INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS

Int. J. Numer. Model. (2015) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/jnm.2137

A parameter extraction method for GaN HEMT empirical largesignal model including self-heating and trapping effects Zhang Wen1, Yuehang Xu1,2,*,†, Changsi Wang1, Xiaodong Zhao1, Zhikai Chen1 and Ruimin Xu1 2

1 School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 611731, China

SUMMARY This paper presents an analytical parameter extraction method for empirical large-signal model of GaN high electron mobility transistors (HEMTs) including self-heating and trapping effects. Every parameter in the model is extracted in an analytic way. An improved Angelov I–V model speciﬁc for GaN HEMTs with 53 parameters is employed. The I–V model parameters are divided into blocks according to their physical meaning, and different blocks are extracted separately by ﬁtting the pulsed I–V transfer characteristic curves of the device at different quiescent bias points. The capacitance model is extracted through mathematical analysis. This method has been implemented in MATLAB (MathWorks, Natick, MA, USA) programming, and good accuracy is obtained between model predictions and experimental results. Copyright © 2015 John Wiley & Sons, Ltd. Received 5 August 2015; Revised 28 October 2015; Accepted 19 November 2015 KEY WORDS:

GaN HEMTs; large-signal model; parameter extraction

1. INTRODUCTION Nowadays, GaN high electron mobility transistors (HEMTs) are known as the promising devices for high-efﬁciency microwave power ampliﬁers [1–4]. Accurate nonlinear transistor models are essential for the design of power ampliﬁers. Compared with physical-based model [5] and table-based model [6], the empirical large-signal equivalent circuit model is simpler and easier to be implemented in commercial simulators and has been widely used in circuit design [7]. Basically, the generation of nonlinear empirical models requires nonlinear functions that account for the current ﬂow (I–V functions) and the charge dynamic variation (Q–V or C–V functions). As for GaN HEMTs, I–V functions are much more complex than those of Si and GaAs devices [8]. This is because the model must import more terms and parameters to account for the self-heating and charge-trapping effects that GaN HEMTs encounter under working conditions. Thus, parameter extraction is a critical problem during the process of building nonlinear models. Angelov nonlinear model [8] has been widely used in GaN HEMT modeling [9–11]. And extraction techniques have been widely exploited in the last decade. For example, some of the parameters in I–V models can be extracted straightforward from measured I–V curves by computing the slopes of speciﬁc regions [12]. Some of the model parameters can be extracted by ﬁtting the low frequency large-signal current waveforms and further numerical optimization [13]. Low and high frequency large-signal measurements with numerical optimizations can also identify the parameters of the I–V and Q–V functions [14]. Nevertheless, the extraction details for self-heating and trapping parameters are not mentioned so *Correspondence to: Yuehang Xu, State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, 611731, China. † E-mail: [email protected]

Copyright © 2015 John Wiley & Sons, Ltd.

Z. WEN ET AL.

much. An organized method to extract these parameters is preferable than putting them into an optimizer and obtaining the values that may be contrary to their physical meaning. However, such organized methods or in other words, the ﬁtting details, are rarely mentioned in most published works. In this paper, a mathematical parameter extraction method for GaN HEMT empirical large-signal model is proposed. Every parameter in the model is extracted in an analytical way. The details to extract each parameter in a modiﬁed Angelov I–V model and capacitance model are presented. The I–V model parameters are ﬁrstly divided into blocks according to their physical meaning to reduce the complexity of the model. Then, different parameter blocks are extracted separately by ﬁtting the pulsed I–V transfer characteristic curves of the device at different quiescent bias points. The proposed parameter extraction method has been implemented in MATLAB (MathWorks, Natick, MA, USA) programming. The extracted large-signal model has been implemented and simulated using the Agilent Advanced Design System (ADS), and good agreement is obtained between model predictions and experimental data.

2. MODEL DESCRIPTION The topology of large-signal model for AlGaN/GaN HEMTs is shown in Figure 1. The main nonlinear elements of the model consist of the nonlinear drain-source current Ids, the gate-source capacitor Cgs, diode characteristics are modeled and the gate-drain capacitor Cgd. The gate-source and gate-drain vD

using the Schottky ideal diode equation I D ¼ I s enV T 1 . The parameters of the diodes were ex-

tracted from forward bias IV measurements (not to damage the gate terminal due to excessive forward current). The parasitic elements are ﬁrstly extracted through ‘cold ﬁeld-effect transistor’ [15, 16] S-parameters by using the extraction method in [17] and numerical optimization. A GaN HEMT with gate length 0.25 μm and gate width 4 × 80 μm is ﬁrstly used in this paper. The extracted parasitic parameters are shown in Table I. The I–V model employed in this paper is a modiﬁed Angelov model [11]. The model equations are written as follows: tanhðαV ds Þ I ds ¼ I pkth 1 þ M ipk V ds ; V gseff tanh Ψ V ds ; V gseff

(1)

Figure 1. The topology of large-signal model for GaN high electron mobility transistors. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

PARAMETER EXTRACTION METHOD FOR GAN HEMT EMPIRICAL LARGE-SIGNAL MODEL

Table I. The extracted parasitic parameters for the 4 × 80 μm device. Cpga 3.48 fF Lg 40.34 pH

Cpda

Cgda

Cpgi

Cpdi

Cgdi

3.93 fF Ld 30.26 pH

2.74 fF Ls 8.35 pH

30.82 fF Rg 2.42 Ω

10.16 fF Rd 1.95 Ω

4.18 fF Rs 0.56 Ω

M ipk V ds ; V gseff ¼ 1 þ 0:5 M ipkbth ðV ds Þ 1 ð1 þ tanhðϕ m ÞÞ ϕ m ¼ Qm ðV ds Þ V gseff V gsm 2 Ψ V ds ; V gseff ¼ P1th ðV ds Þ V gseff V pk1 þ P2th ðV ds Þ V gseff V pk2

(2) (3) (4)

3 þP3th ðV ds Þ V gseff V pk3 where Pith are the coefﬁcients of the Ψ polynomial. The sufﬁx ‘th’ stands for that it is a self-heating effect related term. Mipk is the hyperbolic tangent-based multiplier for Ipkth. ϕ m controls the shape of Mipk as a function of Vgs centered around Vgsm. Qm is the coefﬁcient for ϕ m, and Mipkbth deﬁnes the upper bound limit for Mipk. The temperature-sensitive terms are identiﬁed as Ipkth, Mipkbth, P1th, P2th, and P3th. These terms are represented as a function of temperature change and drain-source voltage by the following relations: I pkth ¼ I pk 1 þ K Ipk ðV ds ÞΔT (5) M ipkbth ðV ds Þ ¼ M ipkb ðV ds Þ 1 þ K Mipkb ðV ds ÞΔT (6) Pith ðV ds Þ ¼ Pi ðV ds Þð1 þ K Pi ðV ds ÞΔT Þ

i ¼ 1; 2; 3

ΔT ¼ Pdiss Rtheq ¼ I ds V ds Rtheq

(7) (8)

Because surface traps reduce the effect of the applied gate bias while substrate traps produce a backgate voltage [11], the charge-trapping effect is described by the term Vgseff, which stands for the effective Vgs. And the Vgseff term is presented as follow: V gseff ¼ V gseff V gs ; V ds ; V dsq ; V gsq ¼ V gs þ γsurf 1 V gsq V gsqpinch V gs V gsqpinch

(9)

þγsubs1 V dsq þ V dssubs0 V ds V dsq According to the aforementioned description, the self-heating effect or charge-trapping effect unrelated parameters are identiﬁed as Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), P3(Vds), α,Vpk1, Vpk2, Vpk3, Ipk, and Vgsm. The self-heating effect related parameters are identiﬁed as KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds). The charge-trapping effect related parameters are identiﬁed as γsurf1, Vgsqpinch, γsubs1, and Vdssubs0. All of the model parameters are divided into three blocks and will be extracted respectively in the next section.

