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2.6 The Deal-Grove Model

A well established model for thermal oxide growth has been proposed by Deal and Grove [51] in the middle of the 60's and because of its simplicity it is still applied frequently. One reason for this simplicity is that the whole physics of the oxidation process is contained in two so-called Deal-Grove parameters, which must be extracted from experiments. Furthermore, it is assumed that the structure is one-dimensional. Therefore, the model can only be applied to oxide films grown on plane substrates.

2.6.1 Concept and Formulation

If one assumes that the oxidation process is dominated by the inward movement of the oxidant species, the transported species must go through the following stages:

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

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

(3) It reacts at the interface with silicon and form a new layer of SiO$ _2$.

Each of these steps can be described as independent flux equation. The adsorption of oxidants is written as

$\displaystyle F_1=h(C^*-C_O),$ (2.8)

where $ h$ is the gas-phase transport coefficient, $ C^*$ is the equilibrium concentration of the oxidants in the surrounding gas atmosphere, and $ C_O$ is the concentration of oxidants at the oxide surface at any given time.

It was found experimentally that wide changes in gas flow rates in the oxidation furnaces, changes in the spacing between wafers on the carrier in the furnace, and a change in wafer orientation (standing up or lying down) cause only little difference in oxidation rates. These results imply that $ h$ is very large, or that only a small difference between $ C^*$ and $ C_O$ is required to provide the necessary oxidant flux.

$ C^*$ is also the solubility limit in the oxide, which is assumed to be related to the partial pressure $ p$ of the oxidant in the gas atmosphere by Henry's law

$\displaystyle C^*=H\cdot p.$ (2.9)

At natural ambient pressure of 1 atm and at a temperature of 1000$ ^{\circ }$C, the solubility limits are 5.2 $ \times$ 10$ ^{16}$ cm$ ^{-3}$ for O$ _2$, and 3.0 $ \times$ 10$ ^{19}$ cm$ ^{-3}$ for H$ _2$O.

The flux $ F_2$ represents the diffusion of the oxidants through the oxide layer to the Si-SiO$ _2$-interface, which can be expressed as

$\displaystyle F_2=D\frac{\partial C}{\partial x} = D\frac{C_O-C_S}{x_O},$ (2.10)

where $ D$ is the oxidant diffusivity in the oxide, $ C_S$ is the oxidant concentration at the oxide-silicon interface, and $ x_O$ represents the oxide thickness. In this expression it is assumed that the process is in steady state (no changing rapidly with time), and that there is no loss of oxidants when they diffuse through the oxide. Under these conditions, $ F_2$ must be constant through the oxide and hence the derivative can be replaced simply by a constant gradient.

The third part of the oxidation process is the flux of oxidants consumed by the oxidation reaction at the oxide-silicon interface given by

$\displaystyle F_3=k_s C_S,$ (2.11)

with $ k_s$ as the surface rate constant. $ k_s$ really represents a number of processes occurring at the Si/SiO$ _2$ interface. These may include oxidant (O$ _2$ $ \to$ 2O), Si-Si bond breaking, and/or Si-O bond formation. The rate at which this reaction takes place should be proportional to the oxidant concentration at the interface $ C_S$.

Figure 2.16: One-dimensional model for the oxidation of silicon.

Deal and Grove assumed that in the steady state condition these three fluxes are equal, which allows to express them as

$\displaystyle F_1=F_2=F_3=F=\frac{C^*}{\frac{1}{k_s}+\frac{1}{h}+\frac{x_0}{D_0}}.$ (2.12)

The rate of oxide growth is proportional to the flux of oxidant molecules,

$\displaystyle \frac{dx_0}{dt}=\frac{F}{N}=\frac{\frac{C^*}{N}}{\frac{1}{k_s}+\frac{1}{h}+\frac{x_0}{D_0}},$ (2.13)

where N is the number of oxidant molecules incorporated per unit volume.

