Interband pairing as the origin of the sublattice dichotomy in monolayer FeSe/SrTiO_3

This paper proposes that interband pairing is the fundamental mechanism responsible for the observed sublattice dichotomy in monolayer FeSe/SrTiO$_3$, whether the underlying symmetry breaking occurs in the normal state or the pairing state.

The Gist

The Big Picture: A Superconducting Mystery

Imagine a tiny, ultra-thin sheet of iron and selenium (FeSe) sitting on top of a special ceramic tile (Strontium Titanate). This setup is a "superconductor," meaning electricity flows through it with zero resistance, and it does so at a surprisingly high temperature for such a small material.

Scientists recently discovered something weird happening inside this sheet. The sheet is made of a grid of iron atoms. In a perfect world, every iron atom should behave exactly the same way. But in this specific material, the iron atoms are split into two teams (let's call them Team A and Team B).

When scientists looked closely at the energy of the electrons, they found that Team A and Team B were acting completely differently. It was like looking at a pair of twins: one was wearing a red shirt and the other a blue shirt, even though they were supposed to be identical. This phenomenon is called "sublattice dichotomy."

The big question was: Why are these two teams behaving so differently?

The Solution: The "Interband Pairing" Dance

The authors of this paper propose that the answer lies in how the electrons pair up to create superconductivity.

In a normal superconductor, electrons usually pair up with a partner from the same group (like two dancers from the same dance troupe). This is called intraband pairing.

However, this paper argues that in this specific material, the electrons are doing something more exotic. They are pairing up with partners from a different group (like a dancer from the Red Troupe pairing with a dancer from the Blue Troupe). This is called interband pairing.

Think of it like a ballroom dance:

• Normal Superconductivity: Dancers only pair with people wearing the same color shoes. Everyone looks the same.
• This Material: Dancers are pairing with people wearing different colored shoes. This creates a mix-up that breaks the symmetry, making the two teams (FeA and FeB) look and act differently.

Two Ways the Mystery Could Be Solved

The paper suggests two possible scenarios for how this "mixed dance" happens.

Scenario 1: The Stage Was Already Crooked (Symmetry Breaking in the Normal State)

Imagine the dance floor itself is slightly tilted before the music even starts.

• The Setup: Before the electrons start dancing (superconducting), the environment is already uneven. The "dance floor" (the Fermi surface) is tilted so that Team A is mostly on one side and Team B is on the other.
• The Dance: When the music starts, the electrons try to dance a specific pattern (a "d-wave" dance). Because the floor was already tilted, this dance naturally forces the two teams to look different.
• The Result: The "interband" nature of the dance (pairing across the tilt) creates the split personalities we see in the experiments.

Scenario 2: The Dancers Change the Rules Mid-Song (Symmetry Breaking in the Pairing State)

Imagine the dance floor is perfectly flat, but the dancers decide to break the rules once the music starts.

• The Setup: The floor is fair. Everyone starts equal.
• The Dance: The electrons decide to mix two types of dances at the same time:
  1. Same-Team Dance: Dancers pair with their own team, but they do it with opposite feelings (one happy, one sad).
  2. Cross-Team Dance: Dancers pair with the other team, and they must share the same feeling (both happy).
• The Result: This specific combination of rules forces the two teams to split apart and act differently, creating the dichotomy.

Why Does This Matter?

You might ask, "So what if the iron atoms act differently?"

This discovery is a key to understanding high-temperature superconductivity.

1. The "Why": It explains a strange experimental observation that previously didn't make sense.
2. The "How": It suggests that the secret to making electricity flow without resistance at higher temperatures might involve these complex, cross-group electron pairings, rather than simple, uniform ones.
3. The Future: By understanding that "interband pairing" is the secret sauce, scientists can design better materials for things like super-fast computers, lossless power grids, and powerful magnets.

The Takeaway

The paper is essentially saying: "The reason the two sides of this iron sheet look different is that the electrons are dancing with partners from the opposite side. Whether the floor was crooked to begin with or the dancers changed the rules mid-song, this 'cross-team' pairing is the key to unlocking the mystery of this superconductor."

Technical Summary

1. Problem Statement

Recent scanning tunneling microscopy and spectroscopy (STM/STS) experiments on monolayer FeSe grown on SrTiO$_3$ substrates have revealed a phenomenon known as sublattice dichotomy. In this system, the two iron sublattices (FeA and FeB), which are structurally equivalent in the pristine lattice, exhibit distinct tunneling spectra within the superconducting gap. Specifically:

• The coherence peaks at positive bias ($+V_i$) on FeA are lower than on FeB, while those at negative bias ($-V_i$) are higher on FeA.
• The intensity difference for the outer gap peaks ($\pm V_o$) shows the opposite behavior.
• This particle-hole asymmetry between the two sublattices suggests a breaking of symmetries that normally exchange FeA and FeB.

