Identification of odd-frequency superconducting pairing in Josephson junctions

Introduction: In a conventional superconductor, electrons form Cooper pairs due to a weak attraction mediated via lattice vibrations[1]. The pairing function between the two electrons within a Cooper pair can be classified into different symmetry classes by means of four permutation operators with respect to spin ($S$), relative coordinate ($P^{*}$), orbital index ($O$), and time coordinate ($T^{*}$), all of which can only have eigenvalues $\pm 1$. Generally, BCS theory of single orbital superconductor ($\expectationvalue{O}=1$) is instantaneous in time ($\expectationvalue{T^{*}}=1$). This leaves two choices for the combinations of eigenvalues of the operators $S$ and $P^{*}$ to satisfy the Berezinskii condition[2] $SP^{*}OT^{*}=-1$ for the pairing function, i.e., $\{\expectationvalue{S},\expectationvalue{P^{*}}\}=\{-1,1\},\{1,-1\}$. These correspond to spin-singlet even-parity (e.g. $s$-wave) and spin-triplet odd-parity[3] (e.g. $p$-wave) symmetries, respectively. If one removes the constraint of instantaneous pairing, the possibility of temporal (non-local in time) Cooper pairing becomes feasible, and both $\expectationvalue{T^{*}}=\pm 1$ can be possible, depending on whether the pairing function is symmetric or anti-symmetric under the permutation of relative time coordinates of the electrons forming the Cooper pair. Temporally symmetric pairing ($\expectationvalue{T^{*}}=1$) does not give rise to any new symmetry classes other than those already described by BCS theory and is generally known as even-frequency (even-$\omega$) pairing. However, temporally anti-symmetric pairing ($\expectationvalue{T^{*}}=-1$), which is known as odd-frequency (odd-$\omega$) pairing[4, 5, 6], opens up possibilities of two new symmetry classes (assuming $\expectationvalue{O}=1$) that cannot be described by BCS theory, namely, spin-singlet odd-parity ($\{\expectationvalue{S},\expectationvalue{P^{*}}\}=\{-1,-1\}$) and spin-triplet even-parity ($\{\expectationvalue{S},\expectationvalue{P^{*}}\}=\{1,1\}$) pairings, consistent with Berezinskii condition. It can also be shown that this odd-$\omega$ (even-$\omega$) pairing is also odd (even) under the sign change of energy[4], which gives the advantage of observing the behavior of this pairing in the energy domain rather than in the temporal domain.

On a historical note, the concept of odd-$\omega$ spin-triplet pairing traces back to Berezinskii’s 1974 proposal in ³He[2] and predictions in disordered systems[7, 8]. Balatsky and Abrahams later indicated the presence of odd-$\omega$ spin-singlet pairing in time-reversal and parity symmetry broken superconductors[9]. Subsequently, the odd-$\omega$ pairing was explored within the framework of a two-channel Kondo system[10], the 1D $t-J-h$ model[11], the 2D Hubbard model[12] and heavy fermion compounds[13]. The bulk odd-$\omega$ pairing has been indirectly hinted theoretically through the Majorana scanning tunneling microscope[14] and also experimentally using the Kerr effect[15, 16], and the paramagnetic Meissner effect[17, 18, 19, 20, 21]. Further, the evidence of bulk odd-$\omega$ pairing due to magnetic impurities has emerged in $s$-wave superconductors[22, 23]. In Ref. [24], a measurement tool was introduced for directly detecting odd-$\omega$ pairing in bulk systems using time- and angle-resolved photoelectron fluctuation spectroscopy. However, this method requires advanced technology beyond the current capabilities of facilities. More recently, in Ref. [25], authors proposed a detection scheme to directly identify odd-$\omega$ pairing in bulk systems using the quasiparticle interference method in the presence of an external magnetic field. Initially, odd-$\omega$ pairing was regarded as an inherent bulk phenomenon[2, 9, 26] but was subsequently acknowledged to manifest in heterojunctions[27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55] and in the systems under the influence of time-dependent fields[56, 57]. Certain theoretical studies have indirectly identified odd-$\omega$ pairing in heterostructures. For instance, this was achieved through phase-tunable electron transport in topological Josephson junctions (JJ)[58], examining Josephson current characteristics on the surface of Weyl nodal loop semimetals[59], and analyzing current noise in JJ[60]. Experimental evidence of odd-$\omega$ pairing in heterostructures emerged through measurements of long-range supercurrents in magnetic JJ[61, 62]. These two experiments utilize a ferromagnetic material to create an odd-$\omega$ pairing effect, which is subsequently identified by measuring long-range superconducting pairing. In contrast, this letter investigates a configuration that inherently supports odd-$\omega$ pairing without the need for any ferromagnet/superconductor junction. Topological JJ at the edge of a two-dimensional (2D) quantum-spin-Hall insulator (QSHI) serves this purpose. It is important to mention here that induced superconductivity in the QSHI edge has already been experimentally demonstrated[63].

