(y 7

CONSIDERATIONS ON SELF-PUMPED PARAMETRIC AMPLIFICATION

IN GUNN-EFFECT AND IMPATT DIODES

by

JAMES STANLEY BLACK, B.S . i n E .E .

A THESIS

IN

ELECTRICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

ELECTRICAL ENGINEERING

Approved

Accepted

May, 1973

AC 80S" T3 1973

ACKNOWLEDGEMENT

I would like to thank Dr. D. K. Ferry for his guidance and his

patience during this investigation. I would also like to thank

Dr. J. C. Prabhakar and Dr. A. G. Walvekar for serving on my committee

11

TABLE OF CONTENTS

ACKNOWLEDGEMENT ii

LIST OF TABLES iv

LIST OF FIGURES v

I. INTRODUCTION 1

A. Basis for Parametric Amplification 1

B. Parametric Amplifier Physical Requirements . . . . 2

C. Previous Work 3

D. The Present Work 4

II. THE PARAMETRIC AMPLIFIER 5

A. Gunn Diode Calculation 12

B. IMPATT Diode Calculation 14

III. EXPERIMENT AND RESULTS 15

A. Procedures 15

B. Gunn Diode Test 17

C. IMPATT Diode Test 2^•

IV. CONCLUSIONS 29

A. Gunn Diode Self-Pumped Parametric Amplifier . . . . 29

B. IMPATT Diode Self-Pumped Parametric Amplifier . . . 31

APPENDIX A. DERIVATION OF THE MANLEY-ROWE RELATIONS 33

APPENDIX B. THE GUNN DIODE 39

APPENDIX C. THE IMPATT DIODE 49

LIST OF REFERENCES 61

111

LIST OF TABLES

Table C.l IMPATT Parameters 53

IV

LIST OF FIGURES

Figure 2.1 Capacitance Mixer With Tuned Input and Tuned Output 8

Figure 2.2 Parametric Amplifier with Tuned Input and Tuned Idler Circuit 11

Figure 2.3 Schematic Circuit Diagram for the Parametric Amplifier 13

Figure 3.1 Self-Pumped Parametric Amplifier and Test

Equipment Arrangement 16

Figure 3.2 Gain vs Frequency for Gunn Diode Amplifier . . . . 19

Figure 3.3 IMPATT Diode Gain vs Frequency Characteristics . . 25

Figure 3.4 IMPATT Dynamic Range 26

Figure A.l Illustrative Circuit for Deriving Manley-Rowe Relations 34

Figure B.l Simplified Diagram of the Energy Band Structure of GaAs 41

Figure B.2 Theoretical and Experimental Velocity-Field

Characteristics 43

Figure B.3 Equivalent Circiiit of a Gunn Diode 45

Figure C.l Read Diode Structure 50

Figure C.2 Boundary Conditions for a General IMPATT Diode . . 51 + +

Figure C.3 Admittance for a Si p -n-n Diode 58

CHAPTER I

INTRODUCTION

The term parametric amplifier is used for those amplifying cir­

cuits wherein the amplifying power is coupled to the signal by the

variation of a circuit parameter. The possibility of parametric

amplification of signals was shown theoretically by Lord Rayleigh in

2 3

1831. Little was done with the idea until 1948 when van der Ziel

pointed out the possible low-noise properties of a microwave paramet­

ric amplifier using a nonlinear capacitance. The realization of a

low-noise parametric amplifier was delayed until 1957 when high quality

semiconductor p-n junction diodes (varactors) became available.

Currently, the parametric amplifier is the most sensitive of the non-

refrigerated microwave amplifiers. When refrigerated, the parametric 5

amplifier can be made almost noisefree.

Most parametric amplifiers in use today are based on the varactor

(contraction for variable reactance), a semiconductor p-n junction

diode designed to provide a variable capacitance under specified

reverse bias conditions. The junction capacitance of the normally

reverse-biased diode varies nonlinearly with the applied voltage, with

the exact form of the nonlinear dependence determined by the junction

construction.

BASIS FOR PARAMETRIC AMPLIFICATION

A set of very general relations derived by J. M. Manley and

H. E. Rowe form a basis for analyzing a parametric amplifier (see

Appendix A for the derivation). One result of their equations is that

if energy at two different frequencies f^ (signal frequency) and f

(pump frequency), f2 > f- . i^ applied to a nonlinear variable reactance,

a third frequency adler .frequency) f = f - f , the difference

frequency, will be generated. Further, if the power at f is much

greater than the power at f^, the Manley-Rowe relations reduce to^

P P

fT" F " ° » (1-1) 1 3

and

P P 2 ^ 3 ^ f-"*" F"" ° • (1-2) 2 ""a

Since P^ is positive, from equation (1.2) P must be negative (f > 0).

Hence, P^ must also be negative from equation (1.1). The negative

sign indicates that power is being radiated from the capacitance. It

is possible to choose f and f such that more power at f is radiated

than is injected at f ; i.e., the signal at f may be amplified. It

is also obvious that to suppress f (the idler) would result in

suppressing the desired amplification. In the special case for which

f is equal to f (f = 2f ), the amplifier is said to be degenerate.

PARAMETRIC AMPLIFIER PHYSICAL REQUIREMENTS

A parametric amplifier needs a source of ac power (the pump

soijrce) and a nonlinear variable reactance in order to amplify the

applied signal. Both the Gunn diode and the IMPATT Cacronym for

IMPact Avalanche Transit Time] diode are such, that both, the pump

source and the nonlinear reactance may be combined in a single semi­

conductor device. Under proper circuit and bias conditions, the

3

transferred electron effect of the Gunn diode (see Appendix B) will

cause conduction electrons to bunch into high electric field domains.

The movement of this space charge constitutes a time varying capaci­

tance. The transit of the domain from the cathode to the anode

induces oscillations in the attached circuit so that microwave pump

power and time varying capacitance are both present in the Gunn effect

diode. The IMPATT diode (see Appendix C) also forms a space-charge

"bunch" (or domain) of carriers which drift through the semiconductor

in such a way as to induce oscillations in the external circuit. In

addition, the p-n junction of the IMPATT diode is normally reverse-

biased and has many of the features of an abrupt-junction varactor

even for voltages below the breakdown voltage. Thus, both the Gunn

effect diode and the IMPATT diode seem to meet the minimum conditions

required for a parametric amplifier, and in fact constitute a self-

pumped reactance, hence a self pumped parametric amplifier.

PREVIOUS WORK

Several investigators have reported the achievement of self-

pumped parametric amplification in both Gunn effect diodes and IMPATT

7 8 9

diodes. DeLoach and Johnston , Clorfeine , and Snapp and Hoefflinger

have demonstrated degenerate parametric amplification using IMPATT

diodes. Brock has demonstrated non-degenerate parametric amplifi­

cation with self-pumped IMPATT diodes. In 1967, Aitchison suggested

that Gunn diodes might be used as self-pumped parametric amplifiers.

Later, he demonstrated degenerate amplification with a self-pumped

12 13

Gunn diode. Kuno demonstrated non-degenerate self-pumped para­

metric amplification of X-band signals with a Gunn diode in 1969.

4

The aforementioned reports have been uniformly short, with few

experimental details presented. Each result reported promising data

in terms of gain, noise figure, and instantaneous gain-bandwidth

products. Each report implied a reduction in required amplifier cir­

cuitry. With these apparent advantages, it would normally be expected

that additional work would have been reported in the past several

years providing more extensive and exact information on both the

promises and pitfalls of the self-pumped technique.

THE PRESENT WORK

The purpose of this work is to investigate the self-pumped

parametric amplifier technique. Self-pumped parametric amplifiers

using both Gunn effect and IMPATT diodes have been constructed and

tested. The results of these tests will be reported and compared

with the results predicted from theoretical considerations. Finally,

those experimental problems felt to be inherent to the technique will

be discussed and the overall practicality of the technique will be

evaluated as it applies to each type of diode.

