Modelling of deviations between static and dynamic ...

Modelling of deviations between static and dynamic drain characteristics in GaAs FETs F.Filicori∗ , G.Vannini∗∗ , A.Mediavilla∗∗∗ , A.Tazon∗∗∗ ∗

Istituto di Ingegneria, Universit`a di Ferrara, Via Scandiana 21 – 44100 Ferrara, Italy.

∗∗

Dipartimento di Elettronica, Informatica e Sistemistica, Universit`a di Bologna, Viale Risorgimento 2 – 40136 Bologna, Italy. ∗∗∗

Departamento de Electronica, Universidad de Cantabria, Avda. de Los Castros – 39005 Santander, Spain. ABSTRACT A new approach is proposed for the accurate modelling of dynamic (e.g., pulsed) drain characteristics in GaAs FETs. It can be easily implemented in the framework of Harmonic-Balance tools for nonlinear circuit analysis and design. The model takes simultaneously into account low-frequency dispersive phenomena due to surface state densities, deep level traps and thermal effects. It is based on mild assumptions confirmed both by theoretical considerations and preliminary experimental results for GaAs MESFETs. INTRODUCTION The effects of low-frequency dispersive phenomena associated with deep level traps and surface state densities in GaAs MESFETs (or other III-V electron devices) have been addressed by many authors [1..11]. From a macroscopic point of view, these phenomena cause important deviations between “statically” and “dynamically” (e.g., pulsed) measured drain current characteristics. In particular, important frequency dispersion of the transconductance, output conductance etc..., has been measured in the range from DC to about some hundreds of KHz. Such a behaviour involves relevant problems for large-signal performance prediction at microwave frequencies, where accurate modelling is needed both for AC and DC components of the drain current. The low-frequency dispersion of electrical characteristics is related to phenomena whose dynamics are described by relatively large time constants (typically from fractions to hundreds of microseconds). In this respect it must be observed that dispersion due to “traps” (with this term hereinafter we intend both surface state densities and deep level traps) cannot always be addressed separately from dynamic phenomena due to thermal effects, which may become relevant in an electron device when large-signal operation involving important power dissipation is concerned. In fact, the time constants associated with thermal effects, although somehow longer, are not always very different from those associated with traps. In this paper a new approach, which takes into account both traps and thermal phenomena, is proposed for the accurate modelling of deviations between static and dynamic drain characteristics in GaAs FETs. The validity of the method has been preliminarily verified for GaAs MESFETs, but the same approach should be valid for other III-V electron devices (e.g., HEMTs). THE NEW MODELLING APPROACH The complexity of low-frequency dispersive phenomena makes it quite difficult to apply a rigorous theoretical approach to electron device modelling for computer-aided analysis and design of microwave integrated circuits. However, for circuit-oriented modelling, a simplified approach can be efficiently adopted as will be shown in the following. Let us consider the low-frequency region where microwave reactive effects, due to those charge-storage and transit-time phenomena which are relevant only at very high frequencies, are still absent in a GaAs FET. In such conditions, the low-frequency drain current response of the device can be expressed in

1

the form: iD (t) = Φ [vG (t), vD (t), x(t), θ(t)]

(1)

where Φ is a purely algebraic nonlinear function and v G , vD (in the following, lower-case letters will denote time-dependent quantities) are the gate and drain voltages. In (1), x represents the set of state variables (e.g., equivalent surface potentials or trap level filling) which describe the “slow” dynamic phenomena associated with the traps, while θ is the intrinsic device temperature which, in this context, is assumed to be uniform In typical microwave applications the smallest frequency of interest, apart from DC, is at least several M Hz, that is, well above the upper cut-off frequency (which is on the order of some hundreds of KHz) associated with low-frequency dispersive phenomena. In such conditions, it can be obviously assumed that x(t) ' X0 and θ(t) ' Θ0 , where the subscript “0 ” denotes the DC components. In our modelling approach the average temperature Θ 0 , for a Rgiven case temperature, is assumed to be dependent only on the average1 ,2 dissipated power P0 = T1 0T vD (t)iD (t)dt. This mild assumption is easily justified if a quasi-linear resistance/capacitance thermal model is adopted 3 to describe the temperature dependence on power dissipation. As far as the state variables associated to the traps are concerned, also the vector X 0 of their DC components is assumed to be dependent only on the DC components V G0 , VD0 of the external voltages vG (t), vD (t). This implies that X0 is not significantly affected by the amplitude and “shape” of the alternate components vG (t) − VG0 and vD (t) − VD0 ; in other words it is assumed that the nonlinear effects possibly related to traps, are not so strong to involve relevant AC to DC conversion in the relation between vG (t), vD (t) and x(t) ' X0 . Such an assumption, which is partially validated by experimental results, can be intuitively justified by observing that the regions of the device where traps are located (i.e., gate-source and gate-drain surface regions, channel-substrate interface) are not directly responsible for important nonlinear effects. Moreover, large-signal equivalent circuits or numerical physics-based models proposed by many authors, directly or indirectly lead to analogous conclusions, and, to our knowledge, no model has been proposed which may really take into account the direct influence of amplitude and “shape” of large AC voltage components on the low-frequency dispersive phenomena associated with the traps. According to the above considerations, eqn. (1) can be expressed in the following form: iD = F[vG , vD , VG0 , VD0 , P0 ]

