Magnetically tunable supercurrent in dilute magnetic topological insulator-based Josephson junctions

The study investigates whether a proximity-induced Fulde–Ferrell–Larkin–Ovchinnikov (pFFLO) state can be realized and tuned in semiconductor-based Josephson junctions by exploiting a controllable spin-splitting (Zeeman) field. FFLO states arise when singlet Cooper pairs acquire finite momentum in a spin-split environment, leading to spatial modulation of the pair amplitude. Prior demonstrations in bulk superconductors were constrained to narrow regions of temperature and field, while hybrid superconductor systems can, in principle, host a tunable pFFLO state at broader conditions. However, unambiguous identification requires observing 0–π ground-state transitions (re-entrant critical current) driven by multiple parameters and linking them to spatial modulation of the induced order parameter in the same device. The authors propose using dilute magnetic semiconductor weak links based on (Hg,Mn)Te, where Mn doping yields a giant effective g factor, enabling strong Zeeman splitting at low applied fields and minimizing Rashba spin-orbit coupling, to demonstrate and map pFFLO behavior versus both in-plane magnetic field and temperature.

Foundational work by Fulde-Ferrell and Larkin-Ovchinnikov predicted inhomogeneous superconducting states under exchange fields. Subsequent observations of FFLO or related pair-density-wave phenomena have been reported in bulk superconductors within restricted parameter ranges. In hybrid structures, spin-splitting (Zeeman or exchange) can induce finite-momentum pairing analogous to FFLO, leading to oscillatory proximity-induced order parameters and 0–π transitions in S/F/S Josephson junctions. Prior experiments in ferromagnetic weak links observed re-entrance versus temperature, thickness, or magnetic configuration, but could not directly confirm spatial modulation within a single device. In non-magnetic semiconductor weak links under in-plane fields, re-entrance could be confounded by Fraunhofer-like interference or Rashba spin-orbit effects, and large fields often suppress supercurrent. The Pientka one-dimensional model has explained certain features (positions of minima in Ic) in non-magnetic devices, but does not capture the richer two-dimensional effects or the full temperature and field dependence observed at higher fields. A tunable platform with strong, low-field Zeeman energy and negligible Rashba coupling is thus needed for unambiguous pFFLO identification.

Device platform and design: The weak link is a 2.3% Mn-doped, 11-nm-thick (Hg,Mn)Te quantum well in (Hg,Cd)Te barriers, grown by molecular beam epitaxy. The structure is highly n-doped, placing it in a regime where topological properties are not central to transport. The design uses a symmetric quantum well and electrostatic layout to minimize the electric field across the well, thereby suppressing Rashba spin-orbit coupling, verified by the absence of beating in Shubnikov–de Haas oscillations from a Hall bar on the same wafer (Supplementary Section 2B). Transport parameters from Hall measurements: carrier density 8.8×10^11 cm^−2, mobility 111,000 cm^2 V^−1 s^−1, corresponding to Fermi wavelength λ ≈ 27 nm and Fermi energy EF ≈ 97 meV. Superconducting contacts and geometry: MoRe leads (Tc ≈ 9.6 K, BCS gap Δ ≈ 1.4 meV, high Hc2) are side-contacted to the quantum well. Josephson junctions have width W = 5.5 μm and lengths L between 600 and 1,300 nm, all below the mean free path l ≈ 1.7 μm (ballistic/clean limit). Leads are long stripe-shaped (4 μm wide). The junctions are measured under perpendicular-to-plane field H⊥ and in-plane field H∥ orthogonal to current. Lithography uses wet etching; device geometry and technology follow prior work. Measurement setup: Quasi-four-probe d.c. I–V measurements are conducted in a dilution refrigerator at base temperatures around 24–25 mK. Current is swept from negative to positive bias. Ic is defined using a 2 μV voltage criterion; with a normal-state resistance ≈60 Ω, this implies a detection threshold ≈30 nA. Fraunhofer patterns under H⊥ assess uniform current distribution; flux focusing from Meissner effect in the leads is accounted for. Great care is taken in aligning the sample plane relative to field directions; since the H⊥ range must be ~1000× larger than H∥ to observe structure in both directions, even ~1° misalignment could distort data (Supplementary Section 2D). Zeeman engineering: Mn substitution provides a giant effective g factor, enhancing Zeeman splitting E_Z = g μB H / 2 and reducing required fields by more than an order of magnitude relative to non-magnetic semiconductors. The effective Zeeman energy in dilute magnetic semiconductors follows a modified Brillouin function B5/2 for Mn2+ spins, parameterized by material constants. Theory and modeling: The clean-limit expression for Ic in S/F/S with nonhomogeneous magnetization (ref. 33) is adapted by mapping the ferromagnetic exchange field to the semiconductor’s giant Zeeman splitting. The effective E_Z(T,H) follows a modified Brillouin function: it depends on an empirical parameter T0 accounting for antiferromagnetic Mn–Mn interactions/clustering, and a saturated spin-splitting energy ΔE_max determined by sp–d exchange. Material parameters: g0 = −20 for undoped HgTe quantum wells; g = 2 for Mn2+; ΔE_max = 4.3 meV for 2.3% Mn; T0 = 730 mK; Fermi velocity v = 6.3×10^5 m s^−1 from band structure; momentum relaxation time τ (as defined in the model); γ as an effective, field- and temperature-independent interface parameter used for overall normalization; anomalous Green’s function f = A/√(A^2 + (ħω)^2) for the superconducting leads with Matsubara frequencies ħω = πkT(2n+1). The finite width of the JJ is incorporated by averaging over injection angles. Using these parameters, Ic(Hx,T)/Ic(0,Tmin) maps are simulated without additional fitting beyond normalization and compared to experiment for different L.

