INTERFEROMETRIC IMPEDANCE MEASUREMENTS

OF IMPATT DIODES AT X-BAND

by

BILLY CHARLES BROCK, B.S. in 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

pprov(

Cnairman of tk'e Commi

^^V^A fy^'r-^'

Accepted

GradiuAte School

Augus t , 1973

No. 92, "-op. 2

ACKNOWLEDGEMENTS

I am deeply indebted to Dr. D. K. Ferry for his aid

and encouragement in the carrying out of this investiga­

tion and to Dr. R. H. Seacat and to Dr. H. D. Meyer for

their service on my committee. I also express my appre­

ciation to E. F. Adamczyk of Alpha Industries, Inc. for

supplying the IMPATT diodes used in this study.

ii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS 11

LIST OF TABLES. iv

LIST OF FIGURES v

I. INTRODUCTION 1

The Theoretical Approach 1

The Measurement Techniques ^

II. THEORY OF THE IMPATT DIODE 8

Avalanche Breakdown in Abrupt p -n Junction 8

Generalized Small-Signal Theory of

the p"*"-n IMPATT Diode 9

III. EXPERIMENTAL ARRANGEMENT AND OBSERVATIONS. . 23

The Microwave Interferometer 23

Additional Diode Measurements 30

Experimental Results 32

IV. CONCLUSION 37

LIST OF REFERENCES 40 APPENDIX 43

A. FORTRAN IV COMPUTER PROGRAM FOR COMPUTING IMPATT DIODE IMPEDANCE 44

B. THEORY OF THE ONE-SIDED ABRUPT p'*'-n JUNCTION 49

C. ALTERNATE METHOD FOR OBTAINING CURRENT DENSITY EXPRESSIONS 55

ill

LIST OF TABLES

Table Page

3.1. ADDITIONAL DIODE DATA 33

IV

LIST OF FIGURES

Figure Page

2.1. Ionization coefficient as a function of distance in an avalanching diode 10

2.2. Generalized IMPATT diode model 12

2.3. IMPATT diode impedance as function of frequency 21

2.4. Diode susceptance plotted against conduc­tance for several values of current 22

3.1. Schematic diagram of microwave inter­

ferometer 24

3.2. Reduced height waveguide and diode mount . . . 26

3.3. Linearly tapered waveguide section 27 3.4. Reflection coefficient as a function o.

frequency for a linearly tapered wave­guide transition section 29

3.5. Schematic diagram for measuring dioae capacitance as a function of reverse bias voltage 31

3.6. Theoretical and experimental impedance . . . . 3^

3.7. Theoretical and experimental impedance . . . . 35

B.l. Doping; profile and electric field for the one-sided abrupt Junction ^1)0

B.2. Depletion capacitance as a function of applied voltage for a one-sided abrupt junction 5^

V

CHAPTER I

INTRODUCTION

The acronym IMPATT stands for IMPact ionization Ava­

lanche Transit Time (which describe the semiconductor prop­

erties that give the IMPATT diode its negative resistance

behavior at microwave frequencies). The use of avalanche

properties to obtain high frequency negative resistance was

first proposed in 1958 by W. T. Read, Jr. [1]. The Read

+ + diode utilized an n ~p-i-p structure. The first IMPATT

operation was obtained in 1964 by Johnston, DeLoach, and

Cohen using a p-n junction diode [2]. The diode structure

can be that of the p-n diode, the p-i-n diode, or the Read

diode [3]. The following work is based on the abrupt p -n

junction.

The Theoretical Approach

The theory proposed by Read [1] in 1958 resulted in a

very complicated expression for the diode impedance. His

work applied to the specialized structure mentioned above

and not to the simple p -n junction. Since 1958, several

approaches have been tried to simplify Read's model and to

develop a more general model that can be applied to other

structures.

1

Gilden and Hines [4] used a space-charge wave approach

to obtain a simplified small signal model which is much

si..pier than the model proposed by Read. Their approach

was applied to a p -n abrupt junction. The diode is con­

sidered to consist of three regions: an avalanche region,

a drift region, and an inactive region. The diode impe­

dance that results from this analysis shows a positive real

part below an avalanche frequency and a negative real part

above this frequency. The reactive part changes from in­

ductive to capacitive at the avalanche frequency. The ava­

lanche frequency is an increasing function of the diode

current.

A detailed study of the IMPATT diode under small sig­

nal conditions, similar to that of Gilden and Hines, was

reported by Gummel and Scharfetter [5]. This analysis

placed more importance on the avalanche region and was ap­

plied to a diode with a localized avalanche region. The

impedance calculated from this approach also shows a reso­

nance effect. In this case, the real part changes from

positive to negative at a frequency below the avalanche

frequency, where the reactive part changes from inductive

to capacitive. The real part remains negative for frequen­

cies above the transition frequency.

A small signal lumped model based on a state-space

analysis was proposed by Stewart, et_ a]^. [6]. This model

J

was applied to the Read structure and to a p -n junction.

As with the previous approaches, the impedance found by

this method shows resonance effects. The results for a

p -n diode show the real part changing from positive to

negative and the reactive part changing from inductive to

capacitive, but the changes do not occur at the same fre­

quency, the transition frequency for the reactance being

higher than the transition frequency for the resistance.

Misawa [7], [8] has reported an approach that allows

an analytical expression to be derived for the IMPATT diode

impedance. This method uses a multiple uniform layer ap­

proximation, similar to that of Gilden and Hines, for the

diode. The avalanche is considered to be uniform and is

limited to a specific region. The resulting impedance be­

haves in much the same way as that resulting from the pre­

viously mentioned methods, except that it shows a positive

real part at some frequencies above the transition fre­

quency. As will be shown in Chapter IV, this result agrees

well with experimental measurements.

The theoretical approach used in this work is that of

Misawa's multiple uniform layer approximation. Chapter II

explains the assumptions made with this approach. The

theory of avalanche multiplication is introduced. An anal­

ysis is then performed on the diode model and an analytical

expression for the diode impedance is derived. This ex-

pression is evaluated numerically for several realistic

diodes by computer calculation. The computer programx,

written in FORTRAN IV, is presented in Appendix A. Real­

istic input data for p -n diodes were obtained from meas­

urements performed on non-avalanching diodes. A discussion

of the theoretical considerations needed to obtain this in­

formation is included in Appendix B.

The Measurement Techniques

The IMPATT diode operates at microwave frequencies so

that conventional impedance measurement techniques cannot

be employed. Because of the difficulties associated with

microwave impedance measurements, some investigators [9],

[10] have been content to measure signal gain as functions

of diode bias current and signal frequency. Small signal

gain and oscillations have been observed in a region where

a negative resistance is predicted by the theories. This

provides some evidence for verification of the IMPATT diode

theory. More precise information is needed, though, to

show any detailed accuracy of the theories.

A measurement technique reported by Gewartowski and

Morris [11] Involved using a computer to reduce measured

data to diode impedance values. This method has also been

applied by Isobe and Nakamura [12]. It requires diode

measurement to be made at low frequency (about 1 MHz.) with

conventional techniques and at microwave frequencies with a

slotted transmission line. Accurate dimensions for the

slotted line and diode mount must be known. This technique

allows the determination of the active diode impedance in­

dependent of parasitic impedance.

Another measurement technique using computer calcula­

tions has been applied by Kuno, Fong, and English [13].