The differential equation can be simplified as

$\displaystyle \frac{dx_0}{dt}=\frac{B}{A + 2 x_0},$ (2.14)

with the physically based parameters

$\displaystyle A = 2D\big(\frac{1}{k_s}+\frac{1}{h}\big),$ (2.15)
$\displaystyle B = 2D\frac{C^*}{N}.$ (2.16)

2.6.2 Analytical Oxidation Relationship

In order to get an analytical relationship between oxide thickness $ x_0$ and oxidation time $ t$ the first order differential equation (2.14) must be solved. For this purpose in the first step (2.14) can be rewritten in the form

$\displaystyle (A + 2 x_0) dx_0=B dt.$ (2.17)

Integration of (2.17) from time 0 to $ t$, with the assumption of an initial oxide thickness $ x_i$ at time 0, yields a quadratic equation for the oxide thickness $ x_0$:

$\displaystyle x_0^2+Ax_0=B(t+\tau),$ (2.18)

where the parameter $ \tau$ is given by

$\displaystyle \tau=\frac{x^2_i+Ax_i}{B}.$ (2.19)

So $ \tau$ takes into account any oxide thickness at the start of the oxidation. It can also be used to provide a better fit to the data in the anomalous thin oxide regime in dry oxidation.

At first with (2.18) the oxidation time for a specific desired oxide thickness can be estimated by

$\displaystyle t=\frac{x_0^2-x_i^2}{B}+\frac{x_0-x_i}{B/A}.$ (2.20)

On the other side solving the quadratic equation (2.18) in regard of x$ _0$ leads to the following explicit expression for the oxide thickness in terms of oxidation time:

$\displaystyle x_0=\frac{A}{2}\Big(\sqrt{1+\frac{4B}{A^2}(t+\tau)} - 1\Big).$ (2.21)

The formulas (2.44) and (2.20) are a real strength of the Deal-Grove model, because the oxide thickness for any oxidation time or the needed time for a specific thickness can be determined in an uncomplicated and fast way. Of course the thickness can be only estimated in one direction on planar structures, but in practice this fast approach is indeed helpful.

It is interesting to examine two limiting forms of the linear-parabolic relationship (2.44). One limiting case occurs for long oxidation times when $ t\gg\tau$ and $ t\gg A^2/4B$

$\displaystyle x_0\cong\sqrt{B\cdot t}, %\thoseblankpages
$ (2.22)

where $ B$ is the so-called parabolic rate constant

$\displaystyle B = \frac{2DC^*}{N}.$ (2.23)

The other limiting case occurs for short oxidation times when $ t\ll A^2/4B$

$\displaystyle x_0\cong\frac{B}{A}(t+\tau), %\clearpage
$ (2.24)

where $ {B/A}$ is the so-called linear rate constant

$\displaystyle \frac{B}{A} = \frac{C^*}{N\big(\frac{1}{k_s}+\frac{1}{h}\big)} \cong \frac{C^*k_s }{N}.$ (2.25)

The linear term (2.24) dominates for small $ x$-values, the parabolic term (2.22) for larger $ x$-values.

The rate constants $ B$ and $ B/A$ are also termed as Deal-Grove-parameters. In most publications which use the Deal-Grove model the oxide growth is described with $ B$ and $ B/A$. The parameters $ B$ and $ B/A$ are normally determined experimentally by extracting them from growth data. The reason for taking this approach is simply that all parameters in (2.23) and (2.25) are not known. $ k_s$ in particular contains a lot of hidden physics associated with the interface reaction.