CHAPTER II

THE PARAMETRIC AMPLIFIER

The computation of parametric amplifier response may be done in

5 2 several ways. Watson and Collin have given equivalent methods for

the circulator coupled varactor reflection amplifier. However,

14

van der Ziel has a quite general derivation of the parametric ampli­

fication process. The following exposition follows van der Ziel's

work quite closely, with such modifications as are necessary to adapt

to the use of Gunn effect or IMPATT diodes.

In nonlinear capacitance mixing, one starts with a device having

a nonlinear charge characteristic Q = f(V) and applies a dc signal V ,

a pump signal AV and a small input signal AV.. The charge Q then is

Q = f(V^ + AVp + AV^) . (2.1)

By making a Taylor expansion of Q with respect to AV., one soon finds

a capacitance C(t) such that

dQ_ dV = C (V^ + AV^) = C(t) . (2,2)

+ AV o

V=V + AV ° P

I f AV = V coso) t , t h e n C ( t ) may be w r i t t e n as a F o u r i e r s e r i e s P P P

C ( t ) = C + 2CT COSO) t + 20^ cos2co t + • • • , ( 2 . 3 ) o 1 p 2 p '

where co = 27rf i s t h e a n g u l a r f r equency . The i n p u t s i g n a l may be

w r i t t e n a s

AV. = V. c o s ( a ) . t + <|).) . ( 2 . 4 ) 1 1 1 1

6

By multiplying equation (2.3) and (2.4) together, the charge AQ is

found to be

00

AQ = C V. cos(co.t + d).) + E C V. o 1 1 ^1 T n 1

n=l

{cos[(na)^ + (jo.)t + c|). ] + cos[(na) - (o.)t - A.]}. (2.5) p 1 1 p 1 1

The significant currents are found by differentiation with respect to

time, as

i(t) = -CD.C V. sin(a).t + 6,)

1 o 1 1 ^1

oo

- Z {(no) + 0).) C V. sin [(nco + a).)t + 6.] T p 1 n 1 p 1 ^1

n=l ^ ^ + (na) - CO.) C V. sin[(nLo - i>.)t - 6.]} . (2.6) p 1 n 1 p ^1 ^1

From equation (2.6), it appears that the nonlinear device repre­

sents a capacitance C to the input. The signal transfer from the

frequency co. to the frequencies (nw + co. ) is represented by the

capacitance C .

The difference frequency oo = noo - oo. is the basis for the ^ - o p i

parametric amplifier. If the input V. is set to zero and a small

signal V = V cos(oo t + *. ) is applied to the output, there will be a ^ o o o ^1

current into C(t) proportional to

- CO C V sin(co t + (|) ) . (2.7) 0 o o o o

Flowing out of the device [see the third term of Eq. (2.6) above] will

be a current proportional to

- co.C V sin(oo.t - <f) ) , (2.8) 1 n o 1 o

where the frequency co. is generated by (noo - co ) and other terms have

been discarded as insignificant.

These can also be expressed in complex phasor notation and with

the complex conjugate denoted by an asterisk. Since (oo + co. = noo ), o 1 p

the input current can be written from Eq. (2.6) as

1. = joo.C V. - jco.C V , (2.9) i - ' i o i - ' i n o ' ^ ^

and Eqs. (2.7) and (2.8) yield

A

i = -jco C V." + jo) C V , (2.10) 0 - ^ o n i - ^ o o o

where the necessary phase reversal has been incorporated into the

equations.

Consider the circuit of Figure 2.1. The input circuit and the

capacitance C [part of C(t)] is tuned to the frequency oo. , and the

output circuit with the capacitance C is tuned to the frequency oo .

The circuit equations for the input and output modes may be

written as:

i. = i - V. (g + -:—r— + joo.C. ) , (2.11) 1 s 1 s 100.L. - 1 1

•^11

i = - V (g, + -.-^ + jo) C_) . (2.12) o o^L IcoL^ -^02

• o 2

The resonant tuning conditions are simply

0) h^ (C^ + C ) = (0. L. (C. + C^) = 1 . (2.13) o 2 2 o 1 1 1 o

The i n p u t c u r r e n t can be e l i m i n a t e d by u s i n g E q . ( 2 . 9 ) i n ( 2 . 1 1 ) , and

i . = joo.C V. - jto.C V " = i - V (g + . . + Jw.C ) , 1 - ' l o i - " i n o s I S D^^L^ 1 1

8

i. — * ^^^^ -•— i

r-^F

Figure 2.1. Capacitance mixer with tuned input and tuned output. (From Ref. 14)

or

i = V.[g + -; ; — + jco.C. + jco.C ] - jOO.C V s I S Tco-L- 1 1 -^lo -"ino -" 1 1

Using the second tuning condition of Eq. (2.13) with n=l, this

becomes

i = v.g - jco.C.v . (2.14) s I S - 1 1 o

The same procedure can be followed for Eqs. (2.10) and (2.12), with

the result:

.'.

0 = -jco C^v " + g^v . (2.15) • o 1 t ''L o

Therefore,

V =

jco C^v. -'oil

SL

and

V

o

* - "o' l i

This may be inserted into Eq. (2.14) to yield

^s = e^v. - 3^.C^ [ — ]

f. 2 00 -00 '-'T

and the input admittance i s

n 2 1 00 co.C, s o i l

i n V. s gj 1 -LI

10

The output circuit thus presents a negative load conductance

2 -00 CO.CT la reflected into the input circuit. This is a direct conse-o 1 1 ^L

quence of the phase reversal effect noted in Eq. (2.5).

The fact that the mixer circuit presents a negative conductance

to the input circuit is used in the parametric amplifier. Figure 2.2

shows a reflection type parametric amplifier equivalent circuit. This

is a mixer circuit with an idler circuit tuned at co = oo - co. , which 2 p 1

2 presents a negative conductance -oo co.C../g_ to the input circuit. The

inductance L. is used to tune the input capacitance C of the mixer

2 2 circuit (oo.L.C = 1). The negative conductance -oo-oo.C-/g_ makes it

i i o 2 i l 2

possible that the power delivered to the load conductance g is

larger than the available power P = |i | /8g of the source. The av s s

-1 r\

power fed into the load is P . = y|v. | g and hence the power gain is

P ^ o out ,, G = p = 4g g ^

av

v. 1 1 s

(2.18)

by the maximum power transfer theorem. Using Eq. (2.16) with g - g^,

this reduces to

(2g,)^ 1 G = L _ __ 1 _ , (2.19)

^i^2^1 ,2 , , ^^2^1 .2 [ 2g ] [ 1 - 2T7 ^

L g^ ^gLg2

The amplifier bandwidth will not be derived for this study. An

2 excellent derivation of bandwidth is found in Collin. He shows that

single tuned high gain varactor parametric amplifiers will usually

have a fractional bandwidth in the 2 to 3% range.

11

I .g.

V cosO) t P P 0

v. -I i ^ C ( t ) ^ L .

f

) \

—AA/V——

Tuned at 0 0 ^ = 00 - 00.

2 P 1

Figiore 2.2. Parametric amplifier with tuned input and tuned idler circuit. (From Ref. 14)

12

The equivalent noise circuit is easily found. Each conductance

will contribute noise to the output and the noise may be considered

uncorrelated. Figure 2.3 is a schematic noise circuit diagram for

the parametric amplifier. The noise from g will be counted against LI

the succeeding stage. With the input shorted, the noise current i

in the input circuit is due to the noise of the source conductance g ,

the noise of input conductance g. and the converted noise of idler

conductance g^. The mean-square value of i is

4k TAf i^ = V ^ s ^ ^ ^ %Tg^Af + ( — ^ ) oj C . (2.20)

Dividing this by the source noise due to g alone, the noise factor is

found to be

2p2 gi ^iC

^s SsS2

= l + _i.+ _E2_iL , (2.21) ^s S2' 2

where -g 9 = -oo.oo C /g„ is the effective negative conductance appearing

at the input.

