Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

and different expressions are needed. For example, in impurity diffusion across the Si02 -Si interface there is a segregation process going on and so the expression for impurity flux must take this effect into consideration. Further complicating the situation is the so-called moving boundary problem which occurs in the process simulation during epitaxy and oxidation. We will not discuss this class of problems in detail, but simply point out that the moving-boundary problems can be handled in a fixed-boundary condition with the modified expression for particle flow [A.l].

The Poisson's equation is written as

It should be noted that there are strong couplings between the Poisson's equation and (charged) particle continuity equation(s). The electrostatic potential distribution is affected by the distribution of mobile carriers, nand p, and in turn the carrier transport is influenced by the electric field, which is determined by the electrostatic potential distribution. Also, particle continuity equations for different species of particles are coupled through the recombination process. In general, in order to obtain solutions for electrostatic potential and particle densities, we need to solve the Poisson's equation and the continuity equations as a complete system of equations.

Because all these equations are partial differential equations, in order to solve them using digital computers, we need to first convert these differential equations to the difference equations. This conversion is completed by a process called discretization, which is the topic of the next section.

A.2 Discretization

To solve a system of nonlinear differential equations in a digital computer, we usually follow one of the two following approaches - either to linearize the system of differential equations first and find a way to solve the resulting system of linear differential equations, or to convert

the system to that of nonlinear difference equations first and then to solve this system of nonlinear, algebraic equations. The latter approach is almost universally used in Ie process/device simulation, and hence we will only discuss this approach. The procedure in this approach is as follows:

•transform the differential equations to difference (i.e. algebraic) equations .

•solve the nonlinear equations by solving a series of linear equations (Newton-Raphson method).

There are two ways of transforming a differential equation into its algebraic counterpart. One is to replace the differential operator with a difference operator. From the physics point of view, this implies that the differential equation must be applicable everywhere. The second way is to integrate the differential equation to obtain an equation with some integral terms. These integrals are then evaluated using various numerical quadrature methods. In either case, the problem domain of interest (spatial and time spaces) must first be discretized, and this process will inevitably incur certain discretization errors. We now use semiconductor equations in one-dimensional space to illustrate the above procedures, and when there are two carriers in place, we use electrons as an example.

The one-dimensional continuity equation for electrons can be written

where the electron current density is expressed using the DD model as

We first consider the use of the difference operator to approximate the differential one. Basically there are three types of difference operators for the approximation of differential operator of order one: forward, backward, and central. They are illustrated below for time t = ti in a

ti+l - ti-l

where ni = n(ti) and etc. The central difference has higher discretization accuracyl than either the forward or backward method when the grid space is uniform, i.e. tiH - ti = ti - ti-l, etc. Mathematically, the forward and backward difference methods essentially use linear segment to represent the functional behavior in that interval, while the central method when used in the uniform grid space can be viewed as the parabolic approximation to the original function in the interval of [ti-I, ti+d with the slope at ti, the midpoint of the interval, equal to that of the linear segment between end points of the interval. However, to initiate a central discretization scheme, one needs to know the values on the grids prior to and after the present one. This is not possible for grids falling on the boundary.

The second approach in transforming a differential equation to an algebraic one is to use the integral form of the original equation and then use numerical quadrature to approximate the integral. We take as an example the electron continuity equation at steady state (ani at = 0

Xi-l/2

The problem then reduces to the following: 1) how to express the values of jn,i-I/2 and jn,i+1/2 in terms of basic variables ('lj;, n, and p) at grids; 2) which integration method to use for the integral. If we choose to use the rectangle integration method [A.2]' we will have

This is similar to using the central difference method in discretizing the differential operator aIax at Xi. Before we further find a discretization form for the current j in the midpoint of the interval, we first consider the discretization scheme used in the time domain.

IThe discretization accuracy of a difference operator is measured by the highest order of derivative in Taylor expansion it retains when the expression for the desired differential operator is obtained. For example, the central difference operator has an accuracy of order two.