2.6.3 Temperature Dependence of $ \boldsymbol{B}$ and $ \boldsymbol{B/A}$

In order to model the corresponding growth rate for different temperatures, the values for $ B$ and $ B/A$ must change with temperature. As explained in Section 2.4.2, the oxidation rate increases with higher temperature, and so the values of $ B$ and $ B/A$ must also increase. It was found experimentally that both $ B$ and $ B/A$ are well described by Arrhenius expressions of the form

$\displaystyle B=C_1 \mathrm{exp}\Big(-\frac{{E_1}}{{kT}}\Big)$ (2.26)
$\displaystyle \frac{B}{A}=C_2 \mathrm{exp}\Big(-\frac{{E_2}}{{kT}}\Big).$ (2.27)

In these expressions, $ E_1$ and $ E_2$ are the activation energies associated with the physical process that $ B$ and $ B/A$ represents, and $ C_1$ and $ C_2$ are the pre-exponential constants. Table 2.2 lists the experimental values for the parameters needed in (2.26) and (2.27) for (111) oriented silicon at one atmosphere. With these values, in Fig. 2.17 the parameters $ B$ and $ B/A$ are plotted over the temperature range 800-1000$ ^{\circ }$C for wet and dry oxidation. In order to get the corresponding values for (100) oriented silicon, only the $ C_2$ values must be divided by the factor 1.68, all the $ E_{1,2}$ and $ C_1$ values are the same.

Table 2.2: Arrhenius parameters for $ B$ and $ B/A$ in (111) oriented silicon [25].
Ambient $ B$ $ B/A$
Dry O$ _2$ $ C_1=7.72 \times 10^2 \mu$m$ ^2$/hr $ C_2=6.23 \times 10^6 \mu$m/hr
  $ E_1= 1.23$ eV $ E_2= 2.00$ eV
Wet H$ _2$O $ C_1=3.86 \times 10^2 \mu$m$ ^2$/hr $ C_2=1.63 \times 10^8 \mu$m/hr
  $ E_1= 0.78$ eV $ E_2= 2.05$ eV

For the parabolic rate constant $ B$ the activation energy $ E_1$ is quite different for O$ _2$ and H$ _2$O ambients. (2.23) suggests that the physical mechanism responsible for $ E_1$ might be the oxidant diffusion through SiO$ _2$, because $ N$ is a constant and $ C^*$ is not expected to increase exponentially with temperature. In fact, independent measurements of the diffusion coefficients of O$ _2$ and H$ _2$O in SiO$ _2$ show that these parameters vary with temperature in the same way as (2.26) and with $ E_1$ values close to those shown in Table 2.2. The clear implication is that $ B$ in the linear parabolic model really represents the oxidant diffusion process.

The $ E_2$ values for $ B/A$ in the table are all quite close to 2 eV. (2.25) suggests that the physical origin of $ E_2$ is likely connected with the interface reaction rate $ k_s$. Traditionally, the 2 eV activation energy has been associated with the Si-O bound formation process because of measurements by Pauling [52] that suggested that the Si-O bond energy was in the correct range to explain the $ B/A$ values. However, the interface reaction is very complex and it is likely that other effects also affect the experimental $ B/A$ values. An additional observation supports the idea that it is somehow associated with the silicon substrate which determines $ E_2$, because $ E_2$ is essentially independent of the oxidation ambient. It is also essentially independent of the substrate crystal orientation, which suggests that $ E_2$ represents a fundamental part of the oxidation process, not something only associated with the substrate.

Figure 2.17: $ B$ and $ B/A$ versus temperature for (111) oriented silicon for wet and dry oxidation.
\includegraphics[width=0.75\linewidth,bb=20 52 706 528, clip]{curv/deal/bandbaparam}

2.6.4 Pressure Dependence of $ \boldsymbol{B}$ and $ \boldsymbol{B/A}$

The linear parabolic model predicts that the oxide growth rate should be directly proportional to the oxidant pressure as shown in (2.9). If Henry's law [53] holds and the concentration of oxidants on the gas/SiO$ _2$ interface $ C^*$ is proportional to the pressure $ p$, then both $ B$ and $ B/A$ are proportional to $ p$ from (2.23) and (2.25), and the oxide growth rate should therefore be proportional to $ p$.