(2)

which shows that, in the presence of low-frequency dispersive phenomena, i D (t) is not only dependent on the instantaneous gate and drain voltages v G (t) and vD (t), but also on their mean values VG0 , VD0 and the average dissipated power P0 . In order to make the experimental characterisation affordable, eqn. (2) can be simplified by linearising with respect to VG0 , VD0 and P0 in the neighbourhood of suitably chosen “nominal” operating ∗ , V ∗ and P ∗ . This is justified by observing that the dynamic phenomena due to traps conditions VG0 0 D0 and device heating, although by no means negligible, are usually not so strong to involve highly nonlinear effects; in fact, the most important nonlinear phenomena in a field-effect electron device mainly derive from direct modulation of the channel conductivity due to v G (t) and vD (t). Moreover, many microwave circuits operate with practically constant values, or with limited variations, of the DC bias voltages on active devices. Linearisation of (2) with respect to V G0 , VD0 and P0 , in the neighbourhood of the “nominal” operating ∗ , V ∗ and P ∗ , leads, after simple algebraic manipulations, to the simplified expression: conditions VG0 0 D0 ∗ iD = FDC [vG , vD ] + fG [vG , vD ](vG − VG0 ) + fD [vG , vD ](vD − VD0 ) + fP [vG , vD ](P0 − P0∗ ) ∗ where FDC

fP = 1



∂F ∂P0 ∗

∂F ∗ ) + = F|∗ + ∂V (vG − VG0 G0 ∗

.

∂F ∗ ∂VD0 (vD − VD0 ), fG = − ∗

∂F ∂VG0 , fD = − ∗

(3)

∂F ∂VD0 and ∗

The contribution to dissipated power deriving from the gate port can be neglected. Periodic steady-state operation is considered in the perspective of Harmonic-Balance circuit analysis. 3 Actually, heat conduction in GaAs is affected by nonlinear phenomena; however, as the validation procedure will confirm, the assumption of average temperature Θ0 depending only on the average dissipated power P0 , is still acceptable when the basic aim is modelling the effects of low-frequency dispersion above cut-off. 2

2

In eqn. (3), the functions fG and fD take into account the deviations between the static and dynamic drain current response due to the traps. The term f D , which is normally dominant, can be assumed to be more directly related to trapping phenomena at the channel-substrate interface; f G , instead, can be somehow associated to surface state densities. In DC steady-state operation, the contribution due to f G and fD is identically zero (since vG (t) = VG0 and vD (t) = VD0 ). In such conditions, by taking into account that i D (t) = ID0 and P0 = VD0 ID0 , eqn. (3) can be rewritten in the following form: ID0 = FDC [VG0 , VD0 ] =

∗ [V , V ∗ FDC G0 D0 ] − fP [VG0 , VD0 ]P0 1 − fP [VG0 , VD0 ]VD0

(4)

