Exciton Oscillator Strength: Envelope Function Approach

Hello! In our last lesson, we derived the optical absorption coefficient for creating an exciton from the semiconductor ground state. We found that the absorption strength is determined by the interplay between the fundamental interband transition probability and the spatial arrangement of the electron and hole within the exciton.

The final expression we derived was:

α (ω) = \frac{4 π ^{2} e ^{2}}{n _{r} c ϵ _{0} m _{0}^{2} ω} ∣ ϵ \cdot p_{c v} ∣^{2} λ \sum ∣ ϕ_{λ} (0) ∣^{2} g (ℏ ω - E_{λ})

Today, we will build directly on this result to meet the learning outcome: to evaluate the oscillator strength for exciton transitions using the envelope function approximation. We will bundle the complex, material-dependent terms of the absorption formula into a single, dimensionless quantity—the oscillator strength. This provides a powerful and convenient way to quantify and compare the "strength" of different optical transitions.

1. Defining the Oscillator Strength

The concept of oscillator strength originated in classical electromagnetism to describe the absorption of light by an electron bound to an atom, modeled as a simple harmonic oscillator. In quantum mechanics, it is generalized to provide a dimensionless measure of the transition probability between two states.

For a transition from an initial state $∣ i ⟩$ to a final state $∣ f ⟩$ , the oscillator strength $f_{i f}$ is defined in terms of the momentum matrix element as:

f_{i f} = \frac{2}{m _{0} ( E _{f} - E _{i} )} ∣ ⟨ f ∣ ϵ \cdot p ∣ i ⟩ ∣^{2}

Here, $m_{0}$ is the free electron mass, $E_{f} - E_{i}$ is the transition energy, $p$ is the momentum operator, and $ϵ$ is the light polarization vector. This definition provides a direct link to our expression for the absorption coefficient.

2. Oscillator Strength of an Exciton Transition

Let's apply this definition to our process of interest: the creation of an exciton from the crystal ground state.

Initial state $∣ i ⟩$ : The crystal ground state, $∣0 ⟩$ , with energy $E_{i} = 0$ .
Final state $∣ f ⟩$ : The exciton state $∣ Ψ_{e x}^{λ} ⟩$ with energy $E_{f} = E_{λ}$ , which is approximately the bandgap energy $E_{g}$ .
Matrix element: In the last lesson, we found this to be $⟨ Ψ_{e x}^{λ, 0} ∣ ϵ \cdot p ∣0 ⟩ = (ϵ \cdot p_{c v}) V ϕ_{λ}^{*} (0)$ .

Substituting these into the definition gives the oscillator strength for creating an exciton of state $λ$ in a crystal of volume $V$ :

f_{e x, λ} = \frac{2}{m _{0} E _{λ}} ∣ (ϵ \cdot p_{c v}) V ϕ_{λ}^{*} (0) ∣^{2} = \frac{2∣ ϵ \cdot p _{c v} ∣ ^{2}}{m _{0} E _{λ}} V ∣ ϕ_{λ} (0) ∣^{2}

This quantity represents the oscillator strength of the entire crystal. A more useful, intensive quantity is the oscillator strength per unit cell. Dividing by the number of unit cells in the crystal, $N = V / V_{ce l l}$ , we get:

f_{e x, λ}^{per cell} = \frac{2∣ ϵ \cdot p _{c v} ∣ ^{2}}{m _{0} E _{λ}} V_{ce l l} ∣ ϕ_{λ} (0) ∣^{2}

This is the central result connecting the microscopic properties of the semiconductor ( $p_{c v}$ , $V_{ce l l}$ ) and the exciton ( $E_{λ}$ , $ϕ_{λ} (0)$ ) to the dimensionless oscillator strength.

