Volume 13 Number 4
August 2016
Article Contents
Imad Benacer and Zohir Dibi. Extracting Parameters of OFET Before and After Threshold Voltage Using Genetic Algorithms. International Journal of Automation and Computing, vol. 13, no. 4, pp. 382-391, 2016. doi: 10.1007/s11633-015-0918-6
Cite as: Imad Benacer and Zohir Dibi. Extracting Parameters of OFET Before and After Threshold Voltage Using Genetic Algorithms. International Journal of Automation and Computing, vol. 13, no. 4, pp. 382-391, 2016. doi: 10.1007/s11633-015-0918-6

Extracting Parameters of OFET Before and After Threshold Voltage Using Genetic Algorithms

Author Biography:
  • Zohir Dibi received the B.Sc.degree in electronics engineering from the University of Sitif, Algeria in 1994, and received the M.Eng.and Ph.D.degrees from the University of Constantine, Algeria, in 1998 and 2002, respectively.He has been the head of Electronics Department.He is currently an assistant professor in Electronics Department and vice-dean of the Faculty of Engineering at Batna University, Algeria.His research interests include neural networks, sensors, smart sensors, and organic devices.E-mail:zohir_dibi@yahoo.fr

  • Corresponding author: Imad Benacer received the B. Eng. and M. Sc. degrees from University of Batna, Algeria in 2006 and 2010, respectively. He is currently a Ph. D. degree candidate related to organic transistor. His research interests include microelectronic devices, organic transistor, modeling, and artificial intelligence. E-mail: benacerimad@gmail.com (Corresponding author) ORCID iD: 0000-0002-3063-9695
  • Received: 2014-03-28
  • Accepted: 2014-07-01
  • Published Online: 2016-06-29
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Extracting Parameters of OFET Before and After Threshold Voltage Using Genetic Algorithms

  • Corresponding author: Imad Benacer received the B. Eng. and M. Sc. degrees from University of Batna, Algeria in 2006 and 2010, respectively. He is currently a Ph. D. degree candidate related to organic transistor. His research interests include microelectronic devices, organic transistor, modeling, and artificial intelligence. E-mail: benacerimad@gmail.com (Corresponding author) ORCID iD: 0000-0002-3063-9695

Abstract: This paper presents a compact analytical model for the organic field-effect transistors (OFETs), which describes two main aspects, the first one is related to the behavior in above threshold regime, while the other corresponds to the below threshold regime. The total drain current in the OFET device is calculated as the sum of two components, with the inclusion of a smooth transition function in order to take into account both regions using a single expression. A genetic algorithm based approach (GA) is investigated as a parameter extraction tool in the case of the compact OFET model to find the parameters' values from experimental data such as: mobility enhancement factor γ, threshold voltage VTh, subthreshold swing S, channel length modulation γ, and knee region sharpness m. The comparison of the developed current model with the experimental data shows a good agreement in terms of the transfer and the output characteristics. Therefore, the GA based approach can be considered as a competitive candidate compared to the direct method.

Imad Benacer and Zohir Dibi. Extracting Parameters of OFET Before and After Threshold Voltage Using Genetic Algorithms. International Journal of Automation and Computing, vol. 13, no. 4, pp. 382-391, 2016. doi: 10.1007/s11633-015-0918-6
Citation: Imad Benacer and Zohir Dibi. Extracting Parameters of OFET Before and After Threshold Voltage Using Genetic Algorithms. International Journal of Automation and Computing, vol. 13, no. 4, pp. 382-391, 2016. doi: 10.1007/s11633-015-0918-6
  • Organic electronics has been evolved as a promising technology in the past three decades with many prototypes of organic devices, such as field-effect transistors (FETs), solar cells and light-emitting diodes[1-3]. The organic transistor offers robustness, low temperature fabrication, low cost and weight. It also consumes much less energy, and it has mechanical flexibility, which are beneficial for the active-matrix backplanes in displays, and wide range of applications, such as memories, sensors and radio frequency identification (RFID) tags[4-6]. Organic field effect transistors have witnessed impressive improvements in performance of the devices. Extensive research in which field, device optimization and process development have shown that both $p$ and $n$ -type organic field-effect transistors (OFETs) can outperform hydrogenated amorphous silicon (a-Si:H) thin film transistors (TFTs)[7, 8].

