# Evidence of robust, universal conformal invariance in living biological
matter

Benjamin H. Andersen1, Francisco M. R. Safara2,3,
Valeriia Grudtsyna1, Oliver J. Meacock4,5,
Simon G. Andersen 3, William M. Durham5∗, Nuno A. M. Araujo2,3∗
Amin Doostmohammadi1∗

1Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
2Departamento de Física, Faculdade de Ciências, Universidade de Lisboa,
Lisboa, Portugal
3Centro de Física Teórica e Computacional, Faculdade de Ciências,
Universidade de Lisboa, Lisboa, Portugal
4Department of Fundamental Microbiology, University of Lausanne, Lausanne,
Switzerland
5Department of Physics and Astronomy, University of Sheffield, Sheffield,
United Kingdom
∗To whom correspondence should be addressed:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Collective cellular movement plays a crucial role in many processes
fundamental to health, including development, reproduction, infection, wound
healing, and cancer. The emergent dynamics that arise in these systems are
typically thought to depend on how cells interact with one another and the
mechanisms used to drive motility, both of which exhibit remarkable diversity
across different biological systems. Here, we report experimental evidence of
a universal feature in the patterns of flow that spontaneously emerges in
groups of collectively moving cells. Specifically, we demonstrate that the
flows generated by collectively moving dog kidney cells, human breast cancer
cells, and by two different strains of pathogenic bacteria, all exhibit
conformal invariance. Remarkably, not only do our results show that all of
these very different systems display robust conformal invariance, but we also
discovered that the precise form of the invariance in all four systems is
described by the Schramm-Loewner Evolution (SLE), and belongs to the
percolation universality class. A continuum model of active matter can
recapitulate both the observed conformal invariance and SLE form found in
experiments. The presence of universal conformal invariance reveals that the
macroscopic features of living biological matter exhibit universal
translational, rotational, and scale symmetries that are independent of the
microscopic properties of its constituents. Our results show that the patterns
of flows generated by diverse cellular systems are highly conserved and that
biological systems can unexpectedly be used to experimentally test predictions
from the theories for conformally invariant structures.

Understanding the collective movement of large populations, and how it arises
from its constituents, is a central problem in biology, ecology, material
science and physics [1, 2, 3, 4]. In these living systems, work is produced at
the level of an individual constituent, and this ‘activity’ is translated into
patterns of collective motion at larger length scales through interactions
between them [5, 1]. However, many of the processes involved in collective
movement, including the mechanisms that individual constituents use to propel
themselves, the processes that give rise to interactions, and the behavioural
responses to stimuli, are incredibly diverse in different biological systems
and are often difficult to decode [6, 7]. While many different models have
been proposed to reproduce the specific pattern of collective movement made by
particular organisms, we lack a general unifying theory or set of principles
that unite the collective movement observed across distinct biological
systems.

In contrast, the study of the complex interactions between the components that
make up certain inanimate materials, like metals and alloys, has led to the
discovery of universal behavior near the so-called critical regimes. In these
conditions, the global macroscopic properties no longer depend on the specific
properties of the individual constituents, but rather exhibit “universal”
behavior [8]. The principles that give rise to this universality in inanimate
materials have been described using the framework of conformal field theory
[9, 10], which predicts how shapes and angles of structures will be locally
conserved across different systems, but not necessarily their length scales or
curvatures. While the techniques used to describe conformally invariant
structures have long been used to make theoretical predictions in statistical
mechanics and condensed matter physics [9, 10] and to establish the
universality of critical phenomena (for example, using numerical studies of
turbulence [11, 12, 13] and rigidity percolation [14, 15]), direct
experimental observation of conformal invariance and robust universal critical
behavior in living matter remains elusive.

In this paper, we experimentally demonstrate that the patterns of collective
movement observed in different types of living matter exhibit universal
characteristics that transcend the particular properties of the cells from
which they are composed. We show that vastly different systems, including
colonies of pathogenic bacteria, groups of collectively moving dog kidney
cells, and human breast cancer cells, all spontaneously generate flows that
exhibit a universal conformal invariance that can be described by the
percolation universality class. This finding suggests that collective cellular
movement, which plays an important role in many biological systems [3, 16,
17], could potentially serve as a fundamental test bed for theories that are
based on conformal symmetry.

Figure 1: High-resolution measurements of the coherent flows from collectives
of eukaryotic and prokaryotic cells. Representative velocity and vorticity
fields observed in monolayers of (a) eukaryotic Madin-Derby Canine Kidney
(MDCK) cells, and (b) prokaryotic wild-type Pseudomonas aeruginosa cells. The
colormap shows the local vorticity and zero-vorticity contours are marked with
black lines. The vorticity is normalised by its maximum value. Insets show a
subset of the cells within a single field of view, which have been overlaid
with green arrows showing the local velocity. Here we have quantified movement
using single cell tracking (PTV), but we have also verified our results using
particle image velocity (PIV) (Materials and Methods).

We made high-resolution measurements of monolayers composed of four different
cellular genotypes, including both prokaryotes and eukaryotes, to resolve
whether we could identify common features in their collective motility. For
prokaryotes, we studied the opportunistic pathogen Pseudomonas aeruginosa,
which uses tiny grappling hooks called pili to pull themselves along solid
surfaces, a process known as twitching motility [18]. We considered two
different strains of this rod-shaped bacteria - wild type (WT) PAO1 and a
deletion mutant $\Delta$pilH lacking one of the response regulators in the
Pil-Chp system, which causes it to become hyperpiliated, move faster, and form
longer cells than its parental WT [18, 19, 20]. For the eukaryotic cells, we
considered the commonly studied Madine-Darby-Canine-Kidney (MDCK) cells [21],
and aggressive human breast cancer cells (MCF-7) [22]. Each of these genotypes
forms monolayers through in-situ growth and exhibits two-dimensional
collective patterns of motion when they reach confluence. Vortical flow
structures, a characteristic feature of the disordered flows observed in wide
diversity of different systems [23], are observed in all four of the cellular
genotypes investigated here (e.g. see Fig. 1). Each vortex either exhibits
clockwise or anti-clockwise rotation and the line that sits at the boundary
between flows that rotate in opposite directions, the zero-vorticity contour,
provides a measure of the underlying structure of the flow.

To compare how the flow structure varies across the four different
experimental systems, we first measure the fractal dimension of the vorticity
contours by plotting the perimeter of closed contours as a function of their
radius of gyration. Surprisingly, without any fitting, special scaling, or
free parameters, the results for all four different experiments collapse on
the same line and exhibit the same power-law behavior (Fig. 2a). This provides
concrete evidence of scale invariance and indicates that the flows generated
by these diverse cellular systems share the same generic features.
Interestingly, the slope of the perimeter-gyration radius plot, or fractal
dimension, is $D=7/4$ for the complete perimeter and $D_{\ast}=4/3$ for the
accessible external perimeter (Fig. S2) and satisfies the duality relation
$4(D-1)(D_{\ast}-1)=1$, conjectured for conformally invariant curves [24].
This finding suggests that these biological flow structures, in addition to
being scale invariant, could exhibit much richer conformal symmetries [25].

Figure 2: Vorticity contours from four distinct cellular systems exhibit the
same patterns of scale and conformal invariance, which is recapitulated using
a continuum model of active fluids. (a) The perimeter of contours as a
function of their radius of gyration for two prokaryotic and two eukaryotic
genotypes, including wild-type P. aeruginosa bacteria (yellow circles) and a
hyperpilated $\Delta$pilH P. aeruginosa mutant (blue squares) that
individually move faster, Madine-Darby Canine Kidney cells (red diamonds), and
MCF-7 human breast cancer cells (purple stars). Here we separately analysed
the complete perimeter and accessible external perimeter of the contours (SI
Fig. S2) - we found that the experimental data for all four genotypes
collapsed onto lines with slopes of approximately $7/4$ and $4/3$ respectively
for the two different perimeter measurements. The flow fields produced by a
numerical model of active fluids (Materials and Methods) generated vorticity
contours with a power-law dependency in close agreement with that observed in
experiments. The radius of gyration is normalised by the maximum system size
(Materials and Methods). (b) The variance in the distribution of the winding
angle, plotted here as a function of distance along the curve for the four
experimental systems and numerical model, all exhibit the same logarithmic
scaling with a slope of $6/7$ (dashed line). (b; inset) In addition, the
distribution of winding angles for a fixed distance along the contour is
closely approximated by a Gaussian (dashed line). Both findings are consistent
with that predicted for conformally invariant curves, which exhibit the same
fractal dimension that we obtained for our data in Fig. 2a. The dashed lines
correspond to the slope $6/7$ and a standard Gaussian distribution,
respectively. In (b; inset) the winding angles are obtained for segments of
contours of length $64$ (filled symbols) and $512$ (empty symbols) and
measured relative to the average angle of the contour. Results are averages
over different samples and error bars represent the standard deviation (see
Materials and Methods).

