Electron-Phonon Interaction: Deformation Potential Approximation

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Hello! Welcome back to the course.

In our previous module, we developed a detailed picture of lattice vibrations, culminating in the concept of the phonon as a quantized normal mode and seeing how these are measured experimentally. So far, however, we have treated the electrons moving through the crystal and the ions vibrating about their lattice sites as two separate, non-interacting systems. This is, of course, a simplification. The periodic potential that gives rise to the electronic band structure is created by the ions themselves. When these ions vibrate, the potential that the electrons feel must change.

This coupling between electrons and phonons is one of the most fundamental interaction mechanisms in condensed matter physics. It is responsible for a vast range of phenomena, including the temperature-dependent electrical resistivity of metals, scattering processes that limit carrier mobility in semiconductors, and even the attractive interaction that leads to conventional superconductivity.

Today, we begin our new module on "Electron-Phonon Interactions." Our goal for this lesson is to derive the interaction Hamiltonian that describes this coupling, using a powerful and widely used framework known as the deformation potential approximation. This will provide the mathematical foundation for much of our subsequent discussion on transport and scattering.

1. The Physical Origin of Electron-Phonon Coupling

The core idea is straightforward. In a rigid, perfect crystal, an electron moves in a perfectly periodic potential, $V_0(\mathbf{r})$. Its states are the well-known Bloch waves. Now, let the ions vibrate, causing a displacement $\mathbf{u}(\mathbf{r},t)$ of the lattice at each point $\mathbf{r}$. This distortion changes the potential to a new, time-dependent potential $V(\mathbf{r},t)$.

The electron-phonon interaction Hamiltonian, $H_{e-ph}$, is defined as the perturbation caused by this change:
$H_{e-ph}(\mathbf{r},t)=V(\mathbf{r},t)-V_0(\mathbf{r})$

For small atomic displacements, we can perform a Taylor expansion of the potential. The first-order term gives:
$H_{e-ph}(\mathbf{r},t)\approx \mathbf{u}(\mathbf{r},t)\cdot \nabla V_0(\mathbf{r})$
While formally correct, this expression is often difficult to work with directly. The deformation potential approximation, introduced by Bardeen and Shockley in 1950, provides a more practical and physically intuitive approach, especially for long-wavelength acoustic phonons.

2. The Deformation Potential Approximation

Instead of tracking the displacement of every single atom, the deformation potential approximation connects the change in an electron's energy to the macroscopic strain induced in the crystal. Strain is a measure of how much the crystal is locally compressed or stretched. For a displacement field $\mathbf{u}(\mathbf{r})$, the local volume change, or dilation, is given by $\Delta (\mathbf{r})=\nabla \cdot \mathbf{u}(\mathbf{r})$.

The central hypothesis of this approximation is that for a slowly varying strain (i.e., from a long-wavelength phonon), the energy of a band edge shifts in direct proportion to the local dilation. For example, consider the bottom of the conduction band, $E_c$. Its energy shift, $\delta E_c$, is given by:
$\delta E_c(\mathbf{r})={\Xi }_d\Delta (\mathbf{r})={\Xi }_d(\nabla \cdot \mathbf{u}(\mathbf{r}))$

This equation defines the deformation potential constant, ${\Xi }_d$.

• It has units of energy (typically expressed in eV).
• It represents the shift in the band-edge energy per unit fractional volume change.
• It's a phenomenological parameter that can be measured experimentally or calculated from first-principles (e.g., using DFT). For silicon, ${\Xi }_d$ is about 9 eV.

This energy shift $\delta E_c(\mathbf{r})$ acts as the effective interaction potential experienced by an electron. This simple, powerful idea connects a microscopic quantum property (band energy) to a macroscopic continuum concept (strain). It's a bit like how the energy levels of a "particle in a box" change if you compress the box.

To get a more formal handle on this concept, let's turn to a more detailed derivation.

As the reading shows, the deformation potential is, in general, a tensor ${\Xi }_{ij}$ that relates the energy shift to the full strain tensor $S_{ij}$. The scalar constant ${\Xi }_d$ we are using is essentially the trace of this tensor, relevant for isotropic volume changes.

