Avalanche injection diodes

Solid-State Electronics Pergamon Press 1960, Vol. 1, pp. 54-69.

Printed in Great Britain

AVALANCHE INJECTION DIODES A. F. GIBSON A N D J. R. M O R G A N

Royal Radar Establishment, Great Malvern, Worcs. (Received 20 June 1959) Abstract--A high-speed two-terminal negative-resistance device is described. A quantitative theory

of the diode characteri§tics is developed and compared with experimental units. R6sum6---On d6crit un 616ment/l deux bomes et ~ r6sistance n6gative susceptible de fonctionner h tr~s haute fr6quence. Une th6orie quantitative des caraet6ristiques de cette diode est d6crite et eompar6e aux r&ultats exp6rimentaux. wird ein Zweielektroden-Bauelement mit negativem Widerstand und hoher Schaltgeschwindigkeit beschrieben. Eine quantitative Theorie der Dioden-Kennlinie wird entwickelt und mit experimentellen Messungen verglichen. Zusarmnenfasstmg--Es

1. INTRODUCTION I f the length of the bar is Lh the voltage drop GUNNa) has described the avalanche injection Vw at or just before the avalanche point will be effect in semiconductors and has shown that a twov~ ~ L1E2 (1) terminal device showing negative-resistance properties can be made in germanium. Physically, where E2 is the critical avalanche field. The rate of this effect can be described most simply by con- ionisation increases extremely rapidly with field for sidering a bar of n-type germanium with n+- fields greater than Es so, after avalanching has (non-injecting) end connections through which is occurred, the field in the avalanche region will passed a steadily increasing current. The electric still be about Eg. Hence the voltage drop across the field E in the specimen rises steadily until a avalanche region will be given approximately by critical field (the avalanche field E2) is reached at V, ~, L2E2 (2) which electron-hole pair production occurs by collision ionisation, as in a gas discharge. In where, as explained above, L2 is less than LI. If, general the cross-section of the bar will not be therefore, the voltage drop across the remainder of uniform and the avalanche will occur at some arbit- the bar is small an extended negative resistance can rary position along the bar. If the specimen is n- be expected. A further property of electrons in germanium type, the holes produced in the avalanche drift which is important in the present connection is the towards the negative terminal and increase the conductivity of the bar near the negative end, thus saturation of the electron drift velocity at high reducing the field at this end. If the externally electric fields first observed by RYDEI~ta).When the applied voltage remains approximately constant, the electric field exceeds about 1 kV/cm the electron field at the positive end is enhanced (by an amount velocity (and hence the current) in n-type gerdepending on the external circuit resistance) and manium becomes substantially independent of the the avalanche generation rate at the positive end electric field, up to the avalanche field (Gtrm~(a)). will be increased. In the limit the field will be con- This effect facilitates the observation of a negative centrated in a thin region ncar the positive elec- resistance due to avalanche injection in two ways, trode, the remainder of the bar being so heavily in- namely: (i) the current density at the avalanche field, at jected that only a small or negligible field exists in least in a linear system of the type considered this region. 54

AVALANCHE INJECTION above, is about 100 times less than that expected from Ohm's law. (ii) The decrease in field per unit carrier injected from the avalanche is very considerably greater than that expected from Ohm's law, thus enhancing the positive feed-back process. In practice a linear structure of the type described is very difficult to fabricate. As the value of E2 is about 105 V/cm in germanium, a bar of length less than 10-8 cm (0.4 thou) must be used if V~, is not to be excessive [equation (1)]. If the current at or about the trigger voltage VT is not to be unduly large, the cross-section must be of similar dimensions. Even so, the power dissipation per unit volume is considerable. In view of these considerations we have studied an alternative structure, namely a small, approximately hemispherical n+-alloy "dot" on a relatively large crystal of n-type germanium or silicon (GIBSON and MORGAN(4)) with a view to assessing the practicability of making a two-terminal negativeresistance device using the avalanche injection effect. It is well known that a small hemispherical contact is the most favourable geometry from the point of view of heat dissipation and the small area ensures a high electric field at the contact for relatively modest currents. In addition this structure is relatively simple to reproduce, as the properties of only one junction are important. In Sections 2 and 3 of this paper we describe the fabrication and general properties of "small dot" avalanche diodes. In Section 4 a theory of the trigger point is developed which indicates how the trigger voltage and current vary with crystal resistivity, alloy dot radius and temperature. The theory is compared with the behaviour of experimental diodes. The sustaining voltage Vs is discussed in Section 5, and in Section 6 the paper concludes with a description of a three-terminal device which is essentially an avalanche diode with an additional triggering electrode. 2. DIODE FABRICATION

Diodes have been made using n-type germanium and n- and p-type silicon ranging from twenty to several hundred ~ - c m resistivity. Early experience showed that the trigger voltage increased with increasing resistivity and that 40 ~ - c m material represented a reasonable compromise. By far the largest number of diodes have been

