Emergence of Charge-Imbalanced BCS State and Suppression of Nonequilibrium FFLO State in Asymmetric NSN Junctions

This paper theoretically demonstrates that in voltage-biased asymmetric NSN junctions, lead-coupling asymmetry and impurity scattering suppress the fragile nonequilibrium FFLO state while splitting the robust nonequilibrium BCS phase into distinct regimes characterized by chemical-potential imbalance, bistability, and phase transitions.

The Gist

Imagine a superconductor as a perfectly synchronized dance floor where electrons pair up (like dance partners) and move in perfect unison without any friction. This is the "superconducting state."

Now, imagine you connect this dance floor to two different rooms (called "leads") and turn on a voltage. This forces electrons to rush in from one side and out the other. The dance floor is no longer in a calm, resting state; it's in a non-equilibrium state, like a party where the music is changing too fast for everyone to settle down.

This paper explores what happens to our electron dance floor when we mess with two specific things:

1. The doors: Are the doors to the two rooms the same size, or is one door much bigger than the other? (This is Lead-Coupling Asymmetry).
2. The obstacles: Is the dance floor clean, or is it littered with random chairs and tables that people trip over? (This is Impurity Scattering).

Here is what the researchers found, explained simply:

1. The "Wobbly" Dance (The NFFLO State)

In a perfectly balanced situation (equal doors, clean floor), the rush of electrons creates a weird phenomenon. Instead of dancing in a straight line, the electron pairs start to wobble back and forth across the floor. They form a pattern that changes from place to place. The scientists call this the NFFLO state.

• The Analogy: Imagine a line of dancers trying to hold hands. If the rhythm is just right, they might start doing a synchronized wave that moves down the line. That's the NFFLO state.
• The Problem: This "wobbly wave" is very fragile.
  • If the doors are uneven: If one door is huge and the other is tiny, the flow of electrons becomes lopsided. The "wave" can't form because the rhythm is too chaotic. The dance floor just becomes a uniform, boring line of dancers.
  • If the floor is messy: If there are obstacles (impurities) on the floor, the dancers trip. The delicate wave pattern breaks immediately. You need a very clean floor for this state to exist.

2. The "Steady" Dance (The NBCS State)

When the "wobbly wave" (NFFLO) is destroyed by uneven doors or messy floors, the electrons don't just stop dancing. They settle into a different, more uniform rhythm called the NBCS state.

• The Analogy: Instead of a wave, everyone is just marching in a straight line at the same speed. It's stable and robust.
• The Surprise: The researchers found that this steady march is immune to the messy floor. Even if you scatter chairs everywhere (non-magnetic impurities), the dancers just step over them and keep marching in perfect sync. This is a modern version of a famous rule in physics called "Anderson's Theorem."
• However: If the obstacles are "magnetic" (like dancers who are spinning wildly and confusing everyone), the steady march does break down.

3. The "Two-Tier" System (Chemical Potential Imbalance)

Here is the most fascinating discovery. When the doors are uneven (asymmetric), the steady march splits into two distinct types of behavior, depending on how hard you push the voltage:

• Type A (Balanced): At lower voltages, the "dance partners" (Cooper pairs) and the "solo dancers" (quasiparticles) are on the same page. They have the same energy level.
• Type B (Unbalanced): At higher voltages, a strange split happens. The solo dancers get pushed to a higher energy level than the pairs.
  • The Analogy: Imagine a two-story building. The pairs are on the ground floor, and the solo dancers are on the second floor. They are no longer communicating directly. This creates a "charge imbalance."
  • The Bistability: In a certain range of voltage, the system can be either on the ground floor or the second floor, and it doesn't want to switch easily. It's like a light switch that gets stuck in the middle; you have to push it hard one way or the other to make it flip. This is called bistability.

Why Does This Matter?

Think of this like tuning a radio.

• If you want to catch the "wobbly wave" signal (NFFLO), you need a very clean antenna and perfectly balanced connections. It's hard to get, but it's a unique signal.
• If you want a strong, reliable signal (NBCS), you don't need a perfect antenna; it works even with static. But if you turn the volume up too high (voltage) and the connections are uneven, you might get a weird "echo" (charge imbalance) where the signal splits into two different frequencies.

In Summary:
The paper tells us that to create exotic, wobbly superconducting states, you need perfect conditions. But if you accept a messier, more realistic setup, you get a very robust state that can surprisingly split into two different "modes" of operation, creating a switch-like behavior that could be useful for future electronic devices.

Technical Summary

1. Problem Statement

The paper investigates nonequilibrium superconductivity in voltage-biased Normal metal–Superconductor–Normal metal (NSN) junctions. While previous work established that nonequilibrium electron distributions (induced by bias voltage) can stabilize an inhomogeneous Fulde–Ferrell–Larkin–Ovchinnikov (NFFLO) state, two critical experimental realities were not fully addressed:

1. Lead-Coupling Asymmetry: In real devices, the coupling strength between the superconductor and the two normal-metal leads ($T_1$ vs. $T_2$) is rarely perfectly symmetric.
2. Impurity Scattering: Real materials contain impurities (both nonmagnetic and magnetic) that affect scattering rates.

The authors aim to determine how these factors modify the nonequilibrium phase diagram, specifically the stability of the NFFLO state versus the uniform nonequilibrium BCS (NBCS) state, and to understand the emergence of charge imbalance phenomena in these systems.

