Ferroelectric polarization controlled orbital Hall conductivity in a higher-order topological insulator: \textit{d1T}-phase monolayer MoS$_2$

This paper theoretically predicts that ferroelectric monolayer $d1T$-phase MoS$_2$ acts as a higher-order topological insulator with quantized corner states and demonstrates that its orbital Hall conductivity can be reversibly modulated by the direction of ferroelectric polarization, offering a promising platform for electric-field-controlled orbitronics.

The Gist

Imagine a piece of material as a bustling city. In most cities, the "traffic" (electrons) flows smoothly on the main roads (the bulk of the material) or gets stuck at the very edges (the boundaries).

This paper introduces a new kind of "city" made of a single layer of Molybdenum Disulfide (MoS₂), but with a very specific, twisted shape called the d1T phase. The researchers discovered that this material is a Higher-Order Topological Insulator (HOTI).

Here is the breakdown of their findings using simple analogies:

1. The "Corner" City (Higher-Order Topology)

Think of a standard topological insulator like a donut. The "magic" happens on the outer ring (the edge), while the inside is boring.

• The New Discovery: The d1T MoS₂ is like a donut where the magic doesn't happen on the ring at all. Instead, the "special traffic" only appears at the four corners of the shape.
• The Evidence: The researchers built a tiny, diamond-shaped (rhombic) model of this material. They found that while the middle and the sides were quiet, the corners were buzzing with special electron states. These corners hold a "fractional charge," which is like having a coin that is worth exactly one-third of a normal coin—something that usually can't happen in standard physics.

2. The "Orbital" Highway (Orbital Hall Effect)

Usually, scientists look for "Spin Hall Effect" to identify these special materials. Imagine "Spin" as cars driving in a circle (spinning) while moving forward.

• The Problem: In this new d1T material, the "Spin" highway is empty. If you looked for the spin traffic, you would see nothing special.
• The Solution: The researchers looked for something else: Orbital Hall Effect. Imagine this not as cars spinning, but as cars carrying a spinning top in their trunk.
• The Result: They found a massive, clear "plateau" (a flat, stable highway) of this "spinning top" traffic flowing across the material. This "Orbital" highway is the unique fingerprint that proves this material is indeed a Higher-Order Topological Insulator. Without looking at this specific traffic, you would miss the material's special nature.

3. The "Light Switch" (Ferroelectric Control)

This material is also ferroelectric, which means it has an internal "arrow" (polarization) pointing either up or down, like a magnet that can be flipped.

• The Magic Trick: The researchers found that if you flip this internal arrow (using an electric field), the direction of the "Orbital Highway" traffic changes.
• The Analogy: Imagine a one-way street. If you flip a switch on the wall, the traffic doesn't stop; it instantly reverses direction.
• The Specifics: They found that flipping the polarization flips the sign of the current flowing in one direction (the x-axis), while leaving the other directions unchanged. This means you can control the flow of this special "orbital" energy just by flipping a switch.

Summary

The paper claims that:

1. d1T MoS₂ is a new type of material where special electron states live only at the corners, not the edges.
2. You can't find this material by looking for "Spin" traffic; you must look for "Orbital" traffic (electrons carrying angular momentum).
3. You can control the direction of this orbital traffic by flipping the material's internal electric "arrow" (ferroelectric polarization).

The authors suggest this gives us a new way to build "orbitronics"—electronics that use this orbital flow, controlled by electric fields, rather than just magnetic fields.

Technical Summary

Technical Summary: Ferroelectric Polarization Controlled Orbital Hall Conductivity in a Higher-Order Topological Insulator: d1T-phase Monolayer MoS2

Problem Statement
While conventional topological insulators (TIs) are well-established, higher-order topological insulators (HOTIs) represent an extended concept where topological boundary states exist in dimensions lower than the standard $(d-1)$ boundary (e.g., corner states in 2D systems). Despite theoretical predictions, the experimental realization and identification of ideal 2D material candidates for HOTIs remain rare. Furthermore, while Spin Hall Conductivity (SHC) has been used as a signature for some HOTIs, recent studies suggest it may not be a universal identifier. The Orbital Hall Effect (OHE), a fundamental origin of spin and anomalous Hall effects, has recently been linked to HOTIs, yet the interplay between ferroelectricity and the Orbital Hall Conductivity (OHC) in these systems remains underexplored. The authors aim to identify a new 2D HOTI material and investigate whether OHC can serve as a robust signature for its topology, specifically exploring the potential for external control via ferroelectric polarization.