DIODES

55

made using 20, 30 or 40 12-cm n-type germanium. Care is taken to use grown crystals having a low radial variation of resistivity, about =1=2 £1-cm. Pieces of germanium about 0-04 in. square by 0.015 in. thick are etched down to 0.012 in. thick in a fast CP4 type etch to remove the worked layer and polish the surface. Using the temperature-reversal technique described by MASONand TAYLOR(5) two gold/0"75 per cent antimony-alloy wires, one 0.010 in. diameter and the other 0-002 in. diameter, are alloyed on to opposite sides of the germanium dice. Diodes have been made on dice cut parallel to <111 > and <100 > crystal planes, the latter proving more suitable for the final etching process as measured in terms of yield. Alloying on a <100) face gives a pyramid-shaped alloy front, bounded by <111) planes, which facilitates a reduction in contact area by etching after alloying. The smaller wire is crimped after alloying and the unit mounted on a suitable header. Using a CP4 type etch incorporating the strongest available hydrofluoric acid, the mounted units are etched for 20 sec, washed in de-ionised water, dried in a stream of hot nitrogen, and their resistance measured at a low voltage. The low-voltage resistance is determined by the spreading resistance of the smaller n+-alloy contact and if the alloy dot is approximately hemispherical is given by p Ro = 2~rro

(3)

where p is the resistivity of the material and ro the radius of the contact. If necessary, etching is continued until a desired value of ro, and hence Ro, is achieved. Units are then dipped in a thick silicone varnish and encapsulated in a can containing a thick silicone grease. 3. DIODE CHARACTERISTICS

Typical characteristics of a germanium and a silicon diode fabricated as described are shown in Figs. 1 and 2 respectively. In this section we shall enumerate briefly the more important properties of such diodes. The trigger point is defined as the point at which the slope resistance dV/dI becomes zero, and subsequently negative, while the portion o f the characteristic in which the current is greater than the trigger current is referred to as the sustaining region. Some diode properties, for example

A. F. GIBSON and J. R. MORGAN

56

the variation of trigger voltage with temperature or resistivity, are amenable to theoretical analysis. Such analysis and the comparison of theory and experiment form the subject matter of subsequent sections. 80 E Germanium

I

70

60

50

t-

u

to

20

50 Voltage,

40

50

60

V

o 30------

2O

Io

FIc. 2. Characteristic of a typical silicon avalanche diode. C o m p a r e d w i t h g e r m a n i u m , silicon u n i t s have a h i g h e r s u s t a i n i n g voltage, lower m i n i m u m s u s t a i n i n g c u r r e n t a n d less curvature near t h e trigger point.

\ J

0

I0

20

30

40

50

60

Voltage, V

FIO. 1. Characteristic of a typical g e r m a n i u m avalanche diode.

(a) The trigger voltage VT is found to increase slowly with crystal resistivity and dot radius r0. The resistance at the trigger point is significantly greater than the resistance at low voltage, typically by a factor of two. This factor arises from a combination of the saturation of the drift velocity at high fields, together with a "majority carrier accumulation" effect (Section 4). In germanium units, the trigger voltage is found to be substantially independent of temperature up to about 140°C and then to fall slowly. For silicon diodes the critical temperature is, of course, considerably

higher. The theoretical interpretation of all these features is given in Section 4. (b) The sustaining voltage at a given current is substantially independent of temperature, dot radius and crystal resistivity, provided the latter is not too low. The fraction of the sustaining current carried by holes can be estimated experimentally and is typically 20-30 per cent. The theory of the sustaining region is appreciably more difficult than that of the trigger voltage as the avalanche is transversely unstable, that is the avalanche does not occur uniformly over the alloy dot but is concentrated into a small region--a phenomenon familiar in gas discharges. These aspects are discussed more fully in Section 5. (c) Considered as bi-stable devices, avalanche injection diodes can be switched "on" and "off" at relatively high speeds with low-energy pulses. Switching speed in bi-stable circuits can be discussed only in terms of the circuits used and some typical circuits are shown in Fig. 3. Fig. 3(a) shows the simplest possible bi-stable circuit, together with a "load-line" characteristic

70

57

AVALANCHE I N J E C T I O N D I O D E S showing the two stable operating points for the circuit. The input and output condensers are kept small ( ~ 100#FF ) and a small resistor RL, ( ~ 100 £~) is included to limit the current surge through the diode on switching. Rc~ is typically a

+

H.T

Output woveform

On pulse

Off

resistor in the earth lead reduces the available output voltage. Two of these disadvantages are overcome by the circuit shown in Fig. 3(b). In this circuit the diode V1 is conducting all the time except when the

l

pulse

Avalanche diode '%

VoltageH.T.+ (o)

H.T.+

Onpul_s]j_e

ONJ~se _AI_

(~ I[

output

~

w(:n~eform

_J--l_

Off pulse

.31_

I:

~Output m

_r--L Off pulse

_n_

II R

H.T,(b)

(c)

H.T.V./otiXR

FTo. 3. Avalanche diode switching circuits. few kilo-ohms. This circuit suffers from a number of practical disadvantages, however, namely that the input and output connections are common, positive- and negative-going switching pulses (not generally available in computer circuits) are required for switching "on" and "off", and the

"turn-on" pulse is applied. The turn-on pulse reverses the bias on V1, so that a high-impedance input is presented to the turn-on pulse, until such time as the avalanche injection diode triggers. When triggering occurs the voltage drop across the avalanche diode decreases sufficiently to bias V1

