Phonon-Assisted Exciton Relaxation: Fermi's Golden Rule

Hello! Welcome to the next lesson in our exploration of exciton physics.

In our last lesson, we investigated the different pathways that lead to an exciton's ultimate demise: radiative and non-radiative recombination. We saw that phonons are deeply involved, either by enabling momentum conservation in indirect-gap materials or by carrying away energy in non-radiative processes.

Today, we shift our focus from the end of an exciton's life to the dynamics that occur during its life. An exciton is rarely created in its lowest possible energy state. It might be formed with significant kinetic energy or in an excited internal state (like n=2, n=3, etc.). Before it recombines, it must relax to lower energy states. The primary mechanism for this relaxation is the interaction with the crystal's vibrational modes—the phonons.

Our learning outcome for this lesson is to analyze phonon-assisted relaxation processes between exciton states using Fermi's golden rule. We will build the theoretical tools to calculate how quickly an exciton scatters from one state to another by emitting or absorbing a phonon.

The Physics of Exciton Relaxation

Imagine an exciton in an initial state $∣ i ⟩$ with energy $E_{i}$ . This could be a state with high kinetic energy or an excited internal state (e.g., $n = 2$ ). It can transition to a final state $∣ f ⟩$ with energy $E_{f}$ by interacting with a phonon.

If $E_{i} > E_{f}$ , the exciton relaxes by emitting a phonon of energy $ℏ ω_{q} = E_{i} - E_{f}$ .
If $E_{i} < E_{f}$ , the exciton can be excited to a higher energy state by absorbing a phonon of energy $ℏ ω_{q} = E_{f} - E_{i}$ .

The rate of these transitions is governed by Fermi's golden rule, a cornerstone of quantum mechanics that should be familiar to you.

Fermi's Golden Rule: Two-Level System — A visual and mathematical representation of Fermi's Golden Rule. The transition rate $\Gamma$ from an initial state $|\psi_i\rangle$ to a set of final states $|\psi_f\rangle$ is proportional to the squared matrix element of the perturbation Hamiltonian $V$ and the density of final states that conserve energy.

To calculate the scattering rate $Γ_{i \to f}$ , we need three ingredients:

The initial and final states, $∣ i ⟩$ and $∣ f ⟩$ .
The perturbation Hamiltonian, $H_{ex-ph}$ , which describes the exciton-phonon interaction.
The density of final states, which we account for by summing over all possible outcomes.

Let's build this framework step-by-step.

1. The Exciton-Phonon Interaction Hamiltonian ( $H_{ex-ph}$ )

The interaction between excitons and phonons arises because lattice vibrations distort the crystal, altering the electronic potential that the electron and hole experience. The two most important mechanisms are:

Deformation Potential Coupling: Caused by acoustic phonons. The compression and rarefaction of the lattice create a slowly varying potential that scatters the exciton. This is the dominant mechanism for scattering at low temperatures.
Fröhlich Coupling: Occurs in polar crystals (like GaAs) and is mediated by longitudinal optical (LO) phonons. The ionic vibrations create a long-range electric field that interacts strongly with the electron and hole. This is a very efficient mechanism for large energy relaxation.

We'll focus on the deformation potential coupling with acoustic phonons to illustrate the calculation. The derivation involves starting with the general electron-phonon Hamiltonian and expressing it in the basis of exciton states.

The following resource provides a clear, step-by-step derivation of the interaction Hamiltonian for a quasi-2D exciton.

DYNAMICS OF EXCITONS IN SEMICONDUCTOR ...

To understand the mathematical form of the interaction, let's turn to Chapter 5 of A. Thilagam's thesis, 'DYNAMICS OF EXCITONS IN SEMICONDUCTOR ...'. This section derives the exciton-phonon interaction operator for acoustic phonons in a quantum well.

