Abstract:
A method of uniquely extracting both intrinsic and parasitic components from a single set of measured S-parameters is useful for extracting a single set of measured S-parameters for the development of non-linear Field Effect Transistor (FET) models. Competitive extraction where multiple trial solutions are attempted spanning a region or space of feedback impedances is used. Extraction is followed by optimization that reduces the extracted values to a model that better fits measured S-parameters. Optimization can be achieved by further evaluating the speed of convergence in an error metric.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is related to the following commonly-owned co-pending patent applications all filed on Apr. 28, 2000, which are incorporated herein: S-PARAMETER MICROSCOPY FOR SEMICONDUCTOR DEVICES, by Roger Tsai, Ser. No. 60/200,307; EMBEDDING PARASITIC MODEL FOR PI-FET LAYOUTS, by Roger Tsai, Ser. No. 60/200,810, SEMI-PHYSICAL MODELING OF HEMT DC-TO-HIGH FREQUENCY ELECTROTHERMAL CHARACTERISTICS, by Roger Tsai, Ser. No. 60/200,648, SEMI-PHYSICAL MODELING OF HEMT HIGH FREQUENCY NOISE EQUIVALENT CIRCUIT MODELS, by Roger Tsai, Ser. No. 60/200,290, SEMI-PHYSICAL MODELING OF HEMT HIGH FREQUENCY SMALL-SIGNAL EQUIVALENT CIRCUIT MODELS, by Roger Tsai, Ser. No. 60/200,666, HYBRID SEMI-PHYSICAL AND DATA-FITTING HEMT MODELING APPROACH FOR LARGE SIGNAL AND NONLINEAR MICROWAVE/MILLIMETER WAVE CIRCUIT CAD, by Roger Tsai and Yaochung Chen, Ser. No. 60/200,622, and PM 2 : PROCESS PERTURBATION TO MEASURED-MODELED METHOD FOR SEMICONDUCTOR DEVICE TECHNOLOGY MODELING, by Roger Tsai, Ser. No. 60/600,302. 
     This application claims benefit of provisional application 60,200,436 filed Apr. 28, 2000. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates generally to the field of high frequency small signal analysis and more specifically to a modeling method for extracting a single set of equivalent circuit model parameters that specifically simulate measured S-parameters, but are also useful in the development of non-linear Field Effect Transistor (FET) models, and analysis of the physical structure of the FET. 
     BACKGROUND OF THE INVENTION 
     There is a need for a high-frequency nonlinear FET modeling algorithm that will produce a unique solution from measured S-parameters. The problem relates to the difficulty in uniquely determining the feedback impedance of a Field Effect Transistor (FET). Present modeling techniques generate non-unique models for fitted S-parameters or require multiple measurements at different bias points (i.e., cold FET measurements) to satisfy uniqueness. Both of these modeling strategies present problems for nonlinear FET modeling. Non-unique models consist of parameters that are not physically consistent with the FETs fabrication parameters, such as layout dimensions and material parameters. In addition, the cold-FET based model extraction methods do not express the bias-dependence of parasitic elements correctly. 
     Another disadvantage of prior art equivalent circuit model techniques is that they assume that the parasitic components are constant with bias when they almost certainly are not. This assumption leads to inaccuracies in the extraction of the intrinsic equivalent circuit. The precise determination of the parasitic, especially the resistive, components directly determines the accuracy of the extraction because the calculated values of the intrinsic components depend on them. In particular, the accurate extraction of feedback resistance in a FET is critical to determining the accurate values for all other components. 
     While most direct extraction techniques rely on cold FET measurements to determine values for the parasitic components, optimization based techniques often use over-dimensioned sets of multi-bias S-parameter (and noise response) delete measurements. In either case, the resistive value of the parasitic components are kept constant even though they can be highly bias-dependent. Because of this, the results of direct extraction based on cold FET parasitics suffer from inaccuracies as bias changes. Although optimized extractions may be attempted to compensate for the parasitic bias dependence, they cannot provide a unique solution to the equivalent circuit because the feedback impedance cannot be uniquely determined with conventional optimization algorithms. This uncertainty often leads to results that are not physically significant, even though they may be accurate for fitting measured S-parameters. 
