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Dislocation behavior in silicon crystal induced by laser shock peening: A
multiscale simulation approach
Article in Scripta Materialia · November 2005
DOI: 10.1016/j.scriptamat.2005.07.014
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Scripta Materialia 53 (2005) 1013–1018
www.actamat-journals.com
Dislocation behavior in silicon crystal induced by laser shock
peening: A multiscale simulation approach
G.J. Cheng *, M.A. Shehadeh
School of Mechanical and Materials Engineering, Washington State University, Pullman 99163, WA, United States
Received 4 February 2005; received in revised form 28 June 2005; accepted 10 July 2005
Available online 5 August 2005
Abstract
This paper presents a modified laser shock peening (LSP) technique to generate plastic deformation in silicon crystal. The
induced dislocation activity in silicon crystal is studied by integrated shock dynamics and multiscale dislocation dynamics plasticity
simulations. This LSP process can also be applied to other brittle materials to generate plastic deformation.
2005 Published by Elsevier Ltd. on behalf of Acta Materialia Inc.
Keywords: Laser shock peening; Silicon; Dislocation dynamics simulation; Hydrodynamics simulation; Brittle materials
1. Introduction
The response of solid to shock compression has been
a topic of interest for more than a century. Nowadays,
laser shock peening (LSP) is widely used for improving
the mechanical properties of metals and getting information about the behavior of materials under high dynamic
pressure. However, it has been always a challenge to
generate plastic deformation in brittle materials through
LSP. The deformation process in silicon has attracted
much interest in the semiconductor industry, as silicon
is the dominant constitutional material for integrated
circuits and micro-machines. In this paper, we will investigate the plastic deformation in silicon crystal generated
by a modified LSP technique through integrated shock
dynamics and multiscale dislocation dynamics plasticity
(MDDP) simulations.
Plastic deformation in crystalline solids is largely
dominated by dislocation nucleation, multiplication
and motion. Dislocations are line defects that move on
*
Corresponding author. Tel.: +1 509 335 7711; fax: +1 509 335
4662.
E-mail address: cheng1@wsu.edu (G.J. Cheng).
certain planes and directions determined by their atomic
arrangement. The difficulties of generating plastic deformation in silicon crystal by LSP include low dislocation
mobility, and not enough shock pressure to move dislocation. At low temperatures, dislocations in silicon
exhibit a very low mobility as a result of the covalent
bonding that gives rise to a high barrier to dislocation
motion [1]. As a result, the dislocations display no activities at low temperature even at high shock pressure.
Loveridge-Smith et al. [2] conducted laser shock peening
experiments on silicon crystals and used X-ray diffraction with sub-nanosecond resolution to measure the lattice parameters. It was found that the response of silicon
crystal to high shock pressure is purely elastic. However,
it has been shown that silicon displays a ‘‘metal-like’’
stress–strain relationship, and high dislocation mobility
at elevated temperature [3–6]. The intrinsic velocity of a
single dislocation has been measured over a range of
temperatures and stresses [7]. The calculations of Cai
et al. [1] showed that dislocation mobility of silicon crystal increases exponentially when temperature is higher
than 850 K.
This paper presents a modified LSP technique in
order to generate dislocation activity in silicon crystal.
1359-6462/$ - see front matter 2005 Published by Elsevier Ltd. on behalf of Acta Materialia Inc.
doi:10.1016/j.scriptamat.2005.07.014
1014
G.J. Cheng, M.A. Shehadeh / Scripta Materialia 53 (2005) 1013–1018
solid target material and the confining medium, respectively. Defining Z = 2/(1/Z1 + 1/Z2) and assuming a
constant fraction a of internal energy is used to increase
the thermal energy of the plasma, the following relations
between shock pressure P(t) and plasma thickness L(t)
can be derived:
dLðtÞ 2P ðtÞ
¼
dt
Z
2
Z
3
dLðtÞ
3Z
d2 LðtÞ
þ
LðtÞ
þ
¼ I p ðtÞ
2 4a
dt
4a
dt2
The schematic of the process is shown in Fig. 1. A short
pulsed excimer laser (k = 248 nm) is used to generate
ablation of the coating material. The silicon target is
heated above 850 K to increase its dislocation mobility.
SiO2 is selected as the confining medium because of its
transparency to excimer laser and high melting point.
