# Ultra high strength and plasticity mechanisms of Si and SiC nanoparticles
revealed by first principles molecular dynamics

L. Pizzagalli<EMAIL_ADDRESS>J. Godet  Institut Pprime,
CNRS UPR 3346, Université de Poitiers, Poitiers, France

###### Abstract

It is now well established that materials are stronger when their dimensions
are reduced to submicron scale. However, what happens at dimensions such as a
few tens of nanometers or lower remains largely unknown, with conflicting
reports on strength or plasticity mechanisms. Here, we combined first
principles molecular dynamics and classical force fields to investigate the
mechanical properties of 1–2 nm Si and SiC nanoparticles. These compression
simulations unambiguously reveal that the strength continues to increase down
to such sizes, and that in these systems the theoretical bulk strength can be
reached or even exceeded in some cases. Most of the nanoparticles yield by
amorphization at strains greater than 20%, with no evidence of $\beta$-tin
phase for Si. Original and unexpected mechanisms are also identified, such as
the homogeneous formation of a dislocation loop embryo for the $\langle
111\rangle$ compression of SiC nanoparticles, and an elastic softening for the
$\langle 001\rangle$ compression of Si nanoparticles.

Exploring materials at the nanoscale is a successful story with the discovery
of new properties and original phenomena in various domains like phononics,
plasmonics, photonics, etc… In the specific case of nanomechanics, pioneering
works by Taylor and Brenner revealed the surprisingly high strength of various
metals at the micron scale [1, 2]. Later investigations at lower dimensions
[3, 4] confirmed the important finding that the reduction of dimensions
considerably increases the strength, with values converging towards the
theoretical limit [5].

These works raise fundamental and unresolved questions. First, one may wonder
whether strength keeps increasing down to the smallest possible sizes. It has
been tentatively proposed that there is a threshold size below which the
strength appears to be constant [6, 7, 8]. However other studies reported a
weakening at sizes equal to a few tens of nm [9, 10]. A second unknown
concerns the maximum strength value that can be attained at small scale. It is
assumed that the intrinsic strength of the perfect bulk crystal represents an
upper bound. Values close to this theoretical limit were recently reported for
several kind of metal nanoparticles (NP) [6, 8, 11]. However, the maximum
strength for even smaller NP made of other materials like Si was determined to
be significantly lower [7]. These conflicting results clearly call for
investigations of strength at the lowest possible scales.

Other unknown factors are plasticity mechanisms and how they are impacted at
low dimensions. It is notorious that materials with covalent bonding become
ductile at low dimensions [12]. But our understanding of the underlying
plastic deformation mechanisms remains limited and controversial. In silicon,
the most studied covalent material, classical molecular dynamics calculations
(MD) predicted either the heterogeneous nucleation of dislocations [13, 10,
14], or the occurrence of phase transition [15, 16] during the compression of
10–50 nm NP. This controversy is possibly caused by classical potentials,
which are unable to provide an accurate description of small and highly
strained NP [17]. Experiments are scarce and the most comprehensive study
suggests that both phase transition and dislocations could concurrently occur
[7]. At the smallest dimensions, i.e. down to a few nm, the situation is even
more complicated since amorphization becomes another competitive deformation
mode [18]. Further investigations are needed to address these issues.

Current experimental apparatuses allow for studying the plastic deformation of
nanostructures with dimensions as small as a few tens of nm [19, 20]. But it
remains difficult to apply a deformation in a controlled manner at lower
sizes. As previously mentioned, the use of classical MD is questionable
because of the inaccuracy of interatomic potentials. In order to circumvent
this issue and obtain key answers concerning the mechanical properties of NP
at the smallest scales, we applied a recently developed approach combining
first principles MD together with planar repulsive force fields [21], thus
allowing for dynamic compression at finite temperature with first principles
accuracy [22, 23]. This approach is used in the present work to investigate
the mechanical properties of small Si and SiC NP, aiming at answering
fundamental and unresolved questions regarding the strength and plasticity
mechanisms in nanometer-scale materials. Our calculations reveal that the
strength of NP reaches or even exceeds the bulk theoretical limit, which is an
original feat to our knowledge. Incidentally, we also demonstrate that the
strength keeps increasing down to a few nm in the case of Si and SiC, at odds
with previous reports [9, 10]. We find that these NP yield mostly by
amorphization, with no transition to a high pressure crystalline phase.
Finally, our simulations also reveal the unexpected homogeneous formation of a
small dislocation embryo in SiC NP. Yet such a mechanism was customarily
acknowledged to be unfavorable, due to the reduced dimensions or with respect
to heterogeneous nucleation.

