyzed GaAs nanowires[END_REF]. The resulting expression in the Au-catalyzed case has the form [START_REF] Dubrovskii | Zeldovich nucleation rate, self-consistency renormalization, and crystal phase of Au-catalyzed GaAs nanowires[END_REF]  . This expression is consistent with the earlier result of Ref. [START_REF] Glas | Predictive modeling of selfcatalyzed III-V nanowire growth[END_REF] for self-catalyzed III-V NWs

                        4 exp 3
              4 exp 2 2 / 1 5 0 a c J J , (3.12) with a S S e h D J            35 2 5 4 / 3 0 3   . ( 3.13)
For a Ga droplet, the surface energy and the diffusion coefficient of group V atoms in liquid Ga depend only on temperature. Therefore, the 0 J coefficient may depend on temperature but neither on   nor 5 c . According to Ref. [START_REF] Glas | Predictive modeling of selfcatalyzed III-V nanowire growth[END_REF], the values of the two unknowns 0 J and eff  [related to a in Eq. (3.12) through Eq. (3.9)] for Ga-catalyzed GaAs NWs equal 7 nm -2 s -1 and 0.123 J  m -2 , respectively, and can be taken independent of temperature within the range of 570-630 o C.

Diffusion coefficient of As in liquid Ga is estimated at 

As D 2 × 10 12 m 2 s -1 for this temperature range [START_REF] Dubrovskii | Zeldovich nucleation rate, self-consistency renormalization, and crystal phase of Au-catalyzed GaAs nanowires[END_REF].

When the critical size c i approaches one III-V pair, CNT is no longer valid [START_REF] Dubrovskii | Nucleation theory and growth of nanostructures[END_REF]. Small Non-stationary equation for dt dN / 3 is separated from the axial growth rate in the latter case and describes the time evolution of the Ga droplet size. The full self-consistency requires determination of (i) chemical potentials as functions of the concentration(s) and temperature, (ii) desorption rate of group V atoms from the droplet, and (iii) diffusion flux of group V atoms into the droplet. This will be discussed in Section 6. Furthermore, relating X N to X c and  to 3 c by simple geometrical expressions enables one to treat interesting not-stationary problems such as formation of axial NW heterostructures in ternary III-V NWs [START_REF] Dubrovskii | Factors influencing the interfacial abruptness in axial III-V nanowire heterostructures[END_REF][START_REF] Dubrovskii | Compositional control of gold-catalyzed ternary nanowires and axial nanowire heterostructures based on IIIP 1-x As x[END_REF], or evolution of the droplet volume under time-dependent fluxes [START_REF] Dubrovskii | Composition and contact angle of Au-III-V droplets on top of Au-catalyzed III-V nanowires[END_REF].

Role of surface energy

Nebol'sin and Shchetinin [START_REF] Nebol'sin | Role of surface energy in the vapor-liquid-solid growth of silicon[END_REF] presented a model for the surface energetics of VLS NWs, in which a stable contact angle of the droplet was introduced related to stable growth of NWs with vertical sidewalls. The model enabled a simple classification of the catalyst metals suitable for the VLS growth of Si NWs [START_REF] Nebol'sin | Role of surface energy in the vapor-liquid-solid growth of silicon[END_REF]. The surface energy criteria were later reconsidered by Tersoff et al. [START_REF] Wen | Periodically changing morphology of the growth interface in Si, Ge, and GaP nanowires[END_REF][START_REF] Jacobsson | Interface dynamics and crystal phase switching in GaAs nanowires[END_REF][START_REF] Ross | Sawtooth faceting in silicon nanowires[END_REF] and Dubrovskii [START_REF] Dubrovskii | Development of growth theory for vapor-liquid-solid nanowires: contact angle, truncated facets and crystal phase[END_REF] and turned out to have an important impact on the droplet stability issues [START_REF] Kim | Bistability of contact angle and its role in achieving quantum-thin self-assisted GaAs nanowires[END_REF][START_REF] Zannier | Nanoparticle stability in axial InAs InP nanowire heterostructures with atomically sharp interfaces[END_REF], NW morphologies [START_REF] Ross | Sawtooth faceting in silicon nanowires[END_REF][START_REF] Dubrovskii | Development of growth theory for vapor-liquid-solid nanowires: contact angle, truncated facets and crystal phase[END_REF], and related crystal phases of III-V NWs [START_REF] Jacobsson | Interface dynamics and crystal phase switching in GaAs nanowires[END_REF][START_REF] Johansson | Effects of supersaturation on the crystal structure of gold seeded III-V nanowires[END_REF][START_REF] Panciera | Phase selection in self-catalyzed GaAs nanowires[END_REF]. In particular, in situ monitoring of the VLS growth of GaAs NWs [START_REF] Jacobsson | Interface dynamics and crystal phase switching in GaAs nanowires[END_REF][START_REF] Panciera | Phase selection in self-catalyzed GaAs nanowires[END_REF] reveal the occurrence of a truncated corner facet at the growth interface at large enough contact angles of the droplet. The amount of truncation oscillates with the period of ML formation. The presence or absence of the truncation is related to the crystal phase of III-V NWs [START_REF] Wen | Periodically changing morphology of the growth interface in Si, Ge, and GaP nanowires[END_REF][START_REF] Jacobsson | Interface dynamics and crystal phase switching in GaAs nanowires[END_REF][START_REF] Panciera | Phase selection in self-catalyzed GaAs nanowires[END_REF],

which can be either cubic ZB or hexagonal wurtzite (WZ). Morphology of the growth interface has an interesting connection with the Glas theory of polytypism of VLS III-V NWs [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF]. All these factors show the importance of the surface energy constrains in the VLS growth of NWs.

Possible geometries of the VLS growth interface are shown in [START_REF] Pankoke | Calculation of the diameter-dependent polytypism in GaAs nanowires from an atomic motif expansion of the formation energy[END_REF][START_REF] Rosini | Surface effects on the atomic and electronic structure of unpassivated GaAs nanowires[END_REF], while the solid-liquid surface energies ) (  SL are usually assumed phase-independent for any  according to Ref. [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF]. Using the results of Ref. [START_REF] Dubrovskii | Development of growth theory for vapor-liquid-solid nanowires: contact angle, truncated facets and crystal phase[END_REF], the excess surface energy (per unit area) of forming NWs with non-wetted facet at the angle  to the vertical with respect to the standard non-wetted vertical facet in the ZB phase is given by

         tan )] cos ) 2 / ( [ ) 0 ( cos ) ( LV SL SV SV W N     G  . ( 4
         sin tan ) 2 / ( ) 0 ( cos ) ( LV tr SL SV tr tr SL W     G . ( 4.2)
We note that these expressions are actually insensitive to whether the VLS growth is considered as instantaneous adding of MLs or at a constant volume of the catalyst droplet [START_REF] Dubrovskii | Development of growth theory for vapor-liquid-solid nanowires: contact angle, truncated facets and crystal phase[END_REF]. [START_REF] Pankoke | Calculation of the diameter-dependent polytypism in GaAs nanowires from an atomic motif expansion of the formation energy[END_REF][START_REF] Rosini | Surface effects on the atomic and electronic structure of unpassivated GaAs nanowires[END_REF]. This feature is essential for the WZ phase formation in III-V NWs (see, for example, Ref. [START_REF] Dubrovskii | Nucleation theory and growth of nanostructures[END_REF] for a detailed review). The calculated surface energies of the solid-vapor facets are given in Table 4.1, along with the corresponding references. The droplet surface energy LV  should be close to that of pure liquid Ga. This is guaranteed for Ga-catalyzed growth and should also be the case for Au-catalyzed due to the lower surface energy of liquid Ga compared to Au. We take the Ga surface energy at 420 o C, corresponding to the growth temperature employed during in situ measurements [START_REF] Panciera | Phase selection in self-catalyzed GaAs nanowires[END_REF]. However, the temperature dependence of the surface energy of liquid Ga is very week [START_REF] Dubrovskii | Gibbs-Thomson and diffusion-induced contributions to the growth rate of Si, InP and GaAs nanowires[END_REF]. All these considerations show that the structural diagram for a given material should not be much affected by the catalyst type and temperature. Calc.

 (J/m 2 ) ) 110 (  (J/m 2 ) ) 00 1 1 (  (J/m 2 ) Ga  (J/m 2 ) min  (deg) max  (deg) w  (deg) tr n    (deg) ) 2 / (  SL (J/m 2 ) ) ( tr SL   (J/m 2 )
[62]

Calc.

[62]

Calc.

[65]

Exp.

[61]

Exp. [START_REF] Panciera | Phase selection in self-catalyzed GaAs nanowires[END_REF] Assum. Exp. Fit Fit corresponds to the VLS growth with vertical sidewalls and planar liquid-solid interface, most probably in the WZ phase. Above max  , the truncated corner facets are developed, growing to a finite length. The crystal phase of such NWs should be cubic ZB. There is a narrow range of contact angles, approximately between 125 o and 127 o , where the truncated top is followed by vertical (110) ZB sidewalls (structure 3).

The contact angle of  127 o can be called the second stable angle for the VLS growth at low V/III flux ratios, because any further increase will be compensated by introduction of the widening facet. If the droplet volume keeps increasing under excessive Ga influx, like in Ga-catalyzed GaAs NWs grown under Ga-rich conditions, the top facet increases its radius. This leads to reverse tapered geometry (structure 4).

Subsequent radial growth on the NW sidewalls may resume cylindrical geometry, with the uniform NW radius increasing with time. Further increase of the contact angle to ~ 137 o (in the kinetic regime) leads to the droplet fall onto the NW sidewalls. The arrows show the transformation of morphology with increasing the contact angle. The increase of surface energy in transition from structure 2 to 3 is explained by the fact that the vertical part of the NW must follow the ZB phase, determined by the center nucleation on the top facet. . These values can be deduced based on the results of in situ monitoring of Ga-catalyzed GaAs NWs [START_REF] Panciera | Phase selection in self-catalyzed GaAs nanowires[END_REF]. These results were obtained under varying Ga and As fluxes, which resulted in different contact angles of Ga droplets. At small contact angles, typically below 100 o , the NW morphology was tapered and the liquid-solid interface was planar. This kinetic regime was achieved under high As and low Ga fluxes where the droplet volume was gradually diminishing. It should be noted that treatment of the crystal phase presented here are based entirely on the surface energetics of ZB or WZ NWs in different morphologies and ignore the influence of chemical potential on the crystal phase switching, which is very important in the Glas theory of polytypism [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF]. In this simplified picture, the preferred crystal phase is controlled by the sole parameter, the droplet contact angle. While including the chemical potential considerations may refine the picture, the existence of the two critical contact angles for the crystal phase switching in GaAs NWs strongly suggests that phase transitions occur quite abruptly whenever allowed by surface energetics. In other words, chemical potential near the transition points quickly exceeds the energy of stacking fault, which is required for switching from the ZB to WZ phase [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF][START_REF] Dubrovskii | Growth rate of a crystal facet of arbitrary size and the growth kinetics of vertical nanowires[END_REF][START_REF] Johansson | Effects of supersaturation on the crystal structure of gold seeded III-V nanowires[END_REF], and do not significantly affect the critical contact angles for the transition. We anticipate that a similar growth picture is relevant for other III-V NWs, but with different contact angles for the morphological transformations and the related crystal phase switching. These contact angles are entirely determined by the surface energies of the relevant facets. It would be very interesting to perform similar in situ growth experiments for other material systems in the future.

Selective area epitaxy of III-V nanowires

Selective area epitaxy (SAE) of III-V NWs, pursued by Fukui et al. [START_REF] Mohan | Controlled growth of highly uniform, axial/radial direction-defined, individually addressable InP nanowire arrays[END_REF][START_REF] Tomioka | Growth of highly uniform InAs nanowire arrays by selective-area MOVPE[END_REF] and other groups [START_REF] Dalacu | Selective-area vapour-liquid-solid growth of InP nanowires[END_REF][START_REF] Hertenberger | Growth kinetics in position-controlled and catalyst-free InAs nanowire arrays on Si(111) grown by selective area molecular beam epitaxy[END_REF][START_REF] Bassett | Evolution of GaAs nanowire geometry in selective area epitaxy[END_REF][START_REF] Robson | Multispectral absorptance from large-diameter InAsSb nanowire arrays in a single epitaxial growth on silicon[END_REF][START_REF] Robson | Conditions for high yield of selective-area epitaxy InAs nanowires on SiOx/Si(111) substrates[END_REF][START_REF] Gao | Simultaneous selective-area and vapor-liquid-solid growth of InP nanowire arrays[END_REF], is another Au-free alternative to the Au-catalyzed VLS growth method. SAE growth of III-V NWs works equally well in MOVPE and MBE techniques. In the true SAE mode with no group III droplets on the NW tops, vertical growth of NWs is directed by the arrays of lithographically defined holes, or pores on patterned substrates such as SiO x /Si(111).

To improve the vertical yield of NWs, a balanced V/III flux ratio should be employed in order to avoid the formation of too large group III droplets in the holes or irregular III-V crystallites [START_REF] Robson | Conditions for high yield of selective-area epitaxy InAs nanowires on SiOx/Si(111) substrates[END_REF].

However, the initial stage of the SAE growth of III-V NWs may have the self-catalyzed VLS nature [START_REF] Robson | Conditions for high yield of selective-area epitaxy InAs nanowires on SiOx/Si(111) substrates[END_REF][START_REF] Gao | Simultaneous selective-area and vapor-liquid-solid growth of InP nanowire arrays[END_REF], with group III droplets disappearing in a later stage due a reduced material supply of group III atoms by surface diffusion.

According to the experimental data [START_REF] Mohan | Controlled growth of highly uniform, axial/radial direction-defined, individually addressable InP nanowire arrays[END_REF][START_REF] Tomioka | Growth of highly uniform InAs nanowire arrays by selective-area MOVPE[END_REF][START_REF] Dalacu | Selective-area vapour-liquid-solid growth of InP nanowires[END_REF][START_REF] Hertenberger | Growth kinetics in position-controlled and catalyst-free InAs nanowire arrays on Si(111) grown by selective area molecular beam epitaxy[END_REF][START_REF] Bassett | Evolution of GaAs nanowire geometry in selective area epitaxy[END_REF][START_REF] Robson | Multispectral absorptance from large-diameter InAsSb nanowire arrays in a single epitaxial growth on silicon[END_REF], both length L and radius R of the SAEgrown NWs usually increase with time, while the radius of Au-catalyzed VLS III-V NWs is fixed by the initial size of the growth seed and stays constant under optimized conditions. Here, we present a model for the time evolution of the mean length and radius of SAE NWs versus the growth parameters [START_REF] Dubrovskii | Evolution of the length and radius of III-V nanowires grown by selective area epitaxy[END_REF]. In particular, we discussed why, after a short incubation stage, the NW length and radius often feature scaling power-law dependences on the growth time, with the power exponents related to the growth conditions, the NW length and the array pitch.

Consider the SAE growth from the group III and V atomic fluxes k as the desorption rates of group V atoms from the NW top and sidewalls, and 35 k and '

35

k as the crystallization rates of III-V pairs on the NW top and sidewalls, respectively. The latter are measured in nm  can be limited by the shadowing effect [START_REF] Sibirev | Influence of shadow effect on the growth and shape of InAs nanowires[END_REF], with

In the equation for the radial growth rate, the L L / * factor accounts for the fact that crystallization of III-V pairs occurs only at the NW areas exposed to the vapor fluxes but contributes into the radial growth of the entire NW (under the assumption of cylindrical NW having a uniform radius from base to top at any moment of time). Kinetic equations (5.1), (5.2) and (5.4) should be valid for the SAE NW growth under rather general conditions. In Ref. [START_REF] Dubrovskii | Evolution of the length and radius of III-V nanowires grown by selective area epitaxy[END_REF], two additional simplifications were used. First, it was assumed that the SAE growth proceeds under group V rich conditions -otherwise, it would quickly transition to a self-catalyzed mode with a group III droplet appearing on the NW top [START_REF] Gao | Simultaneous selective-area and vapor-liquid-solid growth of InP nanowire arrays[END_REF]. Then, much more group V atoms should desorb than crystallize in solid, corresponding to negligible desorption terms in Eqs. (5.1) and (5.2) for group V atoms. Second, the diffusion current 3 j should be presented as a function of 3 n . In the absence of desorption of a group III element, the simplest approximation used in Ref. [START_REF] Dubrovskii | Evolution of the length and radius of III-V nanowires grown by selective area epitaxy[END_REF] writes

) 1 ( 2 3 ' 3 * 3 f I RL j      . ( 5
          k k k k I I f          (5.6)
is given by the ratio of the group III adatom activities on the NW top facet over the sidewalls. Under these assumptions, Eqs. (5.1), (5.2) and (5.4) yield the kinetic equations for the axial and radial NW growth rates of the form

         ) 1 ( 2 * ' 3 3 3 f R L v dt dL    , f L L v dt dR   * ' 3 3  , ( 5.7)
where

3 3 I v S  
is the vapor flux of group III species in nm/s. This model should be relevant for sufficiently long NWs, while for shorter ones we should additionally account for a diffusion flux from the substrate surface [START_REF] Gao | Simultaneous selective-area and vapor-liquid-solid growth of InP nanowire arrays[END_REF]. While negligible for the growth on rough substrates with a parasitic layer [START_REF] Dubrovskii | Role of nonlinear effects in nanowire growth and crystal phase[END_REF], this flux can be much more important in the case of atomically flat patterned SiO x /Si(111) substrates, and essential for understanding of the growth start.

Equation (5.7) for dt dL / is similar to the conventional expression for the axial growth rate of VLS III-V NWs [START_REF] Seifert | Growth of one-dimensional nanostructures in MOVPE[END_REF][START_REF] Dubrovskii | Diffusion-induced growth of GaAs nanowhiskers: theory and experiment[END_REF][START_REF] Dubrovskii | Group V sensitive vapor-liquid-solid growth of Au-catalyzed and selfcatalyzed III-V nanowires[END_REF][START_REF] Dubrovskii | Gibbs-Thomson and diffusion-induced contributions to the growth rate of Si, InP and GaAs nanowires[END_REF][START_REF] Dubrovskii | Role of nonlinear effects in nanowire growth and crystal phase[END_REF][START_REF] Harmand | Growth kinetics of a single InP 1-x As x nanowire[END_REF][START_REF] Dubrovskii | Diffusion-induced growth of nanowires: generalized boundary conditions and self-consistent kinetic equation[END_REF][START_REF] Johansson | Mass transport model for semiconductor nanowire growth[END_REF][START_REF] Fröberg | Diameter-dependent growth rate of InAs nanowires[END_REF] or self-induced GaN NWs [START_REF] Consonni | Quantitative description for the growth rate of self-induced GaN nanowires[END_REF] under group V rich conditions. For small 1  f  , it gives the exponential elongation with time at ) are given by

                     0 0 0 1 1 ) ( t t R t t R L L c a c ,           0 0 1 t t R R , (5.8)
with

f f a   / ) 1 ( 2   , ] ) 1 ( /[ ' 3 3 0      a R R f c , ) /( ' 3 3 0 0  v R t 
, and 0 R , 0 L as the initial NW radius and length at

0  t , respectively. For long NWs ( 3   L
), the solutions change to

1 1 * 3 1             a a t t t L  , 1 1 1 * * 1             a t t t R R , (5.9) with ] ) 2 /[( ' 3 3 * 1   v R t f  
and * R as the NW radius at the moment of time * t where L becomes Next, we discuss the data of Ref. [START_REF] Bassett | Evolution of GaAs nanowire geometry in selective area epitaxy[END_REF] for the length and diameter evolution of GaAs NWs grown by SAE MOVPE at 750 o C on patterned SiO 2 /GaAs substrates, with a 600 nm pitch and variable pore diameter from 125 to 225 nm. These data show that the NW length first increases super-linearly with time, converging to a sub-linear behavior for longer times (Figure 5.

2). This effect should be due to a limited diffusion length of Ga adatoms on the NW sidewalls.

At a high temperature of 750 o C, a fraction of Ga adatoms desorb before reaching the NW top, corresponding to the transition from the growth regime at Other parameters are summarized in Table 5.1 and provide the excellent fits to all data. From the fits, we can deduce the values of  In Ref. [START_REF] Gao | Simultaneous selective-area and vapor-liquid-solid growth of InP nanowire arrays[END_REF], it was shown that true SAE growth without droplets on top can coexist with the In-catalyzed VLS growth for InP NWs by MOVPE even in the same sample. The actual growth mode of NWs depends on the initial radius of the pores, V/III flow ratio and growth time.

The VLS growth mode becomes more probable for thinner NWs (which start from smaller pores), lower V/III flow ratios and shorter times. In particular, Figure 5.

