# Spatiotemporal Vortex Pulses: Angular Momenta and Spin-Orbit Interaction

Konstantin Y. Bliokh  Theoretical Quantum Physics Laboratory, RIKEN Cluster
for Pioneering Research, Wako-shi, Saitama 351-0198, Japan

###### Abstract

Recently, spatiotemporal optical vortex pulses carrying a purely transverse
intrinsic orbital angular momentum were generated experimentally [Optica 6,
1547 (2019); Nat. Photon. 14, 350 (2020)]. However, an accurate theoretical
analysis of such states and their angular-momentum properties remains elusive.
Here we provide such analysis, including scalar and vector spatiotemporal
Bessel-type solutions as well as description of their propagational,
polarization, and angular-momentum properties. Most importantly, we calculate
both local densities and integral values of the spin and orbital angular
momenta, and predict observable spin-orbit interaction phenomena related to
the coupling between the transverse spin and orbital angular momentum. Our
analysis is readily extended to spatiotemporal vortex pulses of other natures
(e.g., acoustic).

Introduction.— Vortex beams carrying intrinsic orbital angular momentum (OAM)
are widely studied and exploited in modern optics Allen  _et al._ (1992,
2003); Bekshaev  _et al._ (2008); Andrews and Babiker (2012); Molina-Terriza
_et al._ (2007); Franke-Arnold  _et al._ (2008); Bliokh and Nori (2015),
acoustics Hefner and Marston (1999); Lekner (2006); Volke-Sepulveda  _et al._
(2008); Demore  _et al._ (2012); Anhauser  _et al._ (2012); Marzo  _et al._
(2018), and electron microscopy Bliokh  _et al._ (2007); Uchida and Tonomura
(2010); Verbeeck  _et al._ (2010); McMorran  _et al._ (2011); Bliokh  _et al._
(2017); Lloyd  _et al._ (2017); Larocque  _et al._ (2018). They have found
numerous applications in a variety of classical and quantum systems. Such
beams are monochromatic, and their intrinsic OAM is produced by a screw phase
dislocation (vortex) aligned with the beam axis Nye and Berry (1974); Berry
(1981). Thus, this OAM is longitudinal, i.e., aligned with the mean momentum
of the beam.

Recently, spatiotemporal analogues of vortex beams, spatiotemporal vortex
pulses (STVPs), were predicted theoretically Sukhorukov and Yangirova (2005);
Dror and Malomed (2011); Bliokh and Nori (2012a) and generated experimentally
in optics Jhajj  _et al._ (2016); Hancock  _et al._ (2019); Chong  _et al._
(2020); Hancock  _et al._ (2021); Wan  _et al._ (2021). Such states are
essentially polychromatic, and they carry intrinsic OAM transverse (or,
generally, tilted) with respect to the propagation direction of the pulse.
This OAM is produced by an edge (or mixed edge-screw) phase dislocation Nye
and Berry (1974); Berry (1981). It is anticipated that the STVPs and
transverse intrinsic OAM can considerably extend functionality and
applications of wave vortices. In particular, while monochromatic vortex beams
are essentially 3D objects, STVPs can exist in 2D spatial geometry. In
addition, they can produce temporal analogues of spatial phenomena known for
monochromatic vortex beams (e.g., time delays instead of beam shifts).

Despite the very recent progress in the generation of STVPs Hancock  _et al._
(2019); Chong  _et al._ (2020); Hancock  _et al._ (2021); Wan  _et al._
(2021), they still lack an accurate theoretical description, including
consistent analysis of their angular-momentum and polarization properties.
Indeed, despite numerous mentionings of the OAM in Refs. Hancock  _et al._
(2019); Chong  _et al._ (2020); Hancock  _et al._ (2021); Wan  _et al._
(2021), its value has not been obtained there. Instead, the topological number
of the phase dislocation, $\ell$, was used, which generically does not
coincide with the normalized OAM value Allen  _et al._ (2003); Berry (1998).
Furthermore, accurate calculations of the OAM are impossible without a full
vector description and separation of the spin and orbital parts of the total
angular momentum Bliokh and Nori (2015); Bliokh  _et al._ (2010); Barnett
(2010); Bialynicki-Birula and Bialynicka-Birula (2011).

In this paper, we fill this gap by constructing simple Bessel-type solutions
for STVPs. We describe their propagational dynamics including ‘temporal
diffraction’ Bliokh and Nori (2012a); Hancock  _et al._ (2019), examine
polarization properties, and calculate both local densities and integral
values of the OAM and spin angular momentum (SAM) of such pulses. We show that
the in-plane linear polarization inevitably produces a longitudinal field
component and a nonzero transverse SAM density. This produces the spin-orbit
interaction effects, known for monochromatic beams Bliokh  _et al._ (2015,
2010), such as observable polarization-dependent intensity distributions of
STVPs. Importantly, an integral value of the transverse SAM vanishes, while
the integral OAM is quantized as $\hbar\ell$ per photon only for circularly-
symmetric pulses with equal width and length. For elliptical STVPs with
different width and length, which were used in experiments Hancock  _et al._
(2019); Chong  _et al._ (2020); Hancock  _et al._ (2021); Wan  _et al._
(2021), the OAM value is larger than $\hbar\ell$ per photon.

Thus, our work provides the self-consistent full-vector description of optical
STVPs. It also allows straightforward extension to analogous acoustic pulses.

Scalar Bessel-type solutions.— We first consider scalar waves and simplest
analytical vortex-beam solutions: Bessel beams Durnin (1987); Durnin  _et al._
(1987); McGloin and Dholakia (2005); Bouchal and Olivik (1995); Jauregui and
Hacyan (2005); Bliokh  _et al._ (2010). Monochromatic Bessel beams propagating
along the $z$-axis can be constructed as a superposition of plane waves with
the same frequency $\omega=\omega_{0}$, wavevectors ${\bf k}$ distributed
within a cone of polar angle $\theta=\theta_{0}$, and with an azimuthal phase
difference $\ell\phi$ ($\phi$ is the azimuthal angle in ${\bf k}$-space)
corresponding to a vortex of integer order $\ell$, Fig. 1(a). In other words,
the wavevectors form a circle in the $k_{z}=k_{\parallel}$ plane, with the
center at $(0,0,k_{\parallel})$ and radius $k_{\perp}$, where
$k_{\parallel}=k_{0}\cos\theta_{0}$, $k_{\perp}=k_{0}\sin\theta_{0}$,
$k_{0}=\omega_{0}/c$, and $c$ is the speed of light. In real space, this
superposition results in the scalar wavefunction $\psi({\bf r},t)\propto
J_{\ell}(k_{\perp}r)\exp(ik_{\parallel}z+i\ell\varphi-i\omega_{0}t)$, where
$(r,\varphi)$ are the polar coordinates in the $(x,y)$-plane and $J_{n}$ is
the Bessel function of the first kind. The transverse intensity and phase
distributions of such Bessel beam are shown in Fig. 1(a).

Figure 1: The plane-wave spectra (left) and phase-intensity distributions of
real-space wavefunctions $\psi({\bf r},t)$ (right) for (a) the monochromatic
Bessel beam with $\ell=2$ and (b) spatiotemporal Bessel pulse with $\ell=2$,
Eqs. (1) and (2). In real-space distributions, the brightness is proportional
to the intensity $|\psi|^{2}$, while the color indicates the phase ${\rm
Arg}(\psi)$.

To construct a Bessel-type STVP with a purely transverse intrinsic OAM, we use
a superposition of plane waves with wavevectors distributed over a circle in
the $k_{y}=0$ plane with the center at $(0,0,k_{0})$ and radius $\Delta k$,
Fig. 1(b). Using the azimuthal angle $\tilde{\phi}$ with respect to the center
of this circle, we introduce the azimuthal phase difference $\ell\tilde{\phi}$
and write the real-space wavefunction as a Fourier-type integral:

$\psi({\bf r},t)\\!\propto\\!\int_{0}^{2\pi}e^{i[{k_{0}z}+\Delta
k\cos\tilde{\phi}\,z+\Delta
k\sin\tilde{\phi}\,x+{\ell\,\tilde{\phi}}-\omega(\tilde{\phi})t]}\,d\tilde{\phi}\,,$
(1)

where $\omega(\tilde{\phi})=c\sqrt{k_{0}^{2}+\Delta k^{2}+2k_{0}\Delta
k\cos\tilde{\phi}}$. Parameter $\Delta k$ determines the degree of paraxiality
and monochromaticity of the Bessel STVP. The maximum polar angle of the
wavevectors in its spectrum is $\sin\theta_{0}=\Delta k/k_{0}$, Fig. 1(b). For
$\Delta k\ll k_{0}$, $\theta_{0}\ll 1$, the pulse can be considered as near-
paraxial and quasi-monochromatic. Below we will use this approximation keeping
terms linear in $\theta_{0}$, which describe some post-paraxial phenomena.

In this approximation, $\omega(\tilde{\phi})\simeq c(k_{0}+\Delta
k\cos\tilde{\phi})$, and the integral (1) results in the analytical Bessel-
pulse solution:

$\psi({\bf r},t)\\!\propto
J_{\ell}(\tilde{\rho})\exp(ik_{0}\zeta+i\ell\tilde{\varphi})\,.$ (2)

Here, $\zeta=z-ct=\tilde{r}\cos\tilde{\varphi}$, $\tilde{\rho}=\Delta
k\tilde{r}$, and $(\tilde{r},\tilde{\varphi})$ are the polar coordinates in
the $(\zeta,x)$ plane. The intensity $I=|\psi|^{2}$ and phase ${\rm
Arg}(\psi)$ distributions for the Bessel STVP (2) are shown in Fig. 1(b). It
has typical Bessel-beam intensity profile
$I\propto|J_{\ell}(\tilde{\rho})|^{2}$ in the $(\zeta,x)$ plane, contains an
edge phase dislocation of order $\ell$, and propagates with the speed of light
along the $z$-axis.

Importantly, the nondiffracting solution (2) is a result of linear expansion
of $\omega(\tilde{\phi})$ with respect to $\Delta k$. The exact solution (1)
evolves in time as shown in Fig. 2. Namely, the $\ell$th order phase
dislocation in the pulse center splits into a raw of $|\ell|$ first-order
dislocations oriented diagonally in the $(\zeta,x)$ plane. This ‘temporal
diffraction’ was predicted in Ref. Bliokh and Nori (2012a) and observed in
Hancock  _et al._ (2019). Akin to the Rayleigh range characterizing spatial
diffraction, a typical scale of the temporal diffraction is given by the
‘temporal Rayleigh range’ $ct_{R}=k_{0}/\Delta k^{2}$, Fig. 2. Notably,
nondiffracting Bessel-like STVPs can be constructed using the wavevectors
distributed along an ellipse in ${\bf k}$ space which could be Lorentz-
transformed to a monochromatic circle Bliokh and Nori (2012a). In our case,
the circular spectrum in Fig. 1(b) should be entended along the $k_{z}$-axis
to become an ellipse with the ratio of semiaxes $\gamma=\sqrt{1+(k_{0}/\Delta
k)^{2}}$.

Figure 2: Temporal diffraction of the spatiotemporal Bessel pulse (1) with
$\ell=3$ and $\Delta k/k_{0}=0.3$. The characteristic time scale
$ct_{R}=k_{0}/\Delta k^{2}$ is analogous to the Rayleigh range of spatially
diffracting beams.

