WNL properties
We start by establishing the characteristic bulk and surface properties of WNLs. In order to clearly isolate the most important features we first use a prototype minimal model that captures the important physical details of a WNL. This is a toy model insofar as it captures an accidental nodal loop without any symmetry protection, but its simplicity significantly aids in understanding the underlying physics of WNL Josephson junctions. We then compare these results with a fully generic low-energy Hamiltonian for a WNL derived by the k ⋅ p method and show how the two models are in excellent agreement for the Josephson current.
A minimal WNL toy model is described by the Hamiltonian22,24:
$${H}_{{\rm{WNL}}}\,=\,{t}_{{\rm{w}}}\left(M{\sigma }_{x}\,+\,2{\alpha }_{2}{k}_{z}{\sigma }_{y}\right)\,-\,\mu ,$$
(1)
where M = α1 − k2, k = (kx, ky, kz) with k = ∣k∣ is the electron wave vector, and σ are the Pauli matrices in spin space. The equivalently minimal Hamiltonian for a DNL has instead σ in Eq. (1) acting in orbital basis as a DNL retains spin degeneracy. Hence the minimal DNL Hamiltonian is described in the same basis as a 4 × 4 Hamiltonian with twofold (spin) degeneracy. Moreover, here tw is the overall hop** amplitude, μ the chemical potential. For simplicity we measure energy in units of tw. The Fermi surface is a nodal line loop at zero do** μ = 0, while it forms a thin torus for nonzero μ, with its shape tuned by the two parameters, α1,2. We primarily use α1 = α2 = 1, which results in an essentially circular Fermi surface, but our results are not sensitive to this particular choice, see Supplementary discussion. The energy dispersion away from zero energy takes the Weyl-like form k ⋅ σ, giving the material class its Weyl nodal loop name14,23.
We choose to illustrate the physics of WNL Josephson junctions using the minimal Hamiltonian in Eq. (1) as it is a very simple model, offer easy comparison between WNLs and DNLs, and is also used in earlier literature discussing the immunity of the surface of WNLs toward conventional superconductivity24. Still, Eq. (1) only uses two out of the three σ-matrices, while in a generic WNL all three σ-matrices will be represented. Thus, in addition to Eq. (1), we present complementary results for a generic low-energy Hamiltonian, which includes all three σ-matrices:
$${H}_{{\rm{g-WNL}}}\,=\,{t}_{{\rm{w}}}\left(M{\sigma }_{x}\,+\,2{\alpha }_{2}{k}_{z}({k}_{x}^{2}\,-\,{k}_{y}^{2}){\sigma }_{y}\,+\,4{\alpha }_{2}{k}_{x}{k}_{y}{k}_{z}{\sigma }_{z}\right)\,-\,\mu ,$$
(2)
which is the same low-energy Hamiltonian as proposed for the WNL HgCr2Se48,10 upon a 90° rotation of the spin axes. The generic Hamiltonian Eq. (2) breaks the \({\mathcal{T}}{\mathcal{I}}\) symmetry, i.e., the product of time-reversal \({\mathcal{T}}\,=\,-{\rm{i}}{\sigma }_{y}{\mathcal{K}}\), where \({\mathcal{K}}\) is the complex conjugation operator, and spatial inversion \({\mathcal{I}}:(x,y,z)\to (-x,-y,-z)\) symmetries, allowing for finite spin polarization along with spin–orbit coupling. It also contain an additional mirror symmetry,\({\mathcal{M}}:(x,y,z)\to (x,y,-z)\), to protect the nodal line8.
To study a finite WNL with its surface states, we place the continuum Hamiltonians Eqs. (1)–(2) on cubic lattice, performing the usual substitution of \(k\to \sin (k)\) and \({k}^{2}\to 2(1\,-\,\cos (k))\). In the x- and y-directions we keep the reciprocal-space description (implemented by applying periodic boundary conditions), while in the z direction we discretize the Hamiltonians to produce a finite slab, see Methods. We assume a lattice with nw = 21 layers along the z-direction, as indicated by black solid circles in Fig. 1a. Note that the distance between lattice sites are that of the full unit cell and therefore nw = 21 junction length is reasonable for the Josephson effect, but we have also checked our results for much longer lengths, see Supplementary discussion.
