Transmission Line on Semiconductor Substrate with Distributed Amplification

Transcription

1 RAIONGINRING, VOL. 19, NO. 2, JUN Trasmissio Lie o Semicoductor Substrate with istributed Amplificatio Michal POKORNÝ, Zbyěk RAIA epartmet of Radio lectroics, Bro Uiversity of Techology, Purkyňova 118, 612 Bro, Czech Republic pokorym@feec.vutbr.cz, raida@feec.vutbr.cz Abstract. I order to compesate losses i metal strips, a active microstrip lie o a semicoductor substrate is proposed, ad its fiite elemet model is preseted. The active medium is provided by A 3 B 5 semicoductor i highitesity electric field. The simple model of the active media is developed ad used for calculatio of the propagatio properties of the fudametal mode ad for the thermal aalysis of the device. The problem of system self-oscillatio is discussed ad empirical stability criteria are itroduced. A proper heat-sik is proposed to provide the operatio i a cotiuous regime. Keywords GaAs, active, mictrostrip, millimeter-wave, Gu s effect, thermal, COMSOL. 1. Itroductio The output power of the covectioal discrete semicoductor devices drastically drops at millimeter-wave frequecies due to their fudametal limitatios. The first oe is associated with the device capacitace which results i impedace decreasig with frequecy ad creates a problem to match the device with a waveguides. Next, the maximal frequecy at which a device acts as a active elemet correspods to the trasit time of charge carriers or high field domais through the structure. Both metioed problems ca be suppressed by the reductio of devices i size. Besides, the miimal dimesios are also limited by several factors; the reduced device surface makes it difficult to effectively dissipate heat [1]. To overcome these fudametal limitatios o output power, the performace of active compoets should be based o the priciples of distributed iteractio of the electromagetic wave with active medium [1], [2]. As the promisig active medium, the bulk egative differetial resistace (BNR) ca be cosidered. BNR arises i A 3 B 5 semicoductors (e.g. GaAs) due to iter-valley scatterig of hot electros i high electric fields. While the wavelegth is short eough for the effective iteractio with the bulk semicoductor, the travelig wave amplificatio is possible [2]. Some authors suggest the cocept of active trasmissio lies combiig the techology of moolithic microwave itegrated circuits with the travelig wave amplificatio provided by a semicoductor substrate. Up to date, maily experimetal results were published i this field. The first successful measuremets of electromagetic wave amplificatio by BNR were performed at the frequecy 15 GHz i coplaar microstrip waveguides o the GaAs substrate [3]. Other experimets deal with the fi-lie ad strip lie structures at the frequecies about GHz ad 16 V bias voltage i the pulse regime [2]. All published structures faced with the problem of device overheatig i the cotiuous regime, so the problem of the heat-sik is very importat. I theoretical works o active structures based o BNR [2], [], a complicated equatio system describig wave propagatio ad its possible aalytical solutio were suggested, but o calculatios were performed. The paper [5] deals with the fiite elemet modelig of active dielectric waveguides based o BNR, however the issue of thermal heatig ad system stability are ot icluded. I this paper, a umerical model of the active BNR based microstrip lie is developed ad its performace ad limits i a cotiuous regime are ivestigated. The origial cotributio is represeted by the itroducig of the simplifyig assumptios which allow us to derive a simple model of the active GaAs medium cosistig of the aalytical form of the differetial coductivity tesor. The we set the procedure of the BNR active device aalysis which beefits from the separate calculatio of active medium parameters ad wave propagatio. Such procedure is easy to implemet i existig commercial software thus eables desig of BNR devices i covectioal idustry egieerig work-flow. Fially the device performace (trasmissio, thermal) is ivestigated by calculatios usig COMSOL Multiphysic computatioal eviromet which is based o fiite elemet method. The proposed active microstrip lie is desiged with respect to empirical criteria of system stability i order to avoid self-oscillatio. 2. Model of Active Medium There are may mechaisms affectig the wave propagatio i bulk semicoductors. However at millime-