58

A. F. GIBSON and J. R. MORGAN

forward again, even if the turn-on pulse is still being applied. The "turn-on" input is therefore very effectively isolated from the output circuit, "on", "off" and output pulses are of the same polarity and if the diode V1 has a low forward resistance the decrease in output amplitude is negligible. Naturally a low hole-storage diode is required for/11 and diodes of the VX3286 type have been found to be satisfactory. At the cost of increased complexity, isolation of both inputs from the output can be obtained as indicated in Fig. 3(c). The diode V2 plays a similar role to that of/71 in as much as it is normally conducting and is closed by the "off" pulse until such time as the avalanche diode turns off. In this arrangement, very good isolation between input and output, together with a high input impedance, has been achieved. Both circuits 3(b) and 3(c) can, of course, be inverted if negative-going trigger pulses are desired. Alternatively both V1 and V2 of Fig. 3(c) may be joined in series and a single input point is used to which both positive (on) and negative (off) pulses may be applied. The switching characteristics of these circuits have been examined experimentally. The following general features have been noted. (i) The time required to turn the circuit "on" is substantially independent of the amplitude of the "on" pulse or the bias conditions and is typically 10-20 m/~sec, as measured on a "Tektronix" type 545 Oscilloscope. This time constant appears to be determined primarily by the circuit used and as the specified rise time of this Oscilloscope is 12 mFsec the avalanche diodes can probably be switched on in less than 10 m/zsec. In practical circuits, however, the switching time is largely determined by the time taken to charge stray capacities associated with succeeding stages. (ii) The time required to turn the circuit "off" decreases with increasing trigger-pulse amplitude and increases slowly with the magnitude of the "on" current of the avalanche diode. A typical turn-off time is 30-40 m/~sec. (iii) The amplitude of the pulses required to turn the diodes "off" and "on" increases and decreases respectively as the H T voltage is increased. It is convenient to operate under conditions such that the "on" and "off" trigger pulses required are o f equal amplitude and about 5 V.

4. THE TRIGGER POINT

In switching circuits, the most important parameter of any two-terminal negative-resistance device is the trigger voltage. (a) The trigger point at room temperature. Theoretical The pre-trigger or "off" state of an avalanche diode may be specified by three quantities, namely the resistance at low voltages R0, the trigger voltage VT and the trigger current IT. We shall now deduce theoretical expressions between these quantities subject to the following assumptions. (i) That the material is n-type germanium and that the current due to thermally generated holes can be ignored. This assumption cannot be justified at high temperatures, as discussed in Section 4(c). (ii) That the electron drift velocity is proportional to the electric field up to E = E1 but independent of field in the range E1 < E < E2. The degree of approximation represented by this assumption may be assessed from GUNI~'s results(3) which indicate that the drift velocity of electrons in germanium increases by a factor of about 2.5 when the field is increased 100 times.

£=E2~ ~'ro //

/

,,'

Fro. 4. Model for theory of avalanche diodes. (iii) That the small alloy dot is hemispherical. This is probably the worst approximation used. With pyramid-shaped alloy junctions, for example, it is found experimentally that the trigger

AVALANCHE I N J E C T I O N D I O D E S voltage decreases if the pyramid point is made sharper. However, three parameters are required to specify any contact geometry other than hemispherical and as only three diode parameters are measured agreement between experiment and a significant.ge°metricallYmore detailed model would not be Referring to Fig. 4, r0 is the radius of the hemispherical alloy dot and y0 is the radius at which the electric field equals El. Let y be any radius. Clearly the current density and electric field increase with decreasing radius, the maximum value occurring at y = r0. To determine the trigger conditions we put the field at ro equal to the avalanche field Eg.. For r0 < y < y0 the electrons drift at a constant velocity vs independent of field. For Y > 3'o the electron drift velocity is proportional to the electric field and Ohm's law applies. The trigger voltage is then given by Ztl

00

Vr = f E(y)dy+ f e ( y ) d y I"o

I Io

(4)

= V~+V2

where the second, "spreading resistance" term is given by v~ = - -

2zry0

If the diffusion term is small, n is given by

n=N(Y°) 2 \7

x

x [ 1 + 21{-( _

1

-- 1 }\

....}] (7)

Clearly the diffusion term is most important when y is small, that is equal to ro. We may neglect diffusion entirely if ~'0V8

D < -3

(8)

The value of v, is about 6X 106 cm/sec (Gmso~i and GaANVlLLE{O, GtmN{Z0 and in practice r0 is not less than 10-s cm. The value of D for electrons at high electric fields is not known but is unlikely to be as large as 3000 cm2/sec (see Section 5) so, in order to proceed, we shall assume that condition (8) is satisfied. Hence

n = N(yo/y) 9"

As the system under consideration has spherical symmetry, Poisson's equation (in rationalised units) may be written in the form

{5)

1 d(y2E)

where Ia, is the trigger current and p is the resistivity of the material. In the volume defined by ro < y < yo, however, there is a significant space charge. This arises from the saturation of the electron drift velocity. The current density must increase as the radius y decreases but the electron drift velocity cannot exceed vs so the electron density n must increase to carry the current. The variation of n with y is given by I = 2~2nqvs+2,ry 2.

59

qDi\dy]l = 27ry°2N~ws

--q'n--N'-( =

)=

Nq

-T

.