It's insightful to relate this to the oscillator strength of the underlying band-to-band transition, $f_{c v} = \frac{2∣ ϵ \cdot p _{c v} ∣ ^{2}}{m _{0} E _{g}}$ . Since $E_{λ} \approx E_{g}$ , we find a beautifully simple relationship:

f_{e x, λ}^{per cell} \approx f_{c v} \cdot V_{ce l l} ∣ ϕ_{λ} (0) ∣^{2}

This equation tells us that the oscillator strength of an exciton transition is the intrinsic interband oscillator strength $f_{c v}$ modulated by the term $V_{ce l l} ∣ ϕ_{λ} (0) ∣^{2}$ . This term represents the probability of finding the electron and hole in the same unit cell, enhanced by the Coulomb interaction that binds them.

3. Evaluating the Oscillator Strength with the Envelope Function

The power of this formalism lies in evaluating the $∣ ϕ_{λ} (0) ∣^{2}$ term using the envelope functions we derived in Module 2.

For 3D Wannier-Mott Excitons (Allowed Transitions)

In most direct-gap semiconductors like GaAs, the $s$ -like states are optically active because their envelope functions are non-zero at the origin ( $r = 0$ ). For a hydrogenic exciton, the envelope function for an $n s$ state at the origin is:

∣ ϕ_{n s} (0) ∣^{2} = \frac{1}{π ( n a _{B} ) ^{3}}

where $a_{B}$ is the exciton Bohr radius and $n$ is the principal quantum number.

Plugging this into our expression for the oscillator strength per unit cell:

f_{n s}^{per cell} \approx f_{c v} \frac{V _{ce l l}}{π ( n a _{B} ) ^{3}}

This result elegantly explains the $1/ n^{3}$ scaling of absorption peak intensities that we noted in the previous lesson. The $1 s$ exciton state, having the largest probability of electron-hole overlap, concentrates the most oscillator strength and produces the most intense absorption peak.

Test your understanding!

Given that the oscillator strength is "borrowed" from the continuum of band-to-band transitions, there is a "sum rule" that states the total oscillator strength over all discrete exciton states plus the continuum must equal the total oscillator strength of the original interband transitions.

\sum_{\lambda} f_{ex, \lambda} + f_{continuum} = f_{cv} \times (\text{Total # of unit cells})

How does the formation of excitons redistribute the oscillator strength compared to the case without Coulomb interaction?

Show answer

In the absence of Coulomb interaction, the absorption begins sharply at the bandgap $E_{g}$ , and all the oscillator strength is in the continuum of interband transitions. When the electron-hole interaction is "turned on," bound exciton states form below the bandgap. These states "pull down" oscillator strength from the continuum and concentrate it into discrete, sharp absorption peaks. The $1 s$ state takes the largest share, followed by the $2 s$ , $3 s$ , etc., in a $1/ n^{3}$ progression. The absorption at the band edge itself is actually suppressed, but the overall strength (integrated over all energies) is conserved.

4. Rewriting the Absorption Coefficient

With the oscillator strength defined, we can rewrite our formula for the absorption coefficient in a more compact and physically transparent form. By substituting the expression for $f_{e x, λ}^{per cell}$ back into the equation for $α (ω)$ , we arrive at:

α (ω) = \frac{2 π ^{2} e ^{2}}{n _{r} c ϵ _{0} m _{0} V _{ce l l}} λ \sum f_{e x, λ}^{per cell} \cdot g (ℏ ω - E_{λ})

This form is extremely useful. It shows that the absorption spectrum is simply a sum of the oscillator strengths of each allowed transition, broadened by the lineshape function $g$ . This cleanly separates the intrinsic strength of the transition ( $f_{e x, λ}^{per cell}$ ) from the broadening mechanisms ( $g$ ).

The following reading reinforces these concepts, directly defining oscillator strength and relating it to the exciton wavefunction and radiative lifetime.

Exciton Decay Lifetime in Monolayer Transition Metal Dichalcogenides

To see a practical application of these ideas, let's look at a paper that connects oscillator strength directly to the exciton's radiative lifetime. This provides a complementary physical interpretation of our key quantity.

Please read section II.A, 'Exciton Decay Lifetime' (page 3). Focus on the definitions in Equations (9) and (10). Equation (9) gives the general formula for oscillator strength ('fval'), and Equation (10) applies it to an exciton, showing its dependence on the interband matrix element and the envelope function at zero separation. This directly mirrors the derivation we just completed.

