Orbital Hall effect from orbital magnetic moments of Bloch states: the role of a new correction term

This paper presents a rigorous derivation of orbital magnetic moment matrix elements for Bloch states by incorporating a previously omitted Berry connection term, revealing two new gauge-covariant corrections that significantly reduce the orbital Hall conductivity in bilayer van der Waals materials.

The Gist

Imagine you are trying to understand how a crowd of people (electrons) moves through a giant, repeating maze (a crystal solid). In the world of "orbitronics," scientists are trying to figure out how these people carry a specific type of "spin" or "twist" called orbital angular momentum. This twist is useful because it can be used to store and process information, much like how we use the spin of a top or the orbit of a planet.

For a long time, scientists had a simplified map to predict how this "twist" moves. They used a rule called the Intra-atomic Approximation. Think of this like assuming every person in the crowd is just spinning in place inside their own small room, ignoring the fact that they are actually walking through hallways and interacting with neighbors. This map worked okay for some things, but it was missing the big picture.

Later, scientists improved the map. They realized that people do walk between rooms, creating a "current" of twist. They developed a new formula (let's call it the Semiclassical Map) that accounted for this movement. This was a huge step forward.

However, this paper says: "We found a missing piece of the puzzle."

The authors, Tarik Cysne, Ivo Souza, and Tatiana Rappoport, went back to the drawing board to derive the math from scratch. They found that in the previous "Semiclassical Map," scientists had accidentally left out a crucial ingredient: the Berry Connection.

The Creative Analogy: The GPS and the Wind

To understand what they found, imagine you are driving a car (an electron) through a city (the crystal).

1. The Old Map (Intra-atomic): You assume the car never leaves its garage. You calculate the twist based only on the engine spinning inside the garage.
2. The Previous New Map (Semiclassical): You realize the car drives on the road. You calculate the twist based on the car's speed and the shape of the road. But, you forgot to account for the wind.
3. The New Discovery (This Paper): The authors realized that the "wind" (the Berry connection) is actually part of the road itself. It's a subtle force that changes how the car turns.

In physics terms, the "wind" is a geometric property of the quantum world. When you calculate how the electron's position changes as it moves, you have to include a term that accounts for how the electron's "wave" twists and turns in the background. Previous calculations ignored this twist when looking at the relationship between different energy levels.

The Two New Corrections

The authors found that when you include this "wind," two new things happen in the math:

1. The "Gauge Fix": The first new term is like a correction factor that ensures the map makes sense no matter how you choose to label the streets (a concept called "gauge covariance"). Without it, the map gives different answers depending on how you draw the grid lines. This term fixes the map so it's consistent.
2. The "Quantum Boost": The second term is the real surprise. It's a new force that actually changes the strength of the "twist" current. It's not just a fix; it's a new physical effect that was previously invisible.

What Happens When You Use the New Map?

The authors tested this new, complete map on two specific materials:

• A Bilayer of 2H-TMD (like a sandwich of Molybdenum Disulfide): A material used in next-gen electronics.
• Biased Bilayer Graphene: Two sheets of graphene (a single layer of carbon atoms) stacked on top of each other with a voltage applied.

The Result: When they used the new map with the extra "wind" terms, the predicted "Orbital Hall Effect" (the flow of the twist) was significantly smaller than what previous maps predicted.

Think of it like this: If you were building a dam to hold back a river of "twist," the old maps told you the river was huge and you needed a massive wall. The new maps say, "Actually, the river is about half that size."

Why Does This Matter?

1. Accuracy: If engineers want to build devices that use orbital currents to store data, they need to know exactly how strong that current is. If they use the old, incomplete math, they might design a device that doesn't work as well as they thought.
2. New Physics: This paper proves that the "twist" of electrons is more complex than we thought. It's not just about the electron spinning in place or moving in a straight line; it's about a subtle, geometric interaction with the entire crystal structure.
3. The Future of Orbitronics: This field is trying to replace or complement traditional electronics (which use charge) with "orbitronics" (which use orbital momentum). By getting the math right, the authors are laying a stronger foundation for the future of computing.

In a Nutshell

The authors found a hidden "wind" in the mathematical description of how electrons move in crystals. By adding this wind back into the equations, they showed that the flow of orbital angular momentum in certain materials is actually weaker than we previously believed. It's a reminder that in the quantum world, even the smallest, most subtle details can change the entire picture.

Technical Summary

1. Problem Statement

The emerging field of orbitronics relies on manipulating the orbital angular momentum (OAM) of electrons in solids. A central quantity in this field is the Orbital Magnetic Moment (OMM) of Bloch states, which drives phenomena like the Orbital Hall Effect (OHE).

Historically, calculations of OMM have faced two main limitations:

1. Intra-atomic Approximation (ACA): Most studies approximate the OMM using atomic operators centered on individual atoms. This neglects "inter-site" contributions arising from electron trajectories spanning multiple atoms, which are crucial in materials like graphene where $s$ and $p_z$ orbitals dominate.
2. Incomplete Gauge Covariance: Previous attempts to go beyond the intra-atomic approximation (e.g., by Bhowal and Vignale) utilized a formula derived from semiclassical wave packets. While valid for nearly degenerate bands, this formula loses gauge covariance when applied to non-degenerate Bloch states across the full Hilbert space. Specifically, previous derivations omitted the Berry connection term in the $k$-derivatives of Bloch states, leading to mathematically inconsistent results for non-diagonal matrix elements.

