Книги2 / 1988 Kit Man Cham, Soo-Young Oh, John L. Moll, Keunmyung Lee, Paul Vande Voorde, Daeje Chin (auth.) Computer-Aided Design and VLSI Device Development 1988

Fig. 4.4. Duality between capacitance and resistance structures.

the surface of the inner sides of the beans.

SCAP2 can calculate the resistance of a two-dimensional geometry as well as its capacitance using the duality between them. A geometry of a resistive material whose resistance we want to calculate is shown in Fig. 4.4(a). Let us assume that the geometry has a unit depth into the paper. Intuitively, the field lines in the resistor would be the same as those in the capacitor shown in Fig. 4.4(b) if the ratio of the dielectric constant of the capacitor to the dielectric constant of the ambient were infinite. This is theoretically true because we solve the Laplace equation with the same boundary conditions inside the resistor and the capacitor if the assumption for the dielectrics is true. The boundary conditions for the resistor are

where ~, ej, eo, and n are potential, dielectric constant of the capacitor, dielectric constant of the vacuum, and a unit normal vector toward outside, respectively. The boundary conditions for resistance and capacitance become identical when eo/ej becomes o. To be practical, we set the relative dielectric constant of the capacitor dielectric to be 1 and that of the ambient to be 10-30 or

smaller. With this condition, we can find the value of the capacitor with almost no field line leaking out of the dielectric. Then we can easily convert the capacitance, C, to the resistance, R, of the same shape in Fig. 4.4(a) using

pej

R= - (4.7)

C

where p and ej are resistivity and dielectric constant, respectively. This method to find a resistance value from the capacitance of the same shape extends itself to three-dimensional cases as discussed in the next section.

4.3 FCAP3: Three-Dimensional Poisson Equation Solver

As discussed in section 4.1, there are cases where three-dimensional simulation is necessary to calculate the parasitic components of the structures. FCAP3 (Fast CAPacitance 3-dimensional program) [4.6] is a Poisson equation solver in three-dimensional space which was developed to attack these kinds of problems. FCAP3 is basically an extension of SCAP2 from two-dimensional space to three-dimensional space. However, the extension is not trivial as readers may expect. In this section, we discuss the algorithm, implementation of user-friendliness, and basic applications of FCAP3.

FCAP3 solves the three-dimensional Poisson equation using the finitedifference method and ICCG method like SCAP2. When a problem is given, FCAP3 generates a semi-uniform rectangular grid with 125,000 grid points. The grid lines are initially allocated at the vertices of the geometries and the rest of grid lines are allocated evenly over the simulation region. After geometries are discretized with the grid, a matrix is set up using the sevenpoint finite-difference equation with the reflective (Neumann) boundary condition applied on the simulation boundary planes. Then the potential distribution of the problem is obtained by solving the matrix using the ICCG method. FCAP3 is capable of reallocating its grid based on the initial potential distribution. The new grid discretizes the geometries more appropriately since there are more grid lines where the potential gradient is large. The description of a geometry or a part of a geometry is poor if the potential gradient in the region is small because the region is discretized with a sparse grid. However, this is

quite reasonable since the region does not have large effect on the amount of charge on the conductors which we want to calculate. This automatic regrid capability of FCAP3 makes the finite-difference method quite useful for three-dimensional problems, even though the finite-difference method is generally inferior to the finite-element method in describing complex geometries. Also, it generates a new grid without any user interaction, whereas a finite element grid is very difficult to automatically reallocate or even allocate, especially in three-dimensional space.

User-friendliness is a very important factor in three-dimensional simulation programs since it is difficult to input the geometries of a problem and to interpret the output of the problem. FCAP3 provides three degrees of complexity in input geometries to make input easy. The first level input deals with box shape geometries which are parallel to the axes of the rectangular coordinates. The second level input includes parallelograms which are parallel to the y-axis and have arbitrary polygonal side walls parallel to the xz-plane. The first and second levels are mainly for the typical geometries which occur in integrated circuits. Examples of two-level interconnect lines implemented with the first and the second level geometries are shown in Fig.4.5(a) and (b), respectively. The third level input is open to users. Users can build their own geometry defining routines according to their needs. The following example of capacitance calculation of concentric sphere is built using the third level input scheme.

The calculation of the capacitance of concentric spheres, for which the finite difference method is the least adequate, supports the argument on the merit of the finite-difference method with the automatic regrid and verifies the accuracy of FCAP3. Fig.4.6(a) shows an agreement within 9 % between the theoretical value (solid line) and FCAP3 results (circles). The optimal finitedifference grid generated by the automatic regrid is shown in Fig. 4.6(b). The grid was taken on the plane which goes through the center of the spheres and parallel to the xz-plane. The potential contour plot on the plane in Fig. 4.6(c) shows almost circular rings. This simulation result demonstrates the accuracy and versatility of FCAP3. An accuracy verification against an experiment is given in section Chapter 15.

