Direct Bandgap Optical Transitions & k-Selection Rule

Introduction

In our last lesson, we determined the number of available electron and hole states for an optical transition of a given energy by formulating the joint density of states (JDOS). We saw that for a 3D direct-gap semiconductor, the JDOS follows a characteristic $\sqrt{E-E_g}$ dependence. However, knowing that states are available doesn't tell us the likelihood that a transition will actually occur. This probability is governed by the fundamental quantum mechanical interaction between light and matter.

Today, we will complete our picture of interband absorption by addressing this crucial point. Our goal is to evaluate the optical transition matrix elements for direct bandgap semiconductors and formally establish the k-selection rule. To do this, we will:

1. Start with Fermi's Golden Rule to define the transition rate.
2. Introduce the Hamiltonian that describes the interaction between an electron and an electromagnetic wave.
3. Calculate the quantum mechanical matrix element for this interaction between valence and conduction band states.
4. Rigorously derive the crystal momentum conservation law and the famous k-selection rule ($\Delta \mathbf{k}\approx 0$) for direct transitions.

This lesson will solidify the concept of "vertical transitions" that we have been using and provide the final piece of the puzzle needed before we introduce the Coulomb interaction to form excitons in the next module.

1. The Framework: Fermi's Golden Rule and the Light-Matter Interaction

The probability of an optical transition is calculated using time-dependent perturbation theory. The central result is Fermi's Golden Rule, which gives the transition rate $W_{i\to f}$ from an initial state $|i⟩$ to a final state $|f⟩$ caused by a perturbing Hamiltonian ${\hat{H}}^{\prime }$:

$W_{i\to f}=\frac{2\pi }{\hslash }|⟨f|{\hat{H}}^{\prime }|i⟩{|}^2\delta (E_f-E_i-\hslash \omega )$

Here, the delta function ensures energy conservation: the energy difference between the final and initial states must equal the absorbed photon energy $\hslash \omega$. Our main task is to determine the interaction Hamiltonian ${\hat{H}}^{\prime }$ and then calculate the matrix element $⟨f|{\hat{H}}^{\prime }|i⟩$.

The interaction of an electron (charge $-e$) with an electromagnetic field described by a vector potential $\mathbf{A}(\mathbf{r},t)$ is introduced through the principle of minimal coupling. The electron's momentum operator $\hat{\mathbf{p}}$ is replaced by $\hat{\mathbf{p}}+e\mathbf{A}$. The full Hamiltonian becomes:
$\hat{H}=\frac{1}{2m_0}(\hat{\mathbf{p}}+e\mathbf{A}{)}^2+V_{crystal}(\mathbf{r})$
where $m_0$ is the free electron mass and $V_{crystal}(\mathbf{r})$ is the periodic potential of the lattice.

Expanding this and keeping only the terms linear in $\mathbf{A}$ (for single-photon processes), we find the interaction Hamiltonian:
${\hat{H}}^{\prime }\approx \frac{e}{m_0}\mathbf{A}\cdot \hat{\mathbf{p}}$
We model the light as a plane wave, so the vector potential has the form $\mathbf{A}(\mathbf{r},t)=A_0\widehat{\mathbf{ϵ}}\cos (\mathbf{q}\cdot \mathbf{r}-\omega t)$, where $\widehat{\mathbf{ϵ}}$ is the polarization vector and $\mathbf{q}$ is the light's wavevector. The part of this Hamiltonian responsible for absorption involves the term $e^{i(\mathbf{q}\cdot \mathbf{r}-\omega t)}$.

2. Deriving the k-Selection Rule

Now we will use this framework to calculate the matrix element between an initial state in the valence band, $|i⟩=|{\psi }_{v,{\mathbf{k}}_i}⟩$, and a final state in the conduction band, $|f⟩=|{\psi }_{c,{\mathbf{k}}_f}⟩$. As you've completed advanced quantum mechanics, the derivation should be a familiar application of these principles.

Let's summarize the key steps of the derivation:

1. The Matrix Element: We need to compute $M_{if}=⟨{\psi }_{c,{\mathbf{k}}_f}|\frac{e{A}_0}{m_0}{e}^{i\mathbf{q}\cdot \mathbf{r}}(\widehat{\mathbf{ϵ}}\cdot \hat{\mathbf{p}})|{\psi }_{v,{\mathbf{k}}_i}⟩$.

2. Substituting Bloch Functions: The initial and final states are Bloch functions:

   • ${\psi }_{v,{\mathbf{k}}_i}(\mathbf{r})=e^{i{\mathbf{k}}_i\cdot \mathbf{r}}{u}_{v,{\mathbf{k}}_i}(\mathbf{r})$
   • ${\psi }_{c,{\mathbf{k}}_f}(\mathbf{r})=e^{i{\mathbf{k}}_f\cdot \mathbf{r}}{u}_{c,{\mathbf{k}}_f}(\mathbf{r})$

   The matrix element becomes an integral over the entire crystal volume:
   $M_{if}\propto {\int }_V{d}^{3}r(e^{i{\mathbf{k}}_f\cdot \mathbf{r}}{u}_{c,{\mathbf{k}}_f}{)}^{\ast }{e}^{i\mathbf{q}\cdot \mathbf{r}}(\widehat{\mathbf{ϵ}}\cdot \hat{\mathbf{p}})(e^{i{\mathbf{k}}_i\cdot \mathbf{r}}{u}_{v,{\mathbf{k}}_i})$

