Roger Johnson Structure and Dynamics: Phonons and electrons Lecture 5

5.1. Summary

In Lecture 2 we derived the normal modes of the water molecule based upon its point group symmetry.

In this lecture we will consider the vibrational normal modes of a crystal that form dispersive, wave-like

patterns of atomic displacements. The respective quasi-particles are known as a phonons, and our analysis

of phonon modes based upon the symmetry of the crystal can be applied to any wave-like material property,

for example electrons and their band structure.

5.2. Vibrations on a 1D monatomic chain

The 1D monatomic chain is comprised of a single-atom basis and an infinite 1D lattice with basis vector a.

If we only consider atomic displacements along the chain direction there are N = 1 degrees of freedom per

atom — a translational normal mode of an isolated atom.

Assuming a classical model with a harmonic potential, it can be shown that the vibrational normal modes

of a crystal (long-range pattern of atomic displacements) correspond to a set of travelling waves:

ur,k = Akei(k·r−ωt)

where k is the propagation vector or wave vector (|k| = 2π/λ), ω is the angular frequency, Ak is the k-

dependent amplitude of the mode, and the atom in the nth unit cell is reached by the vector r = na.

In Lecture 2 we defined a normal mode as ‘a snap-shot of a pattern of atomic displacements’. Similarly, the

vibrational mode in any unit cell of the crystal can be described by the complex mode ur = u0eik·r, where

u0 is the mode of the zeroth unit cell. By applying any translational symmetry operator of the lattice, T ,

we find:

T [ur] = ur+T = u0eik·(r+T ) = u0eik·reik·T = ureik·T

Therefore, the mode ur is symmetric by the translational symmetry of the lattice up to a complex factor

whose effect is to shift the phase of the travelling wave by k · T . The mode ur is known as a Bloch wave.

The vibrational modes of a molecule do not propagate, i.e. k = 0. Vibrational waves of the crystal do

propagate due to interatomic interactions from one unit cell to the next. Furthermore, they propagate with

a strict relationship between the angular frequency and the wave vector that depends upon the physical

nature of the interatomic interactions. This relationship is known as the dispersion relation. By solving

the equations of motion for each Bloch wave, one can derive a general relationship between the angular

frequency and wave vector:

ω = 2√K/m| sin(|k|a/2)|

where K is the elastic force constant.

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Roger Johnson Structure and Dynamics: Phonons and electrons Lecture 5

In the limit of long wavelength, the angular frequency is linear in the wave vector, and has the general form

ω = c|k|, where c = a√K/m is the speed of the wave in the crystal. This dispersion relation is equivalent

to the propagation of sound waves (where c is the speed of sound). The vibrational mode of the crystal is

therefore known as an acoustic mode.

The dispersion of the acoustic mode is periodic in k, repeating every 2π/|a|. This is exactly what is expected

for the reciprocal lattice of the chain, where in 1D a* = 2π/|a|. The reciprocal space periodicity means that

we only ever need to consider wave vectors in the range −π/a < k < π/a. This leads us to the concept of

the First Brillouin zone discussed in detail later in this lecture. We note that one can see directly from

the mode ur = u0eik·r that the phonon modes are periodic in the wave vector. We can write k = τ + k′

where τ is a reciprocal lattice vector and τ · r = 2πn, by definition (see Lecture 4).

One special case to consider is when the wavelength of the mode is exactly two unit cells long, which

corresponds to k = π/a. At this point, ω = 2√K/m — a constant. The group velocity ∂ω/∂k = 0, and

therefore the vibrational phonon mode is a standing wave. Zero group velocity at k = π/|a| is in fact a

strict requirement of symmetry (consider how the group velocity transforms under both the translational

symmetry of the lattice and inversion symmetry at the origin).

By imposing boundary conditions on the lattice the wave vector must take discrete values:

k =2πn

Na

There are therefore N allowed values of k in the range −π/a < k < π/a.

5.3. Vibrations on a 1D diatomic chain

The 1D diatomic chain is comprised of a two-atom basis of distinct masses M and m and an infinite lattice

with basis vector a. The unit cell has doubled compared to the monatomic chain, and accordingly, the re-

ciprocal lattice (a∗ = 2π/|a|) has halved. For a given vibrational normal mode, both atoms vibrate with the

same angular frequency and the same wave vector, but with different amplitudes. Again, only considering

displacements along the chain we have two degrees of freedom — one translational mode and one vibrational

mode of an isolated two-atom basis.

The translational mode of our diatomic basis corresponds to an in-phase translation of both atoms. As

such it leads to an acoustic mode as in the case of the monatomic chain. The vibrational mode of the

diatomic basis corresponds to the two atoms moving out of phase. If the two atoms had opposite charge

(for example in an ionic material), their motion described by this mode is equivalent to that induced by

electromagnetic radiation at optical frequencies. For this reason it is named an optic mode. The two

dispersion relations for the diatomic chain are given by

ω2± = K

M +m

Mm± K

Mm

√(M +m)2 − 4Mm sin2(ka/2)

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Roger Johnson Structure and Dynamics: Phonons and electrons Lecture 5

As before, for small wave vector the acoustic mode satisfies the relation ω− = c|k|, where c = a√K/(2(M +m)).

