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ANALYTICAL SMALL-SIGNAL THEORYOF BARITT DIODES

by

Th. G, van de Roer

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Department of Electrical EngineeringEindhoven University of TechnologyEindhoven, The Netherlands

ANALYTICAL SMALL-SIGNAL THEORYOF BARITT DIODES

byTh. G. van de Roer

TH-Report 74-E-46

May 1974

ISBN 90 6144 046 7

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ABSTRACTI

An analytical theory for the s m a l l - s i g ~ a l impedance and noise of Bari t t I(or punch-through) diodes is presentedl The diode is divided into threeregions. I ~ th e two regions closest to!the inject ing contact th e effects,of thermionic injection and diffusion *re accounted for in an approximateway. In the remaining region diffusion I is neglected but an otherwise exact,solution is given for an arbitrary r e l ~ t i o n s h i p between dr i f t velocity andelec t r ic f ie ld . Results of the calculations are presented in graphical,form and the influence of th e p a r a m e t e ~ s frequency, d.c. current, temperatureand impurity concentration is discussed.

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CONTENTS PageI Introduction 4II General 5II I The Small-Signal Impedance 6

III-I . The Drift Region 6I I I - I . I . D.C. Solution 7I I I - I .2 . A.C. Solution 7III-I .3 . Discussion 9

1II-2. The Diffusion Region IIIII-2 .1 . A.C. Solution IIIII-2.2. D.C. Solution 12III-2.3. Discussion 13

1II-3. The Contact Region 13IV Noise Properties IS

IV-I. Introduction ISIV-2. Shot Noise 16IV-3. Thermal Noise 17

IV-3.1. Introduction 17IV-3.2. Diffusion Region 17IV-J.3. Drift Region 18IV-4. Discussion 19

V Numerical Results 21VI Conclusion 23VII References 24VIII Figures 25IX List of Symbols 37

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1. Introduction

Since Shockley [I] f i r s t proposed the USje of punch-through diodes asnegative-resistance devices for microwav,e frequencies, a number of papershas appeared treating the d.c. and small' signal a.c. theory of thesedevices. Especially since the f i r s t experimental realization by Colemanand Sze [2], the interest in punch-through diodes, and with i t the numberof papers about them, have increased strongly.

Yoshimura [3] has given solutions for the d.c. and small signal a.c .impedances for the case where the mobility is constant throughout thediode. Wright [4], Weller [5], Coleman [,6] and Haus et. a l. [7] havepublished theories for the case of saturated dr i f t velocity throughoutthe device, the main difference between their theories being the boundaryconditions applied at the injecting contact . Vlaardingerbroek and theauthor [8-] have pointed out the importance of the combination of a nonsaturated and a saturated region.Finally, a number of numerical calculations have been published [9,10,11,12].The small-signal noise properties have been discussed in some of the abovementioned papers as well as in a few others [7,12,13,14,15,16].In th e analytical theories published hitherto, diffusion effects on thesmall-signal impedance have -been neglected or represented by a modifiedboundary condition. An exact analysis would require the solution of asecond-order differential equation with 'variable constants which can onlybe done numericallY. I t is fe l t , however, that incorporating diffusion inan analytical theory, although approximate, is s t i l l worthwile because i tcan give more insight than a numerical ~ n a l y s i s . To do this is the scopeof this investigation of which preliminary resul ts already have beenpublished [I 7J.

The approach chosen here rel ies on the fact that the elect r ic f ie ld r isessteadily from the injecting contact to th e other one while the carrierdensity decreases simultaneously. One may then assume that near theinjecting contact the main factors governing carrier transport will bethermionic injection and diffusion whereas in the region of higher fieldstrength the elect r ic f ie ld wil l be dominant.

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The diode is divided into three regions:i . th e contact region where thermionic injection prevails . This region

l ies between the in ject ing contact and the point of zero electr ic f ield(potent ia l maximum).

i i . the diffusion region where car r ier transport is by diffusion mainly.This region stretches from the potent ia l maximum to a point where thed.c . f ield ha s r isen to such a value that diffusion may be neglected.Necessarily the choice of this point will be somewhat arbitrary.

i i i . the dr i f t region, comprising the res t of the diode, where the elec t r icf ield is dominant.

