Vibrational and vibronic structure of isolated point defects - The nitrogen-vacancy center in diamond

At a Glance

Metadata	Details
Publication Date	2021-07-12
Journal	Physical review. B./Physical review. B
Authors	Lukas Razinkovas, Marcus W. Doherty, Neil B. Manson, Chris G. Van de Walle, Audrius Alkauskas
Institutions	University of California, Santa Barbara, Center for Physical Sciences and Technology
Citations	56
Analysis	Full AI Review Included

Technical Documentation & Analysis: Vibrational and Vibronic Structure of $\text{NV}^{-}$ Centers in Diamond

Executive Summary

This research provides a rigorous theoretical framework for understanding the optical lineshapes of the negatively charged Nitrogen-Vacancy ($\text{NV}^{-}$) center in diamond, a critical solid-state system for quantum technologies.

Quantum System Focus: Presents advanced first-principles Density Functional Theory (DFT) analysis of the $\text{NV}^{-}$ center, focusing on the ${}^3\text{E} \leftrightarrow {}^3\text{A}_2$ optical transitions essential for quantum sensing and computing.
Advanced Modeling: Successfully models complex vibronic structure and optical lineshapes (luminescence and absorption) by developing an efficient methodology to solve the multi-mode dynamical Jahn-Teller (JT) problem.
High Accuracy: Achieves converged spectral densities across the full phonon spectrum by employing an embedding methodology utilizing supercells up to $20 \times 20 \times 20$ (64,000 atoms).
Functional Comparison: Confirms that the PBE functional provides better agreement with experimental vibrational frequencies, while the HSE functional offers a more accurate Zero-Phonon Line (ZPL) energy estimate (1.995 eV calculated vs. 1.945 eV experimental).
Material Requirement: The successful experimental validation and extension of this work rely fundamentally on the availability of high-purity, low-strain Single Crystal Diamond (SCD) with precisely controlled nitrogen concentration, a core capability of 6CCVD.

Technical Specifications

The following hard data points were extracted from the theoretical calculations and experimental comparisons:

Parameter	Value	Unit	Context
Target Defect	$\text{NV}^{-}$ Center	N/A	Negatively charged Nitrogen-Vacancy in Diamond
Experimental ZPL Energy	1.945	eV	Zero-Phonon Line (ZPL) for ${}^3\text{E} \leftrightarrow {}^3\text{A}_2$ transition
Calculated $E_0$ (HSE)	1.995	eV	Energy difference between potential energy minima
Total Huang-Rhys Factor ($S_{\text{tot}}$) (Expt, Emission)	3.49	N/A	Average number of phonons created during transition
$e$ Mode Contribution to $S_{\text{tot}}$	14 - 18	%	Contribution of asymmetric modes (Jahn-Teller effect)
Lattice Constant ($a$) (Expt)	3.567	Å	Bulk diamond parameter
Bulk Modulus ($B$) (Expt)	443	GPa	Bulk diamond parameter
Highest Phonon Frequency ($\omega(L)$) (Expt)	153.0	meV	High-symmetry point L
Supercell Size (Max)	$20 \times 20 \times 20$	N/A	Used for embedding methodology (64,000 atomic sites)
Force Convergence Cutoff ($r_{c1}$)	7	Å	Distance from N site for force decay convergence

Key Methodologies

The theoretical approach combines state-of-the-art quantum chemistry and solid-state physics techniques:

