Evidence of Andreev bound states as a hallmark of the FFLO phase in κ-(BEDT-TTF)2Cu(NCS)2

Journal name:
Nature Physics
Volume:
10,
Pages:
928–932
Year published:
DOI:
doi:10.1038/nphys3121
Received
Accepted
Published online

Superconductivity is a quantum phenomenon arising, in its simplest form, from the pairing of fermions with opposite spin into a state with zero net momentum. Whether superconductivity can occur in fermionic systems with an unequal number of two species distinguished by spin or flavour presents an important open question in condensed-matter physics or quantum chromodynamics1. In condensed matter the imbalance between spin-up and spin-down electrons that form the Cooper pairs is induced by the magnetic field. Such an imbalanced system can lead to exotic superconductivity in which pairs acquire finite momentum2, 3. This momentum leads to a spatially inhomogeneous state consisting of periodically alternating ‘normal’ and ‘superconducting’ regions. Here, we establish that the hallmark of this state is the appearance of spatially localized and spin-polarized quasiparticles forming the so-called Andreev bound states (ABS). These are detected through our nuclear magnetic resonance (NMR) measurements.

At a glance

Figures

  1. (H,T) phase diagram of [kappa]-(BEDT-TTF)2Cu(NCS)2.
    Figure 1: (H,T) phase diagram of κ-(BEDT-TTF)2Cu(NCS)2.

    Curves and colour shaded areas sketch the phase diagram based on magnetic torque measurements12 for fields parallel to the conducting planes. Circles denote Tc, the transition temperature from the normal to the SC state, determined from our NMR data, as explained in the text, whereas squares mark the peak in the NMR rate. Diamonds denote the onset H above which (T1T)−1 exceeds the value extrapolated from the low-field SC state (Supplementary Fig. 6 and Supplementary Information) below HP ≈ 20.7 T (ref. 15). Arrows indicate the field (H) and temperature (T) scans covered by NMR relaxation rate measurements.

  2. NMR relaxation rate in the normal and superconducting states.
    Figure 2: NMR relaxation rate in the normal and superconducting states.

    Temperature dependence of 13C NMR (T1T)−1 at fields of 15, 22 and 27 T, applied in the conducting planes (symbols). Solid line denotes the quadratic temperature dependence characteristic for superconductors with a gap having a line of nodes, such as for a d-wave symmetry. The dashed lines are guides to the eye. The parts of the phase diagram explored are shown by horizontal arrows in Fig. 1. Error bars reflect the scattering of measured (T1T)−1 values and the standard error of the mean in fitting the recovery curve. Inset: 13C NMR spectra at 22 T field applied parallel to the conducting planes in the superconducting (T = 1.4K), FFLO (T = 2.6K), and normal state (T = 10.9K). Multiple peaks evident in the normal-state spectrum arise from the eight distinct crystallographic sites of 13C (Methods).

  3. Enhancement of the NMR relaxation rate in the FFLO state.
    Figure 3: Enhancement of the NMR relaxation rate in the FFLO state.

    Magnetic field dependence of 13C NMR (T1T)−1 at various temperatures. The parts of the phase diagram explored are shown by vertical arrows in Fig. 1. Error bars reflect the scattering of measured (T1T)−1 values and the standard error of the mean in fitting the recovery curve.

  4. Field dependence of the electronic spin polarization and NMR relaxation rate at low temperatures.
    Figure 4: Field dependence of the electronic spin polarization and NMR relaxation rate at low temperatures.

    Square root of the second moment, measuring electronic spin polarization, of the 13C NMR spectra (filled symbols, left scale) as a function of magnetic field applied in the conducting planes at different temperatures. Typical error bars are on the order of a few per cent and not shown for clarity. The corresponding field dependence of (T1T)−1 (open symbols, right scale) is shown for comparison. Vertical dashed lines identify a possible transition from the FFLO to the homogeneous SC state. Vertical solid lines denote a transition from the normal to the FFLO state. Other lines are guides to the eye.

  5. Schematic of the properties of the modulated superconductivity.
    Figure 5: Schematic of the properties of the modulated superconductivity.

    a, Sketch of the spatial profile of the order parameter and density of the localized polarized quasiparticles in length units normalized by the superconducting coherence length (ξ). b, Schematic of the spatial structure of the modulated superconducting states with excess polarized quasiparticles in the nodes of the order parameter. c, Bottom panel: Sketch of the spatial profile of the order parameter for two different magnetic fields. In the vicinity of the transition from the SC to FFLO state the nodes in the order parameter form the domain walls14, as illustrated by the blue solid line. With increasing magnetic field the period of the order parameter decreases, as shown by red lines. Yellow circles denote nodal regions. Top panel: Sketch of the local DOS (ABS formed by quasiparticle resonances) at nodes of the order parameter normalized to its normal-state value. Sharp DOS peaks (blue) are expected in magnetic fields where nodes in the order parameter form the domain walls14. As the applied field increases, ABS are broadened and further shifted from zero energy, as depicted by red lines. The vertical arrows denote the spin orientation. The energy difference between spin-up and-down states is proportional to the applied magnetic field. d, Calculated temperature dependence (in arbitrary units (a.u.)) of the (T1T)−1 arising from the Andreev bound states formed at the nodes of the order parameter as sketched in part c. The result is only qualitatively correct, as states are phenomenologically modelled by Gaussians shifted away from the Fermi level by an amount proportional to the applied field14 and the temperature dependence of the gap is neglected.

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Author information

Affiliations

  1. Laboratoire National des Champs Magnétiques Intenses, LNCMI - CNRS (UPR 3228), UJF, UPS and INSA, BP 166, 38042 Grenoble Cedex 9, France

    • H. Mayaffre,
    • S. Krämer,
    • M. Horvatić &
    • C. Berthier
  2. Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan

    • K. Miyagawa &
    • K. Kanoda
  3. Department of Physics, Brown University, Providence, Rhode Island 02912, USA

    • V. F. Mitrović

Contributions

K.M. and K.K. prepared the samples. H.M., S.K., K.M. and V.F.M. performed the experiments. S.K. and M.H. developed and operated the high-field NMR facility. H.M. created software for the spectrometers. H.M. and V.F.M. analysed the data. C.B. provided conceptual advice and contributed to the planning of the project. H.M., C.B., M.H. and V.F.M. developed the data interpretation. V.F.M. wrote the paper and supervised the project. All authors discussed the results and commented on and edited the manuscript.

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The authors declare no competing financial interests.

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