A Fixed Grid Numerical Methodology for Modeling Wet Chemical Etching

A new mathematical model based on the total concentration approach is proposed for modeling wet chemical etching process. The proposed mathematical model is a fixed domain formulation of the etching problem. The governing equation based on the total concentration includes the interface condition too. The total concentration of etchant includes the reacted and the unreacted concentration of etchant. The unreacted etchant concentration is solved in both the etchant solution and the substrate (with zero unreacted etchant concentration). The reacted concentration of etchant is used to capture the etchfront during the progress of etching with time. Unlike the moving grid method, the etchfront is found implicitly with the total concentration method. Finite volume method is used to solve for the transient concentration distribution of etchant. The proposed method is applied from etching simple to complex geometries partially covered with mask. Results from the proposed approach are compared with the existing analytical and numerical solutions. NOMENCLATURE D diffusion coefficient of etchant MSub molecular weight of the substrate X, Y non-dimensional coordinate directions 1 Author for correspondence a coefficient of the discretization equation c unreacted etchant concentration cR reacted etchant concentration cR,max maximum possible value of the reacted concentration cT total concentration m stoichiometric reaction parameter t time t* non-dimensional time n̂ v normal velocity of the etchant-substrate interface x, y coordinate directions Greek Symbols α underrelaxation factor β non-dimensional etching parameter ∇ vector differential operator ∆t time step ρSub density of the substrate Subscripts Et the etchant Sub the substrate o initial P control volume P T total Superscripts m iteration number o previous time step Introduction Wet chemical etching (WCE) process involves the removal of material from the substrate surface by the application of a reactive liquid etchant to form a specific pattern on the substrate surface. This process has potential application in microelectronic industries in the fabrication of integrated circuit devices [1], MEMS devices [2] and sensors [3]. As etching progresses, the etched interface moves. Hence this process is regarded as a moving boundary problem. This process motivates to predict and understand the etching profile growth in designing a specific pattern on the substrate surface. Various mathematical models have been proposed by different researchers to model the WCE process such as the analytical asymptotic solution [4, 5], the moving grid (MG) method [6, 8-10], the level-set method [11, 12], and the total concentration fixed grid (FG) method [14, 15]. Based on the rate of reaction, two possible cases of WCE process can exists namelythe diffusion-controlled (infinite reaction rate) and the reaction-controlled (finite reaction rate) etching. These two cases are studied in the modeling of one-dimensional [7, 9, 14, 15], twodimensional [4-8, 10, 11, 13] and three-dimensional [12] WCE using the above analytical and numerical approaches. The MG method is the widely used numerical method for modeling WCE process. In the MG method, the computational domain is limited to the space occupied by the etchant, which expands with time. Hence the computation mesh has to be regenerated at every time step. The mesh velocities have to be accounted for in the governing equation in terms of an extra convection term [6, 10]. Further an unstructured mesh system or a bodyfitted grid system is needed to model multidimensional WCE. Recently, Chai and co-workers [14, 15] presented a FG approach based on the total concentration of etchant to model WCE process. This method is analogous to the enthalpy method used in the modeling of melting/solidification processes [16, 17]. The total concentration is the sum of the unreacted etchant concentration and the reacted etchant concentration. The governing equation based on the total concentration includes the interface condition. The reacted etchant concentration is the measure of the etchfront position during the etching process. Unlike the MG method, the etchfront is found implicitly with the total concentration method. Since the grids are fixed, hence there is no grid velocity. Therefore, cartesian grid can be used to capture the complicated etchfront in multidimensional etching. In this article, the total concentration FG method is applied to model one-dimensional (1-D), twodimensional (2-D) and three-dimensional (3-D) diffusion-controlled WCE. The governing equation, the boundary conditions and the interface condition are described. A brief description on various ingredients of the proposed FG method is given. A brief description of the numerical method used to solve the governing equation is given. The overall solution procedure is then summarized. Discussions of the results obtained using the proposed FG method are presented. Some concluding remarks are given to conclude this article. Problem Description and Governing Equation A diffusion-controlled etching is studied in this article from simple to complex geometries. The diffusioncontrolled etching is associated with infinitely fast reaction at the interface. Hence the etchant concentration at the etchant-substrate interface closes to zero. The schematic and computational domains for three test problems are shown in Figs. 1-3. In 2-D etching, a gap of width 2a is to be etched in a substrate as shown in Fig. 2a and in 3-D etching a cavity of square cross-section with dimension 2a × 2a is to be etched in a substrate as shown in Fig. 3a. The origin of the coordinate system is set to the etchant-substrate interface at t = 0 for the 1-D problem and at the center of the gap for 2-D and 3-D problems. Since the problem is symmetrical about the origin in Figs. 2 and 3, only half of the domain is considered in 2-D etching (Fig. 2b) and one-quarter is considered in 3-D etching (Fig. 3b). The governing equation, the initial condition, the boundary conditions and the interface condition are presented next. In the absence of convection, the etchant concentration within the etchant domain is governed by the mass diffusion equation given by