We demonstrate that it is possible to read off all the independent, spatially non-local Green’s functions by measuring differential conductance between two spin-polarized probes placed at the junction. The underlying interplay between spin-momentum locking of helical edge states (HES) and spin-polarization of the probes allows for this advantage. Experimentally, the injection of spin-polarized current into the HES of QSHI has already been achieved[64]. By using spin-polarized probes, one can inject (detect) electrons into (from) the helical edge with well-defined position and momentum simultaneously, which, in general, is prohibited due to uncertainty principle[65].To elaborate further, a spin-$\uparrow$ (spin-$\downarrow$) electron injected at position $x$ in the helical edge can only tunnel into the right (left) moving edge mode owing to spin-momentum locking. Keeping the spin-polarization axes of the tunneling probes aligned parallel or anti-parallel to the spin-quantization axis of HES, we study the difference between non-local differential conductance from the left probe to right probe $\kappa^{21}_{s\bar{s}}(V,0)$ and from right probe to left probe $\kappa^{12}_{\bar{s}s}(0,V)$ i.e.

where $\{s,\bar{s}\}\in\{\uparrow,\downarrow\}$ with $s\neq\bar{s}$, and $V_{1}(V_{2})$ denotes the applied voltage at the left (right) tunneling probe $P_{1}$ ($P_{2}$). It can be shown that this quantity $\mathcal{A}^{2,1}_{s,\bar{s}}$ can be written as a product of odd-$\omega$ and even-$\omega$ pairing amplitudes and hence should be odd with respect to incident energy (and hence with respect to the applied voltage $V$). We show that this difference stems from the same origin as that of the odd-$\omega$ part of the non-local differential conductance, which can be accessed via bias reversal i.e. $\mathcal{B}^{2,1}_{s,\bar{s}}=\kappa^{21}_{s\bar{s}}(V,0)-\kappa^{21}_{s\bar{s}}(-V,0)$. This fact leads to

From an experimental perspective, measuring the anti-symmetric behavior of each of these independent quantities and their equality confirms the existence of odd-$\omega$ pairing. In what follows, we will first present our results using an effective 1D analytic model of HES. Following that, we’ll provide a 2D simulation of a realistic setup, in line with HgTe/CdTe quantum well parameters, confirming our predictions.

Model and its motivation: The system, we are interested in, is a JJ of length $L$ ($0\leq x\leq L$), realized in a helical edge state (HES) of a 2D QSHI[66, 67, 68, 69, 70] (1D Dirac fermions) which is proximitized to a conventional $s$-wave superconductor. Schematic of the setup is shown in FIG. 1 (a), while FIG. 1 (b) shows the corresponding simplified cartoon representation of the same setup. The Bogoliubov-de Gennes (BdG) Hamiltonian characterizing such junctions can be written as[71, 72] $H_{J}=\int_{-\infty}^{\infty}dx\,\Psi^{\dagger}\mathcal{H}_{J}\Psi$ where $\Psi=(\psi_{\uparrow},\psi_{\downarrow},\psi_{\downarrow}^{\dagger},-\psi_{\uparrow}^{\dagger})^{T}$ is the Nambu basis and