In order to understand the basic physics of minimal WNLs, we plot in Fig. 1b the energy dispersion along the kx direction for the Hamiltonian in Eq. (1). At zero do** the band dispersion is electron–hole symmetric and two bands, indicated in red, have a vanishing electron group velocity and form zero-energy flat bands in a large region around the Γ point. These two bands reside on the two slab surfaces, as clearly seen in Fig. 1c, where we display the spin-polarized DOS for several different layers. The two surface layers have a very large peak at zero energy, which are fully spin polarized but in opposite directions. The bulk on the other hand has only a small constant DOS at zero energy due to the nodal line Fermi surface and there is no net spin polarization in the middle of the slab. It is the spin–orbit-like interaction term \({\alpha }_{2}{\sigma }_{y}\sin ({k}_{z})\) in the WNL Hamiltonian Eq. (1) that causes the characteristic spin rotation throughout a WNL material. This term is off-diagonal in spin space and hence couples the up and down spins in a spatially dependent manner. In a slab configuration, this term rotates the spins from spin-down polarization at zero-energy for the left surface (layer 1) to spin-up polarization for the right surface (layer nw), which we schematically illustrate with arrows in Fig. 1a. It is only at very large energies that the first (nwth) surface hosts any up- (down-)spin polarization. At finite do** the drumhead surface states remain unchanged with a full spin polarization, but are now located at an energy μ below the Fermi level. At the same time, the DOS at the Fermi level in the bulk increases due to the torus-shaped Fermi surface at finite do**. The spin-polarization stays however almost complete in a rather large energy window around μ = 0 and thus results are not sensitive to the exact tuning of the chemical potential.
Odd-frequency pairing
Next, we place two conventional spin-singlet s-wave SCs of the same superconducting material in proximity to the two surfaces of the WNL slab, see Fig. 1a. The superconducting order parameter amplitude in the SCs is set by Δ, but we allow for different phases, φL,R such that a Josephson current can be generated across the superconducting heterostructure. We couple the WNL and the SCs using a generic spin-independent tunneling amplitude tsc-wσ0, which connects the surface site of the left and right SCs to the left and right surfaces of the WNL, respectively.
To study proximity-induced superconductivity in the WNL we extract the superconducting pair amplitudes in the WNL by calculating the anomalous Green’s function F of the full heterostructure (see “Methods”). Here we only have to focus on isotropic, or equivalently on-site, s-wave pairing as we find that to be the only relevant spatial symmetry of all pair amplitudes. Notably, all the p-wave components in the xy-plane are practically zero for both WNL models; Eq. (1) even hosts an explicit in-plane spatial even parity preventing in-plane odd-parity proximity pairing. In the Supplementary discussion we additionally show that out-of-plane p-wave pairing is also much smaller, stemming from it not being aligned with the superconducting surface, and also that extended s-wave pairing is not relevant. Moreover, we note here that, if we were to add any disorder, that would even further favor isotropic s-wave pairing, thus only strengthening our results. In terms of spin configurations, the Weyl spectrum rotates the spin and can thus allow for both equal- and mixed-spin-triplet pairs. We therefore study all possible spin configurations for the superconducting pairing.
In Fig. 2 we display the real (upper panels, a–d) and imaginary (lower panels, e–h) parts of the anomalous Green’s function F as a function of frequency ω, divided into all possible spin configurations for the minimal WNL model. Each column represent a different layer in the WNL, layers 1, 2, 3, and n = (nw + 1)/2. We note directly that all different spin-triplet components appear throughout the WNL and that they are always odd functions of frequency, as required by the Fermi-Dirac statistics of the Cooper pairs. In the first surface layer there is still notable spin-singlet pairing. This is to be expected since the first layer is directly coupled to the SC and therefore necessarily harbors superconducting pairs of same symmetry as in the SC. However, the spin-singlet amplitude decay extremely quickly into the WNL, such that it has essentially disappeared already in the second layer. This behavior is not surprising when considering that the drumhead states of the WNL are fully spin polarized and thus the tunneling of opposite spins is energetically extremely costly. Similar immediate destruction of spin-singlet amplitudes have previously also been reported for HM junctions42. Despite the complete lack of proximity effect for spin-singlet superconductivity beyond the first surface layer, there is still significant pairing induced in the WNL. It is instead equal-spin-triplet pairing with spins aligned with the spin polarization of the drumhead state that growths and heavily dominates in the subsurface layers. Thus, the WNL essentially becomes an odd-frequency SC beyond the very first surface layer.
In the middle of the sample, Fig. 2d, h, all pair amplitudes are suppressed due to the distance from the SC interface, but notably, the two equal-spin pairing terms have exactly the same magnitude, just mirrored in ω = 0. Plotting the pair amplitudes also for the right half of the WNL, we find exactly the same results as for the left part shown in Fig. 2, only with spin-up and spin-down interchanged. The behavior of the equal-spin pairing is the superconducting equivalent of the spin polarization in the normal state twisting from full spin-down polarization in the left surface layer to full spin-up polarization in the right surface layer. Thus the appearance of large odd-frequency equal-spin-triplet components in WNLs is guaranteed by the intrinsic Weyl spin–orbital structure of the WNL normal state.
