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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 = Ak ei(k·r−ωt) where k is the propagation vector or wave vector (|k| = 2π/λ), ω is the angular frequency, Ak is the kdependent 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 = u0 eik·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 = u0 eik·(r+T ) = u0 eik·r eik·T = ur eik·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: p ω = 2 K/m| sin(|k|a/2)| where K is the elastic force constant. Page 1 of 5 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 p 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 = u0 eik·r that the phonon modes are periodic in the wave vector. We can write k = τ + k0 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 p 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 reciprocal 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 q K M +m 2 ± (M + m)2 − 4M m sin2 (ka/2) ω± =K Mm Mm Page 2 of 5 Roger Johnson Structure and Dynamics: Phonons and electrons Lecture 5 p As before, for small wave vector the acoustic mode satisfies the relation ω− = c|k|, where c =p 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)/(M m). 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 confused. 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 elements of the crystal Laue Class. For example, for space groups P 4mm and P 2mm: Page 3 of 5 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: Page 4 of 5 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 functions and dispersion relation: h̄2 k2 Ψ = eik·r , and E = 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 degenerate. 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 : πx πx Ψ1 = exp i , and Ψ2 = exp −i a 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: πx πx Ψ+ = Ψ1 + Ψ2 = 2 cos , and Ψ− = Ψ1 − Ψ2 = 2i sin a a The real charge density is equal to the square of the wavefunction, giving: πx πx , and ρ− = Ψ− Ψ∗− = 4 sin2 ρ+ = Ψ+ Ψ∗+ = 4 cos2 a a 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. Page 5 of 5