Solid-State Electronics, 1975, Vol. 18, pp. 1123-1130. Pergamon Press. Printed in Great Britain

THE EFFECTS OF CARRIER ACCUMULATION AT THE CATHODE ON THE NEGATIVE RESISTANCE INDUCED

BY AVALANCHE INJECTION IN Si BULK DEVICESt

GIANFRANCO VITALE Istituto Elettrotecnico, Facolth di Ingegneria, via Claudio 21,80125 Napoli, Italy

(Received 26 November 1974; in revised form 28 April 1975)

Abstract--An analytical model for the negative resistance induced by avalanche injection in bulk semiconductors is developed including diffusion, recombination and accounting for the properties of the majority carrier injecting electrode with respect to the avalanche generated carriers.

Two limiting situations are discussed in detail namely that of a n +-n cathode which is blocking for the avalanche generated holes and that of a metal-semiconductor ohmic contact with an infinite surface recombination rate.

The calculations show that in the first case the avalanche ionization is extremely low and the negative resistance occurs because a low-field neutral region creates into the solid thus reducing the voltage; in the other case, the multiplication is relatively high and the negative resistance is due to the lowering of the field in the region of scattering-limited velocity. As a consequence, from the first to the second situation the peak voltage changes by a factor of two and the peak current by a factor of three or more.

These results give a new insight of a number of experimental observations reported in the literature. The performance of devices made with different techniques which approximate one or the other limiting situations mentioned above is accurately predicted by the proposed model.

NOTATION

b i~p/tx, /9,, (Dp) electron (hole) diffusion coefficient (cm2/s)

E electric field (V/cm) E,, E2, E3, EL electric field at x = Xl, x2, x3, L respectively

E, E(v, =v~) g generation rate density (cm -3 s-') J total current density (A]cm 2)

J~, qvsNo J,, (Jo) electron (hole) current density

:.o :o(x = o) L , lp(x = x,) K Boltzmann's constant L cathode-anode spacing (cm)

Lo [2V:gp/(l + b)] ~:2 ambipolar diffusion length L. [D:o] ':2 minority carrier diffusion length

M. multiplication factor (for ionization started by electrons)

ND density of donor levels (cm ~) n electron free carrier density p hole free carrier density

po p(x =0) q absolute value of the electronic charge

(Coulomb) r recombination rate density (cm-3s ')

spo (dp/dxL=o T absolute temperature (°K) V external voltage (volt)

v. (vp) electron (hole) drift velocity (cm/s) v~ electron and hole scattering-limited velocity

Vr KT/q (volt) x space coordinate (cm)

x~, x2, x3 boundary of the first, second, third region a electron ionization rate (cm ') /3 hole ionization rate E permittivity of sample (Farad/cm)

g. (gp) electron (hole) drift mobility (cm2/V/s) cr surface recombination velocity (cm/s) r lifetime at high free carrier levels (s)

rp hole lifetime

~'This work was supported by C.N.R.-Roma, Italy.

I. INTRODUCTION

The term "Avalanche Injection" indicates the avalanche multiplication which takes place in bulk semiconductors due to the high electric field which is produced by the space-charge subsequent to the injection of majority carriers from a contact. The combined effects of injection of majority carriers and internal generation of minority carriers leads to a current instability which was studied in the past years either to explain the current-induced second breakdown in transistors[I,2] or to obtain relaxation oscillators [3-6] or for microwave generation. In fact it has been demonstrated that avalanche injection can lead to transit-time oscillations [7, 8] which have been observed to occur in Silicon devices [9].

Recently[10], a detailed model of the avalanche injection effect has been developed, allowing to quantita- tively account for the relevant features of the I, V characteristics of properly realized devices.

In this work it will be demonstrated that a current instability in a bulk semiconductor terminated by ohmic contacts, may arise due to two different mechanisms which lead to different properties of the external I, V characteristic.

For the sake of definiteness, let's suppose to deal with an n-type semiconductor sample and that the electric field is high enough to give rise to avalanche ionization; the avalanche generated holes travel toward the cathode where one of the following limiting situations may occur:

(i) The cathode has an high capability for absorbing holes (e.g. through an high surface recombination rate) so that any hole density generated by avalanche ionization is absorbed at the cathode. In this case, the negative resistance occurs because the solid changes from single carrier to the two carrier regime with decrease of its resistance.

