Embeddig the Quatum Mixer Theory ito a Time Domai Field Solver Douglas W. Heke, Poma P. M. So, Stéphae M. X. Claude, ad Wolfgag J. R. Hoefer Abstract To completely accout for the quatum-mechaical features of the supercoductor-isulator-supercoductor (SIS) juctio, Tucker s quatum mixer theory must be used. The stadard approach is to perform the computatios i the frequecy domai. To determie the large sigal waveform resultig across the juctio, a time domai formulatio has also bee used i the voltage update method (UM). Eve usig this method, the oliear diode curret (computed usig the time domai equatios) is related to the rest of the circuit i the Fourier domai. A iterative procedure eables the correct juctio voltage to be foud. This paper proposes a variatio to the UM such that all calculatios are performed i the time domai, i a time-steppig fashio, eablig the theory to be implemeted ito a time domai field solver. The mai advatages of performig simulatios withi the time domai are the ability to process arbitrary time sigals applied to the oliear juctio ad to geerate complete iformatio o the mixig products over a wide frequecy rage. It is recogized that a oise aalysis is ot feasible withi the time domai ad would be treated separately. Idex Terms Quatum mixer theory, time domai aalysis, supercoductor-isulator-supercoductor devices I. INTRODUCTION: TIME DOMAIN QUANTUM MIXER THEORY T HE time domai formulatio of the quatum mixer theory has bee preseted i [1]-[3]. Time domai theory has already bee used for certai aspects of simulatio, e.g., determiig the local oscillator (LO) voltage waveform established across the juctio usig the voltage update method [4],[5]. However, a full embeddig of the mixer theory ito a time domai 3D electromagetic field solver has ot yet bee doe. While the curret ad voltage waveforms ca be modeled i the time domai, it is ot feasible to predict the oise, ad so this would eed to be treated separately. Abstract received February 24, 25. This work was supported i part by the Natioal Research Coucil of Caada at the Herzberg Istitute of Astrophysics through research cotributios uder the D. C. Morto Fellowship, ad by the Natural Sciece ad Egieerig Research Coucil of Caada (NSERC). D. W. Heke is with the Departmet of Electrical ad Computer Egieerig, Uiversity of ictoria, ictoria, BC, 8W 3P6, Caada (e-mail: dheke@ece.uvic.ca). P. P. M. So is with the Departmet of Electrical ad Computer Egieerig, Uiversity of ictoria (e-mail: pso@ece.uvic.ca). S. M. X. Claude is with the Herzberg Istitute of Astrophysics, Natioal Research Coucil, ictoria, BC, 9E 2E7, Caada (e-mail: Stephae.Claude@rc.ca). W. J. R. Hoefer is with the Departmet of Electrical ad Computer Egieerig, Uiversity of ictoria (e-mail: whoefer@ece.uvic.ca). The followig summarizes the time domai theory preseted i [1]-[3]. I this first sectio, oly the simple curret-to-voltage relatioship will be reviewed; o source impedace or juctio capacitace is icluded. To provide further isight, a MATLAB [6] implemetatio is used to demostrate juctio currets ad photo-assisted tuelig. 2 15 1 5 2 15 1 5 1 15 5 oltage (m) 1 2 3 4 5 oltage (m) Fig. 1. Upumped I- curves that may be used for aalysis. (a) Measured I- curve for a series array of four Nb juctios. This is the typical respose of the mixers used withi the ALMA Bad 3 receivers [7]. (b) I- curve characteristic mimickig a measured respose for a sigle Nb juctio [9]. All MATLAB results cotaied herei are based o this fitted curve. Recall that the uique result of the quatum mixer theory is that the upumped curve provides sufficiet iformatio to predict the RF performace of the mixer. Fig. 1a shows a measured I- curve for a typical mixer used i the ALMA Bad 3 receivers [7],[8]. These mixers each cotai four juctios arraged i a series array. For simplicity, the I- curve characteristic used withi this paper to demostrate the time domai theory is show i Fig. 1b ad represets data that might be measured for a sigle iobium (Nb) juctio [9]. (a) (b) 27
