Electron States in Solids
QUALITATIVE DESCRIPTION OF SOLIDS AND THEIR ENERGY BANDS
Now that we have developed our quantum mechanical weapons and have a proper theory of atoms, we can move to the description of energy bands in solids, which are the building blocks of micro- and nano-devices. Before we embark into a quantitative analysis, we first give a qualitative description of the formation of energy bands in solids.
Hydrogen is a gas: when two hydrogen atoms come near each other, they form a diatomic gas molecule. But for the purpose of exploring the simplest possible example showing the mechanism behind the formation of any solid from its atoms, imagine the following thought experiment: we have a collection of N hydrogen atoms as an 1-dimensional chain spaced a mile apart from each other. Every electron in each hydrogen atom will see only the attractive potential of its own nucleus and therefore it will be in the Is state as described in the previous chapter and it will have the same Is energy of l3.6eV, see figure 2.1.
Now imagine a godly hand pushing the atoms towards each other along the chain direction. Let the interatomic distance be denoted by d (a variable) and construct a diagram of their energies E as a function of d. Initially all the energies of the N electrons (in Is state) will be at the same level E equal to 3.6eV. But as the atoms approach each other, their electrons will not only see the coulombic attraction of their own nucleus, but also the attraction of the neighbouring nuclei, as well as the repulsion of neighbouring electrons until bonds are formed and d attains a final value at d = a. Then, because of the new coulombic interactions, the electron Is energies will no longer be equal to each other, as they initially were, but will split into N different values and then a band is formed, see figure 2.2. Pauli’s principle prevents the states from being filled by more than 2 electrons (with different spins) so that the end result can be described as follows.
We have N atoms and N electrons in Is states so that we have N states for spin up and N states for spin down, 2N in total. The N electrons will occupy the lowest N states and the remaining N higher states will be empty. As we will see later in the chapter, this is the picture of a metal. If hydrogen atoms had formed a solid, it would have been a metal. The critical reader will notice that we say “states” instead of atomic orbitals. There is a reason
FIGURE 2.1 A linear chain of N hydrogen atoms, each having an electron in an Is state. They are at a variable distance d, which attains a final value a when bands are formed.
FIGURE 2.2 The splitting of the original N Is energies - all equal to Es = 13.6eV - into a band.
for it: although the electrons started as atomic orbitals when they were far apart, they are no longer in atomic orbitals when the atoms end up close together at Angstrom distances in a solid.
The purpose of this chapter is to explore the type of states we have in a solid and set up the ground for their conduction properties. However even before we move on to the next chapter, with just the knowledge of this chapter, we will be able to answer such questions as:
a. Why A1 is a metal while GaAs is a semiconductor
b. Why intrinsic Ge has more carriers (electrons in its conduction band) than intrinsic Si
c. Why GaAs has a higher conductivity than Si given the same amount of doping