  As an emerging complex two-dimensional structure, plane fractal has attracted
much attention due to its novel dimension-related physical properties. In this
paper, we check the feasibility to create an effective Sierpinski carpet (SC),
a plane fractal with Hausdorff dimension intermediate between one and two, by
applying an external electric field to a square or a honeycomb lattice. The
electric field forms a fractal geometry but the atomic structure of the
underlying lattice remains the same. By calculating and comparing various
electronic properties, we find parts of the electrons can be confined
effectively in a fractional dimension with a relatively small field, and
representing properties very close to these in a real fractal. In particular,
compared to the square lattice, the external field required to effectively
confine the electron is smaller in the hexagonal lattice, suggesting that a
graphene-like system will be an ideal platform to construct an effective SC
experimentally. Our work paves a new way to build fractals from a top-down
perspective, and can motivate more studies of fractional dimensions in real
systems.