3. I–V MODEL EXTRACTION 3.1. Extraction of self-heating or charge-trapping unrelated parameters When the quiescent bias point of pulsed measurement is set as Vgsq = 3.0 V ≈ Vgsqpinch and Vdsq = 0 V, the term Vgseff can be regarded as Vgs for approximation. The dissipated power is zero so that the selfheating related parameters can be neglected. As in the last section, the self-heating or charge-trapping effect unrelated parameters are identiﬁed as Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), P3(Vds), α,Vpk1, Vpk2, Vpk3, Ipk, and Vgsm. α and Ipk can be extracted using the method mentioned in [12]. In this section, the functions Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds) are speciﬁed as follows [11]: Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

Z. WEN ET AL.

QM ðV ds Þ ¼ ðPQ0 þ PQ1 V ds ÞtanhðαQ V ds Þ þ PQo

(10)

M ipkb ðV ds Þ ¼ PM0 þ PM1 V ds þ PM2 V ds 2 þ PM3 V ds 3 tanhðαM V ds Þ þ PMo

(11)

Pi ðV ds Þ ¼ ðPi0 þ Pi1 V ds ÞtanhðαPi Þ þ Pio ; i ¼ 1; 2; 3

(12)

Considering that Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds) are functions that use Vds as a single independent variable, the ﬁve functions will become constants when Vds is ﬁxed. Thus, values of Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds) at each Vds can be extracted by ﬁtting the transfer characteristic curves at each Vds. Vpk1, Vpk2, and Vpk3, and Vgsm can be extracted along with Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds) during the transfer characteristic curves ﬁtting progress. In other words, the number of parameters of the I–V function at a constant Vds is reduced to nine. For each constant Vds, the nine parameters are then extracted by ﬁtting the transfer characteristic curve with the least square algorithm in MATLAB program. The whole calculate procedure is implemented in a MATLAB program and does not need intervention by the user. As a complement, if some unreasonable parameter values are acquired during the curve ﬁtting progress, the MATLAB program will shrink the searching interval and recalculate for reasonable parameter values. After the values of Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds) at each Vds are extracted, the inner parameters PQ0, PQ1, αQ,PQo, PM0, PM1, PM2, PM3, αM,PMo, Pi0, Pi1, αPi, and Pio can be extracted by ﬁtting the discrete values of Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds) at each Vds with the least square method. This progress is also programmed in MATLAB. The acquired unreasonable values will also be rejected by the program. Figure 2 shows the ﬁtting details of Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds). Figure 3 shows the ﬁtting result of pulsed I–V curves of Vgsq = 3.0 V and Vdsq = 0 V. The ﬂow chart of the extraction of self-heating or charge-trapping unrelated parameters is shown in Figure 4.

Figure 2. Fitting details of Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds).

Figure 3. Simulated (lines) and measured (circles) pulsed I–V curves at Vgsq = 3.0 V and Vdsq = 0 V for Vgs from 4 V to 0 V with 0.2 V steps and Vds from 0 V to 30 V with 1.25 V steps. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

PARAMETER EXTRACTION METHOD FOR GAN HEMT EMPIRICAL LARGE-SIGNAL MODEL

Figure 4. Flow chart of the extraction of self-heating or charge-trapping unrelated parameters.

3.2. Extraction of charge-trapping related parameters The charge-trapping related parameters are extracted by ﬁtting pulsed I–V curves at different quiescent bias points, which the term Vgseff cannot be regarded as Vgs for simpliﬁcation. The power dissipation should also be negligible at the selected bias points to exclude thermal effects. In this paper, I–V curves with Vgsq = 0 V, Vdsq = 0 V, and Vgsq = 6 V, Vdsq = 30 V were ﬁtted to extract parameters in the term Vgseff. For Vgsq = 0 V and Vdsq = 0 V, because of Vdsq = 0 V, there is no current in the channel, so the dissipated power at this quiescent bias point is negligible. For Vgsq = 6 V and Vdsq = 30 V, because Vgsq = 6 V is below the pinch-off voltage, the dissipated power at this quiescent bias point is also negligible. Because there are only four parameters, γsurf1, Vgsqpinch, γsubs1, and Vdssubs0 need to be extracted; the least square ﬁtting progress is quite simple by programming in MATLAB. The ﬁtting result of pulsed I–V curves is shown in Figure 5. As can be seen, good accuracy is achieved by using the extracted trap-related parameters.

Figure 5. Fitting results of pulsed I–V curves. (a) Vgsq = 0 V and Vdsq = 0 V and (b) Vgsq = 6 V and Vdsq = 30 V. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

Z. WEN ET AL.

3.3. Extraction of self-heating related parameters In this section, parameters extracted in Sections 3.1 and 3.2 are put into the Ids function. There is a critical problem in the process of extracting self-heating related parameters. Because the term ΔT = Pdiss Rtheq = Ids Vds Rtheq contains Ids that has not been totally extracted, the Ids function will become nested after putting ΔT into it. We have considered that only the values of Pdiss need to be put into the Ids function, and Pdiss is a function of Vds and Vgs. So the values of Pdiss at each Vds and Vgs can be calculated and then ﬁtted by using a polynomial. By putting the polynomial into the Ids function, the term ΔT no longer contains Ids that has not been totally extracted. Figure 6 shows the polynomial ﬁtting result of Pdiss. The equivalent thermal resistance Rtheq can be extracted using the method in [18] and its value is 24.5°C/W for the GaN HEMT used in this paper. In this section, the self-heating related functions KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds) are speciﬁed as follows [11]: K Ipk ðV ds Þ ¼ K Ipk0 þ K Ipk1 V ds tanh αKIpk V ds þ K Ipk0

(13)

K Mipkb ðV ds Þ ¼ K Mipkb0 þ K Mipkb1 V ds tanh αKMipkb V ds þ K Mipkb0

(14)

K Pi ðV ds Þ ¼ ðK Pi0 þ K Pi1 V ds ÞtanhðαKPi V ds Þ þ K Pio ; i ¼ 1; 2; 3

(15)

Considering that KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds) are functions that use Vds as a single independent variable, the ﬁve functions will become constants when Vds is ﬁxed. Thus, values of KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds) at each Vds can be extracted by ﬁtting the transfer characteristic curves at each Vds. After the values of KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2 (Vds), and KP3(Vds) at each Vds are extracted, the inner parameters KIpk0, KIpk1, αKIpk,KIpko, KMipkb0, KMipkb1, αKMipkb,KMipkbo, KPi0, KPi1, αKPi, and KPio can be extracted by ﬁtting the discrete values of KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds) at each Vds with the least square method. The static I–V dataset was used to extract these self-heating related parameters because self-heating effect of the device is obvious under this operation condition. Figure 7 shows the ﬁtting details of KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds). Figure 8 shows the ﬁtting result of static I–V curves. Figure 9 shows the ﬂow chart of extraction of self-heating related parameters. Table II shows the extracted 28 self-heating or charge-trapping unrelated parameters. The four charge-trapping related parameters are shown in Table III. Table IV shows the extracted 20 self-heating related parameters.

