This thesis addresses the tunnelling of charge carriers in different materials. First looking at the simplest case of electron tunnelling in metals at zero, then finite temperature,
the current is obtained using the Fermi-Dirac golden rule and then the conductance is
obtained. This is extended to take into account the spatial dependence of one of the
metals being a tip since experimentally this is done by scanning tunnelling microscopy
where a tip traces over the surface of a sample. The next step is to look at tunnelling
between a metal and a semiconductor, again the current is found. Semiconductors can
be doped and the effect this has on tunnelling is examined. Next superconductors are
introduced. The purpose of my research has been to look at the tunnelling spectra of
high-temperature superconducting cuprates for both extrinsic (metal-superconductor)
and intrinsic (superconductor-superconductor) tunnelling. The main features seen experimentally with cuprate tunnelling are identified and then a theory capable of explaining these features is discussed. The theory is compared to experimental results and we find good agreement.
A Doctoral Thesis. Submitted in partial fulfillment of the requirements for the award of Doctor of Philosophy of Loughborough University.