ISSN 10642269, Journal of Communications Technology and Electronics, 2015, Vol. 60, No. 8, pp. 928–935. © Pleiades Publishing, Inc., 2015. Original Russian Text © V.N. Biryukov, A.M. Pilipenko, I.V. Semernik, I.V. Shekhovtsova, 2015, published in Radiotekhnika i Elektronika, 2015, Vol. 60, No. 8, pp. 865–872.
PHYSICAL PROCESSES IN ELECTRON DEVICES
Diagnostics of Differential Parameters in Models of FieldEffect Transistors V. N. Biryukov, A. M. Pilipenko, I. V. Semernik, and I. V. Shekhovtsova Southern Federal University, per. Nekrasovskii 44, Taganrog, 347930 Russia email:
[email protected],
[email protected] Received January 12, 2011
Abstract—The feasibility of increasing the accuracy of simulation of differential parameters in the models of fieldeffect transistors and, therefore, of the frequency characteristics of radio circuits by separate diagnostics of static and differential parameters is considered. The feasibility of reaching the simulation error of the same order for both types of parameters is demonstrated. DOI: 10.1134/S1064226915070037
INTRODUCTION In programs for circuit analysis (electronic simula tors), the frequency characteristics are calculated in three stages. In the first stage, the circuit is analyzed at a DC current in order to determine the operating points of the nonlinear elements of the circuit: primarily, tran sistors. In the second stage, by numerical differentia tion, parameters of the elements of smallsignal equiva lent circuits of transistor—differential transconduc tance and output conductance—are determined. In the third stage, the analysis of the linearized circuit is per formed. The main source of error is in the mathematical model of a transistor. The error of finding the operating points is usually much smaller than the error of simula tion of differential parameters. In the standard case, when the parameters of the transistor model are extracted by minimizing the approximation error of the current–voltage characteristics (CVC), the error of determining the differential parameters can be inadmis sibly large. To exclude such a situation, it is suggested— in addition to the error of simulation of currents—to introduce into the objective function, minimization of which enables one to obtain the sought parameters, the error of simulation of the differential parameters [1]. This palliative measure (the error of finding the operat
ing points in this case increases) makes it possible to increase the accuracy of analysis of the frequency char acteristics. In the present work, we consider the feasibility of increasing the accuracy of simulating the frequency characteristics of circuits by more accurate determina tion of lowsignal parameters of transistors without loss of accuracy of finding the operating points. To this end, it is suggested to use the fact that operating points and lowsignal parameters are determined not simul taneously, which theoretically enables one either to use different models of devices in different stages or to determine the parameters of the same model by differ ent methods. 1. STATIC MODELS OF FIELDEFFECT TRANSISTORS In this work, we use three compact static models of a fieldeffect transistor (FET) described below, des tined for the analysis of electronic circuits in SPICE simulators. The model with a continuous second 2 on the boundary of the lowslope derivative ∂2I/∂V DS and steep parts of the CVC has the form [2]
⎧ ⎪0, for V GS ≤ VT 0; ⎪⎪ I = ⎨β(2V G − V DS )V DS , for V DS ≤ V E ; 2 ⎪ ⎪β(2V G − V E )V E 1 + a(V DS − V E ) + b(V DS − V E ) , for V DS > V E , 1 + c(V DS − V E ) ⎩⎪
where I is the drain current; VGS is the gate–source voltage; VDS is the drain–source voltage; VT0 is the threshold voltage; β is the specific steepness; λ is the
(1)
coefficient taking into account the modulation of the channel length by the drain voltage; VG = VGS – VT0 is the effective gate voltage, equal to the pinchoff volt
