An Efficient Technique for the Optimization of Submillimeter-wave Schottky-diode Harmonic Multipliers

Transcription

1 A Efficiet Techique for the Optimizatio of Submillimeter-wave Schottky-diode Harmoic Multipliers Chih-Chie Lee 1, Boris Gelmot 1 ad Dwight Woolard 1 The Uiversity of Virgiia U.S. Army Research Laboratory Electrical Egieerig Departmet U.S. Army Research Office Charlottesville, VA 903 Research Triagle Park, NC 7709 Abstract A simple ad efficiet modified harmoic-balace techique is preseted. This ew algorithm is suitable for the large-sigal time-depedet aalysis of oliear millimeter wave circuits. The accurate desig ad successful implemetatio of very high frequecy oliear circuits requires both a detailed uderstadig of the physical operatio of the active devices ad of how these oliear devices iteract with the liear embeddig circuit. This work addresses the specific task of establishig a robust simulatio tool that combied this ovel modified harmoic-balace circuit aalysis techique with a physicsbased hydrodyamic trasport model device simulator. A compariso with the accelerated fixed-poit (AFP) harmoic-balace techique has bee made through a evaluatio of a Schottky-diode multiplier. The result shows that this approach ca be used to realize improved multiplier operatio with a miimal time of umerical computatio. I. INTRODUCTION The developmet of computer-aid aalyses of highly oliear circuits requires both a detailed uderstadig of the physical operatio of the active devices ad of how these oliear devices iteract with the liear embeddig circuit. Traditioally, the oliear active devices are modeled by the lumped quasi-static equivalet circuit aalysis. However, this equivalet circuit model has lost validity at higher frequecy millimeter wave where the large-sigal ostatioary pheomea begi to domiate device operatio. It is clearly that a physics-based umerical device model is required for the active device aalysis. I a earlier ivestigatio, drift-diffusio ad Mote Carlo based harmoicbalace algorithms have bee utilized. The drift-diffusio model is a low-order approximatio of the Boltzma trasport equatio, it implies that mobility of the carrier is oly a fuctio of the local electrical field ad it does ot take accout of the ostatioary characteristics such as carrier heatig ad velocity overshoot. The Mote Carlo simulatio has difficulty i treatig low-field regio. Also, the statistical oise i the solutio ad the itesive amout of computatio time required make this model impractical for device desig. A alterative approach is hydrodyamic trasport model, which is obtaied by takig the first three momets of the Boltzma Trasport Equatio [1]. Comparig to quasi-static equivalet circuit ad drift-diffusio model, hydrodyamic model is more accurately i treatig high-field, ostatioary, ad hot-electro effects.

2 The specific focus of this paper is o eergy trasport effects ad their ifluece o the harmoic multiplicatio withi reversed-bias Schottky-barrier varactor diodes. Equally importat to the characterizatio of diode multipliers is the optimizatio of the oliear elemet to the exterally coupled embeddig impedace. Here, the secod-harmoic power (i.e., curret ad voltage) of the matched diode is optimized before costraits o embeddig impedace, local voltage ad dc bias are specified. The key of this approach is the reductio of systems variables by a efficiet mathematical orderig of the optimizatio procedure. Specifically, this ew method reduces the typical optimizatio problem (i.e., where device ad circuit are cosidered simultaeously) for a doubler, where harmoics are cosidered, from + variables to variables. As will be show, this approach leads to a sigificat computatioal advatage for Schottky-diode multiplier desig. I this paper, a oe-dimesioal time-depedet simulatio algorithm is preseted which employs a fully hydrodyamic trasport equatio [1,] with two-valley mobility model to describe the diode physics withi the depletio ad bulk regios. These studies assume a reversed-biased situatio ad employ appropriate boudary coditios at the barrier iterface. I additio, this work combies the physically accurate diode model with a modified harmoic-balace algorithm to determie diode-circuit desigs that maximize power geeratio ad/or power efficiecy i the secod harmoic []. The modified harmoic-balace method utilizes a ovel two-step procedure where the available doublerpower ad the secod-harmoic diode-impedace is first derived as a fuctio of first ad secod harmoic diode excitatio. The harmoic diode-voltages ad secod-harmoic diode-impedace at the optimum power-poit are the used to defie the matched embeddig impedace ad the optimum local-oscillator (LO) voltage. The utility of this simulatio tool is illustrated by comparig to prior studies, by others, that employed traditioal drift-diffusio trasport models, Mote Carlo trasport models ad a covetioal harmoic-balace algorithm. Specifically, this work demostrates a computatioally efficiet ad accurate physical descriptio as well as a more robust approach for circuit optimizatio. II. PHYSICS-BASED DIODE MODEL These multiplier studies utilize a physics-based Schottky-diode model accurate for both mometum ad eergy relaxatio effects. Here, a oe-dimesioal time-depedet simulatio algorithm is implemeted that employs a fully hydrodyamic trasport model (i.e. first three momets of the Boltzma equatio) to describe the diode physics withi the depletio ad eutral regios. The applicatio this type of approach will be ecessary for GaAs diodes operatig at terahertz frequecies because eergy relaxatio is importat above 300 GHz ad mometum relaxatio is importat above 1 THz. While our method is fully ameable to mometum relaxatio these iitial studies will cosider device operatio below 1 THz. These studies cosider the large-sigal operatio of a Schottky-barrier

