Topological insulator nanoribbon Josephson junctions: Evidence for size effects in transport properties

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 two-dimensional (2D) and three-dimensional (3D) Topological Insulators (TIs)^1–5 and Dirac semimetals.^6,7 In these cases, the Josephson transport properties of the hybrid devices will manifest neat fingerprints related to the formation of Majorana bound states, which is of great interest for topological quantum computation.^8–10 Lately, superconductor–TI–superconductor Josephson junctions with 2D and 3D TIs have shown a 4$π$ periodic Josephson current phase relations,^11–14 which could be associated with the presence of Majorana modes¹⁵ and gate-tunable Josephson effects^2,16–19 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 with the Topological Surface States (TSSs).

In this work, we have fabricated Josephson junctions by using $Bi2Se3$ Topological Insulator Nanoribbons (TINRs) with widths spanning from 50 nm to almost a micron. Because the TSSs extend over the entire circumference of the TINR, the superconducting transport associated with them is carried by modes on both the top and bottom surface of the nanoribbon. As shown in Fig. 1(a), 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 of $C=2W+2t$ (where $W$ is the width and $t$ the thickness of the nanoribbon), the transverse momentum $ky$⁠, perpendicular to the current (see Fig. 1), is quantized as

where $n$ is an integer.²¹ Therefore, the modes with $ky∼0$ remain on the top surface, while the modes with $ky≫0$ are winding around the perimeter of the TINR [see Fig. 1(a)]. Here, we show that the value of the Josephson current density strongly depends on the junction width, which in our case corresponds to the width of the nanoribbons [Fig. 1(a)]. We discuss the possible origin of this phenomenology also in connection with the fact that only small $ky$ value modes are involved in the Josephson transport, which are the ones that travel on the top surface of the nanoribbon. This number reduces by reducing the nanoribbon width [see Eq. (1)]. This finding is of great relevance since it indicates that (a) the Josephson current is mainly carried by the surface states and (b) because of the selectivity in $ky$⁠, 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.

Nanoribbons of $Bi2Se3$ with a variation of widths from $∼50nm$ to about $1μm$ were grown as reported in Ref. 22. For the fabrication of Josephson junctions, nanoribbons were transferred to pre-patterned substrates of Si/300 nm $SiO2$ and $SrTiO3$ (STO). We used two different TINR growth batches for the two substrates, respectively. For the two batches, the carrier densities of the topological surface states measured through Shubnikov–de Haas oscillations varied by a factor of 2.²² A standard process of electron beam lithography was used, followed by the evaporation of 3 and 80 nm thick layers of Pt and Al, respectively. Before the evaporation of the metals, the samples were etched for 30 s by Ar ion milling to remove the native oxide layer of the nanoribbons. SEM images of the fabricated $Al/Bi2Se3/Al$ Josephson junctions for different nanoribbon widths are shown in Figs. 1(b) and 1(c). The Pt interlayer between the $Bi2Se3$ and Al has an important role in the formation of a transparent interface, as it was shown in our previous works.^3,23,24

Junctions were measured at a base temperature of 19 mK in an $rf$-filtered dilution refrigerator. The planar Josephson junction can be described as $SIS′I′−N−I′S′IS$⁠. Here, $S′$ is the proximized TINR that lies underneath the superconductor (S)—Al, and N represents the not-covered TINR part between the two Al electrodes [see schematics in the inset of Fig. 1(b)]. I is the interface of the barrier between the Al and the $Bi2Se3$ nanoribbon, while $I′$ represents the barrier interface formed between the $Bi2Se3$ under the Al and the $Bi2Se3$ of the normal metal region N (nanogap). We have previously demonstrated that both I and $I′$ are highly transparent^3,23 and that we have full control of the strength and phenomenology of the proximity effect. When the voltage drop across the $Al/Bi2Se3$ interface is negligible, compared to that across the TI in the nanogap, as in our case,²⁵ the physics of the planar junction is effectively that of a $S′I′−N−I′S′$⁠.