To test whether our experimental data exhibits conformal invariance, we
calculated the winding angle of the vorticity contours across the four
different experimental systems. The winding angle is defined as the angle
between two points on a contour that are separated by a given distance
measured along the contour (Materials and Methods). For conformally invariant
curves (i) the winding angles are Gaussian distributed and (ii) the variance
in the distribution of winding angles increases logarithmically with the
length of the curve [26]. Our experimental data is in close agreement with
both predictions for conformal invariance – with both metrics collapsing the
data from the vorticity contours of the four cellular systems onto the same
line (Fig. 2b and Fig. S4). Moreover, the rate at which the variance of the
winding angle increases with the logarithm of the length is predicted to scale
as $\alpha=6/7=2(D-1)/D$ for conformally invariant curves [26]. Thus, for the
fractal dimension of $D=7/4$ we measured in Fig. 2a, we would predict that
$\alpha=6/7$, which is supported by our direct measurements of the variance
(Fig. 2b).

Our results strongly suggest that the flows spontaneously generated by diverse
cellular genotypes exhibit robust conformal invariance. We next sought to
ascertain we could resolve which universality class these biological flows
belong to. One of the central mathematical breakthroughs of the last few
decades was to demonstrate that certain systems with conformal invariance and
domain Markov property, can be described, in the scaling limit of interfaces,
by a family of planar curves defined by a single parameter $\kappa$. This
formalism is known as the Schramm-Loewner Evolution (SLE) [27, 28] and the
value of the $\kappa$ distinguishes different fundamental statistical
mechanics models at criticality and thus resolves the universality class that
a system belongs to [29, 30, 31]. To determine if the vorticity contours in
the cellular systems are SLE curves, we extracted the $\kappa$ parameter from
the four experimental systems. We used two distinct and independent methods
[32]: (i) directly calculating the driving function [33], and (ii) measuring
the left-passage probability, comparing both to analytic predictions for the
SLE [34] (Materials and Methods). Both methods consistently yielded
$\kappa=6$, for all four cellular genotypes (Fig. 3). The value of $\kappa=6$
is also consistent with the estimated fractal dimension, ($D$, Fig. 2a), which
for SLE is related to $\kappa$, as $D=1+\kappa/8$ [35]. This particular value
of $\kappa$ has an important physical meaning, as it has been uniquely proven
for $\kappa=6$ that SLE curves correspond to the contours of critical
percolation clusters and have the locality property (such that the properties
only depend on the immediate neighborhood) [30, 36]. As such, our analyses
reveal that the vorticity contours found in the four different cellular
systems are not only conformally invariant, but they also all fall into the
universality class of percolation.

Figure 3: Resolving the underlying universality class of vorticity contours
using two independent methods. (a) The left-passage probability is defined as
the probability that a point in space is on the right-hand side of the contour
for a given polar angle. Data from all four cellular genotypes and the results
from the numerical model are in close agreement with Schramm’s formula for
$\kappa=6$ (dashed black line, [34]). (b) Time-dependence of the variance of
driving function obtained from a unique conformal slit map [33]. The dashed
black line shows the result for one-dimensional Brownian motion with
$\kappa=6$. (b, inset) Probability distribution of the driving function,
rescaled by $\kappa t$, where $t$ is the Loewner time. Here the data at two
different times $t=0.25$ and $t=0.75$ is shown, which collapses onto the same
curve. For further details see Materials and Methods.

Our experimental results indicate that diverse cellular types collectively
generate flows with remarkably similar patterns of scale and conformal
invariance, implying that the physical mechanisms that underlie the flow
structures are highly conserved. While many different physical models of
active matter have been developed to approximate specific types of cells and
the processes unique to them [5, 2, 37], we tested whether a generic model
could recapitulate our experimental observations. We used a simple continuum
model in which a nematic order parameter (corresponding to cell orientation)
was coupled to an incompressible velocity field (see Materials and Methods for
further details). The two main parameters are the activity $\zeta$,
represented by active stress generation in the velocity equation, and
elasticity, represented by the elastic constant $K$ that penalizes
deformations. Dimensional analysis of the governing equations yields a
characteristic length scale of $K/\zeta$, which defines the fundamental length
scale of the flow. We find that the vorticity contours of this minimal model
recapitulate each of the measurements observed in our experimental systems,
including the fractal dimension $D=7/4$ (Fig. 2a), winding angle scaling
$\alpha=6/7$ (Fig. 2b), and the scaling of the driving function and left-
passage probability $\kappa=6$ (Fig. 3 and Table 1). While differences in cell
morphology, intercellular adhesion, mechanotransduction, and the mechanisms
that give rise to local alignment can all affect patterns of collective
motility [38, 39, 7], the results of our continuum model imply that such
idiosyncratic characteristics do not materially influence scale and
conformally invariant flow patterns, but rather they are a generic feature of
collective cellular flows.

The observed scaling of the vorticity contours from both experiments and model
are compatible with the Schramm-Loewner evolution (SLE) with $\kappa=6$, for
more than two decades in range and this was confirmed using two independent
methods (Fig. 3 and Table 1). Remarkably, this finding demonstrates that,
although the collective cellular motility spontaneously generates patterns of
flow with lengthscales much larger than that of individual cells (Fig. 1) [38,
3], the associated vorticity contours are local and fall into the same
universality class as those from random percolation [30]. Moreover, the
collective cellular motion we studied here is driven far from equilibrium by
the motility of individual cells that continuously inject energy into the
system at small scales. The observation of conformal invariance in collective
cellular flows that are continuously driven far from thermodynamic equilibrium
presents both challenges and new opportunities for the development of non-
equilibrium conformal field theories [40].

continuum model experimental data percolation universality class bacterial
cells eukaryotic cells WT $\Delta$pilH MDCK MCF7 scale invariance fractal
dimension $D$ $1.75\pm 0.01$ $1.72\pm 0.02$ $1.72\pm 0.04$ $1.74\pm 0.02$
$1.74\pm 0.03$ $7/4=1.75$ conformal invariance winding angle $\alpha$
$0.853\pm 0.005$ $0.86\pm 0.01$ $0.87\pm 0.01$ $0.85\pm 0.01$ $0.87\pm 0.02$
$6/7\approx 0.857$ Schramm-Loewner Evolution left-passage probability $\kappa$
$6.02\pm 0.02$ $5.97\pm 0.05$ $5.96\pm 0.05$ $5.95\pm 0.03$ $5.95\pm 0.06$ $6$
driving function $\kappa$ $5.96\pm 0.05$ $6.03\pm 0.06$ $5.96\pm 0.06$
$5.98\pm 0.04$ $5.93\pm 0.04$ $6$

Table 1: Measurements of the fractal dimension, winding angle, left-passage
probability, and driving function of the four different cellular genotypes and
numerical simulations. The values are calculated from the velocity fields
obtained from single cell tracking (Materials and Methods) and the errors
represent standard deviation about the mean.