3 varactor frequecy multiplier ad will therefore cosider diodes uder time-depedet reverse-bias up to the breakdow voltage. Set of equatios of the hydrodyamic model i the sigle-gas approximatio has bee writte i [1,] together with appropriate boudary coditios. We have used i [] oe valley approximatio. However more adequate descriptio of GaAs Schottky diodes ca be obtaied utilizig a two-valley model [3]. We ll extract the mobility ad the eergy relaxatio time from this model. Cosider a semicoductor material whose eergy bad structure ca be approximated by a two-valley model with a eergy separatio UL betwee the upper ad lower valleys. The collisio terms are modeled by the relaxatio-time approximatio. The eergy collisio term is expressed as ( w w w0 ) c t τ w, (1) where w o 3k B To is the equilibrium eergy at the lattice temperature T o, τ w is the eergy relaxatio time, k B is the Boltzma costat. The average electro total eergy w is defied usig the thermal approximatio 3 w k BTe + η UL () where T e is the electro temperature, η is the fractio of the electros i the upper valley. The eergy separatio betwee the upper ad lower valleys UL is set to 0.9 ev i this work. The total eergy relaxatio time is give by 1 τ w 1 η η +. (3) τ wl τ wu The eergy relaxatio time i the lower-valley ad upper-valley are τ wl 1.5 ps ad τ wu 0.8 ps, respectively. The fractio of the electros i the upper valley ad is give by U r η (4) + r + 1 U L where L ad U are the electro desities i the lower valley ad upper valley. The ratio of electro desities r is defied as

4 r 3 U mu UL m exp k T (5) L L B e where m L ad m U are the desity-of-states effective masses of lower valley ad upper valley are set equal to m 0 ad 0.55 m 0, respectively. The temperature depedet electro mobility is obtaied by + ( 1 η) L η U. (6) where L ad U are the electro mobility i the lower valley ad upper valley, respectively. The mobility ca be approximated by [4,5] ad 0L 0L L, γ L + 3k γ ( T T ) / qτ (7) 1 B L e 0 v wl satl 0U 0U U, γ U + 3k γ ( T T ) / qτ (8) 1 B U e 0 v wu satu where the saturatio velocities for lower-valley v satl ad upper-valley v satu are set equal to m/s ad m/s, respectively. These values were obtaied by fittig Mote Carlo simulatio results. The low-field, dopig-depedet mobility for two valleys are expressed as [1] (9) 00 L 00U 0L, 0U 1+ [log( N T ) / BL ] 1 + [log( N T ) / BU ] where N T is the total dopig desity (here total ioizatio is assumed, so N T N D ). Here, 00L 0.85 m /V-s, 00U 0.6 m /V-s [6], 3, ad B L 3, ad B U 6 are used. As to boudary coditios discussed i [] we have to chage the pheomeological eergy-flux boudary coditio where δ.5 is used S ( 0 ) vm M δ k B T0 vs (0 ) ( δ kb Te (0 ) + η UL ), (10) III. VARACTOR FREQUENCY-MULTIPLIER NUMERICAL SIMULATION While a accurate model for the physical diode is very importat as oe attempts to desig varactor multipliers at very high frequecies, the ability to effectively couple the device to