Vector solutions and spin-orbit effects.— We now examine vector Bessel STVPs.
For simplicity, we consider the electric field ${\bf E}$ of transverse
electromagnetic waves; similar arguments could be applied to the magnetic
field and other types of vector waves. Due to the transversality condition,
the electric field of each plane wave in the pulse spectrum must be orthogonal
to its wavevector ${\bf k}$. This determines two basic polarizations in the
problem: (i) out-of-plane, ${\bf E}$ is directed along the $y$-axis, Fig.
3(a), and (ii) in-plane, ${\bf E}$ lies in the $(z,x)$ plane, Fig. 3(b).

For the out-of-plane polarization, the field has only one component, and the
problem reduces to the scalar case: ${E_{y}}({\bf r},t)\propto\psi({\bf
r},t)$.

For the in-plane polarization, the situation is less trivial. Each plane wave
in the pulse spectrum has two electric-field components, $E_{x}$ and $E_{z}$,
Fig 3(b). The amplitudes and phases of these components depend on the
wavevector ${\bf k}$, which signals the spin-orbit interaction Bliokh  _et
al._ (2015, 2010). To describe the spin-orbit effects in the linear
approximation in $\Delta k$, we write the electric field components for each
plane wave as $E_{x}=E\cos\theta\simeq E$ and
$E_{z}=-E\sin\theta\simeq-E(\Delta k/k_{0})\,{\sin\tilde{\phi}}$, where
$\theta\leq\theta_{0}$ is the polar angle of a given wavevector. Since for the
integrand of Eq. (1), $i\Delta k\,{\sin\tilde{\phi}}=\partial/\partial x$, one
can write the real-space transverse ($x$) and longitudinal ($z$) field
components as

$E_{x}({\bf r},t)\simeq\psi({\bf r},t),\quad E_{z}({\bf
r},t)\simeq\frac{i}{k_{0}}\frac{\partial\psi({\bf r},t)}{\partial x}\,.$ (3)

Figure 3: Electric fields of plane waves in the spectra of Bessel-type vector
STVPs (left) and the corresponding real-space intensity distributions $|{\bf
E}({\bf r},t)|^{2}$ (right) for (a) the out-of-plane polarization and (b) the
in-plane polarization. The parameters are: $\ell=1$ and $\Delta k=0.7$ for
better visibility of nonzero intensity (5) in the center of the pulse in (b).

Substituting Eq. (2) into Eq. (3), we obtain the longitudinal field

$E_{z}\\!\propto\\!\frac{i\Delta
k}{2}e^{ik_{0}\zeta}\\!\left[e^{i(\ell-1)\tilde{\varphi}}J_{\ell-1}(\tilde{\rho})+e^{i(\ell+1)\tilde{\varphi}}J_{\ell+1}(\tilde{\rho})\right].$
(4)

From Eqs. (2)–(4), the total field intensity, $I=|E_{x}|^{2}+|E_{z}|^{2}$, is
given by

$I\propto J_{\ell}^{2}+\frac{\Delta
k^{2}}{4k_{0}^{2}}\\!\left[J_{\ell-1}^{2}+J_{\ell+1}^{2}+2\cos(2\tilde{\varphi})J_{\ell-1}J_{\ell+1}\right].$
(5)

Here and hereafter, for brevity, we omit the Bessel-functions argument
$\tilde{\rho}$.

The intensity distribution of the Bessel STVP, Eq. (5), resembles intensity
distributions of vector Bessel beams in optics Bliokh  _et al._ (2010),
acoustics Bliokh and Nori (2019), and quantum mechanics Bliokh  _et al._
(2011). In particular, the presence of the Bessel functions of orders $\ell\pm
1$ is a typical signature of the spin-orbit interaction. The easiest-to-
observe spin-orbit effect is a nonzero intensity $\sim\theta_{0}^{2}$ in the
center of the in-plane-polarized STVPs with $|\ell|=1$, Fig. 3. For
monochromatic vortex beams, this phenomenon has been observed in optical
experiments Bokor  _et al._ (2005); Gorodetski  _et al._ (2008); Bliokh  _et
al._ (2015). The main difference is that in the case of monochromatic vortex
beams this phenomenon occurs for circular polarizations, corresponding to the
longitudinal SAM of the beam, while for STVPs it takes place for the in-plane
linear polarization. As we show below, this polarization generates a nonzero
transverse SAM density directed along the $y$-axis, i.e., along with the
intrinsic OAM of the pulse.

Spin and orbital angular momenta.— Analysis of the spin and orbital angular
momenta of polychromatic STVPs is a challenging problem because most
theoretical methods are developed for monochromatic fields. Indeed, standard
formulas for the SAM and OAM densities imply averaging over the cycle of
periodic temporal oscillations Bliokh and Nori (2015), and they become ill-
defined in generic polychromatic fields Bialynicki-Birula and Bialynicka-
Birula (2011). Nonetheless, here we can employ the quasi-monochromaticity of
pulses with $\Delta k\ll k_{0}$ and separate fast temporal oscillations with
the central frequency $\omega_{0}=k_{0}c$ and slow temporal evolution with the
inverse temporal scale $\Delta k\,c$. This results in the SAM and OAM
densities given by the standard monochromatic formulas involving canonical SAM
and OAM operators and time-dependent ‘wavefunction’ ${\bf E}({\bf
r},t)e^{i\omega_{0}t}/\sqrt{\omega_{0}}$ Bliokh and Nori (2015):

$S_{y}=\omega_{0}^{-1}{\rm Im}\left({\bf E}^{*}\times{\bf
E}\right)_{y},\leavevmode\nobreak\ \leavevmode\nobreak\
L_{y}=\omega_{0}^{-1}{\rm Im}\\!\left({\bf
E}^{*}\cdot\frac{\partial}{\partial\tilde{\varphi}}{\bf E}\right).$ (6)

Although the angle $\tilde{\varphi}$ in the $(\zeta,x)$ plane involves time,
in Eq. (6) we used the fact that at $t=0$ it becomes the desired azimuthal
angle in the $(z,x)$ plane, whereas in the diffractionless approximation the
SAM and OAM distributions are invariantly translated together with the pulse
along the $z$-axis.

Figure 4: Spatial distributions of (a) the normalized transverse spin angular
momentum density $\omega_{0}S_{y}/I$ and (b) the orbital angular momentum
density (with the subtracted propagational term) $\omega_{0}L_{y}+k_{0}xI$ in
the in-plane-polarized Bessel STVP, Eqs. (Spatiotemporal Vortex Pulses:
Angular Momenta and Spin-Orbit Interaction) and (5). The parameters are:
$\ell=2$ and $\Delta k/k_{0}=0.5$.

For the out-of-plane polarization (the scalar case), Eqs. (6) yield

$S_{y}=0\,,\quad L_{y}=\omega_{0}^{-1}(\ell-k_{0}x)\,I\,.$ (7)

where $I\propto J_{|\ell|}^{2}(\tilde{\rho})$. The OAM density (7) contains
the standard vortex-related term proportional to $\ell$ as well as the
$k_{0}x$-dependent term caused by the $z$-propagation of the pulse. After
integration of the OAM density $L_{y}$ over the $(z,x)$ plane, the
$k_{0}x$-term vanishes, while the vortex-induced term yields the integral
value of $\hbar\ell$ per photon Allen  _et al._ (1992, 2003); Bekshaev  _et
al._ (2008); Andrews and Babiker (2012); Molina-Terriza  _et al._ (2007);
Franke-Arnold  _et al._ (2008); Bliokh and Nori (2015): $\omega_{0}\langle
L_{y}\rangle/\langle I\rangle=\ell$. Here
$\langle...\rangle=\int...\,dz\,dx=\int...\,d\zeta\,dx$ and we imply quasi-
monochromatic quantization of photon’s energy, $\hbar\omega_{0}$.

For the in-plane polarized pulse (3)–(5), Eqs. (6) yield

$\displaystyle S_{y}\propto\omega_{0}^{-1}\frac{\Delta
k^{2}x}{2\ell\,k_{0}}\left(J_{\ell+1}^{2}-J_{\ell-1}^{2}\right),$
$\displaystyle L_{y}=\omega_{0}^{-1}(\ell-
k_{0}x)\,I+\frac{\ell}{2k_{0}x}S_{y}\,.$ (8)

These distributions are depicted in Fig. 4. The nonzero spin density $S_{y}$
is a manifestation of the transverse spin phenomenon, which recently attracted
great attention Bliokh and Nori (2015); Bliokh  _et al._ (2015); Aiello  _et
al._ (2015); Lodahl  _et al._ (2017); Eismann  _et al._ (2021). Here, the
transverse spin arises from the interference of plane waves with different
directions, phases, and in-plane linear polarizations, Fig. 3(b). The
normalized SAM density $\omega_{0}S_{y}/I$ reaches the minimum and maximum
values of $-1$ and $1$, Fig. 4(a), i.e., the polarization becomes left-hand
and right-hand circular in these zones. As is typical for the transverse spin
of free-space propagating waves Bliokh and Nori (2015); Aiello  _et al._
(2015); Eismann  _et al._ (2021), its integral value vanishes: $\langle
S_{y}\rangle=0$.

The OAM density (Spatiotemporal Vortex Pulses: Angular Momenta and Spin-Orbit
Interaction) has a form similar to Eq. (7) with an additional spin-related
term; this is also a singnature of the spin-orbit interaction Bliokh  _et al._
(2015, 2010). Nonetheless, in contrast to the analogous spin-dependent OAM
parts in monochroimatic beams, the integral value of this term vanishes. (This
follows from the relation
$\ell\int_{0}^{\infty}\left(J_{\ell+1}^{2}-J_{\ell-1}^{2}\right)\tilde{\rho}\,d\tilde{\rho}\propto\ell\int_{0}^{\infty}J_{\ell}dJ_{\ell}/d\tilde{\rho}\,d\tilde{\rho}=0$.)
Thus, akin to the scalar case, the integral OAM value corresponds to
$\hbar\ell$ per photon.

We conclude that in spite of local spin-orbit interaction effects, the
integral SAM and OAM of STVPs are rather robust:

$\langle S_{y}\rangle=0\,,\qquad\frac{\omega_{0}\langle L_{y}\rangle}{\langle
I\rangle}=\ell\,.$ (9)

Figure 5: The plane-wave spectrum (left) and the phase-intensity distribution
of the real-space wavefunction $\psi({\bf r},t)$ (right) for an elliptical
Bessel STVP with $\ell=2$ and ratio of principal axes $\gamma=2$. The
normalized integral OAM of this pulse is $\omega_{0}\langle
L_{y}\rangle/\langle I\rangle=2.5$, Eq. (10).

Importantly, the above calculations are made for STVPs with circularly
symmetric intensity profiles in the $(\zeta,x)$ plane. However, in most cases
these profiles are elliptical with some ratio of principal axes $\gamma$ in
the $(\zeta,x)$ plane, as shown in Fig. 5. Such an elliptical STVP is
described by the substitution $x\to\gamma x$ in the scalar wavefucntion (1)
and (2) or, equivalently, by the substitution $k_{x}\to\gamma^{-1}k_{x}$ in
the beam spectrum. For the field of the form
$\propto\exp(i\ell\tilde{\varphi})$ and the OAM operator
$\hat{L}_{y}=-i\partial/\partial\tilde{\varphi}=i\left(x\partial/\partial\zeta-\zeta\partial/\partial
x\right)$, this results in an additional factor in the intrinsic OAM value
Bliokh and Nori (2012a, b):

$\frac{\omega_{0}\langle L_{y}\rangle}{\langle
I\rangle}=\frac{\gamma+\gamma^{-1}}{2}\,\ell\,.$ (10)

This factor is significant: for example, the experiments Hancock  _et al._
(2019); Chong  _et al._ (2020) generated STVPs with $\gamma\simeq 2.5$–$3$,
which yields $(\gamma+\gamma^{-1})/2\simeq 1.5$–$1.7$. Figure 5 shows an
example of the STVPs with $\ell=2$, $\gamma=2$, and $\omega_{0}\langle
L_{y}\rangle/\langle I\rangle=2.5$.