Growth kinetics of III-V nanowires

As already discussed in Section 3, the main kinetic processes governing the VLS growth of III-V NWs include (i) material transport of the group III and V species into the droplet by different kinetic pathways, including surface diffusion of group III atoms, (ii) desorption of group V atoms, and (iii) nucleation of 2D islands on the liquid-solid interface or at the TPL (at low supersaturations) or their regular crystallization (at high supersaturations). Here, we briefly discuss theoretical understanding and modeling methods for these processes achieved to this end.

Transport-limited nanowire growth

Models based on the mass transport equations do not consider 2D nucleation and hence are relevant in the case of Au-catalyzed VLS growth of III-V NWs under group V rich conditions, self-induced growth of GaN NWs or true SAE growth of III-V NWs. The axial NW growth rate in this case is limited by the mass transport of group III atoms into the droplet according to Eq. (3.3). Similar models can also be used for NWs of elemental semiconductors.

The main problem is in finding the diffusion-induced contribution to the axial NW growth rate, given by

L z f f S diff dz dn D R dt dL           2 . (6.1)
Here, subscript "f" is introduced instead of "3" in order to distinguish group III adatoms on the NW sidewalls ("f") or on the substrate surface ("s").

Diffusion-limited growth of NWs was considered in many original papers and reviews, for example, in Refs. [START_REF] Givargizov | Highly Anisotropic Crystals[END_REF][START_REF] Dubrovskii | Nucleation theory and growth of nanostructures[END_REF][START_REF] Dubrovskii | Theory of VLS growth of compound semiconductors[END_REF][START_REF] Dubrovskii | Diffusion-induced growth of GaAs nanowhiskers: theory and experiment[END_REF][START_REF] Dubrovskii | Gibbs-Thomson and diffusion-induced contributions to the growth rate of Si, InP and GaAs nanowires[END_REF][START_REF] Harmand | Growth kinetics of a single InP 1-x As x nanowire[END_REF][START_REF] Dubrovskii | Diffusion-induced growth of nanowires: generalized boundary conditions and self-consistent kinetic equation[END_REF][START_REF] Johansson | Mass transport model for semiconductor nanowire growth[END_REF][START_REF] Fröberg | Diameter-dependent growth rate of InAs nanowires[END_REF][START_REF] Consonni | Quantitative description for the growth rate of self-induced GaN nanowires[END_REF]. The most general result is given by [START_REF] Dubrovskii | Diffusion-induced growth of nanowires: generalized boundary conditions and self-consistent kinetic equation[END_REF] ) / (

) / ( 1 3 f f diff L U C L BU dt dL V            , (6.2)
where

3 3 3 cos S I V   is the 2D equivalent growth rate in MBE, 3
 is the angle of group III beam to the vertical, and f  is the diffusion length of group III adatoms on the NW sidewalls. The functions U and U  are defined as follows

                                   1 cosh sinh ) / ( f s f f L R Q L L U      , (6.3)                             f s f f f L R Q L L d dU L U       sinh cosh ) / ( ) / ( . ( 6.4)
Here,

) / ( / ) / ( ) / ( 0 1 s s s R K R K R      , with ) (x K m
as the modified Bessel functions of order m, R as the radius of the NW top, and s  as the diffusion length of group III adatoms on the substrate surface. The kinetic coefficients B and C stand for the diffusion fluxes into the droplet which originate from the NW sidewalls and the substrate, respectively

) 1 ( tan 2 3 f f R B       , ) 1 ( 2 s s s R R C               . ( 6.5)
The parameters f  and s  describe the differences between chemical potentials of group III atoms on the two surfaces and in liquid, similarly to Eq. (5.5):

           T k B fl f   exp ,            T k B sl s   exp . (6.6)
Here,

L f fl 3       , L s sl 3      
are the chemical potential differences for group III atoms on the surfaces ( f  on the NW sidewalls and s  on the substrate) and in the liquid phase ( L 3  ).

The coefficient Q in Eqs. (6.3) and (6.4) is given by

3 tan      f s f s Q  , ( 6.7)
and does not contain chemical potential in liquid.

The total group III limited axial NW growth rate is obtained by summing up Eq. ( 6.2)

with the flux originating from the direct impingement from vapor. If NWs grow on unpatterned substrates, we should subtract the 2D growth rate of parasitic layer between the NWs. Without desorption, all group III atoms deposited from vapor should be distributed in the parasitic layer and NWs. Then the final result for the axial growth rate takes the form [77]

1 1 ) / ( ) / ( 1 2 3                       NW f f S R L U C L BU A dt dL V   , (6.8)
with NW S as the mean substrate area per NW (the reverse of the NW surface density). The vapor flux in the MBE case is given by

3 3 3 cos ) , (      A , ( 6.9)
where the geometrical function

2 / ) cos 1 ( ) 2 / , ( 3 3 3       at 2 /    ,   2 3 sin / 1  when 2 / 3     
, and has a more complex form for intermediate contact angles  [START_REF] Glas | Vapor fluxes on the apical droplet during nanowire growth by molecular beam epitaxy[END_REF]. Thus, the transport-limited elongation rate generally contains four unknown parameters: the two adatom diffusion lengths on the substrate surface and the NW sidewalls ( s  and f  ), and the two chemical potential differences between the surface and sidewall adatoms and group III atoms in liquid ( sl   and fl   ).

The transport-limited growth model given by Eq. (6.8) was used for modeling the growth kinetics of a single InP 1-x As x NW with modulated composition in Ref. [START_REF] Harmand | Growth kinetics of a single InP 1-x As x nanowire[END_REF]. The NWs were grown at 420°C by the Au-catalyzed MBE on InP (111)B substrates. These growth conditions produced NWs of WZ structure which grow in the [0001] direction without stacking faults, with a uniform diameter. A periodic modulation of the incident As 4 :P 2 flux ratio was achieved by using non-uniformity of the fluxes on the substrate area. The InP 1-x As x NWs were removed from their substrate and imaged by high angle annular dark field scanning TEM (HAADF STEM).

The image of a NW segment of 45 nm diameter is presented in Figure 6.1. The modulation of the HAADF intensity profile integrated over the whole diameter is superimposed [Figure 6.1 (a)].

The FFT-filtered HAADF image with the corresponding intensity profile is shown in Figure 6.1 (b). The oscillation amplitude is about 0.7% of the total HAADF intensity. The corresponding variation x  of the As concentration x was determined to be about 0.03 around the mean value of 0.66. Such a small variation of composition is not expected to influence the NW elongation. The distance between any two concentration extrema equals the increase of the NW length during one time period. Therefore, it is straightforward to determine the instantaneous growth rate at each oscillation. This procedure was applied to a 21 nm diameter NW. The dependence with time of the NW length counted from the cleaved end is shown in Figure 6.2.

The measured NW length clearly displays a nonlinear behavior with time. In this growth experiment, the flux modulation was stopped for 12 time periods. The corresponding region without compositional modulation, indicated in Figure 6.2, may serve as a time reference. At some point during growth, the nominal As 4 :P 2 ratio was changed abruptly from 1:5 to 3:3, while the total average group V flux was kept constant. Thanks to the former time reference, this event can be accurately dated, as shown in Figure 6.2. The impact of this large change of flux ratio is strong and immediate -as seen from the change of the slope in Figure 6.2, the growth rate was suddenly reduced by a factor of 2. The obtained length-time dependence was fitted by Eq. (6.8). The value of f  has little influence on the results, provided that it is larger than the final NW length. Therefore, it was chosen to be 2 m. As regards the length-radius dependences of different semiconductor NWs, they were measured and modeled by many authors (see Refs. [START_REF] Dubrovskii | Nucleation theory and growth of nanostructures[END_REF] and [START_REF] Dubrovskii | Theory of VLS growth of compound semiconductors[END_REF] for a detailed review). The diffusion transport into the droplet given by the general Eq. (6.2) can be reduced to the inverse power law dependence on the NW radius and becomes independent of length in some important particular cases (see Ref. [START_REF] Dubrovskii | Theory of VLS growth of compound semiconductors[END_REF] for systematization of such growth modes). For thin NWs, the chemical potential of group III atoms in the catalyst droplet can be elevated by the Gibbs-Thomson effect due to the curvature of the droplet surface according to [START_REF] Dubrovskii | Gibbs-Thomson and diffusion-induced contributions to the growth rate of Si, InP and GaAs nanowires[END_REF][START_REF] Dayeh | Direct observation of nanoscale size effects in Ge semiconductor nanowire growth[END_REF]

R L LV L eff     sin 2 3 3    . ( 6.10)
Here, L  ~is the elementary volume per group III atom in the liquid phase and

 sin / R R d 
is the droplet radius, with R as the radius of the NW top. Using Eq. (6.6) for the adatoms contributing to the NW elongation in a given growth regime, we obtain

  R R p GT be R A dt dL V / 3 * 3 1 1           . (6.11)
Here, *

A stands for the vapor flux and accounts for possible growth on the substrate surface. The

] / ) exp[( 3 T k b B a L    
is a radius-independent constant (with index "a" corresponding to sidewall or surface adatoms), [START_REF] Zhang | Nanowires: building blocks for nanoscience and nanotechnology[END_REF]  is the effective diffusion length of group III adatoms and p is the growth index which typically ranges from ½ to 2. For example, 1  p corresponds to the collection of group III atoms from the top part of the NW as discussed above, and 2  p to the collection from the substrate surface. The characteristic Gibbs-Thomson radius is given by

T k R B LV L GT   sin 2  , (6.12)
and equals a few nanometers in the typical cases [START_REF] Dubrovskii | Gibbs-Thomson and diffusion-induced contributions to the growth rate of Si, InP and GaAs nanowires[END_REF]. Integration of Eq. ( 6.11) at a time-independent NW radius gives

                    R R p GT be R A H L / 3 * 1 , (6.13)
with t V H 3 
as the effective deposition thickness. This simplified expression shows that the length decreases with their radius for thick enough NWs where the Gibbs-Thomson effect becomes negligible, while for thin NW the growth rate tends to zero at a certain minimum droplet size. NWs cannot be grown thinner that this minimum size due to the Gibbs-Thomson effect [START_REF] Dubrovskii | Gibbs-Thomson and diffusion-induced contributions to the growth rate of Si, InP and GaAs nanowires[END_REF][START_REF] Dayeh | Direct observation of nanoscale size effects in Ge semiconductor nanowire growth[END_REF]. Therefore, the length-radius correlation is generally composed of an increasing part for thin NWs, then reaching a maximum and further decreasing toward larger R in the pure diffusion-controlled regime [START_REF] Dubrovskii | Gibbs-Thomson and diffusion-induced contributions to the growth rate of Si, InP and GaAs nanowires[END_REF]. The probability to observe non-monotonic length-radius correlation increases for smaller radii and larger

            c c c c c c c T k Au Au Au B PL L        , ... ) 1 ( ) ( ) 1 ( ln 5 3 3 3 5 35 2 5 3 5 2 3 35 5 5 5             c c c c c c c T k Au Au Au B PL L        , (6.14)
Here, PL X  are the temperature-dependent chemical potentials of pure liquids and XY  are the binary interaction parameters between X and Y atoms accounting for deviations from ideality, for X , Y = 3, 5 and Au. The written terms with composition-independent interaction constants correspond to the regular solution model. In the so-called sub-regular solution model, XY  are allowed to depend on concentrations, as in Refs. [START_REF] Glas | Chemical potentials for Au-assisted vapor-liquid-solid growth of III-V nanowires[END_REF] or [START_REF] Leshchenko | Nucleation-limited composition of ternary III-V nanowires forming from quaternary gold based liquid alloys[END_REF]. In Ref. [START_REF] Glas | Chemical potentials for Au-assisted vapor-liquid-solid growth of III-V nanowires[END_REF], the XY  were calculated according to Stringfellow [START_REF] Stringfellow | Calculation of ternary phase diagrams of III-V systems[END_REF]. Johansson et al. [START_REF] Leshchenko | Nucleation-limited composition of ternary III-V nanowires forming from quaternary gold based liquid alloys[END_REF] took the XY

 in the form ) ( 1 0 Y X XY XY XY c c      
, where 0 XY  and 1 XY  denote the zero-order and first-order binary interaction parameters, respectively. Higher order terms include ternary interactions, described by the (composition-independent) ternary interaction parameter Au 35  (more parameters appear if ternary interactions are considered in a quaternary liquid during the Au-catalyzed growth of ternary III-V NWs [START_REF] Leshchenko | Nucleation-limited composition of ternary III-V nanowires forming from quaternary gold based liquid alloys[END_REF]). To compute the chemical potential difference with respect to the solid state as given by Eq. (3.7), one needs to know the interaction parameters and the temperature- Refs. [START_REF] Glas | Chemical potentials for Au-assisted vapor-liquid-solid growth of III-V nanowires[END_REF][START_REF] Glas | Predictive modeling of selfcatalyzed III-V nanowire growth[END_REF][START_REF] Leshchenko | Nucleation-limited composition of ternary III-V nanowires forming from quaternary gold based liquid alloys[END_REF] for the parameters of different material systems. and group V concentrations in the range considered. However, this property is not critical and some calculations show that chemical potential may decrease with 3 c in group III rich regions [START_REF] Dubrovskii | Compositional control of gold-catalyzed ternary nanowires and axial nanowire heterostructures based on IIIP 1-x As x[END_REF]. At the fixed concentrations,   always decreases with increasing temperature. These trends remain the same for other Au-III-V systems such as Au-In-As, Au-In-P and Au-Ga-P [START_REF] Glas | Chemical potentials for Au-assisted vapor-liquid-solid growth of III-V nanowires[END_REF]. For WZ III-V NWs, all the curves will be shifted down by a constant value

ZB WZ 35 35     
(   24 meV per pair for GaAs [START_REF] Dubrovskii | Theory of VLS growth of compound semiconductors[END_REF]). island at the TPL. For highly volatile As or P species, the group V concentration is always much lower than the group III one. The island of height h consists of i III-V pairs. The contact angle of the droplet with respect to the horizontal equals  . The droplet base radius equals the NW radius R . The surface energies of interest are those of the liquid-vapor interface ( LV  ), solid-vapor lateral interface for one third of the island perimeter ( SV  ) and solid-liquid lateral interface for two thirds of the island perimeter ( SL  )

Nucleation and growth of 2D islands

    sin LV SV eff  
for that portion of the TPL located at the TPL [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF]; see also Section 11.2) and by γ ΔS = 2γ sin β ΔV/R due to the volume variation ΔV = (Ω Ω )i in the liquid-solid phase transition of i III-V pairs having the elementary volumes S  and L  in the solid and liquid phases, respectively [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF][START_REF] Dubrovskii | Growth kinetics and crystal structure of semiconductor nanowires[END_REF]. This gives the additional Gibbs-Thomson term

R i L S LV / ) ( sin 2      (which is positive when L S   
, as in the Au-Ga-As system [START_REF] Dubrovskii | Growth kinetics and crystal structure of semiconductor nanowires[END_REF]), proportional to the island size i (which is why it modifies the chemical potential). The total surface energy change thus equals [START_REF] Dubrovskii | Growth kinetics and crystal structure of semiconductor nanowires[END_REF] i

R Ph x x G L S LV LV SV TPL SL TPL surf ) ( sin 2 )] sin ( ) 1 [(                . (6.15) For = 1, = 2 and s Ω h R i 2  
, this yields the exact result for the complete ML slice in the non-wetting growth mode (where the droplet sits entirely above the ML), namely [START_REF] Dubrovskii | Nucleation theory and growth of nanostructures[END_REF]). Note that generally changes during the island growth, which is considered in the generalized expression for the surface energy of Ref. [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF]. In any case, the last, Gibbs-Thomson term in Eq. (6.15) is proportional to the difference between elementary volumes in the solid and liquid phase and becomes negligible when , as in the case of GaAs NWs [START_REF] Glas | Predictive modeling of selfcatalyzed III-V nanowire growth[END_REF][START_REF] Dubrovskii | Growth kinetics and crystal structure of semiconductor nanowires[END_REF].

Rh G LV S L SV surf     2 ] sin ) / ( [      (Ref.
Let us now include the volume energy change upon island formation [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF]. For the fast growth stage of island that occurs before refill, let 0 j be the initial number of As atoms in the droplet, the number of As atoms corresponding to its equilibrium concentration with the solid phase, and the number of As atoms (or GaAs pairs) in the single island growing in the mononuclear growth regime. In the closed system, the number of group V (As) atoms remaining in the droplet simply equals = . The concentration of As atoms in the droplet is = ( )/ , where is the total number of all atoms in the droplet, hence / = / .

We introduce the dimensionless free energy of forming an island (formation energy for brevity) consisting of i III-V pairs (size for brevity), ( ) = ( )/ . According to the general result [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF], its derivative with respect to size is given by

( ) = [ ( ) ] + ( ) . (6.16)
Here, ( ) is the chemical potential (in thermal units) of a III-V pair in a droplet containing i j  0 As atoms, eq  the equilibrium chemical potential, and ( ) = / the island surface energy (with the Gibbs-Thomson term now removed from ) which scales as / for small enough islands and in any case does not grow faster than / [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF]. The stopping size introduced in Ref. [START_REF] Dubrovskii | Refinement of nucleation theory for vapor-liquid-solid nanowires[END_REF] corresponds to the minimum of the formation energy [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF], while the critical size of CNT (which is of course smaller) is determined by its maximum. After reaching the stopping size, the island cannot continue growing from liquid; its further extension requires refill from vapor (for wurtzite GaAs NWs) or developing an interface truncation (for zincblende GaAs NWs), as described in Section 4 [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF].

Let us derive Eq. ( 6

( ) = ( ′ ) + ( ′ ) + ( ) + [ ( ′ ) ( )] + ( )[ ( ′ ) ( )]. (6.17)
Differentiating this with respect to i , we get

( ) = [ ( ′ ) + ( ′ ) ] + ( ) + ( ) ′ + ( ) ′ ′ . ( 6.18)
The first term in the right hand side of Eq. (6.18) is indeed the same as in Eq. (6.16). The second bracket in Eq. (6.18) equals zero regardless of the specific form of chemical potentials. This is a consequence of the equality = ( ′ / ′ ) following from the Gibbs-Duhem equation for a binary system [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF]. Using the expressions of ′ and ′ , we get = [( )/( )] and hence the second bracket term cancels. It can be shown that this result remains valid in the more general case of metal (Au)-catalyzed VLS growth of III-V NWs [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF].

We now recall that the chemical potential of a III-V pair in the droplet is well approximated by a logarithmic function of the group V concentration (due to the low value of this concentration) [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF], therefore

  ] / ln[ ) ( ) ( 0 5 5 5 3 eq eq j i j c c          , ( 6.19)
(this also holds for Au-catalyzed VLS growth, but then eq j depends also on the Ga concentration). As recalled above, the stopping size s i corresponds to the minimum of the formation energy [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF][START_REF] Dubrovskii | Refinement of nucleation theory for vapor-liquid-solid nanowires[END_REF]. Neglecting the variation of edge energy Γ (which is a good approximation if s i is much larger than the critical size of CNT and smaller than a full ML [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF]), it simply equals

eq s j j i   0 . (6.20)
This corrects the result of Ref. [START_REF] Dubrovskii | Refinement of nucleation theory for vapor-liquid-solid nanowires[END_REF], obtained under unnecessarily restrictive assumptions.

Using Eqs. (6.18) and (6.19), the formation energy itself is obtained by integration:

) ( ln ) ( 0 0 i j u j du i F i eq G              . (6.21)
The result is: .22) This can be rewritten

) ( ln ln ) ( 0 0 0 0 i i j i j j j i j i i F eq G                        . ( 6
i j i i j i F i F               0 0 1 ln ) ( ) ( ) ( , (6.23) with )
( ) / ln( ) ( 0 i j j i i F eq G     (6.24) ) (i F 
is the CNT result for infinite systems (not affected by depletion), which is indeed recovered from Eq. ( 6.23) at 0 / 0  j i . Eq. (6.23) thus separates the CNT result

) (i F  from a
size-dependent correction which vanishes at 0 / 0  j i .