     Thus, there is a need for a modeling method that can extract an accurate and unique model solution to a single set of measured S-parameters to develop nonlinear FET models that are both accurate and physically significant. 
     SUMMARY OF THE INVENTION 
     The present invention provides a FET modeling algorithm that generates a unique solution for FET feedback impedance, thus determining resistive, capacitive, and inductive equivalent circuit parameter values based on a single set of S-parameter measured values. The subsequent extracted circuit model is a unique global solution rather than a non-unique local minima. In addition, the extracted values satisfy self-consistent checks such as scaling with FET device periphery, expected bias-dependence, and convergence to known fabrication parameters such as gate metalization resistance, material sheet resistance, gate length and recess dimensions. 
     Disclosed in one embodiment is a method of uniquely determining the feedback impedance values for a FET device. The method includes the step of competitive extraction where multiple trial solutions are attempted spanning a region or “space” of feedback impedance values. Each solution in the competitive extraction can be achieved by utilizing the same cycle of conventional direct extraction via Minasian extraction. Next, the extraction step is followed by an optimization step that reduces the extracted values to a model that better fits measured S-parameters. This optimization step can be achieved by conventional optimization algorithms. Last, selection of the most accurate and unique trial solution can be achieved by further evaluating the speed of convergences in an error metric. 
     Also disclosed is a method for unique determination of FET equivalent circuit parameter values, the method comprising the steps of selecting a feedback impedance value from a space of expected parameter values and generating a FET equivalent circuit model. The method further includes the step of establishing those parameter values for the circuit model that converge fastest so that a sufficiently low error fit between the expected S-parameters and the measured S-parameter is obtained, the convergence obtained using multiple Minasian extraction and optimization cycles. 
     A technical advantage of the invention is that by tracking the speed of convergence, the solutions surrounding unique, physically significant models condense faster than the rest. Conventional techniques stop once some least-squares based error metric is minimized. 
     Another technical advantage of the invention, is the development of unique models that best represents the physical structure of a FET. This has been proven through extensive verification and testing. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Other advantages of the invention including specific embodiments are understood by reference to the following detailed description taken in conjunction with the appended drawings in which: 
     FIG. 1 is a process flow diagram illustrating the extraction and convergence method of the present invention according to one embodiment; 
     FIG. 2 shows an example FET equivalent circuit; 
     FIGS. 3 and 4 show plots illustrating measured S-parameters for a FET device; 
     FIGS. 5 and 6 show plots comparing initial modeled S-parameters with measured S-parameters; 
     FIGS. 7 and 8 show plots comparing modeled parameters with measured parameters after application of one embodiment of the method of the present invention; 
     FIG. 9 is a plot illustrating final fitting of extracted parasitic feedback resistance values obtained using the method of the present invention for the circuit of FIG. 2; 
     FIGS. 10,  11  and  12  are plots illustrating scaling of extracted values as a function of device periphery scaling; 
     FIG. 13 illustrates the bias dependence of Rs versus Vgs; 
     FIG. 14 illustrates the bias dependence of C GS  and C GD  in a typical HEMT; and, 
     FIGS. 15 and 16 illustrate inaccuracies and expected errors using constant parasitic model parameter values. 
    
