On the incorporation of cubic and hexagonal interfacial energy anisotropy in phase field models using higher order tensor terms

In this paper, we show how to incorporate cubic and hexagonal anisotropies in interfacial energies in phase field models; this incorporation is achieved by including upto sixth rank tensor terms in the free energy expansion, assuming that the free energy is only a function of coarse grained composition, its gradient, curvature and aberration. We derive the number of non-zero and independent components of these tensors. Further, by demanding that the resultant interfacial energy is positive definite for inclusion of each of the tensor terms individually, we identify the constraints imposed on the independent components of these tensors. The existing results in the invariant group theory literature can be used to simplify the process of construction of some (but not all) of the higher order tensors. Finally, we derive the relevant phase field evolution equations.Comment: About 14 pages; 2 figures, to be submitted for revie

Effect of Epitaxial Strain on Phase Separation in Thin Films

We examine the role of an imposed epitaxial strain e in enhancing or depressing the spinodal instability of an alloy thin film. Since the alloy film starts with an imposed strain, phase separation offers a mechanism to relieve it, but only when the film is elastically inhomogeneous. With composition-dependence of elastic modulus given by y, and that of lattice parameter by {\eta}, our simulations using the Cahn-Hilliard model show (and analytical results for early stages confirm) that, for (ey/{\eta}) > 0, the imposed strain adds to the driving force for phase separation, decreases the maximally growing wave length, and expands the coherent spinodal in the phase diagram. Further, when (ey/{\eta}) > 0.372, it expands to even outside of chemical spinodal. Phase separation produces islands of elastically softer (harder) phase with (without) a favorable imposed strain. These results are in agreement with experimental results in GeSi thin films on Si and Ge substrates, as well as in InGaAs films on GaAs substrates.Comment: 4 pages, 3 figure

Chaos-Logistic Map-Tent Map -Corresponding Cellular Automata

We study the characteristic features of Tent Mop and Logistic Map towards Chaos. Cellular Automata is obtained for Tent Mop and Logistic Map. The behavior of Tent Mop and Logistic Mop is reflected in the Cellular Automata generated

Critical role of electron-phonon interactions in determining the relative stability of Boron Nitride polymorphs

Despite several first principles studies, the relative stability of BN polymorphs remains controversial. The stable polymorph varies between the cubic (c-BN) and hexagonal (h-BN) depending on the van der Waals (vdW) dispersion approximation used. These studies are unable to explain the main experimental results, c-BN is stable, the relative stability order and the large energy difference between h-BN and c-BN (greater than 150 meV/formula unit). In this study, we introduce contributions from electron-phonon interactions (EPI) to the total energy of BN polymorphs. This clearly establishes c-BN is the stable polymorph irrespective of the vdW approximation. Only by including EPI contributions do the ab initio results match, for the first time, the main experimental results mentioned above. The EPI contribution to the total energy is strongly sensitive to chemical bonding (approximately twice in $sp^2$-bonded layered over $sp^3$-bonded polymorphs) and to crystal structure. The crucial role of EPI contributions is seen in $sp^2$-bonded layered BN polymorphs where it is greater than the vdW contribution. Given that h-BN is a prototype layered material, in bulk or 2D form, our results have a broader relevance, that is, including EPI correction, along with vdW approximation, is vital for the study of energetics in layered materials.Comment: 6 pages, 2 figures, 3 table

Theoretical issues in the accurate computation of the electron-phonon interaction contribution to the total energy

The total energy is the most fundamental quantity in ab initio studies. To include electron-phonon interaction (EPI) contribution to the total energy, we have recast Allen's equation, for the case of semiconductors and insulators. This equivalent expression can be computed using available software, leading to more accurate total energy. We calculate the total energies and their differences for carbon-diamond and carbon-hexagonal polytypes. For ab initio investigations on crystalline materials, the accepted norm is to compute important quantities only for the primitive unit cell because per-atom quantities are independent of unit cell size. Our results, unexpectedly, show that the per-atom total energy (EPI included) depends on the unit cell size and violates the unit cell independence. For example, it differs for carbon-diamond by 1 eV/atom between the primitive cell and supercells. We observe that reliable energy differences between polytypes are obtained when, instead of primitive cells, supercells with identical number of atoms are used. A crucial inference of general validity is that any equation which contains a partial Fan-Migdal self-energy term violates the unit cell independence. Further theoretical studies are needed to establish if the total energy (EPI included) is an exception or can be reconciled with the unit cell independence.Comment: submitted for publication. 6 pages, 1 figure, 2 table