To better probe the propagation of Cooper pairs inside the WNL, we plot in Fig. 3a, b the absolute value of the different pair amplitudes for the minimal WNL in Eq. (1), as a function of the layer index n for the whole left side of the WNL. The pair amplitudes in the right part is obtained from the left layers by just interchanging spin-up and spin-down. We display the result for two different chemical potentials, μ = 0, 0.1, respectively, to capture both nodal loop and torus-shaped Fermi surface WNLs. Further, we set ω = 0.5Δ, but keep all other parameters as in Fig. 2. Other choices of ω can be found in the Supplementary discussion, showing no change of trends compared with Fig. 3. We find the same extremely fast suppression of the spin-singlet amplitude for both the undoped and doped case. The mixed-spin-triplet state experiences the same decay, due to the same unfavorable spin alignment as the spin-singlet pairing. Instead, it is spin-down triplet pairing that is clearly dominating, also well into the WNL and for all μ. The finite but still small spin-up triplet pair amplitude is due to probing the pair amplitudes at finite energies, where also bulk states give a finite contribution. Increasing the do** level thus show no significant changes in the relative importance of the different pairing channels. However, the magnitude of the pair amplitudes increases due to the increased bulk DOS in finite doped WNL. Overall this shows that the almost exclusively odd-frequency pairing state in the WNL is not sensitive to the tuning of the do** level. The relative insensitivity to the do** level means no do** fine-tuning is needed nor can the effects be fragile to a finite buckling of the drumhead surface state. In the Supplementary discussion we also plot the pair amplitude propagation for other lengths of the WNL junction and confirm that dominating odd-frequency pairing is also present for much longer junctions. Thus the presence of odd-frequency pairing is not just a simple surface effect, but more appropriately linked to the Weyl spin–orbital structure of the bulk band structure.
Furthermore, in Fig. 3c we present the evolution of pair amplitudes based on a generic WNL given in Eq. (2), also at μ = 0 to compare with panel (a). As seen, the spin-singlet pairing decays very fast, although it still have a finite amplitude in the middle of the sample. This is due to a lack of full spin polarization on the surfaces. Thus the spin-singlet response is rather similar to that of the slightly doped minimal WNL in panel (b). Dominating inside the WNL are clearly the equal-spin-triplet states, and they are even stronger in comparison to the equal-spin-triplet pairing in the minimal WNL in panel (a), which we can attribute to more strongly spin-splitted bands in the generic WNL. At the same time, the bands in the generic WNL are symmetric under interchange of spin and changing k → −k, and thus both equal-spin-triplet states after summation over k are equal.
To demonstrate the remarkable pairing in WNL Josephson junctions we compare the results with the behavior of similar Josephson junctions made of NM (d) and HMs without (e) and with (f) a spin-active interface region. In order to create systems with directly comparable properties we model the NM by removing all spin-dependence from the WNL Hamiltonian in Eq. (1), such that \({H}_{{\rm{NM}}}\,=\,{t}_{{\rm{w}}}(6\,-\,2\cos ({k}_{x})\,-\,2\cos ({k}_{y})\,-\,2\cos ({k}_{z}))\,-\,\mu\). This creates a prototype parabolically dispersive metal, where we set μ = 0.1 to reach a finite bulk DOS. For the NM, no term breaks the spin degeneracy, and thus only spin-singlet pairing is present in the NM. This spin-singlet amplitude experiences a regular slow decay, set by the conventional proximity effect.
For the HM junctions, we add the term mzσz, with mz = 0.5, to the NM Hamiltonian HNM, such that only spin-down electrons are present at zero energy. This strong magnetization causes the same dramatic suppression of the spin-singlet amplitude as in the WNL. It also allows for spin rotation into the mixed-spin-triplet state. However, this odd-frequency spin-triplet pairing state is always small and fast-decaying, as the occurrence of spin-up components is not energetically favored by the magnetization44. To achieve a spin-down triplet component, an additional spin quantization axis has to be present in the HM junction. This is often achieved by introducing a spin-active region at the interface between the SC and HM, see e.g.,42,43. In this case the Cooper pair spin quantization axis rotates between the interface and the HM bulk, with the consequence that spin-equal pairing is induced beyond the interface. In Fig. 3f we therefore add a term mxσx, with mx = 1, to the the two surface layers of the HM to model a strongly spin-active interface. As a result, spin-equal triplet pairing is generated, of which the spin-down component survives throughout the HM region. The spin-up triplet state is also initially generated at the interface, but it is energetically unfavorable and decays very quickly in the HM.