• At zero field and base temperature (~24–25 mK), the junction with L = 950 nm (J2) exhibits Ic ≈ 1.5 μA. Under H⊥, a standard Fraunhofer pattern confirms uniform current distribution across W, with flux focusing due to Meissner effect in the leads.
• Under in-plane field H∥, Ic shows clear re-entrant behavior: Ic decreases from H∥ = 0, reaches a near-zero minimum around ~60 mT, increases again up to ~95 mT, and then decreases to zero at ~195 mT, indicating multiple 0–π transitions. First and second re-entrance nodes occur at H∥ ≈ 60 mT and ≈ 200 mT, largely independent of H⊥, consistent with Zeeman-driven amplitude modulation of the supercurrent density and spatial modulation of the pFFLO order parameter.
• Temperature dependence at finite H∥ is non-monotonic, revealing 0–π transitions: for example, at H∥ = 85 mT, a zero-Ic node appears at about T ≈ 400 mK; as H∥ increases (up to ~100 mT), Ic at base temperature grows and the temperature of the zero-Ic node shifts higher.
• Constructed Ic(Hx,T)/Ic(0,Tmin) maps for devices with L = 950 nm (J2) and L = 730 nm (J1) show that, at the lowest temperatures, the first node shifts to lower Hx as T decreases—explained by the Brillouin-function dependence of E_Z(T,H), requiring lower fields to maintain constant E_Z at lower T.
• Simulations based on the adapted Bergeret–Volkov–Efetov model with giant Zeeman splitting reproduce the measured Ic(Hx,T) color maps, the positions of the first and second re-entrance nodes versus L across six devices, and the amplitude of re-entrant supercurrent at higher fields. Agreement supports a Zeeman-driven pFFLO mechanism.
• Engineering with (Hg,Mn)Te reduces the required in-plane fields by over an order of magnitude relative to non-magnetic semiconductor weak links, enabling direct mapping of Ic versus in-plane field over a substantial temperature range.

The observation of multiple re-entrant nodes (0–π transitions) in the same junction as functions of both in-plane magnetic field and temperature provides strong evidence for a proximity-induced FFLO state. The weak-link design minimizes Rashba spin-orbit coupling, distinguishing these effects from prior non-magnetic semiconductor studies where Rashba and orbital interference could mimic re-entrance. The independence of re-entrance nodes from H⊥ and their systematic dependence on junction length and temperature align with a Zeeman-driven finite-momentum pairing mechanism. Modeling that incorporates a giant Zeeman splitting governed by a modified Brillouin function and accounts for finite junction width reproduces the temperature and field dependence of Ic, including node positions and re-entrant lobe amplitudes, thus linking the re-entrant supercurrent to spatial modulation of the induced order parameter. While one-dimensional models capture some minima positions, the richer two-dimensional features and temperature evolution observed here necessitate the more complete treatment, supporting the identification of pFFLO behavior in this platform.

This work demonstrates magnetically tunable, proximity-induced FFLO behavior in Josephson junctions based on the dilute magnetic topological insulator (Hg,Mn)Te. By leveraging the giant Zeeman effect from Mn dopants, the authors achieve multiple re-entrant critical-current nodes (0–π transitions) at low in-plane fields and map their evolution with temperature and junction length. Agreement between experiment and a microscopic model adapted for giant Zeeman splitting confirms the Zeeman-driven pFFLO nature of the state. The results establish semiconductor weak links as a flexible platform to explore FFLO physics under experimentally accessible conditions. Future work could directly image spatial modulation of the order parameter, refine modeling to include residual Rashba-type spin-orbit contributions at low fields, explore device geometries and gating for enhanced tunability, and investigate potential connections to topological superconductivity.

• The spatial modulation of the order parameter is not directly imaged (no scanning probe), so pFFLO is inferred from transport and modeling.
• The theoretical description assumes E_Z > ħ/τ; this condition is best satisfied at fields higher than ~24 mT, and deviations at lower fields may limit quantitative accuracy.
• The modified Brillouin-function parameterization (with T0) is empirical and applied even below its most validated temperature range.
• Precise alignment of magnetic fields is critical; small misalignments could distort H⊥ vs H∥ mappings.
• The experimental Ic extraction has a ~30 nA resolution limit due to the voltage criterion and normal resistance.
• Possible residual Rashba spin-orbit coupling effects at low fields are not fully disentangled and would require further dedicated studies.