Figure 6. Polynomial ﬁtting result of Pdiss for Vgs from 4 V to 0 V with 0.2 V steps and Vds from 0 V to 30 V with 1.25 V steps. Circles for the calculated values from measured data and lines for the polynomial ﬁt functions. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

PARAMETER EXTRACTION METHOD FOR GAN HEMT EMPIRICAL LARGE-SIGNAL MODEL

Figure 7. Fitting details of KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds). Circles for discrete values at each Vds and lines for the ﬁtting functions.

Figure 8. Fitting result of static I–V curves for Vgs from 4 V to 0 V with 0.2 V steps and Vds from 0 V to 30 V with 0.5 V steps. Circles for measured data and lines for model predictions.

4. CAPACITANCE MODEL EXTRACTION In this section, the capacitances parameters are extracted from the values extracted through multi-bias S-parameter measurements. The bias voltage Vgs ranges from 4 V to 0 V with the step of 0.5 V, and Vds ranges from 0 V to 35 V with the step of 5 V. The Angelov capacitance equations [8] are employed in this paper. The Cgs and Cgd equations are written as follows: C gs ¼ C gsp þ C gs0 1 þ tanh P10 þ P11 V gs ð1 þ tanhðP20 þ P21 V ds ÞÞ

(16)

C gd ¼ C gdp þ C gd0 1 þ tanh P40 þ P41 V gd ð1 þ tanhðP30 P31 V ds ÞÞ

(17)

Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

Z. WEN ET AL.

Figure 9. Flow chart of extraction of self-heating related parameters.

Table II. Extracted self-heating or charge-trapping unrelated parameters. P11

αP1

P1o

4.5511

7.4435e-4

0.4455

3.3784

P20 8.3847 P30 6.8705 PQ0 2.6965 PM0 8.3582 PM2 0.0085 Vpk1 2.0203 V

P21 0.0183 P31 0.0057 PQ1 0.0011 PM1 0.0560 PM3 1.3627e-4 Vpk2 2.0765 V

αP2 0.4889 αP3 0.4764 αQ 2.5820 αM 0.3723 α 1.5015 Vpk3 1.5966 V

P2o 8.1909 P3o 6.7172 PQo 2.2434 PMo 0.7548 Ipk 49.1 mA Vgsm 0.0937 V

P10

Table III. Extracted charge-trapping related parameters. γsurf1 9.6723e-4

Vgsqpinch

γsubs1

Vdssubs0

3.0310 V

0.0352

0.0294

Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

PARAMETER EXTRACTION METHOD FOR GAN HEMT EMPIRICAL LARGE-SIGNAL MODEL

Table IV. Extracted self-heating related parameters. αKIpk

KIpk1

KIpk0

KIpko

0.0034

5.1648e-5

0.4291

0.0020

KP10 0.0027 KP20 0.0048 KP30 0.0051 KMipkb0 0.0033

KP11 3.7697e-5 KP21 3.1019e-4 KP31 1.7130e-4 KMipkb1 4.0203e-5

αKP1 0.2195 αKP2 0.2814 αKP3 0.3053 αKMipkb 0.2960

KP1o 0.0020 KP2o 0.0026 KP3o 0.0026 KMipkbo 0.0018

4.1. Gate-source capacitance extraction The ﬁrst-order partial derivatives of Cgs against Vgs and Vds are written as follows: ∂C gs =∂V gs ¼ P11 C gs0 ð1 þ tanhðP20 þ P21 V ds ÞÞ 1 tanh2 P10 þ P11 V gs

(18)

∂C gs =∂V ds ¼ P21 C gs0 1 þ tanh P10 þ P11 V gs 1 tanh2 ðP20 þ P21 V ds Þ

(19)

As a result of the tanh behavior, the partial derivative ∂Cgs /∂Vgs has a peak when P10 + P11*Vgs = 0 and the partial derivative ∂Cgs /∂Vds has a peak when P20 + P21*Vds = 0. Figure 10 shows the partial derivatives calculated from the extracted Cgs values. Thus P10 = 2P11 and P20 = 10P21 are acquired. The following Equations (20) and (21) can be acquired by putting P10 = 2P11 and P20 = 10P21 into (16). (20) Cgs Vgs ¼ 2V ¼ C gsp þ ð1 þ tanhð25P21 ÞÞC gs0 Vds ¼ 35V

C gs Vgs ¼ 2V ¼ C gsp þ C gs0

(21)

Vds ¼ 10V

Thus, Cgsp and Cgs0 can be written as C gs0 ¼

! C gs Vgs ¼ 2V ; C gs Vgs ¼ 2V ÷ tanhð25P21 Þ Vds ¼ 35V

(22)

Vds ¼ 10V

C gsp ¼ C gs Vgs ¼ 2V C gs0

(23)

Vds ¼ 10V

It is inferred from (18) that the maximum of ∂Cgs /∂Vgs is reached when P10 + P11*Vgs = 0, Vds = 30 V, and its maximum is P11*Cgs0*(1 + tanh(20*P21)). So P11 can be written as

Figure 10. Numerical partial derivatives of Cgs. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

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P11 ¼

∂C gs =∂V gs Vds ¼ 30V Vgs ¼ 2V

ð1 þ tanhð20*P21 ÞÞC gs0

(24)

So far, Cgsp, Cgs0, P10, P11, and P20 are all functions of P21. Once P21 is extracted, the rest of the parameters can be calculated from the aforementioned deduction. Dividing Equation (18) by (19), the following equation can be acquired: ∂C gs =∂V gs P11 1 tanh P10 þ P11 V gs P11 expð2ðP20 þ P21 V ds ÞÞ þ 1 ¼ (25) ¼ ∂C gs =∂V ds P21 1 tanhðP20 þ P21 V ds Þ P21 expð2ðP10 þ P11 V ds ÞÞ þ 1 Putting the values of ∂Cgs /∂Vgs and ∂Cgs /∂Vds at a selected bias point into (25), P21 can be acquired by solving the nonlinear equation. Thus, the six parameters in (16) can be extracted with the described method. 4.2. Gate-drain capacitance extraction To ensure charge conservation, P41 = P11 and P31 = P21 can be obtained [19]. The ﬁrst-order partial derivatives of Cgd against Vgs can be written as follows: ∂Cgd =∂V gs ¼ P41 C gd0 ð1 þ tanhðP30 P31 V ds ÞÞ 1 tanh2 P40 þ P41 V gs V ds (26) As a result of the tanh behavior, the partial derivative ∂Cgd /∂Vgs has a peak when P40 + P41* (VgsVds) = 0 and its maximum is P41*Cgd0*(1 + tanh(P30P31*Vds)). Figure 11 shows the partial derivatives calculated from the extracted Cgd values. Thus, P40 = 2P41 can be acquired, and Cgd0 can be written as follows: ∂C gd =∂V gs Vds ¼ 0 V C gd0 ¼

Vgs ¼ 2V

P41 ð1 þ tanhðP30 ÞÞ

(27)

So far, P40, P41, and P31 have been extracted and Cgd0 is a function of P30. Thus, only Cgdp and P30 need to be extracted. Two equations can be acquired by putting the extracted Cgd values at two bias points into (17). Cgdp and P30 can be obtained by solving the two equations. Considering the uncertainty of measurements, the selected two bias points should not be very close to the endpoint of the measured interval. And the distance between the two selected bias points should not be too close. This will lead to a typical solution instead of a singular solution. In this paper, Vgs = 3 V, Vds = 25 V and Vgs = 1 V, Vds = 10 V were selected.

