Exciton Transitions in Cubic Crystals: A Group Theory Approach

1. The Language of Symmetry: Group Theory in Crystals

Your physics background has undoubtedly exposed you to group theory, particularly in quantum mechanics with rotation groups like SO(3). In solid-state physics, we apply the same principles to the discrete symmetries of a crystal lattice.

A crystal's structure is invariant under a set of symmetry operations (rotations, reflections, inversion), which form a mathematical group called a point group. For Cu₂O, which has a cubic lattice with an inversion center, the point group is Oₕ. The quantum states of any particle or quasiparticle within this crystal must transform according to the irreducible representations (irreps) of this group.

To familiarize yourself with the core concepts and terminology we'll be using, please start by reading an introductory section from a doctoral thesis on Rydberg excitons.

The Master Key: Selection Rules from Group Theory

The central principle for determining if a transition is allowed is elegantly simple. A transition from an initial state $|{\Psi }_i⟩$ to a final state $|{\Psi }_f⟩$ driven by an interaction operator $\hat{O}$ is allowed if, and only if, the matrix element $⟨{\Psi }_f|\hat{O}|{\Psi }_i⟩$ is non-zero.

In the language of group theory, this matrix element is guaranteed to be zero unless the final state's irrep, ${\Gamma }_f$, is contained within the direct product of the operator's irrep, ${\Gamma }_{\hat{O}}$, and the initial state's irrep, ${\Gamma }_i$:

If the initial state is the crystal ground state, which is always totally symmetric (${\Gamma }_i={\Gamma }_1^{+}$), the rule simplifies: a transition to state ${\Psi }_f$ is allowed if its irrep ${\Gamma }_f$ is the same as the operator's irrep ${\Gamma }_{\hat{O}}$.

Our task, therefore, breaks down into three steps:

1. Identify the irrep of the initial state (the crystal ground state).
2. Identify the irrep of the interaction operator (the photon).
3. Identify the irrep of the final state (the exciton).
4. Apply the selection rule.

2. Assigning Symmetries in Cu₂O (Point Group Oₕ)

We'll use the Oₕ character table as our guide. This table is the "dictionary" that translates physical functions and states into the language of irreps. We will use the Koster notation (e.g., Γ⁻₄) which is common in semiconductor physics literature, but the table also shows the equivalent Schoenflies notation (e.g., T₁ᵤ).

Step 1 & 2: Symmetry of the Ground State and the Photon

• Crystal Ground State ($|{\Psi }_i⟩$): The state with no excitons is the vacuum. It is invariant under all symmetry operations of the crystal. It therefore always belongs to the totally symmetric irrep, which is Γ⁺₁ (also A₁g).

• Interaction Operator ($\hat{O}$):

  • Electric Dipole Transition: The interaction operator is proportional to the position vector $\mathbf{r}=(x,y,z)$. Looking at the basis functions in the character table, we see that $(x,y,z)$ transforms as the Γ⁻₄ irrep (T₁ᵤ). The parity is odd, as indicated by the minus sign (or 'u' for ungerade).
  • Electric Quadrupole/Magnetic Dipole Transition: These higher-order transitions involve operators that transform like second-rank tensors, such as $xy,yz,zx$ or $x^2-y^2$. From the table, these functions correspond to the even-parity irreps Γ⁺₃ (E_g) and Γ⁺₅ (T₂g).

Step 3: Symmetry of the Exciton State ($|{\Psi }_f⟩$)

The total symmetry of the exciton is given by the direct product of the symmetries of its components:

where CB is the conduction band, VB is the valence band, and 'env' is the exciton envelope function.

• Band Symmetries in Cu₂O:

  • The lowest conduction band is formed from Cu 4s orbitals and has Γ⁺₆ symmetry.
  • The highest valence band originates from Cu 3d orbitals, split by spin-orbit coupling, and has Γ⁺₇ symmetry.
  • Crucially, both bands have even parity (indicated by the '+'). This means they are symmetric under inversion.

• Envelope Function Symmetry: This depends on the relative angular momentum ($L$) of the electron-hole pair.

  • s-like envelope (L=0): The wavefunction is spherically symmetric (like a hydrogen 1s orbital). It is invariant under all rotations and has even parity. It transforms as Γ⁺₁.
  • p-like envelope (L=1): The wavefunction has three components that transform like the vector $(x,y,z)$. It therefore has the symmetry of the dipole operator: Γ⁻₄. It has odd parity.

3. Applying the Selection Rules: Solving the Cu₂O Puzzle

Now we have all the pieces. We'll use the Oₕ multiplication table (found in Appendix B.3 of the Krueger dissertation you read, LINK) to evaluate the direct products.

