2. Thermal oxidation models for Si and SiC

#### 2.1. Deal-Grove model and its related models

The kinetic model of Si oxidation that is most often taken as a reference is the one so-called Deal-Grove model proposed by Deal and Grove [6, 12]. According to this model, the beginning of oxidation is limited to the interfacial oxidation reaction and, after oxidation proceeds, the rate-imiting process is transferred from the interfacial reaction to the diftusion of oxidants in SiO2. This process is expressed by the following equation given by Deal and Grove as [6, 12]

$$\frac{dX}{dt} = \frac{B}{A + 2X} \tag{1}$$

where B/A and B are denoted as the linear and parabolic rate constants of oxidation, respectively. It is noted that B/A and B are the rate coefficients for the interfacial reaction and the diffusion of oxidants, respectively, i.e.,

$$\frac{B}{A} = \frac{kC\_O^0}{N\_0}, \ B = \frac{2D\_O C\_O^0}{N\_0} \tag{2}$$

where k is the interfacial oxidation rate constant, C the solubility limit in SiO2 No the molecular density of SiO2, D the diffusivity in SiO2, and the subscript means the value for the corresponding atom. By the way, according to the definition in the D-G model, in which the oxide growth proceeds under steady-state and a SiO2 film is grown only at the SiO2-Si interface, the equation (1) can also be written as [6, 12]:

$$\frac{dX}{dt} = \frac{kC\_O^1}{N\_0} \left(= \frac{k}{N\_0} \frac{C\_O^0}{(k/D\_O)X + 1} \right) \tag{3}$$

where C' is the atomic concentration at the interface. Comparing eq. (3) with eqs. (1, 2), C), C is almost constant by CJ while X ≤Do/k and, in turn, it reduces inverse proportional to X when X >Do/k (see the right-end parentheses in eq. (3)). Therefore, we can see that the length Do/k(=A/2) corresponds to the thickness at which the interfacial reaction rate-limiting step transits to the diffusion of oxidants one. On the contrary, Watanabe et al. reconfirmed the validity of eq. (1), but modified the framework of A parameter, based on the consideration that the growth rate at the full range of oxide thickness is limited by the oxygen diffusion, i.e., there is no interfacial reaction-limited step in the Si oxidation process [16].

#### 2.2. Massoud empirical relation

It is well known that, in the case of dry oxidation, the oxidation rate of Si in the thin oxide thickness range cannot be reproduced by the D-G equation [6, 12] and, hence, several models to describe the growth rate enhancement in the thin oxide regime have been proposed. Among them, Massoud et al. [13] have proposed an empirical relation for the oxide thickness dependence of oxidation rate, that is, the addition of an exponential term to the D-G equation,

$$\frac{dX}{dt} = \frac{B}{A + 2X} + C \exp\left(-\frac{X}{L}\right) \tag{4}$$

where C and L are the pre-exponential constant and the characteristic length, respectively. Also in the case of SiC oxidation, we have found that it is possible to fit the calculated values to the observed ones using eq. (4) much better than using eq. (1) regardless of oxidation temperature, as shown in Fig. 1. However, this empirical equation can only reproduce the observed growth rates numerically, but does not provide a physical meaning.

*k* =*k*0(1

(1hSi) + -

0 *X*

where the h is the emission ratio, the e is the oxidation rate of Si interstitials inside SiO2, dis the oxidation rate of Si interstitials on the oxide surface, and the superscript 'S' means the position at the oxide surface (*x=X*). The first, second, and third term in the right-hand side of eq. (6) correspond to the interfacial oxide growth, the oxide growth inside SiO2, and that on the oxide surface, respectively. The concentrations of Si interstitials and O2 moleculers in SiO2 (*C*Si and *C*O\_z.!/,!0%2!(5dz.!z !.%2! z1/%\*#zz\*1)!.%(z(1(0%+\*z/! z+\*z0\$!z %""1¥ sion theory [17f^z \$!z %""1/%+\*z !-10%+\*/z3%((z !z /\$+3\*z%\*z !^z E^G^z "z 0\$!z /0! 5w/00!z ,¥ proximation is assumed for the calculation, the obtained growth rate equation is equivalent