Unlike the acoustic mode, the optic mode has finite frequency at k = 0; in this case ω+(0) =√

2K(M +m)/(Mm).

5.4. Phonons of a 3D crystal

The first Brillouin zone is a Wigner-Seitz cell of the reciprocal lattice. However, the two must not be con-

fused. The Wigner-Seitz cell is a primitive repeating unit cell of any generic lattice, whereas Brillouin zones

are always centred at the origin of the reciprocal lattice, and higher zones are constructed from the next

‘shell’ of reciprocal lattice points. The first Brillouin zone is constructed by drawing connecting lines between

the origin and the nearest set of reciprocal lattice points. The connecting lines are then bisected, forming

the first Brillouin zone boundary:

One can identify symmetry points, directions, and planes in the 3D Brillouin zone from the symmetry ele-

ments of the crystal Laue Class. For example, for space groups P4mm and P2mm:

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Roger Johnson Structure and Dynamics: Phonons and electrons Lecture 5

In 3D there will exist three acoustic normal modes, and (3Z−3) optic modes, where Z is the number of atoms

in the primitive unit cell. For a given wave vector direction one acoustic mode is longitudinal — atomic

displacements are parallel to the propagation wave vector, and two acoustic modes are transverse — atomic

displacements are orthogonal to the wave vector. In general, the force constants of the three acoustic modes

could be different to each other, leading to different dispersion curves. Whether or not acoustic modes have

the same force constant depends upon the symmetry of the crystal. If two vibrational modes are associated

with symmetry equivalent forces, then they will be energetically degenerate. For example, the transverse

modes with wave vector parallel to a four-fold axis will be degenerate.

The optic modes can also be grouped into threes, one longitudinal and two transverse, and similar energetic

degeneracies can be found.

It is important to note that whether or not a phonon is longitudinal or transverse is determined by the

direction of the propagation wave vector in the first Brillouin Zone. Phonons that exist in higher Brillouin

zones with wavevector Q have the same frequency as symmetry equivalent phonons in the first Brillouin

zone with wavevector k = Q− τ , but the polarisation of the phonon must be determined by k, not Q.

As previously discussed, the group velocity is a polar vector. By appreciating how a polar vector transforms

under the symmetry elements of the Laue class, one can determine symmetry constraints on its direction

throughout the Brillouin zone:

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Roger Johnson Structure and Dynamics: Phonons and electrons Lecture 5

5.5. Symmetry and electronic band structure

The phonon case has been derived classically. However, we can draw complete analogy with a quantum

mechanical case. Here, the symmetry of the potential is replaced with the symmetry of the Hamiltonian, and

the symmetry of the modes with the symmetry of the wave functions. Symmetry constraints on electronic

properties such as the group velocity are exactly analogous with phonon dispersion relations. Furthermore,

one can also identify energetic degeneracy that arises due to the symmetry of the reciprocal lattice.

We begin with the free electron model in 1D. We have the following results for the electron wave func-

tions and dispersion relation:

Ψ = eik·r, and E =h̄2k2

2me

No two points in the interior of the first Brillouin zone are separated by a reciprocal lattice vector. At

the zone boundary, however, opposite points can be related by a reciprocal lattice vector. Then, by the

translational symmetry of the reciprocal lattice, the respective wave functions will be energetically degener-

ate. The two degenerate wave functions at the zone boundary are constructed by taking a linear combination

of the plane wave functions for +k and -k, as k = −k + τ :

Ψ1 = exp(iπx

a

), and Ψ2 = exp

(−iπx

a

)As for the case of phonons, the group velocity at the zone boundary is necessarily zero by symmetry. The

wave functions at these points must therefore correspond to standing waves. There are just two solutions:

Ψ+ = Ψ1 + Ψ2 = 2 cos(πxa

), and Ψ− = Ψ1 −Ψ2 = 2i sin

(πxa

)The real charge density is equal to the square of the wavefunction, giving:

ρ+ = Ψ+Ψ∗+ = 4 cos2(πxa

), and ρ− = Ψ−Ψ∗− = 4 sin2

(πxa

)The ρ+ maxima coincide with the positive charge centres of the ionic lattice, whereas the ρ− maxima are

located in between the positive charge centres. The potential energy of ρ+ is therefore reduced and the

potential energy of ρ− is increased with respect to the average travelling wave at the interior of the zone.

This shift in potential energy through perturbation by the periodic potential results in the opening of gaps

in the electron dispersion at the Brillouin zone boundary.

These gaps separate electron states into bands, and are therefore known as band gaps. Band gaps are a

direct result of the symmetry of the crystal.

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