Accounting for the refill from vapor and neglecting As desorption from liquid, the number of As atoms in the droplet by time t equals + , where is the rate of refill and the number of GaAs pairs in the island, as above. Note that between nucleation and time t, the volume enclosed by the droplet surface (liquid and island) increases by

=   S L i i vt     .
Using Eq. (6.22), and presenting the result in terms of the ML coverage = ⁄ (with i = (3√3 2 ⁄ )R / the number of GaAs pairs in the complete ML and the radius of the hexahedral NW), we obtain the normalized formation energy in the form

= = ( ) + 1 + + ( ) (6.25)
Here, = ⁄ is the initial As content at nucleation, = ⁄ the equilibrium As content, = / the coverage equivalent to refill (which is proportional to time), the chemical potential difference per Ga-As pair between vapor and solid, and ( ) = Γ( )/ . In Ref. [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF], we derived a representation of the ML edge energy from modeling in situ experiments [START_REF] Harmand | Atomic step flow on a nanofacet[END_REF], namely: of the formation energy then disappears for large enough , meaning that fractional ML can grow even from equilibrium liquid. This effect is due to a decrease of the surface energy at high coverage, partly explained by a decrease of the island perimeter, which enables fast ML growth from undersaturated liquid [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF]. These questions are investigated in detail in Ref. [START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF], where the present thermodynamic treatment is complemented by a kinetic model taking into account As desorption and a possible hindered incorporation to the growing ML. The stopping effect does not change the NW axial growth rate given by Eq. (3.10), which still equals the nucleation rate times the available area for nucleation at the maximum chemical potential within the ML growth cycle. However, it may affect the oscillations of the truncated facet [START_REF] Wen | Periodically changing morphology of the growth interface in Si, Ge, and GaP nanowires[END_REF] and nucleation statistics in VLS NWs [START_REF] Glas | Nucleation antibunching in catalyst-assisted nanowire growth[END_REF] which in turn influence some important properties such as the preferred crystal phase and length uniformity within the NW ensembles.

( ) = / + (1 ) / + (1 + ) . ( 6
One important conclusion is that the nucleation probability in VLS NWs exponentially decreases immediately after nucleation, which makes the next nucleation improbable until the droplet is refilled from vapor [START_REF] Glas | Nucleation antibunching in catalyst-assisted nanowire growth[END_REF]. In III-V NWs, this nucleation "antibunching" is critically sensitive to the concentration of group V atoms in the catalyst droplet.

Self-consistent growth models

The first fully self-consistent VLS growth model was developed by Glas et al. in Ref.

[40] for Ga-catalyzed GaAs NWs grown by MBE. The NW growth rate given by the As input, desorption and nucleation [stationary Eq. (3.1)] was solved numerically using Eq. (3.10) in the case of center nucleation. The Zeldovich nucleation rate was used given by Eq. (3.12). The model reproduces quantitatively all salient features of Ga-catalyzed VLS growth, namely the variations of the axial NW growth rate with the As flux, temperature, and NW radius. for this MBE system [START_REF] Ramdani | Arsenic pathways in self-catalyzed growth of GaAs nanowires[END_REF]. Experimental data shown in the bottom panel were obtained for NW1 having a constant radius R of 32 nm, with   125 . growth rate with As flux, as observed earlier in Ref. [START_REF] Colombo | Gaassisted catalyst-free growth mechanism of GaAs nanowires by molecular beam epitaxy[END_REF]. The decrease of dt dL / with temperature is stronger than linear. The radius dependence of the axial growth rate will be discussed shortly. In Ref. [START_REF] Dubrovskii | Group V sensitive vapor-liquid-solid growth of Au-catalyzed and selfcatalyzed III-V nanowires[END_REF], an analytic approach was developed for the self-consistent description of Au-catalyzed and self-catalyzed VLS growth of III-V NWs, based on some earlier results [START_REF] Dubrovskii | Diffusion-induced growth of nanowires: generalized boundary conditions and self-consistent kinetic equation[END_REF].

Using the simplified diffusion flux of group III adatoms as given by Eq. (6.11), it is easy to obtain Under certain assumptions, solution to the closed system of stationary Eq. (6.28), Eq. (6.29) and (3.11) can be approximated by the following analytical formula for the growth rate

                             dt dL e h v R v f dt
) 1 )( 1 ( 5 5 G v dt dL      , (6.30)
with

)] 1 ( ) / [( 2 * *     R R W i G , (6.31
)

p p R F v R v v ) / )( / ( 1 ) / ( 3 3 3 53 3 3 3 3 3 5 5              , ( 6
v v F    . The "nucleation" radius * R is given by 2 / 1 0 * 5 5 * ) (              J hi v R ( 6
                              des p p v v h R v R v . (6.34) The ) ( 0 *    c i i
denotes the critical size at this chemical potential.

The axial growth rate given by Eq. (6.30) is reduced to the transport-limited growth rate given when

1   and 1  G , while at 1  G
it yields the nucleation-limited growth rate. In some regimes, this model is further reduced to the result of Ref. [START_REF] Dubrovskii | Diffusion-induced growth of nanowires: generalized boundary conditions and self-consistent kinetic equation[END_REF], providing explicitly the NW length in the form

                           q p n p R R W R H L 3 ~. (6.35)
Here, H is the deposition thickness, 3  is the re-normalized diffusion length of group III adatoms, n R is the re-normalized nucleation radius (which depends on the kinetics of group V species), and the index q equals 2 for the center and 1 for the TPL nucleation position. Similarly to Eq. (6.30), Eq. (6.35) describes the diffusion-limited regime (for sufficiently thin NWs) at small n R , transitioned to the nucleation-limited regime at large n R . The L(R) correlation transitions from non-monotonic to monotonically increasing, as shown in Figure 6.11. The limitation of the axial growth rate at small R is explained, roughly, by a small facet area available for nucleation even without the Gibbs-Thomson effect. phase, fitted by a more general Eq. (6.30), can be found in Ref. [START_REF] Dubrovskii | Group V sensitive vapor-liquid-solid growth of Au-catalyzed and selfcatalyzed III-V nanowires[END_REF]. It turns out that WZ GaAs NWs usually exhibit the transport-limited inverse length-radius correlations, while ZB GaAs NWs often show lengths that increase with the radius, and in this sense become similar to the self-catalyzed case.

p : curve 1- n R 14 nm, 3  =50 nm; curve 2 -n R =40 nm,   3 ~75 nm; curve 3 - n R 75 nm,   3 ~225 nm; curve 4 -  n R 125 nm,   3 ~450 nm.
For self-catalyzed III-V NWs, the analytic approximation for the axial growth rate is

given by [41]

    1 5 5 v dt dL , (6.36) with
] ) / [( 2 * R R W i c    . ( 6.37)
The nucleation radius * R is given by Eq. (6.33) but at the different, radius-independent chemical with the coefficients

                 des v v . ( 6
) ( 5 5 3 3 v v g a     , 3 3 3   v g b  . (6.40)
Here, )] (

/[ 35  f g L   
is the shape constant, which can be treated as time-independent at , the shape of Ga-catalyzed GaAs NWs can be put as

const   . Integrating this with the initial condition 0 0 ) ( R t t R   , one obtains             a b R a b R a b a R R t t / / ln 0 2 0 0 . ( 6
                     s s s R R R R R R R F g F L L 0 0 53 53 0 ln ) 1 ( . ( 6.42)
From Eqs. (6.39), (6.40), we have

) 1 ( 53 3 3 3      F a b R s   . (6.43)
Here, R denotes the NW top radius corresponding to its length L at a given moment of time, but can also be treated as the radius at a given position along the NW ("height") without subsequent VS radial growth on the sidewalls. The radius increases with time, or with the height when 0  a , corresponding to the case of positive Ga to As imbalance (

1 53  F and 0  s R ). When 1 53  F ,
positive s R gives the stable NW radius as will be discussed shortly. In Ref. [START_REF] Leshchenko | Tuning the morphology of self-assisted GaP nanowires[END_REF], the shapes of Ga-catalyzed GaP NWs grown at 600 o C by gas source MBE in patterned arrays on Si(111) substrates was analyzed as a function of V/III flux ratio. The shape of these NWs was not considerably affected by radial VS growth on the sidewalls. In particular, Figure 6.14 shows an example of a NW array grown with a V/III flux ratio of 4, whose shape is tapered toward the top. For high V/III flux ratios, some shorter NWs were systematically observed without any droplets on top. Figure 6.15 shows the measured shapes of NWs for different V/III flux ratios changing from 1 to 6. It is seen that the NWs are reverse tapered for at V/III=1, almost straight at V/III=3, and tapered for V/III>3, with the degree of tapering increasing for higher V/III. These results are fully consistent with the model presented above, and well fitted by Eqs. (6.41), (6.42) with the parameters detailed in Ref. [START_REF] Leshchenko | Tuning the morphology of self-assisted GaP nanowires[END_REF]. In particular, the stable radius s R was found in the range 12-18 nm for the V/III flux ratios from 3 to 6.

Self-equilibration of the radius of self-catalyzed nanowires

Equations (6.41) and (6.42) demonstrate that the radius distribution of self-catalyzed III-V NWs can be narrowed during growth to an almost delta-like shape due to the unusual selffocusing effect. This effect can be observed if the group V atomic flux into the droplet, including re-emitted species, is larger than that of group III ( 1 53  F

). A positive diffusion flux of group III atoms from the NW sidewalls contributes more to the increase of droplet volume for thinner NWs and less for thicker NWs. Therefore, larger droplets will grow and smaller droplets will shrink until the diffusion flux equalizes the positive group V to III imbalance. As a result, the droplet volume will converge to a certain stationary one regardless of the initial size distribution of the growth seeds. For a fixed contact angle of the droplet, increase or decrease of the droplet volume leads to extending or diminishing the radius of the NW top. This was predicted theoretically in Ref. [START_REF] Priante | Stopping and resuming at will the growth of GaAs nanowires[END_REF] and later in Ref. [START_REF] Tersoff | Stable self-catalyzed growth of III-V nanowires[END_REF], and confirmed experimentally in Ref. [START_REF] Dubrovskii | Self-equilibration of the diameter of Ga-catalyzed GaAs nanowires[END_REF] for Ga-catalyzed GaAs NWs. Copyright (2015) American Chemical Society.

These NWs were grown by MBE on Si(111) substrates through three steps: (i) patterning of a hole array in a thin SiO 2 layer, (ii) predeposition of Ga to form droplets in the etched holes, and (iii) growth of NWs at a temperature of 630°C, As/Ga growth rate equivalent ratio of 1.8 and 2D equivalent growth rate of GaAs of 1 ML/s for 5 min. Figure 7.1 shows the SEM images obtained after steps (ii) and (iii) for an array of holes with a diameter of 60 nm and a pitch of 100 nm. It is seen that the Ga predeposition leads to the formation of Ga droplets in the oxide-free openings only, resulting in the growth of NWs at the position of the holes. While the diameter of the Ga droplets varies significantly from hole to hole (the droplet diameter distribution may depend on the droplet preparation process but is never uniform), a high degree of uniformity of the NW diameters is noticed.

This difference between the droplet size distribution and the NW diameter distribution was reproducible, whatever the initial size and the pitch between the holes in the array are. For example, Figure 7.2 compares the size distribution of the Ga droplets with the distribution of the NW diameters for an array with a hole size of 60 nm and a pitch of 250 nm. Again, the NWs in the array are quite uniform and the NW diameter distribution appears clearly much narrower than the hole size distribution. The Ga-catalyzed growth of NWs leads to a focusing effect towards a diameter of ~ 50 nm, slightly bigger than the peak value of the droplet size distribution, but still smaller than the hole size.  .

Length distributions of nanowires

The measured [START_REF] Koivusalo | Sub-Poissonian narrowing of length distributions realized in Ga-catalyzed GaAs nanowires[END_REF][START_REF] Tauchnitz | A simple route to synchronized nucleation of self-catalyzed GaAs nanowires on silicon for sub-Poissonian length distributions[END_REF][START_REF] Dubrovskii | Length distributions of Au-catalyzed and In-catalyzed InAs nanowires[END_REF][START_REF] Dubrovskii | Length distributions of nanowires growing by surface diffusion[END_REF][START_REF] Matteini | Tailoring the diameter and density of self-catalyzed GaAs nanowires on silicon[END_REF] and modeled [START_REF] Koivusalo | Sub-Poissonian narrowing of length distributions realized in Ga-catalyzed GaAs nanowires[END_REF][START_REF] Tauchnitz | A simple route to synchronized nucleation of self-catalyzed GaAs nanowires on silicon for sub-Poissonian length distributions[END_REF][START_REF] Dubrovskii | Length distributions of Au-catalyzed and In-catalyzed InAs nanowires[END_REF][START_REF] Dubrovskii | Length distributions of nanowires growing by surface diffusion[END_REF][START_REF] Dubrovskii | Length distributions of nanowires: effects of surface diffusion versus nucleation delay[END_REF][START_REF] Glas | Self-narrowing of size distributions of nanostructures by nucleation antibunching[END_REF][START_REF] Dubrovskii | Analytic form of the size distribution in irreversible growth of nanoparticles[END_REF][START_REF] Matteini | Tailoring the diameter and density of self-catalyzed GaAs nanowires on silicon[END_REF] NW LDs are commonly compared to this Poissonian case.

Without one specific effect which will be discussed later on, the Poisson LD is the best case regarding the length uniformity. There are several effects contributing into the LD broadening, including (i) a nucleation delay due to difficult nucleation of NWs emerging from the substrate; (ii) continuing nucleation of the seed droplets during the NW growth, as in the Gacatalyzed growth of GaAs NWs without Ga pre-deposition [START_REF] Matteini | Tailoring the diameter and density of self-catalyzed GaAs nanowires on silicon[END_REF], (iii) surface diffusion of group III adatoms, (iv) collective effects in the NW growth such as shadowing in directional MBE method, and (v) initial size inhomogeneity of the droplets.

To account for the nucleation delay for NWs whose growth rate is controlled by the direct impingement from vapor, the probabilities ) (t p s are written in the form

v p   0 , v p s  , ... 3 , 2 , 1 ,  s . (8.7)
Here, 1   describes difficult nucleation of the very first NW ML relative to the upper layers.

The corresponding continuum LD is now given by [START_REF] Dubrovskii | Length distributions of Au-catalyzed and In-catalyzed InAs nanowires[END_REF] (8.9)

                    2 2 ) , ( 2
The mean length and variance of this LD, assuming that it is fully formed ( 0

) , 0 (    s f ), equal   1   s , 2 2 1      , (8.10)
showing that the mean length decreases and the variance increases with respect to the Poisson case. The best fits of the LDs shown in Figs. 8.3.correspond to  between 0.0053 and 0.017, which are on the order of magnitude larger than for the standard annealing step but still two orders of magnitude lower than in the best Poissonian case achieved for thermally dewetted Au film.

In-catalyzed InAs NWs [START_REF] Dubrovskii | Length distributions of Au-catalyzed and In-catalyzed InAs nanowires[END_REF] and Ga-catalyzed GaAs NWs [START_REF] Matteini | Tailoring the diameter and density of self-catalyzed GaAs nanowires on silicon[END_REF], whose growth is accompanied by simultaneous nucleation of the group III droplets, feature much broader LDs than shown here, because the NWs continue to emerge from the newly formed droplets.

In Ref. [START_REF] Dubrovskii | Length distributions of nanowires growing by surface diffusion[END_REF], it was demonstrated experimentally and explained theoretically that NWs growing by surface diffusion of group III adatoms exhibit very broad LDs, with the variance increasing proportionally to the squared mean length. Theoretical work [START_REF] Dubrovskii | Length distributions of nanowires: effects of surface diffusion versus nucleation delay[END_REF] considered simultaneous effects of surface diffusion and nucleation delay on the LDs and predicted even larger broadening than in Ref. [START_REF] Dubrovskii | Length distributions of nanowires growing by surface diffusion[END_REF]. Figure 8.4 shows the model with its main ingredients, a delayed nucleation and the diffusion flux of adatoms collected from the entire length of the NW, which equals s MLs. As a result, the instantaneous growth rates become linear in s . Such size- linear model was discussed earlier in connection with the scaling regime of growth of 2D islands competing for the diffusion flux of adatoms (see Ref. [START_REF] Dubrovskii | Analytic scaling function for island-size distributions[END_REF] and references therein). Introducing the dimensionless time  and the model parameters a and b according to .11) with t as the growth time and other parameters illustrated in Figure 8.4, Eqs. (8.1) take the form

R hvt   2  , h R R a s s      2 ) 2 / ( 2    , hv R p b  2 0  , ( 8
0 0 bf d df    , 1 0 1 ) 1 ( f a bf d df     , s s s f s a f s a d df ) ( ) 1 ( 1        , 2  s . (8.12)
Here, b describes the nucleation probability of the first ML [in a different normalization compared to  in Eq. (8.7)], while a appears due to normalization of the diffusion flux (to make

it equal s at   s
). This a is usually much larger than unity due to the presence of the h R / term in the corresponding Eq. (8.11), which equals the NW radius in the units of lattice spacing.

Exact solution to Eqs. (8.12), given in Ref. [START_REF] Dubrovskii | Length distributions of nanowires: effects of surface diffusion versus nucleation delay[END_REF], is expressed through the incomplete beta-function. The particular case of this LD at b a  is reduced to the Polya distribution [START_REF] Dubrovskii | Length distributions of nanowires growing by surface diffusion[END_REF][START_REF] Dubrovskii | Length distributions of nanowires: effects of surface diffusion versus nucleation delay[END_REF].

In the continuum limit 1  s , the beta-distribution converges to .15) where the approximate expression corresponds to the continuum LD. The variance of the continuum LD is given by Au-catalyzed InAs NWs of Ref. [START_REF] Dubrovskii | Length distributions of nanowires growing by surface diffusion[END_REF] were grown by MOVPE. Before MOVPE growth, patterns of Au particles were prepared by means of electron beam lithography (EBL) followed by thermal evaporation of Au and lift-off. The distance between the Au particles was 1000 nm and by varying the EBL dose the nominal NW diameter was varied between 20 and 100 nm. The NWs were grown at 450°C and the investigated growth times were 7.5, 15, 22.5, 30, and 60 min. SEM images of the NW arrays were recorded at the specific tilt angles and NW length and diameter statistics were extracted using the SEM image analysis. proportional to the mean length for  t 22.5 but becomes sub-linear for longer times. Such changes are attributed to the transition from the length-linear to length-independent NW growth rates, as discussed previously. Therefore, these data were fitted with This solution holds for any   s , while the mean length is given by Eq. (8.17) , showing that a scales as R . This effect was explained by nucleation antibunching, whereby the probability to nucleate the next ML decreases exponentially immediately after nucleation of the previous ML [START_REF] Glas | Nucleation antibunching in catalyst-assisted nanowire growth[END_REF]. Therefore, the next nucleation event becomes improbable until the droplet is refilled from vapor with the missing As. The measure of nucleation antibunching is readily obtained from the In these expressions, the chemical potential, its derivative with respect to 5 c , and 5 c itself should be taken before nucleation. Clearly, the antibunching effect is stronger for thinner NWs and low group V concentrations in the droplet (small R and 5 c ), and disappears at large R . As an example,

) 1 , ( ) ( ) 1 ( ) , ( 1   G  G      b a se se a be s f b     , ( 8
  ce e c s    ) 1 ( , ) 1
( ) 1 (    b b a c , ( 8
2 ) 2 ( ) 1 ( )] 2 ( 2 [ s b b a b b a        . ( 8
) ( 1 max max max t t v C a s s s            ; max 2 s s     . ( 8
             2 2 2 exp 2 1 ) , (    s s s s f ,                max max
              
 equals 0.2 for <111> GaAs NWs ( 02 . 0 ~  L nm 3 , 0452 . 0   S nm 3 , h = 0.326 nm) with  R 50 nm and   127 o if we assume 1 ) / (     c c  ,  c 0.03 and   ) (  c i 5.
Glas [START_REF] Glas | Statistics of sub-Poissonian nucleation in a nanophase[END_REF] presented the exact solution for sub-Poissonian nucleation statistics in individual VLS NWs, controlled by the sole parameter  . Later, Glas and Dubrovskii [START_REF] Glas | Self-narrowing of size distributions of nanostructures by nucleation antibunching[END_REF] considered the effect of nucleation antibunching on the LDs within the ensembles of NWs by using three different methods: (i) numerical simulations in a large ensemble of NWs, (ii) exact analytical solution for the LD and its statistical properties based on the results obtained for individual NW [START_REF] Glas | Statistics of sub-Poissonian nucleation in a nanophase[END_REF], and (iii) simplified analytic approach based on the continuum rate equation. For the exact analytical calculations, the central object is the probability ( | ) that exactly s nucleations occur during a time interval of length , conditional to the fact that the NP was at the beginning of this interval. According to Ref. [START_REF] Glas | Statistics of sub-Poissonian nucleation in a nanophase[END_REF], ( | ) is given by

( | ) = ∑ ( ; ) exp[ ( ) ( ) ⁄ ] . (8.25)
Here, )

exp(    
, ( ; ) = ( 1)( 1) … ( 1) is the q-Pochhammer symbol (for The conditional probability ( | ) is a powerful tool for calculating the time evolution of ensembles of NWs. Indeed, the ensemble LD is simply given by the convolution of this probability with the distribution of times of formation of the first ML, which equals ) exp( 0 0 T p p  . After some calculations (see Ref. [START_REF] Glas | Self-narrowing of size distributions of nanostructures by nucleation antibunching[END_REF] for the details), the LD can be expressed in terms of the incomplete gamma functions ) , ( y x G as

≥ 1, with ( ; ) = 1), = ( 1)
( ) = ∑ ( ; ) exp[ ( ) + ] G , ( ) G( , ) (8.26)
for ≥ 1, with = = ⁄ , = ⁄ and = as the normalized growth time.

