For the first time, a fully measurement based extraction procedure of non linear and non uniform transmission line model of FET devices is proposed. This model describes accurately the distributed nature under the device gate which allows a good distortion prediction (IM3) and promises good perspectives for simulation of noise characteristics in non linear circuits

Camiade, M.

Laloue, A.

Nallatamby, J.C.

Obregon, J.

Prigent, M.

Quéré, R.

Valenza, M.

Vildeuil, J.C.

English

AMS Acta - Alm@DL - Università di Bologna

A MEASUREMENT BASED MODEL OF HEMT TAKING INTO ACCOUNT THE NON
LINEAR, NON UNIFORM TRANSMISSION LINE NATURE OF THE CHANNEL AND ITS
ASSOCIATED LOW FREQUENCY NOISE SOURCES
A. LALOUE*, M. CAMIADE**, M. VALENZA***, J.C. VILDEUIL***,
 J.C. NALLATAMBY* ,M. PRIGENT*, J. OBRÉGON****, R. QUÉRÉ*
* IRCOM - CNRS UMR 6615- University of Limoges - IUT - 7, rue Jules Vallès - 19100 BRIVE - FRANCE
E-mail : laloue@brive.unilim.fr
** UMS - Domaine de Corbeville - 91404 ORSAY - FRANCE
*** CEM2 - CNRS UMR 5507 -University of Montpellier II - Place Eugène Bataillon -
34095 MONTPELLIER Cedex 5 - FRANCE
**** IRCOM - CNRS UMR 6615 - University of Limoges - 123, av. Albert Thomas - 87000 LIMOGES - FRANCE
ABSTRACT
For the first time, a fully measurement based extraction procedure of non linear and non uniform transmission line
model of FET devices is proposed. This model describes accurately the distributed nature under the device gate which
allows a good distortion prediction (IM3) and promises good perspectives for simulation of noise characteristics in non
linear circuits.
INTRODUCTION
With the increase of working frequencies toward millimeter waves, high volume commercially available MMICs
including HEMTs operate yet at several ten GHz.
On the other hand, in FET based non linear circuits, inter-modulation frequencies are generated along the non linear
channel from source to drain, and are directly responsible for many characteristics such as : the distortion in power
amplifiers (IM3) or the PM noise in oscillators.
In order to optimize these features, the development of an accurate distributed model of HEMT is no longer an academic
work [1]. In this communication we present a new non linear distributed model, fully extracted from pulsed I-V and S
parameter measurements, which may be applied to all the field effect devices (MESFET, HEMT and MOSFET).
In this model, channel is considered from source to drain as a non uniform, non linear, active transmission line [2][3][4].
Figure 1 shows the proposed model. It includes N unit cells. Obviously the number of cells N depends on the gate length
L. Practically, we found that for L ≤ 0.3 µm, N = 10 is a good compromise between accuracy and complexity of this
model.
Every unit cell includes :
• On one hand : a non linear gate channel capacitance C = fC (Vgk), in parallel with a schottky diode I = fG (Vgk).
These elements are only function of their own port voltage Vgk.
• On the other hand a non linear channel current controlled source I = fI (Vgk, ∆Vck) which depends on its own port
voltage ∆Vck, and on the controlling voltage of the cell : Vgk.
• At last, two linear fringing capacitances are added at the source and drain end of the channel. The other extrinsic
linear parasitic elements are described by lumped elements as usual [5].
It must be pointed out that the non linear functions fC, fG and fI are identical for all the unit cells. Nevertheless, the unit
cell voltages : Vgk and ∆Vck vary along the channel from source to drain, following the rank k of the cell under
consideration : from 1 to N. It results an accurate non linear, non uniform, distributed model of the channel.
Moreover, while the model is non quasistatic by nature, it does not need the inclusion of any time-delay, nor a
questionable charging resistor Ri [6].
In the first part of this paper, the model extraction procedure from pulsed I-V characteristic and S parameter
measurements is described. As an example a 2 fingers of 100 µm wide PHEMT model is shown. The extracted model
and the measurements are compared. The improvement on the non linear transistor modeling is shown in the third part.
Finally, in the last part, low frequency noise sources are included into the model along the channel.
1. DISTRIBUTED MODEL EXTRACTION PROCEDURE
The extraction procedure is based on pulsed measurement of I-V characteristics and S parameters. The pulsed
measurement allows to keep constant temperature and trap steady state during the characterization [5].
First, the extrinsic linear parasitic elements are obtained by the well known extraction methodology used for the classical
lumped FET model.
The intrinsic part of the model is formed by a chain of N identical cells. Each cell represents the electrical behavior of a
transistor slice. Then, it includes a schottky diode I = fG (Vgk), in parallel with a non linear capacitance C = fC (Vgk), and
a non linear current controlled source I = fI (Vgk, ∆Vck). These elements only depend on the voltage Vgk and ∆Vck,
which are present on the access of cell under consideration. In our model, every cell elements have both same equations
and same parameter values along the channel. Only the controlled voltages Vgk and ∆Vck vary following the k rank of
the cell number. Two edge capacitances are added at source side and drain end. They take into account the capacitive
coupling effects between the metallic electrodes and the edge channel.
In a first step, the parameters of the diode and the controlled non linear current source are extracted. To do this, a
software based on fast simulated diffusion algorithm, allowing to minimize a non linear error function, is used [7].  From
the characteristic measurement Ids (Vgs, Vds), the 13 parameters of the controlled current generator equations, fI (Vgk,
∆Vck) are determined. And from characteristic measurement Igs (Vgd, Vds), the two diode parameters are obtained.
The non linear element equations are fully described in [8].
Finally, the N non linear capacitances, C = fC (Vgk) and the two edge capacitances Cbs and Cbd are determined from the
S parameter measurements. The direct extraction procedure can not be used, here, due to the high element number. Then
