Ortho- and Para-Excitons in Absorption and Photoluminescence

1. Signatures in Absorption Spectra

An absorption spectrum measures the photons that are removed from a beam as it passes through a material, promoting electrons to excited states. Let's see how ortho- and para-excitons manifest here.

Direct and Indirect Absorption

As we established using group theory, the ground state to 1S paraexciton transition is strictly forbidden for single-photon absorption due to spin conservation. The 1S orthoexciton transition is electric-dipole forbidden but weakly electric-quadrupole allowed in Cu₂O.

The most prominent feature in the absorption spectrum of a high-quality semiconductor like Cu₂O is not the ground state exciton, but the Rydberg series of higher-energy P-states (2P, 3P, etc.). These transitions are dipole-allowed and thus much stronger.

So, how can we "see" the dark paraexciton?

1. Phonon-Assisted Absorption: A transition can become weakly allowed if the crystal lattice participates by creating or annihilating a phonon. This process creates broad, continuous absorption features starting at $E_{para}+\hslash {\Omega }_{phonon}$, but it's an indirect signature.
2. Induced Absorption (Pump-Probe Spectroscopy): A much more powerful and direct method is to probe the absorption from an excited population. In this technique, a "pump" laser pulse first creates a population of excitons. Many of the initially-created, higher-energy excitons and orthoexcitons will quickly relax into the lower-energy, long-lived 1S paraexciton state. A second, delayed "probe" pulse then measures the sample's absorption. This probe beam can be absorbed by the existing paraexcitons, promoting them to higher excitonic states (e.g., from 1S-para to 2P-para). Since this is an s→p transition, it is dipole-allowed and produces a strong signal.

The following resource provides a good overview of the theory behind induced absorption and presents experimental data.

This technique results in a "differential transmission" signal ($-\Delta T/T$), which directly maps the population of paraexcitons. The image below shows such spectra for Cu₂O at various temperatures. You can see distinct absorption features corresponding to transitions originating from the populated paraexciton level.

2. Signatures in Photoluminescence (PL) Spectra

Photoluminescence (PL) measures the light emitted when excited states relax back down. It's the inverse process of absorption and is exceptionally sensitive to the nature of the emitting state.

Direct vs. Phonon-Assisted Recombination

• Orthoexciton PL: The 1S orthoexciton can decay radiatively via its weak, quadrupole-allowed channel. This produces a sharp, but often faint, emission line right at the orthoexciton energy, often labeled $X_o$.
• Paraexciton PL: The 1S paraexciton is dark. Its direct radiative decay is forbidden. Therefore, you will not see a sharp emission line at the paraexciton energy.

The dominant PL mechanism, especially in indirect-gap materials but also crucial in Cu₂O, is phonon-assisted recombination. Here, an exciton recombines by emitting both a photon and a phonon. This process relaxes the strict $\mathbf{k}$-selection and spin-selection rules, allowing even dark excitons to emit light.

This gives rise to broad emission bands in the PL spectrum:

• Stokes Line: A band shifted to lower energy than the direct line by one phonon energy ($E_{photon}=E_{ex}-\hslash \Omega$).
• Anti-Stokes Line: A band shifted to higher energy ($E_{photon}=E_{ex}+\hslash \Omega$), requiring thermal energy to absorb a phonon from the lattice.

To understand how these features appear in a real spectrum, please read the following section.

Lifetime Differences

The most dramatic signature distinguishing ortho- and para-excitons is their lifetime.

• Orthoexcitons have multiple fast decay channels: direct radiative recombination (ns scale), non-radiative decay, and—most importantly—fast relaxation to the lower-energy paraexciton state. Their lifetime is typically in the nanosecond range.
• Paraexcitons, unable to decay radiatively, are very long-lived. Their lifetime is limited only by slower processes like phonon-assisted recombination, trapping at defects, or non-radiative recombination. Lifetimes can be in the microsecond to millisecond range.

Time-Resolved PL (TRPL) can beautifully disentangle these. Immediately after excitation, you see PL from the short-lived orthoexcitons. At later times, this signal vanishes, and all that remains is the slow, weak, phonon-assisted PL from the vast population of long-lived paraexcitons. The resource below shows this dynamic explicitly.

3. Advanced Case Study: Dark-to-Bright Conversion in Cu₂O

Selection rules are not absolute laws; they can be bypassed. A fascinating experiment in Cu₂O demonstrates this by using two-photon absorption (TPA) to selectively populate "dark" exciton states and then observing "bright" luminescence. This is a perfect illustration of how different experimental geometries and techniques can be combined to reveal subtle physics.

As the paper explains, even though the laser exclusively creates dark excitons, these can scatter and convert into bright orthoexciton states. It is these subsequently populated bright states that produce the sharp $X_o$ PL line. The experimenters confirm this by analyzing the polarization of the emitted light (Figure 4a), which matches the one-photon selection rules for bright states, not the two-photon rules used for excitation.

This experiment beautifully demonstrates that excitons are not static entities. They are dynamic, and they can change their character through scattering and relaxation processes—a perfect lead-in to our next topic.

Conclusion

In this lesson, we have bridged the gap between the abstract group theory of exciton fine structure and the concrete world of spectroscopy.

Key Takeaways:

• Ortho- and para-excitons leave distinct fingerprints in spectra. Dark paraexcitons are typically invisible in direct absorption and emission.
• Absorption Signatures: Paraexcitons are best observed via induced absorption (pump-probe), where they absorb light to transition to higher, p-like states.
• PL Signatures: The key signatures in PL are the sharp, direct orthoexciton line ($X_o$), the broad, phonon-assisted sidebands (from which both ortho and para PL can be seen), and the stark difference in lifetimes (ns for ortho vs. µs-ms for para).
• A combination of spectroscopic techniques—steady-state absorption/PL, time-resolved PL, and pump-probe—is essential to build a complete picture of the exciton states and their dynamics.

Preview of the Next Lesson

We've seen that excitons are not static; they relax, convert, and recombine. We've talked about their lifetimes, but what happens during that lifetime? How do they move? In the next lesson, we will begin our exploration of exciton dynamics by learning to model exciton transport using a drift-diffusion equation and relate mobility to momentum relaxation time. This will be our first step in understanding how excitons propagate and interact within the crystal.