On the Ruderman-Kittel-Kasuya-Yosida interaction in graphene

The two dimensionality plus the linear band structure of graphene leads to new behavior of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which is the interaction between two magnetic moments mediated by the electrons of the host crystal. We study this interaction from linear response theory. There are two equivalent methods both of which may be used for the calculation of the susceptibility, one involving the integral over a product of two Green's functions and the second that involves the excitations between occupied and unoccupied states, which was followed in the original work of Ruderman and Kittel. Unlike the $J \propto (2k_FR)^{-2} \sin (2k_FR)$ behavior of an ordinary two-dimensional (2D) metal, $J$ in graphene falls off as $1/R^3$, shows the $1 + \cos ((\bm{K}-\bm{K'}).\bm{R})$-type of behavior, which contains an interference term between the two Dirac cones, and it oscillates for certain directions and not for others. Quite interestingly, irrespective of any oscillations, the RKKY interaction in graphene is always ferromagnetic for moments located on the same sublattice and antiferromagnetic for moments on the opposite sublattices, a result that follows from particle-hole symmetry.Comment: 12 pages, 5 figures, submitted to AIP Conference Proceeding

Two Dimensional Spin-Polarized Electron Gas at the Oxide Interfaces

The formation of a novel spin-polarized 2D electron gas at the LaMnO$_3$ monolayer embedded in SrMnO$_3$ is predicted from the first-principles density-functional calculations. The La (d) electrons become confined in the direction normal to the interface in the potential well of the La layer, serving as a positively-charged layer of electron donors. These electrons mediate a ferromagnetic alignment of the Mn t$_{2g}$ spins near the interface via the Anderson-Hasegawa double exchange and become, in turn, spin-polarized due to the internal magnetic fields of the Mn moments.Comment: 5 pages, 6 figure

Nuclear incompressibility: An analytical study on leptodermous expansion

A comparative study of the liquid-drop model (LDM) type expansions of energy $E$ and compression modulus $K_A$ is made within the energy density formalism using Skyrme interactions. As compared to the energy expansion, it is found that, in the pure bulk mode of density vibration, the LDM expansion of $K_A$ shows an anomalous convergence behaviour due to {\it pair \ effect}. A least squares fit analysis is done to estimate the minimum error, one would expect even with synthetic data due to the inherent nature of the LDM expansion of $K_A$ as well as the narrow range of accessible mass number $A$, in the values of the various coefficients. The dependence of the higher-order coefficients like curvature and Gauss curvature on the coupling $\beta_c$ between the bulk and surface parts of the monopole vibrations is analytically studied. It is shown that the $K_A -$ expansion including the dynamical effect ( $A-$ dependence of $\beta_c$ ) shows an `up-turn' behaviour below mass number about 120, suggesting the inapplicability of the LDM expansion of $K_A$ over this mass region.Comment: 30 latex pages, five figures available on request ( to appear in Phy. Rev. C

Some cryptographic algorithms

Cryptography is the practice and study of techniques for secure communication in the presence of third parties, called adversaries. Modern cryptography is heavily based on mathematical theory and computer science practice. Cryptographic algorithms are designed so that in practice they are hard to break by any adversary. In the present thesis consisting of two chapters first we have given a brief review of some important number theoretic concepts and results. Then we have discussed S-DES and DES algorithms for Secret key cryptography, RSA and DSA algorithms for Public key cryptography and at last a brief introduction of elliptic curves and their use in Cryptography is given