another software, based on the fast simulated diffusion algorithm is used to obtain the 6 parameter values of the non
linear capacitances and the two edge capacitance values. The error function is computed with the intrinsic Y parameters
of the model and those calculated from the S parameter measurements. Obviously, this optimization must be performed
with several measurement points chosen on the I-V characteristic load-line zone.
2.  COMPARISON BETWEEN MODEL AND MEASUREMENTS.
As an example, we show the results obtained on a PHEMT of 2 fingers of 100 µm wide. The pulsed I-V and S parameter
measurements are performed on the 1 to 20 GHz frequency range. The bias point is Vgs = 0.21 V, Vds = 2.8 V and Ids
= 41.7 mA.
A 10 cell distributed model seems to be a good compromise between accuracy and model complexity. The table 1 gives
the extrinsic linear parasitic elements. Note that the linear capacitance Cds is considered as an extrinsic element.
The non linear function fI (Vgk, ∆Vck) of one controlled current source is reported on figure 2. The extraction procedure
leads to the edge capacitance values : Cbs = 247 fF and Cbd = 34 fF.
The figure 3 shows a good agreement between whole model and measured transistor I-V characteristics. The figure 4
presents the modeled and measured S parameters for a pulsed point nearby the bias point. These figures demonstrate that
the extracted distributed model can accurately fit the measured data.
To show that the distributed model have physical behavior, the channel voltage Vck is plotted on the figure 5 for this
intrinsic bias condition Vgsi = 0.2 V and Vdsi = 3V. It shows that the voltage Vck is a non linear function along the
channel.
3. APPLICATION TO THE THIRD ORDER INTER-MODULATION SIMULATION
In order to show the model ability to take into account the distributed nature of the field effect transistor, two tone
measurement is performed on the device The chosen bias point is Vgs = 0.2 V and Vds = 2.8 V and the generator
frequencies are f1 = 10.0 GHz and f2 = 10.02 GHz. The load value is 50 Ω.The figure 6 shows a good agreement
between the simulated and measured output power at f1 and 2f1-f2 frequencies.
4. DISTRIBUTED NON LINEAR LOW FREQUENCY NOISE SOURCES
In order to model the low frequency noise sources of the channel, the measurements of the drain noise spectral density
for different bias points (Vgs0, Vds0) are performed from 1 Hz to 1 MHz frequency range. In each unit cell, an
elementary noise source is added, calculated from short circuit noise current extracted from previous measurement.
These elementary Norton current sources are presumed uncorrelated. At a given noise frequency and bias point, the
measured noise spectral density Sidrain (Vgs0 ,Vds0 ) can be related to the local noise sources <ick2> as:
( ) ( )∑
=
=
N
k
kkdrain VdsVgshicVdsVgsSi
1
2
00
2
00 ,,
Where the hk(Vgs0,Vds0) are transfer functions, calculated from the linearized non linear model and N is number of the
cells. To extract the value of every local power spectral density <ick2>, a physically based expression of <ick2> function
of the applied local voltages Vgk and ∆Vck, must be defined. In a first simplified approach <ick2> are taken identical for
all cells by writing :
( )
( )∑==
N
k k
drain
k
VdsVgsh
VdsVgsSiic
1
2
00
002
,
,
As an example, the figure 7 gives the local low frequency noise source density of a cell, versus the local voltages Vgk
and ∆Vck. A more accurate expression may be obtained by taking into account the non-linear expression of the carrier
velocity in function of the local electric field.
CONCLUSION
For the first time to our knowledge, a non linear, non uniform transmission line model of HEMT, fully extracted from
measurements, is presented. Due to the robustness of the extraction procedure, it may be applied without any significant
modification to other field effect devices. This extraction robustness is due to the accurate representation of the
distributed nature of the channel from source to drain, which reproduces naturally the non-quasistatic behavior of the
transistor.
This distributed model from source to drain may be included in a whole model which takes also into account both the
wave propagation along the width of the transistor fingers, and the distributed nature of the metallization [9].
Finally, local non linear noise sources, measurement based, are integrated to every unit-cell. This model should
accurately simulate the PM and AM noise in oscillators, or more generally the conversion noise in non linear circuits.
ACKNOWLEDGEMENT
This work was partly supported by the Centre National d'Etude Spatiale (Convention n°714/CNES/98/7305/00) and the
European Community Program Esprit 38311 Locomotive. Authors thank to J.M NEBUS and C. ARNAUD for
measuring the transistor power and IM3 characteristics.
REFERENCES
[1] J. Portilla, R. Quéré, J. Obregon, "An improved CAD oriented FET model for large-signal and noise
applications", San Diego 1994. IEEE MTTS Symposium, pp 849-852.
[2] D. Sodini, D. Rigaud , "A new approach to model d.c. and a.c. characteristics of junction gate field effect
transistors".
 1981., Solid State Electronics, Vol. 24, p. 635-647,
[3] D. Huang, H. C. Lin, "DC and transmission line models for a high electron mobility transistor",1989., IEEE
Transactions on Microwave Theory and Techniques, Vol. 37, n°9, pp 1361-1370,
[4] B. Mallet-Guy, Z. Ouarch, M. Prigent, R. Quéré, J. Obregon, "A distributed, measurements based, nonlinear
model of FETs for high frequencies applications",
 Denver 1997., IEEE MTTS Symposium, pp 869-872,
[5] J.P. Teyssier, J.P. Viaud, J.J. Raoux, R. Quéré, "Fully integrated nonlinear modeling and characterization
system of microwave transistors with on-wafer pulsed measurements",
 Orlando 1995. IEEE MTTS Symposium, pp
1033-1036,
[6] R. Anholt, "Electrical and Thermal Characterization of MESFETs, HEMTs and HBTs", Artech House Editor.
[7] G.L. Bilbo, M.B. Steer, R.J. Trew, C.R. Chang, S.G. Skaggs, "Extraction of the Parameters of Equivalent Circuit
of Microwave Transistors Using Tree Annealing",
 IEEE Trans on MTT, vol 38, n°11, 1990, pp 1711-1718.
[8] J.P. Teyssier, J.P. Viaud, R. Quéré,
 
