  Graphene membrane irradiated by weak activating periodic electric field in
terahertz range is considered. The corrected formula for the graphene quantum
conductivity is found. The obtained formula gives complex conjugate results
when radiation polarization direction is clockwise or it is opposite clockwise.
The found formula allows us to see that the graphene membrane is an oscillating
contour. Its eigen frequency coincides with a singularity point of the
conductivity and depends on the electrons concentration. So the graphene
membrane could be used as an antenna or a transistor and its eigen frequency
could be tuned by doping in a large terahertz-infrared frequency range. The
obtained formula allows us also to calculate the graphene membrane quantum
inductivity and capacitance. The found dependence on electrons concentration is
consistent with experiments. The method of the proof is based on the study of
the time-dependent density matrix. The exact solution of the von Neumann
equation for density matrix is found for our case in linear approximation on
the external field. On this basis the induced current is studied and then the
formula for quantum conductivity as a function of external field frequency and
temperature is obtained. The method of the proof suggested in this paper could
be used to study other problems. The found formula for quantum conductivity can
be used to correct the SPPs Dispersion Relation and for the description of
radiation process. It would be useful to take the obtained results into account
when constructing devices containing graphene membrane nanoantenna. Such
project could make it possible to create wireless communications among
nanosystems. This would be promising research area of energy harvesting
applications.
