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The static and dynamic properties of the avalanche injection diode
Solid-State
Electronics
Pergamon
Press 1963. Vol. 6, pp. 631-643.
Printed in Great Britain
THE STATIC AND DYNAMIC PROPERTIES OF THE AVALANCHE INJECTION DIODE* J. R. SZEDONt Westinghouse
Electric
Corporation
Research
Laboratory,
Pittsburgh,
Pennsylvania
and A. G. JORDAN Department
of Electrical Engineering, Carnegie Institute Pittsburgh 13, Pennsylvania
(Received
22 February
1963
of Technology,
; in revised form 31 May 1963)
Abstract-Since the proposal of the avalanche injection diode, AID, by Gunn, it has become necessary to incorporate fundamental knowledge regarding collision multiplications and other high field, high current density effects into models that approximate actual devices. Previous approaches to the problem have been inadequate for all but a qualitative appreciation of the behavior of the device. This investigation has produced a more comprehensive model for the AID; one that encompasses to some extent both high and low electric field effects, and explains the observed, low-level dynamic response of the device in terms of conductivity modulation effects. The steps leading to a numerical evaluation of the high frequency behavior of the device in terms of geometric and material parameters have been developed and have been used to explain the characteristics of typical units. R&urn&-Depuis que la proposition concernant la diode d’injection AID B avalanche a et4 faite par Gunn, il est devenu necessiare d’incorporer des connaissances fondamentales ayant trait aux multiplications de collision et autres champs Clev& et aux effets de courant a haute densitt dans les modhles qui sont presque equivalents aux dispositifs actuels. De prC&dentes approches au probleme ont 6th insuffisantes en tout sauf pour l’appr&iation qualitative du comportement du dispositif. Cet examen a produit un modele plus d&ail16 du AID; il comprend jusqu’8 une certaine mesure les effets de champs Clectriques Clev& et bas, et explique la response dynamique observCe B bas niveau du dispositif en termes de la modulation de conductivitC. Les &apes menant B une Cvaluation numCrique du comportement du dispositif B haute frkquence en termes des parametres geometriques et du matCriau ont Ctt5 employees pour expliquer les CaractCristiques d’C16ments typiques. Zusammenfassung-Seit der Einfiihrung der Lawinen-Injektions-Diode (AID) durch Gunn wurde es erforderlich, grundlegende Tatsachen fiber die Vervielfachung durch ZusammenstGsse und andere Effekte eines starken Feldes und einer hohen Stromdichte in die Modelle aufzunehmen, die zur angeniherten Darstellung der wirklichen Gerlte dienen. Frtieren Versuchen in dieser Richtung gelang es nur, eine qualitative Darstellung vom Verhalten des Geriites zu geben. Die vorliegenden Untersuchungen haben ien besseres Model1 fiir die AID geschaffen; es umfasst in gewissem Ausmass die Effekte starker und schwacher elektrischer Felder und erklart die beobachtete schwache dynamische Reaktion des Gerates durch Modulationseffekte der Leitfshigkeit. Die Schritte, die zu einer numerischen Auswertung des Hochfrequenzverhaltens des Gergtes auf Grund der geometrischen und materiellen Parameter ftiren, wurden durchgefiihrt und dann dazu benutzt, die Eigenschaften typischer Gerlte zu erkliiren.
* This paper is a portion of a thesis submitted in partial fulfillment of the Ph. D. requirement in Electrical Engineering at Carnegie Institute of Technology.
7 Formerly in the Department of Electrical Engineering at Carnegie Institute of Technology, Pittsburgh 13, Pennsylvania. 631
632
J.
R. SZEDON
and
1. INTRODUCTION
form the avalanche injection diode,(l) AID, consists of a relatively lightly doped semiconductor with contacts which permit the establishment of high magnitude electric fields in the bulk at low carrier injection levels. In the pre-avalanche mode, also referred to as the off state, the device is characterized by a high incremental resistance. When the field magnitude is sufficiently high over a carrier path length, collision ionization occurs. Significant avalanche multiplication of carriers then permits sustaining IN
ITS
simplest
G.
JORDAN
in fabricating devices for the present investigation.@) The voltage-current characteristics of typical devices are given in Fig. 1. In most cases, negative incremental resistance in the on-state has been observed only at low values of current. Previous work on the properties of the device has been in two general areas. In the first, the prebreakdown analysis has been performed using an idealized hemispherical geometry and the assumption that the drift transport mechanism can be characterized by a constant mobility for field magnitudes less than a critical value E c, and by a
30
30
25
25 20
4
40
20
To-
30
I5
c .-
20
E 2 IO 7 UO
A.
15 I
IO I
5 0
5
IO
IS
20
25
30
5
IO
I5
Voltage, (A) FIG. 1. Current-voltage
5
\
0 L_j 0
20
25
0
0
5
IO
15
20
25
V (Cl
(61 curves of typical injection diodes.
point-contact
avalanche
large currents at voltages ordinarily lower than the constant drift velocity for field magnitudes in maximum, or trigger, voltage in the off state. excess of E,.(3) While the approach appears to The device operates in the sustained avalanche be correct, even though objection may be made to mode with a much lower incremental resistance the assumption of complete drift velocity saturathan in the pre-avalanche state. It has been tion in silicon,(5) the criterion for breakdown demonstrated that the voltage-current characterhas not been examined carefully, heretofore. In istic for a suitable avalanche region model should ‘the second area, the role of avalanche multiplicaexhibit a negative resistance.(l) Furthermore, with tion has been treated for a simplified model in avalanche the dominant process in the operation which collision ionization is assumed to occur of a device, the switching interval between the within a region supporting very high electric pre- and sustained avalanche modes should be fields. Recourse has been made to this model in on the order of the collision ionization time, i.e. explaining the low incremental resistance of the less than 1 x lo-12 sec.(a) Devices exhibiting sustained avalanche state; however, the theory sustaining mode negative resistance and having predicts a pronounced negative resistance which is circuit-limited switching times on the order of not observed. No concrete attempts to explain 10 nanosec have been fabricated.(s) Primarily as a this have been made to date. Transients in the result of this agreement, little additional work has AID have been treated meagerly. Switching been done on the original AID model which has between the stable state of the V-1 characteristic been inadequate for the prediction of the detailed has been reported to require less than 30 nanosec,(a) dynamic behavior of the device. but theoretical treatment of the dynamic behavior HOGARTH’S technique for making small area n+ of the device has been lacking. contacts to an n-type silicon wafer has been used It is the purpose of this work to present a
THE
STATIC
AND
DYNAMIC
PROPERTIES
comprehensive model for the AID; one that encompasses both high and low electric field effects; that includes conductivity modulation and that predicts the dynamic response of the diode, biased in the sustained avalanche state, to low level signals. Support of the theory with observed behavior will be presented where appropriate. 2. BEHAVIOR
UNDER
STATIC
CONDITIONS
A. The pre-avalanche state In general, the V-1 characteristic of the preavalanche state exhibits in incremental resistance of approximately 10,000 R, with breakdown occuring at a current of several milliamperes and a voltage ranging from 20 to 40 V. The breakdown is very abrupt, i.e. there is no noticeable decrease in incremental resistance which would be expected from carrier multiplication. This is reasonable in terms of SHOCKLEY’S model of isolated regions with locally high electric fields(s) since sustained avalanche could be initiated in a region which carried only a small fraction of the total current prior to breakdown. A theoretical treatment for the pre-breakdown state of the device has been given in the literature.(s) At currents less than a value i, given by ie = 2rrqpnNdECR$
(1)
the field magnitude nowhere exceeds EC, the value assumed for saturation of the electron drift velocity. A purely ohmic analysis is appropriate under these conditions. For currents in excess of i,, the ohmic treatment is valid beyond a critical radius, R e, given by equation (1) with R, and iC replaced by R, and i, respectively. In the high field region, the requirements of no minority carrier current and a divergenceless total current permit determination of the spatial dependence of the electric field. For currents in excess of i,, the voltage across the device is
OF THE
AVALANCHE
INJECTION
DIODE
633
given to the production of hole-electron pairs by collision ionization. The usual breakdown condition, modified for the high field region alone, i.e. RC
s
x(E)dr
= 1,
(3)
RP does not appear to be applicable in this situation, since the pre-avalanche model would become invalid long before it were to obtain. Were the integral to achieve a value as high as 0.5, the incremental resistance of the pre-avalanche characteristic would decrease noticeably prior to breakdown; but this behavior is not observed, indicating that breakdown occurs for values of the integral less than 0.5; perhaps as low as 0.2. Such a condition is not unreasonable, since local perturbations to the electric field by dislocations could easily give localized avalanche before its predicted existence according to the uniform treatment.@) The results of the numerical work are plotted in Fig. 2, in terms of the bulk, the SDV region and the total voltage drops as functions of current in the pre-avalanche mode. Maximum voltages 70
I
I
I
I
60
20
-
V = ECRC($-$)
. 0
(2) At electric field magnitudes on the order of 1 x 10s V/cm and higher, consideration must be
2
6 Currentf
6
mA
FIG. 2. Voltage-current characteristics for the saturated drift velocity and the bulk regions and for the AID as a whole in the me-dreakdown state.
634
J.
R.
SZEDON
and
ranging from 25 to 50 V are observed in actual devices. The corresponding breakdown current range in the theoretical results is 4-5 mA. It is important to determine whether or not such currents are compatible with the avalanche requirements just discussed. The ionization coefficient dependence on the field, for both holes and electrons is assumed to be of the form a(E)
= ai exp(XE),
(4)
an approximation to the empirical data which has been used in earlier work. The integration of the ionization coefficient over the SDV region has been performed numerically, with the result given as a function of current in Fig. 3. Included is a curve for the maximum electric field in the
A.