"A new Nonlinear I(V) model for FET including Breakdown Effects", IEEE
Microwave and Guided Wave Letters, , April 1994 Vol 2, No 4, pp 104 - 106
[9] A. Laloue, J.B. David, R. Quéré, B. Mallet-Guy, E. Laporte, J.F. Villemazet, M. Soulard, "Extrapolation of a
Measurement Based Millimeter Wave Non Linear Model of  PHEMT to Arbitrary Shaped Transistors Through
Electromagnetic Simulations",1998., IEEE Transactions on Microwave Theory and Techniques, Vol. 47, n°6, pp 908-
914,
Rg (Ω) Rd (Ω) Rs (Ω) Lg (pH) Ld (pH) Ls (pH) Cpg (fF) Cpd (fF) Cds (fF)
1.0 1.5 1.5 251 213.4 67.4 22.9 46.7 21.4
Table 1 : extrinsic linear parasitic elements
∆Vc1               ∆Vck                 ∆VcN
Vg1            Vgk                     VgNCbs CbdCell k
Drain
Source
Gate
∆Vck
Vgk
fQ (Vgk)
fI (Vgk, ∆Vck)fG (Vgk)
Cell 1 Cell N
Vck
Figure 1 : topology of the intrinsic distributed model
along the gate.
0
0.02
0.04
0.06
0.08
0.1
0.12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Ids (A)
Vds (V)
Vgs = 0.8V, 0.6V, 0.4V, 0.2V, 0.0V, -0.2V, - model
Meas.
Figure 3 : Modeled and measured I-V Characteristics
of the whole transistor
.
0
0.5
1
1.5
2
2.5
3
0 2 4 6 8 10
Vck (V)
k
Vgsi = 0.2 V,  Vdsi=3V
Figure 5 : Vck voltage for Vgsi = 0.2 V and Vdsi = 3V.
0
5e-15
1e-14
1.5e-14
2e-14
2.5e-14
0 0.1 0.2 0.3 0.4 0.5 0.6
∆Vc9 (V)
<Ic92> (A2.Hz-1)
Vg 9 = 0.2V, 0V, -0.2V, -0.4V, -0.6V,
-0.8V, -1V, -1.2V, -1.4V
∆Vck
Vgk
fQ (Vgk)
fI (Vgk, ∆Vck)fG (Vgk)
<Ick2> (Vgk, ∆Vck)
Figure 7 : Local noise source <Ic92> = f(Vg9, ∆Vc9)
at 10 kHz of the cell number 9.
0
0.05
0.1
0.15
0.2
0.25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Idk (A)
∆Vck
Vgk = 1.0V, 0.5V, 0V, -0.5V, -1.0V, -1.5V, -2.0V, -
Figure 2 : Idk = fI (Vgk, ∆Vck).
S11 S12
S21 S22
10
Frequency : 1 - 16 GHz
Bias point : Vgs0 = 0.20 V, Vds0 = 2.81 V
Pulsed Point : Vgs = 0.22 V, Vds = 3.26 V
Measure
Model
Figure 4 : modeled and measured S parameters.
-20
-15
-10
-5
0
5
10
-4 -2 0 2 4
Pout (dBm)
Pin (dBm)
Meas. f1
Mod. f1
Meas. 2f1-f2
Mod .2f1-f2
Figure 6 : Comparison between two tone simulation
and measurement.


A measurement based model of HEMT teking into account the non linear, non uniform transmission line nature of the channel its associated low frequency noise sources

