Parallel Hybrid Evolutionary Algorithm based on Chaos-GA-PSO for SPICE Model Parameter Extraction

Yuping Wu wuyuping@ime.ac.cn

Institute of Microelectronics of CAS, Beijing, China

Abstract—SPICE model is one of the key technical connections between the integrated-circuit technology community and the design community. Design community requires accurate SPICE model parameters so as to make the difference between design spec and practical spec minimized as possible. To get accurate model parameters, optimization algorithms are used for parameter extraction. A parallel hybrid evolutionary algorithm based chaos-GA-PSO is presented for SPICE model parameter extraction, which gets leverage of the advantages from chaos algorithm, genetic algorithm, and particle swarm optimization, overcomes their respective disadvantages, passes good individuals among them, avoids local area optimization efficiently, and gets more accurate global optimized parameter extraction results. Also such algorithm based SPICE model parameter extraction architecture is presented with model convergence checking and derivate ability checking supported.

Keywords-SPICE model; parameter extraction; parallel global optimization; generic alorithm (GA), particle swarm optimization (PSO); Message Passing Interface (MPI)

I. INTRODUCTION

SPICE (simulation program with integrated circuit emphasis) based circuit simulation is one of the most fundamental design verification methods in integrated-circuit (IC) design flow; SPICE models are the primary bases of SPICE simulation and the key technical connections between IC manufacture community and IC design community. To minimize the differences between IC design performance spec and practical performance spec, high accurate device models are required for SPICE simulation during IC design stage, so device modeling needs to get all the device characterizations and the parasitic effects exactly as possible. With the scaling down of the semiconductor technology feature dimensions, the device characterization is more sensitive to high order parasitic effects, the device models are becoming more and more complex so as to characterize the devices exactly, which makes SPICE models have hundreds of device model parameters and very complex model equations, such as BSIM3V3, BSIM4, PSP, etc.

SPICE model parameter extraction is a very important step on device modeling for supporting such technical connection. Due to the device model complexity and the high accuracy requirement, model parameter extractions become more and more complex, both for global model and bin model. Actually, due to the convergence issue on all the bin boundaries, the bin model extraction is not easier than global model extraction; in somehow, it is a bit more difficult which reduces bin model advantage.

There are too many methods for SPICE model parameter extraction, the direct analytical approach cannot work for the device models with complex equations, the most popular methods are least square method, Newton-Raphson iteration method, and Levenberg-Marquardt method, all of them are gradient-based, and such parameter extraction methods have more and more difficulty to deal with the models which have hundreds of parameters. There are more drawbacks, such as extraction results depending on the initial solutions, low flexibility on the extraction strategy and specific extraction steps to be followed, low convergence due to derivate of functions, complexity of computing on model equations, and low source code reusability among different device models. For those reasons, genetic algorithms (GA) are used for device model parameter extractions via selection, crossover, and mutation [9-16], but premature causes that the final solutions may be limited in local optimums rather than global optimization solution, which push the solution combined with simulation annealing, steepest descent, Levenberg-Marquardt gradient-based method, and hill climbing method so as to overcome such intrinsic drawback [13].

Fundamentally, particle swarm optimization can reach the final solutions with fewer generations than generic algorithm in terms of computational time and convergence rate from bench function test [1-6]. To make the initial particle distribution more uniform and escape from local optimums for PSO, chaos is used to improve the capability on robustness, convergence rate and accuracy, and global search for bench functions [7].

Several modifications of the basic PSO algorithm [5-8] were tried to improve the performance of PSO for model parameter extraction: particle are hierarchically arranged and influenced by the position of the local and global leaders; memory loss operation due to which a particle forgets its past best position; intensive local search in which the solution space around the global leader is searched with high resolution; adaptive inertia which causes the inertia of the particles to change adaptively, depending on the fitness of the population [11].

Simulation of a series of benchmark test functions showed integrated PSO and GA algorithm pose better ability to find global optimum than that of the standard PSO algorithm [1-3].