e(*C*O)2*C*Si*dx* + d(*C*<sup>O</sup>

) and the residual carbon is unlikely to exist at the oxide-SiC interface in the

!.(5z/0#!z+"z%z+4% 0%+\*\_z0\$!z/0.!//z\*!.u0z0\$!z%\*0!."!z%/z+\*/% !.! z0+z!z()+/0z% !\*¥ 0%(z0+z0\$!z/!z+"z%z+4% 0%+\*^z\$!.!"+.!\_z%0z%/z,.+(!z0\$0z0+)%z!)%//%+\*z 1!z0+z0\$!z%\*¥ terfacial stress also accounts for the growth enhancement in SiC oxidation. In addition, in the case of SiC oxidation, we should take C emission as well as Si emission into account

!!\*0(5\_z3!z\$2!z,.+,+/! zz%z+4% 0%+\*z)+ !(\_z0!.)! zo%z\* zz!)%//%+\*z)+ !(o\_z0'¥ %\*#z0\$!z%z\* zz!)%//%+\*/z%\*0+z0\$!z+4% !z%\*0+z+1\*0\_z3\$%\$z(! z0+zz.! 10%+\*z+"z%\*0!."¥ cial reaction rate [19]. Figure 2. schematizes the Si and C emission model. Considering Si and C atoms emitted from the interface during the oxidation as well as the oxidation process

S ) 2 *C*Si

<sup>S</sup> (6)

[18]) is almost the same as that in

where *k*0 is the initial interfacial oxidation rate.

*N*0 *dX dt* <sup>=</sup>*kC*<sup>O</sup> I

to the Massoud empirical relation [14].

2.4. Si and C emission model for SiC

because SiC consists of Si and C atoms.

Si (5×1022cm<sup>3</sup>

Since the density of Si atoms in 4H-SiC (4.80×1022cm<sup>3</sup>

of C, the reaction equation for SiC oxidation can be written as,

*C*Si I

*C*Si

In the D-G model and the Massoud empirical relation, it has been considered that oxide #.+30\$z+1./z+\*(5z+.z)%\*(5z0z0\$!z%w+4% !z%\*0!."!^z+3!2!.\_z+. %\*#z0+z0\$!z%\*0!."¥ %(z%z!)%//%+\*z)+ !(zeDGf\_z%z0+)/z.!z!)%00! z%\*0+z0\$!z+4% !z(5!.\_z/+)!z+"z3\$%\$z!\*+1\*¥ ter the oxidant inside the SiO2 layer to form SiO2. In addition to this, when the oxide is very 0\$%\*\_z/+)!z+"z0\$!z!)%00! z%z0+)/z\*z#+z0\$.+1#\$z0\$!z+4% !z(5!.z\* z.!\$z0\$!z+4% !z/1.¥ face, and are instantly oxidized, resulting in the formation of an SiO2z(5!.z+\*z0\$!z+4% !z/1.¥ face. Therefore, there are two oxide growth processes other than the interfacial oxide growth, *i.e.*\_z+4% !z"+.)0%+\*z 1!z0+z+4% 0%+\*z+"z%z%\*0!./0%0%(/z%\*/% !z0\$!z+4% !z\* z+\*z0\$!z+4% !z/1.¥ face. The total growth rate is given by the sum of these three oxidation processes, as [14],

<sup>0</sup> ) (5)

Thermal Oxidation Mechanism of Silicon Carbide

http://dx.doi.org/10.5772/50748

185

Figure 1. Oxide thickness dependences of oxidation rate at various oxidation temperatures on (0001¯ ) C-face. The solid and dashed lines denote the values derived from the Deal-Grove model (eq. (1) [689F<L@GK=>JGEL@==EHAJA;9DJ=D9s tion (eq. (4) [13]), respectively.