AMS Acta

A MEASUREMENT BASED MODEL OF HEMT TAKING INTO ACCOUNT THE NONLINEAR, NON UNIFORM TRANSMISSION LINE NATURE OF THE CHANNEL AND ITSASSOCIATED LOW FREQUENCY NOISE SOURCESA. LALOUE*, M. CAMIADE**, M. VALENZA***, J.C. VILDEUIL***, J.C. NALLATAMBY* ,M. PRIGENT*, J. OBRÉGON****, R. QUÉRÉ** IRCOM - CNRS UMR 6615- University of Limoges - IUT - 7, rue Jules Vallès - 19100 BRIVE - FRANCEE-mail : laloue@brive.unilim.fr** UMS - Domaine de Corbeville - 91404 ORSAY - FRANCE*** CEM2 - CNRS UMR 5507 -University of Montpellier II - Place Eugène Bataillon -34095 MONTPELLIER Cedex 5 - FRANCE**** IRCOM - CNRS UMR 6615 - University of Limoges - 123, av. Albert Thomas - 87000 LIMOGES - FRANCEABSTRACTFor the first time, a fully measurement based extraction procedure of non linear and non uniform transmission linemodel of FET devices is proposed. This model describes accurately the distributed nature under the device gate whichallows a good distortion prediction (IM3) and promises good perspectives for simulation of noise characteristics in nonlinear circuits.INTRODUCTIONWith the increase of working frequencies toward millimeter waves, high volume commercially available MMICsincluding HEMTs operate yet at several ten GHz.On the other hand, in FET based non linear circuits, inter-modulation frequencies are generated along the non linearchannel from source to drain, and are directly responsible for many characteristics such as : the distortion in poweramplifiers (IM3) or the PM noise in oscillators.In order to optimize these features, the development of an accurate distributed model of HEMT is no longer an academicwork [1]. In this communication we present a new non linear distributed model, fully extracted from pulsed I-V and Sparameter measurements, which may be applied to all the field effect devices (MESFET, HEMT and MOSFET).In this model, channel is considered from source to drain as a non uniform, non linear, active transmission line [2][3][4].Figure 1 shows the proposed model. It includes N unit cells. Obviously the number of cells N depends on the gate lengthL. Practically, we found that for L ≤ 0.3 µm, N = 10 is a good compromise between accuracy and complexity of thismodel.Every unit cell includes :• On one hand : a non linear gate channel capacitance C = fC (Vgk), in parallel with a schottky diode I = fG (Vgk).These elements are only function of their own port voltage Vgk.• On the other hand a non linear channel current controlled source I = fI (Vgk, ∆Vck) which depends on its own portvoltage ∆Vck, and on the controlling voltage of the cell : Vgk.• At last, two linear fringing capacitances are added at the source and drain end of the channel. The other extrinsiclinear parasitic elements are described by lumped elements as usual [5].It must be pointed out that the non linear functions fC, fG and fI are identical for all the unit cells. Nevertheless, the unitcell voltages : Vgk and ∆Vck vary along the channel from source to drain, following the rank k of the cell underconsideration : from 1 to N. It results an accurate non linear, non uniform, distributed model of the channel.Moreover, while the model is non quasistatic by nature, it does not need the inclusion of any time-delay, nor aquestionable charging resistor Ri [6].In the first part of this paper, the model extraction procedure from pulsed I-V characteristic and S parametermeasurements is described. As an example a 2 fingers of 100 µm wide PHEMT model is shown. The extracted modeland the measurements are compared. The improvement on the non linear transistor modeling is shown in the third part.Finally, in the last part, low frequency noise sources are included into the model along the channel.1. DISTRIBUTED MODEL EXTRACTION PROCEDUREThe extraction procedure is based on pulsed measurement of I-V characteristics and S parameters. The pulsedmeasurement allows to keep constant temperature and trap steady state during the characterization [5].First, the extrinsic linear parasitic elements are obtained by the well known extraction methodology used for the classicallumped FET model.The intrinsic part of the model is formed by a chain of N identical cells. Each cell represents the electrical behavior of atransistor slice. Then, it includes a schottky diode I = fG (Vgk), in parallel with a non linear capacitance C = fC (Vgk), anda non linear current controlled source I = fI (Vgk, ∆Vck). These elements only depend on the voltage Vgk and ∆Vck,which are present on the access of cell under consideration. In our model, every cell elements have both same equationsand same parameter values along the channel. Only the controlled voltages Vgk and ∆Vck vary following the k rank ofthe cell number. Two edge capacitances are added at source side and drain end. They take into account the capacitivecoupling effects between the metallic electrodes and the edge channel.In a first step, the parameters of the diode and the controlled non linear current source are extracted. To do this, asoftware based on fast simulated diffusion algorithm, allowing to minimize a non linear error function, is used [7].  Fromthe characteristic measurement Ids (Vgs, Vds), the 13 parameters of the controlled current generator equations, fI (Vgk,∆Vck) are determined. And from characteristic measurement Igs (Vgd, Vds), the two diode parameters are obtained.The non linear element equations are fully described in [8].Finally, the N non linear capacitances, C = fC (Vgk) and the two edge capacitances Cbs and Cbd are determined from theS parameter measurements. The direct extraction procedure can not be used, here, due to the high element number. Thenanother software, based on the fast simulated diffusion algorithm is used to obtain the 6 parameter values of the nonlinear capacitances and the two edge capacitance values. The error function is computed with the intrinsic Y parametersof the model and those calculated from the S parameter measurements. Obviously, this optimization must be performedwith several measurement points chosen on the I-V characteristic load-line zone.2.  