# Along the edges of crystal growth: planar, nanostructured and van der Waals epitaxy

Fabrice Oehler

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Fabrice Oehler. Along the edges of crystal growth: planar, nanostructured and van der Waals epitaxy. Physics [physics]. Université Paris - Saclay, 2023. tel-04415900

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# Along the edges of crystal growth: planar, nanostructured and van der Waals epitaxy

Habilitation à diriger des recherches
de l'Université Paris-Saclay

présentée et soutenue à Palaiseau,
le 8 Novembre 2023, par

**Fabrice OEHLER**

## Composition du jury

**Evelyne GIL**

Professeure, Institut Pascal

**Guillaume SAINT-GIRONS**

Directeur de recherche, INL

**Fabrice SEMOND**

Directeur de recherche, CRHEA

**Maria TCHERNYCHEVA**

Directrice de recherche, C2N

**Thierry BARON**

Directeur de recherche, LTM

**Xavier WALLART**

Directeur de recherche, IEMN

Rapportrice

Rapporteur

Rapporteur

Examinatrice

Examinateur

Examinateur
Acknowledgments

To my parents and their efforts into my education.

To my brother for compelling memories of my first experiments.

To my past and present colleagues, for more memories on many experiments.

To my kids, surely the two best outcomes of all my experiments.

To my wife, who listens patiently to my silence and yet understand it all.

Many thanks to Frank Glas and the Jury members for their careful reading of this manuscript.
# Contents

<table><tr><td><b>Acknowledgments</b></td><td><b>2</b></td></tr><tr><td><b>Structure and focus of the manuscript</b></td><td><b>5</b></td></tr><tr><td><b>1 Context</b></td><td><b>6</b></td></tr><tr><td>1.1 On the importance of material research for semiconductors</td><td>6</td></tr><tr><td>1.2 Manufacturing costs of solid-state semiconductors</td><td>6</td></tr><tr><td>1.3 Environmental cost of solid-state semiconductors</td><td>6</td></tr><tr><td><b>2 Practical constraints in planar epitaxy</b></td><td><b>8</b></td></tr><tr><td>2.1 Basic concepts</td><td>8</td></tr><tr><td>2.1.1 Substrates for epitaxial growth</td><td>8</td></tr><tr><td>2.1.2 Typical defects in planar epitaxy</td><td>9</td></tr><tr><td>2.2 Facet and growth in crystalline solids</td><td>11</td></tr><tr><td>2.3 Maintaining a flat surface during growth (GaN/InGaN by MOCVD)</td><td>14</td></tr><tr><td>2.3.1 Miscut and dislocations: c-GaN homoepitaxy on bulk c-GaN</td><td>14</td></tr><tr><td>2.3.2 Miscut and step bunching: m-InGaN on bulk m-GaN</td><td>16</td></tr><tr><td>2.3.3 Roughness from extended defects in active InGaN layers</td><td>18</td></tr><tr><td>2.4 Becoming flat: coalescence in planar epitaxy (GaN MOCVD)</td><td>19</td></tr><tr><td>2.4.1 Coalescence-based defect reduction techniques for c-GaN (0001)</td><td>20</td></tr><tr><td>2.4.2 Coalescence-based defect reduction techniques for a-GaN (1120)</td><td>20</td></tr><tr><td>2.5 Polar and twinned domains (GaAs/Ge by MBE)</td><td>22</td></tr><tr><td>2.5.1 Polarity mismatch and solution for GaAs/Ge (100) epitaxy</td><td>22</td></tr><tr><td>2.5.2 Twinning during GaAs/Ge (111) epitaxy</td><td>23</td></tr><tr><td>2.6 Conclusion</td><td>26</td></tr><tr><td><b>3 Doing without crystalline substrate: oriented growth on amorphous surfaces</b></td><td><b>27</b></td></tr><tr><td>3.1 On the role(s) of the substrate</td><td>27</td></tr><tr><td>3.2 Oriented low temperature crystallization of amorphous Ge on silica</td><td>28</td></tr><tr><td>3.2.1 Metal Induced Crystallization of amorphous Ge</td><td>28</td></tr><tr><td>3.2.2 Oriented Aluminium induced crystallization of amorphous Ge</td><td>29</td></tr><tr><td>3.2.3 In situ observations</td><td>30</td></tr><tr><td>3.3 GaAs epitaxy on Ge(111) pseudo-substrates obtained by ALILE</td><td>34</td></tr><tr><td>3.4 Growth of GaN nanowires on glass</td><td>35</td></tr><tr><td>3.5 Conclusion</td><td>37</td></tr><tr><td><b>4 Getting rid of coalescence: nanowire epitaxy</b></td><td><b>38</b></td></tr><tr><td>4.1 Context</td><td>38</td></tr><tr><td>4.1.1 Axial strain, heteroepitaxy and heterostructures</td><td>38</td></tr><tr><td>4.1.2 Radial strain and heterostructures</td><td>40</td></tr><tr><td>4.2 Vapor Liquid Solid growth</td><td>43</td></tr><tr><td>4.2.1 Self-catalyzed III-V nanowire growth by MBE</td><td>43</td></tr><tr><td>4.2.2 Group III flux to the VLS catalyst</td><td>45</td></tr><tr><td>4.2.3 Group V flux on the VLS catalyst</td><td>48</td></tr><tr><td>4.3 Sidewall passivation of self-catalyzed GaAsP nanowires</td><td>54</td></tr><tr><td>4.3.1 Sidewall passivation and radial growth</td><td>54</td></tr><tr><td>4.3.2 Unintentional GaAsP shell growth</td><td>55</td></tr><tr><td>4.3.3 Homogeneous GaAsP alloys</td><td>56</td></tr></table>
<table><tbody><tr><td>4.4</td><td>Axial heterostructure and catalyst thermodynamics</td><td>58</td></tr><tr><td>4.4.1</td><td>Expressions of thermodynamic potentials</td><td>58</td></tr><tr><td>4.4.2</td><td>Group V concentration in binary self-catalyzed nanowire catalyst</td><td>60</td></tr><tr><td>4.4.3</td><td>Axial group V heterostructures in self-catalyzed nanowires</td><td>62</td></tr><tr><td>4.4.4</td><td>Axial group III heterostructures in self-catalyzed nanowires</td><td>63</td></tr><tr><td>4.4.5</td><td>Catalyst engineering in group III catalyzed nanowires</td><td>64</td></tr><tr><td>4.5</td><td>Conclusion</td><td>66</td></tr><tr><td>5</td><td>Growth without covalent bonding: van der Waals epitaxy</td><td>68</td></tr><tr><td>5.1</td><td>Context</td><td>68</td></tr><tr><td>5.1.1</td><td>The galaxy of 2D materials</td><td>68</td></tr><tr><td>5.1.2</td><td>Layered crystals, van der Waals gap and allotropes</td><td>69</td></tr><tr><td>5.2</td><td>Van der Waals epitaxy</td><td>70</td></tr><tr><td>5.2.1</td><td>Concept</td><td>70</td></tr><tr><td>5.2.2</td><td>Case study: WSe<sub>2</sub> on Se-terminated GaAs(111)B</td><td>71</td></tr><tr><td>5.2.3</td><td>Moiré superlattices</td><td>73</td></tr><tr><td>5.2.4</td><td>Limits of van der Waals epitaxy</td><td>74</td></tr><tr><td>5.3</td><td>Current work: 3R and 2H polytypes of bi-layer WSe<sub>2</sub></td><td>75</td></tr><tr><td>5.3.1</td><td>CVD growth of WSe<sub>2</sub> mono- and bi-layer on amorphous SiO<sub>2</sub> substrates</td><td>75</td></tr><tr><td>5.3.2</td><td>Crystal structure and optical signatures of bi-layer WSe<sub>2</sub> polytypes</td><td>76</td></tr><tr><td>5.4</td><td>Future work: WSeTe alloys</td><td>77</td></tr><tr><td>5.4.1</td><td>Crystal structure of WTe<sub>2</sub></td><td>77</td></tr><tr><td>5.4.2</td><td>Electronic structure of WTe<sub>2</sub></td><td>78</td></tr><tr><td>5.4.3</td><td>Crystal and electronic structure of ternary WSeTe alloy</td><td>80</td></tr><tr><td>5.4.4</td><td>Growth challenge of WSeTe alloys</td><td>80</td></tr><tr><td>5.4.5</td><td>Janus WSeTe monolayers</td><td>81</td></tr><tr><td>5.5</td><td>Conclusion</td><td>82</td></tr><tr><td colspan="2">Final remarks</td><td>83</td></tr><tr><td>6</td><td>References with authorship</td><td>84</td></tr><tr><td>7</td><td>Curriculum Vitæ & Publications</td><td>87</td></tr><tr><td>7.1</td><td>Curriculum vitæ</td><td>87</td></tr><tr><td>7.2</td><td>Publication list with supervised PhD or post-doc</td><td>88</td></tr><tr><td>7.2.1</td><td>Publication with supervised PhD</td><td>88</td></tr><tr><td>7.2.2</td><td>Publications with supervised post-doc</td><td>89</td></tr><tr><td colspan="2">A Complete list of publications by year</td><td>90</td></tr></tbody></table>
# Structure and focus of the manuscript

## Structure

In this memoir, I attempt to present some aspects of crystal growth for semiconductor applications. Elements of conventional planar homo- and heteroepitaxy are presented in the first chapter, whereas the later chapters deal with the practical consequences of circumventing some of the defining traits of epitaxial growth. Namely:

* getting rid of the crystal lattice of the substrate.

* purposely doing without the coalescence of individual crystallites.

* removing the covalent bonds at the initial growth interface.

## References with and without authorship

For the purpose of this memoir, it is of interest to dissociate bibliographic references to which I have directly contributed from the immense body of work in the field.

* References in green such as [Oehler2018] are used to cite work with authorship. All references with authorship are compiled in chapter 6 ‘References with authorship’ (page 84) and listed in alphabetic order.

* As a complement, the complete list of publication with authorship, including those not cited in this memoir, is listed in Annex A (page 90) in reverse chronological order.

* References to work without authorship are marked with a small blue number and referenced as footnote. These appear at the bottom of the relevant page, such as Nakamura *et al.*<fnref n="1" />.

While this system provides an easy visual guide to identify references with and without authorship, the footnote format imposes that the same reference without authorship cited at different locations of this memoir will appear with different numbering.

## Highlights and shadows

In each section of this manuscript, I have tried to based the discussion using references to which I have directly contributed.

The positive result of this approach is the particular focus, which showcases some of the challenges in the field from a personal perspective.

The main drawback of this method is the many blind spots, which are completely left out from this memoir. Please bear in mind that this manuscript is by no mean an exhaustive review, despite the logic in its organization.

<footnote n="1">¹S. Nakamura. "Background story of the invention of efficient blue InGaN light emitting diodes (Nobel Lecture)". *Annalen der Physik* 527.5-6 (2015), pp. 335-349. DOI: 10.1002/andp.201500801.</footnote>
Chapter 1

Context

1.1. On the importance of material research for semiconductors

Due to their extreme tunability in electrical transport and physical interactions with light, heat or strain, solid-state semiconductors form the bulk of the active materials in current electronic devices. While we often reduce "electronics" to the assembly of miniature integrated circuits (ICs), the applications of semiconductors extend to power devices (power conversion) and opto-electronics in small (lasers, LEDs, CCDs) or large (solar panels, displays) package. With solid-state lighting becoming ubiquitous, solar panels spreading on rooftops and house appliances turning to “smart” objects, it is now hard for any technological device to escape solid-state semiconductor physics and their synthesis as materials.

While politics can be lured by the belief that material research should be driven by clear technological objectives, it is the opposite which offers the most prospects. Compact blue lasers used in DVD disks and white-light LEDs can only exist today because the planar growth of Gallium Nitride (GaN) crystals and related compounds was mastered by Nakamura and Ammano in the early 1990s, at a time when GaN was a challenging material, with potential but uncertain technological outcomes<fnref n="2" />.

1.2. Manufacturing costs of solid-state semiconductors

Solid-state semiconductors are usually fabricated in expensive and large devices named ‘growth reactors’ or ‘deposition chambers’. From the manufacturing point of view, *amorphous* semiconductors usually have an edge in ‘large area’ devices (displays, solar cells) due to their lower fabrication costs, while *crystalline* materials are usually more expensive and found in more compact applications (laser, LEDs). However, the commercial success of any particular semiconductor is actually that of a particular technology, which balances the cost of all its components against its total efficiency.

The success of crystalline Si as the active layer of commercial solar cells (i.e. a large area device) is a important
reminder that industrial scaling and other market factors can significantly alter the retail value or manufacturing
costs of any material. For the scientific perspective, this means that material properties are probably the most
robust criteria to qualify as a promising technology, while the estimated cost may vary widely.

1.3. Environmental cost of solid-state semiconductors

While the iconic semiconductor silicon is one of the most abundant chemical elements in the earth crust, it takes a great deal of energy to refine the starting raw silicates into the defect free, ultra-pure crystalline wafers used in the industry. The situation is worse from many other solid-state semiconductors, as most of their composing chemical elements (metal/non-metal) originate from dilute raw deposits, often collected as by-products of the metal mining industry. For example, the abundance of Ge, In, Ga and As is comparable to that of the ‘rare-earth’ elements. That of Te (studied here in bi-dimensional WSeTe alloys) is of the same order of Au.

<footnote n="2">²S. P. Shuji Nakamura and G. Fasol. *The Blue Laser Diode, The Complete Story*. Springer Berlin, Heidelberg, 2000. DOI: 10.1007/978-3-662-04156-7.</footnote>
We note here that abundance is not correlated to market value (in 2023, 100g of 99.9% Te is ~400€, 100g of Au is ~6000€) so that the commercial extraction of Te will require more concentrated deposits than Au.

<figure><img src="image_3.png" /><figcaption>**Figure 1.1:** Abundance of chemical elements in the upper earth continental crust, extracted from 'US Interior and US Geological Survey' (http://geopubs.wr.usgs.gov/fact-sheet/fs087-02/).</figcaption></figure>

Summing up the energy used by the extraction and refining processes, with that dissipated in our growth reactors (high temperature, high purity, vacuum, ...), followed by more fabrication steps (performed by other power-hungry tools in high-maintenance clean-room environments), our final solid-state semiconductors turn out quite environmentally-expensive, most of the time with little recycling perspectives.

It is thus important that we understand and design our crystalline semiconducting materials in the best possible way.
Chapter 2

Practical constraints in planar epitaxy

In this chapter, I attempt to list the some of the constraints relative to the planar growth of one crystal onto
another. Instead of the typical approach based on defect reduction technique which will depend on various
lattice-, thermal-, polarity- or even symmetry-mismatch, this section is organized around the following question:

"How to maintain a planar growth front in the presence of various defects ?"

Examples in this chapter are mainly extracted from III-V semi-conductors, mostly Gallium Nitride, Indium Gallium Nitride alloys and Gallium Arsenide.

This particular focus on the *planar* character of the grown layers eludes many technologically-important aspects such as sheet carrier mobility, carrier concentration or quantum efficiency, which are all critical parameters for the success of any subsequent device fabrication.

**2.1. Basic concepts**

**2.1.1 Substrates for epitaxial growth**

Planar processing and planar growth of crystalline semiconductors are the backbones of the current electronic
industry. Here, the starting substrates are typically solid monocrystals shaped as thin disks called ‘wafers’, on
which one or several subsequent crystal growth steps are performed to obtain the desired structure. The active
material which can be the same as the starting substrate, ‘homoepitaxy’, or of a different nature, ‘heteroepitaxy’.

<figure><img src="image_4.png" /><figcaption>**Figure 2.1:** Si(100) wafers of different diameters.
(from https://www.si-cnx.com/products/wafer-speciality-materials/dummy-wafer/).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 2.2:** Typical structure of a blue InGaN LED grown on sapphire (0001), *Nobel lecture, S. Nakamura (2015)*.</figcaption></figure>

Due to their massive usage in the industry, Si(100) monocrystalline wafers (FIG. 2.1) can reach extraordinarily
large diameters (up to 450 mm) and low prices (4" ~10€), with extreme purity and low defect density. This is
not the case for other semiconductors or substrates (Ge, Al₂O₃, SiC, III-V, II-VI, ...) for which the base wafer price is much higher (4" >100 €) and the diameters much smaller (most III-V are still sold as ~ 2").

As homoepitaxy seems much simpler in notion than heteroepitaxy, the growth of GaN(0001) on sapphire Al₂O₃ (0001) substrate for the blue LED technology may first appear as a bizarre choice (see Fig. 2.2, from the Nobel lecture of S. Nakamura<fnref n="3" />). However monocrystalline GaN substrates were not available from a long time (first commercial appearance in late 2000s), and even today they are prohibitively expensive compared to the retail price of GaN LEDs or lasers grown on sapphire.

Note that typical devices, even simple LEDs, consist in a coherent assembly of several crystalline layers, each
having its own chemical composition and distinct role (active medium, contact, defect reduction, buffer layer).

2.1.2 Typical defects in planar epitaxy

As seen in FIG. 2.2, the commercial growth of GaN is usually performed on another type of crystal, sapphire Al₂O₃(0001), silicon Si(111) or silicon carbide SiC(0001). As there is no reason for an arbitrary couple substrate/epilayer to share the same crystal structure (size, symmetry and geometry), some defects develop naturally to reduce the strain and adapt the different crystal lattices at the substrate/epilayer interface.

<figure><img src="image_6.png" /><figcaption>**Figure 2.3:** Typical defects in planar epitaxy: point defects (0D), dislocations (1D) and domain wall boundaries (2D).</figcaption></figure>

* The rare case of stress-free perfect epitaxy is shown Fig. 2.3.a, in which the substrate and epilayer crystal lattice sizes and geometries are identical (or very close). Even if the two lattices naturally coincide over all their interface, some point defects (misplaced, missing or extra atoms) can still form in the epilayer volume or at the interface due to inadequate growth parameters.

In practice, such coherent growth is commonly observed for epilayers with relatively small lattice mis-
match and thickness. In this situation, the stress energy due to the (small) lattice mismatch is fully
accommodated in the epilayer volume via elastic strain. Above a *critical* thickness, the strained epilayer
accumulates too much energy and the system relaxes through the formation of dislocation networks at
the substrate/epilayer interface (see Fig. 2.3.b).

* A typical case of coherent but defective epilayer growth is shown Fig. 2.3.b. Here the unit cell of the epilayer is chosen larger than that of the substrate but the same happens in the reverse situation. Missing unit cells arranged in line known as a misfit dislocation segment (MD, solid black line) and strain are used to join the two lattices. A more realistic description of dislocation usually requires the addition of theading dislocation (TD, red solid line) segments, which form by curving the dislocation line towards the epilayer surface, to limit the total line energy. As each individual TD core holds a large density of broken bonds and related defects, the emergence of TD through an active layer volume (light green) usually degrades significantly the electrical and optical properties of the device.

<footnote n="3"><sup>3</sup>S. Nakamura. "Background story of the invention of efficient blue InGaN light emitting diodes (Nobel Lecture)". *Annalen der Physik* 527.5-6 (2015), pp. 335–349. DOI: 10.1002/andp.201500801.</footnote>
* A common case of conflicting epitaxial relationships is shown Fig. 2.3.c. We depict here a simplistic rectangular epilayer unit cell growing over a cubic substrate along two non-equivalent orientations (black arrows) to illustrate the case of conflicting epitaxial relationships. This problem occurs when two such non-equivalent crystals come into contact. The resulting 2D wall boundary which form at their interface is usually defective (broken bonds) and may extend through all the epilayer thickness, potentially affecting every layers above, including the active medium. Polarity reversal in III-V semiconductors is common source of such defect.

* An interesting variant of the 2D defective wall boundaries is the case of coherent 2D interfaces, such as stacking faults (SF). Some SF variants such as basal plane stacking fault (BSF) may not even classify as 2D defect because they do not present any broken bonds over the 2D plane, only at the 1D boundary with the matrix (i.e. partial dislocations). BSFs typically occur in case of weak polymorphism when the different crystal structures are closely related (phase 1: ABC-ABC, phase 2: AB-AB), so that we can insert a repeat of one phase into the other (ABC-AB-ABC). While BSF may not present any broken bond, they may still show specific optical or electronic signatures due to carrier confinement or scattering.

* The case of fully incoherent growth (not shown) is found when the lattices of the growing crystal and substrate are so different that the grown crystals are randomly oriented. Such random interfaces are very defective and may combine various types of defects.

* Partial crystalline alignments along one direction also exist, such as fiber texture or axiotaxy<fnref n="4" />.

<figure><img src="image_7.png" /><figcaption>**Figure 2.4:** (a). Lattice constant and band-gap of several semiconductor families: cubic III-V, cubic II-VI, hexagonal III-N, from *Solid-State Physics, H. Ibach, H. Lüth (Springer Verlag, 2009)* with corrected InN band gap from *Monemar (2005)*. (b) Effect of threading dislocation density on the normalized LED efficiency fabricated from various III-V compounds, from *Nakumara Nobel lecture (2015)*.</figcaption></figure>

FIG. 2.4.a shows the iconic lattice-band gap diagram of several semiconductor families: III-V, II-VI, III-N (including a more correct value for the InN band gap, from Monemar et al.<fnref n="5" />). With some notable exceptions ($a_{Ge} = 5.658Å$, $a_{GaAs} = 5.653Å$, $a_{AlAs} = 5.661Å$), most semiconductor crystals present different lattice parameters, with typical couples showing lattice-mismatch of several percents. Most attempts at the direct growth of a pure compound semiconductor crystal onto another will thus lead to defective material, with dense dislocation networks and high threading dislocation densities.

Note that threading dislocations do not affect all semiconductors in the same way. In FIG. 2.4.b from Nakamura<fnref n="6" />, we compare the normalized efficiency of light emitting diodes (LEDs) grown on various III-V substrates with different threading dislocation densities. We see that III-As compounds are the most sensitive to dislocation ($\sim 10^5$ cm⁻² at mid efficiency), followed by III-P and GaN compounds about two decades higher ($\sim 10^7$ cm⁻²). The case of InGaN ($\sim 10^9$ cm⁻²) is remarkable but also quite unique. This extreme resilience of InGaN active layers to threading dislocations is one reason for the commercial success of InGaN-based (blue) LEDs and lasers, despite the enormous lattice mismatch between the sapphire substrate ($a_{Al_2O_3} = 4.785Å$), silicon substrate ($a_{Si(111)} = a_{Si}/\sqrt{2} = 3.83Å$) and the GaN (0001) base layer ($a_{GaN} = 3.186Å$).

<footnote n="4">⁴C. Detavernier, A. S. Özcan, J. Jordan-Sweet, E. A. Stach, J. Tersoff, F. M. Ross, and C. Lavoie. "An off-normal fibre-like texture in thin films on single-crystal substrates". *Nature* 426.6967 (2003), pp. 641–645. DOI: 10.1038/nature02198.</footnote>

<footnote n="5">⁵B. Monemar, P. Paskov, and A. Kasic. "Optical properties of InN-the bandgap question". *Superlattices and Microstructures* 38.1 (2005), pp. 38–56. DOI: 10.1016/j.spmi.2005.04.006.</footnote>

<footnote n="6">⁶S. Nakamura. "Background story of the invention of efficient blue InGaN light emitting diodes (Nobel Lecture)". *Annalen der Physik* 527.5-6 (2015), pp. 335–349. DOI: 10.1002/andp.201500801.</footnote>
## 2.2. Facet and growth in crystalline solids

A crystalline solid is the three-dimensional (3D) ordering of individual atoms, which assemble to repeat an identical unit cell along three non-colinear directions in space to form arbitrarily large volumes. With different type of atoms preferring different bond lengths and geometries, there exists a lot of variety in crystal structures, which all try to minimize their formation energy in some way. However, this variety is nonetheless constrained to one of the 230 3D space groups<fnref n="7" />, which simply translates the mathematical fact that there is a finite number of arrangements which can repeat arbitrarily by translation in 3D space.

Given any 3D regular crystal, broken bonds (covalent or van der Waals) exist at the interface and are associated with an additional energy cost, which is typically referred as the Gibbs excess surface energy. Depending on the crystal structure, one can identify specific crystal planes which minimize the total energy of these broken bonds (density, orientation) per unit of surface area for a given volume. When approximated to constant value, this (positive) excess surface energy is at the origin of the spherical shape of liquid droplet (i.e dense isotropic phase). For crystalline solids (i.e dense anisotropic phases), this value now strongly depends on the local orientation of the surface, which we define mathematically as the orientation of surface normal $\vec{n}$ in 3D space, and that we may express in reciprocal space coordinates $\vec{n}(H, K, L)$.

