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Emergence of Larkin-Ovchinnikov-type superconducting state in a voltage-driven superconductor

Taira Kawamura, Yoji Ohashi, and H. T. C. Stoof
Phys. Rev. B 109, 104502 – Published 12 March 2024
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  • INTRODUCTION
  • FORMALISM
  • NONEQUILIBRIUM PHASE DIAGRAM OF THE…
  • SUMMARY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We theoretically investigate a voltage-biased normal metal–superconductor–normal metal (NSN) junction. Using the nonequilibrium Green's function technique, we derive a quantum kinetic equation to determine the superconducting order parameter self-consistently. The derived equation is an integral-differential equation with memory effects. We solve this equation by converting it into a system of ordinary differential equations with the use of a pole expansion of the Fermi-Dirac function. When the applied voltage exceeds the critical value, the superconductor switches to the normal state. We find that when the voltage is decreased from the normal phase, the system relaxes to a Larkin-Ovchinnikov-type (LO) inhomogeneous superconducting state, even in the absence of a magnetic Zeeman field. We point out that the emergence of the LO-type state can be attributed to the nonequilibrium energy distribution of electrons due to the bias voltage. We also point out that the system exhibits bistability, which leads to hysteresis in the voltage-current characteristic of the NSN junction.

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    • Received 9 January 2024
    • Revised 12 February 2024
    • Accepted 28 February 2024

    DOI:https://doi.org/10.1103/PhysRevB.109.104502

    ©2024 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Exotic phases of matterFFLOSuperconductivity
    1. Physical Systems
    Nonequilibrium systems
    1. Techniques
    Nonequilibrium Green's function
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Taira Kawamura1,2,*, Yoji Ohashi1, and H. T. C. Stoof2

    • 1Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
    • 2Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princentonplein 5, 3584 CC Utrecht, The Netherlands

    • *tairakawa@keio.jp

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    Vol. 109, Iss. 10 — 1 March 2024

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    • Figure 1
      Figure 1

      (a) The device under consideration. A thin-film (l≲ξ and d≫ξ) superconductor is sandwiched between left (L) and right (R) normal-metal leads. The superconductor is driven out of equilibrium by the bias voltage V between the normal-metal leads. (b) Schematic description of our model. The thin-film superconductor (main system) is described by the two-dimensional attractive Hubbard model. The coupling between the superconductor and the α (=L,R) normal-metal lead is modeled by assuming each site of the system is connected to an independent free-fermion α reservoir, which is equilibrated with the chemical potential μα and temperature Tenv. Electrons are injected into (extracted from) the superconductor from (to) the reservoirs. Moreover, −t⊥α=L,R describes the hopping amplitude between the superconductor and the α reservoir, which can be controlled by changing the insulating barrier strength inserted between the superconductor and the α normal-metal lead.

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    • Figure 2
      Figure 2

      Schematic energy band structure of our model. The energy is commonly measured from the average of the Fermi energy levels of the left and right reservoirs. In the left (right) reservoir at Tenv=0, the energy band ξpL(R)(t) is filled up to ±eV(t)/2, respectively. The chemical potential difference μR(t)−μL(t) between the left and right reservoirs equals the applied bias voltage eV(t) between the normal-metal leads.

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    • Figure 3
      Figure 3

      (a) Nonequilibrium phase diagram of the voltage-driven superconductor, when γ/t∥=0.01. NBCS and NLO are the nonequilibrium BCS and Larkin-Ovchinnikov state, respectively. While NBCS is a uniform superconducting state, NLO is a nonuniform superconducting state with a spatially oscillating order parameter. The system exhibits bistability in regions II and III. When the system enters these regions from region I, NBCS is realized in both the regions. On the other hand, the system relaxes to NLO (normal state) in region II (III), when entering from region IV. V0=VNBCS (boundaries between regions I and IV, as well as regions III and IV) and V0=VNLO (boundary between regions II and III) are, respectively, obtained by solving Eqs. (70) and (74) (see the main text for details). (b) Effects of the system-lead coupling strength γ on the phase diagram.

