Exciton Fine Structure: Crystal Field and Spin-Orbit Coupling

Hello! Welcome back to our study of exciton physics.

In the last lesson, we successfully used group theory to establish the fundamental optical selection rules in Cu₂O. We saw that the crystal's Oₕ symmetry, combined with the parity of the electron/hole bands and the exciton envelope function, dictates whether a transition is allowed by the electric dipole operator. This explained why the P-series excitons are optically bright while the 1S exciton is dark.

Today, we're going to dig deeper. The band symmetries we've been using (like Γ⁺₆ and Γ⁺₇) don't just appear out of nowhere. They are the result of fundamental interactions within the crystal. Our goal is to describe how crystal field and spin-orbit coupling modify the exciton fine structure. We will investigate the physical origins of the band structure and see how these effects create the intricate set of energy levels that form the basis for excitonic states.

1. Spin-Orbit Coupling: From Atoms to Crystals

You'll recall from your quantum mechanics courses that spin-orbit coupling (SOC) in an atom arises from the interaction of an electron's spin magnetic moment with the magnetic field it experiences in its rest frame due to its motion through the nucleus's electric field. For a spherically symmetric potential $V (r)$ , this famously leads to an interaction term proportional to $L \cdot S$ .

However, in a crystal, the potential $V (r)$ is periodic, not spherically symmetric. Consequently, the orbital angular momentum $L$ is no longer a conserved quantity, and the $L \cdot S$ formulation is conceptually incorrect. We must start from a more general expression for the SOC Hamiltonian:

H_{so} = \frac{ℏ}{4 m _{0}^{2} c ^{2}} (\nabla V (r) \times p) \cdot σ

where $V (r)$ is the crystal potential. How do we solve this problem without recourse to $L$ ? The following paper provides a brilliant and rigorous explanation. It shows that by using only the symmetry properties of a cubic crystal, one can derive the spin-orbit split eigenstates without ever invoking $L$ or $J$ .

Spin-orbit coupling: atom versus semiconductor crystal

This paper, 'Spin-orbit coupling: atom versus semiconductor crystal' by Combescot et al., directly addresses the common but flawed practice of applying atomic physics concepts to crystals. It offers a clear, first-principles derivation that is essential for a deep understanding.

Please read the following sections: Section II (page 2): Focus on the general expression for the spin-orbit interaction (Eq. 1) and the origin of the Thomas factor of 1/2. Section IV (pages 5-6): This is the core section. Follow the derivation for semiconductor crystals. Note how it handles the periodic potential and, crucially, how the result is independent of the valence state's parity. Section V.B 'Physical understanding' (page 7): This provides a concise physical reason why the spin-orbit eigenstates in a cubic crystal end up having the same structure as in an atom, even though the underlying physics and symmetry are different. Your goal is to understand why we can talk about j=3/2-like and j=1/2-like states in a crystal, even though j is not a valid quantum number.

Key Results of Spin-Orbit Coupling

As the paper demonstrates, for a three-fold degenerate set of p-like orbital states in a cubic crystal, the spin-orbit interaction lifts the degeneracy in a way that is structurally identical to the atomic case.

Let's see this in the language of group theory. The valence band in many zinc-blende and cubic semiconductors (like Ge, or the bands in Cu₂O) is formed from states that have orbital symmetry transforming as a p-like representation (e.g., Γ⁺₅ in Oₕ). The electron spin (s=1/2) transforms as the Γ⁺₆ irrep. The combined system of orbital and spin degrees of freedom has a symmetry given by the direct product:

Γ_{orbital} \otimes Γ_{spin} = Γ_{5}^{+} \otimes Γ_{6}^{+} = Γ_{7}^{+} \oplus Γ_{8}^{+}

(You can verify this using the multiplication table from the previous lesson, found in Appendix B.3 of the Krueger dissertation).

This means the original six-fold degeneracy (3 orbital states × 2 spin states) is lifted by spin-orbit coupling into:

A four-fold degenerate set of states with Γ⁺₈ symmetry (analogous to $j = 3/2$ ).
A two-fold degenerate set of states with Γ⁺₇ symmetry (analogous to $j = 1/2$ ).

The energy difference between these is the spin-orbit splitting energy, $Δ_{so}$ . In many materials like GaAs, the Γ⁺₈ states form the heavy-hole and light-hole bands, while the Γ⁺₇ state is the lower-energy "split-off" band. In Cu₂O, the situation is inverted, and the highest valence band has Γ⁺₇ symmetry.

This splitting is a cornerstone of the fine structure of both the bands and the excitons built from them.

Spin-Orbit Coupling and Band Splitting in ABX3 Perovskites — A schematic diagram illustrating how spin-orbit coupling (SOC) lifts the degeneracy of a band. A single band (left) splits into two sub-bands with distinct spin character and an energy separation at k=0, modifying the electronic structure.

2. Crystal Field Splitting

Before spin-orbit coupling can act on the p-like crystal states, those states must first be formed from the atomic orbitals of the constituent atoms. This is where the crystal field comes in.

Crystal field splitting is the lifting of atomic orbital degeneracy by the electrostatic field of the surrounding ions in a crystal lattice. A free atom has spherical symmetry (group SO(3)), and its orbitals are degenerate (e.g., the five d-orbitals). When this atom is placed in a crystal, the symmetry is reduced to the crystal's point group (e.g., Oₕ for a cubic site).

