Page 206 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Semi-Classical Boltzmann Transport Equation
relaxation time τ k()
k
as a function of may even allow for anisotropic
effects and different relaxation laws for the moments of a specific order.
This is why in the balance equations we might distinguish between parti-
cle relaxation, momentum relaxation and energy relaxation terms. This in
turn makes a difference for the dynamics of momentum and energy and
may allow for further distinguishing timescales of the respective
moments.

Bloch Before proceeding with local descriptions we look at the phenomenon
Oscillations called Bloch oscillations. Therefore, we drop our assumption of para-
bolic band and look at a the entire conduction band periodic in momen-
2
tum space that may be described by E = ( sin —k) ⁄ ( 2m)
2
= ( sin p) ⁄ ( 2m) . Suppose an electron starts at t = 0 with
p = —k = 0 at the conduction band edge at the space point x . Owing
0
to an applied electric field it accelerates and gains momentum accord-
E
ing to Newton’s law p ˙ = F = – qE . For small momenta
2
E = p ⁄ ( 2m) and thus the velocity of the electron is given by
⁄
⁄
v = ∂ E ∂ p = p m . For the given dispersion relation after half of the
Brillouin zone the velocity of the particle starts to decrease with increas-
ing momentum until at the zone edge the velocity is zero. Moreover fur-
ther acceleration results in a negative velocity, i.e., the particle turns back
to its starting point x where the whole process repeats. This is a striking
0
consequence of the conduction band picture of electrons since a constant
electric field applied results in an alternating current. It is contrary to our
experience of electronic behavior under normal circumstances and thus
there must rather be something wrong with the picture. Indeed, a very
important thing is missing in the description above: the interaction of the
electron with phonons. At room temperature the huge number of phonons
provide many scattering events so that the above scenario is not observ-
able. Even at very low temperatures where only a few phonons exist the
emission of phonons impedes this effect. As soon as an electron gains
enough energy it will bounce back in energy emitting a phonon and
restart to accelerate until it newly loses energy to the phonon system.

Semiconductors for Micro and Nanosystem Technology 203