Mechanical and electronic properties of nanoscale materials studied

K. Masuda-Jindo and Vu Van Hung / Ôèçè÷åñêàÿ ìåçîìåõàíèêà 7 4 (2004) 77–81
77
Mechanical and electronic properties of nanoscale materials studied by density
functional molecular dynamics and lattice Green’s function methods
K. Masuda-Jindo and Vu Van Hung1
Department of Materials Science and Engineering, Tokyo Institute of Technology, Yokohama, 226-8503, Japan
1
Hanoi National Pedagogic University, Hanoi, Vietnam
The mechanical and electronic properties of nanoscale materials are studied using the molecular dynamics (MD) and lattice Green’s
function (LGF) methods. The strength and fracture properties are investigated for the nanoscale materials, quantum wires and carbon
related materials like graphenes and nanotubes using the first principles O(N) molecular dynamics method. We compare the mechanical
properties of nanoscale materials with those of the corresponding bulk-size materials. For the study of the defect properties in the nanocrystals
like graphene sheets and nanographites, we also use the LGF method, which allows us to perform the analytic and accurate calculations.
We calculate the Green function for the defective lattice, with dislocation and cracks, by solving the Dyson equation, appropriate for
absolute zero temperature. After the lattice Green functions of the absolute zero temperature have been determined, the lattice parameters
and interatomic force constants are adjusted to fit to materials at finite temperature T, using the statistical moment method.
1. Introduction
Recently, there has been a great interest in the study of
nanoscale materials since they provide us a wide variety of
academic problems as well as the technological applications
[1–14]. In particular, the important experimental findings
in this field are the discovery of carbon nanotubes and the
discovery of superconductivity in the alkali-metal doped
C 60 - systems. The properties of clusters and fine particles
are generally quite different from those of the bulk materials,
e.g., in magnetism, catalytic activities, elastic properties and
optical properties. The discovery of carbon nanotubes (CNT)
by Iijima [3] and subsequent observations of CNT’s unique
electronic and mechanical properties have initiated intensive
research on these quasi-one-dimensional (quasi-1D) structures. CNT’ have been identified as one of the most promising building blocks for future development of functional
nanostructures. One of the purposes of the present paper is
to investigate the plasticity of materials with quasi-1D structures using the quantum ab initio tight-binding molecular
dynamics method. In addition, we also calculate the atomic
configurations and strength properties of nanocrystals including extended defects (dislocations and cracks) using the
temperature dependent LGF method. Some of the calculation
results by temperature dependent LGF method will be given
below.
2. Principle of calculations
For treating nanoscale materials we will use ab initio
tight-binding molecular dynamics methods [13–15], which
© K. Masuda-Jindo and Vu Van Hung, 2004
have been very successful in the various thermodynamic,
lattice defects and nanoscale materials calculations. In the
present article, we also use the lattice Green’s function
(LGF) approach to study the mechanical properties of nanoscale materials, like graphene sheets and nanographites.
In the treatment of LGF, we generalize the conventional
LGF theory to take into account the temperature effects.
Our principle reference to lattice Green’s Functions is Tewary et al. [16], and we will use the terminology of that
paper. If the displacement u(l) of an arbitrary atom at lattice
position l is small, then in a Taylor expansion of the potential
energy V of the lattice can be given by
V = −∑ Fα (l ) uα (l ) +
l,α
1
+ ∑ φαβ (l , l ′) uα (l ) uφ (l ′),
2 l,α
(1)
l,φ
where Fα (l ) and φ αβ (l , l ′) are the externally imposed forces and the internal spring constants, respectively. The equilibrium lattice equation is given in terms of force constant
matrix Φ by
Φu = F.
(2)
The Green’s function is defined from (2) as the inverse
of the force constant matrix,
G = (Φ)–1.
