SHE on Unstructured Grids

Up to now, the SHE equations have been derived and investigated on the continuous level. Generic discretization schemes were considered in the previous chapter and a system matrix compression scheme was developed. It is thus time to consider the discretization of the equations, which is the focus of this chapter. To make the chapter as self-contained as possible, the necessary prerequisites about the box integration scheme and the generation of suitable unstructured grids are presented first.

The box integration scheme is a widely used discretization scheme, particularly for semiconductor device simulation and for fluid dynamics. Outside the field of microelectronics it is typically called finite volume method [4, 64]. The main advantage of the box integration scheme is its local conservation property, thus it is particularly suited for the discretization of conservation equations such as the drift-diffusion equations. The box integration can be derived in different ways. In this section a rather direct approach is taken, while a mathematically cleaner approach can be found in [4], where connections with the finite element method are more pronounced.

In the following, the box integration scheme is explained by example of the discretization of the Poisson equation on the domain . Consider

Δψ | = f in Ξ ⊂ ℝ^{n} , | ||

ψ | = 0 on ∂Ξ , |

∫
_{Bi}ΔψdV | = ∫
_{Bi}fdV on Ξ , ∫
_{∂Bi∩∂Ξ}ψdA = 0 . |

∫
_{∂Bi}∇ψ ⋅ndA | = ∫
_{Bi}fdV on Ξ , ∫
_{Bi∩∂Ξ}ψdA = 0 , |

Let the connecting line of the box centers and of the boxes and be perpendicular to the box boundary, cf. Fig. 5.1. A first-order approximation of the normal derivative of the potential is a differential quotient along the connecting line of length . Thus, one obtains the discrete local conservation equations

To summarize, the main ingredients of a box integration scheme are as follows:

- Integrate the equation to be discretized over each box of the tessellation of the domain .
- Use Gauss’ Theorem to recast the volume integral as an integral over the box boundary.
- Discretize the flux through the box interfaces.
- Compute the interface integrals over the discretized fluxes either directly, or via a suitable numerical integration scheme.
- Solve the resulting system of linear equations.

Two families of tessellations of the domain are frequently used with a box integration scheme. The first family is referred to as vertex-centered, where the unknowns and the boxes are associated with the vertices of another mesh. The second family is based on a cell-centered approach, where the unknowns are directly associated with the cells of the mesh (e.g. triangles, tetrahedra, hexahedra), which also constitute the boxes . Other formulations such as cell-vertex methods exist [4], but are not further addressed in the following. The respective preference of a certain family is often based on the availability of good flux approximation schemes for the problem at hand. Due to the wide-spread use of vertex-centered box integration schemes in the field of semiconductor device simulation, the focus in this section is on this first family. In particular, the box integration scheme for the SHE equations presented in the next section is based on a vertex-centered construction.

While a cell-centered box integration scheme can be carried out on virtually any mesh, the vertex-centered method is a-priori based on a different mesh and the boxes in need to be constructed first. The following two constructions are common [4]:

- When employing Voronoi diagrams (aka. circumcenter method) [71], the boxes are
convex sets associated with a given set of disjoint points . Each
box consists of points closer to than to any other point of ,
i.e.
The advantage of Voronoi diagrams is that the interfaces between boxes are perpendicular to the connections between the respective vertices, thus allowing for the simple approximation of the normal flux. However, the generation of underlying Delaunay triangulations results in additional effort needed for mesh generation on the one hand and mesh refinement on the other hand [94, 113]. In addition, the computation of Voronoi diagrams becomes increasingly challenging in higher spatial dimensions.

- For the barycenter method [4] the barycenters of the triangles are connected with the midpoint of the edges to obtain the dual box tessellation (cf. Fig. 5.2(b)). An advantage is that the method can be, unlike others, algorithmically generalized to higher dimensions with reasonable effort and without further requirements on the mesh. However, in comparison to the circumcenter method the number of facets of each box is larger, and the boxes are not necessarily convex.

In order to employ a box integration scheme, there is usually no need to set up the boxes explicitly. As motivated in Sec. 5.1, it is usually sufficient to have the box volumes and the interface areas available. For the discretization of an additional advective term, the box volume fraction associated with the interface is also required, cf. Fig. 5.3. Since the computation of these values for triangular and tetrahedral meshes is only poorly documented in the relevant literature, the respective algorithms are detailed in the following.

