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

The diffusion equation for each impurity species is obtained by extending Eq. (2.14) to two dimensions as

where the basic models on the diffusivity D and the total/active concentrations are the same as SUPREM II discussed in section 2.2. The potential tP can be eliminated by setting the net carrier concentration (n - p) equal to the net active impurity concentration (U) and then using the relationship np = nf.

Assuming that there is a differentiable relation between the electrically active concentration N and the total chemical concentration C, the expression for flux can be written as

(2.37a)

(2.37b)

Deff is often referred to as the effective diffusivity and will in general be less than the true diffusivity D since only a portion of the total impurity concentration is mobile at high concentration due to clustering. The numerical solution of the diffusion equation is formulated by approximating the continuous concentration profile with its values on a network of nodes within the device boundaries. The grid structure chosen for this numerical simulation is rectangular with nonuniform spacing in both spatial directions. The finite-difference approximation of Eqs. (2.35) at a node surrounded by its four nearest neighbors [2.23) is

(2.38)

where Bm and Em are coefficients related to the grid spaces and the electron concentration [2.24].

For approximating the time derivative 8Co /at, it has been found that a three-level time discretization scheme is advantageous for treating nonlinear parabolic equations. As an approximation,

where k is the value of the time step and the superscripts n +1 and n -1 denote values at the future and previous time levels, respectively.

NMOS Transistor Simulation

This section presents an example of a complete NMOS transistor fabrication process. The field region oxide growth and boron redistribution are simulated analytically, making the assumption that the boron concentration in the field region is low enough that its diffusivity will be constant. The program is then switched to numerical mode where the arsenic source/drain regions are implanted and driven in. In numerical mode the arsenic clustering and impurity interaction are taken into account. The structure file is saved and is used both for two-dimensional plots and to create the complete transistor structure.

a) Analytic Simulation ofa Fully-recessed Oxide Isolation Region

The input command file shown in Fig. 2.10 simulates the initial portion of an NMOS process through the definition of the polysilicon gate. The substrate is p-type with <100> orientation and 20 O-cm resistivity. The structure is 5 I'm deep and 3 I'm wide. The height is specified to be 1 I'm which allocates space for the thermal oxide that will be grown. The horizontal grid has 0.1 I'm spaces at the device edges, decreasing toward the center of the device. Nodes inside the region are automatically generated with a nonuniform but smooth

54 Computer-Aided Design

(actually quadratic) distribution.

The program is started out in analytic mode, since only in analytic mode can the silicon substrate be etched or thermal oxide be grown. After the silicon substrate is etched, an 80 nm pad oxide, a 70 nm nitride layer, and a 2 J.'m photoresist layer are deposited and selectively etched. The field region is implanted with boron and subsequently oxidized for 3 hours at 1000 °C in wet oxygen. The nitride is then stripped and an unmasked enhancement implant is performed. The polysilicon gate is deposited and etched with sloping sides. After switching to the numerical mode, the current result is saved into a file by SAVE card for further simulation. The current result is plotted twodimensionally by using a PLOT.2D card. Equi-concentration lines are plotted within the range specified by FIRST and lAST. The next SUPRA input example file reads the saved result and plots the structure (Fig. 2.11).

b) Numerical Source/Drain Simulation

The input command file shown in Fig. 2.12 performs the numerical source/drain simulation. The initial structure is defined by loading the structure file EX2AST. This will be a high concentration source/drain simulation, so the program is placed in numerical mode. Arsenic is implanted, using the field oxide and polysilicon gate to define where the arsenic enters the silicon. The drive-in is then performed for 30 minutes at 1000 °C in an inert ambient and the resulting structure saved in the structure file EX2NST. The discretized boundaries of the thermal oxide and silicon substrate used in the numerical solution are plotted in Fig. 2.12. The contours of constant arsenic concentration are plotted with a dashed line type to distinguish them from the boron contour. Note that the electric field generated by the arsenic tends to pull the boron into the source/drain region where it segregates into the thermal oxide.

The whole transistor structure can be generated when the output file EX2NST is read twice. The second LOAD card, however, needs a parameter REFLECT to convert the structure of Fig. 2.12 to a mirror image. SUPRA is also capable of extending or shrinking the right or left side of the original structure during loading.

