1674 PROCEEDINGS OF THE IEEE, VOL. 59, NO. 12, DECEMBER 1971

for receiving arrays,” ZEEE Trans. Antennas Propagat. (Commun.), [22] C. J. Drane, Jr., and J. F. McIlvenna, “Gain maximization and VO~.AP-14, NOV.1966, pp. 792-794. controlled null placement simultaneously achievedin aerial array [9] A. I. Uzkov, ‘‘An approach to the problem of optimum directive patterns,” Air Force Cambridge Res. Labs., Bedford,Mass., antenna design,” C. R. Acad. Sci. USSR., vol. 35, 1946, p. 35. Rep. AFCRL-69-0257, June 1969. [lo] A. Bloch, R. G. Medhurst, and S. D. Pool, “A new approach to [23] R. F. Hamngton, “Matrixmethods for field problems,” Proc. the design of superdirective aerial arrays,” Proc. Znst. Elec. Eng., ZEEE, vol. 55, Feb. 1967, pp. 136-149. VO~.100, Sept. 1953, pp. 303-314. [24] J. A. Cummins,“Analysis of a circulararray of antennas by [ll] M.Uzsoky and L. Solymar,“Theory of superdirectivelinear matrix methods,” Ph.D. dissertation, Elec. Eng. Dept., Syracuse arrays,” Acta Phys. (Budapt), vol. 6, 1956, pp. 185-204. University, Syracuse,N. Y., Dec. 1968. [12] C. T. Tai, “The optimum directivity of uniformly spaced broad- [25] B. J. Strait and K. Hirasawa, “On radiation and scattering from sidearrays ofdipoles,” ZEEE Trans. Antennas Propgat., vol. arrays of antennas,” Proc. Nut. Elec. Con$, vol. 25, 1969. AP-12, July 1964, pp. 447-454. [26] A.T. Adams and B. J. Strait, “Modernanalysis methods for [13] D. K. Cheng and F. I. Tseng, “Gain optimization for arbitrary EMC,” ZEEEIEMC Symp. Rec., July 1970, pp. 383-393. antennaarrays,’’ ZEEE Trans. Antennas Propgat. (Commun.), [27] R. F. Harrington, Field Computation by Moment Methods. New VO~.AP-13, NOV.1965, pp. 973-974. York: Macmillan, 1968. [14] -, “Maximisation of directive gain for circular and elliptical [28] E. N. Gilbert and S. P. Morgan, “Optimum design of directive arrays,” Proc. Znst. Elec. Eng., vol. 114, May 1967, pp. 589-594. antenna arrays subjectto random variations,” Bell Syst. Tech. J., [15] E. I. Krupitskii, “On the maximum directivity of antennas con- vol. 34, May 1955, pp. 637-663. sisting of discreteradiators,” SOD.Phys.-Dokl., vol. 7, Sept. [29] M. D. Smaryshev, “Maximizing the directive gain of an antenna 1962, pp. 257-259. array,” Radio Eng. Electron. Phys. (USSR), vol. 9, Sept. 1964, pp. [16] F, R. Gantmacher, The Theory of Matrices, vol. I. New York: 1399-1400. Chelsea Publishing Co., 1960. [30] F. I. Tseng and D. K. Cheng, “Gain optimization for arbitrary [17] B. Das, “On maximumdirectivity of symmetrical systems of antennaarrays subject to randomfluctuations,” ZEEE Trans. Antennas Propgat., vol. AP-15, May 1967, pp. 356-366. radiators.” Radio Ena.I Electron. Phys. (USSR), vol. 10, June 1965, pp. 853-859. [31] D. K. Cheng and F. I. Tseng, “Optimum spatial processing in a [18] D. K. Cheng and F. I. Tseng, “Pencil-beam synthesis for large noisy environment for arbitrary antenna arrays subjectto random circular arrays,” Proc. Inst. Elec. Eng., vol. 117, July 1970, pp. errors,” ZEEE Trans. Antennas Propgat., vol. AP-16, Mar. 1968, 1232-1234. pp. 164-171. [19] P. D. Raymond, Jr., “Optimization of array directivity by phase [32] R. W. P. King, R. B. Mack, and S. S. Sandler, Arrays of Cylihdri- adjustment,” M.Sc. thesis, Elec. Eng. Dept., Syracuse University, cal Dipoles. Cambridge, England: Cambridge University Press, Syracuse, N. Y., June 1970. 1967. [20] F. B. Hildebrand, Methodr of AppIied Mathematics. Englewood [33] R. F. Harrington, “Antenna excitation formaximum gain,” IEEE Cliffs, N. J.: Prentice-Hall, 1965. Trans. Antennas Propgat., vol. AP-13, Nov. 1965, pp. 896-903. [21] E.A. Guillemin, The Mathematics of Circuit Atdysis. New [34] D. K. Cheng, “Antenna optimization criteria,” RomeAir Develop. York: Wiley, 1951. Cent., Cambridge, Mass., Rep. RADC-TR-71-156, Apr. 1971.

Noise in Avalanche Transit-Time Devices

AbstracCThe work performed onnoise in avalanche transit-titno de- of understanding of noise in these devices. It is hoped that this vices is reviewed and presented in detail. Both theoretical and experimen- review does not only have the advantage of consolidating this tal results on the noise mechanisms and performance of these devices when knowledge and arranging it systematically as to put the entire employed asoscillators, ampliflers, and self-oscillatingmixers are in- so cluded. Suggestions for further studies in this area are also given. field in perspective, but it also reveals unresolved problems and suggests new areas of investigation. I. IWRODUCTION The importance of noisestudies in avalanche transit-time A. Object of the Review cannot be overstated for two reasons. First,it is generally believed that the noisiness of avalanche diodes may limit their HE INCEPTION of every new electron device triggers a applications and represents their majorweakness in comparison detailed study of the various noise mechanisms in it and with other devices.On the other hand, the largenoise Tways to reducethe noise. The literature on avalanche generation makes the avalanche diodean attractive noise source transit-time devices, which is nowsix years old, is no exception for microwave frequencies. and it worthwhileis to make a comprehensive review of the status Historically,noise studies in avalanchetransit-time diodes began almost as soon as the devicebecame operational. The Manuscript received May 6, 1971. This work was supported by the earliest reported results of measurements of noise in avalanche- National Aeronautics and Space Administration under Grant NGL andoscillators werethose of DeLoach and 23-005-183. Johnston [l ] who reported1) very large noise figures(50 The author is with the Electron Physics Laboratory, Departmentof Electrical Engineering, the University of Michigan, AM Arbor, Mich. to 60 dB), 2) reduction of noise figure with an increase in bias 48104. current, and 3) very large spectral linewidth (over 100 kHz for GUPTA : NOISE IN AVALANCHE TRANSIT-TIME DEVICES 1675

