Condensed Matter Physics

a) Crystal Structure Periodic array of atoms – Lattice translation vectors. The basis and the crystal structure, primitive lattice cell, unit cell. Lattice symmetry operations – point groups and space groups. Fundamental types of lattices – two dimensional lattice types, three dimensional lattice types, crystal planes indexing. Simple crystal structures – sodium Chloride, Cesium Chloride, hexagonal closed packing , diamond, cubic zinc sulphide and hexagonal zinc sulphide structures, amorphous substances, glasses. Use of X-ray diffraction for structure determination. Scanning electron and transmission electron microscopy. b) Crystal diffraction and the reciprocal lattice Bragg’s Law, Lave equations, reciprocal lattice – its properties, Bragg diffraction, condition in terms of reciprocal lattice vectors, brillouin zones, reciprocal lattice and brillouin zones of bcc, fcc, and hexagonal lattices, symmetry points of brillouin zones.

Free electron Model of Metals Energy levels and density of orbitals in one dimension, effect of temperature on the Fermi- Dirac distribution function, free electron gas in three dimensions. Heat capacity of electron gas, electrical conductivity,. thermal conductivity of metals, Wiedmann- Franz law. Introduction to Hartree and Hartree- Fock methods for solving a many electron problem in metals. Classical Hall effect, Integral quantum Hall effect, Fractional Quantum Hall effect.

Electron energy bands Nearly free electron Model; origin of energy gap , the bloch theorem, Kronig-Penny model, wave equation of electron in a periodic potential, crystal momentum of an electron, reduced zone scheme; approximate solution near a zone boundary ; number of orbitals in a band, density of states; metals, insulators and semiconductors. Construction of Fermi surfaces, electrons, holes and open orbitals, effective mass of electrons in crystals.

a) The Pseudopotential Method The approximations, orthogonalized plane wave method. The pseudopotential formulation, nonlocal and local pseudopotentials, separation of the pseudopotential. Selfconsistent screening of a local pseudopotential- Thomas Fermi method, Hartree Dielectric Screening function , Exchange and Correlation. The diffraction model, Energy Eigen states, scattering, factorization of matrix elements– structure factor and form factors, evaluation of total energy. b) Density functional Theory Thomas-Fermi - Dirac approximation – example of a functional, the Hohenberg - Kohn theorems, formulation of density functional theory ,the Kohn-Sham equations, and advantages of density functional method; Extension of DFT to spin polarized systems. Exchange –correlation potential, Local Density Approximation, Introduction to time dependent density functional theory, local spin density approximation (LSDA), Generalized –gradient approximations, (GGAs).

a) Superconductivity Occurrence of superconductivity, experimental observations, persistent currents, Effect of magnetic fields, Meissner effect, type I and type II superconductors, intermediate states, entropy and heat capacity, Energy gap, isotopic effect, thermal conductivity. Theoretical explanations : London’s equations, penetrations depth, coherence length, cooper pairs, elements of BCS theory, flux quantization, Josephson effect, High Tc superconductors: MgB2 , Cuprate superconductors, Hubbard model. b) Theoretical estimations of superconducting state parameters Electron- phonon coupling strength (λ), Coloumb pseudopotential (μ), Eliashberg gap equations, Mc Millan’s formulation for Tc and isotope effect in low temperature superconductors. Interaction strength, estimation of superconducting state parameters for metallic superconductors and alloys