The analysis wil l s ta r t with the d r i f t region and then work i ts way backto the in ject ing contact. This is done because the d r i f t region makes upthe greatest par t of the diode and i t ca n be treated without approximations.The properties of the diode can then be discussed in terms of th e boundaryconditions a t th e input of the dr i f t region, which in turn are determinedby the in ject ing contact and the diffusion region.The small-signul noise propert ies wil l be discussed af ter th e impedance.Shot noise and thermal noise wil l be taken as the only noise sources.In th e las t section some numerical results and comparison with experimentswill be presented.

11- General

Consider a planar semiconductor s t ructure consist ing of a layer of n-typematerial sandwiched between two metal (o r p+) contacts ( f ig . I ) . The p+(metal)layers form rect i fying contacts (Schottky-barriers) and consequently narrowdepletion layers are formed a t both contacts . When a d.c. voltage is appliedwith the plus on th e l e f t hand contact , th e r ight hand depletion layer wil lwiden but the device draws no current. This goes on unt i l the two depletionlayers meet, a s i tua t ion called reach-through or punch-through. The f ieldand potent ial distr ibut ions are now as shown in f ig . 2. Also shown is th eenergy band diagraro. Any hole that now is injected from the l e f t handcontact with suf f ic ient energy to cross the potent ial bar r ier is picked upby the f ield and transported to the other side. When the voltage now isincreased the potent ial bar r ier i s reduced and the current increases sharply.I t is this feature of the punch-through diode that makes i t s operation as ahigh-frequency negative-resistance device possible.

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For the analysis reference is made to : f ig . 3. The variables to beconsidered are:

the total current density Jthe electr ic f ieldthe dr i f t velocity

Ev

which are a l l three in the x - d i r e c t i o ~ , andthe hole density p.

All these are assumed to be functions: of the space coordinate x and toconsis t of a large d.c. part (with index 0) and a small a .c . part (index 1)with time dependence exp(jwt).

IAlso entering th e equations are the donor density ND, taken constant overthe length of the diode, and the dielectric constant E of the semiconductormaterial .

The position of x.1f ield ha s attainedis defined by specifying thea t this point . I t wil l be in

value E. th e1the order ofd.c . electr icseveral ki lo-

vol ts per cm.

I t wil l be convenient to use reduced quantit ies . These are defined asfollo..s.

n EE 'S i =J

aE 'S, F;

eNb= --xEE s and a = WEav

with E = Ss )1 and a = e)1oND, e being th e elementary unit of charge.. 0In pr1c1ple vs and )1. are arbitrary scaling factors. The most natural ,a

( 1)

however, is to take for v the value of the saturated dr i f t velocity andsfor)1 th e low-field mobility.a

I I I The small-signal Impedance

I I I - I . The dr i f t region

In this region th e dr i f t velocity is a function of the electr ic f ield only.The total current density is no function of x and is given by:

ClEJ = e.p.v(E) + E. Cit (3)

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The f ie ld is given by Poisson's equation:eE (ND + p) (4)

In reduced form these equations read: an+- - .a at (3a)

(4a)

The equations are now spl i t in their d.c. and a.c . parts which are solvedseparately. The a.c . equations are l inearized assuming the a.c . quantit iesto be small.

111-1.1. D.C. Solution

The d.c. parts of (3a) and (4a) yield, eliminating p:

i = - "Vo 0 dno+ v o dsWith the boundary cond:tion nsolution of (5) as:

no

J "0 (n )s - dn + s- i +" (n } 1o 0ni

E.1n. = - at1 Es x = x. one can write th e1

The d.c. voltage over the dr i f t region can also be found directly:

E E 2 1 d n.v (n)Vod LE dx s 0 dn= eND i +" (n)0 o 0n1 1where nd is the value of no at x = L

111-1.2. A.C. Solution

(5)

(6)

(7 )

A f i r s t order pertubation analysis is applied to find the a.c . components.The a .c . componentdvo

of v is obtained from a Taylor-series expansion:(8)

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Combining (I) and (2) and eliminating'the d.c. terms gives in reduced form:

( ja + + " o (9 )

IThis is most easily solved by converting to a new independent variable T,defined by:sI ds'T = '::"-"0""(-s,..,);-s1 (10)Evidently T is the t ransi t- t ime of a hole from x. to x, divided by the1dielectric relaxation time c/o.