Here, $\sigma_{i}$ and $\tau_{i}$ are the Pauli matrices acting respectively on the spin basis and particle-hole basis, $\mu$ is the chemical potential of the HES, $v_{F}$ represents Fermi velocity of the HES, and, $\hat{p}=-i\hbar\partial_{x}$ is the momentum operator. The spin-quantization axis of the HES is considered to be along $z$-direction[71, 72]. The superconducting pairing potential is given by $\Delta(x)=\Delta_{0}[\Theta(-x)+\Theta(x-L)]$. Superconducting leads ($S_{r}$) are identified as $r\in\{1,2\}$ with the corresponding superconducting phases $\varphi_{r}$ such that the superconducting phase difference $\varphi_{21}=\varphi_{2}-\varphi_{1}\neq 0$, in general.

Two spin-polarized tunneling probes[73] $P_{\{1,2\}}$ ($P_{r}$ being closer to $S_{r}$) with spin-polarization axes oriented respectively along $\hat{n}_{1}$ and $\hat{n}_{2}$, at angles $\theta_{1}$ and $\theta_{2}$ with respect to the spin-quantization axis of the HES (say, in the $z-x$ plane)¹¹1The azimuthal angle is of no consequence, e.g., it can also be assumed in the $z-y$ plane, are introduced at positions $x=x_{1}$ and $x=x_{2}$ respectively within the junction region ($0<x_{1}<x_{2}<L$). For most of the paper, we will be only discussing the case for $\{\theta_{1},\theta_{2}\}=\{0,\pi\}$, unless otherwise stated. To keep the algebra simple while retaining the essential physical inputs, we model the probes as a spin-polarized 1D mode. The second quantized Hamiltonians of these probes can be expressed as:

The tunneling Hamiltonian between the HES and the probes can be written as:

with $\alpha,\alpha^{\prime}\in\{\uparrow,\downarrow,\hat{n}_{i}\}$ ($\alpha\neq\alpha^{\prime}$ and $i\in\{1,2\}$). $t^{i}_{\alpha\alpha^{\prime}}=t_{i}\gamma_{\alpha\alpha^{\prime}}$ is the tunneling strength between $\alpha$ and $\alpha^{\prime}$ at $x=x_{i}$ and $\gamma_{\alpha\alpha^{\prime}}$ is the overlap of the spinor part of the first quantized wave function of electrons in the probes and the HES. From this point onward, we shall consider $\hbar=v_{F}=1$. Note that this form of tunneling respects $SU(2)$ symmetry and hence cannot induce spin-flip scattering when $n_{i}$ is parallel or anti-parallel to the spin-quantization axis of the HES. Using the equation of motion approach for the Hamiltonian $H=H_{\text{HES}}+\sum_{i\in\{1,2\}}H_{P_{i}}+H_{T}^{P_{i}}$ (where $H_{\text{HES}}=H_{J}|_{\Delta=0}$), the scattering matrices can be expressed as,

where $\psi^{\text{in(out)}}$ are the corresponding incoming (outgoing) plane wave amplitudes on the HES and the probe. The scattering matrix for holes ($S^{h}$) can be determined by exploiting the particle-hole symmetry (for details, see Supplemental Material (SM) Section A) of the system. We are interested in studying charge transport between the two probes via the junction when a finite voltage bias $V$ is applied such that $-\Delta_{0}\leq eV\leq\Delta_{0}$.