In this equation, FDC represents the actual static drain characteristic of the electron device while ∗ the term FDC is the corresponding equi-thermal static characteristic (i.e., the static characteristic corresponding to constant4 average temperature Θ∗0 ). In eqns. (3) and (4) the term fP takes into account the deviations from the equi-thermal characteristic due to variations in the power P 0 actually dissipated in the device. When the dynamic deviations due to thermal effects are not very relevant (e.g., operating conditions with moderate power dissipation), the term f P can be neglected in eqns. (3) and (4); in such conditions, ∗ since FDC = FDC , the drain current response of the device is described by the following simplified model: iD = FDC [vG , vD ] + fG [vG , vD ](vG − VG0 ) + fD [vG , vD ](vD − VD0 ) (5) It can be shown that a number of approaches [1,9] proposed by different authors for the equivalent circuit modelling of low-frequency dispersive phenomena, are special cases of (5). The identification of the model described by eqn. (5), as will be shown in the following, can be performed very easily, even without using a pulsed measurement set-up. MODEL IMPLEMENTATION IN CIRCUIT ANALYSIS ALGORITHMS In order to apply the above described modelling approach for microwave circuit analysis, eqns. (3) or (5) must be “embedded” in a large-signal model which also includes RF nonlinear dynamic phenomena. To this end both classical equivalent circuits or one of the recently proposed mathematical approaches [12,13] can be adopted. In the former case, model implementation can easily be based on an equivalent circuit of the type T shown in Fig. 1 (for simplicity only the “intrinsic” FET is considered), where the capacitances C G T provide a “macroscopic” modelling of charge trapping. Suitable values for the equivalent and CD T C T and τ = RT C T can easily be chosen when, as it is quite common time constants τ1 = RG 2 D D G in microwave applications, the operating frequency is well above the trap cut-off frequency. The T and RT should assume suitably “large” values, in order to make the currents equivalent resistors RG D T and C T negligible with respect to i and i . The introduction of long across the capacitances CG G D D equivalent time constants through “small” capacitances and “large” resistances is important in order to make the equivalent circuit physically meaningful, as the amount of energy storage associated to charge trapping is very small and practically negligible. This choice leads to low-frequency dynamic drain characteristics which, in accordance with experiment and unlike other similar models [1,9], give zero channel current iCH current at any time instant when vD (t) = 0. The channel current source iCH in the equivalent circuit in Fig. 1 is defined, according to the simplified model (5), by the following expression: T T iCH = F¯ [veCG , vD ] − fG [veCG , vD ]veG − fD [veCG , vD ]vD

(6)

where ve = v(t − τd ) (τd being the well-known delay associated to the transconductance g m in classical equivalent circuits) and F¯ [veCG , vD ] = FDC [veCG , vD ] + fG [veCG , vD ]veCG + fD [veCG , vD ]vD . By taking into account that in the low-frequency range above the trap cut-off frequency veCG = T T = v T (t − τ ) ' V vCG (t − τd ) ' vG (t), veG G0 and vD (t) ' VD0 , it can be easily verified that eqn. (6) d G coincides with the low-frequency dispersion model (5). 4

This characteristic could be, for instance, the “ideal ” DC characteristic obtained by a conventional physics-based numerical device simulation where temperature variations deriving from power dissipation are neglected.

3

When the thermal effects due to power dissipation are not negligible, the term f P [veCG , vD ](P0 − P0∗ ) should also be included in the channel current i CH ; this involves also the computation of P 0 in the framework of the circuit analysis algorithm. When large-signal RF electron device modelling is performed by using the mathematical Nonlinear Integral approach proposed in [12], the low-frequency dispersion model (5) is intrinsically embedded in the RF model. In fact, according to the Nonlinear Integral Model (NIM) [12], the device drain 5 current is described by the following equation: iD = FDC [vG , vD ] +

+M X

k=−M

n

o

jωk t g g Y 21 [vG , vD , ωk ]VGk + Y22 [vG , vD , ωk ]VDk e

(7)

where VGk and VDk are the harmonic components of vG (t) and vD (t) at the angular frequencies ωk (with ω0 = 0). In eqn. (7), the terms Yfij [vG , vD , ωk ] are “dynamic” admittances nonlinearly controlled by the instantaneous values of the gate and drain voltages, which can easily be obtained [12] as the differences between the conventional bias-dependent small-signal Y parameters and the corresponding DC differential conductances (see [14] in these Proceedings). In the low-frequency range above the trap cut-off frequency, the Yfij ’s are real and frequency independent, since microwave reactive effects are not present and the traps do not respond to rapid voltage variations. Under such conditions, it can be easily shown [15] that the drain current response (7) of the NIM practically coincides with the low-frequency dispersion model (5). When low-frequency dispersion due to thermal effects is not negligible, the term f P [vG , vD ](P0 − P0∗ ) can be easily included in eqn. (7). EXPERIMENTAL CHARACTERISATION AND MODEL VALIDATION The experimental characterisation of the simplified model (5) can easily be carried out by means of conventional equipment for DC and small-signal AC measurements. In particular, the term F DC in (5) is simply the measured DC drain characteristic of the electron device, while the functions f G and fD can be identified on the basis of small-signal admittance parameter measurements at a frequency above the cut-off frequency of the traps but below the frequency where microwave reactive effects become significative. In fact, linearisation of (5) with respect to small sinusoidal signals in this frequency range leads, for each bias condition VG0 , VD0 , to the following expressions: fG [VG0 , VD0 ] = gmAC [VG0 , VD0 ] − gmDC [VG0 , VD0 ] fD [VG0 , VD0 ] = goAC [VG0 , VD0 ] − goDC [VG0 , VD0 ]