Conclusions.— We have examined spatiotemporal vortex pulses with purely
transverse intrinsic orbital angular momentum. We provided analytical Bessel-
type solutions, both scalar and vector, and described their propagation,
polarization, and angular-momentum properties. Most importantly, we provided
accurate calculations of the spin and orbital angular momenta of STVPs and
described observable spin-orbit interaction phenomena. Notably, the
polarization and spin-orbit effects manifest themselves locally via observable
intensity and spin-density distributions, while the integral values of the
spin and orbital angular momenta of STVPs are rather robust. At the same time,
the integral OAM value is significantly affected by the elliptical shape of
STVPs with different width and length, which are typically generated in
experiments Hancock  _et al._ (2019); Chong  _et al._ (2020).

The results of our work provide a theoretical platform for investigations of
novel spatiotemporal vortex states and call for experimental measurements of
the predicted polarization and angular-momentum phenomena. We considered
Bessel-type pulses solely for the simplicity of their theoretical description.
The results can be straightforwardly generalized to the Laguerre-Gaussian-type
spectra Allen  _et al._ (1992, 2003); Bekshaev  _et al._ (2008); Andrews and
Babiker (2012), more relevant to typical experimental situations. Furthermore,
a variety of phenomena, well studied for monochromatic vortex beams, can now
be investigated for STVPs: e.g., fractional OAM Berry (2004); Leach  _et al._
(2004), shifts at planar interfaces Bliokh  _et al._ (2009); Merano  _et al._
(2010), etc.

Importantly, our results are applicable to waves of different natures. For
example, STVPs can be generated in sound waves in fluids or gases. In doing
so, one can use the scalar approach (1)–(2) for the pressure wavefield $P({\bf
r},t)$ or the vector approach similar to Eqs. (3)–(5) for the velocity
wavefield ${\bf V}({\bf r},t)$. Since sound waves are longitudinal, i.e.,
${\bf V}\parallel{\bf k}$ for each plane wave in the spectrum, the velocity
field has the $z$ and $x$ components, $V_{z}({\bf r},t)\simeq\psi({\bf r},t)$
and $V_{x}({\bf r},t)\simeq-ik_{0}^{-1}\partial\psi({\bf r},t)/\partial x$,
generating the transverse spin and related spin-orbit phenomena Shi  _et al._
(2019); Bliokh and Nori (2019).

## References

  * Allen  _et al._ (1992) L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
  * Allen  _et al._ (2003) L. Allen, S. M. Barnett, and M. J. Padgett, eds.,  _Optical Angular Momentum_ (Taylor and Francis, 2003).
  * Bekshaev  _et al._ (2008) A. Bekshaev, M. Soskin, and M. Vasnetsov,  _Paraxial Light Beams with Angular Momentum_ (Nova Science Publishers, New York, 2008).
  * Andrews and Babiker (2012) D. L. Andrews and M. Babiker, eds.,  _The Angular Momentum of Light_ (Cambridge University Press, 2012).
  * Molina-Terriza  _et al._ (2007) G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305 (2007).
  * Franke-Arnold  _et al._ (2008) S. Franke-Arnold, L. Allen, and M. J. Padgett, “Advances in optical angular momentum,” Laser & Photon. Rev. 2, 299 (2008).
  * Bliokh and Nori (2015) K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
  * Hefner and Marston (1999) B. T. Hefner and P. L. Marston, “An acoustical helicoidal wave transducer with applications for the alignment of ultrasonic and underwater systems,” J. Acoust. Soc. Am. 106, 3313 (1999).
  * Lekner (2006) J. Lekner, “Acoustic beams with angular momentum,” J. Acoust. Soc. Am. 120, 3475 (2006).
  * Volke-Sepulveda  _et al._ (2008) K. Volke-Sepulveda, A. O. Santillan, and R. R. Boullosa, “Transfer of angular momentum to matter from acoustical vortices in free space,” Phys. Rev. Lett. 100, 024302 (2008).
  * Demore  _et al._ (2012) C. E. M. Demore, Z. Yang, A. Volovick, S. Cochran, M. P. MacDonald, and G. C. Spalding, “Mechanical evidence of the orbital angular momentum to energy ratio of vortex beams,” Phys. Rev. Lett. 108, 194301 (2012).
  * Anhauser  _et al._ (2012) A. Anhauser, R. Wunenburger, and E. Brasselet, “Acoustic rotational manipulation using orbital angular momentum transfer,” Phys. Rev. Lett. 109, 034301 (2012).
  * Marzo  _et al._ (2018) A. Marzo, M. Caleap, and B. W. Drinkwater, “Acoustic virtual vortices with tunable orbital angular momentum for trapping of mie particles,” Phys. Rev. Lett. 120, 044301 (2018).
  * Bliokh  _et al._ (2007) K. Y. Bliokh, Y. P. Bliokh, S. Savel’ev, and F. Nori, “Semiclassical dynamics of electron wave packet states with phase vortices,” Phys. Rev. Lett. 99, 190404 (2007).
  * Uchida and Tonomura (2010) M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464, 737–739 (2010).
  * Verbeeck  _et al._ (2010) J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467, 301–304 (2010).
  * McMorran  _et al._ (2011) B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331, 192–195 (2011).
  * Bliokh  _et al._ (2017) K. Y. Bliokh, I. P. Ivanov, G. Guzzinati, L. Clark, R. Van Boxem, A. Beche, R. Juchtmans, M. A. Alonso, P. Schattschneider, F. Nori, and J. Verbeeck, “Theory and applications of free-electron vortex states,” Phys. Rep. 690, 1–70 (2017).
  * Lloyd  _et al._ (2017) S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: Beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
  * Larocque  _et al._ (2018) H. Larocque, I. Kaminer, V. Grillo, G. Leuchs, M. J. Padgett, R. W. Boyd, M. Segev, and E. Karimi, “‘twisted’ electrons,” Contemp. Phys. 59, 126–144 (2018).
  * Nye and Berry (1974) J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. Ser. A 336, 165–190 (1974).
  * Berry (1981) M. V. Berry, “Les Houches Lecture Series Session XXXV,” (North-Holland: Amsterdam, 1981) Chap. Singularities in Waves, pp. 453–543.
  * Sukhorukov and Yangirova (2005) A. P. Sukhorukov and V. V. Yangirova, “Spatio-temporal vortices: properties, generation and recording,” Proc. SPIE 5949, 594906 (2005).
  * Dror and Malomed (2011) N. Dror and B. A. Malomed, “Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity,” Physica D 240, 526–541 (2011).
  * Bliokh and Nori (2012a) K. Y. Bliokh and F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A 86, 033824 (2012a).
  * Jhajj  _et al._ (2016) N. Jhajj, I. Larkin, E. W. Rosenthal, S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, “Spatiotemporal optical vortices,” Phys. Rev. X 6, 031037 (2016).
  * Hancock  _et al._ (2019) S. W. Hancock, S. Zahedpour, A. Goffin, and H. M. Milchberg, “Free-space propagation of spatiotemporal optical vortices,” Optica 6, 1547 (2019).
  * Chong  _et al._ (2020) A. Chong, C. Wan, J. Chen, and Q. Zhan, “Generation of spatiotemporal optical vortices with controllable transverse orbital angular momentum,” Nat. Photon. 14, 350 (2020).
  * Hancock  _et al._ (2021) S. W. Hancock, S. Zahedpour, and H. M. Milchberg, “Second harmonic generation of spatiotemporal optical vortices and conservation of orbital angular momentum,” Optica 8, 594–597 (2021).
  * Wan  _et al._ (2021) C. Wan, J. Chen, A. Chong, and Q. Zhan, “Experimental demonstration of ultrafast wavepacket containing orbital angular momentum with controllable orientation,” arXiv:2101.04949 (2021).
  * Berry (1998) M. V. Berry, “Paraxial beams of spinning light,” Proc. of SPIE 3487, 6–11 (1998).
  * Bliokh  _et al._ (2010) K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
  * Barnett (2010) S. M. Barnett, “Rotation of electromagnetic fields and the nature of optical angular momentum,” J. Mod. Opt. 57, 1339–1343 (2010).
  * Bialynicki-Birula and Bialynicka-Birula (2011) I. Bialynicki-Birula and Z. Bialynicka-Birula, “Canonical separation of angular momentum of light into its orbital and spin parts,” J. Opt. 13, 064014 (2011).
  * Bliokh  _et al._ (2015) K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photon. 9, 796–808 (2015).
  * Durnin (1987) J. Durnin, “Exact solutions for nondiffracting beams. i. the scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
  * Durnin  _et al._ (1987) J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
  * McGloin and Dholakia (2005) D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
  * Bouchal and Olivik (1995) Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
  * Jauregui and Hacyan (2005) R. Jauregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
  * Bliokh and Nori (2019) K. Y. Bliokh and F. Nori, “Spin and orbital angular momenta of acoustic beams,” Phys. Rev. B 99, 174310 (2019).
  * Bliokh  _et al._ (2011) K. Y. Bliokh, M. R. Dennis, and F. Nori, “Relativistic electron vortex beams: Angular momentum and spin-orbit interaction,” Phys. Rev. Lett. 107, 174802 (2011).
  * Bokor  _et al._ (2005) N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, “Investigation of polarization effects for high-numerical-aperture first-order laguerre–gaussian beams by 2d scanning with a single fluorescent microbead,” Opt. Express 13, 10440–10447 (2005).
  * Gorodetski  _et al._ (2008) Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
  * Aiello  _et al._ (2015) A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photon. 9, 789–795 (2015).
  * Lodahl  _et al._ (2017) P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017).
  * Eismann  _et al._ (2021) J. S. Eismann, L. H. Nicholls, D. J. Roth, M. A. Alonso, P. Banzera, F. J. Rodriguez-Fortuno, A. V. Zayats, F. Nori, and K. Y. Bliokh, “Transverse spinning of unpolarized light,” Nat. Photon. 15, 156–161 (2021).
  * Bliokh and Nori (2012b) K. Y. Bliokh and F. Nori, “Relativistic Hall Effect,” Phys. Rev. Lett. 108, 120403 (2012b).
  * Berry (2004) M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6, 259–268 (2004).
  * Leach  _et al._ (2004) J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
  * Bliokh  _et al._ (2009) K. Y. Bliokh, I. V. Shadrivov, and Y. S. Kivshar, “Goos–Hänchen and Imbert–Fedorov shifts of polarized vortex beams,” Opt. Lett. 34, 389–391 (2009).
  * Merano  _et al._ (2010) M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
  * Shi  _et al._ (2019) C. Shi, R. Zhao, Y. Long, S. Yang, Y. Wang, H. Chen, J. Ren, and X. Zhang, “Observation of acoustic spin,” Natl. Sci. Rev. 6, 707 (2019).