    
     References in the figures correspond to like references in the detailed description unless otherwise indicated. 
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The present invention is a unique and novel method of extracting both intrinsic and parasitic equivalent circuit parameter values for a FET device (not shown). This capability is vital for the generation of accurate high-frequency nonlinear FET circuit models. Accurate bias dependent and linear equivalent circuits are not only important to circuit design, but are also needed to provide a close description of the physical structure of a device for process feedback. 
     The description of the invention is presented in connection with the small signal circuit model upon which the extraction method of the present invention has been used to extract feedback resistance values between the gate, source and drain terminals of the circuit model. A process flow diagram illustrating the competitive extraction method of the invention is shown and described with reference to a specific example. Next, a set of plots are presented for various circuit model parameters such as the source resistance (R s ), the gate-to-source capacitance (C GS ), the transconductance (G M ), the gate-source, gate-drain line capacitance (C GS , C DG ) and the percentage error in the gate source line capacitances as a function of gate-source voltage (V GS ) and total device periphery. Ultimately, the plots are evidence of convergence between a set of measured device parameter values and extracted parameter values so that the error between the two values sets is minimized. 
     Reference is now made to FIGS. 1 and 2 in which FIG. 1 is a process flow diagram of a method, denoted generally as  100 , of determining unique FET parameter values of a FET device (not shown). The method uses S-parameters which are measured on the FET device as well as S-parameters calculated from a circuit model to determine the parameters of the FET device. The present invention provides a method of extracting realistic parameter values for the circuit model  200  that accurately reflect the FET device modeled by the circuit  200 . For clarity of explanation, the method  100  will be described with reference to a specific example. 
     The method  100  begins at step  102  wherein a circuit model which is representative a FET device is generated. Referring to FIG. 2, therein is shown an example equivalent circuit model of a FET device, denoted generally as  200 , on which the extraction algorithm of the present invention can be applied to extract a unique set of circuit parameter values. The topology of circuit model  200  is complete enough to model most FET small-signal characteristics in all bias regimes. The circuit model  200  includes gate  202 , drain  204  and source 206  terminals with an assortment of resistance, impedance and capacitance parameters extending between the gate  202 , drain  204 , and source  206  terminals. These parameters include the source resistance R s , the gate-source capacitance C GS , and the transconductance G M , among others as indicated in the figure. The circuit model  200  represents an ideal FET device suitable for demonstration of the ability to uniquely and accurately determine the FET feedback impedance through a single set of measured S-parameter values according to the invention. 
     Generally, in deriving the circuit model  200 , a designer will consider the operating frequency and bias condition of the FET device to be modeled as carriers enter the device through the source  206  and leave through the drain  204  subject to the control or gating action of the gate terminal  202 . For a FET device, the voltages applied to the gate relative to the source (V GS ) can determine the operating region of the device as V GS  is increased from zero to a threshold voltage level (V T ) that biases the device into its active region. The biasing and operation of devices represented by the circuit model  200  are well understood by those of ordinary skill in the art. 
     The model  200  contains circuit parameters and components, designated as Rs, Ls, Rg, Lg, Rd and Ld. The selection of the initial inductance (L), resistance (R), transconductance (G) and capacitance (C) values for the various components of the circuit model  200  is straightforward when it is assumed the performance of the underlying device is linear with frequency. (Such assumptions, however, prove inaccurate for most FET device models since most are non-linear at high frequencies and given actual device profiles peripheries. This assumption however does provide an initial starting point for the method of the present invention). 
     The next step in the process is to measure the S-parameters for the FET device, step  104 . To do so, the FET device is physically tested and measured and the S-parameters for the FET device are determined. A determination is then made as to whether the measurements and the associated measured S-parameter values are accurate, step  106 . To do so requires the knowledge of one skilled in the art. If the parameters are not accurate, the S-parameters are remeasured until the measurements are acceptable. FIGS. 3 and 4 show visual examples of measured FET S-parameters for a selected FET device. Note that four S-parameter measurements are shown in which three are depicted in a Smith Chart format (FIG. 3) and one is shown in a linear graph format (FIG.  4 ). Assuming the measured S-parameters are accurate, the measured S-parameters are stored, step  108 , for use during subsequent comparison. 
     Referring back to FIG. 1, at some point in the process, the space defining all values for trial feedback impedance solutions is generated, step  110 . For example, the trail feedback impedance space could be as shown in Table 1. The trial values are used to generate a set of feedback resistance values or points. The points can be thought of as extending from one (1) to N, where N is in the Nth trial feedback impedance solution in the space generated at step  110 . The example shown in Table 1 shows thirty (30) trial points such that N=30 for this example. 
     The next step in the process, step  112 , is to extract model parameters for each point, N, in the feedback impedance space generated at step  110 . To do so, using the example trail space shown in Table 1 and the circuit model  200  shown in FIG. 1, the values of Rs and Ls for each impedance point, N=1 to 30, is applied to the circuit model of FIG.  1  and values for Cgs, Rgs, Cgd, Rgd, Cds, Rds, Gm and Tau are extracted for each impedance point, N =1 to 30, using algorithms and techniques well known to one skilled in the art. The extractions are preferably achieved by utilizing Minasian extraction algorithms which are known in the art. A more detailed description of these algorithms and their use can be found in “Broad-Band Determination of the FET Small-Signal Equivalent Circuit” by M. Berroth et al. IEEE-MTT vol. 38, no. 7, July 1990. Example model parameters for one impedance point, N=17, Rs=1.7 ohms and Ls=0.0045 pH (Table 1) are shown in Table 2. Note that a similar table will be generated for each impedance value in the trial solution space. As such, for the trial space depicted by Table 1, thirty (30) tables of model parameter values will be generated. 
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Trial starting feedback impedance space point values. 
               
