Advanced Solid State Physics

5.3 Central Equations

We can also take thecentral equationsto do this numerically. Now we have the wave equation with
the speed of light as a function ofr, which means we assume for example that there are two materials
with different dielectric constants (so there are also different values for the speed of light inside the
material).

c^2 (r)∇^2 u=

d^2 u

dt^2

This is a linear differential equation with periodic coefficients. The standard technique to solve this
is to expand the periodic coefficients in a fourier series (same periodicity as crystal!):

c^2 (r) =

∑

G

UGexp (iG·r) (37)

This equation describes the modulation of the dielectric constant (and so the modulation of the speed
of light).UGis the amplitude of the modulation. We can also write

u(r,t) =

∑

k

ckexp (i(k·r−ωt)). (38)

Putting the eqns. (37) and (38) into the wave equation gives us
∑

G

UGexp (iG·r)

∑

k

ckk^2 exp (i(k·r−ωt)) =

∑

k

ckω^2 exp (i(k·r−ωt)) (39)

On the right side we have the sum over all possiblek-vectors and on the left side the sum over all
possiblek- andG-vectors. It depends on the potential how many terms ofGwe have to take. Eqn.
(39) must hold for everyk, so we take a particular value ofkand then search through the left side of
eq. (39) for other terms with the same wavelength. So we take all those coefficients and write them
as an algebraic equation:
∑

G

(k−G)^2 UGck−G=ckω^2

The vectorsGdescribe the periodicity of the modulation of the material. So we only do not know
the coefficients andωyet and we can write this as a matrix equation.

We look at a simple case with just a cosine-potential in x-direction. This means the speed of light is
only modulated in the x-direction. Looking at just 3 relevant vectors ofGand with

c^2 (r) =U 0 +U 1 exp (iG 0 r) +U 1 exp (−iG 0 r)

we get




(k+G 0 )^2 U 0 k^2 U 1 0

(k+G 0 )^2 U 1 k^2 U 0 (k−G 0 )^2 U 1

0 k^2 U 1 (k−G 0 )^2 U 0





·







ck+G

ck

ck−G





=ω^2







ck+G

ck

ck−G





 (40)

as the matrix equation.GandUcome from the modulation of the material,kis just the value ofk.
So what we do not know are the coefficients andω. The coefficients have to do with the eigenvectors