Solid-State Electronics 51 (2007) 683–690 www.elsevier.com/locate/sse
Optimized parameter extraction using fuzzy logic Rodrigo Picos
a,*
, Oscar Calvo a, Benjamı´n In˜iguez b, Eugeni Garcı´a-Moreno a, Rodolfo Garcı´a b, Magali Estrada c a
University of Balearic Islands, 07122 Palma, Balears, Spain b Universitat Rovira i Virgili, Tarragona, Spain c CINVESTAV, Mexico DF, Mexico
Received 7 July 2006; received in revised form 14 February 2007; accepted 19 February 2007 Available online 16 April 2007
The review of this paper was arranged by Prof. S. Cristoloveanu
Abstract Precise extraction of transistor model parameters is of much importance for modeling and at the same time a difficult and time consuming task. Methods for parameter extraction can rely on purely mathematical basis, calling for intensive use of computational resources, or in human expertise to interpret results. In this work, we propose a method for parameter extraction based on fuzzy logic that includes a precise knowledge about the function of each parameter in the model to create a set of simple fitting rules that are easy to describe in human language. To simplify the computational effort, the parameter fitting rules work using only data at specific points (e.g. the distance between the calculated curve and the measured one at VDS corresponding to 50% of the maximum current). If necessary, a more accurate implementation can be used without altering the basic underlying philosophy of the method. In this work, the method is applied to extract model parameters required by Level 3 bulk MOS model and by a compact model for TFTs used in the Unified Model and Extraction Method (UMEM), which is based on an integral function. Results obtained show that the method is quite insensitive to the initial conditions and that it is also quite fast. Extension of this method for more complex models requires only the creation of the corresponding rule base, using the appropriate measurements. The method is especially useful for production testing or design. 2007 Elsevier Ltd. All rights reserved. Keywords: Thin film transistors; Fuzzy control; Parameter extraction; Compact modeling
1. Introduction Extraction of device model parameters from measured I–V characteristics is a complex task in modern models. It may involve the determination of hundreds of parameters, some of them correlated, requiring global optimization methods and human expertise. One way to simplify this task is to use direct extraction methods for these parameters, or at least for some of them. This last approach eases the entire extraction procedure in *
Corresponding author. Tel.: +34 971173227; fax: +34 971172486. E-mail address:
[email protected] (R. Picos).
0038-1101/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2007.02.031
the case of models with large number of parameters, reducing the iteration time in case of optimization when these values are to be used as starting data. As an example, we can mention the extraction of the threshold voltage [1–5] or the saturation voltage [6–8], that can be considered fundamental. Once the parameters have been extracted, most of the direct extraction methods need a second step to take into account the interactions among the different parameters. This leads to the use of global methods (SaPOSM [9], Fast Diffusion [10], Genetic algorithms [11], etc) to find the set of values that can best fit the experimental data. SaPOSM and Fast Diffusion are based on calculating derivatives and
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are thus computationally expensive and difficult to code. Genetic algorithms, on the other hand are easy to code, but they can present lack of precision. Moreover, all these methods are based only on mathematics and, thus, they do not use of the knowledge a human expert would deploy to do this task. Fuzzy logic is a widely used technique in control and identification applications. Its main advantages over other techniques can be synthesized as follows: it does not require a precise description of the behavior of the system. For example, it only needs to know that increasing the value of the threshold voltage implies that the ids–vgs curve shifts to the right and vice versa. On the contrary, in other methods such as SaPOSM, it is necessary to know the precise behavior of IDS and VTH in order to be able to calculate the derivatives that are the core of the process. In this paper, we present a method to extract the different parameters of transistor models based on fuzzy logic. To show its versatility, the method is demonstrated for determining parameters required by simplified Level 3 MOSFET model and also for the first TFT compact model and extraction procedure called the Unified Modeling and Extraction Method (UMEM), applied to organic TFTs in [12]. UMEM, as any direct extraction method needs some human know-how to be performed successfully. In this paper, we apply fuzzy logic techniques to include expert’s knowledge in a systematic way in the process. The procedure is extremely rapid without loosing precision and repeatability, paving the road to industrial use of these devices. The paper is organized in four sections. Section 2 reviews the basics of fuzzy control and fuzzy logic. Section 3 applies the method to extract Level 3 parameters for a bulk MOSFET. In Section 4, the method is used with the UMEM TFT model and, finally, Section 5 details the main conclusions of the paper. 