High temperature superconductors were discovered in 1986, but despite considerable research efforts,
both experimental and theoretical, these materials remain poorly understood. Because their
electronic structure is both inhomogeneous and highly correlated, a full understanding will require
knowledge of quasiparticle properties both in real space and momentum space.
In this thesis, we will present a theoretical analysis of the scanning tunneling microscopy (STM)
data in BSCCO. We introduce the Bogoliubov-De Gennes Hamiltonian and solve it numerically on
a two-dimensional 20 x 20 lattice under a magnetic field perpendicular to the surface. We consider a
vortex at the center of our model. We introduce a Zn impurity in our lattice as a microscopic probe
of the physical properties of BSCCO. By direct numerical diagonalization of the lattice BogoliubovDe
Gennes Hamiltonian for different positions of the impurity, we can calculate the interaction
between the vortex and the impurity in a d-wave superconductor