G.
JORDAN
voltage in excess of 80 V. From the arguments given above, an appropriate value for ai is approximately 50 cm-l if the magnitude of the integral for the idealized model is as low as O-2 when localized breakdown occurs. An interesting consequence of this reasoning is that the field magnitude at the contact is about 1.4 x 105 V/cm, a value in general agreement with the arbitrary 1 x 105 V/cm value used in earlier work. B. The sustained avalanche state The active avalanche region has been treated on the basis of GUNN’S planar model.(l) Assumptions inherent in the treatment are: (1) Drift velocities of holes and electrons are equal and independent of field. (2) Carrier diffusion and recombination are neglected. (3) The ionization coefficients for holes and electrons are equal, having a field dependence which has been expressed mathematically in equation (4). (4) The avalanche model is used only for electric field magnitudes greater than Ea, this value to be consistent with the adaptation of the model to a finite geometry. The extent of the region is then given by the expression
1
1.i"
nJh(l-[qn(0)/J]+qNdVd,/J}2 __-
KVS~ the electric
xln
exp(X&)
17 "'
(5)
field by
I l+tan2 \
and the majority
ScjTJA
--x)1 ] LC exp(~&))l”‘(r, 2
G
carrier
density
by
FIG. 3. Predicted behavior of the pre-avalanche maximum field and the ionization probability integral.
(7) For the particular parameters used in device. approximating the ionization coefficient data, the integral achieves the value of unity for a current of 9.7 mA which, from Fig. 2, corresponds to a total
The hole profile is given in an equation identical with the last one, except that the second and the third terms are negative.
THE
STATIC
AND
DYNAMIC
PROPERTIES
Integration of the electric field in determining the voltage across the planar region may be performed with the second term in equation (6) treated by the substitution
24 =
tan[Z(~~~‘2.exp~]
.(sa--s)].
(8)
The resulting integral has a current dependent upper limit, but over the range of currents (S-40 mA) which is appropriate, the integral itself is insensitive to current and equal to approximately 75 per cent of the value for an infinite upper limit. With this approach, the avalanche region voltage is given by the expression
A thought experiment leads to the conclusion that a large, but finite avalanching area would be unstable. It would tend to diminish in size under the action of field distortion around the periphery of the active region and of carrier multiplication within the region. The stable situation might be visualized as a thin disk having a thickness given by the extent of the planar avalanche model. GIBSON and MORGAN have suggested that diffusion effects would determine the radial extent of the avalanche region. Requirement of electron and hole currents to be individually zero across the . . cylmdrical walls yields an exponential carrier profile in the radial direction, having a characteristic dimension(s)
4
Dno MO E
VdS
Vd”-dsVds
ya=-=----
for
E > EC.
(10)
The zero subscripts denote low field values. On this basis, with a value for E of 1 x 10s V/cm, the value for ru is about an order of magnitude greater than that for xa. If the thickness of the chip, Rm, is large in comparison with pa, the field pattern in the bulk of the material will not be altered greatly if the cylinder is replaced by a hemisphere of radius, R,,, which preserves the current density. Carrier concentrations and the field magnitude at xa in the planar model will be imposed at R,, while the circuit properties of the avalanche region will be given by the voltagecurrent relationship of the planar model in
OF THE
AVALANCHE
INJECTION
DIODE
63.5
equation (9). The remainder of the device can then by analyzed, taking full advantage of spherical symmetry. The treatment of the saturated drift velocity region for the sustained avalanche case is quite similar to that under pre-breakdown conditions, except for the fact that both carrier species are assumed to support the flow of current, diffusion effects being ignored. Since the extent of the region is small in comparison with a minority carrier diffusion length, recombination is ignored. With additional requirements of zero divergence for both hole and electron currents, the carrier profiles are
(12) The carrier concentrations at R, are determined from equation (7) for the avalanche model. From Poisson’s equation, the electric field is given as function of radial distance by the expression (13) where (14) The outer boundary of this region is at the radius RL, for which the electric field falls to the value E e, below which normal mobility obtains. Integration of the radial field component through the region yields a voltage VSDVR
=
(&+ca)Rax
It will be noted that the voltage contribution of the region in this model is not a function of current.
636
J.
R.
SZEDON
and
As negative charge accumulates with increasing current due to the saturation of the electron drift velocity, a larger quantity of positive charge is injected from the avalanche region, thus the net space charge and the associated electric field are independent of current. At this point it is necessary to consider the problem of specifying the critical avalanche field which is used in the demarcation of the avalanche and the saturated drift velocity (SDV) regions. The effect of the value given to E, is most strongly manifested in the SDV region voltage as given by equation (15). In addition, it enters into the thickness to radius ratio of the quasi-cylindrical view of the planar avalanche region which may be determined from equation (5). A reasonable compromise is achieved for Ea = 1.1 x 105 V/cm, with the SDV region voltage of approximately 8 V and a thickness to radius ratio of approximately 0.5 for the avalanche region. This permits marginal treatment of the avalanche region in terms of Gunn’s model. Although this approach forces the high field model to agree with the observation, it also influences, through the value for the outer radius of the SDV region, the contribution of the normal mobility part of the device. The normal mobility region is bounded by the inner surface, r = R$ at which the electric field has the magnitude EC and by the outer surface at the radius R,, corresponding to the outer contact Major emphasis is placed on of the device. obtaining the minority carrier concentration profile in this region. This is done under the general assumptions of normal mobility and diffusion behavior with recombination entering in terms of a constant minority carrier lifetime. The explicit boundary conditions for this model result from matching the hole density at the inner surface with that for the saturated drift velocity region and requiring the hole density at the outer contact to be at its thermal equilibrium value. After the carrier concentrations are determined, the voltage-current relationship are found in the usual manner, by integrating the electric field. The steps in this process are indicated. Since the hole and electron concentrations at the outer surface of the saturated drift velocity region are virtually equal and much larger than their thermal equilibrium values, it is appropriate to require quasi-space charge neutrality in the normal mobility, or bulk,
A.
G.
JORDAN
region. This condition is imposed independently of Poisson’s equation; i.e. there is no requirement that the divergence of the electric field be identically equal to zero. Since the electric field in the SDV region is nearly divergenceless, there will be no difficulties in matching functions at RL. In the continuity equation, decoupling of the electric field associated with the hole current from the hole density may be accomplished by taking the divergence of the hole current density C . JP = qpPC . (pE)-qD,Wp. The expression for the total current as a solution for the quantity pE :
b
J pE=_-_____ qdl+b)
(16) density
yields
ENd + D, k?Q,. PP (l+b)
l+b
(17) The divergence of this expression, requirement that the total current divergence-free, may be approximated
V . (pE) _N-
with the density is as
D,l-b -Wp. ml+b
(18)
It should be noted that this result does not require zero divergence of the electric field. The carrier concentration is given in terms of hyperbolic function, i.e.
p = p, + [p(R;)
_po]s sinh(RmdL) p-p-. y sinh(R,-
Ri,/L)
(19)
The expression for the minority carrier profile is quite similar to that obtained for a planar geometry with similar boundary conditions, except for the modifying factor of Rl/r. This tends to isolate the conductivity modulation effects nearer the injecting contact than for the rectangular model and has important consequences in the speed with which the device may respond to time dependent excitation. From equation (17) the expression for the electric field dependence on the minority carrier concentration is E
_
1 J+@W -b)vP --. PLuq
The bulk region
Al+b)+bX
contribution
(20)
to the device voltage
THE
STATIC
AND
DYNAMIC
PROPERTIES
is then indicated as R, vB=
i
~
OF THE
AVALANCHE
-dr s ppq27+ p( 1 + b) + bN2
VB = Kial,
RL PO
4
s
l-b p(l+b)+b&
dp.
(21)
p(Rf)
The second term is at most a few kT/q in magnitude and is insignificant for typical values of the current being passed. With the solution for the minority carrier density given in equation (19), the first term in the bulk voltage contribution may be written, after some manipulation, as an integral over the normalized distance variable, r/Rm :
(22)
It is interesting to note that for high injection, the current is linearly related to the hole concentration at the inner surface, and the term, i/p(R& in the coefficient of the integral becomes constant. The V-I behavior is then decidedly non-ohmic; the actual variation of voltage with current is through the factor p(Rz) in the second term of the denominator of the integrand in equation (22). It is the decreasing importance of this term at higher bias levels that gives rise to strong conductivity modulation effects. As regards numerical evaluation of the contributions of the various portions of the sustaining mode model, the SDV region has been mentioned previously. For appropriate values of the parameters, the avalanche region contribution is 0.59
VA = -
$12
637
V Al/z.
The bulk region voltage has been obtained with a
(24)
where al varies from O-4 to O-6 depending on the values assigned to Rb and TV. The complete theoretical I-V relationship for the model of the avalanche injection diode is given in Fig. 4; sustaining mode behavior for two values of 40
I
FIG. 4. Predicted
x?[sinhFil-$)]&,‘di).
DIODE
digital computer program. From the results it is possible to characterize the bulk V-1 behavior by an expression of the type
1
+E
INJECTION
I
1-V
I
curves
I
I
for the AID
model.
minority carrier lifetime is indicated. Comparison with the observed characteristics given in Fig. 1 is invited. It has been demonstrated that, although the avalanche effect causes the two modes of operation of the device and provides negative incremental the bulk region is an important resistance, determinant of the static behavior. In this regard, the dynamic analysis is expected to depend strongly on the contribution of the bulk region. Previous work on the avalanche injection diode has not indicated that this is the case.