(8)

These equations provide the terms f G and fD as the differences between the dynamic (g mAC , goAC ) and static (gmDC = ∂ID0 /∂VG0 , goDC = ∂ID0 /∂VD0 ) differential transconductance and output conductance of the electron device. Equations (5) and (8) show that the large-signal dynamic drain characteristics can be predicted on the basis of measured static characteristics (including DC differential parameters) and bias-dependent small-signal AC parameters. This property makes the model intrinsically coherent both with DC and small-signal RF data. When low-frequency dispersion due to thermal effects is not negligible, the term f P should also be included in the model (see eqn. (3)). In such conditions, also large-signal AC (e.g., pulsed) measure∗ , f , f and f is not so straightforward ments are needed. Moreover, separate identification of F DC G D P as for the model described by eqn. (5). This can be intuitively understood by observing that the variation of the device bias conditions necessarily involves a variation in the dissipated power P 0 . ∗ , f , f For such a reason, it is more convenient to “extract” the four functions F DC G D and fP by a model “fitting” procedure based on a suitably large number (the minimum number required is 4) of static and dynamic (e.g., pulsed) measurements. In particular, for each value of v G (t) and vD (t), a least squares algorithm can be adopted to solve a linear system of equations (of the type (3)) for the ∗ , f , f and f . unknown terms FDC G D P ∗ , f , f and f have been identified on the basis of a suitably chosen Once the four functions FDC P G D set of dynamic (e.g., pulsed) measurements, the model will be able to predict the dynamic response 5

An analogous expression describes the gate current.

4

of the electron device to any gate and drain voltage waveform, by taking also into account the actual power dissipation. A preliminary experimental validation of the proposed model has been carried out on the basis of static and pulsed measurements of the drain current performed on a NEC NE720 GaAs MESFET. The four ∗ , f , f and f have been identified by least square minimisation of the discrepancies functions FDC G D P between the model and pulsed measurements. The results obtained give clear, positive indications about the capability of the model to describe both the static and dynamic drain characteristics. In particular, in Fig. 2, static and pulsed drain current characteristics measured, for the NEC NE720, by superimposing gate and drain voltage pulses (with sufficiently “short” pulse width and “long” repetition time) on the “quiescent” bias voltages V G0 = −1V , VD0 = 5V are shown. Short pulse duration (lower than 1µsec) was chosen in order to characterise the dynamic response of the device above the cut-off frequency of the traps. The same figure also shows the drain current responses computed, under the same condition as for measurements, by means of the proposed model when the temperature effects described by the term f P (P0 −P0∗ ) are neglected or taken into account. The better agreement obtained in the latter case shows that both traps and thermal effects should be taken into account for more accurate large-signal performance prediction. It can also be noted that according to device physics and experimental evidence, and unlike other empirical modelling approaches [1,9], all the predicted dynamic characteristics give i D = 0 for vD = 0. ∗ (both In Fig. 3 the DC drain characteristic F DC and the corresponding equi-thermal characteristic F DC computed according to the model equation (3)) are compared with the actual DC measurements. In addition to overall good agreement, it should be noted that, in accordance with physics-based ∗ models, the equi-thermal characteristic F DC does not show the negative slope which is mainly due to temperature increase deriving from increased power dissipation. For model validation at microwave frequencies either the equivalent circuit in Fig. 1 or the Nonlinear Integral Model defined by eqn. (7) can be used. Results are quite similar, although those provided by the NIM are slightly better since no “lumped-element” approximation is needed. In Fig. 4 the results of performance prediction using the NIM are compared with measured values for an 8GHz large-signal GaAs MESFET amplifier, showing overall good agreement. These preliminary results, together with those provided in Figs. 2 and 3, seem to confirm the validity of our approach for the modelling of dynamic drain characteristics. REFERENCES [1] C.Camacho-Penalosa, C.S.Aitchison, “Modelling frequency dependence of output impedance of a microwave MESFET at low frequencies”, Electronics Letters, vol.21, n.12, pp. 528-529, 1985. [2] P.H.Ladbrooke, S.R.Blight, “Low-field low-frequency dispersion of transconductance in GaAs MESFET’s with implications for other rate-dependent anomalies”, IEEE Trans. on ED, vol.35, pp. 257-267, 1988. [3] N.Scheinberg, R.J.Bayruns, P.W.Wallace, R.Goyal, “An accurate MESFET model for linear and microwave circuit design”, IEEE Journ. of Solid-State Circuits, vol.24, pp. 532-539, 1989. [4] J.A.Reynoso-Hernandez, J.Graffeuil, “Output conductance frequency dispersion and low-frequency noise in HEMT’s and MESFET’s”, IEEE Trans. on MTT, vol.37, pp. 1478-1481, 1989. [5] J.M.Golio, M.G.Miller, G.N.Maracas, D.A.Johnson, “Frequency-dependent electrical characteristics of GaAs MESFET’s”, IEEE Trans. on ED, vol.37, pp. 1217-1227, 1990. [6] A.J.McCamant, G.D.McCormack, D.H.Smith, “An improved GaAs MESFET Model for SPICE”, IEEE Trans. on MTT, vol.38, pp. 822-824, 1990. [7] M.Lee, L.Forbes, “A self-backgating GaAs MESFET model for low-frequency anomalies”, IEEE Trans. on ED, vol.37, pp. 2148-2157, 1990. [8] J.F.Vidalou, J.F.Grossier, M.Chaumas, M.Camiade, P.Roux, J.Obregon, “Accurate nonlinear transistor modeling using pulsed S parameters measurements under pulsed bias conditions”, IEEE MTT-S, 1991. [9] H.Sledzik, I.Wolff, “Large-signal modeling and simulation of GaAs MESFETs and HFETs”, Intern. Journ. of Microwave and Millimeter-wave Computer-Aided Engineering, vol.2, pp. 49-60, 1992. [10] V.Rizzoli, A.Costanzo, A.Neri, “An advanced empirical MESFET model for use in nonlinear simulation”, Proc. of 22nd EUMC, pp. 1103-1108, 1992.

