High aspect ratio structures are essential for the fabrication of various semiconductor devices, where the aspect ratio (AR) of the structure is defined as in case of cylinders and as in case of trenches. One particular example is negative-AND (NAND) flash cell fabrication , where three-dimensional multi-layer designs (3D-NAND) involve vertical holes which require aspect ratios above 40. The accurate simulation of the fabrication process for these high aspect ratio structures is challenging, in particular for etching processes: If the re-emission of particles (of the etching species) is neglected, the error in the results for the surface rates increases towards the bottom of the structure.
For a neutral particle species, the flux originating from multiple reflections dominates the surface rates towards the bottom of high aspect ratio structures; surface properties on the sidewalls of a structure which exhibit a high re-emission probability (i.e., low sticking probability) for a neutral particle species emphasize the importance to model a high number of re-emission events. The computational effort for the three-dimensional particle transport calculation increases with the number of considered re-emission events: In a bottom-up scheme the number of necessary re-distribution iterations is increased. In a top-down Monte Carlo scheme the necessary number of particles is increased in order to obtain an acceptable signal-to-noise ratio at the bottom of a high aspect ratio structure.
In  an approach is presented to calculate the neutral flux in long trenches and holes by exploiting symmetry properties of the structures. The three-dimensional problem is reduced to a line integral and the Nyström method  is used for discretization. Special numerical treatment is needed to handle singularities during the integration. Spikes and oscillations of the solution near corners of the structure were reported, when the resolution is not refined (compared to the resolution required by the Nyström method) at these critical spots. Assumptions for the transport of the neutral particles are ideal diffuse sources/reflections, a locally constant sticking probability, and molecular flow (ballistic transport without considering inter-particle collisions) of the neutral particles; these assumptions also justify radiosity-based approaches.
In the following, a radiosity-based one-dimensional approximation for the surface rates of a neutral particle species is presented for convex holes and trenches . The approximation is eligible to replace the three-dimensional particle transport calculation for neutral particle species. It is applicable to simulations where the three-dimensional geometry can be approximated with a convex rotationally symmetric hole or convex symmetric trenches. Although these constraints seem restrictive, the approach can be an attractive choice in modern Process TCAD, in particular for memory devices where symmetric high aspect ratio structures are utilized to increase the integration density.
First, the details of the approach are presented. Then, the utilized view factors are introduced including a novel combination of analytical view factors to obtain an analytical view factor for coaxial cones. Then, the approximation is validated using the three-dimensional top-down particle transport implementation of ViennaTS. Finally, the capabilities of the presented approach are demonstrated.
For cylindrical holes, the simulation domain is a rotationally symmetric closed convex surface. For trenches, the simulation domain is a trench with a closed convex symmetric cross section. The neutral flux source is modeled by closing the structures at the top. This leads to a disk-shaped source for holes and a strip-shaped source for trenches. Figure 7.1a and Figure 7.1b illustrate the cross sections of domains with vertical walls and with a kink at one half of the depth, respectively.
The surface adsorption is modeled using a locally constant sticking probability . The received flux is split according to into an adsorbed flux and a re-emitted flux as depicted in Figure 7.1c. Source areas additionally emit a flux independent of .
The discrete form of the radiosity equation (2.15) is used. For a surface element the received flux is
where is the self-emitted energy, is the sticking probability, and is the view factor (proportion of the radiated energy, which leaves element and is received by element ).
The linear system of equations is obtained by rewriting (7.1) in matrix notation
and transformation into the standard form
with the vector of emitted flux , a vector of sticking probabilities , and a matrix of view factors (where corresponds to the view factor ).
The solution of the diagonally-dominant linear system of equations (7.3) is approximated using the Jacobi method. The number of performed Jacobi-iterations corresponds to the considered number of re-emissions of each element to all other elements. The adsorbed flux is related to by the corresponding sticking probability of the element
The relation , which holds for closed surfaces, can be used to test the implementation and to define a stopping criterion for the Jacobi iterations.
The approach presented here is based on the discretization of the surface into discrete surface elements along the structure’s line of symmetry. Figure 7.2 shows the cross section of a convex structure and the shape of the resulting surface elements. Two vertical ranges are indicated in Figure 7.2b and the resulting surface elements and are shown for a trench (Figure 7.2a) and a hole (Figure 7.2c). The elements are formed from two strips for the trench and take the form of a sliced cone for the hole.
To assemble the matrix the view factors between all possible pairs of surface elements are required.
