5.5: Vibrational States and Phonon Dispersion Curves

Phonons are vibrational modes of a crystal that form wave packets characterized by a frequency \(\omega\) and momentum \(\hslash k\). In a solid, they can be created by increasing temperature or providing some mechanical stress. As a result, phonons are important quasi-particles in solids that contribute to heat conduction and sound propagation. The nature of phonons in crystals are typically investigated by applying the adiabatic approximation, which assumes that nuclear motion does not disturb electronic degrees of freedom and vice versa. This approximation often works because electrons are very much lighter than nuclei so that their velocities are much higher. As a result, electrons adjust essentially immediately to any changes of nuclear positions and nuclear motions are not strongly influenced by electronic motion. Of course, there are characteristics of certain solids where electron-phonon interactions play significant roles, but we will not consider those properties here.

To show the qualitative features of phonons in crystals, we use a 1-dimensional linear chain of atoms of mass \(M\) and each pair of atoms separated by interatomic distance \(a\):

Figure 5.31

The equilibrium positions of each atom are the lattice sites \(ma\). Vibrational motion involves small displacements of the atoms from their equilibrium positions \(u_{n}(ma);n = (x,y,z)\), which are very small movements compared to the interatomic separation \(a\). Three orthogonal displacements yield three distinct phonon modes per atom of the chain. The interatomic potential consists of pairwise interactions that depend only on the interatomic distance \(\left| u_{n}(ma) - u_{n}\left( m'a \right) \right|\). Because \(u_{n}(ma)\) are small, this potential can be expanded in a Taylor series. Relative to the potential energy of the equilibrium structure and keeping terms up to second order, this potential takes form

\[\frac{1}{2}K_{mm'}\left( u_{n}(ma) - u_{n}\left( m'a \right) \right)^{2}, \nonumber \]

in which \(K_{mn}\) is the force constant between sites \(ma\) and \(m'a\). The resulting potential energy for the atom at \(ma\) is the sum of all pairwise potentials from other atoms in the chain:

\[V_{n}(ma) = \frac{1}{2}\sum_{m'}^{}{K_{mm'}\left( u_{n}(ma) - u_{n}\left( m'a \right) \right)^{2}} \nonumber \]

Now, if just nearest neighbor interactions in the potential are considered, then the expression is

\[V_{n}(ma) = \frac{1}{2}K_{vib}\left\lbrack \left( u_{n}(ma) - u_{n}\left( (m + 1)a \right) \right)^{2} + \left( u_{n}(ma) - u_{n}\left( (m - 1)a \right) \right)^{2} \right\rbrack \nonumber \]

Using this potential, the goal is to evaluate the vibrational frequencies \(\omega_{n}\), which is accomplished by solving Newton’s equation of motion for the displacement at each site \(ma\):

\[F_{n}(ma) = M\frac{d^{2}u_{n}(ma)}{dt^{2}} = - \frac{dV_{n}(ma)}{du_{n}(ma)} \nonumber \]

which is the harmonic restoring force \(F_{n}(ma)\) exerted at the atom at \(ma\) in the 1-d chain. To solve this problem, Bloch’s theorem is applied to the displacement:

\[u_{nk}(ma) = u_{n0}e^{- i\left( \omega_{n}t - kma \right)} = u_{n0}e^{- i\omega_{n}t}e^{ikma};\ \ n = x,y,z \nonumber \]

Using this Bloch function in both sides of Newton’s equation gives

\[M\frac{d^{2}u_{nk}(ma)}{dt^{2}} = - M\omega_{n}^{2}u_{n0}e^{- i\omega_{n}t}e^{ikma} = - M\omega_{n}^{2}u_{nk}(ma) \nonumber \]

and

\[- \frac{dV_{n}(ma)}{du_{nk}(ma)} = - K_{vib}\left\lbrack 2u_{nk}(ma) - u_{nk}\left( (m + 1)a \right) - u_{nk}\left( (m - 1)a \right) \right\rbrack \nonumber \]

\[- \frac{dV_{n}(ma)}{du_{nk}(ma)} = - K_{vib}\left\lbrack 2 - e^{ika} - e^{- ika} \right\rbrack u_{nk}(ma) = - K_{vib}\left\lbrack 2 - 2\cos{ka} \right\rbrack u_{nk}(ma). \nonumber \]

Setting these two expressions equal provides the relationship between phonon frequency and wavevector:

\[\omega_{n}(k) = 2\sqrt{\frac{K_{vib}}{M}}\sin\frac{ka}{2} \nonumber \]

If the atomic displacements are parallel to the chain \(u_{xk}(ma)\), which is assigned to the \(x\)-direction, the resulting phonons are called longitudinal modes. The force constant is \(K_{vib} \equiv K_{||}\) and these phonons are “stretching”-type vibrations. According to the phonon frequency dispersion relation, the maximum frequency occurs at the zone boundary \(X:k = \frac{\pi}{a}\).

Transverse modes

If the atomic displacements are perpendicular to the chain \(u_{yk}(ma)\) and \(u_{zk}(ma)\), the resulting phonons are called transverse modes. The force constant is \(K_{vib} \equiv K_{\bot}\) and these phonons are “bending”-type vibrations. Since the force constant for stretching is larger than for bending, \(K_{||} > K_{\bot}\), the maximum frequency of the doubly degenerate transverse phonon mode is lower than that for the longitudinal mode.

As for electronic states, the results of these calculations can be plotted in phonon dispersion curves and phonon density of states (DOS):

Figure 5.32

The phonon dispersion illustrates \(\omega_{n}(k)\) from the zone center \(\Gamma\ (k = 0)\) to the zone boundary \(X\ \left( k = \frac{\pi}{a} \right)\). The corresponding DOS curve identifies two peaks at the maximum frequencies for stretching and bending phonon modes. In the figures illustrating each phonon mode, the arrows signify the direction and “size” of the atomic displacements, which are, in fact, oscillations about the equilibrium positions of the atoms.

In both dispersion curves, the frequencies \(\omega_{n}(k)\) approach 0 linearly as \(k\) approaches 0. At the zone center, all displacements are completely in phase with each other. Therefore, the phonon modes at \(\Gamma\) are translational modes because all interatomic distances remain at their equilibrium values. Taking the slope of the phonon dispersion relations as \(k\) approaches 0 gives the speed of sound for longitudinal and transverse modes in the crystal:

\[\left( \frac{d\omega_{||}}{dk} \right)_{k \rightarrow 0} = \sqrt{\frac{K_{||}a^{2}}{M}} \nonumber \]

and

\[\left( \frac{d\omega_{\bot}}{dk} \right)_{k \rightarrow 0} = \sqrt{\frac{K_{\bot}a^{2}}{M}} \nonumber \]

As a result, these three modes are called acoustic modes of the crystal.

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1. Resonance integrals are often calculated using overlap integrals, β_m^′m = H_m^′m ∝ S_m^′m. After resonance integrals are determined, then the overlap matrix is set to the identity in Hückel theory.↩