  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. Hall
measurements on microribbon field effect devices show a high bulk charge
carrier concentration and electrostatic simulations show an inhomogeneous gate
potential profile on the perimeter of the TI ribbon. In nanoribbon devices we
identify a magnetic field dependency of the surface state dispersion as it
changes the occupation of transverse subbands close to the Fermi energy. We
quantify the energetic spacing in between these subbands by measuring the
conductance as a function of the applied gate potential and use an
electrostatic model that treats the inhomogeneous gate profile and the initial
charge carrier densities on the top and bottom surface. 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 providing evidence for a magnetic flux dependent
topological phase transition in narrow, selectively deposited TI nanoribbon
devices.
