  Using first-principles density-functional calculations, we explore the
possibility of magnetic order at the rebonded $D_B$ step of the Si(001)
surface. The rebonded $D_B$ step containing threefold coordinated Si atoms can
be treated as a one-dimensional dangling-bond (DB) wire along the step edge. We
find that Si atoms composing the step edge are displaced up and down
alternatively due to Jahn-Teller-like distortion, but, if Si dimers on the
terrace are passivated by H atoms, the antiferromagnetic (AFM) order can be
stabilized at the step edge with a suppression of Jahn-Teller-like distortion.
We also find that the energy preference of AFM order over Jahn-Teller-like
distortion is enhanced in an oscillatory way as the length of DB wires
decreases, showing the so-called quantum size effects.
