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particle swarm optimization approach
Microelectronics Journal 43 (2012) 562–568
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Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
Multibias scalable HEMT small-signal modeling based on a hybrid direct extraction/particle swarm optimization approach Y. Campos-Roca a,n, H. Massler b, A. Leuther b a b
´ceres, Spain Universidad de Extremadura, Escuela Polite´cnica, Avda. de la Universidad, s/n, E-10003, Ca Fraunhofer Institute for Applied Solid State Physics (IAF), Tullastrasse 72, D-79108 Freiburg, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 June 2011 Received in revised form 22 April 2012 Accepted 24 April 2012 Available online 9 May 2012
A new multibias HEMT small-signal model extraction method is proposed. The approach, based on scaling rules, combines direct extraction techniques and a particle swarm optimization algorithm. This method has been successfully tested with PHEMTs and MHEMTs, leading to accurate and scalable models up to 70 and 120 GHz, respectively. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Metamorphic high electron mobility transistor (MHEMT) Particle swarm optimization (PSO) Pseudomorphic high electron mobility transistor (PHEMT) Scalability Small-signal modeling
1. Introduction The design of linear millimeter-wave circuits requires accurate small-signal models of the active components. Besides, smallsignal models are often used as building blocks to generate largesignal ones, by extracting small-signal parameters from S-parameter measurements at different bias points. Small-signal models are mostly based on an equivalent circuit representation. As an alternative, some approaches based on artificial neural networks have been proposed [1]. However, the former type of models is generally preferred because it provides a compact representation and the possibility to link the circuit parameters to the device physical structure. There are two main families of extraction techniques that can be used to determine the model parameters: direct extraction and optimization-based techniques. Direct-extraction methods are typically based on cold-FET S-parameter measurements, used to obtain the parasitic component values. The intrinsic elements are analytically calculated after de-embedding the extrinsic ones. [2,3] are classical references. Other contributions in the area of analytical techniques are [4–6]. This type of methods are faster than optimization-based techniques, but they are susceptible to measurement errors and their implementation is device specific.
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Corresponding author. Tel.: þ34 927 257195x57561; fax: þ 34 927 257203. E-mail address: [email protected] (Y. Campos-Roca).
0026-2692/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mejo.2012.04.008
Often, a subsequent optimization step is performed by hand to improve the fitting in active bias conditions. Optimization-based extraction techniques aim to determine the values of the small-signal model elements by fitting calculated S-parameters to measured ones. Many contributions propose single-bias extraction algorithms, which are not useful when small-signal models are intended to be used as building blocks to generate large-signal ones. Therefore, a multibias solution is required. In general, the main drawback of optimization-based techniques is the convergence to nonphysical element values due to the multiple local minima of the cost function. Additionally, the obtained solution may be sensitive to the optimization starting values. To overcome this problem, several extraction methods based on global optimization techniques (such as genetic algorithms) have been proposed. Several approaches that combine data measured at different bias points have been suggested in the literature [7,8]. To the knowledge of the authors, the multibias extraction algorithms proposed in the literature are not based on scaling rules for the equivalent circuit components, i.e., each optimization problem aims to fit measurements from a single device geometry. This paper proposes a novel extraction procedure, which combines S-parameter measurements from devices with different gate widths and at different bias points into one unified optimization problem. The algorithm has two characteristics that contribute to obtaining physically meaningful values. The first one is the use of high-quality starting values. Direct extraction
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techniques are initially employed to give a rough estimation of the parasitic component values. These rough estimations are used as starting values in the subsequent optimization process. The second characteristic is the use of scaling rules in the definition of the optimization problem. A single model for devices with different gate widths is automatically obtained. The complexity of the optimization problem is kept relatively low, since only a reduced number of measurement points is required. The proposed method uses particle swarm optimization (PSO). Since the pioneering work of Kennedy and Eberhart [9], PSO has been proved to solve many problems. It is a population-based stochastic optimization strategy showing faster convergence speed and simpler implementation than genetic or ant colony algorithms, with comparable or better-quality solutions [10]. The performance of the proposed extraction method is evaluated by using wide-band multibias S-parameter measurements on PHEMTs and MHEMTs. Scalability and good accuracy are demonstrated up to 70 and 120 GHz, respectively. The paper is organized as follows. In Section 2, the extraction process (including the scaling rules) is described. The results are presented in Section 3. Finally, some conclusions are drawn in Section 4.
2. Extraction process In order to keep the extraction method as simple as possible, the ordinary 16-element model topology has been used (see Fig. 1). The equivalent circuit is divided into an extrinsic part (outside the dashed box), whose elements are assumed to be biasindependent, and an intrinsic part (inside the dashed box), whose elements are bias-dependent. The extraction process has the following main characteristics: – – – –
scalability, combination of analytical and optimization-based techniques, particle-swarm optimization, multibias data fitting, based on a reduced number of bias points.
To design millimeter-wave circuits, it is desirable to have scalable models, that provide greater flexibility. The proposed algorithm is based on scaling rules, and aims to obtain a single model for devices with different gate widths. The whole extraction algorithm has been implemented in MATLAB code.
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The multibias procedure is based on a hybrid direct extraction/ optimization approach. The optimization variables are the different parameters in the scaling rules of the extrinsic components (see Section 2.1). In order to limit the dimension of the optimization problem, the intrinsic elements are not optimized. They are analytically calculated after de-embedding the extrinsic ones [2]. However, their scalability must also be guaranteed. This is taken into account by introducing a scalability error term into the cost function (see Section 2.2). As mentioned above, the use of measurements from different device sizes and scaling rules in the definition of the optimization problem also contributes to achieving physically meaningful values. Fig. 2 shows the algorithm pseudocode in a compact way. In order to obtain a rough a priori knowledge of the parameter values, direct extraction techniques are employed. These techniques, based on cold-FET (VDS ¼0) measurements, provide starting values for the optimization variables. The initial values of the parasitic capacitances (Cpgs, Cpgd and Cpds) are determined by performing S-parameter measurements at a pinch-off gate voltage on devices with different gate widths [11]. Also initial values for the extrinsic resistances and inductances are determined by using the method proposed in [2]. Rough estimations are sufficiently good; thus the calculation does not require such a precise knowledge of technological parameters, such as when the technique is applied without a subsequent optimization step. Besides, high gate currents can be avoided in the cold-FET forward measurements if the devices are sensitive to forward gate bias. The calculation of extrinsic resistances and inductances is performed on transistors with different gate widths, so that the initial values for the parameters in the scaling rules can be determined. The optimization step is based on a reduced number of bias points and sizes: two measurements on devices with different gate widths in the saturation region (at the same bias voltages), and two additional measurements (in the pinch-off and ohmic regions) on devices of one size. This reduced set of measurements allows obtaining accurate results at a low computational cost. The optimization procedure is based on a PSO algorithm. PSO [9] is a computation technique inspired by the social behavior of bird flocking or fish schooling. PSO has several attractive attributes, including the fact that the basic algorithm is very easy to understand and implement. It is similar in some ways to genetic algorithms, but it shows superior performance in convergence speed [10]. PSO uses a population of particles that fly through the problem hyperspace with given velocities. Each particle’s position corresponds to a candidate solution to the optimization problem. A cost function, that measures the deviation from the desired solution, is defined. The cost function defined in this approach is presented in Section 2.2. As the particles fly through the problem hyperspace, each particle remembers its own personal best position that it has ever found, called its local best. Each particle also knows the best position found by any particle in the swarm, called the global best. At each iteration, the velocities of the individual particles are updated according to their own best positions and the global best. Further details will be given in Section 2.2. 2.1. Scaling rules The extraction process aims to obtain models that are scalable with the gate width, for a fixed number of gate fingers. The scaling rules the extraction process is based on are the following:
Fig. 1. Equivalent circuit topology for HEMT small-signal modeling. The dashed box contains the intrinsic part of the device.
a) The pad capacitances (Cpgs, Cpgd, Cpds) should take a fixed value; they are not dependent on the gate width [11].
