Pismak and Shukhobodskaia recently proposed a model of a Dirac particle interacting with a material plane. Depending on the interaction constants, the Dirac particle can have bound states moving parallel to the plane and decaying exponentially perpendicular to the plane. Moreover, the waves on the two sides of the plane can move with different velocities. Although states with sharp momenta parallel to the plane do not have currents perpendicular to it, it turns out that the wave packets considered here do have a weak current component in this direction, which keeps the wave packets above and below the plane together.