Dynamical Matrix and Modes in 3D Monatomic Crystals

Hello! Welcome back to our study of lattice dynamics.

In our last lesson, we built our first model of lattice vibrations using a 1D diatomic chain. We introduced the powerful concept of the dynamical matrix, whose eigenvalues gave us the frequencies of the vibrational modes. This led to the fundamental distinction between acoustic and optical phonon branches, which arise whenever there is more than one atom in the crystal's basis.

Today, we will generalize this framework from our simple 1D model to a realistic 3D crystal. This is a crucial step toward understanding the vibrational properties of the materials you encounter in your experimental and computational work. Our goal is to generalize the concept of the dynamical matrix to a 3D monatomic crystal and identify transverse and longitudinal modes. We'll see how the vector nature of atomic displacements in 3D naturally gives rise to different polarizations of the lattice waves.

1. From a 1D Chain to a 3D Crystal

Let's briefly revisit the core idea. The equation of motion for an atom is determined by the forces exerted by its neighbors. In 1D, displacement $u$ was a scalar. In 3D, the displacement of an atom from its equilibrium position ${\mathbf{R}}_l$ is a vector, ${\mathbf{u}}_l$.

The force on atom $l$ now depends on the displacements of all other atoms $l^{\prime }$. In the harmonic approximation, the force component in the $\alpha$-direction ($\alpha \in \{x,y,z\}$) on atom $l$ is a linear combination of the displacement components $\beta$ of all other atoms:

$$F_{l,\alpha }=-\sum _{l^{\prime },\beta }{\Phi }_{\alpha \beta }(l,l^{\prime })u_{l^{\prime },\beta }$$

The quantity ${\Phi }_{\alpha \beta }(l,l^{\prime })$ is the force constant tensor. It represents the negative of the force in the $\alpha$-direction on the atom at site $l$ when the atom at site $l^{\prime }$ is displaced by a unit distance in the $\beta$-direction. Newton's second law for a monatomic crystal (all masses are $M$) becomes:

$$M{\ddot{u}}_{l,\alpha }=-\sum _{l^{\prime },\beta }{\Phi }_{\alpha \beta }(l,l^{\prime })u_{l^{\prime },\beta }$$

Just as in the 1D case, we seek plane-wave solutions, leveraging the crystal's translational symmetry. The ansatz for the displacement vector is now a 3D plane wave:

$${\mathbf{u}}_l(t)=\mathbf{ϵ}{e}^{i(\mathbf{q}\cdot {\mathbf{R}}_l-\omega t)}$$

Here, $\mathbf{q}$ is the 3D wavevector, and $\mathbf{ϵ}$ is the polarization vector, a 3D vector describing the direction of atomic motion.

2. The 3D Dynamical Matrix

Substituting this plane-wave solution into the equation of motion leads us, after some algebra, to an eigenvalue equation for each wavevector $\mathbf{q}$. The following reading formalizes this generalization. It shows how the scalar equation from our 1D model becomes a matrix equation in 3D, and defines the 3D dynamical matrix.

As the notes show, for a monatomic crystal (one atom per primitive cell), the equation simplifies to a 3x3 eigenvalue problem for each $\mathbf{q}$:

$${\omega }^2{ϵ}_{\alpha }=\sum _{\beta =x,y,z}{D}_{\alpha \beta }(\mathbf{q}){ϵ}_{\beta }$$

where the 3x3 dynamical matrix is given by:

$$D_{\alpha \beta }(\mathbf{q})=\frac{1}{M}\sum _{l^{\prime }}{\Phi }_{\alpha \beta }(0,l^{\prime })e^{-i\mathbf{q}\cdot {\mathbf{R}}_{l^{\prime }}}$$

The sum is over all lattice sites $l^{\prime }$, and we have used translational invariance so that $\Phi (l,l^{\prime })$ only depends on the difference ${\mathbf{R}}_{l^{\prime }}-{\mathbf{R}}_l$.

For each wavevector $\mathbf{q}$, solving this 3x3 eigenvalue problem gives us three eigenvalues, ${\omega }_j^2(\mathbf{q})$, and three corresponding eigenvectors, ${\mathbf{ϵ}}_j(\mathbf{q})$, where $j=1,2,3$. These three solutions are the three phonon branches for a monatomic crystal. Since there is only one atom per primitive cell, all three branches are acoustic modes—their frequencies all go to zero as $\mathbf{q}\to 0$.

3. Longitudinal and Transverse Modes

The physical nature of these three acoustic modes is determined by their polarization vectors ${\mathbf{ϵ}}_j(\mathbf{q})$ relative to the wave propagation direction $\mathbf{q}$.

