Transport Phenomena - Mass Transfer Problem :
Oxidation of silicon - a diffusion problem in microelectronics
Problem.
A diffusion problem that occurs in the field of microelectronics is the oxidation of silicon according to the reaction Si + O2
SiO2. Consider the the oxidation of a material M given by the general reaction M + (1/2)x O2 MOx (with x = 2 for silicon oxidation). When a slab of the material is exposed to gaseous oxygen (species A), the oxygen undergoes a first-order reaction with rate constant k1" to produce a layer of the oxide (species B).
The task is to predict the thickness
d
of the very slowly-growing oxide layer as a function of time t using a quasi-steady-state approach (which suggests that the rate of change of the dissolved oxygen content in the layer is small compared to the rate of reaction).
Let the oxygen, whose dissolved concentration is cA0 at the free surface of the oxide layer at z = 0, diffuse through the layer as per Fick's law to reach the reaction surface at z =
d as in the figure below.
Figure. Diffusion of oxygen through an oxide layer formed by oxidation of silicon.
a) Write unsteady-state molar balances on the oxide and oxygen over the thickness of the layer.
b) Find the concentration profile of oxygen in the layer at steady-state.
c) Using quasi-steady-state arguments, show that the oxygen concentration at the reaction plane (z = d)
is given by
where DAB is the oxygen diffusivity through the oxide film and cB is the molar density of the oxide layer.
d) Derive an expression for the thickness of the very slowly-growing oxide layer when the quadratic term on the right-hand side of the above equation is negligible.
e) Reduce the above results for the limiting case of diffusion - controlled oxidation where the reaction rate constant k1" tends to infinity.
Solution.
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a)
Step. Unsteady-state molar balance on oxide over thickness of layer
The unsteady-state balance on the moles of oxide is
rate of moles of oxide generated = rate of moles of oxide accumulated
Thus,
(1)
where cB is the molar density of the oxide layer and S is the cross-sectional area of the slab. Note that (2/x) moles of oxide are generated for each mole of oxygen consumed in accordance with the stoichiometry of the reaction.
Step. Unsteady-state molar balance on oxygen over thickness of layer
The unsteady-state balance on the moles of oxygen is
rate of moles of oxygen in by diffusion - rate of moles of oxygen consumed by reaction = rate of moles of oxygen accumulated
Thus,
(2)
where NAz is the molar flux of oxygen in the z-direction. Dividing throughout by S (since the cross-sectional area is constant) and then using Leibniz formula on the right-hand side of the above equation, we get
(3)
Since the rate of change of the dissolved oxygen content in the layer is small compared to the rate of reaction, the first term on the right-hand side above is neglected. Then, on substituting Fick's law for NAz,
(4)
where DAB is the oxygen diffusivity through the oxide layer.
b)
Step. Steady-state concentration profile of oxygen in layer
At steady-state in the absence of homogeneous reaction, the species continuity equation and the diffusion equation simplify to .NA = 0 (where NA is the molar flux of species A) and 2cA = 0.
Since the oxygen concentration is a function of z only, the differential equation in Cartesian coordinates may be simply integrated twice as follows:
(5)
The integration constants C1 and C2 are determined using the boundary conditions:
(6)
(7)
On substituting the integration constants,
(8)
Thus, the steady-state profile for the dissolved oxygen concentration is linear. The oxygen concentration at the reaction plane is so far unknown and is determined next.
c)
Step. Expression for oxygen concentration at reaction plane
On eliminating dcA/dz (using Eq. 8) and
dd/dt (using Eq. 1), Eq. 4 yields
(9)
On rearranging the above equation, the oxygen concentration at the reaction plane is given by the following expression.
(10)
d)
Step. Solution of differential equation for thickness of oxide layer
The quadratic term on the right-hand side of the above equation has been shown to be negligible in the microelectronics literature. Thus, the oxygen concentration at the reaction plane is simply given by
(11)
It must be emphasized that the above equation can be directly obtained from a simple steady-state oxygen balance (rather than the unsteady-state balances in Eqs. 2 - 4). In other words, the quasi-steady-state arguments could include a steady-state balance where the moles of oxygen in by diffusion are exactly equal to the moles of oxygen consumed by the oxidation reaction.
On combining Eq. 1 with Eq. 11, the following ordinary differential equation
for d(t) is obtained.
(12)
Noting that d
= 0 at t = 0, integration gives
(13)
On solving the above quadratic equation, the oxide layer thickness is obtained as
(14)
e)
Step. Limiting solution for diffusion - controlled oxidation
For the limiting case of diffusion - controlled oxidation, the reaction rate constant tends to infinity and the dissolved oxygen concentration tends to zero at the reaction plane. Thus, the above results at the diffusion - controlled limit yield
(15)
The oxide layer thickness is seen to be predicted by a quadratic law.
Related Problems in Transport Phenomena - Mass Transfer :
SiO2. Consider the the oxidation of a material M given by the general reaction M + (1/2)x O2 
Click here for stepwise solution
.NA = 0 (where NA is the molar flux of species A) and 
