Longitudinal Josephson effect in systems with pairing of spatially separated electrons and holes

This paper investigates longitudinal non-dissipative currents in bilayer electron-hole systems with potential barriers, demonstrating that the critical current scales with the product of barrier transparencies in the high-density limit but becomes inversely proportional to the sum of barrier heights in the low-density limit.

The Gist

The Big Picture: A Dance of Separated Partners

Imagine a ballroom where two groups of dancers are present: Electrons (let's call them the "Left-Handers") and Holes (the "Right-Handers"). In a normal metal, these dancers just bump into each other and move chaotically.

But in this specific system, something magical happens. The Left-Handers and Right-Handers fall in love across a wide gap. They form pairs, but they can't touch because there is a thick, insulating wall (a dielectric layer) between them. Even though they are separated, they move in perfect synchronization. When a Left-Hander moves forward, their partner Right-Hander moves backward. This synchronized movement creates a super-current that flows without any friction or energy loss. This is called Counterflow Superconductivity.

The paper investigates what happens when we try to push this synchronized dance across a second obstacle: a wall that divides the ballroom into a Left Room and a Right Room.

The Main Question: Can the Dance Cross the Wall?

The researchers asked: If we have a wall separating the Left Room from the Right Room, can the "dance" (the super-current) jump across it?

In standard physics, this is known as the Josephson Effect. Usually, this happens when two superconductors are separated by a thin barrier. Here, the setup is more complex because the "partners" are in different layers (one layer of electrons, one layer of holes).

The paper explores two very different scenarios, depending on how crowded the ballroom is.

---

Scenario 1: The Packed Ballroom (High Density)

The Analogy: Imagine a crowded dance floor where the dancers are packed so tightly that they are constantly bumping into each other. In this crowd, the "pairs" are huge and fuzzy; they overlap with many other pairs.

The Physics:
In this high-density state, the electron-hole pairs act like a giant, collective wave (similar to the BCS theory in standard superconductors).

The Result:
To get the dance to jump from the Left Room to the Right Room, both the electron layer and the hole layer must have a "leaky" spot in the wall.

• Think of it like a relay race. The baton (the current) can only pass if the electron runner can jump the gap AND the hole runner can jump the gap.
• If the electron wall is solid (no holes) but the hole wall has a gap, the dance stops.
• The Rule: The current is proportional to the product of the "leakiness" (transparency) of both walls. If one wall is 100% solid, the current is zero.

Key Takeaway: In a crowded system, the two layers are so tightly linked that they need help from both sides to cross the barrier.

---

Scenario 2: The Empty Ballroom (Low Density)

The Analogy: Now imagine the ballroom is almost empty. The dancers are far apart. Each pair is a distinct, tight unit, like a single couple holding hands, walking through a vast empty hall. They don't overlap with other couples.

The Physics:
Here, the pairs behave like a Bose-Einstein Condensate (a state of matter where particles act like a single giant wave). The pairs are strong and self-sufficient.

The Result:
This is where it gets surprising. In this sparse system, the dance can cross the barrier even if only one of the walls has a gap!

• Imagine the electron layer has a solid wall, but the hole layer has a small door. Because the pairs are so tightly bound to each other, if the hole partner squeezes through the door, the electron partner is dragged along with it, even through the solid wall.
• The Rule: The current depends on the sum of the barriers. It's like a harmonic mean. If one wall is infinitely high (impossible to cross), the current stops. But if one wall is low and the other is high, the current can still flow, limited by the "weakest link" in the chain.

Key Takeaway: In a sparse system, the bond between the partners is so strong that they can drag each other through a barrier even if only one of them has a clear path.

---

The "Longitudinal" Twist

The paper distinguishes between two types of currents:

1. Transverse Current: Jumping up or down through the insulating layer between the electron and hole sheets. (The paper ignores this, assuming it's blocked).
2. Longitudinal Current: Jumping sideways from the Left Room to the Right Room through the barrier. This is what the paper is about.

Why Does This Matter?

The authors found that the rules for how this current flows change completely based on how dense the system is:

• Crowded System: Needs a "double pass" (both layers must be permeable).
• Sparse System: Needs a "single pass" (if one layer is permeable, the pair drags the other through).

This is crucial for designing future quantum devices. If you want to build a super-sensitive sensor or a quantum computer component using these electron-hole pairs, you need to know exactly how dense your material is to predict how it will behave when you put a barrier in its way.

Summary in One Sentence

The paper reveals that in a system where electrons and holes are paired across a gap, the ability of their super-current to cross a wall depends on the crowd size: in a crowd, both sides must have a door; in an empty room, if one side has a door, the pair will squeeze through together.