3. Quantizing the Interaction

We now have an expression for the interaction potential, $V_{e-ph}(\mathbf{r})={\Xi }_d(\nabla \cdot \mathbf{u}(\mathbf{r}))$. To get the Hamiltonian, we need to express the lattice displacement field $\mathbf{u}(\mathbf{r})$ in terms of its fundamental quantum excitations: the phonons.

From the previous module, you'll recall that the displacement operator $\hat{\mathbf{u}}$ can be written as a sum over phonon modes, involving their creation ($a_{\mathbf{q},s}^{†}$) and annihilation ($a_{\mathbf{q},s}$) operators. In the continuum limit, this is:
$\hat{\mathbf{u}}(\mathbf{r})={\sum }_{\mathbf{q},s}\sqrt{\frac{\hslash }{2\rho V{\omega }_{\mathbf{q},s}}}{\mathbf{e}}_{\mathbf{q},s}\left(a_{\mathbf{q},s}{e}^{i\mathbf{q}\cdot \mathbf{r}}+a_{\mathbf{q},s}^{†}{e}^{-i\mathbf{q}\cdot \mathbf{r}}\right)$
where $\rho$ is the crystal's mass density, $V$ is the volume, ${\omega }_{\mathbf{q},s}$ is the phonon frequency for mode $s$ with wavevector $\mathbf{q}$, and ${\mathbf{e}}_{\mathbf{q},s}$ is its polarization vector.

To find our interaction potential, we must take the divergence of $\hat{\mathbf{u}}(\mathbf{r})$:
$\nabla \cdot \hat{\mathbf{u}}(\mathbf{r})={\sum }_{\mathbf{q},s}\sqrt{\frac{\hslash }{2\rho V{\omega }_{\mathbf{q},s}}}(i\mathbf{q}\cdot {\mathbf{e}}_{\mathbf{q},s})\left(a_{\mathbf{q},s}{e}^{i\mathbf{q}\cdot \mathbf{r}}-a_{\mathbf{q},s}^{†}{e}^{-i\mathbf{q}\cdot \mathbf{r}}\right)$
This expression reveals a crucial selection rule:

• For longitudinal acoustic (LA) phonons, the polarization ${\mathbf{e}}_{\mathbf{q},LA}$ is parallel to the wavevector $\mathbf{q}$. Therefore, $\mathbf{q}\cdot {\mathbf{e}}_{\mathbf{q},LA}=|\mathbf{q}|\ne 0$. These phonons create compressions and rarefactions, coupling effectively to electrons via the deformation potential.
• For transverse acoustic (TA) phonons, the polarization ${\mathbf{e}}_{\mathbf{q},TA}$ is perpendicular to $\mathbf{q}$. Therefore, $\mathbf{q}\cdot {\mathbf{e}}_{\mathbf{q},TA}=0$. These modes correspond to shear motions, which (to first order) do not change the local volume.

Thus, within the deformation potential approximation for a simple band, electrons couple to LA phonons but not to TA phonons.

Focusing only on the LA modes (and dropping the mode index $s$), our interaction potential becomes:
${\hat{V}}_{e-ph}(\mathbf{r})={\sum }_{\mathbf{q}}i{\Xi }_d|\mathbf{q}|\sqrt{\frac{\hslash }{2\rho V{\omega }_{\mathbf{q}}}}\left(a_{\mathbf{q}}{e}^{i\mathbf{q}\cdot \mathbf{r}}-a_{\mathbf{q}}^{†}{e}^{-i\mathbf{q}\cdot \mathbf{r}}\right)$

4. The Electron-Phonon Hamiltonian in Second Quantization

The final step is to write the total interaction Hamiltonian. In the language of second quantization, an interaction of this form is expressed as:
${\hat{H}}_{e-ph}=\int d^{3}r{\hat{\psi }}^{†}(\mathbf{r}){\hat{V}}_{e-ph}(\mathbf{r})\hat{\psi }(\mathbf{r})$
where ${\hat{\psi }}^{†}(\mathbf{r})$ and $\hat{\psi }(\mathbf{r})$ are the electron field operators that create and annihilate an electron at position $\mathbf{r}$. Expanding these in terms of operators for Bloch states, $c_{\mathbf{k}}^{†}$ and $c_{\mathbf{k}}$, and performing the integration leads to the final, compact form of the Hamiltonian.