3. INVESTIGATION OF DYNAMIC A. Transient behavior
PROPERTIES
The success of the d.c. treatment of the AID suggests that it may be applied, with appropriate modifications, to an investigation of transients. For convenience, a step current driving function is used in this analysis. The exact nature of the step permits several possibilities to be considered. If operated entirely within the pre-breakdown mode, the device would be characterized by dielectric relaxation effects associated with majority
638
J.
R.
SZEDON
and
carrier conduction. Behavior such as this, viewed independently, is not particularly interesting in the AID, hence a detailed analytical treatment of it is not to be undertaken. The transient response of the device biased in the sustained avalanche mode, on the other hand, would be affected by minority carrier effects in the bulk, as indicated by the results of the static treatment. While such general problems have already been investigated,(T) this particular situation is unique because of the extremely high carrier injection levels and the small effective contact area established by the avalanche region model. Furthermore, previous work on the AID has not indicated the significance of these effects. A third possibility is the use of a step current large enough to cause switching between the on and the off states of the device. In terms of dynamic applications of the AID, this behavior is very important. The mathematical treatment of this situation is complicated by the fact that the geometric models for the two states have effective inner contact radii which differ by at least an order of magnitude. For this reason, the execution of a formal analysis of complete switching in the present model is not undertaken. For the planar model, the carrier transit time in the avalanche region is of the order of magnitude of the collision ionization time, i.e. less than 0.1 nanosec, which has been estimated in the literature.(s) The dynamic behavior of this region should be determined by the avalanche process itself. For a practical treatment, the respond avalanche region is assumed to instantaneously to a step change in the current. Such a view of the region will be valid if the time required to produce a significant change in the voltage across the remaining portions of the device is much greater than the collision ionization time. The static model for the saturated drift velocity region predicts the existence of an electric field totally independent of current as long as the transport process permits the establishment within the region of a charge density which is also current independent. Since the carrier transit time for this part of the diode is at most a tenth of a nanosecond, the individual carrier profiles, with recombination effects excluded, may be assumed to adjust immediately to the conditions imposed by the boundary at the avalanche region. Thus,
A.
G.
JORDAN
the saturated drift velocity region will present no transient voltage response on a time scale for which transit effects can be ignored. The bulk region treatment uses a step increase in current at time equal to zero, with the initial carrier profiles determined by the current prior to the application of the step; the steady-state profiles, by the final current. Near the inner boundary, the bulk region must support field magnitudes in excess of the critical drift field, E c, for the interval of time necessary to redistribute the carriers and increase the conductivity of the region. In the strictest sense, the problem should be reformulated with a provision that the critical drift radius instantaneously move into the static bulk region, as required by field magnitude considerations, and then relax to its original location as the bulk carrier profiles achieve their steady-state forms. This approach would predict a time dependent SDV region voltage with an initial spike. There is considerable difficulty in solving this problem, however, since it requires developing an internally self-consistent model for the SDV and bulk regions considered simultaneously. The present, less involved, treatment ignores the velocity saturation effects in the bulk region and predicts the initial voltage spikes as a result of conductivity modulation in the normal mobility region. The minority carrier density must obey the time-dependent differential equation corresponding to that developed for the static analysis
$-PO) =- py) +~p”‘(p-p~).(2,) Prior to time equal to zero, the direct current, 11, passes through the device and the initial minority carrier profile is given by equation (19) in which the hole concentration at the inner surface of the bulk region corresponding to the current 11 is designated as p(Rz, Ii). At time equal to zero, the current is incremented to the value 1s; the steadystate hole profile is characterized by the parameter p(RL, Ia), corresponding to the final current value. With these conditions, the minority carrier distribution is given by the expression
THE
STATIC
AND
sinh&IL[l
sinhR,/L[l
DYNAMIC
PROPERTIES
-(r/&)1 _2k -(Rb/Rm)]
OF THE
oth$
RM ’
L
AVALANCHE
O”7~ - exp{ c n=O I
T(Z) is defined as
INJECTION
- t/r(n)> cos(2n+
DIODE
k] .
1);
nk
639
(26)
1.00 0.00
T(n) =
[(2n+ &;(L,Rm)]2+
1
(27)
Except for the influence of the geometry in the factor of l/r, the solution is identical with that for planar models of the same type. The transient voltage is determined analogously to the static case. The actual evaluation has been done numerically for arbitrary values of time. A current step from 10 to 15 mA has been used as being well within the sustained avalanche portion of the predicted V-I curve. The minority carrier lifetime has been given a range of specific values in the computation to permit an investigation of its effect on the voltage transient. Results are given in Fig. 5. Straight lines corresponding to exponential behavior provide extremely good fits to the predicted points for normalized voltages less than O-1. The fits have been made for the points at which the predicted normalized voltages have values of 0.135, thus defining a time equal to twice the time constant of the exponential. For the geometry used in the analysis, the time constant of the exponential in each case is nearly the same as the minority carrier lifetime. In Fig. 6, typical observed voltage transients are shown with the predicted transients for 0.8 and 2.0 psec lifetime models. The current increments were approximately 30 per cent of the pre-step values for the cases considered. B. Behavior
with respect to alternating
signals
Since the slowest devices require about 1.5 psec to come to steady-state in response to step-current excitation, the bulk region equivalent circuit indicated in Fig. 5 should be valid for sinusoidal signals below 300 kc/s in frequency; over an extended frequency range, the various regions, particularly the bulk, must be examined in more detail. The avalanche region response to a sinusoidal current is assumed to be instantaneous, yielding a voltage in phase with the current as long as the collision ionization process can come to equilibrium
0.60 Approximate
Equivalent
0. IO 0.08 0.06
0.04
$
.c?
0.02
0 E
b
z
0.01 *
Time
4
3
2
0
in 16’
set
The Stroight Lines ore Exponentiol Fits to the Predicted Behovior, Matched ot o Normalized Voltage
Value
of
6’
FIG. 5. Semilogarithmic plots of the predicted, normalized voltage response of the bulk region to a step current, for various minority carrier lifetime values.
within a small fraction of the period of the signal. The avalanche region is characterized by its incremental resistance, as given by differentiating equation (9) with respect to current, viz. RA =
dVA
--_=
di
---.
1
VA
2
i
(28)
As in the transient analysis the voltage contribution of the SDV region is assumed to be zero, as long as the frequency is below the limit that is imposed by the avalanche region. The SDV small signal impedance is thus zero. The hole concentration at the inner surface (Ri)
640
J.
R.
SZEDON
and
A.
G.
JORDAN
excitation, the a.c. component of the hole concentration is in general given by
i
(32)
pat = @ exp(jwt),
where 9 is the spatial part of the a.c. hole concentration. After manipulations similar to those for the transient case, the differential equation for the hole concentration is l+
O=--__ (
Time in Id6 set FIG. 6. Comparison of predicted and observed voltage transients for step current driving functions. Curve (a) is the observed behavior for device No. LJ. Curve (b) is the observed behavior for device No. UE. Curve (c) is the predicted behavior for a lifetime of 0.8 psec. Curve (d) is the predicted behavior for a lifetime of 2.0 psec.
of the bulk region is assumed to be in phase with the current passing across that surface. Neglect of any time lag is consonant with the assumptions regarding the size of the avalanche and bulk regions and the nature of current transport within them. Specifically it is assumed that the current passing through the device is complex and given by
i = I[l+A
exp(j,t)].
(29)
The boundary condition for holes at RL becomes p(RL) = po+p(Ri,
W+A
exp(j41,
(30)
where the excess minority carrier concentration at the inner surface of the bulk region due to the bias current is designated as p(Ri., I). The differential equation in the minority carrrer density, equation (25), must be solved subject to the requirement of equation (30) and the further minority carrier restriction that the excess concentration at the outer contact be zero. This solution may be indicated as a sum of the thermal equilibrium, the d.c. and the a.c. solutions, i.e. p = po+p0%+pac.
(31)
The d.c. portion of the solution has been indicated in the previous section. For steady-state a.c.
L2
d%?
2d9?
jar,
(33)
i 9+yr+F*
The general Bessel form of solution for the spatial dependence is indicated except that the argument of the function, instead of being purely imaginary, i.e. j(r/L), is complex:
The a.c. solution, therefore, must be analogous to the d.c. one, except for this change. The boundary condition in equation (30) yields as the a.c. solution for the hole concentration
pa, = p(RL, I)A exp(j&)s
Y
sinh(R,-r/L2/[1 x sinh(R,-
Ri/Ld[l
+j,r,])
+jwP])’
(34)
The complex value of the radial component of the electric field may be obtained by substituting the instantaneous hole concentration in equation (31) into the electric field expression. The instantaneous bulk voltage due to the current, I[1 +A exp(jwt)], is given by integrating the electric field expression with the bias contribution accounted for by requiring the ratio of a.c. to d.c. current, A, to be zero in a separate integration of the complete electric field expression. The corresponding voltage is designated as VaC; hence the a.c. component of the instantaneous bulk voltage is given as V,, = VB-- Vde = re V,,+j With the a.c. component expression
im Vu,.