#### 2.3. Interfacial Si emission model for Si

Some Si oxidation models that describe the growth rate enhancement in the initial stage of oxidation have been proposed [14, 15, 17]. The common view of these models is that the stress near/at the oxide-Si interface is closely related to the growth enhancement. Among these models, the 'interfacial Si emission model' is known as showing the greatest ability to fit the experimental oxide growth rate curves. According to this model, Si atoms are emitted as interstitials into the oxide layer accompanied by oxidation of Si, which is caused by the strain due to the expansion of Si lattices during oxidation. The oxidation rate at the interface (*k* in eq. (2)) is initially large and is suppressed by the accumulation of emitted Si atoms near the interface with increasing oxide thickness, *i.e.*, the *k* is not constant but a function of oxide 0\$%'\*!//z\* z0\$!z+4% 0%+\*z.0!z%/z\*+0z!\*\$\*! z%\*z0\$!z0\$%\*z+4% !z.!#%)!z10z%/z-1%'(5z/1,¥ pressed with increasing thickness. To describe this change in the interfacial reaction rate, Kageshima *et al.* introduce the following equation as the interfacial reaction rate, *k* [14, 17]:

$$k = k\_0 \left( 1 - \frac{C\_{\rm Si}^{\rm I}}{C\_{\rm Si}^{\rm O}} \right) \tag{5}$$

where *k*0 is the initial interfacial oxidation rate.

In the D-G model and the Massoud empirical relation, it has been considered that oxide #.+30\$z+1./z+\*(5z+.z)%\*(5z0z0\$!z%w+4% !z%\*0!."!^z+3!2!.\_z+. %\*#z0+z0\$!z%\*0!."¥ %(z%z!)%//%+\*z)+ !(zeDGf\_z%z0+)/z.!z!)%00! z%\*0+z0\$!z+4% !z(5!.\_z/+)!z+"z3\$%\$z!\*+1\*¥ ter the oxidant inside the SiO2 layer to form SiO2. In addition to this, when the oxide is very 0\$%\*\_z/+)!z+"z0\$!z!)%00! z%z0+)/z\*z#+z0\$.+1#\$z0\$!z+4% !z(5!.z\* z.!\$z0\$!z+4% !z/1.¥ face, and are instantly oxidized, resulting in the formation of an SiO2z(5!.z+\*z0\$!z+4% !z/1.¥ face. Therefore, there are two oxide growth processes other than the interfacial oxide growth, *i.e.*\_z+4% !z"+.)0%+\*z 1!z0+z+4% 0%+\*z+"z%z%\*0!./0%0%(/z%\*/% !z0\$!z+4% !z\* z+\*z0\$!z+4% !z/1.¥ face. The total growth rate is given by the sum of these three oxidation processes, as [14],

$$kN\_0 \frac{dX}{dt} = k\,\mathrm{C\_O^{I}}(1-\nu\_{\mathrm{Si}}) + \int\_0^X \kappa \langle \mathrm{C\_O} \rangle^2 \mathrm{C\_{Si}} dx + \eta \,\mathrm{(C\_O^{S\_2})^2} \mathrm{C\_{Si}^{S\_{\mathrm{Si}}}} \tag{6}$$

where the h is the emission ratio, the e is the oxidation rate of Si interstitials inside SiO2, dis the oxidation rate of Si interstitials on the oxide surface, and the superscript 'S' means the position at the oxide surface (*x=X*). The first, second, and third term in the right-hand side of eq. (6) correspond to the interfacial oxide growth, the oxide growth inside SiO2, and that on the oxide surface, respectively. The concentrations of Si interstitials and O2 moleculers in SiO2 (*C*Si and *C*O\_z.!/,!0%2!(5dz.!z !.%2! z1/%\*#zz\*1)!.%(z(1(0%+\*z/! z+\*z0\$!z %""1¥ sion theory [17f^z \$!z %""1/%+\*z !-10%+\*/z3%((z !z /\$+3\*z%\*z !^z E^G^z "z 0\$!z /0! 5w/00!z ,¥ proximation is assumed for the calculation, the obtained growth rate equation is equivalent to the Massoud empirical relation [14].

