Detection of second-order topological superconductors by Josephson junctions

We study Josephson junctions based on a second-order topological superconductor (SOTS) which is realized in a quantum spin Hall insulator with a large inverted gap in proximity to an unconventional superconductor. We find that tuning the chemical potential in the superconductor strongly modifies the pairing gap of the helical edge states and leads to topological phase transitions. As a result, the supercurrent in the junction is controllable and a 0-π transition is realized by tuning the chemical potentials in the superconducting leads. These striking features are stable in junctions with different sizes, doping in the normal region, and in the presence of disorder. We propose them as novel experimental signatures of SOTSs. Moreover, the 0-π transition can serve as a fully electric way to create or annihilate Majorana bound states in the junction without magnetic manipulation.

We study Josephson junctions based on a second-order topological superconductor (SOTS) which is realized in a quantum spin Hall insulator with a large inverted gap in proximity to an unconventional superconductor. We find that tuning the chemical potential in the superconductor strongly modifies the pairing gap of the helical edge states and leads to topological phase transitions. As a result, the supercurrent in the junction is controllable and a 0-π transition is realized by tuning the chemical potentials in the superconducting leads. These striking features are stable in junctions with different sizes, doping in the normal region, and in the presence of disorder. We propose them as novel experimental signatures of SOTSs. Moreover, the 0-π transition can serve as a fully electric way to create or annihilate Majorana bound states in the junction without magnetic manipulation.
In this Letter, we investigate superconductor-normal metal-superconductor (SNS) junctions formed by a 2D SOTS. The SOTS is realized in a QSHI with a large inverted gap in proximity to high-temperature cuprate or iron-based superconductors. We find that due to the nontrivial momentum-dependence in the pairing and mass terms, the chemical potential in the SOTS alters the pairing gap opened at the edge states significantly. It can even switch the sign of the pairing gap, leading to a topological phase transition. While the supercurrent across the junction is insensitive to the chemical potential in the N region, it depends strongly on those in the superconductors. Strikingly, tuning the chemical potentials in the superconductors also gives rise to a 0-π transition, which is absent in the junctions with conventional s-wave pairing. These features are robust against disor-der in junctions of different lengths and widths. They could be exploited to detect the SOTS with Majorana corner states. Furthermore, while Majorana bound states (MBSs) emerge in the 0-junction when the phase difference across the junction is φ = ±π, they appear at vanishing φ in the π-junction. Thus, the Josephson junctions with such a doping-induced 0-π transition provide a novel platform to create or annihilate MBSs by purely electric gating in the absence of φ. These predictions are applicable to a number of candidate systems, including high-temperature QSHIs [44][45][46][47][48][49][50][51][52][53][54][55], in proximity to cuprate or iron-based superconductors.
Model for SOTSs.-We consider the minimal model for SOTSs realized in QSHIs in proximity to superconductors, where c † σ,s,k (c σ,s,k ) creates(annihilates) an electron with spin s ∈ {↑, ↓}, orbital σ ∈ {a, b}, and k = (k x , k y ) is the momentum measured from the band inversion point of the QSHI. τ , σ and s are Pauli matrices acting on Nambu, orbital and spin spaces, respectively. m(k) = m 0 − m x k 2 x − m y k 2 y is the mass term of the QSHI and µ is the chemical potential. The band inversion implies the conditions m 0 m x(y) > 0 [56]. The pairing potential is written in general as ∆(k) = ∆ 0 + ∆ 2 k 2 x − k 2 y . When ∆ 0 = 0 and ∆ 2 = 0, it refers to conventional s-wave pairing. When ∆ 0 = 0 and ∆ 2 = 0, the pairing is formally d x 2 −y 2 -wave. It can be induced in a QSHI with the band inversion at the Γ point via the proximity to a cuprate superconductor [4]. When 0 |∆ 0 | < m 0 |∆ 2 |/2m x(y) , the system possesses a mixture of s-wave and d x 2 −y 2 -wave pairing. It can also describe effectively a QSHI with the band inversion at the X point [53][54][55] and s ± -wave pairing induced from an iron-based superconductor [57][58][59][60][61].