    Simulation of OFETs requires detailed models, which should be accurate enough to represent the behavior of the device. The similarity in the current-voltage and other characteristics between the OFET and a-Si:H TFTs or metal oxide semiconductor field effect transistor (MOSFET) has allowed to modify the existing models for a-Si:H TFTs or MOSFET in order to represent the characteristics of OFETs. A universal model for amorphous transistors (a-Si:H TFTs) has been implemented in AIM-Spice level 15 model[9, 10].

    The extraction of the parameters $'$ values for an FET device model is a complex task, traditional model extraction methods are based on a combination of direct parameter extraction that relies on purely mathematical basis. Unified model and parameter extraction method (UMEM) as any direct extraction method, requires human assistance in order to be performed successfully. UMEM has been applied to various transistor types and recently has been introduced practically in OFETs modeling[11, 12]. This method uses analytical expressions for both modeling and parameter extraction from experimental data, which can be seen as a complex task, since it needs a huge amount of parameters (most of them correlated). Therefore, the employment of global optimization methods would be a paramount solution for this purpose.

    Many researchers have studied the application and advancements in evolutionary algorithms, such as simulated annealing pseudo objective function substitution method (SaPOSM)[13], fast diffusion[14], genetic algorithms[15], to find the set of values that can best fit the experimental data, but the genetic algorithms (GAs) remain the most widely used in various applications. Genetic algorithms have been widely known for their robustness in solving tough and miscellaneous problems. The robustness of GA in transistor devices has been shown by a number of researchers[16-18]. Compared with existing conventional techniques, the main advantages of the GA based approach are their simplicity of implementation, easier coding, and being generic and applicable to extract parameters of other devices.

    Our aim in this paper is to present an extraction method of the different parameters in the case of OFET models using GAs. The proposed technique reduces significantly the processing time without losing precision and behavior of the device. This method is applied to a compact model of an OFET, where the model covers all operation regimes of OFETs from OFF to ON-state with smooth transition between the different regimes. To confirm the validity of the model, we fit the experimentally measured I-V characteristics with extracted model parameters, where a sufficient accuracy is needed.

    The paper is organized in five Sections as follows. Section 2 presents the description of the OFET model. Section 3 reviews the parameter extraction method based on GA formalism. In Section 4, the extracted parameters are applied to the OFET model to make a comparison with the experimental data. Finally, Section 5 provides some concluding remarks and future work directions.

  • This model is based on an interpolation approach used to obtain the desired behavior from the subthreshold region to the above threshold region.

  • OFET operates in the accumulation mode, in the linear and saturation regions and in the above threshold regime. The accumulation current $I_{abv}$ can be calculated from the conductance $g_{ch}$ as[19, 20]

    ${I_{abv}}={g_{ch}}{V_{dse}}\left({1 + \lambda {V_{ds}}} \right)$

    (1)

    where λ is the channel length modulation parameter and $V_{dse}$ is the effective drain voltage. Taking into account the resistance of the drain ( $R_{D})$ and source ( $R_{S})$ areas, the conductance $g_{ch}$ is given by

    ${g_{ch}}=\frac{{{g_{chi}}}}{{1 + {R_c}{g_{chi}}}}$

    (2)

    where $R_{c}$ is the resistance of source plus drain areas and $g_{chi}$ is the intrinsic channel conductance. It can be calculated using the physical dimensions of the device, width (W) and length (L), the capacitance per unit area of the insulating layer dielectric ( $C_{i})$ , and the field effect mobility $\mu_{FET }$ as

