5. Lattice Vibrations

Solid State Physics

355

Introduction

Unlike the static lattice model, which deals with average positions of atoms in a crystal, lattice dynamics extends the concept of crystal lattice to an array of atoms with finite masses that are capable of motion.This motion is not random but is a superposition of vibrations of atoms around their equilibrium sites due to interactions with neighboring atoms.A collective vibration of atoms in the crystal forms a wave of allowed wavelengths and amplitudes.

Applications

Lattice contribution to specific heat

Lattice contribution to thermal conductivity

Elastic properties

Structural phase transitions

Particle Scattering Effects: electrons, photons, neutrons, etc.

BCS theory of superconductivity

Normal Modes

http://www-users.york.ac.uk/~dmh8/cymatic/PhysicalModelling.htm

Check this out...

x1

x2

x3

x4

x5

u1

u2

u3

u4

u5

Atoms are bound to each other with bonds that have particular directions and strengths. These bonds may be modeled as ideal springs, as long as the displacements from the equilibrium positions of the atoms is small.

Were going to consider a classical model consisting of masses connected by ideal springs. We can write expressions for the forces, apply Newtons 2nd law and generate a set of differential equations, one for each atom. Our goal is to find solutions for which the atomic displacements have the same frequency.

Quantum mechanics can be used to describe atomic motions. Both approaches lead to the same result.

x1

x2

x3

u1

u2

u3

Consider this simplified system...

Suppose that only nearest-neighbor interactions are significant, then the force of atom 2 on atom 1 is proportional to the difference in the displacements of those atoms from their equilibrium positions.

Net Forces on these atoms...

The forces tend to restore the atoms to their equilibrium separations. If u1 > u2, the force on atom 1 is in the negative x direction.

Normal Modes

Mr. Newton...

To find normal mode solutions, assume that each displacement has the same

sinusoidal dependence in time.

The complex notation is used for later convenience. Of course, displacements are real quantities, so we only really use the real part.

The second derivative of u with respect to t is just

so we can then write...

Normal Modes

Of course, one possible solution is u = o for all of the atoms...boring!

Other solutions only exist if the determinant formed from the coefficients is zero. The determinant will allow us to find the normal mode frequencies.

The bottom equation is satisfied by three values of omega squared.

Normal Modes

Only two equations can be considered independent and the displacements are not completely determined by these equations. However, any two ratios of displacements can be evaluated.

This is all easily extended to the two and three dimensional cases following the same procedure as we did here.

Before we can go further, we need to define a useful construction called the Brillouin Zone.

Brillouin Zones

C

D

O

k1

k2

The value of the Brillouin Zone is that it gives a vivid geometrical interpretation of the diffraction condition:

We divide both sides by 4 to get,

Working in reciprocal space, select a vector G from the origin to a reciprocal lattice point. Construct a plane normal to this vector at its midpoint.

This plane forms a boundary.

An x-ray beam will be diffracted if its wavevector has the magnitude and direction required by the diffraction equation. The diffracted beam will be in the direction k G, so delta k = - G.

The Brillouin construction shows us all wavevectors k which can be Bragg-reflected from the crystal.

Brillouin Zones

The first Brillouin zone is defined to be the Wigner-Seitz primitive cell of the reciprocal lattice.Or, it could be defined as the set of points in k space that can be reached from the origin without crossing any Bragg plane.The second Brillouin zone is the set of points that can be reached from the first zone by crossing only one Bragg plane.The n th Brillouin zone can be defined as the set of points that can be reached from the origin by crossing n - 1 Bragg planes, but no fewer.

The set of planes that are perpendicular bisectors of the reciprocal lattice vectors is of particular interest to us as we move into the study of lattice vibrations and wave propagation in crystals.

These planes divide the reciprocal lattice space into fragments, with the central fragment, the primitive unit cell, called the Wigner Seitz primitive cell for the reciprocal lattice.

Wigner-Seitz Primitive Cell

How can we choose primitive cells?

One algorithm is the Wigner-Seitz cell.

Steps:

Draw lines to connect a given lattice point to all nearby lattice points

(2) At the midpoint and normal to these lines, draw new lines

(or planes in 3D)

(3) The smallest area (or volume in 3D) enclosed in this way is the Wigner-Seitz primitive cell

You can also construct Wigner-Seitz cells for the FCC and BCC lattices. These look a bit strange, but they will be useful.

