Non-monotonic dependence of $T_c$ on the c axis compression in the HTSC cuprate La$_{2-x}$Sr$_x$CuO$_4$

This study investigates the non-monotonic dependence of the critical temperature ($T_c$) on $c$-axis compression in La$_{2-x}$Sr$_x$CuO$_4$ by demonstrating that the competition between a compression-induced increase in the density of states near the Fermi level (enhancing $T_c$ in underdoped regions) and the renormalization of pairing constants (reducing $T_c$) governs the superconducting properties near optimal doping.

The Gist

Imagine a high-temperature superconductor (a material that conducts electricity with zero resistance) as a busy dance floor inside a crystal. The dancers are electrons (or more accurately, "holes," which are the absence of electrons), and the music is the quantum mechanical force that makes them pair up and dance in perfect sync. When they dance in sync, they create a supercurrent.

The paper you provided investigates what happens to this dance floor when you squeeze the ceiling of the room (compressing the crystal along its vertical, or "c-axis," direction).

Here is the story of their findings, broken down into simple concepts:

1. The Setup: The Dance Floor and the Ceiling

The material in question is LSCO (Lanthanum Strontium Copper Oxide).

• The Floor: This is the layer where the magic happens, made of copper and oxygen atoms.
• The Ceiling: Above this floor are "apical" oxygen atoms, like a low-hanging chandelier.
• The Squeeze: The researchers are asking: "What happens to the superconducting dance if we push the ceiling down closer to the floor?"

2. The Two Opposing Forces

The paper discovers that squeezing the ceiling triggers two different effects that fight against each other. Think of it like a tug-of-war between two teams.

Team A: The "Tightrope Walkers" (The Bad News)

When you push the ceiling down, the floor itself has to stretch out sideways (like a balloon being squeezed from the top; it bulges out the sides).

• The Effect: The dancers on the floor get further apart from each other horizontally.
• The Result: It becomes harder for them to hold hands and pass energy to one another. In physics terms, the "hopping" between atoms gets weaker, and the force that pairs them up (superexchange) gets weaker.
• The Outcome: This effect lowers the temperature ($T_c$) at which the material becomes a superconductor. If this were the only effect, squeezing the ceiling would always make the superconductor worse.

Team B: The "New Dancers" (The Good News)

This is the novel part of the paper. The crystal isn't just a simple floor; it has different "levels" of energy, like a multi-story building.

• The Effect: When the ceiling (apical oxygen) gets closer to the floor, it changes the energy levels of the building. Specifically, it lifts up a group of "dancers" (electrons in specific orbitals called $a_{1g}$) that were previously stuck in the basement (deep in the valence band).
• The Result: These basement dancers are now lifted up to the main dance floor, right next to the main dancers. Suddenly, there are many more dancers available to pair up, and they are all crowded into a very small, energetic space.
• The Outcome: This creates a "traffic jam" of potential partners, which boosts the superconducting temperature ($T_c$).

3. The Tug-of-War: Why the Results are "Non-Monotonic"

The title of the paper mentions "non-monotonic dependence." In plain English, this means the result isn't a straight line up or down; it's a rollercoaster.

• At Low Pressure (Light Squeeze): The "Tightrope Walkers" (Team A) win. The floor stretching hurts the pairing more than the new dancers help. So, $T_c$ goes down. This matches what many previous experiments saw.
• At High Pressure (Hard Squeeze): The "New Dancers" (Team B) take over. The ceiling is so low that the basement dancers are fully integrated into the main floor. The sheer number of available partners overwhelms the fact that the floor is stretched. So, $T_c$ starts to go up.
• The Sweet Spot: Near the "optimal doping" (where the material is already a good superconductor), these two effects fight so hard that $T_c$ might drop a little, then rise, then drop again, depending on exactly how hard you squeeze and how many dancers are on the floor.

4. The Analogy of the Crowded Elevator

Imagine an elevator (the crystal) with people (electrons) inside.

