  Negatively biased surface-gates allow electrostatic depletion of selected
regions of a 2DEG, forming confined regions of specific geometry called
billiards, in which ballistic transport occurs. At millikelvin temperatures,
the electron phase coherence length is sufficient that quantum interference
effects produce reproducible magneto-conductance fluctuations (MCF) that act as
a 'magneto-fingerprint' of the scattering dynamics in the billiard. It has been
predicted that billiard MCF are fractal. Fractal MCF in mesoscopic
semiconductor billiards are investigated experimentally. The MCF of a Sinai
billiard displayed exact self-similarity (ESS). A correlation function analysis
is used to quantify the presence of ESS. A model for the Sinai billiard MCF
based on a Weierstrass function is presented. Using a bridging interconnect, a
continuous transition between the Sinai and an empty square geometry is
achieved. The removal of the circle induces a transition from ESS to
statistical self-similarity (SSS), suggesting that ESS is due to the presence
of an obstacle at the center of the billiard. The physical dependencies of SSS
are investigated and show variation in the fractal dimension, rather than the
fractal scaling range. SSS obeys a unified picture where the fractal dimension
depends only on the ratio between the average spacing and broadening of
billiard energy levels, irrespective of other billiard parameters. The
semiclassical origin of SSS is demonstrated and the suppression of SSS is
observed in both the quantum and classical limits. The influence of soft-wall
potential profile on fractal MCF is investigated using double-2DEG billiards.
Detailed reviews of semiconductor billiard fabrication, low-temperature
electrical measurements and fractal analysis are also presented.