These results suggest that the theories used to describe conformally invariant
structures might have a much broader range of applications than previously
anticipated. While collective movement is observed in diverse biological
systems, that observed in microscopic cellular systems is particularly
amenable to experimental analysis because in-situ cell division rapidly gives
rise to large genetically identical populations, two-dimensional movement of
monolayers of cells can be readily imaged, and the environmental conditions
can be carefully controlled. Similar to the collective cellular motility
studied here, many different kinds of living systems are formed of strongly
interacting components driven far from thermal equilibrium and exhibit complex
vortical patterns, including subcellular flows [41, 42], synthetic active
material [43, 1, 44], animal swarms [45, 46], and in-vitro reconstitutions of
cytoskeletal transport systems [47, 48, 49]. In addition, emergent vortical
structures also shape many important processes in biology including cell
differentiation [16], cartilage regeneration [50], embryogenesis [51],
signaling waves that propagate along cell membranes [52] and between cells
[53], as well as vortical waves associated with cardiac arrhythmia [54] and
spiral-like patterns of brain activity linked to cognitive processing [55]. We
speculate that such biological processes might not only serve as novel test
bed to validate predictions based on conformal symmetry, but this robust
symmetry might also lead to the development of new analytical techniques to
identify the fundamental mechanisms that give rise to both function and
dysfunction in complex biological systems.

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## Acknowledgments

We thank Guido Boffetta for comments on a previous version of this manuscript
and Ramin Golestanian for helpful discussions.

## Funding

This work was supported by the Novo Nordisk Foundation (grant no.
NNF18SA0035142 and NERD grant no. NNF21OC0068687) (to AD), Villum Fonden Grant
no. 29476 (to AD), the European Union via the ERC-Starting Grant PhysCoMeT,
grant no. 101041418 (to AD), the Portuguese Foundation for Science and
Technology (FCT) under Contracts no. EXPL/FIS-MAC/0406/2021, UIDB/00618/2020,
and UIDP/00618/2020 (to NA), a Biotechnology and Biological Sciences Research
Council (BBSRC) New Investigator grant (grant no. BB/R018383/1) (to WMD), and
a Human Frontier Science Program grant (grant no. RGY0080/2021) (to WMD).

## Authors contributions

A.D. and N.A.M.A. designed the project. B.H.A. and F.S. performed analyses of
scale and conformal invariance as well as SLE measurements on the experimental
data. B.H.A. implemented the model, analysed data and prepared figures. V.G.
performed the experiments and conducted particle image velocimetry
measurements on MDCK and MCF-7 cells. S.G.A. performed single cell tracking
analyses for MDCK and MCF-7 monolayers. O.J.M. performed the experiments on
bacterial cells and conducted particle image velocimetry and single cell
tracking analyses on bacterial monolayers. W.M.D., N.A.M.A. and A.D. all
contributed to the design of experiments and models, as well as to the
interpretation of results. A.D. prepared the first draft. A.D. and W.M.D.
wrote the paper with input from N.A.M.A. This collaborative effort was led by
A.D.


# Dual-adatom diffusion-limited growth model
for compound nanowires: Application to InAs nanowires

Danylo Mosiiets  Yann Genuist  Joël Cibert  Univ. Grenoble Alpes, CNRS,
Grenoble INP, Institut Néel, 38000 Grenoble, France  Edith Bellet-Amalric
Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, 38000 Grenoble, France
Moïra Hocevar<EMAIL_ADDRESS>Univ. Grenoble Alpes, CNRS, Grenoble
INP, Institut Néel, 38000 Grenoble, France

###### Abstract

We propose a dual-adatom diffusion-limited model for the growth of compound
semiconductor nanowires via the vapor-liquid-solid or the vapour-solid-solid
mechanisms. The growth is catalyzed either by a liquid or a solid
nanoparticle. We validate the model using experimental data from the growth of
InAs nanowires catalyzed by a gold nanoparticle in a molecular beam epitaxy
reactor. Initially, we determine the parameters (diffusion lengths, flux to
the seed, Kelvin effect) that describe the growth of nanowires under an excess
of one of the two beams (for instance, group III or group V atoms). The
diffusion-limited model calculates the growth rate resulting from the current
of atoms reaching the seed. Our dual-adatom diffusion-limited model calculates
for a compound semiconductor, the instantaneous growth rate resulting from the
smallest current of the two types of atoms at a given time. We apply the model
to analyze the length-radius dependence of our InAs nanowires for growth
conditions covering the transition from the As-limited to the In-limited
regime. Finally, the model also describes the complex dependence of the
transition between both regimes on the nanowire radius and length. This
approach is generic and can be applied to study the growth of any compound
semiconductor nanowires.

##  I Introduction

Nanowire growth began in the early 1960s with silicon. Experimental results
[1] and theoretical studies [2] during this period identified the role of the
diffusion of single types of adatoms in the growth process. However, various
mechanisms have been and are currently being invoked and discussed based on
the impact of the nanowire diameter on the growth rate [3]. During the
following decades after, interest in the growth of nanowires made of compound
semiconductors has progressively increased, particularly III-V nanowires such
as GaAs and InAs [4, 5, 6]. For instance, GaAs nanowires provide an
alternative to planar structures for integrating photonics on Si substrates,
and InAs nanowires stand out for their advantages in applications such as
chemical sensors [7] and photodetectors [8]. Today, InAs nanowires contribute
significantly to quantum devices because of their small bandgap, large
mobility and large g-factor. InAs nanowire devices feature not only high
mobility transistors [9] but also serve as a promising platform for various
quantum bit implementations, including spin-based quantum bits (qubits) [10]
and transmon qubits [11].

All those novel devices were enabled not only thanks to better cleanrooms and
better lithography techniques, but also thanks to better materials. Knowledge
and control over growth and crystalline quality have been foundational to the
improvement of nanowires. InAs nanowires grow self-catalyzed and by the gold-
assisted vapour-solid-solid (VSS) or vapour-liquid-solid (VLS) mechanisms. In
the latter case, they are particularly easy to grow thanks to the large range
of parameters for which the VLS mechanism works. Parameters are either
external (In and As fluxes sent by the cells, growth temperature, growth time)
or specific to the sample (size of the Au nanoparticle, substrate orientation,
distance between nanoparticles, surface chemistry).

Understanding and controlling the growth of nanowires is a priori more complex
for a compound semiconductor than for an elemental semiconductor since it
depends on the behavior of the two species. Many studies focused on growth
conditions under a strong excess of one constituent, so that the growth is
essentially controlled by the other one. For instance, the growth of GaAs
nanowires self-catalyzed by a Ga nanoparticle usually reveals the behavior of
the As molecules and adatoms, albeit complex [12]. As detailed below, arsenic
is highly volatile, and its adatoms exhibit a very short diffusion length,
leading to a weak dependence of the growth rate on the nanowire radius. In the
opposite case of a strong arsenic excess, the negligible evaporation of the
type-III adatoms and their diffusion length in the micrometer range result in
a characteristic decrease of the growth rate upon increasing the nanowire
radius [13]. However, a cutoff at small radius is sometimes reported [14] and
attributed either to a size effect (”Gibbs-Thomson effect”) incorporated into
the single-species model, or to the low probability of nucleation on a small
area [15]. We may note here that the notion of a diffusion length,
$\lambda=\sqrt{D\tau}$ may cover different mechanisms: $D$ is a
phenomenological diffusion coefficient, and the residence time $\tau$ may
correspond to the desorption of the adatoms or to their incorporation. Fewer
studies take into account the two species. An experimental diagram of the
growth rate of InAs nanowires seeded by Au colloids of radius 10 nm, vs. As
and In beam equivalent pressures (BEP), was reported in Ref. [16], and
described qualitatively. Indeed, both In and As BEPs play a significant role.
Quantitative models were proposed more recently [17, 18, 19]. They confirm
that in case of a strong excess of one species, the other one controls the
growth. They show also that the dependence of the chemical potentials, and
hence the nucleation rates, on both fluxes may make the transition between the
two limits quite progressive.