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age; VE = VG – ⎡ 1 + λ 2VG2 − 1⎤ λ is the boundary volt ⎣ ⎦ age between the steep and lowslope parts of the CVC; 2 2 c = 1 {λ(1 − d )[VG − (VG – V E ) ]}, a = c + λ, and b = cdλ are formal parameters determined from the conditions of continuity of the first, ∂I/∂VDS, and second, 2 ∂2I/∂V DS , derivatives at VDS = VE; and −1 ⎛ ⎞ d = ⎜ lim ∂I ∂V DS ⎟ ( ∂I ∂ V DS VDS =VE ) ⎝VDS →∞ ⎠ is the variation in the output conductance of the FET in the lowslope part of the CVC, assumed to be con stant d = 0.25. Model (1) has only three parameters: β, λ, and VT0. The parameter β can be expressed through the FET parameters: β = μС0W/(2L), where µ is the charge carrier mobility in the channel; С0 is the gate–channel capacitance per unit area; and W and L are the FET channel width and length, respectively. In [3], a fourparameter FET model was proposed, which is continuous with all derivatives and is described by one analytic expression for the steep and lowslope parts of the CVC:
β⎡ V ⎤ I = ⎢VGVN − (VG − VDS )VD + Vε2 ln N ⎥ VD ⎦ 2⎣ 2 2 2 2 (2) × λ ⎡ Vε + (VDS − VE ) + VDS − Vε + VE ⎤ ⎦ 2⎣ 2 2 2 2 + κ ⎡ Vε + (VDS − VE ) − VDS − Vε + VE ⎤ + 1 , ⎦ 2⎣ where κ is the parameter taking into account the reduction carrier mobility in the longitudinal electric
{
field in the channel, VN =
}
2 2 Vε + VG + VG; VD =
2 2 Vε + (VG − VDS ) + VG − VDS ; Vε = 0.25VG + Vε0; and Vε0 = 10 mV is a formal constant determining the length of the intermediate part of the CVC between the steep and lowslope parts, which serves for limiting the higherorder derivatives. According to the classification with respect of the order of continuity of derivatives, model (1) is С2con tinuous, and model (2) is С∞continuous. The use of С∞continuous models is recommended for increasing the accuracy of analysis of harmonic distortions [4]. Model (2) takes into account the dependence of the charge carrier mobility in the FET channel only on the longitudinal electric field. In [5], a fiveparameter С∞ FET model was proposed, making it possible to take into account the influence of both the longitudi nal and transverse fields on the charge carrier mobility:
1 + λ VDL 1 (3) , 1 + κVDE 1 + θVG where θ is a parameter taking into account the influence of the transverse field on the charge carrier mobility, I = β(2VG − VDE )VDE
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VDE = 0.5⎡VDS − Vε20 + (VDS − VS )2 + Vε20 + VS2 ⎤ ; VDL = ⎦ ⎣ 2 2 2 2 ⎡ 0.5 VDS + Vε1 + (VDS − 0.9VS ) – Vε1 + ( 0.9VS ) ⎤ ; ⎣ ⎦ and VS = 1 + 2κVG − 1 κ ; Vε1 = 10Vε0 is a formal constant serving for compensating the error of the approximate expression for the saturation voltage VS. The additional variable VDE was introduced only for the continuity of the original regional model [4]. The choice of the model is caused by the necessity of obtaining a sufficiently small error of the models and providing the feasibility of measuring the model parameters with a guaranteed accuracy. The simplest Shichman–Hodges model, applied for estimates in the highsignal mode [6], is practically inapplicable for the analysis of lowsignal characteristics, because it employs the linear approximation of the CVC in the working (slowsloped) part. The identification of models with more than seven–ten parameters is lim ited by the accuracy of methods of nonlinear program ming [7].
(
)
2. DIAGNOSTICS OF MODEL PARAMETERS The diagnostics (extraction) of parameters in the static model of the fieldeffect transistor, I (VGS , VDS ), from its experimental current–voltage characteristic by the leastsquare method employs the objective function N
2
M
2
⎡I (VGS,k ,VDS,k ) − I k ⎤ (4) ⎢ ⎥, Ik ⎦ k =1 ⎣ where {Ik, VGS, k , VDS, k}, k = 1, 2, …, N is the experi mental CVC of the FET in the tabulated form. In this work, we suggest to increase the accuracy of analysis of the frequency characteristics by assigning several sets of parameters to each model of the same electronic component. Different sets of parameters can be obtained at the stage of identification from the condition of the minimum of the corresponding objective function. The basic set of parameters was obtained from the condition of minimum of the func tion FI, and additional sets of parameters, by the min imization of the following objective functions: FI =
∑
⎡S(VGS,k ,VDS,k ) − S k ⎤ FS = ⎢ ⎥, Sk ⎦ k =1 ⎣
∑ M
2
(5)
⎡G(VGS,k ,VDS,k ) − G k ⎤ (6) FG = ⎢ ⎥, Gk ⎦ k =1 ⎣ where {Sk , VGS, k, VDS, k} and {Gk , VGS, k, VDS, k} are the measured values of the transconductance S and output conductance G, corresponding to the original tabu lated CVC {I k , VGS, k, VDS, k}, k = 1, 2, …, М, М < N (M is the number of points in the slowsloped part of the CVC; the steep part of the CVC was not considered).
