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, Australian National University
Citations	56
Analysis	Full AI Review Included

Executive Summary

This research presents a significant advancement in the theoretical modeling of optical properties for solid-state quantum defects, specifically the Nitrogen-Vacancy (NV^-) center in diamond.

Core Value Proposition: Developed a first-principles Density Functional Theory (DFT) methodology capable of accurately calculating luminescence and absorption lineshapes by fully incorporating complex electron-phonon coupling mechanisms.
Scale and Convergence: Achieved the dilute defect limit by implementing an embedding technique, allowing the calculation of converged electron-phonon spectral densities using massive supercells containing up to 64,000 atoms.
Jahn-Teller Solution: Introduced an efficient, computationally tractable algorithm to solve the multi-mode E ⊗ e dynamical Jahn-Teller (JT) problem, demonstrating that proper treatment of asymmetric e modes is essential for accurate absorption lineshapes.
Functional Performance: Confirmed that the PBE functional provides superior accuracy for predicting vibrational frequencies (lattice constants and bulk phonons), while the HSE functional yields a more accurate electronic excitation energy (Zero-Phonon Line, ZPL).
Experimental Agreement: Achieved excellent quantitative agreement with experimental luminescence and absorption spectra by scaling the PBE spectral densities by a factor of ζ = 1.2, compensating for slight underestimation of geometry relaxation.
Engineering Impact: The validated methodology provides a robust tool for the identification, characterization, and design of new quantum defects for applications in quantum technology.

Technical Specifications

Parameter	Value	Unit	Context
NV Center Symmetry	C_3v	Point Group	Defect structure in diamond lattice.
Electronic Ground State	³A₂	Triplet	Spin S = 1.
Electronic Excited State	³E	Triplet	Exhibits dynamical Jahn-Teller effect.
Experimental ZPL Energy	1.945	eV	Zero-Phonon Line energy (T = 0 K).
Calculated E₀ (HSE)	1.995	eV	Energy difference between equilibrium configurations.
Experimental Total Huang-Rhys Factor (S_tot)	3.49	Dimensionless	Quantifies electron-phonon coupling strength (emission).
Calculated S_tot (HSE Emission)	3.76	Dimensionless	Overestimated coupling strength.
Calculated S_tot (PBE Emission)	2.91	Dimensionless	Underestimated coupling strength.
e Mode Contribution Ratio (S_e/S_tot)	14 - 18	%	Contribution of asymmetric modes to total coupling.
Maximum Supercell Size	20 x 20 x 20	Cells	Used for converged spectral density (64,000 atoms).
Hessian Matrix Sparsity	99.5	%	Computational efficiency metric for large supercells.
Embedding Cutoff Radius (r_c1)	7	A	Distance from N site where forces are considered converged.
Diamond Band Gap (Experimental)	5.48	eV	Bulk diamond property.
Diamond Lattice Constant (PBE)	3.574	A	Calculated value (closer to experiment than HSE).

Key Methodologies

DFT Functional Selection: Calculations utilized the hybrid HSE functional (α = 1/4 screened Fock exchange) for electronic structure (ZPL energy) and the PBE functional for geometry optimization and vibrational properties (lattice constant, bulk modulus, phonon dispersion).
Excited State Modeling (ASCF): The ³E excited state was modeled using the ASCF (Average of Singlet Configurations) method, which constrains Kohn-Sham orbital occupations to simulate the mixed electronic state (a₁e^1.5e^1.5) while retaining C_3v symmetry for vibrational analysis.
Embedding Technique for Dilute Limit: A two-component Hessian matrix construction was used for large supercells: (1) elements near the defect (within r_c2 = 5.6 A of N or V) were taken from explicit DFT calculations, and (2) elements far from the defect were set to bulk diamond values.
Vibrational Mode Calculation: Dynamical matrices (Hessians) were calculated using the finite-difference approach on 4x4x4 supercells. Diagonalization for large supercells (up to 192,000 x 192,000) was performed using the spectrum slicing technique (SLEPc library).
a₁ Mode Coupling (Huang-Rhys Theory): Coupling to symmetric a₁ modes was calculated using the equal-mode approximation. The partial Huang-Rhys factors (S_k) were derived from the calculated force (F_α) induced by the electronic transition projected onto the normal modes (Q_k).
e Mode Coupling (Multi-mode Jahn-Teller): The complex E ⊗ e JT problem was solved by transforming the Hamiltonian into a basis of “chiral” phonons, which separates the diagonalization problem into independent angular momentum channels (j).
Effective Mode Approximation: To handle the infinite number of e modes participating in the JT effect, the actual spectral density S_e(ħω) was approximated by a sum of N_eff effective Gaussian modes, where N_eff was optimized to minimize the integral difference between the actual and effective spectral densities.

Commercial Applications

Quantum Computing and Memory: NV centers are leading solid-state qubits. Accurate vibronic modeling is crucial for optimizing the optical interfaces used for spin initialization, readout, and entanglement generation, directly impacting qubit fidelity and operational speed.
Nanoscale Quantum Sensing: NV centers are used as highly sensitive sensors for magnetic fields, electric fields, and temperature. The precise calculation of optical lineshapes allows for better calibration and interpretation of spectral shifts observed during sensing operations.
Quantum Communication Networks: The NV center’s stable optical transition makes it a candidate for quantum repeaters and network nodes. The methodology supports the design of defects with narrow, stable Zero-Phonon Lines for efficient photon-spin coupling.
New Quantum Defect Discovery: The computational framework is generalizable, enabling rapid and quantitative prediction of optical signatures for newly proposed or synthesized point defects in materials like SiC, ZnO, and GaN, accelerating the search for next-generation quantum materials.
Advanced Materials Characterization: Provides a high-fidelity tool for materials scientists to understand how lattice vibrations (phonons) interact with electronic states in doped semiconductors and insulators, guiding materials synthesis and processing.

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]