Tunneling spectroscopy of few-monolayer ${\mathrm{NbSe}}_{2}$ in high magnetic fields: Triplet superconductivity and Ising protection

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Tunneling spectroscopy of few-monolayer ${NbSe}_{2}$ in high magnetic fields: Triplet superconductivity and Ising protection

M. Kuzmanović, T. Dvir, D. LeBoeuf, S. Ilić, M. Haim, D. Möckli, S. Kramer, M. Khodas, M. Houzet, J. S. Meyer, M. Aprili, H. Steinberg, and C. H. L. Quay

Phys. Rev. B 106, 184514 – Published 28 November 2022

Abstract

In conventional Bardeen-Cooper-Schrieffer superconductors, Cooper pairs of electrons of opposite spin (i.e., singlet structure) form the ground state. Equal-spin triplet pairs (ESTPs), as in superfluid ${}^{3}{He}$ , are of great interest for superconducting spintronics and topological superconductivity, yet remain elusive. Recently, odd-parity ESTPs were predicted to arise in (few-)monolayer superconducting ${NbSe}_{2}$ , from the noncollinearity between the out-of-plane Ising spin-orbit field (due to the lack of inversion symmetry in monolayer ${NbSe}_{2}$ ) and an applied in-plane magnetic field. These ESTPs couple to the singlet order parameter at finite field. Using van der Waals tunnel junctions, we perform spectroscopy of superconducting ${NbSe}_{2}$ flakes, of 2–25 monolayer thickness, measuring the quasiparticle density of states (DOS) as a function of applied in-plane magnetic field up to 33 T. In flakes $≲ 15$ monolayers thick the DOS has a single superconducting gap. In these thin samples, the magnetic field acts primarily on the spin (vs orbital) degree of freedom of the electrons, and superconductivity is further protected by the Ising field. The superconducting energy gap, extracted from our tunneling spectra, decreases as a function of the applied magnetic field. However, in bilayer ${NbSe}_{2}$ , close to the critical field (up to 30 T, much larger than the Pauli limit), superconductivity appears to be more robust than expected from Ising protection alone. Our data can be explained by the above-mentioned ESTPs.

Received 18 May 2021
Revised 26 July 2022
Accepted 11 October 2022

DOI:https://doi.org/10.1103/PhysRevB.106.184514

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Density of states Mesoscopics Spin-orbit coupling Spin-triplet pairing Spintronics Superconductivity

Transition metal dichalcogenides Tunnel junctions Unconventional superconductors

Dilution refrigerator Eliashberg theory Liquid helium cooling Spectroscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

M. Kuzmanović ^1,2,*, T. Dvir^3,4,*, D. LeBoeuf⁵, S. Ilić^6,7, M. Haim ³, D. Möckli ^3,8, S. Kramer ⁵, M. Khodas³, M. Houzet⁶, J. S. Meyer⁶, M. Aprili¹, H. Steinberg ³, and C. H. L. Quay^1,†

¹Laboratoire de Physique des Solides (CNRS UMR 8502), Bâtiment 510, Université Paris-Saclay, 91405 Orsay, France
²QTF Centre of Excellence, Department of Applied Physics, Aalto University School of Science, P.O. Box 15100, 00076 Aalto, Finland
³Racah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel
⁴QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
⁵Laboratoire National des Champs Magnétiques Intenses (LNCMI-EMFL), CNRS, Université Grenoble Alpes, UPS, INSA, 38042 Grenoble, France
⁶Université Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, 38000 Grenoble, France
⁷Centro de Física de Materiales (CFM-MPC), Centro Mixto CSIC-UPV/EHU, 20018 Donostia–San Sebastián, Spain
⁸Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, Rio Grande do Sul, Brazil

^*These authors contributed equally to this work.
^†Corresponding author: charis.quay@universite-paris-saclay.fr

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Vol. 106, Iss. 18 — 1 November 2022