    ${g_{chi}}=-{\mu _{FET}}K\left({{V_{gs}}-\lambda {V_{Th}}} \right)$

    (3)

    where

    $K=\frac{W}{L}{C_i}.$

    (4)

    Unlike crystalline devices, carrier mobility in OFETs is bias dependent. The dependence of the field effect mobility ${\mu _{FET}}$ on gate voltage can be described as[21]

    ${\mu _{FET}}={\mu _0}{\left({\frac{{\left| {{V_{gs}}-{V_{Th}}} \right|}}{{{V_{aa}}}}} \right)^\gamma }$

    (5)

    where ${\mu _0}$ is the band mobility for the material of OFET, $V_{aa}$ is the mobility enhancement voltage introduced to adjust ${\mu _{FET}}$ to the experimental value of the low field mobility of the device being modeled, and γ is the characteristic mobility exponent, which is related to the conduction mechanism. Parameter γ can be used to describe the behavior of mobility.

    From (2) and (3), the expression of the channel conductance is given by

    ${g_{ch}}=\frac{{-K{\mu _{FET}}({V_{gs}}-{V_{Th}})}}{{1-{R_c}K{\mu _{FET}}({V_{gs}}-{V_{Th}})}}.$

    (6)

    The negative sign is given for a P-channel device OFET, this means that we have accumulating holes in the channel $\left({{V_{gs}}-{V_{Th}}} \right) < 0$ , where $V_{gs}$ is the gate-source voltage and $V_{Th}$ is the threshold voltage.

    The effective drain-source voltage $V_{dse}$ in (1) enables a smooth linear to saturation transition by a transition parameter $m$ (sharpness of the knee region)[21], and it can be expressed by

    ${V_{dse}}={V_{ds}}{\left[{1 + {{\left({\frac{{{V_{ds}}}}{{{V_{sat}}}}} \right)}^m}} \right]^{-\frac{1}{m}}}$

    (7)

    which simply approximates to $V_{ds}$ when $V_{ds} < < V_{sat}$ and to $V_{sat}$ when $V_{ds} >> V_{sat}$ , where $V_{sat}$ is the saturation voltage, defined using the saturation modulation parameter α $_{sat }$ as

    ${V_{sat}}={\alpha _{sat}}({V_{gs}}-{V_{Th}}).$

    (8)

    The above threshold current in the linear and saturation regions $I_{abv}$ is obtained by replacing $g_{ch}$ by (6), $V_{dse}$ by (7), and $V_{sat}$ by (8), so the expression of the drain current is given by

    $\begin{array}{l} {I_{abv}}=\frac{{-K{\mu _{FET}}({V_{gs}}-{V_{Th}})}}{{1-{R_c}K{\mu _{FET}}({V_{gs}}-{V_{Th}})}}{V_{ds}}\left({1 + \lambda \left| {{V_{ds}}} \right|} \right) \times \\ \qquad \quad {\left[{1 + {{\left({\frac{{{V_{ds}}}}{{{V_{sat}}}}} \right)}^m}} \right]^{-\frac{1}{m}}}. \end{array}$

    (9)

    The model correctly predicts the continuous change of the current to both signs of drain-source voltage, for this reason, the absolute value of $V_{ds}$ terms are added in (9) (as depicted in Fig. 5).

  • In the subthreshold region, where the gate voltage is below the threshold voltage, the subthreshold drain current $I_{sub}$ of OFETs is based on the MOSFET theory that accounts for the subthreshold conduction as a diffusion current[22]. The large gradient of the charge concentration between the source and the drain contact regions, gives rise to a diffusion current which is independent of the drain voltage.

    The subthreshold current depends exponentially on the gates-source voltage as

    ${I_{sub(a)}}={I_0}{{\rm{e}}^{-\frac{{q({V_{gs}}-{V_{on}})}}{{n{K_B}T}}}}$

    (10)

    where

    $n=\frac{{Sq}}{{{K_B}T\ln (10)}}$

    (11)

    $K_{B}$ is the Boltzmann constant, $V_{on}$ is the onset voltage, and $I_{0}$ is the off current.