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Fig. 2. Pseudocode of the extraction algorithm.
b) The source and drain resistances vary inversely with the total gate width [12]. Therefore, the following scaling rules are used: Rd ¼ Rd0/Wg and Rs ¼Rs0/Wg, where Wg is the total gate width, and Rd0 and Rs0 are constants. The starting values for Rd0 and Rs0 in the optimization process are determined by using cold-FET forward measurements from devices with different gate widths [2]. c) The gate resistance is considered to comprise three components [13]: the first one is a residual resistance (Rg0), the second one is a classical expression (Rg1Wg) and the third component (Rg2/Wg) is a contact resistance. Therefore, for a fixed number of gate fingers, the scaling rule has the following formulation: Rg ¼Rg0 þ Rg1Wg þ Rg2/Wg, where Rg0, Rg1 and Rg2 are constants. The starting values for these three constants in the optimization process are calculated by fitting this function to the gate resistance values obtained by direct extraction. The extractions are based on cold-FET forward measurements from devices with different gate widths [2]. d) In the case of the extrinsic inductances, different scaling rules have been proposed in the literature. For example, in [14] the authors consider that the values do not vary much for their interdigitated power PHEMT devices and they do not need any scaling. In [12], Lg and Ld are considered to scale proportionally to the gate width. The extraction process proposed here is
based on flexible scaling rules, comprising an independent term and a scaling factor, i.e., Lg ¼Lg0 þLg1Wg; Ld ¼Ld0 þLd1Wg; and Ls ¼ Ls0 þ Ls1Wg. The starting values for the different parameters in the previous scaling rules are also determined by using cold-FET forward measurements from devices with different gate widths [2]. e) For the intrinsic components, the scaling rules are simple and conventional: gm, Cgs, Cds and Cgd scale proportionally to the gate width, whereas Rgs and Rds scale inversely proportional to the gate width. The transconductance delay t is considered constant with respect to the gate width [15].
2.2. Optimization process The optimization process is based on a PSO algorithm [9]. The swarm is initialized with a population of Np random solutions, called particles, and searches for optima by updating generations. Each particle is defined by a position and a velocity, and it moves through the search hyperspace. Each particle’s velocity is dynamically updated according to its own and its neighbors’ previous behavior. The updating process is based on three terms, called the ‘‘social’’, the ‘‘cognitive’’ and the ‘‘inertia’’ terms. The ‘‘social’’ term attempts to guide the particle to the best position found by any
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particle in its neighborhood so far (called the global best, gbest), the ‘‘cognitive’’ term conducts it to the best position found by itself so far (called the local best, pbest), and the ‘‘inertia’’ term is related to the memory of its previous velocity. The following equations describe the updating process of a particle’s velocity and position: vn ¼ ovn1 þ c1 r 1 ðpbest,n xn1 Þ þ c2 r 2 ðg best,n xn1 Þ, xn ¼ xn1 þ vn where vn and xn are the particle’s velocity and position at the nthiteration, respectively; o is the inertia weight; c1 and c2 are acceleration constants (the ‘‘cognitive’’ ratio and the ‘‘social’’ ratio, respectively); and r1 and r2 are random numbers uniformly distributed between 0 and 1. Once the velocity vn has been calculated, the particle moves to its next location xn. The swarm will continue moving until a criterion is met: a sufficiently low cost or a maximum number of iterations. In the considered extraction problem, the optimization variables are the parameters in the scaling rules of the extrinsic components (see Section 2.1). 14 optimization variables have been defined (Cpgs, Cpgd, Cpds, Rd0, Rs0, Rg0, Rg1, Rg2, Lg0, Lg1, Ld0, Ld1, Ls0, and Ls1); therefore, the particles’ positions and velocities are 14-dimensional vectors. As mentioned above, rough estimates of the extrinsic components are initially obtained by applying direct extraction techniques. These estimations allow us to calculate the starting values for the optimization variables. The particles’ positions are initialized by generating random vectors, uniformly distributed, inside a 14-dimensional search hyper-rectangle. Let Pinit i , i¼1,y,14, denote the initial values of the optimization variables. In the proposed approach, the lower Pmin and upper Pmax boundaries are determined by specifying a i i percentage variation (Pi). If any of the parameters had a null starting value Pinit i , the lower and upper boundaries would also be null. In these cases, an arbitrary upper boundary (Mi), empirically determined, is specified. Therefore, the search hyper-rectangle is defined as ½Pmin , Pmax , i ¼ 1,. . .,14, where i i Ymin P , i ¼ 1,. . .,14, ¼ max 0, Pinit 1 i i i 100 and Ymin i
¼
8 Pi < Pinit 1þ 100 i :M
i
9 if Pinit a0 = i , if Pinit ¼0 ; i
Lg0, Lg1, Ld0, Ld1, Ls0, Ls1]. However, the intrinsic components are not optimized but extracted after de-embedding the extrinsic ones. Thus it is necessary to include a second goal related to the scalability of these intrinsic elements. The cost function comprises two terms (e1 and e2), each of them related to one of the previously defined goals, and an additional penalty term (e3), that is introduced to include constraints. The first term represents the error between measured and modeled S-parameters:
Nf meas mod N X 2 X 2 X X S f f ,x S 1 k k
pqn pqn
, e1 ðxÞ ¼
4N f N n ¼ 1 p ¼ 1 q ¼ 1 k ¼ 1
Smeas fk pqn
k ¼max
1,:::,N f where Smeas and Smod pqn pqn represent measured and calculated Sparameters, respectively, Nf denotes the number of frequency points and fk each frequency value. A logarithmic frequency step has been used. In this approach, only a reduced number of Sparameter measurements is required. All the results reported in Section 3 have been obtained by using four frequency sweeps (N ¼ 4): two of them are performed on transistors with different gate widths at the same active bias point in the saturation region, and two additional measurements are performed on only one device size (in the pinch-off and ohmic regions). These measurements allow us to obtain accurate results in a wide range of bias points and gate widths with a low computational cost. The second term in the cost function represents the scalability error of the intrinsic components. The scalability check is performed at an active bias point by using the first two measurements (performed on transistors with different gate widths). Once the intrinsic elements for the two device sizes are extracted, simple straight-line equations are obtained, where the independent variable is always the gate width and each dependent variable is one of the following intrinsic components: Cgs, Cds, Cgd, Ggs ¼1/Rgs, Gds ¼1/Rds and gm. For notation convenience, function names are used for these intrinsic elements at the considered active bias condition, i.e., h1 ¼Cgs(Wg, x), h2 ¼Cds(Wg,x),y,h6 ¼gm(Wg,x). The straight-line equations are the following: hj ðW g ,xÞ ¼ aj ðxÞ þ bj ðxÞW g ,