SPICE model equations are much more complex than test bench functions, and there are hundreds of parameters for a device model, which cause more local optimum points, and the distribution is very complicated, also the convergence become an important issue, so standard GA, PSO, and variations of them cannot handle SPICE model parameter extraction problem effectively. All of these algorithms show some _____________________________

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potential parameter extraction convergence issues, and cannot make the extraction escape from the complicated local optimums, so we need to explore a more efficient evolutionary algorithm for it, which should have exact ability to escape from very complicated local optimums, also must have excellent parameter extraction convergence for the SPICE models of hundreds of parameters.

With the leverage of the public research reports as the guide, we propose a parallel hybrid evolutionary algorithm based on chaos-GA-PSO for SPICE model parameter extraction, the main ideas are: integrating chaos algorithm, GA, and PSO; using chaos for GA and PSO to avoid premature on local optimums; passing good individuals among chaos, GA and PSO to speed up GA and PSO convergence; and using parallel computing to overcome high computations on model equations and convergence checking.

This paper is organized as follows. In Section II, parallel hybrid Chaos-GA-PSO algorithm is described. An overview of SPICE model parameter extraction architecture is given in section IV. Results on a sample device model parameter extraction are presented in section V. Finally, conclusions are presented in section VI along with some comments that need to be addressed in the future.

II. PARALLEL HYBRID CHAOS-GA-PSO ALGORTIHM

A. Hybrid Chaos-GA Algorithm Premature is often appeared in standard GA while there are

too many local optimums for the function curves. Thousands of hundreds of local optimums are very often for functions with hundreds of variables, which make standard GA optimization converged in local optimums very often. Also due to the large number of variables, GA optimization process convergence becomes very slow. Hence, the new GA-based algorithm needs to solve such two issues, one is to escape from local optimums, and another is to speed up GA optimization process for fast convergence.

The new GA-based algorithm is presented in Figure 1. Compared with standard GA, there are two major differences, one is the introducing of good individuals from outside, and another is the introducing of chaos step. Both of them increase the capability to escape from local optimums by generation of new good chromes and chaos-based escaping ability enhancement, and also outside good individuals speeds up the optimization process of GA by new good chromes from outside, which can be good individuals (a normal way to introduce good gene for future crossover) from other integrated algorithms.

One more a bit difference is that conditional washout process is added; it is activated while the number of individuals is over the upper boundary.

Chaos-GA has a bit but not obvious performance improvements on bench functions due to the simplicity of the bench functions compared with model equations of hundreds of model parameters, and because of the limitation of number

of pages, the performance of chaos-GA on bench functions is omitted here.

B. Hybrid Chaos-PSO Algorithm Although standard PSO has faster convergence than GA,

PSO still has no good convergence and optimization speed become very slow while a function have hundreds of variables, because there are too many local optimums for the function with hundreds of variables. Thousands of hundreds of local optimums are very often for functions with hundreds of variables, which also make standard PSO optimization limited in local optimums very often.

In similarity, the new PSO-based algorithm needs to solve two issues, one is to escape from local optimums, and another is to speed up PSO optimization process for fast convergence.

The new PSO-based algorithm is presented in Figure 2. Compared with standard PSO, there are two major differences, one is the introducing of good individuals from outside; another is the introducing of chaos step. Both of them increase the capability to escape from local optimums, also good

Y

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Generation of initial population

Evaluation for each individual

Chaos when needed

Mutation

Crossover

Final solution

Termination criteria met?

Good individuals from outside

Washout process

Selection

Figure 1. Flow of hybrid chaos-GA with outer good individual joining

individuals from outside acting as good particles in PSO speed up the optimization process of PSO.

Chaos-PSO has a bit but not obvious performance improvements on bench functions due to the simplicity of the bench functions compared with model equations of hundreds of model parameters, and because of the limitation of number of pages, the performance of chaos-PSO on bench functions is omitted here.