    The subthreshold swing S is a measure of how easily a transistor can be switched from the OFF-state to the ON-state, $S$ is defined as the inverse of the subthreshold slope (log( $I_{d})$ versus $V_{gs})$ and corresponds to the gate voltage needed to increase the drain current by a factor of 10 in [23], it can be calculated as

    $\begin{array}{l} S=\frac{{\partial {V_{gs}}}}{{\partial ({{\log }_{10}}{I_d})}}=\\ \; \;\; \;\; \frac{{{K_B}T\ln (10)}}{q}\left[{1 + \frac{q}{{{C_i}}}\left({\sqrt {{\varepsilon _{sc}}{N_{bulk}}} + q{N_{int}}} \right)} \right] \end{array}$

    (12)

    where $C_{i}$ is the capacitance of the gate dielectric per unit area, $ q$ is the unit charge, $N_{bulk}$ is the density of deep bulk traps, $N_{int}$ denotes the density of deep interface traps and ${\varepsilon _{sc}}$ is the permittivity of the semiconductor material. By setting $ N_{bulk}$ to zero, the density of deep interface traps at the semiconductor/dielectric interface ( $N_{int}$ )is calculated as

    ${N_{int}}=\frac{{{C_i}}}{{{q^2}}}\left({\frac{{Sq}}{{{k_B}T\ln (10)}}-1} \right).$

    (13)

    Since the organic transistors have leakage current across the gate dielectric, the subthreshold current is composed of two contributions in subthreshold region (off current denoted by $I_{0 }$ and subthreshold current model denoted by I $_{sub(a)})$ , which can be modeled as

    ${I_{sub(a)}}={I_0} + {I_{sub(a)}}={I_0}\left({1 + {{\rm{e}}^{-\frac{{q({V_{gs}}-{V_{on}})}}{{n{K_B}T}}}}} \right).$

    (14)

    So we get a new expression for the subthreshold current.

  • As discussed previously, the modeled $I_{abv}$ and $I_{sub}$ can only describe their associated regimes. Combined equation used for Si-based TFTs is not compatible with the general applicability to OFETs. An approach that is very easy to implement is the use of the hyperbolic tangent function (tanh) as the suitable method for OFETs to get a smooth transition between two regimes[24].

    The OFET total drain current is calculated as the sum of the two components, i.e., above and below threshold voltage. We define our final expression of the drain current as a single compact expression for all operation regimes as indicated below:

    $\begin{array}{l} {I_T}={I_{abv}} \times 0.5 \times \left[{1-\tanh \left({\frac{{{V_{gs}}-{V_{tr}}}}{{{S_l}}}} \right)} \right] + {I_{sub}} \times \\ \; \;\quad 0.5 \times \left[{1 + \tanh \left({\frac{{{V_{gs}}-{V_{tr}}}}{{{S_l}}}} \right)} \right]. \end{array}$

    (15)

    This model remains valid even at points where both regimes do not intersect (Figs. 1(a) and (b)). The smoothing function can fill the gap between $I_{abv}$ and $I_{sub}$ , and $V_{tr}$ is shifted from $V_{Th}$ by a few volts toward the above threshold direction to reach the sufficiently smooth transition.

    Figure 1.  Transition region from above ( $I_{abv})$ to subthreshold ( $I_{sub})$ regime for: a) intersection between $I_{abv}$ and $I_{sub}$ currents; b) without intersection between $I_{abv}$ and $I_{sub}$ currents

    Parameters $V_{tr}$ can be used to define the transition point and $S_{l}$ for the smoothing level. The smoothing level is adjusted with the parameter $S_{l}$ by iterative calculation and comparison to the experimental data, as shown in Fig. 2.

