Page 37 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
α T()
(2.1)
3
–
4
–
= ( 3.725 1 –( exp – ( 5.88×10 ( T – 124))) + 5.548×10 T)×10 – 6
and a lattice parameter (the interatomic distance in ) that varies with
temperature as
aT()
(2.2)
–
5
–
2
9
= 5.4304 + 1.8138×10 ( T – 298.15) + 1.542×10 ( T – 298.15)
We will later take a more detailed look at the thermal strain ε T –( T ) =
0
(
α T() T – T ) . Both α T() and aT() are plotted in Figure 2.3. Both of
0
5.440 – 6
4×10
5.438
– 6
5.436 2×10
400 800
5.434
0
0 600 1200
Temperature in K Temperature in K
Figure 2.3. The thermal expansion properties of Silicon. Shown on the left is the tempera-
ture dependence of the lattice parameter (the size of a unit cell), and to its right is the
temperature dependence of the thermal expansion coefficient. The typical engineering
temperature range for silicon electronic devices is indicated by the background gray
boxes.

these properties are also dependent on the pressure experienced by the
material, hence we should write α Tp,( ) and aT p,( ) . It is important to
note that “technological” silicon is doped with foreign atoms, and will in
general have material properties that differ from the values quoted in
Table 2.1, but see [2.6] and the references therein. Silicon’s phonon dis-
persion diagram is shown in Figure 2.4.

34 Semiconductors for Micro and Nanosystem Technology