# Stiffening of nanoporous Au as a result of dislocation density increase upon
characteristic length reduction

Claudio Melis  Dipartimento di Fisica, Università degli Studi di Cagliari,
Cittadella Universitaria, I-09042 Monserrato (CA), Italy
<EMAIL_ADDRESS>Giorgio Pia  Dipartimento di Ingegneria Meccanica,
Chimica, e dei Materiali, Università degli Studi di Cagliari,via Marengo 2,
I-09123 Cagliari, Italy  Elisa Sogne  NABLA Lab, Biological and Environmental
Sciences and Engineering (BESE) Division,King Abdullah University of Science
and Technology (KAUST), Thuwal, Saudi Arabia  Andrea Falqui  NABLA Lab,
Biological and Environmental Sciences and Engineering (BESE) Division,King
Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia
Stefano Giordano  Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique
Hauts-de-France, UMR 8520,- IEMN - Institut d’Electronique de
Microélectronique et de Nanotechnologie, F-59000 Lille, France  Francesco
Delogu  Dipartimento di Ingegneria Meccanica, Chimica, e dei Materiali,
Università degli Studi di Cagliari,via Marengo 2, I-09123 Cagliari, Italy
Luciano Colombo  Dipartimento di Fisica, Università degli Studi di Cagliari,
Cittadella Universitaria, I-09042 Monserrato (CA), Italy

###### Abstract

Structure is the most distinctive feature of nanoporous metals. The intricate
maze of rounded shapes, where ligaments and pores run after each other
disorderly, strikes imagination no less than it imparts properties that, tuned
by size effects, have no counterpart in the bulk form. Indisputably,
nanoporous Au has been the absolute protagonist of the field of study,
unveiling the disrupting potential of nanoporous metals in areas ranging from
catalysis to energy and sensing. Here, we still focus on nanoporous Au,
addressing the long-standing issue of mechanical properties in nanoporous
metals. In particular, we investigate how Young’s modulus changes with
ligament size, being the porosity the same. Based on atomistic replicas
generated starting from experimental tomographic evidence, atomistic
simulations reveal that nanoporous Au stiffens as ligaments become finer,
reproducing experimental findings obtained by nanoindentation of dealloyed
samples. Ruled out surface stress effects, theoretical considerations relate
stiffening to the dislocation density increase.

##  1 Introduction

Nanoporous metals (NPM) form a broad class of monolithic solids that are
currently under intense scrutiny. With their architecture consisting of an
elegant, and yet confusing, tangle of voids and solid 1, in the past 20 years
these materials have never ceased to fascinate commoners and amaze scientists
from very different backgrounds. Their history contributes to make the class
of materials unique. Manufactured and utilized by ancient civilizations 2, 3,
time by time NP metals, and NP Au in particular, re-surface from the darkness
of alchemical laboratories and the implications of corrosion phenomena 4, 5,
6, 7, 8.

This is not surprising considering that the main synthetic route to NP metals
is based on the selective dissolution of less noble metals from a parent
alloy. Dealloying is, indeed, a relatively simple process that can be mastered
by craftsmen unaware of chemistry fundamentals 9. The alloy is exposed to an
acidic aqueous solution and less noble elements are progressively removed by a
free corrosion process 10, 11, 12. At present, electrochemical methods are
often applied to achieve a better control of the dissolution process 11, 13,
but this leaves the key features of the corrosion process unaffected.

In short, dealloying relies upon the complex interplay between the chemistry
of selective dissolution and the physics of surface rearrangement 10, 11, 12.
As quickly as atoms of less noble species are removed from the surface, the
remaining atoms of most noble species diffuse from low-coordination surface
sites to more stable ones, possibly exposing initially buried less noble
atoms. A gradual change of characteristic lengths accompanies porosity
formation on the nanometer scale.

The result of dealloying is an homogeneous, interpenetrating, bi-continuous
disordered network of ligaments and pores 1, 11, 14. The nature of metallic
bond pervades harmoniously the whole architecture, ensuring that any reduction
of characteristic lengths deeply impacts the physical and chemical properties
of the NPM.

In this regard, NP Au represents the most typical example 3. Easy to obtain
because of Au chemical nobility, it has provided the ideal model system to
explore the structure-property relationships for NPM and has rapidly risen to
the role of absolute protagonist in the field of study. Gradually, it has
shown promise for actuation, catalysis, energy, and sensing-related
applications, just to cite a few 15, 16, 17, 18, 19, 20, 21, 22, 23, 24.

A key enabling technology issue stands, nonetheless, in the way to the
practical exploitation of NP Au, and NPM in general, namely their mechanical
properties 25, 26, 27, 28, 29, 30, 31, 32, 33, 34. Actuators, self-supported
catalysts and scaffolds for surface-enhanced Raman spectroscopy, thermal
insulators and structural materials all need long-term mechanical stability
and, therefore, a satisfactory characterization of the NPM response to elastic
and plastic deformation. However, the subject is also per se extremely
interesting. Porosity degree and characteristic lengths severely affect the
response to mechanical loads. Particularly intriguing is the evidence that the
reduction of ligament and pore size to a few nanometers can induce a
significant hardening and stiffening of NP architectures. Although literature
data are still fragmentary and contradictory, there is no doubt, at least for
NP Au, that size effects tune somehow the mechanical properties.

Concerning the elastic properties in particular 35, 33, 36, 37, 38, 39,
experiments and computer simulations have been used to investigate Young’s
modulus and Poisson ratio. Constitutive models have been mostly based on the
Gibson-Ashby scaling laws 40, 41, but their validity has been questioned.
Typically, ligaments have aspect ratios far from those compatible with Gibson-
Ashby’s models, and other structural and topological factors have been shown
to play a role 36, 42. For instance, several experiments have provided robust
evidence that the NP Au Young’s modulus increases as the ligament thickness is
reduced approximately from 200 nm to 5 nm 35, 36, 43. Surface stress has been
invoked to explain experimental observations as well as the enhanced moment of
inertia of finer structures 36 and the content of lattice defects 44. In this
latter case, it is worth recalling that, while point defects can induce a
significant reduction of the elastic moduli 45, 46, extended defects may
affect local interatomic bonding 44.

Motivated by this intriguing and open scenario, in this work we investigate
the Young’s modulus of NP Au combining experimental, simulative and
theoretical methods. To this aim, we have generated in silico replicas of NP
Au structures with atomistic resolution by mapping the initial volumetric data
obtained by experimental tomographic evidence. Re-scaling the characteristic
lengths of the atomistic sample, we have systematically explored the impact of
size effects, being the porosity degree the same.

Our findings confirm that the Young’s modulus significantly increases as the
ligament size decreases. Further, we provide evidence that the observed
stiffening cannot be ascribed to surface stress or grain-boundary effects.
Rather, our results point out a quantitative correlation with the dislocation
density. Specifically, we show that local stress and strain fields in the
neighbourhood of dislocation cores allow dislocations to work as reinforcing
solutes.

##  2 Results and discussion

### 3D reconstruction and rendering of NP Au samples.

We used serial block-face (SBF) scanning electron microscopy (SEM) to collect
two-dimensional images of NP Au samples through a SEM-dedicated VolumeScope
cutting device (see Materials and Methods for details). A typical SEM image is
shown in Fig 1 (Panel a) for illustration purposes. The typical appearance of
a NP architecture is immediately evident, with irregularly shaped ligaments
and ligament junctions forming condensed open cells finally resulting in an
inherently distorted three-dimensional lattice. The two-dimensional images
were assembled into a single three-dimensional structure via a suitable volume
reconstruction method (see Materials and Methods for details). A
representative tomographic reconstruction of the NP Au structure is shown in
in Fig. 1 (Panel b). It allows clearly appreciating how solid and void give
rise to a maze of rounded shapes, where ligaments interconnecting at massive
junctions form an interpenetrating, bi-continuous network of pores and
ligaments and nodes. Ligaments and pores have similar diameter of about 42
$\pm$ 3 nm. Porosity evaluation throughout the reconstructed sample volume
indicates a solid volume fraction of 0.48 $\pm$ 0.03.

### “In silico” sample generation with atomistic resolution

Starting from the volumetric data previously obtained for the NP Au samples,
we proceeded to decorate the corresponding SBF-SEM images with an atomistic
structure corresponding to face-centered-cubic (fcc) Au lattice with spacing
$a$= 0.40782 nm. The procedure has been performed by following 3 consecutive
steps:

  1. 1.