             
          
           
               
                 Impedance 
                 Resistance 
                 Inductance 
               
               
                 Point (N) 
                 (Rs) 
                 (Ls) 
               
               
                   
               
             
          
           
               
                 1 
                 0.1 Ω 
                 0.0045 pH 
               
               
                 2 
                 0.2 Ω 
                 0.0045 pH 
               
               
                 3 
                 0.3 Ω 
                 0.0045 pH 
               
               
                 4 
                 0.4 Ω 
                 0.0045 pH 
               
               
                 5 
                 0.5 Ω 
                 0.0045 pH 
               
               
                 6 
                 0.6 Ω 
                 0.0045 pH 
               
               
                 7 
                 0.7 Ω 
                 0.0045 pH 
               
               
                 8 
                 0.8 Ω 
                 0.0045 pH 
               
               
                 9 
                 0.9 Ω 
                 0.0045 pH 
               
               
                 10 
                 1.0 Ω 
                 0.0045 pH 
               
               
                 11 
                 1.1 Ω 
                 0.0045 pH 
               
               
                 12 
                 1.2 Ω 
                 0.0045 pH 
               
               
                 13 
                 1.3 Ω 
                 0.0045 pH 
               
               
                 14 
                 1.4 Ω 
                 0.0045 pH 
               
               
                 15 
                 1.5 Ω 
                 0.0045 pH 
               
               
                 16 
                 1.6 Ω 
                 0.0045 pH 
               
               
                 17 
                 1.7 Ω 
                 0.0045 pH 
               
               
                 18 
                 1.8 Ω 
                 0.0045 pH 
               
               
                 19 
                 1.9 Ω 
                 0.0045 pH 
               
               
                 20 
                 2.0 Ω 
                 0.0045 pH 
               
               
                 21 
                 2.1 Ω 
                 0.0045 pH 
               
               
                 22 
                 2.2 Ω 
                 0.0045 pH 
               
               
                 23 
                 2.3 Ω 
                 0.0045 pH 
               
               
                 24 
                 2.4 Ω 
                 0.0045 pH 
               
               
                 25 
                 2.5 Ω 
                 0.0045 pH 
               
               
                 26 
                 2.6 Ω 
                 0.0045 pH 
               
               
                 27 
                 2.7 Ω 
                 0.0045 pH 
               
               
                 28 
                 2.8 Ω 
                 0.0045 pH 
               
               
                 29 
                 2.9 Ω 
                 0.0045 pH 
               
               
                 30 
                 3.0 Ω 
                 0.0045 pH 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Initial “Intrinsic” equivalent circuit parameters. 
               
