# Atomistic deformation mechanism of silicon under laser-driven shock
compression

Silvia Pandolfi  SLAC National Accelerator Laboratory, 2575 Sand Hill Rd.,
Menlo Park, CA 94025, USA  e-mail<EMAIL_ADDRESS>S. Brennan Brown
SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, CA
94025, USA  P. G. Stubley  Department of Physics, Clarendon Laboratory,
Univeristy of Oxford, Parks Road, Oxford, OX1 3PU, UK  A. Higginbotham
Department of Physics, University of York, Heslington York, YO10 5DD, UK  C.
A. Bolme  Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
H. J Lee  SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo
Park, CA 94025, USA  B. Nagler  SLAC National Accelerator Laboratory, 2575
Sand Hill Rd., Menlo Park, CA 94025, USA  E. Galtier  SLAC National
Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, CA 94025, USA  R.
Sandberg  Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Now at: Department of Physics and Astronomy, Brigham Young University, Provo,
Utah 84602, USA  W. Yang  Center for High Pressure Science and Technology
Advanced Research, HPSTAR, Shanghai, China  W. L. Mao  Geological Sciences,
Stanford University, 367 Panama St., Stanford CA 94305, USA  J. S. Wark
Department of Physics, Clarendon Laboratory, Univeristy of Oxford, Parks Road,
Oxford, OX1 3PU, UK  A. E. Gleason  SLAC National Accelerator Laboratory, 2575
Sand Hill Rd., Menlo Park, CA 94025, USA

###### Abstract

Silicon (Si) is one of the most abundant elements on Earth, and it is the most
important and widely used semiconductor, constituting the basis of modern
electronic devices. Despite extensive study, some properties of Si remain
elusive. For example, the behaviour of Si under high pressure, in particular
at the ultra-high strain rates characteristic of dynamic compression, has been
a matter of debate for decades. A detailed understanding of how Si deforms is
crucial for a variety of fields, ranging from planetary science to materials
design. Simulations suggest that in Si the shear stress generated during shock
compression is released inelastically, i.e., via a high-pressure phase
transition, challenging the classical picture of relaxation via defect-
mediated plasticity. However, experiments at the short timescales
characteristic of shock compression are challenging, and direct evidence
supporting either deformation mechanism remain elusive. Here, we use sub-
picosecond, highly-monochromatic x-ray diffraction to study (100)-oriented
single-crystal Si under laser-driven shock compression. We provide the first
unambiguous, time-resolved picture of Si deformation at ultra-high strain
rates, demonstrating the predicted inelastic shear release. Our results
resolve the longstanding controversy on silicon deformation under dynamic
compression, and provide direct proof of strain rate-dependent deformation
mechanisms in a non-metallic system, which is key for the study of planetary-
relevant materials.

The response of materials to ultra-high pressures has been investigated for
more than a century [1, 2, 3, 4, 5, 6, 7, 8, 9], fostering our understanding
of the fundamental properties of matter as well as the phenomena taking place
at the interior of planets within our own solar system and beyond[10, 11, 12,
13]. Si is one of the most abundant elements in our planet, and it has found
wide-ranging application in the semiconductor industry. Because of its
technological interest, Si properties have been extensively studied, both at
ambient and high-pressure (HP) conditions[14, 15, 16]. However, despite
decades of research, there is no consensus around the deformation mechanism
that drives Si transitions at HP. Precise understanding of these
transformation pathways is of interest for fields ranging from planetary
science[17] to the recovery of novel functional materials for industrial and
energetic applications[18, 19, 20, 21]