# Exploring the formation of gold/silver nanoalloys
with gas-phase synthesis and machine-learning assisted simulations

Quentin Gromoff  CEMES, CNRS and Université de Toulouse, 29 rue Jeanne Marvig,
31055 Toulouse Cedex, France  Patrizio Benzo  CEMES, CNRS and Université de
Toulouse, 29 rue Jeanne Marvig, 31055 Toulouse Cedex, France  Wissam A. Saidi
National Energy Technology Laboratory, United States Department of Energy,
Pittsburgh, PA 15236, USA  Department of Mechanical Engineering and Materials
Science, University of Pittsburgh, Pittsburgh, PA 15261, USA  Christopher M.
Andolina  National Energy Technology Laboratory, United States Department of
Energy, Pittsburgh, PA 15236, USA  Department of Mechanical Engineering and
Materials Science, University of Pittsburgh, Pittsburgh, PA 15261, USA  Marie-
José Casanove  CEMES, CNRS and Université de Toulouse, 29 rue Jeanne Marvig,
31055 Toulouse Cedex, France  Teresa Hungria  Centre de MicroCaractérisation
Raimond Castaing, Université de Toulouse, 3 rue Caroline Aigle, F-31400
Toulouse, France  Sophie Barre  CEMES, CNRS and Université de Toulouse, 29 rue
Jeanne Marvig, 31055 Toulouse Cedex, France  Magali Benoit  CEMES, CNRS and
Université de Toulouse, 29 rue Jeanne Marvig, 31055 Toulouse Cedex, France
Julien Lam<EMAIL_ADDRESS>CEMES, CNRS and Université de Toulouse, 29 rue
Jeanne Marvig, 31055 Toulouse Cedex, France  Univ. Lille, CNRS, INRA, ENSCL,
UMR 8207, UMET, Unité Matériaux et Transformations, F 59000 Lille, France

###### Abstract

While nanoalloys are of paramount scientific and practical interests, the main
processes leading to their formation are still poorly understood. Key
structural features in the alloy systems, including crystal phase, chemical
ordering, and morphology, are challenging to control at the nanoscale, making
it difficult to transfer their usage to industrial applications. In this
contribution, we focus on the gold/silver system that has two of the most
prevalent noble metals, and combine experiments with simulations to uncover
the formation mechanisms at the atomic-level. Nanoparticles are produced using
state-of-the-art inert-gas aggregation source and analyzed using transmission
electron microscopy and energy-dispersive x-ray spectroscopy. Machine-
learning-assisted molecular dynamics simulations are employed to model the
crystallization process from liquid droplets to nanocrystals. Our study finds
a preponderance of nanoparticles with five-fold symmetric morphology,
including icosahedron and decahedron which is consistent with previous results
on mono-metallic nanoparticles. However, we observe that gold atoms, rather
than silver atoms, segregate at the surface of the obtained nanoparticles for
all the considered alloy compositions. These segregation tendencies are in
contrast to previous studies and have consequences on the crystallization
dynamics and the subsequent crystal ordering. We finally show that the
underpinnings of this surprising segregation dynamics is due to charge
transfer and electrostatic interactions rather than surface energy
considerations.

##  I Introduction

By combining two or more metallic elements within the same nanoparticle,
synergistic properties can emerge and result into innovative technological
applications.Alloyeau  _et al._ (2012); Calvo (2020); Ferrando (2016);
Ferrando  _et al._ (2008) Numerous research fields including optics,
catalysis, bio-medicine and electronics are already considering these so-
called nanoalloys mainly because they exhibit the fundamental advantage of an
extremely rich structural landscape with a variety of shapes, chemical
orderings and crystalline phases. However, since the physical and the chemical
properties of nanoalloys are intrinsically related to their internal
structure, advances in this field are strictly constrained by synthesis
experiments, and it becomes crucial to better understand the intricate
relationship between the experimental conditions and the obtained structures.
Addressing this pivotal challenge first comes directly from experimental
studies, where two types of complementary approaches are usually considered:
(1) Systematic variations of experimental conditions are followed by post-
mortem structural analysisPenuelas  _et al._ (2008); Amendola  _et al._
(2014); Swiatkowska-Warkocka  _et al._ (2015); Ahmad  _et al._ (2019); Benzo
_et al._ (2019); Wang  _et al._ (2022a) and (2) In situ experiments allow for
a direct observation of the nanoalloys formation.van der Hoeven  _et al._
(2018); Liu and Huang (2013); Carrillo  _et al._ (2018); Ramade  _et al._
(2017); Mertens  _et al._ (2011); Chen  _et al._ (2022); Ahmad  _et al._
(2018)

Computational simulations have been the ideal complementary tool because it
offers an unambiguous atomistic picture and enables standardized examinations
of various experimental parameters. In this context, the dynamics of nanoalloy
formation are usually investigated with classical force fields via Monte Carlo
and molecular dynamics simulations, typically employed to study large-scale
systems.Goudeli and Pratsinis (2017); Nelli  _et al._ (2020); Nelli and
Ferrando (2019); Förster  _et al._ (2019); Hizi  _et al._ (2023) The derived
predictions, however, often lack chemical accuracy and can hardly be used to
draw quantitative conclusions. As an alternative, quantum simulations based on
first-principles density functional theory (DFT) have been performed in order
to address this accuracy issue.Chen and Johnston (2008a); Demiroglu  _et al._
(2016); Barcaro  _et al._ (2011); Tan  _et al._ (2012); Davis  _et al._ (2014)
However, due to their high computational costs, these methods are often
limited to the equilibrium properties of bulk systems or small clusters.
Recently, machine-learning approaches have been proposed to bridge the gap
between these two approaches. Indeed, machine-learning interaction potentials
(MLIP) are constructed by combining a very complex mathematical formulation
with numerous fitting parameters along with an extensive DFT-generated
database composed of different structures in conjunction with their associated
energies, forces, and virials. Evidence of the success of these methods can be
seen through the very diverse types of materials that were so far modeled with
MLIP, including metals, oxides, carbon and silicon-based, organics, and
perovskites.Wisesa  _et al._ (2022); Wang  _et al._ (2022b); Andolina  _et
al._ (2020, 2021a); Bayerl  _et al._ (2022); Erhard  _et al._ (2022); Mocanu
_et al._ (2018); Gartner  _et al._ (2022); Piaggi  _et al._ (2022); Panizon
_et al._ (2015); Benoit  _et al._ (2020); Tallec  _et al._ (2023); Laurens
_et al._ (2021); Goniakowski  _et al._ (2022); Lam  _et al._ (2020); Caro  _et
al._ (2018); Deringer and Csányi (2017) Prompted by these advances, DFT
accurate large scale simulations can finally be carried out to investigate the
intricate formation processes occurring in nanoparticle synthesis.

Herein we studied the formation of AuAg nanoalloys that have been considered
in many applications owing to, in particular, their plasmonic,Tsao  _et al._
(2014); da Silva  _et al._ (2016); Newmai  _et al._ (2022) catalyticTsao  _et
al._ (2014); Newmai  _et al._ (2022); Han  _et al._ (2014); Urzúa  _et al._
(2022) and antibacterialLi  _et al._ (2018); Velmurugan  _et al._ (2022);
Padmos  _et al._ (2015); Ding  _et al._ (2017) properties. In this context,
targeted technological applications require the control over two principle
structural parameters. First, different morphologies can be stabilized with a
competition between truncated octahedron structures and five-fold symmetric
morphology including icosahedron and decahedron that originates from the
balance between cohesive, surface and elastic strain energies.Barnard  _et
al._ (2009); Wang and Palmer (2012); Pohl  _et al._ (2012); Liao  _et al._
(2018); Belić  _et al._ (2011); Casillas  _et al._ (2012); Garden  _et al._
(2018); Wang  _et al._ (2011); Myshlyavtsev  _et al._ (2017); Baletto  _et
al._ (2001, 2002) Second, while metallic species can be found in different
chemical arrangements with the possibility of surface segregation, there is
currently no consensus in the literature on whether gold or silver is more
likely to segregate to the surface. Yen  _et al._ (2009); Liao  _et al._
(2020); Guisbiers  _et al._ (2016); Stein  _et al._ (2022); Liao  _et al._
(2018); Martínez  _et al._ (2012); He  _et al._ (2018); Du  _et al._ (2019);
Bon  _et al._ (2019); Paz-Borbón  _et al._ (2008); Deng  _et al._ (2011);
Cerbelaud  _et al._ (2011); Chen and Johnston (2008a, b); Hoppe and Müller
(2017); Rapetti and Ferrando (2019); Wang  _et al._ (2021); Gould  _et al._
(2014)

In this contribution, we experimentally show that gas-phase synthesis can lead
to decahedral and icosahedral AuAg nanoalloys both displaying unambiguous gold
surface segregation. Our machine-learning assisted simulations confirmed those
experimental findings and enabled investigations over a wider spectrum of
chemical compositions. Moreover, the simulations allow us to go beyond post-
mortem analysis, thus uncovering how gold segregation can affect the
nucleation process triggering the nanoparticle crystallization at the
atomistic level. When compared to the current literature, the novelty of this
work is three-fold: (1) Pentatwinned decahedra/icosahedra are not only present
in mono-metallic systems and can also be stabilized in the case of Ag/Au
nanoalloys, (2) Gold atoms can segregate at the nanoparticle’s surface even
with equimolar and silver-rich compositions and (3) Machine-learning assisted
simulations can be used to model nanoalloys accurately to observe atomic scale
processes occurring during their formation.

##  II Methodology

Two complementary approaches were applied to investigate the formation of AgAu
nanoalloys. We used gas-aggregation magnetron-sputter deposition from two
elemental targets of gold and silver to synthesize the AuAg nanoparticles.
Note that this synthesis process involves the nucleation and growth of the
nanoparticles inside a gas aggregation chamber, before their landing on the
substrateHuttel (2017). These nanoparticles are extremely pure, i.e., free
from any ligand or surfactant. The synthesized nanoparticles were further
investigated using high-resolution transmission electron microscopy (HRTEM)
and high-angle annular dark-field scanning TEM (HAADF-STEM) for uncovering
their structural and morphological details. The chemical distribution of the
two elements inside a given nanoparticle was analyzed by energy-dispersive
x-ray spectroscopy (EDS-STEM). In concert, machine-learning-assisted
simulations were carried out. In particular, we used a deep neural network
potential (DNP) that was previously developed by Andolina et al.Andolina  _et
al._ (2021b) and was further tested under numerous conditions [see SI A]. A
key advantage of using the DNP approach is that, although this is not yet
sufficient for reaching the observed experimental sizes ($\approx$7.5 nm), we
still managed to perform molecular dynamics (MD) with a much longer duration
and a larger number of atoms ($\approx$15000 atoms ie. 6.3 nm) than can be
expected from typical DFT calculations.