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[11] T.Fernandez, Y.Newport, A.Tazon, A.Mediavilla, “Extracting advanced large signal MESFET models from DC, AC and pulsed I/V measurements”, Proc. of MIOP, 1993. [12] F.Filicori, G.Vannini and V.A.Monaco, “A nonlinear integral model of electron devices for HB circuit analysis”, IEEE Trans. on MTT, vol. 40, pp. 1456-1465, 1992. [13] D.E.Root, S.Fan, J.Meyer, “Technology independent large-signal non quasi-static FET models by direct construction from automatically characterized device data”, Proc. of 21st EUMC, pp. 927-932, 1991. [14] F.Filicori, G.Vannini, V.A.Monaco, “Large-signal modelling of Dual-Gate GaAs MESFETs”, these Proceedings. [15] F.Filicori, G.Vannini, A.Mediavilla, A.Tazon, “Large-signal GaAs FET modelling including traps and thermal phenomena”, to be published on Alta Frequenza. iD (mA) 70 60

0

50

G

D

v

T G

T CG

v

-0.25

40 -0.5 30

CG

v

T D

i

-0.75

T CD

20

CH

-1

T

10

T

RG

VGS=-1.25

RD 0

0

1

2

3

4

5

6

VDS (V)

S Fig. 1: Large-signal intrinsic equivalent circuit of a GaAs MESFET including also low-frequency dispersive phenomena. Deviations between static and dynamic drain characteristics are modelled through the nonlinear current source iCH which is nonlinearly controlled by vCG , vD and linearly controlled T T by vG and vD .

Fig. 2: Pulsed DC drain characteristics for a GaAs MESFET NEC NE720 obtained superimposing short duration gate and drain pulses to the quiescent bias condition VG0 = −1V , VD0 = 5V . The agreement between measured (•) and predicted values (—) (- - -) is better when also the thermal term fP (P0 − P0∗ ) is included in the model (—).

POUT (dBm) 80

20

ID (mA)

k=1

0

70 60

10 k=3

-0.25

50

0 -0.5

40 30

k=2

-0.75

-10

NIM measured

20 -1 10 0

VGS=-1.25 0

1

2

3

4

5

-20 6

VDS (V)

Fig. 3: Plots of measured (•) and predicted (—) static DC characteristics for the GaAs MESFET NEC ∗ NE720. The equi-thermal DC characteristic FDC (- -), computed according to the model equation (3), is also shown.

0

5

10 PIN (dBm)

15

20

Fig. 4: Plots of output powers associated with different order harmonics (k=1,2,3) vs input power PIN for a Ku -band GaAs MESFET amplifier (VGS0 = −0.4V , VDS0 = 4V ) with sinusoidal input (f0 = 8 GHz).

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