These formulas are exact for any values of parameters ≠ 0 , , and time and are computationally efficient unless is very small. The LDs so calculated agree extremely well with those obtained from numerical simulations [START_REF] Glas | Self-narrowing of size distributions of nanostructures by nucleation antibunching[END_REF]. This confirms that for identical and noninteracting NWs, the statistics of the ensemble may be calculated from those of the single NW.

The average length (in MLs) of the NW ensemble at time  equals

= ( ; ) ∑ ( ; ) 1 exp[ ( ) + ] G , ( ) G( , ) + 1 . (8.27)
The variance of the LD is given by 1), (8.28) where

= 2( ; ) ∑ ( ) ( ; ) 1 exp[ ( ) + ] G , ( ) G( , ) + 3 2(
( ) = ∑ (1 ) ⁄
is the Lambert series with unit coefficients. It turns out that the parameter I p has no influence on the LDs of sufficiently long NWs of interest, so the statistical properties of the ensemble are controlled by the two parameters  and  .

In the continuum approach, the asymptotic method of Ref. is obtained in the form

          2 2 2 ) ( exp 2 1 ) , (       z z F , (8.30) with    1   s e ,   1   e z , 2 2 2 z z     . ( 8
              2 ) ( ) ( 1 1 exp ) (         s s e e s F
. (8.32) This property directly shows that Green's function does not broaden with time due to nucleation antibunching. For small enough  and large s , it is well approximated by the Gaussian

  2 ) ( exp ) (          s s F (8.33)
Clearly, the mean length of this LD remains  but the variance is time-independent and equals ) 2 /( 1  rather than growing infinitely as in the Poissonian case. The variance ) 2 /( 1  exactly equals the one obtained in Ref. [START_REF] Glas | Statistics of sub-Poissonian nucleation in a nanophase[END_REF] for individual NW.

Convolution of Green's function given by Eq. (8.33) with the nucleation rate

) exp( ) (      J
according to Eq. ( 8.3) yields the asymptotic LD in the form . This effect is stronger for smaller  , while for larger  the LDs converge to a shape that is simply determined by the duration of the NW nucleation step. In both figures, the excellent quantitative agreement is obtained between the exact LDs given by Eq. (8.26) and the continuum LD given by Eq. (8.34). and from Eq. (8.35) are also extremely close (see Ref. [START_REF] Glas | Self-narrowing of size distributions of nanostructures by nucleation antibunching[END_REF] for a more detailed discussion).

                      2 ) ( erfc 2 ) ( ) 4 /( ) (
Summarizing the results of Ref. [START_REF] Glas | Self-narrowing of size distributions of nanostructures by nucleation antibunching[END_REF], it was demonstrated by three different methods that nucleation antibunching in individual NWs, described by the antibunching parameter  , completely suppresses the Poissonian broadening of LDs within ensembles of NWs. The initial nucleation randomness, described by the parameter , affects the LDs forever, which is why the asymptotic variances and the LD shapes depend on the two parameters  and  . However small these parameters may be, the LDs finally acquire time-independent shapes rather than spread infinitely with the growth time. Numerical simulations over large ensembles of NWs and two different analytical approaches based on discrete and continuum rate equations yield very similar results. Recent experimental results confirm qualitatively these predictions by showing markedly sub-Poissonian LDs of Ga-catalyzed GaAs NWs [START_REF] Koivusalo | Sub-Poissonian narrowing of length distributions realized in Ga-catalyzed GaAs nanowires[END_REF][START_REF] Tauchnitz | A simple route to synchronized nucleation of self-catalyzed GaAs nanowires on silicon for sub-Poissonian length distributions[END_REF]. However, quantitative comparison with these data requires some care. In theoretical analysis, we used a time-independent antibunching parameter throughout the entire growth process, while the GaAs NWs of Ref. [START_REF] Koivusalo | Sub-Poissonian narrowing of length distributions realized in Ga-catalyzed GaAs nanowires[END_REF] grow also in diameter, which affects the  value according to Eq. (8.24). In addition, we assumed a time- independent nucleation probability for forming the first NW ML. This corresponds to the VLS regimes with long incubation times for NW growth (small ). Surface diffusion effects may affect the described picture, which seems more relevant for self-catalyzed III-V NWs. All these questions require further studies. We now present one example of sub-Poissonian LDs observed in ensembles of selfcatalyzed GaAs NWs [START_REF] Koivusalo | Sub-Poissonian narrowing of length distributions realized in Ga-catalyzed GaAs nanowires[END_REF] (more examples are given in Ref. [START_REF] Tauchnitz | A simple route to synchronized nucleation of self-catalyzed GaAs nanowires on silicon for sub-Poissonian length distributions[END_REF]). Ga-catalyzed GaAs NWs of Ref. [START_REF] Koivusalo | Sub-Poissonian narrowing of length distributions realized in Ga-catalyzed GaAs nanowires[END_REF] were grown by MBE on p-Si(111) substrates by the procedure which is presented stepby-step in Figure 8.15. Before the first growth step, the Si(111) substrates were dipped in HF in order to remove the native oxide and immediately transferred to vacuum where they were first annealed at 275 °C for 1 h and then at 700 °C for 10 min. Then GaAs nanocrystals were grown by droplet epitaxy; (1) Ga droplets were first formed by depositing 0.09 MLs of Ga at 550 °C and then (2) crystallized into GaAs by a 10 min As 2 exposure. The sample temperature in step 1 is a critical parameter that defines the density of the nucleation sites. In step 3, the sample was removed from vacuum and exposed to air for 18 h in order to form an oxide layer on the Si(111) surface. In the following step (4), the sample was transferred back to vacuum where it was annealed at 660 °C for 30 min in order to remove the GaAs nanocrystals and reveal oxide free nucleation sites. In the final surface preparation steps (5) Ga droplets were formed in the nucleation sites by depositing Ga for 60 s at 640 °C using a growth rate of 0.3 ML/s, and (

subsequently removed by annealing at 660 °C for 10 min. The purpose of steps 5 and 6 is to prepare the nucleation sites for subsequent NW growth. By introducing these additional steps to the process presented in Ref.

[101], a high vertical yield of 70% is obtained, without having to use Ga pre-deposition at the beginning of the NW growth. Finally, in step [START_REF] Dubrovskii | Theory of VLS growth of compound semiconductors[END_REF] the NW growth was initiated by simultaneously opening the Ga shutter and As valve and using sample temperature of 640 °C. The Ga growth rate was 0.3 ML/s and As 2 /Ga flux ratio 9. The NW samples were grown using different growth durations: 3 min, 6 min, 20 min, 40 min, 50 min, and 80 min. The 50 min sample was grown on a template having nucleation site density of 2×10 8 cm - 2 while a slightly higher density of 4×10 8 cm -2 was used for the other samples, resulting in NW densities of 0.7×10 8 cm -2 and 1.5×10 8 cm -2 , respectively. including re-emitted flux, can be understood simply from the fact that longer NWs are less shadowed and hence grow faster, while shorter NWs are more shadowed and grow slower than the longer ones. As a result (see Ref. [START_REF] Koivusalo | Sub-Poissonian narrowing of length distributions realized in Ga-catalyzed GaAs nanowires[END_REF] for the details), the model expression for the variance is given by Overall, the results presented in this Section and in Section 7 suggest that remarkable size uniformity can be achieved within the ensembles of III-V NWs, particularly those obtained in the self-catalyzed approach. Together with a tremendous progress in growing such NWs on Si substrates, and introducing high quality heterostructures in NWs, this may open a new path for monolithically integrated NW-based photonic devices and optoelectronic circuits, with a high degree of NW ordering in terms of both radii and length.

    0 / 0 / 2 * 2 / 1 1 0 * L L e hL e L L L L         . ( 8

Semiconductor alloys in nanowires 9.1. Introduction

In the field of planar (2D) epitaxial structures, it is well known that the range of possible applications is vastly extended by using alloys instead of pure elemental semiconductors or simple stoichiometric binary III-V or II-VI compounds. This allows one to access band gap energies intermediate between those of the end components of the alloy. Moreover, in the frequent case where alloy composition variations induce changes of lattice parameter, one can play on the strain exerted by a substrate or by various parts of an heterostructure on each other to tune further the electronic and optical properties of the alloys. In this respect, the 1D geometry has two important consequences that distinguish NWs from planar structures: first, because of strain relaxation at the free sidewalls, one can associate without creating extended defects, materials that are much more lattice-mismatched than would be possible in the 2D case; second, the strain fields are generally less homogeneous and more complex. These points will be treated in section 10.

Many semiconductor alloys have been fabricated in the form of NWs. This includes alloys between group IV elements, primarily Si-Ge [102] but also Ge-Sn [103], and many III-V and II-VI ternary and even quaternary alloys (for an overview, see e.g. Ref.

[104]). The aim may be to obtain fully or locally homogeneous NWs, or else to create a controlled composition gradient along the axis. Alloys may also form unwantedly when growth fluxes are switched to form an axial heterostructure between nominally non-alloyed sections (e.g. Si x Ge 1-x forms at the interface between pure Si and pure Ge sections) or when, despite steady growth conditions, the relative influx of the various components changes with time (for instance because of different diffusion lengths).

Quite generally, the major question is that of the dependence of the alloy composition on the growth conditions, which here include, in addition to what we considered before, the growth fluxes of all the elements that are alloyed in the NW (or at least their ratios). Alloys may form in NW shells but these grow in the vapor-solid mode, whereas we concentrate on VLS growth. A key point is then to describe theoretically the relationship between liquid and solid compositions.

This in turn is an essential step in the modeling of the effect of the growth conditions (fluxes and temperature) on the solid formed; however, this all-important question requires in addition the description of how the liquid droplet is fed by the external fluxes, which is in itself a complicated question (see section 1.4).

Description of the vapor, liquid and solid phases

For each phase , the number  n of constituents must be distinguished from the number of independent compositional parameters  i required to describe the thermodynamic state of this phase. Usually, S V n n  , since each element of the solid is provided by a separate vapor flux, be it single atoms or molecules. For self-catalyzed III-V growth,

S L n n  but S L n n  when a
foreign catalyst is used. The vapor state is characterized by fluxes or partial pressures, so that in general

V V n i 
(3 for self-catalyzed growth of ternary alloys, 2 for Ge x Si 1x ).

We shall consider the simplest case, 1

 S i
, of a solid composition defined by a single atomic concentration x (for species A, that of the other species B being x  1 ). This holds for binary alloys of elementary semiconductors, such as Ge x Si 1x , and for ternary alloys of compound semiconductors (in the latter case, the numbers of atoms on each sublattice are equal: atomic species A, B sharing a sublattice while the other is fully occupied by species C produce a pseudo-binary

A x B 1 -x C alloy). With 1  S i , we have 3  L n , 2  L i for self-catalyzed growth:
the liquid is a genuine ternary alloy, but its state is defined by two atomic concentrations that add up to 1 (e.g. y and y for Al x Ga 1x As NWs, with y = 1 y y ). With a foreign catalyst,

3  L i
or more (if a mixture of foreign metals is used).

Dependence of alloy composition on vapor fluxes

It is only in the simplest cases that one can model how the alloy composition x depends on the vapor fluxes. In particular, in stationary VLS growth, the rate at which each component enters the droplet must equal its rate of consumption. However, an element may reach the liquid via various pathways (vapor, surface diffusion, reemission; see section 3); conversely, in addition to solid growth, it may be removed from the liquid by desorption or reverse diffusion along the sidewalls.

Following Dubrovskii [105], if the vapor composition is defined by

B A A I I I z   , (9.1)
with A I and B I the atomic contents of the MBE fluxes or CVD flows, the fraction of element A effectively available for growth is:

B A A V V V Z   , (9.2)
where A V or B V is given by the right sides of Eq. (3.4) for group III elements and of Eq. (

for group V elements. The two cases are however quite different. Eq. (3.4) assumes a constant surface diffusion flux to the droplet, independent of the liquid composition. This may be acceptable for mixed group III alloys. In this case, the stationary growth constraint Z x  solves the problem. If however surface diffusion depends on the chemical potentials in the liquid, or if the liquid also loses atoms by desorption, with a desorption flux that depends strongly on the concentration of the desorbing species in the droplet, as in the case of mixed group V alloys, then equation Z x  is still valid, but Z now depends on liquid composition and the equation needs to be solved self-consistently.. The simple group III case is a rare instance where we can ignore the state of the liquid. In general, during steady state growth, the liquid composition will adjust to provide a nucleation rate (or more generally a growth rate) that balances the influx of the various elements, but we have to determine this composition.

Dependence of solid composition upon liquid composition

Except in the simplest cases treated in section 9.3, determining which solid composition a liquid of a given composition will produce is an essential step of VLS growth modeling. This will in particular give the solid composition profile in a NW as a function of evolving liquid composition, for instance during heterostructure formation. This can in turn be related in a selfconsistent fashion to the changes in the growth fluxes, as will be illustrated in section 10. We now review briefly several approaches (items (a) to (d)) for the solid-liquid composition dependence. Note that all quantities of interest, in particular the chemical potentials, depend on temperature in addition to compositions. This dependence is kept implicit, the calculations being performed at a given temperature.

(a) For a ternary compound semiconductor A x B 1-x C, a simple description of the solidliquid dependence is derived in the regular growth model (section 3). Generalizing Eq. (3.14), the rates of formation of the AC and BC pairs from a liquid with atomic concentrations i y are assumed to be Dubrovskii showed that, within this model and assuming that the rates of loss of the various NW components by the droplet (via desorption or diffusion) are proportional to their respective concentrations, the steady state relation between solid ( x ) and vapor ( z ) can be determined analytically [START_REF] Dubrovskii | Fully analytical description for the composition of ternary vapor-liquidsolid nanowires[END_REF].

The other approaches make use, in various ways, of the differences of chemical potential k   of each alloy constituent unit k between liquid and solid phases (these units are atoms for group IV alloys and pairs for ternary compounds; see Eq. (3.7) and ref. [START_REF] Glas | Chemical potentials for Au-assisted vapor-liquid-solid growth of III-V nanowires[END_REF]). The calculation of the chemical potentials of use for VLS growth of III-V NWs, with or without foreign catalyst, has been discussed in section 6.2. We describe briefly these approaches.

(b) It may simply be assumed that the rate of incorporation of each unit is proportional to the relevant difference of chemical potential [107]. A difficulty is that the proportionality constants are a priori unknown. The underlying idea is that ML composition is defined not at nucleation stage (since critical nuclei are very small) but during its extension over the top facet.



for the two pairs. This gives the well-known result that the solid composition x is fixed by a single concentration in the liquid, for instance A y . The method consists in using this equilibrium relationship also during growth. This may look too simplistic, since NWs grow well out of equilibrium. The underlying idea is that, for such systems, the growth rate is set by the group V concentration C y in the liquid whereas the relative incorporation of A and B into the solid is more likely to be governed by group-III concentrations in the liquid phase and barriers for attachment to the solid, which presumably depend only weakly on As y . Note that this does not make any assumption about the composition of the solid nucleus that mediates the formation of each ML, but simply supposes that this nucleus is sufficiently small for the ML composition to be determined by post-nucleation attachment kinetics [108,109]. This effective 'decoupling' between group III and group V atoms was checked experimentally for MBE-grown Al 1-x Ga x As NWs [108,109].

(d) Probably the most sophisticated models proposed so far rely on a version of the CNT (see section 3) developed to handle the nucleation of a phase of a priori unknown composition.

The formation energy F of any 2D island (sections 3 and 6) now depends not only on its size but also on its composition (or alternatively, on the absolute numbers of both types of atoms or pairs that it contains). Accordingly, the critical island is defined by critical size (section 3) and critical composition. In 1950, this problem was solved elegantly by Reiss, albeit in quite a different context [110]. The formation energy of an island of radius r and composition x is: . The conditions on the r derivatives are similar to the standard CNT conditions. In turn, the composition of the critical island minimizes the standard nucleation barrier calculated for any fixed island composition. Note that all derivatives should be taken at given liquid composition. Figure 9.1 gives an example of such a saddle point in the formation energy surface for the self-catalyzed (Al,Ga,As) system. Only introduced in 2017 in the field of alloy NW growth, this nucleation-based approach is now used in systems of ever increasing complexity [START_REF] Leshchenko | Nucleation-limited composition of ternary III-V nanowires forming from quaternary gold based liquid alloys[END_REF]105,106,109,111,112,113].

    h r c x y y h r c x r F e   2 2 1 , , ,     C A , ( 9.4)
However, finding the critical island is not the end of the story, since its size is usually much smaller than that of the NW top facet. What is required is the composition of the full ML that forms by lateral growth of the critical island. The simplest assumption is that the whole ML adopts the critical island composition [105,106,109] (note that this assumption is opposite to that underlying approach (b)). Although difficult to justify, it was shown to lead to accurate predictions of the composition profiles in Al x Ga 1-x As NWs, measured with ML axial resolution

[109] (see section 10). Based on the discussion by Dubrovskii and coworkers of the growth rate of supercritical islands in the non-alloyed case [START_REF] Dubrovskii | Nucleation theory and growth of nanostructures[END_REF][START_REF] Dubrovskii | Zeldovich nucleation rate, self-consistency renormalization, and crystal phase of Au-catalyzed GaAs nanowires[END_REF], Johansson and Ghasemi calculated formally the rates of incorporation to the island of the two species constituting the alloy [114].

However, they concentrated on the limit of large islands. An accurate description of the average composition of the ML will require further investigations.

    BC AC x x x y y          1 , , C A , (9.5)
with:

S AC L C L A AC         , S BC L C L B BC         . ( 9
      x x x x k B     1 ln 1 ln
. Then:

    x T k x x T k x B s BC BC S BC B s AC AC S AC         1 ln , ln 1 2 2       , ( 9
    1 2 1 ln ,           x x x T k y y s B C A   , ( 9.8)
with:

    BC BC AC AC L B L A C A y y          , . (9.9)
Dubrovskii noted that, if the liquid is also assumed to be a regular solution (see section

                          1 2 exp 1 2 exp 1 x T k x x y T k y y y B s B L AB tot , ( 9
     tot BC AC tot AB BC BC AC AC B B A A y y         1         . ( 9
                T k x x x x y B s   1 2 exp 1 , (9.12) with   BC AC AB BC BC AC AC B B A A                
independent on compositions.

For mixed group III systems, the full calculations show that very wide ranges of solid compositions can be achieved with very composition-asymmetric liquids: any Al x Ga 1-x As solid (except nearly pure AlAs) can be obtained with a very low Al concentration in the liquid (Fig. 9.2, left) and any In x Ga 1-x As or In x Ga 1-x P solid (save nearly pure GaAs or GaP) with a very low Ga concentration in the liquid (Fig. 9.2, right). This is related to the very different affinities of the three group III atoms for a given group V atom (P or As), themselves related to the differences of cohesive energies of the three solid binaries (these affinities increase from In to Ga to Al). Moreover, for self-catalyzed systems, 1  tot y . Then, Eq. (9.12) with a constant   is again a very good approximation; specifically,

  AB BC BC AC AC B B A A            
for Al x Ga 1-

x As and

  AB BC BC AC AC B B A A            
for In x Ga 1-x As or In x Ga 1-x P.

Hence, for ternary III-V alloys, in the liquid, one NW element is much more abundant than all the others (namely, the group III element for mixed group V systems, In for mixed In-Ga or In-Al systems, Ga for mixed Al-Ga systems). An important consequence is that the regular solution model (with only pair interactions, possibly in a reduced number) is usually very accurate, since higher order concentration products are extremely small. This analysis was developed rigorously in references [105,106], where more details can be found. instance that corresponding to equilibrium. This explains the near-coincidence between the equilibrium and nucleation models for mixed group III alloys.

Heterostructures in nanowires

NWs offer great flexibility for associating different materials in a single nanostructure, which opens broad fields of basic research and applications. The 1D geometry of NWs lends itself to two elementary types of combinations: in axial heterostructures, material B is stacked on top of material A; in radial heterostructures, a shell of material B is wrapped around a core of material A. These can be elaborated into superlattice-type multiple axial or core-shell structures but also into mixed heterostructures: for instance, a quantum size section inserted axially between two barriers can be wrapped in a barrier shell, to produce a genuine quantum dot. In addition to these compositional heterostructures, crystal phase heterostructures, associating the cubic and hexagonal crystalline forms of the same material can be formed in NWs [116,117].

