Page 165 - Sami Franssila Introduction to Microfabrication

144 Introduction to Microfabrication

Oxygen
Hydrogen
Nitrogen
Burn
box DCE/HCl
3-zone resistive heating
Figure 13.1 Horizontal oxidation furnace: wafers are vertically loaded in quartz boats

z = 0 z = Z Wafer backside This leads to the oxide thickness equation:
C gas t = Z/(KC s υ) + Z /(2DC s υ) (13.9)
2
C s When thin oxides are considered, we can ignore the
second term, and rate is then simply
Z = kC s t (13.10)

SiO film Silicon or growth is linear in time and related to the rate
2
constant k.
Figure 13.2 Model of thermal oxidation: oxygen diffuses
through SiO 2 film and reacts at the SiO 2 /Si interface. For thick oxides, we can ignore the first term, and
Concentration of oxygen inside oxide decreases linearly we get
Z = 2DC s υt (13.11)
The latter equation specifies that all oxygen reaching or growth is parabolic, related to diffusion length √ Dt.
the interface will react there to form oxide: there will be The Deal–Grove model thus predicts linear oxida-
no build-up of unreacted oxygen inside oxide or silicon. tion rate initially, followed by a parabolic behaviour for
For a reaction like Si (s) + O 2 (g) → SiO 2 (s), the thicker oxides, Figure 13.3. The linear regime covers
rate is assumed to be first order, that is, R = kC, directly only the initial stages of oxidation with some success.
related to concentration of reactive species, C, and
The model works much better for thick oxides, and
characterized by a rate constant k. We can then rewrite
theory and experiment agree that doubling oxide thick-
the second boundary condition as
ness requires quadrupling oxidation time in the parabolic
−D(dC/dz) = kC at z = Z (13.4) regime (this can be used as a quick estimate for oxida-
tion time once one process is known and fixed).
A solution that satisfies these conditions is Dry oxidation is slower than wet oxidation
(Figure 13.4) even though diffusion of oxygen molecules
C = C s − (kC s /(kZ + D)) (13.5)
through silicon dioxide is faster than diffusion of water
Rate (at the interface z = Z) is then molecules. But water solubility in silicon dioxide is
4 orders of magnitude larger that oxygen solubility,
R = kC(Z) = kDC s /(kZ + D) (13.6) and therefore, concentration of the oxidant in oxide is
much greater.
To calculate thickness growth rate, we must convert
molar concentration to volume through density:
13.2.1 Oxidation of other materials
(dZ/dt) (13.7)
RM SiO 2 = ρ SiO 2
Very few materials can tolerate oxidizing ambients at
where the molar volume of SiO 2 is υ = M SiO 2 /ρ SiO 2 ◦
3
3
(60 g/mol/2.2 g/cm = 27.3 cm /mol). ca. 1000 C. No metal can withstand such conditions.
When we solve for Z(t) from the rate equation, Silicon and silicon-containing compounds are really
we get exceptional in this respect.
Polysilicon oxidation presents a number of complica-
dZ/dt = (kDC s υ)/(kZ + D) subject to Z = 0 at t = 0 tions compared to single-crystal oxidation. The polysil-
(13.8) icon surface is not smooth like a single-crystal surface