🔬 materials science

Calculation of DFT Spin-Orbit Spillage with Quantum ESPRESSO

This paper presents a workflow for calculating spin-orbit spillage using Quantum ESPRESSO to assess topological character, validates the method against VASP results for the insulator BaMg2Bi2, and discusses its applicability and limitations for semimetals.

Original authors: Duy Quan Nguyen, Paul C. H. Li

Published 2026-09-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Duy Quan Nguyen, Paul C. H. Li

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

In the hidden architecture of solid matter, where atoms arrange themselves into repeating patterns, electrons do not sit still. They flow through the crystal, forming energy bands that determine whether a material is a conductor, an insulator, or something far more exotic. Among these exotic possibilities are topological materials, a class of substances that behave like insulators on the inside but conduct electricity perfectly along their surfaces, a property that could revolutionize future electronics. The key to identifying these materials often lies in a subtle, relativistic interaction called spin–orbit coupling. This effect, which becomes particularly strong in heavy atoms like bismuth, can twist the electron's path and flip the order of its energy levels, a phenomenon known as band inversion. When this inversion happens, the material may possess the unique topological character scientists are eager to find. However, detecting this flip is difficult because it requires comparing two slightly different versions of the same material: one where this relativistic twist is present and one where it is absent.

A team of researchers at Simon Fraser University has developed a new, open-source method to perform this delicate comparison using a widely available software package called Quantum ESPRESSO. Their work focuses on a specific insulating compound, barium magnesium bismuth, to test whether their approach can accurately measure a quantity known as spin–orbit spillage. This number acts as a gauge for how much the electron states change when the relativistic twist is turned on. If the change is small, the material is likely ordinary. If the change is large, it suggests the electrons have swapped places in a way that signals a topological nature. The researchers successfully calculated this value for their test material, finding a result of 2.094. This figure matches remarkably well with a previously published value of 2.075 obtained using a different, proprietary software, differing by less than one percent. This agreement confirms that their open-source workflow is a reliable tool for screening materials, offering a free alternative to expensive commercial software for scientists hunting for the next generation of topological devices.

The process the team developed is essentially a side-by-side comparison of the same crystal under two different physical rules. First, they modeled the material using standard quantum mechanics, ignoring the relativistic spin effects. Then, they ran a second, more complex simulation that included the spin–orbit coupling, which accounts for the interaction between an electron's spin and its motion around the nucleus. In the second run, the electrons behave as if they have two components, spin-up and spin-down, which become entangled. The researchers then took the wavefunctions—the mathematical descriptions of where the electrons are likely to be found—from both simulations and compared them point by point across the crystal's momentum space. They looked specifically at the occupied states, the energy levels filled with electrons, to see how much they shifted or changed character when the relativistic effect was introduced. The degree of this change, or "spillage," is highest at a specific point in the crystal's momentum space, and that peak value is the number reported as the material's signature.

A critical part of their success was choosing the right mathematical tools to represent the atoms, known as pseudopotentials. These are simplified models that replace the complex core of an atom to make the calculations feasible. The researchers discovered that using a specific type of pseudopotential called norm-conserving was essential for getting a clean, accurate result. When they initially tried a different type, known as PAW, the comparison produced nonsensical numbers, likely because the mathematical definition of the electron overlap was incomplete without extra correction steps. By switching to the norm-conserving type, they ensured that the comparison between the two simulations was direct and valid, requiring no complex approximations. This choice allowed them to verify that their code correctly aligned the electron states from the two different runs, ensuring that the final number truly reflected the physical change caused by spin–orbit coupling rather than a flaw in the calculation method.

The study also revealed important limitations when applying this method to different types of materials. While the approach worked perfectly for the insulating compound, which has a clear gap between its filled and empty electron states, the researchers found that the method becomes unstable for semimetals. In these materials, the energy bands touch or overlap, making it ambiguous exactly which states are "filled" and which are "empty." When the team tested a semimetal, the calculated spillage value swung wildly depending on how they defined the boundary between filled and empty states, changing from a low number to a high one just by shifting the count by a few electrons. This finding serves as a crucial warning: while the spillage metric is a powerful tool for identifying topological insulators, it cannot be blindly applied to materials where the electron count is not fixed, as the result may depend more on the arbitrary choice of definition than on the material's true physics.

Ultimately, this work provides a robust, reproducible path for the scientific community to explore topological materials without relying on expensive, closed-source software. By demonstrating that their open-source workflow can reproduce high-precision results for insulators, the authors have validated a new standard for the field. They showed that for materials with a clear energy gap, the spin–orbit spillage is a stable and reliable indicator of topological character. The slight differences they observed between their results and previous data are well within the expected range of variation between different high-quality simulation codes. However, their careful examination of the semimetal case highlights that the method is not a universal magic bullet; it requires a well-defined electronic structure to work. For the insulating compounds that dominate the search for new topological materials, this new workflow offers a clear, accessible, and accurate way to measure the subtle quantum shifts that define the next generation of electronic wonders.