Comparing the WNLs with the NM and HM junctions, we see that the WNLs junction closest resemble that of the HM with an active spin interface, since they both experience a strong proximity effect consisting of odd-frequency equal-spin-triplet pairing. However, WNL Josephson junctions are fundamentally different from HM Josephson junctions as they do not need any additional spin-active interface region added during manufacturing in order to generate equal-spin odd-frequency pairing. In the simplistic WNL junctions based on Eq. (1) we find that the intrinsic Weyl spin–orbit coupling causes the initial spin-down pairing in the left surface to rotate into spin-up pairing in the right surface. As a consequence, there is a clear decay of the spin-up triplet component into the middle of the WNL, which is not present in the HM. It is thus not the distance from the SC that causes the main decay of the equal-spin-triplet components in Fig. 3a, b, but mainly the continuous rotation of the spin orientation of the Cooper pairs. This is also clear when considering other WNL junctions lengths in the Supplementary discussion. For the generic WNL model, we find that both equal-spin-triplet pairing are present at equal amounts in each layer due to its spin structure, and then there is a smaller decay of the odd-frequency state away from the surface of the WNL.
We also note that the size of the pair amplitudes in the WNLs at zero do** is actually of the same order of magnitude as in the NM junctions. This is particularly surprising since the nodal line bulk state has a much smaller DOS at low energies compared with the NM (see Supplementary discussion for a detailed account on the low-energy DOS). We attribute the large pair amplitudes in undoped WNLs to the singular zero-energy DOS of the drumhead surface states, which creates a naturally strong coupling between the WNL and SCs. The large effect of the surface drumhead states is further evident when we compare the results for a DNL Josephson junction. Here only spin-singlet s-wave pairing is present since the DNL Hamiltonian is spin degenerate, but we find a very large conventional proximity effect due to the the (unpolarized) surface drumhead states, see Supplementary discussion for details.
Exotic Josephson current
Having shown how odd-frequency spin-triplet pairing generates large proximity-induced superconductivity in WNLs, despite the incompatibility of spin-polarization and the spin-singlet SCs, we now turn to the possibility of measuring a finite Josephson current in the WNL junctions. For comparison we also calculate the current in DNL, NM, and HM Josephson junctions. While we limited the investigation in Figs. 2–3 to isotropic s-wave pairing, the contributions from all pairing channels are automatically included when calculating the current, although all non-s-wave contributions are small for WNLs.
In Fig. 4 we show in a log-plot the maximum Josephson current J between two conventional spin-singlet SCs as a function of the chemical potential μ, superconducting order parameter Δ, and WNL-SC tunneling tsc-w. The results are obtained for φL = π/2, φR = 0, which to a very good approximation gives the maximum current as \(J \sim \,\sin\, ({\varphi }_{{\rm{L}}}\,-\,{\varphi }_{{\rm{R}}})\), see Supplementary discussion. The most remarkable result is that WNL junctions carry a large Josephson current through all parameter regimes, orders of magnitude higher than the HM junctions, the other odd-frequency systems, but also compared with the NM junctions for a large range of do** levels. The current for both WNL junctions is also always essentially identical, illustrating explicitly that the specific form of the WNL Hamiltonian is irrelevant for the Josephson current. We note particularly that the slightly different spin-singlet pairing in the two different WNL junctions do not influence the Josephson effect. This can also be understood from the results for NM junction, which only contains spin-singlet pairing but where the Josephson current is orders of magnitude smaller than the WNL junctions in the small do** regime. Instead it is the common physical properties of WNLs, their strong spin-polarization (surface and bulk), and drumhead surface states, that generates a unique odd-frequency system with large Josephson currents. WNL Josephson effect thus forms a key example on the importance of odd-frequency pairing in inhomogeneous superconducting systems. If it were not for the odd-frequency correlations, there would be no proximity-induced superconductivity or measurable Josephson effect in WNL junctions.
In particular, we find for increasing μ in Fig. 4a that the current increases for all junctions, while also displaying some overlaid oscillatory behavior. This is expected since the low-energy DOS increases with μ for all junctions with some smaller oscillations due to finite size effects. Most notably, the WNL junctions carry a very large current over a wide range of low to moderate do** levels, even orders of magnitude larger than both the NM and HM junctions. This is really quite exceptional considering the low bulk DOS of the nodal line/thin torus Fermi surface in the bulk of WNLs. In fact, the WNL bulk DOS for Eq. (1) is for the full do** range smaller than that of the bulk NM, for μ ≲ 0.4 it is even several times smaller, see Supplementary discussion. We must therefore accredit the large Josephson current in WNL junctions to a remarkable effect of the drumhead surface states.