[(Yoly)2--1]

(9) where K is the dielectric constant of the material. Integrating equation (9) with the boundary condition that E ----E1 at y = y0 gives

y2E--yo2E1 = ~

(yo--y)(2yo2 - y y o - y ~) (10)

where 1/pl~ has been written for low field electron mobility. V1, equation (4), is given by

Nq, ~ being the

(6)

where N is the density of donor impurities in the bulk material, assumed equal to the electron density in the absence of space charge (y >/y0), q is the electronic charge and D the electron diffusion constant.

r6= f e(y)dy ra

Integration with the boundary condition E = E2

T

105

2xlO 5

5xlO 5

I0~

~1:.o 2~10~

T

5xlO6

107

2xlO7

2xlO4

i

i

5XlO4

i

d

i

FIG. 5(a). Theoretical curves of

104

•

I

C

2xlO5

P

I

[]

x

t

o

F

•S

I

I

S •

x

I06

×

xxx × ~

C/m2

5xlO5 3,o/~.~ '

r_.o

~ ~ ~ °°°°°

80 40 30

V (~) /h

20 16 5.3

[] ~ x

VT/roand I~/ro as functions

zx

x

×

2xlO6

~s S

F

ss S • •

I

5xlO6

I

I

r

P

107

s s S"

sss SS

sS

, SS~ 'S

2xlO?

JTT

1.4 0.4

• •

i

o*S

of ro/3pl~. As ro is deduced from I~, all diodes must fall on this curve. Crystal resistivities in fl-cm:

[3

'o

105

o

SS

S~

i

5xlO7

j S S~

>

©

©

go

O

AVALANCHE INJECTION 40

o ~ @

3O

-L

o-

g8

f

~=_ 2c -~:

f

o~ ~, $

L

I0

rS

-,o

-8

_6

-I4

L1 -:,

~ Index,

4

;

8

DIODES

61

convenient to determine the quantities (VT/ro) and (IT/rO) as functions of ro/3p#~¢; the equations may then be represented by a single curve. In addition we have equation (3) relating to the diode resistance at low voltages, R0, and the alloy dot radius, r0. If we assume that x, p, F, Vs, E1 and E9 are known constants of the material the only unknown diode parameter is r0 and any of the three measured quantities (VT, IT or R0) may be used to determine r0 for each diode. We have chosen to determine r0 from the trigger current IT and use the remaining two quantities to compare theory and experiment. The theoretical curves shown in Figs. 5(a) and 6 assume the following numerical values of the constants: K = 1 6 = l ' 4 X 10 -1° F/m

E1 = lOS V/cm E2 = 106 V/cm

n

FIG. 5(b). The histogram indicates deviation of experimental diodes from theoretical lille for Vf/ro.

vs = 6 × 106 cm/sec /~ = 3800 cm2/V per sec

at y = r0 (the condition for an avalanche) gives

VI = EI ( ~y°2 --Yo ) + \ ro 1 + 7 1 2 y o 3 / r o - - 2yo2 - 3y02 ln(yo/ro)+ ½(Yo2-- r0~) ~p~K

(11)

where y0 is given by 1

ro2E2--yo2E1 = 3--~ (yo--ro)2(2yo+ro) (12) Finally the trigger current IT is given by equation (6) as 2~rvsy02 IT = - (13) where y0 is given by equation (12) as before. (b)

The trigger point at room temperature. Experimental

T h e above equations for VT and IT do not have simple analytical solutions and have been solved numerically.* For graphical representation it is * We are indebted to Mr. P. M. WOODWARDand Miss R. HENSMANfor their assistance in these computations.

The values of E1 and E2 are based on measurements by GUNN(Z) and others. In practice the results are relatively insensitive to the choice of El. The value of vs is a mean of results by RYDEa, GUN~ and others, tabulated by GmsoN and GRANVILLE(6). The value of vs appears linearly in the expression for trigger current but is relatively unimportant in the trigger voltage. There are naturally no experimental points on the theoretical curve for IT/ro as a function of ro/3ptzK given in Fig. 5(a) as all diodes must lie on this curve by the definition of r0. Each experimental point on the V~,/ro curve represents a diode and all recorded diodes have been included. The spread of experimental values is rather wide, but the general agreement between theory and experiment, over a range of nearly a factor of 1000 in ro/3pl~K, is as good as can be expected. That the theoretical line is also the best mean line is shown by the histogram in Fig. 5(b): about 30 per cent of all diodes lie within -4-10 per cent of the theoretical line and about 90 per cent lie within + 6 0 per cent. As the theoretical value of VT/ro for a given ro/3pFK is fairly sensitive to the choice of E2 this result represents an accurate determination of the effective avalanche field for germanium. It has been noticed that, for any given resistivity,

Vr

O'IL 104

0.5

1.0

2-0

5.0

I0.0

5xlO4

I v.

FIo. 6. Theoretical curve of

2xlO 4

•

o0

¥o

x

C/m-Z

5xlO5

a function of

3/o~.~'

2xlO5

T

VT/ITRo as

105

T

ro'3ptt•.

2xlO6

5XlO6

iO"t

2xlO~

Same experimental diodes as s h o w n in Fig. 5.