COMPARISON BETWEEN MODEL AND MEASUREMENTS.As an example, we show the results obtained on a PHEMT of 2 fingers of 100 µm wide. The pulsed I-V and S parametermeasurements are performed on the 1 to 20 GHz frequency range. The bias point is Vgs = 0.21 V, Vds = 2.8 V and Ids= 41.7 mA.A 10 cell distributed model seems to be a good compromise between accuracy and model complexity. The table 1 givesthe extrinsic linear parasitic elements. Note that the linear capacitance Cds is considered as an extrinsic element.The non linear function fI (Vgk, ∆Vck) of one controlled current source is reported on figure 2. The extraction procedureleads to the edge capacitance values : Cbs = 247 fF and Cbd = 34 fF.The figure 3 shows a good agreement between whole model and measured transistor I-V characteristics. The figure 4presents the modeled and measured S parameters for a pulsed point nearby the bias point. These figures demonstrate thatthe extracted distributed model can accurately fit the measured data.To show that the distributed model have physical behavior, the channel voltage Vck is plotted on the figure 5 for thisintrinsic bias condition Vgsi = 0.2 V and Vdsi = 3V. It shows that the voltage Vck is a non linear function along thechannel.3. APPLICATION TO THE THIRD ORDER INTER-MODULATION SIMULATIONIn order to show the model ability to take into account the distributed nature of the field effect transistor, two tonemeasurement is performed on the device The chosen bias point is Vgs = 0.2 V and Vds = 2.8 V and the generatorfrequencies are f1 = 10.0 GHz and f2 = 10.02 GHz. The load value is 50 Ω.The figure 6 shows a good agreementbetween the simulated and measured output power at f1 and 2f1-f2 frequencies.4. DISTRIBUTED NON LINEAR LOW FREQUENCY NOISE SOURCESIn order to model the low frequency noise sources of the channel, the measurements of the drain noise spectral densityfor different bias points (Vgs0, Vds0) are performed from 1 Hz to 1 MHz frequency range. In each unit cell, anelementary noise source is added, calculated from short circuit noise current extracted from previous measurement.These elementary Norton current sources are presumed uncorrelated. At a given noise frequency and bias point, themeasured noise spectral density Sidrain (Vgs0 ,Vds0 ) can be related to the local noise sources <ick2> as:( ) ( )∑==Nkkkdrain VdsVgshicVdsVgsSi1200200 ,,Where the hk(Vgs0,Vds0) are transfer functions, calculated from the linearized non linear model and N is number of thecells. To extract the value of every local power spectral density <ick2>, a physically based expression of <ick2> functionof the applied local voltages Vgk and ∆Vck, must be defined. In a first simplified approach <ick2> are taken identical forall cells by writing :( )( )∑==Nk kdrainkVdsVgshVdsVgsSiic1200002,,As an example, the figure 7 gives the local low frequency noise source density of a cell, versus the local voltages Vgkand ∆Vck. A more accurate expression may be obtained by taking into account the non-linear expression of the carriervelocity in function of the local electric field.CONCLUSIONFor the first time to our knowledge, a non linear, non uniform transmission line model of HEMT, fully extracted frommeasurements, is presented. Due to the robustness of the extraction procedure, it may be applied without any significantmodification to other field effect devices. This extraction robustness is due to the accurate representation of thedistributed nature of the channel from source to drain, which reproduces naturally the non-quasistatic behavior of thetransistor.This distributed model from source to drain may be included in a whole model which takes also into account both thewave propagation along the width of the transistor fingers, and the distributed nature of the metallization [9].Finally, local non linear noise sources, measurement based, are integrated to every unit-cell. This model shouldaccurately simulate the PM and AM noise in oscillators, or more generally the conversion noise in non linear circuits.ACKNOWLEDGEMENTThis work was partly supported by the Centre National d'Etude Spatiale (Convention n°714/CNES/98/7305/00) and theEuropean Community Program Esprit 38311 Locomotive. Authors thank to J.M NEBUS and C. ARNAUD formeasuring the transistor power and IM3 characteristics.REFERENCES[1] J. Portilla, R. Quéré, J. Obregon, "An improved CAD oriented FET model for large-signal and noiseapplications", San Diego 1994. IEEE MTTS Symposium, pp 849-852.[2] D. Sodini, D. Rigaud , "A new approach to model d.c. and a.c. characteristics of junction gate field effecttransistors". 1981., Solid State Electronics, Vol. 24, p. 635-647,[3] D. Huang, H. C. Lin, "DC and transmission line models for a high electron mobility transistor",1989., IEEETransactions on Microwave Theory and Techniques, Vol. 37, n°9, pp 1361-1370,[4] B. Mallet-Guy, Z. Ouarch, M. Prigent, R. Quéré, J. Obregon, "A distributed, measurements based, nonlinearmodel of FETs for high frequencies applications", Denver 1997., IEEE MTTS Symposium, pp 869-872,[5] J.P. Teyssier, J.P. Viaud, J.J. Raoux, R. Quéré, "Fully integrated nonlinear modeling and characterizationsystem of microwave transistors with on-wafer pulsed measurements", Orlando 1995. IEEE MTTS Symposium, pp1033-1036,[6] R. Anholt, "Electrical and Thermal Characterization of MESFETs, HEMTs and HBTs", Artech House Editor.[7] G.L. Bilbo, M.B. Steer, R.J. Trew, C.R. Chang, S.G. Skaggs, "Extraction of the Parameters of Equivalent Circuitof Microwave Transistors Using Tree Annealing", IEEE Trans on MTT, vol 38, n°11, 1990, pp 1711-1718.[8] J.P. Teyssier, J.P. Viaud, R. Quéré, "A new Nonlinear I(V) model for FET including Breakdown Effects", IEEEMicrowave and Guided Wave Letters, , April 1994 Vol 2, No 4, pp 104 - 106[9] A. Laloue, J.B. David, R. Quéré, B. Mallet-Guy, E. Laporte, J.F. Villemazet, M. Soulard, "Extrapolation of aMeasurement Based Millimeter Wave Non Linear Model of  PHEMT to Arbitrary Shaped Transistors ThroughElectromagnetic Simulations",1998., IEEE Transactions on Microwave Theory and Techniques, Vol. 47, n°6, pp 908-914,Rg (Ω) Rd (Ω) Rs (Ω) Lg (pH) Ld (pH) Ls (pH) Cpg (fF) Cpd (fF) Cds (fF)1.0 1.5 1.5 251 213.4 67.4 22.9 46.7 21.4Table 1 : extrinsic linear parasitic elements∆Vc1               ∆Vck                 ∆VcNVg1            Vgk                     VgNCbs CbdCell kDrainSourceGate∆VckVgkfQ (Vgk)fI (Vgk, ∆Vck)fG (Vgk)Cell 1 Cell NVckFigure 1 : topology of the intrinsic distributed modelalong the gate.00.020.040.060.080.10.120 0.5 1 1.5 2 2.5 3 3.5 4 4.5Ids (A)Vds (V)Vgs = 0.8V, 0.6V, 0.4V, 0.2V, 0.0V, -0.2V, - modelMeas.Figure 3 : Modeled and measured I-V Characteristicsof the whole transistor.00.511.522.530 2 4 6 8 10Vck (V)kVgsi = 0.2 V,  Vdsi=3VFigure 5 : Vck voltage for Vgsi = 0.2 V and Vdsi = 3V.05e-151e-141.5e-142e-142.5e-140 0.1 0.2 0.3 0.4 0.5 0.6∆Vc9 (V)<Ic92> (A2.Hz-1)Vg 9 = 0.2V, 0V, -0.2V, -0.4V, -0.6V,-0.8V, -1V, -1.2V, -1.4V∆VckVgkfQ (Vgk)fI (Vgk, ∆Vck)fG (Vgk)<Ick2> (Vgk, ∆Vck)Figure 7 : Local noise source <Ic92> = f(Vg9, ∆Vc9)at 10 kHz of the cell number 9.00.050.10.150.20.250 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Idk (A)∆VckVgk = 1.0V, 0.5V, 0V, -0.5V, -1.0V, -1.5V, -2.0V, -Figure 2 : Idk = fI (Vgk, ∆Vck).S11 S12S21 S2210Frequency : 1 - 16 GHzBias point : Vgs0 = 0.20 V, Vds0 = 2.81 VPulsed Point : Vgs = 0.22 V, Vds = 3.26 VMeasureModelFigure 4 : modeled and measured S parameters.-20-15-10-50510-4 -2 0 2 4Pout (dBm)Pin (dBm)Meas. f1Mod. f1Meas. 2f1-f2Mod .2f1-f2Figure 6 : Comparison between two tone simulationand measurement.