Here, we propose to look deeper into the influence of the flux of each type of
atoms, for instance type-III and type-V fluxes, on the growth of compound
nanowires, both experimentally on InAs nanowires, and theoretically. InAs is
particularly well suited for this study because In and As behave very
differently (In doesn’t re-evaporate, has long diffusion length and a small
direct flux; As re-evaporates, suffers from the Kelvin effect, has limited
diffusion and is re-emitted from surfaces). We synthesized nanowires from gold
catalysts of various radii, using different In and As cell BEPs, at a
temperature chosen within the range
($\mathrm{400^{\circ}\text{C}-420^{\circ}\text{C}}$) where the growth rate of
InAs nanowires was shown to reach a maximum [20]. We found that the nanowire
growth rate shows a maximum for a given radius for all values of the In and As
BEPs. Our scan of the As BEP allows us to observe the evolution of the length
dependence on radius from a dependence typical of an As-limited growth, to a
dependence typical of the growth limited by the In flux. However in all cases
no nanowires are observed below a critical radius, the value of which depends
slightly on the In flux and not on the As flux. We then propose a dual-adatom
diffusion-limited growth model that takes into account the dynamics of both
type-III and type-V adatoms, based on their very different properties, and on
a size effect which is discussed as the Kelvin effect acting on As only. Our
model shows that for a given V/III ratio, the growth rate can be limited by
either one of the two adatom currents, depending on the diameter and the
length of the nanowire. As the currents of both type-III and type-V adatoms
are not balanced, we can describe solely their limiting behaviors. This
enables the calculation for the overall growth of our nanowires by applying
these limiting behaviors at each moment of growth. Finally, we find a set of
parameters that allows us to fit all our experimental data. The diffusion
lengths of In and As on the nanowire sidewall facets are approximately 7000 nm
and 100 nm, respectively, at $\mathrm{420^{\circ}\text{C}}$.

The paper is organized as follows: first, we present the growth conditions of
the nanowires and the results of the experimental growth of InAs nanowires.
Then, we present our nanowire growth model with a particular focus on III-V
compound semiconductors. We demonstrate the dependence of the instantaneous
growth rate of each species on the nanowire length and apply the model to our
experimental data with nanowires grown under various conditions. Furthermore,
we discuss our experimental results that cover the transition between As-
limited and In-limited growth, our dual-adatom diffusion-limited model, and
finally, the effect of a critical radius characterizing the sharp decrease in
growth rate at a small radius.

##  II Experimental results

###  II.1 Growth conditions and characterization

InAs nanowires are synthesized utilizing the VLS mechanism, assisted by gold
catalysts, within a solid-source molecular beam epitaxy (MBE) reactor. Sample
preparation proceeds as follows: several pieces of InAs(111)B substrates are
de-oxidized in a solution of $\mathrm{H_{3}NO_{4}:H_{2}O}$ (1:1) for 5 to 10
minutes, followed by a thorough rinsing in de-ionized water for an additional
10 minutes. Subsequently, Au colloids (purchased from BBI Solution) with radii
ranging from 10 to 50 nm (with a size dispersion of 2%) are deposited onto the
InAs substrates via drop-casting. For each growth experiment, a minimum of
four colloid nominal sizes are assessed.

The samples are then glued on a Si substrate and mounted onto a molybdenum
block, which is subsequently introduced into the MBE reactor. An initial
annealing step is performed at $\mathrm{250^{\circ}\text{C}}$ under a vacuum
of $\mathrm{10^{-9}}$ Torr, to eliminate condensed water. Furthermore, the
molybdenum block is introduced to the main chamber for growth. The substrate
temperature is raised to $\mathrm{500^{\circ}\text{C}}$ to form the Au-In
eutectic under an As BEP of $\mathrm{10^{-5}}$ Torr measured by a Bayard-
Alpert gauge. The As cracker cell is operated at reservoir and cracker
temperatures of $\mathrm{350^{\circ}\text{C}}$ and
$\mathrm{650^{\circ}\text{C}}$, respectively so that the beam is mainly
composed of As4 molecules [21].

After the annealing step, which lasts around 10 min, the sample is cooled down
to the growth temperature ($\mathrm{420^{\circ}\text{C}}$). At this
temperature, after 5 min of temperature stabilization during which the As BEP
is set to the desired value and the In shutter is opened in order to prevent a
burst, growth is initiated by opening the main shutter. Growth is conducted at
constant In and As BEPs. After 30 min of growth, the main shutter is closed
and the sample returned to room temperature.

Table 1: Samples for the study of the As BEP dependence. | I25R60 | I25R20 | I25R10 | I25R05
---|---|---|---|---
In BEP (x$\mathrm{10^{-7}}$ Torr) | 2.5 | 2.5 | 2.5 | 2.5
As BEP (x$\mathrm{10^{-6}}$ Torr) | 15 | 5 | 2.5 | 1.25
As/In BEP ratio | 60 | 20 | 10 | 5
Projected In flux (nm/s) | 0.16 | 0.16 | 0.16 | 0.16
Projected As flux (nm/s) | 1.65 | 0.55 | 0.28 | 0.14
Projected As/In flux ratio | 11 | 4 | 2 | 1
Growth time (min) | 30 | 30 | 30 | 30

Once removed from the MBE reactor, the samples are imaged in a scanning
electron microscope (SEM) at a 45∘ tilt. We use a Zeiss Ultra$+$ Field
Emission SEM with an acceleration voltage of 5 keV. Broad views of the samples
and zoomed views of single nanowires are recorded. The diameters of the
nanowires are measured at the basis and at the tip of the nanowire, just below
the catalyst. This is done on images with a magnification high enough to
obtain an accurate value of the diameter. The length of the nanowire is
measured from the basis (at the top of the pyramid which usually forms around
it), to the top of the nanowire (below the catalyst-nanowire interface), on
images with a lower magnification. We measure a minimum of 30 nanowires per
colloid size to ensure a reliable statistic of the sample. The nanowire height
is corrected by adding the mean thickness of the two dimensional layers
calculated from the growth rate calibrations. We consider this thickness to be
a sufficient approximation for the pyramid’s height.

The values of the flux emitted by the cells have been calibrated through the
measure of the growth rates of InAs layers on a (001) InAs substrate, using
RHEED oscillations. In the following, the term ”flux” will be used for the
flux of atoms, projected onto the substrate surface, and expressed as the
growth rate of InAs in nm/s. The values are given in Tables 1 and 2. Note that
we probably underestimate slightly the As flux. For more details, see
Supplementary Material S1 in ref [22]. As the angle of incidence with the
substrate is larger for our As beam than for our In beam, for a given value of
the As-to-In projected flux ratio, the corresponding ratio for the flux along
the cell axis is larger, and the ratio of flux to the nanowire sidewalls is
even larger.

###  II.2 Results

We show an example of such nanowires grown in our MBE system with a V/III flux
ratio of 11 (corresponding to a BEP ratio of 60) using colloids with sizes
ranging from 10 to 50 nm as initial radius (Fig. 1(a) to 1(d)). The nanowires
are cylinder-shaped, with the same radius at the bottom and at the tip, equal
or slightly smaller that of the gold particle. There is no evidence of radial
growth that could result in a pencil-shaped [20, 23] or tapered (conical) [16,
23] nanowire. It is clear from the images that the nanowires grown from 10-nm
radius colloids show a large dispersion in length (Fig. 1(a)), while the
larger radii nanowires are shorter and more homogeneous in length.

Figure 1: Au-assisted InAs wires grown on an InAs (111)B substrate at large As
flux (sample I25R60, As BEP=15$\times 10^{-6}$ Torr, In BEP=2.5$\times
10^{-7}$ Torr, As/In flux ratio of 11). (a) to (d), SEM images at 45∘ tilt.
Scale bars are 2 $\mu$m. Radius of colloids: (a) 10 nm, (b) 20 nm, (c) 40 nm,
(d) 50 nm. (e) Nanowire length vs. nanowire radius. The initial radius of the
colloids is displayed with an arrow, the corresponding nanowires are
identified by the color of the symbols.

The length-radius dependence is shown in Fig. 1(e). A salient feature is that
each colloid size, symbolized by the colour of the symbols and marked by
arrows, gives rise to a range of the nanowire radius much broader than the
initial range of the colloid radius. This agrees with previous reports.
Smaller radii have been ascribed to splitting [24] or secondary nucleation of
In nanoparticles [25]. Larger radii result from coalescence [26]. The effect
of coalescence is particularly obvious for the smallest crystallites but we
could not observe any trace left behind by the moving nanoparticle, as
reported for the growth of GaAs nanowires [27]. As a result, we obtain a
dependence of the length over a continuum of the radius value, with no clear
dependence on the initial crystallite size. It features an extremum at radii
less than 20 nm. Above 20 nm, the length of the nanowires decreases with
increasing radius. As discussed later on, such a behavior is generally
attributed to a growth limited by diffusion with a large diffusion length. As
in Ref. [14], we also observe a sharp cut-off below which the nanowires do not
grow.