This method involves Using a slotted transmission line and

an adjustable short to measure voltage standing wave ratio

and phase shift. An iterative computer calculation is used

to evaluate circuit model parameters to fit the experi­

mental curves to the model. The package reactance is in­

cluded in this measurement. The resulting impedance has

been successfully used to predict diode performance.

Impedance measurements have been made from measurements

of the reflected wave resulting from the diode impedance.

In the method described by Steinbrecher and Peterson [l4],

a reflection type microwave network analyzer is required.

The impedance as a function of bias current can be plotted

directly on a Smith chart. This impedance consists of the

diode impedance and the network impedance. Measurements

with known impedances in place of the diode are required so

that the actual diode impedance can be separated from the

measured impedance. Another reflected wave measurement,

reported by Misawa [15], uses simpler equipment, but pro­

vides only evidence of the existence of a negative resis-

tance as a function of frequency and bias current. In this

method, the diode, mounted in a coaxial transmission line,

is fed by a leveled microwave power source and the reflected

power is measured. The actual value of the Impedance is

not calculated. A change in the reflection coefficient

shows evidence of the negative resistance.

Accurate impedance values were obtained by van Iperen

and Tjassens [16] with a microwave bridge technique. To

obtain the high accuracy reported, the diode mount was

machined with optically flat surfaces, and the dimensional

requirements were very stringent. The system was precisely

calibrated for a given frequency and results were reported

only for one frequency. Although accurate measurements can

be made with this method, the inconvenience in obtaining

data over a band of frequencies is considerable.

The microwave impedance measuring techniques described

above each have important disadvantages. For some cases

the package reactance can mask the actual active device re­

actance causing inaccurate results unless package para­

meters are known precisely. The conductor supplying dc

bias to the diode results in a transmission line impedance

that must be considered in analyzing the data. Transmis­

sion line lengths must be known accurately. In some cases,

the changing of the measurement frequency requires the

changing of circuit dimensions or calibration. Accurate

7

impedance measurements can be made, but with considerable

inconvenience on the part of the investigator.

The u.se of a microwave interferometer as the measuring

device can provide accurate yet simple measurements. The

Interferometer measures differences between impedances in

two arms of a microwave circuit. Because only difference

measurements are made, any deviation from symmetry in the

two circuit arms can be zeroed out of the measurements.

The effects of the diode package can also be zeroed out of

the measurement so that only the active device parameters

are measured. The interferometer is readjusted for each

new frequency so that any frequency dependence of the meas­

urement accuracy is eliminated. Chapter III describes the

microwave interferometer and the procedure used in making

the impedance measurements reported here.

CHAPTER II

THEORY OF THE IMPATT DIODE

Avalanche Breakdown in Abrupt p -n Junction

When high electric fields are applied to semiconductor

materials, carriers are accelerated to high velocities and,

when the field is high enough, they can knock more carriers

off of the atoms during collisions. Thus a carrier multi­

plication process occurs and the current suddenly increases.

The carrier concentration increases by a factor of M, the

multiplication coefficient. The rate a at which the ioni­

zation process occurs can be expressed as [17]

where m = 1 for silicon, and

A = 3.8-10^cm"i n

b = 1.75-lO^V/cm

for electrons in silicon, and

A = 2.25-10^cm~i P

b = 3.26-lO^V/cm P

for holes in silicon. When the multiplication process is

due primarily to electrons, as in the one-sided p -n

8

junction. Miller [l8] has shown a general result that

1-i = M

rX ^d - (a -a )dx' ^ a^e ^0 " P dx , (2-2)

where M is the carrier multiplication factor, and w. is the

width of the depletion region. The condition for avalanche

breakdown occurs when

M->«, , (2-3)

and the integral in equation (2-2) approaches unity. The

integrand of equation (2-2) is plotted as a function of x

for the avalanche condition in Figure 2.1. The diode

voltage at which the avalanche occurs is called the break­

down voltage V and has been shown by Sze and Gibbons [17]

to be given by the empirical expression

V^ = 60(E^/l.l)3/2(N^/10i6)"3/^ , (2-4)

where E =1.12 eV for silicon and N^ is the donor concen-g D

tration in the abrupt junction. The diode area A, the im­

purity concentration N^, and the n region width w can be

determined from measurements based on equations (B-14) and

(2-4).

Generalized Small-Signal Theory

of; the_ p' -n IMPATT Diode

The generalized theory is based on the multiple uni-

10

A 25 V

4^ G 0

O •H CM CM 0 O

O

0 Xi O G cd

rH cd >

< a > = a e n

a Wi = 1

rX

(a - a )dx o n p

<a>dx = 1

+ Wi w

Distance Into n Region (x)

Fig. 2.1.—-Ionization coefficient as a function of distance x in an avalanching diode. Notice the approxi­mations matlon.

a o w I made in the multiple uniform layer approxi-

11

form layer model of the IMPATT diode suggested by T. Misawa

in 1967 [8]. The model consists of an avalanche region

(x = 0 to X = wj) in which the avalanche coefficient a is

constant, non-zero, and in which 95% of the avalanche oc­

curs and a drift region (x = wj to x = w) in which a = 0.

It is also assumed that the carrier drift velocities are

the saturation drift velocities (in silicon, the saturated

velocity, v^ (= lO" cm/sec), occurs when the electric field S X

exceeds about 10^ V/cm)[3]. The model for a p -n diode is

shown in Figure 2.2. The analysis begins with the applica­

tion of Poisson's equation

If = q/e(Nj^-N^+p-n) , (2-5)

where q is the electronic charge, e is the material perma-

tivity, N. is the accepter concentration, and p and n are

the hole and electron concentrations respectively. The

current density in the diode is composed of two parts such

that

J = J +J , (2-6) p n

where the components Jp and J^ are given by

J = qpvi 2 = -qpVg- , (2-7)

and

J = qny £ = -qnv^. , (2-8) n n ^ si

12

(a) +

p 1 i 2 In

X = 0 Wi w

E(x)

(b)

Wi w <a>

(c)

X

Wi w

Fig. 2.2. —(a)

(b) (c)

Generalized IMPATT diode model 1. Avalanche region 2. Drift region Electric field distribution Avalanche coefficient <a>

13

where u and u are the hole and electron mobilities, re-p n

spectively. Imposing current continuity for the avalanch­

ing junction requires that

3J ! t = ^33r^»^i("^p^ ' (2-9'

and

If = - J al^-si("^P) • (2-10)

A substitution of (2-7) and (2-8) into (2-5), (2-9), and

(2-10) delivers the equations in terms of the quantities of

interest, current densities and electric fields, as follows

|1 = a(N N J + - ^ ( J -J ) , (2-11) ax e D A V , e n p ' si ^

, 9J dJ - - ^ TT^ = r-^ - a(J +J ) , (2-12) V T 8t ax n p ' si '^

and

. aj -aj - -i- — ^ = --^ - a(J^+J^) . (2-13)

V -, at ax n p si

For the small signal case, the following quantities are de­

fined

E(x,t) = Eo(x)+E(x,t)

Jp(x,t) H Jp^(x)+Jp(x,t)

• n ""* ^ ' ' no ' ' ' n ''' ^

14

a(x,t) = a^(x)+a"(x)E(x,t)

a^(x) E ll (2-14)

where the time dependence is assumed to be of the form

e'' . After substituting the relations of (2-14) into

(2-11), (2-12), and (2-13) and ignoring second order pro­

ducts of time varying terms, the following time dependent

equations result:

lf = 7rT(VJp) ' (2-15) si ^

aj . . . . T—^ = - ^r-^ +OL(J +J ) + a'E(J +J ^ , (2-16) ax V , n o n p no po *

and

aJ . , . . ^ = ^ J -a (J +J )-a E(J +J ) . (2-17) ax V -, p o n p no po

S J-

Equations (2-l6) and (2-17) are added together, resulting

in

aj aJ . . . -J^--^ = ^^^{J -J ) , (2-18) ax ax V , p n ' si

which is then substituted into (2-15) to give

^^ . aJ aJ II = ^(jz^^TT') . (2-19) ax £03 ax ax

Integration with respect to x of the above equation gives

15

the following Important result

jeu)E+J +J = constant = J . (2-20) " p n

The time dependent current density is a constant function

of X in the diode. It is composed of hole, electron, and

displacement components.

