Estimation and Validation of Mechanical Properties of Single Crystal
Silicon by Atomic- Level Numerical Model
Chun-Te Lin, Chan-Yen Chou, and Kuo Ning Chiang
Advanced Microsystem Packaging and Nano-Mechanics Research Laboratory
Department of Power Mechanical Engineering, National Tsing Hua University
101, Sec. 2, Kuang-Fu Rd., HsinChu Taiwan 300
[email protected]
Abstract
In this research, an atomistic-continuum mechanics (ACM) conjunction of the finite element method (FEM) is applied to
investigate the elastic constant of the nanoscale single crystal silicon in different crystallography planes of (100), (110), and
(111) under uniaxial tensile and vibration loading. The ACM transfers an originally discrete atomic structure into an equilibrium
continuum model by atomistic-continuum transfer elements. The mean positions of each atom of the metallic elements can be
treated as the positions which achieve the minimum total energy. All interatomic forces, which are described by the empirical
potential function, can be transferred into atomic springs to form the lattice structure. The spring network models were also
widely utilized in finite element method based nanostructure studies [1-3]. The ACM model simplifies the complexities of
interaction forces among atoms, while the calculation accuracy can be remained with affordable computational time. Based on
the atomistic-continuum mechanics, a series of atomic level tensile testing and modal analysis are conducted to estimate the
mechanical properties of single crystal silicon, and the results are validated with bulk properties with good agreements.
Introduction
Single crystal silicon (SCS) is one of the most common structure materials used in microelectro- mechanical and nanoelectromechanical systems (MEMS/NEMS). Rapid development in the file of MEMS/NEMS, particularly in the design of mechanical
devices, requires an accurate value of the mechanical properties include Young’s modulus, Poisson’s ratio, tensile strength
and so on. Tests of MEMS materials have been carried out for evaluating their mechanical properties, such as the tensile tests
[4,5], bending tests [6]. However, the material properties have been estimated only on a micrometer to millimeter scale
structure because of difficulties in fabrication of a nanoscale test specimens and problems associated with measuring ultrasmall physical phenomena in an experiment.
An alternate method, such as simulation techniques, would be used to investigate the nature of deformation and the material
properties of nanoscale devices. The atomistic-continuum mechanics (ACM) [1] conjunction of the classic finite element
method (FEM) is proposed herein to explore the mechanical properties of nanoscale single crystal silicon (SCS) under uniaxial
tensile and vibrational loading condition. Mechanical properties such as the Young’s modulus and Poisson’s ratio are
evaluated from the ACM simulation and compared with the experiment data. Different from the molecular dynamics (MD)
simulation, the ACM considered the minimization of the total potential energy to analyze and simulate the mechanical
behaviors of the nanoscale structure. Compared with MD simulation, The ACM can be very efficient and it can provide results
quickly in an accurate range.
Theory of Atomistic-Continuum Mechanics
Figure1 shows that the schematic of the atomistic- continuum mechanics (ACM) conjunction of the classic finite element
method. This figure represents a simple model for a crystal where an orderly array of spheres represents the atoms linked
one-with-other by nonlinear spring elements to represent the interatomic bonds (interatomic force). Moving one atom aside
from its equilibrium position implies stretching/compressing or attractive/repulsive behavior. The ACM considered the
minimization of the total potential energy. The total potential energy of an elastic body Π is defined as:
Π = U-W
(1)
where U is the strain energy and W is the energy of the external loads, respectively. The minimization of total potential energy
with respect to the nodal displacement {D} requires that:
∂Π
=0
∂{D}
(2)
Atom
Nonliner spring
Fig.1 Schematic of ACM.
Fig.2 Single crystal silicon structure and
Boundary conditions.
