Analytical small-signal theory of baritt diodes

van de Roer, T.G.

Published: 01/01/1974

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Citation for published version (APA):Roer, van de, T. G. (1974). Analytical small-signal theory of baritt diodes. (EUT report. E, Fac. of ElectricalEngineering; Vol. 74-E-46). Eindhoven: Technische Hogeschool Eindhoven.

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Download date: 16. May. 2018

ANALYTICAL SMALL-SIGNAL THEORY

OF BARITT DIODES

by

Th. G, van de Roer

Department of Electrical Engineering

Eindhoven University of Technology

Eindhoven, The Netherlands

ANALYTICAL SMALL-SIGNAL THEORY

OF BARITT DIODES

by

Th. G. van de Roer

TH-Report 74-E-46

May 1974

ISBN 90 6144 046 7

-2-

ABSTRACT

I

An analytical theory for the small-sig~al impedance and noise of Baritt­I

(or punch-through) diodes is presentedl The diode is divided into three

regions. I~ the two regions closest to!the injecting contact the effects ,

of thermionic injection and diffusion *re accounted for in an approximate

way. In the remaining region diffusion I is neglected but an otherwise exact , solution is given for an arbitrary rel~tionship between drift velocity and

electric field. Results of the calculations are presented in graphical , form and the influence of the paramete~s frequency, d.c. current, temperature

and impurity concentration is discussed.

-3-

CONTENTS Page

I Introduction 4

II General 5

III The Small-Signal Impedance 6

III-I. The Drift Region 6

III-I.I. D.C. Solution 7

III-I.2. A.C. Solution 7

III-I.3. Discussion 9

1II-2. The Diffusion Region II

III-2.1. A.C. Solution II

III-2.2. D.C. Solution 12

III-2.3. Discussion 13

1II-3. The Contact Region 13

IV Noise Properties IS

IV-I. Introduction IS

IV-2. Shot Noise 16

IV-3. Thermal Noise 17

IV-3.1. Introduction 17

IV-3.2. Diffusion Region 17

IV-J.3. Drift Region 18

IV-4. Discussion 19

V Numerical Results 21

VI Conclusion 23

VII References 24

VIII Figures 25

IX List of Symbols 37

-4-

1. Introduction

Since Shockley [I] first proposed the USje of punch-through diodes as

negative-resistance devices for microwav,e frequencies, a number of papers

has appeared treating the d.c. and small' signal a.c. theory of these

devices. Especially since the first experimental realization by Coleman

and Sze [2], the interest in punch-through diodes, and with it the number

of 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 the

diode. Wright [4], Weller [5], Coleman [,6] and Haus et. al. [7] have

published theories for the case of saturated drift velocity throughout

the device, the main difference between their theories being the boundary

conditions applied at the injecting contact. Vlaardingerbroek and the

author [8-] have pointed out the importance of the combination of a non­

saturated 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 above­

mentioned papers as well as in a few others [7,12,13,14,15,16].

In the analytical theories published hitherto, diffusion effects on the

small-signal impedance have -been neglected or represented by a modified

boundary condition. An exact analysis would require the solution of a

second-order differential equation with 'variable constants which can only

be done numericallY. It is felt, however, that incorporating diffusion in

an analytical theory, although approximate, is still worthwile because it

can give more insight than a numerical ~nalysis. To do this is the scope

of this investigation of which preliminary results already have been

published [I 7J.

The approach chosen here relies on the fact that the electric field rises

steadily from the injecting contact to the other one while the carrier

density decreases simultaneously. One may then assume that near the

injecting contact the main factors governing carrier transport will be

thermionic injection and diffusion whereas in the region of higher field

strength the electric field will be dominant.

-5-

The diode is divided into three regions:

i. the contact region where thermionic injection prevails. This region

lies between the injecting contact and the point of zero electric field

(potential maximum).

ii. the diffusion region where carrier transport is by diffusion mainly.

This region stretches from the potential maximum to a point where the

d.c. field has risen to such a value that diffusion may be neglected.