Homotetic transformation - We used the Meshmixer code 47 to apply a series of
homothetic transformations to the selected experimental cubic NP Au sub-
volumes. Preserving shape and inner geometries while changing the volume, we
generated 9 NP Au samples with side length $L_{cell}$ ranging from 5.5 to 40.7
nm per each selected sub-volume. Fig. 1 (Panel c) shows the NP Au sample with
$L_{cell}$=19.7;

  2. 2.

Solid phase identification – We used the Nanosculpt software 48 to identify
the NP Au volume occupied by the solid phase, and build the surface mesh that
encloses it;

  3. 3.

Crystallographic information addition - We filled the selected volume with Au
atoms arranged in a perfect fcc lattice. The NP Au samples we obtained had
number of atoms ranging from 4.95$\times$103 up to 1.85$\times$106.

One out of the 9 cubic NP Au samples with $L_{cell}$=19.7 nm we have generated
is shown in Fig. 1 (Panel d). The comparison with the corresponding
experimental SBF-SEM tomographic reconstruction reported in Fig. 1 (Panel e)
clearly shows how accurate is the atomistic decoration in reproducing the
characteristic shapes of NP Au samples.

### Young’s modulus of atomistic NP Au samples

We used the Ovito software 49 to evaluate the specific surface area $\alpha$
of the different NP Au samples. The corresponding values obtained from the 9
different NP Au sub-volumes selected per each NP Au sample size considered are
plotted in Fig. 2 (left) as a function of the side length $L_{cell}$ of the
cubic NP Au sample. It can be seen, in the range of the explored $L_{cell}$
values, that $\alpha$ monotonically decreases as function of $L_{cell}$.

The specific surface area can be used to estimate the apparent ligament
diameter $L_{ap}$. The calculation is immediate if the structure is assumed to
consist of long circular rods. Accordingly,

$L_{ap}=\frac{4}{\alpha}$ (1)

Other estimates can be obtained assuming that the NP Au samples have a
periodic diamond-like cubic structure interconnected by cylindrical ligaments
of diameter $D_{L}$ 50, 39:

$D_{L}=\frac{1.63(1.25-\phi)[1.89+\phi(0.505+\phi)]}{\alpha}$ (2)

where $\phi$ is the solid volume fraction $\sim$0.49 for all samples. A more
quantitative topological measure, which involves an exact relationship with
the specific surface area, is the characteristic spacing $\tilde{L}$ between
the centers of neighbouring ligaments. Since ligaments and pores most often
have the same characteristic length, $\tilde{L}$ can be also regarded as a
measure of the ligament size 51. In particular, $\tilde{L}$ can be expressed
as 51, 33:

$\tilde{L}=\left(\frac{gV_{total}}{G}\right)^{1/3}$ (3)

where $G$ is the topological genus measuring the number of connections in a
specific network, $g$ is the scaled genus representing the number of
connections in a representative volume element having size $\tilde{L}^{3}$ and
$V_{total}$ is the total volume of the sample 33. The calculations of $G$ and
$g$ have been performed using the open-source code CHomP 52 and Eq. (16) of
Ref. $33$, respectively (see Supporting Information for details). The
estimated values of $L_{ap}$, $D_{L}$ and $\tilde{L}$ are compared with each
other in Fig. 2 (right) as a function of $L_{cell}$. The different ligament
size estimates undergo a consistent change with the total sample volume. In
fact, they increase with $L_{cell}$ up to a maximum value of $\sim$16 nm.
Eventually, we decided to use $\tilde{L}$ as the characteristic length of the
selected NP Au structures and to refer to it the Young’s modulus estimates
obtained by subjecting the atomistic NP Au samples to uniaxial compression
(see Supporting Information for details). Performed within the elastic limit,
the simulated uniaxial compression determines a gradual deformation of the NP
Au samples. A typical stress-strain curve is shown in Fig. 3 (right). Almost
perfectly linear, its slope allows to univocally evaluate the Young’s modulus
of the corresponding NP Au structure. The Young’s modulus $E$ is plotted in
Fig. 3 as a function of $\tilde{L}$. It exhibits a clean dependence on the
length scale. In particular, $E$ increases approximately from $18.0$ GPa to
$25.3$ GPa as $\tilde{L}$ decreases.

An experimental estimate of the Young’s modulus, obtained by nanoindentation
of the NP Au sample (see Materials and Methods for details), is also shown in
Fig. 3 (left), shaded area. It is equal to 20.8 $\pm$ 2.3 GPa, i.e. it is
almost coincident with the value of 18.3 GPa estimated by atomistic
simulations for large $\tilde{L}$ values. The agreement between the two
estimates goes beyond the initial expectations, based on the numerous sources
of difference that can bring experimental and numerical values far apart for
almost any physical quantity. In this case, the observed agreement can be
tentatively ascribed to the high purity of the experimental NP Au sample and
to the low level of local stresses probably achieved by the low-temperature
thermal annealing to which the NP Au sample has been subjected.

As outlined in the Introduction, several mechanisms can been taken into
account in order to explain the $E=E(\tilde{L})$ dependence reported in Fig.
3. In the next Section we analyze in detail all of them, by focusing on the
primary NP Au elements, namely the single ligament, which we mimic as Au fcc-
nanowire with a constant diameter $D_{nw}$.

## Discussion

A variation of Young’s modulus similar to the one reported in Fig. 3 has been
already observed in several cases. However, it has never been interpreted
univocally 36, 43, 35. In contrast, various factors have been invoked to
rationalize the experimental and numerical evidence. In the following, we
examine the different possible mechanisms that can be expected to play a role
in determining the observed mechanical response of NP Au structures.

### Effect of surface-stress on fcc-nanowires

Free surfaces are made of low-coordinated atoms. Reduced coordination numbers
imply a redistribution of electronic charge and potential energy, possibly
resulting in a change of local elastic moduli near the surface region compared
with bulk ones. These effects have been extensively reported for clusters,
wires and films 53, 54, 55, 56. While the general formalism for the
theoretical description of surface mechanics is fully established 57, 58, 59,
60, 61, 62, 56, the surface constitutive equation and its parameters are still
a matter of investigation 63. In particular, surface effects, and the
resulting size-dependent elastic behaviour, of Au nanowires remain unexplored.

To elucidate this effect, we theoretically evaluated the Young’s modulus of
three macroscopic Au wires with their main axis aligned, respectively, to the
three main crystallographic directions of the fcc lattice, namely (100), (110)
and (111) (see Supporting Information for details). Being referred to
macroscopic Au nanowires for which surface effects are negligible, the
obtained values have to be compared with the Young’s modulus of Au nanowires
with diameter on the nanometer scale, which can be significantly affected
surface-stress effects. To this aim, we performed molecular dynamics (MD)
simulations on fcc Au nanowires with the main axis parallel to the (100),
(110) and (111) directions. We considered Au nanowires with total length
$L_{z}$ of 20 nm and diameter varying in the range 1.5 nm $\leq$ $D_{nw}$
$\leq$ 15 nm. After a geometry optimization based on the conjugate-gradients
algorithm, we obtained the Young’s modulus values summarized in Fig. 4 (left),
where $E$ is plotted as a function of the nanowire diameter $D_{nw}$ . The
dashed lines account for the Young’s moduli of macroscopic Au wires.

The Young’s moduli of Au nanowires exhibits significant deviation from the
macroscopic counterparts, decreasing down to 60% as the nanowire size
decreases below 10 nm. The observed reduction can be entirely ascribed to
surface effects. This means that surface stresses induce a softening of the Au
nanowire, not a stiffening. It follows that surface effects cannot explain the
Young’s modulus increase observed when the characteristic length of NP Au
structure decreases. Fig. 4 (center) shows a color map of the local von Mises
stress calculated on a Au-fcc nanowires with $D_{nw}$= 3 nm.

### Effect of grain boundaries on fcc Au nanowires

We applied the methodology described above to evaluate the Young’s modulus of
an Au nanowire containing two grain boundaries obtained by the rotation of the
central region of the nanowire with respect to the end regions (see Supporting
Information for details). In this way, grain boundaries are perpendicular to
the nanowire main axis. The Young’s modulus estimates are plotted in Fig.4
(right) as a function of the nanowire diameter for the different rotation
angles considered. It can be seen that the presence of grain boundaries
determines a reduction of the Young’s modulus as the nanowire diameter
decreases. Therefore, similar to surface effects, grain-boundaries cannot
justify the observed stiffening of NP Au structures with increasingly reduced
ligament size.