Standard axial heterostructures may be fabricated in the VLS mode by switching the growth fluxes. This is not possible for radial heterostructures, which are fabricated in the VS mode, with solid B nucleating on an already formed A NW directly from the vapor phase. Since this chapter deals with VLS, we will only consider axial heterostructures.

Heterostructures frequently associate materials with different lattice parameters. We first recall briefly the specific features and advantages of axial heterostructures in terms of strain relaxation, as compared with 2D structures, although these features are not specific to VLSgrown NWs. In the second part of this section, we will then discuss the VLS growth of axial heterostructures.

Elastic and plastic strain relaxation in axial heterostructures -Critical dimensions

The sidewalls of the NWs are free surfaces which permit a very efficient elastic relaxation. This has two main consequences: the strain field is more complex and less homogeneous than in planar heterostructures and the usually deleterious plastic relaxation is hindered. Let us assume that, on top of a NW stem of radius R of material A, we grow a "layer" of thickness h of material B with larger stress-free (intrinsic) lattice parameter (this applies to elastically isotropic or cubic materials; for hexagonal crystals, several lattice parameters come into play). We first assume that strain relaxation remains elastic (no extended defect forms). At the very beginning of growth, since R h  , the system is in a quasi-2D state: the layer contracts in-plane to adapt to the stem lattice parameter and expands quadratically along the NW axis.

Very soon however, strain starts distributing between layer and stem because strain relaxation of both parts is possible at the sidewalls while maintaining a coherent interface; the elastic energy stored in the system is therefore lower than in the 2D case (for the same volume of layer). This makes less likely the formation of interfacial dislocations. When h becomes of the order of R, the layer in-plane parameter approaches its strain-free value and further deposit is virtually unstrained. Following Glas [START_REF] Glas | Critical dimensions for the plastic relaxation of strained axial heterostructures in freestanding nanowires[END_REF][START_REF] Glas | Strain in nanowires and nanowire heterostructures[END_REF], this effect may be quantified by introducing function . By performing numerous finite elements numerical calculations, Glas showed that a very good approximation of function f, for elastically isotropic materials with identical elastic constants, is given by:

           2 3 1 1 exp 1     f . (10.1)
where parameters i

 ( 3 , 2 , 1  i
) depend only on the Poisson's ratio of the materials [START_REF] Glas | Strain in nanowires and nanowire heterostructures[END_REF]118].

This is a refinement of Glas' original approximation, with numerator equal to 1 [START_REF] Glas | Critical dimensions for the plastic relaxation of strained axial heterostructures in freestanding nanowires[END_REF]. The elastic energy is considerably reduced with respect to the 2D case, for instance to about a tenth of its 2D value for 12 . 0   (Fig. 10.1). Let us now consider the plastic relaxation of this system, which may occur via the

  0  c R . When   0  c R R 
, it should be possible to grow a layer of arbitrary thickness without introducing dislocations.

Glas gave a simple analytical formulas that allows one to determine the critical radius as a function of misfit. Since c R typically amounts to several tens of nm even for misfits of several % (for which the 2D critical thickness may be subnanometric), NWs having this radius (or less) are easily fabricated in most systems. Moreover, the very existence of a critical radius can be demonstrated on purely dimensional arguments [START_REF] Glas | Critical dimensions for the plastic relaxation of strained axial heterostructures in freestanding nanowires[END_REF]. We saw that, at large h, the layer (and hence the stem) are substantially strained only over a height of order R. Hence, the limit of NW e W at large h scales with 3 R . We also saw that, on the other hand, NW p W contains a part scaling with R. From these different power-law dependences in R, it follows that, if the layer is thick enough, the dislocation energy always dominates, provided R is small enough. This proves the existence of a critical radius c R below which plastic relaxation is unfavorable, whatever the layer thickness.

To conclude, lateral relaxation at the free sidewalls is a characteristic feature of NWs that enables very efficient stress relaxation. This allows for a much larger range of defect-free structures than in planar heterostructures or even in standard quantum dots structures. In addition to the great flexibility in terms of misfit discussed in section 10.1, axial heterostructures also offer, in principle, freedom of design and ease of control: for instance, a section of material B can be sandwiched between sections of material A, its diameter set by the NW stem diameter, its height by the growth time. In practice, controlling the section height, the sharpness of the interfaces and even the composition of the material in the case of alloys, may however be challenging.

A major difficulty is commonly associated with the "reservoir effect" (Fig. 10.2). We mentioned in section 2 that the liquid droplet acts as a reservoir of NW atoms. Whereas this is usually not a problem during the growth of a homogeneous NW, this may cause trouble for heterostructures. Indeed, during the growth of elementary semiconductor B or compound BC, B atoms reside in the droplet to ensure adequate supersaturation. Even if one switches abruptly the growth flux to A (in order to grow material A or compound AC), B will remain for some time in the droplet before it is eliminated. Elimination occurs via desorption or sidewall surface diffusion but also by incorporation in the growing NW. The joint presence of A and B in the liquid, in a proportion varying over time, thus leads to forming an A x B 1-x or A x B 1-x C alloy, which composition evolves during growth, in other words a composition gradient. As a rule of thumb, before switching to B, the number of A atoms in the droplet scales as 3

R y A

, with A y the (average) steady state concentration for A growth. In the graded solid section, this number will distribute over a length L and hence a volume of order 2 R L . Hence, /R y L Al ~. This readily gives simple recipes for growing sharp interfaces: use small NW radii and elements diluted in the droplet. One way to reduce the solubility and the concentration of a NW element in the seed particle is to use solid catalysts. This is effective but it generates its own problems, in particular a low growth rate. Anyway, the VSS growth mode lies out of the scope of this review. the proportionality constant between R and L was found to be on the order of 1. This confirms that the concentrations of group IV elements in the catalyst are of several tens of percent, as expected from the equilibrium phase diagrams of the Au-Si and Au-Ge systems [145] (see the discussion at beginning of section 2 and generic figure 2.1).

According to the same simple estimate, it was long thought that heterostructures of III-V compounds alternating group V atoms offered the best prospect of interface sharpness, simply because the steady state concentrations of As or P in the liquid catalyst (whatever it is) are much smaller than those of group III elements or elementary semiconductors. Experimentally, sharp interfaces have indeed been obtained in alternating group V systems based on the (P,As) couple [123][124][125]128,131,140,160]. Since P is present at an even lower concentration in the liquid than As (its equilibrium concentration is typically 5 times smaller), the P→As interface tends to be sharper than the As→P interface [131], as expected from our simple estimate of interface width.

Interfaces tend to be broader when one of the group V elements is Sb, especially at the top of the Sb-rich segments [127,139]. This can be related to the much higher concentration of Sb in the catalyst droplet as compared with the other group V elements [146]. Since the group V species Letters 16, 1917Letters 16, -1924Letters 16, (2016)). Copyright (2016) American Chemical Society.

On the other hand, given the high group III concentrations in the liquid, the same simple argument seemed to preclude sharp transitions in mixed-group III systems (and especially so in the self-catalyzed scheme, when the droplet is made of at least 99% group III atoms). Recent experimental work [108] and theoretical studies of the liquid-solid composition dependence, summarized in section 9.3, have demonstrated that this is fortunately not the case: reasonably sharp interfaces may be obtained even in mixed group III heterostructures (Fig. 10.3). This can be understood as follows. As discussed in section 9.4, the liquid is overwhelmingly composed of one of the group III elements, In in the case of (In,Ga) or (In,Al) systems, Ga for the (Al,Ga) system. This means, for instance, that when growing an Al x Ga 1x As section atop a GaAs stem, little Al is needed in the liquid to reach high Al concentrations in the solid. The necessary Al y will thus be reached rapidly and the GaAs → Al x Ga 1x As interface will be sharp. Moreover, since little Al is stored in the droplet during alloy growth, it will be consumed rapidly when forming the reverse Al x Ga 1x As → GaAs interface, which will also be sharp. This is rather

Modeling the formation of axial heterostructures

Modeling the dynamics of heterostructure formation, with the aim of obtaining the axial composition profile generated by non-steady growth conditions and finding conditions insuring sharp interfaces, relies on coupling the dependence of solid composition upon liquid composition (discussed in section 9.4), which in itself is time-independent, with the rates at which the various elements are added to the droplet (by direct fluxes, reemission, surface diffusion...) or removed from it (by growth, desorption...).

Let us consider again a ternary A x B 1-x C compound semiconductor alloy. Very generally, we may write, reasoning on full MLs, equations governing the variations of the numbers i N of each species i in the liquid: ; any solid within this range will tend to decompose in solids of compositions 1

      y y y , A in A ML A dt dN N x dt dN              , (10.2a)         y y y , 1 B in B ML B dt dN N x dt dN               , (10.2b)     y y , C in C ML C dt dN N dt dN              . ( 10
x and 2

x . Consider such a ternary A x B 1-x C system grown self-catalytically below c T (Fig. 10.4, right). Starting from 0  A y , the liquid will be in equilibrium with solids with continuously increasing AC content x up to a certain critical value c A y . This particular liquid is in equilibrium with both solids of compositions 1

x and 2

x . Liquids with A concentration further increasing from c A y will again be in equilibrium with solids with AC contents increasing smoothly from 2

x . It can then be expected that the interface profile will be driven by the reservoir effect for One of the most puzzling features of semiconductor NWs is that they frequently exhibit polytypism. Namely, several crystalline forms with the same composition coexist either in different NWs of the same ensemble or, most often, in a single NW. This is all the more surprising since, for many materials, in particular the III-V compounds, polytypism is specific of NWs: it appears neither in their bulk counterpart (or only in extreme pressure or temperature conditions) nor in other types of nanostructures. As far as growth is concerned, polytypism raises two main questions: Why does it appear? Is it possible to control it?

We will discuss polytypism in NWs of III-V compounds based on P, As or Sb, for which it is best documented. In bulk or quantum dot form, these III-V materials invariably adopt the cubic ZB structure. We exclude N-based compounds, which lowest energy structure is hexagonal WZ and which growth mode is usually non-VLS, catalyst-free. Polytypism also occurs in NWs of II-VI compounds [142], but this is less specific to NWs, and in Si NWs [154]. In addition to ground-state cubic ZB, the other structure most frequently encountered is hexagonal WZ (other hexagonal polytypes may be observed in some systems, but usually as transitory forms between ZB and WZ [155]). The differences of cohesive energy between the ZB and WZ forms are given in Table 10.1 for various semiconductors.

NWs usually grow along a ZB <111> direction (section 2) or WZ <0001> direction (the latter sometimes called c axis). Along these axes, the two structures consist in different stacking of the same biatomic III-V planes (MLs). In ZB (a FCC structure), the three lateral positions of the planes, called A,B,C occur periodically, producing an ABCABC... sequence, while only two positions occur in WZ, which produces an ABABAB... stacking (the A,B,C labelling is arbitrary;

only the relative positions of the plane matter). Hence, these structures are easily distinguished when observing the NWs by high resolution TEM with the electron beam along a properly chosen zone axis normal to the growth axis, or by electron or X-ray diffraction.

The coexistence of ZB and WZ was first observed in InAs and GaAs NWs by Hiruma and coworkers, more than 25 years ago [156,157]. However, explanations of the formation of WZ were first proposed much later and remain few. In the next section, we review the models that have been devised so far. We summarize the main features of these models and the key parameters involved, namely surface, interface and edge energies, supersaturation of the droplet medium, contact angle of the catalyst particle and interfacial structure of the liquid.

Models

In principle, the formation of ZB in NWs of the materials considered does not require any explanation. For any model explaining the formation of WZ, care should however be taken that it does not predict its occurrence in conditions where experiments show ZB to form!

(1) Total energy models

The first model proposed for the formation of WZ in NWs is not growth-related [162,163]. The idea is to consider the total energy of fully formed ZB and WZ NWs. This includes the bulk energy, which favors ZB, and energies associated with the sidewall surfaces and with the vertical edges between these facets. For a section of height H of a NW of radius R , these energies scale respectively as 2 R H , R H and H . For sufficiently small radii, the terms of lowest orders in R dominate the total energy. Hence, if the sidewall surface energy density or the edge line energy density is lower for WZ than for ZB, the total energy of the NW will be lower in the WZ form than in the ZB form, provided R is small enough. Qualitatively, Akiyama et al. argue that the ZB edges have more dangling bonds, and hence a larger specific energy, than the corresponding WZ edges [162]. This is substantiated by calculating the total energies using an empirical interaction potential between neighbor atoms [162,163], with the conclusion that, indeed, for all our III-V binaries, there exists a critical radius c R below which the total energy of the NW is lower in the WZ structure. These ideas were elaborated by others, who performed ab initio calculations for GaAs NWs with different radii. From the variation of the total energy, they could extract the specific energies of facets and edges [START_REF] Pankoke | Calculation of the diameter-dependent polytypism in GaAs nanowires from an atomic motif expansion of the formation energy[END_REF][START_REF] Rosini | Surface effects on the atomic and electronic structure of unpassivated GaAs nanowires[END_REF]. These calculations show that, on purely energetic grounds, narrow NWs should adopt the WZ structure. However, the critical radii thus found are all very small. For GaAs, the three groups find critical radii between 2 and 8 nm [START_REF] Pankoke | Calculation of the diameter-dependent polytypism in GaAs nanowires from an atomic motif expansion of the formation energy[END_REF][START_REF] Rosini | Surface effects on the atomic and electronic structure of unpassivated GaAs nanowires[END_REF]163]. For the nine III-V binaries considered, the critical radii range between 6 and 15 nm [163].

Although the Akiyama-Ito theory was adduced to explain some systematic observations of pure WZ in NWs of very small radii [164], it cannot account for the numerous observations NWs of totally or partially WZ structure, with radii of many tens of nm. This is why, in 2007, Glas proposed a nucleation-based model of the formation of WZ that does not suffer from such NW radius restrictions [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF]. This will be reviewed in the next section.

(

2) Nucleation-based model

From the very early days of the theory of VLS growth, the mechanism of 2D nucleation of new MLs at the solid-liquid interface was deemed to operate at intermediate supersaturations [165]. Many theoretical investigations indicate that 2D nucleation actually mediates NW growth and that the mononuclear regime, whereby each ML requires a single nucleation event, is very frequent (although polynucleation and regular growth cannot be entirely excluded at very high supersaturations; see section 3). Recent in situ TEM observations have brought strong support to this idea: all show NWs growing ML by ML and detailed quantitative analysis of the videos indicates that each ML starts with a nucleation event, even if the event itself is too elusive to be captured [START_REF] Harmand | Atomic step flow on a nanofacet[END_REF].

Glas noted that each new ML could form in ZB or WZ position with respect to the already formed NW: calling AB the sequence of the two last formed MLs, the new ML will be in ZB position if it adopts the C position, in WZ position if it adopts the A position (position B is forbidden). It is reasonable to assume that this position is actually fixed by the position adopted by the nucleus and then adopted by the whole ML that grows laterally from it [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF].

Then, explanation of WZ formation reduces to that of 2D nucleation in WZ position. Glas argued that if the nucleus is formed anywhere at the S-L interface, the ZB and WZ positions are not much different (in terms of atomic environment) so that ZB position should be favored on energetics grounds (interaction with the underlying MLs). However, this might not be the case if the nucleus forms at the TPL: part of its perimeter is then in contact with the vapor instead of the liquid and this might favor the WZ position [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF]. This is akin to the idea at the basis of the total energy models (1), but, since the nucleus is always found to be very small (section 6.4), its perimeter has a very strong impact on its formation energy F (see sections 3 and 9.4).

The expression of the nucleation barrier given in section 6.4 must be adapted to differentiate between nuclei positions (ZB or WZ) and locations with respect to the TPL. Very simply, for ZB and WZ, we find barriers:

    coh WZ WZ c ZB ZB c E A G A G             2 2 , . (11.1)
Here, A is a constant depending on material and nucleus geometry (assumed to be the same for ZB and WZ nuclei) and 0   is the difference of chemical potential between liquid and bulk ZB solid, which has to be diminished for a nucleus in WZ position. This reduction relates to the interaction of the nucleus with the underlying MLs; taking it equal to the difference of cohesive energy between bulk WZ and ZB (counted positive, as in Table 10.1) is only an approximation. According to what we just mentioned, we take This theory was initially developed in ref. [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF], where it was argued that the conditions for WZ formation might easily be met in NW growth. It has become the basis of the vast majority of models discussing the formation of the WZ phase in NWs. It has of course been refined in many ways (for instance, to take into account that, whereas nucleation away from the TPL can occur on most of the NW top facet, nucleation at the TPL can occur only on a small ring at the TPL [START_REF] Dubrovskii | Growth kinetics and crystal structure of semiconductor nanowires[END_REF]). We will not describe these developments, for which the reader may refer to section 6.6 of ref. [START_REF] Dubrovskii | Nucleation theory and growth of nanostructures[END_REF]. Instead, we discuss how theory and experiments have confirmed virtually all the assumptions on which it is founded.

A key condition for WZ nucleation (at the TPL) is that

ZB SV WZ SV   
. Ab initio calculations have indeed confirmed that this is the case at least for several materials, such as GaAs and InAs, and for some of their sidewall facets commonly found in real NWs [START_REF] Pankoke | Calculation of the diameter-dependent polytypism in GaAs nanowires from an atomic motif expansion of the formation energy[END_REF][START_REF] Rosini | Surface effects on the atomic and electronic structure of unpassivated GaAs nanowires[END_REF]166]. Simpler calculations based on broken bonds lead to the same conclusions, although they produce larger surface energies [167]. Even though the surface energy differences so calculated are rather small (typically only a few %), the corresponding critical values of the difference of chemical potential (Eq. (11.4)) seem to be easily attainable given the chemical potentials deduced from comparison of other experiments with models (e.g. regarding the kinetics of self-catalyzed GaAs growth; see ref. [START_REF] Glas | Predictive modeling of selfcatalyzed III-V nanowire growth[END_REF], section 6.4 and Fig. 6.10).

A very striking confirmation of a key point of the model was recently brought by in situ TEM experiments. Harmand et al. observed the VLS growth of Au-catalyzed GaAs NWs by solid source MBE in a dedicated TEM [START_REF] Harmand | Atomic step flow on a nanofacet[END_REF]. Both ZB and WZ MLs were found to grow in the mononuclear regime. Whereas ZB MLs extend very quickly after nucleation, WZ MLs advance slowly (as already reported in ref. [START_REF] Jacobsson | Interface dynamics and crystal phase switching in GaAs nanowires[END_REF]) and, by tilting the solid-liquid interface with respect to the electron beam, it is possible to follow the evolution of the geometry of their edge as the coverage of the NW top facet increases. The first fraction of ML that is observed forms very quickly after nucleation and it is systematically located at the TPL, and even at the corner between two sides of the hexagon that constitutes the TPL. In the experiments, this fraction is about 10% of a full ML, which is probably larger than the nucleus. However, probabilistic arguments indicate that these observations make nucleation at the TPL of the WZ MLs extremely likely [START_REF] Harmand | Atomic step flow on a nanofacet[END_REF].

As already mentioned, Eq. (11.2) predicts that nucleation at the TPL, and hence WZ formation, is more likely to happen for contact angles around 2  . Conversely, in situ TEM reveals another correlation between structure and contact angle. Namely, for GaAs at least, at large contact angle (  over about 125°), the solid-liquid interface is truncated and ZB grows. At angles closer to 2  (although maybe still larger than that value), the solid-liquid interface is flat and WZ grows. A model has been devised that shows that, irrespective of the solid structure and depending on surface and interface energies, the truncated shape may be favored at large  and the flat shape around 2  [START_REF] Jacobsson | Interface dynamics and crystal phase switching in GaAs nanowires[END_REF]. It is then tempting to reason that, for contact angles around 2  , the interface is truncated so that the (111) top facet (where nucleation occurs) is not in contact with the vapor anymore. Obviously, in absence of a proper TPL (as considered in the nucleation-based model; of course, there is still a TPL, but at the edge of the truncation facet (see Fig. 4.1(c)), nucleation cannot occur at the TPL, hence WZ does not form and ZB prevails. This simple interpretation is however questionable. First, it only holds if truncation is present all around the NW, which is not clear at all at the moment. Second, although nucleation at the TPL and planar solid-liquid interface are both predicted to occur for angles around 2  [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF][START_REF] Jacobsson | Interface dynamics and crystal phase switching in GaAs nanowires[END_REF], the corresponding critical angles depend partially on different parameters and have no reason to be equal. This leaves open the question of the exact coincidence between WZ formation and vanishing of truncation observed experimentally. These questions are further discussed in section 4.

