  The topological insulator BiSbTeSe$_2$ corresponds to a compensated
semiconductor in which strong Coulomb disorder gives rise to the formation of
charge puddles, i.e., local accumulations of charge carriers, both in the bulk
and on the surface. Bulk puddles are formed if the fluctuations of the Coulomb
potential are as large as half of the band gap. The gapless surface, in
contrast, is sensitive to small fluctuations but the potential is strongly
suppressed due to the additional screening channel provided by metallic surface
carriers. To study the quantitative relationship between the properties of bulk
puddles and surface puddles, we performed infrared transmittance measurements
as well as scanning tunneling microscopy measurements on the same sample of
BiSbTeSe$_2$, which is close to perfect compensation. At 5.5 K, we find surface
potential fluctuations occurring on a length scale $r_s = 40-50$ nm with
amplitude $\Gamma = 8-14$ meV which is much smaller than in the bulk, where
optical measurements detect the formation of bulk puddles. In this nominally
undoped compound, the value of $\Gamma$ is smaller than expected for pure
screening by surface carriers, and we argue that this arises most likely from a
cooperative effect of bulk screening and surface screening.

Mesoscale and Nanoscale Physics