Chemistry Reference
In-Depth Information
k
z
k
y
k
x
Figure 4.4
Surfaces of constant energy are described by ellipsoids near the lowest
conduction band X points. This occurs because there is a large effective
mass
m
l
along the
directions (pointing towards
, the Brillouin zone
centre), with a smaller effective mass,
m
t
perpendicular to the
direction.
an electron at the bottom of the conduction band as if it were a particle
in free space, with an effective mass
m
c
. If we apply an external electric
field
E
, such that the electron experiences a force
F
=−
e
E
this implies the
electron will move with an acceleration
a
given by
m
c
a
=−
e
E
; that is,
m
c
=
−
F
e
E
m
c
a
=
(4.10)
For small effective mass, the electron then accelerates more rapidly in the
solid than in free space. This is at first surprising, but reflects the fact
that the electron is acted on not only by the external field
E
but also by
the periodic field of the lattice. If we took explicit account of both fields
in discussing the dynamics of the electron it would exhibit its ordinary
mass.
In order to derive the correct form for the electron effective mass, we
consider an electron represented by a wave packet near the bottom of the
conduction band, so that the electron velocity is then given by the group
or energy velocity,
v
g
,definedas
d
d
k
1
dE
d
k
v
g
=
=
(4.11)
where
k
is the Blochwavevector associatedwith the electron state of energy
E, and for simplicity we only consider motion in one dimension. In the
applied field
E
, the electron experiences a force
F
such that its energy E