Please read Section 5.3, 'Derivation of the Exciton-Phonon Interaction Operator for Acoustic phonons in Quasi-2D systems' (starts on pg. 96). Focus on the strategy: The starting point is the 3D electron-phonon Hamiltonian. The key step is to calculate the matrix element of this Hamiltonian between two quasi-2D exciton states (Equation 5.14). By using the explicit form of the exciton envelope function, a final expression for the interaction Hamiltonian is obtained (Equation 5.31). Note how it depends on the electron and hole deformation potentials (D_1 and D_0), the quantum well width (L), and phonon momentum (q_z).

The crucial result from this derivation is an operator, $H_{ex-ph}^{DF (2 D)}$ , that has the form:

H_{ex-ph} \propto K, K^{'}, q \sum M (q) ∣ S^{'}, K^{'} ⟩ ⟨ S, K ∣ (b_{q} + b_{- q}^{†}) δ_{K^{'}, K \mp q}

Here, $M (q)$ is the interaction matrix element, $∣ S, K ⟩$ represents an exciton in state $S$ with momentum $K$ , and $b^{†}, b$ are the phonon creation and annihilation operators. This operator annihilates an exciton in state $∣ S, K ⟩$ and a phonon with momentum $- q$ (or creates a phonon with momentum $q$ ), and creates an exciton in state $∣ S^{'}, K^{'} ⟩$ .

2. Calculating the Scattering Rate

With the Hamiltonian in hand, we can now apply Fermi's Golden Rule to find the total rate of scattering from an initial state $∣ i ⟩ = ∣ S, K ⟩$ to all possible final states. The total rate is the sum of rates for phonon absorption and emission.

Let's consider phonon emission. The initial state is $∣ S, K; n_{q} ⟩$ and the final state is $∣ S^{'}, K - q; n_{q} + 1 ⟩$ . The transition rate for scattering via one specific phonon mode $q$ is:

Γ_{q}^{emis} = \frac{2 π}{ℏ} ∣ ⟨ S^{'}, K - q; n_{q} + 1∣ H_{ex-ph} ∣ S, K; n_{q} ⟩ ∣^{2} δ (E_{S^{'}, K - q} + ℏ ω_{q} - E_{S, K})

The matrix element picks up a factor of $n_{q} + 1$ from the phonon creation operator. To get the total rate, we must sum over all possible phonon modes $q$ that the exciton can emit into.

The same logic applies to phonon absorption, but the matrix element is proportional to $n_{q}$ , and the process is only possible if thermal energy is available to provide a phonon to be absorbed.

Thilagam's thesis again provides a direct and accessible walkthrough of this calculation.

DYNAMICS OF EXCITONS IN SEMICONDUCTOR ...

Let's now see how the Hamiltonian is used to compute the actual scattering rates. Chapter 7 of the same thesis is dedicated to this calculation.

Please read Section 7.5, 'Transition Rates of Exciton Scattering' (starts on pg. 132). Pay attention to: The setup of the initial and final state energies (Equations 7.18a, 7.18b, 7.23). Note the inclusion of the exciton kinetic energy. The use of Fermi's rule (Equation 7.17) to write down the scattering rate. How the summation over final states is converted into an integral over the phonon momentum \mathbf{q}. The final expressions for the rates for scattering with emission and absorption of acoustic phonons (Equation 7.24 and 7.25). Notice how the rate depends on temperature through the phonon occupation number n_q \approx k_B T / \hbar \omega_q.

The key physics embedded in these equations is that the scattering rate is proportional to:

The square of the deformation potentials ( $(D_{c} - D_{v})^{2}$ ), which quantifies the coupling strength.
A form factor related to the exciton and quantum well wavefunctions.
The temperature $T$ , because the number of available acoustic phonons, $n_{q}$ , increases with $T$ .
The density of final states for the exciton and phonon.

Test your understanding!

