- Nano Express
- Open Access
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.
- Josephson effect
- Topological insulator
- Furusaki-Tsukada formula
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.
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).
- Kane CL, Mele EJ: Quantum spin Hall effect in graphene. Phys. Rev. Lett. 2005, 95: 226801.View ArticleGoogle Scholar
- Hasan MZ, Kane CL: Colloquium: topological insulators. Rev. Mod. Phys. 2010, 82: 3045. 10.1103/RevModPhys.82.3045View ArticleGoogle Scholar
- Qi X-L, Zhang S-C: Topological insulators and superconductors. Rev Mod Phys 2011, 83: 1057. 10.1103/RevModPhys.83.1057View ArticleGoogle Scholar
- Chang K, Lou W-K: Helical quantum states in HgTe quantum dots with inverted band structures. Phys Rev Lett 2011, 106: 206802.View ArticleGoogle Scholar
- Zhang D, Lou W-K, Miao M, Zhang S-C, Chang K: Interface-induced topological insulator transition in GaAs/Ge/GaAs quantum wells. Phys Rev Lett 2013, 111: 156402.View ArticleGoogle Scholar
- Bernevig BA, Hughes TL, Zhang S-C: Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 2006, 314: 1757–1761. 10.1126/science.1133734View ArticleGoogle Scholar
- Li J, Chang K: Electric field driven quantum phase transition between band insulator and topological insulator. Appl. Phys. Lett. 2009, 95: 222110. 10.1063/1.3268475View ArticleGoogle Scholar
- Fu L, Kane CL: Topological insulators with inversion symmetry. Phys Rev B 2007, 76: 045302.View ArticleGoogle Scholar
- Yao Y, Ye F, Qi X-L, Zhang S-C, Fang Z: Spin-orbit gap of graphene: first-principles calculations. Phys Rev B 2007, 75: 041401(R).View ArticleGoogle Scholar
- Konig M, Wiedmann S, Brune C, Roth A, Buhmann H, Molenkamp L, Qi X-L, Zhang S-C: Quantum spin Hall insulator state in HgTe quantum wells. Science 2007, 318: 766–770. 10.1126/science.1148047View ArticleGoogle Scholar
- Hsieh D, Qian D, Wray L, Xia Y, Hor Y, Cava R, Hasan M: A topological Dirac insulator in a quantum spin Hall phase. Nature 2008, 452: 970–974. 10.1038/nature06843View ArticleGoogle Scholar
- Fu L, Kane CL, Mele E: Topological insulators in three dimensions. Phys Rev Lett 2007, 98: 106803.View ArticleGoogle Scholar
- Cao T, Wang S: Topological insulator metamaterials with tunable negative refractive index in the optical region. Nanoscale Research Letters 2013, 8: 526. 10.1186/1556-276X-8-526View ArticleGoogle Scholar
- Efimkin DK, Lozovik YE, Sokolik AA: Collective excitations on a surface of topological insulator. Nanoscale Research Letters 2012, 7: 163. 10.1186/1556-276X-7-163View ArticleGoogle Scholar
- Fu L, Kane CL: Superconducting proximity effect and Majorana fermions at the surface of a topological Insulator. Phys Rev Lett 2008, 100: 096407.View ArticleGoogle Scholar
- Chen H-J, Zhu K-D: Nonlinear optomechanical detection for Majorana fermions via a hybrid nanomechanical system. Nanoscale Research Letters 2014, 9: 166. 10.1186/1556-276X-9-166View ArticleGoogle Scholar
- Bolech C, Demler E: Observing Majorana bound states in p-wave superconductors using noise measurements in tunneling experiments. Phys Rev Lett 2007, 98: 237002.View ArticleGoogle Scholar
- Law K, Lee P, Ng T: Majorana fermion induced resonant Andreev reflection. Phys Rev Lett 2009, 103: 237001.View ArticleGoogle Scholar
- Alicea J, Oreg Y, Refael G, Von Oppen F, Fisher M: Non-Abelian statistics and topological quantum information processing in 1D wire networks. Nat Phys 2011, 7: 412417.View ArticleGoogle Scholar
- Fu L: Electron teleportation via Majorana bound states in a mesoscopic superconductor. Phys Rev Lett 2010, 104: 056402.View ArticleGoogle Scholar
