(Received 10 May 1965; in final form 9 September 1965) The thermal-oxidation kinetics of silicon are examined in detail. Based on a simple model of oxidation which takes into account the reactions occurring at the two boundaries of the oxide layer as well as the diffusion process, the general relationship x_o²+Ax₀=B(t+t ) is derived. This relationship is shown to be in excellent agreement with oxidation data obtained over a wide range of temperature (700° --1300° C), partial pressure (0.1-1.0 atm) and oxide thickness (300-20,000A ) for both oxygen and water oxidants. The parameters A, B, and t are shown to be related to the physico-chemical constants of the oxidation reaction in the predicted manner. Such detailed analysis also leads to further information regarding the nature of the transported species as well as space-charge effects on the initial phase of oxidation.

1. INTRODUCTION

Owing to its great importance in planar silicon-device technology, the formation of silicon dioxide layers by thermal oxidation of single-crystal silicon has been studied very extensively in the past several years^1-15 Now, with the availability of large amounts of experimental data, it appears that there is much contradiction and many peculiarities in the store of knowledge of silicon oxidation. For instance, reported activation energies of rate constants vary between 27 and 100 kcal/mole for oxidation in dry oxygen; pressure dependence of rate constants has been reported as linear as well as logarithmic. While most of the data on silicon oxidation have been evaluated using the parabolic rate law, certain authors have taken recourse to using empirical power-law dependence,¹⁴ x_oⁿ = kt, where both n and k were complex functions of temperature, pressure, and oxide thickness.

The problems associated with the latter approach can be illustrated by considering Figs 1 and 2 (not provided in this copy). These figures contain a summary of data obtained in these laboratories which are in good general agreement with the corresponding data of Fuller and Stricter¹’ and of Evitts, Cooper, and Flaschen.¹³ (The experimental methods are dealt with in detail later.) The plots are logarithm of oxide thickness vs the logarithm of oxidation time for dry and wet oxygen (95⁰C H₂O) at various temperatures. The slope of the lines corresponds to the exponent it in the above power law. These values are indicated at the limiting position of some of the curves. In the case of wet oxygen (Fig. 1), n ranges from 2 for thicker oxides at l200⁰C to 1 for the thinner oxide region of the 920⁰C data. However, for dry oxygen (Fig. 2), the value of n at 1200⁰ approaches 2 as the oxide thickness increases above 1.0 um but at lower temperatures and oxide thicknesses the value of n decreases only to about 1.5 and then appears to increase again. Obviously the data cannot be represented by a simple power law.

Most of the previous theoretical treatments of the kinetics of the oxidation of metals emphasize only two limiting types of oxidation mechanisms.¹⁶ In one, the oxide thickness is small in comparison to the extent of possible space-charge regions within the oxide, and the oxidation kinetics is strongly influenced by the space charge or by a voltage drop across the oxide film due to contact potential differences. In the other, the rate of diffusion of either the oxidizing species or the metal across the oxide film determines the oxidation kinetics. This latter condition leads to a parabolic relationship, x_oµ t^1/2. These two approaches have been emphasized to such an extent that deviations from parabolic oxidation have been attributed incorrectly to space-charge effects.

In this paper the oxidation kinetics are examined in greater detail. It is shown that when the reactions occurring at the two boundaries of the oxide layer are taken into account, a general relationship can be obtained with parameters which are explicitly related to the physico-chemical constants of the oxidation system. This general relationship is shown to be in excellent agreement with data obtained by various groups of investigators over a wide range of conditions. It is shown that the parameters of this relationship follow the predictions of the model. In addition, further information is obtained regarding the mechanism of oxidation: the nature of the transported species, and the role of space-charge effects.

2. GENERAL OXIDATION EQUATION

Consider silicon covered by an oxide layer of thickness x_o, as indicated in Fig. 3. In accordance with the experimental evidence for silicon, it is assumed that oxidation proceeds by the inward movement of a species of oxidant rather than by the outward movement of silicon.^2,3,8,17 The transported species must go through the following stages:

(1) It is transported from the bulk of the oxidizing gas to the outer surface where it reacts or is adsorbed.

(2) It is transported across the oxide film towards the silicon.

(3) It reacts at the silicon surface to form a new layer of SiO₂.

 

Fig. 3. Model for the oxidation of silicon.

It is assumed in this paper that the oxidation process is beyond an initial transient period with the consequence that the fluxes of oxidant in each of the above three steps are, at all times, identical. (The flux of oxidant is the number of oxidant molecules crossing a unit surface area in a unit time.) The validity of this steady-state assumption is shown in the Appendix. The steady-state fluxes are approximated as follows.