(ii) The cathode is blocking for holes (it has a very high

1123

1124 G. VITALE

injection ratio and the recombination is small) so the holes created by impact ionization accumulate near the cathode leading to a conductivity modulation region which is almost neutral and has very low field. The negative resistance occurs because this region spreads into the solid to reach an equilibrium between the hole current generated by impact ionization and that coming out through the cathode, while the current is still carried essentially by electrons.

In practice, a cathode of the first kind is well approximated by a metal-semiconductor ohmic contact with an high density of traps at the interface while a cathode of the second kind may be supposed to be an abrupt n +n transition, the n + side being highly doped and having recombination properties similar to those of the semiconductor bulk; both techniques have been used in the past to obtain the emitter electrode of these devices.

In the following sections, these two limiting situations are discussed in detail using the Regional Approximation Method which allows to solve the problem analytically and helps to clarify the involved physical phenomena.

Results are then compared to a number of experimental data obtained on devices made with different techniques.

2. ANALYSIS

The device considered heretofore is made of a bulk n-type semiconductor of constant doping No, bounded by plane-parallel ohmic contacts set at a distance L; the cathode is taken as the origin of the space coordinates x. The basic equations, written for one-dimensional geometry are:

J=J. + L (1)

J,, = qv.n - qD. dn/dx (2)

Jp = qvpp + qD~ dp/dx (3)

1 dJ. q d--x- = g - r (4)

_ 1 dJp = g _ r (5) q dx

edE - n - N D - p ( 6 )

q dx

where g is the generation term due to the avalanche ionization

g = l ( c d . +/3L) (7)

r is a recombination term in the bulk of the semiconduc- tor, the other symbols have their usual meaning. In this section, an approximate solution of (1)+(6) is derived under the following assumptions: (i) The value E, of the electric field for which the electron velocity saturates is much less than the minimum field for the onset of the avalanche ionization. (if) Diffusion terms are sizable only in the region where the carrier velocity v =/xE; hence the diffusion coefficients are calculated as D =/zV~; (VT ~- KT/q). (iii) The hole injection from the anode is neglected. (iv) Currents are well below the cathode

saturation current. (v) We shall consider only relatively "short" samples i.e. the cathode-anode spacing is within a few ambipolar diffusion lengths Lo. (vi) A finite charge density p0 = p (x = 0) is needed to sustain the hole current exiting through the cathode.

In order to illustrate the subdivision of the solid into the various regions, let's consider the qualitative sketch of the electric field in the two carrier regime, Fig. 1, which is appropriate for a cathode which blocks egress of minority carriers.

This plot is justified by the following arguments: in the neighborhood of the cathode holes pile up to sustain the hole current through the cathode and are almost completely neutralized by the electrons the result being a conductivity modulation region with a low and slowly varying field and a diffusion dominated current. Thus the carrier velocity is v =/.rE and there is no carrier generation. The presence of holes near the cathode implies that they have been generated somewhere else into the solid by impact ionization. Thus the field, which increases toward the anode, must largely exceed the saturation value Es. Starting from the cathode we distinguish a first region characterized by charge neutral- ity, the second region characterized by a "tiepid" electron regime; the third region with scattering-limited velocity but with negligible avalanche ionization which in turn dominates the fourth region, near the anode. In detail, the four regions are characterized as follows:

Region 1 (O<- x <- x,) Electrons are injected from the left side and holes from

the right side into this volume leading to a situation similar to that of a classical double injection problem. The common approach to this class of problems consists to combine linearly (5) and (6) and to use (2) and (3) to eliminate J. and Jr,:

V, dd-~x ( p + n ) - q ~x (E d E / d x ) - ND dE/dx

= r(b + 1)]/zp (8)

where b =/zp//x,. The solution to (8) is in general extremely complicated; however, it has been demon- strated by Shilling and Lampert[ll] on the basis of the

I v. = ~.E f(E) - - v, - - I - -

dE = O n-p,-N D n~-p-N D Mn.-N D q dx [n--p) I

I 0 x x 2 x, L

region 1 I

Fig. 1. Schematic representation of the electric field distribution and regional approximation diagram.