2. Fundamentals of fuzzy logic Parameter extraction procedures can be considered a control problem where an optimum operation point has to be determined to achieve a minimum error between model and measured data. For this reason, we can use control procedures to perform parameter extraction. Among control techniques, one that is especially easy to use is the one based on verbal rules that control the behavior of the system, which is called fuzzy control [13–15]. It is based on the theory of fuzzy sets and fuzzy logic [13]. This technique is based on the principle that imprecise data can be classified into sets having fuzzy rather than sharp boundaries, which can be manipulated to provide a framework for approximate reasoning in the presence of imprecise and uncertain information. For instance, given a datum, x, a fuzzy set A is said to contain x with a degree of membership lA(x), where lA(x) can take any real value in the domain [0,1].
Fuzzy sets are often given descriptive names (called linguistic variables) such as POSITIVE; the membership function lPOSITIVE(x) is then used to reflect the similarity between values of x and the contextual meaning of POSITIVE. For example, if x represents the difference between the experimental and calculated currents in the IDS VGS curves at a given value of VDS, and POSITIVE is to be used to determine the fitting, then POSITIVE might have a membership function equal to zero for values below 0 and equal to one for values above 10%, with a curve joining these two extremes for intermediate values. The truth degree of the statement ‘‘the two curves are separated by a positive distance’’ is then evaluated by reading off the value of the membership function corresponding to the distance. Logical operations on fuzzy sets require an extension of the rules of classical logic. The three fundamental Boolean logic operations (intersection, union, and complement) have fuzzy counterparts defined by extension of the Boolean logic rules. A fuzzy expert system uses a set of membership functions and fuzzy logic rules to reason about data. The rules are of the form ‘‘if x is POSITIVE and y is NEGATIVE then z is MEDIUM,’’ where x and y are input variables, z is an output variable, and POSITIVE, MEDIUM, and NEGATIVE are linguistic variables. The set of rules in a fuzzy expert system is known as the rule base, and together with the database of input and output membership functions it comprises the knowledge base of the system. A fuzzy expert system works in four steps. The first step is fuzzification, during which the membership functions defined over the so called universe of discourse (the expected range of variation) of the input variables are applied to their actual values, to determine the degree of truth for each rule. Next is inference, during which the truth-value for the premise of each rule is computed and applied to the conclusion part of each rule. This results in one fuzzy set to be assigned to each output variable for each rule. The third step is composition in which all of the fuzzy sets assigned to each output variable are combined together to form a single fuzzy set for each output variable. Finally comes defuzzification, which converts the fuzzy output set to a crisp (non-fuzzy) number. A fuzzy logic controller may then be implemented as a system performing fuzzy operations on fuzzy sets represented by linguistic variables in a qualitative set of control rules (see Fig. 1). A fuzzy logic controller (FLC) is just a special controller that is used to modify the dynamics of a closed-loop system based on heuristic rules. It elaborates a control law from a set of rules that mimic the reactions of a human expert to various situations, mainly, when the system to be controlled is vaguely defined, is very complex and nonlinear, or when its dynamics is unknown and the sensors provide noisy and incomplete data. In our case, these rules will be provided by the knowledge about the transistor model, which is, obviously, a complex and nonlinear system.