Necessarily the choice of this point will be somewhat arbitrary.

iii. the drift region, comprising the rest of the diode, where the electric

field is dominant.

The analysis will start with the drift region and then work its way back

to the injecting contact. This is done because the drift region makes up

the greatest part of the diode and it can be treated without approximations.

The properties of the diode can then be discussed in terms of the boundary

conditions at the input of the drift region, which in turn are determined

by the injecting contact and the diffusion region.

The small-signul noise properties will be discussed after the impedance.

Shot noise and thermal noise will be taken as the only noise sources.

In the last section some numerical results and comparison with experiments

will be presented.

11- General

Consider a planar semiconductor structure consisting of a layer of n-type

material sandwiched between two metal (or p+) contacts (fig. I). The p+(metal)

layers form rectifying contacts (Schottky-barriers) and consequently narrow

depletion layers are formed at both contacts. When a d.c. voltage is applied

with the plus on the left hand contact, the right hand depletion layer will

widen but the device draws no current. This goes on until the two depletion

layers meet, a situation called reach-through or punch-through. The field

and potential distributions are now as shown in fig. 2. Also shown is the

energy band diagraro. Any hole that now is injected from the left hand

contact with sufficient energy to cross the potential barrier is picked up

by the field and transported to the other side. When the voltage now is

increased the potential barrier is reduced and the current increases sharply.

It is this feature of the punch-through diode that makes its operation as a

high-frequency negative-resistance device possible.

-6-

For the analysis reference is made to:fig. 3. The variables to be

considered are:

the total current density J

the electric field

the drift velocity

E

v

which are all three in the x-directio~, and

the hole density p.

All these are assumed to be functions: of the space coordinate x and to

consist of a large d.c. part (with index 0) and a small a.c. part (index 1)

with time dependence exp(jwt).

I Also entering the equations are the donor density ND, taken constant over

the length of the diode, and the dielectric constant E of the semiconductor

material.

The position of x. 1

field has attained

is defined by specifying the

at this point. It will be in

value E. the 1

the order of

d.c. electric

several kilo-

volts per cm.

It will be convenient to use reduced quantities. These are defined as

follo..s.

n E E ' S

i = J

aE ' S , F;

eNb = --x EE

s and a =

WE

a

v with E = S

s )1 and a = e)1oND, e being the elementary unit of charge. •. 0 In pr1c1ple v

s and )1. are arbitrary scaling factors. The most natural, a

( 1 )

however, is to take for v the value of the saturated drift velocity and s

for)1 the low-field mobility. a

III The small-signal Impedance

III-I. The drift region

In this region the drift velocity is a function of the electric field only.

The total current density is no function of x and is given by:

ClE J = e.p.v(E) + E. Cit (3)

-7-

The field is given by Poisson's equation:

e E (N

D + p) (4)

In reduced form these equations read:

£ an +--. a at (3a)

(4a)

The equations are now split in their d.c. and a.c. parts which are solved

separately. The a.c. equations are linearized assuming the a.c. quantities

to be small.

111-1.1. D.C. Solution

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

i = - "V o 0

dn o + v o ds

With the boundary cond:tion n

solution of (5) as: no

J "0 (n) s - dn + s··

- i +" (n} 1 o 0

ni

E. 1 n. = - at

1 Es x = x. one can write the

1

The d.c. voltage over the drift region can also be found directly:

£ EE21d n.v (n)

Vod L E dx s 0

dn = eND i +" (n) 0 o 0

n· 1 1

where nd is the value of no at x = L

111-1.2. A.C. Solution

(5)

(6)

(7)

A first order pertubation analysis is applied to find the a.c. components.

The a.c. component dv

o

of v is obtained from a Taylor-series expansion:

(8)

-8-

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

(ja + + " o (9 )

I This is most easily solved by converting to a new independent variable T , defined by:

s

I ds' T = '::"-"0""( -s,.., );-

s· 1

(10)

Evidently T is the transit-time of a hole from x. to x, divided by the 1

dielectric relaxation time c/o.

