Abstract
The chiral topological superconductor, which supports propagating nontrivial edge modes while maintaining a gapped bulk, can be realized hybridizing a quantumanomalousHall thin slab with an ordinary swave superconductor. We show that by sweeping the voltage bias in a normalhybridnormal double junction, the pattern of electric currents in the normal leads spans three main regimes. From singlemode edgecurrent quantization at low bias, to doublemode edgecurrent oscillations at intermediate voltages and up to diffusive bulk currents at larger voltages. Observing such patterns by resolving the spatial distribution of the local current in the thin slab could provide additional evidence, besides the global conductance, on the physics of chiral topological superconductors.
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Introduction
The chiral topological superconductor (TSC) is a topologicallynontrivial superconducting state which supports 1D chiral Majorana edge modes (CMEM) while maintaining a full pairing gap in the 2D bulk^{1,2,3,4,5}. A recent proposal^{5,6,7} for the realization of this exotic phase suggests exploiting hybridized quantumanomalousHall (QAH) insulator heterostructures, inducing superconductivity in the QAH chiral edge modes through proximity coupling with an swave superconductor. 3D topological insulators (TIs) like \(\text {Bi}_2 \text {Se}_3\), \(\text {Bi}_2 \text {Te}_3\) and \(\text {Sb}_2 \text {Te}_3\) represent promising materials for such a purpose, because they host nontrivial Diraccone shaped gapless surface states within a bulk gap \(E_g \approx 0.3\) eV^{8,9,10}, which is larger than the typical energy scale of the room temperature^{11}. In a thick slab geometry, these topological boundary modes are welldefined for top and bottom surfaces. However, when placed in an effective 2D thin film, quantum tunnelling couples the Dirac cones localized on opposite surfaces, opening a mass gap in the energy spectrum^{12,13}. Magnetic ordering induced by doping with transition metals (like V or Cr) can give rise to a QAH state with quantized Hall conductance^{14,15,16}.
In this work, we discuss the electric conductance and the current spatial distribution for the specific case of a QAH thin slab containing a central proximityhybridised sector in the chiral TSC phase. The system is, in practice, a normalhybridnormal double junction as sketched in Fig. 1a. We show here that, when the central sector is a chiral TSC, three different regimes can be distinguished depending on the bias applied across the junction: quantized conductance with edge current at low bias (Fig. 1b), biasdependent edge current oscillations at intermediate bias (Fig. 1c) and quasiparticle diffusive transport through delocalized states for higher bias exceeding the surface gap (Fig. 1d).
Model
The 2D minimal model for a full translational invariant TI thin film, which captures the lowenergy physics dominated by the Diraccone shaped topological surface states (TSSs) around \({\textbf{k}}=0\), is given by^{9,12,17}
in the basis \(\psi _{\textbf{k}} = \left( c_{{\textbf{k}},\uparrow }^t, c_{{\textbf{k}},\downarrow }^t, c_{{\textbf{k}},\uparrow }^b, c_{{\textbf{k}},\downarrow }^b\right) \) where \(c_{{\textbf{k}},\uparrow }^\tau \) annihilates an electron of momentum \({\textbf{k}}=\left( k_x,k_y\right) \) with spin \(\sigma =\uparrow , \downarrow \) on the \(\tau =t,b\) layer. The Pauli matrices \(\sigma _{x,y,z}\) \(\left( \tau _{x,y,z}\right) \) act on spin (layer) subspaces, \(v_F\) is the Fermi velocity of the Dirac electrons and \(m({\textbf{k}})=m_0 + m_1\left( k_x^2 + k_y^2\right) \) represents the hybridization between top and bottom TSSs, which at the Dirac point \({\textbf{k}}=0\) opens a finitesize gap \(m_0\)^{18,19}. Magnetic doping breaks timereversal symmetry, coupling the electrons to the magnetization through an exchange interaction \(\lambda \sigma _z\)^{20}. The topological state of a 2D system with broken timereversal symmetry is characterized by the Chern invariant^{1,2}, which for \(\lambda >m_0\) takes the value \({\mathscr {C}} = \lambda /\lambda \), while for \(\lambda <m_0\) becomes \({\mathscr {C}}=0\)^{21}. The \({\mathscr {C}}=0\) phase is a trivialgapped insulator, while the \({\mathscr {C}}=1\) one corresponds to a QAH state with quantized Hall conductance \(\sigma _{xy}=e^2/h\)^{12,22}.