Si exhibits a complex phase diagrams, with several HP and metastable
polymorphs[14]. Under static loading, upon increasing pressure Si-I transforms
into metallic $\beta$-tin Si-II, Imma Si-XI, sh Si-V, Cmca Si-VI, hcp Si-VII
and fcc Si-X[22, 23, 24, 25, 26, 27, 28, 29]. Under dynamic loading, Si
displays a complex response that has been a matter of debate for decades[30,
31, 32], with a multi wave profile emerging between 5.4 and 9.2 GPa depending
upon crystal orientation and strain rate. Velocimetry experiments have
identified the wave following elastic deformation with the onset of a plastic
[33, 34] or inelastic[35, 36, 37] regime, followed by a HP phase transition.
Molecular dynamic (MD) simulations have suggested that, for dynamic
compression of Si, HP phase transitions can occur inelastically rather than
via defect-mediated plasticity - i.e., the shear stress is released via the HP
phase transition, not defect motion per se[38]. The activation of a different
deformation mechanism at high strain rates could explain the different phase
boundaries observed under static-[14] and dynamic-[39] compression.

Recent in situ x-ray imaging and x-ray diffraction (XRD) of laser-driven Si
compression have demonstrated that the emergence of the second wave is due to
the onset of the HP phase transition, supporting the inelastic deformation
model proposed by MD[39, 40]. However, these experiments could not inform the
exact mechanism underlying Si structural transitions, as it requires to
establish the specific orientation dependencies between the two phases. In
situ XRD on shock-compressed single-crystal Si using gas gun was performed by
Turneaure et al. to study the relative orientation between the ambient Si-I
and the HP Si-V phase at 19 GPa[41]. However, not all the reflections could be
indexed in terms of the proposed geometry, preventing an unambiguous
interpretation of the experimental data. Thus, the exact deformation mechanism
driving Si phase transitions under dynamic compression and the nature of the
shear release mechanism (i.e., inelastic vs plastic) remain elusive.

Here, we present a detailed, time-resolved XRD characterization of single-
crystal Si(100) under laser-driven shock compression using an ultra-fast X-ray
probe from the x-ray free electron laser (XFEL). The use of a highly
monochromatic XFEL beam combined with single-crystal starting material ensures
high fidelity for the analysis of single-crystal orientation and texture,
which is key to investigate Si deformation mechanism. By analyzing the
preferred orientation of the HP Si-V phase, we are able to determine the
specific transition pathway and to provide the first direct evidence of the
inelastic shear release in this compression regime. Comparison with gas gun
data resolves previous controversies on the nature of Si deformation,
revealing that Si exhibits a complex response to dynamic compression, as it
can deform plastically or inelastically depending on the strain rate.

Figure 1: XRD data showing the structural evolution of Si(100) during shock
compression for two different peak pressures: 12.5$\pm$0.5 GPa (a-c) and
19.5$\pm$0.5 GPa (d-f). (a,d) XRD data projected onto the $2\theta-\phi$ space
acquired at different times for compression at 12.5 GPa and 19.5 GPa,
respectively. The reflections from Si-I are indexed, and the peak position is
indicated by yellow dashed line. The green lines show the position of the
third harmonic ($3\omega$) reflections; (b,c,e,f) azimuthally-integrated data
are reported with different colors depending on the relative delay between the
drive laser and the XFEL pulse; (b),(e) full XRD profile, Si-I peaks are
indicated; (c),(f) zoomed view of the high-pressure phases structure; for each
pattern the estimated pressure is reported, and the HP phase peaks are
indexed.