After working through the algebra (which involves a momentum-conserving delta function from the spatial integral), we arrive at the standard form of the electron-phonon interaction Hamiltonian:
${\hat{H}}_{e-ph}={\sum }_{\mathbf{k},\mathbf{q}}M(\mathbf{q})\left(c_{\mathbf{k}+\mathbf{q}}^{†}{c}_{\mathbf{k}}{a}_{\mathbf{q}}+c_{\mathbf{k}-\mathbf{q}}^{†}{c}_{\mathbf{k}}{a}_{\mathbf{q}}^{†}\right)$
This elegant expression describes the two fundamental electron-phonon scattering processes:

1. c^\dagger_{\mathbf{k}+\mathbf{q}} c_{\mathbf{k}} a_{\mathbf{q}}: An electron in state $\mathbf{k}$ is annihilated ($c_{\mathbf{k}}$), a phonon with momentum $\mathbf{q}$ is absorbed ($a_{\mathbf{q}}$), and an electron is created in the new state $\mathbf{k}+\mathbf{q}$ ($c_{\mathbf{k}+\mathbf{q}}^{†}$).
2. c^\dagger_{\mathbf{k}-\mathbf{q}} c_{\mathbf{k}} a^\dagger_{\mathbf{q}}: An electron in state $\mathbf{k}$ emits a phonon of momentum $\mathbf{q}$ ($a_{\mathbf{q}}^{†}$) and scatters into a new state $\mathbf{k}-\mathbf{q}$.

The strength of these processes is governed by the coupling matrix element $M(\mathbf{q})$:
$M(\mathbf{q})={\Xi }_d\sqrt{\frac{\hslash |\mathbf{q}{|}^2}{2\rho V{\omega }_{\mathbf{q}}}}$
In the long-wavelength limit for acoustic phonons, the dispersion is linear: ${\omega }_{\mathbf{q}}=v_s|\mathbf{q}|$, where $v_s$ is the speed of sound. Substituting this into the coupling element gives:
$M(\mathbf{q})={\Xi }_d\sqrt{\frac{\hslash |\mathbf{q}|}{2\rho V{v}_s}}$
This reveals that the coupling strength is proportional to $\sqrt{|\mathbf{q}|}$. This means that electrons interact more strongly with shorter-wavelength (higher-momentum) acoustic phonons, but the interaction vanishes for phonons with $\mathbf{q}\to 0$.

Conclusion

Today, we have taken a major step by bridging the worlds of electrons and lattice vibrations. We derived the electron-phonon interaction Hamiltonian, a cornerstone for understanding electron transport and other key phenomena in solids.

Key Takeaways:

• The electron-phonon interaction arises because lattice vibrations distort the periodic potential experienced by electrons.
• The deformation potential approximation provides a powerful model for this interaction, particularly for acoustic phonons. It posits that the band edge energy shifts in proportion to the local lattice strain, quantified by the deformation potential constant ${\Xi }_d$.
• This mechanism leads to a selective coupling: electrons interact with longitudinal acoustic (LA) phonons (which cause volume changes) but not with transverse acoustic (TA) phonons (which cause shear).
• The resulting interaction Hamiltonian, ${\hat{H}}_{e-ph}={\sum }_{\mathbf{k},\mathbf{q}}M(\mathbf{q})(c_{\mathbf{k}+\mathbf{q}}^{†}{c}_{\mathbf{k}}{a}_{\mathbf{q}}+h.c.)$, describes the fundamental processes of phonon absorption and emission by electrons, with a coupling strength $M(\mathbf{q})$ that depends on the phonon wavevector $\mathbf{q}$.

Preview of the Next Lesson

The deformation potential is crucial, but it's not the only mechanism. In polar crystals, such as GaAs or other III-V semiconductors, another long-range interaction becomes dominant, especially for optical phonons. In our next lesson, we will derive the Fröhlich Hamiltonian, which describes the coupling between electrons and longitudinal optical (LO) phonons mediated by long-range Coulomb forces. This will be essential for understanding the electronic and optical properties of many important semiconductor materials.