5 the proper embeddig impedace is of paramout importace. I fact, the majority of the aalytical ad umerical effort ivolved i realizig efficiet multipliers is expeded i the matchig of the oliear device to the embeddig circuit [7]. For example, may harmoic-balace methods have bee derived (e.g., multiple-reflectio algorithm [8] ad accelerated fixed poit [9]) to improve the matchig of the various harmoics betwee oliear device ad liear circuit. Here the difficulty lies i the large umber of iteratios that are required to optimize power-efficiecy of a give oliear elemet over both embeddig impedace ad local oscillator (LO) voltage. This problem becomes compouded for optimizatio over the physical diode desig ad i circuits that cotai may oliear device elemets. Of course, i some situatios where multiple diodes are ivolved equivalet-circuits ca be developed that reduce groups of diodes to a sigle oliear elemet [10]. Whe a covetioal harmoic-balace (HB) algorithm is applied to the aalysis of a varactor-doubler the problem is to optimize the secod-harmoic geeratio. Here, embeddig impedaces are selected for the first- ad secod-harmoic circuits ad shortcircuit coditios are assumed at the higher harmoics. While there are may HB techiques oe of the most popular is the multiple-reflectio algorithm [8] that seeks to smooth trasiets via the itroductio of a loss-less trasmissio-lie sectio. This relaxatio method, which i some cases requires log covergece times, has bee improved by Tait [9]. Regardless of the exact mathematical algorithm employed, the primary challege i the desig of a doubler is to determie the embeddig impedace that will yield the maximum secod-harmoic power. I the covetioal approach, the physical model of the Schottky diode is used i cojuctio with the HB algorithm to optimize the secod-harmoic power. Specifically, the HB techique will iterate betwee the Fourier-domai solutios of the circuit(s) ad the curret/voltage harmoics derived from steady-state time-domai simulatios of the diode. Sice this combied Fourierdomai ad time-domai aalysis ievitably leads to extesive iteratios ad to a large computatioal cost, it is atural to seeks alterative methods that reduce the umerical burde of the optimizatio task. I the simulatios preseted here, a ew two-step procedure has bee utilized to achieve this goal. This modified HB algorithm itroduces a ew optimizatio costrait that eables a atural separatio of the device-circuit problem. Specifically, this procedure allows the curret/voltage harmoics of the oliear device to be optimized idepedetly of the exteral circuit. Subsequetly, the optimum embeddig circuit ca the be derived from a very simple aalysis. Harmoic-Balace Techique The harmoic-balace circuit aalysis applied i this paper is the accelerated fixed-poit (AFP) method [9]. The iterative process is repeated util the harmoic compoets of the voltage coverge to withi a specified tolerace factor of their steady-state value for all harmoics. I this work, the tolerace is set to 0.1%. There is oe thig to remid that AFP algorithm has oe adjustable covergece parameter Z 0, the fictitious trasmissio

6 lie characteristic impedace. The chose value of Z 0 ca affect the covergece property of the iteratio. Modified Harmoic-Balace Techique The key to this simplified approach is to formulate the optimizatio-problem of the varactor-doubler i a completely differet way. Specifically, to recogize that the available-power of the diode at the doubler-frequecy may be optimized a priori (i.e., idepedet of the exteral circuit) ad that the Fourier results ca be used, subsequetly, to determie the embeddig impedace ad LO voltage. While this techique represets, i a maer of speakig, a iverse trasformatio of the covetioal problem, it is completely valid ad offers defiite computatioal advatages. The techique has bee outlied i []. I preset aalysis we have icluded the frequecy-depedet parasitic impedaces as additioal cotributios to the liear device embeddig circuit. It is importat to ote that the parasitic impedaces of iterested here are exteral to the active regio of the oliear device. The matchig embeddig impedaces are give by Z Circuit Z Z. (11) Liear Parasitic Liear where Z are the matchig liear impedaces determied usig equatios from []. The power icidet from the pump at the iput port is Parasitic P V I + ( I + I ) Re{ Z }/. (1) i C1 C1 C1 S1 1 Circuit The power delivered to port with impedace Z is output Parasitic ( V I + V I ) ( I I ) Re{ Z }/ P + (13) C C S S C S where V C1, V C ad V S are the dc, first, ad secod harmoic voltages, respectively. The curret harmoics I C1, I C, I S1 ad I S are supplied by Fourier trasforms of time-domai simulatios from the physical diode model. Note that this ew approach offers a immediate reductio i problem complexity as compared to the covetioal HB approach. Specifically, the traditioal HB method will maximize the oliear diode-circuit problem over LO voltage (dc ad ac) ad over the real ad imagiary parts of the first- ad secod-harmoic embeddig impedace which is a six-dimesioal space. It should be oted that this reductio from N to N- i the double problem also occurs i the aalysis of a varactor-tripler due to symmetry cosideratios. As show, the traditioal HB techique requires two iteratios. For each mappig over the six-dimesioal LO-impedace space, oe must perform coverget iteratio to