A second order differential equation for E is found by

differentiating equation (2-15) with respect to x and mak­

ing substitutions with equations (2-16), (2-17), and (2-20)

as follows

; 2F ,.,2 J2aoa) 2a'Jo ~ ~ 2ao ax" v -. V -, V ,e V ,e v -, e + (- i H- )E = J( - V") , (2-21)

" T^ V T V ,£ V ,£ V T^'~ ' Sl sl sl sl sl

where

Jn = J +J . (2-22) ^ no po

The electric field E will then have the form

i = CieJ' ''+Cae-J' ''+(|H - T^^W ' (2-23) sl sl

where Cj and C2 are coefficients, which can be complex and

k = [-^ - 2oLjLa .i2aj -il/2 (2-24) sl sl sl

is the propagation function. V(x) can be obtained by in­

tegrating E(x)

V(x) = (X~

E(x)dx . (2-c'5) 0

16

Using equations (2-l8), (2-19), (2-20), and (2-23) to solve

for J and J algebraically, the following are obtained:

} = _ i!^l(kWv3,)C,e^'^='.q^(Wv^^)C,e-J'<'^-J?^ ^ sl

(2-26)

and

J„ = i^(k-<./v ,)CieJ'^^-i^(kWv ^)c,e-J'^^-^^:i^ . n 2 s l ^ 2 sl"^ k v , e sl

(2-27)

These results can also be obtained by solving the differ­

ential equations (2-l6) and (2-17). The details are shown

in Appendix C. These results for J and J are similar to p n

those obtained by Sze [3] and Misawa [7], and DeLoach [19].

The solutions obtained for the electric field and cur­

rent densities contain tv;o arbitrary constants that must be

evaluated. The form of the solutions in each region of the

uniform layer model must be considered. In region 1, where

a is a non-zero constant, the solutions are exactly as

above, but in region 2,

ao = a" = 0

and k = oj/v -, si

The results for the two regions are

17

il(x) = Cne^'^^+Ci2e-J'^^+(f^ - ^ ; ^ ) T ^ , (2-23) ^sl^ ^sl ^ ^

• •

T - "^ ^sl/, ^ /„ Np jkx.'^^^sl/, / v^ -jkx Jg'Jn

^ sl

(2-26)

T - ^_^sl/, /, Np jkx "^^^sl/, , / N^ -jkx Jg'Jn

sl

(2-27)

for region 1, and

ECO

(2-28)

J = -jea)C2ie-' ''''' ''sl , (2-29)

P2 " ^ ^

J = -jea)C2 2e'^'''''^''sl (2-30) n2 ^ ^ '

for region 2.

The following boundary information is used to evaluate

the four coefficients:

J - = 0 at X = 0 y ^^ (2-31)

Jpi = Jp2 = 0 at X = wi ^ (2.32)

and

J = J at X = wi . (2-33) ni n2

The resulting coefficients are

11

- j k w i - j 2 k w i c J ( b e +ae )_

- j 2 k w i ( b ^ - a ^ e )

18

( 2 - 3 4 )

- j kw 1 cJ ( b+ae )_

12 - - j 2 k w i ( b 2 - a 2 e )

( 2 -35 )

C21 = 0 ( 2 - 3 6 )

a n d

22

V j b w i V - j a w i : 5 ^ a C i i e + 3 ^ b C i 2 e 2a) ^ - 2(1) ^ ^

J v ^ ce s l 2a)

j a )Wi /Vg^

( 2 -37 )

w h e r e

a = k-a) /v s l

( 2 -38 )

b = k+a}/v s l

( 2 - 3 9 )

and

s l (2 -40 )

The t i m e v a r y i n g v o l t a g e a c r o s s t h e d i o d e i s g i v e n by

V =

rW

' 0

rw

Eidx+ E2dx ( 2 - 4 1 ) Wi

w h i c h r e s u l t s i n

V = C i i j k w i C12 - j k w i

' \ ( e - 1 ) - T ^ ( e - 1 ) j k 3^

19

sl sl

C22V--1 -ja)w/v . -ja)Wi/v . S-L.(e si_g Sl)

_JCw::w^ (2-42) eo) '

where it should be noted that C n , Ci2> O22 are propor­

tional to J. The diode Impedance is defined as

Z = J- , (2-43) JA

where A is the diode area. This gives the following result

for the impedance

-jkwj -j2kwi jkwj r. f s _ c(be +ae )(e -1) ^ ^ = jtol

jkA(b2-a2e )

-jkwj -jkwi ,c(b+ae H e -1)

-jkwi jkA(b2-a2e )

. (2an _ J(o )Wi iw-wi) ^ v .£ V '^e^iPT ea)A Sl" Sl

2 - j kwi - j2kwi jbwi - j kwi - j a w i . ^ s l / ^ r a ( b e +ae )e - b ( b + ae )e T2Z7X'- - j2kwi

(b2-a2e )

ja)Wi/v^-, -joiw/v -Ju)Wi/v +e ^ ^ ] [ e ^ - e ^ ^ ] . ( 2 - 4 4 )

20

The impedance is shown in Figure 2.3 as a function of fre­

quency for a typical diode. The diode susceptance is

plotted against conductance for several current values in

Figure 2.4. The diode exhibits resonance properties with

the reactance inductive below the resonance and capacitive

above the resonance. The reactance also shows the expected

capacitive behavior at frequencies much below the resonant

frequency. The real part of the impedance is negative over

a large range, showing capability for amplification or

oscillation at microwave frequencies. These results are

very similar to those obtained by Misawa [7].

21

a; o C CTJ

- p o Cd 0 Pi

<L) O C cti 4-> (0 . H CO <U PC

X

:ii

o C 0)

o o

o o CO

o o CVJ

o o

o o r-A 1

o o C\J 1

o o en 1

o o I

(suiqo) aouBpaduii

CO 0)

cd >

cd ?-. Q) > 0 CO

o C M

O

:^ cr 0)

«M

«M

o

o -P O :3

«M

CO cd

0) o C cd

<u

B

(D

O •H T5

H EH < cu S M

1 1 •

m •

C\J

. -^^ 4J

c <D U U :^ o

bO 0) •H fc

r ) o •H T?

< M

o

22

Conductance (10 mhos)

Fig. 2.4.—Diode susceptance plotted against conduc­tance for several values of current.

