7 Approximation of Flux in High Aspect Ratio Structures

High aspect ratio structures are essential for the fabrication of various semiconductor devices, where the aspect ratio (AR) of the structure is defined as \( \text {depth}/\text {diameter} \) in case of cylinders and as \( \text {depth}/\text {width} \) in case of trenches. One particular example is negative-AND (NAND) flash cell fabrication [64], 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 [75] 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 [76] 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 [4]. 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[87]. Finally, the capabilities of the presented approach are demonstrated.

7.1 One-Dimensional Radiosity-Based
Particle Transport

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.

(a) Vertical domain

(b) Kinked domain

(c) Surface model

Figure 7.1: Cross sections of simulation domains with vertical walls (a) and with a kink at one half of the depth (b). \( s_s \), \( s_w \), and \( s_b \) designate the sticking probabilities for the source, the wall, and bottom region, respectively. (c) illustrates the surface model showing the relation between the received flux \( R \), the adsorbed flux \( A \), and the re-emitted flux \( RE \); source areas emit a flux \( E \) independent of the received flux \( R \).

The surface adsorption is modeled using a locally constant sticking probability \( s \). The received flux \( R \) is split according to \( s \) into an adsorbed flux \( A \) and a re-emitted flux \( RE \) as depicted in Figure 7.1c. Source areas additionally emit a flux \( E \) independent of \( R \).

The discrete form of the radiosity equation (2.15) is used. For a surface element \( i \) the received flux \( R_i \) is

(7.1) \begin{equation} R_i = \sum _j \left ( E_j F_{j i }\right ) + \sum _j \left (\left (1-s_j\right ) R_j F_{ji }\right ), \label {eq:radiosity2} \end{equation}

where \( E_j \) is the self-emitted energy, \( s_j \) is the sticking probability, and \( F_{ji} \) is the view factor (proportion of the radiated energy, which leaves element \( j \) and is received by element \( i \)).

The linear system of equations is obtained by rewriting (7.1) in matrix notation

(7.2) \{begin}{align}   \vec {R} & = \bm {F}^T\cdot \vec {E} + \diag \left (1-\vec {s}\right )\bm {F}^T\cdot \vec {R} \ , \label {eq:receivedmatrix} \{end}{align}

and transformation into the standard form

(7.3) \{begin}{align}   \left (\bm {I}-\diag \left (1-\vec {s}\right )\bm {F}^T\right )\cdot \vec {R} & = \bm {F}^T\cdot \vec {E} \ , \label {eq:receivedmatrix2} \{end}{align}

with the vector of emitted flux \( \vec {E} \), a vector of sticking probabilities \( \vec {s} \), and a matrix of view factors \( \bm {F} \) (where \( F_{ij} \) corresponds to the view factor \( i\rightarrow j \)).

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 \( A \) is related to \( R \) by the corresponding sticking probability \( s \) of the element

(7.4) \begin{equation} A_i = R_i s_i \ .   \label {eq:adsorbed2} \end{equation}

The relation \( \parallel \vec {A} \parallel - \parallel \vec {E} \parallel =0   \), which holds for closed surfaces, can be used to test the implementation and to define a stopping criterion for the Jacobi iterations.

7.2 View Factors

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 \( a \) and \( b \) 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.

(a) Trench

(b) Domain

(c) Hole

Figure 7.2: Two surface elements, which result when discretizing the domain (b) are displayed: (a) is the side view of two surface elements \( a \) and \( b \), which result from a trench discretization and (c) is the isometric view of two surface elements \( a \) and \( b \), which result from a hole discretization.

To assemble the matrix \( \mathbf {F} \) the view factors between all possible pairs of surface elements are required.

Trench View Factors

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 [77]. 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 [77]

(7.5) \begin{equation} F_{1\rightarrow 2} = \frac {(d_1 + d_2) - (s_1 + s_2) }{2 \cdot a_1}, \label {eq:crossedstring} \end{equation}

where \( d_1 \) and \( d_2 \) denote the lengths of the diagonals, when connecting the cross section of the two strips to form a convex quadrilateral, \( s_1 \) and \( s_2 \) denote the lengths of the sides of that quadrilateral which connects the strips, and \( a_1 \) 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 \( a_r \) towards the other three strips is visualized in Figure 7.3b.

(a) Isometric view


(b) Side view

Figure 7.3: Isometric (a) and side view (b) on the four infinite strips which correspond to the surface elements \( a \) and \( b \) from Figure 7.2a. In (b) the view factors from the top right strip \( a_r \) towards the other three strips are visualized.

The view factor between the two segments \( a \) and \( b \) is

(7.6) \begin{equation} F_{a\rightarrow b} = F_{a_r \rightarrow b_r} + F_{a_r \rightarrow b_l}, \label {eq:strip2strip} \end{equation}

where the subscripts denote the side of the strip according to Figure 7.3b; \( a_l \) can be neglected, as the cross section is symmetric. The view factor from an element to itself is

(7.7) \begin{equation} F_{a\rightarrow a} = F_{a_r \rightarrow a_l}, \label {eq:strip2self} \end{equation}

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).

Hole View Factors

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 \( r_1 \) and \( r_2 \) at a distance \( z \) defined by

(7.8) \begin{equation} F_{1\rightarrow 2} = \frac {1}{2}\left (X-\sqrt {X^2-4(R_1/R_2)^2 }\right ), \label {eq:disk2disk} \end{equation}

where \( R_i= r_i/z \) and \( X = 1 + (1 + R_2^2)/R_1^2 \) [78]. Using this relation and the reciprocity theorem of view factors

(7.9) \begin{equation} S_1 \cdot F_{1\rightarrow 2} = S_2 \cdot F_{2\rightarrow 1}, \label {eq:reciprocity} \end{equation}

where \( S \) 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 \( a \) and \( b \) in such a configuration and denotes the four coaxial disks which represent the apertures of the two elements.

