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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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