i ¼ 1,. . .,14:
Regarding the particles’ velocities, they are also initialized by generating random numbers, uniformly distributed, in a hyperrectangle. This hyper-rectangle is defined as ½vimax ,vimax 14 , i ¼ 1,. . .,14, vimax being the maximum velocity along each dimension of the search hyperspace. In the successive iterations, the algorithm also limits the maximum velocity allowed along each dimension to the value vimax : One of the key aspects in the definition of any optimization problem is to select an appropriate cost function. This function will be evaluated to determine the deviation from the desired solution. In order to obtain a scalable small-signal model, two goals must be achieved: a) Minimum error. the difference between measured and modeled S-parameters, over the considered frequency range, for different bias points and device sizes, must be as small as possible. b) Scalability of the intrinsic components. the scalability of the extrinsic components is taken into account by the use of scaling rules (see Section 2.1). The parameters in these scaling rules are the optimization variables, so that each candidate solution is defined as x¼ [Cpgs, Cpgd, Cpds, Rd0, Rs0, Rg0, Rg1, Rg2,
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j ¼ 1,. . .,6:
The results of each calculation will show an independent term (aj) equal to zero if the corresponding intrinsic component (hj) scales proportionally to the gate width (see the scaling rules in Section 2.1, point (e)). Thus the sum of normalized independent terms represents the scalability error for the six considered intrinsic components. Additionally, the remaining intrinsic element (the transconductance delay, t) is considered to be constant with respect to the gate width. Therefore, the global scalability error is described by the following equation:
6
X aj ðxÞ
e2 ðxÞ ¼
hj W g1 ,x þ hj W g2 ,x =2
j¼1
t W ,x t W ,x
g1 g2
, þ C t
t W g1 ,x þ t W g2 ,x =2
where Wg1 and Wg2 represent the two considered gate widths, and Ct is a weighting factor. To the knowledge of the authors, the inclusion of a term representing the scalability error in a cost function has not been reported in the literature related to smallsignal model extraction so far. Also a penalty method is used to include constraints. In the case of negative values for the intrinsic components, a third term (e3) is added to the cost function that prescribes a very high cost. The final cost function is calculated as the linear weighted sum of the terms e1, e2 and e3.
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3. Experiments and results Two sets of experiments have been performed to validate the proposed extraction method. The first one is applied to double ddoped AlGaAs/InGaAs/GaAs PHEMTs, with 0.15 mm T-shaped gates. These devices exhibit an extrinsic transit frequency fT of 100 GHz and a maximum oscillation frequency fmax of 150 GHz. As mentioned above, the starting values for the optimization variables have been obtained by applying direct extraction techniques. These starting values are the following: Cpgs ¼7 fF, Cpgd ¼2 fF, Cpds ¼ 7 fF, Rd0 ¼0.3 O mm, Rs0 ¼ 0.2 O mm, Rg0 ¼ 3 O, Rg1 ¼0 O/mm, Rg2 ¼0 O mm, Lg0 ¼3 pH, Lg1 ¼70 pH/mm, Ld0 ¼3 pH, Ld1 ¼ 60 pH/ mm, Ls0 ¼0 pH, and Ls1 ¼0 pH/mm. As mentioned above, four Sparameter measurements (frequency sweeps) have been used in the multibias optimization procedure. The first two measurements correspond to the same active bias point (VGS ¼0 V, VDS ¼ 2.6 V) and two different device sizes: 2 30 mm2 and 2 60 mm2. The third and fourth measurements correspond to a 2 60 mm2
Fig. 3. Cost of gbest versus iterations, for different numbers of particles.
geometry at a pinch-off condition (VGS ¼ 1 V, VDS ¼ 2.4 V) and at a bias condition in the ohmic region (VGS ¼ 0.3 V, VDS ¼0.4 V). The percentage variation (Pi) is set to 80% for all optimization variables. Fig. 3 shows the convergence plot for different numbers of particles (Np ¼10, Np ¼12 and Np ¼15). This plot shows the cost function values of the successive gbest positions. As can be observed, a swarm of only Np ¼12 particles allows us to obtain good results. The values of the different PSO parameters that have been used are the following: number of particles Np ¼12, relative maximum velocity virmax ¼2, i¼1,y,14, cognitive ratio c1 ¼1.5, social ratio c2 ¼ 1.5 and inertia weight o ¼1. The final values of the optimization parameters are the following: Cpgs ¼12.6 fF, Cpgd ¼ 1.4 fF, Cpds ¼2.8 fF, Rd0 ¼ 0.2 O mm, Rs0 ¼0.04 O mm, Rg0 ¼3.75 O, Rg1 ¼ 0 O/mm, Rg2 ¼ 0 O mm, Lg0 ¼5.4 pH, Lg1 ¼126 pH/mm, Ld0 ¼0.6 pH, Ld1 ¼32 pH/mm, Ls0 ¼0 pH, and Ls1 ¼3 pH/mm. As mentioned above, the intrinsic components are not optimized but analytically extracted. However, due to the inclusion of a term representing the scalability error into the cost function, the obtained intrinsic components satisfy the scaling rules (Section 2.1, point (e)). The extraction process has been successfully validated by comparison of measured and modeled S-parameters, at different bias settings and for different device sizes. A single scalable model is used for the different geometries. Fig. 4 shows the results for a 2 30 mm device in the following bias setting: VGS ¼ 0.2 V, VDS ¼3 V. As can be observed, a very good agreement has been obtained up to 70 GHz. Figs. 5 and 6 show verification results for a completely new device geometry, which has not been used in the optimization procedure: 2 45 mm. Fig. 5 corresponds to the following bias point: VGS ¼ 0.4 V, and VDS ¼1.5 V, while Fig. 6 was obtained at a cold-FET pinch-off condition (VGS ¼ 1 V, and VDS ¼0 V). Finally, Fig. 7 shows a comparison between modeled and measured S-parameters for a 2 60 mm device at a gate voltage of 0.5 V and a drain voltage of 3 V. This is a completely new bias point, which has not been used in the optimization process for any device geometry. The second set of experiments is based on InAlAs/InGaAs MHEMTs. These devices have 100 nm T-shaped gates and achieve an extrinsic transit frequency of 220 GHz and a maximum oscillation frequency of 300 GHz. More details about this MHEMT process can be found in [16].