arXiv:1905.09308v1 [cond-mat.supr-con] 22 May 2019
In the absence of ∆(k), the system has time-reversal and rotational symmetries. It hosts gapless helical edge states across the bulk gap. The pairing term with ∆ 2 = 0 breaks the rotational symmetry. It induces a pairing gap at the edge states. The gap may switch sign at the corners, giving rise to Majorana corner modes [3][4][5]. This is the essence of the SOTS. The SOTS can be characterized by a topological invariant calculated from the bulk Hamiltonian [62][63][64]. Based on the model (1), we can see the second-order topology more intuitively from the picture of edge states and show that it can be strongly altered by changing µ. Pairing gaps of edge states and topological phase transitions.-To analyze the Majorana corner states and the influence of µ on the SOTS, we analytically derive the effective model for edge states. As in realistic systems, we assume weak pairing. We first calculate the edge states of H 0 , following the approach of Ref. [65]. For illustration, we consider the edge along x direction of the SOTS in the half-plane y 0 and assume hard-wall boundary conditions. k x is a good quantum number. The helical electron and hole edge bands are found explicitly as The wavefunctions in the orbital basis {a, b} read They fulfill Ψ e,↓,kx (r) = Ψ * e,↑,−kx (r) and Ψ h,↓/↑,kx (r) = Ψ * e,↓/↑,−kx (r), due to time-reversal and particle-hole symmetries. Here, λ 1(2) = |v y /2m y |−(+)(v 2 y /4m 2 y −m 0 /m y + m x k 2 x /m y ) 1/2 and N is the normalization factor. The decaying length of edge states is given by ξ edge = max[1/Re(λ 1 ), 1/Re(λ 2 )]. At zero energy, the electron and hole bands touch at k x = ±k c with k c = sgn(m y v y ) µ/v x . For µ = 0, k c = 0. However, for µ = 0, the touching points shift to finite ±k c . Projecting the pairing term onto the edge states, the resulting Bogoliubov de-Gennes (BdG) Hamiltonian for edge states is obtained as in the basis (Ψ e,↑ , Ψ e,↓ , Ψ h,↓ , Ψ h,↑ ), and the pairing gap is given analytically by Here, a real ∆(k) has been assumed without loss of generality [66]. We provide the derivation in detail in the Supplemental Material [67]. Similarly, for an edge along y direction, we find the BdG Hamiltonian of the same form but with a different pairing gap The combination of ∆ x eff and ∆ y eff (with opposite signs) in Eq. (4) mimics the Jackiw-Rebbi model [68] at corners of x-and y-axes. Thus, Majorana corner states with zero energy appear if ∆ x eff ∆ y eff < 0. For s-wave pairing, ∆ x eff and ∆ y eff are identical and constant. Thus, no corner state exists. In contrast, for unconventional pairings with |∆ 0 | < m 0 |∆ 2 |/2m x(y) , we obtain corner states at small µ. When µ = 0 and ∆ 0 = 0, Eq. (5) reproduces the result of Ref. [4]. Interestingly, ∆ x(y) eff can depend strongly on µ. The µ dependence stems from the quadratic terms in the model (1), which are crucial for the topological properties of the SOTS. It surprisingly turns out that the µ-dependence is independent of m x(y) . Moreover, ∆ This behavior indicates that we can change the sign of ∆ x(y) eff by varying µ. Suppose µ c x µ c y without loss of generality. The system is then in a SOTS phase in the regions 0 |µ| < µ c x and µ c y < |µ| < m g with m g being the bulk gap [69], whereas in between it becomes a trivial superconductor without corner state. For isotropic QSHIs with v x(y) = v, m x(y) = m, and ∆ 0 = 0, ∆ x eff and ∆ y eff are always opposite. They both close at µ = ±v m 0 /2m. Thus, there is no space for the trivial phase. Nevertheless, the sign of ∆ This condition indeed corresponds to the QSHI phase with a large inverted gap or equivalently an indirect bulk gap. It is likely realized in the inverted InAs/GaSb bilayer [71][72][73], WTe 2 monolayer [44][45][46][47][48], functionalized MXene [49,50], Bismuthene on SiC [51,52], PbS monolayer [53][54][55], etc.
