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We have used Bi$_2$Se$_3$ nanoribbons, grown by catalyst-free Physical VaporDeposition to fabricate high quality Josephson junctions with Alsuperconducting electrodes. In our devices we observe a pronounced reduction ofthe Josephson critical current density $J_c$ by reducing the width of thejunction, which in our case corresponds to the width of the nanoribbon. Becausethe topological surface states extend over the entire circumference of thenanoribbon, the superconducting transport associated to them is carried bymodes 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 byassuming that only the modes travelling on the top surface contribute to theJosephson transport as we derive by geometrical consideration. This finding isof a great relevance for topological quantum circuitry schemes, since itindicates that the Josephson current is mainly carried by the topologicalsurface states.

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