Finally, we note that the nucleation-based model describes stacking defects in otherwise perfect structures as well as polytypism. In both case, we are concerned with the position (WZ or ZB) adopted by the new ML. For instance, in the ZB structure, a stacking fault correspond to a local ABABC sequence, a twin boundary to a change of stacking direction ABCABCBACBA, abrupt, although mixed structures or higher order polytypes [155] may form between two pure phase sections. Another one is that, for each material, the ZB and WZ forms have different electronic structures, in particular different band gap energies. The phrase 'crystal phase heterostructure' (or 'crystal phase quantum dot') actually appeared when one started studying their optical properties (in GaAs [116] or InP [117]) or their transport properties (in InAs [170]). Recent in situ TEM experiments have demonstrated that, as discussed in section 4, the droplet contact angle is a key parameter for phase selection [START_REF] Jacobsson | Interface dynamics and crystal phase switching in GaAs nanowires[END_REF], with ZB forming at large and small contact angles and WZ in an intermediate range [START_REF] Panciera | Phase selection in self-catalyzed GaAs nanowires[END_REF]. It is likely that the painstaking studies that allowed Dick et al. blindfoldly to form nearly perfect ZB/WZ superlattices [170] amounted primarily to controlling this parameter. As discussed in section 11.2, both the nucleation-based model of WZ formation [START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF][START_REF] Dubrovskii | Growth kinetics and crystal structure of semiconductor nanowires[END_REF] and the more recent models describing the morphology of the NW sidewalls and of the solid-liquid interface (section 4) attribute a prominent role to contact angle, with WZ predicted to form at intermediate angles. As yet, a full

Figure 2 . 1 .

 21 Figure 2.1. Liquid-solid phase diagram of M-Au alloy, characterized by an almost horizontal solidus at the eutectic melting temperature e T , and the two liquidus lines separating a mixture of M-Au liquid with solid Au (for low fractions of M), or solid M (for high fractions of M) for lower temperatures, from a single melt at higher temperatures. The temperatures Au T and M T correspond to melting of pure Au (1064

Figure 2 . 2 .

 22 Figure 2.2. Illustration of the Au-catalyzed VLS growth of GaAs NWs on GaAs(111)B substrate by MBE with pre-deposition of ~ 1 nm thick solid Au layer. Atomic force microscopy (AFM) images show the surface after Au deposition (1 st stage) and three-dimensional (3D) droplets after annealing (2 nd stage). Scanning electron microscopy (SEM) image shows the vertically aligned GaAs NWs (3 rd stage). Reprinted from Semiconductors and Semimetals, vol. 93, V. G. Dubrovskii, Theory of VLS growth of compound semiconductors, pp. 1-78. Copyright (2015), with permission from Elsevier.

Figure 2 . 3 .

 23 Figure 2.3. Tilted SEM images of regular two-dimensional (2D) arrays of InP/GaP core-shell NWs grown by MO-MBE from Au nanoparticles formed prior to epitaxy in the openings of a SiN x mask, defined by electron-beam lithography. The pattern pitches are 0.5 and 1 µm in (a) and (b), respectively. Reprinted from Journal of Crystal Growth, vol. 463, N. N. Halder, A. Kelrich, Y. Kauffmann, S. Cohen and D. Ritter, Growth of wurtzite InP/GaP core-shell nanowires by metal-organic molecular beam epitaxy pp. 10-13, Copyright (2017), with permission from Elsevier.

Figure 2 . 4 .

 24 Figure 2.4. Time series of Ga-catalyzed GaAs NW arrays: (a-i) SEM images of the array grown for 0 min (just after the droplet deposition) up to 10 min. Scale bar is 200 nm, and the tilt angle is 20°; the corresponding height (j) and diameter (k) distributions obtained by AFM. The diameter decrease seen after 4 min of GaAs deposition is due to the shape change as the droplets are lifted from the substrate surface, after which the NWs start to grow radially. Reprinted from J. Vukajlovic-Plestina, W. Kim, V. G. Dubrovskii, G. Tütüncüoğlu, M. Lagier, H. Potts, M. Friedl and A. Fontcuberta i Morral, Engineering the size distributions of ordered GaAs nanowires on silicon, Nano Letters 17, 4101-4108 (2017),

Figure 2 . 5 . 9 10 - 6 I

 2596 Figure 2.5. (a) Schematic morphology evolution of GaAs NWs and 20° tilted SEM images of 800 nm pitch arrays: (b) Standard growth under Ga-rich

Figure 2 .

 2 [START_REF] Seifert | Growth of one-dimensional nanostructures in MOVPE[END_REF] shows the example for Ga-catalyzed GaAs NWs grown in regular arrays on patterned SiO 2 /Si(111) templates[START_REF] Kim | Bistability of contact angle and its role in achieving quantum-thin self-assisted GaAs nanowires[END_REF]. The initial step is the standard growth under negative V/III influx imbalance, achieved at a low As 4 BEP of 2×10 -6 Torr for a fixed Ga flux corresponding to 0.11 nm/s planar growth rate of GaAs. All NWs grow radially in this step, reaching an average radius of 75 nm for 4 μm length. Now, the idea is to obtain quantum-thin nanoneedles (NNs) on top of this NWs by engineering the As flux. Increasing the As 4 BEP to 8.9 10 -6 Torr for 2 min after the standard growth step yields to the structures shown in Figure2.5 (c). Even though the Ga flux is still on, the droplets have decreased to a radius smaller than that of the NW. Keeping these conditions for 3 min longer yields the formation of conical-like NN tops, as shown in Figure2.5 (d). The top radius of conical-like NNs can be decreased to a very small size, about 5 nm. After that, the VLS growth should stop due to the consumption of Ga and NNs cannot grow any longer.If, instead of maintaining the As 4 BEP at a constant value of 8.9 10 -6 Torr in the droplet shrinking stage, it is decreased down to 5.5  10 -6 Torr from the step shown in Figure2.5 (c), the morphology changes dramatically. Figure2.5 (e) shows a representative SEM image of the structures obtained after 20 min of growth at 5.5  10 -6 Torr. It is clearly seen that introducing the third growth step with a lower As flux after shrinking the droplets results in the formation of thin NNs that first taper and then acquire a uniform radius of approximately 10 nm. This stationary radius stays constant along ~ 1 µm of the NN length. These structures show quantum confinement effect as discussed in detail in Ref.[START_REF] Kim | Bistability of contact angle and its role in achieving quantum-thin self-assisted GaAs nanowires[END_REF].

Figure 2 .

 2 Figure 2.6 (a) shows the results of Ref.[START_REF] Priante | Stopping and resuming at will the growth of GaAs nanowires[END_REF], where the Ga-catalyzed VLS growth of

Figure 2 . 7 .

 27 Figure 2.7. Growth of bent Ga-catalyzed GaAs NWs: (a) and (d) Vertical NWs grown for 20 min and rapidly cooled down; (b) and (e) NWs annealed at the growth temperature for 45 s prior to cool down; (c) NWs grown for 5 min after 45 s annealing; Side and top-view images of (f) type 1 horizontal and (g) type 2 downward growth. Scale bars in low magnification 30° tilted images are 1 µm in (a) to (c) and 100 nm in (d) to (g). Reprinted with permission from E. S. Koivusalo, T. V. Hakkarainen, H. V. A. Galeti, Y. G.

  simultaneously providing the Ga and As fluxes. A typical NW sample after 20 min of vertical growth and immediate cool down without any fluxes is shown in Figures 2.7

  (a) and (d). It is clearly seen that the Ga droplets remain stationary on the NW tips just after growth. In contrast, the sample that was annealed for 45 s at the growth temperature prior to cool down [Figure 2.7 (b) and (e)] exhibits 100% yield of droplets falling toward one of the (110) side facets. When the growth is resumed after annealing by simultaneously opening the Ga shutter and As cracker, the NWs continue their growth perpendicular to their initial growth direction [type 1 NWs in Figures 2.7

  (c) and (f)], or slightly downward [type 2 NWs in Figures 2.7

  Figures 2.7 (e) to (g)].

. 2 ) 5 I 5 

 255 Equation (3.1) accounts for (i) the direct ( ] which are entering the droplet seated on the NW top, with accounting for the beam geometry and the droplet contact angle in MBE[START_REF] Glas | Vapor fluxes on the apical droplet during nanowire growth by molecular beam epitaxy[END_REF] or the precursor cracking efficiency in cracking), (ii) the desorption flux des I 5 from the surface area of spherical cap droplet with the contact angle  , and (iii) the sink of group V atoms due to the NW elongation dt dL / , with S  as the elementary volume per III-V pair in solid. We neglect surface diffusion of group V species due to their high volatility. The influx of group III atoms in Eq. (3.2) is the sum of the impinging flux 3 3 I

  group III adatoms on the sidewalls, and 3 n as their surface concentration as a function of the height z . Desorption of group III atoms from the droplet is neglected. The sink due to NW growth should be exactly the same as for group V atoms to ensure stoichiometric III-V compound in the solid NW.

  chemical potentials of group III atoms in the droplet ( L 3

5 I

 5 potential of group V atoms in liquid in thermal units, T is temperature and B k is the Boltzmann constant. The temperature-dependent pre-exponent 0 , is related to the vapor phase. Finding 5 c requires a second equation for dt dL / , which is determined

. 7 ) 35 

 735 Here, the L X  are the chemical potentials of group III and V atoms in liquid, which depend on the two atomic concentrations 3 c , 5 c and temperature T , and S is the temperature-dependent chemical potential per III-V in solid. The number of III-V pairs in the critical 2D island (the critical size for brevity) is given by the standard equation of classical nucleat ion theory (CNT)

   = 1 for the center and R r c / 2   for the TPL nucleation.

5 D

 5 is the diffusion coefficient of group V atoms in liquid,

Figure 4 . 1 .

 41 The NWs can form in (i) inward tapered geometry with non-wetted (N-W) narrowing (n) facets making a positive angle n  with respect to vertical [Figure 4.1 (a)]; (ii) outward tapered geometry with non-wetted widening (w) facets inclined at a negative angle w  with respect to the vertical [Figure 4.1 (b)]; (iii) with truncated (tr) wetted (W) growth interface at an angle tr  with respect to the vertical, which can grow only to a finite length according to Ref. [47] [Figure 4.1 (c)]; or (iv) with non-wetted vertical facets and planar growth interface [Figure 4.1 (d)], which is the standard VLS configuration. The solid-vapor surface energies ) (  SV depend on the crystal phase (ZB or WZ)

Figure 4 . 1 .

 41 Figure 4.1. Non-wetted (a) narrowing and (b) widening facets, yielding an increasing or decreasing contact angle  , respectively. (c) Truncated wetted facet with the angle tr  , which will grow only to a finite length. Configurations (a) to (c) are compared to the standard VLS growth mode with planar liquidsolid interface and vertical side facets, shown in (d). Reprinted with permission from V. G. Dubrovskii, N. V. Sibirev, N. N. Halder and D. Ritter, Classification of the morphologies and related crystal phases of III-V nanowires based on the surface energy analysis, Journal of Chemical Physics C 123, 18693-18701 (2019). Copyright (2019) American Chemical Society.



  for widening facet. The excess surface energy of forming NWs with truncated wetted facet with the angle tr  equals

Figure 4 . 2 .

 42 Figure 4.2. Structural diagram for VLS GaAs NWs, showing the energetically preferred morphology versus the contact angle of the droplet. Zero energy level corresponds to vertical (110) ZB side facets. Horizontal line at -0.098 Jm 2 corresponds to the negative surface energy difference between ) 00 1 1 ( WZ and (110) ZB vertical facets. Structure 1 with tapered NW shape and planar liquid-solid interface is preferred at small contact angles < 100 o . The stable contact angle

  At intermediate contact angles, typically between 100 o and 125 o , stable NW growth was observed, with vertical facets and planar liquid-solid interface. At large contact angles, above 125 o the truncated wetted corner facets developed, oscillating with the period of ML growth and increasing their size toward larger  . Soon after the 125 o contact angle, the NWs became reverse tapered due to the droplet swelling. Very importantly, these changes of the morphology led to the crystal phase switching -the crystal phase was ZB for small (   100 o ) and large   125 o ) and WZ for the intermediate range of  . Fitting the two critical contact angles, the structural diagram of VLS GaAs NWs shown in Figure 4.2. From the fits, deduce the plausible values of SL  =0.593 J/m 2 and L tr   = 0.566 J/m 2 , which appear close to each other. The entire set of the parameters for GaAs NWs is summarized in

  spacing (pitch) P and the angle 3  of the group III beam with respect to the vertical. be used in the first approximation. Then the axial and radial NW growth are given by

5 

 5 negligible radial growth. The smallness of f  for VLS NWs can be due to fast crystallization rate from liquid compared to the VS crystallization rate on the NW sidewalls ( ' 35 35 k k  ), slower desorption of group V atoms from liquid compared to the NW sidewalls ( ' 5 5k k  ) or their better adsorption at the liquid surface ( ' 5   ). The f  values for the SAE NWs are expected to be larger, and highly dependent on the material composition. This explains, for example, why InSb NWs grown on InAs stems become nano-discs at high Sb fluxes[START_REF] Robson | Multispectral absorptance from large-diameter InAsSb nanowire arrays in a single epitaxial growth on silicon[END_REF], that is, grow faster in radius than in length.Solutions to Eqs. (5.7) for short NWs (

equal 3  2 .

 32 . With the known deposition rate 3 v and geometry (the  coefficients and the pore radius 0 R ), these simple expressions are controlled by the single parameter f  . It determines the power exponents in the scaling power-law dependences of the length and radius on the growth time as well as the characteristic times 0 t and 1 t for the axial elongation and radial extension in different stages. After an incubation stage, the NW radius evolves linearly with time as long as 3 We now consider experimental data on the SAE InAs NWs grown by MBE in patterned arrays on SiO 2 /Si(111) at 480 o C, with As-BEP fixed at 3.5  10 -6 Torr, and In flux corresponding to 0.024 nm/s, in 18 nm deep pores with a fixed radius of 40 nm and variable pitches [69].

Figures 5. 1

 1 Figures 5.1(a) and (b) show the time evolution of the NW length and diameter, respectively, for

0 R 1 . 0 L

 010 Figures 5.1(a) and (b) show the time evolution of the NW length and diameter, respectively, for

Figure 5 . 1 .

 51 Figure 5.1. Time evolution of (a) length and (b) diameter of SAE InAs NWs grown by MBE in patterned arrays on SiO x /Si(111) with different pitches from 250 to 3000 nm (symbols)[START_REF] Hertenberger | Growth kinetics in position-controlled and catalyst-free InAs nanowire arrays on Si(111) grown by selective area molecular beam epitaxy[END_REF], fitted by the model (lines). For the largest pitch of 3000 nm, the length evolution is slightly super-linear and the diameter is linear in time in the absence of shadowing. Decreasing the pitch leads to the decrease of the In collection length due to the shadowing effect, from more than 3500 nm for 3000 nm pitch down to 500 nm for 250 nm pitch. This leads to sub-linear behavior of both length and diameter after the NW length exceeds the collection length of In. Reprinted from V. G. Dubrovskii, Evolution of the length and radius of III-V nanowires grown by selective area epitaxy, ACS Omega 4, 8400-8405 (2019), https://pubs.acs.org/doi/10.1021/acsomega.9b00525, with permission from the American Chemical Society, to which further permissions related to the material excerpted should be directed.

  super-linear increase of the NW length for shorter times requires sufficiently large a according to Eq. (5.8). The NW diameter in this stage is linear in time regardless of a . All theoretical curves shown in Figure5.2 correspond to a fixed  a 1.83, or f  =

  , showing that the adsorption of gallium is better on the top NW facet than on the sidewalls. The observed decrease of the NW length for larger pore size is explained simply by the fact that the NW elongation is controlled by surface diffusion of Ga, which contributes more into the axial growth rate of thinner NWs. The Ga diffusion length on the sidewalls appears very close to 2 m in all cases.

Figure 5 . 2 .

 52 Figure 5.2. Length and diameter evolution of SAE GaAs NWs grown by MOVPE in patterned arrays on SiO 2 /GaAs. SEM images in (a) to (d) show the time-dependent morphology of NWs after 20, 40, 60, and 80 min of growth, respectively, for a 600 nm pitch and 225 nm diameter pore array. All scale bars represent 1 m. (e) Time-dependent mean length and diameter in the arrays with different pore sizes from 125 nm to 225 nm (symbols), fitted by the model (solid lines for the length and dashed lines for the diameter) with the parameters summarized in Table 5.1. Time dependence of the NW length converges

Figure 5 . 3 .

 53 Figure 5.3. (a) Average NW length (after 5 min of growth time) of <111>A-oriented InP NWs grown by MOVPE in patterned arrays on SiO x /InP(111)A substrates with variable pore size, as a function of their diameter under different V/III ratios obtained by varying only group V flow. Panels (b, c) and (d, e) show 30° tilted SEM images of the InP NWs grown with a V/III ratio of 81 and 198, respectively, as indicated in (a). The NW diameters are around 60 nm in (b), 150 nm in (c), 60 nm in (d), and 150 nm in (e). Lines in (a) are theoretical fits by the model. Reprinted with permission from Q. Gao, V. G. Dubrovskii, P. Caroff, J. Wong-Leung, L. Li, Y. Guo, L. Fu, H. H. Tan and C. Jagadish, Simultaneous selective-area and vapor-liquid-solid growth of InP nanowire arrays, Nano Letters 16, 4361-4367 (2016). Copyright (2016) American Chemical Society.

  [START_REF] Zhang | Nanowires: building blocks for nanoscience and nanotechnology[END_REF] shows the lengthdiameter correlation for InP NWs grown for 5 min with different V/III ratios, which can only be understood by considering simultaneous SAE and VLS growth. The growth rate of thin, 50-60 nm diameter NWs is tremendously enhanced by the presence of In droplet, forming spontaneously at the tip due to surface diffusion of In. Medium diameter NWs evolve in a combined mode, with In droplets appearing and then disappearing in the course of growth. Only thick, ~ 150 nm diameter NWs grow in the true SAE mode from the very beginning. The coexistence of the VLS and SAE mechanism was also observed in Ref.[START_REF] Robson | Conditions for high yield of selective-area epitaxy InAs nanowires on SiOx/Si(111) substrates[END_REF] for InAs NWs grown by MBE in patterned arrays on SiOx/Si(111) substrates. It was found that most NWs start with In droplets that nucleated in the pores and disappeared in a later stage due to insufficient In supply from the oxide surface.

Figure 6 . 1 .

 61 Figure 6.1. (a) HAADF STEM image of an InP 1- x As x NW segment grown with a modulated As 4 :P 2 flux ratio. Scale bar represents 20 nm. The HAADF signal is integrated over the whole diameter and the profile is plotted along the growth axis. (b) FFT-filtered HAADF image and corresponding profile. Reprinted with permission from J. C. Harmand, F. Glas and G. Patriarche, Physical Review B 81, 235436 (2010), https://doi.org/10.1103/PhysRevB.81.235436. Copyright (2010) by the American Physical Society.

Figure 6 . 2 .

 62 Figure 6.2. Growth chronology of a single InP 1-x As x NW. The length of the NW shown in the inset (Scale bar is 50 nm), is plotted as a function of time, as deduced from the composition oscillations along its axis. The different colors correspond to overlapping HAADF images. During growth, the flux modulation was stopped for 12 time periods. The corresponding region serves as a time reference. At 355 s, the nominal As 4 :P 2 flux ratio was changed abruptly from 1:5 to 3:3. Reprinted with permission from J. C.

Figure 6 . 3 .

 63 Figure 6.3. (a) Experimental NW elongation and (b) axial growth rate (circles), fitted with the diffusion transport model with the four parameters shown in the legends. Different chemical potentials are used for each As 4 :P 2 ratio, before and after t=355 s. Reprinted figure with permission from J. C. Harmand, F.Glas and G. Patriarche, Physical Review B 81, 235436 (2010), https://doi.org/10.1103/PhysRevB.81.235436. Copyright (2010) by the American Physical Society.

  Concerning s  , it was found that very small values (a few nm) are necessary to fit the experiment. The best fit is obtained with s  = 2 nm. This very short length can be understood as a limitation of adatom diffusion by atomic steps of the rough parasitic layer. The abrupt change of As 4 :P 2 flux ratio is expected to modify significantly the composition and chemical potential of the catalyst droplet. Accordingly, two sets of chemical potential differences were adjusted, valid before and after this change. The best fit for the first part of growth at As 4 :P 2 flux ratio of 1:5 was obtained with fl   =270 meV and sl   =18.2 meV. The contribution from the substrate surface is very small due to the combination of low s  and moderate sl   . Very quickly, the contribution from the sidewalls dominates largely. After the change of As 4 :P 2 flux ratio to 3:3, fl   must be reduced to 160 meV, which indicates that the flow from the sidewalls to the droplet is less efficient when P 2 is replaced by As 4 . At the same time, sl   takes a negative value of 145 meV, showing that a significant part of the atoms intercepted by the NW now migrates to the substrate surface. With these parameters, the diffusion transport model describes fairly well the experimental ) (t L dependence, as shown in Figure 6.3. For both As 4 :P 2 flux ratios, NW length increases almost exponentially with time, because the diffusion-induced is proportional to the length at L f   .

