Research Reports from the Ferdinand-Braun-Institut für Höchstfrequenztechnik

from the series:Innovations with Microwaves & LightResearch Reports from the Ferdinand-Braun-Institut fürHöchstfrequenztechnikVolume No. 3R. Doerner, M. Rudolph (Eds.)Selected Topics on Microwave Measurements,Noise in Devices and Circuits, and Transistor ModelingA Festschrift for Peter HeymannSeries Editors: Prof. Dr. Günther Tränkle, Dr.-Ing. Wolfgang HeinrichFerdinand-Braun-Institut Phone +49.30.6392-2600für Höchstfrequenztechnik (FBH)Fax +49.30.6392-2602Gustav-Kirchhoff-Straße 4 Email fbh@fbh-berlin.de12489 BerlinWeb www.fbh-berlin.deGermany

Innovations with Microwaves and LightResearch Reports from the Ferdinand-Braun-Institut für HöchstfrequenztechnikForeword of the Series EditorsNew research ideas and methodologies advance the state-of-the-art in knowledge and technologies.Applying them in products and services yields innovations, those indispensableingredients the development and the future of our modern world is based on.This is the reason why the series „Forschungsberichte aus dem Ferdinand-Braun-Institut fürHöchstfrequenztechnik“ (Research Reports from the Ferdinand-Braun-Institut) was established.It is to document the current research activities at the Ferdinand-Braun-Institut. Makingthem public helps in stimulating discussions of their results and opens new applications.As the title indicates the present volume "Selected Topics on Microwave Measurements,Noise in Devices and Circuits, and Transistor Modeling" has a special scope, it is a Festschriftfor Peter Heymann on the occasion of his 65 th birthday. It is included in the series becauseit provides a good selection of contributions in the microwave measurement and modelingfield, covering the spectrum from plasma diagnostics to III-V transistors, and written byinternationally well-known experts in the field. There is a focus on transistor noise characterizationand low-noise devices, which is well in line with the current trends in today's microwaveand millimeterwave systems.We are grateful to the editors of this issue, Matthias Rudolph and Ralf Doerner, for their effortsin bringing this issue to reality.Prof. Dr. Günther TränkleDirector of the InstituteDr.-Ing. Wolfgang HeinrichVice DirectorThe Ferdinand-Braun-InstitutAt the Ferdinand-Braun-Institut für Höchstfrequenztechnik (FBH), we research cutting-edgetechnologies in the fields of microwaves and optoelectronics. We realize high-frequency devicesand circuits for communications and sensors as well as high-power diode lasers formaterials processing, laser technology, medical applications, and high precision metrology.The FBH is a center of competence for III/V-compound semiconductors and the correspondinghigh-frequency devices and diode lasers. We operate industry-compatible and flexibleclean-room laboratories with gas-phase epitaxy units and a III/V-semiconductor process line.We use advanced methods in simulation and design and are equipped with comprehensivemeasurement techniques for material and device characterization.We work in close collaboration with industrial partners thus ensuring rapid transfer of ourresearch results. Spin-off companies help in bringing innovative product ideas to the market.

Selected Topics onMicrowave Measurements,Noise in Devices and Circuits,and Transistor Modeling—A Festschrift for Peter Heymannedited by R. Doerner and M. Rudolph

IIITable of ContentsTable of Contents ........................................................IIIEditorialR. Doerner, M. Rudolph ...................................................VPeter Heymann — From Plasma Diagnostics to Microwave ElectronicsW. Heinrich .............................................................VIISpectrum Broadening and Fluctuations of Lower Hybrid Waves Observed inCASTOR TokamakF. Žáček, R. Klíma, K. Jakubka, P. Plíšek, S. Nanobashvili, P. Pavlo, J. Preinhaelter,J. Stöckel and L. Kryška ...................................................1Diagnosis of Chemically Reactive PlasmaH. Wittrich, L. Weixelbaum, W. John .......................................13Over-Temperature Noise Modeling of Submicron Devices Brought the Question: Isthe Diffusion Coefficient Temperature Dependent?A. Boudiaf ...............................................................21On Some Errors in Noise Characterization of High Performance SemiconductorDevicesW. Wiatr .................................................................31Low Frequency Noise in Resistive MixersG. Böck ..................................................................49RF Noise Model for CMOS TransistorsI. Angelov, M. Ferndahl, A. Masud .........................................63Extremely Low-Noise Amplification with Cryogenic FET’s and HFET’s: 1970-2004M. W. Pospieszalski .......................................................67Extraction of GaAs-HBT Equivalent Circuit Considering the Impact of MeasurementErrorsF. Lenk, M. Rudolph ......................................................95On the Implementation of Transit-Time Effects in Compact HBT Large-SignalModelsM. Rudolph, F. Lenk, R. Doerner .........................................105

VEditorialTHIS THIRD volume of the Forschungsberichte presents a collection of ninetechnical papers on selected topics of microwave engineering, ranging frominvestigations of the plasma in a Tokamak to the modeling of Heterojunction BipolarTransistors. The main focus, however, is on noise in transistors and circuits, andhow to measure it. Eight of the contributions are original papers, and one is areprint from Plasma Phys. Control. Fusion, it appears by courtesy of the Instituteof Physics Publishing.Since this collection is dedicated to Peter Heymann to celebrate his 65 th birthday,colleagues and former colleagues were invited to contribute either• a review of their main research results,• extended background information to an already published topic, or• details from recent studies.Although we took care not to forget to ask someone who might want to write apaper, we would like to apologize if we did. On the other hand, it is also necessary,if not to apologize, so at least to thank all the contributors who took the challengeto finish an original paper in a very short time.It was a very tight time schedule, indeed. We started to collect names, and to askfor committments to this project in early May. The feedback was overwhelming.None of the former colleagues refused to try to meet the deadline in early July.However, two weeks were lost until the critical mass of papers was collected and theproject could be launched officially. Even though it turned out that the July deadlinewas not absolute, we are indebted to all of the authors who did an excellent job.And did not send their high-quality papers to a conference or a journal but to us.We would thank the Ferdinand-Braun-Institut, too, for making this publicationpossible.In conclusion, this volume comprises a number of excellent technical papers onselected topics related to Peter Heymann’s work over the past decades. We thinkthat all of them are worth reading, and hope that you will find them useful in yourdaily work.Ralf Doerner and Matthias Rudolph

Peter Heymann — From Plasma Diagnosticsto Microwave ElectronicsVIITHE PROFESSIONAL life of Peter Heymannreflects the technical developmentsduring the past 4 decades, and it provesthat diligence and dedication form a soundbasis (not to say the only sound basis) for agood scientist.Peter started his academic career in 1963with his diploma thesis at the Universityof Greifswald “Langmuirsondenmessung instromstarker Glimmentladung”. The gas dischargeand plasma phenomena then went withhim the next 20 years. He worked on plasmadiagnostics using microwaves and on highfrequencyplasma sources, and he took part inexperiments on nuclear fusion at the Tokamakin Prague. The first 4 years he spent as ascientific assistant with the then Heinrich-Hertz-Institut in the eastern part of Berlin,and entered the scientific scene in 1968 with his PhD thesis bearing the impressivelylong title “Der Einfluss der Elektronenverteilungsfunktion auf Mikrowellenemissionund Hochfrequenzleitfähigkeit eines schwach ionisierten Entladungsplasmas” (Theinfluence of the electron-density function on microwave emission and RF conductivityof a weakly ionized discharge plasma). In 1967, he joined the Physikalisch-Technisches Institut of the academy of sciences in the German Democratic Republic,the later Zentralinstitut für Elektronenphysik (ZIE). He stayed with this institute,and since 1992 the Ferdinand-Braun-Institut für Höchstfrequenztechnik (FBH) asone successor of the ZIE, until now.What became obvious already in Peter’s early plasma years is the growingimportance of high-frequency and microwave topics. So, in the view back, it appearsas a straightforward consequence that he changed his research subject in 1983 andmoved into the new and strongly developing field of microwave semiconductors,applying his well-trained expertise in diagnostics now in order to describe andcharacterize GaAs high-frequency electronics. Measurement techniques for suchdevices and circuits dominated the second half of Peter’s professional career. TheGerman reunification in 1989 changed many lives basically, but not as much thefocus of Peter’s work. GaAs FETs and later HBTs remained at the heart of hisscientific interests, and it was this last 15 years when his international reputation

VIIIgrew steadily. Well known are his contributions on FET noise description, on powertransistorload-pull characterization, and on oscillator phase-noise measurements.As leader of the microwave measurement group of the FBH he also took overhierarchical responsibility; as senior scientist, he acted directly and indirectly as aneducator to the PhD students and young scientists in his field.Personally, I know Peter Heymann since the times of reunification in 1989. Ienjoyed working with him at the FBH for now more than 10 years. He is one ofthe assets of the microwave department and the example of a colleague you canfully rely on, who though working successfully and effectively will never make agreat play of his achievements. At the same time, he is an excellent scientist tracingproblems back to their very origin and drawing the necessary conclusions. His lifelongexperience in experimental microwave work makes him the well-known advisorin all practical µ-wave and measurement questions at the Berlin-Adlershof campus.Therefore, it is adequate to celebrate his 65 th birthday in the appropriate wayby collecting scientific contributions of former and nowadays colleagues in thisFestschrift.To conclude, a final question to you, Peter, referring to your personal life andthe development of our high-frequency community: After 20 years of plasma diagnosticsand another 20 years of microwave electronics, what’s on the horizon nowfor the following decades?Wolfgang HeinrichHead of microwave department at the FBH

1Spectrum Broadening and Fluctuationsof Lower Hybrid Waves Observedin the CASTOR TokamakF. Žáček, R. Klíma, K. Jakubka, P. Plíšek, S. Nanobashvili ∗ ,P. Pavlo, J. Preinhaelter, J. Stöckel and L. KryškaInstitute of Plasma Physics, Za Slovankou 3, 182 00 Prague 8,Czech Republic∗ Institute of Physics, Tamarashvili 6, 380077 Tbilisi, Georgiafirst published in Plasma Phys. Control. Fusion 41 (1999) 1221-1230c○Institute of Physics Publishing Ltd., reprinted with permission.AbstractStrong fluctuations of lower hybrid waves (LHW) amplitudes inside the plasmahave been observed in experiments with different types of launchers. The fluctuationsfrequency is much higher than the frequency of plasma density fluctuations.This effect is explained by the presence of plasma fluctuations along the wholeray path. LHW spectrum broadening has been found by means of a doubleradiofrequency probe. This experimental result supports the assumption of spectralgap filling between the electron velocity distribution function and the spectrumlaunched into the plasma.I. INTRODUCTIONRECENTLY, measurements of low-power LHW launched into the tokamakCASTOR plasma by several types of quasi-optical grills were performed [1]–[5]. To fulfil requirements of the quasi-optics a relatively high frequency f =9.3 GHzhas been chosen lying in the upper part of the LHW band. The main goal ofthese investigations at low power level has not been a manifestation of the currentgeneration itself, but a direct proof whether or not such quasi-optical system is ableto excite LHW in tokamak plasma. For this purpose, i.e. for the measurement ofN ‖ inside the plasma, a double radiofrequency (RF) probe movable through thewhole small tokamak cross section has been developed and installed in CASTOR.Here, N ‖ =(c/2πf)k ‖ is the index of refractivity of the launched LHW, c is thevelocity of light and k is the wavevector. The subscript ‖ denotes the component ofthe corresponding quantity in the direction of the magnetic field. Two indicationsthat the wave observed is the LH wave have been gained:(a) a region of enhanced RF field has been found in the cross section where raytracing predicts an occurrence of LH conus [1];

2 F. Žáček et al.(a)Fig. 1. Ray tracing of LHW with f =1.25 GHz launched near CASTORequatorial plane for a parabolic density distribution with n(0) = 7×10 18 m −3(¯n =4× . 10 18 m −3 , see Fig. 4); (a) top view; (b) poloidal cross section.(b)(b) the dependence of the power reflected back into the launching antenna on thedensity of plasma in front of the quasi-optical grill [3] matches very well thatpredicted by the theory of LHW coupling [4].However, strong fluctuations of LHW amplitude and phase observed during allthese measurements precluded the determination of the wave N ‖ , and the main goalof this work could not be reached.Motivation of this work is a natural question arising in this situation: are thesefluctuations specific for the quasi-optical grills, or do they accompany LHW launchedby other types of grills, too? We attempted to answer this question by repeating theexperiments with a multijunction grill working at the frequency f =1.25 GHz anda power up to 50 kW, sufficient for a substantial current drive [6]. Results of theseexperiments are presented in this paper. A short description of the experiment onthe CASTOR tokamak is given in Section II. Section III presents the experimentalresults obtained and, in Section IV, these results are interpreted. We emphasizethat the interpretation does not concern the reason for spectral broadening of thelaunched LHW spectrum [7]–[10]. This effect is asumed here as a (commonly used)hypothesis and it is shown that it implies results which are consistent with the dataobtained in the experiment described below.II. EXPERIMENTAL ARRANGEMENTCASTOR is a small limiter tokamak with the following parameters: major radius,R =0.4 m; wall radius, b =0.1 m; limiter radius, a =0.085 m; magnetic field on

Spectrum Broadening and Fluctuations ... 3Fig. 2.RF coaxial circuit for detection of the LHW phase velocity.the axis, B(0) ≤ 1.5 T; plasma density on the axis, n(0) ≤ 3 · 10 19 m −3 ; length ofthe discharge, τ ≤ 40 ms.To facilitate finding the best position for direct measurement of LHW inside thechamber of the CASTOR tokamak, ray-tracing computations were performed fora frequency 1.25 GHz and for the stationary phase of a chosen CASTOR regime:n e (0) = 7 × 10 18 m −3 , T e (0) = 170 eV, I p =8kA, B T (0) = 1 T. Before givingresults of such computations, a comparison of the relevant wavelengths and plasmadimensions is necessary. Taking N ‖ ≃ 2 for the largest characteristic wavelength andn e ≃ 4 × 10 18 m −3 , we have the LHW wavelengths parallel (λ ‖ ) and perpendicular(λ ⊥ ) to the magnetic field, λ ‖ ≃ 12 cm ≪ πR ≃ 120 cm, and λ ⊥ ≃ 0.8 cm ≪a ≃ 8 cm. The last two inequalities provide information on the applicability of theray-tracing code here. Due to the massive current drive (see Section III), the wavesare damped and, therefore, a radial mode structure is not likely to occur.An example of the ray-tracing computation is given in Fig. 1. Fig. 1(a) (top view)shows the rays of LHW outgoing from the centre (the initial vertical coordinateof the ray z = 0, see Fig. 1(b)), and from the upper (z = +5cm) and lower(z = −5 cm) parts of the grill aperture for N ‖ =4. Fig. 1(b) shows the projectionof the same rays (however, in this case for N ‖ =4and 6) in the poloidal crosssection of the CASTOR tokamak. It is seen (on the right-hand side of Fig. 1(b)) thatthe launching antenna is placed in the low-field-side port and its poloidal shape isaligned to the plasma column cross section. The ray-tracing computation has beenmade for a parabolic plasma density profile with central value n(0) = 7×10 18 m −3 .Obviously, the rays depend only weakly on the N ‖ value.For the measurement of the LHW N ‖ spectrum, a double RF coaxial probe withtwo measuring tips 1 and 2, placed in a poloidal cross section 225 ◦ toroidallyaway from the grill antenna (see Fig. 1(a)), has been used. Receiving tips 1 and 2of the probe are a distant 6 mm toroidally from each other (the distance of 6 mmcorresponds to the phase difference ϕ ≃ 90 ◦ .if a slowed wave with N ‖ = 10 is

4 F. Žáček et al.Fig. 3. The three-waveguide multijunction grill (used in the experiment)power spectrum, computed for the plasma density n =1× 10 18 m −3 andradial density gradient ∇n =10 20 m −4 at the grill mouth; dotted curves areintegrated spectral power densities.supposed). The tips have a length of 5 mm and they are oriented in the supposeddirection of the LHW electric field ⃗ E, i.e. nearly radially. The probe enters theplasma through a lower port and it is movable through the whole poloidal crosssection of the device.A coaxial RF interferometric circuit for determination of the wave phase velocity(i.e. the mutual phase ϕ of the wave at two tips 1 and 2) for the case of a rapidlychanging wave amplitude has been developed and assembled, see Fig. 2. RF1 andRF2 in the figure denote the feeding coaxial lines from the two RF tips 1 and 2,DC are variable directional couplers for measurement of RF powers (squares ofwave electric field) and AT are attenuators. A hybrid ring junction has been usedas a mixer for the phase measurement. Each of the three detecting diodes shown inFig. 2 is absolutely calibrated in the whole range of the power used (to exclude anydifferences in the characteristics and departures from the quadratic dependences ofoutput voltages P 1 , P 2 ,andP ph on the corresponding wave electric fields).For the absolute phase measurements, the difference in electrical lengths of boththe coaxial RF lines 1 and 2 from the respective RF tips 1 and 2 to the mixer hasbeen determined on the test stand. For this purpose a synphase feeding of both RFtips at a frequency f =1.25 GHz has been assured. Using a high-sensitivity phasedevice (HP Network Analyser 8410B), the line 1 has been found to be 10 ◦ ± 1 ◦longer.The data measured have been stored using a transient recorder with samplingrate of up to 5 MS/s, a resolution of 12 bits and memory of up to 128 kB/channel.For evaluation of the mutual phase ϕ between the fast fluctuating RF signalsP 1 and P 2 , the interferometric scheme shown in Fig. 2 has been adjusted (in the

Spectrum Broadening and Fluctuations ... 5Fig. 4. Loop voltage U loop , plasma current I p, line-averaged plasma density¯n, incident LH power P LH and noninductively generated LH current I LH ina typical CASTOR discharge with LHCD (shot #5581).absence of the plasma) in the following way:1) P 1 ≡ P ph for AT2 = max (closing of the second arm of the interferometer),2) P 2 ≡ P ph for AT1 = max(closingofthefirst arm of the interferometer).After opening of both interferometric arms, the phase ϕ can be determined in thefollowing form:ϕ = arccos P ph − (P 1 + P 2 )2 · √P .1 · P 2As a launcher, a three-waveguide multijunction grill (having the phase shiftbetween adjacent waveguides 90 ◦ ) with a relatively broad spectrum 1 ≤ N ‖ ≤ 5 hasbeen used (the output dimensions of the grill mouth are 160 mm in the poloidal and50 mm in the toroidal directions). The spectral power density of the waves launchedby this grill, computed theoretically for the edge plasma density n =1× 10 18 m −3and a radial gradient ∇n =10 20 m −4 at the grill mouth, is given in Fig. 3. Thepower of the RF generator used (several tens of kW) is comparable with the ohmicheating power.III. EXPERIMENTAL RESULTSA typical discharge of the CASTOR tokamak in the lower hybrid current drive(LHCD) regime is shown in Fig. 4, where loop voltage U loop , plasma current I p ,line-averaged density ¯n measured by 4 mm microwave interferometer, and LHWincident power P inc are given. The last trace is the value of the non-inductive

6 F. Žáček et al.(a)Fig. 5. Phase evaluation of LHW launched in CASTOR by three-waveguidemultijunction grill (shot#5581, probes on the radial position r = R − R 0 =−60 mm, z =0).(b)current I RF driven by LHW (evaluated from the relative drop of U loop ). It may beseen that about 60% of the total current is driven by LHW during the RF pulseapplication at the density ¯n =4× 10 18 m −3 .The measurements of LHW inside the plasma, made under the conditions givenin Fig. 4 and using the scheme shown in Fig. 2, yield the following results:1) Amplitudes of the signals from the RF double probe increase from the edgetowards the centre of the plasma.2) Also in this regime with current drive (LHCD), general feature of all RFmeasurements inside the plasma is a strong fluctuating modulation of thewave amplitudes in time, similarly to the case of the quasi-optical grill [1]–[6].3) The relative level of LHW fluctuating modulation does not depend on theprobe position.The situation is illustrated in Fig. 5. In addition to the squares of electric fieldintensities at the two tips, P 1 and P 2 , the square of interference signal P ph ofthe two fields is given in the figure as well, together with the phase difference ϕbetween these two signals (evaluated using the formula given above). The left-handside of Fig. 5 shows the quantities on a longer time scale (during the whole RFpulse), while the right-hand side shows the same on a short time scale (the signalshave been sampled with a rate of 0.5 µs in this case).Fig. 6 gives comparison of the frequency spectrum of LHW power P fluctuationsshown in Fig. 5 (the upper trace in Fig. 6) and the frequency spectrum of plasmadensity fluctuations (the lower trace in Fig. 6). These density fluctuations havebeen measured by a Langmuir probe in the CASTOR tokamak plasma (note thattheir spectrum has only a slight dependence on the place of measurement). Forcomparison, a typical frequency spectrum of LHW launched into the plasma by

Spectrum Broadening and Fluctuations ... 7the quasi-optical grill (f =9.3 GHz) is also shown in Fig. 6 (the middle trace). Arelatively large frequency step in the last spectrum (10 kHz compared with 500 Hzin the other spectra) is given by the short RF pulse length in the experiments withthe quasi-optical grill (150 µs only) as well as by a shorter sampling period (0.2 µs).It is seen that both the LHW fluctuations spectra have quite similar character andthat they contain much higher frequencies than those of the density fluctuations.In general, the following conclusions can be drawn from the measurements.1) All wave signals measured in the plasma strongly fluctuate in time withfrequencies far above 100 kHz, reaching values approximately three to fourtimes higher than the frequencies of the plasma density fluctuations.2) The phase difference ϕ of the waves detected using the RF double probe (i.e.N ‖ ) fluctuates strongly as well.3) However, in this case of relatively low frequency (1.25 GHz, instead of 9.3 GHzused in the experiments with the quasi-optical grill), the phase ϕ is measurable,because its absolute value does not leave the interval 0

8 F. Žáček et al.Fig. 6. Frequency spectra of LHW power fluctuations (upper trace: frequency1.25 GHz; middle trace: frequency 9.3 GHz) and of fluctuations of plasmadensity detected by a Langmuir probe (lower trace).In the case of previous experiments [3], [5] with the 9.3 GHz quasi-optical grill,the length of the wavepacket was ∆s ‖ ≤ 1 cm. Comparing this with the RF probetips mutual distance of 0.6 cm, we can understand why practically no correlation ofthe two probe signals has been found. For the present 1.25 GHz multijunction grill,the length of the shortest wave packet is ∆s ‖ ≈ λ 0 /N ‖max which is approximatelyfour times the RF probe tips mutual distance. This fact yields better conditions forthe wave phase measurements.The above estimates are in accordance with the experimental results, where(i) the wave phase measurements with the 1.25 GHz grill are possible and(ii) the maximum phase difference of the waves detected by the double RF probeis 90 ◦ , corresponding to N ‖ ≃ 10.B. Fluctuations of the RF Probe SignalsThe phase α of a wave at the RF probe can be expressed as an integral alongthe ray path, viz.,∫ s2α = −ωt + [k + κ(t)] ds, (1)s 1where s 1 is some initial position at the grill mouth, s 2 is the RF probe position,k is the wavenumber at zero plasma fluctuations and κ(t) is the oscillating part ofthe wavenumber given by the plasma density fluctuations. We recall that the angle