###  II.1 Experimental setup

AuAg nanoparticles were grown in a water-cooled Nanogen-Trio Gas Aggregation
Source (GAS), from Mantis Deposition Ltd. The nucleation and growth of the
particles are obtained through DC magnetron co-sputtering of extremely pure
(99.99%) Au and Ag targets of diameter equal to $1$ inch, located side by side
on an integrated magnetron sputtering head, positioned at 90 mm from the exit
slit of the aggregation zone. The DC magnetron current, applied independently
to the two targets, was fixed to 40 mA for both gold and silver. With the
synthetic method, it would be possible to control the stoichiometry by using
an alloyed target with precise chemical composition. However, studying the
influence of the chemical composition would then require fabrication of
targets with different stoichiometry. For each of them, the sputtering
property would not necessarily be similar which should change the nanoparticle
size distribution. In order to study the influence of the chemical composition
for a fixed size distribution, we opted instead to use two elemental targets
with a specific set of electrical powers applied in each of them. This allows
us to obtain different chemical compositions in a single experiment all with
the same sputtering conditions. Before depositing on the ultra-thin-carbon
coated copper TEM grid, located in the ultra-high-vacuum deposition Chamber
(with a base pressure of $10^{-9}$ mbar), the nanoparticles were size-selected
by a Quadrupole Mass Filter installed between the GAS and the deposition
chamber. The obtained nanoparticles were studied through high-resolution TEM
(HRTEM), atomically resolved high-angle annular-dark-field scanning TEM
(HAADF-STEM) and energy-dispersive x-ray spectroscopy (EDS) experiments. A Cs
corrected 200 kV FEI Tecnai F20 microscope was used for HRTEM studies, and a
probe corrected Jeol ARM200F microscope was used for HAADF-STEM and EDS
studies. We note that no oxidation of the nanoparticles was observed even if
the EDS analysis was performed several months after the synthesis.

###  II.2 Atomistic simulations

Regarding the interaction potentials, the complexity and non-linearity of deep
neural networks allow the development of interaction potentials capable of
making predictions close to DFT accuracy. The deep neural network interaction
potential for Au-Ag (DNP) developed by Andolina et al.Andolina  _et al._
(2021b) using the DeepPot-SEZhang  _et al._ (2018) method of the DeePMD-
KitWang  _et al._ (2018) was selected for this work because of its excellent
accuracy with pure as well as alloyed systems [See SI A]. All DFT calculations
were performed with VASPKresse and Hafner (1993); Kresse and Furthmüller
(1996a, b) using the Perdew-Burke-ErnzerhofPerdew  _et al._ (1996) (PBE)
functional and the projector augmented wave (PAW) methodKresse and Joubert
(1999) with an energy cutoff of $400\text{\,}\mathrm{e}\mathrm{V}$.

The molecular dynamics (MD) trajectories are obtained using the Large-scale
Atomic/Molecular Massively Parallel Simulator (LAMMPS) coupled to the DNP. We
employed constant volume and constant temperature (NVT) ensemble in these
simulations with a time-step of 1 fs and a damping coefficient of 100 fs.
Liquid droplets are obtained by melting nanoparticles of three sizes (250, 500
and 750 atoms with respectively 1.4 nm, 2.0 nm and 2.4 nm) for three
stoichiometries (Ag3Au, AgAu and AgAu3) at 2000 K. For the freezing
simulations, we utilized a cooling rate of
$2\text{\times}{10}^{11}\text{\,}\mathrm{K}\mathrm{/}\mathrm{s}$ from
$750\text{\,}\mathrm{K}$ to $350\text{\,}\mathrm{K}$. The process is repeated
three times with different initial velocities. The employed numerical setup
that consists in freezing disordered liquid droplets was already used to
simulate nanoparticle formation and mimic gas-phase synthesisAmodeo  _et al._
(2020); Snellman  _et al._ (2021); Nelli  _et al._ (2023). In addition, to
assess the stability of our obtained systems, we also carried out hybrid MD/MC
simulations that consists of 10 Monte Carlo (MC) moves combined with 10 atomic
species swapping every 100 MD time steps.

We note that our choice of simulation protocol assumes that the nanoparticles
are formed from the liquid phase. However, we must note that similar
experiments have also been modeled with atom by atom growth directly from
solid precursors Goniakowski and Mottet (2010); Wells  _et al._ (2015); Xia
_et al._ (2021). At this stage, it remains difficult experimentally to know
which formation pathway is more likely to occur in our experimental protocol.

Finally, simulations are analyzed using OvitoStukowski (2009) built-in
functions. Common neighbor analysis is employed to measure crystal ordering
Stukowski (2012). To obtain the size of the largest ordered cluster, atoms
with crystalline order are grouped together in clusters within a cut-off equal
to 3.5 Å.

##  III Results

###  III.1 Morphology

Figure 1: (Left) HRTEM images of 3 representative AuAg nanoparticles. (a) a
decahedral NP observed along its five-fold symmetry axis, (b) an icosahedral
NP observed along its two-fold symmetry axis, (c) an icosahedral NP observed
close to its 3 fold-symmetry axis. (Right) Typical nanoparticles obtained
after freezing using machine-learning-assisted MD simulations for three
different chemical compositions along with the corresponding crystal analysis
as obtained using common neighbor analysis. Non-crystalline atoms (including
surface atoms) were removed for clarity.

We begin by analyzing the morphologies of both the experimentally synthesized
and the simulated structures. The HAADF-STEM observations show that the
synthesized nanoparticles are well-dispersed on the substrate and display a
narrow size distribution with a mean diameter of $7.4$ nm and a standard
deviation of $0.75$ nm (see SI B). When suitably oriented along (or close to)
one of their low-index zone axes, the synthesized nanoparticles can be clearly
identified as either decahedra or icosahedra,Buffat  _et al._ (1991) both
exhibiting triangular surfaces dominated by (111) orientations. Fig. 1(a-c)
show three different HRTEM images of representative nanoparticles, one $10.5$
nm large decahedron (Fig. 1(a)), one $7.5$ nm large icosahedron (Fig. 1(b))
and one $9.9$ nm large icosahedron (Fig. 1(c)). Similarly, although not with
the same size because of the computational cost of the numerical freezing
setup, our MD simulations in conjunction with the DNP were able to reproduce
the (111) surface preponderance signature of the icosahedron and decahedron
shapes [See Fig. 1(d-f)]. In addition, DNP simulations allow for
characterizing the crystal structure at the atomic scale [Fig. 1(d-f)]. In all
cases, the facets consist of face-centered cubic triangles whose edges are
characterized by hexagonal close-packed atoms which is consistent with the
(111) orientation of the surfaces. To better characterize the crystallinity of
the simulated nanoparticles, we also measured the atomic strain distribution
with respect to the radial position [See SI E]. The strain map shows overall a
very small deviations from the bulk. The latter is slightly more pronounced
for atoms located at droplet surface, which can be attributed to an inherent
surface relaxations. Importantly, out hybrid MD/MC simulations did not observe
any structural modification thus confirming the stability of five-fold
symmetric shapes [See SI C]. Altogether, our results suggest that
icosahedron/decahedron can be stabilized instead of truncated octahedron even
with in Au/Ag nanoparticles in the investigated size regime, i.e. up to $10.5$
nm in experiments and up to $2.4$ nm in simulations.

For metallic systems in general, icosahedron/decahedron and truncated
octahedral shapes are considered more stable respectively at small and large
sizes according to surface vs. volume energy considerationsBarnard  _et al._
(2009); Wang and Palmer (2012); Pohl  _et al._ (2012); Casillas  _et al._
(2012); Garden  _et al._ (2018); Wang  _et al._ (2011); Myshlyavtsev  _et al._
(2017); Baletto  _et al._ (2001, 2002); Pohl  _et al._ (2012) However, the
size threshold allowing for the transition between the two shapes is still
highly debated. In experiments for mono-metallic systems, icosahedron and
decahedron were usually obtained by physical methods of synthesis.Barnard  _et
al._ (2009); Wang and Palmer (2012); Pohl  _et al._ (2012); Belić  _et al._
(2011) Furthermore, previous studies studied gold nanoparticles of diameters
up to 10 nm and showed that the as-obtained icosahedron/decahedron-shaped
nanoparticles remain stable after long periods of electron irradiation in the
TEM experiments Wang and Palmer (2012); Barnard  _et al._ (2009),
demonstrating that, even when obtained under non-equilibrium conditions, both
icosahedron and decahedron are thermodynamically more stable than truncated
octahedron. So far, such experimental results were only obtained with mono-
metallic systems including both silver and gold. Meanwhile, from the
simulation viewpoint, the literature regarding this competition in shape also
focused solely on mono-metallic systems. In this context, early works based on
semi-empirical interactions potentials obtained much smaller crossover sizes
leading to truncated octahedral particles being more preponderant even at
sizes in the 1 nm to 10 nm regimesBaletto  _et al._ (2001, 2002); Myshlyavtsev
_et al._ (2017); Settem  _et al._ (2018). However, most recent results
combining DFT accurate models with equilibrium thermodynamic approaches based
on Helmholtz free energy demonstrated instead the decahedral stability Barnard
_et al._ (2009); Garden  _et al._ (2018) for nanoparticles up to 15 nm.
Similar to previous studies focusing on mono-metallic systems, our study shows
with alloyed AuAg nanoparticles that icosahedron/decahedron shapes are stable
even for diameters up to $10.5$ nm. Altogether, our results are therefore
consistent with the literature obtained in the mono-metallic regime. Finally,
we must note that stability of non-crystalline structures including decahedra
can be favored when mixing different metals in a nanoparticle like in
AgCu.Roncaglia  _et al._ (2021); Langlois  _et al._ (2012) Furthermore, it can
not be ruled out that the observed decahedra may also result from kinetic
trapping at the early stages of the growth processes.

###  III.2 Chemical ordering

By using EDS measurements, we were able to investigate the chemical
distribution inside a given nanoparticle. Fig. 2(a,b,c) displays the chemical
maps of isolated nanoparticles with composition Au_0.58Ag_0.42, Au_0.65Ag_0.35
and Au_0.74Ag_0.26. Interestingly, these different compositions were observed
in the same sample. Such composition variations can be attributed to a slight
evolution of the synthesis conditions such as targets race track and
temperature during the deposition time, which was in our case as long as 15
minutes, a duration necessary to collect a sufficiently large number of
nanoparticles on the TEM grid in our setting. We were thus able to analyze
nanoparticles with different compositions although silver-rich nanoparticles
were not observed experimentally. A striking feature in all these maps is the
occurrence of a non-homogeneous distribution with gold atoms segregating at
the surface for all of synthesized chemical compositions. For a more
quantitative picture, density profiles recorded along the nanoparticles
diameter are presented in Fig. 2(d,e,f). These profiles confirm the gold
surface segregation, the silver atoms being mostly confined in the
nanoparticle core. In greater details, for nanoparticles with composition
Au_0.58Ag_0.42, Au_0.65Ag_0.35 and Au_0.74Ag_0.26, we obtained the following
respective Au/Ag ratio 1.41, 1.29 and 1.10 by averaging the Au content over
the 5 external atomic layers ($\sim$1.25 nm). Note that gold surface
segregation is also qualitatively confirmed by HAADF-STEM observations owing
to the sensitivity of this technique to the atomic number of the encountered
elements (z-contrast) (see SI B).

In order to span a larger range of chemical composition, we complement the
experimental observations with numerical simulations. We separate surface
atoms from the bulk based on their lower coordination numbers ($<10$). Fig.
2(g) shows the ratio between the gold proportion at the surface and in the
entire nanoparticle denoted $\xi_{surf}^{Au}$. As seen in the figure, our
results confirm that gold segregates to the surface even in the regime of
silver-rich nanoparticles [see SI D for the density profile obtained in
simulations].

An additional finding is that the gold surface segregation is larger for
Au_0.25Ag_0.75 than for the less silver-rich nanoparticles. In order to better
understand the gold surface segregation obtained at the end of the freezing
simulation, we also measured the temporal evolution of the mean square
displacement in the initial liquid regime[See Fig. SI F]. While all atoms
located in the center of the droplet exhibit similar behavior for the mean
square displacement, atoms located at the surface possess different diffusion
properties depending on their chemical nature or the chemical composition of
the droplet. In particular, the diffusion at the surface is lowered when
increasing the gold composition. More importantly, silver atoms are always
more diffusive than gold atoms at the surface which is consistent with silver
atoms being less stable at the surface and showing the tendency to migrate
inside the core of the droplet.