The fact that reciprocal space, not direct space, is the ‘natural’ space for expressing surface normal may first appear odd. However one can remark that Miller indices are a universal and compact notation for any crystallographic planes in 3D, with a simple vector representation in reciprocal space (‘the normal of plane (H,K,L) in real space is colinear to the [H,K,L] vector in reciprocal space’).

The following simplistic 2D model is adapted from "Physique de la croissance crystalline" from Vilain & Pinpinelli<fnref n="8" />. It is based on square unit cell of length $a$ connected by a single atomic bond in the $\vec{x}$ and $\vec{y}$ directions, FIG. 2.5(a). Here in 2D, the angle $\theta \in [0, 2\pi]$ can be used to uniquely define the surface normal, $\vec{n} = n(\theta)$.

<figure><img src="image_8.png" /><figcaption>**Figure 2.5:** (a) Simplistic 2D crystal made of identical square unit cells, bonded exclusively along the $x$ and $y$ directions. (b) Resulting Wulff plot of the surface energy and the resulting equilibrium shape (black line), limited by the tangents (red lines) of all $\vec{\Gamma}(\theta)$.</figcaption></figure>

For any segment L connecting two summits of any arbitrary squares in the lattice, the number of broken bonds $N_b$ along the $x$ and $y$ directions along the segment length L is simply:

$$ \left\{ \begin{aligned} N_b x &= \frac{L |\cos(\theta)|}{a} \\ N_b y &= \frac{L |\sin(\theta)|}{a} \end{aligned} \right. \qquad (2.1) $$

In this simple model, all broken bonds have the same energy cost, irrespectively of their orientation. The

<footnote n="7">⁷see: http://img.chem.ucl.ac.uk/sgp/large/sgp.htm</footnote>

<footnote n="8">⁸J. Vilain and A. Pimpinelli. *Physique de la croissance cristalline*. Ed. by Eyrolles. 1999.</footnote>
surface energy $\Gamma(\vec{n}) = \Gamma(\theta)$ is simply proportional to the density of broken bond number can be simply expressed as follows, setting $L = 1$ to normalize the segment length.

$$ \Gamma(\theta) \propto |\cos(\theta)| + |\sin(\theta)| \qquad (2.2) $$

The polar plot of $\Gamma(\theta)$ shown Fig. 2.5.b is known as the Wulff plot of the surface energy. It allows one to determine the shape of the crystal which minimizes its total surface energy by the ‘Wulff’s construction’, in which we determine the inner envelope of the polar plot (thick black square) as the convex boundary delimited by all tangents of $\vec{\Gamma}(\theta)$ (red color). This is known as the Wulff’s theorem (stated without proof by Wulff in 1901 and later proved by Herring in 1953⁹). At constant number of unit cell (i.e given volume), our simplified 2D system will thus try to minimize its total interfacial energy by adopting a square shape limited by the $\vec{n}_x$ and $\vec{n}_y$ normals. The simple 2D square symmetry in real space is also apparent in reciprocal space and the resulting equilibrium shape is limited by the normals of the (01) and (10) planes, respectively. This result is obviously restricted to the continuous limit of $\theta$, which requires a large number of unit cell.

Although the above "broken bond" model in 2D may appear simplistic, Yu and Flodström<fnref n="10" /> have extended it to full 3D and realistic bond configurations, to a point where it can model the surface energy of pure compounds (i.e symmetric bonds between identical chemical element X-X) in cubic configurations (simple, face centered, body and diamond cubic), which applies to most metals and but also diamond cubic semiconductors such as (Fd3m) Si and Ge.

<figure><img src="image_9.png" /><figcaption>**Figure 2.6:** (a) 3D Wulff plot from Yu and Flodström for diamond cubic lattice, exposing the (111) planes with the lowest surface energy. (b) Crystal structure of cubic Si and an example crystal shape, tweaked to display all the (100), (110) and (111) planes (F. Oehler, personal notes).</figcaption></figure>

In the diamond cubic case, Yu and Flodström show that $\Gamma_{100} = \sqrt{3} \Gamma_{111} = \sqrt{2} \Gamma_{110}$, which leads to the prevalence of (111) planes for the complete 3D surface minimization (FIG. 2.6.a). This result is in contrast of the simplistic picture of FIG. 2.5, in which other low index faces, (100) or (110) could be expected (see FIG. 2.6.b). Such "broken bond" model could be further extended to less symmetric crystals such as cubic III-V binaries or hexagonal compounds at the cost of greater complexity. It is interesting to note that the Wulff plot of the surface energy shows low energy facets which may not appear in the equilibrium shape.

In addition to the *thermodynamic* Wulff plot of the surface energy $\Gamma(\vec{n}_i)$, we can define the *kinetic* Wulff diagram which is the polar plot of the facet growth rate $V_i(\vec{n}_i)$.

The *kinetic* Wulff diagram can be used to model the evolving shape of a crystal at different times $t_{n+1} = t_n + \delta t$, during which each local interface advances by $V_i \cdot \delta t \cdot \frac{\vec{n}_i}{||\vec{n}_i||}$ in the 3D space. If the $V_i/\Gamma_i$ ratio is constant for the low energy facets, the nodes of the *thermodynamic* and *kinetic* Wulff plots are homothetic and the asymptotic *kinetic* shape of any starting nucleus will match that given by *thermodynamic* equilibrium.

<footnote n="9">⁹see https://en.wikipedia.org/wiki/Wulff_construction</footnote>

<footnote n="10">¹⁰Z. Yu and A. Flodström. “Orientation of (1x1)-surface free energies of crystals”. *Surface Science* 401.2 (1998), pp. 236–247. DOI: 10.1016/s0039-6028(97)01084-4.</footnote>
<figure><img src="image_10.png" /><figcaption>**Figure 2.7:** (a) Simulated kinetic growth using a level-set method of a starting square 2D crystal nuclei toward a rectangular shape. (b) Corresponding kinetic Wulff plot. Red arrows indicate the magnitude of the (fictive) tangential speed used in the chosen numerical model to maintain sharp edges. (M2 internship of A. Grigore, supervised by F. Oehler, Material Dpt, University of Cambridge, 2013).</figcaption></figure>

FIG. 2.7 shows a simple example in 2D, in which a level-set method is used to represent and evolve arbitrary crystal shape<fnref n="11" /> in combination with a numerical kinetic Wulff plot calibrated to obtain a rectangular ‘equilibrium’ shape. We see that starting nucleus smoothly evolves from the initial square toward the chosen rectangle.

Alternatively, we can tune the *kinetic* Wulff diagram so that it matches the experimental growth rate, and compare simulated and experimental crystal growth<fnref n="12" />. The deviation from *thermodynamic* to *kinetic* Wulff diagram arises from two fundamental reasons : (1) crystal growth necessarily proceed out of equilibrium, (2) surface phenomena, mainly bond reconstruction and passivation via adatom/precursor adsorption, can significantly reshuffle the broken bond density and nature, altering the corresponding facet surface energy.

<figure><img src="image_11.png" /><figcaption>**Figure 2.8:** (a) Cross section SEM image of (112̄2) GaN stripe grown over a patterned (11̄00) sapphire. AlGaN maker layers (black line) are inserted to track the crystal interface position with time. (b) Corresponding facet orientation, the semi-polar (112̄2) is labeled ‘t’. (c-d) Examples of simulated growth using 2D kinetic Wulff diagram. (M2 internship of A. Grigore, supervised by F. Oehler, Material Dpt, University of Cambridge, 2013).</figcaption></figure>

<footnote n="11">¹¹G. Russo and P. Smereka. "A Level-Set Method for the Evolution of Faceted Crystals". *SIAM Journal on Scientific Computing* 21.6 (2000), pp. 2073-2095. DOI: 10.1137/s1064827599351921.</footnote>

<footnote n="12">¹²D. Du, D. J. Srolovitz, M. E. Coltrin, and C. C. Mitchell. "Systematic Prediction of Kinetically Limited Crystal Growth Morphologies". *Physical Review Letters* 95.15 (2005), p. 155503. DOI: 10.1103/physrevlett.95.155503.</footnote>
In Figure 2.8, we see an example of simplified 2D kinetic Wulff simulations in which we try to expand the result of Sun *et al.*<fnref n="13" /> on non-polar (11100) GaN to semi-polar (1122) GaN grown on patterned (11100) sapphire. Here, AlGaN layers are inserted during the Metal Organic Chemical Vapor Deposition (MOCVD) growth to reconstruct the time evolution of the crystal interface after growth, using cross section SEM observations. The positions of AlGaN markers are used to calibrate the growth rates of the kinetic Wulff diagram.

<figure><img src="image_12.png" /><figcaption>**Figure 2.9:** (a) Coalescence of semi-polar (1122) GaN stripes grown on a patterned (11100) sapphire observed by cross section SEM. (b) Corresponding growth simulation obtained using a calibrated kinetic Wulff plot. (M2 internship of A. Grigore, supervised by F. Oehler, Material Dpt, University of Cambridge, 2013).</figcaption></figure>

FIG. 2.9 shows the same GaN (1122) crystals, after much longer growth by MOCVD. The separated GaN stripes have now coalesced in a planar (1122) surface. Note that the semi-polar (1122) facet (labeled ‘t’) is absent at the beginning of the growth and only appears later, before overtaking the other orientations. Such emergence of a new facet takes root from the change of curvature of the surface, from convex (individual stripes, just before coalescence) to concave (merged surface, just after coalescence).

The real case study of FIG. 2.9 highlights the importance of the starting geometry, curvature and choice of growth parameters during the coalescence of individual crystallites to form a planar layer.

**Conclusion:** Low energy facets and specific orientations are the hallmarks of crystalline solids. While some facets may dominate the crystal shape due to their low surface energy at the thermodynamic equilibrium, local geometry and the growth conditions can alter significantly their corresponding growth rate and their actual extension in real samples.

## 2.3. Maintaining a flat surface during growth (GaN/InGaN by MOCVD)

In section 2.2, we have seen that specific crystal orientations (i.e. crystal facets) act as stable growth planes due to their favorable bond termination or low surface energy. Yet, maintaining such a flat growth interface on real surfaces is not guaranteed and requires some experimental developments.

### 2.3.1 Miscut and dislocations : c-GaN homoepitaxy on bulk c-GaN

In the following we study the homoepitaxy of (0001) GaN by MOCVD on commercial bulk GaN (0001) substrates [Oehler2013].

Compared to heteroepitaxial growth on sapphire or silicon substrate, bulk GaN substrates provide a pricey (~1000 €.cm⁻²) but simple alternative to complex defect reduction procedures (such as those detailed in §2.4.2).

<footnote n="13">¹³Q. Sun, C. D. Yerino, T. S. Ko, Y. S. Cho, I.-H. Lee, J. Han, and M. E. Coltrin. "Understanding nonpolar GaN growth through kinetic Wulff plots". *Journal of Applied Physics* 104.9 (2008), p. 093523. DOI: 10.1063/1.3009969.</footnote>
<figure><img src="image_13.png" /><figcaption>**Figure 2.10:** (a) Large scale DIC optical micrograph of the GaN epilayer grown by MOCVD on a bulk GaN substrate (0001). Black rectangles indicate the position of images (b) and (c). The red dotted box highlights a region without any visible defect. (b) Isolated pyramidal defects. (c) More complex defect, close to the substrate edge. (from [Oehler2013])</figcaption></figure>

FIG. 2.10.a shows a top-view optical micrograph of the sample after 3 µm homoepitaxial growth of GaN. While the GaN (0001) surface is relatively smooth, even fully flat over milimetric area (red rectangle), many features can be distinguished. Isolated pyramidal hillocks (FIG. 2.10.b) or more complex structures (FIG. 2.10.c) are easily revealed using differential interference contrast (DIC) optical microscopy.

<figure><img src="image_14.png" /><figcaption>**Figure 2.11:** (a) AFM topography of a single pyramidal hillock (z-scale is 250nm). (b) Corresponding AFM amplitude error image showing slope contrast (facet contrast). (c) Corresponding SEM-CL panchromatic image (room temperature). (d) AFM zoom of the apex of the hillock (z-scale 13nm). (e) Corresponding SEM-CL panchromatic image. (from [Oehler2013])</figcaption></figure>

FIG. 2.11.(a-c) shows a combined AFM and cathodoluminescence (SEM-CL) study of a single hillock, about 250 nm height and tens of microns wide. Clear facets are visible and they align with specific axes of the underlying (hexagonal) GaN (0001) surface. The apex of the pyramidal hillock is detailed FIG. 2.11.d and presents atomic steps arranged in a spiral, terminated by a pit. This surface pit is characteristic of the emergence
of a threading dislocation core when the growth is performed under particular surface termination based on silane<fnref n="14" />. The corresponding SEM-CL at room temperature (FIG. 2.11.e) shows that the apex is dark, which indicates the presence of an increased density of non-radiative recombination centers, at the expected position of the dislocation core. The convex edges of the hillock, marked by red arrows, FIG. 2.11.(a-c) and dotted lines FIG. 2.11.(d-e), also present an increased density of recombination center (dark in SEM-CL).

<figure><img src="image_15.png" /><figcaption>**Figure 2.12:** Model of the hillock shape, relative to the substrate miscut angle $α$. (from Oehler 2013)</figcaption></figure>

From the AFM and CL data (FIG. 2.11), it is clear that the hillock forms due to the emergence of a threading dislocation with a screw component, which creates a spiral staircase at the top of the GaN (0001) surface.

If the dislocation density is too large, the macroscopic surface of the sample becomes rough (see FIG. 2.10.a, top edge) which can cause problems during the later stages of the growth.

The asymmetric shape of the hillock is peculiar and it can be quantified by an inclination angle $α$ (see FIG. 2.12) assuming symmetric shape for $α = 0$. The experimental value is $α ≈ 0.3°$ (from AFM) and corresponds in amplitude and direction to the nominal substrate miscut angle, as measured by XRD.

From these data, we anticipate that substrate miscut above 0.4° from the (0001) normal (toward the in-plane $\vec{m}$ direction) should prevent the formation of the observed pyramidal hillocks on polar GaN (0001) bulk substrate, similarly that reported by Hirai et al.<fnref n="15" /> for non-polar GaN (1100).

The reasoning is that the hillock consists in a spiral of edges, due to the emergence of a dislocation with a screw component. These step edges provide preferred sites for atoms to bond, as in the classic growth theory of Burton, Cabrera and Frank (BCF). If the main surface orientation is offset enough from the (0001) normal, the vicinal plane offers a similar density of step edges than near the spiral and the growth proceed in a homogenous manner. Note that the dislocation core is still a singularity which will locally disrupt the step flow, but the associated macroscopic hillock (several microns wide) will not form anymore.

**Conclusion:** Real substrates always present surface defects such as threading dislocations, which disrupt the facet growth by forcing a local atomic step flow. Imposing a large enough substrate miscut can coerce the step flow along a particular direction, effectively 'overflowing' defect-related surface features, to recover a flat growth interface at the macroscopic scale.

### 2.3.2 Miscut and step bunching: m-InGaN on bulk m-GaN

In the previous section (§2.3.1), we have observed that a minimum substrate miscut is necessary to maintain a flat growth interface, even in the simplest case of homoepitaxial growth, GaN (0001) on bulk c-plane GaN (0001) substrates. In this section, we explore the use of high substrate miscut angle (2°) with the growth of five InGaN quantum wells on bulk m-plane GaN (1100) substrates. This expands on the homoepitaxial results from the previous section, from polar (0001) GaN to non-polar (1100) surfaces, and gives a first flavor of the heteroepitaxy of active layers, with repeat of thin InₓGa₁₋ₓN/GaN quantum wells/barrier (2.6 nm/5.7 nm thick). The InₓGa₁₋ₓN atomic alloy composition is $x ≈ 0.3$ for those non-polar QWs grown on the (1100) surface, to obtain blue LEDs. For m-plane QWs, the two in-plane directions($a$ [1120] and $c$ [0001]) are non-equivalent but the associated in-plane lattice mismatches with respect to GaN are comparable for the two directions ($\sim +1.7°$).

<footnote n="14">¹⁴R. Oliver, M. Kappers, J. Sumner, R. Datta, and C. Humphreys. "Highlighting threading dislocations in MOVPE-grown GaN using an in situ treatment with SiH₄ and NH₃". *Journal of Crystal Growth* 289.2 (2006), pp. 506-514. DOI: 10.1016/j.jcrysgro.2005.12.075.</footnote>

<footnote n="15">¹⁵A. Hirai, Z. Jia, M. C. Schmidt, R. M. Farrell, S. P. DenBaars, S. Nakamura, J. S. Speck, and K. Fujito. "Formation and reduction of pyramidal hillocks on m-plane (1-100) GaN". *Applied Physics Letters* 91.19 (2007), p. 191906. DOI: 10.1063/1.2802570.</footnote>
<figure><img src="image_16.png" /><figcaption>**Figure 2.13:** (a) AFM of top (1100) surface, after growth on the 2°±0.2° off bulk m-plane (1100) substrate. (b) Cross section HAADF-STEM image. (from [Zhu2016])</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 2.14:** (a) Cross section HAADF-STEM image. (b) Chemical analysis of the atomic In/Ga ratio using EDS. The distance is normalized with respect to the length between the first and last InGaN layers along each orientation. (from [Tang2015])</figcaption></figure>

In Fig. 2.13.a we see that the top GaN surface (i.e. the surface of GaN barrier of the last repeat of the five InGaN/GaN pairs) presents large terraces separated by vertical striations aligned along the *a* direction [1120]. Such geometry is the direct consequence of the strong miscut of the substrate (2° along [0001]), which creates a high density of atomic steps on the surface perpendicularly to this direction. During growth, the step flow is difficult along the *c* [0001] direction and atomic steps accumulate locally (step bunching) to create titled semi-polar micro-facets of type (110X), with *X* > 0.

Due to the different atomic numbers of Ga (Z=31) and In (Z=49), the five InGaN/GaN quantum wells (QWs) are well visible in High Angle Annular Dark Field (HAADF, contrast ∝ Z²) scanning transmission electron microscopy (STEM) images Fig. 2.13.b. The change of surface orientation is apparent in the cross section and the inclined normal of the average semi-polar facet is marked with a yellow dotted line. Similar data is shown in Fig. 2.14.a for another location. Corresponding chemical analysis by energy dispersive X-ray spectroscopy (EDS) in Fig. 2.14.b shows that the average In atomic content *x* differs between the InₓGa₁₋ₓN alloy from the main (1100) surface and the inclined semi-polar facet. The relative quantum wells and barrier thicknesses are also slightly different between the two orientations, in addition to the change in total thickness.

<figure><img src="image_18.png" /><figcaption>**Figure 2.15:** Low temperature (10 K) SEM-CL analysis of 5xInGaN QWs grown on (1100) m-plane bulk substrate with strong miscut (2° toward [0001]) showing severe step-bunching. (a) Spatially-averaged CL signal. (b) Spatial intensity of the integrated spectral signal (panchromatic image). (c) Selected spectra for the m-plane and semi-polar step edge. (d-f) Selected mono-chromatic images. (from [Zhu2016])</figcaption></figure>
Due to these differences in alloy content and in quantum well thickness, the emission wavelength differ between the main (1100) InGaN QWs and that grown on the semi-polar facet. This is well demonstrated by low temperature cathodoluminescence (SEM-CL), Fig. 2.15. The average emission properties shows two contributions (Fig. 2.15.a), with the high energy peak (2.92 eV) originating from the main (1100) surface (Fig. 2.15.d) and the low energy line (2.69 eV) from the semi-polar facet (Fig. 2.15.f).

Despite their small footprint on the substrate, the semi-polar stepped regions contribute significantly to the total CL-emission. Their actual emission decomposes into two comparable contributions, actual semi-polar facets (2.69 eV) and m-plane facets (2.89 eV) which are slightly red-shifted compared to the main (1100) m-plane QWs (2.92 eV). The distinction between semi-polar and non-polar QW emission requires time-resolved SEM-CL experiments (see [Zhu2016]) to measure the corresponding decay time and unambiguously assign each emission feature to its corresponding QW type.

**Conclusion:** Substrate miscut is an important parameter for the experimental planar crystal grower. Too much (§2.3.2) or too little (§2.3.1) of it can lead to non-flat growth interfaces and subsequent non-uniformities in active layers. In combination with optimized growth conditions, it is one of the keys to obtain macroscopic smooth surfaces on real (i.e. defective) substrates.

### 2.3.3 Roughness from extended defects in active InGaN layers

The semi-polar InGaN QWs grown on the step-bunched (1100) surface from §2.3.2 also exhibit specific defects known as basal plane stacking faults (BSFs), see Fig. 2.16.

<figure><img src="image_19.png" /><figcaption>**Figure 2.16:** (a) Weak beam dark field TEM image showing the BSF starting from the InGaN QWs (white arrows). (b) HAADF-STEM image zooming on one of the BSFs starting point. Yellow rectangle is Fig. 2.17. (from [Tang2015])</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Figure 2.17:** High resolution HAADF-STEM image showing the partial dislocation bounding the BSF, at the start of the InGaN QW (adapted from [Tang2015]).</figcaption></figure>

For the m-plane GaN substrate orientation, the partial dislocation which bounds the BSF at the lower InGaN/GaN interface has the potential to relieve some of the InGaN epitaxial strain. The atomically resolved HAADF-STEM image Fig. 2.17 shows that an atomic plane is missing in the InGaN (bright region) compared to the GaN barrier (dark region), which is a textbook picture of strain relaxation via misfit dislocation segments (see Fig. 2.3 page 9). Such arrangement shows that strain relaxation does not necessarily proceed using nominally 1D defects but that (edges of) 2D defects can also provide some form of stress relief.

Interestingly, BSF can also form at the lower interface of an InGaN QW grown on c-plane [0001] GaN substrates to form ‘trench defect’ (red arrows, Fig. 2.18). These ‘trench defects’ delimit ‘islands’ inside the InGaN/GaN QW volume within which the material presents emission wavelength different from that of QWs from the main (0001) volume (due to different local growth rates). Structural analysis (Fig. 2.18.b) constructed from TEM observations shows that ‘trench defect’ decomposes into a planar horizontal BSF (I₁-BSF, coherent 2D defect, no broken bonds), surrounded by a vertical stacking mismatch boundary (SMB, defective 2D interface, broken bonds) aligned along specific planes, which is further associated to surface pit when the SMB change direction [Rhode2014]. Such surface pits are usually related to the emergence of threading dislocation core. In this configuration, the BSF is expected to have little to no effect on the overall strain relaxation in the InGaN layer, but the density of ‘trench’ defect is correlated with the average atomic In content in the QWs [Massabuau2014].
<figure><img src="image_21.png" /><figcaption>**Figure 2.18:** (a) AFM image of the top of 5xInGaN/GaN quantum wells grown along [0001] orientation. Trench defects are marked by red arrows (adapted from [Massabuau2014]). (b) Atomic structure of a trench defect (from [Rhode2014]).</figcaption></figure>

**Conclusion:** Hetero-epitaxial strain may be associated the emergence of surface features in the epilayer (pits, trenches) before the critical thickness. In practice, it can be difficult to dissociate between strain-related features and non-optimized growth conditions. For ternary alloys, such as the above InGaN/GaN, such surface features disrupt the local growth conditions and lead to non-uniform alloy content and micro-faceting. Pre-existing defects, from the substrate or other underlying heteroepitaxial interfaces, are prime locations for further non-uniform heteroepitaxial growth, so that the top layer interface may accumulate defects with increasing number of heterointerfaces. Therefore it is best to start from a base layer with the lowest defect density.

## 2.4. Becoming flat : coalescence in planar epitaxy (GaN MOCVD)

While the data from §2.3.3 hint at an ever-increasing surface roughness as the growth proceeds, it is usually possible to 'fill up' the various pits or trenches from the active layer in subsequent growth steps. This is usually done by tailoring the growth conditions (substrate temperature, partial pressure, precursor ratio and flow rates) to favor specific facets orientation (see §2.2).

FIG. 2.19 from Hiramatsu et al.<fnref n="16" /> shows a well-known example on the growth of (0001) oriented GaN stripes over a patterned Al₂O₃ surface. The orientation of the sidewalls (vertical, tilted, mixed) is controlled by the growth temperature and the total reactor pressure.

