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RF Distortion Analysis with Compact MOSFET Models

Peter Bendix, P. Rakers+, P. Wagh

+, L. Lemaitre

+,W. Grabinski

+, C. C. McAndrew

+,

X. Gu*

and G. Gildenblat*

LSI Logic Inc.,+Motorola Inc.,

*The Pennsylvania State University

AbstractThis paper examines the relation between the structure

of a compact MOSFET model and its ability to modelharmonic distortion. It is found that non-singular behaviorat zero drain bias is essential for qualitatively correctsimulations of the third harmonic power dependence.Specifically, nonlinear distortion analysis requires that theGummel symmetry condition be satisfied by the compactmodel. A simple procedure to enforce the Gummelsymmetry without increasing the complexity of the modelis incorporated in an advanced surface-potential-basedMOSFET model to enable correct harmonic distortionmodeling.

Keywords: IM3, distortion, FFT, harmonic analysis, RF

CMOS, MOSFET modeling.

Introduction

The design of RF CMOS circuits for mixed signalapplications is becoming increasingly important. Distortionis a key measure of performance for RF circuits, and formixer design third order intermodulation IM3 ([1], pp. 296-297) is commonly used as a measure to evaluate distortion.Unfortunately, simulation of IM3 can be qualitativelywrong, and therefore cause significant errors for RF design,

because of fundamental problems with models used forsimulation. This has been shown for MESFETs in [2]. Inthis paper we show that a similar problem exists in

MOSFET models; we then describe the reasons for theproblem, and an efficient approach to fix the problem.The IM3 modeling problem is manifest in the simple

situation where a sinusoidal voltage (of zero DC bias andswept amplitude A ) is used to drive the drain of atransistor with the gate biased on, and a fast Fouriertransform (FFT) of the drain current is done to get theamplitude of the various harmonics of the drain current.The MOSFET channel in this case acts as a weakly nonlin-ear resistor, which produces the harmonic distortion beinganalyzed. Both theoretical considerations [1], [2] andanalysis of experimental data indicate that the secondharmonic should be proportional to the square of the inputsignal amplitude

2A , and the third harmonic should be

proportional to the cube of the signal amplitude3

A . As aresult of unphysically asymmetric MOSFET model

behavior around 0dsV , all older and many moderncompact MOSFET models (including source-referencedthreshold-voltage based models like the standard Level 1,Level 2, Level 3, BSIM, BSIM2, BSIM3, and BSIM4models) used in commercial circuit simulators fail theGummel symmetry test [3] and therefore cannot be used forIM3 and related harmonic distortion tests.

The purpose of this paper is to conclusively demonstratethe IM3 modeling problem. For this purpose we work withthe latest generation compact MOSFET models SP [3] andBSIM3 and BSIM4 [4].

Transient Simulation

A transient analysis for the circuit of Fig. 1 was doneusing the Spectre circuit simulator with the BSIM4 and SPMOSFET models (BSIM4 is a threshold voltage-basedmodel while SP is a true surface-potential-based model).The results are not dependent explicitly on the choice ofmodel parameters. The transient analyses were done for 8values of driving sinusoid amplitude (160, 200, 240, 280,320, 360, 400, and 440mV). The current was then analyzed

using an FFT code based on the four1 routine of [5], andthe magnitude of the fundamental, second, and thirdharmonics were plotted as a function of )log(A . In runningthese simulations, tolerances and biasing were chosencarefully to avoid anomalies in the results. In particular,the amplitude of the sinusoid was not too large, there was asufficiently small timestep chosen to provide enough data

points for the specified frequency to get accurate FFTs,and the simulation accuracy tolerances were chosen smallenough so as not to corrupt the results.

Fig. 1 Circuit for distortion analysis, proposed in [2].

Fig. 2 shows the simulation results from BSIM4. Asexpected, on a dB scale the slope of the fundamental is oneand the slope of the second harmonic is two. However, theslope of the third harmonic is two (just as for the second

harmonic) instead of the correct value of three. The reasonfor this problem with BSIM4, and many other MOSFETmodels, is a lack of symmetry in the model. This leads todifferent left and right hand second order derivatives at

0dsV . Consequently, the second and higher orderderivatives do not exist at 0dsV .

As Fig. 3 shows, the SP model gives results identical towhat theory predicts. The slope of the third harmonic isthree, as it should be. This means SP gives correct resultsfor IM3 RF simulations.

)cos( tAVRF gV

bV

dI

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Fig. 2 BSIM4 transient simulation distortion results forthe single transistor circuit of Fig. 1.

Fig. 3 SP transient simulation distortion results for thesingle transistor circuit of Fig. 1.