R. Picos et al. / Solid-State Electronics 51 (2007) 683–690
V gtx ¼ f ðV gs ; V t ; d1 Þ
Knowledge Base
Experimental Curves
Data Base
Fitting
Fuzzifier
Calculated Curves
where function f is defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 f ðx; y; dÞ ¼ x y þ d ðx y dÞ þ 4dx 2
Rule Base
Inference Kernel
Model
685
ð4Þ
ð5Þ
In strong inversion Vgtx ! Vgs–Vt, but as Vgs goes below Vt, Vgtx goes smoothly to zero. The parameter d1 is an empirical smoothing parameter used to describe the rate at which Vgsx goes to Vgs or zero. The expression for Vdsx is similar:
Defuzzifier
New Parameter Set
Fig. 1. Fuzzy logic controller block diagram.
V dsx ¼ V Dsat f ðV Dsat ; V ds ; d2 Þ
The diagram shown in Fig. 1 corresponds to a singleinput–single-output (SISO) controller. The input variable, usually an analog signal, must be sampled and converted to a discrete signal for its further processing. The main elements of the FLC are a fuzzification unit, an inference engine, a knowledge base, and the defuzzification unit. Defuzzification implies a mapping from a fuzzy space into a crisp value. No single optimal strategy exists. There are several methods, and the matter is still a subject of research. Nevertheless, only a few methods cover most of the practical cases. They are the center of area (COA), the center of gravity (COG), center of largest area (COLA), maximum (MAX), and mean of maximum (MOM). In our case, we have selected the COA method, though tests done with other methods provide basically the same results. 3. Example of application I: Bulk MOSFET 3.1. Model description A bulk MOSFET model is used as a first example of application. This model is a modification of the SPICE Level 3, similar to the Power-lane model [16] that introduces smoothing functions for the various transition regions in the operation of the device. This allows the use of the following single equation to model the drain current for all regions of device operation. h i b a Id ¼ ðV gs V t ÞV ds V 2ds 1 þ hðV gs V t Þ 2
ð6Þ
where f is also defined by (4). For low values of Vds, Vdsx ! Vds, but as the device enters in saturation Vdsx ! VDsat. Parameter d2 has a similar role as d1 in (3). Parameter VDsat in (1) is also replaced by Vdsx to remove the channel length modulation term in the triode region. With these changes the drain current expression becomes h i b a Id ¼ V gtx V dsx V 2dsx ½1 þ kðV ds V dsx Þ ð7Þ 1 þ hV gtx 2 This is a very simple expression that is useful for our purposes, although it is not expected to provide good accuracy in weak inversion. In addition, we have considered the bulk impact ionization current, following the classical expression: B I bulk ¼ I d A ðV ds V dsx Þ exp ð8Þ V ds V dsx Finally, the total number of parameters to be extracted is ten: Vt, b, a, h, k, C, /B, A, and B. 3.2. Parameter extraction Parameters b, Vt and h are determined by fitting the experimental IDS VGS data in strong inversion for VDS = 50 mV and VBS = 0 V. To define the extraction rules, we need some knowledge about the how each of these parameters affects the behavior of the Ids Vgs curve.
ð1Þ
1. VT shifts the whole Ids Vgs curve. 2. b provides a scale factor. 3. h describes the curvature of IDS VGS as VGS increases.
ð2Þ ð3Þ
This way, we can define the following rules for the IDS VGS curve:
The model parameters are: Vt (threshold voltage), b (gain factor), a (saturation factor), h (mobility reduction factor), C (bulk factor), 2/B (strong inversion surface potential), and k (channel length modulation factor). No dependence on the device size is included. To extend the above equation to subthreshold and saturation regions, the gate and drain bias are replaced with two modified expressions, Vgtx and Vdsx:
1. If the calculated curve IDS VGS is to the right of the measured data, then decrement Vt, and vice versa. This rule has been implemented by measuring the distance (in V) between the experimental and the calculated curves at IDS equal to 10% of its maximum measured value. 2. If the calculated curve IDS VGS is over the measured data, then decrease b and vice versa. It has been
½1 þ kðV ds V ‘ Þ where V gs V t a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi V t ¼ V t0 þ Cð j2/B j V bs j2/B jÞ
V Dsat ¼
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implemented by measuring the distance (in V) as in the previous case, at IDS is equal to 50% of its maximum value. 3. If the calculated curve IDS VGS is more curved than the measured data, then decrease h, and vice versa. It has been implemented by measuring the distance (in V) as in the previous case, at IDS set equal to 90% of its maximum value.
