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.


I. INTRODUCTION
The study of the proximity effect between a superconductor and a semiconductor or an unconventional metal, has lately received a dramatic boost due to the increasing possibilities to manufacture a larger variety of interfaces and materials. Novel phenomenology of the proximity effect is currently coming from the integration of semiconducting nanowires, with strong spin orbit coupling, as barriers, as well as the edge and surface states of twodimensional (2D) and three-dimensional (3D) Topological Insulators (TIs) [1][2][3][4][5] , and Dirac semimetals 6,7 . In these cases, the Josephson transport properties of the hybrid devices would manifest neat fingerprints related to the formation of Majorana bound states, of great interest for topological quantum computation [8][9][10] . Lately Superconductor-TI-Superconductor Josephson junctions with 2D 11 and 3D TIs have shown a 4π periodic Josephson current phase relations [12][13][14] which could be associated to the presence of Majorana modes 15 and a gate-tunable Josephson effects 2, [16][17][18][19] when the TI is tuned through the TI's Dirac point. Still several aspects of the physics of the Josephson effect related to the topological protected edge/surface states and the contribution of the unavoidable bulk remain to be clarified. In this respect the use of 3D TI nanoribbons could be advantageous because of the reduced number of transport channels involved in the transport. Here the transport is ruled by the quantization of the nanoribbon's propagation modes, which could give new hints about the Josephson phenomenology associated to the Topological Surface States (TSS).
In this work, we have fabricated Josephson junctions by using Bi 2 Se 3 Topological Insulator Nanoribbons (TINR) with a widths spanning from 50 nm to almost a micron. Because the TSS extend over the entire circumference of the TINR, the superconducting transport associated to them is carried by modes on both the top and bottom surface of the nanoribbon. As shown in Fig. 1a, in our TINR Josephson junction the current flows between the superconducting contacts fabricated on the top surface of the nanoribbon. For the TINR with a circumference C = 2W + 2t (where W is the width and t the thickness of the nanoribbon) the transverse momentum k y , perpendicular to the current (see Fig. 1), is quantized as: where n is an integer 21 . Therefore, the modes with k y ∼ 0 remain on the top surface while the modes with k y 0 are winding around the perimeter of the TINR (see Fig. 1a). Here 2 we show that the value of the Josephson current density strongly depends on the junction width, that in our case corresponds to the width of the nanoribbons (Fig. 1a). We discuss the possible origin of this phenomenology also in connection with the fact that only small k y value modes are involved in the Josephson transport, that are the ones which travel on the top surface of the nanoribbon. This number reduces by reducing the nanoribbon width (see Eq. 1). This finding is of a great relevance since it indicates that a) the Josephson current is mainly carried by the surface states and b) because of the selectivity in k y , the number of modes involved in the transport scales much faster compared to the width. This is of great relevance for topological quantum circuitry schemes.  of a dc biased Josephson junction 31,32 . Here the capacitance is dominated by the shunting capacitance through the substrate.

Nanoribbons of Bi
For STO with a relative dielectric contant of 25000 at low temperatures we approximate C 1 pF resulting in a quality factor smaller than one for junctions having a critical current lower than 50 nA. Instead for the Si/SiO 2 substrate it is more difficult to estimate the shunting capacitance due to the conducting substrate. However we expect a much smaller capacitance value resulting in quality factors smaller than one for critical currents already below 500 nA.
The value of the critical current of the junction I c is obtained from the forward scan, and the critical current density J c is calculated accordingly by dividing I c by the width of the nanoribbon (W ). The normal state resistance R N is determined by the inverse of the slope, calculated from the IVC region at voltages above 2∆ in S (represented by the area of Bi 2 Se 3 underneath the Al). Fig. 3a,b show the dependence of the J c as a function of the W for devices with different lengths L fabricated on Si/SiO 2 and STO substrates, respectively. In the length of TINR; for this device no winding modes are expected to contribute to the Josephson transport. One clearly sees that if the length is fixed the J c sharply decreases as a function of the width W of the nanoribbon. We note that the value of J c can change by a factor of 5-6 by going from the narrowest nanoribbons of 60 nm to the widest ones. In contrast, as shown in Fig. 4a,b, the specific resistance, obtained by considering the product R N × (W ) for the devices on the Si/SiO 2 and STO substrate, respectively, is almost independent of W . This fact allows to exclude that the reduction of the J c at small widths W has its origin in strong modifications/deterioration of the junction specific resistance for narrow TINR.

III. DISCUSSION AND CONCLUSIONS
What is the origin of this peculiar J c (W ) phenomenology? As we have discussed earlier, the explanation of the phenomenon can have its grounds in the quantization of the nanoribon's propagation modes.
One can derive that the relative number of modes n top /n tot travelling only on the top surface reduces with the junction width for a fixed junction length L. Here n tot is the total number of modes: and n top is the number of modes travelling only on the top surface of the nanoribbon: The relations above can be obtained from geometric considerations, see Fig. 5a. Here k F is the Fermi vector.
In Fig. 5b  calculation of J C and the relative number of transport modes using equation 2 and 3 give the same qualitative behaviour.
This dependency qualitatively reproduces the measured J c vs width dependence shown in Fig. 3 a,b. Indeed, the solid lines represent the relative number of transport modes for L = 100 nm. This suggests that only the modes travelling on the top surface contribute to the Josephson current. We note that critical current density of the 10 µm wide device is in agreement with the saturation value of the expected current density, where the contribution 8 of winding modes to the total Josephson current is negligible.
A possible explanation for this finding could be related to the lower mobility of the Dirac states at the interface between the TINR and the substrate. Indeed, our magnetotransport measurements have shown the formation of a trivial 2D gas at the interface with the substrate overlapping with the Dirac states at the nanoribbon bottom (see inset Fig. 1b) 20 .
This interaction, which leads to a lower mobility (diffusive transport regime) of the Dirac states, might be responsible for the transport modes winding around the nanoribbon (high k y modes) contributing less to the Josephson transport. Although we can not exclude that the transport through the trivial 2D gas contributes to the Josephson current it would only cause a constant offset of the J C values without affecting the overall width dependence. We observe that the overall values of J C for the devices on the STO substrate are higher than those on Si/SiO 2 . This can be partially attributed to the larger values of the top surface Dirac carrier density of the batch used to realize the devices on the STO substrate. The larger value of the trivial 2D carrier density we typically observe in devices fabricated on STO substrates 33 could be further responsible for the difference observed in the critical current densities between devices on STO and Si/SiO 2 substrates.
To conclude we have fabricated high transparency 3D TINR Josephson junctions showing a peculiar phenomenology that can be associated to the transport through the topological surface states. This is a step forward towards the study of topological superconductivity in few modes devices instrumental for topological quantum computation.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.