Spectrum Broadening and Fluctuations ... 9between the wavefront and the ray direction is rather small for LHW. Denoting ¯kand ¯κ the mean values of k and κ along the integration path and s its length, wehaveα = −ωt + ¯ks +¯κ(t)s. (2)The length s can be expressed in terms of the wavelengths 2π/¯k. Thefluctuatingpart Ψ(t) of the phase is then proportional to the number m of wavelengths alongthe ray path,Ψ(t) =¯κ(t)s =2πm¯κ/¯k. (3)Assume that there is a characteristic angular frequency Ω of the plasma densityfluctuations. Then, the characteristic frequency of ¯κ(t) is also Ω. According to (3),the same frequency governs the time dependence of Ψ, when|Ψ(t)| ≪2π. A quitedifferent situation arises when Ψ > 2π. This can occur at sufficiently long ray paths(m ≫ 1). Suppose that Ψ increases by 2π during a time τ ≪ 2π/Ω. According to(3),2π ≃ 2πm¯kd¯κτ. (4)dtFor the RF field, Ψ + 2π = Ψ and, consequently, the value of 2π/τ is thecharacteristic angular frequency Ω Ψ of the Ψ(t) oscillations. In (4), we can estimatethe magnitude of |d¯κ/dt| ≃ Ωκ max ,whereκ max is the amplitude of the ¯κ(t)oscillations. Consequently, we haveΩ Ψ ≃ 2πm κ maxΩ. (5)¯kThis rough physical estimate is refined by the following algebra. According to(2) and (3), the fluctuating factor in the expression for the measured RF electricfield is exp[iΨ(t)]. We put ¯κ(t) =κ max sin Ωt and use (3) to obtain(e iΨ = exp 2πmi κ )maxsin Ωt , (6)¯kwhere Ω is the characteristic frequency of plasma density fluctuations, cf. SectionIII. Due to a familiar formula with Bessel functions J l (x), we can writee iΨ =+∞∑l=−∞J l (a)e −ilΩt , (7)where a =2πmκ max /¯k. It follows from the plots of J l (a) as functions of l thatthe values of J l (a) are small for l>1.1a.The expression for the measured electric field component E = E 0 exp(iα) withamplitude E 0 is given by (2), (3), and (7),∑E = E 0 J l (a)e i[¯ks−(ω+lΩ)t] . (8)l

10 F. Žáček et al.This is a superposition of waves with frequencies ω + lΩ, where|lΩ| ≪ω andl =0, ±1, ±2, ... with |l| ≤1.1a. SinceJ −l (a) =(−1) l J l (a), we obtain[E = E 0 e i(¯ks−ωt) J 0 (a) − 2i ∑J l (a) sin(lΩt) + 2 ∑]J l (a) cos(lΩt) , (9)l oddl evenwhere the first and the second summations are over odd and even values of l>0,respectively, and of course l ≤ 1.1a. Note that the highest frequency lΩ approximatelycoincides with the value of Ω ψ in (5). The RF probe signal P (i.e., P 1 orP 2 in previous sections) is proportional to [R{E}] 2 = EE ∗ /2, viz.⎡( 2P ∝ E 0 E0∗ ⎣ 1 ∑2 J 0 2 (a) + 2 J l (a) sin(lΩt))++2( ∑l evenJ l (a) cos(lΩt)l odd) 2+ 2J 0 (a) ∑l even⎤J l (a) cos(lΩt) ⎦ . (10)The last expression is a sum of terms oscillating with frequencies lΩ and (l 1 ± l 2 )Ω,where 0 ≤ l 1,2 ≤ 1.1a. Thus, the maximum frequency of the probe signal fluctuationsisΩ max ≃ 4πmΩκ max /¯k. (11)The value of mκ max /¯k may vary in broad limits. Plausible values m =10andκ max /¯k = 0.03 give Ω max ≃ (3–4)Ω. This relation shows why the RF probesignals fluctuate with much higher frequencies than the plasma density, as it is seenin Fig. 6.V. CONCLUSIONSStrong fluctuations of LHW amplitudes observed in the former experimentswith quasi-optical grills [1]–[5] have also been found in the present experimentusing the multijunction grill. The characteristic frequency of these wave amplitudefluctuations is much higher than the characteristic frequency of plasma densityfluctuations. This effect is explained by the presence of plasma density fluctuationsalong the whole path of waves from the grill to the RF probe.Correlation measurements performed by means of the double RF probe confirmthe assumption of LHW spectrum broadening up to the value of N ‖ =10,correspondingto a wave phase velocity ω/k ‖ ≃ (3–4)v Te . This result supports thehypothesis of “spectral gap filling” [11] between the electron velocity distributionfunction and the spectrum launched by the grill. However, the present experimentdoes not imply that the plasma density fluctuations are the main cause of spectrumbroadening in question. Physical mechanisms resulting in this effect [7]–[10] arenot considered here.

Spectrum Broadening and Fluctuations ... 11Similar measurements of the N ‖ values were not realizable in previous experiments[1], [3], [5], because the wave phase correlation distance was apparentlycomparable with the mutual distance of the double RF probe tips.ACKNOWLEDGMENTSThe authors thank the referees for valuable comments and suggestions. Work hasbeen supported by the grants of GA-CR No. 202/96/1355, No. 202/97/0778 andGA-AS No. 143405, No. 1043701.REFERENCES[1] Žáček F, Jakubka K, Kletečka P, Klíma R, Krlín L,Kryška L, Nanobashvili S, Pavlo P, Preinhaelter Jand Stöckel J 1995 22nd EPS Conf. on Controlled Fusion and Plasma Physics (Bournemouth, 1995)vol 19C ed Keen B E, Stott P E and Winter J (Geneva: EPS) part IV, p 373.[2] Žáček F, Stöckel J, Jakubka K, Klíma R, Kryška L, Nanobashvili S, Nanobashvili I, Pavlo P andPreinhaelter J 1996 International Conference on Plasma Physics (Nagoya, 1996) vol I ed Sugai Hand Hayashi T (The Japan Society of Plasma Science and Nuclear Fusion Research) p 1030.[3] Žáček F, Stöckel J, Jakubka K, Klíma R, Kryška L, Nanobashvili S, Nanobashvili I, Pavlo P andPreinhaelter J 1996 Detection of LHW in tokamak CASTOR IAEA TCM on Research Using SmallTokamaks (Prague 1996) (Vienna: IAEA).[4] Preinhaelter J 1996 Nucl. Fusion 36 593.[5] Žáček F, Jakubka K, Klíma R, Kryška L, Nanobashvili S, Pavlo P, Preinhaelter J and Stöckel J1997 24th EPS Conf. on Controlled Fusion and Plasma Physics (Berchtesgaden, 1997) vol 21A(Geneva: EPS) part II, p 629.[6] Žáček F, Klíma R, Jakubka K, Plíšek P, Nanobashvili S, Pavlo P, Preinhaelter J, Stöckel J andKryška L 1998 ICPP’98 combined with 25th EPS Conf. on Controlled Fusion and Plasma Physics(Prague, 1998) vol 22C ed Pavlo P (Mulhouse: EPS) p 1414.[7] Bonoli P T and Ott E 1982 Phys. Fluids 25 359.[8] Pericoli-Ridolfini V and Cesario R 1991 18th EPS Conf. on Controlled Fusion and Plasma Physics(Berlin, 1991) vol 15C ed Bethge K (Geneva: EPS) part III, p 397.[9] Pericoli-Ridolfini V, Bartiromo R, Tucillo A A, Leuterer F, Söldner F X, Steuer K H and Bernabei S1992 Nucl. Fusion 32 286.[10] Vahala G, Vahala L and Bonoli P T 1992 Phys. Fluids B 4 4033.[11] Fisch N J 1987 Rev. Mod. Phys. 59 175.

13Diagnosis of Chemically Reactive PlasmaH. Wittrich, L. Weixelbaum, W. JohnFerdinand-Braun-Institut für Höchstfrequenztechnik (FBH),Gustav-Kirchhoff-Str. 4, D-12489 Berlin, GermanyAbstractProcess control is a fundamental demand in solid-state technology. For developmentand control of plasma-enhanced processes, like etching or deposition,the diagnosis of special features of the chemically reactive plasma, like plasmacolor or mass spectra, has often proved to be more meaningful for process controlthan the measurement of summarizing plasma parameters like electron density n eor electron temperature T e. As examples for modern process control techniques,we explain end point detection during polymer etching by means of plasma colormeasurements and mass spectrometric investigations of GaAs etching in fluorineand chlorine containing gases, respectively.I. INTRODUCTIONWITHIN the scope of this paper we are defining a low-temperature plasmato be chemically reactive, if the components of the plasma react with eachother and/or with surfaces exposed to the plasma in order to create new compounds.In solid-state technology, non-thermal plasma processes are used for well definedand reproducible removal of wafer material by transforming it into gaseous reactionproducts (dry etching) and for deposition of reaction products on the wafer surface(plasma-enhanced chemical vapor deposition). The plasma provides both the reactivespecies and the necessary energy for the reaction and thus enables the specificreaction. For process control, the diagnosis of such plasma processes should befocused more on the result of the particular processing step than on the plasmaproperties themselves. Nevertheless, important findings for the process can also begained from investigations, which are refer to electrical and/or optical parametersor mass distribution in the plasma.We would like to report on two methods that are applied at FBH to control, monitorand improve plasma etching processes for microelectronic and optoelectronicdevices in three-five semiconductor technology.II. HISTORICAL EXCURSIONAlready in 1971, the former ”Deutsche Akademie der Wissenschaften“ (GermanAcademy of Science) realized the importance of plasma assisted processes andconducted a study entitled ”Material transformation in non-thermal plasma andplasma chemical production of special materials“. At that time, Helmut Prinzler

14 H. Wittrich et al.Fig. 1. Microwave interferometer for the measurement of the electron density.Aromatic hydrocarbons were admixed to argon in an RF plasma.and Peter Heymann took up the latest trend in plasma diagnostics in their scientificpublication ”Microwave diagnostic of reactive, non-thermal plasma“. Profitingfrom their experience in weakly ionized plasmas, they modified their method forchemically reactive plasmas and applied microwave methods to enable contactlessmeasurement of electron concentration, electron energy and collision frequency ofthe electrons with the gas atoms. Microwave cavities and interferometers were usedtogether with vacuum chambers made of glass. At that time, the vacuum was createdby mercury diffusion pumps (Fig. 1).Today, the basic ideas of this work on the characterization of electron densityand electron/gas collision frequency in a chemically reactive plasma are successfullyexploited by other groups to get predictions on process stability and status of theplasma reactor (”Hercules“-APC of Advanced Semiconductor Instruments GmbH,Berlin).III. MASS SPECTROMETRY OF BCL 3 /SF 6 MIXTURES FOR GAAS DRY ETCHINGReactive gas mixtures containing chlorine are used for plasma etching of GaAsand other III-V semiconductors, because the arising chlorides as reaction productsshow a sufficiently high vapor pressure to be efficiently removed from the reactionchamber by vacuum pumps. In contrast, fluorine containing gases (SF 6 ,CF 4 )astheonly reactant are not suitable for the etching of III-V materials, because the resultingfluorine compounds are not or only little volatile. If the gases or mixtures of gasescontain fluorine as well as chlorine, such as CF 2 Cl 2 , BCl 3 /SF 6 or SiCl 4 /SiF 4 ,the

Diagnosis of Chemically Reactive Plasma 15etching rate (nm/min)60040020000 0.5 1 1.5 2 2.5 3gas flux SF 6(sccm)15 Pa10 Pa5 Pa1 PaFig. 2. Dependence of the GaAs etch rate in a BCl 3 plasma on the admixtureof SF 6 .etch characteristics depend both on the plasma (process) conditions and on the material.Often these mixtures are used for material selective dry etch processes wherethe selectivity depends on the formation of a practically non-volatile compound onthe appearing surface during etching. An example is the selective etching of GaAsto AlGaAs in fluorine and chlorine containing plasmas where the selectivity is dueto the formation of highly non-volatile AlF on top of the AlGaAs surface.The goal of the mass spectrometric investigations shown here was to analyze thestable reaction products in the vapor phase of a BCl 3 /SF 6 /Ar plasma in order toderive conclusions for an improved process control.As shown in Fig. 2, the etch rate of GaAs in a BCl 3 /Ar plasma rises if SF 6 isadded. The increase of the etch rate depends on the amount of added SF 6 as wellas on the process pressure. In the whole parameter range, smoothly etched surfaceswith almost perpendicular sidewalls (Fig. 3) are obtained. The mass spectra ofthe inspected vapor phases show lots of occupied mass numbers (Fig. 4). Even ina pure BCl 3 /Ar plasma (Fig. 4(a)) we found a small amount fluorine containingfragments. The memory effect of the reactor surface is responsible for that. In theBCl 3 /SF 6 /Ar plasma, we found the typical fragments of BCl 3 and SF 6 ,butalsoadditional components resulting from a Cl-F exchange (Fig. 4(b) and Table I).Fig. 5 shows the time behavior of some selected species and indicates whathappens when the discharge is switched on. While the portion of BCl 3 and itsfragments (m/e 116, 81, 46) decreases in the discharge the portion of hybrid Cl-F compounds (m/e 65 BClF, 84 BClF 2 , 100 BCl 2 F) increases strongly. The reasonsare volume reactions as well as an increased activation of the reactor surface. Thehigher GaAs etching rate after adding fluorine containing gases to chlorine-based

16 H. Wittrich et al.TABLE ITYPICAL SPECIES IN A BCL3/SF 6 /AR PLASMA.BCl 3 m/e SF 6 m/e Cl-F Exchange m/eB + 11 SF + 51 BF + 30BCl + 46 SF 2+4 54 BF + 2 49BCl + 2 81 SF + 2 70 BClF + 65BCl + 3 116 SF + 3 89 BClF + 2 84SF + 4 108 SFCl + 86SF + 5 127 BCl 2 F + 100etch gases is caused by a greater amount of highly reactive chlorine compoundscreated by fluorine-chlorine exchange. The etch rate can be modified in a widerange by variation of the portion of the fluorine component in the gas mixture.IV. OPTICAL CONTROL OF ETCH PLASMA BY COLOR MEASUREMENTSThe measurement of plasma color is a classical example of the fact that in thisparticular case even summarizing plasma parameters may offer extremely sensitiveprocess control. A simple measurement of the plasma radiation in the wavelengthranges red/green/blue combined with subsequent signal processing is used to reliablydetermine the endpoint of an etch process in an oxygen plasma used for polymeretching. Figs. 6 and 7 show the sensitivity of the method. Even minimal variationsof gas pressure or plasma power are recorded. Fig. 8 shows the time behavior ofone channel (blue) during a etching of polymer in a oxygen plasma. The radiationdetector is arranged perpendicular to the surface of the wafer. In this alignment,additional information about detraction or growth of layers is gained because ofinterference effects. The plasma light is reflected at the surface of the polymeras well as at the wafer surface and interferes on its way to the receiver. Theresulting minima and maxima of the light intensity are a measure of the polymerthickness during etching. The endpoint of the etch process, e.g. when the polymeris completely removed from the wafer is clearly indicated.V. CONCLUSIONThe importance of reliable diagnostics for the control of plasma processes is evergrowing with the higher demands on structure dimensions, process reproducibilityand automatic process control. Plasma diagnostics more and more matures to anessential and indispensable element of process control during solid-state devicefabrication.

Diagnosis of Chemically Reactive Plasma 17Fig. 3. Dry-etched structure in GaAs using BCl 3 ,SF 6 and Ar reactive gaschemistry.

18 H. Wittrich et al.E-05(a)ion current (A)E-06E-07BCl + BCl 2+BCl 3+E-08E-090 20 40 60 80 100 120 140 160 180E-05m/e (amu)(b)ion current (A)E-06E-07BClF + BClF 2+SFCl +E-08E-090 20 40 60 80 100 120 140 160 180m/e (amu)Fig. 4. Mass spectrum of the stable species in (a) a BCl 3 /Arand(b)aBCl 3 /SF 6 /Ar plasma.−710Plasma switched on5 10 −8BCl2ion current (A)10 −85 10 −9BClFBClF2BClBCl3BCl 2 FSFCl−9101 234time (min)Fig. 5.Time behavior of selected mass numbers in a BCl 3 /SF 6 /Ar mixture.

Diagnosis of Chemically Reactive Plasma 192.11 Paintensity2.01.92 Pa1.83 Pa0 2040 60 80 100time (s)Fig. 6. Dependence of the over-all intensity of the plasma light (sum of thethree channels) on changes of pressure between 1 Pa and 3 Pa.2.2100 Wintensity2.12.095 W1.91.80 50 100time (s)90 W150 200Fig. 7. Dependence of the over-all intensity of the plasma light (sum of thethree channels) on changes of power between 90 W and 100 W.

20 H. Wittrich et al.0.080.07intensity0.060.050.040 1020time (min)Fig. 8. The endpoint of a polymer etching in an oxygen plasma. Only theblue channel is evaluated. The radiation detector is arranged perpendicular tothe wafer surface.

Over-Temperature Noise Modeling ofSubmicron Devices Brought the Question:Is the Diffusion Coefficient TemperatureDependent?Ali BoudiafMaury Microwaveemail: aboudiaf@maurymw.com21AbstractA new procedure is presented for modeling the variations with temperatureof the noise source coefficients related to the gate and the drain of a field effecttransistor (FET). The experimental results obtained for a temperature range over−60 ◦ Cto140 ◦ C are compared to two recent similar studies, using a PseudomorphicHEMT. It is demonstrated that good agreement can be obtained for someof the temperature coefficients for both the small-signal model and the internalnoise sources. Further in the analysis of the drain noise source, which comesfrom fluctuations in the electron velocity, which in turn is related to the electrondiffusion constant D, we were able to quantify the temperature dependence ofthis coefficient.I. INTRODUCTIONVERY low noise figures with high associated gain performance requirements arenow being met by the sub-micrometer PHEMT transistors, which are replacingMESFETs because of their lower noise performance for the same gate length. Tosupport the design of communication systems, such as satellite systems, operatingin varied environments, accurate noise models are required which can predict allthe noise parameters of the transistor over a wide frequency range [1], but also overwide temperature variations.Although there are several ”Noise Temperature” models in the literature [2], [3],these models are not predictive in the sense that they can be used to computenoise parameters over different operating temperatures. Gate temperatures are approximatelyequal to the operating temperature, but very large drain temperaturesare observed, which leaves us with unanswered question: How does the draintemperature change with the operating temperature? Thereisnomuchdatainthe scientific literature that allows us to answer this question [1].In this paper, we present a procedure for modeling the temperature dependenceof both the small-signal model and the noise coefficients that characterize theequivalent gate and drain noise sources. Experimental results are reported for 0.5 µm

22 A. BoudiafLgRgCgdRdLdCgsGdsYmCdsRsLs-jwYm=g emFig. 1.Small-signal and noise equivalent circuit parameter model.gate length PHEMTs, the extracted temperature dependent noise model is comparedto two previous works [1], [2], [3] in the same area using the same kind of device.II. MODELING VERSUS TEMPERATUREThe procedure is based on Pucel et al. model [5], shown in Fig. 1. The intrinsicnoise sources are represented by a drain current source: 〈i 2 d 〉, in parallel with theoutput conductance G ds , and by a gate current source: 〈i 2 g 〉 in parallel with C gs andR i . These two equivalent noise sources are correlated and can be expressed withthe dimensionless constants P , R, andC, defined by:P =R =jC =〈i 2 d 〉4kT∆f g mg m 〈i 2 g 〉4kT∆f C 2 gs ω2 (1)〈i ∗ g i d〉√〈i 2 g 〉〈i2 d 〉This noise modeling procedure requires the measurement of the S-parametersand the noise parameters, to extract the small-signal equivalent circuit parametersand the noise coefficients, defined by P , R and C, together versus the temperaturevariation. The relation of noise model parameters versus temperature is supposedto be quasi-linear and can be approximated by the following relation [1]:Pr(T )=Pr(T 0 ) · [1 + B(T − T 0 )] (2)

Over-Temperature Noise Modeling ... 23Fig. 2.Measured and simulated S-parameters 0.5 × 200 µm PHEMT.where Pr(T ) is the parameter value at the temperature of interest, Pr(T 0 ) is thereference temperature parameter value (T 0 = 296 K), and B is the linear fittingcoefficient to be determined.Pospieszalski [6] developed an intrinsic noise model in terms of two temperatures,called T g and T d . These can be related to Pucel’s PRC factors using:P =T dT 0 g m R ds(3)R i T gR = g m (4)T 0The origin of the drain noise source comes from fluctuations in the electronvelocity, which in turn is related to the electron diffusion constant D and the electronmobility through the Einstein relationship:D = kTµ nq(5)So whether P and R scale as T/T 0 depends on D/µ being equal to kT/q. Theobjective of this study is to try to confirm or deny this statement.g mv sat = L g(6)C gs + C gd

24 A. BoudiafFig. 3.Measured and simulated noise parameters of 0.5 × 200 µm PHEMT.B ft =1 ∂f Tf T (T 0 ) ∂T = 1 ∂v satv sat (T 0 ) ∂T(7)B ft = −1.44 [−1.0 − 2.5] · 10 −3 /K (8)The electron velocity v sat can be estimated from equation (6), then the thermalcoefficient B ft can be calculated with equation (7). The obtained result in (8) iswithin the range of the published values obtained by other means.1 ∂S idS id (T 0 ) ∂T≈1 ∂g mg m (T 0 ) ∂T + 1 ∂PP (T 0 ) ∂T(9)≈1 ∂D ||D || (T 0 ) ∂T − 1 ∂v satv sat (T 0 ) ∂T(10)B D|| =0.77 · 10 −3 /K (11)Equations (9) and (10) report the relationships between the parallel diffusionconstant, electron velocity, the transconductance and P extracted noise coefficient.The result obtained in (11) suggests that the main origin of noise variation withtemperature is due to the diffusion coefficient, it is proportional to temperaturevariation.