I the time domai, the respose fuctio is defied i the followig maer: χ 2 π = I ( ω e) DC hω h siωt dω (1) er where e is the electro charge magitude ad h is Plack s costat scaled by 2π. The fuctio, I DC (), is the equatio that describes the measured upumped I- curve ad is assumed to be a odd fuctio that teds toward the ormal state coductace, 1 R, at large bias voltages. Sice I () DC is a fuctio of voltage, it is evaluated at the correspodig photo voltages for each ω. The respose fuctio characterizes the quatum-mechaical features of the SIS juctio; it is based etirely o the I- curve ad is idepedet of ay applied voltage. The respose fuctio oly eeds to be calculated oce ad, otig the bracketed expressio of (1), computatio is see to coverge whe I DC() /R. Fig. 2 depicts the iitial time segmet of the respose fuctio that was calculated for the I- curve data of Fig. 1b. The respose fuctio will oscillate at a frequecy proportioal to the gap eergy, 2. Depedig o the quality of the juctio, i.e., the sharper the oliearity of the I- curve, the loger it will take for the respose fuctio to decay [1]. 6.E+7 χ(t) 5 1 15 2 25 ad t e φ = ( t ) dt. (4) h As poited out i [1], the expected curret of (2) shows a istataeous respose (the first term) ad a delayed compoet due to the covolutio. I this way, the curret depeds o the past history of the applied voltage. This maifests itself i the frequecy domai i the form of a quatum susceptace [3]. Eve though the time domai implemetatio ca accept a arbitrary applied voltage, i order to provide cotiuity with the frequecy domai approach, cosider the large sigal voltage defied as ( t) DC + LO cos( ω LO t) =. (5) This waveform shows a DC bias voltage compoet combied with a local oscillator voltage sigal of magitude LO. It is assumed that (t) is the voltage that falls across the juctio ad, therefore, o LO source impedace has bee take ito accout. DC 14 12 1 8 6 4 alpha = alpha = 1 alpha = 2 I- Curve Respose 2-1.4E+8 Time (psec) Fig. 2. The respose fuctio, χ(t), derived from the I- curve data show i Fig. 1b. The respose fuctio characterizes the oliearities ad quatum-mechaical effects of the SIS juctio i the time domai. Cosider the full time domai expressio for the curret across the juctio for a give applied voltage, (t). The expected quasiparticle tuelig curret is give as where I U t () ( ) ( ) * = + Im U t χ t t U t dt. (2) R is the phase factor ad is give as iφ ( t ) U = e (3) 1.5 2 2.5 3 3.5 DC oltage (m) Fig. 3. Photo-assisted tuelig demostrated with pumped I- curves. Each curve was geerated by extractig the DC compoet from the time domai respose. No source impedace has bee take ito accout. A iterestig verificatio of the time domai equatios is to demostrate the quatum-mechaical effect of photoassisted tuelig, as show most clearly i the pumped I- curve. The time-varyig curret was calculated at icremetal DC bias voltages, from which the DC compoet of the curret was the extracted through a Fourier trasform. Fig. 3 shows the pumped curves that were computed for differet pumpig stregths, as defied i (9). Each step has a approximate width of.4 m, which correctly correspods to the photo voltage of the local oscillator (f LO = 1 GHz). Sice, i practice, the juctio is most ofte biased ear the midpoit of the first photo step below the gap voltage, it serves well to look at the trasiet curret for this coditio. Fig. 4a shows the iitial trasiet behavior ad how, after 28