A.2.1 Time Integration for Initial Value Problems

An equation which involves the time derivative on one and only one side of the equation is an initial value problem. That is, we start from a known solution (or an initial value) at the starting time instant, and the evolution of the solution (time derivative) can be obtained from its spatial variation. More specifically, we may use the electron continuity equation as a model problem in which

where we did not write down the explicit dependence offunction f on 'Ij;

and p and note that there is no explicit dependence of f on t. Starting from an initial instant, say to, and a known solution no = n(to), there

are two possible ways to proceed as time advances. In either way, the forward difference operator is used to discretize the time derivative. But in writing the right-hand-side term, we have two choices:

(A.13)

or

(A.14)

where L1ti = ti - ti-1. For the first approach, the solution at the next instant, ni, can readily be computed using the previous solution and hence the method is explicit. It is called the forward Euler method. The second approach needs to solve a nonlinear equation because f is generally a nonlinear function with its argument, ni, an unknown. This method is of implicit nature and is called the backward Euler method. Apparently the forward Euler method needs much less computation time to advance the solution but this comes at a great expense in terms of the numerical stability. Namely, the error caused by the discretization of the time derivative can be accumulated in the forward Euler method and is not contained as time develops. So as time goes on, this accumulated error grows steadily and will eventually lead to a meaningless, unphysical solution. On the other hand, the backward Euler method has an implicit constraint to the solution at the next time step: the increment of the solution is limited by the requirement that L1.njL1t must be equal to the slope of the solution curve at the level of the next solution. The solution cannot go wildly since the discretization errors are

partially canceled from time to time, and thus the accumulated error is confined to a certain value.

In the numerical simulation, stability issue always prevails the other issues such as the computation efficiency. So in practice, the backward Euler method is a preferred one to the forward Euler method. In the next section, we will discuss the discretization of the current expression, which has a profound effect in stability of the device simulation.

A.2.2 Space Discretization

As we have learned from the discussion in the previous chapters, in semiconductor devices different quantities may change in a widely different range and manner. For example, electrostatic potential and quasi-Fermi levels change in space nearly linearly, whereas carrier concentrations may change several orders of magnitude in a small distance and in general change exponentially over the space. The different discretization schemes discussed so far essentially determine the functional behavior of variables in a small interval. Taking as an example the current expression, if we apply the central discretization scheme to Eq. (A.6), one would obtain the following equation:

(A.15)

where ~Xi = Xi - Xi-I' This discretization scheme has the accuracy of order one because the approximation of ni-I!2 = (ni + ni-I)/2 is based on the linear interpolation, which is a rather poor assumption considering the exponential change of carrier concentration over the interval. Moreover, early device simulators employing this scheme showed very unstable numerical behavior. Namely, it was observed [A.3] that when the potential difference between two adjacent grids exceeds two times of the thermal voltage, the simulation either gives spatially oscillated solution (wiggles) or the iterations in the Newton-Raphson method do not converge if variable values are forced to be in a physically meaningful range [A.4]. The reason for this instability actually arises from the existence of the drift term in the current expression (Eq. (A.6)). We now analyze this phenomenon and through the analysis hope to find a cure for the problem.

The carrier continuity equation, with the carrier flow consisting of diffusion and drift components, can be converted to a form of the stan-

dard advection-diffusion equation [A.5]. At the steady state, the electron continuity equation Eq. (A.5) becomes

(A.16)

Assuming f.ln, and hence Dn, is constant, and for simplicity further assuming that E is also constant over the space, one obtains the following equation:

(A.17)

where v = -f.lnE is the electron drift velocity. The above equation is an advection-diffusion equation with v being the advective velocity. When the equation is solved in the discrete form, it has been shown in [A.5] that the relative importance of the advective term will determine the behavior of the solution. For a uniformly spaced mesh of spacing h, it has been shown that the quantity vhf Dn plays a critical role, and this quantity is named the cell Reynolds number,

discretization scheme is used, when Re > 2, Le., the potential drop across one spacing (cell) is greater than 2Vt, a grid-to-grid oscillated solution of carrier concentration will result. This is not necessarily the situation if some other discretization scheme is used. In fact, for the one-sided discretization scheme, if the difference operator D..nj D..x is chosen in such a way that the panel D..x is always facing the incoming direction of v (to be exemplified below), the solution is absolutely stable no matter how big the cell Reynolds number is. For example, if v > 0, then

This discretization scheme is called the upwind method, for the discretization panel always faces the incoming wind (v). In terms of the

discretization form of the current expression, Eq. (A.I5) becomes

(A.2I)

for 'l/Ji > 'l/Ji-l (i.e., v > 0), and

(A.22)

for 'l/Ji < 'l/Ji-l (Le., v < 0).

