Page 142 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Periodic Potentials in Crystal
The main features of a band structure as discussed in Figure 3.13 are
experimentally well known and are therefore the corner pillars that any
calculation must reproduce.
3.3.4 Effective Mass Approximation

Tensor A closer look to the band extremes indicates that electrons close to those
points are well described with a quadratic energy dispersion as for the
free electron if we approximate the local behavior around these points by
a parabolic band structure. The center of the Brillouin zone in the con-
duction band as shown in Figure 3.13 shows this quadratic dependence of
k
E
the energy on the wave vector . The energy of a free electron is given
2 2
as E = — k ⁄ ( 2m) . Hence we can approximate the electronic behavior
at point C in Figure 3.13 by taking the second derivative of the energy
with respect to k

∂ 2 2 1
m∗
∂ k k∂ j E k() = — ---------- ij (3.98)
i
2
⁄
(3.98) allows us to use the dispersion relation E k() = ( — k k ) 2m∗
i j ij
near the conduction band minimum. The mass calculated by (3.98) in
general is not equal to the free electron mass. Moreover, it is a tensor–
like quantity that depends on the direction in k-space where the respec-
tive derivative is taken. This is why it is called the effective mass. There-
fore, in this sense the electron is a quasi–particle, as will be discussed in
Section 5.2.4, that only behaves like a free electron but with a changed
mass. The importance of this approximation will become clear with the
fact that most electrons are to be found at the band edge minimum at
T = 300K . Therefore, it will be the effective mass that enters all rela-
tions of electronic transport.

Anisotropy The definition given in (3.98) immediately suggests that there must be
some remainders from the crystal anisotropic structure inside the elec-
tronic mass. Indeed, the masses in different semiconductors depend

Semiconductors for Micro and Nanosystem Technology 139