15 Summary: Silicon
Silicon is the second most common element in the earth’s crust (in the form of rocks, sand, glass, etc...)
and becauseSiO 2 is such a good insulator, silicon is the most important semiconducting material in
the world today. Most of the time, when silicon is used as a semiconductor, it is doped with other
elements to obtain certain properties of the material.
Figure 147: Crystal structure of siliconThe silicon crystal has a diamond structure, as depicted in fig. 147, the primitive lattice vectors are:
a 1 =
a
2Y+
a
2
Z a 2 =
a
2X+
a
2
Z a 3 =
a
2X+
a
2Y
and the basis vectors therefor are:
B 1 =−
a
8X−
a
8Y−
a
8Z B 2 =
a
8X+
a
8Y+
a
8Z
whereais the lattice constant
The first Brillouin zone for a silicon crystal is illustrated in fig. 148 and the phonon dispersion rela-
tionship is shown in fig. 149. Because silicon has two atoms per unit cell, there are 6 branches in the
dispersion relation, 3 acoustic and 3 optical branches. (see chapter 6).
How would you calculate the dispersion relation? (see chapter 6)
By knowing the crystal structure, one can write down Newton’s laws for the case that the atoms
are connected to each other by linear forces. Solving the resulting system of equations by assuming
harmonic solutions, one can plot the phonon dispersion curve for different directions and different
values ofk.
What is the Debye frequency? (see chapter 6)
The Debye frequency is the highest frequency of the phonons possible in the crystal. The highest