Figure 11. Numerical partial derivatives of Cgd. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

PARAMETER EXTRACTION METHOD FOR GAN HEMT EMPIRICAL LARGE-SIGNAL MODEL

To this step, the six parameters in (17) can be extracted with the described method. The extracted capacitance model parameters are shown in Table V.

5. MODEL VERIFICATION The developed model is embedded in ADS software by using the Symbolically Deﬁned Devices (SDD), and the extracted parameters are put into the SDD model. Figure 12 shows the comparison between S-parameter measurements and large-signal simulations at two bias points up to 40 GHz. Figure 13 shows the one-tone power sweep measurements and simulations of the device at Vgs = 2.5 V and Vds = 28 V with f0 = 10, 12, and 14 GHz. At this bias condition, the device works in class AB. The load impedances for f0 = 10, 12, and 14 GHz are 34.82 + 90.75j, 26.17 + 76.17j, and 14.52 + 71.32j, respectively. Figure 14 shows the same results of another device with gate width 4 × 100 μm and gate length 0.25 μm. The load impedances for this device at f0 = 10, 12, and 14 GHz

Table V. Extracted capacitance model parameters. Cgsp(fF)

Cgs0(fF)

P10

P11

P20

P21

339.3952 Cgdp(fF) 56.191

238.2186 Cgd0(fF) 93.34

3.7813 P30 1.353

1.6229 P31 0.0091

0.5206 P40 3.7813

0.0091 P41 1.6229

Figure 12. Comparison between S-parameter measurements and large-signal simulations. (a) Vgs = 2.5 V and Vds = 28 V and (b) Vgs = 4 V and Vds = 10 V.

Figure 13. One-tone power sweep measurements (circles) and simulations (lines) of the device with gate width 4 × 80 μm and gate length 0.25 μm at Vgs = 2.5 V and Vds = 28 V with f0 = 10 GHz (a), 12 GHz (b), and 14 GHz (c). Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

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Figure 14. One-tone power sweep measurements (circles) and simulations (lines) of the device with gate width 4 × 100 μm and gate length 0.25 μm at Vgs = 2.5 V and Vds = 28 V with f0 = 10 GHz (a), 12 GHz (b), and 14 GHz(c).

are 33.29 + 76.22j, 24.29 + 64.37j, and 9.99 + 59.24j, respectively. Figure 15 shows the power sweep measurements and simulations of the 4 × 100 μm device at Vgs = 2.0 V and Vds = 20 V. At this bias condition, the device works in class AB. The load impedances for f0 = 10, 12, and 14 GHz are 33.45 + 64.43j, 20.49 + 53.50j, and 19.07 + 48.98j, respectively. Figure 16 shows the simulated and measured DC current versus input power for the 4 × 100 μm device. It is clearly seen that the extracted model can accurately predict the large-signal characteristics of the device. Figure 17 shows the ﬁtting results of PIV curves at Vgsq = 2.5 V, Vdsq = 28 V and Vgsq = 2.0 V, Vdsq = 28 V. As can be seen, the extracted model can accurately reproduce the pulsed measurements.

Figure 15. One-tone power sweep measurements (circles) and simulations (lines) of the device with gate width 4 × 100 μm and gate length 0.25 μm at Vgs = 2.0 V and Vds = 20 V with f0 = 10 GHz (a), 12 GHz (b), and 14 GHz(c).

Figure 16. Simulated and measured DC current versus input power for the 4 × 100 μm device. (a) Vgs = 2.5 V and Vds = 28 V and (b) Vgs = 2.0 V and Vds = 20 V. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

PARAMETER EXTRACTION METHOD FOR GAN HEMT EMPIRICAL LARGE-SIGNAL MODEL

Figure 17. Fitting results of pulsed I–V curves for the 4 × 80 μm device. (a) Vgsq = 2.5 V and Vdsq = 28 Vand (b) Vgsq = 2.0 V and Vdsq = 28 V.

6. CONCLUSION An analytical parameter extraction method for GaN HEMTs empirical large-signal model is presented in this paper. A modiﬁed Angelov drain-source current model with 53 parameters and the capacitance model are extracted in an analytical way. Good accuracy of large-signal characteristics is achieved. The proposed parameter extraction method is implemented in MATLAB programming, and the whole extraction procedure is automatically performed by the program. It is highly efﬁcient when used for modeling of large quantities of devices. ACKNOWLEDGEMENTS

The authors would like to thank Dr Kelvin S. Yuk, Department of Electrical and Computer Engineering, University of California, Davis, USA, for his useful advices and suggestions for the work in this paper.

REFERENCES 1. Kobayashi K, Chen Y, Smorchkova I, Tsai R, Wojtowicz M, Oki A. A 2 watt sub-db noise ﬁgure GaN MMIC LNA-PA with multioctave bandwidth from 0.2–8 GHz. IEEE MTT-S Int Microw Symp Dig 2007;619–622. 2. Gustafsson D, Cahuana J, Kuylenstierna D, Angelov I, Rorsman N, Fager C. A wideband and compact GaN MMIC Doherty ampliﬁer for microwave link applications. IEEE Trans Microw Theory Techn 2013; 61(2):922–930. 3. Qiu Y, Xu Y, Xu R, Lin W. Compact hybrid broadband GaN HEMT power ampliﬁer based on feed-back technique. Electron Lett 2013; 49(5):372–374. 4. Klockenhoff H, Behtash R, Wurﬂ J, Heinrich W, Trankle G. A compact 16 watt X-band GaN MMIC power ampliﬁer. IEEE MTT-S Digest 2006;1846–1849. 5. Mari D, Bernardoni M, Sozzi G, et al. A physical large-signal model for GaN HEMTs including selfheating and trap-related dispersion. Microelectron Reliab 2011; 51(2): 229–234.