3. Separating Periodic and Non-Periodic Parts: The integral can be rewritten as:
   $M_{if}\propto {\int }_V{d}^{3}r{e}^{i(-{\mathbf{k}}_f+{\mathbf{k}}_i+\mathbf{q})\cdot \mathbf{r}}\left[u_{c,{\mathbf{k}}_f}^{\ast }(\widehat{\mathbf{ϵ}}\cdot \hat{\mathbf{p}})u_{v,{\mathbf{k}}_i}\right]$
   The term in the square brackets is a periodic function with the periodicity of the lattice. The integral of a plane wave $e^{i\mathbf{K}\cdot \mathbf{r}}$ multiplied by a lattice-periodic function is non-zero only if $\mathbf{K}$ is a reciprocal lattice vector (including zero). For interband transitions within the first Brillouin zone, this requires:
   $-{\mathbf{k}}_f+{\mathbf{k}}_i+\mathbf{q}=0⟹{\mathbf{k}}_f={\mathbf{k}}_i+\mathbf{q}$
   This is the law of crystal momentum conservation for optical transitions. It states that the electron's change in momentum must be supplied by the absorbed photon.

3. The Dipole Approximation and Vertical Transitions

The final step is to compare the magnitude of the photon's wavevector $\mathbf{q}$ to the typical crystal momentum of an electron.

As the video explains, the magnitude of a photon's wavevector is $|\mathbf{q}|=2\pi /\lambda$. For visible light, $\lambda \approx 500$ nm, so $|\mathbf{q}|\sim 10^7{\text{m}}^{-1}$. In contrast, the size of the first Brillouin zone is on the order of $\pi /a$, where $a\approx 0.5$ nm is a typical lattice constant. This gives a maximum crystal momentum of $k_{max}\sim 10^{10}{\text{m}}^{-1}$.

Since $|\mathbf{q}|\ll k_{max}$, the photon's momentum is negligible compared to the electron's crystal momentum. This is the dipole approximation. Under this approximation, the momentum conservation law simplifies to the k-selection rule:

${\mathbf{k}}_f\approx {\mathbf{k}}_i$

This is the reason optical transitions in direct bandgap semiconductors are depicted as vertical transitions on an E-k diagram.

4. The Momentum Matrix Element

With the k-selection rule established, the matrix element simplifies significantly. The transition is only possible between states with the same $\mathbf{k}$, and its strength is determined by:

$M_{cv}(\mathbf{k})\propto ⟨u_{c,\mathbf{k}}|\widehat{\mathbf{ϵ}}\cdot \hat{\mathbf{p}}|u_{v,\mathbf{k}}⟩=\widehat{\mathbf{ϵ}}\cdot {\mathbf{p}}_{cv}(\mathbf{k})$

where ${\mathbf{p}}_{cv}(\mathbf{k})=⟨u_{c,\mathbf{k}}|\hat{\mathbf{p}}|u_{v,\mathbf{k}}⟩$ is the momentum matrix element between the periodic parts of the Bloch functions.

The total transition rate will be proportional to $|M_{cv}(\mathbf{k}){|}^2$. This has two profound implications:

1. Selection Rules: If ${\mathbf{p}}_{cv}(\mathbf{k})$ is zero for a given pair of bands (e.g., due to symmetry), the transition is "forbidden". Your familiarity with k.p theory from our first lesson is relevant here; ${\mathbf{p}}_{cv}$ is precisely the interband coupling term in that theory. Transitions are allowed between bands of different parity (e.g., s-like and p-like), where this matrix element is non-zero.
2. Polarization Dependence: The dot product with the light's polarization vector $\widehat{\mathbf{ϵ}}$ means that the absorption strength can depend on the orientation of the light's electric field relative to the crystal axes. This gives rise to optical anisotropy and polarization-dependent selection rules, which we will explore in detail in a later module.

Conclusion

This lesson concludes our foundational module on semiconductor physics. We have established the final theoretical pillar needed to understand optical absorption.

Key Takeaways:

• The interaction between light and electrons is described by the $\mathbf{A}\cdot \mathbf{p}$ term in the Hamiltonian.
• Applying Fermi's Golden Rule with Bloch wavefunctions rigorously yields the crystal momentum conservation law: ${\mathbf{k}}_f={\mathbf{k}}_i+\mathbf{q}$.
• The dipole approximation ($\mathbf{q}\approx 0$) is justified because photon momentum is negligible compared to the Brillouin zone scale. This leads to the k-selection rule for direct optical transitions: ${\mathbf{k}}_f\approx {\mathbf{k}}_i$, meaning transitions are vertical in k-space.
• The probability of a vertical transition is proportional to the squared momentum matrix element, $|\widehat{\mathbf{ϵ}}\cdot {\mathbf{p}}_{cv}(\mathbf{k}){|}^2$, which connects states in the valence and conduction bands and underlies fundamental optical selection rules.

Preview of the Next Module

So far, we have treated the optically excited electron and the hole it leaves behind as independent particles. This is a valid picture for high photon energies, deep in the absorption continuum. However, we have neglected a crucial interaction: the Coulomb attraction between the negatively charged electron and the positively charged hole. This attraction can bind them together to form a new quasiparticle—an exciton.

In our next module, "The Wannier-Mott Exciton Model," we will begin by reformulating the electron-hole problem to account for this interaction. Our first step will be to separate the motion of the pair's center-of-mass from their relative motion, a classic technique from two-body problems that perfectly sets the stage for solving the exciton's hydrogen-like structure.