A MEASUREMENT BASED MODEL OF HEMT TAKING INTO ACCOUNT THE NON
LINEAR, NON UNIFORM TRANSMISSION LINE NATURE OF THE CHANNEL AND ITS
ASSOCIATED LOW FREQUENCY NOISE SOURCES
A. LALOUE*, M. CAMIADE**, M. VALENZA***, J.C. VILDEUIL***,
 J.C. NALLATAMBY* ,M. PRIGENT*, J. OBRÉGON****, R. QUÉRÉ*
* IRCOM - CNRS UMR 6615- University of Limoges - IUT - 7, rue Jules Vallès - 19100 BRIVE - FRANCE
E-mail : laloue@brive.unilim.fr
** UMS - Domaine de Corbeville - 91404 ORSAY - FRANCE
*** CEM2 - CNRS UMR 5507 -University of Montpellier II - Place Eugène Bataillon -
34095 MONTPELLIER Cedex 5 - FRANCE
**** IRCOM - CNRS UMR 6615 - University of Limoges - 123, av. Albert Thomas - 87000 LIMOGES - FRANCE
ABSTRACT
For the first time, a fully measurement based extraction procedure of non linear and non uniform transmission line
model of FET devices is proposed. This model describes accurately the distributed nature under the device gate which
allows a good distortion prediction (IM3) and promises good perspectives for simulation of noise characteristics in non
linear circuits.
INTRODUCTION
With the increase of working frequencies toward millimeter waves, high volume commercially available MMICs
including HEMTs operate yet at several ten GHz.
On the other hand, in FET based non linear circuits, inter-modulation frequencies are generated along the non linear
channel from source to drain, and are directly responsible for many characteristics such as : the distortion in power
amplifiers (IM3) or the PM noise in oscillators.
In order to optimize these features, the development of an accurate distributed model of HEMT is no longer an academic
work [1]. In this communication we present a new non linear distributed model, fully extracted from pulsed I-V and S
parameter measurements, which may be applied to all the field effect devices (MESFET, HEMT and MOSFET).
In this model, channel is considered from source to drain as a non uniform, non linear, active transmission line [2][3][4].
Figure 1 shows the proposed model. It includes N unit cells. Obviously the number of cells N depends on the gate length
L. Practically, we found that for L ≤ 0.3 µm, N = 10 is a good compromise between accuracy and complexity of this
model.
Every unit cell includes :
•  On one hand : a non linear gate channel capacitance C = fC (Vgk), in parallel with a schottky diode I = fG (Vgk).
These elements are only function of their own port voltage Vgk.
•  On the other hand a non linear channel current controlled source I = fI (Vgk, ∆Vck) which depends on its own port
voltage ∆Vck, and on the controlling voltage of the cell : Vgk.
•  At last, two linear fringing capacitances are added at the source and drain end of the channel. The other extrinsic
linear parasitic elements are described by lumped elements as usual [5].
It must be pointed out that the non linear functions fC, fG and fI are identical for all the unit cells. Nevertheless, the unit
cell voltages : Vgk and ∆Vck vary along the channel from source to drain, following the rank k of the cell under
consideration : from 1 to N. It results an accurate non linear, non uniform, distributed model of the channel.
Moreover, while the model is non quasistatic by nature, it does not need the inclusion of any time-delay, nor a
questionable charging resistor Ri [6].
In the first part of this paper, the model extraction procedure from pulsed I-V characteristic and S parameter
measurements is described. As an example a 2 fingers of 100 µm wide PHEMT model is shown. The extracted model
and the measurements are compared. The improvement on the non linear transistor modeling is shown in the third part.
Finally, in the last part, low frequency noise sources are included into the model along the channel.
1. DISTRIBUTED MODEL EXTRACTION PROCEDURE
The extraction procedure is based on pulsed measurement of I-V characteristics and S parameters. The pulsed
measurement allows to keep constant temperature and trap steady state during the characterization [5].
First, the extrinsic linear parasitic elements are obtained by the well known extraction methodology used for the classical
lumped FET model.
The intrinsic part of the model is formed by a chain of N identical cells. Each cell represents the electrical behavior of a
transistor slice. Then, it includes a schottky diode I = fG (Vgk), in parallel with a non linear capacitance C = fC (Vgk), and
a non linear current controlled source I = fI (Vgk, ∆Vck). These elements only depend on the voltage Vgk and ∆Vck,
which are present on the access of cell under consideration. In our model, every cell elements have both same equations
and same parameter values along the channel. Only the controlled voltages Vgk and ∆Vck vary following the k rank of