3-dBwidth). The first theories of noisein avalanche-diode TABLE I oscillators and amplifiers were both due to Hines [2], (31. Sub- APPLICATIONSOF AVALANCHE TRANSIT-TIMEDIODESAND stantial improvements have been madesince, both indevice noise CORRESPONDING CHARACTERIZATIONS OF NOISE performance and in sophistication of theories of noise in these PERFORMANCE UTLIZEDIN THISPAPER devices. Application of OneCommon Characteri- B. Sources of Noise in Avalanche Diodes Avalanchezation of Noise N etwork Diode PerformanceType Diode of Network The noise inan avalanche transit-time diode is primarily made up of three parts which are, in the order of their significance: Passive one-portnoise generator spectrum of current avalanche noise, frequency-converted noise, and thermal noise. fluctuations Avalanche noise, which is a type of shot noise, is the noise in Active one-portoscillator AMnoise FMspectraand carriercurrent generated by the inherently noisy avalanche Lineartwo-port small-signal amplifier noise figure multiplication process. Avalanche multiplicationacts much like a self-oscillatingmixer noise figure noisy amplifier,not only multiplyingthe original signal noise by aNonlinear two-port large-signal amplifier AM and FM noise of factor M (called the multiplication factor), but also by addingan output appreciable amount of noise to it. This noise addition is best described in terms of the statistical fluctuationsof M. To under- stand the origin of fluctuations in M, the avalanche mechanism combination noise is expectedto be small because the transit time must be considered in detail. Following the discussion of Hines of the carriers through thespace-charge region is of the order of [3], consider an avalanche region with a self-sustaining constant s, while the carrier lifetimeof excesscarriers isof the orderof current so that each carrier pair produces, on the average, exactly10-6 seven in direct gap . one carrier pair by impact ionization.N Letbe the total numberof Flicker noise, on the other hand, has mostly been neglected carrier pairs present in the avalanche region on the average. Ifuntil the now on experimental grounds. Early workby Josenhans [4] history of one carrier pair in time is followed, it is found that its on X-band avalanche diodes showedno l/f effect in noise closeto “offsprings” are generated and swept out of the avalanche region the carrier, leading himto the observation that “no flicker noise at time intervals of T, on the average, where T, is the mean time was present in Read oscillators.” Ondria and Collinet [5] have interval between successive ionizations todue a single carrier pair. also reported the absence of 1,” behavior in FM noise spectra The chain ionization leadsto a pulse-train-like currentat the out- close to the carrier frequency, indicating that no flicker noise is put. The total current through the avalancheregion is then given present in avalanche diodes at low frequencies. More recently, by Ashley and Palka [6]have observed a 3-dB/octave slope in noise spectraclose to the carrier in someKu-band avalanche-diode I = Nqs/r+ (1) oscillators with a knee below 5 kHz, although the data are not where qsequals theelectronic charge. This current would appear consistent enoughto be conclusive. to be a constantif N is large, andT, is very small comparedto the Obviously, when the avalanche transit-time diodeused is as an period of the frequencies of interest. Now the sources of fluctua- amplifier or an oscillator the circuit in which itis embedded also tions inZare looked for. contributes to the totalnoise. Circuitlosses, impedance mismatch First of all fluctuations inM should be expected because each in the circuit, and noise in the nonreciprocal element (e.g., cir- carrier pair has a “probability”of ionization; i.e., the ionization culator)are all known to cause additional noise.However, process is notdeterministic. In terms of the previously mentionedavalanche diodes are very noisy devices, and it is generally be- model, this means that not all chainsof ionization are intennin- lieved that thenoise contributions fromall sources other than the able and uniform; some of them terminate (when a carrier pair diode can safely be neglected in estimatingthe noise behavior. produces no ionizations before being swept outof the avalanche region) and some branch (when a carrier pair produces more thanC. Plan of the Paper one ionization), even though on the average there is only one The fact that avalanche transit-time diodes have more than impact ionization per carrier pair. This random beginning and one circuit application makes the work of studying noise more ending of pulse trains causes fluctuations in the output current. difficult. Table I shows the various applications of an avalanche Again, since T, is only the average time interval between ioniza- diode anda corresponding commonly employed measureof noise tions in a single chain, the random fluctuation of this interval used to describe noise quantitatively.Of course many other mea- would give rise to another sourceof noise inZ. sures of noise are also used;e.g., the noise of a passive one-port The secondsource of noise is frequency-convertednoise, may also be measured in terms of an equivalent saturated diode which is primarily the low-frequency noise upconverted to the current, noise temperature, an equivalent noiseresistor, available neighborhoodof the carrier frequency by beatingwith the noisepower, noise ratio, or otherparameters. The difficulty carrier. Thesources of this low-frequency noiseand its naturewill arises when the noise behaviorof the device in one applicationhas be discussed later in Section111-B. Finally, there is thermal noise to be correlated with its noise behavior in another application. generated in thediode, which is very muchsmaller thanthe The noise behaviorof avalanche diodesin the various applications avalanche noise, and has therefore been neglected in all noise will be surveyed. analyses to date. The plan of this paper is as follows. Noise mechanisms inan It can be seen that generation recombinationnoise, diffusion are discussed in the next section. Early workson noise, and flicker noise are all missing from the preceding list. In random pulse-typenoise dueto microplasmasin avalanche practice they would all be present; however, these have not nor- breakdown are briefly touched upon. Thereafter, two theoriesof mallybeen accounted forin avalanche diodes, because their noise, one dueto Tager [7]and McIntyre[SI, and the other due to contribution is expected to be unimportant either from theoreticalHines [3] and Gummel and Blue [9] are summarized. Experi- considerations or from experimental evidence. Generation re- mental work verifying various aspects of these theories is then 1676 PROCEEDINGS OF THE IEEE, DECEMBER 1971 summarized. Inthe third section, noise inavalanche-diode T~ through it is much shorter than the carrier lifetime, and there- oscillators is considered. Theoretical analysesof both a small- and forethermal generation and recombination in thelayer are large-signal nature arereviewed. Experimental results underCW negligible. Atthe same time the layer width ismuch larger and pulsed conditions on unsynchronized aswell as phase-locked (>10-5 cm in ) than the mean free paths so that only the oscillators, lockedwith the fundamental orwith the subharmonic average motion (“drift”)of carriers need be considered. frequency,are discussed, asare the effects of the circuit on 5) The ionization rates (number of electron-hole pairs pro- oscillator noise. SectionIV is concernedwith the theory of noise duced by impact ionization per unit length per carrier) are the in small-signal avalanche-diodeamplifiers and its comparison same for electrons and holes.If a, and ah,the ionization rates for with experimental results is reported. Noise in avalanchediode electronsand holes, respectively, areunequal an equivalent mixers is briefly discussed in Section V. Finally, in Section VI, ionization rate maybe defined a4 some suggestions are made concerning possible of areas investiga- tion.

II. NOBEp1 AVALANCHEDIODES A. Noise in Microplasmas This ionization rate is assumedto be uninfluenced by temperature and is an instantaneous functionof the electric field. Pearson and Sawyer[lo] reported a current instability related 6) Fluctuations of the fickereffect-type and thermalfluctua- to the onset of avalanche breakdown in a reverse-biased p-n tions areneglected. junction. McKay [ll] showed that the breakdown current con- 7) The primary carrier current ZO entering the avalanching sisted of a train of rectangular pulses of equal amplitude but layer contains full shot noise (the mean square noise current randomly varyingpulse length and repetition rate, andexplained equals 2q,Z& whereq, is the chargeof an electron, lo the primary this phenomenon by predicting the existenceof regions of local- current, andB the bandwidth). ized ionization in areverse-biased silicon p-n junction. Such non- 8) The primary current is not rapidly time varying;i.e., uniform carrier multiplicationover the junctionarea was detected by Chynoweth and McKay[ 121, and was called a microplasmaby Rose [13]. Champlin [14] made detailed experimental studiesof this breakdown and deduced that the noise is caused by current fluctuations among stable levels. A simple account of a theory With these assumptions, taking into account 1) the fluctua- a random on-off model is given by van der Ziel based on tions in the entering primary current, 2) the fluctuations in the [IS]. Haitz [16] later improved the model and characterized the multiplication factorM of the avalanche layer, and3) the depres- microplasma by four quantities: 1) an extrapolated breakdown sion of current fluctuations dueto the space-chargeof the moving voltage; 2) a series resistance; 3) a turnon probability; and 4) a carriers, Tager obtainedthe following results. turnoff probability. A summary of this and some later work is 1) The mean square fluctuations in the multiplication factor given by Monch [17]. This phenomenon will not be discussed in M depend upon the ratio of ionization rates of the two types of great detail as thereviews of van derZiel and Monch cited previ- carriers. Whenthis ratiois unity (equal ionization rates) ously provide an excellent survey of this field, and also because advances in the technologyof preparation of p-n junctions have made it possible to make microplasma-free or “uniform” junc- tions. Good microwave avalanche diodes are usually fabricated while for aninfinite ratio (only one speciesof carrier ionizing) with such uniform junctions. Prior to the use of avalanche diodes as devices at microwave frequencies, theywere being used, based on the previously stated current fluctuation mechanism, RF as noise 2) The mean square value of M is equal to the cube of the sources (up to several hundred MHz), and were commercially average valueof M;@ = (M)“. available (Penfield [18]) to replace gas dischargetube noise 3) Thespectrum of carrier current leaving theavalanche sources. Haitz [19], [20] has discussed at length the design of layer is uniform avalanche diodes (using guard rings)as noise sources and has described their spectrum, amplitude distribution, and tem- perature dependence.

