--- title: Nonreciprocal Josephson Transport url: https://www.emergentmind.com/topics/nonreciprocal-josephson-transport type: topic --- # Nonreciprocal Josephson Transport Searching arXiv for the cited papers to ground the article in current literature. arXiv search: 2605.17066 "Supercurrent spin Hall effect enabled nanopillar Josephson diodes" Nonreciprocal Josephson transport denotes superconducting transport regimes in which reversing current bias, exchanging active ports, or reversing propagation direction does not produce a response related by simple odd symmetry. In the two-terminal case its canonical manifestation is the Josephson diode effect, for which the forward and backward critical currents are unequal, \(I_c^+ \neq |I_c^-|\), commonly quantified by \(\eta = (I_c^+ - |I_c^-|)/(I_c^+ + |I_c^-|)\). Closely related phenomena include anomalous Josephson currents at zero phase bias, transverse Hall-like supercurrents, direction-dependent depinning of vortices, and nonreciprocal microwave transmission in extended junctions. Across recent work, the field has evolved from Rashba- and field-driven short junctions to intrinsically noncentrosymmetric barriers, multi-terminal phase networks, textured magnets, topological and helical channels, and explicitly nonequilibrium dynamical schemes [2103.06984]. ## 1. Symmetry conditions and operational definitions The standard symmetry logic is that a Josephson diode requires the simultaneous breaking of inversion symmetry and time-reversal symmetry. In highly transparent InAs-based Josephson junctions, inversion breaking is supplied by Rashba spin-orbit coupling and time-reversal breaking by an in-plane magnetic field, which yields an asymmetric current-phase relation and unequal critical currents [2103.06984]. The same symmetry principle is formulated in vertical van der Waals junctions with Td-WTe\(_2\), where the non-centrosymmetric crystal structure breaks inversion symmetry and an in-plane magnetic field breaks time reversal, producing an anomalous phase shift and supercurrent asymmetry [2303.13049]. In heavy-metal nanopillars, inversion symmetry is argued to be broken naturally at the Nb/Pt interfaces, while an in-plane field breaks time reversal and converts a current-induced spin moment into a diode response [2605.17066]. This two-symmetry rule is not exhausted by the Rashba-plus-field paradigm. Several platforms realize time-reversal breaking without an external magnetic field. In a graphene Josephson triode, a control supercurrent through one branch biases the common superconducting island and thereby breaks the time-reversal symmetry of the transport response, enabling field-free nonreciprocity with efficiencies above \(90\%\) [2210.02644]. In a conical magnet, the helical spin rotation generates Rashba-like inversion breaking while the out-of-plane canting acts as a Zeeman-like time-reversal-breaking component, so a single magnetic material supplies both ingredients [2409.00223]. In planar \(s\)-wave/\(d\)-wave/\(s\)-wave junctions, asymmetric interface coupling can induce a spontaneous \(\pm \pi/2\) phase difference and thus spontaneous time-reversal-symmetry breaking without a magnetic field, although the resulting \(\phi_0\)-junction state alone does not guarantee a diode effect [2512.02752]. Operational definitions also vary with geometry. In ordinary two-terminal junctions, nonreciprocity is usually identified from \(I_c^+ \neq |I_c^-|\). In multi-terminal graphene devices this criterion is insufficient: a reciprocal device can appear asymmetric in a single-port measurement if an additional bias current is present. The appropriate quantity is then a two-port reciprocity relation defined through a critical-current tensor \(I_{c,jk}\), with nonreciprocity meaning \(I_{c,jk} \neq I_{c,kj}\) rather than merely \(I_c^+ \neq I_c^-\) [2301.05081]. In transverse Josephson transport, the asymmetry is instead between oppositely directed transverse critical currents, \(j_{y,c}^+ \neq |j_{y,c}^-|\), quantified by a transverse diode quality factor \(\mathcal{Q}_t\) [2412.01394]. ## 2. Current-phase relations, anomalous phases, and harmonic content The current-phase relation is the central organizing object of nonreciprocal Josephson transport. A purely sinusoidal CPR, \(I \propto \sin\varphi\), is reciprocal. A simple phase shift \(I = I_c \sin(\varphi+\phi)\) is also insufficient by itself, because it can be absorbed