Effect of external magnetic and electric fields and strains on diamond with negatively charged nitrogen vacancies

At a Glance

Metadata	Details
Publication Date	2025-06-03
Journal	Physical review. B./Physical review. B
Authors	A. A. Zvyagin, G. A. Zvyagina
Institutions	V. N. Karazin Kharkiv National University, National Academy of Sciences of Ukraine
Analysis	Full AI Review Included

Technical Documentation & Analysis: Magneto-Electro-Elastic Effects in Diamond $\text{NV}^{-}$ Centers

Executive Summary

This research analyzes the critical magneto-electro-elastic coupling effects in diamond containing negatively charged nitrogen vacancy ($\text{NV}^{-}$) centers, providing essential data for advanced quantum sensing and spintronics applications.

Core Mechanism: Analytical calculation of how external strains and electric fields couple with the spin degrees of freedom ($S=1$) of $\text{NV}^{-}$ centers, leading to measurable changes in magnetic, elastic, and electric properties.
Key Finding 1 (Magnetic Response): External strain or electric fields reduce the local $C_{3v}$ symmetry, causing the onset of magnetic moment projections perpendicular to the applied magnetic field.
Key Finding 2 (Elastic Renormalization): Spin quadrupole moments renormalize the elastic modules ($C_{11}, C_{44}, C_{66}$), showing temperature- and magnetic-field-dependent softening and hardening effects.
Key Finding 3 (Electric Anisotropy): The spin quadrupole moments induce anisotropy in the electric permittivity ($\epsilon$), which is dependent on external magnetic fields and temperature.
Application Relevance: These predicted effects are crucial for the precise determination of local magnetic, electric, and elastic characteristics when using diamond nanocrystals as high-sensitivity sensors in nanoelectronics and quantum computing.
6CCVD Value Proposition: 6CCVD provides the necessary high-purity Single Crystal Diamond (SCD) substrates with controlled nitrogen doping required to replicate and extend this research, ensuring long coherence times and stable $\text{NV}^{-}$ performance.

Technical Specifications

The following hard data points were extracted from the analytical calculations and experimental context cited in the paper, defining the material and operational parameters for $\text{NV}^{-}$ center research.

Parameter	Value	Unit	Context
$\text{NV}^{-}$ Spin State	$S = 1$	Dimensionless	Lowest energy term $\text{}^3\text{A}_2$
Crystalline Electric Field ($D$)	$0.1377$	K	Zero-field splitting parameter
g-Factor Components ($g_{x,y,z}$)	$\approx 2.0028$	Dimensionless	Close to free electron value (2.0023)
Hyperfine Interaction	$\approx 10^{-4}$	K	Negligible above this temperature range
Dipole-Dipole Interaction	$\approx 10^{-3}$	K	For vacancy separation distance of $1.5$ nm
Diamond Debye Temperature	$\approx 1900$	K	Reference for low-temperature strain analysis
Strain Value Used ($\epsilon$)	$0.001$	Dimensionless	Externally imposed parameter for calculation
Diamond Permittivity ($\epsilon$)	$5.7$	Dimensionless	Bulk value for diamond
Spin-Electric Coupling ($d_{\parallel}$)	$0.168 \times 10^{-6}$	K cm/V	Parallel component
Spin-Electric Coupling ($d_{\perp}$)	$8 \times 10^{-6}$	K cm/V	Perpendicular component

Key Strain Coupling Constants ($h_{ij}$):

Parameter	Value	Unit	Context
$h_{41}$	$-0.308$	K/strain	Density functional theory input
$h_{43}$	$0.110$	K/strain	Density functional theory input
$h_{15}$	$0.274$	K/strain	Density functional theory input
$h_{16}$	$-0.943$	K/strain	Density functional theory input
$h_{25}$	$-0.125$	K/strain	Density functional theory input
$h_{26}$	$-0.136$	K/strain	Density functional theory input

Key Methodologies

The research employed analytical calculations based on the theoretical framework of the $\text{NV}^{-}$ spin subsystem Hamiltonian ($H_s$) coupled with external fields and strains ($H_{se}$).

Hamiltonian Definition: The spin subsystem Hamiltonian ($H_s$) was defined using the crystalline electric field ($D$) and the Zeeman term ($\hat{g}\mu_B \mathbf{H} \cdot \mathbf{S}$), assuming $C_{3v}$ local symmetry along the [111] axis.
Spin-Elastic Coupling ($H_{se}$): The coupling between the elastic subsystem (strain $\epsilon_{ij}$) and the spin subsystem was introduced using Stevens equivalent operators ($O^2, O^2_2, \Omega^2_2$) and coupling constants ($h_{ij}$).
Spin-Electric Coupling ($H_{es}$): The interaction with the external electric field ($\mathbf{E}$) was introduced using spin-electric coupling parameters ($d_{\parallel}, d_{\perp}$).
Gibbs Distribution Calculation: Expectation values of spin projections and quadrupole operators were calculated using the Gibbs distribution as a function of temperature ($T$) and external magnetic field ($\mathbf{H}$).
Renormalization Calculation: The changes (renormalization) of the elastic modules ($\Delta C_{ij}$) and electric permittivity ($\Delta \epsilon$) were derived based on the interaction with the spin quadrupole moments, demonstrating dependence on $\mathbf{H}$ and $T$.
Parameter Input: Calculations utilized density functional theory (DFT) derived values for strain coupling constants ($h_{ij}$) and literature values for the zero-field splitting ($D$) and g-factors.