#### 2.4. Si and C emission model for SiC

Figure 1. Oxide thickness dependences of oxidation rate at various oxidation temperatures on (0001¯ ) C-face. The solid and dashed lines denote the values derived from the Deal-Grove model (eq. (1) [689F<L@GK=>JGEL@==EHAJA;9DJ=D9s

Some Si oxidation models that describe the growth rate enhancement in the initial stage of oxidation have been proposed [14, 15, 17]. The common view of these models is that the stress near/at the oxide-Si interface is closely related to the growth enhancement. Among these models, the 'interfacial Si emission model' is known as showing the greatest ability to fit the experimental oxide growth rate curves. According to this model, Si atoms are emitted as interstitials into the oxide layer accompanied by oxidation of Si, which is caused by the strain due to the expansion of Si lattices during oxidation. The oxidation rate at the interface (*k* in eq. (2)) is initially large and is suppressed by the accumulation of emitted Si atoms near the interface with increasing oxide thickness, *i.e.*, the *k* is not constant but a function of oxide 0\$%'\*!//z\* z0\$!z+4% 0%+\*z.0!z%/z\*+0z!\*\$\*! z%\*z0\$!z0\$%\*z+4% !z.!#%)!z10z%/z-1%'(5z/1,¥ pressed with increasing thickness. To describe this change in the interfacial reaction rate, Kageshima *et al.* introduce the following equation as the interfacial reaction rate, *k* [14, 17]:

tion (eq. (4) [13]), respectively.

2.3. Interfacial Si emission model for Si

184 Physics and Technology of Silicon Carbide Devices

Since the density of Si atoms in 4H-SiC (4.80×1022cm<sup>3</sup> [18]) is almost the same as that in Si (5×1022cm<sup>3</sup> ) and the residual carbon is unlikely to exist at the oxide-SiC interface in the !.(5z/0#!z+"z%z+4% 0%+\*\_z0\$!z/0.!//z\*!.u0z0\$!z%\*0!."!z%/z+\*/% !.! z0+z!z()+/0z% !\*¥ 0%(z0+z0\$!z/!z+"z%z+4% 0%+\*^z\$!.!"+.!\_z%0z%/z,.+(!z0\$0z0+)%z!)%//%+\*z 1!z0+z0\$!z%\*¥ terfacial stress also accounts for the growth enhancement in SiC oxidation. In addition, in the case of SiC oxidation, we should take C emission as well as Si emission into account because SiC consists of Si and C atoms.

!!\*0(5\_z3!z\$2!z,.+,+/! zz%z+4% 0%+\*z)+ !(\_z0!.)! zo%z\* zz!)%//%+\*z)+ !(o\_z0'¥ %\*#z0\$!z%z\* zz!)%//%+\*/z%\*0+z0\$!z+4% !z%\*0+z+1\*0\_z3\$%\$z(! z0+zz.! 10%+\*z+"z%\*0!."¥ cial reaction rate [19]. Figure 2. schematizes the Si and C emission model. Considering Si and C atoms emitted from the interface during the oxidation as well as the oxidation process of C, the reaction equation for SiC oxidation can be written as,

$$\begin{aligned} \text{SiC} + \left(2 - \nu\_{\text{Si}} - \nu\_{\text{C}} - \frac{\alpha}{2}\right) \text{O}\_2 &\rightarrow \left(1 - \nu\_{\text{Si}}\right) \text{SiO}\_2 + \nu\_{\text{Si}} \text{Si} + \nu\_{\text{C}} \text{C} + \alpha \text{CO} \\ &\star \text{(1} - \nu\_{\text{C}} - \alpha) \text{CO}\_2 \end{aligned} \tag{7}$$

Diffusion equations for Si and C interstitials, and oxidants can be written by modifying the

*<sup>x</sup>* ) *R*<sup>1</sup> *R*2,

, *R*<sup>2</sup> =e1*C*Si*C*<sup>O</sup> + e2*C*Si(*C*O)2,

*C*C*C*<sup>O</sup> + e<sup>2</sup>

*<sup>x</sup>* ) *R*<sup>1</sup> *R*<sup>2</sup> *R*<sup>1</sup>

sorption of interstitials inside the oxide and they are assumed to be consist of two terms, as suggested by Uematsu *et al.* [20]. From eq. (7), the boundary conditions at the interface are