Fig. 4. Comparison of measured and modeled S-parameters of a 2 30 mm AlGaAs/InGaAs PHEMT. Frequency range: 0.5–70.5 GHz. Bias point: VGS ¼ 0.2 V, and VDS ¼ 3 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig.5. Comparison of measured and modeled S-parameters of a 2 45 mm AlGaAs/InGaAs PHEMT. Frequency range: 0.5–50 GHz. Bias point: VGS ¼ 0.4 V, and VDS ¼1.5 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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was set to 50% for the pad capacitances and 100% for the other optimization variables. Again four S-parameter measurements have been used in the multibias optimization procedure. Two of the measurements correspond to an active bias condition in the saturation region (VGS ¼0.2 V, and VDS ¼1 V) on two different device sizes: 2 30 mm and 2 45 mm. The pinch-off measurement (VGS ¼ 1 V, and VDS ¼1.5 V) was performed on a 2 45 mm transistor. Finally, the fourth measurement corresponds to the ohmic region (VGS ¼ 0.2 V, and VDS ¼0.3 V) and was performed also on the 2 45 mm device. The final values of the optimization parameters are the following: Cpgs ¼15.7 fF, Cpgd ¼4 fF, Cpds ¼5.25 fF, Rd0 ¼0 O mm, Rs0 ¼0 O mm, Rg0 ¼ 6.8 O, Rg1 ¼14 O/mm, Rg2 ¼ 0 O mm, Lg0 ¼Ld0 ¼Ls0 ¼0 pH, Lg1 ¼100 pH/mm, Ld1 ¼0 pH/mm, and Ls1 ¼ 0 pH/mm. Figs. 8–11 show the results at different bias points and for different device sizes. Fig. 8 presents a comparison between modeled and measured S-parameters, for a 2 45 mm device and a bias point in the ohmic region (VGS ¼0 V, and VDS ¼0.1 V), up to 120 GHz. Fig. 9 refers to the same geometry and a different bias condition: VGS ¼ 0.4, and VDS ¼1.5. Note that this is a completely new bias point, regarding the measurements that have been used in the optimization process. Fig. 10 corresponds to a 2 60 mm device and an active bias point (VGS ¼ 0.2 V, and VDS ¼1 V). Finally, Fig. 11 corresponds to a 2 15 mm MHEMT, in a cold-FET pinchoff condition (VGS ¼ 1.5, and VDS ¼0). These two geometries
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Fig. 6. Comparison of measured and modeled S-parameters of a 2 45 mm AlGaAs/InGaAs PHEMT. Frequency range: 0.5–50 GHz. Bias point: VGS ¼ 1 V, and VDS ¼ 0 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 8. Comparison of measured and modeled S-parameters of a 2 45 mm InAlAs/InGaAs MHEMT. Frequency range: 0.25–120 GHz. Bias point: VGS ¼0 V, and VDS ¼ 0.1 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 7. Comparison of measured and modeled S-parameters of a 2 60 mm AlGaAs/InGaAs PHEMT. Frequency range: 0.5–50 GHz. Bias point: VGS ¼ 0.5 V, and VDS ¼ 3 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The same equivalent circuit topology was used (see Fig. 1). The starting values for the optimization variables (obtained by direct extraction) are the following: Cpgs ¼10.5 fF, Cpgd ¼3.5 fF, Cpds ¼6 fF, Rd0 ¼0.19 O mm, Rs0 ¼0.13 O mm, Rg0 ¼5 O, Rg1 ¼10 O/mm, Rg2 ¼0.2 O mm, Lg0 ¼Ld0 ¼Ls0 ¼0 pH, Lg1 ¼50 pH/mm, Ld1 ¼ 60 pH/ mm, and Ls1 ¼0 pH/mm. In this case, the percentage variation (Pi)
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Fig. 9. Comparison of measured and modeled S-parameters of a 2 45 mm InAlAs/InGaAs MHEMT. Frequency range: 0.25–120 GHz. Bias point: VGS ¼ 0.4 V, and VDS ¼ 1.5 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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problem has not been reported in the literature so far. A single model for devices with different sizes is obtained. By the use of a reduced number of S-parameter measurements in the optimization procedure, the computational effort is kept low. The extraction procedure has been verified with measurements on PHEMTs and MHEMTs. The reproducibility of the results has been successfully checked. Good agreement between measured and calculated S-parameters is achieved in a wide range of bias points and up to 70 and 120 GHz, respectively.
Acknowledgment
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Fig. 10. Comparison of measured and modeled S-parameters of a 2 60 mm InAlAs/InGaAs MHEMT. Frequency range: 0.25–120 GHz. Bias point: VGS ¼0.2 V, and VDS ¼ 1 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The authors would like to thank the Fraunhofer IAF, in Freiburg (Germany), for providing the PHEMT and MHEMT measurements. This work has been partially funded by the Junta de Extremadura (Project GR10126). References
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Fig. 11. Comparison of measured and modeled S-parameters of a 2 15 mm InAlAs/InGaAs MHEMT. Frequency range: 0.25–50 GHz. Bias point: VGS ¼ 1.5 V, and VDS ¼ 0 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(2 60 and 2 15 mm) have not been used in the optimization process. The results prove the scalability of the extracted model. The results show a very good agreement between measured and calculated S-parameters. All the experiments have been repeated many times to guarantee the reproducibility of the results. The experiments have been carried out on a 2 GHz Intel Core 2 Duo processor with 3 GB of RAM.
4. Conclusion A novel multibias approach for HEMT small-signal equivalentcircuit model extraction has been proposed. The method is based on scaling rules and uses a hybrid direct extraction/particle swarm optimization-based approach. To the knowledge of the authors, the combination of measurements on different device sizes and at different bias points into one unified optimization
[1] Z. Marinkovic, G. Crupi, A. Caddemi, V. Markovic, Comparison between analytical and neural approaches for multibias small-signal modelling of microwave-scaled FETs, Microwave Opt. Technol. Lett. 52 (2010) 2238–2244. [2] G. Dambrine, A. Cappy, F. Heliodore, E. Playez, A new method for determining the FET small-signal equivalent circuit, IEEE Trans. Microwave Theory Tech. 36 (1988) 1151–1159. [3] M. Berroth, R. Bosch, High-frequency equivalent circuit of GaAs FET’s for large-signal applications, IEEE Trans. Microwave Theory Tech. 39 (1991) 224–229. [4] A. Caddemi, G. Crupi, N. Donato, A robust and fast procedure for the determination of the small signal equivalent circuit of HEMTs, Microelectron. J. 35 (2004) 431–436. [5] S. Yanagawa, H. Ishihara, M. Ohtomo, Analytical method for determining equivalent circuit parameters of GaAs FET’s, IEEE Trans. Microwave Theory Tech. 44 (1996) 1637–1641. [6] R. Tayrani, J.E. Gerber, T. Daniel, R.S. Pengelly, U.L. Rhode, A new and reliable direct parasitic extraction method for MESFETs and HEMTs, in: Proceedings of the 23rd European Microwave Conference, Madrid, 1993, pp. 451–453. [7] A. Jarndal, G. Kompa, A new small-signal modeling approach applied to GaN devices, IEEE Trans. Microwave Theory Tech. 53 (2005) 3440–3448. [8] C. van Niekerk, J.A. du Preez, D.M.M.–P. Schreurs, A new hybrid multibias analytical/decomposition-based FET parameter extraction algorithm with intelligent bias point selection, IEEE Trans. Microwave Theory Tech. 51 (2003) 893–902. [9] J. Kennedy and R. Eberhart, Particle swarm optimization, in: Proceedings of IEEE International Conference on Neural Networks IV, Washington, USA, 1995, pp. 1942–1948. [10] V. Zuniga, A.T. Erdogan, and T. Arslan, Adaptive radiation pattern optimization for antenna arrays by phase perturbations using particle swarm optimization, in: Proceedings of the NASA/ESA Conference on Adaptive Hardware and Systems, Anaheim, California, USA, 2010, pp. 209–214. [11] R. Anholt, S. Swirhun, Measurement and analysis of GaAs MESFET parasitic capacitances, IEEE Trans. Microwave Theory Tech. 39 (1991) 1247–1251. [12] J. Wood, D.E. Root, Bias-dependent linear scalable millimeter-wave FET model, IEEE Trans. Microwave Theory Tech. 48 (2000) 2352–2360. [13] H. Rohdin, N. Moll, C. Su, G.S. Lee, Interfacial gate resistance in Schottkybarrier-gate field-effect transistors, IEEE Trans. Electron Devices 45 (1998) 2407–2416. [14] S.-W. Chen, O. Aina, W. Li, L. Phelps, T. Lee, An accurately scaled small-signal model for interdigitated power PHEMT up to 50 GHz, IEEE Trans. Microwave Theory Tech. 45 (1997) 700–703. [15] A. Caddemi, G. Crupi, A. Macchiarella, On wafer-scaled GaAs HEMTs: direct and robust small signal modelling up to 50 GHz, Microwave Opt. Technol. Lett. 51 (2009) 1958–1963. ¨ [16] M. Schlechtweg, I. Kallfass, A. Tessmann, C. Schworer, and A. Leuther, Millimeter-wave front-end components in metamorphic HEMT technology, in: Proceedings of the 1st European Microwave Integrated Circuits Conference, Manchester, UK, 2006, pp. 31–34.
