Feature Article - Thermal Processes, Part 3 - Oxidation and Kinetics - By Christopher Henderson

Last month we explored the topic of oxidation and kinetics in thermal processing. We’ll continue our discussion this month. There are three steps in the oxidation model. The first step is the gas transfer of oxygen from the ambient environment to the gas/oxide interface. The second step is the diffusion of oxygen across the oxide layer. The third step is the chemical reaction that occurs at the silicon surface. The equations for each step are shown here.

F1 is the gas transport equation, F2 is the oxide diffusion equation, and F3 is the oxygen/silicon reaction equation. H sub G is the mass transfer coefficient in gas, and H is Henry’s law constant. C star is defined as H times p sub G, where p sub G is the partial pressure of the oxidizing species in the bulk of the gas. D is the diffusivity of the oxidizing species in SiO2 and k sub s is the chemical surface reaction rate constant for oxidation.

In the steady state condition F1 should equal F2 which should also equal F3. When the equations are set equal to one another, one can solve for the concentration at the surface of the oxide and the concentration at the interface between the silicon and the silicon dioxide. When the diffusion constant D is very small, the flux of oxygen through the silicon dioxide is small compared to the reaction rate at the interface. This is known as the diffusion rate controlled case, indicated by the line shown here.

When the diffusion constant D is very large, an ample supply of oxygen can reach the interface. This is a reaction rate controlled case, indicated by the line shown here.

If one solves for equation F3 (the reaction rate at the silicon/silicon dioxide surface) after applying the appropriate boundary conditions, one can obtain a solution for X, or the thickness of the oxide. This is a somewhat complicated differential equation boundary value problem, so we will not work it here. Basically, the thickness can be described under two different conditions: one where the time scale is quite long, and one where the time scale is short. The short time scale describes the initial growth region, or the reaction rate limited case, and the long time scale describes the diffusion limited case. For short times where t plus tau is much less than A squared over 4B, X sub O equals B over A times the quantity t plus tau, where B over A is called the linear rate constant. For very long times where t is much greater than A squared over 4B, X sub O squared equals B times t, where B is called the parabolic rate constant. These are the fundamental pieces of the Deal-Grove oxidation model, which we will cover next. The graph here shows the composite of the two growth regimes, where oxide thickness is plotted against oxidation time.

Scientists have studied thermal oxidation for many years. Bruce Deal and Andrew Grove developed the first model to explain thermal oxidation in the mid-1960s while working at Fairchild Semiconductor. Here we show the basic model, where x is the oxide thickness, B is the parabolic rate constant, B over A is the linear rate constant, and tau is a factor to account for oxide present at the start of the oxidation.

Deal and Grove developed their model to agree with the two regimes, the linear growth regime and the parabolic growth regime. In the linear growth regime, the rate of the oxidation reaction dominates. This regime occurs while the oxide layer is still thin, and diffusion of oxygen to the surface is unimpeded. So if B is much greater than B over A, the Deal-Grove equation reduces to this formula.

In the parabolic growth regime, the diffusion of oxygen to the silicon-silicon dioxide interface is increasingly limited. This regime occurs when the oxide increases beyond 500 angstroms. So in this case, if B over A is much greater than B, the Deal-Grove equation reduces to this formula.

These graphs show the Deal-Grove oxidation model applied to both wet oxidation and dry oxidation examples at various temperatures. We show the experimentally-derived constants in the tables below the graphs. Notice that the Deal-Grove model shows a clear oxidation rate driven regime as well as a diffusion rate limited regime.

This image shows a cross-section of the silicon dioxide growth in a LOCOS (pronounced “low-cos”) technology. In a LOCOS technology, process engineers grow a thermal oxide that serves as the isolation between transistors. A silicon nitride mask prevents the oxidation in the active region areas. Notice that the silicon dioxide has consumed a portion of the silicon, expanding down into the silicon itself. The original silicon surface is shown by the dotted line. There is even some growth of the oxide beneath the edge of the silicon nitride mask.

In conclusion, we discussed thermal oxidation and the kinetics of the reaction. While thermal oxidation is primarily driven by temperature, there are two regimes that also control the growth of the oxide: the oxidation rate-limited, or linear regime, and the diffusion rate-limited, or parabolic regime. Deal and Grove developed a model for thermal oxidation in the mid-1960s, and that model is still the prevailing model to explain and understand the first-order effects associated with oxidation.