A New Small Signal Model Parameter Extraction Method Applied to GaN Devices A. Jarndal and G. Kompa University of Kassel, Fachgebiet Hochfrequenztechnik, Wilhelmshöher Allee 73, D-34121 Kassel, Germany, Tel: +49-561-804-65 09, Fax: -6529, Email: jarandal@uni-kassel.de Abstract — A new parasitic elements extraction method applied to GaN devices is presented. First, using cold Sparameter measurements, high quality starting values for the extrinsic parameters are generated that would place the extraction close to the global minimum of the objective function for the distributed equivalent circuit model. In a second step, the optimal model parameter values are searched through optimization using the starting values already obtained. The validity of the developed method and the proposed small-signal model is verified by comparing the simulated wide-band smallsignal S-parameter, over a wide bias range, with measured data of a 0.5 µm GaN HEMT with 2x 50 µm gate width. Index Terms — GaN HEMT, semiconductor device modeling, computer aided analysis, parameter extraction. I. INTRODUCTION GaN HEMT is supposed to be an excellent candidate for high power applications. Where both the high frequency response and high breakdown voltage for this device make it ideally suited for high power application. GaN amplifier design requires accurate large signal model (LSM) for the GaN HEMT. This accurate model should simulate the breakdown, forward conduction, and frequency dispersion effects. In bottom-up modeling technique a multi-bias small– signal measurement is carried out over a range of bias points, and LSM is then determined from small signal model (SSM) derived at each of these bias points. Therefore, the accuracy of the constructed LSM depends on the accuracy of the biasdependent SSM, which should reflect the electrical and physical characteristics of the device. Accurate determination for the intrinsic bias-dependent circuit of GaN HEMT SSM requires an efficient extraction method for the parasiticelements of the device. Due to high contact resistance of GaN device, the standard circuit model extraction method [1-3] cannot be performed in ordinary way [4]. Chigaeva et al. [5] showed that the series elements of the equivalent circuit model, for GaN HEMT, could be extracted from cold Sparameter measurement at high gate-forward voltage. But there is a problem concerning the reliability of some extracted elements, especially the extrinsic feedback inductance (Ls). Therefore, a special extraction method should be developed to extract the parasitic-elements of GaN device. In this paper a new reliable parameter extraction approach was developed, for GaN-based HEMT, using only cold S-parameter measurement. The main advantage of this method is that it 0-7803-8846-1/05/$20.00 (C) 2005 IEEE 1423 gives reliable values for the parasitic-elements of the device without need for additional measurements or separate test pattern. II. A GENERAL DISTRIBUTED SMALL-SIGNAL EQUIVALENT CIRCUIT MODEL Because the knowledge of distributed effects is important to identify the device parasitic-elements for further minimization, 22-element distributed model shown in Fig. 1 is used as SSM for GaN HEMT. The main advantage of this model is that it accounts for all expected parasitic-elements of the device. And so it is more suitable for scalable LSM construction. Intrinsic FET Fig. 1. 22-element distributed model for active GaN HEMT. In this model Cpgi, Cpdi, and Cgdi account for the interelectrode and crossover capacitances (due to air-bridge source connections) between gate, source, and drain. While Cpga, Cpda, and Cgda account for parasitic-elements due to the pad connections, measurement equipment, probes, and probes tipto-device contact transitions. III. EXTRINSIC PARAMETER EXTRACTION This extraction method is an optimisation-based method. The efficiency of this extraction type depends on the quality of starting values and the number of optimisation variables. In our case high quality measurement-correlated starting values for the extrinsic elements are generated [6]. To minimize the number of optimisation variables only the extrinsic elements of the SSM will be optimised, while the intrinsic elements are determined quasi-analytically from the de-embedded Z- parameter [7]. For consistent extraction of the parasitic elements of the SSM, broadband S-parameter measurements (up to 60 GHz) are needed for the analysed 2x50 µm device. This reduces significantly for larger devices, e.g. to 20 GHz for 8x125 µm device. A. Generation of starting value of SSM parameters The starting values of the SSM parameters are generated from cold S-parameter measurements. The starting values of parasitic capacitances and inductances are generated from cold pinch-off measurements, while those of the parasitic resistances are generated from cold forward