It is predicted that Majorana zero modes can be produced in condensed matter systems.^[1–8] To search for Majorana zero modes, various experiments have been carried out on topological insulator (TI) or related, material-based devices, reporting the observation of, for examples, a zero bias conductance peak (ZBCP),^[9–16] and a component of 4π-periodic current-phase relations.^[17–23] However, so far there is still a lack of a consensus that Majorana fermions have been found. More experimental evidence is still needed to fully confirm the existence of Majorana zero modes.

For S–TI–S type Josephson junctions (where S denotes s-wave superconductor), the Andreev reflection of helical electrons on the TI surface is topologically protected to be fully transparent. When the phase difference of the junction reaches π, the two lowest-energy branches of Andreev bound states cross with each other at the fermi level, resulting in the closing of the minigap^[7,24] and thus the formation of 4π-periodic energy-phase relations (EPRs).^[25–32] Previously, we have already observed gap closing and 2π-periodic but fully skewed EPRs in rf-SQUIDs^[33] as well as in single Josephson junction^[34] constructed on the surface of Bi₂Te₃. However, the location of gap closing, which is of key importance in braiding Majoranas for the purpose of quantum computation, still needs to be investigated.

In this paper, we report our investigations on the locations of gap closing as a function of magnetic flux in single Josephson junctions constructed on a Bi₂Te₃ surface. The results verify that the positions of gap closing, hence the Majoranas, can be manipulated by varying magnetic flux, in a manner proposed in our early studies.^[33,34]

Pb Josephson junction was fabricated on the surface of Bi₂Te₃ with five normal metal Pd electrodes distributed along the junction area to measure the local proximity induced minigap. The device was cooled down to 10 mK in a dilution refrigerator (Triton 200, Oxford Instruments) and measured by using lock-in amplifier technique. Fraunhofer diffraction pattern was measured with ac excitation current of 0.5 μA, and local contact resistance of Pd electrodes was measured by using a three-terminal measurement configuration.

The devices used in this experiment contain single Josephson junction constructed on the surface of Bi₂Te₃, with normal-metal electrodes attached to the junction area to detect the local contact resistance. Figure 1(a) shows the scanning electron microscope (SEM) image of such a typical device. Two Pb electrodes separated by 1.35 μm were firstly fabricated on the surface of exfoliated Bi₂Te₃ flake of ∼100 nm in thickness. Then, over-exposed PMMA with evenly-spaced five windows of 600 nm in diameter was fabricated on the top of the Bi₂Te₃ surface and the Pb electrodes. Finally, Pd electrodes were deposited and contacted with the Bi₂Te₃ surface through the windows of the PMMA. For the device shown in Fig. 1(a), four of the five Pd electrodes worked well at low temperatures, labeled as A, B, C, and D. Electrodes A and B are located at the two ends of the junction. Electrodes C and D are located at 1/4 and 3/4 positions of the junction along the width direction.

Fig. 1.

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Fig. 1. (a) SEM image of a typical device used in this experiment, containing a Pb-Bi2Te3-Pb Josephson junction and five Pd electrodes connected to Bi2Te3 in the junction area. The black area is over-exposed PMMA to isolate the rest part of the Pd electrodes from touching the Pb films and the Bi2Te3 surface. (b) 2D plot of dV/dIJJbias of the Josephson junction as a function of magnetic field and IJJbias, the bias supercurrent of the Josepshon junction, showing a Fraunhofer-like pattern. The red color represents the zero-resistance state.

Figure 1(b) shows the 2D color plot of the differential resistance of Pb-Bi₂Te₃-Pb Josephson junction as a function of magnetic field B and bias current I_JJbias passing through the junction. The red-colored area represents the zero-resistance state (≤ 0.02 Ω). This demonstrates an equal-period oscillation, indicating that the Josephson junction is a large junction compared to the Josephson penetration depth, so that the supercurrent flows mostly near the two edges of the junction, interfering like in a dc SQUID. The oscillation period is ΔB = 1.3±0.3 G, corresponding to an effective area of S_eff = ϕ₀/ΔB = 15.9 μm² (where ϕ₀ = h/2e is flux quantum, e is the electron charge, and h is Planck’s constant). This estimated area S_eff is in good agreement with the geometric area of the junction 6.7 × 2.4 = 16.08 μm² after considering flux compression (see the supplementary materials in Refs. [33] and [34].

