## Energy Bands

2.2.1. Insulators, Semiconductors, and Conductors

When a solid is formed the energy levels of the atoms broaden and form bands with forbidden gaps between them. The electrons can have energy values that exist within one of the bands, but cannot have energies corresponding to values in the gaps between the bands. The lower energy bands due to the inner atomic levels are narrower and are all full of electrons, so they do not contribute to the electronic properties of a material. They are not shown in the figures. The outer or valence electrons that bond the crystal together occupy what is called a valence band. For an insulating material the valence band is full of electrons that cannot move since they are fixed in position in chemical bonds. There are no delocalized electrons to carry current, so the material is an insulator. The conduction band is far above the valence band in energy, as shown in Fig. 2.1 la, so it is not thermally accessible, and remains essentially empty. In other words, the heat content of the insulating material at room temperature T - 300 K is not sufficient to raise an appreciable number of electrons from the valence band to the conduction band, so the number in the conduction band is negligible. Another way to express this is to say that the value of the gap energy Eg far exceeds the value kBT of the thermal energy, where kB is Boltzmann's constant.

In the case of a semiconductor the gap between the valence and conduction bands is much less, as shown in Fig. 2.1 lb, so Eg is closer to the thermal energy kBT, and

Conduction Band

Conduction Band

Valence Band

### Valence Band

Figure 2.11. Energy bands of (a) an insulator, (b) an intrinsic semiconductor, and (c) a conductor. The cross-hatching indicates the presence of electrons in the bands.

the heat content of the material at room temperature can bring about the thermal excitation of some electrons from the valence band to the conduction band where they carry current. The density of electrons reaching the conduction band by this thermal excitation process is relatively low, but by no means negligible, so. the electrical conductivity is small; hence the term semiconducting. A material of this type is called an intrinsic semiconductor. A semiconductor can be doped with donor atoms that give electrons to the conduction band where they can carry current. The material can also be doped with acceptor atoms that obtain electrons from the valence band and leave behind positive charges called holes that can also carry current The energy levels of these donors and acceptors lie in the energy gap, as shown in Fig. 2.12. The former produces n-type, that is, negative-charge or electron, conductivity, and die latter produces p-type, that is, positive-charge or hole,

Figure 2.12. Sketch of the forbidden energy gap showing acceptor levels the typical distance Aa above the top of the valence band, donor levels the typical distance AD below the bottom of the conduction band, and deep trap levels nearer to the center of the gap. The value of the thermal energy /%7" is indicated on the right.

Figure 2.12. Sketch of the forbidden energy gap showing acceptor levels the typical distance Aa above the top of the valence band, donor levels the typical distance AD below the bottom of the conduction band, and deep trap levels nearer to the center of the gap. The value of the thermal energy /%7" is indicated on the right.

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 full valence band and a conduction band partly full 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 2.11c shows the energy bands for this case.

In actual crystals the energy bands are 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 / = (a/In, the wavelength A, and the velocity v of a wave is If = v. It is convenient to define the wavevector k = 2n/X to give / = (k/2n)v. For a matter wave, or the wave associated with conduction electrons, the momentum p = mv of an electron of mass m is given by p = (h/2n)k, where Planck's constant A is a universal constant of physics. Sometimes a reduced Planck's constant fi = h/2n is used, where p = hk. Thus for this simple case the momentum is proportional to die 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 = 10a, 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,..., 20n/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 kx 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 kx is proportional to the momentum px of the conduction electron in the x direction, and similarly for the relationship between ky and py.

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