Substituting (10) eq. (9) becomes:

( I I )

The general solution of the homogeneous form ( i l = 0) of (II) is well known.I t reads:exp -J ( j a+ ( 12)o.

Using (4a) and (10) this can be simplified to:i

= ( I + 0) .- eXp-]aT" ( 13)oThe complete solution of (II) now can'be found by substi tuting

and solving for F. The boundary condition for DI at T = 0(x = x.) is formally put down as1

P.i1 Iwhere p. has to be determined from an analysis of the preceding region.1

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The following express10n then is obtained:i [P'v, IT0) . :;.1...:.1,--_+ - exp-Jel, "'" +\) 0 10 + V i

o

(14)

Here v. = v (n.) .1 0 1Calculation of the a.c . impedance Zd of the dr i f t region now is s t ra ight forward. With A the diode area and T = T() we have:

with Z ~ ovSa2A

Substi tuting (14) one obtains:T

Z S( io 0o,

+ v ) exp- j el T { , , ~ : : : i _ V : : : i __ +So ~ + \I .o 1 o v ' }. 0 ,expjal'dt' dT1 +vo 0

(15 )

(16)

A similar expres, . >n valid for majori ty-carr ier current (Gunn-diodes) hasbeen obtained by D c s c ~ l u [18]. This expression is obtained from (16) bychanging the sign of 1 .o

111-1.3 Discussion

Expression (16) can be evaluated further without specifying th e v-Erelat ionship. In the following this has been done in such a way that theinfluence of the non-saturated part of the dr i f t region is separated out.One then arrives at the expression:

V . ( I + i ) {+ 1 0 P _\,). + i i1 0

~ ( i +v )exp-jelTd,i ['Jel 0 0

~ el ----.j-el--:::. +}l-exp-jelT

- v )exp-jelTdToTj exPjelT IVi + io 0 0

+

( 17)

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(17) can be interpreted as follows:the f i r s t term clear ly represents the ia t t ice capacitance of the d r i f tspace. The l a s t two terms are zero i f the dr i f t velocity is saturatedthroughout the dr i f t region.In thisarg (P.1

caseI+ - )ato obtain a negative resistance p. must satisfy the condition1

In Impatt diodes arg (p . + 1)1 athe avalanche frequency which

TI= - - when the2is th e optimal

signal frequency is abovecondition for negative

resistance. In Baritt-diodes Pi has a posit ive real part in most cases andnegative resis tance in possible too.

However, even in case arg (P .1r .h . s . of (17) may contribute

+ 1) = ~ ' t h e thi rd and fourth term of th ea 2a negative real part when the d r i f t velocityis not saturated, as has been pointed out in [8] for a specif ic v-Erelat ionship. A general proof is hard to give but looking at the third termi t can be seen that when I - v is a monotonously decreasing function of To(which is theI - VO{T) is

case when v increases monotonously with n ) and wheno 0so small that the upper l imit can be extended to inf ini ty

the integral has a negative imaginary part so that the whole thi rd termhas a negative rea l part .

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111-2 The diffusion region111-2.1 A.C. Solution

The car r ier transport in this region is governed to a large extent bydiffusion. An exact analysis would require th e solution of second-order nonl inear differential equations, which in general can only be done by numericalmethods. In this work we will res t r i c t ourselves to a simplified analysis, whichalthough less exact, can provide better insight than a numerical calculation.

