🔬 materials science

Pseudogap in $Sr_{2-x}La_{x}IrO_{4}$ : Gor'kov-Teitel'baum thermal activation model

This paper applies the Gor'kov-Teitel'baum thermal activation model to Lanthanum-doped Strontium Iridate ( $Sr_{2-x}La_{x}IrO_{4}$ ) to determine its pseudogap phase boundary and map the evolution of Fermi arcs, thereby confirming the existence of a pseudogap state ending near a doping concentration of $x \simeq 0.16$ and providing an updated phase diagram consistent with previous experimental signatures.

Original authors: Jalaja Pandya, Devarshi Dave, Navinder Singh Bathinda

Published 2026-09-07

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CC BY 4.0

Original authors: Jalaja Pandya, Devarshi Dave, Navinder Singh Bathinda

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep within the world of materials science, researchers are constantly searching for new ways to understand how electricity flows through solids, particularly those that behave strangely when cooled or mixed with other elements. For decades, a family of copper-based minerals known as cuprates has captivated scientists because they can conduct electricity without any resistance at all, a phenomenon called superconductivity. However, before these materials become superconductors, they often pass through a mysterious state called a "pseudogap." In this state, the material acts as if it is missing some of its electrical carriers, even though it is not a full insulator. This behavior is so peculiar that it has become a central puzzle in physics: understanding why these electrons disappear and reappear could unlock the secrets of high-temperature superconductivity. Recently, scientists have turned their attention to a different family of materials based on iridium, a heavy metal, which shares a similar crystal structure and electronic behavior with the copper-based ones. The question is whether these iridium compounds follow the same rules, and if so, where the boundary lies between their strange pseudogap state and their normal conducting state.

A team of researchers in India has taken a fresh look at a specific iridium compound, strontium iridate, which has been chemically tweaked by replacing some of its strontium atoms with lanthanum. This process, known as doping, changes how many electrons are available to carry current. Previous experiments had hinted that something dramatic happens in this material when the amount of lanthanum reaches a certain level, around 16 percent. At this point, the number of charge carriers seems to jump suddenly, suggesting a major shift in the material's internal state. However, the exact temperature at which the pseudogap state begins and ends for different levels of doping remained a mystery. To solve this, the researchers applied a proven mathematical framework originally developed for the copper-based cuprates. This model treats the movement of electrons as a thermal activation process, where electrons need a certain amount of heat energy to jump across an energy gap. By fitting this model to existing experimental data on how the material responds to magnetic fields, the team was able to map out the invisible boundaries of the pseudogap state with much greater precision.

The results reveal a clear and consistent pattern. At very low levels of doping, the energy gap required to activate electrons is enormous, corresponding to a temperature of about 1200 Kelvin. As the researchers increased the amount of lanthanum, this energy gap shrank steadily and almost linearly. The gap continued to decrease until it vanished completely at a doping level of approximately 16 percent. This finding confirms that the pseudogap state exists in this iridium compound and defines its limits, showing that the state closes out at the same critical doping point where previous studies observed a sudden change in carrier density. The researchers also checked their findings against other experimental measurements, such as those taken with light-based spectroscopy, and found that their calculated energy gaps matched the observed values very well. This agreement suggests that the model is not just a theoretical exercise but a reliable tool for describing the real physics of the material.

Beyond mapping the boundaries, the study also explored how the "Fermi arcs" in the material change with temperature. In simple terms, a Fermi arc is a broken segment of the path that electrons follow, which appears in the pseudogap state instead of a complete circle. Using the data from their model, the team calculated how these arcs grow as the material gets warmer. They found that as the temperature rises, the arcs extend further, meaning the gap that was blocking the electrons at the edges of the material begins to close. When they compared these calculations to actual measurements taken on similar samples, the shapes and sizes of the arcs matched closely. This provides a coherent picture of how the electronic structure evolves, linking the macroscopic behavior of the material to the microscopic arrangement of its electrons.

The work culminates in an updated map of the material's phases, combining the newly defined pseudogap boundary with known magnetic properties. This diagram shows that the pseudogap state coexists with a specific type of magnetic order at lower doping levels, creating a complex landscape where different states of matter overlap. The researchers emphasize that while their map is a significant step forward, it is a starting point for future investigation. The boundaries they have drawn are based on the best available data and a robust model, but they acknowledge that further experiments will be needed to refine the details, especially at the lowest doping levels where the material becomes inhomogeneous. By successfully applying a model from one family of materials to another, the study strengthens the idea that the strange physics of high-temperature superconductors may be governed by universal principles, regardless of whether the atoms involved are copper or iridium.