parameters involved. More complex models would require using more curves. In any case, as before, we first have to know the general rules of behavior of the parameters:
In order to implement the above rules, we have used the fuzzy sets shown in Fig. 2, while defuzzification is done through the method of center of area (COA), as said before. To use them, we have first to normalize the input. This is done by using the measured curves as reference. The controller will provide us a response among 1 and 1. This response is then used as the relative change between the old value of the controlled parameter and the new value. When the threshold voltage for VBS = 0 V is found, we can then extract C and /B using different values of VBS. We know that the effect of these two parameters is to control the spacing among the IDS VGS curves when the bulk voltage is varied. We can describe the extraction rules as:
1. If the IDS VDS calculated curve saturates too soon, decrease alpha, and vice versa. This can be done by comparing the values of IDS for the voltage corresponding to 50% of the maximum experimental current. 2. If the slope of the calculated IDS VDS curve is too small, then increase k, and vice versa. This can be done by comparing the values of IDS at maximum VDS and VGS.
1. If the IDS VGS calculated curve is to the left of the corresponding experimental curve for the minimum value of VBS, then increase C, and vice versa. This can be done by comparing the values of IDS for the voltage corresponding to 10% of the maximum experimental current. 2. If the IDS VGS calculated curve for an intermediate value of VBS is to the left of the corresponding measurements, then increase /b, and vice versa. This can be done by comparing the values of IDS for the voltage corresponding to a 10% of the maximum experimental current. Once the above parameters have been found after an iterative process, we can extract the two parameters related to the saturation region (a and k) from the IDS VDS curve for a given VGS. In the present case, we are using only the curve corresponding to the maximum value of the gate voltage (VGS = 3.3 V), because of the low number of
1 Right
Too negative
-0.5
Too positive
0
0.5
Fig. 2. Fuzzification functions.
1. a establishes the relation between VGS and VDsat. 2. k controls the slope of the IDS VDS curve in saturation. Then, we can define the following extraction rules:
Finally, the only remaining parameters to determine are A and ld, corresponding to the bulk current. Their effect can be measured in both the drain and bulk current. In order to get a more explicative example, we will take into account both the drain-source current and the bulk current, for the same voltage sweeps as before. The effect of the above parameters can be described as: 1. A provides a scaling factor. 2. B describes the dumping of the bulk current with VDS. Then, we can define the following extraction rules: 1. If the Ibulk VDS calculated curve is higher than the measured one, decrease A, and vice versa. This can be done by comparing the values of Ibulk for the voltage corresponding to the maximum experimental current, and for the maximum VGS. 2. If the Ibulk VDS calculated curve rises faster than the measured one, decrease B, and vice versa. This can be done by comparing the values of Ibulk for the voltage corresponding to 50% of the maximum experimental current for an intermediate value of VGS. Once all the parameters have been estimated, the method will then iterate from an initial point (a set of parameters) until a stopping condition is found. In each iteration a new value for the parameters will be calculated by the fuzzy controller. The global rms error between the measured and the calculated curves, considering all points on the curve, will be updated. In our case, we have used a stopping condition of global rms error less than a 1%. The method has been tested by fitting the model presented before to experimental measurements of a W = 4 lm and L = 0.6 lm transistor, made using a 0.35 lm technology. Figs. 3 and 4 show an excellent agreement between modeled and experimental curves. Table 1a and b show that variation of the initial values have little influence on the final result. Processing time is
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also shown in these tables, and it is around 25–30 s using a 2.0 GHz PC, with 512 MB RAM running W2K.
