Exciton Absorption Coefficient from Fermi's Golden Rule

1. The Quantum Framework for Optical Transitions

From your advanced quantum mechanics course, you'll recall that time-dependent perturbation theory provides the framework for describing how a quantum system responds to an external field, like light. The cornerstone of this framework is Fermi's Golden Rule.

Fermi's Golden Rule

Fermi's Golden Rule gives the transition rate $W_{i\to f}$ from an initial state $|i⟩$ with energy $E_i$ to a final state $|f⟩$ with energy $E_f$, induced by a weak, oscillating perturbation $H^{\prime }$ with frequency $\omega$. For absorption, the rule is:

This powerful equation contains three key components:

• The Matrix Element $⟨f|H^{\prime }|i⟩$: This term determines the "selection rules"—it quantifies how strongly the perturbation couples the initial and final states. If it's zero, the transition is forbidden.
• The Density of States: Implicitly, we sum over all possible final states, which brings in the density of states.
• The Delta Function $\delta (E_f-E_i-\hslash \omega )$: This enforces the conservation of energy. The energy of the absorbed photon must precisely match the energy difference between the final and initial states.

The Light-Matter Interaction Hamiltonian

To use Fermi's rule, we need to specify the perturbation $H^{\prime }$. The following reading explains how to derive the interaction Hamiltonian for an electron in an electromagnetic field.

As the text shows, by using the minimal coupling $\mathbf{p}\to \mathbf{p}+e\mathbf{A}$ in the electron's kinetic energy term and taking the weak field limit, the primary interaction term is:

where $m_0$ is the free electron mass. For a monochromatic plane wave, the vector potential is $\mathbf{A}(\mathbf{r},t)={\mathbf{A}}_0{e}^{i(\mathbf{q}\cdot \mathbf{r}-\omega t)}+c.c.$. A crucial simplification is the electric dipole approximation. The wavelength of light ($\sim 500$ nm) is vastly larger than the size of an exciton ($\sim 10$ nm). Therefore, the spatial variation of the field across the exciton is negligible, and we can set $e^{i\mathbf{q}\cdot \mathbf{r}}\approx 1$. This simplifies the interaction to:

The matrix element in Fermi's rule becomes proportional to $⟨f|ϵ\cdot \mathbf{p}|i⟩$, where $ϵ$ is the polarization vector of the light.

2. The Exciton Creation Matrix Element

Now we apply this formalism to our specific process: creating an exciton.

• Initial State $|i⟩$: This is the semiconductor ground state, $|0⟩$, where the valence band is full and the conduction band is empty. Its energy is $E_i=0$.
• Final State $|f⟩$: This is the semiconductor containing one exciton, $|{\Psi }_{ex}⟩$. The exciton state is a coherent superposition of electron-hole pairs, described by a center-of-mass momentum $\mathbf{K}$ and an envelope function ${\varphi }_{\lambda }(\mathbf{r})$ for the relative motion (where $\lambda$ represents the set of quantum numbers like $n,l,m$). Its energy is $E_f=E_g+E_{\lambda }+\frac{{\hslash }^2{K}^2}{2M}$, where $E_{\lambda }$ is the binding energy relative to the bandgap $E_g$.

The following reading provides the key steps to evaluate the matrix element for exciton creation.

Let's summarize the essential logic of the calculation:

1. The interaction operator $\mathbf{p}$ acts on the ground state, promoting an electron from a valence band Bloch state $|v,{\mathbf{k}}^{\prime }⟩$ to a conduction band Bloch state $|c,\mathbf{k}⟩$.
2. The matrix element for this single-particle transition is $⟨c,\mathbf{k}|\mathbf{p}|v,{\mathbf{k}}^{\prime }⟩\approx {\delta }_{\mathbf{k},{\mathbf{k}}^{\prime }}⟨u_c|\mathbf{p}|u_v⟩$, where $⟨u_c|\mathbf{p}|u_v⟩={\mathbf{p}}_{cv}$ is the momentum matrix element between the Bloch periodic parts. The Kronecker delta enforces that optical transitions are vertical in k-space.
3. The full matrix element involves an overlap between this created electron-hole pair and the final exciton state. The dipole approximation ($\mathbf{q}\approx 0$) implies that the center-of-mass momentum of the created exciton must be $\mathbf{K}\approx 0$. Only excitons that are nearly stationary can be created by single-photon absorption.
4. The final result for the matrix element connecting the ground state to an exciton state $|{\Psi }_{ex}^{\lambda ,\mathbf{K}=0}⟩$ is: $$⟨{\Psi }_{ex}^{\lambda ,0}|H^{\prime }|0⟩=\frac{e{A}_0}{m_0}(ϵ\cdot {\mathbf{p}}_{cv})\sqrt{V}{\varphi }_{\lambda }^{\ast }(0)$$ where $V$ is the crystal volume.

This is a beautiful and intuitive result. The strength of optical absorption into a particular exciton state $\lambda$ is directly proportional to ${\varphi }_{\lambda }(0)$, the value of the relative motion envelope function at zero separation. In other words, the transition strength depends on the probability of finding the electron and the hole in the same unit cell. This makes perfect sense: to create an electron-hole pair with a single photon, the electron and hole must originate from the same point.

For hydrogen-like s-orbitals, ${\varphi }_{ns}(0)$ is non-zero, while for states with non-zero angular momentum ($l>0$), ${\varphi }_{nl...}(0)=0$. This immediately gives us a powerful selection rule: in the dipole approximation, only s-like excitons can be created by single-photon absorption.

