Gauge theory of defects in the elastic continuum

A gauge theory of defects in an elastic continuum is developed after providing the necessary background in continuum elasticity and gauge theories. The gauge group is the three-dimensional (3D) Euclidean group [semi-direct product of the translation group T (3) with the rotation group SO (3)]. We obtainboth dislocations and disclinations by breaking of the translational invariance. Breaking of the rotational invariance is shownnot to lead to any interesting effects in a linear analysis. These results are shown to be consistent with the topological analysis which is briefly discussed at the end of the paper. Any defect given by the present theory acquires acore which removes the singularity of the stress field at the origin. The stress field agrees with the continuum result asymptotically, as is expected. Geometrical aspects of the deformed state of condensed matter are also briefly touched upon.     

Dislocations and gauge invariance

The paper deals with gauge invariance applied to dislocations in their field theory formulation. By comparison with electromagnetism, the role of distortion and velocity fields as potentials for dislocation density and dislocation current is shown. The gauge transformation involving one vector field for those potentials is given and the equilibrium equation is recognized to be a guage condition. The constitutive laws are shown to form a basis of this gauge condition.     

On the traction boundary value problem in the linearized gauge theory of dislocations

The static traction boundary value problem for finite material bodies is shown to be well posed in the linearized gauge theory of dislocations. The dislocation field variables assume the roles of generalized stress potentials that satisfy a system of fourth order linear partial differential equations. Accordingly, the stress distributions may be calculated directly from the traction boundary data without solving for the elastic displacement fields. Satisfaction of appropriate gauge conditions are shown to lead to significant simplifications and certain systems of first integrals of the governing equations are exhibited. The important thing here is that the gauge theory of dislocations provides direct means of calculating the distributions of dislocations that arise from given systems of boundary tractions. This is in sharp contrast with previous theories in which the distributions of dislocations are calculated from given distributions of dislocation densities.     

A yang-mills type minimal coupling theory for materials with dislocations and disclinations

Noting that the group $\text{S0(3) ? T(3)}$ may be viewed as a 6-parameter gauge group that leaves the Lagrangian of elasticity theory invariant, the Yang-Mills universal gauge theory construction is used to erect a complete continuum theory of material bodies with dislocation and disclination fields. Breaking of the homogeneity of the action of $\text{S0(3)}$ is shown to give rise to disclinations and rotational dislocations while homogeneity breaking of $\text{T(3)}$ gives rise to translational dislocations. A rigorous justification for replacing displacement gradients by the components of the distortion tensor and Newtonian kinematic velocity by distortional velocity is obtained. Exact determinations are made of the elastic excess forces, the forces on dislocations and the forces on disclinations, and these forces are shown to be totally equilibrating in all instances. Implications of the theory are given and an analysis is made of the field equations and associated dispersion relations that obtain in a disclination free material in the linear elasticity approximation.     

Dislocation and displacement fields of finite bodies with given boundary tractions

The traction boundary value problem for spatially finite material bodies is examined in the context of the gauge theory of dislocations. In contrast with classical theory of dislocations in infinite bodies, the boundary conditions for the dislocation fields are shown to have pronounced effects. Expansion in the load parameter that is naturally associated with the applied loading shows that dislocation effects are essentially nonlinear. If the dislocation coupling constant is of the order of the shear modulus or larger, the dislocation density tensor vanishes throughout the body in the linear engineering approximation. A sequence of well-posed linear boundary value problems are shown to provide approximate solutions to any desired degree of accuracy in the load parameter.     

Quantum Nature of Dislocations in Pure bcc Helium

Recent experiments show the thermal growth of dislocation lines in ultra-pure bcc ³He. The activation energy for the growth of the dislocation lines is found to agree with the activation energy of mass diffusion. We propose that these dislocations are topological defects in the phase of the complex order-parameter, which describes the dynamic zero-point atomic correlations, unique to the bcc phase. There is also a shear field associated with these topological defects. We show that the smallest topological defect is a localized excitation, a loop-defect, which leads to the exponential growth of the dislocation lines with temperature.     

A comparative discrete-dislocation/crystal-plasticity analysis of bending of a single-crystalline microbeam

Bending of a micron-size single-crystalline beam is analyzed using both discrete-dislocation plasticity and crystal-plasticity formulations. Within the discrete-dislocation plasticity formulation, dislocations are treated as infinitely long straight-line defects residing within a linear elastic continuum. Evolution of the dislocation structure during bending is simulated by allowing the dislocations to glide in response to long-range interactions between different dislocations, and between dislocations and the applied stresses, and by incorporating various short-range reactions which can result in dislocation nucleation, annihilation or pinning. At each stage of bending, the stress and deformation fields are obtained by superposing the dislocation fields and the complementary fields obtained as a solution of the corresponding linear-elastic boundary value problem. The results obtained show that there is a continuing accumulation of geometrically necessary dislocations during bending which is expected due to the gradient in the strain throughout the beam height. In addition, it is found that localization of plastic flow into slip bands is a salient feature of materials deformation at the micron-length scale. Within the crystal-plasticity analysis, of beam bending, a small displacement gradient formulation is used and the material parameters selected in such a way that plastic flow localizes into deformation bands at low strains. It is found that, while the global response of the beam predicted by the two approaches can be quite comparable, fine details of the dislocation-based stress and deformation fields cannot be reproduced by the continuum crystal-plasticity model.     