GUNN DIODE CALCULATION

The Gunn diode used in the experiments had an oscillation thresh­

old of 115 ma at 3 V, a low field resistance of 13 J, and maximum

ratings of 10 V, 150 ma. The diode yielded 8.8 milliwatts at an

operating frequency of 8.57 GHz with a bias of 80 ma at 8 V. Further

characterization of the diode requires certain estimates to be made.

C(t)

13

T ii A Sg, B sg- "3

A = C4k^Tg^Af)^ C = (4kgTg2Af)"

B = (4k^Tg^Af)'

Figure 2.3. Schematic circuit diagram for the parametric amplifier. (From Ref. 14)

14 15

Khandelwal and Curtice used similar diodes and their estimates have

1 c

been used as a guide along with the data reported by Hasty, et al.

Appendix B details the estimation of the diode equivalent circuit

parameters. The parameters to be used in the equivalent circuit

include a low field resistance R of 13 ohms, static diode capacitance

C of 0.0477 pf, dynamic negative resistance of -145 ohms, and dynamic

space charge capacitance of 0.372 pf. The predicted power gain of the

Gimn diode amplifier is found by using Eq. (2.19). The transfer

capacitance C, is one-half the domain capacitance. The load is taken

as 50 ohms because of the circulator. The idler cavity is tuned to

obtain maximum gain. At maximum current, the idler admittance is

estimated to be 1.32 millimhos. The calculated parametric gain is

3.3 dB. Using Eq. (2.21) and assuming g. is zero, the noise figure

of the amplifier is 8.38 dB without including the excess noise from

the Gunn diode oscillation itself.

IMPATT DIODE CALCULATION

The IMPATT diode used in the test had a maximum bias rating of

20 ma at 60 V. The operating point was chosen as 19 ma at 59 V,

which gave an impedance of 3100 n. Measured oscillator power output

was 9 milliwatts. Efficiency was 0.8% with 1% efficiency achievable

with optimizing the tuning. By the method detailed in Appendix C,

the capacitance at 9.12 GHz was calculated as 0.162 pf and the nega­

tive resistance as -447 ohms. Using these values in Eq. (2.19), the

estimated gain is 5.7 dB. Using Eq. (2.21) the noise figure is 7.84 dB,

without including the excess noise of the IMPATT diode oscillation.

CHAPTER III

EXPERIMENT AND RESULTS

The experimental arrangement used in building the self-pumped

parametric amplifiers is shown in Figure 3.1. The standard General

Radio bias insertion unit furnished connections for both the regulated

power supply and the variable coaxial 20 cm shorting stub line used as

an idler cavity. The X-band signal generator with externally sensed

leveling circuitry f\irnishes measured 0.0 j^ 0.5 dBm signal at port 1

of the circulator over the 8 to 12.4 GHz range of the signal generator.

The power was adjusted with the calibrated variable attenuator set at

0 dB attenuation. The test method was chosen to allow elimination of

other test circuit variations.

PROCEDURES

With both the Gunn and IMPATT diodes, it was found that the E-H

tuner provided many settings that would allow oscillation. However,

the Gunn diode would seldom return to the same frequency and power

when the bias supply was cycled off and on and would sometimes refuse

to oscillate at all until tuning adjustments were made. The IMPATT

diode had similar characteristics except that it was generally more

reliable and the tuning changes were smoother. Consequently, the

tuning slugs of the E-H tuner were adjusted to their innermost position

and the slugs were adjusted until a position was found which gave the

highest repeatable frequency and power output at the specified

operating bias point. The E-H tuner adjustments then remained at this

setting for the remainder of a data run. Frequency of oscillation was

verified with the absorption wavemeter.

15

Power Supply

15

Bias Limiter

Bias Insertion Unit i

Variable Short

Diode Mount

E - H Tuner

T ^ i n l ^"fTiY /v

3dB

ALC

X-Band G e n e r a t o r

20dB

Power Meter

Spectrum Analyzer

Figure 3.1. Self-pumped parametric amplifier and test equipment arrangement.

17

The principal measuring instrument used for the gain tests was

the Tektronix 491 Spectrum Analyzer. It had been factory calibrated

one month prior to the experiment. The X-band signal was applied to

the circuit and the gain of the spectrum analyzer (SA) was adjusted

for a given deflection as measured against the SA graticle. The

diode bias was then applied and the power meter checked for proper

reading. The stub tuner was varied through its length to find the

highest gain point (if any) as displayed on the SA. The calibrated

attenuation switches of the SA were used to return the trace to the

first adjustment (gain adjustment not changed) and the amount of

indicated gain was recorded. The signal was then changed 25 MHz and the

process repeated. Later, the diode mount was removed and the same

relative indicated gain measurement was made using the diode mount

and a copper shorting plate to find the relative reflection between

the two. The indicated gain was then adjusted to the known -1 reflec­

tion coefficient of a waveguide short.

Once the peaks in real gain were found, the signal power was

varied at the peak gain point to find amplifier sensitivity and

dynamic range within the limits of the circuit and the measuring

equipment. In addition, the signal power available at the diode

mount was measured to find the transmittance of the circulator and

E-H tuner at that frequency.

GUNN DIODE TEST

The Gunn diode used had maximum ratings of 10 volts and 150 ma

bias. The bias operating point selected was 8 volts and 80 ma. To

prevent inadvertent destruction of the device, a voltage limiter

18

consisting of a 130 ohm series resistor and a shunt 9 volt 1 watt

Zener diode were connected prior to the bias insertion unit. The

power supply limiter was preset to limit at 150 ma. The voltage

across the diode did not exceed 9.2 volts under maximum drive condi­

tions. The gain curve for the Gunn diode is shown in Figure 3.2.

The oscillator frequency of 11.8 GHz was maintained within limits

of +0.12, -0.16 GHz. As the signal frequency approached the oscillator

frequency from the lower frequency side, the gain first increased then

decreased before entering the region of frequency pulling on the

oscillator. A strong idler frequency at 80-100 MHz was noted (three

measurements at different signal power levels indicating oscillator

pulling with signal power) but adjustment of the idler stub had no

discernible effect. The peak power gain of 8 dB then decreased

before rising again as the oscillation frequency decreased and

other harmonic frequencies were noted above the oscillator frequency.

At this point the SA had too many spurious responses to determine

whether a difference frequency was present, although the spectrum

around the oscillator frequency indicated the signal and oscilla­

tor were generating an idler frequency and harmonics of the idler

which were then adding to the oscillator to produce lower ampli­

tude peaks above the oscillator frequency. Above the oscillation

frequency, no regions of gain were found. The frequencies above 12

GHz were not checked because of decreased response of the connecting

circuitry. Below 11.525 GHz, no significant gain peaks were found

until the 10.65 and 10.5 GHz region. Only the peak centered about

19

PQ

P •H

o u

a

6 P

5 -

0

-1 I ± ± 10.25 10.75 11.25

± 11.75

Signal Frequency - GHz

Figure 3.2. Gain vs frequency for Gunn diode amplifier.

20

10.65 GHz was found to have a significant idler power. At the highest

measured gain point on the 10.55 GHz peak it was found that adjusting

the idler stub would vary the gain by approximately 1 dB. The idler

frequency was checked by observing adjustment points on the idler

tuning stub which would suppress the idler signal. Below 10.1 GHz,

absorption of the signal increased and no measurements were made below

8.95 GHz. The dynamic range is not linear. The gain was 7 dB with

the signal power 12.5 dB below oscillator. The gain increased to

8 dB with signal 15.5 dB below oscillator and decreased to 5 dB with

signal 18.12 dB below oscillator. The signal was lost in the SA

noise level below this point and further measurements could not be

made. Oscillator measured power was 2.55 dBm.

