  The Bardeen-Cooper-Schrieffer (BCS) condensation and Bose-Einstein
condensation (BEC) are the two limiting ground states of paired Fermion
systems, and the crossover between these two limits has been a source of
excitement for both fields of high temperature superconductivity and cold atom
superfluidity. For superconductors, ultra-low doping systems like graphene and
LixZrNCl successfully approached the crossover starting from the BCS-side.
These superconductors offer new opportunities to clarify the nature of
charged-particles transport towards the BEC regime. Here we report the study of
vortex dynamics within the crossover using their Hall effect as a probe in
LixZrNCl. We observed a systematic enhancement of the Hall angle towards the
BCS-BEC crossover, which was qualitatively reproduced by the phenomenological
time-dependent Ginzburg-Landau (TDGL) theory. LixZrNCl exhibits a band
structure free from various electronic instabilities, allowing us to achieve a
comprehensive understanding of the vortex Hall effect and thereby propose a
global picture of vortex dynamics within the crossover. These results
demonstrate that gate-controlled superconductors are ideal platforms towards
investigations of unexplored properties in BEC superconductors.