2 38 M. POKORNÝ, Z. RAIA, TRANSMISSION LIN ON SMICONUCTOR SUBSTRAT WITH ISTRIBUT AMPLIFICATION ter-wave frequecies, oly free charge cocetratio is relatively sigificat, so the diffusive processes ad the space-charge wave are ivolved. The static ad trasiet behavior of semicoductor devices ca be described by a covetioal drift-diffusio scheme [6]. The scheme should be exteded by the Helmholtz equatio to defie the wave propagatio. The resultat system basically describes two kids of waves, the slow space-charge wave with speed give by carriers drift velocity, ad fast wave with speed of light [2]. Sice the space charge is preset at iterfaces ad defects of uiformly doped semicoductor oly, we eglect the slow space-charge wave ad assume the fast wave oly. The, we ca itroduce the simplificatio by the separate calculatio of media properties ad wave propagatio i order to obtai a easily computable model of electromagetic wave propagatio i BNR media. The illustrative descriptio of such model follows. The discussed structure is depicted i Fig. 1. The sample of -doped GaAs is biased via ohmic cotacts by a exteral voltage source which excites the electrostatic field i volume of a sample. The harmoic propagatio of liearly polarized wave p is superposed o, so the total electric field itesity is give by their additio. Fig. 1. Illustrative sketch of wave propagatio i biased GaAs sample. If the dopig profile of the GaAs sample is uiform ad low perturbatio of free space charge by p is assumed, the gradiet of the carrier cocetratio is zero; thus the diffusive processes ca be eglected. The the free carrier cocetratio c ca be assumed to be always equal to the dopig cocetratio N (all impurities are ioized). Uder these simplifyig cosideratios, the properties of the medium ca be simply described by the coductivity give by qn μ. (1) Here, q deotes the elemetary charge ad μ is the electro mobility. If the exteral voltage is applied ad the electrostatic field arises, the coductivity becomes to be aisotropic due to the reductio of the mobility caused by the trasfer of electros from the lowest valley i the coductio bad to the valleys of higher eergy (Gu s effect). For bulk GaAs, Gu s effect ca be described by [6] 3 vsat 1 crit crit. (2) Here, represets the origial mobility, v sat is the saturatio velocity, is the compoet of i the directio of curret flow ad crit is the threshold electric field. As a result, the coductivity is expressed as a uiaxial tesor σ. I order to obtai small-sigal approximatio, << p is assumed [2]. That itroduces the differetial form of the coductivity σ. Istead of usig the piecewise liear approximatio of the differetial coductivity supposed i [2] which is ivariat with electric field ad dopig cocetratio, we derived the aalytical form which ca be computed from the drift compoet of curret desity formula [6] J qn. (3) Substitutig the mobility formula (2) ito (3) ad differetiatig with respect to the electric field itesity, the aalytical form of the differetial coductivity is obtaied crit sat 2 3 crit qn v 3μ μ ( ) =. () The formula is advatageous because it ca be evaluated for various temperatures, dopig cocetratios etc. by proper modelig of the physical quatities of GaAs. Fially, the elemets of the desired tesor σ are give by () evaluated for the correspodig compoet of the electrostatic field x y (5) z If the Poisso equatio is solved i the domai of the active semicoductor medium, the the solutio defies the compoets of the biasig electric field ad the differetial coductivity tesor ca be evaluated ad used for the purpose of wave propagatio simulatios i BNR devices. Such medium descriptio is used i the followig calculatio of the active trasmissio lie. 2.1 Thermal Properties of GaAs The mai physical parameters of semicoductors sigificatly deped o the temperature. The iteractio of electros with the thermally geerated vibratios of the crystal lattice ca be simply modeled by the reductio of carrier mobility [6] crit