17 H. W. Thim performed a similar experiment to the one reported

here except that his oscillator was designed to prohibit the existence

of an idler current. He measured the amplification due to the nega­

tive conductance of the Gunn diode while it was oscillating without

parametric effects. Thim measured maximum gains between 3 and 4 dB

13 with a high degree of frequency dependence in the curve. Kuno also

noted 4 to 5 dB gain, prior to tuning, in his experiment with self-

pumped parametric amplification. It should be noted that Kuno's

report is the most complete published report on the self-pumped

parametric amplifier with a Gunn diode.

The measured gain of the Gunn diode amplifier reported here was

5 dB in the single region where parametric effects could be positively

identified. It was noted that tuning the idler stub would lower the

gain to about 5 dB and that the null points were very close to a

21

half-wavelength at the predicted idler frequency, implying that the

1.0 dB variance came from the parametric effects. The circuit used

for prediction of gain yielded a predicted gain of 3.3 dB due to

parametric effects. This implies that, other factors being equal, the

domain capacitance was computed high by a factor of 1.89. Any change

in the assumed values of load or idler conductances would also affect

the discrepancy between predicted and measured gain.

The 5 dB gain not affected by the idler cavity adjustment 17 undoubtedly comes from the mechanism reported by Thim. Using the

standard formula for reflection amplifiers, with an estimated negative

resistance of -145 ohms and load of 50 ohms, the amplifier would

deliver 6.24 dB gain. The total gain from both amplifying effects

could be predicted as being from 7 to 9.54 dB, depending on the

resulting phase relations.

The relatively low value of negative resistance, -145 ohms, is

17 18

less than that estimated by either Thim or Hobson and occurs

because of the comparatively large (estimated 5000 square micrometers)

cross-sectional area of the diode coupled with the fairly wide domain.

Since the input circuit had no provision for impedance transformation

other than the E-H tuner (Kuno used a variable coupler) used to

control the oscillation frequency, the 50 ohm load could not be

more closely matched.

The space charge capacitance estimated at 0.372 pf was based on

rather gross approximations. For example, the frequency used for the transit time frequency may not have been the true transit time

5 frequency. The value of domain velocity used, 10 m/sec, is commonly

22 4 19

used although other values down to 5 x 10 m/sec have been reported. 20

If the threshold field is taken as 3200 volts/cm, the active region

width would be 9.4 microns instead of the 11.5 micron used. If the

21 -3 active region were uniformly doped to a density of 10 m as

assumed, (approximate resistivity of 1 ohm-cm) the low field resis­

tance would be greater than 20 ohms instead of the 13 ohms stated.

The verified 13 ohm low field resistance corresponds to a resistivity

21 -3 of about 0.5 ohm-cm (3 x 10 m doping) if the contacts and case

are considered perfect conductors. Variations in mobility also would

affect the low field resistance. Since the large change in doping

would affect the relative velocities of the domain electrons and out-

20 side electrons, the oscillations would be difficult to control, whica

was not observed experimentally (a subjective judgement). Finally,

simply reversing the polarity of the bias voltage to the diode

resulted in a change of oscillation frequency from 11.8 GHz to

9.53 GHz. Clearly, the diode active region is not the homogenous

entity of theory. Hasty used similar diodes and found evidence

that they were not homogenous. Thus, the estimation of space charge

capacitance was actually an educated guess, and an exact design match

would have been merely fortuitous.

Many odd effects were found in working with the Gunn diode.

Most of these effects have been reported by one or more investigators

21 in the extensive literature available. Mode jumping was particularly

bothersome since the diode would operate at one frequency, be cycled

off then on, to operate at a different power and frequency. It was

found that in a few instances the desired mode would repeat for as

23

many as six cycles, then a new mode would appear on the seventh cycle.

Mode jumping would also occur spontaneously for no apparent reason.

The bias voltage could be used to tune the device frequency some,

higher voltage resulting in a lower frequency, but the rate of tuning

was inversely proportional to the amount of power coupled to the

power meter, indicating cavity Q dependence. The frequency spectrum

of the oscillation was also wider at lower voltages and at higher

delivered powers, indicating a noise content dependent on cavity Q and

operating voltage. At lower bias voltages, multiple frequencies

were obtained with one instance of a spectrum linewidth greater than

75 MHz. This effect also occurred at bias voltages higher than 8

volts at a few tuning conditions. At the frequency where parametric

amplification was found (10.55 GHz), the oscillator frequency would

vary discernibly when following the sweeping signal. This also

occurred near 11.55 GHz where the next gain peak was located. At

11.55 GHz, parametric amplification was suspected but the idler

frequency was too low to be affected by the tuning stub. The idler

was probably resonated by the bias insertion unit and the decrease

in gain just prior to the oscillator frequency could be due to the

lack of a resonator for the idler circuit at the frequency. Note

that the oscillator and first peak gain are separated by 150 MHz as

are the 10.55 and 10.5 peaks lower on the gain curve in Figure 3.2.

The spectrum analyzer responses noted above the oscillation frequency

just before frequency locking are probably due to multiple mixing

products which occurred in the analyzer passband. This phenomenon

22 has been reported also by Hakki.

24

IMPATT DIODE TEST

The IMPATT diode was adjusted to oscillate at 9.12 GHz with a

bias of 59 V and 19 ma. Output power was 9 milliwatts, although it

would tend to increase slightly when an idler signal was detectable.

No tests were made at signal frequencies above the oscillator

frequency.

The curve of gain vs frequency for the IMPATT diode is shown as

Figure 3.3. When the signal frequency was near the oscillator

frequency, the oscillator tended to synchronize with the signal as

expected. As the signal and oscillator lost synchronism, the signal

was absorbed greatly. With further decrease in signal frequency, the

gain rose to zero followed by an absorption region at 8.925 GHz. The

amount of loss, though small, was variable with the idler stub. A

region of small gain was found at 8.85 GHz, then a period of no gain.

The major region of amplification is found from 8.225 to 8.5 GHz, with

the peak gain at 8.3 GHz. The variation of the idler tuning stub pro­

vided control of the amplification throughout this region. It was

noted that the position of the tuning stub which gave maximum ampli­

fication was fairly broad while the adjustment to suppress amplifica­

tion was sharp. The 20 cm stub could not find two null points without

the addition of fixed stub line. Below 8.2 GHz, the signal was

absorbed down to the 8 GHz limit of the signal generator.

Figure 3.4 shows the dynamic range of the amplifier. The curve

shows the 1 db compression point to be about -5 dBm or 12.33 dB below

the oscillator power. Gain drops off linearly until the 3 dB gain

25 dB

m)

o CN

1

w •H

O

pq xi

P •H

Gai

n

8 7 6

5

4

3

2 1 0

- 1 - 2

- 3 - 4

-5 -5

- 7 - 8

8.1 8.2 8.3 8.4 8.5 8.5 8.7 8.8 8.9 9.0 9.1

Frequency in GHz

Figure 3.3. IMPATT diode gain vs frequency characteristic.

pq

I

p •H fd CD

m (0

25

-35 -30 -25 -20 -15 -10 -5 0

Signal Input Power - dBm (NOTE: Oscillator (3 7.33 dEm)

Figure 3.4. IMPATT dynamic range. Adjusted for transmission losses and reflection coefficient.

27

point at -29 dBm, 36.33 dB below the oscillator power. The oscilla­

tor showed signs of frequency pulling at signal power -6 dBm.

The parameters estimated in Chapter II and Appendix C result

in an estimated gain of 5.7 dB against a peak measured gain of 5 dB.

Negative resistance amplification alone would contribute another

0.9 dB of gain, although no evidence of negative resistance amplifica­

tion was found as all amplification detected was tunable with the

idler stiib.

23

Ku and Scherer have developed a theory and have applied it to

the optimization of the gain and bandwidth of avalanche-diode negative

resistance amplifiers under small-signal conditions. They found that

the diode susceptance and conductance were both functions of frequency,

bias current, and the amplitude and waveform of the applied signal. 24 Scherer considered the large-signal case but uses the approximation

that the susceptance is "tuned out". Both report the sensitivity of

the diode characteristics to large signals and to harmonics of the

signals present. Scherer has developed a computer technique for the

solution of large signal problems but cautions that there are still

approximations in the program. It is difficult to determine if the

applied signal principally affected the particle current by changing

the timing of the avalanche or if the additional signal voltage

modified the varactor capacitance being pumped by the avalanche.