B. Theory of Avalanche Noise Tager [7] presented thefirst theory of current fluctuations in Le., at low frequency it is just the shotnoise multiplied by (My, semiconductors under impact ionization and avalanche break- while at high frequencies it inverselyis proportional to the square down conditions. He made the following assumptions in his in- of the frequency. vestigation of the statisticalcharacteristics of theavalanche Further developmentsof this theorywere primarily dueto an process. interest inthe noise properties of avalanchephotodiodes. 1) Avalanche breakdown occurs ina plane layer of the semi- McIntyre [8] derived a general expression for the spectral density conductor. of noise generated in uniformly multiplying p-n junctions for any 2) A large uniform electricfield is applied across this layer. distribution of injected carriers which is valid at frequencies that 3) Carriers of both species move with the same saturated are low compared to the inverse of the transit time through the velocity, which is independent of the number of ionizations pro- avalanche layer. He used the same set of assumptions as Tager, duced by a given carrier. except that unequal ionization rates for holes and electrons and 4) The width of the avalanching layerof the is any field profile are permitted. The expression for the spectral small enough(< 10-2 cm in silicon) so that the carrier transit timedensity at low frequencies(f<

2) All noise is assumed to originate in the avalanche region. si(.) =2qe 2 I,(O)M2(0)+In(ur)MZ(w)+ J"'"~~z(z)dz] 3) All theassumptions of Gildenand Hines' small-signal {[ theory are carried over; namely, the assumptions of saturated drift velocities which are thesame for electronsand holes; a thin +Io [2 JWaJ!f'(z)dz-M2(w)I> (7) avalanche region at one side of a high-field region; equal ioniza- tion rates for electrons and holes; constant electric field in the where w is the width of the avalanche layer, ZP(0) is the primary avalanche region; the current, electric field, and ionization rate hole current entering at one end (x=O) of the avalanche layer, being expressed as a constant dcvalue plus a small sinusoidalac Zn(w) is the primary electron current entering the other(x = end w) variation;neghgible thermally generated reverse saturation of the layer, M is the multiplication factorof the avalanche layer current; the drift region current entirely due to one species, the which is a function ofx because the electric field is a function ofx, majority carriers;no fixed space charge in the drift region; punch- and g is the rate of thermal or optical generation of carriers in through operation at all times; negligible series resistance and pairs per unit length. Whilea. occurs explicitly in theexpression, neghgible displacement currentso that the total carrier current is both a. and ah are present through M.Also note that theexpres- independent of distance alongthe diode length. sion is independentof frequency (whichis a direct consequenceof 4) The ionization rate a varies as a power of the electric omitting all time dependence), resultingin a white noise spectrum field E". at low frequencies. 5) The noise due to statistical variation of the interval T+ Underthe additional assumption that the electric field is between ionizations in anygiven chain of ionizationsis negligible. uniform over the entire avalancheregion, the previous expression 6) Carrier inertial effects are negligibleso that the probability reduces to of ionization by a given carrier during its transitis constant and consequently the Poisson distribution canbe used. Under these assumptions, Hines found that the mean square value of total noise current is

where 2d& Rd 1 Vll InyW)= ~ -.--.- (9) w2r22 Zt2 wa2 2m10 l-- w2

or where ld 1 COS ah - - e A=- Rd=q d2 ae -- ] when, respectively, holes or electrons alone are injected into the avalanching layer. For the special case ofa.=ah, this reduces to Zo is the dc bias current,id is the width of the drift region,0 is the S,(w)=2q&W, which is identical with Tager's result at low fre- transit angle through the drift region at frequency w, e is the quencies. permittivity of the semiconductor material,A is the area of the the basis of these results McIntyre concluded that for a On cross section of the diode, Wa is the avalanche frequency of the given M,noise could be reduced if the ionization were caused diode, Z, is the total impedance around theRF loop containing primarily by the more ionizing speciesof carriers. McIntyre also the diode including the load impedance, andVa is the dc voltage estimated the effect of timedependent mechanisms which intro- across the avalanche region. Under the special case of low fre- duce frequency dependence in the noise spectrum at high fre- quencies (w <