Fig. 4 Harmonic balance simulation of circuit of Fig. 1.

Harmonic Balance Simulations

A single tone harmonic balance simulation of the circuitof Fig. 1 was run in the Mica simulator, for BSIM3,BSIM4, and SP models. Different model parameters wereused than for the transient simulations of Fig. 2 and Fig. 3,

but the parameters used for all the harmonic balancesimulations were extracted for the same 0.18m CMOStechnology. Fig. 4 shows the results of the harmonic

balance simulations. The third harmonic component againhas an incorrect slope of two for BSIM3 and BSIM4, butSP has the correct slope of three.

CMOS RF Mixer Simulation

A three tone intermodulation distortion harmonicbalance simulation was also run on a simple CMOS RFmixer, to investigate the effect of the singularity at 0dsVon a circuit, and not just a single device used as a voltagecontrolled resistor. Fig. 5 shows the topology of the mixer.

Fig. 5 Simple CMOS RF mixer. Besides the RF signalsthere are DC offsets to bias the MOSFETs on and avoidforward biasing of source-bulk and drain-bulk junctions.

The local oscillator frequency LOf was 1GHz, and twoRF signals at 1RFf 910 and 2RFf 920 MHz were fed into the mixer and the power level ramped. The first orderintermodulation (IM1) frequencies are LORF ff 1 and

LORF ff 2 . The second order intermodulation (IM2)frequencies are 21 RFRF ff and 21 RFRF ff . The IM3frequencies are LORFRF fff 212 and

LORFRF fff 122 (the other IM3 frequenciesLORFRF fff 212 and LORFRF fff 212 are

considered out of band and are not analyzed).The power levels of the IM1 at 90MHz and the secondand third intermodulation products at 10MHz and 100MHzwere recorded. As Fig. 6 shows, the IM3 slope for SP isvery nearly three (it was calculated from a linear fit to thelowest power data points as 2.98), whereas for BSIM3 andBSIM4 it is again close to two. As far as we are aware, thisis the first presentation of incorrect simulation of IM3 at acircuit level, as opposed to the individual device level,caused by the singularity at 0dsV .

16 15 14 13 12 11 10 9 8 7 680

70

60

50

40

30

20

10

VRF

(dB)

Id

(dB)

Fundamentalslope=1

2nd Harm

onicslope=2

3rd Harm

onicslope=2(should

be3)

16 15 14 13 12 11 10 9 8 7 6100

90

80

70

60

50

40

30

20

10

VRF

(dB)

Id(dB)

Fundamentalslope=1

2ndHarmo

nicslope=2

3rd Har

monicslo

pe=3

40 38 36 34 32 30 28 26 24 22 20160

140

120

100

80

60

40

Vrf

(dB)

Id(dB)

FundamentalBSIM

3,BSIM4andSP

2nd Harm

onicBSIM

3,BSIM4a

ndSP

3rd Ha

rmonicSP

3rd Harm

onicBSIM3an

dBSIM4

BSIM3BSIM4SP

LOV

LOV

RFV

RFV

outV

DCgV ,

DCgV ,

DCbV ,

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Fig. 6 Intermodulation products for the simple RFCMOS mixer circuit of Fig. 5.

Gummel Symmetry Test

To corroborate the reason for the success or failure ofmodeling IM3, the models were subjected to the Gummelsymmetry test (as described in [6]). For this test a MOSFETis biased symmetrically with respect to source and drain,with xd VVV 0 and xs VVV 0 , see Fig. 7. Thismakes the drain current dI an odd function of xV .

Consequently, all odd order derivatives of )( xd VI withrespect to xV should be continuous at 0xV , and all evenorder derivatives, including

22xd VI , should exist and

should be equal to zero at 0xV .

Fig. 7 Circuit for Gummel symmetry test.

Fig. 8 shows the results of this simulation for the SP,

BSIM3, and BSIM4 models. SP has the desired behavior,with no kinks or discontinuities in

22xd VI at 0dsV .

BSIM3 and BSIM4 however both exhibit a singularity at0xV . More specifically, the left and right hand

derivatives of xd VI with respect to xV are different at0dsV , so

22xd VI does not exist at 0dsV .

Consequently BSIM3 and BSIM4, and many otherMOSFET compact models, fail to simulate third orderharmonic distortion properly.

Fig. 8 Gummel symmetry test results for the secondderivative.

Ensuring Non-Singular Behavior at 0ds

V

The violation of the Gummel symmetry test in BSIMand other MOSFET models has been traced to the use of asource-referenced threshold voltage and to the singularityof the velocity-field relation [6]. In body-referenced modelslike SP only the second issue needs to be examined.