Methodology
The study employs Density Functional Theory (DFT) calculations using the Vienna ab initio Simulation Package (VASP) with the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation.

• Structural Stability: Phonon spectra were calculated to verify the dynamic stability of the $d1T$-phase monolayer MoS$_2$, confirming the absence of imaginary frequencies.
• Topological Characterization: The authors utilized symmetry indicators based on $C_3$ symmetry eigenvalues at high-symmetry points (HSPs) to calculate the topological index. They constructed finite rhombic nanoflakes (75 Mo and 150 S atoms) to visualize and analyze the energy spectrum and charge density distributions of corner states.
• Transport Properties: Spin Hall Conductivity (SHC) and Orbital Hall Conductivity (OHC) were calculated using maximally localized Wannier functions (MLWFs) generated via Wannier90 and the WannierTools package. The Kubo formula was applied with a dense $k$-point sampling ($200 \times 200 \times 1$) in the 2D Brillouin zone.
• Ferroelectric Control: To investigate polarization dependence, a polarization-reversed structure was constructed via mirror symmetry operations in the $xy$ plane, and OHC was recalculated for this configuration.

Key Results

1. Identification of $d1T$-MoS$_2$ as a HOTI: The authors predict that monolayer MoS$_2$ in the $d1T$ phase is a higher-order topological insulator. This phase arises from the trimerization of Mo atoms in a $\sqrt{3} \times \sqrt{3}$ supercell, which breaks central inversion symmetry and introduces out-of-plane ferroelectric polarization.
2. Topological Index and Corner States: The calculated nontrivial topological index is $[-2, 1]$, predicting a fractional corner charge of $Q_{corner} = e/3$. Energy spectrum calculations on finite rhombic nanoflakes reveal gapless corner states localized at the acute angles of the rhombus, situated near the Fermi level, confirming the hallmark of a second-order topological insulator.
3. Orbital Hall Conductivity as a Signature: The study finds that within the band gap, the SHC of $d1T$-MoS$_2$ is negligible, similar to trivial insulators. In contrast, a significant OHC plateau exists within the insulating gap. The absolute value of the OHC is calculated to be $1.303 (e/2\pi)$, with specific components $\sigma^z_{OH} = 1.160$ and $\sigma^x_{OH} = 0.594$. This indicates that OHC, rather than SHC, serves as the distinguishing signature for this HOTI system.
4. Ferroelectric Modulation of OHC: The direction of the out-of-plane ferroelectric polarization was found to modulate the OHC. Specifically, flipping the polarization direction (via mirror symmetry operation) reverses the sign of the $\sigma^x_{OH}$ component while leaving $\sigma^z_{OH}$ and $\sigma^y_{OH}$ (which remains zero due to symmetry constraints) unchanged. This modulation is attributed to the change in sign of the orbital angular momentum operator components ($l_x, l_y$) under mirror symmetry, while the velocity operator components remain invariant.

Significance and Claims
The paper claims to provide a theoretical foundation and a specific material candidate ($d1T$-phase MoS$_2$) for a ferroelectricity-tunable orbital Hall effect. The primary significance lies in demonstrating that:

• $d1T$-MoS$_2$ is a robust 2D HOTI with quantized fractional corner charges.
• OHC is a more effective signature than SHC for identifying this specific class of HOTIs.
• The orbital Hall effect in this system can be directionally controlled by switching the ferroelectric polarization.

The authors conclude that these findings offer a pathway toward realizing external electric field-controllable orbitronics, leveraging the coupling between ferroelectricity and higher-order topology. The work is presented as a theoretical prediction supported by first-principles calculations, aiming to guide future experimental efforts in orbitronic applications.