Oxidation of Silicon by Water
1773
and Oxygen
Oxidation of Silicon by Water and Oxygen and Diffusion in Fused Silica R. H. Doremus Rensselaer Polytechnic Institute, Materials EngineeringDepartment, Troy, New York 1218 I
(Received January 29, 1976)
Publication costs assisted by Rensselaer Polytechnic Institute
A model for the oxidation of silicon by diffusion of molecular oxygen or water through the silicon oxide layer is compared to experimentally measured oxidation rates, using diffusion coefficients and solubilities measured for fused silica. Good agreement between calculated and measured rates results in three respects: the absolute rate, and its pressure and temperature dependence. The model of molecular diffusion therefore provides the most satisfactory explanation for diffusion-controlled oxidation of silicon.
Introduction Diffusion of components through a coherent film of oxide often controls the rate of oxidation of metals and semiconductors; then the square of the oxide thickness is proportional to time (parabolic rate). Diffusion of metallic or oxygen ions is usually considered to control the oxidation rate; however, diffusion of the oxidizing species themselves (water or oxygen molecules) through the oxide can control the rate of oxidation if such molecular diffusion is rapid enough. In this paper evidence is presented that the diffusion-controlled oxidation of silicon by water or oxygen results from molecular diffusion of these gases through the silicon dioxide layer. First experimental results on the oxidation of silicon are briefly reviewed, and then molecular diffusion of gases in fused silica is discussed. X-ray diffraction experiments have shown that the oxide layer on silicon is amorphous and similar to fused si1ica;l the density of the layer is also about that of fused silica, so the film is probably quite similar to fused silica. Models for the oxidation of silicon by molecular diffusion are presented and compared to oxidation rates and diffusion coefficients of water and oxygen in fused silica. Good agreement between the models, oxidation experiments, and these diffusion coefficients is found in three respects: the pressure dependence, the absolute value of the oxidation rate, and the temperature dependence. Thus molecular diffusion of water and oxygen in the oxide layer provide the most satisfactory explanation for diffusion-controlled oxidation of silicon. Oxidation of Silicon by Dry Oxygen Early results were reviewed by Deal and Grove,2 and they reported results of their own. More recent reports are in ref 3-5. These authors all agree that there are two different time dependencies during the oxidation of silicon by dry oxygen: initially there is a linear region; then the rate becomes parabolic. The layer thickness X as a function of oxidation time t is given by the equation
+
X2 AX = B(t
+ r)
(1)
where A and B are coefficients independent of time and r provides for a thin layer present before experimental oxidation begins. At long times and large thicknesses 2X dX/dt = B (2) B is the parabolic rate constant related to diffusion-controlled oxidation. Deal and Grove found that B = 3.2(10)-14 cm2/s a t 1000 "C and 1 atm of oxygen with a temperature dependence
B = BOexp(-Q/RT)
(3)
and the activation energy Q was 28 kcal/mol. B was directly proportional to oxygen pressure. Similar results were found by most other investigators, Growth occurred at the siliconoxide interface, so one can conclude that an oxidizing species must diffuse through the oxide film to this interface. Oxidation of Silicon by Water Deal and Grove discussed their own and earlier work;2later reports are in ref 3,5, and 6. Again there was an initial linear dependence of layer thickness on time and subsequently parabolic dependence; eq 1reasonably represented the results. B was proportional to pressure, and oxidation occurred at the silicon-oxide interface, so an oxidizing species was diffusing through the oxide layer. Deal and Grove found B = 9.45(1 0 p cm2/s at lo00 "C and 1 atm of water, and Ota and Butler agreed closely with this result. The temperature dependence followed eq 3 with Q = 16 kcal/mol. Diffusion in Fused Silica Fused silica contains silicon-oxygen tetrahedra linked together strongly in a three-dimensional network. The resulting amorphous structure shows considerable short-range order but no long-range order beyond a few tetrahedral distances. The structure is quite open, and can be thought of as containing interstices or holes a few angstrom units in diameter and 5 to 10 8, apart. Gas molecules dissolve and diffuse easily in fused silica.? The solubilities of helium, neon, and hydrogen in fused silica are about the same, are almost independent of temperature, and are proportional to gas pressure. The solubility of oxygen is somewhat lower, but the concentration dissolved is still about 1%of the surrounding concentration of oxygen gas.8 The activation energies Q for molecular diffusion in fused silica are related to molecular sizes, and are given by the equation7