several local oscillator periods, the sigal will reach a steady state. The ext figure displays a steady state sample of this same waveform separated ito the idividual compoets of (2): the istataeous curret ad the curret arisig from the covolutio term. 15 1 5-5 5 1 15 2 25 3 Time (psec) 15 1 5-5 -1 Quatum Covolutio part term of Curret of curret Istataeous part of Curret curret -15 2 25 21 215 22 225 23 Time (psec) Fig. 4. (a) Trasiet respose of the juctio curret whe the DC bias voltage is cetered o the first photo step ad the LO is pumped at α = 1. (b) Compariso of the two cotributios of curret oce it has reached a steady state. II. ACCOUNTING FOR THE SOURCE IMPEDANCE To solve for the large sigal waveform across the juctio withi a embeddig etwork, the time domai equatios have bee used i the voltage update method (UM) [4],[5]. Eve usig this method, however, the oliear diode curret (computed usig the time domai equatios) is related to the rest of the circuit i the Fourier domai. A short review of this techique is give ad the a variatio is preseted, allowig full computatio of the curret i the time domai. A. oltage Update Method (UM) Durig the large sigal aalysis of the quatum mixer theory, approximatios are made with respect to the LO ad its harmoics. For example, whe usig a three-port approximatio, all harmoics are cosidered to be shorted out, ad the LO is treated as a pure sie wave. As higher-order harmoics are icluded, the resultig LO waveform across the (a) (b) mixer becomes icreasigly complex. A further complicatio is due to the dispersive ature of the LO source impedace so that higher-order harmoics see a differet impedace tha the fudametal. Fig. 5 demostrates how the UM allows the user to Z ω, at each harmoic specify the source impedace, ( ) frequecy. The geerator voltage, source ge, is represeted by a DC bias ad a pure siusoidal source at the LO frequecy. The dotted lie divides the oliear ad liear parts of the circuit. The correct voltage fallig across the juctio is determied through a iterative process desiged to match the termial voltage of the liear etwork, LIN, to the oliear part, NL ( t). The procedure starts with a iitial guess o the voltage, NL ( t), fallig across the juctio ad the correspodig diode curret is foud. Usig circuit theory, the liear etwork voltages at the iterface are foud by where LIN e LIN = Z I + ge, (6) =, 1, 2,..., N deotes the idex of the Fourier amplitude correspodig to the th e oscillator, ad ( Z Z ) Z source, C, harmoic of the local = is the equivalet source impedace that icludes the juctio capacitace. The Fourier amplitudes of the curret are foud through trasformatio. ge, LO ge, DC Z source ( ω ) C J Liear Network I LIN ( t) I NL LIN ( t) NL SIS Juctio Fig. 5. Circuit describig the relatio of the oliear diode curret to the rest of the circuit i the voltage update algorithm [5]. The oliear curret is trasformed ito the frequecy domai ad the liearly related to each harmoic of the LO. Oce the Fourier amplitudes, voltages have bee foud, iverse Fourier trasform. If LIN, of the liear etwork LIN LIN NL ( t) is calculated through the =, the the juctio voltage has bee foud. Otherwise, the voltage may be updated accordig to NL LIN NL ( t) p ( t) + ( p) ( t) = 1 (7) where p is a covergece parameter valid over the rage of < p < 1. Covergece is better for the UM whe the termiatig impedaces approach a short-circuit, but may become a problem whe the source impedace is high compared with the iput impedace of the juctio [5]. 29
B. ariatio: Time Domai oltage Update Method (TDUM) Sice the UM computes the liear etwork voltages i the frequecy domai, it is assumed that the voltage waveforms oly cotai harmoics of the LO ad are i a steady state. If the UM is modified so that all calculatios are performed i the time domai, deoted as the time domai voltage update method (TDUM) for compariso purposes, true trasiet behavior ad arbitrary time sigals ca be aalyzed [11]. To be compatible with time domai circuit ad field solvers, the ew algorithm must be able to evolve with each additioal time step, yet still be accurate to the complexities of (2). ge t = k t R ge cir f ( ) I = mix mix CONOLUTION Fig. 6. Circuit describig the time domai voltage update method (TDUM). The source impedace is represeted here