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

The research explores the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, a spatially modulated superconducting order parameter predicted to occur in superconductors exposed to a spin-exchange field. The FFLO state arises from the interplay between the exchange splitting (or Zeeman field) and the singlet-state Cooper pairs, resulting in a finite momentum pairing state and a spatially oscillating order parameter. Early observations of the FFLO state were limited to narrow temperature and magnetic field ranges in materials like κ-(BEDT-TTF)₂Cu(NCS)₂. Recent studies have demonstrated similar behavior in various bulk superconductors. An analogous state, the proximity-induced FFLO (PFFLO) state, can occur in hybrid superconductor systems where superconducting correlations are induced in a non-superconducting material in the presence of a spin-splitting field. The PFFLO state, unlike the bulk FFLO state, exhibits tunability, making it potentially more applicable. This paper aims to demonstrate the PFFLO state using a dilute magnetic topological insulator (Hg,Mn)Te-based Josephson junction, achieving tunability via magnetic field control and observing re-entrant superconductivity as a hallmark of the PFFLO state. The use of a dilute magnetic semiconductor enhances the Zeeman effect, significantly reducing the magnetic fields needed to observe the re-entrant behavior, which is typically a challenge in previous studies using non-magnetic semiconductors.

The paper reviews existing literature on the FFLO state, highlighting its initial prediction and limited observations in specific materials. It discusses the analogous PFFLO state observed in hybrid superconductor-ferromagnet systems, emphasizing the tunability advantage over the bulk FFLO state. Previous experimental attempts to observe PFFLO are reviewed, noting limitations such as the inability to finely tune the exchange field in ferromagnetic weak links and the insufficient Zeeman effect in non-magnetic semiconductors requiring impractically high magnetic fields. The authors contrast their approach using dilute magnetic semiconductors, which allows for precise control over the Zeeman effect, mitigating the limitations of earlier studies. They reference multiple studies supporting the understanding of FFLO and PFFLO states in various materials and configurations.

The researchers employ Josephson junctions engineered from (Hg,Mn)Te topological insulator weak links. Unlike previous work with HgTe heterostructures focusing on strong spin-orbit coupling, this study minimizes Rashba spin-orbit coupling using a highly doped symmetric quantum well and an electrostatic layout to reduce the electric field across the quantum well. The negligible spin-orbit coupling is confirmed through Shubnikov-de Haas analysis. The introduction of Mn dopants significantly enhances the Zeeman effect due to the giant effective g-factor. The superconducting leads are made of MoRe, characterized by a high upper critical field. A side-contact geometry is used to connect the MoRe leads to the quantum well. The junctions' width and length are carefully controlled to be below the mean free path in the semiconductor. Measurements are performed under both perpendicular and in-plane magnetic fields using a quasi-four-probe configuration. The critical current (Ic) is extracted from I(V) curves using a voltage criterion of 2 μV. The experimental setup allows for precise control and measurement of the magnetic field's impact on the supercurrent, especially its in-plane component which is crucial for observing the re-entrant behavior characteristic of the PFFLO state. A detailed analysis of the alignment between the sample plane and the magnetic field is emphasized to avoid distortions in the data, highlighting the precision required for this experiment. Furthermore, the theoretical analysis involves adapting a model from existing literature to account for the effects of magnetic dopants, incorporating parameters like the effective Zeeman energy (EZ) based on a modified Brillouin function reflecting the behavior of the Mn spins. The model is used to simulate the temperature and magnetic field dependence of the critical current, which is then compared to the experimental findings to validate the PFFLO state.

The experimental results reveal a clear re-entrant behavior of the critical current (Ic) as a function of the in-plane magnetic field (H||). Ic initially decreases, reaches zero, and then increases again before decreasing to zero at a higher field. This re-entrant superconductivity is observed at a remarkably low in-plane magnetic field of 60 mT. The observed Fraunhofer pattern under a perpendicular magnetic field confirms a uniform current distribution. The re-entrant behavior is mapped in detail as a function of both H|| and temperature, showing that these re-entrant features exhibit a 0-π transition both as a function of temperature and magnetic field. The temperature dependence of Ic at different H|| values confirms a non-monotonic behavior, further supporting the PFFLO state. The authors present detailed color plots illustrating the evolution of Ic with both magnetic field and temperature, clearly showing the re-entrant regions. A theoretical model, adapted to account for the giant Zeeman splitting caused by Mn doping, accurately predicts the observed behavior of Ic as a function of magnetic field and temperature. The model accounts for the shifting of the re-entrant nodes towards lower magnetic fields at lower temperatures due to the temperature dependence of Mn spin alignment. The agreement between experimental data and theoretical simulation, particularly regarding the position and amplitude of the re-entrant features, strongly supports the conclusion that the observed re-entrant superconductivity is indeed due to the PFFLO state. The study also compares the results to a simpler one-dimensional model, demonstrating that a more comprehensive model considering the finite width of the junction is necessary to capture the richness of the observed phenomena, such as the temperature and magnetic field dependence within the re-entrant lobes.

The findings directly address the research question by providing strong evidence for the existence of a PFFLO state in the (Hg,Mn)Te-based Josephson junction. The observation of re-entrant superconductivity, accurately predicted by a theoretical model incorporating the giant Zeeman effect from Mn doping, is crucial evidence. The ability to tune the PFFLO state using an in-plane magnetic field represents a significant advancement. The results demonstrate the feasibility of studying FFLO physics in a highly controllable semiconductor system, offering advantages over previous methods that relied on less tunable materials. The successful use of the dilute magnetic semiconductor provides a superior platform for future research focusing on the FFLO state. The study showcases the importance of precise experimental control over parameters like magnetic field alignment and a robust theoretical model that incorporates specific material properties.

This research successfully demonstrates the tunable PFFLO state in a (Hg,Mn)Te-based Josephson junction. The re-entrant behavior of the critical current, accurately captured by a theoretical model, provides compelling evidence. The use of dilute magnetic semiconductors offers a highly controllable platform for exploring FFLO physics, overcoming limitations of previous methods. Future research could explore the effects of varying Mn concentration, junction geometry, and the role of Rashba spin-orbit coupling in finer detail.

The theoretical model used relies on several assumptions, including the temperature and magnetic field independence of certain parameters within the experimental range. While the agreement between the model and experiment is excellent, the model's accuracy beyond this range is not fully tested. The study focuses on a specific material system and junction geometry; further investigation is needed to determine the generality of these findings. The 2 μV voltage criterion used for determining Ic might introduce a small degree of uncertainty, although it’s carefully stated and addressed in the paper. The study does not directly measure the spatial modulation of the order parameter; the re-entrant behavior serves as indirect evidence.