• Normal State: People are spread out.
• Squeezing the Ceiling (Low Pressure): The elevator gets shorter. People bump into each other and get annoyed (repulsion), making it hard to cooperate. The elevator stops working well.
• Squeezing the Ceiling (High Pressure): Suddenly, the pressure forces a hidden group of people (who were hiding in a storage closet) to jump into the main cabin. Now, the cabin is packed. Even though the people are annoyed by the crowding, the fact that there are so many of them allows them to form a massive, efficient human chain that moves the elevator perfectly.

5. Why Does This Matter?

For a long time, scientists thought squeezing these materials would only make them worse (or only better, depending on who you asked). This paper explains why the experiments were contradictory.

• If you squeeze a little bit, you see the "bad" effect (temperature drops).
• If you squeeze a lot, you see the "good" effect (temperature rises).
• The "winner" depends on exactly how much you squeeze and how many "dancers" (doping level) are already on the floor.

The Bottom Line:
By understanding this tug-of-war, scientists can now predict that if they apply the right amount of pressure (around 10 GPa, which is about 100,000 times atmospheric pressure), they might be able to boost the superconducting temperature of these materials even higher, potentially getting closer to room-temperature superconductivity. It's not just about squeezing; it's about squeezing just enough to wake up the hidden dancers without crushing the dance floor.

Technical Summary

1. Problem Statement

High-temperature superconducting (HTSC) cuprates exhibit complex responses to pressure. While hydrostatic pressure generally increases the critical temperature ($T_c$) by reducing intraplane Cu-O distances, the effect of uniaxial compression along the c-axis ($P_c$) remains controversial and contradictory in experimental literature.

• Some experiments suggest $T_c$ increases with c-axis compression (implying a correlation with the distance between the CuO$_2$ plane and apical oxygens).
• Others suggest $T_c$ decreases or shows negligible change.
• The Core Issue: Existing theoretical models often rely on simplified two-band Hubbard models that ignore excited orbital states. They fail to explain why experimental results vary so significantly with doping levels and pressure ranges, particularly near optimal doping ($x_{opt} \approx 0.15$). The authors hypothesize that the non-monotonic behavior of $T_c$ arises from a competition between two distinct mechanisms: the renormalization of pairing constants and the reconstruction of the electronic band structure involving higher-energy orbitals.

2. Methodology

The authors employ a sophisticated theoretical framework to model the electronic structure and superconducting properties of La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) under c-axis compression (0 to 10 GPa).

• Model: An effective five-band Hubbard model is used. Unlike standard two-band models (which only consider $d_{x^2-y^2}$ and planar oxygen $p_{x,y}$ orbitals), this model includes:
  • Copper $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals.
  • Planar oxygen $p_x, p_y$ orbitals.
  • Apical oxygen $p_z$ orbitals.
  • This allows for the inclusion of $a_{1g}$ symmetry orbitals (involving $d_{3z^2-r^2}$ and $p_z$), which become relevant under c-axis compression.
• Basis States: The model is built on local multiparticle states of the CuO$_6$ cluster, including:
  • Vacuum ($n_h=0$).
  • Single-hole doublet ($n_h=1$, ground state Zhang-Rice singlet).
  • Two-hole states ($n_h=2$): The ground state Zhang-Rice singlet ($AS$), excited triplet states ($BT$), and excited singlet states ($BS$).
• Calculation Technique:
  • Generalized Tight-Binding (GTB) Method: Used to construct quasiparticle excitations from local multiparticle states, accounting for strong local Coulomb interactions exactly.
  • Equation of Motion (EOM) for Green's Functions: Applied to Hubbard operators to derive the electronic structure and superconducting gap.
  • Pairing Mechanism: Superconductivity is driven by a superexchange mechanism involving not just the Zhang-Rice singlet, but also excited two-hole triplet and singlet states. The gap is assumed to have $d$-wave symmetry.