The dr i f t velocity now is given by:

v = v(E) - ~ ~ w h e r e D is the diffusion constant. In reduced form this becomesp ax8 apv=v(n)--p a ~ where 8 = aD2EV Swhich together with (I), (2), (3) and (7) gives for the a.c . quanti t ies:

i dvo 0vdi l )o 0

(18)

(1 9 )

(20)

This equation is simplified by replacing v by an average value v and assumingo dv athe mobility constant at i t s low-field value so ~ = 1. The solution of (20)then becomes:

withBo 1/( i Iv + jet)o a

a -v {= 28 + 48 io+-(-+2 vva a

(21 )

(22)

(23)

Eq. (21) reveals the existence of two waves, one forward t ravell ing withamplitudecoefficient B1 and one backward t ravell ing with amplitude B2 . Thela t ter usually is called a diffusion wave because i t does not show up in ananalysis where diffusion is neglected. But even in the present case i t isdoubtful that this wave will be excited. The point where i t would be excited,x ., is an ar t i f ic ia l boundary created to simplify th e analysis but not existing1 .in rea l i ty . Therefore i t is considered appropriate, although i t is not correctmathematically, to leave out this wave so that th e influence of diffusion only

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1S to change the character of the forward wave.

The amplitude BI then is found by matching theth e relation between field and current densityI

f ie ld a t x .It ismcan be written as:

assumed that

(24 ) Iwhere a has to be found from an analys1s. of the contact region. Section III-3cwill be devoted to this analysis.

For BI one thus finds:B = - B1 a + ja 0c (25)

Finally, the impedance Z. of the diffusion region and the boundary condition1parameter Pi are found as:

where

_E _ B - BaC. 0 11e x p - Y I ( ~ ' - ~ ) - 11 m

C.1 x. - X1 mis the "cold" capacitance of the diffusion region.

III-2.2 D.C. Solution

(26)

(27)

The quanti t ies ~ . - [ and v , occurring in the preceding paragraph, have to be1 "m afound from a d.c. analysis . Again, approximations have to be introduced becauseof the non-lineari ty of the equations. The approximation used now is that thed. c. current is carried by diffusion only:. This gives:i o =

o dpo- ND (ff" (28)

With the boundary condition that p be continuous at x. the solution of (28) is :o 11 oV.1

i oTWith Poisson's equation then the electr ic field is found:n = ( I +o

i o-+v '1

i ~ . ...2...2:.)o ( ~ - ~ ) -mi o26

(29)

(30)

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Demanding continuity of E a t x. then yields C - ~ :0 1 1 m

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-14- Iin this case our analysis is not valid Ifor high current densit ies . Onth e other hand, th e screening effect d e ~ c r i b e d in the previous sectionbecomes more pronounced a t higher currept densit ies , so the error introducedprobably remains small.

For the following derivation reference is made to f ig. 3. The d.c. holedistr ibution for x < x is given by :mV + ljJ'm kTexp - - : :V-T-= wi th VT = ep = No vDefine ~

Then Poisson's equation gives:d 2 ~ -1 - I; exp - ~ dl;2

N oj! ,with v mI; = - exp -ND VT2

. d ~ Wr1t1ng - '- =ds 2

d! d ~ integrate (32) to:

Sm =lm ___ --'dl.Jl$'--_____-'r'{ 2 ( ~ - ~ + 1; exp -4>-l;exp - ~ )}!m mo,

(34).

(35)

(36)

(37)

(38)

The largest contribution comes from the region ~ ~ ~ where the integrandmhas 'an integrable singular i ty, so substi tute

Then one finds(. 2 ~ -'mYsm m= + 1; exp (39)~ m is calculated from the thermionic emission formula:

(40)

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The quantity 0 introduced in the preceding section is calculated thecsame way as by Haus et .a l . [7) , v i z . by a perturbation .of (36)assuming th e a. c. convection current to be small compared to thedielectricJ xom

current. The result is :oc = oVTThe a.c . voltage drop over the contact region is equal to E1xm so thatthe impedance of th e contact region becomes:

zc =xm

0 (0 +ja)Ac = Z~ m

o a +jac

IV - Noise PropertiesIV-I. Introduction------------

(41 )

(42)

To discuss the small-signal noise properties of Baritt-diodes the opencircui t noise voltage VN is calculated. From this a noise measure M canbe defined by the following expression:

M v2N (43)

M is directly related to the noise figure F of a reflection-amplifier usingth e diode [19). The relationship can be expressed as follows:F = 1 + M( 1 - 2..)Gwhere G is th e gain of the amplifier.

(44 )

In a Baritt-diode, where carrier multiplication and intervalley-scatter ingare absent, the main sources of noise are:

shot noise, originating in s ta t i s t ica l fluctuations of the injectedcarrier current, andthermal nois due to random thermal motion of th e carr iers .