Over-Temperature Noise Modeling ... 2543T=-60T=23T=140Fmin (dB)2100 2 4 6 8 10 12 14Frequency (GHz)Fig. 4. Measured temperature dependent variation of F min at I d = I dss .III. EXPERIMENTAL RESULTSTo demonstrate the noise modeling procedure, S-parameter and noise parametermeasurements were made on a 0.5×200 µm PHEMT (AlGaAs/InGaAs/GaAs). Themeasurements were made using an on-wafer probe station with Cascade-MicrotechHF probes, over a temperature range from −60 ◦ Cto140 ◦ C performed by theThermo-Jet system (SAGEM). The S-parameter measurements were made from100 MHz to 26.5 GHz using the HP 8510 network analyzer, and the noise parametersmeasurements were performed between 2 and 12 GHz using Maury MicrowaveElectronic Tuner System NP5. The calibrations were performed each time at themeasurement temperature after chuck temperature stabilization.Results of this noise modeling procedure are reported in Table I, at I ds = I dss ,with results from other two references. We have observed that the minimum noisefigure (F min ) Fig. 4, and the equivalent resistance (R n ) Fig. 5, exhibit a largerrelative increase with increasing temperature than the optimum reflection coefficient(Γ opt ) Fig. 6. The same effect is observed at I d = 50%I dss .The extracted values of the temperature coefficients B[10 −3 / ◦ C], TableIandFigs. 7 – 10, show that for the parameters f T , g m , C gs , R ds ,andP , they are of thesame order of magnitude compared to the results in [2], [3]. It demonstrates thatfor these parameters the extracted temperature coefficients are less sensitive to themodeling method and to the measurement errors than for the rest of the parameters.The temperature coefficients for R and C were expected to be different since thenoise model is different between reference [2] and our work.

26 A. Boudiaf4035Rn (Ohm)T=-60 T=23 T=14030252015100 2 4 6 8 10 12 14Frequency(GHz)Fig. 5. Measured temperature dependent variation of R n at I d = I dss .IV. CONCLUSIONA new noise modeling procedure has been introduced for FET’s that is useful forthe temperature dependent modeling of PHEMT noise parameters. A comparisonwith two previous works shown that the extraction of the temperature coefficientsis less sensitive to the used method for some of the model parameters, but stillcritical for some others.V. DEDICATIONI dedicate this paper to Dr. Peter Heymann for his contribution to the field ofnoise parameter measurements, noise modeling and low phase noise VCO design.REFERENCES[1] R. E. Anholt et al., “Experimental investigation of the temperature dependence of GaAs FETequivalent circuits,” IEEE Trans. Electron Dev., vol. 39, no. 9, pp. 2029–2035, Sept. 1992.[2] J. Pence et al., “High frequency over temperature. Measurements and modeling,” Application NoteCascade Microtech, 1994.[3] S. Lardizabal et al., “Experimental investigation of the temperature dependence of PHEMT noiseparameters,” in IEEE MTT-S Int. Microwave Symp. Dig., pp. 845–848, 1994.[4] A. Boudiaf et al., “An accurate and repeatable technique for noise parameter measurements,” IEEETrans. Instrum. Meas., vol. 42, no. 2, pp. 532–537, April 1993.[5] R. A. Pucel, H. A. Haus, and H. Statz, “Signal and noise properties of gallium arsenide microwavefield-effect transistors,” in Advances in Electronics and Electron Physics, vol. 38, New York:Academic Press, 1975, pp. 195–265.[6] M. W. Pospieszalski, “Modeling of noise parameters of MESFETs and MODFETs and theirfrequency and temperature dependence,” IEEE Trans. Microwave Theory Tech., vol. 37, pp. 1340–1350, 1989.

Over-Temperature Noise Modeling ... 27Γ optFrequency 2-12 GHzFig. 6. Measured temperature dependent variation of Γ opt at I d = I dss .TABLE IREFERENCE TEMPERATURE PARAMETER VALUE P (T 0 )(LINEAR FITTING COEFFICIENT B[10 −3 / ◦ C])Parameter T1 T2 T3f T [GHz] 36.1 (-1.44) — 68 (-1.43)g m [mS] 75 (-0.97) 171 (-1) 150 (-1.03)C gs [fF] 330 (0.43) 397 (-0.13) 276 (0.31)C gd [fF] 46 (-0.38) 65 (-0.36) 40 (0.10)C ds [fF] 41 (-1.08) 64 (-0.78) 48 (0.10)R ds [Ω] 324 (0.31) 93 (0.41) 154 (0.46)R i [Ω] 2 (-2.81) 1.9 (2.85) —τ [ps] 1.5 (4.81) 0.36 (-0.18) 0.48 (-0.64)P 1.34 (3.18) 0.69 (3.5) —R 0.29 (6.31) 0.13 (38.6) —C 0.24 (-1.41) 0.82 (-8.5) —T1: PHEMT (PML) 0.5 × 200 µm; V gs =0V, V ds =3V.T2: PHEMT [3] 0.25 × 300 µm.T3: PHEMT [2] 0.25 × 200 µm; V gs =0V, V ds =2V.

28 A. BoudiafPRCPRC1402321,81,61,41,210,80,60,40,20-60Fig. 7. Extracted noise coefficients P , R, and C versus temperature atI ds = I dss .PCRPCR14023-601,41,210,80,60,40,20Fig. 8. Extracted noise coefficients P , R, and C versus temperature atI ds = 50%I dss .

Over-Temperature Noise Modeling ... 29B (10^-3 /°C)3,532,521,510,50-0,5-1-1,5T1T2T3Ft gm Cgs Cgd Cds Rds PFig. 9.Thermal coefficient comparison.Fig. 10.Thermal coefficient comparison.

On Some Errors in Noise Characterization ofHigh Performance Semiconductor DevicesWojciech WiatrWarsaw University of Technology, Institute of Electronic SystemsNowowiejska 15/19, 00-665 Warszawa, Polande-mail: wiatr@ise.pw.edu.pl31AbstractThe paper discusses actual problems faced in noise characterization of modernmicrowave semiconductor devices whose dimension steadily decrease, and thenoise and frequency performance improve. It is focused on evaluation of measurementerrors related to finite bandwidth of the noise measuring receiver andsystematic effects of erroneously measured source impedances on the determinationof the four noise parameters. Both error factors have been so far disregardedin the noise characterization, but presented results show their significance to highlymismatched low-noise devices, particularly at lower microwave frequencies. Toenhance accuracy of the characterization, these effects should be accounted for innovel noise measurement techniques to be developed in future.I. INTRODUCTIONNOISE characterization relies on measuring noise powers of a device undertest (DUT) in certain conditions and then determining parameters of a noisemodel by solving an inverse problem, i.e. by fitting the model to measurements.The determined parameters may, in general, represent both the noise and signalparameters of the DUT.Due to numerous errors, noise parameter measurement has been considered asone of the most difficult tasks in microwave metrology. Its errors result from variouscauses. Firstly, the noise power to be measured is typically low and thus may easilysubject to any interference. Since direct power measurement at this level is not feasible,the noise requires to be amplified, but this entails an additional noise (systemnoise) that corrupts the noise to be measured. Secondly, due to the nature of noise,the power measured fluctuates and thus the measurement results are always random.Thirdly, there are various peculiarities of the noise measurement instrumentation,which cause errors, if not properly accounted for in underlying models. All thoseerrors strongly influence the noise parameter determination, spoiling it completelyin many cases. Right understanding their sources enables one to avoid some pitfallsof the noise characterization or start seeking new better solutions of the problems.Despite the significant progress in the noise metrology that brought about higherthroughput of the noise measurement systems, accurate noise characterization in

32 W. Wiatrfrequency is still a tedious and time consuming job involving labor calibrations,necessary to precisely define the test conditions. Usually, number of measurementssurpasses well the number of unknown parameters and this redundancy serves as asimple means for reducing effects of measurement errors and sometimes overcomingdeficiencies of employed models. Unfortunately, all this is in conflict with the callof rapid DUT noise characterization, which expressed over twenty five years ago in[1] still remains an actual goal for efficient noise testing of semiconductor devices.There are also new challenges ahead the noise metrology, which emerge nowadayswith the progress of semiconductor technology. Dimensions of modern devicescontinuously shrink and their frequency and noise performance steadily improve.Consequently, the noise characterization of such devices becomes more difficultthan in the past, since some phenomena, disregarded so far, grow in significanceat very low noise levels measured. This causes errors that hamper reliable noiseparameter determination.The rapid progress of wafer-level measurement technology has brought aboutfresh metrology problems. Some of the problems result from the DUT measuredon wafer, which along with its coplanar waveguide (CPW) pads and connected tothem launching air-coplanar lines of microwave probes form an open system. Insuch system, various wave modes may propagate as the wave propagation conditionsare not completely defined and may even alter when manipulating the probes.Therefore, any electrical DUT characterization made on wafer is, to some extent,uncertain. This uncertainty needs to be considered when interpreting the DUTmeasurement results. A recent analysis [2] of errors in the multiline calibration[3] of vector network analyzer (VNA) has shown significance of the microstrip-likemode propagation to the accuracy of S matrix measurement at millimeter wavefrequencies. This issue is of paramount importance to noise characterization thatassumes S parameter measurements to be perfect. Since the current noise modelsalso stem from the underlying postulate on propagation of just one mode, they needto be modified in the future to account for other modes, too. Theoretical foundationsfor this have been already laid in [4] with the modal description of noise transferproposed for passive waveguides.This work presents some phenomena, which have been so far disregarded, butnow may significantly influence the determination of the DUT noise parameters,particularly at lower microwave frequencies. It considers measurement errors relatedto finite bandwidth of the noise measuring receiver and effects in the determinationof the four noise parameters due to residual errors in the source impedance measurementusing vector network analyzer. It shows effects of these errors calculatedfor a low-noise modern pseudomorphic high-electron mobility transistor (PHEMT).

On Some Errors in Noise Characterization ... 33II. GENERAL CONSIDERATIONSTwo-port’s noise properties are typically defined using either noise figure F oreffective input noise temperature T e , which depend on operation conditions of thetwo-port. Since F and T e are linearly related to each other, the further discussionwill be confined just to the effective input noise temperature and statements madeon it afterwards will regard the noise figure as well.As it is well known, dependence of T e on the source (generator) reflection coefficientcan be described analytically using four real quantities [5]. In practice, thereis a broad diversity of different noise parameter sets that result from a particularchoice of relevant formula describing the noise characteristic. Since all such setsare equivalent and can be easily converted to each other, the further discussion willbe related solely to the dependence of T e on the generator reflection coefficient Γ gas follows [6], [7]|Γ g − Γ opt | 2T e (Γ g )=T min + T N(1 −|Γ g | 2 )(1−|Γ opt | 2 (1))with parameters referred to as the four noise parameters:T min minimum value of the noise temperature T e (Γ g ),optimum source reflection coefficient (complex number) for which theΓ optT Nminimum occurs,temperature determining how rapidly (1) increases when moving withΓ g away from the minimum.Besides the four noise parameters, terms of noise correlation matrices [8] orparameters linearly related to them [7], [9] are frequently used to describe thenoise transfer with a linearized form of (1), useful when determining two-port noiseparameters. However, they are not so valued by circuit designers and semiconductorcustomers as the four noise parameters that possess a clear interpretation as thecoordinates of the noise temperature minimum. Moreover, for any real two-port,the noise temperatures fulfill the inequality [7]T min ≤ T N (2)which results from the fundamental requisite that the correlation coefficient ofany two partially correlated noise sources have to not exceed unity [10]. Thetemperatures T min and T N are also invariant to lossless cascade embedding of twoport[7], [11]. For all these reasons, formula (1) will be applied here as superior toother similar formulae and sets of the relevant parameters e.g., used in [5] or [12].A general setup for measuring the four noise parameters is depicted as a blockdiagram in Fig. 1. It comprises the Variable Noise Generator block separated fromthe DUT and Receiver blocks with the dash line marking the input referenceplane for the noise characterization. The parameters of the generator, the reflectioncoefficient Γ g and noise temperature T g , can be varied in controllable manners.

34 W. WiatrVariableNoiseGenerator gT g iDUTReceiverT eTwo-port under measurementpFig. 1.General measurement setup for the device noise characterization.The Receiver block represents a narrow-band RF receiver that operates along witha power-level meter at its output as a RF total power radiometer tuned in broadfrequency range of interest. The DUT and the receiver are marked in Fig. 1 as onesection, the two-port under measurement, for contributing together to the total noisepower measured at the output.The noise power level p indicated by the meter in the setup from Fig. 1 dependsgenerally on Γ g and T g . If the noise power density is constant over the radiometer’sbandwidth, the noise level measured can be expressed as follows [13]p = G · (T g + T min ) ( 1 −|Γ g | 2) + T q |Γ g − Γ opt | 2T r |1 − Γ i Γ g | 2 (3)where T r is a system constant expressed as a temperature(characterizing the radiometer,while the parameters T min , Γ opt , T q = T N 1 −|Γopt | 2) −1, the powergain G and the input reflection coefficient Γ i represent the cascade of DUT andreceiver, i.e., the two-port measured. The total set of parameters in (3) comprisesseven real quantities: the four noise parameters and three others, T r ,Re{Γ i } andIm{Γ i }, related to the gain variation of the two-port.Today, the noise characterization is typically based on source-pull noise measurementprocedure, i.e., power levels measured for different known states of thenoise generator, and (3) as the underlying model for describing observed noisebehavior of the two-port tested. Way, in which the model is dealt with, to accountfor the input mismatch of the two-port, enable splitting known noise parametermeasurement methods into two groups. Older methods [14]–[16] employ a VNA tomeasure the input reflection coefficient Γ i of the two-port, while others, the complexnoise characterization method [13] and the seven-state method [17], determine thisjust from the noise measurements. Therefore, the former group yields five whilethe later seven parameters of the model (3) determined in course of the noisecharacterization. The later group is more versatile for not using any VNA in themeasurement system and thus is discussed here.To facilitate the noise characterization, (3) is typically converted to a linear formas regards the unknown parameters. The eight-term linear model, derived in this

On Some Errors in Noise Characterization ... 35TABLE IPARAMETERS OF THE TWO-PORT WITH A 4 × 40 µm GATE-WIDTH PHEMT AT THE FRONT.OPERATING POINT: I D =20MA, V DS =3V ,FREQUENCY f =2GHZ.T min T N Γ opt Γ i T r(K) (K) mag/ang mag/ang (K)18.3 36.0 0.874/11.8 ◦ 0.977/-21.3 ◦ 47.5way, is written in matrix form [18]:pxβ g − xβ n = T ge (4)where the row vector x comprises terms dependent on Γ gx = [ 1 −|Γ g | 2 1+|Γ g | 2 2 Re{Γ g } 2 Im{Γ g } ] ,the vector β n represents the linear noise parameters of the two-port:[βn T = T min − T N T q (1+|Γopt | 2) ]− T q Re{Γ opt } −T q Im{Γ opt } ,2 2while the vector of gain parametersβgT = T r [1 −|Γi | 2 1+|Γ i | 2 − 2 Re{Γ i } 2 Im{Γ i } ] ,2and T ge = x 1 T g is the effective noise temperature of the generator.Model (4) comprises eight real parameters in both vectors β g and β n ,whichorigin from seven real parameters of (3), three related to gain and four to noise.This means that one of the β g parameters can be expressed by its counterparts andthat nonlinear relationship is [13], [18]β T g Dβ g =0 (5)where D = diag{1, −1, 1, 1}. This relationship supplements the model (4) as aconstraint imposed on the determined parameters [13], [18].The model (4) along with (5) is capable also of representing the comparablemodel employed by the seven-state method [17], which stems from the same underlyingmodel (3), too. This means that (5) can explain specific features of that methoddisregarding its original model introduced in the admittance notation. Similaritiesof both methods were discussed in [19].With the use of (3)–(5), it is possible to discuss errors that arise in noise characterizationof low-noise semiconductor devices. To make this discussion explicit,the error analysis will regard a modern PHEMT with the gate be 0.2 µm long and4 × 40 µm wide. The parameters of the two-port with the PHEMT at the front areshowninTableI.ValuesofΓ i and Γ min evidence high input mismatch for bothnoise and signal optimum transfer conditions of the two-port.

36 W. WiatrFig. 2. Dependence of the output noise power level upon the reflectioncoefficient Γ g for T g = 290 K.It is interesting to look at a potential noise level response calculated for the twoportusing (3) and shown in Fig. 2. This response exhibits very strong variationof the output noise power peaking at the reflection coefficient Γ g ≈ Γ ∗ i , the pointof the two-port’s maximum power gain. The power changes over 25 dB and thesteepest slope of the response takes place between the points of maximum powergain and minimum of the input noise temperature Γ g ≈ Γ opt .Suchalargepowervariation may bring about serious problems in the noise characterization, if thereflection coefficient Γ g either alters in frequency within the measurement bandof the receiver or has been measured with errors. An analysis of the problems ispresented in the next two sections.III. FINITE BANDWIDTH RELATED ERRORSThe eight-term linear model, introduced for the complex noise characterizationin [13], assumes narrow-band measurement of noise power at the radiometer output.This assumption is valid as far as the generator reflection coefficient and thetwo-port’s parameters, can be assumed invariant across the receiver’s bandwidth.However, severe measurement errors arise if this requirement is not fulfilled, as ithappens when testing a highly mismatched transistor [20], due to a resonant interactionbetween a mismatched generator and the transistor, both placed in a distanceeach other. This is a frequent situation met in on-wafer measurement systems thatuse cables to connect the microwave probes with measurement instrumentation, e.g.,

On Some Errors in Noise Characterization ... 37Fig. 3. Dependence of the errors in the noise level measurement upon Γ gat T g = 290 K. The errors are caused by finite bandwidth of the receiver(B ≈ 4 MHz) and group time delay of 1.15 ns in the phase variation (3).an impedance tuner. Due to the cable’s delay, one observes fast phase variation ofthe generator reflection coefficient that can be expressed with the formula:Γ g (ω) ≈ Γ gc e −j2ωτ , (6)where Γ gc is the reflection coefficient measured at the center frequency f c inthe receiver’s pass band [f c − B/2,f c + B/2], ω =2π(f − f c ) represents theangular frequency within the bandwidth B and τ is a group time delay, whilethe approximation sign refers to assumption of constant magnitude of Γ g in thebandwidth. In context of the response shown in Fig. 2, this fast phase variationmeans that the noise level measured changes across that bandwidth and this maycause errors.The error analysis, reported here, was aimed at evaluation of effects of the grouptime delay and finite receiver bandwidth on the noise level measurements. The timedelay τ and frequency characteristic of the IF section of a HP 8790A noise figuremeter were determined in a real measurement system and then used in numericalsimulations of the noise measurements with Γ gc as a parameter. The simulationsrelied on integrating the noise power density across the bandwidth to determinethe power indicated at the output and then comparing it with its counterpart calculateddirectly from (3). Dependence of the relative errors on Γ g , calculated forB ≈ 4 MHz and τ =1.15 ns, is shown in Fig. 3. The diagram evidences largeerrors that occur for high reflection coefficients lying in the proximity of the peak

38 W. Wiatr0.03Error (dB)0.020.010-0.01 g0.800.750.70-0.02-0.03PHEMTf = 2 GHz-0.04-10° 0° 10° 20° 30° 40° 50°Arg gFig. 4. Dependence of the errors in the noise power level measurements versusthe phase of Γ g for |Γ g| ≤ 0.85 as a parameter (T g = 290 K).and associated with very steep slopes of the response seen in Fig. 2. In the realmeasurement system, however, the magnitudes of Γ g hardly excess 0.85 due tolosses in the tuner and cable. Then, the errors are considerably smaller as it is shownin Fig. 4. Nevertheless, they increase very rapidly with the product of Bτ and thusmay disturb the determined parameters, if a longer cable is used. Unfortunately,it is unknown yet how the determined parameters of the two-port depend on sucherrors. This problem still requires a more detailed study.The above discussion showed that the product Bτ needs to be small enoughto maintain the finite bandwidth related errors negligible. Therefore, one needsto narrow the measurement bandwidth, if the cables are long. This can be easilyperformed in high-performance spectrum analyzers, especially dedicated fornoise measurements, like that described in [21]. For few years, there is anotheroption, a new line of Agilent’s noise analyzers that provide high accuracy ofnoise measurements and adjustment of the radio bandwidth [22]. However, oneneeds to keep in mind that the narrower the bandwidth the longer integration timeis necessary to maintain the same measurement variance. So, the measurementthroughput decreases.IV. EFFECTS OF ERRONEOUS SOURCE IMPEDANCE MEASUREMENTSeveral papers [23]–[25] analyzed errors in the determination of four-noise parametersusing Monte Carlo simulations. Though the measurement errors were

On Some Errors in Noise Characterization ... 39' g " g gE rE lFig. 5. The error box as a cascade of two-ports representing fractionalimpedance transforms: E l represents the lossless transform (8a) and E rrepresents (8b) and is modeled with a resistive network inside.assumed random without any correlation, they demonstrated importance of accuratemeasuring of the generator reflection coefficient. However, besides random, there arestill systematic factors affecting VNA measurements of Γ g , albeit their effects arestrongly reduced in course of a valid VNA calibration. The VNA systematic errorsarise mainly due to imperfections of the calibration standards, random measurementerrors during the calibration and any change in the measurement conditions after thecalibration. They are referred to as residual errors for remaining in measurementsperformed after a VNA calibration.Though the residual VNA errors are usually very small, their consequences tothe parameters determined during the noise characterization may be serious as itresults from a very recent study [26]. An outline of the approach applied in thisoriginal study along with some selected results is presented below, while the fullmaterial is to be published in [27].Due to the residual errors, the measured reflection coefficient Γ ′ g usually differsfrom its true value Γ g and this can be expressed with the bilinear formulaΓ ′ g = e 1 Γ g + e 2(7)1 − e 3 Γ gwhere e i (i =1, 2, 3) represent residual VNA errors. After a valid calibration Γ ′ g ≈Γ g , and the errors are approximately |e 1 |≈1, |e 2 |≈0 and |e 3 |≈0.The transform (7) is typically depicted with an error box that can be generallyinterpreted as a network composed of a resistive two-port with a lossless embeddingcircuit in series with its input and output [28]. Using this interpretation, the networkcan be decomposed into two cascaded two-ports, as shown in Fig. 5, from which thefirst one comprises the resistive two-port, while the other is lossless. Transformationof Γ g through this cascade results in the fractional transformsΓ ′′g = Γ g − e ∗ 31 − e 3 Γ ge jψ (8)Γ ′ g = r Γ ′′g + c

40 W. Wiatr0.1950.860.870.880.1900.850.18512°j Im opt0.1800.1750.1700.165|e | = 0.0023|e | = 0.0043|e | = 0.006311°0.840 0.850 0.860 0.870Re optFig. 6. Effects of the lossless transform (8a) on Γ opt shown as loci of constantmagnitude of e 3 in the complex reflection coefficient plane.where Γ ′′gis an intermediate reflection coefficient, andr = |e 1 + e 2 e 3 |1 −|e 3 | 2 , c = e 2 + e 2 e ∗ 31 −|e 3 | 2 , and ψ = arg(e 1 + e 2 e 3 ).In the complex Γ ′ g plane, the parameters r and c are interpreted as the radiusand center of the |Γ ′′g| = |Γ g | =1circle, respectively. These two linear fractionaltransforms (8) enable one to decompose the residual VNA errors into two relevantfactor sets and then analyze how each affects the determined parameters.Results of the error analysis, presented here, regard the complex noise characterizationmethod employing measurements performed using cold-source procedurethat has become very popular for its simple and very effective scheme. The procedurewas initiated by Engen [29], who determined the DUT noise parametersby observing the output noise power when manually tuning sliding short at theDUT input. Then, Adamian and Uhlir in [14] extended this procedure for the useof any passive source terminations. To reduce labor necessary for characterizationof all the generator states, they proposed measuring power levels for many ’cold’passive source terminations at ambient temperature T a and just one terminationhaving a different noise temperature T g ≠ T a . This single termination (state) isusually referred to as the ’hot’ one for the standard noise source switched on to

