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Three-dimensional topological insulators (TIs) host helical Dirac surfacestates at the interface with a trivial insulator. In quasi-one-dimensional TInanoribbon structures the wave function of surface charges extendsphase-coherently along the perimeter of the nanoribbon, resulting in aquantization of transverse surface modes. Furthermore, as the inherentspin-momentum locking results in a Berry phase offset of $\pi$ ofself-interfering charge carriers an energy gap within the surface statedispersion appears and all states become spin-degenerate. We investigate andcompare the magnetic field dependent surface state dispersion in selectivelydeposited Bi$_2$Te$_3$ TI micro- and nanoribbon structures by analysing thegate voltage dependent magnetoconductance at cryogenic temperatures. While inwide microribbon devices the field effect mainly changes the amount of bulkcharges close to the top surface we identify coherent transverse surface statesalong the perimeter of the nanoribbon devices responding to a change in topgate potential. We quantify the energetic spacing in between these quantizedtransverse subbands by using an electrostatic model that treats an initialdifference in charge carrier densities on the top and bottom surface as well asremaining bulk charges. In the gate voltage dependent transconductance we findoscillations that change their relative phase by $\pi$ at half-integer valuesof the magnetic flux quantum applied coaxial to the nanoribbon, which is asignature for a magnetic flux dependent topological phase transition in narrow,selectively deposited TI nanoribbon devices.

Gate-induced decoupling of surface and bulk state properties in selectively-deposited Bi$_2$Te$_3$ nanoribbons