46 CHAPTER 2. ELECTRONIC LEVELS IN SEMICONDUCTORS
ithasapositivecharge. It therefore responds to external electric and magnetic fieldsEandB,
respectively, according to the equation of motion
dkh
dt=e[E+vh×B] (2.5.4)wherekhandvhare the momentum and velocity of the hole.
Thus the equation of motion of holes is that of particles with apositive chargee. The mass of
the hole has a positive value, although the electron mass in its valence band is negative. When
we discuss the valence band properties, we refer to holes. This is because in the valence band
only the missing electrons or holes lead to charge transport and current flow.
2.6 BANDSTRUCTURE OF SOME IMPORTANT
SEMICONDUCTORS
In this section we will examine the band structure near the band edges for several important
materials. To represent the bandstructure on a figure that is two-dimensional, we draw theE-k
diagram in several panels wherekgoes from zero to its maximum value along the (100) direction
or the (111) direction, etc within the Brillouin zone. As shown in figure 2.6 for the fcc lattice,
the maximumk-value along the (100) direction is 2 π/a(1, 0 ,0). This point is called theX-point
and there are five other equivalent points, due to the cubic symmetry of the lattice. Similarly,
along the (111) direction, the maximumk-point isπ/a(1, 1 ,1)and seven other similar points.
This point is called theL-point. Thus we commonly display theE-kdiagram withkgoing from
the origin (called theΓ-point) to theX-point and from the origin to theL-point.
2.6.1 Direct and indirect semiconductors
Two types of band structures arise in semiconductors- direct and indirect. The top of the
valence band of most semiconductors occurs at effective momentum equal to zero. A typical
bandstructure of a semiconductor near the top of the valence band is shown in figure 2.11. We
notice the presence of three bands near the valence bandedge. These curves or bands are labeled
I, II, and III in the figure and are called the heavy hole (HH), light hole (LH), and the split off
hole bands.
The bottom of the conduction band in some semiconductors occurs atk=0. Such semicon-
ductors are called direct bandgap materials. Semiconductors, such as GaAs, InP, GaN, InN, etc.,
are direct bandgap semiconductors. In other semiconductors, the bottom of the conduction band
does not occur at thek=0point, but at certain other points. Such semiconductors are called
indirect semiconductors. Examples are Si, Ge, AlAs, etc.
Due to the law of momentum conservation, direct gap materials have a strong interaction with
light. Indirect gap materials have a relatively weak interaction with electrons.
When the bandedges are atk=0it is possible to represent the bandstructure by a simple
relation of the form
E(k)=Ec+^2 k^2
2 m∗