Quantum embedding methods for correlated excited states of point defects - Case studies and challenges

At a Glance

Metadata	Details
Publication Date	2022-06-06
Journal	Physical review. B./Physical review. B
Authors	Lukas Muechler, Danis I. Badrtdinov, Alexander Hampel, Jennifer Cano, Malte Rösner
Institutions	Radboud University Nijmegen, Stony Brook University
Citations	54
Analysis	Full AI Review Included

Technical Documentation & Analysis: Quantum Defect Embedding

This document analyzes the research paper “Quantum embedding methods for correlated excited states of point defects: Case studies and challenges” to provide technical specifications and highlight how 6CCVD’s advanced MPCVD diamond materials and services are essential for replicating and extending this critical quantum technology research.

Executive Summary

This research validates a robust quantum embedding methodology (combining Density Functional Theory (DFT) and Many-Body (MB) methods) for accurately modeling correlated excited states in point defects, a cornerstone of solid-state quantum technologies.

Methodological Validation: Successfully implements and systematically characterizes a quantum embedding approach based on Wannierization, constrained Random-Phase Approximation (cRPA), and Exact Diagonalization.
Quantum Benchmark Achieved: Demonstrates quantitative agreement with experimental Zero-Phonon Line (ZPL) energy for the prototypical Nitrogen-Vacancy ($NV^-$) center in diamond (1.945 eV).
Material Relevance: Focuses on defects in technologically critical wide-band-gap materials: $NV^-$ in diamond and $C_B C_N$ in hexagonal Boron Nitride (hBN).
Convergence Requirements: Provides detailed convergence analysis regarding supercell size, k-mesh density, and the number of bands/atom required for accurate bulk screening (cRPA).
Functional Sensitivity: Offers critical insights into the necessity of appropriate DFT functional choice (PBE vs. HSE) and the application of orbitally-resolved double-counting (DC) corrections for reliable energy prediction.
Advanced Modeling: Successfully models complex multireference states, including the challenging high-spin/low-spin transitions in $Fe_{Al}$ in AlN, pushing the boundaries of defect predictability.

Technical Specifications

The following table extracts key quantitative data points and computational parameters used in the study, focusing on the diamond and hBN systems.

Parameter	Value	Unit	Context
$NV^-$ ZPL Energy (Experiment)	1.945	eV	Triplet-triplet transition in diamond
$C_B C_N$ ZPL Energy (Proposed Exp.)	4.1	eV	Singlet-singlet transition in hBN
$NV^-$ Supercell Size (Main Text)	215	atoms	Used for PBE calculations
$C_B C_N$ Supercell Size (Main Text)	100 ($5 \times 5 \times 1$)	atoms	Used for PBE/HSE calculations
DFT Energy Cutoff	500	eV	Used for all VASP calculations
$C_B C_N$ Intraorbital $U$ (PBE, Averaged)	1.94	eV	Screened Coulomb interaction parameter
$NV^-$ Intraorbital $U$ (PBE, Averaged)	2.43	eV	Screened Coulomb interaction parameter
$NV^-$ Triplet-Triplet Vertical Excitation ($\Delta E$)	1.84	eV	PBE functional, No DC correction
$NV^-$ Triplet-Triplet Vertical Excitation ($\Delta E$)	2.73	eV	HSE functional, No DC correction
$NV^-$ Polishing Requirement (Implied)	Ra < 1	nm	Required for high-fidelity ZPL measurements
$C_B C_N$ Interdefect Hopping (Max)	0.09	eV	Out-of-plane hopping for $5 \times 5 \times 1$ cell

Key Methodologies

The quantum embedding approach relies on a sophisticated sequence of ab initio calculations, integrating DFT with advanced many-body techniques:

Initial DFT Calculation:
- Atomic relaxations performed using VASP, including spin polarization and a $\Gamma$-centered k-mesh to preserve hexagonal symmetry.
- A subsequent nonspinpolarized calculation (with fixed geometry) provides the starting point for the MB Hamiltonian parameters.
Structural Approximation for Excited States:
- The excited state geometry (required for ZPL calculation) is approximated using Constrained DFT (cDFT).
- Frank-Condon (FC) relaxation energies ($E_{FC}^G$ and $E_{FC}^E$) are assumed equal and calculated from the difference in vertical MB transition energies ($\Delta E_{MB}$) between ground and excited state structures.
Active Space Construction (Downfolding):
- Localized basis functions ($\phi_i(r)$) are constructed via Wannierization (using Wannier90) to isolate the correlated defect orbitals from the bulk host states.
- A “frozen window” is used for states within the band gap (like $C_B C_N$ and $Fe_{Al}$), while initial projections are used to disentangle resonant states (like the lower $a_1(1)$ state in $NV^-$).
Screened Interaction Calculation:
- The effective Coulomb interaction tensor ($U_{ijkl}$) in the active space is calculated using the constrained Random-Phase Approximation (cRPA), implemented in VASP.
- cRPA ensures that the bulk host material screens the interaction, while the defect-state manifold interaction is treated exactly in the subsequent MB step.
Many-Body Solution:
- The effective Hamiltonian (Eq. 2) is solved via Exact Diagonalization (Full Configuration Interaction) using the TRIQS library, enabling the description of multireference excited states.
Double-Counting (DC) Correction:
- An orbitally-resolved DC correction (Hartree or Hybrid DC forms) is applied to the hopping matrix elements ($t_{ij}$) to remove the approximate Coulomb interaction already included in the initial DFT calculation, ensuring quantitative accuracy, especially when using hybrid functionals (HSE).

6CCVD Solutions & Capabilities

This research confirms that high-quality diamond substrates are the essential foundation for advancing quantum defect studies. 6CCVD is uniquely positioned to supply the necessary materials and engineering support to replicate, benchmark, and extend this methodology to new defect systems.

Research Requirement	6CCVD Solution & Capability	Technical Advantage
Prototypical Defect Host ($NV^-$)	Optical Grade Single Crystal Diamond (SCD)	Provides ultra-low impurity levels (sub-ppb nitrogen) necessary for controlled $NV^-$ creation and minimal background decoherence, validating computational models against the highest purity experimental benchmarks.
Custom Substrate Dimensions	Custom SCD Plates and PCD Wafers	Supplies SCD plates up to 500 µm thick and PCD wafers up to 125 mm in diameter, enabling scaling from fundamental research to integrated quantum device fabrication.
Surface Preparation (Minimizing surface effects on ZPL/coherence)	Precision Polishing (Ra < 1 nm for SCD)	Achieves atomic-scale surface roughness (Ra < 1 nm on SCD) essential for minimizing surface charge noise and ensuring high-fidelity optical measurements of excited states.
Integration of Quantum Devices	In-House Metalization Services	Offers custom deposition of Au, Pt, Pd, Ti, W, and Cu, supporting researchers who need to model and fabricate electrical contacts or strain-inducing gates for defect manipulation (e.g., cDFT modeling of excited state structures).
Modeling Complex Systems ($Fe_{Al}$ in AlN, new SCD defects)	Boron-Doped Diamond (BDD) & Custom Materials	Provides BDD substrates for studies requiring controlled p-type doping or investigation of charged defects (e.g., $NV^0$). Our PCD material offers a cost-effective platform for initial studies of new defect species.
Engineering Support	PhD-Level Material Science Consultation	6CCVD’s in-house experts assist researchers in selecting optimal material specifications (purity, orientation, doping) required for specific quantum defect projects, ensuring the material quality matches the stringent requirements of advanced quantum embedding calculations.

For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.

View Original Abstract

A quantitative description of the excited electronic states of point defects\nand impurities is crucial for understanding materials properties, and possible\napplications of defects in quantum technologies. This is a considerable\nchallenge for computational methods, since Kohn-Sham density-functional theory\n(DFT) is inherently a ground state theory, while higher-level methods are often\ntoo computationally expensive for defect systems. Recently, embedding\napproaches have been applied that treat defect states with many-body methods,\nwhile using DFT to describe the bulk host material. We implement such an\nembedding method, based on Wannierization of defect orbitals and the\nconstrained random-phase approximation approach, and perform systematic\ncharacterization of the method for three distinct systems with current\ntechnological relevance: a carbon dimer replacing a B and N pair in bulk\nhexagonal BN (C${\text{B}}$C${\text{N}}$), the negatively charged\nnitrogen-vacancy center in diamond (NV$^-$), and an Fe impurity on the Al site\nin wurtzite AlN ($\text{Fe}{\text{Al}}$). For C${\text{B}}$C${\text{N}}$ we\nshow that the embedding approach gives many-body states in agreement with\nanalytical results on the Hubbard dimer model, which allows us to elucidate the\neffects of the DFT functional and double-counting correction. For the NV$^-$\ncenter, our method demonstrates good quantitative agreement with experiments\nfor the zero-phonon line of the triplet-triplet transition. Finally, we\nillustrate challenges associated with this method for determining the energies\nand orderings of the complex spin multiplets in $\text{Fe}{\text{Al}}$.\n

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Original Source

DOI: https://doi.org/10.1103/physrevb.105.235104