  By combining $n$-type $\mathrm{Bi_2Te_3}$ and $p$-type $\mathrm{Sb_2Te_3}$
topological insulators, vertically stacked $p$-$n$ junctions can be formed,
allowing to position the Fermi level into the bulk band gap and also tune
between $n$- and $p$-type surface carriers. Here we use low-temperature
magnetotransport measurements to probe the surface and bulk transport modes in
a range of vertical $\mathrm{Bi_2Te_3/Sb_2Te_3}$ heterostructures with varying
relative thicknesses of the top and bottom layers. With increasing thickness of
the $\mathrm{Sb_2Te_3}$ layer we observe a change from $n$- to $p$-type
behavior via a specific thickness where the Hall signal is immeasurable.
Assuming that the the bulk and surface states contribute in parallel, we can
calculate and reproduce the dependence of the Hall and longitudinal components
of resistivity on the film thickness. This highlights the role played by the
bulk conduction channels which, importantly, cannot be probed using surface
sensitive spectroscopic techniques. Our calculations are then buttressed by a
semi-classical Boltzmann transport theory which rigorously shows the vanishing
of the Hall signal. Our results provide crucial experimental and theoretical
insights into the relative roles of the surface and bulk in the vertical
topological $p$-$n$ junctions.