The IMPATT diode presented a few problems in the initial

circuit operation. The only major problem was in achieving a

repeatable state of the oscillation. As with the Gunn diode, the

cycling off and on of the bias voltage would result in different

28

oscillation states. Once started, however, the diode behaved

reasonably, showing smooth transitions between frequency and power

states with E-H tuner changes. The frequency spectrum linewidth was

variable with tuning, also showing smooth transitions between states.

The oscillation frequency chosen for the test provided high output

power with narrow spectrum linewidth.

The frequency of maximum gain, 8.3 GHz, is related to the

oscillation frequency, 9.12 GHz, in the ratio 10:11. The existence

of major harmonic effects on the gain curve seems improbable. The

idler frequency, 0.82 GHz, does not appear to be harmonically

related to either oscillator or signal frequency.

The dynamic range of the amplifier appears to be a function of

the power in the signal. The point where the gain began dropping

was also the point where the oscillator frequency began to show signs

of synchronization with the input signal.

CHAPTER IV

CONCLUSIONS

GUNN DIODE SELF-PUMPED PARAMETRIC AMPLIFIER

It is clearly possible to build a self-pumped parametric amplifier

using a Gunn diode as the active element. There have been no published

reports of the technique being used other than in a laboratory experi­

ment situation. Based on the results of this experiment, and on

literature available, it seems that the technique has too few benefits

to outweigh its many shortcomings. The principal benefit of the

technique is the elimination of one active component (the varactor)

and some circuitry in addition to the varactor bias supply. These

components are not extremely expensive nor are they difficult to

design.

The shortcomings of the technique begin with the difficulty of

arranging for a single device to resonate at three incommensurate

frequencies simultaneously. The oscillator tank circuit should be

high-Q to lower noise levels and to improve the stability of the

oscillator. The idler circuit should be low-Q to improve bandwidth

but with only a small resistance to add to the noise. Ideally, the

idler is designed to store little energy and to use the device

internal resistance as its load. The signal circuit must be wide

bandwidth but also should have rejection filters at the pump and

idler frequencies to eliminate spurious responses from succeeding

circuits. These filters should not interfere with coupling the

signal power into the device. The dynamic domain capacitance should

29

30

be high and the device low field resistance should be low to increase

dynamic quality factor and reduce noise, while also making the

matching of the device to standard 50 ohm circuits easier. Increasing

doping level and larger cross-sectional area to the active region

accomplishes this but the result is larger dc currents and there are

limits on the thermal dissipation of GaAs. ' Increased doping

levels also affect domain velocity and excess voltages, ultimately

causing the domain field to reach the avalanche breakdown point.

Should it be desired to use microstrip or stripline construction for

size, weight, and cost factors, the limited range of impedances

obtainable and the lack of tuning techniques add more difficulties to

the design problem. The technology of Gunn diodes is still such that

the manufactured units are far from uniform in static and dynamic

characteristics. Individual selection of diodes would be costly and

would probably destroy any cost advantages gained in using stripline

or microstrip construction. In standard waveguide construction, with

more normal frequency relationships than used here, the interacting

adjustments required would cause enough problems to negate the cost

savings on the varactor. The exact design can only be done with

computer techniques since the device equations are nonlinear and only

approximate models exist even in present computer programs.

The logical competitor to the self-pumped parametric amplifier

would be a simpler amplifier using a Gunn diode oscillator to pump a

standard varactor. Okean, et al.^"^ have built three S-band parametric

amplifiers using a Gunn diode as a source pump and a varactor for the

reactance. Both semiconductor devices were standard commercial units.

31

Using waveguide construction for the Gunn oscillator and microstrip

for the varactor unit, the amplifiers occupy 6 cubic inches each and

weigh under 0.6 pounds each. The only additional equipment required

is input and output signal connections and a power supply providing

8 volts at 0.5 amperes. The amplifiers deliver 15 dB centerband gain,

bandwidth greater than 76 MHz and a noise temperature of 170° K for

the noisiest unit.

Considering the above, the self-pumped parametric amplifier

with a Gunn diode does not seem to have sufficient advantages over

simpler techniques to warrant further commercial development at this

time.

IMPATT DIODE SELF-PUMPED PARAMETRIC AMPLIFIER

As with the Gunn diode, the IMPATT diode used as a self-pumped

parametric amplifier has been investigated and ignored. The tech­

nique , while possible, has numerous shortcomings and few benefits.

The technique would allow the varactor diode and some small circuitry

o

to be discarded. Clorfeine feels that the noise figure could be

better than the avalanche diode negative resistance amplifier if the

frequency ratios were optimized as in normal varactor parametric

amplifiers. There is no doubt that the design of narrowband amplifiers

of this type could be systematized although optimum wideband circuits

would probably require computer aids to the design.

The question is if the potential gains are worth the design and

development effort. At this time, it appears that the benefits are

not worth the effort. For example, the coupling of a signal into an

32

oscillating IMPATT diode results in the conductance and susceptance

(upon which the technique depends), as well as the frequency of

27 oscillation and possibly the bias ciu?rent, being changed. The

exact effect will be determined by the amplitude, and waveform of the

signal. The amplification achieved is adjustable by the required

idler cavity and the gain is a function of the idler impedance

characteristic. The stated first step in all of the published

amplifier design articles, negative resistance or parametric, has

been to test the device for the parameters to be used to design the

amplifier. This is seldom possible for commercial applications. The

IMPATT diode is inherently noisy and the noise would be coupled at

least in part into the succeeding amplifier stages. The tuning of an

amplifier could probably be optimized at any frequency over a wide

band but would be narrow-band at any single set of adjustments.

The conclusion is that the IMPATT diode as a self-pumped

parametric amplifier has too few advantages to recommend it for

further development.

APPENDIX A

DERIVATION OF THE MANLEY-ROWE RELATIONS

Manley and Rowe derived a set of general equations relating

power flowing out of nonlinear elements which form a basis for

parametric amplification. Watson gives a concise guide to the

derivation of the Manley-Rowe relations. Assume that the characteris­

tic of a nonlinear capacitor is specified by the voltage as an arbi­

trary function of the charge q, or

V = f(q) , (A.l)

where q is the charge on the nonlinear capacitor and v is the voltage

across the capacitor. The shape of the capacitance characteristic

must be single-valued but is otherwise arbitrary. If the capacitor

is now excited with two non-harmonically related frequencies f and

f , it is expected that, in general, mixing action in the capacitor

will result in sidebands mf + nf being generated, where m and n are p s

integers. This is symbolized with a circuit as in Figure A.l. Each

box represents a filter which has zero impedance at the frequency

indicated and infinite impedance at all other frequencies. Each ac

source generates voltage at the frequency indicated by the series

filter. In any practical case, a source is set to zero if the cir­

cuit conditions do not allow existence of the frequency current.