Figure 6 . 4 .

 64 Figure 6.4. Length-diameter correlation of Aucatalyzed WZ GaAs NWs grown by MBE on a GaAs(111)B substrate at 585 o C, with Ga flux

.

  The latter values values are usually referred to the standard reference states of pure elements at temperature 0 T = 298.15 K. The reader may refer to

Figure 6 . 6 .

 66 Figure 6.6. Variation with the atomic Figure 6.7. Variation with the atomic

Figure 6 . 8 .

 68 Figure 6.8. III-V VLS system with an Au-III-V droplet having the atomic concentrations 3 c of group III

[ 86 ,

 86 87]. Reprinted with permission from V. G. Dubrovskii, Refinement of nucleation theory for vaporliquid-solid nanowires, Crystal Growth & Design 17, 2589-2593 (2017). Copyright (2017) American Chemical Society.

Figure 6 .

 6 Figure 6.8 illustrates the VLS growth model for Au-catalyzed III-V NWs in the case of

. 26 )At ≪ 1 ,

 261 with = 4 √3 ⁄ and the effective surface energy of the island edge defined earlier. Eq. (6.26) correctly accounts for the non-monotonous variation of the edge energy with ML size and for leading terms ~ / and ~ ( ) / at small and large sizes, respectively[START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF]. This functional form is expected to be general but coefficients A =3.3748 and B = 2.2993 should vary with catalyst and contact angle (fits to in situ experiments on Au-catalyzed NWs[START_REF] Harmand | Atomic step flow on a nanofacet[END_REF] yield A =3.3748 and B = 2.2993[START_REF] Glas | Energetics and kinetics of monolayer formation in vapor-liquidsolid nanowire growth[END_REF]).The NW ML is completed at = 1 whereas the ML formation cycle ends at = 1, after a full equivalent ML of GaAs is added to the droplet from vapor. Since = ⁄ ⁄ , the minimum of the formation energy obtained by differentiating Eq. (6.25) at given refill yields the size of the island as a function of time. The extrema of the formation energy are given by corresponding to large NW radius and low values of , the influence of the surface energy term on island size is negligible. In this case, this size equals = + , which is reduced to = for instantaneous growth from liquid.

Figure 6 .

 6 9 (a) shows the behavior of + ( 2 Ω sin ⁄ ) as a function of at different (and hence times) for a 20 nm radius GaAs NW, at a growth temperature of 610 o C, in the case of a low surface energy =0.037 J/m 2 . For instance, values of = 0.34 and =0.7 yield the stopping size =0.36 in the absence of refill. Then, the minimum of the formation energy shifts to the right as increases. This describes the slow growth stage at the rate of refill. The minimum of the formation energy is well approximated by the volume part of the formation energy shown by the dashed lines, which indicates a negligible effect of the island surface energy. The situation changes at a high surface energy, e.g. = 0.25 J/m 2 . According to Fig. 6.9 (b), the minimum

Figure 6 . 9 .

 69 Figure 6.9. (a) Normalized island formation energy + ( 2 Ω sin ⁄ ) versus for different indicated near each curve (solid lines). The curves are obtained from Eqs. (6.25) and (6.26) for a GaAs NW with = 20 nm, at =610 o C, = 0.34, =0.7 at a low =0.037 J/m 2 , corresponding to =0.01584. Dotted lines show the volume part of the formation energy and correctly describe the stopping size in all cases. (b) Same as (a) at a high =0.25 J/m 2 , corresponding to =0.1056. A minimum of the formation energy is observed only at small and disappears above = 0.5, due to a decrease of the ML surface energy [86]. The volume part of the formation energy gives an incorrect behavior at large .

Figure 6 .

 6 Figure 6.10 (a) shows the calculated nucleation barrier

Figure 6 . 4 AsP = 4 . 2 ( 3 .

 64423 10 (b) shows the same quantities for NW2 grown under 8×106 Torr and different temperatures. For this NW,  R 37 nm and β =125 . The crystal phase of both NWs was ZB. It is interesting to note that this contact angle is close to the one corresponding to vertical growth of ZB GaAs NWs in Figure4.between 125 o and 127 o ). The best fits to the data were obtained with the re-emission coefficient  r 25 and temperature-independent 0 J and eff  given in Section 3. According to these calculations, the As concentration increases almost linearly with the As flux and temperature, and varies only from ~ 0.01 to 0.015 in the experimental range. The chemical potential also does not vary much over the experimental range, remaining within the interval of 115-160 meV. These small variations result in almost linear correlation of the axial

Figure 6 .

 6 Figure 6.10. (a) Bottom panel: growth rates measured (triangles) and calculated (disks) for NW1, as a function of incoming As flux, at temperature T= 883 K. Three upper panels: As concentration in the droplet, difference of chemical potential between liquid of composition c As and solid ZB GaAs, and nucleation barrier, calculated for each discrete data point of bottom panel. In the two bottom panels, the dashed lines are calculations extended to the whole pressure range, made for NW1 using the best model parameters. The As flux is expressed as BEP (lower scale) and as absolute atomic flux. (b) Same as (a) for NW2, as a function of growth temperature. Reprinted figure with permission from F. Glas, M. R. Ramdani, G. Patriarche and J. C. Harmand, Physical Review B 88, 195304 (2013), https://doi.org/10.1103/PhysRevB.88.195304. Copyright (2013) by the American Physical Society.

3  3 L

 33 [START_REF] Dubrovskii | Self-equilibration of the diameter of Ga-catalyzed GaAs nanowires[END_REF] describes the time evolution of the NW radius (at a fixed contact angle  ), with with 3 v as the deposition flux in nm/s and as the characteristic time constant for the reverse diffusion flux from the droplet to the NW sidewalls, which is proportional to ) exp[START_REF] Bastiman | Growth map for Ga-assisted growth of GaAs nanowires on Si(111) substrates by molecular beam epitaxy[END_REF] gives chemical potential of group V atoms in the droplet, for gaseous precursor incorporating  As atoms, with 5 v as the vapor flux including re-emitted species and des v 5 as the pre-exponent in the desorption rate. The NW elongation rate in the mononuclear regime is given by Eq. (3.10), with the Zeldovich nucleation rate of CNT )Eq. (3.11).



  containing characteristics of the vapor phase and becoming radiusindependent in many practical cases,

Figure 6 . 11 .

 611 Figure 6.11. Transition from non-monotonic to increasing length-diameter correlation with increasing n R , obtained from Eq. (6.35) at

  Reprinted from Journal of Crystal Growth, vol. 401, V. G. Dubrovskii and Yu. Yu. Hervieu, Diffusioninduced growth of nanowires: generalized boundary conditions and self-consistent kinetic equation, pp. 431-440, Copyright (2014), with permission from

Figure 6 . 12 .

 612 Figure 6.12. Length-radius correlation of MOVPE-grown InP NWs [89], fitted by Eq. (6.35) (solid line) and by the Gibbs-Thomson curve [65] (dashed line). Reprinted from Journal of Crystal Growth, vol. 401, V. G. Dubrovskii and Yu. Yu. Hervieu, Diffusion-induced growth of nanowires: generalized boundary conditions and self-consistent kinetic equation, pp. 431-440, Copyright (2014), with permission from Elsevier.

Figure 6 .R

 6 [START_REF] Harmand | Analysis of vapor-liquid-solid mechanism in Au-assisted GaAs nanowire growth[END_REF] shows the example of non-monotonic length-radius curve for Au-catalyzed InP NWs grown by MOVPE at 470 o C[START_REF] Moewe | Growth mechanisms and crystallographic structure of InP nanowires on lattice-mismatched substrates[END_REF]. The NW length first increases with R , then saturates at about 22 nm, and starts to slightly decrease after that. The fit obtained from Eq. (6.35) with 2 23 nm provides better fit than the Gibbs-Thomson curve[START_REF] Dubrovskii | Gibbs-Thomson and diffusion-induced contributions to the growth rate of Si, InP and GaAs nanowires[END_REF], particularly for thinner NWs. More examples of different length-radius correlations of III-V NWs, related to their crystal

Figure 6 . 3 .5 which decreases the 0 

 630 Figure 6.13 shows the excellent correlation between the model given by Eqs. (6.36) and

Figure 6 . 13 .

 613 Figure 6.13. Growth rate of Ga-catalyzed GaAs NWs grown by MBE versus the NW radius at two different temperatures. Numerical calculations of Ref. [40] (symbols) and fits by Eqs. (6.36), (6.37) with the parameters listed in the insert [90]. Reprinted from Journal of Crystal Growth, vol. 440, V. G. Dubrovskii, Group V sensitive vapor-liquid-solid growth of Au-catalyzed and self-catalyzed III-V nanowires, pp. 62-68, Copyright (2016), with permission from Elsevier.

Figure 6 . 14 .

 614 Figure 6.14. 30 tilted SEM image of an array of self-catalyzed GaP NWs for a V/III flux ratio of 4. Two types of NWs, with (red box) and without (blue box) Ga droplets are indicated. Scale bar is 440 nm. Reprinted from E. D. Leshchenko, P. Kuyanov, R. R. LaPierre and V. G. Dubrovskii, Tuning the morphology of selfassisted GaP nanowires. Nanotechnology 29, 225603 (2018); https://doi.org/10.1088/1361-6528/aab47b. © IOP Publishing. Reproduced with permission. All rights reserved.

Figure 6 . 15 .

 615 Figure 6.15. Experimental (symbols) and theoretical (lines) morphologies of GaP NWs at a fixed pitch of 360 nm and different V/III flux ratios. Experimental morphologies correspond to a fixed growth time of 870 s under different V/III flux ratios after the initial growth step of 435 s at a fixed V/III=2 to form the NW stems. The measured and calculated shapes correspond only to the upper NW parts, with the initial conditions ) ( 0 0 L R relating to the initial stems. Reprinted from E. D. Leshchenko, P. Kuyanov, R. R. LaPierre and V. G. Dubrovskii, Tuning the morphology of self-assisted GaP nanowires. Nanotechnology 29, 225603 (2018); https://doi.org/10.1088/1361-6528/aab47b. © IOP Publishing. Reproduced with permission. All rights reserved.

Figure 7 . 1 .

 71 Figure 7.1. 30° tilted SEM image of Ga droplets obtained after predeposition of Ga onto a hole array in a SiO 2 on Si(111) substrate. The diameter of the holes is 60 nm and the pitch between the holes is 100 nm. (b) and (c) show top view and 30° tilted SEM images of self-catalyzed GaAs NWs grown from the array of Ga droplets shown in (a). Scale bar corresponds to 100 nm. Reprinted with permission from V. G. Dubrovskii, T. Xu, A. Díaz Álvarez, G. Larrieu, S. R. Plissard, P. Caroff, F. Glas, B. Grandidier, Selfequilibration of the diameter of Ga-catalyzed GaAs nanowires, Nano Letters 15, 5580-5584 (2015).

Figure 7 . 2 .

 72 Figure 7.2. Histograms of the Ga droplet size distribution and the NW diameter distribution. Inset: 30°tilted SEM images of the corresponding array of GaAs NWs with the hole size and pitch of 60 nm and 250 nm, respectively. Scale bar in the insert corresponds to 100 nm. Reprinted with permission from V. G. Dubrovskii, T. Xu, A. Díaz Álvarez, G.

Figure 7 . 3 .R

 73 Figure 7.3. Time dependences of the NW diameters obtained from Eq. (6.41) with

Figure 7 . 7 . 2 .

 772 Figure 7.3 shows the time evolution of the NW diameters for the array shown in Figure

Figure 8 . 1 .

 81 Figure 8.1. (a) Normalized LDs (histograms) fitted by the Poissonian shapes (lines) for Au-catalyzed InAs NWs grown by CBE for 20 and 45 min from Au nanoparticles obtained by thermally dewetting of 0.5 nm thick Au film at 510 °C for 20 min, along with their plan view SEM images. Reprinted from V. G. Dubrovskii, N. V. Sibirev, Y. Berdnikov, U. P. Gomes, D. Ercolani, V. Zannier and L. Sorba, Length distributions of Au-catalyzed and In-catalyzed InAs nanowires, Nanotechnology 27, 375602 (2016); doi:10.1088/0957-4484/27/37/375602. © IOP Publishing. Reproduced with permission. All rights reserved.

Figure 8 . 2 .

 82 Figure 8.2. Normalized LDs (histograms) of Au-catalyzed InAs NWs grown from 40 nm colloids annealed at 500 °C for 60 and 130 min, along with their plan view SEM images. The fits (lines) are obtained from Eq. (8.8) with  = 0.0025 for 60 min and 0.0013 for 130 min. Reprinted from V. G. Dubrovskii, N. V. Sibirev, Y. Berdnikov, U. P. Gomes, D. Ercolani, V. Zannier and L. Sorba, Length distributions of Au-catalyzed and In-catalyzed InAs nanowires, Nanotechnology 27, 375602 (2016); doi:10.1088/0957-4484/27/37/375602. © IOP Publishing. Reproduced with permission. All rights reserved.

Figure 8 . 3 .

 83 Figure 8.3. 45° tilted SEM image of (a) Au colloidal nanoparticles after the standard annealing step (20 min at 500 °C under TBAs=1.0 Torr), compared to (b) Au colloidal nanoparticles with just ramping the temperature up to 380 °C under TBAs=4.0 Torr and removing the annealing step; (c) the reference sample with thermally dewetted 0.5 nm thick Au film after the standard annealing step; (d)-(f) slow emerging of InAs NWs during the early stages of NW growth (3-12 min); (g) normalized LDs (histograms) of Au-catalyzed InAs NWs grown from Au colloids at 380 °C without the annealing step for 30, 60, 90 and 130 min respectively. The fits (lines) are obtained from Eq. (8.8) with  ranging from 0.0053 to 0.017. Reprinted from V. G. Dubrovskii, N. V. Sibirev, Y. Berdnikov, U. P. Gomes, D. Ercolani, V. Zannier and L. Sorba, Length distributions of Au-catalyzed and In-catalyzed InAs

Figure 8 . 4 .

 84 Figure 8.4. Schematics of the VLS NW growth from pre-existing Au droplets with the radius 0 R . The

. 16 )Figure 8 ./ 2 ,

 1682 Figure 8.5 shows how the symmetric Polya LD transitions to much broader LDs, which

Figure 8 . 5 .

 85 Figure 8.5. Discrete beta-distribution at a fixed a of 100, which is very well approximated by the continuum LD given by Eq. (8.13), transitioning from the symmetric Polya LD at b a  to much broader and asymmetric LDs with decreasing b , and finally becoming monotonically decreasing at 1  b . Reprinted from Journal of Crystal Growth, vol. 463, V. G. Dubrovskii, Length distributions of nanowires: effects of surface diffusion versus nucleation delay, pp. 139-144, Copyright (2017), with permission from

Figure 8 .

 8 [START_REF] Dubrovskii | Nucleation theory and growth of nanostructures[END_REF] shows the typical SEM images of differently sized InAs NWs after 7.5 min of growth. It is clearly seen that the NW length decreases for larger diameters, which confirms that their VLS growth is controlled by surface diffusion of In adatoms.

Figure 8 . 6 .

 86 Figure 8.6. 30 o tilted SEM images of InAs NW arrays with 1000 nm pitch, growth time of 7.5 min and NW diameters around (a) 53 nm, (b) 75 nm, and (c) 90 nm. Scale bar is the same in all three images. Reprinted with permission from V. G. Dubrovskii, Y. Berdnikov, J. Schmidtbauer, M. Borg, K. Storm, K. Deppert, J. Johansson, Length distributions of nanowires growing by surface diffusion, Crystal Growth & Design 16, 2167-2172 (2016). Copyright (2016) American Chemical Society.

Figure 8 . 7 .

 87 Figure 8.7. Transition from the exponential to linear growth regime of the 100-110 nm diameter InAs NWs (symbols), the fitting parameters in Eq. (8.19) for the lines are:  a 237, v  = 34 mim -1 and C = 0.247. Reprinted with permission from V. G. Dubrovskii, Y. Berdnikov, J. Schmidtbauer, M. Borg, K. Storm, K. Deppert, J. Johansson, Length distributions of nanowires growing by surface diffusion, Crystal Growth & Design 16, 2167-2172 (2016). Copyright (2016) American Chemical Society.

Figure 8 . 8 .

 88 Figure 8.8. Best fits by Eqs. (8.19) (lines) of the experimental standard deviations of the 100-110 diameter NW LDs (symbols) versus the mean length at different growth times. Reprinted with permission from V. G. Dubrovskii, Y. Berdnikov,

Figure 8 .

 8 Figure 8.9 shows the experimental LDs and their best fits by Eq. (8.20) for NWs with

Figure 8

 8 Figure 8.9. Normalized experimental LDs (histograms) of differently sized InAs NWs grown for 7.5 min. The data for 52-55, 64-70, 72-80, 82-90, 90-100 and 100-110 nm diameter NWs are fitted by Eq. (8.20) (lines) with = 140,   s 3098,  a 160   s 2231 MLs,  a 180, s =1915 MLs,  a 200,

Figure 8 . 10 .

 810 Figure 8.10. Broadening of the LD of 100-110 nm diameter InAs NWs with the growth time: experimental LDs (histograms), fitted by the Gaussian distributions (lines) with 237  a and s = 1000, 2110, 4400, 5400, and 10600 MLs

Figure 8 .

 8 11 (b) compares the distribution of the oscillation lengths with a Poisson distribution, normalized to the same total number of oscillations. The nucleation statistics are markedly sub-Poissonian. In three NW segments studied (see Ref.[START_REF] Glas | Nucleation antibunching in catalyst-assisted nanowire growth[END_REF] for the details), the measured standard deviations are less than half the Poisson standard deviations. This proves that the nucleation events are not independent.

Figure 8 .

 8 Figure 8.11. (a) Length of successive oscillations. The dashed line gives the mean length; the solid line is a linear regression t dt dL  / . (b) Histogram of the numbers of nucleation events per oscillation. The two curves represent distributions calculated for a large number of nucleation events and normalized to the total number of experimental counts: Poissonian statistics (triangles), self-regulated statistics (disks). Reprinted figure with permission from F. Glas, J. C. Harmand and G. Patriarche, Physical Review Letters 104, 135501 (2010), https://doi.org/10.1103/PhysRevLett.104.135501. Copyright (2010) by the American Physical Society.

  following considerations. The chemical potential decrease immediately after nucleation of one ML equals barrier under the exponent of the Zeldovich nucleation rate [Eq.(3.11)] as

  . (3.8) for the critical size c i , we note that the decrease in the nucleation probability immediately after nucleation equals )  is called the antibunching parameter in what follows. It can equivalently be presented in

  The main goal was to obtain the LD as a function of two parameters,  for antibunching and delay, assuming the NW nucleation rate in the form of exponential decay

  , and ( ) = e 1. The probability I p corresponds to the beginning of growth, just after forming the first ML of the NW (the third parameter which is not required in the continuum theory).

.

  The instantaneous growth rates in the continuum Eq. (8growth of the nucleation probability with time due to incoming flux and its exponential decrease immediately after the ML formation. Green's function of Eq.(8.2)

. 31 )

 31 This solution is quite interesting as it combines the Poissonian LD at 0   with selfregulated LD at 0   . In the latter case, Green's function becomes a universal function of   s :

35 )

 35 shape does not change with time. The mean length and variance of the fully formed LD given by Eq. (8.34) equal It is interesting to note that the Poissonian growth with nucleation delay can be resumed from these results by setting ) Eqs. (8.34) and (8.35) to Eqs. (8.8) and (8.10), respectively.

Figure 8 .

 8 Figure 8.12 shows how the universal LD ) (x f at a fixed  of 0.1 is broadened for

Figure 8 . 12 .

 812 Figure 8.12. Shapes of the universal LD at a fixed   0.1 and different  ranging from 0.01 to 0.2. Lines: continuum approximation [Eq. (8.34)]. Symbols: exact solution [Eq. (8.26)], for v p s  . Reprinted figure with permission from F. Glas and V. G. Dubrovskii, Physical Review Materials 1, 036003 (2017), https://doi.org/10.1103/PhysRevMaterials.1.03600 3. Copyright (2017) by the American Physical Society.