the cell number. Two edge capacitances are added at source side and drain end. They take into account the capacitive
coupling effects between the metallic electrodes and the edge channel.
In a first step, the parameters of the diode and the controlled non linear current source are extracted. To do this, a
software based on fast simulated diffusion algorithm, allowing to minimize a non linear error function, is used [7].  From
the characteristic measurement Ids (Vgs, Vds), the 13 parameters of the controlled current generator equations, fI (Vgk,
∆Vck) are determined. And from characteristic measurement Igs (Vgd, Vds), the two diode parameters are obtained.
The non linear element equations are fully described in [8].
Finally, the N non linear capacitances, C = fC (Vgk) and the two edge capacitances Cbs and Cbd are determined from the
S parameter measurements. The direct extraction procedure can not be used, here, due to the high element number. Then
another software, based on the fast simulated diffusion algorithm is used to obtain the 6 parameter values of the non
linear capacitances and the two edge capacitance values. The error function is computed with the intrinsic Y parameters
of the model and those calculated from the S parameter measurements. Obviously, this optimization must be performed
with several measurement points chosen on the I-V characteristic load-line zone.
2.  COMPARISON BETWEEN MODEL AND MEASUREMENTS.
As an example, we show the results obtained on a PHEMT of 2 fingers of 100 µm wide. The pulsed I-V and S parameter
measurements are performed on the 1 to 20 GHz frequency range. The bias point is Vgs = 0.21 V, Vds = 2.8 V and Ids
= 41.7 mA.
A 10 cell distributed model seems to be a good compromise between accuracy and model complexity. The table 1 gives
the extrinsic linear parasitic elements. Note that the linear capacitance Cds is considered as an extrinsic element.
The non linear function fI (Vgk, ∆Vck) of one controlled current source is reported on figure 2. The extraction procedure
leads to the edge capacitance values : Cbs = 247 fF and Cbd = 34 fF.
The figure 3 shows a good agreement between whole model and measured transistor I-V characteristics. The figure 4
presents the modeled and measured S parameters for a pulsed point nearby the bias point. These figures demonstrate that
the extracted distributed model can accurately fit the measured data.
To show that the distributed model have physical behavior, the channel voltage Vck is plotted on the figure 5 for this
intrinsic bias condition Vgsi = 0.2 V and Vdsi = 3V. It shows that the voltage Vck is a non linear function along the
channel.
3. APPLICATION TO THE THIRD ORDER INTER-MODULATION SIMULATION
In order to show the model ability to take into account the distributed nature of the field effect transistor, two tone
measurement is performed on the device The chosen bias point is Vgs = 0.2 V and Vds = 2.8 V and the generator
frequencies are f1 = 10.0 GHz and f2 = 10.02 GHz. The load value is 50 Ω.The figure 6 shows a good agreement
between the simulated and measured output power at f1 and 2f1-f2 frequencies.
4. DISTRIBUTED NON LINEAR LOW FREQUENCY NOISE SOURCES
In order to model the low frequency noise sources of the channel, the measurements of the drain noise spectral density
for different bias points (Vgs0, Vds0) are performed from 1 Hz to 1 MHz frequency range. In each unit cell, an
elementary noise source is added, calculated from short circuit noise current extracted from previous measurement.
These elementary Norton current sources are presumed uncorrelated. At a given noise frequency and bias point, the
measured noise spectral density Sidrain (Vgs0 ,Vds0 ) can be related to the local noise sources <ick
2> as:
( ) ( )∑
=
=
N
k
kkdrain VdsVgshicVdsVgsSi
1
2
00
2
00 ,,
Where the hk(Vgs0,Vds0) are transfer functions, calculated from the linearized non linear model and N is number of the
cells. To extract the value of every local power spectral density <ick
2>, a physically based expression of <ick
2> function
of the applied local voltages Vgk and ∆Vck, must be defined. In a first simplified approach <ick2> are taken identical for
all cells by writing :
( )
( )∑=
=
N
k
k
drain
k
VdsVgsh
VdsVgsSi
ic
1
2
00
002
,
,
As an example, the figure 7 gives the local low frequency noise source density of a cell, versus the local voltages Vgk
and ∆Vck. A more accurate expression may be obtained by taking into account the non-linear expression of the carrier
velocity in function of the local electric field.
CONCLUSION
For the first time to our knowledge, a non linear, non uniform transmission line model of HEMT, fully extracted from