If all frequencies of the form f = mf + nf can exist, the mn p "

capacitor receives charge and current from each source, which may be

noted as

00 00 j2ir[mf t + nf t] q = Z Z Q e P " , (A.2)

mn ni=-oo n=-~

33

34

Nonlinear Capacitor

0 0

O

^

Ideal Bandpass Filter Impedance = 0 at f

= 00 at other frequencies

Figure A.l. Illustrative Circuit for Deriving Manley-Rowe Relations. (From Ref. 5)

35

~ «> j2TT[mf t + nf t] p s i: I J^e " , . (A.3)

ni=-oo n=-«>

and a voltage develops across the capacitor, as

00 00 j27T[mf t + nf t] p s - v = E Z V e ^ " . (A.4)

mn m=-«> n=-«>

The charge q, current i, and voltage v must be real which means that

the coefficients Q_ , I , and V must satisfy the relations inn mn mn -

U^ ^ = U* , (A. 5) +m,+n -m,-n

U = ^1 ^ <> (A-6) -m,-n +m,+n

where the asterisk indicates the complex conjugate and U is one of the

above coefficients. Since the current i equals the total derivative

of the charge with respect to time, the coefficients are related by

(co = 2irf)

I = i[moo + noo ] Q . (A.7) m,n - p s m,n

The average power flowing into the nonlinear capacitor at the

frequencies +_ |moo + noo | is

p = p = V I" + V" I m,n -m,-n m,n mn mn mn

= V [-j(mu) + noo )] Qj^ mn p s mn

+ V|^ [j(mu3p + nco^)] Q^„

= -2 (mu p + nu,^) Re (j V^^ Q; ) , (A. 8)

where Re(x) indicates the real part of x. The nonlinear capacitor

36

is taken to be lossless, so that the total power into the capacitor

must be zero. Then

00 00

Z Z P = 0 . (A 9) ni=-oo n=-«> '

Equation (A.9) may be rewritten as

moo + noo «> <»

0 = C ; ; T T ^ - T — ^ 3 ^ 5: P moo + noo mn

p s m=-'» n=-«'

00 00 mP 00 00 nP = (0^ E Z M + ^ 2 2 EiB . (A . IO)

P ^_ ^ .„_ mco + noo s _ _ _ moo + noo m—oo n--oo p s m=-oo n=-«> p s

Taking the first double sum of (A.IO) only, one finds

00 00 m p

Z Z "^"^ moo + noo m=-oo n=-oo p s

00 00 - m P 00 jnp Y. ( z ""^»"^ + j ; m>ri . _

^ -mco -noo , moo + noo n=-oo m=0 p s m=0 p s

00 00 J Q P 2 Z Z 2 L 2 l L _ (A .11 )

^ _^ mco + noo n=-oo m=0 p s

where Eqs. (A. 5) and (A.6) have been used. The second double sum of

Eq. (A.IO) may be similarly treated to yield

00 00 nP oo 00 nP Z Z 2 2 = 2 Z Z ^ . ( A . 1 2 )

moo + noo ^ moo + noo m=-oo n=-oo p s m=-oo n=0 p s

An energy quantum at frequency f is hf or fioo, where h is Planck's

28 constant and is h/2TT. Using the method of Brown , the power in

Eq. (A.8) may also be written as

37

^m.n=*^m (% + ""s^ ' (A.13)

where A has the units of quanta per second, with each quanta carry­

ing energy'6(mco + nco ). Each of the quanta A must then be an

p s ^ mn integer. Expressing Eq. (A.IO) in these terms yields

00 CO (j) « 00

Z Z mA + (- ) Z Z nA = 0 . (A.14) mn CO mn m=-oo n=-oo p m=-oo n=-oo

Earlier it was assumed that the frequencies co and oo were not harmo-s p

nically related. Therefore, the ratio co /oo must be irrational. s p

Since m, n, and A are all integers, Eq. (A.14) can only be true if mn to 5 -a J

the terms are separately equal to zero, and

00 CO m P

£ z HE = 0 , (A.15) mco + noo m=-oo n=-oo p s

00 oo n P

^ 2 E2 = 0 . (A.15) mco + noo

m=-oo n=-oo p s

Using Eqs. (A.11) and (A.12), Eqs. (A.15) and (A.15) may be written

as

00 oo j n p

^ ^ m , n _ ^ Q (^^j^7) mco + noo n=-oo ni=0 p s

00 00 n P

Z Z 22 = 0 . (A.18) ^ moo + noo

m=-oo n=0 p s

which are the more usual forms for the Manley-Row relations.

In this study it is of interest to find the case where a third

frequency f. will be generated as the difference between f and f .

38

Equation (A. 17) may be evaluated with m equal to zero and one and

with n equal to zero and minus one, respectively. The result is

1 n Pi 1 P P . CO ^ CO -co ^ 0) CO. • l A . i y ;

p p s P 1

Evaluation of Eq. (A. 18) with m equal to zero and minus one, and with

n equal to zero and one, respectively, yields

-^^t-^:if^= 0 = - ^ - - ± . (A.20) CO -00 +00 CO 00. S P S S I

In a physically realizable circuit, frequency cannot be negative.

Since we have defined pump power p to be flowing into the capacitor,

Eq. (A. 19) says that power at the idler frequency oo. must be coming

out of the capacitor, symbolized by a negative sign. Then, Eq. (A.20)

says that power at the signal frequency must be also flowing out of

the capacitor. Thus, the Manley-Rowe relations state that power

pumped into the capacitor at the pump frequency is converted to the

idler and signal frequencies and is radiated from the capacitor along

with power converted from the pump. It is important to note that the

Manley-Rowe relations do not demand 100% conversion of all pump

energy into signal and idler energy.

APPENDIX B

THE GUNN DIODE

Since J. B. Gunn reported the first experimental findings of

29 the Gunn effect, much work has been reported on various facets of

the device. A complete listing of all references known is not

30 31 feasible in this paper but partial listings have occurred. '

Gunn reported that when a dc voltage was applied to a bar of n-type

gallium arsenide, microwave oscillations were detected when the

average electric field rose above a threshold of several kilovolts

29 per centimeter. The time period of this oscillation was found to

be approximately equal to the transit time of the carriers from

32 cathode to anode of the sample. Further experiments with sensitive

capacitive probes showed that high field domains were being formed

at the cathode and were traveling to the anode inducing current

33 oscillations in the external circuitry. Other experiments proved

Oil ' R

Kroemer was correct when he noted that Ridley and Watkins and

Hilsum had predicted just such effects as experimentally found by

Gunn.

Hilsum logically used the term "transferred electron effect"

to describe what has become known as the Ridley-Watkins-Hilsum (RWH)

mechanism. The terms Gunn effect and Gunn diode usually refer only

to the transit time oscillatory mode which will be discussed later.

The RWH mechanism is a field-induced transfer of conduction-

band electrons from a low-energy, high mobility, valley to higher

energy, lower mobility, satellite valleys. There are several

39

40

semiconductors which possess suitable energy band satellite valleys

(InSb, InP, GaAs, CdTe) but for various reasons, primarily GaAs has

been investigated and used more extensively than the other III-V and

II-VI semiconductor compounds.

Figure B.l is a simplified diagram of the energy band structure

of GaAs. Only one of three satellite valleys is shown. The lower

valley is the <000> minimum while the upper valley represents the

collective effects of all the satellite valleys. At room temperature,

electrons in the lower valley have an effective mobility of about

2 7500 cm /volt-sec and electrons in the upper valley have an effective

2 37 mobility of about 180 cm /volt-sec. The energy difference between

the upper and lower valleys is 0,35 eV. With no applied electric

field, the relative electron populations of the two valleys are

determined by the energy difference and the electron temperature T .

At room temperature (290°K), the average energy of the electrons is

determined by k_T , approximately 0.025 eV, where k is the Boltzri ann

constant. This is much less than the energy difference between the

valleys so that the upper valley will be essentially empty.

As the electric field across the sample is increased, the

electrons will be accelerated, gaining kinetic energy from the

electric field. This increase in energy, reflected as an increase of

electron temperature, causes a redistribution of electrons between

the two valleys , with many more electrons scattering into the upper

valley. When the average energy gained by the electrons from the

applied field becomes comparable to the energy difference between

upper and lower valleys, the rate of transfer from lower to upper

41

Gap 1.4 eV

0

35 eV

• ^ k

wave number

Figure B.l. Simplified diagram of the energy band structure of GaAs. (From Ref. 37)

42

valley increases and the average carrier velocity reaches a peak.