I06

° 0

5xlo~

0

0

0 :Z

0 ~=~

AVALANCHE I N J E C T I O N

the experimental diodes lie on a line of somewhat steeper slope than the theoretical line and an empirical curve may be drawn through the points such that the spread from this line is much less than that indicated by Fig. 5(a). Hence at least part of the spread aroUnd the theoretical VT/ro curve in Fig. 5(a) is not random but due to the inadequacy of the theory to give an accurate description of the variation of VT/r0 with r0. Presumably this arises from the assumption of a simple geometry. Thus the variation of V~, with r0, predicted by theory to be slower than linear, is in fact very slow. This is at least fortunate from the practical point of view as it increases the reproducibility of VT from unit to unit and in practice VT is certainly the most important diode parameter. In Fig. 6 we show the quantity VT/ITRo as a function of ro/3pl~K. This quantity measures the ratio of the resistance at the trigger point to that at low voltage and is a measure of the curvature due to the saturation of the electron drift velocity. In a linear structure VT/ITRo would be about 100 (Section 1) but is much lower, both theoretically and experimentally, in a radial structure owing to the accumulation of electrons near the contact, which effectively reduces the resistivity at high voltages. The experimental points in Fig. 6 are, of course, the same diodes as those of Fig. 5. Again agreement between theory and experiment is adequate, though the experimental points appear to be generally below the theoretical curve. It is believed that this is due to a small amount of avalanche generation of holes at voltages below the trigger voltage. Certainly the current starts to increase rapidly with voltage very near the trigger point even though the slope resistance remains positive (see Fig. 1). This feature is not included in the theoretical treatment. Two final points should be noted, namely: (i) that the theoretical equations given above are still valid for near intrinsic material, provided that the resistivity is interpreted as 1/Nqlz, where N is the number of impurities. Some of the diodes shown in Figs. 5 and 6 were made on material having an effective resistivity of 80 ~-cm. (ii) n- or p-type silicon diodes have very similar trigger characteristics (V~, and IT) t o germanium units of comparable resistivity and dot radius. The values of the electron or hole drift velocities in silicon are not so well known as those in german-

63

DIODES

ium, so no quantitative analysis has been attempted. However the availability of very high resistivity silicon* ( ~ 5000 ~-cm) has allowed the fabrication of diodes with trigger voltages in excess of 200 V, with quite modest trigger currents. (c) The trigger point at elevated temperatures.

Theoretical At high temperatures the generation rate and density of holes in the space-charge region can no longer be neglected in a calculation of the trigger voltage. As the dot is positive on n-type material the generated holes are swept away from the dot at their saturated drift velocity and we may estimate the hole density p(y) by equating the rate of generation between r0 and y to the rate at which holes leave an element dy thick at radius y. The rate of generation of holes per unit volume in n-type material is given approximately by n~2/N • % where n~ is the intrinsic electron or hole density and r is the carrier lifetime, n~2 varies exponentially with temperature and is the dominant temperature-dependent term. The generation rate between y and r0 is then N~" ~Tr (y3--roa) which may be equated to the number leaving an element at radius y per unit time, given by

p . 27ry2 . dy

= 2~yZv~. p.

dy/vs Therefore

p(y) =

2n~2

[y--ro3/y 2]

3 r " N "v , =

(a4)

f2(y-ro3/y9

where Q has been written for brevity. If the saturated drift velocity vs is the same for electrons and holes ( R ~ r ~ (2)) and the diffusion term is neglected as before [Section 4(a)], the current ! at y and at y0 is now given by

I = 2¢ry~. qvs(n+p) = 2try02" q" vs(no+po)

05) * We wish to thank Standard Telecommunications Laboratories, Enfield for supplying this material.

64

A.

F.

GIBSON

and

equation (15) being equivalent to equation (6) of Section 4(a). At y = Y0 there is no space charge so, by electrical neutrality,

no =

=

R.

where P0 is given by equation (14) with y = yo. Equations (15) and (16) give the variation of n with y, as in section 4(a). Poisson's equation now becomes

--q'n--p--N'L K

J

(17)

Integrating twice, with the same boundary conditions as before [Section 4(a)], gives the voltage

K

(19)

The trigger current Ia, is now given by

IT

=

2zrqv,[yo~N+2Q(yoS--roS)]

°C

,oo ,20 ,4,o ,~,o

60

4o

6~0

80

I00

120

,so 2oo 140

22o

160 12

4C

8

30

4

20 i0 r9

] Ii J 1020

(20)

Equations (18), (19) and (20) reduce to (11), (12) and (13) respectively when (2 = 0, corresponding to a low temperature at which n, 2 is very small. The numerical solution of equations (17) and (18) is tedious, and has only been attempted for p = 40 Q-cm and one value of r0, corresponding to ro/3pt~K= 1.6 X 105 C/m 2 and a trigger voltage at low temperatures of about 47 V. In addition we have assumed that %/z, vs, Ez and E2 are invariant

Temperature scale, For~=lO/~sec 4 0 11 For z'=l/2 sec 2 0 5C

MORGAN

q-[N(rS/3--yo~,+], yo3)+2Q(r414--yo3r+]yo4)]

(16)

po+ N

1 d(y2E) y~ dy

J.

IO21

IO22

1023

iO24

o I0 z5

"o:' m4 FIG. 7. T h e o r e t i c a l curves of VT a n d I T as f u n c t i o n s of t e m perature for a g e r m a n i u m avalanche diode m a d e with 40 £)-crn material.

drop across the space-charge region as V = gl(yo~/ro--yo)+

+ ~[2(y03/r--y02) -- 3y02 ln(yo/r)+ ½(yo~

r02)]

+ Qq [{(yo4/ r - yoal- 2yo3 ln(yo/rl + {(y£~- #1] K

(18)

where y0 is given by ro2E2--yo~E1

with temperature, only Q being temperature dependent. This approximation allows V to be deduced as a function of Q without any a priori knowledge of the lifetime -r. Clearly the relevant value of ~- is the lifetime of the material within a radius 2 or 3 r0 of the dot, and this may be markedly less than the initial lifetime of the crystal before alloying. The result of this computation* is shown in * W e are i n d e b t e d to M r . J. B. ARTHUR for c o m p u t i n g this curve.