Interatomic Potential for Silicon
The single crystal silicon structure can be represented as two interpenetrating face-centered cubic (FCC) lattices with one
displaced (1/4 1/4 1/4) crystal lattice with respect to the other, as shown in Fig. 2. The chemical bonding forces considered for
the interaction between the silicon atoms in this investigation are a general form of Stillinger-Weber potential function [7]. The
Stillinger-Weber potential is a combination of two-body and three-body potentials, f2 and f3. The two-body potential describes
the formation of a chemical bond stretching between two atoms while the three-body potential represents the angle formation
between two bonds. The two-body potential is described as a Lennard-Jones potential form as shown in eq. (3):
(
) [
]
⎧A Br -p - r -q exp (r - a )-1 , r < a
f 2 (r) = ⎨
0, r ≥ a
⎩
(3)
Note that f2(r) is a function of the rescaled bond length r only, and it vanishes without discontinuities at r=a. this function
1/6
1/6
exhibits a minimum which equals to -1 at r=2 , (i.e. the ideal bond length is given by R0=2 σ). Here the distance a sets the
range of potential, B is adjusted so that the minimum potential occurs at desired value, and A is set so that the value of f2(r) at
the potential minimum is -1.
The three-body potential f3 which describes the interaction of atoms i, j, and k is given by:
f 3 (ri , rj , rk ) = h (rij , rik ,θ jik ) + h (rji , rjk ,θijk ) + h (rki , rkj ,θikj )
(4)
Originally, the parameters of the Stillinger-Weber potential function were fitting to the simulation results including the total
energy and the lattice constant from experimental data. The energy and the distance units are deduced from the observed
atomic energy and lattice constant at 0 K in the silicon diamond structure: σ=0.20951 nm. The parameters A, B, p, q, a, λ and
γ are constant used in the Stillinger-Weber potential function for single crystal silicon (see Table1 for detail) [7].
Table1: Parameters used in the Stillinger-Weber potential function for silicon [7]
Parameter
Parameter
A
7.049556277
a
1.8
B
0.602224558
λ
21
p
4
γ
1.2
q
0
SCS Atomistic-Continuum Mechanics Models
The ACM method transfers the interatomic potential function into a force-displacement curve so as to create an equivalent
atomistic-continuum transfer element. Afterward, the equivalent nano-scaled model could be analyzed by FEM.
According to the theory of ACM simulation described above, the nonlinear spring elements replace the covalent bond s
(interatomic forces) between two atoms while the silicon atoms were described as the node position. Moreover, in order to
describe the two-body and three-body interaction completely in ACM simulation, two kinds of nonlinear spring elements of
different behaviors were applied to represent the chemical bonding interactions. One illustrates the stretching motion, and the
other defines the bond angle behavior between the two atoms and two covalent bonds, i.e., the spring constant was derived
from second order derivative of the two-body and three-body terms of the Stillinger-Weber potential. The stretch motion
between the two atoms is defined by Eq. (3), and the configuration of stretch mechanisms is shown in Fig. 3. Similarly, the
formation of the angle motion of the two covalent bonds is defined by Eq. (4), and it is shown in Fig. 4.
2.5
Interatomic energy of three-body
6
4
2
0
-0.02
0.00
-2
-4
-6
Interatomic force of two-body (nN)
-0.04
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Elongation (nm)
2.0
109.50
1.5
1.0
0.5
0.0
0
20
40
60
80
100
120
140
160
180
200
Angle (Degree)
Fig.3 The mechanisms of stretch motion between two
silicon atoms
Fig. 4 The mechanisms of angle motion between two
silicon covalent bonds
In this research, a straight spring element is applied to equivalent the in-plane angle motion based on the small deformation
assumption. Therefore, the rotational movement was transformed into translational motion. The geometry arrangement of
transformation is shown in Fig. 5. According to the law of cosine, this transformation is given by:
L2 = 2r12 - 2r22 cos(θ )
(5)
where L is the length of equivalent spring which corresponds to angle θ, the r1 and r2 represent the bond length, and θ is the
angle between the two bonds. According to small deformation assumption, Eq. (5) can be rewritten with r1 = r2 = r as shown in
Eq. (6):
cos(θ ) = 1 -
L2
2r 2
(6)
Fig.5 The geometry arrangement of transformation from angle motion to stretch motion
According to the above-mentioned transformation, the original single lattice crystal silicon structure in Fig. 2 was reconstructed
to the new model which has two kinds of nonlinear spring element in ACM simulation.
