# Spin orbit coupling from spin polarized density of states?

From what I read online, spin orbit coupling is how the angular momentum of an electron w.r.t. the nucleus interacts with its spin. The word 'relativistic' came up every time. My quantum-dot sized brain already heats up at the mention of 'quantum mechanics'. What is spin orbit coupling (soc) and what does soc strength mean?

I was told:

$$\begin{equation} \textrm{soc}\:\textrm{strength} = \textrm{DoS}_{(\textrm{spin-up})_{E_F}} -\: \textrm{DoS}_{(\textrm{spin-down})_{E_F}} \end{equation}$$ where $$E_F$$ = Fermi energy.

Questions on this SE (here and here) suggest soc calculations are an entirely different class of calculations that build upon a spin polarised calculation.

• +1. This reached 100 views despite never becoming a HNQ, that's quite rare! Clearly the posts on FB groups helped to reach 100 views, but now I wonder how a question with so many upvotes on both the Q and the A, and 100 views, didn't make it to the HNQ list! Oct 22, 2020 at 21:19

I'll try to be as basic as I can in regard to explaining the stuff youve posted.

From what I read online, spin-orbit coupling is how the angular momentum of an electron w.r.t. the nucleus interacts with its spin.

Yes, and there are two types interactions Russell Saunders coupling(LS coupling) and the j-j coupling. The electron has an orbital angular momentum(let's call it L) as it revolves around the nuclei and a spin angular momentum (Let's call it S) (This is where the relativistic stuff comes in !!. The spin is not a natural consequence of the Schrodinger equation, it's a consequence of the relativistic Dirac equation)

LS coupling: This happens when the L and S of an individual electron interact strongly with the L and S of another electron respectively (i.e, L1 of electron 1 interacts with L2 of electron 2 and S1 & S2 does the same thing. But here L1 and S1 wont interact strongly). Then the total L and the total S interacts to give you the J value (which is the total angular momentum).

J-J coupling: Here the L1 and S1 of electron 1 interact to give you j1 and then the j1, j2, j3.. etc of electrons 1 ,2 ,3 respectively interact together to give you the total J value. In such cases, the total L and the total S are not explicitly mentioned.

The word 'relativistic' came up every time.

Thats because the spin is a relativistic concept as its obtained from the Dirac equation.

What is spin orbit coupling (soc) and what does soc strength mean?

You could think of SOC as the interaction of the particles spin with its motion in the presence of a potential. And about SOC strength a rough estimate would be the atomic number of the element.

$$\textrm{soc}\:\textrm{strength} \; \alpha \; Z^{4}$$

That's why SOC is mostly seen in heavy elements.

$$\begin{equation} \textrm{soc}\:\textrm{strength} = \textrm{DoS}_{(\textrm{spin-up})_{E_F}} -\: \textrm{DoS}_{(\textrm{spin-down})_{E_F}} \end{equation}$$

Another method :

You can study SOC if you could simulate the Electron paramagnetic resonance(EPR) spectra. If the EPR spectra can be simulated. Then we can extract important Hamiltonian parameters such as D and E which are the zero-field splitting terms. D and E quantify the extent of zero-field splitting.

$$\zeta = \; \textrm{spin orbit coupling constant}$$ $$D = \frac{\zeta}{6[g_{zz} - \frac{1}{2(g_{xx} + g_{yy})}]}$$ $$E = \frac{\zeta}{12(g_{xx}-g_{yy})}$$

Thus, EPR spectra can be used to extract the spin-orbit coupling parameter provided that your material is EPR active

• Thank you for the simplification. I'll look further into EPR. Oct 21, 2020 at 22:20
• +1. But please use \textrm{spin orbit coupling constant} to get non-math text out of math mode (same for soc-strength, spin-up, etc.) Oct 21, 2020 at 22:51

What is spin-orbit coupling (soc) and what does soc strength mean?

The spin-orbital coupling (SOC) is a relativistic effect. Mathematically, it can be represented as: $$\vec{L} \cdot \vec{S}$$ in which $$\vec{L}$$ is orbital angular momentum and $$\vec{S}$$ is spin angular momentum. How to identify the strength of SOC?

• Taking the Hamiltonian without the consideration of SOC as $$H$$, you can always obtain the energy dispersion as $$E_1(\vec{k})$$.
• Taking the $$H$$ and SOC term $$\vec{L}\cdot \vec{S}$$ at the same time, you will obtain a new energy dispersion $$E_2(\vec{k})$$.
• Compare the $$E_1(\vec{k})$$ and $$E_2({\vec{k}})$$ and find the difference. Usually, you will find the band splitting due to SOC. If the splitting is large, you can think SOC is strong.
• Thank you, Jack. In VASP output for an SoC calculation, they write the energy of SoC for each ion (see here: vasp.at/wiki/index.php/LSORBIT), whereas your answer is about energies at each $\vec{k}$. Oct 24, 2020 at 10:39
• Yes, I am considering the effect of SOC on the whole electronic structure. Inversion symmetry breaking with SOC in materials will lead to the band splitting, we can identify the strength of SOC in terms of the splitting.
– Jack
Oct 25, 2020 at 11:16