In order to understand this process, shock dynamics
and MDDP simulations are conducted here to investigate dislocation activity in silicon crystal in response
to high strain rate, and high pressure.
2. Simulation background and methodology
The simulation consists of two parts. A coupled laser
radiation and hydrodynamic simulation is conducted to
obtain shock wave history. The shock wave is then
applied to a MDDP code to simulate the dislocation
activities.
2.1. LSP physics and MDDP modeling
2.1.1. Hydrodynamic simulation
Different models applicable to LSP have been used
over the years [8,9]. Fabbro et al. [9] developed a model
for LSP assuming that laser irradiation is uniform, and
shock propagation in both the confining medium and
the target is one-dimensional. In this paper, the 1-D
model is applied assuming that the laser beam size is
much larger than the coating thickness. When the plasma is formed at the interface of the solid and the confining medium, its volume expands, its pressure increases
and shock waves propagate into the sample and the confining medium. A portion of the incident laser intensity
I(t) is absorbed by the plasma as
I p ðtÞ ¼ AP ðtÞIðtÞ;
ð1Þ
where AP(t) is the absorption coefficient and t is time.
The shock wave impedance is expressed as Zi = qiDi,
i = 1, 2, where q is the mass density and D is the shock
propagation velocity. The subscripts, 1 and 2, denote the
ð3Þ
Dependence of shock pressure on laser intensity is calculated and shown in Fig. 2 where the laser intensity varies
from 1.33 to 6 GW/cm2 while a is kept as 0.2 and the
pulse duration is 30 ns.
2.1.2. MDDP
In this paper, a multiscale dislocation dynamics
model is used to investigate laser shock peening in silicon single crystals. This model merges two length scales,
nano- microscale and continuum scale. In the nanomicroscale, dislocation dynamics (DD) analyses are
used to determine the plasticity of single crystals by
explicit three-dimensional evaluations of dislocations
motion and interaction among themselves and other defects that might be present in the crystal. In DD, dislocations are discretized into segments of mixed character.
The dynamics of the dislocation is governed by a ‘‘Newtonian’’ equation of motion, consisting of an inertia
term, damping term, and driving force arising from
short-range and long-range interactions. In the continuum scale, energy transport is based on basic continuum
mechanics laws. The result is a hybrid elasto-viscoplastic
simulation model coupling discrete DD with finite
element analysis (FE). More details of the model can
tp = 30ns hCu = 0.2mm
28
I0 = 1.33GW/cm2
24
Shock pressure (GPa)
Fig. 1. Schematic model of laser shock processing.
ð2Þ
I0 = 2GW/cm2
I0 = 3GW/cm2
20
I0 = 4GW/cm2
16
I0 = 6GW/cm2
12
8
4
0
-4
-30
0
30
60
Time (ns)
90
120
Fig. 2. Shock pressure history under five laser intensity levels with
pulse width of 30 ns and copper film thickness of 0.2 mm.
G.J. Cheng, M.A. Shehadeh / Scripta Materialia 53 (2005) 1013–1018
be found in a series of papers by Zbib and co-workers
[10,11].
As mentioned above, the velocity v of a dislocation
segment ÔsÕ is governed by a first order differential equation consisting of an inertia term, a drag term and a
driving force vector such that
1
1 dW
ms v_ þ
v ¼ F s with ms ¼
ð4Þ
M s ðT ; pÞ
v dv
F s ¼ F Peirels þ F D þ F Self þ F External þ F Obstacle
þ F Image þ F Osmotic þ F Thermal
ð5Þ
In the above equation the subscript s stands for the segment, ms is defined as the effective dislocation segment
mass density, Ms is the dislocation mobility which could
depend both on the temperature T and the pressure p,
and W is the total energy per unit length of a moving
dislocation (elastic energy plus kinetic energy). As implied by (5), the glide force vector Fs per unit length
arises from a variety of sources described above. The
following relations for the mass per unit dislocation
length have been suggested by Hirth et al. [12] for screw
(ms)screw and edge (ms)edge dislocations when moving at
a high speed:
W0
ðc1 þ c3 Þ
v2
W 0C2
¼
ð16cl 40c1
l
v4
3
þ 8cl þ 14c þ 50c1 22c3 þ 6c5 Þ
ðms Þscrew ¼
ðms Þedge
1
ð6Þ
1
where cl ¼ ð1 v2 =C 2l Þ2 ; c ¼ ð1 v2 =C 2 Þ2 ; C l is the longitudinal sound velocity, C is the transverse sound
2
velocity, m is PoissonÕs ratio, W 0 ¼ Gb
ln ðR=r0 Þ is the rest
4p
energy for the screw per unit length, G is the shear modulus. The value of R is typically equal to the size of the
dislocation cell, or in the case of one dislocation is the
shortest distance from the dislocation to the free surface.