measurements. The whole starting values generating procedure can be summarized as follows: 1) The equivalent circuit in Fig. 1 of the active device can be used for a cold pinch-off device (VDS = 0.0 V and VGS < Vp) if the drain current source, the gate forward conductance and the output channel conductance are excluded. Moreover, at low frequencies (below 5 GHz) this circuit reduces to a capacitive network shown in Fig. 2 and the Y-parameters of this equivalent circuit can be written as: Y11 = jω (C gso + C gdo ) (1) Y22 = jω (C dso + C gdo ) (2) Y12 = Y21 = jω C gdo (3) C gdo = C gda + C gdi + C gd (4) C gso = C pga + C pgi + C gs (5) C dso = C pda + C pdi + C ds . (6) specified ranges. Here for each assigned values of the capacitances, the other model parameters are determined from the de-embedded Y-parameter. Finally the minimum error model parameters are obtained. Cpga and Cpda are scanned from 0 to 0.5Cdso while Cgda is scanned from 0 to 0.5Cgdo. During the scanning process, Cpga is assumed to be equal to Cpda [6]. C pga ≈ C pda . The gate-drain inter-electrode capacitance Cgdi is assumed to be twice of pad capacitance Cgda value. C gdi ≈ 2C gda . For symmetrical gate-source and gate-drain spacing, the depletion region will be uniform under pinch-off, so that C gs ≈ C gd = C gdo − C gdi − C gda . The value of Cpgi is calculated using C pgi = C gso − C gd − C pga . (11) Cgda Intrinsic FET G Lg Rg δLg δRg C g Cd δRd δLd Rd Ld D Cs δRs Intrinsic FET C pga D δL s C pda Rs Cgd S (10) With the GaN device under analysis Cpdi is a significant part of the total drain-source capacitance. Therefore, it is found that the assumption C gdi C gs (9) minimizes the error between the simulated and measured Sparameters. For medium and high frequency range, the intrinsic transistor of the pinch-off model is represented in Tnetwork as shown in Fig. 3 where the inter-electrode capacitances (Cpgi, Cpdi, and Cgdi) have been absorbed in the intrinsic capacitances (Cgs, Cds, and Cgd). The values for Cpga, Cpda, and Cpda are de-embedded from Y-parameter and then converted to Z-parameter. 3) Estimation of inductances from the stripped Z-parameter data. C gda Cpga C pgi (8) C pdi = 3 C pda where G (7) Ls C ds Cpdi C pda S S Fig. 2. Cold pinch-off equivalent circuit for the GaN HEMT at low frequency. The total capacitances for gate-source, gate-drain, and drain source branches are determined from the low frequency range of pinch-off S-parameter measurements, which are converted to Y- parameter. 2) The next step is searching for the optimal distribution of the total capacitances, which gives the minimum error between the measured and simulated S-parameters. This is achieved by scanning Cpga, Cpda, and Cgda values within the 1424 S Fig. 3. T-network representation of a cold pinch-off FET equivalent circuit. The stripped Z-parameter can be written as: Z 11 = R g + Rs + jω (L g + Ls ) + 1  1 1  + δZ g (12) +  jω  C g C s  Z 22 = Rd + Rs + jω (Ld + Ls ) + 1  1 1    + δZ d + jω  C d C s  (13) (14) where δZ d = δR d + δR s + jω (δLd + δL s ) (16) δZ g = δR s + jωδL s . (17) δZg, δZd, and δZs represent correction terms related to the intrinsic parameters of the model. Ignoring the correction terms and multiplying the Z-parameters by ω and then taking the imaginary parts give,  1 1  Im[ωZ 11 ] = (L g + L s )ω 2 −  +  C  g Cs  (18)  1 1   + Im[ωZ 22 ] = (Ld + L s ) ω 2 −  C C s   d (19) Im[ωZ 12 ] = Lsω 2 − 1 . Cs Extrinsic Parameters Lg =39.50 pH Cpga =7.00 fF Ld =46.40 pH Cpda =7.00 fF Ls =5.25 pH Cgda =0.50 fF Cpdi =21.00 fF Rg =5.20 Ω Cpgi =4.90 fF Rd =9.20 Ω Cgdi =1.00 fF Rs =6.60 Ω (20) Hence, the values of Lg, Ld and Ls can be extracted from the slope of Im[Zij] versus ω2 curve. 4) The next step is estimation of resistances from the stripped Z-parameter data. First, the estimated values of inductances described above and the inter-electrode capacitances (Cpgi, Cpdi, and Cpdi) are de-embedded. But the incomplete de-embedding of the outer capacitances and the inductances introduce non-linear frequency dependence in the real part of de-embedded Z-parameters. By multiplying the de-embedded Z-parameter by ω2, this effect is reduced [6]. Ignoring the correction terms and multiplying the deembedded Z-parameter by ω2 and then taking the real part of this Z-parameter, gives ω 2 Re[Z 11 ] = ω 2 (R g + Rs ) ω Re[Z 22 ] = ω 2 2 (Rd + Rs ) ω 2 Re[Z 12 ] = ω 2 Rs . TABLE I STARTING VALUES FOR 22-ELEMENT EQUIVALENT CIRCUIT MODEL OF A 0.5 µm GaN HEMT WITH 2×50 µm GATE WIDTH (21) A good agreement between the measured and simulated Sparameters, shown in Fig. 4, verifies the high quality of these starting values, particularly at low and medium frequencies. 