> *C*<sup>C</sup> *x*

<sup>2</sup> )*kC*<sup>O</sup> I

It has been believed that the oxidation rate in the thick oxide regime is solely limited by the in-diffusion of oxidant and the diffusivity of CO in SiO2 is much larger than that of O2. Thus, 3!z//1)! z0\$0z0\$!z %""1/%+\*z,.+!//z+"zz%/z%\*/!\*/%0%2!z0+z0\$!z+4% !z#.+30\$z.0!^z\$!z+4¥ ide growth rate is described as eq. (6), but the second term in the right-hand side (*i.e.* the absorption-term of Si interstitials inside the oxide) is modified to be the combination of e<sup>1</sup> and e2 shown in eq. (9) (see eq. (6) in Ref. [19]). We approximated this integral term by the

0\$!z% !z0\$0z%\*0!./0%0%(/z.!z1/1((5z %/0.%10! z+. %\*#z0+z3\$0z.!/!)(!/zz+),(!)!\*¥ tary error function or exponential function, and their areal density is given approximately

Equations (8-10) were solved numerically using the partial differential equation solver ZOMBIE [22]. The oxide thickness, *X*\_z0z!\$z 0%)!z/0!,z3/z+0%\*! z ".+)z!-^z cId^z\$!z,¥

ues as those obtained for Si oxidation [17]. The parameters concerning C interstitials

*R*<sup>2</sup> '

,

'

*C*C(*C*O)2,

'

*R*<sup>2</sup> '


<sup>I</sup> in height and *C*Si/ *<sup>x</sup>*|*x*=0 / <sup>2</sup> in gradient of hypotenuse, on the basis of

I ,

0

) as well as the values of *k*0 and hSi were determined by fitting the

dz3!.!z/!0z0+z0\$!z/)!z2(¥

*R*3,

(9)

187

¦

Thermal Oxidation Mechanism of Silicon Carbide

http://dx.doi.org/10.5772/50748

z)!\*z¥

(10)

those given by the interfacial Si emission model [17], that is,

*<sup>x</sup>* (*D*Si

*<sup>x</sup>* (*D*<sup>C</sup>

*<sup>x</sup>* (*D*<sup>O</sup>

<sup>S</sup> *C*<sup>O</sup> 0 )


rameters related to the properties of SiO2 (*D*Si,*D*O , d, e1, e2and*C*Si

calculated oxide growth rates to the measured ones.

<sup>|</sup> *<sup>x</sup>*=0 =(2*v*Si*v*<sup>C</sup> \_

*C*Si

*C*<sup>C</sup> *<sup>x</sup>* ) *R*<sup>1</sup> '

> =e<sup>1</sup> '

*C*<sup>O</sup>

where the prime means the variation for C atoms. It is noted that the *R*2 and *R*<sup>2</sup>

I , *D*<sup>C</sup>

*C*Si *<sup>t</sup>* <sup>=</sup>

*C*<sup>C</sup> *<sup>t</sup>* <sup>=</sup>

*R*1 '

*D*Si *C*Si *x*

*D*<sup>O</sup> *C*<sup>O</sup> *x*

given as,

area of a triangle *C*Si

(\_, <sup>h</sup>C, *<sup>D</sup>*C, <sup>d</sup> ¦

by the area of the triangle [21].

, e<sup>1</sup> ¦ , e<sup>2</sup> ¦ and*C*<sup>C</sup> 0

*C*<sup>O</sup> *<sup>t</sup>* <sup>=</sup>

*R*<sup>1</sup> =d*C*<sup>O</sup> S *C*Si S

> =d ' *C*O S *C*C S , *R*<sup>2</sup> '

*R*<sup>3</sup> =*h* (*C*<sup>O</sup>

where \_ denotes the production ratio of CO.

