Книги+1 / 2013 [Chandan_Kumar_Sarkar]_Technology_CAD

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convergence. Say, for example, to achieve a convergence for an equation a large number of iterations are needed. This will severely affect the speed and hence the time consumed. Conversely, a high-speed solver may not achieve a convergence. In the following sections the different iterative techniques implemented in technology computer aided design (TCAD) to achieve convergence efficiently are described in detail.

3.2  Numerical Solution Methods

For semiconductor devices, different solution methods are used depending upon the situation. It is also possible to use several different numerical methods to obtain solutions. And in addition, different combinations of models for a particular numerical method are also required for solving equations. There are three different types of solution techniques commonly used for obtaining solutions for semiconductor devices. These are represented by (a) de-coupled (GUMMEL), (b) fully coupled (NEWTON), and

(c) BLOCK. The de-coupled technique like the Gummel method solves for each unknown in an equation while keeping the other variables constant and repeats the process until a stable solution is achieved [1]. In fully coupled techniques such as the Newton method, the total system of unknowns are solved together. Finally, in the combined or block method, the solution is obtained by solving some equations by the fully coupled method, while others are solved by the de-coupled method. Both techniques mentioned are broadly classified under the non-linear iteration method where the method converges to a solution nonlinearly and provides a more accurate result than its linear counterpart. In the next section these techniques are discussed in detail.

3.3  Non-Linear Iteration

3.3.1  Newton Iteration

For the solution of nonlinear systems, the scheme developed by Bank and Rose is applied. This scheme tries to solve the nonlinear system by using the Newton method: Each iteration of the Newton method solves a linearized version of the entire non-linear algebraic system. The size of the problem is relatively large, and each iteration takes a relatively long time. However, the iteration will normally converge quickly (in about three to eight iterations) so long as the initial guess is sufficiently close to the final solution. Strategies that use automatic bias step reduction in the event of

non-convergence loosen the requirement of a good initial guess. Newton’s method is the default for drift-diffusion calculations in ATLAS. There are several calculations for which ATLAS requires that Newton’s method be used. These are DC calculations that involve lumped elements, transient calculations, curve tracing, and when frequency-domain small-signal analysis is performed. The Newton-Richardson method is a variant of the Newton iteration that calculates a new version of the coefficient matrix only when slowing convergence demonstrates that this is necessary. An automated Newton-Richardson method is available in ATLAS, and it improves performance significantly on most problems. The automated Newton-Richardson method is enabled by specifying the AUTO parameter of the METHOD statement [2]. If convergence is obtained only after many Newton iterations, the problem is almost certainly poorly defined. The grid may be very poor (i.e., it contains many obtuse or high aspect ratio triangles), or a depletion region may have extended into a region defined as an ohmic contact, or the initial guess may be very poor.

3.3.2  Gummel Iteration

Each iteration of Gummel’s method solves a sequence of relatively small linear sub-problems. The sub-problems are obtained by linearizing one equation of the set with respect to its primary solution variable, while holding other variables at their most recently computed values. Solving this linear sub-system provides corrections for one solution variable. One step of Gummel iteration is completed when the procedure has been performed for each independent variable. Gummel iteration typically converges relatively slowly, but the method will often tolerate relatively poor initial guesses. The Gummel algorithm cannot be used with lumped elements or current boundary conditions. Two variants of Gummel’s method can improve its performance slightly. These both limit the size of the potential correction that is applied during each Gummel loop.

The first method, called damping, truncates corrections that exceed a maximum allowable magnitude. It is used to overcome numerical ringing in the calculated potential when bias steps are large (greater than 1 V for room temperature calculations). The maximum allowable magnitude of the potential correction must be carefully specified: too small a value slows convergence, while too large a value can lead to overflow. The DVLIMIT parameter of the METHOD statement is used to specify the maximum allowable magnitude of the potential correction. By default, the value of this parameter is 0.1 V. Thus, by default Gummel iterations are damped. To specify undamped Gummel iterations, the user should specify DVLIMIT to be negative or zero.

The second method limits the number of linearized Poisson solutions per Gummel iteration, usually to one. This leads to under-relaxation of the potential update. This “single-Poisson” solution mode extends the usefulness of

Gummel’s method to higher currents. It can be useful for performing low current bipolar simulations, and simulating MOS transistors in the saturation region. It is invoked by specifying the SINGLEPOISSON parameter of the METHOD statement.

