Spin-Orbit Coupling

Learning Objectives

By the end of this section you should be able to:

1. Tell the story of where SOC comes from in two sentences — a moving electron sees the nuclear electric field as a magnetic field, and that field couples to the electron's own magnetic moment.
2. Write down the SOC Hamiltonian $\hat{H}_{\text{SOC}} = \xi(r)\,\mathbf{L}\cdot\mathbf{S}$ and use the identity $2\,\mathbf{L}\cdot\mathbf{S} = \mathbf{J}^2 - \mathbf{L}^2 - \mathbf{S}^2$ to predict the splittings of any LS-coupled atomic shell.
3. Read off the canonical p-shell partition: a 6-fold degenerate $p$ level breaks into a 4-fold $p_{3/2}$ at $+\xi/2$ and a 2-fold $p_{1/2}$ at $-\xi$, with fine-structure gap $\tfrac{3}{2}\xi$.
4. Trace the same partition into a band picture: the 6-fold p-derived valence top of a zincblende crystal becomes a 4-fold $\Gamma_8$ (HH + LH) and a 2-fold $\Gamma_7$ (split-off) separated by $\Delta_{\text{SO}}$.
5. Predict the order of magnitude of $\Delta_{\text{SO}}$ for a new material from a quick $Z^4$ argument and the chemistry of its anion.
6. Switch on SOC in VASP with the right INCAR tags and read $\Delta_{\text{SO}}$ straight out of vasprun.xml with a few lines of pymatgen.

One-line preview: SOC is the relativistic correction that turns spin from a passive label into an actor in the band Hamiltonian. Without it, every band has a spurious 2-fold spin degeneracy; with it, valence-band tops split, conduction bands twist, and quantum dots like Mn:CdSe acquire the precise emission energies we want them to have.

The Story — Why Should an Electron Care About Its Own Spin?

In the non-relativistic Schrödinger equation, spin is a spectator. You attach a 2-component spinor to every wavefunction, sum probabilities at the end, and never let $\hat{S}$ appear in the Hamiltonian. Every eigenstate is automatically Kramers-degenerate: spin-up and spin-down of the same orbital share an energy.

The Dirac equation tells a different story. When you reduce it to its non-relativistic limit (the Pauli equation) and keep terms of order $(v/c)^2$, three corrections fall out: the kinetic-energy correction, the Darwin term, and spin–orbit coupling. The first two shift levels rigidly. SOC is the only one that splits degeneracies.

The size of the effect is set by $(v/c)^2$. Inner-shell electrons of a heavy atom orbit at a respectable fraction of the speed of light — for an innermost $1s$ in lead, $v/c \sim Z\alpha \approx 0.6$, where $\alpha \approx 1/137$ is the fine-structure constant. SOC is therefore a property of the chemistry, and it grows spectacularly fast with atomic number.

The L·S Hamiltonian

From the Pauli reduction of Dirac, the SOC operator inside an atom is

$\hat{H}_{\text{SOC}} \;=\; \frac{1}{2 m_e^2 c^2}\,\frac{1}{r}\,\frac{dV(r)}{dr}\;\mathbf{L}\cdot \mathbf{S} \;\equiv\; \xi(r)\,\mathbf{L}\cdot\mathbf{S}.$

The function $\xi(r)$ is sharply peaked near the nucleus where $dV/dr$ is largest — SOC is overwhelmingly an atomic, near-nucleus phenomenon. Inside a crystal, the lattice is just a perturbation on top: the SOC matrix elements are dominated by single-atom integrals, and to a very good approximation

$\langle n l j \,|\, \xi(r) \,|\, n l j\rangle \;\equiv\; \xi_{nl}\quad \text{(an atomic constant)}.$

For a single open shell with orbital angular momentum $L$ and spin $S$, the operator $\mathbf{L}\cdot \mathbf{S}$ is diagonal in the coupled basis $|J, m_J\rangle$ via the simple algebraic identity

$2\,\mathbf{L}\cdot \mathbf{S} \;=\; \mathbf{J}^{2} - \mathbf{L}^{2} - \mathbf{S}^{2},$

with eigenvalues $J(J{+}1) - L(L{+}1) - S(S{+}1)$. The same one line generates all the textbook fine-structure formulas of atomic physics. For a p-electron ($L=1, S=1/2$):

