Advanced Solid State Physics

6 Phonons

In the long wavelength limit (at low frequencies), sound obeys the wave equation:

c^2 ∇^2 u(x,t) =

∂^2 u(x,t)

∂t^2

(42)

By substituting the speed of light with the speed of sound we can make use of all the previously stated
formulas with which we computed the properties of photons, to compute the properties of phonons in
the long wavelength limit.

6.1 Linear Chain

We will now take a closer look at two simple models, the first being theLinear Chain. In this model,
we have a chain of identical masses, which are connected to their neighbors by a linear force.

m m m m

Figure 12: Linear chain of atoms

Thus,Newtons lawfor thesthmass reads as follows:

m

∂^2 us(x,t)

∂t^2

=C(us+1− 2 us+us− 1 )

with a spring constantCand the lattice constanta.
By assuming harmonic solutions like

us=Akei(ksa−ωt)

we obtain the following equation:

−ω^2 m=C(eika−2 +e−ika)

With using the Euler identities this equation simplifies to

−ω^2 m= 2C(1−cos(ka)).

Hence, the solutions forωare:

ω=

√

4 C

m

∣∣

∣∣sin

(

ka

2

)∣∣

∣∣

Now it is possible to draw thedispersion relationfor this 1-dimensional chain.

TheDebye-frequency, which is the highest frequency of the normal mode, can be computed very
easily: The shortest wavelength we can create in our chain isλ= 2aandkcan be expressed like
k=^2 λπ, sokD=πa. ThiskDalso gives us theBrillouin zone boundaries. It also should be noted,