Electron-Phonon Scattering Rate: Fermi's Golden Rule

Hello! Welcome to our next lesson.

In the past two lessons, we derived the Hamiltonians for the two principal electron-phonon interaction mechanisms: the short-range deformation potential for acoustic phonons and the long-range Fröhlich interaction for LO phonons in polar crystals. These Hamiltonians, $H_{e-ph}$, represent the "rules of engagement" between electrons and lattice vibrations.

Today, we will put these rules into action. Our goal is to use these Hamiltonians to calculate a crucial physical quantity: the rate at which an electron scatters from one momentum state to another by absorbing or emitting a phonon. This scattering rate is the microscopic origin of electrical resistance and limits carrier mobility in pure crystals. Our primary tool for this task will be Fermi's Golden Rule.

1. The Framework: Fermi's Golden Rule

From your quantum mechanics courses, you'll recall that when a system is subjected to a weak, time-dependent perturbation $H^{\prime }$, it can cause transitions between the system's stationary states. Fermi's Golden Rule gives us the transition rate from an initial state $|i⟩$ to a final state $|f⟩$:

In our context:

• The system is the entire crystal, so the states $|i⟩$ and $|f⟩$ must describe both the electron and the phonon field.
• The perturbation $H^{\prime }$ is the electron-phonon interaction Hamiltonian, $H_{e-ph}$, that we've derived.
• The matrix element $⟨f|H^{\prime }|i⟩$ quantifies the strength of the coupling between the specific initial and final states.
• The Dirac delta function, $\delta (E_f-E_i)$, ensures that the total energy is conserved during the scattering event.

The key idea, as mentioned in the video, is that we often want the total scattering rate for an electron in a state $\mathbf{k}$, which we can write as $1/{\tau }_{\mathbf{k}}$. To get this, we must sum the transition rates over all possible final electron states ${\mathbf{k}}^{\prime }$ and consider all possible phonon absorption and emission processes that can take the electron out of state $\mathbf{k}$.

In the limit of a large crystal, this sum over final states becomes an integral: ${\sum }_{{\mathbf{k}}^{\prime }}\to \frac{V}{(2\pi {)}^3}\int d^3{k}^{\prime }$. Let's now apply this framework to a concrete and important case.

2. Case Study: Electron-LO Phonon Scattering Rate

We'll start with the Fröhlich interaction. The calculation is somewhat involved, but it beautifully demonstrates the interplay of quantum mechanics, solid-state physics, and electrostatics. We will follow a detailed derivation presented in the context of Quantum Cascade Lasers (QCLs), which provides a nice link to an experimental application area. The derivation considers a system that is quantized in the z-direction (like a quantum well) but free in the x-y plane. The core steps, however, are general.

The resource below walks through the entire calculation. Instead of reproducing it here, I will guide you through its structure, highlighting the key physical steps. Your task is to follow the mathematics in the document.

The Derivation Roadmap

Step 1: Setting the Stage (Sections 2.1 - 2.2)
The derivation begins by writing down Fermi's Golden Rule (Eq. on page 1). The states are defined: $|i,{\mathbf{k}}_i,n_{i,q}⟩$. The Fröhlich interaction Hamiltonian is then substituted into the matrix element. The key move here is to apply the phonon creation ($b^{†}$) and annihilation ($b$) operators to the phonon states. This immediately separates the calculation into two cases:

• Emission: Final phonon number $n_f=n_i+1$. Matrix element $\propto \sqrt{n_{i,q}+1}$.
• Absorption: Final phonon number $n_f=n_i-1$. Matrix element $\propto \sqrt{n_{i,q}}$.

Step 2: Momentum Conservation (Sections 2.3 - 2.4)
With the phonon part handled, the focus shifts to the electronic part of the matrix element. The electron states are plane waves in the x-y plane. Evaluating the spatial integral $⟨{\mathbf{k}}_f|e^{\mp i\mathbf{q}\cdot \mathbf{r}}|{\mathbf{k}}_i⟩$ naturally produces Dirac delta functions in momentum space, e.g., $\delta (k_{f,x}\pm q_x-k_{i,x})$. This is the statement of in-plane momentum conservation: ${\mathbf{k}}_{f,\parallel }={\mathbf{k}}_{i,\parallel }\mp {\mathbf{q}}_{\parallel }$.

Step 3: Summing Over Final States & Energy Conservation (Sections 2.5 - 2.6)
To get the total rate, we integrate over all possible final states ${\mathbf{k}}_f$. This integral now contains the energy-conserving delta function $\delta (E_f-E_i\pm E_{LO})$. The standard procedure is to use the properties of the delta function to perform this integral. This step fixes the magnitude of the final wavevector ${\mathbf{k}}_f$ that satisfies energy conservation for a given initial state ${\mathbf{k}}_i$.

Step 4: The Final Integrations (Sections 2.7 - 2.11)
After the previous steps, we are left with integrals over the phonon momentum component $q_z$ and the angle $\theta$ between the initial and final electron wavevectors. The derivation shows how to solve these integrals, introducing a "form factor" $A(q)$ that contains the electron wavefunctions in the confined direction. After much algebra, this leads to the final expression for the scattering rate $W_{i\to f}({\mathbf{k}}_i)$. The final expression (at the top of page 7) depends on material parameters ($m^{\ast },e,E_{LO},{ϵ}_{\infty },{ϵ}_s$), the temperature through the Bose-Einstein factor $n_{LO}$, and the overlap of the electron wavefunctions.