3 RAIONGINRING, VOL. 19, NO. 2, JUN L 3K 3 T. (6) The depedecy of the saturatio of carrier drift velocity o the temperature is give by [6] v sat v T 3K 1 A A 3 sat. (7) A sufficiet approximatio of the thermal coductivity depedece ca be modeled by [6] T k k 3. (8) T T 3K The geerated heat for a o-degeerated semicoductor is give by [6] H J. (9) I equatios (6) to (9), the quatities with idex 3 deote the values at 3 K; coefficiets, β ad A are obtaied by fittig experimetal data ad T is the temperature. Typical values of physical quatities of GaAs are give i Tab. 1 [6]..8 [m 2 V -1 s -1 ] kt 3 6 [Wm -1 K -1 ] 3 1 [-] 1.25 [-] 3 vsat [ms -1 ] A.56 [-] Tab. 1. Physical quatities of GaAs. The evaluated differetial coductivity () for the door cocetratio N = 1 19 m -3 ad various temperatures is depicted i Fig. 2. Obviously, the BNR behavior of the medium arises at 3 kvm -1, ad the strogest BNR is reached at about 8 kvm -1 of the electric field magitude. 3. Active Microstrip Lie Calculatios The dimesios of the ivestigated structure were adopted from [7]. The structure represets the 5 Ω microstrip lie o a GaAs wafer with the thickess of.1 mm ad relative permittivity 13. The strip is made of gold with coductivity of 5.2 MSm -1, ad trasverse dimesios are 7 μm 2. The 2 fiite-elemet model of the microstrip lie was implemeted i COMSOL Multiphysics. Sice the fiite-elemet models have to be eclosed, the microstrip lie was iserted ito the waveguide mm 2 with walls defied as the perfect electric coductor. The size of such a waveguide is more tha te times larger tha the thickess of the substrate i order to miimize its ifluece o the microstrip lie parameters [8]. If sufficiet voltage for creatig a BNR is applied, it may lead to curret istabilities (self oscillatios). The stable operatio of the device ca be achieved if the electro cocetratio c (N respectively) ad the distace betwee the ohmic cotacts (thickess of substrate h) are chose accordig to the criterio [2] 15 N h 1. (1) This criterio implies that the maximum value of the door cocetratio N for a give h is 1 19 m -3. First, the validity of the model was verified by calculatios of the atteuatio by ohmic losses for the domiat mode usig a aalytical formula [8] based o the coformal mappig method (CMM) ad by direct compariso with experimetal data of the same microstrip lie o a GaAs substrate preseted i [7]. For this purpose, the model of a passive microstrip lie o a lossless substrate was aalyzed (N = m -3 ). Fig. 3 shows good agreemet of the umerical, aalytical ad experimetal results. Fig. 2. ifferetial coductivity characteristics of the bulk GaAs at differet temperatures. Fig. 3. Frequecy respose of the atteuatio for passive ad active microstrip lie; a) HTC = 1 kwm -2 K -1, b) HTC = 1 kwm -2 K -1. The, the model of the active versio was aalyzed with the uiformly doped substrate. However, the presece

4 31 M. POKORNÝ, Z. RAIA, TRANSMISSION LIN ON SMICONUCTOR SUBSTRAT WITH ISTRIBUT AMPLIFICATION of impurity icreases its coductivity, so the effect of BNR is suppressed by additioal losses i the semicoductor. The calculatio of resistive heatig by propagatig wave i the cross sectio of the device (see Fig. ) reveals the high-loss areas located straight uder the metal lie edges. If these areas are kept udoped or doped lightly, the loss i the substrate ca be sigificatly reduced. Sice the trasitio betwee two dopig cocetratios caot be arbitrarily steep due to the limitatios of the maufacturig techology ad atural diffusio of impurities i bulk material, we use a typical soft trasitio with the dopig cocetratio differece of two orders. Fig. 5 shows the used dopig profile, where the substrate dopig cocetratio is 117 m-3 ad the active area is formed by a highly doped regio