3. Key Contributions

1. Inclusion of Excited States: The paper explicitly demonstrates that excited two-hole states (triplets and singlets with $a_{1g}$ character) play a crucial role in the electronic structure under pressure, a factor neglected in standard two-band models.
2. Identification of Competing Mechanisms: The study isolates and quantifies two competing effects of c-axis compression on $T_c$:
   • Mechanism (i): Renormalization of superexchange pairing constants ($J$).
   • Mechanism (ii): Reconstruction of the electronic band structure leading to an increased Density of States (DOS).
3. Resolution of Experimental Contradictions: The authors provide a theoretical explanation for why experimental results on c-axis pressure vary: the behavior of $T_c$ is highly sensitive to the doping level ($x$) and the specific pressure range, leading to non-monotonic behavior near optimal doping.

4. Key Results

Electronic Structure Reconstruction

• Orbital Energy Shifts: Under c-axis compression, the energy of $a_{1g}$ orbitals (involving apical oxygen $p_z$ and Cu $d_{3z^2-r^2}$) increases, while $b_{1g}$ orbitals (planar) decrease relative to the Fermi level.
• Band Hybridization: As pressure increases, the $a_{1g}$ quasiparticle bands ($q_3, q_4, q_5$) rise and strongly hybridize with the $b_{1g}$ valence band top ($q_2$).
• DOS Enhancement: This hybridization flattens the dispersion of the valence band top near the $K_1$ and $K_2$ points (near $(\pi, 0)$ and $(0, \pi)$). Consequently, the Density of States (DOS) near the Fermi level increases significantly, particularly in underdoped and optimally doped regions.

Critical Temperature ($T_c$) Behavior

The behavior of $T_c$ is determined by the competition between two mechanisms:

1. Mechanism (i) - Pairing Constant Renormalization: Compression increases intraplane Cu-O distances (due to Poisson effect), which reduces the superexchange interaction $J_{22}$ (between Zhang-Rice singlets). This tends to decrease $T_c$.
2. Mechanism (ii) - DOS Reconstruction: The hybridization-induced flattening of the band increases the DOS and the number of states participating in pairing. This tends to increase $T_c$.

Doping-Dependent Outcomes:

• Underdoped Region ($x < 0.14$): Mechanism (ii) dominates. $T_c$ monotonically increases with pressure.
• Overdoped Region ($x > 0.2$): The band reconstruction is weak (large energy gap between peaks). Mechanism (i) and (ii) largely compensate, or (i) dominates slightly, resulting in $T_c$ being insensitive to pressure.
• Optimal Doping ($x \approx 0.15$): The region of greatest interest. The paper predicts a non-monotonic dependence of $T_c$ on pressure:
  • Low Pressure (0–3 GPa): Mechanism (i) dominates; $T_c$ decreases.
  • Intermediate Pressure (3–6/8 GPa): Complex competition leads to oscillations or plateaus.
  • High Pressure (>6–8 GPa): Mechanism (ii) becomes dominant due to strong band hybridization; $T_c$ increases.

Comparison with Two-Band Model

Calculations using a standard two-band Hubbard model only show a monotonic decrease in $T_c$ with pressure (due to Mechanism i). The five-band model is essential to capture the increase in $T_c$ observed in some experiments and the non-monotonic behavior.

5. Significance

• Theoretical Advancement: This work bridges the gap between simplified models and the complex reality of HTSC materials by rigorously including excited orbital states and their pressure dependence.
• Explanation of Experimental Discrepancies: It resolves the "contradictory" experimental data on c-axis pressure. The variation in reported $dT_c/dP_c$ is attributed to slight differences in doping levels in different samples and the specific pressure ranges investigated.
• Guidance for Future Research: The study suggests that to maximize $T_c$ in cuprates via pressure, one should target high uniaxial c-axis compression (>6 GPa) in optimally doped samples, where the electronic structure reconstruction (Mechanism ii) can overcome the reduction in pairing strength.
• Methodological Impact: It validates the use of the five-band Hubbard model with GTB and EOM methods for studying pressure effects in strongly correlated systems where orbital degrees of freedom are critical.

In conclusion, the paper demonstrates that the non-monotonic behavior of $T_c$ under c-axis compression is a result of the delicate balance between the suppression of the superexchange interaction and the enhancement of the density of states via orbital hybridization, a phenomenon that can only be captured by a multi-band model including excited states.