7 balace the harmoics betwee the embeddig impedace ad the oliear device. O the other had, the modified method ivokes allowable costraits o the available diode power ad the embeddig impedace(s) (i.e., are always matched to the diode impedace) to reduce the optimizatio space by two variables. Furthermore, this method oly requires a trivial calculatio i the secod step to derive the fial impedace(s) ad the LO voltage. I simplest terms reducig the degrees of freedom allows for a sigificat reductio i computatio. IV. SIMULATION RESULTS Both harmoic-balace ad modified harmoic-balace simulatios were utilized with the hydrodyamic diode model to study Schottky diode multiplier operatio at a doublerfrequecy of 00 GHz. Here, the earlier varactor diode ivestigatios that were preseted i [11] were cosidered for compariso purposes. The GaAs Schottky diode cosidered 16 was UVA-6P4 with a epitaxial dopig desity of D cm -3, epitaxial thickess of 1.0 m, aode diameter 6.3 m. Experimetal results icludig iput power ad dc bias were obtaied from [1] ad were used as the iput data for both algorithms. Frequecydepedet parasitic impedaces of the UVA 6P4 varactor we used here are similar to those of [8]. Harmoic-Balace Techique Sice we wat to compare our result with Mote Carlo trasport model, we have the same costraits as [11]. The followig costraits are set o the embeddig circuit impedaces Circuit Circuit Circuit Circuit Circuit Circuit X 1 Z1 R1 + jx1, Z R1 + j. (14) The embeddig impedaces of higher harmoics were take equal to zero. The optimized results obtaied from the hydrodyamic model for four simulated iput powers are show i Table Ι. The simulated result for output power versus icidet pump power is show i Fig. 1. The experimetal result ad simulated results obtaied from the drift-diffusio (DD) ad Mote Carlo simulators [11] are also show i this figure. As see i this figure, the drift-diffusio model with dc field-depedet mobility is ot suitable for high frequecy. Sice the drift-diffusio model with costat ac mobility did ot iclude the velocity saturatio effects, it overpredicted at high iput power. The optimal power results of the hydrodyamic model are i good agreemet with the result obtaied from the simulatio of the drift-diffusio model with costat ac mobility at low iput power ad the result derived from the Mote Carlo simulatio at high iput power. The output curret ad voltage waveforms for 18.8 mw pump excitatio are show i Fig. Table I. AFP Harmoic-Balace simulatio result by hydrodyamic model

8 Icidet Pump Power (mw) Output Power (mw) P out /P i (%) Z 1 (Ω) Z (Ω) j j j j j j j j17 Modified Harmoic-Balace Techique The optimized results obtaied from the modified harmoic-balace techique hydrodyamic model for four simulated iput powers are give i Table ΙΙ. The optimized results are better tha the results from the harmoic-balace (AFP) techique as show i Fig 3. As see i the figure, the modified harmoic-balace techique got higher optimized results tha the harmoic-balace (AFP) techique. Ulike the traditioal HB techique where the fictitious trasmissio lie characteristic impedace has to be carefully chose, there is o fictitious impedace eeded i the modified HB techique. The output curret ad voltage waveforms for Modified Harmoic-balace method is show i Fig.4. Table II. Modified Harmoic-Balace simulatio result by hydrodyamic model Icidet Pump Power (mw) Output Power (mw) P out /P i (%) V LO (V) Z 1 (Ω) Z (Ω) j j j j j j j j133.6 The compariso of results by two techiques shows that the efficiecy is a rather flat fuctio of the secod harmoic embeddig resistace if the first harmoic resistace is matched i appropriate way. V. CONCLUSIONS I coclusio, a simple ad efficiet modified harmoic-balace techique is preseted. This ew algorithm is suitable for the large-sigal time-depedet aalysis of oliear millimeter wave circuits. A robust simulatio tool that combied this ovel modified harmoic-balace circuit aalysis techique with a temperature-depedet hydrodyamic trasport model device simulator has bee established i this paper. I compariso to the drift-diffusio model, this temperature-depedet hydrodyamic model ca more accurately describe the ostatioary high frequecy dyamics ad velocity saturatio of

