Optical conductivity of exfoliated Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ nanocrystals

We report on infrared spectromicroscopy of mechanically exfoliated under-doped and optimally-doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ thin crystals on SiO$_2$/Si substrates. The infrared reflectance and transmission was measured for samples of various thicknesses and the the optical conductivity extracted in the frequency range 0.15 eV to 1 eV. Trends in the optical conductivity with thickness are discussed. In particular, we observe that the conductivity of thicker ($\sim$ 100 nm) samples is comparable to bulk while that of thinner ($\sim$ 20 nm) samples is markedly suppressed.