The flux of the oxidant from the gas to the vicinity of the outer surface is taken to be

where h is a gas-phase transport coefficient, Co is the concentration of the oxidant at the outer surface of the oxide at any given time, and C* is the equilibrium concentration of the oxidant in the oxide. (This linearized approximation to the transport rate is analogous to Newton’s law of cooling.) Note that when C₀ = C*, F₁ = 0. The equilibrium concentration of the oxidant is assumed to be related to the partial pressure of the oxidant in the gas by Henry’s law,

C*= Kp.

Henry’s law only holds in the absence of dissociation or association of the oxidant at the outer surface. Thus, the oxidant is assumed to be molecular O₂, and H₂O in the cases of dry- and wet-oxygen oxidation, respectively. The experimental evidence will be seen to justify this assumption.

The flux of the oxidant across the oxide layer is assumed to be given by Fick’s law,

F₂ = —D_eff (dC/dx), (3)

 at any point x within the oxide layer, where D_eff is the effective diffusion coefficient and dC/dx is the concentration gradient of the oxidizing species in the oxide.

It follows from the assumption of steady-state oxidation that F₂, must be the same at any point x within the oxide film, or that dF₂/dX = 0. Consequently, the concentration of the oxidant within the oxide layer is linear as indicated in Fig. 3, and the flux F₂ is given by

F,= D_eff (C₀— C_i)/x₀, (4)

where Ci is the concentration of the oxidant near the oxide-silicon interface.

If the transported species are ionic, they set up a space charge within the oxide film. The effective diffusion coefficient D_eff incorporates the effect of such space charges in enhancing the rate of transport. It was shown by Cabrera and Mott,¹⁶ as well as by Bardeen, Brattain, and Shockley,¹⁸ that if the oxide thickness is large enough in comparison to the thickness of the space-charge regions, D_eff is approximately twice the actual diffusion coefficient. The extent of the space-charge region in SiO₂ films is shown later to be of the order of 200 A, or less, so that this condition is met in most practical cases.

Finally, the flux corresponding to the oxidation reaction is expressed by the first-order relation

F₃ = kC_i. (5)

Noting that F₁=F₂, and F₂=F₃ due to the steady-state condition, and solving for C_i and C_o between these two equations, one obtains

  (6) and (7)

Note that in the limit where the diffusivity becomes very small relative to the rate constants associated with surfaces, k and h (i.e., D_eff/kx_o —> 0), C_i —> 0 and C_o —> C*. This condition is commonly referred to as "diffusion controlled."

With C_i and C_o eliminated, the flux is given by

 (8)

If N₁ is the number of oxidant molecules incorporated into a unit volume of the oxide layer, the rate of growth of the oxide layer is described by the differential equation

 (9)

To arrive at a general initial condition to this equation, the total oxide thickness x₀ is taken to consist of two parts: an initial layer of oxide x_i that might have been present on the silicon prior to the oxidation step under consideration, and the additional thickness grown during the step under consideration. Thus the initial condition for this step is:

x_o= x_i at t=0, (10)

where C4 is the concentration of the oxidant near the oxide—silicon interface.

Such division of the oxide layer is not only important because it permits consideration of multiple oxidations, but also because xi can be regarded as the thickness of the layer grown before the approximations of the above treatment become valid. Thus, an initial oxide layer may be formed by mechanisms involving fields and space charges within the oxide layer.¹⁶

The solution of the differential equation, Eq. (9), subject to the initial condition, Eq. (10), obtained by a straightforward integration, is

(11)

which can be rewritten in the form

(12a)

where

(12b)

and

(12c)

The quantity tau corresponds to a shift in the time coordinate which corrects for the presence of the initial oxide layer x_i. The form of this mixed parabolic relationship was first proposed by Evans.^19,20

Solving the quadratic equation (12) yields the form

(13)

which gives the oxide thickness as a function of time.

It is interesting to examine two limiting forms of Eq. (13). At relatively large times, i.e., t³ >A²/4B, and also t>>t ,

or

(13)

Thus, the general relation, Eq. (13), reduces to the well-known parabolic oxidation law for relatively long times. The coefficient B, as given by Eq. (12b), is identical with that derived by Cabrera and Mott.¹⁶ It is referred to as the parabolic rate constant.

At the other extreme, for relatively small oxidation times, i.e., ,

(15)

Thus, the general relationship reduces to a linear law. The coefficient

is referred to as the linear rate constant. The general relationship and its two limiting forms are shown in Fig. 12, together with experimental points which are discussed later. It should be noted at this point, however, that for most of the range the parabolic limit does not provide a good approximation. For this reason, simply dividing x₀² by t does not necessarily yield the true parabolic rate constant B, but rather it gives some quantity which may or may not approximate B depending on the oxidation conditions and the time. Thus, it appears that many of the contradictions referred to in the Introduction may well have been due to forcing the parabolic law in regions where it could not provide a meaningful fit.