Notice the bookkeeping: 4 + 2 = 6 (the original p-shell degeneracy survives) and the weighted sum $4(\xi/2) + 2(-\xi) = 0$ — SOC redistributes the levels but does not move their centre of mass. The fine-structure gap is

$\Delta_{\text{fs}} \;=\; E(p_{3/2}) - E(p_{1/2}) \;=\; \tfrac{3}{2}\,\xi.$

Interactive — Splitting a p-Shell

Drag the slider through ξ. Six degenerate ticks at zero break into the$p_{3/2}$ quadruplet at $+\xi/2$ (amber) and the $p_{1/2}$ doublet at $-\xi$ (cyan). The dashed grey line marks the unsplit centre of mass.

Two checks worth doing in your head as you drag:

1. Centre-of-mass conservation. The amber states sit at $+\xi/2$, the cyan ones at $-\xi$. Verify that $4(\xi/2) + 2(-\xi) = 0$ for any ξ. SOC never moves the average energy of a closed shell — it just spreads it.
2. Linear scaling. Both levels move linearly with ξ, and the gap $\tfrac{3}{2}\xi$ grows linearly too. SOC is a first-order effect on the atomic energies; everything that looks non-linear in a real material (e.g. how the conduction-band Bychkov–Rashba splitting evolves with strain) comes from how ξ couples to the surrounding orbital structure, not from the L·S operator itself.

The Z⁴ Scaling — Why Heavy Atoms Matter

The atomic SOC parameter $\xi_{nl}$ is the expectation value of $\xi(r)$ over a hydrogenic-like radial wavefunction. Plugging in the hydrogenic forms gives the textbook estimate

$\xi_{nl} \;\sim\; \frac{Z^4 \alpha^2}{n^3\,l(l+1/2)(l+1)}\,\text{Ry}.$

The headline message is the $Z^4$: doubling the nuclear charge multiplies SOC by sixteen. Real screening flattens the scaling somewhat — outer-shell electrons see an effective charge well below $Z$ — but the qualitative trend survives, and it is the only rule of thumb you need when judging whether SOC matters for a new material.

Two observations worth internalising:

• The anion dominates. For a II-VI semiconductor like ZnSe, CdSe, or CdTe, the valence-band top is built from anion-p orbitals. The SOC at the VBM scales with $\xi_p$ of the anion, not the cation. Replace Se by Te (Z = 34 → 52) and $\Delta_{\text{SO}}$ jumps from ≈ 0.42 eV to ≈ 0.94 eV.
• Z⁴ overshoots. The dashed reference curve, pinned at Si, predicts much larger splittings for heavy elements than what real materials show. Screening by valence electrons reduces the effective nuclear charge that the relevant p-orbital sees; self-consistent DFT-PAW calculations capture this naturally and land much closer to experiment.

From Atomic Splittings to Band Splittings

The same algebra carries straight into the solid. Consider the top of the valence band of a zincblende II-VI or III-V crystal (CdSe, GaAs, InP, ZnSe). At the Γ-point — the centre of the Brillouin zone — the crystal's point group is $T_d$. Without SOC, the valence-band top is derived from anion-p orbitals and transforms as the 3-dimensional $\Gamma_{15}$ representation of $T_d$. Including spin (which we have ignored until now) doubles every state, so the manifold is 6-fold degenerate.

Switching on $\hat{H}_{\text{SOC}}$ reduces the effective symmetry to the double group $T_d^*$, whose representations are labelled by half-integer J. The 6-D representation reduces as

$\Gamma_{15} \otimes \Gamma_{1/2} \;=\; \Gamma_{8}\;\oplus\;\Gamma_{7}.$

Translation: the 6-fold splits into a 4-fold $\Gamma_{8}$ (J = 3/2, the HH and LH bands) and a 2-fold $\Gamma_{7}$ (J = 1/2, the split-off band). It is the same $6 = 4 + 2$ partition you saw in the atomic splitter; only the names of the labels have changed.