Contents lists available at SciVerse ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
Multibias scalable HEMT small-signal modeling based on a hybrid direct extraction/particle swarm optimization approach Y. Campos-Roca a,n, H. Massler b, A. Leuther b a b
´ceres, Spain Universidad de Extremadura, Escuela Polite´cnica, Avda. de la Universidad, s/n, E-10003, Ca Fraunhofer Institute for Applied Solid State Physics (IAF), Tullastrasse 72, D-79108 Freiburg, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 June 2011 Received in revised form 22 April 2012 Accepted 24 April 2012 Available online 9 May 2012
A new multibias HEMT small-signal model extraction method is proposed. The approach, based on scaling rules, combines direct extraction techniques and a particle swarm optimization algorithm. This method has been successfully tested with PHEMTs and MHEMTs, leading to accurate and scalable models up to 70 and 120 GHz, respectively. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Metamorphic high electron mobility transistor (MHEMT) Particle swarm optimization (PSO) Pseudomorphic high electron mobility transistor (PHEMT) Scalability Small-signal modeling
1. Introduction The design of linear millimeter-wave circuits requires accurate small-signal models of the active components. Besides, smallsignal models are often used as building blocks to generate largesignal ones, by extracting small-signal parameters from S-parameter measurements at different bias points. Small-signal models are mostly based on an equivalent circuit representation. As an alternative, some approaches based on artificial neural networks have been proposed [1]. However, the former type of models is generally preferred because it provides a compact representation and the possibility to link the circuit parameters to the device physical structure. There are two main families of extraction techniques that can be used to determine the model parameters: direct extraction and optimization-based techniques. Direct-extraction methods are typically based on cold-FET S-parameter measurements, used to obtain the parasitic component values. The intrinsic elements are analytically calculated after de-embedding the extrinsic ones. [2,3] are classical references. Other contributions in the area of analytical techniques are [4–6]. This type of methods are faster than optimization-based techniques, but they are susceptible to measurement errors and their implementation is device specific.
n
Corresponding author. Tel.: þ34 927 257195x57561; fax: þ 34 927 257203. E-mail address: [email protected] (Y. Campos-Roca).
0026-2692/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mejo.2012.04.008
Often, a subsequent optimization step is performed by hand to improve the fitting in active bias conditions. Optimization-based extraction techniques aim to determine the values of the small-signal model elements by fitting calculated S-parameters to measured ones. Many contributions propose single-bias extraction algorithms, which are not useful when small-signal models are intended to be used as building blocks to generate large-signal ones. Therefore, a multibias solution is required. In general, the main drawback of optimization-based techniques is the convergence to nonphysical element values due to the multiple local minima of the cost function. Additionally, the obtained solution may be sensitive to the optimization starting values. To overcome this problem, several extraction methods based on global optimization techniques (such as genetic algorithms) have been proposed. Several approaches that combine data measured at different bias points have been suggested in the literature [7,8]. To the knowledge of the authors, the multibias extraction algorithms proposed in the literature are not based on scaling rules for the equivalent circuit components, i.e., each optimization problem aims to fit measurements from a single device geometry. This paper proposes a novel extraction procedure, which combines S-parameter measurements from devices with different gate widths and at different bias points into one unified optimization problem. The algorithm has two characteristics that contribute to obtaining physically meaningful values. The first one is the use of high-quality starting values. Direct extraction
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techniques are initially employed to give a rough estimation of the parasitic component values. These rough estimations are used as starting values in the subsequent optimization process. The second characteristic is the use of scaling rules in the definition of the optimization problem. A single model for devices with different gate widths is automatically obtained. The complexity of the optimization problem is kept relatively low, since only a reduced number of measurement points is required. The proposed method uses particle swarm optimization (PSO). Since the pioneering work of Kennedy and Eberhart [9], PSO has been proved to solve many problems. It is a population-based stochastic optimization strategy showing faster convergence speed and simpler implementation than genetic or ant colony algorithms, with comparable or better-quality solutions [10]. The performance of the proposed extraction method is evaluated by using wide-band multibias S-parameter measurements on PHEMTs and MHEMTs. Scalability and good accuracy are demonstrated up to 70 and 120 GHz, respectively. The paper is organized as follows. In Section 2, the extraction process (including the scaling rules) is described. The results are presented in Section 3. Finally, some conclusions are drawn in Section 4.
2. Extraction process In order to keep the extraction method as simple as possible, the ordinary 16-element model topology has been used (see Fig. 1). The equivalent circuit is divided into an extrinsic part (outside the dashed box), whose elements are assumed to be biasindependent, and an intrinsic part (inside the dashed box), whose elements are bias-dependent. The extraction process has the following main characteristics: – – – –
scalability, combination of analytical and optimization-based techniques, particle-swarm optimization, multibias data fitting, based on a reduced number of bias points.
To design millimeter-wave circuits, it is desirable to have scalable models, that provide greater flexibility. The proposed algorithm is based on scaling rules, and aims to obtain a single model for devices with different gate widths. The whole extraction algorithm has been implemented in MATLAB code.