9 the electro. I additio, the computatioal time is much reduced over that of the Mote Carlo method. The results ad simulatio-time requiremets for a Schottky-diode multiplier study have bee compared to that utilizig the harmoic-balace algorithms. These results show that this approach ca be used to realize improved multiplier operatio with a miimal time of umerical computatio. This model will be used i the future to optimize multiplier operatio at terahertz frequecies. The authors wish to ackowledge the support of the U.S. Army Research Laboratory, Army Research Office. REFERENCES [1] H. Hjelmgre, Numerical Modelig of Hot Electros i -GaAs Schottky-Barrier Diodes, IEEE Trasactios o Electro Devices, 37, pp , [] C. C. Lee, B. L. Gelmot, D. L. Woolard ad C. Fazi, A Modified Harmoic- Balace Aalysis of Schottky Diode Multipliers Based upo A Hydrodyamic Trasport Model, Proc. Teth It. Symp. o Space Terahertz Techology, Charlottesville, USA, pp , [3] K. Bløtekjær, Trasport Equatios for Electros i Two-Valley semicoductors, IEEE Trasactios o Electro Devices, ED-17, pp , [4] W. Häsch ad M. Miura-Mattausch, The Hot-Electro Problem i Small Semicoductor Devices, J. Appl. Phys., 60, pp , [5] Y. Zhag ad M. E. Nokali, A Hydrodyamic Trasport Model ad Its Applicatios i Semicoductor Device Simulatio, Solid-State Electroics, 36, pp , [6] M. Shur, GaAs Devices ad Circuits, (Pleum Press, 1987). [7] S. A. Maas, Noliear Microwave Circuits, (Artech House, MA, 1988). [8] P. H. Siegel, A. R. Kerr, ad W. Hwag, Topics i the Optimizatio of Millimeter-Wave Mixers, NASA Techical Papers, o. 87, [9] G. B. Tait, Efficiet Solutio Method for Uified Noliear Microwave Circuit ad Numerical Solid-State Device Simulatio, IEEE Microwave Guided Wave Letters, 4, pp. 40 4, [10] M. T. Faber, J. Chramies ad M. E. Adamski, Microwave ad Millimeter-Wave Diode Frequecy Multipliers, (Artech House, MA, 1995). [11] R. E. Lipsey, S. H. Joes, J. R. Joes, T. W. Crowe, L. F. Horvath, U. V. Bhapkar, ad R. J. Mattauch, Mote Carlo Harmoic-Balace ad Drift-Diffusio Harmoic-Balace Aalysis of GHz Schottky Barrier Varactor Frequecy Multipliers, IEEE Trasactios o Electro Devices, 44, pp , [1] T. W. Crowe, W. C. B. Peatma, R. Zimmerma, ad R. Zimmerma, Cosideratio of velocity saturatio i the desig of GaAs varactor diodes, IEEE Microwave Guided Wave Lett., Vol. 3, pp , 1993.

10 0 15 Experimetal Data Our Hydrodyamic HB Mote Carlo HB DDHB w/ Costat A.C. Mobility DDHB w/ D.C. Field-Depedet Mobility Output Power (mw) Icidet Pump Power (mw) Figure 1. Experimetal ad AFP theoretical secod harmoic output power versus icidet pump power for the UVA 6P4 frequecy doubler subject to 100 GHz excitatio Voltage (V) Curret (A) Time (ormalized to period) Figure. AFP Harmoic-balace curret ad voltage waveforms for 18.8 mw pump excitatio.

11 14 Modified Harmoic-Balace 1 Harmoic-Balace (AFP) Output Power (mw) Icidet Pump Power (mw) Figure 3. Theoretical secod harmoic output power versus icidet pump power for the Modified Harmoic-Balace method ad Harmoic-Balace (AFP) method Voltage (V) Curret (A) Time (ormalized to period) Fig.4. Modified Harmoic-balace curret ad voltage waveforms for 18.8 mw pump excitatio.