Under high strain rate loading, the elastic properties
of metals depend on the applied pressure [8,9]. In order
to better simulate the physical problem, the effects of
pressure on the elastic constants and PoissonÕs ratio
are taken into account in our constitutive equation.
2.2. Simulation setup
The simulated matrix is a single crystal silicon, having
a face-centered cubic structure, of a block 10,000b ·
10,000b · 100,000b, where b = bSi = 0.384 nm is the
magnitude of a Burgers vector. In the simulations,
eleven Frank–Read dislocation sources of 1000b in
length were used (see Fig. 1 of crystal in 3-D and front
views). Given the volume of the simulated block, this
contains an initial density of Frank–Read sources of
1E10m2, which is a normal initial dislocation density
in a commercial silicon single crystal.
1015
Since the DD simulations are very intensive computationally, the computation cell selected in this study is in
the microscale. It is reasonable to assume that the scaling down from macroscale to microscale does not affect
the study of the interaction between the shock wave and
dislocation sources, and between the dislocations.
Although the shock wave only propagates for 38.4 lm
(100,000b), it is reasonable to assume that the dislocation only interacts with other dislocations within this
100,000b range. Therefore, statistically, the macrobehavior of the dislocation structure, such as plasticity
and hardening effect, can be simulated within the microscale computation cell. A similar multiscale approach
has been applied to study the generation and propagation of dislocation [10,11].
Our system of axes is taken such that x, y and z axis
are coincident with [1 0 0], [0 1 0] and [0 0 1]. In order to
achieve the uniaxial strain involved in shock loading,
the four sides are confined so that they can only move
in the loading direction. The bottom surface is rigidly
fixed. A velocity-controlled boundary condition (vp)
with finite rise time (trise) is applied on the upper surface
in the [0 0 1] direction for a short period of time (t*) to
generate the shock wave. The upper surface is then released and the simulations continue for the transmitted
elastic wave to interact with the dislocationsÕ sources.
Periodic boundary condition of dislocation lines is used
in the representative volume element to ensure both the
continuity of the dislocation curves and the conservation
of dislocation flux across the boundaries. The value of
dislocation mobility is calculated from the dislocation
velocity results from Cai et al. [1] and Eq. (4).
3. Results and discussion
3.1. Dislocation evolution
The application of high rate uniaxial strain loading
condition results in the propagation of shock waves that
advance in the material. Fig. 3 presents snapshots of
dislocation–shock wave interactions in silicon. These
snapshots show that dislocations start to emit from the
sources when the wave impact these sources. As the
wave progresses into the material, the dislocations
arrange themselves in certain morphologies as time
evolves.
3.2. Effects of laser processing conditions
3.2.1. Laser power intensity
Shock wave histories at five laser intensity levels with
pulse duration of 30 ns and copper film thickness of
0.2 mm is shown in Fig. 2. The laser intensity has to
be in a certain range to generate plasma and avoid
breakdown in SiO2. For the laser pulse width of 30 ns,
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G.J. Cheng, M.A. Shehadeh / Scripta Materialia 53 (2005) 1013–1018
Fig. 3. Dislocation dynamics simulation results of dislocation structure (mobility = 0.052/pa s1, strain rate = 5E6 s1, pulse width = 10 ns).
it is found in this simulation that there is no plasma generated when the laser intensity is less than 1.1 GW cm2.
It has been reported that the breakdown threshold
intensity of fused silica is 9 GW cm2, at which the
absorption of the laser mainly occurs inside the SiO2
glass [13]. In this study, the laser intensity is chosen to
vary from 1.33 to 6 GW cm2. It is seen from Fig. 2 that
the peak shock pressure increases with laser intensity. It
is noticed that the shock pressure can reach 16–26 GPa,
while the shock pressure using water as the confining
medium is in the range of 1.7–3.5 GPa using the same
laser intensities [14]. The shock pressures are increased
by 7–10 times. There are two reasons for this increase.
