Spread-balanced Wannier functions - Robust and automatable orbital localization

At a Glance

Metadata	Details
Publication Date	2021-09-27
Journal	Physical review. B./Physical review. B
Authors	P. Fontana, Ask Hjorth Larsen, Thomas Olsen, Kristian S. Thygesen
Institutions	Technical University of Denmark
Citations	8
Analysis	Full AI Review Included

Technical Documentation & Analysis: Robust Wannier Function Localization

Reference: Fontana, P. F., Larsen, A. H., Olsen, T., & Thygesen, K. S. (2021). Spread balanced Wannier functions: Robust and automatable orbital localization. arXiv:2107.01722v3.

Executive Summary

This research introduces a highly robust and automatable method for generating maximally localized Wannier functions (WFs), addressing critical limitations in high-throughput computational materials science.

Core Innovation: Introduction of the variance-reducing functional ($\Omega_{var}$), which explicitly penalizes the delocalization of individual WFs, ensuring a more balanced spread distribution across the entire set.
Robustness Improvement: $\Omega_{var}$ resolves the common issue in standard localization ($\Omega$) where one or a few WFs become poorly localized (“sacrificed”) to minimize the overall quadratic spread.
Direct Diamond Application: The method was successfully demonstrated on the Nitrogen-Vacancy (NV) center in diamond, a key quantum material system.
Quantified Performance Gain: For the NV center, $\Omega_{var}$ reduced the maximum spread ($\text{s}_{max}$) of the least localized WF by approximately 34% (from 6.8 Å² to 4.46 Å²) compared to the standard $\Omega$ functional.
High-Throughput Capability: A fully automated protocol was developed for selecting the optimal number of WFs ($N_{w}$) and generating the initial guess, enabling reliable high-throughput screening (validated across 30 2D materials).
Broad Applicability: The method is confirmed to accurately calculate physical properties, such as the spontaneous polarization of ferroelectrics (BaTiO₃), matching established Berry-phase methods.

Technical Specifications

The following data points highlight the performance improvements achieved by the new $\Omega_{var}$ functional, particularly focusing on the diamond NV center and general material robustness.

Parameter	Value	Unit	Context
Test System	NV Center in Diamond	N/A	64-atom cubic unit cell
Minimal $N_{w}$	127	WFs	Number of WFs required for NV center
$\Omega$ Max Spread ($\text{s}_{max}$)	6.8 ± 0.3	Å²	Standard functional result (average over 5 runs)
$\Omega_{var}$ Max Spread ($\text{s}_{max}$)	4.46 ± 0.04	Å²	Variance-reducing functional result (34% reduction)
$\Omega$ Max Interpolation Error ($\eta_{max}$)	8	meV	NV Center in diamond
$\Omega_{var}$ Max Interpolation Error ($\eta_{max}$)	8	meV	NV Center in diamond
BaTiO₃ Spontaneous Polarization	45.4	µC/cm²	Calculated using $\Omega_{var}$ (matches Berry phase result)
Average $\Omega$ Max Spread (Optimal $N_{w}$)	2.9 ± 0.2	Å²	Average across 30 2D materials
Average $\Omega_{var}$ Max Spread (Optimal $N_{w}$)	2.5 ± 0.1	Å²	Average across 30 2D materials (Improved robustness)
Ru(111) Slab Adsorbate $N_{w}$ Range	133 - 135	WFs	H, N, O adsorbates on Ru(111) surface slab

Key Methodologies

The automated Wannierization protocol relies on a combination of Density Functional Theory (DFT) calculations and the new variance-reducing optimization scheme.

DFT Calculation: Self-consistent PBE calculations using the GPAW code were performed with a minimum k-point density of 5 k-points per Å^-1 and a real-space grid spacing of 0.2 Å.
Initial Guess Generation (Automated):
- For atoms with d-states (e.g., transition metals), a group of 5 atom-centered d-orbitals were included.
- The remaining $N_{s}$ s-orbitals were placed at random positions within a 1.5 Å radius of an atom, acting as “nucleation centers” for s, p, or sp-like WFs.
- All orbitals were set with a Gaussian half-width of 1 Å.
Wannier Function (WF) Selection: The Partly Occupied Wannier Function (POWF) formalism was used, defining the localization subspace by M eigenstates below an energy threshold ($E_{0}$) plus L extra degrees of freedom (EDF).
Optimal $N_{w}$ Selection: The optimal number of WFs ($N_{w}$) was determined by iterating $N_{w}$ from $N_{min}$ up to $N_{min} + 5$ and selecting the solution that yielded the lowest maximum spread ($\text{s}_{max}$).
Localization Functional: The iterative localization procedure maximized the new spread-balanced functional, $\Omega_{var}$, which includes a penalty term proportional to the variance of the spread distribution: $$\Omega_{var} = \Omega - w_{var} \text{Var} \left[ \sum_{\alpha=1}^{N_{G}} W_{\alpha} |Z_{\alpha, nn}|^2 \right]$$ Note: $w_{var}$ was consistently set to $N_{w}$ for all materials.
Robustness Check: Five independent localizations were performed for each $N_{w}$ value using different random initial guesses, and the best solution (lowest $\text{s}_{max}$) was selected.

