3. Coulomb blockade in gold nanoparticles. The capacitance of a nan...

3. Coulomb blockade in gold nanoparticles. The capacitance of a nanoparticle is given as 𝐶 = 𝜋𝜋𝜀0 𝑟 𝑑 (𝑟 + 𝑑). where 𝜀 is the dielectric constant of the ligand shell of thickness d, and r is the core radius. (a) For a single-nanoparticle STM experiment, the charging energy, 𝑈 = 𝑒2 2𝐶 . In order to observe the Coulomb blockade, thermally-driven electron tunneling must be prevented, which is the requirement 𝑘𝐵𝑇 ≪ 𝑈. Practically, this requirement means𝑘𝐵𝑇𝑚𝑚𝑚 ≈ 𝑈/25, where 𝑇𝑚𝑚𝑚 is the maximum temperature at which the Coulomb blockade is observable. Find an expression for 𝑇𝑚𝑚𝑚 in terms of 𝑟, 𝑑 and 𝜀. (b) Consider octanethiolate-stabilized Au nanoparticles, for which 𝜀 = 0.8 and 𝑑 = 1.0 inn. Plot 𝑇𝑚𝑚𝑚 vs 𝑟 for radii from 1 nm to 10 nm. (c) For which radius is 𝑇𝑚𝑚𝑚 room temperature (293 K)? (An estimate from your graph is OK.)