Negative resistance in Si bulk devices 1125

numerical solutions obtained by Baron, that the second term in (8) can be safely neglected; furthermore in the vicinity of the injecting contact the electric field varies slowly due to the quasi neutrality condition accompanying the hole pile up so the Poisson equation may be replaced by:

integrated with boundary conditions set at the cathode and at the plane x = x~; the latter condition depends on the solutions of the other regions that finally must obey the following condition at the anode:

p(x = L) = 0 (15)

n - p - N o =0

with these assumptions (8) becomes,

(9) therefore it seems more convenient for calculations to solve (14) with the following couple of conditions set at the cathode x = 0:

2 VT~--~x2 d2p- No dE=dx r(b + 1)/p,p

which has to be coupled to the current equation

(10)

J = qlx, E [No + (b + 1)p ] + q Vrp~n (b - 1 ) dp/dx. (11)

However, as pointed out in[12] the diffusion term in (11) is important only for calculating the field itself while it can be neglected When eliminating E from (10). Therefore we shall use the following

p(O) = po (16a)

dpl _ (1 + b)Jpo - bJ -~x [ x~o = spo - 2qD, (16b)

The latter condition is obtained by eliminating the field between (3) and (11) and by defining Joo = Jo(x = 0). This condition is completely specified through the structure of the cathode which allows to relate the current Jpo to the charge density po at x = 0. The correct value of po is the one leading to a complete solution that satisfies (15).

The solution to (14) with the boundary conditions (16) is:

J = ql~oE[No +(b + 1)p] (12)

to obtain from (10) an equation for p(x) whose solution is then inserted in (11) to calculate the field distribution E(x).

However, in contrast with the behaviour of a plasma injected by the electrodes, the neutrality condition (9) does not hold all along the solid because the double injection regime requires also the existence of a space-charge region with saturated drift velocity; thus the first region is terminated at the plane x = x~ where the neutrality condition (9) is no longer self-consistent, in particular we have chosen the following condition to define the value of x~

q--~x E ..... = p(x,). (13)

The solution to (10) + (12) is carried out for two current regimes namely:

Regime (la). The high current regime characterized by po > No ; this situation probably leads to a large volume of this region so that the recombination process has to be taken into account.

Regime (lb). The low current regime, with po < No, in which the terms involving No must be retained while the volume of this region is quite small thus allowing to neglect the recombination term.

Regime (la). In the high current regime, (9) reduces to n = p, and (10) becomes:

o-Dd 2 :2 Vr ~-~. ~ = r(b + 1)l/x, (14)

l a x

and the recombination process can be described as r = p[z where r is the common value of the lifetime in the limit of high free carrier density. This equation should be

p(x)=po cosh(x/La)+spoLa sinh(x/L,). (17)

Where L, =[2V~rl~p/(l+b)] 'n. The electric field is obtained from (11):

1 F J VT E ( b + l ) p ( x ) [ - ~ - - ~ ( b - 1 )

[p° sinh (x/La) + s,oLa cosh (x/L,)]]. (18)

Substitution of (17) and (18) in (13) leads to an equation for x whose solution is the boundary of the first region xt.

If the solid is short with respect to L~ the above solution simplifies greatly because for x~ ~ La (17) and (18) become:

P = po + SooX (19)

E = [J/qtz, - Vr(b - l)spo]/[(b + l)(po + SooX)] (20)

which are independent of the recombination term; the condition (13) leads to:

x, = [eVr(b - l)[qspo -EJ/s~oq2tz,] 't3-po[spo. (21)

This extremely simple solution gives a clear picture of the phenomena which occur in this region; in fact the hole density decays linearly (spo < 0) leading to a constant diffusion current; the drift current must overcome the diffusion current, the difference being the current exiting through the cathode; this drift current is sustained by an electric field which increases as an hyperbola with its asymptote located at Y,=-po/S~o where the charge density goes to zero. The solution is truncated slightly on the left of £ in order to allow the correct connection with the following region.