Further increases in applied field result in more electrons in the

upper valley (lower mobility) and therefore the average electron

velocity decreases with increasing field. The result is a negative

differential mobility over a portion of the GaAs velocity vs field

characteristic. Figure B.2 shows a theoretical and experimental

20 velocity-field characteristic of GaAs, as presented by Sze. The

precise calculation of the velocity-field characteristic must take

into consideration the various relaxation and scattering mechanisms,

a task much beyond the scope of this work. A relatively concise

38 discussion of the calculation may be found elsewhere.

The v(E) vs E curve (figure B.2) for GaAs may be approximated

K 39

by

y^E[l+b(|-)^]

v<^) = , .E.k ' ''-'' It(^)

c

where v(E) is the average velocity for a given field E, M^ is the

average low-field mobility, b is the ratio of upper valley mobility

to lower valley mobility, E is defined to be the field such that

the number of electrons in the upper valley is equal to the number of

electrons in the lower valley when the applied field E is equal to

E and (E/E )^ is the ratio of electrons in the upper valley to the c' c

number of electrons in the lower valley. Best fit for Eq. (B.l) to

Figure B.2 occurs for b = 0.05, E^ = 4 kilovolts/cm, k=4, and y^ = 2

8000 cm /volt-sec.

43

o <D CO

""^ 6 o

o H

X

I

-M • H O O

H Q)

>

• H

Q

T 1 1 r I r

Theory

Experiment

0 8 10 11 12

Electric Field (Kv/cm)

Figure B.2. Theoretical and experimental velocity-field characteristic of GaAs. (From Ref. 20)

44

A small signal equivalent circuit of a Gunn diode may be

obtained from the stable domain dynamics of Copeland or Butcher

40 and Fawcett. By making the assumption that the domain is thin

compared to the sample length, Hobson and Knight and Peterson

have modelled the Gunn diode as two series resistors shunted by

capacitances, as shown in Figure B.3.

Using the curves of Figure B.2 and assuming that diffusion is

independent of electric field, the conductance representing the

domain may be calculated as a function of the outside field E

using the following expressions for a unit cross-sectional area:

dE

I = qyn E + 6 -j-^ , (B.2) • o o dt

V = V^ + E L , (B.3) D o

%-§-^- ^-o AV ' ''•''

where € is the dielectric constant of the material, E is the field

outside the domain, V is the domain excess voltage, and L is the

sample length of active material. Domain excess voltage is

^D =

00

f (E - E ) dx . (B.5)

—oo

17 Numerical values of AE /AV^ are obtained from Figure 2 of Thim ;

o D 20

a similar curve is found in Sze. These curves were drawn from

numerical solutions of the coupled differential equations formulated

LLO 40 by McCumber and Cheynoweth and Butcher and Fawcett. Domain

capacitance C will be estimated later.

C A

45

qyn A o

Figure B.3. Equivalent circuit of a Gunn diode. (From Ref. 17)

45

The small signal impedance of the device may be found beginning

with the equivalent circuit of Figure B.3. The domain width is

considered to be thin compared to the sample length which allows the

use of Eqs. (B.2), (B.3), and (B.4). The total impedance of the

device then becomes

^^"^ = X ^ qwn \ ja.e A T ^ ^ ' ( .6) o o . ^

qyn Trr-- + 1C0€ C_ ^^ o AV -" D

where A is the cross-sectional area of the diode and the other

factors have been previously defined. This expression holds during

the time the domain is fully formed and in transit. The cyclic

nucleation and extinction of the domain at the contacts will modulate

this impedance.

It is necessary to estimate the domain capacitance of the Gunn

diode in order to compute the response of the parametric amplifier.

90 18 The procedure that follows uses information from Sze , Hobson and

r 43 Ferry.

Assume the active region is n-doped to a concentration of

21 3 1 X 10 donors/m ; this allows the assumption that domain velocity

5 40 will be about 1 x 10 m/sec. Then the active region of the device

is

5 V 10 m/sec ^, ^ TA"^ „, L = p = p- = 11.5 X 10 m

8.57 X 10

20 Using the curve of domain excess voltage vs outside field from Sze,

the domain excess voltage is estimated to be

47

Vjj = Vg - E^L = 8 - (1.5 X 10^ v/m)(11.5 x lO"^ m)

= 6.15 V .

The operating point of the diode is expressed as

P o = ^J = V [qn Av(E)] = 8(.08) , dc o o o ^ o

-19 where q is electronic charge 1.6 x 10 coul, n is doping density,

A is the active region cross-sectional area and v(E) is the domain

5 velocity assumed to be 1 x 10 m/sec. From this the estimated area

is

I -9 2 A = TTTT = 5 X 10 m

qn^v(E)

39 The domain is represented as triangular in shape, so that as in

E -E Vj = [ -^Y^ ]d = 5.15 V ,

where E is the peak domain field, E^ is the outside field, and d is

the domain width. Maxwell's equations give

^ -_ P—_ ^ JjL^ -. 1.46 X 10 2 /„/„ . AX € € d

Substituting back into the V equation and solving for the result

gives

d = 2.91 X 10~ m = 2.91 microns .

The charge Q in the domain is

-12 Q = Adn q = 2.3 X 10 coul ^ o^

because of current continuity. The capacitance is then

48

C = 2. = 0.372 pf ,

17 18 a value consistent with the findings of Thim and Hobson. The

static capacitance is

€ e A C = ^,^ = 0.0477 pf , S L

where the relative dielectric constant of GaAs is taken as 12.4.

17 20 Using the method of Thim with the figure by Sze , the negative

conductance of the device is estimated as

I (.088-.08) _ _ T -3 , G- = —rr— - -~ .. = -6.9 X 10 mhos . D V 5 - 6.16

APPENDIX C

THE IMPATT DIODE

45 The Read diode structure is shown in Figure C.l. The Read

structure has been difficult to construct while other more general

structures have been found to have IMPATT qualities. Physically, the

Read structure depends on increasing the reverse bias across the p-n

junction until avalanche breakdown occurs. Each electron collides

with structural molecules, causing them to ionize. The ionization

rapidly builds to a large domain of carriers which drift toward the

end contacts. One of the groups of carriers (either electrons or

holes) will enter the low doping density drift region where it will

take it a finite time to drift from the junction to the contact. In

normal IMPATT operation, the carriers move at the saturated drift

velocity. Impressing an ac voltage of selected frequency on the dc

level will result in periodic increases in avalanching. At a

sufficiently high frequency, it will be found that on the upper half-

cycle of voltage, avalanching continues and the space charge continues

to grow until the rf voltage passes through zero again. The space

charge is then at its maximum, so that the now drifting charge lags

the voltage by a nominal 90°. If the transit time of the charge

across the drift region is also 90 electrical degrees, then when the

current reaches the contacts it will be 180° in phase from the

impressed voltage. This results in the effective negative resistance

of the device and explains the source of the space charge capacitance.

A more general structure for an IMPATT diode is shown in

Figure C.2. Since the ionization is no longer confined to the junction,

49

(a)

50

0 +

p n V +

n +v

+v-

(b)

+ n P IT +

P 0

Figure C.l. Read diode structure. (a) Structure with electrons drifting; (b) Structure with holes drifting. (From Ref. 45)

51

J^(o).

J (0,t) n

J w ns

ps J (X,,t) p L

J^M ¥ irl

I 0 X,

Figure C.2. Boundary conditions for a general IMPATT diode. (From Ref. 46)

5:

the analysis of this diode can be transferred to particular diodes

more easily than the analysis of the Read structure.

The basic equations governing the operation of an IMPATT diode

are Poisson's equation and the one-dimensional continuity equations

for electrons and holes with terms included for impact ionization.

Several assumptions may be made to simplify the equations:

CD The velocity of holes and electrons in the device may be

assumed to be equal and field independent.

C2) The effects of diffusion and recombination may be

neglected.

(3) The ionization rates for holes and electrons are assumed

equal to the average or effective rate for the two

carriers. This effective rate is obtained by solving for

the dc breakdown voltage of the diode with unequal rates

and then obtaining an equal rate for both carriers which

gives the same breakdown voltage.