The Josephson penetration depth of the device is^[35] λ_J = (4πeμ₀dj/h)^−1/2 ≈ 3.8 μm, where μ₀ is the magnetic constant, d is the distance between the two Pb electrodes, j = I_c/W_t is the supercurrent density, I_c = 9 μA is the critical supercurrent of the Josephson junction extracted from Fig. 1(b), W = 6.7 μm is the width of the junction, and t ≈ 100 nm is the thickness of the Bi₂Te₃ flake. It can be seen that the λ_J is indeed less than the width of the junction, leading the junction to be marginally a large junction. The penetrated area defines a Fraunhofer-like envelope for the interference pattern, with the first node field located at ∼2.3 G, as marked by the black arrows in Fig. 1(b).

The contact resistance between Pd electrodes and Bi₂Te₃ at positions A, B, C, and D was measured by using a three-terminal measurement configuration. Depending on detailed combination of the electrodes, the signal we measured might not only contain the contact resistance but also a part of the sample resistance. Nevertheless, the sample resistance was several orders of magnitude smaller than the contact resistance and thus its contribution could be neglected. Figs. 2(a)–2(d) show the 2D color plots of the contact resistance dV/dI_b measured at positions A, B, C, and D, respectively, as functions of magnetic field B and bias current I_b. Clear oscillations can be seen at all positions, with the same period as the Fraunhofer pattern shown in Fig. 1(b), indicating that the contact resistance is influenced by the EPR of the junction.

Fig. 2.

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Fig. 2. (a)–(d) 2D color plots of the contact resistance dV/dIb of positions A, B, C, and D, respectively, as functions of magnetic field and bias current, when IJJbias = 0 and T ≈ 10 mK. (e)–(h) The vertical line cuts in panels (a)–(d), respectively, at places marked by the arrows of the same color in corresponding color plots. The unit of the number in the legends is Gauss. (i)–(l) Horizontal line cuts in panels (a)–(d), respectively, at places marked by the arrows of the same color in corresponding color plots.

According to our previous studies,^[33,34] the measured oscillations of the contact resistance reflect the oscillation of the superconducting minigap in Bi₂Te₃ beneath the Pd electrodes. According to the theory,^[7] the energies of the Andreev bound states in a Josephson junction constructed on TI, including the lowest-energy bound states (i.e., the minigap), oscillate against the flux. We believe that the oscillation of contact resistance is caused by the oscillation of the minigap in Bi₂Te₃.

From Fig. 2 we see that the contact resistance that oscillates can either be suppressed (i.e., the whitish area in Figs. 2(a), 2(b), and 2(d)), or be enhanced (i.e., the dark-blue area in Fig. 2(c)). This is understandable within the Blonder–Tinkham–Klapwijk (BTK) theory.^[36] When a normal-metal electrode is attached to the surface of a superconductor, the contact resistance within the gap energy can be either high or low depending on the interfacial barrier. In either case, the oscillation of the gap can be reflected on the contact resistance. From the data shown in Fig. 2, the size of the minigap in Bi₂Te₃ ranges from 20 to 50 μV, which is consistent with our previous observations on similar devices.^[33,34] Apart from this, we notice from Figs. 2(b) and 2(d) that there is a zero-bias resistance peak which barely oscillates. This resistance peak can be more clearly seen from the vertical line cuts (Figs. 2(f) and 2(h)). It is likely caused by the reentrant resistance effect.^[37]

Here we note that the current applied to the Pd electrodes for measuring the minigap are small enough in both the high- and low-contact resistance cases compared to the critical supercurrent of the Josephson junction. For position D whose contact resistance is much smaller than that of the other three Pd electrodes and thus the applied measuring current is relatively lager, the bias current I_b (≤ 0.15 μA) and the ac excitation current (5 nA for position D) were both much less than the critical supercurrent of the Josephson junction (9 μA). Such small probing current would not change the current or phase distribution of the Josephson junction and the minigap.