Experimental measurements have shown that for wet oxidation this prediction is correct, and for H$ _2$O ambients the pressure dependence of the parabolic and linear rate constants are [36]

$\displaystyle B(P) = B(1\mathrm{atm}) \cdot p, %\label{wpres}
$ (2.28)
$\displaystyle B/A(P)= B/A(1\mathrm{atm})\cdot p.$ (2.29)

In contrast to wet oxidation for dry oxidation the pressure dependence is inconsistent with the linear parabolic model. A considerable body of data has consistently shown that dry oxidation can only be modeled with a linear parabolic equation, where $ B \propto p$ and $ B/A \propto p^\mathrm{n}$ with $ \mathrm{n} \approx 0.7 - 0.8$ [35]. Hence, to use the model for O$ _2$ ambients at any pressure $ p$ the parabolic and linear rate constants should be

$\displaystyle B(P) = B(1\mathrm{atm}) \cdot p$ (2.30)
$\displaystyle B/A(P)= B/A(1\mathrm{atm})\cdot p^{0.75}.$ (2.31)

Within the context of the model it can be inferred that the pressure dependence of $ B \propto p$ comes exclusively from $ C^*$, because $ C^*$ as determined in Henry's law (2.23) must be $ C^* \propto p$. Therefore, the diffusion coefficient $ D$ for the oxidants in the solid phase can be assumed constant.

If $ B/A$ is not linearly proportional to $ p$, $ k_s$ from (2.25) must depend on $ p$ in a non-linear fashion. Considering the pressure dependence of $ B/A$ and $ C^*$ above, the chemical surface reaction must depend on pressure in the way $ k_s \propto p^{-0.25}$.

2.6.5 Dependence of $ \boldsymbol{B}$ and $ \boldsymbol{B/A}$ on Crystal Orientation

Even before the development of the Deal-Grove model, it has been observed that crystal orientation affects the oxidation rate [39]. The crystal effects can be incorporated in the following way: Except perhaps in the region very near the Si/SiO$ _2$ interface, the oxide grows on silicon in an amorphous way. So it does not incorporate any information about the underlying silicon crystal structure. Therefore, the parabolic rate constant B should not be orientation dependent, since B represents the oxidant diffusion through the SiO$ _2$. If the oxide structure is unrelated to the underlying substrate, there should be no crystal orientation effect on B. In fact it was found experimentally by extracting growth data [38], that in context of the model there is no crystal effect on the rate constant B. The B values are the same for all orientations.

On the other hand B/A should be orientation dependent, because it involves the reaction at the Si/SiO$ _2$ interface. This reaction surely involves silicon atoms and should be affected by the number of available reaction sites. It was found experimentally [38], that there are two extremes of the linear rate constant B/A. The minimum was found for (100) oriented silicon whereas the maximum is at (111) orientation, and all other orientation are normally between these two extremes. In the context of the model the orientation effect must be incorporated for the rate constant B/A in the following way [38]:

$\displaystyle B/A\langle 111\rangle = 1.68\cdot B/A\langle 100\rangle.$ (2.32)

2.6.6 Thin Film Oxidation with Deal-Grove Model

It has been observed in many experiments that there is a rapid and non-linear oxide growth in the initial stage of dry oxidation [54], as presented in Fig. 2.18. One weakness of the model is the impossibility to predict the initial stage of the oxidation growth. As shown in Fig. 2.18, even with the best fit, the approximately first 30 nm of the oxide thickness can not be forecasted with the linear parabolic model, because the oxide growth is fast and non-linear but the model offers only a linear fit for such thin thicknesses [55] .

Figure 2.18: Rapid, non-linear growth rate in the inital stage of dry oxidation.

next up previous contents
Next: 2.7 The Massoud Model Up: 2. Physics of Thermal Previous: 2.5 Nitrided Oxide Films

Ch. Hollauer: Modeling of Thermal Oxidation and Stress Effects