Gate-tuned Josephson effect on the surface of a topological insulator
© Bai and Yang; licensee Springer. 2014
Received: 22 July 2014
Accepted: 13 September 2014
Published: 20 September 2014
In the study, we investigate the Josephson supercurrent of a superconductor/normal metal/superconductor junction on the surface of a topological insulator, where a gate electrode is attached to the normal metal. It is shown that the Josephson supercurrent not only can be tuned largely by the temperature but also is related to the potential and the length of the weak-link region. Especially, the asymmetry excess critical supercurrent, oscillatory character, and plateau-like structure have been revealed. We except those phenomena that can be observed in the recent experiment.
Since the pioneering work of Kane and Mele , there has been a great deal of theoretical and experimental investigations concerning the exotic new phase of condensate matter-topological insulator (TI) [2–5]. Originally, a TI state (first termed as the quantum spin Hall phase) is prophesied in graphene based on the spin-orbit interaction and time-reversal symmetry . Shortly after TI state was first proposed in a 2-dimensional (2D) graphene, the amazing quantum state was theoretically proposed independently in HgTe quantum wells [6, 7] and the alloy Bi1-xSb x with a special range of x. Unlike the weak intrinsic spin-orbit coupling in graphene , the amazing TI states have been observed experimentally soon after the theoretical prediction [10, 11]. In general, we can first divide TI into two broad classes in a real-space picture: the 2D TI holding a pair of 1D edge states with Dirac-like dispersion and the 3D TI hosting the 2D massless Dirac fermion states on the surface. In the 3D case, the weak TI and strong TI correspond to the even and odd number of Dirac cones on the surface [12–14]. Because the weak TI is adiabatically connected to stacked layers of 2D TI, the strong TI has received a surge of research activities due to the robustness of its surface states as a genuine new state of matter [2–5, 12–14].
An intriguing pitch came into the field when a superconductor is proximate to the surface of strong TI. By the proximity effect of superconductor on the surface state, Majorana fermions are predicted to occur [15, 16]. The appearance of such Majorana fermions is expected to lead to a number of unusual electronic properties such as zero-bias conductance anomalies [17, 18], non-Abelian statistics , electron teleportation , and so on. It is also interesting that the appearance of such quasiparticles in the surface of strong TI can also provide us with a smoking gun experimental setup to diagnose them by the fractional Josephson effect [21, 22]. The Josephson effect describes a phenomenon of supercurrent through a device known as a Josephson junction . It is a two-particle process in which a Cooper pair in one superconductor can across a weak link into the other superconductor without any voltage applied. Recently, Josephson effect on the surface of strong TI has attracted a lot of attention about the peculiar Majorana fermion [12–14, 21, 22, 24–31]. In most of the conditions, it is assumed that the Fermi level is close to the Dirac point. However, the chemical potential of TI does not always certainly reside at the Dirac point from an experimental point of view. Also, in those studies, a linear junction is generally analyzed by the discretized bound states in the superconductor gap (without the consideration of the continuous spectrum above the gap). But for a finite length scale junction, the continuous spectrum begins to play a partial role to the supercurrent. In addition, in those calculations, it is assumed that a ferromagnet lead is sandwiched between the two superconductor leads to exploit Majorana fermion. Besides charming of Majorana fermion, Josephson junction has an important application in quantum-mechanical circuits, such as superconducting quantum interference device, superconducting qubits, and rapid single flux quantum digital electronics, and so on . Thus, it is also an important thing to understand the fundamental properties of the Josephson effect on the surface of TI.
Hence, in this work, we study the Josephson effect through a Josephson junction on the surface of a strong TI where a gate voltage is exerted on the central normal lead with a finite width. Here, we adopt the Furusaki-Tsukada method [33, 34], which is applicable to any length of the junction and any potential strength of the weak link. Based on the method, it will allow us to reveal a number of characteristics, such as the dependence of the supercurrents on relevant variables, such as the length between the two superconductor leads, the temperature, the phase bias, and the gate voltage of the central weak-link region.
where and v F is the Fermi velocity, is the Pauli matrices, the four-dimensional spinor Ψ contains for the electron-like quasiparticle and for the hole-like quasiparticle, and E is the quasiparticle energy measured from E F . In the following, we set ℏ = v F = 1.
where , the coherence factors are given by and ζ = L or R.
where r1 and are the amplitudes of normal and Andreev reflections, respectively, f, g, m, and n are the corresponding transmission and reflection amplitudes in NM, and t1 and are the amplitudes of electron-like and hole-like quasiparticles in the right superconductor lead.