(a) Cone/cone


(b) Cone/annulus


(c) Cone/disk

Figure 7.4: Three possible pairs of segments as they result from discretizing the hole. For each pair, the near apertures \( a_n \) and \( b_n \), and the far apertures \( a_f \) and \( b_f \) are denoted: (a) two cone-like segments, (b) cone and annulus, and (c) cone and disk. The far aperture is treated as an infinitely small element.

The final goal to compute the view factor between two elements \( a \) and \( b \) is divided into multiple inexpensive view factor computations between coaxial disks. First, the difference of the view factors from \( b_f \) towards the two disks of \( a \) is computed, and the reciprocity theorem (7.9) is applied to obtain \( F_{ab_f} \) (red indicates sending and blue receiving areas).

(7.10) \{begin}{align}   F_{b_{f} a} =F_{b_{f} a_{n}}-F_{b_{f} a_{f}}\label {eq:c2c:a} \Rightarrow \frac {S_{b_{f}}}{S_a} \cdot F_{b_{f} a} = F_{a b_{f}} \{end}{align}

The same is done for \( b_n \) to obtain \( F_{ab_n} \).

(7.11) \{begin}{align}   F_{b_{n} a} = F_{b_{n} a_{n}} -F_{b_{n} a_{f}}\label {eq:c2c:b} \Rightarrow \frac {S_{b_{n}}}{S_a} \cdot F_{b_{n} a} = F_{a b_{n}} \{end}{align}

Finally \( F_{ab} \) is obtained by subtracting \( F_{ab_f} \) from \( F_{ab_n} \).

(7.12) \{begin}{align}    F_{a b_n} - F_{a b_{f}} =F_{a b} \label {eq:c2c:c} \{end}{align}

The view factor of an element to itself \( F_{aa} \) is computed by subtracting the flux leaving through the two apertures from unity.

(7.13) \begin{equation} F_{a a} = 1 - F_{a a_{n}} - F_{a a_{f}} \label {eq:vfself} \end{equation}

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.

7.3 Validation and Results

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 [79] (IECE) environment. A sticking probability \( s_s=1 \) 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 \( s_b=1 \). A constant sticking probability \( s_w \) is used for the walls of the structures.

The results for cylindrical holes with different aspect ratios (\( 5 \) to \( 45 \); cf. Figure 7.5a), sticking probabilities \( s_w \) (\( 0.02 \) to \( 0.2 \)), and geometries (cf. Figure 7.5b-7.5e) are compared with the reference results obtained using ViennaTS [87], which uses a three-dimensional top-down Monte Carlo approach for the particle transport.

(a) Aspect ratios

(b) Vertical

(c) Extended

(d) Tapered

(e) Kinked

Figure 7.5: (a) True to scale aspect ratios from 5 to 45. (b)- (e) Cross sections of the geometric variations of the wall for holes and trenches (shown for AR=3); the resulting angle \( \alpha   \), which is identical for all three variations, is depicted.

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 \( s_w=0.2 \) and \( s_w=0.01 \): All results are in agreement with the result obtained by the reference simulation.

(a) Cylinder wall, AR=5

(b) Cylinder bottom, AR=5

(c) Cylinder wall, AR=45

(d) Cylinder bottom, AR=45

Figure 7.6: Normalized flux distributions along the wall and at the bottom for cylinders of aspect ratio 5 (a)(b) and aspect ratio 45 (c)(d). The one-dimensional radios- ity approach (circles) is compared to the results of a three-dimensional ray tracing simulator (lines). The sticking probability of the wall \( s_w \) is varied between 0.02 and 0.2. The deviations between ray tracing and radiosity towards the wall-bottom interface are due to the resolution of the ray tracing simulator. In (c) the ray tracing results are plotted using the minimum and maximum along the cylinder radius, particularly visible for \( s_w=0.2 \).

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 [75]. 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 \( s_b=1 \) (cf. Figure 7.7a7.7b) and \( s_b = s_w \) (cf. Figure 7.7c7.7d). The total flux (solid lines) and the flux originating from re-emission (indirect flux, dashed lines) is plotted for aspect ratios between \( 0.1 \) and \( 50 \), and different sticking probabilities \( s_w \). 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 \( 50 \) and \( s_w=0.02 \), the ratio indirect/total flux is 0.33 and 0.8 for \( s_b = 1 \) and \( s_b = 0.02 \), respectively.

(a) Hole, \( s_b=1 \)

(b) Trench, \( s_b=1 \)

(c) Hole, \( s_b=s_w \)

(d) Trench, \( s_b=s_w \)

Figure 7.7: Total flux (solid) and indirect flux (dashed) at the bottom center of a vertical hole and trench structure for various aspect ratios (\( 0.1 \) to \( 50 \)) and different sticking probabilities (\( s_w \)) of the wall (\( 0.02 \) to \( 1 \)). The ratio between indirect and total flux (dotted) is plotted additionally using the right y-axis. (a)- (b) Result for a sticking probability \( s_b=1 \) at the bottom. (c)- (d) Result for a sticking probability \( s_b=s_w \) at the bottom.

7.4 Summary

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.