C. Hybrid Chaos-GA-PSO Algorithm Hybrid GA-PSO is better than both standard GA and

standard PSO on speed and convergence according to bench function, which is useful for SPICE model parameter extraction, but that is not enough, SPICE model equations are very complicated than bench functions, there are hundreds of model parameters, very complex local optimums, which can cause local optimum search rather than global optimum and long time to convergence, for this reason, we use the chaos-GA to replace standard GA, use chaos-PSO to replace standard PSO, and add the communication between chaos-GA and chaos-PSO for exchanging good individuals, which improves the optimization convergence and overcome local optimums efficiently because the introductions of good individuals from GA and PSO respectively and the new strong gene pieces can spread through the whole society. Also chaos added on GA and PSO increases the ability to escape from the local optimums. Figure 3 presents the flow for hybrid chaos-GA-PSO algorithm.

D. Fast Evaluation with Parallel Computing There are a huge number of equation evaluations for model

parameter extractions. To speed up device model parameter extraction process, full leverage of state of art computing technology is required. Each model evaluation for a set of given parameter values is regarded as a unit task, there are huge number of such unit tasks, they will be scheduled and assigned to a specific computing resource through Message Passing Interface (MPI) for parallel computing on model evaluation adaptively. Also looking-up-table technique and Bessel interpose are used for speeding up while there are enough points evaluated before.

III. SPICE MODEL PARAMETER EXTRACTION

A. General Iuuses of SPICE Model Parameter Extraction With scaling down of the technology feature dimensions,

the characterizations of devices become more and more sensitive to parasitic effects, and there are many high order effects need to be considered, so SPICE models have two trends, first the number of model parameters in a model is increased to more than one hundreds, even more than three

Y

Initialization for chaos-GA Initialization for chaos-PSO

chaos-GA

op.

Termination criteria met?

Hybrid process

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Figure 3. Flow of hybrid chaos-GA-PSO algorithm

Passing good individuals between chaos-GA and

chaos-PSO

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Initialization for PSO and Chaos

Stan

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PSO

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Passing good individuals between chaos and PSO

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Good individuals from outside

Figure 2. Flow of hybrid chaos-PSO with outer good individual joining

hundreds; second the number of model equations is increased much more than before, and model equations become much more complicated than before.

All these cases make SPICE model have much more local optimums, make standard GA and PSO much slower on model parameter extraction, and also make model convergence and derivate ability become a serious issue, which need to be put more attention on verification than before.

Hence, we must to find out much more valid optimization algorithm for speed and ability to escape from many more local optimums, and efficient SPICE model parameter extraction architecture to assure the model convergence and derivate ability.

B. Architecture for SPICE Model Parameter Extraction Figure 4 presents a model-independent architecture for

SPICE model parameter extraction program, which consists of 4 parts: input, output, interface, and core engine. The input part includes at least those parts: model equations, measurement I-V and C-V data, list of parameters to be extracted, model parameter value ranges, list of SPICE model convergence checking, list of derivate checking, priority table of model parameter extraction, and output model card format; the output part includes at least the extracted parameter values and the model generated data; the interface part includes GUI, parser of input data, model parameter extraction flow controller, and task scheduler for parallel computing; and the core engine is mainly built with chaos-GA-PSO based parallel hybrid evolutionary algorithm.

Model parameter value ranges are used to specify the value range for each model parameters, which will used to control the generation of initial particles, genes, and variations of them on evolutionary, also for checking the validation of model parameter values.

Output model card format will used to control the format of target SPICE model card, for hspice, for spectre, for eldo, or for other circuit simulators.

Priority table of model parameter extraction specifies the extraction priority for each model parameter; it determines the clustering of model parameters and the sequence for extracting the cluster of model parameters, which reduces the complexity of model parameter extraction, and speed up the extraction procedure.