The authors aim to rigorously derive the OMM matrix elements for non-degenerate Bloch states, ensuring gauge covariance, and to quantify the impact of previously neglected terms on the OHE.

2. Methodology

The authors employ a rigorous quantum mechanical derivation rather than relying solely on semiclassical approximations.

• Formal Derivation: They start from the definition of the electronic OMM operator: $\hat{m}_z = -\frac{e}{4} [(\hat{\mathbf{r}} \times \hat{\mathbf{v}}) - (\hat{\mathbf{v}} \times \hat{\mathbf{r}})]_z$.
• Inclusion of Berry Connection: A critical step in their methodology is the consistent inclusion of the Berry connection ($\mathbf{A}_{nk} = i\langle n | \nabla_k n \rangle$) in two key identities:
  1. The derivative of the Bloch state $|\nabla_k n\rangle$ (Eq. 3).
  2. The matrix element of the position operator $\langle n | \hat{\mathbf{r}} | n' \rangle$ (Eq. 8).
     Previous works often dropped these terms for non-degenerate states, assuming they vanish for diagonal elements. The authors demonstrate that for non-diagonal elements (crucial for OHE), these terms are essential for restoring gauge covariance.
• Decomposition of the OMM: The derived matrix elements $m_{nn'k}$ are decomposed into three distinct contributions:
  1. $m^{SR}_{nn'k}$: The term analogous to the semiclassical self-rotation formula (Eq. 14).
  2. $g^I_{nn'k}$: A correction term involving the sum of Berry connections and velocity (Eq. 15), which restores gauge covariance to the semiclassical term.
  3. $g^{II}_{nn'k}$: A new correction term (Eq. 16) involving the cross product of the average velocity of the bands and the derivative of the wavefunction. This term was previously neglected in OHE studies.
• Numerical Application: The authors apply this complete formalism to two specific bilayer systems using linear response theory (Kubo formula):
  1. Bilayer 2H-Transition Metal Dichalcogenide (TMD): Specifically 2H-MoS$_2$.
  2. Biased Bilayer Graphene (BLG): Graphene with a perpendicular electric field breaking inversion symmetry.

3. Key Contributions

• Rigorous Derivation: The paper provides the first consistent derivation of OMM matrix elements for non-degenerate Bloch states that maintains gauge covariance across the entire Hilbert space.
• Identification of a New Term ($g^{II}$): The authors identify a previously overlooked term, $g^{II}_{nn'k}$, which arises from the Berry connection in the position operator matrix elements. They prove that the sum $m^{SR} + g^I + g^{II}$ is the physically correct, gauge-covariant OMM.
• Quantitative Correction: They demonstrate that neglecting the $g^{II}$ term leads to significant quantitative errors in the calculated Orbital Hall Conductivity (OHC).
• Validation via Multipole Expansion: The authors note that their derived formula coincides with recent results obtained via a completely different approach (multipole expansion of optical matrix elements), validating the robustness of their derivation.

4. Results

The authors calculated the Orbital Hall Conductivity ($\sigma_{OH}$) for the two model systems, comparing three approaches:

1. Intra-atomic approximation (ACA).
2. Bloch OMM with only $m^{SR} + g^I$ (previous literature standard).
3. Complete Bloch OMM including the new $g^{II}$ term.

Key Findings:

• Bilayer 2H-MoS$_2$:
  • The inclusion of the $g^{II}$ term reduces the height of the orbital Hall conductivity plateau by a factor of 1/2 compared to the previous semiclassical approach ($m^{SR} + g^I$).
  • The complete formula yields $\bar{\sigma}_{OH} \approx 1.26 (e/2\pi)$, whereas the previous approach yielded $\approx 2.52 (e/2\pi)$.
  • The intra-atomic approximation significantly overestimates the effect, predicting a quantized value of $4(e/2\pi)$.
• Biased Bilayer Graphene:
  • Unlike TMDs, BLG has zero OHE in the intra-atomic approximation (due to $p_z$ orbital dominance).
  • The $m^{SR} + g^I$ approach predicts a non-zero OHE plateau.
  • Including the $g^{II}$ term reduces the OHE plateau height by approximately 30% compared to the $m^{SR} + g^I$ result.
  • The magnitude of the correction depends on the band gap ($\Delta$); the correction is more pronounced for smaller gaps (weak interlayer hopping).

General Trend: In both systems, the new correction term acts to suppress the orbital Hall conductivity compared to calculations that omit it. This suggests that multi-layered van der Waals materials are particularly susceptible to these corrections due to their non-degenerate, weakly coupled band structures.

5. Significance

• Theoretical Foundation: This work establishes a rigorous, gauge-invariant foundation for calculating OMM in orbitronics, correcting a long-standing oversight in the treatment of non-degenerate bands.
• Impact on Orbitronics: The findings imply that previous theoretical predictions of OHE in bilayer materials may have been overestimated by factors of 2 to 3. Accurate prediction of OHE is critical for designing orbitronic devices for information storage and processing.
• Material Design: The results suggest that the "new" correction term is not a minor perturbation but a dominant factor in systems with weak interlayer coupling and small energy separations between bands.
• Future Directions: The authors highlight that while their derivation assumes non-degenerate bands, extending this to degenerate bands and studying the interplay of these gauge-covariant terms with disorder (vertex corrections) are vital next steps for the field.

In summary, the paper corrects the theoretical framework for orbital transport, revealing that previously accepted values for the Orbital Hall Effect in bilayer materials are significantly inflated due to the omission of a specific Berry-connection-dependent correction term.