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Infrared electrodynamics and ferromagnetism in the topological semiconductors Bi$_2$Te$_3$ and Mn-doped Bi$_2$Te$_3$

We report on infrared (IR) optical experiments on Bi$_2$Te$_3$ and Mn-doped Bi$_2$Te$_3$ epitaxial thin films. In the latter film, dilute Mn doping (4.5\%) of the topologically nontrivial semiconductor host results in time-reversal-symmetry-breaking ferromagnetic order below $T_C$=15 K. Our spectroscopic study shows both materials share the Bi$_2$Te$_3$ crystal structure, as well as classification as bulk degenerate semiconductors. Hence the Fermi energy is located in the Bi$_2$Te$_3$ conduction band in both materials, and furthermore, there is no need to invoke topological surface states to describe the conductivity spectra. We also demonstrate that the Drude oscillator strength gives a simple metric with which to distinguish the possibility of topological surface state origins of the low frequency conductance, and conclude that in both the pristine and Mn-doped Bi$_2$Te$_3$ samples the electromagnetic response is indeed dominated by the bulk material properties, rather than those of the surface. An encouraging aspect for taking advantage of the interplay between nontrivial topology and magnetism, however, is that the temperature dependence of the Mn-doped Bi$_2$Te$_3$ film suggests bulk charge carriers do not play a significant role in mediating ferromagnetism. Thus, a truly insulating bulk may still be suitable for the formation of a ferromagnetic ground state in this dilute magnetic topological semiconductor.