Two Si and four SiC NP models, with cuboctahedron or truncated octahedron
shapes and with sizes ranging from 1.1 nm to 1.8 nm, are studied [24]. Car-
Parrinello molecular dynamics (CPMD) calculations are performed using the
Quantum Espresso code [25]. All NP are located in supercells with dimensions
adjusted to ensure a minimum vacuum distance of 10 Å between replicas. The
electronic structure is computed using a 25 Ry plane wave cutoff,
$\gamma$-point sampling, the PBE exchange correlation functional [26], and
ultrasoft pseudopotentials [27]. Other parameters specific to the CPMD method
are similar to the ones used in Ref. [22]. The CPMD timestep is 0.2 fs and the
compression speed is 0.1 Å/ps ensuring a reasonable strain rate [17] but at
the cost of typically $3\times 10^{5}$ ionic iterations. Compression is done
at 300 K in a controlled strain mode along the $\langle 111\rangle$ or the
$\langle 001\rangle$ crystallographic orientations.

Figure 1: Force (nN, blue), energy (eV, red), stress (GPa, orange), and
contact surface (Å2, green) as a function of strain, for a Si79C68 NP
compressed along $\langle 111\rangle$. The displayed contact surface and
stress are obtained using the $S_{1}$ surface definition (see Supp. Inf.).
Dashed lines are linear interpolation of force and stress in the strain range
0.05–0.1 where contact surface is constant. The pink strip shows the strain
region where plastic deformation starts.

We first analyze how properties like energy and force applied on the NP vary
as a function of compression strain $\varepsilon$. Both are directly obtained
from the simulation. The energy averaged before compression is used as the
energy reference. The zero strain is estimated from a linear extrapolation of
the contact force in the elastic regime, with an accuracy at best of $\pm
0.01$. We also determine the contact stress as the ratio of the force to the
contact surface. The latter is inherently prone to uncertainty, a fortiori in
small systems. We used different calculation methods [28] to determine a
meaningful range of possible contact surface area. Accordingly in the
following all stress-related quantities are given as a range of values.

Figure 1 shows results for Si79C68 as an example. At low strains, energy and
force exhibit a quadratic and linear variation, respectively, as expected for
an elastic deformation. The contact surface is constant in the 0.04–0.11
strain range, leading to an almost linear increase of contact stress. The
elastic regime is characterized by a stiffness of 303 N/m, and an elastic
modulus of 411–780 GPa (depending on the surface calculation method) in good
agreement with the $\langle 111\rangle$ bulk SiC value of 541 GPa. The data
and curves for all systems are included in Supplementary Material [29]. It is
found that elastic moduli of NP are close to the corresponding bulk value (for
a given material and orientation). As expected, larger moduli are obtained for
SiC and $\langle 111\rangle$ than for Si and $\langle 001\rangle$. For
$\varepsilon>0.10$, small ripples are observed in the force curves, in
association with weak variations of contact surfaces, while the energies grow
smoothly. Similar observations are made for all studied systems.

Figure 2: Plasticity mechanisms occurring in the Si79C68 NP compressed along
$\langle 111\rangle$ (a), and in the Si123 NP compressed along $\langle
001\rangle$ (b). In (a), pictures on the left show the NP structure at
compression strains of 0.223 and 0.225, the compression axis being the
vertical of the figure (Si in gold and C in black). Pictures on the right show
a flat view of atoms contained in a $\langle 111\rangle$-oriented slice
(limited by the two red lines in the left pictures). Red arrows indicate the
main atomic displacements leading to the formation of a dislocation dipole
(emphasized by the green thick lines). In (b), the left picture displays the
NP structure at a strain of 0.26, with atoms colored according to a PTM
analysis [30] (blue: cubic diamond, orange: hexagonal diamond, white:
unidentified structure), whereas the right graph compares RDF for uncompressed
and compressed NP.