Substituting (10) eq. (9) becomes:

(II )

The general solution of the homogeneous form (i l = 0) of (II) is well known.

It reads:

exp -J (ja+ ( 12)

o.

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

i = (I + 0) . - eXp-]aT

" ( 13)

o

The complete solution of (II) now can'be found by substituting

and solving for F. The boundary condition for DI at T = 0

(x = x.) is formally put down as 1

P.i 1 I

where p. has to be determined from an analysis of the preceding region. 1

-9-

The following express10n then is obtained:

i [P'v, IT 0) . :;.1...:.1,--_ + - exp-Jel, "'" + \)0 10 + Vi

o

(14)

Here v. = v (n.). 1 0 1

Calculation of the a.c. impedance Zd of the drift region now is straight­

forward. With A the diode area and T£ = T(£) we have:

with Z ~ o

£v S

a2A

Substituting (14) one obtains:

T£

Z S (i o 0

o

, + v ) exp- j el T{,,~:::i_V:::i __ + S

o ~ + \I. o 1

o

v' } . 0 ,expjal'dt' dT 1 +v

o 0

(15 )

(16)

A similar expres,. >n valid for majority-carrier current (Gunn-diodes) has

been obtained by Dcsc~lu [18]. This expression is obtained from (16) by

changing the sign of 1 . o

111-1.3 Discussion

Expression (16) can be evaluated further without specifying the v-E­

relationship. In the following this has been done in such a way that the

influence of the non-saturated part of the drift region is separated out.

One then arrives at the expression:

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

1 0

~ (i +v )exp-jelTd, i ['£ Jel 0 0

~ el ----.j-el--:::. + }

l-exp-jelT£

- v )exp-jelTdT o

T

j exPjelT I

Vi + i o 0 0

+

( 17)

-10-

(17) can be interpreted as follows:

the first term clearly represents the iattice capacitance of the drift

space. The last two terms are zero if the drift velocity ·is saturated

throughout the drift region.

In this

arg (P. 1

case I

+ -) a

to obtain a negative resistance p. must satisfy the condition 1

In Impatt diodes arg (p. + 1) 1 a

the avalanche frequency which

TI = - - when the

2 is the optimal

signal frequency is above

condition for negative

resistance. In Baritt-diodes Pi has a positive real part in most cases and

negative resistance in possible too.

However, even in case arg (P. 1

r.h.s. of (17) may contribute

+ 1) = ~'the third and fourth term of the a 2

a negative real part when the drift velocity

is not saturated, as has been pointed out in [8] for a specific v-E­

relationship. A general proof is hard to give but looking at the third term

it can be seen that when I - v is a monotonously decreasing function of T o

(which is the

I - VO{T£) is

case when v increases monotonously with n ) and when o 0

so small that the upper limit can be extended to infinity

the integral has a negative imaginary part so that the whole third term

has a negative real part.

-11-

111-2 The diffusion region

111-2.1 A.C. Solution

The carrier transport in this region is governed to a large extent by

diffusion. An exact analysis would require the solution of second-order non­

linear differential equations, which in general can only be done by numerical

methods. In this work we will restrict ourselves to a simplified analysis, which

although less exact, can provide better insight than a numerical calculation.

The drift velocity now is given by:

v = v(E) - ~ ~where D is the diffusion constant. In reduced form this becomes p ax 8 ap

v=v(n)--­p a~

where 8 = aD 2

EV S

which together with (I), (2), (3) and (7) gives for the a.c. quantities:

i dv o 0

vdil) o 0

(18)

(19 )

(20)

This equation is simplified by replacing v by an average value v and assuming o dv a

the mobility constant at its low-field value so ~ = 1. The solution of (20)

then becomes:

with

B o

1/(i Iv + jet) o a

a -v { = 28 + 48 io

+-(-+ 2 v

va a

(21 )

(22)

(23)

Eq. (21) reveals the existence of two waves, one forward travelling with

amplitudecoefficient B1 and one backward travelling with amplitude B2 . The

latter usually is called a diffusion wave because it does not show up in an

analysis where diffusion is neglected. But even in the present case it is

doubtful that this wave will be excited. The point where it would be excited,

x., is an artificial boundary created to simplify the analysis but not existing 1 .

in reality. Therefore it is considered appropriate, although it is not correct

mathematically, to leave out this wave so that the influence of diffusion only

-12-

1S to change the character of the forward wave.