The shot noise will be calculated following the method of Haus et .a l . [7).For the thermal noise th e impedance-field method will be used [13,20).

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IV-2. Shot noise

The shot noise is calculated under the assumption that a t xcurrent J is injected whose mean squ.ire amplitude is givens

= x a noisemby the wellknown formula12 = 2eJ AMs 0To obtain the open-circuit noise v o l t ~ g e we have to solve for the a.c.electric field under the assumption t ~ a t the to ta l current J 1 is zero.So at x the sum of injected current and field-induced current has tombe zero:

i + (0s cwhere i s

+ j a ) ' n ( ~ ) = 0mI s

= crE AsIn the diffusion region we now have with (46)

-1 Sns= +. e x p { - Y l ( ~ - ~ ) } o c Ja mFrom this the noise voltage

exp { -y 1 ( ~ . - ~ ) }V Z I 1 ms i = (oe+ja)o s Yl

over the diffus ion- 1

region follows:

(45)

(46)(47)

(48)

(49 )

Equation (48) also gives us the boundary condition for the dr i f t regionby inser t ing ~ = ~ . In th e dr i f t region we then have:1

- i s- - = - - ~ . - exp { -Y 1 ( ~ . - ~ )cr + Ja: ~ mcThe noise voltage over this region thus becomes:

TR," i 1Vsd = - Z I exp{ - Y l ( ~ . - ~ )} . (" +i ) exp-os 1 m \ l . + 1 0 0t 0 o

(50)

(51)

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IV-3. Thermal noise

The thermal noise is calculated with the impedance-field method [20] . Itgives the expression:

7 = 4e 2 Ath IdZ 12[ d;X D(x) p (x) dx Mo (52 )Here D(x) is a quantity dependent on th e specific noise-generation mechanismconsidered. For thermal noise i t can be identif ied with the diffusion constant D.

The quanti tyby f ig. 4.

is the impedance-field vector. I ts meaning is i l lus t ra ted

Suppose a noise current o ~ is injected at a plane X and extracted again a ta plane X + ~ X . This current produces a voltage OVN across the terminals ofthe diode. No," th" impedance-field is defined by

(53)

Assuming that th e noise currents in the different parts of the diode areuncorrelated the noise voltages have to be summed quadratically which leadsto (52). Evidently in ~ 2 i s formula

. h rx 1nOlse current w ereas ~ re atesvoltage across the terminals.

D(x) represents the actual nature of theto the way this current produces a

. dZ rxTo obtaln ~ on e has to solve th e same equations as before but assuminga total current o ~ between X and X + dX and zero total current in theres t of the diode.

When the plane X is in the diffusion region we assume the following fieldsto exist (Xr is the reduced value of X): For ~ m < ~ < Xr a backwardtraveling wave:

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Ut = A exp Y2 ~ (54)For X < ~ < X + llX the complete soll,ltion of the inhomogeneousr r rdifferential equation:

where Dit is th e reduced injected noiseas in section 111-2. For Xr + llXr < ~ Xr andintegrate to obtain the noise voltage. The result is :

exp j elT JT R.aA - V - - ; ( - T ~ ) +_x"'i- o x 0 ~ x

IV-3.4. Discussion

(v + i ) exp - j ~ T d T ,o 0

(60)

(61 )

From th e expressions for the noise voltage derived in the preceding sectioni t is hard to draw any general conclusions. However, one may note that (51)and (58) l ,ed to terms of th e same form in th e expressions for the meansquare voltage. If we take (51) and (58) as representative for the shotnoise and the thermal noi se , respect ive ly , then we are able to get animpression of the relat ive magnitude of both noise sources. Taking thesquare of the absolute value of (51) and substituting (58) in (52) anddividing the results we obtain:

2 exp 2 Re YI(s.-s ) - I1 m

Substituting Bo' YI and Y2 this reduces to:V2t h ~ 2 6 exp 2 Re YI (si -sm)- ~ 40i 2 2 2 Re YIv2 ,,3 { (I + 0 ) + ( 4 o ~ ) }s a -3- 2v vaaInserting representative numbers e. g. :N ~ 10 21 -3 D = 10-3 2 -ID m m s

0.05 2 -1-1 = 0.4~ = m V s V.1lOS -I 10- 10 A -Iv = m s ~ = s Vs

(62)

- I (63 )

-Im

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we find at low current densit ies: 0 = 0.008 and v = 0.03 so that theaf i r s t factor of (63) is a large number Mhereas the second factor is inth e order of one. At increasing current densit ies the f i r s t factor decreasesbut the second one increases. One m i g h ~ therefore conclude that th e thermalnoise is the dominant noise source a t a l l current densit ies .