Figure 2:  (Left) (a-c) Experimental EDS measurements for three nanoparticles
with different chemical compositions. Au and Ag elements are respectively
colored in red and green. (d-f) Distribution of atomic species for the
corresponding nanoparticles. (Right) (g) Surface chemical composition obtained
in machine-learning assisted MD simulations. Results are shown for different
chemical compositions and nanoparticle sizes. Error bars are obtained from
averaging over three independent initial conditions.

Because these first numerical results were obtained from fully freezing liquid
droplets, we could only consider systems not larger than 750 atoms which
correspond to diameters of $2.4$ nm. Therefore, we also performed simulations
with an alternative protocol to study larger nanoparticles of diameters up to
6.3 nm. The systems are initialized in the icosahedral structures that
circumvent the necessity of having to simulate the whole freezing mechanisms
while enabling for directly reaching the previously observed morphology. The
atomic chemical species between gold and silver are then randomly assigned to
correspond to the three studied stoichiometries of Ag-Au alloys. MD
simulations are combined with MC moves and atomic species swapping at 600 K,
which is large enough for atomic swap to operate while maintaining the crystal
ordering and overall morphology. By starting with ordered structures yet with
chemical disorder, we can focus on the temporal evolution of the surface
composition while simulating the larger nanoparticles. In particular, we
studied nanoparticles made of 923, 5083 and 14993 atoms, which correspond
respectively to 2.5 nm, 4.4, nm and 6.3 nm. Fig.3 shows that convergence is
already obtained after 10 ps for the smallest studied systems while the others
would require unreasonably higher computational times. Yet, it remains that in
all cases, while starting at a value of $1$, $\xi_{surf}^{Au}$ monotonically
increases with time indicating that similar to the smaller nananparticles gold
segregation at the surface exists at the larger sizes, which are consistent
with the experimental results. Moreover, similar to previous results obtained
after the freezing simulations for the smaller nanoparticles, the gold surface
segregation remains higher in the silver-rich stoichiometries.

Altogether, both our experimental and our numerical results demonstrate that
gold surface segregation can be obtained in Au/Ag nanoparticles regardless of
the size and the composition. In experiments, we show that it happens for
gold-rich systems while in simulations, we confirm the experimental results
and predict that it remains true even for silver-rich systems.

Figure 3: Temporal evolution of the surface chemical composition obtained in
machine-learning assisted MD simulations combined with Monte-Carlo and atomic
species swap initialized with icosahedron nanoparticles for three different
nanoparticles sizes 923, 5083 and 14993 atoms which correspond respectively to
2.5 nm (a), 4.4, nm (b) and 6.3 nm (c).

###  III.3 Study of the crystallization dynamics

Figure 4: (a-c) Temporal evolution obtained in MD freezing with nanoparticles
of 750 atoms. $N_{crys}$ is the number of atoms within the biggest crystalline
cluster. $\xi_{surf}^{Au}$ (resp. $\xi_{crys}^{Au}$) denotes the ratio between
the gold proportion at the surface (resp. in the biggest crystalline cluster)
and in the entire nanoparticle. We note that for $\xi_{crys}^{Au}$ results are
only shown when the crystalline cluster is big enough ie. $N_{crys}>50$. (d)
Corresponding images obtained during the crystallization. Non-crystalline
atoms are rendered in transparency and grey (resp. yellow) spheres correspond
to silver (resp. gold) atoms.

A key advantage of our DNP simulations over experiments, and also DFT
calculations that usually employ 0 K minimization to explore the structural
landscape, is that we can retrieve the crystallization dynamics at the
atomistic scale. To this end, we follow the evolution of the crystal ordering
by measuring the size of the biggest ordered cluster [$N_{crys}$ in Fig. 4(a)]
and observe that crystallization starts at different temperatures depending on
the chemical composition. In particular, silver-rich nanoparticles crystallize
first and also reach a higher final value for $N_{crys}$. Fig. 4(b) shows the
temporal evolution of $\xi_{surf}^{Au}$. While oscillations are observed,
$\xi_{surf}^{Au}$ consistently remains greater than one, indicating that gold
segregation already occurs in the liquid regime at the early stages of
freezing. A plateau in $\xi_{surf}^{Au}$ is also reached after 1 ns and bellow
550 K. In the same intermediate times, according to Fig. 4(a), the plateau for
$N_{crys}$ is not yet fully achieved, suggesting that chemical ordering
happens beforehand. We further define $\xi_{crys}^{Au}$ as the relative
chemical composition that was previously measured, but within the largest
crystalline cluster as opposed to the surface. Fig. 4(c) shows that
$\xi_{crys}^{Au}$ also quickly converges when compared to $N_{crys}$ thus
confirming that chemical ordering occurs first. The value of the observed
plateaus are all below or close to $1$, which is consistent with
$\xi_{surf}^{Au}$ being above $1$. Indeed, the more gold atoms are present at
the surface, the more silver atoms are within the crystalline core of the
nanoparticles. For this analysis, while we only showed the temporal evolution
of one nanoparticle per chemical composition, SI G shows two additional cases
that exhibit similar behavior. Finally, the associated snapshots for different
sizes of the largest crystalline cluster can be retrieved from those temporal
evolutions [see Fig. 4(d)]. From these images, one can observe that
crystallization emerges first at the periphery of the droplet instead of its
core and slowly grows towards the core [See SI H for a more quantitative
confirmation].

Altogether, our findings show that the gold surface segregation occurs already
in the liquid regime which has consequences on the subsequent crystallization
process. First, we note that gold possesses a larger melting temperature both
in bulk and at nanoscale thus indicating that gold crystals are more stable
than silver ones. However, in the silver-rich system, the atoms available to
trigger crystallization are mostly silver since the majority of gold atoms are
at the surface. On the contrary, in the gold rich system, there is a mixture
of the remaining gold atoms with all the silver atoms. In one case, a nearly
pure metal crystallizes, but in the other, nucleation occurs in an alloying
regime which is less favorable. As such, the fact that silver-rich
nanoparticles are more crystalline in our simulations is a consequence of the
initial chemical ordering that is established already in the liquid regime.

##  IV Discussion

The surface segregation in gold/silver nanoparticles has been the subject of
numerous studies with experimentalYen  _et al._ (2009); Liao  _et al._ (2020);
Guisbiers  _et al._ (2016); Stein  _et al._ (2022); Martínez  _et al._ (2012)
and numericalHe  _et al._ (2018); Du  _et al._ (2019); Bon  _et al._ (2019);
Deng  _et al._ (2011); Cerbelaud  _et al._ (2011); Chen and Johnston (2008b);
Rapetti and Ferrando (2019); Wang  _et al._ (2021); Gould  _et al._ (2014);
Hoppe and Müller (2017); Paz-Borbón  _et al._ (2008); Moreira  _et al._ (2023)
approaches. Experimentally, chemically-synthesized nanoparticles are found to
exhibit silver surface segregation,Yen  _et al._ (2009); Liao  _et al._
(2020); Guisbiers  _et al._ (2016) which was explained in part by the presence
of the oxidized surfaces Hoppe and Müller (2017). To the best of our
knowledge, only two experimental studies observed surface segregation in Ag-Au
nanoalloys made with physical routes of synthesis which is crucial for
comparison with our results since they allow for the effects of the ligands
and the liquid solvent to be ruled out Martínez  _et al._ (2012); Liao  _et
al._ (2018). The first study shows surface segregation occurring for the most
preponderant chemical species and explained these observations by kinetic
trapping at the early stages during growthLiao  _et al._ (2018). On the one
hand, in the gold-rich system, their result is similar to ours although we
will show later that kinetic trapping is not the only possible stabilizing
effect and that charge transfer can also stabilize the gold segregated
structures. On the other hand, in the silver-rich system, difficulty in
interpreting the experimental results can be raised because oxidation seems to
be present and to provoke the apparent silver segregation. Similarly to our
work, the second study also used EDS to characterize the chemical
orderingMartínez  _et al._ (2012) and reported Au surface segregation.
However, the results were obtained only with one nanoparticle that exhibited
equal amounts of gold and silver ($Ag_{0.51}Au_{0.49}$).

In the present study, while we consistently observed gold segregation, we did
not manage to generate silver-rich nanoparticles using our experimental
approach and it is therefore difficult to be assertive that gold segregation
remains in the whole composition range of the alloy. However, our machine-
learning assisted simulations predict that Au would segregate even for Ag rich
compositions. While further experimental studies are required to confirm these
trends, we posit that these predictions can be rationalized by combining a
literature review of the numerical simulations along with additional
calculations, as discussed below. Indeed, the gold segregation at the surface,
even for silver-rich nanoparticles, might appear counterintuitive at first
sight because it is not found in all experiments, and because it is in
contradiction with the surface energy hierarchy ($\gamma$(Ag) is slightly
lower than $\gamma$(Au)) and with the atomic size (Ag is slightly bigger than
Au). Furthermore, silver surface segregation was consistently found in
simulations when using empirical force fields He  _et al._ (2018); Du  _et
al._ (2019); Deng  _et al._ (2011); Paz-Borbón  _et al._ (2008); Moreira  _et
al._ (2023). However, in this context, the simplicity of the employed
empirical force fields when compared to our MLIP may lead to inaccurate
modeling. In particular, in the work of Paz-Borbon et al.Paz-Borbón  _et al._
(2008), the Gupta potential seemed at first to demonstrate silver segregation
but when further optimization was made at the DFT level, the authors found
that gold segregation is favored instead. Similarly, we tested an embedded-
atom model (EAM) which was used in the recent paper of Moreira et al.Moreira
_et al._ (2023) and demonstrated that it leads to much more energetic
nanoparticles when exhibiting silver surface segregation [See SI J].

Meanwhile, all of the DFT studies also show the gold segregation both for
extended surfaces or for nanoparticles up to few hundreds of atoms.Cerbelaud
_et al._ (2011); Chen and Johnston (2008b); Rapetti and Ferrando (2019); Wang
_et al._ (2021); Paz-Borbón  _et al._ (2008); Gould  _et al._ (2014); Hoppe
and Müller (2017) As an explanation, it has been shown from electronic
structure investigations that the segregation of the Au atoms at the surface,
regardless of composition, is caused by electrostatic forces rather than
surface energy. In order to confirm this hypothesis, we performed single-point
DFT calculations initialized with the 250-atom nanoparticles found with our
freezing simulations and computed the Bader charges. In Figure 5(a), the total
charge on the different layers from the center of the NP (index 1) to the
surface of the NP (index 4) are shown for the three compositions. It is
interesting to note that the surface is always negatively charged and the
subsurface is positively charged, by the same magnitude in all the studied
chemical compositions. The charge distribution on the different atom types is
depicted in Fig. 5(b,c,d) as color maps and in Fig. 5(e,f,g) as histograms of
averaged values taken over atoms in the different layers. A significant charge
transfer from Ag atoms to Au atoms is thus observed with Au and Ag atoms being
always respectively negatively and positively charged. A striking observation
is that the charge transfer is more observed for the silver rich
nanoparticles. Indeed, in the Au0.25Ag0.75 nanoparticles (Fig. 5(b) and (e)),
a small number of surface Au atoms bear a negative charge of $\approx$ -0.3 e
whereas the Au atoms at the Au0.75Ag0.25 nanoparticles surface carry a much
lower average charge of $\approx$ -0.1 e (Fig. 5(d) and (g)). Overall, the
total charge of the surface layer remains the same for all of the studied
compositions but it is carried by a few but highly charged Au atoms in
Au0.25Ag0.75 nanoparticles and by many but less charged Au atoms in
Au0.75Ag0.25 nanoparticles. These results show that the same effects observed
in previous DFT studies on small clusters are also present in larger
nanoparticles obtained in our own machine-learning based simulations. They
also show that charge transfer between Ag and Au always yields the same
surface charge irrespective of the alloy composition. We therefore hypothesize
that, as Au atoms become negatively charged when alloyed with silver, they
will tend to move towards the surface due to Coulomb repulsion. Further,
because the nanoparticle’s surfaces then become negatively charged, positively
charged Ag atoms are attracted to the subsurface. As the charge transfer from
Ag atoms to Au atoms is more pronounced for Ag-rich compositions, the
segregation of gold at the surface will be favored even for such compositions,
over that of the majority species, as could be intuitively expected. We note
that, even if it did not explicitly taking into account the atomic charges,
the DNP model as it was fitted to DFT calculations was still able to
remarkably translate such complex charge transfer mechanisms into
corresponding changes in forces.