For the system showcased in Fig. 1, the force, energy and stress exhibit large
drops when the strain exceeds 0.21–0.23, suggesting the initiation of plastic
relaxation. Figure 2-a shows the compressed NP just before
($\varepsilon=0.223$) and after ($\varepsilon=0.225$) the activation of the
first plasticity mechanism. The main structural changes concern atoms in a
$\langle 111\rangle$-normal strip, bordered by the two red lines in the
figure. We observe that the stress relaxation mechanism essentially consists
of the concerted motion of four atoms along $\langle 011\rangle$, Si and C
atoms moving in opposite directions (red arrows in the right side of Fig.
2-a). It leads to the formation of a point defect dipole (green lines in Fig.
2-a). Interestingly, this configuration is also equivalent to a dislocation
core dipole embryo with a minimal expansion of the dislocation loop. In fact,
each core defect exhibits the 5/7-ring structure typical of a $1/2\langle
110\rangle$ 60∘ glide dislocation core [31]. An atomic displacement analysis
reveals that the Burgers vector magnitude is about 80% of the value expected
for a dislocation in bulk SiC. This is coherent with previous analyses of the
homogeneous nucleation of a dislocation loop at low dimensions [32, 33]. At
$\varepsilon=0.242$ one of these cores glides in the Burgers vector direction
and annihilates itself at the NP surface, which further confirms the
dislocation nature of the defect. The homogeneous nucleation of a dislocation
loop in such a small volume was unexpected [15] and never reported to our
knowledge, and thus constitutes one of the highlights of the present study. A
similar mechanism is also identified for Si61C61, which suggests it is
specific to the $\langle 111\rangle$ compression of SiC NP.

Table 1: Identified mechanisms and corresponding strains and stresses (GPa) during the compression. Model | Strain | Stress (GPa) | Mechanism
---|---|---|---
Si123 $\langle 001\rangle$ | 0.2 | 19–43 | softening
| 0.26 | 10.7–24 | amorphization
Si148 $\langle 111\rangle$ | 0.195 | 20.6–39 | amorphization
Si71C56 $\langle 001\rangle$ | 0.30 | 64–107 | amorphization
Si80C92 $\langle 001\rangle$ | 0.24 | 76–170 | NP rotation
| 0.30 | 64–116 | amorphization
Si61C61 $\langle 111\rangle$ | 0.25 | 85–146 | dislocation embryo
| 0.33 | 47–82 | amorphization
Si79C68 $\langle 111\rangle$ | 0.21 | 80–135 | dislocation embryo
| 0.37 | 49–72 | amorphization

The structural changes associated with large stress drops in all compressed NP
were all thoroughly examined and the results reported in Tab. 1. In most
cases, Polyhedral Template Matching (PTM) and Radial Distribution Functions
(RDF) analyses reveal that a partial crystal-amorphous transition occurs as
primary or secondary stress relaxation mechanism (Fig. 2-b). This is in
agreement with the generally accepted notion of an improved stability of
disordered phases at the nanoscale [18]. Two other events are noteworthy.
First, we observe a force/stress maximum at 0.20 strain for the $\langle
001\rangle$ compression of Si123, with only a small inflection of the energy
curve (Tab. 1 and Fig. S9). Atomic and electronic structure analyses, detailed
in Supplementary Material, were carried on, which indicate a relatively
homogeneous deformation and the absence of amorphous or $\beta$-tin phases
formation [34]. The important finding is that the onset of this $\langle
001\rangle$ softening would appear as the ultimate yield stress of the NP by
examining the stress–strain curve, although the deformation remains purely
elastic, with the first irreversible event occurring at $\varepsilon=0.26$.
The second interesting event is observed during the $\langle 001\rangle$
compression of Si80C92, at $\varepsilon=0.24$. Compared to the previous case,
the stress reaches a maximum but not the force or the energy (Fig. S7 in Supp.
Inf.). The stress drop is due to an increase of contact surface, which is a
consequence of the disorientation of the NP relatively to the compression axis
(Fig. S12 in Supp. Inf.).

Figure 3: Maximum stress (GPa) versus size (nm) plots for Si (top) and SiC
(bottom), with (a) data from Tab. 1, or compiled from the literature (b) [14],
(c) [16], (d) [10], (e) [15], (f) [35], (g) [36], (h) [37], (i) [38], (j)
[39]. The orange squares show the averages of the four stress values
(depending on surface definition) for each NP. The orange lines run from the
lowest to the highest of these stress values for each NP. The dashed grey
lines show a power fit to the data (see text for details).

We now focus on the highest contact stress $\sigma$ achieved during
compression. In the investigated strain range, $\sigma$ is associated with
primary events as reported in Tab. 1. It is well acknowledged that decreasing
dimensions leads to an increase of strength for sizes down to a few tens of nm
[17]. Conflicting propositions were made for smaller dimensions, with either
constant [6, 7] or decreasing [9] strength when size is reduced. Our maximum
stress values are represented in Fig. 3, together with data from the
literature for other Si and SiC systems. Considering the ranges due to surface
area definitions, it is clear that our values are overall significantly
greater than those computed or measured for larger systems. It suggests that
for Si and SiC, and also likely for other zinc-blende or wurtzite materials,
the strength keeps increasing down to a few nm. Fitting all data with a power
law expression $\beta d^{-\alpha}$, $d$ being the size (Fig. 3), $\alpha$
exponents of 0.16 (0.28) are obtained for Si (SiC). The Si value is close to
0.08–0.11, the estimated exponents for much larger nanopillars [36], and in
the expected range for ceramic-like materials [40]. Note that there is a
debate about the value and meaning of these scaling exponents [41, 42, 43].