Regime (lb). In the low current regime, the recombina-

1126

tion term is neglected in (10); this leads to the following first order equation:

dp N o E / 2 V r = so,, (22) dx

by eliminating the field using (12) and by defining

p* = ND +(b + 1)p

p * = Nr, +(b + l)p,,

n,,= N e ' / ( b +b 1Jpoj l )

G. VITALE

where the velocity is saturated only over a small fraction of the volume and the avalanche ionization is either absent or very low. At higher currents, this region affects only slightly the J-V external characteristic so in either case assumption (iv) leads to small errors. With the above assumptions, (2) (3) and (6) are combined to give:

dE e~x = ( J - Jp~)(l/tx, E + l ive , ) - qNo -J~,/vs. (27)

The boundary condition comes from the requirement of continuity of the electric field across the plane x = xj, E(x~ +) = E(x~-) = E,. By defining

and using the boundary condition (16a), one obtains:

p * - p * - no ln [(p*/no + l)/(p*/n,, + 1)1 = sp,,x(b + 1). (23)

The electric field is obtained from (11):

(24)

Application of the condition (13) leads to an equation for p* whose solution is p* =p*(x0; this quantity is then inserted in (23) and (24) to obtain x, and E, = E(xO respectively.

With these solutions we can describe the whole range of currents provided the device length L is within a few Lo (assumption v). It may exist also a range of medium currents where the solutions (la) and (lb) are equivalent and in practice the simplified equations (19)+ (21) hold,

At the boundary x, of the first region, the hole current density takes the value,

Jp(x = x ~ ) = Jp~ = Jp,, + qp°L"sinh(x,[L°) Y

2 + qsp,,L ° [cosh ( x , / L o ) - 1] (25)

7

in the (lb) regime, of course, L.~ = J~o.

J~r = qv~No

E* = v~ltx,

Vs (L, + Jo~)/(I- L~)

s, = (J - J,s)/e.g,E*

the solution to (27) is:

E - E~ - E* In [(1 + E/E*)/(1 +Et /E*)] = se(x - xO. (28)

This region is terminated at the plane x = x2 where the velocity saturates: E(x2)= Es.

Region 3 (x2 <- x <- x3) Electron and hole velocities are both saturated but the

carrier generation can be neglected when calculating the electric field, so the linear solution

E = E~ + (J - 2Jps - L,)(x - Xz)/,Evs (29)

obtains. The third region is terminated where the avalanche generated carriers can no longer be neglected in the Poisson equation. The boundary x3 is calculated by equating the total generated carrier density to the net carrier density accounted for in this region:

fo x3 2 g dx = (J - 2J~,s - L,)lqs. (30)

Region 2 (x, <- x <- x2) This is a transition region from the "cold" electron

regime of the first region to the "hot" electron regime of the following regions. The basic assumptions are: (i) Diffusion terms can be neglected. (ii) Generation and recombination terms are neglected so that Jp takes the constant value Jp~ as defined by (25). (iii) The electron velocity is related to the field through:

l /v, = lllx,E + llv.~. ; E < Es (26)

Now, it has been demonstrated [13] that in the limit of low multiplications, the integral on the left-hand side of (30) can be approximately solved in closed form in terms of the electron ionization rate a = a~. exp ( - b . /E ) yielding,

f0 x3 JO~ 2 g dx = S--~b E3 exp ( -b , /E3 ) (31)

where S is the slope of the field and E3 = E(x3). By approximating S = E 3 / ( L - xl) (30) and (31) yield

where Es is the field for which the velocity takes the scattering limited value vs, v~, is an adjustable parameter which is chosen to obtain v,,(E~)= vs. (iv) In the Poisson equation the excess hole density is neglected with respect to the density p, = J,,,/qvs entering from the third region.

This region is of importance only at low current values

E3 = b,,/ln (J / L ) (32)

where Jo = ~vsb,,/2a=(L - xl) 2.

Region 4 (x3 <- x <- L ) Avalanche ionization dominates this region, causing the

Negative resistance in Si bulk devices 1127

slope of the field to increase toward the anode. On the basis of the numerical solutions obtained in [10], the field has been linearized with a slope having the correct value at x =L. By defining E~ = E ( L ) , this approximation yields

tion centers thus leading to a vanishingly small value of

po. (2) The cathode is made of an abrupt n+n transition. Intermediate situations may be roughly described by

defining a surface recombination velocity cr as the ratio:

EL :: E3 + (L - x3)(J - J~r)/~.t)s. (33) o- = Jpo/qpo (36)

If the ionization is very low, this region may fall outside of the solid; however this causes the slopes of the third and fourth regions to coincide because Jp ~ J,, = J. Thus in either case the multiplication factor M, may be calculated according to the expression found in[13] valid for a ionization started by electrons and in the range of low multiplications:

1 M,1 ( J - ~ b , EL2exp(-b, /EL) (34)

independently on whether or not the fourth region or only a part of the third region fall within the solid.