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V Numerical Results

For a numerical calculation the v-E-relationship must be specified. Themeasured v-E-characteristics of Canali e t .a l . [22] were used as a start ingpoint. They can very well be approximated by th e function:

JJ Eov = -"--"")1 Eo1 + VsFor instance, at room7v = 0.9.10 cm/s.s

(64)

temperature one finds )1. 0 = 450 cm2/Vs and

Unfortunally i t is not possible to evaluate the expressions for impedanceand noise obtained in the previous sections for this v-E-relationship.Therefore the curve has been approximated by three straight lines (fig. 5).The f i r s t intersection is chosen at the electric f ie ld Ei ' which alsomarks th e end of th e diffusion region. When the values of )J ,v and E.o s 1have been selected, th e value of )12 is taken such that at zero d.c. currentth e t r a n s i t ~ t i m e from xi to i is the same as i t would be for the v-Erelationship of (64).

Firs t i t was tried to reproduce the experimental results of Bjorkman andSnapp [23]. The following parameter values were used:)Jo = 450 cm2 /Vs= 0.075.107 cm/sv sE. 7 kV/cm1

= 3.10-4 cm2A2 7.9)JmND 1 .2.10 15T = 17 0 C 12o

-3cm

The results are shown in fig . 6. I t ~ u r n s out that the agreement ia 880da t th e low current of 5 mA but a t the higher currents th e calculated valuesof the negative conductance are higher and occur at higher frequencies thanth e measured ones.

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-22-One might wonder whae-influence th e l'aramtiter'1!''has"on, 1 the result . An

varied fromimpression of this influence is given in fig . 7 where E. is1/ '.6 to 8 kV cm with the d. c. current at 5 mA. EV1dently there is anappreciable influence. At higher currents however the change of the curveswith E. becomes less and at 40 mA i t is insignificant.1As a second step i t was tr ied to find out i f th e temperature r ise of th ediode at high currents could be responsible for th e discrepancy betweentheory and experiment. The temperature enters explici t ly 'in the formulasfor the contact region. Furthermore i t is assumed that th e low-fieldmobility varies as:

T )-2,3~ o - ~ o T T, 0 0

From th e data given in [22] one may conclude that th e saturated dr i f tvelocity varies l i t t l e with temperature, so i t was held constant. Thediffusion constant to o was kept constant.

(65)

The results of this calculation are shown in f ig . 8 for currents of 20 and40 mA and various temperatures. E v i d e ~ t l y th e frequency shif t (o r betterth e lack of frequency shif t) of the negative conductance can be explainedby a temperature change but not the variation of the magnitude of th enegative conductance. Not shown is the variation of the susceptance. Itdecreases wi th increas ing current , but increases with increas ing temperature.

Finally, the influence of donor density was examined. A typical result isoshown in fig . 9 for a current of 20 mA and a temperature of 50 C. Twoconclusions may be drawn from this figure: f irs t ly , the frequency regionof negative conductance shif ts to higher frequencies with increasing donordensity and secondly, the magnitude of the negative conductance decreasessharply when th e donor density drops below a certain value. Furtherinvestigation showed that th e las t phenomenon is dependent on the lengthof the diode and i t seems that t h e r ~ is something like a minimum N D ~ product for good operation of this type of diode.

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VI Conclusion

An analytical theory for the small-signal character is t ics 6f Baritt-diodeshas been developed. I t takes into account the influences of th e nonsaturated dr i f t velocity, diffusion and the properties of the injectingcontact.

The theory gives insight into the physical behaviour of the diode as wellas numerical values for the impedance and noise that f i t well to experimentalresults.

