Dynamical electron compressibility in the 3D topological insulator Bi$_2$Se$_3$

Measurements of the quantum capacitance $c_q$, related to the electron compressibility $\chi=c_q/e^2$ is a sensitive tool to probe the density of states. In a topological insulator (TI) the situation is enriched by the coexistence and the interplay of topologically protected surface states and massive bulk carriers. We investigate top-gate metal-oxyde-TI capacitors using Bi$_2$Se$_3$ thin crystals at GHz frequencies. These measurements provide insight into the compressibillity of such a two electron-fluid system. Furthermore, the dynamical response yields information about electron scattering properties in TIs. More specifically, in our measurements we track simultaneously the conductivity $\sigma$ and the compressibility as a function of a DC-gate voltage. Using the Einstein relation $\sigma = c_q D$, we have access to the gate dependence of the electron diffusion constant $D(V_g)$, a signature of the peculiar scattering mechanisms in TIs.