Exciton Formation and Exchange Splitting

1. Spin States of an Electron-Hole Pair

In our previous discussions, we largely ignored the spin of the electron and hole. Let's now build our exciton model by including them. We have two spin-1/2 particles: the electron in the conduction band ($s_e=1/2$) and the hole in the valence band ($s_h=1/2$).

You might recall that a hole is the absence of an electron. The "hole spin" is a convenient and formally correct way to describe the properties of the partially filled valence band. If an electron with spin $m_s$ is removed from a filled band (which has total spin zero), the remaining system behaves as a particle with spin $-m_s$.

Combining two spin-1/2 particles is a standard exercise in quantum mechanics. The total spin $\mathbf{S}={\mathbf{s}}_e+{\mathbf{s}}_h$ can take on two possible values, according to the rules of angular momentum addition: $1/2\otimes 1/2=0\oplus 1$.

• Singlet State (Total Spin S=0): This state has a single magnetic sub-level ($M_S=0$) and an antisymmetric spin wavefunction. This is known as the para-exciton. $$|S=0,M_S=0⟩=\frac{1}{\sqrt{2}}\left(|{\uparrow }_e{\downarrow }_h⟩-|{\downarrow }_e{\uparrow }_h⟩\right)$$
• Triplet State (Total Spin S=1): This state has three magnetic sub-levels ($M_S=-1,0,+1$) and symmetric spin wavefunctions. These are the ortho-excitons. $$\begin{array}{rl}|S=1,M_S=1⟩& =|{\uparrow }_e{\uparrow }_h⟩\\ |S=1,M_S=0⟩& =\frac{1}{\sqrt{2}}\left(|{\uparrow }_e{\downarrow }_h⟩+|{\downarrow }_e{\uparrow }_h⟩\right)\\ |S=1,M_S=-1⟩& =|{\downarrow }_e{\downarrow }_h⟩\end{array}$$

In the absence of any spin-dependent interactions, these four states (one para, three ortho) are degenerate. This is where the exchange interaction comes into play.

2. Exchange Splitting of Spin States

The exchange Hamiltonian we derived, $H_{\text{exch}}$, acts on both the spatial and spin coordinates of the particles. While the underlying Coulomb potential is spin-independent, the antisymmetry requirement for the total wavefunction of the original many-electron system makes the exchange energy dependent on the relative spin orientation of the electron and hole.

For excitons with an s-like envelope function (orbital angular momentum L=0), the spatial part of the wavefunction is symmetric under particle exchange (${\mathbf{r}}_e\leftrightarrow {\mathbf{r}}_h$). To ensure the total exciton wavefunction (a proxy for the underlying fermionic system) has the correct symmetry, the spin part must be antisymmetric. This seems paradoxical until we remember we are dealing with an electron and a hole, not two electrons. The proper many-body treatment shows that the exchange interaction splits the states according to their total spin.

A very useful effective Hamiltonian for the exchange interaction, which captures this physics, can be written as:

where $J$ is an energy constant proportional to the exchange integral, which in turn depends on the overlap of the electron and hole wavefunctions, particularly $|F_{env}(0){|}^2$. Using the operator identity ${\mathbf{S}}_e\cdot {\mathbf{S}}_h=\frac{1}{2}({\mathbf{S}}^2-{\mathbf{S}}_e^2-{\mathbf{S}}_h^2)$, we can find the energy shift for our singlet and triplet states.

• For the singlet (para-exciton, S=0):
  The eigenvalue of ${\mathbf{S}}^2$ is $S(S+1){\hslash }^2=0$. The energy shift is:
  $\Delta E_S=\frac{J}{2}(0-\frac{3}{4}{\hslash }^2-\frac{3}{4}{\hslash }^2)=-\frac{3}{4}J{\hslash }^2$
• For the triplet (ortho-excitons, S=1):
  The eigenvalue of ${\mathbf{S}}^2$ is $1(1+1){\hslash }^2=2{\hslash }^2$. The energy shift is:
  $\Delta E_T=\frac{J}{2}(2{\hslash }^2-\frac{3}{4}{\hslash }^2-\frac{3}{4}{\hslash }^2)=+\frac{1}{4}J{\hslash }^2$

The total energy splitting between the triplet and singlet states is therefore:
$\Delta E_{T-S}=\Delta E_T-\Delta E_S=J{\hslash }^2$
Since the exchange integral $J$ is positive, the triplet ortho-exciton states are shifted to a higher energy than the singlet para-exciton state.

