SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

3.3. TRANSPORT AND SCATTERING 95

Crystalline Solid

E-field

Electron

TIME

F ree el ectron path

DISTANCE

d = vt

v = μE

Mobility= VE

Figure 3.3: A typical electron trajectory in a sample and the distance versus time profile.

We can see that in the absence of any applied electric field, the occupation of a state with
momentum+kis the same as that of a−kstate. Thus there is net cancellation of momenta
and there is no net current flow. The distribution function in momentum space is shown schemat-
ically in figure 3.4a. The question we would like to answer is the following: If an electric field
is applied, what happens to the free electrons (holes)? When a field is applied the electron dis-
tribution will shift, as shown schematically in figure 3.4b, and there will be a net momentum of
the electrons. This will cause current to flow. If the crystal is rigid and perfect, according to the
Bloch theorem the electron states are described by

ψk(r, t)=ukexpi(k·r−ωt) (3.3.1)

whereω=E/is the electron wave frequency. There is no scattering of the electron in the
perfect system. Also, if an electric fieldEis applied, the electron behaves as a “free” space
electron would, obeying the equation of the motion

dk

dt

=Fext=−eE (3.3.2)

According to this equation the electron will behave just as in classical physics (in absence of
scattering) except the electron will gain energy according to the appropriate bandstructure rela-
tion.
In a real material, there are always imperfections which cause scattering of electrons so that the
equation of motion of electrons is not given by equation 3.3.2. A conceptual picture of electron
transport can be developed where the electron moves in space for some time, then scatters and