    Figure 2.  Effect of smoothing level parameter $S_{l}$ on the complete model

  • Unified model and parameter extraction method (UMEM) depends on purely mathematical basis and would typically take more time, with a human expertise to interpret results. To overcome this difficulty, we propose a genetic algorithm (GA) based parameter extraction method for the previous OFET model. The GA is a highly flexible method for solving a tough and miscellaneous problems based on function optimization through evolutionary computation. In the algorithm, each unknown parameter is called gene and each vector of these parameters is called a chromosome[25, 26].

    The algorithm starts by generating a population of individuals from random variations of a seed, and each individual is rated on its ``fitness'' or ability to satisfy a set of criteria specified by the user. Each of these individuals is a list of parameters values that are to be extracted. It uses the operations of selection, crossover, and mutation as its steps iteratively, the GA flow is shown in Fig. 3. The purpose of the genetic algorithm is to determine the best fitted individual which optimizes (evaluation) the defined fitness function. This cycle is repeated until the termination criteria are reached.

    Figure 3.  Flowchart diagram for the genetic algorithm

    We define the fitness function mean-squared error (MSE) as

    $f=\frac{1}{M}\sum\limits_{{V_{gs}}} {{{\sum\limits_{{V_{ds}}} {\left[{\frac{{{I_{d, Exp}}-{I_{d, GA}}}}{{{I_{d, Exp}}}}} \right]} }^2}} $

    (16)

    where $I_{d, GA }$ is the predicted drain current based on GA, $I_{d, Exp }$ represents the target function (experimental measures), and $M$ represents the number of samples (database size). The GA parameters are depicted in Table 1. Our objective is to minimize the fitness function, in order to obtain the best configuration.

    Table 1.  Parameters used for GA based computation

  • We have implemented the method by fitting the previous model to experimental data corresponding to three different organic TFTs, and whose technological characteristics are shown in Table 2.

    Table 2.  Technological parameters of the adopted organic transistors

    Transistor T1 uses 3.6 nm Aluminum oxide and 2.1 nm of an organic self-assembled monolayer (SAM) as a gate dielectric. These devices have channel length $L$ of 10 $\mu $ m and a channel width $W$ of 100 $\mu $ m, with 30 nm thick film of pentacene and 30 nm thick gold source/drain contacts as indicated in [27].

    T2 type transistor[28], uses oxygen-plasma-grown AlOx layer (3.6 nm thick) and a solution-processed SAM of $n$ -tetradecylphosphonic acid (1.7 nm thick) as a gate dielectric layer, 30 nm thick aluminum gate, 11 nm thickness of dinaphtho-thieno-thiophene (DNTT) as active layer, and 25 nm thick gold source/drain contacts, $W$ =400 $\mu $ m and $L$ =200 $\mu $ m.

    Transistor T3 previously reported in [24], uses both poly methyl methacrylate (PMMA) and poly(3, 4-ethylenedioxythiophene) poly(styrenesulfonate) (PEDOT:PSS) as a gate dielectric, 100 nm thick film of pentacene, 60 nm of Au (gold) to define source/drain electrodes, $W$ =1 000 $\mu $ m and $L$ =40 $\mu $ m.

    Table 3 summarizes the parameters values (onset voltage $V_{on}$ , off current $I_{0}$ , and band mobility ${\mu _0})$ used for the modeling of the three types of transistors.

    Table 3.  OFETs parameters used for simulation

    The considered types of devices are all bottom-gate top-contact configuration but differ in terms of materials used in the active layer and dielectric gate.

    For simplicity, we only need to extract eight different parameters: $V_{Th}$ , γ, $V_{aa}$ , λ, $m$ , $R_{c}$ , α $_{sat}$ and $S$ , the remaining parameters can be calculated from the values extracted. Interface trap density $N_{int}$ is calculated from (13) by using the extracted value of $S$ , the field effect mobility ${\mu _{FET}}$ using each of γ, $V_{aa}$ and $V_{Th}$ to calculate its value from (5), the transition point $V_{tr}$ previously mentioned is determined by the point of intersection between $I_{abv}$ and $I_{sub}$ or by minimizing the vertical distance between the two currents as previously highlighted in Figs. 1(a) and 1 (b).