CHAPTER III

EXPERIMENTAL ARRANGEMENT AND OBSERVATIONS

The Microwave Interferometer

The IMPATT diode impedance measurements were made at

X-band frequencies (8.0 GHz. to 12.0 GHz.) using a wave­

guide microwave interferometer. A diagram of the inter­

ferometer is shown in Figure 3.1. The signal from the

microwave sweep oscillator is divided into two parts of

equal amplitude and phase by the tee junction. One part

of the signal follows a waveguide path containing a vari­

able attenuator and a variable phase shifter, while the

other part travels through a waveguide section containing

the IMPATT diode. The resulting signals are amplitude

and phase mixed through another tee junction. The rf

electric field is converted to a dc voltage by a crystal

diode detector. When the two signals are exactly in phase,

the dc voltage is a maximum. If the two signals have

identical amplitudes and differ in phase by exactly pi

radians, the dc voltage is zero. The magnitude and phase

of the impedance due to the diode are obtained in the fol­

lowing way. The rf source is set to the desired frequency.

The attenuator and phase shifter are then adjusted to give

23

24

u r-i O cd -P -P O t o <1> >s-P

u o O Q

PQ

cri

o TA

bO cd

/sA/\ c o

-d cd G -^ -H G

zi Q) O 'd s o •H <D Q 73

.H

<D G Xi -H o B 4J ? cd 0) S EH

E-i EH

M

:3 hO 0) > cd

\ > > < ^

T3 Cd 0)

o -p

G • H

s Cd <D S EH

(U -P r-A Cd

cd G

U -P cd -p > <

s^^ 0) 0)

EH

o •H bO cd

o

cd -a 0)

a

O «M

0) CO

^

• P <U

e o <D

CM ?H 0) -P G

• H

0 >

Cd

o

o

0 Xi -p

<M

o B cd fn bO cd

TA

O •H • P Cd B 0

o CO

I I

• to en ^

G

bO •H

0

0

w cd 0

25

a zero voltage null at the crystal detector, as measured

with a microvolt meter. The required attenuation and phase

shift are noted. Bias voltage is then applied to the diode

to obtain the avalanche condition. The attenuator and

phase shifter are readjusted to obtain a zero voltage null

at the crystal detector and the attenuation and phase are

noted. The change in attenuation and phase are functions

of the impedance magnitude and phase due to the diode.

These values are normalized to the characteristic Impedance,

Z , of the waveguide in which the diode is situated. The

diode is placed in a reduced height waveguide to obtain a

Z near the magnitude of the diode impedance to produce

more accurate measurements.

The reduced height waveguide with the diode mount is

matched to the standard waveguide through matching sections

of tapered waveguide. The diode mount is shown in Figure

3.2 and the tapered section is shown in Figure 3.3. When

they are properly matched so that there are no reflections

due to the transition sections, the impedance magnitude and

phase as measured by the interferometer are that of the

diode impedance. If reflections do occur, the measured

impedance would have to be corrected by taking these re­

flections into account and correcting the measurements ac­

cordingly.

A linearly tapered section of waveguide with impedance

26

— --•\ \

\ J-"

Xi 0 -P o Xi

TA

p bO

:3 bO 0 -O -rA 0 0

Qd Xi

> cd ^

-p G Zi O B

0 Ti O •H n G cd 0

bO 0 > Cd

-p

bO .H 0

Xi

0 o zi xi 0 I I •

C\J m

bO TA PH

Xi

27

S O S o o i n Lnoo c\j

• C\J o CO . • rH C\J ^^

11 ii II 1—4

•J X) cd

s C)

vo -=r C\J

•

II <M

Cd

• ^ - ' — t -

<N Cd

o -p Xi 0

G O •H -P O 0 CO

-^

I T3 0

0

& -p

I I

on c>n

28

Z such that

Z(x) = Zi+(Z2-Zi)x/L (3-1)

where L is the length of the taper, Zi and Z2 are the ini­

tial and final Impedances, is used to match the reduced

height waveguide to the standard waveguide. The reflec­

tion coefficient, r, has been shown [20] for a tapered

waveguide section to be given by

-I 0

e^^^^A.(lnZ)dx , (3-2)

where 3 is the propagation function and

32 = a)2y e - Tr2/a2 . ( 3 - 3 ) 0 0

The characteristic impedance of the rectangular waveguide

is [21]

1/2 p, (u^/e )

^ /l-xV^a^

where a is the waveguide width and b is the waveguide

height. The dimensions of the taper section are given in

Figure 3.3. A numerical integration of r shows it small

enough to be neglected in these measurements. The reflec­

tion coefficient as a function of frequency across the

X-band is shown in Figure 3.^.

An absorbing ring is placed around the diode to pre­

vent oscillations when the diode is biased to avalar^che.

29

8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency in GHz

Fig. 3.4.—Reflection coefficient as a function of frequency for a linearly tapered waveguide transition.

30

The dc bias current is applied through a low pass filter

consisting of a capacitor to ground and a ferrite ring

series inductor in the mount, and reaches the diode through

a conducting post. The effects of the absorber and post

cancel at each frequency since the interferometer is used

only to measure changes in Impedance.

Additional Diode Measurements

The doping concentration in the n region, the width

of the n region, and the diode area are obtained from meas­

urements of the diode capacitance and reverse breakdown

voltage. The capacitance is measured at 1 kHz. using a

capacitance meter in the arrangement shown in Figure 3.5.

The bias source has a very low impedance. Therefore a

series resistor and inductor are used to isolate it from

the capacitance meter. The diode package capacitance is

removed from the measurement by zeroing the meter with an

empty package in the diode mount. Measuring the capaci­

tance at a known reverse voltage and also at the saturation

point of the capacitance-voltage curve allow the determina­

tion of the diode area and the n region width in terms of

the n layer doping from equation (B-l4). The breakdown

voltage is taken as the voltage at which the reverse cur­

rent increases to 0.05 mA. This voltage allows the n layer

doping to be obtained from equation (2-4). These parameters

are used in the theoretical calculations of the diode

31

Cd

0 to

0 > 0 u

<M

o G O

'rA 4 ^ O G : 3

<M

CO cd

0 o C cd

4-> • H O Cd

cd o

0

o

•H

bO

c: • H : 3 CO cd 0

o <M

H 4 o

• H - P Cd 6 0 o CO

(r;

HI CO Cd

• H X3

o

:3 o •H o

I I «

bO •H

32

Impedances.

Experimental Results

Measurements of the breakdown voltage V^, capacitance

at 5 V reverse bias Cc, capacitance at saturation C ^, n sat

region doping N^, n region width w, and diode area A are

recorded in Table 1 for typical diodes. Magnitude and

phase measurements were made with the interferometer at

0.5 GHz. intervals from 8.0 GHz. to 12.0 GHz. Having ob­

tained the magnitude and phase of the diode impedances, the

real and imaginary parts are calculated. Experimental

values and theoretical values are plotted as a function of

frequency for the diodes listed in Table 1 in Figure 3.6

and Figure 3.7.

The impedance function shows a resonance that is pre­

dicted by the theory. The diode reactance changes from in­

ductive to capacitive at the resonant frequency. As pre­

dicted, the real part of the impedance is negative over a

range of frequencies below the resonant frequency. Above

the resonant frequency, the real part is positive and de­

creases toward zero as the frequency is increased. The

peak magnitude of the negative real part occurs at a fre­

quency just below the resonant frequency. The occurrence

of the negative real part shows the capability of the IMPATT

diode to function as an oscillator or amplifier.