Gummel and Blue [9] have generalized the analysis of Hines AVALANCHE FREQUENCY, OH2 by removing some of the assumptions made, in particular the 1.08 1.532.15 3.4 4.8 7.5 IO+ assumptions of an idealizedRead structure, equal saturated Ill I1 I velocities for electrons and holes, and equal ionization rates for holes andelectrons. The onedimensional model and small-signal analysis are maintained. In addition it is assumed that the con- duction current in the diode has full shot noise; this noise is in- jected into the avalancheregion of the diode, and solutionof the carrier transport equations then yields the open-circuit voltage fluctuations caused by the shot noise. A rigorousjustification for such an approach has been given by Blue [23] for a stationary quasi-linear system. The mean squared voltage (u*) is calculated analytically foran idealizedRead-type diode structureand I I I Ill I I numerically for a realistic p-n junction diode with a wide av- 0.5 I 2 4 7 IO 4520 alanche region. The expression for (uz) shows the following be- BIAS CURRENT, mA havior of the mean square noise voltage. Fig. 1. Mean square noise current of an avalanche diodeas a function 1) For a ked bias current this voltageis constant at low fre- of bias current and frequency. The avalanche frequency calculated from bias current also marked on the et ul. [29].) quencies, then it increases and attainsa maximum at a resonance is axis. (Constant frequency, and falls approximately at the fourth power of fre- quency above resonance. 2) The spectraldensity of the noise voltage is proportional to 2) At low frequencies (below resonance)(D*) is inversely pro- the breakdownvoltage for several different doping profilesof the portional to the bias current, the resonance frequency is propor-diode :n+-p, p+-n,and n-p-u-pf. tional to the square root of the bias current density, and in the 3) At high frequenciesthe spectral density varies linearly with l/04 domain it is approximately directly proportionalto the bias bias current for small bias currents (where the avalanche fre- current. quency& is small), attains a peak, and decreases thereafter as the These conclusions agreewith the experimental observations of bias current (andfa ) mcreases.' hence Haitz andVoltmer [24], [25] described inthe next section. 4) For low bias currents (and hence small &), the spectral density of noise shows l/f4 variation with frequency, while at C. Experimental Results larger bias currentsthe frequency dependenceis much smaller, as Tager's result thatthe mean square value of fluctuations expected theoretically. equals M has been experimentally verified by Naqvi et af. [26]. They have measured multiplication noise as a function of the D. Aoalanche- Diode Microwave Noise Generators multiplication factorand have plotted the spatial variationof both Several workers have explored the possibility of using av- over the cross sectionof the diode. Theirexperimental setup con- alanche diodes as microwave noise sources. Such noise sources sisted of a sharply focused chopped laser beam, scanning the wouldhave the followingadvantages over gas-discharge-type reverse-biaseddiode junction, and aphase-sensitive lock-in noise sources. amplifier to measure the avalanche multiplied photocurrent. They 1) Much larger noise output, eliminating the need of ampli- found that:1) multiplication noiseat a multiplication of88 was as fication in many applications. high as 37 dB above the full shot noise in the junction current; 2) Small size, small weight, low power requirements,and low and 2) agreement with McIntyre's theorysuggests that there isno cost, making it possibleto use them for inservice measurementof space-charge correlation effect up to a current density of 100 critical parameters in microwave equipment (Chasek [31]). A/cm*, which wasthe maximum usedin theirexperiments. 3) Possibility of pulse modulation with very narrow pulses (of Baertsch [27], [28] made measurements of noise in a silicon the orderof 1 ps). avalanche current asa function of the multiplication The use of flat-topped random pulses of microplasma break- factor and foundgood agreement with McIntyre's theory. down in p-n junctions as a source of noise has been mentioned. Constant et al. [29] have reported experimental studies of Minden [32] has carried out some experimental work demon- and microwave noise in avalanching p-n junc- strating that noise in avalanche diodes is of two distinct types: tions. Their results on the observed variationof the mean square that associated with microplasma breakdown,and thatassociated noise current with bias current and frequency are shown 1,in Fig.with uniform junction breakdown. He pointed out the following and show agreement with Hines' theory. difference between the two. Microplasma breakdown occurs at Haitz and Voltmer [24], [25] and Haitz [30] have reported very low breakdown current(1 to 300 PA), and thenoise level as a experimental measurements of the spectral density of a silicon function of bias current shows several peaks. The uniform junc- avalanche-diode noise voltage, both at audio frequencies (1 kHz tion breakdown noise becomes appreciable at larger reverse bias to 11 kHz) andat microwave frequencies(3 GHz to9 GHz), and current and attainsa maximumat a valueof bias current whichis have verified Hines' theory of noise in Read diodes as modified lower than the threshold for coherent microwave oscillations. for p-n junctions. The main results of this experimental study Minden also found that high-output avalanche-diode oscillators follow. show no microplasma noise, while diodes which generated larger 1) At low frequencies the spectral density varies inverselyas uniform junction noise also generated large microwave output the square root of the dc bias current as expected. There is an when operated asoscillators. excess noise at high frequencies due to thermal effects, and an Val'd Perlov et al. [33] described a microwave noise source, excess noise at low frequencies due to local variations of the verifying some of the predictions of Tager [7]. In particular they brakdown voltage (as evidenced bythe fact that theexcess noise found the following. increases with areaand is not reproducible from diode to diode). 1) For diodes with breakdown voltagesof 10 to 15 V, and at GUPTA: NOlSE IN AVALANCHE TRANSIT-TIhE DEVICES 1679 bias currentsof 0.5 to 10 mA, the noise temperatureof the source strongly dependent upon the load, particularly near the oscilla- was 105 to 106'K in the centimeter band, and 106 to lO7OK in the tionthreshold. Therefore, for high-frequency noise generation meter band. the should be kept low so that the threshold 2) The long-term stability of noise level was very good, with a oscillation frequency is high. temperature coefficient of spectral density of noise power being 5) A change in temperatureaffects the junction impedance and 0.02 dB/OK. hencethe noise power output.Thermal compensation can be 3) The noisesource could be modulated by microsecond achieved by using a to control the diode bias current. pulses. Tager's result (6) shows that athigh frequencies the noiselevel III. NOISEIN AVALANCHE TRANSIT-TIME depends only upon frequency and bias current,and not upon the DIODEOSCILLATORS multiplicationfactor M. Hence theoutput is independent of A. Theory of Oscillator Noise reverse saturation current of the diode. At low frequencies, how- Hines [2] presented thefirst noise theory for avalanche transit- ever, it does depend upon M, and consequently upon reverse time diode oscillators. His analysis, which can only be considered saturation current and temperature. as a first-order approximation, is based upon results found in his Dalman and Eastman [34] reported the following results of small-signalanalysis presented in Section 11-B. Two sets of noise measurementson diffused p+-n-n+Si avalanche diodes. assumptions are carried over in this work. 1) The avalanchediode itself is an inherentlybroad-band 1) The assumptions madeby Edson [38] in his analysisof the noise source and the circuit largely determined the bandwidth of perturbation of an oscillator by thermal noise. the noise source. 