The amplitude BI then is found by matching the

the relation between field and current density I

field at x .It is m

can be written as:

assumed that

(24 )

• I where a has to be found from an analys1s. of the contact region. Section III-3 c

will be devoted to this analysis.

For BI one thus finds:

B = - B 1 a + ja 0 c

(25)

Finally, the impedance Z. of the diffusion region and the boundary condition 1

parameter Pi are found as:

where

_E_ B - B aC. 0 1

1

exp-YI(~' - ~ ) - 1 1 m

C. 1 x. - X

1 m is the "cold" capacitance of the diffusion region.

III-2.2 D.C. Solution

(26)

(27)

The quantities ~. - [ and v , occurring in the preceding paragraph, have to be 1 "m a

found from a d.c. analysis. Again, approximations have to be introduced because

of the non-linearity of the equations. The approximation used now is that the

d. 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 1

1 o

V. 1

i o

T

With Poisson's equation then the electric field is found:

n = (I + o

i o

-+ v' 1

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

m

i o

26

(29)

(30)

-13-

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

<I t -I i

~. - ~m 0

(31 ) = - -+ 1 i v.

0 1

The average reduced drift-velocity is defined such that it gives the same

transit-time from x to X. as the non-averaged velocity: m 1

c - ~

=t d~ 1 m which v v a 0 ~ m v.

1 V = --=--'-;:--:0-, a v. (~.-~ )

1 + 11m 2 <I

111-2.3 Discussion

gives for v : a

Eq. (21) can, using (25), be written as:

(32)

(33)

Apparently nl

consists of a constant term and and a wave propagating in the

direction of the drift which is damped by velocity modulation and diffusion.

The result of the damping is that as the wave progresses the influence of

the contact region, repre.sented by the first term in (33), is decreased whereas

the influence of the injection region itself increases. The result is that,

depending on the degree of damping, the contact region is more or less

screened off by the injection region. Situations are

there is hardly any screening at all. Specifically,

of the wave is small and the a.c. electric field is

possible,.however. where 10

when a ~v the amplitude c a

constant throughout the

injection region. In this case the boundary condition (24) can be applied

without modification at x .• 1

111-3 The contact region

The contact region is analysed under the assumption that the flat-band situation

is not reached, i.e: the zero-electric-field point (potential-minimum) lies at a

finite distance from the contact. Numerical analysis [15,21] has shown that in

m-s-m diodes flat-band can be reached at relatively low current densities so that

-14- I

in this case our analysis is not valid Ifor high current densities. On the other hand, the screening effect de~cribed in the previous section

becomes more pronounced at higher currept densities, so the error introduced

probably remains small.

For the following derivation reference is made to fig. 3. The d.c. hole

distribution for x < x is given by: m

V + ljJ' m kT

exp - -::V-T-= wi th V T = e p = N

o v

Define ~

Then Poisson's equation gives:

d2~ -1 - I; exp - ~

dl;2 N oj! ,

with v m I; = - exp -

ND VT

2 •. d ~ Wr1t1ng -' - =

ds2

d ! d~ integrate (32) to:

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

o

,

(34).

