| Silicon, graphene and carbon nanotube are three covalent materials. Silicon is an important semiconductor, it has been widely used in the manufacture of electronic devices. Graphene and carbon nanotubes possess the unique physical and chemical properties, and thus have very good prospects. Dislocations in these three materials will affect their magnetic, optic, electronic, especially the mechanical properties. The crucial problems of dislocation theory are the core structure and the Peierls stress. The classical dislocation model is Peierls-Nabarro (P-N) model. Althought it can determine the core structure and Peierls stress quantitatively, it is based on the elastic theory. The theoretical predictions of the core structure and Peierls stress given by the classical P-N model have larger deviations. Especially for the narrow dislocations in covalent materials, the dislocation width is only a fraction of the lattice constant. Therefore the lattice discrete effect must be considered. In this paper, the core structure and Peierls barrier and Peierls stress of the dislocations in semiconductor silicon, graphene and zigzag single-walled carbon nanotubes (SWCNTs) are studied based on the lattice theory of dislocation which has considered the lattice discrete effect. Including the 30 and 90 partials, shuffle 60 and shuffle screw dislocations in semiconductor silicon, the glide dislocation and shuffle dislocation in graphene and the pentagon-heptagon pair dislocation in zigzag single-walled carbon nanotubes (SWCNTs). The main work and results include:(1)The discrete parameter in dislocation equationIn the dislocation equation given by the lattice theory of dislocation, the discrete effect correction term is respresented in the form of second derivative of the displacement field. It mainly results from the surface effect. When a crystal is viewed as a set of parallel lattice planes, the surface layer is not equivalent to those in the interior. As is known, the misfit planes become the surface layers of the semi-crystals while the crystal is cut into two semi-crystals. The discrete parameter (coefficient of the discrete effect correction term) is related to the acoustic velocities of the decoupled surface layer (the surface layer that has been removed the coupling with the internal layers). For simple crystal lattice structure, the surface layer is single layer of atoms and the discrete parameter can be approximately given by the structural and elastic constants. However, the results for simple crystals are no longer valid for complex lattices due to the surface layers are composed of multi layers of atoms for complex lattices. For silicon and graphene, the decoupled surface layers (surface chains) of glide dislocations are composed of two head to head atom layers (atom chains); and those of shuffle dislocations are staggered two atom layers (atom chains). In order to investigate the discrete parameters in dislocation equations of silicon and graphene, a simple dynamis model includes the interaction energy attributed to variation of bond length and angle is constructed. According to the Hamiltonian, the relation between model parameters and macro-parameters and the relation between acoustic velocities of decoupled surface layer and model parameters can be determined, hence the discrete parameters can be given by the macro-parameters. Firstly, one needs to find the matrices of atomic force constants. And then determine the relative displacement of two atoms in one primitive cell on the basis of equilibrium condition of forces under homogeneous deformation. According to the relation between strain and elements of deformation matrix, the energy density can be expressed by strain. For the semiconductor silicon, the relation between the model parameters and elastic constants can be obtained according to the definition of energy density. For graphene, the energy density can be written as an isotropic one and the model parameters can be given by shear modulus and Possion's ratio. The relations between model parameters and macro-parameters are determined. Next we will find the relations between acoustic velocities of decoupled surface layer and model parameters. For silicon, we need to decouple the surface layer firstly. To do this, the interaction force is expressed in terms of relative displacements of paired atoms and each individual term is interpreted as an interaction between the paired atoms. Then only keep the terms that represent theinteraction between the atoms in the surface layer. Because only the plane deformation is of interest, the normal displacement is fixed to be zero. The next step is to find the dynamical equation and obtain the relation between acoustic velocities and model parameters under quasi-continuum approximation finally. For graphene, according to the Hamiltonian, the effective interaction constants between atoms along and perpendicular to the decoupled surface layer can be obtained. Therefore the dynamical equation for longitudinal vibration of decoupled surface layer can be determined (we only need the vibration modes along the decoupled surface layer) and then the relations between acoustic velocities and model parameters can be obtained under quasi-continuum approximation. Combining the relations between model parameters and macro-parameters obtained previously, the discrete parameters can be given by macro-parameters. Because the atoms in decoupled surface layers (surface chains) of glide dislocations in silicon and graphene interact through bond angles, while the atoms in decoupled surface layers (surface chains) of shuffle dislocations interact through covalent bond directly. The discrete parameters of shuffle dislocations should be larger than those of glide dislocations.