  We consider a typical heterostructure domain patterned ferroelectric
film/ultra thin dielectric layer/ semiconductor, where the semiconductor can be
an electrolyte, paraelectric or multi layered graphene. Unexpectedly we have
found that the space charge modulation profile and amplitude in the
semiconductor, that screens the spontaneous polarization of a 180-degree domain
structure of ferroelectric, depends on the domain structure period, dielectric
layer thickness and semiconductor screening radius in a rather non-trivial
nonlinear way. Multiple size effects appearance and manifestation are defined
by the relationship between these three parameters. In addition, we show that
the concept of effective gap can be introduced in a simple way only for a
single domain limit. Obtained analytical results open the way for understanding
of current AFM maps of contaminated ferroelectric surfaces in ambient
atmosphere as well as explore the possibilities of conductivity control in
ultra-thin semiconductor layers.