In this regard, it is worth noting that the case study we examined is the most
favourable to sizeably affect the elastic properties. For instance, let us
consider the uniaxial compression of the fcc Au nanowire along its main axis.
It is intuitively clear that grain boundaries perpendicular to the main axis
have a minor impact compared with grain boundaries with a different
orientation. In particular, oblique grain boundaries are likely to give rise
to grain boundary sliding events upon compression that make the Au nanowire
much softer and deformable. Similar behaviour has been reported for several
metal nanowires 64, 65, 66. For this reason, we did not investigate further
along this direction.

### Effect of extended defects: dislocations

Grain-boundaries and dislocations can be also expected to affect the elastic
properties of NP metals 67, 68, 69. Many experimental works pointed out the
effects of extended defects on the Young’s modulus of nanowires 70, 71, 72,
73. For instance, it has been shown that a high density of stacking faults in
GaAs nanowires can determine a Young’s modulus enhancement up to 13% compared
with the one of defect-free structures 71.

Dislocations are also expected to affect the elastic properties of NP Au
structures because of the associated stress and strain fields. We tried to
identify the presence of dislocations in the 9 NP Au structures previously
generated by means of the dislocation extraction algorithm 74, 75. Fig. 5
(left) shows the dislocation density $\rho_{D}$ as a function of the the
characteristic spacing between local centers of the solid or the pore space
$\tilde{L}$ calculated for the 9 NP Au samples under investigation. We clearly
observe a sizeable $\rho_{D}$ reduction by increasing $\tilde{L}$ suggesting
that the occurrence of large $\rho_{D}$ values, observed for samples with
relatively small $\tilde{L}$, could in principle give rise to a stiffening
effect, in turn resulting as an increase of the Young’s modulus. This is fully
confirmed in Fig. 5 (right), where a reasonable linear correlation is observed
between the Young’s modulus and the corresponding $\rho_{D}$ value, indicating
that high dislocation densities lead to a sample stiffening.

In order to make the above conclusion more robust, we further investigated the
effect of dislocations on the mechanical response of individual ligaments, by
considering an Au nanowire with the main axis oriented along the (100)
crystallographic direction, a diameter of 3 nm and a total length of 20 nm.
Specifically, we evaluated the Young’s modulus of the nanowire decorated with
an increasing number of dislocations. Although different types of dislocations
can be present in a typical NP Au sample, we restricted our investigation to
edge dislocations, which can be easily generated in any specific position of
the crystalline lattice 76. We carried out calculations using two different
dislocation arrangements. In one case, dislocations were suitably placed to
maximize the distance between them. In the other, their positions were
selected randomly. The obtained results are shown in Fig. 6. In both cases,
the Young’s modulus $E$ undergoes a marked increase up to $\sim$25% for
dislocation densities of $\sim$2$\times$10-8 nm-2, clearly showing that
dislocations can induce a significant stiffening of NP Au ligaments.

We investigated the stiffening effect further estimating the local Young’s
modulus of a fcc Au nanowire containing a single edge dislocation (see
Supporting Information for details). To this aim, we calculated the per-atom
stress tensor and the corresponding per-atom elastic strain tensor. Their
ratio provides a local measure of the Young’s modulus. The map of local
Young’s modulus in a 1-nm-thick nanowire slab is shown in Fig. 7 (top). The
Young’s modulus is approximately equal to 20.5 GPa relatively far from the
dislocation, but it increases up to 40 GPa nearby the dislocation. This is a
clear evidence that dislocation effects stem from the local enhancement of
elastic properties.

### Theoretical considerations on dislocation effects

Based on the numerical evidence concerning individual dislocations, we
developed an elastic model to estimate the effects of multiple dislocations.
We considered a nanowire with $n$ edge dislocations generating a surface
$s_{d}$ over the total wire section $S$. We supposed that the effective
longitudinal width of each dislocation is $\ell_{d}$ over the total wire
length $L$. Additionally, we assumed that the effective volume of the
dislocation $\ell_{d}s_{d}$ corresponds to an inhomogeneity with Young’s
modulus $E_{d}>E$. A procedure of elastic homogenization 77, 78 of the
nanowire elastic response with the population of dislocations eventually
leading to the effective Young’s modulus as

$\displaystyle\frac{1}{E_{eff}}=\frac{1}{E}\left(1-n\frac{\ell_{d}}{L}\right)+\frac{1}{E_{d}\frac{s_{d}}{S}+E\left(1-\frac{s_{d}}{S}\right)}n\frac{\ell_{d}}{L},$
(4)

Eq. 4 is correct for $n\ll L/\ell$ as far as interactions between the elastic
fields of nearby dislocations are negligible.

Theoretical predictions are compared with numerical findings in Fig. 7
(bottom). Data refer to an Au nanowire with total length $L$=20 nm and
diameter $D_{nw}$=3 nm, dislocation width $\ell_{d}$=2 nm, surface associated
with dislocations $s_{d}$=14.1 nm2, bulk Au Young’s modulus $E$=20.5 GPa and
locally enhanced Young’s modulus $E_{d}$=40 GPa. It can be seen that
homogenized elastic theory satisfactorily best fits the atomistic data, thus
supporting the hypothesis that edge dislocations act as reinforcing
dispersoids in the Au nanowire. We note that atomistic simulations involving
more than 7 dislocations indicate that the Young’s modulus reaches a plateau
value of about 24 GPa. This can be ascribed to the fact that interactions
among dislocations, not included in the homogenized elastic model, become
increasingly more intense as the dislocation density increases, thus
depressing the reinforcing effect.

The very good agreement between the atomistic data and the analytical model
further confirms the idea that dislocations act as reinforcing elements in
single metal ligaments. This result ultimately explains the experimentally
observed Young’s modulus increases upon reduction of the average metal
ligament thickness.

##  3 Conclusions

In conclusion, we used experimental and theoretical methods to investigate the
mechanical behaviour of nanoporous Au by addressing the long-standing debate
of the observed Young’s modulus increases upon characteristic length
reduction. To this aim we have used an experimental tomographic reconstruction
to generate NP Au samples with atomistic resolution by mapping the initial
volumetric data. Atomistic simulations confirm that nanoporous Au stiffens as
ligaments become finer, reproducing the experimental results obtained by
nanoindentation. A combination of numerical and theoretical findings allow to
rule out the effect of surface stress and grain boundaries. In contrast, we
have clear evidence that dislocations act as strengthening agents that enhance
the nanoporous Au Young’s modulus. The results obtained emphasize the
importance of line defects and can address future research to investigate the
mechanical properties of nanoporous metals.

##  4 Materials and methods

### Fabrication

Nanoporous (NP) gold samples have been fabricated by electrochemical
dealloying of a homogeneous chemically disordered polycrystalline Ag70Au30
solid solution. The parent alloy was prepared by mechanical alloying of
Ag50Au50 powder mixtures followed by melting in an induction furnace and
magnetic stirring. Elemental Ag and Au powders with particle size below 45
$\mu$m and 99.99% purity were manually mixed to obtain powder mixtures with
Ag50Au50 stoichiometry. Mechanical alloying was carried out on powder batches
of 10 g. The powders were sealed in a hardened-steel cylindrical reactor
together with two 8-g stainless-steel balls under inert Ar atmosphere with
oxygen and humidity contents below 2 ppm. The reactor was clamped to the
mechanical arm of a SPEX Mixer/Mill 8000 ball mill and swung back and forth
along a three-dimensional trajectory at about 875 rotations per minute. The
mechanical processing resulted in the mutual dissolution of Ag and Au and the
formation of a homogeneous chemically disordered nanocrystalline solid
solution with Ag50Au50 chemical composition.

The obtained powders were placed in a quartz crucible and molten inside a
laboratory Nabertherm N60/ER furnace. The melt was kept at 850 $\degree$C for
5 days under magnetic stirring and Ar flux conditions. Then, the molten phase
was cast into cylindrical pellets about 1 mm high and with a diameter of about
1 cm. The surfaces of cylindrical pellets were finely polished and annealed
for 10 h at about 400 $\degree$C K to relieve residual stresses.

Structural and microstructural transformations of powders and bulk solid along
the different steps of the preparation process of the Ag50Au50 solid solution
were investigated by X-ray diffraction (XRD). To this aim, a Rigaku SmartLab
diffractometer equipped with a PhotonMax high-flux 9 kW rotating anode X-ray
source was used. Not shown for brevity, XRD patterns were analyzed according
to the Rietveld method. Mechanical alloying confirms its ability to induce the
formation of homogeneous nanocrystalline solid solutions. Although we do not
have direct evidence in this regard, melting and magnetic stirring can be
expected to enhance the homogeneity of the final polycrystalline Ag50Au50
solid solution on the atomic scale.