It is important to note that such facet-favoring growth conditions are usually outside the safe boundaries of 'good quality material' and may lead to excessive residual doping or high point defect densities.

Therefore, they typically cannot be used during the active layer growth, which is often a ternary alloys of some sort and hold some growth challenge of its own (see §2.3.3), but before and after these critical layers, to 'prepare' or 'smooth' the growth interface.

<figure><img src="image_22.png" /><figcaption>**Figure 2.19:** Effect of pressure and temperature on (0001) GaN stripes aligned along [1100], from Hiramatsu et al.</figcaption></figure>

<footnote n="16">¹⁶K. Hiramatsu, K. Nishiyama, A. Motogaito, H. Miyake, Y. Iyechika, and T. Maeda. "Recent Progress in Selective Area Growth and Epitaxial Lateral Overgrowth of III-Nitrides: Effects of Reactor Pressure in MOVPE Growth". *physica status solidi* (a) 176.1 (1999), pp. 535-543. DOI: 10.1002/(sici)1521-396x(199911)176:1<535::aid-pssa535>3.0.co;2-i.</footnote>
2.4.1 Coalescence-based defect reduction techniques for c-GaN (0001)

The coalescence of individual crystallites to obtain a smooth layer is also an important tool in defect reduction techniques, known as the 3D-2D growth transition. It is mostly used for (0001) GaN grown on Al₂O₃, to obtain buffer layers with ‘reduced’ threading dislocation densities (mid 10⁷ cm⁻²).

<figure><img src="image_23.png" /><figcaption>**Figure 2.20:** 3D nucleation seed layer used 3D-2D GaN (0001) defect reduction on Al₂O₃ wafers. (Adapted from *Gibart*)</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>(d) 3D island growth at low V/III ratio, 1050 °C</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>(b) GaN nucleation layer deposited at 540 °C</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>(e) 3D island coalescence at higher V/III ratio, 1050 °C</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>(c) Anneal at 1020°C to form 3D islands</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>(f) TEM of dislocation bending in 3D island</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>**Figure 2.21:** Main growth steps along the 3D-2D transition (from *Zhu and Oliver*).</figcaption></figure>

The FIG. 2.20 from Gibart<fnref n="17" /> displays such a 3D-2D growth transition used to obtain a ‘low defect density’ (0001) GaN buffer layer on (0001) Al₂O₃ substrates. Here, the inclined facets of individual crystallites help to curve the threading dislocation sideways, i.e. perpendicular to substrate normal, so that they reduce their total line energy. When such individual crystallites merge, the likelihood of TD interaction and annihilation is increased, which lowers the overall TD dislocation density after the complete layer coalescence (see FIG. 2.21 from Zhu and Oliver<fnref n="18" />).

This 3D-2D coalescence process can be repeated several times, using successive nano-masks grown *in situ* from thin silicon nitride layers (SiNₓ) to localize the 3D growth and further reduce the dislocation density. The main limitation is the extra-thickness and growth duration required to coalesce the 3D crystallites into a smooth planar layer. Note that the growth over these additional SiNₓ layers can also be performed without the 3D step to minimize the coalescence thickness (i.e. simple nano-masking).

2.4.2 Coalescence-based defect reduction techniques for a-GaN (1120)

The adaptation of the above c-plane GaN defect reduction techniques for the growth of non-polar (1120) GaN layers grown on r-plane Al₂O₃ substrates was investigated in [Oehler2014].

<figure><img src="image_30.png" /><figcaption>**Figure 2.22:** Different coalescence based defect reduction growth techniques for the growth of non-polar (1120) GaN layers, using Low-Temperature (LT) nucleation layer, SiN nano-mask or direct 3D (d3D) growth variants (i.e. 3D crystallites grown at high temperature *without* LT layer), from [Oehler2014].</figcaption></figure>

<footnote n="17">¹⁷P. Gibart. “Metal organic vapour phase epitaxy of GaN and lateral overgrowth”. *Reports on Progress in Physics* 67.5 (2004), pp. 667–715. DOI: 10.1088/0034-4885/67/5/r02.</footnote>

<footnote n="18">¹⁸T. Zhu and R. A. Oliver. “Unintentional doping in GaN”. *Physical Chemistry Chemical Physics* 14.27 (2012), p. 9558. DOI: 10.1039/c2cp40998d.</footnote>
The FIG. 2.22 shows different 3D-2D strategies to obtain (112̄0) GaN layers with reduced defect densities, starting from crystallite of various size and shape, which are obtained using different growth parameters. The case of LT-SiN (FIG. 2.22.a) is different and uses a single SiNₓ nano-mask, directly followed by 2D growth conditions (i.e. coalescence condition, no crystallite enlargement), in which the TDs are simply blocked by the SiN nano-mask.

<figure><img src="image_31.png" /><figcaption>**Figure 2.23:** (a) XRD analysis of the full width at half maximum (FWHM) of the symmetric (110) reflection. Note the anisotropic in-plane FWHM with the Phi angle (Phi=±90° aligned with c). (b) Low temperature (6K) photoluminescence of the grown a-plane GaN layers, showing the donor-bond exciton line $D^0F$ as well as the BSF contribution and other defect. (adapted from [Oehler2014].)</figcaption></figure>

XRD and photoluminescence (FIG. 2.23) show that the quality of the grown GaN buffer layers depends on the chosen growth technique and that all the 3D-2D grown layers exhibit degraded properties compared to the SiN method (LT-SiN). The later shows the narrowest and most intense (110) XRD reflection, *without* in-plane anisotropy (FIG. 2.23.a) and the best donor bound exciton ($D^0F$) over BSF intensity ratio (FIG. 2.23.b).

The physical origin of the degraded properties of the a-plane layer grown using 3D crystallites can be revealed by growing a thin additional InGaN top layer, to highlight specific surface features (see §2.3.3).

After a thin InGaN active layer capping, SEM-CL characterization reveals that all samples present ‘bright regions’ which are characterized by intense defect related emission (yellow band, $\lambda \simeq 550$nm), with the LT-SiN sample showing the smallest surface density (FIG. 2.24.a). Each ‘bright’ region is separated by a trench from the main matrix (FIG. 2.24.b), showing as a dark line in SEM-CL (FIG. 2.24.a), which indicates its defective nature.

Further investigation by AFM (FIG. 2.24.c) shows that surface pits in the InGaN layer are rotated by 180° in the ‘bright’ region compared to those in the main matrix, indicative of different in-plane orientation.

<figure><img src="image_32.png" /><figcaption>**Figure 2.24:** (a) Low temperature SEM-CL panchromatic image of specific defect (bright region, defect emission). (b) Corresponding AFM images. (c) Zoomed-in AFM data. (from [Oehler2014]).</figcaption></figure>

From FIG. 2.24, we see that the 3D-2D growth transition in a-plane GaN favors the formation of antiphase domains<fnref n="19" />, i.e opposite orientations of crystallites relative to the *in plane* c-direction. The resulting antiphase domains boundaries (APB) at their interface cannot be defect-free (broken or metal-metal bonds) and extend through all the layer thickness. Nano-masking, in the form of SiN surface treatment followed directly by 2D growth conditions *without* 3D growth step, provides here the best compromise, with the lowest surface density of antiphase domains [Oehler2014].

Obviously, any effective dislocation reduction technique should avoid creating other type of defect in the

<footnote n="19">¹⁹'antiphase domains' may be referred to as 'inversion domains' in the III-Nitrides community.</footnote>
grown material. Overall, masking or nano-masking seems to be the most robust concept for defect reduction across crystal orientations and materials, with layer overgrowth implemented in GaN (0001)<fnref n="20" />, GaN (1122) see FIG. 2.9 page 14, GaN (1120)[Reid2015] or various III-V<fnref n="21" />.

**Conclusion:** The 3D-2D dislocation reduction technique, which is iconic to polar (0001) GaN growth on c-plane Al₂O₃ (see §2.4.1) cannot be transferred directly to non-polar (1120) GaN on r-plane Al₂O₃. While threading dislocation densities may be reduced by the 3D facets overgrowth, another type of defect appears in the form of antiphase domains. Each of those bring an increased defect-related emission in the volume (yellow band) and is bounded by a very defective antiphase boundary (2D defect), which extends through all the layer thickness.

## 2.5. Polar and twinned domains (GaAs/Ge by MBE)

Section §2.4.2 introduces antiphase domains as 3D defects specific to the growth of non-polar (1120) GaN layers. However such 3D inclusions can occur in any non centro-symmetric crystal, which includes most III-V semiconductors, cubic or hexagonal. In all cases, some sections of the 2D interface between the inverted domain and the main matrix are defective (broken or metal/metal bonds), which degrades the overall material quality.

Antiphase domain in (0001) GaN are relatively uncommon but can happen due to the over-nitridation of the initial Al₂O₃ substrate<fnref n="22" />. Still, antiphase-free (0001) GaN layer can be usually obtained without severe experimental constraints (which is *not* the case for pure (0001) GaN on (0001) Al₂O₃).

The case of antiphase domains in cubic III-Vs is vastly publicized in the literature, as the direct epitaxy of cubic III-V semiconductors on Ge(100) or Si (100) substrates is of major commercial interest for the semiconductor industry or solar cell manufacturers.

### 2.5.1 Polarity mismatch and solution for GaAs/Ge (100) epitaxy

We study here the model case of GaAs/Ge grown along the (100) orientation. Here, the Ge(100) substrate and GaAs(100) epilayer are near lattice-matched (and thermal expansion-matched) with $a_{Ge} = 5.658Å$, $a_{GaAs} = 5.653Å$ ($\delta a/a \simeq -0.08\%$), so that epitaxial (and thermal) strain can be neglected for thin epilayers.

<figure><img src="image_33.png" /><figcaption>**Figure 2.25:** (a) SEM view of the top surface of a GaAs (100) layer grown by MBE on a Ge(100) substrate. (b) Schematic cross section of the atomic details of the GaAs/Ge(100) epitaxial relationship. (from PhD thesis of D. Pelati[thesis_pelati_2018]).</figcaption></figure>

<footnote n="20">²⁰P. Gibart. "Metal organic vapour phase epitaxy of GaN and lateral overgrowth". *Reports on Progress in Physics* 67.5 (2004), pp. 667-715. DOI: 10.1088/0034-4885/67/5/r02.</footnote>

<footnote n="21">²¹D. J. Ironside, A. M. Skipper, A. M. García, and S. R. Bank. "Review of lateral epitaxial overgrowth of buried dielectric structures for electronics and photonics". *Progress in Quantum Electronics* 77 (2021), p. 100316. DOI: 10.1016/j.pquantelec.2021.100316.</footnote>

<footnote n="22">²²M. Sumiya, M. Tanaka, K. Ohtsuka, S. Fuke, T. Ohnishi, I. Ohkubo, M. Yoshimoto, H. Koinuma, and M. Kawasaki. "Analysis of the polar direction of GaN film growth by coaxial impact collision ion scattering spectroscopy". *Applied Physics Letters* 75.5 (1999), pp. 674-676. DOI: 10.1063/1.124478.</footnote>
FIG. 2.25 shows the top surface of a GaAs (100) layer directly grown by Molecular Beam Epitaxy (MBE) on a commercial Ge(100) wafer, with no specific treatment. The top surface is very rough, with meandering trenches (over 10 occurrences per linear micron, see FIG. 2.25.a). These trenches are the surface emergence of anti-phase domain boundaries, which separates domains with two opposite orientations of the GaAs dipole (FIG. 2.25.b, red circles).

On such Ge(100) surface, there are two possible origins for the formation of antiphase domains:

1. the starting adatom on the Ge(100) surface can be either be As or Ga, imposing the GaAs dipole direction.

2. the surface step of Ge(100) are half-height of the GaAs dipole, so crystallites grown on neighboring terraces show reversed polarity (FIG. 2.25.b, step A vs. step B).

While the origin (1) may be fixed by carefully tuning the growth conditions at the start of the experiment<fnref n="23" /> (i.e. pre-exposure of the Ge surface to either As or Ga), it is important to note that origin (2) occurs even in the case of deterministic initial growth (As adatom first on FIG. 2.25.b).

To eliminate the mechanism (2), delicate surface annealings and miscut substrates along specific direction<fnref n="24" /> are required to create double steps Ge(100), or step heights of multiple of two.

**Conclusion:** Although Ge and GaAs are both cubic crystals and nearly lattice-matched, the polarity mismatch between the non-polar Ge and polar GaAs crystals leads to the growth of severely defective along the standard (100) direction, in the form of antiphase domains and the associated antiphase boundaries. Solutions exist in the guise of surface treatment but require specific substrates (high miscut) and delicate annealings.

## 2.5.2 Twinning during GaAs/Ge (111) epitaxy

As the direct growth of GaAs/Ge(100) requires very specific double step surfaces, it is reasonable to explore the growth along other low index orientations.

Compared to the Ge(100) surface, the Ge(111) surface only shows steps of a full GaAs dipole height (FIG. 2.26), so the antiphase origin(2) should not exist.

Consequently the direct growth of GaAs/Ge(111) should only have to deal with antiphase origin (1), which requires the deterministic start of the first atomic layer with a single adatom type, As or Ga.

In the simplified picture (FIG. 2.26), the polarity of the whole layer, GaAs(111)A or GaAs(111B), is now determined by the position of the first As adatom on the Ge(111) surface, independently of any surface steps.

<figure><img src="image_34.png" /><figcaption>**Figure 2.26:** GaAs growth on Ge (111), with only full height surface steps (from [Pelati2019]).</figcaption></figure>

In principle, we could also pre-expose the Ge(111) surface to either Ga or As. However a Ga treatment must be well calibrated to avoid excessive Ga deposition and the formation of liquid Ga droplets on Ge(111). Such Ga droplets usually lead to the formation of various defect and are detrimental to crystal quality. The case of excess group V pre-deposition does not occur as it is not possible to form or deposit thick clusters of As on Ge, at the typical substrate temperatures used for GaAs growth<fnref n="25" />.

<footnote n="23">²³R. M. Sieg, S. A. Ringel, S. M. Ting, E. A. Fitzgerald, and R. N. Sacks. "Anti-phase domain-free growth of GaAs on offcut (001) Ge wafers by molecular beam epitaxy with suppressed Ge outdiffusion". *Journal of Electronic Materials* 27.7 (1998), pp. 900-907. DOI: 10.1007/s11664-998-0116-1.</footnote>

<footnote n="24">²⁴P. Pukite and P. Cohen. "Suppression of antiphase domains in the growth of GaAs on Ge(100) by molecular beam epitaxy". *Journal of Crystal Growth* 81.1-4 (1987), pp. 214-220. DOI: 10.1016/0022-0248(87)90393-9.</footnote>

<footnote n="25">²⁵The operating temperature of solid source As evaporation MBE cell is about 300°C, that of P cell varies between 70 and 150°C. In contrast, the growth of GaAs or GaP proceeds between 500°C and 650°C substrate temperature.</footnote>
<figure><img src="image_35.png" /><figcaption>**Figure 2.27:** GaAs(111) growth on Ge (111) with varying As:Ga precursor ratio (from [Pelati2019]).</figcaption></figure>

In FIG. 2.27, we use different As over Ga precursor ratios (expressed in terms of beam equivalent pressure) to obtain the growth of GaAs(111) layer on the top of Ge(111) which has been thermally de-oxidized (450°C) and As-treated (610°C, 5min).

The precursor ratio has a strong influence on the layer morphology and produces:

* pyramids at low As:Ga ratio (FIG. 2.27.a)

* a relatively smooth surface at intermediate As:Ga ratio (FIG. 2.27.b)

* complex flat terraces at high As:Ga ratio (FIG. 2.27.c).

The peculiar two-domain contrast on the titled SEM view FIG. 2.27.c recalls antiphase domains on Ge(100) of FIG. 2.25.a but those should *not* occur on Ge(111). Complementary analyses are therefore required.

In FIG. 2.28, we use a particular XRD technique, pole figure of the 111 reflection, to better characterize the type of defect in our GaAs/Ge(111) layers. FIG. 2.28.d demonstrates the two domains to be twins, i.e. obtained by rotation of 180° along particular axes (here the <111> directions), of first or multiple order.

<figure><img src="image_36.png" /><figcaption>**Figure 2.28:** XRD pole figure of the (111) reflection for GaAs(111) /Ge (111) with varying As:Ga precursor ratio (from [Pelati2019]).</figcaption></figure>

It is interesting to note that twinning along the substrate normal [111] and inversion domains [111] ↔ [111] would give the same pole figure for the primary 111 reflections, i.e. 6 poles instead of 3. However twinning along other directions such as [111], as well as multiple order twinning (i.e. ‘twinning convolution’), is the only possible explanation for the reflections families observed at high As:Ga ratio (FIG. 2.28.d).

Complementary TEM analysis (FIG. 2.29) on a small twinned region obtained at intermediate As:Ga ratio, confirms the XRD analysis with twinning, *not* antiphase, being responsible for the 6-fold pole figure of FIG. 2.28.b.
<figure><img src="image_37.png" /><figcaption>**Figure 2.29:** TEM analysis of twinned region on GaAs(111)A on Ge(111). (from [Pelati2019]).</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>**Figure 2.30:** EBSD and SEM-CL analysis of twinned region on GaAs(111)A on Ge(111). (from [Pelati2019]).</figcaption></figure>

The atomic details of a twinned domain (FIG. 2.29.f-g) are interesting and show a coherent interface perpendicular to the twin axis with no broken bonds (blue line, stacking-fault like) and defective along the twin axis (red line). Consequently [111]-twinned domains on Ge(111) present defective (vertical) sidewalls, which can be directly observed by SEM-CL, FIG. 2.30.

The polarity of the layer is determined to be single-phase GaAs(111)A from further TEM analysis (not shown), which is rather unexpected, but matches the patterns observed by reflection high-energy electron diffraction (RHEED) during growth [Pelati2019].

**Conclusion:** While antiphase domain do not form on Ge (111) during the growth of GaAs(111)A layers, other 3D defects appear in the form of twinned domains. These twinned regions present defective vertical sidewalls, very similar to the antiphase boundaries observed during the GaAs/Ge(100) heteroepitaxy. It is thus critical to minimize the twin density and it possible to obtain twin-free GaAs(111)A/Ge(111) layers by adapting the growth conditions.

Compared to Ge(100), the growth of single phase GaAs layer on Ge(111) can be achieved without delicate surface treatment or specific substrate miscut.
## 2.6. Conclusion

Planar growth on crystalline substrate involves a lot of constraints, which we can separate into three groups:

* Existence and stability of the facet: first, the crystal surface must exist as a facet (§2.2) and persist during growth in the presence of competing alternative orientations. Then, the surface must remain a smooth plane in the proximity of extended defects, such as threading dislocations with a screw component (§2.3.1). Controlled substrate miscut is an important tool but it should be used carefully to avoid step-bunching or micro-faceting (§2.3.2).

* Growth parameter window of the active layer: the heteroepitaxy of active layers, notably ternary alloys in the form of thin quantum wells, imposes even more constraints on the growth parameters to control the alloy composition and limit the point defect density. The subsequent difficulties in adapting the growth parameters lead to compromises on the surface roughness, with enlarging pits and trenches (§2.3.3). Any surface defect existing prior to the active layer is likely to be amplified during the active layer growth and to modify the local properties (§2.3.2).

* Ability to avoid or heal extended defects: any dissimilarity (lattice, polarity, symmetry) between the substrate and the epilayer can lead to the formation of varied extended defects: dislocation networks, antiphase domains (§2.5.1), twinned domains (§2.5.2) and many others. All these extended defects will usually intersect some active layer and degrade the device properties. Here, there is no generic solution and all particular substrate/epilayer affinities must be investigated to limit or avoid defect formation. For example the controllable (reversible) 3D roughness may have its use to obtain layers with reduced dislocation density in GaN(0001) (§2.4.1) but fails for GaN (1120) (§2.4.2). In contrast, dislocation filtering nano-masks perform more evenly across different substrate orientation (§2.4.2).

Another important criteria in the controlled growth of semiconductors is the possibility to engineer the carrier concentration and type, usually carried out by introducing chemical impurities which act as active electrical dopants in the valence or conduction bands.

While we have not discussed this aspect here, it is a most critical feature which will decide the actual prospect of a semiconductor material/substrate orientation for device fabrication. A notorious example is the elusive stable p-type doping in the ZnO/ZnMgO system<fnref n="26" />. Closer to this work, the growth of GaAs(111)A on Ge(111) combines relaxed substrate specifications, smooth layers and low defect densities[Pelati2019], but it is difficult to obtain n-type GaAs(111)A with decent carrier densities using the standard (amphoteric) Si impurities[thesis_pelati_2018], therefore sealing the fate of GaAs/Ge(111) heteroepitaxy in the current state of the art.

<footnote n="26">²⁶ J. Fan, K. Sreekanth, Z. Xie, S. Chang, and K. Rao. "p-Type ZnO materials: Theory, growth, properties and devices". *Progress in Materials Science* 58.6 (2013), pp. 874–985. DOI: 10.1016/j.pmatsci.2013.03.002.</footnote>
Chapter 3

Doing without crystalline substrate:
oriented growth on amorphous surfaces

**3.1. On the role(s) of the substrate**

The only part of the substrate actually required for epitaxial growth is the final top surface, all the rest is
technically a costly mechanical support, most often in the form of an exquisite defect-free crystalline material.
From this perspective, conventional epitaxy techniques are actually wasting the quasi-totality of the crystal
mass (and price) to only use the final few Angströms.

Such dichotomy between the ‘surface for crystal growth’ and the ‘volume for mechanical strength’ may be exploited in conventional crystalline epitaxy in the form ‘substrate recycling’ procedures, in which the epilayers are separated from their crystalline substrates so that the later can be re-used for further epitaxial growth. Such recycling procedures are mostly confined to ‘large area’ devices grown on ‘expensive’ crystalline wafers, such as III-V solar cells<sup>2728</sup>. Still these recycling techniques may also be relevant to devices based on ‘flip chip’ technology<sup>29</sup>. Indefinite substrate recycling is technically difficult as the successive epilayer separation steps usually create some form of cumulative surface damage, which requires delicate (or costly) cleaning procedures.

Given the many constraints imposed by crystalline substrates on the epitaxial layers (see §2.6), it is interesting
to investigate crystal growth on amorphous supports.

Due to the absence of the crystal lattice, there is obviously no definite epitaxial relationship between the
support and layer. This lack of ordered chemical bond immediately relaxes several of the standard epitaxy
constraints. First, lattice-matching is now irrelevant but comes at the price of a ‘naturally’ defective interface
between the amorphous support and the crystalline layer. However this defective boundary also provides a
‘natural’ location for various defects to adapt any strain between the (amorphous) support and the (crystalline)
layer.

Another obvious advantage is the price of amorphous materials compared to that of thick crystalline wafers, especially bulk III-V, SiC or GaN substrates. Considering that we can further limit the thickness of the amorphous layer to a thin coating (SiO₂, SiN) on an even cheaper metallic or plastic foil, this economic advantage can be pushed to rather extreme limits. Note that such mass (thickness) reduction has also a strong ecological edge, and it is much less wasteful than conventional epitaxy.

The main drawback of the absence of crystal lattice in the substrate is that we need another physical effect to orient the crystal lattice of the growing crystallite. In theory, the minimization of the surface energy (see §2.2 page 11) between the growing crystallite and its amorphous support could replace the local ordering of chemical bond of crystalline substrates. However such energy minimization is a less stringent constraint and may allow the growth to proceed not only at the energy minima but in a small range of orientation.

