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.