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In this paper a methodology for the three dimensional (3D) modeling and simulation of the profile evolution during anisotropic wet etching of silicon based on the level set method is presented. Etching rate anisotropy in silicon is modeled taking into account full silicon symmetry properties, by means of the interpolation technique using experimentally obtained values for the etching rates along thirteen principal and high index directions in KOH solutions. The resulting level set equations are solved using an open source implementation of the sparse field method (ITK library, developed in medical image processing community), extended for the case of non-convex Hamiltonians. Simulation results for some interesting initial 3D shapes, as well as some more practical examples illustrating anisotropic etching simulation in the presence of masks (simple square aperture mask, convex corner undercutting and convex corner compensation, formation of suspended structures) are shown also. The obtained results show that level set method can be used as an effective tool for wet etching process modeling, and that is a viable alternative to the Cellular Automata method which now prevails in the simulations of the wet etching process.

Micro- and Nano Electro Mechanical Systems (MEMS and NEMS) represent a rapidly expanding field of semiconductor fabrication technologies for producing micro and nano scale mechanical, electric, optical, fluidic, and other devices [

Actually, two types of simulations exist [

The most common type of the engineering simulators are so called geometrical simulators [

The level set method for evolving interfaces [

The paper is organized as follows: in Section 2 some aspect of the silicon wet etching process are discussed. After that, the relations describing the angular dependence of the etching rates, based on an interpolation procedure and silicon crystal symmetry properties, are derived. In Section 3 the necessary details for the implementation of the sparse field method for solving the level set equations in the case of etching rates defined in Section 2, are described. Section 4 contains simulation results for some interesting initial 3D shapes (cube and sphere), as well as some more practical examples illustrating anisotropic etching simulation in the presence of masks (simple square aperture mask, convex corner undercutting and convex corner compensation, formation of suspended structures).

Although silicon etching techniques are currently undergoing a revolution driven by the incorporation of plasma etching process, anisotropic wet chemical etching is still the most widely used processing technique in silicon technology [

The anisotropy of the etching process is actually the orientation dependence of the etch rate. Regardless of the great amount of work done in this field, there is no generally accepted single theory for a mechanism that explains the great anisotropy in silicon wet etching. It is accepted [

As stated earlier, in order to simulate the time evolution of 3D etching profiles it is essential that exact etch rates in all directions are known. In this paper we shall use the experimental values of etching rate for silicon in KOH solutions for three different etchant concentrations (30%, 40% and 50%) for etching temperature of 70 °C, and they are listed in

The etching rates for only limited number of directions are known, but they can be used to determine rate value in an arbitrary direction by an interpolation procedure. The problem of etching rate interpolation is equivalent to function interpolation over a sphere in 3D. For accuracy, the etching rate model must interpolate through the given etching rates and directions while maintaining its continuity, since possible requirement that the first derivative must be continuous also, is too high, as empirical studies have shown cusps in etching rate diagrams. Here we shall use etching rate model developed by Hubbard [

It is supposed that _{x}_{y}_{x}_{z}

The second step is to divide this angular section into three equivalent (curved) triangles as it is shown in

The simplest method is to use only the experimental rate values for the principal directions [100], [110] and [111], since in three dimension three independent vectors are needed to define a basis. In that case, the interpolation region is shown in _{x}_{y}_{z}_{hkl} is etching rate in [hkl] direction. Details of this interpolation procedure can be found in [

In _{x}_{y}_{z}

In

It is important to remember that all physical aspects of the etching process are contained in these angular dependences, and that they determine time evolution of the feature profile completely, appearance and disapearrance of particular planes and the final profile. For different values of the parameters these shapes look different. Inclusion of additional planes will also change the shape of angular dependences.