2. Methodology

The authors employ a rigorous theoretical framework based on the Keldysh Green's function technique to extend thermal-equilibrium mean-field BCS theory to the nonequilibrium steady state.

• Model Hamiltonian: They consider a thin superconducting layer sandwiched between two normal-metal leads. The system is driven out of equilibrium by a bias voltage $V$ applied between the leads. The Hamiltonian includes:
  • Kinetic energy and attractive interaction ($U$) for Cooper pairing.
  • Tunneling terms coupling the superconductor to the leads (with asymmetric strengths $\gamma_1$ and $\gamma_2$).
  • Scattering terms for nonmagnetic and magnetic impurities.
• Theoretical Tools:
  • Nonequilibrium Thouless Criterion: Used to analyze the onset of superconductivity and the stability of the NFFLO state (characterized by a finite center-of-mass momentum $Q$) against the normal state.
  • Nonequilibrium Mean-Field BCS Theory: Used to study the symmetry-broken phase (NBCS state, $Q=0$) and calculate the order parameter $\Delta$ and pair chemical potential $\mu_{pair}$.
  • Self-Consistent Born Approximation: Applied to handle impurity scattering self-consistently, including vertex corrections to satisfy the Ward–Takahashi identity.
• Key Variables: The study focuses on the bias voltage $V$, lead-coupling asymmetry ($P_{lead} = |\gamma_1 - \gamma_2|/(\gamma_1+\gamma_2)$), and impurity scattering rates ($\tau_{imp}^{-1}$).

3. Key Contributions and Results

A. Suppression of the NFFLO State

The paper demonstrates that the inhomogeneous NFFLO state is highly fragile under realistic conditions:

• Effect of Impurities: Similar to the equilibrium FFLO state, the NFFLO state is strongly suppressed by nonmagnetic impurity scattering. The critical temperature for the NFFLO transition ($T_{env}^{c}$) vanishes when the impurity scattering rate exceeds a critical threshold ($\tau_{imp}\Delta_0 \lesssim 0.028$). This is because impurity scattering disrupts the specific momentum pairing between electrons on different effective Fermi surfaces required for the NFFLO state.
• Effect of Coupling Asymmetry: The NFFLO state is also detrimental to lead-coupling asymmetry. In the symmetric case, the nonequilibrium distribution creates two effective Fermi surfaces (FS1 and FS2) of comparable size, favoring inter-surface pairing (NFFLO). As asymmetry increases, the population on one Fermi surface becomes significantly smaller than the other. This limits the inter-surface pairing channel, causing the system to revert to the uniform NBCS state driven by the dominant Fermi surface.

B. Emergence of Charge-Imbalanced NBCS States

In the presence of asymmetric lead couplings, the uniform NBCS state exhibits a novel classification:

• Two Distinct Regimes: The NBCS phase splits into two types based on the relationship between the quasiparticle chemical potential ($\mu_{qp}$) and the pair chemical potential ($\mu_{pair}$):
  1. Balanced Regime: $\mu_{qp} = \mu_{pair}$. This occurs at lower bias voltages or in symmetric junctions.
  2. Charge-Imbalanced Regime: $\mu_{qp} \neq \mu_{pair}$. This emerges at high bias voltages in asymmetric junctions.
• Bistability: In the intermediate voltage range, the system exhibits bistability, where both the balanced and imbalanced NBCS states are dynamically stable. The transition between these states is first-order-like, accompanied by a discontinuous jump in the order parameter amplitude and hysteresis.
• Microscopic Origin: The authors provide a microscopic derivation of the charge imbalance using the "branch-imbalance" picture (Tinkham). They show that the nonequilibrium distribution function creates an asymmetry between electron-like and hole-like quasiparticle populations. To maintain charge neutrality, the condensate distribution shifts, resulting in a non-zero electrostatic potential $\phi$ and a chemical potential difference $\mu_{qp} - \mu_{pair} \neq 0$.

C. Robustness of NBCS vs. Sensitivity to Magnetic Impurities

• Nonmagnetic Impurities: Consistent with Anderson's theorem, the uniform NBCS state is robust against nonmagnetic impurity scattering. The critical temperature and the order parameter remain unchanged because nonmagnetic scattering does not alter the superconducting density of states or the nonequilibrium distribution function in the $Q=0$ channel.
• Magnetic Impurities: In contrast, magnetic impurities strongly suppress the NBCS state, destroy the bistability phenomenon, and eliminate the charge imbalance. This is because magnetic scattering breaks time-reversal symmetry and modifies the superconducting density of states.

4. Significance

• Unified Framework: The paper provides a unified microscopic understanding of nonequilibrium superconductivity in NSN junctions, bridging the gap between idealized theoretical models and experimentally relevant conditions (asymmetry and disorder).
• Experimental Guidance: The results suggest that observing the NFFLO state requires extremely clean samples and nearly symmetric lead couplings. Conversely, the observation of charge imbalance and bistability in the NBCS state is feasible in asymmetric, clean junctions but requires the minimization of magnetic impurities.
• Broader Impact: The theoretical framework developed here is applicable to a broad class of hybrid structures, including ferromagnet-superconductor-ferromagnet (FSF) junctions used in superconducting spintronics, where the interplay between charge and spin imbalance is crucial.

In summary, this work clarifies that while the exotic NFFLO state is easily destroyed by realistic disorder and asymmetry, the uniform nonequilibrium state offers rich new physics, including charge imbalance and bistability, which are robust against nonmagnetic disorder but sensitive to magnetic impurities.