I-Engage UTA Summer 2026

This repository contains the source code developed for the I-Engage Summer 2026 project, "Comparative Study of Spin Hamiltonian due to Zero-Field Splitting and Zeeman Effect in Silicon Vacancy Defects Across 3C, 4H, and 6H-SiC", conducted under the mentorship of graduate student Sadia Rahman Jhilik and faculty sponsor Dr. Muhammad N. Huda.

The project provides a Python framework for constructing and analysing spin Hamiltonians of silicon-vacancy $V_{\text{Si}}$ defects in 3C-, 4H-, and 6H-SiC, computing their energy eigenvalues under an applied magnetic field, and reproducing published zero-field splitting (ZFS) parameters.

Author: Nafisa Nawrin Labonno
Contact: nafisanlab5@gmail.com

For questions, corrections, or suggestions, please feel free to reach out.

Table of Contents

1. Project Overview
2. Features
3. Physics Background
4. Literature Parameters
5. Repository Structure
6. Installation
7. Running the Project
8. Testing
9. References
10. Future Work

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Project Overview

Silicon carbide (SiC) is a wide-bandgap semiconductor that hosts optically addressable silicon-vacancy $V_{\text{Si}}$ spin defects with applications in quantum sensing, quantum communication, and quantum information processing.

Different SiC polytypes and defect configurations produce different zero-field splitting (ZFS) parameters, resulting in distinct spin-energy structures. This repository provides a unified Python framework for constructing spin Hamiltonians, computing energy eigenvalues under an applied magnetic field, and reproducing published literature values for multiple SiC polytypes and charge states.

The repository implements a unified computational framework for:

• constructing spin matrices for arbitrary spin quantum numbers;
• building Zero-Field Splitting (ZFS) and Zeeman Hamiltonians from published literature parameters;
• computing energy eigenvalues as a function of the applied magnetic field using numerical diagonalization;
• generating energy-level diagrams for 3C-, 4H-, and 6H-SiC; and
• validating simulated zero-field splittings against reported experimental values. :contentReference[oaicite:1]{index=1}

The implementation is modular, making it straightforward to compare different SiC polytypes, defect sites, and charge states within a single framework and to extend the code for future studies involving additional spin interactions.

Features

• General spin-matrix generation for arbitrary spin quantum number (S)
• Construction of Zero-Field Splitting (ZFS) Hamiltonians
• Construction of Zeeman Hamiltonians
• Eigenvalue computation over user-defined magnetic-field ranges
• Literature validation for multiple SiC polytypes
• Energy-level plots
• Modular code suitable for extension to additional defect systems (Work-In-Progress along with D & E parametrization)

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Physics Background

The spin Hamiltonian implemented in this repository consists of two physical contributions:

• Zero-Field Splitting (ZFS)
• Zeeman interaction

For a magnetic field applied along the principal $z$-axis:

$$H_{\mathrm{spin}} = D\left(S_z^2 - \frac{1}{3}S(S+1)\right) + \frac{E}{3}\left(S_x^2 - S_y^2\right) + g_e\mu_B B_0 S_z$$

where:

• $D$ — axial zero-field splitting parameter
• $E$ — rhombic zero-field splitting parameter
• $g_e$ — electron g-factor
• $\mu_B$ — Bohr magneton
• $S$ — total spin quantum number
• $B_0$ — externally applied magnetic field

For defects with approximately $C_{3v}$ symmetry (4H- and 6H-SiC sites), the rhombic parameter is negligible ($E \approx 0$), so the Hamiltonian reduces to its axial form. Lower-symmetry defect environments may require both $D$ and $E$ parameters.

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Literature Parameters

Polytype	Site	Spin	Symmetry	D (MHz)	E (MHz)
4H-SiC	h	1	(C_{3v})	65.9	0
6H-SiC	h	1	(C_{3v})	128.3	0
6H-SiC	k	1	(C_{3v})	26.9	0
3C-SiC	Neutral	1	Literature values	0.45	-0.09
3C-SiC	Charged	3/2	Literature values	0.10	-0.03

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Repository Structure

vsi-spin-hamiltonian/
├── README.md
├── requirements.txt
├── src/
│   ├── spin_matrices.py
│   ├── hamiltonian.py
│   ├── parameters.py
│   ├── compute_eigenvalues.py
│   └── plot_energy_levels.py
├── tests/
│   └── test_hamiltonian.py
├── notebooks/
│   └── exploration.ipynb
├── figures/
└── run_analysis.py

Source Files

File	Description
spin_matrices.py	Generates angular momentum matrices for arbitrary spin.
hamiltonian.py	Builds Zero-Field Splitting and Zeeman Hamiltonians.
parameters.py	Stores literature parameters for each polytype and charge state.
compute_eigenvalues.py	Computes eigenvalues as a function of magnetic field.
plot_energy_levels.py	Produces energy-level plots.
run_analysis.py	Main entry point for the project.

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Installation

Windows

python -m venv venv
venv\Scripts\activate
pip install -r requirements.txt

macOS

python3 -m venv venv
source venv/bin/activate
pip install -r requirements.txt

Linux

python3 -m venv venv
source venv/bin/activate
pip install -r requirements.txt

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Running the Project

Run the complete analysis:

python run_analysis.py

The program will

• construct the spin Hamiltonian,
• compute eigenvalues,
• compare against literature values, and
• generate energy-level plots in the figures/ directory.

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Testing

Execute the unit tests:

pytest tests/

The test suite verifies

• spin-matrix construction,
• Hamiltonian assembly,
• zero-field energy levels, and
• consistency with literature parameters.

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References

1. Soltamov et al., Room Temperature Coherent Spin Alignment of Silicon Vacancies in 4H- and 6H-SiC, Physical Review Letters, 108, 226402 (2012).

2. Christle et al., Isolated Electron Spins in Silicon Carbide with Millisecond Coherence Times, Nature Materials (2015).

3. Fazio et al., Stability and Decoherence Analysis of the Silicon Vacancy in 3C-SiC, Physical Review A, 109, 022603 (2024).

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Future Work

Potential extensions include

• Hyperfine interactions with nearby $$^{29}Si$$ and $$^{13}C$$ nuclei
• Arbitrary magnetic-field orientations
• Anisotropic $$g$$-tensor support
• Strain-dependent spin Hamiltonians
• Automatic fitting to experimental ODMR spectra
• Interactive visualization tools