Technical Summary: Calculation of DFT Spin–Orbit Spillage with Quantum ESPRESSO

Problem Statement
The identification of topological materials often relies on detecting band inversion, a phenomenon where the character of occupied and unoccupied electronic bands swaps due to spin–orbit coupling (SOC). The "spin–orbit spillage" metric, introduced by Liu and Vanderbilt, quantifies the change in occupied electronic states when SOC is included. A large spillage indicates significant band rearrangement and potential topological character. While this metric has been used for high-throughput screening of databases like JARVIS, the standard implementation relies on the VASP code, which requires proprietary software and specific wavefunction file formats (WAVECAR). There was no reusable, open-source workflow to calculate spillage directly from Quantum ESPRESSO (QE), nor was there a clear understanding of how pseudopotential choices (specifically norm-conserving vs. PAW) and material classes (insulators vs. semimetals) affect the calculation's validity and numerical accuracy.

Methodology
The authors developed a complete computational workflow to calculate spin–orbit spillage using Quantum ESPRESSO. The procedure involves two distinct Density Functional Theory (DFT) Self-Consistent Field (SCF) calculations on the same crystal structure:

Non-SOC Run: A scalar-relativistic calculation using spin-degenerate wavefunctions and scalar-relativistic (SR) pseudopotentials.
SOC Run: A fully relativistic, non-collinear calculation using spinor wavefunctions and fully relativistic (FR) pseudopotentials.

Key technical constraints and choices include:

Consistency: To ensure the wavefunctions are directly comparable, symmetry reduction (nosym=.true.) and time-reversal folding (noinv=.true.) are disabled. This ensures both runs utilize the identical k-point mesh and plane-wave basis set.
Pseudopotentials: The workflow explicitly utilizes norm-conserving (NC) pseudopotentials. The authors demonstrate that using Projector Augmented Wave (PAW) pseudopotentials without the proper augmentation correction leads to unphysical overlap values (e.g., negative spillage). With NC pseudopotentials, the overlap is simply the dot product of the plane-wave coefficients.
Post-Processing: A custom Python script (compute_spillage.py) reads the HDF5 wavefunction files generated by QE. It aligns the plane-wave coefficients (Miller indices) between the two runs, constructs the overlap matrix between the occupied states of the non-SOC and SOC calculations, and computes the spillage $\gamma(k)$ at each k-point using the formula:
$\gamma(k) = N_{occ} - \sum_{m=1}^{N_{occ}} \sum_{a=1}^{N_{occ}} |\langle \psi_m^{so}(k) | \psi_a^{no}(k) \rangle|^2$
The material's spillage $\eta$ is defined as the maximum $\gamma(k)$ over the Brillouin zone.
Occupation Handling: For insulators with a finite band gap, the number of occupied states ( $N_{occ}$ ) is fixed and unambiguous. The script uses a fixed count derived from the valence electron count rather than relying on fractional occupation numbers generated by smearing schemes, which can be misleading near small gaps.

Key Results
The workflow was applied to the insulating compound BaMg $_2$ Bi $_2$ (JVASP-4053) and compared against published VASP results:

Numerical Agreement: The QE calculation yielded a spillage of 2.094, compared to the reference VASP value of 2.075, a difference of only 0.9%. This agreement extends to other insulators like Bi $_2$ Se $_3$ and Bi $_2$ Te $_3$ , where differences were also under 1%.
Band Structure Validation: The calculated band gaps (both with and without SOC) matched the reference values within 1%, confirming the electronic structure accuracy of the QE setup.
Semimetal Limitations: The study examined the semimetal Ba $_3$ BiSb, which has a degenerate manifold at the Fermi level. Here, the spillage value proved highly sensitive to the definition of the occupied subspace. Changing the assumed number of occupied electrons by just two bands altered the spillage from 0.08 to 2.12. This indicates that for semimetals, the spillage metric is ill-conditioned and the numerical value depends heavily on the specific convention used to define occupied states, rather than solely on the computational code.
Pseudopotential Impact: Initial attempts using PAW pseudopotentials resulted in unphysical raw overlap values ( $\gamma \approx -17$ ). Switching to norm-conserving pseudopotentials resolved this, validating the necessity of NC potentials for this specific overlap metric without requiring uncontrolled augmentation corrections.

Significance and Claims
The paper claims to provide the first reusable, open-source implementation of the spin–orbit spillage calculation using Quantum ESPRESSO, removing the dependency on VASP for this specific screening metric. The authors emphasize that their work validates the QE implementation against established VASP benchmarks for insulators, showing that the difference in results is negligible (<1%) and likely due to minor systematic offsets between codes rather than fundamental errors.

Crucially, the paper modestly frames its findings regarding semimetals: it does not claim to have solved the ambiguity of spillage in zero-gap systems but rather identifies the "occupied-band ambiguity" as the primary source of discrepancy in such cases. The work establishes that for materials with well-defined band gaps, the QE workflow is robust and reproducible, while for semimetals, the metric's sensitivity to occupation conventions must be carefully considered. The authors do not propose new applications or future experimental directions, focusing strictly on the computational reproducibility and the methodological constraints of the spillage metric itself.

More like this