Figure 2. Schematic illustration of the Si and C emission model [19]. It is to be noted that Si and C atoms are practically emitted into not only to the SiO2-side, but also the SiC-side, as shown in the figure.

In the case of Si oxidation, the interfacial reaction rate (eq. (5)) is introduced by assuming that the value of *C*Si <sup>I</sup> does not exceed the *C*Si <sup>0</sup> though the reaction rate decreases with increase of*C*Si I ^z-/! z+\*z0\$%/z% !\_z0\$!z%\*0!."%(z.!0%+\*z.0!z"+.z%z%/z0\$+1#\$0z0+z!z#%2!\*z5z)1(0%¥ plying decreasing functions for Si and C [19]:

$$k = k\_0 \left( 1 - \frac{C\_{\rm Si}^{\rm I}}{C\_{\rm Si}^{\rm O}} \right) \left( 1 - \frac{C\_{\rm C}^{\rm I}}{C\_{\rm C}^{\rm O}} \right) \tag{8}$$

This equation implies that the growth rate in the initial stage of oxidation should reduce by two steps because the accumulation rates for Si and C interstitials should be different from each other, and hence, the oxidation time when the concentration of interstitial saturates should be different between Si and C interstitial. This prediction will be evidenced in the next section.

Diffusion equations for Si and C interstitials, and oxidants can be written by modifying the those given by the interfacial Si emission model [17], that is,

SiC <sup>+</sup> (2hSih<sup>C</sup> \_

186 Physics and Technology of Silicon Carbide Devices

where \_ denotes the production ratio of CO.

<sup>2</sup> )O2¨(1hSi)SiO2 <sup>+</sup> <sup>h</sup>SiSi <sup>+</sup> <sup>h</sup>CC <sup>+</sup> \_CO

Figure 2. Schematic illustration of the Si and C emission model [19]. It is to be noted that Si and C atoms are practically

In the case of Si oxidation, the interfacial reaction rate (eq. (5)) is introduced by assuming

/! z+\*z0\$%/z% !\_z0\$!z%\*0!."%(z.!0%+\*z.0!z"+.z%z%/z0\$+1#\$0z0+z!z#%2!\*z5z)1(0%¥

*C*C I

*C*C

emitted into not only to the SiO2-side, but also the SiC-side, as shown in the figure.

<sup>I</sup> does not exceed the *C*Si

*k* =*k*0(1

*C*Si I

*C*Si <sup>0</sup> )(1

This equation implies that the growth rate in the initial stage of oxidation should reduce by two steps because the accumulation rates for Si and C interstitials should be different from each other, and hence, the oxidation time when the concentration of interstitial saturates should be different between Si and C interstitial. This prediction will be evidenced in the

plying decreasing functions for Si and C [19]:

that the value of *C*Si

of*C*Si I ^z-

next section.

+(1h<sup>C</sup> \_)CO2

<sup>0</sup> though the reaction rate decreases with increase

<sup>0</sup> ) (8)

(7)

$$\begin{aligned} \frac{\partial \mathbf{C}\_{\rm Si}}{\partial t} &= \frac{\partial}{\partial x} \left( D\_{\rm Si} \frac{\partial \mathbf{C}\_{\rm Si}}{\partial x} \right) - R\_1 - R\_2 \\ R\_1 &= \eta \mathbf{C}\_{\rm C}^{-} \mathbf{C}\_{\rm Si}^{-}, R\_2 = \kappa\_1 \mathbf{C}\_{\rm Si} \mathbf{C}\_{\rm O} + \kappa\_2 \mathbf{C}\_{\rm Si} (\mathbf{C}\_{\rm O})^2, \\ \frac{\partial \mathbf{C}\_{\rm C}}{\partial t} &= \frac{\partial}{\partial x} \left( D\_{\rm C} \frac{\partial \mathbf{C}\_{\rm C}}{\partial x} \right) - R\_1^{\top} - R\_2^{\top}, \\ R\_1 &= \eta \mathbf{C}\_{\rm C}^{-} \mathbf{C}\_{\rm C}^{-}, R\_2 = \kappa\_1 \mathbf{C}\_{\rm C}^{-} \mathbf{C}\_{\rm C} + \kappa\_2 \mathbf{C}\_{\rm C} (\mathbf{C}\_{\rm O})^2, \\ \frac{\partial \mathbf{C}\_{\rm C}}{\partial t} &= \frac{\partial}{\partial x} \left( D\_{\rm O} \frac{\partial \mathbf{C}\_{\rm O}}{\partial x} \right) - R\_1 - R\_2 - R\_1^{\top} - R\_2^{\top} - R\_3. \\ R\_3 &= h \left( \mathbf{C}\_{\rm O}^{-} - \mathbf{C}\_{\rm O}^{0} \right) \end{aligned} \tag{9}$$