Level set method, introduced by Osher and Sethian [^{n} |

The ^{n}_{xi}_{i}

Several approaches for solving level set equations exist which increase accuracy while decreasing computational effort. They are all based on using some sort of adaptive schemes. The most important are narrow band level set method, widely used in etching process modeling tools, and recently developed sparse-field method [

The non-convex Hamiltonians are characteristic for anisotropic etching and deposition simulations [_{x}_{y}_{z}

The terms on the second row of the above equation are the smoothing terms. They are similar to the second derivatives in each variable. In general, these terms need not be calculated exactly. Overestimated values will produce non-realistic smoothing of the sharp corners in the implicit surfaces. Too little smoothing usually leads to numerical instabilities in calculations. In Reference [

It is essential to express the etching rates in terms of the level set function itself in order to obtain level set equation in Hamilton-Jacobi form. To accomplish this goal, we start from the facts that the unit vector normal to the zero level set is given by _{y}

If the high index planes {311} are included, the expressions become more complicated because the interpolation region is then divided in two subregions. Since one of the goals of this study is investigate the influence of these planes on the final outcomes, we shall write them explicitly. The Hamiltonian corresponding to the relation (3) has the form:

The same procedure can be used to derive relations when thirteen experimental values of the etching rates for interpolation are used (eleven subregions,

The necessary first derivatives can be derived directly from (18), but the corresponding expressions are too cumbersome to be stated here.

The second derivatives of the Hamiltonian appearing in (11) and (15) are also needed for checking their convexity condition (7). Actually, it is not necessary as the

Potassium hydroxide (KOH) is the most common and the most important chemical etchant, because of its excellent repeatability and uniformity in fabrication, and its low production cost. In actual calculations we made use of etching rates listed in

Here we shall present some results obtained using previously described methodology. In view of the fact that we were not able neither to compare our results with cellular automata simulations nor to perform any experimental work, we have carefully chosen set of examples for which the outcomes (final profiles) are known, either from simple theoretical considerations or from published experimental results. For example, it is expected that the final shape after the etching of any 3D object must be composed of the fastest etching planes. Similarly, etching through the aperture of any form ends with a cavity bounded by the slowest {111} planes [

Since the cube is the simplest isometric crystal form [

It is obvious that the initial cube shape gradually transforms to the final tetrahexahedron, consisting of 24 triangles belonging to the {012} family of planes, through the combinations of these shapes. It is expected given that tetrahexahedron is the only isometric form made of {012} principal planes. If the fastest planes are not {012} family, as in the case of 40% etchant concentration, the final profile shape will change accordingly. We shall pay more attention to this in the next, more interesting case.

In order to test the strength of the method we have chosen to simulate etching of the silicon ball in KOH etchant. The initial spherical surface contains all possible velocity directions, so it is expected that the anisotropy of the etching process will produce the most dramatic changes of the initial shape. This shape, or more precisely hemisphere, is used in the experimental setup [

In

The final shapes are the same in both cases; it is octahedron made of the slowest {111} family of planes. It can be obtained [

It can be easily shown that applying transform (20) on etching rates defined by (1), (3) and (4) lead to octahedrons. This example shows that possibilities for using the inverse modeling for design purposes are very limited, since the information about the initial shape is lost quickly. Actually, it is connected with the detailed knowledge of the etching rate anisotropy, as well as with the mask orientation, and probably models based only on relatively small number of parameters (like one presented in this paper) are not sufficient for this purpose.

The first example in which the maks are used is etching through square openings in the {100} silicon plane with edges aligned to <100> (

Real applications very often require that mask includes, instead of only concave as in the previous case, many convex corners. In that case typical undercutting faceted shape beneath such a corner appears [

In order to avoid the effects of convex corner undercutting various compensation techniques are widely used. One of the most effective compensation structure is <100> oriented beam [

As it is mentioned in section 2, the anisotropic etching of sacrificial layers is usually used for manufacturing of suspended structures using. In