It should be noted that in this problem one must also account for extremely low multiplications because even though the resulting hole current is a small fraction of the total current, its effect may be relevant in the vicinity of the cathode if it is blocking for holes.

The total generated hole current is:

J~= J( l - 1/M,). (35)

tr increases as the cathode is more effective in absorbing holes.

Application of the above extreme situations to the model of Section 2, leads in the former case (labelled as ~r--, oo) to the disappearance of the first region since the condition (13) is met at x =0. Then calculations are carried out starting from the second region, with p = ps, and the hole density will fall to zero within a very narrow volume close to the cathode. The solution so obtained is very similar to that developed in [10], differing from it only for a better description of the velocity-field curve leading to more accurate results in the low currents range of the J-V characteristic.

In the case of the ideal n ÷n transition, we can assume that the current is purely diffusive inside the n ÷ material and is continuous across the x = 0 plane. The value of Jp,, is obtained from the well-known expression of Jp inside the n ÷ material:

Jp = q~(p(O-) exp (x/L,,) (37)

In general this quantity will not coincide with the current Jps that we have supposed to enter the second region, unless the value of p,, is exactly the one needed to obtain a solution consistent with the boundary condition at the anode (15). Hence the problem must be solved iteratively for different po values until the equality Jps = J~ is met.

The total voltage is then calculated by integrating the appropriate expressions of the electric field in each region, accounting only for regions or part of them, which fall within the solid.

3. THE INFLUENCE OF THE INJECTING CONTACT

AS pointed out in Section 1, the negative resistance may be induced by different phenomena depending on the ability of the majority-carrier injecting contact to absorb the carriers generated by impact ionization. With refer- ence to n-type material, this feature of the electron- injecting cathode is described by relating the hole density po to the hole current Jpo at x = 0.

Independently of the detailed phenomena which occur at the cathode, po will be small if holes are effectively absorbed by the contacts and this leads to a small volume of the first region; if oppositely the hole absorption is poor, a larger po value is needed to sustain a given hole current; region 1 will be large and the phenomenon is dominated by conductivity modulation.

In order to express these behaviors in a quantitative way, we discuss the operation of two kinds of cathode which are representative of fairly extreme situations:

(1) The cathode is characterized by a very large density of defect states, located at x = 0 and acting as recombina-

and remembering that at the n+n transition the Boltzmann condition p2(0+)= p(0-)n(0 ) holds, we ob- tain at x = 0 ÷, where p (0 ÷) = po,

Jpo = qDppo2/L,,n (0-). (38)

In practice, the value of n(0-) will coincide with the value of the doping concentration No + of the n ÷ material.

The influence of the cathode termination is apparent from the plots of the electric field obtained by the model of Section 2 and reported on Fig. 2. The calculations have been carried out for n +n cathode (2a); o- = I0 ~ cm/sec (2b) and tr-~ c, (2c). The parameters used for calculation are listed on Table 1 and correspond to Silicon; for n+n the doping concentration of the n + layer was No ÷ = 1019 c m -3.

On each curve, circles indicate the boundaries of the regions in which the solid has been divided.

From the general features of these figures it can be noted that increasing current causes the volume of the first region to increase but this region enters the solid at higher currents as the recombination is higher.

Furthermore, in the situations of Fig. 2a and 2c, the

Table 1.

U n = 1500 cm2/V/s a = 3.80 × 106 c m - I

Up = 600 em2/V/s b n = 1.75 x 106 V/cm

L = 10 -3 cm P

E s = 2-5 × i0 ~ V/cm

u 8 = l.O × 107 em/s

UBn= 1.37x 107 cm/s T = i0 -s s

1128 G. V~TALE

~E2 .o >

w" 1

l I I I I ]

J= 2.5.103A.cm-21 3 ~ 4 .Q ,

2 4 6 8 10 X, pm

J n.......~s

J

1

0.5

I 10 3

i i I I

n + n o, ~ cr= 10 5

I I I I 104 j, A.cm -2 105

Fig. 3. Electron current density in the saturated drift velocib. region, vs the total current for different surface recombination

rates ~r. The circle indicates, on each curve, the peak current.