On Some Errors in Noise Characterization ... 4160| c| = 0.002T N50Temperature (K)403020T min100r = 0.998r = 1.000r = 1.002-150° -100° -50° 0° 50° 100° 150°Phase of cFig. 7. Effects of the transform (8b) on the noise temperatures T min and T N :graphs depicted versus the phase angle of c at constant magnitude |c| =0.002and for r as parameterproduce an elevated ’hot’ noise temperature. Since the majority of states feature the’cold’ noise temperature, the procedure is named the cold-source technique [30].After a subsequent improvement in [16], it has been successfully implemented invarious noise measurement systems and nowadays is the most commonly utilizedprocedure for the noise characterization.The determination of noise model parameters from the cold-source measurementscan be split into two steps [16], [31], each related to the noise temperature of thegenerator states, i.e. the ’cold’ and ’hot’ ones. Majority of the parameters: Γ opt ,Γ i and the temperatures T min and T N normalized however to T r , are extractedin the first step from the set of measurements made for many ’cold’ states. Theextraction is based on the isothermal noise model introduced in [31]. In the secondstep, the remaining parameter, T r , is determined from the single ’hot’ state andthen utilized to retrieve the temperature scale for the both normalized quantities[27], [31]. This clear extraction scheme facilitates analysis of the error effects onthe extracted parameters.The error analysis, presented in [27], lend itself to verifying how the residual

42 W. Wiatr0.280.2614°1.00j Im opt0.240.220.200.180.160.9013°12°11°0.140.12| c| = 0.002r = 0.998r = 1.000r = 1.0020.100.80 0.85 0.90 0.95Re optFig. 8. Effects of the transform (8b) on the optimum source reflectioncoefficient Γ min : loci of |c| = 0.002 drawn in the complex reflectioncoefficient plane Γ min for r as a parameter.VNA errors propagate to the linear parameters of the eight-term model (4). It hasturned out that in the above parameter determination scheme the errors are fullyaccommodated by these parameters. Therefore, using this scheme, one may do noteven notice any bad symptoms of those errors in ones results or any worsening ofthe fit, as far as the errors spoil the parameters to such an extent that the fundamentalcondition (2) is not fulfilled. Effects of the errors are described below in two stepsrelated to sequential transforms given in (8) and depicted with relevant error boxesin Fig. 5.A. Effects of the First TransformAs mentioned earlier, the lossless transform (8a), governed by finite e 3 , does notaffect the temperatures T min and T N , but modifies only the reflection coefficientsΓ ′′opt = Γ opt − e ∗ 31 − e 3 Γ opte jψ (9)Γ ′′i = Γ i − e 31 − e ∗ 3 Γ e −jψi

On Some Errors in Noise Characterization ... 43-0.348-0.3500.970-21°| c|=0.002-0.3520.980-0.354j Im i-0.356-0.358-0.360-0.362-0.364-22°r = 0.998r = 1.000r = 1.002-0.3660.900 0.905 0.910 0.915 0.920Re iFig. 9. Effects of the transform (8b) on the input reflection coefficient Γ i :loci of |c| =0.002 drawninthecomplexreflection coefficient plane Γ i for ras a parameter.where double prime at the reflection coefficients refers to the input of the losslesstwo-port in Fig. 5. Since the magnitude of Γ i is equal almost one, (9b) describesin fact phase variations of Γ i confined to ±2|e 3 | boundaries. Similarly, variationsof the optimum reflection coefficient shown in Fig. 6 with ellipses in the complexΓ opt plane as loci of |e 3 | = const, take place also mostly along the phase direction.In general, this type of errors does not influence Γ opt nor Γ i much. Hence,residual errors represented by the lossless transform (8a) do not preclude correctcharacterization of basic two-port noise properties. This is not, however, the casefor the transform (8b) described below.B. Effects of the Second TransformThe analysis in this subsection regards the parameters c and r of the transform(8b), while e 3 related to the previous transform is assumed to be zero as a consequenceof the latest conclusion above. The results are shown in Fig. 7 and Fig. 8.Fig. 7 evidences large variations of T min , T N versus the phase of c at relativelysmall changes of r. The largest deviations happen at angles of c coinciding with thegradient of T e in the complex reflection coefficient plane Γ g , which is determinedby the phase angle of Γ opt (see Table I). The extreme deviations are -16.4 K and

44 W. Wiatr+7.6 K in T min , and approximately twice as large in T N , since the relevant curvepairs for given r maintain this ratio.Fig. 8 depicts variation of Γ opt with plane curves of shapes tending to ellipses,drawn in the complex reflection coefficient plane. The curves exhibit high sensitivityof Γ opt to c and r. The largest changes of Γ opt occur also along the T e gradient,manifesting in the magnitude deviations -0.046 and +0.113 from the value given inTable I and a low sensitivity in the phase.It is worth to emphasize that the results shown in Fig. 7 and Fig. 8 have beencalculated for very small VNA errors. Surely, larger errors occur when measuringimpedance on wafer. However, simulating the effects at larger magnitudes of c orincrements of r ends in nonphysical results due to violation of (2). Already, theextreme values of T min and T N approach almost zero in Fig. 7, and Γ opt goestoward the unity circle in Fig. 8, i.e., the boundaries for the noise parameters.In contrast to the behavior of Γ opt , the input reflection coefficient Γ i is subjectto only small variations and this is the consequence of the small errors. The lociof |c| = const, shown in Fig. 9, are circles of the same radius that is equal |c|.Such small changes confirm the high accuracy of Γ i determination from the noisepower measurements reported previously in [32]–[34]. This entitles one to excludeVNA from the noise measurement systems and thus cut down costs of the noisecharacterization.All results presented here explain serious problems faced usually when measuringnoise parameters of contemporary semiconductor devices as PHEMTs andmonolithic microwave integrated circuits (MMICs) manufactured with the use ofleading-edge technologies. In course of the error analysis, erroneous measurement ofthe generator reflection coefficients Γ g and the cold-source measurement techniqueemploying just one ’hot’ state sources turned out to be main source of theseproblems. Although the problems have been exemplified for the complex noisecharacterization method based on the eight-term model [13], the same results andconclusions regard also the seven-state method [17] for the similarity explained in[19]. Comparable errors may occur when using other more popular methods [14],[16], exploiting the five-term model. Due to the use of a VNA for measuring theinput reflection coefficient Γ i , the errors may even increase, since this measurementbrings about an additional error. This problem has not been, however, searched yet.V. CONCLUSIONSThe paper has introduced new errors that occur in the noise characterizationof low-noise microwave two-ports using source-pull measurement procedures. Itconsidered two different error sources: finite bandwidth of the noise measuringreceiver and residual VNA errors in the source impedance measurement. Bothsources cause errors, which have been so far disregarded, but nowadays grow insignificance when characterizing highly mismatched modern devices, e.g., PHEMTs

On Some Errors in Noise Characterization ... 45or MMICs, at low microwave frequencies. The paper presents errors evaluated fora low-noise PHEMT and data taken from a real noise measurement system in orderto verify their effects on measured results.The bandwidth related errors, observed in the measured noise power at the output,result from variation of the noise power density across the receiver band due toresonant interaction between the mismatched DUT and the source termination.Calculations showed that the errors grow rapidly with the magnitude of the sourcereflection coefficient, and the product of the bandwidth and the group time delayrelated to the phase variation of this coefficient. Such errors need to be accountedfor in systems with long interconnecting cables.The residual VNA errors in the source impedance measurements affect the determinationof device parameters. The paper showed effects of such errors, calculatedfor the complex noise characterization method employing the eight-term linearmodel and noise levels measured using the cold-source procedure. The resultsshowed that even small residual errors can seriously disturb the determined parameters,explaining the difficulties faced in the noise characterization of modernsemiconductor devices.In order to avoid these problems, one need to keep the measurement bandwidthnarrow enough and provide the highest accuracy available for the source reflectioncoefficient measurement. Accomplishment of these requirements is essential forcorrect noise characterization of such devices. Since the high measurement accuracyis not attainable in on-wafer measurement systems due to effects attributed to highermodes [2], one needs to seek other options to the noise characterization methodsapplied presently.New solutions can be found on the base of novel models accounting for thephenomena described in this paper. Those models will help developing new measurementprocedures and parameter extraction techniques in order to improve theaccuracy of the four-noise parameter determination. Work towards these goals isbecoming now a necessity in order to meet fresh problems that will certainly bringabout the advent of nanoscale devices.REFERENCES[1] R. Q. Lane, “Derive noise and gain parameters in 10 seconds,” Microwaves, pp. 53–57, 1978.[2] A. Lewandowski, W. Wiatr, “Errors in on-wafer measurements due to multimode propagation,” inProc. 15th Int. Microwave Conf. (MIKON-2004), Warsaw, Poland, May 2004, pp. 759–763.[3] R.B.Marx,“A multiline method of network analyzer calibration”, IEEE Trans. Microwave TheoryTech., vol. 39, pp. 1205–1215, July 1991.[4] D. F. Williams, “Thermal noise in lossy waveguides,” IEEE Trans. Microwave Theory Tech., vol. 44,pp. 1067–1073, July 1996.[5] H. Rothe, W. Dahlke, “Theory of noisy fourpoles,” Proc. of IRE, vol. 44, pp. 811–818, June 1956.[6] G. Caruso, M. Sannino, “Computer-aided determination of microwave 2-port noise parameters,”IEEE Trans. Microwave Theory Tech., vol. 26, pp. 639–642, Sept. 1978.[7] W. Wiatr, “A method for estimating noise parameters of linear two-ports in radio frequency range,”(in Polish), Ph.D. dissertation, Warsaw Univ. of Technology, Warsaw, Poland, 1980.

46 W. Wiatr[8] H. Hillbrand, P. H. Russer, “An efficient method for computer aided noise analysis of linearamplifier networks,” IEEE Trans. Circuits Syst., vol. 23, pp. 235–238, Apr. 1976.[9] R. Q. Lane, “The determination of the device noise parameters,” Proc. IEEE, vol. 57, pp. 1461–1462, Aug. 1969.[10] M. W. Pospieszalski, W. Wiatr, “Comments on ‘Design of microwave GaAs MESFET’s for broadbandlow-noise amplifiers’,” IEEE Trans. Microwave Theory Tech., vol. 34, p. 194, Jan. 1986.[11] J. Lange, “Noise characterization of linear two-ports in terms of invariant parameters,” IEEE J.Solid-State Circuits, vol. 2, pp. 37–40, Jan. 1967.[12] R. P. Meys, “A wave approach to the noise properties of linear microwave devices,” IEEE Trans.Microwave Theory Tech., vol. 26, pp. 34–37, Jan. 1978.[13] W. Wiatr, “Characterization of radiometer using eight-term linear model,” IEEE Trans. Instrum.Meas., vol. 44, pp. 343–346, Apr. 1995.[14] V. Adamian, A. Uhlir, “A novel procedure for receiver noise characterization,” IEEE Trans. Instrum.Meas., vol. 22, pp. 181–182, 1973.[15] M. S. Gupta, “Impossibility of linear two-port noise-parameter measurement with single temperaturenoise source,” IEEE Trans. Instrum. Meas., vol. 32, pp. 443–445, Sep. 1983.[16] A. C. Davidson, B. W. Leake, E. Strid, “Accuracy improvements in microwave noise parametermeasurements,” IEEE Trans. Microwave Theory Tech., vol. 37, pp. 1973–1978, Dec. 1989.[17] I. Rolfes, T. Musch, B. Schiek, “Cryogenic noise parameter measurements of microwave devices,”IEEE Trans. Instrum. Meas., vol. 50, pp. 373–376, Apr. 2001.[18] W. Wiatr, M. Schmidt-Szalowski, “The multi-state radiometer: A novel means for impedance andnoise temperature measurement,” IEEE Trans. Instrum. Meas., vol. 46, pp. 486–489, Apr. 1997.[19] M. W. Wiatr, “Comments on ’Cryogenic noise parameter measurements of microwave devices’,”IEEE Trans. Instrum. Meas., vol. 53, Apr. 2004, p. 619.[20] E. C. Valk, et.al., “Microwave noise measurement errors caused by frequency discrepancies andnonzero bandwidth,” IEEE Trans. Instrum. Meas., vol. 42, pp. 983–989, Dec. 1993.[21] P. Heymann, W. Wiatr, “Measuring noise parameters of twoports with Spectrum Analyzer FSM,”News from Rohde & Schwarz, vol. 37, pp. 20–23, Jan. 1997.[22] Agilent Technologies Inc., “Noise Figure Analyzers – NFA series – Brochure,” Publication No.5980-0166E, Palo Alto, CA, USA.[23] A. J. McCamant, “Error analysis aids measurement of noise parameters,” Microwaves & RF,pp. 109–118, June 1989.[24] M. L. Schmatz, H. R. Benedickter, “Accuracy improvements in microwave noise parameterdetermination,” in 51st ARFTG Conf. Dig., Baltimore, MD, USA, 1998.[25] J. Randa, W. Wiatr, “Monte Carlo estimation of noise-parameter uncertainties,” IEE Proc.-Sci.Meas. Technol., vol. 149, pp. 333–337, Nov. 2002.[26] W. Wiatr, D. K. Walker, “Systematic errors of noise parameter determination due to imperfectsource impedance measurement,” in Conf. on Precision Electromagnetic Measurements,CPEM’2004, London, UK, 2004, pp. 416–417.[27] W. Wiatr, D. K. Walker, “Systematic errors of noise parameter determination caused by imperfectsource impedance measurement,” to be published in IEEE Trans. Instrum. Meas., vol. 54, Apr. 2005.[28] A. Weissfloch, Schaltungstheorie und Meßtechnik des Dezimeter- und Zentimeterwellengebietes.Basel: Birkhäuser Verlag, 1954.[29] G. F. Engen, “A new method of characterizing amplifier noise performance,” IEEE Trans. Instrum.Meas., vol. 19, pp. 344–349, Nov. 1970.[30] R. Meierer, Ch. Tsironis, “An on-wafer noise parameter measurement technique with automaticreceiver calibration,” Microwave Journal, vol. 38, no. 3, pp. 22–37, Mar. 1995.[31] W. Wiatr, M. Schmidt-Szalowski, “A simplified procedure for impedance measurement using multistateradiometer,” in 27th European Microwave Conf. Dig., Jerusalem, Israel, 1997, pp. 897–902.[32] W. Wiatr, ”High accuracy characterization of two-ports in noise measurement system,” in Proc.MIOP’93, Sindelfingen, Germany, 1993, pp. 65–69.[33] W. Wiatr, “Accuracy verification of a technique for noise and gain characterization of two-ports,”in Proc. 10th Int. Microwave Conf. (MIKON’94), Ksiaz, Poland, 1994, pp. 525–529.

49Low-Frequency Noise in Resistive MixersGeorg BöckTechnische Universität BerlinMicrowave Engineeringhttp://www-mwt.ee.tu-berlin.deAbstractLow-frequency noise phenomena of resistive FET mixers are the scope of thispaper. Because commercially available low-frequency (LF) noise models are DCcurrent related and otherwise the mixing process in a resistive mixer is not coupledto a FET DC current, a new model based on resistance fluctuations is presentedfor field-effect transistors in the ohmic bias region (V ds ≈ 0). LF noise generationas a consequence of the self-mixing process by excitation of a local oscillator atthe gate of a single FET device is described and calculated quantitatively. It willbe shown, that a high measure of LF noise compensation is possible by properchoice of the mixer circuitry.I. INTRODUCTIONCURRENT development trends in the mobile communication area are towardssystems of higher integration density and lower power consumption, size,weight, and cost. Because of these requirements direct conversion architectures arein the scope of interest. Critical system parameters of such systems are second orderinter-modulation and low-frequency noise. The mixer is one of the key functionblocks inside a receiver. Compared with diode and active mixers, resistive FETmixers seem to be very superior candidates especially for direct conversion wirelessterminals because of the following advantages: no DC power consumption, low LOpower consumption, low LF noise, high linearity, high port isolation, compatibilitywith current integration trends, e.g. RF-CMOS technology [1]–[5]. Besides of wirelesssystems, resistive mixers have been applied very successfully also in millimetrewave range [6], [7].Besides all advantages, the analysis and optimisation of LF noise in resistivemixers is not possible with current commercially available microwave CAE tools,e.g, Agilent ADS, Ansoft designer, Microwave Office etc. The reason for that is,that the LF noise models of these tools are DC current based. This approach failsin resistive FET mixers where no DC voltage is applied and therefore, no DCpower is consumed. It will be shown that the noise sources in resistive mixershave their origin in resistance fluctuations of the FET channel activated by the LOcoupling to the drain. In this paper this mechanism is called ‘self-mixing process’.Although flicker (1/f-) noise and generation-recombination (gr-) noise are present

50 G. Böckat low frequencies [8]–[10], from a circuit point of view both phenomena can bemodelled by a similar model. The fundamentals of low-frequency noise and generalmodelling aspects are described in Section II. LF cold FET modelling is addressedin Section III. In Section IV, LF noise generation in resistive mixers is discussedand in Section V, this behaviour is modelled. Section VI shows possibilities ofLF noise compensation and their limits are presented. Section VII concludes thework.II. MODELLING OF LF RESISTANCE FLUCTUATIONSThe two main contributions to low-frequency (LF) noise are:1) Flicker (1/f-) noise,2) Generation-recombination (gr-) noise.Although the physical origin of flicker noise is not completely understood anddiffers from device to device (density or mobility fluctuations), the empirical Hoogeequation holds for all electronic 1/f-noise sources [10]:∣S V ∣∣∣I=constV 2 = S ∣I ∣∣∣VI 2 = S G=constG 2 = S RR 2 = α(1)N · fwhere S X is power spectral density (PSD) of the quantity X, V voltage, I current,R resistance, G conductance, α Hooge parameter, N number of free charge carrier,f frequency. The equality of all PSDs (voltage, current, conductance, and resistance)is a necessary criterion for resistance fluctuations and therefore, it is valid for grnoiseas well. Although the Hooge parameter originally was introduced empirically,it seems that there exist some correlations with physical parameters of the materialas charge mobility, temperature and lattice quality [10]–[12].The physical origin of gr-noise is the trapping and de-trapping of electrons,leading to a statistical variation of the free carrier density and thus to a resistancefluctuation. The normalized PSD is given by [8]S GG 2 = S NN 2 = ∆N 2 4τN 2 ·1+ω 2 · τ 2 (2)with 〈∆N 2 〉 variance of N, τ relaxation time, ω angular frequency.From a circuit point of view flicker noise and gr-noise can be interpreted asequilibrium fluctuations of resistance. Because of this fact, both noise phenomenacan be included in one circuit oriented LF noise model. In an electronic circuitresistance fluctuations can be measured as fluctuations of voltage, current or power.E.g., if a noiseless DC current is injected into a fluctuating resistance, a noisevoltage is generated along the resistor. Therefore, most LF noise models are voltageor current based and work well as long as a DC current is flowing. However incases, where LF noise occurs without any DC current (e.g. passive circuits with ACexcitation [13]) these models fail [14]. A simple test, e.g. whether a low-frequency

Low-Frequency Noise in Resistive Mixers 51Fig. 1.Implementation of a fluctuating resistance in ADSnoise model of a CAE-tool works properly or not, could be to look on the generationof AM noise sidebands in the case of a pure AC excitation. If no noise sidebandsoccur in this case, the model fails.To overcome the problem, the model has to be related closer to the physicalnature of the noise sources. In time domain, the equality of the normalized noisequantities (2) can be written as∆vv∣ = − ∆ii=consti ∣ = ∆gv=constg= ∆rr=∆k (3)where ∆x is the fluctuating part of quantity x and v, i, g, r are non-fluctuating partsof voltage, current, conductance and resistance, respectively. ∆k is the normalizednoise quantity. Its PSD equals the right term of (1) (1/f-noise) or (2) (gr-noise). Toemphasize that the electrical quantities in (3) are not necessarily constant ones (DCexcitation) small letters have been used. Using (3) we can now formulate Ohm’slaw for a fluctuating conductance:i = v · ˜g = v · (g +∆g) =v · g · (1 − ∆k) (4)where ˜g is overall conductance (fluctuating and non-fluctuating part) and all otherquantities are according to (3). Equation (4) can be put into a circuit simulator inorder to realize a fluctuating resistor.Fig. 1 shows an realization example for Agilent’s Advanced Design System(ADS). A ‘symbolically defined device’ (SDD) is used for the implementation ofthe current equation (4). The term ‘I[2,0]’ represents the current into port 2, theterms ‘ v1’ and ‘ v2’ represent the voltages across port 1 and 2, respectively. Thenoise source on the left is based on the equationS v =KA 0 + A 1 · f Fe (5)

52 G. BöckFig. 2. Intrinsic equivalent circuit of a cold GaAs-FET with low-frequencynoise sources (hatched areas).where S v is PSD of voltage v, K = α/N, measure of PSD for a given device,f frequency and A 0,1 , Fe, parameters for the colour of the PSD, respectively. Inthe given example with the parameters A 0 =0, A 1 =1and Fe =1, 1/f-noise isgenerated.III. COLD FET MODELLINGThe modelling work is based on different GaAs devices (FETs and HEMTs).For simplification, only the gate-source voltage dependence was taken into accountwith drain-source voltage in the cold FET state (V ds =0V). Thus, the non-linearbehaviour like compression of conversion loss and inter-modulation behaviour ofmixers can not be studied with this model. Figs. 2 and 3 show the in- and extrinsicequivalent circuits with parameter values for a Fujitsu HEMT FHC40LF.Parameters were determined based on cold FET S-parameter characterization,application of de-embedding methods and LF noise measurements. All parametersof the intrinsic circuit are constant as well as the gate charging resistors and thedrain-source capacitance C ds . For the drain-source resistance r ds , the gate-draincapacitance C gd , and the gate-source capacitance C gs the following equations were

Low-Frequency Noise in Resistive Mixers 53used [17]:r ds = R ds · [tanh(k r · (v gs − v r )) + 1] +R dsgtanh[k g · (v gs − v g )] + 1(6)C gd =C gd11+(Q gd · v gd + K gd ) 2 + C gdo (7)C gs = C gs0 ·{arctan[Q gs (v gs − v gs0 )] + K gs } (8)v gs is the intrinsic gate-source voltage and v gd the intrinsic gate-drain voltage,respectively.The gate resistance R g models the metallic and contact resistance of the gate electrodeand is assumed being LF-noiseless as a first-order approach. All semiconductorresistances produce LF noise (hatched areas in Figs. 2 and 3), i.e., they fluctuate. Thenoise contribution of the resistors between drain and source electrode can easily bemeasured by injecting a small DC current into the channel and measuring the noisevoltage. The channel has to remain in its linear region during this measurement.For the FHC40LG the LF noise can be described by the empirical functionS N,ch = S ∣R ∣∣∣f=1HzR 2 = α N = S 0,dB + S 1,dB · tanh(k n · v gs + v 0 ) (9)The noise contribution of the extrinsic series resistances R s and R d can beneglected in a first-order approach. However it has been shown [15], [16], thatit can be determined separately, if necessary.S N,par = S ∣R ∣∣∣f=1HzR 2 = α = −99.2 dB (10)NA normalized PSD of -99.2 dB has been found experimentally for these twoparasitic resistances. The noise contribution of intrinsic gate charging resistors R isand R id is neglected for the first time. A method for its determination will be givenlater on.The model parameters of eqns. (6) to (9) are summarized in Table I. The Hoogeequation (1) and (9), (10), as well, already show the scaling of the noise properties.The number of free charge carriers N increases proportionally with the gate areaof FETs. Measurements at other MESFET and HEMT devices confirmed this fact.This means, that the normalized noise PSD decreases by 3 dB if the gate widthdoubles. Therefore, small devices (e.g. millimetre wave PHEMTs) suffer more fromLF noise.IV. LF NOISE GENERATION IN RESISTIVE MIXERSA single-ended resistive FET mixer has been used for the theoretical and experimentalinvestigations. The topology is given in Fig. 4.