First-Principles DFT: Calculations were performed using the Vienna Ab-initio Simulation Package (VASP) employing both PBE (Generalized Gradient Approximation) and HSE (Hybrid Functional) methods to model electronic structure and forces.
Excited State Modeling: The ${}^3\text{E}$ excited state was modeled using the $\Delta$SCF (Constrained Orbital Occupation) method, standard for defect calculations, to handle the electronic configuration $\text{a}_1\text{e}^3$.
Vibrational Analysis: Phonon dispersion curves and Hessian matrices were calculated using the PHONOPY software package and a finite-difference approach, primarily in $4 \times 4 \times 4$ supercells (512 atoms).
Embedding Methodology: An embedding technique was applied to construct effective models for large supercells (up to 64,000 atoms), enabling the calculation of converged spectral densities across the entire phonon spectrum, including low-frequency acoustic modes.
Jahn-Teller Solution: A novel, efficient algorithm was developed and tested to solve the multi-mode $E \otimes e$ Jahn-Teller problem, crucial for accurately modeling the contribution of asymmetric $e$ modes to the absorption lineshape.
Spectral Function Generation: Luminescence and absorption lineshapes were calculated via the convolution of spectral functions ($A_{a1}$ and $A_e$), using the calculated Huang-Rhys factors ($S_k$) derived from atomic forces and displacements.

6CCVD Solutions & Capabilities

The successful experimental realization and extension of this critical quantum defect research require diamond materials with exceptional purity, precise defect control, and advanced surface engineering—all core competencies of 6CCVD.

Research Requirement / Challenge	6CCVD Solution & Capability	Technical Advantage
Ultra-Low Defect Density Host (Minimizing strain and unwanted background defects)	Optical Grade Single Crystal Diamond (SCD)	SCD wafers available in thicknesses from 0.1µm to 500µm, providing the highest purity and lowest strain environment necessary to isolate and study single $\text{NV}^{-}$ centers.
Precise Nitrogen Concentration (Achieving the “dilute limit” required by the theory)	Custom Nitrogen Doping during MPCVD Growth	6CCVD offers highly controlled nitrogen incorporation during growth, enabling researchers to specify the exact concentration needed to create dilute $\text{NV}^{-}$ ensembles or isolated centers for fundamental studies.
Large-Scale Integration (Need for larger wafers for device fabrication)	Large Format PCD and SCD Wafers	We supply PCD plates/wafers up to 125mm diameter, and SCD substrates up to 10mm thick, supporting both fundamental research and scalable quantum device manufacturing.
Surface and Interface Quality (Minimizing surface effects on ZPL and coherence)	Precision Polishing Services	SCD wafers polished to achieve surface roughness $R_a$ < 1nm, and inch-size PCD polished to $R_a$ < 5nm, ensuring minimal surface scattering and strain for near-surface $\text{NV}^{-}$ applications.
Device Architecture Integration (Need for electrical contacts or strain tuning layers)	Custom Metalization and Patterning	In-house capability for depositing standard and custom metal stacks (Au, Pt, Pd, Ti, W, Cu), allowing for immediate integration into device architectures for electrical readout or controlled strain application.
Material Optimization Support (Translating theoretical parameters into physical materials)	In-House PhD Engineering Team Support	6CCVD’s expert material scientists can assist researchers in selecting the optimal SCD grade, thickness, and doping level required to replicate or extend the findings of this advanced vibronic structure analysis.

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

View Original Abstract

We present a theoretical study of vibrational and vibronic properties of a point defect in the dilute limit by means of first-principles density functional theory calculations. As an exemplar we choose the negatively charged nitrogen-vacancy (NV) center, a solid-state system that has served as a testbed for many protocols of quantum technology. We achieve low effective concentrations of defects by constructing dynamical matrices of large supercells containing tens of thousands of atoms. The main goal of the paper is to calculate luminescence and absorption lineshapes due to coupling to vibrational degrees of freedom. The coupling to symmetric a1 modes is computed via the Huang-Rhys theory. Importantly, to include a nontrivial contribution of e modes we develop an effective methodology to solve the multimode E - e Jahn-Teller problem. Our results show that for NV centers in diamond a proper treatment of e modes is particularly important for absorption. We obtain good agreement with experiment for both luminescence and absorption. Finally, the remaining shortcomings of the theoretical approach are critically reviewed. The presented theoretical approach will benefit identification and future studies of point defects in solids.

Tech Support

Original Source

DOI: https://doi.org/10.1103/physrevb.104.045303

References

2001 - Theory of Defects in Solids [Crossref]
2004 - Electronic Structure: Basic Theory and Practical Methods [Crossref]