NP Au structures were obtained from the parent alloy by electrochemically-
assisted dealloying. The cylindrical pellets were exposed to an aqueous
solution 0.75 M in HNO3 and 0.01 M in AgNO3. Selective Ag dissolution was
carried out at room temperature inside a three-electrode electrochemical cell
controlled by a Metrohm Autolab 302N potentiostat-galvanostat. A potential of
about 600 mV was applied, using a Pt wire and an Ag electrode as counter and
reference electrodes respectively. Dealloying was interrupted after 10 days.
To this aim, the cylindrical pellets were repeatedly rinsed with high-purity
water and, finally, dried using an Ar flux at room temperature for 4 days.
X-ray fluorescence measured a residual Ag content lower than 1 at %.To
eliminate internal residual stresses due to the dealloying process, the NP Au
structures were subjected to a 5-day long annealing at 400 K under Ar flux
conditions.

### Serial Block Face-SEM sectioning and imaging

The NP gold sample was embedded in Epon epoxy resin using small silicon mold
by polymerization at 65 $\degree$C for 72h. Resin blocks were mounted on
aluminum specimen pins (10-006002-10, Micro to Nano) using cyan acrylic glue
and trimmed with a glass knife to a rectangle 0.5 $\times$ 0.75 mm, with the
gold sample exposed on all four sides. Silver paint (16 062-PELCO Conductive
Silver Pain, TedPella) was used to favor the electrically grounding of the
block edges to the aluminum pin. The entire, glued block was then sputter
coated with a thin (25 nm) layer of gold (Cressington 208 HR sputter coater)
to make its surface reflecting. After the block alignment into the SEM
chamber, block cutting and serial block-face (SBF)-SEM images collection was
performed using a dedicated VolumeScope (Thermo Scientific) cutting device
mounted in a Teneo (Thermo Scientific) variable Pressure SEM, the latter
operating at an acceleration voltage of 2 kV, and with a probe current of 50
pA and an in-chamber pressure of 20 Pa of water vapor. The sample was imaged
at a magnification and with a pixel number that could allow obtaining an
ultimate voxel size of 6 $\times$ 6 $\times$ 50 nm (x-y-z), where the z-one is
the thickness chosen for the block cutting. The SEM images were collected
using a dedicated backscattered electron (BSE) detector, provided for working
with the VolumeScope cutting device.

### 3D reconstruction and rendering

Serial SBF-SEM images were assembled into volume files using AVIZO software
package (Thermo Scientific). NP gold ligaments and pores were segmented using
Ilastik software30. The models extracted were assembled into volume and
rendered using again AVIZO software package. By using the reconstructed volume
the porosity (i.e, the volume of the pores) was also determined slice by slice
over the whole thickness of the reconstructed sample.

### Nanoindentation

The Young’s modulus of NP Au structures obtained by dealloying was measured
using depth-sensing nanoindentation. A calibrated three-sided pyramid diamond
Berkovich tip with radius of about 200 nm was used. Indentation was carried
out using a constant loading rate of 500 $\mu$N s-1. The load was varied
between 200 and 2000 $\mu$N. Up to 100 loading-unloading curves were measured
per sample. The Young’s modulus was estimated measuring the slopes of the
linear part for any unloading curve and averaging over all unloading curves.

### Molecular dynamics simulations

Once the atomistic NP Au structures have been generated, we performed a series
of molecular dynamics (MD) simulations aimed at calculating their elastic
properties as a function of the actual atomic scale structure. MD runs have
been performed using the LAMMPS 79 simulation code and modelling the
interatomic interactions by means of an embedded-atom model (EAM potential) 80
which has been previously validated as a good quantitative predictor of the
elastic properties of gold systems 39. Atomic trajectories have been aged by
the velocity-Verlet algorithm with a discretization of time evolution based on
a time-step as small as 1.0 fs; temperature and pressure control was operated
by a Nosé-Hoover thermostat and barostat, respectively.

Upon “in silico” generation, NPM samples have been at first relaxed by
performing a conjugate-gradient minimization, until the modulus of the maximum
force on each atom was less than 10-6 eV/Å. The samples were then equilibrated
at temperature T=300 K and pressure P=1 Atm for 100 ps and eventually cooled
down to zero temperature by a gentle simulated annealing. After such a careful
equilibration procedure, we statically applied a uniaxial tensile strain
$\epsilon_{ii}$ (where $i$=$x$, $y$ or $z$) on each NPM sample in the range
$0\%\leq\epsilon_{ii}\leq 2$% at incremental steps as small as
$\Delta\epsilon_{ii}=0.1\%$. After another conjugate-gradient minimization of
each deformed configuration, we calculated the resulting stress value
$\sigma_{ii}$ by averaging the per-atom stress tensor 81 throughout the whole
simulated structure. Due to the relatively small maximum strain applied, we
assume a regime of linear elastic response and we accordingly estimate the NPM
Young’s modulus by fitting the stress-strain curve. The inset of Fig.3 shows a
typical stress-strain curve with the corresponding fitting line obtained for a
cubic NPM Au sample with 40.7 nm lateral size. We acknowledge financial
support by “Fondazione di Sardegna” under project ADVANCING (ADVAnced
Nanoporous materials for Cutting edge engineerING ), call 2018 for basic
research projects.

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Figure 1: Panel (a), a typical SEM two-dimensional image of an NP Au sample.
Panel (b), representative tomographic reconstruction of an NP Au structure.
Panel (c), cyan color: volumetric structure of an NP Au sample with
$L_{cell}$=19.7 nm. Panel (d), yellow color: corresponding structure obtained
by using the “in silico” procedure described in the main text. Panel (e):
superposition of the volumetric and atomistic pictures, proving the almost
perfect matching between the two structures.

Figure 2: Left: specific surface area $\alpha$ as a function of the side
length $L_{cell}$ of the cubic NP Au sample. Right: apparent ligament diameter
($L_{ap}$), characteristic spacing between local centers of the solid or the
pore space ($\tilde{L}$) and average ligament diameter ($D_{L}$) as a function
of the cubic NP Au sample side length $L_{cell}$.

Figure 3: Left: calculated Young’s modulus $E$ for the computer-generated NP
Au gold samples as a function of the characteristic spacing between local
centers of the solid or the pore space $\tilde{L}$; the shaded black area
represents the experimental estimate of the Young’s modulus obtained by
nanoindentation. Right: a typical calculated linear elastic stress-strain
curve (back squares) for a sample with $L_{cell}$= 40.7 nm; the Young’s
modulus is obtained by linear interpolation (thin red line).

Figure 4: Left: Young’s modulus $E$ of gold-fcc nanowires with length
$L_{z}$=20 nm oriented along the (100), (110) and (111) directions as a
function of the diameter $D_{nw}$. Center: color map of the local von Mises
stress calculated on a gold-fcc nanowires with $D_{nw}$=3 nm. Right: Young’s
modulus $E$ vs. $D_{nw}$ of gold-fcc nanowires containing two grain-boundaries
obtained by rotating by an angle $\theta$ the central part (half) of the NW
with respect to the two top and bottom quarters.

Figure 5: Left: dislocation density $\rho_{D}$ as a function of the the
characteristic spacing between local centers of the solid or the pore space
$\tilde{L}$. Right: Young’s modulus as a function of the dislocation density
$\rho_{D}$ showing a reasonable linear dependence (red dashed line).

Figure 6: Young’s modulus $E$ of a gold fcc nanowire (oriented along the (100)
direction with Dnw= 3 nm and Lz=20 nm) as a function of the edge-dislocation
density $\rho_{D}$. Black dots and line: dislocations are positioned so as to
maximize their distance. Red dots and lines: dislocations are placed in
totally random positions.

Figure 7: Top: map of the Young’s modulus, calculated as the ratio between the
local stress and local strain values on a gold-fcc nanowire (oriented along
the (100) direction with Dnw= 3 nm and Lz=20 nm) with a single edge-
dislocation positioned at $z$ = 6.3 nm. Bottom: black squares: Young’s modulus
$E$ of a gold fcc nanowire (oriented along the (100) direction with Dnw= 3 nm
and Lz=20 nm) as a function of the number of edge-dislocations. Red line:
effective Young’s modulus obtained from Eq. 4 by considering $L=20$nm,
$\ell_{d}/L\simeq 1/10$, $s_{d}/S\simeq 1/2$, $E=20.5$ GPa and $E_{d}=40$ GPa.


