  We study transport across either a potential or a magnetic barrier which is
placed on the top surface of a three-dimensional thin topological insulator
(TI). For such thin TIs, the top and bottom surfaces interact via a coupling
$\lambda$ which influences the transport properties of junctions constructed
out of them. We find that for junctions hosting a potential barrier, the
differential conductance oscillates with the barrier strength. The period of
these oscillations doubles as the coupling $\lambda$ changes from small values
to a value close to the energy of the incident electrons. In contrast, for
junctions with a magnetic barrier, the conductance approaches a non-zero
constant as the barrier strength is increased. This feature is in contrast to
the case of transport across a single TI surface where the conductance
approaches zero as the strength of a magnetic barrier is increased. We also
study the spin currents for these two kinds of barriers; in both cases, the
spin current is found to have opposite signs on the top and bottom surfaces.
Thus this system can be used to split applied charge currents to spin currents
with opposite spin orientations which can be collected by applying opposite
spin-polarized leads to the two surfaces. We show that several of these
features of transport across finite width barriers can be understood
analytically by studying the $\delta$-function barrier limit. We discuss
experiments which may test our theory.