2.5 2.0
IDS (uA)
4. Application example II: Organic TFT model 1.5
4.1. Model description 1.0 0.5 0.0 0.0
1.0
2.0
3.0
4.0
VGS (V) Fig. 3. Experimental () and calculated (line) IDS VGS curves for a Bulk MOSFET.
The fuzzy parameter extraction technique has been also applied to an organic TFT model [12] to determine its parameters, and then they are compared with those calculated in [12] through the integral function and a multi-step extraction procedure. In this model, the mobility dependence with gate voltage is described as: c V GS V T a c lFET ¼ l0 ¼ lFET0 ðV GS V T Þ a ð9Þ V aa where ca is a fitting parameter, VT is the threshold voltage and l0 is the value of band mobility for the material used in the TFT. However, since this last parameter is usually estimated, a parameter Vaa is introduced to adjust lFET to the experimental value of the low field mobility of the device being modeled, after ca is extracted. Drain current in the linear and saturation regions in the above threshold regime is modeled as: lFET ðV GS V T Þ 1 þ K R lFET ðV GS V T Þ V DS ð1 þ kV DS Þ m 1=m þ I 0 V DS 1 þ V Dsat
I DS ¼ K
Fig. 4. Experimental () and calculated (lines) IDS VGS curves for a Bulk MOSFET.
Table 1 Extracted parameters for initial point: (a) 2 V and (b) 5 V Parameter
Initial value
Extracted value
In the above equation, R is the source plus drain resistance, I0 is the leakage current and m and k are fitting parameters related to the sharpness of the knee region between linear and saturation and to channel length modulation, respectively. In addition, K is:
Direct method
K¼
For 2 V VT b H k a C /b A B Time
2V 0.15 0.1 0.01 1 1 V1/2 2V 3.0 V1 30 V 26 s
0.6459 V 21.2 lA/V2 0.1298 V1 2.95 · 1010 V1 1.183 0.65 V1/2 0.40 V 0.606 V1 14.2 V
For 5 V VT b H k a C /b A B Time
5V 0.001 0.01 1 · 1010 0.1 4 V1/2 1V 1.5 V1 1.0 V 29 s
0.6457 V 21.1 lA/V2 0.1295 V1 2.95 · 1010 V1 1.181 0.66 V1/2 0.40 V 0.604 V1 14.1 V
ð10Þ
0.6397 V 21.0 lA/V2 0.1260 V1 0 1.17 0.65 V1/2 0.40 V 0.61 V1 14.1 V
0.6397 V 21.0 lA/V2 0.1260 V1 0 1.17 0.65 V1/2 0.40 V 0.61 V1 14.1
W C diel L
ð11Þ
where W is the channel width, L is its length and Cdiel is the gate capacitance. In addition, The saturation voltage is defined using the parameter aS as: V Dsat ¼ aS ðV GS V T Þ
ð12Þ
If a non-ohmic contact is present at drain and source, the external bias Vext will fall part on this non-ohmic contact (diode) and part on the transistor itself (VDS), including its series resistance. This can be modeled as: V DSext ¼ V DS þ V diode
I DS kT I DS log þn ¼ q GðV GS ; V DS Þ I d0 I DS kT I DS log þn ¼ q GðV GS ; V DSext Þ n I d0
ð13Þ
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where function G has been defined as: GðV GS ; V DSext Þ ¼
K lFET ðV GS Þ ðV GS V T Þ ð1 þ k V DSext Þ m 1=m ð1 þ K R lFET ðV GS Þ ðV GS V T ÞÞ 1 þ VVDSext Dsat
ð14Þ
and n is a fitting parameter that takes into account the real voltage drop across the transistor when the diode resistance is important. 4.2. Parameter extraction For simplicity we will analyze devices with ohmic contacts, so we only need to extract six different parameters: aS (saturation effect), Vaa (mobility fitting parameter), VT (threshold voltage), k (channel length modulation), ca (effect of vertical electrical field on mobility), and m knee sharpness of the saturation transition). Parameters Vaa, ca and VT are determined by fitting the experimental IDS VGS data in strong inversion with low VDS. These parameters produce the following effects on the device characteristics: 1. VT shifts the whole IDS VGS curve. 2. Vaa provides a scale factor. 3. ca takes into account the effect of vertical electrical field on mobility. We define the following rules to calculate the IDS VGS curve (similar to the case of the bulk MOSFET): 1. If the calculated curve IDS VGS is to the right of the measured data, then decrement VT, and vice versa. This rule has been implemented by measuring the distance (in V) between the experimental and the calculated curves at IDS equal to 10% of its maximum value. 