             
          
           
               
                   
                 Intrinsic Equivalent 
                   
               
               
                   
                 Circuit Parameter 
                 Initial Solution 
               
               
                   
                   
               
               
                   
                 Cgs 
                 0.23595 pF 
               
               
                   
                 Rgs 
                 91826 Ω 
               
               
                   
                 Cgd 
                 0.0177 pF 
               
               
                   
                 Rgd 
                 100000 Ω 
               
               
                   
                 Cds 
                 0.04045 pF 
               
               
                   
                 Rds 
                 142.66 Ω 
               
               
                   
                 Gm 
                 142.1025 mS 
               
               
                   
                 Tau 
                 0.1 pS 
               
               
                   
                   
               
             
          
         
       
     
     The next step in the process, step  118 , is to compare the measured S-parameters to the modeled S-parameters. To do so, an initial guess at the model solution is made for the initial parasitic values. This initial guess is an informed guess made by one skilled in the art. An example of an initial guess for the parasitic circuit parameters Rs, Lg, Rs, Ls, Rd and Rd, for model circuit  200  are shown in Table 3. The intrinsic and extrinsic values are applied to the circuit model and the S-parameters are calculated for the model circuit. For example, the intrinsic values shown in Tables 2 and 3 as well as the values for Rs and Ls for each impedance point, N shown in Table 1, is applied to the circuit model  200  and the S-parameters are calculated for each impedance point, N, preferably using a Minasian extraction algorithm mentioned above. 
     After the S-parameters are calculated for the model circuit, the next step in the process, step  118 , is to compare the calculated S-parameters to the measured S-parameters for each impedance point, N, where the measured S-parameters were stored in step  108 . FIGS. 5 and 6 are shown to visually illustrate the comparison of calculated and measured S-parameters for one of the impedance points (N=17, Rs=1.7 and Ls-0.0045). The measured values for the S 11 , S 12  and S 22  parameters are depicted by the curves identified by the squares, circles and inverted diamonds, respectively. And, the simulated or modeled values for S 11 , S 12  and S 22  are depicted by the lines marked by the triangles, diamonds and stars, respectively. It is noted that in actuality, the difference between the measured and modeled S-parameters would typically not be easily discernable on a Smith Chart or a linear plot due to the lack of fidelity of these types of plots. As such, a computer program is used to compare the difference between the measured and modeled S-parameter values. One typically measure of the closeness of the measured and modeled values is the best fit error which is easily calculable with the aid of a computer. A low best fit error indicates that the measured and modeled S-parameter values closely align. 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Initial “Parasitic” equivalent circuit parameters. 
               
             
          
           
               
                   
                 Intrinsic Equivalent 
                   
               
               
                   
                 Circuit Parameter 
                 Initial Solution 
               
               
                   
                   
               
               
                   
                 Rg 
                 3.0 Ω 
               
               
                   
                 Lg 
                 0.014 nH 
               
               
                   
                 Rs 
                 1.7 Ω 
               
               
                   
                 Ls 
                 0.0045 nH 
               
               
                   
                 Rd 
                 2.5 Ω 
               
               
                   
                 Ld 
                 0.024 nH 
               
               
                   
                   
               
             
          
         
       