Figure 2 also shows that the contact resistance dV/dI_b at positions A, B, C, and D all jumps simultaneously at the node fields of the Fraunhofer pattern. The jump can be clearly seen at the horizontal line cuts shown in Figs. 2(i)–2(l). At positions A and B, dV/dI_b shows a fully skewed behavior, i.e., the minigap decreases gradually with varying field, and finally gets closed at the node fields of the Fraunhofer pattern, then jump abruptly to a pronounced value. The evolution of minigap against magnetic field can also be clearly seen from the vertical line cuts shown in Figs. 2(e) and 2(f). The red curves in these figures were taken at the fields indicated by the red arrows in the corresponding 2D color plots, showing nearly a gapless feature. The result is consistent with our previous finding.^[34]

In contrast to the minigap at positions A and B, the minigap at positions C and D exhibits a step-like dependence against magnetic field, with jumps at the node fields of the Fraunhofer pattern, as shown in the 2D color plots (Figs. 2(c) and 2(d)) and the corresponding horizontal line cuts (Figs. 2(k) and 2(l)). The minigap fades out beyond the 2^nd node of the Fraunhofer pattern. The red curves in Figs. 2(g) and 2(h) are the vertical line cuts taken at the fields indicated by the red arrows in the corresponding 2D color plots, demonstrating a relatively weak gap structure.

To calculate the minigap, we need to know the phase difference of the Josephson junction. When I_JJbias = 0, the local phase difference can be written as^[33,34]

Figures 3(a)–3(c) show the phase distribution lines of the junction near the first, the second, and the third node of the Fraunhofer pattern, respectively. The gray arrows indicate the expected π phase shift of the phase distribution lines near the nodes of Fraunhofer pattern.^[34] The red circles represent the position where φ(x) reaches odd multiples of π, so that the local minigap is expected to be closed. With the variation of flux in the junction area, the phase distribution line rotates about the center of the junction, resulting in the change of the minigap as well as the motion of the gap-closing points in the junction. It can be seen that the minigap at the center of the junction is unchanged with varying flux (the minigap is either maximized or close). At positions A and B (x = ±W/2), φ(x) reaches odd multiples of π at every node of the Fraunhofer pattern. At positions C and D (x = ±W/4), φ(x) reaches π only beyond the second node of the Fraunhofer pattern.

Fig. 3.

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Fig. 3. (a)–(c) The phase distribution of the junction as a function of position x before/after the jumping when the flux in the junction reaches ϕ0, 2ϕ0, and 3ϕ0, respectively. The phase distribution lines undergo π phase shift, as indicated by the gray arrows. The red circles represent the place where the local φ(x) hits odd multiples of π so that the minigap is closed. (d) The calculated minigap Δ at position A/B (i.e., at x = ±W/2). (e) The calculated Δ at position C/D (i.e., at x = ±W/4).

It is known that the minigap is defined by the energy separation between the two lowest-energy Andreev bound state, the latter has the form of:^[38,39] E_n = ±Δ₀[1 – D₀sin²(ϕ/2)]^1/2, where E_n and D_n are the energy and transmission coefficient of the n^th mode of quasiparticles, respectively. As has been pointed out,^[33,34] the time-reversal symmetry protected helical surface states are immune from backscattering, leading to the full transparency (i.e., D_n = 1) and the appearance of topologically-protected perfect Andreev reflection. In this case, the minigap can be fully closed and described by the following expression:

Figures 3(d) and 3(e) show the calculated minigap at positions A/B and C/D, respectively, by using Eqs. (1) and (2). Since the strength of the minigap decays with magnetic field, in these two figures we have phenomenologically imposed a Gaussian-like envelope e^{−γ(ϕ/ϕ₀)2} (where γ = 0.2) for the minigap. The results excellently mimic the data shown in Fig. 2. This confirms that the Josephson junctions constructed on TI surface are indeed fully transparent to quasiparticle transport so that Majoranas can be hosted, and that the gap-closing points in the junction, where Majoranas are supposed to be held, can be predicted and manipulated by varying magnetic flux via the formulas listed above.

For topologically-trivial states such as the bulk states of Bi₂Te₃, their Andreev bound states are located at relatively higher energies compared to that of the topologically-nontrivial surface states, thus will not influence the process of minigap closing.

We note that the observed gap closing is often incomplete. This is because that the minigap closes completely only at certain x coordinates precisely, whereas the local Pd electrodes used to detect gap closing has a finite size (600 nm in diameter).^[33] Position uncertainty of gap closing caused by thermal smearing at ∼10 mK places additional difficulty to align the Pd electrodes to the gap-closing points.^[33]

To conclude, we have measured the local minigap in single Josephson junctions constructed on Bi₂Te₃ surface, as a function of position x and magnetic flux. Gap closing is observed and can only be explained within the scenario that the quasiparticle transport is fully transparent in the junction area. The results support the theoretical proposals that Majorana zero modes can be hosted in Josephson junctions constructed on the TI surface. Our results further verify that the places of gap closing can be predicted by using Eqs. (1) and (2), based on which future braiding of Majorana fermions could be designed.