Vibrationally resolved optical excitations of the nitrogen-vacancy center in diamond

At a Glance

Metadata	Details
Publication Date	2022-11-15
Journal	npj Computational Materials
Authors	Yu Jin, Marco Govoni, Giulia Galli
Institutions	University of Chicago
Citations	39
Analysis	Full AI Review Included

Executive Summary

This analysis summarizes the first-principles investigation into the vibrationally resolved optical excitations of the Nitrogen-Vacancy (NV) center in diamond, focusing on the challenging singlet shelving states (¹E and ¹A₁).

Core Achievement: Developed and validated a general theoretical framework based on Spin-Flip Time-Dependent Density Functional Theory (SF-TDDFT) with analytical forces to accurately model the Potential Energy Surfaces (PESs) of highly correlated singlet excited states.
Spectral Prediction: Successfully predicted the infrared vibrationally resolved absorption spectrum (¹E -> ¹A₁ transition), achieving very good agreement with experimental measurements.
Key Phonon Identification: Revealed that the ¹E -> ¹A₁ transition is primarily driven by coupling with e type phonon modes, resulting in a main absorption peak at 73 meV and a sharp local mode peak at 170 meV.
Non-Adiabatic Coupling: Demonstrated the critical role of non-adiabatic interactions in determining the PES anharmonicity and increasing the phonon energies of the ¹A₁ state relative to the triplet ground state (³A₂).
Magnetometry Insight: Calculated a high Debye-Waller Factor (34%) for the ¹E -> ¹A₁ absorption, confirming that the Zero-Phonon Line (ZPL) is significantly more absorptive than the phonon sideband, making it better suited for infrared-absorption-based magnetometry.
General Applicability: The developed SF-TDDFT framework is robust and general, applicable to modeling the optical cycles and shelving states of a variety of other spin defects in semiconductors and insulators.

Technical Specifications

Parameter	Value	Unit	Context
Zero-Phonon Line (ZPL) Energy	1.19	eV	Experimental ZPL for ¹E -> ¹A₁ transition.
Main Absorption Peak Energy	73	meV	Calculated energy shift from ZPL (quasi-local e mode).
Sharp Absorption Peak Energy	170	meV	Calculated energy shift from ZPL (local e mode).
Debye-Waller Factor (DWF)	34	%	Calculated DWF for ¹E -> ¹A₁ absorption line shape.
Experimental DWF (Inferred)	~40	%	Inferred experimental DWF for ¹E -> ¹A₁ absorption.
Effective Phonon Energy (ħw_e)	63	meV	Parameter derived from fitting the effective Hamiltonian.
Energy Gap (A)	821	meV	Energy gap between ¹A₁⁽⁰⁾ and degenerate ¹E⁽⁰⁾ states (TDDFT@PBE).
Geometry Optimization Threshold	0.01	eV/A	Maximum nuclear force allowed during optimization.
Supercell Size (Dilute Limit)	12x12x12	Cells	Used for extrapolation, containing 13,824 atomic sites.
Calculation Temperature (Absorption)	10	K	Used for calculating the absorption line shape (matching experiment).

Key Methodologies

The theoretical framework relies on advanced first-principles calculations to accurately model excited state properties:

Electronic Structure Calculation:
- Ground state (³A₂) obtained using Density Functional Theory (DFT) with the semi-local PBE functional and the dielectric-dependent hybrid (DDH) functional.
- Excited states (³E, ¹E, ¹A₁) computed using Spin-Flip Time-Dependent DFT (SF-TDDFT) within the Tamm-Dancoff approximation (implemented in the WEST code).
Geometry Optimization and Forces:
- Equilibrium geometries for all excited states were obtained by minimizing nuclear forces calculated analytically using the Lagrangian formulation of TDDFT.
- This use of analytical forces is critical for accurately determining the complex, symmetry-breaking geometries of the singlet states (e.g., the “tricorn Mexican hat” PES of the ¹E state).
Phonon and Dilute Limit Extrapolation:
- Phonon modes were computed using the frozen phonon approach (PHONOPY package) at the PBE level.
- Results were extrapolated to the dilute limit using a large (12x12x12) supercell approximation (13,824 atoms) and a force constant matrix embedding approach.
Spectral Calculation:
- The vibrationally resolved absorption spectrum (¹E -> ¹A₁) was calculated using the Huang-Rhys (HR) theory and the generating function approach.
- HR factors were derived from the forces of the ¹A₁ state evaluated at the equilibrium geometry of the ¹E state, coupled with the phonon modes of the ¹A₁ state.
Non-Adiabatic Coupling Analysis:
- An effective Hamiltonian was defined to model the electron-phonon interaction and non-adiabatic coupling between the ¹A₁ and ¹E states, allowing for the calculation of quantized vibronic levels.

Commercial Applications

The accurate modeling of the NV center’s optical cycle, particularly the singlet shelving states, directly impacts the design and optimization of diamond-based quantum technologies.

Application Area	Specific Relevance to Research Findings
Quantum Sensing (Magnetometry)	The high calculated Debye-Waller Factor (34%) confirms that the ZPL of the ¹E -> ¹A₁ transition is the optimal target for infrared-absorption-based magnetometry, enabling improved sensitivity and signal-to-noise ratios.
Quantum Computing (Qubits)	Provides a robust, first-principles understanding of the shelving states critical for the initialization and readout of the NV center spin qubit, necessary for defining optimal control pulses.
Optical Pumping Schemes	Insights into electron-phonon coupling and non-adiabatic interactions allow for the optimization of optical pumping protocols used to polarize the NV spin state, potentially leading to faster or more efficient quantum operations.
Solid-State Spin Defect Engineering	The general SF-TDDFT framework is immediately applicable to predicting the optical properties and stability of other promising spin defects (e.g., SiV, GeV, SnV centers) in diamond and other wide-bandgap semiconductors.
Infrared Spectroscopy Calibration	Provides highly accurate theoretical spectral densities, which can be used to calibrate and interpret complex experimental infrared absorption measurements of diamond defects.

Tech Support

Original Source

DOI: https://doi.org/10.1038/s41524-022-00928-y