Interactive — HH, LH, and the Split-Off Band

Drag $\Delta_{\text{SO}}$ from 0 (light-element limit) up to past 1 eV (heavy elements like CdTe, PbTe). At Γ, the amber circle is the $\Gamma_8$ manifold and the cyan circle is the $\Gamma_7$ split-off band. Away from Γ you can already see the curvature difference between the heavy and light holes — the heavy hole is the flatter band.

Things to verify visually:

1. At $\Delta_{\text{SO}} = 0$ the SO band (cyan) collapses onto the LH band (orange) — they share the same curvature, after all.
2. At Γ, HH and LH are exactly degenerate. The classic ring-shaped isoenergy surface near the VBM is born here: two bands of the same energy but different curvatures.
3. For CdSe ($\Delta_{\text{SO}} \approx 0.42$ eV), the split-off band sits well below the optical gap — visible photons couple only to HH/LH states. For HgTe, the SO splitting is large enough to invert the band order: the s-like Γ_6 actually falls below Γ_8, producing a topological insulator.

Where the SOC Lives in CdSe

SOC is an atomic effect glued onto the band structure by the chemistry. For CdSe in the zincblende phase below, the cation Cd (Z = 48) and the anion Se (Z = 34) both contribute, but the valence-band top is almost entirely Se 4p. To the bands at the gap, only the anion's ξ_p matters.

Bonus — Inversion Asymmetry, Rashba and Dresselhaus

Everything above lived at $\mathbf{k} = 0$. Away from Γ in a crystal with broken inversion symmetry, SOC produces a more subtle effect: a k-linear spin splitting of bands that would otherwise be Kramers-degenerate. Two famous flavours:

Both terms split each band into two spin-textured copies of energy$E_\pm(\mathbf{k}) = E_0(k) \pm |\boldsymbol{\Omega}(\mathbf{k})|$, where $\boldsymbol{\Omega}$ is an effective magnetic field that depends on $\mathbf{k}$. In Mn-doped CdSe, the Dresselhaus term is what eventually limits electron spin coherence — Section 5.8 will use this fact when we compute spin lifetimes for Mn impurity states.

VASP — Switching On Spin–Orbit Coupling

VASP turns SOC on as a single, inexpensive switch. The cost is that the calculation becomes non-collinear — every band is now a 2-component spinor, the spinor density requires complex arithmetic, and the basis size doubles. Plan for ~3–4× longer SCF cycles compared to a collinear spin-polarised run.

The three INCAR tags you actually need

1MAGMOM = 15*0 0 0   0 0 5     # 15 (Cd,Se) atoms ×  zero, then Mn = (0,0,5)

A complete CdSe SOC INCAR

1# INCAR — CdSe non-collinear band structure (step 2 of the recipe)
2SYSTEM   = CdSe zincblende, SOC, line-mode bands
3
4# Plane-wave + accuracy
5PREC     = Accurate
6ENCUT    = 400         # eV — set by ENCUT convergence test
7EDIFF    = 1E-6        # SCF tolerance
8
9# Smearing — semiconductor
10ISMEAR   = 0
11SIGMA    = 0.05
12
13# Spin–orbit coupling
14LSORBIT  = .TRUE.      # MASTER SWITCH
15SAXIS    = 0 0 1       # quantisation axis along z (cubic, take any)
16GGA_COMPAT = .FALSE.
17
18# Initial moments — three per atom; CdSe is non-magnetic
19MAGMOM   = 24*0  0 0   # 8 atoms (zincblende cell) × (0 0 0)
20
21# Read pre-converged charge density, do not update it
22ICHARG   = 11
23
24# Symmetry off (SOC + non-trivial SAXIS can clash with crystal symmetry)
25ISYM     = 0
26
27# Output the projections so we can colour the bands
28LORBIT   = 11
29
30NBANDS   = 64          # spinor count = 2 × old NBANDS — at least double it

VASP Walkthrough — Reading Δ_SO From the Output

After the non-collinear band run finishes, the eigenvalues at every k-point of the line-mode path live in vasprun.xml. The script below extracts the spin–orbit splitting at Γ and compares it to the experimental CdSe value of 0.42 eV. Every line is annotated — click any line of code on the right to see what is happening to the variables at that step.