into a redefinition of the phase origin and does not force unequal critical currents. What repeatedly produces diode behavior in the literature is the coexistence of a phase-shifted first harmonic with higher harmonics or other nontrivial anharmonic structure [2504.09453]. In short, high-transparency junctions this structure commonly emerges from Andreev bound states. In InAs quantum-well junctions, cosine components enter the CPR when both inversion and time-reversal symmetries are broken, and in the highly transparent regime these cosine terms cannot simply be removed by a \(\varphi_0\) shift, leading to a genuinely asymmetric CPR. Josephson inductance measurements were used there to connect the diode effect directly to CPR asymmetry and to extract the supercurrent magnetochiral anisotropy coefficient \(\gamma_L \approx 0.76\times 10^6\,\mathrm{T^{-1}A^{-1}}\) [2103.06984]. In Ge-based SQUIDs embedding JoFETs, the single-junction CPR contains up to three harmonics with gate-tunable amplitude; the reported ratios reach \(A_2/A_1 \approx 0.13\) and \(A_3/A_1 \approx 0.03\), and flux and gate tuning drive the device between a Josephson-diode regime with \(27\%\) asymmetry and an effectively \(\pi\)-periodic \(\sin(2\varphi)\) regime associated with charge-\(4e\) transport [2311.15371]. Higher harmonics are also central in magnetically textured junctions. For a conical magnet, the numerically extracted CPR is written as \[ J(\Delta\varphi)=J_1\sin(\Delta\varphi-\varphi_1)+J_2\sin[2(\Delta\varphi-\varphi_2)], \] and a finite diode effect requires \(J_2\neq 0\). Near the \(0\)-\(\pi\) transition, \(J_1\) becomes small, \(J_2\) becomes comparatively important, and the diode efficiency becomes large [2409.00223]. An analogous crossover physics is invoked in Nb/Pt/Nb nanopillars, where the thickness dependence of the normalized transition temperature is fit by \(e^{-d/\xi_1}\sin[(d-d_0)/\xi_2]\), and in a \(30\) nm Pt device a sudden change in the slope of \(I_cR_N(T)\) near \(2.6\) K is accompanied by a Fraunhofer-pattern bump interpreted as a second-harmonic Josephson contribution [2605.17066]. Several papers sharpen the distinction between anomalous phase and true diode behavior. In planar \(s\)-\(d\)-\(s\) junctions, spontaneous time-reversal breaking can generate a \(\phi_0\)-junction-like state, but diode transport appears only when single-Cooper-pair tunneling is symmetry-allowed and coexists with even-harmonic terms, giving a CPR of the form \[ I(\phi_1) = -\frac{2e}{\hbar}\left[2E_{J,1}^{(2)}\sin 2\phi_1 + E_{J,1}^{(1)}\sin \phi_1 \pm E_{J,s}^{(1)}\cos \phi_1\right], \] for which \(I_{c+}\neq |I_{c-}|\) and \(\eta_{\max}\approx 1/3\) are reported [2512.02752]. In a topological diode based on a conventional \(2\pi\)-periodic channel in parallel with a \(4\pi\)-periodic Majorana channel, \[ I(\theta)=I_{c1}\sin(\theta+\varphi)+I_{c2}\sin\!\left(\frac{\theta}{2}\right), \] and it is precisely the coexistence of the \(2\pi\) and \(4\pi\) contributions that makes a Zeeman-induced phase offset unabsorbable and the critical current nonreciprocal [2504.09453]. ## 3. Microscopic routes in two-terminal junctions One major route is the Rashba-plus-Zeeman mechanism. In Al/InAs junction arrays, strong Rashba SOC, a sizeable in-plane \(g^*\)-factor, and high transparency produce short, ballistic SNS junctions whose diode response is strongest for transverse in-plane field and vanishes for field parallel to the current. The same work measured a finite-bias interval satisfying \(|I_c^-| < I < I_c^+\), directly identifying a superconducting diode regime [2103.06984]. A distinct route replaces interfacial Rashba engineering with intrinsic SOC in the barrier. In Nb/Pt/Nb vertical nanopillars, first-principles calculations show that broken inversion symmetry at the Nb/Pt interfaces and strong atomic SOC in Pt split interfacial bands near the Fermi level by about \(0.3\)–\(0.5\) meV and generate spin textures. The resulting singlet-triplet mixing is argued to produce a nonequilibrium spin accumulation in the Pt barrier via a supercurrent spin-Hall effect. Under an in-plane field, the field-induced phase \(\theta_H\) and the spin-moment phase \(\theta_M\) combine as \[ I_c = I_{c0}\sin(\theta_H \pm \theta_M), \] with opposite signs for opposite current directions. Experimentally, the reported diode efficiency reaches about \(17\%\), strongest for \(d_{\rm Pt}\sim 10\)–15 nm, with the best