Superconducting coupling
Proximity coupling to an ordinary swave superconductor induces finite pairing amplitudes \(\Delta _{1,2}\) to the electrons in the top and bottom layers of the magnetic TI thin film, respectively^{6,7}. The effective description of the superconducting system can be carried out through the Bogoliubov de Gennes (BdG) Hamiltonian, which in the basis \(\Psi _{\textbf{k}} = (\psi _{\textbf{k}}, \psi ^\dagger _{{\textbf{k}}})\) takes the following form^{23,24}
where
and \(\mu \) represents the chemical potential. The BdG Hamiltonian preserves particlehole symmetry \({\mathscr {P}}=\delta _x {\mathscr {K}}\), as \({\mathscr {P}} \, {\mathscr {H}}_{BdG}({\textbf{k}}) \, {\mathscr {P}}^{1} = {\mathscr {H}}_{BdG}({\textbf{k}})\) by construction. Here \(\delta _{x,y,z}\) stands for a set of Pauli matrices acting in particlehole space and \({\mathscr {K}}\) is complex conjugation. The topological state of the proximitized thin film is thus characterized by an integer topological invariant \({\mathscr {N}}\) analogous to the Chern invariant^{1,2}. In presence of a finite pairing amplitude, the \({\mathscr {C}}=1\) QAH state can be viewed as a TSC with \({\mathscr {N}}=2\) Majorana edge modes^{5,6}, where the doubling of the edge modes is due to the electronhole degeneracy introduced by the choice of the wavefunction in Nambu space. Asymmetric pairing amplitudes on top and bottom surfaces are required to lift the degeneracy^{7}, leading to a chiral TSC with \({\mathscr {N}}=1\) unpaired CMEMs propagating over the boundaries. For this reason, our numerical results were obtained for fixed \(\Delta _1=1.5\) meV and \(\Delta _2=0\), meaning that the superconducting pairing acts only on electrons in TSSs on the upper interface of the thin film. Assuming that the penetration length of Cooper pairs into the TI film is smaller than the thickness^{25}, interlayer pairing terms in Eq. (3) can be neglected.
Edge picture
A simple way to understand the emergence of the chiral TSC is to observe the evolution of the edge modes of a QAH thin film under the effect of opposite pairing amplitudes \(\Delta _1 = \Delta _2 = \Delta \)^{5,6}. In the QAH phase, the electric current is transported through chiral edge modes, whose \({\textbf{k}}=0\) localization length is \(\xi \propto 1/m_0 + \lambda \)^{26}. In the simple limit of \(\mu = 0\) and in the absence of superconductivity \(\Delta =0\), \({\mathscr {H}}_{BdG}\) reduces to two identical copies of \({\mathscr {H}}_0\) with opposite sign, thus each chiral edge mode is doubled into a pair of identical electron and hole quasiparticle states^{27}. A finite pairing amplitude couples electron and hole excitations, opening a gap in one of the two edge crossings. In these conditions, the localization length of the doubled edge modes at \({\textbf{k}}=0\) can be estimated as \(\xi _1 \propto 1/m_0 + \lambda + \Delta \) and \(\xi _2 \propto 1/m_0 + \lambda  \Delta \)^{5,6}, which means that the coupled edge states become delocalized as \(\Delta \xrightarrow {} \pm ( m_0 + \lambda )\), moving toward the bulk of the film. The corresponding eigenstates are pushed to higher energy.
The effect of superconducting coupling is explicitly shown in Fig. 2a, which displays the lowenergy eigenvalues of a magnetic TI thin film for an increasing upper pairing amplitude \(\Delta _1\) and \(\Delta _2=0\). In the absence of superconductivity, the QAH insulatorsuperconductor heterostructure hosts a pair of perfectly degenerate edge modes on each side. A finite pairing amplitude \(\Delta _1\) splits them apart, until a single gapless CMEM remains within the bulk gap, achieving the \({\mathscr {N}}=1\) phase. As explained, the second pair of edge crossings becomes gapped and delocalized due to superconducting coupling. These gapped modes are strongly coupled around \({\textbf{k}}=0\), but maintain their edge character away from the Dirac point. This fact is shown in Fig. 2b, which displays the lowenergy band structure of an infinite TI thin film with superconducting pairing \(\Delta _1=1.5\) meV and \(\Delta _2=0\). As can be noted, at higher energy and for longitudinal wavenumber \(k_x \ne 0\), it is still possible to find a pair of chiral modes with different wavenumbers \(k_1, k_2\) welllocalized on the same edge of the system. A gray horizontal line is drawn at \(E=1\) meV, within the energy range where such a pair of degenerate edge states can be found. Due to the spatial interference induced by such a finite momentum difference, the electric current resulting from quasiparticle transport through the edge of a proximitized magnetic TI exhibits characteristic oscillations with respect to propagation length and energy of the incident quasiparticles^{28,29}.
