Introduction to Nanotechnology

22 INTRODUCTION TO PHYSICS OF THE SOLID STATE
conductivity, as will be clarified in Section 2.3.1. These two types of conductivity in
semiconductors are temperature-dependent, as is the intrinsic semiconductivity.
A conductor is a material with a fill valence band, and a conduction band partly
fill with delocalized conduction electrons that are efficient in carrying electric
current. The positively charged metal ions at the lattice sites have given up their
electrons to the conduction band, and constitute a background of positive charge for
the delocalized electrons. Figure 2.1 IC shows the energy bands for this case.
In actual crystals the energy bands arZ much more complicated than is suggested
by the sketches of Fig. 2.11, with the bands depending on the direction in the lattice,
as we shall see below.
2.2.2. Reciprocal Space
In Sections 2.1.2 and 2.1.3 we discussed the structures of different types of crystals
in ordinary or coordinate space. These provided us with the positions of the atoms in
the lattice. To treat the motion of conduction electrons, it is necessary to consider a
different type of space that is mathematically called a dual space relative to the
coordinate space. This dual or reciprocal space arises in quantum mechanics, and a
brief qualitative description of it is presented here.
The basic relationship between the frequency f = 0/2q the wavelength A, and
the velocity u of a wave is Af = u. It is convenient to define the wavevector k = 2x11
to give f = (k/2n)u. For a matter wave, or the wave associated with conduction
electrons, the momentum p = M U of an electron of mass M is given by p = (h/2n)k,
where Planck’s constant h is a universal constant of physics. Sometimes a reduced
Planck’s constant h = h/2n is used, where p = hk. Thus for this simple case the
momentum is proportional to the wavevector k, and k is inversely proportional to the
wavelength with the units of reciprocal length, or reciprocal meters. We can define a
reciprocal space called k space to describe the motion of electrons.
If a one-dimensional crystal has a lattice constant a and a length that we take to
be L = loa, then the atoms will be present along a line at positions x = 0, a,
2a, 3a,. . . , 10a = L. The corresponding wavevector k will assume the values
k = 2n/L, 4n/L, 6n/L, . . . ,2On/L = 2n/a. We see that the smallest value of k is
2n/L, and the largest value is 2n/a. The unit cell in this one-dimensional coordinate
space has the length a, and the important characteristic cell in reciprocal space,
called the Brillouin zone, has the value 2n/a. The electron sites within the Brillouin
zone are at the reciprocal lattice points k = 2nn/L, where for our example n =
1,2,3, . . . , 10, and k = 2n/a at the Brillouin zone boundary where n = 10.
For a rectangular direct lattice in two dimensions with coordinates x and y, and
lattice constants a and b, the reciprocal space is also two-dimensional with the
wavevectors k, and ky. By analogy with the direct lattice case, the Brillouin zone in
this two-dimensional reciprocal space has the length 2n/a and width 2n/b, as shown
sketched in Fig. 2.13. The extension to three dimensions is straightforward. It is
important to keep in mind that k, is proportional to the momentum p, of the
conduction electron in the x direction, and similarly for the relationship between
5. and Py.