54 G. BöckFig. 3. Extrinsic equivalent circuit of Fujitsu HEMT FHC40LG with lowfrequencynoise sources (hatched areas).The FHC40LG was mounted on a microstrip PCB. Coaxial bias tees perform gatebiasing as well as RF (>1 GHz) and IF (

Low-Frequency Noise in Resistive Mixers 55TABLE IMODEL PARAMETERS OF THE FUJITSU HEMT FHC40LGr ds R dsr R dsg k r v r k g v g2220 Ω 7.4 Ω -12.5/V -0.58 V 5.1/V -0.18 VC gd C gd1 C gd0 Q gd K gd40 fF 66 fF 3.5/V 0.95C gs C gs0 Q gs K gs v gs046 fF 6.9/V 3.16 -0.21 VS N,ch S 0 ,dB S 1 ,dB k n v 0-85 dB -25 dB 3/V 0.165Both contribute to the frequency conversion, and thus, all resulting mixing productsexhibit amplitude noise sidebands. Hence, the self-mixing product, whichresults in a DC current (mixing product with frequency zero) additionally createsLF noise. A strongly simplified calculation should illustrate the described mechanism.Applying the approach from (4), the total IF current (signal and noise) canexpressed asi IF = v LO,c · ˜g ch = v LO,c · g ch · (1 + ∆k)(11)= i| f=0Hz · (1 − ∆k) =i DC − i noisewhere i IF is IF current, v LO,c LO voltage coupled from gate to drain, ˜g ch totalchannel conductance, ∆k normalized fluctuating part of ˜g ch and g ch non-fluctuatingpart of channel conductance. Because g ch varies with the same frequency as v LO,c ,the multiplication creates a current whose frequency f is zero. As a consequence, aDC current i DC and a LF noise current i noise are generated. Although both currentsare proportional to the same quantities (e.g. the magnitude of v LO,c ), they areindependent of each other! The process can be compared also to a (noiseless)resistive down converting FET mixer pumped by an LO signal (at the gate) withAM noise sidebands. In this case, the FET channel varies not only periodicallywith the LO frequency, but fluctuates additionally in its amplitude. This noise isconverted to the low-frequency domain, too. In fact, because of the simplifications,there is no difference between both cases. The term ‘(1 + ∆k)’ in (11) describesthis random amplitude modulation.The low-frequency noise measurements have to be carried out very carefully.Otherwise it is not possible to distinguish between the two mentioned noise contributions.E.g., the AM noise of the LO source can be considerably lowered by alimiter, and from the frequency dependence of both noise contributions, criterionsfor the separation of both can be deduced [17].

56 G. BöckFig. 4. Single ended resistive FET mixer for theoretical and experimentalLF noise investigations.V. MIXER LF NOISE MODELUp to now, two ways of creating LF noise power stemming from the FET channelhave been discussed: The pure LO self-mixing process (according to (11)) and a DCcurrent flowing through the channel. The origin of both are the same fluctuationsof the drain-source (channel) resistance r ch . Therefore, they produce completelycorrelated noise voltages at a load resistance if they appear simultaneously. Thebehaviour can be modelled by the approach given in Fig. 5. The LF noise modelconsists of a DC voltage source V dc (producing the self-mixing DC current), the intrinsicnoisy drain-source resistance ˜r ch , a noise voltage source ∆v sm,ch (producingthe self-mixing LF noise of the FET channel), and a noise voltage source ∆v sm,Ri(noise contribution of the gate charging resistors), which we will discuss in the nextsection and neglect for now. If no DC current is flowing (switch S 1 in Fig. 4 open)only the noise voltage source generates LF noise (self-mixing LF noise). Hence,within this constellation the pure self-mixing noise can be measured. If switchS 1 is closed, a self-mixing DC current flows through the noisy channel resistanceand produces an additional, correlated LF noise voltage source ∆v ch .Itisveryinteresting, that this additional noise contribution reduces the total noise power atthe load resistance due to the opposite flow direction as indicated in Fig. 5.This results from a correlation coefficient of minus one between the two noisevoltages ∆v sm,ch and ∆v ch . Because of this reason but nevertheless surprisinglyon a first sight, the noise level (at the load resistance) with IF DC path is lower thanwithout DC path. This can be taken from Fig. 6 where simulated and measured PSD

Low-Frequency Noise in Resistive Mixers 57Fig. 5. Simplified LF noise model of the mixer IF output port with self-mixingproducts (LF noise and DC voltage).are given versus frequency. Similar investigations carried out at about 10 differentFETs/HEMTs showed in all cases an improvement of up to 15 dB in case of a DCpath. This result will be discussed in the next section quantitatively.VI. INFLUENCE OF DC CURRENT AND LOADING RESISTANCESBecause of the compensation effect described above it becomes clear, that theLF noise of the channel can be compensated completely by driving an additionalDC current into the drain electrode. This method can be used to reduce the lowfrequencynoise of a resistive mixer considerably. Not less important is the fact,that by application of this method a separation of channel noise from noise contributionof the loading resistors R is and R id becomes possible [18]: By varying theinjected drain DC current, a noise minimum occurs. The LF noise observed at thisminimum contains no noise from the channel because of the compensation of thetwo correlated channel noise contributions. The amount of DC current to achievethe compensation depends primarily on FET size, LO power and frequency and wasin all our experiments in the order of a few hundreds of µA. The remaining noisestems from the loading resistors R is and R id and the parasitic series resistances

@?= >0

Low-Frequency Noise in Resistive Mixers 59Fig. 7. LF noise PSD of the intrinsic gate charging resistors at 1 kHz, LOfrequency 9 GHz, LO power 0 dBm; lines: simulations, stars: measurement.Only the gate charging resistors contribute to the IF LF noise at this operating point.From this experiment it becomes obvious, that IF DC load plays an important role.Thus, the behaviour improves if a DC short circuit is used at the IF output port ofthe mixer (e.g. a large inductor or a parallel resonant circuit). The results in thiscase are shown in Fig. 9.The LF noise minimum becomes broader and appears at a more positive biasvoltage. The position of the minimum may be shifted towards a more negative gatevoltage by using a small resistance within the DC path (instead of a short).VII. CONCLUSIONA comprehensive analysis concerning low-frequency noise in resistive mixershas been performed. A noise model for cold FETs was developed using fluctuatingresistances of the channel and of the intrinsic gate charging resistors. Detailedinvestigations proved that this approach explains all phenomena existing in realityand that it is consistent with observations made by authors of other publications. Themodel uses universal methods that may be applied to all kinds of field-effect transistors(MESFET, HEMT, JFET, MOSFET). Principles of minimizing the LF noiseof single ended mixers are given. The work also revealed a new method to measurethe LF noise properties of the gate charging resistors.

60 G. BöckFig. 8. LF noise PSD at 1 kHz with dc output resistance 50 Ω, LO frequency9 GHz, LO power 0 dBm; lines: simulations, stars: measurement.Fig. 9. LF noise PSD at 1 kHz with dc output resistance 50 Ω, LO frequency9 GHz, LO power 0 dBm; lines: simulations, stars: measurement.

Low-Frequency Noise in Resistive Mixers 61REFERENCES[1] S.A.Maas,Microwave Mixers, Artech House Inc., 1986.[2] K. Schmidt von Behren, M. Tempel, G. Boeck, W. Schwartz, D. Leipold, “5–6 GHz ResistiveImage Reject Mixer in a 2.5 V CMOS Technology,” in IP2000 Europe – New Technologies for RFCircuits, Scotland, 23–24 October 2000.[3] S.A.Maas,“A GaAs MESFET Mixer with Very Low Intermodulation,” IEEE Trans. MicrowaveTheory Tech., vol. 35, no. 4, Apr. 1987.[4] F. Ellinger, R. Vogt, W. Baechtold, “Compact Monolithic Integrated Resistive Mixers With LowDistortion for HIPERLAN,” IEEE Trans. Microwave Theory Tech., vol. 50, no. 1, Jan. 2002.[5] R. Circa, D. Pienkowski, G. Boeck, “Double Balanced Resistive Mixer for Mobile Applications,”Proc. 15th Int. Microwave Conf. (MIKON-2004), Poland, Warsaw, May 17–19, 2004, pp. 347–350.[6] U. Schaper, A. Schäfer, A. Werthof, H. J. Siweris, H. Tischer, L. Klapproth, G. Böck, W. Kellner,“70–90 GHz balanced resistive PHFET mixer MMIC,” Electron. Letters, vol. 34, no. 14, pp. 1377–1379, July 1998.[7] H.J.Siweris,H.Tischer,“Monolithic Coplanar 77 GHz Balanced HEMT Mixer with very SmallChip Size,” in IEEE MTT-S Int. Microwave Symp. Dig., pp. 125–128, 2003.[8] R. Müller, Rauschen, 2nd ed, Berlin, Heidelberg, New York, London, Paris, Tokyo: Springer, 1990.[9] X.-Y. Chen, “Lattice scattering and 1/f noise in semiconductors,” Ph.D. dissertation, TechnischeUniversiteit Eindhoven, 1997.[10] F. N. Hooge, “1/f Noise Sources,” IEEE Trans. Electron Devices, vol. 41, no. 11, pp. 1926–1935,Nov. 1994.[11] A. van der Ziel, “Unified Presentation of 1/f Noise in Electronic Devices: Fundamental 1/f NoiseSources,” Proc. IEEE, vol. 76, no. 3, Mar. 1988.[12] P. H. Handel, “The Nature of Fundamental 1/f Noise,” in Conf. Proc. “Noise in Physical Systemsand 1/f Fluctuations”, American Institute of Physics, 1993.[13] F. N. Hooge, T. G. M. Kleinpenning, and L. K. J. Vandamme, “Experimental studies on 1/f noise,”Rep. Prog. Phys., vol. 44, 1981.[14] M. Margraf and G. Boeck, “A New Scaleable Low Frequency Noise Model for Field-EffectTransistors Used in Resistive Mixers,” in IEEE MTT-S Int. Microwave Symp. Dig., pp. 559–562,June 2003.[15] J.-M. Peransin, P. Vignaud, D. Rigaud, and L. K. J. Vandamme, “1/f Noise in MODFET’s atLowDrain Bias,” IEEE Trans. Electron Devices, vol. 37, no. 10, pp. 2250–2253, Oct. 1990.[16] J. Berntgen, K. Heime, W. Daumann, U. Auer, F.-J. Tegude, and Matulionis, “The 1/f Noise of InPBased 2DEG Devices and Its Dependence on Mobility,” IEEE Trans. Electron Devices, vol. 46,no. 1, pp. 194–203, Jan. 1999.[17] M. Margraf, “Niederfrequenz-Rauschen und Intermodulationen von resistiven FET-Mischern”,Ph.D. dissertation, Dept. Elect. Eng., Berlin Univ. of Technology, Berlin, Germany, 2004.[18] M. Margraf and G. Boeck, “1/f Noise Optimum for Field-Effect Transistors in Single-EndedResistive Mixers,” in 33rd European Microwave Conf. Dig., Munich, Germany, Sept. 2003,pp. 1015–1018.

63RF Noise Model for CMOS TransistorsI. Angelov, M. Ferndahl, A. MasudMicrowave Electronics Laboratory,Chalmers University of Technology,SE-412 96 Göteborg, SwedenAbstractThe RF Noise model for submicron CMOS transistors was proposed andimplemented in commercial CAD tool. The model was experimentally evaluatedand shows a good correspondence between the measurements and the model.I. INTRODUCTIONTHE refinement of the technology and scaling the gate to submicron size greatlyimproved the RF performance of the CMOS transistors. Now these transistorscan operate at frequencies up to 60 GHz showing very promising performance. Animportant issue is existence of easy to understand and extract models for thesetransistors. Despite the maturity of the subject still there is a need for simple,compact, well converging Large-Signal Model (LSM) for CMOS transistors. Forthe circuit designers it is important to unite the noise model with the LSM in orderto make the design easier and more complete.II. CMOS RF NOISE MODELA good candidate for a frame which can be used for the CMOS noise modelare the noise model ideas proposed by Marian Pospieszalski [1]. The basic ideais that the channel resistance (i.e. respective drain current) is producing morenoise than it should if this was only a Johnson noise. This can be reflected witha higher temperature of the channel resistance T d . In the ordinary case, whendealing with a good quality transistors, the gate current is small and the gate partresistances contribute noise, which can be considered as the noise generated fromthese resistances at ambient temperature T g .His approach is intended to be used in a small-signal case, but if proper biasdependencies of gate part T g and T d for channel resistance are found, then this canbe transformed to work with large-signal cases. For submicron CMOS devices thenoise power P Pout generated by the I ds and from gate currents I gs , I gd , can bearranged in the following way:

64 I. Angelov et al.CgdpGateLgRgRgdCgdRdLdDrainCthermRthermTg RiCgsCC10C=CgsTdCdsCrfCrfVrfCinRFRinRFRsLsSourceFig. 1.Equivalent circuit of the CMOS.I dn1 = |I ds | + |I gs | (1)T di = T d · [1 + tanh(|V ds − V kn |)] (2)P Pout = L w 4 kT amb∣ ∣∣∣ T diT amb· I dn1 + T d1 (I dn1 − I dnmin ) 2 ∣ ∣∣∣(1 + K LF0 )()1K LF0 = K LF11+f F + K LF2fe1+(f/f gr ) 2〈|i gs | 2 I A fgs〉 = 2q e I gs ∆f + K ff F (5)fe ∆fT g,Ri = T g (1 + tanp) tanh(α p V ds )(1 + λV ds ) (6)(3)(4)I A f〈|i gd | 2 gd〉 = 2q e I gd ∆f + K ff F (7)fe ∆fwhere T d is the hot temperature of the channel resistance, T g is the gate temperature,K f , A f , F fe are parameters responsible for the low frequency noisecontributions.The drain current equations for the LS model are as they are described in [2].There are several important issues with the sub-micron CMOS noise model:

RF Noise Model for CMOS Transistors 65Fig. 2.Measured and modelled minimum noise figure vs. frequency.Fig. 3.Measured and modelled noise figure vs. bias.1) There is a stronger dependence of the noise figure vs. gate bias voltage. I.e.there is an optimum drain current at which we have minimum of the noisefigure. The deviation from this optimum bias are more pronounced and aremodelled with the coefficients T d1 and I dnmin .2) The gate resistances R g , R i are larger for the submicron CMOS devices andthey contribute much more noise in comparison with the ordinary GaAs FET.That is why, the noise generated from the gate part cannot be consideredbias independent and temperature of R i should be bias dependent. There isa stronger dependence of the noise on the V ds which can be attributed to hot

66 I. Angelov et al.electron effects.The noise model was implemented in ADS as a user defined model and evaluatedexperimentally. Figs. 2 and 3 show the bias dependence of the measured and modelledRF noise. Despite the simplicity, the model is quite accurate. The evaluationwith amplifiers in the frequency range 20–40 GHz shows that the model can beused in the practical work for RF designs.III. CONCLUSIONSA simple noise model for advanced CMOS RF devices was proposed and implementedin a commercial CAD tool. The model shows a good agreement betweenthe measurements and simulations and was evaluated with several designs of highfrequency CMOS amplifiers.REFERENCES[1] M. W. Pospieszalski, “Modeling of noise parameters of MESFET’s and MODFET’s andtheirfrequency and temperature dependence,” IEEE Trans. Microwave Theory Tech., vol. 37, no. 9,pp. 1340–1350, Sept. 1989.[2] I. Angelov, M. Fernhdal, F. Ingvarson, H. Zirath, H. O. Vickes, “CMOS large signal model forCAD,” in IEEE MTT-S Int. Microwave Symp. Dig., pp. 643–646, 2003.

67Extremely Low-Noise Amplification withCryogenic FET’s and HFET’s: 1970-2004Marian W. PospieszalskiNational Radio Astronomy Observatory ∗2551 Ivy Road, Charlottesville, VA 22901, USAhttp://www.nrao.edu, e-mail: mpospies@nrao.eduAbstractImprovements in the noise temperature of field-effect transistors (FET’s) and,later, heterostructure field-effect transistors (HFET’s) over the last several decadeshave been quite dramatic. In 1970, a noise temperature of 120 K was reported at1 GHz and physical temperature of 77 K; in 2003, noise temperatures of 2, 8 and35 K were reported at 4, 30 and 100 GHz, respectively, for physical temperaturesof 14 to 20 K. The paper reviews developments in this field and attempts to identifyimportant milestones within the broader context of technological developments.Examples of experimental results obtained with different generations of FET’s(HFET’s) are compared with the model predictions. Some gaps in our currentunderstanding of experimental results are emphasized, and some comments onpossible future developments are offered.I. INTRODUCTIONTHE QUEST for ultra-low-noise reception is as long as the history of radiocommunication. A list of devices which at one time provided the lowest reportednoise temperature in some frequency band is very long: vacuum tubes, crystalmixers, tunnel diode amplifiers, parametric amplifiers, solid-state masers, Schottkydiode mixers, superconductor-insulator-superconductor (SIS) mixers, GaAs fieldeffecttransistors (FET’s) and heterostructure field-effect transistors (HFET’s), hotelectron bolometers (HEB’s), etc. Okwit gives a very interesting historical reviewof the pre-1970’s evolution of low-noise concepts and techniques [1]. Relevantinformation concerning developments in the 1960’s and 1970’s can also be foundin special issues of IEEE MTT Transactions on Noise (September 1968) and onLow Noise Technology (April 1977).Three of the low-noise devices mentioned in the previous paragraph, namelysolid-state masers, SIS mixers, and HEB’s, will operate only at cryogenic temperatures.Most other devices (with the exception of vacuum tubes!) have also beencooled to cryogenic temperatures for two main reasons: improvement in the device∗ The National Radio Astronomy Observatory is a facility of the National Science Foundation operatedunder cooperative agreement by Associated Universities, Inc.

68 M. W. PospieszalskiFig. 1. Equivalent circuit of a FET (HFET) chip. The parasitic elements areshown disconnected from the intrinsic chip. Noise properties of the intrinsicchip are represented by equivalent temperatures T g of r gs and T d of g ds .Noisecontributions of ohmic resistances r s, r g,andr d are determined by physicaltemperature T a of a chip.performance, usually due to the improvement of the electron transport properties,and reduction of the influence of thermal noise generated by parasitic elements [2],[4], [6].Cryogenic cooling of receivers to reduce their noise temperature is especiallyimportant in satellite and space communication, and in radio astronomy. This isbecause the antenna noise is determined by celestial sources and the atmosphere.In the absence of strong celestial sources (Sun, Moon, planets, Cassiopeia, Cygnus,Taurus, Virgo, Orion, galactic plane) in an antenna beam, an antenna “looks” ata very cold sky: 2.725 K of the cosmic microwave background radiation modifiedby the presence of atmosphere [3]. The antenna temperature is typically one orderof magnitude or more less than in terrestrial applications (300 K). Consequently, areceiver noise temperature typically would constitute a large part of a system noisetemperature, and its reduction by cryogenic cooling offers a very effective way ofimproving receiver sensitivity.In the early 1970’s, the ultra-low-noise receiving systems for deep space and radioastronomy employed mainly solid-state masers, cryogenically-cooled parametricamplifiers (or converters) and Schottky diode mixers. At the end of that decade,advances in GaAs FET technology made the noise performance of GaAs FET amplifierscompetitive with the performance of parametric amplifiers [5]. Also, a new mixingelement, the SIS tunnel junction capable of almost quantum-limited detection,

Extremely Low-Noise Amplification ... 69Fig. 2. Example of measured and model-predicted noise parameters of asample MESFET. Points marked by “∗”, “×”, “◦” and“+” are for T min ,X opt , R opt ,andg n, respectively. Lines are for model prediction.was developed [7]–[10]. In the 1980’s and 1990’s, FET amplifiers were graduallyreplaced by HFET amplifiers of different generations. The first generation was usingAlGaAs/GaAs HFET’s, the second generation AlGaAs/InGaAs/GaAs HFET’s andthe third AlInAs/InGaAs/InP HFET’s [2], [4], [55]. These amplifiers have becomethe low-noise technology of choice for frequencies up to W-band (3 mm), althoughruby masers are still sometimes used in Deep Space Network antennas at X- andK a -band frequencies [11]. At W-band frequencies, HFET receivers now competein performance with SIS/HFET mixer-preamplifiers. At frequencies above 120 GHzup to about 1 THz, SIS mixers demonstrate the best noise performance. Above1 THz, cooled Schottky diode mixers and HEB mixers provide the lowest noisetemperatures [12].Section II of this paper reviews noise models of FET’s andHFET’s andtheirexperimental verification from the point of view of their application at cryogenictemperatures. Progress in the noise performance of cryogenic FET’s andHEMT’sfrom the first attempts at cryogenic cooling in 1970 until 1993 is addressed inSection III. This section covers the milestones demonstrated with FET’s andtwogenerations of HFET’s, conventional and pseudomorphic, all based on GaAs substrates.Certain rather old results are interpreted from the point of view of theoreticalunderstanding developed much later. Also, the issue of what makes a good cryogenicdevice is addressed, and certain gaps in our understanding of HFET properties atcryogenic temperatures are pointed out. Finally, Section IV covers developments