Next, we assess how reducing the As beam equivalent pressure impacts the
growth rate of the nanowires. To do this, we grow a series of samples using
identical growth conditions to the previous sample, with the exception of
lowering the As BEP. We use V/III flux ratios of 4, 2, and 1, respectively.
Fig. 2 illustrates the behavior of a sample grown under a V/III flux ratio of
1 (see the additional samples in Fig. S2 [22]). The nanowire shape is
unchanged, see Fig. 2(a) to (d). For all the samples of the series, there is
an extremum in the dependence of the length on the radius; however, its
magnitude increases and shifts towards smaller radii as the V/III flux ratio
increases (Fig. 3). Also, the increase in nanowire length becomes more and
more noticeable as the radius decreases further. The cut-off is still present
and at the same radius.

Figure 2: Same as Fig. 1 for a low As flux (sample I25R05, As BEP= 1.25
$\times 10^{-6}$ Torr, In BEP=2.5$\times 10^{-7}$ Torr, As/In flux ratio=1).

Nanowires grown with a different In flux (increased or decreased by a factor
of 1.5, respectively) still have a cylinder shape (see Fig. S4 [22]). Examples
of their length vs. radius dependence are shown in Fig. 4(a). The most
noticeable change is a decrease of the cut-off radius upon increasing the In
flux.

Figure 3: Length-radius dependence for different values of the As flux. The symbols show the experimental data at an As/In flux ratio equal to (a) 1, sample I25R05; (b) 2, sample I25R10, same as in Fig. 2; (c) 4, sample I25R20 and (d) 11, sample I25R60, same as in Fig. 1. (e)-(h) Same as (a) to (d) with the solid blue lines showing the fit calculated using the model. The grey dotted lines are the limits due to In and As, respectively. Figure 4: (a) Length-radius dependence for different values of the In flux. The symbols show the experimental data at an In growth rate equal to 0.11 nm/s (sample I18R100), 0.16 nm/s (sample I25R60), and 0.21 nm/s (sample I38R42). Note the 30% larger growth time for sample I18R100 (b) Same as (a) with the solid lines calculated using the model and the parameters of Tables 3, 2, but for fitting values for $R_{0}$ (Table 6). Table 2: Samples for the study of the In BEP dependence. | I18R100 | I25R60 | I38R42
---|---|---|---
In BEP (x$\mathrm{10^{-7}}$Torr) | 1.8 | 2.5 | 3.8
As BEP (x$\mathrm{10^{-6}}$Torr) | 18 | 15 | 16
As/In BEP ratio | 100 | 60 | 42
Projected In flux (nm/s) | 0.11 | 0.16 | 0.21
Projected As flux (nm/s) | 1.99 | 1.65 | 1.77
Projected As/In flux ratio | 18 | 11 | 8
Growth time (min) | 40 | 30 | 30

##  III Growth model

###  III.1 III-V nanowires growth model

The VLS (and the VSS as well) growth of compound semiconductor nanowires takes
place because two types of atoms (such as type-III and type-V) are
incorporated into the seed nanoparticle. The nanoparticle is liquid in the
case of the VLS growth, while it is solid in the case of the VSS. Hence the
relevant quantities are not the flux from the cells, but the currents to the
seed, due to diffusion of adatoms from the substrate ($J_{sub}$), diffusion of
atoms impinging the sidewall facets ($J_{f}$), direct impingement onto the
seed ($J_{Au}$), and re-evaporation ($J_{evap}$):
$J=J_{sub}+J_{f}+J_{Au}-J_{evap}$.

In a simple approach, $J$ is determined for each species by solving a
diffusion equation for the adatom density on the substrate and along the
nanowire sidewall facets, assuming that the seed is a perfect trap [28]. In a
one-species model, the growth rate of the nanowire of radius $R$
is$:\frac{dL}{dt}=\frac{J}{\pi R^{2}}$.

In our model, we evaluate separately the diffusion-limited currents of type-
III and of type-V atoms, $J_{III}$ and $J_{V}$, under the fluxes $F_{III}$ and
$F_{V}$, and assume that the nanowire instantaneous growth rate is determined
by the smallest current:

$\frac{dL}{dt}=\frac{\min(J_{III},J_{V})}{\pi R^{2}}.$ (1)

It is quite convenient [28] to consider the growth rate of a two-dimensional
layer under the same flux $F$ from the cell, $\frac{dH}{dt}=F\cos\alpha$,
where $\alpha$ is the incidence angle of the incoming beam with respect to the
normal to the substrate. This results in expressing the nanowire growth rate
as $\frac{dL}{dH}$.

Type-III adatoms are known to be non-volatile [29, 30, 31] at the usual growth
temperatures. Hence, we let $J_{III_{evap}}=0$. This assumption can be
revisited if the growth temperature is high, or for different materials (the
evaporation rate of Ga from the seed corresponds to 12% of the deposition rate
at most [32]). When sending a quantity of type-III atoms corresponding to a 2D
layer thickness $dH_{III}=F_{III}\cos\alpha_{III}dt$, the maximum increase in
the nanowire length, $dL$, is obtained [33] as follows:

$dL_{III}=\left[\frac{2}{R}\frac{1}{U^{\prime}(\frac{L}{\lambda_{f,III}})}\lambda_{s,III}\frac{K_{1}(\frac{R}{\lambda_{s,III}})}{K_{0}(\frac{R}{\lambda_{s,III}})}+\frac{2}{R}\frac{U(\frac{L}{\lambda_{f,III}})}{U^{\prime}(\frac{L}{\lambda_{f,III}})}\frac{\tan\alpha_{III}}{\pi}\lambda_{f,III}+(1+\kappa_{III})\right]dH_{III}$
(2)

The first term in the right-hand side of Equation 2 is due to the flux to the
substrate diffusing to the nanowire, the second one to the flux to the
nanowire sidewalls, and the third one to the flux to the seed. The flux is
characterized by its intensity $F_{III}$ and the incidence angle
$\alpha_{III}$ of the incoming beam with respect to the normal to the
substrate, and $dH_{III}=F_{III}\cos\alpha_{III}dt$. The diffusion lengths of
the type-III adatoms are written $\lambda_{f,III}$ and $\lambda_{s,III}$ for
diffusion along the nanowire sidewall facets and on the substrate,
respectively. The two dimensional character of the diffusion on the substrate
is taken into account by the ratio of Bessel functions of the second kind of
order $i$, $K_{i}(x)$, evaluated at $x=\frac{R}{\lambda_{s,III}}$. The lattice
parameters (unit cell volume, surface unit cell area) are incorporated into
the diffusion lengths. Finally, the functions $U(x)$ and $U^{\prime}(x)$ are
given by $U(x)=\sinh
x+\frac{\lambda_{f}}{\lambda_{s}}\frac{K_{1}}{K_{0}}(\cosh x-1)$ and
$U^{\prime}(x)=\cosh x+\frac{\lambda_{f}}{\lambda_{s}}\frac{K_{1}}{K_{0}}\sinh
x$, respectively. The direct flux to the seed contains a parameter
$\kappa_{III}$ which takes into account the shape of the seed and the
incidence of the beam [34]. It vanishes when the seed is nearly flat or when
it adopts a shape resembling a half-sphere at small angles of $\alpha_{III}$.

Type-V atoms are known to be highly volatile, and to feature a diffusion
limited in length, if any. Then Eq. 2 has to be modified.

  * •

The current into the seed is shared between the growth of the nanowire, and
the re-evaporation ($J_{V_{evap}}\neq 0$). Thus, we rewrite the flux equation
for type-V adatoms as
$J=\left(J_{Sub}+J_{f}+J_{Au}\right)(1-\theta_{evap}^{in})$, where
$\theta_{evap}^{in}$ is the ratio of the evaporation current to the input
current. Similar terms were derived in Ref. [17] at every point of the
nanowire-seed system, and shown to be related to the chemical potential. As
discussed below, we will keep the term $\theta_{evap}^{in}$ fixed when the
current from the group V is the minority current, and keep only the factor
$\exp(\frac{R_{0}}{R})$ which describes size effects.