Model equations will be translated into the internal functions by the interface program automatically, and they will co-work with core engine parts under the management of flow controller and task scheduler, which make such SPICE model parameter extraction program independent with specific device model, i.e. generalization for model parameter extraction.

List of model convergence check and list of model derivate check are to specify the contents of model related convergence check and derivate checks. Model convergence check and derivation check are very important for complex model

equations especially while the technology feature dimensions becoming smaller and smaller and closing to 12nm-22nm level, which is useful for avoiding model induced simulation convergence issues on circuit simulation stage. Model convergence check is implemented as: a) scanning the values of device geometry parameters; b) scanning the control voltages; c) doing fine perturbation of extracted model parameters; d) calculating I-V and C-V characterization curve; e) calculating the derivate. If there are no failures, the model convergence check and derivate check are passed; otherwise, there are potential model convergence issue for failure on d), and derivate issue for failure on e). Hence, both model convergence check and derivate check related computation are time consuming, so the state of art parallel computing technology is required for such checks because of parallel computing can reduce the time spent on those checks.

After the model parameter extracting, the system will output the I-V and C-V data which is generated based on the SPICE model and the final extracted model parameters so as to show the consistency between measurement data and model generation data.

IV. EXPERIMENTS

To test such algorithm based SPICE model parameter extraction strategy, MOSFET BSIM4 model is used. BSIM4 includes various complex physical effects, besides 23 parameters for mode control and 17 parameters for process, there are 82 basic model parameters, 14 parameters for asymmetric and bias-dependent Rds model, 8 parameters for gate-induced drain leakage model, 26 parameters for gate

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List of parameters to be extracted

List of SPICE model convergence check

List of derivation checking

Modelparameter value

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Model generated data

Extracted parameter

values

Figure 4. Architecture for SPICE model parameter extraction program

Priority table for model parameter extraction

Core Engine with Chaos-GA-PSO based optimizer

Circuit simulator Output model card format

dielectric tunneling current model, 20 parameters for charge and capacitance model, 33 parameters for high-speed/RF model, 25 parameters for flicker and thermal noise model, 12 parameters for layout-dependent parasitic model, 32 parameters for asymmetric source/drain junction diode model, 22 parameters for temperature dependence, 24 parameters for stress effect model, 10 parameters for well-proximity effect, and 16 parameters for dW and dL, so there are more than 300 model parameters totally in BSIM4 model, which determines that model parameter extraction for BSIM4 is a high multi-dimensional optimization problem, and there are hundreds of thousands of local optimums.

Compared with the standard GA and PSO, due to the communication among chaos, GA, and PSO, the extraction is fast converged in the global points. The extraction result is shown in Figure 5. The global optimum point is found, where the RMS is 0.39% for Id-Vg@Vds=0.10V, 0.47% for Id-Vg@Vds=1.54V, 0.51% for Id-Vd@vbs=0.00V, and Id-Vd@Vbs=-0.77V, which means that this set of model parameter values is the global optimum point. Also the model convergence is checked for such set of model parameter values.

Figure 5. Curves from model extraction for BSIM4 (a) Id-Vg@Vds=0.10V (b) Id-Vg@Vds=1.54V (c) Id-Vd@Vbs=0.00V (d) Id-Vd@Vbs=-0.77V

V. CONCLUSIONS

SPICE model equations are much more complicated than all the bench functions for traditional evolutionary algorithm test. Local optimization issues are more prominent in SPICE

model parameter extraction than in the classic bench functions. Parallel hybrid evolutionary algorithm based on chaos-GA-PSO gets the leverage of all the advantages of chaos algorithm, genetic algorithm, and particle swarm optimization, avoids their respective disadvantages, overcomes the local area optimization solution, and gets more accurate device model parameter extraction results. The speed of model parameter extraction is a bit slow than traditional PSO because of the extra computation increasing due to GA and chaos, but it is valuable for getting more accurate extraction results, and reduce the potential simulation errors for future design.

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