Concurrently coupled atomistic and XFEM models for dislocations and cracks

A method for the modeling of dislocations and cracks by atomistic/continuum models is described. The methodology combines the extended finite element method with the bridging domain method (BDM). The former is used to model crack surfaces and slip planes in the continuum, whereas the BDM is used to link the atomistic models with the continuum. The BDM is an overlapping domain decomposition method in which the atomistic and continuum energies are blended so that their contributions decay to their boundaries on the overlapping subdomain. Compatibility between the continua and atomistic domains is enforced by a continuous Lagrange multiplier field. The methodology allows for simulations with atomistic resolution near crack fronts and dislocation cores while retaining a continuum model in the remaining part of the domain and so a large reduction in the number of atoms is possible. It is applied to the modeling of cracks and dislocations in graphene sheets. Energies and energy distributions compare very well with direct numerical simulations by strictly atomistic models. Copyright © 2008 John Wiley & Sons, Ltd.     

Edge and screw dislocations in the linearized gauge theory of defects

The field equations governing static pure dislocation states in the linearized gauge theory of defects are obtained. Explicit solutions are exhibited. The fields of stress associated with these solutions are shown to agree with known stress fields for screw and edge dislocations in the near field but decay exponentially m the farfield.     

On the gauge transformations admitted by the equations of defect dynamics

This paper deals with gauge invariance applied to dislocations and disclinations in the field theory formulation given by Kossecka and de Wit[1,2]. The gauge transformations of the strain, linear velocity, bend-twist, and rotational velocity that leave the dislocation and disclination densities and currents invariant are obtained explicitly, and the equations of balance of linear momentum are shown to lead to a natural gauge condition. These transformations are generated by a vector field that depends upon position and time and two vector fields that depend on time alone. The latter two fields are shown to realize superimposed rigid body motions as part of the generalized gauge transformation structure of the theory. Representation of the space and time generating vector in terms of a gradient plus the curl of another vector is shown to lead to several useful decompositions.     

Atomistic simulation of dislocation core structure and dynamics in Fe–Ni–Cr–N austenite

Atomistic computer simulations based on the use of the conjugate gradient and molecular dynamics methods were employed to determine the core structure and dynamics of the a/2 <1 0 0> edge and screw dislocations in Fe-Ni-Cr and Fe-Ni-Cr-N austenites. The embedded-atom method was used to quantify the interactions between iron, nickel, chromium and nitrogen atoms. In Fe-Ni-Cr austenite, both the edge and screw dislocations dissociate along one of the {1 1 1} planes, forming stacking fault ribbons. The ribbon widths were found to be comparable to their values calculated using the continuum theory. The analysis of dislocation dynamics showed that the phonon drag interferes more with the motion of screw dislocations, reducing their mobility in comparison with the mobility of edge dislocations. In Fe-Ni-Cr-N austenite, the structure of the dislocation core of the a/2 <1 1 0> edge dislocation does not seem to be significantly affected by the presence of nitrogen. In sharp contrast, the core structure of the dissociated a/2 <1 1 0> screw dislocation undergoes a major change, resulting in spreading of the core on to two or more non-parallel planes. As a result, mobility of the screw dislocations is substantially lower than that of the edge dislocations. This finding is consistent with the experimental observations that the dislocations are predominantly of the screw character in Fe-Ni-Cr-N austenite. This revised version was published online in November 2006 with corrections to the Cover Date.     

On the formation and stability of dislocation patterns—I: One-dimensional considerations

By distinguishing among mobile and immobile dislocations and operating within the framework of continuum mechanics it is possible to derive a set of partial differential equations of the diffusion-reaction type for the evolution of dislocation species. On examining the competition between gradient dependent terms modelling the motion of dislocations and nonlinear terms modelling their interactions, it is shown that stable solutions are possible. The wavelength turns out to be a material property in agreement with observations. The discussion is limited to one dimension, that is to glide of straight dislocations in the slip direction, and the model corresponds physically to the ladder-like structure of persistent slip bands.     