(35)

(36)

(37)

(38)

The largest contribution comes from the region ~ ~ ~ where the integrand m

has 'an integrable singularity, so substitute

Then one finds

(. 2~

-'m Y sm m

= + 1; exp (39)

~m is calculated from the thermionic emission formula:

(40)

-15-

The quantity 0 introduced in the preceding section is calculated the c

same way as by Haus et.al. [7) ,viz. by a perturbation .of (36)

assuming the a·. c. convection current to be small compared to the

dielectric J x om

current. The result is:

o c

= oVT

The a.c. voltage drop over the contact region is equal to E1xm so that

the impedance of the contact region becomes:

z c

= x

m 0(0 +ja)A

c = Z

~m o a +ja

c

IV - Noise Properties

IV-I. Introduction ------------

(41 )

(42)

To discuss the small-signal noise properties of Baritt-diodes the open­

circuit noise voltage VN

is calculated. From this a noise measure M can

be defined by the following expression:

M v2

N (43)

M is directly related to the noise figure F of a reflection-amplifier using

the diode [19). The relationship can be expressed as follows:

F = 1 + M( 1 - 2..) G

where G is the gain of the amplifier.

(44 )

In a Baritt-diode, where carrier multiplication and intervalley-scattering

are absent, the main sources of noise are:

shot noise, originating in statistical fluctuations of the injected

carrier current, and

thermal nois due to random thermal motion of the carriers.

The shot noise will be calculated following the method of Haus et.al. [7).

For the thermal noise the impedance-field method will be used [13,20).

-16-

IV-2. Shot noise

The shot noise is calculated under the assumption that at x

current J is injected whose mean squ.ire amplitude is given s

= x a noise m

by the well

known formula

12 = 2eJ AM s 0

To obtain the open-circuit noise volt~ge we have to solve for the a.c.

electric field under the assumption t~at the total current J1

is zero.

So at x the sum of injected current and field-induced current has to m

be zero:

i + (0 s c

where i s

+ ja)'n(~ ) = 0 m

I s

= crE A s

In the diffusion region we now have with (46)

-1 S

ns= +. exp{-Yl(~-~)} o c Ja m

From this the noise voltage

exp { -y 1 (~. -~ )} V Z I 1 m si =

(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 drift region

by inserting ~ = ~ .• In the drift region we then have: 1

- i s --=--~.- exp { -Y 1 (~. -~ ) cr + Ja: ~ m

c

The noise voltage over this region thus becomes: TR,

"i 1 Vsd = - Z I exp{ -Yl(~.-~ )} . (" +i ) exp-os· 1 m \l. + 1 0 0 t 0

o

(50)

(51)

-17-

IV-3. Thermal noise

The thermal noise is calculated with the impedance-field method [20]. It

gives the expression:

7 = 4e2

A th

£ IdZ 12 [ d;X D(x) • p (x) dx • M

o (52 )

Here D(x) is a quantity dependent on the specific noise-generation mechanism

considered. For thermal noise it can be identified with the diffusion constant D.

The quanti ty

by fig. 4.

is the impedance-field vector. Its meaning is illustrated

Suppose a noise current o~ is injected at a plane X and extracted again at

a plane X + ~X. This current produces a voltage OVN

across the terminals of

the diode. No," th" impedance-field is defined by

(53)

Assuming that the noise currents in the different parts of the diode are

uncorrelated the noise voltages have to be summed quadratically which leads

to (52). Evidently in ~2is formula . h rx 1 nOlse current w ereas ~ re ates

voltage across the terminals.

D(x) represents the actual nature of the

to the way this current produces a

. dZrx To obtaln ~ one has to solve the same equations as before but assuming

a total current o~ between X and X + dX and zero total current in the

rest of the diode.

When the plane X is in the diffusion region we assume the following fields

to exist (Xr is the reduced value of X): For ~m < ~ < Xr a backward

traveling wave:

-18-

Ut = A exp Y2 ~ (54)

For X < ~ < X + llX the complete soll,ltion of the inhomogeneous r r r differential equation:

where Dit is the reduced injected noise

as in section 111-2. For Xr + llXr

< ~ <

(55)

current and B has the same value o

~. a forward travelling wave: 1

( 56)

The backward wave in this region is neglected on the same grounds as in 111-2.

Imposing continuity of u1

and

llX + 0:

and X r

+ llX we find in the limit r

r

c = (57)

The calculation of the. impedance-field is simylified considerably if we

assume that the voltage drop across the diffusion region itself may be

neglected, the diffusion region being short compared to the drift region,

so that the voltage across the latter only has to be calculated. The

boundary condition for the drift region follows from (56) and (57).