2) The assumptions made by Hinesin his analysisof the noise 2) Noise temperatures from15 to 25 dB greater than an argon in the avalanche diode(given in Section11-B). gas discharge tube (30 to 35 dB over 290'K) are available, depend- Thereafter, it is assumed that (9) accurately described noise ing upon the bandwidth ofthe diodecircuit. current in an oscillator and that thermal noise in Edson's theory Chadelas et al. [35] subsequentlysuggested an automatic can be directly replaced by this shot noise. Using these assump- radiometer utilizing an avalanche-diodenoise generator, and tions, Hines found expressions for the AM and FM noise. made two observations. Obviously, the assumptions made are quite restrictive. Edson 1) The noise temperature of the source (lo5 to 106'K) varies analyzed an oscillator consisting of a simple resonant circuit and linearly with the diode bias current(1 to 10 mA). a nonlinear conductance only. Therefore, only the device con- 2) The noise source wouldbe usable to Q band (35 GHz). ductance is assumed to be voltage dependent and not the suscep- Haitz and Opp [36] have designed an avalanche-diode noise tance, and the nonlinearity of the active device conductance is source integrated in a microstrip-line circuit, and reported the described by using a constant s factor, defined as following results with the diodes biased at 30 mA (20- to 35-V AG breakdown voltage). s=- 1) The noise source had a potential range fromdc to 18 GHz. G 2) The spectral noise power density was in excess of 30 dB above kTo from 1 to 11 GHz, with variations within 0.7 dB over where AG is the small-signal conductance, and G is the total con- this frequency range.The variations are furtherreduced by tuning ductance of the device. out reflections from connectorsand adaptors. Within the limitations of themodel, Hines found an AM 3) The source has a very good long-term stability (k 0.1 dB noise spectrum that is white for low-Qcircuits, given by over a 500-h period). 4) The temperaturedependence of the spectral power density (measured at 2.5 GHz) was less than -0.002 dB/"C. 5) The bias voltage dependence of the spectral density was better than -0.005 dB/V. and a constant rmsfrequency deviation 6) Output variations from diode to diodewere small. Bareysha et af. [37] discussed in detail the design of a micro- wave using diffused Ge avalanche diodes. They have used a circuit model of the noise generator to study the effect of mismatch between the diode and the load upon noise where power output, and the bandwidth of the generator. In addition results of experimental measurements of the dependence of noise spectra on bias current, junction capacitance, breakdownvoltage, and temperature are alsogiven. Their main findings follow. 1) Noise-power spectral densityat a given frequency attains a broad maximum as a function of diode bias current; the higher Td is the transit time in the drift region, and the other symbols the frequency, the larger the bias current required for maximum have the same meaning as in (9). noise power and the lower the magnitude of this maximum. (See Inkson [39] published the first large-signal theory of noise Fig. 1 due to Constantet al.) generation in avalanchetransit-time diode oscillators. Hecon- 2) The noise-power spectral density remains practically un- sidered Read's model of an avalanche diode; i.e., he assumed a changed as the junction capacitancevaries. thin avalanche region, equal drift velocities and ionization rates 3) Noise power is slightly dependent on breakdown voltage for electrons and holes, and a power law dependenceof ionization for large breakdown voltages where Zener current is negligible. rate a on electric field, and heused the sharppulse approximation 4) The operation of the noise generator is very unstable if the also first suggested by Read [40]. The "sharp pulse assumption" dynamic resistance of the junction is negative, and its output is maintains that the carrier currentconsists of one sharp pulse of 1680 PROCEEDINGS OF THE IEEE, DECEMBER 1971 chargeper cycle travelingin thedrift region at asaturated 1) Only the FM noise depends upon circuit Q, as evidenced velocity with only one pulse being present at any time, and the by the aZL/aw term in (14). space charge of the avalanche current being nonzero only when a 2) A minimum of noise occurswhen += 90’ which can be ob- pulse is present in the avalanche region. It is further assumed tained by a proper circuit arrangement, and does not require a that the electrons and holes are always in equilibrium with the high-Q cavity. field, and hence a Poisson distribution is used for the number of Experimental results verifying these conclusionsare described ionizations. Inkson then calculated the jitter in pulse amplitude in a later section. and interpulse interval due to the random process of ionization, Vlaardingerbroek [43] found expressions for the outputpower and calculated the AM and FM noise spectra therefrom. The spectrum and the width of the output spectrum of an avalanche analytical expressions obtainedare quite complicatedand arenot transit-time diode oscillator operating at low and intermediate reproduced here; however,the following conclusions can be output levels. He followed the assumptions and methodof Hines, drawn from Inkson’s results. discussed in Section11-B, in that the “primary”noise current due 1) As the bias current of the diode is increased, the output of to fluctuations in the random avalanche processis related to the the oscillator rapidly becomes very noisy. noise in the Read diode terminal current; however, nonlinearity 2) A larger area avalanche diode would make the oscillator is included, as opposed to Hines’ small-signal approximations. less noisy (the diode model being one-dimensional, nonunifor- The output power spectrum is then expressed in terms of the mities are assumed to be absent). spectrum of the primary noise current. 3) A wider avalanche region in the diode (;.e., appreciable ionization over a larger part of the depletion layer) leads to a B. Upconversion of Law-Frequency Noise lower noise output from the oscillator (although the analysis is The nonlinearity of avalanchediodes, which makesthem not applicable to such diodes). useful as mixers, introduces another sourceof noise in avalanche 4) FM noise (rms frequency deviation) variesinversely as the transit-timediode oscillators; namely, upconversion of low- square root of the loaded Q of the microwave cavity (ratherthan frequency noise to frequencies near the carrier. Experimental as inversely with Qas in Edson’s model),so that thedevice should be well as theoretical results obtained by Evans and Haddad [44] coupled to a high-Q cavity. have shown that upconversion of low-frequency signals to fre- Ulrich [41] has also found expressions for AM andFM noise quenciesin theneighborhood of thecarrier frequency in an spectra of an avalanche transit-time diodeoscillator, but in terms avalanche diode is accompanied by a large gain. This upconver- of the nonlinearity of the diode impedance rather than the diode sion gain therefore makes low-frequency noisean important con- material,structure, and operating point parameters as in the tributor to total noise.2 In addition to thermal noise in the FtF case of earlier works. Ulrich used an approach similar to Kuro- circuit, this low-frequency noise could arise from at least two kawa’s general analysis [42], modeling the oscillator by a series different sources, both of them stronger than the circuit thermal circuit containingthe device (with RF amplitude-dependent noise. impedance), the impedance as seen by the device (which is fre- 1) Low-frequency noise (both avalanche and thermal noise quency dependent), and anall-inclusive noise source independent with the avalanche noise predominating) generated within the of RF amplitude or frequency. He found that the AM and FM diode. noise spectraare 2) Bias current fluctuations (dueto supply fluctuations, ther- mal noise, or llfnoise) generated inthe dcbias circuit. Goedbloed [45] camed out a theoretical analysis of upcon- verted noise in an avalanche transit-time diode oscillator using the Readmodel of the diode(with infinitely thin avalanche region and infinitely highmultiplication factor) and a passivesingle and resonant network model for the oscillatorcircuit. He has shown that if both of the previously mentioned assumptionsof the Read model are maintained, there is upconversion of low-frequency noise to AM noise of the oscillator, but not to FMnoise. How- ever, if one of these assumptions is discarded, FM noise is en- hanced due to upconversion of low-frequency noise. This ob- and their correlation coefficient is servation should serve as a warning for caution in model selec- tion before studying noise in such nonlineardevices as avalanche c = - cos 4 (15) diodes. Scherer [a]has taken the effect of low-frequency noise into where I el * is the mean square noise voltage of the noise source account by calculatingthe components of AM and FMnoise due previously mentioned, including all fluctuations producedby the to fluctuations in the bias current. These fluctuations are assumed device; A and w arethe current amplitude and frequency of to be due to avalanche noise alone, and are estimated using a oscillation with mean valuesAo and wo, respectively; ZL= RL+]’XL low-frequencynoise equivalent circuit of the avalanche diode is the impedance of the load (including Circuit) as seen by the first given by Haitz [30] and an impedance representing the bias diode; Zd = RdfjXd is the impedance of the diode, and circuit. The low-frequencynoise current, which is calculated using this model, modulates the dc bias current, and is trans- formed into the high-frequency AM and FM noise using ampli-