<footnote n="27"><sup>27</sup>A. van Geelen, P. Hageman, G. Bauhuis, P. van Rijsingen, P. Schmidt, and L. Giling. "Epitaxial lift-off GaAs solar cell from a reusable GaAs substrate". *Materials Science and Engineering: B* 45.1-3 (1997), pp. 162-171. DOI: 10.1016/s0921-5107(96)02029-6.</footnote>

<footnote n="28">²⁸G. Bauhuis, P. Mulder, E. Haverkamp, J. Huijben, and J. Schermer. "26.1% thin-film GaAs solar cell using epitaxial lift-off". *Solar Energy Materials and Solar Cells* 93.9 (2009), pp. 1488-1491. DOI: 10.1016/j.solmat.2009.03.027.</footnote>

<footnote n="29">²⁹O. B. Shchekin, J. E. Epler, T. A. Trottier, T. Margalith, D. A. Steigerwald, M. O. Holcomb, P. S. Martin, and M. R. Krames. "High performance thin-film flip-chip InGaN–GaN light-emitting diodes". *Applied Physics Letters* 89.7 (2006), p. 071109. DOI: 10.1063/1.2337007.</footnote>
## 3.2. Oriented low temperature crystallization of amorphous Ge on
silica

### 3.2.1 Metal Induced Crystallization of amorphous Ge

Since the direct growth of GaAs on a SiO₂ surface is usually not oriented, we turn to another strategy (FIG. 3.1). Here, we first fabricate a thin oriented Ge layer on an amorphous SiO₂ support, which is later used as pseudo-substrate for the epitaxial growth of GaAs and subsequent device fabrication. As seen in §2.5 (page 22), the GaAs/Ge epitaxy is a model case with near lattice-matched crystals but may produce anti-phase domains on Ge(100) or twinned regions on Ge(111) (see §2.5.1 and §2.5.2).

<figure><img src="image_39.png" /><figcaption>**Figure 3.1:** Proposed workflow for the fabrication of low-cost Ge pseudo-substrates on amorphous SiO₂ supports for the epitaxial growth of GaAs, toward solar cell devices. (from [thesis_pelati_2018])</figcaption></figure>

The fabrication of an oriented crystalline Ge (c-Ge) layer on amorphous SiO₂ can be realized from the controlled crystallization of an amorphous germanium layer (a-Ge) using the Metal Induced Crystallization (MIC) process. Many metals can be used to promote MIC but the particular case of Aluminium Induced Crystallization (AIC) has attracted much interest<fnref n="30" />. The low cost of aluminium and the low annealing temperatures required for the AIC of amorphous Ge are also attractive commercially.

<figure><img src="image_40.png" /><figcaption>**Figure 3.2:** Temperature operating range of various crystallization processes of amorphous Ge: metal induced crystallization (MIC), solid phase crystallization (SPC) and growth from melt (from [thesis_pelati_2018])</figcaption></figure>

FIG. 3.2 compares different crystallization techniques for amorphous Ge. The MIC domain is restricted to the temperature below the metal-Ge eutectic (421°C for Al-Ge), while the (metal-free) solid phase crystallization (SPC) of amorphous Ge occurs below the corresponding melting temperature (947°C).

The atomistic mechanism of the SPC relies on bond rearrangement at defects sites in the amorphous layer to initiate the solid-solid crystallization reaction. The corresponding SPC activation energy in amorphous Ge is rather high (2.0 eV) compared to the SPC annealing temperatures (typically 450-600 °C, kT~ 60 meV) but still permits measurable crystallization rates<fnref n="31" />.

In this context, ‘Aluminium Induced Layer Exchange’ (ALILE) is a refined MIC (or AIC) process. Furthermore, MIC can be also seen as a metal-catalyzed version of SPC, which operates at lower annealing temperatures with extra benefits (see §3.2.2)

<footnote n="30">³⁰O. Nast and S. R. Wenham. "Elucidation of the layer exchange mechanism in the formation of polycrystalline silicon by aluminum-induced crystallization". *Journal of Applied Physics* 88.1 (2000), pp. 124-132. DOI: 10.1063/1.373632.</footnote>

<footnote n="31">³¹S. Yamaguchi, N. Sugii, S. K. Park, K. Nakagawa, and M. Miyao. "Solid-phase crystallization of Si1-xGex alloy layers". *Journal of Applied Physics* 89.4 (2001), pp. 2091-2095. DOI: 10.1063/1.1344215.</footnote>
The progression of the ALILE process, initially observed using amorphous silicon (a-Si) by Nast *et al.*, can be described as follows (FIG. 3.3):

<figure><img src="image_41.png" /><figcaption>FIG. 1. Cross-section FIB micrographs of the a-Si/Al/glass structure: (a) before annealing, (b) after annealing for 5 min, (c) 10 min, and (d) 60 min at 500 °C; the samples are tilted by 45°.</figcaption></figure>

<caption>**Figure 3.3:** Main steps of the ALILE process (from [thesisfurati 2018]) and corresponding SEM cross-sections from Nast *et al*.</caption>

(a). a-Si is deposited on the top of the Al layer.

(b). Si atoms diffuse from the a-Si into the metal layer through grain boundaries

The Si concentration in selected grain boundaries reaches a critical value sufficient to initiate the crystallization

(c). The Si crystallite grows inside the Al grain boundary, creating significant stress

The Al material diffuse out of the stressed area, in the void created above the crystallized Si layer

(d). Comparing the initial and final stages, we can see that the amorphous Si layer (a-Si) is now crystallized (c-Si) and that the Al and Si layers have exchanged their positions, which is at the origin of the ALILE acronym (Aluminum-Induced Layer Exchange).

(e). A final step (not shown) consists in the selective chemical etching of the top Al layer in order to expose the crystallized Si surface.

### 3.2.2 Oriented Aluminium induced crystallization of amorphous Ge

A better understanding and control of the ALILE phenomena is required to improve the control over the crystallization process. The first improvement is the addition of an *interlayer* between the Al and amorphous Ge layers (see FIG. 3.4):

* This *interlayer* regulates the influx of Ge atoms inside the Al grain boundaries and controls the final size of the crystallized Ge domains. Native $GeO_x$ can be used<fnref n="32" /> but aluminium oxide ($AlO_x$) is the most common choice.

* The thickness and chemical composition of the $AlO_x$ *interlayer* can be tailored to move the initial Ge nucleation from an arbitrary location within the Al grain boundaries to the bottom $Al/SiO_2$ interface, where (111) oriented Ge crystals are produced<fnref n="33" />.

* A similar strategy can be used for Si-ALILE with a $SiO_x$ interlayer, favoring Si(100) orientation for nucleations occurring inside the Al grain boundaries and Si(111) when they are localized at the $Al/SiO_2$ interface<fnref n="34" />.

A further improvement of ALILE can be obtained by adding a very thin amorphous Ge *underlayer* between the Al and $SiO_2$ surface (see FIG. 3.4), to speed up the nucleation process and 'pre-saturate' the $Al/SiO_2$ interface with Ge. This enhances the crystallization rate and yields large-grained Ge(111) on silica<fnref n="35" /> (see FIG. 3.5).

<footnote n="32">³²S. Hu, A. F. Marshall, and P. C. McIntyre. "Interface-controlled layer exchange in metal-induced crystallization of germanium thin films". Applied Physics Letters 97.8 (2010), p. 082104. DOI: 10.1063/1.3480600.</footnote>

<footnote n="33">³³K. Toko, K. Nakazawa, N. Saitoh, N. Yoshizawa, N. Usami, and T. Suemasu. "Orientation control of Ge thin films by underlayer-selected Al-induced crystallization". CrystEngComm 16.13 (2014), p. 2578. DOI: 10.1039/c3ce42057d.</footnote>

<footnote n="34">³⁴K. Toko, R. Numata, N. Saitoh, N. Yoshizawa, N. Usami, and T. Suemasu. "Selective formation of large-grained, (100)- or (111)-oriented Si on glass by Al-induced layer exchange". Journal of Applied Physics 115.9 (2014), p. 094301. DOI: 10.1063/1.4867218.</footnote>

<footnote n="35">³⁵R. Numata, K. Toko, K. Nakazawa, N. Usami, and T. Suemasu. "Growth promotion of Al-induced crystallized Ge films on insulators by insertion of a Ge membrane below the Al layer". Thin Solid Films 557 (2014), pp. 143-146. DOI: 10.1016/j.tsf.2013.08.040.</footnote>
<figure><img src="image_42.png" /><figcaption>**Figure 3.4:** ALILE process in complete a-Ge/Al sample with the $AlO_x$ interlayer and a-Ge underlayer (from Numata *et al.*).</figcaption></figure>

<caption>**Figure 3.5:** Experimental observation of the preferential Ge(111) crystallization and lateral crystal size by electron backscatter diffraction (EBSD), after selective etching of the Al layer (from Numata *et al.*).</caption>

**Conclusion:** The fabrication of a thin (50 nm) crystallized and (111)-oriented Ge layer on the amorphous $SiO_2$ surface can be obtained from the ALILE crystallisation of an amorphous Ge layer. This process occurs at relatively low temperature (250-300°C) and requires annealing during a few hours.

However, the fine tuning of all the ALILE parameters: namely annealing temperature, duration, a-Ge thickness, Al thickness, $AlO_x$ interlayer composition and underlayer thickness is necessary to obtain the largest Ge crystals and uniform (111) orientation.

### 3.2.3 In situ observations

Given the number of experimental parameters and the duration of individual growth experiments (several hours), conventional experimental plans in which samples are analyzed after the growth are relatively inadequate. For example, if a particular parameter speeds up the crystallization rate, we not only need to vary this parameter but also fabricate and analyze samples with different annealing times, which may take about one or two weeks.

The simplicity of the schematic FIG. 3.4 is interesting, but it is an idealized scenario with well identified steps. In actual experiments, several processes can compete or occur in parallel, As each of these can be rate-limiting, it is very difficult to infer the actual key moments of the ALILE process by only studying the final crystallized layers, after the samples have been cooled to room temperature.

In this context, the ability to monitor and quantify the nuclei density or the crystallization rate *in situ*, during the annealing, is unmatched compared to time series in which thermal treatments are stopped and resumed at regular intervals. *In situ* growth monitoring is well known in MBE (RHEED, curvature measurements) or MOCVD (optical reflectance, ellipsometry, curvature) but such tools usually provide an average picture of the surface, whereas we are interested here in actual image of the growth front which can only be achieved by implementing *in situ* microscopy.

Dedicated tools such as in situ TEM can now obtain atomic resolution images during the growth of nanostructures<fnref n="36" /> but for the ALILE process we can already learn a lot using conventional optical microscopy. Compared to in situ TEM, a particular advantage of optical microscopy is that we do not need to prepare a dedicated sample or restrict ourselves to a specific geometry. In the context of Ge-ALILE, optical in situ microscopy allows one to follow the crystallization of standard samples, or even to compare in real time different samples annealed side by side in the same conditions.

<footnote n="36">³⁶J.-C. Harmand, G. Patriarche, F. Glas, F. Panciera, I. Florea, J.-L. Maurice, L. Travers, and Y. Ollivier. "Atomic Step Flow on a Nanofacet". *Physical Review Letters* 121.16 (2018), p. 166101. DOI: 10.1103/physrevlett.121.166101.</footnote>
<figure><img src="image_43.png" /><figcaption>**Figure 3.6:** Picture and schematic of the in situ optical microscope mounted on the programmable hot plate in a controlled atmosphere (from [thesis_pelati_2018]).</figcaption></figure>

FIG. 3.6 presents the experimental setup used during the experiments. The most important optical element is the high quality 50x long distance (13 mm) microscope objective. However, several key elements are needed so that hot samples (250-300°C) can be effectively imaged over several hours at the same optical resolution than room-temperature imaging:

1. Infrared filter(s) must be installed along the optical chain to avoid concentrating heat on the CMOS sensor.

2. Thermal turbulence between the objective front lens and the lid windows needs to be reduced by imposing an forced air flow.

3. An autofocus routine must be used at regular intervals to compensate the thermal dilatation of the various elements during the many hours of annealing.

The FIG. 3.7 presents in situ optical images of a Ge-ALILE sample during a 15h annealing at 247°C. As we use transparent SiO₂ substrates (for visible wavelength), we actually image the bottom of the (reflective) Al layer, through the back face of the (double side polished) SiO₂ support.

The optical resolution is about 0.5-1µm. Due to the layer inversion during the ALILE process (see FIG. 3.4), optical contrast variations (Al bright, Ge dark) are indicative of the crystallization progress, which needs to be confirmed ex situ with other techniques (TEM, EBSD, XRD).

We see that the crystallization process is not as simple as sketched in FIG. 3.4 or FIG. 3.3. Two distinct crystallization processes occur at different times, the second one (t=5.42h, red arrows) being the expected ALILE mechanism.

Separate kinetic studies performed from in situ image sequences (see [Pelati2017]) show that the two processes have different activation energies and rate limiting steps.

<figure><img src="image_44.png" /><figcaption>**Figure 3.7:** In situ optical microscopy during Ge ALILE. Two distinct crystallization processes occurs at t=0.05-0.19h (black arrows) and t=5.42h (red arrows). (from [Pelati2017])</figcaption></figure>
With the information from FIG. 3.7, we can also stop the annealing at a specific time and study the structure of the sample *ex situ* using more specialized techniques. Using a combination of SEM-EDX, cross-section TEM-EDX (FIG. 3.8) and EBSD (not shown), we are able to resolve the corresponding chemical distribution and crystal structure to understand the difference between the two distinct MIC processes (referred as M1 and M2 in FIG. 3.8).

<figure><img src="image_45.png" /><figcaption>**Figure 3.8:** Combination of SEM (a), EDX (b) and cross-section TEM-EDX (c-h) showing the chemical and structural differences between the two crystallization processes (M1 and M2). (from [Pelati2017])</figcaption></figure>

* The MIC-1 process is characterized by a double layer of Ge in the cross section TEM (FIG. 3.8.f). The corresponding zone in the top view EDX also show the largest Ge signal (FIG. 3.8.b).

Additional EBSD characterizations (not shown) reveal that the top layer still consists of amorphous Ge, while the bottom is crystallized Ge.

In summary, the MIC-1 corresponds to Ge crystallization inside the Al layer, with Ge replacing Al material, but without the layer inversion. The corresponding Ge grains are randomly oriented.

* In contrast, the MIC-2 process corresponds the expected ALILE process with a clear inversion of the Al and Ge layer positions between the reference stack (R) and the M2 region in FIG. 3.8.(c-d).

The Ge grains obtained by the above ALILE process show a preferred (111) orientation.

This findings are summarized in FIG. 3.9, which compares the two processes. Given that the Ge grains from the MIC-1 do not show a preferred (111) orientation, contrary to those obtained from the MIC-2/ALILE, the MIC-1 is considered a parasitic process in our samples. With hindsight from the literature<fnref n="37" />, we propose that the starting nucleation of MIC-1 occurs at random position inside Al grain boundaries (despite the use of optimized interlayer and underlayer), while that of MIC-2/ALILE takes place at the lower Al/SiO₂ interface.

<footnote n="37">³⁷K. Toko, K. Nakazawa, N. Saitoh, N. Yoshizawa, N. Usami, and T. Suemasu. "Orientation control of Ge thin films by underlayer-selected Al-induced crystallization". *CrystEngComm* 16.13 (2014), p. 2578. DOI: 10.1039/c3ce42057d.</footnote>
<figure><img src="image_46.png" /><figcaption>**Figure 3.9:** Difference between the (parasitic) MIC-1 and the expected ALILE/MIC-2 crystallization processes. (from [thesis_pelati_2018]).</figcaption></figure>

Further developments in Ge-ALILE may require the use of patterned layers to shape and limit the lateral size of the Ge crystals. The FIG. 3.10 shows in situ imaging of the annealing of such patterned sample (square pattern 40 µm width). From the in situ movies, we clearly identify an increased probability for the parasitic MIC-1 to nucleate at the defective edges of the sample, which concentrate much of the etching damage. Optimized sample processing allows one to limit the MIC-1 nucleation and to maximize the ALILE/MIC-2 surface coverage (FIG. 3.10 bottom).

<figure><img src="image_47.png" /><figcaption>**Figure 3.10:** In situ and ex situ imaging of patterned MIC-substrates and optimization of lithographic processing. (adapted from [thesis_pelati_2018]).</figcaption></figure>

**Conclusion:** In situ microscopy allows one to isolate and quantify the effect of individual parameters in efficient manner in order to obtain a good overview of the important processes at play.

On the particular topic of Ge-ALILE, in situ microscopy was a key technique which revealed the existence of a parasitic crystallization process and how to minimize its footprint on the total crystallized area.
3.3. GaAs epitaxy on Ge(111) pseudo-substrates obtained by ALILE

After the optimization of the sample structure and ALILE parameters (see §3.2.2), we obtain large (several tens of microns) islands of (111) oriented 20 nm thick Ge crystals on amorphous SiO₂ substrates, which can be used for the epitaxial growth of GaAs(111) layers by MBE (see §2.5.2 page 23).

The corresponding growth results are shown in FIG. 3.11. SEM imaging shows that the initial MIC-Ge pseudo-substrate mostly presents a flat Ge(111) surface (FIG. 3.11.a). For a GaAs thickness of about 40 nm, we maintain a relatively smooth surface and the GaAs growth versus the SiO₂ substrate remain selective (FIG. 3.11.b). Triangular hillocks, typical of GaAs(111) orientation, are visible all over the grown surface. At intermediate thickness, the selectivity against SiO₂ is lost and the GaAs layer becomes rough (FIG. 3.11.c). At larger thicknesses (several microns), the GaAs surface shows an ever enlarging roughness (FIG. 3.11.d).

<figure><img src="image_48.png" /><figcaption>**Figure 3.11:** SEM bird view of GaAs(111) epitaxy on Ge(111) ALILE pseudo substrates. (from [Pelati2020])</figcaption></figure>

<figure><img src="image_49.png" /><figcaption>**Figure 3.12:** EBSD local crystal orientation map from thin 40 nm GaAs layer grown on Ge(111)-ALILE pseudo-substrate. (from [Pelati2020])</figcaption></figure>

The local crystal orientation analysis by EBSD of a thin GaAs layer (40 nm) grown on a Ge-ALILE pseudo-substrate is shown in FIG. 3.12. The (111) orientation of the GaAs layer indicates that it is growing epitaxially on the Ge(111) platelet (FIG. 3.12.d-e, blue color). This confirms that the initial GaAs/Ge growth interface is free of major defects and contamination. However, there exists some minor orientation variations, visible as small color shifts in the in-plane (FIG. 3.12.c), and out of plane (FIG. 3.12.d-e) maps. Many of these crystal misorientations are associated to grain boundaries and twins. Note that these defects are often not visible when using the default (coarse grained) EBSD coloring scheme (FIG. 3.12.b).

While the loss of GaAs growth selectivity on SiO₂ may be unavoidable for extended MBE growth duration, the
impossibility to maintain a flat GaAs(111) surface (FIG. 3.11.c) is indicative of a serious defective micro-structure and/or very poor growth conditions. As seen in §2.5.2, the reference growth of GaAs(111) on commercial Ge(111) substrates leads to a smooth surface when using the same growth parameters. We can formulate two hypotheses on the origin of our defects: (1) either they are created in the GaAs layer due to poor growth conditions or (2) they pre-exist in the Ge-ALILE pseudo-substrate and are transferred in the GaAs due to the (perfect) epitaxial relation and (good) growth conditions.

FIG. 3.13 presents images of TEM cross-sections obtained by focused ion beam on the zone previously investigated by EBSD (FIG. 3.12). This particular imaging mode (dark field $g_{002}$) is very sensitive to compositionnal changes and we observe a contrast between the GaAs and Ge layers, despite their similar cubic structure, lattice parameter and average atomic number. The surface consists in flat regions separated by hillocks, which origin is linked to vertical defects that can be traced down to the underlying Ge-ALILE layer (FIG. 3.13.c). Further analysis by HRTEM (see [Pelati2020]) shows these defects are low angle grain boundaries (LAGB), which separate crystals with minor angular misorientations (about $\pm 2^\circ$). Such small misalignments may be smoothed out in standard EBSD analysis (our FIG. 3.12 was acquired with specific care) and may be close to the experimental detection limit of some low-end EBSD setups. Note that we could not detect any other defect by TEM over several hundred manometers in lateral length (up to a micron, see [Pelati2020]). Overall, the density of LAGB in our Ge-ALILE is simply too high and the (perfect) epitaxy from the Ge-ALILE to the GaAs layer transfers too many defects (such as LAGB-related hillock lines) for the growing GaAs(111) layer to maintain a flat surface.

<figure><img src="image_50.png" /><figcaption>**Figure 3.13:** $g_{002}$ dark field TEM images of a cross section from a GaAs/Ge-ALILE epitaxial layer (from [Pelati2020]).</figcaption></figure>

**Conclusion:** The growth of oriented crystals on amorphous substrates is possible. We have obtained very flat thin Ge platelets (20 nm) on amorphous $SiO_2$ support with a clear (111) orientation using the ALILE process. However the spread in the obtained crystalline (111) alignment on amorphous substrates is much wider than the one from epitaxial growth using (111) monocrystalline substrates. These local misorientations (in plane and out of plane) cause neighboring crystallites to form low angle grain boundaries. These defective 2D interfaces disrupt the growth of planar layers by creating rough surfaces and significantly degrade the quality of the grown material in all its volume.

## 3.4. Growth of GaN nanowires on glass

A possible solution to the above problem (§3.2.3) is to use a cheap amorphous substrate to align each crystallite individually but to stop the growth before these individual crystallites coalesce in a uniform layer. This way, no grain boundary is created.

Such as particular variation was not explored using Ge-ALILE but with the growth of (crystalline) hexagonal GaN nanowires on (amorphous) $SiO_2$ substrates [Kumaresan2016b]. As will be seen in Chapter 4, nanowires can provide an alternative to planar epitaxy on lattice-mismatched substrates. Here the GaN nanowires can grow perpendicularly to a crystalline Si(111) surface, using the particular epitaxial relationship $GaN(0001) \parallel Si(111)$, but such 'vertical' orientation can also occur on an amorphous flat $SiO_2$ surface (see FIG. 3.14). Such particular $GaN(0001)$ alignment on $SiO_2$ was also observed by others<fnref n="38" />.

Considering that cubic (111) and hexagonal (0001) planes are both dense planes of the diamond cubic and wurtzite crystal structures, we could propose some kind of surface energy minimization as underlying mechanism

<footnote n="38">³⁸S. Zhao, M. G. Kibria, Q. Wang, H. P. T. Nguyen, and Z. Mi. "Growth of large-scale vertically aligned GaN nanowires and their heterostructures with high uniformity on SiOx by catalyst-free molecular beam epitaxy". *Nanoscale* 5.12 (2013), p. 5283. DOI: 10.1039/c3nr00387f.</footnote>
for the preferred crystal orientation of the starting nuclei. However TEM studies of GaN/SiO₂ interface reveal
that other compounds (SiNₓ) may be involved in the alignment mechanism<fnref n="39" />.

<figure><img src="image_51.png" /><figcaption>**Figure 3.14:** SEM top view and cross section of GaN nanowires grown by MBE on various substrates. (a,d) Si(111), (b,e) thermal SiO₂ on Si (100), (c,f) commercial fused SiO₂ substrates. (adapted from [Kumaresan2016b])</figcaption></figure>

As seen on FIG. 3.14, the GaN nanowires maintain a vertical orientation on the two amorphous SiO₂ surfaces, either obtained by the thermal oxidation of Si or directly as bulk SiO₂ substrate. To better quantify the vertical orientation of the NWs, XRD pole figures of the GaN (11̄01) reflection are shown in FIG. 3.15.

<figure><img src="image_52.png" /><figcaption>**Figure 3.15:** (a) XRD pole figure of the GaN (11̄01) reflection of GaN NWs on various substrate. (b) Linewidth of (11̄01) along the Ψ angle, which quantify the angular spread of the ('vertical') NW axis relative the surface normal. (adapted from [Kumaresan2016b])</figcaption></figure>

The profile (FIG. 3.15.b) shows that the ‘vertical’ angular spread in case of the epitaxial growth GaN(0001)∥Si(111) is less than 0.5° (FWHM), which is the experimental XRD resolution in this configuration. The ‘vertical’ angular spread on bulk SiO₂ substrates is about 2.1° and 5.2° on thermal SiO₂/Si(100) and fused silica, respectively. The in-plane orientation is fully random on SiO₂ (isotropic circles in FIG. 3.15.a, ii and iii) and fully determined on Si(111), as GaN[11̄20]∥Si[11̄0] (six individual spots in FIG. 3.15.a, i).

**Conclusion:** The preferential alignment of GaN (0001) facet on the SiO₂ surface is robust (various type of silica) and allows one to obtain near vertical GaN nanowires in absence of epitaxial relationship. However, the inclination the growth axis relative to the substrate normal is not uniquely defined (as in conventional epitaxy) and it can varies within a ±2° interval (narrowest interval obtained using bulk commercial silica substrates).