3. Optical Activity: "Bright" vs. "Dark" Excitons

This energy splitting has a profound consequence for the optical properties of the excitons. An optical transition (absorption or emission) involves a photon, which carries one unit of angular momentum. The selection rule for an electric dipole transition is $\Delta J=\pm 1$.

Let's assume the crystal ground state (no excitons) has zero total angular momentum, $J_{g.s.}=0$.

• Para-exciton (Singlet, S=0): For a 1s state, the orbital angular momentum is $L=0$. The total angular momentum is $J=L+S=0$. A transition from $J=0$ to the ground state $J_{g.s.}=0$ is a $0\to 0$ transition, which is strictly forbidden for a single photon process. Therefore, the para-exciton cannot be created or destroyed by a single photon. It is optically inactive, or "dark".
• Ortho-exciton (Triplet, S=1): Again for a 1s state, $L=0$. The total angular momentum is $J=L+S=1$. A transition from $J=1$ to the ground state $J_{g.s.}=0$ is allowed by the $\Delta J=\pm 1$ selection rule. Therefore, the ortho-exciton can couple to light. It is optically active, or "bright".

So, the general rule emerging from this simple model is: the exchange interaction splits the degenerate exciton states into a low-energy, dark singlet (para-exciton) and a higher-energy, bright triplet (ortho-exciton).

4. A Real-World Example: The Case of Cu₂O

This simple model provides a powerful framework, but the rich physics of real materials often adds fascinating complexity. Cuprous oxide (Cu₂O), which you are studying, is a classic example.

Experiments on Cu₂O find that the lowest-energy 1s "yellow" excitons are split into two states separated by about 12 meV. The lower state is identified as the para-exciton and the upper state as the ortho-exciton, consistent with our energy level predictions.

However, the optical properties are reversed! In Cu₂O, the lowest 1s ortho-exciton is dipole-forbidden and only very weakly active through a quadrupole transition, while the 1s para-exciton is even more strongly forbidden. What causes this departure from the simple model? The answer lies in the specific symmetries of the crystal lattice and the electronic bands involved.

The following paper by Kavoulakis et al. provides a clear explanation, making it an excellent bridge between our simplified model and the rigorous group-theoretical treatment we will undertake in the next lesson.

Let's summarize the key insights from the paper:

• The simple s_h = 1/2 hole spin model is an oversimplification. The hole states in the Γ⁺₇ valence band of Cu₂O are complex mixtures of orbital (d-like) and spin components.
• The paper provides the explicit total angular momentum wavefunctions for the triplet (ortho) and singlet (para) states in Equations (39)-(44).
• Due to the specific character of these wavefunctions, the exchange interaction (visualized as a virtual annihilation and re-creation of the exciton) has a non-zero matrix element only for the ortho-exciton states.
• This explains both the energy splitting (the ortho-exciton is pushed to higher energy) and the optical properties: the same mechanism that gives rise to the energy shift also makes the direct recombination of the para-exciton highly forbidden. The ortho-exciton transition is also forbidden in the dipole approximation because the conduction (Γ⁺₆) and valence (Γ⁺₇) bands have the same parity.

This detailed analysis of Cu₂O shows that while our simple spin model gives the right qualitative picture for the energy ordering, predicting optical activity requires a careful consideration of the specific symmetries of the crystal and the bands—a perfect motivation for our next topic.

Conclusion

In this lesson, we have connected the abstract concept of the exchange Hamiltonian to observable physical phenomena: the fine structure splitting and optical activity of excitons.

Key Takeaways:

• The exchange interaction lifts the degeneracy of the four spin states of an electron-hole pair.
• It splits them into a lower-energy para-exciton (singlet, S=0) and a higher-energy, triply degenerate ortho-exciton (triplet, S=1).
• In a simplified model, ortho-excitons are "bright" (optically active) while para-excitons are "dark" (optically inactive).
• In real materials like Cu₂O, the detailed band structure and crystal symmetry determine the precise nature of the exchange interaction and the optical selection rules, sometimes leading to outcomes (like forbidden ortho-exciton transitions) that differ from the simple model.

Preview of the Next Lesson

We've seen that understanding the "rules" in a specific material like Cu₂O requires us to talk about symmetries like Γ⁺₇, Γ⁺₅, etc. In the next lesson, we will formalize this by learning how to apply group theory and character tables to determine dipole-allowed and forbidden exciton transitions in crystals with cubic symmetry (e.g., Cu₂O). This will give us a powerful and systematic tool to predict which excitonic states can be observed in an optical experiment.