where the prime means the variation for C atoms. It is noted that the *R*2 and *R*<sup>2</sup> ¦ z)!\*z¥ sorption of interstitials inside the oxide and they are assumed to be consist of two terms, as suggested by Uematsu *et al.* [20]. From eq. (7), the boundary conditions at the interface are given as,

$$\begin{aligned} \left. D\_{\rm Si} \frac{\partial \mathcal{L}\_{\rm Si}}{\partial \boldsymbol{x}} \right|\_{\boldsymbol{x} = 0 = -\boldsymbol{v}\_{\rm Si} k \, \mathcal{L}\_{\rm O}^{\rm I}} D\_{\rm C} \frac{\partial \mathcal{L}\_{\rm C}}{\partial \boldsymbol{x}} \Big|\_{\boldsymbol{x} = 0 = -\boldsymbol{v}\_{\rm C} k \, \mathcal{L}\_{\rm O}^{\rm I}} \\ \left. D\_{\rm O} \frac{\partial \mathcal{L}\_{\rm O}}{\partial \boldsymbol{x}} \right|\_{\boldsymbol{x} = 0 = \left( 2 - \boldsymbol{v}\_{\rm Si} - \boldsymbol{v}\_{\rm C} - \frac{\alpha}{2} \right) k \, \mathcal{L}\_{\rm O}^{\rm I}} \right| \end{aligned} \tag{10}$$

It has been believed that the oxidation rate in the thick oxide regime is solely limited by the in-diffusion of oxidant and the diffusivity of CO in SiO2 is much larger than that of O2. Thus, 3!z//1)! z0\$0z0\$!z %""1/%+\*z,.+!//z+"zz%/z%\*/!\*/%0%2!z0+z0\$!z+4% !z#.+30\$z.0!^z\$!z+4¥ ide growth rate is described as eq. (6), but the second term in the right-hand side (*i.e.* the absorption-term of Si interstitials inside the oxide) is modified to be the combination of e<sup>1</sup> and e2 shown in eq. (9) (see eq. (6) in Ref. [19]). We approximated this integral term by the area of a triangle *C*Si <sup>I</sup> in height and *C*Si/ *<sup>x</sup>*|*x*=0 / <sup>2</sup> in gradient of hypotenuse, on the basis of 0\$!z% !z0\$0z%\*0!./0%0%(/z.!z1/1((5z %/0.%10! z+. %\*#z0+z3\$0z.!/!)(!/zz+),(!)!\*¥ tary error function or exponential function, and their areal density is given approximately by the area of the triangle [21].

Equations (8-10) were solved numerically using the partial differential equation solver ZOMBIE [22]. The oxide thickness, *X*\_z0z!\$z 0%)!z/0!,z3/z+0%\*! z ".+)z!-^z cId^z\$!z,¥ rameters related to the properties of SiO2 (*D*Si,*D*O , d, e1, e2and*C*Si 0 dz3!.!z/!0z0+z0\$!z/)!z2(¥ ues as those obtained for Si oxidation [17]. The parameters concerning C interstitials (\_, <sup>h</sup>C, *<sup>D</sup>*C, <sup>d</sup> ¦ , e<sup>1</sup> ¦ , e<sup>2</sup> ¦ and*C*<sup>C</sup> 0 ) as well as the values of *k*0 and hSi were determined by fitting the calculated oxide growth rates to the measured ones.