<footnote n="39">³⁹A. L. Mulyo, Y. Konno, J. S. Nilsen, A. T. van Helvoort, B.-O. Fimland, H. Weman, and K. Kishino. "Growth study of self-assembled GaN nanocolumns on silica glass by plasma assisted molecular beam epitaxy". *Journal of Crystal Growth* 480 (2017), pp. 67–73. DOI: 10.1016/j.jcrysgro.2017.10.009.</footnote>
**3.5. Conclusion**

In Chapter 3, we have investigated the oriented crystal growth of selected compounds on amorphous supports as a substitute for the conventional epitaxy on mono-crystalline substrates.

For GaN nanowires grown on SiO₂, we have seen that the nanowires show a very strong preferential orientation with the GaN(0001) face parallel to the silica surface. The resulting angular spread is narrow (2° FWHM) but still at least four times larger than GaN nanowires grown epitaxially on a reference Si(111) substrate (0.5° FWHM, limited by instrumental resolution). In plane, each GaN nanowire is randomly oriented, so that the coalescence of individual nanowire into a planar layer would yield a very defective material (if it can be obtained).

We can draw a similar picture for crystalline Ge platelets obtained by ALILE on amorphous SiO₂. At large scale, we observe large ‘islands’ (i.e. several microns wide) with preferential out-of-plane Ge(111) orientation and random in-plane alignment, as in the case of the individual single GaN nanostructures. However the in-situ study[Pelati2017] shows that this large Ge islands are not grown in a single nucleation event but created by successive crystallization steps. While this growth process may appear epitaxial (‘perfect’ alignment of the growing Ge crystallite with the preceding Ge(111) grain), it is only strongly preferential and small misalignments (± 2°) are created (in plane and out of plane). As these nearly-aligned Ge(111) grains coalesce, the resulting layer exhibits a large density of low angle grain boundaries, which severely degrade the overall quality of the Ge(111) layer or that of any subsequent epilayer[Pelati2020].

It is thus possible to obtain oriented crystal growth on amorphous substrates but the constraint on the relative alignment of individual crystallites is less stringent that on mono-crystalline substrate. In the systems investigated, preferential orientation could only be achieved within ±2°. Such ‘strong’ preferential orientation is very impressive from the thermodynamic point of view, but it cannot match the near perfect crystal orientation obtained at large distance on monocrystalline substrates.

Here, the *oriented* crystal growth on *amorphous* substrates only yields *well-aligned polycrystals* in the studied systems. The small misalignments between different nuclei prohibits a defect-free coalescence and the fabrication of good quality planar layers.
Chapter 4

Getting rid of coalescence: nanowire
epitaxy

**4.1. Context**

In Chapter 3, we have seen that individual nanostructures are a promising way to obtain ensembles of *separate*
nano-crystals of good quality on mismatched substrates (crystalline or amorphous). For example, (near) vertical
GaN nanowires of comparable crystal quality grow indiscriminately on amorphous SiO<sub>2</sub> substrates, crystalline
Si(111)[Kumaresan2016b] or even graphene[Kumaresan2016]. Unless the individual crystallites merge or coa-
lesce, the overall crystal quality is maintained, at the cost of the lateral surface exposed by each nanostructure.
This lateral surface is the hallmark of the nanowire geometry, bringing new constraints (§4.3.2), but also new
opportunities (§4.1.1 and §4.1.2).

Compared to amorphous substrates presented in Chapter 3, crystalline substrates allow one to obtain a
more varied collection of oriented (crystalline) nanostructures. For example, ‘vertical’ GaAs nanowires, i.e.
with the [111]B growth axis aligned with the normal of the substrate, can be obtained at least on two cubic
substrates (Si[111], GaAs[111]B) whereas GaAs nanowires grown on SiO₂ are reported not to show a preferred
orientation<fnref n="40" />. The strength of the nanowire geometry is to relax the lattice-match constraint while still making
use of a well-defined epitaxial relationship.

**4.1.1 Axial strain, heteroepitaxy and heterostructures**

<figure><img src="image_53.png" /><figcaption>**Figure 4.1:** (a) Conventional planar heteroepitaxy. (b) Nanowire epitaxy below the critical radius. (c) Nanowire epitaxy above the critical radius.</figcaption></figure>

<footnote n="40">⁴⁰V. Dhaka, T. Haggren, H. Jussila, H. Jiang, E. Kauppinen, T. Huhtio, M. Sopanen, and H. Lipsanen. "High Quality GaAs Nanowires Grown on Glass Substrates". *Nano Letters* 12.4 (2012), pp. 1912–1918. DOI: 10.1021/nl204314z.</footnote>
FIG. 4.1 compares the conventional case of planar heteroepitaxy on a mismatched substrate with those of nanowires of different diameters.

* In FIG. 4.1.a, the thickness of the planar epilayer (grey color) is above the critical limit and the epilayer strain is (partially) relaxed via the creation of a misfit dislocation (MD) array, parallel to the interface$^{41,42}$. As the layer thickness increases, some of these MD segments will continue as threading dislocation (TD) segments and emerge through the active layer (green color), which is very likely to degrade the final device performance.

* In FIG. 4.1.b, we show the schematics of an individual nanowire. As each nanowire remains isolated, they present a large lateral surface. In these conditions, the heteroepitaxial strain at the nanowire/substrate interface is still present, but it can be fully or partially relaxed by the free surface (black arrows). In the ideal picture of FIG. 4.1.b, the nanowire diameter is below the critical limit, so that all the strain from the mismatched heterointerface is accommodated elastically$^{43}$.

* The case of a nanowire above the critical diameter limit is shown FIG. 4.1.c. As in the 2D case, MD segments are introduced to accommodate the strain at the interface. These MD segments have a finite length and some can continue as TD segments in the nanowire volume. In contrast with the planar layer, the length of these TD segments is now much limited, as all TDs can now emerge at a short distance through the lateral surface, far from the active layer (green color). Whether TDs are created at the heterointerface or pre-exist in the substrate, the volume affected remains extremely limited, close to the base of the nanowire.

* The case of heteroepitaxial strain between an active region (green) and the nanowire matrix (grey) is a variant of FIG. 4.1.c. In such *axial* heterostructure, the layer composition gradient aligns with the NW axis. The heteroepitaxial strain in these axial structures obeys similar rules to those for a nanowire and a lattice-mismatched substrate. All the strain can be elastically accommodated below a critical radius (see FIG. 4.1.b arrows) and MD segments are introduced appearing above this radius. Similarly to FIG. 4.1.b, the MD segments will typically emerge through the lateral surface, with short or no TD segment at all.

<figure><img src="image_54.png" /><figcaption>**Figure 4.2:** Cross-section TEM analysis of GaAs nanowires grown on a patterned Si(111) substrate (3.6% mismatch). The nanowire diameter is in the right range for a fully elastic relaxation but individual dislocations are observed at the interface, possibly due to the presence of the SiO$_2$ mask. (a) Large view STEM-HAADF image. (b) Atomically resolved STEM-HAADF image, showing the position of a single dislocation at the interface. (F. Oehler, unpublished data)</figcaption></figure>

<footnote>$^{41}$J. Matthews and A. Blakeslee. "Defects in epitaxial multilayers". *Journal of Crystal Growth* 27 (1974), pp. 118-125. DOI: 10.1016/s0022-0248(74)80055-2.</footnote>

<footnote>$^{42}$S. M. Hu. "Misfit dislocations and critical thickness of heteroepitaxy". *Journal of Applied Physics* 69.11 (1991), pp. 7901-7903. DOI: 10.1063/1.347476.</footnote>

<footnote>$^{43}$F. Glas. "Critical dimensions for the plastic relaxation of strained axial heterostructures in free-standing nanowires". *Physical Review B* 74.12 (2006). DOI: 10.1103/physrevb.74.121302.</footnote>
4.1.2 Radial strain and heterostructures

As seen above (§4.1.1), the lateral surface around each nanowire changes the rules of strain relaxation in heteroepitaxy. This lateral surface also offers a major opportunity for the growth of active layer(s), as shell(s) of a different material wrapped around the nanowire core. Such radial heterostructures, also referred to as *core-shell*, are not available in planar geometries and offer new perspectives for crystal growth and devices.

First, *core-shell* structures offer lateral crystal facets with normal directions perpendicular to the substrate orientation. For polar semiconductors, such as III-Vs or III-N, this has direct implications on the physical properties of the active layer (quantum wells). For example, the direct growth of non-polar planar GaN layers on Al<sub>2</sub>O<sub>3</sub> is difficult (§2.4.2 page 20), and the cost of bulk (11&#x2126;00) substrates (§2.3.2 page 16) may be prohibitively expensive, while the combined lateral sidewalls of a GaN<0001> nanowire ensemble grown on a conventional Al<sub>2</sub>O<sub>3</sub> (0001) or Si(111) substrate may offer a comparable non-polar area (11&#x2126;00) of good crystal quality.

Second, *core-shell* structures theoretically permit to extend the actual active layer volume beyond the planar limit. For planar geometries, the volume of a quantum well of a given thickness is limited by the total area of the wafer, whereas core-shell nanowire (theoretically) offers an unrestricted volume as the nanowire height is a free parameter. In practice, high densities of nanowires combined to long nanowire heights quickly bring experimental challenges in the form of excessive shadowing (see §4.2.2, page 45).

Finally, *core-shell* nanowires also follow different strain relaxation rules than conventional planar heteroepitaxy. Similarly to nanowire axial heterostructures, the strain relaxation criteria now depends on the respective diameters of the core and shell regions. In conventional planar heteroepitaxy, the strain concentrates in the epilayer due to the very large thickness of the substrate. Here, the relative thicknesses of the *shell* (epilayer) and *core* (substrate) can be comparable or even reversed. The association of a very thin *core* to a thicker *shell* is emblematic of nanowire geometries, as the strain will tend to localize in the nanowire *core* (i.e. the ‘substrate’) while avoiding the shell volume (i.e. the ‘epilayer’), which is the exact opposite of the conventional planar case.

<figure><img src="image_55.png" /><figcaption>**Figure 4.3:** Strained core-shell structures with different relative core-shell thicknesses. (a-c) Conventional case, ‘thin shell / thick core’ (d-e) Reverse case ‘thick shell / thin core’, specific to the nanowire geometry. The colored maps are the diagonal strain tensor components $\epsilon$ in cylindrical coordinates. (adapted from [thesis_pishchagin_2021])</figcaption></figure>

FIG. 4.3 shows the elastic modeling of a GaAs nanowire core ($a_{GaAs} = 5.6533\AA$) associated to an homogeneous epitaxial GaP shell ($a_{GaP} = 5.4505\AA$), taken from [thesis_pishchagin_2021]. The nanowire is considered infinite along the $z$ axis and the problem is treated in cylindrical coordinates. In the conventional case, the GaP shell is thinner than the GaAs core, FIG. 4.3.(a-c). If we ‘unroll’ the nanowire along $\theta$, we have a similar situation to
the planar case, with a positive tensile 'in-plane' strain in the GaP shell along $\epsilon_{zz}$ and compressive strain 'out-of-plane' along $\epsilon_{rr}$. The strain is exclusively localized in the GaP shell and the GaAs core is nearly strain-free (cyan color, $\epsilon_{ii} \simeq 0$).

The nanowire-specific combination of a 'thin' core and 'thicker' shell is shown Fig. 4.3.(d-f). Along the (infinite) direction $z$, the now negative strain $\epsilon_{zz}$ is almost fully localized in the core (dark blue, Fig. 4.3.f). Unrolling the nanowire, this would correspond to (compressive) strain in the GaAs 'substrate' to accommodate the (mostly) strain-free GaP 'epilayer'. For the other strain components ($\epsilon_{rr}, \epsilon_{\theta\theta}$), the strain is localized close to the heterointerface.

The cylindrical geometry shown Fig. 4.3 also opens interesting theoretical grounds for strain relaxation. In contrast to the planar geometry, in which the in-plane strain is accommodated by arrays of misfit dislocation along geometrically-equivalent directions in the plane, the core-shell geometry now distinguishes between two types of (pure-edge) misfit dislocation.

<figure><img src="image_56.png" /><figcaption>**Figure 4.4:** (a) Line and (b) Loop misfit dislocation in core-shell nanowire structure. (from [thesis_pishchagin_2021])</figcaption></figure>

FIG. 4.4.a shows the conventional 'line' dislocation. This is a pure edge dislocation (Burgers vector **b** perpendicular to dislocation line) running along the infinite $z$ axis of the heterointerface, similarly to the planar case. In isotropic media and axisymmetric geometries, the Burgers vector $b_{line}$ is simply taken tangential. Real (anisotropic) crystals will show specific sidewall facets instead of a curved interface surface and the Burgers vectors will take specific orientations.

The nanowire specific 'loop' dislocation (FIG. 4.4.b) is also a pure edge misfit dislocation running along the rolled-up dimension of the nanowire. In the case of the cubic <111>-oriented nanowire, the addition or removal of a single (111) plane along the <111> growth axis is actually a stacking fault (intrinsic if a plane is missing, extrinsic if a plane is added). The full restoration of the crystal ordering along the cubic <111> axis would impose not one, but three planes to added or removed ('ABC'-type stacking), which is unlikely to happen as a single defect. Technically, the depicted pure edge-dislocation of FIG. 4.4.a thus qualifies as a Frank partial dislocation<sup>44</sup> in our cubic <111> oriented nanowires. Whether such ideal partial dislocations can form in a real cubic crystal is debatable (same extra plane missing/added left and right of the core) and was not observed in experiments, which favor more complex structures<sup>45</sup>.

Whether 'loop' or 'line' dislocation presents the lowest energy threshold for partial relaxation is a difficult but interesting theoretical question. While the case of the simple 'loop' respects the cylindrical symmetry of the nanowire, the introduction of the 'line' dislocation at the radial heterointerface breaks the cylindrical symmetry and requires special attention[thesis_pishchagin_2021].

For a given lattice mismatch (and elastic coefficients), the core-shell geometry is characterized by a critical radius $r_\infty$ below which the shell of arbitrary thickness can be grown. This is a direct consequence of the strain localization shown FIG. 4.3. We consider in the following a fixed core radius (below the critical value) and a growing shell:

* At low shell thickness, the strain is localized in the shell.

<footnote n="44"><sup>44</sup>Frank partial dislocations are distinguished from Shockley partial dislocations, depending on the orientation of their Burger's vector relative to the stacking fault plane.</footnote>

<footnote n="45"><sup>45</sup>S. A. Dayeh, W. Tang, F. Boioli, K. L. Kavanagh, H. Zheng, J. Wang, N. H. Mack, G. Swadener, J. Y. Huang, L. Miglio, K.-N. Tu, and S. T. Picraux. "Direct Measurement of Coherency Limits for Strain Relaxation in Heteroepitaxial Core/Shell Nanowires". Nano Letters 13.5 (2012), pp. 1869-1876. doi: 10.1021/nl3022434.</footnote>
* At intermediate shell thickness, the strain is balanced between core and shell regions.

* At large shell thickness, the strain is nearly totally localized in the core

Any subsequent shell growth now proceeds strain-free so that no extra energy accumulates in the system and arbitrary shell thicknesses can be obtained. In these conditions, the only limit is the (volumic) strain energy accumulated into the nanowire core, which must be small enough so that the introduction of misfit dislocations is not energetically favored. A small enough core volume (i.e. core radius) will always be able to accommodate strain elastically, without dislocation. Unpublished data from A. Pishchagin PhD[thesis_pishchagin_2021] shows that the experimental limit for GaP shell on GaAs core is about $r_{\infty}^{GaAs} \simeq 15$ nm (30 nm diameter) for ~3.6% lattice mismatch (FIG. 4.5.a).

Comparison of critical energy for the introduction of the first ‘loop’ or ‘line’ dislocation in theoretical GaAs/GaP core-shell structures indicates that ‘line’ dislocations should be introduced first (FIG. 4.5.b, light blue). However the observed experimental threshold (FIG. 4.5.b, dashed green) is closer the theoretical limit imposed by the ‘loop’ dislocations (FIG. 4.5.b, black). Note that the experimentally observed dislocations are also much more complex than the idealized ‘loop’ or ‘line’ dislocations studied theoretically.

<figure><img src="image_57.png" /><figcaption>**Figure 4.5:** (a) Scatter plot showing the fully-strained (blue) or plastically relaxed (red) character of GaP/GaAs core-shell heterostructures for various shell (H) and core ($r_c$) thicknesses. (b) Comparison of theoretical shell limits for the formation of isolated ‘loop’ and ‘line’ misfit dislocations. (c-e) Selected TEM images used to populate the experimental data in (a). (adapted from [thesis_pishchagin_2021])</figcaption></figure>

**Conclusion:** Nanowires offer the promises of good quality semiconductor on mismatched substrates, which is very interesting for the direct growth of (expensive) III-V semiconductors on (much cheaper) silicon substrates. For many cubic III-V systems, the growth of nanowires usually proceeds along the <111> direction, so that Si(111) substrates can be used to obtain vertical nanowires. The lattice mismatch between individual nanowire and substrate can be accommodated elastically if the nanowire radius is small enough. Above this critical radius, misfit dislocations can appear but their spatial extension is often limited since they may emerge on the neighboring sidewalls, close to the mismatched interface, far from the active regions (FIG. 4.2).

Nanowires also offer interesting perspectives for the growth of radial heterostructures, making use of their large lateral surface. In this geometry, defect-free shells of arbitrary thickness can also be obtained, provided the nanowire core is smaller than (another) critical radius (FIG. 4.5).
## 4.2. Vapor Liquid Solid growth

Contrary to the GaN nanowires of Chapter 3 (page 36) which grow directly from the vapor phase without a catalyst, the III-V nanowires (GaAs, GaP) discussed in this chapter are obtained by the Vapor-Liquid-Solid (VLS) growth mechanism$^{46}$. The VLS mechanism can be seen as a catalyst-assisted vapor-solid growth, in which the precursor first accumulates into a liquid (nano-)particule, i.e. the catalyst, before crystallizing into a solid column, i.e. the nanowire (FIG. 4.6.a). The flux of mass between the different phases, from the Vapor to the Liquid, and from the Liquid to the final Solid is at the origin of the VLS acronym.

If the catalyst is in the solid state, the growth mechanism is named Vapor-Solid-Solid (VSS)$^{47}$. Such naming scheme is so generic so that ALILE crystallization (Chapter 3) could be labeled ‘Solid-Solid-Solid’ (SSS) and the conventional planar growth (Chapter 2) is often referred as Vapor-Solid (VS). The VLS growth of nanostructures is not limited to III-V semiconductors or elementary Si or Ge, but also includes many metal oxides$^{48}$. FIG. 4.6.(b-c) shows examples of Au-catalyzed silicon nanowires grown by VLS using Molecular Beam Epitaxy (MBE) or Chemical Vapor Deposition (CVD) reactors from the author’s PhD thesis[thesis\_oehler\_2010].

<figure><img src="image_58.png" /><figcaption>**Figure 4.6:** (a) Schematic view of the Si mass transport during the Au-catalyzed VLS growth of Si nanowires. (b-c) Example of Au-catalyzed Si nanowires grown by VLS using Molecular Beam Epitaxy (MBE) and Chemical Vapor Deposition (CVD). (d) (adapted from the author’s PhD thesis [thesis\_oehler\_2010])</figcaption></figure>

### 4.2.1 Self-catalyzed III-V nanowire growth by MBE

A particular flavor of III-V VLS nanowire growth is the Ga-catalyzed VLS growth of GaAs$^{49}$ or GaP [Prante2015] nanowires by MBE. Here the liquid catalyst mainly consists in group III metal (Ga), complemented by group V at concentration below a few percents. Such growth scheme is rather unique as it does not introduce any other element than those used directly for the growth of the solid III-V phase, in contrast to other VLS growth variants based on gold, silver or platinum catalyst (see FIG. 4.6).

For III-V nanowires, this Ga-catalyzed growth mechanism is thus referred to as *self-catalyzed* VLS growth. It brings the following advantages:

1. Possible deep trap states in the III-V semi-conductor originating from foreign metal elements are avoided.

2. The Ga catalyst can be localized on Si(111) substrates using a SiO$_2$ mask, which results in the selective growth of vertical GaAs nanowires at specific locations, determined by standard lithography techniques.

3. The Ga catalyst can be created *in situ*, from the same high purity Ga sources as used during the III-V growth.

<footnote>$^{46}$R. S. Wagner and W. C. Ellis. "Vapor-Liquid-Solid Mechanism of single crystal growth". *Applied Physics Letters* 4.5 (1964), pp. 89-90. DOI: 10.1063/1.1753975.</footnote>

<footnote>$^{47}$S. Kodambaka, J. Tersoff, M. C. Reuter, and F. M. Ross. "Germanium Nanowire Growth Below the Eutectic Temperature". *Science* 316.5825 (2007), pp. 729-732. DOI: 10.1126/science.1139105.</footnote>

<footnote>$^{48}$A. Klamchuen, M. Suzuki, K. Nagashima, H. Yoshida, M. Kanai, F. Zhuge, Y. He, G. Meng, S. Kai, S. Takeda, T. Kawai, and T. Yanagida. "Rational Concept for Designing Vapor-Liquid-Solid Growth of Single Crystalline Metal Oxide Nanowires". *Nano Letters* 15.10 (2015), pp. 6406-6412. DOI: 10.1021/acs.nanolett.5b01604.</footnote>

<footnote>$^{49}$C. Colombo, D. Spirkoska, M. Frimmer, G. Abstreiter, and A. F. i Morral. "Ga-assisted catalyst-free growth mechanism of GaAs nanowires by molecular beam epitaxy". *Physical Review B* 77.15 (2008), p. 155326. DOI: 10.1103/physrevb.77.155326.</footnote>
4. The same Ga catalyst can be destroyed (i.e. consumed) *in situ* by saturating it with group V atoms to create III-V solid.

Combining (3) and (4), we see that the growth conditions can be chosen to either maintain a stable catalyst shape or to increase or decrease its volume (see FIG. 4.7).

<figure><img src="image_59.png" /><figcaption>**Figure 4.7:** (a) SEM view (45° tilt) of vertical self-catalyzed (Ga-catalyzed) GaAs nanowires of near constant diameter grown by MBE inside patterned apertures etched in a SiO₂/Si(111) wafer. (b) SEM view (45° tilt) of self-catalyzed GaAs NWs of varying diameter on a similar patterned SiO₂/Si(111) wafer. The diameter variations are obtained by changing the Ga droplet size *in situ*. (F. Oehler, unpublished data).</figcaption></figure>

Another advantage of self-catalyzed VLS is found in the modeling of this growth mechanism, as we only need to consider the interaction between the two chemical species present in the liquid catalyst. This way, one can attempt the (complex) modeling of this particular growth process, as proposed by Glas *et al.*<fnref n="50" />.

<figure><img src="image_60.png" /><figcaption>**Figure 4.8:** (a) Parameters used for the VLS growth modeling. (b) Variations of the growth rate $\rho_n$, As evaporation rate $\rho_e$ and As concentration $c_{As}$ in the catalyst with the impinging As flux. (from Glas *et al*.)</figcaption></figure>

Taking into account the near linear experimental dependence of the axial growth rate of self-catalyzed GaAs nanowires with the impinging As flux (FIG. 4.8.b, red curve), Glas *et al.* provide a model of the self-catalyzed VLS growth combining thermodynamics, nucleation theory and experiments.

They distinguish two As incoming fluxes, $\rho_d$ the ballistic direct flux from the As source to the liquid catalyst surface, and $\rho_r$ the indirect flux due to As re-emission from neighboring surfaces.

Similarly two outgoing As fluxes are taken into account, $\rho_n$ the As material used for growth using nucleation theory and $\rho_e$ the evaporating rate from the droplet back into the vapor.

Then they balance the variations of the As chemical potential $\mu_{As}$ inside the As-Ga liquid with respect to these positive and negative As fluxes. The numerical applications are key, based on existing Ga-As reference thermodynamic data<fnref n="51" />, inferred parameters and experiments.