Experiments were conducted at the Matter in Extreme Conditions (MEC)
endstation of the Linac Coherent Light Source (LCLS)[42, 43]; single-crystal
XRD enabled characterization of the crystal structure and microstructural
changes of Si(100) under laser-driven shock compression. XRD data were
acquired at time delays of 2, 5, 8 and 20 -ns for two compression regimes,
i.e., compression up to 12.5 GPa and 19.5 GPa (Fig.1). Upon compression,
alongside the broad peaks of Si-I, new peaks corresponding to HP phases are
visible (Fig.1(b,e)); these peaks and those of the compressed Si-I were
indexed using whole-profile fit. The coexistence of Si-I and HP structures is
consistent with MD simulations suggesting a mixed phase region[38] and with
recent in situ XRD that reported the presence of Si-I above 19.5 GPa and well
beyond the Hugoniot elastic limit[39, 40]. The intensity of the XRD signal
from the HP phases increases with time, while the peak width decreases,
indicating that the HP crystalline domains grow as the shock wave propagates
through the material (Fig.1(c,f)). At 12.5 GPa, shock compression results in a
mixed HP phase comprised of both Si-XI and Si-V (Fig.1(c)) in agreement with
previous studies[39]. The XRD peaks observed at 2-ns and 5-ns delay fit as
both Si-XI and Si-V reflections yielding consistent densities, but a precise
deconvolution of the two phases is not possible due to peak superposition. At
19.5 GPa the HP structure is consistent with a pure Si-V phase, even if the
asymmetry of the Si-V(010) peaks suggests that small amounts of Si-XI may be
present (Fig.1(f), see also Supplementary Information Sec.5[44]).

We characterized the structural transitions during both compression and
decompression acquiring XRD data up to 20-ns time delay. As inferred from
hydrocode simulations (Supplementary Information, Sec.3[44]), pressure release
starts at 5.5 ns for the experiments at 12.5 GPa ($43\mu$m-thick samples), and
at 13 ns for those at 19.5 GPa ($100\mu$m-thick samples). During release from
12.5 GPa, we observe the crystallization of Si-II at 8 ns time delay, but the
phase is not preserved and it transforms into Si-I upon complete release
(Fig.1(c)). The Si-II formed during decompression has a surprisingly low
density, i.e., the measured atomic volume of 15.14 ${\textup{\AA}}^{3}$/atom
would corresponds to a negative pressure value from extrapolation of Si-II
Hugoniot curve[45]. This suggests the presence of tensile strain, most likely
due to the rarefaction wave generated at the free surface of the sample after
5.5 ns. When releasing from 19.5 GPa, low-density Si-II is observed to be
metastable down to ambient pressure, with a measured atomic volume of 15.16
${\textup{\AA}}^{3}$/atom (Fig.1(f)). Despite the ultra-high strain rate,
which, according to recent DAC experiments[46], should favour amorphization,
we mainly observe Si-I recrystallization upon decompression. Most likely, the
crystal-crystal phase transitions are favoured by the high temperature
conditions during isentropic pressure release. It is interesting to notice
that none of the metastable phases that have been synthesized in static
compression experiments (e.g., Si-III[19] or a-Si[46]) was observed. The
extended metastability of metallic Si-II at ultra-high strain rates presents
opportunities for future recovery experiments that could extend the
capabilities of Si HP synthesis. Indeed, the quench and recovery of HP phases
from laser-driven shock compression has been demonstrated for the $\omega$
phase of zirconium[47]; our results suggest that this experimental approach
may be applied also for the recovery of metastable metallic Si-II.

Figure 2: Schematic view of the orientation relationship. (a) Preferential
orientation for the nucleation of Si-II (orange) from shock-compressed Si-I
(blue): $[001]_{II}//[100]_{I}$, i.e,. the compression direction. (b)
Perpendicularly to the shock: Si-I (blue) and 4 equivalent Si-II orientations,
each of which is indicated with a different color. (c) Orientation
relationship between the ambient pressure Si-I (blue) and the HP Si-V phase
(red), for which two non-equivalent domains, Si-$V_{0^{\circ}}$ and
Si-$V_{90^{\circ}}$ are indicated.