     
     The next step in the process, step  122 , is to select a test parameter criteria and determine if that criteria has been met. The test parameter criteria can be selected amount of time or a selected number of runs. The selected criteria will be the same for each trial value. The preselected criteria is preferably a preselected number of cycles of the extraction-optimization routine indicated by the box marked  121 . In this manner, each feedback resistance value (1-N, where for the example of Table 1, N=30) is subjected to the same criteria. As such, the feedback resistance value which starts off closest to the unique solution will contain the least amount of error and converge faster and obtain a low best fit value quicker. For our example, the preselected criteria was completion of six (6) extraction-optimization cycles where one cycle is depicted by the subroutine indicated by reference number  121 . 
     The next step in the process, step  122 , is to optimize the model parameters. To do so, known optimization programs are used. For example, a commercially available computer program, Libra 3.5, produced by HP EEsof could be used as the optimizer program. The optimization is additionally performed per preselected restrictions. For example, the optimization program Libra 3.5 could be used with the restrictions shown in Table 4 for a fixed value of Rs. These constraints are in accordance with the overall modeling algorithm. By fixing the value of Rs, for example to a value of 1.7 ohms, the optimization of the model parameters is confined to creating a trial model solution for only the trial feedback impedance point, Rs. The optimized values for the equivalent circuit parameters are output from the optimization program. Tables 5 and 6 show example optimized equivalent circuit parameters for the circuit  200  (FIG.  2 ), are output from the Libra 3.5 computer program. As previously mentioned, Rs is a fixed value for the cycles. 
     
       
         
               
             
               
               
             
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 Environment used for competitive solution strategy, as implemented 
               
               
                 in this example. 
               
             
          
           
               
                   
                 Implementation Parameter 
               
               
                   
                   
               
             
          
           
               
                   
                 Circuit Simulator and Optimizer 
                 Libra 3.5 
               
               
                   
                 Optimization Algorithm 
                 Gradient 
               
               
                   
                 Optimization Error Metric 
                 Mag and angle of 
               
               
                   
                   
                 S11, S21, S12, and S22 
               
               
                   
                   
                 from 4 to 40 GHz 
               
               
                   
                 Number of Iterations 
                 60 
               
               
                   
                 Number of Extraction/Optimization 
                 6 
               
               
                   
                 Cycles 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 5 
               
             
             
               
                   
               
               
                 Optimized “Intrinsic” equivalent circuit parameters. 
               
             
          
           
               
                   
                 Intrinsic Equivalent 
                   
               
               
                   
                 Circuit Parameter 
                 Initial Solution 
               
               
                   
                   
               
               
                   
                 Cgs 
                 0.227785 pF 
               
               
                   
                 Rgs 
                 65247 Ω 
               
               
                   
                 Cgd 
                 0.017016 pF 
               
               
                   
                 Rgd 
                 130820 Ω 
               
               
                   
                 Cds 
                 0.047521 pF 
               
               
                   
                 Rds 
                 160.18 Ω 
               
               
                   
                 Gm 
                 135.74 mS 
               
               
                   
                 Tau 
                 0.446 pS 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 6 
               
             
             
               
                   
               
               
                 Optimized “Parasitic” equivalent circuit parameters. 
               
             
          
           
               
                   
                 Intrinsic Equivalent 
                   
               
               
                   
                 Circuit Parameter 
                 Initial Solution 
               
               
                   
                   
               
               
                   
                 Rg 
                 4.715 Ω 
               
               
                   
                 Lg 
                 0.02903 nH 
               
               
                   
                 Rs* 
                 1.7 Ω 
               
               
                   
                 Ls 
                 0.002102 nH 
               
               
                   
                 Rd 
                 3.2893 Ω 
               
               
                   
                 Ld 
                 0.0317 nH 
               
               
                   
                   
               
               
                   
                 *Remains fixed for each trial model solution for feedback impedance space.  
               
             
          
         
       