as a simple resistor, but may be a more complicated etwork. For each time step, t, the mixer voltage is iteratively matched to the circuit voltage at the termial iterface. However, sice the quatum mixer curret requires a covolutio of previous voltage values, each iteratio also requires a covolutio. Note that ge icludes the DC voltage bias. Fig. 6 depicts a simplified circuit represetatio for the TDUM such that the source impedace is give by a simple resistor. The geerator voltage is assumed to iclude the DC bias. As with the UM, the termial voltages are matched through a iterative process. Give a specified geerator voltage, a iitial guess o the mixer voltage,, is made. mix The oliear diode curret is calculated, ad the circuit voltage is foud by where t = k t cir = ge I Rge (8) ad is evaluated for the k th time step. If the termial voltages match, the the algorithm is complete, otherwise, mix is updated accordig to (7). However, this is ot as simple as it appears, because the curret, foud by (2), requires a covolutio operatio that depeds o the past history of the voltage across the juctio. Therefore, ot oly is the covolutio operatio called durig each time step, but also for umerous times withi each iteratio of that time step creatig a very demadig computatioal algorithm. Oe simplificatio to the covolutio procedure is that the storage of the etire voltage time sample is ot required; oly a time sample equal to the legth of the respose fuctio, calculated by (1), is ecessary. This, of course, assumes that the time sample of the respose fuctio is log eough so that it has coverged (e.g., loger tha what is show i Fig. 2). Furthermore, the covolutio withi each k th time step may be broke up ito two parts: a term depedig o the most recet k th voltage guess ad a term that makes use of the previous time samples already determied, deoted as the k 1 covolutio term. By re-usig this latter term durig the iteratio process, the computatioal effort is greatly dimiished. Implemetig these cocepts, a MATLAB algorithm was programmed i the followig steps: Iitialize parameters the I- curve parameters ad time step are determied ad the respose fuctio is calculated. The for each k th time step: Guess iitial voltage mix = ge. Compute curret the phase factor ad curret are calculated for the iitial voltage guess ad the k 1 covolutio term is stored. Perform iteratio ad voltage update the circuit of Fig. 6 is solved for each ew test voltage. Each voltage guess requires re-calculatig the k th term of the phase factor. The curret is calculated for the ew test voltage ad the k 1 covolutio term is re-used each time. If =, the the correct voltage has bee foud, cir mix otherwise, the ew test voltage is modified by (7). Next time step yes oltage update o Iitialize parameters I- curve, time step, respose fuctio Guess iitial mixer voltage mix = ge Compute curret k 1 covolutio term Termial voltages match? Solve circuit voltage Fig. 7. Algorithm of the time domai variatio of the oltage Update Method. As a cosequece of workig withi the time domai, the source impedace of Fig. 6 must be implemeted usig real elemets. The implicatio is that if a resistor is used, it is cosidered ideal ad preset for all frequecies icludig DC. I reality, mixer desigs have DC bias etworks, dispersive trasmissio lie impedaces ad juctio capacitace. More complex source etworks, icludig DC bias elemets, ca be modeled, but the itetio of this paper is to keep the demostratio simple. 3