Technical Summary

Technical Summary: Longitudinal Josephson Effect in Systems with Pairing of Spatially Separated Electrons and Holes

1. Problem Statement
The paper investigates the longitudinal Josephson effect in bilayer systems where spatially separated electrons and holes form bound pairs (excitons) that transition into a superfluid state. Unlike the conventional transverse Josephson effect (tunneling between layers), this study focuses on the non-dissipative current flowing along the layers (longitudinally) when the system is divided into left and right macroscopic regions by potential barriers.

The central problem is to determine the dependence of the critical longitudinal current on the properties of the potential barriers and the carrier density. Specifically, the authors aim to distinguish between two regimes:

• High-density limit: Where the pair size is small compared to the inter-pair distance (BCS-like regime).
• Low-density limit: Where the system behaves as a dilute Bose gas of excitons (BEC-like regime).

A key challenge addressed is the "congruence problem" in high-density systems, where the dispersion laws of electrons and holes must be nearly identical for pairing to occur, a constraint relaxed in strong magnetic fields or specific graphene-based systems.

2. Methodology
The authors employ two distinct theoretical frameworks tailored to the density regimes:

• High-Density Limit (BCS Regime):

  • Tunneling Hamiltonian Method: The system is modeled using a Hamiltonian split into isolated left/right subsystems ($H_1, H_2$) and a tunneling term ($H_T$). The order parameter phases ($\phi_1, \phi_2$) are explicitly separated.
  • Bogoliubov Transformation: The effective Hamiltonian is diagonalized to define quasiparticle operators. The current is calculated as the rate of change of particle number in one half-space, utilizing off-diagonal correlation functions derived from the Heisenberg equations of motion.
  • Verification (t-representation method): The Appendix provides a rigorous derivation using Gorkov's equations and the "t-representation method" (expanding Green's functions in eigenfunctions of the single-particle Hamiltonian with a barrier). This confirms the validity of the simpler tunneling Hamiltonian approach for first-order perturbation theory.

• Low-Density Limit (BEC Regime):

  • Gross-Pitaevskii Equation: The system is treated as a dilute Bose gas of electron-hole pairs. The ground state is described by a generalized coherent state.
  • Wave Function Approximation: The problem reduces to solving the Gross-Pitaevskii equation for the center-of-mass wave function of the pairs in the presence of a potential barrier (approximated as a delta-function).
  • Phase Matching: The solution involves matching the wave function and its derivative at the barrier to determine the phase jump and the resulting current-phase relationship.

3. Key Contributions and Results

The paper derives distinct current-phase relationships ($I \propto \sin(\Delta\phi)$) and critical current dependencies for the two density limits:

• High-Density Limit (BCS-like):

  • The critical current ($I_c$) is proportional to the product of the tunneling matrix elements (transparencies) of the barriers in the electron and hole layers: $I_c \propto T_e \times T_h$.
  • This implies that if either layer has a perfectly opaque barrier ($T=0$), the longitudinal Josephson current vanishes. Both layers must be weakly coupled for the effect to occur.
  • The result includes a new energy term $\eta_p$ in the denominator, arising from the asymmetry in electron and hole effective masses. If masses are equal, the result matches conventional superconductors.

• Low-Density Limit (BEC-like):

  • The critical current is proportional to the harmonic mean (or sum in the denominator) of the barrier transparencies: $I_c \propto 1/(1/T_e + 1/T_h)$.
  • Crucial Finding: The current depends on the sum of the barrier potentials, not their product. Consequently, a longitudinal Josephson effect can persist even if one layer has no barrier (perfect transparency), provided the other layer has a barrier. This is due to the strong coupling of the pair components (the electron and hole move together as a single boson).
  • Separated Barriers: If the barriers in the electron and hole layers are spatially separated by a distance $l$ exceeding the coherence length, the system behaves as two successive Josephson junctions. The total current is determined by the combined phase drops, leading to a modified critical current formula involving the individual transparencies.

4. Significance

• Fundamental Physics: The work clarifies the mechanism of longitudinal supercurrents in excitonic insulators and bilayer electron-hole systems, distinguishing clearly between BCS and BEC regimes.
• Experimental Relevance: The results provide specific predictions for experimental setups involving:
  • Graphene bilayers separated by hBN (high density).
  • Transition metal dichalcogenides (e.g., $WSe_2/MoSe_2$) or semiconductor heterostructures (low density).
  • Systems under strong magnetic fields where the total filling factor is unity.
• Design Implications: The finding that low-density systems allow for longitudinal currents even with a barrier in only one layer offers new possibilities for designing excitonic circuits and Josephson junctions where perfect alignment of barriers in both layers is difficult to achieve.
• Theoretical Validation: The successful cross-verification of the high-density results using both the tunneling Hamiltonian and the more rigorous t-representation method strengthens the theoretical foundation for studying non-equilibrium and tunneling phenomena in paired fermion systems.

In summary, the paper establishes that the nature of the longitudinal Josephson effect in electron-hole bilayers is fundamentally dictated by the carrier density, transitioning from a product-dependence on barrier transparencies in the high-density limit to a sum-dependence in the low-density limit.