∑
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Table 1. Results of MOSFET parameter identification Model
Objective function
β, μA/V2
VT0, V
λ, V–1
κ, V–1
θ, V–1
FI
20.01
– 2.120
0.2231
–
–
FS
20.62
– 2.174
0.1178
–
–
FG
16.79
– 2.102
0.3127
–
–
FI
22.95
– 2.186
0.05843
0.158
–
FS
22.81
– 2.219
0.04121
0.127
–
FG
21.71
– 2.168
0.04723
0.133
–
ptype transistor (1)
(2)
ntype transistor (1)
(3)
FI
36.33
0.8428
0.04272
–
–
FS
27.54
0.8149
0.2802
–
–
FG
34.64
0.5991
1.128
FI
88.27
1.251
0.04129
0.2237
0.06942
FS
96.18
1.597
0.04750
0.2293
0.08699
1.384
0.02236
0.1865
0.9893
FG
318.1
The extraction of parameters was later performed by a program of optimization by the Levenberg–Mar quardt method with additional control of accuracy by variation of the initial vector of variables. To prove the efficiency of the proposed algorithm for extraction of parameters, identification of parameters of a complementary pair of test metaloxidesemicon ductor field effect transistors (MOSFET) with the channel length L ≈ 0.8 μm and width W ≈ 20 μm was performed for the case when the substrate was con nected with the source. The results of identification of parameters and simulation errors of a pchannel MOSFET were partly presented in [8, 9]. Table 1 presents the results of parameter identifica tion, and Table 2, the error of simulation of p and nchannel MOSFETs. For both the transistors, at first, the diagnostics of the parameters of model (1) was per formed, because the parameters of this model are determined most quickly and can be used as the initial approximation in the parameters identification in more complex models. For the pchannel MOSFET, the table also present the results of parameter identifi cation by model (2), which is most accurate in the case when the charge carrier mobility weakly depends on the electric field, i.e., for relatively long channels and/or the hole conduction of FET’s channel. The fiveparameter model (3) is adequate when the charge carrier mobility strongly depends on the electric field; it is used below for the parameter identification of an nchannel MOSFET. For obtaining more complete information on the accuracy of the model, for each set of parameters, two types of error were calculated: (1) σI, σS, and σG—the meansquare CVC simulation error, the transconduc
–
–
tance, and the output conductance, respectively; (2) δImax, δSmax, and δGmax —the maximum relative errors of simulation of the same characteristics. It follows from Table 2 that the use of an additional set of parameters for model (2) made it possible to reduce by the factor of 2.2 the meansquare error of approximating the transconductance of the pchannel MOSFET and, by the factor of 1.5, the error of approximation of the conductance. The use of addi tional sets of parameters for the nchannel FET made it possible to increase by 2–3 fold the accuracy of sim ulating its differential characteristics. 3. ANALYSIS OF THE FREQUENCY CHARACTERISTICS OF AMPLIFIERS Let us consider the practical application of the pro posed algorithm for the analysis of RF circuits. As test circuits, let us choose a broadband amplifier whose circuit diagram is presented in [10] and a resonance amplifier described in [11]. Figure 1 shows the circuit of the amplifiers; in each of them, the active element is an nchannel MOSFET (VT1). In the case when a pchannel MOSFET is employed, the only things changing in the circuits are the polarity of the feed voltage EDD and the gate bias circuit, which practically do not affect the frequency characteristics of the amplifiers. The parameters of the passive elements in the broadband amplifier are as follows (Fig. 1a): С1 = С2 = 10 μF; R1 = 99 kΩ, R2 = 20 kΩ, RD = 17.5 kΩ, RS = 2 kΩ, and RL = 175 kΩ. The parameters of the passive elements in the resonance amplifier are as fol lows (Fig. 1b): С1 = 0.01 μF, С2 = 0.1 μF, С = 40.5 pF,
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Table 2. MOSFET simulation error Model
(1)
(2)
(1)
(3)
Objective function
σI, %
FI FS FG FI FS FG
0.706 1.259 1.624 0.238 0.711 0.370
FI FS FG FI FS FG
2.66 3.58 11.5 0.461 3.59 4.23
σS, %
σG, %
ptype transistor 1.311 3.056 1.123 – – 2.427 0.4251 2.147 0.1949 – – 1.451 ntype transistor 5.62 24.7 4.06 – – 7.86 2.16 10.2 0.852 – – 3.88
δI max, %
δS max, %
δG max, %
11 29 24 4.7 17 6.6
26 23 – 12 4.0 –
64 – 52 55 – 32
130 63 – 43 27 –
43 51 93 18 66 110
670 – 110 360 – 100
Remark. Bold font emphasizes the errors of simulation of each MOSFET characteristic obtained in the optimal choice of objective func tions (5) and (6).