# Lattice-plasmon-induced asymmetric transmission
in two-dimensional chiral arrays

N. Apurv Chaitanya  Tecnologico de Monterrey, School of Engineering and
Sciences, Ave. Eugenio Garza Sada 2501, Monterrey, N.L., Mexico, 64849.  M. A.
Butt  Max Planck Institute for the Science of Light, Staudtstrasse 2, D-91058
Erlangen, Germany  Institute of Optics, Information and Photonics, University
Erlangen-Nuremberg, Staudtstrasse 7/B2, D-91058 Erlangen, Germany  School in
Advanced Optical Technologies, University Erlangen-Nuremberg, Paul-Gordan-
Strasse 6, D-91052 Erlangen, Germany  Max Planck, University of Ottawa Centre
for Extreme and Quantum Photonics, University of Ottawa, Ottawa, ON, K1N6N5
Canada  O. Reshef  Department of Physics, University of Ottawa, Ottawa,
Ontario K1N 6N5, Canada  Robert W. Boyd  Department of Physics, University of
Ottawa, Ottawa, Ontario K1N 6N5, Canada  Max Planck, University of Ottawa
Centre for Extreme and Quantum Photonics, University of Ottawa, Ottawa, ON,
K1N6N5 Canada  The Institute of Optics, University of Rochester, Rochester,
NY, 14627, USA  P. Banzer  Max Planck Institute for the Science of Light,
Staudtstrasse 2, D-91058 Erlangen, Germany  Institute of Optics, Information
and Photonics, University Erlangen-Nuremberg, Staudtstrasse 7/B2, D-91058
Erlangen, Germany  Max Planck, University of Ottawa Centre for Extreme and
Quantum Photonics, University of Ottawa, Ottawa, ON, K1N6N5 Canada  Institute
of Physics, University of Graz, NAWI Graz, Universitätsplatz 5, 8010 Graz,
Austria  Israel De Leon<EMAIL_ADDRESS>Tecnologico de Monterrey, School of
Engineering and Sciences, Ave. Eugenio Garza Sada 2501, Monterrey, N.L.,
Mexico, 64849.  Max Planck, University of Ottawa Centre for Extreme and
Quantum Photonics, University of Ottawa, Ottawa, ON, K1N6N5 Canada  School of
Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON,
K1N6N5 Canada

# Lattice-plasmon-induced asymmetric transmission in two-dimensional chiral
arrays

N. Apurv Chaitanya  Tecnologico de Monterrey, School of Engineering and
Sciences, Ave. Eugenio Garza Sada 2501, Monterrey, N.L., Mexico, 64849.  M. A.
Butt  Max Planck Institute for the Science of Light, Staudtstrasse 2, D-91058
Erlangen, Germany  Institute of Optics, Information and Photonics, University
Erlangen-Nuremberg, Staudtstrasse 7/B2, D-91058 Erlangen, Germany  School in
Advanced Optical Technologies, University Erlangen-Nuremberg, Paul-Gordan-
Strasse 6, D-91052 Erlangen, Germany  Max Planck, University of Ottawa Centre
for Extreme and Quantum Photonics, University of Ottawa, Ottawa, ON, K1N6N5
Canada  O. Reshef  Department of Physics, University of Ottawa, Ottawa,
Ontario K1N 6N5, Canada  Robert W. Boyd  Department of Physics, University of
Ottawa, Ottawa, Ontario K1N 6N5, Canada  Max Planck, University of Ottawa
Centre for Extreme and Quantum Photonics, University of Ottawa, Ottawa, ON,
K1N6N5 Canada  The Institute of Optics, University of Rochester, Rochester,
NY, 14627, USA  P. Banzer  Max Planck Institute for the Science of Light,
Staudtstrasse 2, D-91058 Erlangen, Germany  Institute of Optics, Information
and Photonics, University Erlangen-Nuremberg, Staudtstrasse 7/B2, D-91058
Erlangen, Germany  Max Planck, University of Ottawa Centre for Extreme and
Quantum Photonics, University of Ottawa, Ottawa, ON, K1N6N5 Canada  Institute
of Physics, University of Graz, NAWI Graz, Universitätsplatz 5, 8010 Graz,
Austria  Israel De Leon<EMAIL_ADDRESS>Tecnologico de Monterrey, School of
Engineering and Sciences, Ave. Eugenio Garza Sada 2501, Monterrey, N.L.,
Mexico, 64849.  Max Planck, University of Ottawa Centre for Extreme and
Quantum Photonics, University of Ottawa, Ottawa, ON, K1N6N5 Canada  School of
Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON,
K1N6N5 Canada

###### Abstract

Asymmetric transmission – direction-selective control of electromagnetic
transmission between two ports – is an important phenomenon typically
exhibited by two-dimensional chiral systems. Here, we study this phenomenon in
chiral plasmonic metasurfaces supporting lattice plasmons modes. We show, both
numerically and experimentally, that asymmetric transmission can be achieved
through an unbalanced excitation of such lattice modes by circularly polarized
light of opposite handedness. The excitation efficiencies of the lattice
modes, and hence the strength of the asymmetric transmission, can be
controlled by engineering the in-plane scattering of the individual plasmonic
nanoparticles such that the maximum scattering imbalance occurs along one of
the in-plane diffraction orders of the metasurface. Our study also shows that,
contrary to the case of a non-diffractive metasurface, the lattice-plasmon-
enabled asymmetric transmission can occur at normal incidence for cases where
the metasurface is composed of chiral or achiral nanoparticles possessing
4-fold rotational symmetry.

††preprint: APS/123-QED

##  I Introduction

Chirality refers to an intrinsic sense of handedness of a three-dimensional
(3D) structure, which remains invariant regardless of the direction of
observation. Because of this property, the structure and its mirror image
cannot be brought into congruence by a translation and rotation operation [1].
Proteins, chemical systems and many bio-molecules are known to be chiral,
exhibiting contrasting functions and properties between its mirror-symmetric
pair (enantiomer)[2, 3, 4]. 3D chiral systems yield a different optical
response when interacting with either handedness of circularly polarized
light, leading to phenomena such as circular dichrosim and optical activity.
Both phenomena are widely used for characterizing the optical response of
chiral systems [5]. Chirality could also be defined for two-dimensional (2D)
structures. However, as planar structures do not possess an intrinsic
handedness, chirality is defined by the inability of a 2D structure and its
mirror image to be brought into congruence unless lifted from its plane [6].
Systems composed of achiral structures are known to exhibit chiral effects as
well, provided that the surface normal and the electric field vector form a
chiral triad [6, 7, 8]. This phenomenon, usually called _extrinsic chirality_
, is indistinguishable from standard (intrinsic) chirality [9], and has been
reported in a variety of systems, including single nanoparticles [10, 11].

Figure 1: Schematic illustration of the metasurfaces under investigation. The
metasurface is illuminated at normal incidence by circularly polarized light
propagating in the positive $z$ direction. The metasurface consists of a
square array of gold quadrumers embedded in a homogeneous medium. The
arrangement and dimensions of the four nanoantennas composing the quadrumer
are shown in the inset. All dimension are given in nanometers.

The sense of handedness of a 2D chiral structure is reversed when it is
flipped with respect to an in-plane axis [9]. Because of this property, 2D
chiral structures can show asymmetric optical transmission, i.e., for a given
circular polarization state, the optical transmittance depends on the side of
the sample that is being illuminated [12, 13, 14, 15]. This is in compliance
with the Lorentz reciprocity theorem and does not require magnetic fields as
for the Faraday effect [12]. The phenomenon can be also observed as a
difference in reflection and/or a difference in absorption. Previous reports
of asymmetric transmission in planar arrays of chiral elements lacking 4-fold
rotational symmetry have attributed the phenomenon to the simultaneous
presence of _anisotropy_ and _losses_ [12, 13, 16]. When excited at normal
incidence, 4-fold symmetric chiral holes appear not to exhibit any asymmetric
transmission [17, 18, 19]. However, this is not necessarily true for a
periodic arrangement of such chiral nanostructures, because of the possibility
of loss channels associated with higher-order diffraction [20].

Chiroptical phenomena have been extensively explored in plasmonic systems such
as individual nanostructures [21, 22, 23, 24], 2D array of plasmonic
nanostructures (metasurfaces) [25, 26, 27, 28] and plasmonic nanostructures in
suspension (metafluids) [29]. Of particular interest is the transmittance for
circular polarized excitation of plasmonic metasurfaces with periodic inter-
particle spacing of the order of the optical wavelength. These structures can
support lattice plasmon modes enabled by diffractive coupling of localized
plasmons [30, 31]. The propagation losses of lattice plasmon modes can act as
a channel for _losses_ , while the topography of the individual plasmonic
nanostructure or the array as a whole can act as a source of _anisotropy_ ,
both fundamental for asymmetric transmission [13]. While previous works on
asymmetric transmission in 2D chiral structures have greatly contributed to
our understanding of low-dimensional chirality, an investigation of the
effects of lattice plasmon modes in metasurfaces is still lacking.