6. Jarndal A, et al. Large-signal model for AlGaN/GaN HEMTs suitable for RF switching-mode power ampliﬁers design. Solid State Electron 2010; 54:696–700. 7. Sang L, Xu Y, Cao R, Chen Y, Guo Y, Xu R. Modeling of GaN HEMT by using an improved k-nearest neighbors algorithm. J Electromagn Waves Appl 2011; 25:949–959. 8. Angelov I, Zirath H, Rorsman N. A new empirical nonlinear model for HEMT and MESFET devices. IEEE Trans Microw Theory Tech 1992; 40(12):2258–2266. 9. Wang C, Xu Y, Yu X, et al. An electrothermal model for empirical largesignal modeling of AlGaN/GaN HEMTs including self-heating and ambient temperature effects. IEEE Trans Microw Theory Techn 2014; 62:2878–2887. 10. Xu Y, Fu W, Wang C, et al. A scalable GaN HEMT large-signal model for high efﬁciency RF power ampliﬁer design. J Electromagn Waves Appl 2014; 28:1888–1895. 11. Yuk K, Branner G, McQuate D. A wideband multiharmonic empirical large-signal model for high-power GaN HEMTs with self-heating and charge-trapping effects. IEEE

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Trans Microw Theory Techn 2009; 57(12):3322–3332. Angelov I, Thorsell M, Kuylenstierna D, Avolio G, Schreurs D, Raffo A, Vannini G. Hybrid measurementbased extraction of consistent large signal models for microwave FETs. Eur Microw Conference 2013; 267– 270. Avolio G, Raffo A, Angelov I, Crupi G, Vannini G, Schreurs D. A novel technique for the extraction of nonlinear model for microwave transistors under dynamic bias operation. IEEE MTT-S Int Microw Symp Dig 2013;1–3. Avolio G, Schreurs D, Raffo A, et al. Identiﬁcation technique of FET model based on vector nonlinear measurements. Electron Lett 2011; 47(24):1323–U37. Dambrine G, Cappy A, Heilodore F, Playez E. A new method for determining the FET small-signal equivalent circuit. IEEE Trans Microw Theory Techn 1988; 36(7);1151–1159. White P, Healy R. An improved equivalent circuit for determination of MESFET and HEMT parasitic capacitances from ‘Cold-FET’ measurements. IEEE Microw Guided Wave Lett 1993; 3:453–454. Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

Z. WEN ET AL. 17. Jarndal A, Kompa G. A new smallsignal modeling approach applied to GaN devices. IEEE Trans Microw Theory Techn 2005; 53(11): 3440–3448. 18. Baylis C, Dunleavy L, et al. Direct measurement of thermal circuit

parameters using pulsed IV and the normalized difference unit. IEEE MTT-S Int Microw Symp Dig 2004:1233–1236. 19. Avolio G, Raffo A, Angelov I, Crupi G, Caddemi A, Vannini G, Schreurs D. Small-versus large-

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signal extraction of charge models of microwave FETs. IEEE Microw Wireless Compon lett 2014; 24(6): 394–396.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

Int. J. Numer. Model. (2015) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/jnm.2137

A parameter extraction method for GaN HEMT empirical largesignal model including self-heating and trapping effects Zhang Wen1, Yuehang Xu1,2,*,†, Changsi Wang1, Xiaodong Zhao1, Zhikai Chen1 and Ruimin Xu1 2

1 School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 611731, China

SUMMARY This paper presents an analytical parameter extraction method for empirical large-signal model of GaN high electron mobility transistors (HEMTs) including self-heating and trapping effects. Every parameter in the model is extracted in an analytic way. An improved Angelov I–V model speciﬁc for GaN HEMTs with 53 parameters is employed. The I–V model parameters are divided into blocks according to their physical meaning, and different blocks are extracted separately by ﬁtting the pulsed I–V transfer characteristic curves of the device at different quiescent bias points. The capacitance model is extracted through mathematical analysis. This method has been implemented in MATLAB (MathWorks, Natick, MA, USA) programming, and good accuracy is obtained between model predictions and experimental results. Copyright © 2015 John Wiley & Sons, Ltd. Received 5 August 2015; Revised 28 October 2015; Accepted 19 November 2015 KEY WORDS:

GaN HEMTs; large-signal model; parameter extraction

1. INTRODUCTION Nowadays, GaN high electron mobility transistors (HEMTs) are known as the promising devices for high-efﬁciency microwave power ampliﬁers [1–4]. Accurate nonlinear transistor models are essential for the design of power ampliﬁers. Compared with physical-based model [5] and table-based model [6], the empirical large-signal equivalent circuit model is simpler and easier to be implemented in commercial simulators and has been widely used in circuit design [7]. Basically, the generation of nonlinear empirical models requires nonlinear functions that account for the current ﬂow (I–V functions) and the charge dynamic variation (Q–V or C–V functions). As for GaN HEMTs, I–V functions are much more complex than those of Si and GaAs devices [8]. This is because the model must import more terms and parameters to account for the self-heating and charge-trapping effects that GaN HEMTs encounter under working conditions. Thus, parameter extraction is a critical problem during the process of building nonlinear models. Angelov nonlinear model [8] has been widely used in GaN HEMT modeling [9–11]. And extraction techniques have been widely exploited in the last decade. For example, some of the parameters in I–V models can be extracted straightforward from measured I–V curves by computing the slopes of speciﬁc regions [12]. Some of the model parameters can be extracted by ﬁtting the low frequency large-signal current waveforms and further numerical optimization [13]. Low and high frequency large-signal measurements with numerical optimizations can also identify the parameters of the I–V and Q–V functions [14]. Nevertheless, the extraction details for self-heating and trapping parameters are not mentioned so *Correspondence to: Yuehang Xu, State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, 611731, China. † E-mail: [email protected]

Copyright © 2015 John Wiley & Sons, Ltd.

Z. WEN ET AL.

much. An organized method to extract these parameters is preferable than putting them into an optimizer and obtaining the values that may be contrary to their physical meaning. However, such organized methods or in other words, the ﬁtting details, are rarely mentioned in most published works. In this paper, a mathematical parameter extraction method for GaN HEMT empirical large-signal model is proposed. Every parameter in the model is extracted in an analytical way. The details to extract each parameter in a modiﬁed Angelov I–V model and capacitance model are presented. The I–V model parameters are ﬁrstly divided into blocks according to their physical meaning to reduce the complexity of the model. Then, different parameter blocks are extracted separately by ﬁtting the pulsed I–V transfer characteristic curves of the device at different quiescent bias points. The proposed parameter extraction method has been implemented in MATLAB (MathWorks, Natick, MA, USA) programming. The extracted large-signal model has been implemented and simulated using the Agilent Advanced Design System (ADS), and good agreement is obtained between model predictions and experimental data.

2. MODEL DESCRIPTION The topology of large-signal model for AlGaN/GaN HEMTs is shown in Figure 1. The main nonlinear elements of the model consist of the nonlinear drain-source current Ids, the gate-source capacitor Cgs, diode characteristics are modeled and the gate-drain capacitor Cgd. The gate-source and gate-drain vD

using the Schottky ideal diode equation I D ¼ I s enV T 1 . The parameters of the diodes were ex-

tracted from forward bias IV measurements (not to damage the gate terminal due to excessive forward current). The parasitic elements are ﬁrstly extracted through ‘cold ﬁeld-effect transistor’ [15, 16] S-parameters by using the extraction method in [17] and numerical optimization. A GaN HEMT with gate length 0.25 μm and gate width 4 × 80 μm is ﬁrstly used in this paper. The extracted parasitic parameters are shown in Table I. The I–V model employed in this paper is a modiﬁed Angelov model [11]. The model equations are written as follows: tanhðαV ds Þ I ds ¼ I pkth 1 þ M ipk V ds ; V gseff tanh Ψ V ds ; V gseff

(1)

Figure 1. The topology of large-signal model for GaN high electron mobility transistors. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

PARAMETER EXTRACTION METHOD FOR GAN HEMT EMPIRICAL LARGE-SIGNAL MODEL

Table I. The extracted parasitic parameters for the 4 × 80 μm device. Cpga 3.48 fF Lg 40.34 pH