70 M. W. PospieszalskiFig. 3. Equivalent gate and drain temperatures versus drain current for thesample MESFET in Fig. 2. Data are given for three different devices denotedby different symbols [35].from 1993 to the present. This period coincides with the introduction of InP HFET’sinto cryogenic receiver technology. Some thoughts on possible future developmentsare offered in Section V.II. NOISE MODELS OF FET’S AND HFET’S AND THEIR EXPERIMENTALVERIFICATIONThe noise performance of field-effect transistors (FET’s) was first modeled byA. van der Ziel [13] and has since been a subject of intensive study. Publishedstudies of noise properties of FET’s (HFET’s) may be divided into two distinctivegroups. The first group, as a starting point of analysis, considers the fundamentalequation of transport in semiconductors [13]–[23]. Most papers in this category publishedover the years may be viewed as progressively more sophisticated treatmentsof the problem originally tackled by van der Ziel [13], [14], [22]. Although theHFET’s wafer structure is different than that of a MESFET, the methods employedin noise studies were basically the same [19], [21], [23], [24] as those applied toMESFET’s [17], [20], [23], [24]. To the best of the author’s knowledge, none ofthese models have been used to explain the performance of FET’s at cryogenictemperatures.The second group of published studies [25]–[35], [40]–[44] addresses the issueof what needs to be known about the device, in addition to its equivalent circuit, topredict noise performance. Until the 1990’s, the most often used method was thesemi-empirical approach originated by Fukui [25]–[27] in which relations between

Extremely Low-Noise Amplification ... 71Fig. 4. Minimum noise temperature T min (at 10 GHz), equivalent gatetemperature T d , and intrinsic cut-off frequency f T as a function of drain currentforthesampleMESFETinFigs.2and3[35].the minimum noise figure at a given frequency and the values of transconductanceg m , gate-to-source capacitance C gs , and source and gate resistances r s and r g ,areestablished (Fig. 1). A quantitative agreement may be obtained only after the properchoice of fitting factor [25], [26], [28] or fitting factors [29]. The extension of thisapproach to other noise parameters [26] results quite often in non-physical twoport[27]. The Fukui approach, although very widely used by device technologists,does not provide any insight into the nature of the noise-generating mechanism inaFETasthefitting factors do not possess physical meaning. Nevertheless, devicetechnologists pursuing the goal of lowest-noise FET relied on the semi-empiricalexpression of Fukui [25], [26] relating the minimum noise temperature with theelements of an equivalent circuit of a FET:√fT min ≈ T 0 K f g m (r g + r s ) (1)f Twhere f is the frequency, T 0 is the standard temperature 290 K, f T is the intrinsiccut-off frequency, g m is the transconductance, r g and r s are gate and drain parasiticresistance and K f is the fitting factor, assuming values between 1.2 and 2.5 [24].Over the years, the most referenced treatment of signal and noise propertiesof a MESFET is that published by Pucel et al. [17]. It quite often serves as abridge between different approaches to noise treatment as many other results are

72 M. W. PospieszalskiFig. 5. The minimum noise temperature T min (at 40 GHz), dc measuredtransconductance and equivalent drain temperature vs. drain current at theambient temperature T a =18Kof0.1 × 50 µm InP HEMT [49].compared to it or adopt similar computational techniques [19]–[21], [23], [24],[26]. The method of Pucel et al. [17] calls for three frequency-independent noisecoefficients P , R, andC to be known in addition to small-signal parameters of anintrinsic FET in order to determine four noise parameters at any given frequency.The model proposed by the author [33], [34] has been shown to describe very wellthe noise properties of FET’s versus frequency, temperature and transistor bias [33]–[47]. The principle of this model is illustrated in Fig. 1 which shows the equivalentcircuit of a FET chip with its noise sources. Parasitic resistances contribute onlythermal noise and, with knowledge of the ambient temperature T a , their influencecan easily be taken into account. The noise properties of an intrinsic chip are thentreated by assigning equivalent temperatures T g and T d to the remaining resistive(frequency-independent) elements of the equivalent circuit r gs and g ds , respectively.No correlation is assumed between noise sources represented by the equivalenttemperatures T g and T d . Consequently, in addition to elements of an equivalentcircuit T g , T d ,andT a need to be known to predict all four noise parameters atany frequency in the frequency range in which 1/f noise and noise caused by thegate leakage current are negligible. For the detailed treatment, the reader is referredto the original papers. However, for the purpose of the discussion in this paper,it is useful to recall the approximate expressions for four noise parameters of an

Extremely Low-Noise Amplification ... 73intrinsic chip:R opt∼ =f TfX opt =√r gs T gg ds T d(2)1(3)ωC gswhich are valid ifwhereorg n =( ff T) 2g ds T dT 0(4)T min∼ = 2ff T√gds T d r gs T g (5)T min∼ =ff max√Td T g (6)ff T≪√T gT d1r gs g ds(7)g mf T =(8)2πC gs√1f max = f T (9)4 g ds r gsand R opt and X opt are the real and imaginary parts of the optimum source impedance,T min is the minimum effective noise temperature of a chip, g n is the noise conductance,f T is the transistor cut-off frequency, and f max is the maximum frequency ofoscillations. Since the model’s introduction in 1988, its validity has been verified bya number of researchers in the field, among those are works by researchers from theFraunhofer Institut [41], [42], Ferdinand-Braun-Institut [43], [44], and the ChalmersUniversity of Technology [45]–[48]. An example of a typical fit between measuredand modeled noise parameters based on the authors own work [35] is shown inFig. 2. A direct confirmation of the model’s validity at cryogenic temperatures hasbeen given in only two studies [33], [34], [47], although there have been a numberof published papers demonstrating an excellent agreement between measured andmodeled results for the amplifiers designed under the assumption of model validity[36]–[39], [47]–[54].There are several important observations concerning the model’s behavior. Theseare illustrated in Table I [34] and Fig. 3 [35], Fig. 4 [35] and Fig. 5 [48]. First,the equivalent drain temperature is linearly dependent on the drain current and,

74 M. W. PospieszalskiTABLE ICOMPARISONS OF NOISE PARAMETERS OF FHR01 INTRINSICCHIP(f =8.5 GHZ)T a Comment T min R opt X opt g n T g T dK K Ω Ω mS K KMeasured 65.6 26.3 59.5 3.0 — —297 Model Best Fit for r gs =2.5Ω 58.7 28.4 66.9 3.27 304 5514Model Best Fit for r gs =3.5Ω 59.6 28.2 66.9 3.24 210 5468Measured 8.2 11.4 65.2 0.80 — —12.5 Model Best Fit for r gs =2.5Ω 7.4 12.3 66.9 0.87 14.5 1406Model Best Fit for r gs =3.5Ω 7.7 12.0 66.9 0.85 9.3 1379within measurement errors, a single function may describe its dependence for alldevices having the same gate length, the same semiconductor structure and thesame current density per unit gate width. Second, the equivalent drain temperature isusually not a strong function of the device ambient temperature, except for very lowdrain current densities per unit gate width. Third, the equivalent gate temperature islightly dependent on drain current and, within measurement errors, is equal to theambient temperature of a device. This observation is clearly illustrated by the dataof Table I (obtained for an AlGaAs/GaAs HFET) and Figs. 3 and 4 (obtained for aGaAs MESFET) and was recently corroborated by data published by Angelov et al.(obtained for a PM HFET) [47]. Another example of the dependence of minimumnoise temperature, transconductance, and equivalent drain temperature on the draincurrent for a modern InP device at cryogenic temperatures [48], [49] is shown inFig. 5. The values of transconductance were dc measured, and the values of theminimum noise temperature were measured at 40 GHz. In this case, it was assumed,in computing the values of equivalent drain temperature, that the only element ofthe HFET equivalent circuit varying with bias and temperature was transconductanceg m . This approach, necessarily oversimplified in the absence of sufficientlyaccurate cryogenic S-parameter measurements, nevertheless demonstrates a verysimilar dependence of T d vs. drain current as observed in much more accurateroom temperature experiments. This data was used to develop a series of cryogenicamplifiers [49]–[53], and an excellent agreement between measured and modeledresults (see Section IV) over a broad temperature range indirectly confirms thisapproach.As was mentioned earlier, the model of Pucel et al. requires knowledge of threefrequency-independent parameters, R, P ,andC [17], in addition to knowledge ofa FET equivalent circuit to predict noise parameters at any frequency. The modeldeveloped by the author requires two frequency-independent parameters T g and T d .Although the starting points in developing both models are entirely different, both

Extremely Low-Noise Amplification ... 75Fig. 6. Minimum noise temperature T min at 8.4 GHz and dc measuredtransconductance g m at 297 K and 12.5 K for two different General ElectricAlGaAs/GaAs HFET’s with and without spacer layer. The devices have similarroom temperature performance but vastly different cryogenic performance [57].Fig. 7. Comparison of g m(I d ) characteristics for three HFET’s from a singlewafer having very similar room temperature performance and very differentcryogenic performance [59].can be made equivalent if one of the parameters in Pucel’s model is determined bythe other two as given by equation:√RC =P(10)The experimental evidence confirming that relationship since it was first notedin 1989 [33], [34] has been quite overwhelming, and papers by P. Heymann etal. [43],I.Angelovet al. [47] and P. Tasker et al. [42] are especially noteworthy.Furthermore, experimental evidence that one of the parameters, the equivalent gatetemperature, follows within the accuracy of experiments the ambient temperatureof a FET device leads to a noise model with only one parameter T d . In fact, thisis the only approach successfully used in the design of amplifiers at cryogenictemperatures [36]–[39], [45], [48]–[53].It is interesting to note that device technologists, in their efforts to improve thenoise performance of FET’s in the 80’s and early 90’s, relied mostly on the guidancegiven by the Fukui equation (1). Their progress was achieved by addressing twoissues in FET (HFET) design:1) maximization of intrinsic cut-off frequency f T = g m /(2πC gs ),and

76 M. W. PospieszalskiFig. 8. Noise temperature and gain of typical Voyager/Neptune amplifier at14 K. The resulting noise temperature of the whole receiver is also plotted forcomparison [58].2) minimization of parasitic resistances of gate and source, r g and r s , respectively.The first issue was addressed by progress in the technology of artificially structuredsemiconductors on which FET structures are built, and progress in the definitionand fabrication of submicrometer length gates (see, for instance, [55]). Theepitaxial GaAs material used exclusively for low-noise FET fabrication in the 1970’swas replaced by progressively more complex heterostructures: AlGaAs/GaAs, Al-GaAs/InGaAs/GaAs and AlInAs/ GaInAs/InP. The gate length of low-noise GaAsFET’s in the 1970’s was about 1 µm [18]and< 0.1 µm in the 1990’s [55]asaresult of the development of electron beam lithography.The second issue was addressed by improvements in FET layout, fabricationof “mushroom” or T-shaped gates, reduction in drain-to-source separation (“selfaligned”ohmic contacts) and progress in the technology of ohmic contacts [55].A corresponding approximate expression for the minimum noise temperature ofa FET chip, which is based on equation (5) can be written as:T min∼f √= 2 rt T g g ds T d (11)f Twhere r t = r s + r g + r gs . Thus, both expressions (1) and (11) can explain theimprovement in noise temperature resulting from technological improvements. Expression(11), however, allows for a deeper insight into how the FET bias affects theminimum noise temperature and how its value is minimized. Only T d and f T are

Extremely Low-Noise Amplification ... 77Fig. 9. Photograph of 8.4 GHz Voyager/Neptune amplifier with coverremoved [58].Fig. 10. Example of noise temperature of AlGaAs/GaAs GE HFET at 8.4 GHzamplifier vs. ambient temperature for two different bias currents illustratingunexpected behavior around 175 K.strong functions of transistor bias. Consequently, the bias optimal from the point ofview of minimum noise temperature at any frequency is that minimizing the valueof√Td g dsf(V ds ,I ds )=(12)f Tor alternativelyf(V ds ,I ds ) ∼ =√Idsg m(13)

78 M. W. PospieszalskiFig. 11. Comparison of model-predicted and measured performance of 40–45 GHz amplifier at 297 K. The amplifiers were built in 1991 for the VeryLarge Baseline Array Q-band receivers with 0.1 µm gate length pseudomorphicHFET’s.as C gs is not a strong function of gate bias (and drain current I ds )andT d isto a first approximation proportional to I ds . This observation is clearly illustratedin both Figs. 4 and 5. Consequently, an excellent low-noise FET, in addition tohaving parasitic resistances as small as possible, should have as large as possiblecut-off frequency f T at as small as possible current I ds . This property has longbeen observed, especially at cryogenic temperatures, and is sometimes referred to as“quality of pinch-off.” It even served as a criterion for a selection of good cryogenicdevices even before the nature of this relationship was understood [57]–[59].III. CRYOGENIC FET’S AND HFET’S: 1970–1993The first attempt at investigating the noise properties of GaAs FET’s at cryogenictemperatures was described in [60] in 1970 by Loriou et al. The experiment wasdone at the frequency of 1 GHz. The change in noise temperature T e from 360 Kat room temperature to 120 K at the temperature of 77 K was observed. Two yearslater somewhat similar results of T e =93K at 77 K ambient temperature at 1.5 GHzwere reported [61]. The next important results were published in 1976 when Liechtiand Larrick [62] reported T e =60KatT a =90K at 12 GHz for a device andT e = 130 K for a three-stage, 31 dB gain amplifier at T a =60K.A more systematic study of the cryogenic noise behavior of FET’s was undertakenby Weinreb and collaborators [63], [64] who investigated the noise propertiesof GaAs FET’s at the very important radio astronomy frequencies in L- and C-

Extremely Low-Noise Amplification ... 79Fig. 12. Comparison of model-predicted and measured performance of 40–45 GHz amplifier at 18 K. The amplifiers were built in 1991 for the VeryLarge Baseline Array Q-band receivers with 0.1 µm gate length pseudomorphicHFET’s.bands. In 1980, noise temperature T e =20K was reported at 5 GHz and 20 Kambient temperature; in 1982, T e =7K was reported at 1.4 GHz and 20 K ambienttemperature.In 1984, the author undertook the study of noise parameters of different commercialFET’s at X-band [65]. A noise temperature of 20 K was repeatedly demonstratedfor packaged commercial FET’s (Fujitsu FSC10FA) at the frequency of 8.4 GHz andambient temperature of 14 K, although noise temperatures as small as 15 K wereobserved for individual devices under the same conditions [65].The first experimental HFET using AlGaAs/GaAs heterostructure was demonstratedby several laboratories in 1980 [55]. The first cryogenic testing of a 0.25 µmgate length HFET, developed by Cornell University, was performed at NRAO in1985 [66] and a record low-noise performance of T e =10K at 8.4 GHz was reported[66]. Shortly thereafter, a consortium of the NRAO Central Development Laboratory(CDL), GE’s Military Electronics Division and the Jet Propulsion Laboratory starteda program to develop very low-noise cryogenic HFET’s for the purpose of equippingthe NRAO Very Large Array (VLA) with low-noise receivers to provide receptionfor the Voyager spacecraft during its encounter with Neptune during August 1989.This very successful program produced devices with T e =6K at 8.4 GHz [57]–[59].A group of studies published by the author between late 1984 and early 1988[57]–[59], [65]–[68] identified a number of effects observed at cryogenic temperaturesfor which only some, later on, had satisfactory explanations.

80 M. W. PospieszalskiFig. 13.Photograph of 40–45 GHz VLBA amplifier with cover removed.It was observed that room-temperature performance is not always a good indicatorof cryogenic performance. It certainly can be understood in terms of the modelpresented in Section II and especially equation (11). Relatively large parasiticresistances could mask otherwise excellent noise behavior of an intrinsic chip atroom temperature. Excellent noise properties of an intrinsic chip could only berevealed at cryogenic temperatures as the thermal sources are reduced by as muchas a factor of 25.It was observed that vastly different performance at cryogenic temperatures fordevices with similar room-temperature performance can usually be traced to poorpinch-off characteristics at cryogenic temperatures. It is interesting to recall the dataon the GE HFET’s shown in Fig. 6 and published in 1986 [57]. Devices with verysimilar room-temperature performance differed greatly in their transconductancevs. drain current characteristics at cryogenic temperatures and small currents. Thisobservation proved to be extremely useful in diagnosing amplifier performance,even in the case where the amplifiers were built with devices from the same wafer.As an example, a set of g m vs. I ds characteristics for several devices from thesame wafer, taken at room and cryogenic temperatures, with corresponding values ofmeasured minimum noise temperature, are shown in Fig. 7 [59]. If the performanceof an amplifier was subpar from the point of view of noise temperature, the dccharacteristics of a first-stage device were measured. That simple dc measurementallowed for a screening of bad cryogenic performers, irrespectively of any otherproblem possibly arising with amplifier construction. Obviously, this behavior cannow be easily interpreted in terms of the noise model and its dependence on bias

Extremely Low-Noise Amplification ... 81Fig. 14. A minimum noise measure of 0.1 µm gate length AlInAs/GaInAs/InPHFET [38]. The best experimental results from different laboratories are alsoshown [72]–[74].as discussed in Section II. As follows from equation (13), a larger drain current fora given transconductance indicates higher minimum noise temperature.A typical performance of Voyager/Neptune amplifiers is shown in Fig. 8, whilea photograph of the amplifier itself is shown in Fig. 9. It is interesting to note thatthe VLA 8.4 GHz system, which even today is considered to be the most sensitiveradio astronomy instrument in the world, is still operating using amplifiers designedalmost 20 years ago.It has been observed that the dc, RF, and noise performances of AlGaAs/GaAsHFET’s at cryogenic temperatures are sensitive to light [57]–[59], [65]–[68]. Almostall devices, if kept under dark conditions, exhibited memory at cryogenic temperatures.That is to say, their performance depended not only on current bias conditionsbut also on the device history, for example, previous bias conditions and temperature.Quite often, a device, if kept dark at cryogenic temperatures, would exhibit anincrease in noise temperature on time scales of hours and days, showing degradationin noise of 50 percent and more with no significant change in RF characteristics [57],[58], [65], [66]. Furthermore, the characteristic of minimum noise temperature vs.ambient temperature would show an anomalous behavior in the temperature rangeof 150 to 175 K. This effect, sometimes referred to as a “camel-hump” effect,is illustrated in Fig. 10. It was present in all AlGaAs/GaAs HFET’s, although

82 M. W. PospieszalskiFig. 15. An illustration of the influence of the gate-to-drain leakage currentgenerating pure shot noise on the minimum noise measure of an InP HFETwith 0.1 µm × 80 µm gate dimensions [69] at T a =18K.not observed in GaAs FET’s. None of these effects were satisfactorily explained.These effects are thought to be related to a charge-trapping mechanism which hasbeen reported to be the cause of the collapse of I-V characteristics at cryogenictemperatures. Trapping, however, was never considered as a possible influence onnoise performance at frequencies as high as X-band, and a convincing explanationof these effects still remains a mystery. An explanation of the phenomena at thattime did not seem to be of particular priority as these were present only outsideof normal operating conditions and illumination with red light made the devices“well-behaved” at cryogenic temperatures [57]–[59], [65]–[68].Conventional HFET’s with about 0.2 µm gate length, which became available inthe 1980’s, allowed for construction of cryogenic amplifiers with sufficient gain andnoise temperature to be competitive with other low-noise technologies up to K a -band frequencies [36], [58], [59]. By the end of that decade, two great technologicaladvances had been made. First, the concept of a pseudomorphic HFET, introduced in1985, was reduced to practice. Second, the manufacture of gates as short as 0.1 µmbecame possible [55]. Pseudomorphic devices using 0.1 µm gate length made lownoiseamplification possible up to 95 GHz, although for cryogenic applicationsthe noise performance at W-band frequencies was not yet competitive with wellestablishedSIS mixer technology [38]. Pseudomorphic HFET’s with 0.1 µm gatelengths made possible the development of cryogenic receivers covering 40–45 GHzfor the Very Large Baseline Array (VLBA) [38], [69]. Typical characteristics of aNRAO-designed amplifier developed in 1991, its gain and noise performance, and

Extremely Low-Noise Amplification ... 83Fig. 16. A comparison of measured gain and noise characteristics of a W-band amplifier with model prediction at room temperature. Measured noisetemperature includes the contribution of pyramidal horn and receiver (T r =2000 K).a comparison between measured and modeled results are shown in Figs. 11 and 12.A photograph of a completed amplifier having WR22 waveguide input and outputis shown in Fig. 13.Since 1991, the pseudomorphic AlGaAs/InGaAs/GaAs HFET’s have replacedconventional AlGaAs/GaAs HFET’s in all commercial low-noise applications. However,these devices also suffer from a similar lack of repeatability of performance atcryogenic temperatures, as was the case with conventional HFET’s. For example,the best noise performance at that time, measured at 40 GHz using 0.1 µm PM-HFET’s, was about 20 K [38], but it was not uncommon to measure twice thisnumber for devices with very similar room-temperature performance (T e ≈ 200 K)from other manufacturers.IV. INP HFET’S ATCRYOGENIC TEMPERATURESA promise of excellent microwave performance from an AlInAs/InGaAs/InPHFET, usually referred to as InP HFET, was demonstrated in 1987 [70]. Two yearslater, Mishra and colleagues at the Hughes Research Laboratories, incorporating thetechnology of 0.1 µm-long mushroom T-gates and AlInAs/InGaAs/InP wafer structure,shattered records for low-noise performance at room temperature [71]. They

84 M. W. PospieszalskiFig. 17. A comparison of measured gain and noise characteristics of a W-bandamplifier with model prediction at cryogenic temperature T a =20K. Measurednoise temperature includes the contribution of dewar window, pyramidal hornat T a =20K, and room temperature receiver (T r = 2000 K).demonstrated a noise figure of 0.9 dB (67 K noise temperature) at 63 GHz. Severallaboratories in 1990 and 1991 demonstrated similar results and as little as 1.2 dBnoise figure (93 K noise temperature) at 94 GHz [72]–[74]. In 1991, the authorpredicted the behavior of noise performance of these InP HFET’s vs. temperature[38]. It was possible by combining the knowledge of an equivalent circuit of a stateof-the-artInP HFET with the knowledge of equivalent gate and drain temperaturesvs. temperature and current gained from evaluation of PM HFET’s with 0.1 µmgate length (which was at that time routinely used at NRAO). The results werepublished in [38] in the form of graphs of minimum noise measure vs. frequencyexpected of InP HFET’s at different cryogenic temperatures. The minimum noisemeasure of a device determines the minimum possible noise temperature that canbe exhibited by an amplifier with sufficiently large gain using this device. Theresults of this calculation, done in 1991, are shown in Fig. 14. The prediction ofattainable noise temperatures at cryogenic temperatures for InP HFET amplifiershas held up remarkably well over the last decade. Some recent experimental resultswere included in this figure to illustrate this point.Generally, InP HFET’s at cryogenic temperatures are much less sensitive toillumination than conventional HFET’s. These devices do not seem to exhibit anymemory at cryogenic temperatures if not illuminated. The unrepeatable noise performanceat cryogenic temperatures can usually be traced to the behavior at pinchoff,in the similar way as it was discussed for conventional devices, and/or to thepresence of gate leakage. For InP HFET’s, the gate leakage at room temperature is

