Diffusion of O 2 and H 2 O in SiO 2 What the semiconductor community learned from the oxidation of silicon Deal-Grove * analysis * Andy Grove, early work before he became CEO of Intel

In thick oxide, moving species in SiO(2) is the neutral oxygen molecule, O 2. In wet oxidation, moving species is H 2 O Si diffusivity is orders of magnitude lower Oxidation rate for very thin oxides is enhanced (explanations very controversial) Solubility for H 2 O in SiO(2) is 3 orders of magnitude larger than that of O 2. Diffusivity of H 2 O is lower than that of O 2 Na increases oxygen diffusion

Deal Grove Model 3 limiting cases exist: Gas phase limited, diffusion across oxide limited, incorporation limited. Oxide Silicon Oxygen Profile If most of the oxygen (O(2)) profile drops across the stagnant layer, process limited by diffusion in the gas phase. The transport coefficient is denoted as h. C= 0

Below is a close up showing the discontinuity in O(2) profile at the outer oxide interface. It reflects the fact that there is more oxygen per ccm in the gas phase, then can be dissolved in silicon dioxide. Incorporation is very slow, so concentration is pegged at solubility limit. Silicon Silicon Dioxide Oxygen Concentration Profile. Cg

In practice, the flux in the gas phase is rarely rate limiting. At 1000 C, and 1 atm, there are about 5E18 O(2) molecules per cubic centimeter (That is C g = 5E18/ccm) The solubility of oxygen in Si at a 1000 C is only 5.2E16 (Deal Grove analysis, same value is obtained by direct permeation experiments). I.e., C* = 5.2E16/ccm If the flux is expressed as F 1 = h (C* - C 0 ) than the values are as follows (dry, 1000 C): F 1 (um/hr) = 1E 8 (um/hr) ( 5.2E16 - C 0 )/(2.2E22) where 2.2E22 is the number of SiO(2) molecules per ccm in the oxide.

The large value of h has the consequence that even a very slight concentration gradient across the stagnant layer will transport enough oxygen to make the oxide grow. Take a C 0 value of 5.0E16 (just 0.2 E 16 smaller than the maximum solubility C* ) and you still get a flux that would correspond to a growth rate (at a 1000 C) of F 1 = 9.09 um/hr If one looks up experimental values, one finds something like 0.1 um for dry oxidation in reasonable thick films. Some other process must be rate limiting.

C = 0 In this case, the oxygen concentration profile is flat in the gas phase. At the oxide/silicon interface the concentration is zero, since incorporation is much faster than transport across the oxide B) Diffusion across the oxide limits the growth rate.

The flux across the oxide is given by the diffusion coefficient times the gradient. If we want the flux in units of oxide growth rate per time we need to divide by the numbers of SiO(2) molecules per ccm. (Otherwise, the flux will be in units of atoms per area and time) F 2 = D * [ C o - C I ] / t where t is the thickness of the oxide. C o is the concentration of oxygen on the outer side of the oxide (actually oxygen molecules dissolved in the Si lattice). C i is the concentration of oxygen dissolved in the Si lattice very close to the oxide/silicon interface). In general, C o is close to C* (the maximum solubility of oxygen, if there is no pressure drop across the stagnant layer), see previous slide

At 1000 C, the diffusivity, D, of oxygen molecules in SiO(2) is about 8E-9 cm 2 /sec. Activation energy 1.24 eV A typical oxide thickness would be 1000 A = 1E-5 cm C i is zero, since in such an oxide incorporation is much faster than diffusion of O 2 across the oxide. C 0 is very close to C*, since diffusion across the gas phase is not limiting so we take the value for C*. With these choices, we get for the flux (in units of cm/s) F 2 = (8E-9/1E-5) * ( 5E16 / 2.2E22) = 1.8E-9 which translates into 0.07 um/hr (close to what one sees). Note, however, that F goes as (1/thickness of oxide). Thus for a 10 A oxide the flux would be a 100 times larger. In this case, incorporation becomes rate limiting.

C) Incorporation is rate limiting C* C= 0 In this case, the concentration of O(2) molecules is flat up to the oxide/silicon interface. The discontinuity at the outer oxide surface reflects, again, the fact that in the gas there are 5E18 O 2 molecules per cc, and in the oxide there are 5.2E16 (maximum solubility).

The process of incorporating the O(2) molecule into Si, forming SiO(2) is described by F 3 = k s * C i where C i is the concentration of O(2) at the Si/SiO(2) interface. This process is rate limiting in very thin oxides. In this case, the concentration of O(2) is essentially flat through the gas phase and oxide and, at the inner oxide interface equal to the oxygen solubility in Si at 760 Torr, ie. 5.2E16. At 1000 C, and dry oxidation, k s is about 3.6E-4 um/hr. Thus we get F 3 = 3.6E4 * (5.2E16 /2.2E22) = 0.8 um/hr

Note that k s is about 4 orders of magnitude smaller than the other linear transfer coefficient, h which controls F 1. This difference in the magnitude of k and h is the reason that transfer through the gas phase is rarely limiting. More specifically, if you derive the kinetics by setting all fluxes equal, you find that the initial linear growth rate is described by: F (linear) = (1/h + 1/k s ) -1 (C* / 2.2 E22) [um/hr] Since h is so much larger (4 orders of magnitude at 1000C) than k s, 1/h is a very much smaller number than 1/k s and the equation reduces for practical purpose to the one given on the previous slide

Deal Grove model predicts d 2 + Ad = Bt B/A is called the linear rate constant B is called the parabolic rate constant A = 2D[(1/k s ) + (1/h)] B = 2DC * /N l D = Diffusivity k s = SiO 2 /Si surface reaction rate constant h = gas transportation mass transfer coefficient N l = 2.2 E 22/ccm (conversion factor) C * = Solubility in oxide

The Deal-Grove law is famous not just for fitting the oxidation data but because the numbers one extracts for the MAXIMUM SOLUBILITY OF OXYGEN IN SI (C*) and the DIFFUSIVITY OF O(2) IN SOLID SiO(2) agree with independent measurements of these quantities. In addition, the model explains the temperature and pressure dependence of A and B.

And here are the numbers C* values in SiO(2) at 1000 C (solubility) O E 16/ccm H 2 O 3.0 E 19/ccm Activation Energy for Diffusion at 1000 C O eV H 2 O 0.71 eV Prefactor, D o, at 1000 C O E-4 cm 2 /s H 2 O1.8 E-7 cm 2 /s Diffusivity at 1000 C O 2 6.9E-9 cm 2 /s H 2 O 2.9E-10 cm 2 /s