  We investigate the transport of excitons and trions in monolayer
semiconductor WS$_2$ subjected to controlled non-uniform mechanical strain. We
actively control and tune the strain profiles with an AFM-based setup in which
the monolayer is indented by an AFM tip. Optical spectroscopy is used to reveal
the dynamics of the excited carriers. The non-uniform strain configuration
locally changes the valence and conduction bands of WS$_2$, giving rise to
effective forces attracting excitons and trions towards the point of maximum
strain underneath the AFM tip. We observe large changes in the
photoluminescence spectra of WS$_2$ under strain, which we interpret using a
drift-diffusion model. We show that the transport of neutral excitons, a
process that was previously thought to be efficient in non-uniformly strained
2D semiconductors and termed as "funneling", is negligible at room temperature
in contrast to previous observations. Conversely, we discover that
redistribution of free carriers under non-uniform strain profiles leads to
highly efficient conversion of excitons to trions. Conversion efficiency
reaches $\simeq 100\%$ even without electrical gating. Our results explain
inconsistencies in previous experiments and pave the way towards new types of
optoelectronic devices.