Extremely Low-Noise Amplification ... 85Fig. 18.Photograph of W-band MAP amplifier with cover removed.of the order of several µA, and for good devices decreases at cryogenic temperaturesby at least an order of magnitude. However, if the gate current of the order of severalµA is still present at cryogenic temperatures, it can greatly influence the cryogenicnoise performance to the point that it may completely erase any advantage thatan InP HFET may have over a conventional or PM device. An illustration of thisobservation is shown in Fig. 15 which shows a comparison of the minimum noisemeasure of 80 µm-wide devices [69] computed with and without the presence ofgate leakage current. For the model, it was assumed that the noise influence of thegate leakage could be represented by an ideal shot noise current source.Current knowledge about noise and signal models of “well-behaved” InP HFET’sat cryogenic temperatures is sufficiently accurate to allow for computer-aided designof cryogenic amplifiers with optimal, according to some criterion, noise bandwidthperformance. Usually an amplifier has to satisfy other requirements, such as inputreturn loss, unconditional stability, and minimum gain and gain flatness, etc. Allofthese parameters can now be reliably investigated using CAD tools.An example of noise and gain characteristics and a comparison with modelprediction for a room-temperature InP HFET, six-stage, W-band amplifier are shownin Fig. 16. The devices have gate dimensions 0.1 × 50 µm and are biased atV ds =1.0 VandI ds =5mA. For the purpose of modeling, the equivalent circuitgiven in [49] is used. The noise model of [33], [34] is assumed for noise computationwith T g = 297 KandT d = 1500 K (compare Fig. 5). An example of noise and gaincharacteristics and comparisons with the model prediction for a cryogenic amplifier

86 M. W. PospieszalskiFig. 19. A comparison of measured gain and noise characteristics of a K-band amplifier with model prediction at room temperature. Measured noisetemperature includes the contribution of pyramidal horn and receiver (T r =2000 K).are shown in Fig. 17. The transistors were biased at V ds =0.9 VandI ds =3mA inthe first two stages and I ds =5mA in the last three stages. For the model data, theonly changes from room temperature were: T g = T a =20KandT d = 500 Kandabout a 20 percent increase in transconductance g m . A photograph of the amplifieris shown in Fig. 18.Examples of the comparisons between measured and modeled data for K-bandamplifiers, under exactly the same assumptions, are shown in Figs. 19 and 20. Aphotograph of the amplifier is shown in Fig. 21 [51].There have only been several wafers of InP devices which exhibited an excellentcryogenic performance since their introduction in 1993. The most recent best resultsare usually demonstrated with devices produced at TRW Space Technology Division(now Northrop Grunman Space Technology) in collaboration with JPL [56]. A summaryof what is currently believed to be the best results at cryogenic temperaturesis shown in Fig. 22, together with the author’s 1992 prediction. The experimentalresults are those published by NGST, JPL and Chalmers University [45], [54], [56],[75]–[76] and some measured at X-, K-, K a - and Q-bands at NRAO.V. CONCLUDING REMARKSThe noise performance of cryogenic FET’s andHFET’s has made tremendousprogress over the last several decades. The most rapid advances seem to haveoccurred between 1980 and 1995. Three different generations of HFET’s wereproposed and reduced to practice during that period. Since 1995, no significant

Extremely Low-Noise Amplification ... 87Fig. 20. A comparison of measured gain and noise characteristics of a K-bandamplifier with model prediction at cryogenic temperature T a =20K. Measurednoise temperature includes the contribution of dewar window, pyramidal hornat T a =20K, and room temperature receiver (T r = 2000 K).new ground has been broken in device technology. However, InP HFET technologyhas matured and allowed construction of extremely low-noise amplifiers. Theseamplifiers were used in the construction of several instruments for investigation ofthe physics of the early Universe. Among those one should mention the WilkinsonMicrowave Anisotropy Probe (WMAP) launched in June 2001, the Cosmic BackgroundExplorer (CBI), the Degree Angular Scale Interferometer (DASI), and theVery Small Array (VSA). Also, the performance of receivers installed on existingradio astronomy telescopes has been greatly improved by the availability ofbroadband InP HFET cryogenically-coolable amplifiers, especially in the K u -, K-,K a - and Q-bands, on NRAO’s Very Large Array and the Green Bank Telescope.This technology made possible the installation of 3-mm wavelength receivers onthe Very Large Baseline Array and the MPI Effelsberg Radio Telescope.It is difficult to predict at this moment whether further dramatic improvementsare possible. Each new generation of HFET’s has required huge investments inthe establishment of repeatable device technology, driven mostly by commercialand military applications. Radio astronomy has been able to derive great benefitsfrom these advances. Still, even today, InP HFET technology has not been ableto consistently reproduce noise performance at cryogenic temperatures, althoughperformance at room temperature has been sufficiently reproducible. Furthermore,performance at room temperature in commercial and military applications below50 GHz has been sufficiently good for any terrestrial system as the system noisetemperature tends to be dominated by the antenna temperature (about 300 K).

88 M. W. PospieszalskiFig. 21.Photograph of K-band MAP amplifier with cover removed.Consequently, a very costly investment in device technology would return verylittle in performance for terrestrial systems. Therefore, not unexpectedly, researchinto new structures from the point of view of noise performance has lost momentum.In the author’s opinion, only InAs/AlSb HFET’s offer a possibility offurther improvements in the noise temperature at cryogenic temperatures. So far,only devices with 0.25 µm gate length have been demonstrated [78]. It remains tobe seen whether the potential for very low-noise amplification using this technologywill ever materialize.Most of the low-noise receivers below W-band are now taking advantage of theextremely low noise performance of cryogenic InP HFET’s. At W-band frequencies,HFET’s compete with SIS mixers, and the choice between these two technologiesis less likely to be determined by noise performance alone but rather by othersystem requirements. Above W-band frequencies, the noise performance of SISmixers is considerably better than that of InP amplifiers. For example, the best-everHFET amplifier at 185 GHz exhibited noise performance of about 150 K [75], [76]which is very much in line with expectations published in 1992 [38], while SISmixer receivers with 60 K SSB noise temperatures are now routinely built for 210–270 GHz (for example, [77]). The difference in performance is so large that, forterrestrial systems in which cooling requirements for an SIS mixer can be easilymet, it is very unlikely that, even with advances in HFET technology, they will everdisplace SIS mixers as a preferred technology for frequencies above 150 GHz.

Extremely Low-Noise Amplification ... 89Fig. 22. Comparison between the predication for the minimum noise measureof a 0.1 µm gate length cryogenic InP HFET (1992) and the best resultsreported to date for cryogenic amplifiers employing NGST/JPL devices. Below50 GHz, the noise temperatures for the Chalmers 4–8 GHz [45] and NRAO X-,K-, K a- and Q-band amplifier designs are shown. For higher frequencies, thenoise temperatures reported for the NGST/JPL MMIC designs are shown [54],[56], [76].REFERENCES[1] S. Okwit, “A Historical View of the Evolution of Low-Noise Concepts and Techniques,” IEEETrans. Microwave Theory Tech., vol. 32, pp. 1068–1082, 1984.[2] J. J. Whelehan, “Low-Noise Amplifiers — Then and Now,” IEEE Trans. Microwave Theory Tech.,vol. 50, pp. 806–813, 2002.[3] J.D.Kraus,Radio Astronomy, Cygnus Quasar Books, 2nd Edition, 1986.[4] J. C. Webber and M. W. Pospieszalski, “Microwave Instrumentation in Radio Astronomy,” IEEETrans. Microwave Theory Tech., vol. 50, pp. 986–995, 2002.[5] S. Weinreb, “Low-noise, Cooled GASFET Amplifiers,” IEEE Trans. Microwave Theory Tech.,vol. 28, pp. 1041–1054, Oct. 1980.[6] S. Weinreb and A. Kerr, “Cryogenic Cooling of Mixers for Millimeter and Centimeter Wavelength,”IEEE Journal of Solid-State Circuits, vol. 8, pp. 58–61, 1973.[7] P. L. Richards et al., “Quasi-Particle Heterodyne-Mixing in SIS Tunnel Junctions,” Appl. Phys.Lett., vol. 34, pp. 345–347, 1979.[8] G. J. Dolan, T. G. Phillips, and D. P. Woody, “Low-Noise 115 GHz Mixing in SuperconductingOxide-Barrier Tunnel Junctions,” Appl. Phys. Lett., vol. 34, pp. 347–349, 1979.[9] J.R.Tucker,“Quantum-Limited Detection in Tunnel Junction Mixers,” IEEE Journal of QuantumElectron., vol. 15, pp. 1234–1258, 1979.[10] A. R. Kerr, S.-K. Pan, M. J. Feldman, and A. Davidson, “Infinite Available Gain in a 115 GHzSIS Mixer,” Physica, vol. 108B, pp. 1369–1370, 1981.[11] J. Shell and D. Neff, “A 32 GHz Reflected-Wave Maser Amplifier with Wide InstantaneousBandwidth,” in IEEE MTT-S Int. Microwave Symp. Dig., New York, NY, pp. 789–792, June 1988.[12] P. Siegel, “Terahertz Technology,” IEEE Trans. Microwave Theory Tech., vol. 50, pp. 910–928,2002.[13] A. van der Ziel, “Thermal Noise in Field-Effect Transistor,” Proc. IRE, vol. 50, pp. 1808–1812,1962.

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Extraction of GaAs-HBT Equivalent CircuitConsidering the Impact ofMeasurement ErrorsF. Lenk, M. RudolphFerdinand-Braun-Institut für Höchstfrequenztechnik (FBH),Gustav-Kirchhoff-Str. 4, D-12489 Berlin, Germany.95AbstractA new algorithm for extraction of the small-signal equivalent circuit elementsof HBTs is presented. An analytical non-iterative approach is used in order toensure physical significance of the extracted parameters. In order to enhance therobustness and reliability of the extraction routine, a simplified formula to determinethe intrinsic base resistance R b2 is presented. An error analysis is performedto prove why the analytical approach fails and to assess the robustness of theapproximated formulas. The algorithm is verified by extraction of GaInP/GaAsHBT equivalent-circuit elements.I. INTRODUCTIONSINCE a few years, GaAs-based HBT MMIC technology has become a matureand widely available standard technology. On the other hand, many issuesof modeling and parameter extraction are still under discussion. So far, even acommonly accepted standard method to extract the small-signal equivalent-circuitparameters does not exist, although huge efforts were spent on this issue. This documentsitself in the large amount of papers published recently on the topic. However,reliable extraction of the small-signal equivalent-circuit parameters remains a keyissue regarding device modeling and technology monitoring.The main difficulty of parameter extraction results from the topology of theintrinsic HBT’s equivalent circuit, and from the relative value of its parameters. Inaddition to the elements describing the active HBT (marked Y II in Fig. 1), we havefurther bias-dependent elements describing the extrinsic base and collector regionsunder the base contact, which are part of the intrinsic equivalent circuit. It turnsout that the crucial part of the extraction is the determination of the intrinsic baseresistance R b2 . This quantity is important since it is required to distinguish properlybetween the base-collector capacitances C bc and C ex . The main difficulty lies inthe fact that R b2 in state-of-the-art GaAs HBTs is significantly reduced (but stillcannot be neglected). This improves HBT behavior, but makes extraction difficult.These difficulties manifest themselves in an instability of extraction algorithms. Thismeans that measurement errors or incertainties in the deembedding of the extrinsic

96 F. Lenk et al.C exCB R b2R bcC bcCbeRbeI eY III eEFig. 1.Equivalent circuit of intrinsic HBT in common emitter configuration.parameters may prevent any extraction. The robustness of an algorithm thereforehas to be proven.In order to ensure physical significance of the extracted parameters, analyticalalgorithms are favorable over algorithms involving numerical optimizations. Also,a-priori knowledge such as technological details should not be required for extraction.However, most analytical algorithms rely on simplifications [1]–[7] instead ofexact formulations, in order to avoid inaccuracies and accumulation of errors insubsequent calculations.While the extraction of the intrinsic parameters is still under discussion, the determinationof the extrinsic parameters is performed commonly using open-collectorand off-state bias points [8].In this paper, a new direct algorithm is presented for extraction of the intrinsicparameters after deembedding of the parasitics. For this approach, first exact formulasare derived for all parameters. Since it turns out that this algorithm fails todetermine R b2 correctly in some cases, an approximation is given that has provento yield good results even if the exact solution fails.In order to assess the robustness of the extraction algorithm, a sensitivity analysisis performed. This is carried out by analytical and numerical calculation of theimpact of errors in the S-parameters on the determination of R b2 and C ex .Thisinvestigation also addresses the so far open question, why analytical HBT extractionalgorithms fail in some cases, leading to the fact that no standard algorithm has

Extraction of GaAs-HBT ... 9710e3µ 11µ ii5e3µ 22µ 21µ 12Fig. 2.0 10 20 30 40 50freq (GHz)Sensivities µ ii according to (10) of R b2 for the closed-form solution.been established so far.In order to give an example and to verify the algorithm, state-of-the-art InGaP/GaAsHBTs are investigated.II. THE EXTRACTION ROUTINEThe extrinsic elements are determined from open-collector and off-state measurements[8]. After deembedding, one usually ends up with the description ofthe intrinsic HBT by admittance parameters in the common-emitter configurationY . The solution presented in [10] uses the chain matrix parameters in commoncollectorconfiguration to get more simple equations. To avoid this additional matrixcalculation, we present here equations for Y . The routine is split in two steps.First, the weakly bias-dependent elements R b2 and C ex are extracted. Because theclosed-form solution of this step may fail, we show an alternative way to determinethe parameters. Implicit deembedding of these parameters yields the remainingelements. The equations for Y can be found in [3]. The following abbreviations areused:a = Y 12 + Y 22b = Y 11 + Y 21c = Y 11 + Y 12 + Y 21 + Y 22|Y | = Y 11 Y 22 − Y 12 Y 21

98 F. Lenk et al.ratio of relativ errors ε / ε Sii15001000500ε Rb2/ε Siiε Cex/ε Sii00 10 20 30 40 50freq (GHz)Fig. 3. Ratio of relative error ɛ/ɛ Sii according to (12) of R b2 and C ex forthe closed-form solution.A. Closed-Form SolutionThe following equations determine R b2 and C ex :Re {ac ∗ }R b2 =Re {|Y | c ∗ }C ex = −Im { a|Y | ∗}ωRe {ac ∗ }(1)B. Practical ApproachBecause the closed-form solution often fails for extraction from measured data,the practical approach presented in [10] is translated to admittance parameters incommon-emitter configuration, which leads to:{ } 1R b2 = ReY 11⎛{ } ⎞C ex = 1 { } |Y | Im ab⎝Im − { } ⎠ (2)ω b Re 1Y11C. Remaining ParametersAfter extraction of R b2 and C ex from (1) or (2) the deembedding of theseelements leads to an overdetermined matrix with admittance parameters Y II ,which

Extraction of GaAs-HBT ... 99ratio of relativ errors ε / ε Sii42ε Rb2/ε Siiε Cex/ε Sii00 10 20 30 40 50freq (GHz)Fig. 4. Ratio of relative error ɛ/ɛ Sii according to (12) of R b2 and C ex forthe practical approach.contains the remaining elements Y be , Y bc ,andα:[ ]Ybc + YY II =be (1 − α) −Y bcαY be − Y bc Y bc(3)The best way to overcome this overdetermination is to use the matrix A II from[10], where an unambiguous relation for Y be and Y bc exists. The deembedding isdone implicitely, so that the following equations in admittance form are obtained:Y bc = |Y | − jωC excY 11 + Y 21(4)α = Y 21 − Y 12Y 11 + Y 21(5)Y be =Y 11 + Y 211+R b2 (jωC ex − Y 11 )(6)For α this is the same equation as presented in [3].III. ERROR ANALYSISEquations (1) and (4) to (6) provide a closed-form solution for HBT extractionas it is available for FETs for a long time [11]. But, in contrast to the FET case, theHBT closed-form solution often fails, when it is applied to measured S-parameterdata. Especially, the extracted values for R b2 are often not constant with frequency.

100 F. Lenk et al.4030closed formR b22010exact valueFig. 5.in S 11 .practical approach00 10 20 30 40 50freq (GHz)Extraction of R b2 from synthetic S-parameter data with 0.1% errorApart from C ex and α all other parameters depend on these values of R b2 , henceits extraction is crucial for the overall extraction accuracy.To examine this phenomenon we calculate the error propagation from the measuredS-parameters to the extraction formulas of R b2 and C ex . For this purpose, weuse synthetic S-parameters. The extrinsic parameters are assumed to be frequencyindependent and known exactly. Then all calculations deal with the S-parametersof the intrinsic device. This is not true in the real measurement situation, but theS-parameters of the intrinsic device should not have too much difference to thewhole HBT S-parameters. In oder to validate the analytical calculation of the errorpropagation, numerical simulations are performed by adding synthetic error to S 11 .All results are obtained for a typical HBT in a typical operation point.The calculation is done for R b2 and C ex in a similar way. In the following, itis derived for R b2 as an example. In a first step the equations (1) are rewritten interms of real and imaginary parts of S-parameters. WithS 11re = Re {S 11 } ,S 11im = Im {S 11 } ,S 12re = Re {S 12 } , ···one can write (1) asR b2 = f (S 11re ,S 11im ,S 12re , ···) (7)and obtains a very complicated expression, which can only be handled by means

Extraction of GaAs-HBT ... 101of a mathematics program. With a Taylor series one can write the error in R b2 :∆R b2 = δR b2∆S 11re +δR b2∆S 11im + δR b2∆S 12re + ···δS 11re δS 11im δS 12reusing the partial derivations of (7) with respect to real and imaginary parts of theS-parameters. This leads to the standard deviation:√ ( ) 2 ( ) 2 ( ) 2 δRb2σ Rb2 =σS 2 δRb2δS 11re+σS 2 δRb211re δS 11im+σS 2 11im δS 12re+ ··· (8)12reThe relative error ɛ 11 of S 11 is the same for real and imaginary part, but dependson the absolute value |S 11 |:ɛ 11 = σ S 11re= σ S 11im|S 11 | |S 11 |⇒ σ S11re = σ S11im = ɛ 11 |S 11 | (9)Insertion of (9) in (8) gives⎡( ( ) 2 ( ) ) 2σ Rb2 =ɛ 2 11⎢|S 11| 2 δRb2 δRb2++δS 11re δS 11im⎣ } {{ }µ 2 11which leads to a relative error ɛ Rb2 of:( ( ) 2 ( ) )2 +ɛ 2 12 |S 12| 2 δRb2 δRb2++ ···δS 12re δS 12im ⎥} {{ } ⎦µ 2 12ɛ Rb2 = σ R b2R b2√ɛ2= 11 µ 2 11 + ɛ2 12 µ2 12 + ···(11)R b2The sensivities µ ii measure the error propagation of the error in S-parameter S ii tothe extracted value of R b2 . They are described as functions of the real and imaginaryparts of the measured S-parameters. Assuming equal relative errors for all measuredS-parameters,ɛ Sii = ɛ 11 = ɛ 12 = ɛ 21 = ɛ 22⎤12(10)

102 F. Lenk et al.TABLE IPARAMETER SET OF HBT FOR ERROR ANALYSIS.I c V c R b2 C be R be C bc R bc C ex α 0 τ f α(mA) (V) (Ω) (pF) (Ω) (fF) (kΩ) (fF) (1) (ps) (GHz)39.03 3.000 6.964 4.233 1.065 9.81 100.0 66.25 0.99225 1.987 98.68the relative error of R b2 simplifies with the geometric sum of the sensivities µ ii to:ɛ Rb2 = ɛ √Sii µ211 + µ 2 12 + µ2 21 + µ2 22(12)R b2IV. RESULTSTo analyze the errors, synthetic S-parameters were used. In Table I the correspondingHBT parameters from a typical 1×(3×30) µm 2 device in class A operationare listed. The extrinsic parameters are assumed to be independent of frequency andknown exactly, so that they are not included in the calculation. In Fig. 2 the resultsfor the sensivities µ ii according to (10) are shown. The errors in R b2 are dominatedby the error in the input reflection S 11 .To compare the reliability of R b2 with C ex extraction, the simplified model ofequal relative S-parameter errors from (12) is used. The ratio of ɛ Rb2 /ɛ Sii describesthe increase in R b2 error caused by error propagation. In Fig. 3 this is plotted forthe parameters R b2 and C ex of the closed-form solution. As expected, the error inC ex extraction is much lower, than it is in case of R b2 .The same data is shown in Fig. 4, but for the practical approach according to(2). The error propagation is significantly less than that of the closed-form solution.It should be mentioned, that only the error propagation of the S-parameters in theextraction formula is considered, but not the errors due to the approximations inthe formulas.To validate the results, synthetic S-parameters with 0.1% error in S 11S 11error = S 11ideal − 1 ∣ 1000 (1 + j) ∣∣S11 ∣ (13)idealare calculated. The two different R b2 extraction routines ((1) and (2)) are applied.Fig. 5 shows the results: The closed-form solution fails in the entire frequencyrange, while being applied to the S-parameters with errors in S 11 .Theshapeofthis curve resembles the shape of µ 11 in Fig. 2. The practical approach providesnearly the same results for ideal and disturbed S-parameters. In the lower frequencyrange, the approximation is not valid [10], but from about 10 GHz and beyond agood estimation of R b2 is obtained.

Extraction of GaAs-HBT ... 103V. CONCLUSIONA robust algorithm for extraction of the HBT intrinsic equivalent-circuit parametersis presented in this paper. Exact formulas for all parameters are given.However, it turnes out that the exact solution fails in some cases. Therefore, arobust approximated algorithm is presented that has proven to yield reliable results.This finding is supported by an analysis of the influence of S-parameter measurementerrors on the extraction accuracy. It turnes out, that the relative error inR b2 , that determines the accuracy of the whole extraction, is in orders of magnitudehigher than the initial S-parameter error in case of the exact solution. The relativeerror is significantly reduced in case of the approximated algorithm. It thereforehas to be concluded that the latter algorithm, despite the approximations, will yieldaccurate results.REFERENCES[1] C.-J. Wei, J. C. M. Hwang, “Direct extraction of equivalent circuit parameters for heterojunctionbipolar transistors,” IEEE Trans. Microwave Theory Tech., vol. 43, pp. 2035–2039, Sept.1995.Corrections: July 1996, p. 1190.[2] A. Kameyama, A. Massengale, Ch. Dai, J. S. Harris Jr., “Analysis of device parameters for pnptypeAlGaAs/GaAs HBTs including high-injection using new direct parameter extraction,” IEEETrans. Microwave Theory Tech., vol. 45, no. 1, pp. 1–10, 1997.[3] M. Rudolph, R. Doerner, P. Heymann, “Direct extraction of HBT equivalent circuit elements,”IEEE Trans. Microwave Theory Tech., vol. 47, no. 1, pp. 82–84, 1999.[4] M. Sotoodeh, L. Sozzi, A. Vinay, A. H. Khalid, Z. Hu, A. A. Rezazadeh, R. Menozzi, “Steppingtoward standard methods of small-signal parameter extraction for HBT’s,” IEEE Trans. ElectronDev., vol. 47, pp. 1139–1151, June 2000.[5] Y. Suh, E. Seok, J.-H. Shin, B. Kim, D. Heo, A. Raghavan, J. Laskar, “Direct extraction methodfor internal equivalent circuit parameters of HBT small-signal hybrid-pi model,” IEEE MTT-S Int.Microwave Symp. Dig., vol. 2, pp. 1401–1404, 2000.[6] M. Hattendorf, D. Scott, Q. Yang, M. Feng, “Method to determine intrinsic and extrinsic basecollectorcapacitance of HBTs directly from bias-dependent S-parameter data,” IEEE Electron.Dev. Lett., vol. 22, no. 3, pp. 116–118, Mar. 2001.[7] T. S. Horng, J. M. Wu, H. H. Huang, “An extrinsic-inductance independent approach for directextraction of HBT intrinsic circuit parameters,” IEEE MTT-S Int. Microwave Symp. Dig., vol.3,pp. 1761–1764, 2001.[8] Y. Gobert, P. J. Tasker, K. H. Bachem, “A physical, yet simple, small-signal equivalent circuit forthe heterojunction bipolar transistor,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 149–153,1997.[9] H. Kuhnert, F. Lenk, J. Hilsenbeck, J. Würfl, W. Heinrich, “Low Phase-Noise GaInP/GaAs-HBTMMIC Oscillators up to 36 GHz,” IEEE MTT-S Int. Microwave Symp. Dig., vol. 3, pp. 1551–1554,2001.[10] F. Lenk, M. Rudolph, “New Extraction Algorithm for GaAs-HBTs With Low Intrinsic BaseResistance,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 725–728, 2002.[11] M. Berroth and R. Bosch, “High-Frequency Equivalent Circuit of GaAs FET’s for Large SignalApplications,” IEEE Trans. Microwave Theory Tech., vol. 39, pp. 224–229, 1991.