Case 1: The 1s Yellow Exciton

1. Find the Exciton Symmetry:
   ${\Gamma }_{1s}={\Gamma }_{\text{CB}}\otimes {\Gamma }_{\text{VB}}\otimes {\Gamma }_{\text{env(s-like)}}$
   ${\Gamma }_{1s}={\Gamma }_6^{+}\otimes {\Gamma }_7^{+}\otimes {\Gamma }_1^{+}$

   From the multiplication table, ${\Gamma }_6^{+}\otimes {\Gamma }_7^{+}={\Gamma }_2^{+}\oplus {\Gamma }_5^{+}$.
   Since ${\Gamma }_1^{+}$ is the identity, the final symmetry is ${\Gamma }_{1s}={\Gamma }_2^{+}\oplus {\Gamma }_5^{+}$.
   (As we learned last time, ${\Gamma }_2^{+}$ corresponds to the para-exciton and ${\Gamma }_5^{+}$ to the ortho-exciton). Both states have even parity.

2. Apply Dipole Selection Rule:

   • Initial State: Γ⁺₁
   • Operator: Γ⁻₄
   • Allowed Final States: Γ⁻₄ ⊗ Γ⁺₁ = Γ⁻₄
   • Conclusion: The dipole operator has odd parity (Γ⁻₄), but the 1s exciton states have even parity (Γ⁺₂ and Γ⁺₅). A transition between states of the same parity is forbidden by an odd-parity operator. Therefore, the 1s exciton is dipole-forbidden.

3. Apply Quadrupole Selection Rule:

   • Initial State: Γ⁺₁
   • Operator: Γ⁺₃ ⊕ Γ⁺₅
   • Allowed Final States: (Γ⁺₃ ⊕ Γ⁺₅) ⊗ Γ⁺₁ = Γ⁺₃ ⊕ Γ⁺₅
   • Conclusion: The final state must have Γ⁺₃ or Γ⁺₅ symmetry. Our 1s ortho-exciton has Γ⁺₅ symmetry! Therefore, the 1s ortho-exciton transition is quadrupole-allowed. This explains why a very weak absorption line for this state can be measured. The 1s para-exciton (Γ⁺₂) remains forbidden.

Case 2: The 2p Yellow Exciton

1. Find the Exciton Symmetry:
   ${\Gamma }_{2p}={\Gamma }_{\text{CB}}\otimes {\Gamma }_{\text{VB}}\otimes {\Gamma }_{\text{env(p-like)}}$
   ${\Gamma }_{2p}=({\Gamma }_6^{+}\otimes {\Gamma }_7^{+})\otimes {\Gamma }_4^{-}=({\Gamma }_2^{+}\oplus {\Gamma }_5^{+})\otimes {\Gamma }_4^{-}$

   Using the multiplication table again:

   • ${\Gamma }_2^{+}\otimes {\Gamma }_4^{-}={\Gamma }_3^{-}$
   • ${\Gamma }_5^{+}\otimes {\Gamma }_4^{-}={\Gamma }_1^{-}\oplus {\Gamma }_3^{-}\oplus {\Gamma }_4^{-}\oplus {\Gamma }_5^{-}$

   So, the 2p excitons form a set of states with symmetries ${\Gamma }_1^{-},{\Gamma }_3^{-},{\Gamma }_4^{-},{\Gamma }_5^{-}$. All have odd parity.

2. Apply Dipole Selection Rule:

   • Initial State: Γ⁺₁
   • Operator: Γ⁻₄
   • Allowed Final States: Γ⁻₄
   • Conclusion: The manifold of 2p exciton states includes one with Γ⁻₄ symmetry. Therefore, the 2p exciton is dipole-allowed.

This perfectly explains the experimental absorption spectrum of Cu₂O, where one sees a series of strong peaks starting from n=2 (the P-series), but the n=1 line is conspicuously absent.

The following resource provides a practical walkthrough of this exact process for one-photon and two-photon transitions in Cu₂O.

Conclusion

Today, we have demystified the optical selection rules in Cu₂O by employing the powerful and rigorous framework of group theory. We've moved beyond simple spin models to a complete description that accounts for the crystal's symmetry.

Key Takeaways:

• Quantum states in a crystal are classified by the irreducible representations (irreps) of the crystal's point group.
• A transition is allowed if the final state's irrep is contained in the direct product of the operator's and initial state's irreps.
• In Cu₂O (Oₕ symmetry), the electric dipole operator has odd parity (Γ⁻₄). Since the valence and conduction bands both have even parity, an exciton's parity is determined solely by its envelope function.
• s-like excitons (1s, 2s, ...) have even-parity envelopes (Γ⁺₁), leading to even-parity exciton states. These are dipole-forbidden.
• p-like excitons (2p, 3p, ...) have odd-parity envelopes (Γ⁻₄), leading to odd-parity exciton states. These are dipole-allowed.
• The 1s ortho-exciton (Γ⁺₅) is weakly allowed via the electric quadrupole interaction (operator contains Γ⁺₅).

Preview of the Next Lesson

We have now established the "zero-order" energy levels and selection rules based on the crystal's symmetry. However, finer effects can further split these levels and modify their properties. In our next lesson, we will investigate these effects by learning how crystal field and spin-orbit coupling modify the exciton fine structure. This will add another layer of detail to our model, bringing us even closer to a complete picture of exciton physics in real materials.