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The multibias procedure is based on a hybrid direct extraction/ optimization approach. The optimization variables are the different parameters in the scaling rules of the extrinsic components (see Section 2.1). In order to limit the dimension of the optimization problem, the intrinsic elements are not optimized. They are analytically calculated after de-embedding the extrinsic ones [2]. However, their scalability must also be guaranteed. This is taken into account by introducing a scalability error term into the cost function (see Section 2.2). As mentioned above, the use of measurements from different device sizes and scaling rules in the definition of the optimization problem also contributes to achieving physically meaningful values. Fig. 2 shows the algorithm pseudocode in a compact way. In order to obtain a rough a priori knowledge of the parameter values, direct extraction techniques are employed. These techniques, based on cold-FET (VDS ¼0) measurements, provide starting values for the optimization variables. The initial values of the parasitic capacitances (Cpgs, Cpgd and Cpds) are determined by performing S-parameter measurements at a pinch-off gate voltage on devices with different gate widths [11]. Also initial values for the extrinsic resistances and inductances are determined by using the method proposed in [2]. Rough estimations are sufficiently good; thus the calculation does not require such a precise knowledge of technological parameters, such as when the technique is applied without a subsequent optimization step. Besides, high gate currents can be avoided in the cold-FET forward measurements if the devices are sensitive to forward gate bias. The calculation of extrinsic resistances and inductances is performed on transistors with different gate widths, so that the initial values for the parameters in the scaling rules can be determined. The optimization step is based on a reduced number of bias points and sizes: two measurements on devices with different gate widths in the saturation region (at the same bias voltages), and two additional measurements (in the pinch-off and ohmic regions) on devices of one size. This reduced set of measurements allows obtaining accurate results at a low computational cost. The optimization procedure is based on a PSO algorithm. PSO [9] is a computation technique inspired by the social behavior of bird flocking or fish schooling. PSO has several attractive attributes, including the fact that the basic algorithm is very easy to understand and implement. It is similar in some ways to genetic algorithms, but it shows superior performance in convergence speed [10]. PSO uses a population of particles that fly through the problem hyperspace with given velocities. Each particle’s position corresponds to a candidate solution to the optimization problem. A cost function, that measures the deviation from the desired solution, is defined. The cost function defined in this approach is presented in Section 2.2. As the particles fly through the problem hyperspace, each particle remembers its own personal best position that it has ever found, called its local best. Each particle also knows the best position found by any particle in the swarm, called the global best. At each iteration, the velocities of the individual particles are updated according to their own best positions and the global best. Further details will be given in Section 2.2. 2.1. Scaling rules The extraction process aims to obtain models that are scalable with the gate width, for a fixed number of gate fingers. The scaling rules the extraction process is based on are the following:
Fig. 1. Equivalent circuit topology for HEMT small-signal modeling. The dashed box contains the intrinsic part of the device.
a) The pad capacitances (Cpgs, Cpgd, Cpds) should take a fixed value; they are not dependent on the gate width [11].
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Fig. 2. Pseudocode of the extraction algorithm.
b) The source and drain resistances vary inversely with the total gate width [12]. Therefore, the following scaling rules are used: Rd ¼ Rd0/Wg and Rs ¼Rs0/Wg, where Wg is the total gate width, and Rd0 and Rs0 are constants. The starting values for Rd0 and Rs0 in the optimization process are determined by using cold-FET forward measurements from devices with different gate widths [2]. c) The gate resistance is considered to comprise three components [13]: the first one is a residual resistance (Rg0), the second one is a classical expression (Rg1Wg) and the third component (Rg2/Wg) is a contact resistance. Therefore, for a fixed number of gate fingers, the scaling rule has the following formulation: Rg ¼Rg0 þ Rg1Wg þ Rg2/Wg, where Rg0, Rg1 and Rg2 are constants. The starting values for these three constants in the optimization process are calculated by fitting this function to the gate resistance values obtained by direct extraction. The extractions are based on cold-FET forward measurements from devices with different gate widths [2]. d) In the case of the extrinsic inductances, different scaling rules have been proposed in the literature. For example, in [14] the authors consider that the values do not vary much for their interdigitated power PHEMT devices and they do not need any scaling. In [12], Lg and Ld are considered to scale proportionally to the gate width. The extraction process proposed here is
based on flexible scaling rules, comprising an independent term and a scaling factor, i.e., Lg ¼Lg0 þLg1Wg; Ld ¼Ld0 þLd1Wg; and Ls ¼ Ls0 þ Ls1Wg. The starting values for the different parameters in the previous scaling rules are also determined by using cold-FET forward measurements from devices with different gate widths [2]. e) For the intrinsic components, the scaling rules are simple and conventional: gm, Cgs, Cds and Cgd scale proportionally to the gate width, whereas Rgs and Rds scale inversely proportional to the gate width. The transconductance delay t is considered constant with respect to the gate width [15].
2.2. Optimization process The optimization process is based on a PSO algorithm [9]. The swarm is initialized with a population of Np random solutions, called particles, and searches for optima by updating generations. Each particle is defined by a position and a velocity, and it moves through the search hyperspace. Each particle’s velocity is dynamically updated according to its own and its neighbors’ previous behavior. The updating process is based on three terms, called the ‘‘social’’, the ‘‘cognitive’’ and the ‘‘inertia’’ terms. The ‘‘social’’ term attempts to guide the particle to the best position found by any
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particle in its neighborhood so far (called the global best, gbest), the ‘‘cognitive’’ term conducts it to the best position found by itself so far (called the local best, pbest), and the ‘‘inertia’’ term is related to the memory of its previous velocity. The following equations describe the updating process of a particle’s velocity and position: vn ¼ ovn1 þ c1 r 1 ðpbest,n xn1 Þ þ c2 r 2 ðg best,n xn1 Þ, xn ¼ xn1 þ vn where vn and xn are the particle’s velocity and position at the nthiteration, respectively; o is the inertia weight; c1 and c2 are acceleration constants (the ‘‘cognitive’’ ratio and the ‘‘social’’ ratio, respectively); and r1 and r2 are random numbers uniformly distributed between 0 and 1. Once the velocity vn has been calculated, the particle moves to its next location xn. The swarm will continue moving until a criterion is met: a sufficiently low cost or a maximum number of iterations. In the considered extraction problem, the optimization variables are the parameters in the scaling rules of the extrinsic components (see Section 2.1). 14 optimization variables have been defined (Cpgs, Cpgd, Cpds, Rd0, Rs0, Rg0, Rg1, Rg2, Lg0, Lg1, Ld0, Ld1, Ls0, and Ls1); therefore, the particles’ positions and velocities are 14-dimensional vectors. As mentioned above, rough estimates of the extrinsic components are initially obtained by applying direct extraction techniques. These estimations allow us to calculate the starting values for the optimization variables. The particles’ positions are initialized by generating random vectors, uniformly distributed, inside a 14-dimensional search hyper-rectangle. Let Pinit i , i¼1,y,14, denote the initial values of the optimization variables. In the proposed approach, the lower Pmin and upper Pmax boundaries are determined by specifying a i i percentage variation (Pi). If any of the parameters had a null starting value Pinit i , the lower and upper boundaries would also be null. In these cases, an arbitrary upper boundary (Mi), empirically determined, is specified. Therefore, the search hyper-rectangle is defined as ½Pmin , Pmax , i ¼ 1,. . .,14, where i i Ymin P , i ¼ 1,. . .,14, ¼ max 0, Pinit 1 i i i 100 and Ymin i