To test the analytical results, we discretize the model (1), put it on a lattice, choose a proper set of parameters (satisfying inequality (8)) and calculate the energy spectrum in a ribbon geometry. For concreteness, we consider ∆ 0 = 0 and set the lattice constant a to unity. As shown in Fig. 1(a), the edge states for µ = 0 open a gap at k x = 0. As µ is raised, the gap splits to two points away from k x = 0. The magnitude of the gap first decreases, closes at a critical µ and then reopens, which explicitly demonstrates a topological phase transition. Its behavior is in perfect agreement with Eq. (5), see Fig. 1(b).
0-π transition and its robustness in SNS junctions.-We now consider an SNS junction in which two SOTSs (also called S leads below) are connected by a QSHI with the length L in x direction. The width of the junction ribbon is W . For simplicity, we assume the chemical and pairing potentials in step-like forms. µ L(R) and µ N denote the chemical potentials in the left(right) S lead and N (QSHI) region, respectively. φ is the phase difference across the junction. We calculate the supercurrent J s by the lattice Green's function technique [74][75][76] and provide the details of calculation in the Supplemental Material [67].
At low temperatures, the transport in the junction is conducted dominantly by the helical edge channels, and perfect Andreev reflection occurs at the NS interfaces. Thus, the current-phase relation (CPR) takes a sawtooth shape with a sudden jump, see Fig. 2(a). The sawtoothlike CPR is insensitive to µ N and stays stable in junctions of different sizes (L and W ), provided that the two edges at y = ±W/2 are well separated, W ξ edge . The sudden jump can be related to the fermion parity anomaly at each edge [77,78]. They also indicate the formation of degenerate MBSs in the junction, which we will discuss later. J s decreases monotonically with increasing L, see Fig. 2(b). The critical current J c (maximal value of J s ) decays as ∼1/L in long junctions, similar to junctions based on conventional s-wave pairing. In short junctions, J c is of the same order of magnitude but always smaller than e |∆ L ∆ R |/ even in the short junction limit, in contrast to the case of s-wave pairing. In this estimate, ∆ L(R) is the pairing gap of edge states in the left(right) S lead and determined by Eq. (5). This behavior can be attributed to the fact that ∆ L(R) vary smoothly when approaching the interfaces.
The CPRs for a fixed µ L and various values of µ R are displayed in Fig. 2(a). Since J s is even in µ L(R) , we present only the results for µ L(R) > 0. While J s is insensitive to µ N , it decreases significantly when we increase µ R . This can be understood as a result of the reduction of |∆ R | by µ R , see Eq. (5). Strikingly, increasing µ R further, we observe a clear 0-π transition for the parameters satisfying inequality (8). While J s (φ) in the region 0 < φ < π is positive for µ R < µ c , it becomes negative for µ R > µ c . We coin the former case a 0-junction and the latter one a π-junction. Meanwhile, the sudden jump of the CPR is switched to φ = 0 in the π-junction, which is in strong contrast to the 0-junction where the jump is at φ = ±π. In Fig. 2(b), we plot J c as a function of µ R . The critical value µ c for the transition is given approximately by v m 0 /2m, in accord with our analytical result. Close to µ c , J c drops quickly and switches sign. These features are generic and apply to junctions of different lengths and widths. They are also robust with respect to nonmagnetic disorder in the N region. To illustrate this, we model the disorder as random on-site potentials in the range [−V dis /2, V dis /2] [67, 79] and calculate 200 random disorder configurations in the inset of Fig. 2(b). There is no qualitative difference in the features compared to those in clean junctions. This can be expected since the helical edge channels which mediate the transport are less sensitive to backscattering. Similar effects can be observed by tuning µ L and fixing µ R . Finally, it is important to note that the variation of J s and 0-π transition by tuning µ L(R) are directly related to the strong µ L(R) -dependence in ∆ L(R) in the SOTS, and absent in conventional junctions based on s-wave pairing.