<footnote n="50">⁵⁰F. Glas, M. R. Ramdani, G. Patriarche, and J.-C. Harmand. “Predictive modeling of self-catalyzed III-V nanowire growth”. *Physical Review B* 88.19 (2013), p. 195304. DOI: 10.1103/physrevb.88.195304.</footnote>

<footnote n="51">⁵¹J. Ansara, C. Chatillon, H. Lukas, T. Nishizawa, H. Ohtani, K. Ishida, M. Hillert, B. Sundman, B. Argent, A. Watson, T. Chart, and T. Anderson. “A binary database for III-V compound semiconductor systems”. *Calphad* 18.2 (1994), pp. 177–222. DOI: 10.1016/0364-5916(94)90027-2.</footnote>
4.2.2 Group III flux to the VLS catalyst

At the end of the previous section (§4.2.1), we have seen that the axial growth rate of self-catalyzed GaAs nanowires is mostly governed by the As flux (group V) reaching the VLS catalyst and the As evaporation from the droplet. This means that the Ga flux (group III) is not limiting the nanowire vertical growth rate in the usual growth conditions, so that some balancing mechanism adapts the Ga input to match the amount of As material consumed during the GaAs crystallization. This adaptation may persist nearly over one order of magnitude in growth rate variation, so that Tersoff<fnref n="52" /> and Dubrovskii et al.<fnref n="53" /> have looked into the stable growth of such self-catalyzed III-V nanowires and how Ga is collected both by the nanowire sidewalls and catalyst to support the axial growth.

In the above attempts, the very existence of ‘stable’ growth can be questioned and the present section tries to solve an even simpler question : ‘Where does the Ga material comes from and where does it go ?’.

For planar layers, the answer is nearly as short as the question ‘Ga comes from the Knudsen cell, hits the surface, forms GaAs or re-evaporates’ and it can be verified experimentally by measuring the in situ planar growth rate from the intensity oscillations of reflection high energy electron diffraction patterns (RHEED).

The same question in the context of tri-dimensional high-aspect ratio nanostructures (aka. nanowires) is
more demanding:

* In absence of in situ growth rate measurement for nanowires inside conventional MBE setups, we have to rely on ex situ observations. The *only* correct way to measure the actual growth rate of nanowire is to then insert marker layers at regular intervals. This way, we can evaluate the volume added between two consecutive markers by ex situ microscopy, determine the exact shape of the nanowire at any given time and compute the corresponding growth rate.

* The environment surrounding each nanostructure must be *completely* known to quantify the exact fluxes reaching all the relevant surfaces using a full three-dimensional treatment (sidewalls, substrate, droplet). In practice, this imposes the nanowire to be either completely isolated, or confined to particular geometries such as regular arrays.

* The actual atomic fluxes must be calibrated beforehand, on reference planar substrates using RHEED oscillations measurements or other in situ means.

Gibson et al.<fnref n="54" />, Dalacu et al.<fnref n="55" /> or Keilrich et al.<fnref n="56" /> have proposed approximate treatments of the flux impinging the nanowire, with various assumptions. In [Oehler2018], we propose a full tri-dimensional numerical treatment of the Ga flux impinging a regular nanowire array. Matching experimental data are provided to test and assess the validity of various scenarios.

Figure FIG. 4.9 presents the ex situ analysis of arrays of Ga-catalyzed GaP nanowires grown on a patterned Si(111) substrate. The hexagonal array of nanowires is relatively dense: pitch is 500 nm with final nanowire diameter and length of 140 nm and 3.2 µm respectively. Cross-section TEM analysis (FIG. 4.9.c-g) reveals the GaAsxP1-x marker layers (x ~ 0.04) which are used to reconstruct the axial and radial growth rates (FIG. 4.10.a).

A most interesting feature of the GaAsxP1-x marker layers is that their lateral extension gives the instanta-
neous nanowire diameter, which is very different from the *final* diameter measured after growth (FIG. 4.10.b).
The diameter at the start of the growth (~ 40 nm) is about three times smaller than that of the final markers
(~ 140 nm). The actual GaAs volume created at each step is thus a combination of axial growth (FIG. 4.10.c,
grey) and radial growth (cyan), which distribute so that the nanowire sidewall remains (nearly) vertical.

<footnote n="52">⁵²J. Tersoff. “Stable Self-Catalyzed Growth of III-V Nanowires”. *Nano Letters* 15.10 (2015), pp. 6609–6613. DOI: 10.1021/nan012386.</footnote>

<footnote n="53">⁵³V. G. Dubrovskii, T. Xu, A. D. Álvarez, S. R. Plissard, P. Caroff, F. Glas, and B. Grandidier. "Self-Equilibration of the Diameter of Ga-Catalyzed GaAs Nanowires". *Nano Letters* 15.8 (2015), pp. 5580–5584. DOI: 10.1021/nan012226.</footnote>

<footnote n="54">⁵⁴S. J. Gibson and R. R. LaPierre. “Model of patterned self-assisted nanowire growth”. *Nanotechnology* 25.41 (2014), p. 415304.
DOI: 10.1088/0957-4484/25/41/415304.</footnote>

<footnote n="55">⁵⁵D. Dalacu, A. Kam, D. G. Austing, X. Wu, J. Lapointe, G. C. Aers, and P. J. Poole. “Selective-area vapour-liquid-solid growth of InP nanowires”. *Nanotechnology* 20.39 (2009), p. 395602. DOI: 10.1088/0957-4484/20/39/395602.</footnote>

<footnote n="56">⁵⁶A. Kelrich, Y. Calahorra, Y. Greenberg, A. Gavrilov, S. Cohen, and D. Ritter. “Shadowing and mask opening effects during selective-area vapor–liquid–solid growth of InP nanowires by metalorganic molecular beam epitaxy”. *Nanotechnology* 24.47 (2013), p. 475302. DOI: 10.1088/0957-4484/24/47/475302.</footnote>
The importance of FIG. 4.10.c relative to growth modeling should not be understated: a 'constant' axial growth rate (FIG. 4.10.a) is obtained in an 'unstable' geometry, with varying nanowire diameters and lengths (FIG. 4.10.b).

<figure><img src="image_61.png" /><figcaption>**Figure 4.9:** (a-b) SEM bird eye and top view of a hexagonal array of Ga-catalyzed GaP nanowire grown by MBE. (c-g) Cross-section TEM along an individual nanowire in the array. The zoomed views reveal the location and spatial extension of the GaAsP marker layers. (from [Oehler2018]).</figcaption></figure>

<figure><img src="image_62.png" /><figcaption>**Figure 4.10:** Reconstructed axial (a) and radial (b) growth rates. (c) Overall growth schematic. (from [Oehler2018]).</figcaption></figure>

FIG. 4.11.a evaluates quantitatively the instantaneous Ga amount consumed by the axial (dashes) and radial (squares) growth, and by catalyst enlargement (dots). We see that the axial and radial grown volumes are *comparable* for most of the growth duration.

This statement is qualitatively represented in FIG. 4.10.c, stating that it may be a *gross* approximation to neglect the radial growth rate in nanowire.

<figure><img src="image_63.png" /><figcaption>**Figure 4.11:** (a) Ga flux consumed by axial and radial growth, and by catalyst enlargement. (b) Ga flux arriving to the catalyst, directly hitting the droplet (blue) and the nanowire sidewalls (pink). (adapted from [Oehler2018]).</figcaption></figure>
FIG. 4.11.b compares the instantaneous Ga fluxes which reach the catalyst, either by direct impingement (blue curve) or direct flux hitting the nanowire sidewall followed by a total migration to the catalyst (pink curve). We assume here that all Ga material can freely distribute in the axial or radial grown volumes, with no physical constraint (i.e. unlimited surface migration length).

* The hashed-pink area is the result of the conventional Ga balance equation, which is limited to the axial growth. Here, the Ga flux directly impinging on the catalyst (blue curve) is vastly insufficient to support the Ga lost through axial growth (black dots). Most of the Ga used for axial growth is actually collected by the nanowire sidewalls (pink), before it migrates to the catalyst (hashed-pink area).

* The full-pink area is determined using the extended Ga balance equation proposed in [Oehler2018], which now includes the Ga consumption from both axial and radial growth. We see that there is not quite enough Ga from the direct fluxes considered so far (catalyst and sidewalls) to account for all the created GaAs volume, axial and radial, so that additional Ga sources have to be considered.

The computation of directly impinging Ga fluxes in a dense nanowire array is a complex 3D problem. Instead of guessing simplifying assumptions, which may be limited to particular ranges of nanowire diameter or length, we implement here a numerical 3D simulation of ballistic trajectories for arbitrary diameter and height. This method is universal and gives an unbiased account of all the experimental nanowire geometries without resorting to simplifying assumptions.

FIG. 4.12.a shows the surface density of impinging Ga atoms for an isolated vertical nanowire (i.e. nanowire height and diameter small compared to the array pitch size) obtained by the numerical simulation of ballistic Ga trajectories. The Ga density is normalized so that it is unity on the horizontal substrate, far from the nanowire (FIG. 4.12.a, dark red). Due to substrate rotation and the cell tilt from the substrate normal (32°), the nanowire sidewalls locally collect much less Ga than the substrate (FIG. 4.12.a, blue). The Ga collected by the sidewalls is now missing on the substrate and the corresponding area appears as the shadow of the nanowire on the substrate (FIG. 4.12.a, light red).

As the nanowire height increases, the shadowing of the substrate becomes more complex (FIG. 4.12.c) and we now have to implement periodic boundaries (FIG. 4.12.b) in the simulated trajectories to correctly account for the overlapping multiple shadows.

Shadowing also occurs on the nanowire sidewalls (FIG. 4.12.d) due to the inclined Ga effusion cell. The shadows of the first, second and third rings of nearest neighbor nanowires are visible (FIG. 4.12.e).

A possible source for the 'missing' Ga is the re-evaporation of Ga from the silica mask. As the GaP growth is selective on SiO₂, all impinging Ga atoms on the mask must either stick and desorb (high sticking coefficient), or simply bounce off the mask (low sticking coefficient). These two extreme cases are investigated considering a Lambertian re-emission (cosine law) or the simple specular reflection of Ga from the surface.

<figure><img src="image_64.png" /><figcaption>**Figure 4.12:** Surface Ga density from direct impingement in arrayed 3D nanostructure. (a) Simple case. (b) Boundary condition in simulation. (c) Substrate shadowing. (d) Sidewall shadowing. (e) First and second neighbors in hexagonal array. (from [Oehler2018]).</figcaption></figure>
<figure><img src="image_65.png" /><figcaption>**Figure 4.13:** (a) Ga mass balance with Lambertian surface re-emission. (b) Ga mass balance from specular reflection. Ga currents are normalized with respect to the total Ga current flown in the unit cell of the nanowire array. (adapted from [Oehler2018]).</figcaption></figure>

FIG. 4.13 presents the Ga currents for the two reemission schemes considered, obtained from 3D trajectory simulations (FIG. 4.12). In both cases, the extra Ga flux ‘reflected’ from the SiO₂ substrate is enough to account for the missing Ga (dashed magenta curves). The specular reflection (FIG. 4.13.b) is nearly spot on but the Lambertian distribution also gives a decent approximation. Note that there are no free parameters in these geometrical simulations.

For dense enough nanowire arrays, nearly all the Ga arriving in the array unit cell is captured and feeds the GaP growth (Ga current ≈ 1 in FIG. 4.13). Most of the Ga atoms are either directly captured by nanowire sidewalls directly (direct flux, pink full line) or after their reemission from the surface (indirect flux, pink dashed line). Shadowing (wire to wire, wire to surface, ...) is the physical mechanism which distributes the available Ga to the various surfaces inside the unit cell.

**Conclusion:** Far away from the model picture of a 'stable growth', we see that the nanowire axial growth rate can be constant over a long period in spite of a quickly varying nanowire geometry (height and diameter). To correctly model the evolution of the nanowire geometry, both the axial and radial growth rates must be taken into account. This requires an improved version of the Ga mass balance, in which an additional Ga flux, 'reflected' from the substrate, must be taken into account. Here, the use of a well-controlled arrayed environment allows one to compute numerically the impinging Ga fluxes without limiting approximations. The specular re-emission of Ga on the flat SiO₂ mask is a good candidate to explain the experimental data.

### 4.2.3 Group V flux on the VLS catalyst

After §4.2.2, we now have a much better picture of the group III fluxes during the growth of self-catalyzed GaP nanowires, including shadowing in dense nanowire arrays. We see that the axial growth proceeds at a constant rate, unaffected by the delicate balance of the captured group III atoms between axial and radial growth. This raises questions on the surfaces capturing the group V fluxes, which we try to investigate in this section.

To improve over the numerical approach of §4.2.2, we set up a local environment for individual nanowire with minimalist shadowing (see FIG. 4.12) so that we can derive analytic formula. Two conditions are required:

1. Nanowires are grown in arrays of large pitch, so that each nanowire can be considered isolated from the shadow of its nearest neighbor (large pitch compared to nanowire height and diameter).

2. The nanowire shape is not very much during the experiment (diameter and height) so that selected approximations can be performed with comparable error for all geometries.

Under assumptions (1) and (2), we can obtain analytic expressions for the ‘direct’ and ‘indirect’ impinging group V fluxes on the sidewalls of an isolated nanowire vertically oriented on a planar mask.
Experimentally, a possible solution to condition (2) is to stop the group III flux, so that the only source of group III element is the Ga contained in liquid VLS catalyst. If the nanowire height is large enough, this finite Ga amount represents a small extra III-V volume relative to that of the total nanowire. Similarly, the shrinking of catalyst droplet by VLS consumption should only lightly affect to total capture section. Yet, all these changes can be finely modeled and accounted for by tracking the variation of the droplet contact angle.

Following the results of §4.2.2, two configurations are investigated, namely *specular reflection* and *Lambertian re-emission* of the group V element from the SiO₂ mask. These correspond to two extreme cases: no interaction of the group V with the mask (specular) and strong interaction with the mask (Lambertian). Alternatively, we can also describe these model cases as full-memory (specular) and no-memory (Lambertian) of the particle trajectory prior to its contact with mask.

*Note that the case of 'isotropic' re-emission (investigated in [Oehler2018]) is not considered here because it yields a non-physical result (infinite flux for an infinite mask).*

<figure><img src="image_66.png" /><figcaption>**Figure 4.14:** Direct and indirect fluxes on the nanowire sidewall, approximated as a cylinder. (a) Direct flux from the effusion cell. (b) Flux from the mask, after specular reflection. (c) Flux from the mask after Lambertian re-emission.</figcaption></figure>

With $\Phi_{\text{side}}^s$ the total group V current on the nanowire surface after integration on the relevant surface ($r$ and $H$, the radius and height of the wire) and $\phi_{2D}$ a reference ‘planar’ atomic flux, we find:

$$ 
\begin{align} 
\Phi_{\text{direct}}^{\text{side}} &= \phi_{2D} \cdot \tan(\alpha) \cdot 2rH \nonumber \\ 
\Phi_{\text{specular}}^{\text{side}} &= \Phi_{\text{direct}}^{\text{side}} \tag{4.1} \\ 
\Phi_{\text{Lambert}}^{\text{side}} &= \phi_{2D} \cdot \pi r H \nonumber 
\end{align} 
$$

All currents are normalized with respect to the ‘planar’ atomic flux ($\phi_{2D}$), which is an equivalent planar growth rate experimentally determined using RHEED oscillations. The angle $\alpha$ is the tilt of the effusion cell with respect to the substrate normal. Substrate rotation is included in the calculation. All formulas for fluxes impinging vertical nanowire sidewalls modeled as cylinder have already been demonstrated by several authors, notably the Lambertian case by Dalacu et al.<fnref n="57" /> or Glas & Harmand<fnref n="58" />.

The calculation of the flux hitting the VLS catalyst, approximated as a spherical cap is more mathematically involved. For direct fluxes, formula are given by Glas<fnref n="59" /> for cell inclination angles $\alpha \in [0, \pi/2]$. They are extended in [Pishchagin2021] to account for specular reflection from the mask with $\alpha \in [\pi/2, \pi]$.

<footnote n="57">⁵⁷D. Dalacu, A. Kam, D. G. Austing, X. Wu, J. Lapointe, G. C. Aers, and P. J. Poole. “Selective-area vapour-liquid-solid growth of InP nanowires”. *Nanotechnology* 20.39 (2009), p. 395602. DOI: 10.1088/0957-4484/20/39/395602.</footnote>

<footnote n="58">⁵⁸F. Glas and J.-C. Harmand. “Calculation of the temperature profile in nanowhiskers growing on a hot substrate”. *Physical Review B* 73.15 (2006), p. 155320. DOI: 10.1103/physrevb.73.155320.</footnote>

<footnote n="59">⁵⁹F. Glas. “Vapor fluxes on the apical droplet during nanowire growth by molecular beam epitaxy”. *physica status solidi* (b) 252.8 (2015), pp. 1897–1897. DOI: 10.1002/pssb.201552440.</footnote>
The formula now depends on three parameters, the cell tilt angle $\alpha$ and the contact angle $\beta$ of the VLS catalyst, approximated as a spherical cap of radius $R$ sitting at the top of a cylinder.

Note that this is a 3D geometry, which requires a full 3D treatment to obtain the correct formula. All formulas for the direct flux to catalyst have the same structure, with projected capture section $A(\alpha, \beta, R)$.

$$ \Phi_{\text{direct}}^{\text{catalyst}}(\alpha, \beta, R) = A(\alpha, \beta, R) \cdot \frac{\phi_{2D}}{\cos \alpha} \quad (4.2) $$

We choose here to classify the cases depending on $\beta$ with $(\beta \le \pi/2)$ and $(\beta \ge \pi/2)$, respectively.

The formula in blue color, eq.(4.6), eq.(4.7) and eq.(4.11), have been derived in [Pishchagin2021].

$$ (\beta \le \pi/2) $$

$$ \bullet \ 0 \le \alpha \le \pi/2 - \beta: $$

$$ A = R^2 \pi \sin^2 \beta \cos \alpha \quad (4.3) $$

$$ \bullet \ \pi/2 - \beta \le \alpha \le \pi/2: $$

$$ A = R^2 \cdot \left[ \arcsin \left( \frac{\sqrt{\sin^2 \beta - \cos^2 \alpha}}{\sin \alpha} \right) + \sin^2 \beta \cos \alpha \cdot \left( \pi - \arcsin \left( \frac{\sqrt{\sin^2 \alpha - \cos^2 \beta}}{\sin \alpha \sin \beta} \right) \right) - \cos \beta \cdot \sqrt{\sin^2 \beta - \cos^2 \alpha} \right] \quad (4.4) $$

$$ \bullet \ \alpha = \pi/2: $$

$$ A = R^2 \cdot \left( \beta - \cos \beta \sin \beta \right) \quad (4.5) $$

$$ \bullet \ \pi/2 \le \alpha \le \pi/2 + \beta: $$

$$ A = R^2 \left[ \arcsin \left( \frac{\sqrt{\sin^2 \alpha - \cos^2 \beta}}{\sin \alpha} \right) + \sin^2 \beta \cos \alpha \cdot \arcsin \left( \frac{\sqrt{\sin^2 \alpha - \cos^2 \beta}}{\sin \alpha \sin \beta} \right) - \cos \beta \sqrt{\sin^2 \beta - \cos^2 \alpha} \right] \quad (4.6) $$

$$ \bullet \ \pi/2 + \beta \le \alpha \le \pi: $$

$$ A = 0 \quad (4.7) $$

<figure><img src="image_67.png" /><figcaption>Figure 4.15: Contact angle $\beta$ and cell tilt $\alpha$.</figcaption></figure>

$$ (\beta \ge \pi/2) $$

$$ \bullet \ 0 \le \alpha \le \beta - \pi/2 $$

$$ A = R^2 \pi \quad (4.8) $$

$$ \bullet \ \beta - \pi/2 \le \alpha < \pi/2: $$

$$ A = R^2 \left( \pi/2 - \arcsin \left( \frac{\cos \beta}{\sin \alpha} \right) + \sin^2 \beta \cos \alpha \cdot \arcsin \left( \frac{\sqrt{\sin^2 \beta - \cos^2 \alpha}}{\sin \alpha \sin \beta} \right) - \cos \beta \cdot \sqrt{\sin^2 \beta - \cos^2 \alpha} \right) \quad (4.9) $$

$$ \bullet \ \alpha = \pi/2: $$

$$ A = R^2 \cdot \left( \beta - \cos \beta \sin \beta \right) \quad (4.10) $$

$$ \bullet \ \pi/2 \le \alpha \le \pi: $$

$$ A = R^2 \left( \beta + \cos \beta \sin \beta \cdot (1 - 2 \sin \alpha) + \frac{\pi}{2} \sin^2 \beta \cdot \cos \alpha \right) \quad (4.11) $$
Despite the compact form of eq.(4.1) for sidewall current, the calculation of the Lambertian re-emitted current from the infinite mask to the spherical cap is (even more) mathematically challenging.

<figure><img src="image_68.png" /><figcaption>**Figure 4.16:** Lambertian emitters problem. [F. Oehler, personal notes]</figcaption></figure>

Again, we insist this is a 3D problem, which requires careful parameterization and 3D integration schemes (see FIG. 4.16).

Contrary to direct or specularly re-emitted fluxes, the Lambertian emitters on the mask have to deal with the *self-shadowing* of the VLS catalyst by itself and by the nanowire stem. This *self-shadowing* is due to the blocking of the line of sight between some mask emitters and the considered surface elements of the VLS catalyst due to geometrical constraints. We distinguish:

1. Self-shadowing by the VLS catalyst, leading to a tangential plane shadow on the mask

2. Self-shadowing by the nanowire stem, leading to a triangular-like shadow on the mask

The concept of self-shadowing is important for the correct derivation of the integrated Lambertian re-emitted flux. FIG. 4.17 gives an annotated schematic of the mask emitters which can "see" a particular surface element of the VLS catalyst (cyan), opposed to those without line of sight (pink).

<figure><img src="image_69.png" /><figcaption>**Figure 4.17:** Annotated schematic of Lambertian re-emitters from the mask impinging on a particular catalyst surface element. (a) Mask surface elements with (cyan) or without (pink) line of sight to the considered catalyst surface element (small red cross). (b) Relative contribution of individual mask elements to the total current on the considered catalyst surface element. [F. Oehler, personal notes].</figcaption></figure>

* The tangential line (black), due to self-shadowing (1) first restrict the integration domain to a half plane.

* Self-shadowing (2) further encroaches on the possible emitters (pink triangle), depending on the droplet contact angle.
* For dense arrays, some of nearest nanowire stems may also shadow the remaining mask emitters. These contributions are not taken into account in the analytic formula eq(4.12), which is strictly valid for an isolated nanowire.

* For dense arrays, numerical integration is then the only solution and FIG. 4.17 gives an example of a simple numerical cutoff (dash blue line) used to limit the integration scheme.

In the limit of sparse nanowire arrays, the self-shadowing (1) can be fully taken into account to obtain a compact analytic formula, while self-shadowing (2) requires approximation if $\beta > \pi/2$.

The final result reads as follows, with R the radius of the VLS catalyst, $\beta$ the contact angle and $r = R \sin \beta$ the radius of the nanowire stem.

$$ \Phi_{\text{Lambert}}^{\text{catalyst}} = \phi_{2D} \cdot \frac{\pi R^2}{2} (1 - \cos \beta)^2 = \phi_{2D} \cdot \frac{\pi r^2}{2} \tan^2 \left( \frac{\beta}{2} \right) \quad (4.12) $$

Equation (4.12) is exact if $\beta \le \pi/2$ and constitutes a decent approximation for $\beta > \pi/2$, in the typical experimental range ($\beta_{max} \approx 120-130^{\circ}$). Curious readers may find (better) approximations in [Pishchagin2021] derived by F. Glas and A. Pishchagin, respectively.

The final formula for the integrated flux on the VLS catalyst (spherical cap) from an infinite plane of Lambertian re-emitters is remarkably compact in comparison with the direct or specular flux formulas, eq.(4.3) to eq.(4.11). Such compact formulation was also obtained for Lambertian emitters impinging the nanowire sidewalls, eq.(4.1).

<figure><img src="image_70.png" /><figcaption>Figure 4.18: (a) Ga droplet consumption under P exposure at the top of self-catalyzed GaP nanowires. (b) Final GaP nanowire array after complete catalyst consumption. (c) Pathways for the liquid Ga catalyst consumption. (adapted from [Pishchagin2021]).</figcaption></figure>

Now that we can evaluate quantitatively the direct and re-emitted fluxes of group V on the nanowire catalyst and sidewalls, we can model the catalyst consumption at the top of a self-catalyzed GaP nanowire under P exposure (i.e no Ga supplied during growth).