device operating around \(5.3\)–\(5.46\) K and the asymmetry absent at zero field [2605.17066]. The paper explicitly distinguishes this induced spin moment from a normal quasiparticle spin current. Vertical van der Waals junctions provide an intrinsic crystal-symmetry route. In NbSe\(_2\)/Td-WTe\(_2\)/NbSe\(_2\) junctions, the non-centrosymmetric Td-WTe\(_2\) crystal furnishes a polar axis and strong spin-orbit interaction. The diode efficiency obeys magneto-linear and magneto-chiral relations, \[ \eta \propto \mu_0 H_{\parallel}\sin(\theta_{MC}), \] and increases with barrier thickness up to a critical regime before deviating from linearity, interpreted as a ballistic-to-diffusive transition [2303.13049]. Twist-angle engineering in a double-barrier structure further modifies the effective polar axis and hence the magneto-chirality. Magnetic-texture mechanisms can encode both required symmetry breakings within a single barrier. In a conical magnet, the helical rotation produces quasi-1D Rashba-like splitting inversely proportional to the helix period \(\lambda_h\), while the canting term \(h\sin\theta\) breaks time reversal and shifts the superconducting gap differently for opposite directions. The diode efficiency is largest near \(0\)-\(\pi\) transitions and changes sign when either \(h\sin\theta\) or the helix rotation direction is reversed [2409.00223]. In antiferromagnetic planar junctions, by contrast, nonreciprocity arises from triplet Cooper diffusion modes that are odd under sublattice exchange; if tunneling into the two antiferromagnetic sublattices is different, sublattice-odd propagators survive and generate \(\cos\phi\) terms so that \(j(\phi)\neq -j(-\phi)\) [2410.12429]. The microscopic origin need not lie in the equilibrium supercurrent branch. Atomic-scale Pb–Pb STM junctions are reciprocal when stabilized by a single Pb atom, but insertion of a single Cr or Mn atom produces a bias-direction-dependent retrapping current. The asymmetry is traced not to an intrinsically asymmetric CPR, but to asymmetric quasiparticle dissipation through Yu–Shiba–Rusinov states inside the gap. The dominant signature is retrapping-current asymmetry, with opposite signs for Cr and Mn, reproduced by an extended RCSJ model with asymmetric damping [2212.04432]. This establishes that nonreciprocal Josephson transport can be fundamentally dynamical and dissipative even in atomically small junctions. ## 4. Multiterminal, transverse, and network-based nonreciprocity Multi-terminal geometries generalize nonreciprocity beyond the two-terminal critical-current asymmetry. In the graphene Josephson triode, three graphene weak-link junctions share a common superconducting island, so the phase drops are linked by current conservation and flux-free constraints. A control current through one branch reshapes the phase landscape seen by another branch, generating nonreciprocal supercurrent transport without external magnetic field. The control branch can remain in a dissipationless regime, and square-wave rectification is demonstrated for amplitudes down to \(10\) nA, with efficiencies above \(90\%\) [2210.02644]. A related but conceptually distinct case is the three-terminal graphene multi-terminal Josephson junction treated as a two-port network. There, nonreciprocity is governed by broken spatial mirror symmetry between the active ports, not by the standard two-terminal diode criterion. In the asymmetric configuration, the measured critical currents differ substantially, with \(|I_{c,12}^{\pm}| = 94.25\) nA and \(|I_{c,21}^{\pm}| = 57.25\) nA, corresponding to \(\eta_{12}^{+} \approx \eta_{12}^{-} \approx 24\%\) at \(B=0\) [2301.05081]. This is polarity-independent port nonreciprocity rather than the usual \(I_c^+\) versus \(I_c^-\) asymmetry. Four-terminal geometries allow simultaneous longitudinal and transverse Josephson responses. In a square-lattice junction with a spin-orbit-coupled central region, four superconducting leads, and an in-plane Zeeman field \(\mathbf b = b(\cos\phi,\sin\phi)\), breaking \(k_x\leftrightarrow -k_x\) generates longitudinal anomalous Josephson effect and diode effect, while breaking \(k_y\leftrightarrow -k_y\) generates transverse current, transverse AJE, and transverse JDE. For \(\alpha=0.5t\), \(b=0.1t\), \(\Delta=0.06t\), \(\phi=\pi/3\), and \(\mu_s=\mu_c=-3.6t\), the transverse CPR is reported to be unidirectional over the range studied, and for \(0