     
     These new optimized parameters are input into step  112  and the process is repeated. The steps  112 - 122  are repeated until the preselected criteria is met at step  120  for each trial solution point. With each repetition of the process, the modeled values more closely match with the measured values. For example, FIGS. 7 and 8 visually depict a comparison of the measured and simulated S-parameter values after six (6) extraction-optimization cycles  121  for the circuit model  200  of FIG.  2 . When FIGS. 7 and 8 are compared to FIGS. 4 and 5, respectively, it can be seen that the measured and modeled values converge with application of the process  100 . 
     The next step in the process is to form a new error metric, step  124 . To do so, the final error and convergence speed for each of the trial solution points are compared to determine which trial solution produced the lowest final error. For example, the graph of FIG. 9 shows the final fitting error for the circuit model  200  as a function of Rs, where Rs is plotted along the horizontal axis with the appropriate scaling factors for fitting the error indicated along the vertical axis. Not that a final fitting error is generated for each trail solution point, N. 
     The next step in the process is to select the model solution which resulted in the minimal error in the new error metric, step  126 . Since each trial solution point was subjected to the same preselected criteria, for example, six (6) cycles at step  120 , the implementation of the extraction-optimization cycles results in the best and the fastest solving model solution appear as the global minima, depicted by reference number  127 , for the final fitting errors of all trial impedance points. For the example depicted by the graph of FIG. 9, the minima  127  occurs for an Rs value of approximately 1.7 ohms. The minima  127  in final error indicates that the algorithm of the present invention converges to a unique solution as opposed to conventional algorithms which would produce near flat dependencies or local minima solutions which do not usually obey device scaling realities. Thus, with unique solutions for the feedback resistance, the rest of the equivalent circuit model  200  can be uniquely determined. 
     The next step in the process is to output the circuit parameters corresponding to the selected model solution. These circuit parameters were calculated at step  112  for the last cycle of process  121 . For example, Tables 7 and 8 depict the final model equivalent circuit parameters for this solution of Rs =1.7 ohms for the circuit model  200  (FIG.  2 ). By selecting those values which produced the minimal error in the error metric and fast convergence, step  126 , an accurate model of the FET device that is unique is obtained at step  128 . 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 7 
               
             
             
               
                   
               
               
                 Global solution for “Intrinsic” equivalent circuit parameters. 
               
             
          
           
               
                   
                 Intrinsic Equivalent 
                   
               
               
                   
                 Circuit Parameter 
                 Initial Solution 
               
               
                   
                   
               
               
                   
                 Cgs 
                 0.227745 pF 
               
               
                   
                 Rgs 
                 64242 Ω 
               
               
                   
                 Cgd 
                 0.017019 pF 
               
               
                   
                 Rgd 
                 133450 Ω 
               
               
                   
                 Cds 
                 0.047544 pF 
               
               
                   
                 Rds 
                 160.1791 Ω 
               
               
                   
                 Gm 
                 135.7568 mS 
               
               
                   
                 Tau 
                 0.443867 pS 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 8 
               
             
             
               
                   
               
               
                 Global solution “Parasitic” equivalent circuit parameters. 
               
             
          
           
               
                   
                 Intrinsic Equivalent 
                   
               
               
                   
                 Circuit Parameter 
                 Initial Solution 
               
               
                   
                   
               
               
                   
                 Rg 
                 4.711895 Ω 
               
               
                   
                 Lg 
                 0.029314 nH 
               
               
                   
                 Rs 
                 1.7 Ω 
               
               
                   
                 Ls 
                 0.002104 nH 
               
               
                   
                 Rd 
                 3.309899 Ω 
               
               
                   
                 Ld 
                 0.031671 nH 
               
               
                   
                   
               
             
          
         
       