FIG. 4.18.a shows the corresponding experimental data, obtained in a (sparse) array of GaP nanowires grown on a patterned SiO$_2$/Si(111) substrate (FIG. 4.18.b). The different pathways for group III consumption are shown FIG. 4.18.c, including group III diffusion on the sidewalls and III-V growth using the direct, re-emitted and diffusing group V fluxes.
<figure><img src="image_71.png" /><figcaption>**Figure 4.19:** Evaluation of Ga volume lost by Ga surface migration (radial growth) and VLS (axial) growth during P exposure. (from [Pishchagin2021])</figcaption></figure>

As in [Oehler2018] (see §4.2.2), we also have used GaAsP marker layers during the GaP stem growth to evaluate the Ga volume lost by the catalyst through the Ga surface migration (which will create GaP on the nanowire sidewalls by radial growth) and the Ga volume lost by VLS axial growth during exposure of the catalyst to P.

With $V_0$ the equivalent GaP volume corresponding to the total amount of Ga atoms in the initial droplet, the GaP volume created by the *axial* (VLS) growth during the Ga catalyst consumption is evaluated to $V_{ax} = 0.53 V_0$. This value is comparable to that of the GaP volume $V_{rad} = 0.44 V_0$ created by *radial* (VS) growth from the Ga migrating away from the catalyst droplet.

Together $V_{rad}$ and $V_{ax}$ account for ~96% of the equivalent GaP catalyst volume $V_0$. Sidewall migration from the Ga catalyst under P exposure is therefore *not* negligible and accounts for nearly half of the volume loss of the Ga catalyst in our experimental conditions!

The axial growth is sustained by the group V fluxes reaching the catalyst surface. They include the direct and re-emitted group V fluxes on the droplet (see eq. (4.11) and eq. (4.12)), as well as an hypothetical group V diffusing on the GaP lateral sidewalls (see eq. (4.1)) within a given surface migration length to the catalyst $\lambda_V$.

<figure><img src="image_72.png" /><figcaption>**Figure 4.20:** (a) Group III atomic loss currents in the case of specular group V re-emission from the mask. (b) Similar computation for Lambertian re-emission from the mask. (adapted from [Pishchagin2021])</figcaption></figure>

FIG. 4.20 investigates the various group V fluxes required to account for the total volume decrease of the Ga droplet, with respect to the P exposure time. In FIG. 4.20.a, we consider a specular group V re-emission from the SiO₂ mask and we see that a significant group V surface migration is required to sustain the axial growth
(λ~V~ ≃ 400 nm). The outcome of the Lambertian re-emission scheme is shown FIG. 4.20.b, the group V surface migration is reduced (λ~V~ ≃ 80 nm) but still necessary to account for the observed Ga depletion rate.

Conclusion: The mass balance of group V atoms during the Ga catalyst consumption of self-catalyzed GaP nanowires under P exposure underlines again the simultaneous actions of the axial (VLS) and radial (VS) growth.

The quantitative modeling of the Ga volume loss indicates that the group V flux directly impinging on the VLS catalyst is far too small to account for the observed droplet depletion rate, even when considering Ga loss through surface migration and subsequent radial growth.

Additional group V fluxes, notably those re-emitted from the surface are required. We find that Lambertian re-emission is about 5 times more efficient than specular reflection for typical cell inclination angles (30-45°), but the cases investigated reveal that the indirect fluxes are still not sufficient to support the formed GaP volume. An extra group V flux to the catalyst in the form of surface migration is thus proposed.

## 4.3. Sidewall passivation of self-catalyzed GaAsP nanowires

Given the high surface to volume ratio of nanowires, surface effects (dangling bonds, Fermi level pinning, high recombination rates) can degrade the optical and electrical performances of nanowire-based devices. In these conditions, it may be interesting to deposit a thin shell of higher bandgap material around the nanowire core, to shield the carriers in the nanowire volume from the defective surface. Mastering such surface passivation process is an important step towards the fabrication of nanowire-based devices.

As the axial (VLS) growth rate is usually much faster than the radial (VS) growth rate, it may be necessary to remove the catalyst to grow a VS shell of *uniform* thickness. For self-catalyzed III-V nanowires, this can be done *in situ* by providing group V flux only, until the (group III) droplet is consumed (see §4.2.3).

### 4.3.1 Sidewall passivation and radial growth

In [Himwas2017], we investigate the surface passivation of GaAs~0.74~P~0.26~ nanowires with a GaP shell (E~GaP= 2.24 eV). The GaAsP alloy composition is chosen to obtain a (direct) band gap energy around 1.7 eV, which is the optimum value for a top III-V solar cell combined with a Si bottom cell in a tandem configuration~<fnref n="60" />.

FIG. 4.21 shows a GaAs~0.74~P~0.26~ nanowire (~57 nm radius) passivated with a thin GaP shell (~5 nm) obtained by VS growth after the consumption of the Ga catalyst. Compared to unpassivated GaAs~0.74~P~0.26~ nanowires, the room temperature photoluminescence of these core/shell nanowires is enhanced by a factor ×200 [Himwas2017].

<figure><img src="image_73.png" /><figcaption>**Figure 4.21:** Chemical imaging of a GaAsP/GaP core/shell nanowire. (a) Large view STEM-HAADF image. (b) Corresponding EDX map of the As signal, showing the GaAsP core. (c) EDX map of the P signal, showing the extra thickness due to the GaP shell. (adapted from [Himwas2017])</figcaption></figure>

Due to the lattice mismatch (~2.30%) between the core ($a_{GaAsP} \simeq 5.6004$ Å) and shell materials ($a_{GaP} = 5.4505$ Å), strain-induced defects can develop at the heterointerface between core and shell (see §4.1.2, page

<footnote n="60">⁶⁰R. R. LaPierre. "Theoretical conversion efficiency of a two-junction III-V nanowire on Si solar cell". Journal of Applied Physics 110.1 (2011), p. 014310. doi: 10.1063/1.3603029.</footnote>
40). There may be an optimum between the efficient carrier confinement and screening from the surface defects, requiring thicker GaP shells, and defect creation at the heterointerface due to excessive epitaxial strain, which is avoided by thinner GaP shells.

Given the large radius of our nanowires ($r_{core} \approx 50$ nm), the localization of epitaxial strain is comparable to that found in planar configurations (see Fig. 4.5 page 42) and all the GaP shells are expected to be fully strained (i.e. below the critical thickness).

<figure><img src="image_74.png" /><figcaption>**Figure 4.22:** (a) Normalized room temperature photoluminescence of GaAsP/GaP core/shell nanowires with increasing GaP shell thickness. (b) Corresponding peak position and intensity. (from [Himwas2017])</figcaption></figure>

FIG. 4.22 shows that the optimum region is rather large and that efficient confinement is already achieved for a GaP shell thickness of 4 nm. Thicker shells do not improve the room temperature photoluminescence intensity (FIG. 4.22.b, red points) but blue-shift the peak position, which is likely due to strain in the structure (FIG. 4.22.a). Such strain-related shifts of the nanowire core emission are surprising as the epitaxial strain should mostly localize in the GaP shell, as already mentioned above.

### 4.3.2 Unintentional GaAsP shell growth

A possible explanation of the observed shift of the GaAsP emission wavelength with increasing GaP shell thickness (FIG. 4.22) is that the *nominally* homogeneous GaAsP core hosts a more complex structure. This was already suggested in [Himwas2017] and it is studied in more details in [Himwas2019].

<figure><img src="image_75.png" /><figcaption>**Figure 4.23:** (a) Overview of GaAsP/GaP core shell nanowire by STEM-HAADF. (b-d) Details of selected axial positions along the nanowire height. (from [Himwas2019])</figcaption></figure>

The TEM analysis of FIG. 4.23 shows clearly that the GaAsP/GaP core shell structure comprises an extra *unintentional* GaAsP shell between the *intentional* GaAsP core and the GaP shell (FIG. 4.23.b, blue color). This unintentional GaAsP shell has a higher As atomic content than the GaAsP core and thus appears brighter in STEM-HAADF image due to the higher atomic number of As relative to P. Note that both the thickness of the unintentional GaAsP and intentional GaP shells vary largely between the top and bottom sections of the nanowire.
<figure><img src="image_76.png" /><figcaption>**Figure 4.24:** Spatially resolved low temperature (10K) cathodoluminescence spectroscopie of an individual GaAsP/GaP core/shell nanowire. (from [Himwas2019])</figcaption></figure>

FIG. 4.24 shows low temperature (10 K) cathodoluminescence (CL) spectra taken at regular intervals along the nanowire growth axis. Compared to photoluminescence (PL) results from [Himwas2017], the CL data is spatially resolved and we can follow the optical response along the nanowire height. It is now obvious that the PL signal corresponds to the spatial average of several contributions, from different locations in the GaAsP core and unintentional GaAsP shell.

In practice, the CL signal from the unintentional GaAsP shell dominates at the bottom of the nanowire and we only observe the optical contribution from the GaAsP core near the top of the nanowire. Such spatial localization of the optical emitters is consistent with the TEM observations of FIG. 4.23. The association between the localization of the GaAsP unintentional shell by TEM and the corresponding emission features by CL gives a physical ground to the observed PL blue-shift with increasing GaP shell (§4.3.1). Since one of the active emitters is now identified as a thin layer sandwiched between the intentional GaAsP core and GaP shell, its strain state may now depend on the GaP shell thickness.

The overall structure of the nominal GaAsP/GaP core/shell nanowire is shown FIG. 4.24.e. The varying thickness of the unintentional GaAsP shell strongly resembles that obtained from concurrent VLS axial and VS radial growth modes in self-catalyzed binary GaP nanowires (FIG. 4.10.c page 46).

### 4.3.3 Homogeneous GaAsP alloys

The lower As content in the unintentional shell identified in §4.3.2 is actually a major concern, as the lower band gap of the shell effectively drives the carriers out of the nanowire core, closer to the defective surface.

In [Lepinau2019], we study in more detail the origin of this unintentional As-rich GaAsP shell in self-catalyzed GaAsP nanowires. Compared to [Himwas2019] and [Himwas2017], the nanowires are now grown on patterned SiO₂/Si substrates. This controlled environment improves the wire-to-wire homogeneity so that small composition changes can be reliably detected and analyzed.

<figure><img src="image_77.png" /><figcaption>**Figure 4.25:** Structure of a self-catalyzed GaAsP nanowire. (a) Nominal and actual core shell structures, without and with unintentional shells. (b-c) Chemical maps and details of the lower and upper heterostructures. (adapted from [Lepinau2019])</figcaption></figure>
To prove that the unintentional radial shell is growing at the same time as than the axial segment, we devise a growth sequence with three axial segments of different nominal composition: pure GaP, intermediate $GaAs_{0.56}P_{0.44}$ and low P-content $GaAs_{0.82}P_{0.18}$. The corresponding nominal structure is presented Fig. 4.25.a (left). From TEM chemical analyses at the different axial interfaces, we demonstrate that each axial segment can be associated to an unintentional radial shell produced by VS growth (FIG. 4.25.b-c).

Compared to self-catalyzed binary GaP nanowires which require marker layers (see Fig. 4.10 page 46), ternary GaAsP alloys allow for an easier identification of the VS and VLS contributions due to their different chemical compositions.

<figure><img src="image_78.png" /><figcaption>**Figure 4.26:** (a) Angle of As and P cell flux against various surface normals (catalyst, sidewall, substrate). (b) Theoretical ternary composition computed using the geometry in (a) for *directly* impinging fluxes only (i.e no re-emission from the mask). $\epsilon_P$ is the ratio of the atomic flux of P with respect to the total atomic flux of group V (As+P). (from [Lepinau2019])</figcaption></figure>

The finding that the As-rich (unintentional) VS-grown GaAsP shell has chemical composition different from the VLS-grown GaAsP core is not surprising. Given the different growth mechanisms (VS vs. VLS), crystal facets ((111) vs (110)), chemical nature of the group V molecular beam ($As_4$ vs. $P_2$), cell inclination angles and other geometrical features (see Fig. 4.26.a), it is actually surprising that the measured compositions are so similar. At 590°C substrate temperature, we only measure $\Delta x \simeq -0.1$ between $x_{core} = 0.29$ and $x_{shell} = 0.20$ for the corresponding $GaAs_{1-x}P_x$ alloys. Simplistic modeling (FIG. 4.26.b), restricted to directly impinging group V fluxes, produces larger differences, which underlines again the important contribution of the indirect group V fluxes re-emitted from the mask to the final composition of the VS and VLS grown volumes (see §4.2.3, page 48).

We can actually make use of the differences between the VS and VLS growth mechanisms to obtain an *homogeneous* ternary composition by adjusting the substrate temperature. From 590°C to 620°C, the As-content of the VLS-grown nanowire core decreases ($x_{core} = 0.29 \rightarrow 0.24$), while that of the VS-grown shell increases ($x_{shell} = 0.20 \rightarrow 0.235$). It is thus possible to obtain an homogeneous self-catalyzed $GaAs_{0.76}P_{0.24}$ nanowires at 620°C, comprised of both VS (radial) and VLS (axial) grown volumes with similar atomic composition.

Such adjustments may not be possible over the whole alloy composition range ($x \in [0-1]$), as 620°C is close to the upper substrate temperature limit for the self-catalyzed growth of GaP, GaAs or GaAsP nanowires.

**Conclusion:** The growth of self-catalyzed ternary GaAsP nanowires investigated in §4.3 closely resembles that of self-catalyzed binary GaP nanowires (§4.2.2). The main feature of group V ternary alloys is that the VLS and VS-grown GaAsP volumes typically exhibit different chemical compositions, which results in different emission wavelengths.

Due to the different growth mechanisms (VS vs. VLS), the fine tuning of the substrate temperature may allow one to obtain near-homogeneous ternary GaAsP nanowires (core and unintentional shell) for selected atomic compositions.
# 4.4. Axial heterostructure and catalyst thermodynamics

In this section, we now explore the interplay between the gas phase, the liquid catalyst and the solid phase. In the context of binary or ternary mixtures, we need to define the relevant thermodynamic potential in each phase.

## 4.4.1 Expressions of thermodynamic potentials

### a. Chemical potential in an ideal solution

For a system at fixed temperature $T$ and pressure $P$, the relevant thermodynamic potential is the Gibbs energy $G$, defined from the enthalpy $H$ and entropy $S$. For a given compound $i$, with partial pressure $P_i$ and number of mole $n_i$

$$G_i(T, P_i, n_i) = H_i(T, P_i, n_i) - T \cdot S_i(T, P_i, n_i) \\ dG = V \cdot dP - S \cdot dT \qquad (4.13)$$

Using the non-interacting perfect gas as the model system for ideal solutions, the perfect gas law $V = \frac{n_i RT}{P_i}$ can be used to integrate eq. (4.13) at constant temperature, from an arbitrary reference pressure $P_0$ (identical for all components $i$, $P_0^i = P_0$) to the partial pressure $P_i$.

$$G_i(T, P_i, n_i) = G_i(T, P^0, n) + \int_{P^0}^{P_i} \left( \frac{dG}{dP} \right)_T \cdot dP \\ G_i(T, P_i, n_i) = G_i(T, P^0, n_i) + \int_{P^0}^{P_i} n_i \frac{RT}{P} \cdot dP \qquad (4.14) \\ G_i(T, P_i, n_i) = G_i(T, P^0, n_i) + n_i RT \ln \left( \frac{P_i}{P^0} \right)$$

The thermodynamic potential $G_i$ being extensive, it simply scales up with the number of mole $n_i$, so we can define a molar version of $g_i$.

$$G_i(T, P^0, n_i) = n_i \cdot g_i(T, P^0) \\ G_i(T, P_i, n_i) = n_i \cdot g_i(T, P^0) + n_i \cdot RT \ln \left( \frac{P_i}{P^0} \right) \\ g_i(T, P_i) = g_i(T, P^0) + RT \ln \left( \frac{P_i}{P^0} \right) \qquad (4.15)$$

For ideal (non-interacting) mixtures, we simply sum the Gibbs energy contributions of the individual components. The total pressure $P$ is also the sum of the individual partial pressure $P_i$,

$$P = \sum_{i=0}^{k} P_i \quad \text{and} \quad P_i = \frac{n_i}{\sum_{l=0}^{k} n_l} \cdot P \\ G(T, P, n_0, \dots, n_k) = \sum_{i=0}^{k} G_i(T, P_i, n_i) = \sum_{i=0}^{k} n_i \cdot g_i(T, P_i) \\ G(T, P, n_0, \dots, n_k) = \sum_{i=0}^{k} n_i \cdot \left( g_i(T, P^0) + RT \ln \left( \frac{P}{P^0} \right) + RT \ln \left( \frac{n_i}{\sum_{l=0}^{k} n_l} \right) \right) \qquad (4.16) \\ G(T, P, n_0, \dots, n_k) = \sum_{i=0}^{k} n_i \cdot \left( g_i(T, P) + RT \ln x_i \right) \quad \text{and} \quad x_i = \frac{n_i}{\sum_{l=0}^{k} n_l}$$
At this point we need to introduce the chemical potential $\mu_i$ as the partial derivative of $G$ along $n_i$ at constant $T$, $P$ and $\{n_j\}_{j \neq i}$.

$$ 
\begin{aligned}
\mu_i(T, P, n_0, \dots, n_k) &= \frac{\partial}{\partial n_i} G(T, P, n_0, \dots, n_k) \\
\mu_i(T, P, n_0, \dots, n_k) &= \left( g_i(T, P) + RT \ln x_i \right) + \sum_{j=0}^{k} n_j \left( \frac{\partial}{\partial n_i} \left( g_j(T, P) + RT \ln x_j \right) \right) \\
\mu_i(T, P, n_0, \dots, n_k) &= \left( g_i(T, P) + RT \ln x_i \right) + RT \sum_{j=0}^{k} n_j \frac{\partial}{\partial n_i} \left( \ln \left( \frac{n_j}{\sum_{l=0}^{k} n_l} \right) \right) \\
\mu_i(T, P, n_0, \dots, n_k) &= \left( g_i(T, P) + RT \ln x_i \right) + RT \left( \frac{n_i}{n_i} - \frac{\sum_{j=0}^{k} n_j}{\sum_{l=0}^{k} n_l} \right)
\end{aligned}
$$

$$ \mu_i(T, P, n_0, \dots, n_k) = g_i(T, P) + RT \ln x_i \quad (4.17) $$

We retrieve the well-known result, which is that the chemical potential $\mu_i$ in an ideal mixture is the molar Gibbs energy $g_i$ of the isolated compound $i$, combined with contribution of the mixing entropy ($RT \ln x_i$). This result is often abbreviated in the following form:

$$ \mu_i(T, P, n_0, \dots, n_k) = \mu_i(T, P, x_i) = \mu_i^0(T, P) + RT \ln x_i \quad (4.18) $$

Given that $G = \sum_i n_i \cdot \mu_i$, we could have directly identified the chemical potential from eq. (4.16), avoiding the tedious derivation in eq. (4.17) and jumping directly to the final result of eq. (4.18). However this shortcut has two major drawbacks:

* it hides that the true variables of the chemical potential are the ($n_0, \dots, n_k$), and not $x_i$.

* the dependency of the $x_j$ derivative against $n_i$, which cancels out for ideal solution, is thus not apparent.

b. Chemical potential in non-ideal solutions

At this point it is convenient to write a molar version of eq. (4.16), in which we only refer to the individual concentrations ($x_0, \dots, x_k$) and not on the total system size ($n$).

$$ G(T, P, n_0, \dots, n_k) = n \cdot g(T, P, x_0, \dots, x_k) \quad \text{and} \quad n = \sum_{i=0}^{k} n_i \quad (4.19) $$

In non-ideal solutions, interaction terms are introduced between individual components. These interactions bring extra energy to the total Gibbs energy of the mixture, as positive or negative contributions. The most simple modification is a binary contribution of the following form, with $L_{ij} < 0$ for stabilizing interaction:

$$ g(T, P, x_0, \dots, x_k) = g^{\text{ideal}}(T, P, x_0, \dots, x_k) + \sum_{i \neq j} L_{ij} x_i x_j \quad (4.20) $$

Remembering the chemical potential must be derived using $n_i$ (not $x_i$!), we can compute how this extra contribution in the total Gibbs free energy of the system is distributed in individual chemical potentials $\mu_i^{\text{extra}}$.
$$
\begin{align}
\mu_i^{\text{extra}}(T, P, n_0, \dots, n_k) &= \frac{\partial}{\partial n_i} \left( n \cdot \sum_{j \neq k} L_{jk} x_j x_k \right) \nonumber \\
&= \frac{\partial}{\partial n_i} \left( \sum_{j \neq k} L_{jk} \frac{n_j n_k}{n} \right) \nonumber \\
&= \left( \sum_{j \neq k, k \neq i} L_{jk} \frac{n_j n_k}{n^2} \cdot (-1) \right) + \left( \sum_{j \neq k, k=i} L_{ij} \left( \frac{n_j}{n} - \frac{n_i n_j}{n^2} \right) \right) \tag{4.21} \\
\mu_i^{\text{extra}}(T, P, n_0, \dots, n_k) &= \sum_{j \neq i} L_{ij} x_j - \sum_{j \neq k} L_{jk} x_j x_k \nonumber
\end{align}
$$

For two components ($x_0, x_1$) we have :

$$
\begin{align}
\mu_0^{\text{extra}}(T, P, x_0, x_1) &= L_{01} x_1 - L_{01} x_0 x_1 \nonumber \\
\mu_0^{\text{extra}} &= L_{01} x_1(1 - x_0) \tag{4.22} \\
\mu_0^{\text{extra}} &= L_{01} x_1^2 \nonumber
\end{align}
$$

Leading to the standard formulation for the non-ideal chemical potential with ($x_0 + x_1 = 1$)

$$
\left\{
\begin{array}{l}
\mu_0(T, P, x_0) = \mu_0(T, P) + RT \ln(x_0) + L(1-x_0)^2 \\
\mu_1(T, P, x_0) = \mu_1(T, P) + RT \ln(1-x_0) + L x_0^2
\end{array}
\right.
\qquad (4.23)
$$

For three components ($x_0, x_1, x_2$) we have :

$$
\begin{align}
\mu_0^{\text{extra}} &= L_{01} x_1 + L_{02} x_2 - (L_{01} x_0 x_1 + L_{02} x_0 x_2 + L_{12} x_1 x_2) \nonumber \\
\mu_0^{\text{extra}} &= L_{01} x_1(1-x_0) + L_{02} x_2(1-x_0) - L_{12} x_1 x_2 \tag{4.24} \\
\mu_0^{\text{extra}} &= L_{01} x_1(x_1+x_2) + L_{02} x_2(x_1+x_2) - L_{12} x_1 x_2 \nonumber \\
\mu_0^{\text{extra}} &= L_{01} x_1^2 + L_{02} x_2^2 + (L_{01} + L_{02} - L_{12}) x_1 x_2 \nonumber
\end{align}
$$

Conclusion: The simplest non-ideal deviations from ideal solutions in the form of binary interactions ($L_{ij} x_i x_j$) affect the chemical potentials of individual compounds in a non-trivial way, with parabolic contributions completely foreign to the standard logarithms formulation of ideal mixtures (RT ln x).

* For binary mixtures ($x_0, x_1$), the extra contribution to $\mu_0$ simply depends on the concentration of the other compounds $x_1$ in a parabolic manner, eq. (4.22)
In this simple case ($x_0 + x_1 = 1$), we still retrieve a pure $x_0$ dependence of $\mu_0$ but with mixed logarithmic and parabolic contributions, eq. (4.23).

* For ternary mixtures ($x_0, x_1, x_2$), the extra contribution on $\mu_0$ also depends on the other compounds concentration ($x_1, x_2$) in a parabolic fashion, but the formulation is now more involved, eq. (4.24).
In addition to the expected $L_{01} x_1^2$ and $L_{02} x_2^2$ additions, $\mu_0$ is also complemented by a negative $L_{12}$ contribution, showing the trace of all possible binary interactions ($L_{01}, L_{02}, L_{12}$) in the chemical potential of any individual component.

4.4.2 Group V concentration in binary self-catalyzed nanowire catalyst

Using the thermodynamic potentials described above, we can now study the following equilibrium between a solid phase of GaAs and a liquid mixture of (Ga,As).