Majorana bound states.-Next, we discuss the Andreev bound states (ABSs) formed in the junction, which can be obtained from the lattice Green's function. In short junctions, there are two bands of ABSs with opposite energies, see Fig. 3(a,c). When the sudden jump of the CPR occurs, the positive and negative bands touch at zero energy. This degeneracy is robust and protected by time-reversal and particle-hole symmetries. Most interestingly, it corresponds to Kramers pairs of MBSs. This can be better understood from the effective model (4) for edge states. In the short junction limit, two ABS bands at a given edge can be found explicitly as Notably, the ABSs are confined in the pairing gaps for φ satisfying (cos φ−∆ L /∆ R )(cos φ−∆ R /∆ L ) > 0, as verified in Fig. 3(a,c). Noticing ∆ L ∆ R > 0 in the 0-junction for µ R < µ c , whereas ∆ L ∆ R < 0 in the π-junction for µ R > µ c , we can see that E ± (φ) touch at φ = ±π and 0, respectively. Using the formula at zero temperature, J s (φ) = ∂|E + (φ)|/∂φ [80], we also reproduce the sudden jump in the CPR. The wavefunctions of the zero modes can be written as where Ψ ± = sgn(∆ L )Ψ h/e,↓ ∓ iΨ e/h,↑ and the spatial de- . Since Ψ h,↓/↑ = Ψ * e,↓/↑ , the zero modes have self-adjoint wavefunctions, γ ± (x) = γ * ± (x), and behave like Majorana fermions. Under the time-reversal operation T , T Ψ e,↓/↑ = ±Ψ * e↑/↓ and T Ψ h,↓/↑ = ±Ψ * h,↑/↓ . Therefore, γ ± are related by time-reversal symmetry, T γ + = γ − . A similar analysis can be applied to the other edge where another Kramers pair of MBSs locate. In long junctions, all features persist but with more pairs of discrete ABS bands emerging from the continuum spectrum, see Fig. 3(b,d).
At φ = 0, the MBSs emerge for µ > µ c , whereas they disappear for µ < µ c . In this sense, we are able to switch between the presence and absence of MBSs by gating the S leads in the absence of φ. As an advantage compared to the Josephson junctions with conventional s-wave pairing [23,77,[81][82][83][84], our setup realizes fully electrically controllable MBSs without a fine tuning of magnetic field or threaded flux. Moreover, since the localization lengths of the MBSs in the S leads are determined by ξ L(R) = |v/∆ L(R) |, we are also able to control the spatial profiles of the MBSs by µ L(R) .
We note that there have been experimental efforts trying to incorporate unconventional superconductivity in topological systems [38][39][40][59][60][61], although the proximity of unconventional superconductivity in the aforementioned QSHIs has yet to be achieved. Moreover, large proximity-induced pairing gaps in 2D systems from unconventional superconductors have been probed [38][39][40][41]. This together with the large inverted gap of the host QSHIs holds promise for high-temperature and tunable MBSs in our setup.
In summary, we have found that the chemical potentials in the superconductors can be used to modulate the supercurrent and realize a 0-π transition in the Josephson junction based on SOTSs. These behaviors are attributed to the dependence of the pairing gap of edge states on the chemical potential in the SOTSs. We have shown that these transport properties are robust against disorder in junctions of different sizes and with different doping in the normal region. We have also predicted the 0-π transition as a fully electric way to create or annihilate MBSs at elevated temperatures.
We thank Fernando Dominguez, Gaomin Tang and Xi-