* The reasons why the downconverted high-frequency noise is not Two conclusions, both of which have been experimen,tally veri- consjdered are twofold. 1) The high-frequency noise can only be gen- fied, can be drawn from the previous analysis concerning the erated in the diode and not in the bias circuit, and 2) it suffers a down- circuit dependence of noise behavior. conversion loss, consequently it is small. GUFTA: NOISE IN AVALANCHE TRANSIT-TME DEVICES 1681

---I 1

s d -130- W 2 W y2 -140- P I I I I 2 0.1 I to IO2 10' FREWENCY OFF CARRIER, hnz -1507 d () Fig. 3. FM noise of avalanche-diode oscillators. All results have been (Si. UNSTABILIZEDI h (si tiffin BIAS CIRCUIT reduced to double sideband FM noise expressed in rms frequency I (si, CAVITY STABILIZED) IMPEDANCE' -160 deviationfor a 1WHz bandwidth.Curve a gives theFM noise g (Si,I LOW BIAS I CIRCUIT LGUNN 0:l '' I spectrum for a Si Read diode oscillator withQext = 1100 [4]. Curves I IMPEDANCE) b and c give these spectra for Si p+-n-n+ structures QeIt for = 20 [67] and Qext=900 [5], respectively. Curve d is the FM noise spectrum -170 10-1 I 10 IO2 103 IO' I( for a GaAs avalanche-diode oscillator for Pert=50 [57]. Curves e FREQUENCY OFF CARRIER. kHz andf giving FM noise for an X-13 klystron [5] and a [4] are included for comparison. Fig. 2. AM noise spectra of avalanche-diode oscillators. All results have been reduced to double sideband AM noise power to carrier power ratio in a 100-Hz bandwidth. Curves a and b give noise 3) The AM noisespectrum is flat andthe rms frequency spectra for a Si Read structure [4] and a GaAs p+-n-n+ structure deviation is constant in the frequency range measured (upto 100 [57], respectively. Point c represents AM noise in a pulsed GaAs diode oscillator [59]. Curves d and i giving AM noise for an X-13 kHz away from the carrier frequency). klystron [5] and a Gunn diode [4] are included for comparison. 4) Noise measurements were performed on some avalanche Curves e andfshow AM noise spectra fora Si diode oscillator with transit-time diodes operated both as amplifiers and oscillators. and without stabilizationusing a transmission cavity 1631. Curves g The noise behavior of the diode used as an oscillator was also and h are AM noise spectra for a Si diode oscillator when the dc estimated using the results of measurements on the diode as biasing circuit impedancewas very low and very high, respectively, used [46]. Output power, bias current density, and frequency of oscilla- an amplifier. The estimated oscillator noise was considerably lower tion are different for the various curves. than the measured oscillator noise, indicating a large-signal be- havior of noise sources inoscillator applications. Cook [SO] reported on avalanche-diode oscillators in which tudeand frequencymodulation sensitivities. Usingvalues of AM noise dropped at 2 dB/octave. Goldwasser et al. [51] re- avalanche current noise from the computer calculated results of ported on experimentalmeasurements of FM noise of CW Gummel andBlue and experimentally measured valuesof modu- avalanchediode oscillators close to the carrier frequency (2 to 10 lation sensitivities, he obtained the following. kHz away from the carrier) and made two important observa- AM Noise: noise-to-signal ratio= -108 dB; tions : FM Noise: rms frequency deviation= 10.6 Hz 1) The FM noise was strongly dependent upon the dc bias current, and for a given diode and circuit, the current could be in a 100-Hz bandwidth under a typical (given) set of operating optimized to reduce the noise sharply. conditions for a Si avalanche diode. 2) Upon such optimization, theFM noise in Schottky barrier GaAs diodeoscillators was much lower(0.7 Hz rms for a 100-Hz C. Measurements of Oscillator Noise bandwidth) than in diffused p-n-n+ Si diodes (80 Hz rms for the No attempts are made to discuss the techniques of measure- same bandwidth). An explanation of the superiority of Schottky ment of AM and FM noise in microwave oscillators; areview on barrier diodes was alsogiven by Goldwasser. this subject appeared elsewhere[47], and some more recent work Cowley [52] experimentallyobserved the effect of power will simply be referred to [48], [49]. The noisebehavior of saturation on avalanche transit-time diode oscillator noise, and "typical" avalanche-diode oscillators is shown in Figs. 2 and 3. concluded that: 1) the Occurrence of power saturation or the in- Measurements of avalanche-diode oscillator noisehave been crease of biascurrent beyond the saturation level makesthe reported by a rather large number of workers, and no attempt is oscillator very noisy. The valueof load resistance RL should made to catalog all of them. Instead, only those experimental therefore be kept somewhat greater than the value that leads to reports that have studied the dependence of noise behavior of maximum power to achieve better noise behavior; and 2) when avalanchediode oscillators on various parameters such as diode saturation occurs, bias current oscillations canarise in the power material, structure, circuit Q, bias current, power saturation, and supply circuit under certain conditions, and would appear AM as bias circuit are included. and FM modulations in theoutput, degradingthe oscillator The first measurements of AM and FM noise in avalanche spectrum. transit-time diodeoscillators were reportedby Josenhans [4]. His In another communicationCowley et al. [53] also reported a results on silicon Read-type diodes follow. large increase in noise dueto a very small increase in the dc bias 1) The FM noise spectra are symmetrical about the camerto current. within & 1 dB (found using a spectrum analyzer). The dependence of FM noise of avalanche transit-time diode 2) FM noise measurements up to as close as 400 Hz away oscillators on the externalQ of the cavity was studied over wide a from the camerfrequency showed no flicker noise effects. range by Harth and Ulrich [54]. They also observed the depen- 1682 PROCEEDINGS OF THE IEEE, DECEMBER 1971 dence of rms frequency deviation on dc bias current and the diode oscillator operating in the same circuit withthe same correlation coefficient of the AM and FMnoise. Theirresults can external Q. be summarized as follows. 2) The FM noise falls off as separation from the carrier fre- 1) The rms frequency deviafion varies as l/QexdG. quency is increased. 2) The rms frequency deviation is constant in the frequency 3) The AM noise sideband-to-carrier ratios were similar for range 100Hz to1 MHz away from the carrier frequency, and the GaAs and Si avalanche-diode oscillators. contribution of low-frequency upconverted noiseto it is negligible Levine [58], [59] reportedmeasurement of AM noise on in this frequency range as verified by varying the low-frequency vapor-grown GaAs avalanchetransit-time diodes operated impedance in the diode biascircuit. pulsed at high current densities of up to about 2ooo A/cmf, and 3) FM noise increases rapidly withan increase of the dc bias obtainedAM noise levels lower than in and GUM current. In the rangeof bias current between the onsetof oscilla- diodes. The best double sideband AM noise-to-signal ratio was tion and maximum admissible current, the excess noise tempera- -152 dB in a 100-Hz bandwidth30 MHz away from the carrier ture over Towas between30 and 50 dB, and varied approximately frequency (11.1 GHz) for a 25-V breakdown diode operated at as thefifth power of current density independentlyof the circuit Q. 1940 A/cmZ. In general, AM noise was found to be lower under 4) The correlation coefficient of FM and AM noise was be- conditions of high-bias current density, and GaAs diodes hada tween 0.8 and l, indicating a commonmechanism fortheir noise level 20 dB lower than Si diodes. generation. Avalanche-diode oscillator noise measurements in terms of Scherer [46] has made detailed observations on the effect of parameters other than AM and FM noise have also often been the impedance in the dc bias circuit and the bias current of the reported. Thus DeLoach and Johnston[ 1 ] have measured spectral diode upon the AM noise of the oscillator and reached the fol- linewidth, Johnstonand Josenhans [60], Scherer [MI,and lowing conclusions. Baranowski et al. [57] have measured the noise figures of mixers 1) The AM noise spectrum at frequencies close to the carrier and receivers using avalanche diodes as local oscillators, while (less than 1 kHz away from the carrier) is not affected by the Kramer et al. [61] have measured the totalnoise power spectrum. impedance of the bias circuit ;for large frequency separationAM These results will not be discussedin detail. noise is increased by lowering the bias circuit impedance. The difference in AM noise between the cases of very small and very D. Noise Reduction by Circuit Techniques largebias circuit impedances(compared to diodeimpedance) In thissection some measurements of noisein avalanche increases as the frequency separation increases to 100 kHz and transit-time diodeoscillators operated with specialcircuit arrange- then becomes constant. ments intended to reduce noise will be reviewed. It is not the in- 2) Some of thediodes show an excess noisewith a I/' tention hereto review the theory ortechniques of injection phase dependence. locking or other noise reduction methods; such details may be 3) For the high bias-circuit impedance case, the measured found in references [62]-[67]. The present discussion is confined noise agreeswell with the calculated RF noise, indicatingthat the to the improvement of noisebehavior achieved by various upconverted component of AM noise is not significant. methods, such as cavity stabilizationand injection locking; these 4) Noise performance deteriorates with increasing powerout- results areshown in Fig.4. put, increasing coupling of the load and increasing bias current. The simplest method of reducing the noise of an avalanche Ulrich [41] observed the effect of RF circuit impedance varia- transit-time diode oscillator is by frequency stabilization using a tion on AM and FMnoise and their correlation, and substantiatedhigh-Q transmission cavity. Gilden [62] reported noise measure- his theory [(13)-(15)]. All three equations involve the rate of ments on an avalanchetransit-time diode oscillator stabilized change of circuit impedance with frequency,which was controlled with a tunable transmission cavity and made the following ob- by weakly coupling a high-Q cavity to the oscillator circuit by a servations. magic tee. Tuning this cavity frombelow to above the oscillation 1) Even though the bandwidth of the cavity was 3 MHz, the frequency changesthe angle I$ by 360". Two maxima of both AM width of the oscillator output spectrum was reducedfrom 300 kHz and FMnoise occurred whenI$ was 0 and 180", and two minima to 2 kHz as observed on a spectrum analyzer. This indicatesthat occurred whenI$ was 90 and 270". The correlationCoefficient was the cavity does not merely act as a filter; instead, the diode noise also maximum in magnitude at the noise maxima and minimum depends strongly uponthe frequency dependence of the external at the minima. circuit reactance as seen by the diode. This fact was further proved FM noise measurements on Ge avalanche transit-time diode by introducing an isolatorbetween the diode mount and the oscillators were first reported by Rulison et af. [55]. They found stabilizing cavity, in which case no noise improvement occurred. that the rms frequency deviation is constant as a function of 2) Measurement of the AM noise of the oscillator, by a mixer frequency separation from the carrier, as forSi diodes. However, crystal and an IF amplifier, showed a reduction of 15 dB in the for a loaded Qm = 11, in a 1-kHz bandwidth, the rms frequency. noise at the outputof the IF amplifier. deviationwas only 300 Hz. In general, Ge avalanche-diode Ondria and Collinet [5]on the other handmeasured FM noise oscillators showed a 10- to 14dB improvement in noise-to-carrier in cavity-stabilized avalanche-diodeoscillators. They useda Stalo ratio over Si diodes operating in the same circuit with the same cavity and reduced the rms frequency deviation of an avalanche- loaded Q-. Barber [56] also reported on the EM noise in Ge diode oscillator by a factor of 10. Ashley and Searles [63] ob- diode oscillators to be 15 dB lower than in Si diode oscillators in served the effectof cavitystabilization on AM noise of an cavities of the same Qext for frequency separations of 1 kHz to 2 avalanche-diode oscillator as a function of frequency separation MHz from the carrier frequency. GaAs avalanche transit-time from the carrier frequency. They found that for large separations diode oscillators have also been reported, by Baranowski et al. (above 500 kHz) the AM noise is reducedby cavity stabilization, [57], to be superior to Si diode oscillators in their noise behavior. as expected, by an amount that increases with frequency separa- Their findingsmay be summarizedas follows. tion. At 10 MHz, the AM noise reduction is approximately 20 1) The FM noise sideband-to-carrier ratio for a GaAs diode dB. However, for small frequency separations (below 500 kHz), oscillator was at least 10 dB lower when compared to a good Si AM noise actually increased when cavity stabilization was used. GUPTA: NOISE IN AVALANCHE TRANSIT-TIME DEVICES 1683

