This is a long post. On mobile some equations may go overfull.

The goal of this post is to elegantly program the Spin Hamiltonian governing recombination in Silicon Carbide. Our system involves two electrons and two nuclei.

We define our orthonormal basis as follows:

The ${\uparrow, \downarrow}$ spin basis is called the Zeeman basis. I define the basis with the two electrons coupled and the nuclei in the Zeeman Basis as the coupled basis. Every state in our two-electron + two-nuclei system is given by

This post is math heavy. I’ll try to walk through it all elegantly though. The important equations are boxed if you’d like to skip through. On mobile some equations may go overfull.

In Part 1 we derived the Zeeman Hamiltonian and the Hyperfine Hamiltonian. In Part 2 we derived the Zero-Field Splitting Hamiltonian analogously. The goal of this post is to derive the last sub-Hamiltonian for the Exchange Interaction and consequently derive the complete Spin Hamiltonian $\mathscr{H}$.

This post is math heavy. I’ll try to walk through it all elegantly though. The important equations are boxed if you’d like to skip through. On mobile some equations may go overfull.

In Part 1 we derived the Zeeman Hamiltonian and the Hyperfine Hamiltonian. The goal of this post is to provide a concise derivation of the next Hamiltonian term for Zero-Field Splitting. The next post will do the last Exchange Interaction term and finally give a complete Spin Hamiltonian description.

This post is math heavy. I’ll try to walk through it all elegantly though. The important equations are boxed if you’d like to skip through. On mobile some equations may go overfull.

The Spin Hamiltonian $\mathscr{H}$ governs the spin-physics of recombination. It’s what we need to program before starting to simulate EDMR.

$\mathscr{H}$ is a combination of the Zeeman Effect, Hyperfine Interactions, the Zero-Field Splitting Effect, and the Exchange Interaction.

My work revolves around Electrically Detected Magnetic Resonance (EDMR). It’s a method to detect small magnetic fields electrically. In principle it’s very simple.

Say we have a semiconductor, like silicon carbide. This semiconductor, made by nature or in a lab, is never perfect. There may exist some missing atoms, or extra atoms, deep in its molecular structure. These defects provide some extra electrons, or extra holes, which can be used for quantum sensing. In EDMR, they are used to measure magnetic fields electrically. I’ll now walk through how this is done in silicon carbide (4H-SiC) specifically.