with the door cocetratio 119 m-3. The active area is located straight uder the metal strip ad is arrow eough ot to overlap the high-loss areas. The simulatio results of the described structure for two heat dissipatio coditios (described later) at bias voltage 5 V are depicted i Fig. 3 i compariso with the results of passive model. The sigificat reductio of atteuatio is observed, eve the gai is provided at lower frequecies, where the total losses are lower tha the amplificatio provided by the semicoductor substrate. ture is K while the maximum workig temperature of GaAs is about 723 K [8], so the device ca be fuctioal whe the used heat sik provides a sufficiet heat trasfer. Fig. 6. The solutio of Poisso equatio. Surface: lectric field itesity, z compoet [kvm-1] Arrow: lectric field itesity, trasversal vector. Fig. 7. Temperature distributio i cotiuous regime. Surface: temperature [K]. Arrow: Heat flux vector. Fig.. istributio of resistive heatig [Wm -1] iside the GaAs substrate uiformly doped by doors at cocetratio 119 m-3. Next, the differetial coductivity tesor σ is evaluated usig the curret solutio ad formula (5) with respect to the computed temperature distributio. Fig. 8 represets the resultat distributio of z compoet of σ. The active regio exhibits by BNR up to.38 Sm-1. The maximal coductivity does ot exceed the value.128 Sm-1, so the additioal loss i the substrate is lower tha the gai i the active regio. Fig. 5. The profile of the used door cocetratio [m -3]. The stepwise descriptio of preseted calculatios i COMSOL Multiphysics follows. First, the Poisso equatio is solved while the bias voltage is applied o the gold strip. The groud plae was set to zero potetial. Solutio of the static electric field is depicted i Fig. 6. Obviously, the z compoet is domiat uder the gold strip i the substrate. The, the heat geeratio is evaluated ad the heat trasport problem is solved i the domai of the GaAs substrate. The heat trasfer coefficiet (HTC) is set to 5 Wm-2K-1 at boudaries where the air without covectio is assumed [9]. The efficiet heat sik is provided at the groud plae iterface, so the HTC is set to 1 kwm-2k-1 at this boudary. The ambiet temperature is set to 3 K. The heat geeratio is give by equatios (3) ad (9). The solutio of the temperature distributio i the cotiuous operatio regime is depicted i Fig. 7. The peak tempera- Fig. 8. istributio of differetial coductivity, z compoet [Sm-1]. Fially, the eigevalue solver is used to fid the modes propagatig at a certai frequecy. Oly the fudametal hybrid mode exists up to frequecy 56 GHz. From this frequecy, the secod mode give by shieldig PC waveguide dimesios starts to propagate. The electric field distributio of the fudametal mode at 25 GHz is preseted i Fig. 9. The temperature ad atteuatio variatio with the applied bias voltage at 25 GHz is depicted i Fig. 1. The ohmic losses i doped GaAs cause the extreme atteuatio

5 RAIONGINRING, VOL. 19, NO. 2, JUN of the trasmissio lie util the sufficiet voltage is applied to create the BNR domai i the substrate. The, the atteuatio is sigificatly reduced with the icreasig voltage to saturate miima at 5 V. The temperature is icreased expoetially at a lower bias voltage ad chages gradually to liear depedecy. Fig. 9. lectric field itesity of the fudametal mode at 25 GHz. Surface: lectric field itesity, x compoet [Vm -1 ] Arrow: lectric field itesity, trasversal vector. frequecy bad. The model of the active medium represeted by the tesor of the differetial coductivity was deduced for A 3 B 5 semicoductor materials ad aalytically formulated for GaAs usig the empirical formula of the electro mobility reductio i high itesity electric field. The fiite elemet model of a active microstrip lie was aalyzed i COMSOL Multiphysics. The atteuatio of the fudametal mode ad thermal characteristics of a active device were calculated ad discussed. The results show the sigificat reductio of atteuatio with the applied bias voltage. ve the gai was observed at lower frequecies about 5 GHz. The thermal problem requires