Metals and nonmetals: Consider a one-dimensional chain of N atoms with one atom
ID: 1769908 • Letter: M
Question
Metals and nonmetals: Consider a one-dimensional chain of N atoms with one atom per unit cell. Assume periodic boundary conditions and that each atom has Z valence electrons. (a) Show that you can fill exactly Z/2 bands with these electrons or, equivalently, that each band can accommodate 2N electrons. (b) Figure 6.13 shows that Si has four filled bands (for some values of k, the energies of the bands are degenerate, but not for all). There are also four electrons per Si atom (not eight!). Explain why this is so. (c) Having an even number of electrons per unit is necessary but not sufficient for a solid to be a semiconductor/insulator. Give an example for an elemental solid that is a metal despite having an even number of electrons per unit cel. 7)Explanation / Answer
a. Each band can accomodate 2N electrons because each k state can accomodate one spin up and one spin down electron. Conversely only Z/2 bands are required for Z elecrons.
b. The first 2 shells in silicon has 2,8 electrons. But since there are only 14 electrons, only 4 are left for the last shell. So the last shell is only half filled and that is why is has 4 and not electrons.Si atom will look for ways to fill it's last shell by sharing of electrons.
c. Iron can be an example of the above condition. It has 26 electrons with [Ar] 3d6 4s2 configuraion.