Some important results are:i . the region of negative resistance shifts to higher frequencies with

increasing current, but to lower frequencies with increasingtemperature. A similar feature in the susceptance could be useful forstabil izat ion of Barit t oscil lators.

i i . the region of negative resistance shifts to higher frequencies withincreasing donor density. A minimum density (at a given diode length)is necessary to obtain a useful negative resistance.

i i i . thermal noise is th e dominant noise source.

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REFERENCES

[I]. W. Shockley - BSTJ 12, 799-826 ( 1 9 5 4 ~ . [2]. D.J. Coleman, S.M.Sze - BSTJ 50, 1695-1699 (1971).[3]. H. Yoshimura - IEEE Trans. ED-II, 414-422 (1964).I[4]. G.T. Wright - Electron. Lett. ~ , 449+451 (1971).[5]. K.P. Weller - RCA Rev. ~ , 373-382 (1971).[6]. D.J. Coleman - J.A.P. 43, 1812-1818 (1972).[7] .[8] .

H.A. Haus, H. Statz, R.A. Pucel - Electron.Lett. ~ , 667-669 (1971).M.T. Vlaardingerbroek, T.G. v.d. Roer -

[9]. E.P. Eer Nisse - Appl. Phys. Let t . , 20,Appl. Phys. Lett.301-304 (1972).

~ , 146-148 (1973).

[10]. J.A.C. Stewart, J . Wakefield - Electron. Lett. ~ , 378-379 (1972).[II] . M. Matsumura - IEEE Trans. ED-19, 1131-1133 (1972).[12]. A. Sjolund - Solid State El. ~ , 5 5 9 ~ 5 6 9 (1973).[13]. H. Statz, R.A. Pucel, H.A. Haus - Proc. IEEE 60, 644-645 (1972).[14]. A. Sjolund - Electron. Lett. 2., 2 - ' (1973).[IS]. J. Christie, B.M. Armstrong,

Microwave Conference, A.IO.2J.A.C. Stewart - Proc. 1973 European,(Brusseis, 1973).

[16]. A. Sjolund, F. Sellberg - Proc. 1973 E.M.C. A.IO.3. (Brussels,1973).[17]. T.G. v.d. Roer, Proc. 1973, E.M.C. A.I1.2. (Brussels, 1973).[18]. A. Dascalu - IEEE Trans. ED-19, 1 2 3 9 ~ 1 2 5 1 (1972).[19]. M.E. Hines - IEEE Trans. ED-13, 158-;163 (1966).[20]. W. Shockley, J.A. Copeland, R.P. James - in Quantum Tbeory of Atoms,

[21] .122J.

Molecules and the Solid-State (P.O. Lowdin,ed.),M. El-Gabaly, J .Nigr inand P.A. Goud - J . Appl.C. Canali, G. Ottaviani, A. Alberighi Quaranta -~ , 1707-1720 (1971).

537-563, Ac.Press,N.York 1966.Phys., 44, 4672-80 .. .- .J . Phys. Chem. Sol.

123]. G. Bjorkman, C.P. Snapp - Electron. iLett. ~ , 501-503 (1972).

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-25-CAPTIONS TO THE FIGURES

Fig. I . Structure of a Baritt-diode.Fig. 2. Field distr ibutions a t punch-through.

a. elect r ic f ie ld .b. electric potential .c. energy-band diagram.Fig. 3. Division of the diode into three regions:

I . contact region.I I . diffusion region.I I I . dr if t region.

Fig. 4. Il lustrat ion of the impedance-field method.Fig. 5. v-E-characteristics

------measured by Canali et .a l . [22] at room temperatureapproximation by eq (64).

- - - - - - three- l ine approximation used in the calculations:I.I I .

v =dvdE =~ Eo~ 2

III . v = v sFig. 6a. Comparison of calculated real admittance and noise figure withexperiments (Bjorkman and Snapp [23]).- - - - - - calculated- - - - - - experimental

oI = SmA, T = 17 C.Fig. 6b. As 6a. I = 20mA, T = 17C.Fig. 6c. As 6a. I = 40mA, T = 17oC.Fig. 7. Influence of

I = SmA, T =the parameter17 oC.