Since Voronoi quantities are required based on connections between and , it is sufficient to provide an algorithm for the computation of the respective quantities for each edge . The Voronoi information for the full domain is then obtained by an iteration over all edges of the domain.

Algorithm 3 (Computation of Voronoi quantities for a triangular Delaunay mesh).
Requirements: Triangular Delaunay mesh with all circumcenters in the interior of the mesh

Input: edge

- If there are two adjacent cells, compute the two circumcenters and .
- If there is one adjacent cell, compute the circumcenter and set to the midpoint of the edge.
- Compute the interface area .
- Compute the edge length .
- Compute the box volume fraction .

Return .

The box volumes for the box associated with vertex is obtained by a summation over all box fractions, i.e. . The three-dimensional equivalent is slightly more involved:

Algorithm 4 (Computation of Voronoi quantities for a tetrahedral Delaunay mesh).
Requirements: Triangular Delaunay mesh with all circumcenters in the interior of the mesh

Input: edge

- Compute the edge length .
- Set .
- Set .
- For each adjacent cell , compute the circumcenter .
- Set , where is the number of adjacent cells.
- For each facet adjacent to edge , do
- If there are two adjacent cells, let and denote the two circumcenters.
- If there is one adjacent cell, let the circumcenter of the cell be given by and set to the circumcenter of the facet.
- Compute the area of the triangle .
- Update the interface area .
- Update the box volume fraction .

Return .

Note that the point is used for the computation of the polygon defined by the circumcenters. Again, the box volumes are obtained by a summation over all related to a vertex .

It has to be noted that the Delaunay property does not ensure that the circumcenters of all cells are inside the mesh, i.e. located inside or at the boundary of any cell of the mesh. Consequently, cells with regular shapes are required near the mesh boundary in order to fulfill the requirements of the two algorithms above, cf. Fig. 5.4. If this criterion is not strictly fulfilled, the system matrix will be disturbed by, say, a matrix . Similar to the analysis of round-off errors, the difference of the solution of the perturbed system with respect to the true solution is bounded by

The generation of Delaunay triangulations of good quality for arbitrary domains is a challenging task already in two dimensions [94], and particularly in three dimensions [17]. The meshes used throughout this thesis are generated by Netgen [91], which is capable of generating two- and three-dimensional Delaunay triangulations. A further discussion of the peculiarities of Delaunay mesh generation and refinement is, however, beyond the scope of this thesis.

The SHE method requires an energy coordinate in addition to the spatial coordinate. In principle, one may generate a mesh directly for the -space, but this is impractical for two reasons. First, the author is not aware of the existence any four-dimensional mesh generators capable of handling complex domains. Second, the essence of the -transform in Sec. 2.3 is the possibility to align the grid with the trajectory of electrons in free flight given by planes of constant total energy . Consequently, the embedding of a mesh from the spatial domain to the -space by a tensorial prolongation is of advantage. Given discrete total energies , every vertex in the -dimensional mesh is first embedded in the -dimensional space at location . The full -dimensional mesh of prismatic cells is then obtained by repeatedly shifting the three-dimensional mesh along the energy axis to obtain the points , , , . The procedure is illustrated in Fig. 5.5 for the case of a spatially two-dimensional mesh. Clearly, the energy spacing does not need to be equidistant. Moreover, the computation of Voronoi quantities for the mesh in -space is obtained from Voronoi information in -space by a multiplication of the energy spacing.

There is no need for setting up the mesh in -space explicitly, because the additional coordinate can also be handled implicitly. Since the coupling of different energy levels is local in space, it is more memory efficient to stick with the spatial mesh and to store the total energy values in an array with, say, elements. The respective array index can then be used to identify e.g. the vertex , even though only the vertex is explicitly stored. Since this approach leads to better compatibility with other macroscopic transport models, which do not require an additional energy coordinate, this implicitly tensorial representation is used in ViennaSHE. The price to pay is an additional effort required for for a visualization of the computed distribution function in one and two spatial dimensions, because the mesh in -space needs to be made available then.

With the discussion of the box integration scheme in Sec. 5.1 and the computation of Voronoi information from Sec. 5.2, the discretization of the SHE equations (2.34) and (2.35) can now be tackled. For the sake of clarity, the SHE method for spherical energy bands in steady-state is considered. The general case is obtained without additional difficulties.