The final result is:

llZTX (f, < X < <,)

~x m r ~,

,

-j(l,d,

When X is in the drift region we assume as before that only one, forward r traveling, wave exists. Then for f, < X there is no electric field. For

r Xr < f, <X + llX we have the differential equation: r r

i dv dUI

" (' 0 0) ( ) Ul t = J" + v di) u1 + dT 59 o 0

(58)

-19-

As nl

= 0 at Xr the contribution of the term with nl

in the above

equation is of second order, so we find for the field at Xr + ~Xr:

~x r

All we have to do now is calculate the field in the region S > Xr and

integrate 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~TdT , o 0

(60)

(61 )

From the expressions for the noise voltage derived in the preceding section

it is hard to draw any general conclusions. However, one may note that (51)

and (58) l,ed to terms of the same form in the expressions for the mean

square voltage. If we take (51) and (58) as representative for the shot

noise and the thermal noise, respectively, then we are able to get an

impression of the relative magnitude of both noise sources. Taking the

square of the absolute value of (51) and substituting (58) in (52) and

dividing the results we obtain:

2 exp 2 Re YI(s.-s ) - I 1 m

Substituting Bo' YI and Y2 this reduces to:

V2 th~ 2 6 exp 2 Re YI (si -sm) -~ 40i 2 2 2 Re YI v2 ,,3 { (I + 0 ) + (4o~) }

s a -3- 2 v va a

Inserting representative numbers e. g. :

N ~ 10 21 -3 D = 10-3 2 -I

D m m s

0.05 2 -1-1 = 0.4 ~ = m V s V.

1

lOS -I 10- 10 A -I v = m s ~ = s V s

(62)

- I (63 )

-I m

-20-

we find at low current densities: 0 = 0.008 and v = 0.03 so that the a first factor of (63) is a large number Mhereas the second factor is in

the order of one. At increasing current densities the first factor decreases

but the second one increases. One migh~ therefore conclude that the thermal

noise is the dominant noise source at all current densities.

-21-

V Numerical Results

For a numerical calculation the v-E-relationship must be specified. The

measured v-E-characteristics of Canali et.al. [22] were used as a starting

point. They can very well be approximated by the function:

JJ E o v = -"--"" )1 E

o 1 +

Vs

For instance, at room 7 v = 0.9.10 cm/s.

s

(64)

temperature one finds )1 . 0

= 450 cm2/Vs and

Unfortunally it is not possible to evaluate the expressions for impedance

and noise obtained in the previous sections for this v-E-relationship.

Therefore the curve has been approximated by three straight lines (fig. 5).

The first intersection is chosen at the electric field Ei' which also

marks the end of the diffusion region. When the values of )J ,v and E. o s 1

have been selected, the value of )12 is taken such that at zero d.c. current

the transit~time from xi to i is the same as it would be for the v-E­

relationship of (64).

First it was tried to reproduce the experimental results of Bjorkman and

Snapp [23]. The following parameter values were used:

)Jo = 450 cm2

/Vs

= 0.075.107 cm/s v s

E. 7 kV/cm 1

= 3.10-4 cm2 A

2 7.9)Jm

ND 1 .2.1015

T = 170 C

£ 12£ o

-3 cm

The results are shown in fig. 6. It ~urns out that the agreement ia 880d at the low current of 5 mA but at the higher currents the calculated values

of the negative conductance are higher and occur at higher frequencies than

the measured ones.

-22-

One might wonder whae-influence the l'aramtiter·'1!·'·'has"on , 1

the result. An varied from impression of this influence is given in fig. 7 where E. is

1

/ '. 6 to 8 kV cm with the d .• c. current at 5 mA. EV1dently there is an

appreciable influence. At higher currents however the change of the curves

with E. becomes less and at 40 mA it is insignificant. 1

As a second step it was tried to find out if the temperature rise of the

diode at high currents could be responsible for the discrepancy between

theory and experiment. The temperature enters explicitly 'in the formulas

for the contact region. Furthermore it is assumed that the low-field

mobility varies as:

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

, 0 0

From the data given in [22] one may conclude that the saturated drift

velocity varies little with temperature, so it was held constant. The

diffusion constant too was kept constant.