(3)
This is the Green’s function for the perfect lattice and
can be found by conversion to reciprocal space in the standard manner. The force constant matrix Φ* of the cracked
lattice is obtained from that of the perfect crystal by intro-
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K. Masuda-Jindo and Vu Van Hung / Ôèçè÷åñêàÿ ìåçîìåõàíèêà 7 4 (2004) 77–81
ducing the force terms on the cleavage surface that annihilate
the bonds there. These corrections to Φ are simply the negative of the perfect crystal forces from the second term in
V evaluated on the crack surfaces. Thus one can write
Φ* = Φ − δΦ, δΦ = [Φ]crack faces.
(4)
The formal solution of the problem is then given by the
Dyson equation,
G* = G + GδΦG*,
(5)
together with the “master equation” for the Green’s function,
u = G*F.
(6)
Figure 1(a) shows the crack geometry of the present
study. The crack is a “double ended” crack of length
a
b
c
2 Lx + 1. The crack is periodic in the z direction, with repeat
distance 2L z + 1, which allows us to work with an infinite
crack in the z direction. We will call that set of atoms with
broken bonds lying on the crack cleavage plane, and which
constitutes a complete repeating cell along the z axis, a basis
set of atoms. The kinks are symmetrically disposed at the
ends of the crack on the x axis and repeated in the z direction.
The kink pairs are each 2Lk + 1 in length. Another special
feature of the problem in Fig. 1(a) is that the “real” external
force distribution is a single force dipole situated at the origin
and repeated along the z axis with the repeat distance. This
choice is made for analytic convenience. If the kink length
Lk is small compared to the half crack length L x , then a
stress intensity K field is well defined over the entire kink
region, so the crack problem is well defined. In order for
one kink pair not to interfere with another, we also require
that the repeat distance 2L z + 1 be large compared with the
kink size. After the appropriate lattice Green’s functions of
the cracked lattice are obtained, it is straightforward to investigate the crack extension events, i. e., kink nucleation
and kink migration processes, by solving the coupled linear
equations, with temperature dependent force constants, nonlinear cohesive forces and surface tensions.
Treatment at finite temperature
The present study includes the temperature effects on
the defect properties: For the LGF treatment at finite temperatures, we take explicitly account the changes in the lattice
spacing, interatomic force constants and non-linear cohesive
forces near the crack tip region, simultaneously. To derive
the temperature dependent ingredients in the LGF theory,
we use the moment expansion method in the quantum statistical mechanics [17–19]. This method allows us to take into
account the anharmonicity effects of thermal lattice vibrations on the thermodynamic quantities in the analytic formulations. The LGF method outlined above can be applied
straightforwardly to the defect calculations in 2D graphene
sheets and nanographites. The detailed analysis of this application will be presented elsewhere.
3. Results and discussions
Fig. 1. Sketch of atomic geometry of kinked crack (a) for l x = 8, l z = 12
and lk = 3. Lower (b) and (c) are the perfect and crack Green’s functions,
respectively
In Fig. 1(a), we present the atomic geometry of kinked
crack in simple cubic lattice with dimensions of l x = 8, l z =
= 12 and lk = 3. The lower Figures 1(b) and (c) are the
Green’s function of unperturbed perfect lattice and that of
kinked crack, respectively. One can see in Fig. 1 that the
atomic displacements due to the external force dipoles F0
are much smaller and more localized in the perfect lattice
than those in the cracked lattice. The calculated temperature
dependence of the kink formation and migration energies
(modeling of silica) are shown in Fig. 2. One can see in
Fig. 2 that both kink formation and migration energies have
the weak temperature dependence, decreasing function of
the temperature.
K. Masuda-Jindo and Vu Van Hung / Ôèçè÷åñêàÿ ìåçîìåõàíèêà 7 4 (2004) 77–81
Fig. 2. Temperature dependence of formation and migration energies of
kinks in cracked lattice
Nanowires and nanocrystals
Firstly, we have performed MD simulations for the caxis edge dislocation in two-dimensional (2D) planar graphene sheets. The core structure of the edge dislocation is
characterized by the five- and seven-membered rings in the
2D graphene sheets, as shown in Fig. 3(a). We have found
that the pair of edge dislocations are most stable in the
nearest-neighhbour configuration. The excess energies due
to introduction of the edge dislocations are estimated by
comparing the energies of graphene sheets with and without
the edge dislocations. We also calculated the relative stability (excess energies) of the carbon nanocrystallites with
spherical shapes.