THEORY FeAP3

o

.lOE+02CAPACITANCE ItR

OUTER SHELL RADIUS ~ 5 micro-meter

o

.1OE+O

o

RADIUS OF INNER SHELL imlcro-meter)

(a)

y

L -_________-1 X

Fig. 4.6. Capacitance calculation of two concentric conductor spheres by FCAP3. (a) Comparison between theory and FCAP3 results. (b) Automatically generated grid based upon the initial calculation with a uniform grid. (c) Equipotential contours on a plane which goes through the center.

FCAP3 can also be used for the resistance calculation of complex threedimensional resistive paths. The same argument in the previous section for the SCAP2 applies when it is extended to three-dimensional space. The same conversion equation (4.7) is valid for three-dimensional cases. An example of

three-dimensional resistance calculation is given in section 15.3.

References

[4.1] Special Issue on Interconnections and Contacts for VLSI, IEEE Trans. Electron Devices, ED-34, no.3, Mar. 1987.

[4.2] K. Lee, Y. Sakai, and A.R. Neureuther, "Topography Dependent Electrical Parameter Simulation for VLSI Design," IEEE Trans. Electron Devices, ED-30, no.11, pp.1469-1474, Nov. 1983.

[4.3] D. S. Kershaw, "The Incomplete Cholesky - Conjugate Gradient Method for the Iterative Solution of Systems of Linear Equations,"

Journal of Computational Physics 26, pp.43-65, 1978.

[4.4] HP Internal Software.

[4.5] W. G. Oldham, A. R. Neureuther, C. Sung, J. L Reynolds, and S. N. Nandgaonkar, "A General Simulator for VLSI Lithography and Etching Processes: Part II - Application to Deposition and Etching," IEEE Trans. Electron Devices, ED-27, no. 8, pp. 1455-1459, Aug. 1980.

[4.6] HP Internal Software.

PARTB

Applications and Case Studies

Chapter 5

Methodology in ComJ!uter-Aided Design

for Process and Device Development

The previous chapters have presented an overview of computer-aided design (CAD) in VLSI development, as well as the simulation tools currently used at Hewlett-Packard Laboratories. In this chapter, CAD is discussed from the user point of view. The methodology for using the sim'uation tools in the most effective way is presented. Then case studies will be presented in the following chapters which show in detail how simulation tools are used in device designs.

5.1 Methodologies in Device Simulations

Simulation tools should be used in the most efficient way, such that time and effort in doing the simulations are minimized. This is especially true in process development where time is of major concern. The goal is to provide process parameters in the shortest time. Ideally, one would like to have CAD tools which can produce the desired output in minimal time. In real life, this is usually not possible, mostly because of limitations in software and hardware capabilities, or due to too many users on the system. In the following

paragraphs, we discuss the simulation methodologies which will use the CAD tools most effectively:

1)Before simulations are to be performed, it is always a good idea to look at the problem from the simplest and most intuitive point of view, and to try to get some basic idea of the problem. For example, in the case of counterdoping for the p-channel transistors with n+ polysilicon gate (to be discussed in detail in Chapter 12), the dose can be roughly estimated by simple arguments, and it turns out to be not very different from the results of twodimensional analysis. This simple procedure not only provides a first estimate of the magnitude of the process parameters to be used, but also provides a simple picture of the physics behind the technique. Numerical analysis is sometimes difficult to interpret unless one has a basic idea of the physics involved. Also by gaining some initial knowledge of the problem, the simulation work can be better planned and executed, with a minimum range of parameter values, instead of using trial and error.

2)A simulation should be considered as an experiment itself. This means that a systematic procedure should be used rather than shooting for a particular number. For example, it may be desired to develop a p-channel MOSFET with a particular long channel threshold voltage, say, -0.7 V, and the thres-

hold voltage is dependent on the counter-doping of boron [5.1]. In the simulation work, a factorial experiment [5.2] should be set up with the boron implant dose and energy as parameters. The simulated results are shown in Fig. 5.1. This figure provides an overall picture of the dependence of the threshold voltage on the two parameters. Also it provides an idea of the reasonable range of the parameters that should be used in the fabrication experiment. To be more complete, the n-well impurity surface concentration should be included in the simulation. In this case, the simulations become a 23 factorial experiment. The results, when arranged in a 3- dimensional form as shown in Fig. 5.2, show the dependence of the threshold voltage on the three parameters. The threshold voltage change due to the change in a combination of the parameters can be estimated quickly from the figure. Several observations can be made from this simple example. Fig. 5.3 shows the threshold voltage (VT ) as a function of the counter-