~E2 tO

t.dl

j°3.103~ A.cm -2 4.10~.~

2 4 6 8 10 X, pm

I I I I ~

"zE 2 J = 4"103A'cm-2

,,,- 1

I I I I 2 4 6 8 10

X, pm

Fig. 2. Field distribution for different current densities: (a) n +n : (b) ~r = 10 ~ cm/s; (c) ~--*:c. Circles indicate on each curve the

boundariesofthevariousregions. No = 10 '~ cm 3.

negative resistance is due to two different phenomena; in the former case, n-'n cathode, the hole accumulation is large, the phenomenon is dominated by conductivity modulation and the negative resistance occurs because the first region widens rapidly as current increases. In the latter case, cr-*o% the voltage is reduced because the conductivity of the third region increases due to the holes which are injected into this region by the multiplication region. This is due to the different ionization regime which occurs at the peak point in these two situations as shown in Fig. 3 where it is plotted how the current shares between electrons and holes in the region of saturated drift velocity. In fact near the peak value (indicated by a

circle on each curve) the current is carried almost completely by electrons for the n +n cathode while it is carried by 26% by holes if o - ~ . This leads to a decreasing slope of the electric field in third region as current tends to share itself equally between the two carriers (~-~ m) while this slope is still increasing at the onset of negative resistance for the n+n case.

Therefore the main effect of a cathode which blocks the egress of minority carriers is to reduce the avalanche ionization rate to very low values so that the negative resistance is no longer controlled by the third region but by the first one.

The J-V characteristics, Fig. 4, show that the range of low currents where ionization is absent, is not affected by the structure of the cathode, while the peak voltage doubles and the peak current is three times higher in passing from one limiting situation to the other.

The peak voltage vs the device length is reported on Fig. 5a; it is seen that both limiting models lead to an almost linear increase of the peak voltage with L, with slope of - 1 0 V/micron in the limit of infinite cathode recombination rate (dashed lines) and - 5 V/micron for the n +n cathode (solid lines).

For a fixed device length these models behave

10

8 (2"=

o

2

1 5O

V,V

\

I

100 150

Fig. 4. Current-voltage curves for different ~ and n ~n cathode. N,, = 10 '~ cm 3, L = l0/xm.

300

>

d

200 -5 >

O..

100

Negative resistance in Si bulk devices

, >

/ / / /

ND= I0~ / /

N ~ O _ ~ / / ~ ' ~ v

' 2' ' 10 0 30 L, [am

' ' ' ' ' ' , 1 ' ' ' ' / / /

10 4 ~ / / / •

_ - c c 2 o . - ~ - / / / /

20 ~ i = 3 0 , , , , l l , , , , , , , ,

1-101403 2 5 10 TM 2 5 10 TM

No, cm -3

Fig. 5. (a) peak voltage vs device length. (h) peak current vs doping concentration. ( ), n+n; (- . . . . . ) o ~ . On Fig. (b), the vertical line across each experimental point indicates by its

extreme the corresponding theoretical value.

differently with respect to the doping concentration in that as tr ~ the peak voltage decreases as doping increases while the opposite dependence occurs for n +n. Therefore the curves corresponding to N o = 0 are also the absolute limits of the peak voltage for this semiconductor. In either case, the peak voltage is weakly dependent on doping.

Table 2.

N D L Peak voltage

NO, Symbol i015cm-3 ~m theory n IExp. theory o~

: i 17 90 145 192 - - /', o.a 2 24.5 132 i92 26i

3 & . i.~ 6.8 3 5 79 86

4 io. 5 56 86 i23

5 ~7 i. 65 20 iii 162 2 i 3

6 30 171 P44 301

7 • 5.0 11.5 66 76 127

1 1 2 9

The peak currents Fig. 5b, show the same functional dependence on length and doping in both the limiting models, while the numerical values are very different.

4 . E X P E R I M E N T A L

In this section we compare the model developed in Sections 2 and 3 to some experimental data obtained on devices made with different techniques and reported in the literature existing on this subject.