The reason for the upwind method to result in an absolute stability can be understood from the following analysis. With v positive, and using the upwind discretization scheme for dn/dx, Eq. (A.17) at Xi becomes

(A.23)

where Ui = U(Xi). ni-l can be expanded at Xi using the Taylor series, and the difference operator in the above equation becomes

(A.24)

where we have kept the terms only up to the second-order derivative. Substituting the above expression into Eq. (A.23) results in

where tJ.Xi has been replaced by h. Note that the additive term, ~vh, to the physical diffusivity, D n , is positive. Compared to Eq. (A.I7), the above equation shows that the upwind scheme actually enhances the diffusivity used in the numerical analysis and the magnitude of the enhanced part is proportional to the spacing. We may name Dn + ~vh as the numerical diffusivity in distinction from the physical diffusivity, and the additive part as the artificial diffusivity. The introduction of the additional diffusivity in the original continuity equation is the cause of the stable numerical behavior in the upwind scheme. Using the concept of Reynolds number, the numerical diffusivity keeps the effective

Reynolds number small. We can now show that with the upwind method the effective Reynolds number can never exceed two:

It is obvious that no matter how big vh might be, Reef! is always less than two, which explains why the upwind method is absolutely stable.

Two comments are in order. One is that even though the one-sided discretization scheme always introduces an extra term with the secondorder derivative, it is not necessary to stablize the numerical property. What enhances the stability is that the sign of the introduced coefficient of the term must be in line with the diffusivity. Simple mathematical manipulation similar to Eq. (A.24) shows that in fact the down-stream

effective diffusivity, making the numerical property even less stable than that of the central difference scheme.

The second comment is that even though the upwind method is absolutely stable, the amount of artificial diffusivity introduced is most often excessive. After all, this method only has accuracy of order one. One may seek a balance between the accuracy of the central difference method and the stability of the upwind method. This can be done in an explicit manner as proposed by Kreskovsky in [A.4]. The principle of that approach is that when the cell Reynolds number exceeds two a certain amount of artificial diffusion, which is related to the original Reynolds number, is artificially introduced so that the effective Reynolds number is just less than two while the amount of additive diffusivity is kept as small as possible.

However, there exists an elegant discretization scheme which preserves the order of accuracy two (the same as the central difference method) and at the same time is absolutely stable just like the upwind scheme. This scheme was first proposed by Scharfetter and Gummel (SG) [A.3] and has since become the classic way to discretize the current expression with the drift (or more generally, advective) term in it. In the following, we will first derive the SG scheme in a way which is different from the original approach used by Scharfetter but which allows us to estimate the accuracy of the scheme and to expose the assumptions made in an explicit way. We will then describe how the

SG scheme was introduced in its original form and use the Bernoulli function to express the formulation concisely.

We start from the expression of electron current using the quasiFermi level,

and note that n = niee(..p-<I>n)/Vt by using the Boltzmann statistics, where nie is the effective intrinsic carrier concentration after taking into consideration effects such as bandgap narrowing. The above equation can be written as

Integrating the above equation in the interval of [Xi-I, Xi], one obtains

where we have made all the position-dependent variables explicit. We can now apply the rectangular quadrature rule to numerically evaluate the integral on the left-hand side and obtain the following expression:

which has accuracy of order two. It is fairly easy to express J.tn,i-l/2 and nje,i-l/2 using the averages of their respective node values and still to have accuracy of order two. To achieve the accuracy of order two for the expression e-..p·-l/2/Vt , we use the following scheme [A.6] that

At this point, we can finally express the current density at Xi-l/2 as

This form of the expression can actually be derived by solving Eq. (A.6) for n in the interval of [Xi-I, Xi], assuming J-ln, £, and jn are constant, and then expressing jn using the node values of n. This is the original approach used by Scharfetter [A.3]. But as we have pointed out earlier, in this same expression, if used only for current density at the midpoint of the interval, the order of accuracy is actually two.

Now let us analyze the stability of this SG scheme. Substituting expression Eq. (A.33) into Eq. (A.ll) and arranging the terms in an order that the central difference method would have, one will find the effective Reynolds number ([AA]) is

Again, the effective Re is always smaller than 2, which means that the method is absolutely stable. Compared to the upwind scheme, however, this discretization has second order of accuracy. We conclude that the SG scheme as represented by Eq. (A.33) and Eq. (A.32) has numerical advantage over both the central difference method (Eq. (A.IS)) and the upwind method (Eqs. (A.2I-A.22)) in either the stability or the accuracy. This is the reason why this current discretization scheme is universally used in all device simulators.

We can simplify the above expression by introducing the Bernoulli function, which is defined as

Eq. (A.33) becomes