The experimental measurements show a smaller than pre-

33

TABLE 3.1

ADDITIONAL DIODE DATA

V3 C^ Cg N^ W A

Diode (Volts) (pF) (pF) (lO cm' ) (lO cm) (lO' cm )

Fig. 3.6 68 .30 .08 .8874 .1078 .2063

Fig. 3.7 71 .50 .25 .8283 .1268 .4247

dieted change in resonant frequency for a given change in

diode current. (The impedance measured at 10 mA. deviates

only slightly from the impedances shown in Figures 3.6 and

3.7 which were measured at 15 mA. For this reason only

one set of data is shown.) It is believed that this result

is due to a cavity effect of the diode mount. Placing an

adjustable short at the standard waveguide end of a transi­

tion section attached to the diode mount forms a microwave

cavity. Adjustment of the waveguide short produced only

small changes in the oscillation frequency. This oscilla­

tion frequency was centered approximately at 11.0 GHz.

(there was some variation from diode to diode). This re­

sult implies a tuning effect of the diode mount on the

diode. The interferometric measurements will not include

any impedance due to the cavity, but it cannot diminish

its effect on the diode. The tuning effects found in the

oscillator circuit and the smaller than predicted change

34

400

300

200

100 -

Theoretical Reactance

— — Theoretical Resistance • Experimental Reactance X Experimental Resistance

CO

O

-100 -

/ ^

I

I I

I .

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency in GHz

Fig. 3.6.—Theoretical and experimental diode impe­dance .

35

CO

O

400

300

2001-

100

0.0

-lOOh

-200

-300

-400

1 1 1

1 1

1 1

1 1

/ /

—-1

1

1

1 **

• Experimental Reactance

* Experimental Resistance

•

It -^ *

/ «* ** / ^ / ^

J • / / / / 1 / / t

1 •

1 J - I « « 1 • 8.0 8.5 9.0 9.5 10.0 10.5 H.O 11.5 12.0

Frequency in GHz

Fig. 3.7.-—Theoretical and experimental diode impe­dance.

36

in the resonant frequency for a diode current change im­

ply the cavity effect does exist.

The accuracy of phase and attenuation measurements

which affects the plots of impedance shown in Figures 3.6,

3.7 are determined by the instrument accuracy (+ 3% for the

phase shifter and + 2% for the attenuator) added to the un­

certainty of the interferometric measurements. The uncer­

tainty for the interferometer is due to the ambiguity of

the phase and attenuation near the null. This ambiguity is

not constant for all frequencies, but is approximately 5*

in phase and 1/2 dB in attenuation.

CHAPTER IV

CONCLUSION

The theoretical calculations predict a resistance that

is negative over a range of frequencies and then becomes

positive or near zero for higher frequencies. The reac­

tance shows the expected capacitive reactance at lower fre­

quencies. The reactance becomes inductive in a large por­

tion of the spectrum where the negative resistance exists.

A resonance effect appears at a frequency near where the

negative resistance becomes positive, and here the reac­

tance again becomes capacitive. As can be seen in Figures

3-6, 3-7, the experimental data clearly shows the tendency

for the negative resistance to become positive or near zero

at frequencies above the frequencies where the useful nega­

tive resistance occurs. This effect has been reported [12],

[15], [22] before but has largely been ignored. It repre­

sents an important limitation on the upper operating fre­

quency of the IMPATT diode. From the evidence presented

here, it can be seen that the effect is predicted by the

theory and is a physical reality.

The theoretical curves are based on Indirect measure­

ment of the diode parameters (the n side doping level,

37

38

diode area, and length of the n side). These measurements

are based on the breakdown voltage and capacitance measure­

ments. The capacitance magnitudes are near the resolution

of the capacitance meter available and some uncertainty

concerning the accuracy of the values exists. The satura­

tion capacitance is difficult to determine since the deple­

tion region will continue to spread into the highly doped

p and n regions. These diode parameters are very signifi­

cant in determining the resonant frequency of the diode,

and the uncertainty accounts for the experimental and

theoretical resonances not occurring at the same frequen­

cies. For these reasons, the comparison between experi­

ment and theory should be viewed as qualitative rather than

quantitative.

The theoretical calculations show a strong dependence

of the resonant frequency on the diode current. This has

been observed in reported measurements [4], [15]. The cur­

rent tuning effects in the measurements reported here are

not as strong as predicted. The reduced height waveguide

transitions demonstrated a resonance effect with a diode

operating as an oscillator. An adjustable short placed on

one transition section proved ineffective in significantly

tuning the cavity. Also decreases in the diode current did

not strongly affect the oscillation frequency although the

output magnitude decreased. This result is expected be-

39

cause the diode negative resistance becomes closer to zero

at the oscillation frequency as the current decreases. The

interferometer will not measure the impedance of the cavity,

but the effect on the diode cannot completely be discounted.

The interferometric impedance measurements are thus still

representative of the diode impedance.

In summary, the data presented shows agreement in

theoretical and experimental impedance values. Of special

interest is the appearance of a positive resistance at fre­

quencies above the frequencies at which the useful negative

resistance occurs.

The measurements show good qualitative agreement with

theory. As mentioned above, the lack of quantitative agree­

ment is thought to be caused by the lack of accuracy of the

capacitance measurements which determine the input data for

the theoretical calculation and by the cavity effect of the

diode mount. The interferometric measurement technique

holds promise if these problems are overcome. It is a much

simpler measurement technique than the other techniques re­

ported and shows potential for accurate results.

LIST OF REFERENCES

1. Read, W. T., Jr. "A Proposed Hlgh-Frequency, Negative-Resistance Diode." The Bell System Technical Journal, Vol. 37, No. 2, March 1958, pp. 401-

2. Johnston, R. L., DeLoach, B. C , and Cohen, B. G. "A Silicon Diode Microwave Oscillator." The Bell System Technical Journal. Vol. 44, No. 2, Febru-ary 1965, pp. 369-372.

3. Sze, S. M. Physics of Semiconductor Devices. New York: John Wiley and Sons, Inc., 1969.

4. Gilden, M. and Hines, M. E. "Electronic Tuning Ef­fects in the Read Microwave Avalanche Diode." IEEE Transactions on Electron Devices, Vol. ED-13, No. 1, January 1966, pp. 169-175.

5. Gummel, H. K. and Scharfetter, D. L. "Avalanche Re­gion of IMPATT Diodes." The Bell System Tech­nical Journal, Vol. 45, December 1966, pp. lT97-1827.

6. Stewart, J. A. C , Conn, D. R. , and Mitchell, H. R. "State-Space Analysis of General IMPATT Diode Small-Signal Lumped Models." IEEE Transactions on Microwave Theory and Techniques. Vol. MTT-18. No. 11, November 1966, pp. 835-842.

7. Misawa, T. "Negative Resistance in pn Junctions Under Avalanche Breakdown Conditions, Part I, and Part II." IEEE Transactions on Electron De­vices, Vol. ED-13, No. 1, January 1966, pp. 137-151.

8. Misawa, T. "Multiple Uniform Layer Assproimations in Analysis of Negative Resistance in pn Junc­tions in Breakdown." IEEE Transactions on Electron Devices, Vol. ED-14, No. 12, December 1967, pp. 795-808.

9. Takayama, Yoichiro. "Power Amplification with IMPATT Diodes in Stable and Injection-Locked Modes."

40

41

IEEE Transactions on Microwave Theory and Tech­niques, Vol. MTT-20, No. 4, April 1972, pp. 266-272.