The view factor between two segments of a symmetric convex trench with a constant cross section, as depicted in Figure 7.2a, is derived using the crossed-strings method . This method computes the view factor between two surfaces with a constant cross section and infinite length utilizing a two-dimensional re-formulation of the problem. For two mutually completely visible strips of infinite length the view factor is 
where and denote the lengths of the diagonals, when connecting the cross section of the two strips to form a convex quadrilateral, and denote the lengths of the sides of that quadrilateral which connects the strips, and denotes the length of the side of the quadrilateral which represents the emitting strip.
Figure 7.3a is an isometric view of the four strips from Figure 7.2a. The view factors from the top right strip towards the other three strips is visualized in Figure 7.3b.
The view factor between the two segments and is
where the subscripts denote the side of the strip according to Figure 7.3b; can be neglected, as the cross section is symmetric. The view factor from an element to itself is
where again the other direction can be neglected due to symmetry. Equation (7.5) is used to compute the view factors between individual strips in (7.6) and (7.7).
A general formulation to compute the view factors between two segments of a rotationally symmetric convex hole (cf. Figure 7.2c) is derived. It is based on the view factor between two coaxial disks of nonequal radii and at a distance defined by
where and . Using this relation and the reciprocity theorem of view factors
where is the element area, a general formulation for the view factor between the inner wall surfaces of two coaxial cone-like segments (whose surfaces are mutually completely visible) is obtained. Figure 7.4a shows two segments and in such a configuration and denotes the four coaxial disks which represent the apertures of the two elements.
The final goal to compute the view factor between two elements and is divided into multiple inexpensive view factor computations between coaxial disks. First, the difference of the view factors from towards the two disks of is computed, and the reciprocity theorem (7.9) is applied to obtain (red indicates sending and blue receiving areas).
The same is done for to obtain .
Finally is obtained by subtracting from .
The view factor of an element to itself is computed by subtracting the flux leaving through the two apertures from unity.
If an element is an annulus or a disk (see Figure 7.4b and Figure 7.4c, respectively), the general formulation still applies. For a disk, the far aperture is treated as an infinitely small element.
The sticking probabilities for the wall and the bottom of the structures are selected to represent a reasonable approximation to the prevalent conditions for the neutral particles in an ion-enhanced chemical etching  (IECE) environment. A sticking probability is used for source areas which do not have any reflections originating from these artificial areas; the bottom is modeled as a fully adsorbing area with a sticking probability . A constant sticking probability is used for the walls of the structures.
The results for cylindrical holes with different aspect ratios ( to ; cf. Figure 7.5a), sticking probabilities ( to ), and geometries (cf. Figure 7.5b-7.5e) are compared with the reference results obtained using ViennaTS , which uses a three-dimensional top-down Monte Carlo approach for the particle transport.
The results show good agreement (cf. Figure 7.6) aside from the deviation at the wall/bottom interface, caused by the discretization which is used in the reference simulation. Figure D.1 in Appendix D plots the flux distributions for a hole and a trench of aspect ratio 25 along the wall and at the bottom for sticking probabilities and : All results are in agreement with the result obtained by the reference simulation.
The neutral particle flux at the bottom of a structure is an important parameter during an IECE process, as it determines the etch rate of the process . Figure 7.7 shows results obtained with the presented approach for the flux at the bottom center of a hole and a trench structure. The sticking probability of the bottom is set to (cf. Figure 7.7a, 7.7b) and (cf. Figure 7.7c, 7.7d). The total flux (solid lines) and the flux originating from re-emission (indirect flux, dashed lines) is plotted for aspect ratios between and , and different sticking probabilities . The results reveal the effect of a high sticking probability at the bottom for high aspect ratio structures: The bottom adsorbs more particles, which leads to a higher contribution of the direct flux. For instance, for a hole with aspect ratio and , the ratio indirect/total flux is 0.33 and 0.8 for and , respectively.
A computationally inexpensive radiosity-based approximation of the local neutral flux for three-dimensional plasma etching simulations of high aspect ratio holes and trenches was presented. All relevant view factors for holes are computed by establishing an inexpensive general formulation for the view factor between coaxial cone-like segments.
It can be used as a drop-in replacement for the neutral flux computation during three-dimensional IECE simulations of high aspect ratio structures offering an underlying symmetry, as shown here for holes and trenches, to significantly reduce simulation times in practical simulation cases — or as a stand-alone tool which provides fast results for exploratory investigations.
Comparing the results for various convex configurations using a rigorous three-dimensional Monte Carlo ray tracing simulation shows good agreement.