  We consider the Schr\"odinger operator on nanoribbons (tight-binding models)
in an external electric potentials $V$. The corresponding electric field is
perpendicular to the axis of the nanoribbon. If V=0, then the spectrum of the
Schr\"odinger operator consists of two spectral bands and the flat band (i.e.,
the eigenvalue with infinite multiplicity) between them. If we switch on an
weak electric potential $V\to 0$, then we determine the asymptotics of the
spectral bands for small fields. In particular, we describe all potentials when
the unperturbed eigenvalue remains the flat band and when one becomes the small
band of the continuous spectrum.