In this work we show, both numerically and experimentally, that lattice
plasmon modes can play a key role in enabling asymmetric transmission in
diffractive chiral metasurfaces. This asymmetric transmission mechanism relies
on different lattice plasmon mode excitation efficiencies for left circularly
polarized (LCP) and right circularly polarized (RCP) light. Such excitation
efficiencies can be controlled by tailoring the nanoparticle’s in-plane
distribution of scattered light for circularly polarized excitation and its
alignment with the in-plane diffraction orders of the metasurface. Contrary to
the case of non-diffractive chiral metasurfaces, the lattice-plasmon assisted
mechanism described here leads to asymmetric transmission even for
nanostructures with 4-fold rotational symmetry and normal incidence
illumination. For simplicity we analyze this phenomenon for a metasurface
composed of an array of achiral nanoparticles - a quadrumer consisting of four
nanoantennas as shown in inset of Fig. 1. We show that near the Rayleigh
anomaly condition, where the asymmetric transmission effect is strongest, the
metasurface supports lattice plasmon modes that respond selectively to the
polarization handedness, confirming the underlying asymmetric transmission
mechanism enabled by the lattice plasmon modes. These results are of general
relevance for understanding the role of diffracted waves in 2D chiral systems
and in particular for the design of chiral plasmonic metasurfaces.

Figure 2: Optical response of isolated nanostructure obtained through FDTD
simulations. (a) The polar plot of the far field scattered intensity as a
function of in-plane angle for an individual nanostructure. $I^{{\rm R}}$,
$I^{{\rm L}}$ are the normalized scattered intensity for RCP (solid blue) and
LCP (solid red) excitation. (b) The normalized plots of differential in-plane
scattering, $\Delta I=\left|I^{{\rm R}}-I^{{\rm L}}\right|,$ for 620 nm
excitation wavelength. Superimposed on the plots are the isolated
nanostructure for reference. (c) The variation of $\Delta I$ as a function of
wavelength for the nanostructure, in normalized units.

##  II Asymmetric transmission mechanism and numerical analysis

Consider the metasurface depicted in Fig. 1, which consists of a planar array
of quadrumer nanostructures. The quadrumers are made of gold, arranged into a
squared lattice, and surrounded by a homogeneous medium with refractive index
$n=1.51$ (i.e., glass). The array lies on the ($x$, $y$) plane, having a
lattice spacing of $\Lambda=600$ nm. The dimensions of each quadrumer are
indicated in the inset of Fig. 1. The metasurface is illuminated by a
circularly polarized planewave propagating in the positive $z$ direction and
impinging at normal incidence onto the surface. Note that although the
individual quadrumers are achiral, it is possible to create a 2D-chiral
metasurface by rotating the quadrumer at each lattice point around its center.

The metasurface can support lattice plasmon modes when $\Lambda$ is of the
order of the optical wavelength. These modes result from the coupling of light
scattered from neighboring particles through grazing diffraction orders, or
Rayleigh anomalies [32, 30]. For normally incident excitation, such Rayleigh
anomalies occur at wavelengths given by [33],

$\lambda_{p,q}=n\Lambda\frac{\sqrt{\left(p^{2}+q^{2}\right)-q^{2}}\pm
p}{p^{2}+q^{2}},$ (1)

where $p$ and $q$ are integers indicating the in-plane diffraction orders
along the $x$ and $y$ directions, respectively. The Rayleigh anomaly condition
for the first diffraction order, $\left\langle p,q\right\rangle=\left\langle
0,1\right\rangle=\left\langle 1,0\right\rangle$, is degenerate and occurs at
the wavelength $\lambda_{1,0}=906$ nm. Similarly, the next Rayleigh anomaly,
which corresponds to the diffraction order $\left\langle 1,1\right\rangle$,
occurs at $\lambda_{1,1}=640$ nm. Lattice plasmon modes propagating along the
different in-plane $\left\langle p,q\right\rangle$ diffraction orders are
supported at wavelengths close to those described by Eq. (1). The direction of
the three main $\left\langle p,q\right\rangle$ diffraction orders are depicted
in Fig. 1.

To understand the asymmetric transmission mechanism exhibited by this
metasurface, we first analyze the optical scattering properties of the
individual quadrumers for LCP and RCP plane-wave illumination. For this, we
use a fully vectorial Maxwell equation solver based on the finite difference
time domain (FDTD) method. The permittivity of gold was taken from Ref. [34].
Fig. 2a shows the obtained in-plane angular distribution of the scattered
intensity at the wavelength $\lambda=620$ nm, which is close to the particle’s
resonance. Here, $I^{{\rm L}}$ and $I^{{\rm R}}$ are the in-plane scattered
intensities for LCP and RCP excitation, respectively. The difference between
these intensities, $\Delta I=|I^{{\rm R}}-I^{{\rm L}}|$, will be referred to
as the differential in-plane scattering. This quantity is plotted in Fig. 2b
for the same wavelength as Fig. 2a, and as a function of wavelength in Fig.
2c. Observe that regardless of the optical wavelength, the differential in-
plane scattering has maxima ($\Delta I_{\rm max}$) and minima ($\Delta I_{\rm
min}$) at particular azimuthal angles, whose values are odd and even integer
multiples of $22.5^{\circ}$, respectively.

Figure 3: (a) Numerically calculated transmittance spectra for RCP excitation
for the metasurface as a function of rotation of the individual nanostructure.
The arrows on the top of the results denote the wavelengths at which the
Rayleigh anomalies occur, corresponding to a lattice period of $\Lambda=600$
nm in a homogeneous medium with $n=1.51$. (b) The variation in asymmetric
transmission calculated using Eq. 2 as a function of the quadrumer’s rotation
angle. Asymmetric transmission occurs when the maxima of the differential in-
plane scattering are aligned along the propagation direction of the grazing
diffraction order. (c-f) Show the schematic illustration of the metasurface
when the individual nanostructure at each lattice point is rotated around its
center.

The individual quadrumer exhibits a non-zero differential in-plane scattering
in a spectral range overlapping with the Rayleigh anomalies of the array.
Consequently, the lattice plasmon excitation efficiency will vary depending on
whether the quadrumer array is illuminated with RCP or LCP light, leading to
different transmittances for each case. In turn, this leads to asymmetric
transmission, as reversing the polarization handedness is equivalent to
illuminating the metasurface from the opposite side. Moreover, since $\Delta
I$ varies along the in-plane angle, the differential excitation of lattice
plasmon modes by RCP and LCP light also vary with the relative angle between
$\Delta I$ and the lattice plasmon mode propagation direction. In particular,
when $\Delta I_{{\rm max}}$ ($\Delta I_{{\rm min}}$) is along the propagation
direction of a lattice plasmon mode, then the difference in efficiency of
excitation of that lattice plasmon mode will be maximized (minimized).

To analyze this effect, the transmittance of the metasurface as a function of
quadrumer’s rotation angles is computed and plotted in Fig. 3a for RCP
illumination; a similar result is obtained for LCP illumination (not shown).
The relative orientation of the nanostructure for different rotation angles
with respect to the lattice is illustrated in Figs. 3c-f. Fig. 3b shows the
asymmetric transmission calculated from the numerical results as

$AT=\frac{T^{{\rm R}}-T^{{\rm L}}}{T^{{\rm R}}+T^{{\rm L}}},$ (2)

where $T^{{\rm R}}$ and $T^{{\rm L}}$ are the transmittance for RCP and LCP
excitation, respectively. The asymmetric transmission vanishes for wavelengths
beyond 906 nm because the Rayleigh anomaly condition is not satisfied, and
hence lattice plasmon modes are not supported. On the other hand, the
asymmetric transmission is maximized at wavelengths close to the Rayleigh
anomaly conditions ($\lambda_{10}$ and $\lambda_{11}$), where lattice plasmon
modes are most efficiently excited.