3.3.3  Block Iteration

ATLAS offers several block iteration schemes that are very useful when lattice heating or energy balance equations are included. Block iterations involve solving subgroups of equations in various sequences. The subgroups of equations used in ATLAS have been established as a result of numerical experiments that established which combinations are most effective in practice.

In non-isothermal drift-diffusion simulation, specifying the BLOCK method means that Newton’s method is used to update potential and carrier concentrations, after which the heat flow equation is solved in a de-coupled step. When the carrier temperature equations are solved for a constant lattice temperature, the BLOCK iteration algorithm uses Newton’s method to update potential and concentrations [2]. The carrier temperature equation is solved simultaneously with the appropriate continuity equation to update the carrier temperature and carrier concentration.

When both the heat flow equation and the carrier temperature equations are included, the BLOCK scheme proceeds as described previously for the carrier temperature case, and then performs one de-coupled solution for lattice temperature as a third step of each iteration.

3.3.4  Combining the Iteration Methods

It is possible to start with the GUMMEL scheme and then switch to BLOCK or NEWTON if convergence is not achieved within a certain number of iterations. One circumstance where this can be very helpful is that the Gummel iteration can refine an initial guess to a point from which a Newton iteration can converge [3].

The number of initial GUMMEL iterations is limited by GUM.INIT. It may also be desirable to use BLOCK iteration and then switch to NEWTON if convergence is not achieved. This is the recommended strategy for calculations that include lattice heating or energy balance. The number of initial BLOCK iterations is limited by NBLOCKIT.

Any combination of the parameters GUMMEL, BLOCK, and NEWTON may be specified on the METHOD statement. ATLAS will start with GUMMEL if it is specified. If convergence is not achieved within the specified number of iterations, it will then switch to BLOCK if BLOCK is specified; if convergence is still not achieved the program will then switch to NEWTON.

3.4  Convergence Criteria for Non-Linear Iterations

After a few non-linear iterations, the errors will generally decrease at a characteristic rate as the iteration proceeds. Non-linear iteration techniques typically converge at a rate that is either linear or quadratic. The error decreases linearly when Gummel iteration is used (i.e., it is reduced by about the same factor at each iteration). For Newton iteration the convergence is quadratic (i.e., small errors less than one are approximately squared at each iteration). The non-linear iteration is terminated when the errors are acceptably small. The conditions required for terminations are called convergence criteria. Much effort has gone into developing reliable default convergence criteria for ATLAS [3]. The default parameters work well for nearly all situations, and most users will never need to change them.

3.5  Initial Guess Requirement

Non-linear iteration starts from an initial guess. The quality of the initial guess (i.e., how close it is to the final solution) affects how quickly the solution is obtained, and whether convergence is achieved. Users of ATLAS are not required to specify an initial guess strategy. If no strategy is defined, ATLAS follows certain rules that implement a sensible, although not necessarily optimum, strategy. There is some interaction between the choice of non-linear iteration scheme and the initial guess strategy. De-coupled iteration usually converges linearly, although perhaps slowly, even from a relatively poor initial guess. Newton iteration converges much faster for a good initial guess but fails to converge if started from a poor initial guess. One simple initial guess strategy is to use the most recent solution as the initial guess. Of course, there is no previous solution for the first calculation in a series of bias points. In this case, an initial solution is obtained for equilibrium conditions [3]. There is no need to solve the current continuity equations at equilibrium, and a solution of Poisson’s equation is quickly obtained. It is also possible to modify the initial guess in a way that makes some allowance for the new bias conditions. Typical strategies include

•Using two previous solutions and interpolation to project a new solution at each mesh point.

•Solving a form of current continuity equation with carrier concentrations held constant. This strategy yields an improved estimate of new potential distribution.

•Modifying the majority carrier quasi-Fermi levels by the same amount as the bias changes.

150Technology Computer Aided Design: Simulation for VLSI MOSFET

•Parameters on the SOLVE statement can be used to specify an initial guess strategy. Five initial guess strategies are available.

•INITIAL starts from space charge neutrality throughout the device. This choice is normally used to calculate a solution with zero applied bias.

•PREVIOUS uses the currently loaded solution as the initial guess at the next bias point. The solution is modified by setting a different applied bias at the contacts.

•PROJECTION takes two previous solutions whose bias conditions differ at one contact and extrapolates a solution for a new applied bias at that contact. This method is often used when performing a voltage ramp.

