We analyse the electric transport through a normal-superconductor-normal (NSN) junction between normal (N) and proximitized (S) magnetic TIs, with the central sector grounded. The normal leads are held into a QAH phase12, while the central sector is coupled to a s-wave superconductor, such that a pairing amplitude \(\Delta _1\) is induced into the TSSs yielding a \({\mathscr {N}}=1\) phase.
Within the Blonder-Tinkham-Klapwijk formalism, the electric current \(I_i\) in the normal terminals \(i=1,2\) of a double junction can be expressed as30,31
$$\begin{aligned} I_i = \int _0^{+\infty } dE \sum _a s_a\, [J_i^a (E) – K_i^a (E) ], \end{aligned}$$
(4)
where we defined the incoming and outgoing fluxes of quasiparticles injected into the superconductor as
$$\begin{aligned} J_i^a (E)= & {} \frac{e}{h} N_i^a(E)\, f_i^a(E) , \end{aligned}$$
(5)
$$\begin{aligned} K_i^a (E)= & {} \frac{e}{h} \sum _{jb} P_{ij}^{ab}(E)\, f_j^b(E) . \end{aligned}$$
(6)
Here \(a,b \in \{e,h\}\) label electron and hole states, \(N_i^a\) is the number of propagating modes in each terminal and \(f_i^a\) is the Fermi distribution function. The fluxes of injected quasiparticles are expressed in terms of the scattering amplitudes \(P_{ij}^{ab}\), which represent the transmission probability of a quasiparticle of type b in lead j to a quasiparticle of type a in lead i. Given a voltage drop \(V_i\) between the i-th terminal and the grounded superconductor, the electric conductance in the normal sectors can be defined as \(G_i = I_i / V\), where \(V=V_1-V_2\) is the total bias across the junction. A symmetric bias configuration \(V_1 = -V_2\) implies opposite terminal conductances \(G \equiv G_1=-G_2\)32. In terms of scattering amplitudes, the conductance can thus be written as
$$\begin{aligned} \begin{aligned} G(E)&= \frac{e^2}{2h} \left[N_1^e(E) – P_{11}^{ee}(E) + P_{11}^{he}(E) \right]\\&\quad + \frac{e^2}{2h} \left[P_{12}^{hh}(E) – P_{12}^{eh}(E) \right], \end{aligned} \end{aligned}$$
(7)
where \(E=eV/2\) is the energy of the quasiparticles.
The electric conductance G computed in the normal-hybrid-normal double junction is shown in Fig. 3 as a function of the total bias V. Here, the three different bias-dependent regimes can be clearly distinguished:
-
(a)
a low-bias conductance plateau \(G=e^2/2h\), with small oscillations due to finite \(L_y\);
-
(b)
an intermediate-bias regime with large conductance oscillations \(0<G<e^2/h\);
-
(c)
a metallic-like behaviour at larger biases, with quasiparticle diffusive transport.
In the low-bias regime, a single CMEM can be found within the surface gap of the \({\mathscr {N}}=1\) nontrivial superconductor. In a junction with non-proximitized TI films in the QAH state, the electric conductance is expected to be half-quantized at \(G=e^2/2h\)7,33. Oscillations around the plateau are due to finite-size coupling between edge states on opposite y sides34. With a higher bias, but still within the surface energy gap, the normal leads remain in the QAH phase, while a pair of chiral edge states with different wavenumbers can be found in the superconducting sector. The quasiparticle propagation through the proximitized TI is affected by the spatial interference produced by the phase difference acquired along the propagation length \(L_x\)28,29, resulting in an oscillatory conductance between \(G=0\) (perfect crossed Andreev reflection) and \(G=e^2/h\) (perfect normal transmission). Lastly, a further increase in the voltage bias leads to a metallic-like behaviour, due to the activation of delocalized excited states both in the normal and the proximitized sectors. The quasiparticles transport becomes diffusive rather than ballistic and the electric conductance is expected to be strongly affected by disorder.
Current distributions
The above different regimes can be further characterized in terms of the current densities along the transverse section of the junction. Resolving the spatial distribution of currents in proximitized TI thin slabs can provide detailed information on the quasiparticle transport processes occurring in the junction. Conversely, conductance measurements alone have proven unable to provide an unambiguous signature of chiral TSCs in proximitized QAH film35,36,37,38. Experimentally, the local probing of currents is in principle possible with magnetic imaging techniques using miniaturized squids39.