L = 1 μH, QL = 100, R1 = 100 kΩ, R2 = 50 kΩ. The gate bias circuit corresponds to one of the CVC’s points. The equivalent circuit of the MOSFET (dynamic model) is shown in Fig. 2. The dynamic model of the MOSFET has the following parameters [10]: Сgd = Сbd = Сbs = Сbg = 20 fF, Сgs = 12.9 fF, rd = rs = 1 Ω, I S bs = 10 −16 А, I bs = I S bs[exp (Vbs ϕT ) − 1], I bd = I S bd ×
−14 А, and ϕT = [exp(Vbd ϕT ) − 1], I S bd = 6.95 × 10 27 mV. Below, the controlled current source I(VGS, VDS) entering into the circuit diagram in Fig. 2 corresponds to one of the static models (1)–(3).
In the analysis of amplifier’s frequency characteris tics, the nonlinear models (1)–(3) describing the con trolled source is replaced by the linearized (lowsig
(a)
(b) + EDD
C2
L
C
+ EDD
RD
R1
VT1 C1
C2 d
R2
VT1
b
g
s V1
R1
RS
C1 RL
V2 V1
g
d b s
R2
Fig. 1. Circuit diagram of a (a) broadband and (b) resonance amplifier. JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 60
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calculated at the points corresponding to the low sloped part of the tabulated CVC {Ik, VGS, k, VDS, k}, because this part is the operating region for amplifiers. A detailed quantitative characteristic of the accuracy of analysis of the frequency characteristics of the p and nchannel MOSFETs can be represented by histograms of the errors δK and δΠ shown in Figs. 3 and 4. The ordi nate axis is the probability that the error will fall within a certain interval (the probability PK corresponds to the error δK, and the probability PΠ corresponds to the error δΠ). The calculated values of the differential parameters for the pchannel transistor were deter mined via model (2) and, for the nchannel transistor, via model (3). The histograms denoted by 1 were obtained when S and G were calculated with the use of additional model parameters, which were determined from the condition of minimum of the functionals FS and FG, respectively. The histograms denoted by 2 were obtained when S and G were calculated with the use of the basic set of parameters, determined from the con dition of minimum of the functional FI. The error distribution shows that, if we use addi tional sets of parameters of the MOSFET model, the probabilities of small errors δK and δΠ increase and the maximum errors decrease, which is especially pro nounced for circuits with a nchannel MOSFET. For example, from Figs. 4a and 4b it follows that, in the analysis of a broadband amplifier with an nchannel MOSFET, the maximum values of δΚ and δΠ calcu lated with additional sets of parameters decrease by the factors of 3 and 4, respectively. Since the parameter diagnostics was performed by the leastsquare method, it is expedient to present also the data on the rootmeansquare (rms) error of the fre quency characteristics. Table 3 presents the rms errors
d
Cgd
rd
Cbd
VD1
Ibd g
I(VGS , VDS ) Ibs
b
VD2
Cbs
Cgs Cbg
rs s
Fig. 2. Equivalent circuit of MOSFET.