On a proposal for a continuum with microstructure

Summary A recently proposed model for a continuum with microstructure is further substantiated by identifying the microstructure with dislocations. In particular, the continuum is viewed as a superimposed state composed of a perfect lattice state, an immobile dislocation state, and a mobile dislocation state. It is assumed that each state evolves continuously in space-time and transitions from one state to another take place spontaneously according to the balance laws of effective mass and momentum. When the constitutive equations are subjected to the requirements of invariance, familiar statements from dislocation dynamics are deduced. When plastic strain and yield are identified in terms of the parameters characterizing the dislocation states, familiar flow rules and yield surfaces are produced. The capability of the model to predict not only Tresca and Von-Mises plastic behavior but also phenomena such as kinematic hardening, different responses in tension and compression, latent hardening, and the Bauschinger effect, is shown. Finally, the appropriateness of our equations to model creep, cyclic plasticity, and fatigue, is illustrated.With 2 Figures     

Effect of crystallographic dislocations on the reverse performance of 4H-SiC p-n diodes

A quantitative study was performed to investigate the impact of crystallographic dislocation defects, including screw dislocation, basal plane dislocation, and threading edge dislocation, and their locations in the active and JTE region, on the reverse performance of 4H-SiC p-n diodes. It was found that higher leakage current in diodes is associated with basal plane dislocations, while lower breakdown voltage is attributed to screw dislocations. The above influence increases in severity when the dislocation is in the active region than in the JTE region. Furthermore, due to the closed-core nature, the impact of threading edge dislocation on the reverse performance of the p-n diodes is less severe than that of other dislocations although its density is much higher.     

Characterization of 4H semi-insulating silicon carbide single crystals using electron beam induced current

In this work we demonstrate the electron beam induced current (EBIC) contrast of dislocations in semi-insulating (SI) bulk SiC single crystals. Our investigations revealed that the screw dislocations produce dark EBIC contrast indicating high leakage current in the defective regions. This type of screw dislocations are harmful for high resolution radiation detectors and should be eliminated by improving the crystal qualities. Chemical etching in molten KOH was used for dislocation mapping in the test structures and to correlate EBIC contrast with dislocations. The EBIC contrast found in our study in SI SiC is discussed.     

Self-organized pattern formation of biomolecules at silicon surfaces: Intended application of a dislocation network

Defined placement of biomolecules at Si surfaces is a precondition for a successful combination of Si electronics with biological applications. We aim to realize this by Coulomb interaction of biomolecules with dislocations in Si. The dislocations form charged lines and they will be surrounded with a space charge region being connected with an electric field. The electric stray field in a solution of biomolecules, caused by dislocations located close to the Si surface, was estimated to yield values up to few kVcm^− 1.A regular dislocation network can be formed by wafer direct bonding at the interface between the bonded wafers in case of misorientation. The adjustment of misorientation allows the variation of the distance between dislocations in a range from 10 nm to a few μm. This is appropriate for nanobiotechnology dealing with protein or DNA molecules with sizes in the nm and lower μm range. Actually, we achieved a distance between the dislocations of 10–20 nm. Also the existence of a distinct electric field formed by the dislocation network was demonstrated by the technique of the electron-beam-induced current (EBIC). Because of the relatively short range of the field, the dislocations have to be placed close to the surface. We positioned the dislocation network in an interface being 200 nm parallel to the Si surface by layer transfer techniques using hydrogen implantation and bonding. Based on EBIC and luminescence data we postulate a barrier of the dislocations at the as bonded interface < 100 meV. We plan to dope the dislocations with metal atoms to increase the electric field.We demonstrated that regular periodic dislocation networks close to the Si surface formed by bonding are realistic candidates for self-organized placing of biomolecules. Experiments are underway to test whether biomolecules decorate the pattern of the dislocation lines.     

On the properties of <111>{110} dissociated superdislocation in B2 structure YAg and YCu: Core structure and Peierls stress

The simplified one-dimensional dislocation equation for mixed dislocations is derived briefly from the two-dimensional modified Peierls-Nabarro equation taking into account the discreteness effect of crystals. The collinear dissociated core structure of <111>{110} superdislocations in the novel B2 structure YAg and YCu are investigated with the simplified equation. Both the core width and the dissociated width are increasing with the increases in the dislocation angle of superdislocations. The dissociated width determined by continuum elastic theory is inaccurate for the high antiphase boundary energy but is recovered for the low antiphase boundary energy. The Peierls stress of the dissociated dislocation is replaced by that of superpartials. The results show that both the unstable stacking fault energy and the core width are crucial for the Peierls stress in the case of a narrow core structure. However, the core width becomes the main factor in controlling the Peierls stress in the case of a wide core.     

On the properties of ?111? {110} dissociated superdislocation in B2 structure YAg and YCu: Core structure and Peierls stress

The simplified one-dimensional dislocation equation for mixed dislocations is derived briefly from the two-dimensional modified Peierls-Nabarro equation taking into account the discreteness effect of crystals. The collinear dissociated core structure of 〈111〉 {110} superdislocations in the novel B2 structure YAg and YCu are investigated with the simplified equation. Both the core width and the dissociated width are increasing with the increases in the dislocation angle of superdislocations. The dissociated width determined by continuum elastic theory is inaccurate for the high antiphase boundary energy but is recovered for the low antiphase boundary energy. The Peierls stress of the dissociated dislocation is replaced by that of superpartials. The results show that both the unstable stacking fault energy and the core width are crucial for the Peierls stress in the case of a narrow core structure. However, the core width becomes the main factor in controlling the Peierls stress in the case of a wide core.