measurements, is presented. Due to the robustness of the extraction procedure, it may be applied without any significant
modification to other field effect devices. This extraction robustness is due to the accurate representation of the
distributed nature of the channel from source to drain, which reproduces naturally the non-quasistatic behavior of the
transistor.
This distributed model from source to drain may be included in a whole model which takes also into account both the
wave propagation along the width of the transistor fingers, and the distributed nature of the metallization [9].
Finally, local non linear noise sources, measurement based, are integrated to every unit-cell. This model should
accurately simulate the PM and AM noise in oscillators, or more generally the conversion noise in non linear circuits.
ACKNOWLEDGEMENT
This work was partly supported by the Centre National d'Etude Spatiale (Convention n°714/CNES/98/7305/00) and the
European Community Program Esprit 38311 Locomotive. Authors thank to J.M NEBUS and C. ARNAUD for
measuring the transistor power and IM3 characteristics.
REFERENCES
[1] J. Portilla, R. Quéré, J. Obregon, "An improved CAD oriented FET model for large-signal and noise
applications", San Diego 1994. IEEE MTTS Symposium, pp 849-852.
[2] D. Sodini, D. Rigaud , "A new approach to model d.c. and a.c. characteristics of junction gate field effect
transistors". 1981., Solid State Electronics, Vol. 24, p. 635-647,
[3] D. Huang, H. C. Lin, "DC and transmission line models for a high electron mobility transistor",1989., IEEE
Transactions on Microwave Theory and Techniques, Vol. 37, n°9, pp 1361-1370,
[4] B. Mallet-Guy, Z. Ouarch, M. Prigent, R. Quéré, J. Obregon, "A distributed, measurements based, nonlinear
model of FETs for high frequencies applications", Denver 1997., IEEE MTTS Symposium, pp 869-872,
[5] J.P. Teyssier, J.P. Viaud, J.J. Raoux, R. Quéré, "Fully integrated nonlinear modeling and characterization
system of microwave transistors with on-wafer pulsed measurements", Orlando 1995. IEEE MTTS Symposium, pp
1033-1036,
[6] R. Anholt, "Electrical and Thermal Characterization of MESFETs, HEMTs and HBTs", Artech House Editor.
[7] G.L. Bilbo, M.B. Steer, R.J. Trew, C.R. Chang, S.G. Skaggs, "Extraction of the Parameters of Equivalent Circuit
of Microwave Transistors Using Tree Annealing", IEEE Trans on MTT, vol 38, n°11, 1990, pp 1711-1718.
[8] J.P. Teyssier, J.P. Viaud, R. Quéré, "A new Nonlinear I(V) model for FET including Breakdown Effects", IEEE
Microwave and Guided Wave Letters, , April 1994 Vol 2, No 4, pp 104 - 106
[9] A. Laloue, J.B. David, R. Quéré, B. Mallet-Guy, E. Laporte, J.F. Villemazet, M. Soulard, "Extrapolation of a
Measurement Based Millimeter Wave Non Linear Model of  PHEMT to Arbitrary Shaped Transistors Through
Electromagnetic Simulations",1998., IEEE Transactions on Microwave Theory and Techniques, Vol. 47, n°6, pp 908-
914,
Rg (Ω) Rd (Ω) Rs (Ω) Lg (pH) Ld (pH) Ls (pH) Cpg (fF) Cpd (fF) Cds (fF)
1.0 1.5 1.5 251 213.4 67.4 22.9 46.7 21.4
Table 1 : extrinsic linear parasitic elements
∆Vc1               ∆Vck                 ∆VcN
Vg1            Vgk                     VgNCbs CbdCell k
Drain
Source
Gate
∆Vck
Vgk
fQ (Vgk)
fI (Vgk, ∆Vck)
fG (Vgk)
Cell 1 Cell N
Vck
Figure 1 : topology of the intrinsic distributed model
along the gate.
0
0.02
0.04
0.06
0.08
0.1
0.12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Ids (A)
Vds (V)
Vgs = 0.8V, 0.6V, 0.4V, 0.2V, 0.0V, -0.2V, - model
Meas.
Figure 3 : Modeled and measured I-V Characteristics
of the whole transistor
.
0
0.5
1
1.5
2
2.5
3
0 2 4 6 8 10
Vck (V)
k
Vgsi = 0.2 V,  Vdsi=3V
Figure 5 : Vck voltage for Vgsi = 0.2 V and Vdsi = 3V.
0
5e-15
1e-14
1.5e-14
2e-14
2.5e-14
0 0.1 0.2 0.3 0.4 0.5 0.6
∆Vc9 (V)
<Ic9
2> (A2.Hz-1)
Vg 9 = 0.2V, 0V, -0.2V, -0.4V, -0.6V,
-0.8V, -1V, -1.2V, -1.4V
∆Vck
Vgk
fQ (Vgk)
fI (Vgk, ∆Vck)fG (Vgk)
<Ick
2> (Vgk, ∆Vck)
Figure 7 : Local noise source <Ic9
2> = f(Vg9, ∆Vc9)
at 10 kHz of the cell number 9.
0
0.05
0.1
0.15
0.2
0.25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Idk (A)
∆Vck
Vgk = 1.0V, 0.5V, 0V, -0.5V, -1.0V, -1.5V, -2.0V, -
Figure 2 : Idk = fI (Vgk, ∆Vck).
S11 S12
S21 S22
10
Frequency : 1 - 16 GHz
Bias point : Vgs0 = 0.20 V, Vds0 = 2.81 V
Pulsed Point : Vgs = 0.22 V, Vds = 3.26 V
Measure
Model
Figure 4 : modeled and measured S parameters.
-20
-15
-10
-5
0
5
10
-4 -2 0 2 4
Pout (dBm)
Pin (dBm)
Meas. f1
Mod. f1
Meas. 2f1-f2
Mod .2f1-f2
Figure 6 : Comparison between two tone simulation
and measurement.


A measurement based model of HEMT teking into account the non linear, non uniform transmission line nature of the channel its associated low frequency noise sources

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