The original irreducible representation of SO(3) corresponding to the atomic orbital (e.g., $L = 2$ for d-orbitals) becomes a reducible representation in the lower symmetry group. It splits into a direct sum of the new group's irreps.

The Krueger dissertation you have previously studied contains a Compatibility table that shows exactly how this works.

Rydberg excitons in external fields - RosDok

Let's return to F. Krüger's dissertation to see how group theory formally describes crystal field splitting. This section provides the direct mapping from atomic angular momentum states to crystal symmetry states.

Please review the following parts in Appendix B: Table B.4 in Section B.4 (page 110): This is the 'Compatibility table of Oₕ with respect to O(3)'. Look at the row for Γ₂ (which corresponds to L=2, or d-states). You will see it splits into Γ₃ ⊕ Γ₅. Note that parity must be added; since d-orbitals are even, the split states are Γ⁺₃ ⊕ Γ⁺₅. Section B.1 (page 104): Read the short text at the bottom. It explicitly states that in Cu₂O, the valence band is formed from copper 3d orbitals (which have L=2, Γ⁺₂ symmetry) and that the crystal field splits them, with one of the resulting Γ⁺₅ manifolds forming the valence band.

This establishes the full hierarchy of effects:

Free Atom: Start with degenerate atomic orbitals (e.g., Cu 3d-orbitals, $L = 2$ ).
Crystal Field: The cubic crystal field of Cu₂O reduces the symmetry from SO(3) to Oₕ, splitting the d-orbitals into manifolds with Γ⁺₃ and Γ⁺₅ symmetry.
Band Formation: These crystal-field-split states hybridize and form the energy bands. The highest valence band originates from the Γ⁺₅ manifold.
Spin-Orbit Coupling: This final interaction acts on the Γ⁺₅ band, splitting it into the Γ⁺₇ and Γ⁺₈ bands that we use to construct our excitons.

Test your understanding!

An impurity atom with a filled f-shell ( $L = 3$ ) is placed at a site with Oₕ symmetry in a crystal. According to the compatibility table (Table B.4 in Krueger's dissertation), what irreducible representations will the f-levels split into? (Assume even parity for simplicity).

Show answer

Looking at the row for Γ₃ ( $L = 3$ ) in Table B.4, we see that it splits as:
$Γ_{3} \to Γ_{2} \oplus Γ_{4} \oplus Γ_{5}$
Adding the even parity (+), the f-levels split into states with Γ⁺₂, Γ⁺₄, and Γ⁺₅ symmetries.

3. Modifying the Fine Structure with Strain

The fine structure we have described is characteristic of a perfect, unstrained cubic crystal. Any perturbation that reduces the crystal's symmetry will further modify this structure, lifting remaining degeneracies and mixing states. A common and experimentally relevant example is applying mechanical strain.

Applying uniaxial stress along a specific crystal axis lowers the symmetry. For example, stress along the [001] direction changes the symmetry from cubic (Oₕ) to tetragonal (D₄ₕ). The degeneracies that were protected by cubic symmetry may now be lifted.

The effect of strain can be described by an effective Hamiltonian, derived using the same group-theoretical "method of invariants" that is used in k·p theory.

Rydberg excitons in external fields - RosDok

This section of the Krueger dissertation describes how to mathematically model the effect of strain on the band structure. It formalizes the idea that strain acts as a symmetry-lowering perturbation.

Please read Section 4.2, 'Band structures of strained crystals' (pages 57-59). Focus on: The concept that strain modifies the crystal potential and can be treated as a perturbation. The form of the effective strain Hamiltonian (Eq. 4.23). Notice its structural similarity to the k-dependent terms in the k·p Hamiltonian—this is because the symmetric strain tensor \epsilon_{ij} transforms in the same way as the symmetric tensor k_i k_j. The rule that the new point group of the strained crystal consists of the symmetry elements that commute with the strain tensor.

This strain Hamiltonian adds new terms to the overall exciton Hamiltonian. When diagonalized, these terms will shift the energies of the exciton states and can split levels that were previously degenerate. For instance, the three-fold degenerate Γ⁺₅ ortho-exciton state could be split into a non-degenerate state and a two-fold degenerate state under uniaxial strain. This is a key technique used in spectroscopy to identify the symmetry of excitonic states.

Conclusion

In this lesson, we have peeled back another layer of complexity in the world of excitons, uncovering the origins of their fine structure.

Key Takeaways:

The fine structure of excitons is a result of a hierarchy of interactions: the crystal field, spin-orbit coupling, and electron-hole exchange.
Crystal field splitting arises because the non-spherical potential of the crystal lattice lifts the orbital degeneracy of the constituent atoms. This determines the fundamental symmetry of the electron and hole bands.
Spin-orbit coupling in crystals must be treated using the general $(\nabla V \times p) \cdot σ$ form, as $L$ is not a good quantum number. In cubic crystals, symmetry dictates that p-like bands split into $j = 3/2$ -like (e.g., Γ⁺₈) and $j = 1/2$ -like (e.g., Γ⁺₇) states.
External perturbations like mechanical strain reduce the crystal symmetry, acting as a modification to the crystal field. This can lift remaining degeneracies and provides a powerful tool for probing the symmetry of exciton states experimentally.

Preview of the Next Lesson

We have now assembled a detailed theoretical picture of the exciton energy level structure. The next logical step is to see how this structure manifests in experiments. In our next lesson, we will analyze experimental signatures of ortho- and para-excitons in absorption and photoluminescence spectra, connecting our theoretical framework directly to measurable quantities and learning how to interpret real spectroscopic data.

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