4.3 Segregation Interface Condition

If at the Si/SiO$_2$ interface there is a dopant concentration $C_{I,O}$ and $C_{I,Si}$ on the oxide and silicon side, respectively, as illustrated in Fig. 4.1, the segregation coefficient can be written as [83]

$$m=\frac{C_{I,Si}}{C_{I,O}}$$ (4.23)

If it is assumed that $C_{I,O}>C_{I,Si}$ the flux of dopants from the SiO$_2$ segment to the silicon segment through the interface is [83]

$$J_S=k_O C_{I,O}-k_{Si} C_{I,Si}=k_O\Big(C_{I,O}-\frac{k_{Si}}{k_O}\Big)=h\Big(C_{I,O}-\frac{C_{I,Si}}{m}\Big),$$ (4.24)

where $k_O$ and $k_{Si}$ are the reaction rate coefficients in SiO$_2$ and silicon, respectively. $h$ is the interface transfer coefficien which has units of velocity.

In the steady state the interface flux $J_S = 0$ and (4.24) can be transformed to the relationship

$$\frac{C_{I,O}}{C_{I,Si}}=\frac{k_{O}}{k_{Si}}.$$ (4.25)