4 Poisson's equation and the continuity equation in one dimension

are

and

§ = 3 . C N ^ - N ^ . p - n ] , (C.l)

% = - - - ^ + g + - [a |J I + cx |J I ] , (C.2) at q 8X d q "- n' n' p' p'

~ = - - ^ + g, + - [a |J I + a |J I ] , (C.3) at q ax ^d q n' n' p' p'

where

53

Jp = qpPpE - kgTPp , ^c.^)

Sn J^ = qny^E - kgTy^ , (C.5)

b' m'

% = K ^ P - ^^^ ""^ ' (C.6)

b' m'

"p = Ap ^^P t- (• ) P] . CC.7)

The parameters used are defined in Table C.l.

Table C.l

Njj, N^ = donor and acceptor densities (cm~ )

PJ n = hole and electron densities (cm )

2 J 9 J = hole and electron current densities (A/cm )

gn = hole and electron generation-recombination

rates through a single-level trapping

center

= hole and electron ionization rates (cm )

2 = hole and electron mobilities (cm /volt-sec)

= Boltzmann constant

o T = Absolute temperature ( K)

p , n = hole and electron concentrations in the con­

duction band when the single-level trap and

the Fermi level are equal

A', b', m' = hole ionization rate constants P P P

A', b', m' = electron ionization rate constants n n n

"p'

^p'

•^B

a n

^n

54

With the assumptions described earlier, the continuity equation

reduces to

1 c v at ax n (J - J ) + 2aJ (C.8)

where J is the conduction current density, a is the ionization rate,

and V is the saturated drift velocity (cm/sec). Substituting the

total current density J less the displacement current density in

Eq. (0.8) and integrating over the entire length of the diode yields

'A^T V

r L 2 a^E 3t2

dx (C.9)

n p

r L

+ 2J, adx - 2 f L

9E ,

where x is the transit time through the diode (=X,/v) and X, is the LI LI

total width of the diode (cm). Integrating over the entire length of

the diode rather than just the avalanche region allows impact ioni-

45 zation to occur anywhere in the diode.

The order of integration and differentiation may be inter­

changed in the second term of Eq. (C.9) to give a term proportional

to the second derivative of the voltage across the diode. The sign

convention observed is

X.

V(t) = E dx . (CIO)

From the boundary conditions shown in Figure C.2, the term

involving the hole and electron currents reduces to

55

n p

X,

0 = ^^sat - 2 1 ^ ^ 37 ^^°'^) ^ ^^^L'^^^ ' ( -ll)

where J is the total reverse saturation current density (A/cm ).

The voltage across the diode is assumed to be of the form

m V(t) = V^ + Z V sin (noot + 0 ) ,

B . n n n=l

(C.12)

where V^ is the bias voltage (volts), n is the harmonic number of the

operating frequency, m is the number of harmonics present, V is the

magnitude of the n harmonic voltage (volts), and 9 is the phase

angle of the n harmonic voltage (radians). Considerable labor

reduces Eq. (0.9) to

^ J^(t) + A(t) J^(t) = Y(t) , (C.13)

where

r L

A(t) = 7 [1 - a dx] , (C.14)

and

2J . ^ . Y(t) = - ~ ^ + 7 [E(o,t) + E(X^,t)]

X,

- 2 . ,„ m 2 2 a Hii dx - Z " V sin(niot + e ) .(C.15) dt T X n n

n=l L

By assuming an initial electric field which is independent of

the total current, the solution to Eq. (C.13) is

56 1

J^(t) = exp[- I ACY)dYJ

o

xCJ^(O) + Y(Y) exp( A(3)d3)dY] , (C.16)

which is periodic with a period T; i.e., J^(t+T) = J^(t). Knowing

the conduction current, the continuity equations can be solved for

the hole and electron current densities to yield

and

X

J (x,t) = J + P ps

aJ (c,t)dx (C.17)

dx = vdt

J (x,t) = J + n ' ns

r L

X

aj (x,t)dx c

(C.18)

dx = -vdt

where J and J are the respective reverse saturation current ps ns ^

densities (A/cm ).

From Poisson's equation a new electric field is calculated which

includes the space charge of the carriers. The field is given by

X

E(x,t) = E^(x) + J (x,t) - J (x,t)

n £ V

dx + E(o,t), (C.19)

where E (x) is the electric field due to the doping profile, indepen­

dent of time, and E(o,t) is the electric field at x = 0; an arbitrary

constant for each t. The voltage across the diode is considered to

be given by Eq. (0.12) and is equal to the integral of the field in

Eq. (0.10). The field at x = 0 is thus given by

57

X, , m /• L

E(o,t) = -£ [v_, + Z V sinCnoot + 6 ) - E^Cx)dx -X, a . n n D L n=l •'

r \ f^ J (Y,t) - J (Y,t) ( -^ ^-~ dY)dx] , (C.20)

o o

where V is selected to give the proper dc total current density for

this electric field.

The conductance and susceptance at each frequency of rf voltage

may be found by Fourier analysis of the total field. For an assumed

electric field at total current density, conductance and susceptance

are obtained from Eq. (C.13) through (0.20). The result must then be

modified to include the effect of the space charge of the carriers

at each point in the diode and every instant of time. Equations

(0.17) through (C.20) may be numerically solved by setting the double

integral space charge term of Eq. (C.20) to zero for the first

iteration. For the succeeding iterations, the space charge of the

previous result is used. When the change in admittance from one

iteration to the next is less than 1-1/2 percent, the solution may

be assumed to have converged.

Greiling and Haddad have used this method to compute the

large signal admittance of several diodes. One result is shown in

Figure 0.3. The diode has an active region of 5 microns width and

the assumed doping density in the p and n regions is high enough

for the field to be zero at the boundary. V^p is defined as 1/2

the peak-to-peak rf single frequency voltage sine-wave present across

RF = 1 V

1 200

- 120

150

80

40

-25 -20 •15 -10 0

Conductance, mho/cm'

CM 6 o

O P rd +J ft 0)

o w C/D

58

+ + Figure C.3. Admittance for a Si p -n-n diode,

(J, =500 A/cm2 and n =3 X lO^^cm"^.) (From Ref. 46) dc o

59

the device terminals. Inherent in this single frequency assumption

is that all the harmonics, of the rf voltage are shorted by the

47 microwave circuit. Similar results were obtained by Mouthaan.

ESTIMATION OF EXPERIMENTAL IMPATT DEVICE CAPACITANCE

The information available on the diode used did not provide a

means of reasonably estimating the parameters to use in computing

the amplifier gain. The logic finally used in the computation follows

a heuristic approach.

The operating point chosen was 19 ma and 59 V. The power

delivered was variable from 7.2 to 11 milliwatts, the most common

power being 9 milliwatts. The efficiency is then 0.8% at 9.12 GHz

and the apparent dc resistance is 3100 ohms. The maximum ratings

were 60 volts and 20 ma, and the breakdown point was noted at

40 volts. It was felt that the diode was probably operating at the

higher end of the susceptance range of the conductance vs susceptance

23 characteristic. Ku and Scherer, in working with the negative

resistance properties of the IMPATT diode, have tested an IMPATT

diode at 20 ma bias and found it to have a capacitance of 0.195 pf.

Greiling and Haddad show the experimentally determined conductance,

susceptance, frequency, and bias conditions for a particular diode

under small signal conditions with an additional note on the doping

density of the diode. The same article has the theoretical

characteristics of a unit diode with the same doping density (see

Figure 5 of Ref. 45 and Figure C.3 of this paper), showing the varia­

tion of conductance and susceptance with frequency and signal

60

conditions. The small signal parameters found experimentally are

graphically extrapolated to large signal conditions for an estimate

of the parameters to be used in our calculation. Therefore, the

experimental conductance of -.4 millimhos is scaled by 7/12 to

-.233 millimhos. The experimental susceptance of 15.5 millimhos is

scaled by 55/90 to 11.3 millimhos or 0.152 pf at 9.12 GHz.

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64

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