Appling the continuity boundary conditions of the wave functions at the boundary Ψ L (0) = Ψ M (0) and Ψ M (l) = Ψ R (l), the amplitudes and can be obtained directly. As the analytical results for these coefficients are tedious, we only give the numerical results in the following section.
Note that where W is the width of the junction. Certainly, using (7) the dc Josephson current for the present junction can be obtained easily by the numerical calculations.
Results and discussion
From an experimental point of view, due to the lattice mismatch between the bulk superconductor and the TI, the induced superconducting gap on the surface state of the TI can be expected to be substantially reduced in magnitude. In general, for a conventional s-wave superconductor such as Al or Nb, the gap and critical temperature can be assumed to Δ ~ 0.1 meV and T C = 23 K, respectively. Here, we estimate the Fermi velocity as v F ≈ 1 × 105 m/s. The superconducting coherence length is then ξ ~ 600 nm. In practice, the bulk band gap of TI opening can be observed on the order of 20 to 300 meV that depends on the material . Moreover, the Fermi energy E F can be tuned arbitrarily by either using the electric field effect or local chemical doping . Therefore, such a junction with E F = 10 ~ 103Δ can be experimentally achieved within the present-day technique. Meanwhile, the requirement of the transport inside the bulk gap can be fulfilled. Therefore, the parameters and the results in this study are authentic.
In Figure 2b,c, we show the dependence of critical supercurrent on potential U/E F for different temperatures with E F /Δ(0) = 103. The other parameters are shown in the figure. Figure 2b,c shows the calculated results of critical supercurrent for cases of l/ξ = 0.005 and l/ξ = 0.2, respectively. It can be seen clearly that both of them exhibit the monotonic decay feature with increasing temperature T. This is due to the thermal effect on Andreev bound states, which tends to reduce Andreev levels in the superconductor gap with increasing temperature T. The less number of Andreev levels contributing to the supercurrent thus leads to the suppressed feature. Besides the similar features, it is worthwhile to note that it also exhibits some different characters between the cases of l/ξ = 0.005 and l/ξ = 0.2. First, the minimum of critical supercurrent is shown as a decay function of T, while it remains nearly constant when the chemical potential of NM is at the Dirac point. Second, it is found that oscillation amplitude of critical supercurrent will disappear with increasing l. The physical origin for those phenomena can be given as follows. In central NM well which is formed by the two superconductor leads, the electron-like and hole-like quasiparticles inside the two interfaces will coherently interfere with each other which results in the formation of Andreev bound states. Through those Andreev levels, critical supercurrent will exhibit an oscillation feature for a short junction (l/ξ = 0.005). On the other hand, for a long junction (l/ξ = 0.2), as the interference effect decays in NM well, the present structure degenerates into a single junction case and then leads to the disappearance of the oscillation feature. In particular, the decay effect of interference exhibits much remarkable for the case of the evanescent mode (chemical potential of NM at the Dirac point). Thus, the minimum of critical supercurrent remains nearly constant.
We now proceed to investigate the temperature dependence of the critical supercurrent in Figure 3c,d. The parameters are shown in the figure. It is shown that the critical supercurrent can be modulated largely by the temperature T and a plateau-like structure can be yielded. In particular, the plateau-like behavior may be achieved even when the chemical potential of the NM is precisely at the Dirac point. However, the plateau-like structure washes out for a long junction. The plateau-like structure disappearance behavior can be explained by considering the decay of the quasiparticles interference effect in the NM. Although the plateau-like structure tuned by the temperature T is very similar to the case of the length of NM l, they have different physical origins. The novel behaviors now can be elucidated as follows. Remember that the temperature dependence of Δ is given by . As a result, the critical supercurrent decreases with increasing temperature because the number of Andreev levels (within the superconductor gap) which contributes to the critical supercurrent decreases with increasing temperature. Especially, the interplay between the restriction in the number of the Andreev levels and thermal average of all Andreev levels results in the plateau-like structure.
To conclude, we have shown that the Josephson supercurrent not only can be tuned largely by the temperature but also is related to the potential and the length of the weak-link region. Compared to the results that have been obtained, there are some pronounced deviations revealed. Based on the Furusaki and Tsukada formula adopted here where both discretized bound states and the continuum spectrum are included, we can expect that our findings will shed more light on the details of the Josephson supercurrent. With the rapid experimental advance in TI, we can suppose a very efficient Josephson device should be realized in the near future.
This work was supported by the National Natural Science Foundation of China (Grant No. U1204110). This project was also supported by China Postdoctoral Science Foundation (Grant No. 2013 M540126). CB also acknowledges the partial support from the Program of Young Core Teachers in Higher Education Institutions of Henan Province, China (Grant No. 2013GGJS-148).
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