CHAPTER 4:  Oxidation
Oxidation of silicon is an important process in VLSI.  The typical roles of SiO2 are:
1.  mask against implant or diffusion of dopant into silicon
2.surface passivation
3.device isolation
4ponent in MOS structures (gate oxide)
5.  electrical isolation of multi-level metallization systems
There are several techniques to form oxide layers, namely thermal oxidation, wet anodization, chemical vapor deposition, and plasma oxidation.  Of the four techniques, thermal oxidation tends to yield the cleanest oxide layer with the least amount of interfacial defects.
4.1Theory of Oxide Growth
The chemical reactions describing thermal oxidation of silicon in dry oxygen or water vapor are:
Si (solid) + O2 (gas)  →  SiO2 (solid)
Si (solid) + 2H2O (gas)  →  SiO2 (solid) + 2H2 (gas)
During the course of the oxidation process, oxygen or water molecules diffuse through the surface oxide into the silicon substrate, and the Si-SiO2 interface migrates into the silicon (Figure 4.1).  Thermal oxidation of silicon results in a random three-dimensional network of silicon dioxide constructed from tetrahedral cells.  Since the volume expands, the external SiO2 surface is not coplanar with the original silicon surface.  For the growth of an oxide of thickness d, a layer of silicon equal to a thickness of 0.44d is consumed.
Figure 4.1: Growth of SiO2.
A model elucidating the kinetics of oxide growth has been developed by Deal and Grove.  The model, which is generally valid for temperatures between 700o C and 1300o C, partial pressure between 0.2 and 1.0 atmosphere, and oxide thickness between 30 nm and 2000 nm for oxygen and water ambients, is schematically illustrated in Figure 4.2.
Figure 4.2: Basic model for thermal oxidation of silicon.
In steady state, the three fluxes, F1 (flux of oxidizing species transported from the gas phase to the gas-oxide interface), F2 (flux across the existing oxide toward the silicon substrate), and F3 (flux reacting at the Si-SiO2 interface) must be equal.  F1 can be approximated to be proportional to the difference in concentration of the oxidizing species in the gas phase and on the oxide surface.
F1 = h G(C G - C S)                                              (Equation 4.1)
where h G is the gas-phase mass-transfer coefficient, C G is the oxidant concentration in the gas phase, and C S is the oxidant concentration adjacent to the oxide surface.  Substituting C = P/kT into Equation 4.1,
F1 = (h G/kT)(P G - P S)                                        (Equation 4.2)
Henry's Law states that, in equilibrium, the concentration of a species within a solid is proportional to the partial pressure of that species in the surrounding gas.  Thus,
C o = HP S(Equation 4.3)
where C o is the equilibrium concentration of the oxidant in the oxide on the outer surface, H is the Henry's Law constant, and P S is the partial pressure of oxidant in the gas phase adjacent to the oxide surface.  Furthermore, we denote the equilibrium concentration in the oxide, that is, the concentration which would be in equilibrium with the partial pressure in the bulk of the gas P G by the symbol C*, and
C* = HP G(Equation 4.4)
Hence,
C* - C o = H (P G - P S), and
F1 = (h G/HkT)(C* - C o) = h (C* - C o)              (Equation 4.5)
where h =h G/HkT is the gas-phase mass-transfer coefficient in terms of concentration in the solid.
Oxidation is thus a non-equilibrium process with the driving force being the deviation of concentration from equilibrium.  Henry's Law is valid only in the
absence of dissociation effects at the gas-oxide interface, thereby implying that the species diffusing through the oxide is molecular.
The flux of the oxidizing species across the oxide is taken to follow Fick's Law at any point d in the oxide layer.  Hence,
F2 = D(C o - C i)/d o(Equation 4.6)
where D is the diffusion coefficient, C i is the oxidizer concentration in the oxide adjacent to the SiO2/Si interface, and d o is the oxide thickness.
The chemical reaction rate at the SiO2/Si interface is assumed to be proportional to the reactant concentration.  Therefore,
F3 = k S C i                                            (Equation 4.7)
where k S is the rate constant.
Under steady-state conditions, F1 = F2 = F3.  Thus,
h(C* - C o) = D(C o - C i)/d o = k S C i
C i = DC o/(k S d o + D)                                        (Equation 4.8)
C i = C*/[1 + k S/h + k S d o/D], and                      (Equation 4.9)
C o = [(1 + k S d o/D)C*]/[1 + k S/h + k S d o/D]      (Equation 4.10)
When D is large, Equation 4.8 becomes C i= C o, implying that the oxidation rate is controlled by the reaction rate constant k S and by C i (=C o), that is, a reaction-controlled case.  When D is very small, h(C* - C o) = 0 = k S C i.  Therefore, C* = C o and C i = 0.  The latter case is called diffusion-controlled case, as the oxidation rate depends on the supply of oxidant to the interface.
In order to calculate the oxide growth rate, we define N1 as the number of oxidant molecules incorporated into a unit volume of the oxide layer.  If oxygen is the reactant, N1 = 2.2 x 1022 atoms/cm3 because the density of SiO2 is 2.2 x 1022 cm-3.  If water is used, N1 becomes 4.4 x 1022 cm-3 as two H2O molecules are incorporated into each SiO2 molecule.  The differential equation for oxide growth is given by
(Equation 4.11)
With an initial condition of d o(t = 0) = d i, the solution of Equation 4.11 may be written as:
d o2 + Ad o = B (t + τ)                                          (Equation 4.12)
where A≡ 2D [1/k S + 1/h]                                          (Equation 4.13)
B≡ 2DC* / N1(Equation 4.14)
τ≡ (d i2 + Ad i) / B(Equation 4.15)
The quantity τrepresents a shift in the time coordinate to account for the presence of the initial oxide layer d i.  Solving Equation 4.12 for d o as a function of time gives
reaction diffusion
(Equation 4.16)
For long oxidation times, i.e., t >> τand t >> A2/4B, d o2≅Bt.  B is therefore called the parabolic rate constant.  For short times, i.e., (t + τ) << A2/4B, d o≅ [B/A](t + τ), and B/A is referred to as the linear rate constant.

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