Cpda

Cgda

Cpgi

Cpdi

Cgdi

3.93 fF Ld 30.26 pH

2.74 fF Ls 8.35 pH

30.82 fF Rg 2.42 Ω

10.16 fF Rd 1.95 Ω

4.18 fF Rs 0.56 Ω

M ipk V ds ; V gseff ¼ 1 þ 0:5 M ipkbth ðV ds Þ 1 ð1 þ tanhðϕ m ÞÞ ϕ m ¼ Qm ðV ds Þ V gseff V gsm 2 Ψ V ds ; V gseff ¼ P1th ðV ds Þ V gseff V pk1 þ P2th ðV ds Þ V gseff V pk2

(2) (3) (4)

3 þP3th ðV ds Þ V gseff V pk3 where Pith are the coefﬁcients of the Ψ polynomial. The sufﬁx ‘th’ stands for that it is a self-heating effect related term. Mipk is the hyperbolic tangent-based multiplier for Ipkth. ϕ m controls the shape of Mipk as a function of Vgs centered around Vgsm. Qm is the coefﬁcient for ϕ m, and Mipkbth deﬁnes the upper bound limit for Mipk. The temperature-sensitive terms are identiﬁed as Ipkth, Mipkbth, P1th, P2th, and P3th. These terms are represented as a function of temperature change and drain-source voltage by the following relations: I pkth ¼ I pk 1 þ K Ipk ðV ds ÞΔT (5) M ipkbth ðV ds Þ ¼ M ipkb ðV ds Þ 1 þ K Mipkb ðV ds ÞΔT (6) Pith ðV ds Þ ¼ Pi ðV ds Þð1 þ K Pi ðV ds ÞΔT Þ

i ¼ 1; 2; 3

ΔT ¼ Pdiss Rtheq ¼ I ds V ds Rtheq

(7) (8)

Because surface traps reduce the effect of the applied gate bias while substrate traps produce a backgate voltage [11], the charge-trapping effect is described by the term Vgseff, which stands for the effective Vgs. And the Vgseff term is presented as follow: V gseff ¼ V gseff V gs ; V ds ; V dsq ; V gsq ¼ V gs þ γsurf 1 V gsq V gsqpinch V gs V gsqpinch

(9)

þγsubs1 V dsq þ V dssubs0 V ds V dsq According to the aforementioned description, the self-heating effect or charge-trapping effect unrelated parameters are identiﬁed as Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), P3(Vds), α,Vpk1, Vpk2, Vpk3, Ipk, and Vgsm. The self-heating effect related parameters are identiﬁed as KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds). The charge-trapping effect related parameters are identiﬁed as γsurf1, Vgsqpinch, γsubs1, and Vdssubs0. All of the model parameters are divided into three blocks and will be extracted respectively in the next section.

3. I–V MODEL EXTRACTION 3.1. Extraction of self-heating or charge-trapping unrelated parameters When the quiescent bias point of pulsed measurement is set as Vgsq = 3.0 V ≈ Vgsqpinch and Vdsq = 0 V, the term Vgseff can be regarded as Vgs for approximation. The dissipated power is zero so that the selfheating related parameters can be neglected. As in the last section, the self-heating or charge-trapping effect unrelated parameters are identiﬁed as Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), P3(Vds), α,Vpk1, Vpk2, Vpk3, Ipk, and Vgsm. α and Ipk can be extracted using the method mentioned in [12]. In this section, the functions Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds) are speciﬁed as follows [11]: Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

Z. WEN ET AL.

QM ðV ds Þ ¼ ðPQ0 þ PQ1 V ds ÞtanhðαQ V ds Þ þ PQo

(10)

M ipkb ðV ds Þ ¼ PM0 þ PM1 V ds þ PM2 V ds 2 þ PM3 V ds 3 tanhðαM V ds Þ þ PMo

(11)

Pi ðV ds Þ ¼ ðPi0 þ Pi1 V ds ÞtanhðαPi Þ þ Pio ; i ¼ 1; 2; 3

(12)

Considering that Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds) are functions that use Vds as a single independent variable, the ﬁve functions will become constants when Vds is ﬁxed. Thus, values of Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds) at each Vds can be extracted by ﬁtting the transfer characteristic curves at each Vds. Vpk1, Vpk2, and Vpk3, and Vgsm can be extracted along with Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds) during the transfer characteristic curves ﬁtting progress. In other words, the number of parameters of the I–V function at a constant Vds is reduced to nine. For each constant Vds, the nine parameters are then extracted by ﬁtting the transfer characteristic curve with the least square algorithm in MATLAB program. The whole calculate procedure is implemented in a MATLAB program and does not need intervention by the user. As a complement, if some unreasonable parameter values are acquired during the curve ﬁtting progress, the MATLAB program will shrink the searching interval and recalculate for reasonable parameter values. After the values of Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds) at each Vds are extracted, the inner parameters PQ0, PQ1, αQ,PQo, PM0, PM1, PM2, PM3, αM,PMo, Pi0, Pi1, αPi, and Pio can be extracted by ﬁtting the discrete values of Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds) at each Vds with the least square method. This progress is also programmed in MATLAB. The acquired unreasonable values will also be rejected by the program. Figure 2 shows the ﬁtting details of Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds). Figure 3 shows the ﬁtting result of pulsed I–V curves of Vgsq = 3.0 V and Vdsq = 0 V. The ﬂow chart of the extraction of self-heating or charge-trapping unrelated parameters is shown in Figure 4.

Figure 2. Fitting details of Qm(Vds), Mipkb(Vds), P1(Vds), P2(Vds), and P3(Vds).

Figure 3. Simulated (lines) and measured (circles) pulsed I–V curves at Vgsq = 3.0 V and Vdsq = 0 V for Vgs from 4 V to 0 V with 0.2 V steps and Vds from 0 V to 30 V with 1.25 V steps. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

PARAMETER EXTRACTION METHOD FOR GAN HEMT EMPIRICAL LARGE-SIGNAL MODEL

Figure 4. Flow chart of the extraction of self-heating or charge-trapping unrelated parameters.

3.2. Extraction of charge-trapping related parameters The charge-trapping related parameters are extracted by ﬁtting pulsed I–V curves at different quiescent bias points, which the term Vgseff cannot be regarded as Vgs for simpliﬁcation. The power dissipation should also be negligible at the selected bias points to exclude thermal effects. In this paper, I–V curves with Vgsq = 0 V, Vdsq = 0 V, and Vgsq = 6 V, Vdsq = 30 V were ﬁtted to extract parameters in the term Vgseff. For Vgsq = 0 V and Vdsq = 0 V, because of Vdsq = 0 V, there is no current in the channel, so the dissipated power at this quiescent bias point is negligible. For Vgsq = 6 V and Vdsq = 30 V, because Vgsq = 6 V is below the pinch-off voltage, the dissipated power at this quiescent bias point is also negligible. Because there are only four parameters, γsurf1, Vgsqpinch, γsubs1, and Vdssubs0 need to be extracted; the least square ﬁtting progress is quite simple by programming in MATLAB. The ﬁtting result of pulsed I–V curves is shown in Figure 5. As can be seen, good accuracy is achieved by using the extracted trap-related parameters.

