  Starting from the Dirac equation, the relativistic quantum hydrodynamic
equations for Dirac electrons on a surface of a three dimensional-topological
insulator (TI) are derived and numerically solved to study the spin plasmons.
The surface of the TI is modulated by a perpendicular magnetic field and a
pulsed exchange field provided by an array of ferromagnetic insulating (FI)
stripes. The collective density and velocity oscillations in the
two-dimensional Dirac electron gas are simulated and analyzed. It is shown that
in spin plasmons, the collective electron density and momentum are modulated by
the spin state. Direct experimental observations of such oscillations are
possible at laboratory conditions.