The longitudinal component of electric-current density is40
$$\begin{aligned} j_x(x,y) = -e\ \Re [\Psi ^*(x,y)\, {\hat{v}}_x\delta _z\,\Psi (x,y)], \end{aligned}$$
(8)
with the quasi-particle velocity operator given by
$$\begin{aligned} {\hat{v}}_x\equiv \frac{\partial {\mathscr {H}}_{BdG}}{\partial p_x} = \frac{2}{\hbar ^2} m_1 p_x \tau _x\delta _z – v_F\,\sigma _y \tau _z\delta _z\;. \end{aligned}$$
(9)
Equation (8) can be straightforwardly evaluated once the real-space wavefunction \(\Psi (x,y)\) is obtained by discretizing the original BdG Hamiltonian \({\mathscr {H}}_{BdG}\) in a 2D lattice. Imposing the continuity of the wavefunction at the interfaces between normal and hybridized sectors yields the full wavefunction in the junction for a fixed energy eV.
The current distribution along the transverse section in the two normal leads of the NSN junction is displayed in Fig. 4 for the three different bias-dependent regimes. The current density is computed along y at \(x=\pm 15\) \(\upmu \)m, assuming that the proximitized sector has dimensions \(L_x=20\) \(\upmu \)m and \(L_y=1\) \(\upmu \)m and the reference \(x=0\) is set in the middle of it. However, as long as the normal leads are held into a QAH state, the current density pattern remains unaffected by the particular choice of x at which the measurement is conducted.
Figure 4a–b display the density current profiles for a bias \(eV \in [0.1,0.3 ]\) meV, which ensures a single CMEM on each side of the superconducting sector. As expected for a QAH phase12, the current is well-localized along the y boundaries: the peaks in the current density profile correspond to the injected quasiparticles, localized on opposite sides and propagating in opposite directions. Apart from small finite-size (\(L_y\)) effects, no electric current is transmitted or reflected by the superconductor33, as evidenced by \(j_x = 0\) on the edge opposite to the one hosting the injected current.
A different situation is shown in Fig. 4c–d, which display the same current profiles for a higher bias \(eV \in [0.6,0.8 ]\) meV still lower than the surface gap. As in the previous case, the normal sectors are in the QAH phase, with edge-localized current and ballistic transport. Due to the pair of degenerate edge modes in the superconductor, the quasiparticles propagate across the central sector with oscillating probability of normal transmission and crossed Andreev reflection, yielding an oscillatory current density on the edge opposite to the one hosting the injected current. As can be noted, the current density profile is strongly affected by small variations of the bias, as the transmission amplitudes depend on the momentum difference \(\delta _k = k_1-k_2\) between the edge channels, which is set by the energy of the quasiparticles. These bias-induced edge-current reversals are a conspicuous manifestation of the interference of chiral modes in the TSC.
Finally, Fig. 4e–f show the metallic-like behaviour corresponding to a bias \(eV \in [1.7,1.9 ]\) meV, which is larger than the surface gap. Both in the normal leads and in the proximitized sector, the quasiparticles can propagate through delocalized modes, which make the transport diffusive rather than ballistic. Despite some peaks near the edges of the system, the current is transmitted through the whole section of the junction and disorder is expected to play an important role in the electric transport.
A similar analysis of the transverse current density can also be carried out in the proximitized section of the TI thin film, computing the average value \(j_x\) in the central sector of the junction. Qualitatively similar results are to be expected for the characteristic oscillations of the two-modes regime, even though in the hybridized sector the quasiparticle current is screened by the condensate. We clarify that the characteristic oscillatory regime requires the quasiparticle excited states in the superconductor to be mixtures of electrons and holes. This requirement is always satisfied in the limit of a small chemical potential \(\mu = 0\), meaning that the Fermi energy should coincide with the Dirac point of the TSSs. Although these fine-tuned conditions constitute a limitation for the validity of our analysis, the band structure of the bismuth-telluride-based TI compounds can be engineered41,42, allowing the manipulation of the Dirac surface states without altering the chemical potential of the bulk crystals.