Fundamental physical mechanisms in deposition and etching processes generate both desired and undesired topographic features. In this chapter we provide a basis for understanding and modeling these effects on topography. A common framework for modeling etching and deposition is given, including the terminology used for describing the various physical phenomena and effects.
The starting point for both is the nature of flux which arrives from a source above the wafer. The second commonality is how the self-shadowing of an existing profile affects the visibility of the source and how the re-emission from the profile or radiosity affects other points of the profile. The third common aspect is that mechanical and chemical reactions at the surface determine the local advancement rate and are one of the most challenging aspects of deposition and etching. The final common characteristic of deposition and etching is the time-advance of the surface profile. In this chapter we focus on this final common aspect.
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The local fluxes and etch or deposition reactions provide sufficient information on how the surface is to be deformed for a short time step. The time evolution of the profile requires many time increments which allow the change of the profile shape to influence its future shape through shadowing, etc. As the surface evolves there may be major topological changes. For example, outgrows of the material from sides of a trench might collide and close off a void, or etch fronts from two trenches might move laterally, collide and form a tunnel or an air bridge.
To ease the presentation a simpler two-dimensional case will be used. In two dimensions the profile is sometimes said to be made of line segments and intersections points which may also be called nodes. Figure 3.3 shows the simulation result of an isotropic deposition of a material into a rectangular structure. It can be seen easily that the boundary is expanded radially outward for a time step . The top right convex outward corner point causes a fan (cf. Figure 3.3) of facets to be generated which creates a rounded section in the profile at time . At the concave outward corner the planar fronts from the horizontal and vertical facets intersect in a line which is the location of a new corner point. These collisions where overlapping fronts occur and some information about the initial profile is lost are said to be shocks (cf. Figure 3.3). Note that shocks and fans play a very important role in determining the changes of topography during processing. For isotropic etching the profile moves from the surface into the material as shown in Figure 3.4. As expected the convex outward corner now produces a shock and the concave corner point produces a fan.
To understand faceting of the initial profile the advancement of two neighboring facets as shown in Figure 3.5 is considered. The direction and rate of advance of a vertex where two facets intersect is of key interest. The two facets have downward normals which form angles and with respect to the vertical downward direction. They advance a distance which is the product of the planar etch rate for their orientation and time step as and , respectively. The vertex moves at an angle and a rate and therefore a distance . The angles of the path of this vertex with respect to the normal of the two facets are and which are and , respectively. Projecting the distance of movement of this vertex onto the distances advanced by the facets gives the two following equations:
Taking the ratio between (3.1) and (3.2) and eliminating gives
Once is known the associated rate can be found from (3.1) and (3.2) and is of course higher than and .
(3.3) makes a number of interesting predictions about the physical nature of the motion of the vertex. Assuming that and is not a function of the facet angle , then the right hand side of (3.3) is unity, forcing or which means that the intersection vertex moves along the bisecting angle.
Another important case is that increases with the angle . This makes the right hand side of (3.3) which leads to or . This means that the faster moving facet will encroach into the region of the slower moving facet and likely expands in size.
To further explore how the movement of a facet depends on the shape of the rate function, the direction of travel of a facet at angle with rate function is now derived in a limiting case. Assuming that the angle between a facet pair is small enough, their intersection vertex will propagate in the direction of motion of the facet. This limiting case can be determined from (3.3) by taking the special case of and and letting go to zero. Substituting and in (3.3) and expanding in a Taylor series gives
Letting go to zero in (3.4) gives
(3.5) indicates that a facet at angle moves laterally with a rate component proportional to the slope of the rate function as well as normal to the surface at the usual rate .
Regarding the evolution of facets each facet has a plane associated with it. The plane moves with a given normal speed which may be different for different facets. The boundaries of the facets are determined by the intersection of the planes. The two-dimensional evolution near a corner is shown in Figure 3.6. If we use a level-set method where the normal speed is a simple interpolation between the two normal speeds, then the corners would be rounded, as shown in Figure 3.7 (in particular, an arc of a circle would arise, if the two speeds are the same). In order to keep flat facets and sharp corners, the following precautions must be made. The evolution of a smooth curve only depends on the normal speed, the addition of a tangential component only has the effect of changing the parameterization of the curve. With a velocity that has a proper tangential component to the facet and is directed towards a corner, the facets will evolve maintaining sharp
In order to minimize the corner rounding during the movement of a boundary the approach introduced by Russo and Smereka [71] can be used where the speed of each facet must be specified. Consider an interface consisting of facets, with normals and absolute value of normal speeds for . Let be the normal at a given point of the interface.
In order to compute the proper velocity of the surface at a given point, first the facet is selected which is closest in direction to , i.e., for which is a maximum,
(3.6) |
Next the following velocity is defined
Some simple geometric considerations as shown in Figure 3.8 and the following equations help us to find the tangential speed that must be added to keep a sharp corner.
(3.10) |
(3.11) |
The quantity can then be chosen equal to , where
(3.12) |
For a faceted interface , because and the facet will evolve with the standard normal speed. On the other hand, if we evolve the whole interface with the velocity , then the corners get rounded as shown in Figure 3.7. When the corners become slightly rounded, then becomes directed towards the corners and the surface will move with a velocity that will try to keep the corners sharp.
Choosing a small value for , the vector is basically a unit vector whenever is not aligned with the facets. Neglecting , the normal speed can be calculated from (3.7) with the following equation
(3.14) |
Substituting (3.15) in (3.13) gives the normal speed of our model for the level set function
(3.16) |
In Chapter 7 we will present the application of this method to get the trenches with a minimal corner rounding during the simulation of an etching process.