Figure 5. Fitting results of pulsed I–V curves. (a) Vgsq = 0 V and Vdsq = 0 V and (b) Vgsq = 6 V and Vdsq = 30 V. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

Z. WEN ET AL.

3.3. Extraction of self-heating related parameters In this section, parameters extracted in Sections 3.1 and 3.2 are put into the Ids function. There is a critical problem in the process of extracting self-heating related parameters. Because the term ΔT = Pdiss Rtheq = Ids Vds Rtheq contains Ids that has not been totally extracted, the Ids function will become nested after putting ΔT into it. We have considered that only the values of Pdiss need to be put into the Ids function, and Pdiss is a function of Vds and Vgs. So the values of Pdiss at each Vds and Vgs can be calculated and then ﬁtted by using a polynomial. By putting the polynomial into the Ids function, the term ΔT no longer contains Ids that has not been totally extracted. Figure 6 shows the polynomial ﬁtting result of Pdiss. The equivalent thermal resistance Rtheq can be extracted using the method in [18] and its value is 24.5°C/W for the GaN HEMT used in this paper. In this section, the self-heating related functions KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds) are speciﬁed as follows [11]: K Ipk ðV ds Þ ¼ K Ipk0 þ K Ipk1 V ds tanh αKIpk V ds þ K Ipk0

(13)

K Mipkb ðV ds Þ ¼ K Mipkb0 þ K Mipkb1 V ds tanh αKMipkb V ds þ K Mipkb0

(14)

K Pi ðV ds Þ ¼ ðK Pi0 þ K Pi1 V ds ÞtanhðαKPi V ds Þ þ K Pio ; i ¼ 1; 2; 3

(15)

Considering that KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds) are functions that use Vds as a single independent variable, the ﬁve functions will become constants when Vds is ﬁxed. Thus, values of KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds) at each Vds can be extracted by ﬁtting the transfer characteristic curves at each Vds. After the values of KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2 (Vds), and KP3(Vds) at each Vds are extracted, the inner parameters KIpk0, KIpk1, αKIpk,KIpko, KMipkb0, KMipkb1, αKMipkb,KMipkbo, KPi0, KPi1, αKPi, and KPio can be extracted by ﬁtting the discrete values of KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds) at each Vds with the least square method. The static I–V dataset was used to extract these self-heating related parameters because self-heating effect of the device is obvious under this operation condition. Figure 7 shows the ﬁtting details of KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds). Figure 8 shows the ﬁtting result of static I–V curves. Figure 9 shows the ﬂow chart of extraction of self-heating related parameters. Table II shows the extracted 28 self-heating or charge-trapping unrelated parameters. The four charge-trapping related parameters are shown in Table III. Table IV shows the extracted 20 self-heating related parameters.

Figure 6. Polynomial ﬁtting result of Pdiss for Vgs from 4 V to 0 V with 0.2 V steps and Vds from 0 V to 30 V with 1.25 V steps. Circles for the calculated values from measured data and lines for the polynomial ﬁt functions. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

PARAMETER EXTRACTION METHOD FOR GAN HEMT EMPIRICAL LARGE-SIGNAL MODEL

Figure 7. Fitting details of KIpk(Vds), KMipkb(Vds), KP1(Vds), KP2(Vds), and KP3(Vds). Circles for discrete values at each Vds and lines for the ﬁtting functions.

Figure 8. Fitting result of static I–V curves for Vgs from 4 V to 0 V with 0.2 V steps and Vds from 0 V to 30 V with 0.5 V steps. Circles for measured data and lines for model predictions.

4. CAPACITANCE MODEL EXTRACTION In this section, the capacitances parameters are extracted from the values extracted through multi-bias S-parameter measurements. The bias voltage Vgs ranges from 4 V to 0 V with the step of 0.5 V, and Vds ranges from 0 V to 35 V with the step of 5 V. The Angelov capacitance equations [8] are employed in this paper. The Cgs and Cgd equations are written as follows: C gs ¼ C gsp þ C gs0 1 þ tanh P10 þ P11 V gs ð1 þ tanhðP20 þ P21 V ds ÞÞ

(16)

C gd ¼ C gdp þ C gd0 1 þ tanh P40 þ P41 V gd ð1 þ tanhðP30 P31 V ds ÞÞ

(17)

Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

Z. WEN ET AL.

Figure 9. Flow chart of extraction of self-heating related parameters.

Table II. Extracted self-heating or charge-trapping unrelated parameters. P11

αP1

P1o

4.5511

7.4435e-4

0.4455

3.3784

P20 8.3847 P30 6.8705 PQ0 2.6965 PM0 8.3582 PM2 0.0085 Vpk1 2.0203 V

P21 0.0183 P31 0.0057 PQ1 0.0011 PM1 0.0560 PM3 1.3627e-4 Vpk2 2.0765 V

αP2 0.4889 αP3 0.4764 αQ 2.5820 αM 0.3723 α 1.5015 Vpk3 1.5966 V

P2o 8.1909 P3o 6.7172 PQo 2.2434 PMo 0.7548 Ipk 49.1 mA Vgsm 0.0937 V

P10

Table III. Extracted charge-trapping related parameters. γsurf1 9.6723e-4

Vgsqpinch

γsubs1

Vdssubs0

3.0310 V

0.0352

0.0294

Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

PARAMETER EXTRACTION METHOD FOR GAN HEMT EMPIRICAL LARGE-SIGNAL MODEL

Table IV. Extracted self-heating related parameters. αKIpk

KIpk1

KIpk0

KIpko

0.0034

5.1648e-5

0.4291

0.0020

KP10 0.0027 KP20 0.0048 KP30 0.0051 KMipkb0 0.0033

KP11 3.7697e-5 KP21 3.1019e-4 KP31 1.7130e-4 KMipkb1 4.0203e-5

αKP1 0.2195 αKP2 0.2814 αKP3 0.3053 αKMipkb 0.2960

KP1o 0.0020 KP2o 0.0026 KP3o 0.0026 KMipkbo 0.0018

4.1. Gate-source capacitance extraction The ﬁrst-order partial derivatives of Cgs against Vgs and Vds are written as follows: ∂C gs =∂V gs ¼ P11 C gs0 ð1 þ tanhðP20 þ P21 V ds ÞÞ 1 tanh2 P10 þ P11 V gs

(18)

∂C gs =∂V ds ¼ P21 C gs0 1 þ tanh P10 þ P11 V gs 1 tanh2 ðP20 þ P21 V ds Þ

(19)

As a result of the tanh behavior, the partial derivative ∂Cgs /∂Vgs has a peak when P10 + P11*Vgs = 0 and the partial derivative ∂Cgs /∂Vds has a peak when P20 + P21*Vds = 0. Figure 10 shows the partial derivatives calculated from the extracted Cgs values. Thus P10 = 2P11 and P20 = 10P21 are acquired. The following Equations (20) and (21) can be acquired by putting P10 = 2P11 and P20 = 10P21 into (16). (20) Cgs Vgs ¼ 2V ¼ C gsp þ ð1 þ tanhð25P21 ÞÞC gs0 Vds ¼ 35V

C gs Vgs ¼ 2V ¼ C gsp þ C gs0

(21)

Vds ¼ 10V

Thus, Cgsp and Cgs0 can be written as C gs0 ¼

! C gs Vgs ¼ 2V ; C gs Vgs ¼ 2V ÷ tanhð25P21 Þ Vds ¼ 35V

(22)

Vds ¼ 10V

C gsp ¼ C gs Vgs ¼ 2V C gs0

(23)