AVALANCHE INJECTION Fig. 7, in which two temperature scales, corresponding t o ~ - = 1 and 10/zsec respectively, have been included. It will be seen that the temperature at which a significant fall in VT occurs is relatively insensitive to the value of ~-. The inflexion in the curve relating VT with Q at about Q = 5 x 1022 is believed to be real and not a feature of computational or other approximations. No comparable inflexion occurs in curve of trigger current with temperature.

DIODES

65

is about 10/~sec. The lifetime of the starting material is several hundred/zsec so this value does not seem unreasonable. From the practical point of view the slow variation in VT or IT with temperature is extremely important. If a 10 per cent decrease in Va, may be tolerated, operation up to 130-140°C ambient is permissible. Alternatively a very high power dissipation can be achieved at moderate ambient temperatures by the use of an efficient heat sink,

10

60

~ o

~-----o----~ ,-------o---.~ ~ . . . . o _ . . _ "

o

o

50

-~

8

> o"

40

$ v30

I__T_x_T__ x

\2

2(

30

50

70

90

I10

Arab. temperature,

130

150

170

180

*C

FIo. 8. Experimental curves of YT and IT as functions of temperature for typical germanium avalanche diodes made with 40 f~-cmmaterial. (d) The trigger point at elevated temperatures.

Experimental Measurements of the variation of trigger voltage and trigger current with temperature have been made using a number of 40 f2-cm germanium diodes and some typical results are shown in Fig. 8. The false origin of the trigger-voltage scale is the same in Figs. 7 and 8 so they are directly comparable. It will be seen that the agreement between theory and experiment is good and in particular there is experimental evidence of the inflexion in the VT curve around 120°C. Comparison of the theoretical and experimental curves suggest that in the immediate neighbourhood of the alloy dot ~-

the junction temperature being permitted to rise to 130-140°C. Considerably higher temperature ratings can, of course, be achieved by using silicon, in which n~2 is about 106 times smaller than in germanium. 5. T H E

SUSTAINING

REGION

(a) General characteristics For diode currents greater than the trigger current the slope resistance (dV/dI) is negative up to at least 100 mA and generally higher. At low currents (dV/dI) is typically --10kf2 and the characteristic is obscured by relaxation oscillations. The sustaining current at which I(av/az)l falls

66

A. F. G I B S O N and J. R. M O R G A N

sufficiently for oscillations to cease, is a function of the test equipment, but is typically 2IT. At high currents (greater than 200 mA) the slope resistance is small but generally still negative, though a positive parasitic resistance, which can usually be ascribed to the base contact, is sometimes important. The extended range of the negative slope resistance of the diode can be useful in some switching circuits (see, for example, Tow~rsEr¢o~7)). The value of the sustaining voltage at any specified current is found to be substantially independent of the alloy dot radius. For 95 per cent of diodes made on 20, 30 or 40 t2-crn germanium the sustaining voltage at 25 mA lies between 8 and 11 V. The value of the sustaining voltage falls slightly at low resistivities and for diodes made with 1 f~-cm material the average sustaining voltage is between 7.5 and 8 V. The sustaining voltage is thus inherently more reproducible than the trigger voltage. (b) Theoretical discussion A theoretical analysis of an injecting avalanche, which can be applied to the present problem, has been given by GUNNtS). His assumptions were as follows: (i) Planar geometry. The thickness of the avalanche region [L~ of equation (2)] must be of the order of 1/~, where ~ is the avalanche ionisation constant (McKAy,(9) MILLERS0)) and hence of the order of 10-5 cm. This is much less than the radius of any practical alloy dot, so planar geometry may be assumed to apply in the present case. (ii) Carrier space charge in the avalanche region much greater than the impurity density, i.e. I/xrea.% >~ N. This assumption is justified in our case. (iii) ~ = ~¢n = ~ and vsp = vs, = vs. (iv) The variation of ~ with field to be of the form = ~0 exp(AE) (21) (v) Lateral stability of the avalanche, i.e. a uniform current density across the surface of the contact. With these assumptions GuNN obtained, as an approximate solution, Vs =

0"414(4"rrKvs'tt .J-~ \ Aa~o /

(22)

where A and ¢¢0 are defined by equation (21) and J is the current density. Hence, at a given current, Vs is proportional to the alloy dot radius to. This is contrary to the experimental result stated above. The crystal resistivity does not appear in equation (22) because of assumption (ii) above but it is clear that, on GUNN's model, Vs would decrease slightly in low-resistivity material, as observed. That the sustaining voltage is approximately inversely proportional to the square root of the current is confirmed experimentally. All diodes examined can be represented, at least in the range 10-100 mA, by a relation of the form Vs = const.I -n