Note that, the spatial non-locality of the probes, along with the helical nature of the edge state, opens up the possibility of detecting only a hole at $P_{2}$ when an electron is injected into HES through $P_{1}$ provided the polarization of the probes is tuned such that $\theta_{1},\theta_{2}=0\text{ or }\pi$ and $\theta_{1}\neq\theta_{2}$. This leads to the fact that the differential conductance between the probes is directly proportional to the square of the modulus of anomalous Green’s functions. This can be understood in terms of a simple process, as described below. If the spin polarization of $P_{1}$ is set to spin-$\uparrow$ (i.e., $\theta_{1}=0$), it can only inject a right-moving electron to the HES due to the spin-momentum locking of the edge states. In the absence of spin-flip scattering within the HES, this electron will either remain as a spin-$\uparrow$ electron after an even number of Andreev reflections (including the possibility of no Andreev reflection) or will convert to a spin-$\downarrow$ hole after an odd number of Andreev reflections when it reaches $x=x_{2}$. Thus, $P_{2}$ will detect only a hole for $\theta_{2}=\pi$. Along the same line of argument, it is straightforward to see that for $\theta_{2}=0$, $P_{2}$ will detect only an electron. By definition, the probability amplitude of these non-local transmissions will be proportional to the corresponding anomalous Green’s functions, namely $t_{ee,\uparrow\uparrow}^{21}\propto G^{r(a)}_{ee,\uparrow\uparrow}(x_{2}>x_{1})$ and $t_{he,\downarrow\uparrow}^{21}\propto G^{r(a)}_{he,\downarrow\uparrow}(x_{2},x_{1})$, where $t_{ee,\uparrow\uparrow}^{21}$ and $t_{he,\downarrow\uparrow}^{21}$ are the propagation amplitudes for spin up electron and spin down hole respectively from the left probe to the right probe and, $G^{r(a)}_{ee,s_{2}s_{1}}(x_{2}>x_{1})$ represents the retarded (advanced) normal Green’s functions while $G^{r(a)}_{he,s_{2}s_{1}}(x_{2},x_{1})$ ($s_{i}\in\{\uparrow,\downarrow\}$) denotes the retarded (advanced) anomalous Green’s functions calculated at $x=x_{2}$. Here retarded and advanced Green’s functions are defined in the appropriate frequency domain[38]. The above discussion is summarized in the form of TABLE. 1.

Using Landauer’s formula[75], we can calculate the differential conductance from $P_{1}$ to $P_{2}$, given by $\kappa_{\downarrow\uparrow}^{21}=\kappa_{ee,\downarrow\uparrow}^{21}-\kappa_{he,\downarrow\uparrow}^{21}$ and from $P_{2}$ to $P_{1}$, given by $\kappa_{\uparrow\downarrow}^{12}=\kappa_{ee,\uparrow\downarrow}^{12}-\kappa_{he,\uparrow\downarrow}^{12}$, where $\kappa_{p_{i}p_{j},s_{i}s_{j}}^{ij}=(e^{2}/h)|t_{p_{i}p_{j},s_{i}s_{j}}^{ij}|^{2}$ ($p_{i(j)}\in\{e,h\}$, $s_{i(j)}\in\{\uparrow,\downarrow\}$, $\{i,j\}\in\{1,2\}$). The difference between these two non-local differential conductances at energy (frequency) $\omega$ is given by:

where we have used the property $G^{r}_{he,s\bar{s}}(x,x^{\prime},\omega)=G^{a}_{he,\bar{s}s}(x^{\prime},x,-\omega)$ ($\{s,\bar{s}\}\in\{\uparrow,\downarrow\}$ and $s\neq\bar{s}$). Here we have used only retarded Green’s functions to express $\kappa$. However, a proof involving advanced Green’s functions follows in a similar way (See SM Section B). In Eq. (7), $\mathcal{A}^{2,1}_{\downarrow,\uparrow}$ is anti-symmetric with respect to $\omega$, and if measured, it can be treated as a direct signature of odd-$\omega$ pairing subjected to the condition that even-$\omega$ pairing within the junction is non-zero. In Eq. (7), $\mathcal{O}$ and $\mathcal{E}$ denote the odd-in-$\omega$ (anti-symmetric with respect to $\omega$) and even-in-$\omega$ (symmetric with respect to $\omega$) parts of the corresponding Green’s functions which are in turn directly related to odd-$\omega$ and even-$\omega$ pairing amplitudes respectively. Note that, the quantity in Eq. (7) can also be expressed as $\mathcal{B}^{2,1}_{\downarrow,\uparrow}=[\kappa_{\downarrow\uparrow}^{21}(\omega)-\kappa_{\downarrow\uparrow}^{21}(-\omega)]$. The voltage configuration necessary for measuring $\mathcal{A}$ and $\mathcal{B}$ is discussed above Eq. (2). It is worth mentioning that the Green’s functions discussed above are not, in general, the free Green’s function of the JJ, i.e. the Green’s functions in the absence of tunneling probes. However, in the weak tunneling limit ($t_{1},t_{2}\ll 1$), different inter-lead transmission amplitudes can be written in terms of free Green’s functions (for details see SM Section C). Next, we will proceed to put the above qualitative discussion on firm grounds by calculating the quantity $\mathcal{A}^{2,1}_{\downarrow,\uparrow}$ from numerical analysis using a 2D model, and comparing the theoretical results discussed above (and also results presented in the SM).

Numerical simulation using 2D model: In this part, we conduct a numerical simulation of a 2D lattice model depicted in Fig. 1 using the KWANT package [76] to further analyse the effective 1D model discussed above. The system is mainly a QSH insulator which is described by a real space configuration of the Bernevig-Hughes-Zhang (BHZ) model[77] mapped on a square lattice of dimension $140a\times 140a$ with additional terms incorporated as needed. Here $a(=3\,nm$) is the lattice spacing. Two superconducting leads are attached to the bottom side ($S_{1}$) and to the right side ($S_{2}$) of the system, which are defined by the $s$-wave superconductivity-proximitized QSH insulators[78] having a finite phase difference ($\phi_{21}=\phi_{2}-\phi_{1}$) between them and thus, defining a 1D JJ extended over the left and top edges of the QSH region (see Fig. 1(a)). On the other hand, two spin-polarized normal leads, which can be described by two quantum anomalous Hall insulators carrying chiral edge states of opposite chiralities, are tunnel-coupled via tiny insulating sections to the left side ($P_{1}$: having the $\uparrow$-spin channel only) and to the top side ($P_{2}$: having only the $\downarrow$-spin channel) of the QSH region. The QAH state of a spin-polarized lead is obtained by applying an exchange field to the QSH system, which oppositely affects the effective bulk gaps for the two spin states, eventually, at sufficient strength, destroying the topological state for one of them [79]. A strong onsite ferromagnetic impurity term is added over a small section of dimension $14a\times 14a$ at the extreme bottom-right corner (denoted by $M_{x}$) of the QSH region to ensure that the current is allowed to propagate only through the left and top edges of the QSH region in-between $S_{1}$ and $S_{2}$. A detailed insight into the 2D simulation is closely comparable to the same discussed in Ref. [80] along with a comprehensive understanding of how particle-hole symmetry can be incorporated into the lattice model [81]. For the numerical calculation, we consider the value of chemical potential ($\mu$) to be zero and the $s$-wave superconducting pairing potential ($\Delta_{0}$) as 2.0 meV. The values of the remaining parameters related to the BHZ model closely resemble those describing HgTe/CdTe quantum wells[77, 80, 81]. Finally, by conducting a numerical simulation with this setup, we compute the quantity $\mathcal{A}^{2,1}_{\downarrow,\uparrow}$ for $\phi_{21}=\pi/2$ as shown in FIG. 2 (solid cream line). The insulating necks of $P_{1}$ and $P_{2}$ are considered to be identical.