• Longitudinal Mode (LA): A mode is called longitudinal if the atoms vibrate parallel to the direction of wave propagation.
  $\mathbf{ϵ}(\mathbf{q})\parallel \mathbf{q}$
• Transverse Modes (TA): A mode is called transverse if the atoms vibrate perpendicular to the direction of wave propagation.
  $\mathbf{ϵ}(\mathbf{q})\perp \mathbf{q}$

Since the two directions perpendicular to $\mathbf{q}$ form a plane, there are two transverse modes for every longitudinal mode. Thus, for any given $\mathbf{q}$, a 3D monatomic crystal has one Longitudinal Acoustic (LA) branch and two Transverse Acoustic (TA) branches.

The following video offers a great conceptual visualization of this idea.

In general, for an arbitrary direction of $\mathbf{q}$ in the Brillouin zone, the modes may not be purely longitudinal or transverse. However, along high-symmetry directions of the crystal (like [100], [110], [111] in a cubic crystal), the eigenvectors often align perfectly, and the modes can be cleanly identified as LA or TA. The two TA branches are often, but not always, degenerate due to symmetry.

Test your understanding!

Consider a hypothetical crystal with a simple cubic lattice and a basis of two different atoms per unit cell. How many distinct phonon branches would you expect to find in its dispersion relation? How would you classify them?

Show answer

A crystal with a basis of $p$ atoms has $3p$ phonon branches in total. Here, $p=2$, so there are $3\times 2=6$ branches.

• 3 of these are acoustic branches: one longitudinal (LA) and two transverse (TA).
• The remaining 3 are optical branches: one longitudinal (LO) and two transverse (TO).

So, in total, you would expect to see LA, TA1, TA2, LO, TO1, and TO2 branches.

4. Visualizing Phonon Dispersion in Real Crystals

To see how these concepts manifest in a real material, we look at phonon dispersion curves measured experimentally, typically using inelastic neutron or X-ray scattering. These plots show $\omega (\mathbf{q})$ along paths connecting high-symmetry points in the Brillouin zone (e.g., $\Gamma ,\mathrm{X},\mathrm{K},\mathrm{L}$).

The following reading shows and explains the dispersion curves for real monatomic crystals, connecting our theoretical framework directly to experimental results.

As you can see from the dispersion of Neon, along the $[\zeta ,0,0]$ direction (from $\Gamma$ to X), the two TA branches are degenerate, so we see only two curves (LA and TA). However, along other directions, this degeneracy can be lifted.

To bring this closer to your interests in semiconductors, let's look at silicon. Silicon has a diamond crystal structure, which has a basis of two identical atoms in the primitive cell. Therefore, like the 1D diatomic chain, it has both acoustic and optical modes.

In this diagram of silicon's dispersion:

• You can clearly see the three acoustic branches (LA, TA) originating from $\omega =0$ at the $\Gamma$ point ($\mathbf{q}=0$).
• You can also see the three optical branches (LO, TO) at higher energies. These exist because the diamond lattice has a two-atom basis.
• Notice how the two TA branches are degenerate along the $\Lambda$ direction (from $\Gamma$ to L), but they are not degenerate along the $\Delta$ direction (from $\Gamma$ to X). This is a consequence of the crystal's symmetry in those directions.

Conclusion

Today we have successfully generalized our understanding of lattice dynamics to three dimensions. We established the formal procedure for constructing and solving the 3D dynamical matrix and learned to interpret its solutions.

Key Takeaways:

• In 3D, atomic displacements and polarizations are vectors, and the force constants form a tensor.
• For a monatomic crystal, the dynamical matrix $\mathbf{D}(\mathbf{q})$ is a 3x3 matrix whose three eigenvalues ${\omega }_j^2(\mathbf{q})$ give the frequencies of the three acoustic phonon branches.
• The corresponding eigenvectors ${\mathbf{ϵ}}_j(\mathbf{q})$ are the polarization vectors, which determine the character of the mode.
• Longitudinal (LA) modes have polarization parallel to the wavevector $\mathbf{q}$.
• Transverse (TA) modes have polarization perpendicular to $\mathbf{q}$. There are two TA branches for every one LA branch.
• Phonon dispersion curves plot these branches along high-symmetry directions in the Brillouin zone, and features like mode degeneracy depend on this direction.

Preview of Next Lesson:
So far, our treatment has been purely classical. We found the normal modes of the lattice, but we haven't yet described them quantum mechanically. Your background in quantum field theory will be very useful here. In the next lesson, we will quantize the lattice vibrations. We will express the lattice Hamiltonian in terms of these normal modes and introduce creation and annihilation operators, formally defining the phonon as the quantum of lattice vibration.