where n lies between 0.4 and 0.6, depending on the diode. Numerical values of A and ¢t0 can be estimated by fitting equation (21) to the experimental data on ionisation rate given by MILLER(t0). The absolute magnitude of the sustaining voltage at 25 mA may then be calculated from equation (22) assuming a typical alloy dot radius of 10- a c m and avalanching to take place uniformly across the contact surface. This procedure indicates a sustaining voltage of about 70 V, which is an order of magnitude too large. This discrepancy, together with the invariance of Vs with r0, can be explained if it is assumed that, as in a gas discharge, the avalanche is laterally unstable and covers only a small fraction of the area of the contact. To obtain a sustaining voltage of about 10 V at 25 mA requires an effective avalanche area of radius about 10-4 cm and unless the alloy dot is smaller than this Vs will be independent of r0. (c) The effective area of an injecting avalanche That the avalanche will in fact shrink to a small area may be shown by consideration of the equipotential lines around the contact. If the degree of avalanche injection is not uniform across the surface, the enhanced conductivity due to injection from a highly injecting region will so distort the equipotential lines as to increase the field, and hence the injection, in this region. Hence the stable configuration is a restricted area of avalanche. It may be anticipated that, by analogy with a gas discharge, the effective area of the avalanche will increase with current.

AVALANCHE INJECTION

DIODES

67

The effective avalanche area cannot shrink indefinitely but will be limited either by diffusion or space-charge considerations, whichever gives the greater radius. Space-charge effects will be important if the diameter of the avalanche region is about the same as its thickness in the field direction, i.e. 10-5 cm, but such a small area would lead to a much smaller sustaining voltage than observed. Hence we conclude that the area of the avalanche is diffusion-limited. If so, the radius of the avalanche region is simply given by equating the inward current due to the field (qnvs) to the outward flow due to diffusion [qD(dn/dr)]. Hence n =noexp(---Vsr/D) and we may define the effective radius of the avalanche region as D/vs. However the diffusion constant of carriers in germanium in the presence of a high electric field has not been estimated, so far as we are aware, either experimentally or theoretically. It would appear that a study of the sustaining region of an avalanche diode represents a method of studying this quantity so we shall now give a rough estimate of the diffusion constant at high electric fields.

At low electric fields, acoustic mode scattering dominates the rate of momentum loss. Hence:

(d) The diffusion constant of hot electrons in ger-

~" oc IlV(Te)

manium

-~

= Zl~ oc l/V(T~)

where l is the mean free path and ~ is the average thermal velocity. Hence

i~kT6

/9/= --

q

E

= Do-/z0-v

(23)

where Do and /z0 are the diffusion constant and mobility at zero field and v is the drift velocity in a field E. At high electric fields, however, optical mode or intervalley scattering are dominant and ~ is of the form

1 ex+ ) T~

(HEm~NG(I~)), but at high fields A e > ~ k T and hence ~- is still approximately given by

At high electric fields, electrons and holes in germanium gain energy from the electric field more rapidly than they can lose it to the lattice. Hence their average energy, or effective temperature, rises (SHOCKLEY(ll)).We shall assume that the carriers have sufficient collisions between themselves to ensure that the energy distribution of the carriers is Maxwellian at all fields: at low fields this is obviously a good approximation, and in an avalanche region the carrier density is very high indeed, so that carrier collisions are very frequent. Hence we may use the Einstein relation

where I is the mean free path under high-field conditions. Analysis of the experimental data on the variation of the ionisation constant 0¢by MILLER(1°) indicates that l is of about the same magnitude under avalanche conditions as at low fields. Hence equation (23) is still a fair approximation at the very highest fields. We shall therefore assume it to be valid at all fields. The radius of the avalanche region is then given by

D : = ~kT,/q

104 cm when E ~ E2 = 105 V/cm

where D I is the effective diffusion constant, k Boltzmann's constant and Te t h e electron or carrier temperature. The mobility/~ is determined by the rate of loss of momentum of the carriers and /~ ~ q~/m where ~r is the momentum relaxation time and m the carrier effective mass.

DI

r = --

°0e

DolzoE

~ - -

(24)

Vs2

and is clearly of the required order of magnitude to explain the observed value of the sustaining voltage. As the area of the contact covered by the avalanche region is now a function of E (and hence current, as in a gas discharge) the integrations upon which equation (22) are based are strictly no longer valid. If the effective radius from equation (24) is included no simple analytical solution can

68

A.

F.

GIBSON

and J.

be obtained when ~ is defined by equation (21). As good a fit to MILLER'S experimental data for can be obtained however by writing =

MORGAN

50

~IE m

where m = 6 for germanium and o(i = 1 "6 X 10-38 m 6 V -6 f r o m MILLER's data. Then, including an effective area defined by equation (24) we obtain, following GO~zN's method, [ I'vs3 ](1-~a)l(m+3)X

/ m + 1 ,~ 4 / ( 3 + r a ) ×

R.

.