In this paper we have shown that the profile evolution during anisotropic wet etching of silicon can be described by the non-convex Hamiltonian arising in the Hamilton-Jacobi equation for the level set function. Angular dependence of the etching rate is calculated on the base of full silicon symmetry properties, by means of the interpolation technique using experimentally obtained values of the etching rates for principal and some of high order planes (totally thirteen) in KOH solutions. The resulting level set equations are solved by applying the sparse field method extended for the case of non-convex Hamiltonians. The simulation results showing profile evolution in some interesting 3D cases are presented: cube and sphere for various etchant concentrations. Also, examples are given showing that the method can be used to model etching process in the presence of masks. These include simple square aperture masks differently oriented, convex corner undercutting and convex corner compensation, formation of suspended structures. The results obtained so far show that level set method can be used as an effective tool for wet etching process modeling on the device level, and that it is a viable alternative to the cellular automata method which now prevails in the simulations of the wet etching process both in microscopic (atomistic) and engineering applications. However, much work is still needed to improve it to the level of the current commercial cellular automata simulators (for example, IntelliEtch [^{3} (N-resolution in one spatial direction). We believe that it is also possible to use the level set method to analyze microscopic etching mechanisms on atomistic level, especially in conjunction with kinematic wave theory [

This work was supported by the MNTRS 141025 and MNTRS 141027 projects.

Etching rate interpolation region: (a) The angular section defined by the planes (0 ≤ _{x}_{y}_{x}_{z}

The (a) three- , (b) four- and (c) thirteen-parameters inetrpolation subregions.

The angular dependence of the etching rate calculated using interpolation formulas with three (a), four (b) and thirteen (c) interpolation parameters (for 30% etchant concentration) listed in

The angular dependence of the etching rate calculated using interpolation formulas with thirteen-parameters for three etchant concentrations: (a) 30%, (b) 40% and (c) 50%.

Etching profiles of the silicon cube with initial edge of 30 μm after 0 s, 120 s, 240 s, 360 s and 480 s, obtained using the thirteen-parameters interpolation formula for 30% etchant concentration.

Etching profiles of the sphere with initial radius of 25 μm after 0 s, 100 s, 200 s, 300 s, 400 s, 500 s and 600 s, obtained using thirteen-parameters interpolation formulas for (a) 40% and (b) 50% etchant concentrations.

Artificial growth of the cube (upper row) and the sphere (lower row) with the growth rates given by the thirteen-parameters etching rate interpolation formulas for 30% etchant concentration.

Etching through a square aperture mask in {100} plane aligned to <100> (upper row) and <110> (lower row) directions. Profiles after 100 s, 300 s, 600 s and 900 s, obtained using thirteen-parameter interpolation formulas.

Convex corner undercutting. Initial aperture is aligned to <110> directions. Evolution of the profiles for the square (upper row) and the circular (lower row) masks, with the dotted masks superimposed, are presented.

Convex corner compensation in mesa structure fabrication. Compensation mask is shown in the upper left corner. Initial aperture is aligned to <110> directions. Profiles at six equidistant time moments, with the dotted mask superimposed, are presented.

Formation of a system of suspended cantilever beams. Initial aperture is aligned to <110> directions.

Formation of a released suspended plate. Initial aperture is aligned to <110> directions.

Etching rates of silicon at different KOH concentrations at 70 °C [

| |||
---|---|---|---|

(100) | 0.797 | 0.599 | 0.539 |

(110) | 1.455 | 1.294 | 0.870 |

(210) | 1.561 | 1.233 | 0.959 |

(211) | 1.319 | 0.950 | 0.621 |

(221) | 0.714 | 0.544 | 0.322 |

(310) | 1.456 | 1.088 | 0.757 |

(311) | 1.436 | 1.067 | 0.746 |

(320) | 1.543 | 1.287 | 1.013 |

(331) | 1.160 | 0.800 | 0.489 |

(530) | 1.556 | 1.280 | 1.033 |

(540) | 1.512 | 1.287 | 0.914 |

(111) | 0.005 | 0.009 | 0.009 |

(411) | 1.340 | 0.910 | 0.660 |