E .1

E. in kV/cm is indicated a t th e curves.1Fig. 8a. Influence of diode temperature.

I = 20mA.The temperature in degrees centigrade i s indicated at the curves .

Fig. 8b . As 8a. I = 40mA.Fig. 9. Influence of donor density.

I = 20mA, T = SOoC.N . lOIS - 3 . . d' dD 1n em 18 1n lcate at the curves.

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I-26-1

metal

or n - t y p ~ orp+ - type semiconductor p+ - type

semiconductor semiconductor

o lI I E x

Fig. 'I .

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-27-

E

L , . . ~ - - - - - - + - x v

J C = : : : : : : : ~ - - I - X

1 - - - - - - - - - - - - - - - - - - - ~ I - - - X o

Fig. 2.

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-28-

~ { ; ~ )metal metal (Ior I n Dr or

!p+ , p+I (!l ~ I - /' ~ )- ,I II II I

E

4 - - - ~ - - ~ - - - - - - - - - - - - - - - - - - - + - - - - x

o 1m xi l

Fig. 31.

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-29-

~ _ _ - - ~ ~ ~ - - - - ~ - - _ _ Xo x X+ aX l

Fig. 4.

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vem/.

107

o Ej 20 40

-30-

E60 80 100 (KVlem)

Fig. 5. I

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HF(dB)

20

10

-ReY( 0-1)

4 5

-31-

6 7 8 9F

(GHz)

FO + - - - - - - - ~ - - - - ~ - - - - ~ - - - - ~ L - - - ~ - -4 5 6 7 8 9 (GHz)

Fig. 6a.

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NF(dB)

20

10

-ReV( n-1)

10-3

0

4 5

4 5

-32-

J//,./F

6 7 ' 8 9 (GHz)

I .\/ , \/ \/ \/ \/ \/ \/I \I \I \I \I \ F

6 7 8 9 (GHz)

Fig. 6b.

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NF(dB)

20

10

-ReV(n-1)10-3

0

4 5

4 5

-33-

\ \ \ \

6

""- .....-7

/ -//I/IIII

6 7

Fig. 6c.

8

\ \ \\\\\\8

9

9

F(GHz)

F(GHz)

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NF(dB)

20

10

-ReY(n-1)

4 5

-35-

6 7 8 9F

(GHz)

0 + - - - - . ~ - - 1 _ ~ _ . - - - - , _ ~ ~ - L - - - F4 5 6 7 8 9 (GHz)

Fig. 8a.

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NF(dB)

20

10

-ReV(0 -1)

4 5

-36-

6 7 8 9F

(GHz)

0 + - - - - - r . . . . L . - - 4 - - - ' - - - - . l , - - . . , . . - - - 1 . . " " T ' " " - ~ . . . u - - - - - F4 5 6 7 8 9 (GHz)

Fig. 8b.

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NF(dB)20

10

-ReV(n-1 )

-310

4 5

-37-

6 7 8 9F

(GHz)

o +-- -Lr. . . .J- -L-T""" '"""- . .L, - - , l - - -+- - - F4 5 6 7 8 9 (GHz)Fig. 9.

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I-38-1

ILIST OF SYMBOLS

A

DEE. ,E1 seFMGI serNiio , i 1i sDi tJ

pPotVVsVTVthVodVsdV S lVN

diode areaintegration constant" "diffusion region capacitancediffusion constantelectric f ieldparameterelementary chargeauxiliary functionbandwidthamplifier gainnoise current

" "reduced current densityd.c . , a .c . components of ireduced noise current

" " "current densityd.c . , a.c . components of Jparameterimaginary uni tBoltzmann's constantdiode lengthnoise measuredonor concentrationvalence band density of stateshole concentrationd.c. component of ptimepotentialshot noise voltagethermal voltagethermal noise voltaged.c . voltage over drif t i regionshot noise voltage, dr i f t region

" " " diffusion regionnoise voltage, open cir*ui t

9I I1812116668

161516176

7,816186

14 ,9146

147

156

146

126

141914177

161615

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v sxXrxX.1xmZoZIZcZdZiZTXCI

r I ,r2

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p.1a

a ct

t '

Til,TXVi'mw

-40-:

parameter""

new coordinateauxiliary variablevalue of t at I