(if m > S)

(25)

7 V at 25 mA, which is of the required magnitude. It will be noticed that Vs is now proportional to (current)-sm which is experimentally indistinguishable from the --½ law predicted by G u M . GUNN'S analysis indicates that the width of the avalanche region in the field direction decreases and the field increases slowly with increasing current. Similar results are obtained when the effective area of the avalanche region is assumed to be determined by equation (24), putting m = 6, namely: The width of the avalanche oc 1-213 The electric field oc I i/o The effective area of the avalanche oc I2l 9 6. AVALANCHE TRIODES

By adding a p+-alloy junction to an otherwise conventional n-type avalanche diode structure a number of new circuit arrangements become possible, which we have examined. In general any three-terminal minority carrier device will be limited in speed by carrier transit times between any pair of electrodes, and the present structure is no exception. Thus one virtue of the diode structuremhigh speed coupled with simplicity of fabrication-is lost. However, the development of modern transistor technology has largely overcome the difficulties of close junction spacings and the addition of a p+-junction can considerably increase the versatility of the device so that some sacrifice in speed and increased cost may be justified. To illustrate the properties of a p+-n-n+ structure (of which the n+-junction is of small area) the

3O E

~' zo

l0

~v

0

I0

20 Volloge,

IJ~ v

30

40

50

v

F I o . 9. " T h 3 ~ a t r o n " characteristic. T h e effect o f m i n o r i t y carrier injection from a p-n junction on the trigger

characteristics of an avalanche diode, ( p - n junction currents indicated on diagram).

following arrangements may be quoted as examples: (i) "Thyratron" type characteristic. The n+avalanche dot is biased positive in the usual avalanche direction and the p+-junction is also biased positive. Injection of holes from the p+-junction increases the conductivity of the base region and hence decreases the trigger voltage of the avalanche diode. A typical static characteristic is shown in Fig. 9, each curve being an avalanche diode characteristic at a given value of p+-junction current. Clearly positive pulses applied to the p+-electrode will turn on the diode. The sensitivity change in V~, (V/mA) is such that appreciably lower power is required and the overall power gain is thereby increased. The unit may also be switched off by the application of a negative pulse to the p+-junction. (ii) "Transistor" characteristic. If the p+junction is operated as a transistor collector (negative bias) it will collect avalanche injected holes when the diode portion of the device is turned on. The avalanche dot then serves as an emitter with two stable states and the structure is similar to a transistor with a "built-in" memory. Experiments

AVALANCHE INJECTION with this type of structure have allowed the injection efficiency of an injecting avalanche to be estimated. The ratio of hole current to total current is found to have a maximum value of 20-30 per cent at low "on" currents and to decrease slowly with increasing current. (iii) High current-gain transistor. If the avalanche dot is biased negatively to collect holes injected by the p+-junction, the structure is essentially the same as that described by GR~a~rVXLLE(s),namely a transistor with high current gain. As shown by GR~,NVILLE, however, the minority carrier accumulation at the n+-collector, which is the origin of the current gain, also severely limits the frequency response of this arrangement. 7. S U M M A R Y A N D CONCLUSIONS W e have demonstrated the feasibility of making a

two-terminal negative-resistance device employing the avalanche injection effect, and indicated how two-state circuits may be constructed using this device. The merits of avalanche injection diodes are as follows: (i) Simplicity of fabrication. The properties of only one junction, made by conventional alloying techniques, need be controlled, and a small spacing between two junctions is not required for highspeed operation. This may be compared with a p-n-p-n diode where three junctions and two spacings between junctions are important. Simplicity of fabrication implies high reproducibility and it should be noted that the histogram given in Fig. 5 does not indicate the reproducibility but merely the degree of agreement between a simplified theory and experiment. (ii) High speed. Though the lowest turn-on times measured by us are about 3 m/~sec, this figure is believed to be determined by the test equipment. Estimates by Gt~vN(1) indicate turn-on times of about 3 mt~sec. Theoretically a turn-on time of the order of yo/vs (Section 4) would be expected, which is appreciably less than 1 m/~sec.

DIODES

69

(iii) Independence of temperature. This feature, discussed in Section 4, arises essentially from the absence of p--n junctions in the device. The high limiting temperature for germanium (--~ 140°C) and even higher value for silicon (N350°C) allows very high power dissipation if an efficient heat sink is used. The chief disadvantage of an avalanche injection diode, when compared with a p-n-p-n, is the high sustaining voltage. Fortunately the insensitivity of the device to temperature allows a high power dissipation in the on state and the power drain from the supplies is independent of Vs, so this feature is not unduly serious. Furthermore, Va is so reproducible from unit to unit that it can, in some circuits, be "backed off" by a biased p--n junction diode. Acknowledgements--For the supply of germanium and silicon crystals we are indebted to Mr. P. J. HOYLAND and Miss J. HUTCHEONrespectively. Mr. J. L. CLARKE made a large number of the diodes and triodes, particularly silicon units, and Mr. A. C. BAYNI-IAMdid much of the circuit work. The paper is published by permission of the Controller, H.M. Stationery Office.

REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

J. B. GUNN,Proc. Phys. Soc. Lond. B66, 781 (1956). E. J. RYDER,Phys. Rev. 90, 766 (1953). J. B. GtrNN,J. Electronics 2, 88 (1956). A. F. GIBSONand J. R. MORGAN,Brit. Pat. Appl. 29800/57 (1957). D. E. MASONand D. F. TAYLOR,Progr. Semiconductors 3, 85 (1958). A. F. GIBSONand J. W. GRANVILLE,J. Electronics 2, 259 (1956). M.A. TOWNSEND,Bell Syst. Tech. J. 36, 755 (1957). J. B. Gtrr~, Progr. Semiconductors 2, 213 (1957). K. G. McKAY, Phys. Rev. 94, 877 (1954). S. L. MILLaR,Phys. Rev. 99, 1234 (1955). W. SHOCK~Y,Bell Syst. Tech. J. 30, 990 (1951). C. HEARING,Bell Syst. Tech. J. 34, 237 (1955).