For circuit simulation, well extracted model parameters are as important as the device model itself. A rough extraction procedure can lead erroneous simu-lation results and eventually circuit design failure. According to the conven-tional extraction method, the parameters are first extracted from the specific measured characteristics. Using numerical curve fitting [20], the model pa-rameters can be estimated so that the calculated outcome will fit the measured data quit well in each parameter’s characteristic. However, in practical cases, this set of parameters is not good enough for modelling the whole device be-havior, such as family curves. The experienced engineer will spend much time to get a set of useful parameters. Because of the imperfect device model, the engineer should ignore some misfit data and focus on the critical parts of the characteristic. Although we use CAD tools to fit the curves, the fine-tuning of a whole set of parameters still needs to be done semi-manually. This procedure is called parameter optimization and is a time-consuming task in microelectronic industry.
To optimize the parameters for the various types and considerable quan-tities of devices, a robust and automatic technology CAD (TCAD) tool is necessary both in academic research and industry applications. The genetic algorithm (GA) is a global optimal strategy used in a wide rage of applica-tions [64],[85]-[89]. In microelectronics it has been applied to various aspects
in VLSI designs, such as cell placement, channel routing, test pattern gen-eration and design for test. In this work, we applied the GA in model pa-rameters optimization for semiconductor device models. The GA is based on floating-point operators and suitable for solving this numerical optimization problem. In the previous sections, the MI and WR method are proposed and implemented for circuit simulation. Because MI method is fast convergent and highly accurate, we can optimize the parameters efficiently with combin-ing MI method with the GA. In this section, we take the optimization for DC
V
CC(V)
0.2 0.4 0.6 0.8 1.0 1.2
I
C(A)
0.12 0.16 0.20 0.24 0.28
Weight value, W
0 50 100 150
V
IN= 1.49 V
1.47 V 1.45 V
Figure 3.5: The proposed weight value W for HBT model parameter extraction.
family curves as an example. As mention in Sec. 3.1.1, Eq. (3.3)-(3.9) are the governing nonlinear algebraic equations to be solved for the circuit operated at the steady state condition (DC). The current models I1, I2, IBL1, IBL2, ICTare the functions of VB, VC, and VE, respectively. For a given set of parameters, there is a corresponding set of I-V curves. Table 3.1 shows a part of param-eters to be extracted. The paramparam-eters resolution and numeric range are also included. The final result is a set of DC I-V curves as shown in Fig. 3.5.
Variable Range Resolution Unit
RB 2.0 ∼ 6.0 0.00001 Ω
IS 1.0e−25 ∼ 1.0e−24 1.0e−25 A
BR 0.2 ∼ 0.3 0.02 −
NR 1.0 ∼ 1.1 0.001 −
Table 3.1: A list of parameters to be extracted.
The genetic algorithm is a self-adaptive optimization strategy that mimics a living system. We briefly state the GA method for the HBT circuit optimal characterization.
• Step 1: Problem Definition. The relationship between modelling in-put parameter (VCC and VIN) and simulated result collector current ICcan be written as follows:
f (VCC,VIN, ˜P) = IC. (3.31) The function f can be regarded as a nonlinear equations
solver, and ˜Pcontains all parameters to be extracted. The set of IC data points represents an I-V curve. The goal of evolvement is to minimize the difference between a set of targets and simulated I-V curves, and to find out their corre-sponding model parameters.
• Step 2: Encoding Method. The design of gene encoding strategy depends on the property of problem. In this problem model, there are totally 15 parameters for DC simulation and all variables are point numbers. We transform the floating-point numbers into bit strings instead of real numbers. The bit string has strongly combinatorial property, and we have found this representing has better results in crossover and mutation.
• Step 3: Fitness function (F ) Calculation. We consider the follow-ing function:
F = W ∗p
(ICT − ICS)2, (3.32)
where the W is weight value shown in Fig. 3.5. ICT and ICS
are the sets for target and simulated I-V points,respectively.
Because the saturation region of the HBT I-V curve is rather sensitive, we define W to emphasize the relative importance of each I-V point. It decreases as the applied voltage VCC
increases. We evaluate the discrepancy for two I-V sets with
the F .
• Step 4: Reproduction. We adopt the tournament selection with floating-point operators as the selection strategy and this hybrid strat-egy not only selects better chromosomes but also keeps weak ones for few generations to achieve higher population diver-sity. For the crossover scheme, in HBT device model, all parameters to be optimized can be classified into four cate-gories which represent different numerical constraints. We take an uniform crossover scheme [85, 86, 90]; and based on our simulation experience, it is more effective than sin-gle and two-point crossover schemes. Finally, the mutation strategy changes the mutation rate dynamically to keep the population diversity.
Fitness function (F) calculation Achieve Goal ?
YES NO
I-V curves simulation by MI Method
Parameters Verification I-V measurement data
Parameters reproduction by GA, including tournament selection , uniform
crossover, and mutation strategy .
Figure 3.6: The employed flowchart of parameters optimization in our study.
The illustration of above steps are plotted in flowchart, Fig. 3.6. We should simulate all data points with respect to the measured data. The numerical ef-ficiency of our circuit simulator can reduce the consumed time for GA pa-rameter extraction. In Fig. 3.6, we note that the extracted papa-rameters should be verified by physical checks after GA procedure. It is because that we can achieve the evolution goal by more than one set of parameters. Figure 3.7 shows the comparison between multiple I-V curves evolution with and with-out the weight value after 300 generations. The evolutionary I-V curves with applying weight value achieve the target I-V curves rapidly, and it has a good evolution behavior. The importance of weight value introduced here not only
V
CC(V)
0.2 0.4 0.6 0.8 1.0 1.2
I
C(A)
0.12 0.16 0.20 0.24 0.28
V
IN=1.49 V
V
IN=1.47 V
Target I-V curve I-V curve with W I-V curve without W
Figure 3.7: A comparison of the method with and without the proposed weight value for multiple I-V curves optimization.
redirects the evolution trend to better direction but also reduces the searching space in this multiple objective evolution problem, and the efficiency can be obtained.
In addition, Fig. 3.8 shows the fitness score convergence behavior for the multiple I-V curves evolution with or without the dynamic mutation technique.
Generation
0 100 200 300 400 500 600
Fitness score (Log)
1e-5 1e-4 1e-3 1e-2 1e-1
A: without dynamic mutation rate B: with dynamic mutation rate
A
B
Figure 3.8: A comparison of the method with and without the proposed dynamic mutation rate scheme.
The results suggest that the dynamic mutation scheme with floating-point op-erators keeps the population diversity and finally has better evolutionary re-sults.
Figure 3.9 shows the evolutions of the I-V curve. Starting from an arbitrary I-V curves, it approaches to the desired final I-V curve step by step. Based on the MI and GA methods, the above process, from a given I-V curve to the final optimal I-V curve, is solved and evolved automatically. Compared with
0.0 0.2 0.4 0.6 0.8 1.0 I
C(A)
-0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12
Target curve
Initial evolutionary curve Final evolutionary curve
Migration
V
CC(V)
Figure 3.9: An illustration of the migration processes for the I-V curve optimization.
the conventional trial-and-error methodology to extract optimal parameters of the circuit model, our approach successfully reduces the complex procedures and simulation time significantly.