  Tensile strained bulk HgTe is a three-dimensional topological insulator.
Because of the energetic position of its surface state Dirac points relative to
its small bulk gap, the electronic properties of the relatively thin MBE-grown
films used to study this material experimentally are quite sensitive to details
of its electrostatic band-bending physics. We have used an 8-band $\mathbf{k}
\cdot \mathbf{p}$ model to evaluate the gate voltage dependence of its
thin-film two-dimensional subbands and related thermodynamic and transport
properties in films with thicknesses between 30 and 70nm, accounting
self-consistently for gate field screening by the topologically protected
surface states and bulk state response. We comment on the effective dielectric
constant that is appropriate in calculations of this type, arguing for a
smaller value $\varepsilon_r$ $\approx$ 6.5 than is commonly used. Comparing
with recent experiments, we find that our fully microscopic model of gate field
screening alters the interpretation of some observations that have been used to
characterize strained HgTe thin films.