(2)The glide partials and shuffle dislocations in semiconductor siliconAs a complex lattice, silicon has two different {111} misfit planes: called glide set; and shuffle set respectively. These two different slip planes induce different kinds of dislocations: glide dislocation and shuffle dislocation. The glide dislocation can dissociates into two partials separated by a low-energy stacking fault. Glide dislocations include 30 and 90 partials along < 112> direction in {111} plane; there is no intrinsic stacking fault and therefore the shuffle dislocations can not dissociate. Shuffle dislocations include 60 and screw dislocations along < 110> direction in {111} plane. Using the generalized stacking fault energies (γ? surface) obtained from first-principles density-functional calculations, the dislocations in silicon are studied based on the classical P-N model by Joos et al.. The calculated Peierls stresses are ( )22×10 ?2 eV/ ? 335GPa and ( )18×10 ?2 eV/ ? 329GPa respectively for 30 and 90 partials, ( )3.0×10 ?2 eV/ ? 34.8GPa and ( )4.1×10 ?2 eV/ ? 36.6GPa respectively for shuffle 60 and screw dislocations. The Peierls stresses calculated from classical P-N model are larger than the experiment and numerical calculation results. It may because of neglect of lattice discrete effect and the contribution of elastic strain energy. In order to consider the infection of discrete effect to dislocation, Bulatov et al. have given the semidiscrete theory and calculated the Peierls stress of glide screw dislocation in silicon. It is found that the results of considering the discrete effects are very different from those did not consider the discrete effects. The dislocation width and Peierls barrier and Peierls stress for silicon have been calculated based on the lattice theory of dislocation. It is found that the corrections from discrete effects to shuffle dislocations are larger than to glide dislocations. The widths of shuffle dislocations are broadened doubly by the discrete effect, and Peierls stresses calculated from total energies are greatly lowered. For glide dislocations, the corrections are relatively small, especially for 30 partial dislocation. In evaluating the energy and stress, the contribution of elastic strain energy is also considered except the misfit energy in classical P-N model. The Peierls barrier and Peierls stress calculated from total energy are greatly lowered. The Peierls barrier calculated here is in the range of 0.12-0.28 eV/? for 30 partial dislocation and 8.5-28 meV/? for shuffle dislocations. The Peierls stress is in the range of 0.065-0.16 eV/?3 for 30 partial dislocation and 3.4-8.4 meV/?3 for shuffle dislocations. (In order to make the glide dissociated dislocation move, we must make the two partials move at the same time. Therefore, the Peierls stresses of glide dissociated dislocations should be identical to the larger one of their two partials, namely identical to the Peierls stress of 30 partial. ) The Peierls stresses calculated here are better meet the experiment and numerical calculation results than those given by classical P-N theory. It is interesting that the Peierls stress for glide 30 dislocation is 0.065-0.16 eV/?3 (10-26GPa), which coincides with the critical stress at low temperature 0.043-0.215 eV/?3 (6.9-34GPa) extrapolated from experimental data. The Peierls stress for shuffle dislocation is 3.4-8.4 meV/?3 (0.54-1.3GPa) which coincides with the critical stress at high temperature (～1GPa) observed in experiments. The calculated results indicate that the energy of the shuffle dislocation is higher than that of the glide dislocation, therefore, shuffle dislocation can not exist stably at low temperature, and glide dislocation may be responsible for the low-temperature plasticity; while at high temperature, shuffle dislocation appears and responsible for high-temperature plasticity. The transition from brittle to ductile is probably related to excitation of shuffle dislocations.(3) The glide dislocation and shuffle dislocation in grapheneGraphene has the hexagonal honeycomb lattice structure. Similar as the semiconductor silicon, graphene also has two different {111} misfit planes, called glide set and shuffle set respectively. These two different misfit planes induce two main dislocations: one is a pentagon-heptagon pair, it is refered to as glide dislocation; another is an octagon, it is refered to as shuffle dislocation. The difference between the dislocations in silicon and graphene is that there has no low-energy stacking fault in the glide plane of graphene and therefore the glide dislocation in graphene can not dissociate. In fact, there still lack of studies on the core strctures and Peierls stresses of dislocations in graphene due to it is a relatively new material. Carpio et al. have studied the stability of glide dislocation, shuffle dislocation, S-W defect and dislocation dipoles in graphene and the electronic properties of most stable configurations. Glide and shuffle dislocations are found to be stable. They have also gave a rough estimate for the Peierls stress of glide dislocation: 10 ?3μ<σp< 10?1μ. Ewels et al. have calculated the Peierls barriers of glide dislocation and shuffle dislocation, about 7.64eV and 2.22eV , respectively. The studies of Carpio et al. and Ewels et al. show that shuffle dislocation moves more easily than the glide dislocation. As we know, there is no quantitative calculation about the core structure and Peierls stress of dislocation in graphene through theoretical method. The dislocation width and Peierls barrier and Peierls stress have been calculated based on the lattice theory of dislocation. Because there is no calculated result of theγ? surface for graphene, we calculated it by the first-principles density functional theory. It's found that in analogy to the other covalent crystals, the width of dislocation in graphene is narrow and the Peierls barrier and stress are high. After considering the contribution of the elastic strain energy, the Peierls barrier and Peierls stress are lowered obviously. The Peierls barrier and Peierls stress of shuffle dislocation are one order of magnitude smaller than those of glide dislocation. The widths of glide dislocation and shuffle dislocation are respectively 0.20a and 0.55a , the Peierls barrier are respectively 4.42eV and 0.38eV , and the Peierls stress are respectively 0.29μand 0.021μ. The calculated Peierls stress of glide dislocation is in the range of estimation given by Carpio et al., the Peierls barriers are lower than those given by Ewels et al.. Our results also show that the shuffle dislocation moves more easily than the glide dislocation.