6CCVD Solutions & Capabilities

The research highlights the critical need for robust computational methods to accurately model complex quantum materials and interfaces, particularly the NV center in diamond. 6CCVD is uniquely positioned to supply the high-quality MPCVD diamond materials necessary to experimentally validate and extend these computational findings.

Applicable Materials for Replication and Extension

Research Requirement	6CCVD Material Solution	Technical Specification Match
NV Center in Diamond	Optical Grade Single Crystal Diamond (SCD)	High-purity, low-defect SCD substrates are essential for creating and studying stable NV centers. We offer SCD up to 500 µm thick and substrates up to 10mm.
High-Throughput Screening	Polycrystalline Diamond (PCD) Wafers	For large-scale experimental validation of 2D materials or interface studies, our PCD wafers offer cost-effective, large-area coverage up to 125mm in diameter.
Ferroelectric/Interface Studies	Boron-Doped Diamond (BDD)	BDD provides a conductive, chemically inert platform ideal for electrochemical or interface studies, such as those involving spontaneous polarization or metal adsorbates.
Quantum Surface Quality	Ultra-Low Roughness SCD	The stability and coherence of near-surface NV centers depend critically on surface quality. We guarantee SCD polishing to Ra < 1nm.

Customization Potential for Advanced Research

The paper investigates complex systems involving defects (NV center) and metal/adsorbate interfaces (Ru slab). 6CCVD provides the necessary engineering services to translate these computational models into experimental reality:

Custom Metalization Services: The study of metal slabs and interfaces (like Ru(111)) often requires subsequent experimental deposition of contacts or capping layers. 6CCVD offers in-house metalization capabilities, including Au, Pt, Pd, Ti, W, and Cu, allowing researchers to define precise contact geometries directly on SCD or BDD substrates.
Precision Dimensioning: While the paper uses a 64-atom cubic cell computationally, experimental setups require specific dimensions. We provide custom laser cutting and shaping of diamond plates and wafers up to 125mm, ensuring compatibility with unique experimental apparatus.
Thickness Control: We offer precise control over SCD and PCD layer thickness, from 0.1 µm to 500 µm, crucial for optimizing quantum device performance or thermal management applications derived from these localization studies.

Engineering Support

The robust localization achieved by the $\Omega_{var}$ functional is vital for accurately modeling electron-phonon coupling, Berry curvatures, and transport calculations—all key areas for diamond-based quantum and power electronics.

6CCVD’s in-house PhD team specializes in the material science of MPCVD diamond and can assist researchers in selecting the optimal diamond grade (SCD vs. PCD, doping level, orientation) required for experimental validation of similar Wannier Function Localization and Quantum Defect projects.

Call to Action: For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly. We offer global shipping (DDU default, DDP available) to support your international research efforts.

View Original Abstract

We introduce a new type of Wannier functions (WFs) obtained by minimizing the\nconventional spread functional with a penalty term proportional to the variance\nof the spread distribution. This modified Wannierisation scheme is less prone\nto produce ineffective solutions featuring one or several poorly localized\norbitals, making it well suited for complex systems or high-throughput\napplications. Furthermore, we propose an automatable protocol for selecting the\ninitial guess and determine the optimal number of bands (or equivalently WFs)\nfor the localization algorithm. The improved performance and robustness of the\napproach is demonstrated for a diverse set of test systems including the NV\ncenter in diamond, metal slabs with atomic adsorbates, spontaneous polarization\nof ferroelectrics and 30 inorganic monolayer materials comprising both metals\nand semiconductors. The methods are implemented in Python as part of the Atomic\nSimulation Environment (ASE).\n

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Original Source

DOI: https://doi.org/10.1103/physrevb.104.125140