•LOCAL sets the applied bias to the specified values and changes the majority carrier quasi-Fermi levels in heavily doped regions to be equal to the bias applied to that region. This choice is effective with Gummel iteration, particularly in reverse bias. It is less effective with Newton iteration.

•MLOCAL starts from the currently loaded solution and solves a form of the total current continuity equation that provides an improved estimate of the new potential distribution. All other quantities remain unchanged. MLOCAL is more effective than LOCAL because it provides a smooth potential distribution in the vicinity of p-n junctions. It is usually more effective than PREVIOUS because MLOCAL provides a better estimate of potential. This is especially true for highly doped contact regions and resistor-like structures.

When a re-grid is performed, the solution is interpolated from the original grid onto a finer grid. This provides an initial guess that can be used to start the solution of the same bias point on the new grid.

Although the initial guess is an interpolation of an exact solution, this type of guess does not provide particularly fast convergence.

3.6  Numerical Method Implementation

The Gummel method is effective where the system of equations is weakly coupled, with only linear convergence. The Newton method is effective where the system of equations is strongly coupled and has quadratic convergence. However, it may so happen that in a system of equations some of the quantities are weakly coupled while others are strongly coupled, or we can say a

mixed coupled system. In such cases the Newton method consumes extra time solving for quantities that are essentially constant or weakly coupled. In accession to that, the Newton method requires a more accurate initial guess for the problem to obtain convergence. In order to compensate for the issues associated with the Newton method, it is better to use the block method to achieve convergence efficiently. The block method provides a comparatively faster simulation time in the mixed case over the Newton method [4]. Because Gummel can often furnish better initial guesses to problems, it is more appropriate to start a solution with a few Gummel iterations in order to generate a better initial guess and then switch to Newton to attain the final solution. The different solution methods are carried out by including the following statement for simulation:

METHOD GUMMEL BLOCK NEWTON

The exact meaning of the statement and the combination of the solution method depend upon the particular models to which it is applied. In the following sections the insights of this statement with respect to different models are described in detail.

3.7  Basic Drift Diffusion Calculations

The isothermal drift diffusion model requires the solution of three equations for potential, electron concentration, and hole concentration. Specifying GUMMEL or NEWTON alone on METHOD statement will produce simple Gummel or Newton solutions of the equations as described earlier. For almost all cases the Newton method is preferred because of its quadratic convergence to produce more accurate solution; hence, it is set as the default method. However, there are also alternative methods such as specifying:

METHOD GUMMEL NEWTON

This will cause the solver to initially start with Gummel iterations and then switch to Newton, if convergence is not achieved. This method is very robust, even though it consumes more time to obtain solutions for any device. However, this method is highly recommended for all simulations with floating regions such as silcon on insulator (SOI) transistors. A floating region is defined as an area of doping that is separated from all electrodes by a p-n junction. It may also be noted that BLOCK is equivalent to NEWTON for all isothermal drift-diffusion simulations.

3.8  Drift Diffusion Calculations with Lattice Heating

When the lattice heating model is added to drift diffusion, an extra equation is added. The BLOCK algorithm solves the three drift diffusion equations as a Newton solution and follows this with a de-coupled solution of the heat flow equation. The NEWTON algorithm solves all four equations in a coupled manner. NEWTON is preferred once the temperature is high; however, BLOCK is quicker for low temperature gradients. Typically the combination used is

METHOD BLOCK NEWTON

3.9  Energy Balance Calculations

The energy balance model requires the solution of up to five coupled equations. GUMMEL and NEWTON have the same meanings as with the drift diffusion model (i.e., GUMMEL specifies a de-coupled solution, and NEWTON specifies a fully coupled solution). However, BLOCK performs a coupled solution of potential, carrier continuity equations followed by a coupled solution of carrier energy balance, and carrier continuity equations. It is possible to switch from BLOCK to NEWTON by specifying multiple solution methods on the same line. For example,

METHOD BLOCK NEWTON

This will begin with BLOCK iterations and then switch to NEWTON if convergence is still not achieved. This is the most robust approach for many energy balance applications. The points at which the algorithms switch is predetermined but can also be changed on the METHOD statement. The default values set by TCAD tool Silvaco work well for most circumstances.

3.10  Energy Balance Calculations with Lattice Heating

When non-isothermal solutions are performed in conjunction with energy balance models, a system of up to six equations must be solved. GUMMEL or NEWTON solve the equations iteratively or fully coupled, respectively. BLOCK initially performs the same function as with energy balance calculations, and then solves the lattice heating equation in a decoupled manner.