The effect of the drumhead states is even more obvious when we compare to a DNL Josephson junction. Here we use a DNL with the same band structure and total DOS as the WNL in Eq. (1), but keep a fully spin-degenerate ground state. Thus the Josephson effect and current are entirely conventional and carried by even-frequency spin-singlet s-wave pairs in a DNL, while the coupling to the SCs is determined by the same surface DOS peak as in the WNL junction. We find that the DNL junction carries a current that is consistently large overall parameter regimes investigated in Fig. 4. In the simplest of cases, the maximum Josephson current in a junction is proportional to the normal current, which in turn is proportional to e2ρvF, where ρ is the DOS and vF is the Fermi velocity in the direction of the current. While the DOS varies within the junction for both DNLs and WNLs, this still demonstrates that the drumhead surface state should dramatically enhance the current in both cases. We here note that in general flat band systems the current usually suffers from the diminishing Fermi velocity. However, for the DNL and WNL junctions considered here it is only the Fermi velocity in-plane that is quenched due to the flat dispersion, while it is the velocity in the out-of-plane direction that determine the current, which is still large. We also note that the DNL current is larger than the WNL current, which we can easily attribute to the DNL junction only hosting conventional pairing and thus does not have to transform to other pair amplitude symmetries to create a viable Josephson effect.
Moving on to compare with the HM junctions that also host odd-frequency spin-triplet pairs, we find that the HM without a spin-active interface cannot effectively carry Josephson current, which is a direct consequence of the lack superconducting pair amplitudes in the HM. Introducing a spin-active interface layer, we find a significantly increased current due to the perseverance of odd-frequency equal-spin-triplet pairs inside the HM. Yet, it is only at extremely large μ that the HM system carries a similarly sized current to the WNL junction. For large μ the drumhead surface states are located very far from the Fermi level and are thus less active in electric transport. Thus, the large current in heavily doped WNL junctions is instead primarily a manifestation of how powerful the Weyl spin–orbit interaction is in generating odd-frequency equal-spin pairing to carry the current, and thus it is natural that the WNL and HM junction behave similarly in this extreme do** limit.
We here note that enhancement of Josephson current in junctions with odd-frequency pairing has previously been associated with generation of zero-energy Andreev bound states, as for example in SC-insulator-SC junctions45,46,47,48. In fact, zero-energy Andreev bound states have been found to be very common in systems with odd-frequency pairing49,50,51, although not always present52,53,54,55. In the Supplementary discussion we show that zero-energy Andreev boundary states also exists at the WNL-SC interface. However, considering the much larger current for the WNL junction as compared with the HM junctions, zero-energy Andreev bound states alone cannot alone explain the results, but instead the drumhead surface states are more important.
Finally, turning Fig. 4b, c, we explore the other parameter dependencies of the Josephson current. Again note that both models of WNL are in a excellent agreement with each other, and thus our main conclusions are not dependent on any specific model. Figure 4b shows the variation of the Josephson current with respect to the superconducting gap parameter Δ, kee** the chemical potential of NM and HM at μ = 0.1, while the WNLs are both undoped. Thus, both the NM and HM have a large metallic DOS at low energies, while the DNL and WNL have only 1D Fermi nodal loops. Despite this we see how the Josephson current is actually larger in WNL junctions compared with the NM and HM junctions for all Δ values. Notably, the choice of Δ = 0.01 in Figs. 2 and 3 gives by no means the maximum Josephson current, for example choosing Δ ≈ 0.005 gives approximately three times larger current. In Fig. 4c we then tune the tunneling between the SCs and the junction, tsc-w. Usually this parameter is lower than the hop** inside the junction and in the SCs, limiting it to tsc-w < 1, and in practice it is tuned by modifying the interfaces. Larger tunneling clearly enhances the Josephson current in all junctions, but we find that the WNL junctions again carry larger Josephson current than the NM and HM junctions for all values of tsc-w. In particular, a large Josephson current is present even for small tsc-w ~ 0.1, where the overall effect of the SCs on the electronic structure of the WNL is necessarily quite limited. We provide additional plots for other choices of chemical potential in NM and HM in the Supplementary discussion, verifying that the large Josephson currents through the WNL junctions in Fig. 4 are generic results and not restricted to a narrow range of physical parameters.