We characterize the relative orientation of the ambient- and high-pressure
phases to gain new insight into the atomistic deformation mechanisms that
drive Si phase transitions. Indeed, a given deformation mechanism will result
in a specific orientation relationship, which can be identified in dynamic
compression experiments because shock-compressed materials, being inertially
confined, typically exhibit a lower mosaicity compared to static loading. In
this study, the use of a highly monochromatic XFEL beam combined with single-
crystal starting material ensures an ultra-fast probe for time-resolved study
and high fidelity for single-crystal characterization. Upon compression, the
XRD signal intensity is not uniform along the Debye-Scherrer rings, confirming
the low mosaic spread and suggesting that the HP crystallites grow along
preferred directions (Fig.1(a,d)). In particular, Si-I(220), Si-V(001) and
Si-V(010) reflections are centered around the same $\phi$ angle (Fig.1(d)).
The observed relative orientation was reproducible and obtained under
different compression regimes, suggesting that it is intrinsic to the
transition mechanism (Supplementary Information, Sec.4 [44]). If dynamic
compression resulted in the growth of single-crystal HP phases, Si-V(001) and
Si-V(010) reflections would appear at 90∘ in reciprocal space, as they
correspond to perpendicular planes. Thus, Si-V reflections are from two
distinct crystalline domains with highly reproducible relative orientation.
With the new insights from our XFEL-based experimental approach, we have
analyzed several potential orientation relationships, including those proposed
by Turneaure et al.[41], to explain the highly preferred orientation of Si HP
phases under dynamic compression.

Figure 3: Schematic view of our experimental configuration and deformation
mechanism. The XRD pattern calculated using the proposed orientation
relationship between Si-I and Si-V fits our experimental data (right upper
panel).

We tested the specific orientation relationship dictated by inelastic shear
release in Si(100) to verify the deformation mechanism proposed by MD
simulations[38]. The initial elastic compression occurs uniaxially along
$[100]_{I}$, i.e., the [100] direction of the Si-I cubic crystal, generating
shear stress in the sample. In order for the system to reach the hydrostat
inelastically, as part of the sample changes structure to Si-II, the remaining
Si-I must experience a compressive stress perpendicularly to the shock in the
$(011)_{I}$ plane. For the Si-I$\rightarrow$Si-II transition, this is possible
if the tetragonal HP phase crystallizes along a specific orientation:
$[001]_{II}//[100]_{I}$, i.e., the c axis of the HP phase, with shorter inter-
atomic distances, is parallel to compression axis (Fig.2(a)). Perpendicularly
to the shock, in the $(011)_{I}$ plane, Si-II inter-atomic distances are
higher, as shown in Fig.2(b), where the basis vectors of Si-II are rotated
through an angle of 45∘ with respect to the cubic Si-I cell. As a result, if
the transition happens with this specific orientation relationship, the
relaxation toward the hydrostat (i.e., release of shear stress) is caused by
the transition to the highly-oriented HP Si-II phase, and it does not require
the generation or motion of crystallographic defects. It is worth noting that
this Si-I - Si-II orientation relationship has also been investigated by ab
initio calculations, and found good agreement with static compression
experiments[48]. Because of the four-fold symmetry of the tetragonal phase,
there are 4 equivalent crystalline orientations for Si-II in the $(011)_{I}$
plane (Fig.2(b)). As Si-II transforms into Imma Si-XI and hexagonal Si-V via
continuous deformation, the crystal structure is distorted and the four-fold
degeneracy is lost ($a_{II}=b_{II}\rightarrow a_{XI}\neq b_{XI}$). The
transition to Si-V ultimately results in the formation of crystalline domains
with two non-equivalent orientations, as shown in Fig.2(c): (i)
Si-$V_{0^{\circ}}$, with $[110]_{V}//[100]_{I}$ and
$[001]_{V}//[0\overline{1}1]_{I}$; (ii) Si-$V_{90^{\circ}}$, with
$[110]_{V}//[100]_{I}$ and $[001]_{V}//[011]_{I}$. The single-crystal XRD
pattern calculated using this orientation relationship fits our experimental
data well (Fig.3), while the other geometries we tested could not explain our
results. We are thus able to unambiguously interpret our XRD data in terms of
a specific deformation mechanism. The sequence of HP phase transitions under
uniaxial shock compression results in two non-equivalent Si-V orientations at
an angle of 90∘, which explains the observation of both Si-V(001) and
Si-V(100) around the same $\phi$.