Kurokawa; and2) the reduction of FM noise by synchronization is less than that predicted by Kurokawa near the carrier fre- quency, but more than the predicted value far from it. Anoise loading test foran avalanchetransit-time diode oscillator, phaselocked by an externalfrequency modulated signal has been reported by Isobe and Tokida [68]. Perichon [69] has reported measurement of FM noise of an b (CAVITY STABILIZED) avalanche-diode oscillator injectionphase locked by asub- harmonic signal, using thethird and fourth subharmonics as synchronizing signals. The conclusions are in each case identical 14.9 dB. 50 MHz OFF to those of Ondriaand Collinet obtained with fundamental

d (STABILIZING KLYSTRON1 frequency injection. In addition it is noted that the higher the harmonic ratio, the smaller the bandwidth overwhich FM noise I I 0.1 I I I i reductionoccurs as a result of synchronization.Stated differ- 0.1 I 10 IO2 10' Io4 10 ently, a smaller phase locking gainis available for the same noise FREQUENCY OFF CARRIER. LHl reduction. Scherer [70] has suggestednegative feedback methods for Fig. 4. FM noise reduction by circuit techniques [5]. Curves a and b are FM noise spectra of an unstabilized and a cavity-stabilized av- reducing oscillator AM and FM noise output. He has pointed alanche-diode oscillator. Curve c is the FM noise spectrum of an- out that dc bias current cannot be used for correction of ampli- other free-runningavalanche-diode oscillator which is injection tudeand frequency, asit would affect both simultaneously. phase locked by a klystron withFM noise spectrum shownby curve Instead AM noise can be reduced by using a pin modulator for d. Curves e,f,and g are FM noise spectraof the phase-locked oscil- amplitude correction, andFM noise can be reduced by controlling lator with locking gains of 7, 13, and 20 dB, respectively, when free- running and klystron frequencieswere identical. Curve h is the FM the frequency using varactor tuning. However, no experimental noise spectrum for a phase-locking gain of 14.9 dB when the two results using these techniques havebeen reported. frequencies were 50 MHz apart. Power output, bias current, and oscillation frequency in cavity-stabilization and phase-locking ex- IV. NOISE IN AVALANCHE-DIODEAMPLIFIERS periments are different ; rms frequency deviation is for one sideband in a 1O@Hz bandwidth. A. Theory of Amplifier Noise The first small-signalnoise analysis of a reflection-type Nagano also measured FM noise in cavity-stabilized oscilla- avalanche transit-time diode amplifier was presented by Hines [a] [3]. Neglecting the diode series resistance R, and assuming that tors and reporteda stabilization factorof 50. fluctuations in theavalanche current are theonly sourceof noise, The use of a transmission cavitycauses a loss of approximately he expressed the noise figureas 6 dB in the signal strength; a method of compensating for lossthis - has been reported by Ashley and Palka [6]. They used a cavity- In2RL stabilized Ku-band avalanche-diode oscillator with FM noise of F=l+- GkTJ3 0.2 Hz rms in a bandwidth of 100 Hz to phase lock another avalanche-diode oscillator, obtaining a lockinggain of 10 to where given in (9) isthe meansquare noise current in the 20 dB. external circuit due to fluctuations in the avalanche, RL is the Ondria and Collinet [65], [5] have reported the following re- load resistance as Seen by the device, k is the Boltrma~constant, sults on effects of phase locking on the noise spectraof avalanche G is the amplifier power gain, B is the bandwidth, and TOthe transit-time diode oscillators. These results include the experi- reference temperatureof 290°K. Since an amplifier power gainis mental work which wasalso reported by Hineset al. [MI. given by 1) For phase locking bya small signal (20 dB below carrier), AM noise is reduced by approximately 3 dB; this effect is not evident from the theory of noise in synchronized oscillators and is attributed to the reduction of FM noise. Consequently, the component of AM noise dueto FM to AM conversion is reduced. andunder highgain conditions (&= -RL), the noisefigure 2) For larger injected signals (10 dB below the carrier), AM expression simplifies to noise spectra show degradation of 1 to 2 dB over that of free- running oscillators, a fact predicted theoretically. 3) In an avalanche-diode oscillator phase locked by a klys- tron, theFM noise level close to the carrier decreasedto the same low level as thelocking source, but atfrequencies fartherfrom the carrier, FM noise increased and asymptotically approached the Using typical valuesof the parameters, a noise figure of approxi- noise level of free-running avalanche-diode oscillators. mately 40 dB is predicted by the analysis. Hines also observed 4) The reductionof FM noise extendedfarther away from the that the carrier currentbecomes very small over part of the cycle carrier frequency for smaller phase-locking gain (larger injected and then builds up to the peak value again by avalanche. Since signal) and for smaller frequency separation between the free- fluctuations become significant when the total current is small, running frequency and the reference frequency. and are multiplied by avalanche, Hines concluded that: 1) the Ashley and Palka [67] phased locked an X-band avalanche noise output may be large at higher signal levels (when current transit-time diode oscillatorby a stabilized klystronand measured minima are smaller); and 2) a diode with large leakage current the AM and FM noise of the oscillator. Two major conclusions should be quieter at high signal levels than one with a sharp from their experiments are: 1) no increase of AM noise due to breakdowncharacteristic, because the 'leakage current would synchronization was detected, as is predicted by the theory of limit the minimum current. 1684 PROCEEDINGS OF THE IEEE, DECEMBER 1971 [a]measured a noise figureof 39 dB at5.5 GHz for a Read-type structure, Josenhans and Misawa [72]reported a noise figure of 45 dB at 7.5 GHz fora p-i-n structure, and DeLoach and Johnston [l] found the lowest noise figureof 49 dB at 11.5 GHz for a p-n structure. These results should not be directly compared because they were obtained at different frequencies, amplifier gains, and bias currentdensities. Dehach andJohnston also found that the noisefigure decreased with increasingbias current and with decreasing amplifier gain. Scherer [73]has reported an X-band avalanche-diode ampli- fier with three cascaded stageswhich has anoveall gain of 36 dB and a noisefigure of 31 dB. Rulison et al. [55] measured the noisefigure of X-band microwave amplifiers usingboth p-n and n-p-i-p+ (or Read-type) avalanche diodes fabricated with Ge under high gain conditions, and reported noise figures of30 f 1 dB, which is an improvement FREWNCY, Gnz of approximately 10 dB over the results obtained with comparable Si devices reported earlier by Johnston and Josenhans Fig. 5. Optimum noise measure for a pn avalanche-diode amplifier [a]. [9].Curve u gives the noise measure as a function of frequency for Kuno et al. [74] have measured noise figures of GaAs av- a Ge diode with a parasitic resistance of 1 Q, dc bias current density alanche transit-time diode reflection-type amplifiers. They reached of loo0 A/crn*, and equal electron and hole ionization rates. All a lowest noise figureof 17 dB under high gain conditions, which other curves are also drawn under the same conditions, except as is lower than any reported using Si or Ge diodes. Two general noted for each curve: b, Si diode (unequal ionization rates); c, zero parasitic resistance;d, 300 A/crnz current density; e, electron ioniza- conclusions can be drawn from their work. tion rate reduced by a factor of 10; and6 hole ionization rate re 1) The noise figure decreases monotonically with increasing dc duced by a factor of 10. bias current of the diode for a constant amplifier gain. 2) For a constant bias current, the noise figureof an amplifier slowly increases with increasing power gain, attains a maximum Gummel and Blue have used their results of open-circuit [9] at a gain somewhere between 10 and 15 dB, and then falls for mean square noise voltage (described in Section11-B) to calculate larger gains. the noisemeasure M of a small-signalavalanche transit-time Noise figure measurements on GaAs avalanche-diode reflec- diode amplifier. The noise measure of a linear two-port amplifier tion-type amplifiers were also reported by Baranowski et af. [57] was defined by Haus and Adler as [71] under high-gain conditions (10 to 30 dB) and 10- to 15-dB im- F-1 provement was found over similar measurements using Si diodes. M= They also found a considerable reduction of noise figure with 1 1-- increasing diode bias current. G Typicalreflection-type amplifier noise figures reported are shown in Fig.6 as a function of diode bias current. where F is the noise figure ofthe amplifier and G its gain, to serve Although both degenerate [75] and nondegenerate [76] as a figure of merit of amplifiers. The optimumvalue of this noise parametric amplifiers using avalanche transit-time diodes have measure for a negative conductance amplifier hasbeen found by long been reported, no measurements of noise figures in these DeLoach to be amplifiers havebeen reported. On the other hand,noise figuresof sidebandamplifiers (“amplifiers” in which oscillations are simultaneously occurring at a nearby frequency) have been re- ported by Evans and Haddad[MI, but will not be discussed here. where B is the bandwidth, and R,+ the real part of the diode Noisten [77] has measured amplifier noise in terms of AM and impedance. Gummel and Blue have numerically calculated Mopt FM noise of output; these results are also not discussed here. for an avalanche-diode amplifier as a function of operating fre- V. NOISE IN AVALANCHE-DIODE MIXERS quency, parasitic resistance, diode material, and ratioof electron- to-hole ionizationrates. Some of theirresults are shown in Fig.5. Avalanche transit-time diodes have beenused as self-oscillat- They have reached the following general conclusions; ing frequency convertersin two different modes:that inwhich the 1) The noise measure shows sharpa minimum as a function of signal is uncorrelated to the “local oscillator” output (the usual frequency, the frequency of minimum Moptbeing higher-for lower mixer application), and that in which the signal is derived from parasitic resistance and for larger bias current density. the local oscillator output. While the noisiness of the former is 2) The lower the parasitic resistance, the smaller theMopt. adequately described by the noise figure of the mixer, perhaps a 3) The larger the bias current density, the smaller the Mop$. more appropriate characterization of noise performance in the 4) Ge avalanche diodes have lower Mop$than Si avalanche second case is the frequency stability or the minimum detectable diodes. In general, Mopt is lowest for equal ionization rates for signal for the oscillator-mixer combination. holes and electrons, and deterioratesif either a, or ah is reduced. No theoretical results on noise in either type of mixer have been reportedso far. The mathematical treatmentof noise in such B. Measurements of Amplijier Noise a mixer is difficult for two reasons. First, the noise of the local Most earlier work reported noise figure measurementson Si oscillator is large and cannot be neglected, and second, the local avalanche transit-time diode amplifiers. Johnston and Josenhans oscillator noise is correlatedto the fluctuations inthe periodically GUPTA : NOISE IN AVALANCHE TRANSIT-TIME DEVICES 1685

