  We revealed that 180 degree ferroelectrics domain walls (FDWs) in a
ferroelectric substrate, which induce p-n junctions in a graphene channel, lead
to the nontrivial temperature and gate voltage dependences of the perpendicular
and parallel modes of the integer quantum Hall effect. In particular the number
of perpendicular modes v, corresponding to the p-n junction across the graphene
channel varies with gate voltage increase from small integers to higher
fractional numbers, e.g.v=1, 1.5, 2, ... 5.1,..9.1, 23 in the vicinity of the
transition from ferroelectric to paraelectric phase.These numbers and their
irregular sequence principally differ from the sequence of fractional numbers
v=1.5,2.5,... reported earlier. The origin of the unusual v-numbers is
significantly different numbers of the edge modes, v1 and v2, corresponding to
significantly different concentration of carriers in the left (n1) and right
(n1) domains of p-n junction boundary. The concentrations n1 and n2 are
determined by the gate voltage and polarization contributions, and so their
difference originates from the different direction of spontaneous polarization
in different domains of ferroelectric substrate. The difference between n1 and
n2 disappears with the vanishing of spontaneous polarization in a paraelectric
phase. The phase transition from the ferroelectric to paraelectric phase can
take place either with the temperature increase (temperature-induced phase
transition) or with the decrease of ferroelectric substrate thickness
(thickness-induced phase transition).