Figure 8 . 13 .

 813 Figure 8.13. Shapes of the universal LD at a fixed   0.07 and different  ranging from 0.01 to 0.2. Lines: continuum approximation [Eq. (8.34)] Symbols: exact solution for v p s  , using Eq.

  /doi.org/10.1103/PhysRevMaterials.1.036003 . Copyright (2017) by the American Physical Society.

Figure 8 . 14 .

 814 Figure 8.14. Discrete (symbols) and continuum (lines) LDs obtained from Eq. (8.26) and Eq. (8.34),

Figure 8 .

 8 Figure 8.14 shows the comparison between the LDs obtained from discrete Eq. (8.26) and

Figure 8 . 15 .

 815 Figure 8.15. Description of the NW growth process. Steps 1 and 2: formation of GaAs nanocrystals on oxide-free Si(111) by droplet epitaxy; Step 3: spontaneous oxidation of the Si surface by air exposure; Step 4: removal of the GaAs nanocrystals by annealing; Steps 5 and 6: formation of Ga droplets inside the holes and subsequent evaporation of the Ga droplets by annealing; Step 7: Ga-catalyzed NW growth. Reprinted with permission from E. S. Koivusalo, T. V. Hakkarainen, M. Guina and V. G. Dubrovskii, Sub-Poissonian narrowing of length distributions realized in Ga-catalyzed GaAs nanowires, Nano Letters 17, 5350-5355 (2017). Copyright (2017) American Chemical Society.

Figure 8 . 16 .

 816 Figure 8.16. Cross-sectional SEM images of Ga-catalyzed GaAs NWs grown for 3 min (a), 6 min (b), 20min (c), 40 min (d), 50min (e), and 80 min (f). Scale bar is 1 µm. Reprinted with permission from E. S. Koivusalo, T. V. Hakkarainen, H. V. A. Galeti, Y. G. Gobato, V. G. Dubrovskii and M. D. Guina, Deterministic switching of the growth direction of self-catalyzed GaAs nanowires, Nano Letters 19, 82-89 (2019). Copyright (2019) American Chemical Society.

Figure 8 .

 8 Figure 8.16 shows cross-sectional SEM images of the NWs after different growth

Figure 8 . 17 .

 817 Figure 8.17. Measured length histograms of Gacatalyzed GaAs NWs at  t 3, 6, 20, 40, 50 and 80 min. Reprinted with permission from E. S. Koivusalo, T. V. Hakkarainen, H. V. A. Galeti, Y. G. Gobato, V. G. Dubrovskii and M. D. Guina, Deterministic switching of the growth direction of self-catalyzed GaAs nanowires, Nano Letters 19, 82-89 (2019). Copyright (2019) American Chemical Society.

Figure 8 . 18 . 2 

 8182 Figure 8.18. Measured variance of the GaAs NW LDs versus the growth time (symbols). Solid linear line shows the corresponding Poisson variance hVt L h P   2  , with V = 62.8. The red line shows theoretical fit by Eq. (8.37) which includes the shadowing effect. The dashed blue line corresponds to the Poissonian growth with shadowing. Reprinted with permission from E. S. Koivusalo, T. V. Hakkarainen, H. V. A. Galeti, Y.

. 37 ) 2 * 0 L 2 *

 37202 This equation shows that, in the absence of Poissonian broadening, the length variation is determined by the two factors -the initial nucleation randomness of the NWs emerging from the substrate (leading asymptotically to the time-independent variance with the characteristic saturation length * L ) and the additional broadening due to the shadow effect (which becomes important for long NWs having the mean length on the order of the shadowing length 0 L ).The line in Fig.8.18 demonstrates the excellent fit to the measured variance obtained from Eq. (8.37) with  6000 nm. The dashed line in Fig. 8.18 corresponds to the Poissonian growth influenced by the shadow effect, with the same 0 L of 6000 nm and shows that the LDs without nucleation antibunching would be even broader than Poissonian in this case. Saturation of the variance in the intermediate growth stage requires that the nucleation step is fast (a short * L ) and the shadowing length 0 L is large. The nucleation variance depends on the preparation technique used for Ga droplets, the type of the substrate/oxide and the nucleation rate of stable NWs emerging from Ga droplets which may be changed by the As flux. In comparison to the previous work [101], by eliminating the Ga predeposition step in the beginning of the NW growth we were able to reduce the incubation time and significantly reduce the saturation level of the variance.

.

  Eq. (9.3) is analogous to the Everett isotherm for adsorption and becomes the Langmuir isotherm for 1   .

( c )

 c Another approach, on the contrary, makes use of equilibrium thermodynamics. Let us illustrate this in the case of mixed group III self-catalyzed III-V NWs, such as A x B 1-x As. in total three independent compositions for the L-S system.

  on nucleus geometry and e  the surface (or edge) energy of the 2D island, introduced in section 3. Note that, in this out of equilibrium situation, we retain the full dependence of   on three independent compositions. Reiss shows that the size and composition of the critical island are the coordinates of the saddle point, solutions of

Figure 9 . 1 .

 91 Figure 9.1. Variation of the formation energy F of an Al x Ga 1-x As nucleus as a function of its size r

. 6 )

 6 In these expressions, at a given temperature, the chemical potentials L i  in the liquid (Eq (6.14)) depend only on liquid composition whereas the chemical potentials III-V pairs in the solid depend only on x. In the regular solution model (see section 6.2), which has been widely used for compound semiconductors, the mixing enthalpy of the solid alloy simply writes the interaction parameter, and the entropy of mixing is

6. 2 )

 2 , which means (in the self-catalyzed case) that its mixing enthalpy contains only the three interaction terms associated to the three pairs of unlike atoms (withL ij the interaction parameter for pair   j i , ), Eq. (9.9) becomes a simple implicit relation linking the compositions of liquid and solid:

Figure 9 . 2 .0. 99 .

 9299 Figure 9.2. Left: (Al,Ga,As) system at T =873 K. Variation of the Al concentration in the critical Al x Ga 1x As nucleus as a function Al concentration in liquid, for various As concentrations As y in the liquid. The dashed curve gives the variation at equilibrium. Reprinted with permission from F. Glas, Comparison of modeling strategies for the growth of heterostructures in III-V nanowires, Crystal Growth & Design 17, 4785-4794 (2017). Copyright (2017) American Chemical Society. Right: (Ga,In,As) system. Variation of the In concentration in the critical In x Ga 1x As nucleus as a function In concentration in liquid at different temperatures for

  elastic energy in a cylinder of radius R cut in a 2D system with the same layer thickness h (for a semi-infinite substrate, D e W 2 is entirely stored in the layer) (Fig.10.1). Then,   1

Figure 10 . 1 .

 101 Figure 10.1. Full symbols, left scale: Variation with layer aspect ratio  of the total elastic energy stored in the NW (without plastic relaxation), for an isotropically elastic system with identical elastic parameters in stem and layer, and for three values of Poisson ratio  . This energy is normalize as indicated, with E the Young's modulus, R the NW radius and 0  the relative misfit between stem and layer. Empty symbols, right scale: Ratio between this energy and the energy in the same volume of a 2D system. Reprinted from Semiconductors and Semimetals, vol. 93, F. Glas, Strain in nanowires and nanowire heterostructures, pp. 79-123, Copyright (2015), with permission from Elsevier.

Figure 10 . 2 .

 102 Figure 10.2. Schematics of the reservoir effect, illustrating the time variations of compositions of vapor, liquid and solid.

  desorb from the droplet, growth interruptions further reduce the amount of species A stored in the droplet before introducing species B and hence sharpen the interfaces. This was demonstrated in the case of GaAs-GaP heterostructures [131]. Technical specificities may however degrade interface sharpness even in this favorable case, for instance the relatively long residence time of residual group V species in the MBE growth chamber after shutter closure, due to poor pumping efficiency [131].

Figure 10 . 3 .

 103 Figure 10.3. STEM-HAADF intensities measured across three Al x Ga 1x As axial insertions grown by MBE for 5 s in a GaAs NW. Each oscillation corresponds to a single ML and the intensity averaged over an oscillation decreases when x increases. Sample A and B were grown by simply switching the Al flux on and off while maintaining the Ga and As fluxes (the latter lower for sample B than for sample A)? Sample C was grown using the pre-filling technique described in the text. Reprinted with permission from G. Priante, F. Glas, G. Patriarche, K. Pantzas, F. Oehler, J.-C. Harmand, Sharpening the interfaces of axial heterostructures in self-catalyzed AlGaAs nanowires: experiment and theory, Nano Letters 16,

Figure 10 . 4 .

 104 Figure 10.4. Left: Composition profiles through a GaAs-Al x Ga 1x As-GaAs heterostructure calculated numerically using the nucleation model with different values of

2 x

 2 scheme (c) but also in the nucleation scheme (d)[105,106]. In this case, for each liquid

  energies of the step bordering the nucleus (in line with the CNT, these are usually taken as energies per unit area, even though the nucleus is only 1 ML high).



  being the energy of a step in contact with the liquid (Fig.11.1(a)). If however the nucleus forms at the TPL, a fraction TPL x of its perimeter is in contact with the vapor (section 6.3), which should have a different energy depending on the type of the nucleus (Fig.11.1(b)). A key point of the model is that this energy is not simply the phase-dependent energy j SV  of the interface between the sidewall of a NW of structure j (ZB or WZ) and vapor: when forming at the TPL, the nucleus indeed acquires an interface with the vapor but this interface "replaces" part of the liquid-vapor that preexisted (this part is indicated by a thick curved line in Fig.11.1(a)), the energy of which should be subtracted from the formation energy. We are thus lead to affect to this portion of the nucleus perimeter a phase-dependent effective energy, a simple expression of which is[START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF]: energy of the liquid-vapor interface, a portion of which is deleted for TPL nucleation (hence the minus sign), and this portion depends on the contact angle  of the droplet[START_REF] Glas | Why does wurtzite form in nanowires of III-V zincblende semiconductors?[END_REF]. Finally, the edge energies of ZB and WZ nuclei (to be inserted in Eq. (11.1)), with standard contributions from those parts of the perimeter in contact with the liquid and phasedependent effective contributions (11.2) from those parts at the TPL, are: interface energies are probably larger than the energies e  of edges in contact with the liquid[START_REF] Glas | Predictive modeling of selfcatalyzed III-V nanowire growth[END_REF]. If so, it is only thanks to the destruction of part of the liquid-vapor interface that TPL nucleation may be favored. Eq. (11.2) predicts that this is more likely to happen for contact angles around 2  .

Figure 11 . 1 .

 111 Figure 11.1. Schematics of nucleation away from the TPL (a) and at the TPL (b), with relevant interface energies indicated. The crystal phase is not specified.

  The control of the lengths of the sections of each phase became a new challenge and spectacular early results were obtained in fabricating periodic ZB/WZ stacks controlled to the ML [170] (Fig.11.2).

Figure 11 . 2 .

 112 Figure 11.2. Periodic crystal phase heterostructure fabricated by Dick et al. in an InAs NW (growth axis is horizontal). The characteristic periods of 2 MLs for WZ (strong contrast) and 3 MLs for ZB (lower contrast). Scale bar is 20 nm. Reprinted with permission from K. A. Dick, C. Thelander, L. Samuelson, P. Caroff, Crystal phase engineering in single InAs nanowires, Nano Letters 10, 3494-3499 (2010).Copyright (2010) American Chemical Society.

Table 4 .

 4

		1. Parameters of GaAs used for modeling the NW morphology
	) 111 (	B
	Parameter

Table 4

 4

	.1, where "Calc."

Table 5 .

 5 1. Parameters of GaAs NWs used for the fits in Figure 5.2

	Pore size		f	R c	0 / R	0 t (min)	1 t (min)	* t (min)
	0 2R (nm)
	125	0.522	3.73	22	44	44
	150	0.522	3.73	27	46	46.5
	175	0.522	3.73	37	47	57.2
	200	0.522	3.73	43	47	60
	225	0.522	3.73	53.5	47	69

  now relates to pure Ga droplet and hence cannot regulate the overall diffusion flux of Ga adatoms.Because of that, there is no steady state solution to Eq. (6.28) for R and instead the NW

	Here, the value of	) 1 (
	top radius becomes time-dependent. When			1	, Eq. (6.36) yields	dL	/	dt			5	5 v	, that is, the
	NW growth rate is simply proportional to the total As influx, including re-emitted flux. If we
	also assume	p		1	and		3	3 v 	h	exp(		3	)	/		3	, Eq. (6.28) yields the result of Ref. [25]
	dR		a		b	,												(6.39)
	dt				R
																		.38)

3



  Ga 1x As MLs were richer in Al. This produced a spectacular decrease of interface width to about 2 MLs [108] (Fig. 10.3, sample C).

	Other methods of manipulation of the liquid composition have been devised to improve interface
	abruptness [147].

satisfactory, but the interface widths measured by Priante et al. in samples with simple Ga-Al switches were still about 10 MLs (for rather thick NWs) 108] (Fig. 10.3, samples A and B). The interface sharpness can be improved by implementing appropriate growth strategies. In this case, Priante et al. performed a growth interruption after GaAs growth, during which the droplet was pre-filled with Al so that the first subsequently grown Al x

  Note also that the rate of ML formation may depend on NW radius (this is the case for nucleation-mediated growth; see section 3); the rates of input/output via diffusion or desorption also depend on NW radius.Solving system[START_REF] Plante | Growth mechanisms of GaAs nanowires by gas source molecular beam epitaxy[END_REF] yields the time dependence of the composition of the liquid and hence of the solid via the   the total number of atoms in the liquid. In separate experiments, the growth rate  was checked to be independent of solid composition for long sections of Al x Ga 1x As NWs, and this was assumed to hold during heterostructure formation. For an Al x Ga 1 -

	L i independent compositions characterizing a given the number of atoms of each group (i.e. III or V) in a ML, and the time-independent function relating solid composition to liquid compositions, discussed liquid; see section 9.2), ML N   y x in section (9.4). In turn,   in i dt dN is the net rate at which species i is added to the droplet, or removed from it, by any process except incorporation to the solid. Each rate may depend not only on the corresponding vapor flux i  but also on the composition of the liquid, which governs desorption and diffusion. All these contributions were discussed in the previous sections. If the NW radius and the droplet contact angle are constant, the numbers i N relate straightforwardly to concentrations i y . A time-dependent geometry can be accounted for by x relation. One is usually more interested in the variation of the NW composition along the NW axis. For this, in principle, the absolute growth rate  and input rates   in i dt dN are not needed, but their ratios must be known. Discrete versions giving the various compositions at each ML step are sometimes used [107]. As will be seen below, additional hypotheses may reduce the number of equations. We now briefly illustrate various ways of modeling the composition profile in heterostructures, either numerically or analytically, and using any of the four types (a)-(d) of liquid-solid composition relationship reviewed in section 9.4.   L Al Al Al N I y x g dt y d     , (10.3) adding extra equations. y with Al I the Al current (atoms per unit time) entering the droplet, g a geometrical factor and

.2c)

The first terms on the right hand side are the rates at which each species is removed from the liquid by incorporation into the solid, with  the rate of ML formation (which depends on liquid composition y , this vector representing the System (9) (or its equivalent in simpler or more complex cases) can be solved in a purely numerical fashion. This was done by Periwal et al. for Ge x Si 1x heterostructures using a type (b) composition relationship [107]. The authors show that the Si/Ge x Si 1x and Ge x Si 1x /Si are always asymmetric, irrespective of the Ge concentration or NW diameter. Another numerical solution was developed by Priante et al. for heterostructures based on self-catalyzed Al x Ga 1x As alloys [108]. Within the type (c) equilibrium scheme, a single concentration, e.g Al y , defines the state of the liquid at a given temperature. From Eq. (10.2a), it obeys equation: L N x As barrier inserted in a GaAs NW, post-growth ML-resolved STEM-HAADF composition measurements (see section 6.1 and circles in Fig. 10.4, left) yielded precise values of growth rate and of total number of Al

Table 10 .

 10 1. Difference of cohesive energy coh E  between the ZB and WZ structures for various materials, in meV per atom pair. Second row: ab initio calculations from references [158] (a) and [159] (b) (for GaSb, see ref. [151]). Third row: estimate based on the measured stacking fault energies collected in ref. [160] (see ref. [161]). Fourth row: estimated errors on third row values.

	Material	GaP	InP	GaAs InAs GaSb InSb	ZnS	ZnSe ZnTe	Si
	E 	coh	18.4 a	5.8 b	24.0 a 12.4 b	29.8	16.2 b	6.2 b	10.6 b 12.8 b 23.4 b
	E 	coh	21.5	9	22.5	15	26.5	19	≤ 3	6.5	8	27.5
			1.5	1.5	3.5	1.5	3.5	2		0.5	1	3.5

J. Schmidtbauer, M. Borg, K. Storm, K. Deppert, J. Johansson, Length distributions of nanowires growing by surface diffusion, Crystal Growth & Design 16, 2167-2172 (2016). Copyright (2016) American Chemical Society.

Acknowledgments

The long-established collaboration between the authors is currently supported by InternationalResearch Project "Physics of nanostructures and innovative devices based on compound semiconductors" (PHYNICS) established between Centre National de la Recherche Scientifique (CNRS) and the Russian Foundation for Basic Research (RFBR). The authors acknowledge fruitful discussions with their colleagues Jean-Christophe Harmand, Gilles Patriarche, Fabrice Oehler, Federico Panciera Giacomo Priante, George Cirlin and Nikolai Sibirev. VGD thanks the Russian Science Foundation for financial support under the Grant 19-72-30004. FG acknowledges financial support from Agence Nationale de la Recherche within projects ESPADON ANR-15-CE24-0029 and NanoMAX 10-EQPX-0050, and from the European Commission within Marie Skodowska-Curie Innovative Training Network INDEED (No 722176).

atoms delivered to the system, from which Al I is easily obtained, knowing the delivery time.

The observed NW geometry gave access to parameters g and Turning now to analytical solutions, the simplest of these is derived within the regular growth scheme (a). Dubrovskii showed that, for simple A-B vapor flux commutation (without growth interruption), the concentrations of A and B in the solid vary exponentially at the interface (and independently of each other when a foreign catalyst is used). This solution can easily incorporate growth interruptions, which are confirmed to sharpen the interfaces in mixed group V systems (as discussed in section 10.2).

Within the nucleation scheme (d), numerical solutions of system (10.2), as discussed above, may easily handle the accurate thermodynamic descriptions of the liquid available in the literature for some systems, in particular, the self-catalyzed ones [START_REF] Leshchenko | Nucleation-limited composition of ternary III-V nanowires forming from quaternary gold based liquid alloys[END_REF]109,[111][112][148][149][150].

However, as mentioned in section 9.4 and discussed in depth by Dubrovskii et al.

[106], for many III-V alloys at least, using such multi-parameter descriptions may be unnecessary given the specificities of these systems. Instead, analytical liquid-solid relationships based on the regular solution model, such as discussed in section 9.4, may lead to analytical descriptions of the axial composition profile in the NW.

with only the single ML in bold wrongly stacked. Of course, forming a WZ segment requires that all successive form in WZ position.

(

3) Ordering at the solid-liquid interface

The only other explanation of WZ formation seems to be that of Algra et al. [168]. These authors propose that WZ formation in Au-catalyzed InP NWs results from the peculiar arrangement adopted by the heavy Au and In atoms in the liquid in the very vicinity of the top facet of the NW. Such liquid ordering is a well-known phenomenon [169]. Typically, the density of the liquid is modulated over a few atomic layers in the direction normal to the interface but there is also some solid-like lateral ordering, especially in the first layer.

Based on surface X-ray diffraction measurements on macroscopic interfaces between solid InP and (Au,In,P) liquids supplemented by ab initio calculations, the authors were able to show that, in the liquid, the heavy atoms close to the solid tend to occupy specific positions.

They argue that, when a new InP ML forms, this hinders the atom to occupy the ZB positions and channels them to the WZ positions.

This mechanism is not specific to NWs. It should produce WZ crystals even if growth of wide crystals is carried out from the appropriate liquids. However, such growth has not been carried out in these experiments. Somewhat surprisingly, this interesting alternative route to WZ formation during VLS growth has not been further investigated.

Crystal phase heterostructures

Polytypism enable a NW-specific type of heterostructure very different from those that associate different materials or different alloy compositions (section 10). Instead, these heterostructures combine different crystalline structures of the same material. Such ZB/WZ mixed structures have long been observed to form spontaneously and more or less randomly in III-V NWs [156] and for many years, it was rather the fabrication of single phase NWs that was a challenge. One of the interests of such heterostructures is that the interfaces may be ideally understanding of the interplay between these various parameters and of the role of the chemical potential, has however not been achieved.