  * •

A portion of the flux reaching the substrate is re-emitted, leading to a
significant flux towards both the nanowire sidewalls and the seed [35]. This
can be taken into account by introducing two enhancement factors $\kappa_{V}$
to the seed and $\kappa^{\prime}_{V}$ to the sidewalls.

Altogether, the equation for type-V atoms is written:

$\displaystyle
dL_{V}=\left[\frac{2}{R}\frac{1}{U^{\prime}(\frac{L}{\lambda_{f,V}})}\lambda_{s,V}\frac{K_{1}(\frac{R}{\lambda_{s,V}})}{K_{0}(\frac{R}{\lambda_{s,V}})}+\frac{2}{R}\frac{U(\frac{L}{\lambda_{f,V}})}{U^{\prime}(\frac{L}{\lambda_{f,V}})}\frac{\tan\alpha_{V}}{\pi}\lambda_{f,V}(1+\kappa^{\prime}_{V})+(1+\kappa_{V})\right]$
(3)
$\displaystyle\times\left[1-\theta_{evap}^{in}\exp(\frac{R_{0}}{R})\right]dH_{V}$

The size effect characterized by the parameter $R_{0}$ describes the
enhancement of the evaporation from a small seed (or nanoparticle) and it is
often coined as a ”Gibbs-Thomson effect”. According to Ref. [36], the term
”Kelvin effect” is more appropriate, as the Gibbs-Thomson effect impacts
actually the nucleation rate at the seed-nanowire interface and can be
neglected in our case. Anyway, the characteristic radius is

$R_{0}=\frac{2\gamma\Omega}{k_{B}T},$ (4)

where $\gamma$ is the surface energy of the seed, $\Omega$ the volume of the
desorbed atom or molecule (for instance, As2 in the case of arsenides), and
$T$ the temperature. The values of $\gamma$ for seeds made of Au, In or Ga, or
their alloys, are given in Ref. [33], together with the calculated value of
the characteristic radius $R_{0}$ (assuming the evaporation of atoms, not
molecules).

Finally, according to Eq. 1, the growth rate of the nanowire during the time
interval $dt$ is written

$dL=\min(dL_{III},dL_{V}).$ (5)

###  III.2 Dependence on nanowire length

The diffusion-limited growth rates, Eqs. 2 and 3, depend on the radius $R$ and
the instantaneous length $L$ of the nanowire. The non-linear character of
$L(t)$ has been often commented [37, 38, 39]. In the present description, it
means that the growth rate of a nanowire of a given radius can be limited by
either type-III or type-V atom species, depending on the growth conditions,
but also that the limiting species can change during the growth. This is
exemplified in Fig. 5 for an InAs nanowire with a radius of 25 nm, using the
parameters determined in the next section (see Table 3). The In-limited growth
rate is plotted versus the nanowire length in green in Fig. 5(a), while the
As-limited growth rate is plotted in orange in Fig. 5(b). For each type of
adatom, the different contributions are plotted in grey (substrate, sidewall,
seed).

In the case of In, the growth rate decreases very rapidly with increasing
nanowire length, reaches a minimum and then increases slowly. This behaviour
is explained by the initial contribution of the substrate diffusion which is
predominant for nanowire lengths below 1 $\mu$m (due to a short diffusion
length on the substrate). Above this value, the current due to the flux onto
the sidewalls is predominant and increases even for nanowire lengths beyond
$\lambda_{f,In}$.

Figure 5: Diffusion limited currents, as a function of the actual nanowire
length, for a radius of 25 nm, calculated with the parameters of Table 3. (a)
Nanowire growth rate limited by the current of In adatoms, divided by the
growth rate of a 2D layer limited by the In flux, $\frac{dL_{In}}{dH_{In}}$
(thick solid green line). The individual contributions of the adatoms
diffusing from the substrate (thin solid line), of the adatoms impinging on
the sidewalls (dashed line) and of the direct flux to the seed (dotted line)
are also shown. (b) Same quantity $\frac{dL_{As}}{dH_{As}}$ for As (thick
solid orange line). (c) shows $\frac{dL_{i}}{dH_{In}}$, for $i=$In (green
solid line), $i=$As with an As-to-In flux ratio =1 (orange solid line), and
$i=$As for an As-to-In flux ratio = 4 (orange dashed line).

In the case of As, the instantaneous growth rate reaches a minimum for very
short nanowire lengths (not visible), and increases until reaching rapidly a
constant value. Below 200 nm in length, diffusion of As from the substrate and
the flux to the nanoparticle are predominant, while above this value, the
growth rate due to the seed and the diffusion of atoms impinging the sidewalls
(with a very small diffusion length) contribute constantly to the growth rate.

Under those conditions, one sees that the 25 nm radius nanowire is always
limited by As for a V/III flux ratio = 1. For a V/III flux ratio of 4, this
same nanowire sees its growth rate limited alternatively by As and In (first
As, then In, then As again), as shown in Fig. 5(c). Finally, at sufficiently
large V/III flux ratios and typical final lengths of several micrometers, the
nanowire elongation rate will be primarily limited by In during growth.

Table 3: Fitting parameters for the study of the As BEP dependence.  In data |  | As data |
---|---|---|---
$\lambda_{s_{In}}$ (nm) | 200 |  $\lambda_{s_{As}}$ (nm) | 50
$\lambda_{f_{In}}$ (nm) | 7500 |  $\lambda_{f_{As}}$ (nm) | 100
$\kappa_{In}$ | 0 | $\kappa_{As}$ | 6
|  | $\theta_{evap}^{in}$ | 0.6
|  |  $R_{0}$ (nm) | 8

The limiting aspect of In and As during growth needs to be re-evaluated for
each radius. Indeed, the contribution of the sidewalls plays an important
role, as the In-limited growth rate curve has a large slope increase with
radius reduction. Interestingly, the contribution of the seed to the As-
limited growth rate becomes predominant with larger radii (see Fig. S5 in
[22]).

###  III.3 Fitting the experimental data

We now return to Fig. 3 which shows the length-radius dependence of four
samples grown under the same condition but for the As BEP (see Table I). The
sample with the lowest As flux (Fig. 3(a)) displays a length dependence on
radius which is characteristic of an As-limited growth all along the growth.
We use it to determine the values of the As-related parameters. A good fit is
obtained (Fig. 3(e)) with the parameters of Table 3 and the known experimental
parameters displayed in Table 4. Similarly, the sample with the highest As
flux (Fig. 3(d)) shows a rapid decrease of the length when the radius is
increased above 20 nm: this is characteristic of a diffusion-limited growth
with a long diffusion length, and a good fit is obtained (Fig. 3(h)) with the
parameters attributed to In in Table 3.

By keeping this set of parameters unchanged and adjusting only the As-flux
($H_{As}$, known from thorough calibration performed for each sample), we
achieve a good fit for all samples for all values of the nanowire radius.
Thus, we confirm the values of the parameters in Table 3, and the validity of
the approach synthesized in Eq.5.

Table 4: Known experimental parameters for the study of the As BEP dependence.  | I25R60 | I25R20 | I25R10 | I25R05
---|---|---|---|---
In data |  |  |  |
$\alpha_{In}$ (∘) | 12 | 12 | 12 | 12
$H_{In}$ (nm) | 280 | 280 | 280 | 280
As data |  |  |  |
$\alpha_{As}$ (∘) | 38 | 38 | 38 | 38
$H_{As}$ (nm) | 3000 | 1000 | 500 | 250

In Fig. 3(e)-(h), the dotted lines show the limits of the nanowire elongation
versus radius allowed by the As and In currents, respectively, as calculated
with our fixed set of parameters and the calibrated values of the fluxes
(Table 5). When the As flux (V/III flux ratio of 1) is low enough, the
elongation due to As remains smaller than the elongation due to In for all
radii (Fig. 5(e)). On the other hand, when the As flux increases to 2, the
elongation curve due to As crosses the one due to In. Fig. 3(f) shows that
below 30 nm radius, the elongation allowed by As is systematically smaller
than the In one. Above 30 nm radius, the two limits cross and the elongation
of the nanowire due to As solely becomes greater with respect to the one due
to In. For even higher As fluxes, the critical radius at which the transition
between As and In limited regimes shifts toward lower values and the maximum
nanowire length increases further.

