**Low-energy Electrodynamics of 3D Topological Insulators**

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Topological insulators (TIs) are a recently discovered state of matter characterized by an “inverted” band structure driven by strong spin-orbit coupling. One of their most touted properties is the existence of robust “topologically protected” surface states. I will discuss what topological protection means for transport experiments and how it can be probed using the technique of time- domain THz spectroscopy applied to 3D TI thin films of Bi2Se3. By measuring the low frequency optical response, we can follow their transport lifetimes as we drive these materials via chemical substitution through a quantum phase transition into a topologically trivial regime [1]. I will then discuss our work following the evolution of the response as a function of magnetic field from the semi-classical transport regime [2] to the quantum regime [3]. In the semi-classical regime, an anomalous increase of the transport scattering rate in cyclotron resonance measurements was observed at high field, which contribute from electron-phonon interaction [2]. In the highest quality samples [3,4], we observe a continuous crossover from a low field regime where the response is given by semi-classical transport in the form of cyclotron resonance to a higher-field quantum regime[3]. In the later case, although DC transport is still semi-classical, we find evidence for Faraday and Kerr rotation angles quantized in units of the fine structure constant [3]. A nontrivial offset of these values provides a measure of the axion angle and therefore an evidence for a novel magneto-electric of the TI’s surface e.g. the much heralded axion electrodynamics of 3D topological insulators. Among other aspects this give a purely solid-state measure of the fine structure constant as a topological invariant for the first time [3].

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