Topological insulator nanoribbon Josephson junctions: evidence for size effect in transport properties

We have used Bi$_2$Se$_3$ nanoribbons, grown by catalyst-free Physical Vapor Deposition to fabricate high quality Josephson junctions with Al superconducting electrodes. In our devices we observe a pronounced reduction of the Josephson critical current density $J_c$ by reducing the width of the junction, which in our case corresponds to the width of the nanoribbon. Because the topological surface states extend over the entire circumference of the nanoribbon, the superconducting transport associated to them is carried by modes on both the top and bottom surfaces of the nanoribbon. We show that the $J_c$ reduction as a function of the nanoribbons width can be accounted for by assuming that only the modes travelling on the top surface contribute to the Josephson transport as we derive by geometrical consideration. This finding is of a great relevance for topological quantum circuitry schemes, since it indicates that the Josephson current is mainly carried by the topological surface states.