Wigner-Seitz Primitive Cells

Brillouin Zones

BCC

Brillouin Zones

Brillouin Zones

Back to Normal Modes

Longitudinal Wave

u represents the displacement of atoms on plane n

q is the wavevector

a is the spacing between planes of atoms and will depend on the particular value of q

Transverse Wave

Back to Normal Modes

Traveling wave solutions

Dispersion Relation

Dispersion Relation

q

0.6

The boundary of the first Brillouin zone lies at q = +/- pi/a

The slope of w versus q is 0 at the boundary.

The dashed line represents the continuum limit (interatomic distance much smaller than wavelength).

The special significance of the phonon wavevectors that lie on the boundary will be developed shortly.

First Brillouin Zone

What range of qs is physically significant for elastic waves?

The range to + for the phase qa covers all possible values of the exponential. So, only values in the first Brillouin zone are significant.

qa varies between pi and +pi, so q ranges from -pi/a to +pi/a, which is the first Brillouin zone of the linear lattice

First Brillouin Zone

There is no point in saying that two adjacent atoms are out of phase by more than . A relative phase of 1.2 is physically the same as a phase of 0.8 .

The wave represented by the solid curve conveys no information not given by the dashed curve. Only wavelengths longer than 2a are needed to represent the motion.

We need both positive and negative values of q because waves can propagate to the left or to the right.

We can treat values of q outside this zone by subtracting multiples of 2pi/a to give an equivalent wavevector inside the first Brillouin zone

First Brillouin Zone

At the boundaries q = /a, the solution

Does not represent a traveling wave, but rather a standing wave. At the zone boundaries, we have

Alternate atoms oscillate in opposite phases and the wave can move neither left nor right.

This situation is equivalent to what happens in Bragg reflection of x-rays. When the Bragg condition is satisfied a traveling wave cannot propagate in the lattice, but through successive reflections back and forth, a standing wave is set up.

The critical value at plus/minus pi/a found here satisfies the Bragg condition: theta = pi/s, d = a q = 2pi/lambda, n=1, so lambda = 2a. With x-rays it is possible to have n greater than 1 because the amplitude of the wave has meaning in between the atoms, whereas the displacement amplitude of an elastic wave only has meaning at the atoms themselves.

Group Velocity

The transmission velocity of a wave packet is the group velocity, defined as

The gradient of the frequency with respect to q gives the velocity for the transmission of energy through the medium.

Note that the group velocity is zero at the zone boundary and we expect zero transmission at the boundary.

In separate papers, Sommerfeld and Brillouin wrote that, in anomalous dispersion, the group velocity cannot be the signal velocity.1 Indeed, in anomalous dispersion, the group velocity goes through both negative and positive infinite values. It also goes through values greater than the speed of light2 (as does the phase velocity).

Group Velocity

Phase Velocity

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave will propagate. You could pick one particular phase of the wave (for example the crest) and it would appear to travel at the phase velocity. The phase velocity is given in terms of the wave's angular frequency and wave vector k by

Note that the phase velocity is not necessarily the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will propagate.

Long Wavelength Limit

When qa

Force Constants

and integrate

The integral vanishes except for p = r. So, the force constant at range pa is

for a structure that has a monatomic basis.

In metals the effective forces may be quite long range...interactions have been found that extend more than 20 atomic planes.

We can generalize the dispersion relation to p nearest planes as shown.

r is some integer

Diatomic Coupled
Harmonic Oscillators

Consider a cubic crystal where atoms of mass m1 and m2 lie on planes that are interleaved.

It is not essential that the masses be different, but if the two atoms of the basis are on two non-equivalent sites, then either the masses are different or the force constants are different.

As before, assume only nearest-neighbor interactions and the force constants are all the same.

For each q value there are two values of .

These branches are referred to as acoustic

and optical branches. Only one branch

behaves like sound waves ( /q const. For q0).

For the optical branch, the atoms are oscillating

in antiphase. In an ionic crystal, these charge

oscillations (magnetic dipole moment) couple to

electromagnetic radiation (optical waves).

Definition: All branches that have a frequency

at q = 0 are optical.

Diatomic Coupled
Harmonic Oscillators

q

The lower curve, corresponding to the minus sign, is the acoustic branch.

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