1"""
2read_soc_bands.py
3=================
4Extract the spin-orbit splitting Delta_SO at Gamma from a VASP
5non-collinear band-structure calculation (LSORBIT = .TRUE.).
6
7Run AFTER the band-structure step (ICHARG = 11, line-mode KPOINTS).
8"""
9
10import numpy as np
11from pymatgen.io.vasp import BSVasprun
12from pymatgen.electronic_structure.core import Spin
13
14# 1.  Parse vasprun.xml (line-mode bands, with projections)
15vr = BSVasprun("vasprun.xml", parse_projected_eigen=True)
16bs = vr.get_band_structure(line_mode=True)
17
18# 2.  Locate the index of the Gamma k-point on the path
19labels = bs.kpoints
20gamma_idx = next(
21    i for i, k in enumerate(labels) if k.label in ("\\Gamma", "GAMMA", "G")
22)
23
24# 3.  In a non-collinear run, eigenvalues live under Spin.up only
25#     (each band is doubly Kramers-degenerate, so n_bands counts spinors)
26eigs = bs.bands[Spin.up][:, gamma_idx]   # shape: (n_bands,)
27
28# 4.  Find the valence-band maximum and the six p-like states beneath it
29vbm_band = bs.get_vbm()["band_index"][Spin.up][0]
30valence_six = np.sort(eigs[vbm_band - 5 : vbm_band + 1])[::-1]   # top 6
31
32# 5.  Top-4 = Gamma_8 (HH + LH degenerate at Gamma)
33#     Bottom-2 of these six = Gamma_7 (split-off)
34gamma8 = valence_six[:4].mean()
35gamma7 = valence_six[4:].mean()
36delta_so = gamma8 - gamma7
37
38print(f"Gamma_8 (HH + LH) = {gamma8:.4f} eV")
39print(f"Gamma_7 (SO)      = {gamma7:.4f} eV")
40print(f"Delta_SO          = {delta_so:.4f} eV   (CdSe expt: 0.42 eV)")

Summary

• Origin. SOC is the relativistic correction in the Pauli reduction of the Dirac equation. Physically, an electron orbiting a nucleus sees a magnetic field; its own magnetic moment couples to that field. The result is $\hat{H}_{\text{SOC}} = \xi(r)\,\mathbf{L}\cdot \mathbf{S}$.
• Master identity. $2\,\mathbf{L}\cdot\mathbf{S} = \mathbf{J}^2 - \mathbf{L}^2 - \mathbf{S}^2$. Every fine-structure splitting in atomic and condensed-matter physics is a one-line application of this identity.
• p-shell partition. 6 → 4 + 2: a degenerate p-shell breaks into a J = 3/2 quadruplet at +ξ/2 and a J = 1/2 doublet at −ξ. Gap = (3/2) ξ. Centre of mass preserved.
• In a crystal. The band-edge p-manifold of a zincblende semiconductor splits as Γ_15 ⊗ Γ_½ = Γ_8 ⊕ Γ_7. The Γ_8 quadruplet contains HH (m_J = ±3/2) and LH (m_J = ±1/2) bands, degenerate at Γ; Γ_7 is the split-off band, lower by Δ_SO.
• Z⁴ scaling. Atomic ξ_p grows as Z⁴ (screening softens this somewhat). For II-VI semiconductors, the anion sets Δ_SO at the VBM.
• Inversion-asymmetric crystals. Bulk inversion asymmetry (Dresselhaus) and structural inversion asymmetry (Rashba) lift the spin degeneracy of bands at finite $\mathbf{k}$. Necessary for spin textures, topological surface states, and finite spin-relaxation times.
• VASP. LSORBIT = .TRUE., set MAGMOM with three components per atom, choose SAXIS, turn ISYM off. Cost ~3–4× collinear; bands need to be doubled.
• Reading the output. In a non-collinear run all eigenvalues live under Spin.up. The six topmost valence states at Γ split into a 4-fold Γ_8 and a 2-fold Γ_7. Their average difference is Δ_SO.

Coming next: Section 5.8 — Defects and Doping — where we put a Mn atom on a Cd site, watch the d-shell of Mn split into $t_2$ and $e$ mid-gap states under the tetrahedral ligand field, and use SOC + crystal field together to predict the emission energy of a Mn:CdSe quantum dot.