(35)
of current
given by the
iae = IA(cos wt+ j sin wt),
(36)
THE
STATIC
AND
a quasi-impedance,
DYNAMIC
PROPERTIES
2, may be defined
This is not a true impedance, since there will be, in general, voltage components at frequencies higher than that of the alternating current. In such a case, the quasi-impedance will vary with electrical angle, wt, and the actual impedance at the current frequency will be given by the average of the quasi-impedance over a period of the current function. The theoretical bulk region impedance from 10 kc/s to 10 km+ has been obtained by numerical analysis for various minority carrier lifetime, bias current and geometric values. In a first order approach which neglects the complications in the avalanche model at very high frequencies, the total impedance of the AID model may be obtained by algebraic addition of the avalanche region resistance to the bulk region impedance. The results of this treatment are given in Fig. 7. The curve for 5 mA bias with a minority
OF THE
AVALANCHE
INJECTION
DIODE
641
very pronounced and gives rise to a rapidly varying slope in that vicinity. The slope at that bias should correspond to a negative incremental resistance. Of major significance is the fact that the total impedance curves are roughly concentric. In addition, over all but the lowest frequency ranges, the predicted behavior does not indicate very large differences with bias in the magnitude of the total impedance or of its component parts. The predicted AID impedance is the most interesting in the mid-frequency range, i.e. 0.3 to 10 me/s. The results of measurements on two devices for this range are given in Fig. 8. The frequencies at which maximum reactances occur 600 500 400 c
300
-200
0
Reol
Port
200
of the
400
600
Impedance,
600
51
~~~~
;
ZOO
0
z
Reol
500
I Frequencp
Port
400
'
600
I I inMc/s-2
I
1000
600
of the Impedance,
L?,
0
I
’
I
’
’
200
Real
’
400
Port
of the
600
Impedance,
800
1000
a
FIG. 8. Measured, mid-frequency impedance behaviour of typical avalanche injection diodes
400
600
Reol
600
Port
of the
1000
1200
Impedance,
1400
1600
Cl
FIG. 7. The predicted AID impedance behavior. The upper set of curves is for a minority carrier lifetime of 3.0 /wx.; the lower one, 0.2 ~~sec.
carrier lifetime of 1.0 psec has a negative resistance at zero frequency. This seems to be in disagreement with the static curve of Fig. 4. Careful numerical consideration of the static model shows that the avalanche contribution near 5 mA is
agree very well with those for the 0.2 psec model in Fig. 7. The data of Fig. 8 are interesting due to the appearance of the negative resistance component below 2 me/s at 7 mA bias. In this respect, the device behavior is very like that of the 1 psec lifetime model in Fig. 7. The measurements in Fig. 8 demonstrate the tendency at higher frequencies for the diminution of differences resulting from bias changes. A proposed application of the AID is the switching of high frequency signals; hence, admittance measurements have been made in the 200 to 900 me/s range. Further, since the use of
642
J. R.
SZEDON
and
the diode is expected to be limited by the susceptance of the encapsulation, the measurements have been made of the behavior of the device as a whole, instead of correcting for the holder susceptance. The device has been examined for zero bias and for several values of current, well into the on-state. As a result of the measurements in the biased off-state, the device may be characterized by a capacitance of approximately 1 pF shunting a resistance of SOOO-10,000 Q. With the . . application of heavy bias, the resistance decreases by an order of magnitude and the capacitive susceptance is reduced by about 10 per cent at 300 me/s. On-state measurements for various levels of bias are virtually identical, substantiating the predicted high frequency behavior in Fig. 7. Near 600 me/s the measured resistance achieves its maximum value of approximately 1400 Q. This is in fair quantitative agreement with the predicted behavior. The latter calls for a small change in the value of the resistance about its maximum, while the observed decrease is roughly 10 per cent for a 200 mcjs change in frequency. An experimental evaluation of the a.c. switching properties of the avalanche injection diodes discussed in the present work has been published.@) 4.
A.
G.
JORDAN
Although the dynamic analysis concentrates on the effects of the bulk region, it should be kept in mind that the important considerations of contact geometry, carrier injection levels and space charge requirements in this portion of the device are determined by the models for the high field region. As a result, there is tighter coupling between the various parts of the model than is first apparent. This constitutes additional support of the validity of the present approach. Since the model in question precludes the existence of important junction effects and of difficulties arising from the avalanche and high drift velocity regions, it serves as an excellent tool for predicting dynamic bulk effects due to conductivity modulation. While the transient analysis has not involved anything novel in this area, except for the geometric complications and the avalancheimposed boundary conditions at the inner contact, it has permitted the investigation of transients an order of magnitude faster than those treated previously. The small signal a.c. investigation has yielded agreement between predicted and observed conductivity modulation effects on the impedance behavior to frequencies in the neighborhood of a kilomegacycle per second.
SUMMARY
The proposed model for the AID, in treating both the pre-breakdown and the sustained avalanche modes of operation, has produced results in quantitative agreement with observations. The agreement in the pre-breakdown model is due, in part, to a more flexible interpretation of the ionization collision data which is supported by Schockley’s recent work in the area. The sustained avalanche model, on the other hand, is thought to be a novel treatment of the device. Although it is similar in concept, it is not as complicated as earlier approaches. The static application of the model reveals the greatest sensitivity to the interactions between the avalanche, the carrier drift velocity and the conductivity modulation effects. In this respect the model, as applied to the on-state static behavior presents some problems in obtaining a balance between the multiplication and the bulk processes. Confirmation of the predicted dynamic response supports the choice of parameters required by the static application.
REFERENCES 1. J. B. GUNN, Progress in Semiconductors, (edited by A. F. GIBSON), 2, p. 217. Heywood, London (1957). 2. D. J. ROSE, Pkys. Rev. 105,413 (1957). 3. A. F. GIBSON and J. R. MORGAN, Solid-State Electron. 1, 54 (1960). 4. C. A. HOGARTH, Solid-State Electron. 1, 70 (1960). 5. A. K. JONSCHER,Progress in Semiconductors, (edited by A. F. GIBSON), 6, 143. Heywood, London (1962). 6. W. SHOCKLEY, Solid-State Electron. 2, 35 (1961). 7. L. I. BARANOV, Radiotekh. Elektron. 5, 1002 (1960). (Translated in Electron. Express; July 30 (1960). 8. J. R. SZEDON and A. G. JORDAN,Proc. of the National Efect. Conf., 18,166 (1962).
LIST al
=
A
=
OF
SYMBOLS
dimensionless constant used in characterizing the current dependence of the under static region voltage bulk conditions. the ratio of the alternating current magnitude to the bias current magnitude in the a.c. treatment.
THE
STATIC
AND
DYNAMIC
PROPERTIES
= ratio of electron to hole drift mobility at low values of electric field. = diffusion coefficient for electrons D~@P) holes. (ems set-i V-r). = diffusion coefficient for electrons for DflO low values of electric field. = the electric field magnitude (vector). E,(E) (V cm-l). = the electric field at the boundary EdEd separating the avalanche and the saturated drift velocity (the saturated drift velocity and the bulk) regions. the = instantaneous current through i diode. (A). = value of current below which normal ie mobility obtains throughout the diode in the pre-avalanche model. = initial (final) current in the transient 4SI2) analysis. = bias (alternating) component of current Z,(iCz C) in the small signal analysis. = the instantaneous total current density J,(J) magnitude (vector). (A cm-s). = the instantaneous hole current density JP,(JP) magnitude (vector), k = Boltzmann constant. (J deg-l). = a constant used in characterizing the K current dependence of the bulk region voltage under static conditions. See above. (V A-l). = effective diffusion lengthforholes. (cm). L = effective donor doping density. (cmm3). Nd = instantaneous electron concentration. It (cme3). = electron concentration at the plane 40) x = 0 in the planar avalanche model. = electron concentration at the boundary n(Rd between the avalanche and the SDV regions. = instantaneous (thermal equilibrium) P,(n) hole concentration. p(Ra), (p(Rz)) = instantaneous hole concentration at the boundary between the avalanche and the SDV (the SDV and the bulk) regions. = hole concentration at the boundary P(% I) between the SDV and the bulk reeions as a result of bias current in the a.c. treatment. P(R;, h), (p(R) 1~)) = hole concentration at the boundary between the SDV and the bulk regions as a result of the initial (final) current in the transient analysis. = hole concentration due to the bias Pat, (Pa,) (alternating) component of current in the a.c. treatment of the bulk region. = magnitude of the electronic charge. (C). 4 r = radial distance in spherical co-ordinates. (cm).
b
OF
THE
AVALANCHE
INJECTION
DIODE
643
XC2
= effective radius of the cylindrical (spherical) avalanche region model. = incremental resistance of the avalanche region. (a). = radius of the boundary separating the SDV and the bulk regions in the preavalanche (sustained avalanche) model. (cm). = effective radius of the inner contact of the diode. = maximum radial extent of the bulk region. = complex function giving the spatial dependence of the hole concentration in the a.~. treatment of the bulk region. (cmm3). = absolute temperature. (“IQ. = time. (set). = dimensionless function used to facilitate integration. = value of the saturated carrier drift the model. assumed in velocity, (cm set-l). = total voltage across the device in the pre-avalanche state. (volt). across the voltage = instantaneous avalanche (bulk) region. across the saturated drift = voltage sustained region in the velocity avalanche model. = alternating (bias) component of the bulk region voltage in the a.c. treatment. = rectangular co-ordinate in the planar avalanche model. (cm). = extent of the planar avalanche model.
z
(cm). = quasi-impedance
~a, (Ra) RA R C,(Ri)
RP R?73 w
T t 24 vzs
V VA, (VB) VS
Vat, (VdC) x
a(E) ‘%
= =
=
K h
= =
= = = = =
of the bulk
region.
(0). avalanche ionization coefficient. (cm-‘). constant used in obtaining a mathematical fit to experimental data for the avalanche ionization coefficient. (cm-i). constant relating to geometric and material parameters in the treatment of the saturated drift velocity region. (V cm-l). permittivity of silicon. (Fcm-l). constant used in obtaining a mathematical fit to the experimental data for the avalanche ionization coefficient. (cm V-i). drift mobility of electrons (holes). (cm? see-l V-l). drift mobility of electrons at low values of electric field. hole lifetime. (XC). effective lifetime parameter in the transient analysis. (set). radian frequency. (set-l).
