Figure 1.16 Subshell electron capacity. Notice that the number of sites in each level increases as the energy level, n, increases. Also notice that the 3d level has lower energy than the 4s level.
Appendix 1.2 Semiconductor Materials
Figure 1.17 shows the portion of the periodic table that contains the elements used in semiconductors, showing how many electrons are in the upper shells. All the lower shells are full. Silicon, for example, has 14 electrons, so the electrons fill up bands 1s (2), 2s (2), 2p (6), 3s (2), and 2p (the last 2). If you superimpose these numbers into Figure 1.16, you see that the two lowest energy levels are full, but energy level n = 3 (s + p) has four electrons, with the possibility of accepting another four (in the 2p level) to complete its orbit.
Consider another element that is used a great deal in semiconductors: antimony, Sb. It has 51 electrons. By looking again at Figure 1.16, we find the last occupied level is 5p, with three electrons. All the lower levels are full. Thus, the last occupied energy level, level 5, has five electrons – two in 5s and three in 5p levels – which gives it a chemical valence of 5. We will use these numbers in the next chapter to explain the difference between insulators, conductors, and semiconductors.
Figure 1.17 Portion of the periodic table emphasizing elements used in semiconductors.
Appendix 1.3 Calculating the Rydberg Constant
To complete some of the details of this chapter, the Rydberg constant can be calculated from more basic units. It is
where me is the mass of the electron (me = 9.1 × 10−31 kg), e is the electronic charge (e = 1.6 × 10−19 C), ε is the permittivity of the material, that is the product of the relative permittivity of the material (e.g. for hydrogen εr = 253.8) times the permittivity of free space (ε0 = 8.85 × 10−12 m−3 kg−1 s2 C2), h is Planck's constant (h = 1.06 × 10−34 J s), and c is the speed of light (c = 3 × 108 m s−1). All of these quantities are fundamental physical constants.
One thing that we often forget or do not check is the fact that any measurement is composed of a numerical value and a unit. In Eq. (1.6) I show several quantities, each of which has a number and unit(s). The result has to agree with the numerical calculation and the units. Very few people check the units when they perform an operation, which can result in mistakes (remember the fiasco when the Mars orbiter failed because of the confusion between metric and English units). So, let me do this with Eq. (1.6). First the units:
Look at the three terms in the denominator. There are meter (m) terms in the denominator with exponents −6 + 6 + 1 = 1 m, so only one meter unit remains in the denominator. Similarly, with the exponents of the kilograms, there are −2 + 3 = 1 in the denominator. There are four Qs in the denominator, and the seconds in the denominator cancel out (4 – 3 – 1 = 0). Now the kilograms and the coulombs in the numerator cancel those in the denominator, leaving only the reciprocal of a meter as the remaining unit, in agreement with Eq. (1.6).
Now let's do the numbers.
This agrees with the published result.
2 Energy Bands
OBJECTIVES OF THIS CHAPTER
We saw in the previous chapter that an atom's electrons have precise energy values (we represent them as orbits or levels). We also saw that electrons must have distinct quantum numbers (designations), which limits the number of electrons in an atom that can have a given energy. As atoms have more and more electrons, the electrons have to occupy higher and higher energy levels. An electron must absorb from somewhere the exact energy needed to jump from one level to a higher one. When it falls back to a lower level, it donates the same amount of energy. This is what happens in a gaseous state where the atoms are separated by large distances and do not interact with each other. This model beautifully explains the absorption and emission spectral lines of the elements, the sun, and the stars.
In this chapter, we are going to push the atoms closer and closer together until we form a solid. Now the atoms and electrons start interacting with each other and forming bonds, which is what keeps them together in a crystallographic structure. As we push them together, the energy levels have to separate because, in a system, according to Pauli's exclusion principle, no two electrons can have the same quantum number. The levels split into bands. Depending on how the bands spread, the material behaves like a conductor, an insulator, or a semiconductor.
Finally, we will analyze the specific case of semiconductors and how the electrons fit into the bands. We will also see how the lack of an electron, which we call a hole, is equivalent to a positive charge.
2.1 Bringing Atoms Together
From the previous chapter, we know that an atom has discrete energy levels, and electrons – unless disturbed by a packet of energy – stay in a stable orbit. We show these energy levels as lines, as in Figure 2.1.
We also saw in the previous chapter that the Pauli exclusion principle states that no two electrons can share the same quantum numbers. If we push together two hydrogen atoms so that they interact and form a single system, the energy levels of the electrons in one atom must have slightly different values than the electrons in the other atom. I show this in Figure 2.2. What was a single level in hydrogen gas, with all the atoms separated by very large distances and acting as independent systems, now becomes two slightly different levels – yes, they are very, very close to each other, but they are still different. This ensures that all the electrons in the system, which is now composed of two atoms, have different quantum numbers.