  Three-dimensional topological insulators (TIs) host helical Dirac surface
states at the interface with a trivial insulator. In quasi-one-dimensional TI
nanoribbon structures the wave function of surface charges extends
phase-coherently along the perimeter of the nanoribbon, resulting in a
quantization of transverse surface modes. Furthermore, as the inherent
spin-momentum locking results in a Berry phase offset of $\pi$ of
self-interfering charge carriers an energy gap within the surface state
dispersion appears and all states become spin-degenerate. We investigate and
compare the magnetic field dependent surface state dispersion in selectively
deposited Bi$_2$Te$_3$ TI micro- and nanoribbon structures by analysing the
gate voltage dependent magnetoconductance at cryogenic temperatures. While in
wide microribbon devices the field effect mainly changes the amount of bulk
charges close to the top surface we identify coherent transverse surface states
along the perimeter of the nanoribbon devices responding to a change in top
gate potential. We quantify the energetic spacing in between these quantized
transverse subbands by using an electrostatic model that treats an initial
difference in charge carrier densities on the top and bottom surface as well as
remaining bulk charges. In the gate voltage dependent transconductance we find
oscillations that change their relative phase by $\pi$ at half-integer values
of the magnetic flux quantum applied coaxial to the nanoribbon, which is a
signature for a magnetic flux dependent topological phase transition in narrow,
selectively deposited TI nanoribbon devices.