6CCVD Solutions & Capabilities

6CCVD is uniquely positioned to supply the high-specification diamond materials necessary to replicate, verify, and advance the magneto-electro-elastic research presented in this paper, particularly for applications in quantum sensing and spintronics.

Applicable Materials

The core requirement for this research is high-purity diamond with controlled nitrogen incorporation to create stable, high-coherence $\text{NV}^{-}$ centers.

Optical Grade Single Crystal Diamond (SCD):
- Requirement: Ultra-low strain and high purity are essential to maintain the intrinsic $C_{3v}$ symmetry and maximize the coherence time ($T_2^*$) of the $\text{NV}^{-}$ centers, crucial for high-sensitivity sensing.
- 6CCVD Solution: SCD plates grown via MPCVD, offering extremely low defect density and superior crystalline quality compared to natural or HPHT diamond.
Controlled Nitrogen Doping:
- Requirement: The predicted effects are linearly proportional to the concentration of $\text{NV}^{-}$ centers. Precise control over nitrogen concentration is necessary for predictable sensor performance.
- 6CCVD Solution: Custom SCD growth recipes allowing for precise, tunable nitrogen incorporation during the MPCVD process, enabling optimization of $\text{NV}^{-}$ density for specific sensing applications (e.g., high-density for bulk sensing or low-density for single-spin studies).
Polycrystalline Diamond (PCD) Substrates:
- Requirement: For large-area sensor arrays or integrated devices, large substrates are needed.
- 6CCVD Solution: PCD wafers available up to $125$mm diameter, providing a scalable platform for manufacturing diamond nanocrystals or large-scale integrated quantum devices.

Customization Potential

6CCVD’s in-house engineering and fabrication capabilities directly address the needs of advanced quantum material research:

Research Requirement	6CCVD Customization Capability	Technical Specification
Low Surface Strain/Noise	High-precision polishing services.	SCD Polishing: Ra < $1$nm
Integrated Nanoelectronics	Custom metalization services.	Au, Pt, Pd, Ti, W, Cu deposition
Specific Sensor Dimensions	Custom laser cutting and shaping.	Plates/wafers up to $125$mm (PCD)
Thickness Control	Precise layer growth control.	SCD/PCD Thickness: $0.1\mu$m to $500\mu$m
High-Quality Bulk Material	Thick substrate growth.	Substrates up to $10$mm thickness

Engineering Support

The analytical results show that even small changes in external strains and electric fields drastically affect the magnetic response of $\text{NV}^{-}$ centers. This complexity necessitates expert material selection.

Strain Management: 6CCVD’s in-house PhD team specializes in minimizing residual strain in MPCVD diamond, which is critical for maintaining the long coherence times required for high-fidelity quantum sensing.
Material Consultation: We offer comprehensive engineering support to assist researchers in selecting the optimal diamond grade (SCD vs. PCD), nitrogen concentration, and crystal orientation (e.g., fixed orientation [58] mentioned in the paper) for specific local magnetic, electric, and elastic sensing projects.
Global Logistics: We ensure reliable, global delivery of sensitive diamond materials (DDU default, DDP available) to facilitate international research collaborations.

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

View Original Abstract

Characteristics of magneto-electro-elastic effects in a diamond with negatively charged nitrogen vacancy centers are calculated analytically. It is predicted that strains or the external electric field can cause the onset of projections of the magnetic moment of those vacancies perpendicular to the applied magnetic field. It is shown how spins of those vacancies produce the anisotropy of the electric permittivity of a diamond, which depends on the external magnetic field and temperature. The temperature- and magnetic-field-dependent renormalization of elastic modules of a diamond caused by spin quadrupole moments of negatively charged nitrogen vacancy centers is calculated. The predicted effects can be important in the practical use of diamond nanocrystals with color vacancies as local sensors in modern nanoelectroncs.

Tech Support

Original Source

DOI: https://doi.org/10.1103/physrevb.111.214406

References

1984 - Electrodynamics of Continuous Media
2010 - Quantum Theory of One-Dimensional Spin Systems
1983 - Quantum Theory of Magnetism [Crossref]
1960 - Paramagnetic Resonance in Solids
1964 - Group Theory and Quantum Mechanics
2005 - Physical Acoustics in the Solid State [Crossref]