Figure 5: (a) Total Bader charge distribution averaged over the three
nanoparticles per chemical composition. (b-d) Slices of typical nanoparticles
where atoms are colored following the Bader charges. (e-g) Absolute value of
the Bader charge distribution per atomic species averaged over three
nanoparticles per chemical composition. We note that the charges are always
negative for gold and positive for silver and that only the nanoparticles made
of 250 atoms were studied because of the large computational costs associated
with DFT calculations.

##  V Conclusion

The main goal of this work was to study the formation of gold/silver
nanoalloys. Our results in terms of experimental synthesis in the gas phase
produced two remarkable observations. First, we found five-fold symmetric
particles, including both icosahedron and decahedron with nanoparticles of
diameters up to 10 nm. Second, while surface oxidation can induce silver
segregation, only gold segregation was observed in our synthesis results that
were obtained in vacuum conditions. These experimental findings were first
confirmed by machine-learning assisted simulations. Then, we further explored
the chemical phase space by reaching different chemical compositions and
confirming the gold segregation and the stability of the five-fold symmetric
morphology, even in silver-rich systems. We emphasize that contrary to
previous works using classical interaction potentials, the use of MLIP to
provide quantum accurate modeling was key to reproduce the gold surface
segregation. An additional advantage of our simulation approach was that we
managed to explore the crystallization dynamics that required large-scale
simulations with unprecedented chemical accuracy and found that the gold
surface segregation occurs before crystal ordering and leads to better
crystallization in silver-rich composition.

By showing that gold surface segregation can be observed in vacuum while
silver surface segregation is usually found in more reactive conditions, our
study highlights the importance of environmental effects on the chemical
distribution of species in a bi-metallic nanoparticle. It also demonstrates
the need to study nanoparticles using advanced experimental observations to be
able to harness these effects. Further, our results show the importance of
taking into account electronic structure effects in nanoalloys, which are
impossible to reproduce with conventional empirical potentials. To this end,
we demonstrate the tremendous power of MLIP-type potentials for studies of
this kind, making it possible to combine the modeling of nucleation processes
in realistically-sized systems with DFT precision calculations.

## Acknowledgment

This study was supported by the French National Research Agency (ANR) in the
framework of its “Jeunes chercheuses et jeunes chercheurs” program,
ANR-21-CE09-0006 and by the EUR grant NanoX n° ANR-17-EURE-0009 in the
framework of the ”Programme des Investissements d’Avenir”. Computational
resources have been provided by CALMIP, by Jean Zay and by TGCC. WAS and CA
acknowledge support through the U.S. National Science Foundation (Award No.
CSSI-2003808). Computational support was provided in part by the University of
Pittsburgh Center for Research Computing through the resources provided on the
H2P cluster, which is supported by NSF (Award No. OAC-2117681).