Figure 4: Microstructure analysis. (a,d) $\phi$ lineouts for Si-V(001) and
Si-V(01) peaks at different times. Experimental data are reported and
overlapped with peak fitting results (continuous line and FWHM values form
fit).(b,c,e,f) time evolution of peak width along $2\theta$ and $\phi$ for
both analyzed peaks.

We analyzed the XRD peak shapes to gain insight on the microstructure of
shock-compressed Si, e.g., changes in mosaicity, grain size, crystallographic
defects. The Si-V(001) and Si-V(010) reflections observed around the same
$\phi$ value necessarily come from two distinct crystal domains, which
prevents analysis of the microstructure via whole-profile refinement (e.g., Le
Bail); therefore, we used single-peak fitting projecting the XRD spots along
both the $2\theta$ and $\phi$ axis (Fig.4). The Full Width at Half Maximum
(FWHM) along $\phi$ gives an estimate of the disorder of and within the
crystalline domains, i.e., a combination of the mosaicity (the angular spread
of the orientations of the crystallites[49]) and rotation of the lattice
planes[50] (Fig.4(a,d)). The $\phi$ lineouts of Si-V peaks at 19.5 GPa show
that the peak width along $\phi$ increases with time (Fig.4(c,f)) in
conjunction with the growth of the HP crystallites, i.e., with the decrease of
the $2\theta$ FWHM (Fig.4(b,e)). At 12.5 GPa we observe a similar trend, and
the mosaicity increase results in a powder-like texture when Si-II is formed
during decompression (Fig.1(a)). In this regime, the XRD peaks are generally
broader and less intense, preventing a quantitative analysis of mosaicity. XRD
analysis suggests thus nucleation of highly-oriented HP crystalline domains,
with a gradual loss of preferred orientation as the crystalline domains grow.
These trends are consistent with the model of HP nano-crystalline domains
originating from shear bands along specific crystalline direction, e.g., the
Si-I[111] suggested by MD[38]. It is worth noting that previous ex situ
characterization of shock-compressed Si(100) also showed that amorphous bands
originate along {111} slip planes, and, as they grow, they tend to deviate
from this orientation and align with the direction of maximal shear (i.e..
35.3∘ with shock direction in Si(100))[51, 52].

Figure 5: Analysis of the inelastic and plastic deformation regimes, and their
XRD signature. (a,c) schematic view of the two deformation mechanisms. (b,d)
corresponding XRD signature from forward calculations using the orientation
relationship proposed in this work and a pink beam (Supplementary Information,
Sec.6[44])

Comparison with previous gas-gun experiments provide additional insights into
Si deformation under dynamic compression. The differences between our results
and previous experiments by Turneaure et al.[41] could be due to the specifics
of the gas gun experiment, e.g., the use of a pink beam, or by the lower
strain rate of gas-gun compression, that may activate different deformation
mechanisms[37]. In order to perform a direct comparison with the data by
Turneaure et al., we performed forward XRD calculations using the orientation
relationship proposed in this work and the experimental parameters of previous
gas-gun studies (for details, see Supplementary Information, Sect.6[44]). XRD
patterns were simulated for both inelastic and plastic deformation. As shown
by the schematics in Fig.5(a), inelastic deformation is characterized by shear
release via a phase transition, followed by the generation of defects and
texture evolution; the growth of the HP phase happens only along a specific
orientation, resulting in highly localized XRD signal (Fig.5(b)). Instead,
when the sample deforms plastically, the system relaxes toward the hydrostat
via generation and motion of defects before the phase transitions take place
(Fig.5(c)). As the compression is not uniaxial, the HP phase can crystallize
isotropically, which results in the additional visible reflections shown in
Fig.5(d), where the XRD signal was calculated including all the equivalent
variants of the given orientation relationship. Thus, our calculations
demonstrate that inelastic and plastic deformation result in different
crystalline geometries, with clearly distinguishable XRD signatures, which
explains the differences between our results and previous XRD studies[41]. At
lower strain rates, like in gas gun experiments, shock-compression results in
defect-mediated plasticity, while the inelastic deformation characteristic of
laser-ablation experiments, induces a strongly preferred orientation for the
growth of the HP phases. With the new insight provided by our time-resolved
single-crystal analysis, we can explain the discrepancies in previous studies
on shock-compressed Si, resolving a long standing debate.

