  We study properly embedded and immersed p(pseudohermitian)-minimal surfaces
in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang,
Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There
are two types of such surfaces: band type and annulus type according to their
topology. We give an explicit expression for these surfaces. Among band types
there is a class of properly embedded p-minimal surfaces of so called helicoid
type. We classify all the helicoid type p-minimal surfaces. This class of
p-minimal surfaces includes all the entire p-minimal graphs (except contact
planes) over any plane. Moreover, we give a necessary and sufficient condition
for such a p-minimal surface to have no singular points. For general complete
immersed p-minimal surfaces, we prove a half space theorem and give a criterion
for the properness.