DC Fourier Amplitude * 1E12 14 12 1 8 6 4 2 alpha = 2 alpha = 3 I- Curve Respose 1.5 2 2.5 3 3.5 DC oltage (m) 2 15 1 5 IF Compoet of Power Through Mixer alpha = 2 alpha = 3 1.5 2 2.5 3 3.5 DC oltage (m) (a) (b) Fig. 8. (a) Pumped I- curve respose for two differet source pumpig stregths. The source impedace has bee implemeted as a ideal resistor of 1 Ω. (b) Two small sigal toes, spaced at ±1 GHz from the LO ad with amplitude of hf LO / 2e, were combied with the LO across the juctio. The Fourier amplitude of the IF compoet of the power through the juctio was extracted ad plotted with respect to the DC voltage bias. Usig the I- curve of Fig. 1b, the optimum source resistace is foud to be R S,opt 1 Ω [12]. Fig. 8 demostrates the resposes for the simple source resistace show by the circuit of Fig. 6. It is appropriate to defie a source pumpig parameter α = (9) source ege, AC hω ge where the distictio is made from [3] i that the voltage,, is the amplitude of the AC geerator voltage, ot the ge, AC voltage established across the juctio. By addig two small sigal toes to the source, each of amplitude hflo / 2e ad set at ±1 GHz from the LO, the full spectrum of mixig products are available. Fig. 8b displays the IF compoet of the power through the mixer ad shows the familiar peaks correspodig to each photo step, with the greatest amplitude at the first photo step below the gap voltage. III. IMPLEMENTATION OF TDUM INTO 3D EM FIELD SOLER (MEFISTO) After demostratig the algorithm of the TDUM, the groudwork has bee laid for a full embeddig ito a circuit solver or field solver. MEFiSTo-3D Pro [13] is a full wave 3D electromagetic field solver that is based o the trasmissio lie matrix (TLM) method [14],[15]. A itercoectio betwee SPICE [16] circuit models ad the field has already bee developed i MEFiSTo by meas of represetig the TLM etwork by a equivalet Thévei or Norto source ad impedace [17]. The trasmissio lie impedace is foud through a equivalet combiatio of iput lik lies. A similar techique was used to implemet the SIS quatum theory via a TLM-MATLAB coectio, such that the full time domai SIS mixer algorithm was implemeted withi the MATLAB elemet cotaiig the TDUM scripts writte to calculate the SIS juctio curret. MEFiSTo 3D Model Z TLM TLM, k t = k t source P3 P2 P1 Z,γ TLM-MATLAB Iterface k MATLAB I = f ( ) Elemet MATLAB Diode Fig. 9. MEFiSTo implemetatio of a parallel plate waveguide termiated with a MATLAB diode. The 3D model shows a distributed source coected to a short sectio of trasmissio lie with three probes placed to moitor the voltage ad curret at each locatio. The lower figure describes the TLM-MATLAB iterface where the distributed TLM mesh has bee reduced to a Thévei source ad resistace. Fig. 9 illustrates this cocept whereby a short sectio of parallel plate waveguide is termiated by a shut diode. The distributed field compoets at the TLM-MATLAB coectio are reduced to a simple Thévei source ad resistace. Sice the sigals are withi the time domai, the source impedace is real ad the phase iformatio is preserved withi the sigal itself. The result is that eve though complex source ad dispersive tuig etworks surroud the juctio, they may be reduced at the juctio termial to the simple circuit show i Fig. 6. I cases where either the voltage or curret teds to zero at the iterface, extrapolatio is ecessary to obtai the true respose. Accordig to [17], the actual TLM-MATLAB coectio is made halfway betwee the MATLAB cell boudary ad the MATLAB ode. Aother ote is that sice 31
the time step withi MEFiSTo is depedet o the mesh resolutio, it is importat that the mesh be discretized uiformly i all directios ad that the time step is kept the same withi the MATLAB code. A great advatage of embeddig the SIS model ito a time domai field solver is the ability to visualize the fields, icludig the trasiet ad stadig waves. Fig. 1 shows the field aimatio alog a parallel plate waveguide desiged to have a characteristic impedace of 1 Ω. The trasmissio lie is termiated by a SIS MATLAB elemet ad excited usig a DC ad LO voltage geerator (which may be defied idepedetly i MEFiSTo). Fig. 1. MEFiSTo model of a parallel plate waveguide termiated with a SIS mixer diode. (a) Trasiet view capturig the DC bias voltage applied from the geerator (left) toward the SIS juctio (right). (b)-(c) Steady state view