  We present a new model to explain the difference between the transport and
spectroscopy gaps in samarium hexaboride (SmB$_6$), which has been a mystery
for some time. We propose that SmB$_6$ can be modeled as an intrinsic
semiconductor with a depletion length that diverges at cryogenic temperatures.
In this model, we find a self-consistent solution to Poisson's equation in the
bulk, with boundary conditions based on Fermi energy pinning due to surface
charges. The solution yields band bending in the bulk; this explains the
difference between the two gaps because spectroscopic methods measure the gap
near the surface, while transport measures the average over the bulk. We also
connect the model to transport parameters, including the Hall coefficient and
thermopower, using semiclassical transport theory. The divergence of the
depletion length additionally explains the 10-12 K feature in data for these
parameters, demonstrating a crossover from bulk dominated transport above this
temperature to surface-dominated transport below this temperature. We find good
agreement between our model and a collection of transport data from 4-40 K.
This model can also be generalized to materials with similar band structure.