$GaAs^s \rightleftharpoons Ga^l + As^l$

The equilibrium is described by balance of thermodynamic potentials:
$$ \mu_{GaAs}^{s}(T, P) = \mu_{Ga}^{l}(T, P, x_{As}, x_{Ga}) + \mu_{Ga}^{l}(T, P, x_{As}, x_{Ga}) \quad (4.25) $$

Regarding condensed (liquid and solid) phases close to atmospheric pressure, the pressure variation is neglected. Given that $x_{Ga} + x_{As} = 1$, we can define $x = x_{As}$ the concentration in group V element to obtain:

$$ \left\{ \begin{aligned} \mu_{GaAs}^{s}(T, P) &= \mu_{GaAs}^{s}(T, P_0) \\ \mu_{Ga}^{l}(T, P, x_{As}, x_{Ga}) &= \mu_{Ga}^{l}(T, P_0, x) = \mu_{Ga}^{l}(T, P_0) + RT \ln(1-x) + L_{AsGa} x^2 \\ \mu_{As}^{l}(T, P, x_{As}, x_{Ga}) &= \mu_{As}^{l}(T, P_0, x) = \mu_{As}^{l}(T, P_0) + RT \ln(x) + L_{AsGa} (1-x)^2 \end{aligned} \right. \quad (4.26) $$

Leading to the following master equation:

$$ \begin{aligned} \mu_{GaAs}^{s}(T, P_0) - \mu_{Ga}^{l}(T, P_0) - \mu_{As}^{l}(T, P_0) &= RT \ln((1-x)x) + L_{AsGa}(x^2 + (1-x)^2) \\ \Delta\mu_{GaAs}(T, P_0) &= RT \ln((1-x)x) + L_{AsGa}(x^2 + (1-x)^2) \end{aligned} \quad (4.27) $$

If the solution is considered ideal ($L = 0$) we find the analytical solution:

$$ K_{GaAs}(T) = \exp\left(\frac{\Delta\mu_{GaAs}(T, P_0)}{RT}\right) $$

$$ x_{ideal} = \frac{1 - \sqrt{1 - 4K_{GaAs}}}{2} \approx K_{GaAs}(T) \quad (K_{GaAs} \ll 1) \quad (x_{ideal} \ll 1) \quad (4.28) $$

The tabulated values of the various chemical potentials and the (non-zero) interaction parameter $L_{GaAs} = L(T)$ can be found in Ansara et al.<fnref n="61" /> for a large range of temperature.

<figure><img src="image_79.png" /><figcaption>**Figure 4.27:** Equilibrium $x_V$ concentration in the droplet of self-catalyzed GaP and GaAs nanowires. Ideal solutions are computed using $L_{III V} = 0$. Real solutions are based on the reference data of Ansara et al. (F. Oehler, personal notes)</figcaption></figure>

Given the form of equation (4.27), solving the system with $L_{AsGa} \neq 0$ requires numerical routines, with the minimization of $F = 0$.

$$ F = x \cdot (1-x) - K \exp\left(-\frac{L}{RT}(x^2 + (1-x)^2)\right) $$

(this is the recommended form to avoid $\ln(x)$ issue with $x < 0$)

However we expect $x \ll 1$ so that we find the following (quick) approximations:

$$ \begin{aligned} x_{regular} &\approx \exp\left(\frac{\Delta\mu_{GaAs}(T, P_0) - L_{AsGa}(T)}{RT}\right) \\ x_{regular} &\approx x_{ideal} \cdot \exp\left(\frac{-L_{AsGa}(T)}{RT}\right) \end{aligned} $$

The FIG. 4.27 confirms the validity of the proposed approximations in the considered temperature range.

<footnote n="61">⁶¹I. Ansara, C. Chatillon, H. Lukas, T. Nishizawa, H. Ohtani, K. Ishida, M. Hillert, B. Sundman, B. Argent, A. Watson, T. Chart, and T. Anderson. “A binary database for III-V compound semiconductor systems”. *Calphad* 18.2 (1994), pp. 177–222. DOI: 10.1016/0364-5916(94)90027-2.</footnote>
### 4.4.3 Axial group V heterostructures in self-catalyzed nanowires

Given that the equilibrium concentration of group V in a (Ga,As) or (Ga,P) liquid respectively in contact with a GaAs or GaP solid is small, it may be reasonable to consider that this remains true during the (out of equilibrium) growth of self-catalyzed GaP or GaAs nanowires. Under this assumption, the growth of sharp axial transitions for GaAs to GaP seems possible, as the group V composition in the liquid droplet will quickly react to any change in vapor phase, resulting in rapid composition changes in the solid.

<figure><img src="image_80.png" /><figcaption>**Figure 4.28:** (a) SEM images self-catalyzed GaP nanowires. (b-c) STEM images of GaAs nanowire with GaP marker layers. (d) Axial extension of GaP (GaAs) insertions in GaAs (GaP) matrix, depending on the growth conditions described in Table 1. (adapted from [Priante2015])</figcaption></figure>

<table><caption>**Table 1. List of Samples**</caption><thead><tr><th>sample</th><th>S</th><th>V</th><th>VI</th><th>VLI</th></tr></thead><tbody><tr><td>Main material</td><td>GaAs</td><td>GaP</td><td>GaP</td><td>GaP</td></tr><tr><td>Insertions</td><td>GaP</td><td>GaAs</td><td>GaAs</td><td>GaAs</td></tr><tr><td>P/Ga BPR</td><td>7</td><td>7</td><td>7</td><td>15</td></tr><tr><td>As/Ga BPR</td><td>10</td><td>11</td><td>11</td><td>15</td></tr><tr><td>Group V switch using</td><td>Shutters</td><td>Shutters and valves</td><td>Shutters, valves, main shutter</td><td>Shutters, valves, main shutter</td></tr><tr><td>Flux interruption</td><td>No</td><td>No</td><td>20 s</td><td>60 s</td></tr></tbody></table>

<figure><img src="image_81.png" /></figure>

FIG. 4.28 adapted from [Priante2015] shows that the actual implementation is not straightforward. We are here mainly limited by the experimental means, more specifically by the type of group V cells used in our MBE system. Both As and P are provided from conventional valved-cracker cells evaporating solid As or P sources. The cells are equipped with individual shutters, in addition to the main (global) shutter.

* The marker layers (As→P) in sample S (see Table 1 in FIG. 4.28) are grown simply by commuting the As and P shutters. The mechanical activation of the shutter is fast, which leads to relatively narrow insertions (~10 nm axial width, see [Priante2015]). However, our shutters are not efficiently blocking the group V fluxes and instead of the planned binary GaAs matrix and GaP markers, we end up with a GaAs<sub>0.7</sub>P<sub>0.3</sub> matrix and GaAs<sub>0.15</sub>P<sub>0.85</sub> marker layers, which is not acceptable for any serious application.

* To efficiently block the group V flux, we have to close the shutter *and* the valve of each group V cell. The valves have longer transient regimes (~1 s) compared to the shutter (~0.1 s). A pure GaP matrix is obtained in this manner (samples V, VI and VLI).

* Using valves, the first marker interface (GaP→GaAs) is sharp. However the second interface (GaAs→GaP) is much broader (sample V, ~10 nm) and a growth interruption of several tens of seconds (i.e. no Ga flux, no group V flux) needs to be introduced to obtain a sharper transition.

* The sharpest interface is obtained for the longest growth interruption (VLI, 60 sec), which also includes the actuation of the main shutter during the growth interruption.
4.4.4 Axial group III heterostructures in self-catalyzed nanowires

Given that group III elements make the most of the VLS catalyst of III-V nanowires, there may be a
strong ‘reservoir’ effect which limits the theoretical sharpness of group III axial heterostructures. However, the
experimental study and detailed modeling of [Priante2016] show that very sharp transitions (a few monolayer
thick) can be obtained in self-catalyzed (AlGa)As nanowires.

<figure><img src="image_82.png" /><figcaption>**Figure 4.29:** (a) Atomically resolved STEM-HAADF image of an AlGaAs insertion in a self-catalyzed GaAs nanowire. (b) Atomically resolved AlGaAs marker intensity profile. (c) Equilibrium concentration of the Al content in the droplet ($y_{Al}$) relative to that of the solid $Ga_{(1-x)}Al_xAs$. (adapted from [Priante2016])</figcaption></figure>

* FIG. 4.29 show an AlGaAs insertion embedded in a self-catalyzed GaAs nanowire using shutters. The composition profile is much sharper than those obtained for axial group V heterostructures (see §4.4.3).

* The width of the interface increases with the growth rate (see [Priante2016]).

* Stopping the group V flux, while providing Al only (i.e. Al ‘pre-filling’ of the VLS catalyst) leads to the sharpest first interface (GaAs→AlGaAs), FIG. 4.29.b.

* The second interface (AlGaAs→GaAs) is always wider than the first.

Equilibrium thermodynamics (see §4.4.1) of the ternary $Ga_{(1-x)}Al_xAs^s \leftrightarrow (Al, Ga, As)^l$ system indicates that the growth of AlGaAs solid actually requires only a very small Al content in the liquid (see FIG. 4.29.c). For example, $Ga_{0.5}Al_{0.5}As$ only requires $y_{Al} \approx 0.002$.

During growth, any Al in excess of the (very small) equilibrium content will thus quickly incorporate into the solid. From the thermodynamic point of view, the present AlGaAs markers in GaAs actually shares the same ‘low reservoir’ property as the ‘sharp’ group V heterostructures (from §4.4.3): few Al or group V species are present in the VLS droplet, which consists of nearly pure Ga.

In addition to slowing the growth rate (i.e reducing the group V flux), we can also stop the growth (i.e. no
group V flux). Providing the catalyst with Al only, before resuming the growth without Al, allows one to obtain
the sharpest profile for the first interface.

<figure><img src="image_83.png" /><figcaption>**Figure 4.30:** Simulation an AlGaAs insertion pro-
file. (from [Priante2016])</figcaption></figure>

The second interface is wider, due to the
(small) *reservoir* effect in the droplet by
which the remaining Al in the liquid is
progressively consumed in the solid phase.

An interesting development of [Priante2016]
is the approximation of the Al partition
curve by a (well-chosen) analytical expres-
sion (FIG. 4.29.c, green curve) to obtain
numerical and analytical models of the Al-
GaAs composition along the insertion width
(FIG. 4.30). Thin nanowires are expected to
show the sharpest second interface.

Note that these results based on equilibrium thermodynamics were further investigated by F.Glas in the
context of out-of-equilibrium growth and nucleation theory<sup>62</sup>.

<footnote n="62">⁶²F. Glas. “Comparison of Modeling Strategies for the Growth of Heterostructures in III-V Nanowires”. *Crystal Growth & Design* 17.9 (2017), pp. 4785–4794. doi: 10.1021/acs.cgd.7b00732.</footnote>
4.4.5 Catalyst engineering in group III catalyzed nanowires

The unequal partition between the solid phase and liquid phases, either between Ga and group V (§4.4.3), or between Al and Ga (§4.4.4) is actually found in many other ternary mixtures, notably GaₓIn₁₋ₓP and GaₓIn₁₋ₓAs.

<figure><img src="image_84.png" /><figcaption>**Figure 4.31:** Distribution coefficient of the Ga fraction in the solid GaₓIn₁₋ₓZ or in the liquid (Ga,In,Z), for Z=[P,As]. Equilibrium (SET) is compared to nucleation theory (NT). (from [Scaccabarozzi2020])</figcaption></figure>

The use of In as VLS catalyst for GaAs or GaP nanowires growth is experimentally interesting, as In can be produced from high-purity effusion cells, similarly to Ga in the self-catalyzed growth of [Ga]-GaAs or [Ga]-GaP nanowires. Contrary to Ga catalysts, In catalysts may also be consumed by direct evaporation inside the MBE at the typical substrate temperature (500-600°C).

FIG. 4.32 shows experimental results, proving that regular arrays of [In]-GaP and [In]-InGaAs can be obtained at 520°C on patterned SiO₂/Si(111) substrates. The lower substrate temperature compared to standard [Ga]-GaP growth (600°C) is required to limit In evaporation.

The chemical analysis (FIG. 4.32.e) indicates that nearly pure GaP and Ga-rich Ga₀.₈In₀.₂As solid phases could be obtained, while the respective VLS catalysts present an overwhelming In content, in line with the theoretical predictions (see FIG. 4.31).

FIG. 4.31 compares equilibrium calculations (see §4.4.1) and nucleation theory for the ternary systems [GaₓIn₁₋ₓP]ₛ/(Ga,In,P)l and [GaₓIn₁₋ₓAs]s/(Ga,In,As)l. Liquids with low Ga contents (yGa ≪ 1) correspond to high Ga contents x in the solid, due to a very asymmetric partition of the alloy content.

This effect is stronger for phosphides than for arsenides (about 1 order of magnitude for yGa at x ≈ 0.5) but the shape of the curves is similar. (Note that these variations closely resemble those of the AlGaAs system studied in §4.4.4).

In practice, this indicates that nearly pure GaAs or GaP nanowires could be obtained from VLS catalysts containing mostly In. Therefore, we now have to distinguish the standard Au-catalyzed VLS growth of GaP nanowires ([Au]-GaP), from the self-catalyzed VLS growth ([Ga]-GaP) and this hypothetical indium catalyzed VLS growth ([In]-GaP).

<figure><img src="image_85.png" /><figcaption>**Figure 4.32:** SEM images and EDS chemical analysis of [In]-GaP and [In]-InGaAs nanowires. (from [Scaccabarozzi2020])</figcaption></figure>
<figure><img src="image_86.png" /><figcaption>**Figure 4.33:** (a-b) Contact angle and crystal structure of [Ga]-GaP nanowires. (c-d) Similar analysis for [In]-GaP nanowires, showing the standard cubic (Zinc-Blende) structure with twinned segments. (from [Scaccabarozzi2020])</figcaption></figure>

FIG. 4.33 investigates the contact angle $\beta$ and the crystal structure of [In]-GaP nanowires. As expected from the vertical orientation the nanowire on Si(111) substrates (FIG. 4.32.a), the axial growth of our [In]-GaP nanowires proceeds along the [111] direction, similarly to [Ga]-GaP nanowires. They also share the same twinned cubic structure.

The main difference is the value of the contact angle $\beta$, measured here *ex situ* (after growth, at room temperature), with $\beta_{Ga} \approx 130^{\circ}$ and $\beta_{In} \approx 95^{\circ}$. Such differences can be ascribed to the different surface energies of In and Ga. Given that the contact angle is expected to control the crystal phase of the nanowire$^{63}$, it is interesting to compare catalyst consumption experiments between [Ga]-GaP and [In]-GaP nanowires.

<figure><img src="image_87.png" /><figcaption>**Figure 4.34:** (left) Crystal structure of [Ga]-GaP nan wires, which Ga catalyst has been consumed by exposure to group V only. (right) Similar analysis for [In]-GaP nanowires, which In catalyst has been consumed in a comparable way. ZB relates to the conventional cubic zinc-blende phase of GaP, while WZ describes the uncommon hexagonal wurtzite phase. I is an intrinsic stacking fault and I+T is the combination of stacking fault and twinning. (adapted from [Scaccabarozzi2020])</figcaption></figure>

<footnote>$^{63}$F. Panciera, Z. Baraissov, G. Patriarche, V. G. Dubrovskii, F. Glas, L. Travers, U. Mirsaidov, and J.-C. Harmand. “Phase selection in self-catalysed GaAs nanowires”. *Nano Letters* (2020), pp. 1669–1675. DOI: 10.1021/acs.nanolett.9b04808.</footnote>
FIG. 4.34 compares the crystal structure of [Ga]-GaP and [In]-GaP nanowires, which [Ga] or [In] catalysts have been consumed by exposure to a pure P flux. For the [Ga]-case (FIG. 4.34.left), we observe the existence of a short hexagonal WZ segment, just before the structure reverts to cubic for the terminal section. This WZ segment is absent in the case of [In]-GaP (FIG. 4.34.right) and the whole wire only consists of (twinned) ZB material.

The crystal phase variations of our [Ga]-GaP nanowires are in line with the Ga-based models from the literature$^{64,65}$ and Ga-based experimental data$^{66}$ based on [Ga]-GaAs nanowires. The absence of WZ phase during all the catalyst consumption of our [In]-GaP nanowires (over a large range of contact angles) is more surprising. Such resilience of the ZB crystal phase could be compatible with some of the existing models, if the lower surface tension of liquid In compared to Ga is taken into account.

## 4.5. Conclusion

Some of the difficulties in the growth of group III-catalyzed III-V nanowires by MBE may be summarized by the following figure (from the Supplementary Information of [Scaccabarozzi2020]).

<figure><img src="image_88.png" /><figcaption>**Figure 4.35:** (a-e) TEM chemical analysis by EDX of [In]-InGaAs nanowires which In catalyst has been consumed by exposure to As. (f-i) TEM structural analysis of the same nanowire, showing a twinned cubic ZB crystal structure without extended WZ segment. (from [Scaccabarozzi2020])</figcaption></figure>

1. As in §4.2.2 or §4.3.2 we see the combined actions of radial VLS growth and axial VS growth. Due to the different growth mechanisms and impinging fluxes, the various InGaAs volumes present different In composition depending on their origin (VLS vs. VS), see FIG. 4.35.(a-e)

2. The nanowire geometry (height and diameter) also evolves with the growth duration. This leads to further changes in the capture rate of group III and V with time, and further gradients of In composition, within the individual VS- or VLS-grown volumes (FIG. 4.35.e).

3. Any change of the VLS catalyst composition potentially affects the composition of the ternary solid phase. The possibility to consume *in situ* the VLS catalyst, which is one of the hallmarks of group-III catalyzed

<footnote>$^{64}$V. G. Dubrovskii. “Development of Growth Theory for Vapor–Liquid–Solid Nanowires: Contact Angle, Truncated Facets, and Crystal Phase”. *Crystal Growth & Design* 17.5 (2017), pp. 2544–2548. DOI: 10.1021/acs.cgd.7b00076.</footnote>

<footnote>$^{65}$F. Glas, J.-C. Harmand, and G. Patriarche. “Why Does Wurtzite Form in Nanowires of III-V Zinc Blende Semiconductors?” *Physical Review Letters* 99.14 (2007), p. 146101. DOI: 10.1103/physrevlett.99.146101.</footnote>

<footnote>$^{66}$J.-C. Harmand, G. Patriarche, F. Glas, F. Panciera, I. Florea, J.-L. Maurice, L. Travers, and Y. Ollivier. “Atomic Step Flow on a Nanofacet”. *Physical Review Letters* 121.16 (2018), p. 166101. DOI: 10.1103/physrevlett.121.166101.</footnote>
III-V nanowires growth, does not escape this rule. Here, the final conical section is obtained by only supplying As to the shrinking catalyst and we obtain a InGaAs solid phase with yet another In atomic content (FIG. 4.35.e).

4. Similarly to planar growth (see §2.3.2 page 16), local facet orientations can further impact the alloy composition inside the VS-grown shell volume. For <111> oriented cubic nanowires with (11̄0) sidewalls, this effect is mostly visible at the (11̄2) corner sidewall<fnref n="67" />.

5. All previous remarks (1-4) on composition gradients can be directly transferred to substitutional doping distribution inside the nanowire volume. Any external chemical impurity introduced intentionally during growth (Si, Se, Te, Be, ...) can distribute with the same diversity than the showcased ternary alloys.

6. Similarly to the difficult n-type doping of GaAs(111)A on planar Ge(111) using Si (see §3.5 page 37), the efficient n-type doping of the GaAs(111)B core of self-catalyzed GaAs nanowires also poses comparable challenges [Chen2021].

7. The VS shell often grows in perfect epitaxy over the VLS core, replicating the complete crystal structure of the core in the shell, stacking faults, twins and WZ segments included. This can be exploited to obtain ‘large’ volumes of the uncommon (and sometimes direct band-gap) hexagonal phase of selected III-V compounds<fnref n="68" />.

8. The crystal structure of the core is usually pure cubic (ZB) for VLS-grown (Al)GaAs(P) nanowires. Twins are a common defect and the [In]-InGaAs nanowires shown Fig. 4.35 exhibits a rather high twin density. Considered as a small insertion of hexagonal WZ in a cubic ZB matrix, these twin defects affect the local electronic structure, which can be evidenced by low temperature spectroscopy<fnref n="69" />.

Still, the VLS-growth of III-V nanowires presents several strong advantages. The large strain accommodation of nanowires growing on mismatched substrates (§4.1.1), or axial heterostructures (§4.1.2), leads to the synthesis of good quality III-V material nearly independently of the chosen substrate. Although its experimental control may be challenging, the VLS-growth of the uncommon hexagonal phase of several III-V compounds is also a significant asset.

Yet the VLS axial growth mechanism does *not* operate in full isolation and concurrent VS crystal growth occurs on all accessible surfaces, especially on the nanowire sidewalls (§4.2.2 and §4.3). In practice, this leads to the formation of complex unintentional core-shell structures in all ternary nanowires investigated here (GaAsP, AlGaAs, InGaAs), which in turn affects the quality of the nano-object (inhomogeneous alloy composition, multiple peak emission or wide emission linewidth).

Efficient sidewall passivation (§4.3.1) and device fabrication from regular nanowire arrays is possible, as shown recently in the case of GaAs/GaInP radial junction solar cells [Tong2022]. However, efficient carrier collection is still challenging due to difficulties in the growth of highly-doped nanowire shells or axial segments.

<footnote n="67">⁶⁷M. Heiss, Y. Fontana, A. Gustafsson, G. Wüst, C. Magen, D. D. O'Regan, J. W. Luo, B. Ketterer, S. Conesa-Boj, A. V. Kuhlmann, J. Houel, E. Russo-Averchi, J. R. Morante, M. Cantoni, N. Marzari, J. Arbiol, A. Zunger, R. J. Warburton, and A. F. i Morral. "Self-assembled quantum dots in a nanowire system for quantum photonics". *Nature Materials* 12.5 (2013), pp. 439–444.
DOI: 10.1038/nmat3557.</footnote>

<footnote n="68">⁶⁸S. Assali, I. Zardo, S. Plissard, D. Kriegner, M. A. Verheijen, G. Bauer, A. Meijerink, A. Belabbes, F. Bechstedt, J. E. M. Haverkort, and E. P. A. M. Bakkers. "Direct Band Gap Wurtzite Gallium Phosphide Nanowires". *Nano Letters* 13.4 (2013), pp. 1559–1563. DOI: 10.1021/nl304723c.</footnote>

<footnote n="69">⁶⁹D. Spirkoska, J. Arbiol, A. Gustafsson, S. Conesa-Boj, F. Glas, I. Zardo, M. Heigoldt, M. H. Gass, A. L. Bleloch, S. Estrade, M. Kaniber, J. Rossler, F. Peiro, J. R. Morante, G. Abstreiter, L. Samuelson, and A. F. i Morral. "Structural and optical properties of high quality zinc-blende/wurtzite GaAs nanowire heterostructures". *Physical Review B* 80.24 (2009). DOI: 10.1103/physrevb.80.245325.</footnote>
Chapter 5

Growth without covalent bonding: van
der Waals epitaxy

While any material can be thinned down to the atomic limit, the most convenient choices are those whose (bulk) crystal structure already presents a layered arrangement. Layered van der Waals solids, which expose parallel planes of covalently-bonded atoms with no dangling bonds, are prime candidates. Therefore many bidimensional (2D) materials are layered van der Waals crystals, which thickness has been reduced to the smallest covalent unit.

**5.1. Context**

5.1.1 The galaxy of 2D materials

The most iconic 2D material is surely graphene (i.e. ‘monolayer’ graphite), which offers a direct experimental access to the unconventional physics of Dirac fermions<sup>70</sup>. Other well-known examples include high band gap Boron Nitride (~ 6 eV) and anisotropic black phosphorus. However, the 2D family hosts many other compounds, such as Transition Metal Dichalcogenides (TMD) such as MoS<sub>2</sub> or WSe<sub>2</sub>, group III monochalcogenide (GaSe, InSe), silicene, and many more<sup>71</sup>.

<figure><img src="image_89.png" /><figcaption>**Figure 5.1:** ‘The world of two-dimensional materials’, from Ajayan *et al*..</figcaption></figure>

<footnote n="70"><sup>70</sup>K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov. "Two-dimensional gas of massless Dirac fermions in graphene". *Nature* 438.7065 (2005), pp. 197–200. DOI: 10.1038/nature04233.</footnote>

<footnote n="71">⁷¹P. Ajayan, P. Kim, and K. Banerjee. “Two-dimensional van der Waals materials”. *Physics Today* 69.9 (2016), pp. 38–44. DOI: 10.1063/pt.3.3297.</footnote>