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    • Figure 4
      Figure 4

      Time evolution of the order parameter under the voltage V(t)=ΔVΘ(t). The initial state at t=0 is the thermal equilibrium BCS state (V0=0). We set γ/t∥=0.01 and Tenv/t∥=0.05. (a) Schematic diagram of how we quench the voltage. The regions I–IV are shown in Fig. 3. (b)–(d) Show the time evolution of |Δ¯(t)| in Eq. (76) and Δqx,qy=0(t) in Eq. (77) for each voltage quench depicted in (a). Here, Δ0 is the order parameter in the case that the superconductor is isolated from the normal-metal leads and is in the BCS ground state.

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    • Figure 5
      Figure 5

      Self-consistent solutions of the nonequilibrium gap equation (70), when γ/t∥=0.01. V0=VNBCS gives the boundaries between regions I and IV, as well as regions III and IV in the nonequilibrium phase diagram in Fig. 3. While the solid line corresponds to NBCS, the dashed line corresponds to an unstable gapless superconducting state.

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    • Figure 6
      Figure 6

      Same plots as Fig. 4 for the voltage eV(t)/t∥=1.6+eΔVΘ(t)/t∥. The initial state at t=0 is the normal steady state (eV0/t∥=1.6).

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    • Figure 7
      Figure 7

      (a) Superconducting transition line obtained from Eq. (74). The system transitions to a uniform NBCS (nonuniform NLO) on the solid V0=VNBCS (dashed V0=VNLO) line. (b) The intensity of the T matrix χr(q,0) at the positions (b1) and (b2) shown in (a).

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    • Figure 8
      Figure 8

      The voltage dependence of the amplitude |Δj| of the NLO order parameter. The intensity is normalized by Δ0. We set γ/t∥=0.01 and Tenv/t∥=0.05.

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    • Figure 9
      Figure 9

      Nonequilibrium momentum distribution nkneq for different system-lead coupling strength γ. We set Tenv=0 and eV0/t∥=1 for both panels. Two Fermi edges (dashed and dotted lines) imprinted on nkneq work like two “Fermi surfaces” (FS1 and FS2) of different sizes.

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    • Figure 10
      Figure 10

      Voltage-current characteristic of the NSN junction. The solid (dashed) line shows the steady-state current I through the junction, when we adiabatically vary the voltage along path A (path B) in Fig. 3. The dotted line shows the result when the system is the normal state (Δj=0). Rn is the normal resistance of the junction.

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    • Figure 11
      Figure 11

      Time evolution of the order parameter Δj(t) under the voltage eV(t)/t∥=1.6−0.8Θ(t). We set γ/t∥=0.01 and Tenv/t∥=0.05, and Lx×Ly=31×31. At t=0, the system is in the normal steady state [region IV in Fig. 3]. (a) The time evolution of |Δ¯(t)| in Eq. (76). (b) The spatial profile of the order-parameter amplitude |Δj(t)| at each point shown in (a).

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    • Figure 12
      Figure 12

      Stability analysis of 2D-NLO for various voltages and system sizes. We set γ/t∥=0.01 and Tenv/t∥=0.05. We set (a) Lx×Ly=31×31, (b) Lx×Ly=33×33, and (c) Lx×Ly=35×35.

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    • Figure 13
      Figure 13

      Diagrammatic representation of the 4×4 Nambu-Keldysh self-energy corrections. (a) Σ̌int,jk describes effects of the onsite pairing interaction −U in the mean-field BCS approximation. The solid line is the dressed Nambu-Keldysh Green's function Ǧjk given in Eq. (A1). The wavy line is the pairing interaction −U, being accompanied by the vertices τs⊗νζ± at both ends. (b) Σ̌lead,jk describes the effects of the system-lead couplings in the second-order Born approximation with respect to the hopping amplitude t⊥. The dashed line denotes the noninteracting Green's function Ďα=L,R in the α reservoir given in Eq. (A9). The solid square represents the hopping amplitude −t⊥ between the system and the α reservoir.

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    • Figure 14
      Figure 14

      Typical NF dependence of ∑j=1N|ΔjNF−Δj|/|Δj| for several steady-state solutions. Here, Δj and ΔjNF are, respectively, obtained with Eqs. (45) and Eq. (D4). We set Lx=Ly=31, γ/t∥=0.01, eV0/t∥=1.1, and Tenv/t∥=0.05.

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