A free boundary value problem modeling thermal oxidation of silicon (Q909144)

The authors consider a one-dimensional model of the thermal oxidation of silicon which describes the diffusion of an oxidant through an oxide layer and the growth of this layer caused by the oxidation of silicon at the oxide silicon interface. The governing equation: \(v_{\tau}-D\cdot v_{\xi \xi}=0,\) and the conditions on the free boundary b(\(\tau)\) are: \[ b(\tau)=m\cdot v(\tau,b(\tau)),\quad D\cdot v_{\xi}(\tau,b(\tau))+(b(\tau)+k)\cdot v(\tau,b(\tau))=0. \] The existence and uniqueness of a weak solution, the regularity of the solution and the estimates of the order of \(\tau^{1/2}\) for the growth of the oxide thickness b(\(\tau)\) are discussed. To obtain the results above, the standard results on evolution equations in Hilbert space and the technique of integral estimates for the solutions are well exploited.