2. If the calculated curve IDS VGS is over the measured data, then increase Vaa, and vice versa. It has been implemented using the distance (in V) at IDS equal to 50% of its maximum value. 3. If the calculated curve IDS VGS is more curved than the measured data, then increase ca, and vice versa. It has been implemented using the distance (in V) at IDS equal to 90% of its maximum value. In order to implement the above rules, we have used again the same procedure as in the previous section. That is, after normalization, we use the fuzzifier in Fig. 2, while defuzzification is also done through the method of center of area (COA). The output of the controller is used as the relative change between the old value of the controlled parameter and the new value. Once the above parameters have been found, we can extract the remaining ones (aS, m and k). Since they affect the saturation region, we will use the IDS VDS curve, corresponding to the maximum value of VGS. The effect of the
involved parameters on the transistor behavior is as follows: 1. aS establishes the relation between VGS and VDsat. 2. k sets the slope of the IDS VDS curve in saturation. 3. m adjusts the sharpness of the knee region. Then, we can define extraction rules: 1. If the IDS VDS calculated curve saturates too soon, decrease alpha, and vice versa. This can be done by comparing the values of IDS for the voltage corresponding to 50% of the maximum experimental current. 2. If the slope of the calculated IDS VDS curve in saturation is too small, then increase lambda, and vice versa. This can be done by comparing the values of IDS at maximum VDS and VGS. 3. If the knee is too sharp, reduce m, and vice versa. This can be done by comparing the values of IDS at the point VDS = aS(VGS VT). The method will then iterate until a global error less than a 1% is achieved. The method is applied to two OTFTs fabricated as reported in [17], and whose technological data are shown in Table 2. Figs. 5 and 6 show the excellent agreement between experimental and modeled curves for T1; Figs. 7 and 8 show the same for T2. Values for the extracted parameters are shown in Tables 3 and 4, along with the extraction time. For comparison, transistor T2 is the same presented as T3 in [12]. Tables 3 and 4 also show a column with the values of the parameters obtained using UMEM, in order to compare results obtained by both methods. As can be seen, agreement between modeled and measured curves is excellent, with the enormous advantage of taking only around 15–25 s, when using a 2.0 GHz PC with a 512 MB RAM. The physical relation of the parameters extracted with their physical or modeling representation is maintained due to the calculation rules introduced. For the above reasons the method can be useful not only in industry but also for analysis of the physical behavior of devices. This parameter extraction method can be applied to other models provided the corresponding rule base is created using the appropriate measurements and the corresponding know-how. For instance, in order to implement the scaling rules built-in in BSIM the following procedure can be followed. First, extract the relevant parameter for a set of different channel lengths. Then, using fuzzy logic, the corresponding function of the BSIM could be fitted Table 2 Technological data for the OTFT transistors
Width (W) Length (L) Gate dielectric Gate dielectric thickness
Transistor T1
Transistor T2
170 lm 130 lm SiO2 100 nm
500 lm 50 lm PVP 120 nm
R. Picos et al. / Solid-State Electronics 51 (2007) 683–690
9
689
60
8 50
7
VDS =-20V
40
IDS ( μA)
IDS ( u A)
6 5 4 3
30 20
2 10
1 0
VDS=-6V
0
-30.00 -20.00 -10.00
0.00
10.00
20.00
30.00
-5
40.00
0
5
10
15
20
25
VGS (V)
V GS (V) Fig. 5. Experimental () and calculated (lines) IDS curves for OTFT T1. Experimental curves are for VDS = 10 V, while calculated curves show both VDS = 0.1 V and 10 V.