Tensile Test for SCS Structure by ACM Models
®
In this research, the SCS structure was built by the commercial finite element code (ANSYS ). Half symmetry models of the
(100), (110) and (111) crystallographic planes are established. The SCS model composes of two parts, one is the two-body
spring element characterization, and the other is the three-body spring element description. Besides, the tensile prescribed
displacement loading ε is applied as shown in Fig.6, and the Young’s modulus Et of the SCS structure could be estimated by
the following equation:
Et =
σ Ftotal
=
ε A ⋅ε
(7)
where Ftotal is the total reaction force, ε represents the applied tensile strain loading, and A represents the equivalent area in
the ACM model.
Fig.6 Schematic of the model used in the ACM simulations of uniaxial tensile testing of SCS
structure (a=5.43Å: Lattice constant)
Modal Analysis for SCS Structure by ACM Models
Following, the modal analysis for SCS structure is accomplished to validate the results of tensile loading test. Through modal
analysis, the elastic modulus of single crystal silicon is calculated by first mode resonant frequency. The SCS structure is built
as a cantilever beam with fixed end, and let mass element joint into the position of silicon atom (atomic mass of silicon is
4.665*10-26 kg). Fig. 7 shows the ACM model and the boundary condition of the SCS structure. After the resonant frequency is
obtained, the Young’s modulus E of this structure could be calculated by the following equation from the Euler-Bernoulli beam
theory as shown in eq. (8). Notably, the atomic level modal analysis differs from those simulations of continuum structure. We
could estimate the Young’s modulus of the nano cantilever beam by the input of two-body and three-body potentials rather
than the simulations of continuum models which input the Young’s modulus from experimental data.
⎡ 2π f 1 ⎤
12 ρL ⎢
(1.875)2 ⎥⎦
⎣
E=
t2
2
4
(8)
where ρ is the density, L represents the length, and f1 represents the first resonant frequency of the SCS structure.
Fig.7 The ACM model and boundary condition of the SCS structure modal analysis
Results and Discussions
Figure8 shows the simulation results and table2 lists the simulated elastic modulus from both tensile test and modal analysis
and then compared with those values in the literature measured for bulk silicon [8]. It could be observed that the estimated
elastic modulus of SCS along the (100), (110), and (111) planes are in agreement with the values reported in the literature
based on macro- and micro-scale experimental work. There are two reasons why the Young’s modulus of the simulation
method agreed with the experiment data. First, the single crystal silicon was defect free material as well as the structure
constructed in the ACM model. Therefore, the silicon material reveal identical behaviors whether in the simulation method or in
experimental testing. Second, the parameters of the Stillinger-Weber potential function were reliable, since the parameters
were fitted and deduced from the experimental data of bulk material and quantum mechanics calculation. Furthermore, the
simulation results of resonant frequency analysis also agree with the analytical solution of resonant frequency based on the
Euler-Bernoulli beam theory. Through these results, one could reduce the complexity of the real bulk silicon and an agreement
between the predicted elastic modulus and the corresponding bulk value is achieved.
(b)
(a)
Fig.8 (a) Displacement distribution of tensile static testing. (b) First mode shape in modal analysis.
Table2. Calculation results comparison. The bulk value are obtained from Sato et al. [8]
Crystallography planes
Tensile testing results
(GPa)
Modal analysis results
(GPa)
Bulk Value (GPa)
Plane (100)
121.8
128.7
125
Plane (110)
153
154.7
140
Plane (111)
174.6
179.8
180
In this research, a novel atomistic-continuum mechanics (ACM) conjunction of the finite element method (FEM) has been
proposed to investigate the mechanical properties of a nanoscale single crystal silicon in the different crystallography planes of
(100), (110), and (111) under uniaxial tensile and vibration loading. One thing should be noted that the same simulation model
is applied in both tensile testing and modal analysis and then the Young’s modulus of the nano SCS structure is estimated in
reasonable range compared with bulk values. According to the methodology of ACM, one could conduct a systematic
approach, which does not involve any parameter fitting, to estimate mechanical properties of nano structures by an appropriate
potential energy.
References
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