First, the confining medium is SiO2 glass instead of
water. SiO2 glass has much higher atomic weight and
solid density than water. SiO2 glass also has higher
breakdown laser intensity than that of water, which is
4 GW cm2 for laser wavelength of 355 nm [15]. The
strain rates caused by the shock pressures at different
laser intensities are plotted in Fig. 4. Similar to shock
pressure, the strain rate is found to be proportional to
the square root of laser intensity. The rate of dislocation
multiplication increases with strain rate because dislocation velocity is proportional to glide force vector Fs
Peak strain rate (1/s)
3.0x10
dislocation density rate (d /dt)
6x1021
strain rate ( )
5x1021
2.5x106
4x1021
2.0x106
1.5x10
6
1.0x10
6
3x1021
2x10
21
1x10
21
Dislocation density rate (1/m 2.s)
7x1021
6
0
1
2
3
4
5
(Eq. 4), which increases with shock pressure. The dislocation density rate increases from 1.3E20 m2 s1 to
6.1E21m2 s1 with strain rate increasing from 1E6 s1
to 3.1E6 s1 as illustrated in Fig. 4. MDDP calculations
show that the dislocation density increases from
2.86E11m2 to 5.75E12m2 for shocks launched at different intensities ranging from 1.33 GW/cm2 to 6 GW/
cm2 for the same pulse duration of 15 ns. However, the
dislocation microstructures appeared to be similar [16].
3.2.2. Pulse duration
Shock pulse duration is another important parameter
in controlling the microstructure of dislocations. Pulse
duration is related to the time required for dislocations
to reorganize. From the shock dynamics simulation, it
is found that the peak shock pressure keeps constant
around 15 GPa with five laser pulse widths from 10 ns
to 50 ns, at laser intensity of 2 GW/cm2 and copper film
thickness of 0.2 mm. This is consistent with results from
Devaux et al. [13], which showed that peak pressure is
independent of pulse width. However, as the pulse duration increased from 10 to 50 ns, the strain rate decreased
from 2.1E6s1 to 1.2E6s1. Consequently the rate of
dislocation multiplication decreased as shown in
Fig. 5. The decrease in the values of strain rate and
dislocation multiplication with pulse duration can be
attributed to the increase in the wave rise time and the
time for reorganization of dislocations. The dislocation
morphologies at different pulse durations show dislocation entanglements and deformation bands coincident
with the slip planes. Plastic deformation occurring as a
result of fast moving dislocations is often localized in
a narrow band-like structure [17]. The dislocation
density within these bands seems to correlate inversely
with the pulse duration.
3.3. Effects of ablative coating conditions
6
Laser intensity (GW/cm2 )
Fig. 4. Effect of laser intensity on the peak strain rate and dislocation
density rate (at t = 15 ns) in silicon crystal with pulse width of 30 ns
and copper film thickness of 0.2 mm.
3.3.1. Coating materials
Selecting the coating material for the target is a very
important matter for LSP. In most LSP, a coating is
used to protect the target from thermal effects so that
G.J. Cheng, M.A. Shehadeh / Scripta Materialia 53 (2005) 1013–1018
1017
Fig. 5. Effect of pulse duration on dislocation multiplication rate and
dislocation structure (laser intensity is 2 GW/cm2, hCu is 0.2 mm,
silicon target temperature is 1073 K).
Fig. 6. Effect of copper film thickness (hCu) on peak strain rate and
dislocation structure with pulse width of 30 ns (silicon target temperature is 1073 K).
nearly pure mechanical effects are induced. The coating
materials could be metallic foil, organic paints or adhesives. These coatings can modify the surface loading
transmitted to the substrate by acoustic impedance mismatch effects at the coating–substrate interface, and an
additional 50% increase in the peak stress values can
be achieved. In this study, copper is selected as the material for the coating thin film instead of aluminum for the
LSP because the silicon substrate is heated above 873 K
and copper has a much higher melting temperature than
aluminum. From shock dynamics simulation, the peak
shock pressures are 16.5 GPa and 22 GPa, respectively,
for aluminum and copper thin films with the same conditions (laser intensity of 4 GW/cm2, pulse duration of
30 ns and thin film thickness of 0.2 mm). As a result,
the strain rate using copper is higher than when using
aluminum as the coating material. The dislocation structures using the two coating materials are similar, except
that the dislocation density is higher with copper than
with aluminum as the coating material. Dislocation density history shows that multiplication of dislocation is
faster with copper than with aluminum as the coating
material.