In general, we have found that there are no marked differences in the stability between the crystallites with and
without edge dislocations. This indicates that the self-energy
of the edge dislocation is very small and may become even
negative for the certain clusters and nanocrystallites. Then,
we come to the conclusion that the dislocation can be generated spontaneously without sizeable activation energy in the
small semiconductor clusters. In other words, the semiconductor clusters such as the 2D graphene, can be mechanically
deformed (rolled up) more easily compared to the corresponding bulk materials. The continuum elasticity theory also
predicts that the elastic distortion energy of lattice defects
depends on the existence of the free surfaces due to the socalled image effect [20], compared to those in the infinite
crystals. However, the above mentioned cluster size dependence of the self-energy of the dislocation, i.e., microscopic
“image effects”, can not be explained within the linear elasticity theory.
In addition, we have also studied the motion of the dislocation under applied stresses. In Fig. 3(b), we show the
core structures of the c-axis edge dislocation in the graphene
sheets. Under the application of shear stresses, the dislocation core structure changes so as to minimizes the excess
energies of the dislocations, but simple shear dislocation
motion is not observed in the present calculations. As shown
79
in Fig. 3(b), when the external shear loadings exceed certain
critical value, the five-seven membered rings are tripled
and a disordered (‘amorphous’ like) region appears. This
may indicate that the edge dislocation has a sessile structure
in graphene sheets, and lattice resistance for the dislocation
motion is extremely high.
In addition, we also considered the fictious Si nanowires
composed of six-membered Si rings (Si 6 ) n as shown in
Fig. 4(c–e), whose initial structure is suggested from the
structure of type IV Si 45 cluster. Under the tensile stress,
this type of nanowire exhibits the non-uniform deformation,
and certain necking occurs at the several portions of the
nanowire, as shown in Fig. 4. The appearance and location
of the necking depends sensitively on the size of the wire,
and we have found the necking occurs near the center of
wires for smaller size ones. This characteristic fracture
behavior of the Si nanowire is considerably different from
these of carbon nanowires with the similar structure. As
shown in Fig. 4(b), the carbon nanowires show very brittle
fracture behavior without producing neckings.
We have also performed tension and compression tests
both for single wall and double wall carbon nanotubes, as
shown in Fig. 4. These tubes can be visualized as graphitic
sheets rolled up into cylinders giving rise to quasi-one dimensional structures. We have found that the compression
strength of both CNT depend strongly on the exsistence of
vacancy type defects as well as on the bond rotation defects
(pair of 5–7 membered rings). In the nonlinear response
regime, locally deformed structures such as pinches, kinks,
and buckles have been observed [12]. Under the compressive stress, the nanotube exhibits the drastic change of the
bonding geometry, from a graphite (sp2) to a localized diamond like (sp3) reconstruction, at the critical stress
(≈ 153 GPa) [11]. In a recent experiment, large compressive
strains were applied to CNT dispersed in composite polymeric films. It has been observed that there are two distinct
deformation modes, sideways buckling of thick tubes and
collapse/fracture of thin tubes without any buckling. The
compressive strain in the experiment is estimated to be larger
than 5 %, and critical stress for inward collapse or fracture
is expected to be 100–150 GPa for thin tubes.
a
b
Fig. 3. Atomic configurations around the edge dislocation in graphene
sheets under applied shear stresses
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K. Masuda-Jindo and Vu Van Hung / Ôèçè÷åñêàÿ ìåçîìåõàíèêà 7 4 (2004) 77–81
a
b
c
e
d
Fig. 4. Atomic structures of stretched Si (a) and C (b) nanowires. (c), (d), and (e) are the (9, 0) CNT, (9, 0) CNT containing SW defects and double wall
(9, 0)–(18, 0) tubules, respectively
Using the ab initio TBMD scheme, the plasticity of the
single-wall carbon nanotube containing dislocations
(Fig. 4(d)), whose core is characterized by the pentagonheptagon pairs, is investigated. The atomic structure of carbon nanotube containing the bond rotation defects shows a
stepwise change diameter near the dislocation. We have
found that the CNT containing the edge dislocation exhibits
the critical stress far below (~ 80 GPa) than that (153 GPa)
of the CNT without bond rotation defect. The c-axis edge
dislocation provides the efficient center for stress concentration and gives rise to the failure of the CNT. The details of
the plastic flow and failure depend on the symmetry of CNT
and will be presented elsewhere.