An exciton is in the ground state ( $n = 1$ ) at the bottom of its dispersion curve ( $K = 0$ ). Can this exciton relax to an even lower energy state by emitting an acoustic phonon? What about by absorbing one? Explain the implications for the exciton lifetime at very low temperatures (T → 0 K).

Show answer

Phonon Emission: No. Since the exciton is already in the lowest possible energy state of its sub-band, there are no lower energy states to scatter into. Therefore, it cannot emit an acoustic phonon for intraband relaxation.
Phonon Absorption: Yes. By absorbing a phonon, the exciton can gain energy and momentum, scattering to a state with $K \neq = 0$ and higher kinetic energy.
Implications for T → 0 K: As the temperature approaches zero, the thermal population of phonons ( $n_{q}$ ) also goes to zero. Since phonon absorption requires an existing phonon, this scattering channel closes. As phonon emission is already forbidden from the ground state, an exciton at $K = 0$ becomes unable to scatter via acoustic phonons. Its lifetime is then limited only by other processes, such as radiative recombination or scattering from impurities, but not by phonon-assisted relaxation.

Connecting to a Modern View: The Exciton Self-Energy

In advanced many-body theory, these scattering processes are elegantly captured by the concept of the exciton self-energy, $Σ$ . The imaginary part of the self-energy is directly related to the total scattering rate out of a state, and thus to its lifetime, $τ$ .

\frac{1}{τ} = Γ_{scatter} = \frac{2}{ℏ} Im [Σ (E)]

The paper by Antonius and Louie provides a state-of-the-art derivation of this self-energy. You don't need to follow the derivation, but looking at their result is insightful.

Theory of exciton-phonon coupling

The paper 'Theory of exciton-phonon coupling' presents a rigorous first-principles approach to this problem. Let's look at their final expression for the imaginary part of the exciton-phonon self-energy.

Please locate Section III.B and find Equation (56). Compare its structure to the Fermi's Golden Rule expression. You should be able to identify: The squared matrix element, |g_{SS''\lambda}|^2, which is the coupling strength between initial exciton state S and final state S'' via a phonon of mode \lambda. The delta functions, \delta(\omega - \epsilon_{S''} \mp \omega_\lambda), which enforce energy conservation for phonon emission and absorption. The phonon occupation numbers, N_B(\omega_\lambda), which bring in the temperature dependence.

This shows that even in the most advanced formalisms, the underlying physics is precisely that of Fermi's Golden Rule: a transition rate determined by the coupling strength and the conservation of energy.

Summary and Conclusion

We have established the formal framework for analyzing how excitons relax by interacting with phonons. This is a critical piece of the puzzle of exciton dynamics, bridging the gap between their initial formation and their eventual recombination.

Key Takeaways:

Phonon-assisted scattering is the primary mechanism for excitons to relax in energy, both by changing their kinetic energy (intraband relaxation) and by moving between internal states (interband relaxation).
The rate of these processes is calculated using Fermi's Golden Rule, which requires the exciton-phonon interaction Hamiltonian and the initial/final state wavefunctions.
Acoustic phonons, interacting via the deformation potential, are responsible for thermalizing the exciton population within a sub-band. The scattering rate increases with temperature.
Optical phonons, interacting via the Fröhlich mechanism, provide a very fast relaxation pathway for excitons to cascade down between different internal energy levels (e.g., from n=2 to n=1).

Preview of the Next Lesson

These very same scattering processes that allow excitons to relax have a direct and measurable consequence on their optical properties. Each scattering event interrupts the phase coherence of the exciton's wavefunction. A higher scattering rate implies a shorter coherence time. According to the time-energy uncertainty principle, a shorter lifetime for a quantum state leads to a broader energy level.

In our next lesson, we will explore this effect. We will calculate the temperature-dependent homogeneous line broadening of an excitonic transition due to exciton-phonon scattering. This will connect the microscopic scattering rates we calculated today to the shape of the absorption and photoluminescence peaks observed in experiments.

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