Advanced Solid State Physics

10 Transport Regimes

Before we start with Transport Regimes we will discuss the Linear Response Theory which we need
to understand.

10.1 Linear Response Theory

What we want to look at is the linear response of a solid to a time dependent force for example
electric field or stress. Linear response means that the response of the solid is proportional to the
force. An other important consideration is causality. Causality requires that effects follow causes this
means that e.g. a bell rings when you hit it and not the other way around. This will lead us to the
Kramers-Kronig relations and the fluctuation dissipation theorem.
To start with the linear response theory we first look at an ordinarylineardifferential equation:

L g(t,t′) = δ(t−t′) (94)

Wheregis called Green’s function, which is the solution to a linear differential equation to aδ-function
driving force (Be careful: the name Green’s function has different meanings!). For instance a damped
spring-mass system:

m

d^2 g

dt^2

+b

dg

dt

+kg = δ(t) (95)

withmthe mass,bthe damping andkthe spring constant. This differential equation has the solution

g(t) =

1

b

exp(

−bt

2 m

) sin(

√

4 mk−b^2

2 m

t), t > 0 (96)

Whereg(t)is the Green’s function for this system which is a decaying sinusoidal function. Green’s
functions are also called impulse response functions.

Figure 68: Example of a responseg(t)toδ(t)driving force (eqn. (96))

The advantage of Green’s functions is that every function can be written as a sum overδpeaks:

f(t) =

∫

δ(t−t′)f(t′)dt′

This means that the solution to a linear differential equationLu(t) = f(t)can be found by a
superposition of Green’s functions.

u(t) =

∫

g(t−t′)f(t′)dt′