Anandh Subramaniam

Critical thickness of equilibrium epitaxial thin films using finite element method

On growth beyond critical thickness, interfacial misfit dislocations nucleate, and this partially relaxes the strain due to lattice mismatch in epitaxially grown thin films. It has been observed that, in metallic thin films, global minimization of energy can explain the experimentally determined critical thickness values and these are referred to as equilibrium thin films. In this work, using a combination of finite element simulation of a coherently strained growing thin film and simple edge dislocation, the critical thickness for dislocation nucleation is determined. The edge dislocation and the growing thin film are modeled by feeding in the appropriate stress free Eshelby misfit strain in suitable regions of the model. The critical thickness of various metallic thin films is calculated and compared with standard theory and with available experimental results.

Critical Size for Edge Dislocation Free Free-Standing Nanocrystals by Finite Element Method

As the size of a free-standing crystal approaches a few tens of nanometers, the image force experienced by a dislocation can exceed the Peierls force. This will lead to dislocations leaving the nanocrystal without the application of an external stress and thus making it dislocation free. In this investigation a finite element methodology is developed for the calculation of the critical size at which a free-standing crystal becomes edge dislocation free. A simple edge dislocation is simulated using Finite Element Method (FEM) by feeding-in the appropriate stress-free strain in an idealized domains corresponding to the introduction of an extra half-plane of atoms. The image force experienced by the edge dislocation is calculated as the gradient of the plot of the energy of the system as a function of the position of the simulated dislocation. In nanocrystals, due to the proximity of multiple surfaces, the net image force due to multiple images has to be calculated. Additionally, surface or/and domain deformations have to be taken into account in nanocrystals; which can drastically alter the image force. For the crystal to become dislocation free, the minimum image force experienced by the dislocation, has to exceed the Peierls force. Minimum image force values calculated from the FEM models are compared with the Peierls stress values obtained from literature to determine the critical domain size at which crystal becomes edge dislocation free.

Analysis of thin film growth using finite element method

The properties and the performance of epitaxial semiconductor thin films depend on the stress-state and the defect structure in the film. When the film is grown layer by layer, the accumulated elastic strain energy due to misfit strain between the substrate and the film is partially released by the formation of misfit dislocations at a threshold thickness. This investigation pertains to finite element analysis of the stress-state in epitaxial thin films as a function of the thickness and the release of the elastic energy by dislocation nucleation. To begin with, stress contours associated with the epitaxial growth and the nucleation of the dislocation are studied independently and the results are compared with the available analytical and experimental data. Subsequently, the above two are combined to analyze the effective potential energy state of the system. Essentially, the energy minimization process that involves accommodation of the misfit strains at a threshold film thickness is studied.

Thermodynamic Rationalization of the Microstructures of CrFeNi & CuCrFeNi Alloys

In high entropy alloys (HEA) a disordered solid solution is entropically stabilized, in competition with possible intermediate compounds or phase segregation. It has been reported that disordered solid solutions are stabilized by the presence of five or more elements in the alloys; typically in an equimolar ratio. In the current investigation, the effect of Cu addition on the microstructure of CrFeNi alloy is rationalized by Gibbs free energy calculations. Two phase (both are FCC) solid solution are seen to form in CuCrFeNi alloy. The alloys are prepared by induction melting and are characterized by x-ray diffraction and scanning electron microscopy (in as-cast and annealed state). Enthalpies of mixing were calculated using Miedemas model and the regular solution model. Entropy of mixing is determined by using the Boltzmanns hypothesis.

Aspects regarding measurement of thickness of intergranular glassy films

Materials such as Si3N4, SiC and SrTiO3 can have grain boundaries characterized by the presence of a thin intergranular amorphous film of nearly constant thickness, in some cases (e.g. Si3N4) almost independent of the orientation of the bounding grains, but dependent on the composition of the ceramic. Microscopy techniques such as high‐resolution lattice fringe imaging, Fresnel fringe imaging and diffuse dark field imaging have been applied to the study of intergranular glassy films. The theme of the current investigation is the use of Fresnel fringes and Fourier filtering for the measurement of the thickness of intergranular glassy films. Fresnel fringes hidden in high‐resolution micrographs can be used to objectively demarcate the glass–crystal interface and to measure the thickness of intergranular glassy films. Image line profiles obtained from Fourier filtering the high‐resolution micrographs can yield better estimates of the thickness. Using image simulation, various kinds of deviation from an ideal square‐well potential profile and their effects on the Fresnel image contrast are considered. A method is also put forth to objectively retrieve Fresnel fringe spacing data by Fourier filtering Fresnel contrast images. Difficulties arising from the use of the standard Fresnel fringe extrapolation technique are outlined and an alternative method for the measurement of the thickness of intergranular glassy films, based on zero‐defocus (in‐focus) Fresnel contrast images is suggested. The experimental work is from two ceramic systems: Lu‐Mg‐doped Si3N4 and SrTiO3 (stoichiometric and nonstoichiometric). Further, a comparison is made between the standard high‐resolution lattice fringe technique, the standard Fresnel fringe extrapolation technique and the methods of analyses introduced in the current work, to illustrate their utility and merits. Taking experimental difficulties into account, this work is intended to be a practical tool kit for the study of intergranular glassy films.