(4) The pentagon-heptagon pair dislocation in zigzag SWCNTsSWCNTs can be viewed as rolled up graphene. They possess curvature effects and size effects. According to the different diameter and the chiral, they can be either metallic or semiconducting. Similar as graphene, there also exist pentagon-heptagon pair and octagon defect. However, pentagon-heptagon pairs are more important due to they can connect carbon nanotubes with different diameters and chiralities to form metal - metal, metal - semiconductor and semiconductor - semiconductor junctions. When ( N ,0) and ( N + 1,0) SWCNTs joint together, misfit interactors will accur due to the different diameters. It makes the larger one shrink and smaller one expand, and the pentagon-heptagon pair will be formed at the joint. From the large semi-tube to small semi-tube, the circumference length changes gradually from ( N + 1)a to Na . It is rational to assume that the circumferential length at the joint is the averaged length( N + 1/ 2)a. As we know, there is no detail description about the structural structure of the pentagon-heptagon pair. Charlier et al. calculated the energy of pentagon-heptagon pair in (11,0) ? (12,0)carbon nanotube, about 30eV . The pentagon-heptagon pair can be regarded as the core of an edge dislocation. Therefore, from the viewpoint of dislocation, a pentagon-heptagon pair defect in a zigzag SWCNT is studied. A dislocation equation that takes the discrete effects, size effects and curvature effects into account is presented for the pentagon-heptagon pair. It is interesting that only the primitive displacement related to the misfit interaction and dominates the intrinsic structure and properties of the pentagon-heptagon pair. In dislocation equation, the primary primitive displacement (circumferential displacement) is not coupled with the other components. This surprising result originates from the intrinsic symmetry of the ziazag SWCNTs. The secondary displacement, which is not directly related to the misfit interaction, is induced by the primitive displacement. Therefore, the primitive displacement is the key of the issue in dislocation theory. According to our model, there is no primitive displacement in axial direction, i.e. the space of the zigzag chains is not affected by appearance of pentagon-heptagon pair. The primitive displacement in radical direction is a / (4π), which is a rational consequence of the fact that the diameter changes gradually along the axis. The main change of the primitive displacement takes place in circumferential direction. Using theγ? surface obtained from the first-principle calculation, it is found that primary primitive displacement is not sensitive to the diameter as long as the SWNT is not too thin. As for the secondary displacement, the main effects are: the zigzag chain including pentagon-heptagon pair is bent slightly towards the pentagon, and the curvature becomes smaller at where the pentagon-heptagon pair located.The bond variation and shape distortion caused by pentagon-heptagon pair have been evaluated. It is found that when the atom number in the circumference is larger than 20, i.e. the diameter is larger than 1.5 nm, the bond length of pentagon-heptagon in different zigzag SWCNTs is almost a constant. Our calculation shows that the neck bond becomes longer while the shoulder bonds become shorter comparing with the one of the hexagons. And the averaged bond length in heptagon is slightly shorter. Furthermore, the dislocation energy, Peierls barrier and stress of pentagon-heptagon pair dislocation in (11,0) ? (12,0)zigzag SWCNT have also been investigated analytically in the framework of lattice theory of dislocation. It is found that the misfit energies are weakly dependent on the perimeters, while the strain and total energies have logarithmic behaviors with the perimeters of larger SWCNTs that N > 10. For smaller SWCNTs that N≤10, because the curvature and size effects, the strain and total energies have some deviation from the logarithmic behaviors. The calculated Peierls barriers and Peierls stresses for different zigzag SWCNTs are about 4.2eV-4.8eV and 0.29μ? 0.31μ, respectively.If the relaxedγ? surface should be used when study the dislocations in covalent materials and how to relax still exist uncertainty. For investigating the infection of the modification factorΔinγ? surface to the dislocation energy, Peierls barrier and stress, the relations between dislocation energy, Peierls barrier and stress of pentagon-heptagon pair dislocation in (11,0)-(12,0) carbon nanotube and the modification factorΔhave been calculated. It is found that when modification factor changes from 0.10 to 0.45, the dislocation energy almost invariant, about18eV, it is smaller than the 30eV given by Charlier et al.. The Peierls barrier and stress increase linearly. Peierls barrier changes from 3.6eV to7.4eV , and Peierls stress changes from 0.2μto0.5μ. The Peierls barrier and stress are not so sensitive to the modification factor as thought.
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