avalanche diodes are someof the possible areas of study. That the work done is not complete is evidenced by the fact that no gen- eral set of rules for optimization of the diode structure, material, operating point, andcircuit can be stated asyet. APPENDIX Since this paperhas been written(in October 1970) severalnew results in this area have been published. Some of these are sum- .” marized below. Naqvi [81] has extended the analysis of McIntyre 0 10 20 30 [8] to higher frequencies by taking into account the finite time GAIN, dB required for multiplication.Vlaardingerbroek andGoedbloed (a) [82] have calculated the width of the oscillator output spectrum using a Read model IMPATT diode and taking equal ionization rates for electrons andholes. They reach the conclusion that the p+-n diode structure leads to a lower noise oscillator than the complementaryn+-p diode structure, which has smallera avalanche-region width. Sjolund [83] has carried out a numerical large-signal analysisof IMPATT oscillators assuming a rectangular voltage waveform acrossthe diode, and has found that themean 34 30 42 44 square noise current in the diodeis inversely proportional to the BIAS CURRENT, mA ratio Zmin/Zo, where Zmin is the minimum conduction current, and (b) Io the dc bias current for the diode. Ulrich and co-workers [84], Fig. 6. Single sideband noise figure of GaAs avalanche-diode amplifier [85] have derived expressions for AM and FM noise spectra of at 8.75 GHz [74]. (a) function of amplifier gain.(b) Bias current (for free-running and synchronized IMPATI oscillators by a method constant gain of 20 dB). Diode diameter is 125 pm. similar to the one used for obtaining (13) and (14) and report goodagreement between theoretical and experimental results. varying device admittance as they botharise from thesame physi- Josenhans [86] useda series combination of three Si IMPATT cal mechanism. In addition, a conventional diode mixer noise diodes in an oscillator and found the oscillator FM noise to be analysis for a nonlinear conductanceor nonlinear reactance type 30 Hz rms in a 1-kHz bandwidth as compared to 90 Hz rms for of mixer cannot be simply extended to avalanche-diode mixers, single diode oscillators under identical conditions, and ascribed asthe conductance and susceptance areboth nonlinear and this noise reductionto theaccompanying power increase.Earlier, cannot be expressed as functions of voltage independent of fre- Fukui [87] had also reported an improvement over the average quency(because the I-V relationship for the diode is not in- FM noise of individual diodeoscillators in an oscillator consisting stantaneous). of eight IMPATT diodes mutually synchronized by a hybrid power Grace [78], whoreported on a Siavalanche transit-time combiner.Tatsuguchi et al. [88] havereported experimental diode downconverter (Xband to 30 MHz) measured a conversion measurements of the noise measure of a phase-locked IMPATT loss of 7 dB, and estimated the noise figure to be approximately oscillator as a function of cavity load resistance and oscillator 50 dB by measuring the minimumdetectable signal power. output power. Evans [89] has measured the noise figure of Ge Evans and Haddad [44] also carried out noise figure measure- TRAPATT diode amplifiers to be about 120 dB in the stable mode, ments on GaAs avalanche diode frequency converters using a but only about 60 dB when TRAPATT mode oscillations are oc- noise figure meter, and reported a conversion loss of 6 dB and a curring simultaneously at a different frequency. The FM noise noise figureof 55 dB. No later measurements using better diodes, of Ge TRAPATT oscillators at UHFwas earlier reported[90] to be particularlySchottky barrier avalanche diodes, have been re- between 8 and 10 Hz rms in a 1-kHz bandwidth and for an ex- ported, and the above figures are poor compared to the Gunn- ternal Q of 170 to 350. Claassen [91] has presented an approxi- diode self-oscillating mixers [79]. mate expression for the noise figure of an IMPATT diode amplifier Measurements of minimumdetectable signal on avalanche in terms of small-signal parameters which can be calculated as a transit-time diode mixers in doppler radar applications have beenfunction of the avalanche-region widthI,, and has compared the carried out [80]. 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