nal) model. In this case, the complex value of the cur rent has the form ID = SVGS + GVDS , where the differential parameters S = ∂I ∂VGS and G = ∂I ∂V DS can be obtained in the explicit form from (1)–(3). For estimating the accuracy of the analysis of ampli fier’s frequency characteristics, the relative errors of the gain factor δ K ,k = K 1,k – K 2,k K 1,k and the passband δ Π,k = Π1,k – Π 2,k Π1,k , were calculated, where K1, k and Π1, k are the gain factor and the passband obtained with the measured values Sk and Gk; K2, k and Π2, k are the gain factor and the passband width obtained with the use of the calculated values S(VGS,k , VDS,k ) and G(VGS,k , V DS,k ). The relative errors δK, k and δΠ, k were
of the gain factor σK =
∑
∑
M k =1
δ 2K ,k M and passband
M
σΠ = δ 2 M . It follows from Table 3 that the k =1 Π,k use of additional sets of parameters substantially increases the accuracy of analysis of the frequency char acteristics. For example, for a broadband amplifier with
Table 3. Simulation error of amplifiers’ frequency characteristics Model
(1) (2)
(1) (3)
Objective function
Broadband amplifier σK, %
FI F S, F G FI F S, F G
1.93 1.21 1.02 0.65
FI F S, F G FI F S, F G
5.00 3.19 3.22 0.89
σΠ, % ptype transistor 1.44 1.02 1.05 0.78 ntype transistor 19.0 2.99 14.6 1.31
Resonance amplifier σK, %
σΠ, %
2.86 1.77 1.35 0.71
1.21 1.05 1.01 0.65
10.1 5.04 6.11 2.65
27.9 2.99 11.0 1.98
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PK 0.8
(b)
PΠ 0.8
1
1
2 0.6
0.6
0.4
0.4
0.2
0.2
0
933
4
8
12 δK, %
16
20
24
(c)
PK 0.8
2
0
4
8
12 δΠ, %
16
20
24
16
20
24
(d)
PΠ 0.8
1
1
2 0.6
0.6
2
0.4
0.4
0.2
0.2
0
4
8
12 δK, %
16
20
24
0
4
8
12 δΠ, %
Fig. 3. Histograms of the (a, c) gain factor and (b, d) passband estimation errors for the (a, b) broadband and (c, d) resonance p channel MOSFET amplifiers, obtained (1) by the method proposed in the present work and (2) by the currently employed method.
an nchannel MOSFET, the rms error of the gain factor calculated using the suggested approach decreases by the factor of 3–4 and the corresponding error of the bassband width decreases by the factor of 10–15. 4. DISCUSSION OF RESULTS From the general consideration it follows that sys tematic errors of approximation of an experimental function and its derivative for a particular model, in the general case, are different. Therefore, by changing the objective function one can increase the accuracy of simulation of differential parameters. The results of this work fully confirm this conclusion and enable one to estimate the gain obtained in this case. Changing the objective function reduces not only the rms error but also the maximum relative error, which suggests that the observed effect is not caused by the choice of
the norm of the error. Our results for other MOSFET models do not differ qualitatively from the results pre sented above. It should be noted that the practical application of the separate parameter diagnostics requires additional testing of the conditionality of the model, because a model wellconditioned for the basic characteristic of an element can be illconditioned for its derivative [12]. For the analysis in this work we used relatively sim ple FET models. Naturally, it is of interest to find out how efficient the method under consideration is in the case of more complex models: primarily, the BSIM models. Unfortunately, in addition to measured elec tric parameters, these models involve design and engi neering parameters and, which is most important, the model parameters are determined from measurements not uniquely, because the standard optimization by extraction is performed over groups of parameters
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PK 0.8
(b)
PΠ 0.8
1
1 0.6
0.6
0.4
0.4
2
0.2
2
0.2
8
0
16
24 δK, %
32
40
48
(c)
PK 0.3
32
48 δΠ, %
64
80
96
64
80
96
(d)
PΠ 0.6
1 0.2
16
0
1 0.4
2
2
0.2
0.1
0
8
16
24 δK, %
32
40
48
0
16
32
48 δΠ, %
Fig. 4. Histograms of the (a, c) gain factor and (b, d) passband width estimation errors for the (a, b) broadband and (c, d) reso nance nchannel MOSFET amplifiers, obtained (1) by the method proposed in the present work and (2) by the currently employed method.