5. Advanced Case: Parity-Forbidden Transitions (Cu₂O)

Your work involves Cu₂O, which presents a fascinating and more complex case. In Cu₂O, the conduction and highest valence bands at the $Γ$ -point both have even parity. Because the momentum operator $p$ has odd parity, the dipole matrix element $⟨ u_{c} ∣ p ∣ u_{v} ⟩$ is zero by symmetry. These are called parity-forbidden transitions.

So, where does the famous yellow exciton series in Cu₂O come from? The absorption is not strictly zero because our dipole approximation ( $e^{i q \cdot r} \approx 1$ ) was only the first term in an expansion. The next order terms in the light-matter interaction can enable transitions.

The following resource provides a detailed theoretical treatment that is highly relevant to your research interests.

Wavefunctions and oscillator strengths of Rydberg excitons...

For a more rigorous look at how oscillator strengths are calculated in materials like Cu₂O, we can turn to this recent theoretical paper. It starts from a many-body framework and uses group theory to navigate the complexities of forbidden transitions.

Please read the beginning of Section II.2, 'Oscillator strengths' (pages 3-5), up to and including Equation (14). Don't worry about following every single step of the k.p expansion. Instead, focus on the overarching logic: The standard matrix element (first term in Eq. 12) vanishes due to parity. Higher-order terms, which depend on the momentum q of the exciton's relative motion, must be considered. This leads to a new selection rule (Eq. 13), where the oscillator strength is now proportional to the gradient of the envelope function at the origin, i.e., (| abla \phi_\lambda(\mathbf{r})|_{\mathbf{r}=0}|^2). This explains why P-excitons (with odd parity envelope functions, for which the gradient is non-zero at the origin) are the ones observed in the absorption spectrum of Cu₂O, not S-excitons.

The key insight from this advanced treatment is that for parity-forbidden transitions, the selection rules change. Instead of probing the value of the envelope function at the origin, $∣ ϕ_{λ} (0) ∣^{2}$ , optical transitions probe the gradient of the envelope function, $∣\nabla ϕ_{λ} (0) ∣^{2}$ .

For s-states, which are spherically symmetric, the gradient is zero at the origin. Thus, s-excitons remain optically dark in Cu₂O.
For p-states, the envelope function is zero at the origin, but its gradient is not. Therefore, p-excitons become the lowest-energy optically active states.

The paper shows that for the yellow P-exciton series in bulk Cu₂O, the oscillator strength scales as $f_{r e l} (n) \propto \frac{n ^{2} - 1}{n ^{5}}$ , which for large $n$ still approaches the familiar $1/ n^{3}$ dependence.

Conclusion

This lesson has formalized the concept of oscillator strength, a cornerstone for quantifying and understanding optical transitions in solids.

Key Takeaways:

The oscillator strength is a dimensionless quantity that bundles the transition matrix element and energy into a single measure of an optical transition's probability.
For an exciton, the oscillator strength per unit cell is given by $f_{e x, λ}^{per cell} \approx f_{c v} \cdot V_{ce l l} ∣ ϕ_{λ} (0) ∣^{2}$ .
Using hydrogenic envelope functions for allowed transitions, we can evaluate this term, correctly predicting the $1/ n^{3}$ scaling of absorption peak intensities for the $n s$ exciton series.
For materials with forbidden transitions like Cu₂O, higher-order terms in the light-matter interaction become dominant. This changes the selection rules, making the oscillator strength proportional to the gradient of the envelope function at the origin, $∣\nabla ϕ_{λ} (0) ∣^{2}$ , and allowing p-excitons to be optically active.

Preview of the Next Lesson

We have now established that the exciton absorption spectrum consists of a series of discrete lines. However, we also know that there is a continuum of states above the bandgap. In the next lesson, we will analyze the Elliott-Toyozawa formula, which provides a complete description of the excitonic absorption spectrum, unifying the discrete Rydberg series with the enhanced absorption continuum above the bandgap.

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