¼
8 Pi < Pinit 1þ 100 i :M
i
9 if Pinit a0 = i , if Pinit ¼0 ; i
Lg0, Lg1, Ld0, Ld1, Ls0, Ls1]. However, the intrinsic components are not optimized but extracted after de-embedding the extrinsic ones. Thus it is necessary to include a second goal related to the scalability of these intrinsic elements. The cost function comprises two terms (e1 and e2), each of them related to one of the previously defined goals, and an additional penalty term (e3), that is introduced to include constraints. The first term represents the error between measured and modeled S-parameters:
Nf meas mod N X 2 X 2 X X S f f ,x S 1 k k
pqn pqn
, e1 ðxÞ ¼
4N f N n ¼ 1 p ¼ 1 q ¼ 1 k ¼ 1
Smeas fk pqn
k ¼max
1,:::,N f where Smeas and Smod pqn pqn represent measured and calculated Sparameters, respectively, Nf denotes the number of frequency points and fk each frequency value. A logarithmic frequency step has been used. In this approach, only a reduced number of Sparameter measurements is required. All the results reported in Section 3 have been obtained by using four frequency sweeps (N ¼ 4): two of them are performed on transistors with different gate widths at the same active bias point in the saturation region, and two additional measurements are performed on only one device size (in the pinch-off and ohmic regions). These measurements allow us to obtain accurate results in a wide range of bias points and gate widths with a low computational cost. The second term in the cost function represents the scalability error of the intrinsic components. The scalability check is performed at an active bias point by using the first two measurements (performed on transistors with different gate widths). Once the intrinsic elements for the two device sizes are extracted, simple straight-line equations are obtained, where the independent variable is always the gate width and each dependent variable is one of the following intrinsic components: Cgs, Cds, Cgd, Ggs ¼1/Rgs, Gds ¼1/Rds and gm. For notation convenience, function names are used for these intrinsic elements at the considered active bias condition, i.e., h1 ¼Cgs(Wg, x), h2 ¼Cds(Wg,x),y,h6 ¼gm(Wg,x). The straight-line equations are the following: hj ðW g ,xÞ ¼ aj ðxÞ þ bj ðxÞW g ,
i ¼ 1,. . .,14:
Regarding the particles’ velocities, they are also initialized by generating random numbers, uniformly distributed, in a hyperrectangle. This hyper-rectangle is defined as ½vimax ,vimax 14 , i ¼ 1,. . .,14, vimax being the maximum velocity along each dimension of the search hyperspace. In the successive iterations, the algorithm also limits the maximum velocity allowed along each dimension to the value vimax : One of the key aspects in the definition of any optimization problem is to select an appropriate cost function. This function will be evaluated to determine the deviation from the desired solution. In order to obtain a scalable small-signal model, two goals must be achieved: a) Minimum error. the difference between measured and modeled S-parameters, over the considered frequency range, for different bias points and device sizes, must be as small as possible. b) Scalability of the intrinsic components. the scalability of the extrinsic components is taken into account by the use of scaling rules (see Section 2.1). The parameters in these scaling rules are the optimization variables, so that each candidate solution is defined as x¼ [Cpgs, Cpgd, Cpds, Rd0, Rs0, Rg0, Rg1, Rg2,
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j ¼ 1,. . .,6:
The results of each calculation will show an independent term (aj) equal to zero if the corresponding intrinsic component (hj) scales proportionally to the gate width (see the scaling rules in Section 2.1, point (e)). Thus the sum of normalized independent terms represents the scalability error for the six considered intrinsic components. Additionally, the remaining intrinsic element (the transconductance delay, t) is considered to be constant with respect to the gate width. Therefore, the global scalability error is described by the following equation:
6
X aj ðxÞ
e2 ðxÞ ¼
hj W g1 ,x þ hj W g2 ,x =2
j¼1
t W ,x t W ,x
g1 g2
, þ C t
t W g1 ,x þ t W g2 ,x =2
where Wg1 and Wg2 represent the two considered gate widths, and Ct is a weighting factor. To the knowledge of the authors, the inclusion of a term representing the scalability error in a cost function has not been reported in the literature related to smallsignal model extraction so far. Also a penalty method is used to include constraints. In the case of negative values for the intrinsic components, a third term (e3) is added to the cost function that prescribes a very high cost. The final cost function is calculated as the linear weighted sum of the terms e1, e2 and e3.
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3. Experiments and results Two sets of experiments have been performed to validate the proposed extraction method. The first one is applied to double ddoped AlGaAs/InGaAs/GaAs PHEMTs, with 0.15 mm T-shaped gates. These devices exhibit an extrinsic transit frequency fT of 100 GHz and a maximum oscillation frequency fmax of 150 GHz. As mentioned above, the starting values for the optimization variables have been obtained by applying direct extraction techniques. These starting values are the following: Cpgs ¼7 fF, Cpgd ¼2 fF, Cpds ¼ 7 fF, Rd0 ¼0.3 O mm, Rs0 ¼ 0.2 O mm, Rg0 ¼ 3 O, Rg1 ¼0 O/mm, Rg2 ¼0 O mm, Lg0 ¼3 pH, Lg1 ¼70 pH/mm, Ld0 ¼3 pH, Ld1 ¼ 60 pH/ mm, Ls0 ¼0 pH, and Ls1 ¼0 pH/mm. As mentioned above, four Sparameter measurements (frequency sweeps) have been used in the multibias optimization procedure. The first two measurements correspond to the same active bias point (VGS ¼0 V, VDS ¼ 2.6 V) and two different device sizes: 2 30 mm2 and 2 60 mm2. The third and fourth measurements correspond to a 2 60 mm2
Fig. 3. Cost of gbest versus iterations, for different numbers of particles.
geometry at a pinch-off condition (VGS ¼ 1 V, VDS ¼ 2.4 V) and at a bias condition in the ohmic region (VGS ¼ 0.3 V, VDS ¼0.4 V). The percentage variation (Pi) is set to 80% for all optimization variables. Fig. 3 shows the convergence plot for different numbers of particles (Np ¼10, Np ¼12 and Np ¼15). This plot shows the cost function values of the successive gbest positions. As can be observed, a swarm of only Np ¼12 particles allows us to obtain good results. The values of the different PSO parameters that have been used are the following: number of particles Np ¼12, relative maximum velocity virmax ¼2, i¼1,y,14, cognitive ratio c1 ¼1.5, social ratio c2 ¼ 1.5 and inertia weight o ¼1. The final values of the optimization parameters are the following: Cpgs ¼12.6 fF, Cpgd ¼ 1.4 fF, Cpds ¼2.8 fF, Rd0 ¼ 0.2 O mm, Rs0 ¼0.04 O mm, Rg0 ¼3.75 O, Rg1 ¼ 0 O/mm, Rg2 ¼ 0 O mm, Lg0 ¼5.4 pH, Lg1 ¼126 pH/mm, Ld0 ¼0.6 pH, Ld1 ¼32 pH/mm, Ls0 ¼0 pH, and Ls1 ¼3 pH/mm. As mentioned above, the intrinsic components are not optimized but analytically extracted. However, due to the inclusion of a term representing the scalability error into the cost function, the obtained intrinsic components satisfy the scaling rules (Section 2.1, point (e)). The extraction process has been successfully validated by comparison of measured and modeled S-parameters, at different bias settings and for different device sizes. A single scalable model is used for the different geometries. Fig. 4 shows the results for a 2 30 mm device in the following bias setting: VGS ¼ 0.2 V, VDS ¼3 V. As can be observed, a very good agreement has been obtained up to 70 GHz. Figs. 5 and 6 show verification results for a completely new device geometry, which has not been used in the optimization procedure: 2 45 mm. Fig. 5 corresponds to the following bias point: VGS ¼ 0.4 V, and VDS ¼1.5 V, while Fig. 6 was obtained at a cold-FET pinch-off condition (VGS ¼ 1 V, and VDS ¼0 V). Finally, Fig. 7 shows a comparison between modeled and measured S-parameters for a 2 60 mm device at a gate voltage of 0.5 V and a drain voltage of 3 V. This is a completely new bias point, which has not been used in the optimization process for any device geometry. The second set of experiments is based on InAlAs/InGaAs MHEMTs. These devices have 100 nm T-shaped gates and achieve an extrinsic transit frequency of 220 GHz and a maximum oscillation frequency of 300 GHz. More details about this MHEMT process can be found in [16].