Electronics
Pergamon
Press 1963. Vol. 6, pp. 631-643.
Printed in Great Britain
THE STATIC AND DYNAMIC PROPERTIES OF THE AVALANCHE INJECTION DIODE* J. R. SZEDONt Westinghouse
Electric
Corporation
Research
Laboratory,
Pittsburgh,
Pennsylvania
and A. G. JORDAN Department
of Electrical Engineering, Carnegie Institute Pittsburgh 13, Pennsylvania
(Received
22 February
1963
of Technology,
; in revised form 31 May 1963)
Abstract-Since the proposal of the avalanche injection diode, AID, by Gunn, it has become necessary to incorporate fundamental knowledge regarding collision multiplications and other high field, high current density effects into models that approximate actual devices. Previous approaches to the problem have been inadequate for all but a qualitative appreciation of the behavior of the device. This investigation has produced a more comprehensive model for the AID; one that encompasses to some extent both high and low electric field effects, and explains the observed, low-level dynamic response of the device in terms of conductivity modulation effects. The steps leading to a numerical evaluation of the high frequency behavior of the device in terms of geometric and material parameters have been developed and have been used to explain the characteristics of typical units. R&urn&-Depuis que la proposition concernant la diode d’injection AID B avalanche a et4 faite par Gunn, il est devenu necessiare d’incorporer des connaissances fondamentales ayant trait aux multiplications de collision et autres champs Clev& et aux effets de courant a haute densitt dans les modhles qui sont presque equivalents aux dispositifs actuels. De prC&dentes approches au probleme ont 6th insuffisantes en tout sauf pour l’appr&iation qualitative du comportement du dispositif. Cet examen a produit un modele plus d&ail16 du AID; il comprend jusqu’8 une certaine mesure les effets de champs Clectriques Clev& et bas, et explique la response dynamique observCe B bas niveau du dispositif en termes de la modulation de conductivitC. Les &apes menant B une Cvaluation numCrique du comportement du dispositif B haute frkquence en termes des parametres geometriques et du matCriau ont Ctt5 employees pour expliquer les CaractCristiques d’C16ments typiques. Zusammenfassung-Seit der Einfiihrung der Lawinen-Injektions-Diode (AID) durch Gunn wurde es erforderlich, grundlegende Tatsachen fiber die Vervielfachung durch ZusammenstGsse und andere Effekte eines starken Feldes und einer hohen Stromdichte in die Modelle aufzunehmen, die zur angeniherten Darstellung der wirklichen Gerlte dienen. Frtieren Versuchen in dieser Richtung gelang es nur, eine qualitative Darstellung vom Verhalten des Geriites zu geben. Die vorliegenden Untersuchungen haben ien besseres Model1 fiir die AID geschaffen; es umfasst in gewissem Ausmass die Effekte starker und schwacher elektrischer Felder und erklart die beobachtete schwache dynamische Reaktion des Gerates durch Modulationseffekte der Leitfshigkeit. Die Schritte, die zu einer numerischen Auswertung des Hochfrequenzverhaltens des Gergtes auf Grund der geometrischen und materiellen Parameter ftiren, wurden durchgefiihrt und dann dazu benutzt, die Eigenschaften typischer Gerlte zu erkliiren.
* This paper is a portion of a thesis submitted in partial fulfillment of the Ph. D. requirement in Electrical Engineering at Carnegie Institute of Technology.
7 Formerly in the Department of Electrical Engineering at Carnegie Institute of Technology, Pittsburgh 13, Pennsylvania. 631
632
J.
R. SZEDON
and
1. INTRODUCTION
form the avalanche injection diode,(l) AID, consists of a relatively lightly doped semiconductor with contacts which permit the establishment of high magnitude electric fields in the bulk at low carrier injection levels. In the pre-avalanche mode, also referred to as the off state, the device is characterized by a high incremental resistance. When the field magnitude is sufficiently high over a carrier path length, collision ionization occurs. Significant avalanche multiplication of carriers then permits sustaining IN
ITS
simplest
G.
JORDAN
in fabricating devices for the present investigation.@) The voltage-current characteristics of typical devices are given in Fig. 1. In most cases, negative incremental resistance in the on-state has been observed only at low values of current. Previous work on the properties of the device has been in two general areas. In the first, the prebreakdown analysis has been performed using an idealized hemispherical geometry and the assumption that the drift transport mechanism can be characterized by a constant mobility for field magnitudes less than a critical value E c, and by a
30
30
25
25 20
4
40
20
To-
30
I5
c .-
20
E 2 IO 7 UO
A.
15 I
IO I
5 0
5
IO
IS
20
25
30
5
IO
I5
Voltage, (A) FIG. 1. Current-voltage
5
\
0 L_j 0
20
25
0
0
5
IO
15
20
25
V (Cl
(61 curves of typical injection diodes.
point-contact
avalanche
large currents at voltages ordinarily lower than the constant drift velocity for field magnitudes in maximum, or trigger, voltage in the off state. excess of E,.(3) While the approach appears to The device operates in the sustained avalanche be correct, even though objection may be made to mode with a much lower incremental resistance the assumption of complete drift velocity saturathan in the pre-avalanche state. It has been tion in silicon,(5) the criterion for breakdown demonstrated that the voltage-current characterhas not been examined carefully, heretofore. In istic for a suitable avalanche region model should ‘the second area, the role of avalanche multiplicaexhibit a negative resistance.(l) Furthermore, with tion has been treated for a simplified model in avalanche the dominant process in the operation which collision ionization is assumed to occur of a device, the switching interval between the within a region supporting very high electric pre- and sustained avalanche modes should be fields. Recourse has been made to this model in on the order of the collision ionization time, i.e. explaining the low incremental resistance of the less than 1 x lo-12 sec.(a) Devices exhibiting sustained avalanche state; however, the theory sustaining mode negative resistance and having predicts a pronounced negative resistance which is circuit-limited switching times on the order of not observed. No concrete attempts to explain 10 nanosec have been fabricated.(s) Primarily as a this have been made to date. Transients in the result of this agreement, little additional work has AID have been treated meagerly. Switching been done on the original AID model which has between the stable state of the V-1 characteristic been inadequate for the prediction of the detailed has been reported to require less than 30 nanosec,(a) dynamic behavior of the device. but theoretical treatment of the dynamic behavior HOGARTH’S technique for making small area n+ of the device has been lacking. contacts to an n-type silicon wafer has been used It is the purpose of this work to present a
THE
STATIC
AND
DYNAMIC
PROPERTIES
comprehensive model for the AID; one that encompasses both high and low electric field effects; that includes conductivity modulation and that predicts the dynamic response of the diode, biased in the sustained avalanche state, to low level signals. Support of the theory with observed behavior will be presented where appropriate. 2. BEHAVIOR
UNDER
STATIC
CONDITIONS
A. The pre-avalanche state In general, the V-1 characteristic of the preavalanche state exhibits in incremental resistance of approximately 10,000 R, with breakdown occuring at a current of several milliamperes and a voltage ranging from 20 to 40 V. The breakdown is very abrupt, i.e. there is no noticeable decrease in incremental resistance which would be expected from carrier multiplication. This is reasonable in terms of SHOCKLEY’S model of isolated regions with locally high electric fields(s) since sustained avalanche could be initiated in a region which carried only a small fraction of the total current prior to breakdown. A theoretical treatment for the pre-breakdown state of the device has been given in the literature.(s) At currents less than a value i, given by ie = 2rrqpnNdECR$
(1)
the field magnitude nowhere exceeds EC, the value assumed for saturation of the electron drift velocity. A purely ohmic analysis is appropriate under these conditions. For currents in excess of i,, the ohmic treatment is valid beyond a critical radius, R e, given by equation (1) with R, and iC replaced by R, and i, respectively. In the high field region, the requirements of no minority carrier current and a divergenceless total current permit determination of the spatial dependence of the electric field. For currents in excess of i,, the voltage across the device is
OF THE
AVALANCHE
INJECTION
DIODE
633
given to the production of hole-electron pairs by collision ionization. The usual breakdown condition, modified for the high field region alone, i.e. RC
s
x(E)dr
= 1,
(3)
RP does not appear to be applicable in this situation, since the pre-avalanche model would become invalid long before it were to obtain. Were the integral to achieve a value as high as 0.5, the incremental resistance of the pre-avalanche characteristic would decrease noticeably prior to breakdown; but this behavior is not observed, indicating that breakdown occurs for values of the integral less than 0.5; perhaps as low as 0.2. Such a condition is not unreasonable, since local perturbations to the electric field by dislocations could easily give localized avalanche before its predicted existence according to the uniform treatment.@) The results of the numerical work are plotted in Fig. 2, in terms of the bulk, the SDV region and the total voltage drops as functions of current in the pre-avalanche mode. Maximum voltages 70
I
I
I
I
60
20
-
V = ECRC($-$)
. 0
(2) At electric field magnitudes on the order of 1 x 10s V/cm and higher, consideration must be
2
6 Currentf
6
mA
FIG. 2. Voltage-current characteristics for the saturated drift velocity and the bulk regions and for the AID as a whole in the me-dreakdown state.
634
J.
R.
SZEDON
and
ranging from 25 to 50 V are observed in actual devices. The corresponding breakdown current range in the theoretical results is 4-5 mA. It is important to determine whether or not such currents are compatible with the avalanche requirements just discussed. The ionization coefficient dependence on the field, for both holes and electrons is assumed to be of the form a(E)
= ai exp(XE),
(4)
an approximation to the empirical data which has been used in earlier work. The integration of the ionization coefficient over the SDV region has been performed numerically, with the result given as a function of current in Fig. 3. Included is a curve for the maximum electric field in the
A.