Most importantly, we note that the asymmetric transmission depends on the in-
plane rotation of the individual quadrumers. Clearly, the largest asymmetric
transmission is obtained when the individual nanostructure is rotated by odd
integer multiples of $22.5^{\circ}$, which coincides with the angles for
$\Delta I_{{\rm max}}$ of the isolated nanostructure (see Fig. 2b). Detailed
analysis of the asymmetric transmission arising from an array of chiral
nanostructures - gammadia is given in the supplementary information. From
these results, we conclude that the value of the differential in-plane
scattering of individual nanostructure, $\Delta I$ along the diffraction
orders at the Rayleigh anomaly condition determines the strength of the
asymmetric transmission. Furthermore, we note that when lattice plasmon modes
are excited, the metasurface can exhibit asymmetric transmission as long as
the individual nanostructure exhibit differential in-plane scattering along
the lattice plasmon mode propagation direction. As such it is irrelevant
whether the nanostructure is chiral or achiral and if it has 4-fold rotational
symmetry or not.

Finally, an analysis of the transmission characteristics for RCP and LCP
excitation along different diffraction orders, away from the Rayleigh anomaly
condition, reveals that the transmission into the zeroth order is degenerate
for both excitations irrespective of the wavelength and nanostructure rotation
angles. Thus, the asymmetric transmission observed in Fig. 3b, is caused
solely by the difference in transmission of higher diffraction orders.

Figure 4: (a) Sketch of the experimental setup. A broadband light source is
filtered to perform spectrally resolved measurements. A broadband polarizer
followed by a quarter-wave plate are used to prepare the polarization state of
the excitation beam. The beam is weakly focused, producing a focal spot of
approximately 50 $\mu$m diameter on the quadrumer metasurface. The transmitted
light is collected with a microscope objective of $\mathrm{NA}=1.3$. The
objective’s back focal plane, containing the $k$-space information, is imaged
onto a CCD camera. (b) The $k$-space image recorded in transmission. Zeroth
and first diffraction orders are visible, which were evaluated for
contributions to asymmetric transmission. (c) Direct imaging of the quadrumer
metasurface in reflection at a wavelength near the Rayleigh anomaly, clearly
showing the leaky propagating first diffraction orders.

##  III Experimental results

Based on the numerical analysis, the fabrication of quadrumer metasurfaces
with required orientation of individual structures was carried out. Details
about the fabrication process can be found in supplementary information. A
schematic illustration of the experimental setup used for the investigation is
shown in Fig. 4a. A filtered broadband light source is used to carry out
spectrally resolved measurements in a range of 630 nm – 950 nm. A broadband
polarizer followed by a quarter-wave plate are used to prepare the
polarization state of the excitation beam. The beam is then weakly focused to
a diameter of approximately 50 $\mu$m with the help of a convex lens ($f=300$
mm), and the metasurface is precisely placed in the focal spot with the help
of a 3-axis piezo stage. Detailed description of experimental setup can be
found in supplementary information.

We consider three metasurfaces with in-plane rotation of nanostructures at
$22.5^{\circ}$, $-22.5^{\circ}$ and $45^{\circ}$ to measure the lattice-
plasmon induced asymmetric transmission. For the present metasurface designs,
a significant fraction of the transmitted power remains in the zeroth order,
hindering the experimental observations of the effect. Hence, we perform
diffraction-resolved measurements to identify the independent power
contributions of the transmitted zeroth and first diffraction orders. To this
end, the transmitted light is collected with a microscope objective
($\mathrm{NA}=1.3$), with its back focal plane imaged onto a CCD camera. The
back focal plane gives us access to the wave-vector space ($k$-space) of the
transmitted light from which we obtain the diffraction-resolved power
contributions (see supplementary information). The zeroth order is measured
for all wavelengths. The first diffraction orders can be measured only for
wavelengths shorter than 780 nm because of the limitation imposed by the
maximum collection angle of the microscope objective ($60.25^{\circ}$). This
limitation prevents us from measuring the power coupled to lattice plasmon
modes (near the Rayleigh anomaly wavelengths). For such cases, we employ
direct imaging of the metasurface in reflection via leakage radiation
microscopy. This technique allows us to obtain qualitative information of the
power coupled to lattice plasmon modes by visualizing light leaked as they
propagate. Figs. 4b and 4c illustrate, respectively, typical $k$-space and
leakage radiation microscopy images recorded using the above mentioned
techniques.

The experimental measurements of lattice-plasmon induced asymmetric
transmission obtained for the achiral ($45^{\circ}$ rotation) and chiral
($22.5^{\circ}$ and $-22.5^{\circ}$ rotations) metasurfaces are summarized in
Fig. 5. The diffraction resolved asymmetric transmission for the zeroth and
first diffraction orders are plotted in Fig. 5a as a function of the
wavelength; both, measurements (dashed lines) and simulations (solid lines)
are included in the plots. Diffraction resolved asymetric transmission is
calculated using Eq. 2, but $T_{R}$ and $T_{L}$ replaced with
$T_{R}^{\left\langle p,q\right\rangle}$ and $T_{L}^{\left\langle
p,q\right\rangle}$ which are the transmittance in the $\left\langle
p,q\right\rangle$ diffraction order for RCP and LCP excitation, respectively.
Leakage radiation microscopy images are used to study the phenomenon at a
wavelength close to the Rayleigh anomaly (marked by dashed grey line in
spectral plots), by constructing differential measurements obtained from a
pixel-by-pixel subtraction of two camera shots taken for excitation with
opposite polarization handedness. Such differential leakage radiation
microscopy images, shown in Fig. 5b, allow us to visualize the asymmetric
excitation of lattice plasmon modes responsible for the asymmetric
transmission.

First, we discuss the case of the metasurface with quadrumers rotated by
$45^{\circ}$ in-plane (see Fig. 5a top-left), which constitutes an achiral
metasurface. As expected, this metasurface results in vanishing asymmetric
transmission in first diffraction orders, which is confirmed by both
experimental and simulation results. The differential-leakage radiation
microscopy image recorded for this metasurface, also confirms vanishing
asymmetric propagating modes (see Fig. 5b, marked in green box). Next,
consider the metasurface constituted by quadrumers with in-plane rotations of
$22.5^{\circ}$ and $-22.5^{\circ}$ (see Fig. 5a top-center and top-right,
respectively). As expected from simulation results shown in Fig. 3d, the two
corresponding cases show opposing signs of asymmetric transmission for first
order. This is confirmed by the experimental results as shown in the spectral
plot in Fig. 5a. The differential leakage radiation microscopy image recorded
for the chiral arrangements depict asymmetrically-excited lattice plasmon
modes with opposing signs (see Fig. 5b, marked in blue and red box).

To demonstrate and validate the sole contribution of the first diffraction
orders to the observed asymmetric transmission, we also show the corresponding
simulation and experimental results for the zeroth diffraction order not
featuring any significant spectral dependence for all three studied
metasurfaces (see Fig. 5a center). The absence of asymmetric transmission for
all three quadrumer metasurfaces for the zeroth order underscores the
essential role of lattice plasmon modes in the assymmetric transmission
mechanism.

Figure 5: Experimental and simulated diffraction resolved asymmetric
transmission results for achiral ($45^{\circ}$ rotation) and chiral
($22.5^{\circ}$ $-22.5^{\circ}$ rotations) metasurfaces. (a) Wavelength versus
diffraction resolved asymmetric transmission for zeroth (top panel) and first
diffraction order (bottom panel) for the achiral (green) and chiral (blue and
red) metasurfaces. (b) Differential leakage radiation microscopy images
recorder at a wavelength of 905 nm, showing asymmetric excitation of lattice
plasmon modes for chiral ($22.5^{\circ}$ $-22.5^{\circ}$ rotations)
metasurfaces with opposing signs, and symmetric excitation of lattice plasmon
modes for the achiral metasurface ($45^{\circ}$).

##  IV Conclusions

In conclusion, we studied the role of lattice plasmon modes on the phenomenon
of asymmetric transmission in chiral 2D arrays of plasmonic nanoparticles. We
showed that the difference in the nanoparticle’s in-plane scattering of LCP
and RCP light can contribute to asymmetric transmission in 2D metasurfaces
through the excitation of polarization-handedness selective lattice plasmon
modes. The different excitation efficiencies of the lattice plasmon modes,
produced by unbalanced in-plane scattering of LCP and RCP light and its
coupling to grazing diffraction orders of the periodic lattice, result in
asymmetries in transmission, reflection and absorption. As the difference in
the in-plane scattered intensity varies as a function of the in-plane angle
around the nanostructure, rotating the nanostructure in the plane of the
metasurface can act as a handle to control the metasurface’s asymmetric
transmission. Using diffraction-order resolved transmission measurements, we
demonstrated that the asymmetric transmission results solely from
contributions of higher-order diffracted waves, while the zeroth-order
transmission in not asymmetric. Furthermore, we showed that, even though the
isolated nanostructure has a 4-fold rotational symmetry, the diffractive
metasurfaces studied here exhibit asymmetric transmission for normal incidence
illumination within the spectral range for which lattice plasmon modes are
supported. This behavior is contrary to what occurs in a non-diffractive
chiral metasurface. Our study sheds new light on asymmetric or chiral
phenomena in metasurfaces and the important role of diffraction effects such
as the excitation of lattice plasmon modes.