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Figure 1
(a) At zero magnetic field, singlet Cooper pairs are composed of electrons at opposite corners of the hexagonal ${NbSe}_{2}$ Brillouin zone ( $K$ and $K^{'}$ points). Their spins are pinned out of plane by the Ising field. (b) An in-plane magnetic field partially aligns electron spins orthogonal to the Ising field and gives rise to odd-parity equal-spin triplet pairs [5]. (c) Theoretical superconducting gap as a function of the in-plane magnetic field. Compared with the case where only the Ising field is considered (black curve), superconductivity is even more robust to the in-plane magnetic field, and the $Δ$ vs $H_{| |}$ curve has a “flattened” shape at intermediate fields (blue curve). The triplet component of the order parameter (red curve) with spin structure $Φ_{t B}$ survives disorder through its coupling to the singlet component, which has spin structure $Φ_{s}$ . The temperature used in the calculations is $0.5 T_{c}$ , the critical temperature. (See Fig. 4 and the text for details and comparison with data.)
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Figure 2
Tunneling spectroscopy of bulk and few-monolayer ${NbSe}_{2}$ through van der Waals barriers. (a) Schematic drawing of the tunnel junction: few-monolayer ${NbSe}_{2}$ , covered with few-monolayer ${WSe}_{2}$ (or ${MoS}_{2}$ ) and a Ti/Au electrode. A voltage $V$ is applied between the Ti/Au electrode and the ${NbSe}_{2}$ , and the current $I$ is measured. (b) Differential conductance ( $G = d I / d V$ ) as a function of $V$ across junction J2 (blue) and $d^{2} I / d V^{2}$ vs $V$ of junction J2 (red). A double peak and a double dip can be seen in $d^{2} I / d V^{2}$ , due to the presence of two superconducting gaps. (c) Same as (b), but for junction J6. (d)–(h) Color maps of the magnetic field dependence of the $d^{2} I / d V^{2}$ curves for junctions J1–J5. The double-dip-and-double-peak feature (yellow and blue regions) disappears in thin samples; a single gap is left. Measurements were taken at temperatures of 30–70 mK.
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Figure 3
Differential conductance $G = d I / d V$ as a function of the bias voltage $V$ and of the in-plane magnetic field $H_{| |}$ of junction J6 [six monolayers, (a)–(c)] and junction J7 [bilayer, (d)–(f)]. The tunneling spectra are normalized by the normal state conductance, $G_{N} (V)$ , measured above $H_{c}$ . (a) and (d) Color map of $G (V)$ as a function of field at 1.3 K. The dashed lines indicate the critical fields. (b) and (e) Horizontal slices of the data in the color maps (a) and (d), respectively, showing $G (V)$ at different fields, vertically displaced for clarity. The black curves are fits to an Abrikosov-Gor'kov-like (A-G-like) density of states, with the energy gap and A-G broadening parameter as fitting parameters. (The gap is not determined self-consistently.) (c) and (f) Data at 50 mK and zero magnetic field (red curves) together with the fits obtained using a BCS DOS and an effective temperature (black curves). The superconducting gaps obtained from the fits are 800 and 400 $μ eV$ , respectively, while the effective temperatures are 0.9 and 1 K, respectively.
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Figure 4
Normalized superconducting gap as a function of the in-plane magnetic field $Δ (H_{| |})$ obtained from the fits of the quasiparticle density of states in Fig. 3. The error bars were obtained following the procedure described in Supplemental Material Sec. I A. The blue dashed curves are a fit of experimental data using the Ising theory alone. Here, $E_{SO} = 14.45 T_{c s}$ (with $T_{c s} = 2.6 K$ ) for the bilayer sample and $E_{SO} = 2.21 T_{c s}$ (with $T_{c s} = 5.4 K$ ) for the six-monolayer (6ML) sample. In brown is the Ising theory with an equal-spin triplet component of the order parameter as described in the text. Here, $E_{SO} = 9.62 T_{c s}$ and $T_{c t} = 0.05 T_{c s}$ , with $T_{c s} = 2.6 K$ . Finally, the solid black curve is calculated using the Ising theory with strong disorder (equivalent to the Abrikosov-Gor'kov theory). In all cases, the critical field is constrained to be the experimental one.
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Physical Review B

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Tunneling spectroscopy of few-monolayer ${NbSe}_{2}$ in high magnetic fields: Triplet superconductivity and Ising protection

M. Kuzmanović, T. Dvir, D. LeBoeuf, S. Ilić, M. Haim, D. Möckli, S. Kramer, M. Khodas, M. Houzet, J. S. Meyer, M. Aprili, H. Steinberg, and C. H. L. Quay

Phys. Rev. B 106, 184514 – Published 28 November 2022

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Physical Review B

covering condensed matter and materials physics

Tunneling spectroscopy of few-monolayer NbSe2 in high magnetic fields: Triplet superconductivity and Ising protection

M. Kuzmanović, T. Dvir, D. LeBoeuf, S. Ilić, M. Haim, D. Möckli, S. Kramer, M. Khodas, M. Houzet, J. S. Meyer, M. Aprili, H. Steinberg, and C. H. L. Quay

Phys. Rev. B 106, 184514 – Published 28 November 2022

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Tunneling spectroscopy of few-monolayer ${NbSe}_{2}$ in high magnetic fields: Triplet superconductivity and Ising protection