Table 5: Known experimental parameters for the study of the In BEP dependence.  | I18R100 | I25R60 | I38R42
---|---|---|---
In data |  |  |
$\alpha_{In}$ (∘) | 12 | 12 | 12
$H_{In}$ (nm) | 260 | 280 | 330
As data |  |  |
$\alpha_{As}$ (∘) | 38 | 38 | 38
$H_{As}$ (nm) | 4800 | 3000 | 3200

Finally, we test our model for different values of the In flux. Fig. 4(b)
shows the length-radius dependence of three different samples with In fluxes
changed by a factor of 1.5 (see Tables 2 and 5 as well as Fig. S4 in [22]). We
observe that it is possible to keep the same fitting parameters in our model,
but for the critical radius $R_{0}$ (Table 6). An increase in the In flux by a
factor of 1.5 leads to the reduction of $R_{0}$ from 11 nm to 6 nm. In other
terms, larger indium fluxes lead to the possibility to grow thinner nanowires.

Table 6: List of $R_{0}$ for various In fluxes.  | I18R100 | I25R60 | I38R42
---|---|---|---
$R_{0}$ (nm) | 11 | 8 | 6

##  IV Discussion

The present study embeds three original aspects that will be discussed now:

  1. 1.

we experimentally explore a range of growth conditions (As and In flux) which
covers the transition from As-limited to In-limited growth, for a broad range
of the nanowire radius;

  2. 2.

we propose a dual-adatom diffusion-limited model which describes the
transition, without adjustable parameter, thanks to a systematic preliminary
calibration of the In and As flux (see S1 in [22]); the V/III BEP ratio at the
transition strongly depends on the nanowire radius, but also on the
instantaneous length;

  3. 3.

the critical radius $R_{0}$ which characterizes the sharp decrease of the
growth rate at small radius is clearly ascribed to the Kelvin effect on As and
its value depends on the In flux.

It is interesting to compare the present study to the phase diagram of the
growth rate of thin InAs nanowires with respect to the In and As BEP values,
Fig. 1 of Ref. [16]. Keeping in mind that for the same values of BEPs, the
values of the projected flux $F_{i}\cos\alpha_{i}$ may differ between two MBE
machines, that it depends on the cell positions and, for As, on the nature of
the molecular beam, we note that we obtain similar values of the growth rate.
Our range of As BEPs covers a similar ratio, but we explore a definitely
narrower range of the In flux (a factor of 2). However, by measuring the
length-radius dependence for different values of the In and As flux, we obtain
a more informative set of experimental data which allows us to discuss
quantitatively the mechanisms involved in the growth process.

The dispersion in the length-radius dependence is much bigger than that
expected for the Poisson distribution [25, 40, 41], which may be expected,
even under perfectly controlled growth conditions, due to the random nature of
step nucleation. The stronger dispersion we observe may be due to an initial
delay in the nucleation of nanowires. Such a delay was reported in Ref [23]
for very short growth times, of the order of the minute, with a dependence on
the In flux. Regarding our case, we observe no delay in the apparition of the
Bragg peaks in the RHEED pattern, and samples grown for a shorter time (7 min
instead of 30 min, not shown) display no nanowires of vanishing length
(excepted close to the cut-off) which would have started freshly. The broad
range of the nanowire radius associated to each size of the nanocristallite
(as mentioned above) suggests that at least the smaller gold nanoparticles
move on the substrate (some nanoparticles may even be buried and re-emerge as
in GaAs [25, 42]) before growth starts. Hence, we cannot rule out an initial
delay, but it is at most of the order of a couple of minutes.

The dispersion may also be due to some roughness of the substrate surface as
the growth starts, or to the presence of the 2D layer which progressively
grows between the nanowires. Both may influence our determination of the
substrate diffusion length $\lambda_{s}$ which must be considered as an
effective capture length. Our value for In ($\lambda_{s}\approx$ 200 nm) is
intermediate between the 10 to 95 nm reported for Ga or In in MBE [39, 29, 33]
and the hundreds of nanometers for chemical beam epitaxy [14, 13].
Nonetheless, Fig. 5(a), for In, and even more Fig. 5(b), for As, show that the
substrate contribution is significant only for short nanowires.

Another important contribution is the flux to the seed, either directly from
the cell, or due to re-emission from the substrate. The shape of the seed may
enhance the contribution of the direct flux, both for In and As. This is the
case, for instance, if the shape of the nanoparticle is close to a full
sphere. Then the flux collected by the nanoparticle with a large cross-section
is used to feed the growth of the nanowire with a much smaller area [34].
Although indirectly, post-growth SEM images and TEM images suggest that in the
present case, the nanoparticle is approximately half-spherical, independently
from the V/III ratio (not shown): then the collection cross-section and the
nanowire area are equal.

As the re-emission of type-III atoms is likely to be negligible (or vanishing
[29, 30, 31]), we keep $\kappa_{In}=0$. This can be revisited for substrates
on which the metal doesn’t stick [43, 44, 45]. The case of As is quite
different. The projected flux of As, $F_{As}\cos\alpha_{As}$, is consistently
larger than that of In. Consequently, a significant amount of As is not frozen
during the growth of the 2D layer. The geometry of the type-V flux has been
detailed in Ref. [46] for As and in more details in Ref [35] for P. The effect
on the flux to the gold seed may be significant, but the most spectacular
aspect from Ref [35] is that the diffusion of the type-V atoms along the
sidewalls is not as vanishing as it was assumed previously (with the exception
of Ref. [47]), and the corresponding contribution can be very strongly
enhanced by the re-emission from the substrate after adsorption. Our values
$\lambda_{f,As}$ and $\kappa_{As}$ (100 nm, and 6, respectively) fit
relatively well the experimental data. Nevertheless, $\lambda_{f,As}$ and
$\kappa_{As}$ (380 nm, and 0.5, respectively) fit as well the data (see Fig.
S3 in [22]) but do not represent the physical behaviour of As (re-emission
here is neglected and compensated by a larger (effective) As diffusion length.

We chose a simple description of the size effect, with an overall factor
$\left[1-\theta_{evap}^{in}\exp(\frac{R_{0}}{R})\right]$ in Eq.3. A size
effect appropriate to each elementary surface of the nanostructure was
considered in Ref. [33] but the experimental data do not allow one to
disentangle the different contributions. The characteristic radius $R_{0}$ for
the Kelvin effect in the seed is calculated in Ref. [33] for mono-atomic
evaporation. Using the same parameters, and Eq. 4 for the evaporation of a di-
atomic molecule (As2) [36], we obtain $R_{0}\approx$ 8 nm for a Au particle.
Due to a lower surface energy, it is expected to be 2.6 times smaller for pure
In, and intermediate for a Au-In alloy. The type-III content in the particle
is difficult to measure during growth, yet is assumed to be as high as several
20%-30% from post-growth characterization, in strong contrast to a small As
content of the order of a few %, negligible in the calculation of $R_{0}$.

The other parameter characterizing the size effect is $\theta_{evap}^{in}$,
which is introduced in Ref. [33] as a ratio of activities of the As in the
vapour and in the seed. Its value should be calculated from a complete
thermodynamical approach [48, 36, 12, 47], which is beyond the scope of the
present study. Based on Fig. 6 of Ref. [36], which shows that under As-limited
conditions the re-evaporated flux is proportional to the incident current to a
good approximation, we keep a constant value for $\theta_{evap}^{in}$ under
such conditions. This is comforted by the stable value, upon changing the As
flux, of the threshold radius below which no nanowire is observed, equal to
$\frac{R_{0}}{\ln(\theta_{evap}^{in})}$.