Assessing thermodynamic properties of amorphous nanostructures by energy-filtered electron diffraction

Since the discovery of the existence of 0.5 – 5nm wide amorphous intergranular films (IGF) in many ceramic materials almost 30 years ago [1] a lot of effort has been put into understanding why they exist as well as into the development of new experimental techniques to probe their properties. A recently developed phase field model [2] which includes energetic contributions from chemical, structural, as well as electrostatic effects able to describe the structural transition at the crystal/amorphous interfaces as well as the role of dopant cations as glass network modifiers relies on a set of thermodynamic parameters (e.g. gradient energy penalties for various fields) which are not accessible in bulk structures. It is therefore important to obtain as much experimental information from these structures under investigation, as possible. However, the small width of these films of 1-2nm and the even narrower interface regions are pushing the limits of today’s high resolution (TEM) imaging techniques. Even the width of these glassy films, one of only a few observables which the theory may be compared against cannot be determined without ambiguity [3,4], a fact which can be attributed to the unavoidable imperfections of electron lenses as well as the TEM image formation process itself. Electron diffraction is the perfect tool to identify the crystallinity of a structure and is also unaffected by phase shifts produced by lens aberrations. Using the Koehler illumination system provided in LEO instruments such as the LEO912, or the SESAMe, which has recently been installed in our lab, in conjunction with an in-column energy filter, we can record the small angle scattering produced by an almost parallel electron beam with a diameter of 100-200nm diameter incident on a specimen area including the IGF (see figure 1a). Because of the low signal produced by the scattering off the amorphous IGF we use imaging plates as the recording medium (fig. 1b). While taking account of dynamical scattering effects a functional form inspired by diffuse interface theory [5,6] of the cross-sectional density α(x) of the amorphous IGF could be fitted to the diffraction data with high precision (fig.1c), thus providing the crystallinity η(x)=1-α(x), one of the phase fields included in the model mentioned above[2]. Using energy-loss spectroscopic profiling (ELSP) [7] and electron holography, additional fields, such as chemical concentration profiles and charge density distribution can be determined. This allows thermodynamic parameters in the model to be fitted very precisely and thus the theory to be extended to new systems. This work was done in collaboration with the other partners in the EU/NSF funded NANOAM project. We would like to particularly thank W.C. Carter and C.M. Bishop at MIT for developing the relevant phase field model. We kindly acknowledge financial support through the EU/NSF NANOAM project (EU project Nr. GRD2-CT-2000-30351).

Orthorhombic rational approximants for decagonal quasicrystals

An important exercise in the study of rational approximants is to derive their metric, especially in relation to the corresponding quasicrystal or the underlying clusters. Kuo’s model has been the widely accepted model to calculate the metric of the decagonal approximants. Using an alternate model, the metric of the approximants and other complex structures with the icosahedral cluster are explained elsewhere. In this work a comparison is made between the two models bringing out their equivalence. Further, using the concept of average lattices, a modified model is proposed.

Stacking sequences and symmetry properties of trigonal vacancy-ordered phases (τ phases)

Abstract The vacancy-ordered phases known as τ phases are described and the literature dealing with the observed stacking sequences is reviewed. It is shown that the stacking sequences along the threefold axis can be derived from a projection method involving projection on to an axis of type [rrq]. The structure has alternating filled and empty lamellae parallel to planes of type (rrq). The particular cases in which r and q are consecutive numbers of the Fibonacci sequence can be regarded as rational approximants to a one-dimensional quasiperiodic structure. Some mathematical properties of the sequences, and their relationship with the three-dimensional structures, are presented.

Critical sizes for coherent to semicoherent transition in precipitates

Abstract A coherent precipitate, on growth beyond a critical size, can become semicoherent through the formation of interfacial misfit dislocations. This investigation pertains to the finite element simulation of the state of stress of a coherent precipitate, its growth and the change in state of stress on the formation of an interfacial misfit dislocation loop. Critical radii are determined from the simulations based on: (i) global energy minimum (r*) and (ii) local force balance along the radial direction (rc). The concept of local force balance as existing in literature is extended to the circumferential direction, to calculate a new critical size (rt). Local force balance gives radii at which the interface is the stable position for the dislocation loop. Off-interface stability of the dislocation loops is also investigated. The Cu–γFe system is used as an example to illustrate the new methodology developed and validate the results of the simulation. The power of the methodology is shown by considering a configuration (precipitation in a thin disc), where standard theoretical formulations are inadequate.

Stable edge dislocations in finite crystals

Dislocations have been considered as mechanically unstable defects in bulk crystals, ignoring the Peierls oscillations. Eshelby [J. Appl. Phys. 24 (1953) p.176] had showed that a screw dislocation can be stable in a thin cylinder. In the current work, considering Eshelbys example of an edge dislocation in a single crystalline plate, we show that an edge dislocation can be stable in a finite crystal. Using specific examples, we also show that the position of stability of an edge dislocation can be off-centre. This shift in the stability from the centre marks the transition from a stable dislocation to an unstable one. The above-mentioned tasks are achieved by simulating edge dislocations using the finite element method.