Due to the use of MEDS for stabilization, cf. Sec. 2.3, two different sets of equations (2.34) and (2.35) need to be considered. The even-order equations are discretized directly on the boxes of the underlying Voronoi tessellation , while the odd-order equations are associated with the dual tessellation centered at the interfaces between the boxes, cf. Fig. 5.6. Note that the set is obtained by taking the union over all adjoint boxes of all boxes . The volume of adjoint the boxes overlapping the neighboring boxes and is given by , cf. Fig. 5.3 and Sec. 5.2. Consequently, the even-order expansion coefficients are associated with the vertices of the mesh, while the odd-order expansion coefficients are associated with the edge midpoints. More precisely, the discretization of the expansion coefficients reads

Following the box integration procedure from Sec. 5.1, integration over energy from to and over a box given on the spatial grid leads to

| (5.6) |

The first term on the left hand side is transformed to a surface integral using Gauss’ Theorem, and then the integrals on the left hand side are decomposed into the individual intersections with the neighboring adjoint boxes :

| (5.7) |

where denotes the outward pointing unit vector with respect to and perpendicular to the common interface of and . Since for even the coupling terms and take according to Thm. 1 nonzero values for odd only, and the scattering operator exclusively couples terms of the same order , the discretization (5.5) can directly be inserted:

| (5.8) |

where , as well as the scattering rates are assumed to be homogeneous over the device, the force is approximated by a constant vector on and where refers to a suitable index shift such that . Typically, discrete energies are chosen as fractions of such that is again a discrete energy. It is noteworthy that the approximation of the force term by a constant vector over is mostly a matter of convenience. One may also use more accurate representations, but the price to pay is increased computational effort for the evaluation of the integral over . A dimensional splitting has been used by Hong et al. [42], which can be seen in this context as considering only the projection for the integration over .

A slight rearrangement for better exposition of the unknowns and yields

| (5.9) |

In the case of spherical energy bands, the integrals on the left hand side can be rewritten using (4.2) and (4.3) as

The discretization of the odd-order projected equations (2.35) is carried out using essentially the same steps as for the even-order equations, but differs in a few technical details. Integration over the energy interval from to and over an adjoint box overlapping the boxes and leads to

| (5.12) |

Assuming to be piecewise constant in each of the adjoint boxes , application of Gauss’ Theorem to the first term, and splitting the second integral into the two overlaps with and leads to

| (5.13) |

For odd , the terms and are nonzero for even only. Inserting the expansion (5.4) on the left hand side and (5.5) on the right hand side, one arrives at

| (5.14) |

where the scattering rates and the density of states are assumed to be independent of the spatial variable. Since the current density expansion terms are also assumed to be constant within the box , an application of Gauss’ Theorem shows

| (5.15) |

which allows to replace the integration over the boundary of by two integrations over the interface of the boxes and . This leads with constant force approximations over the box to

| (5.16) |

The integrals over the coupling coefficients can again be further simplified as in (5.10) and (5.11).

A comparison of the number of vertices required for a MOSFET and for a FinFET using structured and unstructured grids is given in the following. A comparison for a one-dimensional -diode is omitted since there the structured and unstructured meshes coincide.

The schematic MOSFET layout shown in Fig. 5.7 consists of nodes. A much more
aggressive coarsening could have been applied to the left and the right of the device, as well as deep
inside the bulk of the device (bottom). Nevertheless, the structured grid shown at the right, which
has the same grid spacing as the triangular mesh in the channel region, consists of nodes,
which consequently leads to about percent more unknowns. Note that the high resolution
inside the channel induces an increased solution deep in the substrate, because so-called hanging
nodes^{1}
are prohibited for the box integration scheme. Therefore, a lower resolution, which would be
typically sufficient deep in the substrate, cannot be obtained there. For demonstration purposes,
only a coarse mesh has been chosen at the contacts. A finer grid at the contacts would result
again in additional grid nodes in the bulk for the structured grid.

For a fully three-dimensional layout such as that of a FinFET, the difference in the number of grid points between structured and unstructured grids becomes much larger. Since the current flow is predominantly near the surface of the fin, a coarser mesh can be chosen in the center of the fin when using unstructured grids. At the transition from the source to the channel and particularly from the channel to the drain, a fine resolution is necessary in order to account for the high electric fields and the heated carriers in the latter case. The sample meshes of a FinFET depicted in Fig. 5.8 show that with only nodes a fine mesh can be obtained in the channel and towards the drain region, while a structured grid with comparable resolution in the channel leads to nodes, hence the difference is a factor of six. This difference mostly stems from the spurious high resolution of the structured grid in the source and drain regions.