     
     The extracted results are fully consistent with device periphery scaling as shown in FIGS. 10,  11 , and  12 , respectively. FIG. 10 is a plot of the source resistance as a function of gate width (W G ) in 2-finger, 4-finger and 8-finger FET configurations. For all configurations, the source resistance R S  follows roughly the dashed line  70  indicating a direct dependence with 1/W G  as would be expected. 
     FIG. 11 illustrates a plot of the gate source capacitance C GS  (indicated along the vertical axis) as a function of 1/W G . The extracted values for C GS  are plotted for both a 0.25 μm (indicated by the oval A) and a 0.12 μm GaAs pseudomorphic High Electron Mobility Transistor (HEMT, indicated by the oval B). The lines surrounding each device model (either A or B) are indicated from top to bottom as 2, 4, and 8-finger configurations with unit gate periphery indicated along the horizontal axis. As expected, C GS  follows a direct dependence with the total gate width periphery W G  for the extracted values. 
     Further verifications that the extracted parameter values obtained with the method of the present invention is illustrated in FIG. 12, which is a plot of the transconductance G M  as a function of total gate periphery W G . As before, the extracted values are shown for G M  scaled for both in a 0.12 mm (oval A) and a 0.25 mm GaAs pseudomorphic HEMT (oval B). As shown, the transconductance G M  follows a direct dependence with the total gate with W G  as expected. 
     What is also expected is that C GS  would follow closely with gate length W G  because the contact area between the gate and channel mostly determines C GS . For a GaAs pseudomorphic HEMT, R S  and, to a large extent G M , do not strongly follow gate length W G  because the recess dimensions are similar, material profiles are identical, and the effect of velocity overshoot is not so pronounced at a given bias. This is verified by the plot of FIG.  11 . The robustness and computational accuracy of the algorithm can also be verified using simulated sets of S-parameters generated from known model parameters. 
     With the improved accuracy of the parameter extraction method of the invention, circuit models can be extracted with physically significant parameters and subtle Physics-based phenomena in FETs can be observed from measured bias dependent S-parameters. In several HEMT model-scaling studies performed with our extraction, the expected gate metal resistance could be determined from well-known scaling rules. These results indicate that the extraction of the parameter R G  was not only accurate, but also physically significant. 
     FIG. 13 is a plot of the source resistance R s  versus the gate bias voltage V GS  plotted along the horizontal axis. In the bias dependent S-parameters of 0.12 μm GaAs pseudomorphic HEMTs, the extraction of both parasitic and intrinsic. equivalent circuit parameter values provide a close view of the physical structure of the device. The lateral extension of the space-charge region and the modulation of surface state density in the recess can be observed in the bias dependence of R S  vs. V GS . 
     The bias dependence of C GS  and C GD  is typical of HEMTs as confirmed by the reports of many researchers. This bias dependence is illustrated in the bias plot of FIG.  14 . At cold FET conditions, C GS  and C GD  are equal and then gradually C GS  increases while C GD  decreases as the FET is biased into its linear region. Typically, C GS  follows a hyperbolic tangent-like function in saturation with a sharp increase occurring near pinch-off voltage and then saturating to a peak value. Increasing C GS  as V GS  moves positive is indicative of charge spilling out of the channel and into the donor or Schottky Barrier layer, amounting to an effect sometimes called parasitic conduction. 
     Increasing C GS  as V DS  increases is indicative of so-called Drain-Induced-Barrier-Lowering (DIBL). DIBL often occurs in submicron FET devices because the short gate lengths of these devices allow drain potential to influence the channel potential well past the beginning of the gate. The inaccuracies associated with a constant parasitic model extraction can also be investigated. By using the extracted parameters as a control, the intrinsic model parameters can be re-extracted for a 4×200 μm  0.12 Φm GaAs pseudomorphic HEMTs, with the new R   S  and R D  set at cold-FET, on-state extracted values of  240  and 250 Ωμm, respectively. 
     Inaccuracies with bias and periphery scaling caused by this assumption are shown in FIGS. 15 and 16, respectively. In particular, FIG. 15 is a plot showing a percentage error in the gate-source capacitance C GS  as a function of the gate source voltage V GS . FIG. 16 illustrates percent error in the scaled gate-source capacitance C GS  versus scaled device periphery W G , as indicated along the horizontal axis. As can be seen, constant parasitic assumptions become more inaccurate when the FET is biased with V GS  approaching pinch-off and as V DS  decreases into the linear region. Also, small errors in the fitting of the equivalent circuit caused by assuming constant parasitics with bias are greatly magnified as the device model is scaled. Roughly, a 10% error/20 μm of device scaling occurs simply because the intrinsic model of the original device started with a small error. 
     While the invention has been described in connection with specific embodiments, it should be understood that variations and modifications of the invention will become apparent to those of ordinary skill in the art without departing from the true scope and spirit of the invention. It is therefore intended that such variations and modification be part of the present invention and encompassed by the following claims.