the efficiet heat sik defied by the trasport coefficiet. The optimal value that guaratees the proper heat dissipatio is about 1 1 kwm -2 K -1. Further work will be focused o the experimetal verificatio of preseted calculatios ad developmet of more complex devices as active ateas ad active feedig etworks. Recetly, aother physical mechaism, which ca provide the travelig wave amplificatio at lower electric fields, seems to be more perspective e.g. the slow-wave iteractio with driftig charge flow [1]. Ackowledgemets The research described i this cotributio was fiacially supported by the Czech Sciece Foudatio uder the grats o. 12/7/688 ad 12/8/H18. The research is a part of the COST project IC63 ad the research cetre program LC671. Refereces [1] LYUBCHNKO, V.. The Sciece ad Techology of Millimetre Wave Compoets ad evices. lectrocompoet Sciece Moographs, vol. 12. Lodo: Taylor & Fracis, 23. Fig. 1. The depedecy of temperature ad atteuatio o applied bias voltage. Other computatios were doe for HTC coefficiet equal to 1 kwm -2 K -1 i order to ivestigate the effect of a less efficiet heat sik. ve i this case, the maximum device temperature 557 K does ot exceed the limit for the GaAs material. The limit is reached at 7 V of the bias voltage. The compariso of device performace at differet coolig coditios ca be see i Fig. 3 ad Fig. 1.. Coclusios I the paper, the active microstrip lie was proposed i order to compesate the ohmic losses which are the serious problem of trasmissio lies i millimeter-wave [2] LIOUBTCHNKO,., TRTYAKOV, S., UOROV, S. Millimeter-Wave Waveguides. Bosto: Kluwer Academic Publishers, 23. [3] FLMING, P. L. The active medium propagatio device. I Proceedigs of the I. 1975, vol. 63, o. 8, p [] GOLOVANOV, O.A., LYUBCHNKO, V.., MAKVA, G. S. Computer modelig of active fi-lies o GaAs with high-field domai. lekroaja Tekhika, ser.1, o. 12, 1978, p [5] POKORNÝ, M., RAIA, Z. Active semicoductor waveguides ad ateas. I Proceedigs of th Workshop o Atea Systems ad Sesors for Iformatio Society Techologies. ubli, Triity College (Irelad), 28, p [6] SLBRHRR, S. Aalysis ad Simulatio of Semicoductor evices. Heidelberg: Spriger-Verlag, 198. [7] ZHANG, M., WU, C., WU, K., LITVA, J. Losses i GaAs microstrip ad coplaar waveguide. I Proceedigs of Microwave Symposium igest, 1992, I MTT-S Iteratioal, 1-5 Jue 1992, vol. 2, p

6 312 M. POKORNÝ, Z. RAIA, TRANSMISSION LIN ON SMICONUCTOR SUBSTRAT WITH ISTRIBUT AMPLIFICATION [8] SVAČINA, J. Microwave Itegrated Techique (Lecture otes). Textbook FC BUT. Bro: MJ Servis, 28 (i Czech). [9] COMSOL Multiphysics Model Library. Stockholm: COMSOL, 26. [1] POUSI, P., LIOUBTCHNKO,., UOROV, S., RAISANN, A.V. ielectric rod waveguide travellig wave amplifier based o AlGaAs/GaAs heterostructure. I Proceedigs of 38th uropea Microwave Coferece 28. umc Oct. 28, p About Authors... Michal POKORNÝ was bor i He is Ph studet at the ept. of Radio lectroics, Bro Uiversity of Techology. His research iterests are atea desig ad modelig of microwave semicoductor devices. Zbyěk RAIA received Ig. (M.Sc.) ad r. (Ph..) degrees from the Bro Uiversity of Techology (BUT) i 1991 ad 199, respectively. Sice 1993, he has bee with the ept. of Radio lectroics of BUT as the assistat professor (1993 to 98), associate professor (1999 to 23), ad professor (sice 2). From 1996 to 1997, he spet 6 moths at the Laboratoire de Hyperfrequeces, Uiversite Catholique de Louvai, Belgium as a idepedet researcher. Prof. Raida has authored or coauthored more tha 8 papers i scietific jourals ad coferece proceedigs. His research has bee focused o umerical modelig ad optimizatio of electromagetic structures, applicatio of eural etworks to modelig ad desig of microwave structures, ad o adaptive ateas. He is a member of the I Microwave Theory ad Techiques Society. From 21 to 23, he chaired the MTT/AP/ joit sectio of the Czech-Slovak chapter of I. I 23, he became the Seior Member of I. Sice 21, Prof. Raida is editori-chief of the Radioegieerig joural (publicatio of Czech ad Slovak Techical Uiversities ad URSI committees).