In this study, we propose the first unified model for Si deformation under
shock-compression, which has been the matter of vigorous debate for decades.
Time-resolved XRD is used to characterize, in situ, the phase transitions of
single-crystal Si(100) under laser-driven shock compression. With this
experimental approach, we provide the first unambiguous, atomistic picture of
the deformation mechanism driving phase transitions under laser-driven
compression. We observe crystallization of the Si-V HP phase above 12 GPa, and
provide evidence of extended metastability of metallic Si-II upon release. The
use of a monochromatic XFEL probe with sub-picosend temporal resolution allows
us to determine the orientation relationship between the Si-V and Si-I phases,
connecting it with the specific deformation mechanism and confirming the
inelastic shear release predicted by MD simulations[38]. Analysis of the
microstructural evolution demonstrates that at ultra-high strain rates, the HP
Si phases form along specific directions with a loss of preferred orientation
and generation of defects during crystal growth. Crucially, comparison with
previous gas gun experiments demonstrates that at different strain rates the
deformation mechanism and the geometry of the phase transition can be
substantially different, resolving the ongoing debate around shock-compressed
Si.

## Methods

Single-crystal, double-side parallel-polished platelets of Si(100) (either
43$\mu$m or 100$\mu$m thick) were mounted such that the starting orientation
with respect to the probe direction and the shock propagation direction could
be tracked. The LCLS delivered 60-fs duration quasi-monochromatic X-ray pulses
of energy 7.952 keV ($\Delta{E}$ = 15-40 eV[53], $\Delta{E}/E=0.2\textendash
0.5\%$). The XRD signal was recorded on two Cornell-SLAC Pixel Array Detectors
covering range of scattering angle $20^{\circ}<2{\theta}<70^{\circ}$ and
azimuth $-25^{\circ}<\phi<25^{\circ}$[43]. In this experimental geometry, no
XRD signal from the single-crystal starting material is observed for the
monochromatic 7.952 keV X-ray probe; spots from Si-I are visible only from
crystallites in appropriate orientation and upon compression, when mosaicity
and strain increase. However, additional spots due to the parasitic third-
harmonic beam are always visible and allow us to evaluate changes in the
orientation of the starting material. Shock compression was achieved by direct
irradiation of the sample by a 527-nm laser with a quasi-flattop pulse profile
of 10-ns duration [54] with 1$\times 10^{12}$ $W/{cm}^{2}$ intensity. We
collected data at time delays up to 20 ns, which enabled characterization of
the transformations taking place during both compression and release, i.e.,
after the shock wave has traversed the sample. Pressure was estimated from the
experimental density obtained from XRD and the multi-phase equation of state
of Si[45]. Velocimetry data obtained using the velocimetry interferometry
system for any reflector (VISAR) diagnostic and hydrocode simulations were
used to confirm pressure and strain-rate estimation (Supplementary Information
Sec.2-3[44]). By varying the time delay between the drive laser and the XFEL
beam, time-resolved XRD data were collected at different pressures. For more
details, see Supplementary Information Sec.1[44].

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## Acknowledgements

This work was performed at the Matter at Extreme Conditions(MEC) instrument of
LCLS, supported by the US DOE Office of Science, Fusion EnergyScience under
contract No. SF00515, and was supported by LCLS, a National UserFacility
operated by Stanford University on behalf of DOE-BES. AEG and WLM are
supported in part by the Geophysics Program at NSF (EAR0738873) with
additional support from LANL Reines LDRD for AEG. AEG and SP also acknowledge
support from 2019 DOE FES ECA. JSW is grateful for support from the UK EPSRC
under grant number EP/S025065/1.

## Author contributions

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