10. DeLoach, B. C. and Johnston, R. L. "Avalanche Transit-Time Microwave Oscillators and Amplifiers." IEEE Transactions on Electron Devices, Vol. ED-13, No. 1, January 1966, pp. 181-186.

11. Gewartowski, J. W. and Morris, J. E. "Active IMPATT Diode Parameters Obtained by Computer Reduction of Experimental Data." IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-18, No. 3, March 1970, pp. 157-161.

12. Isobe, Toyosaku and Nakamura, Tetsuo. "Admittance Measurements of IMPATT Diodes at X-Band." IEEE Transactions on Microwave Theory and Tech­niques, Vol. MTT-18, No. 11, November 1970, pp. 993-995.

13. Kuno, H. J., Fong, T. T., and English, D. L. "Charac­terization of IMPATT Diodes at Millimeter-Wave Frequencies." IEEE Transactions on Electron Devices. Vol. ED-19, No. 6, June 1972, pp. 752-757.

14. Steinbrecher, D. H. and Peterson, D. F. "Small-Signal Model with Frequency-Independent Elements for the Avalanche Region of a Microwave Negative Resistance Diode." IEEE Transactions on Elec­tron Devices. Vol. ED-17, No. 10, October 1970, pp. 883-891.

15. Misawa, T. "Silicon pn Avalanche Transit-Time Diodes." IEEE Transactions on Electron Devices, Vol. ED-17, No. 4, April 1970, pp. 299-304.

16. Iperen, B. B. van and Tjassens, H. "An Accurate Bridge Method for Impedance Measurements of IMPATT Diodes." Microwave Journal, Vol. 15, No. 11, November 1972, pp. 29-33.

17. Sze, S. M. and Gibbons, G. "Avalanche Breakdown Volt­ages of Abrupt and Linearly Graded p-n Junc­tions in Ge, Si, GaAs, and GaP." Applied Phys­ics Letters, Vol. 8, No. 5, March 1966, pp. 111-113.

42

18. Miller, S. L. "Avalanche Breakdown in Germanium." Physical Review, Vol. 99, No. 4, August 1955, pp. 1234-1241.

19. Watson, H. A., ed. Microwave Semiconductor Devices and Their Circuit Applications. New York: McGraw-Hill Book Co., 1969.

20. Collin, Robert E. Foundations for Microwave Engineer­ing. New Yorlcl McGraw-Hill Book Co., 1966.

21. Montgomery, C. G., Dicke, R. H., and Purcell, E. M., eds. Principles of Microwave Circuits. New York: McGraw-Hill Book Co., 1948.

22. Irvln, J. C , Coleman, D. C , Johnson, W. A., Tatuguchi, I., Decker, D. R., and Dunn, C. N. "Fabrica­tion and Noise Performance of High Power GaAs IMPATTS." Proceedings of IEEE, Vol. 59, No. 8, August 1971, pp. 1212-1221.

APPENDIX

A. Fortran IV Computer Program for Computing IMPATT

Diode Impedance

B. Theory of the One-Sided Abrupt p -n Junction

C. Alternate Method for Obtaining Current Density

Expressions

43

APPENCU A

FC^T-^AN IV C r y i M l i r ' * PRnCRAf-' Prvt CALCULATING IVPATT OIGUR I^'PF^A^CG

^ A I N P W O G R A M

c c r.

n(:\F M T M nri.cLf: P'^tcisifN. sui'Roun^ji: P R N P L T IS LTiLi/ro f l ]R DISPLAYING HATA

C A L C L L A T l f J S M'T LIH:-^/iRY S 'JP 'JLIFr - iRAPHICALLY.

I N P L I C l r '>E AL*R ( A - H , C - Z ) l^f^AL^'^'i X <( '»(:C) . Y Y C C C ) , 7,?(^CC) , i - F ,FL - ( ^CO) .AM'- fOO) , X X X l ' » 0 0 ) Rf-AL'-^ \ L , I , J C , lisTGRL CCM-LN/ I APiJS/ NCLT CnVV( ' \ ' : f^,U1 ,l^0 , A \ ' , A P , H N ,nP,'/« C r v P L r X U i ^ / 1 , Z 2 , 7 3 , / ' . , / 5 CCf^i^LFK :=l!S J , K , A , P , C , / , E N 2KW , EMKK , G P K W ,CI'X!', C SORT , C M P L X . C O S U R T ,

icnExp^pc.'^r'Lx.r. 11 ,r, i ? , c ? i r x r U R N A L A L F A . F ? SCi' T ( M ) ^ ^ S ' : R T ( A l ) r,SuRT( ^) ^ c o s r j u ( A ) c/' 'pL X ( M , nK,) =nc N cL X ( A I , R e) CfEXP( / ) = C r F X P ( A) /i (' S ( X ) = C A P S ( X )

I N I T I A L I Z ' ^ CnN'STA^irS. ALL PHYSICAL CATA F C S I L I C O N IS OHTAINLC FROM Rf-F':RtNCF ( 3 ) .

V S L 1 . L- 7 EPSR-=1 1.8 FPSn=P.f?5'.E-l''« A N = 3 . H F 6 AP = -?. 25(:7 H-J=l . 7 5 F 6 HP=3.?'5E6 EG = 1 .1? C=1.6C2P-iq

C C

C C

INPUT rFVICL- PARAVtTERS I N REGIHN DUPING LEVEL, LENGTH CF N REGION,

AREA CF C i r C F ) . PF An( '"> , 1 ) ND,Vs , AREA FPRrAT ( :M7?C. *= ) IF(Mr.FC.O) CALL EXIT

CALCLLATE »';<FAKf:CWN VCLTAGE ANC CHPLETICM WIOTH FOR CASE WHERE N PEGir\ LENGTH IS LCN^^E'^ TFAN THE CtPLEIICN ViICTH.

VR = .0.*<LG/l . ! )**!.«:'(NC/l.Fl6)*«(-.7 5) ^^c=S(.RT(?.*EPSC/f:*F"S"/Nr:=>vh')

CALCLLATE NAvi^U'N ELECTRIC F IELC PRESENT. EN = ? . « v b / v > n W l = 0 .

44

45

C C A L C c r Q (: A C AVAL

9

10 C CALC C RESF

C TNPL

3

fcl

62 I

C C r C

63

CALC ECUA PART THE

I

X = I.LAT r la^• A N C F C A L L ir ( A U l = W cc T U 1-U L L A T ECT A L F A A L F A J = CN T ni READ FCRV V, R I T FOR?' V' R I T v ^ R I r FORP-: •AC ' ) l . ••( I T F C R f' j r = T ^ R IT r c p y 1 1 1 1 l i L A T T I C N S . CONS

nc F = I . 1 1 1 1 FF = F

K-C^' SL) K = CS A = K-P=K + C = J *

E W K U F P K ' A

C l l C 1 2 C 2 2 =

I . E - 6 F W i r i H CF A V A L A N C H E P E G I C N .