Two-Dimensional Oxidation Model

The two-dimensional oxidation model introduced in 1982 [2.27] is based on the physical behavior that the oxide can flow at high temperature. The viscous flow phenomenon was first observed in an experiment that a silicon wafer with thermal oxide on one side is heated in a nonoxidizing condition [2.28,29). Since the thermal expansion of oxide is larger than silicon, the wafer bends toward the silicon substrate when the temperature is lower than 960 0c. However, the wafer remains flat above this critical temperature, which suggests that stress due to the thermal mismatch is relaxed through viscous flow of oxide. The viscosity of oxide has been measured from the curvature of wafers as

where J.l.o = 1.586ElO poises, and E", = 5.761 eV. This viscous flow is a dominant mechanism for the relaxation of stress induced by volume expansion during thermal oxidation. It should be noted that the viscosity increases tremendously as temperature goes down. Stress, therefore, may be insufficiently relaxed at temperatures below the glass-transition temperature (960 °C) causing defect generation. The two-dimensional oxidation model in SOAP is based on viscous flow [2.30).

As discussed in section 2.2, the oxidant diffuses in steady state. The flux conservation of Eq. (2.4) and Eq. (2.5) is replaced by a generalized diffusion equation as

(2.41)

where the effective diffusivity Deff is assumed to be independent of oxidant concentration C or stress as in the linear-parabolic model. Boundary conditions for Eq. (2.41) are also obtained from the flux conditions as

OXIDE

55I

SILICON

Fig. 2.14. Boundary conditions of the 2-D oxidation.

where F is the oxidant flux and n is a unit vector normal to the oxide surface (positive in the outward direction from the oxide bulk). The boundaries S1, S2 and S3 are the oxide/silicon interface, free-oxide surface, and nitride boundary, respectively, as defined in Fig. 2.14. ex is the oxygen concentration in the gaseous ambient. These non-homogeneous conditions cause the oxidant distribution at the silicon/oxide interface to be non-uniform when a nitride layer is involved as in Eqs. (2.42) or the normal vector n changes direction. The oxidation rate, as a consequence, is different from place to place at the interface. The volume expansion rate, equivalent to the velocity of an oxide element close to the interface, can be given as

where the minus sign indicates that the velocity is toward the oxide bulk and a is the ratio of the consumed silicon volume to the grown oxide volume. N 1 is the number of oxidant molecules per unit volume of the oxide. Every element in the already existing oxide bulk moves because of the newly grown oxide layer. The velocity, however, is very slow because the oxidation rate is at most a few angstrom per second even in a steam ambient.

Due to the unique slow-viscous flow behavior of oxidation, the motion of oxide elements can be described by a simplified Navier-Stoke's equation as

(2.44)

where p. is the oxide viscosity, V is the velocity, and P is the pressure. This first equation indicates that the viscous damping force is balanced by the pressure gradient. Since the compressibility of oxide is very small (2.7 X 10 -u cm2 jdyne), the oxide is assumed to be incompressible. This incompressibility characteristics requires that the density remain constant with respect to time. The continuity equation, consequently, is approximated below as

where p is the oxide density. The velocity obtained from the O2 concentration in Eq. (2.42) serves as a boundary condition on Sl. On the oxide surfaces, boundary conditions are derived from an assumption that the internal pressure is balanced with external stress. Namely P becomes the ambient pressure on S2 and the nitride stress on S3, added by surface tension.

In the two-dimensional oxidation system, the ultimate solution we are looking for is the oxide boundary position that moves continuously. This moving-boundary problem is extremely difficult to solve with a conventional numerical method based on finite-difference or finite-element. The SOAP program adopted a different approach known as a boundary-value technique in which grid points are located on the boundaries and an integral form of Eqs. (2.41) and Eqs. (2.44) is solved by using Green's function. Initially SOAP takes an initial guess that the pressure distribution is constant and thus VP = o. The velocity field calculated from this guessed pressure distribution would not satisfy the zero divergence of Eq. (2.45) everywhere. The resulting V·V is fed back to obtain a new pressure distribution and then the velocity field is again calculated from the new pressure. This velocityjpressure iteration scheme continues until the incompressibility condition is satisfied within an allowable range. The flow chart of this algorithm is given in Fig. 2.15. Details of the numerical method as well as the physical model are discussed elsewhere [2.30].