The last parameter is the diffusion length of In along the nanowire sidewalls.
All studies report a long diffusion length, $\lambda_{f,In}$ in the several
micrometer range. Our value, $\lambda_{f,In}=7.5\leavevmode\nobreak\ \mu$m, is
based on the non-linear growth rate of long nanowires under As excess (see
Fig. 5(a)) and provides a lower bound to $\lambda_{f}$. The determination of
an upper bound would require to put into evidence the saturation of the growth
rate for nanowires definitely larger than $\lambda_{f}$. The observation of
cylinder-shaped nanowires is not a relevant criterion: the application of the
Burton-Cabrera-Frank model to the propagation of steps along the nanowire
sidewalls shows that if the sticking probability of an adatom to a step
present on the sidewalls is below a threshold value, cylinder-shaped nanowires
of arbitrary length, larger than the diffusion length limited by desorption,
can be obtained [49]. The other parameters needed (Table 4), are known from
the position of the cells and substrate in the chamber, and from measurements
of the growth rates of InAs layers (see Fig. S1 in [22]).

Having in hand a set of fitting parameters which describes the growth limited
by the As minority current, whatever the nanowire radius, and the growth
limited by the In minority current, for large values of the radius, we can
address the main goal of our study: how do we switch from one limit to the
other one, as a function of the two projected fluxes, but also of the nanowire
radius and nanowire length, and what is the impact on the growth rate. The
calculated limits in Fig. 5(a) confirm that for such a V/III flux ratio (=1)
the As current is smaller than the In current, whatever the nanowire radius.
Fig. S5 in [22] confirms that this remains true whatever the nanowire length.

For large nanowire radii, the deep minimum in the In current (which reflects
the long diffusion length of In) may fall below the almost constant As current
at low V/III flux ratio, as shown in Fig. S5. Hence the growth starts as As-
limited, then switches to In-limited, and back to As-limited for long enough
nanowires.

For a smaller radius, see also Fig. S5 in [22], the As-current is made smaller
by the Kelvin effect and the In-limited section shrinks: the growth is As-
limited whatever the nanowire length. Thus, without any further assumption, we
obtain that for a large enough V/III flux ratio, the growth is indeed In-
limited for a large nanowire radius, but remains As-limited for thin
nanowires. Thus the length-radius dependence combines the Kelvin effect on As
at small radius and the long-diffusion of In at large radius.

At intermediate radius, the limiting current depends on the nanowire length,
so that the transition in the length-radius plot is smooth (see Figs. 3(b) and
(c)). This remains true when the V/III flux ratio is further increased, as
shown in Figs. 5(a-b) and Fig. S5 in [22]. We note that the minimum radius,
given by $\frac{R_{0}}{\ln(\theta_{evap}^{in})}$, is sharply defined and
remains unchanged when changing the As flux. This comforts our assumption that
both $R_{0}$ and $\theta_{evap}^{in}$ do not change under constant In flux and
growth temperature.

The evolution of the length-radius dependence, when changing the In flux (Fig.
4(b)), is also well described with the same set of parameters, but for
$R_{0}$, see Table 6. Indeed a decrease of $R_{0}$ is expected if the In
content in the nanoparticle increases, as a result of a smaller surface energy
for an In-rich alloy. Unfortunately, although all determinations of the post-
growth In content conclude to high values [6, 50, 51, 52], typically 30% or
more, it is quite difficult to assess what was the indium content during the
growth. Regarding GaAs nanowires grown by metal-organic chemical vapor
deposition, the high value of the Ga content in the Au nanoparticle, and its
increase upon increasing the Ga precursor flux, have been confirmed by an in
situ study of the growth [53]. A practical result of the shift of the critical
radius with In flux is that long, thin nanowires can be grown at high In flux,
as already reported for instance in Ref. [24].

A crucial aspect of our approach is that the growth rate, as calculated using
Eq. 5, depends only on the minority current to the nanoparticle. The currents
calculated in Eqs. 2 and 3 with the fitting parameters determined under a
strong excess of the opposite species are a good estimate of the maximum
current that one can anticipate for each species. While the minority current
defines the nanowire elongation through Eqs. 5, no assumption is made for the
majority species. Of course, the majority current is expected to ultimately
adjust through a specific elimination process. If the minority species is the
metal, the As content in the nanoparticle will increase to a value such that
the excess As current is eliminated by evaporation. The increase of the As
content maintains As-rich conditions, and the increase of $\theta_{evap}^{in}$
above its value in Table 3 adjusts the As current contributing to the growth
to the same value as the (minority) metal current. The opposite case of As-
limited growth is more complex. A symmetric process can take place in gold-
seeded growth with an increase of the type-III metal content in the droplet
[53]. It is probably less efficient than the adjustment of As, and totally
ruled out in the case of self-catalyzed growth [54]. An efficient mechanism of
back diffusion of Ga along the sidewalls has been identified in the self-
catalyzed growth of GaP [35] and may be present in gold-seeded growth as well.
To first order, these two processes do not modify the minority As current and
the growth rate. Special care should be taken in more involved case, for
instance if the large type-III content in the gold droplet impacts the
activity of As, or if a modification in the droplet size occurs during self-
catalyzed growth.

The present model describes efficiently our experimental data and helps
understanding the mechanisms involved in the growth of compound nanowires. It
can be applied to the growth of nanowires made of compound semiconductors,
through the VLS, the VSS and the solution-liquid-solid [55] processes, once
the equations and parameters describing the dynamics of the two species have
been established using strongly unbalanced growth conditions. Indeed, a sharp
maximum in the length-radius dependence has been reported for various
materials: III-Vs where the type-V atome is likely to be the volatile species,
such as GaAs [15], InP [15], InAs [14], or the extreme case of GaN [56]; but
also II-VIs such as CdSe [55], ZnSe [55] [57] and ZnS [57] where the cation
may be the volatile species like Cd in CdTe [58]. The role of the Kelvin
effect, of nucleation [15] and of individual constituents, must be identified
thanks to studies under broad growth conditions.

An obvious extension would be the introduction of equations based on the
chemical potentials instead of the adatom density. Note that this should
affect primarily the diffusion equation for the majority species (which has to
evaporate if not fully consumed by the nanowire growth) while our model uses
mainly the diffusion equation for the minority species. However, it would
allow us to take into account the nucleation rate [18, 19] and could possibly
make the transition in the length-radius plot smoother than in the present
model. It should also be possible to detect the effect of the nucleation rate
on the growth rate as a function of length measured on a single nanowire with
markers [38, 39, 59], by choosing a radius at the transition as in Fig. 5(c).
Another aspect is the decrease of the nucleation probability on a small
interface area, which may also play a role in the absence of growth seeded by
small Au nanoparticles, although the calculated cut-off appears to be less
sharp in the absence of Kelvin effect [15, 60]. The approach should also be
used to describe the nucleation of steps at the nanowire-seed interface, and
their propagation [61, 62]. Finally, a full thermodynamic assessment of the
parameters [12, 36, 47] would be clearly a welcome improvement of the present
description.

##  V Conclusion

We developed a generic dual-adatom diffusion-limited model for the VLS and VSS
growth of nanowires made of compound semiconductors, that implements the
distinct contributions of each species to the nanowire elongation. Knowing the
growth parameters of the nanowires in the case each species is in excess, the
model considers the growth under intermediate flux ratio by calculating the
maximum current to the seed for each species. It allows us to describe the
nanowire elongation, when growth is limited alternatively by one of each
species - the one which has the smaller current, depending on the radius and
the length of the nanowire at the moment of its growth. The model is applied
and tested on InAs nanowires, where the growth rate is governed by the smaller
current between In and As. It is relevant for other compound semiconductors,
self-assembled and gold-seeded nanowires, and should be considered also for
understanding the nucleation and propagation of steps at the interface between
the seed and the nanowire.

###### Acknowledgements.

This work was supported by ANR HYBRID (ANR-17-PIRE-0001), ANR IMAGIQUE
(ANR-42-PRC-0047), IRP HYNATOQ, the Transatlantic Research Partnership and the
LANEF IRGA LABEX BICOQ. We thank Frank Glas for extended discussions on
modeling.

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