## References

  * Alloyeau  _et al._ (2012) D. Alloyeau, C. Mottet, and C. Ricolleau,  _Nanoalloys: Synthesis, Structure and Properties_ (Springer London Ltd, London, England, UK, 2012).
  * Calvo (2020) F. Calvo,  _Nanoalloys_ (Elsevier, Walthm, MA, USA, 2020).
  * Ferrando (2016) R. Ferrando,  _Structure and Properties of Nanoalloys_ (Elsevier, Walthm, MA, USA, 2016).
  * Ferrando  _et al._ (2008) R. Ferrando, J. Jellinek, and R. L. Johnston, Chem. Rev. 108, 845 (2008).
  * Penuelas  _et al._ (2008) J. Penuelas, P. Andreazza, C. Andreazza-Vignolle, H. C. N. Tolentino, M. De Santis, and C. Mottet, Phys. Rev. Lett. 100, 115502 (2008).
  * Amendola  _et al._ (2014) V. Amendola, S. Scaramuzza, S. Agnoli, S. Polizzi, and M. Meneghetti, Nanoscale 6, 1423 (2014).
  * Swiatkowska-Warkocka  _et al._ (2015) Z. Swiatkowska-Warkocka, A. Pyatenko, F. Krok, B. R. Jany, and M. Marszalek, Sci. Rep. 5, 1 (2015).
  * Ahmad  _et al._ (2019) N. Ahmad, M. Bon, D. Passerone, and R. Erni, ACS Nano (2019), 10.1021/acsnano.9b06614.
  * Benzo  _et al._ (2019) P. Benzo, S. Combettes, B. Pecassou, N. Combe, M. Benoit, M. Respaud, and M. J. Casanove, Phys. Rev. Mater. 3, 096001 (2019).
  * Wang  _et al._ (2022a) Y. Wang, W. Nong, N. Gong, T. Salim, M. Luo, T. L. Tan, K. Hippalgaonkar, Z. Liu, and Y. Huang, Small 18, 2203340 (2022a).
  * van der Hoeven  _et al._ (2018) J. van der Hoeven, T. A. J. Welling, T. A. G. Silva, J. E. van den Reijen, C. La Fontaine, X. Carrier, C. Louis, A. van Blaaderen, and P. E. de Jongh, ACS Nano 12, 8467 (2018).
  * Liu and Huang (2013) Y. Liu and C. Z. Huang, ACS Nano 7, 11026 (2013).
  * Carrillo  _et al._ (2018) P. Carrillo, R. Shi, K. Teeluck, S. D. Senanayake, and M. G. White, ACS Catal. 8, 7279 (2018).
  * Ramade  _et al._ (2017) J. Ramade, C. Langlois, M. Pellarin, L. Piccolo, M.-A. Lebeault, T. Epicier, M. Aouine, and E. Cottancin, Nanoscale 9, 13563 (2017).
  * Mertens  _et al._ (2011) S. F. L. Mertens, M. Gara, A. S. Sologubenko, J. Mayer, S. Szidat, K. W. Krämer, T. Jacob, D. J. Schiffrin, and T. Wandlowski, Adv. Funct. Mater. 21, 3259 (2011).
  * Chen  _et al._ (2022) X. Chen, S. Zhang, C. Li, Z. Liu, X. Sun, S. Cheng, D. N. Zakharov, S. Hwang, Y. Zhu, J. Fang, G. Wang, and G. Zhou, Proc. Natl. Acad. Sci. U.S.A. 119, e2117899119 (2022).
  * Ahmad  _et al._ (2018) N. Ahmad, G. Wang, J. Nelayah, C. Ricolleau, and D. Alloyeau, J. Microsc. 269, 127 (2018).
  * Goudeli and Pratsinis (2017) E. Goudeli and S. E. Pratsinis, ACS Nano 11, 11653 (2017).
  * Nelli  _et al._ (2020) D. Nelli, A. Krishnadas, R. Ferrando, and C. Minnai, J. Phys. Chem. C 124, 14338 (2020).
  * Nelli and Ferrando (2019) D. Nelli and R. Ferrando, Nanoscale 11, 13040 (2019).
  * Förster  _et al._ (2019) G. D. Förster, M. Benoit, and J. Lam, Phys. Chem. Chem. Phys. 21, 22774 (2019).
  * Hizi  _et al._ (2023) A. Hizi, G. D. Forster, R. Ferrando, Y. Garreau, A. Coati, C. Andreazza-Vignolle, and P. Andreazza, Faraday Discuss. 242, 35 (2023).
  * Chen and Johnston (2008a) F. Chen and R. L. Johnston, ACS Nano 2, 165 (2008a).
  * Demiroglu  _et al._ (2016) I. Demiroglu, Z. Y. Li, L. Piccolo, and R. L. Johnston, Catal. Sci. Technol. 6, 6916 (2016).
  * Barcaro  _et al._ (2011) G. Barcaro, A. Fortunelli, M. Polak, and L. Rubinovich, Nano Lett. 11, 1766 (2011).
  * Tan  _et al._ (2012) T. L. Tan, L.-L. Wang, D. D. Johnson, and K. Bai, Nano Lett. 12, 4875 (2012).
  * Davis  _et al._ (2014) J. B. A. Davis, S. L. Horswell, and R. L. Johnston, J. Phys. Chem. A 118, 208 (2014).
  * Wisesa  _et al._ (2022) P. Wisesa, C. M. Andolina, and W. A. Saidi, J. Phys. Chem. Lett. 14, 468 (2022).
  * Wang  _et al._ (2022b) B. Wang, W. Chu, Y. Wu, D. Casanova, W. A. Saidi, and O. V. Prezhdo, J. Phys. Chem. Lett. 25, 5946–5952 (2022b).
  * Andolina  _et al._ (2020) C. M. Andolina, P. Williamson, and W. A. Saidi, The Journal of Chemical Physics 152, 154701 (2020), https://doi.org/10.1063/5.0005347 .
  * Andolina  _et al._ (2021a) C. M. Andolina, J. G. Wright, N. Das, and W. A. Saidi, Phys. Rev. Mater. 5, 083804 (2021a).
  * Bayerl  _et al._ (2022) D. Bayerl, C. M. Andolina, S. Dwaraknath, and W. A. Saidi, Digital Discovery 1, 61 (2022).
  * Erhard  _et al._ (2022) L. C. Erhard, J. Rohrer, K. Albe, and V. L. Deringer, npj Comput. Mater. 8, 1 (2022).
  * Mocanu  _et al._ (2018) F. Mocanu, K. Konstantinou, T. H. Lee, N. Bernstein, V. L. Deringer, G. Csányi, and S. R. Elliott, J. Phys. Chem. B 122, 8998 (2018).
  * Gartner  _et al._ (2022) T. E. Gartner, P. M. Piaggi, R. Car, A. Z. Panagiotopoulos, and P. G. Debenedetti, Phys. Rev. Lett. 129, 255702 (2022).
  * Piaggi  _et al._ (2022) P. M. Piaggi, J. Weis, A. Z. Panagiotopoulos, P. G. Debenedetti, and R. Car, Proc. Natl. Acad. Sci. U.S.A. 119, e2207294119 (2022).
  * Panizon  _et al._ (2015) E. Panizon, J. A. Olmos-Asar, M. Peressi, and R. Ferrando, Phys. Chem. Chem. Phys. 17, 28068 (2015).
  * Benoit  _et al._ (2020) M. Benoit, J. Amodeo, S. Combettes, I. Khaled, A. Roux, and J. Lam, Mach. Learn.: Sci. Technol. 2, 025003 (2020).
  * Tallec  _et al._ (2023) G. Tallec, G. Laurens, O. Fresse-Colson, and J. Lam, in  _Quantum Chemistry in the Age of Machine Learning_ (Elsevier, Walthm, MA, USA, 2023) pp. 253–277.
  * Laurens  _et al._ (2021) G. Laurens, M. Rabary, J. Lam, D. Peláez, and A.-R. Allouche, Theor. Chem. Acc. 140, 1 (2021).
  * Goniakowski  _et al._ (2022) J. Goniakowski, S. Menon, G. Laurens, and J. Lam, J. Phys. Chem. C 126, 17456 (2022).
  * Lam  _et al._ (2020) J. Lam, S. Abdul-Al, and A.-R. Allouche, J. Chem. Theory Comput. 16, 1681 (2020).
  * Caro  _et al._ (2018) M. A. Caro, V. L. Deringer, J. Koskinen, T. Laurila, and G. Csányi, Phys. Rev. Lett. 120, 166101 (2018).
  * Deringer and Csányi (2017) V. L. Deringer and G. Csányi, Phys. Rev. B 95, 094203 (2017).
  * Tsao  _et al._ (2014) Y.-C. Tsao, S. Rej, C.-Y. Chiu, and M. H. Huang, J. Am. Chem. Soc. 136, 396 (2014).
  * da Silva  _et al._ (2016) A. G. M. da Silva, T. S. Rodrigues, V. G. Correia, T. V. Alves, R. S. Alves, R. A. Ando, F. R. Ornellas, J. Wang, L. H. Andrade, and P. H. C. Camargo, Angew. Chem. Int. Ed. 55, 7111 (2016).
  * Newmai  _et al._ (2022) M. B. Newmai, M. Verma, A. Dahiya, and P. S. Kumar, J. Phys. Chem. Solids 161, 110371 (2022).
  * Han  _et al._ (2014) C. Han, X. Yang, G. Gao, J. Wang, H. Lu, J. Liu, M. Tong, and X. Liang, Green Chem. 16, 3603 (2014).
  * Urzúa  _et al._ (2022) E. Urzúa, F. Gonzalez-Torres, V. Beltrán, P. Barrias, S. Bonardd, A. M. R. Ramírez, and M. Ahumada, Nanoscale Adv. 4, 4789 (2022).
  * Li  _et al._ (2018) Q. Li, F. Lu, H. Ye, K. Yu, B. Lu, R. Bao, Y. Xiao, F. Dai, and G. Lan, ACS Sustainable Chem. Eng. 6, 9813 (2018).
  * Velmurugan  _et al._ (2022) P. Velmurugan, V. Mohanavel, P. Sekar, S. Vijayanand, A. Chinnathambi, C. Govindasamy, and S. Sivakumar, Mater. Lett. 310, 131471 (2022).
  * Padmos  _et al._ (2015) J. D. Padmos, M. Langman, K. MacDonald, P. Comeau, Z. Yang, M. Filiaggi, and P. Zhang, J. Phys. Chem. C 119, 7472 (2015).
  * Ding  _et al._ (2017) X. Ding, P. Yuan, N. Gao, H. Zhu, Y. Y. Yang, and Q.-H. Xu, Nanomed. Nanotechnol. Biol. Med. 13, 297 (2017).
  * Barnard  _et al._ (2009) A. S. Barnard, N. P. Young, A. I. Kirkland, M. A. van Huis, and H. Xu, ACS Nano 3, 1431 (2009).
  * Wang and Palmer (2012) Z. W. Wang and R. E. Palmer, Phys. Rev. Lett. 108, 245502 (2012).
  * Pohl  _et al._ (2012) D. Pohl, A. Surrey, L. Schultz, and B. Rellinghaus, Appl. Phys. Lett. 101, 263105 (2012).
  * Liao  _et al._ (2018) T.-W. Liao, A. Yadav, K.-J. Hu, J. van der Tol, S. Cosentino, F. D’Acapito, R. E. Palmer, C. Lenardi, R. Ferrando, D. Grandjean, and P. Lievens, Nanoscale 10, 6684 (2018).
  * Belić  _et al._ (2011) D. Belić, R. L. Chantry, Z. Y. Li, and S. A. Brown, Appl. Phys. Lett. 99, 171914 (2011).
  * Casillas  _et al._ (2012) G. Casillas, J. J. Velázquez-Salazar, and M. Jose-Yacaman, J. Phys. Chem. C 116, 8844 (2012).
  * Garden  _et al._ (2018) A. L. Garden, A. Pedersen, and H. Jónsson, Nanoscale 10, 5124 (2018).
  * Wang  _et al._ (2011) B. Wang, M. Liu, Y. Wang, and X. Chen, J. Phys. Chem. C 115, 11374 (2011).
  * Myshlyavtsev  _et al._ (2017) A. V. Myshlyavtsev, P. V. Stishenko, and A. I. Svalova, Phys. Chem. Chem. Phys. 19, 17895 (2017).
  * Baletto  _et al._ (2001) F. Baletto, C. Mottet, and R. Ferrando, Phys. Rev. B 63, 155408 (2001).
  * Baletto  _et al._ (2002) F. Baletto, R. Ferrando, A. Fortunelli, F. Montalenti, and C. Mottet, J. Chem. Phys. 116, 3856 (2002).
  * Yen  _et al._ (2009) C.-W. Yen, M.-L. Lin, A. Wang, S.-A. Chen, J.-M. Chen, and C.-Y. Mou, J. Phys. Chem. C 113, 17831 (2009).
  * Liao  _et al._ (2020) S. Liao, Z. Luo, J. B. Metternich, R. Zenobi, and F. Stellacci, Nanoscale 12, 22639 (2020).
  * Guisbiers  _et al._ (2016) G. Guisbiers, R. Mendoza-Cruz, L. Bazán-Díaz, J. J. Velázquez-Salazar, R. Mendoza-Perez, J. A. Robledo-Torres, J.-L. Rodriguez-Lopez, J. M. Montejano-Carrizales, R. L. Whetten, and M. José-Yacamán, ACS Nano 10, 188 (2016).
  * Stein  _et al._ (2022) F. Stein, S. Kohsakowski, R. Martinez-Hincapie, S. Reichenberger, C. Rehbock, V. Colic, D. Guay, and S. Barcikowski, Faraday Discuss. (2022), 10.1039/D2FD00092J.
  * Martínez  _et al._ (2012) L. Martínez, M. Díaz, E. Román, M. Ruano, D. Llamosa P., and Y. Huttel, Langmuir 28, 11241 (2012).
  * He  _et al._ (2018) X. He, S.-E. Zhang, F. Cheng, and Z.-X. Chen, Chem. Commun. 54, 638 (2018).
  * Du  _et al._ (2019) R. Du, S. Tang, X. Wu, Y. Xu, R. Chen, and T. Liu, R. Soc. Open Sci. 6, 190342 (2019).
  * Bon  _et al._ (2019) M. Bon, N. Ahmad, R. Erni, and D. Passerone, J. Chem. Phys. 151, 064105 (2019).
  * Paz-Borbón  _et al._ (2008) L. O. Paz-Borbón, R. L. Johnston, G. Barcaro, and A. Fortunelli, J. Chem. Phys. 128, 134517 (2008).
  * Deng  _et al._ (2011) L. Deng, W. Hu, H. Deng, S. Xiao, and J. Tang, J. Phys. Chem. C 115, 11355 (2011).
  * Cerbelaud  _et al._ (2011) M. Cerbelaud, R. Ferrando, G. Barcaro, and A. Fortunelli, Phys. Chem. Chem. Phys. 13, 10232 (2011).
  * Chen and Johnston (2008b) F. Chen and R. L. Johnston, Acta Mater. 56, 2374 (2008b).
  * Hoppe and Müller (2017) S. Hoppe and S. Müller, J. Appl. Phys. 122, 235303 (2017).
  * Rapetti and Ferrando (2019) D. Rapetti and R. Ferrando, J. Alloys Compd. 779, 582 (2019).
  * Wang  _et al._ (2021) Y. Wang, L. Zhang, B. Xu, X. Wang, and H. Wang, Model. Simul. Mater. Sci. Eng. 30, 025003 (2021).
  * Gould  _et al._ (2014) A. L. Gould, C. J. Heard, A. J. Logsdail, and C. R. A. Catlow, Phys. Chem. Chem. Phys. 16, 21049 (2014).
  * Huttel (2017) Y. Huttel,  _Gas-Phase Synthesis of Nanoparticles_ (2017).
  * Andolina  _et al._ (2021b) C. M. Andolina, M. Bon, D. Passerone, and W. A. Saidi, J. Phys. Chem. C 125, 17438 (2021b).
  * Zhang  _et al._ (2018) L. Zhang, J. Han, H. Wang, W. A. Saidi, R. Car, and W. E, arXiv (2018), 10.48550/arXiv.1805.09003, 1805.09003 .
  * Wang  _et al._ (2018) H. Wang, L. Zhang, J. Han, and W. E, Comput. Phys. Commun. 228, 178 (2018).
  * Kresse and Hafner (1993) G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).
  * Kresse and Furthmüller (1996a) G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996a).
  * Kresse and Furthmüller (1996b) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996b).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Amodeo  _et al._ (2020) J. Amodeo, F. Pietrucci, and J. Lam, J. Phys. Chem. Lett. 11, 8060 (2020).
  * Snellman  _et al._ (2021) M. Snellman, N. Eom, M. Ek, M. E. Messing, and K. Deppert, Nanoscale Adv. 3, 3041 (2021).
  * Nelli  _et al._ (2023) D. Nelli, E. Y. El Koraychy, M. Cerbelaud, B. Crespin, A. Videcoq, A. Giacomello, and R. Ferrando, ACS Nano 17, 587 (2023).
  * Goniakowski and Mottet (2010) J. Goniakowski and C. Mottet, Phys. Rev. B 81, 155443 (2010).
  * Wells  _et al._ (2015) D. M. Wells, G. Rossi, R. Ferrando, and R. E. Palmer, Nanoscale 7, 6498 (2015).
  * Xia  _et al._ (2021) Y. Xia, D. Nelli, R. Ferrando, J. Yuan, and Z. Y. Li, Nat. Commun. 12, 1 (2021).
  * Stukowski (2009) A. Stukowski, Model. Simul. Mater. Sci. Eng. 18, 015012 (2009).
  * Stukowski (2012) A. Stukowski, Model. Simul. Mater. Sci. Eng. 20, 045021 (2012).
  * Buffat  _et al._ (1991) P.-A. Buffat, M. Flüeli, R. Spycher, P. Stadelmann, and J.-P. Borel, Faraday Discuss. 92, 173 (1991).
  * Settem  _et al._ (2018) M. Settem, M. Islam, and A. K. Kanjarla, Comput. Mater. Sci. 148, 266 (2018).
  * Roncaglia  _et al._ (2021) C. Roncaglia, D. Rapetti, and R. Ferrando, Phys. Chem. Chem. Phys. 23, 23325 (2021).
  * Langlois  _et al._ (2012) C. Langlois, Z. L. Li, J. Yuan, D. Alloyeau, J. Nelayah, D. Bochicchio, R. Ferrando, and C. Ricolleau, Nanoscale 4, 3381 (2012).
  * Moreira  _et al._ (2023) M. Moreira, L. C. Felix, E. Cottancin, M. Pellarin, D. Ugarte, M. Hillenkamp, D. S. Galvao, and V. Rodrigues, J. Phys. Chem. C 127, 1944 (2023).