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105On the Implementation of Transit-TimeEffects in Compact HBT Large-SignalModelsM. Rudolph, F. Lenk, R. DoernerFerdinand-Braun-Institut für Höchstfrequenztechnik (FBH)Gustav-Kirchhoff-Str. 4, D-12489 Berlin(e-mail: rudolph@fbh-berlin.de).AbstractDifferent approaches to include transit-time effects into Π-topology HBTequivalent circuits are investigated in order to assess their compatibility with thephysics-based T topology. The aim is to find an implementation that not only yieldsan exact model but also has a unique set of parameters in both Π and T case.This is of prime importance for reliable parameter extraction and thus physicalsignificance of the model. It is achieved using a transcapacitance approach. In caseof other approaches, the parameters have to be approximated. Basic considerationsare made with respect to the small-signal case. The results are then transferred tothe large-signal model. A relatively simple implementation for the transit-times isproven to yield the best agreement with measured data.I. INTRODUCTIONALTHOUGH GaAs-based HBTs nowadays are widely available in standardMMIC technology, circuit designers are still waiting for a commonly availablestandard model for circuit design. But not only the large-signal model, even themethod how to determine the small-signal equivalent circuit parameters is still underdiscussion in the respective publications.One finds that most papers dealing with small-signal parameter-extraction techniquesrely on T-circuit topology, usually with the argument that it is “more physical”.On the other hand, it is attractive to maintain backward compatibility withthe standard SPICE Gummel-Poon model, at least for the isothermal DC case. Thisrequires compact large-signal models to be formulated in Π-topology. However, bothformulations are not equivalent in all cases. The crucial issue is the implementationof transit time that is associated with the current source in the model. This startingpoint leads to a well-known paradoxon: One may be able to determine the smallsignalparameters quite accurately, but those parameters cannot directly be translatedinto large-signal parameters. The extracted physically significant parameters becomemere starting points for a global optimization of the large-signal model. Obviously,this is also a problem when developing a compact model from measurement data.

106 M. Rudolph et al.R b2C exCollector(a)BaseC beG’ beC bcβ I bI bEmitterR b2C exC bcCollector(b)BaseC beG beI eα I eEmitterFig. 1.shown.Intrinsic equivalent circuits of HBT. (a) Π-topology, (b) T-topology. Extrinsic elements notAlthough the reported large-signal models yield good results that match measureddata very well, we will stress the question of compatibility with T-topology for thefollowing reasons. Firstly, it does not seem to be of any advantage to abandon theT-description with its physical meaning completely and to switch to Π-topologyin small-signal considerations, too. Secondly, reliability of the parameter extractionprocedure is most important, since it determines the basic accuracy of the model.Asimplified model with a stable parameter-extraction routine is always preferableto a sophisticated one that leaves one alone with a huge number of unknowns.This paper reconsiders different ways to implement transit times. First, fourdifferent methods how to introduce transit time into Π-topology large-signal modelsare investigated. The aim is to find the formulation for transit times in Π-topologythat describes the extracted T-topology data most accurately. This part will berestricted to the small-signal case where a single parameter τ is used to accountfor transit-time effects. In large-signal models, however, this parameter is split intodifferent functions representing the different physical origin and bias dependenceof the parts of the total transit time. Therefore superpositions of the differentimplementations are used. We will focus on GaAs HBTs, where a bias-dependentcollector transit-time and an almost constant base transit-time has to be considered.Three different approaches of implementation are discussed.

On the Implementation ... 107II. BASIC ANALYSISIn this section, the parameters of the T-topology equivalent circuit according toFig. 1(b) will be derived from the Π-topology circuit of Fig. 1(a). Besides thedifferent location of the current source, also the driving current is different in thetwo cases, as indicated in the respective figures. Calculation of the T-topology baseemitteradmittance Y be and α from Π-topology yields:Y be = jωC be + G ′ be (1 + β) (1)α =β(1 + β)+jω(C be /G ′ be )It is obvious that an arbitrary choice of the frequency dependence of β in Π-topology will result in a complex, frequency-dependent G be , C be ,andα in T-topology. Only if β has a constant real value β 0 , Π– and T-topology are equivalent:Y be = jωC be + G be (2)1α = α 0 (3)1+jω/ω αWith ω α =1/(C be /G be ), G be = G ′ be (1 + β 0) and α 0 = β 0 /(1 + β 0 ).However, in the microwave range, transit times cannot be neglected. And, insteadof (3), it also is usual to model the T-topology current source by:α = α 0 e −jωτ(4)1+jω/ω αIn the following, four different implementations of transit time in the Π-circuitwill be investigated. These are• use C be to tune ω α properly• apply an excess-phase network• introduce a time delay: β = β 0 e −jωτ• a base-collector transcapacitanceSince it is desirable that a model yields physically meaningful results for arbitrarilychosen positive parameters as (4) does, it is useful to consider the impact ofthe different transit-time implementations on the loci of α. It is not ensured in allcases that α shows a low-pass behavior for C be > 0. While the overall behaviorof the equivalent circuit still might yield HBT-like characteristics, the unexpectedfrequency dependence, especially for large values of τ, may cause confusion.In order to estimate the Π-topology parameters from extracted T-topology equivalentcircuit elements, it has been observed that proper approximation of the phaseis much more important than that of the frequency dependence of the absolutevalue of α [1]. Therefore, expressions are given that allow for estimation of theΠ-topology parameters that yield an approximation to ∠α and assure a low-passbehavior of |α|.

108 M. Rudolph et al.I’ bL’I b C’1Fig. 2.Excess-phase network.A. Manipulation of ω αThis method is the simplest way to introduce a time delay. It does not needchanges in circuit topology. Hence, T- and Π-scheme are equivalent. In this case,C be is modified in order to change the time constant of the base-emitter admittanceto 1/ω α′ !=1/ω α + τ. Theaimistouse(3)asafirst-order approximation of (4)[2]. Thereby, one sacrifices accuracy in input return loss for the description of thecurrent source.B. Excess-Phase NetworkIn order to gain an additional degree of freedom compared to the previousimplementation, an excess-phase network, Fig. 2, may be introduced. In this case,the current Ib ′ is used as driving current of the current source. β thereby gets abessel-filter-like frequency dependence instead of a single-pole low-pass, and thenetwork easily can be implemented in circuit-simulation software. The values of L ′and C ′ are derived from a single time constant: L ′ = τ/3, C ′ = τ [3], [4]. However,this implementation can not directly be translated into T-topology. In contrast, theT-topology parameters have the following form:Y be = jωC be + G ′ β 0 G ′ bebe +1+jωτ− ω 2 τ 2 (5)/31α = β 01+jωτ− ω 2 τ 2 /3 · G′ be(6)Y beAlthough α has a low-pass characteristic, it might exceed the initial value α 0for lower frequencies. If C be is assumed to be negligibly small, |α| √ β 0 − 1/2. In the other case, the inequality√(3β0 + 3)(3β 0 − 1) − 3β 0τ ≤(2β 0 − 1)G ′ be /C (7)bemust be satisfied.In order to estimate the parameters ω β = G ′ be /C be and τ β from the parametersof the T-topology equivalent circuit, ∠α calculated by (6) is approximated for low

On the Implementation ... 10910.90.8|α| 0.70.60.50.400Phase( α ) (Degree)-20-40-60-80-10010(1)(0)2030 40f (GHz)(3)(1)0 10 20 30 40f (GHz)(0)(2)(3)(4)50(2)60(4)7050 60 70Fig. 3. Magnitude and phase of α. Measured (symbols) and modeled (lines) data with: (0) T-topology, (1) modified ω α, (2) excess-phase network, (3) time delay, (4) transcapacitance. I c =18mA,V ce =3V .frequencies:⎛ ( ) (1⎜ωω β+ τ β − ω 3 τ3) ⎞β3ω β ⎟∠α = −arctan ⎝( )(1 + β 0 ) − ω 2 τβω β+ τ ⎠ (8)β2 3(1≈ −ωω β (1 + β 0 ) + τ )β(9)1+β 0Calculating τ β and ω β from ∠α estimated from (4), one has to observe the condition(7). Therefore, τ is held constant for both topologies, and ω β is reduced.τ β = τ α (10)ω β =ω α1+β 0 + ω α β 0 τ α(11)Hence, the approximation leads to a variation of C be in the Π-topology.For high frequencies, real(Y be ) approaches G be due to the −ω 2 term in thedenominator. Besides that reduction, real(Y be ) might swing into the negative region.

110 M. Rudolph et al.C. Time DelayIntroducing a time delay of the form β = β 0 e −jωτ is problematic in a largesignalmodel. Since those models usually have to be formulated in the time domain,a previous time step (t − τ) has to be accessed. This is not always possible,or, e.g. in case of user-compiled models in Agilent’s ADS, only a constant τis allowed and a bias-dependence cannot be considered. Besides that, the baseemitteradmittance cannot be described by frequency-independent parameters [5].Calculating T-topology parameters from this equivalent circuit yields:Y be = jωC be + G ′ be (1 + β 0e −jωτ ) (12)β 0 e −jωτα =jωC be /G ′ be +1+β 0 e −jωτ (13)In this formulation, low-pass behavior is caused only by C be . In the worst case,C be → 0, α will oscillate and describe circles from α 0 to the maximum pointβ 0 /(β 0 − 1) for ωτ =(2n − 1)·π. Therefore, it is necessary that τ is small enoughcompared to C be /G ′ be. More precisely, in order to ensure a low-pass behavior, onehas to choose√1/α0 − 1τ ≤G ′ be /C (14)beFor this formulation, the approximation of ∠α for low frequencies yields:( ω)ω∠α = −ωτ β − arctanβ− β 0 sin(ωτ β )(15)1+β 0 cos(ωτ β )()1≈ −ω τ β −+ α 0 τ β (16)(1 + β 0 ) ω βAs in the excess-phase network approach, τ β and ω β can be estimated from theT-topology parameters by (10) and (11). This settings approximate the phase of αwell and fulfill relation (14) at the same time.D. TranscapacitanceIn case of the transcapacitance approach [6]–[8], a charge Q b = f(V be ) is insertedin parallel with C bc . This quantity represents the charge stored in base and collector.This leads to a transcapacitance, i.e. a capacitance driven by the voltage of anotherbranch. The transcapacitance is introduced automatically, when C bc is consideredto be a function of collector current [9]–[13]. In general, transcapacitances aredangerous since they act like voltage-driven current sources with a weighting factorjω, and thereby may cause excessive gain at high frequencies. In HBTs, however,it turns out, that a transcapacitance C tr modifies α as follows:α = α 0 − jω(C tr /G be )(17)1+jω(C be /G be )

On the Implementation ... 1111180Abs(S11)0.8900.600.4-900.200 10 20 30 40-18050f (GHz)Arg(S 11 ) (Degree)Fig. 4. Magnitude and phase of S 11 . Measured (symbols) and modeled (lines) data with: (—)standard implementation, (- - -) excess-phase network, (– ––) transcapacitance. I c = 18kA/cm 2 ,V ce =1.5, 3, 6 V.In order to ensure α ≤ α 0 , it is necessary to set C tr ≤ C be α 0 . In case of thisformulation, the transit time is modeled indirectly. α is described by (3), and therebyT- and Π-topology are equivalent.E. VerificationThe previous theoretical considerations will now be applied to a practical example,a 3×30 µm 2 GaInP/GaAs-HBT fabricated on the 4 ′′ process line of theFerdinand-Braun-Institut [14]. The elements of the small-signal equivalent-circuitin T-topology are determined by an analytical algorithm [15]. In this bias point, theS-parameters are well modeled by the T-topology equivalent circuit with τ =1.6 psand ω α =2π 67 GHz. For the transcapacitance approach, C tr = τG be is chosen,while the parameters are adjusted according to the approximations given above incase of the time-delay and excess-phase approaches.In order to evaluate the accuracy of the modeling approaches, the frequencydependence of the T-topology current source α is calculated from the differentΠ-circuits. Fig. 3 provides the results. Except for the approach with modified ω α(curve marked 1), α is modeled well if the appropriate approximations are used,even beyond f t =40GHz.III. LARGE-SIGNAL CONSIDERATIONSIn the large-signal case, one has to take into account the bias dependence of theparameters as well as the physical origin of the transit-time components. Additionally,it is not possible in all cases to distinguish properly between the contributions ofτ and ω α to the total transit time τ tot . Therefore, the extracted parameter in this case

112 M. Rudolph et al.20180Abs(S21)161284001020 30f (GHz)4090050Arg(S 21 ) (Degree)Fig. 5. Magnitude and phase of S 21 . Measured (symbols) and modeled (lines) data with: (—)standard implementation, (- - -) excess-phase network, (– ––) transcapacitance. I c = 18kA/cm 2 ,V ce =1.5, 3, 6 V.is τ tot = τ +1/ω α , which is to be split into its components and modeled accordingly.Concerning the physical origin, it has been shown for standard GaAs-HBTs, thatthe total transit time is given by the emitter charging time, a bias-independent basetransit-time τ B and the collector transit-time component τ C [11]. In order to modelτ C , the transcapacitance approach is best suited, since it accounts also for the biasdependence of C bc . A unified collector charge model can be applied to describe bothquantities by a single formula [13]. The emitter charging time is determined by thebase-emitter depletion capacitance and the base-emitter resistance. The remainingquestion therefore is how to implement τ B . Three implementations have been tested:• an additional transcapacitance,• a ‘pseudo-Π’ approach, which means that an excess phase network approach isapplied to a T-topology circuit — and only the DC parameters are formulatedlike those of the corresponding Π-circuit, and• the standard case, where τ B simply is modeled by the base-emitter junction’stime constant.It is the benefit of all three approaches, that they are equivalent to the T-descriptionand therefore preserve the physical significance of the parameters.Comparison of measurements and simulations with a large-signal model accountingfor τ B according to the three approaches are shown in Figs. 4–7. Measurementsat three different bias points are shown, with V ce =1.5, 3, 6 VandI c =18mA.It turns out that only the standard implementation yields good agreement withthe measurement. This is most pronounced seen in S 22 , Fig. 7, where the otherapproaches only show reasonable results below 10 GHz, or f t /4. Thisfinding canbe explained from careful extraction of the small-signal parameters of α: Althoughτ B is larger than 1/ω α in the bias points, this time-constant is necessary to ensure

On the Implementation ... 1131180Abs(S12)0.80.60.40.2900Vce-9000 10 20 30 40-18050f (GHz)Arg(S 12 ) (Degree)Fig. 6. Magnitude and phase of S 12 . Measured (symbols) and modeled (lines) data with: (—)standard implementation, (- - -) excess-phase network, (– ––) transcapacitance. I c = 18kA/cm 2 ,V ce =1.5, 3, 6 V.the low-pass behavior of α. Further splitting of τ B into different parameters or amore sophisticated model for the contribution of base-emitter junction and neutralbase to the total transit time is not necessary.IV. CONCLUSIONSFirst, different methods to implement transit times into compact Π-topology HBTmodels are investigated for the small-signal case. The leading question was how tokeep the physical significant parameters of the T-topology equivalent circuit in theΠ-topology. The following conclusions can be drawn:Neglecting the transit time can partly be compensated by properly adjusting thebase-emitter pn-junctions time constant. However, this is possible only in the lowerfrequency range.If transit times are modeled by introducing a time delay or by using an excessphasenetwork in the Π-topology, even reasonably chosen parameters may lead tonon-physical (and therefore unexpected) behavior in terms of the corresponding T-topology. Especially current gain α may exceed unity at high frequencies. In orderto prevent this, the time constant has to be set within the bounds defined in (7),(14). The other drawback of these implementations is that parameters extracted forthe T-topology equivalent circuit have to be approximated by the formulas (10),(11) in order to be compatible with the Π-description.The transcapacitance approach, therefore, turns out to be the most promisingapproach to implement transit times into compact HBT models.Concerning the large-signal description, one has to account for the mutual dependenceof the small-signal parameters which originate from derivatives of thesame charge or current. In case of GaAs-based HBTs as those under investigation

114 M. Rudolph et al.1Abs(S22)0.80.60.40.2Vce00-90V ce0 10 20 30 40-18050f (GHz)Fig. 7. Magnitude and phase of S 22 . Measured (symbols) and modeled (lines) data with: (—)standard implementation, (- - -) excess-phase network, (– ––) transcapacitance. I c = 18kA/cm 2 ,V ce =1.5, 3, 6 V.Arg(S 22 ) (Degree)in this paper, this means that the collector transit-time τ C is already defined by atranscapacitance resulting from a current-dependence of C bc . The remaining questiontherefore is, how to implement the almost bias-independent part τ B . It turnsout, that best accuracy is obtained from the simplest implementation, i.e. if τ B isused to define the base-emitter junction’s time constant.In conclusion, it is shown in this paper that transit-time effects in state-of-the-artGaAs HBTs can accurately be accounted for in a compact large-signal model bya standard SPICE Gummel-Poon like constant parameter τ B and an appropriatemodel for the collector charge. Thereby, the physical significance of the parametersis kept from the T-topology, while Π-topology is used for the convenience of circuitdesign. As shown, a very simple implementation of τ B yields best agreement withmeasurement data.ACKNOWLEDGMENTSThe authors would like to thank the material and process technology departmentsof the FBH for providing the HBTs, S. Schulz for performing measurements, andDr. W. Heinrich for helpful discussions and continuous encouragement.

On the Implementation ... 115REFERENCES[1] M. Rudolph, F. Lenk, R. Doerner, P. Heymann, “Towards a unified method to implement transit-timeeffects HBT large-signal compact models,” in IEEE MTT-S Int. Microwave Symp. Dig., pp. 997–1000, 2002.[2] I.E.Getreu,Modeling the bipolar transistor, Amsterdam: Elsevier, 1978.[3] P.B.Weil,L.P.McNamee,“Simulation of excess phase in bipolar transistors,” IEEE Trans.Circuits Syst., vol. 25, no. 2, pp. 114–116, 1978.[4] C.C.McAndrew,et.al.,“VBIC95, the vertical bipolar inter-company model,” IEEE Journ. Solid-State Circ., vol. 31, no. 10, pp. 1476–1483, 1996.[5] D. A. Teeter, W. R. Curtice, “Comparison of hybrid pi and tee HBT circuit topologies and theirrelationship to large signal modeling,” in IEEE MTT-S Int. Microwave Symp. Dig., pp. 375–378,1997.[6] J. C.-N. Huang, I. M. Abdel-Motaleb “Small-signal non-quasi-static model for single and doubleheterojunction bipolar transistors,” Solid State Electron., vol. 36, no. 7, pp. 1027–1034, 1993.[7] J. Ph. Fraysse, D. Floriot, Ph. Auxémery, M. Campovecchio, R. Quéré, J. Obregon, “A non-quasistaticmodel of GaInP/AlGaAs HBT for power applications,” in IEEE MTT-S Int. Microwave Symp.Dig., vol. 1, pp. 379–382, 1997.[8] A. Ouslimani, H. Hafdallah, J. Gaubert, M. Medjnoun, “Time-domain nonlinear HBT modelincluding non-quasi-static effects,” Electron. Lett., vol. 36, no. 17, pp. 1497–1499, 17th Aug. 2000.[9] Q. M. Zhang, H. Hu, J. Sitch, R. K. Surridge, J. M. Xu, “A new large signal HBT model,” IEEETrans. Microwave Theory Tech., vol. 44, pp. 2001–2009, Nov. 1996.[10] C.-J. Wei, J. C. M. Hwang, W.-J. Ho, J. A. Higgins, “Large-signal modeling of self-heating,collector transit-time, and RF-breakdown effects in power HBT’s,” IEEE Trans. Microwave TheoryTech., vol. 44, pp. 2641–2647, Dec. 1996.[11] M. Rudolph, R. Doerner, K. Beilenhoff, P. Heymann, “Scalable GaInP/GaAs HBT Large-SignalModel,” IEEE Trans. Microwave Theory Tech., vol. 48, pp. 2370–2376, Dec. 2000.[12] M. Iwamoto, P. M. Asbeck, T. S. Low, C. P. Hutchinson, J. B. Scott, A. Cognata, X. Qin,L. H. Camnitz, D. C. D’Avanzo, “Linearity characteristics of GaAs HBTs and the influence ofcollector design,” IEEE Trans. Microwave Theory Tech., vol. 48, pp. 2377–2388, Dec. 2000.[13] M. Rudolph, R. Doerner, K. Beilenhoff, P. Heymann, “Unified model for collector charge inheterojunction bipolar transistors,” IEEE Trans. Microwave Theory Tech., vol. 50, pp. 1747–1751,July 2002.[14] M. Achouche, Th. Spitzbart, P. Kurpas, F. Brunner, J. Würfl, G.Tränkle, “High performanceInGaP/GaAs HBTs for mobile communications,” Electronics Lett., vol. 36, pp. 1073–1075,June 2000.[15] M. Rudolph, R. Doerner, P. Heymann, “Direct extraction of HBT equivalent circuit elements,”IEEE Trans. Microwave Theory Tech., vol. 47, no. 1, pp. 82–84, 1999.

Festschrift for Peter HeymannThis volume of the Forschungsberichte is publishedon the occasion of Peter Heymann celebratinghis 65 th birthday. It presents a collectionof nine technical papers, contributed by expertsin their respective fields from Europe and theUS. The topics adressed range from investigationsof the plasma in a Tokamak to the modelingof Heterojunction Bipolar Transistors. There is afocus on transistor noise characterization andlow-noise circuits, which is well in line with thecurrent trends in today’s microwave and millimeterwavesystems.ISBN 3-86537-328-3Cuvillier Verlag