- Fu L, Kane CL: Josephson current and noise at a superconductor/quantum-spin-Hall-insulator/superconductor junction. Phys Rev B 2009, 79: 161408(R).View ArticleGoogle Scholar
- Lutchyn R, Sau J, Das Sarma S: Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures. Phys Rev Lett 2010, 105: 077001.View ArticleGoogle Scholar
- Josephson BD: Possible new effects in superconductive tunneling. Physics Letters 1962, 1: 251–253. 10.1016/0031-9163(62)91369-0View ArticleGoogle Scholar
- Tanaka Y, Yokoyama T, Nagaosa N: Manipulation of the Majorana fermion, Andreev reflection, and Josephson current on topological insulators. Phys Rev Lett 2009, 103: 107002.View ArticleGoogle Scholar
- Linder J, Tanaka Y, Yokoyama T, Sudbo A, Nagaosa N: Interplay between superconductivity and ferromagnetism on a topological insulator. Phys Rev B 2010, 81: 184525.View ArticleGoogle Scholar
- Akhmerov A, Nilsson J, Beenakker CWJ: Electrically detected interferometry of Majorana fermions in a topological insulator. Phys Rev Lett 2009, 102: 216404.View ArticleGoogle Scholar
- Fu L, Kane CL: Probing neutral Majorana fermion edge modes with charge transport. Phys Rev Lett 2009, 102: 216403.View ArticleGoogle Scholar
- Law K, Lee PA: Robustness of Majorana fermion induced fractional Josephson effect in multichannel superconducting wires. Phys Rev B 2011, 84: 081304.View ArticleGoogle Scholar
- Olund C, Zhao E: Current-phase relation for Josephson effect through helical metal. Phys Rev B 2012, 86: 214515.View ArticleGoogle Scholar
- Snelder M, Veldhorst M, Golubov AA, Brinkman A: Andreev bound states and current-phase relations in three-dimensional topological insulators. Phys Rev B 2013, 87: 104507.View ArticleGoogle Scholar
- Tkachov G, Hankiewicz EM: Helical Andreev bound states and superconducting Klein tunneling in topological insulator Josephson junctions. Phys Rev B 2013, 88: 075401.View ArticleGoogle Scholar
- Makhlin Y, Schön G, Shnirman A: Quantum-state engineering with Josephson-junction devices. Rev Mod Phys 2001, 73: 357. 10.1103/RevModPhys.73.357View ArticleGoogle Scholar
- Furusaki A, Tsukada M: Dc Josephson effect and Andreev reflection. Solid State Commun 1991, 78: 299–302. 10.1016/0038-1098(91)90201-6View ArticleGoogle Scholar
- Furusaki A, Tsukada M: Current-carrying states in Josephson junctions. Phys Rev B 1991, 43: 10164. 10.1103/PhysRevB.43.10164View ArticleGoogle Scholar
- Muhlschlegel B: Die thermodynamischen funktionen des supraleiters. Z Phys 1959, 155: 313–327. 10.1007/BF01332932View ArticleGoogle Scholar
- Zhang H, Liu C-X, Qi X-L, Dai X, Fang Z, Zhang S-C: Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat Phys 2009, 5: 438–442. 10.1038/nphys1270View ArticleGoogle Scholar
- Sacepe B, Oostinga JB, Li J, Ubaldini A, Couto NJG, Giannini E, Morpurgo AF: Gate-tuned normal and superconducting transport at the surface of a topological insulator. Nat Commun 2011, 2: 575.View ArticleGoogle Scholar
- Blonder G, Tinkham M, Klapwijk TM: Transition from metallic to tunneling regimes in superconducting microconstrictions: excess current, charge imbalance, and supercurrent conversion. Phys Rev B 1982, 25: 4515. 10.1103/PhysRevB.25.4515View ArticleGoogle Scholar
- Chen X, Tao JW: Design of electron wave filters in monolayer graphene by tunable transmission gap. Appl Phys Lett 2009, 94: 262102. 10.1063/1.3168527View ArticleGoogle Scholar
- Linder J, Black-Schaffer AM, Yokoyama T, Doniach S, Sudbø A: Josephson current in graphene: role of unconventional pairing symmetries. Phys Rev B 2009, 80: 094522.View ArticleGoogle Scholar
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