Vds ¼ 10V

It is inferred from (18) that the maximum of ∂Cgs /∂Vgs is reached when P10 + P11*Vgs = 0, Vds = 30 V, and its maximum is P11*Cgs0*(1 + tanh(20*P21)). So P11 can be written as

Figure 10. Numerical partial derivatives of Cgs. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

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P11 ¼

∂C gs =∂V gs Vds ¼ 30V Vgs ¼ 2V

ð1 þ tanhð20*P21 ÞÞC gs0

(24)

So far, Cgsp, Cgs0, P10, P11, and P20 are all functions of P21. Once P21 is extracted, the rest of the parameters can be calculated from the aforementioned deduction. Dividing Equation (18) by (19), the following equation can be acquired: ∂C gs =∂V gs P11 1 tanh P10 þ P11 V gs P11 expð2ðP20 þ P21 V ds ÞÞ þ 1 ¼ (25) ¼ ∂C gs =∂V ds P21 1 tanhðP20 þ P21 V ds Þ P21 expð2ðP10 þ P11 V ds ÞÞ þ 1 Putting the values of ∂Cgs /∂Vgs and ∂Cgs /∂Vds at a selected bias point into (25), P21 can be acquired by solving the nonlinear equation. Thus, the six parameters in (16) can be extracted with the described method. 4.2. Gate-drain capacitance extraction To ensure charge conservation, P41 = P11 and P31 = P21 can be obtained [19]. The ﬁrst-order partial derivatives of Cgd against Vgs can be written as follows: ∂Cgd =∂V gs ¼ P41 C gd0 ð1 þ tanhðP30 P31 V ds ÞÞ 1 tanh2 P40 þ P41 V gs V ds (26) As a result of the tanh behavior, the partial derivative ∂Cgd /∂Vgs has a peak when P40 + P41* (VgsVds) = 0 and its maximum is P41*Cgd0*(1 + tanh(P30P31*Vds)). Figure 11 shows the partial derivatives calculated from the extracted Cgd values. Thus, P40 = 2P41 can be acquired, and Cgd0 can be written as follows: ∂C gd =∂V gs Vds ¼ 0 V C gd0 ¼

Vgs ¼ 2V

P41 ð1 þ tanhðP30 ÞÞ

(27)

So far, P40, P41, and P31 have been extracted and Cgd0 is a function of P30. Thus, only Cgdp and P30 need to be extracted. Two equations can be acquired by putting the extracted Cgd values at two bias points into (17). Cgdp and P30 can be obtained by solving the two equations. Considering the uncertainty of measurements, the selected two bias points should not be very close to the endpoint of the measured interval. And the distance between the two selected bias points should not be too close. This will lead to a typical solution instead of a singular solution. In this paper, Vgs = 3 V, Vds = 25 V and Vgs = 1 V, Vds = 10 V were selected.

Figure 11. Numerical partial derivatives of Cgd. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

PARAMETER EXTRACTION METHOD FOR GAN HEMT EMPIRICAL LARGE-SIGNAL MODEL

To this step, the six parameters in (17) can be extracted with the described method. The extracted capacitance model parameters are shown in Table V.

5. MODEL VERIFICATION The developed model is embedded in ADS software by using the Symbolically Deﬁned Devices (SDD), and the extracted parameters are put into the SDD model. Figure 12 shows the comparison between S-parameter measurements and large-signal simulations at two bias points up to 40 GHz. Figure 13 shows the one-tone power sweep measurements and simulations of the device at Vgs = 2.5 V and Vds = 28 V with f0 = 10, 12, and 14 GHz. At this bias condition, the device works in class AB. The load impedances for f0 = 10, 12, and 14 GHz are 34.82 + 90.75j, 26.17 + 76.17j, and 14.52 + 71.32j, respectively. Figure 14 shows the same results of another device with gate width 4 × 100 μm and gate length 0.25 μm. The load impedances for this device at f0 = 10, 12, and 14 GHz

Table V. Extracted capacitance model parameters. Cgsp(fF)

Cgs0(fF)

P10

P11

P20

P21

339.3952 Cgdp(fF) 56.191

238.2186 Cgd0(fF) 93.34

3.7813 P30 1.353

1.6229 P31 0.0091

0.5206 P40 3.7813

0.0091 P41 1.6229

Figure 12. Comparison between S-parameter measurements and large-signal simulations. (a) Vgs = 2.5 V and Vds = 28 V and (b) Vgs = 4 V and Vds = 10 V.

Figure 13. One-tone power sweep measurements (circles) and simulations (lines) of the device with gate width 4 × 80 μm and gate length 0.25 μm at Vgs = 2.5 V and Vds = 28 V with f0 = 10 GHz (a), 12 GHz (b), and 14 GHz (c). Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

Z. WEN ET AL.

Figure 14. One-tone power sweep measurements (circles) and simulations (lines) of the device with gate width 4 × 100 μm and gate length 0.25 μm at Vgs = 2.5 V and Vds = 28 V with f0 = 10 GHz (a), 12 GHz (b), and 14 GHz(c).

are 33.29 + 76.22j, 24.29 + 64.37j, and 9.99 + 59.24j, respectively. Figure 15 shows the power sweep measurements and simulations of the 4 × 100 μm device at Vgs = 2.0 V and Vds = 20 V. At this bias condition, the device works in class AB. The load impedances for f0 = 10, 12, and 14 GHz are 33.45 + 64.43j, 20.49 + 53.50j, and 19.07 + 48.98j, respectively. Figure 16 shows the simulated and measured DC current versus input power for the 4 × 100 μm device. It is clearly seen that the extracted model can accurately predict the large-signal characteristics of the device. Figure 17 shows the ﬁtting results of PIV curves at Vgsq = 2.5 V, Vdsq = 28 V and Vgsq = 2.0 V, Vdsq = 28 V. As can be seen, the extracted model can accurately reproduce the pulsed measurements.

Figure 15. One-tone power sweep measurements (circles) and simulations (lines) of the device with gate width 4 × 100 μm and gate length 0.25 μm at Vgs = 2.0 V and Vds = 20 V with f0 = 10 GHz (a), 12 GHz (b), and 14 GHz(c).

Figure 16. Simulated and measured DC current versus input power for the 4 × 100 μm device. (a) Vgs = 2.5 V and Vds = 28 V and (b) Vgs = 2.0 V and Vds = 20 V. Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Model. (2015) DOI: 10.1002/jnm

PARAMETER EXTRACTION METHOD FOR GAN HEMT EMPIRICAL LARGE-SIGNAL MODEL

Figure 17. Fitting results of pulsed I–V curves for the 4 × 80 μm device. (a) Vgsq = 2.5 V and Vdsq = 28 Vand (b) Vgsq = 2.0 V and Vdsq = 28 V.

6. CONCLUSION An analytical parameter extraction method for GaN HEMTs empirical large-signal model is presented in this paper. A modiﬁed Angelov drain-source current model with 53 parameters and the capacitance model are extracted in an analytical way. Good accuracy of large-signal characteristics is achieved. The proposed parameter extraction method is implemented in MATLAB programming, and the whole extraction procedure is automatically performed by the program. It is highly efﬁcient when used for modeling of large quantities of devices. ACKNOWLEDGEMENTS

The authors would like to thank Dr Kelvin S. Yuk, Department of Electrical and Computer Engineering, University of California, Davis, USA, for his useful advices and suggestions for the work in this paper.

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