Figure 4: Critical resolved shear stress (GPa) versus strain at the first
contact stress maximum (brown squares for Si and blue spheres for SiC). For
each system, the different CRSS values correspond to the stress values
associated with each surface definition. The lower and upper yellow strips
show the ranges of theoretical shear strength, built from extreme values
reported for Si [44, 45] and SiC [46, 47], respectively.

Assuming an homogeneous deformation inside the NP during compression, we
determined the critical resolved shear stress (CRSS) $\tau$, which is the
projected shear stress for a given slip plane and direction and is obtained
using $\tau=m\sigma$, with $m$ the Schmid factor (see Supp. Inf.). $m$ is
computed from atomic positions at each compression step for all systems, for a
$\langle 110\rangle$ slip in {111} planes. Initial values are 0.408 and 0.272
for the $\langle 001\rangle$ and $\langle 111\rangle$ compression
orientations, respectively. At strains associated with the maximum contact
stresses, $m$ slightly increases up to 0.41–0.42 for $\langle 001\rangle$ and
0.30–0.32 for $\langle 111\rangle$. The corresponding CRSS data are shown in
Fig. 4. As expected, CRSS values are greater for SiC than for Si. Also, they
are relatively close for all clusters, except for Si80C92 which appears as an
outlier. Next we compare our computed values with theoretical shear strength
data (TSS) from the literature (Fig. 4). TSS, also called ideal shear stress,
is defined as the maximum stress in a perfect crystal under a uniform shear
deformation. Assuming the upper limit for surface areas ($S_{3}$ in Supp.
Inf.), i.e. considering the lowest strength values for each case, we find that
the computed CRSS are close or only slightly lower than the TSS. This suggests
that in 1–2 nm NP the theoretical shear strength is almost reached. This is
even more remarkable since our calculations are carried out at 300 K while TSS
is a 0 K calculated quantity. Now considering the other surface definitions
described in Supp. Inf., the CRSS values are greater than the TSS in several
cases. This surprising finding might be explained by the large compression
strains, since it is known that normal stresses can significantly change the
TSS [47]. Another rationale is that our original assumption is not correct
i.e., the stress inside the NP is not homogeneous and then locally not equal
to the projection of the compression stress. Finally, quantum confinement
effects might also play a role at small sizes.

Overall our CPMD simulations reveal a rich and unexpected picture of the
mechanical properties of 1-2 nm Si and SiC NP. Very high strain/stress are
required to reach the plastic regime, which is in most cases initiated by
amorphization. We also identified the homogeneous formation of a seemingly
dislocation embryo as first plastic event. This finding conflicts with the
broad consensus that homogeneous dislocation nucleation is prevented in such
small volumes. Another interesting and original finding is the occurrence of a
softening along the $\langle 001\rangle$ orientation in Si associated with a
compression stress maximum, although no phase formation occurs and the
deformation remains elastic. The maximum stress values suggest that the NP
strengths, computed at 300 K, are close to the theoretical strength of the
ideal bulk material at 0 K. It also confirms that for covalent materials the
strength keeps increasing when dimensions are reduced to a few nm. Note that
in this work we investigated $\langle 001\rangle$ and $\langle 111\rangle$
compression orientations, which correspond to the normals of well defined
facets in Wulff-like NP. Compression along other orientations like $\langle
110\rangle$ is likely to lead to NP rotation and reorientation of the
compression axis along $\langle 001\rangle$ or $\langle 111\rangle$. To
conclude, this study opens the way towards a better understanding of
mechanical properties at a previously inaccessible scale. The originality and
importance of our findings is a clear incentive to apply an equivalent
approach to other class of materials, like FCC and BCC metals.

###### Acknowledgements.

The computer time for this work was provided by several sources: the Spin
Center at the University of Poitiers, the MCIA (Mésocentre de Calcul Intensif
Aquitain), and GENCI-CINES (Grant 2020-A0090912035). This work pertains to the
French Government program “Investissements d’Avenir” (EUR INTREE, reference
ANR-18-EURE-0010, and LABEX INTERACTIFS, reference ANR-11-LABX-0017-01).

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