Fig. 7. Experimental () and calculated (lines) IDS curves for OTFT T2. Experimental curves are for VDS = 6 V, while calculated curves show both VDS = 6 V and 20 V. 50
10
45
9 8
35
IDS (μA)
7 IDS (uA)
VGS=20V
40
6 5 4
30
VGS=15V
25 20 15
3
VGS=10V
10
2
5
1
VGS=5V
0
0
0
0
5
10
15
20
25
5
30
in a similar fashion as we have done to extract the behavior of the threshold voltage with the bulk bias in the first example. 5. Conclusions In this work, we present a new method of extraction of model parameters based on fuzzy logic, which takes into account expert knowledge about the model, that is, the general rules of behavior of each parameter of the model. This is a huge advance over other global parameter extraction methods. This implies that instead of calculating the Jacobian of the fitting function (calculating around the order of N2 second derivatives), or the fitting function for a population of hundreds of individuals (in the case of genetic algorithms), we only need to use discrete values of the parameters at specific operation points to perform the implementation of the fitting rules (e.g. the distance between the value of the calculated current and the measured one at the VDS such that the measured current is the 50% of the maximum current). Since the behavior of
15
20
25
30
-VDS (V)
-VDS (V) Fig. 6. Experimental () and calculated (lines) IDS curves for OTFT T1.
10
Fig. 8. Experimental () and calculated (lines) IDS curves for OTFT T2. Table 3 Extracted parameters value for pentacene OTFT T1 with W = 170 lm, L = 130 lm; gate dielectric is SiO2 100 nm thick Parameter
Initial value
Extracted value
Direct method [12]
VT Vaa ca k aS m Time
1.0 10 V 0.5 0.01 0.5 1.0 19 s
11.2 V 49.2 V 1.01 6.27 · 1010 V1 0.383 2.31
12.1 55.4 1.3 0 0.40 2.5
the parameters has been embedded into the procedure itself, the need for computational power is significantly reduced without affecting fitting capabilities. The parameter extraction procedure was tested in two different kinds of semiconductor devices: in a 0.6 lm channel length bulk MOSFET and in two pentacene OTFTs. In this later case, extracted parameters using this method were compared with those obtained using UMEM and reported in [12], showing an excellent agreement. The method is quite insensitive to initial conditions and fast.
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Table 4 Extracted parameters value for pentacene OTFT T2 with W = 500 lm, L = 50 lm; gate dielectric is PVP 120 nm thick, having er = 3.9 Parameter
Initial value
Extracted value
Direct method [12]
VT Vaa ca k aS m Time
1.0 10 V 0.5 0.01 0.5 1.0 21 s
3.11 V 1600 V 0.17 1.05 · 104 V1 1.25 3.14
3.9 V 1400 V 0.15 9.6 · 105 V1 1.4 2.97
This parameter extraction method can be applied to other models (like BSIM, EKV or PSP) provided the corresponding rule base is created using the appropriate measurements and the corresponding know-how. Acknowledgements The authors wish to thank J. Deen, from McMaster University and M. Halik and H. Klauk from Infineon Technologies for providing OTFTs measurements. This work has been partially supported by the Spanish TEC2006-04103 project, the CONACYT project 45689 in Mexico, and the SINANO European NoE. The authors wish also to thank the reviewers for their helpful feedback and comments. References [1] Ortiz-Conde A, Rodrı´guez J, Garcı´a Sa´nchez FJ, Liou JJ. An improved definition for modeling the threshold voltage of MOSFETs. Solid State Electron 1998;42:1743–6. [2] Benson J, D’Halleweyn NV, Redman-White W, Easson CA, Uren MJ, Faynot O, et al. A physically based relation between extracted threshold voltage and surface potential flat band voltage for MOSFET compact modelling. IEEE Trans Electron Dev 2001;48:1019–21.
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