G.
JORDAN
voltage in excess of 80 V. From the arguments given above, an appropriate value for ai is approximately 50 cm-l if the magnitude of the integral for the idealized model is as low as O-2 when localized breakdown occurs. An interesting consequence of this reasoning is that the field magnitude at the contact is about 1.4 x 105 V/cm, a value in general agreement with the arbitrary 1 x 105 V/cm value used in earlier work. B. The sustained avalanche state The active avalanche region has been treated on the basis of GUNN’S planar model.(l) Assumptions inherent in the treatment are: (1) Drift velocities of holes and electrons are equal and independent of field. (2) Carrier diffusion and recombination are neglected. (3) The ionization coefficients for holes and electrons are equal, having a field dependence which has been expressed mathematically in equation (4). (4) The avalanche model is used only for electric field magnitudes greater than Ea, this value to be consistent with the adaptation of the model to a finite geometry. The extent of the region is then given by the expression
1
1.i"
nJh(l-[qn(0)/J]+qNdVd,/J}2 __-
KVS~ the electric
xln
exp(X&)
17 "'
(5)
field by
I l+tan2 \
and the majority
ScjTJA
--x)1 ] LC exp(~&))l”‘(r, 2
G
carrier
density
by
FIG. 3. Predicted behavior of the pre-avalanche maximum field and the ionization probability integral.
(7) For the particular parameters used in device. approximating the ionization coefficient data, the integral achieves the value of unity for a current of 9.7 mA which, from Fig. 2, corresponds to a total
The hole profile is given in an equation identical with the last one, except that the second and the third terms are negative.
THE
STATIC
AND
DYNAMIC
PROPERTIES
Integration of the electric field in determining the voltage across the planar region may be performed with the second term in equation (6) treated by the substitution
24 =
tan[Z(~~~‘2.exp~]
.(sa--s)].
(8)
The resulting integral has a current dependent upper limit, but over the range of currents (S-40 mA) which is appropriate, the integral itself is insensitive to current and equal to approximately 75 per cent of the value for an infinite upper limit. With this approach, the avalanche region voltage is given by the expression
A thought experiment leads to the conclusion that a large, but finite avalanching area would be unstable. It would tend to diminish in size under the action of field distortion around the periphery of the active region and of carrier multiplication within the region. The stable situation might be visualized as a thin disk having a thickness given by the extent of the planar avalanche model. GIBSON and MORGAN have suggested that diffusion effects would determine the radial extent of the avalanche region. Requirement of electron and hole currents to be individually zero across the . . cylmdrical walls yields an exponential carrier profile in the radial direction, having a characteristic dimension(s)
4
Dno MO E
VdS
Vd”-dsVds
ya=-=----
for
E > EC.
(10)
The zero subscripts denote low field values. On this basis, with a value for E of 1 x 10s V/cm, the value for ru is about an order of magnitude greater than that for xa. If the thickness of the chip, Rm, is large in comparison with pa, the field pattern in the bulk of the material will not be altered greatly if the cylinder is replaced by a hemisphere of radius, R,,, which preserves the current density. Carrier concentrations and the field magnitude at xa in the planar model will be imposed at R,, while the circuit properties of the avalanche region will be given by the voltagecurrent relationship of the planar model in
OF THE
AVALANCHE
INJECTION
DIODE
63.5
equation (9). The remainder of the device can then by analyzed, taking full advantage of spherical symmetry. The treatment of the saturated drift velocity region for the sustained avalanche case is quite similar to that under pre-breakdown conditions, except for the fact that both carrier species are assumed to support the flow of current, diffusion effects being ignored. Since the extent of the region is small in comparison with a minority carrier diffusion length, recombination is ignored. With additional requirements of zero divergence for both hole and electron currents, the carrier profiles are
(12) The carrier concentrations at R, are determined from equation (7) for the avalanche model. From Poisson’s equation, the electric field is given as function of radial distance by the expression (13) where (14) The outer boundary of this region is at the radius RL, for which the electric field falls to the value E e, below which normal mobility obtains. Integration of the radial field component through the region yields a voltage VSDVR
=
(&+ca)Rax
It will be noted that the voltage contribution of the region in this model is not a function of current.
636
J.
R.
SZEDON
and
As negative charge accumulates with increasing current due to the saturation of the electron drift velocity, a larger quantity of positive charge is injected from the avalanche region, thus the net space charge and the associated electric field are independent of current. At this point it is necessary to consider the problem of specifying the critical avalanche field which is used in the demarcation of the avalanche and the saturated drift velocity (SDV) regions. The effect of the value given to E, is most strongly manifested in the SDV region voltage as given by equation (15). In addition, it enters into the thickness to radius ratio of the quasi-cylindrical view of the planar avalanche region which may be determined from equation (5). A reasonable compromise is achieved for Ea = 1.1 x 105 V/cm, with the SDV region voltage of approximately 8 V and a thickness to radius ratio of approximately 0.5 for the avalanche region. This permits marginal treatment of the avalanche region in terms of Gunn’s model. Although this approach forces the high field model to agree with the observation, it also influences, through the value for the outer radius of the SDV region, the contribution of the normal mobility part of the device. The normal mobility region is bounded by the inner surface, r = R$ at which the electric field has the magnitude EC and by the outer surface at the radius R,, corresponding to the outer contact Major emphasis is placed on of the device. obtaining the minority carrier concentration profile in this region. This is done under the general assumptions of normal mobility and diffusion behavior with recombination entering in terms of a constant minority carrier lifetime. The explicit boundary conditions for this model result from matching the hole density at the inner surface with that for the saturated drift velocity region and requiring the hole density at the outer contact to be at its thermal equilibrium value. After the carrier concentrations are determined, the voltage-current relationship are found in the usual manner, by integrating the electric field. The steps in this process are indicated. Since the hole and electron concentrations at the outer surface of the saturated drift velocity region are virtually equal and much larger than their thermal equilibrium values, it is appropriate to require quasi-space charge neutrality in the normal mobility, or bulk,
A.
G.
JORDAN
region. This condition is imposed independently of Poisson’s equation; i.e. there is no requirement that the divergence of the electric field be identically equal to zero. Since the electric field in the SDV region is nearly divergenceless, there will be no difficulties in matching functions at RL. In the continuity equation, decoupling of the electric field associated with the hole current from the hole density may be accomplished by taking the divergence of the hole current density C . JP = qpPC . (pE)-qD,Wp. The expression for the total current as a solution for the quantity pE :
b
J pE=_-_____ qdl+b)
(16) density
yields
ENd + D, k?Q,. PP (l+b)
l+b
(17) The divergence of this expression, requirement that the total current divergence-free, may be approximated
V . (pE) _N-
with the density is as
D,l-b -Wp. ml+b
(18)
It should be noted that this result does not require zero divergence of the electric field. The carrier concentration is given in terms of hyperbolic function, i.e.
p = p, + [p(R;)
_po]s sinh(RmdL) p-p-. y sinh(R,-
Ri,/L)
(19)
The expression for the minority carrier profile is quite similar to that obtained for a planar geometry with similar boundary conditions, except for the modifying factor of Rl/r. This tends to isolate the conductivity modulation effects nearer the injecting contact than for the rectangular model and has important consequences in the speed with which the device may respond to time dependent excitation. From equation (17) the expression for the electric field dependence on the minority carrier concentration is E
_
1 J+@W -b)vP --. PLuq
The bulk region
Al+b)+bX
contribution
(20)
to the device voltage
THE
STATIC
AND
DYNAMIC
PROPERTIES
is then indicated as R, vB=
i
~
OF THE
AVALANCHE
-dr s ppq27+ p( 1 + b) + bN2
VB = Kial,
RL PO
4
s
l-b p(l+b)+b&
dp.
(21)
p(Rf)
The second term is at most a few kT/q in magnitude and is insignificant for typical values of the current being passed. With the solution for the minority carrier density given in equation (19), the first term in the bulk voltage contribution may be written, after some manipulation, as an integral over the normalized distance variable, r/Rm :
(22)
It is interesting to note that for high injection, the current is linearly related to the hole concentration at the inner surface, and the term, i/p(R& in the coefficient of the integral becomes constant. The V-I behavior is then decidedly non-ohmic; the actual variation of voltage with current is through the factor p(Rz) in the second term of the denominator of the integrand in equation (22). It is the decreasing importance of this term at higher bias levels that gives rise to strong conductivity modulation effects. As regards numerical evaluation of the contributions of the various portions of the sustaining mode model, the SDV region has been mentioned previously. For appropriate values of the parameters, the avalanche region contribution is 0.59
VA = -
$12
637
V Al/z.
The bulk region voltage has been obtained with a
(24)
where al varies from O-4 to O-6 depending on the values assigned to Rb and TV. The complete theoretical I-V relationship for the model of the avalanche injection diode is given in Fig. 4; sustaining mode behavior for two values of 40
I
FIG. 4. Predicted
x?[sinhFil-$)]&,‘di).
DIODE
digital computer program. From the results it is possible to characterize the bulk V-1 behavior by an expression of the type
1
+E
INJECTION
I
1-V
I
curves
I
I
for the AID
model.
minority carrier lifetime is indicated. Comparison with the observed characteristics given in Fig. 1 is invited. It has been demonstrated that, although the avalanche effect causes the two modes of operation of the device and provides negative incremental the bulk region is an important resistance, determinant of the static behavior. In this regard, the dynamic analysis is expected to depend strongly on the contribution of the bulk region. Previous work on the avalanche injection diode has not indicated that this is the case.
