Exciton Selection Rules: Parity & Angular Momentum

1. The Quantum Mechanical Origin of Selection Rules

At its heart, the absorption of a photon is a quantum transition from an initial state $|i⟩$ to a final state $|f⟩$. According to Fermi's golden rule, the rate of this transition is proportional to the square of the transition matrix element:

where ${\hat{H}}_{int}$ is the Hamiltonian describing the interaction between the electron system and the electromagnetic field of the light. For optical wavelengths, which are much larger than atomic or excitonic dimensions, we can use the electric dipole approximation. In this approximation, the interaction Hamiltonian simplifies to ${\hat{H}}_{int}=-e\mathbf{E}\cdot \mathbf{r}$, where $\mathbf{E}$ is the electric field vector and $e\mathbf{r}$ is the electric dipole operator. The matrix element becomes:

A transition is allowed if this matrix element is non-zero and forbidden if it is zero. Therefore, all the physics of selection rules is contained within the integral $⟨f|\mathbf{r}|i⟩$, known as the transition dipole moment.

To build intuition, let's first review how these principles lead to the well-known selection rules in atomic physics. The following video provides a clear and concise derivation.

2. The Parity Selection Rule

The video's derivation for $\Delta L=\pm 1$ is based on the properties of spherical harmonics. A more general and powerful way to frame the primary selection rule is through the concept of parity.

Parity refers to the behavior of a wavefunction under spatial inversion, $\mathbf{r}\to -\mathbf{r}$. The parity operator $\hat{\Pi }$ acts on a wavefunction as $\hat{\Pi }\psi (\mathbf{r})=\psi (-\mathbf{r})$. If $\psi (-\mathbf{r})=\psi (\mathbf{r})$, the state has even parity ($\pi =+1$). If $\psi (-\mathbf{r})=-\psi (\mathbf{r})$, the state has odd parity ($\pi =-1$). For example, atomic s-orbitals have even parity, while p-orbitals have odd parity.

Now, let's analyze the transition dipole moment integral $\int {\psi }_f^{\ast }(\mathbf{r})\mathbf{r}{\psi }_i(\mathbf{r})d^{3}r$ under a parity transformation. The volume element $d^{3}r$ is unchanged. The dipole operator $\mathbf{r}$ is an odd-parity operator, since $\mathbf{r}\to -\mathbf{r}$. For the integral to be non-zero, the entire integrand must have even parity. Let's see what this implies:

For the integrand to be even, this product must be $+1$. This requires ${\pi }_f\times {\pi }_i=-1$, which means the initial and final states must have opposite parity.

This leads to the fundamental electric dipole selection rule:
A transition is dipole-allowed only if it involves a change in parity.
(${\pi }_f\ne {\pi }_i$)

This rule is the parent of the $\Delta l=\pm 1$ rule for atoms, since changing the orbital angular momentum quantum number by an odd integer (e.g., from $l=0$ to $l=1$) flips the parity of the wavefunction.

The following document provides a concise, formal derivation of this rule in the context of semiconductors, starting from the interaction Hamiltonian you may have seen in your advanced QM or E&M courses.

3. Selection Rules for Excitons

Now, let's apply these rules to excitons. An optical transition creates an exciton from the crystal's ground state.

• Initial state $|i⟩$: The crystal ground state $|g⟩$, where all electrons fill the valence band. This state has even parity by definition, ${\pi }_i=+1$.
• Final state $|f⟩$: The exciton state $|{\Psi }_{exc}⟩$.

For the transition to be dipole-allowed, the exciton state must have odd parity: ${\pi }_f=-1$.

What determines the parity of the exciton? Within the envelope function approximation, the exciton wavefunction is a product:

The total parity of the exciton state is the product of the parities of its constituent parts:

1. Parity of the Envelope Function (${\pi }_{env}$): This is determined by its angular momentum, $l$. It has parity $(-1{)}^l$. So, s-states ($l=0$) are even, p-states ($l=1$) are odd, d-states ($l=2$) are even, and so on.
2. Parities of the Bloch Functions (${\pi }_c,{\pi }_v$): These are the parities of the periodic parts of the Bloch functions at the band edge (e.g., at the $\Gamma$-point). These are fixed properties of the material's crystal structure and electronic orbitals.

The total parity of the exciton state is ${\pi }_{exc}={\pi }_{env}\times {\pi }_c\times {\pi }_v$. The selection rule ${\pi }_{exc}=-1$ becomes:

This single equation elegantly explains the difference between materials like GaAs and Cu₂O.

Case 1: Allowed Interband Transition (e.g., GaAs)

In many direct-gap semiconductors like GaAs, the conduction band minimum is formed from s-like atomic orbitals (even parity, ${\pi }_c=+1$) and the valence band maximum is from p-like orbitals (odd parity, ${\pi }_v=-1$).
The product of band parities is ${\pi }_c\times {\pi }_v=(+1)\times (-1)=-1$.
Our selection rule becomes ${\pi }_{env}\times (-1)=-1$, which requires ${\pi }_{env}=+1$.
This means the envelope function must have even parity. The lowest-energy, strongest transitions will be to s-like exciton states ($l=0$), which is exactly what is observed.

Case 2: Forbidden Interband Transition (e.g., Cu₂O)

This is the case relevant to your research. In Cu₂O, due to its specific crystal symmetry, both the lowest conduction band (${\Gamma }_6^{+}$) and the highest valence band (${\Gamma }_7^{+}$) have even parity.
The product of band parities is ${\pi }_c\times {\pi }_v=(+1)\times (+1)=+1$.
Our selection rule becomes ${\pi }_{env}\times (+1)=-1$, which requires ${\pi }_{env}=-1$.
This means the envelope function must have odd parity. The allowed transitions are to p ($l=1$), f ($l=3$), etc., exciton states. The 1s exciton state is dipole-forbidden because its total parity is even. The lowest-energy allowed transition is to the 2p state.

This is precisely why the beautiful Rydberg series you saw in the previous lesson's paper on Cu₂O consists of P-excitons. Let's confirm this with a resource focused on Cu₂O.

Conclusion

Today, we have connected fundamental principles of quantum mechanics to the practical interpretation of exciton spectra. By starting with Fermi's golden rule and the dipole approximation, we derived the powerful selection rules based on parity and angular momentum conservation.

Key Takeaways:

• Optical selection rules originate from the symmetries of the states and the interaction operator involved in the transition integral $⟨f|\mathbf{r}|i⟩$.
• The primary electric dipole selection rule is that the initial and final states must have opposite parity.
• For an exciton created from the ground state, this means the exciton's total wavefunction must have odd parity.
• The exciton's parity is the product of the parities of the band-edge Bloch functions (${\pi }_c,{\pi }_v$) and the exciton envelope function (${\pi }_{env}$).
• This framework perfectly explains why materials with opposite-parity bands (like GaAs) exhibit s-exciton absorption, while materials with same-parity bands (like Cu₂O) exhibit p-exciton absorption.

Preview of the Next Lesson

So far, we have treated light as a perturbation that creates excitons. But what happens when the interaction between light and excitons is so strong that they can no longer be considered separate entities? In our next lesson, we will introduce the concept of exciton-polaritons and derive the polariton dispersion relation, entering the fascinating regime of strong light-matter coupling.