1 0.5 0.95 0.9 Simulated S 11 Simulated S 22 Measured S 11 Measured S 0.85 0.8 20 0.4 0.3 0.2 Simulated S 12 Simulated S 21 Measured S 12 Measured S 0.1 22 0 40 0 60 21 0 Frequency [GHz] 20 40 60 Frequency [GHz] 0 100 -20 -40 -60 -80 Simulated S 11 Simulated S 22 Measured S 11 Measured S -100 -120 (22) Intrinsic Parameters Cgs =20.37 fF Gm =0.0 mS Gds =0.0 mS Cds =8.47 fF Cgd =20.01 fF Ggsf =0.0 mS Ggdf =0.0 mS Ri =0.0 Ω Rgd =0.0 Ω τ =0.0 ps Magnitude (15) Phase [°] δZ g = δR g + δR s + jω (δL g + δLs ) model parameters, P(εmin), corresponding to the lowest error, εmin, is then taken as the appropriate starting value. 7) Because of unavoidable high measurement uncertainty for cold pinch-off device, the determination of a reliable starting value for the extrinsic resistances is difficult if not impossible [6]. More reliable starting value was generated using a gate-forward measurement at high gate voltage (>2.0 V). This is due to the higher conduction band of GaN-based HEMT with respect to the corresponding GaAs-based HEMT. Therefore, significantly higher voltages have to be applied to reach the condition when the influence of the gate capacitance is negligible. The complete starting values for the pinch-off device model parameters are tabulated in Table I. Magnitude 1 + δZ s jωC s Phase [°] Z12 = Z 21 = Rs + jωLs + 22 0 20 40 60 50 Simulated S 12 Simulated S 21 Measured S 12 Measured S 0 -50 21 0 Frequency [GHz] 20 40 60 Frequency [GHz] Fig. 4. S-parameter fitting with starting element values for 22element equivalent circuit model of a 0.5 µm GaN HEMT. (23) By linear regression, the value of Rg+Rs, Rd+Rs and Rs can be extracted from the slope of ω2Re[Zij] versus ω2 curve. 5) The resulting estimated parameters are used to simulate the device S-parameters, which are then compared with the measured ones to calculate the residual fitting error (ε). 6) The outer capacitances (Cpga, Cpda, and Cgda) are incremented and the procedure is repeated until Cpga (Cpda) equals to 0.5Cdso and Cgda equal to 0.5Cgdo. The vector of 1425 B. Final Model Parameter Optimization The procedure for the generation of starting values of the model parameters was discussed in part A. In this part the results of the optimal value for each model parameter is presented. Model parameter optimization is done based on the principle of bi-directional optimization technique proposed by Lin [7]. Using this technique, the extrinsic parameters are optimized using a modified Simplex algorithm proposed in [8], while the intrinsic parameters are optimized by means of data fitting. The optimized pinch-off device parameters are listed in Table II. TABLE II OPTIMIZED PINCH-OFF DEVICE PARAMETERS OF A 0.5 µm GaN HEMT WITH 2×50 µm GATE WIDTH Extrinsic Parameters Cpga =9.97 fF Lg =46.55 pH Ld =47.90 pH Cpda =7.13 fF Ls =6.25 pH Cgda =0.47 fF Cpdi =29.42 fF Rg =4.80 Ω Cpgi =7.09 fF Rd =11.80 Ω Cgdi =0.86 fF Rs =5.47 Ω Intrinsic Parameters Cgs =15.38 fF Gm=0.0 mS Gds=0.0 mS Cds =0.0 fF Cgd =20.17 fF Ggsf =0.0 mS Ggdf =0.0 mS Ri =0.0 Ω Rgd =0.0 Ω τ =0.0 ps After de-embedding the extracted extrinsic parameters in section III, the bias-dependent intrinsic parameters can be extracted. Fig. 5 shows the extracted intrinsic capacitances and conductances, at VGS = -1.0 V and VDS = 10.0 V, versus frequency. The frequency-independence of these intrinsic elements verifies the validity of the proposed SSM topology and the developed extraction method. Fig. 6 shows the results of S-parameter simulation at different bias points, in linear and saturation regions, for a wideband frequency range. Very good agreement is achieved between measurements and simulations. (fF) gd C , C C gs The authors gratefully acknowledge the support from the German Ministry of Education and Research (BMBF), contract No. 01BU385, and from the Top Amplifier Research Groups in a European Team (TARGET), contract No. 507893. The authors also thank FBH for supplying the GaN HEMTs and IAF for performing the wide-band S-parameter measurements. REFERENCES [1] [2] [3] 150 gs 100 0 10 20 30 40 F re q ue n c y (G H z) (mS) 40 ds 50 60 [4] Gm 30 m G , G C gd 50 0 20 10 0 [5] G ds 0 10 20 30 40 F re q ue nc y (G H z) 50 60 Fig. 5: Extracted intrinsic capacitances and conductances versus frequency, at VGS= -1.0 V & VDS = 10.0 V, for a 0.5 µm GaN HEMT with 2x5 µm gate width. 901 120 S21/3 0.8 0.8 0.6 0.6 0.4 150 30 -3xS12 2xS22 S11 0.0 60 [7] 30 0.4 0.2 180 [6] 901 120 60 150 0 180 0.2 [8] 0.0 0 2xS12 210 An improved reliable model parameter extraction method, for GaN-based HEMT, is presented. First, a high quality starting value for the extrinsic elements was generated using cold S-parameter measurements. Next, searching for the optimal values through optimization was done. Simulation results, for a 0.5 µm GaN HEMT with 2x50 µm gate width, verify the validity of our developed extraction method for both small and large signal modelling applications. ACKNOWLEDGEMENT IV. INTRINSIC PARAMETER EXTRACTION 250 200 V. 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