The atomistic and mechanical properties of nanographites are also investigated using the ab initio TBMD method. Nanographites are nanometer-sized graphite fragments
that represent a new class of a mesoscopic system intermediate between aromatic molecules and extended graphite
sheets [21]. In Fig. 5, we present the calculated atomic configuration of nanographites with “zigzag” structure
(Fig. 5(b)), in comparison with the unrelaxed initial atomic
configuration of simple AAA… stacking (Fig. 5(a)). One
can see in Fig. 5 that the certain atomic relaxation occurs
so as to minimize both the surface energy and the structural
energies of the whole crystallites, reducing more or less the
periodicity of ABABAB… stackings. In these systems, the
boundary regions play an important role so that edge effects can influence strongly the π-electron states near the
Fermi energy.
a
b
c
Fig. 5. Atomic structures of nanographite with AAA… stacking (a), with
ABAB… structure (b), and ABAB… structure (c) in applied (tensile)
stresses
K. Masuda-Jindo and Vu Van Hung / Ôèçè÷åñêàÿ ìåçîìåõàíèêà 7 4 (2004) 77–81
We have also investigated the properties of dislocations
in Si nanocrystallites [22–32], in comparison with the lattice dislocation in diamond cubic Si crystal [12–14]. In a
diamond cubic crystal, the important dislocations are the
60°, screw and 90° (edge) perfect dislocations [20]. The
first one dissociates into a 30° and 90° partial dislocations
while the others split into a pair of 30° and 60° partial dislocations, respectively. All the partials are separated by intrinsic stacking faults. These partials, which have line directions along <110> are believed to be reconstructed into
a structure with no dangling bonds. The atomic configurations of 30° partial dislocations in Si and in bulk Si crystal
are compared, and it has been found that the reconstruction
defects “solitons” can be seen near the center of the nanocrystallite [13]. These point singuralities “solitons” in the
small crystallites are formed by the atomic reconstruction,
which are initiated from the crystallite surface, and nanocrystallites different in nature from those appearing along
the dislocation line in the bulk crystal, which are thermodynamic reconstruction defects.
We have also considered Si quantum wires with extended defects. This is simply because the Si quantum wires
synthesized by laser ablation often contain the kinks, twins
and grain boundaries [32, 33]. It is also known that the quality of the polycrystalline Si films depends on their texture
[33] because the tilt grain boundaries are generally believed
to be electrically inactive. In this respect, it is of great significance to investigate the atomistic and electronic structures of dislocations and grain boundaries in microcrystalline
semiconductors.
Electronic and electrical properties
In the present study, we have also investigated the electronic and electrical transport properties of nanowires and
nanocrystals in conjunction with the mechanical deformations. We use the finite element technique to study the strain
and deformation effects on the electronic and transport properties of the deformed quantum wires and nanotubes. In
general, we have found that the electrical conductance depend strongly on the mechanical deformation, compressed,
stretched or bending conditions. Again, the details of our
calculation results will be published elsewhere.
4. Conclusions
We have studied the atomistic and mechanical properties
of nanoscale materials like carbon nanotubes, nanographite,
nanographene and semiconductor nanocrystallites, using the
O(N) ab initio TBMD method. This method is very efficient
and reliable scheme to study properties of large scale systems. The properties of extended lattice defects like dislocations and grain boundaries in nanoscale materials depend
sensitively on the size of the crystallites and differ signifi-
81
cantly from those of the bulk materials, especially due to
the nanoscale “image effects”. It has also been found that
the edge-type dislocation whose core is characterized by
pentagon-heptagon pair acts as the center for the stress concentration and contribute to the plastic deformation far below the stress level than the critical stress of CNT including
no defects.
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