2-2 IS L S E C TC C A L C L L A T E M L T H CF RFGir\' V.HERF '15% CF THE i: rCCURS. INT (ALFA .O.C'.IA, .95,nx )

B S ( M A - M ) . L r . . CI * H A ) G C T C 10 1 A C 0 lA

t TH-: AVERAGE AVALAMCHE COEFFICIENT AND THE DERIVATIVE WITH rc FLFCTRIC FIELC. C^l ./IAI

P = l!\'--MLFAa/FM4«2 FLX(0.,l . ) CViE P IAS C L R R E M . ( 5 , 3 ) I AT ( F I O . O ) E ( 6 , 61 ) A T ( l H l , q x , ' N n » , l O X , ' l > • , 1 7 X , « A R E A » , 1 7 X , M » ) I; ( 6 , 6 ) MC, W, AREA, I E ( 6 , ^ 2 ) A T ( I H C , 9 X , M ; n » , l ^ X , ' E f " , 1 7 X , ' A L F A C ' , l 6 X , ' A L F A P « , I 7 X , ' W l ' , l 8 X , ' W

E (6 ,f: ) VP.Er^, ALFAC, ALFAP.Wl.WO A T ( lHn,r>D20.5) /AREA E (f, c. n AT(ll'n,0X,«F •,16X,»REAL Z',1AX,«IHAG /. • ) ^C E TfU; CCNPLEX IMPEDANCE FOR VARICLS FRICOEN'C IES. 2-''. IS USEi: TC CALCULATE THE IKPECANCF UUT IS EVALUAfEO IN

TAN'TS IN 2-3A, 2-3A, 2-3^ ARE EVALUATEC SEPARATELY. 30 11^=1,200 n + C >} x I I

= 1 m n

PLX(Cr'GA**2/(VSL**2)-2*ALFAP*JC/(VSL'>EPSR*EPSO) , 2*aLF AO*OMGA / V

CRTIK)

Cf'Ci/VSL C N G W S L 2.*ALFAP*JC/((K*VSL*EFSR*EPSC)**2) K = CF.<D(-J*2*K«Wl ) = CE<P(-.J*K«V,1) = Cr XP( +.|«K«Vv 1 ) = C^(P*F^'K'W + A->F»'2KV^)/('}**2-A**2*Ef'2KW) = (;*(P*A«FyKU)/(P**2-A**2*E'''2KW) -VSL*&*Cll*CFXP(J*H*l^l)/(2.*C^'GA)•VSL•e•Cl2«CEXP(-J•A*Wl)/(2.•

46

C THE 30 5

111 11?

113

16

15

C THE C FUN

17

200 210

211

212

iny.GA )-v

7 1 = \ C U /.?=-Cl? Z3=-C22

lA) Z'. = M ? .

I/AREA /'-;=(-)-:' Z = Z1 M 2 X X ( II I I V Y ( T I rI ? ? ( I I I I I f' P F C A N V R I T E I t F O R M A T ( F O R M A T ! F C R ' A T ( V> R I T F U . C A L L P R V-. o I T F ( 6 FCRf-'A T ( V R I T E ( : C A L L F ••' > R I T E ( I I I I T - C X = 0 . V, f' I T F ( r> FOR'''AT ( I X = W l / l Y l = A L r A Y2 = E( F I I I 1= I F L E C T R I

C T I C N S C EEl I I I I AA(I I I I XXX (III WRI T E ( 6 F O R M A T ! X = X + U X I F I X . G T GO TO I W R I T F ( 6 F C R N A T ( C A L L F- W R I T E ! 6 FCR^'AT( U R I T E ( 6 FORf''AT( C A L L PR

S L *C *CF X P ( .) •/r ,v 1A * U 1 / V SL ) / ( ?. *r 'G A ) MCFXP(.J*K=^.J l)-I . )/( J * K * A R r A ) ) •• ( C P < P ( - J '•.• K v,s 1 ) - 1 . ) / ( J * X « A R E )

- V S L - = ( C £ x P ( - J v r V G A * W / V S L ) - C E X F ( - J > C N G A * W l / V S L ) ) / ( J «Of^GA • A R E

•:' A L F A C / ( V ': L 1" F P S R « F P S C ) - J : (Jf' G A/ ( V SL « * 2 'S'E P SR F P SU ) ) * W 1 /K * • 2

(\v-Kl )/(f'VGA I'f.PSO^'rPSR) ) / A R E A

• Z 3 » / 4 • Z '") ) = F F ) = R E A L ( Z ) )= AIN^AGIZ ) C E IS PR I NT EC IN r a j M E R I C A L F O R M A N O IS P L O T T E D B Y P R N P L T . , 5 ) F , Z IH , lPn2C.5, 1^020.5,IPD2C.5) IF 1 , 'REAL PA ?r OF Z)• ) IHl ,' IMAGINARY PART CF Z« ) ,111) NPLT(YY,XX,-S0G.,';CC.,l.Ed,2C.ES,Iin,l,l,50,2) ,113) IHC,6CX,'FRFCL-NCY l\fl,)*) ,112) NPlT(ZZ,XX,-l.E3,l.F3,l.Ert,20.E9,niI,l,l,50,2) . ] 1 3)

,16) IHC,////,1H ,')X,'X«,16X,«ALFA(X)«,15X,'E(X)') C.

(X) X) I IT I + l C F I F L P ANO THE I O N I Z A T I O N C O E F F I C I E N T ARE P L O T T E O AS F C I S T A N C E IN T H E N R E G I O N . I ) = E ( X ) I ) = A L F A { X ) I I )=X , 1 7 ) X , Y 1 , Y 2 IH , 3 ( I P 0 2 0 . 5 ) )

.V.) GC TC 2 0 0 c

,210) IHl,'ELECTRIC FIELC (VOLTS/CK . ) • ) NPLT(EE,XXX,-C.,-C.,0.,V',niII»lfl,50,2)

, 2 U ) l F n , 6 0 X , T I S T A f v C F INTO N REGION ( C M . ) ' ) . 2 1 2 ) n - ) , • I C M Z A T I C N C C E F F I C I f c N T * ) r.'PLT ( A A , X X X , - C . , - C . » C . , W , I I I I I , I , 1 . 5 0 , 2 )

47

V R I T E ( 6 ,2 M ) GC TO 2 END

SUBROUTINES

C C c

SO SUPRCU I N T E G R C O N D T T

IV AP P = S = < -

I

IF IF P = CC <; =

rx CO EN

BR OUT TINE AL IS ION. PLICI S(X) = A 4-0 0.

S+(F( ( ARS ( ( S . G T

(; + nx TC 2

S-(F( = 0X/2 TO 2

0

INE IMT(F, A . H . C C X ) INT INTEGRATES A fUNCT ICN F FCUAL TC C a,\r: RfTLRNS THE

FROM LIMIT A UNTIL THE UPPER LIMIT REClJIREO FOR THIS

T i?E^.L«8 C A P S ( X ) X

(A-F,0-Z)

P ) tFIP-CX ) ) «CX/2. S-C).LE.C*.OI ) RETURN .C) GO TC 3

! ) * F (P-CX ) )*rx/2.

rCUPLF PRFCISION FUVCTION F(X) C FUNCTILN F(X) IS THE FLFCTRIC FIELC

INFLICIT ''EAL + b (A-H,C-Z) COMMON' E M , H ,VNU ,AN',AP,BN ,eP,W IF(X.LC.V.l) F = EM IF (X.GT.V.I ) E.= EM*(WC*W1-X )/WC RETURN END

IN THE N REGION.