+
Q1I2 = a br (4) where a and b are constants and r is the molecular radius. This equation holds well for gases with molecular radii from helium (r = 1.0 A) to xenon (r = 2.5 A). Activation energies for molecular diffusion of oxygen8 and watergJOare consistent with their molecular sizes. Thus these gases can diffuse readily through fused silica, and analogously through the silica layer on oxidized silicon. The high silicon-oxygen bond energy of 106 kcal/mol leads The Journal of Physical Chemistv, Vol. BO, No. 16, 1976
1774
R. H. Doremus
one to predict a low concentration of oxygen vacancies and a low diffusion coefficient of lattice oxygen in silica. The activation energy for diffusion of lattice oxygen in fused silica has not been reliably measured, but in soda-lime glass, in which Q should be lower than in fused silica, the activation energy for diffusion of lattice oxygen was measured to be about 66 kcal/mol.ll Therefore it is highly unlikely that, diffusion of lattice oxygen contributes appreciably to the oxidation of silicon, with its relatively high rate and low activation energy. Furthermore direct proportionality of oxidation rate to gas pressure would not be expected for diffusion of lattice oxygen. One might also imagine that oxygen ions could be introduced into the silica from the gas phase. However, to introduce a concentration of oxygen ions sufficient to give the experimental rates of oxidation would require the formation of an impossibly large space charge in the oxide. Thus compensating positive ions would have to be introduced along with the negative oxygen ions. There are no likely candidates for these positive ions, or evidence for their existence. The diffusion of water in fused silica is complicated by the reaction of water with silicon-oxygen bonds: H2O
+ Si-0-Si
= 2SiOH
(5)
The apparent solubility of water in fused silica is much higher than the molecular solubilities of other gases because of this reaction, and the solubility is proportional to the square root of the water vapor pressure, as expected from eq 5. Reaction 5 also influences the diffusion of water in fused silica, leading to a concentration-dependent diffusion coefficient.9 These complications are taken into account in a model of molecular diffusion plus reaction to immobile SiOH groups.lO The concentration of reacted water S at any point in the glass is much greater than the concentration of molecularly dissolved water c, and these concentrations are related by S = KC1'2, as derived from reaction 5 , where K is a concentration-independent coefficient. Under these conditions diffusion is controlled by an effective diffusion coefficient De: where D is the diffusion coefficient of molecularly dissolved water. The profile of S vs. penetration distance derived from eq 6 is in good agreement with experimental data for water in fused silica.10 Thus it is possible to derive values of the diffusion coefficient of molecular water from profiles of the reacted water 5' vs. distance. There are several printing errors in ref 7 and 10. The diffusion coefficient of molecular water at 1000 "C in Table 11,p 133, ref 7, and Table I, p 670, ref 10, should be 3.6(10)-7 cm2/s, as derived in the Appendix. Models for Diffusion-Controlled Oxidation of Silicon The diffusion of oxygen in silica is not complicated by its reaction with the silica lattice. The equation for the diffusion-controlled moving boundary problem for formation of an oxide layer of thickness X is12
dX x-=-
c~D (7) dt P where D is the diffusion coefficient of the oxidizing species in the layer, ci is its concentration at the outer oxide surface, and p is the concentration of oxygen in the film. For eq 7 to be valid ci > c allows one to neglect dclat compared to dS/at, with eq A2 as the result. For fused silica at 1000 "C the equilibrium solubility Si = l.l(10)-4 mol of HzOlcm3 and the concentration of molecularly dissolved water should be about 0.01 the concentration in the gas phase, or about 9.6(10F8mol of HzO/cm3 for 1 atm pressure, so the inequality is valid. These values give a molecular diffusion coefficient D for water of about 3.6(10)-7cm2/s at 1000 O C from eq 6. In the present case a solution of eq A2 for a film of increasing dimension is desired. For diffusion without reaction (i.e., oxygen in silica) the condition c