of the resultig voltage ad curret field compoets, respectively, due to a applied local oscillator. I. CONCLUSION AND FUTURE WORK The time domai theory of the quasiparticle tuel juctio has bee reviewed ad a ew time-steppig algorithm, based o the voltage update method, has bee demostrated usig MATLAB. The ew time domai variatio of the voltage update method (TDUM), has immediate applicatio i determiig large sigal voltage developed across the juctio while accoutig for the source impedace. It has also bee show that the TDUM algorithm is well suited for embeddig withi a time domai field solver where the fields may be reduced to a Thévei or Norto equivalet at the juctio termials. Future work would iclude implemetig a 3-termial, 2- port MATLAB elemet. I the above results, a 2-termial, 1- port elemet was used to termiate a sectio of trasmissio lie. Usig the 2-port elemet, a output IF sectio ca be modeled as it appears o the mixer chip. MEFiSTo already has this feature for SPICE elemets. Further experimetatio with complex geometries, icludig biasig etworks, juctio capacitace, ad tuig elemets must also be completed. REFERENCES [1] J. R. Tucker, Quatum limited detectio i tuel juctio mixers, IEEE J. of Quatum Electroics., vol. QE-19, o. 11, pp. 1234-1258, November 1979. [2] J. R. Tucker, The quatum respose of oliear tuel juctios as detectors ad mixers, Reviews of Ifrared ad Millimeter Waves, New York: Pleum, vol. 1, pp. 1-46, 1983. [3] J. R. Tucker ad M. J. Feldma, Quatum detectio at millimeter wavelegths, Rev. Mod. Phys., vol. 57, o. 4, pp. 155-1113, October 1985. [4] R. G. Hicks ad P. J. Kha, Numerical aalysis of oliear solid-state device excitatio i microwave circuits, IEEE Tras. Microwaves Theory Tech., vol. 3, o. 3, pp. 251-259, March 1982. [5] R. G. Hicks, M. J. Feldma, ad A. R. Kerr, A geeral umerical aalysis of the supercoductig quasiparticle mixer, IEEE Tras. Mag., vol. MAG-21, o. 2, pp. 28-211, March 1985. [6] MATLAB, The MathWorks, Ic., 3 Apple Hill Drive, Natick, MA 176-298, USA. Available: http://www.mathworks.com, April, 25. [7] S. -K. Pa, A. R. Kerr, M. W. Pospieszalski, E. F. Lauria, W. K. Crady, N. Horer, Jr., S. Srikath, E. Bryerto, K. Saii, S. M. X. Claude, C. C. Chi, P. Dido, G. Rodrigues, D. Derdall, J. Z. Zhag ad A. W. Lichteberger, A fixed-tued SIS mixer with ultra-wide-bad IF ad quatum-limited sesitivity for ALMA Bad 3 (84-116 GHz) receivers, 15 th It. Symp. o Space Terahertz Tech., Northampto, MA, April 24. [8] http://www.alma.ifo/, April, 25. [9] A. Navarrii, Developmet of DSB ad SSB SIS mixers for radio astroomy i the frequecy bad 25-37 GHz, Ph. D. dissertatio, Uiversité Joseph Fourier, Greoble, Frace, 22. [1] R. E. Harris, Itrisic respose time of a Josephso tuel juctio, Phys. Rev. B, vol. 13, o. 9, pp. 3818-3821, May 1976. [11] D. W. Heke, Measuremet ad time domai modelig of supercoductor-isulator-supercoductor (SIS) mixig juctios for radioastroomy, M. A. Sc. dissertatio, Uiversity of ictoria, ictoria, Caada, 25. [12] A. R. Kerr, S. -K. Pa, A. W. Lichteberger, ad D. M. Lea, Progress o tuerless SIS mixers for the 2-3 GHz bad, IEEE Microwave Guided Wave Lett., vol. 2, o. 11, pp. 454-456, November 1992. [13] MEFiSTo-3D Pro, FAUSTUS Scietific Corporatio, 1256 Beach Drive, ictoria, BC, 8S 2N3, Caada. Available: http://www.faustcorp.com, April, 25. [14] P, B. Johs, The art of modelig, IEE Tras. Electro. Power, vol. 25, o. 8, pp. 565 569, August 1979. [15] W. J. R. Hoefer, The trasmissio-lie matrix method Theory ad applicatios, IEEE Tras. Microwave Theory Tech., vol. MTT-33, pp. 882-893, October 1985. [16] SPICE, Uiversity of Califoria, Berkeley, USA. Available: http://bwrc.eecs.berkeley.edu/classes/icbook/spice/, April, 25. [17] P. P. M. So ad W. J. R. Hoefer, A TLM-SPICE itercoectio framework for coupled field ad circuit aalysis i the time domai, IEEE Tras. Microwaves Theory Tech., vol. 5, o. 12, pp. 2728-2733, December 22. 32