This detailed process gives us the scattering rate for a specific electron-LO phonon interaction, a cornerstone for modeling transport in polar semiconductors.

3. Case Study: Acoustic Phonon Scattering

Next, we'll tackle scattering by acoustic phonons via the deformation potential. This process is generally weaker than the Fröhlich interaction in polar materials but is dominant in non-polar semiconductors like Silicon and Germanium. The calculation is simpler due to some useful approximations.

Key Steps and Approximations

1. The Matrix Element (Page 2-3): The process is analogous to the LO phonon case. We take the matrix element of the deformation potential Hamiltonian. The result for the squared matrix element is:

   $$|M_{fi}{|}^2=\frac{{\hslash }^2{\Xi }^2{q}^2}{2\rho V{\omega }_q}\times (n_q+1/2\mp 1/2)$$

   where $\Xi$ is the deformation potential constant, $\rho$ is the crystal density, and ${\omega }_q=v_{s}q$ is the acoustic phonon frequency. The $(n_q+1)$ term corresponds to emission and $n_q$ to absorption.

2. Approximation 1: Elastic Scattering: The maximum energy of an acoustic phonon involved in scattering an electron is typically very small ($\sim 1$ meV) compared to the electron's kinetic energy and the thermal energy $k_{B}T$ ($\sim 25$ meV at room temp). Therefore, we can assume the scattering is elastic, meaning the electron's energy does not change: $E_{{\mathbf{k}}^{\prime }}\approx E_{\mathbf{k}}$. This greatly simplifies the energy conservation delta function.

3. Approximation 2: Equipartition: In the high-temperature limit, where $k_{B}T\gg \hslash {\omega }_q$, the Bose-Einstein distribution for phonon occupation can be approximated as $n_q\approx \frac{k_{B}T}{\hslash {\omega }_q}$. Since $n_q\gg 1$ in this limit, we also have $n_q+1\approx n_q$. This is the equipartition approximation.

Applying these approximations, the calculation proceeds as follows (detailed on page 3 of the PDF):

• The total scattering rate $1/\tau$ is found by integrating $W_{\mathbf{k}\to {\mathbf{k}}^{\prime }}$ over all final states.
• Under the elastic approximation, the integration over $q$ (the momentum transfer) runs from 0 to $2k$.
• Plugging in the matrix element and the equipartition approximation, we find that $|M{|}^2\propto T$.
• After performing the final integral, we arrive at a classic result: $$\frac{1}{{\tau }_{ac}}=\frac{2\pi }{\hslash }\frac{(m^{\ast }{)}^{3/2}{\Xi }^2{k}_{B}T}{\sqrt{2}\pi {\hslash }^3\rho v_s^2}\sqrt{E_k}$$

The scattering rate from acoustic phonons is directly proportional to temperature $T$ and the square root of the electron's energy $E_k$.

4. Visualizing the Scattering Rates

We've derived expressions for the scattering rates from two different phonon types. They have very different dependencies on electron energy.

• Acoustic Phonons: The rate $1/{\tau }_{ac}\propto \sqrt{E}$ starts at zero and increases smoothly with electron energy. This is because the density of available final states increases with energy.
• LO Phonons: An electron cannot emit an LO phonon unless its kinetic energy is greater than the phonon energy, $E_k>\hslash {\omega }_{LO}$. Therefore, the emission scattering rate is zero below this threshold and then turns on abruptly. Absorption is possible at any energy, provided there are thermal phonons to absorb.

The image below shows a typical calculation of these rates as a function of carrier energy in a semiconductor.

This plot is a powerful visual summary of our findings. It shows how different scattering mechanisms dominate in different energy ranges, a crucial concept for understanding device performance, especially for "hot" (high-energy) electrons.

Conclusion

In this lesson, we have taken a major step from formal Hamiltonians to calculating concrete physical observables. Using Fermi's Golden Rule, we have sketched out the process for finding the rate at which electrons are scattered by lattice vibrations.

Key Takeaways:

• Fermi's Golden Rule is the central tool for calculating transition rates caused by a weak perturbation, such as the electron-phonon interaction.
• The calculation involves finding the matrix element of the interaction Hamiltonian between initial and final states of the combined electron-phonon system.
• The total scattering rate out of a state $\mathbf{k}$ is obtained by summing/integrating over all possible final states, subject to energy and momentum conservation.
• Acoustic phonon scattering, in the high-temperature elastic approximation, leads to a rate $1/{\tau }_{ac}\propto T\sqrt{E_k}$.
• LO phonon scattering is a threshold process. Emission is only possible when the electron energy $E_k>\hslash {\omega }_{LO}$, leading to a sharp onset in the scattering rate.

Preview of the Next Lesson

We've now calculated the microscopic scattering rate $1/\tau$. The mini-exercise already hinted at the next logical step: connecting this microscopic quantity to a macroscopic transport property like electrical resistivity. In the next lesson, we will formalize this connection and derive the temperature dependence of phonon-limited resistivity, exploring both the high-temperature ($T>{\Theta }_D$) and low-temperature ($T<{\Theta }_D$) limits.