3. INVESTIGATION OF DYNAMIC A. Transient behavior
PROPERTIES
The success of the d.c. treatment of the AID suggests that it may be applied, with appropriate modifications, to an investigation of transients. For convenience, a step current driving function is used in this analysis. The exact nature of the step permits several possibilities to be considered. If operated entirely within the pre-breakdown mode, the device would be characterized by dielectric relaxation effects associated with majority
638
J.
R.
SZEDON
and
carrier conduction. Behavior such as this, viewed independently, is not particularly interesting in the AID, hence a detailed analytical treatment of it is not to be undertaken. The transient response of the device biased in the sustained avalanche mode, on the other hand, would be affected by minority carrier effects in the bulk, as indicated by the results of the static treatment. While such general problems have already been investigated,(T) this particular situation is unique because of the extremely high carrier injection levels and the small effective contact area established by the avalanche region model. Furthermore, previous work on the AID has not indicated the significance of these effects. A third possibility is the use of a step current large enough to cause switching between the on and the off states of the device. In terms of dynamic applications of the AID, this behavior is very important. The mathematical treatment of this situation is complicated by the fact that the geometric models for the two states have effective inner contact radii which differ by at least an order of magnitude. For this reason, the execution of a formal analysis of complete switching in the present model is not undertaken. For the planar model, the carrier transit time in the avalanche region is of the order of magnitude of the collision ionization time, i.e. less than 0.1 nanosec, which has been estimated in the literature.(s) The dynamic behavior of this region should be determined by the avalanche process itself. For a practical treatment, the respond avalanche region is assumed to instantaneously to a step change in the current. Such a view of the region will be valid if the time required to produce a significant change in the voltage across the remaining portions of the device is much greater than the collision ionization time. The static model for the saturated drift velocity region predicts the existence of an electric field totally independent of current as long as the transport process permits the establishment within the region of a charge density which is also current independent. Since the carrier transit time for this part of the diode is at most a tenth of a nanosecond, the individual carrier profiles, with recombination effects excluded, may be assumed to adjust immediately to the conditions imposed by the boundary at the avalanche region. Thus,
A.
G.
JORDAN
the saturated drift velocity region will present no transient voltage response on a time scale for which transit effects can be ignored. The bulk region treatment uses a step increase in current at time equal to zero, with the initial carrier profiles determined by the current prior to the application of the step; the steady-state profiles, by the final current. Near the inner boundary, the bulk region must support field magnitudes in excess of the critical drift field, E c, for the interval of time necessary to redistribute the carriers and increase the conductivity of the region. In the strictest sense, the problem should be reformulated with a provision that the critical drift radius instantaneously move into the static bulk region, as required by field magnitude considerations, and then relax to its original location as the bulk carrier profiles achieve their steady-state forms. This approach would predict a time dependent SDV region voltage with an initial spike. There is considerable difficulty in solving this problem, however, since it requires developing an internally self-consistent model for the SDV and bulk regions considered simultaneously. The present, less involved, treatment ignores the velocity saturation effects in the bulk region and predicts the initial voltage spikes as a result of conductivity modulation in the normal mobility region. The minority carrier density must obey the time-dependent differential equation corresponding to that developed for the static analysis
$-PO) =- py) +~p”‘(p-p~).(2,) Prior to time equal to zero, the direct current, 11, passes through the device and the initial minority carrier profile is given by equation (19) in which the hole concentration at the inner surface of the bulk region corresponding to the current 11 is designated as p(Rz, Ii). At time equal to zero, the current is incremented to the value 1s; the steadystate hole profile is characterized by the parameter p(RL, Ia), corresponding to the final current value. With these conditions, the minority carrier distribution is given by the expression
THE
STATIC
AND
sinh&IL[l
sinhR,/L[l
DYNAMIC
PROPERTIES
-(r/&)1 _2k -(Rb/Rm)]
OF THE
oth$
RM ’
L
AVALANCHE
O”7~ - exp{ c n=O I
T(Z) is defined as
INJECTION
- t/r(n)> cos(2n+
DIODE
k] .
1);
nk
639
(26)
1.00 0.00
T(n) =
[(2n+ &;(L,Rm)]2+
1
(27)
Except for the influence of the geometry in the factor of l/r, the solution is identical with that for planar models of the same type. The transient voltage is determined analogously to the static case. The actual evaluation has been done numerically for arbitrary values of time. A current step from 10 to 15 mA has been used as being well within the sustained avalanche portion of the predicted V-I curve. The minority carrier lifetime has been given a range of specific values in the computation to permit an investigation of its effect on the voltage transient. Results are given in Fig. 5. Straight lines corresponding to exponential behavior provide extremely good fits to the predicted points for normalized voltages less than O-1. The fits have been made for the points at which the predicted normalized voltages have values of 0.135, thus defining a time equal to twice the time constant of the exponential. For the geometry used in the analysis, the time constant of the exponential in each case is nearly the same as the minority carrier lifetime. In Fig. 6, typical observed voltage transients are shown with the predicted transients for 0.8 and 2.0 psec lifetime models. The current increments were approximately 30 per cent of the pre-step values for the cases considered. B. Behavior
with respect to alternating
signals
Since the slowest devices require about 1.5 psec to come to steady-state in response to step-current excitation, the bulk region equivalent circuit indicated in Fig. 5 should be valid for sinusoidal signals below 300 kc/s in frequency; over an extended frequency range, the various regions, particularly the bulk, must be examined in more detail. The avalanche region response to a sinusoidal current is assumed to be instantaneous, yielding a voltage in phase with the current as long as the collision ionization process can come to equilibrium
0.60 Approximate
Equivalent
0. IO 0.08 0.06
0.04
$
.c?
0.02
0 E
b
z
0.01 *
Time
4
3
2
0
in 16’
set
The Stroight Lines ore Exponentiol Fits to the Predicted Behovior, Matched ot o Normalized Voltage
Value
of
6’
FIG. 5. Semilogarithmic plots of the predicted, normalized voltage response of the bulk region to a step current, for various minority carrier lifetime values.
within a small fraction of the period of the signal. The avalanche region is characterized by its incremental resistance, as given by differentiating equation (9) with respect to current, viz. RA =
dVA
--_=
di
---.
1
VA
2
i
(28)
As in the transient analysis the voltage contribution of the SDV region is assumed to be zero, as long as the frequency is below the limit that is imposed by the avalanche region. The SDV small signal impedance is thus zero. The hole concentration at the inner surface (Ri)
640
J.
R.
SZEDON
and
A.
G.
JORDAN
excitation, the a.c. component of the hole concentration is in general given by
i
(32)
pat = @ exp(jwt),
where 9 is the spatial part of the a.c. hole concentration. After manipulations similar to those for the transient case, the differential equation for the hole concentration is l+
O=--__ (
Time in Id6 set FIG. 6. Comparison of predicted and observed voltage transients for step current driving functions. Curve (a) is the observed behavior for device No. LJ. Curve (b) is the observed behavior for device No. UE. Curve (c) is the predicted behavior for a lifetime of 0.8 psec. Curve (d) is the predicted behavior for a lifetime of 2.0 psec.
of the bulk region is assumed to be in phase with the current passing across that surface. Neglect of any time lag is consonant with the assumptions regarding the size of the avalanche and bulk regions and the nature of current transport within them. Specifically it is assumed that the current passing through the device is complex and given by
i = I[l+A
exp(j,t)].
(29)
The boundary condition for holes at RL becomes p(RL) = po+p(Ri,
W+A
exp(j41,
(30)
where the excess minority carrier concentration at the inner surface of the bulk region due to the bias current is designated as p(Ri., I). The differential equation in the minority carrrer density, equation (25), must be solved subject to the requirement of equation (30) and the further minority carrier restriction that the excess concentration at the outer contact be zero. This solution may be indicated as a sum of the thermal equilibrium, the d.c. and the a.c. solutions, i.e. p = po+p0%+pac.
(31)
The d.c. portion of the solution has been indicated in the previous section. For steady-state a.c.
L2
d%?
2d9?
jar,
(33)
i 9+yr+F*
The general Bessel form of solution for the spatial dependence is indicated except that the argument of the function, instead of being purely imaginary, i.e. j(r/L), is complex:
The a.c. solution, therefore, must be analogous to the d.c. one, except for this change. The boundary condition in equation (30) yields as the a.c. solution for the hole concentration
pa, = p(RL, I)A exp(j&)s
Y
sinh(R,-r/L2/[1 x sinh(R,-
Ri/Ld[l
+j,r,])
+jwP])’
(34)
The complex value of the radial component of the electric field may be obtained by substituting the instantaneous hole concentration in equation (31) into the electric field expression. The instantaneous bulk voltage due to the current, I[1 +A exp(jwt)], is given by integrating the electric field expression with the bias contribution accounted for by requiring the ratio of a.c. to d.c. current, A, to be zero in a separate integration of the complete electric field expression. The corresponding voltage is designated as VaC; hence the a.c. component of the instantaneous bulk voltage is given as V,, = VB-- Vde = re V,,+j With the a.c. component expression
im Vu,.