(65)

The results of this calculation are shown in fig. 8 for currents of 20 and

40 mA and various temperatures. Evide~tly the frequency shift (or better

the lack of frequency shift) of the negative conductance can be explained

by a temperature change but not the variation of the magnitude of the

negative conductance. Not shown is the variation of the susceptance. It

decreases with increasing ·current, but increases with increasing temperature.

Finally, the influence of donor density was examined. A typical result is o shown in fig. 9 for a current of 20 mA and a temperature of 50 C. Two

conclusions may be drawn from this figure: firstly, the frequency region

of negative conductance shifts to higher frequencies with increasing donor

density and secondly, the magnitude of the negative conductance decreases

sharply when the donor density drops below a certain value. Further

investigation showed that the last phenomenon is dependent on the length

of the diode and it seems that ther~ is something like a minimum ND~

product for good operation of this type of diode.

-23-

VI Conclusion

An analytical theory for the small-signal characteristics 6f Baritt-diodes

has been developed. It takes into account the influences of the non­

saturated drift velocity, diffusion and the properties of the injecting

contact.

The theory gives insight into the physical behaviour of the diode as well

as numerical values for the impedance and noise that fit well to experimental

results.

Some important results are:

i. the region of negative resistance shifts to higher frequencies with

increasing current, but to lower frequencies with increasing

temperature. A similar feature in the susceptance could be useful for

stabilization of Baritt oscillators.

ii. the region of negative resistance shifts to higher frequencies with

increasing donor density. A minimum density (at a given diode length)

is necessary to obtain a useful negative resistance.

iii. thermal noise is the dominant noise source.

-24-

REFERENCES

[I]. W. Shockley - BSTJ 12, 799-826 (1954~.

[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. Lett., 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. ~, 559~569 (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.2

J.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, 1239~1251 (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.Nigrinand 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).

-25-

CAPTIONS TO THE FIGURES

Fig. I. Structure of a Baritt-diode.

Fig. 2. Field distributions at punch-through.

a. electric field. b. electric potential. c. energy-band diagram.

Fig. 3. Division of the diode into three regions:

I. contact region. II. diffusion region. III. drift region.

Fig. 4. Illustration of the impedance-field method.

Fig. 5. v-E-characteristics

------measured by Canali et.al. [22] at room temperature

approximation by eq (64).

------ three-line approximation used in the calculations:

I.

II.

v = dv dE =

~ E o

~2

III. v = v s

Fig. 6a. Comparison of calculated real admittance and noise figure with experiments (Bjorkman and Snapp [23]).

------ calculated

------ experimental o

I = SmA, T = 17 C.

Fig. 6b. As 6a. I = 20mA, T = 17°C.

Fig. 6c. As 6a. I = 40mA, T = 17 oC.

Fig. 7. Influence of

I = SmA, T = the parameter

17o

C.

E .• 1

E. in kV/cm is indicated at the curves. 1

Fig. 8a. Influence of diode temperature.

I = 20mA.

The temperature in degrees centigrade is indicated at the curves.

Fig. 8b. As 8a. I = 40mA.

Fig. 9. Influence of donor density.

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

I -26-

1

metal

or n - typ~ or

p+ - type semiconductor p+ - type

semiconductor semiconductor

o l I I E x

Fig. 'I.

-27-

E

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

v

JC=:::::::~--I-X

£

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

Fig. 2.

-28-

~

{ ; ~

) metal

· metal (

I

or I n Dr or • !

p+ ,

p+ I ( !

· l~ I - /' ~

) - ,

I I

I I I I

E

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

o 1m xi l

Fig. 31.

-29-

~ __ --~~~----~--__ X

o x X+ aX l

Fig. 4.

v em/.