To compare the numerical result with the theoretical result, one needs to estimate the values of $L$, $t_{1}$, and $t_{2}$. To estimate the value of $L$, we analyzed the positions of the peaks and dips of $\mathcal{A}^{2,1}_{\downarrow,\uparrow}$. By mapping the corresponding energy values to the ABS energies, we estimate $L=4.35\xi$ where $\xi=\hbar v_{F}/\Delta_{0}$ is the superconducting coherence length. Plugging this value of $L$ into theoretical calculations we numerically calculated the following quantity:

for different values of $t_{1}$ and $t_{2}$. Here $N$ is the number of numerical grids within the energy window $-\Delta_{0}\leq\omega\leq\Delta_{0}$. Keeping the tunneling strengths same, it turns out $\delta_{\text{min}}\approx 2\times 10^{-3}$ for $t_{1}=t_{2}=0.37$ (dashed brown line in FIG. 2). Although, this plot is a good fit to the numerical results, if we relax the condition of $t_{1}=t_{2}$. then the value of $\delta$ can be even minimized giving rise to a better fit to the numerical results. It turns out, $\delta_{\text{min}}\approx 1\times 10^{-4}$ for $t_{1}=0.73$ and $t_{2}=0.32$ (black dot-dashed line in FIG. 2). Dependence of $\mathcal{A}$ on the system parameters $t_{1}$, $t_{2}$ and $L$ is shown in SM Section D.

Discussion: The above results rely on the fact that the spin-polarization axes of $P_{1}$ and $P_{2}$ can be adjusted exactly parallel or anti-parallel to the spin-polarization axis of HES. For any other values of $\theta_{1}$ and $\theta_{2}$, $\mathcal{A}$ will have contributions from interference effects between the spin channels and the electron-hole channels due to the presence of more than one Green’s function in the expressions of $t^{ij}_{p\bar{p}}$. These interference contributions will lead to the deviation from perfectly anti-symmetric features. Also, the detection of odd-frequency pairing through $\mathcal{A}$ does not require the probes $P_{1}$ and $P_{2}$ to be placed at different positions. This is due to the fact that the information of position enters into the calculation as pure phase contributions to $t^{ij}_{p\bar{p},s\bar{s}}$ ($\{s,\bar{s}\}\in\{\uparrow,\downarrow\}$, $s\neq\bar{s}$).

In our proposed set-up, the size of the sample (i.e. the length of the junction $L$) should be large enough to accommodate two spin-polarized probes. In Ref. [64], the width of the spin-polarized probe tip is $0.77$ $\mu m$, which sets the minimum dimension of the quantum spin-Hall sample (length of the edge) to be $1.54$ $\mu m$. However, in the same experiment, the size of the sample (length of the edge) is $4.1$ $\mu m$. Thus, it should be possible to experimentally fabricate a similar sample to realize our proposal. The values of other model parameters, e.g. Fermi velocity $v_{F}$, the chemical potential $\mu$, and pairing term $\Delta$ can be estimated from the experiment[63].

Lastly, note that $\kappa_{he,\downarrow\uparrow}^{21}$ is related to the ABS corresponding to the shuttling of a Cooper pair from left to right, while $\kappa_{he,\uparrow\downarrow}^{12}$ is related to the ABS corresponding to the shuttling of Cooper pair in the opposite direction. Thus, in the weak tunneling limit, at $\varphi_{21}=0$ or $\pi$, (when these two ABS become degenerate), the differential conductances $\kappa_{he,\downarrow\uparrow}^{21}$ and $\kappa_{he,\uparrow\downarrow}^{12}$ becomes equal, leading to the vanishing of $\mathcal{A}^{2,1}_{\downarrow,\uparrow}$.

Acknowledgements: S.P. and A.M. acknowledge the Ministry of Education, India, and IISER Kolkata for funding. V.A. would like to acknowledge Korea NRF (SRC Center for Quantum Coherence in Condensed Matter, Grant No. RS-2023-00207732) for funding.

Author contribution: S.P. and A.M. contributed equally to this work.