###### Acknowledgements.

IDL acknowledges the financial support from the Federico Baur Endowed Chair in
Nanotechnology. RWB acknowledges support through the Natural Sciences and
Engineering Research Council of Canada, the Canada Research Chairs program,
the US ARO Grant W911NF-18-1-0337, and US DARPA Award W911NF1810369

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Supplemental Materials: Lattice-plasmon induced asymmetric transmission in
two-dimensional chiral arrays

##  V Simulation Results of Gammadion

Simulations were also carried out for metasurface composed of squared array of
gammadion nanostructure. Although the isolated gammadion has four fold
rotation symmetry, it is 2D chiral. Unlike the quadrumer metasurface discussed
previously, this makes the gammadion metasurface 2D chiral irrespective of the
in-plane rotation of the isolated nanostructure. Fig. S1a shows the normalized
far field scattered intensity for RCP and LCP excitation at 620 nm. $\Delta
I$, which quantifies the difference in angular scattering due to LCP and RCP
excitation is shown in Fig. S1b. Note that, for this metasurface in the entire
spectral range that $\Delta I_{{\rm max}}$ occurs around odd integer multiples
of $45^{\circ}$. As in the case of quadrumer metasurface discussed previously,
here too we would see that the angle along which $\Delta I_{{\rm max}}$ occurs
play a significant role in the generation of asymmetric transmission.

Figs. S2a and b show the simulated transmission and asymmetric transmission
results for gammadion metasurface as a function of the in-plane rotation of
gammadion. The schematic illustration of the metasurfaces obtained by in-plane
rotation is given in Figs. S2c and d. Although the gammadion metasurface shows
varying strengths of asymmetric transmission for different in-plane rotation
angles of the individual gammadion, the highest asymmetric transmission is
obtained when the individual structure is rotated by odd integer multiples of
$45^{\circ}$, which coincides with the angles for $\Delta I_{{\rm max}}$ of
the isolated gammadion nanostructure (see Fig. 2d). Also, the gammadion
rotation angles for which $AT=0$ vary with wavelength, mimicking the
characteristics of $\Delta I$ (see in Fig. 2d) of an individual gammadion.
From the analysis of both quadrumer and gammadion metasurface it is now clear
that, irrespective of the individual nanostructure being chiral or achiral,
the maximum (minimum) asymmetric transmission is observed when the $\Delta
I_{{\rm max}}$ ($\Delta I_{{\rm min}}$) is aligned along the corresponding in-
plane diffraction orders of the metasurface.

##  VI Fabrication of metasurface

The quadrumer metasurfaces investigated in this paper were fabricated
following a standard metal lift-off procedure. A glass cover slip (170 $\mu$m
thick, BK7) is thoroughly cleaned and treated with an oxygen plasma. PMMA is
spun onto the sample, the pattern is exposed using e-beam lithography, and is
subsequently developed at room temperature. The shapes of the masks were
optimized using shape-correction [35]. A thin layer of gold (20 nm) is
deposited using thermal evaporation. Note that no additional adhesion layer is
used. The lift-off procedure consists of an overnight soak in acetone at
$70^{\circ}$C.

Figure S1: Optical response of isolated gammadion obtained through FDTD
simulations. (a) The polar plot of the farfield scattered intensity as a
function of azimuthal angle for a individual gammadion nanostructure. $I^{{\rm
R}}$, $I^{{\rm L}}$ are the normalized scattered intensity for RCP (solid
blue) and LCP (solid red) excitation. (b) The normalized plots of differential
in-plane scattering, $\Delta I=\left|I^{{\rm R}}-I^{{\rm L}}\right|,$ for 620
nm excitation wavelength. Superimposed on the plots are the isolated gammadion
nanostructure for reference. (c) The variation of $\Delta I$ as a function of
wavelength for the nanostructure, in normalized units.

##  VII Experimental methods for the measurements of asymmetric transmission

For experimentally observing asymmetric transmission, we use a customized
experimental setup operating in a mode similar to leakage radiation microscopy
[36, 37]. We perform measurements in real and Fourier space to detect the
propagating surface modes as well as the diffraction orders. To this end, we
perform back focal plane (k-space) polarimetry [38, 39, 40] with a
polarimetric analysis. We use a broadband supercontinuum light source (NKT
Photonics SuperK Extreme) with wavelength selective filtering accomplished by
an acousto-optical tunable filter in the range of 630 nm – 950 nm (AOTF; Gooch
& Housego AOTFnC-NIR; spectral width $\sim$12 nm) as shown in Fig. 4a.

The spectrally filtered light is fed into a single-mode optical fiber, which
acts as a mode filter. The resulting beam with a near fundamental Gaussian
profile is then further expanded with the help of two lenses. We use a
combination of a broadband polarizer and a quarter-wave plate to control the
polarization of the incoming light. The incoming circularly polarized beam is
focused using a lens with a focal length of 60 mm, resulting in a focal spot
of $\sim$50 $\mu$m in diameter. The fabricated sample consists of quadrumer
arrays with lateral dimensions of 200 $\mu$m by 200 $\mu$m with a periodicity
of $\Lambda=600$ nm. The sample is immersed in oil with refractive index
matching that of the glass substrate (N-BK7), providing a symmetric refractive
index environment for the metasurface. The precise positioning of the sample
at the focal plane was done by using a 3D piezo scanning system. The
transmitted light from the metasurface was collected by an oil-immersion
microscope objective with a numerical aperture (NA) of 1.3. The back focal
plane of this collection lens is then imaged onto a CCD-camera with 12-bit
dynamic range to get access to the angular spectrum (Fourier space) of the
transmitted light using an additional achromatic lens ($f=300$ mm). A sketch
of the experimental setup is shown in Fig. 4a. In the Fourier space image,
transmitted diffraction orders are naturally separated. We record Fourier
space images for incident light of both polarization handedness to extract
asymmetric transmission based on diffraction orders. To extract meaningful
data with respect to the asymmetric transmission from the quadrumer array, the
focused beam must be significantly smaller than the lateral extent of the
array itself to avoid edge effects from the boundaries of quadrumer array and
at the same time sufficiently paraxial to not illuminate the metasurface with
a large angular spectrum of plane waves. In the recorded back focal plane
images (see Fig. 4c), the large intensity in the central angular range,
corresponds to the zeroth diffraction order mode. We also find wavelength-
dependent higher diffraction orders in the back focal plane images. As
expected, the NA of the collection microscopic objective sets an upper bound
with respect to the collectible diffraction orders. For a sample immersed into
index matching oil ($n=1.51$), we define the maximum normalized transverse
wave-vector component as $k_{{\rm max}}/k_{0}=1.3/1.51=0.861$, equivalent to a
maximum collection angle of microscope objective of $60.25^{\circ}$. It is
important to highlight here that we expect to observe certain diffraction
orders depending on the periodicity, the operating wavelength and the
microscope objective collection angle. As mentioned earlier, our experimental
scheme is similar to leakage radiation microscopy in the Fourier domain,
performed for a normally incident weakly focused beam. Due to the angular
limitation of the collection microscope objective, the asymmetric transmission
effects in higher diffraction orders near the wavelength for the Rayleigh
anomaly cannot be observed, as they propagate parallel to the metasurface (in-
plane). Hence, we observe the asymmetric transmission at wavelengths around
the Rayleigh anomaly qualitatively by performing leakage radiation microscopy
in real space in reflection by imaging the metasurface plane. The asymmetric
transmission effect can be observed as a difference in intensity of surface
waves generated by right- and left-hand circularly polarized light. This was
done by placing an imaging lens in the reflection arm and recording the real
space image with a CCD camera.

Figure S2: Optical response of isolated nanostructure obtained through FDTD
simulations. (a) The polar plot of the far field scattered intensity as a
function of azimuthal angle for an individual nanostructure. $I^{{\rm R}}$,
$I^{{\rm L}}$ are the normalized scattered intensity for RCP (solid blue) and
LCP (solid red) excitation. (b) The normalized plots of differential in-plane
scattering, $\Delta I=\left|I^{{\rm R}}-I^{{\rm L}}\right|,$ for 620 nm
excitation wavelength. Superimposed on the plots are the isolated
nanostructure for reference. (c) The variation of $\Delta I$ as a function of
wavelength for the nanostructure, in normalized units.