Electric transport
We analyse the electric transport through a normalsuperconductornormal (NSN) junction between normal (N) and proximitized (S) magnetic TIs, with the central sector grounded. The normal leads are held into a QAH phase^{12}, while the central sector is coupled to a swave superconductor, such that a pairing amplitude \(\Delta _1\) is induced into the TSSs yielding a \({\mathscr {N}}=1\) phase.
Within the BlonderTinkhamKlapwijk formalism, the electric current \(I_i\) in the normal terminals \(i=1,2\) of a double junction can be expressed as^{30,31}
where we defined the incoming and outgoing fluxes of quasiparticles injected into the superconductor as
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 ith terminal and the grounded superconductor, the electric conductance in the normal sectors can be defined as \(G_i = I_i / V\), where \(V=V_1V_2\) 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
where \(E=eV/2\) is the energy of the quasiparticles.
The electric conductance G computed in the normalhybridnormal double junction is shown in Fig. 3 as a function of the total bias V. Here, the three different biasdependent regimes can be clearly distinguished:

(a)
a lowbias conductance plateau \(G=e^2/2h\), with small oscillations due to finite \(L_y\);

(b)
an intermediatebias regime with large conductance oscillations \(0<G<e^2/h\);

(c)
a metalliclike behaviour at larger biases, with quasiparticle diffusive transport.
In the lowbias regime, a single CMEM can be found within the surface gap of the \({\mathscr {N}}=1\) nontrivial superconductor. In a junction with nonproximitized TI films in the QAH state, the electric conductance is expected to be halfquantized at \(G=e^2/2h\)^{7,33}. Oscillations around the plateau are due to finitesize coupling between edge states on opposite y sides^{34}. 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 metalliclike 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 film^{35,36,37,38}. Experimentally, the local probing of currents is in principle possible with magnetic imaging techniques using miniaturized squids^{39}.
The longitudinal component of electriccurrent density is^{40}
with the quasiparticle velocity operator given by
Equation (8) can be straightforwardly evaluated once the realspace 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 biasdependent 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 phase^{12}, the current is welllocalized 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 finitesize (\(L_y\)) effects, no electric current is transmitted or reflected by the superconductor^{33}, 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 edgelocalized 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 \(\delta _k = k_1k_2\) between the edge channels, which is set by the energy of the quasiparticles. These biasinduced edgecurrent reversals are a conspicuous manifestation of the interference of chiral modes in the TSC.
Finally, Fig. 4e–f show the metalliclike 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 twomodes 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 finetuned conditions constitute a limitation for the validity of our analysis, the band structure of the bismuthtelluridebased TI compounds can be engineered^{41,42}, allowing the manipulation of the Dirac surface states without altering the chemical potential of the bulk crystals.
Conclusion
In conclusion, we characterized the electric quasiparticle transport through a NSN junction made by magnetic TI thin slabs, where the normal leads are kept into a QAH state while the proximitized sector realizes a \({\mathscr {N}}=1\) chiral superconductor. Three different electric regimes can be observed by increasing the total bias across the junction. In the limit of \(\mu =0\), we predicted peculiar oscillations in the electric conductance due to the coexistence of a pair of degenerate edge modes in the proximitized sector of the junction. The emergence of this oscillatory regime is strictly related to the physics of the chiral TSC.
Data availability
All data generated or analysed during this study are included in this published article.
Change history
23 January 2024
A Correction to this paper has been published: https://doi.org/10.1038/s41598024524688
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Acknowledgements
This project is financially supported by the QuantERA grant MAGMA by MCIN/AEI/10.13039/501100011033 and by European Union NextGenerationEU/PRTR under project PCI2022132927. L.S. acknowledges support from Grants No. PID2020117347GBI00 funded by MCIN/AEI/10.13039/501100011033 and No. PDR202012 funded by GOIB.
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L.S. Conceptualization, supervision; D.D. research, initial draft; D.D. and L.S. analysis, manuscript reviewing.
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The original online version of this Article was revised: The Acknowledgements section in the original version of this Article was incomplete. It now reads: “This project is financially supported by the QuantERA grant MAGMA by MCIN/AEI/10.13039/501100011033 and by European Union NextGenerationEU/PRTR under project PCI2022132927. L.S. acknowledges support from Grants No. PID2020117347GBI00 funded by MCIN/AEI/10.13039/501100011033 and No. PDR202012 funded by GOIB.”
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Di Miceli, D., Serra, L. QuantumanomalousHall current patterns and interference in thin slabs of chiral topological superconductors. Sci Rep 13, 19955 (2023). https://doi.org/10.1038/s41598023472863
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DOI: https://doi.org/10.1038/s41598023472863
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