107

o Ej 20 40

-30-

E 60 80 100 (KVlem)

Fig. 5. I

HF (dB)

20

10

-ReY ( 0-1)

4 5

-31-

6 7 8 9

F

(GHz)

F O+-------~----~----~----~L---~--

4 5 6 7 8 9 (GHz)

Fig. 6a.

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.

NF (dB)

20

10

-ReV (n-1 )

10-3

0

4 5

4 5

-33-

\ \ \

\

6

" "- ..... -

7

/-

/ /

I /

I I

I I

6 7

Fig. 6c.

8

\ \ \ \ \ \ \ \

8

9

9

F (GHz)

F

(GHz)

NF (dB)

20

10

-ReV (0-1)

o

4 5

4 5

-34-

6

6 7

Fig. 7.

8 9

8 9

F (GHz)

F (GHz)

NF (dB)

20

10

-ReY (n-1)

4 5

-35-

6 7 8 9

F (GHz)

0+----.~--1_~_.----,_~~-L--- F 4 5 6 7 8 9 (GHz)

Fig. 8a.

NF (dB)

20

10

-ReV (0 -1)

4 5

-36-

6 7 8 9

F (GHz)

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

Fig. 8b.

NF (dB)

20

10

-ReV (n-1 )

-3 10

4 5

-37-

6 7 8 9

F (GHz)

o +---Lr....J--L-T"""'"""-..L,--,l---+--- F 4 5 6 7 8 9 (GHz)

Fig. 9.

I

-38-1 I

LIST OF SYMBOLS

A

D

E

E. ,E 1 s

e

F

M

G

I s

erN i

io,i 1 i

s Di

t

J

p

Po t

V

V s

VT

Vth

Vod

Vsd V •

Sl

VN

diode area

integration constant

" " diffusion region capacitance

diffusion constant

electric field

parameter

elementary charge

auxiliary function

bandwidth

amplifier gain

noise current

" " reduced current density

d.c., a.c. components of i

reduced noise current

" " " current density

d.c., a.c. components of J

parameter

imaginary unit

Boltzmann's constant

diode length

noise measure

donor concentration

valence band density of states

hole concentration

d.c. component of p

time

potential

shot noise voltage

thermal voltage

thermal noise voltage

d.c. voltage over driftiregion

shot noise voltage, drift region

" " " diffusion region

noise voltage, open cir*uit

9

I I

18

12

11

6

6

6

8

16

15

16

17

6

7,8

16

18

6

14 ,9

14

6

14

7

15

6

14

6

12

6

14

19

14

17

7

16

16

15

v s

x X

r x

X. 1

x m

Z o

ZI

Z c

Zd

Zi

ZTX

CI

r I ,r 2

<5

E

n no,n l , nlh,no nd

ni

ns 1;

AD

).10

v

'Va'V]

v' o

V. 1

Va

S

si sR, sm

noise voltage

drift velocity

-39-

" " , saturated value

space coordinate

reduced value of X

space coordinate

beginning of drift region

potential maximum

normalizing impedance

small-signal impedance

impedance, contact region

"

" , drift region

diffusion region

transfer impedance

reduced frequency

propagation constants

reduced diffusion constant

dielectric constant

reduced electric field

d.c., a.c. component of n auxiliary variable

value of n at diode end " II 11 II X.

1

reduced noise field

parameter

debye length

low field mobility

reduced drift velocity

d.c., a.c. component of v

auxiliary variable

parameter

" reduced space coordinate x

reduced value of x. 1

" " " " " "

17

6

6

17

17

6

7

fig. 3.

9

15

15

9

12

17

6

I I

II

6

6

7

8

7

7

16

14

14

6

6

7

8

9

II

6,14

7

9

II

p. 1

a

a c

t

t'

Til,

T X

¢

Vi' m w

-40-:

parameter

"

" new coordinate

auxiliary variable

value of t at I<

auxiliary variable

reduced potential

potential barrier for Iholes

angular frequency

8

6

12

8

8

9

19

14

14

6