Axion electrodynamics in topological matter

In this thesis we search for the topological magnetoelectric effect. We first review the theoretical aspects of axion electrodynamics in topological matter. It was found that for such materials, an additional term can be added to the Lagrangian density, which leads to the modification of Maxwell's equations. These modifications arise due to spatial or temporal changes in the axion angle, which is non-zero in topological materials. Several experiments have been proposed to measure the resulting topological magnetoelectric effects, of which we consider three experiments which are related to electronic properties. Two experiments, the search for the magnetic monopole and the interference experiment for fractional charge, are related to a spatial gradient in the axion angle, while the chiral magnetic effect relates to temporal changes in the axion angle of Weyl semimetals. In the rest of this thesis, we use thin films, deposited by molecular beam epitaxy (MBE), and exfoliated flakes of topological materials to study the experiments that were proposed above. For the magnetic monopole experiment, we fabricate top gate samples of magnetically doped topological insulator films. We measure the magnetic response as a function of applied gate voltage via a SQUID sensor in a dilution refrigerator.  We find that heat dissipation in the sample plays a major role at ultra low temperatures. The signature of fractional charges is measured in an interference experiment where we make use of a Josephson junction array that is fabricated on both topological and reference samples. While no topological signature was observed in the measurements, we find a good agreement between the experimental results and numerical simulations. For the chiral magnetic effect, we used exfoliated flakes of a Dirac semimetal, of which we fabricated non-local nanostructures to measure the diffusion length of the chiral charge polarization. We find that the relaxation rate of the chiral charge is slower than the Ohmic relaxation rate.