  Electronic properties of FeSe can be tuned by various routes. Here, we
present a comprehensive study on the evolution of the superconductivity and
nematicity in FeSe with thickness from bulk single crystal down to bilayer
($\sim$ 1.1 nm) through exfoliation. With decreasing flake thickness, both the
structural transition temperature $T_{\rm s}$ and the superconducting
transition temperature $T_{\rm c}^{\rm zero}$ are greatly suppressed. The
magnetic field ($B$) dependence of Hall resistance $R_{xy}$ at 15 K changes
from $B$-nonlinear to $B$-linear behavior up to 9 T, as the thickness ($d$) is
reduced to 13 nm. $T_{\rm c}$ is linearly dependent on the inverse of flake
thickness (1/$d$) when $d\le$ 13 nm, and a clear drop of $T_{\rm c}$ appears
with thickness smaller than 27 nm. The $I$-$V$ characteristic curves in
ultrathin flakes reveal the signature of Berezinskii-Kosterlitz-Thouless (BKT)
transition, indicating the presence of two-dimensional superconductivity.
Anisotropic magnetoresistance measurements further support 2D superconductivity
in few-layer FeSe. Increase of disorder scattering, anisotropic strains and
dimensionality effect with reducing the thickness of FeSe flakes, might be
taken into account for understanding these behaviors. Our study provides
systematic insights into the evolution of the superconducting properties,
structural transition and Hall resistance of a superconductor FeSe with flakes
thickness and provides an effective way to find two-dimensional
superconductivity as well as other 2D novel phenomena.