3. The Excitonic Absorption Coefficient

We are now ready to assemble the final piece: the absorption coefficient $\alpha (\omega )$. This is defined as the energy absorbed per unit time, per unit volume, divided by the incident energy flux (intensity, $I$).

1. Total Transition Rate ($W_{tot}$): We sum the rates for all possible final exciton states $\lambda$ (with $\mathbf{K}=0$) per unit volume.

   $$W_{tot}=\frac{1}{V}\sum _{\lambda }{W}_{i\to \lambda }=\frac{1}{V}\sum _{\lambda }\frac{2\pi }{\hslash }|⟨{\Psi }_{ex}^{\lambda ,0}|H^{\prime }|0⟩{|}^2\delta (E_{\lambda }-\hslash \omega )$$

   where $E_{\lambda }=E_g-E_b^{\lambda }$ is the exciton energy.

2. Intensity ($I$): The intensity of the light wave is $I=\frac{1}{2}{n}_{r}c{ϵ}_0{E}_0^2$, where $n_r$ is the refractive index. Using $\mathbf{E}=-\frac{\partial \mathbf{A}}{\partial t}=i\omega \mathbf{A}$, we have $E_0=\omega A_0$, so $I=\frac{1}{2}{n}_{r}c{ϵ}_0{\omega }^2{A}_0^2$.

3. Putting it all together:
   Substitute the matrix element into $W_{tot}$:

   $$W_{tot}=\frac{1}{V}\sum _{\lambda }\frac{2\pi }{\hslash }{\left|\frac{e{A}_0}{m_0}(ϵ\cdot {\mathbf{p}}_{cv})\sqrt{V}{\varphi }_{\lambda }^{\ast }(0)\right|}^2\delta (E_{\lambda }-\hslash \omega )$$ $$W_{tot}=\frac{2\pi }{\hslash }{\left(\frac{e{A}_0}{m_0}\right)}^2|ϵ\cdot {\mathbf{p}}_{cv}{|}^2\sum _{\lambda }|{\varphi }_{\lambda }(0){|}^2\delta (E_{\lambda }-\hslash \omega )$$

   Now, we find $\alpha (\omega )$:

   $$\alpha (\omega )=\frac{W_{tot}\hslash \omega }{I}=\frac{\left[\frac{2\pi }{\hslash }{\left(\frac{e{A}_0}{m_0}\right)}^2|ϵ\cdot {\mathbf{p}}_{cv}{|}^2{\sum }_{\lambda }|{\varphi }_{\lambda }(0){|}^2\delta (E_{\lambda }-\hslash \omega )\right]\hslash \omega }{\frac{1}{2}{n}_{r}c{ϵ}_0{\omega }^2{A}_0^2}$$

   After cancelling terms ($A_0^2$, one $\omega$), we arrive at the absorption coefficient:

   $$\alpha (\omega )=\frac{4{\pi }^2{e}^2}{n_{r}c{ϵ}_0{m}_0^2\omega }|ϵ\cdot {\mathbf{p}}_{cv}{|}^2\sum _{\lambda }|{\varphi }_{\lambda }(0){|}^2\delta (E_{\lambda }-\hslash \omega )$$

   In a real system, scattering and finite lifetimes broaden the sharp delta functions into a lineshape function, typically a Lorentzian $g(\hslash \omega -E_{\lambda })$ with a width $\Gamma$. The final, more realistic expression is:

   $$\alpha (\omega )=\frac{4{\pi }^2{e}^2}{n_{r}c{ϵ}_0{m}_0^2\omega }|ϵ\cdot {\mathbf{p}}_{cv}{|}^2\sum _{\lambda }|{\varphi }_{\lambda }(0){|}^{2}g(\hslash \omega -E_{\lambda })$$

This equation is the central result of our lesson. It predicts that the optical absorption spectrum of a semiconductor will consist of a series of discrete peaks corresponding to the creation of different s-like exciton states ($1s,2s,3s,...$) at energies just below the bandgap.

Conclusion

In this lesson, we have built a bridge from fundamental quantum mechanics to a key experimental observable in semiconductor physics.

Key Takeaways:

• The absorption of light to create an exciton is governed by Fermi's Golden Rule.
• The strength of the transition is determined by a matrix element $⟨{\Psi }_{ex}|ϵ\cdot \mathbf{p}|0⟩$, which simplifies under the dipole approximation.
• The transition strength is directly proportional to $|{\varphi }_{\lambda }(0){|}^2$, the probability of finding the electron and hole at the same position. This leads to the selection rule that only s-like excitons are optically active in single-photon absorption.
• The final expression for the absorption coefficient, $\alpha (\omega )$, predicts a Rydberg-like series of discrete peaks just below the bandgap, with intensities decreasing as $1/n^3$ for the $ns$ states.

Preview of the Next Lesson

We have successfully derived the absorption coefficient. The expression we found contains a collection of fundamental constants and two key physical quantities: the interband momentum matrix element ${\mathbf{p}}_{cv}$ and the envelope function term $|{\varphi }_{\lambda }(0){|}^2$. It is common practice to group these terms into a single, convenient parameter called the oscillator strength. In our next lesson, we will formally define and evaluate the oscillator strength for exciton transitions, providing a quantitative measure of how "strong" an optical transition is. This will be our next step in fully characterizing the optical properties of excitons.