    Table 4 illustrates the parameters values extracted using the GA and compares the obtained results with experimental data[24, 27, 28]. Resulting output and transfer characteristics plots for simulated and experimental results are shown in Figs. 4 and 5 for the transistors T1, T2, and T3, respectively, where the transfer characteristics of T1, T2, and T3 in saturation regions are shown, in semi-logarithmic plots.

    Table 4.  Parameters extracted using the GA method with experimental data

    Figure 4.  Comparison of experimental and simulated transfer characteristics of: a) T1; b) T2; c) T3

    Figure 5.  Comparison of experimental and simulated output characteristics of: a) T1; b) T2; c) T3

    Figs. 4 and 5 present comparisons between experimental data[24, 27, 28] and the drain current calculated from the extracted parameters by GA for the OFET. It can be observed that we have a good agreement between experimental and simulated values for T1 type transistor for both transfer and output characteristics (Figs. 4 (a) and 5 (a)), Figs. 4 (b), 4 (c), 5 (b) and 5 (c) compare experimental results and modeled current-voltage characteristics ( $I_{ds}-V_{ds}$ ) and ( $I_{ds}-V_{gs}$ ) curves for T2 and T3 type transistors by the use of two values of $V_{ds}$ , one in the linear region ( $V_{ds(T2)}$ =-0.1 V, $V_{ds(T3)}$ =-2 V) and the other in saturation, ( $V_{ds(T2)}$ =-2 V, $V_{ds(T3)}$ =-50 V), where a good agreement is also observed.

    Agreement between modeled and measured curves is good, this results in 70 000 parameter set evaluations, and takes about 30 min for each sample (three samples of transistor) to complete, when using a 2.7 GHz PC with a 2 GB RAM (Pentium IV with Windows XP).

    Figs. 6 (a)-6 (c) show the variation of the fitness function as a function of generation number, where the minimum of objective function can be reached for 500--700 iterations for T1, T2 and T3 respectively. If the fitness function remains constant for about 200 iterations, the genetic algorithm halts executing and the best value is displayed.

    Figure 6.  Variation of fitness function with generations for: a) T1; b) T2; c) T3

    The field effect mobility ${\mu _{FET}}$ extracted is 0.42 cm $^{2}$ /Vs for the transistor T1, 1.9 cm $^{2}$ /Vs for T2, and 0.16 cm $^{2}$ /Vs for T3 at maximum value of gate-source voltage. Additional extracted parameters such as, $\gamma $ , $V_{aa}$ , $V_{Th}$ are summarized in Table 4. As can be noticed from Fig. 7, the behavior of mobility for transistors T1, T2 and T3 increases with $V_{gs}$ , which is calculated using (5). The mobility dependence with $V_{gs}$ is similar to that observed for amorphous silicon (a-Si:H) transistors. Experimental and modeled curves for all characteristics are well established, including the above and subthreshold regions, thus the model is proven to be quite accurate in all regimes.

    Figure 7.  Behavior of mobility for transistors: a) T1; b) T2; c) T3

  • The presented model in this paper showed the output and the transfer characteristics of OFET device in two regimes, which were described by different current models. In addition, our expression of the drain current was deduced by using a hyperbolic tangent transition function as a suitable method for OFET device to get a single compact expression for all operation regimes.

    We have applied a parameter extraction technique based on the use of genetic algorithms to deduce the parameters of the compact model for OFET. The extracted parameters have been introduced in the model to simulate the electric characteristics of the three different types of OFETs, where a good agreement has been obtained between the simulated and the experimental results. As a future perspective, we intend to extend the present model including doping concentration and other design parameters such as overlap, for other OFETs with various materials and geometries.

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