Fig. 4. Comparison of measured and modeled S-parameters of a 2 30 mm AlGaAs/InGaAs PHEMT. Frequency range: 0.5–70.5 GHz. Bias point: VGS ¼ 0.2 V, and VDS ¼ 3 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig.5. Comparison of measured and modeled S-parameters of a 2 45 mm AlGaAs/InGaAs PHEMT. Frequency range: 0.5–50 GHz. Bias point: VGS ¼ 0.4 V, and VDS ¼1.5 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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was set to 50% for the pad capacitances and 100% for the other optimization variables. Again four S-parameter measurements have been used in the multibias optimization procedure. Two of the measurements correspond to an active bias condition in the saturation region (VGS ¼0.2 V, and VDS ¼1 V) on two different device sizes: 2 30 mm and 2 45 mm. The pinch-off measurement (VGS ¼ 1 V, and VDS ¼1.5 V) was performed on a 2 45 mm transistor. Finally, the fourth measurement corresponds to the ohmic region (VGS ¼ 0.2 V, and VDS ¼0.3 V) and was performed also on the 2 45 mm device. The final values of the optimization parameters are the following: Cpgs ¼15.7 fF, Cpgd ¼4 fF, Cpds ¼5.25 fF, Rd0 ¼0 O mm, Rs0 ¼0 O mm, Rg0 ¼ 6.8 O, Rg1 ¼14 O/mm, Rg2 ¼ 0 O mm, Lg0 ¼Ld0 ¼Ls0 ¼0 pH, Lg1 ¼100 pH/mm, Ld1 ¼0 pH/mm, and Ls1 ¼ 0 pH/mm. Figs. 8–11 show the results at different bias points and for different device sizes. Fig. 8 presents a comparison between modeled and measured S-parameters, for a 2 45 mm device and a bias point in the ohmic region (VGS ¼0 V, and VDS ¼0.1 V), up to 120 GHz. Fig. 9 refers to the same geometry and a different bias condition: VGS ¼ 0.4, and VDS ¼1.5. Note that this is a completely new bias point, regarding the measurements that have been used in the optimization process. Fig. 10 corresponds to a 2 60 mm device and an active bias point (VGS ¼ 0.2 V, and VDS ¼1 V). Finally, Fig. 11 corresponds to a 2 15 mm MHEMT, in a cold-FET pinchoff condition (VGS ¼ 1.5, and VDS ¼0). These two geometries
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Fig. 6. Comparison of measured and modeled S-parameters of a 2 45 mm AlGaAs/InGaAs PHEMT. Frequency range: 0.5–50 GHz. Bias point: VGS ¼ 1 V, and VDS ¼ 0 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 8. Comparison of measured and modeled S-parameters of a 2 45 mm InAlAs/InGaAs MHEMT. Frequency range: 0.25–120 GHz. Bias point: VGS ¼0 V, and VDS ¼ 0.1 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 7. Comparison of measured and modeled S-parameters of a 2 60 mm AlGaAs/InGaAs PHEMT. Frequency range: 0.5–50 GHz. Bias point: VGS ¼ 0.5 V, and VDS ¼ 3 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The same equivalent circuit topology was used (see Fig. 1). The starting values for the optimization variables (obtained by direct extraction) are the following: Cpgs ¼10.5 fF, Cpgd ¼3.5 fF, Cpds ¼6 fF, Rd0 ¼0.19 O mm, Rs0 ¼0.13 O mm, Rg0 ¼5 O, Rg1 ¼10 O/mm, Rg2 ¼0.2 O mm, Lg0 ¼Ld0 ¼Ls0 ¼0 pH, Lg1 ¼50 pH/mm, Ld1 ¼ 60 pH/ mm, and Ls1 ¼0 pH/mm. In this case, the percentage variation (Pi)
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Fig. 9. Comparison of measured and modeled S-parameters of a 2 45 mm InAlAs/InGaAs MHEMT. Frequency range: 0.25–120 GHz. Bias point: VGS ¼ 0.4 V, and VDS ¼ 1.5 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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problem has not been reported in the literature so far. A single model for devices with different sizes is obtained. By the use of a reduced number of S-parameter measurements in the optimization procedure, the computational effort is kept low. The extraction procedure has been verified with measurements on PHEMTs and MHEMTs. The reproducibility of the results has been successfully checked. Good agreement between measured and calculated S-parameters is achieved in a wide range of bias points and up to 70 and 120 GHz, respectively.
Acknowledgment
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Fig. 10. Comparison of measured and modeled S-parameters of a 2 60 mm InAlAs/InGaAs MHEMT. Frequency range: 0.25–120 GHz. Bias point: VGS ¼0.2 V, and VDS ¼ 1 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The authors would like to thank the Fraunhofer IAF, in Freiburg (Germany), for providing the PHEMT and MHEMT measurements. This work has been partially funded by the Junta de Extremadura (Project GR10126). References
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Fig. 11. Comparison of measured and modeled S-parameters of a 2 15 mm InAlAs/InGaAs MHEMT. Frequency range: 0.25–50 GHz. Bias point: VGS ¼ 1.5 V, and VDS ¼ 0 V. Blue solid-lines: calculated values. Red crosses: measured values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(2 60 and 2 15 mm) have not been used in the optimization process. The results prove the scalability of the extracted model. The results show a very good agreement between measured and calculated S-parameters. All the experiments have been repeated many times to guarantee the reproducibility of the results. The experiments have been carried out on a 2 GHz Intel Core 2 Duo processor with 3 GB of RAM.
4. Conclusion A novel multibias approach for HEMT small-signal equivalentcircuit model extraction has been proposed. The method is based on scaling rules and uses a hybrid direct extraction/particle swarm optimization-based approach. To the knowledge of the authors, the combination of measurements on different device sizes and at different bias points into one unified optimization
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