Over the years several physically motivated andempirical equations have been suggested for the driftvelocity dv of mobile carriers in the channel of MOStransistors. The most popular form, routinely encounteredin compact models,

(1)

nn

c

d

E

E

Ev

1

0

1

(where 0 is the low-field effective mobility, E is thelateral component of the electric field, and cE is thecritical field), has two drawbacks. First, the )(Evddependence for 1n is too weak for n-channel transistors.Further, for 1n

(2)

02

20

02

22

E

d

cE

d

dE

vd

EdE

vd

so the 2nd and higher order derivatives at 0dsV do notexist. This is the root cause of the singularity in the

)( dsd VI characteristics at 0dsV , i.e. why models thatuse this form of velocity saturation model fail the Gummelsymmetry test [6].

As noted in [6], the expression (1) with 2n is moreaccurate for n-channel transistors and does not have thesingularity at 0E . However, setting 2n results in acomplicated expression for the drain current. SP removes

50 45 40 35 30 25 20 15 10200

180

160

140

120

100

80

60

40

20

Vrf

(dBm)

Vout

(dBm)

Fundamental

BSIM3,BSIM

4andSP

2nd Har

monicB

SIM3,B

SIM4an

dSP

3rd Ha

rmonicS

P

3rd Ha

rmonicB

SIM3an

dBSIM4

BSIM3BSIM4SP

gV

bV

dI

xVV 0 xVV 0

10 8 6 4 2 0 2 4 6 8 100.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

Vx

(mV)

2Id/V

x2(

S/V)

BSIM3BSIM4SP

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this singularity via a semi-empirical correction factor 0 .In this approach [7], [8]

(3)

c

d

E

E

Ev

0

0

1

where

(4)LEgV

V

chds

ds

0 ,

L is the effective channel length, and hg is an empiricallyadjustable parameter.

Expression (3) has several advantages for the purpose ofcompact modeling. First, since 0 does not depend on

position along the channel, it is as easy to use as (1).Further, the dsV dependence of 0 sharpens the )(Evddependence, which was the original purpose of introducing

0 in [7], [8]. Finally, when used in an otherwisesymmetric compact model it leads to [3]

(5) 0

02

2

xVx

d

V

I.

Hence (3) will not lead to a violation of the Gummelsymmetry test.

Conclusions

We have presented transient and harmonic balancesimulations that show that compact MOSFET models with

a singularity at 0dsV , as evidenced by failing theGummel symmetry test, are not able to model distortionproperly. Even for a symmetric, bulk-referenced modelformulation, the use of a velocity saturation model with asingularity at 0E will still cause problems for distortionmodeling, and this can be overcome, as in SP, by using (3)in place of (1).

As far as we are aware, this is first time that distortionanalysis results that show the problem caused by asingularity at 0dsV have been presented for MOSFETs,as opposed to MESFETs in [2], and the first time that suchresults have been presented for a mixer circuit, as comparedto a single transistor simulation.

References

[1] T. H. Lee, The Design of CMOS Radio-Frequency

Integrated Circuits, Cambridge University Press, 1998.

[2] N. Scheinberg and A. Pinkhasov, A Computer

Simulation Model for Simulating Distortion in FET

Resistors,IEEE Trans. CAD, vol. 19, no. 9, pp. 981-

989, Sep. 2000.

[3] G. Gildenblat, T.-L. Chen, X. Gu, H. Wang, and X.Cai, SP: An advanced Surface-Potential Based

Compact MOSFET Model, Proc. IEEE CICC, pp.

233-240, 2003.

[4] W. Liu, MOSFET Modeling for SPICE Simulation

Including BSIM3v3 and BSIM4, John Wiley and Sons,

2001.

[5] W. H. Press, B. P. Flannery, S. A. Eukolsky, and W. T.

Vettering, Numerical Recipes in C, Cambridge

University Press, 1988.

[6] K. Joardar, K. K. Gullapalli, C. C. McAndrew, M. E.

Burnham, and A. Wild, An Improved MOSFET

Model for Circuit Simulation, IEEE Trans. Electron

Devices, vol. 45, no. 1, pp. 134-148, Jan. 1998.

[7] T. Grotjohn and B. Hoefflinger, A Parametric Short-

Channel Transistor Model for Subthreshold and Strong

Inversion Current,IEEE J. Solid-State Circuits, vol. 9,

no. 1, pp. 100-112, Feb. 1984.

[8] N. D. Arora, R. Rios, C.-L. Huang, and K. Raol,

PCIM: A Physically Based Continuous Short-Channel

IGFET Model for Circuit Simulation, IEEE Trans.

Electron Devices, vol. 41, no. 6, pp. 988-997, Jun.

1994.

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