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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 C60 - 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
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 Te-wary 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 = -E Fa (l) Ua (l) +
l,a
+ 1 (l, l) Ua (l) ^ (0, (1)
2 l,a
where Fa (l) and ^ap (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 O by
Ou = F. (2)
The Green’s function is defined from (2) as the inverse of the force constant matrix,
G = (O)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 O* of the cracked lattice is obtained from that of the perfect crystal by intro-
© K. Masuda-Jindo and Vu Van Hung, 2004
ducing the force terms on the cleavage surface that annihilate the bonds there. These corrections to O are simply the negative of the perfect crystal forces from the second term in
V evaluated on the crack surfaces. Thus one can write
O* = O - SO, SO = [O]crack faces' (4)
The formal solution of the problem is then given by the Dyson equation,
G* = G + GSO GT, (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
2/v+1
Fig. 1. Sketch of atomic geometry of kinked crack (a) for lx = 8, lz = 12 and lk = 3. Lower (b) and (c) are the perfect and crack Green’s functions, respectively
2Lx + 1. The crack is periodic in the z direction, with repeat distance 2Lz +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 Lx, 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 2Lz +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
In Fig. 1(a), we present the atomic geometry of kinked crack in simple cubic lattice with dimensions of lx = 8, lz = = 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.
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 c-axis 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 so-called 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
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 (Si6)n as shown in Fig. 4(c-e), whose initial structure is suggested from the structure of type IV Si45 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.
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Fig. 3. Atomic configurations around the edge dislocation in graphene sheets under applied shear stresses
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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 pentagon-heptagon 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 n-electron states near the Fermi energy.
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Fig. 5. Atomic structures of nanographite with AAA. stacking (a), with ABAB... structure (b), and ABAB... structure (c) in applied (tensile) stresses
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-
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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