A typical current–voltage characteristic (IVC) of one of the TINR based Josephson junctions is shown in Fig. 2(a). The IVCs of our junctions have a hysteresis that points to an increased electron temperature as the junction is switched to the resistive state.²⁶ A clear gap structure is seen at $2Δ′$ (where $Δ′$ is the induced gap of the order of $170μV$⁠); several bumps in the IVC at $2Δ′/n$ are also visible: as we have previously demonstrated they are connected to multiple Andreev reflections that are made possible by the high transparency of the I and $I′$ barriers.²³ The IVCs measured at various temperatures are shown in Fig. 2(b). At higher temperatures, we observe a finite resistance in the supercurrent branch. This could be attributed to premature switching and consecutive retrapping of the superconducting phase difference.^27–30 The so-called phase diffusion regime is characteristic for a moderate to low quality factor Josephson junction. The quality factor can be estimated using $Q=ωPRC$⁠, with the plasma frequency $ωP=2πIc/Φ0C$⁠, where $Φ0=h/2e$ is the superconductive flux quantum, and $R≃100Ω$ is the real part of the shunting admittance of a dc biased Josephson junction.^31,32 Here, the capacitance is dominated by the shunting capacitance through the substrate. For STO with a relative dielectric constant of 25 000 at low temperatures, we approximate $C≃1pF$ resulting in a quality factor smaller than one for junctions having a critical current lower than 50 nA. Alternatively, for the $Si/SiO2$ 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 $Ic$ is obtained from the forward scan, and the critical current density $Jc$ is calculated accordingly by dividing $Ic$ by the width of the nanoribbon $(W)$⁠. The normal state resistance $RN$ is determined by the inverse of the slope, calculated from the IVC region at voltages above 2$Δ′$ in S$′$ (represented by the area of $Bi2Se3$ underneath the Al). Figures 3(a) and 3(b) show the dependence of the $Jc$ as a function of the $W$ for devices with different lengths $L$ fabricated on $Si/SiO2$ and STO substrates, respectively. In Fig. 3(b), we have included a $10μm$ wide junction where the Al electrodes are patterned along 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 $Jc$ sharply decreases as a function of the width $W$ of the nanoribbon. We note that the value of $Jc$ 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 Figs. 4(a) and 4(b), the specific resistance, obtained by considering the product $RN×(W)$ for the devices on the $Si/SiO2$ and STO substrate, respectively, is almost independent of $W$⁠. This fact allows one to exclude that the reduction of the $Jc$ at small widths $W$ has its origin in strong modifications/deterioration of the junction specific resistance for narrow TINR.

What is the origin of this peculiar $Jc$(⁠$W$⁠) phenomenology? As we have discussed earlier, the explanation of the phenomenon can have its ground in the quantization of the nanoribbon’s propagation modes. One can derive that the relative number of modes $ntop/ntot$ traveling only on the top surface reduces with the junction width for a fixed junction length $L$⁠. Here, $ntot$ is the total number of modes,

and $ntop$ is the number of modes traveling only on the top surface of the nanoribbon,

The relations above can be obtained from geometric considerations; see Fig. 5(a). Here, $kF$ is the Fermi vector.

In Fig. 5(b), we show the width dependence of the relative number of modes $ntop/ntot$ for three different junction lengths. The curves of Fig. 5(b) have been obtained by considering $kF=0.55nm−1$⁠, a value that is typical for our nanoribbons.²⁰ For $L=100nm$⁠, we obtain a reduction of the relative number of modes traveling only on the top surface by a factor of 5 when reducing the junction width going from 900 nm to 50 nm.

For comparison in Fig. 5, we also show the full calculation of the supercurrent density following Ref. 7 taking into consideration also the angle dependence of the transmission coefficients of each transport mode traveling only on the top surface. We see that the full calculation of $JC$ and the relative number of transport modes using Eqs. (2) and (3) give the same qualitative behavior.

This dependency qualitatively reproduces the measured $Jc$ vs width dependence shown in Figs. 3(a) and 3(b). Indeed, the solid lines represent the relative number of transport modes for $L=100nm$⁠. This suggests that only the modes traveling on the top surface contribute to the Josephson current. We note that the critical current density of the $10μm$ wide device is in agreement with the saturation value of the expected current density, where the contribution 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 the inset of Fig. 1(b)].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 $ky$ modes) contributing less to the Josephson transport. Although we cannot exclude that the transport through the trivial 2D gas contributes to the Josephson current, it would only cause a constant offset of the $JC$ values without affecting the overall width dependence. We observe that the overall values of $JC$ for the devices on the STO substrate are higher than those on $Si/SiO2$⁠. 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 substrates³³ could be further responsible for the difference observed in the critical current densities between devices on STO and $Si/SiO2$ substrates.

To conclude, we have fabricated high transparency 3D TINR Josephson junctions showing a peculiar phenomenology that can be associated with the transport through the topological surface states. This is a step forward toward the study of topological superconductivity in a few mode devices instrumental for topological quantum computation.

This work has been supported by the European Union’s Horizon 2020 Research and Innovation Program (Grant Agreement No. 766714/HiTIMe) and by the European Union’s project NANOCOHYBRI (Cost Action CA 16218). G.K. acknowledges the European Regional Development Fund project (No. 1.1.1.2/VIAA/1/16/198). This work was supported by the European Union H2020 under the Marie Curie Actions (No. 766025-QuESTech).

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