OCLPLF Pf'FCISICN FUNCTION ALFA(X) C FUNCTIO-^' ALrA(X) GIVES THE ICNIZATICN COEFFICIENT IN THE N REGION

IMPLICIT REAL«rt (A-H,0-Z) PFAL*R INTCRL

CCMMCN P M . M ,U0, AN,AP,PN ,BP,V» EXTERNAL Fl LXP(AA)=nrxP(AA) 7 = - BN/r(X) - INTCRL(Fl,0.,X) ALFA = AN'*EXP(Z)

48

RETURN ENC

I^CUPLF P R r C I S l O N F i r 'CT i rV . ' INTG'JLIF , A , P ) C F l INCTICN INTG) G I V L S THE VALUE CF THE iNfEGRAL C T R A p r Z C 1 n ^ L I N T I C R A r I r N .

I M P L I C I T Ri-AC'S ( A - H , r - Z ) R E A L * " 1\TCRL S = 0 . 0 X = ( n - A ) / 1 0 0 . X=A-DX CC 1 .J = i , i c n x = x + c x

1 S = S + ( F ( X ) t F ( X + UX) ) - 0 X / 2 . C INTGRL^S RrTlJRN

ENO

F UJM A TO B CF F BY

C C

rCUPLE PRFCISICN r t lNCTICN F l ( X ) FUNCTIC.N F l IS THE C I F F i ; i r \ C E IN FLECTRCK ANC HOLE lO^ I IZAT lUN COEFr iCTENTS LSCO IN TMI- EXPONENTIAL IN FCUATICN 2 - 2 .

I M P L I C I T '^EAL'fH { \ - h , r - l \ CCMMCN FM.V.l TVC, A N , AP,r .N,PP,W F X P ( \ ' i ) = P f ' X P ( A A ) F 1 = AN ^E < P ( - ( ' f ' / r ( X ) ) - A P * E X P ( - B P / E ( X ) ) RETURN END

APPENDIX B

THEORY OF THE ONE-SIDED ABRUPT p" -n JUNCTION

The abrupt p-n junction occurs when the impurity con­

centration changes abruptly from net acceptor concentration,

N -Nj. = N., to net donor concentration, Nj.-N = N . When

N

profile for the one-sided abrupt junction is shown in

Figure B.l. Applying Gauss' law for the electric field in

the diode, Poisson's equation in x is obtained

.. >> N- , a one-sided abrupt junction occurs. The doping

1^ = p/e . (B-1) 3X

For the abrupt junction case

-qN^ -w 1 X <_ 0

p(x) = < . (B-2) qNj) o < X £ w^

The solution obtained for E(x) is

-q/eN (x+w ) -w^ < x < 0 ^ a p p - -

_E + — — o < X < w ' m e — n

where

qN^w N^w

19

50

» ^D-^A

N D

-w -N A

(a)

w n

E(x)

X

Fig. B.l.— (a) Doping profile for the one-sided abrupt junction.

(b) Electric field inside the one­sided abrupt junction.

51

The electric field is also shown in Figure B.l. Integra­

tion of the electric field across the diode gives the diode

potential,

V = 1/2 E^w^ (B-5)

where w^ is the total depletion width. Solving for w, and

eliminating E^ results in

2c V ^ D 1/2

-d = T N77 ^ • ^^-'^ The depletion width is thus a function of the diode poten­

tial V. The diode potential includes the applied potential

V^ and the built-in potential V . that arises from equilib­

rium requirements with zero applied voltage. The potential

arises to oppose the tendency for the diffusion current to

flow due to the impurity gradient across the junction. The

current is composed of a drift component and a diffusion

component such that

J^(x) = q(y^pE(x)-D^|^) = 0 , (B-7) P P po A

and

J^(x) = q(y^nE(x)+D^|^)= 0 , (B-8) n n iio A

where u and u are the mobilities of holes and electrons p n

respectively and D and D are the diffusion constants for '^ p n

holes and electrons respectively. Using the Einstein

52

relation

^ = q/KgT , (B-9)

and solving for E(x) and integrating across the junction

for the case of no current flow through the junction, there

results in an expression for the built in potential V j_-

K^T n K^T p V,, = - ^ m - ^ = - ^ i n ^ . (B-10) bi q np q p^

The total diode potential used in equation (B-6) is given

by

V = V,,-V^ , (B-11)

where V is the applied voltage, referenced to the forward a

direction (voltage on p side is positive with respect to

the n side). The sign of V^ can be verified by considering a

that a positive voltage tends to force carriers into the

junction region while a negative voltage tends to deplete the junction region.

A capacitance due to the depletion region can be de­

fined as

C = , (B-12) ^ - dV

so that

d(qN^w^A) C - D ^ . (B-13)

53

For an abrupt one-sided junction where w « w^ n d

C « ^^^VdA> eA ,<1^VN^/^

d( ilw 2)

(B-14)

The capacitance per unit area is plotted In Figure B.2 as

a function of voltage. The capacitance essentially satu­

rates when the depletion region extends to the boundary of

the n-region.

51

Fig. B.2.—Depletion capacitance for an abrupt one­sided pn junction diode as a function of bias voltage. Notice the saturation effect due to the depletion region extending through the entire n region of the diode.

APPENDIX C

ALTERNATE METHOD FOR OBTAINING CURRENT

DENSITY EXPRESSIONS

In Chapter II, the expressions for J and J given by n p ° '^

equations (2-26) and (2-27) are obtained through algebraic

manipulations. These results can be obtained directly by

solving the two following differential equations:

and

3J . . . . r-^ = ^^^^J -a (J +J )-a E(J +J ) . (2-1?) V -. p o n p no po '

Substituting equations (2-20) and (2-22) into (2-l6) and

(2-17), the following are obtained

3 J jojJ ^ . •r-^ ^ = a J+E(J a'-jwea ) , (C-1) 3x V T o o " o ' sl

and

3J jcjJ . . r-^" 2. = Qt J+E(ja)ea -J a') . (C-2) 3X V , O 0 0

sl Particular solutions of the following form are assumed

55

56

and

The X dependence is assumed to be that of E(x) since E(x)

appears in the differential equations. These solutions are

now substituted into (C-1) and (C-2) and, using (2-23) for

E(x), the constants can be evaluated as follows:

Ci (J^a^-jtoea^)

^ m " j(k+a3/v J > ^"5^ sl

Cgd.a'-jooea ) A ^ = r-r-1 T T ^ . (C-6) n2 J (to/v -j -k) *

Ja +(J^a--j(oea^)(-^ - -iiii^)J/k2 o o o v - , e v^'^e A , = , ; ^^ ^= , (C-7) n3 "^^/^sl

Cj(jwea^-J^a")

C2(ja,sap-J^a-)

and

Ja^HJ^a-j.sa^)(^-/^)J/k2

A = r-^ ^ . (C-10) P3 J' /Vg-L

Using equation (2-24), it is found that

(k2.co2/v^^2)!^. J

57

(C-11)

Equation (C-11) is used to reduce the expressions of (C-5)

through (C-10) to the following forms:

^ni = jCi(k-co/v^3_)I^ (C-12)

^sl^ ^nz = -JCaCkWv^,)-^ (C-13)

-J a'J

'n3 " vTrTk^ sl (C-lM

A = - C i ( k W v ^ , ) ^ (C-15)

Ap, = jC,(k../v^^)^ (C-16)

and

-J a'J

PS Vg^ek^ (C-17)

These results, when substituted into equations (C-3) and

(C-4) produce expressions that are Identical to equations

(2-26) and (2-27) that were obtained in Chapter II.