Quantum embedding methods for correlated excited states of point defects - Case studies and challenges
At a Glance
Section titled âAt a Glanceâ| 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 | Flatiron Health (United States), Radboud University Nijmegen |
| Citations | 54 |
| Analysis | Full AI Review Included |
Executive Summary
Section titled âExecutive SummaryâThis research validates a quantum embedding methodology for accurately predicting the correlated excited electronic states of point defects in semiconductors and insulators, crucial for quantum technologies.
- Core Methodology: The approach combines Kohn-Sham Density Functional Theory (KS-DFT) for the bulk host material with Many-Body (MB) methods (Exact Diagonalization) for a small, correlated defect active space.
- Key Components: The method relies on Wannierization to define the active space, Constrained Random-Phase Approximation (cRPA) for calculating screened Coulomb interactions (Uijkl), and a functional-dependent Double-Counting (DC) correction (HDC).
- Quantitative Successes: The method achieved good quantitative agreement with experimental Zero-Phonon Line (ZPL) energies for the prototypical NV- center in diamond (benchmark) and the CBCN dimer in hexagonal BN (SPE candidate).
- Methodological Robustness: Applying the appropriate orbitally-resolved DC correction significantly reduced the dependence of the final MB energies and ZPLs on the choice of initial DFT functional (PBE vs. HSE).
- Convergence Improvement: Localized Wannier basis sets improved convergence with respect to supercell size compared to previous methods, though convergence with respect to the number of empty bands and k-mesh density remains critical.
- Identified Challenge: The methodology failed to predict the expected high-spin ground state (S = 5/2) for the complex FeAl impurity in AlN, indicating a need for further development in DC correction schemes for systems highly sensitive to crystal field splitting (CFS).
Technical Specifications
Section titled âTechnical Specificationsâ| Parameter | Value | Unit | Context |
|---|---|---|---|
| NV- ZPL (Exp.) | 1.945 | eV | Triplet-triplet transition benchmark in diamond. |
| CBCN ZPL (Exp.) | 4.1 | eV | Singlet-singlet transition attribution in hBN. |
| DFT Energy Cutoff | 500 | eV | Standard VASP PAW potential cutoff for all defects. |
| CBCN Active Space | 4 | Spin-orbitals | Minimal space (bonding/antibonding C pz states). |
| NV- Active Space | 8 | Spin-orbitals | Dangling sp3 bonds of N and C atoms. |
| FeAl Active Space | 10 | Spin-orbitals | Fe 3d states (d5 configuration). |
| CBCN Intraorbital U (PBE) | 1.94 | eV | Averaged screened Coulomb interaction (orbital basis). |
| CBCN Intraorbital U (HSE, alpha=0.4) | 2.73 | eV | Increased U due to reduced screening from larger HSE band gap. |
| NV- Supercell Size | 215 | Atoms | Used for final MB calculations (rapid convergence observed). |
| FeAl Convergence Requirement | > 200 | Atoms | Required for quantitative accuracy of MB states. |
| FeAl Expected Ground State | High-spin S = 5/2 | Spin (S) | Neutral Fe3+ in AlN (6A1 state). |
| CBCN Convergence Metric | Atoms x k-points | Effective Size | Required acceleration of bulk screening convergence in layered hBN. |
Key Methodologies
Section titled âKey MethodologiesâThe quantum embedding approach follows a multi-step computational recipe:
- Structural Relaxation: Atomic positions are optimized using spin-polarized DFT (PBE or HSE functionals) within a large supercell to obtain the equilibrium geometry for the defect ground state.
- Noninteracting Hamiltonian (tij): A subsequent nonspinpolarized DFT calculation is performed on the fixed geometry to generate the single-particle hopping matrix elements (tij) for the active space.
- Active Space Construction (Wannierization): Maximally-Localized Wannier Functions (MLWFs) are constructed (using Wannier90) from the DFT Bloch states to define the localized, correlated active space (e.g., d-orbitals or dangling bonds).
- Screened Interaction (cRPA): The full four-index Coulomb tensor (Uijkl) is calculated using the Constrained Random-Phase Approximation (cRPA) within VASP, effectively screening the defect interaction using the bulk host materialâs polarization.
- Double-Counting (DC) Correction: An orbitally-resolved DC correction (HDC) is applied to the tij terms to compensate for the approximate Coulomb interaction already present in the initial DFT calculation (forms explored include Hartree and Hybrid functional-dependent corrections).
- Many-Body Solution: The resulting correlated Hamiltonian (H) is solved using Exact Diagonalization (ED) via the TRIQS library to obtain the Many-Body (MB) excited state energies and wavefunctions.
- Excited State Geometry (cDFT): Excited state geometries, necessary for calculating the ZPL, are approximated using constrained DFT (cDFT) by fixing the occupation of the relevant single-particle defect orbitals.
Commercial Applications
Section titled âCommercial ApplicationsâThe methodology and results directly support the engineering and optimization of materials for next-generation electronic and quantum devices.
- Quantum Computing:
- Spin Qubits: Accurate prediction of excited state energy levels (e.g., 3A2 to 3E transition in NV-) is essential for designing optical manipulation protocols and optimizing qubit coherence.
- Quantum Communication:
- Single-Photon Emitters (SPEs): Precise calculation of ZPL energies (e.g., CBCN in hBN) allows for tuning defect creation or host materials to match desired communication wavelengths.
- Power Electronics and Optoelectronics:
- Wide-Band-Gap Semiconductors (AlN, GaN): Characterization of transition metal impurities (like FeAl) is critical for understanding and mitigating defect-mediated recombination (Shockley-Read-Hall, SRH), which limits device efficiency in LEDs and power converters.
- Materials Design and Screening:
- Predictive Modeling: The validated embedding approach provides a robust, first-principles tool for screening novel defect candidates in various host materials (insulators, semiconductors) for specific quantum or electronic functionalities.
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