The central problem addressed by the authors is identifying the microscopic mechanism responsible for this dichotomy. While previous theories often focused on normal-state symmetry breaking, this paper proposes that interband pairing is the fundamental origin, regardless of whether the symmetry breaking occurs in the normal state or the superconducting pairing state.

2. Methodology

The authors employ a combination of symmetry analysis and low-energy effective modeling:

• Symmetry Analysis: They analyze the space group $G = P4/nmm$ of the pristine monolayer FeSe. They identify that the symmetry operations exchanging FeA and FeB belong to specific irreducible representations (specifically $B_{2u}$ in the context of the BdG Hamiltonian). To observe dichotomy, these symmetries must be broken.
• $k \cdot p$ Modeling: They utilize a low-energy effective Hamiltonian based on the $d$-orbital basis near the $M$ point (Brillouin zone corner), where electron pockets reside. The model includes spin-orbit coupling and sublattice degrees of freedom.
• Perturbation Theory:
  • Case 1 (Normal State Breaking): They introduce $B_{2u}$ perturbations into the normal-state Hamiltonian to break sublattice equivalence, creating sublattice-polarized Fermi surfaces. They then apply a nodeless $d$-wave pairing order.
  • Case 2 (Pairing State Breaking): They construct a two-band BdG Hamiltonian to analyze the coexistence of intraband and interband pairing orders. They perform a perturbation analysis to determine the phase constraints required to reproduce the experimental dichotomy.
• Density of States (DOS) Calculation: They calculate the sublattice-projected DOS for both scenarios to compare with experimental STM/STS data.

3. Key Contributions

The paper makes three primary theoretical contributions:

1. Identification of Interband Pairing as the Mechanism: The authors establish that the observed sublattice dichotomy is a direct signature of interband pairing. They demonstrate that the particle-hole asymmetry arises because Cooper pairs are formed between bands with different effective masses (or distinct orbital characters), inherently breaking particle-hole symmetry in the subsystem.
2. Dual Scenarios for Symmetry Breaking: The paper proposes two distinct pathways to achieve the observed dichotomy, both relying on interband pairing:
   • Scenario A: Symmetry breaking occurs in the normal state. This leads to Fermi surfaces that are polarized toward specific sublattices. The intersublattice $d$-wave pairing then acts effectively as interband pairing.
   • Scenario B: Symmetry breaking occurs in the pairing state. This requires a specific coexistence of intraband and interband pairing orders with strict phase constraints.
3. Derivation of Phase Constraints: For the pairing-state scenario, the authors derive a rigorous condition: Interband pairings must share the same sign, while intraband pairings must carry opposite signs ($\Delta_{inter}^{(1)} = \Delta_{inter}^{(2)}$ and $\Delta_{intra}^{(1)} = -\Delta_{intra}^{(2)}$). This specific configuration is shown to be the only one yielding the correct sublattice dichotomy.

4. Results

• Normal State Breaking Case:

  • Introducing $B_{2u}$ perturbations ($h'_M$) to the normal Hamiltonian redistributes the sublattice weights on the Fermi surfaces (inner FS dominated by FeA, outer by FeB).
  • When combined with a nodeless intersublattice $d$-wave pairing, the calculated DOS reproduces the experimental observation: the coherence peaks on FeA and FeB shift in opposite directions relative to the gap center, creating the observed dichotomy.
  • Note: The authors acknowledge a potential conflict with vortex-core STM data (which suggests non-$d$-wave features), indicating this scenario requires further investigation.

• Pairing State Breaking Case:

  • Using a two-band model, the authors prove that only the configuration where interband pairings have the same sign and intraband pairings have opposite signs generates the dichotomy.
  • In the FeSe/SrTiO$_3$ context, this corresponds to a mixture of $A_{1g}$ (intraband) and $B_{2u}$ (interband) pairing orders.
  • The $B_{2u}$ order corresponds to nearest-neighbor intersublattice pairing with a uniform sign, while the $A_{1g}$ order corresponds to intrasublattice pairing with opposite signs on FeA and FeB.
  • Numerical simulations of the DOS for this specific combination (Fig. 3a) successfully reproduce the experimental spectra, whereas other combinations (e.g., same-sign intraband pairing) fail to show the dichotomy.

5. Significance

• Unifying Framework: The paper provides a unified theoretical framework explaining the sublattice dichotomy, suggesting that interband pairing is an indispensable ingredient for superconductivity in monolayer FeSe/SrTiO$_3$.
• Mechanism for High-$T_c$: By linking the dichotomy to interband pairing, the work reinforces the idea that the unique high-$T_c$ properties of this system may stem from unconventional pairing mechanisms involving multiple bands and orbital degrees of freedom, rather than simple Fermi surface nesting.
• Experimental Guidance: The derived constraints on pairing phases (same sign for interband, opposite for intraband) offer specific targets for future experimental probes and theoretical modeling of the pairing symmetry in iron-based superconductors.
• Resolution of Asymmetry: It resolves the puzzle of why two structurally equivalent sublattices show different spectral weights, attributing it not just to static structural distortions but to the dynamic nature of the superconducting order parameter itself.