Technical Summary: Pseudogap in Sr $_{2-x}$ La $_x$ IrO $_4$ : Gor'kov-Teitel'baum Thermal Activation Model

Problem Statement
The paper addresses the characterization of the pseudogap (PG) phase in the electron-doped iridate Sr $_{2-x}$ La $_x$ IrO $_4$ , a 5d analogue of high- $T_c$ cuprates. While recent Hall effect measurements by Hsu et al. [1] identified a critical doping concentration ( $x \simeq 0.16$ ) where the effective carrier density ( $n_H$ ) undergoes a crossover from $n_H \sim x$ to $n_H \sim 1+x$ , the precise boundary of the pseudogap phase in the doping-temperature plane remains unknown. Existing literature has reported signatures of a PG state and "Fermi arcs" in this system [2, 3, 27], but a comprehensive phase diagram defining the PG crossover line is lacking.

Methodology
The authors apply the Gor'kov-Teitel'baum Thermal Activation (GTTA) model, a phenomenological framework originally developed for La $_{2-x}$ Sr $_x$ CuO $_4$ [29], to the Sr $_{2-x}$ La $_x$ IrO $_4$ system. The rationale for this application rests on the structural and electronic similarities between Sr $_2$ IrO $_4$ and La $_2$ CuO $_4$ , as well as the observed similarities in Hall coefficient temperature dependence.

The core of the methodology involves fitting the experimental Hall coefficient data ( $R_H$ ) to the GTTA equation:
$R_H = \frac{1}{n_{Hall} e}$
$n_{Hall}(x, T) = n_0(x) + n_1(x)e^{-\Delta(x)/T}$
where:

$n_0(x)$ represents the temperature-independent carrier density from external doping.
$n_1(x)$ is a roughly constant term for low doping.
$\Delta(x)$ is the activation energy, interpreted as the energy gap between the "Fermi arc" states and the band bottom.

The authors extracted Hall effect data for various doping concentrations ( $x$ ) from Hsu et al. [1] and private communications. They utilized this data to extract the doping-dependent parameters $n_0(x)$ , $n_1(x)$ , and $\Delta(x)$ . Furthermore, using the carrier density derived from the GTTA model, they calculated the evolution of "Fermi arcs" ( $\Delta\phi$ ) as a function of temperature for a specific doping concentration ( $x \simeq 0.08$ ) using a free-electron model relation.

Key Results

PG Phase Boundary: The analysis yields a doping-dependent activation energy $\Delta(x)$ that serves as a proxy for the pseudogap magnitude. For low doping ( $x \simeq 0.036$ ), $\Delta(x)$ is found to be approximately 1200 K. As doping increases, $\Delta(x)$ decreases roughly linearly, vanishing at a critical doping of $x \simeq 0.16$ . This defines the PG phase boundary (crossover line) in the temperature-doping phase diagram.
Consistency with Literature: The extracted $\Delta(x)$ values show strong agreement with previously reported PG magnitudes obtained via Angle-Resolved Photoemission Spectroscopy (ARPES) [3] and infrared spectroscopy [2] for specific doping levels ( $x \simeq 0.10$ and $x \simeq 0.134$ ).
Carrier Density Evolution: The parameter $n_0(x)$ exhibits a transition from linear behavior ( $n_0 \sim x$ ) at low doping to a non-linear increase ( $n_0 \sim 1+x$ ) as $x$ approaches 0.14, consistent with the Hall effect crossover reported in [1].
Fermi Arcs: For $x \simeq 0.08$ , the calculated Fermi arc length ( $\Delta\phi$ ) increases with temperature (e.g., $\Delta\phi \simeq 30^\circ$ at 80 K). This trend qualitatively matches ARPES observations for 7% electron doping [27], where arc length increases with temperature.

Significance and Claims
The paper claims to provide the first application of the GTTA model to Sr $_{2-x}$ La $_x$ IrO $_4$ , successfully extracting a pseudogap phase boundary that aligns with existing experimental signatures. The primary contribution is the construction of an updated phase diagram (Fig. 5) that overlays the newly determined PG boundary onto the known magnetic phase diagram (Canted Anti-Ferromagnetic order).

The authors emphasize that the PG boundary in Sr $_{2-x}$ La $_x$ IrO $_4$ exhibits behavior strikingly similar to that of hole-doped cuprates, opening at high temperatures near the Mott insulating phase and closing at a critical doping ( $x \simeq 0.16$ ) just above the optimal doping regime. They note that in the underdoped regime, the extracted gap represents an average value due to the likely coexistence of Mott and PG states (phase separation). The work concludes that while the current results are consistent with known data, the PG boundary requires further refinement through future experiments, particularly in the low-doping regime.

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