with the film thickness to a certain value after which the
strain rate becomes invariant. This suggests the existence
of a saturation thin film thickness (hCu ) beyond which the
peak pressure (strain rate) becomes thickness independent. This implies that we can limit the amount of the
coating material to hCu under certain laser intensity. With
thicker coating material, the amount of generated plasma does not increase. For laser intensities of 2, 3, and
4 GW/cm2 the values of hCu are 0.12, 0.18, and 0.3 mm,
respectively. This increase in hCu can be related to the increase in the amount of generated plasma with laser
intensity. From MDDP simulation, it is found that when
hCu is less than hCu the dislocation multiplication is faster
with increasing coating thin film thickness. The dislocation densities make no difference with coating thicknesses larger than hCu [15].
We conclude by emphasizing that the present work is
the first attempt to theoretically show that strong dislocation activity can be generated in silicon crystal by the
LSP process presented, by which dislocation mobility of
silicon and shock pressure is sufficiently high to generate
and transport dislocation. An integrated shock dynamics and MDDP simulations have been conducted to predict the response of silicon crystal to LSP. It is found
that dislocation density, multiplication rate, and microstructure are strongly dependent on laser processing
conditions and coating conditions. This LSP process
can also be applied to other brittle materials to generate
plastic flow.
3.3.2. Coating thickness
Selection of an appropriate coating thickness is also
important for LSP of silicon. As the laser beam passes
through the SiO2 glass, the laser energy is absorbed by
the coating material and a thin layer of this sacrificial
material is vaporized. Moreover, as the generated plasma
continues to absorb the rest of the laser energy, it is readily heated and its expansion between the target and the
confining medium is retarded, thus generating higher
pressures than in the direct ablation mode. The effect
of copper thin film thickness (hCu) on peak strain rate
is shown in Fig. 6. As illustrated, the strain rate increases
References
[1] Cai W, Bulatov VV, Justo JF, Argon AS, Yip S. Phys Rev Lett
2000;84(15):3346.
[2] Loveridge-Smith A, Allen A, Belak J. Phys Rev Lett
2001;86(11):2349–52.
1018
G.J. Cheng, M.A. Shehadeh / Scripta Materialia 53 (2005) 1013–1018
[3] Siethoff H. J Phys 1983;C4:217–25.
[4] Siethoff H, Ahlborn K, Schroeter W. Philos Mag A
1984;50(1):L1–6.
[5] Rabier J, Demenet JL. Phys Stat Sol B 2000;222(63):63–74.
[6] Yang EH, Fujika H. Jpn J Appl Phys 1999;38:1580–3.
[7] Schröter W, Brion HG, Siethoff H. J Appl Phys
1983;54(4):1816–20.
[8] Clauer AH, Holbrook JH. Shock waves and high strain phenomena in metals—concepts and applications. New York
(NY): Plenum; 1981. pp. 675–702.
[9] Fabbro R, Fournier J, Ballard P, Devaux D, Vinmont J. J Appl
Phys 1990;68(2):775.
[10] Zbib HM, de la Rubia TD. Int J Plast 2002;18:1133–63.
View publication stats
[11] Zbib HM, Shehadeh M, Khan SM, Karami G. Int J Multiscale
Computat Eng 2003;1:73.
[12] Hirth JP, Zbib HM, Lothe J. Model Simul Mater Sci Eng
1998;6:165.
[13] Devaux D, Fabbro R, Tollier L, Bartnicki E. J Appl Phys
1993;74(4):2268.
[14] Berthe L, Fabbro R, Peyre P, Bartnicki E. J Appl Phys
1999;85:7552.
[15] Zhang W, Yao YL. ASME Trans J Manuf Sci Eng
2002;124(2):369.
[16] Cheng GJ, Shehadeh MA. Int J Plast, submitted for publication.
[17] Shehadeh MA, Zbib HM, de la Rubia TD. Philos Mag
2005;85(15):1667–85.