[13]. Nevertheless, we may assume that, since the reduction in the error of determination of the differen tial parameters on separate diagnostics of these parameters weakly depends on the accuracy of the models employed, the results obtained by models more accurate than (1)–(3) will be close to the results obtained in this work. However, the parametric identi fication of the models from measured differential characteristics of transistors when the number of parameters is more than ten requires developing a sep arate method of diagnostics. It should be noted that models (2) and (3) with the MOSFET channel length about 1 µm make it possible to obtain the rms error on the same order of magnitude as the error of the BSIM3 model [14]. The later version of the BSIM models, developed for technologies with an especially short channel, do not guarantee improvement of the accuracy of simulation for transistors with a long channel [1].
CONCLUSIONS It has been shown that diagnostics of parameters of known models with respect to measured differential parameters of simulated devices makes it possible to improve the accuracy of analysis of the frequency characteristics of electronic circuits by assigning sev eral sets of parameters to each model of the same ele ment—separately for each static and differential char acteristic. The resulting reduction in the error of dif ferential parameters is essential for practical purposes. The improvement of accuracy is observed not only for one particular model or one type of fieldeffect tran sistor but for all studied models and transistors. This fact suggests that the method can also be applied with other compact models and other types of transistor with various channel sizes. The parameter identifica tion procedure becomes slightly longer but, as it is per
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formed only once, the total time of the circuit analysis remains practically the same. ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research, project no. 100700162, and a Scholarship of the President of the Russian Federa tion for young scientists and postgraduate students no. SP398.2012.5. REFERENCES 1. Y. P. Tsividis and K. Suyama, IEEE J. Sol.St. Circuits 29 (3), 210 (1994). 2. V. N. Biryukov and A. M. Pilipenko, Izv. Vyssh. Uchebn. Zaved. Elektron. 8 (6), 22 (2003). 3. V. N. Biryukov, Izv. Vyssh. Uchebn. Zaved. Elektron. 13 (4), 69 (2008). 4. C. C. McAndrew, IEEE J. Sol.St. Circuits 33 (3), 439 (1998). 5. A. M. Pilipenko and V. N. Biryukov, Usp. Sovremen. Radioelektron. No. 9, 66 (2011). 6. Q. Huang, in Proc. 1997 IEEE Int. Symp. on Circuits and Systems (ISCAS'97), HongKong, June 9–12, 1997 (IEEE, New York, 1997), Vol. 2, p. 1401.
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7. V. V. Denisenko, Compact Models of MOPTransistors (MOSFET) for SPICE in Micro and Nanoelectronics (Fizmatlit, Moscow, 2010) [in Russian]. 8. I. V. Semernik and I. V. Shekhovtsova, in Proc. 10th All Russia Sci. Conf. on Engineering Cybernetics, Radioelec tronics and Control Systems, Taganrog, 2010 (Techn. Inst. Yuzhn. Federal. Univ., Taganrog, 2010), Vol. 1, p. 12. 9. V. N. Biryukov, Izv. Vyssh. Uchebn. Zaved. Elektron. 15 (5), 22 (2010). 10. E. Gad, M. Nakhla, R. Achar, and Y. Zhou, IEEE Trans. Comput.Aided Des. Integr. Circ. Syst. 28, 1359 (2009). 11. P. Horowitz and W. Hill, The Art of Electronics (Cam bridge Univ. Press, Cambridge, 1989; Mir, Moscow, 1993), Vol. 3. 12. V. N. Biryukov, J. Commun. Technol. Electron. 54, 1087 (2009). 13. D. P. Foty, MOSFET Modeling with SPICE: Principles and Practice (Prentice Hall, Upper Saddle River, 1997). 14. M. Kondo, H. Onodera, and K. Tamaru, IEEE Trans. Comput.Aided Des. Integr. Circ. Syst. 17 (5), 400 (1998).
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