The check is carried out in terms of peak voltage and current which have been demonstrated to be closely related to the different modes of operation of those devices and then are the most significant parameters to compare with the theoretical values. In fact it should be noted that the part of the I-V characteristic above the peak point cannot be correctly analyzed in terms of the above developed one-dimensional model; this is due to current filaments which are created in these bulk structures above the peak point, leading to substantial modifications of the static I-V characteristic [14].

The data reported on Fig. 5a and b, for comparison to the theoretical curves, are identified on Table 2 where the theoretical values are also reported; these data corres- pond to two groups of silicon n-type devices namely:

(a) Devices with the cathode obtained by evaporation of an Au-Sb alloy directly on the n material followed by alloying process. This technique leads to an ohmic contact very close to the surface of the crystal so it is likely that there is a large number of lattice defects; therefore those contacts should behave as a recombining surface with very large ~. These data are taken from reference [i0] and indicated by triangles on Fig. 5.

(b) In these devices, the cathode is obtained by forming an n * layer by standard phosphorus diffusion technique so the cathode is the n * - n transition which is located inside the crystal where the density of the recombination centers is of the same order as in the bulk of the n material. These data are taken from Ref. [1]; they are indicated by squares on Fig. 5.

] Peak current(303A cm -2)

thoory ~'4 E~Pl~h°°rY o~ 2 . 0 I~.n 4.4

1.8 2 .5 3 .4 ~-

3,6 9.7 11.o

3.8 6.4 8.5

3.2 4.3 5-7

3 ,0 3,7 4.7

9.1 9.0 14.i

8 6o -- • 0,7 i0 51 9 63

i0 i0 53 82 -- [] 2 ii ii 59 120

12 • 2.5 5. t 28 50

13 <> 3.5 12. ~ 72 130

? ~ 7 9 52 86

%5 121 2.3 7.0

2.9 @ o

118 4.31 h.9 9.5

F 128 h.3o ~.7 8.9 v

69 6.2 9.8 16.1

141 6.6 5-5 10.8

102 12.7 12.6 19.2

1130 G. VITALE

All these devices are obtained from epitaxial material, the substrate acting as the anode.

From the plot of peak voltage vs device length Fig. 5a, we note that all the experimental points fall within the curves corresponding to the two limiting models; moreover some points from group (b) devices are located near the lower limit curve and, except one case, all these devices meet the requirement of increasing the peak voltage with increasing doping concentration, at fixed length.

On the contrary, most data from group (a) tend toward the upper limit curve thus indicating a fairly high surface recombination rate at the cathode. The experimental peak voltage of those devices decreases with increasing doping as was predicted by the theory appropriate to this group of devices.

On the plot of peak current vs doping concentration, Fig. 5b, it has been reported one experimental point for each doping concentration; because the theoretical curves corresponding to different lengths are quite spread over the plot, each experimental data has been connected by a vertical line to the appropriate theoretical point i.e. that corresponding to or ~oo for group (a) devices (triangles) and that corresponding to n+n for group (b) devices (squares). The agreement is excellent in most cases thus confirming that the discrepancy of the experimental results obtained by different authors is essentially due to the different techniques used to make the devices.

5. CONCLUSIONS

It has been demonstrated that the negative resistance induced by avalanche injection in bulk semiconductors may be caused by two rather different phenomena according to the properties of the contact which injects the primary carriers; these are: conductivity modulation near the cathode which dominates if the egress of minority carriers is blocked, and space-charge reduction in the region of saturated drift velocity if the carrier are effectively absorbed by the cathode.

Near the turnover point the current can either be carried essentially by electrons either be divided

between electrons and holes; these two effects lead to different shapes in the I-V curve because if the instability is due to conductivity modulation the electric field is lower and then a lower peak voltage is reached; in this case, the field shape causes the carriers to travel at saturated drift velocity only over a small fraction of the solid.

The properties of the cathode, which injects the primary carriers and must absorb the holes generated by avalanche ionization, will control both the amount of conductivity modulation near the cathode and the amount of avalanche ionization thus determining which phenome- non will dominate.

Therefore rather different dynamic properties are to be expected from these devices, according to the techniques used to make the contacts, and this could explain the problems which arise in the experimental work on those devices to obtain transit-time oscillations[9]. In either case, results show that the structure of the cathode must be carefully accounted for in both the theoretical and experimental work on this class of phenomena.

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