(35)
of current
given by the
iae = IA(cos wt+ j sin wt),
(36)
THE
STATIC
AND
a quasi-impedance,
DYNAMIC
PROPERTIES
2, may be defined
This is not a true impedance, since there will be, in general, voltage components at frequencies higher than that of the alternating current. In such a case, the quasi-impedance will vary with electrical angle, wt, and the actual impedance at the current frequency will be given by the average of the quasi-impedance over a period of the current function. The theoretical bulk region impedance from 10 kc/s to 10 km+ has been obtained by numerical analysis for various minority carrier lifetime, bias current and geometric values. In a first order approach which neglects the complications in the avalanche model at very high frequencies, the total impedance of the AID model may be obtained by algebraic addition of the avalanche region resistance to the bulk region impedance. The results of this treatment are given in Fig. 7. The curve for 5 mA bias with a minority
OF THE
AVALANCHE
INJECTION
DIODE
641
very pronounced and gives rise to a rapidly varying slope in that vicinity. The slope at that bias should correspond to a negative incremental resistance. Of major significance is the fact that the total impedance curves are roughly concentric. In addition, over all but the lowest frequency ranges, the predicted behavior does not indicate very large differences with bias in the magnitude of the total impedance or of its component parts. The predicted AID impedance is the most interesting in the mid-frequency range, i.e. 0.3 to 10 me/s. The results of measurements on two devices for this range are given in Fig. 8. The frequencies at which maximum reactances occur 600 500 400 c
300
-200
0
Reol
Port
200
of the
400
600
Impedance,
600
51
~~~~
;
ZOO
0
z
Reol
500
I Frequencp
Port
400
'
600
I I inMc/s-2
I
1000
600
of the Impedance,
L?,
0
I
’
I
’
’
200
Real
’
400
Port
of the
600
Impedance,
800
1000
a
FIG. 8. Measured, mid-frequency impedance behaviour of typical avalanche injection diodes
400
600
Reol
600
Port
of the
1000
1200
Impedance,
1400
1600
Cl
FIG. 7. The predicted AID impedance behavior. The upper set of curves is for a minority carrier lifetime of 3.0 /wx.; the lower one, 0.2 ~~sec.
carrier lifetime of 1.0 psec has a negative resistance at zero frequency. This seems to be in disagreement with the static curve of Fig. 4. Careful numerical consideration of the static model shows that the avalanche contribution near 5 mA is
agree very well with those for the 0.2 psec model in Fig. 7. The data of Fig. 8 are interesting due to the appearance of the negative resistance component below 2 me/s at 7 mA bias. In this respect, the device behavior is very like that of the 1 psec lifetime model in Fig. 7. The measurements in Fig. 8 demonstrate the tendency at higher frequencies for the diminution of differences resulting from bias changes. A proposed application of the AID is the switching of high frequency signals; hence, admittance measurements have been made in the 200 to 900 me/s range. Further, since the use of
642
J. R.
SZEDON
and
the diode is expected to be limited by the susceptance of the encapsulation, the measurements have been made of the behavior of the device as a whole, instead of correcting for the holder susceptance. The device has been examined for zero bias and for several values of current, well into the on-state. As a result of the measurements in the biased off-state, the device may be characterized by a capacitance of approximately 1 pF shunting a resistance of SOOO-10,000 Q. With the . . application of heavy bias, the resistance decreases by an order of magnitude and the capacitive susceptance is reduced by about 10 per cent at 300 me/s. On-state measurements for various levels of bias are virtually identical, substantiating the predicted high frequency behavior in Fig. 7. Near 600 me/s the measured resistance achieves its maximum value of approximately 1400 Q. This is in fair quantitative agreement with the predicted behavior. The latter calls for a small change in the value of the resistance about its maximum, while the observed decrease is roughly 10 per cent for a 200 mcjs change in frequency. An experimental evaluation of the a.c. switching properties of the avalanche injection diodes discussed in the present work has been published.@) 4.
A.
G.
JORDAN
Although the dynamic analysis concentrates on the effects of the bulk region, it should be kept in mind that the important considerations of contact geometry, carrier injection levels and space charge requirements in this portion of the device are determined by the models for the high field region. As a result, there is tighter coupling between the various parts of the model than is first apparent. This constitutes additional support of the validity of the present approach. Since the model in question precludes the existence of important junction effects and of difficulties arising from the avalanche and high drift velocity regions, it serves as an excellent tool for predicting dynamic bulk effects due to conductivity modulation. While the transient analysis has not involved anything novel in this area, except for the geometric complications and the avalancheimposed boundary conditions at the inner contact, it has permitted the investigation of transients an order of magnitude faster than those treated previously. The small signal a.c. investigation has yielded agreement between predicted and observed conductivity modulation effects on the impedance behavior to frequencies in the neighborhood of a kilomegacycle per second.
SUMMARY
The proposed model for the AID, in treating both the pre-breakdown and the sustained avalanche modes of operation, has produced results in quantitative agreement with observations. The agreement in the pre-breakdown model is due, in part, to a more flexible interpretation of the ionization collision data which is supported by Schockley’s recent work in the area. The sustained avalanche model, on the other hand, is thought to be a novel treatment of the device. Although it is similar in concept, it is not as complicated as earlier approaches. The static application of the model reveals the greatest sensitivity to the interactions between the avalanche, the carrier drift velocity and the conductivity modulation effects. In this respect the model, as applied to the on-state static behavior presents some problems in obtaining a balance between the multiplication and the bulk processes. Confirmation of the predicted dynamic response supports the choice of parameters required by the static application.
REFERENCES 1. J. B. GUNN, Progress in Semiconductors, (edited by A. F. GIBSON), 2, p. 217. Heywood, London (1957). 2. D. J. ROSE, Pkys. Rev. 105,413 (1957). 3. A. F. GIBSON and J. R. MORGAN, Solid-State Electron. 1, 54 (1960). 4. C. A. HOGARTH, Solid-State Electron. 1, 70 (1960). 5. A. K. JONSCHER,Progress in Semiconductors, (edited by A. F. GIBSON), 6, 143. Heywood, London (1962). 6. W. SHOCKLEY, Solid-State Electron. 2, 35 (1961). 7. L. I. BARANOV, Radiotekh. Elektron. 5, 1002 (1960). (Translated in Electron. Express; July 30 (1960). 8. J. R. SZEDON and A. G. JORDAN,Proc. of the National Efect. Conf., 18,166 (1962).
LIST al
=
A
=
OF
SYMBOLS
dimensionless constant used in characterizing the current dependence of the under static region voltage bulk conditions. the ratio of the alternating current magnitude to the bias current magnitude in the a.c. treatment.
THE
STATIC
AND
DYNAMIC
PROPERTIES
= ratio of electron to hole drift mobility at low values of electric field. = diffusion coefficient for electrons D~@P) holes. (ems set-i V-r). = diffusion coefficient for electrons for DflO low values of electric field. = the electric field magnitude (vector). E,(E) (V cm-l). = the electric field at the boundary EdEd separating the avalanche and the saturated drift velocity (the saturated drift velocity and the bulk) regions. the = instantaneous current through i diode. (A). = value of current below which normal ie mobility obtains throughout the diode in the pre-avalanche model. = initial (final) current in the transient 4SI2) analysis. = bias (alternating) component of current Z,(iCz C) in the small signal analysis. = the instantaneous total current density J,(J) magnitude (vector). (A cm-s). = the instantaneous hole current density JP,(JP) magnitude (vector), k = Boltzmann constant. (J deg-l). = a constant used in characterizing the K current dependence of the bulk region voltage under static conditions. See above. (V A-l). = effective diffusion lengthforholes. (cm). L = effective donor doping density. (cmm3). Nd = instantaneous electron concentration. It (cme3). = electron concentration at the plane 40) x = 0 in the planar avalanche model. = electron concentration at the boundary n(Rd between the avalanche and the SDV regions. = instantaneous (thermal equilibrium) P,(n) hole concentration. p(Ra), (p(Rz)) = instantaneous hole concentration at the boundary between the avalanche and the SDV (the SDV and the bulk) regions. = hole concentration at the boundary P(% I) between the SDV and the bulk reeions as a result of bias current in the a.c. treatment. P(R;, h), (p(R) 1~)) = hole concentration at the boundary between the SDV and the bulk regions as a result of the initial (final) current in the transient analysis. = hole concentration due to the bias Pat, (Pa,) (alternating) component of current in the a.c. treatment of the bulk region. = magnitude of the electronic charge. (C). 4 r = radial distance in spherical co-ordinates. (cm).
b
OF
THE
AVALANCHE
INJECTION
DIODE
643
XC2
= effective radius of the cylindrical (spherical) avalanche region model. = incremental resistance of the avalanche region. (a). = radius of the boundary separating the SDV and the bulk regions in the preavalanche (sustained avalanche) model. (cm). = effective radius of the inner contact of the diode. = maximum radial extent of the bulk region. = complex function giving the spatial dependence of the hole concentration in the a.~. treatment of the bulk region. (cmm3). = absolute temperature. (“IQ. = time. (set). = dimensionless function used to facilitate integration. = value of the saturated carrier drift the model. assumed in velocity, (cm set-l). = total voltage across the device in the pre-avalanche state. (volt). across the voltage = instantaneous avalanche (bulk) region. across the saturated drift = voltage sustained region in the velocity avalanche model. = alternating (bias) component of the bulk region voltage in the a.c. treatment. = rectangular co-ordinate in the planar avalanche model. (cm). = extent of the planar avalanche model.
z
(cm). = quasi-impedance
~a, (Ra) RA R C,(Ri)
RP R?73 w
T t 24 vzs
V VA, (VB) VS
Vat, (VdC) x
a(E) ‘%
= =
=
K h
= =
= = = = =
of the bulk
region.
(0). avalanche ionization coefficient. (cm-‘). constant used in obtaining a mathematical fit to experimental data for the avalanche ionization coefficient. (cm-i). constant relating to geometric and material parameters in the treatment of the saturated drift velocity region. (V cm-l). permittivity of silicon. (Fcm-l). constant used in obtaining a mathematical fit to the experimental data for the avalanche ionization coefficient. (cm V-i). drift mobility of electrons (holes). (cm? see-l V-l). drift mobility of electrons at low values of electric field. hole lifetime. (XC). effective lifetime parameter in the transient analysis. (set). radian frequency. (set-l).